Handbook Of Coastal And Ocean Engineering

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HANDBOOK OF COASTAL AND OCEAN ENGINEERING

edited by

Young C Kim California State University, Los Angeles, USA

World Scientific NEW JERSEY

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LONDON

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SINGAPORE

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BEIJING

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SHANGHAI

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HONG KONG

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TA I P E I

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CHENNAI

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Published by World Scientific Publishing Co. Pte. Ltd. 5 Toh Tuck Link, Singapore 596224 USA office: 27 Warren Street, Suite 401-402, Hackensack, NJ 07601 UK office: 57 Shelton Street, Covent Garden, London WC2H 9HE

British Library Cataloguing-in-Publication Data A catalogue record for this book is available from the British Library.

HANDBOOK OF COASTAL AND OCEAN ENGINEERING Copyright © 2010 by World Scientific Publishing Co. Pte. Ltd. All rights reserved. This book, or parts thereof, may not be reproduced in any form or by any means, electronic or mechanical, including photocopying, recording or any information storage and retrieval system now known or to be invented, without written permission from the Publisher.

For photocopying of material in this volume, please pay a copying fee through the Copyright Clearance Center, Inc., 222 Rosewood Drive, Danvers, MA 01923, USA. In this case permission to photocopy is not required from the publisher.

ISBN-13 978-981-281-929-1 ISBN-10 981-281-929-0

Typeset by Stallion Press Email: [email protected]

Printed in Singapore.

YHwa - Hdbk of Coastal & Ocean.pmd

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Preface Although coastal and ocean engineering is a very ancient field with the construction of Port A-ur near the mouth of the Nile in 3,000 BC, significant advances in this field have been made in the last several decades. The rise of interest in this field can be seen from the number of attendees by academics and practitioners in international conferences. The first International Conference on Coastal Engineering was held in Long Beach, California in 1950 with less than 100 people. When the same conference was held in San Diego, California, in 2006, over 1000 delegates attended. In the last several decades, the world has seen significant coastal and ocean engineering projects, one of which is the Delta Project in the Netherlands. This project was designed to shorten and strengthen the total length of coast and dykes washed by the sea by closing off the sea arms in the Delta region. Other noteworthy coastal engineering projects include the Kansai Airport Project in Japan and, in recent years, the construction of mobile barriers at inlets to regulate tides in the Venice Lagoon known as the Venice Project. Interest in coastal and ocean engineering has arisen in recent years as humankind experiences coastal disasters that derive from coastal storm, hurricane and coastal flooding and seismic activities such as tsunamis, and the impacts of climate change which result in sea-level rise. The tsunami activity in Sumatra in December 2004 affected countries throughout the Indian Ocean and resulted in the loss of thousands of lives. Hurricane Katrina in New Orleans also claimed many lives with property damage exceeding $63 billion. Global warming and sea-level rise will affect shoreline retreats, inundate low coastal areas, damage coastal structures, and accelerate beach erosion. The need for better understanding of our coastal and ocean environment has risen considerably in recent years. This handbook contains a comprehensive compilation of topics that are the forefronts of many technical advances in ocean waves, coastal and ocean engineering. It represents the most comprehensive reference available on coastal and ocean engineering to date, and it also provides the most up-to-date technical advances and latest research findings on coastal and ocean engineering. More than 70 internationally recognized authorities in the field of coastal and ocean engineering contributed papers on their areas of expertise to this handbook. These international luminaries are from highly respected universities and renowned research and consulting organizations from all over the world.

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Preface

This handbook provides a comprehensive overview of shallow-water waves, water-level fluctuations, coastal and offshore structures, ports and harbors, coastal sediment processes, environmental problems, sustainable coastal development, coastal hazards, physical modeling, and coastal engineering practice and education. This book is an essential source of reference for professionals and researchers in the areas of coastal engineering, ocean engineering, oceanography, meteorology, and civil engineering, and as a text for graduate students in these fields. This handbook will be of immediate, practical use to coastal, ocean, civil, geotechnical, and structural engineers, and coastal planners and managers as well as marine biologists and oceanographers. It will also be an excellent source book for educational and teaching purposes, and would be a good reference book for any technical library. I would like to express my indebtedness to those who guided me and supported me as a mentor and a colleague throughout my professional life. They are: Professor Robert L. Wiegel, University of California, Berkeley Professor Joe W. Johnson, University of California Berkeley Professor Robert G. Dean, University of Florida Professor Fredric Raichlen, California Institute of Technology Professor Raymond C. Binder, University of Southern California Professor Shoshichiro Nagai, Osaka City University Dr Basil Wilson, Science Engineering Associates Dr Lars Skjelbreia, Science Engineering Associates Dr Bernard LeMehaute, University of Miami Professor Richard Silvester, University of Western Australia Mr Orville T. Magoon, Coastal Zone Foundation Professor Billy L. Edge, Texas A&M University Professor Michael E. McCormick, US Naval Academy Professor Yoshimi Goda, Yokohama National University and ECOH Corporation Professor Philip L.F. Liu, Cornell University Professor Forrest M. Holly, The University of Iowa Dr Etienne Mansard, National Research Council, Canada Professor J. Richard Weggel, Drexel University Mr Ronald M. Noble, Noble Consultants, Inc. I also wish to express my indebtedness to those who nurtured me from my early teen years and changed my course of life. They are: Dr Helen Miller Bailey, East Los Angeles College Mr H. Karl Bouvier, Jet Propulsion Laboratory I extend my gratitude to my wife, Janet, for her constant support, encouragement, patience, and understanding while I was undertaking this task and to my daughter, Susan Calix, for proofreading some of the materials.

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Finally, I wish to express my deep appreciation to Ms Kimberley Chua of World Scientific Publishing Company who gave me invaluable support and encouragement from the inception of this handbook to its realization. Young C. Kim Los Angeles, California January 2008

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Contents

Preface. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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Contributors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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Section 1: Shallow-Water Waves 1. Wave Setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . R. G. Dean and T. L. Walton

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2. Wavemaker Theories . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . R. T. Hudspeth and R. B. Guenther

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3. Analyses by the Melnikov Method of Damped Parametrically Excited Cross Waves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . R. B. Guenther and R. T. Hudspeth

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4. Random Wave Breaking and Nonlinearity Evolution Across the Surf Zone . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Y. Goda

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5. Aeration and Bubbles in the Surf Zone . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . N. Mori, S. Kakuno and D. T. Cox

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6. Freak Wave . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . N. Mori

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7. Short-Term Wave Statistics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A. Kimura

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Section 2: Water-Level Fluctuations 8. Generation and Prediction of Seiches in Rotterdam Harbor Basins . . . . . M. P. C. de Jong and J. A. Battjes

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9. Seiches and Harbor Oscillations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A. B. Rabinovich

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10. Finite Difference Model for Practical Simulation of Distant Tsunamis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . S. B. Yoon

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Section 3: Coastal Structures 11. Tsunami-Induced Forces on Structures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . I. Nistor, D. Palermo, Y. Nouri, T. Murty and M. Saatcioglu

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12. Nonconventional Wave Damping Structures . . . . . . . . . . . . . . . . . . . . . . . . . . . H. Oumeraci

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13. Wave Interaction with Breakwaters Including Perforated Walls . . . . . . . . K.-D. Suh

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14. Prediction of Overtopping . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . J. van der Meer, T. Pullen, W. Allsop, T. Bruce, H. Sch¨ uttrumpf and A. Kortenhaus

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15. Wave Run-Up and Wave Overtopping at Armored Rubble Slopes and Mounds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . H. Sch¨ uttrumpf, J. van der Meer, A. Kortenhaus, T. Bruce and L. Franco

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16. Wave Overtopping at Vertical and Steep Structures . . . . . . . . . . . . . . . . . . . T. Bruce, J. van der Meer, T. Pullen and W. Allsop

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17. Surf Parameters for the Design of Coastal Structures . . . . . . . . . . . . . . . . . . D. H. Yoo

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18. Development of Caisson Breakwater Design Based on Failure Experiences . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . S. Takahashi

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19. Design of Alternative Revetments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . K. Pilarczyk

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20. Remarks on Coastal Stabilization and Alternative Solutions . . . . . . . . . . . K. Pilarczyk

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21. Geotextile Sand Containers for Shore Protection. . . . . . . . . . . . . . . . . . . . . . . H. Oumeraci and J. Recio

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22. Low Crested Breakwaters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A. Lamberti and B. Zanuttigh

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23. Hydrodynamic Behavior of Net Cages in the Open Sea . . . . . . . . . . . . . . . . Y.-C Li

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Section 4: Offshore Structures 24. State of Offshore Structure Development and Design Challenges . . . . . . . S. Chakrabarti

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Section 5: Ports and Harbors 25. Computer Modeling for Harbor Planning and Design . . . . . . . . . . . . . . . . . . J.-J. Lee and X. Xing

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26. Prediction of Squat for Underkeel Clearance . . . . . . . . . . . . . . . . . . . . . . . . . . . M. J. Briggs, M. Vantorre, K. Uliczka and P. Debaillon

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Section 6: Coastal Sediment Processes 27. Wave-Induced Resuspension of Fine Sediment . . . . . . . . . . . . . . . . . . . . . . . . . M. Jain and A. J. Metha

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28. Suspended Sand and Bedload Transport on Beaches . . . . . . . . . . . . . . . . . . . N. Kobayashi, A. Payo and B. D. Johnson

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29. Headland-Bay Beaches for Recreation and Shore Protection . . . . . . . . . . . J. R.-C. Hsu, M. M.-J. Yu, F.-C. Lee and R. Silvester

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30. Beach Nourishment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . R. G. Dean and J. D. Rosati

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31. Engineering of Tidal Inlets and Morphologic Consequences . . . . . . . . . . . . N. C. Kraus

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Section 7: Environmental Problems 32. Water and Nutrients Flow in the Enclosed Bays . . . . . . . . . . . . . . . . . . . . . . . Y. Koibuchi and M. Isobe

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Section 8: Sustainable Coastal Development 33. Socioeconomic and Environmental Risk in Coastal and Ocean Engineering . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . M. A. Losada, A. Baquerizo, M. Ortega-S´ anchez, J. M. Santiago and E. S´ anchez-Badorrey 34. Utilization of the Coastal Area . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . H.-H. Hwung

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Section 9: Coastal Hazards 35. Ocean Wave Climates: Trends and Variations Due to Earth’s Changing Climate . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . P. D. Komar, J. C. Allan and P. Ruggiero

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36. Sea Level Rise: Major Implications to Coastal Engineering and Coastal Management . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . L. Ewing

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37. Sea Level Rise and Coastal Erosion. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . M. J. F. Stive, R. Ranasinghe and P. J. Cowell

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38. Coastal Flooding: Analysis and Assessment of Risk . . . . . . . . . . . . . . . . . . . . P. Prinos and P. Galiatsatou

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Section 10: Physical Modeling 39. Physical Modeling of Tsunami Waves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . M. J. Briggs, H. Yeh and D. T. Cox

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40. Laboratory Simulation of Waves. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . E. P. D. Mansard and M. D. Miles

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Section 11: Coastal Engineering Practice and Education 41. Perspective on Coastal Engineering Practice and Education . . . . . . . . . . . J. W. Kamphuis

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Contributors Jonathan C. Allan Coastal Field Office Oregon Department of Geology and Mineral Industries Newport, Oregon [email protected] William Allsop Technical Director HR Wallingford Wallingford, UK [email protected] Elena Sanchez Badorrey Associate Professor CEAMA — Universidad de Granada Granada, Spain [email protected] Asuncion Baquerizo Associate Professor CEAMA — Universidad de Granada Granada, Spain Jurjen A. Battjes Emeritus Professor Environmental Fluid Mechanics Section Delft University of Technology Delft, The Netherlands [email protected] Michael J. Briggs Research Hydraulic Engineer Coastal and Hydraulics Laboratory U.S. Army Engineer Research and Development Center Vicksburg, Mississippi [email protected]

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Tom Bruce School of Engineering and Electronics University of Edinburgh Edinburgh, UK [email protected] Subrata Chakrabarti Joint Professor, Civil and Mechanical Engineering University of Illinois at Chicago Chicago, Illinois [email protected] Peter J. Cowell Associate Professor School of Geosciences Institute of Marine Science University of Sydney Sydney, Australia Daniel T. Cox Professor School of Civil and Construction Engineering Oregon State University Corvallis, Oregon [email protected] Robert G. Dean Graduate Research Professor of Coastal Engineering, Emeritus Department of Civil and Coastal Engineering University of Florida Gainesville, Florida [email protected]fl.edu Pierre Debaillon Research Hydraulic Engineer Centre d’Etudes Techniques Maritimes Et Fluviales (CETMEF) Compiegne, France [email protected] Martijn P. C. de Jong Formerly at Environmental Fluid Mechanics Section Delft University of Technology Presently at Delft Hydraulics Delft, The Netherlands

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Lesley Ewing Senior Coastal Engineer California Coastal Commission San Francisco, California [email protected] Leopoldo Franco Professor of Coastal Engineering Department of Civil Engineering University of Rome 3 Rome, Italy [email protected] Panagiota Galiatsatou Research Associate Department of Civil Engineering Aristotle University of Thessaloniki Thessaloniki, Greece [email protected] Yoshimi Goda Professor Emeritus Yokohama National University Adviser to ECHO Corporation Tokyo, Japan [email protected] Ronald B. Guenther Professor Emeritus Department of Mathematics Oregon State University Corvallis, Oregon [email protected] John Rong-Chung Hsu Professor Department of Marine Environment and Engineering National Sun Yat-sen University Kaohsiung, Taiwan Honorary Research Fellow School of Civil and Resource Engineering University of Western Australia Nedland, Australia [email protected]

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Robert T. Hudspeth Professor and Director, Emeritus Coastal and Ocean Engineering Program Oregon State University Corvallis, Oregon [email protected] Hwung-Hweng Hwung Professor of Hydraulic and Ocean Engineering Director of Tainan Hydraulics Laboratory Department of Hydraulic and Ocean Engineering National Cheng Kung University Tainan, Taiwan [email protected] Masahiko Isobe Professor and Special Adviser to the President Department of Sociocultural Environmental Studies Graduate School of Frontier Sciences The University of Tokyo Chiba, Japan [email protected] Mamta Jain Coastal Engineer Halcrow Inc. Tampa, Florida [email protected] Bradley D. Johnson Coastal and Hydraulics Laboratory U.S. Army Engineer Research and Development Center Vicksburg, Mississippi Shohachi Kakuno Professor and Vice President Department of Civil Engineering Osaka City University Osaka, Japan [email protected] J. William Kamphuis Professor of Civil Engineering, Emeritus Department of Civil Engineering Queen’s University Kingston, Ontario, Canada [email protected]

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Akira Kimura Professor Department of Social Systems Engineering Tottori University Tottori, Japan [email protected] Nobuhisa Kobayashi Professor and Director Center for Applied Coastal Research University of Delaware Newark, Delaware [email protected] Yukio Koibuchi Assistant Professor Department of Sociocultural Environmental Studies Graduate School of Frontier Sciences The University of Tokyo Chiba, Japan [email protected] Paul D. Komar Professor of Oceanography College of Oceanic and Atmospheric Sciences Oregon State University Corvallis, Oregon [email protected] Andreas Kortenhaus Leichtweiss-Institute for Hydraulics Technical University of Braunschweig Braunschweig, Germany [email protected] Nicholas C. Kraus Senior Scientist Coastal and Hydraulics Laboratory U.S. Army Engineer Research and Development Center Vicksburg, Mississippi [email protected] Alberto Lamberti Professor Department of Civil Engineering University of Bologna Bologna, Italy [email protected]

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Fang-Chun Lee Department of Marine Environment and Engineering National Sun Yat-sen University Kaohsiung, Taiwan Jiin-Jen Lee Professor of Civil and Environmental Engineering Sonny Astani Department of Civil and Environmental Engineering University of Southern California Los Angeles, California [email protected] Yu-Cheng Li Professor School of Civil Engineering Dalian University of Technology Dalian, China [email protected] Miguel A. Losada Professor Research Group on Environmental Flux Dynamics CEAMA — Universidad de Granada Granada, Spain [email protected] Etienne P. D. Mansard Executive Director Canadian Hydraulics Centre National Research Council Canada Ottawa, Ontario, Canada [email protected] Ashish J. Mehta Professor of Coastal Engineering Department of Civil and Coastal Engineering University of Florida Gainesville, Florida [email protected]fl.edu Michael D. Miles Canadian Hydraulics Centre National Research Council Canada Ottawa, Ontario, Canada

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Nobuhito Mori Associate Professor Disaster Prevention Research Institute Kyoto University Kyoto, Japan [email protected] Tad S. Murty Adjunct Professor Department of Civil Engineering University of Ottawa Ottawa, Ontario, Canada [email protected] Ioan Nistor Assistant Professor Department of Civil Engineering University of Ottawa Ottawa, Ontario, Canada [email protected] Younes Nouri Department of Civil Engineering University of Ottawa Ottawa, Ontario, Canada Miquel Ortega Associate Professor CEAMA — Universidad de Granada Granada, Spain [email protected] Hocine Oumeraci University Professor Leichtweiss-Institute for Hydraulic Engineering and Water Resources Technical University of Braunschweig Braunschweig, Germany [email protected] Dan Palermo Assistant Professor Department of Civil Engineering University of Ottawa Ottawa, Ontario, Canada [email protected]

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Andres Payo Graduate School of Science and Technology University of Kumamoto Kumamoto, Japan Krystian W. Pilarczyk (Former) Manager, Research and Development Hydraulic Engineering Institute Rykswaterstaat Delft, The Netherlands HYDROpil Consultancy Zoetermeer, The Netherlands [email protected] Panayotis Prinos Professor of Hydraulic Engineering Department of Civil Engineering Aristotle University of Thessaloniki Thessaloniki, Greece [email protected] Tim Pullen Senior Engineer HR Wallingford Wallingford, UK [email protected] Alexander B. Rabinovich P.P. Shirshov Institute of Oceanology Russian Academy of Sciences Moscow, Russia Department of Fisheries and Oceans Institute of Ocean Sciences Sidney, B.C., Canada [email protected]. Roshanka Ranasinghe Associate Professor UNESCO-IHE/Delft University of Technology Delft, The Netherlands [email protected] Juan Recio Leichweiss-Institute for Hydraulic Engineering and Water Resources Technical University of Braunschweig Braunschweig, Germany

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Julie D. Rosati Research Hydraulic Engineer Coastal and Hydraulics Laboratory U.S. Army Corps of Engineers Mobile, Alabama [email protected] Peter Ruggiero Assistant Professor Department of Geosciences Oregon State University Corvallis, Oregon [email protected] Murat Saatcioglu Professor Department of Civil Engineering University of Ottawa Ottawa, Ontario, Canada [email protected] Juan M. Santiago Associate Professor CEAMA — Universidad de Granada Granada, Spain [email protected] Holger Schuttrumpf Professor and Director Institute of Hydraulic Engineering and Water Resources Management RWTH — Aachen University Aaachen, Germany [email protected] Richard Silvester Professor Emeritus School of Civil and Resource Engineering University of Western Australia Nedland, Australia [email protected]

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Marcel J. F. Stive Professor and Director Delft Water Research Centre Department of Hydraulic Engineering Delft University of Technology Delft, The Netherlands [email protected] Kyung-Duck Suh Professor Department of Civil and Environmental Engineering Seoul National University Seoul, Korea [email protected] Shigeo Takahashi Executive Researcher and Director Tsunami Research Center Port and Airport Research Institute Yokosuka, Japan takahashi [email protected] Klemens Uliczka Research Hydraulic Engineer Federal Waterways Engineering and Research Institute (BAW) Hamburg, Germany [email protected] Jentsje van der Meer Principal Van der Meer Consulting Heerenveen, The Netherlands [email protected] Marc Vantorre Professor Division of Maritime Technology Ghent University Ghent, Belgium [email protected]

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Todd L. Walton Director Beaches and Shore Resource Center Florida State University Tallahassee, Florida [email protected] Xiuying Xing Graduate Research Assistant Sonny Astani Department of Civil and Environmental Engineering University of Southern California Los Angeles, California Harry Yeh Professor School of Civil and Construction Engineering Oregon State University Corvallis, Oregon [email protected] Dong Hoon Yoo Professor Department of Civil Engineering Ajou University Suwon, Korea [email protected] Sung Bum Yoon Professor Department of Civil and Environmental Engineering Hanyang University Ansan, Korea [email protected] Melissa Meng-Jiuan Yu Department of Marine Environment and Engineering National Sun Yat-sen University Kaohsiung, Taiwan Barbara Zanuttigh Assistant Professor Department of Civil Engineering University of Bologna Bologna, Italy [email protected]

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The Editor Young C. Kim, PhD, is currently a Professor of Civil Engineering, Emeritus at California State University, Los Angeles. Other academic positions held by him include a Visiting Scholar of Coastal Engineering at the University of California, Berkeley (1971); a NATO Senior Fellow in Science at the Delft University of Technology in the Netherlands (1975); and a Visiting Scientist at the Osaka City University for the National Science Foundations’ US–Japan Cooperative Science Program (1976). For more than a decade, he served as Chair of the Department of Civil Engineering and recently he was Associate Dean of the College of Engineering. For his dedicated teaching and outstanding professional activities, he was awarded the university-wide Outstanding Professor Award in 1994. Dr Kim was a consultant to the US Naval Civil Engineering Laboratory in Port Hueneme and became a resident consultant to the Science Engineering Associates where he investigated wave forces on the Howard-Doris platform structure, now being placed in Ninian Field, North Sea. Dr Kim is the past Chair of the Executive Committee of the Waterway, Port, Coastal and Ocean Division of the American Society of Civil Engineering (ASCE). Recently, he served as Chair of the Nominating Committee of the International Association of Hydraulic Engineering and Research (IAHR). Since 1998, he served on the International Board of Directors of the Pacific Congress on Marine Science and Technology (PACON). He currently serves as the President of PACON. Dr Kim has been involved in organizing 10 national and international conferences, has authored three books, and has published 52 technical papers in various engineering journals.

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Chapter 1

Wave Setup Robert G. Dean Department of Civil and Coastal Engineering University of Florida, Gainesville, FL, USA [email protected] Todd L. Walton Beaches and Shores Resource Center Florida State University, Tallahassee, FL, USA [email protected] Wave setup is the increase of water level within the surf zone due to the transfer of wave-related momentum to the water column during wave-breaking. Wave setup has been investigated theoretically and under laboratory and field conditions, and it includes both static and dynamic components. Engineering applications include a significant flooding component due to severe storms and oscillating water levels that can increase hazards to recreational beach goers and can contribute to undesirable oscillations of both constructed and natural systems including harbors and moored ships. This chapter provides a review of the knowledge regarding wave setup and presents preliminary recommendations for design. It will be shown that wave setup is not adequately understood quantitatively for engineering design purposes.

1.1. Introduction Wave setup was brought to the attention of coastal engineers and scientists in the 1960s (i.e., see Ref. 1, p. 245) after the initial theoretic developments of LonguetHiggins2 and Longuet-Higgins and Stewart3,4 along with limited field observations and laboratory studies supported the existence of wave setup, the magnitude of which was observed to be in the order of 10–20% of the incident wave height. It was noted in early field observations that water levels on the beach were higher than those recorded by a tide gauge at the end of a pier suggesting a wave setup physically forced by wind waves and swell. 1

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Fig. 1.1. Definition sketch. Energy and momentum are transferred from winds to waves in the generating area. The waves convey energy and momentum to the surf zone where the waves break. Upon breaking, the energy is dissipated and the momentum is transferred to the water column resulting in longshore and onshore forces exerted on the water column.

Wave setup is the additional water level that is due to the transfer of wave-related momentum to the water column during the wave-breaking process. As waves approach the shoreline, they convey both energy and momentum in the wave direction. Upon breaking, the wave energy is dissipated, as is evident from the turbulence generated; however, momentum is never dissipated but rather is transferred to the water column resulting in a slope of the water surface to balance the onshore component of the flux of momentum (see Fig. 1.1). If waves are irregular, in addition to a steady wave setup, the setup includes a dynamic component that oscillates with the wave group period and there may be a weak resonance within the nearshore amplifying this oscillating component. These have been termed infragravity waves and are more dominant for narrow banded spectra both in frequency and in directional spreading. The oscillatory component is denoted “dynamic wave setup” in this chapter. This chapter discusses the significance of wave setup to coastal engineering design, provides a review of the classical linear wave theory of wave setup, reviews results from laboratory and field studies, summarizes results and recommends preliminary design approaches for the static component. To provide a “look ahead,” we will see that the phenomenon of wave setup is not yet adequately understood for satisfactory engineering calculations and that the effects of profile slope are very

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significant. The interested reader is also referred to an earlier review article on wave setup by Holman.5 1.2. Engineering Significance of Wave Setup Wave setup (both static and dynamic components) is relevant to a number of engineering applications. The contributions of wave setup under extreme storm events can be substantial, adding several feet to the elevated water levels. The interaction of wave setup with vegetation differs from wind surge and thus it is important to differentiate the two components, for example in ascertaining the benefits of wetlands in reducing wave setup. Finally, the oscillating component of wave setup is relevant to beach safety in some locations and to many natural and constructed coastal systems that have the capability to resonate including harbors and moored ships. 1.3. Terminology and Related Considerations Standard terminology defines the water level in the absence of wave effects as “still water level,” whereas wave setup will cause a departure from the still water level and this water level including the effects of the waves is the “mean water level.” As implied, the mean water level is determined as the average of the fluctuating water level over a suitable time frame usually taken as a number of multiples of the short wave period, say the spectral peak. In considering wave setup, often the location of interest is that of the maximum wave setup at the shoreline. This raises the question of whether wave setup is defined at elevations above the maximum rundown, say on the beach face where the water is present over only a portion of the wave period. Since wave setup is defined as the mean water level, over what period should the water surface be averaged on the beach face which is “wetted” over only a portion of the wave period? If the time average is over only the portion of the period that water is present, in the upper limit, the maximum setup will be the maximum runup. For purposes here, wave setup will usually be defined only for conditions where water is present over a full wave period. When calculating wave runup on a structure such as a levee or revetment, the question arises whether it is appropriate to first calculate wave setup and then add the wave runup which is usually empirically based on model results. In the more recent empirical results (e.g., the TAW method, see Ref. 6), the runup is expressed as a proportion of the significant wave height at the base of the steeper slope (e.g., at a revetment or levee). The wave runup determined in the model on which the method was based generally included some wave setup (or setdown) seaward of the toe of the slope and included wave setup landward of the toe of the slope. Thus, in the application of interest, the most appropriate approach is to calculate and include wave setup at the toe of the slope; however, recognizing that the measured landward runup includes setup, no additional setup should be added explicitly landward of the toe of the slope.

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1.4. A Brief Review of Wave Setup Mechanics 1.4.1. Static wave setup for monochromatic waves Longuet-Higgins2 and Longuet-Higgins and Stewart3,4 were the first to formalize the notion of wave momentum flux and its relationship to wave setup. The momentum flux, Sij , is a second-order tensor given by 
1 , Sxx = E n(cos2 θ + 1) − 2 
1 2 (1.1) Syy = E n(sin θ + 1) − , 2 E sin 2θ, 2 where E is the wave energy density, n is the ratio of wave group velocity to wave celerity and θ is the angle between the wave direction and the x-axis. The term Sxy reads “the flux per unit width, in the x-direction, of the y-component of momentum,” etc. The steady-state equations of motion obtained by time averaging over the short wave period are, including the effects of wind stress and bottom friction:   1 ∂Sxx ∂Sxy ∂ (η wind + ηwave ) =− + − τsx + τbx ∂x ρg(h + η) ∂x ∂y Sxy = Syx =

and ∂ (η wind + η wave ) 1 =− ∂y ρg(h + η)





(1.2)

∂Syy ∂Syx + − τsy + τby . ∂y ∂x

In the above, ηwind is the surge component due to the wind stress, ηwave is the wave setup, ρ is the mass density of water, g is the gravitational constant, h is the local water depth, τsx and τbx are the surface and bottom shear stresses, respectively, and similarly for the y-direction. The coordinate direction, x is oriented shoreward and a right-handed coordinate system is considered. The most simple solution is for waves propagating directly shoreward (Sxy = 0) in which the surface and bottom stresses are considered negligible, and all variables are considered uniform in the y-direction. The resulting equation is   1 ∂Sxx ∂η wave =− . (1.3) ∂x ρg(h + η) ∂x To proceed, we need to determine a boundary condition for η wave ,a at the seaward end of the surf zone. Longuet-Higgins7 has shown that in the absence of energy dissipation, the following general relationship for η applies η=C−

1 2 (u − w2 )η=0 , 2g

(1.4)

a For purposes of convenience, hereafter the subscript on η wave will be omitted such that the wave setup is simply denoted as η.

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where u2 and w2 represent the time averages of the square of the first-order horizontal and vertical wave velocities evaluated at the mean water surface, respectively. Equation (1.4) is a type of a Bernoulli equation for unsteady flows which, when evaluated at the break point and considering no wave setup in deep water to evaluate the constant, C = 0, the setup is negative (setdown) and given by ηb = −

Hb2 kb , 8 sinh 2kb hb

(1.5)

where Hb is the breaking wave height and kb is the wave number at breaking. For shallow water conditions and depth limited breaking (Hb = κ(hb + η b )), Eq. (1.5) yields ηb = −

κHb . 16

(1.6)

As an example, for a κ value of 0.78, the wave setdown is approximately 5% of the breaking wave height. With the seaward boundary condition now established, for the case of shallow water wave-breaking and the consideration of depth limited breaking across the surf zone, the wave setup is η=−

3κ2 /8 κHb + (hb − h) . 16 (1 + (3κ2 /8))

(1.7)

It is noted that in the above equation, the bottom shear stress has been taken as zero and that a shoreward directed bottom shear stress on the water column as would occur due to undertow would increase the wave setup. As examples, the ratio of wave setup to breaking height at the still water line (h = 0) and at the location of maximum wave setup (η = − h) for a κ value of 0.78 are
 5κ η(h = 0, κ = 0.78) = = 0.198 (1.8) F0 |κ=0.78 ≡ Hb 16(1 + (3κ2 /8)) κ=0.78 and Fmax |κ=0.78

η(h = − η, κ = 0.78) ≡ = F0 Hb

   3κ2  5κ  1+ = = 0.244 . 8 κ=0.78 16 κ=0.78 (1.9)

It is seen that the wave setup is strongly dependent on the value of the breaking ratio κ which will be shown to decrease with decreasing beach slope. Figure 1.2 presents the ratios, F0 and Fmax versus κ. It is useful to relate κ in an approximate manner to beach slope. Although there is not a one-to-one correspondence, Fig. 1.3 is based on the Dally et al.8 wave-breaking model and provides an approximate correspondence between uniform profile slope and the associated κ value. It is evident that the Dally et al. model provides reasonable κ values for smaller beach slopes (say less than about 0.06), but the κ values are too large for steeper slopes.

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0.4

Fo Fmax

Fo and Fmax

0.3

0.2

0.1

0.0 0.0

0.2

0.4

0.6

0.8

1.0

Kappa Fig. 1.2.

Values of F0 and Fmax versus wave-breaking index, κ (kappa).

2.0

Kappa

1.5

1.0

0.5

0.0 0.00

0.02

0.04

0.06

0.08

0.10

Profile Slope

Fig. 1.3. Relationship between profile slope and κ (kappa) value. Based on Dally et al.8 wavebreaking model.

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1.4.2. Effects of wave nonlinearity Wave nonlinearity depends on the following parameters: H/L0 and h/L0 . The nonlinearity is exhibited in the wave profile by peaked crests and flatter troughs and increases with wave height and shallow water. Somewhat surprisingly, the momentum fluxes in shallow water are less for nonlinear waves than for linear waves of the same height. This is primarily because the momentum fluxes are proportional to wave energy (Eq. (1.1)) and the wave energy is proportional to the root-mean square of the water surface displacement that is less for nonlinear waves with long troughs and peaked wave crests. Figure 1.4 presents the ratio of nonlinear to linear momentum fluxes as determined by Stream Function wave theory.9–11 The reason that the quantities for nonlinear waves are greater in deep water than for linear waves is that the nonlinear calculations extend up to the actual free surface whereas the linear quantities only extend up to the mean free surface. 1.4.3. Role of wave directionality

1.2

b =1

H/ H

b

/H H

b=

b =0

0.6

H/ H

.2 5

0. 50

=0 .7 5

0.8

.0

1.0

H/ H

Ratio of Nonlinear to Linear Momentum Flux

Equation (1.1) demonstrates that for a given wave height, the maximum shoreward flux of onshore momentum occurs for normally incident waves (θ = 0◦ ). Thus as expected for directional waves, the Sxx term is reduced. However, this reduction is relatively small as can be demonstrated by considering a breaking wave direction of 30◦ relative to a beach normal (this represents a reasonably large wave obliquity

0.4

0.2

0.0 10-3.000

2

3

4 5 6

10-2.000

2

3

4 5 6

10-1.000

2

3

4 5 5

100.000

2

3

4 5 6

101.000

h/Lo Fig. 1.4. Ratio of nonlinear to linear wave momentum flux, Sxx , for forty stream function wave combinations.12

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at breaking). The reduction in Sxx for a wave of given height and for shallow water conditions is 16.7%. 1.4.4. Effects of vegetation The effects of vegetation have been shown to result in a reduced setup and, in some cases, may cause a setdown.12 For linear waves, vegetation protruding through the water surface experiences a net drag force (quadratically related to velocity) on the vegetation in the direction of wave propagation and, of course, there must be an equal and opposing force exerted on the water column. This opposing force acting on the water column partially counteracts the force due to momentum transfer and thus reduces the wave setup (similar to an offshore directed wind stress). For linear waves and vegetation which is submerged during the entire wave passage, no net vegetation-related force exists on the water column and thus there is no effect on the wave setup. However, due to the character of nonlinear waves with higher and shorter shoreward velocities under the wave crests, even if the vegetation is fully submerged during the passage of the wave, a net drag force is induced on the vegetation in the wave propagation direction again resulting in a reduction in the wave setup and, for some cases, a wave setdown. 1.4.5. Dynamic wave setup It is noted that theoretical formulations of the dynamic wave setup must include the time dependent terms in the counterparts of Eq. (1.2). The dynamic wave setup or “surf beat” was first identified through field observations and measurements by Munk13 and Tucker.14 A number of theoretical treatments of dynamic wave setup based on various hypotheses have been developed with each focusing on a different mechanism. These include Symonds et al.15 (time-varying breakpoint), Symonds and Bowen16 (trapping of long waves by longshore bars), Schaffer and Svendsen17 (reinforcement of incoming and reflected long waves), etc. Kostense18 conducted laboratory experiments to investigate the dynamic setup component and found that the results were in qualitative agreement with the theory of Symonds et al.15 We can apply the results for monochromatic waves to investigate the approximate dynamic wave setup for a simple irregular wave case. Consider a bichromatic wave system with wave heights H1 and H2 (H1 > H2 ) and a small frequency difference between the two components. If the resulting wave group varies so slowly that static conditions occur within the surf zone, Eq. (1.7) applies and is written as η = F Hb ,

(1.10)

where Hb is the breaking wave height and F is a proportionality factor depending on whether the referenced setup is at the still water shoreline or the maximum wave setup (see Eqs. (1.8) and (1.9)). The maximum and minimum wave setup values are: η max =

F (H1 + H2 ) ,

ηmin =

F (H1 − H2 ) .

(1.11)

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Table 1.1. Static and dynamic wave setup characteristics for a biharmonic wave system. H2 /H1 0.2 0.4 0.6 0.8 1.0

η max /F H1

η avg /F H1

(η max − η avg )/η avg

1.2 1.4 1.6 1.8 2.0

1.01 1.04 1.09 1.16 1.27

0.19 0.35 0.47 0.55 0.57

It can be shown that the average wave setup depends on the ratio H2 /H1 as shown in Table 1.1. The fourth column presents the ratios of the maximum dynamic wave set amplitude to the average wave setup component. In the case with H2 = H1 , the dynamic wave setup displacement from the mean setup equals 57% of the average wave setup (Table 1.1, Column 4). In the above, we have examined the dynamic wave setup for the case of a simple bichromatic wave system in which the difference in frequencies of the two components was fairly small. For the case of a wave spectrum, the situation is much more complex with, for the case of a narrow spectrum, the group envelope varying according to the Rayleigh distribution. For the case of a wide spectrum, the dynamic component is reduced considerably.

1.5. Laboratory and Field Measurements of Wave Setup Having reviewed the theory of wave setup and its relationship to various factors, the two sources available for evaluation are laboratory and field data. 1.5.1. Laboratory experiments on wave setup Many laboratory investigations of static and dynamic wave setup have been conducted. The results of an early laboratory investigation with monochromatic waves by Bowen et al.19 are shown in Fig. 1.5. For this study, the ratio of maximum wave setdown and wave setup on the beach face to breaking wave height are − 0.035 and + 0.316, respectively, compared with − 0.049 and + 0.244 on the beach face for a κ value of 0.78. The effect of beach slope has been noted earlier and the relatively large beach slope of 0.082 in these experiments is undoubtedly a contributor to the large setup value. Later, laboratory investigations have included examination of irregular waves including measurements of water particle velocities and pressures which form the basis of the Sxx momentum flux component. Battjes20 conducted one of the earliest laboratory studies of wave setup due to irregular waves. Setup was measured through bottom mounted manometers and it was found that the wave setup was less than predicted. It was hypothesized that this difference was possibly due to air in the water column of the manometers. The entire setup was shifted landward relative to the theoretical and this delay was later attributed to a “roller” that is transported along with the wave crest region and

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Fig. 1.5.

Measured wave setup and setdown in the laboratory.19

conveys wave energy and momentum landward prior to transfer of the momentum to the water column and the associated wave setup.21 Later, Stive and Wind22 conducted a very detailed laboratory investigation in which they demonstrated the role of wave nonlinearity. In this study, the momentum flux components (velocities and pressures) were measured to the degree possible and it was found that the calculated wave setup based on nonlinear wave theories was in much better agreement with measured wave setup than calculations based on linear wave theories. In these comparisons, it was not necessary to introduce the roller concept. The two laboratory studies reviewed above have focused on static setup and it has been noted that irregular waves also produce dynamic wave setup. Hedges and Mase23 have presented an interesting reanalysis of earlier runup laboratory measurements by Mase24 in which irregular waves provided the forcing.b The planar slopes represented in the data were: 1:5, 1:10, 1:20, and 1:30. Figure 1.6 presents an example of the form in which the data were plotted where the horizontal axis is b Walton25

was the first to analyze the Mase data to extract the static wave setup.

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Fig. 1.6.

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Variation of nondimensional runup with Iribarren number.23

the Iribarren number, ξ0 defined as tan α ξ0 =  H/L

(1.12)

in which tan α is the profile slope. The interpretation of Fig. 1.6 is that for a zero slope (zero Iribarren number), there would be no short wave runup; therefore, the intercept represents the sum of the static and dynamic components of wave runup. Equations of the following form were fit to plots of the type of Fig. 1.6: Rchar Schar = + cchar ξ0 , H1/3 H1/3

(1.13)

where the subscript “char” refers to the percent associated with the variable; for example, the 2% runup is defined as R2% . It was found that both Schar and cchar were Rayleigh distributed with S1/3 and c1/3 equal to 0.27 and 1.04, respectively, where only the first term represents wave setup and is of interest here. The results for Schar can be interpreted in terms of the static and dynamic wave setup components. As an example, Smean = 0.17 and S2% = 0.37. Thus, the 2% value of the nondimensional dynamic setup defined here as ∆S2% is ηdyn,2% = (S2% − Smean ) = (0.37 − 0.17) = 0.20 . (1.14) ∆S2% = H1/3 Thus, the mean setup at the still waterline is 17% of the significant wave height measured at the toe of the slope and the 2% dynamic component at the still waterline is 20% of the significant wave height at the toe of the slope or slightly larger than the mean wave setup. These results are interesting and of reasonable magnitudes; however, there are two problems with recommending them for universal application. First, we know that the mean setup depends on the slope (through the κ dependency

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as discussed earlier), and the second is that the oscillating wave setup component should depend on the width of the input spectrum. Referring to Fig. 1.6, which is one of several similar plots presented in the Hedges and Mase paper, since each plot may include a mix of beach slopes, the slope dependency is not resolved in the Smean results which of course are derived from the y-intercept of these graphs. Secondly, these experiments were not designed to evaluate the effect of spectral width and the spectral characteristics included in the experiments are not known. However, it is of interest to identify the “representative” κ value and beach slope associated with a Smean = η avg /H1/3 of 0.17. Referring to Fig. 1.2, we see that the associated κ value is approximately 0.63 for F0 . Based on the Dally et al.8 breaking wave model, the associated beach slope from Fig. 1.3 is 1:29 compared to the beach slopes in the Mase experiments ranging from 1:30 to 1:5. 1.5.2. Field experiments on wave setup The paragraphs below describe several field experiments and observations of wave setup. An early study of wave setup comprised a pair of observations at an exposed coastal site (Narragansett Pier, RI) and a calmer water site (Newport, RI) where, at the latter, wave action was assumed not a factor and was found to show an approximate 3 foot water level difference during the peak of the 1938 hurricane storm surge.26 In a second early field experiment on wave setup at Fernandina Beach, Florida, Dorrestein27 placed transparent plastic tubes with lightweight floats to track the water surface on the beach in the zone of wave setup. To obtain the setup records, 16 mm movie film recorded the tracked surface of the floats. A float type tide gage on the end of a fishing pier provided offshore water level records. Through analysis of the tide gage records and the beach placed setup gages, Dorrestein27 evaluated the setup (with respect to the end of the pier) and compared observational results to existing setup theory. He found the measured setup in four of five experiments to be larger than the computed setup. One shortcoming of Dorrestein’s work is that the water level records were only 72 s in length and thus subject to considerable scatter and large standard deviation as later noted by Holman and Sallenger.28 Although rationale was provided by Dorrestein27 for possible differences between measured and computed setup in this early experiment, large discrepancies between measured and analytically or numerically computed setup still exist today. A North Sea field wave setup experiment was conducted on the Island of Sylt by Hansen.29 Utilizing a combination of ultrasonic wave gages and pressure sensor wave gages out to a distance of 1280 m from shore (10 m depth), Hansen29 found good correspondence of data to an empirical expression provided by: η = 0.3Hos = 0.42Horms .

(1.15)

Hansen also noted the maximum wave setup to be approximately 50% of the significant breaking wave height. It is not clear as to the methodology utilized to obtain η max in this field experiment.

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A wave setup field experiment was conducted as part of the Nearshore Sediment Transport Study at Torrey Pines Beach, San Diego, California by Guza and Thornton.30 The Torrey Pines Beach face was gently sloping (beach slope ≈ 0.02) and the beach material was a moderately sorted fine grain sediment (≈ 0.1 mm). A dual wire resistance runup meter was used for the recording and estimation of the wave setup. It should be noted that the measurements of the wave setup were considered to be the average runup determined from wires placed approximately 3 cm above the beach level rather than an actual water level at one location in these experiments. Offshore pressure sensors outside the surf zone at mean depths of 7 to 10.5 m were used for estimating wave height with recording lengths of 4096 s. Guza and Thornton30 note specific problems in the data set, which are typical of field measurements, i.e., the difficulty in obtaining a common datum for the offshore wave measurements and the beach wave setup measurements. Results of their measurement program suggest an empirical relationship as follows: η = 0.17Hos = 0.24Horms

(1.16)

with scatter that suggests η/Hos ranging approximately from 0.05 to 0.50 for individual experiments. Holman and Sallenger28 conducted a field experiment for measuring wave setup as well as other surf zone parameters at the U.S. Army Corps of Engineers field research pier in Duck, NC, USA. Data on water level at the shoreline were collected using longshore looking time lapse photography from Super-8 movie cameras mounted on the research pier scaffolding. The beach at the experiment site had a very steep foreshore (∼ 1 on 10) while the offshore profile slope is much milder (∼ 1 on 100). Beach material was bimodal in size with a median sand size of 0.25 mm and a coarse fraction of 0.75 mm. Results of the experiments showed considerable scatter and dependence on tide level. Regression lines were fit to the data (segmented by tide levels) with results as follows for high tide and mid-tide data: η = 0.35ξ0 + 0.14 (high tide) , (1.17) Hs η = 0.46ξ0 + 0.06 (mid-tide) . (1.18) Hs As most of the data fell in a range of ξ0 = 1 to 2, the maximum setup was noted to be of the same order as the significant wave height in many of the experiments, much higher than theoretically suggested values. Note that in terms of Horms (based on consideration of monochromatic theory results) the setup would be much higher than most other studies show or suggest. Although Holman and Sallenger28 conclude from their experiments that the setup is dependent on the Iribarren number, it is not entirely clear from their data, especially for higher waves (i.e., see Fig. 1.4, Ref. 28). An additional problem that must be considered when computing the Iribarren number for real beaches and irregular waves is how to define beach slope. It should be noted that video camera (visual) approaches estimate setup via the measurement of the water surface elevation on the beach (similar to the Guza and Thornton measurements) rather than an actual vertically fluctuating water level. The anomaly between dependence of

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setup on Iribarren number as noted by Holman and Sallenger28 is likely due to the aforementioned relationship between the wave-breaking coefficient, κ, and beach slope. Nielsen31 and Davis and Nielsen32 conducted a novel setup experiment on Dee Why Beach, in New South Wales, Australia using a set of manometer tubes as shown in Figs. 1.7 and 1.8 from Davis and Nielsen.32 The tubes were deployed throughout the beach face and surf zone. A total of 120 setup profiles were measured in 11 days. Wave heights Horms ranged from 0.6 to 2.6 m in height and significant wave periods (Ts ) ranged from 5.8 to 12.1 seconds. A shoreline setup of about 40% of Horms was found although Davis and Nielsen32 point out that there is reason to believe that the surf zone characteristics influence the relationship between wave height and setup magnitude, and also note a problem of defining beach slope via the Iribarren number. Nielsen31 and Davis and Nielsen32 also observe that a major portion of the setup occurred on the beach face as shown in Fig. 1.9. Nielsen31 points out that previous field investigations have typically measured the mean water level elevation on the beach as opposed to the average fluctuating mean water level in the vertical plane (i.e., the wave setup as usually defined), and that the two measurements are often different in part due to the beach permeability, which in turn is related to beach material size. The issue of extracting wave setup from runup and rundown on the beach is illustrated in Fig. 1.10. King et al.33 collected wave setup data at Woolacombe Beach in North Devon, U.K. which faces the North Atlantic Ocean. The beach face slope varied between 1 on 40 at high tide and 1 on 70 at mid-tide level with a tidal range of 3 m at neap and as much as 9 m at springs. Beach face material consisted of fine sand with

Fig. 1.7.

Manometer setup of Davis and Nielsen32 for measuring setup.

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Fig. 1.8.

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Schematic diagram of apparatus (from Ref. 32).

Fig. 1.9. Dimensionless setup versus total depth where much of the setup occurs on the beach face (from Ref. 32). In this figure, B and D are equal to η and h as used in this chapter, respectively.

90% in the 0.125 mm to 0.25 mm size range. Pressure transducers were utilized to collect wave and setup information at various stations across the beach and also in a longshore direction to assess the spatial variability of the mean setup. Both tripod mounted and buried pressure transducers were utilized. The buried pressure transducers were 50 to 80 cm below the beach surface and were protected by a porous cover. Instruments collected pressure data which were then transformed to water level data over 4096 second intervals. Data did not include sampling in very shallow water and the maximum wave setup was estimated by extrapolating the

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Fig. 1.10. Illustration of differences between mean water level (MWL) shoreline and mean water line on beach.

water surface from the most shoreward water stations. Wave setup estimated from the data showed the wave setup to be roughly: η = 0.10H os = 0.14H orms

(1.19)

with most of the values of η/H orms between 0.11 and 0.15. The authors do not speculate as to why such low values of setup (compared to analytical results) were found in this measurement program. Yanagishima and Katoh34 discuss field measurements of mean water level near the shoreline on the Pacific Coast of Japan as measured by an ultrasonic wave gage mounted on a pier where the mean depth of water was ∼ 0.4 m. The setup was determined via a multiple regression approach on 1305 sets of (20 minute records) data taking into account astronomical tide, wind setup, and atmospheric pressure head components of mean water level. Their data included 91 records in which the offshore wave height was above 3 m. Yanagishima and Katoh’s34 regression analysis suggested the following relationship:  0.2 Los η = 0.0520H os , (1.20) Hos which can be formulated in terms of Iribarren number for their beach slope (1 on 60) to the following: η = 0.27H os (ξ0 )0.4 = 0.38H orms (ξ0 )0.4 .

(1.21)

Yanagishima and Katoh34 noted reasonable agreement with the theory of Goda35 (to be discussed later). Even higher values of setup would be expected on the beach face in accord with theory and findings of other researchers. Greenwood and Osborne36 conducted field measurements on a Georgian Bay Beach, in Lake Huron, Ontario, Canada. Lake Huron has no measurable tide and the beach profile at the site had a slope of 0.015 with a steeper sloped (0.031 to 0.047) inshore bar. Setup was measured using surface piercing resistance wire wave staffs with the shoreward most gage being in approximately 0.4 m of water depth. Measured setup values were found as follows: η = 0.19H os = 0.27H orms .

(1.22)

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It is again noted that even higher values of setup would be expected on the beach face in accord with theory and experience of other researchers. Further work by Hanslow and Nielsen37,38 utilized the manometer tube deployment shown in Fig. 1.7 on three additional beaches (Seven Mile, Palm, and Brunswick) in New South Wales, Australia. With beach face slopes ranging from 0.03 to 0.16 and mean grain sizes of swash zone beach material ranging from 0.18 to 0.5 mm, shoreline beach setup was measured using 20 minute record averages. Using the data from these three beaches as well as earlier measurements at Dee Why Beach (see Refs. 31 and 32), linear least square relationships were fit to the data as follows: η = 0.27H os = 0.38H orms

with R = 0.65

(1.23)

  η = 0.040 Hos L0 = 0.048 Horms L0

with R = 0.77 ,

(1.24)

or

where a somewhat higher value of explained regression was noted using wave height and wave period. Data and regression lines for these two relationships are shown in Figs. 1.11 and 1.12. The improvement in fit due to inclusion of the deep water wavelength is not evident visually. A significant finding of these studies was that a major portion of the setup occurred on the beach face (see Fig. 1.9). Further measurements on wave setup at two river entrances is also discussed in Hanslow and Nielsen37 and Dunn et al.39 with the result that the wave setup at river entrances was found to be (somewhat surprisingly) negligible. Lentz and Raubenheimer40 report on a field experiment at the U.S. Army Field Research Pier in Duck, NC, USA where 11 pressure sensor gages and 10 sonar altimeters extended across the surf zone from 2 to 8 m of water depth. Close agreement with Longuet-Higgins radiation stress theory for wave setup was noted

Fig. 1.11.

Empirical relationship between setup and wave height (from Ref. 38).

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Fig. 1.12.

Empirical relationship between setup and wave parameters (from Ref. 38).

although the lack of setup measurements in shallow water (< 2 m) did not allow conclusions regarding the maximum setup that might be expected on the beach. Raubenheimer et al.41 report on a second field experiment at the U.S. Army Field Research Pier in Duck, NC, USA where 12 buried pressure sensor gages were employed across the surf zone from the shoreline to 5 m of water depth. Again good agreement with Longuet-Higgins and Stewart42 radiation stress theory was noted by integration of the cross-shore momentum equation to estimate the wave setup for water depths greater than 1 m but the theory was found to under-predict wave setup in shallow water (h < 1 m). The lack of setup measurements on the beach face did not allow conclusions regarding the maximum setup that might be expected on a beach although an empirical equation was provided to estimate wave setup at the SWL line as follows: η SWL −1/3 = 0.019 + 0.003βf , Hos

(1.25)

where βf is the average slope across the surf zone. Raubenheimer et al.41 suggest that theory under-predicts the setup by a factor of 2 for water depths less than 1 m. Stockdon et al.43 using video shoreline water level time series determined wave setup and wave runup results during 10 diverse field experiments (four from Duck, NC, USA; four from West Coast beaches in California/Oregon, USA; and two from Terschelling, The Netherlands). These wave setup results were analyzed to provide empirical parameterizations for wave setup under many natural beach conditions as follows:  η = 0.385βf H0 L0 , (1.26) which, assuming that tan βf ≈ βf can be expressed in terms of the Iribarren number as η = 0.385ξ0 H0

(1.27)

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and for extremely dissipative beaches η 0.043 = , H0 H0 /L0

(1.28)

where H0 is the effective deep water wave height, L0 is the deep water wave length associated with the peak spectral period and βf is the average slope over a depth range defined in terms of the standard deviation of the water surface displacement. It should again be noted that the video camera (visual) approach estimates setup via the mean of measurements of the water surface elevation on the beach rather than the mean of fluctuating water levels at one location. Results from nine of the field experiments presented here have been analyzed to determine the average ratio of wave setup at the still water line to significant wave height and its associated standard deviation. The ratios at the still water line were determined to be 0.191±0.100. Several caveats apply to these results. In cases where the beach slope and/or the deep water wave steepness was incorporated into the expression presented, these were taken as 0.01 and 0.04, respectively. Some of the published expressions were in terms of the breaking wave height and some in terms of the deep water wave height and no attempt was made to differentiate between breaking and deep water wave heights. The Holman and Sallenger results were not included in these results as they appeared to be anomalously high. Finally, the wave setup ratio at the intersection of the mean water line intersection with the beach profile would be greater than the average ratio (0.191) above. Also, although not examined in detail here, the dynamic wave setup which increases with energetic narrow spectra, would also contribute to the total wave setup. It is relevant to note that results from field measurements are often not consistent, possibly due to: (1) Limited measurement distances across the nearshore. (2) Use of many different approaches to measure/evaluate setup (i.e., videos, pressure sensors, runup gages, manometers, etc.). (3) Inherent difficulties in obtaining a consistent datum for nearshore measurements and offshore measurements. (4) A clear definition of setup on the beach face is lacking due to the nature of the permeable beach and the difficulty of sub-aerial setup measurements. 1.6. Published Guidance on Wave Setup for Engineering Applications Several sources of wave setup recommendations are available; two are reviewed here. The U.S. Army Corps of Engineers 1984 Shore Protection Manual (SPM) presents a graphical method to calculate wave setup at mid-depth of the surf zone. This method, developed for irregular waves, is presented in Fig. 1.13 in which the normalized setup has been multiplied by a factor of 2 to transfer approximately the results to the still water shoreline. The effect of beach slope and deep water wave steepness in Fig. 1.13 are evident.

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Fig. 1.13. Nondimensional wave setup versus deep water wave steepness and profile slope by the 1984 Shore Protection Manual recommendations as incorporated in Appendix D of FEMA44 Guidelines. Note that the normalized setup has been multiplied by a factor of 2 to transfer the setup from the mid-depth of the surf zone as it appears in SPM to the approximate still water level contour. Note: S in this figure is equal to η in this chapter.

Fig. 1.14. Nondimensional wave setup by Goda versus deep water wave steepness and relative water depth within the surf zone. Profile slope = 1:100.

Goda35 has presented guidance for static and dynamic wave setups due to irregular waves. The guidance for static setup and a profile slope of 1:100 is shown in Fig. 1.14. The effects of various deep water wave steepness values are illustrated in Fig. 1.14.

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Table 1.2. Comparison of nondimensional wave setup by SPM and Goda methods for irregular waves. η/H0 H0 /L0

SPM

Goda

0.005 0.01 0.02 0.04 0.08

0.154 0.135 0.120 0.103 0.097

0.122 0.102 0.083 0.065 0.049

Note: Values in SPM method have been multiplied by 2.0 to transfer from surf zone mid-depth to still water line.

Table 1.2 presents a comparison of ratios of nondimensional wave setup values at the still water shoreline as recommended by SPM and Goda. In examining the results in Table 1.2, recall that an additional wave setup occurs from the still water line to the location where the maximum setup intersects the beach profile.

1.7. Summary and Recommendations The reviews of theory, laboratory and field data, and published guidance for engineering applications presented here have identified static and dynamic wave setup components as contributing to the deviation from still water level in the surf zone and their relevance to engineering design. Examination of the static wave setup has reinforced the effect of beach slope on wave setup. The theory presented here does not account for the onshore bottom stress acting on the water column due to undertow. The available field measurement results exhibit a wide range of wave setup to wave height ratios. Some of this variability is undoubtedly due to the effect of profile slope, which is not accounted for explicitly in some of the analyses and part is due to the effect of wave-breaking in depths greater than the shallow water limit. Design methodology should account for the static and dynamic wave setup components. In determining the wave setup to include in design, the characteristics of the particular application of interest should be compared with those of the various field and laboratory experiments available including those referenced here. The dominant role of beach slope should be recognized. The preliminary results presented here of η/Hs = 0.191 ± 0.100 may serve as a useful guide for the static wave setup component. It is hoped that further research with improved instrumentation, modern surveying techniques, and more diverse field site studies will help to clarify both the static and dynamic wave setup components for future design applications.

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References 1. A. T. Ippen, Estuary and Coastline Hydrodynamics (McGraw-Hill Book Company, 1966), 744 pp. 2. M. S. Longuet-Higgins, Radiation stress and mass transport in gravity waves with application to “surf beats”, J. Fluid Mech. 13(4), 481–504 (1962). 3. M. S. Longuet-Higgins and R. W. Stewart, A note on wave set-up, J. Marine Res. 21, 4–10 (1963). 4. M. S. Longuet-Higgins and R. W. Stewart, Radiation stresses in water waves; a physical discussion, with applications, Deep-Sea Res. 11, 529–562 (1964). 5. R. A. Holman, Wave set-up, Handbook of Coastal and Ocean Engineering, Vol. 1, ed. J. Herbich (Gulf Publishing, 1990), Chapter 11, pp. 635–646. 6. van der Meer, TAW Technical Report: Wave Run-up and Wave Overtopping at Dikes, Technical Advisory Committee for Flood Defence, Delft (2003). 7. M. S. Longuet-Higgins, On the wave-induced difference in mean sea level between the two sides of a submerged breakwater, J. Marine Res. 25(2), 148–153 (1967). 8. W. R. Dally, R. G. Dean and R. A. Dalrymple, Wave height variation across beaches of arbitrary profile, J. Geophys. Res. 90(C6), 11917–11927 (1985). 9. R. G. Dean, Stream function representation of nonlinear ocean waves, J. Geophys Res. 79(30), 4489–4504 (1965). 10. R. G. Dean, Evaluation and Development of Water Wave Theories for Engineering Application; Volume I: Presentation of Research Results; Volume II: Tabulation of Dimensionless Stream Function Variables, Special Report No. 1, Published by U.S. Army Corps of Engineers, Coastal Engineering Research Center (1974). 11. R. A. Dalrymple, A finite amplitude wave on a linear shear current, J. Geophys. Res. 87(C1), 483–491 (1974). 12. R. G. Dean and C. J. Bender, Static wave setup with emphasis on damping effects by vegetation and bottom friction, Coast. Eng. 13, 149–156 (2006). 13. W. R. Munk, Surf beats, Trans. Am. Geophys. Union 30, 849–854 (1949). 14. M. J. Tucker, Surf beats: Sea waves of 1 to 5 minutes period, Proc. Roy. Soc. A 202, 565–573 (1950). 15. G. Symonds, D. A. Huntley and A. J. Bowen, Two-dimensional surf beat: Long wave generation by a time-varying breakpoint, J. Geophys. Res. 87(C1), 492–498 (1982). 16. G. Symonds and A. J. Bowen, Interaction of nearshore bars with wave groups, J. Geophys. Res. 89(C2), 1953–1959 (1984). 17. H. A. Schaffer and I. A. Svendsen, Surf beat generation on a mild-slope beach, Proc. ASCE Int. Conf. Coastal Engineering (1988), pp. 1058–1072. 18. J. K. Kostense, Measurements of surf beat and set-down beneath wave groups, Proc. ASCE Int. Conf. Coastal Engineering (1984), pp. 724–740. 19. A. J. Bowen, D. L. Inman and V. P. Simmons, Wave set-down and set-up, J. Geophys. Res. 73, 2569–2577 (1968). 20. J. A. Battjes, Set-Up Due to Irregular Waves, Report No. 72–2, Communications on Hydraulics, Delft University of Technology, Department of Civil Engineering (1972), 13 pp. (Paper also presented at the 13th Int. Conf. Coastal Engineering, Vancouver, B.C.). 21. I. A. Svendsen, Wave heights and set-up in a surf zone, Coast. Eng. 8, 302–329 (1984). 22. M. J. F. Stive and H. G. Wind, A Study of Radiation Stress and Set-Up in the Nearshore Zone, Publication No. 267, Waterlopkundig Laboratorium, Delft Hydraulics Laboratory (1982), 25 pp.

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23. T. S. Hedges and H. Mase, Modified Hunt’s equation incorporating wave setup, J. Waterway Port Coast. Ocean Eng. 130(3), 109–113 (2004). 24. H. Mase, Random wave runup height on gentle slopes, J. Waterway Port Coast. Ocean Div. 115(WW2), 649–661 (1989). 25. T. L. Walton, Interim guidance for prediction of wave run-up on beaches, Ocean Eng. 19(2), 199–207 (1999). 26. T. Saville, Experimental determination of wave set-up, Proc. 2nd Technical Conf. Hurricanes, Miami Beach, FL, National Hurricane Research Project Report No. 50, U.S. Department of Commerce Washington, D.C. (1961), pp. 242–252. 27. R. Dorrestein, Wave set-up on a beach, Proc. 2nd Technical Conf. Hurricanes, Miami Beach, FL, National Hurricane Research Project Report No. 50, U.S. Department of Commerce, Washington, D.C. (1961), pp. 230–241. 28. R. A. Holman and A. H. Sallenger, Setup and swash on a natural beach, J. Geophys. Res. 90(C1), 945–953 (1985). 29. U. A. Hansen, Wave setup and design water level, J. Waterway Port Coast. Ocean Div. 104(WW2), 227–240 (1978). 30. R. T. Guza and E. B. Thornton, Wave set-up on a natural beach, J. Geophys. Res. 96(C2), 4133–4137 (1981). 31. P. Nielsen, Wave setup: A field study, J. Geophys. Res. 93(C12), 15643–15652 (1988). 32. G. A. Davis and P. Nielsen, Field measurement of wave set-up, ASCE Int. Conf. Coastal Engineering, Malaga, Spain (1988), Chapter 38, pp. 539–552. 33. B. A. King, M. W. L. Blackley, A. P. Carr and P. J. Hardcastle, Observations of wave induced setup on a natural beach, J. Geophys. Res. 95(C12), 22289–22297 (1990). 34. S. Yanagishima and K. Katoh, Field observation on wave setup near the shoreline, Proc. 22nd Int. Conf. Coastal Engineering, Vol. 1, ASCE, New York, N.Y. (1990), Chapter 7, pp. 95–108. 35. Y. Goda, Random Seas and Design of Maritime Structures (World Scientific Publishing Co., 2000), 443 pp. 36. B. Greenwood and P. D. Osborne, Vertical and horizontal structure in cross-shore flows: An example of undertow and wave setup on a barred beach, Coast. Eng. 14, 543–580 (1990). 37. D. J. Hanslow and P. Nielsen, Wave setup on beaches and in river entrances, 23rd Int. Conf. Coastal Engineering, Venice, Italy (1992), pp. 240–252. 38. D. J. Hanslow and P. Nielsen, Shoreline set-up on natural beaches, J. Coast. Res. SI15, 1–10 (1993). 39. S. L. Dunn, P. Nielsen, P. A. Madsen and P. Evans, Wave setup in river entrances, Proc. 27th Int. Conf. Coastal Engineering, ASCE, New York, Sydney, Australia (2000), pp. 3432–3445. 40. S. Lentz and B. Raubenheimer, Field observations of wave setup, J. Geophys. Res. 104(C11), 25867–25875 (1999). 41. B. Raubenheimer, R. T. Guza and S. Elgar, Field observations of wave-driven setdown and setup, J. Geophys. Res. 106(C3), 4629–4638 (2001). 42. M. S. Longuet-Higgins and R. W. Stewart, Radiation stresses in water waves: A physical discussion with applications, Deep Sea Res. 11(4), 529–562 (1964). 43. H. F. Stockdon, R. A. Holman, P. A. Howd and A. H. Sallenger, Jr., Empirical parameterization of setup, swash, and runup, Coast. Eng. 53, 573–588 (2006). 44. FEMA, Guidelines and Specifications for Flood Hazard Mapping Partners, Appendix D: Guidance for Coastal Flooding Analysis and Mapping, Map Modernization Program, Washington, D.C. (2003). Also available at: www.fema.gov/fhm/ dl cgs.shtm.

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Chapter 2

Wavemaker Theories Robert T. Hudspeth School of Civil and Construction Engineering Oregon State University, Corvallis, OR 97331, USA [email protected] Ronald B. Guenther Department of Mathematics Oregon State University, Corvallis, OR 97331, USA [email protected] The fundamental solutions to the wavemaker boundary value problem (WMBVP) are given for 2D channels, 3D basins, and circular basins. The solutions are given in algebraic equations that replace integral and differential calculus. The solutions are generic and apply to both full- and partial-draft piston and hinged wavemakers; to double-articulated wavemakers, and to directional wave basins. The WMBVP is solved by conformal mapping and by domain mapping. The loads on a wavemaker are connected to the radiation boundary value problem for semiimmersed bodies and demonstrate the connection of these loads to the added mass and radiation damping coefficients required to compute the dynamic response of large Lagrangian solid bodies.

2.1. Introduction Wavemaker theories play several important roles in coastal and ocean engineering. The most important role is the application to laboratory wavemakers for both wavemaker designs and wave experiments. A second role for wavemaker theories is to compute a scalar radiated wave potential to compute the wave-induced loads on large solid bodies applying potential wave theory. The displacements and rotations of a semi-immersed six degrees-of-freedom large Lagrangian solid body are related to the displacements and rotations of wavemakers. The boundary between a planar wavemaker and an ideal fluid requires special care because the fluid motion

25

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is an Eulerian field with time and space as the independent variables, and the planar wavemaker is a Lagrangian solid body with time and the wavemaker as the independent variables. Consequently, the kinematic boundary condition will be different from the free surface boundary that separates two Eulerian fluid fields of air and water. The boundary between the fluid and wavemaker separates an Eulerian field (the fluid) from a Lagrangian body (the wavemaker), and the wavemaker kinematic boundary condition (WMKBC) must convert the Lagrangian wavemaker motion to a Eulerian field motion in order that the independent variables for both dependent motion variables are equivalent. This may be accomplished by multiplying the Lagrangian motion of the wavemaker by the unit normal to the boundary. Because the unit normal is a function of space and the Lagrangian wavemaker motion is a function of time, the product will produce a motion that is a function of both space and time that are the independent variables of the Eulerian fluid field. Although this fact is not central to the WMBVP, it is an important connection between the WMBVP and the radiation potential boundary value problem for semi-immersed large Lagrangian solid bodies.1 The formulae for computing the two fundamental fluid unknowns for an incompressible fluid of the velocity q(x, z, t) and the pressure p(x, z, t) from a scalar velocity potential Φ(x, z, t) are given first. The classical linear WMBVP for dimensionless 2D planar wavemaker is reviewed for two types of double-articulated planar wavemakers. The sway X1 (t) displacement of a full-draft piston wavemaker and the roll Θ5 (t) rotation of a hinged wavemaker are connected directly to the sway displacement and the roll rotation of a semi-immersed large Lagrangian solid body. In this review, integral calculus formulae for computing the integrals that are required to compute the coefficients of the eigenseries for the fluid motion, to compute the loads on the wavemaker and the average power required to generate the propagating waves are replaced by generic algebraic formulae. For example, an integral equation that is required to compute the nth eigenseries coefficient Cn for the nth eigenfunction Ψn (Kn , z/h) from a wavemaker shape function χ(z/h) may be computed symbolically and expressed by a dimensionless algebraic formula In (α, β, b, d, Kn ), that is given by
Cn = h

0

−1

χ(z/h)d(z/h)Ψn (z/h)d(z/h) = In (α, β, b, d, Kn ).

(2.1)

The coefficient in (2.1) may then be computed very efficiently by substitution into algebraic formulae in all subsequent applications. Next a dimensionless theory for both amplitude-modulated (AM) and phase-modulated (PM) circular wavemakers is reviewed. Then, a dimensionless theory for double-actuated wavemakers is reviewed. Following that, a dimensionless directional wavemaker theory for large wave basins based on a WKBJ approximation1 is reviewed. Next, a theory for sloshing waves due to transverse motions of a segmented wavemaker in a narrow wave channel is reviewed. Then, 2D planar wavemakers are mapped to a unit circle by conformal mapping and to a fixed rectangular domain by domain mapping; and both the linear and nonlinear wavemaker solutions are computed numerically.

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2.2. Planar Wavemaker in a 2D Channel Two generic planar wavemaker configurations are shown in Figs. 2.1(a) and 2.1(b). The fluid motion may be obtained from the negative gradient of a dimensional scalar velocity potential Φ(x, z, t) according to q(x, z, t) = u(x, z, t)ex + w(x, z, t)ez = −∇2 Φ(x, z, t), where the 2D gradient operator in (2.2a) is given by ∇2 (•) =

∂(•) ∂(•) ex + ez . ∂x ∂z

Fig. 2.1(a).

Definition sketch for a Type I planar wavemaker.

Fig. 2.1(b).

Definition sketch for a Type II planar wavemaker.

(2.2a)

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The total pressure field P (x, z, t) may be computed from the unsteady Bernoulli equation according to 

 ∂Φ(x, z, t) 1 2 P (x, z, t) = p(x, z, t) + pS (z) = ρ − |∇Φ(x, z, t)| + Q(t) − ρgz, ∂t 2 (2.2b) where Q(t) = the Bernoulli constant; and the free surface elevation η(x, t) for zero atmospheric pressure according to 1 η(x, t) = g



 ∂Φ(x, η, t) 1 2 − |∇Φ(x, η, t)| + Q(t) ; ∂t 2

x ≥ ξ(η, t);

z = η(x, t). (2.2c)

The scalar velocity potential must be a solution to the Laplace equation ∇22 Φ = 0;

x ≥ ξ(z, t);

−h ≤ z ≤ η(x, t),

(2.3a)

with the following boundary conditions: Kinematic Bottom Boundary Condition (KBBC): ∂Φ = 0; ∂z

x ≥ ξ(−h, t);

z = −h.

(2.3b)

Combined Kinematic and Dynamic Free Surface Boundary Condition (CKDFSBC):   ∂ 2Φ 1 ∂ ∂Φ dQ − − ∇Φ · ∇ |∇Φ|2 + = 0; + g ∂t2 ∂z ∂t 2 dt

x ≥ ξ(η, t);

z = η(x, t). (2.3c)

Kinematic WaveMaker Boundary Condition (KWMBC): A Stokes material surface for planar wavemaker is W (x, z, t) = x − ξ(z, t), and the Stokes material derivative gives the KWMBC from ∂Φ ∂ξ ∂Φ ∂ξ DW = + − = 0; Dt ∂x ∂t ∂z ∂z

x = ξ(z, t);

−h ≤ z ≤ η(t).

(2.3d)

Kinematic Radiation Boundary Condition (KRBC): A KRBC is required as x → +∞ for uniqueness to insure that propagating waves are only right progressing or that evanescent eigenmodes are bounded. For a temporal dependence proportional to exp ±iωt, the KRBC may be expressed by  lim

x→+∞

 ∂ ± iKn Φ(x, z, t) = 0. ∂x

(2.3e)

A velocity potential ϕ(x, z) may be defined by the real part of Φ(x, z, t) = Re{ϕ(x, z) exp −i(ωt + ν)},

(2.4)

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where Re{•} means the real part of {•}; and ν = arbitrary phase angle. The linearized WMBVP for kh = O(1) is1 ∇22 ϕ(x, z) = 0;

0 ≤ x < +∞;

∂ϕ(x, z) = 0; ∂z

0 ≤ x + ∞;

∂ϕ(x, z) − k0 ϕ(x, z) = 0; ∂z  lim

x→+∞

η(x, t) = Re

z = −h,

(2.5b)

z = 0,

 ∂ − iKn ϕ(x, z) = 0, ∂x x = 0;

 −iω ϕ(x, 0) exp −i(ωt + ν) ; g

p(x, z, t) = ρ

(2.5a)

0 ≤ x < +∞;

∂ϕ(x, z) ∂ξ(z, t) exp −i(ωt + ν) = − ; ∂x ∂t 

−h ≤ z ≤ 0,

∂Φ(x, z, t) ; ∂t

(2.5d)

−h ≤ z ≤ 0, x ≥ 0;

0 ≤ x < +∞;

(2.5c)

z = 0,

z = 0,

(2.5e)

(2.5f)

(2.5g)

where k0 = ω 2 /g. Because the boundary conditions defined by (2.5b)–(2.5e) are prescribed on boundaries with constant values of the independent variables x and z, a solution by the method of separation of (independent) variables may be computed.1 The instantaneous wavemaker displacement ξ(z, t) from its mean position x = 0 is assumed to be strictly periodic in time with period T = 2π/ω, and may be expressed by     S χ(z/h) exp −i(ωt + ν) ξ(z/h, t) = Re i (∆/h)   S = χ(z/h) sin(ωt + ν). (2.6) (∆/h) The specified shape function χ(z/h) for the Type I wavemaker shown in Fig. 2.1(a) is valid for either a double-articulated piston or hinged wavemaker of variable draft and is given by the following dimensionless equation for a straight line2 : χ(z/h) = [α(z/h) + β][U (z/h + 1 − d/h) − U (z/h + b/h)],

(2.7a)

where α, β = dimensionless constants; U (•) = the Heaviside step function with two boundary conditions given by [S/(∆/h)]χ(z/h = −1 + d/h + ∆b /h + ∆/h) = S,

(2.7b)

[S/(∆/h)]χ(z/h = −1 + d/h + ∆b /h) = Sb ,

(2.7c)

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that may be solved simultaneously for the dimensionless coefficients α, β to obtain α = (1 − Sb /S);

β = ∆/h + α(1 − d/h − ∆b /h − ∆/h).

(2.7d,e)

The coefficients α and β for the specified shape function χ(z/h) in (2.7a) may be obtained for the Type II wavemaker shown in Fig. 2.1(b) by substituting ˆ S = S¯ + S;

¯ Sb = S;

∆=h−b−d

into the following boundary conditions2 in (2.7b) and (2.7c):   ˆ (S¯ + S) ˆ χ(z/h = −b/h) = S¯ + S, 1 − b/h − d/h 

 ˆ (S¯ + S) ¯ χ(z/h = −1 + d/h) = S, 1 − b/h − d/h

(2.8a)

(2.8b)

that may be solved simultaneously for the constant coefficients α, β to obtain  ¯ d b S Sˆ ; β =1− − . (2.8c,d) α= ¯ ˆ ¯ ˆ h S+S S+S h 2.2.1. Eigenfunction solution to the WMBVP Because all of the boundary conditions defined by (2.5b)–(2.5e) are now prescribed for constant values of the independent variables (x, z) and the dimensionless parameter kh = O(1), a solution by separation of independent variables1 is suggested according to ϕ(x, z) = X(x) • Z(z).

(2.9)

The eigenseries solution may be written compactly as1,3,4

Φ(x, z, t; Kn ) = Cn cosh Kn (z + h) exp +i(Knx − ωt + ν),

(2.10a)

n=1

where Kn = k for n = 1 and Kn = +iκn for n ≥ 2 provided that ko h − kh tanh kh = ko h + κn h tan κn h = 0;

n > 2.

(2.10b)

The eigenseries (2.10a) may be separated into a propagating Φp (x, z, t; k) and evanescent eigenmodes Φe (x, z, t; κn) or “local ” wave components3 according to Φ(x, z, t; Kn ) = Φp (x, z, t; k) + Φe (x, z, t; κn ) 

Cn cos κn (z + h) exp +i(Kn x − ωt + ν). = C1 cosh k(z + h) + n=2

(2.10c)

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The wave number k = 2π/λ where λ = wavelength. Because the numerical value of kh must be computed from an eigenvalue problem in the vertical z coordinate, equivalence of the eigenvalue k to the wave number 2π/λ requires a pseudo-horizontal boundary condition of periodicity given by k = 2π/λ and ϕ(x + λ, z) = ϕ(x, z). It is computationally efficient to normalize the eigenseries in (2.10a) according to cosh Kn h(1 + z/h) ; n = 1, 2, . . . , Nn where the nondimensional normalizing constant Nn is  2kh + sinh 2kh  
0 ; n = 1,  4kh Nn2 = cosh2 Kn h(1 + z/h)d(z/h) =  2κn h + sin 2κn h −1   ; n ≥ 2. 4κn h Ψn (Kn , z/h) =

(2.11a)

(2.11b) (2.11c)

The eigenseries in (2.10a) may be written as an orthonormal eigenseries by Φ(x, z, t; Kn ) =

∞

Cn Ψn(Kn , z/h) exp i(Knx − ωt − ν),

(2.12)

n=1

where the orthonormal eigenfunction Ψn (•,•) is dimensionless. 2.2.2. Evaluation of Cn by WM vertical displacement χ(z/h) The following dimensionless coefficient computed from (2.5e) will replace integral calculus with algebraic substitution for the coefficients Cn in the eigenseries (2.12):
−b/h [α(z/h) + β]Ψn (Kn , z/h)d(z/h) In (α, β, b, d, Kn ) = −1+d/h

    d b     sinh K K h 1 − d − K b sinh K h 1 −   n n n  n h h  α =   (Kn h)2 Nn  b     − cosh Kn h 1 − + cosh Kn d   h    b β (2.13) + sinh Kn h) 1 − − sinh Kn d (Kn h)Nn h that is dimensionless when α and β are given by (2.7d) and (2.7e) or (2.8c) and (2.8d). The coefficients Cn may be computed algebraically by (2.13) from the KWMBC (2.5e) to obtain Sωh In (α, β, b, d, Kn ), Kn ∆ and the orthonormal eigenseries (2.12) is given by Cn = i

(2.14)

Φ(x, z, t; Kn ) =

∞

iSωh In (α, β, b, d, Kn )Ψn (Kn , z/h) exp i(Kn x − ωt − ν). Kn ∆ n=1

(2.15)

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2.2.3. Decay distance of evanescent eigenmodes n ≥ 2 Numerical solutions and experimental measurements of ocean and coastal designs require that the KRBC (2.5d) be applied far enough away so that only the propagating eigenmode for n = 1 in (2.12) is measurable. The evanescent eigenseries in (2.12) for n ≥ 2 will decay spatially at least as fast as the smallest evanescent eigenvalue κ2 . This eigenvalue must be κ2 h > (n − 3/2)π = π/2. If the smallest value for κ2 h > π/2, then κ2 > π/2h and ϕ(x, z) ∝ exp −(πx/2h). For the values of the evanescent eigenseries to be less than 1% of their values at the wavemaker, ϕ(x, z) ∝ exp −(πxd /2h) = 0.01 and πxd /(2h) = 4.6 ≈ 3π/2, and the minimum decay distance is xd ≥ 3h. 2.2.4. Transfer function for wave amplitude from wavemaker stroke The average rate of work or power done by a wavemaker of width B is1
˙ τ = Pτ = B W

τ +1




0

h

p(x, z, τ )u(x, z, τ )d(z/h)dτ,

τ

(2.16a)

−1

where the temporal averaging operator is defined by
•τ =

τ +1

(•)dτ,

(2.16b)

τ

and  ˙ τ = Pτ = W

ρω 3 S 2 Bh4 ∆2 2kh

 I12 (α, β, b, d, k),

(2.16c)

so that all of the average power from a wavemaker is transferred to only the propagating eigenmode. The average energy flux in a linear wave is given by1  ˙ τ = E

ρgBA2 2

CG ,

(2.16d)

where the group velocity CG is given by1 CG =

  2kh C 1+ . 2 sinh 2kh

(2.16e)

Equating (2.16c) to (2.16d) gives the following transfer function for a planar wavemaker:  A ko h = Ψ1 (k, 0)I1 (α, β, b, d, k). (2.16f) S kh

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2.2.5. Hydrodynamic pressure loads (added mass and radiation damping) The wave loads on a planar wavemaker may be estimated by integrating the total pressure over the wetted surface of the wavemaker, i.e., 

 F n = dS, (2.17a,b) P M r×n 0

S

where the outward pointing unit normal n points from the wavemaker into the fluid, and the pseudo-unit normal n for the rotational modes is given by n = r × n = (z + h − d)nx ey = ny ey .

(2.17c)

Force. For the Type I piston wavemaker of total width B, the horizontal component of the pressure force on the fluid side only may be computed from the real part of 
−b/h

Cn Ψn (Kn , z/h)d(z/h) exp −i(ωt + ν) F1 (t) = Re iρωBh n=1

−1+d/h

= −F1 cos(ωt + ν − α1 ),

(2.18a)

where the static component of the pressure force on the fluid side only is Fs = −

ρgBh2 [1 − 2(d/h) + (d/h)2 − (b/h)2 ]. 2

(2.18b)

The hydrodynamic component of F1 (t) may be separated linearly into a propagating and an evanescent component that are related to the piston wavemaker translational velocity and acceleration, respectively, from the real part of F1 (t) = −Re{[λ11 (Sω) + µ11 (−iSω 2 )] exp −i(ωt + ν)} = −µ11 (−Sω 2 sin(ωt + ν)) − λ11 (Sω cos(ωt + ν))

(2.18c)

¨ 1 (t) − λ11 X˙ 1 (t)}, = Re{−µ11X

(2.18d)

where the added mass coefficient µ11 may be computed from the evanescent eigenmodes only, and the radiation damping coefficient λ11 may be computed from the propagating eigenmode only. The average power may be computed from λ11 (Sω)2 . −F1 X˙ 1 t = 2

(2.19a)

Equating (2.19a) to (2.16d) yields  λ11 =

A1 S1

2

ρBh CG , ko h

(2.19b)

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that relates the radiation damping coefficient to the square of the ratio of the radiated wave amplitude to the amplitude of the wavemaker displacement. Moment. For the Type I wavemaker of width B, the dynamic pressure moment on one side only of the wavemaker may be computed from the real part of  
 −b/h 

d z 2 Cn M5 (t) = Re iρωBh 1+ − h h −1+d/h n=1   z  × Ψn (Kn , z/h)d exp −(ωt + ν) h = −M5 cos(ωt + ν − α5 ),

(2.20a)

and the static component of the pressure moment on the fluid side only is       2   3  2  3 2 d d b ρgBh3 b d b −3 . 1− 1− +2 +3 − Ms = 6 h h h h h h (2.20b) The pressure moment M5 (t) in (2.20a) may be separated linearly into a propagating and an evanescent component that are related to the rotational velocity and acceleration from the real part of1   Sω 2 M5 (t) = −Re µ55 −i ∆(1 + ∆b /∆)    Sω + λ55 −i exp −i(ωt + ν) , ∆(1 + ∆b /∆) ¨ 5 (t) − λ55 Θ ˙ 5 (t), M5 (t) = −µ55 Θ

(2.21a)

where µ55 = ρBh4

I 2 (α, β, b, d, κn ) n , κn h n=2

λ55 = ρωBh4

I12 (α, β, b, d, k) . kh

(2.21b)

(2.21c)

2.3. Circular Wavemaker Havelock5 applied Fourier integrals to develop a theory for surface gravity waves forced by circular wavemakers in water of both infinite and finite depth. The fluid motion may be obtained from the negative gradient of a scalar velocity potential Φ(r, θ, z, t) according to q(r, θ, z, t) = −∇Φ(r, θ, z, t),

(2.22a)

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where the 3D gradient operator ∇(•) in polar coordinates is  ∇(•) =

∂ er + ∂r

  1 ∂ ∂ eθ + e3 (•). r ∂θ ∂z

(2.22b)

The total pressure field P (r, θ, z, t) may be computed from the unsteady Bernoulli equation in polar coordinates according to P (r, θ, z, t) = p(r, θ, z, t) + pS (z)   ∂Φ(r, θ, z, t) 1 − |∇Φ(r, θ, z, t)|2 + Q(t) − ρgz, =ρ ∂t 2

(2.22c)

where Q(t) = the Bernoulli constant, and the free surface elevation η(r, θ, t) for zero atmospheric pressure according to 1 η(r, θ, t) = g



Q(t) ∂Φ(r, θ, η, t) 1 − |∇Φ(r, θ, η, t)|2 + ∂t 2 ρ

r ≥ b + ξ(θ, η, t);

 ;

z = η(r, θ, t).

(2.22d)

The scalar velocity potential Φ(r, θ, z, t) must be a solution to the continuity equation ∇2 Φ =

1 ∂ r ∂r

 ∂Φ 1 ∂2Φ ∂2Φ r + 2 + = 0, ∂r r ∂θ2 ∂z 2

r ≥ b + ξ(θ, z, t);

0 ≤ θ ≤ 2π;

−h ≤ z ≤ η(r, θ, t),

(2.23a)

with the following boundary conditions: Kinematic Bottom Boundary Condition (KBBC): ∂Φ = 0; ∂z

r ≥ b + ξ(θ, −h, t);

0 ≤ θ ≤ 2π;

z = −h.

(2.23b)

Combined Kinematic and Dynamic Free Surface Boundary Condition (CKDFSBC):   ∂Φ dQ ∂ 1 ∂ 2Φ + g − − ∇Φ · ∇ |∇Φ|2 + = 0; ∂t2 ∂z ∂t 2 dt r ≥ b + ξ(θ, η, t);

0 ≤ θ ≤ 2π;

z = η(r, θ, t).

(2.23c)

Kinematic WaveMaker Boundary Condition (KWMBC): ∂Φ ∂ξ 1 ∂Φ ∂ξ ∂Φ ∂ξ + − − = 0; ∂r ∂t r2 ∂θ ∂θ ∂z ∂z

r = ξ(θ, z, t);

−h ≤ z ≤ η(b, θ, t).

(2.23d)

Two types of circular cylindrical wavemaker displacements ξ(θ, z, t) may be analyzed, viz., amplitude-modulated (AM) and phase-modulated (PM) wavemakers.

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The distinction between these two types is in the azimuthal θ dependency of the wavemaker displacement ξ(θ, z, t) from its mean position r = b, given by   cos mθ sin(ωt + ν) mS (2.23e) χ(z/h) . ξ(θ, z, t) = (∆/h) sin(ωt + ν + mθ) (2.23f) Kinematic Radiation Boundary Condition (KRBC):   ∂ ± iKn Φ(r, θ, z, t) = 0; r → ∞. √ lim |Kn r|→+∞ ∂r

(2.23g)

Finally, physically realizable solutions to (2.23a) must be periodic in θ; i.e., Φ(r, θ, z, t) = Φ(r, θ + 2π, z, t).

(2.23h)

1

The dimensional WMBVP may be scaled and linearized by expanding the variables in perturbation series with a dimensionless perturbation parameter ε = kA. A scalar radiated velocity potential ϕ(r, θ, z) may be defined by the real part of Φ(r, θ, z, t) = Re{ϕ(r, θ, z) exp −i(ωt + ν)}.

(2.24)

A linearized WMBVP may be obtained by setting the dimensionless parameter kA = ε = 0 and by requiring that kh = O(1). This linearized WMBVP is ∇2 ϕ(r, θ, z) = 0;

b ≤ r < +∞; 0 ≤ θ ≤ 2π; −h ≤ z ≤ 0,

∂ϕ(r, θ, z) = 0; ∂z

b ≤ r < +∞; 0 ≤ θ ≤ 2π; z = −h,

∂ϕ(r, θ, z) − ko ϕ(r, θ, z) = 0; ∂z  √

lim

|Kn r|→+∞

b ≤ r < +∞; 0 ≤ θ ≤ 2π; z = 0,  ∂ − iKn ϕ(r, θ, z) = 0, ∂r

∂ϕ(r, θ, z) ∂ξ(θ, z, t) exp −i(ωt + ν) = − ; ∂r ∂t  η(r, θ, t) = Re

(2.25b)

(2.25c)

(2.25d)

r = b; 0 ≤ θ ≤ 2π; −h ≤ z ≤ 0,

 −iωϕ(r, θ, z) exp −i(ωt + ν) ; g

(2.25e) b ≤ r < ∞; 0 ≤ θ ≤ 2π; z = 0,

P (r, θ, z, t) = {p(r, θ, z, t)} + ps (z) = Re{−iωρϕ(r, θ, z) exp −i(ωt + ν)} − ρgz, ϕ(r, θ, z) = ϕ(r + λ, θ, z);

(2.25a)

ϕ(r, θ, z) = ϕ(r, θ + 2π, z).

(2.25f) (2.25g) (2.25h,i)

The specified wavemaker shape function χ(z/h) is valid for either a doublearticulated piston or hinged circular AM or PM wavemaker of variable draft that

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37

Definition sketch for circular wavemaker.

is shown in Fig. 2.2 is identical to (2.7) for a 2D planar wavemaker with the dimension b replaced with a and the stroke S replaced with the azimuthal stroke m S. The solution to the WMBVP (2.25) may be compactly expressed by the following orthonormal eigenseries: m ϕ(r, θ, z)

=

∞

(1) Cmn Ψn (Kn , z/h)Hm (Kn r)MA(P ) (mθ),

(2.26a,b)

n=1

where the azimuthal mode function is  cos mθ ; MA(P ) (mθ) = exp −imθ

m ≥ 0 and integer,

(2.26c,d)

and where (2.26a) represents an AM wavemaker; (2.26b) represents a PM wavemaker; Ψn (Kn , z/h) = the orthonormal eigenseries defined in (2.11); (1) Hm (Kn r) = the Hankle function of the first kind. When K1 = k and Kn = iκn for n ≥ 2 and integer, (1) Hm (iκn r) = Jm (iκn r) + iYm (iκn r) =

2 −(m+1) i Km (κn r), π

(2.26e)

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where Km (•) = the Modified Bessel (or Kelvin) function of the second kind of order m. The coefficients Cmn may be computed by expanding the KWMBC in an eigenseries following the procedure in (2.14) and obtaining Cmn = −

m Sj hω

In (α, β, a, d, Kn )

(1) Kn ∆ Ln (Hm (Kn b))

Ln (Zm (ζn )) =

;

n ≥ 1 and integer,

(2.26f)

dZm (ζn ) 1 = {Zm−1 (ζn ) − Zm+1 (ζn )}; dζn 2

(1) (ζn ). Zm (ζn ) = Jm (ζn ), Ym (ζn ), Km (ζn ), Hm

(2.26g)

The solution to (2.25) is given by the real part of the following eigenseries expansion: m Φj (r, θ, z, t)

[m Sj hω]

=

m Φpj (r, θ, z, t)

+ m Φej (r, θ, z, t) [m Sj hω]

  (1)   I1 (α, β, a, d, k) Ψ1 (k, z/h)Hm (kr)    (1)  k∆   L1 (Hm (kb))   = −Re  ∞ 

In (α, β, a, d, κn ) Ψn (κn , z/h)Km (κn r)       +  κn ∆ Ln (Km (κn b)) n=2       −i(ωt+ν) . (2.27a,b) × MA(P ) (mθ)e      Because the asymptotic behavior of the evanescent eigenseries Km (κn r) depends on the mode m(1) , it is not possible to specify a minimum distance from the wavemaker equilibrium boundary at r = b where the evanescent eigenvalues are less than 1% of their value at the circular wavemaker boundary. The wave field must be computed far away from the wavemaker, and it is understood that far away must be computed uniquely for each radial mode m for either an AM or PM circular wavemaker. The evaluation of the power, forces, and moments, and added mass and radiation damping coefficients for both AM and PM circular wavemakers are given by Hudspeth.1 2.4. Directional Wavemakers Directional wavemakers are vertically segmented wavemakers that undulate sinuously and, consequently, are also called snake wavemakers. Segmented directional wavemakers may be driven either in the middle of each vertical segment or at the joint between vertical segments. Because of these two methods of wave generation, parasitic waves are formed along the wavemaker due to either the discontinuity of

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the wavemaker surface (middle segment driven) or of the derivative of the wavemaker surface (joint driven). A dimensional scalar spatial velocity potential ϕ(x, y, z) may be defined by the real part of Φ(x, y, z, t) = Re{ϕ(x, y, z) exp −i(ωt + ν)}.

(2.28a)

The dimensional linear fluid dynamic pressure field p(x, y, z, t) and 3D fluid velocity vector field may be computed from p(x, y, z, t) = ρ

∂Φ(x, y, z, t) , ∂t

q(x, y, z, t) = −∇3 Φ(x, y, z, t), ∇3 (•) =

∂(•) ∂(•) ∂(•) ex + ey + ez . ∂x ∂y ∂z

(2.28b) (2.28c) (2.28d)

The dimensional WMBVP for directional waves is given by ∇23 ϕ(x, y, z) = 0;

x ≥ 0;

B ≤ y ≤ +B;

−h(x, y) ≤ z ≤ 0,

(2.29a)

∂ϕ(x, y, 0) − ko ϕ = 0; x ≥ 0; −B ≤ y ≤ +B; z = 0, ∂z

(2.29b)

∂ϕ = 0; x ≥ 0; y = ±B; −h(x, y) ≤ z ≤ 0, ∂y  x = 0, ∂Φ(x, y, z, t) ∂ξ(y, z, t)  =− ; −B ≤ y ≤ +B,  ∂x ∂t −h(0, y) ≤ z ≤ 0,  lim

x→+∞

 ∂ − iKn ϕ(x, y, z) = 0, ∂x

∂ϕ(x, y, z) = −∇2 ϕ(x, y, z) · ∇2 h(x, y); z = −h(x, y), ∂z   2 ∂ ∂2 (•), ∇22 (•) = , ∂x2 ∂y 2  ˆ z) U(y, ξ(y, z, t) = Re i [∆U (y, a)][∆U (z, b, d)] exp −i(ωt + ν) , ω  ˆ (y, z) = U

Sω ∆/ho

(2.29c)

(2.29d)

(2.29e)

(2.29f) (2.29g)

(2.29h)

Γ(y)χ(z/ho ),

∆U (y, a) = U (y + a− ) − U (y − a+ ), ∆U (z, b, d) = U (z + h − d) − U (z + b),

(2.29i)

(2.29j)

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Fig. 2.3.

Definition sketch for rectangular directional wave basin.

where ko = ω 2 /g and where a± denotes the (possibly nonsymmetric) transverse ends of the directional wavemaker in the transverse y-direction in Fig. 2.3. The solution to the WMBVP in (2.29) is given by the following set of orthonormal eigenfunctions: ϕ(x, y, z) = i

g

ζn (x, y)Υn (Kn , z/h), ω n=1

  Ψ1 (k, z/h) ; Ψ1 (k, 0) Υn (Kn , z/h) =  Ψ1 (κn , z/h); ko h = Kn h tanh Kn h = 0;

(2.30a)

n = 1,

(2.30b)

n ≥ 2,

(2.30c)

n = 1, 2, 3, . . . ,

(2.30d)

where K1 = k and Kn = +iκn for n ≥ 2 and Ψn (•) is defined in (2.11). The orthonormal eigenfunctions (2.30b) and (2.30c) are applicable strictly only for constant depth wave basins; however, they may be applied to slowly varying depth wave basins if (2.30b) and (2.30c) are considered to be evaluated only locally over relatively small horizontal length scales (e.g., several wavelengths λ), where the depth may be considered to be locally equal to a constant by a Taylor series expansion of the depth.1 Substituting (2.30a) into (2.29a) yields the following 2D Helmholtz equation: ∇22 ζ(x, y) + Kn2 ζ(x, y) = 0;

x ≥ 0;

−B ≤ y ≤ +B.

(2.31)

Alternatively, for wave basins with mildly sloping bottoms, the mild slope equation may be applied according to1 ∇2 • (CCG ∇2 ζ(x, y)) + Kn2 CCG ζ(x, y) = 0,

(2.32)

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where the wave group velocity CG is given by (2.16e). If the product CCG is a constant, (2.32) reduces to the 2D Helmholtz equation (2.31). Applying the WKBJ approximation1 for the x-dependent solution in the method of separation of variables to (2.32) yields the following solution1 :   g Φ(x, y, z, t) = Re i ζ(x, y)Υn (Kn , z/h) exp −i(ωt + ν) ω    M   g

[C(x)CG (x)]x=0     Ξm (µm , y/B) Amn i  ω m=0 n=1 C(x)CG (x) = Re 
x        × Υn(Kn , z/h) exp i Qmn dξ exp − i(ωt + ν)  (2.33a)

Ξm (µm , y/B) =

cos µm B(y/B − 1) ; Mm 

Qmn =

Mm = 1 + δm0 ;

Kn2 − µ2m ;

m > 0;

µm =

mπ 2B

K1 = k > µ2m .

(2.33b–d)

(2.33e)

The coefficients Am may be computed from (2.29d) by expanding the wavemaker shape function in orthonormal eigenfunctions1 and are given by k

Amn

S Qmn (∆/hx0 )

= −Ψn (Kn , 0)In (α, β, b, d, k)
×

+

a+ /B

− a /B −

Γj (qj , y/B)Ξm (µm , y/B)d(y/B).

(2.34)

2.4.1. Full-draft piston wavemaker The prescribed transverse y-component of the snake displacement of a full-draft (b = d = 0) piston (α = 0 and β = 1) wavemaker may be expressed as Γj (qj , y/B) =

+∞

c˜j exp i[qj B(y/B + νy )],

(2.35a)

j=−∞

where the coefficients c˜j may be computed from the integral in (2.34) by
cmj = =

+

a+ /B

− a /B −

Γj (qj , y/B)Ξm (µm , y/B)d(y/B)

Ra+ ,a− + iIa+ ,a− . mB(qj2 − µ2m )(1 + δm0 )

(2.35b)

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If the full-draft piston snake wavemaker spans the entire width of the wave basin so that a± = ±B, then (2.35b) reduces to the integral in (2.34) and  4qj B cmj =

sin[qj B(νj − 1)] + (−1)m sin[qj B(νj + 1)] −i{cos[qj B(νj − 1)] + (−1)m cos[qj B(νj + 1)]} ((qj B − (mπ)2 )2 )(1 + δm0 )

.

(2.35c)

Values for cmj for (possibly nonsymmetric) values for a± are given by Hudspeth.1

2.5. Sloshing Waves in a 2D Wave Channel A long rectangular wave channel with a horizontal flat bottom, two rigid vertical sidewalls, and a wavemaker may generate either 2D, long-crested progressive waves or two types of transverse waves, viz., (1) sloshing waves that are excited directly by transverse motion of the wavemaker or (2) cross waves that are excited parametrically by the progressive waves at a sub-harmonic of the wavemaker frequency. The WMBVP for 3D sloshing waves is identical to (2.5) for planar 2D wavemakers except for the KWMBC at x = 0 and an additional kinematic boundary condition on the sidewalls of the 2D wave channel at y = ±B in Fig. 2.4. The kinematic and dynamic wave fields may be computed from a dimensional 3D scalar

Fig. 2.4.

Definition sketch for a sloshing wave channel.

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velocity potential Φ(x, y, z, t). The fluid velocity q(x, y, z, t) may be computed from the negative directional derivative of a scalar velocity potential by q(x, y, z, t) = −∇Φ(x, y, z, t). (2.36a) The dimensional fluid dynamic pressure field p(x, y, z, t) may be computed from p(x, y, z, t) = ρ

∂Φ(x, y, z, t) . ∂t

(2.36b)

A spatial velocity potential ϕ(x, y, z) may be defined by the real part of Φ(x, y, z, t) = Re{ϕ(x, y, z) exp −i(ωt + ν)}.

(2.36c)

The WMBVP for sloshing waves is given by the following: ∇2 ϕ(x, y, z) = 0;

x ≥ 0; −B ≤ y ≤ +B; −h ≤ z ≤ 0,

(2.37a)

∂ϕ(x, y, −h) = 0; x ≥ 0; −B ≤ y ≤ +B; z = −h, ∂z

(2.37b)

∂ϕ(x, y, 0) − ko ϕ = 0; x ≥ 0; −B ≤ y ≤ +B; z = 0, ∂z

(2.37c)

∂Φ(x, y, z, t) ∂ξ(y, z, t) =− ; x = 0; −B ≤ y ≤ +B; −h ≤ z ≤ 0, ∂x ∂t ∂ϕ = 0; x ≥ 0; y = ±B; −h ≤ z ≤ 0, ∂y  ∂ lim − iKn ϕ(x, y, z) = 0, x→+∞ ∂x     x ≥ 0, −iω −B ≤ y ≤ +B, Φ(x, y, 0, t) ; η(x, y, t) = Re  g z = 0.

(2.37d)

(2.37e)



A solution to (2.37) is given by the following eigenfunction expansions1 : 

˜ Φ(x, y, z, t) = Re ψn (x, y)Ψn (Kn , z/h) exp −i(ωt + ν) ,

(2.37f)

(2.37g)

(2.38a)

n=1

ϕ(x, y, z) = ψ˜n (x, y)Ψn (Kn , z/h), 

η(x, y, t) = Re ζn (x, y) exp −i(ωt + ν) ,

(2.38b) (2.38c)

n=1

ω ζn (x, y) = −i ψ˜n (x, y)Ψn (Kn , 0), g

(2.38d)

g ζn (x, y) , ψ˜n (x, y) = i ω Ψn (Kn , 0)

(2.38e)

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where ζn (x, y) is sometimes referred to as a displacement potential. The scalar potential (2.38a) may be expressed from (2.38d) and (2.38e) as 

g Ψn (Kn , z/h) Φ(x, y, z, t) = Re exp −i(ωt + ν) . (2.39) i ζn (x, y) ω Ψn (Kn , 0) n=1 The WMBVP may be expressed in terms of a displacement potential ζn (x, y) by   x ≥ 0, 2 2 ∂ ζn (x, y) ∂ ζn (x, y) 2 −B ≤ y ≤ +B, (2.40a) + + K ζ (x, y) = 0; n n  ∂x2 ∂y 2 −h ≤ z ≤ 0.   x = 0,

∂ζn (x, y) Ψn (Kn , z/h) ω (2.40b) = i U (y, z) B ≤ y ≤ +B,  ∂x Ψn (Kn , 0) g n=1 −h ≤ z ≤ 0.   ∂ (2.40c) lim − iKn ζn (x, y) = 0. x→+∞ ∂x ∂ζn (x, y) = 0; x ≥ 0; y = ±B; −h ≤ z ≤ 0, ∂y

(2.40d)

where (2.40a) is the 2D Helmholtz equation.9,10 Because the boundary conditions are prescribed on boundaries that are constant values of (y,z), a solution to the WMBVP (2.40) may be computed by the method of separation of variables and is given by1 Φ(x, y, z, t)                   g = Re i  ω                

 M

 Ψ1 (k, z/h)   Cm1 Υm (y/B)  exp iPm1 x   Ψ1 (k, 0)  m=0    

 Ψ1 (k, z/h)   exp + Cm1 Υm (y/B)    Ψ (k, 0)                 

                 

             1 m=M+1 exp −i(ωt + ν) ,   − Ξm1 x          

  Ψn(κn , z/h)    exp + Cmn Υm (y/B)       Ψ (κ , 0) n n  m=0 n=2     − Qmn x (2.41a)

where Υm (µm , y/B) =

cos µm B(y/B − 1) ; Mm µm =

mπ , 2B

2 Mm = 1 + δm0 ;

(2.41b)

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Pm1 =

K12 − µ2m = Ξm1 =



k 2 − µ2m ;

 µ2m − k 2 ;

Q2mn = µ2m − Kn2 ; n ≥ 1: K1 = k > µm : Qm1 = i

k < µm

45

k > µm ; m ≤ M,

k < µm ; m ≥ 0;

(2.41d)

n ≥ 1,

(2.41e) m ≤ M,

n = 1: k = µm : Qm1 = 0  : Qm1 = µ2m − k 2 = Ξm1 > 0; m > M, 

(2.41c)

m > M,

 k 2 − µ2m = iPm ;

n ≥ 2 : Kn = iκn : Qmn =

FA

µ2m + κ2n > 0,

(2.41f)

(2.41g) (2.41h)

where M is the maximum integer value for m in order for µm < k. Substituting (2.41a) into (2.40b) yields the following solutions for Cmn 1 :

  ω Ψ1 (Kn , 0) +1  y  0  z   z  z y d d U y, Ψ1 K n , Υ m µm ; Cm1 = g Pm1 B −1 h h h B −1

Cm1 = −i

Cmn = −i

ω Ψ1 (Kn , 0) g Ξm1 ω Ψn (Kn , 0) g Qmn




+1

d

y B

−1




+1

d

y

−1

B




m≤M (2.42a) 0   z   z  z y U y, Ψ1 K n , Υm µm , ; d h h h B −1




m>M +1 (2.42b) z   z  0   z y U y, Ψn K n , Υ m µm , ; d h h h B −1 m ≥ 0, n ≥ 2.

(2.42c)

The first three transverse eigenmodes are illustrated in Fig. 2.5. 2.6. Conformal Mappings Conformal and domain mappings are applications of complex variables to solve 2D boundary value problems. Conformal mapping is an angle preserving transformation that will compute exact nonlinear solutions for surface gravity waves of constant

Fig. 2.5.

First three transverse eigenmodes in a 2D wave channel.

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form that may be treated as a steady flow following a Galilean transformation from a fixed inertial coordinate system to a noninertial moving coordinate system. Domain mapping is a transformation of the wavemaker geometry into a fixed computational domain where a solution may be computed efficiently. 2.6.1. Conformal mapping1 Conformal mapping of the WMBVP provides a global solution that accurately accounts for the singular behavior at all irregular points. The irregular points in the physical wavemaker domain are transformed into both weak and strong singular kernels in a Fredholm integral equation. The two irregular points on the WMBVP boundary are located at (1) the intersection between the free-surface and the wavemaker boundary and (2) the intersection between the horizontal bottom and the wavemaker boundary. These two irregular points exhibit integrable weakly singular kernels. The far-field radiation boundary exhibits a strongly singular kernel and significantly affects the solution. The irregular frequencies3,4 are included in the global solution by the Fredholm alternative. A theory for the planar WMBVP computes a global solution by a conformal mapping of the physical wavemaker boundary to a unit disk that includes the motion of an inviscid fluid near irregular points that are illustrated in Fig. 2.6. A numerical solution to Laplace’s equation in a transformed unit disk may be computed from a Fredholm integral equation. The WMBVP defined by (2.5) is transformed into complex-valued analytical functions where the complex-valued coordinates are defined as z = x + iy. The coordinates for the semi-infinite wave channel in Fig. 2.1(a) must be transformed to complexvalued coordinates z. The fluid velocity q(x, y, t) and dynamic pressure p(x, y, t) may be computed from a scalar velocity potential Φ(x, y, t) according to q(x, y, t) = −∇Φ(x, y, t);

p(x, y, t) = ρ

∂Φ(x, y, t) . ∂t

(2.43a,b)

The WMBVP and Type I wavemaker shape function are given by (2.5)–(2.7). There are both Irregular (I) and Regular (R) points at the intersections between the Smooth (S) and Critical (C) boundaries B1 and B2 in the WMBVP as illustrated in Fig. 2.6 where these two boundary intersection points are identified as P1 and P2 . The classification of the boundary points P1 and P2 in Fig. 2.6 depends on (1) the boundary conditions ϕi (Pj ) and (2) the continuity of the boundaries Bm and their derivatives where i, j, and m = 1 or 2. A conformal mapping of the semi-infinite wave channel strip in the physical plane will yield a Fredholm integral equation,6,7 where these critical points may be transformed to singular points that are integrable over a smooth continuous mapped boundary. 2.6.1.1. Conformal mapping to the unit disk 1 Two conformal mappings are: (i) the physical z-plane to a semi-circle in the Z-plane shown in Fig. 2.7; and (ii) a semi-circle in the upper Z-plane to a unit disk in the

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Fig. 2.6. Combinations of Irregular (I) and Regular (R) boundary points P1 and P2 between Smooth (S) and Critical (C) boundaries B1 and B2 intersections in the WMBVP.8

Fig. 2.7. Mapping of the semi-infinite strip in the lower half x–y-plane in the physical z-plane to the upper half X–Y -plane in the Z-plane.8

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Fig. 2.8.

Mapping of the upper half-plane in the Z-plane to the unit disk in the Q-plane.

Q-plane shown in Fig. 2.8. The Schwarz–Christoffel transformation dz C1 √ =√ dZ Z +1 Z −1

(2.44a)

may be integrated to obtain z = x + iy  h = − Log[−Z − Z 2 − 1], π

(2.44b)

where the Log[•] denotes the principal value of Log[•] and the argument of the Log[•] is −π ≤ arg < π. Inverting (2.44b) yields Z = X + iY = − cosh(πz/h),

(2.44c)

where ! πy " cos , h h ! πx " ! πy " Y = − sinh sin . h h

X = − cosh

! πx "

(2.44d) (2.44e)

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The radiation boundary in the z-plane transforms to a semi-circle in the Z-plane by     1 2πy 2πx cos + cosh , (2.44f) R2 = X 2 + Y 2 = 2 h h  πx   πy  Y tan θ = = tanh tan . (2.44g) X h h Details of the transformation of the WMBVP are given by Hudspeth.1 2.6.1.2. Mapping Z-plane to a unit disk The upper-half-plane of the Z-plane may be mapped into a unit disk shown in Fig. 2.8 by the following bilinear transformation:   i−Z Q = ξ + iζ = exp(iθ0 ) , (2.45a) i+Z that may be inverted to obtain  1−Q , Z = X + iY = i 1+Q 

(2.45b)

and the mapping function coordinates are ξ=

1 − X2 − Y 2 , X 2 + (Y + 1)2

ζ=

2X , X 2 + (Y = 1)2

(2.45c,d)

that may be transformed into the cylindrical coordinates for the unit disk in Fig. 2.8 by   2X (X 2 + Y 2 − 1)2 − 4X 2 ζ . = arctan r2 = ξ 2 +ζ 2 = ;  = arctan (X 2 + (Y + 1)2 )2 ξ 1 − X2 − Y 2 (2.45e,f) Details of the transformation are given by Hudspeth.1 A numerical solution to the transformed WMBVP may be computed from the following Fredholm integral equation1 :   
+π  ˆ r , ) ∂G(r, r ˆ , , ) ˆ ∂ Φ(ˆ ˆ 2π ˆ ˆ r , ) ˆ + G(r, rˆ, , ) ˆ rˆd, ˆ π Φ(r, ) = − −π Φ(ˆ ∂ˆ r ∂ˆ r (2.46a) where G(r, rˆ, , ) ˆ = − ln[ρ(r, rˆ, , )]; ˆ

ρ2 (r, rˆ, , ) ˆ = r2 − 2rˆ r cos( ˆ − ) + rˆ2 . (2.46b,c)

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Fig. 2.9. Nodal points on the unit disk in the Q-plane and the corresponding nodal points on the wavemaker in the physical z-plane.8

Numerical solutions to (2.46a) may be computed by discretizing the unit disk boundary as shown in Fig. 2.9. The numerical details regarding the evaluation (2.46a) at the two weakly singular irregular points at B and D in the physical z-plane in Fig. 2.7 and the strongly singular point at ±π that is the vertical radiation boundary A–E at +∞ in the physical z-plane in Fig. 2.7 are tedious.8 Global numerical solutions may be computed for both the linear and the nonlinear WMBVPs. 2.6.1.3. Conformal mapping to the unit disk 2 The wavemaker geometry shown in Fig. 2.10 is mapped to the unit disk by two transformations. The WMBVP is given by (2.5) and the WM shape function is χ(y/h) = [α(y/h) + β][U (y/h + 1 − b0 /h) − U (y/h + a0 /h)].

(2.47)

In order to transform the wavemaker geometry to a Jacobian elliptic function, it must be rotated and translated as shown in Fig. 2.11. The 90◦ rotation to the z  -plane is given by z  = x + iy  = iz = −y + ix.

(2.48a)

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Fig. 2.10.

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51

WMBVP11 with the six critical boundary points at a–a0 –b–b0 –c–d.

Fig. 2.11. Rotation and translation of the physical wavemaker rectangular strip in the z-plane to the w-plane.11

The horizontal shift to the left in the z  -plane is given by z  = x + iy  = w − h/2 = −y − h/2 + ix.

(2.48b)

In order to map the WM geometry in the z-plane to the semi-circle in the Z-plane shown in Fig. 2.12 as a Jacobian elliptic function, the rotated and translated strip in the z  -plane must have the dimensions of −K ≤ ξ ≤ +K and 0 ≤ ζ ≤ K  , where K = h/2 and K  = 3h = 6K. This requires a coordinate amplification given by 2K  (x + iy  ) h  h 2K −y − + ix . = h 2

w=

(2.48c)

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Fig. 2.12.

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Mapping of the wavemaker semi-circle in the Z-plane to the unit disk in the Q-plane.11

The Schwarz–Christoffel transformation from the w-plane to the Z-plane is dw = 

ˆ CkdZ (a − Z)(b − Z)(c − Z)(d − Z)

.

(2.48d)

The following change of variables: ˆ dZ = adZˆ : κ = a/c; Cˆ = c, Z = aZ; modifies (2.48d) to the following Jacobian elliptic function:
Z dZˆ = sn−1 [Z, κ], w= 1 [(1 − Zˆ 2 )(1 − κ2 Zˆ 2 )] 2 0

(2.48e)

where sn[Z, κ] = the Jacobian elliptic function of modulus κ or sine amplitude function.9 Define kˆ = sn−1 [1, κ],

(2.48f)

and the mapping of the rectangle {x1 , x2 ; y1 , y2 } = {0, 3h; 0, h} is given by Z = X + iY "   # $ % & ! ˆ  sn − 2K  y + h2 , κ dn 2Kx h h ,k " ! =  1 − dn2 #− 2K $y + h % , κ& sn2 2Kx , kˆ  h 2 h ! " ! "  # $ % & # $ % & h 2K h 2Kx ˆ 2Kx ˆ   cn − 2K y + , κ dn − y + , κ , sn , k cn , k h 2 h 2 h h " ! , +i # 2K $ % &   h 2Kx ˆ 2 2 1 − dn − h y + 2 , κ sn h ,k (2.48g) where sn[•, •] in the copolar half-period trio in (2.48g) is defined in (2.48e), and cn[•, •] and dn[•, •] are defined by cn 2 [•, •] = 1−sn 2 [•, •], dn 2 [•, •] = 1−κ2 sn2 [•, •]. The mapping from the semi-circle in the Z-plane to the unit disk in the Q-plane

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Fig. 2.13.

53

Transformed boundary conditions mapped to arcs on the perimeter of the unit disk.11

is shown in Fig. 2.12; and the mapping to the unit disk in the Q-plane is shown in Fig. 2.13. The mapping of the Z-plane to the Q-plane is given by Q=

ˆ )2 + 2iZ(1 − α ˆ )2 i−Z −α ˆ (i + Z) (1 − α ˆ )2 − Z 2 (1 + α , = i+Z −α ˆ (i − Z) (1 − α ˆ 2 ) + Z 2 (1 + α) ˆ 2

(2.49a)

where −1 < α ˆ < +1. Changing variables to circular cylindrical coordinates by (1 − α ˆ 2 ) − 2Y (1 − α ˆ 2 ) + (X 2 + Y 2 )(1 + α ˆ )2 , (1 − α ˆ )2 − 2Y (1 − α ˆ 2 ) + (X 2 + Y 2 )(1 + α ˆ2 )   2X(1 − α ˆ2) , θ(X, Y ) = arctan (1 − α) ˆ 2 − (X 2 + Y 2 )(1 + α) ˆ 2

R2 (X, Y ) =

(2.49b) (2.49c)

the unit disk may be transformed into functions of the copolar trio of Jacobian elliptic functions. The transformed WMBVP in circular cylindrical coordinates is given by Hudspeth.1 A general solution to the transformed WMBVP may be written as10 ϕ(R, θ) =

N

n=0

R

n



 a ˆn ˆ cos nθ + bn sin nθ , 1 + δn0

(2.50)

where δij is the Kronecker delta function. Substituting (2.50) into the generic boundary conditions on each of the six arcs on the perimeter of the unit disk illustrated in Fig. 2.13, multiplying each of these six boundary conditions by a member of the set of the orthogonal series in (2.50), integrating over the interval of orthogonality −π ≤ θ ≤ +π yields the following matrix equation for each of the coefficients a ˆn and ˆbn AB = H.

(2.51)

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Fig. 2.14.

Physical fluid domain.12

2.7. Domain Mapping Domain mapping of the WMBVP12 follows the theory by Joseph.13 The physical fluid domain shown in Fig. 2.14 for the fully nonlinear WMBVP is mapped to a fixed computational fluid domain, and the discretized coupled free-surface boundary conditions are computed by an implicit Crank–Nicholson (C–N) method.14,15 At each iteration of the C–N method, the potential field is computed by the conjugate gradient method.15 The wavemaker motion Ξ(y/h, t) is assumed to be periodic with period T = 2π/ω, and the WMBVP with the surface tension Tˆ is given by  0 ≤ y ≤ Γ(x, t), 2 ∇ Φ(x, y, t) = ∆Φ(x, y, t) = 0; (2.52a) Ξ(y/h, t) ≤ x ≤ L. Tˆ ∂ 2 Γ(x, t) ∂Φ(x, y, t) 1 ρ ∂x2 + |∇Φ(x, y, t)|2 −  + gΓ(x, t) = 0.  2 3/2 ∂t 2 ∂Γ(x, t) 1+ ∂x

(2.52b)

∂Φ(x, y, t) ∂Γ(x, t) ∂Φ(x, y, t) ∂Γ(x, t) − + = 0; ∂y ∂x ∂x ∂t Ξ(Γ(x, t), t) ≤ x ≤ L; ∂Φ(x, y, t) = 0; ∂y ∂Φ(x, y, t) = 0; ∂x

y = Γ(x, t).

Ξ(Γ(x, t), t) ≤ x ≤ L;

x = L;

(2.52c) y = 0.

0 ≤ y ≤ Γ(L, t).

∂Φ(x, y, t) ∂Ξ(y/h, t) ∂Ξ(y/h, t) ∂Φ(x, y, t) =− + ; ∂x ∂t ∂y ∂y



x = Ξ(y/h, t), 0 ≤ y ≤ Γ(0, t).

(2.52d)

(2.52e)

(2.52f)

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The initial conditions for t = 0 are Γ(x, 0) = H(x); ∂Γ(x, t) = 0;

(2.52g)

 Ξ(x, 0) ≤ x ≤ L.

Φ(x, Γ(x, 0), 0) = 0;

(2.52h) (2.52i)

The physical fluid domain shown in Fig. 2.14 may be mapped into a dimensionless fixed rectangle of dimensions 0 ≤ ξ ≤ 1 by 0 ≤ ζ ≤ 1 by the transforms ξ=

x ; L

ζ=

y ; Γ(x, t)

τ = ωt;

Γ(x, t) , h

(2.53a–d)

P (x, y, t) , ρA2 ω 2

(2.53e,f)

γ(ξ, τ ) =

and dimensionless variables by q(ξ, ζ, τ ) = −

ϕ(ξ, ζ, τ ) =

∇Φ(x, y, t) ; Aω

Φ(x, y, t) ; Ahω

w(ζ, τ ) =

p(ξ, ζ, τ ) =

Ξ(y/h, t) ; S

ˆ= T

˜ T . ρALhω 2

(2.53g–i)

Because ζ is a function of both x and y in (2.53b), transforming partial derivatives with respect to x must be done with some care.12 Details of these lengthy transformations and the transformed WMBVP in the fixed mapped domain are given by Hudspeth.1 References 1. R. T. Hudspeth, Waves and Wave Forces on Coastal and Ocean Structures (World Scientific, Singapore, 2006). 2. R. T. Hudspeth, J. M. Grassa, J. R. Medina and J. Lozano, J. Hydraulic Res. 32, 387 (1994). 3. F. John, Commun. Pure Appl. Math. 2, 13 (1949). 4. F. John, Commun. Pure Appl. Math. 3, 45 (1950). 5. T. H. Havelock, Phil. Mag. 8, 569 (1929). 6. P. R. Garabedian, Partial Differential Equations (Wiley, Inc., New York, 1964). 7. P. M. Morse and H. Feshbach, Methods of Theoretical Physics (McGraw-Hill Book Company, New York, 1953). 8. Y. Tanaka, Irregular points in wavemaker boundary value problem, PhD thesis, Oregon State University (1988). 9. G. F. Carrier, M. Krook and C. E. Pearson, Functions of a Complex Variable, Theory and Technique (McGraw-Hill Book Co. Inc., New York, 1966). 10. R. B. Guenther and J. Lee, Partial Differential Equations of Mathematical Physics and Integral Equations (Dover Publications, Inc., New York, 1996). 11. P. J. Averbeck, The boundary value problem for the rectangular wavemaker, MS thesis, Oregon State University (1993). 12. S. J. DeSilva, R. B. Guenther and R. T. Hudspeth, Appl. Ocean Res. 18, 293 (1996).

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13. D. D. Joseph, Arch. Rational Mech. Anal. 51, 295 (1973). 14. B. Carnahan, H. A. Luther and J. O. Wilkes, Applied Numerical Methods (John Wiley and Sons, New York, 1965). 15. R. Glowinski, Numerical Methods for Nonlinear Variational Problems (SpringerVerlag, 1984).

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Chapter 3

Analyses by the Melnikov Method of Damped Parametrically Excited Cross Waves Ronald B. Guenther Department of Mathematics Oregon State University, Corvallis, OR 97331, USA [email protected] Robert T. Hudspeth School of Civil and Construction Engineering Oregon State University, Corvallis, OR 97331, USA [email protected] The Wiggins–Holmes extension of the generalized Melnikov method (GMM) to higher dimensions and the extension of the Generalized Herglotz Algorithm (GHA) to nonautonomous systems are applied to weakly damped parametrically excited cross waves with surface tension in order to demonstrate that cross waves are chaotic. The GMM is a global perturbation analysis about a manifold of fixed points that are connected by separatrices for higher dimensional nonlinear dynamical systems. The Luke Lagrangian, density function for surface gravity waves with surface tension and dissipation is expressed in three generalized coordinates that are the time-dependent components of three velocity potentials that represent three standing waves. The Hamiltonian for these cross waves is homomorphic to the Hamiltonian for a parametrically excited pendulum that is an example of a Floquet oscillator that may be approximated by the Mathieu equation. Neutral stability curves measured from wave tank data motivated the inclusion of dissipation in the Luke Lagrangian density function for cross waves. An integral containing a generalized dissipation function that is proportional to the Stokes material derivative of the free surface is added to the Luke Lagrangian integral so that dissipation is correctly incorporated into the dynamic free surface boundary condition. The generalized momenta are computed from the Lagrangian; and the Hamiltonian is computed from a Legendre transform of the Lagrangian. This Hamiltonian contains nonautonomous components and is transformed by three canonical transformations in order to obtain a suspended system for the application of the Kolmogorov–Arnold–Moser (KAM) averaging

57

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58

operation and the GMM. The system of nonlinear nonautonomous evolution equations determined from Hamilton’s equations of motion of the second kind must be averaged in order to obtain an autonomous system that may be analyzed by the GMM. Hyperbolic saddle points that are connected by heteroclinic orbits are computed from the unperturbed autonomous system. The nondissipative perturbed Hamiltonian system with surface tension satisfies the KAM nondegeneracy requirements; and the Melnikov integral is calculated to demonstrate that the motion is chaotic. For the perturbed dissipative system with surface tension, the only hyperbolic fixed point that survives the averaged equations is a fixed point of weak chaos that is not connected by a homoclinic orbit; and, consequently, the Melnikov integral is identically zero. The chaotic motion for the perturbed dissipative system with surface tension is demonstrated by numerical computation of positive Liapunov characteristic exponents. A chaos diagram of the largest Liapunov exponent demonstrates regions in the Floquet forcing parameter space of possible chaotic motions; and the range of values of the Floquet parametric forcing parameter ε and of the wavemaker dissipation parameter α in the α−ε space where chaotic motions may exist.1,3

3.1. Introduction The cross waves shown in Fig. 3.1 are excited parametrically by progressive wavemaker waves at a sub-harmonic of the wavemaker frequency.1 Parametrically excited standing cross waves that oscillate in a direction transverse to the wavemaker forcing with crests perpendicular to the wavemaker are analyzed by the generalized Melnikov method (GMM) and by the Generalized Herglotz Algorithm (GHA) extended to nonautonomous systems. The cross wave wavelengths and wave modes

Fig. 3.1.

Mode 2 cross wave in a 2D wave channel.1–3

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possible are determined by the channel width L
c = 2(channel width)/n, where n = the mode number and is equal to the number of half-wavelengths across the channel. Floquet instabilities occur when the cross wave wavelengths also satisfy the frequency dispersion equation for surface gravity waves. Energy is transferred from the progressive wavemaker waves to the parametrically forced cross waves through the spatial mean motion of the free surface and their growth is due to the rate of working of the transverse stresses of the progressive wavemaker waves. The cross wave instability is shown to be homomorphic to the Floquet oscillator instability2 by constructing the neutral Floquet stability diagram shown in Fig. 3.2. The simultaneous generation of a primary resonance (ωp : ωc = 2 : 1) and a secondary resonance (ωp : ωc = 1 : 1) may be observed in the cross wave data. The criteria for horseshoe maps and for homoclinic/heteroclinic orbits are selected to test whether or not cross waves are a chaotic dynamical system. Specifically, the GMM provides local criterion for the transverse intersection of stable and unstable manifolds of the perturbed system and for the resulting chaotic motion near the unperturbed (undamped and unforced) homoclinic/heteroclinic orbits. In order to apply the GMM to a suspended dynamical system that will survive the KAM averaging operation, the nonautonomous Hamiltonian is transformed by three separate canonical transformations by applying the GHA for nonautonomous transformations. Three canonical transformations are applied in order to (1) eliminate cross product terms by a rotation of axes; (2) to eliminate two degrees of freedom, and (3) to suspend the nonautonomous terms in the wavemaker forcing component of the Hamiltonian by applying the Hamilton–Jacobian transformation. The Liapunov characteristic exponents represent an alternative criterion for diagnosing the chaotic behavior of a dynamical system by measuring the mean rate of exponential divergence of nearby trajectories, and is computed numerically for the perturbed dissipative Hamiltonian when the GMM fails to predict chaos.

Fig. 3.2. Neutral Floquet stability diagram for mode 2 cross waves (o = mode 2 cross waves and
= no cross waves).1,2

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Fig. 3.3.

Definition sketch of fluid domain.

3.2. Hamilton’s Principle for Cross Waves The fluid is assumed to be incompressible and inviscid and the flow to be irrotational. The dimensional fluid particle velocities u
and the dimensional total pressure in the fluid P
are computed from ¯
ϕ
; u
= − ∇

P
1 ¯

2 = −g
z
+ ϕ
t
− |∇ ϕ| ; ρ
2

(3.1a,b)

where ϕ
= a velocity potential; ρ
= the fluid mass density; and g
= the gravitational acceleration. The fluid domain is the 3D rectangular wave channel shown in Fig. 3.3. 3.2.1. Variational principle The Lagrangian L
with kinematic surface tension T
is given by L
=




V
(t
)


(ζ
− 1)
1 ¯

2 dSη
; |∇ ϕ | − ϕ
t
+ g
z
dV
+ T
2 ζ
S



(3.2a)

η

where η
= the free surface profile and where dSη

= ζ
dx
dy
;

¯
ϕ
|2 ]1/2 . ζ
= [1 + |∇

(3.2b,c)

Generalized Hamilton’s principle with dissipation.4 The first variational of (3.2) is
δ

t
2 t
1

L
dt
+




t
2

t
1

F




Dη
Dt




dt
= 0;

(3.3a)

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where the generalized dissipation function per unit mass density F
(Dη
/Dt
) is F






Dη
Dt






= −α

g
κ



Sη



Dη
δη

dSη
; Dt
ζ


(3.3b)

where α = a dimensionless damping parameter and κ
= the cross wave wavenumber. The dimensional boundary-value-problem may be obtained by requiring that the variation of φ
and η
vanish at the arbitrary values t
1 and t
2 , and is given by δL
+

d dt









 1

2 |∇ ϕ | −ϕ
t
+g
η
−T
∇
(∇
η
/ζ
) δη
dx
dy


−b
χη 2 z =η


2



ˆ s
) − (∇ ϕ )δϕ dV + (ϕn
+ Us
· n V
(t
) S








ηn (3.4) × δϕ dSs
+ T
δη /ζ dSη
,


δϕ
dV
=

(t )

b


Sη



where n
denotes a normal derivative. The Lagrangian in (3.2) reduces to1  1

ˆ s
ϕ
dS
ϕ ∇ ϕ dV + ϕ
+ Us
· n 2 n V
S





b


1 b + g
η
2 dx
dy
+ T
(ζ
− 1)dx
dy
. 2 −b
χη
−b
χ



1 L =− 2








2










(3.5)

η

The velocity potential ϕ
is the field variable, and the free surface η
may be expressed in terms of ϕ
by the free surface boundary conditions and the contact line conditions at the vertical sidewalls.1,3 The velocity potential ϕ
and the free surface displacement η
are assumed to be linear combinations of the progressive wavemaker wave and the parametrically excited cross wave given by the following: ϕ
(x
, y
, z
, t
) = ϕ
p (x
, z
, t
) + ϕ
c (y
, z
, t
),











η (x , y , t ) =

ηp
(x
, t
)

+

ηc
(y
, t
).

(3.6a) (3.6b)

The dimensional variables may be scaled by the following scales1,5: x
=

x ; k


y
=



h=κh; ωp
2 = g
k
τλ ; τ12 = 1 + τ ;

y ; κ


z
=

z ; κ






t ; g
κ


(3.7a–g)





ξ=k;

b=κb; τ = T
κ
2 /g
;

τλ = 1 + (τ /λ4r ); λ2r = κ
/k
;

t
= √

ω
c = g
κ
τ12 ; 2

(3.7h–m)

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 ϕ
p

=

ϕp a
p

ηc


=

g
; k


a
c ηc ;

 ϕ
c

=




χ =

ϕc a
c

a
w χ;

g
; κ


ηp
= a
p ηp ;

La
2 g
L =
c
. kκ

(3.7n–s)




Scaling by (3.7) yields the following dimensionless scaling parameters1,3: ε = κ
a
c ;

β = ωc
/ωp
;

γ = k
a
w ;

Γ = a
p /a
c ;

(3.8a–d)

and the following dimensionless field variables: ϕ(x, y, z, t) = ϕc (y, z, t) + Γλr ϕp (x, z, t); η(x, y, t) = ηc (y, t) + Γηp (x, t).

(3.9a,b)

The independent variation of ϕ
and η
vanish at arbitrary temporal values t
1 and t
2 in Hamilton’s principle (3.3) yielding the following scaled boundary value problem: 1 ϕxx + ϕyy + ϕzz = 0; γχ ≤ x ≤ ξ, y ≤ |b|, −h ≤ z ≤ εη, (3.10a) λ4r   1 z = εη, γχ ≤ x ≤ ξ, y ≤ |b|, ϕ η + ϕ η ϕz = −τ1 ηt + ε x x y y ; λ4r (3.10b)     ε 1 2 1 ϕ + ϕ2y + ϕ2z − τ1 ϕt + η = τ ηxx + ηyy − αϕz ; 2 λ4r x λ4r  γχ ≤ x ≤ ξ, (3.10c) y ≤ |b|, z = εη  ˆ = 0; ∇ϕ · n

y = |b|,

−h ≤ z ≤ εη,

γχ ≤ x ≤ ξ

z = −h,

y ≤ |b|,

γχ ≤ x ≤ ξ,

γ ϕx = − λ4r τ1 χt + γλ4r ϕz χz ; ε

x = γχ,

y ≤ |b|,

−h ≤ z ≤ εη,

(3.10d,e)

(3.10f)

plus an appropriate radiation condition at x = ξ. The free surface curvature requires a dynamical constraint that is given either by the contact line condition ηn

= 0 or by the edge constraint boundary condition η
= 0.1,3 The wavemaker perturbation forcing parameter γ is smaller than the Floquet parametric forcing parameter ε because experiments demonstrate that the standing cross wave amplitude becomes larger than the wavemaker forcing amplitude as t → ∞. The parameter ordering is 0 < γ 2 < εγ < ε2 < γ < ε < 1 or

0<

γ2 γ < γ < < 1. ε ε

(3.11a,b)

The higher order terms that will be neglected are O(ε2 ), O(εγ), and O(γ 2 ); but terms O(γ/ε) will be retained. Neglecting these higher order terms, the nondimensional Lagrangian integrals may be approximated by the following linear

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combination1,3 : Lχ = −

γτ1 2ε
b

+ −b







b −b

0

−h

[(ϕc + Γλr ϕp )χt ]x=0 dzdy

[εχt (ηc + Γηp )(ϕc + Γλr ϕp )]x,z=0 dy,


b
0 Γ [ϕpx (ϕc + Γλr ϕp )]x=ξ dzdy Lξ = − 3 2λr −b −h
b + [εϕpx (ηc + Γηp )(ϕc + Γλr ϕp )]x=ξ,z=0 dy, Lη =

1 2




−b b

−b




ξ

(3.12a)

(3.12b)

{τ1 (ηc + Γηp )t [(ϕc + Γλr ϕp )

0

+ ε(ηc + Γηp )(ϕc + Γλr ϕp )r ] + (ηc + Γηp )2 }z=0 dxdy
γ b − {χ[τ1 (ϕc + Γλr ϕp )(ηc + Γηp )t + (ηc + Γηp )2 ]}x,z=0 dy, (3.12c) 2 −b  


b   2 τ b ξ Γ2 2 Γ 2 2 2 Lτ = η + ηcy dxdy − η + ηcy dy. γχ 2 −b 0 λ4r px λ4r px −b x,z=0 (3.12d) The wavemaker forcing χ is specified to be t χ = f (z) sin β



f (z) = 1 + f (z) = 1,

z , for a full draft hinge, h for a full draft piston;

(3.13a,b)

and the deepwater cross wave and wavemaker wave are approximated by ϕc (y, z; t) = q(t) cos(y − b) exp[z], ϕp (x, z; t) = [Q1 (t) cos x +

Q2 (t) sin x] exp[z/λ2r ],

(3.14a) (3.14b)

where the variables q(t), Q1 (t), and Q2 (t) are the generalized coordinates. The freesurface displacement η is a solution to the linearized inhomogeneous boundary value problem given by L η η = [Lpη ηp − fp ] + [Lcη ηc − fc ] ≡ 0; Lpη ηp − fp = 0; ηpx = 0;

x = 0,

(3.15a)

Lcη ηc − fc = 0;

(3.15b,c)

ηcy = 0;

(3.15d,e)

y = |b|,

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where (3.15d and 3.15e) are the contact line conditions and where Lpη ηp =

τ ηp − ηp , λ4r xx

fp = αλr ϕpz − τ1 λr ϕpt ;

Lcη ηc = τ ηcyy − ηc ,

fc = αϕcz − τ1 ϕct ;

z = 0,

z = 0.

(3.15f,g) (3.15h,i)

Substituting (3.14a) and (3.15h,i) into (3.15c) yields ηc =

1 (−qα + τ1 q) ˙ cos(y − b). τ12

Substituting (3.14b) and (3.15f,g) into (3.15b) yields   −Q2 α + λ2r τ1 Q˙ 2 −Q1 α + λ2r τ1 Q˙ 1 cos x + ηp = λr τλ λr τλ  

√ λ2p τ . × sin x + 2 exp − √ x λp τ

(3.16)

(3.17)

The Lagrangian may be decomposed into the following components: L = L0 + Lε + Lγ + O(ε2 , εγ, γ 2 );

(3.18a)

where   bΓ2  2    (t) + Q22 (t)) + Q1 (t)Q2 (t)] [ξ(Q −   1   2         2 2 bτ Γ 1 4 2 2 ˙ ˙ ˙ ˙ L0 = ; + [ξλ τ ( Q (t) + Q (t)) + 4τ Q (t) Q (t)] 1 2 r λ 1 2   2λ2r τλ2             bξ 2 2   − (q (t) − q˙ (t)) 2 εbΓτ1 τ εbΓ Lε = − 3 q 2 Q˙ 2 − q qQ ˙ 2; 2λr τλ 2τ1 λ3r

(3.18b)

(3.18c)

   √ t q q˙ Γλr τ1 Q1 Γ2 τ12 2 ˙ ˙ Q1 (λr Q1 + τ Q2 ) + f1 + Lγ = −γb cos β εβ βτλ 2β    √ t Γ2 τ 2 −q˙2 + q2 Γ2 Q21 − 2 12 (λ4r Q˙ 21 + λ2r τ Q˙ 1 Q˙ 2 + τ Q˙ 22 ) + , + γb sin β λr τλ 2 (3.18d) where f1 is an integral of (3.13a or 3.13b), L0 represents the free oscillations, Lε represents the Floquet parametric forcing of the cross wave by the progressive wave, and Lγ represents the completely nonautonomous perturbation. The analogy to the nonautonomous Hamiltonian for a Floquet oscillator follows.

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3.2.2. Legendre transform The Legendre transform of the Lagrangian yields the following Hamiltonian: ˙ t), H(q, p, t) = pq˙ + P1 Q˙ 1 + P2 Q˙ 2 − L(q, q,

(3.19)

where p = (p, P1 , P2 ) are the conjugate momenta corresponding to the generalized coordinates q = (q, Q1 , Q2 ). The conjugate momenta are computed from pi =

∂(L0 + Lε ) ; ∂ q˙i

p1 = p = bξ q˙ −

i = 1, 2, 3, εbΓ qQ2 ; 2τ1 λ3r

  2τ ξ Q˙ 1 + 4 Q˙ 2 ; λr τλ   2 2 2 εbτ1 Γτ 2 bτ1 Γ λr 2τ ˙ ˙ ξ Q2 + 4 Q1 . p3 = P2 = − 3 q + 2λr τλ τλ λr τλ p2 = P1 =

bτ12 Γ2 λ2r τλ

(3.20a) (3.20b) (3.20c) (3.20d)

Inverting (3.20b–3.20d) yields q˙1 = Q˙ 1 =

p εΓqQ2 ; + bξ 2τ1 ξλ3r

λ4r τλ P1 −

Q˙ 2 =

2τ ξ P2 2 bτ1 Γ2 λ6r ξ

λ4r τλ P2 −

−

2τ ξ P1 2 2 bξτ1 Γ λ6r

(3.21a)

ετ 2 q 2 ; Γλ9r ξ 2 τ1 τλ

(3.21b)

ετ q 2 . 2Γξλ5r τ1

(3.21c)

+

Substituting (3.18), (3.20), and (3.21) into (3.19) yields the following Hamiltonian: H(q, p, t) = H0 (q, p) + Hε (q, p) + Hγ (q, p, t) + O(ε2 , εγ, γ 2 ),

(3.22a)

where the free oscillation component is H0 =

bΓ2 1 p2 (ξ(Q21 + Q22 ) + Q1 Q2 ) + bξq 2 + 2 2 2bξ +

2τ P1 P2 (P12 + P22 ) − 2 2 6 2; 2bβ 2 Γ2 ξ bτ1 Γ λr ξ

(3.22b)

the Floquet parametric forcing component is Hε =

εΓ ετ ετ 2 2 qpQ + q P − q 2 P1 ; 2 2 2τ1 ξλ3r 2τ1 Γξλ5r τ1 Γξ 2 λ9r τλ

(3.22c)

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and the nonautonomous perturbed component from the wavemaker forcing is   √   Q1 (λ2r P1 + τ P2 ) bΓλr τ1 Q1     +    f1      εβ βξλ2r   t       γ cos   √     2 β   2τ Q1 (λr P2 + τ P1 ) pq − + . Hγ = 2 6 βξ λr τλ 2βξ            √      (λ2r P1 + τ P2 )2 1 2 p2 t   2 2  −γ sin  + bq − bΓ Q1 − 2 2 2 6 β bτ1 Γ ξ λr 2 2bξ 2

(3.22d)

Although the nonautonomous Hamiltonian in (3.22d) is homomorphic to a nonautonomous Floquet Hamiltonian, a subtle but important distinction exists. The nonautonomous component in a Floquet Hamiltonian is due to parametric forcing; while, in contrast, the nonautonomous component in (3.22d) is due to external wavemaker boundary forcing that generates the progressive wave that then parametrically forces the cross waves. This sequence of external forcing to parametric forcing distinguishes cross wave parametric forcing from the simple pendulum Floquet parametric forcing. Damping forces. The scaled Hamilton’s principle (3.3) is given by1,3
δ

t2

t1


L dt =

t2

t1

3 


Di δqi dt = −

i=1

t2




t1

b −b




ξ

(αϕz δη)dxdydt,

(3.23a)

0

where D = (D1 , D2 , D3 ) is a set of generalized components of the damping force corresponding to the set of generalized coordinates q = (q, Q1 , Q2 ) and where

δη =

3   ∂η i=1

∂qi

δqi +

 ∂η δ q˙i . ∂ q˙i

(3.23b)

Integrating by parts, the last term1,3 reduces D to the canonical variables q, p   ∂ ∂η ∂η + Di = α −ϕz dxdy; i = 1, 2, 3, ϕz ∂qi ∂t ∂ q˙i −b 0      2  αεbΓ α α bξ p+ qQ2 + q, D1 = τ1 2τ12 λ3r τ12         2 α 2ατ αεbΓτ 2 α bξΓ2 2 D2 = P1 − P2 − q + Q1 , λ2r τ1 ξλ6r τ1 τλ λ9r ξτλ λ2r τλ    2  2     α αεbΓτ 2α bτ Γ2 α bξΓ2 2 P2 + q + Q1 + Q2 . D3 = τ1 λ2r 2λ5r τλ λ6r τλ2 λ2r τλ


b




ξ



(3.24a) (3.24b) (3.24c) (3.24d)

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3.3. Hamilton’s Dynamic Equations and Canonical Transformations Hamilton’s equations of motion are given by q˙i =

∂H ; ∂pi

p˙ i = −

∂H − Di ; ∂qi

i = 1, 2, . . . , N.

(3.25a,b)

If the canonical transformation Q = Q(q, p, t), P = P(q, p, t) is computed from a generating function F (u, U, t) computed from the GHA-Type I (Appendix A), then the Hamiltonian K for the new set of variables is given by K(Q, P, t) = H[q(Q, P, t), p(Q, P, t), t] +

∂F (u, U, t) ; ∂t

(3.26a)

and the transformed Hamilton’s equations of motion are N

∂K  ˜ ∂qj Q˙ i = + ; Dj ∂Pi j=1 ∂Pi

i = 1, 2, . . . , N,

N  ∂K ˜ j ∂qj ; P˙i = − − D ∂Qi j=1 ∂Qi

i = 1, 2, . . . , N,

˜ i (Q, P, t) = Di [q(Q, P, t), p(Q, P, t)]; D

i = 1, 2, . . . , N.

(3.26b)

(3.26c) (3.26d)

3.3.1. Three canonical transformations and analyses by the GMM In order to apply the GMM to the dynamical system in (3.26), canonical transformations are required first to simplify terms in H0 ; second to simplify terms in Hε ; and third to suspend nonautonomous terms in Hγ . The GHA will be applied to two of the following three canonical transformations, because the rotation of axes transformation is well known so that the GHA will not be applied for that transformation even though it is still applicable. (1) Rotation of axes. The cross product terms in (3.22b) may be eliminated by q = q˜,

p = p˜,

(3.27a,b)

˜ 1 cos θ + Q ˜ 2 sin θ, Q1 = Q

P1 = P˜1 cos θ + P˜2 sin θ,

(3.27c,d)

˜ 2 cos θ − Q ˜ 1 sin θ, Q2 = Q

P2 = P˜2 cos θ − P˜1 sin θ,

(3.27e,f)

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that satisfy the Poisson bracket requirements in Appendix A. The rotation angle θ = π/4 eliminates the cross product terms, and the transformed Hamiltonian is given in component form by ˜ε + H ˜ γ (t) + O(ε2 , εγ, γ 2 ), ˜ =H ˜0 + H H→H where

   2   bξ 2 p˜2 P˜12 bΓ ξ ˜ 2     q˜ + + Q1 +    2 2   2 2bξ 2 2bβ Γ ξ            

    2 2 ˜2 P ξ bΓ 2 ˜ ˜ , + Q2 + H0 =   2 2bβ 2 Γ2 ξ          

    2 2 2 ˜ ˜   τ ( P − P ) bΓ   1 2 2 2  ˜  ˜ + (Q2 − Q1 ) +    2 4 bτ1 Γ2 λ6r ξ 2 

(3.28a)

(3.28b)

4 ˜2 − Q ˜ 1 ) − ετ√(2τ + ξλr τλ ) q˜2 P˜1 ˜ ε = √ εΓ q˜p˜(Q H 2 2τ1 ξλ3r 2 2τ1 Γξ 2 λ9r τλ

+

ετ (−2τ + ξλ4r τλ ) 2 ˜ √ q˜ P2 , 2 2τ1 Γξ 2 λ9r τλ

(3.28c)

   ˜1 + Q ˜ 2) bΓλr τ1 (Q     √ f   1      2εβ               √ √     2 2  ˜ 2 )((λr − τ )P˜1 + (λr + τ )P˜2 )  ˜1 + Q t ( Q       γ cos     +   2     β 2βξλ   r           √ √   2 2 ˜ ˜ ˜ ˜ ˜ τ (Q1 + Q2 )((λr + τ )P2 + (−λr + τ )P1 ) p˜q˜ . Hγ = − +   2 λ6 τ   βξ 2βξ   r λ         √ √     2 2 2 2 ˜ ˜   ((λ − τ ) P + (λ + τ ) P ) bΓ   1 2 r r 2   ˜ ˜    + Q ) −  ( Q 1 2   2 2 2 6   2 t   2bτ1 Γ ξ λr       −γ sin     2   β 1 p ˜   2   + b˜ q − 2 2bξ 2 (3.28d) The first three energy square brackets [•] in (3.28b) are the action variables in the next canonical transformation. (2) Action/Angle Transformation. The three new canonical variables are given by pˆ =

bξ 2 p˜2 q˜ + , 2 2bξ

bΓ2 ξ ˜ 2 P˜12 Pˆ1 = , Q1 + 2 2bβ 2 Γ2 ξ

bΓ2 ξ ˜ 2 P˜22 , Q2 + Pˆ2 = 2 2bβ 2 Γ2 ξ

(3.29a–c)

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that satisfy the Poisson brackets. The Herglotz auxiliary functions Xi are the ratios p; X1 = q˜/˜

˜ 1 /P˜1 ; X2 = Q

˜ 2 /P˜2 . X3 = Q

(3.30a–c)

The angle variables are qˆi =

3 

q˜j

j=1

∂F ∂ p˜j + ; ∂ pˆi ∂ pˆi

i = 1, 2, 3,

(3.31a)

where qˆ = tan−1 (bξX1 );

ˆ 1 = β tan−1 (bβξΓ2 X2 ); Q

ˆ 2 = β tan−1 (bβξΓ2 X3 ). Q (3.31b–d)

The canonical transformation to action/angle variables is given by the following:  q˜ =

 ˜1 = Q

 ˜2 = Q

2Pˆ1 sin bξΓ2

2Pˆ2 sin bξΓ2





ˆ1 Q β

ˆ2 Q β

2ˆ p sin qˆ, bξ

p˜ =





2bξ pˆ cos qˆ,



 ,

P˜1 = Γβ

,

P˜2 = Γβ



2bξ Pˆ1 cos



 2bξ Pˆ2 cos

(3.32a,b)

ˆ1 Q β

ˆ2 Q β

,

(3.32c,d)

,

(3.32e,f)



that satisfy the Poisson brackets.1,3 The transformed Hamiltonian is ˜ ⇒H ˆ =H ˆ0 + H ˆε + H ˆ γ (t) + O(ε2 , εγ, γ 2 ), H     ˆ1 2β 2 τ Q 2β 2 τ 2 ˆ P1 cos − λ6r ξτ12 β λ6r ξτ12      ˆ2 ˆ1 ˆ2 ˆ1 ˆ2 Q Q Q P P 2 2 2 × Pˆ2 cos − sin + sin , β 2ξ β 2ξ β

(3.33a)



ˆ 0 = [ˆ p + Pˆ1 + Pˆ2 ] + H

(3.33b)

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    

ˆ2 ˆ2   Q Q   ˆ   P2 cos 2ˆ − cos 2ˆ q+ q−     β β   εˆ p ˆ Hε =   

   4 bξ 3 λ3r τ1    ˆ1 ˆ1   Q Q   ˆ   − cos 2ˆ q+ − P1 cos 2ˆ q−   β β εβτ pˆ −  2 bξ 5 λ9r τ1 τλ     

ˆ1 ˆ1   Q 1 Q   4   (2τ + ξλr τλ ) Pˆ1 cos − cos 2ˆ q+       β 2 β          

       ˆ2  ˆ1 Q 1 Q 4 ˆ , × q− + (2τ − ξλr τλ ) P2 cos − cos 2ˆ  2 β β                   ˆ ˆ   Q 1 Q 1   2 2   cos 2ˆ q + − cos 2ˆ q − −     2 β 2 β (3.33c)    √     ˆ1 ˆ2 bλr f1 τ1  Q Q     √ + Pˆ2 sin Pˆ1 sin       β β βε ξ                    √      2 4   ˆ   + τ )(−2τ + ξλ τ ) (λ Q   λ 1 r r   ˆ     P + cos 1     6 2     λ ξ τ β λ   t r     γ cos           β   √      2 4   ˆ + τ )(−2τ + ξλ τ )  (λ  Q λ 2   r r   ˆ   P + cos   2   6 ξ2τ  λ β    λ  r                  pˆ sin(2ˆ q)   + ˆ . Hγ = 2βξ    

        ˆ   √ −β 2   2   ˆ1 cos Q1 P (λ − τ )     r   6τ 2   λ β     r 1               2        ˆ √   Q   2   2   ˆ  γ  + τ ) cos + (λ P   2 t   r     β   − sin       ξ β       

  2         ˆ ˆ     Q Q 1 2   ˆ1 sin ˆ2 sin     + + P P        β β            2 2 + pˆ(sin qˆ − cos qˆ) (3.33d)

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The nonautonomous perturbed component in (3.33d) will not survive the KAM averaging theorem, and some of the nonautonomous terms may be suspended by applying the following Hamilton–Jacobian nonautonomous canonical transformation. (3) Hamilton–Jacobian transformation. The following nonautonomous transformations: qˆ =

t + q + Q1 , 2β

ˆ 1 = t + 2βQ1 , Q

ˆ 2 = 3t + 2β(q + Q1 + Q2 ) Q

(3.34a–c)

satisfy the Poisson brackets. The Herglotz auxiliary functions Xi = Xi (p, q) satisfy the nonzero Jacobian condition given by ∂(ˆ q, X) 4β 2 = 0. = ∂(p, q) qQ1 Q2

(3.35)

Solving (3.35) for qi = qi (p, X), substituting qi (p, X) into (3.34), and inverting yields p=

X1 ˆ 1 ); P1 = −X2 (t − Q ˆ 1 ); (2β qˆ − Q 2β 2β −X3 ˆ 2 ). P2 = (2t + 2β qˆ − Q 2β

(3.36a–c)

The generating function F (•, •, •) for the nonautonomous canonical transformation may be computed from the following indefinite integral: F (ˆ q, p(ˆ q, X, t), t)  3
Xi   ∂P1 ∂P2 ∂p
=− q
+ Q1 ∂X
+ Q2 ∂X
dXi ∂X i i i i=1 =

−1 ˆ 1 )2 + X2 (t − Q ˆ 1 )2 + X3 (2t + 2β qˆ − Q ˆ 2 )2 ]. [X1 (2β qˆ − Q 4β 2

(3.37)

Computing the three remaining conjugate momenta from pˆi = −

3  j=1

qj

∂F ∂pj − ; ∂ qˆi ∂ qˆi

i = 1, 2, 3

(3.38a)

1 Pˆ2 = P2 . 2β

(3.38b–d)

gives pˆ = p − P2 ,

1 Pˆ1 = (P1 − p), 2β

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The transformations in (3.34) and (3.38) satisfy the Poisson brackets, and the first square bracket term [•] in (3.33b) may be transformed to [ˆ p + Pˆ1 + Pˆ2 ] =

  1 P1 1− (p − P2 ) + . 2β 2β

(3.39)

The primary Floquet resonance condition is, approximately, ωc
− ωp
≈ O(ε) so that a detuning parameter Ω may be defined by     ωp
1 1− = 1 − = εΩ; 2ωc
2β

(3.40)

and the primary parametric Floquet resonance is ωp
: ωc
= 2:1. A generating function is now given by   1 ˆ 1 ˆ 1 ) + 1 P2 (2t + 2β qˆ − Q ˆ 2 ); F (ˆ q, p, t) = p −ˆ q+ P1 (t − Q Q1 + 2β 2β 2β

(3.41)

and the transformed Hamiltonian H(p, q, t) with autonomous and nonautonomous terms by ˆ − ∂F ⇒ H = H0 + H0 (t) + Hε + Hε (t) H ∂t + Hγ + Hγ (t) + O(ε2 , εγ, γ 2 ),

(3.42a)

where     βτ 1 βτ 1 1 (P P2 , + + − − − p) + − 1 8βξ 2ξλ6r τ12 β 8βξ 2ξλ6r τ12 √ ε(p − P2 ) P1 − p Hε = εΩ(p − P2 ) +  [(2βτ 2 − ξλ4r τλ (λ2r − βτ )) cos(2q)], 4 2bβξ 5 λ9r τ1 τλ  √   γ bf1 λr τ1  2β − 1  P1 − p sin(2Q1 ) + γ Hγ = 4βξ 2ε 2ξβ 3 H0 =

× (P2 − p) sin 2(q + Q1 ),

(3.42b) (3.42c)

(3.42d)

and where the nonautonomous terms are given by Fadel.5 (4) Transformation to original variable (qorig , porig ) ⇒ (q, p). In order to compute the generalized damping forces in (3.24), the original canonical variables (qorig , porig ) must be expressed as functions of the final transformed canonical

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variables (q, p) by

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  2 t , p − P2 sin q + Q1 + bξ 2β     t , porig = 2bξ p − P2 cos q + Q1 + 2β     1 t Q1orig = √ P1 − p sin 2Q1 + β Γ 2bβξ    3t + P2 sin 2(q + Q1 + Q2 ) + , β     bβξ  t P1orig = Γ P1 − p cos 2Q1 + 2 β    3t , + P2 cos 2(q + Q1 + Q2 ) + β     1 t √ Q2orig = − P1 − p sin 2Q1 + β Γ 2bβξ    3t + P2 sin 2(q + Q1 + Q2 ) + , β      bβξ t − P1 − p cos 2Q1 + P2orig = Γ 2 β    3t + P2 cos 2(q + Q1 + Q2 ) + . β qorig =

(3.43a) (3.43b)

(3.43c)

(3.43d)

(3.43e)

(3.43f)

3.3.2. Transformed damping forces The transformed damping forces are1,3    √ t 2bβξα      p − P2 cos q + Q1 +     τ1 2β         √       2    t 2bξα     + p − P sin q + Q +   2 1 2   τ 2β   1           εα t , D1 = √ + p − P sin q + Q + 2 1  2β  2λ3r ξ βτ12                  3t     P sin 2(q + Q + Q ) +   2 1 2     β         ×              t     − P1 − p sin 2Q1 + β

(3.44a)

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    √ t bβαΓ(2τ + λ4r ξτλ )      √ P − p cos 2Q + 1 1   6 2ξ   β τ τ λ   1 λ r       √         4    3t bβαΓ(−2τ + λ ξτ ) λ   r   √ + P cos 2(q + Q + Q ) +   2 1 2   6 2ξ β   τ τ λ 1 λ r             2 2Γεατ t − 9 2 2 (p − P2 ) sin 2(q + Q1 ) + , D2 =   λr ξ τλ 2β         √     2    Γ bξ α t     √   + P − p sin 2Q + 1 1   2τ   β λ 2β   r λ       √     2      Γ bξ α 3t     + √ P sin 2(q + Q + Q ) + 2 1 2 2 β λr τλ 2β (3.44b)        t     − P − p cos 2Q + 1 1  √      β   αΓ bβξ      √          2τ      2λ   3t r 1     + P cos 2(q + Q + Q ) +   2 1 2   β                 εαΓτ t   + ) sin 2(q + Q ) + (p − P 2 1 5 D3 = , λ ξτ 2β r λ          √       t α2 Γ b(2τ − λ4r ξτλ )       √  P − p sin 2Q + + 1 1   2 6τ   β λ 2βξ   r λ          √     2 4    3t Γ b(2τ + λ ξτ ) α   λ r     √ P sin 2(q + Q + Q ) + + 2 1 2  2 6 β  λr τλ 2βξ (3.44c) that are completely autonomous and will survive the KAM averaging operation. Averaging H(q, p, t) over the dimensionless cross wave period 2π yields

H = H0 (p, P1 , P2 ) + Hε (q, p, P1 , P2 ) + Hγ (q, p, P1 , P2 , Q1 )     βτ 1 βτ 1 1 (P P2 + + − = − − p) + − 1 8βξ 2ξλ6r τ12 β 8βξ 2ξλ6r τ12

 √ (p − P2 ) P1 − p 2 4 2 + ε Ω(p − P2 ) +  [(2βτ − ξλr τλ (λr − βτ )) cos(2q)] 4 2bβξ 5 λ9r τ1 τλ 

 √   2β − 1 bf1 λr τ1   (P2 − p) sin 2(q + Q1 ) . P1 − p sin(2Q1 ) + +γ 4βξ 2ε 2ξβ 3 (3.44d)

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The averaged system of autonomous equations is1,3   2P1 + P2 − 3p     √ −a cos[2q] + εΩ + εa 1 3     P − p   1         sin[2Q1 ] 2 − γc2 sin[2(q + Q1 )] + α d1 , − γc1 √ q˙ = ε P1 − p               d (P − p) + d (P − p)   2 1 3 2     √ sin[2q] − αε P1 − p      4εα3 P1 − p(p − P2 ) sin[2q] − 2γc2 (P2 − p) cos[2(q + Q1 )]   p˙ = ,  α     + αd4 P2 − p + 2αεd2 P1 − p(p − P2 ) cos[2q] τ1    4γ     − P − p cos[2Q ] − 2γc (P − p) cos[2(q + Q )] c 1 1 2 2 1   ε 1 ˙ , P1 = 2  (4λr − 1)(P2 − p) − P1       +α − αεd P − p(p − P ) cos[2q] 5 1 2 2λ2r τ1   τ − λ4r ξτλ P˙ 2 = αP2 , λ6r ξτ1 τλ   (p − P2 ) sin[2Q1 ]     √ √ cos[2q] + γc + εa a 3 1   1 P1 − p ε P1 − p ˙ , Q1 =   P2 − p     sin[2q] + α2 d6 + αεd3 √ P1 − p     a2 − εΩ − 2εa3 P1 − p cos[2q] + γc2 sin[2(q + Q1 )]  , Q˙ 2 =    + α2 d7 + αεd2 P1 − p sin[2q]

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(3.45a)

(3.45b)

(3.45c)

(3.45d)

(3.45e)

(3.45f)

where ai , ci , and di are tabulated in Appendix B. 3.4. Application of the GMM and Largest Liapunov Characteristic Exponent The GMM determines the existence of transverse homoclinic/heteroclinic points that are transverse intersections between the stable and unstable manifolds to any invariant sets of the perturbed system when a homoclinic/heteroclinic orbit exists to a hyperbolic invariant manifold in the unperturbed (undamped α = 0 and unperturbed γ = 0) system. The unperturbed vector field may be computed by setting the perturbation (wavemaker forcing) parameter γ = 0 and the dissipation parameter α = 0 in (3.45), and are q˙ = −a1 + εΩ + εa3

2P1 + P2 − 3p √ cos(2q), P1 − p

(3.46a)

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p˙ = 4εa3



P1 − p(p − P2 ) sin(2q),

P˙1 = P˙2 = 0,

(3.46b) (3.46c,d)

(p − P2 ) Q˙ 1 = a1 + εa3 √ cos(2q), P1 − p  Q˙ 2 = a2 − εΩ − 2εa3 P1 − p cos(2q),

(3.46e) (3.46f)

where the coefficients a1 , a2 , and a3 are defined in Appendix B. The unperturbed vector field α = γ = 0 in (3.46) is a three degrees of freedom Hamiltonian system with (q, p, P1 , P2 , Q1 , Q2 ) ∈ T1 × R1 × R2 × T2 . An important consequence of the Hamilton–Jacobian canonical transformation is that the unperturbed Floquet Hamiltonian H(γ = 0) = H0 (p, P1 , P2 ) + Hε (q, p, P1 , P2 ) is independent of Q1 and Q2 . Hyperbolic saddle points. For every (P1 , P2 ) ∈ R2 , the (q − p) components of the unperturbed vector field α = γ = 0 in (3.46) possess a hyperbolic saddle point that varies smoothly with P1 and P2 and is given by p0 (P2 ) = P2 ;

2q0 = nπ;

n = 0, 1, . . . ,

  a1 − εΩ 1 √ , cos−1 2 2εa3 P1 − P2  2 a1 − εΩ P1 ≥ P2 + . 2εa3

q0 (P1 , P2 ) =

(3.47a,b)

(3.47c)

(3.47d)

The fixed point (q0 , p0 ) is a hyperbolic saddle point if the determinant satisfies # # # ∂ q˙ ∂ q˙ # # # # ∂q ∂p # # # = (a1 − εΩ)2 − 4(P1 − P2 )ε2 a23 < 0. (3.48) # ∂ p˙ ∂ p˙ # # # # ∂q ∂p # (q=q0 ,p=p0 ) The symmetry properties of Hamiltonian systems imply that the stable and the unstable manifolds of the hyperbolic saddle point (q0 , p0 ) have equal dimensions in the full 6D phase space (q, p, P1 , P2 , Q1 , Q2 ) ∈ T1 × R1 × R2 × T2 . The unperturbed system α = γ = 0 has a 4D (R2 × T2 ) normally hyperbolic invariant manifold given by the union of the hyperbolic saddle points (q0 , p0 ) according to M = {(q, p, P1 , P2 , Q1 , Q2 ) ∈ T1 × R1 × R2 × T2 : q = q0 (P1 , P2 ); p = p0 (P2 )}. (3.49) The normally hyperbolic invariant manifold M has 5D (R1 ×R2 ×T2 ) stable manifold W s (M) and unstable manifold W u (M) that coincide along the 5D heteroclinic manifold H = W s (M) ∩ W u (M) − M, where W s (M) and W u (M) are the set of

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initial conditions that approach the hyperbolic saddle points as t → ±∞ under the action of the unperturbed flow. Dynamics on M. The unperturbed vector field α = γ = 0 restricted to the normally hyperbolic invariant manifold M may be determined from1,3 P˙1 = 0;

P˙2 = 0;

Q˙ 1 = a1 ;

Q˙ 2 = a2 − a1 .

(3.50a–d)

The unperturbed vector field α = γ = 0 restricted to M has the form of two degrees of freedom, 4D phase space (T1 × R1 × T2 ), completely integrable Hamiltonian system with the Hamiltonian given by the level energy surfaces H(P1 , P2 ) = H(q = q0 , p = p0 ; γ = 0) = a1 P1 + (a2 − a1 )P2 = E,

(3.51)

where E is a constant energy that allows the phase space motion to be reduced from four dimensions (R2 × T2 ) to three dimensions. Because P1 = P2 = 0, the constant energy surfaces may be reduced to a 1D surface in the 3D constant energy space. On this 1D surface, the angular motion is parameterized by the two frequencies σ1 = a1 , σ2 = a2 − a1 .

(3.52a,b)

The angular components of the motion on the normally hyperbolic invariant manifold M are given by Q1 (t) = a1 t + Q1 (0);

Q2 (t) = (a2 − a1 )t + Q2 (0).

(3.53a,b)

The normally hyperbolic invariant manifold M is a two-parameter (P1 , P2 ) family of 2D tori shown in Fig. 3.4. A 2D torus on M may be defined by  $ (q, p, P1 , P2 , Q1 , Q2 ) ∈ T1 × R1 × R2 × T2 : q = q0 (P¯1 , P¯2 ); ¯ ¯ Υ(P1 , P2 ) = , p = p0 (P¯2 ); P1 = P¯1 ; P2 = P¯2 (3.54) where each 2D torus is invariant. The level energy surface is a family of concentric tori as shown in Fig. 3.5. Because the frequencies of motion σ1 and σ2 in (3.52) are independent of P1 or P2 , they do not change from one concentric torus to another concentric torus. The motion on the surface of the invariant torus Υ ⊂ M is quasiperiodic. When the frequency ratio σ1 /σ2 = a1 /(a2 − a1 ) is an irrational number, the motion on the surface of the 2D invariant nonresonant torus may no longer be periodic; i.e., trajectories wind densely on the surface of the torus and never close on themselves. This 2D torus has a 3D (R1 × T1 ) stable manifold W S and unstable manifold W U that coincide along the 3D (R1 × T1 ) heteroclinic manifold H defined for fixed values of P1 and P2 , as shown in Fig. 3.4. On a constant level energy surface, neither the nonresonant invariant tori nor the stable and unstable manifolds may be isolated. In addition, the 2D nonresonant invariant torus has a 2D center manifold corresponding to nonexponentially expanding or contracting directions tangent to the normally hyperbolic invariant manifold M.

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Fig. 3.4.

3D unperturbed heteroclinic manifold.1

Fig. 3.5. Motion of a phase space point for an integrable Hamiltonian system with two degrees of freedom. (a) Invariant tori in a 3D constant energy space E; and (b) the flow on a 2D torus.1,3

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Heteroclinic orbits. For fixed values of (P1∗ , P2∗ ) ∈ R2 , the (q, p) components of the unperturbed vector field α = γ = 0 in (3.46) possess a 1D heteroclinic orbit connecting the hyperbolic saddle points (q0 , p0 ). The heteroclinic orbit lies on each of the level energy surfaces defined by (3.51) and are solutions to

H(γ = 0) − (a1 P1∗ + (a2 − a1 )P2∗ ) = 0.

(3.55)

Values for q on the heteroclinic orbit are given by & %  1 q = cos−1 (a1 − εΩ)/2εa3 P1∗ − p . 2

(3.56)

In the full 6D phase space (q, p, P1 , P2 , Q1 , Q2 ) ∈ T1 ×R1 ×R2 ×T2 , the heteroclinic manifold H may be determined by substituting   2   a1 − εΩ a1 − εΩ √ √ , sin[2q] = 1 − , (3.57a,b) cos[2q] = 2εa3 P1 − p 2εa3 P1 − p into the unperturbed vector field α = γ = 0 in (3.46), and integrating to obtain1,3 √ A ph (t) = P2 + sech2 [ At] + p(0), (3.58a) B 

√ √ 1 A −1 (3.58b) − qh (t) = tan tanh( At) + q(0), 2 a1 − εΩ P1h (t) = P1 (0), P2h (t) = P2 (0),

(3.58c,d)

Q1h (t) = a1 t − qh (t) + Q1 (0) + q(0) 

√ √ 1 A −1 tanh( At) + Q1 (0), = a1 t − tan − 2 a1 − εΩ Q2h (t) = (a2 − a1 )t + Q2 (0), A = [−(a1 − εΩ)2 + (P1 − P2 )B] > 0,

(3.58e) (3.58f)

B = 4ε2 a23 ,

(3.58g,h)

π q = qn (0) = (2n + 1) ; n = 0, 1, 2, . . . . (3.58i,j) 2 The trajectories of the unperturbed system α = γ = 0 along the 5D phase space (R1 × R2 × T2 ) heteroclinic manifold H may be expressed as p(0) = 0,

Ψ(P1 ,P2 ) = {qh (t), ph (t), P1 (0), P2 (0), Q1h (t), Q2h (t)}.

(3.59)

The 6D phase space (q, p, P1 , P2 , Q1 , Q2 ) ∈ T × R × R × T is a direct product between a 4D space with coordinates (q, p, P1 , P2 ) and a 2D torus with angular 1

1

2

2

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Fig. 3.6.

Unperturbed 4D phase space.

coordinates (Q1 , Q2 ). Because P1 and P2 are constants, the motion in the 6D phase space is reduced to the four dimensions (q, p, Q1 , Q2 ) shown in Fig. 3.6. Small perturbations are anticipated to break up the geometric structure of the unperturbed system α = γ = 0 and to separate the manifolds. The behavior of the perturbed systems γ > 0 and α > 0 near the unperturbed heteroclinic manifold H is required in order to apply the GMM. The distance between the stable and unstable manifolds of any surviving invariant set in the perturbed system must be computed at a point P on the unperturbed heteroclinic manifold H. Two perturbed systems are evaluated: (1) γ > 0, α = 0 and (2) γ > 0 and α > 0. 3.4.1. Geometric structure for γ > 0 and α = 0 The perturbed vector field for nonzero wavemaker forcing γ > 0 and no dissipation α = 0 is given by q˙ = −a1 + εΩ + εa3

2P1 + P2 − 3p √ cos[2q] P1 − p

sin[2Q1 ] − γc1 √ − γc2 sin[2(q + Q1 )], ε P1 − p

    4εa3 P1 − p(p − P2 ) sin[2q] − 2γc2 (P2 − p) cos[2(q + Q1 )] ,  p˙ =  +αd4 P2 − α p + 2αεd2 P1 − p(p − P2 ) cos[2q] τ1  4γ P˙1 = − c1 P1 − p cos[2Q1 ] − 2γc2 (P2 − p) cos[2(q + Q1 )], ε

(3.60a)

(3.60b)

(3.60c)

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P˙2 = 0, (p − P2 ) sin[2Q1 ] Q˙ 1 = a1 + εa3 √ cos[2q] + γc1 √ , P1 − p ε P1 − p  Q˙ 2 = a2 − εΩ − 2εa3 P1 − p cos[2q] + γc2 sin[2(q + Q1 )],

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(3.60d) (3.60e) (3.60f)

where (q, p, P1 , P2 , Q1 , Q2 ) ∈ T1 × R1 × R2 × T2 . The perturbed vector field is computed from 5D level energy surfaces defined by

H = H0 (p, P1 , P2 ) + Hε (q, p, P1 , P2 ) + Hγ (q, p, P1 , P2 , Q1 ).

(3.61)

Because the perturbation is Hamiltonian, the 3D level energy surfaces are preserved. In the 4D normally hyperbolic invariant manifold of the unperturbed space, the locally stable and unstable manifolds and the flow describe the geometric structure of the perturbed phase space given by the perturbed normally hyperbolic locally invariant manifold, the locally stable and unstable manifolds, and the persistence of the 2D nonresonant invariant tori Υγ (P1 , P2 ). Persistence of M. The perturbed system γ > 0 and α = 0 possesses a 4D normally hyperbolic locally invariant manifold Mγ given by   (q, p, P1 , P2 , Q1 , Q2 ) ∈ T1 × R1 × R2 × T2 :     Mγ = q = q˜0 (P1 , P2 , Q1 , Q2 ; γ) = q0 (P1 , P2 ) + O(γ); .     p = p˜0 (P1 , P2 , Q1 , Q2 ; γ) = p0 (P2 ) + O(γ)

(3.62)

On Mγ there are locally stable and unstable manifolds that are of equal dimensions and are close to the unperturbed locally stable and unstable manifolds. The perturbed normally hyperbolic locally invariant manifold Mγ intersects each of the 5D level energy surfaces in a 3D set of which most of the two-parameter family of 2D nonresonant invariant tori persist by the KAM theorem.1,3 The Melnikov integral may be computed to determine if the stable and unstable manifolds of the KAM tori intersect transversely. KAM Theorem.1,3 The KAM theorem determines whether the recurrent motions occur on the perturbed normally hyperbolic locally invariant manifold Mγ and whether any of the two parameter families of 2D nonresonant invariant tori survive the Hamiltonian perturbation. The unperturbed Floquet Hamiltonian H(γ = 0) = H0 (p, P1 , P2 ) + Hε (q, p, P1 , P2 ) satisfies the following nondegeneracy (or nonresonance) condition: # 2 # ∂ H # # ∂P 2 # 1 # 2 # ∂ H # # ∂P ∂P 2

1

# # # # (a1 − εΩ)2 # =− < 0. # 4(P1 − P2 )2 ∂ 2 H ## ∂P22 #(q=q0 ,p=p0 ;γ=0)

∂ 2 H ∂P1 ∂P2

(3.63)

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Most of the 2D nonresonant invariant tori Υ(P1 , P2 )) that persist are only slightly deformed on the perturbed normally hyperbolic locally invariant manifold Mγ and are KAM tori. In the phase space of the perturbed system γ > 0 and α = 0, there are invariant tori that are densely filled with winding trajectories that are conditionally periodic with two independent frequencies σ1 and σ2 . The resulting conditionally-periodic motions of the perturbed system are smooth functions of the perturbation γ. A generalization of the KAM theorem states that the KAM tori have both stable and unstable manifolds by the invariance of manifolds.1,3 In order to determine if chaos exists, two measurements are required in order to determine whether or not W s (Υγ ) and W u (Υγ ) intersect transversely. Melnikov integral. The distance between W s (Υγ ) and W u (Υγ ) at any point P ∈ H may be computed from


βn

∂H γ (P1 ,P2 ) (Ψ )dt, ∂Q1

(3.64a)

2c1  P1 − p sin(2Q1 ) + c2 (P2 − p) sin[2(q + Q1 )], ε

(3.64b)

M(Q1 (0)) = −

lim

αn ,βn →∞

−αn

where Hγ =

where Hγ = γH γ in (3.61), and where1,3 M(Q1 (0)) = cos[2(Q1 (0) + q(0))]     4c1 
βn   P1 − ph (cos(2qh − 2a1 t)) −  dt  ε lim   αn ,βn →∞ −αn − 2c2 (P2 − ph ) cos(2a1 t) + sin[2(Q1 (0) + q(0))]     4c1 
βn   P1 − ph (sin(2qh − 2a1 t)) −  dt ,  ε lim   αn ,βn →∞ −αn + 2c2 (P2 − ph ) sin(2a1 t) (3.64c) that after retaining only even integrands reduces to1,3 M(Q1 (0)) = cos[2(Q1 (0) + q(0))][I1 + I2 + I3 ],  $
−2c1 (a1 − εΩ) βn I1 = lim cos(2a1 t)dt , n→∞ ε 2 a3 −αn ' (
√ 2c2 A ∞ cos(2a1 t)sech2 ( At)dt , I2 = B −∞  $ √
∞ √ −2c1 A I3 = sin(2a1 t)tanh( At)dt . ε 2 a3 −∞

(3.65a) (3.65b)

(3.65c)

(3.65d)

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The Melnikov integral M(Q1 (0)) = 0 when ¯ 1n (0) = (2n + 1) Q1 (0) = Q

π − q(0); 4

n = 0, 1, 2, . . . ,

¯ 1n (0)) ∂M(Q ¯ 1n (0) + q(0))][I1 + I2 + I3 ] = 0, = −2 sin[2(Q ∂Q1 (0)

(3.66a)

n = 0, 1, 2, . . . . (3.66b)

Consequently, the stable W s (Υγ (P1 , P2 )) and unstable W u (Υγ (P1 , P2 )) manifolds of the KAM tori Υγ (P1 , P2 ) intersect transversely yielding Smale horseshoes1,3 on the appropriate 5D level energy surfaces. This implies multiple transverse intersections and the corresponding existence of chaotic dynamics in the perturbed system γ > 0 and α = 0. 3.4.2. Geometric structure for γ > 0 and α > 0 The perturbed dissipative system α > 0 and γ > 0 possesses a 4D normally hyperbolic locally invariant manifold Mγα that is given by

Mαγ

  (q, p, P1 , P2 , Q1 , Q2 ) ∈ T1 × R1 × R2 × T2 :       = q = q˜0 (P1 , P2 , Q1 , Q2 ; γ) = q0 (P1 , P2 ) + O(γ), .       p = p˜0 (P1 , P2 , Q1 , Q2 ; γ) = p0 (P2 ) + O(γ)

(3.67)

The manifold Mγα has locally stable and unstable manifolds that are close to the unperturbed locally stable and unstable manifolds; and if these manifolds intersect transversely, then the Smale–Birkhoff theorem predicts the existence of horseshoes and their chaotic dynamics in the perturbed dissipative system. A 2D hyperbolic invariant torus Υγα (P1 , P2 ) may be located on Mγα by averaging the perturbed dissipative vector field γ > 0 and α > 0 restricted to Mγα over the angular variables Q1 and Q2 . The averaged equations have a unique stable hyperbolic fixed point (P1 , P2 ) = (0, 0) with two negative eigenvalues provided that the determinant1,3 # # # ∂ P˙1  ∂ P˙ 1  # # # # ∂P ∂P2 ## ν 2 γ 2 (λ4r ξτλ − τ ) 1 # > 0. (3.68) # #= # ∂ P˙  ∂ P˙  # 2λ8r ξτ12 τλ 2 2 # # # # ∂P1 ∂P2 The Melnikov method fails to predict chaos for this dissipative system when α > 0.1,3 Liapunov characteristic exponents (LCE). Dissipative systems are characterized by the attraction of all trajectories passing through a certain domain toward an invariant surface or an attractor of lower dimensionality than the original space.

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For the 6D phase space (q, p, P1 , P2 , Q1 , Q2 ) ∈ T1 × R1 × R2 × T2 , there are six real exponents that may be ordered as µ1 ≥ µ2 ≥ µ3 ≥ µ4 ≥ µ5 ≥ µ6 , where µ1 is the largest Liapunov characteristic exponent (LCE). The LCE is calculated from the first variation of the perturbed dissipative vector field γ > 0 and α > 0 according to  ∂ q˙  ∂q   ∂ p˙     δ q˙  ∂q     δ p˙   ˙    ∂ P1  ˙     δ P1   ∂q    δ P˙  =  ∂ P˙2  2      δ Q˙   ∂q  1    ∂ Q˙ 1  δ Q˙ 2  ∂q    ∂ Q˙ 2 ∂q

∂ q˙ ∂p

∂ q˙ ∂P1

∂ q˙ ∂P2

∂ q˙ ∂Q1

∂ p˙ ∂p

∂ p˙ ∂P1

∂ p˙ ∂P2

∂ p˙ ∂Q1

∂ P˙ 1 ∂p

∂ P˙1 ∂P1

∂ P˙ 1 ∂P2

∂ P˙1 ∂Q1

∂ P˙ 2 ∂p

∂ P˙2 ∂P1

∂ P˙ 2 ∂P2

∂ P˙2 ∂Q1

∂ Q˙ 1 ∂p

∂ Q˙ 1 ∂P1

∂ Q˙ 1 ∂P2

∂ Q˙ 1 ∂Q1

∂ Q˙ 2 ∂p

∂ Q˙ 2 ∂P1

∂ Q˙ 2 ∂P2

∂ Q˙ 2 ∂Q1

∂ q˙  ∂Q2   ∂ p˙    ∂Q2    ∂ P˙ 1   ∂Q2    ∂ P˙ 2   ∂Q2    ˙ ∂ Q1   ∂Q2    ˙ ∂Q 



δq



   δp       δP1     δP  .  2    δQ   1 δQ2

(3.69)

2

∂Q2

For 20 different values of the dimensionless damping parameter and of the dimensionless Floquet parametric forcing parameter, the largest LCEs were calculated and a chaos diagram for the positive values of the largest Liapunov exponents identified the parameter space in which chaotic motions exist.1,3

Appendix A The extension of the Herglotz algorithm to nonautonomous dynamical systems (GHA) significantly reduces the effort required to suspend the nonautonomous Hamiltonian component in (3.33d).

Generalized Herglotz algorithm (GHA) The Herglotz algorithm for autonomous dynamical systems4 may be generalized by: (1) including time t; and (2) defining a generating function with a nonzero determinant of second derivatives. The GHA transforms a set of 2N variables (u, v) to a set of 2N new variables (U, V) by choosing N new variables Ui and then computing the remaining N new variables Vi uniquely from the chosen Ui so that the transformation (u, v) → (U, V) is canonical as shown by satisfying the Poisson bracket conditions.

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GHA:Type I Equate N new variables to N old variables Ui = Ui (u, v, t) such that the Poisson brackets satisfy [[Ui , Uj ]]uv = 0 and the determinant # ∂U # 1 # # ∂v1 # # # · # # # # ∂UN # ∂v1

·

·

·

∂U1 ∂vN

·

∂Ui ∂vj

·

·

·

·

·

∂UN ∂vN

# # # # # # # = 0. # # # # #

(3.A.1)

Equate N Herglotz auxiliary functions Xi (u, v) to the absolute value of the ratios of the old variables in either of the following forms:  |ui /vi | Xi = Xi (u, v) = ; i = 1, 2, . . . , N, (3.A.2a,b) |vi /ui | such that the Jacobian of the new variables U and the Herglotz auxiliary functions X is nonzero ∂(U, X)/∂(u, v) = 0. Solve (3.A.2) for vi = vi (u, X); substitute vi (u, X) into Ui (u, v, t) and invert to obtain ui = ui (U, X, t). Compute the generating function F (u, U, t) from dF (u(U, X, t), U, t) =

N 

(vi dui − Vi dUi ),

(3.A.3)

i=1

 N   ∂F ∂F dXi + dUi ∂Xi ∂Ui i=1   N N N    ∂uj ∂uj  vj dXi + vj dUi − Vi dUi  . = ∂X ∂U i i i=1 j=1 j=1

(3.A.4)

Equate the coefficients of like differentials and compute: (1) the generating function F (u, U, t) =

N
 i=1

N Xi  j=1

vj

∂uj dXi
+ C(U, t), ∂Xi


(3.A.5)

and (2) the new N variables from Vi (u, v, t) =

N  j=1

vj

∂F ∂uj − ; ∂Ui ∂Ui

i = 1, 2, . . . , N.

(3.A.6)

In order to compute the transformed Hamiltonian in terms of the new variables (U, V), the inverse canonical transformation (u(U, V, t), v(U, V, t)) must be computed,

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and the new Hamiltonian K(Q, P, t) is given by K(Q, P, t) = H(q(Q, P, t), p(Q, P, t), t) +

∂F (u, U, t) . ∂t

(3.A.7)

The GHA may also be applied to equate N old variables ui = ui (U, V, t) to obtain N new variables Ui = Ui (u, v, t).1,3 Appendix B

τ1 =

√ 1+τ

κτ = κ(1 + τ )

τλ = 1 + (τ /λ4r )

    h  2  λ 1 − exp − ; for a full draft piston,   
0  r λ2r z f1 = f (z) exp 2 ds =     λr h −h  λ4r h  ; for a full draft hinge − 1 + exp − 2  h λ2r λr a1 = −

1 βτ + −8βξ 2ξλ6r τ12

c1 =

√ bf λ τ 1 r 1 4 2ξβ 3

d1 =

a2 = −

1 βτ 1 + − β 8βξ 2ξλ6r τ12

c2 =

2β − 1 4βξ

1 d2 =  8 2bβξ 3 λ3r τ12

2βτ 2 − ξλ4r τλ (λ2r − βτ )  8 2bβξ 5 λ9r τ1 τλ

d3 =

τ (2τ + λ4r ξτλr )  8 2bβξ 5 λ9r τλ2

d4 =

a3 =

τ τ 2 − λ2r τλ d5 =  1 2 2bβξ 3 λ5r τλ τ12

−τ + λ4r ξτλ d6 = 4βλ6r ξτλ2

1 4



τ − λ4r ξτλr 2 2 + τ1 βλ6r ξτλ2



τ + λ4r ξτλ (λ2r − 1) λ6r ξτ1 τλ

1 d7 = 4



−2 τ + λ4r ξτλ + τ12 βλ6r ξτλ2



References 1. R. T. Hudspeth, Waves and Wave Forces on Coastal and Ocean Structures (World Scientific, Singapore, 2006). 2. C. M. Bowline, R. T. Hudspeth and R. B. Guenther, Applicable Anal. 72, 287 (1999). 3. R. T. Hudspeth, R. B. Guenther and S. Fadel, Acta Mechanica 175, 139 (2005). 4. R. B. Guenther, H. Schwerdtfeger and G. Herglotz, Vorlesungen u ¨ber die Mechanik der Kontinua (1985). 5. S. Fadel, Application of the generalized Melnikov method to weakly damped parametrically excited cross waves with surface tension, PhD dissertation, Oregon State University, USA (1998).

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Chapter 4

Random Wave Breaking and Nonlinearity Evolution Across the Surf Zone Yoshimi Goda ECOH CORPORATION, 2-6-4 Kita-Ueno Taito-ku, Tokyo 110-0014, Japan [email protected] A review is made on the statistical features of breaking waves in the nearshore waters. Inherent variability of the breaker index for regular waves is examined with the revised Goda’s formula. The incipient breaking height of significant wave is about 30% lower than that of regular waves. Nonlinearity of random waves is strongest at the outer edge of surf zone, but it is destroyed by wave-breaking process inside the surf zone. The wave height distribution is the narrowest in the middle of the surf zone, but it returns to the Rayleigh near the shoreline. Large differences among various wave models are noted for prediction of wave heights in the surf zone.

4.1. Introduction Breaking waves exert strong actions on maritime and coastal structures, while wave dissipation through breaking plays a major role in the generation of nearshore currents. Without good understanding of wave-breaking process, we cannot pursue any study for coastal engineering works. Nevertheless, wave breaking is an elusive phenomenon. Not many people spend enough time to observe wonderful pictures of wave deformation by breaking and regeneration after breaking. One needs some kind of a pier at a beach to have a good look of waves that break and rush toward the shore. Otherwise, one should work for some hours in a laboratory to measure waves at various locations along a wave flume. Our knowledge on wave breaking mostly comes from various literature based on previous research works. Quite a number of people use the formulas, diagrams, and other information listed there without examining the credibility of the information. For example, many people regard the breaker index, or the ratio of wave height to water depth at breaking, as a deterministic value without paying consideration to the fact that the breaker height exhibits large fluctuations even for a given wave 87

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condition and that the breaker index has been obtained by drawing a mean curve among scattered data. Nowadays, many researchers are developing various numerical models of random wave transformations. They have to adopt some kind of wave-breaking criteria and energy dissipation mechanism so that the model can reproduce wave deformation in the nearshore waters. However, modelers seem to pick up whatever is readily available without deliberation on the physical features of the wave-breaking process and the appropriateness of the breaking model. The present chapter is a revised version of the author’s paper1 presented at the 4th International Conference on Asian and Pacific Conference 2007, in Nanjin, China. It aims at providing coastal engineers and researchers with the most advanced knowledge on the statistics of wave breaking in the nearshore waters so that they can make a correct approach to the problems related to wave-breaking processes.

4.2. Physical Definitions of Wave Breaking Waves are defined as breaking when the crest starts to contain foams or when water particles jump out from the wave crest. It is a disruption of smooth water surface, and breaking waves are often classified into three types: spilling, plunging, and collapsing breakers. In the sea, there is no pure spilling breaker, because it is always accompanied with a small portion of plunging water. For example, we can observe from a window of an airplane such a scene of wave breaking in deep water that a crest of large wave makes plunging, leaves a patch of white foam behind, and moves forward with blue color. The following wave grows in height and breaks. It is a manifestation of wave energy being transported by the group velocity, which is one-half the phase velocity in deep water. Theoretically, three criteria can be cited. The first is the condition that the horizontal velocity of the water particle at the wave crest becomes equal to or greater than the phase speed of wave profile. The second is the upward vertical acceleration of the water particle at the wave crest to be equal to or to exceed the gravitational acceleration. The third is the vertical gradient of the total pressure at the wave crest to be zero or negative. The first criterion has been employed by mathematicians to find out the limiting waves of permanent form on a horizontal bed. Conventional perturbation techniques are ineffective to derive the limiting waves, and the approach specific to the limiting form needs to be employed. These limiting waves are characterized with the angular crest having the angle of 120◦ . Yamada and Shiotani2 have produced the most reliable computation results so far, which are summarized by Goda3 as reproduced in Table 4.1. The symbol H denotes the wave height, h is the water depth, L is the wavelength, L0 is the deepwater wavelength, C is the wave celerity, and ηc is the crest elevation. The subscripts “b” and “A” refer to the quantities at breaking and those of the small amplitude waves (Airy’s wave), respectively.

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Table 4.1. Characteristics of breaking waves of permanent type (after Goda3 based on Yamada and Shiotani2 ). Hb /L0

hb /LA a

hb /Lb

Cb /CA a

Hb /Lb

Hb /hb

ηc /Hb

infinity 0.935 0.471 0.286 0.1856 0.1117 0.0763 0.0474 0.0284 0.01669 0.01095 0.00575 0.00239 0.001144 0.000437 0

infinity 0.935 0.474 0.300 0.216 0.1510 0.1198 0.0915 0.0694 0.0525 0.0422 0.0306 0.01953 0.01351 0.00833 0

infinity 0.7686 0.4011 0.2597 0.1885 0.1331 0.1050 0.07915 0.05909 0.04398 0.03499 0.2483 0.01570 0.01075 0.00660 0

1.193 1.189 1.181 1.154 1.143 1.134 1.141 1.156 1.174 1.193 1.207 1.231 1.244 1.257 1.263 1.285b

0.1412 0.1409 0.1386 0.1277 0.1115 0.08997 0.07410 0.05771 0.04430 0.03371 0.02720 0.01962 0.01260 0.00871 0.00538 0

0 0.1791 0.3456 0.4919 0.5912 0.6683 0.7059 0.7293 0.7496 0.7666 0.7774 0.7904 0.8028 0.8099 0.8160 0.8261b

0.6706 0.6765 0.6908 0.7165 0.7619 0.7939 0.8392 0.8766 0.9061 0.9242 0.9453 0.9649 0.9757 0.9849 1.0000b

Notes: a LA and CA denote the wavelength and celerity of small amplitude waves. b These values are those of solitary wave computed by Yamada et al.4

The limiting height of solitary wave is (H/h)b = 0.8261 by Yamada et al.,4 instead of the often-quoted value of (H/h)b = 0.78 by McCowan.5 The limiting wave steepness of (H/L)b = 0.142 by Miche6 is calculated for waves of nonlinear waves, and it becomes Hb /(L0 )A = 0.1684 when the linear deepwater wavelength (L0 )A is used. The second criterion is for the breaking of standing waves, but no theoretical computation of limiting standing waves has been made. The third criterion has been proposed by Nadaoka et al.,7 for defining wave breaking in numerical time-domain computation. Zero vertical gradient of total pressure implies no presence of water below the wave crest, i.e., wave breaking.

4.3. Parameters Governing Breaker Index Because wave breaking attracts attention of many researchers, there have been proposed a number of formulas to describe the ratio of wave height to water depth. Kaminsky and Kraus8 called this ratio as the breaker height-to-depth index, but the present chapter employs the term of breaker index for the ratio of wave height to water depth at breaking for the sake of simplicity. Kamphuis9 has compared 11 formulas for testing of goodness-of-fit with his 225 sets of hydraulic model tests. Rattanapitikon and Shibayama10 have collected 574 data points from 24 papers/reports and calculated the root-mean-square errors of 24 breaking index formulas. Both authors have proposed their own formulas by modifying some of the previous ones.

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Rattanapitikon et al.11 have further added 121 data points in large-scale flume tests and proposed revised formulas. Similar to the approach by Rattanapitikon et al.,11 the breaker index formulas can be categorized into the following four functional forms: Hb /hb = f1 (0) = constant ,

(4.1)

Hb /hb = f2 (hb /L0 or hb /Lb ) ,

(4.2)

Hb /hb = f3 (s) ,

(4.3)

Hb /hb = f4 (s, hb /L0 or hb /Lb ) ,

(4.4)

where s denotes the bed slope. The wavelength L0 is the deepwater wavelength given by L0 = g/(2π)T 2 , where T is the wave period. Because the relative water depth hb /L0 is easily converted to hb /Lb through the dispersion relationship, the two relative depths hb /L0 and hb /Lb are interchangeable. The formula of Hb /hb = 0.78 is a typical example of Eq. (4.1). There are some other formulas using the parameter of deepwater wave steepness H0 /L0 . They can be useful for predicting the breaker height of regular waves. Nevertheless, they cannot be applied for breaking of random waves, because individual zero-crossing waves in a train of random waves are unrelated to individual waves in deepwaters. Thus, there is little room for the parameter H0 /L0 to function in the breaker index for random waves. Performance of a breaker index formula can be judged by the magnitude of the bias of the predicted breaker height from the observed heights. It should also be examined with either the root-mean-square error of predicted breaker heights or the correlation coefficient between prediction and observation. The root-mean-square error analysis by Rattanapitikon and Shibayama10 is not conclusive in differentiating the merits of four functional forms, but they recommend a certain modification of the slope effect in the function f4 (s, hb /L0 ), apparently indicating their preference of this functional form. Kamphuis9 calculated the correlation coefficients between 11 formulas and his laboratory data. By assigning the best-fitting value to the proportionality coefficient of each formula, he obtained the determination coefficient R2 = 0.69 to f1 (0), R2 = 0.67 to f2 (hb /L0 ), R2 = 0.84 to f3 (s), and R2 = 0.88 to f4 (s, hb /L0 ). His result clearly suggests the necessity to include both the parameters of bed slope and relative water depth in the breaker index formula.

4.4. Breaker Index for Regular Waves and Its Scatter 4.4.1. Scatter of regular breaking waves In 1970, Goda3 presented a diagram of breaker index curves of regular waves for four bed slopes, based on the laboratory data from eight sources, which included his own large-scale tests with Hb = 0.43 to 0.93 m; Rattanapitikon et al.11 did not

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analyze this data set. Later, Goda12,13 approximated the breaker index curves with the following empirical formula:
  Hb hb  A 4/3 1 − exp −1.5π 1 + 15s : A = 0.17 . (4.5) = hb hb /L0 L0 Equation (4.5) was derived by a graphical curve-fitting technique without direct comparison with the original laboratory data. Rattanapitikon and Shibayama10 have recommended a modification of the slope effect term of (1 + 15s4/3 ) into (1.033 + 4.71s − 10.46s2 ). Upon reexamination of the original laboratory data, the author has also recognized a necessity of modifying the slope effect term. The revised formula is as follows:
  hb  Hb A 1 − exp −1.5π 1 + 11 s4/3 : A = 0.17 . (4.6) = hb hb /L0 L0 Muttray and Oumeraci14 found the best-fitting coefficient of Eq. (4.5) being 0.167 instead of 0.170 for the slope of 1/30. When Eq. (4.6) is applied to their data, the coefficient would have the value of 0.173. Comparison of the laboratory data of breaker index with Eq. (4.6) is shown in Fig. 4.1 for five groups of bed slopes, i.e., 1/7 to 1/12, 1/20, 1/30, 1/50, and 1/200 to 1/200. (See Goda3 for description of the laboratory data employed here.) Recent data sets by Li et al.15 for s = 1/30 and s = 1/50, Li et al.16 for s = 1/200, and Lara et al.17 for s = 1/20 are also added. It is clear in Fig. 4.1 that the value of the breaker index increases as the bed slope becomes steep. Thus, it is absolutely necessary to incorporate the slope effect into the breaker index formula. Because the experimental data are scattered around the index curves, the upper and lower bound curves with the range of 87%–115% of the value by Eq. (4.6) are drawn in Fig. 4.1. A quantitative evaluation of the degree of the scatter is made by means of the relative error of the breaker index, i.e., E = (1 − γmeas/γest ), where γmeas is the measured value of Hb /hb and γest is the predicted value by the breaker index formula of Eq. (4.6). The mean and the standard deviation of the relative error are calculated for each group of the bed slope. The mean Emean indicates a bias of the breaker index and the standard deviation of E represents the degree of scatter of the breaker index. Because E is defined as the relative error, the standard deviation σ(E) is equivalent to the coefficient of variation (CoV). A positive bias indicates a tendency of overestimate, while a negative bias shows an underestimate. Table 4.2 lists the bias and CoV of the breaker index of Eq. (4.6) for the data of various bed slopes. The slope data of 1/9 and 1/12 are excluded from the analysis because of their small sample sizes. The bias varies from −2.9% to +6.2% depending on the bed slope, but the formula of Eq. (4.6) can be regarded as yielding reasonable estimates of the breaker heights. The scatter of the data as expressed by CoV is about 5% to 7% for the bed slope of 1/200 to 1/50; it increases as the slope becomes steep, and it takes the value of 14% for the bed slope of 1/10. Such scatter of data represents an inherent stochastic nature of wave-breaking phenomenon. It resides in the data set itself, being independent of the breaker index formula being applied.

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2.0

Breaker Index, γ = Hb /hb

Slope = 1/9 to 1/12

1.0 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.001

Kishi-Iohara 1/9 Iversen 1/10 Goda et al. 1/10 Galvin 1/10 Bowen et al. 1/12 Index curve 1/10 Upper 115% Lower 87%

0.01

0.1

Relative water depth, hb /L0

(a) Breaker index data for the slope of 1/9−1/12. 1.5 Breaker Index, γ = Hb /hb

Slope = 1/20 1.0 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.001

Iversen 1/20 Galvin 1/20 Toyoshima 1/20 Lara 1/20 Index curve 1/20 Upper 115% Lower 87%

0.01 Relative water depth, hb /L0

0.1

(b) Breaker index data for the slope of 1/20.

Breaker Index, γ = Hb /hb

1.2

Slope = 1/30

1.0 0.9 0.8 0.7 0.6 0.5 0.4

0.3 0.001

Iversen 1/30 Mitsuyasu 1/30 Toyoshima 1/30 Li Y.C. et al. 1/30 Index curve 1/30 Upper 115% Lower 87%

0.01 Relative water depth, hb /L0

0.1

(c) Breaker index data for the slope of 1/30. Fig. 4.1.

Comparison of breaker index formula Eq. (4.6) with laboratory data of regular waves.

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Breaker Index, γ = Hb /hb

1.0 0.9 0.8 0.7

Slope = 1/50

0.6

Iversen 1/50 Mitsuyasu 1/50 Li Y.C. et al. 1/50 Index curve 1/50 Upper 115% Lower 87%

0.5 0.4

0.3 0.001

0.01 Relative water depth, hb /L0

0.1

(d) Breaker index data for the slope of 1/50.

Breaker Index, γ = Hb /hb

1.0 0.9 0.8 0.7 0.6

Slope = 1/100 to 1/200

0.5 0.4

0.3 0.001

Goda 1/100 Li Y.C. et al. 1/200 Index curve 1/100 Index curve 1/200 Upper 115% Lower 87%

0.01 Relative water depth, hb /L0

0.1

(e) Breaker index data for the slope of 1/100–1/200. Fig. 4.1.

Table 4.2.

(Continued )

Bias and CoV of the breaker index formula of Eq. (4.6).

Bed slope

No. of data

Bias = Emean (%)

CoV = σ(E) (%)

1/10 1/20 1/30 1/50 1/100 1/200

29 47 73 28 32 19

−0.5 +3.9 +1.4 −2.9 +6.2 +3.5

14.0 11.3 8.6 4.8 5.5 7.4

Sources Iversen, Goda et al. Iversen, Galvin, Toyoshima et al., Lara et al. Iversen, Mitsuyasu, Toyoshima et al., Li et al. Iversen, Mitsuyasu, Li et al. Goda Li et al.

Even under a well-controlled laboratory test, the breaking point fluctuates over some distance and the breaker height varies from wave to wave. Smith and Kraus18 reported on their regular wave tests that “despite care in conducting the tests and use of the average value of the given quantity (i.e., over 10 waves), wide scatter appeared in some quantities and must be considered inherent to the breaking process of realistic waves.” One cause of the data scatter is the presence of small-amplitude, long-period oscillations of water level in a laboratory flume, but the breaking process itself is triggered by many small factors beyond the control of experimenters. We should regard the wave-breaking phenomenon as stochastic one and accept a certain range of natural fluctuation. As listed in Table 4.2, the coefficient of

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variation is large for steep slope and it becomes small for gentle slope. In any research work involving wave-breaking phenomena, due consideration should be given to such stochastic nature of breaker heights. 4.4.2. Scatter of random breaking wave heights The first effort to identify the breaker heights of individual waves among trains of irregular waves was undertaken by Kimura and Seyama19 and Seyama and Kimura,20 who analyzed about 1000 individual breaking waves for each slope of 1/10, 1/20, 1/30, and 1/50 with the aid of a video camera and wave gauges. The breaker index of individual waves varied over a wide range, which was equivalent to CoV of 18% to 23% (the present author’s visual inspection of scatter diagrams). In order to reduce the scatter of data, they proposed to employ an artificial water depth below the mid-level between the wave crest and trough of individual breaking waves and succeeded in reducing CoV to the values between 8% and 11%. Because Eq. (4.5) yielded the breaker index larger than most of the observed value, they proposed its modified version. When the revised breaker index formula of Eq. (4.6) is employed, however, the center line of the scatted data appears to be at the level of 85% (s = 1/10) to 95% (for other slopes) of the predicted value. Black and Rosenberg21 made observation of individual breaking waves on a natural beach with a depth of 1.0–1.5 m at Apollo Bay in southern Australia. The median value of the breaker index was about 84% and 87% of those by Eqs. (4.5) and (4.6), respectively. Another observation was made by Kriebel22 in a large wave flume with a bed slope of 1/50 for waves with a significant height of 0.46 m and peak period of 2.9 s. When he applied Eq. (4.5) to individual breaker heights, he found that the value of the proportionality coefficient A fitted to the data varied from about 0.09 to 0.18 (against the original value of A = 0.17) with the mean of 0.142 (84% of Eq. (4.5)) and the standard deviation of 0.017 (equivalent to CoV of 12%). Li et al.23 also reported the result of their measurements of random breaking waves on the slope of 1/50 and 1/200, recommending the coefficient value of A = 0.150 with the standard deviation of 0.031. It should be recalled that the random wave breaking model by Goda24 in 1975 had already incorporated the variability of breaker heights by assigning a variable probability of individual wave breaking. The probability was assumed to increase linearly from 0 to 1 in the range of the wave height from 71% to 106% of the height predicted by Eq. (4.5), corresponding to A = 0.12 to 0.18. Therefore, it is expected that the median value of individual breaker heights would be smaller than those predicted by Eq. (4.5) or (4.6). 4.5. Breaker Index for Random Waves 4.5.1. Incipient breaking index of significant wave Equations (4.5) and (4.6) are examples of the breaker index for regular waves. There are some people who try to apply such breaker index formulas to coastal

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waves or random waves, but such application does not yield correct answers. The breaker index for regular waves may be utilized for the highest wave in an irregular wave train, but it cannot be applied for the significant wave, the root-mean-square wave, or any other characteristic wave. When random waves approach the shore, breaking of individual waves occurs gradually with large waves first at the far distance, medium waves next at some distance from the shore, and small waves near the shoreline. The variation of significant wave height from the offshore toward the shore is so gradual that we cannot employ the concept of wave-breaking line, which is so obvious in the case of regular waves. Against such difficulty of defining the breaking point of significant wave, Kamphuis9 introduced the definition of “incipient wave breaking.” He measured cross-shore variations of significant wave height beyond and across the surf zone, drew a curve of wave shoaling trend in the outside of the surf zone and a curve of wave height decay within the surf zone, and called the condition at the cross-point of the two curves as the incipient wave breaking. By using the data of the significant wave height at incipient breaking, he calibrated 11 breaker index formulas and determined the best-fitting proportionality coefficient. For the formula of Eq. (4.5), he obtained the proportionality coefficient of A = 0.12 for significant wave height. Li et al.23 have also presented a data of the breaker index of (H1/3 /h)b on the slope of 1/200, which is fitted to Eq. (4.5) with a modified constant value of 0.12 for the initial stage of breaking. Their breaking condition was some observation of large individual breaking waves in an irregular wave train. Goda24 has prepared a set of diagrams depicting variations of significant wave height across the surf zone (reproduced as Figs. 3.29 to 3.32 in Ref. 25). The boundary lines of 2% decay in these diagrams approximately correspond to the breaker index with A = 0.11, and the water depth (h1/3 )peak at which the significant wave takes a peak value within the surf zone (Fig. 3.34 in Ref. 25) corresponds to A = 0.11–0.13. Therefore, the incipient breaker index of the significant wave can be expressed with the following formula: 
   H1/3, b 0.12 (hb )incipient  4/3 = 1 − exp −1.5π 1 + 11 s . (4.7) hb hb /L0 L0 incipient Thus, the incipient breaker index of significant wave is about 30% lower than that of regular waves. The incipient breaking of significant wave corresponds to the condition that the high waves of upper several percent among individual waves have begun to break. 4.5.2. Laboratory data of breaker index of random waves After incipient breaking, the percentage of wave breaking increases as waves proceed across the surf zone. The ratio of the significant wave height to the water depth gradually increases toward the shoreline. Ting26,27 made detailed laboratory investigations of random wave deformations on a uniform slope of 1/35, using frequency spectra of broad- and narrow-band with the peak enhancement factor of 3.3 and 100, respectively. Waves had the significant height of Hs = 0.15 m and the spectral

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Breaker indices, H1/3 /h & Hrms /h

1.0

0.6 0.4 0.3 0.2

0.10

Fig. 4.2.

Slope = 1/35

0.8

Broad spec. H1/3 /h Narrow spec. H1/3 /h Breaker envlp. H1/3 /h Broad spec. Hrms /h Narrow spec. Hrms /h Breaker envlp. Hrms /h

0.01

0.1 0.03 Relative depth, h/L0

0.3

Breaker indices for H1/3 and Hrms on s = 1/35 with the data by Ting.26,27

peak period of Tp = 2.0 s. He recorded wave profiles at an offshore station with the depth of 0.457 m and at six stations on the slope with the depth of 0.27–0.0625 m. Waves at the six stations on the slope had the percentage of breaking ranging from 5% to 94% (the case of broad-band spectrum). Wave records were analyzed by the zero-downcrossing method, and calculated results of characteristic wave heights and periods are presented in tabular forms. From these results, the ratios of H1/3 and Hrms to the local depth (inclusive of mean water level change) are calculated and plotted against the relative water depth h/L0 , as shown in Fig. 4.2. The curves denoted as breaker envelopes are calculated by Eq. (4.6) for s = 1/35 with the proportionality coefficient of A = 0.145 (85% of regular waves) for the significant height H1/3 and to A = 0.111 (65%) for the root-mean-square height Hrms . Because the percentage of breaking waves is high in these data, an A value higher than that for incipient breaking fits to the data. The breaker index data for Hrms by Tick is higher than the value proposed by Sallenger and Holman,28 who gave an expression of Hrms /h = 3.2s + 0.32 without inclusion of the relative depth (h/L0 ) term. They converted the orbital velocity spectra to the surface wave spectra with the transfer function based on the linear theory, and estimated the energy-based Hrms , which must have been smaller than the statistical Hrms value based on direct measurement data of surface profiles. 4.5.3. Description of field data employed for analysis In the field wave observation at a fixed station, it is not feasible to judge whether individual waves are at the stage of breaking or not, unless simultaneous measurements with video cameras are taken. However, we may find out an upper limit of significant wave height for a given water depth by taking an envelope of many data at different relative water depths. For this type of analysis, stationary wave

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Summary of stationary coastal wave data employed in the present analysis.

Observation station Rumoi Port Yamase-Domari Port Tomakomai Port

Kanazawa Port Caldera Port, Costa Rica Sakata Port

97

Type of wave gauge

Water depth (m)

Sampling interval ∆t (s)

Significant height H1/3 (m)

Significant period T1/3 (s)

No. of data

Step-resistance ”

−11.5 −12.7

0.5 0.5

2.2–7.1 1.9–6.2

5.9–11.7 7.7–15.6

44 9

” ” Ultrasonic ” ”

−10.8 −13.8 −20 −20 −18

1.0 0.5 0.5 1.0 0.5

2.9–5.8 2.6–2.8 2.4–2.5 1.0–6.8 1.5–3.6

7.7–10.9 6.7 – 7.5 6.9–7.4 1.0–6.8 14.2–18.4

9 2 2 13 50

Pressure

−14.5 −10.5

0.5 0.5

1.7–9.7 1.7–6.1

6.3–13.4 6.5–15.0

123 123

(Source: Goda and Nagai,29 Goda24,30 ).

observation data analyzed by Goda and Nagai29 and the data of long-traveled swell recorded with an ultrasonic wave sensor reported by Goda30 were utilized. Table 4.3 lists the characteristics of these field data. Waves were recorded by means of either step-resistance gauges or ultrasonic wave sensors so that they were reliable registration of surface wave profiles. The data at Tomakomai and Kanazawa as well as Caldera were measured with ultrasonic wave sensors. They are not analyzed for breaker limits but for wave nonlinearity effects to be described in Sec. 4.6. Table 4.3 also lists the wave records at Sakata Port measured by means of pressure gauges, which were utilized by Goda24 for calibration of his random wavebreaking model. Although there remains a problem of pressure conversion to surface profiles, the conversion error would have been small because of the relatively shallow water depth at Sakata stations (10.5 and 14.5 m). They were included in the present analysis to increase the size of database. Other sources of nearshore waves are the photogrametric measurement data by Hotta and Mizuguchi31,32 as well as by Ebersole and Hughes.33 Hotta and Mizuguchi mobilized 11 motion-picture cameras set on top of a coastal observation pier at Ajigaura Beach, Ibaragi, Japan. They took film pictures of instantaneous water surfaces simultaneously at some 60 surveyor’s poles erected in the nearshore waters on a line perpendicular to the shoreline stretched over a distance of about 120 m. Films of surface wave records were taken every 0.2 s for an effective duration of 760 s. The beach profile in September 1978 had a trough at the distance of 25 m from the shoreline and the slope of about 1/60 offshore of the trough. The beach profile in December 1978 was somewhat uniform without any bar or trough, and the slope was about 1/70. The water depth inclusive of tides at the poles varied from 0.1 to 2.7 m. Photogrametric measurements of nearshore waves were also executed by Ebersole and Hughes33 during the DUCK85 campaign in Duck, North Carolina, USA with the cooperation of Dr. Hotta who brought twelve cameras with him and took charge of filming. They referred to this technique as “the photopole method”; this terminology is employed in the present chapter. Over the distance of 64 m,

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Summary of photopole measurements.

Date

No. of data

h (m)

H0 (m)

T1/3 (s)

H0 /L0

1978/09/05 1978/12/13–14 1985/09/04–05 1986/09/11–19

54 175 99 140

0.6–2.7 0.1–1.8 0.5–2.4 0.4–3.7

0.7 0.5–0.7 0.3–0.5 0.5–1.1

8.4 7.2–7.9 10.3–11.1 5.4–11.5

0.0064 0.0059–0.0081 0.0017–0.0028 0.0027–0.0254

12 poles were erected in the initial depth ranging from 0.4 to 1.9 m. Measurements were taken for nine runs during the fourth and fifth days of September 1985. With the variation of the tide level, the actual water depth varied from 0.5 to 2.4 m. The effective duration of wave recording was about 650 s, judging from the number of waves and average periods listed by Ebersole and Hughes.33 The beach profile was nearly flat for about 25 m from the shoreline with the depth of about 0.5 m below the mean sea level, and it had the slope of about 1/30 beyond that. Another series of photopole measurements were carried out during the SUPERDUCK campaign in 1986. Dr Hughes kindly supplied the author with the data files of measured wave statistics. The number of poles was increased to 20 and the water depth inclusive of tides varied from 0.4 to 3.7 m. The beach profile during SUPERDUCK is not known, but it would have been nearly the same as DUCK85 because of the same season. All the photopole measurement data were analyzed by the zero-downcrossing method, and various statistical wave characteristics were calculated. Table 4.4 lists the summary of the photopole wave measurement conditions. The significant wave period T1/3 has been converted from the pole-averaged values of either the mean period Tmean or the spectral period Tp by assumption of T1/3 = 1.05Tmean or T1/3 = 0.95Tp, which would be appropriate for swell of very low steepness. The offshore wave height H0 was converted from the significant height H1/3 measured at the most offshore pole using the shoaling coefficient; no refraction effect was taken into consideration as no information of wave direction was available. All the waves were swell of very low steepness ranging from 0.0017 to 0.0081, except one case of SUPERDUCK with H0 /L0 = 0.0254. 4.5.4. Field data of breaker index for energy-based significant waves Coastal surface wave data, pressure-converted wave data, and three sets of photopole data are plotted together in Fig. 4.3 in the form of Hm0 /h versus h/L0 . The energybased significant wave height Hm0 defined by Eq. (4.8) is employed here instead of the zero-crossing height H1/3 , because the latter is greatly enhanced over Hm0 by strong effects of wave nonlinearity and it is not representative of breaking-dissipated wave energy level; this aspect is discussed in Sec. 4.6. √ (4.8) Hm0 = 4.0ηrms = 4.0 m0 , where m0 denotes the zeroth moment of frequency spectrum.

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Step-resistance ( h=11-14m) Presssure gage (h=10-14m) Photopole Ajigaura (h=0.1-2.7 m) Photopole DUCK85 (h=0.5-2.4m) Photopole SUPERDUCK (h=0.5-3.2 m) Breaker envelope (80% limit)

Wave height ratio, Hm0 /h

0.8 0.6 0.4

0.2

0.1 0.001

Fig. 4.3.

0.01 0.1 Relative depth, h/L0

1

Breaker index for Hm0 based on the field wave data.

Coastal wave data, on the other hand, were not much affected by wave nonlinearity effects, and the zero-crossing significant height H1/3 was almost the same as Hm0 . Many data points in Fig. 4.3 belong to nonbreaking condition, but what interests us is the upper envelope which provides an estimate of the upper limit of breaking wave height. The curve of dash-dot line in Fig. 4.3 has been calculated by Eq. (4.6) for the slope of s = 0.0143 (1/70) with the coefficient being reduced to A = 0.136 (80%). Similar with the laboratory data shown in Fig. 4.2, the wave height ratio Hm0 /h is higher than the incipient breaker index of significant wave expressed by Eq. (4.7). It is because the breaker index increases inside the surf zone as the percentage of breaking waves increase. It is seen that the energy-based significant wave height Hm0 on gentle slopes does not exceed 0.7 times the local water depth except for the low-steepness swell in very shallow water. For the range of h/L0 > 0.03, the upper limit of significant wave height is about 0.6 times the local water depth. Some data points above the dash-dot curve are those of DUCK85 and SUPERDUCK, which were conducted on the beach steeper than the beach in Ajigaura.

4.6. Evolution of Wave Nonlinearity Across Surf Zone 4.6.1. Variations of skewness and kurtosis across the surf zone Ocean waves are characterized with almost linear property, as evidenced by the Gaussian distribution of surface elevation. Wave linearity is the basis of spectral

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representation and analysis. Deviation from the linearity is measured with the values of the skewness and kurtosis of the surface elevation with reference to the mean water level during a wave record. The skewness is zero when a distribution is symmetric with respect to the mean, and takes a positive value when a distribution is asymmetric with a long tail toward the right side (large value). The kurtosis takes a value of 3.0 for the Gaussian distribution. When the mode of distribution has a sharp peak and the distribution has long tails in both the left and right sides, the value of kurtosis becomes much larger than 3.0. The degree of positive skewness and the deviation of kurtosis from 3.0 are the measure of the strength of wave nonlinearity. The skewness of ocean waves is less than 0.5, and the kurtosis is below 4.0 for most cases, and thus the nonlinearity of waves in deepwater is weak. The variations of the skewness and kurtosis of field waves are examined with coastal surface waves listed in Table 4.3 (excluding the pressure sensor data of Sakata Port), and the photopole data in Ajigaura, DUCK85, and SUPERDUCK listed in Table 4.4. The data of skewness and kurtosis of September 5th in Ajigaura were kindly provided by Dr Hotta. The data of the photopole measurements of SUPERDUCK provided by Dr Hughes had the data of skewness only. As waves approach the shore, wave nonlinearity is enhanced and both the skewness and kurtosis increase significantly. Figure 4.4(a) exhibits the increase of the skewness with the nonlinearity parameter Π1/3 , which was introduced by Goda34 with the following definition: Π1/3 =

H1/3 2πh coth3 , LA LA

(4.9)

where LA denotes the wavelength calculated by small amplitude wave theory or Airy’s theory. The data are grouped by the range of the offshore wave steepness H0 /L0 : the first group for 0.001 < H0 /L0 < 0.0029, the second group for 0.003 < H0 /L0 < 0.0049, the third group for 0.0050 < H0 /L0 < 0.0099, the fourth group for 0.010 < H0 /L0 < 0.029, and the fifth group for 0.030 < H0 /L0 < 0.049 (Legends are shown with abbreviated figures). The data shown in the left diagram of Fig. 4.4 are those outside the surf zone. Because the boundary of surf zone is difficult to be set for random waves, an arbitrary boundary of h/H0 = 2.5 is employed here to separate the wave data outside and inside the surf zone. As shown in the left diagram, the skewness outside the surf zone shows a clear correlation with the wave nonlinearity parameter. The skewness begins from the value of zero at Π1/3 = 0, increases almost linearly with Π1/3 , and attains the value of 2.0 around Π1/3 = 4. The dashed line represents a semitheoretical relationship, which is based on the analysis of finite amplitude regular wave profiles by Goda34 with the consideration of the probability of individual wave heights according to the Rayleigh distribution. Variation of the skewness inside the surf zone (h/H0 < 2.5) is shown in Fig. 4.4(b). The ordinate is the ratio of offshore wave height to water depth, H0 /h, which increases rapidly as waves approach the shore. There is a clear trend of skewness decreasing toward β1 = 0 with the increase of the height-to-depth ratio

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Random Wave Breaking and Nonlinearity Evolution Across the Surf Zone

2.5

2.5 0.001
h/H0 < 2.5

h/H0 > 2.5 2.0

1.5

Skewness, β 1

Skewness, β 1

2.0

1.0

1.0

0.5

0.0

0.0

0.1 1 Nonlinearity parameter, Π1/3

0.6

0.8

1

1.2

1.4

Height-to-depht ratio, H0 /h

(a) O utside the surf zone (h/H0 > 2.5) Fig. 4.4.

-0.5 0.4

10

0.001
1.5

0.5

-0.5 0.01

(b) Inside the surf zone (h/H0 < 2.5)

Variation of the skewness of surface elevation outside and inside the surf zone.

10.5

10.5

7.5

h/H0 > 2.5

h/H0 < 2.5 9.0

Kurtosis, β 2

0.001
9.0 Kurtosis, β 2

101

6.0 4.5 3.0

0.001
7.5 6.0 4.5 3.0

1.5 0.01

0.1 1 Nonlinearity parameter, Π1/3

(a) Outside the surf zone (h/H0 > 2.5) Fig. 4.5.

10

1.5 0.4

0.6

0.8 1.2 1.0 Height-to-depht ratio, H0 /h

1.4

(b) Inside the surf zone (h/H0 < 2.5)

Variation of kurtosis of surface elevation outside and inside the surf zone.

H0 /h. Use of the height-to-depth ratio in Fig. 4.4(b) is to provide a kind of contrast of the increase and decrease of skewness in the outside and inside of the surf zone, respectively. While the increase of skewness outside the surf zone seems indifferent to the wave steepness, the value of skewness inside the surf zone is much affected by the wave steepness; waves of low steepness maintain large values of skewness, while waves of high steepness have small values only. Variation of the kurtosis of surface elevation is shown in Fig. 4.5; Fig. 4.5(a) shows kurtosis outside the surf zone and Fig. 4.5(b) shows kurtosis inside the surf zone. The pattern of variation is the same as that of skewness, though the available number of kurtosis data is smaller than the skewness data. The kurtosis starts from the value of 3.0 at Π1/3 = 0, increases as the nonlinearity parameter increases, and attains the value of 9 around Π1/3 = 4. In Fig. 4.5(b), the kurtosis inside the surf

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zone decreases toward β2 = 3.0 as waves approach the shoreline (i.e., increase of H0 /h). Waves with the steepness larger than 0.01 exhibit the kurtosis value much smaller than swell with very low steepness. It is because waves with large steepness cannot experience the full process of nonlinear shoaling owing to the early start of wave breaking. 4.6.2. Variations of zero-crossing wave heights across the surf zone Wave nonlinearity is also reflected in individual wave heights defined by the zerocrossing method. In the offshore, individual wave heights almost follow the Rayleigh distribution. Close examination of wave height distribution has revealed a distribution slightly narrower than the Rayleigh with the result of the mean ratio of H1/3 /Hm0 being 0.95. The difference is due to the energy spread over frequency as represented by wave spectrum, the degree of which can be measured with the spectral shape parameter as discussed by Goda and Kudaka.35 As waves propagate toward the shore, however, waves undergo nonlinear shoaling. In this process, wave profiles become skewed with high and sharp crests and low and flat troughs. Because of skewed wave profile, the potential energy contained in such a profile is smaller than the energy of sinusoidal wave with the same height. In other words, nonlinear waves can have the height much larger than the linear (sinusoidal) waves for the same potential energy. Owing to the nonlinear shoaling, individual wave heights increase rapidly with the rate greater than that of linear shoaling. Significant wave heights and other characteristic wave heights also grow rapidly. The nonlinearity effect becomes most conspicuous around the outer edge of the surf zone. After waves enter the surf zone and begin to be attenuated through breaking process, the wave nonlinearity is gradually lessened. The degree of nonlinear shoaling effect may be judged by a departure of the statistical wave height from the theoretical prediction by the Rayleigh distribution, which is expressed as below: H1/10 = 1.27Hm0 ,

H1/3 = Hm0 ,

Hrms = 0.707 Hm0 .

(4.10)

Figures 4.6–4.8 show the evolutions of the wave height ratios H1/10 /Hm0 , H1/3 /Hm0 , and Hrms /Hm0 , respectively. The difference between H1/3 and Hm0 in the nearshore waters has been pointed by several researchers such as Thompson and Vincent36 and Ebersole and Hughes.33 In Figs. 4.6–4.8, the (a) parts are for those outside the surf zone and the (b) parts are for those inside the surf zone, similar to Figs. 4.4 and 4.5. The boundary of the surf zone is subjectively set at h/H0 = 2.5 in Figs. 4.6 and 4.7 and h/H0 = 1.5 in Fig. 4.8. In case of the root-mean-square wave height Hrms , the surf zone should be defined in a shallower area than for the significant wave, because Hrms is calculated with all individual waves. Thus, the boundary of Hrms was set at h/H0 = 1.5. The dashed lines in the left diagrams are semi-theoretical predictions based on the potential energy calculation of finite amplitude waves and the Rayleigh distribution of wave heights, by referring to the methodology employed by Longuet-Higgins.37

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2.2

Wave height ratio, H1/10 /Hm0

1.8

h/H0 > 2.5

h/H0 < 2.5 2.0 Wave height ratio, H1/10 /Hm0

0.001
2.0

1.6

1.4

1.2

0.001
1.8

1.6

1.4

1.2

1.0 0.01

1.0 0.1 1 Nonlinearity parameter, Π1/3

10

0.4

(a) Outside the surf zone (h/H0 > 2.5)

4 1 Height-to-depht ratio, H0 /h

10

(b) Inside the surf zone (h/H0 < 2.5)

Variation of wave height ratio H1/10 /Hm0 outside and inside the surf zone.

Fig. 4.6.

1.75

1.75 0.001
1.50

h/H0 > 2.5

h/H0 < 2.5 Wave height ratio, H1/3/Hm0

Wave height ratio, H1/3 /Hm0

103

1.25

1.00

0.75 0.01

0.1

1

Nonlinearity parameter, Π1/3

(a) Outside the surf zone (h/H0 > 2.5) Fig. 4.7.

10

1.50

0.001
1.25

1.00

0.75

0.4

1

4

10

Height-to-depht ratio, H0 /h

(b) Inside the surf zone (h/H0 < 2.5)

Variation of wave height ratio H1/3 /Hm0 outside and inside the surf zone.

Outside the surf zone shown in the left diagrams of Figs. 4.6–4.8, the wave height ratios H1/10 /Hm0 , H1/3 /Hm0 , and Hrms /Hm0 increase with the increase of the wave nonlinearity parameter. Although there is much scatter of data, they follow the trend of semi-theoretical curves. The one-tenth highest wave height H1/10 in Fig. 4.6 is given the initial value of 1.20Hm0 based on the trend of the data in consideration of the spectral width effect. The maximum values of the ratios of the statistical wave heights to the energybased significant wave height for swell of very low steepness are around 2.1 for H1/10 /Hm0 , 1.6 for H1/3 /Hm0 , and 1.15 for Hrms /Hm0 . With reference to the wave height ratios at the weak nonlinearity state, the statistical wave heights are

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1.2

1.2 h/H0 > 1.5

h/H0 < 1.5

1.1 Wave height ratio, Hrms /Hm0

0.001
1.1 Wave height ratio, Hrms /Hm0

FA

1.0

0.9 0.8

0.7

0.001
1.0

0.9

0.8

0.7

0.6 0.01

0.6 0.1 1 Nonlinearity parameter, Π1/3

(a) Outside the surf zone (h/H0 > 1.5) Fig. 4.8.

10

0.4

1 4 Height-to-depht ratio, H0 /h

10

(b) I nside the surf zone (h/H0 < 1.5)

Variation of wave height ratio Hrms /Hm0 outside and inside the surf zone.

enhanced by 1.75 times for H1/10 , 1.68 times for H1/3 , and 1.64 times for Hrms . Such enhancement of statistical wave heights are apparent ones without real increase of wave energy as discussed earlier. Inside the surf zone shown in the right diagrams of Figs. 4.6–4.8, the data exhibit large scatter but they indicate a clear tendency of decrease with the height-to-depth ratio H0 /h; the wave height ratios H1/10 /Hm0 , H1/3 /Hm0 , and Hrms /Hm0 converge to 1.27, 1.0, and 0.70, respectively, as H0 /h becomes larger than 2. As for the effect of wave steepness, waves with low steepness in the range of 0.001 to 0.0049 have the mean value of H1/3 /Hm0 = 1.4 around H0 /h = 0.4, while waves with the medium steepness of 0.005 to 0.009 starts from the mean value of H1/3 /Hm0 = 1.2. Waves with a high steepness of 0.010 to 0.049 have the H1/3 /Hm0 value less than 1.1. Decrease of the wave height ratio with decreasing water depth has been pointed out by Thompson and Vincent,36 who suggested earlier start of decrease for waves of high steepness compared with waves of low steepness. Such effects of wave steepness on the nonlinear features of waves inside the surf zone are originated from the fact that waves of low steepness experience a high degree of nonlinear wave shoaling before they are attenuated by breaking, while waves of high steepness are attenuated much earlier before they experience strong nonlinear shoaling. It is seen in the left diagrams of Figs. 4.6–4.8 that waves with the steepness of 0.010–0.049 have the nonlinearity parameter up to Π1/3 = 0.3 only and waves are transferred into the right diagrams which represent waves inside the surf zone. Waves with the steepness of 0.005 to 0.0099 have the nonlinearity parameter up to Π1/3 = 1.2 and moves into the group of those inside the surf zone. Waves with the steepness below 0.0029 can have the nonlinearity parameter up to Π1/3 = 4 before they enter into the surf zone. Thus, the evolution of wave nonlinearity expressed in terms of the skewness β1 , kurtosis β2 , and wave height ratios H1/10 /Hm0 , H1/3 /Hm0 , and Hrms /Hm0 is summarized as follows. It is weak in the offshore, increases gradually as waves propagate shoreward, becomes strongest in the outer half of the surf zone, begins

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to decrease as breaking process of random waves become intensive, and is nearly lost at the shoreline.

4.7. Changes of Wave Height Distribution and Probability Density Function Across Surf Zone 4.7.1. Histogram of wave height distribution The first formulation of random wave-breaking process in a numerical model was made by Collins38 by modifying the probability density function (pdf ) of the Rayleigh distribution. He assumed that all waves beyond the breaking limit will reduce their heights to the breaker height Hb , i.e., the pdf is truncated at H = Hb , and it has an infinite intensity of delta function there. An early work of Battjes39 adopted the same approach, while Kuo and Kuo40 redistributed the portion of broken waves over the range of 0 < H ≤ Hb in proportion to the remaining probability density. On the other hand, Goda24 proposed to make a gradual cut over an upper one-third range of wave height below the breaker height as explained in Sec. 4.4.2. Goda’s proposal was based on the field data of wave height histograms, which was analyzed by himself. Figure 4.9 is a replotting of the data of surface wave records of Rumoi, Yamase-Domari, Tomakomai, and Kanazawa Ports listed in Table 4.3. The data is presented in four groups according to the value of relative water height H1/3 /h. The group with H1/3 /h < 0.4 demonstrates the wave heights being in good agreement with the Rayleigh distribution. Though not discernible in the figure, there is a small number of waves in the class of H/Hmean = 3.25–3.50, corresponding to the probability density of the Rayleigh. The next group with 0.4 ≤ H1/3 /h < 0.5 does not have any wave in the class of H/Hmean > 3.25, and the histogram indicates a slight leftward shift. As the relative wave height increases, disappearance of large individual waves becomes clearer and so is the deviation from the Rayleigh distribution. However, there is no concentration of wave heights at the upper end as assumed by Collins38 and Battjes.39 Changes of the pdf of wave heights within the surf zone have been observed in laboratory data such as Goda,24 Baldock et al.,41 and Ting.42,43 Figure 4.10 is an example of the cumulative distribution of wave heights in the surf zone, which has been prepared for the data presented in Fig. 4.5 of Ting.42 The abscissa of Fig. 4.10 is the relative wave height H/H1/3 and the ordinate is the nonexceedance probability. When wave heights follow the Rayleigh distribution, their cumulative distribution is aligned on the heavy straight line of Fig. 4.10. The data at the relative depth of h/H0 = 1.90 deviates only slightly from the straight line of the Rayleigh. As the relative depth decreases, the data of cumulative distribution shift leftward with the largest relative height being Hmax /H1/3 = 1.3 or less. At the relative depth of h/H0 = 0.69 and 0.48, however, the cumulative distribution exhibits a tendency of returning to the Rayleigh with Hmax /H1/3 = 1.5. By referring to the original diagram of Fig. 4.5 of Tick,42 it is observed that deviation from the Rayleigh distribution is largest around h/H0 = 1.0. Closer to the

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1.0

1.0

H1/3/h = 0.400 - 0.499 17 records with 1398 waves

0.8

Probability density, p (x)

Probability density, p (x)

H1/3/h = 0 - 0.399 52 records with 5111 waves

Observation Rayleigh

0.6

0.4

0.2

0.8 Observation Rayleigh

0.6

0.4

0.2

0

0 0

1.0 1.5 2.0 2.5 3.0 0.5 Normalized wave height, x = H /Hmean

0

3.5

0.5

1.0

1.5

2.0

2.5

3.0

3.5

Normalized wave height, x = H /Hmean

1.0

1.0

H1/3/h = 0.500 - 0.599 7 records with 534 waves Observation Rayleigh

0.6

H1/3/h = 0.600 - 0.630 3 records with 188 waves

0.8 Probability density, p (x)

0.8 Probability density, p (x)

FA

0.4

0.2

Observation Rayleigh

0.6

0.4

0.2

0

0 0

0.5

1.0

1.5

2.0

2.5

3.0

Normalized wave height, x = H /Hmean

Fig. 4.9.

3.5

0

0.5

1.0

1.5

2.0

2.5

3.0

3.5

Normalized wave height, x = H /Hmean

Histograms of field wave data in four groups of relative water depth (after Goda24 ).

shore, the range of wave height begins to spread owing to regeneration of waves after breaking, and the wave heights show a tendency to approach the Rayleigh again. Some researchers such as Battjes and Groenendijk44 have tried to apply certain statistical distributions to the wave heights in the surf zone and to empirically formulate the parameter values. As demonstrated in Figs. 4.9 and 4.10, however, the wave height distribution continues to vary throughout the surf zone, and any effort to describe it with a combination of multiple distribution functions will be futile. 4.7.2. Variation of wave height ratios across the surf zone Change in the pdf of wave heights also yields variation of the ratios among characteristic wave heights. General tendency in the surf zone is that the values of Hmax /H1/3 , H1/20 /H1/3 , and H1/10 /H1/3 decrease as waves enter the surf zone and take the minimum values around at its middle, while the values of Hrms /H1/3 and Hmean /H1/3 increase. However, the wave height ratios come back to the values predicted by the Rayleigh distribution as waves approach the shoreline. Goda45 has made a prediction of such variation of wave height ratios by means of a numerical model based on the Parabolic Equation with a Gradational Breaker

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2.5 0.99

Rayleigh h/H0'=1.90 h/H0'=1.60 h/H0'=1.28 h/H0'=0.97 h/H0'=0.69 h/H0'=0.48

2.0

1.5

0.95 0.90 0.75

1.0 0.50 0.5

0.25

0

0.10 0.05 0.01 0

0.5

1.0

1.5

Nonexceedance Probability, P

Reduced variate, SQRT [ -ln (1 - P)]

Ting's Lab. Data

2.0

Relative wave height, H/H1/3 Fig. 4.10. Evolution of cumulative distribution of wave heights in the surf zone based on the data by Ting.42

Index for Spectral waves (PEGBIS). It has been demonstrated that the difference among characteristic wave heights gradually decreases as the relative water depth becomes shallow down to around h/H0 = 1, but the range of wave heights expands again as the water becomes much shallower toward the shoreline. Laboratory tests by Ting42,43 on the wave breaking on a uniform slope of 1/35 show a variation of the wave height ratio Hrms /H1/3, following the general trend described above. Similar variations of the wave height ratios H1/10 /H1/3 and Hrms /H1/3 are found in the photopole data at Ajigaura, DUCK85, and SUPERDUCK. Figure 4.11 presents a compilation of the laboratory and field data concerning the wave height ratio changes. Also included in Fig. 4.11 is the prediction of the wave height ratios across the surf zone by means of the PEGBIS model applied for the Ajigaura beach condition. Although there are large scatters of data, the ratio H1/10 /H1/3 takes the minimum value of about 1.15 around h/H0 = 1.4 for the field data, which indicate slightly larger values than the PEGBIS prediction, probably owing to the wave nonlinearity effect. The ratio Hrms /H1/3 takes the maximum value of about 0.81 around h/H0 = 1.0 for both the laboratory and field data. The DUCK85 data exhibit the Hrms /H1/3 being closer to 1 than the Ajigaura data. It might be due to lower wave steepness of the swell at DUCK85. Such changes of the wave height ratios across the surf zone are the result of the evolution of the pdf of wave heights discussed in Sec. 4.7.1. Narrowing of the wave height distribution at the middle of the surf zone provides a larger safety margin in the design of breakwaters and other structures than the case of applying the Rayleigh distribution, because the design wave height is specified with either Hmax (composite breakwaters) or H1/20 (mound breakwaters) for a prescribed offshore

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H1/10 /H1/3: Ajigaura H1/10 /H1/3: DUCK85 H1/10 /H1/3: SUPERDUCK H1/10 /H1/3: PEGBIS Hrms /H1/3: Ajigaura

Hrms /H1/3: DUCK85 Hrms /H1/3: SUPERDUCK Hrms /H1/3: Ting's Lab. Hrms /H1/3: PEGBIS

Wave height ratios, H1/10 /H1/3 & Hrms /H1/3

1.50

1.25

1.00

0.75

0.50 0.0

1.0

2.0

3.0

4.0

5.0

6.0

Relative depth, h/H0 Fig. 4.11. Variations of wave height ratios H1/10 /H1/3 and Hrms /H1/3 by photopole field data and Hrms /H1/3 in laboratory tests by Ting together with prediction by PEGBIS model.

significant wave height. Therefore, reliable information on the evolution of characteristic wave heights, not just Hm0 or Hrms , needs to be provided by numerical models of wave transformation with random breaking.

4.8. Incorporation of Wave Breaking Process in Numerical Wave Transformation Models For engineering applications, numerical wave transformation models are indispensable tools, and there have been many models, proposed by various researchers and institutions. The models must be capable of dealing with random waves and the process of individual wave breaking, because the majority of engineering problems are concerned with the situations within the surf zone. For the simplest case of random waves on a uniform slope, Goda46 compared performance of seven models listed in Table 4.5, which include those by Battjes,39 Kuo and Kuo,40 Goda,24,45 Battjes and Janssen,47 Thornton and Guza,48 and Larson and Kraus,49 though there are many more models for random wave breaking. All the models introduced the breaker index Hb /hb , but the degree of inclusion of the parameters of the relative depth hb /L0 and the slope s differs among them. The inherent variability of breaker heights is taken into account by the models by

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109

Main characteristics of random wave-breaking models for comparative study. Factors affecting breaking wave height

Models Battjes39

Energy dissipation mechanism

PDF of broken wave

pdf Delta function deformation at breaker height Kuo and ” Remove and Kuo40 adjust the remainder Goda24 ” ” Bore model Delta function Battjes at breaker and height Janssen47 Thornton ” Adjust with and weight Guza48 function Larson and Dally Delta function Kraus49 model at breaker height ” Remove and Goda45 (PEGBIS) adjust the remainder

Breaker variability

Waves spectrum

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Goda24,45 only. The energy dissipation is dealt directly with either the bore analogy or the model by Dally et al.50 or indirectly through modification of the pdf of wave heights. Figure 4.12 is an example of differences in the variations of the root-mean-square wave height in the nearshore waters predicted by the seven models on the slope of s = 0.02 (Goda46 ). Differences are very large beyond expectation and Goda46 shows other examples on the slope of s = 1/10, 1/30, and 1/100. Because the generation of longshore currents is governed by the spatial gradient of the radiation stresses, these models produce much different cross-shore profiles of longshore current velocities as demonstrated by Goda.46 Smith51 has also demonstrated different performances of five wave-breaking models with different breaker indices against field measurement of spectral waves. Because of the large difference in the prediction of nearshore wave heights, one should be careful in selecting a numerical model for engineering applications. Development of numerical models for time-domain wave transformation has been tried by many researchers, but random wave-breaking process has not been well reproduced in these models. One of the exceptions is the Boussinesq-type model developed by Hirayama et al.52 and Hirayama and Hiraishi,53 who employed the breaking criterion of the vertical pressure gradient by Nadaoka et al.7 They raised the threshold gradient from 0 to 0.5 to compensate the insufficiency in numerical accuracy due to the features of weak nonlinearity inherent to the Boussinesq equation. They have succeeded in reproducing the pdf variation across the surf zone.

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Depth, h (m) Fig. 4.12. Variations of wave heights Hrms in the surf zone on planar beaches computed by seven random wave-breaking models for (H1/3 )0 = 2.0 m, T = 8.004 s, and θ0 = 30◦ (Goda46 ).

Good reproduction of random wave-breaking process having the features discussed in the present chapter is a key factor for future development of reliable and practical numerical models for engineering applications.

4.9. Summary Wave-breaking phenomenon in the nearshore waters is characterized by the following features: (1) The breaker index formula by Goda13 for regular waves is revised by reducing the bed slope effect for better agreement with laboratory data. (2) Depth-limited breaker heights have inherent variability with the coefficient of variation of about 6% for the slope of 1/100 and 14% for the slope of 1/10, which increases as the bed slope becomes steep. (3) Breaker index or the ratio of breaker height to water depth is governed by both the relative water depth hb /L0 and the bed slope. (4) Incipient breaker index for the significant height of random waves is smaller than that for regular waves by about 30%, and the significant wave height has an upper limit of about 0.6 times the water depth on gentle slopes. (5) Wave nonlinearity expressed in terms of skewness, kurtosis, and the wave ratios H1/10 /Hm0 , H1/3 /Hm0 , and Hrms /Hm0 increases with wave propagation toward the shore and is most enhanced just outside the surf zone. For swell of very low steepness, the skewness, kurtosis, and wave height ratio H1/3 /Hm0 may go up to 2.0, 9.0, and 1.6, respectively, at the wave nonlinearity parameter around 4.0.

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(6) For wind waves and swell of relatively large steepness, random wave breaking begins at large depths compared with swell of small steepness. Thus, the former waves do not exhibit conspicuous wave nonlinearity. (7) Wave nonlinearity is weakened by wave breaking inside the surf zone and is eventually lost near the shoreline with return to linear wave features. (8) Probability density function (pdf ) of wave heights gradually deviates from the Rayleigh as waves propagate toward the shore, with the largest deviation taking place at the middle of the surf zone. Near the shoreline, the wave height distribution approaches the Rayleigh again. (9) Differences between characteristic wave heights such as H1/10 , H1/3 , and Hrms decrease in the middle of the surf zone, but they return to those by the Rayleigh near the shoreline. The PEGBIS model can simulate such changes of characteristic wave heights. (10) Various existing models of random wave breaking produce quite different prediction of wave heights in the surf zone. Careful selection of a random wave-breaking model should be made so that the model will yield reliable information for engineering applications.

Acknowledgment The author much appreciates kind cooperation by Dr Steven Hughes and Dr Shintaro Hotta for providing him with their valuable wave data.

References 1. Y. Goda, How much do we know about wave breaking in the nearshore waters, Proc. 4th Int. Conf. Asian and Pacific Coast (APAC 2007), Nanjin, China, CD-RM (2007). 2. H. Yamada and T. Shiotani, On the highest water waves of permanent type, Bull. Disaster Prevention Res. Inst., Kyoto Univ. 18–2(135), 1–22 (1968). 3. Y. Goda, A synthesis of breaker indices, Trans. Japan Soc. Civil Eng. 2(2), 39–49 (1970). 4. H. Yamada, G. Kimura and J. Okabe, Precise determination of the solitary waves of extreme height on water of a uniform depth, Rep. Res. Inst. Appl. Mech., Kyushu Univ. XVI(52), 15–32 (1968). 5. J. McCowan, On the highest waves in water, Phil. Mag. Ser. 5, 36, 351–358 (1894). 6. R. Miche, Mouvements ondulatoires de lamer en profondeur ou d´ecroissante, Annales de Ponts et Chauss´ees 19, 370–406 (1944). 7. K. Nadaoka, O. Ono and H. Kurihara, Near-crest pressure gradient of irregular water waves approaching to break, Proc. Coastal Dynamics ’97 (1997), pp. 255–264. 8. G. M. Kaminsky and N. C. Kraus, Evaluation of depth-limited wave breaking criteria, Second Int. Symp. Ocean Waves Measurement and Analysis, WAVES 1995 (1995), pp. 180–193. 9. J. W. Kamphuis, Incipient wave breaking, Coastal Eng. 15, 185–203 (1991). 10. W. Rattanapitikon and T. Shibayama, Verification and modification of breaker height formulas, Coastal Eng. J. 42(4), 389–406 (2000).

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11. W. Rattanapitikon, T. Vivattanasirisak and T. Shibayama, A proposal of new breaker height formula, Coastal Eng. J. 45(1), 29–48 (2003). 12. Y. Goda, A new method of wave pressure calculation for the design of composite breakwaters, Rept. Port Harbour Res. Inst. 14(3), 59–106 (1973) (in Japanese). 13. Y. Goda, New wave pressure formulae for composite breakwaters, Proc. 14th Int. Conf. Coastal Eng., Copenhagen, ASCE (1974) pp. 1702–1720. 14. M. Muttray and H. Oumeraci, Wave transformation on the foreshore of coastal structure, Coastal Eng. 2000 (Proc. ICCE) (2000) pp. 2178–1191. 15. Y. C. Li, G. H. Dong and B. Teng, Wave breaker indices in finite water depth, China Ocean Eng. 5(1), 51–64 (1991). 16. Y. C. Li, Y. Yu, L. F. Cui and G. H. Dong, Experimental study of wave breaking on gentle slope, China Ocean Eng. 14(1), 59–67 (2000). 17. J. L. Lara, I. J. Losada and P. L.-F. Liu, Breaking waves over a mild gravel slope: Experimental and numerical analysis, J. Geophys. Res. 111(C11019), 1–26 (2006). 18. J. W. Smith and N. C. Kraus, Laboratory study of wave-breaking over bars and artificial reefs, J. Waterway, Port, Coastal, Ocean Eng. ASCE, 117(4), 307–325 (1991). 19. A. Kimura and A. Seyama, Breaking limit of irregular waves on slopes, Proc. 33rd Japanese Coastal Eng. Conf. (1986), pp. 174–178 (in Japanese). 20. A. Seyama and A. Kimura, The measured properties of irregular wave breaking and wave height change after breaking on the slope, Proc. 21st Int. Conf. Coastal Eng. (1988), pp. 419–497. 21. K. P. Black and M. A. Rosenberg, Semi-empirical treatment of wave transformation outside and inside the breaker line, Coastal Eng. 16, 313–345 (1992). 22. D. Kriebel, Breaking waves in intermediate-depths with and without current, Coastal Eng. 2000 (Proc. ICCE), (2000) pp. 203–215. 23. Y. C. Li, Y. Yu, L. F. Cui and G. H. Dong, Transformation and breaking of irregular waves on very gentle slope, China Ocean Eng. 14(3), 261–278 (2000). 24. Y. Goda, Irregular wave deformation in the surf zone, Coastal Eng. Jpn, JSCE 18, 13–26 (1975). 25. Y. Goda, Random Seas and Design of Maritime Structures, 2nd Edn. (World Scientific, Singapore, 2000). 26. F. C. K. Ting, Laboratory study of wave and turbulence velocities in a broad-banded irregular wave surf zone, Coastal Eng. 43, 183–208 (2001). 27. F. C. K. Ting, Laboratory study of wave and turbulence characteristics in a narrowbanded irregular breaking waves, Coastal Eng. 46, 291–313 (2002). 28. A. H. Sallenger and R. A. Holman, Wave energy saturation on a natural beach of variable slope, J. Geophys. Res. 90(C6), 11939–11944 (1985). 29. Y. Goda and K. Nagai, Investigation of the statistical properties of sea waves with field and simulation data, Rept. Port Harbour Res. Inst. 13(1), 3–37 (1974) (in Japanese). 30. Y. Goda, Analysis of wave grouping and spectra of long-travelled swell, Rept. Port Harbour Res. Inst. 22(1), 3–41 (1983). 31. S. Hotta and M. Mizuguchi, A field study of waves in the surf zone, Coastal Eng. Jpn, JSCE 23, 59–79 (1980). 32. S. Hotta and M. Mizuguchi, Statistical properties of field waves in the surf zone, Proc. 33rd Jpn Coastal Eng. Conf. (1986) pp. 154–157 (in Japanese). 33. B. A. Ebersole and S. A. Hughes, DUCK85 photopole experiment, US Army Corps of Engrs., WES, Misc. Paper, CERC-87-18, (1987) pp. 1–165. 34. Y. Goda, A unified nonlinearity parameter of water waves, Rept. Port Harbour Res. Inst. 22(3), 3–30 (1983).

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35. Y. Goda and M. Kudaka, On the role of spectral width and shape parameters in control of individual wave height distribution, Coastal Eng. J. 49(3), 311–335 (2007). 36. E. F. Thompson and C. L. Vincent, Significant wave height for shallow water design, J. Waterway, Port, Coastal, Ocean Eng. ASCE 111(5), 828–841 (1985). 37. M. S. Longuet-Higgins, On the distribution of the heights of sea waves: Some effects of nonlinearity and finite band effects, J. Geophys. Res. 85(C3), 1519–1523 (1980). 38. J. I. Collins, Probabilities of breaking wave characteristics, Proc. 12th Int. Coastal Eng. Conf., Washington, DC, ASCE (1970) pp. 399–414. 39. J. A. Battjes, Setup due to irregular wave, Proc. 13th Int. Conf. Coastal Eng., Vancouver, ASCE (1972) pp. 1993–2004. 40. C. T. Kuo and S. T. Kuo, Effect of wave breaking on statistical distribution of wave heights, Proc. Civil Eng. Ocean, ASCE (1974) pp. 1211–1231. 41. T. E. Baldock, P. Holmes, S. Bunker and P. van Weert, Cross-shore hydrodynamics with an unsaturated surf zone, Coastal Eng. 34(3–4), 173–196 (1998). 42. F. C. K. Ting, Laboratory study of wave and turbulence velocities in a broad-banded irregular wave surf zone, Coastal Eng. 43, 183–208 (2001). 43. F. C. K. Ting, Laboratory study of wave and turbulence characteristics in a narrowbanded irregular breaking waves, Coastal Eng. 46, 291–313 (2002). 44. J. A. Battjes and H. W. Groenendijk, Wave height distributions on shallow foreshores, Coastal Eng. 40, 161–182 (2000). 45. Y. Goda, A 2-D random wave transformation model with gradational breaker index, Coastal Eng. J. 46(1), 1–38 (2004). 46. Y. Goda, Examination of the influence of several factors on longshore current computation with random waves, Coastal Eng. 53, 156–170 (2006). 47. J. A. Battjes and J. P. F. M. Janssen, Energy loss and set-up due to breaking of random waves, Proc. 16th Int. Conf. Coastal Eng., Hamburg, ASCE (1978), pp. 1–19. 48. E. B. Thornton and R. T. Guza, Transformation of wave height distribution, J. Geophys. Res. 88(C10), 5925–5938 (1983). 49. M. Larson and N. C. Kraus, Numerical model of longshore current for bar and trough beaches, J. Waterway, Port, Coastal, Ocean Eng., ASCE 117(4), 326–347 (1991). 50. W. R. Dally, R. G. Dean and R. A. Darlymple, Wave height variation across beaches of arbitrary profile, J. Geophys. Res. 90(C6), 11917–11927 (1985). 51. J. M. Smith, Breaking in a spectral wave model, Ocean Wave Measurement and Analysis (Proc. WAVES 2001), ASCE (2001) pp. 1022–1031. 52. K. Hirayama, N. Hara and T. Hiraishi, Application of bore model to non-linear wave transformation, Proc. 13th Int. Offshore Polar Eng. Conf. (2003), pp. 796–801. 53. K. Hirayama and T. Hiraishi, A Boussinesq model for wave breaking and run-up in a coastal zone, 1D, Fifth Int. Symp. Ocean Waves Measurement Analysis, WAVES 2005, CD-ROM, No. 151 (2005), pp. 1–10.

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Chapter 5

Aeration and Bubbles in the Surf Zone Nobuhito Mori Disaster Prevention Research Institute Kyoto University, Gokasho, Uji, Kyoto 611-0011, Japan [email protected] Shohachi Kakuno Osaka City University 3-3-138 Sugimoto, Sumiyoshi-ku, Osaka 558-8585, Japan [email protected] Daniel T. Cox Oregon State University, Corvallis, OR 97331-4501, USA [email protected] This chapter presents a brief summary of aeration in the surf zone, beginning with a review of air–water characteristics in surf zone waves. Second, measurements techniques of the bulk of air and bubbles induced by breaking waves in the surf zone are described, and third, the bulk of air and bubble characteristics are summarized based on the in situ and visualization laboratory measurements. Finally, the gas transfer in the surf zone is described and related to the wave characteristics.

5.1. Introduction Ocean surface gravity wave propagation from offshore to shoreline is often regarded as a single phase flow using potential flow theories or solving the Navier–Stokes equation. Generally, the single phase flow approach to ocean waves is successful for simulating wave transformation in the coastal area. However, this assumption breaks down because waves become steep and start to break owing to the bottom bathymetric effects in the near shore. The wave-breaking process creates dense bubble plumes, dissipates energy and generates turbulence in the surf zone. Figure 5.1 shows an example of the crest of wave breaking on a gentle slope. Most of the waves 115

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Fig. 5.1.

Image of breaking wave crest at an initiation of breaking in the surf zone.

can be regarded as a single (liquid) phase flow, and only the near-crest region is aerated by the jet attachment to the water surface. An accurate estimation of the bulk of the void fraction, bubble size, and population distributions in the surf zone are important for understanding the two-phase flow characteristics, solving engineering problems, and elucidating environmental mechanisms of the coastal area.1 For example, Peregrine2 summarized that the entrapped air of breaking wave reduces the wave impact pressure owing to its greater compressibility compared with the single-phase approach. There is also a significant difference between salt water and fresh water experiments on the wave impact pressure (bubbles generated by turbulence in saltwater are smaller than fresh water), and the entrapped air reduced the pressure by approximately 10%.3 The compressibility owing to air– water mixture decreases the velocity of sound and is being used to estimate largescale prototype impacts, since the usual Froude scaling is unlikely to be correct for engineering problems. Therefore, the connection between air-mixture, bubble distribution, and wave-breaking-induced turbulence is essential to understand gas– liquid interactions in the surf zone. The air bubble plume is always present in the surf zone breaking waves. Experimental data show that the entrained air is proportional to the energy dissipation.4 For these breaking waves, entrained air enhances the gas-exchange at the water surface. For example, the dissolved oxygen is associated with the breaking waves in the coastal waters,5,6 although the carbon dioxide gas transfer can be regarded as a function of wind speed. Understanding air entrainment and bubble distribution for the depth-limited breaking of surf zone waves is limited in comparison with the wind-wave breaking. The air bubble distributions under the wind-wave breaking in the upper layer of the ocean surface has been summarized in detail by Thorpe.7 The wind-wave bubble formation and its dependence on the turbulent dissipation rate of fluids have been discussed,8 and it is proposed a −2 power-law scaling with bubble diameter. The −10/3 power-law scaling is alternatively proposed based on the discussion of bubble fragmentation due to strong turbulent shear flow. These bubble size scaling laws are still under development.9 However, the knowledge of wind-wave breaking induced

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bubbles and associated gas transfer has become well-known issue over the last decade.10 Despite the fruitful knowledge of air entrainment of wind-wave breaking, few studies of air entrainment for surf zone wave breaking exist. It has been found that there is little difference in the bubble populations beneath the mechanically generated surface waves in saltwater and fresh water.11 On the other hand, it has been reported that the void fraction depends on the turbulent intensity in the bore region of surf zone waves.12 The large-scale bubbles entrained by breaking waves in the surf zone are split by strong local turbulent shear at the bubble scale.13 However, neither qualitative nor quantitative bubble characteristics in the surf zone and connections between bubble characteristics and wave breaking are well known owing to the lack of detailed observations. This detailed information of two-phase flow characteristics is required for mathematical modeling to solve both basic scientific interests and applied engineering problems.

5.2. Measurements Techniques There are several measurement methods for bubble population and size distributions, i.e., in situ measurements,14,15 video or photographic measurements,16 laser measurements,17 and acoustic measurements.18 The bubble size measurements using lasers probe such as a phase Doppler anemometer show high accuracy in small-scale bubble measurements smaller than 500 µm but is of limited use in the presence of high void fractions and large-scale bubble measurements due to fundamental instrument limitations. Acoustic measurements of bubbles are useful in deepwater but have limitation for very shallow water region owing to the multi-reflection of sound beams. Therefore, conventional optical or resistivity type void probes are useful for the surf zone breaking waves. The electrical conductivity probe (so-called void probe) has been generally used for the high void fractions.19 This type of void probe is robust and easy to handle for both wave flume experiments and field observations. Therefore, the void probe measurements have been widely used for the twophase flow measurement in the surf zone.12,14,20 However, the in situ measurement has clear limitation for inhomogeneous surf zone breaking phenomena. The photographic or imaging technique has been advanced to in situ measurements which can cover the spatial and temporal variations of the two-phase flow characteristics in the surf zone. Recent photographic studies have illustrated the disintegration of entrapped air cavities divided into bubbles.13,21 Several imaging techniques have been proposed to measure the bubble characteristics in the surf zone. One is an application of Particle Image Velocimetry (PIV) method with forward and backward lighting.22,23 This method basically correlates the scattered light intensity and bubble plume intensity, and it can be used for high void fraction region over 20%, although it cannot measure the bubble shape itself. The other imaging technique is a bubble shape tracking method with backlighting.24 This method is a so-called shadow graph method because the bubble shape is enhanced by a backlight source toward a camera. The shadow graph method can measure both bubble shape and velocity simultaneously. However, the density of the bubbles has to be within a certain limit to separate individual bubbles from the image by the shadow graph method.

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In summary, there is no ideal method that can be used for universal purpose of bubble measurements and one can choose appropriate method(s) for a particular purpose.

5.3. Bulk Aeration Characteristics in the Surf Zone Figure 5.2 shows successive images of breaking wave taken by high-speed video camera in the surf zone. Figure 5.2(a) is just after attachment of jet on the water surface which corresponds to the initial stage of bubble generation. Figures 5.2(b) and 5.2(c) are 42 ms and 108 ms after image (a), respectively. The lifetime of the large volume of entrained air can be categorized into two phases. The first phase is the initial stage of breaking waves [Figs. 5.2(a) and 5.2(b)]. The newly injected air column produces pluses of sound and, therefore, this phase is regarded as the acoustically active phase.21 In the active phase, the trapped volume of air between the overturning jet and water surface has fragmented and injected into the deeper region, and the air pocket is fragmented into small size bubbles. This phase occurs within 1 s of the initial stage of wave breaking. The second phase is the acoustically silent phase. Most of the bubbles are advected by the surface roller of the carrier waves with buoyancy, and some of them are trapped by oblique descending eddy in this phase [Figs. 5.2(c)]. In the active phase, the void fraction ratio is relatively high and there is no clear formation of bubbles in that region. Therefore, the only possible approach to this phase is to account for the total amount of air, which is the summation of the air pocket and bubbles. The silent phase has a lower void fraction and is more dispersive, and it is possible to discriminate individual bubbles. These injected air penetrate into a deeper region during the active phase. After that, the volume of air and/or bubbles are advected by the wave motion due to their buoyancy. Therefore, the gas phase characteristics in the surf zone, such as the void fraction, are time–space variables and are inhomogeneous in both time and space. The next part of this chapter discusses the total amount of air, bubble size distributions, and gas-transfer in the surf zone, respectively. 5.3.1. Temporal characteristics The aeration process by breaking wave in the surf zone is obviously time-dependent. Therefore, the temporal transition is important in understanding the phenomena in the beginning. Figure 5.3 shows an example of phase-averaged time series of surface elevation and turbulence components of the mechanically generated periodic wave breaking on a slope.20 The notations z and Hb denote the vertical axis positive upward from the mean water level and wave height at the breaking point, respectively. The initial wave height and period are 16.3 cm and 1.6 s, respectively. Using the phase-averaged method, the velocity data were divided into the mean, periodic, 2 and turbulence components. The notation of σu2 , σv2 , and σw are the turbulent stresses for the velocity components (u, v, w). The peak of the turbulent stress is located at the front of the crest and decreases linearly as time passes. This is a typical time series of turbulence characteristics of depth-limited breaking waves.

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(a)

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Fig. 5.2. Formation of bubble plume under the breaking wave in the surf zone. (a) Attachment of jet on the surface (initiation of bubble plume generation); (b) Active phase of bubble plume generation [42 ms after (a)]; (c) Bubble plume in silent phase [108 ms after (a)].

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η and turbulence components 1.0 m from B.P. in Case 1. (a) Free surface η (solid) and standard deviation η ± ση ; (b)–(d) temporal variation of σu (thick solid), σv (thin solid), and σw (dotted) and (e) the phase-averaged void fraction
α.20

The highest void fraction occurs at the wave front, and the void fraction is decreased rapidly in time. The temporal variation of the void fraction depends on the air supply from the surface, advection, and buoyancy, and these effects are difficult to discriminate from each other. Although the process is complex, the normalized temporal characteristics show a universal behavior. Similar laboratory experiments were observed by several researchers.12,25,26 Furthermore, Cox and Shin12 found the self-similar exponential decay of the temporal series of void fraction normalized by the averaged value in different types of breaking waves. In addition, they reported that the maximum void fraction in the time series was three to four times higher than its average value. 5.3.2. Vertical distribution of void fraction The vertical distribution of void fraction of the surf zone breaking waves was first systematically measured by Hoque and Aoki.27 They showed a simple description of vertical void fraction distribution as follows. If a void fraction α simply follows a

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Vertical distribution of void fraction for plunging breaker.27

diffusion equation in steady state, we have ∂ ∂(αu) = ∂z ∂z


 ∂α D , ∂z

(5.1)

where z is the vertical axis, u is the water velocity, and D is the turbulent diffusivity of air bubble plume. Assuming a uniform velocity distribution and D independent in the vertical direction, we have ∂ 2α u0 ∂α , = 2 ∂z D ∂z

(5.2)

where u0 denotes the uniform velocity. The solution of the above equation is α(z) = α0 exp(kz),

(5.3)

where k = u0 /D is the newly introduced decay parameter of the vertical void fraction distribution and α0 denotes the void fraction at the mean water level (z = 0). Figure 5.4 shows the comparison between experimental data and Eq. (5.3) for plunging breakers on a gentle slope. The experimental data shows an exponential decay of the void fraction in the vertical direction. Furthermore, introducing k0 = kh, where h is the local depth, k0 can be regarded as a constant value ranging approximately from 3.75 to 4.0, independent of breaker type. Similar results were measured by a visualization technique23 after that. In addition, the linear decay of the maximum water depth of penetrated bubble shoreward is also observed as shown in Fig. 5.4.27 5.3.3. Horizontal distribution of void fraction The characteristics of the volume of air changes both horizontally, vertically and temporally. The void fraction and total kinetic energy show similar distributions, and Cox and Shin12 also reported the dependence of void fraction on turbulent intensity in the bore region of surf zone waves. The entrapped air induced by wave breaking generates strong turbulence near the free surface. Therefore, the relationship between void fraction and turbulence should be connected to each other.

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×10

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xb/Hb Fig. 5.5. Horizontal relationship between time-averaged void fraction α (solid line) and timeaveraged TKE k (dotted line).20

Figure 5.5 shows experimental results of void fraction and total kinetic energy (TKE) variations from the breaking point xb to the shore line.20 The spatial variations of the time-averaged void fraction α and time-averaged TKE, k show similar tendency, although the peak of α is located approximately 0.2 × Hb ∼ 2 × Hb seaward of the peak of k, as shown in Fig. 5.5. The short spatial lag between α and k might be associated with the spatial difference of entrapped air and associated turbulence. The spatial variations of the turbulence and void fraction are highly correlated near the mean water level. The entrained bulk of air and bubbles generate turbulence, and turbulence splits the coarse bubbles into fine bubbles in this area. Moreover, the TKE and void fraction increased by an order of magnitude from the trough level (z/Hb = −0.013) to above the mean water level (z/Hb = 0.033); thus, the two-phase flow characteristics are remarkably strong near the wave crest. Neglecting the short spatial lag between the void fraction and TKE, the direct comparison between the time-averaged void fraction α and normalized timeaveraged TKE k of all the measuring locations is shown in Fig. 5.6. For small values of α or k, the relationship between the two shows a high correlation and becomes scattered at high void fraction or TKE region. Overall, the relationship between time-averaged void fraction and time-averaged TKE shows high correlation12 as does the incident wave-dependence on the relationship between α and k.

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0.14

0.12

0.1

α

0.08

0.06

0.04

Case1 Case2 Case3

0.02

0

0

0.1

0.2 ( k/gh )

0.3

0.4

1/2

Fig. 5.6. Relationship between time-averaged TKE k and time-averaged void fraction α. (•: Case 1, +: Case 2,
: Case 3).

The relation between the void fraction and TKE, the scale-dependence of void fraction, and further theoretical considerations will be required to find the universal relationship between the turbulence characteristics of depth-limited breaking waves and air entrainment.

5.4. Bubble Characteristics in the Surf Zone Bubble generation is a small-scale process of aeration in the surf zone.28 There is a large body of literature dealing with the distribution of bubbles in wind-wave breaking.7,8,29,30 From the knowledge of wind-wave breaking, there are two modes of bubble generation.31 One is entrapment of air under the falling water jet by the wave overturning and another is air entrainment at the point where the falling water jet meets the vortex formed at the rear of the splash-up. Figure 5.7 shows the time series of bubble diameter distribution just beneath the mean water level near the breaking point under the same wave condition as in Fig. 5.5.24 The bubble diameter here means the averaged value of the major and minor diameter. The horizontal axis is the characteristic bubble diameter, the vertical axis is the temporal change, and the contour indicates number of bubbles per unit area. The plunging wave breaking not only entrains the bulk of air but also generates a large number of bubbles in comparison with the spilling breaker. As discussed earlier, the void fraction decreases exponentially in time. The time series

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Time series of bubble size distribution.24 (a) Spilling breaker; (b) Plunging breaker.

of the bubble number density distribution (Fig. 5.7) shows a similar tendency, but the small-scale bubbles survive longer than large-scale bubbles. Generally, entrapped air by breaking waves is split into small bubbles owing to turbulence shear of surrounding flows. The early work of this problem for a general condition is discussed by Kolmogorov32 and Hinze33 in the middle of the last century, and a large bibliography has been generated.34 Longuet-Higgins35 proposed a crushing air cavity model for the bubble distributions after the formation of a bubble plume and shows good agreement of bubble aeration in general condition, qualitatively. Figure 5.8 shows the instantaneous and time-averaged bubble size spectra with the estimated power law measured by an imaging technique.24 The instantaneous bubble size spectra corresponds to the time of the maximum void fraction observed. There is no significant difference of bubble size distribution

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Power law of bubble size distribution.

between instantaneous and time-averaged spectra at small scale (diameter < 1 mm). On the other hand, two scaling laws can be seen in the bubble size distribution. The small and large size bubble power scaling laws are d−1 and d−3–−4 , respectively. Garrett et al.9 proposed semi-empirical −10/3 power-law scaling for the bubble which is larger than the Hinze scale, based on the discussion of bubble fragmentation in the strong turbulence flow below the trough level. On the other hand, Deane and Stokes21 empirically proposed a −3/2 power-law scaling, smaller than the Hinze scale in the acoustically active phase near the crest. The node point between the two power scaling corresponds to the Hinze scale of the bubble splitting theory.33 The measured power-law scalings of the small size bubble shown in Fig. 5.8 are close to −3/2 and −10/3 power laws, respectively. These power-law scalings can be seen in different conditions in mechanically generated waves in the flume.13,20,21 The Hinze scale dH 21 of bubble splitting can be given by
 ρw ε−2/5 , dH = 2−8/5 c (5.4) γ where ρw is the water density, γ is the surface tension coefficient, ε is turbulence energy dissipation at the bubble diameter, and c = (W ec )3/5 , where W ec is the critical Weber number. The Weber number W e is defined as ρw 2 δu d, We = (5.5) γ where δu is turbulence velocity at the bubble scale d. The turbulence velocity δu at the bubble scale in the inertial range can be given by δu = 2ε2/3 d2/3 . If the energy dissipation rate ε can be estimated by wave energy dissipation, the Hinze scale can be obtained. For example, if ε = 12 W/kg, the Hinze scale is 2 mm in diameter. These theoretical predictions agree fairly well with the experimental data,20,21 although the Hinze scale is formulated under the assumption of isotropic and homogeneous turbulence flow.

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5.5. Gas Exchange in the Surf Zone For moderately solvable gases such as carbon dioxide and oxygen, the gas transfer across the air–sea interface is controlled by the aqueous condition, and is parameterized as a function of a gas transfer velocity kL kL = αSc−1/2 u∗ ,

(5.6)

where Sc = νκ is the Schmidt number, α is the empirical coefficient, and u∗ is the friction velocity, respectively.36 The role of bubbles in the exchange of gas in the surf zone is poorly understood in comparison to the wind-wave breaking. An analogy of wind-wave breaking to surf zone breaking waves is similar, but surface renewal process and the bubble-mediated gas transfer are different from each other. For example, the surface renewal by the wind stress is important for the windwave scenario, but the turbulence generated by the wave overturning is dominant in the surf zone. Moreover, the bubble-mediated gas transfer process depends on the bubble population and size distribution in the surf zone. For the wind-wave case, the large-scale bubbles with a diameter greater than approximately 1 mm play an important role in air–sea gas exchange.37,38 However, there are a few confident prediction models that exist for macro or microscopic bubble-mediated gas transfer for surf zone breaking waves. The oxygen gas transfer is important for fish and marine mammals near the coastal region. Therefore, oxygen gas transfer, rather than carbon dioxide, plays a vital role in aquatic environment in the surf zone.39 There are significant difference between the wind-waves and surf zone waves generally. For example, the spatial homogeneity can be assumed in wind-waves, but there is a strong spatial dependence for surf zone waves. Therefore, a different approach to predict the gas transfer is required. One may assume that the gas transfer is related to the total water mass volume of surf zone (Fig. 5.9). The general expression of the gas transfer in an aeration tank of constant volume can be formulated: ∂C A (5.7) = kL ∆C = k2 ∆C, ∂t V where C is the gas concentration, ∆C is the concentration difference between liquid and gas phase, A is the aeration (surface) area, kL is the gas transfer velocity,

Fig. 5.9.

Illustration of gas transfer in the surf zone.

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and k2 is the reaeration coefficient. Eckenfelder40 proposed the bubble-mediated gas transfer velocity kL for aeration tank as kL =

αh2/3 Qa , A dm Sc1/2

(5.8)

where α is the empirical coefficient, h is the water depth of aeration tank, Qa is the total amount of air flow rate, dm is mean bubble diameter. Multiplying the surf zone area A by Eq. (5.8), it can be formulated for aeration by surf zone breaking waves as41 2/3

kL A =

αHb Vg 1/2

dm Sc T

,

(5.9)

where Hb is the wave height at breaking point, Vg is the total entrained air volume per wave period, and T is the wave period (Fig. 5.10). In this formulation, the gas transfer in the surf zone linearly depends on the total entrained air volume. Now, we introduce the gas transfer flow rate kL A instead of the gas transfer velocity kL in Eq. (5.9). This is because the aeration area A is difficult to be estimated in the surf zone, but the total water volume V is easily obtained. Assuming a uniform bottom slope, Kakuno41 empirically obtained a relationship between Vg and surf zone parameter as Vg ∝ Hb hb / tan θ, where hb is the water depth at the breaking point, and θ is the bottom slope. Substituting this expression into Eq. (5.9), we have the gas transfer flow rate coefficient kL A for slope beach41 : 5/3

kL A =

α2 Hb hb 1/2

T dm Sc

tan θ

.

(5.10)

Kakuno compared Eq. (5.10) with the fresh water experimental data and found that the empirical coefficient α2 is approximately 2.3 × 10−3 ∼ 2.8 × 10−3 as shown in Fig. 5.11.41 Equation (5.10) expresses the gas transfer flow rate as a function of breaking wave height Hb , water depth at breaking point hb , wave period T , mean bubble diameter dm , bottom slope θ, and Schmidt number, although salinity and scaling effects are neglected. Furthermore, Kakuno41 obtained an empirical relation between the gas transfer velocity and wave energy dissipation as kL ∝ ε1/3 . wave1

wave2

wave3

Vg XTi Xb Fig. 5.10. Total bubble volume distribution in the surf zone (Xb : length of surf zone, Vg : bubble volume per wave period).

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Fig. 5.11. Gas transfer flow rate and bottom bathymetry in the surf zone (•: 1/10 slope, ◦: 1/20 slope, : 1/30 slope).

Research to estimate the gas transfer velocity kL from the wave characteristics through the wave energy dissipation is ongoing.

5.6. Conclusions and Future Study In this chapter, we summarize our general review of gas–water two-phase flow characteristics in the surf zone. There are several measurement methods for the aeration in the surf zone, such as in situ measurements, video or photographic measurements, laser measurements, and acoustic measurements. There is no one appropriate method to measure the volume of air and bubbles in the surf zone. The spatial and temporal characteristics of void fraction are summarized, and the bubble size distribution is described with simple theories. Furthermore, the gas transfer prediction method is presented as a function of the bottom slope and wave characteristics for the surf zone waves. There are many other problems such as salinity effect, scale effects, and more mathematical modeling should be investigated in the future.

References 1. M. Donelan, W. Drennan, E. Saltzman and R. Wanninkhof, Gas Transfer at Water Surfaces (AGU, Washington, DC, 2002). 2. D. Peregrine, Water wave impact on walls, Ann. Rev. Fluid Mechanics 35, 23–43 (2003). 3. B. G. N. A. Crawford, P. Hewson and P. Bird, Experiments on full-scale wave impact pressures, Coastal Eng. 8, 331–346 (1999). 4. S. Longo, M. Petti and I. J. Losada, Turbulence in the swash and surf zones: A review, Coastal Eng. 45, 129–147 (2002).

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5. D. Farmer, C. McNeil and B. Johnson, Evidence for the importance of bubbles in increasing air-sea gas flux, Nature 361, 620–623 (1993). 6. D. Wallace and C. Wirick, Large air-sea fluxes associated with breaking waves, Nature 356, 145–147 (1992). 7. S. Thorpe, On the clouds of bubbles formed by breaking wind waves in deep water, and their role in air-sea gas transfer, Phil. Trans. R. Soc. London A 304, 155–210 (1982). 8. S. Baldy, A generation-dispersion model of ambient and transient bubbles in the close vicinity of breaking waves, J. Geophys. Res. C10, 18277–18293 (1993). 9. C. Garrett, M. Li and D. Farmer, The connection between bubble size spectra and energy dissipation rates in the upper ocean, J. Phys. Oceanogr. 30, 2163–2171 (2000). 10. B. J¨ ahne and H. Haußecker, Air-water gas exchange, Ann. Rev. Fluid Mechanics 30, 443–468 (1998). 11. M. Loewen, M. O’Dor and M. Skafel, Bubbles entrained by mechanically generated breaking waves, J. Geophys. Res. 101, 20759–20769 (1996). 12. D. Cox and S. Shin, Laboratory measurements of void fraction and turbulence in the bore region of surf zone waves, J. Eng. Mechanics, ASCE 129(10), 1197–1205 (2003). 13. G. B. Deane and M. D. Stokes, Air entrainment processes and bubble size distributions in the surf zone, J. Phys. Oceanogr. 29, 1393–1403 (1999). 14. H. Chanson, S. Aoki and M. Maruyama, Unsteady air bubble entrainment and detrainment at plunging breaker: Dominant time scales and similarity of water level variation, Coastal Eng. 46, 139–157 (2002). 15. H. Chanson, Comments on fiber optic refectometer for velocity and fraction ratio measurements in multiphase flows, Rev. Sci. Instrum. 75(1), 284–285 (2003). 16. I. Leifer and G. de Leeuw, Bubble measurements in breaking-wave generated bubble plumes during the LUMINY wind-wave experiment, in Gas Transfer at Water Surfaces, eds. M. Donelan, W. Drennan, E. Saltzman and R. Wanninkhof (AGU, Washington, DC, 2002), pp. 303–310. 17. N. Mori, Experimental study of air bubble distributions induced by wind wave breaking, in Proc. Asia Pacific Coast 2003, JSCE, Makuhari (2003), CD–ROM. 18. G. B. Deane, Sound generation and air entrainment by breaking wave in the surf zone, J. Acoustic Soc. Am. 102, 2671–2689 (1997). 19. S. Vagle and D. Farmer, A comparison of four methods of bubble measurement, IEEE Oceanic Eng. 23(3), 211–222 (1998). 20. N. Mori, T. Suzuki and S. Kakuno, Experimental study of air bubbles and turbulence characteristics in the surf zone, J. Geophys. Res. 112(C05014) (2007), doi: 10.1029/2006JC003647. 21. G. B. Deane and M. D. Stokes, Scale dependence of bubble creation mechanisms in breaking waves, Nature 418, 839–844 (2002). 22. Y. Ryu, K.-A. Chang and H. J. Lim, User of bubble image velocimetry for measurement of plunging wave impinging on structure and associated greenwater, Meas. Sci. Technol. 16, 1945–1953 (2002). 23. O. Kimmoun and H. Branger, A particle image velocimetry investigation on laboratory surf-zone breaking waves over a sloping beach, J. Fluid Mech. 588, 353–397 (2007), doi: 10.1017/S0022112007007641. 24. N. Mori and S. Kakuno, Aeration and bubble measurements of the coastal breaking waves, Fluid Dynamics Res. 40(7–8), 616–626 (2008). 25. F. Ting and J. Kirby, Observation of undertow and turbulence in a laboratory surf zone, Coastal Eng. 24, 51–80 (1994).

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26. D. Cox, N. Kobayashi and A. Okayasu, Vertical variations of fluid velocities and shear stress in surf zones, Proc. 24th Int. Conf. on Coastal Eng., ASCE (1994), pp. 98–112. 27. A. Hoque and S. Aoki, Distributions of void fraction under breaking waves in the surf zone, Ocean Eng. 32, 1829–1840 (2005). 28. M. Loewen, Inside whitecaps, Nature 418, 830 (2002). 29. R. Cipriano and D. Blanchard, Bubble and aerosol spectra produced by a laboratory breaking waves, J. Geophys. Res. C86, 8085–8092 (1981). 30. C. Garrett, M. Li and D. Farmer, The connection between bubble size and energy dissipation rates in the upper ocean, J. Phys. Oceanogr. 30, 2163–2171 (2000). 31. P. Bonmarin, Geometric properties of deep-water breaking waves, J. Fluid Mech. 209, 405–433 (1989). 32. A. Kolmogorov, On the breakage of drops in a turbulent flow, Dokl. Akad. Navk. SSSR 66, 825–828 (1949). 33. J. Hinze, Fundamentals of the hydrodynamic mechanism of splitting in dispersion processes, Am. Inst. Chem. Eng. J. 1, 289–295 (1955). 34. C. Mart´ınez-Baz´ an, J. Montan´es and J. Lasheras, On the breakup of air bubble injected into a fully developed turbulent flow. Part I: Breakup frequency, J. Fluid Mech. 401, 157–182 (1999). 35. M. Longuet-Higgins, The crushing of air cavities in a liquid, Proc. Roy. Soc. London A 439, 611–626 (1992). 36. W. Melville, The role of surface-wave breaking in air-sea interaction, Ann. Rev. Fluid Mechanics 28, 279–321 (1996). 37. L. Memery and L. Marlivat, Modeling of gas flux through bubbles at the air-water interface, Tellus 37B, 272–285 (1985). 38. R. Keeling, On the role of large bubbles in air-sea gas exchange and supersaturation in the ocean, J. Marine Res. 51, 237–271 (1993). 39. C. Moutzouris and E. Daniil, Water oxygenation in the vicinity of coastal structures due to wave breaking, Proc. 24th Int. Conf. Coastal Eng., ASCE (1994), pp. 3167–3177. 40. W. Eckenfelder Jr., Absorption of oxygen from air bubbles in water, J. Sanitary Eng. Div. Proc. ASCE, 89–99 (1959). 41. S. Kakuno, D. B. Moog, T. Tatekawa, K. Takemura and T. Yamagishi, The effect of bubble on air-water oxygen transfer in the breaker zone, in Gas Transfer at Water Surfaces, eds. M. Donelan, W. Drennan, E. Saltzman and R. Wanninkhof (AGU, Washington, DC, 2002), pp. 265–277.

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Chapter 6

Freak Wave Nobuhito Mori Disaster Prevention Research Institute Kyoto University, Gokasho, Uji, Kyoto 611-0011, Japan [email protected] In this chapter, we describe mechanisms that lead to the occurrence of freak waves in the ocean. It is now generally recognized that freak waves can be generated by any one of four possible mechanisms. The traditional mechanism is a simple linear superposition of waves, the theory of which is reviewed here. The newest mechanism attempts to include the third-order nonlinearities that depart from linear wave theory. Therefore, we present here a state-of-the-art review that is based on nonlinear wave dynamics.

6.1. Introduction The history contains many reports of encounters with monstrous waves.1 For example, the captain of the cargo vessel Junior reported a wave estimated to be 100 ft high and the famous reliable report was that of the wave encountered by the U.S.S. Ramapo in the North Pacific in 1993; that wave was estimated to be 112 ft high. There are many more reports of encounter of similar waves in the history of the seas. Over the last decade, more evidence has accumulated for such extreme waves. This may be because of an increase in the number of measurement locations together with the development of more accurate field measurements, for example, several extreme wave events, the so-called freak waves, recorded in the North Sea.2 A sample wave profile is shown in Fig. 6.1, which was extracted from the data provided by wave radar shooting downward from the platform to the sea surface in the North Sea. The data were collected over 12–15 hours at a depth of 40 m. Since the early 1990s such an event has come to be called a freak wave or a rogue wave. More evidence has been obtained by analyzing field data from the North Sea,3,4 the Sea of Japan,5,6 and the Gulf of Mexico.7 The first clear definition of a freak wave was proposed by Klinting and Sand2 ; in its original form, the definition required a wave to satisfy three conditions to be

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Fig. 6.1. Freak wave recorded on the 24th November, 1981 at the Gorm Field in the Danish Sector of the North Sea.8

considered “freak wave”: (1) It has a wave height higher than twice the significant wave height. (2) Its wave height is larger than two times of the foregoing and the following wave heights. (3) Its wave crest height is larger than 65% of of its wave height. Subsequently, these conditions were found to be overly stringent, and today the first condition is generally considered to be sufficient to identify a freak wave.8 Freak wave research has two basic objectives. The first is an understanding of the mechanisms that create freak waves; several mechanisms have been proposed to explain why extreme-wave events occur in the ocean. The second is to establish a reliable statistical model for the occurrence of freak waves. However, the problem is difficult because a freak wave is only one wave, or perhaps just a few waves, in a wave train. Therefore, its occurrence exhibits statistical sensitivity. Moreover, an extreme wave event is transient: it forms and disappears quickly in both time and space. To address these problems, we should study wave populations that contain freak waves, rather than concentrate on the features of an individual wave profile. (This is not to say, however, that observations of single events are unimportant.) Once such wave populations can be characterized, then it should be possible to estimate how often waves of any given size will occur. Over the last decades, researchers in coastal and ocean engineering, and the oceanography have dedicated much efforts to the study of extreme wave events. Detail on the state of the art are available in the conference proceedings of Rogue Waves 2004.9 In this chapter, we provide an overview of freak waves. First, we review the four mechanisms that are now thought to generate freak waves in Sec. 6.2. Next, we describe the probability distribution for the populations of linear waves in Sec. 6.3. Third, we show how that probability distribution changes when nonlinear effects are taken into account in Sec. 6.4. Fourth, we discuss the accuracy with which extreme wave events can be predicted in Sec. 6.5. Finally, we make some brief suggestions about future work. 6.2. Mechanisms for Generating Freak Waves It is now accepted that at least four mechanisms can explain the formation of freak waves. A problem common to these four is whether a freak wave obeys an ordinary

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statistical law or whether other physical mechanisms come into play. That is, if we are to be able to predict freak waves, then any underlying mechanism must lead to reliable statistical model for extreme wave events. First Mechanism. The first mechanism is simply that a freak wave can be caused by a linear superposition of waves. In this case, the probability distribution for wave height in the limit of the narrow-band approximation obeys a Rayleigh distribution10 ; corrections due to finite spectral band width have been obtained.11–13 Wave crest statistics can be obtained by using a second-order theory.14 The probability distribution for wave crests has been found in a narrow-band approximation15 and for finite band width.16 This mechanism is theoretically well established and relatively easy to follow; the basic theory will be reviewed in Sec. 6.3. Second mechanism. The second mechanism is that freak wave can result when waves interact with currents. When this situation occurs, ray theory can be used to explain the formation of extreme waves. However, it is not yet known how the statistical properties of surface are related to the properties of currents. From the studies of freak waves generated over a random current, it has been found that wave focusing with 1 m/s velocity fluctuations occurs in the uniform current.17 Third Mechanism. A third mechanism has its basis in crossing-sea states (i.e., two-sea systems). An example is a swell and a wind sea in which the two have different directions coexisting over some region of the ocean, or diffraction of waves behind an island. Usually, both the wave-current mechanism and the two-sea state mechanism occur in specific locations, such as along the Angus current; therefore, freak waves caused by either of these mechanisms are relatively predictable. Fourth Mechanism. In the fourth mechanism, extreme wave events result from the high-order nonlinear interactions; specifically, the underlying mechanism is fourwave interactions.18,19 Numerical and experimental studies have demonstrated that freak-like waves can be generated frequently in a two-dimensional wave flume, even in the absence of a current, or wave refraction, or wave diffraction18,20–22 ; see Fig. 6.2. Further, numerical studies clearly indicate that a freak wave having a single, steep crest can be generated in deep-water by the third-order nonlinear interactions.18 Freak wave generation is sometimes discussed in the context of the Benjamin– Feir instability in deep-water waves because of the similarity in the steep wave

Fig. 6.2.

Freak wave simulated by the high-order spectrum method.18

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profile.18,22 Over the last two decades, the Benjamin–Feir instability of deepwater gravity waves has been studied using the nonlinear Schr¨ odinger-type equations,23–25 mode-coupling equations,26 pseudo-spectral methods,27 and experiments.28 However, the Benjamin–Feir instability vanishes when the wave spectrum is sufficiently broad. This means, there is a discrepancy between the nonlinear behaviors of periodic waves and random waves. Janssen19 investigated freak waves that occur as a consequence of four-wave interactions, including the effects of nonresonant four-wave interactions. Using Monte Carlo simulations of the Zakharov equation, he found that homogeneous nonlinear interactions give rise to deviations from the Gaussian distribution for the surface elevation. The fourth mechanism for freak wave generation requires additional knowledge before predictions can be made. Relationship between kurtosis and freak wave events has been studied using laboratory data,29 numerical results,30 and field data.7 North Sea data shows that the maximum crest, normalized by its wave heights, correlates with skewness, while the wave heights, normalized by significant wave height, correlate with the kurtosis.7 This is because the kurtosis is the fourth moment of the probability density function; hence, it measures the relevance of tails in a distribution. These results explain the general characteristics of a freak wave. (a) A secondorder nonlinearity causes wave asymmetry over the short-time evolution, O(1/ε2 ), of a wave train (Here ε is a small parameter that corresponds to the steepness of deep-water waves). (b) But the kurtosis is associated with the third-order nonlinear interactions.14 Therefore, it may be that third-order nonlinear interactions are the fundamental factors contributing to the occurrence of freak waves. However, the timescales for such interactions are longer than those for second-order interactions, O(1/ε4 ), and they are two orders of magnitude longer than those for second-order nonlinearities. The formal relation between the kurtosis and the exceedance probability for wave height have been discussed (Mori and Janssen31 ; see also Tayfun and Lo32 ; Mori and Yasuda33 ). The kurtosis enters into the distribution function as a correction to the Rayleigh distribution: when the kurtosis tends to three, the expected Gaussian values, and then the distribution function tends to the Rayleigh distribution. In this context, changes in the kurtosis can be evaluated once the evolution of the wave spectrum is known.19 The wave height exceedance probability has been derived as a function of the number of waves.31 This relation will be explained in a later part in this chapter. In all these studies, an ultimate goal is the determination of the shape of the exceedance probability function P (H). Once an analytical form for P (H) is found for all sea conditions (for example, as a function of the wave spectrum), then the probability of a wave exceeding a certain height can be estimated directly. One difficulty in studying extreme wave events is the verification of theory. Statistical theories are evaluated under the assumptions of stationarity; however, the real seas change states both spatially and temporally. Therefore, statistical theories are difficult to verify.34 In addition, freak wave prediction requires us to predict the maximum wave height distribution. However, although study of the maximum wave height distribution is necessary to verify a theory for freak wave

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Surface Elevation (m)

8 6 4 2 0 −2 −4 45000

45100

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Wavelet Spectrum of Yura Y88121401 Surface Waves

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Surface Elevation (m)

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Wavelet Spectrum of Yura Y88010901 Surface Waves

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0.6 0.5 0.4 0.3 0.2 0.1

10

Fig. 6.3.

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30

40

50

60

70

Freak wave recorded in 1988 in the Sea of Japan.34

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prediction, verification of wave height distribution is not sufficient to completely verify a theory.3,31 Samples of freak wave profiles and their wavelet spectrum, recorded in the Sea of Japan, are shown in Fig. 6.3.34 The wave profile shown in Fig. 6.3(a) is a 10 min time-series segment that contains a freak wave along with the corresponding contour plot for the wavelet spectrum. It appears that for the well-defined freak wave shown in Fig. 6.3(a), the occurrence of the freak wave can be readily identified from the wavelet spectrum: the strong energy density in the spectrum instantly increases at the onset of the freak wave, at the same time, that increase in energy appears to carry over into high frequency components. However, for another freak wave time series shown in Fig. 6.3(b), no corresponding focused surge in energy appears in the wavelet spectrum. So in this case, it is uncertain whether a freak wave is begun, generated by linear or nonlinear wave focusing. Nevertheless, at the onset of the event in Fig. 6.3(a), the freak wave profile appears rather asymmetric with respect to the mean level, whereas the freak wave profile in Fig. 6.3(b) is generally symmetric. So, these differences in wavelet spectra might result from differences in the processes that generated these freak waves. In any case, time series alone may or may not be reliable guides for identifying freak wave events. These observations suggest that linear and nonlinear focusing effects are coupled (that is, linear and nonlinear processes do not belong to totally different populations), and this coupling must be understood if we are to be able to predict freak waves. 6.3. Linear Wave The surface gravity waves’ celerity depends on the wave length owing to dispersive effects in deep-water. The nonlinear effects slightly modify the dispersion relation in the order of wave steepness.35 The order of nonlinear effects, both dispersion and wave profile, are square or cubic of the wave steepness if there is no resonant interactions in the system. Therefore, in general, the ocean waves can be regarded as a linear, ergodic, homogeneous, stationary, Gaussian processes. Because of these assumptions, the probability density function for water surface elevation of unidirectional waves can be expressed as the Gaussian distribution
 1 η2 p(η) = √ exp − , (6.1) m0 2πm0  ∞ 2 2 S(ω)dω , (6.2) σ = ηrms = m0 = √

0

and η is the surface elevation while σ = m0 is its variance. Let η(t) be the sea surface elevation at time t and let ζ(t) be an auxiliary variable such that η(t) and ζ(t) are not correlated. Assuming both η(t) and ζ(t) are real zero-mean functions with variance σ, we have Z(t) = η(t) + iζ(t) = A(t)eiφ(t) ,  A(t) = η 2 (t) + ζ 2 (t) ,

(6.3) (6.4)

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 ,

(6.5)

where A is the envelope of the wave train and φ is the phase. As there is no correlation between η and ζ, the joint probability density function of η and ζ becomes   1 1 p(η, ζ) = exp − (η2 + ζ 2 ) . (6.6) 2π 2 √ All variables will be normalized by the variance of the surface elevation σ = m0 . The probability density function (PDF) for the envelope A can now be obtained by integrating the joint probability distribution p(A, φ) = A p(η, ζ) ,

(6.7)

over φ; hence, 

2π

p(A) =

dφ p(A, φ).

(6.8)

0

The results are the usual Rayleigh distributions for wave amplitude p(A) = Ae−(1/2)A

2

(6.9)

and wave height p(H) =

2 1 He−(1/8)H . 4

(6.10)

In the narrow-band approximation, the wave height H = 2A. The exceedance probability PH (H) for wave height now follows by integrating (6.10) from H to ∞: 2

PH (H) = e−(1/8)H .

(6.11)

√ In an exact, linear narrow-band random wave theory, H1/3 = 4.004 m0 , but for √ simplicity we will assume H1/3 = 4 m0 . In this notation, the freak wave condition √ H/H1/3 > 2 corresponds to H/ m0 > 8. According to Eq. (6.11), the probability of occurrence for a freak wave in a wave train is P (8) = e−8 = 3.35 × 10−4 . This is equivalent to a freak wave occurring once in every 2981 waves. Assuming a characteristic wave period of 10 s, this corresponds to a storm having a duration of 8.3 h. When this linear wave theory is extended to 3D space, the appearance probability for fully directional conditions (Smax = 10) increases by 45% over that for the unidirectional condition.36 The maximum wave height distribution becomes important when we examine statistical properties of freak waves over a series of recorded data. This is because a freak wave prediction is equivalent to estimating the maximum wave height and the exceedance probability (that is, Eq. (6.11) is insufficient for prediction). The PDF for the maximum wave height pm in wave trains can be obtained once the

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PDF of wave height p(H) and exceedance probability of the wave height P (H) are known.37 Thus, pm (Hmax )dHmax = N [1 − P (Hmax )]N −1 p(Hmax )dHmax ,

(6.12)

where N is the number of waves in the wave train. For sufficiently large N , we may use the approximation N

N

N →∞

N →∞

lim [1 − P (Hmax )]N  lim exp [−N P (Hmax )] .

(6.13)

Substituting Eqs. (6.11) and (6.13) into Eq. (6.12), gives the PDF for the maximum wave height pm ,   2 2 N (6.14) pm (Hmax )dHmax = Hmax e−Hmax /8 exp −N e−Hmax /8 dHmax , 4 and the exceedance probability of maximum wave height Pm ,   2 (6.15) Pm (Hmax ) = 1 − exp −N e−Hmax /8 . Note that Eqs. (6.14) and (6.15) are evaluated as functions of N number of waves √ in the train. The freak wave condition in this study becomes Hmax / m0 > 8, and we obtain from Eq. (6.15) the following simple formula for probability of a freak wave occurring as a function of N ,

(6.16) Pfreak = 1 − exp −e−8 N . This is a general expression for the expected probability of freak waves in unidirectional linear random waves. Therefore, in the linear wave case, the occurrence probability for a freak wave depends only on the number of waves N . For example, if the number of waves equals 2981, then Eq. (6.16) gives Pfreak = 0.632, that is, the expected probability of a freak wave is 63.2% when we randomly sample from a population of 2981 waves. For N = 120 waves, the expected probability of a freak wave is 3.9% and for N = 10,000 waves, it is 96.5%. These results show that the number of waves in a wave train (or, equivalently, the duration of a storm) is important for predicting freak waves.

6.4. Nonlinear Interactions 6.4.1. Role of nonlinear interactions Since the beginning of freak wave research, high-order nonlinear effects have been considered.18,30 As a result, the nonlinear occurrence of freak waves, including their mechanisms and detailed dynamic properties, are becoming better understood.21,22,34,38–40 The state of the art has been summarized at the Rogue Wave Conference, held in 2000 and 2004.9,41 The consensus from those conferences is that third-order nonlinear interactions not only enhance the appearance of freak wave, but they are also the primary cause of freak waves. The only exceptions occur during strong wave–current interactions or wave diffraction behind an island.

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6.4.2. Quasi-resonant four-wave interactions and high-order moments Investigations have been carried out on the creation of freak waves in response to four-wave interactions, including the effects of nonresonant four-wave interactions.19,31 Those studies culminated in the formulation of a probability for the occurrence of freak waves in nonlinear wave fields. However, the modulational instability is a quasi-resonant interaction process, i.e., wave numbers and frequencies satisfy the following conditions: k1 + k2 − k3 − k4 = 0 , ω(k1 ) + ω(k2 ) − ω(k3 ) − ω(k4 ) ≤ 2 ,

(6.17) (6.18)

where  is a small parameter that corresponds to the steepness of deep-water waves. The modulational instability takes place when two wave numbers are the same, k1 = k2 , while k3 and k4 are side bands separated from k1 by ∆k, which should be small in order to satisfy the conditions Eqs. (6.17) and (6.18). The standard kinetic equation that describes the evolution of the wave spectrum in time42 is formally valid only for exact resonances and large times, O(−4 ω0−1). It has been extended to include quasi-resonant interactions,19 where a kinetic equation, which should also be valid on the timescale of the modulational instability, has been derived. If we then consider the evolution of higher order moments, such as the kurtosis µ4 , we find that the quasi-resonant interactions are responsible for deviations from the Gaussian distribution. The explicit relation between nonlinear interactions of free waves and the fourth-order moment of the surface elevation η(x, t) in deep-water has been investigated.19 The result can be expressed in terms of κ40 , which is the fourth-order cumulant of the surface elevation (κ40 = µ4 − 3), η 4  −3 m20  √ 12 dk1,2,3,4 T1,2,3,4 ω1 ω2 ω3 ω4 = 2 2 g m0 × δ1+2−3−4 Rr (∆ω, t)N1 N2 N3 .

κ40 =

√

(6.19)

Here, m0 is the variance of the surface elevation η, g is the acceleration of gravity, k is the wave number, ω is the angular frequency, N is the wave action spectral density, and T1,2,3,4 is the coupling coefficient in the Zakharov equation (see Krasitskii43 for its analytical form), δ1+2−3−4 = δ(k1 + k2 − k3 − k4 ), dk1,2,3,4 = dk1 dk2 dk3 dk4 . The resonant function is Rr =

1 − cos(∆ωt) , ∆ω

(6.20)

with ∆ω = ω1 + ω2 − ω3 − ω4 .

(6.21)

In the limit of large times, Rr → P/∆ω, where P denotes the principle value of the integral to avoid singularity in the integral.

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In the narrow-band approximation, we assume that the spectrum E(ω) has a Gaussian shape, E(ω) =

2 m0 √ e−(1/2)ν , σω 2π

(6.22)

where ν = (ω − ω0 )/σω and σω is the spectral band width; the integral in Eq. (6.19) for large times becomes  2 2 2 242 dν1,2,3 e−(1/2)[ν1 +ν2 +ν3 ] P . (6.23) κ40 = ∆2 (2π)3/2 (ν1 + ν2 − ν3 )2 − ν12 − ν22 + ν32 √ Here,  = k0 m0 is the steepness parameter and ∆ = σω /ω0 is the relative spectral band width. The integral can be evaluated analytically to obtain π κ40 = √ BF I 2 , 3

(6.24)

where BFI is defined as by Janssen19: BFI =

√ 2. ∆

(6.25)

Equation (6.24) is the simplified predictive equation for the kurtosis of the surface elevation assuming a narrow-band, unidirectional wave train, but a full description requires evaluation of six-dimensional integral in wave number space Eq. (6.19). 6.4.3. Wave height distributions in a nonlinear wave field To include nonlinear effects in the wave height distribution function, thereby possibly obtaining deviations from Rayleigh statistics, the standard approach is to use the Edgeworth series.44 The theory for wave height has been described,31–33 and the resulting distribution has been named the Modified Edgeworth Rayleigh (MER) distribution. The MER wave height and exceedance wave height distributions are given by Mori and Janssen31 : p(H)dH =

1 −(1/8)H 2 He [1 + κ40 AH (H)] dH , 4

(6.26)

2

(6.27) PH (H) = e−(1/8)H [1 + κ40 BH (H)] . √ Here, H is the wave height normalized by ηrms = m0 , κ40 is defined in Eq. (6.23), while AH (H) and BH (H) are polynomials defined as 1 (H 4 − 32H 2 + 128) , 384 1 BH (H) = H 2 (H 2 − 16) . 384 AH (H) =

(6.28) (6.29)

Note that the distributions describe the deviations from linear statistics under the hypothesis of a narrow-band, weakly nonlinear wave train. Further, in the case of linear waves (κ40 = 0), these distributions are equivalent to Eqs. (6.10) and (6.11).

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Second-order contributions, which are important for the distribution of wave crests, can be included by using an approach like that used by Tayfun.15,45 The probability distribution function, pm , and the exceedance probability, Pm , of the maximum wave height can also be expressed as a function of the fourth cumulant of the surface elevation κ40 and the number of waves N in the wave train, pm (Hmax ) =

2 N Hmax e−Hmax /8 [1 + κ40 AH (Hmax )] 4   2 × exp −N e−Hmax /8 [1 + κ40 BH (Hmax )] ,

  2 Pm (Hmax ) = 1 − exp −N e−Hmax /8 [1 + κ40 BH (Hmax )] ,

(6.30) (6.31)

where Hmax is the maximum wave height normalized by ηrms . Qualitative agreement has been found between the theoretical wave height distribution, given by Eq. (6.30), and field data.31 Equations (6.30) and (6.31) are evaluated as functions of N and κ40 (or µ4 ). When κ40 = 0, Eqs. (6.30) and (6.31) reduce to Eqs. (6.14) and (6.15), which are provided by the Rayleigh distribution. The occurrence probability of a freak wave as a function of N and κ40 , taking into account nonlinear effects, now becomes  (6.32) Pfreak = 1 − exp −e−8 N (1 + 8κ40 ) . Using Eq. (6.31) we see that the effect of kurtosis becomes of the same order as the linear theory for κ40 = 1/8. This corresponds to µ4 = 3.125, which is not a strong nonlinear condition. Hence, both the effects of finite kurtosis and the number of waves N are important for determining the probability of maximum wave height in the nonlinear wave train. It should be emphasized here that the above discussion does not require any empirical, ad hoc parameters. A different approach, also based on the dynamics of a cubic nonlinearity, has been used for finding the exceedance probability for wave height.46 That approach involves an optimization problem, requiring that at a specific space and time, the Fourier phases are all equal, under the constraints that the conservation laws of the Zakharov equation are satisfied. To summarize, quasi-resonant four-wave interactions introduce deviations from linear statistics for surface elevation; in particular, for weakly nonlinear, narrowbanded and long-crested wave trains, the kurtosis evolves according to Eq. (6.19). In the narrow-band approximation, the kurtosis is related to the BFI [defined in Eq. (6.25)]; the tail of the wave height distribution depends on the kurtosis/BFI and it increases as the kurtosis increases. Finally, the maximum wave height distribution depends on the number of waves in the wave train (record length) and on the kurtosis of the nonlinear wave field, as shown in Eq. (6.30). 6.5. Accuracy of Freak Wave Prediction Figure 6.4 compares the linear (Rayleigh) theory with those from the present theory for the occurrence probability of a freak wave, Pfreak ; the plot shows how Pfreak

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0

Pfreak

10

−1

10

Rayleigh µ4 = 3.1 µ4 = 3.2 µ4 = 3.3 µ4 = 3.4 µ4 = 3.5 −2

10

2

3

10

10 N

4

10

Fig. 6.4. Occurrence probability for freak waves as a function of number of waves N and kurtosis µ4 .31

responds to changes in the number of waves N and in the strength of nonlinearities, as measured by the kurtosis µ4 . For N = 100 waves, the occurrence probability of a freak wave is predicted to be 3.3% by the linear theory, while it is 15.4% according to Eq. (6.30) with µ4 = 3.5. For N = 1000 waves, the occurrence probability of a freak wave is 28.5% according to linear theory, while it is 81.3% according to Eq. (6.31) with µ4 = 3.5. The number of waves N = 1000 corresponds to a duration of about 3 hours, assuming T1/3 = 10 s, which is not unusual in stormy conditions. Alternatively, defining the threshold value for the occurrence probability of a freak wave as 50%, the expected number of waves that would include at least one freak wave is predicted to be 2000 waves from linear theory, but only 500 waves when Eq. (6.31) is used with µ4 = 3.5. Thus, in a strong nonlinear wave field, freak waves can occur several times more frequently than in a linear field. Figure 6.5 shows freak wave occurrence probability as a ratio Rfreak of the value predicted by the present approach to that predicted by the Rayleigh theory (κ40 = 0); the ratio depends on the kurtosis µ4 , Rnonlinear =

Pfreak −1. Pfreak |κ40 =0

(6.33)

For a small number of waves N ≤ 250, the ratio Rfreak depends linearly on µ4 . If µ4 = 3.1 and N ≤ 500, the occurrence probability of freak waves is 50% more from the nonlinear theory than from the linear theory. However, the rate of increase in Rfreak decreases as the number of waves increases. This is because, for very large number of waves, the maximum wave height almost always exceeds 2H1/3 , even in linear wave trains.

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350 300

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N=100 N=250 N=500 N=1000 N=5000

Rfreak [%]

250 200 150 100 50 0 3

Fig. 6.5.

3.1

3.2

µ4

3.3

3.4

3.5

Ratio of freak wave occurrence predicted by Eq. (6.31) to the Rayleigh theory.31

The consequences of the maximum wave height distribution [Eq. (6.30)] are generally hard to verify, not only because Eq. (6.30) depends on the number of waves but also because large amount of data are required to characterize nonlinear effects. In the following discussion, we compare the above theories with a set of long time-series experimental data47 ; these data appear to be sufficiently long to make useful comparisons. Figure 6.6 shows the probability of occurrence of freak waves Hmax /ηrms ≥ 8 for short time records, as a function of kurtosis. We assumed 100 waves per record and counted the number of freak waves. In the experimental data, the occurrence probability for freak waves shows a clear dependence on the kurtosis (this effect of course is not described by the Rayleigh distribution). But the experimental data appear to depend quadratically on the kurtosis, while the nonlinear theory Eq. (6.31) predicts a linear dependence. This is probably because the nonlinear theory includes only the lowest-order correction for nonlinearity: it excludes high-order cumulants. Figure 6.7 compares the maximum wave height distribution from the experimental data with linear, Eq. (6.14), and nonlinear theories, Eq. (6.30), for N = 150. In Fig. 6.7(a), the comparison between theory and experiment is shown for a nearby linear wave condition (µ4 = 3.06). The peak in the observed maximum wave height distribution is larger than the Rayleigh Hmax distribution and the MER Hmax distribution with µ4 = 3.06, but the observed distribution is not as wide (we ascribe this difference to the effects of the finite width of the spectrum). The maximum wave height distribution from the laboratory data departs from the Rayleigh distribution [Figs. 6.7(a)–6.7(c)] at some distance from the wavemaker, and the peak of the observed distribution shifts to larger wave heights. While the Rayleigh Hmax distribution, Eq. (6.14), is independent of the kurtosis, the MER Hmax distribution,

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N=11900 0.45 Exp. MER Rayleigh

0.4

Probability of Freak Wave [%]

0.35 0.3 0.25 0.2 0.15 0.1 0.05 0 3

3.5

4

4.5

µ4 Fig. 6.6. Occurrence probability for freak waves in long recorded wave data: N = 11, 900 (•: Experimental data, solid line: MER distribution from Eq. (6.31), dashed line: Rayleigh distribution).

Eq. (6.30), follows the behavior of the experimental data. The MER Hmax distribution seems to reproduce the experimental data reasonably well; for very small kurtosis, it overestimates the experimental data, and for large kurtosis it slightly underestimates the data. This is consistent with what we have seen in Fig. 6.6 where the occurrence probability of freak waves is plotted as a function of the kurtosis. Similar results have been obtained in other studies.31 We also consider the general behavior of the probability density function of maximum wave height in a nonlinear wave field. To do so, we plot in Fig. 6.8 the expected value of the maximum wave height, indicated by angle brackets, as a function of µ4 and N . In the figure, we show the numerically integrated value of Eq. (6.30) together with the ensemble average of the experimental data. The quantity Hmax /H1/3  in Fig. 6.8 from the Rayleigh theory corresponds to √ H/ m0 > 8. The dependence of Hmax /H1/3  on µ4 and N is clear in both the experimental data and Eq. (6.30). Overall, Hmax /H1/3  from the experimental data is smaller than that from the MER Hmax distribution; nevertheless, the theoretical and experimental data show similar trends. The above evidence shows that freak waves can be predicted for given strengths of nonlinearities, as measured by the kurtosis. The remaining problems, therefore,

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0.8 Exp. MER Rayleigh

(a) µ4 =3.06 0.7

0.5

P(H

max

/η

rms

)

0.6

0.4 0.3 0.2 0.1 0 4

5

6

7

8

H

9

10

11

12

/η

max

rms

0.8 Exp. MER Rayleigh

(b) µ4 =3.43 0.7

rms

0.4

P(H

/η

0.5

max

)

0.6

0.3 0.2 0.1 0 4

5

6

7

8

H

9

10

11

12

/η

max

rms

0.8

Exp. MER Rayleigh

(c) µ4 =4.10 0.7

P(Hmax/ηrms)

0.6 0.5 0.4 0.3 0.2 0.1 0 4

5

6

7

8

9

10

11

12

H max/ηrms

Fig. 6.7. Comparison of maximum wave height distribution Hmax /ηrms with N = 150 (•: experimental data, solid line: MER Hmax distribution, dashed line: Rayleigh Hmax distribution).

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2.2



2.1 2

1.9 1.8 1.7 1.6 1.5

1.4 1000 800 600 400 200 N

3

3.2

3.4

3.6

3.8

4

µ4

2.2



2.1 2

1.9 1.8 1.7 1.6 1.5

1.4 1000 800 600 400 N

Fig. 6.8.

200

3

3.2

3.4

3.6

3.8

4

µ4

Comparison of expected value of
Hmax /H1/3 .48

are developing accurate methods for estimating the kurtosis and learning how to take directional effects into account for the application in the ocean. 6.6. Conclusion and Future Study This chapter summarized generally accepted mechanisms that cause freak waves and we have described linear and nonlinear mechanisms in more detail. Four-wave

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quasi-resonant interactions have been shown to play an important role in determining the statistical properties of surface elevation. This discovery, which is fairly new, is of some relevance to extreme wave forecasting. Previously, deviations from Gaussian behavior have been attributed only to bound modes. However, even in the most severe sea states, the kurtosis obtained from second-order theory rarely reaches values above 3.15. Moreover, contributions to wave height distribution from second-order theory are practically negligible (in the narrow-band approximation that are exactly zero). If waves are long-crested and sufficiently steep, the dynamics of the free waves can cause very strong departures from Gaussian behavior. Large values of the kurtosis can substantially change the tails of the probability density function for wave height and can increase the occurrence of freak waves. Although nonlinear effects have been found to be important, some issues remain unresolved. First, we need an evaluation of the accuracy with which the kurtosis is estimated from spectra. BFI originally describes the asymptotic behavior, given an initial value, but in reality, there is no initial condition in the ocean. This should be verified in the near future. Second, we need to account for wave directionality. In real sea states, directional effects are important. Directional contributions to the evolution of the kurtosis, using a modified nonlinear Schr¨ odinger equation, have been described,49 and the effects of four-wave interactions only appear close to longcrested waves conditions.50 A quantitative discussion of directional effects should be presented in the near future.

References 1. L. Draper, “Freak” ocean waves, Marine Observer 35, 193–195 (1965). 2. P. Klinting and S. Sand, Analysis of prototype freak waves, in Coastal Hydrodynamics, ed. R. Darlymple (ASCE, 1987), pp. 618–632. 3. P. Stansell, J. Wolfram and S. Zachary, Horizontal asymmetry and steepness distributions for wind-driven ocean waves, Appl. Ocean Res. 25, 137–155 (2003). 4. G. Guedes Soares, Z. Cherneva and E. Ant¨ ao, Characteristics of abnormal waves in North Sea storm sea states, Appl. Ocean Res. 25, 337–344 (2003). 5. T. Yasuda and N. Mori, Occurrence properties of giant freak waves the sea area around Japan, J. Waterway, Port, Coastal Ocean Eng. 123(4), 209–213 (1997). 6. T. Yasuda, N. Mori and S. Nakayama, Characteristics of giant freak waves observed in the Sea of Japan, in Waves97 (ASCE, 1997), pp. 482–495. 7. G. Guedes Soares, Z. Cherneva and E. Antao, Abnormal waves during hurricane Camille, J. Geophys. Res. Ocean 109, C08008 (2004). 8. S. Sand, N. O. Hansen, P. Klinting, O. Gudmestad and M. Sterndorff, Freak Wave Kinematics (Kluwer Academic Pub., 1990), pp. 535–549. 9. M. Olagnon (ed.), Rogue Waves 2004 (IFREMER, 2004). 10. M. Longuet-Higgins, On the statistical distribution of the heights of sea waves, J. Marine Res. 11, 245–266 (1952). 11. A. Næss, On the distribution of crest-to-trough wave heights, Ocean Eng. 12(3), 221–234 (1985). 12. P. Boccotti, On Mechanics of Irregular Gravity Waves, Chap. VIII (Atti Accademia Nazionale dei Lincei, Memorie, 1989), pp. 111–170.

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13. M. A. Tayfun, Distribution of crest-to-trough wave heights, J. Waterway, Port, Coastal Ocean Eng. 107(3), 149–158 (1981). 14. M. Longuet-Higgins, The effect of non-linearities on statistical distirbutions in the theory of sea waves, J. Fluid Mechanics 17, 459–480 (1963). 15. M. A. Tayfun, Narrow band nonlinear sea waves, J. Geophys. Res. Ocean 85(C3), 1548–1552 (1980). 16. F. Fedele and F. Arena, Weakly nonlinear statistics of high non-linear random waves, Phys. Fluids 17, 026601 (2005). 17. B. White and B. Fornberg, On the chance of freak wave at sea, J. Fluid Mechanics 355, 113–138 (1998). 18. T. Yasuda, N. Mori and K. Ito, Freak waves in a unidirectional wave train and their kinematics, Proc. 23rd Int. Conf. Coastal Eng., Vo. 1, ASCE, Venice (1992), pp. 751–764. 19. P. Janssen, Nonlinear four-wave interactions and freak waves, J. Phys. Oceanogr. 33(4), 863–884 (2003). 20. C. Stansberg, Extreme waves in laboratory generated irregular wave trains, in Water Wave Kinematics, eds. A. Tørum and O. Gudmestad (Kluwer Academic Pub., 1990), pp. 573–590. 21. K. Trulsen and K. Dysthe, Freak waves — A three-dimensional wave simulation, Proc. 21st Symp. Naval Hydrodynamics (National Academy Press, 1997), pp. 550–558. 22. M. Onorato, A. Osborne, M. Serio and S. Bertone, Freak waves in random oceanic sea states, Phys. Rev. Lett. 86(25), 5831–5834 (2001). 23. H. C. Yuen and B. M. Lake, Nonlinear dynamics of deep-water gravity waves, Adv. Appl. Mech. 22, 67–229 (1982). 24. E. Caponi, P. Saffman and H. Yuen, Instability and confined chaos in a nonlinear dispersive wave system, Phys. Fluids 25(12), 2159–2166 (1982). 25. K. Dysthe, Note on modification to the nonlinear Schr¨ odinger equation for application to deep water waves, Proc. Roy. Soc. Lond. A 369, 105–114 (1979). 26. M. Stiassnie and L. Shemer, Energy computations for evolution of class I and II instabilities of Stokes waves, J. Fluid Mechanics 174, 299–312 (1987). 27. T. Yasuda and N. Mori, Roles of sideband instability and mode coupling in forming a water wave chaos, Wave Motion 26(2), 163–185 (1997). 28. M. Su, Three-dimensional deep-water waves. Part 1. Experimental measurement of skew and symmetric patterns, J. Fluid Mechanics 124, 73–108 (1982). 29. N. Mori, T. Yasuda and K. Kawaguchi, Breaking effects on waves statistics for deepwater random wave train, in Waves97 (ASCE, 1997), pp. 316–328. 30. T. Yasuda and N. Mori, High order nonlinear effects on deep-water random wave trains, Int. Symp. Waves-Physical and Numerical Modelling, Vol. 2, Vancouver (1993), pp. 823–832. 31. N. Mori and P. Janssen, On kurtosis and occurrence probability of freak waves, J. Phys. Oceanogr. 36(7), 1471–1483 (2006). 32. M. A. Tayfun and J.-M. Lo, Nonlinear effects on wave envelope and phase, J. Waterway, Port, Coastal Ocean Eng. 116(1), 79–100 (1990). 33. N. Mori and T. Yasuda, A weakly non-Gaussian model of wave height distribution for random wave train, Ocean Eng. 29(10), 1219–1231 (2002). 34. N. Mori, P. Liu and T. Yasuda, Analysis of freak wave measurements in the Sea of Japan, Ocean Eng. 29(11), 1399–1414 (2002). 35. M. Longuet-Higgins and O. Phillips, Phase velocity effects internary wave interactions, J. Fluid Mechanics 12, 133–135 (1962).

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36. A. Kimura, Probability of the freak wave appearance in a 3-dimensional sea condition, Proc. 24th Int. Conf. Coastal Eng., Vol. 1, ASCE (1994), pp. 356–369. 37. Y. Goda, Random Seas and Design of Maritime Structures, 2nd Edn. (World Scientific, 2000). 38. I. Lavrenov, The wave energy concentration at the Agulhas current off South Africa, Natural Hazard 17, 117–127 (1998). 39. A. Osborne, M. Onorato, M. Serio and S. Bertone, The nonlinear dynamics of rogue waves and holes in deep water gravity wave trains, Phys. Lett. A 275(5–6), 386–393 (2000). 40. S. Haver, Evidences of the existence of freak waves, in Rogue Waves, eds. M. Olagnon and G. Atanassoulis (IFREMER, 2001), pp. 129–140. 41. M. Olagnon and G. Athanassoulis (eds.), Rogue Waves 2000 (IFREMER, 2000). 42. K. Hasselmann, On the nonlinear energy transfer in gravity-wave spectrum. I. General theory, J. Fluid Mechanics 12, 481–500 (1962). 43. V. Krasitskii, On reduced equations in the Hamiltonian theory of weakly nonlinear surface waves, J. Fluid Mechanics 272, 1–20 (1994). 44. F. Edgeworth, On the representation of statistical ferquency by a series, J. Roy. Statist. Soc. 70, 102–106 (1907). 45. M. A. Tayfun, Statistics of nonlinear wave crests and groups, Ocean Eng. 33(11–12), 1589–1622 (2006). 46. F. Fedele and M. A. Tayfun, Explaining freak waves by a stochastic theory of wave groups, 25th Int. Conf. Offshore Mechanics and Arctic Engineering 1 (2006). 47. M. Onorato, A. R. Osborne, M. Serio, L. Cavaleri, C. Brandini and C. T. Stansberg, Extreme waves, modulational instability and second order theory: Wave flume experiments on irregular waves, Eur. J. Mech. B/Fluids 25(5), 586–601 (2006). 48. N. Mori, M. Onorato, P. A. E. M. Janssen, A. R. Osborne and M. Serio, On the extreme statistics of long crested deep water waves: Theory and experiments, J. Geophys. Res., Ocean 112, C09011 (2007), doi: 10.1029/2006JC004024. 49. H. Socquet-Juglard, K. Dysthe, K. Trulsen, H. Krogstad and J. Liu, Probability distribution of surface gravity waves during spectral changes, J. Fluid Mechanics 542, 195–216 (2005). 50. O. Gramstad and K. Krulsen, Influence of crest and group length on the occurrence of freak waves, J. Fluid Mechanics 582, 463–472 (2007).

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Chapter 7

Short-Term Wave Statistics Akira Kimura Department of Social Systems Engineering Faculty of Engineering, Tottori University, 680-8552 Koyama Minami 4-101, Tottori, Japan [email protected] This chapter deals with the short-term statistical properties of irregular sea waves. It begins with a description of an early study by Rice [Bell Syst. Tech. J. 23, 282–332; 24, 46–156 (1944, 1945)] about the relation between wave spectra and statistics of wave amplitudes. Next, an explanation is given of studies which deal with the effect of spectrum width and wave nonlinearity on wave statistics as well as with joint statistical properties between wave heights and periods. Finally, the chapter closes with a look at the statistics of wave direction, length of wave crest, and spatial maximum amplitude (which are the 3D wave’s properties) as well as the time series of wave height.

7.1. Introduction Waves in sea conditions have certain irregular properties. These comprise properties such as wave heights and wave periods. For many years, engineers have known that it is not sufficient to know only statistically representative values such as the mean value. Larger wave heights can induce damage on coastal structures while longer wave periods may induce resonance phenomena. To understand their overall statistical properties, engineers and researchers have tried to clarify the statistical characteristics of populations for these properties with the intention of finding out their probability density function (pdf). In particular, their pdf have been studied in relation to the wave spectrum. The study of this development was initiated primarily by Rice.20 The application of this idea in coastal engineering was started notably by Longuet-Higgins.15 He developed pdf for surface elevation and gradient, maxima and minima of water surface, number of zero-crossings along a line, etc. Of course, sailors have carried out visual observations of waves since ancient times. From the theoretical and empirical knowledge of wave statistics, two types of statistical study have appeared: shortterm and long-term wave statistics. Since waves must be stationary stochastic to 151

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study short-term statistics, the duration of concern is 1 or 2 days at most. On the other hand, long-term statistics cover waves for time periods of at least dozens of years. In this type, the time series of water surface elevation is not used. Instead, representative wave properties such as significant wave height and period are utilized. The wave height/period is estimated for coastal structures within their lifetime from long-term statistics, for example. The effectiveness of results from long-term statistics is heavily dependent on short-term statistics. Both are inseparable for the efficient use of wave statistics in coastal engineering.

7.2. Rice Model Rice20 proposed a model of the Fourier sum for irregular signals. The irregular signal I(t) is expressed by the summation of trigonometric functions as I(t) =

N


(an cos σn t + bn sin σn t) =

n=1

N


cn cos(σn t + εn ),

(7.1)

n=1

in which σn is the angular frequency and εn (= tan−1 (−bn /an )) is the phase of the nth component wave, respectively. an , bn , and cn are amplitudes of nth component waves which are distributed normally with zero mean. Their variances are  2 cn 2 2
an  =
bn  = = S(σ)dσ , (7.2) 2 where
 means the ensemble mean, S(σ) is the energy spectrum of I(t). Rice defined the envelope function for I(t) as I(t) =

N


cn cos(σn t + εn ) = Ic (t) cos σt − Is (t) sin σt ,

(7.3)

n=1

where σ is the mean frequency of S(σ), and Ic (t) and Is (t) are Ic (t) =

N


cn cos(σn t − σt + εn ),

n=1

Is (t) =

N


cn sin(σn t − σt + εn ) .

(7.4)

n=1

The envelope function R(t) is R(t) = {Ic2 (t) + Is2 (t)}1/2 .

(7.5)

The nth component wave in Eq. (7.4), cn cos(σn t − σt + εn ) or cn sin (σn t − σt + εn ), has a U-shape pdf of finite width.14 From the central limit theorem, the infinite sum (N → ∞) of the trigonometric functions, Ic (t) and Is (t), are distributed normally. Their joint pdf is   2 1 Ic + Is2 p(Ic , Is )dIc dIs = dIc dIs , exp − (7.6) 2πϕ0 2ϕ0

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since Ic (t) and Is (t) are independent (
Ic (t) · Is (t) = 0). ϕ0 is the variance of Ic (t), Is (t), and I(t), respectively:
Ic2  =
Is2  =
I 2  = ϕ0 .

(7.7)

The variables R(t), I(t), Ic (t), and Is (t) are represented as R, I, Ic , and Is , respectively. It is convenient to introduce the variable θ, where Ic = R cos θ,

Is = R sin θ .

By substituting Eq. (7.8) into Eq. (7.6), the joint pdf of R and θ becomes   R R2 dRdθ . p(R, θ)dRdθ = exp − 2πϕ0 2ϕ0

(7.8)

(7.9)

Since R and θ are independent and θ is distributed uniformly over the range 0 to 2π, the pdf of R is obtained as   R R2 p(R)dR = exp − dR . (7.10) ϕ0 2ϕ0 Figure 7.1(a) shows the relation between I and R. The spectrum S(σ) for I is the following normalized Wallops type (r = 10)8 :    −4   −r r σ σ exp . (7.11) · 1− S(σ) = σp 4 σp The maxima or minima of I almost coincide with R within a “zero-up/downcrossing” interval of I. If I takes extremes (∂I/∂t = 0) at t = t∗1 , t∗2 , . . . , wave amplitude can be approximated by R|t∗i , (i = 1, 2, . . .) [Fig. 7.1(b)]. The intervals between t∗1 , t∗2 , . . . are almost constant when the spectrum is very narrow. The pdf for R|t∗i is also the Rayleigh distribution approximately, since the local maxima/ minima is equal to the wave amplitude. Equation (7.10) becomes the pdf for wave amplitude14 approximately.

7.3. Pdf of Wave Heights 7.3.1. Nonlinear effect on the pdf Since the study by Longuet-Higgins,14 the pdf for wave amplitudes has been used extensively as pdf for wave heights. Many measurements and numerical simulations have supported the Rayleigh distribution for the wave height distribution. The standard form of the widely used Rayleigh distribution is   π H π H2 p(H)dH = exp − dH , (7.12) 2 H2 4 H2 in which H is the mean wave height. When the wave spectrum is very narrow, wave height distribution can be the Rayleigh distribution, since wave height can be approximated as H ≈ 2R .

(7.13)

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Fig. 7.1. (a) Wave profile I(t) (solid line) and its envelope R(t) (dotted line). (b) Wave profile I(t) (solid line) and envelope R(t) (dotted line) (small discrepancies between I(t) and R(t) at t = t∗1 , t∗2 , · · · ).

However, the zero-crossing wave height is defined by H = Il max − Il min , where Il max and Il min are the consecutive local maximum and minimum of the wave profile, respectively. The pdf for H may not be the Rayleigh distribution if the pdf for Il max and Il min are the Rayleigh distribution. Field measurements have shown that there are small discrepancies between the Rayleigh distribution p(H) and frequency distributions of measured data F (H): F (H) < p(H) for H < Hp and F (H) > p(H) for H > Hp , where p(Hp ) is a peak of p(H). In particular, as nonlinearity on the wave profile becomes prominent, this characteristic may bring a considerable effect on the pdf. Forristall3 used the following Weibull distribution to improve the goodness of fit:

α −ς , (7.14) P (ς) = exp β where P (ς) is the Weibull distribution function and α = 2.126, β = 8.42, and ς=

H∗ 1/2

m0

,

(7.15)

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in which H ∗ is the difference between the heights of consecutive wave crest and trough within a zero-crossing interval of the wave profile. Longuet-Higgins17 pointed out that the discrepancy should be improved using rms (root mean square) wave amplitude. The ith moment of a spectrum is determined as ∞ mi = σ i S(σ)dσ (i = 1, 2, . . .) . (7.16) 0

From the Khintchine theorem, the 0th moment is also defined in another way as 2 , m0 = ηrms

(7.17)

where η is the wave profile and ηrms is its rms. When the spectrum is narrow and the nonlinearity on the wave profile is negligible, (arms )2 , (7.18) 2 in which arms is the rms of wave amplitude a. The Rayleigh distribution function given by Longuet-Higgins is

 a2 P (a) = exp − 2 . (7.19) arms m0 =

a2rms can be replaced by 2m0 if the nonlinearity is negligible. He discussed the nonlinearity effect on the wave profile as follows. Applying the third-order Stokes wave theory, the 2nd moment of the periodic wave profile is derived from Eq. (7.17) as

 1 1 2 19 3077 2 4 6 (ak) − · · · , (7.20) m0 = a 1 − (ak) − (ak) − 2 2 12 720 in which k is the wave number. m0 for irregular waves is calculated by ∞ m0 = 2V (a)dp(a) ,

(7.21)

0

in which V (a) denotes the potential energy (2V = η 2 = m0 ) of the zero-crossing wave with an amplitude a given as Eq. (7.20), p(a) is the pdf of a. Applying Eq. (7.19), m0 becomes

  amax a2 2V (a)d exp − 2 m0 = +δ, (7.22) arms 0 in which amax is the maximum wave amplitude of nonbreaking waves. He used the same V for the waves (ak)2 ≥ 0.196. δ is the remainder and is exponentially small for the large value of a2max /a2rms . The new m0 is

 1 19 3077 (arms k)6 − · · · . (7.23) m0 = a2rms 1 − (arms k)2 − (arms k)4 − 2 2 30 Since arms k for irregular waves distributes roughly 0.158 < arms /k < 0.203,13 0.935 <

2m0 < 0.968 . a2rms

(7.24)

If wave profiles have a nonlinearity, rms wave amplitude between 2.066m0 < a2rms < 2.140m0 may give a correction for the Rayleigh distribution of wave amplitudes.

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7.3.2. Effect of spectrum width Since the zero-crossing method for irregular waves does not have a physical basis, a mathematical treatment is difficult except for the very narrow band spectrum. Cartwright and Longuet-Higgins2 derived a spectrum width parameter as

 m22 ε= 1− : 0 < ε < 1, (7.25) m0 m4 in which m0 , m2 , and m4 are determined by Eq. (7.16). Pdf of amplitudes (local maxima) is the Rayleigh distribution when ε → 0. If ε = 0, however, waves must be periodic. Longuet-Higgins17 investigated the effect of the spectrum width on pdf of wave amplitudes. He used a periodic wave profile with small disturbance given as
y = b cos(σt + ε) + an cos(σn t + εn ) , (7.26) n

in which an , σn , and εn are the amplitude, angular frequency, and phase of the disturbance, respectively. Their amplitudes are b >> an . The spectrum of the disturbances S  (σ) is defined as σn +dσ
σn

The rms of y is

2
yrms  = b2rms + 0

∞

1 2 a = S  (σn )dσ . 2 n

(7.27)



πσ  σ 2  sin2 S  (σ)dσ , + 2σ σ

(7.28)

and the 0th moment of the spectrum of y is ∞ ∞ b2rms  {S(σ) + S (σ)}dσ = S  (σ)dσ , + m0 = 2 0 0

(7.29)

where S(σ) is the spectrum of the periodic wave and brms is the rms of the periodic wave profile. From Eqs. (7.28) and (7.29), yrms is  ∞  2

 σ 2 2 πσ  = 2m0 + − 1 − cos (7.30)
yrms S  (σ)dσ. 2σ σ2 0 Using the relations   2   σ 2

σ σ    −1 + −1 −1 =2   σ σ σ2  

πσ  1 − cos π(σ/σ − 1) 2 − 1) π(σ/σ 2 cos = = + ···  2 4  2σ  ∞   i   (σ − σ) S (σ)dσ , µi = 

(7.31)

0

the next relation is obtained. 2
yrms (π 2 /8 − 1/2)µ2 µ1  − =1+ . 2m0 m0 σ m0 σ 2

(7.32)

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If the spectrum is symmetry for σ ¯, 2 
yrms =1− 2m0



1 π2 − 8 2

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157

 ν2 ,

(7.33)

µ2 m0 m2 − m21 = , m21 m0 σ 2

(7.34)

where ν2 =

in which ν is also the spectrum width parameter.18 Since b  an , yrms ≈ brms , Eq. (7.33) may give a correction for the Rayleigh distribution of wave amplitudes when the spectrum is not a narrow band. Tayfun25 also investigated the effect of spectrum width. A better definition for the zero-crossing wave height than Eq. (7.13) may be given as Hi = Rmi + Rmi+1 ,

(7.35)

where Rmi (i = 1, 2, . . .) are the amplitudes of the wave envelope when the wave profile I takes the local extreme values (t = t∗1 , t∗2 , . . .) [Fig. 1(b)]. The joint pdf for the consecutive amplitudes Rmi , Rmi+1 is     πκξ1 ξ2 π(ξ12 + ξ22 ) π2 ξ1 ξ2 p(ξ1 , ξ2 ) = exp − I , (7.36) 0 4(1 − κ2 ) 4(1 − κ2 ) 2(1 − κ2 ) in which I0 [ ] is the 0th order modified Bessel function, ξi = Rmi /(πm0 /2)1/2 , ξi+1 = Rmi +1 /(πm0 /2)1/2 . They are replaced as ξi = ξ1 and ξi+1 = ξ2 , respectively for simplicity. κ is the correlation parameter given by  1/2 , (7.37) k = ρ2 + λ2 in which ρ=

1 m0



∞ 0

τ S(σ) cos(σ − σ) dσ , 2

−1 ∞ τ λ= S(σ) sin(σ − σ) dσ , m0 0 2 m1 τ = , m0 where τ is the mean wave period. Normalized wave height is given as H ξ1 + ξ2 . ς= = Hm 2

(7.38)

Substituting the relation ξ2 = 2ς − ξ1 into Eq. (7.36) and the integration in terms of ξ1 from 0 to 2ς gives the pdf for ς: 2ς p(ξ1 , 2ς − ξ1 )dξ1 . (7.39) p(ς) = 0

Figure 7.2 shows p(ς) for two different wave spectra [Wallops-type, Eq. (7.11): r = 5 (dotted line), r = 10 (broken line)] and the Rayleigh distribution (solid line). Kimura et al.10 showed that the discrepancy in Fig. 7.2 comes from the difference between I and R at (t = t∗1 , t∗2 , . . .) [Fig. 7.1(b)].

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1.2 1.0

r=5 r = 10

p (ζ)

0.8

Rayleigh 0.6 0.4 0.2 0.0 0.0

0.5

1.0

Fig. 7.2.

1.5 ζ

2.0

2.5

3.0

Pdf of wave height.

7.4. Maximum Wave Height Zero-crossing wave amplitude in Eq. (7.19) extends the range from 0 to infinity if the number of waves is infinity. When the population of waves is finite, such as data from 20 min measurements every 2 h, maximum wave amplitude may be finite. If a time series of wave amplitudes a1 , a2 , . . . , aN , aN +1 , . . . , a2N , a2N +1 , . . . , a3N , a3N +1 . . . , a4N ,             Group No. 1

2

3

4

is divided into groups with N waves, the maximum values (amax ) from individual groups show irregular properties. The pdf of the maximum wave amplitude is derived by Longuet-Higgins.14 The probability that one wave amplitude a is greater than a0 is P (a0 ) [Eq. (7.19)]. When the size of the group is N , the probability that all the wave amplitudes are a < a0 is {1 − P (a0 )}N . The probability that one amplitude, at least, exceeds a0 is 1 − {1 − P (a0 )}N . Therefore, the probability that maximum wave amplitude lies in the range amax ∼ amax + da is 1 − {1 − P (amax )}N − 1 + {1 − P (amax + da)}N = −d[1 − {1 − P (amax )}N ] dP (amax ) = −N {1 − P (amax )}N −1 da . da The pdf for the maximum wave amplitude is     2  2  N −1 amax −amax amax exp p(amax ) = 2N 1 − exp − . arms arms a2rms

(7.40)

(7.41)

Its mean value is derived as E(amax ) = (ln N )1/2 + γ(ln N )−1/2 + · · · arms

(7.42)

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with an approximation   2 N −θ ≈ e−e , 1 − e−(amax /arms )

(7.43)

in which θ =



amax arms

2 − ln N ,

(7.44)

and γ is the Euler constant (γ = 0.5772 . . .). Maximum wave height E(Hmax )/Hrms is usually compared also with Eq. (7.42).

7.5. Zero-Crossing Wave Period The problem of deriving the pdf of zero-crossing wave periods is very difficult. Rice derived a pdf for the zero-crossing interval. He treated the interval where I(t) [Eq. (7.1)] crosses the 0 line upward at t = 0 and crosses the 0 line downward between τ and τ + dτ . 1 τ 1 τ τ p(τ ) = p0 (τ ) − p1 (r, τ )dr + p2 (r, s, τ )drds 1! 0 2! 0 0 1 τ τ τ p3 (r, s, u, τ )drdsdu + · · · , (7.45) − 3! 0 0 0 where p0 (τ ) is the probability that I(t) = 0 at t = 0 and I(t) = 0 lies between τ < t < τ + dτ , p1 (r, τ ) is the probability that I(t) = 0 at t = 0 and I(t) = 0 lies between r < t < r + dr and τ < t < τ + dτ (r < τ ), p2 (r, s, τ ) is the probability that I(t) = 0 at t = 0, and I(t) = 0 lies between s < t < s + ds, r < t < r + dr, and τ < t < τ + dτ (s < r < τ ), and so on. Since this series of integrations is inexcusable, Rice21 showed p(τ ) only applying the first term of Eq. (7.45) as p(τ ) =

dτ 2π

1/2

 −3/2 M23  2 ψ0 −  ψ0 − ψτ2 {1 + H cot−1 (−H)} , ψ0 H

(7.46)

with  2  2 −1/2 − M23 , H = M23 M22 where M22 and M23 are the cofactors of elements µ22 and µ23 of the matrix, 

µ11  µ21 M =  µ31 µ41

µ12 µ22 µ32 µ42

µ13 µ23 µ33 µ43

  µ14 ψ0  0 µ24  = µ34   ψτ ψτ µ44

0 −ψ0 −ψτ −ψτ

ψτ −ψτ −ψ0 0

 ψτ −ψτ   0  ψ0

(7.47)

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and 1 ψτ = 2π ψτ = − ψτ = −



∞

S(σ) cos στ dσ ,

0

1 2π 1 2π



∞

σS(σ) sin στ dσ ,

0

(7.48) ∞

σ 2 S (σ) cos στ dσ ,

0

ψ0 = ψτ |τ =0 ,

ψ0 = ψτ |τ =0 .

Equation (7.46) has a peak around the mean value of τ . The second and third peaks appear for larger τ . This is because the higher terms in Eq. (7.45) are neglected. No successful analytical solution for the zero-crossing wave period has been derived at the present time. Instead of wave periods, Longuet-Higgins16 derived the pdf for wave lengths. The wave profile of unidirectional irregular waves is expressed as ς = R cos {χ} ,

(7.49)

where χ = ux + φ,

χx = u + φx ,

(7.50)

in which R is the amplitude of the wave envelope, φ is the phase, u is the mean wave number in x-direction, and subscript x means a differentiation in terms of x. Joint pdf for χ and χx was derived by Longuet-Higgins15 as p (χ, χx ) = with m∗n



∞

=

µ∗2 = u=

(m∗0 /µ∗2 )

1/2

&3/2 , % 2 4π 1 + (χx − u) · m∗0 /µ∗2

(7.51)

E (u) un du ,

(7.52)

(n = 0, 1, . . .)

0



∞

2

E (u) (u − u) du ,

0 m∗1 /m∗0

,

(7.53) (7.54)

in which u is the wave number in the x-direction and E (u) is its spectrum. The event ς = 0 is equivalent to χ = (2r − 1/2) π or χ = (2r + 1/2) π (r: integer). The probability that χ = (2r − 1/2) π between x and x + dx is 1/2  ∞ 1 m∗2 H(χ)dx = p(χ, χx )χ=(2r−1/2)π |χx | dxdχx = dx . (7.55) 2π m∗0 −∞ Since the probability that χ = (2r + 1/2)π between x and x + dx is given by the same equation, Eq. (7.55), the probability that ς = 0 between x and x + dx is 2H(χ)dx. The average zero-crossing interval is  ∗ 1/2 1/2 m0 π 1 + ν ∗2 =π = , (7.56) ∗ m2 u

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with ν ∗2 =

m∗0 m∗2 − m∗2 1 = m∗2 1



µ∗2 u2 µ∗0

 .

(7.57)

If ν ∗ is small, ≈

π . u

(7.58)

The zero-up-crossing event of ς takes place when χ = (2r − 1/2)π with ∂χ/∂x > 0 or χ = (2r + 1/2)π with ∂χ/∂x < 0. The probability of the latter condition is

∞

−∞

p (χ, χx )χ=(2r+1/2)π |χx | dxdχx 1 = 4



m∗2 m∗0

1/2  −1/2   1 − 1 + ν ∗2 dx .

(7.59)

This probability is negligible for small values of ν ∗ . Between any two successive zero-crossing points x1 and x2 , χ must increase by π. Hence,   1 π = χ(x2 ) − χ(x1 ) = χx + 2 χxx + · · · . 2 The terms on the left of this equation, Taylor series expansion of χ(x2 ) around x = x1 is used. Since χxx = O(ν 2 ), χxxx = O(ν 3 ), . . . and they are very small, neglecting these terms, ≈

π . χx

(7.60)

Applying the same method by Rice,20 a mean zero-crossing number of ς per unit length, the conditional pdf of χx is p(χx )χ =

1 p(χ, χx )|χx | = H(χ) 2



m∗0 m∗2

1/2

|χx |(m∗0 /µ∗2 )1/2 , {1 + (χx − u)2 (m∗0 /µ∗2 )}3/2

(7.61)

in which |χx |/H(χ) is the probability that χ falls between x and x + dx. Since Eq. (7.61) does not involve χ, p(χx )χ directly becomes the pdf for χx . Substituting Eqs. (7.57), (7.58), and (7.60) into Eq. (7.61), p(ξ) =

1 , 2ν{1 + (ξ/ν ∗ )2 }3/2

(7.62)

in which ξ = ( − )/. If the frequency spectrum S(σ) is used instead of the wave number spectrum E(u), the pdf for the zero-crossing interval (half-wave period) is derived following the same method.

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7.6. Joint Pdf for Wave Height and Period Longuet-Higgins18 derived the joint pdf for wave amplitude and period. He used the following expression for the wave profile: ς = Re {A exp (iσt)} ,

(7.63)

where Re means a real part and A = a exp{iφ} ,

(7.64)

in which a is the wave amplitude, φ is the phase of A, and σ is the mean angular frequency given by m1 σ= , (7.65) m0 where m0 and m1 are calculated by Eq. (7.16). The phase of ς is χ = σt + φ. Differentiation of the phase with t becomes χt = φt + σ .

(7.66)

When the spectrum is very narrow, the envelope of ς changes very slowly and φt  σ. The wave period can be approximated as 2π 2π τ= = . (7.67) χt σt + φt Longuet-Higgins derived a joint pdf for a and φt : 



2 a2 a (1/µ0 + φ2t /µ2 ) , p(a, φt ) = exp − 2 (2πµ20 µ2 )1/2

(7.68)

where µ0 and µ2 are calculated with Eq. (7.31), and ν is calculated by Eq. (7.34). With new variables τ a , T = , R= τ (2m0 )1/2 Eq. (7.68) is expressed as ' ' 

2 2 2 ' ∂(a, φt ) ' ' = 2R L exp − R [1 + (1 − 1/T ) ] , p(R, T ) = p(a, φt ) '' ∂(R, T ) ' π 1/2 νT 2 ν2

(7.69)

in which L is the correction factor to neglect the p(R, T ) in the region T < 0, L=

2 . 1 + (1 + ν 2 )−1/2

(7.70)

Figure 7.3 shows the contour map of p(R, T ). Solid curves are contours for p(R, T )/pmax = 0.99, 0.90, 0.70, 0.50, 0.30, 0.10 from inside to outside, respectively. The marginal distribution for normalized wave amplitudes and periods are R/ν 2L p(R) = 1/2 R exp(−R2 ) exp(−β 2 )dβ, (7.71) π −∞

−3/2 (1 − 1/T )2 L . (7.72) 1+ p(T ) = 2νT 2 ν2

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Joint pdf for wave amplitude and period Eq. (7.69) (after Longuet-Higgins18 ).

Fig. 7.4.

Pdf for wave amplitude p(R) Eq. (7.71) (after Longuet-Higgins18 ).

Figures 7.4 and 7.5 show p(R) and p(T ), respectively. If ν = 0, p(R) is not the Rayleigh distribution. Peak of the p(R) slightly shifts to the larger side of R with increasing spectrum width ν. When ν = 0, p(T ) becomes a straight line (broken line, T : constant). Figure 7.6 shows the comparison of data and Eq. (7.69) by Goda.8 The number of waves within a grid of ∆H = 0.2H and ∆T = 0.2T are shown. H and T are the mean values of wave height and period, respectively. Many data sets, their correlation coefficients between H and T distributed from 0.19 to 0.25, are used. The spectrum width parameter in Eq. (7.69) is v = 0.26.

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Fig. 7.5.

Fig. 7.6.

Pdf for wave period p(T ) Eq. (7.72) (after Longuet-Higgins18 ).

Joint distribution of wave height and period (after Goda5 ).

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7.7. Statistical Properties of 3D Irregular Waves Statistical properties of waves, such as length and direction of wave crest, are important in the design of coastal structures. Wave height and period are determined from a record of point measurement in the ordinary method. In contrast, a 3D wave profile is necessary for the definition of wave crest length, direction, etc. As a result, very few studies have examined their properties. 7.7.1. Pdf for wave directions Kimura12 introduced a theoretical pdf for the wave direction. Figure 7.7 shows axes and wave crest lines (dotted line). Two wave gauges are installed along the y-axis. One is on the origin y1 and the other is on y2 (0, y0 ). Figure 7.8 shows the wave profiles measured at these points. ∆t1 , ∆t2 , ∆t3 , . . . are the time lags between corresponding wave crests. The phase lag of individual waves is determined as   ∆ti εi = 2π , (i = 1, 2, . . .) , (7.73) Ti in which Ti is the zero-crossing wave period. The angle between the crest line and y-axis (Fig. 7.7) is

 εi Li /2π −1 θi = sin , (i = 1, 2, . . .) , (7.74) y0 where Li is the wave length corresponding to the zero-crossing wave period Ti and θi is the direction of wave propagation measured from the x-axis.

Fig. 7.7.

Wave crest lines (dotted line) and axes.

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Fig. 7.8.

Wave profiles (solid line) at y1 and y2 .

Assuming y0 u < 1 in which u is the mean wave number in the x direction, wave profiles at these points are η1 (t) =

∞


ci cos(σi t + δi ) = ηc1 cos σt − ηs1 sin σt ,

(7.75)

ci cos(σi t + δi + vi y0 ) =ηc2 cos σt − ηs2 sin σt ,

(7.76)

i=1

η2 (t) =

∞
i=1

in which ci , σi , δi , and vi are the amplitude, the angular frequency, the initial phase, and the wave number in the y-direction of the ith component wave, respectively. σ is the mean angular frequency. ηc1 , ηs1 , ηc2 , and ηs2 are derived in the same way as Eq. (7.4). Their pdfs are the normal distribution with 0 mean, respectively. Their joint pdf is 1 p(ηc1 , ηs1 , ηc2 , ηs2 ) = (2π)2 (m20 − µ23 − µ24 )

1 2 − 2µ3 ηc1 ηc2 − 2µ4 ηc1 ηs2 × exp − (m0 ηc1 2(m20 − µ23 − µ24 )  2 2 2 + m0 ηs1 + 2µ4 ηs1 ηc2 − 2µ3 ηs1 ηs2 + m0 ηc2 + m0 ηs2 ) , (7.77)

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with 2 2 2 2  =
ηs1  =
ηc2  =
ηs2 , m0 =
ηc1

µ3 =
ηc1 ηc2  =
ηs1 ηs2  , µ4 =
ηc1 ηs2  = −
ηs1 ηc2  ,

(7.78)


ηc1 ηs1  =
ηs2 ηc2  = 0 . From the translations ηc1 = R1 cos(σt + φ1 ) = R1 cos(χ1 ) , ηs1 = R1 sin(σt + φ1 ) = R1 sin(χ1 ) , ηc2 = R2 cos(σt + φ2 ) = R2 cos(χ2 ) ,

(7.79)

ηs2 = R2 sin(σt + φ2 ) = R2 sin(χ2 ) , the joint pdf for R1 , R2 , χ1 , and χ2 is 1 p(R1 , R2 , χ1 , χ2 ) = (2π)2 (m20 − µ23 − µ24 )

1 (m0 (R12 + R22 ) × exp − 2(m20 − µ23 − µ24 )

 − 2R1 R2 [µ3 cos(χ1 − χ2 ) − µ4 sin(χ1 − χ2 )]) ,

(7.80)

in which Ri and χi are the amplitude and phase of ηi (i = 1, 2), respectively.15 The crest of the wave profile passes the origin of axes when χ1 = 2nπ (n = 0, 1, 2, . . .). The probability of the event that χ1 = 2nπ between t and t + dt is H(χ1 )dt = m1 /m0 dt. The conditional pdf for R1 , R2 , and χ2 when χ1 = 2nπ is ' σp(R1 , R2 , χ1 , χ2 ) '' p(R1 , R2 , χ2 | χ1 = 2nπ) = ' H(χ1 ) χ1 =2nπ

=

R1 R2 exp 2πm20 (1 − κ2 )



−

1 2 m20 (1

− κ2 )

[m0 (R12 + R22 )

 − 2R1 R2 (µ3 cos(2nπ − χ2 ) − µ4 sin(2nπ − χ2 ))] , (7.81) in which κ2 =

µ23 + µ24 ≈ 1 − y02 u2 γ∗2 m20

(7.82)

and γ∗ is the long-crestedness parameter.15 Putting χ2 = 2nπ + ε, pdf for ε is p(R1 , R2 , ε|χ1 = 2nπ) p(R1 , R2 )  ( 

R1 R2 κ R1 R2 κ cos(ε − β) 2πI0 , = exp m0 (1 − κ2 ) m0 (1 − κ2 )

p(ε|R1 , R2 ) =

(7.83)

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where I0 [ ] is the 0th order modified Bessel function and β = tan−1 (µ4 /µ3 ). β is an angle of the dominant wave direction from x-axis. From Eq. (7.74), the pdf for the wave direction from x-axis becomes 2π y0 cos θ L  (  

2πy0 R1 R2 ) R 1 R2 ) 2 cos 2 , 1 − y 1 − y sin θ − β 2πI × exp 0 ∗ ∗ m0 y∗2 m0 y∗2 L (7.84)

p(θ|R1 , R2 ) =

in which L is the wave length corresponding to σ, and * ) µ2 + µ2 y∗ ≈ 1 − κ2 = 1 − 3 2 4 . m0

(7.85)

If the interval between two wave gauges is infinitesimally small, i.e., y∗ → 0, R1 = R2 → R. Putting β = 0 and applying the relation z → ∞ ⇒ I0 (z) →

exp(z) , 2πz

(7.86)

Eq. (7.84) reduces to p(θ|R)dθ =

 1 R2 R cos θ √ exp − 2 sin2 θ dθ . 2γ∗ m0 γ∗ 2πm0

(7.87)

Figure 7.9 shows p(θ|R) for several R. R is the mean of R. The 3D spectrum is the Bretschneider–Mitsuyasu type with Mitsuyasu-directional function modified by Goda.8 The peakedness parameter for the directional function is Smax = 10. The pdf for larger R shows a narrower width. 7.7.2. Mean length of 3D wave crest The length of 3D irregular wave crests may affect the total wave force on coastal structures with finite width such as caisson-type breakwaters, for example. Since a definition of the wave crest length is difficult, there are very few studies about its statistical properties. Goda6 studied its statistics numerically. He calculated a spatial wave profile η(x, y) of 3D irregular waves. The black and white pattern in Fig. 7.10 shows  η(x, y) > ηth , Black area , (7.88) η(x, y) < ηth , Other , in which ηth is the threshold value (ηth = 0.1H1/3 in Fig. 7.10). The 3D wave spectrum is the Bretschneider–Mitsuyasu type (H1/3 = 2.0 m, T1/3 = 8.01 s) with Mitsuyasu-directional function modified by Goda, and Smax = 25. The pattern recognition method is applied to determine the wave crest lengths. Four threshold values ηth = 0.2 m, 0.5 m, 1.0 m, and 2.0 m were used. The calculated

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Pdf for wave directions: Eq. (7.87).

crest lengths were arranged in descending order. The top 0.4%, 1%, 5%, and 10% of total data for each ηth were used in the analysis, respectively. Figure 7.11 shows the distribution of the mean length of wave crests. The four lines in the figure show the results from the above four different data. 7.7.3. Pdf of the local maximum of 3D wave amplitudes Figure 7.12 shows a short crested spatial wave profile and a wave gauge. The fixedtype wave gauges usually fail to record the peak wave profile in the neighborhood. Wave height changes along a large object such as a ship, caisson-type breakwater, etc. If a large wave hits any part of these objects, it is recognized that the large wave attacked it regardless of its position. Since local damage on the caisson-type breakwater, for example, may lead to the whole failure, the size effect on the pdf becomes important. Kimura et al.11 showed the pdf of the maximum wave amplitudes within a finite interval. x , y  , and vertical z  -axes are taken in the 3D wave field. The vertical plane H is installed parallel to x -axis. Figure 7.13 schematically shows the envelope R(x ) (solid line) for the cross section of the 3D wave profile on H. A wave gauge is installed at x = 0. The width of the object is 2∆L (−∆L < x < ∆L). R∗ is the local maximum amplitude of R(x ) (−∆L < x < ∆L), and R(0) is the amplitude of the envelope at x = 0. ∆R is the difference between R∗ and R(0). ∆R is approximately

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Fig. 7.10.

Fig. 7.11.

Wave crests (after Goda6,7 ).

Mean length of wave crest (after Goda6,7 ).

given as '  ' ' R (0) ' ' ' ' R (0) ' ≤ ∆L : ' '  ' R (0) ' ' ' ' R (0) ' > ∆L

∆R = −

{R (0)}2 , 2R (0)

(7.89)

R (0)(∆L)2 , 2

R (0) ≥ 0 :

∆R = R (0)∆L +

R (0) < 0 :

∆R = −R (0)∆L +

R (0)(∆L)2 , 2

(7.90) (7.91)

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Fig. 7.12.

Fig. 7.13.

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3D wave profile and the vertical plane H.

Wave envelope (solid line) for 3D wave profile.

in which R (0) and R (0) are the first and second derivatives of R(x ) at x = 0, respectively. The probability that R∗ falls between c < R∗ < c + dR on condition R(0) is p[R (0), R (0) : R(0)]dS , (7.92) Prob(c < R∗ < c + dR)|R(0) = S

in which p[R (0), R (0) : R(0)] is a conditional joint pdf of R (0) and R (0) on condition R(0), and S is the region in which c < R∗ < c + dR in the R ∼ R plane. The probability that R∗ falls between c < R∗ < c + dR regardless of R(0) is Prob(c < R∗ < c + dR) = 0

c

{Prob(c < R∗ < c + dR)|R(0) }p[R(0)]dR(0) ,

(7.93)

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Fig. 7.14. Pdf of local maximum wave amplitudes (dotted lines) within three finite width ∆L = 0 (Rayleigh dist.), 0.05L1/3 , and 0.10L1/3 .

where p[R(0)] is the pdf of R at x = 0 (Rayleigh distribution). Connecting Eqs. (7.92) and (7.93), the pdf of the local maximum amplitudes within the finite width is given as R∗ p(R∗ )dR∗ = p[R(0), R (0), R (0)]dGdR(0) , (7.94) 0 

G



in which p[R(0), R (0), R (0)] is the joint pdf of R(0), R (0), and R (0), and G is the region in which the local maximum amplitude falls within R∗ and R∗ + dR∗ on condition R(0). Rice20 formulated p[R(0), R (0), R (0)]. Figure 7.14 shows p(R∗ ) (dotted lines). The spectrum is the Bretschneider– Mitsuyasu-type with H1/3 = 5.5 m and T1/3 = 10 s and Goda’s directional function with Smax = 10, the water depth is h/L1/3 = 1.0, and the half-object width is ∆L = 0.050L1/3 and ∆L = 0.10L1/3, respectively. The Rayleigh distribution (solid line ∆L = 0) is also shown in the figure. Broken lines are the Weibull distribution with m ≈ 2.5 (shape factor) shows good agreement when ∆L/L1/3 > 0.05 (h/L1/3 = 1.0, Smax = 10). 7.8. Statistics of the Time Series of Wave Heights Sawhney23 and Goda4 started to investigate the statistical properties of the time series of irregular wave heights, . . . , Hi−2 , Hi−1 , Hi , Hi+1 , Hi+2 , . . . ,

(7.95)

since this property may affect wave overtopping, slow drift oscillation of floating bodies, etc. This property is analyzed by applying the concept of “run.” Figure 7.15 shows a time series of wave heights. The run of high waves is defined by waves, which are Hi > H∗ consecutively, in which H∗ is the threshold wave height. In the case shown in Fig. 7.15, the length of the high wave run is 3.

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Fig. 7.15.

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Run of wave heights (high waves, low waves, and total run).

Goda4 gave the pdf for the run of high waves as p(j1 ) = pj1 −1 q ,

(7.96)

in which j1 is the length of the high wave run, p is the probability that the event H > H∗ takes place and q = 1 − p. Since the pdf for wave heights is the Rayleigh distribution, p and q are given as 

H2 , p = exp − ∗ 8m0 (7.97) q = 1−p. The mean and standard deviation of j1 are j1 =

1 , q

√

σ(j1 ) =

p , q

(7.98)

respectively. The recurrence interval (total run) of high waves is defined as the sum of the high wave run and the consecutive low wave run. The run of low waves is defined by waves, which are Hi < H∗ , consecutively. The total run in Fig. 7.15 is jt = 3 (j1 : high wave run) + 4 (j2 : low wave run) = 7. The pdf for the total run is pq p(jt ) = (pjt −1 − q jt −1 ) . (7.99) p−q The mean and standard deviation of jt are jt =

1 1 + , p q

σ(jt ) =

+

q p + 2, 2 q p

(7.100)

respectively. Equations from (7.96) to (7.100) are derived assuming that the wave height is independent: γ(Hm Hm+n ) = 0,

m, n = 1, 2, 3, . . . ,

(7.101)

where γ( ) is the correlation coefficient of wave heights. From observation, Rye22 showed that there is a correlation between consecutive wave

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heights γ(Hm Hm+1 ) = 0. From numerical calculations, Kimura9 showed that γ(Hm Hm+1 ) = 0 but γ(Hm Hm+n ) ≈ 0 (n ≥ 2). He assumed the time series of wave heights from a Markov chain9 and introduced the pdf for the high wave run: p(j1 ) = pj221 −1 (1 − p22 ) .

(7.102)

p22 is the transition probability from the event Hi > H∗ to Hi+1 > H∗ . Its mean and standard deviation are √ p22 1 j1 = , σ(j1 ) = . (7.103) 1 − p22 1 − p22 The pdf for the total run is p(jt ) =

(1 − p22 )(1 − p11 ) jt −1 (p22 − pj11t −1 ) . p22 − p11

(7.104)

p11 is the transition probability from the event Hi < H∗ to Hi+1 < H∗ . The mean and standard deviation of jt are 1 1 jt = + , 1 − p22 1 − p11 (7.105) 1/2

p11 p22 σ(jt ) = + , 2 (1 − p11 ) (1 − p22 )2 where p11 and p22 are given by H∗ H∗ p(H1 , H2 )dH1 dH2 / p11 = 0



p22 =

0 ∞ H∗



p(H1 )dH1 ,

0

∞ H∗

p(H1 , H2 )dH1 dH2 /

with p(H1 , H2 ) =

H∗



H12 + H22 2 (κ2 − 1)Hrms 

2H1 H2 . p(H1 ) = 2 exp − 21 Hrms Hrms 4H1 H2 exp 4 (1 − κ2 )Hrms

∞

H∗



 I0

(7.106) p(H1 )dH1 ,

2κH1 H2 2 (1 − κ2 )Hrms

 ,

(7.107) (7.108)

Equations (7.107) and (7.108) are the 2D Rayleigh and Rayleigh distributions, respectively. k is the correlation parameter. The relation between κ and the correlation coefficient γ(Hm Hm+1 ) is

 1 π 1 2 γ(Hm Hm+1 ) = E(κ) − (1 − κ )K(κ) − , (7.109) 1 − π/4 2 4 in which E( ) and K( ) are complete elliptic integrals of the first and second kind, respectively. Battjes et al.1 and Longuet-Higgins19 showed κ and the relation between κ and γ(Hm Hm+1 ) as   2  ∞ 2  ∞ 1 2 S (σ) cos στ0 dσ + S (σ) sin στ0 dσ (7.110) κ = m0 0 0

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Fig. 7.16. (a) Mean length of high wave runs (solid line: Eq. (7.103); circles: after Goda8 ). (b) Mean length of total runs (solid line: Eq. (7.105); circles: Goda used j2 for the total run, after Goda8 ).

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and κ2 ≈ γ(Hm Hm+1 ) κ2 < 0.6 , γ(Hm Hm+1 ) ≈ 1 −

1 − κ2 4−π

κ: very close to 1 .

Figures 7.16(a) and 7.16(b) show the comparison between j1 and Eq. (7.103), and jt and Eq. (7.105), respectively. The threshold wave height is H1/3 .

7.9. Remarks Statistical studies have shown us the entire structures of wave irregularities in sea conditions. A knowledge about their pdf has brought us an advance not only for designing coastal structures but also for understanding nearshore hydraulics of the wave and current system. However, we have to keep several things in mind when we use these wave statistics. The Rayleigh distribution for wave height is derived on the condition that the wave spectrum is narrow. The problem still remains to clarify a good correspondence between measured distributions and the Rayleigh distribution regardless of the spectrum. The assumption of the narrow band spectrum gives a considerable effect on the pdf of wave periods. Although the theoretical models have the spectrum parameter in the equation, we have to be careful about the stretch use for the spectrum with finite band width. The range of the pdf may be changed by wave dynamics such as wave breaking. Extensive application of the theoretical pdf is sometimes risky. The applications of the periodic wave theory for zero-crossing wave properties raise problems. There have been very few studies about the physical properties of zero-crossing waves, of wave period–wavelength/celerity, of wave height/period — water particle velocities, etc. Finally, we have to take into account the statistical properties for 3D irregular waves. Common definitions for wave properties, first of all, become necessary: for wave height, wave period, wavelength, celerity, wave direction, wave crest length, etc.

References 1. J. A. Battjes and G. Ph. van Vledder, Verification of Kimura’s theory for wave group statistics, Proc. 19th ICCE (1984), pp. 642–648. 2. D. E. Cartwright and M. S. Longuet-Higgins, The statistical distribution of the maxima of a random function, Proc. R. Soc. Lond. A 237, 212–232 (1956). 3. G. Z. Forristall, On the statistical distribution of wave heights in a storm, J. Geophys. Res. 83(C5), 2353–2358 (1978). 4. Y. Goda, Numerical experiments on wave statistics with spectral simulation, Report PHRI 9(3), 3–57 (1970).

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5. Y. Goda, On the joint distribution of measured wave heights and periods, Tec. Note of PHRI, No. 272, (1977) 19pp. (in Japanese). 6. Y. Goda, Statistics of wave crest lengths based on directional wave simulations, J. Offshore Mech. and Arctic Eng., Trans. ASME 116, 239–245 (1994). 7. Y. Goda, Numerical simulation of ocean waves for statistical analysis, J. Marine Res. 33, 5–14 (1999). 8. Y. Goda, Random Sea and Design of Maritime Structures, 2nd Edn. (World Scientific, 2000), p. 443. 9. A. Kimura, Statistical properties of random wave groups, Proc. 17th ICCE (1980), pp. 2955–2973. 10. A. Kimura and T. Ohta, Statistical property of the irregular wave heights defined by the zero-crossing method, Proc. Coastal Eng. 40, 146–150 (1993) (in Japanese). 11. A. Kimura and T. Ohta, Probability distribution of the maximum wave height along a sea wall with finite width, Proc. 25th ICCE (1996), pp. 2272–2283. 12. A. Kimura, Probability distribution of 3-D irregular wave directions, Proc. Waves 97, 295–306 (1997). 13. B. M. Lake and H. C. Yuen, A new model for nonlinear wind waves, 1, Physical model and experimental evidence, J. Fluid Mech. 88, 33–62 (1978). 14. M. S. Longuet-Higgins, On the statistical distribution of the heights of sea waves, J. Marine Res. IX(3), 245–266 (1952). 15. M. S. Longuet-Higgins, The statistical analysis of a random, moving surface, Phil. Trans. Roy. Soc. Ser. A 249(966), 321–387 (1957). 16. M. S. Longuet-Higgins, On the intervals between successive zeros of a random function, Proc. Roy. Soc. Ser. A 246, 99–118 (1958). 17. M. S. Longuet-Higgins, On the distribution of the heights of sea waves: Some effects of nonlinearity and finite band width, J. Geophys. Res. 85(C3), 1519–1523 (1980). 18. M. S. Longuet-Higgins, On the joint distribution of wave periods and amplitudes in a random wave field, Proc. Roy. Soc. Ser. A 389, 241–258 (1983). 19. M. S. Longuet-Higgins, Statistical properties of wave groups in a random sea state, Phil. Trans. Roy. Soc. Ser. A 312, 219–250 (1984). 20. S. O. Rice, The mathematical analysis of random noise, Bell System Tech. J. 23, 282–332, 24, 46–156 (1944). 21. S. O. Rice, Selected papers on Noise and Stochastic Processes, ed. N. Wax (Dover, 1945), pp. 133–294. 22. H. Rye, Wave group formation among storm waves, Proc. 14th ICCE (1974), pp. 164–183. 23. M. D. Sawhney, A study of ocean wave amplitudes in terms of the theory of runs and a Markov chain process, Tech. Rep. of New York University (1962), p. 29. 24. R. J. Sobey, Correlation between individual waves in a real sea states, Coastal Eng. 27, 223–242 (1996). 25. M. A. Tayfun, Effects of spectrum band width on the distribution of wave heights and periods, Ocean Eng. 10(2), 107–118 (1983).

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Chapter 8

Generation and Prediction of Seiches in Rotterdam Harbor Basins M. P. C. de Jong Formerly at Environmental Fluid Mechanics Section Delft University of Technology; Presently at Deltares |Delft Hydraulics P. O. Box 177, 2600 MH, Delft, The Netherlands J. A. Battjes Environmental Fluid Mechanics Section Delft University of Technology, P. O. Box 5048, 2600 GA Delft, The Netherlands A resume is presented of recent work that has identified the mechanism through which the seiches are generated that occasionally occur in Rotterdam Harbor basins, and that has led to the development and implementation of an operational warning system for the occurrence of significant seiche events. The generation is due to moving systems of atmospheric convection cells that can arise over the relatively warm water of the North Sea, behind a cold front. These cause fluctuations in wind speed and atmospheric pressure that can generate long waves at sea, as these cells move toward the coast. The average variance spectrum of these long-wave surface elevations is found to have an approximately f −1.5 tail. As these long waves at sea approach the harbor mouth, they can be resonantly amplified inside certain semi-closed basins. The ratio of seiche amplitudes in two different basins, derived through numerical simulations based on a 2DH mildslope model and an incident f −1.5 long-wave elevation spectrum, is in excellent agreement with the mean ratio derived from observations for the same locations. Finally, the principle of a possible system of prediction of the occurrence of seiche events is described. Its predictive potential is proven through a series of hindcasts. Based on these promising results, an operational forecasting system has been developed and implemented by the authorities involved.

8.1. Introduction Resonant harbor oscillations (seiches) occur in a number of basins of the Port of Rotterdam, The Netherlands, during rough weather episodes. Within the Rotterdam 179

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Fig. 8.1.

The Port of Rotterdam.

Harbor system, the highest seiches occur in the Caland Canal, with measured crest heights over 1 m at the closed end near Rozenburgse Sluis, ROZ (see Fig. 8.1). This semi-closed basin has a length of approximately 20 km and a depth of approximately 20 m. For almost all of the observed seiche events at ROZ, the dominant frequency (determined through wavelet analysis) equals the lowest eigenfrequency (quarter-wavelength mode), equivalent to a period of approximately 90 min. Smaller-amplitude seiches, though still quite significant, also occur in other harbor basins. The seiche oscillations in Rotterdam Harbor give rise to some practical concerns. First, water levels used for the design of dikes (in particular their crest height) are required by law to contain an allowance for seiches, superposed on the astronomical high water level (spring tide) and wind setup. The appropriate measure of this allowance is not well known. Second, seiches affect the design and management of a movable storm surge barrier (MSSB) in Rotterdam Waterway (Fig. 8.1), which protects the low areas behind it against flooding. The MSSB must be closed during extreme storm conditions. In that case a semi-closed basin is created on its seaside. Seiches are expected to occur in this temporary basin, with a dominant eigenperiod of approximately 30 min. These contribute to the failure probability of the barrier, but their expected heights were not known. Third, the Rotterdam Port Management needs to guarantee a minimum depth (“nautical depth”) in the approach channel and throughout the harbor for deepdraught vessels. To this end, water levels are predicted operationally including tides and wind-induced setup. Until recently, no method was available for inclusion of seiche events in this short-range prediction, despite their practical importance: the trough of a seiche could cause ships to run aground in case of a narrow predicted tidal window. The aforementioned practical applications in which seiches play a role have led to a recent study on the origin of the seiche events in Rotterdam. The present chapter summarizes the findings. Results are also presented of an analysis of the response of Rotterdam Harbor basins to incoming wave energy in the seiche frequency band using measured and theoretical ratios of seiche amplitudes at the closed ends of two basins. Finally, a new system of routine prediction of seiche events is described,

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based on the newly identified generation mechanism, as well as an indication of its forecasting skill. The above-mentioned research and results have been more fully described by de Jong and Battjes.1,2

8.2. Measurements 8.2.1. Data acquisition The sea surface elevation measurements used in this study were obtained at three offshore platforms located approximately 20–40 km from the harbor mouth: Europlatform (EUR), Lichteiland Goeree (GOE), and Meetpost Noordwijk (MPN). Inside the harbor, surface elevation measurements are available from multiple locations (more than 10 key-locations are monitored). These have been taken continuously since 1995 with floaters, sampled at a 60 s interval. In addition, sea surface temperature measurements have been obtained from 1995 to 2001 at platforms located several hundreds of kilometers to the north and north-west of the harbor mouth in the North Sea (along the path of the relevant storm systems approaching the Port of Rotterdam). These were available as 24-h averaged daily values. Vertical profiles of air temperature and other meteorological parameters, obtained from balloon measurements, were available at the North Sea at Ekofisk platform every 12 or 24 h. 8.2.2. Data analysis To identify seiche events, records of the surface elevations at ROZ were Fourier band-passed filtered, passing the seiche frequency band (0.1–2.0 mHz). A wavelet analysis based on the Morlet wavelet3−6 has been applied to the filtered surface elevation data and meteorological data. This technique is suited for the detection of fluctuations that come in bursts. It has been used to identify the seiche events inside the harbor and the corresponding periods of increased levels of low-frequency energy at sea. Following first identification from the wavelet spectra, the time intervals with seiche events were reviewed in more detail.

8.3. Origin of the Seiches in Rotterdam Harbor 8.3.1. Generation of seiche events by atmospheric convection cells In this section, a short description is given of the mechanism generating the long waves at sea that cause seiche episodes inside the harbor (as mentioned above, a more detailed description can be found in the work of de Jong and Battjes1,2 ). Meteorological measurements and visual inspection of weather charts show that all seiche episodes in Rotterdam Harbor (arbitrarily defined as a seiche-event in which the crest height exceeded 0.25 m at ROZ, a criterion that was applied in several previous studies of seiches at this location) coincided with low-pressure

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systems with cold fronts crossing the southern North Sea toward the Dutch coast. However, the converse does not hold: not all such cold front passages give rise to seiche events. Analysis of measurements and numerical simulations showed that the seiches in Rotterdam Harbor result from resonant amplification of oscillations at sea, which in turn are generated by fluctuations in atmospheric pressure and wind speed following a moving cold front. For the large majority of the seiche events, these fluctuations are caused by atmospheric convection cells that can arise in the area behind a cold front. These cells are a mesoscale atmospheric phenomenon, with typical horizontal spatial scales of 30–100 km.7 Conditions conducive to the formation of convection cells occur when cold (polar) air moves over relatively warm sea water, which is likely to occur in fall and winter (also the seasons of significant seiche episodes). Especially in the area behind the cold front, the relatively warm sea water causes heating of the air in the lower layers, which results in an unstable lower atmosphere that causes air to rise to higher altitudes in narrow plumes, (self-)organized into a more or less regular pattern of convection cells. If sufficient moisture is present in the air, condensation of water vapor can locally cause a release of latent heat, further driving the circulation of air. Narrow bands of clouds can then be formed surrounding clear skies, which can be seen on satellite images (an example is shown in Fig. 8.2) and, in case of precipitation,

Fig. 8.2. Satellite image of infrared frequency band (10:12 GMT, 11 January 1995), taken during a seiche event in the Port of Rotterdam. The image shows a pattern of convection cells that passes the southern North Sea from the northwest. The location of the Port of Rotterdam is indicated by the black and white dot. Source image: Dundee Satellite Receiving Station, Dundee University, Scotland.

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also on radar precipitation images. These satellite and precipitation images showed that the episodes of convection cells passing the southern North Sea more or less coincided with the seiche episodes. The convection cells cause more or less spatiallyperiodic fluctuations in atmospheric pressure and wind speed, which in a fixed point will manifest themselves as temporal fluctuations as the cells are advected with the weather system. Because the system of convection cells moves, these fluctuations can generate low-frequency waves at sea, which in turn can cause seiches in the harbor basins. A sketch of the sea surface elevation response to the wind speed changes caused by the convection cells is depicted in Fig. 8.3. A 1D model based on the linearized shallow water equations was applied for a transect across the North Sea along the main path of the convection cells, forced by sinusoidal variations in atmospheric pressure and wind speed translating at a constant speed, with values that are typical for those observed. Negligibly low responses were found in the more remote, deeper parts of the North Sea. However, the model showed that the fluctuations in wind speed and atmospheric pressure caused by the passing convection cells are particularly efficient at generating long waves with significant amplitudes in the relatively shallow part of the North Sea, the southernmost 400 km near the Dutch coast. This is because in this region, with typical depths of approximately 30 m, the phase speed of shallow-water waves is approximately 17 m/s, i.e., near the advection velocity of the system of cold front and convection cells (typically 15–20 m/s). The result is a near-resonant response of the long waves at sea to the moving pattern of atmospheric convection cells, with amplitudes up to 1–2 dm near the Dutch coast. These waves can in turn be amplified in semi-closed harbor basins through (a quarter-wave) resonance, occasionally forming seiches with observed amplitudes (crest heights) up to a meter. 8.3.2. Energy density spectrum of surface elevation at sea Most previous studies of the seiches in the Port of Rotterdam focused on the relative responses in different basins. To this end, and for lack of better information, those studies assumed a uniform incident energy density spectrum (a “white noise” spectrum). Veraart8 first studied the spectra of measured surface elevations at the

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southern North Sea in the seiche frequency band, obtained during seiche events at ROZ. Based on a relatively small number of seiche episodes he found that, on average, the variance density spectrum in the seiche frequency band approximately followed a f −1.2 trend. From June 2000, numerous processed and validated surface elevation data have become available for a number of storm events. These have now been used to determine the spectrum in the seiche frequency band at sea in more detail (results described here are based on measurements from the platform EUR). The variance density spectra have been estimated from filtered records (0.1– 2.0 mHz) of 4096 min (212 values) approximately centered around the seiche-event, covering the complete storm duration of each event. Spectra of a number of events normalized by the energy content of the seiche frequency band are shown in Fig. 8.4. A least-squares fit of a power law to the averaged normalized spectral densities results in a f −1.5 trend.

8.4. Seiche Amplitudes at Different Locations 8.4.1. Seiche events in different basins of the Port of Rotterdam A total of 49 seiche events were registered at ROZ in the studied time interval (1995–2001). For this location, each such event is defined as a period of enhanced

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seiche activity with amplitudes (crest heights) exceeding 0.25 m. During most of these events, the response at the lowest eigenperiod of the Caland Canal (90 min) was dominant. Surface elevation measurements at Europe Harbor (EH, see Fig. 8.1) have also been studied for the same time interval as the observations at ROZ (1995–2001). These time series were filtered for the frequency band 0.3–2.0 mHz in order to isolate the main eigenfrequencies of this (shorter) harbor basin. Based on a wavelet analysis, and using 0.1 m as an amplitude threshold, 46 seiche events at EH were identified, all at the dominant eigenperiod of 50 min. Measurement data at ROZ were available during 43 of the 46 events that were identified at EH. Of these 43 events at EH, 86% coincided with a seiche-event at ROZ. This percentage depends to some extent on the thresholds for the identification of a seiche-event. These have been chosen subjectively: in case a threshold of 0.15 m is chosen for the identification of seiche events at EH, 100% of the events at EH coincide with a seiche-event at ROZ (with amplitude exceeding 0.25 m). Conversely, of the 49 events at ROZ, 82% coincided with a seiche-event at EH. In this case too, this result depends on the threshold of the definition of a seicheevent. In case a threshold of 0.40 m is applied at ROZ, 100% of the events at ROZ coincided with an event at EH (with a threshold of 0.10 m). The fact that large-amplitude seiche events simultaneously occur in basins of different dimensions indicates that the same mechanism generates seiches in basins of different natural frequencies. 8.4.2. Empirical amplitude ratio Surface elevation signals from ROZ and EH have been filtered for the complete seiche frequency band (0.1–2.0 mHz) in order to compare the crest heights during coinciding seiche events in the Caland Canal and the Europe Harbor. For this purpose, a typical local seiche amplitude could be based on, for example, the standard deviation of the filtered surface elevation signals. However, the time span for which this value needs to be determined (for each event separately) is not obvious. This is problematic in view of the transient nature of the seiche events. Therefore, the maximum crest height of an event at each location has been chosen. Moreover, in order to get a more robust estimate, the average of the five largest crest heights and the five largest trough depths of an event has also been used. The latter parameter is taken as a typical average seiche amplitude of an event. Figure 8.5 shows these average seiche amplitudes for the 30 coinciding events in 1995–2001 for ROZ and EH for which the average amplitude could be determined from at least five crests and five troughs. These data have a correlation coefficient of 0.92. A linear trend line was least-squares fitted to the data, indicating that on average the seiche amplitudes at EH are a factor of approximately 0.56 smaller than those at ROZ. A scatter plot based on the single maximum crest height for all coinciding events (not shown) gave a similar result, though with some more scatter (correlation coefficient 0.7).

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Fig. 8.5. Scatter plot of average seiche amplitudes (the average of the five highest crests and five deepest troughs of an event) of coinciding seiche events at ROZ and Europe Harbor. Trend line based on least-squares estimate.

8.4.3. Theoretical amplitude ratio The calculation of theoretical seiche amplitude ratios (r) at two locations a and b is based on a combination of the energy supply at the harbor mouth and its amplification according to
 S(f )Ra2 (f )df , (8.1) rab =  S(f )Rb2 (f )df where S(f ) is the (normalized) energy spectrum incident from the sea at the harbor mouth and Ra (f ) and Rb (f ) are the amplitude amplification spectra for the considered locations a and b. These had been determined previously using a 2DH mildslope numerical wave model (PHAROS).9 Effects of bottom stress, inflow losses, and seaward radiation were included. Applying these response functions and the empirical seiche energy spectrum with a f −1.5 tail, the theoretical amplitude ratio between EH and ROZ is found to be 0.57, nearly equal to the value of 0.56 that was estimated from the measurements. This result, based on 30 seiche events, supports the validity of the theoretical approach, and it confirms that the energy density at sea in the seiche frequency band typically follows a f −1.5 trend, not just in the three events from which it was estimated.

8.5. Prediction of Occurrence of Seiche Events 8.5.1. Principle of prediction and first verification As noted in the Introduction, practical needs related to shipping or to the movable storm surge barrier require real-time knowledge of expected occurrences of seiches

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in Rotterdam Harbor. However, operational prediction of these cannot be based on measurements at available platforms at the North Sea, because the response at the remote stations is too low, whereas the nearby platforms are too close to allow a useful prediction window. Since moving convection cells are at the basis of the generating mechanism of seiche events in Rotterdam Harbor, circumstances needed for these cells to arise can, in principle, be used to predict the occurrence, and also the absence, of seiche events. The feasibility of this approach was shown by de Jong and Battjes2 using historical data. Based on this promising academic result, a pilot study has been set up using real-time data and weather predictions, as a precursor of a fully operational system. This confirmed that the method was indeed feasible and that it had sufficient forecasting skill to warrant implementation of an operational version. Such a system, in which predictions with windows of 12 h and 24 h are provided to relevant parties throughout the harbor, has now been in operation since 2005 and so far has functioned to satisfaction. The basis for the prediction system is as follows. Convection cells of sufficient strength arise only if the lower atmosphere is (sufficiently) unstable. The criterion for this is that the adiabatic lapse rate (rate of decrease in temperature from the sea surface upward) exceeds some threshold, the value of which depends on the moisture content because of its consequence for possible cloud formation. In case the lapse rate is smaller than 5.5 to 7.5◦ C/km, the lower atmosphere is stable; here 6.5 C/km is used as a representative critical value. Between approximately 10◦ C/km and (say) 6.5◦ C/km, it is conditionally unstable (depending on the moisture content), and a temperature lapse rate larger than 10◦ C/km results in an unstable atmosphere.10 Prediction of the lapse rate and moisture content using multi-layered atmospheric models is part of routine weather forecast systems. This, therefore, also allows operational prediction of the occurrence of significant seiche events on a routine basis, just like common weather prediction. To test the feasibility of this approach, several months of atmospheric temperature and pressure data as well as sea surface water temperature data have been used for “predictions” in a hindcast mode, comparing the outcomes to observations of occurrences (or not) of seiche events. Results of a three-month period are shown here as a typical example of the outcome. The lapse rate can in practice be estimated using the air temperature at sea level (provisionally taken equal to the sea surface water temperature) and at some higher elevation. In meteorology, standard reference levels in the boundary layer are used corresponding to an atmospheric pressure of 925 hPa (average altitude of about 700 m) and 850 hPa (average altitude of about 1500 m). At the latter altitude, cloud formation can occur, which would result in the release of latent heat, implying influence of moisture content on the possible generation of convection cells. Indeed, preliminary tests showed that correct predictions as well as false predictions are found if the average altitude of the reference pressure of 850 hPa is used and moisture is not taken into account. However, generally speaking, the air will not become saturated below the level of the 925 hPa pressure. Therefore, cloud formation will not occur below this level. In this case, no information about moisture content of the air is required because

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the uncertain range of conditional stability is excluded. A lower limit of the lapse rate for unconditional instability of 10◦ C/km then applies. The following “predictions” use this criterion and the estimated temperature gradient between sea level and the observed level of 925 hPa pressure. For comparison, results based on the average elevation of this pressure level (approximately 700 m) are shown as well. The latter would be simpler operationally, albeit at the cost of enhanced errors in the verification of the temperature difference criterion. The results of the aforementioned approach are illustrated by presenting the data of January till March 1999. This time interval was selected because the quality of the data was relatively high during these months (only a small percentage of missing data due to, e.g., atmospheric soundings that were not available on certain days). Weather maps have been used to identify the days during which cold fronts have passed the North Sea that resulted in a flow toward the Dutch coast (including possible successive days during which this flow direction continued). This is obviously needed for seiche events to be generated in Rotterdam Harbor. Figure 8.6 shows the results. The bars in the top panel show the observed difference between the air temperature at the (observed) altitude of the reference pressure 925 hPa (at platform EKO) and the sea surface water temperature (at platform AUK) for each day (not restricted to days of a front passage). Gaps are present for those days for which no data were available. The lower panel shows the

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same data, now restricted to days during which cold fronts have passed the North Sea toward the Dutch coast. The straight dashed lines in both panels indicate the critical temperature difference for an unstable atmosphere based on the average altitude of the pressure level of 925 hPa (∆T = 7◦ C). The wiggly curves in both panels indicate the value of the critical temperature difference for an unstable atmosphere from day to day, based on the temperature at the measured altitude of the reference pressure of 925 hPa for each day. A seiche-event is predicted for those days in the lower panel for which the observed temperature difference (the bar) exceeds the critical one (indicated by the wiggly curve). The asterisks indicate the days for which seiche events have been identified at ROZ. (Consecutive days with seiche events indicated represent a single event that started during the first day and ended on the last day indicated.) The bottom panel shows that an unstable atmosphere is found during a number of potentially seiche-prone cold-front situations, correctly predicting a seicheepisode. All the seiche events that occurred in this time interval are correctly predicted (no misses). A stable atmosphere is found during most other days indicated in the bottom panel and no convection cells are expected during these days (regardless of the availability of moisture in the air), which agrees with the fact that no seiche events occurred. Nevertheless, a limited number of false predictions is also found (17 January and 28 February). On these days, the temperature difference just slightly exceeded the critical value. The resulting false predictions could therefore easily be due to the criterion not being strict enough, or to the spatial variability of the temperature of the sea surface water and the air at the reference levels: although the nominal criterion is met locally, this does not necessarily have to be representative for the complete southern North Sea region since only observations from sparse locations are used. This will not be the case during the correctly predicted seiche events in view of the fact that the prediction on these days is based on relatively large temperature differences, which far exceed the critical values. 8.5.2. Operational seiche prediction The evidence presented above indicates that conditions for the existence of convection cells, together with the information regarding the passage and direction of cold fronts, can be used to identify and predict seiche-prone situations. In an operational setting, the predicted weather chart for the next 24 h is available with relatively high reliability, and the predicted air temperature can be obtained from numerical weather prediction models. The moisture content of the air could also be taken into account. Moreover, a more thorough analysis of the stability of the lower atmosphere is possible, since information of the complete vertical and for the complete southern North Sea area is available (possibly avoiding false alarms due to criteria that are only met locally). This makes the use of specific reference levels unnecessary. Therefore, all relevant parameters are available for the prediction of seiche-prone situations with a 24-h prediction window, sufficient for the port

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authorities and for the closure management of the movable storm surge barrier of Rotterdam (which typically require prediction windows of, say, 6 h). Based on the aforementioned findings, a pilot project was implemented, set up by the Ministry of Public Works and Waterways in cooperation with the Royal Dutch Meteorological Institute (KNMI) for operational prediction of the occurrence of seiche events in the Port of Rotterdam, using parameter values derived from operational numerical weather prediction models. This pilot project has shown that the system has sufficient predictive skill for operational use. Such a system has since been implemented on an operational basis. Experience shows that all major seiche events (above the seiche-amplitude threshold) have been successfully predicted. However, in some cases a false prediction was made in the sense that a warning was issued while no significant seiches occurred. This is a drawback that needs repair, but even as it is, the system has merit because it does not miss significant events. The false alarms mentioned above are related to an important limitation of the above-described method, viz, the fact that so far it only predicts seiche-prone situations but not the expected energy level of the seiche-event. In case additional parameters are taken into account, such as the (range in) sizes of the atmospheric convection cells, the advection velocity, the excess of the temperature difference above the critical value (not just the question whether it exceeds a critical value or not), and the direction of movement of the weather systems, prediction of the seiche energy level might become possible, for example, through the use of a neural network that can be trained with a set of observed values of these quantifiable parameters. This may open the possibility of operational forecasting of seiche-prone conditions together with expected seiche crest heights. In that case, a seiche prediction (warning) would only be issued when seiche-conducive atmospheric conditions are expected that can lead to a seiche-event with amplitudes which justify a warning.

8.6. Conclusions Fluctuations in atmospheric pressure and wind speed, induced by mesoscale atmospheric convection cells that can arise following a cold front passing southward over the North Sea, generate low-frequency gravity waves in the North Sea. These low-frequency waves can subsequently induce a resonant response in coastal harbor basins, thus causing the so-called seiche events. During episodes of enhanced seiche activity in Rotterdam Harbor, the surface elevation energy spectrum at sea in the frequency band corresponding to the typical eigenfrequencies in Rotterdam Harbor basins, say, 0.1–2.0 mHz, varies with frequency (f ) approximately as f −1.5 . An episode of enhanced wave energy at sea in the seiche frequency band causes simultaneous resonant seiche responses in basins with different natural frequencies. The ratio between seiche amplitudes in two different basins can be accurately predicted using an f −1.5 excitation spectrum and numerically calculated spectral transfer functions. Operational prediction of the occurrence of a seiche-episode cannot be based on measurements at available platforms at sea, because at the more remote stations

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the signal is too weak, and for the nearby stations the available prediction window is too short to be of practical use. A hindcast study has shown that the occurrence of significant seiche episodes in Rotterdam Harbor can be accurately predicted on the basis of the prediction of the occurrence of meteorological conditions conducive to the formation of mesoscale atmospheric convection cells behind a cold front. This is presently realized using output of air temperatures and moisture content from numerical atmospheric models that are used in operational weather forecasting. The outcomes of this prediction method are used by the port authorities for ship traffic control and, after further development and testing, can eventually be incorporated into the closuremanagement system of the storm surge barrier in Rotterdam Waterway.

Acknowledgments The first author acknowledges the financial support from the Dr Ir Cornelis Lely Foundation. A part of this study was financed by the Directorate-General for Public Works and Water Management, South-Holland Directorate (Contract number: ZHA 23735). Furthermore, the authors thank the Directorate-General for Public Works and Water Management and the Municipal Harbor Organization, Rotterdam for providing the surface elevation measurements at sea and in the harbor, respectively. The authors thank Dr A. J. van Delden of Utrecht University, The Netherlands, for his extensive information on convection cells. Atmospheric sounding data have been obtained from the Department of Atmospheric Science, University of Wyoming, USA. Wavelet software was provided by Torrence and Compo,6 through URL: http://paos.colorado.edu/research/wavelets/. The figures in this chapter are reprinted from de Jong and Battjes,2 with permission from Elsevier.

References 1. M. P. C. de Jong and J. A. Battjes, Low-frequency sea waves generated by atmospheric convection cells, J. Geophys. Res. 109(C1), C01011 (2004), doi: 10.1029/2003JC001931. 2. M. P. C. de Jong and J. A. Battjes, Seiche characteristics of Rotterdam Harbour, Coastal Eng. 51(5–6), 373–386 (2004). 3. J. Morlet, G. Arens, E. Furgeau and D. Giard, Wave propagation and sampling theory — Part 2: Sampling theory and complex waves, Geophysics 47(2), 222–236 (1982). 4. M. Farge, Wavelet transforms and their applications to turbulence, Ann. Rev. Fluid Mech. 24, 395–457 (1992). 5. S. Mallat, A Wavelet Tour of Signal Processing (Academic Press, San Diego, 1998). 6. C. Torrence and G. P. Compo, A practical guide to wavelet analysis, Bull. Am. Meteorol. Soc. 79, 61–78 (1998). 7. B. W. Atkinson and J. W. Zhang, Mesoscale shallow convection in the atmosphere, Rev. Geophys. 34(4), 403–431 (1996).

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8. C. Veraart, Events of Long Waves in the North Sea, Technical Report RIKZ-94.034, Rijkswaterstaat/RIKZ and Delft University of Technology, Delft, The Netherlands (1994). 9. P. Van den Bosch and J. J. Veldman, Europoortkering-Beerdam, Technical Report H2438, WL | Delft Hydraulics, Delft, The Netherlands (1995). 10. J. R. Holton, An Introduction to Dynamic Meteorology, 3rd edn. (Academic Press, London, 1992).

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Chapter 9

Seiches and Harbor Oscillations Alexander B. Rabinovich P.P. Shirshov Institute of Oceanology, Russian Academy of Sciences 36 Nakhimovsky Prosp., Moscow, 117997 Russia Department of Fisheries and Oceans, Institute of Ocean Sciences 9860 West Saanich Road, Sidney, BC, Canada V8L 4B2 [email protected] [email protected] This chapter presents an overview of seiches and harbor oscillations. Seiches are long-period standing oscillations in an enclosed basin or in a locally isolated part of a basin. They have physical characteristics similar to the vibrations of a guitar string or an elastic membrane. The resonant (eigen) periods of seiches are determined by basin geometry and depth and in natural basins may range from tens of seconds to several hours. The set of seiche eigen frequencies (periods) and associated modal structures are a fundamental property of a particular basin and are independent of the external forcing mechanism. Harbor oscillations (coastal seiches) are a specific type of seiche motion that occur in partially enclosed basins (bays, fjords, inlets, and harbors) that are connected through one or more openings to the sea. In contrast to seiches, which are generated by direct external forcing (e.g., atmospheric pressure, wind, and seismic activity), harbor oscillations are mainly generated by long waves entering through the open boundary (harbor entrance) from the open sea. Energy losses of seiches in enclosed basins are mostly due to dissipative processes, while the decay of harbor oscillations is primarily due to radiation through the mouth of the harbor. An important property of harbor oscillations is the Helmholtz mode (pumping mode), similar to the fundamental tone of an acoustic resonator. This mode is absent in a closed basin. Harbor oscillations can produce damaging surging (or range action) in some ports and harbors yawing and swaying of ships at berth in a harbor. A property of oscillations in harbors is that even relatively small vertical motions (sea level oscillations) can be accompanied by large horizontal motions (harbor currents), resulting in increased risk of damage of moored ships, breaking mooring lines as well as affecting various harbor procedures. Tsunamis constitute another important problem: catastrophic destruction may occur when the frequencies of arriving tsunami waves match the resonant frequencies of the harbor or bay. Seiches, as natural resonant oscillations, are generated by a wide variety of mechanisms, including tsunamis, seismic ground waves, internal ocean waves, and 193

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jet-like currents. However, the two most common factors initiating seiches are atmospheric processes and nonlinear interaction of wind waves or swell. At certain places in the World Ocean, waves due to atmospheric forcing (atmospheric gravity waves, pressure jumps, frontal passages, squalls) can be responsible for significant, even devastating harbor oscillations, known as meteorological tsunamis. They have the same temporal and spatial scales as typical tsunami waves and can affect coasts in a similar damaging way.

9.1. Introduction Seiches are long-period standing oscillations in an enclosed basin or in a locally isolated part of a basin (in the Japanese literature they are commonly known as “secondary oscillations (undulations) of tides”).30,56,59 The term “seiches” apparently originated from the Latin word siccus which means dry or exposed (from the exposure of the littoral zone at the down-swing).31,95 Free-surface oscillations, known as seiches or seiching in lakes and harbors or as sloshing in coffee cups, bathtubs, and storage tanks, have been observed since very early times; a vivid description of seiching in Lake Constance, Switzerland, was given in 1549, and the first instrumental record of seiches obtained in 1730 in Lake Geneva.46,95 Korgen34 describes seiches as “the rhythmic, rocking motions that water bodies undergo after they have been disturbed and then sway back-and-forth as gravity and friction gradually restore them to their original, undisturbed conditions.” These oscillations occur at the natural resonant periods of the basin (so-called “eigen periods”) and physically are similar to vibrations of a guitar string and an elastic membrane. The resonant (eigen) periods of seiches are determined by the basin geometry and depth94,95 and in natural basins may be from a few tens of seconds to several hours. The oscillations are known as natural (or eigen) modes. The mode with the lowest frequency (and thus, the longest period) is referred to as the fundamental mode.41 The set of seiche eigen frequencies (periods) and associated modal structures are a fundamental property of a particular basin and are independent of the external mechanism forcing the oscillations. In contrast, the amplitudes of the generated seiches strongly depend on the energy source that generates them, and can therefore have pronounced variability.31 Resonance occurs when the dominant frequencies of the external forcing match the eigen frequencies of the basin. Harbor oscillations (coastal seiches according to Ref. 19 are a specific type of seiche motion that occur in partially enclosed basins (gulfs, bays, fjords, inlets, ports, and harbors) that are connected through one or more openings to the sea.41,94 Harbor oscillations differ from seiches in closed water bodies (for example, in lakes) in three principal ways68 : (1) In contrast to seiches generated by direct external forcing (e.g., atmospheric pressure, wind, and seismic activity), harbor oscillations are mainly generated by long waves entering through the open boundary (harbor entrance) from the open sea.

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(2) Energy losses of seiches in closed basins are mostly associated with dissipation, while the decay of harbor oscillations is mainly due to radiation through the mouth of the harbor. (3) Harbor oscillations have a specific fundamental mode, the Helmholtz mode, similar to the fundamental tone of an acoustic resonator.54 This modes is absent in closed basins. Because harbor oscillations can produce damaging surging (or range action) — yaw and swaying of ships at berth in a harbor — this problem has been extensively examined in the scientific and engineering literature.4,11,12,36,41,46,47,62,67,68,70,78,79, 94,95 One of the essential properties of oscillations in harbors is that even relatively small vertical motions (sea level oscillations) can be accompanied by large horizontal water motions (harbor currents); when the period of these motions coincides with the natural period of sway, or yaw of a moored ship, further resonance occurs, which can result in considerable motion and possible damage of a moored ship.82,94 Harbor oscillations can also break mooring lines, cause costly delays in loading and unloading operations at port facilities, and seriously affect various harbor procedures.79,80 Tsunamis constitute another important problem that have greatly stimulated investigations of harbor oscillations. Professor Omori (Japan) was likely the first to notice in 1902 that the dominant periods of observed tsunami waves are normally identical to those caused by ordinary long waves in the same coastal basin (see Ref. 30). His explanation was that the bay or portion of the sea oscillates like a fluid pendulum with its own period, i.e., the arriving tsunami waves generate similar seiches as those generated by atmospheric processes and other types of external forcing (see also Ref. 30). Numerous papers on the spectral analysis of tsunami records for various regions of the world ocean have confirmed this conclusion.13,48,69,74,75,84,90 Catastrophic destruction may occur when the frequencies of arriving tsunami waves match the resonant frequencies of the harbor or bay. One of the best examples of strong tsunami amplification due is the resonant response of Port Alberni (located at the head of long Alberni Inlet on the Pacific coast of Vancouver Island, Canada) to the 1964 Alaska tsunami.28,54 9.2. Hydrodynamic Theory The basic theory of seiche oscillations is similar to the theory of free and forced oscillations of mechanical, electrical, and acoustical systems. The systems respond to an external forcing by developing a restoring force that re-establishes equilibrium in the system. A pendulum is a typical example of such a system. Free oscillations occur at the natural frequency of the system if the system disturbed beyond its equilibrium. Without additional forcing, these free oscillations retain the same frequencies but their amplitudes decay exponentially due to friction, until the system eventually comes to rest. In the case of a periodic continuous forcing, forced oscillations are produced with amplitudes depending on friction and the proximity of the forcing frequency to the natural frequency of the system.84

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9.2.1. Long and narrow channel Standing wave heights in a closed, long and narrow nonrotating rectangular basin of length, L, and uniform depth, H, have a simple trigonometric form35,95 : ζ(x; t) = A cos kx cos ωt,

(9.1)

where ζ is the sea level elevation, A is the wave amplitude, x is the along-basin coordinate, t is time, k = 2π/λ is the wave number, λ is the wavelength, ω = 2π/T is the angular wave frequency, and T is the wave period. The angular frequency and wave number (or the period and wavelength) are linked through the following well-known relationships: ω = kc, λ T = , c

(9.2a) (9.2b)

√ where c = gH is the long-wave phase speed and g is the gravitational acceleration. The condition of no-flow through the basin boundaries (x = 0; x = L) yields the wave numbers: k=

π , L

2π , L

3π nπ ,..., , L L

(9.3)

which are related to the specific oscillation modes [Fig. 9.1(a)], i.e., to the various eigen modes of the water basin. The fundamental (n = 1) mode has a wavelength equal to twice the length of the basin; a basin oscillating in this manner is known as a half-wave oscillator.34 Other modes (overtones of the main or fundamental

Closed basin

Open-ended basin

n=1

n=0

n=2

n=1

n=3

n=2

n=4

n=3

0.00

1.00 0.00

L

1.00

L

Fig. 9.1. Surface profiles for the first four seiche modes in closed and open-ended rectangular basins of uniform depth.

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√ Table 9.1. Normalized periods, Tn∗ = Tn gH/(2L), for a closed and open-mouth rectangular basin of uniform depth. Mode Basin

n=0

n=1

n=2

n=3

n=4

— 2

1 2/3

1/2 2/5

1/3 2/7

1/4 2/9

Closed Open-mouthed

“tone”) have wavelengths equal to one-half, one-third, one-fourth, and so on, of the wavelength of the fundamental mode (Fig. 9.1(a), Table 9.1). The fundamental mode is antisymmetric: when one side of the water surface is going up, the opposite side is going down. Maximum sea level oscillations are observed near the basin borders (x = 0; x = L), while maximum currents occur at the nodal lines, i.e., the lines where ζ = 0 for all time. Positions of the nodal lines are determined by xm n =

(2m − 1)L , 2n

m, n = 1, 2, . . . ; m ≤ n.

(9.4)

Thus, for n = 1, there is one nodal line: x11 = L/2 located in the middle of the basin; for n = 2, there are two lines: x12 = L/4 and x22 = 3L/4; for n = 3: x13 = L/6, x23 = 3L/6 = L/2 and x33 = 5L/6 and so on. The number of nodal lines equals the mode number n [Fig. 9.1(a)], which is why the first mode is called the uninodal mode, the second mode is called binodal mode, the third mode the trinodal mode, etc.31,95 The antinode positions are those for which ζ attains maximum values, and are specified as xjn =

jL , n

j = 0, 1, 2, . . . , n.

(9.5)

For example, for n = 2 there are three antinodal lines: x02 = 0, x12 = L/2, and = L. Maximum currents occur at the nodal lines, while minimum currents occur at the antinodes. Water motions at the seiche nodes are entirely horizontal, while at the antinodes they are entirely vertical. The relationships (9.2) and (9.3) yield the well-known Merian’s formula for the periods of (natural) in a rectangular basin of uniform depth68,78 : x22

2L , Tn = √ n gH

(9.6)

where n = 1, 2, 3, . . .. Merian’s formula (9.6) shows that the longer the basin length (L) or the shallower the basin depth (H), the longer the seiche period. The fundamental (n = 1) mode has the maximum period; other modes — the overtones of the main fundamental — “tone” — have periods equal to one-half, one-third, one-fourth, and so on, of the fundamental period (Fig. 9.1(a), Table 9.1). The fundamental mode and all other odd modes are antisymmetric, while even modes are symmetric; an antinode line is located in the middle of the basin.

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The structures and parameters of open-mouth basins are quite different from those of closed basins. Standing oscillations in a rectangular bay (harbor) with uniform depth and open entrance also have the form (9.1) but with a nodal line located near the entrance (bay mouth). In general, the approximate positions of nodal lines are determined by the following expressions (Fig. 9.1(b), Table 9.1): xm n =

(2m + 1)L , 2n + 1

m, n = 0, 1, 2, . . . ; m ≤ n,

(9.7)

j, n = 0, 1, 2, . . . ; j ≤ n.

(9.8)

while antinodes are located at xjn =

2jL , 2n + 1

In particular, for n = 1 there are two nodal lines: x01 = L/3 and x11 = L and two antinodal lines: x01 = 0 and x11 = 2L/3; for n = 2 there are three nodal lines: x02 = L/5, x12 = 3L/5 and x22 = L, and three antinodal: x02 = 0, x12 = 2L/5, and x22 = 4L/5. The most interesting and important mode is the lowest mode, for which n = 0. This mode, known as the Helmholtz mode, has a single nodal line at the mouth of the bay (x = L) and a single antinode on the opposite shore (x = L). The wavelength of this mode is equal to four times the length of the bay; a basin oscillating in this manner is known as a quarter-wave oscillator.34 The Helmholtz mode, which is also called the zeroth mode,a the gravest mode and the pumping mode (because it is related to periodic mass transport — pumping — through the open mouth,36,41 is of particular importance for any given harbor. For narrow-mouthed bays and harbors, as well as for narrow elongated inlets and fjords, this mode normally dominates. The periods of the Helmholtz and other harbor modes can be approximately estimated as84,95 Tn =

4L √ , (2n + 1) gH

for mode n = 0, 1, 2, 3, . . . .

(9.9)

Using (9.9) and (9.6), the fundamental (Helmholtz) mode in a rectangular open√ mouth basin of uniform depth H is found to have a period, T 0 = 4L/ gH, which √ is double the period of the gravest mode in a similar but closed basin, T 1 = 2L/ gH. Normalized periods of various modes (for n ≤ 4) are shown in Table 9.1. Expressions (9.4)–(9.9), Table 9.1, and Fig. 9.1 are all related to the idealized case of a simple rectangular basin of uniform depth. This model is useful for some preliminary estimates of seiche parameters in closed and semi-closed natural and artificial basins. Analytical solutions can be found for several other basins of simple geometric form and nonuniform depth. Wilson95 summarizes results that involve common basin shapes (Tables 9.2 and 9.3), which in many cases are quite a In many papers and text books,8,94,95 this mode is considered the “first mode”. However, it is more common to count nodal lines only inside the basin (not at the entrance) and to consider the fundamental harbor mode as the “zeroth mode”.41,68,78,80,84 This approach is physically more sound because this mode is quite specific and markedly different from the first mode in a closed basin.

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good approximations to rather irregular shapes of natural lakes, bays, inlets, and harbors. The main concern for port operations and ships and boats in harbors is not from the sea level seiche variations but from the strong currents associated with the seiche. As noted above, maximum horizontal current velocities occur at the nodal lines. Therefore, its locations in the vicinity of the nodes that are potentially most risky and unsafe. Maximum velocities, Vmax , can be roughly estimated as84 :
g , (9.10) Vmax = An H where An is the amplitude of the sea level oscillation for the mode. For example, if An = 0.5 m and H = 6 m, Vmax ≈ 0.64 m/s. 9.2.2. Rectangular and circular basins If a basin is not long and narrow, the 1D approach used above is not appropriate. For such basins, 2D effects may begin to play an important role, producing compound or coupled seiches.95 Two elementary examples, which can be used to illustrate the 2D structure of seiche motions, are provided by rectangular and circular basins of uniform depth (H). Consider a rectangular basin with length L (x = 0, L) and width l (y = 0, l). Standing oscillations in the basin have the form35,41 ζ(x, y; t) = Amn cos

nπy mπx cos cos ωt, L l

where m, n = 0, 1, 2, 3, . . .. The eigen wave numbers (kmn ) are  1/2 mπ 2  nπ 2 kmn = + , L l and the corresponding eigen periods are78 −1/2   m 2  n 2 2 Tmn = √ + . L l gh

(9.11)

(9.12)

(9.13)

For n = 0 expression (9.13) becomes equivalent to the Merian’s formula (9.6); the longest period corresponds to the fundamental mode (m = 1, n = 0) which has one nodal line in the middle of the basin. In general, the numbers m and n denote the number of nodal lines across and along the basin, respectively. The normalized eigen ∗ periods Tmn = Tmn /T10 and spatial structure for the different modes are shown in Table 9.4. For oscillations in a circular basin of radius r = a, it is convenient to use a polar coordinate system (r, θ) with the origin in the center: x = r cos θ,

y = r sin θ,

where θ is the polar angle. Standing oscillations in such basins have the form ζ(x, y; t) = Js (kr)(As cos sθ + Bs sin sθ) cos ωt,

(9.14)

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Table 9.2.

Modes of free oscillations in closed basins of simple geometric shape and constant width (after Ref. 95). Periods of free oscillation Mode ratios Tn /T1 values for n

Basin type Description

Dimensions

2

3

4

2L/(gh0 )1/2

1.000

0.500

0.333

0.250

h(x) = h0 (1 − 2x/L)

1.305[2L/(gh0 )1/2 ]

1.000

0.628

0.436

0.343

h(x) = h0 (1 − 4x2 /L2 )

1.110[2L/(gh0 )1/2 ]

1.000

0.577

0.408

0.316

h(x) = h0 (1 − 4x2 /L2 )2

1.242[2L/(gh0 )1/2 ]

1.000

0.686

0.500

0.388

L 0

Triangular (isosceles)

h0

h0

L0 h(x)

h0

Quartic

h0

x

∠ 2

L Parabolic

x

h(x)

0 h(x)

x

∠ 2

L

0 h(x)

∠ 2

x

h(x) = h0

(Continued)

A. B. Rabinovich

Rectangular

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Fundamental T1

Profile equation

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Table 9.2.

(Continued ) Periods of free oscillation Mode ratios Tn /T1 values for n

Basin type Dimensions

Profile equation

2

3

4

1.640[2L/(gh1 )1/2 ]

1.000

0.546

0.377

0.288

1.000

0.546

0.377

0.288

L1 /L2 = 1/2 4L2 /(gh2 )1/2

1.000

0.500

0.250

0.125

L1 /L2 = 1/3 3.13L2 /(gh2 )1/2 L1 /L2 = 1/4 2.73L2 /(gh2 )1/2 L1 /L2 = 1/8 2.31L2 /(gh2 )1/2

1.000 1.000 1.000

0.559 0.579 0.525

0.344 0.367 0.371

0.217 0.252 0.279

L 0

Triangular (right-angled)

h1

Trapezoidal

h0

h(x)

h1

h(x) = h1 x/L

L

0

h(x) = h0 + mx m = (h1 − h0 )/L

x h(x) h1

L1 Coupled, rectangular

x

L2 0

x h(x) h 2

h(x) = h1 (x < 0) h(x) = h2 (x < 0) h1 /h2 = 1/4

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Description

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Modes of free oscillations in semi-closed basins of simple geometric shape (modified after Ref. 95).

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202

Table 9.3.

Periods of free oscillation Mode ratios Ts /T1 [n = (s + 1)/2]

Basin type Description

Dimensions 0

0

b(x)

b1

x

h(x)

n=0

1

2

3

h(x) = h1

2.000[2L/(gh1 )1/2 ]

1.000

0.333

0.200

0.143

h(x) = h1 x/L

2.618[2L/(gh1 )1/2 ]

1.000

0.435

0.278

0.203

h(x) = h1 (1 − x2 /L2 )

2.220[2L/(gh1 )1/2 ]

1.000

0.409

0.259

0.189

b(x) = b1 x/L h(x) = h1

1.308[2L/(gh1 )1/2 ]

1.000

0.435

0.278

0.230

x

h1

0

b(x)

b1

x

0

h1

h(x)

L Rectangular

Triangular L

x

b(x)

b1

0

x

h1

h(x)

L Rectangular

Semi-parabolic

b1

L Triangular

x

0

h(x)

h1

x

Rectangular

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L

b(x)

0

0

A. B. Rabinovich

L x

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L Rectangular

L Rectangular

Fundamental T0

Profile equation

(Continued)

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Table 9.3.

(Continued ) Periods of free oscillation Mode ratios Ts /T1 [n = (s + 1)/2]

Basin type Description

Dimensions

1

2

3

b(x) = b1 x/L h(x) = h1 x/L

1.653[2L/(gh1 )1/2 ]

1.000

0.541

0.374

0.283

b1 /L = 2 = 4/3 =1 = 2/3

2.220[2L/(gh1 )1/2 ]

1.000

0.707 0.554 0.447 0.317

0.578 0.493 0.468 0.455

0.378 0.323 0.264 0.185

h(x) = h1 (1 − r 2 /L2 )

2.220[2L/(gh1 )1/2 ]

1.000

0.707

0.578

0.500

L

b(x)

0

b1

x

0

h1

h(x)

L Triangular

x

Triangular L

0

b1

x

0

L Semi-elliptic

h1

x

Semi-paraboloidal

L

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n=0

Seiches and Harbor Oscillations

Fundamental T0

Profile equation

L 0 b 1

0

r h(x)

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h1 r

Semi-circular

Semi-paraboloidal

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Mode parameters for free oscillations in uniform depth basins of rectangular and circular geometric shape.

Rectangular basin (l = 0.5L)

1

2

1

2

1

2

0.500

2

0

0.500

0

1

0.447

1

1

0.354

2

1

+

+

+ + + +

+

+

0

2

0.243

1

2

ωa/c

Tsm /T10

—

—

1.841

1.000

—

—

3.054

0.603

+

0.628

—

3.832

0.480

0.719

—

5.331

0.345

0.766

—

6.706

0.275

0.343

0.787

7.016

0.262

0.449

0.820

8.536

0.216

+

+

+

+

0.250

r2

+ +

+

+ +

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0

0

Relative period

+

1

1

Normalized frequency

r1

2a

+

1

1.000

l

+

Circular nodal lines

+

0

m

+

0

s

+

2

Tmn /T10

+

0

L

Mode forms

+

1

Relative period

9.75in x 6.5in

n

Mode numbers

Mode forms

A. B. Rabinovich

m

Circular basin

+

Mode numbers

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Table 9.4.

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where Js is the Bessel function of an order s, As and Bs are arbitrary constants, and s = 0, 1, 2, 3, . . .35,41 These oscillations satisfy the boundary condition: Js (kr)|r=a = Js (ka) = 0.

(9.15)

The roots of this equation determine the eigen values ksm (m, s = 0, 1, 2, 3, . . .), with corresponding eigen modes described by Eq. (9.14) for various k = ksm . Table 9.4 presents the modal parameters and the free surface displacements of particular modes. As illustrated by Table 9.4, there are two classes of nodal lines, “rings” and “spokes” (diameters). The corresponding mode numbers m and s give the respective exact number of these lines. Due to mass conservation, the mode (0, 0) does not exist in a completely closed basin.41 For the case s = 0, the modes are symmetrical with respect to the origin and have annular crest and troughs.35 In particular, the first symmetrical mode (s = 0, m = 1) has one nodal ring r = 0.628a (Table 9.4). When the central part of the circular basin (located inside of this ring) is going up, the marginal part (located between this ring and the basin border) is going down, and vice versa. The second symmetrical mode (s = 0, m = 1) has two nodal rings: r = 0.343a and r = 0.787a. For s > 0, there are s equidistant nodal diameters located at an angle ∆θ = π/s from each other; i.e., 180◦ for s = 1, 90◦ for s = 2, 60◦ for s = 3, etc. Positions of these diameters are indeterminate, since the origin of θ is arbitrary. The indetermatability disappears if the boundary deviates even slightly from a circle. Specifically, the first nonsymmetrical mode (s = 1, m = 0) has one nodal diameter (θ = π/2), whose position is undefined; but if the basin is not circular but elliptical, the nodal line would coincide with either the major or minor axis, and the corresponding eigen periods would be unequal.35 The first unsymmetrical mode has the lowest frequency and the largest eigen period (Table 9.4); in this case, the water sways from one side to another relative to the nodal diameter. This mode is often referred to as the “sloshing” mode.78 Most natural lakes or water reservoirs can support rather complex 2D seiches. However, the two elementary examples of rectangular and circular basins help to understand some general properties of the corresponding standing oscillations and to provide rough estimates of the fundamental periods of the basins. 9.2.3. Harbor resonance Let us return to harbor oscillations and consider some important resonant properties of semi-closed basins. First, it is worthy to note that expressions (9.7)–(9.9) and Table 9.3 for open-mouth basins give only approximate values of the eigen periods and other parameters of harbor modes. Solutions of the wave equation for basins of simple geometric forms are based on the boundary condition that a nodal line (zero sea level) is always exactly at the entrance of a semi-closed basin that opens onto a much larger water body. In this case, the free harbor modes are equivalent to odd (antisymmetric) modes in a closed basin, formed by the open-mouth basin and its

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mirror image relative to the mouth.b However, this condition is not strictly correct because it does not take into account wave energy radiation through the mouth into the open sea. The exact solutions may be obtained based on the Sommerfeld radiation condition of free wave radiation through the open boundary.35,41 Following application of the appropriate mouth correction (α), the nodal line is located close to but outside the entrance. In other words, the effect of this correction is to increase the effective length of the basin.95 The mouth correction depends on two parameters: the basin aspect ratio q = l/L, which relates the width of the basin (l) to its length (L); and the aperture ratio ϑ = b/l, in which b is the actual width of the mouth. Mathematical determination of α is rather complicated but, as a rule, it increases with increases of q and ϑ. For example, the fractional correction to Eq. (9.9) for the fundamental mode in a rectangular basin of uniform depth and open mouth (ϑ = 1.0) is determined as30,95     q 3 πq α= − γ − ln , (9.16) π 2 4 where γ = 0.5772 is Euler’s constant. Roughly speaking, radiation into the external basin and the mouth correction are important when the semi-closed basin is broad and has a large open entrance, and negligible when the basin is long and narrow (i.e., when q is small); in the latter case, expressions (9.7)–(9.9), as well as those presented in Table 9.3, are quite accurate. The character of natural oscillations in a bay or harbor is strongly controlled by the aperture ratio ϑ = b/l, which can vary from ϑ = 1.0 to ϑ = 0.0. These two asymptotic cases represent a fully open harbor and a closed basin, respectively. It is evident that the smaller is ϑ (i.e., the smaller the width of the entrance), the slower water from the external basin (open sea) penetrates into the harbor. Thus, as ϑ decreases, the periods of all harbor modes for n ≥ 1 in Table 9.1 increase, tending to the periods of the corresponding eigen modes for a closed basin, while the period of the fundamental (Helmholtz) harbor mode tends to infinity.c This is one of the important properties of harbor oscillations. Another important property is harbor resonance. The amplification factor for long waves impinging on a harbor from the open sea is H 2 (f ) =

(1 − f /f0

)2

1 , + Q−2 (f /f0 )2

(9.17)

where f is the frequency of the long incoming waves, f0 is the resonant frequency of the harbor, and Q is the quality factor (“Q-factor”), which is a measure of energy damping in the system.47,95 Specifically, Q−1 =

b This

c This

dE/dt = 2β, ωE

approach is used for numerical computation of eigen modes in natural 2D basins.70 is the reason for calling this the “zeroth mode.”

(9.18)

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where E = E0 e−2βωt is the energy of the system as it decays from an initial value E0 , β is a dimensionless damping coefficient, and ω = 2πf is the angular frequency. The power amplification factor attains the value Q2 at resonance (f = f0 ), decreases to unity at f = 0 and goes to zero as f goes to infinity. Therefore, Q for harbor oscillations plays a double role: as a measure of the resonant increase of wave heights for waves arriving from the open ocean and as an index of the time decay rate of wave heights inside the harbor. The higher the Q, the stronger will be the amplification of the incoming waves and the slower the energy decay, i.e., the longer the “ringing” of seiche oscillations inside the harbor. In closed basins, like lakes, bottom friction is the main factor controlling energy decay. Normally, it is quite small, so in lakes with fairly regular topographic features (low damping), a high Q-factor may be expected. Consequently, even a small amount of forcing energy at the resonant frequency can produce significant seiche oscillations that persist for several days.31,95 In contrast, the main factor of energy decay in semi-closed water basins, such as gulfs, bays, fjords, inlets and harbors, is wave radiation through the entrance. In their pioneering work, Miles and Munk47 concluded that narrowing the harbor entrance would increase the quality factor Q and, consequently, the amplification of the arriving wave. This means that the construction of dams, dikes, and walls to protect the harbor from wind waves and swell could so constrict the entrance width that it leads to strong amplification of the resonant seiche oscillations inside the harbor. Miles and Munk47 named this harbor paradox. As pointed out by Miles and Munk,47 there are two limitations to the previous conclusions: (1) A time of order Q/π cycles is necessary for the harbor oscillations to adjust to the external forcing. This means that harbors with high Q would not respond to a strong but short-lived incoming disturbance. In most cases, this limitation is not of major concern because atmospheric disturbances (the major source of open-sea long waves inducing harbor oscillations) are likely to last at least for several hours. Even tsunami waves from distant locations “ring” for many hours, resonantly “feeding” harbor seiches and producing maximum oscillations that have long (12–30 h) durations that persist well after the arrival of the first waves.74,75 This contrasts with the case for near-field sites, where tsunamis normally arrive as short-duration impulsive waves. Such tsunamis are much more dangerous at open coastal regions than in bays or harbors, as was observed for the coast of Thailand after the 2004 Sumatra tsunami.86 (2) As the harbor mouth becomes increasingly narrower, the internal harbor dissipation eventually exceeds energy radiation through the mouth. At this stage, further narrowing does not lead to a further increase in the Q-factor. However, normally internal dissipation is small compared to the typical radiative energy losses through the entrance. Originally, Miles and Munk47 believed that their “harbor paradox” concept was valid for every harbor mode provided the corresponding spectral peak was sharp and well defined. Further thorough examination of this effect37,46,78 indicated that the harbor paradox is only of major importance for the Helmholtz mode, while for higher modes frictional and nonlinear factors, not accounted for in the theory,

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dampen this effect.95 However, the Helmholtz mode is the most important mode in natural basins and is normally observed in bays, inlets, and harbors with narrow entrance, i.e., in semi-closed basins with high Q-factor. Significant problems with the mooring and docking of ships (and the loading and unloading of their cargo) in ports and harbors are often associated with this fundamental mode and most typically occur in ports with high Q.4,6,41,62,66–68,78–80 Rabinovich67 suggested reducing these negative effects in ports by artificially increasing the internal dissipation. The idea is the same as that widely used in rocket technology to damp eigen oscillations in fuel tanks.44,45 Radial piers in ports and harbors play the same role as internal rings and ribs in rocket tanks, efficiently transforming wave energy into vortical motions which reduce the wave energy and therefore the intensity of the seiches and their associated horizontal currents. As shown by Rabinovich,67 the logarithmic attenuation factor, δ0 = π/Q, for the Helmholtz mode associated with the jth pier, is given by  ˆ   ∆E0j 1 j bj ω0 ζ0 j = Cx , (9.19) δ0 = 2E0 6 r0 h0 σ0 where E0 is the energy of the mode inside the harbor, ∆E0j is the energy dissipated at the pier over the mode period (T0 = 2π/ω0 ), bj is the length of the pier, r0 and h0 are the mean radius and depth of the harbor, ζˆ0 is the mean amplitude of the Helmholtz mode in the harbor, Cxj is a dimensionless resistance coefficient, and σ0 = (gh0 )1/2 /(πr0 ). Thus, the rate of damping of oscillations in a harbor depends on the number of piers (N ) and a number of dimensionless parameters: specifically, the relative amplitudes of the oscillations, ξ0 = ζˆ0 /h0 ; the normalized harbor frequency, Ω0 = ω0 /σ0 ; the relative lengths of the piers, Bj = bj /r0 ; and the coefficient Cxj . The parameter ξ0 depends on the intensity of the external forcing while the two other parameters Ω0 and Bj do not depend on forcing but only on the characteristics of the harbor. The coefficient Cxj strongly depends on the Keulegan– Carpenter (KC) number which relates hydraulic resistance in oscillating flows to those for stationary currents.32 For typical values Bj = 0.3, ξ0 = 0.1, Ω0 = 1.0, N = 8, and Cxj = 10, we find δ0 ≈ 0.4 and Q ≈ 8. Another important aspect of the harbor oscillation problem is that changes in port geometry, and the construction of additional piers and dams can significantly change the natural (eigen) periods of the port, thereby modifying considerably the resonant characteristics of the basin.6 Helmholtz resonators in acoustics are used to attenuate sound disturbances of long wavelengths, which are difficult to reduce using ordinary methods of acoustical energy dissipation. Similarly, side channel resonators are suggested as a method for attenuating incident wave energy in harbors.6,66,78 In general, estimation of the Q-factor is a crucial consideration for ports, harbors, bay, and inlets. For a rectangular basin of uniform depth and entirely open mouth (ϑ = b/l = 1.0), this factor is easily estimated as: Q=

L , l

(9.20)

which is inversely proportional to the aspect ratio q = l/L. This means that high Q-factors can be expected for long and narrow inlets, fjords, and waterways.

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Honda et al.30 and Nakano and Unoki59 examined coastal seiches at more than 110 sites on the coast of Japan and found that strong and highly regular seiche oscillations are most often observed in such elongated basins and that the periods of these oscillations are in good agreement with the approximate period (9.9) for the Helmholtz mode (n = 0): 4L T0 = √ . gH

(9.21)

If the aperture ratio ϑ < 1.0, corresponding to a partly closed entrance, it is more difficult to estimate the Q value and the resonant mode periods analytically. In practice, special diagrams for a rectangular basin with various q and ϑ are used for these purposes.80,84 For natural basins, these parameters can be estimated numerically or from direct observations. If the respective spectral peak in observational data is isolated, sharp and pronounced enough, then we can assume that Q  1. In this case, it follows from (9.17) that the half-power frequency points (f1/2 ) are given by the following expression47:   1 ± f1/2 = f0 1 ± , (9.22) 2Q and the relative frequency bandwidth is simply ∆f = Q−1 , f0

(9.23)

+ − where ∆f = f1/2 − f1/2 and f0 = 1/T0 is the resonant frequency. This is a useful practical method for estimating the Q-factor and amplification for coastal basins based on results of spectral analysis of observational data. However, the spatial structure of different modes, the distribution of currents, and sea levels inside a natural basin, influence harbor reconstruction based on changes in these characteristics, and many other aspects of harbor hydrodynamics, are difficult to estimate without numerical computations. Numerical modeling has become a common approach that is now widely used to examine harbor oscillations.4,14,70,92

9.2.4. Harbor oscillations in a natural basin Some typical features of harbor oscillations are made more understandable using a concrete example. Figures 9.2 and 9.3 illustrate properties of typical harbor oscillations and results of their analysis and numerical modeling. Several temporary cable bottom pressure stations (BPS) were deployed in bays on the northern coast of Shikotan Island, Kuril Islands in 1986–1992.13,14,68,70 All BPSs were digital instruments that recorded long waves with 1-min sampling. One of these stations (BPS-1) was situated at the entrance of False Bay, a small bay with a broad open mouth [Fig. 9.2(a)]. The oscillations recorded at this site were weak and irregular; the respective spectrum [Fig. 9.2(b)] was “smooth” and did have any noticeable peaks, probably because of the closeness of the instrument position to the position of the entrance nodal line. Two more gauges (BPS-2 and BPS-3) were located inside Malokurilsk Bay, a “bottle-like” bay with a maximum width of about

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Sea of Okhotsk

45°

K

Hokkaido I.

sla il I ur

s nd

Iturup I.

4

Kunashir I. Shikotan I. 5

1 2

False Bay

3

Malokurilsk Bay

Pacific Ocean

Shikotan I. 0.5 km

(a) 40° 140°

145°

150° 10 5

Spectra (cm 2 cpm -1)

10 4

18.6

10 4 1

10 4 2 10 3

10 2

95%

10 3

18.6 3

10 2

4

10 0

10 -2 10 -3

95%

95%

10 2

(b)

10 1 10 -2

10 -1

10 -3

10 1

(c)

10 0 10 -2

10 -1

(d) 10 -2

10 -1

Frequency (cpm)

Fig. 9.2. (a) Location of cable bottom pressure stations near the northern coast of Shikotan Island (Kuril Islands) and sea level spectra at (b) BPS-1, (c) BPS-2 (both in autumn 1986) and (d) BPS-3 and BPS-4 (October–November 1990).

1300 m and a narrow neck of 350 m [Fig. 9.2(a)]. The oscillations recorded by these instruments were significant, highly regular and almost monochromatic; the corresponding spectra [Figs. 9.2(c) and 9.2(d)] have a prominent peak at a period of 18.6 min. An analogue tide gauge [#5 in Fig. 2(a)] situated on the coast of this bay permanently measure oscillations with exactly the same period.70 It is clear that this period is related to the fundamental mode of the bay. The Q-factor of the bay, as estimated by expression (9.23) based on spectral analysis of the tide gauge data for sites BPS-2 and BPS-3, was 12–14 and 9–10, respectively. The high Qfactors are likely the main reason for the resonant amplification of tsunami waves that arrive from the open ocean. Such tsunami oscillations are regularly observed in this bay.13,69 In particular, the two recent Kuril Islands tsunamis of 15 November 2006 and 13 January 2007 generated significant resonant oscillations in Malokurilsk Bay of 155 cm and 72 cm, respectively, at the same strongly dominant period of 18.6 min.76

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n=0 18.9 min

n=1 6.5 min

n=2 3.8 min

n=3 3.4 min

n=4 2.6 min

n=5 2.3 min

Fig. 9.3. Computed eigen modes and periods of the first six modes in Malokurilsk Bay (Shikotan Island). Black triangles indicate positions of the BPS-2 and BPS-3 gauges (from Ref. 70).

Figure 9.3 shows the first six eigen modes for Malokurilsk Bay.70 The computations were based on numerical conformal mapping of the initial mirror reflected domain on a circular annulus (for details, see Ref. 77) and the following application of Ritz’s variational method to solve the eigenvalue problem. The computed period of the fundamental (Helmholtz) mode (18.9 min) was close to the observed period of 18.6 min. The spectra at BPS-2 and BPS-2 indicate weak spectral peaks (three orders of magnitude less than the main peak) with periods 4.1, 3.3, and 2.9 min (the latter only at BPS-3), thought to be related to modes n = 2, 3, and 4. The first mode (n = 1), with period of 6.5 min, was not observed at these sites apparently because the nodal line for this mode passes through the positions of BPS-2 and BPS-3. Thus, the computed periods of the bay eigen modes are in good agreement with observation; plots in Fig. 9.3 give the spatial structure of the corresponding modes. However, this approach does not permit direct estimation of the bay response to the external forcing and the corresponding amplification of waves arriving from the open ocean. In actuality, the main purpose of the simultaneous deployments at sites BPS-3 and BPS-4 [Fig. 9.2(a)] in the fall of 1990 was to obtain observed response parameters that could be compared with numerically evaluated values.14 The spectrum at BPS-4, the station located on the outer shelf of Shikotan Island near the entrance to Malokurilsk Bay [Fig. 9.2(d)], contains a noticeable peak with period of 18.6 min associated with energy radiation from the bay. This peak is about 1.5 orders of magnitude lower than a similar peak at BPS-3 inside the bay. The amplification factor for the 18.6 min period oscillation at BPS-4 relative to that at BPS-3 was found to be about 4.0. Numerical computations of the response characteristics for Malokurilsk Bay using the HN-method14 gave resonant periods which

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were in close agreement with the empirical results of Rabinovich and Levyant70 (indicated in Fig. 9.3). Resonant amplification of tsunami waves impinging on the bay was found to be 8–10. 9.2.5. Seiches in coupled bays A well-known physical phenomenon are the oscillations of two simple coupled pendulums connected by a spring with a small spring constant (weak coupling). For such systems, the oscillation energy of the combined system systematically moves from one part of the system to the other. Every time the first pendulum swings, it pulls on the connecting string and gives the second pendulum a small tug, so the second pendulum begins to swing. As soon as the second pendulum starts to swing, it begins pulling back on the first pendulum. Eventually, the first pendulum is brought to rest after it has transferred all of its energy to the second pendulum. But now the original situation is exactly reversed, and the first pendulum is in a position to begin “stealing” energy back from the second. Over time, the energy repeatedly switches back and forth until friction and air resistance eventually remove all of the energy out of the pendulum system. A similar effect is observed in two adjacent bays that constitute a coupled system. Nakano56 was probably the first to investigate this phenomenon based on observations for Koaziro and Moroiso bays located in the Miura Peninsula in the vicinity of Tokyo. The two bays have similar shapes and nearly equal eigen periods. As was pointed out by Nakano, seiches in both bays are very regular, but the variations of their amplitudes are such that, while the oscillations in one bay become high, the oscillations in the other become low, and vice versa. Nakano56 explained the effect theoretically as a coupling between the two bays through water flowing across the mouths of each bay. More than half a century later Nakano returned to this problem60 and, based on additional theoretical studies and hydraulic model experiments, demonstrated that two possible regimes can exist in the bays: (1) co-phase oscillations when seiches in the two bays have the same initial phase; and (2) contraphase when they have the opposite phase. The superposition of these two types of oscillations create beat phenomenon of time-modulated seiches, with the opposite phase modulation, such that “while one bay oscillates vigorously, the other rests”. Nakano and Fujimoto suggested the term “liquid pendulums” for the coupled interaction of two adjacent bays. A more complicated situation occurs when the two adjacent bays have significantly different eigen periods. For example, Ciutadella and Platja Gran are two elongated inlets located on the west coast of Menorca Island, one of the Balearic Islands in the Western Mediterranean [the inlets are shown in the inset of Fig. 9.5(a)]. Their fundamental periods (n = 0) are 10.5 min and 5.5 min, respectively.50,71,73 As a result of the interaction between these two inlets, their spectra and admittance functions have, in addition to their “own” strong resonant peaks, secondary “alien” peaks originating from the other inlet.38 This means that the mode from Ciutadella “spills over” into Platja Gran and vice versa. The two inlets are regularly observed to experience destructive seiches, locally known as “rissaga”.18,22,49–51,85 Specific aspects of rissaga waves will be discussed later (in

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Sec. 9.4); however, it is worth noting here that the coupling between the two inlets can apparently amplify the destructive effects associated with each of the inlets individually.38

9.3. Generation Because they are natural resonant oscillations, seiches are generated by a wide variety of mechanisms (Fig. 9.4), including tsunamis,14,28,54,69 seismic ground waves,2,15,34,40 internal ocean waves,8,19,21 and jet-like currents.30,54,57 However, the two most common factors initiating these oscillations in bays and harbors are atmospheric processes and nonlinear interaction of wind waves or swell (Fig. 9.4).62,68,95 Seiches in lakes and other enclosed water bodies are normally generated by direct external forcing on the sea surface, primarily by atmospheric pressure variations and wind.31,95 In contrast, the generation of harbor oscillations is a two-step process involving the generation of long waves in the open ocean followed by forcing of the

Period (sec) 10 5

10 4

10 3

10 2

101 sec

AT M O S P H E R E

Storm surges

Scattering nonstationarity

Nonlinearity

Long waves

Meteorological waves

Swell Wind waves

Infragravity waves

Tsunami waves

OCEAN Seismicity

BOTTOM 10 6

10 5

10 4

10 3

10 2

10 1

10 0

Wavelength (m) Fig. 9.4.

Sketch of the main forcing mechanisms generating long ocean waves.

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harbor oscillations as the long waves arrive at the harbor entrance where they lead to resonant amplification in the basin. Seiche oscillations produced by external periodic forcing can be both free and forced. The free oscillations are true seiches (i.e., eigen oscillations of the corresponding basin). However, if the external frequency (σ) differs from the eigen frequencies of the basin (σ = ω), the oscillations can be considered forced seiches.95 Open-ocean waves arriving at the entrance of a specific open-mouth water body (such as a bay, gulf, inlet, fjord, or harbor) normally consist of a broad frequency spectrum that spans the response characteristics of the water body from resonantly generated eigenfree modes to nonresonantly forced oscillations at other frequencies. Following cessation of the external forcing, forced seiches normally decay rapidly, while free modes can persist for a considerable time. Munk53 jokingly remarked that “the most conspicuous thing about long waves in the open ocean is their absence.” This is partly true: the long-wave frequency band, which is situated between the highly energetic tidal frequencies and swell/wind wave frequencies, is relatively empty (Fig. 9.5). For both swell/wind waves and tides, the energy is of order 104 cm2 , while the energy contained throughout the entire intermediary range of frequencies is of order 1–10 cm2 . However, this particular frequency range is of primary scientific interest and applied importance (Walter Munk himself spent approximately 30 years of his life working on these “absent” waves!). Long waves are responsible for formation and modification of the coastal zone and shore morphology;5,68 they also can strongly affect docking and loading/unloading of ships and construction in harbors, causing considerable damage.41,78,79,96 Finally,

100

24 12 5

hr

Period

2 1 30 10 5

Tides

min

2 1 30 10 5

sec

2 1

Swell, wind waves

10 3

10 1 -2

10 0 Storm surges

Meteorological waves IG- waves

10 -1

Variance (cm 2 )

10 2 -5

10 -2 Tsunami 10 -4

10 -3

10 -2

10 -1

10 0

10 1

10 2

10 -3

Frequency (cpm) Fig. 9.5. Spectrum of surface gravity waves in the ocean (modified from Ref. 68). Periods (upper scale) are in hours (h), minutes (min), and seconds (s).

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and probably the most important, are tsunamis and other marine hazardous long waves, which are related to this specific frequency band. The recent 2004 Sumatra tsunami in the Indian Ocean killed more than 226,000 people, triggering the largest international relief effort in history and inducing unprecedented scientific and public interest in this phenomenon and in long waves in general.86 Because of their resonant properties, significant harbor seiches can be produced by even relatively weak open ocean waves. In harbors and bays with high Qfactors, seiches are observed almost continuously. However, the most destructive events occur when the incoming waves have considerable energy at the resonant frequencies, especially at the frequency of the fundamental mode. Such a situation took place in Port Alberni located in the head of long Alberni Inlet on Vancouver Island (Canada) during the 1964 Alaska tsunami, when resonantly generated seiche oscillations in the inlet had trough-to-crest wave heights of up to 8 m, creating total economic losses of about $10 million (1964 dollars).28,54 9.3.1. Meteorological waves Long waves in the ocean are the primary factor determining the intensity of harbor oscillations. If we ignore tsunamis and internal waves, the main source of background long waves in the ocean are atmospheric processes (Fig. 9.4).10,53 There are three major mechanisms to transfer the energy of atmospheric processes into long waves in the ocean68 : (1) Direct generation of long waves by atmospheric forcing (pressure and wind) on the sea surface. (2) Generation of low-frequency motions (for example, storm surges) and subsequent transfer of energy into higher frequencies due to nonlinearity, topographic scattering and nonstationarity of the resulting motions. (3) Generation of high-frequency gravity waves (wind waves and swell) and subsequent transfer of energy into larger scale, lower frequency motions due to nonlinearity. Long waves generated by the first two mechanisms are known as atmospherically induced or meteorological waves.d Typical periods of these waves are from a few minutes to several hours, typical scales are from one to a few hundreds of kilometers. The first mechanism is the most important because it is this mechanism that is responsible for the generation of destructive seiche oscillations (meteorological tsunamis) in particular bays and inlets of the World Ocean (Sec. 9.4). “Meteorological waves” can be produced by the passages of typhoons, hurricanes or strong cyclones. They also have been linked to frontal zones, atmospheric pressure jumps, squalls, gales, wind gusts and trains of atmospheric buoyancy waves.10,59,68,71,87,95 The most frequent sources of seiches in lakes are barometric fluctuations. However they can also be produced by heavy rain, snow, d The Russian name for these waves is “anemobaric”68 because they are induced by atmospheric pressure (“baric”) and wind (“anemos”) stress forcing on the ocean surface.

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or hail over a portion of the lake, or flood discharge from rivers at one end of the lake.27,31,95 9.3.2. Infragravity waves Long waves generated through the nonlinear interaction of wind waves or swell are called infragravity waves.5,63 These waves have typical periods of 30 s to 300–600 s and length scales from 100 m to 10 km. The occurrence of relatively high-frequency long waves, highly correlated with the modulation of groups of wind or swell waves, was originally reported by Munk52 and Tucker.89 Because the waves were observed as sea level changes in the nearshore surf zone, they became known as surf beats. Later, it was found that these waves occur anywhere there are strong nonlinear interacting wind waves. As a result, the more general term infragravity waves (proposed by Kinsman33 ) became accepted for these waves. Recent field measurements have established that infragravity waves (IG waves) dominate the velocity field close to the shore and consist of superposition of free edge waves propagating along the shore, free leaky waves propagating in the offshore direction, and forced bound waves locked to the groups of wind waves or swell propagating mainly onshore.3,5,68 Bound IG waves form the set-down that accompanies groups of incident waves, having troughs that are beneath the high short waves of the group and crests inbetween the wave groups.39 They have the same periodicity and the same lengths as the wave groups and travel with the group velocity of wind waves, which is significantly smaller than the phase speed of free long waves with the same frequencies. Free edge IG waves arise from the trapping of swell/wind wave generated oscillations over sloping coastal topography, while free leaky waves are mainly caused by the reflection of bound waves into deeper water.5,63 The general mechanisms of the formation of IG waves are shown in Fig. 9.6.e IG waves are found to be responsible for many phenomena in the coastal zone, including formation of rip currents, wave setup, crescentic bars, beach cusps and other regular forms of coastal topographies, as well as transport of sediment materials. Being of high-frequency relative to meteorological waves, IG waves can induce seiches in comparatively small-scale semi-closed basins, such as ports and harbors, which have natural periods of a few minutes and which may pose a serious threat for large amplitude wave responses. Certain harbors and ports are known to have frequent strong periodic horizontal water motions. These include Cape Town (South Africa), Los Angeles (USA), Dakar (Senegal), Toulon and Marseilles (France), Alger (Algeria), Tuapse and Sochi (Russia), Batumi (Georgia) and Esperance (Australia). Seiche motions in these basins create unacceptable vessel movement which can, in turn, lead to the breaking of mooring lines, fenders and piles, and to the onset of large amplitude ship oscillations and damage.62,67,68,82,94–96 Known as surging or range action,78,79 this phenomenon has well-established correlations with (a) harbor oscillations, (b) natural oscillations of the ship itself, and (c) intensive swell or wind waves outside the e Figure 9.6 does not include all possible types of IG waves and mechanisms of their generation; a more detailed description is presented by Bowen and Huntley5 and Battjes.3

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Fig. 9.6.

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Generation mechanisms for infragravity waves in the coastal zone.

harbor. Typical eigen periods of a harbor or a moored ship are the order of minutes. Therefore, they cannot be excited directly by wind waves or swell, having typical periods on the order of seconds.96 However, these periods exactly coincide with the periods of wave groups and IG waves. So, it is conventional wisdom that surging in harbors is the result of a triple resonance of external oscillations outside the harbor, natural oscillations within the harbor, and natural oscillations of a ship. The probability of such triple resonance is not very high, thus surging occurs only in a limited number of ports. Ports and harbors having large dimensions and long eigen periods (>10 min) are not affected by surging because these periods are much higher than the predominant periods of the IG waves and the surging periods of the vessels. On the other hand, relatively small vessels are not affected because their natural (eigen) periods are too short.82 The reconstruction of harbors and the creation of new harbor elements, can significantly change the harbor resonant periods, either enhancing or, conversely, reducing the surging.f Another important aspect of the problem is that ship and mooring lines create an entirely separate oscillation system.79 Changing the material and the length of the lines and their position, f A famous example of this kind is the French port Le Havre. Before World War II it was known for very common and strong surging motions that created severe problems for ships. During the war a German submarine torpedoed by mistake a rip-rap breakwater, creating a second harbor opening of 20–25 m width. After this, the surging in the port disappeared.68

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changes the resonant properties of the system (analogous to changing the material and the length of a pendulum). It is important to keep in mind that each oscillation mode has a specific spatial distribution of sea level variability and associated current (as emphasized in Sec. 9.2.1, maximum currents are observed near the nodal lines). The intensity of the currents varies significantly from place to place. Moreover, topographic irregularities within the harbor and the presence of structure elements (dams, dykes, piers and breakwaters) can create intense local vortexes that may significantly affect the ships.67 So, the effect of surging on a ship strongly depends on the exact location of the ship, and even on its orientation, in the harbor. In summary, harbor oscillations arise through co-oscillation of sea surface elevations and currents in the harbor with those at the entrance to the harbor. Seichegenerating motions outside the harbor typically have periods of several minutes and most commonly arise from bound and free long waves that are incident on the harbor entrance. 9.3.3. Tsunami Tsunami waves are the main factor creating destructive seiche oscillations in bays, inlets and harbors.30,41,53,54,95 Tsunamis can produce “energies” of 103 –105 cm2 , although such events are relatively rare (depending on the region, from once every 1–2 years to once every 100–200 years). The main generation mechanisms for tsunamis are major underwater earthquakes, submarine landslides and volcanic explosions. Great catastrophic trans-oceanic tsunamis were generated by the 1946 Aleutian (magnitude Mw = 7.8), 1952 Kamchatka (Mw = 9.0), 1960 Chile (Mw = 9.5), and 1964 Alaska (Mw = 9.2) earthquakes. The events induced strong seiche oscillations in bays, inlets, and harbors throughout the Pacific Ocean.90 The magnitude Mw = 9.3 earthquake that occurred offshore of Sumatra in the Indian Ocean on 26 December 2004 generated the most destructive tsunami in recorded history. Waves from this event were recorded by tide gauges around the world, including near-source areas of the Indian Ocean (Fig. 9.7), and remote regions of the North Pacific and North Atlantic, revealing the unmatched global reach of the 2004 tsunami.42,74,86,88 In general, the duration of tsunami “ringing” increased with increasing off-source distance and lasted from 1.5 to 4 days.74,75 The recorded oscillations were clearly polychromatic, with different periods for different sites, but with clear dominance of 40–50 min waves at most sites. The analysis of various geophysical data from this event indicates that the initial tsunami source had a broad frequency spectrum, but with most of the energy within the 40–50 min band. Therefore, although tsunami waves at different sites induced local eigen modes with a variety of periods, the most intense oscillations were observed at sites having fundamental periods close to 40–50 min. Differences in spectral peaks among the various tide gauge records are indicative of the influence of local topography. For example, for the Pacific coast of Vancouver Island (British Columbia), the most prominent peaks in the tsunami spectra were observed for Winter Harbor (period ∼30–46 min) and Tofino (∼50 min). In fact, the frequencies of most peaks in the tsunami spectra invariably coincide with

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West and Central Indian Ocean 200 cm

E

Colombo (SL)

Hanimaadhoo (Md)

Sea level

Male (Md)

Salalah (Om)

Pointe La Rue (Sl)

Port Louis (Mt)

26

27 December 2004 (UTC)

28

Fig. 9.7. Tsunami records in the Indian Ocean for the 2004 Sumatra tsunami for six selected sites: Colombo (Sri Lanka); Male and Gan (both Maldives); Salalah (Oman); Pointe La Rue (Seychelles); and Port Louis (Mauritius). Solid vertical line labeled “E” denotes the time of the main earthquake shock (from Ref. 74).

corresponding peak frequencies in the background spectra. This result is in good agreement with the well-known fact that periods of observed tsunami waves are mainly related to the resonant properties of the local/regional topography rather than to the characteristics of the source, and are almost the same as those of ordinary (background) long waves for the same sites. For this reason, the spectra of tsunamis from different earthquakes are usually similar at the same location.30,48,69,g It is therefore difficult to reconstruct the source region spectral characteristics based on data from coastal stations. Rabinovich69 suggested a method for separating the effects of the local topography and the source on the resulting tsunami wave spectrum. This method can be used to reconstruct the open-ocean spectral characteristics of tsunami waves. The approach is based on the assumption that the spectrum S(ω) of both the tsunami and background sea level oscillations near the coast can be represented as S(ω) = W (ω)E(ω),

(9.24)

g The resonant characteristics of each location are always the same; however, different sources induce different resonant mode, specifically, large seismic sources generate low-frequency modes and small seismic sources generate high-frequency modes.

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where W (ω) = H 2 (ω), H(ω) is the frequency admittance function describing the linear topographic transformation of long waves approaching the coast, and E(ω) is the source spectrum. It is assumed that the site-specific properties of the observed spectrum Sj (ω) at the jth site are related to the topographic function Hj (ω) for that site, while all mutual properties of the spectra at all sites are associated with the source (assuming that the source is the same for all stations). For typical background oscillations the source spectrum has the form, E(ω) = S0 (ω), where S0 (ω) = Aω −2 , and A = 10−3 −10−4 cm2 .68,69 During tsunami events, sea level oscillations observed near the coast can be represented as ζobs (t) = ζt (t) + ζb (t),

(9.25)

where ζt are the tsunami waves generated by an underwater seismic source and ζb are the background surface oscillations. If the spectra of both tsunami, St (ω), and background oscillations, Sˆb (ω) and Sb (ω) (during and before the tsunami event, respectively) have the form (9.24), and the admittance function, W (ω), is the same for the observed tsunami and the background long waves, then the spectral ratio R(ω), is estimated as 

ˆ0 (ω) E(ω) + S ˆ St (ω) + Sb (ω) (9.26) R(ω) = = = A−1 ω 2 E(ω) + 1.0. Sb (ω) S0 (ω) The function R(ω), which is independent of local topographic influence, is determined solely by the external forcing (i.e., by tsunami waves in the open ocean near the source area) and gives the amplification of the longwave spectrum during the tsunami event relative to the background conditions. The close similarity of Rj (ω) for various sites confirms the validity of this approach.69 The topographic admittance function Hj (ω), which is characteristic of the resonant properties of individual sites, can be estimated as

1/2

1/2  1/2 j Sb (ω) Sbj (ω) Sj (ω) = =ω . (9.27) Hj (ω) = E(ω) S0 (ω) A The same characteristic can be also estimated numerically. 9.3.4. Seismic waves There is evidence that seismic surface ground waves can generate seiches in both closed and semi-closed basins. In particular, the Great 1755 Lisbon earthquake triggered remarkable seiches in a number of Scottish lochs, and in rivers and ponds throughout England, western Europe and Scandinavia.95 Similarly, the Alaska earthquake of 27 March 1964 (Mw = 9.2) induced seismic surface waves that took only 14 min to travel from Prince Williams Sound, Alaska, to the Gulf Coast region of Louisiana and Texas where they triggered innumerable seiches in lakes, rivers, bays, harbors, and bayous.15,34 Recently, the 3 November 2002 Denali earthquake (Mw = 7.9) in Alaska generated pronounced seiches in British Columbia and Washington State.2 Sloshing oscillations were also observed in swimming pools during

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these events.2,15,40 The mechanism for seiche generation by seismic waves from distant earthquakes is not clear, especially considering that seismic waves normally have much higher frequencies than seiches in natural basins. McGarr40 concludes that there are two major factors promoting efficient conversion of the energy from distant large-magnitude earthquakes into seiches: (1) A very thick layer of soft sediments that amplify the horizontal seismic ground motions. (2) Deeper depths of natural basins, increasing the frequencies of eigen periods for the respective water oscillations. It should be noted, however, that seismic origins for seiches must be considered as very rare in comparison, for example, with seiches generated by meteorological disturbances.95 9.3.5. Internal ocean waves In some regions of the World Ocean, definitive correlation has been found between tidal periodicity and the strong seiches observed in these regions. For example, at Palawan Island in the Philippines, periods of maximum seiche activity are associated with periods of high tides.21 Bursts of 75-min seiches in the harbor of Puerto Princesa (Palawan Island) are assumed to be excited by the arrival at the harbor entrance of internal wave trains produced by strong tidal current flow across a shallow sill located about 450 km from the harbor.8 Internal waves can have quite large amplitudes; furthermore, they can travel over long distances without noticeable loss of energy. Internal waves require 2.5 days to travel from their source area in the Sulu Sea to the harbor of Puerto Princessa, resulting in a modulation of the seiche oscillations that are similar to those of the original tidal oscillations. Similarly, large amplitude seiches on the Caribbean coast of Puerto Rico are also related to tidal activity and are usually observed approximately seven days after a new or full moon (syzygy). Highest seiches in this region occur in late summer and early fall, when thermal stratification of the water column is at its annual maximum. The seven-day interval between syzygy and maximum seiche activity could be accounted for in terms of internal tidal soliton formation near the southwestern margin of the Caribbean Sea.34 A theoretical model of seiche generation by internal waves, devised by David Chapman (Woods Hole Oceanographic Institution), demonstrated that both periodic and solitary internal waves can generate coastal seiches.8,20 Thus, this mechanism can be responsible for the formation of seiches in highly stratified regions. 9.3.6. Jet-like currents Harbor oscillations (coastal seiches) can also be produced by strong barotropic tidal and other currents. Such oscillations are observed in Naruto Strait, a narrow channel between the Shikoku and Awaji islands (Japan), connecting the Pacific Ocean and

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the Inland Sea. Here, the semi-diurnal tidal currents move large volumes of water back and forth between the Pacific and the Inland Sea twice per day with typical speed of 13–15 km/h. This region is one of the greatest attractions in Japan because of the famous “Naruto whirlpool”, occurring twice a month during spring tides, when the speed of tidal flow reaches 20 km/h. Honda et al.30 noticed that flood tidal currents generate near both coasts significant seiche oscillations with a period of 2.5 min, which begin soon after low tide and cease near high tide; the entire picture repeats with a new tidal cycle. No seiches are observed during ebb tidal currents (i.e., between high and low water) when the water is moving in the opposite direction. Nakano57 explained this phenomenon by assuming that a strong current passing the mouth of a bay could be the source of bay seiches, similar to the way that a jet of air passing the mouth piece of an organ pipe produces a standing oscillation within the air column in the pipe. Special laboratory experiments by Nakano and Abe58 demonstrated that jet-like flow with a speed exceeding a specific critical number generates a chain of antisymmetric, counter-rotating von Karman vortexes on both sides of the channel. The checker-board pattern of vortexes induce standing oscillations in nearby bays and harbors if their fundamental periods match the typical vortex periods, Tvor =

l , u

(9.28)

where l is the distance between vortexes, and u is the speed of the vortexes (u = 0.4V to 0.6V , where V is the speed of the tidal currents). For the parameters of the Naruto tidal currents, the laboratory study revealed that values of Tvor agreed with the observed seiche period of 2.5 min. Apparently, the same mechanism of seiche generation can also work in other regions of strong jet currents. 9.3.7. Ice cover and seiches It seems clear that ice cannot generate seiches (except for the case of calving icebergs or avalanches that generate tsunami-like waves). However, an ice cover can significantly impact seiche motions, suppressing them and impeding their generation. At the same time, strong seiches can effectively break the ice cover and promote polynya creation. Little is known on the specific aspects of ice cover interaction with seiche modes. Hamblin26 suggested that the ice cover in Lake Winnipeg influences the character of seiche activity. Schwab and Rao83 assumed that absence of certain peaks in the sea level spectra for Saginaw Bay (Lake Huron) in winter may have been due to the presence of ice cover. Murty55 examined the possible effect of ice cover on seiche oscillations in Kugmallit Bay and Tuktoyaktuk Harbor (Beaufort Sea) and found that the ice cover reduces the effective water depth in the bay and harbor and in this way diminishes the frequency of the fundamental mode: in Kugmallit Bay from 0.12 cph (ice-free period) to 0.087 cph (ice-covered); and in Tuktoyaktuk Harbor from 1.0 cph to 0.9 cph.

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9.4. Meteorological Tsunamis As discussed in Sec. 9.3.3, tsunamis are the main source of destructive seiches observed in various regions of the World Ocean. However, atmospheric disturbances (atmospheric gravity waves, pressure jumps, frontal passages, squalls) can also be responsible for significant, even devastating, long waves, which have the same temporal and spatial scales as typical tsunami waves. These waves are similar to ordinary tsunami waves and can affect coasts in a similar damaging way, although the catastrophic effects are normally observed only in specific bays and inlets. Nomitsu,61 Defant,10 and Rabinovich and Monserrat 71,72 suggested to use the term “meteorological tsunamis” (“meteotsunami”) for this type of waves. At certain places in the World Ocean, these hazardous atmospherically induced waves occur regularly and have specific local names: “rissaga” in the Balearic Islands, “ˇs´ciga” on the Croatian coast of the Adriatic Sea, “marubbio” (“marrobio”) in Sicily, “milghuba” in Malta, “abiki” and “yota” in Japan, “Seeb¨ ar” in the Baltic Sea, “death waves” in Western Ireland, “inchas” and “lavadiads” in the Azores and Madeira islands. These waves are also documented in the Yellow and Aegean seas; the Great Lakes; northwestern Atlantic; Argentina and New Zealand coastal areas; and Port Rotterdam.7,9–12,16,27,29,30,34,43,51,64,68,71,72,91,92 Table 9.5 gives a list of destructive harbor oscillations, which apparently have the same atmospheric origin and similar resonances due to similarities in the characteristics of the atmospheric disturbances and local geometry and topography of the corresponding basins. Because of the strong likeness between “meteotsunamis” and seismically generated tsunamis,51,88 it is quite difficult sometimes to recognize one from another. Catalogues of tsunamis normally contain references to numerous “tsunami-like” events of “unknown origin” that are, in fact, atmospherically generated ocean waves. “Rissaga” (a local Catalan word that means “drying”, similar to a Spanish word “resaca”) is probably the best known example of meteorological tsunamis.h These significant short-period sea level oscillations regularly occur in many bays and harbors of the Catalan and Valencian coasts of the Iberian Peninsula, and on the coast of the Balearic Islands. The waves in Ciutadella Harbor, Menorca Island [Fig. 9.8(a)] are particularly high and occur more frequently than in any other location.18,22,49–51,71–73,81,85 Ciutadella Inlet is about 1 km long, 100 m wide, and 5 m deep; the harbor is located at the head of the inlet [Fig. 9.8(a)]. The fundamental period of the inlet (Helmholtz mode) is approximately 10.5 min [Figs. 9.8(b,c)]. Due to the particular geometry of Ciutadella Inlet, it has a large Q-factor, which results in significant resonant amplification of long-wave oscillations arriving from the open sea. Seiche h For

this reason Derek Goring, a wave specialist from New Zealand, suggested to apply the term “rissaga” to all rissaga-like meteorological seiches in other areas of the World Ocean.23 However, if we were to adopt this term, then we would loose information on the cause of the oscillations and the fact that they are part of a family of events that include seismically generated tsunamis, landslide tsunamis, volcanic tsunamis, and meteotsunamis.

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224 Table 9.5.

Region

Extreme coastal seiches in various regions of the World Ocean.

Local name

Typical peroid

Maximum observed height

References

Nagasaki Bay, Japan Pohang Harbor, Korea Longkou Harbor, China Ciutadella Harbor, Menorca I., Spain

Abiki

35 min

4.78 m

—

25 min

>0.8 m

Honda et al.30 Amano [1957], Akamatsu1 , Hibiya and Kajiura29 Chu [1976], Park et al. [1986]

—

2h

2.93 m

Wang et al.93

Rissaga

10.5 min

>4.0 m

Gulf of Trieste, Italy West Sicily, Italy

—

3.2 h

1.6 m

Marrubio (Marrobbio)

∼15 min

>1.5 m

Malta, Mediterranean West Baltic, Finland coast Croatian coast East Adriatic Newfoundland, Canada Western Ireland Azores Is and Madeira Is, East Atlantic Rotterdam Harbor, The Netherlands

Milghuba

∼20 min

∼1.0 m

Fontser´e,17 Tintor´e et al.,88 Monserrat et al.,49−51 Gomis et al.,22 Garcies et al.,18 Rabinovich and Monserrat,71,72 Rabinovich et al.73 Caloi [1938], Greco et al. [1957], Defant,10 Wilson95 Plattania [1907], Oddone [1908], Defant,10 Colucci and Michelato,9 Candela et al.7 Airy [1878], Drago16

Seeb¨ ar

—

∼2.0 m

ˇciga S´

10–30 min 10–40 min ? ?

∼6.0 m

— Death Waves Inchas, Lavadiads

85–100 min

2.0–3.0 m

Doss [1907], Meissner [1924], Defant,10 Credner [1988], Hodˇzi´ c [1979/1980]; Orli´c60 ; Vilibi´ c et al.92 ; Monserrat et al.51 Mercer et al. [2002]

? ?

Berninghausen [1964], Korgen34 Berninghausen [1964], Korgen34

>1.5 m

de Looff and Veldman [1994], de Jong et al.,11 de Jong and Battjes12 [2005]

Comment: Exact references can be found in: Wiegel (1964), Korgen (1995), Rabinovich and Monserrat (1996), de Jong et al. (2003) and Monserrat et al. (2006).

oscillations of duration ranging from a few hours to several days and wave heights exceeding 0.5 m recur in Ciutadella every summer. However, rissaga events (largeamplitude seiches) having wave heights more than 3–4 m, with dramatic consequences for the harbor, usually take place once in 5–6 years. During the rissaga of 21 June 1984 (Fig. 9.9), about 300 boats were destroyed or strongly damaged.71 More recently, on 15 June 2006, Ciutadella Harbor was affected by the most dramatic rissaga event of the last 20 years, when almost 6-m waves were observed in the harbor and the total economic loss was of several tens millions of euros.51 Fontser´e,17 in the first scientific paper on extreme seiches for the Catalan coast, showed that these seiches always occur from June to September and first suggested their atmospheric origin. This origin of rissaga waves was supported by

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Fig. 9.8. A map of the Balearic Islands and positions of four tide gauges (M0, M1, M2 and MW3) deployed in Ciutadella and Platja Gran inlets and on the shelf of Menorca Island during the LAST97 experiment.50 The arrow shows the predominant direction of propagation of atmospheric waves during “rissaga” events. (b) The strong “rissaga” event recorded in Ciutadella Inlet on 31 July 1998 by a tide gauge located at position M0. (c) Spectra for “rissaga” of 24 July 1997 (solid line) and background oscillations (dashed line) for four tide gauges indicated in (a). The actual four-day records during this event are shown in the insets.

Ramis and Jans`a81 based on observed oscillations on the Balearic Islands. These authors also defined a number of typical synoptic atmospheric conditions normally associated with rissaga events. The atmospheric source of rissaga is now well established.18,22,49,50,85 During late spring and summer, meteorological conditions in

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Fig. 9.9. Ciutadella Harbor during the rissaga of 21 June 1984. Photo by Josep Gornes (from Ref. 71).

the western Mediterranean are favorable for the formation of high-frequency atmospheric pressure disturbances with parameters promoting the generation of rissaga waves. These conditions include the entrance of warm air from the Sahara at nearsurface levels, and relatively strong middle level winds from the southwest. When this synoptic meteorological situation exists, trains of atmospheric pressure gravity waves (with periods of minutes) are reported traveling from SW to NE.49 If these atmospheric pressure disturbances propagate from SW to NE with a phase speed of about 22–30 m/s, resonant conditions are set up for the southeastern shelf of Mallorca Island (“Proudman resonance”) and dynamic energy associated with the atmospheric waves is efficiently transferred into the ocean waves. When these waves reach the coast of Menorca Island, they can generate significant (and sometimes even hazardous) seiche oscillations inside Ciutadella and other inlets due to harbor resonance. The Q-factor for the fundamental Helmholtz mode in Ciutadella Inlet (10.5 min), roughly estimated by Eq. (9.20), is about 9. Spectral estimates based on Eq. (9.23) give a similar value, Q ≈ 10.73 As shown in Fig. 9.8(b), rissaga oscillations in Ciutadella Inlet have a very regular monochromatic character. Maximum wave heights occur during the fourth to the sixth oscillations, in good agreement with the criterion by Miles and Munk47 that time of the order of Q/π cycles is necessary for

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the harbor oscillations to adjust themselves to external forcing. The peak period of 10.5 min for the Helmholtz mode strongly dominates the spectra for the M0 and M2 gauges located in Ciutadella Inlet [Fig. 9.8(c)] both for rissaga and background spectra, while in the adjacent inlet Platja Gran (M1), where rissaga waves are also observed but weaker than in Ciutadella, the dominant peak associated with the Helmholtz mode is 5.5 min. In contrast, on the shelf (MW3) both peaks are absent and oscillations are significantly weaker. Spectral analysis results [Fig. 9.8(c)] reveal that harbor resonance is a crucial factor in the formation of rissaga waves, as well as “meteorological tsunamis” in other bays, inlets, and harbors of the World Ocean. Barometric data from the Balearic Islands49–51,81 as well as from Japan,29 and Eastern Adriatic Sea64,91,92 demonstrate that generation of these destructive waves is associated with strong atmospheric disturbances, e.g., trains of atmospheric gravity waves, or isolated pressure jumps. These atmospheric disturbances may have different origin: dynamic instability, orographic influence, frontal passages, gales, squalls, storms, tornados, etc.24 However, even during the strongest events, the atmospheric pressure oscillations at the meteotsunami scales (from a few minutes to a few hours) reach only 2–6 hPa, corresponding to only a 2–6 cm change in sea level. Consequently, these atmospheric fluctuations may produce significant sea level response only when resonance occurs between the ocean and the atmosphere. During the resonance process, the atmospheric disturbance propagating above the ocean surface generates significant long ocean waves by continuously pumping additional energy into these waves. Possible resonances that are responsible for the formation of meteorological tsunamis are68 : disturbance speed (U ) • Proudman resonance,65 when U = c, i.e., the atmospheric √ equals to the long-wave speed of ocean waves c = gh; • “Greenspan resonance”,25 when Ul = cj , the alongshore component (Ul ) of the atmospheric disturbance velocity equals the phase speed of the jth mode of edge waves (cj ); • “shelf resonance,” when the atmospheric disturbance and associated atmospherically generated ocean wave have a period/wavelength equal to the resonant period/length of the shelf. These resonant effects may significantly amplify ocean waves approaching the coast. Nevertheless, even strong resonant amplification of atmospherically generated ocean waves normally cannot produce waves with sufficient energy to extensively affect the open coast (for example, a 3–4 hPa pressure jump and a factor of 10 resonant amplification, will only produce ocean wave heights of 30–40 cm). It is when energetic ocean waves arrive at the entrance of a semi-closed coastal basin (bay, inlet, fjord or harbor) that they can induce hazardous oscillations in the basin due to harbor resonance. On the other hand, intense oscillations inside a harbor (bay or inlet) can only be formed if the external forcing (i.e., the waves arriving from the open sea) are energetic enough. Seismically generated tsunami waves in the open ocean can

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be sufficiently energetic even in the absence of additional resonant effects (thus, according to satellite altimetry measurements, tsunami waves generated by the 2004 Sumatra earthquake in the open Indian Ocean had trough-to-crest wave heights of approximately 1.0–1.2 m,86 while atmospherically generated tsunami-like can reach such potentially dangerous levels only in the case of some external resonance. This is an important difference between tsunami waves and meteotsunamis. It follows from expression (9.17) that a large Q-factor is critical but that anomalously pronounced harbor oscillations can only be produced when there is resonant matching between the dominant frequency (f ) of the arriving (external) waves and an eigen frequency f0 of the harbor (normally, the eigen frequency of the fundamental — Helmholtz — harbor mode). This means that catastrophic harbor oscillations are the result of a double resonance effect 51,68 : (a) external resonance between the moving atmospheric disturbances and open-ocean waves; and (b) internal resonance between the arriving open-ocean waves and the fundamental eigen mode of the harbor (bay, inlet). An additional favorable factor is the specific direction of the propagating atmospheric waves (and corresponding open-ocean waves) toward the entrance of the harbor (bay). Summarizing what has been presented above, we can formulate the particular conditions promoting creation of extreme atmospherically induced oscillations near the coast (meteotsunamis) as follows: • A harbor (bay, inlet or fjord) with definite resonant properties and high Q-factor. • The occurrence of strong small-scale atmospheric disturbance (a pressure jump or a train of internal atmospheric waves). • A propagation direction that is head-on toward the entrance of the harbor. • The occurrence of an external resonance (Proudman, Greenspan or shelf) between the atmospheric disturbance and ocean waves. • The occurrence of internal resonance between the dominant frequency of the incoming open-ocean waves and the fundamental harbor mode frequency. Due to these necessary levels of matching between the atmospheric disturbance, the open-ocean bathymetry and the shelf-harbor geometries, the direction and speed of the atmospheric disturbance probably are even more important than the actual energy content of the incoming waves. In any case, the necessary coincidence of several factors significantly diminishes the possibility of these events occurring, and is the main reason why this phenomenon is relatively rare and restricted to specific locations.68 Honda et al.30 and Nakano and Unoki59 investigated more than 115 gulfs, bays, inlets, and harbors of the Japanese coast and found that highly destructive seiches (not associated with tsunami waves) occur only in a few of them. Extremely strong seiche oscillations (so-called “abiki” waves) are periodically excited in Nagasaki Bay. In particular, the abiki waves of 31 March 1979 with periods of about 35 min reached wave heights of 478 cm at the northern end of the bay and killed three people.1,29 High meteotsunami risk in certain exceptional locations mainly arises from the combination of shelf topography and coastline geometry coming together to create a multiple resonance effect. The factors (internal and external) of critical importance

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are: (1) well-defined resonant characteristics of the harbor (bay, inlet, etc.); and (2) specific properties of the shelf favorable for external resonance (between atmospheric and open-ocean waves) and internal resonance (between arriving open-ocean waves and harbor oscillations). The combination of these factors for some particular sites is like a “time-bomb”: sooner or later it will explode (when the atmospheric disturbance is strong enough and the parameters of disturbance coincide with the resonant parameters of the corresponding topography/geometry). Locations with known regular extreme seiches are just the places for these “time-bombs”.51 The catastrophic abiki wave event of 31 March 1979 best illustrates the physical mechanisms responsible for the generation of meteotsunamis [Fig. 9.10(a)]. Hibiya and Kajiura29 (HK in the following text) examined this event in detail and constructed an effective numerical model that agrees well with observational data. Nagasaki Bay is a narrow, elongated bay located on the western coast of Kyushu Island, Japan [Fig. 9.10(b)]; the length of the bay is about 6 km, the width is 1 km, and the mean depth is 20 m. The fundamental period of the bay (Helmholtz mode) is 35 min, and this period prevails in seiche oscillations inside the bay (95% of all observed events) and it was specifically this period that was observed on 31 March 1979.1 HK noticed that almost all known cases of significant abiki waves are associated with pressure jumps. In the case of the 1979 event, there was an abrupt pressure jump (∆Pa ) of 2–6 hPa (according to the observations at several sites) that propagated eastward (more precisely, 5.6◦ north of east) over the East China Sea with an approximate mean speed U = 31 m/s (Fig. 9.5). HK approximated this jump as ∆Pa = 3 hPa over a linear increase distance L1 = 28 km and a linear decrease distance L2 = 169 km. So, the corresponding static inverted barometer response of sea level was ∆ζ¯ ≈ −3 cm [Fig. 9.10(a)]. Moreover, the depth of the East China Sea between mainland China and Kyushu Island is between 50 and 150 m, and the corresponding long-wave speed c ≈ 22–39 m/s. Thus, it was a classical example of Proudman resonance. HK presented a simple expression describing resonant amplification of forced open-ocean long waves as: ∆ζ =

∆ζ¯ xf , L1 2

(9.29)

where xf = U t is the distance traveled by the pressure jump during time t. If L1 = 28 km and xf = 300 km [from the source area to the Goto Islands — see Fig. 9.10(b)], then ∆ζ ≈ 16 cm. More precise numerical computation with realistic two-dimensional bathymetry gives the resonant factor ε = ∆ζ/∆ζ¯ = 4.3 and ∆ζ ≈ 12.9 cm in good agreement with observation. Therefore, due to the resonance, the initial disturbance of 3 cm increased in the open sea by four to five times [Fig. 9.10(a)]. It is interesting to note that the resonant amplification is inversely proportional to L1 [see Eq. (9.29)], so the faster the change in atmospheric pressure (the more abrupt is the pressure jump), the stronger is the amplification of the generated waves (HK). According to the HK computations, the outer shelf region between the Goto Islands and the mainland of Kyushu (“Goto Nada”) has resonant periods of 64, 36, and 24 min. The second period (36 min) almost coincides with the fundamental period of Nagasaki Bay (35 min). The Goto Nada shelf did not significantly amplify

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U = 31 m/s

(a)

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4.8 m measured at tide gauges

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3

air-pressure disturbance

45 cm

16 cm

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Nagasaki Bay harbor resonance

(b)

0

a East China Se

shelf amplification

Korea

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Japan

100

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km 50 0

Kyushu I.

Goto Is

Sea level (cm)

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10 0

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China Kyushu I. 3 Nagasaki 1 2 2 Bay

-50 -100 -150

Nezumi (1)

50 0 -50

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12

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31 March 1979

Fig. 9.10. (a) A sketch illustrating the physical mechanism for formation of the catastrophic meteotsunami at Nagasaki Bay (Japan) on 31 March 1979. Numbers “1”, “2”, and “3” correspond to locations shown in (b). (b) Map of Nagasaki Bay and the initial atmospheric pressure disturbance (shaded rectangular region). (c) Tide records of the catastrophic “abiki waves” of 31 March 1979 at Nezumi (9.1) and Nagasaki (2); positions of the tide gauges are shown in the inset in panel (b).

the incoming wave (the first crest height was 16 cm at the shelf depth of 60 m) but it selected and amplified waves with specific periods, in particular those with a period of 36 min. Between the outer sea (depth 60 m) and the head of Nagasaki Bay, the arriving waves were amplified by a factor of 2.4 due to the combined effects of topographic convergence, partial reflection, and shoaling inside the bay. Finally, resonant amplification in Nagasaki of incoming wave train with a period of about 35 min formed catastrophic oscillations within the bay with a maximum recorded

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wave height of 278 cm [as measured by a tide gauge located in the middle of the bay — see Fig. 9.10(c)] and an estimated wave height in the head of the bay of 478 cm.1 Thus, for this extreme event, we observe the full combination of “hazardous” conditions (factors) responsible for the formation catastrophic oscillations inside Nagasaki Bay: (1) A pronounced atmospheric disturbance (pressure jump of 2–6 hPa), (2) propagating toward the bay with (3) near-resonant phase speed of 31 m/s; this disturbance resonantly generated open-sea long waves with selected (over the shelf) 36 min period that matched (4) the fundamental 35-min period of the bay that has (5) high Q-factor and well-defined resonant properties. As a result, 3 cm ocean waves in the source area resulted in 478 cm waves at the head of the bay (Fig. 9.4). Analysis of destructive meteotsunami events in the Mediterranean18,22,50,51,64, 71,72,91,92 indicate that the physical mechanisms of these events were similar to those for Nagasaki Bay event. Tides in the Mediterranean are small; consequently, harbors are not designed to accommodate large amplitude sea level changes associated with occasional meteotsunamis. Consequently, it is atmospherically generated phenomena (not ordinary tsunamis) that are normally responsible for significant flooding and damage in this region. However, the main reason for the damaging nature of meteotsunamis is likely due to the strong currents in the harbor that accompany the sea level oscillations. Seiches with a 10 min period give raise to currents that are 70 times stronger than semi-diurnal tides having the same amplitude. Acknowledgments This work was initiated by Professor Fred Raichlen (CalTech, Pasadena, CA); the author sincerely appreciates his help and friendly support. He is also very grateful to Professor Young Kim (California State University, Los Angeles), the Editor of this Handbook, for his patience and useful comments and to Drs. Sebastian Monserrat (Universitat de les Illes Balears, Palma de Mallorca, Spain) and Ivica Vilibi´c (Institute of Oceanography and Fisheries, Split, Croatia) for their assistance and providing various observational data. Dr. Richard Thomson (Institute of Ocean Sciences, Sidney, BC, Canada) did tremendous work editing this chapter and encouraging the author, Patricia Kimber (Sidney, BC) helped to draft the figures. Partial financial support was provided by the Russian Foundation on Basic Research, grants 06-0565210a, 08-05-13582, 09-05-00599 and 09-05-01125a and by NATO Science for Peace project SfP-981382. References 1. H. Akamatsu, On seiches in Nagasaki Bay, Pap. Meteor. Geophys. 33(2), 95–115 (1982). 2. A. Barberopoulou, A. Qamar, T. L. Pratt and W. P. Steele, Long-period effects of the Denali Earthquake on water bodies in the Puget lowland: Observations and modeling, Bull. Seism. Soc. Amer. 96(2), 519–535 (2006).

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3. J. A. Battjes, Surf-zone dynamics, Ann. Rev. Fluid Mech. 20, 257–293 (1988). 4. W. A. M. Botes, K. S. Russel and P. Huizinga, Modeling harbor seiching compared to prototype data, Proc. 19th Int. Coastal Eng. Conf., Houston (1984), pp. 846–857. 5. A. J. Bowen and D. A. Huntley, Waves, long waves and nearshore morphology, Marine Geol. 60, 1–13 (1984). 6. E. C. Bowers, The modelling of waves and their effects in harbors, in Hydraulic Modelling in Maritime Engineering (Thomas Telford, London, 1982), pp. 121–127. 7. J. Candela, S. Mazzola, C. Sammari, R. Limeburner, C. J. Lozano, B. Patti and A. Bonnano, The “Mad Sea” phenomenon in the Strait of Sicily, J. Phys. Oceanogr. 29, 2210–2231 (1999). 8. D. C. Chapman and G. S. Giese, Seiches, Encyclopedia of Ocean Sciences, eds J. H. Steele, K. K. Turekian and S. A. Thorpe, Vol. 5 (Academic Press, London, 2001) pp. 2724–2731. 9. P. Colucci and A. Michelato, An approach to study of the ‘Marubbio’ phenomenon, Boll. Geofis. Theor. Appl. 13, 3–10 (1976). 10. A. Defant, Physical Oceanography, Vol. 2 (Pergamon Press, Oxford, UK, 1961), 598 pp. 11. M. P. C. de Jong, L. H. Holthuijsen and J. A. Battjes, Generation of seiches by cold fronts over the southern North Sea, J. Geophys. Res. 108, 3117 (2003), doi:10.1029/2002JC001422. 12. M. P. C. de Jong and J. A. Battjes, Low-frequency sea waves generated by atmospheric convection cells, J. Geophys. Res. 109, C01011 (2004), doi:10.1029/2003JC001931. 13. V. A. Djumagaliev, E. A. Kulikov and S. L. Soloviev, Analysis of ocean level oscillations in Malokurilsk Bay caused by tsunami of 16 February 1991, Science Tsunami Hazards, 11, 47–58 (1993). 14. V. A. Djumagaliev, A. B. Rabinovich and I. V. Fine, Theoretical and experimental estimation of transfer peculiarities of the Malokurilsk Bay coast, the Island of Shikotan, Atmosph. Oceanic Phys. 30, 680–686 (1994). 15. W. L. Donn, Alaskan earthquake of 27 March 1964: Remote seiche simulation, Science 145, 261–262 (1964). 16. A. F. Drago, A study on the sea level variations and the ‘Milghuba’ phenomenon in the coastal waters of the Maltese Islands, Ph.D. thesis, University of Southampton (1999). 17. E. Fontser´e, Les ‘seixes’ de la costa catalana. Servei Meteorol` ogic de Catalunya, Notes d’Estudi, 58 (1934) (in Catalan). 18. M. Garcies, D. Gomis and S. Monserrat, Pressure-forced seiches of large amplitude in inlets of the Balearic Islands. Part II: Observational study, J. Geophys. Res. 101, 6453–6467 (1996). 19. G. S. Giese and D. C. Chapman, Coastal seiches, Oceanus 36, 38–46 (1993). 20. G. S. Giese, D. C. Chapman, P. G. Black and J. A. Fornshell, Causations of largeamplitude harbor seiches on the Caribbean coast of Puerto Rico, J. Phys. Oceanogr. 20, 1449–1458 (1990). 21. G. S. Giese and R. B. Hollander, The relation between coastal seiches at Palawan Island and tide-generated internal waves in the Sulu Sea, J. Geophys. 92, 5151–5156 (1987). 22. D. Gomis, S. Monserrat and J. Tintor´e, Pressure-forced seiches of large amplitude in inlets of the Balearic Islands, J. Geophys. Res. 98, 14437–14445 (1993). 23. D. G. Goring, Rissaga (long-wave events) on New Zealand’s eastern seaboard: A hazard for navigation, Proc. 17th Australasian Coastal Ocean Eng. Conf., 20–23 September 2005, Adelaide, Australia (2005), pp. 137–141. 24. E. E. Gossard and B. H. Hooke, Waves in Atmosphere (Elsevier, New York, 1975), 456 pp.

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9.75in x 6.5in

Seiches and Harbor Oscillations

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233

25. H. P. Greenspan, The generation of edge waves by moving pressure disturbances, J. Fluid Mech. 1, 574–592 (1956). 26. P. F. Hamblin, Seiches, circulation and storm surges of an ice-free Lake Winnipeg, J. Fisheries Res. Board Canada 33, 2377–2391 (1976). 27. D. L. Harris, The effect of a moving pressure disturbance on the water level in a lake, Meteorol. Monogr. 2, 46–57 (1957). 28. R. F. Henry and T. S. Murty, Tsunami amplification due to resonance in Alberni Inlet: Normal modes, inTsunami: Progress in Prediction, Disaster Prevention and Warning, eds. Y. Tsuchiya and N. Shuto (Kluwer Acad. Publ., Dordrecht, The Netherlands, 1995), pp. 117–128. 29. T. Hibiya and K. Kajiura, Origin of ‘Abiki’ phenomenon (kind of seiches) in Nagasaki Bay, J. Oceanogr. Soc. Japan 38, 172–182 (1982). 30. K. Honda, T. Terada, Y. Yoshida and D. Isitani, An investigation on the secondary undulations of oceanic tides, J. College Sci., Imper. Univ. Tokyo (1908), 108 pp. 31. G. E. Hutchinson, A Treatise on Limnology. Volume 1. Geography, Physics and Chemistry (J. Wiley, New York, 1957), 1015 pp. 32. G. H. Keulegan and L. H. Carpenter, Forces on cylinders and plates in an oscillating fluid, J. Res. Nat. Bureau Stand. 60, 423–440 (1958). 33. B. Kinsman, Wind Waves: Their Generation and Propagation on the Ocean Surface (Prentice Hall, Englewood Cliffs, 1965), 676 pp. 34. B. J. Korgen, Seiches, Amer. Sci. 83, 330–341 (1995). 35. H. Lamb, Hydrodynamics, 6th Edn. (Dover, New York, 1945), 738 pp. 36. J. J. Lee, Wave induced oscillations in harbors of arbitrary geometry, J. Fluid Mech. 45, 375–394 (1971). 37. B. Le M´ehaut´e and B. W. Wilson, Harbor paradox (discussion), J. Waterways Harbor Division, ASCE 88, 173–195 (1962). 38. P. L.-F. Liu, S. Monserrat, M. Marcos and A. B. Rabinovich, Coupling between two inlets: Observation and modeling, J. Geophys. Res. 108, 3069 (2003), doi:10.1029/ 2002JC001478, 14-(1–10). 39. M. S. Longuet-Higgins and R. W. Stewart, Radiation stress and mass transport in gravity waves, with application to ‘surf-beats’, J. Fluid Mech. 13, 481–504 (1962). 40. A. McGarr, Excitation of seiches in channels by seismic waves, J. Geophys. Res. 70, 847–854 (1965). 41. C. C. Mei, The Applied Dynamics of Ocean Surface Waves (World Scientific, London 1992), 740 pp. 42. M. A. Merrifield et al., Tide gage observations of the Indian Ocean tsunami, December 26, 2004, Geophys. Res. Lett. 32, L09603 (2005), doi:10.1029/2005GL022610. 43. M. Metzner, M. Gade, I. Hennings and A. B. Rabinovich, The observation of seiches in the Baltic Sea using a multi data set of water levels, J. Marine Systems 24, 67–84 (2000). 44. G. N. Mikishev and B. I. Rabinovich, Dynamics of a Solid Body with Cavities Partially Filled with Liquid (Mashinostroyeniye Press, Moscow, 1968), 532 pp. (in Russian). 45. J. W. Miles, Ring damping of free surface oscillations in a circular tank, J. Appl. Mech. 25, 26–32 (1958). 46. J. W. Miles, Harbor seiching, Ann. Rev. Fluid Mech. 6, 17–36 (1974). 47. J. Miles and W. Munk, Harbor paradox, J. Waterways Harbor Division, ASCE 87, 111–130 (1961). 48. G. R. Miller, Relative spectra of tsunamis, HIG-72–8 Hawaii Inst. Geophys (1972), 7 pp.

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9.75in x 6.5in

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49. S. Monserrat, A. Ibberson and A. J. Thorpe, Atmospheric gravity waves and the “rissaga” phenomenon, Q. J. R. Meteorol. Soc. 117, 553–570 (1991). 50. S. Monserrat, A. B. Rabinovich and B. Casas, On the reconstruction of the transfer function for atmospherically generated seiches, Geophys. Res. Lett. 25, 2197–2200 (1998). 51. S. Monserrat, I. Vilibi´c and A. B. Rabinovich, Meteotsunamis: Atmospherically induced destructive ocean waves in the tsunami frequency band, Nat. Hazards Earth Syst. Sci. 6, 1035–1051 (2006). 52. W. H. Munk, Surf beats, Eos, Trans. Amer. Geophys. Un. 30, 849–854 (1949). 53. W. H. Munk, Long waves, The Sea (J. Wiley, New York, 1962), pp. 647–663. 54. T. S. Murty, Seismic Sea Waves — Tsunamis (Bull. Fish. Res. Board Canada 198, Ottawa, 1977), 337 pp. 55. T. S. Murty, Modification of hydrographic characteristics, tides, and normal modes by ice cover, Marine Geodesy 9, 451–468 (1985). 56. M. Nakano, Secondary undulations in bays forming a coupled system, Proc. Phys. Math. Soc. Japan 3, 372–380 (1932). 57. M. Nakano, Possibility of excitation of secondary undulations in bays by tidal or oceanic currents, Proc. Imp. Acad. Japan 9, 152–155 (1933). 58. M. Nakano and T. Abe, Standing oscillation of bay water induced by currents, Records Oceanogr. Works in Japan, Spec. No. 3, 75–96 (1959). 59. M. Nakano and S. Unoki, On the seiches (secondary undulations of tides) along the coast of Japan, Records Oceanogr. Works Japan, Spec. No. 6, 169–214 (1962). 60. M. Nakano and N. Fujimoto, Seiches in bays forming a coupled system, J. Oceanogr. Soc. Japan 43, 124–134 (1987). 61. T. Nomitsu, A theory of tsunamis and seiches produced by wind and barometric gradient, Mem. Coll. Sci. Imp. Univ. Kyoto A 18, 201–214 (1935). 62. M. Okihiro, R. T. Guza and R. J. Seymour, Excitation of seiche observed in a small harbor, J. Geophys. Res. 98, 18201–18211 (1993). 63. J. Oltman-Shay and R. T. Guza, Infragravity edge wave observations on two California beaches, J. Phys. Oceanor. 17, 644–663 (1987). 64. M. Orli´c, About a possible occurrence of the Proudman resonance in the Adriatic, Thalassia Jugoslavica 16, 79–88 (1980). 65. J. Proudman, The effects on the sea of changes in atmospheric pressure, Geophys. Suppl. Mon. Notices R. Astr. Soc. 2, 197–209 (1929). 66. D. Prandle, The use of wave resonators for harbor protection, Dock Harbour Author (1974), pp. 279–280. 67. A. B. Rabinovich, Possible vorticity effect in longwave motions (surging) in harbors, Trans. (Doklady) Russian Acad Scie, Earth Sci. Sec. 325, 224–228 (1992). 68. A. B. Rabinovich, Long Ocean Gravity Waves: Trapping, Resonance, and Leaking (Gidrometeoizdat, St. Petersburg, 1993), 325 pp. (in Russian). 69. A. B. Rabinovich, Spectral analysis of tsunami waves: Separation of source and topography effects, J. Geophys. Res. 102, 12663–12676 (1997). 70. A. B. Rabinovich and A. S. Levyant, Influence of seiche oscillations on the formation of the long-wave spectrum near the coast of the Southern Kuriles, Oceanology 32, 17–23 (1992). 71. A. B. Rabinovich and S. Monserrat, Meteorological tsunamis near the Balearic and Kuril Islands: Descriptive and statistical analysis, Nat. Hazards 13, 55–90 (1996). 72. A. B. Rabinovich and S. Monserrat, Generation of meteorological tsunamis (large amplitude seiches) near the Balearic and Kuril Islands, Nat. Hazards 18, 27–55 (1998).

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73. A. B. Rabinovich, S. Monserrat and I. V. Fine, Numerical modeling of extreme seiche oscillations in the region of the Balearic Islands, Oceanology 39, 16–24 (1999). 74. A. B. Rabinovich, R. E. Thomson and F. E. Stephenson, The Sumatra tsunami of 26 December 2004 as observed in the North Pacific and North Atlantic oceans, Surveys Geophys. 27, 647–677 (2006). 75. A. B. Rabinovich and R. E. Thomson, The 26 December 2004 Sumatra tsunami: Analysis of tide gauge data from the World Ocean Part 1. Indian Ocean and South Africa, Pure Appl. Geophys. 164, 261–308 (2007). 76. A. B. Rabinovich, L. I. Lobkovsky, I. V. Fine, R. E. Thomson, T. N. Ivelskaya and E. A. Kulikov, Near-source observations and modeling of the Kuril Islands tsunamis of 15 November 2006 and 13 January 2007, Adv. Geosci. 14, 105–116 (2008). 77. B. I. Rabinovich and Y. V. Tyurin, Numerical Conformal Mapping in TwoDimensional Hydrodynamics (Space Research Institute RAS, Moscow, 2000), 312 pp. 78. F. Raichlen, Harbor resonance, Estuary and Coastline Hydrodynamics, ed. A. T. Ippen (McGraw Hill Book Co., New York, 1966), pp. 281–340. 79. F. Raichlen, The effect of ship-harbor interactions on port operations, Maritime Technol., 11–20 (2002). 80. F. Raichlen and J. J. Lee, Oscillation of bays, harbors and lakes, Handbook of Coastal and Ocean Engineering, ed. J. B. Herbich (Gulf Publishing Company Houston, Texas, 1992), pp. 1073–1113. 81. C. Ramis and A. Jans` a, Condiciones meteorol´ ogicas simult´ aneas a la aparici´ on de oscilaciones del nivel del mar de amplitud extraordinaria en el Mediterr´ aneo occidental, Rev. Geof´ısica 39, 35–42 (1983) (in Spanish). 82. T. Sawaragi and M. Kubo, Long-period motions of a moored ship induced by harbor oscillations, Coast. Eng. Japan 25, 261–275 (1982). 83. D. J. Schwab and D. B. Rao, Gravitational oscillations of Lake Huron, Saginaw Bay, Georgian Bay and the North Channel, J. Geophys. Res. 82, 2105–2116 (1977). 84. R. M. Sorensen and E. F. Thompson, Harbor hydrodynamics, Coastal Engineering Manual, Part II (U.S. Army Corps. of Engineers, Washington, D.C., New York, 2002), Chapter 7, pp. 1–92. 85. J. Tintor´e, D. Gomis, S. Alonso and D. P. Wang, A theoretical study of large sea level oscillations in the Western Mediterranean, J. Geophys. Res. 93, 10797–10803 (1988). 86. V. V. Titov, A. B. Rabinovich, H. Mofjeld, R. E. Thomson and F. I. Gonz´alez, The global reach of the 26 December 2004 Sumatra tsunami, Science 309, 2045–2048 (2005). 87. R. E. Thomson, P. W. Vachon and G. A. Borstad, Airborne synthetic aperture radar imagery of atmospheric gravity waves, J. Geophys. Res. 97, 14249–14257 (1992). 88. R. E. Thomson, A. B. Rabinovich and M. V. Krassovski, Double jeopardy: Concurrent arrival of the 2004 Sumatra tsunami and storm-generated waves on the Atlantic coast of the United States and Canada, Geophys. Res. Lett. 34, L15607 (2007), doi:10.1029/2007GL030685. 89. M. J. Tucker, Surf beats: Sea waves of 1 to 5 minute period, Proc. Roy. Soc. London Ser. A 202, 565–573 (1950). 90. W. G. Van Dorn, Some tsunami characteristics deducible from tide records, J. Phys. Oceanogr. 14, 353–363 (1984). 91. I. Vilibi´c and H. Mihanovi´c, A study of resonant oscillations in the Split harbor (Adriatic Sea), Estuar. Coastal Shelf Sci. 56, 861–867 (2003). 92. I. Vilibi´c, N. Domijan, M. Orli´c, N. Leder and M. Pasari´c, Resonant coupling of a traveling air-pressure disturbance with the east Adriatic coastal waters, J. Geophys. Res. 109, C10001 (2004), doi:10.1029/2004JC002279.

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93. X. Wang, K. Li, Z. Yu and J. Wu, Statistical characteristics of seiches in Longkou harbor, J. Phys. Oceanogr. 17, 1063–1065 (1987). 94. R. L. Wiegel, Tsunamis, storm surges, and harbor oscillations, Oceanographical Engineering (Prentice-Hall, Englewood Cliffs, NJ, 1964), Ch. 5, pp. 95–127. 95. B. Wilson, Seiches, Advances in Hydrosciences 8, 1–94 (1972). 96. J.-K. Wu and P. L.-F. Liu, Harbor excitations by incident wave groups, J. Fluid Mech. 217, 595–613 (1990).

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Finite Difference Model for Practical Simulation of Distant Tsunamis Sung Bum Yoon Department of Civil and Environmental Engineering Hanyang University Ansan, 426-791, Korea [email protected] A practical numerical model based on the dispersion-correction finite difference scheme of Yoon et al. [Terres. Atmos. Oceanic Sci. 18(1), 31–53 (2007)] equipped with the grid nesting scheme of Lim et al. [Natural Hazards 47(1), 95–118 (2008)] is introduced. The model is applied to simulate the propagation of a historical tsunami event that attacked the east coast of Korea. The calculated free surface displacements are compared with the observations at two tidal stations along the east coast of Korea. The comparison shows that the results agree well with the observations. The analyses of the simulated results show that the underwater topography such as submerged rises and ridges plays an important role in the propagation of tsunamis in this region.

10.1. Introduction Tsunamis are the ocean water surface waves generated by an undersea earthquake, a landslide, a volcanic eruption, or even a meteoric impact on water surface. As crust movement of the earth became active, the region along the rim of the Pacific Ocean experienced large-scale tsunamis including the 1946 Alaskan tsunami, the 1960 Chilean tsunami, and the 1964 Alaskan tsunami, for instance. These large-scale tsunamis propagated across the Pacific Ocean, and attacked the coastal area along the Pacific rim. The damage including loss of human lives caused by these catastrophic tsunamis was well documented by Lander and Lockridge.3 More recent tsunamis in this area have been investigated by many scientists, and reported by Satake et al.4 for the 1992 Nicaragua tsunami, and Hokkaido Tsunami Survey Group5 for the 1993 Hokkaido tsunami. On the other hand, large-scale tsunamis including the 1992 and 1994 Indonesia tsunamis, 1995 Philippine tsunami, 1996 Indonesia tsunami, and 1998 Papua New Guinea tsunami were also generated along the rim of the Indian Ocean. In particular, the tsunami of magnitude 9.3 237

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was generated off the west coast of Sumatra in the Indian Ocean on 26 December 2004. The west coast of northern Sumatra was most severely damaged by this tsunami. The coastal areas along the rim of the Indian Ocean including Malaysia, Thailand, Myanmar, Bangladesh, India, Sri Lanka, Arabian, and African countries were attacked by this tsunami. More than 290,000 people were killed due to this tsunami event. And the property damages and injuries cannot be countable. In order to reduce such damages caused by tsunamis, a proper monitoring, forecasting, and warning system should be established. For this we need to fully understand the mechanism such as generation, propagation, and inundation of tsunamis, and predict the possible quantities such as arrival times and run-up heights of a tsunami event. Thus, the development of an accurate and efficient numerical model for computing all the aspects of tsunami propagation is indispensable. When the tsunamis are generated by an earthquake, they have a shape of solitary wave with a long crest. After they propagate a long distance from the source area over a deep ocean, the tsunamis evolve into a train of waves due to the dispersion effect of waves. When the source area is narrow, the dispersion effect plays an important role in the deformation of tsunamis also in the relatively shallow water area. Boussinesq equations are one of the best choices for governing equations to construct a numerical model for the far-field tsunamis, because they can take into account the dispersion effects. The numerical models such as FUNWAVE6 and COULWAVE7,8 were developed based on a fully nonlinear and dispersive Boussinesq equations, and were applied to simulate the 26 December 2004 Indian Ocean tsunami.9,10 However, these Boussinesq models require a small mesh size to suppress the numerical dispersion error. The small mesh system consumes huge computer resources due to the implicit nature of the solution technique to deal with dispersion terms. Thus, the Boussinesq model is not preferred for the simulation of the far-field tsunamis, and the shallow-water equations are generally employed instead. Considerable efforts to construct a numerical model for the simulation of tsunamis have been made. For example, finite difference models were developed by Hwang and Divoky,11 Goto and Shuto,12 Kowalik and Murty,13 and Mader and Curtis.14 Most of the existing numerical models are based on the shallow-water equations and have been successfully applied to the near-field tsunamis which have little dispersion effect of waves. However, distant tsunamis generated far from the region of interest are mainly transformed due to the accumulation of dispersion effects during the propagation. Thus, the numerical models based on the shallowwater equations will suffer from the lack of accuracy for the simulation of distant tsunamis. Imamura et al.15 presented a finite difference model for the simulation of transoceanic propagation of tsunamis. The model solves the shallow-water equations using a leap-frog scheme. The physical dispersion is compensated by the numerical dispersion introduced by the truncation error of the numerical scheme. This can be done only if the grid size is appropriately selected for the given water depth and the time step satisfying the criterion proposed by Imamura et al.15 Cho16 improved the numerical model of Imamura et al.15 for tsunamis obliquely propagating to

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the principal axes of computational grids. However, strictly speaking, those models have no capability to consider the dispersion effect of waves in the case of practical application. The use of numerical models developed by Imamura et al.15 and Cho16 is limited to the case of constant water depth. Yoon17 developed a finite-difference scheme that uses a uniform grid, but the actual computations are made on a hidden grid of variable size determined from the condition proposed by Imamura et al.15 This model satisfies local dispersion relationships of waves for a slowly varying topography. However, the accuracy of the model becomes degraded when applied to short waves due to numerical errors associated with the interpolation of quantities at hidden grid points. Yoon et al.1 developed an efficient and relatively accurate finite-difference model to simulate the propagation of distant tsunamis over varying topography. This finite-difference model solves a linear Boussinesq-type wave equation employing a uniform finite-difference grid. A considerable accuracy is achieved by eliminating the numerical dispersion error and by providing the physical dispersion. The dispersioncorrection is performed by controlling the parameters determined from the given time step, grid size, and local water depth. As a result, an accurate dispersioncorrection is possible although the grid size does not meet Imamura’s mesh condition. As the tsunamis approach the coastal area, they experience a shoaling due to decreasing water depth over a continental shelf. The wavelength becomes short, and the grid resolution becomes low near the coastline. Thus, the accuracy of the farfield model proposed by Yoon et al.1 is degraded. This difficulty can be solved by nesting a finer grid system to the coarse grid trans-oceanic model. As the tsunamis approach the coastline, nonlinearity and bottom friction play a significant role in the transformation of tsunami waves. In this case, a near-field model such as full Boussinesq or shallow-water equation model can be nested to the trans-oceanic model as described in Yoon17 and Lim et al.2 Thus, various nesting schemes between systems with different grid sizes or governing equations are indispensable to extend the applicability of the far-field numerical model. The objective of this chapter is to review a recent development of a practical finite-difference model to simulate tsunami propagation and runup for constructing inundation maps. The contents in this chapter are not an original work but a remake of research works presented by Yoon et al.1 and Lim et al.2,18 Section 10.2 describes governing equations and numerical scheme employed in the dispersion-correction finite-difference model developed by Yoon et al.1 and Lim et al.2,18 Section 10.3 presents the test of numerical model for uniform and variable depth cases. Section 10.4 describes the near-field model. Section 10.5 introduces the grid nesting scheme developed by Lim et al.2,18 to extend the applicability of the numerical model. In order to evaluate the applicability of the present model to real tsunami events, a historical tsunami is simulated and the results are compared with the observations at two harbors along the east coast of Korea in Sec. 10.6. Then, the effect of underwater topography on the propagation of tsunamis in this sea is analyzed based on the simulated results. Finally, this study is summarized in Sec. 10.7.

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10.2. Governing Equations and Numerical Scheme Tsunamis are generally regarded as long waves in comparison with the windgenerated waves. However, tsunamis can be short when the tsunami source is narrow. In this case the dispersion effect can play an important role in the transformation of tsunami waves. The dispersion effect, rather than the nonlinear or the bottom friction effects, dominates the transformation of the far-field tsunamis. For the far-field tsunami, the linear Bousinessq equations, which neglect the nonlinear effects but include the dispersion effects, can be used as a governing equation. On the other hand, the nonlinear effect and the bottom friction effect become important for the transformation of tsunamis in the near-field where the water is shallow near the coastal area. For the near-field tsunami, the nonlinear shallowwater equations are generally employed as a governing equation. In the present numerical model, the linear Boussinesq-type wave equation proposed by Yoon et al.1 is employed for the far-field tsunami propagation, and the nonlinear shallow-water equations with the bottom friction term are used for the near-field tsunamis. For trans-oceanic propagation of tsunamis, the nonlinearity of waves can be neglected because the free surface displacement is much smaller in comparison with the water depth. However, the dispersion effect of tsunami waves should be considered properly for the far-field tsunamis. Thus, the following linear Boussinesq equations can be used as governing equations. ∂ζ ∂P ∂Q + + = 0, ∂t ∂x ∂y
    ∂ζ h2 ∂ ∂ ∂P ∂ ∂Q ∂P + gh = + ∂t ∂x 2 ∂x ∂x ∂t ∂y ∂t
2     ∂ P ∂2 Q h3 ∂ + , − 6 ∂x ∂t∂x h ∂t∂y h
    ∂Q ∂ζ h2 ∂ ∂ ∂P ∂ ∂Q + gh = + ∂t ∂y 2 ∂y ∂x ∂t ∂y ∂t
2     ∂ P ∂2 Q h3 ∂ + , − 6 ∂y ∂t∂x h ∂t∂y h

(10.1)

(10.2)

(10.3)

where ζ represents the free surface elevation from still water level (m), and P and Q are the depth-integrated volume fluxes (m2 /s) in the x- and y-directions, respectively, g is the acceleration of gravity (m/s2 ), and h is the still water depth (m). The equations of motion (10.2) and (10.3) include the dispersion terms in the right-hand side. These dispersion terms cause numerical difficulties in practice because of the mixed form of differentiations with respect to both time and space. Consequently, it calls for the use of implicit scheme, which solves a matrix system. The implicit scheme requires very fine grids to reduce the numerical dispersion errors inherent in the numerical scheme such as finite-difference method. A fine grid

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needs the tremendous computational effort with a huge computer memory space and excessive execution time. To develop an efficient and relatively accurate numerical model for the propagation of dispersive tsunamis over slowly varying topography, Yoon et al.1 derived the following linear Boussinesq-type wave equation (LBTWE) from the linear Boussinesq equations (10.1)–(10.3) as     
   
∂2ζ ∂ ∂ζ ∂ ∂ζ h2 ∂ 2 ∂ζ ∂ ∂ζ ∂ gh + gh − gh + gh − ∂t2 ∂x ∂x ∂y ∂y 3 ∂x2 ∂x ∂x ∂y ∂y      ∂ ∂ζ ∂ ∂ζ ∂2 gh + gh = 0, (10.4) + ∂y 2 ∂x ∂x ∂y ∂y where the water depth is assumed to be slowly varying in the computational domain. This equation includes the dispersion terms in the last part. Equation (10.4) can be split into two first-order partial differential equations in time as    2  ∂h ∂ζ ∂h ∂ζ ∂ ζ ∂v ∂ 2ζ =g + + gh + 2 , (10.5) ∂t ∂x ∂x ∂y ∂y ∂x2 ∂y  2  ∂ 2v ∂ v ∂ζ + , (10.6) = v − γ∆x2 ∂t ∂x2 ∂y 2 where v denotes an auxiliary variable introduced for computational convenience, and γ is the dispersion-correction parameter. The solution of (10.5) and (10.6) is identical with that of (10.4) if γ is set to be −h2 /3∆x2 . Since the linear Boussinesqtype wave equations (10.5) and (10.6) can be solved by a pure explicit numerical scheme, the dispersive tsunami propagation over a large domain can be simulated with low computational cost. Yoon et al.1 developed a dispersion-correction finite-difference model which solves the equations (10.5) and (10.6) using a relatively simple explicit numerical scheme given by
 n  n  n−1/2 n − vi,j hi+1,j − hni−1,j ζi+1,j − ζi−1,j +g ∆t 2∆x 2∆x  n  n  n hi,j+1 − hni,j−1 ζi,j+1 − ζi,j−1 + 2∆y 2∆y
 n  n n n n n ζi,j+1 + ζi−1,j − 2ζi,j + ζi,j−1 ζi+1,j − 2ζi,j + − gh (1 − α) ∆x2 ∆y 2   n n n n n n ζi+1,j+1 − 2ζi,j−1 + ζi−1,j−1 − 2ζi,j+1 + ζi−1,j+1 α ζi+1,j−1 + + 2 ∆x2 ∆x2   n n n n n n ζi+1,j+1 − 2ζi−1,j + ζi−1,j−1 − 2ζi+1,j + ζi+1,j−1 α ζi−1,j+1 + + = 0, 2 ∆y 2 ∆y 2 (10.7)

n+1/2

vi,j

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and n+1 n − ζi,j ζi,j n+1/2 − vi,j + γ∆x2 ∆t   n+1/2 n+1/2 n+1/2 n+1/2 n+1/2 n+1/2 + vi−1,j + vi,j−1 vi+1,j − 2vi,j vi,j+1 − 2vi,j = 0, × + ∆x2 ∆y 2

(10.8)

where indices i (and j) and n represent a spatial grid point and a time level, respectively, as shown in Fig. 10.1. ∆x(= ∆y) is the spatial grid size, and ∆t denotes the time step. The weighting factor α to get orientation-free solution is given as 1/6, and the dispersion-correction parameter γ is given by γ=

∆x2 − (4h2 + gh∆t2 ) . 12∆x2

(10.9)

If the dispersion-correction parameter γ calculated by (10.9) for given local water depth, spatial grid size, and time step are used, a considerably accurate solution can be achieved for the linear Boussinesq wave equation (10.4). Thus, the selection of grid size ∆x is free from the condition proposed by Imamura et al.15 This means that the present dispersion-correction scheme is much more flexible in the selection of grid size than the conventional scheme. The stability analysis of the dispersion-correction numerical scheme shows that the stability criterion is Cr ≤ 0.67 and −0.125 ≤ γ ≤ 0.083, where Cr is the Courant number. The lower limit of γ imposes a limitation on the ratio of grid size to water depth as 1.27. This means that the uniform grid size greater than approximately 1.27 times of a local water depth must be employed for the stability of the present dispersion-correction scheme. To satisfy the stability criterion of the dispersion-correction finite-difference model, the range of applicable water depth is limited if a uniform grid size is used for varying water depth. For practical purposes, this stability criterion can be solved by imposing intentionally a limitation on the

Fig. 10.1. Sketch of the arrangement of variables and grid points; (a) grid system in space, (b) grid system in time.

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γ value within the range of −0.125 ≤ γ ≤ 0.083. This means that the γ value is set to be −0.125 if the γ calculated by (10.9) is less than −0.125. The γ value is always less than 0.083. Thus, the upper limit of γ is automatically satisfied for very large grid size. 10.3. Test of Numerical Scheme 10.3.1. Tsunami propagation over constant depth region In order to test the applicability of the present model, the propagation of tsunamis is first simulated with an initial Gaussian shape of water surface for the case of various constant water depths, and the computed free surface displacements are compared with the analytical solutions of the linear Boussinesq equations.19 The initial free surface profile and the velocity of free surface movement are described by (10.10) and (10.11), respectively, as 2

ζ(r, θ) = 2e−(r/a) ,

(10.10)

∂ζ(r, θ) = 0, ∂t

(10.11)

where a is the characteristic radius of Gaussian function, r(= (x2 + y 2 )1/2 ) denotes the distance from the center of the Gaussian hump, and θ is the angle from the x-axis, as shown in Fig. 10.2. The analytical solution of the linear Boussinesq equations is given by Carrier19 as   √ ∞ 2 ghkt  ζ(r, t) = a2 e−(ak) /4 k cos  (10.12) J0 (kr)dk, 2 0 1 + (kh) 3 where J0 is the 0th order Bessel function of the first kind and g = 9.81 m/s2 . The parameters used for the solution are a = 7500 m, ∆x = 2086 m, and ∆t = 6 s. The numerical simulations using the present scheme are performed for various constant water depths of h = 500 m, 1000 m, and 1500 m. Figures 10.3 and 10.4 present the comparisons of analytical solutions and numerical results calculated considering or neglecting the dispersion-correction

Fig. 10.2. Coordinate system and initial free surface profile to test the accuracy of the present numerical scheme.

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Fig. 10.3. Comparison of free surface calculated using FDM and analytical solution for the case of h = 500 m (∆xIm = 1085 m, ∆x = 2086 m): (a) γ = 0.0, α = 0.015 ; (b) γ = 0.0, α = 1/616 ; (c) γ = 0.061, α = 1/6 (present).1

parameters γ and the weighting factor α. We remark here that the numerical scheme of Imamura et al.15 is recovered if we set intentionally γ = α = 0.0. The orientation-free scheme of Cho16 is obtained if γ = 0.0 and α = 1/6. These figures show a time history of free surface displacements at the location of 150∆x(r = 312, 900 m) far from the center of the initial Gaussian hump. The free surface profiles calculated by the present finite-difference model considering or neglecting dispersion-correction are compared with the analytical solution of Carrier19 for the case of 500 m water depth in Fig. 10.3.

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Fig. 10.4. Comparison of free surface calculated using FDM and analytical solution for the case of h = 1500 m (∆xIm = 3087 m, ∆x = 2086 m): (a) γ = 0.0, α = 0.015 ; (b) γ = 0.0, α = 1/616 ; (c) γ = −0.099, α = 1/6 (present).1

Since the grid size ∆x(= 2086 m) employed in the computationis larger than ∆xIm (= 1085 m) evaluated by Imamura condition, i.e., ∆xIm = 4h2 + gh∆t2 , the free surface profiles calculated without dispersion-correction are more dispersive than those of the analytical solutions as shown in Fig. 10.3(a). Especially, the numerical model gives a less dispersive profile along the diagonal direction (θ = 45◦ ) than along principal axes (θ = 0◦ , 90◦ ), while the numerical solution with α = 1/6 shows no directional dependency as shown in Fig. 10.3(b). The present model with the dispersion-correction parameter γ = 0.061 and the weighting factor α = 1/6

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gives a good agreement with the analytical solution in the surface profiles in all directions as shown in Fig. 10.3(c). On the other hand, for the case of 1500 m water depth, ∆x(= 2086 m) is smaller than ∆xIm (= 3087 m). Thus, the numerical solutions calculated with γ = 0.0 show less dispersive nature in the surface profiles than the analytical solutions based on the linear Boussinesq equations as shown in Figs. 10.4(a) and 10.4(b). The present numerical model using γ = −0.099 and α = 1/6, however, still gives a correct dispersion effect in each direction as shown in Fig. 10.4(c). From the results discussed above, it is concluded that the present numerical model is less sensitive to the choice of grid size than the models of Imamura et al.15 and Cho.16 As a result, the present model gives more flexibility in selecting grid size. 10.3.2. Tsunami propagation over a submerged circular shoal In this section, the validity of the present finite-difference model (FDM) in the case of slowly varying topography is tested using a submerged circular shoal. Figure 10.5 shows a computational domain with a submerged circular shoal with a conical frustum shape. The center of the shoal is located at x0 = 500 km and y0 = 250 km. The base radius R1 and the top radius R2 are 150 km and 86 km, respectively. The bathymetry of the computational domain is given by  r ≥ R1 ,  1500 2 2 1500r /R1 R2 < r < R1 , h(r) = (10.13)  500 r ≤ R2 ,  where r = (x − x0 )2 + (y − y0 )2 is the distance from the center of the submerged circular shoal. Along x = 0, the initial water surface displacement in the form of a solitary wave is prescribed as 2

ζ(x) = 2e−(x/7500) .

(10.14)

Four wave gages are installed to measure the free surface profile. Wave gages ➀ and ➂ are positioned at the front and back slopes of the shoal, respectively, where

Fig. 10.5.

Schematic diagram of a submerged circular shoal, wave gages, and initial condition.

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the water depth is 1000 m. Gage ➁ is located at the center of the shoal where the water depth is 500 m. Gage ➃ is placed behind the shoal. Sponge layers are placed along both ends of the computational domain to absorb the energy of outgoing waves. Since no analytical solution is known for this case, numerical solutions using the present dispersion-correction finite-difference model are compared with those of the linearized Nwogu’s Boussinesq equations20 implemented in the FUNWAVE code.6,21 FUNWAVE is a fully nonlinear Boussinesq wave model with improved dispersion relationships for short waves. The accuracy of FUNWAVE has been verified for various coastal problems such as shoaling, refraction, diffraction, and breaking of waves. The numerical simulation using FUNWAVE is performed with a uniform grid size of 500 m, the finest grid allowable on the personal computer employed in this study, to minimize the numerical dispersion. An additional computation using FUNWAVE is also conducted with 2000 m grid size to test the sensitivity of the grid resolution. On the other hand, the numerical simulation employing the present dispersion-correction model is made with a uniform grid size of 2000 m. The time step ∆t is determined from the stability criteria for each wave model. The algorithms to compute various physical processes, such as nonlinear advection, nonlinear dispersion, and wave dissipation due to bottom friction and breaking, are eliminated from the source code of the FUNWAVE model. Thus, the computational time for FUNWAVE model to calculate only the propagation step of small-amplitude waves can be measured for fair comparison with that of the present model. The numerical simulation is conducted for 9000 s after the initial water surface displacement imposed along x = 0 is released. The computational time elapsed for different models is presented in Table 10.1. The FUNWAVE model employing a predictor-corrector scheme consumes a long computational time, while the present fully explicit model takes only 1/10 of the computational time required for FUNWAVE in the case of using the same grid size, i.e., ∆x = 2000 m. The computational efficiency of the present model can be realized even more dramatically if the computational time is compared with that of the FUNWAVE using finer grid of ∆x = 500 m. The present model is approximately 2200 times faster than the FUNWAVE model. If the accuracy of the present model is comparable to that of FUNWAVE, it can be concluded that the present model is highly efficient for practical problems. Figure 10.6 presents the comparison of the free surface time history at various gage locations computed by FUNWAVE and the present FDM. Figure 10.6(a) shows

Table 10.1.

Comparison of computational time for each simulation.1

Model

FUNWAVE

FUNWAVE

Present FDM

∆x (m) Number of grids ∆t (s) Number of time steps CPU time (s)

500 3001×1001 2 4500 156,888

2000 751×251 8 1125 770

2000 751×251 4 2250 72

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Fig. 10.6. Comparison of time history computed by FUNWAVE and the present FDM: (a) location ➀ (h = 1000 m); (b) location ➁ (h = 500 m); (c) location ➂ (h = 1000 m); (d) location ➃ (h = 1500 m).1

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the free surface time history recorded at the wave gage ➀ located on the front slope of the shoal. This figure shows that the initial solitary wave evolves into a group of waves due to dispersion of waves. Figure 10.6(b) presents the time history at the location ➁ on the top of the shoal. The free surface calculated by the present model shows a good agreement with that of the FUNWAVE with fine grid of ∆x = 500 m. The results calculated using FUNWAVE with 2000 m grid size, however, show a considerable phase lag in the tail of wave train. This phase lag is caused by the accumulation of numerical dispersion errors due to poor resolution of computational grid near the top of the shoal. Although the same grid size of 2000 m is employed, the present model still gives more accurate results. This implies that the present model is capable of minimizing the numerical dispersion efficiently through the dispersion-correction algorithm. Figure 10.6(c) shows the time history at location ➂ on the back slope of the shoal. Two trains of waves arrive here. The first train of waves propagating around the shoulders of the shoal meets at this location at t = 6000 s. The second train of waves traveling over the top of the shoal arrives here at t = 6300 s. A good agreement between the results calculated using the present model and the FUNWAVE with fine grids is achieved. The results calculated using the FUNWAVE with coarse grids, however, suffers from the numerical dispersion. Figure 10.6(d) shows the time history at location ➃ behind the shoal where the water depth h is 1500 m. The agreements between the numerical solutions are reasonable. In summary, the present FDM is proven to be sufficiently accurate in comparison with the FUNWAVE model which can deal with a full coverage of dispersion effects for the varying water depth region. On top of this, the present model is shown to be highly efficient. Thus, the present model can be used as a practical numerical model to simulate the propagation of trans-oceanic tsunamis over slowly varying topography.

10.4. Near-Field Model As the tsunamis propagate over a continental shelf and approach a coastal area, the dispersion effect of waves becomes weak, and the nonlinearity and bottom friction of waves dominantly influence the transformation of the tsunamis. However, the linear Boussinesq-type wave equation does not include the nonlinear and bottom friction terms. Thus, the nonlinear shallow-water equations (NSWE) are employed for the near-field transformation of tsunamis. ∂ζ ∂P ∂Q + + = 0, ∂t ∂x ∂y     ∂P ∂ P2 ∂ PQ ∂ζ gn2  + + + gD + 7/3 P P 2 + Q2 = 0, ∂t ∂x D ∂y D ∂x D    2 ∂ PQ ∂ Q ∂ζ gn2  ∂Q + + + gD + 7/3 Q P 2 + Q2 = 0, ∂t ∂x D ∂y D ∂y D

(10.15) (10.16)

(10.17)

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where n denotes the Manning’s roughness coefficient (sec/m1/3 ) for the bottom friction, and D(= h + ζ) is the total water depth. A conventional leap-frog finite difference scheme with wet–dry technique17 can be employed.

10.5. Grid Nesting Scheme In the central area of the East Sea of Korea, there exists a huge submerged shoal called Yamato Rise. The depth difference between shallow and deep areas is significant. To achieve the best performance from the dispersion-correction model, the grid size for deep area of 4 km water depth should be approximately 6 km. This grid size causes a problem of poor resolution for the shallow water area near the top of submerged shoal, because the wavelength becomes shorter due to shoaling. Thus, the accuracy of the simulation will be seriously degraded. To overcome this difficulty, a nesting between coarser grid for deep area and finer grid for shallow area is indispensable. Yoon17 developed a dynamic grid nesting scheme for the finite-difference model to solve shallow-water equations. This grid nesting scheme uses a second-order interpolation in space and a first-order interpolation in time. Numerical tests of the nesting scheme developed by Yoon17 show some reflection for short waves along the nesting line. To improve the performance of the nesting scheme, a new dynamic grid nesting scheme has been developed by Lim et al.2,18 for the present finite-difference model based on the linear Boussinesq-type wave equations. They employed a third-order interpolation in space and a second-order interpolation in time. The higher order interpolation helps short waves propagate with weak reflection from the nesting line. Figures 10.7(a) and 10.7(b) show the layout of variables in space and time, respectively, to nest the coarse and fine grid regions. The

Fig. 10.7. Layout of variables for grid nesting; (a) grid nesting system in space, (b) grid nesting system in time.

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grid size and the time step for the fine grid region are 1/3 of the values of the coarse grid region. Along the boundary between two regions, the information calculated in one region should be transferred to another region for each time step. When the information is transferred from the coarse grid region to the fine grid region, both spatial and temporal interpolations of the variables of the coarse grid region are required. Lim et al.2 tested their grid nesting scheme for the propagation of a linear solitary wave over a submerged circular shoal with a significant depth difference between the top and the base of the shoal. Numerical solutions calculated using their nested model were compared with those of the linearized Nwogu’s Boussinesq equations20 implemented in the FUNWAVE code.6,21 They showed that the present model system is approximately 4000 times faster than the FUNWAVE model, while the accuracy of the present model is comparable to that of FUNWAVE. Thus, it can be concluded that the present model system is highly efficient for practical problems. To link the near-field model using nonlinear shallow-water equations to the farfield model employing linear Boussinesq-type wave equations, special care should be taken because the governing equations are different between two regions. If the dispersion-correction is neglected along the nesting line, i.e., γ = 0.0, (10.6) can be simplified to give ∂ζ = v. ∂t

(10.18)

Furthermore, the continuity equation (10.15) of the nonlinear shallow-water equations can be rewritten as   ∂ζ ∂P ∂Q =− + . (10.19) ∂t ∂x ∂y The right-hand side of (10.18) and (10.19) should be identical. Thus, the auxiliary variable v can be approximated as   ∂Q ∂P + . (10.20) v=− ∂x ∂y The auxiliary variable v calculated in the far-field propagation model using LBTWE is transferred to the near-field model in the form of the divergence of flow rate as a boundary condition to solve the continuity equation of NSWE. The auxiliary variable v required for the far-field model can be computed by (10.20) using the flow rates P and Q calculated in the near-field model.

10.6. Numerical Simulations of Historical Tsunami Event A historical tsunami event, i.e., the 1993 Hokkaido Tsunami, is selected to verify the nested dispersion-correction finite-difference model (NDCFDM). The calculated free surface displacements are compared with the tide gage records obtained at two harbors: Mukho and Ulsan harbors, along the east coast of Korea.

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10.6.1. Computational information For the simulation of the historical tsunami event, the computational domain bounded by four latitudinal and longitudinal lines, i.e., 127◦ E, 142◦ E, 33◦ N, and ◦ 45 N for west, east, south, and north boundaries, respectively, are selected as shown in Fig. 10.8. This figure presents the bathymetry of the East Sea. For the finitedifference model, the computational domain is divided into 900 × 931 square grids, for which the grid spacing is 1 min in the longitudinal direction. To make square grids, the grid spacing in the latitudinal direction is adjusted to give the same grid size of 1 min grid spacing in the longitudinal direction at each grid location.18 This can be done if the following condition is satisfied. ∆φ = cos φ∆ψ,

(10.21)

where φ and ψ represent the latitude and longitude of grid point, respectively. ∆φ and ∆ψ are the radian grid size in latitudinal and longitudinal directions, respectively. Since ∆ψ is chosen as 1 min, ∆φ is obtained by multiplying ∆ψ by cos φ, and ∆x is identical with ∆y everywhere. Thus, the actual grid size is given by ∆x = ∆y = R cos φ∆ψ,

(10.22)

where R(= 6.378 × 106 m) is the radius of the earth. Subsequently, the finitedifference mesh system taken into account the curvature of the earth can be regarded

Fig. 10.8.

Computational domain and bathymetry of the East Sea (depth unit: m).

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as locally uniform, and the actual grid size slowly changes in association with the latitude of grid point. It is approximately 1.4 km around the central part of computational domain. The time step ∆t is chosen as 3 s, and the actual grid size ∆x slowly varies in the range of 1312–1556 m. The dispersion-correction parameters γ calculated by given conditions such as time step ∆t, actual spatial grid ∆x, and local depth h, give the values between −2.93 and 0.08 in the whole computational domain. As mentioned above, the minimum value of γ is restricted to −0.125 to maintain the stability for the deep areas of the East Sea. Due to this limitation, the dispersion effects are underestimated in the numerical solutions. However, the present scheme gives more accurate solutions than the conventional models which use a uniform γ value, i.e., γ = 0.0, everywhere. As they approach the shallow-water region, the wavelength of tsunamis becomes short due to shoaling, and the resolution of grids becomes lower. Thus, a set of finer grid system, B, C, D, and E, is nested dynamically to the coarse grid region A. As shown in Fig. 10.8, two subregions, Mukho (C1) and Ulsan (C2) harbors, are selected for computations with finer grid system, because tide gage records are available at these harbors. The computational information such as grid size and time step employed for the nesting is listed in Table 10.2. The subregions are coupled by the nesting scheme as proposed above. The linear Boussinesq-type wave equation (LBTWE) is employed to compute the regions, A, B, C, and D. For the finest grid regions, E1 and E2, the nonlinear shallow-water equations (NSWE) are solved because the nonlinearity increases as the water depth decreases. Manning-type energy dissipation for bottom friction is implemented in the nonlinear model. The Manning’s roughness coefficient is set to be equal to 0.05 s/m1/3 . To verify the applicability of the present model to the real topography, the calculated tsunami-free surface elevation is compared with the elevation measured by tide gage records at two harbors, Mukho and Ulsan harbors. The bathymetry of computational subregions and the locations of tide gage at Ulsan harbor are presented in Fig. 10.9. Two kinds of boundary conditions, no-flux boundary and absorbing boundary, are mainly used in the tsunami propagation model. No-flux boundary condition is employed to express the rigid impermeable wall, and an absorbing boundary condition proposed by Larsen and Darcy22 is placed in front of the open-sea boundary. The fault parameters of the historical tsunami event, such as epicenter location ◦ ◦ in latitude ( N) and longitude ( E), depth of upper rim of fault H(km), orientation

Table 10.2.

Computational information for subregions of each harbor.

Region

Grid spacing ∆ψ

Time step ∆t (s)

Remark

A B C1, C2 D1, D2 E1, E2

1 min 20 s 6.67 s 2.22 s 0.74 s

3.0 1.0 0.33 0.11 0.037

LBTWE LBTWE LBTWE LBTWE NSWE

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Fig. 10.9.

Computational subregions and detailed bathymetry near Ulsan harbor (depth unit: m).

Fig. 10.10.

Definition sketch for fault parameters of three historical tsunami events.

◦

◦

◦

of fault with strike angle θ( ), dip angle δ( ), and slip angle λ( ) from the north, length of fault L(km), width of fault W (km), and dislocation of fault D(m), are defined in Fig. 10.10. It is assumed that the water surface displacement created by the earthquake is identical to the vertical displacement of sea bed induced by the ground motion.

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The displacement field can be determined by using the fault model proposed by Mansinha and Smylie.23 10.6.2. Simulation of the 1993 Hokkaido Tsunami The magnitude of the 1993 Hokkaido Tsunami which originated from the deep sea near the Okushiri Island of Hokkaido on 12 July 1993 was recorded as M7.8. The source is composed of three faults. The south fault has the largest dislocation of 12 m. The fault parameters for the tsunami event proposed by Takahashi et al.24 are listed in Table 10.3. Figure 10.11 presents the initial free surface profile of the 1993 Tsunami. The 1993 Tsunami is simulated. Figure 10.12 shows the propagation map with 5 min time interval obtained from the numerical simulation. Figure 10.13 shows the distribution of the highest water levels obtained from the numerical simulation Table 10.3.

Fault parameters for the 1993 Tsunami.24

Fault No.

Lat. ◦ ( N)

Long. ◦ ( E)

H (km)

θ ◦ ( )

δ ◦ ( )

λ ◦ ( )

L (km)

W (km)

D (m)

1 2 3

42.10 42.34 43.13

139.30 139.25 139.40

5 5 10

163 175 188

60 60 35

105 105 80

24.5 30 90

25 25 25

12.00 2.50 5.71

Fig. 10.11.

Initial free surface profile of the 1993 Tsunami (unit: m).2

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Fig. 10.12.

Fig. 10.13.

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Distribution of calculated highest water levels due to the 1993 Tsunami (unit: m).2

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Fig. 10.14. Comparison of calculated and measured time history of free surface displacements of the 1993 Tsunami at (a) Mukho harbor and (b) Ulsan harbor.2

of the 1993 Tsunami. The Shimane Peninsula (JF) of Japan and the Imwon area (KA) of the east coast of Korea are shown to be vulnerable to the tsunami attacks generated in the East Sea because of the large wave heights due to the wave trapping over the Yamato Rise (YM) and the submerged ridges. The calculated time histories of free surfaces at two harbors are compared with the tide gage records of the 1993 Tsunami in Fig. 10.14. The calculated results are generally in good agreement with measured data in many respects even though there are slight phase and height differences. 10.6.3. Effect of topography on propagation of tsunamis As shown in Fig. 10.15, the underwater topography of the East Sea of Korea is complicated. In the central part of the sea there exists a huge submerged shoal called Yamato Rise (YM). This rise is connected to the Shimane Peninsula (JF) of Japan by a submerged ridge. Two submerged ridges which protrude eastward from the Imwon area (KA) on the east coast of Korea form a K-shaped underwater topography. These underwater rises and ridges play a significant role of a waveguide for the tsunamis generated in this sea. Figure 10.15 shows a sketch of paths along which the main tsunami energy is transmitted for the case of the 1993 Tsunami. The tsunami waves are first captured by Yamato Rise due to topographical lens

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Fig. 10.15.

Schematic diagram of energy stream path for the 1993 Tsunami.2

effect, and then trapped again by the underwater ridge connecting the Yamato Rise and the Shimane Peninsula of Japan. As a result, the main stream of tsunami energy radiated toward the open sea turns back to the Shimane Peninsula (JF) of Japan. On the other hand, the K-shaped submerged ridges catch the tsunami energy propagating toward the east coast of Korea and carry it along the two branch ridges toward the Imwon area (KA) where two ridges meet. Thus, the tsunami energy is focused at this spot. Even though these regions, i.e., Shimane Peninsula (JF) of Japan and Imwon (KA) of Korea, are located far from the tsunami source area, these two regions are vulnerable to the attack of various tsunamis generated in the East Sea of Korea.

10.7. Conclusions A practical finite-difference model developed by Yoon et al.1 and Lim et al.2,18 for the efficient and relatively accurate simulation of tsunami propagation is introduced. This model solves a linear Boussinesq-type wave equation to achieve proper dispersion effect of waves. The present model is applied to simulate the propagation of a historical tsunami event occurred in the East Sea of Korea. The calculated free surface displacements for the 1993 Hokkaido Tsunami are compared with the observations at two tidal stations along the east coast of Korea. The comparison shows

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that the results agree well with the observations. Based on the simulated results of the historical tsunami that occurred in the East Sea of Korea, it is found that the underwater topography, such as submerged rises and ridges, plays an important role in the propagation of tsunamis in this region. The Yamato Rise located in the central part of the sea behaves as a topographic lens to focus the tsunami energy, and the submerged ridge connecting the rise to the Shimane Peninsula of Japan behaves like a waveguide along which the focused energy is trapped and carried to the Shimane Peninsula. The K-shaped submerged ridges exert a similar influence on the tsunami incident on the Imwon area in the east coast of Korea.

Acknowledgment This research was supported by a grant (No. NEMA-06-NH-06) from the Natural Hazard Mitigation Research Group, National Emergency Management Agency of Korea.

References 1. S. B. Yoon, C. H. Lim and J. Choi, Dispersion-correction finite difference model for simulation of transoceanic tsunamis, Terres. Atmos. Oceanic Sci. CGU 18(1), 31–53 (2007). 2. C. H. Lim, J. S. Bae, J. I. Lee and S. B. Yoon, Propagation characteristics of historical tsunamis that attacked the east coast of Korea, Natural Hazards 47(1), 95–118 (2008). 3. J. F. Lander and P. A. Lockridge, United States Tsunamis (including United States Possessions) 1690–1988 (National Geophysical Data Center, Boulder, Colorado, 1989). 4. K. Satake, J. Bourgeois, K. Abe, Y. Tsuji, F. Imamura, Y. Iio, H. Katao, E. Noguera and F. Estrada, Tsunami field survey of the 1992 Nicaragua Earthquake, EOS Trans. AGU 74, 156–157 (1993). 5. Hokkaido Tsunami Survey Group, Tsunami devastates Japanese coastal region, EOS Trans. AGU 74, 417, 432 (1993). 6. J. T. Kirby, G. Wei, Q. Chen, A. B. Kennedy and R. A. Dalrymple, FUNWAVE 1.0 Fully nonlinear Boussinesq wave model, Documentation and User’s Manual (CACR98-06) Technical report, Center for Applied Coastal Research, Ocean Engineering Laboratory, University of Delaware (1998). 7. P. Lynett and P. L.-F. Liu, A two-layer approach to water wave modeling, Proc. Roy. Soc. London A 460, 2637–2669 (2004). 8. S.-C. Hsiao, P. Lynett, H.-H Hwung and P. L.-F. Liu, Numerical simulations of nonlinear short waves using the multi-layer model, J. Eng. Mech. ASCE 131(3), 231–243 (2005). 9. M. Ioualalen, J. Asavanant, N. Kaewbanjak, S. T. Grilli, J. T. Kirby and P. Watts, Modeling the 26 December 2004 Indian Ocean tsunami: Case study of impact in Thailand, J. Geophys. Res. 112, C07024 (2007), doi:10.1029/2006JC003850. 10. S. T. Grilli, M. Ioualalen, J. Asavanant, F. Shi, J. T. Kirby, P. Watts, Source constraints and model simulation of the December 26, 2004, Indian Ocean tsunami, J. Waterway, Port, Coastal Ocean Eng. ASCE 133(6), 414–428 (2007).

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11. L. S. Hwang and D. Divoky, Tsunami generation, J. Geophys. Res. 75, 6802–6817 (1970). 12. C. Goto and N. Shuto, Numerical simulation of tsunami propagations and run-up, Tsunami — Their Science and Engineering, eds. K. Iida and T. Iwasaki (Terra Science Publishing Company, Tokyo, 1983), pp. 439–451. 13. Z. Kowalik and T. S. Murty, Computation of tsunami amplitudes resulting from a predicted major earthquake in the Shumagin seismic gap, Geophys. Res. Lett. 11, 1243–1246 (1984). 14. C. L. Mader and G. D. Curtis, Numerical modeling of tsunami inundation of Hilo harbor, JIMAR Contribution No. 91–251, University of Hawaii, Honolulu (1991). 15. F. Imamura, N. Shuto and C. Goto, Numerical simulation of the transoceanic propagation of tsunamis, Proc. 6th Congress Asian and Pacific Regional Division, IAHR, Japan (1988), pp. 265–271. 16. Y. S. Cho, Numerical simulations of tsunami propagation and run-up, PhD thesis, School of Civil and Environmental Engineering, Cornell University, Ithaca, NY (1995). 17. S. B. Yoon, Propagation of distant tsunamis over slowly varying topography, J. Geophys. Res. 107(C10), 3140 (2002), doi:10.1029/2001JC000791. 18. C. H. Lim, J. S. Bae, Y. J. Jeon and S. B. Yoon, Grid nesting for dispersion-correction finite difference model for tsunami simulation, Proc. 32nd Congress of the International Association of Hydraulic Engineering and Research, Venice, Italy, Paper No. D2.b-050-O (2007). 19. G. F. Carrier, Tsunami propagation from a finite source, Proc. 2nd UJNR Tsunami Workshop, NGDC, Hawaii (1991), pp. 101–115. 20. O. Nwogu, Alternative form of Boussinesq equations for nearshore wave propagation, J. Waterway, Port, Coastal Ocean Eng. 119(6), 618–638 (1993). 21. G. Wei and J. T. Kirby, A time-dependent numerical code for the extended Boussinesq equations, J. Waterway, Port, Coastal Ocean Eng. 121(5), 251–261 (1995). 22. J. Larsen and H. Dancy, Open boundaries in short wave simulations — A new approach, Coastal Eng. 7, 285–297 (1983). 23. L. Mansinha and D. E. Smylie, The displacement fields of inclined faults, Bull. Seismol. Soc. Am. 61, 1433–1440 (1971). 24. T. Takahash, N. Shuto, F. Iammura and M. Ortis, Fault model to describe Hokkaido Nansei offshore earthquake for tsunami, J. Coastal Eng. 41, 251–255 (1994) (in Japanese).

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Chapter 11

Tsunami-Induced Forces on Structures Ioan Nistor∗ , Dan Palermo, Younes Nouri, Tad Murty and Murat Saatcioglu Department of Civil Engineering, University of Ottawa 161 Louis Pasteur, CBY, A115, Ottawa, Canada ∗ [email protected] This chapter deals with the estimation of tsunami-induced hydrodynamic forces on infrastructure located in the vicinity of the shoreline. While extensive research has been conducted on the impact of hydrodynamic forces on classical coastal protection works (breakwaters, seawalls, reefs, etc.), there is limited research on their impact on structures such as buildings and bridges located inland. The devastation brought by the 26 December 2004 Indian Ocean Tsunami on coastal communities in Indonesia, India, Sri Lanka, Thailand, and other countries outlined the urgent need for research on the evaluation of structural resilience of infrastructure located in tsunami-prone areas. This chapter summarizes the state-ofthe-art knowledge with respect to forces generated by tsunami-induced hydraulic bores, including debris impact. Further, sample calculations of tsunami loading on a prototype structure are presented.

11.1. Introduction Tsunami waves represent extreme, often catastrophic events, which significantly and adversely impact coastal areas. In spite of the lower frequency of occurrence comparing to storms and storm-induced surges, tsunami-induced coastal flooding often leads to massive casualties and tremendous economic losses.1–3 Hence, tsunamis are rare events, high-impact natural disasters. The devastating effects of the 26 December 2004 Tsunami on many countries bordering the Indian Ocean raised public concern and revealed existing deficiencies within the current warning and defense systems against tsunamis. One of the important elements that needs significant improvement is the estimation of forces generated by tsunami-induced bores, as well as water-borne debris. Before the 2004 Indian Ocean Tsunami, the design of structures against tsunami-induced forces was considered of minor importance when compared to the attention given to tsunami warning systems. This was due to the assumption that tsunamis are 261

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(a)

(b)

(c)

(d)

Fig. 11.1. Tsunami damage in Thailand and Indonesia (December 2004 Indian Ocean Tsunami): (a) severe structural damage, Khao Lak, Thailand; (b) column failure of a reinforced concrete frame, Phuket, Thailand; (c) column failure due to debris impact, Banda Aceh, Indonesia; (d) punching failure of infill walls, Banda Aceh, Indonesia.4

rare events, with significantly high return periods (sometimes more than 500 years). Reconnaissance missions of the December 2004 Indian Ocean Tsunami disaster revealed that tsunami-induced forces can lead to severe damage or collapse of structures as shown in Fig. 11.1.3–11 Therefore, these forces should be properly accounted for in the design of infrastructure built within a certain distance from the shoreline in tsunami-prone areas. The design of coastal structures such as breakwaters, jetties, and groins against waves is typically based on considering the effect of breaking waves and their associated forces, and is well established. Unlike coastal structures, the evaluation of tsunami-induced hydrodynamic forces on structures used for habitation and/or economic activity, received little attention by researchers and engineers. Results of field surveys conducted in the aftermath of the December 2004 Indian Ocean Tsunami in Indonesia and Thailand showed that poorly detailed concrete structures experienced severe damage.3,4 This highlighted the fact that the current structural design codes do not account for tsunami-induced forces and the impact of associated debris. Reinforced concrete structures have been observed to withstand tsunamis with acceptable low levels of damage.12 However, as shown in

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: The 2004 Indian Ocean Tsunami : Past Tsunamis Damage Partial Damage Withstand

0

1

2

3

4

5

6

7

Lack of data

20

Inundation Depth (m) Fig. 11.2. Relation between the inundation depth and degree of damage to reinforced concrete buildings.13

Fig. 11.2, inundation depths of more than 5 m can induce partial damage to concrete structures. Currently, there are no clearly established procedures to address the aforementioned forces. Moreover, significant disagreement on existing empirical formulae fostered new research interest in an effort to properly address the inclusion of both tsunami-induced forces and the impact of debris into design codes. Aspects related to these forces are discussed in this chapter. Some of the shortcomings and inconsistencies of existing codes are also highlighted.

11.2. Literature Review 11.2.1. Tsunami-induced hydraulic bores As tsunami waves advance toward the shoreline and water depth decreases, their wave height increases while celerity decreases. Tsunami waves may break offshore or at shoreline, inundating low-lying coastal areas in the form of a hydraulic bore. On the other hand, tsunami inundation can also occur as a gradual rise and recession of the sea level for the case of nonbreaking tsunami waves. The width of the continental shelf, the initial tsunami wave shape, the beach slope, and the tsunami wave length are the parameters which govern the breaking pattern of tsunami waves.2,15 Figure 11.3 shows a tsunami wave approaching the Khao Lak Beach in Thailand during the December 2004 Indian Ocean Tsunami. Tsunami waves have a larger horizontal length scale compared to the vertical. Consequently, implementing shallow water wave theory (i.e., depth-integrated equations of momentum and mass conservation with the assumption of a hydrostatic pressure field) seems to be a reasonable method for representing tsunami wave propagation. Using these equations, it has been shown that the behavior and runup of nonbreaking tsunami waves can be predicted with acceptable accuracy. However, disagreement has been observed between experiment and prediction, in terms of behavior and runup of breaking waves and the resulting bore.16,17

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(a)

(b)

(c)

(d)

Fig. 11.3. Tsunami wave in Khao Lak, Thailand (December 2004 Indian Ocean Tsunami): (a) water recedes; (b) waves approach the shoreline; (c) tsunami waves break close to the shoreline; (d) tsunami waves inundate the shoreline.14

Significant efforts were directed toward the experimental investigation of the mechanisms of tsunami bore runup.2,18,19 Although a two-dimensional dam-break phenomenon was used in the experiments, the bore motion was observed to be fully three-dimensional and highly turbulent. This is in agreement with other observations regarding the irregularity of the bore front in transverse direction and the noticeable fluctuation of the front propagation.19 The onshore propagation of the tsunami wave is similar to the classical dambreak problem. Chanson20 compared the instantaneous free surface flow profiles of a tsunami-induced bore with floating bodies to a dam-break flow on a horizontal bed. A frame-by-frame analysis of a video recording taken during the Indian Ocean Tsunami in Banda Aceh, Indonesia was used to obtain the flow profile of the tsunami-induced bore. The agreement between the tsunami field data and the dam-break analytical formulation demonstrated the analogy between propagation of tsunami-induced bores and dam-break flow. Direct estimation of tsunami inundation bore velocities is limited. Tsutsumi et al.21 estimated the nearshore flow velocity of the tsunami caused by the Southwest Hokkaido Earthquake of 12 July 1993. Forces exerted on a bent iron handrail and an iron guardrail were estimated using in situ strength tests. Then, the velocity of the tsunami flow was calculated from the forces using Morison’s equation.22

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Matsutomi et al.13 and Chanson20 estimated inundation flow velocities in terms of inundation depth. Frame-by-frame analysis of video recordings from the 2004 Indian Ocean Tsunami in Banda Aceh was conducted. The videos showed that the inundation flow carried numerous floating bodies with approximately the same speed as that of the bore. This may have influenced the bore characteristics. Furthermore, Matsutomi et al.13 developed an empirical model that predicts the bore front velocity. Also, surveyed data were used to improve the criterion for estimating the degree of structural damage.

11.2.2. Tsunami-induced forces on structures Snodgrass et al.23 noticed that broken waves induced larger hydrodynamic horizontal forces on a test pile compared to waves breaking at the pile location. As previously mentioned, broken tsunami waves inundate shoreline in the form of a hydraulic bore, which is a fast moving body of water with an abrupt front. However, mechanisms of impingement of broken tsunami waves on structures located inland are not yet well understood. Pioneering analytical and experimental attempts to quantify forces due to bores date back to Stoker,24 Cumberbatch,25 Fukui,26 Cross,27 and Dames and Moore.28 Comprehensive experimental investigation of the interaction of bores and drybed surges with a vertical wall was performed by Ramsden and Raichlen29 and Ramsden.30,31 In these experiments, three flow conditions were analyzed: (1) turbulent bores (initial still water downstream of the gate); (2) dry-bed surges (no initial water depth downstream of the gate); and (3) solitary waves. Forces and overturning moments due to bores and dry-bed surges were recorded and calculated, respectively. The results of Ramsden’s studies are not applicable to the estimation of impulsive forces.31 It was observed that the pressure distribution during impact is essentially nonhydrostatic. The experiment also demonstrated that the transition from undular to turbulent bores led to a discontinuous increase in water-surface slope, followed by an increase in measured runup, pressure head, and exerted forces and moments. Figure 11.4 shows the difference between a strong turbulent bore and a dry-bed surge with approximately the same celerity. It was shown that recorded forces gradually increased to an approximately constant value for both the case of a surge and a bore. No impulsive (shock) force exceeding the hydrodynamic force was observed. However, an initial impulsive pressure equal to three times the pressure head, corresponding to the measured runup, was recorded. Ramsden31 further derived empirical formulae for the maximum force and moment exerted on a vertical wall due to the bore impact [Eqs. (11.1) and (11.2)].
3 H 1 + , 7160 h
2
3
 1 H H M H 1 + = 1.923 + 0.454 + , Ml h 8.21 h 808 h F = 1.325 + 0.347 Fl




H h



1 + 58.5




H h

2

(11.1) (11.2)

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Fig. 11.4. Comparison of: (a) wave profile; (b) runup; (c) pressure head; (d) force due to a strong turbulent bore and a dry-bed surge.31

where F is the force on the wall; Fl is the force on the wall due to a runup equal to twice the wave height, assuming hydrostatic pressure; H is the wave height at the wall; h is still water depth; M is the moment on the wall; and Ml is the moment corresponding to Fl . Okada et al.32 conducted a survey of previous studies on tsunami wave forces and pressures, and identified five empirical formulae for tsunami-induced forces or pressures. It was found that calculation of tsunami load on structures using these formulae would result in approximately the same magnitude of load. These formulae are as follows: • • • • •

Tsunami wave pressure without soliton breakup,33 Tsunami wave pressure with soliton breakup,33 Tsunami wave pressure without soliton breakup,34 Tsunami-induced wave forces on houses,35 Tsunami force exerted on houses.36

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Fig. 11.5.

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Measured and nondimensional force history for a square column.37

Arnason37 measured forces exerted on rectangular, rhomboidal, and circular structures due to a hydraulic bore on a dry bed. It was observed that the surge force overshot the hydrodynamic force in the case of a square column for small bore heights (Fig. 11.5). However, no overshoot was recorded for the case of circular and rhomboidal columns. This is in agreement with the results obtained by Nouri et al.,38 where similar experiments with larger bore heights, up to three times those of Arnason,37 were performed (Fig. 11.6). Nouri et al.38 conducted experiments with the objective of estimating boreinduced forces on free-standing structural components. The effect of other parameters such as upstream obstacles, flow constrictions, and debris impact was also investigated. The structural components were subjected to a dam-break flow generated by impoundment depths of 0.5, 0.75, and 0.85 m.

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350

Force (N)

250

150 h0 = 1.00 m h0 = 0.85 m

50

-50 10

Fig. 11.6.

12

14

16 Time (s)

22

60

Height of the structure (cm)

Height of the structure (cm)

20

Time history of exerted forces on a circular structure38 ; h0 is the impoundment depth.

60 50

t=0.000 s t=0.006 s t=0.006 s t=0.009 s

40 30 20 10

50

t=0.160 s t=0.170 s t=0.180 s t=0.190 s

40 30 20 10

0 -2

18

0

3

8

13

18

-2

3

8

13

18

P (kPa)

P (kPa)

Fig. 11.7. Variation of pressure distribution on the front face — circular structure38 ; t = 0.0 s is the instant when the bore impacts the structure.

The variation of the vertical distribution of pressure was measured. Selected snapshots from the variation of pressure distribution due to bore impact generated by an impoundment depth of 1.0 m are shown in Fig. 11.7. 11.2.3. Debris impact force Matsutomi39 performed small- and full-scale experiments on impact forces generated by driftwood on rigid structures. Dam-break waves generated in a small flume carried pieces of lumber to the point of impact on a downstream wall. Also, full-scale experiments in which wooden logs impacted a frame were conducted in open air, and impact forces were measured. An empirical formula for estimating the impact force, F , was derived using regression analysis of collected data [Eq. (11.3)]. F = 1.6CM γw D 2 L




u √ gD

1.2


σf γw L

0.4 ,

(11.3)

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Fm/γ D2L

σf / γL = 2000 1500 120

1000 500 80

200 100 40

20

0 0.0

1.0

Fig. 11.8.

2.0

3.0

0.5

VA0/(gD)

Impact forces of wood logs for bores and surges.39

where γw is the specific weight of wood, D is the diameter of the log, L is the length of the log, CM is a coefficient which depends on the flow passing around the receiving wall (≈ 1.7 for bore or surge, and 1.9 for steady flow), u is the velocity of the log at impact, and σf is the yield stress of the log. Figure 11.8 shows the design chart based on Eq. (11.3). Currently, three basic models are proposed for estimating the forces due to the impact of debris on structures, which are used by a few design codes. In these models, the impact force is calculated based on the mass and velocity of debris, while ignoring the mass and rigidity of the structure. However, other than the mass and velocity of debris, each model needs an additional parameter. The three models and their corresponding additional parameters are • Contact stiffness — stiffness between debris and structure, • Impulse–momentum — stopping time of debris after impact and time history of impact, • Work energy — distance traveled from where initial contact occurs, to where debris stops. Haehnel et al.40,41 used a single-degree-of-freedom model with effective collision stiffness as an additional parameter. They reviewed the current models discussed above and demonstrated that none of the additional parameters are independent. Hence, the three models are equivalent, provided that additional parameters are appropriately selected. Further, small- and large-scale experiments were performed in order to develop the single-degree-of-freedom model. Small-scale tests were performed in a flume where wooden logs were released into the flow and impacted a load frame located further downstream. Large-scale experiments were performed in

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Fig. 11.9.

Effect of impact orientation on force.40

a large basin where water was stationary and logs were placed on a movable carriage. The effect of parameters such as added mass of the water and the eccentricity and obliqueness of the collision were also considered. Figure 11.9 shows the effect of impact orientation on the measured force. It was found that the maximum impact force, Fi,max , can be calculated using Eq. (11.4).  ˆ =u Fi,max = Maxkx

ˆ 1, km

(11.4)

where u is the impact velocity of the log, kˆ is the constant effective stiffness between the log and the structure, and m1 is the mass of the log. Based on experiments, Haehnel et al.40 found the value of kˆ = 2.4 MN/m to be the representative for the upper envelope of the collected data.

11.3. Existing Design Codes The design of structures in flood-prone areas has previously been investigated. However, few existing codes specifically address the design of onshore structures built in tsunami-prone areas. Design codes that specifically address tsunami-induced forces were introduced in order to suggest provisions for designing infrastructure in tsunami-prone areas. Post-tsunami field investigations of the December 2004 Indian Ocean Tsunami are indicative of the extreme loads generated by tsunamiinduced floods,3,4 and have outlined the need for developing new design guidelines. Recent research work42 indicated that tsunami-induced loads are comparable or can exceed earthquake loads. Tsunami-induced forces and the impact of debris are not properly accounted for in the current codes, and significant improvement is needed.

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At present, only four design codes and guidelines specifically account for tsunamiinduced loads as listed below: • FEMA 55: The code is adopted by the Federal Emergency Management Agency, the United States, and recommends formulae for tsunami-induced flood and wave loads.43 • The City and County of Honolulu Building Code (CCH): The code, developed by the Department of Planning and Permitting of Honolulu, Hawaii, United States, makes provisions for regulations that apply to districts located in flood and tsunami-risk areas.44 • Structural Design Method of Buildings for Tsunami Resistance (SMBTR): The code is proposed by the Building Center of Japan32 and outlines the structural design for tsunami refuge buildings. • Development of Guidelines for Structures that Serve as Tsunami Vertical Evacuation Sites: The guidelines were prepared by Yeh et al.45 for estimating tsunamiinduced forces on structures for the Washington State Department of Natural Resources. There are several other design codes (sometimes country-specific) which contain prescriptions and design guidelines for flood-induced loads. Examples of widely used codes are indicated below: • 1997 Uniform Building Code, Appendix 33, proposed by the International Conference of Buildings Officials (ICBO),47 • ASCE 7-05 of the American Society of Civil Engineers,48 • 2006 International Building Code by the International Code Council.49 However, none of the above codes address directly the tsunami-induced forces, which represent the focus of this chapter. The reader is advised to refer to these codes when seeking guidance for the design of structures subjected to flood-induced loads other than tsunamis: coastal flooding due to storm surges, flooding of river banks above bank-full conditions, etc. 11.3.1. Tsunami-induced forces A broken tsunami wave running inland generates forces which affect structures located in its path. Three parameters are essential for defining the magnitude and application of these forces: (1) inundation depth, (2) flow velocity, and (3) flow direction. The parameters mainly depend on: (a) tsunami wave height and wave period; (b) coast topography; and (c) roughness of the coastal inland. The extent of tsunami-induced coastal flooding, and therefore the inundation depth at a specific location, can be estimated using various tsunami scenarios (magnitude and direction) and modeling coastal inundation accordingly. However, the estimation of flow velocity and direction is generally more difficult. Flow velocities can vary in magnitude from zero to significantly high values, while flow direction can also vary due to onshore local topographic features, as well as soil cover and obstacles. Forces associated with tsunami bores consist of: (1) hydrostatic force, (2) hydrodynamic

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(drag) force, (3) buoyant force, (4) surge force, and (5) impact of debris. A brief description of these forces is further presented. 11.3.1.1. Hydrostatic force The hydrostatic force is generated by still or slow-moving water acting perpendicular onto planar surfaces. The hydrostatic force per unit width, FHS , can be calculated using Eq. (11.5), where ρ is the seawater density, g is the gravitational acceleration, ds is the inundation depth, and up is the normal component of flow velocity. Equation (11.5) is proposed by CCH and accounts for the velocity head. Alternatively, FEMA 55 does not include the velocity head in its formulation since it is assumed to be a negligible component of the hydrostatic force:

FHS

 2 u2p 1 = ρg dS + . 2 2g

(11.5)

The point of application of the resultant hydrostatic force is located at one-third distance from the base of the triangular hydrostatic pressure distribution. In the case of a broken tsunami wave, the hydrostatic force is significantly smaller than the drag and surge forces. Conversely, Dames and Moore28 noted that the hydrostatic force becomes important when tsunami is similar to a rapidly-rising tide. 11.3.1.2. Buoyant force The buoyant force is the vertical force acting through the center of mass of a submerged body. Its magnitude is equal to the weight of the volume of water displaced by the submerged body. The effect of buoyant forces generated by tsunami flooding was clearly evident during post-tsunami field observations.1,5,6 Buoyant forces can generate significant damage to structural elements, such as floor slabs, and are calculated as follows: FB = ρgV,

(11.6)

where V is the volume of water displaced by submerged structure. 11.3.1.3. Hydrodynamic (drag) force As the tsunami bore moves inland with moderate to high velocity, structures are subjected to hydrodynamic forces caused by drag. Currently, there are differences in estimating the magnitude of the hydrodynamic force. The general expression for this force is shown in Eq. (11.7). Existing codes use the same expression, but different drag coefficient values (CD ). For example, values of 1.0 and 1.2 are recommended for circular piles by CCH and FEMA 55, respectively. For the case of rectangular

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piles, the drag coefficient recommended by FEMA 55 and CCH is 2.0: ρCD Au2 , (11.7) 2 where FD is the drag force acting in the direction of flow, A is the projected area of the body normal to the direction of flow, and u is the tsunami-bore velocity. The flow is assumed to be uniform, and therefore, the resultant force will act at the centroid of the projected area. As indicated, the hydrodynamic force is directly proportional to the square of the tsunami-bore velocity. The estimation of the bore velocity remains one of the critical elements on which there is significant disagreement in literature. A brief discussion on the tsunami-bore velocity is presented below. FD =

Tsunami-bore Velocity. Previous research shows that significant differences in estimating forces exerted on structures by tsunami bores, as well as impact of debris, are due to differences in estimating bore velocity. Since the hydrodynamic force is proportional to the square of the bore velocity, uncertainties in estimating velocities induce large differences in the magnitude of the resulting hydrodynamic force. Tsunami-bore velocity and direction can vary significantly during a major tsunami inundation. Current estimates of the velocity are crude; a conservatively high flow velocity impacting the structure at a normal angle is usually assumed. Also, the effects of runup, backwash, and direction of velocity are not addressed in the current design codes. Although there is certain consensus in the general form of equation for the hydrodynamic force, several researchers proposed different empirical coefficients. The general form of the bore velocity is shown below [Eq. (11.8)]:  u = C gds , (11.8) where u is the bore velocity, ds is the inundation depth, and C is a constant coefficient. Various formulations were proposed by FEMA 55 (based on Dames and Moore28 ), Iizuka and Matsutomi,35 CCH,44 Kirkoz,50 Murty,51 Bryant,52 and Camfield53 for estimating the velocity of a tsunami bore in terms of inundation depth (Fig. 11.10). Velocities calculated using CCH and FEMA 55 represent a lower and upper boundary, respectively. 11.3.1.4. Surge force The surge force is generated by the impingement of the advancing water front of a tsunami bore on a structure. Due to lack of detailed experiments specifically applicable to tsunami bores running up the shoreline, the calculation of the surge force exerted on a structure is subject to substantial uncertainty. Accurate estimation of the impact force in laboratory experiments is a challenging and difficult task. CCH recommends using Eq. (11.9), based on Dames and Moore28 : FS = 4.5ρgh2 , where FS is the surge force per unit width of wall and h is the surge height.

(11.9)

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16

CCH: u=h FEMA 55: u=2(gh)^0.5 Iizuka: u=1.1(gh)^0.5 Kirkoz: u=(2gh)^0.5 Murty: u=1.83(gh)^0.5 Bryant: u=1.67h^(0.7)

V (m/s)

12

8

4

0 0

2

4

6

d S (m) Fig. 11.10.

Comparison of various tsunami-bore velocities as a function of inundation depth.42

building qx Design inundation depth

3h z h

z 3ρgh x Fig. 11.11.

Tsunami wave pressure for structural design recommended by SMBTR.32

The point of application of the resultant surge force is located at a distance h above the base of the wall. This equation is applicable to walls with heights equal to, or greater than 3h. Structural walls with height less than 3h require surge forces to be calculated using an appropriate combination of hydrostatic and drag force for each specific situation. SMBTR recommends using the equation for tsunami wave pressure without soliton breakup derived by Asakura et al.33 [Eq. (11.10)]. The equivalent static pressure resulting from the tsunami impact is associated with a triangular distribution where water depth equals three times the tsunami inundation depth (Fig. 11.11): qx = ρg(3hmax − z),

(11.10)

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where qx is the tsunami wave pressure for structural design, z is the height of the relevant portion from ground level (0 ≤ z ≤ 3h), ρ is the mass per unit volume of water, and g is the gravitational acceleration. Integration of the wave pressure formula for walls with heights equal to or greater than 3h results in the same equation as the surge force formula recommended by CCH [Eq. (11.9)]. The magnitude of the surge force calculated using Eqs. (11.9) and (11.10) will generate a value equal to nine times the magnitude of the hydrostatic force for the same flow depth. However, a number of experiments31,37 did not capture such differences in magnitude. Yeh et al.45 commented on the validity of Eq. (11.9) and indicated that this equation gives “excessively overestimated values.” On the other hand, Nakano and Paku46 conducted extensive field surveys in order to examine the validity of the proposed tsunami wave pressure formula [Eq. (11.10)]. The coefficient 3.0 in Eq. (11.10) was taken as a variable, α, and was calculated such that it could represent the boundary between damage and no damage in the surveyed data. A value of α equal to 3.0 and 2.0 was found for walls and columns, respectively. The former is in agreement with the proposed formulae by both CCH and SMBTR [Eqs. (11.9) and (11.10)]. The tsunami wave force may be composed of drag, inertia, impulse, and hydraulic gradient components. However, SMBTR does not specify different components for the tsunami-induced force, and the proposed formula presumably accounts for other components. 11.3.1.5. Debris impact force A high-speed tsunami bore traveling inland carries debris such as floating automobiles, floating pieces of buildings, drift wood, boats, and ships. The impact of floating debris can induce significant forces on a building, leading to structural damage or collapse.5,6 Both FEMA 55 and CCH codes account consistently for debris impact forces, using the same approach and recommend using Eq. (11.11) for the estimation of debris impact force: Fi = mb

dub ui =m , dt ∆t

(11.11)

where Fi is the impact force, mb is the mass of the body impacting the structure, ub is the velocity of the impacting body (assumed equal to the flow velocity), ui is approach velocity of the impacting body (assumed equal to the flow velocity), and ∆t is the impact duration taken equal to the time between the initial contact of the floating body with the building and the instant of maximum impact force. The only difference between CCH and FEMA 55 resides in the recommended values for the impact duration which has a noticeable effect on the magnitude of the force. For example, CCH recommends the use of impact duration of 0.1 s for concrete structures, while FEMA 55 provides different values for walls and piles for various construction types as shown in Table 11.1. According to FEMA 55, the impact force (a single concentrated load) acts horizontally at the flow surface or at any point below it. Its magnitude is equal to

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Table 11.1. Impact duration of floating debris (FEMA 55). Impact duration (s) Type of construction

Wall

Pile

Wood Steel Reinforced concrete Concrete masonry

0.7–1.1 N.A. 0.2–0.4 0.3–0.6

0.5–1.0 0.2–0.4 0.3–0.6 0.3–0.6

the force generated by 455 kg (1000-pound) of debris traveling with the bore and acting on a 0.092 m2 (1 ft2 ) surface of the structural element. The impact force is to be applied to the structural element at its most critical location, as determined by the structural designer. It is assumed that the velocity of the floating body goes from ub to zero over some small finite time interval (∆t). Finding the most critical location of impact is a trial and error procedure that depends, to a large extent, on the experience and intuition of the engineer. 11.3.1.6. Breaking wave force Tsunami waves tend to break offshore and approach shoreline as a broken hydraulic bore or a soliton, depending on the wave characteristics and coastal bathymetry. Consequently, classic breaking wave force formulae are not directly applicable to the case of tsunami bores. Hence, this chapter does not discuss the estimation of breaking wave forces. 11.3.2. Loading combinations for calculating tsunami-induced forces Based on the location and type of structural elements, appropriate combinations of tsunami-induced force components (hydrostatic, hydrodynamic, surge, buoyant, and debris impact force) should be used in calculating the total tsunami force. This is due to the fact that a certain element may not be subjected to all of these force components simultaneously. Loading combinations can significantly influence the total tsunami force and the subsequent structural design. Unlike the case of tsunami waves, loading combinations for flood-induced surges are well-established and have been included in design codes. The literature review revealed that proposed tsunami loading combinations must be significantly improved and incorporated into new design codes. Tsunami-induced loads are different from flood-induced loads. Therefore, load combinations based on flood surges are not directly applicable to tsunamis. Loading combinations proposed in the literature are as follows: (i) FEMA 55 does not provide loading combinations specifically for calculation of tsunami force. However, flood load combinations can be used as guidance. Flood load combinations for piles or open foundations, as well as solid walls (foundation) in flood hazard zones and coastal high hazard zones are presented

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as follows: Pile or open foundation: Fbrkp (on all piles) + Fi (on one corner or critical pile only), or Fbrkp (on front row of piles only) + Fdyn (on all piles but front row) + Fi (on one corner or critical pile only). Solid (wall) foundation: Fbrkw (on walls facing shoreline, including hydrostatic component) + Fdyn (assumes one corner is destroyed by debris), where Fbrkp , Fi , Fdyn , and Fbrkw refer to breaking force on piles, impact force, hydrodynamic force, and breaking force on walls, respectively. The reader can refer to FEMA 5543 for more details. (ii) Yeh et al.45 modified flood load combinations provided by FEMA 55 and adapted them for tsunami forces as follows: Pile or open foundation: Fbrkp (on column) + Fi (on column), or Fd (on column) + Fi (on column), where Fd is the drag force. Solid (wall ) foundation (perpendicular to flow direction): Fbrkw (on walls facing shoreline) + Fi (on one corner), or Fs (on walls facing shoreline) + Fi (on one corner), or Fd (on walls facing shoreline) + Fi (on one corner), where Fs is the surge force on walls. (iii) Dias et al.54 proposed two load combinations called “point of impact” and “post-submergence/submerged” (Fig. 11.12). These load combinations are based on two conditions: (i) the instant that tsunami bore impacts the structure, and (ii) when the whole structure is inundated. Point of impact: Fd (on walls facing shoreline) + Fs (on walls facing shoreline), where Fs is defined as the hydrostatic force by Dias et al.54 Post-submergence/submerged: Fd (on walls facing shoreline) + Fb (on submerged section of the structure). The net hydrostatic force is zero and Fb (γV) is the buoyant force. (a)

(b)

Fd

Fd

Fs

Fs W

Fs W- V

Fig. 11.12. Loading combinations: (a) point of impact/not submerged; and (b) post-submergence/ submerged.54

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(a)

Fi FS

h

(b)

Fi Fd

dS

FHS Fig. 11.13.

FHS

Proposed loading conditions: (a) point of impact; and (b) post-impact.42

(iv) Nouri et al.42 proposed two new load combinations based on the two conditions considered by Dias et al.,54 as shown in Fig. 11.13. The proposed load combinations by Nouri et al.42 are adapted to follow a consistent format as the above combinations: Columns: Fs (on front row of piles only) + Fi (on one corner or critical column in the front row only), or Fd (on all piles) + Fi (on one corner or critical column only), where Fs is the surge force on walls. Solid (wall) foundation: Fs (on walls facing shoreline) + Fi (on walls facing shoreline), or Fd (on walls facing shoreline) + Fi (on one critical wall facing shoreline) + Fb (on submerged section of the structure). 11.4. Design Example Building codes provide guidance for the design of lateral force resisting systems subjected to wind and seismic excitations. Tsunami-induced loading is normally not considered. The objective of this example is to demonstrate the levels of lateral loading associated with tsunamis for a prototype reinforced concrete building located in a tsunami-prone area. Specifically, the loads generated by a tsunami bore are addressed. Other researchers have provided comparisons between tsunami loading and other lateral loads (Okada et al.,32 Pacheco and Robertson,55 Nouri et al.,42 and Palermo et al.56 ).

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6.0

6.0

6.0 m

279

6.0

6.0

6.0

A

B 6.0

Tsunami

6.0

C

D 1 Fig. 11.14.

2

3

4

5

6

Plan view of structural layout of reinforced concrete moment-resisting frame.56

The following example consists of a moment-resisting frame with simple geometry, as shown in Fig. 11.14. The thickness of the slab is assumed to be 200 mm, and the beams are 450 mm deep (including the thickness of the slab) and 300 mm wide. The exterior and interior columns are 450 mm and 500 mm square sections, respectively. The center-to-center storey heights are 3.65 m, and a 10-storey structure is considered. The components of the tsunami-induced forces are calculated based on CCH, FEMA 55, and SMBTR. The calculation of tsunami-induced loads require a number of assumptions, together with engineering judgment and lessons from reconnaissance missions in regions affected by tsunamis. The authors assume that the net hydrostatic force exerted on the lateral system is zero in calculating the base shear. The surge and drag forces require an effective area for load transfer to the lateral force resisting system. In this example, two scenarios are considered: (1) 100% breakaway walls, which expose the structural elements; and (2) nonbreakaway walls. In the latter, the exterior nonstructural elements remain intact. For breakaway walls, the external elements will be damaged and all the columns of the structure will be subjected to drag force. The forces are calculated for inundation depths of 1–5 m, and the structure is oriented with its short side parallel to the shoreline. When designing structures located in tsunami-prone areas, the inundation depth at a specific location should be obtained from tsunami inundation hazard maps, when available. Otherwise, numerical modeling based on various tsunami scenarios should be conducted in order to estimate the tsunami inundation depth. 11.4.1. Hydrodynamic (drag) forces For the case where 100% breakaway walls are assumed and the columns are exposed to the hydraulic bore, the drag forces are based on a drag coefficient of 2.0 for square columns, as recommended by CCH and FEMA 55. For the second case, nonbreakaway walls are assumed to remain intact and the hydraulic bore impacts

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the entire surface of the building. In this situation, the drag coefficients are taken as 1.5 and 1.25 for CCH and FEMA 55, respectively. 11.4.2. Debris impact forces To calculate debris impact, a mass of 455 kg is used to represent a floating object at the water surface. The mass used is consistent with the recommendations of CCH and FEMA 55. In this example, it is assumed that the debris will impact a single reinforced concrete column. CCH assumes a duration of 0.1 s for concrete structures. FEMA 55 recommends an impact duration ranging from 0.3 to 0.6 s for reinforced concrete piles or columns. Hence, impact duration of 0.3 s is assumed in the design example. 11.4.3. Surge forces In this example, the surge force is applied over the full length of the building in the direction of the tsunami for nonbreakaway walls. For the case of 100% breakaway walls, it is assumed that the surge force will develop on the four exterior columns which face the hydraulic bore. Note that the surge force is not applicable for FEMA 55 and that SMBTR assumes a different surge force per unit width for columns and walls, as mentioned in Sec. 11.3.1.4. 11.4.4. Sample calculation: Breakaway walls The following is a sample calculation for breakaway walls for the given structure subjected to a tsunami inundation depth of 5 m: g = 9.81

m ; s2

ρ = 1030

kg ; m3

ds = 5 m.

Surge Force:
 kg  m 9.81 2 (5 m)2 FS = 2.0ρgh = 2.0 1030 3 m s 2

N ; SMBTR m
 6N (4(0.45 m)) = 909 × 103 N = 909 kN; FS × width = 0.505 × 10 m 
m kg  2 FS = 4.5 ρgh = 4.5 1030 3 9.81 2 (5 m)2 m s = 0.505 × 106

N ; CCH m 
6N (4(0.45 m)) = 2046 × 103 N = 2046 kN. FS × width = 1.14 × 10 m = 1.14 × 106

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Drag Force: ρCD Au2 2

FD =  u = C gds = 2

 m m 9.81 2 (5 m) = 14 ; s s

u = ds = 5

FEMA 55

m ; s

CCH

CD = 2.0 Rectangular columns; A = (5 m)((16 × 0.45 m) + 8(0.50 m)) = 56 m2 ;

FD =

 2 1030 kg/m3 (2)(56 m2 ) (14m/s) = 11,317 × 103 N = 11,317 kN; FEMA 55 2  2 1030 kg/m3 (2)(56 m2 ) (5 m/s) = 1442 × 103 N = 1442 kN. = 2

FD

CCH

Debris Impact Force: ui Fi = m = 455 kg ∆t




14m/s 0.3 s


Fi = 455 kg



5 m/s 0.1 s

= 21.2 × 103 N = 21 kN;

FEMA 55

= 22.8 × 103 N = 23 kN.

CCH



11.4.5. Sample calculation: Nonbreakaway walls The following is a sample calculation for nonbreakaway walls for the given structure subjected to a tsunami inundation depth of 5 m. g = 9.81

m ; s2

ρ = 1030

kg ; m3

ds = 5 m

Surge Force:
FS =

1.14 × 106

N m

 (18 m + 0.45 m)

= 20,973 × 103 N = 20,973 kN

SMBTR/CCH

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Drag Force: FD =

ρCD Au2 2

CD = 1.25 Walls

FEMA 55

CD = 1.5 Walls

CCH

A = (5 m)(18 m + 0.45 m) = 92 m2 ;

FD

 1030 kg/m3 (1.25)(92 m2 ) (14 m/s)2 = 2 = 11,652 × 103 N = 11,652 kN;

FEMA 55



 2 1030 kg/m3 (1.5) 92 m2 (5 m/s) = 1782 × 103 N = 1782 kN. = 2

FD

Debris Impact Force: Fi = m

 14 m/s = 21.2 × 103 N = 21 kN; 0.3 s 
5 m/s = 22.8 × 103 N = 23 kN. Fi = 455 kg 0.1 s

ui = 455 kg ∆t

CCH




FEMA 55 CCH

11.4.6. Results Tables 11.2 through 11.7 provide the results for the calculation of the individual force components for the structure considered using CCH, FEMA 55, and SMBTR, respectively. Given the force components, a loading combination must be specified in order to evaluate the maximum tsunami load that would be used for either design or analysis purposes. Yeh et al.45 suggested loading combinations that are applicable for tsunami loading. CCH does not specifically provide guidance to evaluate the maximum tsunami load. Nouri et al.42 proposed a two-part loading combination: Initial impact and Post-impact flow. For this example, these loading combinations are similar to those of Nouri et al.42 Table 11.8 provides the results of the tsunami Table 11.2. Tsunami-induced force components based on CCH for breakaway walls.

Code CCH

Inundation depth (m)

Velocity (m/s)

Surge (kN)

Drag (kN)

Debris impact (kN)

1 2 3 4 5

1 2 3 4 5

82 327 737 1310 2046

12 92 311 738 1442

5 9 14 18 23

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Table 11.3. Tsunami-induced force components based on CCH for nonbreakaway walls.

Code

Inundation depth (m)

Velocity (m/s)

Surge (kN)

Drag (kN)

Debris impact (kN)

1 2 3 4 5

1 2 3 4 5

839 3356 7550 13,423 20,973

14 114 385 912 1782

5 9 14 18 23

CCH

Table 11.4. Tsunami-induced force components based on FEMA 55 for breakaway walls.

Code FEMA 55

Inundation depth (m)

Velocity (m/s)

Drag (kN)

Debris impact (kN)

1 2 3 4 5

6 9 11 13 14

453 1811 4074 7243 11,317

10 13 16 19 21

Table 11.5. Tsunami-induced force components based on FEMA 55 for nonbreakaway walls.

Code FEMA 55

Inundation depth (m)

Velocity (m/s)

Drag (kN)

Debris impact (kN)

1 2 3 4 5

6 9 11 13 14

466 1864 4195 7457 11,652

10 13 16 19 21

load calculation for CCH, FEMA 55 and SMBTR for an inundation depth of 5 m based on loading combinations of Nouri et al.42 For the prototype moment-resisting frame structure with the short side perpendicular to the advancing bore, it is apparent that nonbreakaway walls or rigid exterior nonstructural components can lead to large design base shears. CCH and SMBTR estimate significantly larger base shears relative to FEMA 55 due to the omission of a surge component in FEMA 55. It is evident that the width of exposed surfaces affects the magnitude of total forces exerted on a structure. Therefore, it would be prudent to orient buildings such that the short side is placed parallel to the shoreline. Furthermore, using breakaway or flexible walls at the lower level would reduce the lateral force that is transmitted to the lateral force resisting system. Note that although the debris impact force is a negligible component in the base shear,

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Table 11.6. Tsunami-induced force components based on SMBTR for breakaway walls. Code SMBTR

Inundation depth (m)

Surge (kN)

1 2 3 4 5

36 146 327 582 909

Table 11.7. Tsunami-induced force components based on SMBTR for nonbreakaway walls. Code SMBTR

Inundation depth (m)

Surge (kN)

1 2 3 4 5

839 3356 7550 13,423 20,973

Table 11.8. Tsunami-induced load based on CCH, FEMA 55, and SMBTR for 5 m inundation depth.

Standard

Case

CCH CCH FEMA 55 FEMA 55 SMBTR SMBTR

Breakaway Nonbreakaway Breakaway Nonbreakaway Breakaway Nonbreakaway

Surge + Impact (kN)

Drag + Impact (kN)

Tsunami load (kN)

2069 20,995 N.A. N.A. 909 20,973

1465 1804 11,338 11,673 N.A. N.A.

2069 20,995 11,338 11,673 909 20,973

it could be critical in the design of individual structural components subjected to the debris impact.

Acknowledgment Special thanks to Dr Andrew Cornett, Group Leader — Coastal, Ports and Offshore at the Canadian Hydraulics Centre, National Research Council of Canada in Ottawa, Canada, for his valuable advice.

References 1. D. C. Cox and J. F. Mink, Bull. Seism. Soc. Am. 53, 1191–1209 (1963). 2. H. Yeh, Natural Hazards 4, 209–220 (1991).

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3. A. Ghobarah, M. Saatcioglu and I. Nistor, Eng. Struct. 28, 312–326 (2006). 4. I. Nistor, M. Saatcioglu and A. Ghobarah, Ann. Conf. Canadian Society for Civil Eng. (2006). 5. M. Saatcioglu, I. Nistor and A. Ghobarah, Earthquake Spectra (Earthquake Engineering Research Institute, 22, 2006), pp. 295–320. 6. M. Saatcioglu, I. Nistor and A. Ghobarah, Earthquake Spectra (Earthquake Engineering Research Institute, 22, 2006), pp. 355–375. 7. C. V. R. Murty, D. C. Rai, S. K. Jain, H. B. Kaushik, G. Mondal and S. R. Dash, Earthquake Spectra (Earthquake Engineering Research Institute, 22, 2006), pp. 321–354. 8. A. Ruangrassamee, H. Yanagisawa, P. Foytong, P. Lukkunaprasit, S. Koshimura and F. Imamura, Earthquake Spectra (Earthquake Engineering Research Institute, 22, 2006), pp. 377–401. 9. B. K. Maheshwari, M. L. Sharma and J. P. Narayan, Earthquake Spectra (Earthquake Engineering Research Institute, 22, 2006), pp. 475–493. 10. Y. Yamamoto, H. Takanashi, S. Hettiarachchi and S. Samarawickrama, Coastal Eng. J. 48(2), 117–145 (2006). 11. T. Tomita, F. Imamura, T. Arikawa, T. Yasuda and Y. Kawata, Coastal Eng. J. 48(2), 99–116 (2006). 12. N. Shuto, Tsunami Engineering Technical Report No.11, DCRC, Tohoku University (in Japanese) (1994). 13. H. Matsutomi, T. Sakakiyama, S. Nugroho and M. Matsuyama, Coastal Eng. J. 48(2), 167–195 (2006). 14. CNN, http://www.cnn.com/interactive/world/0502/gallery.tsunami.photos/frameset. exclude.html (2007). 15. H. Yeh, Int. Workshop on Fundamentals of Coastal Effects of Tsunamis, Hilo, Hawaii (2006). 16. S. Hibberd and D. H. Peregrine, J. Fluid Mech. 95, 323–345 (1979). 17. A. R. Packwood and D. H. Peregrine, Report AM-81-07, University of Bristol (1981). 18. H. Yeh, A. Ghazali and I. Marton, J. Fluid Mech. 206, 563–578 (1989). 19. H. Yeh and K. M. Mok, Phys. Fluids A 2, 821–828 (1990). 20. H. Chanson, Coastal Eng. J. 48(4), 355–370 (2006). 21. A. Tsutsumi, T. Shimamoto, E. Kawamoto and J. M. Logan, J. Waterway, Port, Coastal, Ocean Eng., ASCE 126(3), 136–143 (2000). 22. J. R. Morison, M. P. O’Brien, J. W. Johnson and S. A. Schaaf, Petroleum Trans., AIME 189, 149–157 (1950). 23. F. E. Snodgrass, E. K. Rice and M. Hall, University of California, Berkeley, CA, Inst. of Engineering Research, Technical Report Series 35, Issue 4, June (1951). 24. J. J. Stoker, Water Waves (Interscience Publishers, New York, 1957). 25. E. Cumberbatch, J. Fluid Mech. 7(3), 353–373 (1960). 26. Y. Fukui, M. Nakamura, H. Shiraishi and Y. Sasaki, Coastal Eng. Jpn. 6, 67–82 (1963). 27. R. H. Cross, PhD thesis, Dept. Civil Engineering, University of California, Berkeley, CA (1966), p. 106. 28. Dames and Moore, in Design and Construction Standards for Residential Construction in Tsunami-Prone Areas in Hawaii (Prepared for the Federal Emergency Management Agency) (1980). 29. J. D. Ramsden and F. Raichlen, J. Waterway Port, Coastal and Ocean Eng., ASCE 116(5), 592–613 (1990). 30. J. D. Ramsden, Report No. KH-R-54, W. M. Keck Laboratory, California Institute of Technology, Pasadena, California (1993), p. 251.

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31. J. D. Ramsden, J. Waterways, Port, Coastal and Ocean Eng. 122(3), 134–141 (1996). 32. T. Okada, T. Sugano, T. Ishikawa, S. Takai and T. Tateno, The Building Centre of Japan (2005). 33. R. Asakura, K. Iwase and T. Iketani, Proc. Coastal Eng. JSCE 47, 911–915 (2000). 34. M. Ikeno, N. Mori and H. Tanaka, Proc. Coastal Eng. JSCE 48, 846–850 (2001). 35. H. Iizuka and H. Matsutomi, Proc. Conf. Coastal Eng., JSCE 47 (2000) (in Japanese). 36. M. Omori, N. Fujii and O. Kyotani, Proc. Coastal Eng. JSCE 47, 376–380 (2000). 37. H. Arnason, PhD thesis, University of Washington, Seattle (2005), p. 172. 38. Y. Nouri, I. Nistor, D. Palermo and A. Cornett, Coastal Structures 2007, Venice, Italy (2007). 39. H. Matsutomi, J. Hyd. Coastal Environ. Eng. JSCE, No. 621/II-47, 111–127 (1999) (in Japanese, with English abs.). 40. R. B. Haehnel and S. F. Daly, Technical Report: ERDC/CRREL TR-02-2, US Army Corps of Engineers (2002), p. 40. 41. R. B. Haehnel and S. F. Daly, J. Hydraul. Eng. 130(2), 112–120 (2004). 42. Y. Nouri, I. Nistor, D. Palermo and M. Saatcioglu, 9th Canad. Conf. Earthquake Engineering, Ottawa, Canada, June (2007). 43. Federal Emergency Management Agency, Coastal Construction Manual (3 Vols.), 3rd edn. (FEMA 55) (Jessup, MD, 2003). 44. Department of Planning and Permitting of Honolulu Hawai, Chapter 16, City and County of Honolulu Building Code, Article 11 (2000). 45. H. Yeh, I. Robertson and J. Preuss, Report No 2005-4, Washington Dept. of Natural Resources (2005). 46. Y. Nakano and C. Paku, Summaries of technical papers of Annual Meeting Architectural Institute of Japan (Kinki) (2005). 47. UBC, Int. Conf. Building Officials, 1997 Uniform Building Code, California (1997). 48. ASCE Standard, Minimum design loads for buildings and other structures, SEI/ASCE 7-05, 424 (2006). 49. International Code Council (INC), 2006 International Building Code 2006, Country Club Hills, IL (2006), p. 675. 50. M. S. Kirkoz, 10th IUGG Int. Tsunami Symposium, Sendai-shi/Miyagi-ken, Japan (Terra Scientific Publishing, Tokyo, Japan, 1983). 51. T. S. Murty, Bull. Fisheries Res. Board of Canada, No. 198, Department of Fisheries and the Environment, Fisheries and Marine Service, Scientific Information and Publishing Branch, Ottawa, Canada (1977). 52. E. A. Bryant, Tsunami: The Underrated Hazard (Cambridge University Press, London, UK, 2001), p. 320. 53. F. Camfield, Tsunami Engineering (Coastal Engineering Research Center, US Army Corps of Engineers, 1980), p. 222, Special Report (SR-6). 54. P. Dias, L. Fernando, S. Wathurapatha and Y. De Silva, Int. Conf. Disaster Reduction on Coasts, Melbourne (2005). 55. K. H. Pacheco and I. N. Robertson, Evaluation of Tsunami Loads and Their Effect on Reinforced Concrete Buildings. University of Hawaii Research Report (2005). 56. D. Palermo, I. Nistor, Y. Nouri and A. Cornett, PROTECT 2007, Whistler, Canada (2007).

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Chapter 12

Nonconventional Wave Damping Structures Hocine Oumeraci Leichtweiss-Institute for Hydraulic Engineering and Water Resources Technical University Braunschweig Beethovenstr. 51a, 38106 Braunschweig, Germany [email protected] After a brief discussion of the necessity (i) to consider in addition to the reduction of wave heights in the sheltered area further aspects of the hydraulic performance when developing and designing new coastal structures for the protection against wave action and (ii) to develop a systematic “roadmap” of the existing concepts and types of structures together with their hydraulic performance characteristics as an important tool for practicing engineers and decision-makers, five examples from selected research studies performed in the last years by the Leichtweiss-Institute are presented in order to illustrate the development of nonconventional structures with substantially improved performance as compared to their conventional counterparts. These examples include (i) a multi-chamber structure to overcome the drawbacks of the perforated JARLAN-type breakwater concept, (ii) an artificial reef made of successive submerged permeable screens to increase the wave damping performance and to better control both wave reflection and wave transmission, (iii) a “High Mound Composite Breakwater” (HMCB) concept to decrease breaking wave loads, wave reflection, overtopping, and spray generation at the structure, (iv) an onshore wave damping barrier made of staggered walls, and (v) a rubble mound breakwater with a core made of geotextile sand containers.

12.1. Introduction In the past, the development of wave damping structures used in coastal protection and harbor engineering was rather directed toward achieving a maximum reduction of wave height in the sheltered area, while less attention was generally paid to further performance aspects such as (i) the reduction of wave loads, wave reflection, and wave overtopping, (ii) the reduction of environmental impacts (degradation of the marine landscape and of the water quality in the sheltered area, erosion of the foreshore, down coast erosion, etc.), 287

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(iii) the control of wave periods in the sheltered area, (iv) the reduction of spray generation at the structure, (v) the adaptability of the wave damping structure for multi-purpose use (recreational activities, wave power generation, etc.), and (vi) the reversibility of built structures if not performing as expected. In the future, the increasing interest in the practical implementation of the sustainability principles in coastal engineering18 will require that more effort should be put on the development of innovative low impact structures with better hydraulic performance characteristics, lower capital and maintenance costs for the proper sheltering of harbor and other facilities as well as for the proper protection of threatened coasts against storm surges. The main objective of this chapter is to illustrate, by means of three selected example research studies performed in the last years by the Leichtweiss-Institute (LWI), the process of developing and testing nonconventional structures. This process particularly consists of (i) a better understanding of the hydraulic functioning of the existing concepts, (ii) a clear identification of their drawbacks with respect to the commonly accepted and new emerging performance characteristics/requirements (see above), and (iii) a better control of the physical processes and structure parameters which contribute to the improvement of the hydraulic performance and to the reduction of wave loads. However, before starting with these example studies, a first attempt is made to discuss, why and how a parameterization of the existing concepts and structures for wave damping might help to develop a systematic “roadmap” of these concepts as an important decision tool for practicing engineers and coastal managers. 12.2. Parameterization of Wave Damping Structures: A First Attempt A large variety of concepts and structures for wave damping is available. However, the designer and coastal manager faced with the problem of finding the concept mostly suited for the intended use have no proper tool to quickly make the right choice before embarking into detailed investigations. For this purpose, a compact overview of the most important concepts/types of structures is needed, including the basic information on their hydraulic functioning (“roadmap”). The roadmap should not only provide the required performance characteristics. Moreover, the unwanted effects and impacts may also be directly provided or easily derived for each type of structure. The latter issue becomes increasingly important, because the choice will not, as in the past, mainly or exclusively be dictated by the wave damping efficiency, but also by further criteria associated with the sustainable protection of coastal zones.18 Unfortunately, such a practical decision tool, which is expected to substantially help both engineers and coastal managers, is still missing. Due to the considerable

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number of existing concepts/types of structures, it will certainly be very difficult or even impossible to elaborate any useful compact overview if a parameterization of the existing types of structures is not undertaken. In order to illustrate this first requirement, a very tentative parameterization is proposed for a single structure as a first starting base in Fig. 12.1. Such a parameterization may then lead to the very tentative compact overview in a parameterized form in Fig. 12.2.

Ei

Er

Et

SWL

H

h1

L ε

hs

h

h2 b

Fig. 12.1.

Tentative parameterization of single wave damping structure.

hs < h

hs > h

1.1

h1 = 0 h2 > 0

h1 > 0 h2 = 0

h1 = 0 h2 = 0 1.2

hs = 0

1.3

h1 > 0 h2 > 0 1.4

h1 + h2 = h 1.5

ε=0 impermeable

no structure

b=0 thin walls

2.1

2.2

2.3

2.4

2.5

ε≠ 0 permeable

no structure 3.1

3.2

3.3

3.4

3.5

4.1

4.2

4.3

4.4

4.5

ε=0 b>0

impermeable

wide structures

ε≠ 0 permeable

Fig. 12.2.

Tentative compact overview of existing wave damping structures in parametrized form.

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It is seen from Figs. 12.1 and 12.2 that only five parameters (structure width B, porosity ε, structure height dB , submergence depth h1 , and clearance h2 ) together with the water depth h are sufficient to achieve a parametric description of most types of existing wave damping structures. But this represents only a first step. The next important step is to provide the characteristic curves describing the hydraulic performance for each of the parametrized structure types in Fig. 12.2. For this purpose, the simplest procedure is to use the fact that the incident wave energy Ei is converted into a reflected (ER ), a dissipated (ED ), and a transmitted (ET ) energy component: Ei = ER +E D +E T .

(12.1)

Equation (12.1) written in terms of the reflection (KR ), dissipation (KD ), and transmission (KT ) coefficients yields 2 2 1 = KR + KD + KT2 .

(12.2)

If only the wave damping performance PW = 1 − KT2

(12.3)

is of interest, then the functioning curves PW = f (T ) such as those qualitatively described in Fig. 12.3 would be useful, because they clearly illustrate how each type of structure will respond over the full range of incident wave periods T . For instance, it is seen that the curtain wall founded on piles (Type 1.3) performs well for shorter wave periods T , but has a very low wave damping performance

Pw = K

2 R

+K

2 D

(

= 1 +KT

2

)

KR = Reflection Coeff.; KD = Dissipation Coeff.; KT = Transmission Coeff.

Wave damping Performance Pw

Vertical Breakwater (3.1)

1.0

Horizontal Breakwater on Piles (3.5) Curtain-Wall on Piles (1.3) Caisson Breakwater on Piles (3.3)

0.5 Submerged Breakwater (3.2)

0.0

0

Wave Period T [s] Fig. 12.3.

Wave damping performance (principle sketch!).

16

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Fig. 12.4. Hydraulic functioning of a single perforated wall under nonbreaking and nonovertopping wave conditions (adapted from Ref. 4).

for longer periods. Widening the wall to a caisson (Type 3.3) will considerably improve the wave damping performance over a wider range of wave periods. Very often, however, the wave damping performance alone is not sufficient and a “full” description of the hydraulic functioning as a function of the most relevant parameter governing wave reflection, wave transmission, and energy dissipation [see Eq. (12.1)] is required. For instance, in the case of a single surface piercing perforated screen (Type 2.1) and for nonbreaking and nonovertopping wave conditions, a kind of “dynamic porosity” (here reflection number RN ) which combines the influence of incident wave height Hi , water depth h, and wall porosity ε, is found to be the most appropriate parameter to describe the reflected, transmitted, and dissipated components of the total incident wave energy (Fig. 12.4). For most of the types of structure shown in Fig. 12.2, curves describing the hydraulic performance in terms of wave reflection, transmission, and damping already exist which are derived empirically, analytically, or numerically.2,11,19,21,22

12.3. Selected Example Studies for the Development of Nonconventional Wave Damping Structures In order to illustrate the processes associated with the development of nonconventional wave damping structures, the main results from three selected example research studies performed in the last years by LWI are briefly discussed below, including

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a multi-chamber caisson structure, an artificial reef made of a submerged progressive filter, a high mound composite breakwater (HMCB-concept), an onshore wave damping barrier, and a rubble mound breakwater with a core made of geotextile sand containers (GSC).

12.3.1. Multi-chamber caisson structure One of the most useful concepts to cope with the high reflection induced by vertical face breakwaters and seawalls is the perforated Jarlan-type breakwater which was introduced in 1960 in Canada.9 It consists of a single dissipating chamber bounded seaward by a perforated front wall (porosity ε ≈ 20%) and shoreward by an impermeable back wall (One Chamber System). The incident wave energy is partly reflected at the front wall and partly transmitted through the perforations into the wave chamber, where a certain amount of the incident wave energy is reflected by the back wall while a large part is dissipated due to resonance phenomena, vortices, and friction losses. The relative importance of the reflected and dissipated part of the total incident wave energy, and thus the hydraulic performance, depends on the porosity of the front wall, but is essentially governed by the ratio of the chamber width B and the wave length L of the incident waves (B/L). Although the JARLAN-type breakwater concept has been used more or less successfully worldwide, it has a basic drawback (see Fig. 12.6) which requires a further development of this concept. For this purpose, it was necessary to investigate first the key processes which contribute to the wave damping by friction (local losses and vortices) and by destructive interference of the incident and reflected waves over the full range of B/L ratios (i.e., over the full range of incident wave periods). The experimental results in Fig. 12.5 well illustrate how a traditional JARLAN-type caisson works. As shown by the upper curve in Fig. 12.6, the traditional JARLAN-type caisson (OCS) has, at its optimal working point (B/L ≈ 0.2), a much lower reflection coefficient (and thus a much larger energy dissipation) than a vertical impermeable wall. However, the response is very selective with respect to the incident wave periods; i.e., it performs satisfactorily only within a very narrow range of the B/Lratios. In order to overcome this drawback, a new Multi-Chamber System (MCS) was developed and tested in the Large Wave Flume of Hannover. As shown by the lower curve in Fig. 12.6, the new MCS concept not only provides a lower reflection coefficient; moreover this reflection coefficient is kept at its lowest level over the full range of practical B/L ratios (i.e., for B/L > 0.25, where B is defined as the overall width of the Multi-Chamber System. In addition to the substantial improvement of the hydraulic performance, which is achieved by the new multi-chamber concept, the new concept has also the + advantage to strongly reduce the resulting horizontal wave forces Ftotal (obtained by superposition of the forces acting simultaneously on each wall). Because wave forces are directly related to water surface elevation and thus to wave reflection,3,6 + + the total force Ftotal (related to force F0% on a single impermeable vertical wall

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impermeable back wall incident wave energy

B

Ei

Wave damping relative to incident wave energy Ei

Er

reflected wave energy

100%

wave chamber

perforated front wall

total wave damping 80%

wave damping by destructive interference

60% wave damping by friction

40% 20% 0% 0

0.1

0.2

0.3

0.4

0.5

relative chamber width B/L [-] Fig. 12.5.

Wave damping friction and destructive interference (Adapted from Ref. 5).

with zero-porosity) exhibits a very similar behavior to the reflection coefficient with respect to the B/L ratio (compare Figs. 12.6 and 12.7). This first example study has illustrated how a detailed insight into the physical processes responsible for the hydraulic functioning of an existing concept may lead to a clear identification of the drawbacks of this concept by indicating how to overcome them and how to achieve substantial improvements through the introduction of additional structure members. Given its potential to substantially reduce and better control wave reflection (less risk to navigation and less sea bed scour), wave loads, wave runup and overtopping, spray generation, etc., the new MultiChamber System represents an ideal alternative as a breakwater, jetty, and quay wall as well as a seawall for the protection of reclaimed sea fronts and artificial islands. Due to the flexibility of caisson structures to allow any shape and size, the seawalls can be adapted to incorporate promenades and any further facility for recreation activities, etc. 12.3.2. Submerged wave absorber as artificial reef for coastal protection An interesting cost effective and soft alternative to conventional seawalls for coastal protection against erosion are artificial reefs which have the advantage

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Fig. 12.6. Reflection coefficient versus relative chamber width for a traditional JARLAN-type caisson and for the new multi-chamber system (Adapted from Ref. 4).

Fig. 12.7. Resulting horizontal wave force versus relative chamber width for a traditional JARLAN-type caisson and for the new multi-chamber system (Adapted from Ref. 4).

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(i) to attenuate the waves before they reach the shoreline, (ii) to be invisible for viewers from the beach and therefore do not affect the marine landscape, (iii) to reduce the morphological impact on the foreshore (erosion) and on the neighboring coast (down-coast erosion), and (iv) to ensure a sufficient water exchange between the open sea and the sheltered area. However, the existing artificial reef concepts have serious drawbacks: the wave damping performance is limited and sometimes not sufficient. Moreover, the overall hydraulic performance is difficult to control, due to the limitations associated with the variation of the structure parameters, etc. Therefore, a reef concept made of submerged permeable screens with predetermined porosity and spacing has been experimentally tested in the Large Wave Flume of Hannover (GWK). As schematically shown in Fig. 12.8 for a three-filter system, this new reef concept is particularly appropriate for the protection of such coastal areas which are frequently used for recreation activities.12−14 Before starting with the systematic study on the hydraulic performance of this new reef concept, it was important to demonstrate first its efficiency with respect to the protection against beach erosion. For this purpose, a submerged two-filter system with porosities ε = 11% (front screen) and ε = 5% (back screen), spacing B = 10.3 m and height dB = 4 m was installed in front of a beach profile. The same beach profile was previously tested in the Large Wave Flume of Hannover17 without any protection under the same storm surge conditions (storm surge water level h = 5.0 m, TMA-wave spectrum with significant wave height Hs = 1.20 m and peak

Fig. 12.8.

Submerged three-filter system as an artificial reef for coastal protection.23

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Beach profile development under storm wave attack with and without reef protection.

period TP = 6.6 s, storm duration t = 10 h). The comparison of the results related to the development of the beach profile at the different stages of the tests with and without the reef structure as shown in Fig. 12.9 provides a good demonstration of the efficiency of the new reef concept as a soft protection alternative. In both cases (i.e., with and without protection), one can indeed observe a beach erosion, a net seaward transport, and a sand bar formation. However, the net seaward transport rate is about twice larger for the unprotected beach than for the protected beach. In addition, the reef causes the seaward transport to occur only within a limited narrow zone, so that this sand bar does not extend further seaward as in the case of an unprotected beach. Moreover, the eroded volume above the storm water level (h = 5.0 m) for the protected beach is only half as much as the eroded volume for the beach without any protection. As a result, the recession of the shoreline of the protected beach is only half as much as the recession of the unprotected beach (Fig. 12.10). Regarding the hydraulic performance of submerged wave absorbers, a brief illustration is given in Fig. 12.11, showing how a two-filter system damps the waves over the full range of B/L ratios. The contribution of the seaward screen to the total wave damping of the system varies between 30% for B/L ≈ 0.5 and 85% for B/L ≈ 0.3. The maximum wave damping performance of the system also occurs at B/L = 0.3, while the minimum value is at B/L = 0.5. These and further results with a three-filter system clearly show that for a given submergence depth (Rc /Hi ) the relative spacing B/L represents the most decisive parameter to describe the hydraulic performance of wave absorbers.23

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Fig. 12.10.

2.5

Beach profile after storm with and without reef (Adapted from Ref. 11).

dB/h = 0.98 wave damping at filter 1

2 relative wave height

Hi/ Ht

rel. damping [%]

297

1.5 H

1

wave damping at filter 2

Ht1/ Ht

i

H

0.5

H

r

t1

H

t

h dB =

Hi/Ht (measurement) Ht1/Ht (measurement)

0 100

B

3 .9 4 m

10m 11%

F i l ter 1

5% F i l ter 2

50 wave damping at filter 1/total wave damping

0 0

0.25

0.5

0.75

1

relative spacing B/L Fig. 12.11.

Wave damping of a submerged two-filter system (Adapted from Ref. 11).

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Fig. 12.12.

Hydraulic performance of submerged single screens and filter systems.

From the comparative analysis of the hydraulic performance shown in Fig. 12.12 for a submerged single screen with different porosities (ε = 0%, 5%, and 11%) and submerged two- or three-filter systems it is seen that (i) using a filter system instead of a single screen substantially increases the amount of dissipated energy, (ii) unlike a single screen, a filter system can substantially reduce and simultaneously control both wave reflection and wave transmission, (iii) using an optimized three-filter system, more than 80% of the incident wave energy can be dissipated, (iv) the relative submergence depth Rc /Hi is an important parameter for the wave damping performance of both single screens and filter systems, and (v) for the range of practical submergence depths (Rc /Hi ≈ −1), the highest improvement of the wave damping performance is achieved when using a twofilter system instead of a single filter. A further increase of the number of filters would lead to comparatively less improvement of the wave damping performance. Further results have indeed shown that the relative spacing B/L is much more important than the number and porosity of the filters to control the reflected,

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transmitted, and dissipated wave energy. A more detailed discussion on these issues is provided by Oumeraci and Koether23 and Koether.11 As mentioned in the introduction, there are many cases where the reduction of wave periods in the sheltered area may also be very important. Such situations occur, for instance, when the reef is principally used to control sediment transport or to reduce wave loads and overtopping at a seawall located behind the reef. An example result to illustrate the efficiency of the new reef concept by reducing both transmitted wave periods and heights is shown in Fig. 12.13 for two relative heights of a three-filter system. This second example study has shown again that substantial improvements of the existing concepts can only be achieved through a better insight into the mechanisms and processes governing the hydraulic performance. In fact, using a single submerged screen, only a very limited amount of incident wave energy can be dissipated, i.e., a decrease of wave transmission can only be achieved at the cost of the increase of wave reflection (see Ref. 23 for more results). Using a conventional reef made of rubble material would require a very wide structure and a progressive decrease of the porosity in wave direction in order to achieve a satisfactory wave damping performance. This is not only costly and difficult to construct and to maintain, but it is also very difficult to control the hydraulic performance by means of a variation of the structure parameters as it is the case for this new artificial reef concept.

Fig. 12.13.

Reduction of wave periods by a three-filter system.

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The understanding of the underlying mechanisms has shown that a new reef concept made of two or three submerged thin filters is an elegant and costeffective alternative to overcome most of the drawbacks of the existing reef concepts, including a substantial reduction and a better control of the reflected and transmitted components of the incident wave energy by means of the variation of the structure parameters (submergence depth, porosity, number and spacing of submerged slit walls.) A theoretical model to optimize submerged wave absorbers has been developed by Koether.11,23 The model has been successfully validated by largescale experimental data for regular and irregular waves as well as for submerged single-slit wall and wave absorbers with two and three filters. Preliminary tests in the Large Wave Flume of Hannover using comparatively a single-wall as well as a two-wall and a three-wall submerged wave absorber subject to 1 m high solitary waves have shown that this reef concept might also be applied for the protection against tsunami. In fact, the part of wave energy of the incident solitary waves dissipated is more than 75% and 85% for a two-wall and a threewall system, respectively. The experimental results showing the incident, reflected, and transmitted waves for a single-, two-, and three-wall systems are plotted and discussed by Oumeraci and Koether.23

12.3.3. High mound composite breakwater (HMCB) Based on a historical concept which was first used in 1830 in Cherbourg, France, and in 1890 in Alderney, UK, a new HMCB-concept has extensively been investigated within two joint research projects by LWI together with the Port and Harbor Research Institute (PHRI), Yokosuka, Japan and with the Civil Engineering Research Institute (CERI), Sapporo, Japan (Fig. 12.14).

Fig. 12.14.

Old and new high mound breakwater concepts.

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The new HMCB-concept was intended to be used mainly as a seawall for the protection of artificial islands (offshore airport) and roads with heavy traffic along the coast, but also as a breakwater [Fig. 12.14(c), 12.14(d)] in harbors. This concept has recently been implemented at MORI Port in Hokkaido, Japan.14 The governing characteristic of the HMCB-concept is to cause the highest waves in the spectrum to break before reaching the crest structure by means of a relatively flat slope (about 1:3). This concept has the following advantages: (i) the required amount of rubble material is much less than for a conventional rubble mound breakwater, (ii) the required armor units are smaller since they are all placed below still water level, and (iii) the required crest structure is much smaller than a conventional caisson breakwater. In order to further substantially reduce the breaking wave impact loads on the superstructure and to overcome further drawbacks of the old HMCB-concept (excessive wave reflection, overtopping, spray generation, etc.), a major innovation was introduced to improve the performance of the concrete superstructure: a front slit wall made of piles (porosity of about 30%) and a relatively short dissipation chamber behind it. If a breaking wave reaches the structure, the total wave force is split spatially and temporarily into the following force components (Fig. 12.15): (i) a force component F1 on the permeable front wall, (ii) a force component F2 on the impermeable back wall, and (iii) a stabilizing downward force F3 on the bottom slab of the dissipation chamber.

Fig. 12.15.

New HMCB-concept.

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In addition to the reduction of wave loads and the subsequent reduction of the required size of the concrete superstructure, a substantial improvement of the overall hydraulic performance characteristics is also achieved by using the new HMCB-concept. A summary of the improvements in comparison to the older concept (vertical impermeable superstructure) as obtained from extensive hydraulic model investigations is given in Table 12.1. With respect to the wave loads, it is seen that a substantial reduction of the horizontal and uplift forces is achieved, which would result in a reduction of about 50% of the required weight of the superstructure to ensure sliding stability. With respect to the hydraulic performance, it is seen that wave reflection is reduced by about 25%, and as a result of the reduction of wave overtopping the required crest level above still water level is reduced by about 40%. Even without using any splash reducer at the front and back wall of the concrete superstructure, the splash/spray heights are reduced by half. Further details on the results in Table 12.1 are given by Oumeraci and Muttray,26 Muttray et al.,15 Oumeraci et al.,25 Takahashi et al.,33 Sch¨ uttrumpf et al.,31 Muttray et al.,16 and Oumeraci et al.28 It might be important to stress that the reduction of spray generation at wave damping structures is becoming an increasingly important design requirement, particularly when the structures are used for the protection of offshore airports, the protection of roads with heavy traffic along the coastline, etc. The growing importance of this relatively new emerging issue may be explained by the detrimental effects that spray might have on inland and nearshore infrastructures, operations behind the structure as well as on vegetation. In fact, spray can be transported by wind up to 30 km inland and a flux of spray salt up to 400 µg/m2 ·s may result. Large quantities of saltwater dispersed over wide coastal areas may result in the following effects: (i) short-term detrimental effects such as disturbance/stoppage of car traffic,10 electric power supply, port, and airport operations and Table 12.1.

Main improvement by the new HMCB-concept.

Improved performance of new HMCB with slit front wall (without splash reducers) Hydraulic performance

rx

CoV [%]

Transmission coefficient Ct Reflection coefficient Cr Mean overtopping rate Maximum overtopping rate qmax Number of overtopping waves Now Splash and spray height Rw , Rs Required freeboard Rc

1.0 0.75 0.5 0.6 0.7 0.5 0.6

50 30 30 25 25 60 30

Wave Forces and Stability Horizontal forces Fh Uplift forces Fu Required weight for sliding stability W

0.4 0.6 0.5

50 40 50

CoV = x¯ σx (Coeff. of vacation); τ¯x = mean reduction factor)

x ¯new HMCB x ¯old HMCB

(mean

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(ii) longer-term impacts such as salt corrosion of buildings and other facilities, damage to agriculture and inland vegetation, etc. Therefore, one of the most challenging tasks to cope with salt spray consists of the development of innovative shapes of the structure crest to substantially reduce the splash/spray height induced by the breaking waves at the structure. For this purpose, special tests were performed to analyze the effectiveness of various alternatives to reduce splash/spray height.8,25,28 As shown in Fig. 12.16, the splash/spray height observed in the Large Wave Flume tests for different structure alternatives is related to the corresponding value observed for the new HMCB without any splash reducer as a reference structure (Type 1). Even without any splash reducer, the new HMCB-concept is capable of reducing the splash/spray height by half as compared to a conventional impermeable wall (Type 0). Adding a conical splash reducer on each pile of the front slit wall and a splash reducer on the back wall will further reduce the splash/spray height by about half (Type 3). The same improvement was achieved by using a continuous slab on the crest of the front wall (Type 4), but this has the drawback to substantially increase the uplift force.28 Since the efficiency of Type 2, both with respect to the reduction of splash heights and overtopping, was found very low, no further analysis was performed. This third example study has highlighted how a very old concept can basically be improved to provide a new solution with a considerable improvement of the wave

Fig. 12.16.

Efficiency of various alternatives for the reduction of splash and spray height.

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loading conditions and of the hydraulic performance characteristics such as wave reflection, wave overtopping and spray generation. The study also allowed to identify the reduction of spray generation as a new emerging criterion for the development and design of novel wave damping structures. 12.3.4. Onshore wave damping barrier (OWBD) A nonconventional permeable barrier to damp storm waves running up a historical promenade on the North Sea Island of Norderney was developed and tested in the Large Wave Flume (GWK), Hannover.29 The barrier consists of 1.30 m high straight and curved staggered wall elements with a length of 5.5 m and 7.7 m, respectively (Fig. 12.17). The relatively low crest level and the discontinuous nature of the barrier resulted from the requirements that, for the tourists on the promenade the view to the sea and the direct access to the shoreline should not be obstructed. Moreover, the barrier should architecturally and aesthetically fit into the local landscape, so that it will not necessarily be perceived as a coastal protection structure [Fig. 12.18(a)]. The efficiency of the OWBD concept in terms of wave overtopping reduction was successfully tested in the Large Wave Flume (GWK) in Hannover [Figs. 12.18(b) and 12.18(c)] and implemented in Norderney [Fig. 12.18(d)], where it withstood without any damage several storm surges for more than five years.

Fig. 12.17. Ref. 29).

Onshore wave damping barrier for the North Sea Island Norderney (Adapted from

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Fig. 12.18.

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Onshore wave damping barrier (OWBD).

Since broadening the crest of the embankment (Alternative 4) was practically not feasible, the OWBD concept (Alternative 2) proved the most performant in reducing wave overtopping (by a factor of 5) as compared to other conventional alternatives (Fig. 12.19). More details on the results can be found in Refs. 29 and 32. The fourth example study has shown that under some circumstances storm waves can be damped effectively by a discontinuous low barrier made of aesthetically and functionally well-conceived staggered wall elements (Fig. 12.20). This concept can be adapted for the protection against tsunami by using a much wider barrier and more robust wall elements. In fact, the field evidence experienced with seawalls and breakwaters during the 2004 Indian Ocean Tsunami clearly suggests that protective structures should not be designed to completely stop the tsunami. Indeed, this is neither economically justifiable nor environmentally and socially acceptable. Therefore, protective structures would be preferable which aim at progressively weakening the tsunami power without blocking completely the inundation and which have the overall additional benefit of broadly blocking floating debris in a rather soft manner. Such a concept would particularly be appropriate for urbanized and touristic coastal areas where forests (bioshields) cannot be planted due to unfavorable local conditions and must therefore be replaced by a man-made barrier fitting architecturally in the local marine landscape.20

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Fig. 12.19. Reduction of wave overtopping by conventional and nonconventional alternatives (Adapted from Refs. 29 and 32).

Fig. 12.20.

Details of wall elements of OWBD in Norderney.

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12.3.5. Rubble mound breakwater with a core made of geotextile sand containers 12.3.5.1. Motivations There are several reasons which might lead the engineer and other decision-makers in practice to use sand instead of conventional quarry run for the core of rubble mound breakwaters and structures, including among others: (i) nonavailability of rock material in sufficient quantities and at affordable costs; (ii) sediment infiltration through rubble mound structures which may result in the shoaling of navigation channels and harbor basins, and thus in higher maintenance dredging costs; and (iii) reduction of wave transmission through the structure which might particularly be crucial in the case of long waves. On the other hand, the use of sand as a quasi-impervious core instead of quarry stone would result in an increase of (i) wave setup and runup at the structure, (ii) wave overtopping, and (iii) wave reflection, which might be detrimental to the stability of the structure, to the operations on and behind the breakwater (due to excessive overtopping) as well as to navigation and seabed stability (due to excessive wave reflection). Moreover, serious difficulties arise in practice when trying to design and construct the filter to protect the sand core against washout by wave action. Applying geometrically closed filter criteria would result in very complex, multiple, and relative thin filter layers which will not only be very costly and very difficult to build in larger water depths, but also might certainly fail due to the almost unpredictable very complex loading conditions of the sand core under cyclic pulsations by waves and entrained air at the interface with the last filter layer. Such failures have indeed been observed under both laboratory and field conditions in the past. Laboratory evidence has also shown that introducing the so-called “geometrically open filter” criteria to design a “hydraulic sand-tight filter” may indeed reduce the number of filter layers. However, the main practical difficulties mentioned above will remain, including those associated with the long-term stability of the sand core due to the high complexity of the loading and its uncontrollability during the entire storm duration and the life cycle of the structure. Geotextile filters might present themselves as an alternative to the very complex, costly, and uncertain filter made of multiple layers of granular material. However, geotextile mats are not only difficult to install under waves and currents, but also may introduce a shear surface which might be detrimental to the stability of the armor layer. A more feasible alternative is to use a core made of geotextile sand containers (geocore). This will not only allow to overcome the aforementioned core stability problems, but also to provide (i) a better erosion stability of the core and (ii) an

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Fig. 12.21.

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Class of geocore structures in comparison to conventional rubble mound structures.

increased stability against seismic loads as compared to a core simply made of loose sand. However, many of the drawbacks mentioned above remain with respect to wave setup, runup, overtopping, reflection, and armor stability in comparison to a conventional breakwater core. Therefore, an extensive research program has been initiated at LWI to study both the hydraulic performance and armor stability, including the processes involved and the development of prediction formulae for the design of a class of rubble mound structures with a core made of geotextile sand containers (Fig. 12.21). In the following, only the case “rubble mound breakwater with GSC-core” in Fig. 12.17(a) is briefly addressed. Based on comprehensive hydraulic model tests performed in the twin-wave flumes of LWI, a comparative analysis of the two breakwater types was first performed with respect to wave reflection, wave transmission, wave runup, and wave overtopping performance. As a result, prediction formulae were proposed for the geocore breakwater and compared with those for the conventional rubble mound breakwater alternative.24 12.3.5.2. Wave reflection performance Using the reflection model proposed by Oumeraci and Muttray,27 which accounts for the relative depth k0 d as the primary influencing parameter, the following prediction formula was derived for both breakwaters (Fig. 12.22): tan α Kr = 0.43 · √ , ko d

(12.4)

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Fig. 12.22.

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Reflection coefficient Kr versus relative water depth k0 d.

with ko = 2π/L0 (wave number in deepwater); tan α = steepness of the seaward slope of the structure; d = water depth in front of the structure. 12.3.5.3. Wave transmission performance Several models have been proposed in the literature7,24 to predict wave transmission through a rubble mound structure which, however, do not explicitly account for the effect of the core permeability. The parameter which mostly affects the transmission coefficient KT has been found to be a function of the relative freeboard Rc /Hs and the steepness sm = Hs /L0 of the incident waves1: Rc R = Hs ∗




sm . 2π

(12.5)

In fact, the best-fit in analyzing the experimental results for the geocore and conventional breakwater was found by using the modified freeboard R∗ according to Eq. (12.5). As a result, the following formula was obtained for the transmission coefficient (Fig. 12.23): KT = ar · (Rc∗ )br ,

(12.6)

where ar and br are constants, which depend on the permeability of the structure. The structure parameters ar and br are obtained for the conventional and the geocore breakwater (Fig. 12.23).

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Fig. 12.23.

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Wave transmission coefficient for conventional breakwater and geocore breakwater.

12.3.5.4. Wave runup performance Based on the model proposed in CEM7 for the conventional breakwater type, the following runup formula was determined (CoV = 2.8%): 0.274 Ru2% = 1.217 · ξom Hmo

for ξom = 3.3 − 7.0,

(12.7)

while for the geocore breakwater the following formula was obtained (CoV = 3.4%): 0.274 Hmo Ru2% = 1.415 · ξom

with ξom = surf similarity parameter:  ξom = tan α

for ξom = 4.1 − 6.7, 

H/

(12.8)

 2 gTm−1.0 . 2π

The difference between the two breakwater types is in the order of 20%; i.e., the required crest level of a geocore breakwater should be increased accordingly if wave runup or/and wave overtopping is an important issue (Fig. 12.24). 12.3.5.5. Wave overtopping performance The best-fit of the experimental data for the conventional breakwater and the geocore breakwater was obtained by using the model proposed by TAW (2002)34 :   b · R∗ , (12.9) Q∗ = a · exp − γf

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Fig. 12.24.

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Runup for conventional breakwater versus runup for geocore breakwater.

where Q∗ =

q , 3 )0.5 (g · Hm0

q = average overtopping discharge [m3 /s · m] and R∗ = Rc /Hm0 = relative freeboard [-]. Using a = 0.2 and b = 2.6 in Eq. (12.9) for both breakwater types, the correction factor γf , initially intended to account for the surface roughness effect, is used to distinguish between the effect of permeability on the average overtopping rate q (Fig. 12.25): • γf = 0.52 for the conventional breakwater (CoV = 12%), • γf = 0.60 for the geocore breakwater (CoV = 20%). Based on the results in Fig. 12.25 and the results of further analysis,30 the geocore breakwater is associated with three to four times higher overtopping discharges than the conventional breakwater. 12.3.5.6. Lessons learned from the study The fifth example study has shown that using sand encapsulated in geotextile containers for the core of rubble mound breakwaters, which is about one order of magnitude less permeable than a conventional core made of quarry run, may provide many advantages. This is particularly the case when rock material (quarry run) is

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Fig. 12.25.

Average wave overtopping for conventional and geocore breakwater.

not available in sufficient quantities and at affordable costs or when the reduction of wave transmission and sediment infiltration through the rubble mound structure is an important issue. However, one should keep in mind that using a GSC-core instead of a conventional core will result in larger runup heights and higher wave overtopping rates. Moreover, larger armor units are required to ensure hydraulic stability. More results are given by Oumeraci et al.24 The advantages of the nonconventional “geocore” concept are expected to be particularly revealed when applied to a seawall protecting reclaimed land (Fig. 12.17(d)), due to the difficulties to protect the latter from being washed out through a conventional core made of quarry run.

12.4. Concluding Remarks and Perspectives This chapter has attempted to show that one of the challenges of a sustainable protection against wave attack is the development of innovative structures/concepts which are not only capable to further reduce the wave heights in the sheltered area, but also to achieve substantial improvements with respect to the reduction of wave loads, wave reflection, and overtopping. Moreover, the novel types of structures should also be able to cope with new emerging requirements such as (i) a better control of the wave periods in the sheltered area, (ii) a reduction of spray generation at the structure, and (iii) a better adaptability to multi-purpose use. Since the importance of these new emerging design criteria is expected to considerably increase in the context of the sustainable development of coastal zones, they might

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represent challenging candidate topics for future research. A further important candidate issue for applied research will certainly consist of the elaboration of a kind of “roadmap” of the existing wave damping concepts (possibly in a parametrized form!), together with the associated characteristic curves which not only summarize the associated hydraulic performance, but also the possible negative impacts and further information required for a fast decision-making.

Acknowledgments Most of the ideas, material, and illustrative examples used herein are based on inputs taken from completed research projects supported by the German Research Council (DFG, Bonn), the Federal Ministry for Education and Research (BMBF, Bonn), the Science and Technology Agency of Japan (STA, Tokyo), and the Fifth Port Construction Bureau of the Ministry of Transport (MOT, Tokyo), the European Community within the EU-MAST research program, NLWK Norden and BBG Bauberatung GmbH & Co. KG, Germany. The author gratefully acknowledges the financial support. Since most of these projects were joint projects with Port and Harbor Research Institute (PHRI, Yokosuka), Civil Engineering Research Institute (CERI, Sapporo), the Institute of Naval Architecture and Ocean Engineering, TU Berlin, and many further European Institutes, the author also gratefully acknowledges the fruitful cooperation with these partners.

References 1. N. W. H. Allsop, Low-crest breakwaters, studies in random waves, Proc. Coastal Structures, Specialty Conference on the Design, Construction, Maintenance and Performance of Coastal Structures, Arlington, Virginia, USA (1983). 2. H. Bergmann, Hydraulische Wirksamkeit und Seegangsbelastung senkrechter Wellenschutzbauwerke mit durchl¨ assiger Front, Dissertation, Mitteilungen des LeichtweissInstituts f¨ ur Wasserbau, Heft 147 (2001) (in German). 3. H. Bergmann and H. Oumeraci, Wave loads at perforated caisson structures, Proc. 27th Int. Conf. Coastal Engineering (ICCE’2000), Sydney, Vol. 2 (2000), pp. 1622–1635. 4. H. Bergmann and H. Oumeraci, Digue innovante en caissons multi-chambres — Fonctionnement et sollicitations hydrauliques, Revue fran¸caise de g´enie civil. 5(7), 973–993 (2001) (Sp´ecial issue) (in French). 5. H. Bergmann and H. Oumeraci, Hydraulische Wirksamkeit und Wellenbelastung senkrechter Wellenschutzbauwerke mit durchl¨assiger Front, Hansa (2002), pp. 1–8 (in German). 6. H. Bergmann and H. Oumeraci, Wave-induced water levels and pressure distribution at perforated wall, Proc. COPEDEC VII, Dubai, UAE (2008), Paper No. B-11 (in print). 7. CEM, Coastal Engineering Manual, Engineer Manual 1110-2-1100 (US Army Corps of Engineers, Washington, DC, USA, 2003).

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8. T. Hayakawa, K. Kimura, S. Takahashi, M. Muttray, M. Kudella and H. Oumeraci, Wave splash height on a high-mound composite breakwater, Proc. 4th Conf. Hydrodynamics (ICHD’2000), Yokohama, Japan, Vol. 2 (2000), pp. 671–676. 9. G. E. Jarlan, A perforated vertical wall breakwater, The Dock and Harbour Authority (1961), pp. 394–398. 10. K. Kimura, T. Fujiike, K. Kamikubo, R. Abe and K. Ishimoto, Damages to vehicles on a coastal highway by wave action, Proc. Conf. Coastal Structures’99, Santander, Spain, Vol. 2 (2000), pp. 1009–1016. 11. G. Koether, Hydraulische Wirksamkeit getauchter Einzelfilter und Filtersysteme — Prozessbeschreibung und Modellbildung f¨ ur ein innovatives Riffkonzept, Dissertation an der TU Braunschweig, Leichtweiss-Institut f¨ ur Wasserbau (2002), p. 163 (in German). 12. G. Koether, H. Bergmann and H. Oumeraci, Wave attenuation by submerged filter systems, Proc. 4th Int. Conf. Hydrodynamics (ICHD’2000), Yokosuka, Japan, Vol. 2 (2000), pp. 711–716. 13. G. Koether and H. Oumeraci, Schutzbauwerke f¨ ur Sandstr¨ ande mit touristischer und o ¨kologischer Bedeutung — Unterwasser-Filtersysteme, Hansa, Heft, 11 (2001), pp. 93–98. 14. M. Mori, Y. Yamamoto and K. Kimura, Wave force and stability of upright section of high mound composite seawall, Proc. Abstracts Int. Conf. Coastal Eng., Hamburg (2008), p. 206. 15. M. Muttray, H. Oumeraci, K. Shimosako and S. Takahashi, Hydraulic performance of high mound composite breakwater, Proc. 26th Int. Conf. Coastal Eng. (ICCE‘98) (1998), pp. 2207–2220. 16. M. Muttray, H. Oumeraci, K. Shimosako and S. Takahashi, Wave load on an innovative high mound composite breakwater: Results of large scale experiments and tentative design formulae, Proc. Conf. Coastal Structures’99, Santander, Spain, Vol. 1 (2000), pp. 353–362. 17. J. Newe, Methodik f¨ ur großmaßst¨ abliche 2D-Experimente zum Strandverhalten unter Sturmflutbedingungen, Dissertation an der TU Braunschweig, Leichtweiss-Institut f¨ ur Wasserbau (2002) (in German). 18. H. Oumeraci, The sustainability challenge in coastal engineering, Keynote Lecture, Proc. 4th Int. Conf. Hydrodynamics (ICHD), eds. Y. Goda et al., Yokohama, Japan, Vol. 1 (2000), pp. 57–83. 19. H. Oumeraci, Breakwaters, Part 2, Planing and Design of Ports and Marine Terminals, ed. H. Agerschou (Thomas Telford, London, UK, 2004), pp. 155–262. 20. H. Oumeraci, Nearshore and onshore tsunami effects, Background paper for DFG-Round Table Discussion, Hannover, http://www.fzk.uni-hannover.de/323.html (2006). 21. H. Oumeraci, G. F. Clauss, R. Habel and G. Koether, Unterwasser Filtersysteme zur Wellend¨ ampfung, Final Report of BMBF-Research Project (2001), p. 311 (in German). 22. H. Oumeraci, A. Kortenhaus, N. W. H. Allsop, M. B. De Groot, R. S. Crouch, J. K. Vrijling and H. G. Voortman, Probabilistic Design Tools for Vertical Breakwaters (Balkema, Rotterdam, The Netherlands, 2001), pp. 392. 23. H. Oumeraci and G. Koether, Hydraulic performance of a submerged wave absorber for coastal protection, Advances in Coastal Ocean Engineering (2008), pp. 36 (in print). 24. H. Oumeraci, A. Kortenhaus, H. Breustedt and P. Schley, Hydraulic model investigations on breakwaters with a core made of geotextile sand containers, Research Report No. 933, Report Leichtweiß-Institut f¨ ur Wasserbau, TU Braunschweig (2007), pp. 71 and Annexes.

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25. H. Oumeraci, M. Kudella, M. Muttray, K. Kimura and T. Hayakawa, Wave run-up and wave overtopping on a high mound composite breakwater, Final Report No. 831, Leichtweiss-Institut, TU Braunschweig (1998), pp. 88. 26. H. Oumeraci and M. Muttray, Large scale model tests on a high-mound composite type breakwater, Research Report No. 818, Leichtweiss-Institute for Hydraulics, TU Braunschweig (1997), pp. 115. 27. H. Oumeraci and M. Muttray, Bemessungswellenparameter vor Strukturen mit verschiedenen Reflexionseigenschaften, Abschlussbericht DFG-Projekt, OU 1/3-3, Braunschweig, Germany (2001), 93 S (in German). 28. H. Oumeraci, M. M. Muttray, K. Kudella, K. Kimura and S. Takahashi, Wave loading of a high mound composite breakwater (HMCB) with splash reducers, Proc. 4th Int. Conf. Hydrodynamics (ICHD’2000), Yokohama, Japan, Vol. 2 (2000), pp. 659–670. 29. H. Oumeraci, H. Sch¨ uttrumpf, A. Kortenhaus, M. Kudella, J. M¨ oller and M. Muttray, Untersuchungen zur Erweiterung bzw. zum Umbau des Deckwerks am Nordstrand von Norderney, Research Report No. 853, Leichtweiss-Institute for Hydraulics, TU Braunschweig (2000b) (in German). 30. P. Schley, Hydraulische Wirksamkeit von geschuetteten Wellenbrechern mit herkoemmlichen Kern und einem Kern aus geotextilen Sandcontainern — eine experimentelle Vergleichsstudie, Diplomarbeit am Leichtweiss-Institut f¨ ur Wasserbau, Fachbereich Bauingenieurwesen, TU Braunschweig, Germany (2006) 80 S, 8 Anlagen (in German). 31. H. Sch¨ uttrumpf, H. Oumeraci, K. Kimura, T. Hayakawa and J. Moeller, Wave overtopping on a high mound composite breakwater, Proc. Conf. Coastal Structures’99, Santander, Spain, Vol. 1 (2000), pp. 397–404. 32. H. Sch¨ uttrumpf, H. Oumeraci, F. Thorenz and J. M¨ oller, Reconstruction and rehabilitation of a historical seawall at Norderney, ICE, Proc. Breakwaters, Coastal Structures Coastlines, ed. N. W. H. Allsop (Thomas Telford, London, 2002), pp. 257–268. 33. S. Takahashi, K. Shimosako, H. Oumeraci, M. Muttray and K. Kimura, Reduced wave overtopping characteristics of a new high mound composite breakwater: Results of small-scale experiments, Proc. Conf. Coastal Structures’99, Santander, Spain, Vol. 1 (2000), pp. 389–396. 34. TAW: Technical Report on wave run-up and wave overtopping at dikes, Techn. Advisory Committee on Flood Defence, Delft, The Netherlands.

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Chapter 13

Wave Interaction with Breakwaters Including Perforated Walls Kyung-Duck Suh Department of Civil and Environmental Engineering Seoul National University, San 56-1, Shillim-dong, Gwanak-gu Seoul 151-744, Republic of Korea [email protected] During the past several decades, breakwaters including perforated walls have been introduced to resolve various problems associated with gravity-type breakwaters. In this chapter, first, the mathematical models are described that predict various hydrodynamic characteristics of single- or multiple-row curtain-wall-pile breakwaters, the upper part of which is a curtain wall and the lower part consisting of an array of vertical piles. Their extension to irregular waves is also described. These models can be used for curtain-wall breakwaters by just removing the piles. They can also be used for pile breakwaters by removing the curtain-wall and extending the piles to the water surface. Second, the mathematical model to predict wave reflection from a fully-perforated-wall caisson mounted on a rubble foundation is described, and its applicability to a partially-perforated-wall caisson and irregular waves is described. Third, a discussion is given for the calculation of the so-called permeability parameter, which represents the energy dissipation and phase-shift of flows passing across a perforated wall.

13.1. Introduction Gravity-type breakwaters using rubble mound or vertical caissons have been widely used to provide a calm basin for ships and to protect harbor facilities from rough seas. In general, the width of these breakwaters increases with water depth, leaving a large footprint and requiring a great amount of construction material, especially when built in deeper water. Often, they block littoral drift and cause severe erosion or accretion in neighboring beaches. In addition, they prevent the circulation of water, thus deteriorating the water quality within the harbor. In some places, they obstruct the passage of fishes and bottom-dwelling organisms. A solid soil foundation is also needed to support such heavy structures. 317

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In order to resolve the above-mentioned problems, perforated wall structures have been introduced especially in small craft harbors. The simplest perforated wall structure may be a curtain-wall breakwater (sometimes called wave screen or skirt breakwater), which consists of a vertical wall extending from the water surface to some distance above the seabed.1−3 Recently, Isaacson et al.4 proposed a slotted curtain-wall breakwater. Another simple perforated wall structure may be an array of vertical piles, which is called a pile breakwater in this chapter. The closely spaced piles induce energy dissipation due to viscous eddies formed by the flow through the gaps. To examine the wave scattering by vertical piles, hydraulic model tests have been used.5−8 Efforts toward developing analytical models to calculate the reflection and transmission coefficients have also been made.8−11 Recently, Suh et al.12 introduced a curtain-wall-pile breakwater, the upper part of which is a curtain wall and the lower part consisting of an array of vertical piles. They developed a mathematical model that predicts various hydrodynamic characteristics of a curtain-wallpile breakwater. More recently, Suh and Ji13 extended the model to a multiple-row breakwater. Another type of breakwaters including perforated walls is a perforated-wall caisson breakwater, even though it is still a gravity-type breakwater. It reduces not only wave reflection, but also wave transmission due to overtopping. It also reduces wave forces, especially impulsive wave forces, acting on the caisson.14,15 In order to examine the reflection characteristics of a perforated-wall caisson breakwater, hydraulic model tests have been used.16−19 Mathematical models for predicting the reflection coefficient have also been developed.10,20,21 On the other hand, Fugazza and Natale22 proposed a closed-form solution for wave reflection from a perforatedwall caisson. The aforementioned mathematical approaches dealt with the case in which the waves are normally incident to the perforated-wall caisson lying on a flat sea bottom. To resolve these limitations, Suh and Park23 developed a mathematical model that can predict the wave reflection from a fully-perforated-wall caisson mounted on a rubble mound foundation when regular waves are obliquely incident on the breakwater at an arbitrary angle. Recently, Suh et al.24 described how to apply this model to a partially-perforated-wall caisson and irregular waves. In this chapter, first, the mathematical models of Suh et al.12 and Suh and Ji13 are described. These models were developed to predict the hydrodynamic characteristics of curtain-wall-pile breakwaters. However, they can also be used for curtainwall breakwaters or pile breakwaters by just removing the piles or by removing the curtain wall and extending the piles to the water surface, respectively. Second, the mathematical model of Suh and Park23 and its applicability to a partiallyperforated-wall caisson and irregular waves is described. Third, a discussion is given for the calculation of the so-called permeability parameter, which represents the energy dissipation and phase-shift of flows passing across a perforated wall. 13.2. Curtain-Wall-Pile Breakwaters 13.2.1. Single-row curtain-wall-pile breakwater A curtain-wall-pile breakwater is sketched in Fig. 13.1, in which h = constant water depth in still water; d = height of the curtain wall below the still water level;

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Fig. 13.1.

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A definition sketch of a curtain-wall-pile breakwater: (a) side view; and (b) front view.

b = thickness of the wall. A Cartesian coordinate system (x, z) is defined with the positive x directing downwave from the crest line of the breakwater and the vertical coordinate z being measured vertically upward from the still water line. The distance between the centers of two neighboring piles is denoted as 2A, and the width of an opening is 2a so that the porosity of the lower part of the breakwater at x = 0 is defined as r0 = a/A. A regular wave train with wave height Hi is incident on the positive x-direction. The fluid domain is divided into region 1 (x ≤ 0) and region 2 (x ≥ 0). Assuming incompressible fluid and irrotational flow motion, the velocity potential exists, which satisfies the Laplace equation. Linearizing the free-surface boundary conditions, the following boundary value problem for the velocity potential Φ(x, z, t) is obtained: ∂ 2Φ ∂ 2Φ + = 0, ∂x2 ∂z 2

(13.1)

∂Φ ω 2 − Φ = 0 at z = 0 , ∂z g

(13.2)

∂Φ = 0 at z = −h , ∂z

(13.3)

where ω = wave angular frequency, and g = gravitational acceleration. Assuming periodic motion in time t, we can assume the solution to the above problem as 
1 igHi φ(x, z) exp(−iωt) , (13.4) Φ(x, z, t) = Re − 2ω cosh(kh) √ where i = −1; and the symbol Re represents the real part of a complex value. The wave number k must satisfy the dispersion relationship, ω 2 = gk tanh(kh). The spatial variation of the velocity potential φ(x, z) should be determined in each region.

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We assume that the length scale of the flow near the breakwater is of the order of the wall thickness, which is much smaller than the far-field length scale of O(k−1 ), so that the wall has no thickness mathematically and the 3D feature near the breakwater does not significantly affect the 2D far-field solutions. Then, φ1 (x, z) and φ2 (x, z) must satisfy the following matching conditions at x = 0: ∂φ1 ∂φ2 = = 0 for − d ≤ z ≤ 0, x = 0 , ∂x ∂x ∂φ1 ∂φ2 = = iG(φ1 − φ2 ) for − h ≤ z ≤ −d, x = 0 , ∂x ∂x

(13.5) (13.6)

where the subscripts indicate the regions of the fluid domain. The first matching condition describes that the horizontal velocities vanish on both sides of the curtain wall of the breakwater. The second one for the lower part of the breakwater describes that the horizontal mass fluxes (or indirectly horizontal velocities) in the two regions must be the same at the breakwater and that the horizontal velocity at the opening is proportional to the difference of velocity potentials, or the pressure difference, across the breakwater. The proportional constant G, often called permeability parameter, is in general complex, which will be discussed in detail later. To solve the boundary value problem [Eqs. (13.1) to (13.3)], we use the eigenfunction expansion method of Isaacson et al.4 The velocity potential is expressed in a series of infinite number of solutions: φ1 = φi −

∞ 

Am cos[µm (h + z)] exp(µm x) ,

(13.7)

Am cos[µm (h + z)] exp(−µm x) ,

(13.8)

m=0

φ2 = φi +

∞  m=0

where φi = cosh[k(h+z)] exp(ikx) is the incident wave potential. The wave numbers µm are the solution of the dispersion relation, ω 2 = −gµm tan(µm h), which has an infinite discrete set of real roots ±µm (m ≥ 1) for nonpropagating evanescent waves and a pair of imaginary roots µ0 = ±ik for propagating waves. We take µ0 = −ik so that the propagating waves in Eqs. (13.7) and (13.8) correspond to the reflected and transmitted waves, respectively. We also take the positive roots for m ≥ 1 so that the nonpropagating waves die out exponentially with the distance from the breakwater. Now, Eqs. (13.7) and (13.8) satisfy the free-surface boundary condition in Eq. (13.2) and the bottom boundary condition in Eq. (13.3). Also, they automatically satisfy the requirement that the horizontal velocities must be matched at the breakwater. In order to solve for the unknown coefficients, Am s, we use the matching conditions at the breakwater. First, Eqs. (13.7) and (13.8) are substituted into Eqs. (13.5) and (13.6), respectively. Multiplying each resulting equation by cos[µn (h + z)], integrating with respect to z over the appropriate domain of z (i.e., z = −d to 0, or z = −h to −d), and finally adding them, we obtain a matrix

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equation for Am : ∞ 

Cmn Am = bn

for n = 0, 1, 2, . . . , ∞ ,

(13.9)

m=0

where

Cmn = µm fmn (−d, 0) + (µm − 2iG)fmn (−h, −d) ,

(13.10)

bn = −µ0 [f0n (−d, 0) + f0n (−h, −d)] ,

(13.11)

 fmn (p, q) =

q p

cos[µm (h + z)] cos[µn (h + z)]dz



q 1 sin[(µm + µn )(h + z)] sin[(µm − µn )(h + z)]    + for m = n , 2 µm + µn µm − µn p =  1   [2µm (h + z) + sin[2µm (h + z)]]qp for m = n .  4µm

(13.12)

Note that the mathematical model developed for a curtain-wall-pile breakwater can be used for a pile breakwater just by setting d = 0. It can also be used for a curtain-wall breakwater by using the permeability parameter G = 1/b, which was derived from the energy dissipation formula for a curtain-wall breakwater of Kriebel.25 Once the wave potentials are calculated, we can obtain various engineering wave properties. The reflection and transmission coefficients are given by Cr = |A0 |

(13.13)

Ct = |1 + A0 | ,

(13.14)

and

respectively. The wave runup on the upwave face of the breakwater is given by   ∞   Hi  1  Am cos(µm h) . (13.15) Ru = 1 −  2  cosh(kh) m=0 In the limiting case of a full-depth impermeable vertical wall (d → h or r0 → 0), Am = 0 for all m ≥ 1 and A0 = −1, so that Cr = 1.0, Ct = 0.0, and Ru = Hi as expected. Since the vertical distributions of wave pressure on both upwave and downwave sides of the breakwater are known, the wave force can also be calculated. The maximum horizontal wave force Fmax per unit width of the breakwater is given by  ∞   A  1   m {sin(µm h) − r0 sin[µm (h − d)]} Fmax = ρgHi   cosh(kh)  µm m=0

2 ∞  1 Am cos(µm h) 1− cosh(kh) m=0 2  ∞  1 , − 1+ Am cos(µm h) cosh(kh) m=0

ρg + Hi2 8



(13.16)

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where ρ = density of fluid. The second term on the right-hand side represents the second-order force contribution of the wave crest regions on the upwave and downwave sides of the breakwater.26 Without this term, in the limiting case of a full-depth impermeable vertical wall (d → h or r0 → 0), Am = 0 for all m ≥ 1 and A0 = −1 so that the preceding equation becomes s = ρgHi Fmax

1 sinh(kh) , k cosh(kh)

(13.17)

where the superscript s stands for the standing wave in front of an impermeable vertical wall. In another limiting case of no breakwater (d → 0 and r0 → 1), Fmax becomes zero, as expected. 13.2.2. Extension to multiple-row breakwaters The single-row curtain-wall-pile breakwater still gives large transmission for longperiod waves. To reduce the wave transmission, the draft of the curtain wall must be increased, or the porosity between the piles must be decreased. Then, however, the wave reflection and the wave forces and moments acting on the breakwater may increase. Therefore, it is difficult to increase the draft or decrease the porosity beyond certain limits. A multiple-row breakwater may be a solution for these problems. It may also be advantageous for providing a space on top of the breakwater for fishing, walking, and maintenance of the breakwater, and so on. Suh and Ji13 extended the mathematical model to a multiple-row breakwater, whose definition sketch is shown in Fig. 13.2, in which dj = height of the jth curtain wall below the still water level; and bj = thickness of the jth wall. The center of the jth wall is located at x = xj . The distance between the centers of two neighboring piles is denoted as 2Aj , and the width of the gap between the piles is 2aj so that the porosity of the lower part of the jth wall is defined as rj = aj /Aj . The fluid domain is divided into J + 1 regions by the J walls. The upwave and downwave regions of the jth wall are defined as Ωj−1 and Ωj , respectively. Using the same assumptions as those made for a single-row breakwater, the same boundary value problem as that given by Eqs. (13.1)–(13.3) is obtained for the velocity potential Φj (x, z, t) in each region, which can be assumed as
 1 igHi φj (x, z) exp(−iωt) . Φj (x, z, t) = Re − 2ω cosh(kh)

(13.18)

The spatial variation of the velocity potential φj (x, z) should be determined in each region. The solutions in the regions of Ω0 and ΩJ are given by φ0 = A00 cos[µ0 (h + z)] exp[−µ0 (x − x1 )] +

∞  m=0

B0m cos[µm (h + z)] exp[µm (x − x1 )] ,

(13.19)

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z Hi

Ω1

Ω0

Ωj

Ω2 Ωj-1

ΩJ-1

ΩJ x bj

h

d1

d2

dj

dJ

b1

b2

bj

bJ

r1

r2

rj

rJ

x1

xj

x2

xJ

dj

2aj 2Aj

(a)

(b)

Fig. 13.2. A definition sketch of a multiple-row curtain-wall-pile breakwater: (a) side view; and (b) bird’s-eye view of jth row.

φJ =

∞ 

AJm cos[µm (h + z)] exp[−µm (x − xJ )] ,

(13.20)

m=0

respectively. Here, Ajm and Bjm are the coefficients of the component waves propagating forward and backward, respectively. The first subscript (j) indicates the row of the wall, while the second one (m) indicates the wave component. The wave numbers µm are the same as those used for a single-row breakwater. Taking A00 = 1, the first term on the right-hand side of Eq. (13.19) represents the incident wave potential, and |B00 | and |AJ0 | are the reflection and transmission coefficients, respectively. On the other hand, the solutions in the regions from Ω1 to ΩJ−1 are given by φj =

∞ 

{Ajm exp[−µm (x − xj )] + Bjm exp[µm (x − xj+1 )]} cos[µm (h + z)] ,

m=0

j = 1, 2, . . . , J − 1.

(13.21)

The matching conditions at each row of the breakwater are the same as Eqs. (13.5) and (13.6), i.e., ∂φj ∂φj−1 = = 0 for − dj ≤ z ≤ 0, x = xj , ∂x ∂x ∂φj−1 ∂φj = = iGj (φj−1 − φj ) for − h ≤ z ≤ −dj , x = xj . ∂x ∂x

(13.22) (13.23)

These matching conditions are used to solve for the unknown coefficients Ajm s and Bjm s. First, for x = x1 , Eqs. (13.19) and (13.21) are substituted into Eqs. (13.22) and (13.23). Multiplying each resulting equation by cos[µn (h + z)], integrating with

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respect to z over the appropriate domain of z (i.e., z = −d1 to 0, or z = −h to −d1 ), and finally adding them, we obtain ∞ 

[{µm fmn(−d1 , 0) + (µm − iG1 )fmn (−h, −d1 )}B0m

m=0

+ iG1 fmn (−h, −d1 )A1m + iG1 exp(−µm ∆x1 )fmn (−h, −d1 )}B1m ] = µ0 A00 {f0n (−d1 , 0) + f0n (−h, −d1 )} + iG1 A00 f0n (−h, −d1 ) for n = 0, 1, 2, . . . , ∞

(13.24)

and ∞ 

[µm {fmn (−d1 , 0) + fmn (−h, −d1 )}B0m

m=0

+ µm {fmn (−d1 , 0) + fmn(−h, −d1 )}A1m − µm exp(−µm ∆x1 ){fmn (−d1 , 0) + fmn (−h, −d1 )}B1m ] = µ0 A00 {f0n (−d1 , 0) + f0n (−h, −d1 )}

for n = 0, 1, 2, . . . , ∞ ,

(13.25)

where ∆xj = xj+1 − xj ,

(13.26)

and fmn (p, q) are the same as given in Eq. (13.12). For x = xj (j = 2 to J − 1), Eq. (13.21) is substituted into Eqs. (13.22) and (13.23). Again, multiplying each resulting equation by cos[µn (h + z)], integrating with respect to z over the appropriate domain of z, and finally adding them, we obtain ∞ 

[exp(−µm ∆xj−1 ){−µm fmn (−dj , 0)+µm fmn (−h, −d1 )

m=0

+ iGj fmn (−h, −dj )}Aj−1,m + {µm fmn (−dj , 0) − µm fmn (−h, −dj ) + iGj fmn (−h, −dj )}Bj−1,m − iGj fmn (−h, −dj )Ajm − iGj exp(−µm ∆xj )fmn (−h, −dj )Bjm ] = 0 for n = 0, 1, 2, . . . , ∞

(13.27)

and ∞ 

[µm exp(−µm ∆xj−1 ){fmn(−dj , 0)+fmn(−h, −dj )}Aj−1,m

m=0

− µm {fmn(−dj , 0) + fmn (−h, −dj )}Bj−1,m − µm {fmn(−dj , 0) + fmn (−h, −dj )}Ajm + µm exp(−µm ∆xj ){fmn (−dj , 0) + fmn (−h, −dj )}Bjm ] = 0 for n = 0, 1, 2, . . . , ∞ .

(13.28)

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Finally, for x = xJ , Eqs. (13.20) and (13.21) are substituted into Eqs. (13.22) and (13.23). Using the similar procedure as above, we obtain ∞ 

[exp(−µm ∆xJ−1 ){−µm fmn(−dJ , 0) + µm fmn (−h, −dJ )

m=0

+ iGJ fmn (−h, −dJ )}AJ−1,m + {µm fmn (−dJ , 0) − µm fmn (−h, −dJ ) + iGJ fmn (−h, −dJ )}BJ−1,m − iGJ fmn (−h, −dJ )AJm ] = 0

for n = 0, 1, 2, . . . , ∞

(13.29)

and ∞ 

[µm exp(−µm ∆xJ−1 ){fmn (−dJ , 0)+fmn (−h, −dJ )}AJ−1,m

m=0

− µm {fmn (−dJ , 0) + fmn(−h, −dJ )}BJ−1,m − µm {fmn (−dJ , 0) + fmn (−h, −dJ )}AJm ] = 0

for n = 0, 1, 2, . . . , ∞ .

(13.30)

If we take M wave modes (i.e., one progressive and M − 1 evanescent modes), the number of unknown coefficients is 2JM . Equations (13.24), (13.25), and (13.27)– (13.30) give 2JM equations, which can be solved for the unknown coefficients. For a multiple-row breakwater, the reflection and transmission coefficients are given by Cr = |B00 | ,

(13.31)

Ct = |AJ0 | ,

(13.32)

and

respectively. The wave runup on the upwave side of the first row of the breakwater is given by   ∞    1 Hi   Ru = (13.33) B0m cos(µm h) . A00 cos(µ0 h) +  2 cosh(kh)  m=0

The wave force on each wall can be calculated by integrating the wave pressure acting on upwave and downwave sides of the wall. The maximum horizontal wave force Fmax per unit width of the first wall of breakwater is given by 
 A00 ρgHi 1  sin(µ0 h) Fmax = 2 cosh(kh)  µ0  ∞  1 + {B0m − A1m − B1m exp[µm (x1 − x2 )]} sin(µm h) µ m=0 m
A00 −r1 sin[µ0 (h − d1 )] µ0

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∞  1 + {B0m − A1m − B1m exp[µm (x1 − x2 )]} sin[µm (h − d1 )] µ m=0 m  2 ∞   1 ρgHi2 + A00 cos(µ0 h) + B0m cos(µm h) 8 cosh2 (kh) 

    

m=0

 −

∞ 

m=0

2   {A1m + B1m exp[µm (x1 − x2 )]} cos(µm h) .  (13.34)

Again, the second term on the right-hand side represents the second-order force contribution of the wave crest regions on the upwave and downwave sides of the wall. Without this term, in the limiting case of a full-depth impermeable vertical wall, B0m = 0 for all m ≥ 1, B00 = A00 = 1, and A1m = B1m = 0 for all m ≥ 0, so that the preceding equation becomes Eq. (13.17). 13.2.3. Extension to irregular waves Using the above regular wave models, the reflection and transmission coefficients can be calculated differently for each frequency component, i.e., Cr (f ) and Ct (f ), where f = wave frequency. In the computation of the permeability parameter G in Eqs. (13.6) and (13.23), the root-mean-squared (rms) wave height is used in place of the incident wave height Hi , because the flow separation due to irregular waves and the resulting energy dissipation are induced not by the individual component waves but by the superposition of the component waves. The rms wave height is √ calculated by Hrms = Hs / 2, where Hs is the significant wave height. The spectral densities of the reflected and transmitted waves, respectively, are calculated for a particular frequency component by Sη,r (f ) = |Cr (f )|2 Sη,i (f ) , 2

Sη,t (f ) = |Ct (f )| Sη,i (f ) ,

(13.35) (13.36)

where Sη,i (f ) = incident wave energy density. The frequency-averaged reflection and transmission coefficients are then calculated as27  m0,r Cr = , (13.37) m0,i  m0,t Ct = , (13.38) m0,i where m0,i , m0,r , and m0,t = zeroth moments of the incident, reflected, and transmitted wave spectra, respectively. Note that the reflection and transmission coefficients calculated by the conventional method, as the ratio of the reflected and

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transmitted rms wave heights to the incident one, are equivalent to the frequency√ averaged coefficients, because the rms wave height is proportional to m0 . The force spectrum, SF (f ), is calculated by 2

SF (f ) = |TF (f )| Sη,i (f ) ,

(13.39)

where TF (f ) is the frequency-dependent transfer function between wave amplitude and force amplitude, which can be computed by using the linear term in Eq. (13.16) as TF = 2ρg

∞  1 Am {sin(µm h) − r0 sin[µm (h − d)]} . cosh(kh) m=0 µm

The zero-moment force, Fm0 , can then be determined as  ∞

1/2 Fm0 = 2 SF (f )df .

(13.40)

(13.41)

0

Note that a factor of 2 instead of 4 was used in order to represent the force amplitude.

13.3. Perforated-Wall Caisson Breakwaters 13.3.1. Fully-perforated-wall caisson breakwater Let us consider the perforated-wall caisson breakwater sketched in Fig. 13.3, in which θ1 = incident wave angle; and B = wave chamber width. The x-axis and y-axis are taken to be normal and parallel, respectively, to the breakwater crest line, and the water depth is assumed to be constant in the y-direction. The vertical

Fig. 13.3. A schematic diagram and coordinate system for calculation of wave reflection form a perforated-wall caisson breakwater.

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coordinate z is measured vertically upward from the still water line. In region 2 (−b ≤ x ≤ 0), the water depth h(x) is a varying function of x. For x ≤ −b (region 1) and 0 ≤ x ≤ B (region 3), the water depth is constant and equal to h1 and h3 , respectively. The rubble mound foundation is assumed to be impervious. Assuming incompressible fluid and irrotational flow motion, the velocity potential Φ(x, y, z, t) for the monochromatic wave propagating over the water depth h(x) with the angular frequency ω and wave height H can be expressed as
 igH Φ(x, y, z, t) = Re − φ(x, y, z) exp(−iωt) . (13.42) 2ω Linearizing the free-surface boundary conditions, the following boundary value problem for the potential φ(x, y, z) is obtained: ∇2 φ +

∂ 2φ = 0, ∂z 2

∂φ − λφ = 0 at z = 0 , ∂z ∂φ ∂φ dh + = 0 at z = −h(x) , ∂z ∂x dx

(13.43) (13.44) (13.45)

where ∇ represents the horizontal gradient operator, and λ = ω 2 /g is a wave number in deepwater. The Galerkin-eigenfunction method is used to formulate the problem. The Galerkin approach assumes that the function φ(x, y, z) can be expanded in terms of M + 1 depth-dependent functions Zm (x, z): φ(x, y, z) =

M 

ϕm (x, y)Zm (x, z) .

(13.46)

m=0

The functions Zm (x, z) are taken as Zm (x, z) =

cos[µm (z + h)] , cos(µm h)

(13.47)

so as to form a complete orthogonal set of eigenfunctions in the domain [−h(x), 0]. The wave numbers µm are the solution of the dispersion relation, λ + µm tan(µm h) = 0. The function Z0 (x, z) represents the free propagating wave mode, while the functions Zm (x, z) (m ≥ 1) correspond to the nonpropagating evanescent modes. The functions Zm (x, z) satisfy the free surface boundary condition in Eq. (13.44) and do not satisfy the bottom boundary condition in Eq. (13.45) individually. However, the global set of orthogonal functions should satisfy this condition. This is known as a tau method. In the tau method, a sufficient number of functions ϕm (x, y) in the approximated solution of Eq. (13.46) is chosen to ensure exact satisfaction of the bottom boundary condition. Turning back to the breakwater problem sketched in Fig. 13.3, the solution of the boundary value problem given by Eqs. (13.43)–(13.45) can be constructed from

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the particular solutions in each region of the fluid domain: φ1 (x, y, z) = {exp[ik1 (x + b) cos θ1 ] − exp[−ik1 (x + b) cos θ1 ]} × exp(iχy) +



cosh k1 (z + h1 ) cosh k1 h1

Rm exp[ν1,m (x + b)] exp(iχy)Z1,m (h1 , z),

(13.48)

µ1,m

φ2 (x, y, z) =



ϕ˜m (x) exp(iχy)Z2,m (h2 , z) ,

(13.49)

Tm exp(−ν3,m x) exp(iχy)Z3,m (h3 , z) ,

(13.50)

µ2,m

φ3 (x, y, z) =



µ3,m

where νj,m =

 µ2j,m + χ2 ,

χ = kj sin θj = k1 sin θ1 = constant (j = 1, 2, 3) .

(13.51)

If m = 0 (propagating mode), the following relationships are obtained: λ = kj tanh(kj hj ),

νj,0 = ±ikj cos θj .

(13.52)

For νj,0 , we take the negative sign for the reflected wave in region 1, while we need both positive and negative signs for the waves inside the wave chamber (i.e., region 3). The potential φj (x, y, z) must satisfy the matching conditions which provide continuity of pressure and horizontal velocity, normal to the vertical planes separating the fluid regions and no-flux condition at the wall on the downwave side of the wave chamber, i.e., ∂φ1 ∂φ2 (13.53) = (x = −b, −h1 ≤ z ≤ 0) , ∂x ∂x i ∂φ2 ∂φ2 ∂φ3 φ3 = φ2 + (13.54) , = (x = 0, −h3 ≤ z ≤ 0) , G ∂x ∂x ∂x ∂φ3 = 0 (x = B, −h3 ≤ z ≤ 0) . (13.55) ∂x As shown in Eq. (13.46), the Galerkin solution consists of a free propagating wave mode and nonpropagating evanescent modes. The evanescent wave modes would be of importance in the region near the breakwater. However, since we are interested in the reflected wave far from the breakwater, we focus on the solution for the propagating mode. Massel28 showed that in region 2, the function ϕ˜0 (x) satisfies the following ordinary differential equation: φ1 = φ2 ,

dϕ˜0 d2 ϕ˜0 + E(x)ϕ˜0 = 0 , + D(x) dx2 dx where



2τ k dh 2 D(x) = 1 − 3τ + , τ + kh(1 − τ 2 ) τ + kh(1 − τ 2 ) dx

  2 u1 dh u 2 d2 h 2 − χ2 , + 2 E(x) = k 1 + 2 k u0 dx k u0 dx2

(13.56)

(13.57) (13.58)

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where τ = tanh(kh), and u0 , u1 , and u2 are given by

K 1 tanh(kh) 1 + , u0 = 2k sinh K u1 =

sech2 (kh) (sinh K − K cosh K) , 4(K + sinh K)

u2 =

ksech2 (kh) [K 4 + 4K 3 sinh K − 9 sinh K sinh(2K) 12(K + sinh K)3 + 3K(K + 2 sinh K)(cosh2 K − 2 cosh K + 3)] ,

(13.59) (13.60)

(13.61)

where the abbreviation K = 2kh was used. In order to solve Eq. (13.56) in region 2, we need the boundary conditions at x = −b and x = 0. Suh and Park23 derived these boundary conditions as dϕ˜0 (−b) = i[2 − ϕ˜0 (−b)]k1 cos θ1 , dx

i dϕ˜0 (0) 1 exp(−ν3,0 B) + exp(ν3,0 B) − . ϕ˜0 (0) = ν3,0 exp(−ν3,0 B) − exp(ν3,0 B) G dx

(13.62) (13.63)

The differential equation (13.56) with the preceding two boundary conditions can be solved using the finite-difference method. Using the forward-differencing for dϕ˜0 (−b)/dx, backward-differencing for dϕ˜0 (0)/dx, and central-differencing for the derivatives in Eq. (13.56), the boundary value problem of Eqs. (13.56), (13.62), and (13.63) can be approximated by a system of linear equations, AY = B, where A is a tridiagonal-band-type matrix, Y is a column vector, and B is also a column vector. The subroutines given in the book of Press et al.29 can be used to solve this matrix equation. In particular, we are interested in ϕ˜0 (−b), from which the reflection coefficient Cr is calculated as follows: Cr = ϕ˜0 (−b) − 1 .

(13.64)

As stated in the introduction, the advantage of this model is that it can be used for a perforated-wall caisson mounted on a rubble mound foundation when waves are incident at an arbitrary angle. In order to examine the effect of rubble mound foundation upon wave reflection, Suh et al.30 calculated the reflection coefficients by changing the height and slope of the rubble mound. Total water depth in front of the breakwater h1 = 10 m, wave period T = 8 s, incident wave height H = 2 m, thickness of the perforated wall l = 0.6 m, porosity of the wall r = 0.3, and wave chamber width B = 10 m were used. Figure 13.4 shows the ratio of the reflection coefficient of the breakwater with a mound to that of the breakwater without a mound (i.e., lying on a flat sea bottom). When the mound height is approximately less than 0.3 times the total water depth, the reflection coefficient of the breakwater with a mound is less than that of the breakwater without a mound. For the mound height greater than this value, however, the wave reflection from the mound slope becomes significant so that the reflection coefficient becomes greater than that of the breakwater without a mound. In addition, the reflection coefficient increases with

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1.0 0.9

Mound Slope

0.8 0.7 0.6 0.5 0.4 0.3 0.0

0.1

0.2

0.3

0.4

0.5

(h1-h3)/h1 Fig. 13.4. Contours of the ratio of the reflection coefficient of the breakwater with a mound to that of the breakwater without a mound as a function of mound height and slope.

the height and slope of the mound as expected. Within the range of the practical design situations, however, the rubble mound effect on wave reflection is not so large. However, it should be noted that the effect of permeability of the rubble mound was not take into account because the rubble mound was assumed to be impervious. Suh et al.30 also examined the effect of angle of incidence upon wave reflection. The reflection coefficients were calculated by changing the wave angle from 0◦ to 80◦ and B cos θ/L from 0.1 to 0.5. The caisson was situated in a constant depth without a mound so that h1 = h2 = h3 = 30 cm. Wave height H = 3 cm, wave period T = 1.554 s, porosity of the wall r = 0.25 were used, while the perforated wall thickness was taken to be zero (i.e., the inertia effect of the perforated wall was neglected). The calculated results are shown in Fig. 13.5. For normal incidence of waves, the wave reflection is minimum at B cos θ/L = 0.25 and increases as it gets away from this value. The same trend is kept for oblique incidence of waves, but the solution exhibits singular behavior (i.e., the reflection coefficient shows a sharp peak) at B/L = 0.5, 1.0, 1.5, and so on, as shown in Fig. 13.5. In the practical design of a perforated-wall caisson, the value of B/L is usually less than 0.5. Thus, when the waves are obliquely incident to the breakwater, to minimize the wave reflection, the value of B cos θ/L should be around 0.25, but the condition B/L ∼ = 0.5 should be avoided. 13.3.2. Application to partially-perforated-wall caisson breakwater A conventional perforated-wall caisson consists of a front wave chamber and a back wall as shown in Fig. 13.6(a), and the water depth inside the wave chamber is the same as that on the rubble foundation. The weight of the caisson is less than that of a vertical solid caisson with the same width, and moreover, most of this

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0.5 1.0

B/L = 0.5

1.5

Bcos θ / L

0.4

0.3

0.2

0.1 0

20

40

60

80

Wave Angle (degree)

Fig. 13.5. Contours of the reflection coefficient as a function of wave incident angle and B cos θ/L; the lines of B/L = 0.5, 1.0, and 1.5 are also included.

weight is concentrated on the rear side of the caisson. Therefore, difficulties are sometimes met in the design of a perforated-wall caisson to satisfy the design criteria against sliding and overturning. In addition, particularly in the case where the bearing capacity of the seabed is not large enough, the excessive weight on the rear side of the caisson may have an adverse effect. In order to solve these problems, a partially-perforated-wall caisson as shown in Fig. 13.6(b) is often used, which provides an additional weight to the front side of the caisson. In this case, however, other hydraulic characteristics of the caisson such as wave reflection and overtopping may become worse than a fully-perforated-wall caisson. The mathematical model described in Sec. 13.3.1 assumes that the water depth inside the wave chamber is the same as that on the mound berm as in a fullyperforated-wall caisson breakwater shown in Fig. 13.6(a). However, for a partiallyperforated-wall caisson [see Fig. 13.6(b)], these water depths are different from each other, having depth discontinuity at the location of the perforated wall. In order to apply the model to the case of a partially-perforated-wall caisson, it can be assumed that the lower part of the front face of the caisson (below the perforated wall) is not vertical but has a very steep slope. As can be seen in Eq. (13.58), the model includes the terms proportional to the square of the bottom slope and to the bottom curvature, so that it can be applied over a bed having substantial variation of water depth. In order to examine the effect of the slope of the lower part of the caisson (which is infinity in reality), Suh et al.24 calculated the reflection coefficient by changing the slope from 0.1 to 10. They found that the reflection coefficient

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(a)

(b) Fig. 13.6. Bird’s-eye views of (a) a fully-perforated-wall caisson breakwater and (b) a partiallyperforated-wall caisson breakwater.

virtually does not change for slopes greater than 2.0, and they used the slope of 4.0 in their comparison between model prediction and experimental results. 13.3.3. Extension to irregular waves As for irregular waves, as explained in Sec. 13.2.3, the reflection coefficient is calculated differently for each frequency component. The wave period is determined according to the frequency of the component wave, while the rms wave height is used for all the component waves to compute the energy dissipation at the perforated wall. The spectral density of the reflected waves is then calculated for a particular frequency component as in Eq. (13.35), and the frequency-averaged reflection coefficient is calculated as in Eq. (13.37). 13.4. Calculation of Permeability Parameter Equations (13.6) and (13.23) state that the fluid velocity normal to the perforated wall is proportional to the pressure difference across the wall with a complex

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constant of proportionality. To elucidate the physical meaning of the permeability parameter G, however, it is more convenient to express the equations in the form of Eq. (13.54). Then, the real part of i/G is associated with the phase difference between the velocity and the pressure due to inertial effects, and the imaginary part of i/G corresponds to the resistance of the wall. There are several approaches to calculate the permeability parameter G. Mei et al.31 expressed the resistance and the inertial effects in terms of a head-loss coefficient and an effective orifice length, respectively. On the other hand, Sollitt and Cross32 expressed them in terms of a friction coefficient and an added mass coefficient, respectively. 13.4.1. Mei et al.’s method The permeability parameter based on Mei et al.31 can be expressed as G=

1 β ω

− i

,

(13.65)

where β = energy dissipation coefficient derived by linearizing the nonlinear convective acceleration term in the equation of motion; and  = effective orifice length, which is the length of the jet flowing through the perforated wall. The linearized dissipation coefficient β for a pile breakwater is given by Kim33 as β=

1 8α 5 + cosh(2kh) Hw ω  , 9π (R + 2)2 + P 2 2kh + sinh(2kh)

(13.66)

where Hw = incident wave height at the perforated wall; P = k; R = βk/ω; and α = head-loss coefficient. The preceding equation was derived for a pile breakwater. However, it could be used for a curtain-wall-pile breakwater as well, because the mechanism of energy dissipation between piles must be similar for these two breakwaters. A pile breakwater has energy dissipation associated with the free surface, but it may be small when compared with the energy dissipation between piles. On the other hand, for a perforated-wall caisson (see Fig. 13.3), Fugazza and Natale22 gave β as β=

W 5 + cosh(2k3 h3 ) 8α Hw ω  , 2 2 2 9π 2k W (R + 1) + F 3 h3 + sinh(2k3 h3 )

(13.67)

where W = tan(k3 B), R = βk3 /ω, F = 1 − P W , and P = k3 . Rearrangement of Eq. (13.66) or (13.67) gives a quartic polynomial of β, which can be solved by the eigenvalue method (e.g., Press et al.29 ). Suh et al.34 showed that the effective orifice length  is related to the blockage coefficient C by  = 2C .

(13.68)

For rectangular piles, Flagg and Newman35 proposed the blockage coefficient as

  b 1 2A 1 2 281 4 (13.69) C= 1 − log(4r0 ) + r0 + r . −1 + 2 r0 π 3 180 0

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The head-loss coefficient α could be given by the plate orifice formula36 :  2 1 α= −1 , r0 cos θCc

335

(13.70)

where Cc = empirical contraction coefficient, for which Mei et al.31 suggested using the formula: Cc = 0.6 + 0.4r02 ,

(13.71)

and r0 cos θ denotes the effective ratio of the opening of the perforated wall taking into account the oblique incidence of the waves to the wall. Very recently, on the other hand, Yoon et al.37 proposed the head-loss coefficient taking into account the effect of wall thickness as  2 1 −2.5(b/2a) ) −1 , (13.72) α = (0.6 + 0.4e r0 Cc for normally incident waves. For circular piles, Kakuno and Oda38 proposed the blockage coefficient as C=

1 π A(1 − r0 )2 ; 4 1−ξ

ξ=

π2 (1 − r0 )2 . 12

(13.73)

Park et al.11 proposed a formula for the head-loss coefficient for circular piles as  2 1 α= −1 , (13.74) rCc where Cc is the contraction coefficient given by Eq. (13.71), and the ad hoc porosity r is given by  D/2 1 1 dx = , (13.75) D −D/2 r(x)2 r2 with the spatially varying porosity   D 2 r(x) = 1 −

2 D 2

− x2

+a

,

(13.76)

where D = diameter of piles. In the case of a perforated-wall caisson mounted on a rubble foundation as in Fig. 13.3, the incident wave height at the perforated wall Hw is a priori unknown. In the case where the caisson does not exist and the water depth is constant as h3 for x ≥ 0, Massel28 has shown that the transmitting boundary condition at x = 0 is given by iϕ(0)k ˜ 3 cos θ3 =

dϕ(0) ˜ . dx

(13.77)

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The governing equation (13.56) and the upwave boundary condition in Eq. (13.62) do not change. After solving this problem, the transmission coefficient Ct is given by Ct = Re{ϕ(0)}, ˜ from which Hw is calculated as Ct times the incident wave height on the flat bottom. The above method to compute the permeability parameter is advantageous compared with other methods in that all the related parameters are known, i.e., incident wave height and period and the geometrical parameters of the barrier. However, Suh et al.12 showed that this method gives a wrong result for very long waves. As can be seen in Eq. (13.66), as kh goes to zero, β goes to infinity, indicating complete dissipation of the long waves. This leads to zero transmission for long waves, which is definitely wrong because in the limit of long waves the breakwater is invisible to the waves so that complete transmission of the waves should occur. This method can be used for irregular waves as well. Using the regular wave model, the reflection and transmission coefficients are calculated differently for each frequency component. In the computation of the energy dissipation coefficient β in Eq. (13.66), however, the rms wave height should be used instead of the height of the component wave, because the flow separation due to irregular waves and the resulting energy dissipation are induced not by the individual component waves but by the superposition of the component waves. 13.4.2. Sollitt and Cross’s method The permeability parameter based on Sollitt and Cross32 can be expressed as G=

r0 , b (fc − is)

(13.78)

where fc = friction coefficient, and s = inertia coefficient given by s = 1 + Cm

1 − r0 , r0

(13.79)

where Cm = added mass coefficient. In the method of Sollitt and Cross,32 the values of friction and added mass coefficients are not known a priori, and they are estimated on the basis of a best fit between measurement and prediction. Many researchers (e.g., Sollitt and Cross32 ; Losada et al.39 ; Yu40 ; Isaacson et al.4,41 ; Zhu and Chwang42; Hossain et al.43 ) suggested to use the value of Cm = 0, i.e., s = 1 by comparing their numerical predictions of reflection and transmission coefficients of perforated structures with experimental results. As for the friction coefficient, recently, Li et al.44 proposed the following empirical equation for estimating fc in terms of b/h, along with the use of Cm = 0: fc = −3338.7

 2   b b b + 82.769 + 8.711 for 0.0094 ≤ ≤ 0.05 . h h h

They also recommended using fc = 2.0 if b/h ≥ 0.1.

(13.80)

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13.4.3. Inertia effect on wave reflection from perforated-wall caisson It is well known that wave reflection from a perforated-wall caisson breakwater depends on the width of the wave chamber relative to the wavelength. For regular waves, neglecting the influence of the inertia term, Fugazza and Natale22 showed analytically that the resonance inside the wave chamber is important so that the reflection is minimal when B/L = (2n + 1)/4 (n = 0, 1, 2, . . .) in which L is the wavelength. However, the resonant mode of practical interest is the fundamental mode (n = 0), i.e., B/L = 0.25 because of the width limit of the breakwater. This is also shown in Fig. 13.5. In reality, however, the reflection coefficient becomes minimal at the value of B/L somewhat smaller than 0.25 because of the influence of the inertia term.24,27 In front of a perforated-wall caisson breakwater, a partial standing wave is formed due to the wave reflection from the breakwater. If there were no perforated wall, the node would occur at a distance of about L/4 from the back wall of the wave chamber, and hence the largest energy loss may occur at this point because the horizontal velocity becomes a maximum at a node. But, in reality, there exists inertia resistance at the perforated wall, which interrupts the smooth pass of the wave, thus creating pressure (or free surface) difference between the front and back of the perforated wall. Consequently, the location of the maximum horizontal velocity will move toward the breakwater, and the point where the maximum energy loss is gained becomes smaller than L/4. Thus, the minimum reflection occurs at a value of B/L smaller than 0.25. In the methods of Mei et al.31 and Sollitt and Cross,32 the effective orifice length  and inertia coefficient s, respectively, represents the inertia effect. Acknowledgments This chapter was prepared while the author was supported by the Korea Sea Grant Program and the Project for Development of Reliability-Based Design Method for Port and Harbor Structures of Korea Ministry of Marine Affairs and Fisheries. References 1. R. L. Wiegel, J. Waterways Harbors Coastal Eng. Div., Am. Soc. Civ. Eng. 86, 1 (1960). 2. D. L. Kriebel and C. A. Bollmann, Proc. 25th Int. Conf. Coastal Eng., ASCE (1997), p. 2470. 3. D. L. Kriebel, C. Sollitt and W. Gerken, Proc. 26th Int. Conf. Coastal Eng., ASCE (1999), p. 2069. 4. M. Isaacson, S. Premasiri and G. Yang, J. Waterways, Port, Coastal, Ocean Eng. 124, 118 (1998). 5. T. Hayashi, M. Hattori, T. Kano and M. Shirai, Proc. 10th Int. Conf. Coastal Eng., ASCE (1966), p. 873. 6. H. Kojima, M. Utsunomiya, T. Ijima, A. Yoshida and T. Kihara, Proc. 35th Jpn Conf. Coastal Eng., ASCE (1988), p. 542.

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7. T. Uda, A. Omata and T. Kawamura, Rep. Public Works Res. Inst., 2891, Japan Ministry of Construction (1990) (in Japanese). 8. S. Kakuno and P. L.-F. Liu, J. Waterways, Port, Coastal, Ocean Eng. 119, 302 (1993). 9. K. Hagiwara, Proc. 19th Int. Conf. Coastal Eng., ASCE (1985), p. 2810. 10. G. S. Bennett, P. McIver and J. V. Smallman, Coastal Eng. 18, 231 (1992). 11. W. S. Park, B. H. Kim, K. D. Suh and K. S. Lee, Coastal Eng. J. 42, 253 (2000). 12. K.-D. Suh, S. Shin and D. T. Cox, J. Waterways, Port, Coastal, Ocean Eng. 132, 83 (2006). 13. K.-D. Suh and C.-H. Ji, Proc. 30th Int. Conf. Coastal Eng., ASCE (2006), p. 4303. 14. S. Takahashi and K. Shimosako, Proc. Hydro-Port ’94, Port and Harbour Res. Inst., Yokosuka, Japan (1994), p. 747. 15. S. Takahashi, K. Tanimoto and K. Shimosako, Proc. Hydro-Port ’94, Port and Harbour Res. Inst., Yokosuka, Japan (1994), p. 489. 16. G. E. Jarlan, The Dock and Harbour Authority, Vol. XII(486) (1961), p. 394. 17. M. Marks and G. E. Jarlan, Proc. 11th Int. Conf. Coastal Eng., ASCE (1968), p. 1121. 18. F. L. Terret, J. D. C. Osorio and G. H. Lean, Proc. 11th Int. Conf. Coastal Eng., ASCE (1968), p. 1104. 19. K. Tanimoto, S. Haranaka, S. Takahashi, K. Komatsu, M. Todoroki and M. Osato, Technical Report, Port and Harbour Res. Inst., Min. of Transport, Japan, No. 246 (1976) (in Japanese). 20. H. Kondo, Proc. Coastal Structures ’79, ASCE (1979), p. 962. 21. S. Kakuno, K. Oda and P. L.-F. Liu, Proc. 23rd Int. Conf. Coastal Eng., ASCE (1992), p. 1258. 22. M. Fugazza and L. Natale, J. Waterways, Port, Coastal, Ocean Eng. 118, 1 (1992). 23. K. D. Suh and W. S. Park, Coastal Eng. 26, 117 (1995). 24. K.-D. Suh, J. K. Park and W. S. Park, Ocean Eng. 33, 264 (2006). 25. D. L. Kriebel, Proc. Coastal Structures ’99, Balkema, Rotterdam (2000), p. 525. 26. R. G. Dean and R. A. Dalrymple, Water Wave Mechanics for Engineers and Scientists (World Scientific, River Edge, New Jersey, 1991). 27. Y. Goda, Random Seas and Design of Maritime Structures (World Scientific, Singapore, 2000). 28. S. R. Massel, Coastal Eng. 19, 97 (1993). 29. W. H. Press, S. A. Teukolsky, W. T. Vetterling and B. P. Flannery, Numerical Recipes in FORTRAN: The Art of Scientific Computing, 2nd edn. (Cambridge University Press, Cambridge, 1992). 30. K. D. Suh, W. S. Park and K. D. Yum, Proc. Int. Conf. Coastal and Port Eng. Developing Countries (1995), p. 2311. 31. C. C. Mei, P. L.-F. Liu and A. T. Ippen, J. Waterways, Harbors Coastal Eng. Div., Vol. WW3, ASCE (1974), p. 217. 32. C. K. Sollitt and R. H. Cross, Proc. 13th Int. Conf. Coastal Eng., ASCE (1972), p. 1827. 33. B. H. Kim, Interactions of waves, seabed and structures, PhD. dissertation, Seoul National University, Seoul, Korea (1998). 34. K. D. Suh, S. Y. Son, J. I. Lee and T. H. Lee, Proc. 28th Int. Conf. Coastal Eng., ASCE (2002), p. 1709. 35. C. N. Flagg and J. N. Newman, J. Ship Res. 15, 257 (1971). 36. C. C. Mei, The Applied Dynamics of Ocean Surface Waves (Wiley, New York, 1983). 37. S. B. Yoon, J. I. Lee, D. H. Nam and S. H. Kim, J. Korean Soc. Coastal Ocean Engrs. 18, 321 (2006) (in Korean). 38. S. Kakuno and K. Oda, J. Jpn. Soc. Civil Engrs. 369, 213 (1986) (in Japanese).

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39. 40. 41. 42. 43. 44.

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I. J. Losada, M. A. Losada and A. Baquerize, Appl. Ocean Res. 15, 207 (1993). X. Yu, J. Waterway, Port, Coastal, Ocean Eng. 121, 275 (1995). M. Isaacson, J. Baldwin, S. Premasiri and G. Yang, Appl. Ocean Res. 21, 81 (1999). S. Zhu and A. T. Chwang, J. Eng. Mech. 127, 326 (2001). A. Hossain, W. Kioka and T. Kitano, Coastal Eng. J. 43, 83 (2001). Y. Li, Y. Liu and B. Teng, Coastal Eng. J. 48, 309 (2006).

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Chapter 14

Prediction of Overtopping Jentsje van der Meer Van der Meer Consulting P. O. Box 423, 8440 AK Heerenveen, The Netherlands [email protected] Tim Pullen HR Wallingford Howbery Park, Wallingford, Oxon, OX10 8BA, UK [email protected] William Allsop HR Wallingford Howbery Park, Wallingford, Oxon, OX10 8BA, UK [email protected] Tom Bruce School of Engineering, University of Edinburgh King’s Buildings, Edinburgh, EH9 3JL, UK [email protected] Holger Sch¨ uttrumpf RWTH-Aachen University Institute of Hydraulic Engineering and Water Resources Management Mies-van-der-Rohe-Str. 1, 52065 Aachen, Germany [email protected] Andreas Kortenhaus Leichtweiss-Institute for Hydraulics, Technical University of Braunschweig Beethovenstr, 51a, 38106 Braunschweig, Germany [email protected]

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This chapter describes the processes of wave overtopping at sea defense and related coastal or shoreline structures. It introduces a range of methods to calculate mean overtopping discharges, individual and maximum overtopping volumes, and the proportion of waves overtopping a seawall. It describes the principal hazards from wave overtopping and will help engineers by suggesting limiting tolerable discharges for frequent, design, and extreme wave conditions. This chapter is supported by more detailed material in Chaps. 15 and 16 which focus on the methods to predict overtopping for rubble mound structures (with partly sloping embankments), and on vertical structures and battered walls. All of these three chapters have been based closely on the new EurOtop Overtopping Manual.5

14.1. Introduction 14.1.1. Wave overtopping Wave overtopping has always been of principal concern for coastal structures constructed to defend against flooding: often termed sea defenses. Similar structures may also be used to provide protection against coastal erosion: sometimes termed coast protection. Other structures may be built to protect areas of water for ship navigation or mooring within ports, harbours, or marinas: often formed by breakwaters or moles. Within harbours, or along shorelines, reclaimed areas must be defended against both erosion and flooding. Some structures may be detached from the shoreline, often termed offshore, nearshore, or detached, but most structures used for sea defense or similar function form a part of the shoreline. Sloping dikes have been widely used for sea defenses along the coasts of the Netherlands, Denmark, Germany, and the United Kingdom. Dikes or embankment seawalls are also used to defend low-lying areas in the Far East, including China, Korea, and Vietnam. Historically, dikes or embankment seawalls were built along many North Sea coastlines, sometimes subsuming an original sand dune line, protecting the land behind from flooding, and sometimes providing additional amenity value. Similar structures have been formed by clay materials or even from a vegetated shingle ridge, in both instances allowing the side slopes to be steeper. All such embankments need some degree of protection against direct wave erosion, often using a revetment facing on the seaward side (Fig. 14.1). Revetment facing may take many forms, but may commonly include closely-fitted concrete blockwork, cast in situ concrete slabs, or asphaltic materials. Embankment or dike structures are generally most common along rural frontages. A second type of coastal structure consists of a mound or layers of quarried rock fill, protected by rock or concrete armour units (Fig. 14.2). The outer armour layer is designed to resist wave action without significant displacement of armour units. Under-layers of quarry or crushed rock support the armour and separate it from finer material in the embankment or mound. These porous and sloping layers dissipate a proportion of the incident wave energy in breaking and friction. Simplified forms of rubble mounds may be used for rubble seawalls or as protection to vertical walls or revetments. Rubble mound revetments may also be used to protect embankments

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Fig. 14.1.

Fig. 14.2.

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Wave overtopping on a revetment seawall.

Wave overtopping on a rubble mound breakwater.

formed from relict sand dunes or shingle ridges. Rubble mound structures tend to be more common in areas where harder rock is available. Along urban frontages, especially close to ports, erosion or flooding defense structures may include vertical (or battered/steep) walls (Fig. 14.3). Such walls

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Fig. 14.3.

Wave overtopping on a battered/vertical seawall.

may be composed of stone or concrete blocks, mass concrete, or sheet steel piles. Typical vertical seawall structures may also act as retaining walls to material behind. Shaped and recurved wave return walls may be formed as walls in their own right, or smaller versions may be included in sloping structures. Some coastal structures are relatively impermeable to wave action. These include seawalls formed from blockwork or mass concrete, with vertical, near-vertical, or steeply sloping faces. Such structures may be liable to intense local wave impact pressures, may overtop suddenly and severely, and will reflect much of the incident wave energy. Reflected waves cause additional wave disturbance and/or may initiate or accelerate local bed scour. It is worth noting that developments along waterfronts are highly valued with purchase or rental prices substantially above those for properties not on the waterfront. Yet, direct (or indirect) effects of wave overtopping have the potential to generate significant hazards to such developments and their users. Residential and commercial properties along a waterfront will often be used by people who may be unaware of the possibility, of the severity, or of the effects of wave overtopping in storm conditions. Regulatory authorities may therefore wish to impose onerous flood defense requirements on new developments. For instance, protection against flooding (including wave overtopping) for any new developments in the United Kingdom is now required to be 0.5% annual probability, equivalent to 1:200 year return. Exposure to overtopping of many coastal sites will, however, be influenced by climate change, probably increasing wave heights and periods as well as sea level rise.

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14.1.2. Predicting wave overtopping A number of different methods may be available to predict overtopping of particular structures (usually simplified sections) under given wave conditions and water levels. Each method will have strengths or weaknesses in different circumstances. In theory, an analytical method can be used to relate the driving process (waves) and the structure to the response through equations based directly on the knowledge of the physics of the process. It is, however, extremely rare for the structure, the waves, and the overtopping process to all be so simple and well-controlled that an analytical method on its own can give reliable predictions. Analytical methods are not therefore discussed further in this chapter. The primary prediction methods are therefore based on empirical methods that relate the overtopping response to the main wave and structure parameters. These are by far the most commonly used methods to predict overtopping. Two other methods have been derived during the CLASH European project3 based on the use of measured overtopping from model tests and field measurements. The first of these techniques uses the CLASH database of structures, waves, and overtopping discharges, with each test described by 31 parameters. Using the database is, however, potentially complicated, requiring some familiarity with these type of data. A simpler approach, and much more rapid, is to use the Neural Network tool that has been trained using the test results in the database. The Neural Network tool can be run automatically on a computer as a stand-alone device, or embedded within other simulation methods. For situations for which empirical test data do not already exist, or where the methods above do not give reliable enough results, then two alternative methods may be used, but both are more complicated than the methods above. A range of numerical models can be used to simulate the process of overtopping. All such models involve some simplification of the overtopping process and are therefore limited to particular types of structure or types of wave exposure. They may, however, run sequences of waves giving overtopping (or not) on a wave-by-wave basis. Generally, numerical models require more skill and familiarity to run successfully. They will not be described in this chapter. The final method to be mentioned is physical modeling in which a scale model is tested with correctly scaled wave conditions. Typically, such models may be built to a geometric scale in the range 1:10 to 1:60. Waves will be generated as random wave trains each conforming to a particular energy spectrum. The model may represent a structure cross section in a 2D model tested in a wave flume. Structures with more complex plan shapes, junctions, transitions, etc., may be tested in a 3D model in a wave basin. Physical models can be used to measure many different aspects of overtopping such as wave-by-wave volumes, overtopping velocities and depths, as well as other responses.

14.1.3. Performance requirements Most sea defense structures are constructed primarily to limit overtopping volumes that might cause flooding. For defenses that protect people living, working, or

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enjoying themselves, designers and owners of these defenses must, however, also deal with potential direct hazards from overtopping. This requires that the level of hazard and its probability of occurrence be assessed, allowing appropriate action plans to be devised to ameliorate risks arising from overtopping. Section 14.6 deals with tolerable wave overtopping.

14.2. Empirical Models, Including Comparison of Structures 14.2.1. Mean overtopping discharge Empirical methods use a simplified representation of the physics of the process presented in (usually dimensionless) equations to relate the main response (overtopping discharge, etc.) to key wave and structure parameters. The form and coefficients of the equations are adjusted to reproduce results from physical model (or field) measurements of waves and overtopping. Empirical equations may be solved explicitly, or may occasionally require iterative methods to solve. Historically, some empirical methods have been presented graphically, although this is now very rare. The mean overtopping discharge, q, is the main parameter in the overtopping process. It is, of course, not the only measure of overtopping, but it is relatively easy to measure in a laboratory wave flume or basin (or even in the field), and most other parameters are related in some way to this overtopping discharge. The overtopping discharge is generally calculated in m3 /s per m width, but in practical applications it may be quoted as liter/s per m width. Although it is given as a discharge, it is usually very far from a steady discharge as the actual processes of wave overtopping are much more dynamic. For most defenses, only large waves will reach the crest of the structure and will overtop, but they may do so with a lot of water in a few seconds. The individual volumes in wave-by-wave overtopping are more difficult to measure in a laboratory than the mean discharge; so data on wave-by-wave volumes are much rarer. As mean overtopping discharges are relatively easy to measure, many physical model tests have been performed all over the world, both for idealized structures and real applications or designs. The European CLASH project3 collected a large database worldwide with more than 10,000 wave overtopping test results on all kinds of structures (see Sec. 14.5). Some series of tests have been used to develop empirical methods for the prediction of overtopping. Such empirical methods or formulae are, however, only directly applicable to idealized structures, like smooth slopes (dikes, sloping seawalls), simple rubble mound structures or vertical structures (caissons) or walls, and may require extrapolation when applied to many existing structures.

14.2.2. Comparing overtopping performance Chapters 15 and 16 will describe overtopping formulae for different kinds of structures, based on the EurOtop Overtopping Manual.5 In this section, an overall view is given to compare the performance of different structure types and to give insight

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into how wave overtopping behaves for different structures. Those structures considered here are: smooth sloping structures (dikes, seawalls); rubble mound structures (breakwaters, rock armored slopes); and vertical structures (caissons, sheet pile walls). The principal prediction formula for many types of wave overtopping is


q 3 gHm0

= a exp(−bRc /Hm0 ).

(14.1)

It is an exponential function with the dimensionless overtopping discharge 3 )1/2 and the relative crest freeboard Rc /Hm0 . This type of equation shows q/(gHm0 a straight line on a log-linear graph, which makes it easy to compare formulae for different structures. Specific equations are given in Chaps. 15 and 16. Two equations are considered for pulsating waves on a vertical structure. Allsop et al.1 consider relatively shallow water and Franco et al.6 more deep water (caissons). Vertical structures in shallow water, and often with a sloping foreshore in front, may become subject to impulsive forces, i.e., high impacts and water splashing high up into the air. Specific formulae have been developed for these kinds of situations. For easy comparison of different structures, like smooth and rubble mound sloping structures and vertical structures for pulsating and impulsive waves, some simplifications will be assumed. In order to simplify the smooth structure, no berm is considered (γb = 1), only a normal wave attack is considered (γβ = 1), and the sloping seawall does not feature any wavewall on top (γv = 1). As the slope is smooth and impermeable, γf = 1. This limits the structure to a smooth and straight slope with normal wave attack. The slope angles considered for smooth slopes are cotα = 1−8, which means from very steep to very gentle. If relevant, a wave steepness of sm−1,0 = 0.04 (steep storm waves) and 0.01 (long waves due to swell or wave breaking) will be considered. Detailed definitions are given in Chap. 15. The same equation as for smooth slopes is applicable for rubble slopes, but now with a roughness factor of γf = 0.5, simulating a rock armoured structure. Rubble mound structures are often steep, but rock armoured slopes may also be gentle. Therefore, slope angles with cotα = 1.5 and 4.0 are considered. For vertical structures under pulsating waves, both formulae of Allsop et al.1 and Franco et al.6 will be compared, together with the formula for impulsive waves. Impulsive waves can only be reached with a relatively steep foreshore in front of the vertical wall. For comparison, values of the breaker ratio (wave height/water depth) of Hm0 /hs = 0.5, 0.7, and 0.9 will be used. These will be discussed further in Chap. 16. Overtopping on smooth slopes can be compared with rubble mound slopes and with vertical structures under pulsating or impulsive conditions. First, the traditional graph is given in Fig. 14.4 with the relative freeboard Rc /Hm0 versus the 3 1/2 logarithmic dimensionless overtopping q/(gHm0 ) . In most cases the steep smooth slope gives the largest overtopping. Steep means cotα < 2, but also a little gentler if long waves (less steepness) are considered. Under these conditions, waves surge up the steep slope. For gentler slopes, waves break as

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Dimensionless overtopping q/(gH

3 0.5 m0 )

1.E+00 steep smooth slopes, cota<2 gentle smooth slope, cota=4, so=0.04 steep rubble mound slope, gf=0.5 gentle rubble mound slope, gf=0.5 vertical structure Allsop (1995) vertical structure Franco et al. (1994) impulsive vertical Hm0/hs=0.5 impulsive vertical Hm0/hs=0.7 impulsive vertical Hm0/hs=0.9

1.E-01

1.E-02

1.E-03

1.E-04

1.E-05

1.E-06 0

0.5

1

1.5

2

2.5

3

3.5

4

Relative freeboard Rc/Hm0 Fig. 14.4.

Comparison of wave overtopping formulae for various structure types.

plunging waves and this reduces wave overtopping. The gentle slope with cotα = 4 gives much lower overtopping than the steep smooth slopes. Both slope angle and wave period have influence on overtopping for gentle slopes. The high roughness and permeability of a rubble mound can reduce overtopping substantially (Fig. 14.4). A roughness factor of γf = 0.5 was used here although γf = 0.4 (two layers of rock on a permeable under layer) would reduce the overtopping further. A gentle rubble mound slope with cotα = 4 gives very low overtopping. Vertical structures under pulsating waves1,6 give lower overtopping than steep smooth slopes, but more than a rough rubble mound slope. The impulsive conditions give a different trend. First of all, the influence of the relative water depth is fairly small as all curves with different Hm0 /hs are quite close. For low vertical structures (Rc /Hm0 < 1.5), there is hardly any difference between pulsating and impulsive conditions. The large difference is present for higher vertical structures and certainly for the very high structures. With impulsive conditions, water can be thrown high into the air, which means that overtopping occurs even for very high structures. The vertical distance that the discharge travels is more or less independent of the actual height of the structure. For Rc /Hm0 > 3 the curves are almost horizontal. Another way of comparing the effectiveness of structure types is to show the influence of slope angle on wave overtopping, as in Fig. 14.5. A vertical structure means cotα = 0. Battered walls have 0 < cotα < 1. Steep slopes are generally described by 1 ≤ cotα ≤ 3. Gentle slopes have roughly cotα ≥ 2 or 3. Overtopping

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1.E+00

Dimensionless overtopping q/(gHm03)0.5

vertical, Allsop (1995) vertical, Franco et al. (1994)

1.E-01

impulsive vertical, Hm0/hs=0.9, so=0.04 smooth slope, so=0.04

1.E-02

smooth slope, so=0.01

Rc/Hm0=1.5

1.E-03

Rc/Hm0=3.0

1.E-04

1.E-05

1.E-06

1.E-07 0

1

2

3

4

5

6

7

8

Slope angle cot α Fig. 14.5.

Comparison of wave overtopping as a function of slope angle.

prediction curves for two relative freeboards: Rc /Hm0 = 1.5 and 3.0 are shown in Fig. 14.5. Of course, similar conclusions can be drawn as for the previous comparison. Steep slopes give the largest overtopping, which reduces for gentler slopes, for a given wave condition and water level. Vertical slopes give less overtopping than steep smooth slopes, except for a high vertical structure under impulsive conditions. This graph gives also the method to calculate for a battered wall: interpolate between a vertical wall and a slope 1:1 with cotα as the parameter to interpolate. Details of all equations used here are described in more detail in Chaps. 15 and 16 (sloping smooth structures, rubble mound structures, and vertical structures). 14.2.3. Overtopping volumes and Vmax Wave overtopping is a dynamic and irregular process and the mean overtopping discharge, q, does not cover this aspect. But by knowing the storm duration, t, and the number of overtopping waves in that period, Now , it is possible to give some description of this irregular and dynamic overtopping, if the overtopping discharge, q, is known. Each overtopping wave gives a certain volume of water, V , and these can be described by a distribution. The two-parameter Weibull distribution can be fitted to many distributions. This equation has a shape parameter, b, and a scale parameter, a. The shape parameter gives a lot of information on the type of distribution. Figure 14.6 gives an overall view of some well-known distributions. The horizontal axis gives the probability of

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b=3 b=2 b=1 b=0.75 b=0.65 b=0.85

14

3

overtopping volume (m /m)

16

12 10 8 6 4 2 0 100

90

70 50

30 20 10

5

2

1 0.5

0.1

probability of exceedance (%) Fig. 14.6. Various distributions on a Rayleigh scale graph. A straight line (b = 2) is a Rayleigh distribution.

exceedance and has been plotted according to the Rayleigh distribution. It is known that wave heights in deep water generally conform to a Rayleigh distribution; so, responses governed by deep water wave conditions will plot on or close to a straight line, whilst shallow water effects will show deviations from the Rayleigh distribution. When waves enter shallow water and the highest waves break, wave heights more closely match a Weibull distribution with b > 2. An example with b = 3 is shown in Fig. 14.6, and this indicates that there are more large waves of similar height. The exponential distribution (often found for extreme wave climates) has b = 1, and shows that extremes become larger compared to most of the data. Such an exponential distribution would give a straight line in a log-linear graph. The distribution of overtopping volumes for all kinds of structures has average values even smaller than b = 1. Such a distribution is even steeper than an exponential distribution. It means that the wave overtopping process can be described by many fairly small overtopping volumes and a few very large volumes. The EAmanual 2 gives values for b and a, based on limited data sets. The b-values are mostly within the range 0.6 < b < 0.9. For comparison, curves with b = 0.65 and b = 0.85 are given in Fig. 14.6. The curves are very similar, except that the extremes differ a little. It is for this reason that an average value of b = 0.75 was chosen for smooth slopes and not different values for various subsets. The same average value has been used for rubble mound structures, which makes smooth and rubble mound structures easily comparable. The exceedance probability, PV , of an overtopping volume per wave is then similar to     0.75 V PV = P (V ≤ V ) = 1 − exp − , (14.2) a with a = 0.84 · Tm ·

q = 0.84 · Tm · q · Nw /Now = 0.84 · q · t/Now . Pov

(14.3)

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Equation (14.3) shows that the scale parameter a depends not only on the overtopping discharge, q, but also on the mean period, Tm , and the probability of overtopping, Now /Nw , or (similarly) on storm duration, t, and the actual number of overtopping waves Nw . The maximum overtopping during a certain event is fairly uncertain as most maxima, but depends on the duration of the event. In a 6-h period, individual peak volume is likely to be larger than that in 15 min. The maximum overtopping volume by a single wave during an event depends on the actual number of overtopping waves, Now , and can be calculated by Vmax = a · [ln(Now )]4/3 .

(14.4)

A comparison can be made between the mean overtopping discharge, q, and the maximum overtopping volume in the largest wave. Note that the mean overtopping is given in l/s per m width and that the maximum overtopping volume is given in liters per m width. Again, three example structures are considered: a smooth 1:4 slope; a rubble slope at 1:1.5; and a vertical wall, and again specific equations are given in Chaps. 15 and 16. The storm duration has been assumed as 2 h (the peak of the tide), and a fixed wave steepness of sm−1,0 = 0.04 has been considered. Figure 14.7 gives the q − Vmax lines for the three structures and for relatively small waves of Hm0 = 1 m and for fairly large waves of Hm0 = 2.5 m.

Maximum volume in overtopping wave (l per m)

100000 Smooth slope Hs=2.5m Rubble mound Hs=2.5m Vertical Hs=2.5m Smooth slope Hs=1m Rubble mound Hs=1m Vertical Hs=1m

10000

1000

100 0.1

1

10

100

1000

Overtopping discharge q (l/s per m) Fig. 14.7. Relationship between mean discharge and maximum overtopping volume in one wave for smooth, rubble mound, and vertical structures for wave heights of 1 m and 2.5 m.

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A few conclusions can be drawn from Fig. 14.7. First of all, the ratio q/Vmax is about 1000 for small q (roughly around 1 l/s per m), and about 100 for large q (roughly around 100 l/s per m). So, the maximum volume (in liters per m width) in the largest wave is about 100–1000 times larger than the mean overtopping discharge (in l/s per m). Secondly, the lines for a 1 m wave height are lower than for a larger 2.5 m wave height, which means that for lower wave heights, but similar mean discharge, q, the maximum overtopping volume is also smaller. For example, a vertical structure with a mean discharge of 10 l/s per m gives a maximum volume of 1000 l/m for a 1 m wave height and a volume of 4000 l/m for a 2.5 m wave height. Finally, the three different structures give different relationships, depending on the equations to calculate q and the equations to calculate the number of overtopping waves. 14.2.4. Wave transmission by wave overtopping For structures like nearshore breakwaters, large overtopping can be allowed as this overtopping simply falls into the water behind the structure causing new waves behind the structure. This is termed wave transmission and is most easily defined by the wave transmission coefficient Kt = Hm0,t /Hm0,i , with Hm0,t = transmitted significant wave height, and Hm0,i = incident significant wave height. The limits of wave transmission are Kt = 0 (no transmission) and 1 (no reduction in wave height). If a structure has its crest above water, the transmission coefficient will never be larger than about Kt = 0.4–0.5. Wave transmission was studied in the European DELOS project,4 and the following prediction formulae were derived for smooth sloping structures:  Kt = −0.3 ·

 Rc 2/3 + 0.75 · (1 − exp (−0.5 · ξop )) · (cos β) , Hm0,i

(14.5)

with a minimum Kt = 0.075 and maximum Kt = 0.8; and limitations 1 < ξop < 3; 0◦ ≤ β ≤ 70, and 1 < B/Hm0,i < 4, where β is the angle of wave attack and B is the crest width (and not berm width). The transmission coefficient Kt is shown in Fig. 14.8 as a function of the relative freeboard Rc /Hm0 and for a smooth structure with slope angle cotα = 4 (a gentle smooth low-crested structure). Three wave steepnesses have been used: sop = 0.01 (long waves), 0.03 and 0.05 (short wind waves). Again, normal wave attack has been assumed. Wave transmission decreases for increasing crest height, but longer waves give more transmission. Wave overtopping can be calculated for the same structure and wave conditions (see Chap. 15 and Fig. 14.9). Wave overtopping and transmission can be related if Figs. 14.8 and 14.9 are combined, and Fig. 14.10 shows this relationship. For convenience, the graphs are not dimensionless, but for a wave height of Hm0,i = 3 m. A small transmitted wave height of Hm0,t = 0.1 m is found if overtopping exceeds q = 30−50 l/s per m. In order to reach a transmitted wave height of Hm0,t = 1 m (Kt ≈ 0.33), the overtopping discharge should exceed q = 500−2500 l/s per m or

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0.6 so=0.01

0.5

so=0.03 so=0.05

Kt

0.4 0.3 0.2 0.1 0.0 0

Fig. 14.8.

0.5

1 Rc /Hm0

1.5

2

Wave transmission for a gentle smooth structure of 1:4 and for different wave steepneses.

1.000 so=0.01

q/(gHm0 )

3 0.5

so=0.03 so=0.05

0.100

0.010

0.001 0

0.5

1 Rc /Hm0

1.5

2

Fig. 14.9. Wave overtopping for a gentle smooth structure of 1:4 and for different wave steepnesses.

0.5–2.5 m3 /s per m. One may conclude that any significant wave transmission is always associated with (very) large wave overtopping. Wave transmission for rubble mound structures has also been investigated in the European DELOS project,4 and the following prediction formulae were derived for wave transmission: Kt = −0.4Rc /Hm0 + 0.64B/Hm0 − 0.31(1 − exp(−0.5ξop )) for 0.075 ≤ Kt ≤ 0.8.

(14.6)

Wave overtopping for a simple trapezoidal rubble mound can be calculated by methods presented in Chap. 15. A typical rubble mound structure has been used as an example, with cotα = 1.5, armour rock of 6–10 ton (Dn50 = 1.5 m), and crest

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3500 so=0.01

3000

so=0.03 so=0.05

q (l/s per m)

2500 2000 1500 1000 500 0 0

0.5

1 Hm0 transmitted (m)

1.5

2

Fig. 14.10. Wave transmission versus wave overtopping for a smooth 1:4 slope and a wave height of Hm0 = 3 m.

3500 3000

so=0.01 so=0.03

q (l/s per m)

2500

so=0.05 2000 1500 1000 500 0 0

0.5

1 Hm0 transmitted (m)

1.5

2

Fig. 14.11. Wave transmission versus wave overtopping discharge for a rubble mound structure, cotα = 1.5, 6–10 ton rock, B = 4.5 m, and Hm0 = 3 m.

width of Bw = 4.5 m (3Dn50 ). A wave height of Hm0,i = 3 m has been assumed with the following wave steepness: sm−1,0 = 0.01 (long waves), 0.03 and 0.05 (short wind waves). In the calculations, the crest height has been changed to calculate wave transmission as well as wave overtopping. These are compared in Fig. 14.11 which shows that longer waves (sm−1,0 = 0.01) give more wave transmission, even for similar overtopping discharge. The reason is probably the effect of wave action penetrating through the permeable upper layers, easier for long waves, thus contributing to the waves behind the structure. So, for permeable mounds, there may still be significant wave transmission through the structure even without substantial overtopping discharge. In this

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example, transmitted wave heights between Hm0,t = 0.5 m and 1 m are found for overtopping discharges smaller than q = 100−200 l/s per m. A simple equation for wave transmission at vertical structures has been given by Goda,7 although again more complete methods are given in Chap. 16: Kt = 0.45 − 0.3Rc /Hm0

for 0 < Rc /Hm0 < 1.25.

(14.7)

For vertical walls, transmission is primarily governed by the relative crest height with little or no effect of wave period or steepness. A simple vertical structure has been used as an example with Hm0,i = 3 m. Overtopping and wave transmission are compared in Fig. 14.12, where in the calculations the crest height has been changed to calculate wave transmission as well as wave overtopping. For comparison, the same rubble mound structure has been used as the example in Fig. 14.11, with cotα = 1.5, 6–10 ton rock (Dn50 = 1.5 m) as armour, a crest width of 4.5 m (3Dn50 ), and a wave steepness sop = 0.03. The curve for a smooth structure (Fig. 14.10) and for sop = 0.03 has been given too in Fig. 14.12. A rubble mound structure gives more wave transmission than a smooth structure for similar overtopping discharge, but a vertical structure can give even more transmission. The reason may be that overtopping water over the crest of a vertical breakwater always falls in a mass from a distance into the water, rather than flowing relatively smoothly over or through a sloping structure. One may conclude that even without considerable wave overtopping discharge at the crest of a vertical structure, there still might be significant wave transmission. In this example of a vertical structure, transmitted wave heights between 0.5 m and 1 m are found for overtopping discharges smaller than 100–200 l/s per m. Finally, an example is shown of both wave overtopping and wave transmission on a rubble mound breakwater in Fig. 14.13. 3500

q (l/s per m)

vertical 3000

rubble mound (so=0.03)

2500

smooth (so=0.03)

2000 1500 1000 500 0 0

0.5

1 Hm0 transmitted (m)

1.5

2

Fig. 14.12. Comparison of wave overtopping and transmission for a vertical, rubble mound, and smooth structure.

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Fig. 14.13.

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Wave overtopping and transmission at breakwater IJmuiden, the Netherlands.

14.3. PC-OVERTOPPING The computer program pc-overtopping was created using the results of the TAW8 Report “Wave runup and wave overtopping at dikes” and is used for the five-yearly safety assessment of all water defenses in the Netherlands. The TAW8 Report has now been replaced by Chap. 5 (dikes and embankments) in the EurOtop Overtopping Manual5 and extended for rubble mound and vertical structures in Chaps. 6 and 7 of that manual. pc-overtopping has been translated into English, and is available from the EurOtop Overtopping Manual5 web site. The program was based on dike-type structures. The structure should be sloping, although a small vertical wall on top of the dike may be included. Some effects of roughness and/or permeability can be included, but not a crest with permeable and rough rock or armour units. In such a case, the structure should be modeled up to the transition to the crest, and other formulae should be used to take into account the effect of the crest. pc-overtopping was set up so that almost every sloping structure can be modeled by an unlimited number of sections. Each section is given by x–y coordinates, and each section can have its own roughness factor. The program calculates most relevant overtopping parameters (except flow velocities and flow depths), such as: • • • •

2% runup level; mean overtopping discharge; percentage of overtopping waves; overtopping volumes per wave (maximum and for every percentage defined by the user); • required crest height for given mean overtopping discharges (defined by the user).

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The main uses of pc-overtopping are: • modeling of overtopping on any sloping structure, including different roughnesses along the slope; • calculation of most overtopping parameters, not only the mean discharge. A disadvantage of pc-overtopping is that it does not calculate overtopping for vertical structures or for rubble structures with rough/permeable crest. An example illustrates some of its capabilities. An example dike cross section with the design water level at +1 m CD is shown in Fig. 14.14. Different materials are used on the slope: rock, basalt, concrete asphalt, open concrete system, and grass on the upper part of the structure. The structure has been schematized in Fig. 14.15 by x–y coordinates and selection of the material of the top layer. The program selects the right roughness factor. The input parameters are wave height, wave period (either spectral period Tm−1,0 or peak period Tp ), wave obliquity, water level (with respect to the same level as used for the structure geometry), and finally, number of waves (derived from the storm duration and mean period) for the calculation of overtopping volumes, etc. Fig. 14.16 gives the input file.

Fig. 14.14.

Fig. 14.15.

Example cross section of a dike.

Input of geometry by x–y coordinates and choice of top material.

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Fig. 14.16.

Fig. 14.17.

Input file.

Output of pc-overtopping.

The output is given in three columns (Fig. 14.17). The left column gives the 2% runup level, the mean overtopping discharge, and the percentage of overtopping waves. If the 2% runup level is higher than the actual dike crest, this level is calculated by extending the highest section in the cross section. The middle column gives the required dike height for the given mean overtopping discharges. Again, the highest section is extended if required. Finally, the right-hand column gives the number of overtopping waves in the given storm duration, together with the maximum overtopping volume, and volumes for specified overtopping percentages, given as a percentage of the total number of overtopping waves. pc-overtopping also provides a check on whether the results for the 2% runup level and mean overtopping discharge fall within the measured ranges. Results on which the formulae were based are shown in the runup or overtopping graphs (see Figs. 14.18 and 14.19). These show the measured runup or overtopping, including effects of reductions due to roughness, berms, etc. The curve gives the maximum for smooth straight slopes with normal wave attack. The program then plots the calculated point in these graphs (the point within the circle).

14.4. Neural Network Tools Artificial neural networks are tools that allow meaning to be extracted from very large quantities of data. Neural networks (NN) are organized in the form of layers, within which there are one or more processing elements called “neurons.” The first layer is the input layer, and the number of neurons in this layer is equal to the number of input parameters. The last layer is the output layer and the number of neurons in this layer is equal to the number of output parameters to be predicted. The layers between the input and output layers are the hidden layers, consisting of a number of neurons to be defined in configuring the NN. Neurons in each layer receive

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Fig. 14.18.

Fig. 14.19.

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Check on 2% runup level.

Check on mean overtopping discharge.

information from the preceding layer, carry out a standard operation, and produce an output. Each connection has a weight factor assigned from the calibration of the NN. The input of a neuron consists of a weighted sum of the outputs of the preceding layer. This procedure is followed for each neuron; the output neuron generates the final prediction of the NN. Artificial NNs have applications in many fields including coastal engineering where examples have been applied to predicting armour stability, forces on walls, wave transmission, and wave overtopping. The development of an artificial NN is useful, where: • the process to be described is complicated with many parameters involved, • there is a large amount of data.

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β

h Hm0,toeTm-1,0,toeht

Bt

f

cot

d

cot

u

Rc

B

hb tan

B

Ac

Gc

INPUT LAYER

HIDDEN LAYER

OUTPUT LAYER q (m3/s/m) Fig. 14.20.

Configuration of the NN for wave overtopping.

It has already been seen that overtopping cannot be predicted by a single formula, but requires a number of different formulae. A single NN can, however, cover the full range of structures, provided that sufficient data are available to “train” the NN. If too few data are available, predictions in the less-well populated regions will be unreliable, particularly where the prediction is trying to extrapolate out of range. Providentially, international cooperation supported by the European CLASH research project3 collected many test results on wave overtopping for all kinds of coastal structures and embankments. Within the CLASH project, two NNs were developed, one within the main project, and one alongside a PhD project.9 In both cases, the NN configuration was based on Fig. 14.20, where the input layer has 15 input parameters (β, h, Hm0toe , Tm−1,0toe, ht , Bt , γf , cotαd , cotαu , Rc , B, hb , tanαb , Ac , Gc ) and one output parameter in the output layer (i.e., mean overtopping discharge, q). CLASH was focused on a three-layered NN, where a configuration with one single hidden layer was chosen. Development of an artificial NN requires that all data be checked thoroughly (rubbish in = rubbish out), and that training be done by those with appropriate skills. Using NN as a prediction tool, however, is easy for most practical engineers! It is for this reason that the CLASH NN was adopted as part of the EurOtop Overtopping Manual.5 Applying NN requires an Excel or ASCII input file with parameters, run the program (push a button), and get a result file with mean overtopping discharge(s). It is therefore as easy as using a formula programmed in Excel and does not need knowledge about NNs. The advantages of the neural network are: • it works for almost every structure configuration; • it is easy to calculate trends instead of just one calculation with one answer.

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The input exists of 10 structural parameters and four hydraulic parameters. The hydraulic parameters are wave height, wave period, and wave angle and water depth just in front of the structure. The structural parameters describe almost every possible structure configuration by a toe (two parameters), two structure slopes (including vertical and wave return walls), a berm (two parameters), and a crest configuration (three parameters). The tenth structural parameter is the roughness factor for the structure (γf ) and describes the average roughness of the whole structure. Although guidance is given, estimation is not easy if the structure has different roughness on various parts of the structure. An overall view of possible structure configurations is shown in Fig. 14.21. It clearly shows that the NN is able to cope with most structure types.

Fig. 14.21.

Overall view of possible structure configurations for the NN.

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Gc = 5 m Rc = 4 m

Ac = 5 m Hs = 5 m; Tp = 10 s β = 0 Xbloc 1:1.5 h = 12 m

ht = 9 m Bt = 4 m

Fig. 14.22.

Example cross section with parameters for application of NN.

Very often one is not only interested in one calculation, but in more. As the input file has no limitations in the number of rows (= number of calculations), it is easy to incrementally increase one or more parameters and to find a trend for a certain (design) measure. As an example for the calculation of a trend with the NN tool, an example cross section of a rubble mound embankment with a wave wall has been chosen (Fig. 14.22). If, for example, the cross section in Fig. 14.22 experiences too much overtopping, the following measures could be considered: • • • •

increasing the crest; applying a berm; widening the crest; increasing only the crest wall.

An example input file with the first six calculations is shown in Table 14.1, where incremental increase of the crest will show the effect of raising the crest on the amount of wave overtopping. Calculations will give an output file with the mean overtopping discharge q (m3 /s per m width) and with confidence limits. Table 14.2 shows as an example the output related to the input in Table 14.1. Assembling the input file for this example took 1 h and resulted in 1400 rows or calculations. The calculation of the NN took less than 10 s. The results were copied into the Excel input file, and the resulting graph was plotted within Excel, which took another hour. Figure 14.23 gives the final result, where the four trends are Table 14.1.

Example input file for NN with first six calculations.

β

h

Hm0

Tm−1,0

ht

Bt

γf

cotαd

cotαu

Rc

B

hb

tan αB

Ac

Gc

0 0 0 0 0 0

12 12 12 12 12 12

5 5 5 5 5 5

9.1 9.1 9.1 9.1 9.1 9.1

9 9 9 9 9 9

4 4 4 4 4 4

0.49 0.49 0.49 0.49 0.49 0.49

1.5 1.5 1.5 1.5 1.5 1.5

1.5 1.5 1.5 1.5 1.5 1.5

4 4.05 4.1 4.15 4.2 4.25

0 0 0 0 0 0

0 0 0 0 0 0

0 0 0 0 0 0

5 5.05 5.1 5.15 5.2 5.25

5 5 5 5 5 5

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Output file of NN with confidence limits.

2.50%

5.00%

25.00%

50.00%

75.00%

95.00%

97.50%

2.45E−02 2.35E−02 2.26E−02 2.19E−02 2.07E−02 1.99E−02

2.77E−02 2.64E−02 2.49E−02 2.39E−02 2.27E−02 2.18E−02

4.15E−02 3.99E−02 3.82E−02 3.69E−02 3.55E−02 3.38E−02

5.91E−02 5.58E−02 5.33E−02 5.08E−02 4.85E−02 4.62E−02

8.35E−02 7.91E−02 7.52E−02 7.17E−02 6.89E−02 6.60E−02

0.1299 0.1246 0.119 0.1133 0.1079 0.1033

0.1591 0.1516 0.1448 0.1383 0.1324 0.1265

Fig. 14.23.

Results of a trend calculation.

shown. The base situation had an overtopping discharge of 59 l/s per m. The graph clearly shows what measures are required to reduce the overtopping by, for example, a factor 10 (to 5.9 l/s per m) or to only 1 l/s per m. It also shows that increasing the structure height is most effective, followed by increasing only the crest wall. Within the CLASH research project, two different NNs exist. One is the official NN developed by Delft Hydraulics in the main part of the CLASH project. It runs as executable and can be downloaded from the CLASH web site or the EurOtop Overtopping Manual web site.5 The other NN has also been developed within CLASH, but as part of a PhD R thesis at Gent University.9 The network was developed in MatLab and so it can R  only be run if the user has a licence for MatLab . The advantage of this NN is that it first decides whether there will be overtopping or not (classifier). If there is no overtopping, it gives q = 0. If there is overtopping, it will quantify the overtopping with a similar network as the CLASH network (quantifier). This use of both “qualifier” and “quantifier” NNs is certainly an advantage over the single-stage CLASH

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NN. The CLASH network was only trained with overtopping data (tests with “no overtopping” were not considered) and, therefore, this network always gives a prediction of overtopping, even in the range where no overtopping should be expected.

14.5. Use of CLASH Database The EU research project CLASH generated an extensive database of overtopping tests from data submitted around the world. Each test was described not only by 31 parameters as hydraulic and structural parameters, but also parameters describing the reliability and complexity of the test and structure. The database includes more than 10,000 tests and was set up as an Excel database. At its simplest, the database is no more than a matrix with 31 columns and more than 10,000 rows. It can be downloaded from the CLASH or Manual web site. If a user has a specific structure, there is a possibility to look into the database and find more or less similar structures with measured overtopping discharges. It may even be possible that the structure has already been tested with the right wave conditions! Finding the right tests can be done by using filters in the Excel database. Every test of such a selection can then be studied thoroughly. One example will be described here in depth. Suppose one is interested in the improvement of a vertical wall with a large wave return wall. The wave conditions are Hm0 toe = 3 m, the wave steepness sm−1,0 = 0.04 (Tm−1,0 = 6.9 s), and the wave attack is perpendicular to the structure. The design water depth h = 10 m and the wave return wall starts 1 m above design water level and has a height and width of 2 m (the angle is 45◦ seaward). This gives a crest freeboard Rc = 3 m, equal to the wave height. Have tests been performed which are close to this specific structure and given wave conditions? The first filter selects data with a vertical down slope, i.e., cotαd = 0. The second filter could select data with a wave return wall overhanging more than about 30◦ seaward. This means cotαu < −0.57. In the first instance every large wave return wall can be considered, say, at least 0.5Hm0 wide. This gives the third filter, selecting data with −cotαu∗ (Ac + hb )/Hm0 ≥ 0.5. With these three filters, the database gives 212 tests from four independent test series. These data are summarized in Fig. 14.24 with the expression of Franco et al.6 for a vertical wall. There are 22 tests which gave no measurable overtopping. These 3 1/2 results are represented in the figure with a value of q/(gHm0 = 10−7. The toe ) majority of the data are situated below the simple prediction curve for a vertical wall, indicating that a wave return wall is efficient, but the data are too scattered to be decisive. The next step in the filtering process could be that only wave return walls overhanging more than 45◦ seaward are selected. This means cotαu < −1. The water depth is relatively large for the considered case, and shallow water conditions are excluded if h/Hm0 toe > 3. Figure 14.25 shows this further filtering process. The number of data has been reduced to 78 tests from only two independent series. In total, 12 tests result in no measureable overtopping. The data show the trend that the overtopping discharges are on average about 10 times smaller than for a vertical

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1.E+00 series 508 series 914

1.E-01

series 113

q/(gHm0 toe )

3 0.5

1.E-02

series 033 Franco et al.,(1994)

1.E-03 1.E-04 1.E-05 1.E-06 1.E-07 0

0.5

1

1.5

2

2.5

3

3.5

4

Rc /Hm0 toe Fig. 14.24.

Overtopping for large wave return walls — first selection.

1.E+00 series 914 series 113

1.E-01

Franco et al.,(1994)

q/(gHm0 toe )

3 0.5

1.E-02 1.E-03 1.E-04 1.E-05 1.E-06 1.E-07 0

0.5

1

1.5

2

2.5

3

3.5

4

Rc/H m 0 toe Fig. 14.25.

Overtopping for large wave return walls — second selection with more criteria.

wall, given by the dashed line. But for Rc /Hm0 toe > 1, there are quite some tests without any overtopping. As still quite some data are remaining in Fig. 14.25, it is possible to narrow the search area even further. With a wave steepness of sm−1,0 = 0.04 in the considered case, the wave steepness range can be limited to 0.03 < sm−1,0 < 0.05. The width

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1.E+00 series 113

1.E-01

Franco et al.,(1994)

q/(gH m0 toe )

3 0.5

1.E-02 1.E-03 1.E-04 1.E-05 1.E-06 1.E-07 0

0.5

1

1.5

2

2.5

3

3.5

4

R c/Hm 0 toe Fig. 14.26. Overtopping for wave return wall with so = 0.04, seaward angle of 45◦ , a width of 2 m and a crest height of Rc = 3 m. For Hm0 toe = 3 m, the overtopping can be estimated from Rc /Hm0 toe = 1.

of the wave return wall of 2 m, with the wave height of Hm0 = 3 m, gives a relative width of 0.67. The range can be limited to 0.5 < −cotαu∗ (Ac + hb )/Hm0 < 0.75. Finally, the transition from vertical to wave return wall is 1 m above the design water level, giving hb /Hm0 toe = −0.33. The range can be set at −0.5 < hb / Hm0 toe < −0.2. The final selection obtained after filtering is given in Fig. 14.26. Only four tests remain from one test series, one of which gave no measurable overtopping. The data now give a clear picture. For a relative freeboard lower than about Rc /Hm0 toe = 0.7, the overtopping will not be much different from the overtopping at a vertical wall. The wave return wall, however, becomes very efficient for large freeboards and effectively prevents any measurable overtopping for Rc /Hm0 toe > 1.2. For the structure considered with Rc /Hm0 toe = 1, the wave overtopping will be 20–40 times less than that for a vertical wall and will probably amount to q = 0.5 − 2 l/s per m width. In this particular case, it was possible to find four tests in the database with very close similarities to the considered structure and wave conditions. 14.6. Tolerable Discharges 14.6.1. Hazards from overtopping Most sea defense structures are constructed primarily to limit overtopping that might cause flooding. Over a particular storm or tide, the overtopping volumes that can be tolerated will be site-specific as the overall volume of water that can be accepted will depend on the size and use of the receiving area, extent, and magnitude

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of drainage ditches, damage versus inundation curves, and return period. Guidance on modeling inundation flows is being developed within the Floodsite project (see: http://www.floodsite.net/html/project overview.htm), but flood volumes, per se, are not distinguished further in this chapter. Instead, advice here focuses on direct hazards from wave overtopping. For sea defenses that protect people living, working, or enjoying themselves, designers and owners of these defenses must deal with potential direct hazards from overtopping. This requires that the level of hazard and its probability of occurrence be assessed, allowing appropriate action plans to be devised to ameliorate risks arising from overtopping. The main hazards on or close to sea defense structures are of death, injury, property damage, or disruption from direct wave impact or by drowning. On average, approximately 2–5 people are killed each year in each of the United Kingdom and Italy through wave action, chiefly on seawalls and similar structures (although this rose to 11 in the United Kingdom during 2005). It is often helpful to analyze direct wave and overtopping effects, and their consequences under three general categories: • direct hazard of injury or death to people immediately behind the defense; • damage to property, operation, and/or infrastructure in the area defended, including loss of economic, environmental, or other resource, or disruption to an economic activity or process; • damage to defense structure(s), either short-term or longer-term, with the possibility of breaching and flooding. The character of overtopping flows or jets, and the hazards they cause, also depend upon the geometries of the structure, the hinterland behind the seawall, and the form of overtopping. Rising ground behind the seawall may allow people at potential risk to see incoming waves, and the slope will slow overtopping flows. Conversely, a defense that is elevated significantly above the land defended will obscure visibility of incoming waves (Fig. 14.27), and post-overtopping flows may increase in speed rather than decreasing. Hazards caused by overtopping therefore depend upon both the local topography and structures as well as on the direct overtopping characteristics. It is not possible to give unambiguous or precise limits to tolerable overtopping for all conditions. Some guidance is, however, offered here on tolerable mean discharges and maximum overtopping volumes for a range of circumstances or uses, and on inundation flows and depths. These limits may be adopted or modified depending upon the circumstances and uses of the site. 14.6.1.1. Wave overtopping processes and hazards Overtopping hazards can be linked to a number of simple direct flow parameters (Fig. 14.28): • mean overtopping discharge, q; • individual and maximum overtopping volumes, Vi and Vmax ;

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Fig. 14.27.

Defended area below seawall and foreshore (saltmarsh) level.

Fig. 14.28.

Overtopping on embankment and promenade seawalls.

• overtopping velocities over the crest, horizontally and vertically, vxc and vzc or vxp and vzp ; • overtopping flow depth, again measured on crest or promenade, dxc or dxp . Less direct responses (or similar responses, but farther back from the defense) may be used to assess the effects of overtopping, perhaps categorized by: • overtopping falling distances, xc ; • post-overtopping wave pressures (pulsating or impulsive), pqs or pimp ; • post-overtopping flow depths, dxc or dxp ; and horizontal velocities, vxc or vxp . The main response to direct overtopping hazards has most commonly been the construction of new defenses, but should now always consider three options, in increasing order of intervention: • move human activities away from the area subject to overtopping or flooding hazard, thus modifying the land-use category and/or habitat status;

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• accept hazard at a given probability (acceptable risk) by providing for temporary use and/or short-term evacuation with reliable forecast, warning and evacuation systems, and/or use of temporary/demountable defense systems; • increase defense standard to reduce risk to (permanently) acceptable levels probably by enhancing the defense and/or reducing loadings. For any structure expected to reduce overtopping, the crest level and/or the front-face configuration will be dimensioned to give acceptable levels of overtopping under specified extreme conditions or combined conditions (e.g., water level and waves). Setting acceptable levels of overtopping depends on: • • • •

use of the defense structure itself; use of the land behind; national and/or local standards and administrative practice; economic and social basis for funding the defense.

Under most forms of wave attack, waves tend to break before or onto sloping embankments with the overtopping process being relatively gentle (see Fig. 14.1). Relatively few water levels and wave conditions may cause “impulsive” breaking where the overtopping flows are sudden and violent. Conversely, steeper, vertical, or compound structures are more likely to experience intense local impulsive breaking, and may overtop violently and with greater velocities (see Fig. 14.3). The form of breaking will therefore influence the distribution of overtopping volumes and their velocities, both of which will have impact on the hazards that they cause. Additional hazards that are not dealt with here are those that arise from wave reflections, often associated with steep-faced defenses. Reflected waves increase wave disturbance, which may cause hazards to navigating or moored vessels; may increase waves along neighboring frontages, and/or may initiate or accelerate local bed erosion thus increasing depth-limited wave heights. 14.6.1.2. Form of overtopping hazard Wave overtopping which runs up the face of the seawall and over the crest in (relatively) complete sheets of water is often termed “green water.” In contrast, “white water” or spray overtopping tends to occur when waves break seaward of the defense structure or break onto its seaward face, producing noncontinuous overtopping, and/or significant volumes of spray. Overtopping spray may be carried over the wall either under its own momentum, or assisted and/or driven by an onshore wind. Additional spray may also be generated by wind acting directly on wave crests, particularly when reflected waves interact with incoming waves to give severe local “clapoti.” This type of spray is not classed as overtopping nor is it predicted by the methods described in this manual. Without a strong onshore wind, spray will seldom contribute significantly to overtopping volumes, but may cause local hazards. Light spray may reduce visibility for driving, important on coastal highways, and will extend the spatial extent of salt spray effects such as damage to crops/vegetation, or deterioration of buildings. The effect of spray in reducing visibility on coastal highways (particularly when

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intermittent) can cause sudden loss of visibility in turn leading drivers to veer suddenly. Effects of wind and generation of spray have not often been modeled. Some research studies have suggested that effects of onshore winds on large green water overtopping are small, but that overtopping under q = 1 l/s per m might increase up to four times under strong winds, especially where much of the overtopping is as spray. Discharges between q = 1 and 0.1 l/s per m are, however, already greater than some discharge limits suggested for pedestrians or vehicles, suggesting that wind effects may influence overtopping at and near acceptable limits for these hazards. This is discussed further in Sec. 14.7. 14.6.1.3. Return periods Return periods at which overtopping hazards are analyzed, and against which a defense might be designed, may be set by national regulation or guidelines. As with any area of risk management, different levels of hazard are likely to be tolerated at inverse levels of probability or return period. The risk levels (probability × consequence) that can be tolerated will depend on local circumstances, local and national guidelines, the balance between risk and benefits, and the level of overall exposure. Heavily trafficked areas might therefore be designed to experience lower levels of hazard applied to more people than lightly used areas, or perhaps the same hazard level at longer return periods. Guidance on example return periods used in evaluating levels of protection suggest example protection levels versus return periods as shown in Table 14.3. In practice, some of the return periods in Table 14.3 may be too short. National guidelines have recommended lower risk, e.g., a low probability of flooding in the United Kingdom is now taken as <0.1% probability (1:1000 year return), and the medium probability of sea flooding as between 0.5% and 0.1% (1:200 to 1:1000 year return). Many existing defenses, however, offer levels of protection far lower than these. In the Netherlands, where two-thirds of the country lies below the storm surge level, protection was substantially improved after the flood in 1953 where almost 2000 people drowned. Standards of protection for large rural areas are currently 1:10,000 years, less densely populated areas at 1:4000 years, and protection for high river discharge (without threat of storm surge) is given to 1:1250 years. Table 14.3.

Hazard types.

Hazard type and reason Temporary or short-term measures Majority of coast protection or sea defenses Flood defenses protecting large areas Special structure, high capital cost Nuclear power stations, etc. a Total

probability return period.

Design life (years)

Level of protectiona (years)

1–20 30–70 50–100 200 —

5–50 50–100 100–10,000 Up to 10,000 10,000

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The design life for flood defenses like dikes which are fairly easy to upgrade, is taken in the Netherlands as 50 years. In urban areas, where it is more difficult to upgrade a flood defense, the design life is taken as 100 years. This design life increases for very special structures with high capital costs, like the Eastern Scheldt storm surge barrier, Thames barrier, or the Maeslandtkering at the entrance to Rotterdam. A design life of around 200 years is then usual. Variations from simple “acceptable risk” approach may be required for publicly funded defenses based on benefit — cost assessments, or where public aversion to hazards causing death requires greater efforts to ameliorate the risk, either by reducing the probability of the hazard or by reducing its consequence. 14.6.2. Tolerable mean discharges and overtopping simulator Guidance on overtopping discharges that can cause damage to seawalls, buildings, or infrastructure, or danger to pedestrians and vehicles have been related to mean overtopping discharges or (less often) to peak volumes. Guidance quoted previously were derived initially from the analysis in Japan of overtopping perceived by port engineers to be safe. Further guidance from Iceland suggests that equipment or cargo might be damaged for q ≥ 0.4 l/s per m. Significantly, different limits are discussed for embankment seawalls with back slopes, or for promenade seawalls without back slopes. Some guidance distinguishes between pedestrians or vehicles, and between slow and faster speeds for vehicles. Tests on the effects of overtopping on people suggest that information on mean discharges alone may not give reliable indicators of safety for some circumstances, and that maximum individual volumes may be better indicators of hazard than average discharges. The volume (and velocity) of the largest overtopping event can vary significantly with wave condition and structure type, even for a given mean discharge. There remain, however, two difficulties in specifying safety levels with reference to maximum volumes rather than to mean discharges. Methods to predict maximum volumes are available for fewer structure types, and are less well-validated. Secondly, data relating individual maximum overtopping volumes to hazard levels are still very rare. In most instances, the discharge (or volumes) discussed here are those at the point of interest, e.g., at the roadway or footpath or building. It is noted that the hazardous effect of overtopping waters reduces with the distance away from the defense line. As a rule of thumb, the hazard effect of an overtopping discharge at a point x m back from the seawall crest will be to reduce the overtopping discharge by a factor of x; and so the effective overtopping discharge at x (over a range of 5–25 m), qeffective is given by qeffective = qseawall /x.

(14.8)

The overtopping limits suggested in Tables 14.4–14.7 derive from a generally precautionary principle informed by previous guidance and by observations and measurements made by the CLASH partners and other researchers. Limits for pedestrians in Table 14.4 show a logical sequence, with allowable discharges reducing

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372 Table 14.4.

Limits for overtopping for pedestrians.

Hazard type and reason Trained staff, well shod and protected, expecting to get wet, overtopping flows at lower levels only, no falling jet, low danger of fall from walkway. Aware pedestrian, clear view of the sea, not easily upset or frightened, able to tolerate getting wet, wider walkway.b

Mean discharge q (l/s per m)

Max volumea Vmax (l/m)

1–10

500 at low level

0.1

20–50 at high level or velocity

a These limits relate to overtopping velocities well below vc ≤10 m/s. Lower volumes may be required if the overtopping process is violent and/or overtopping velocities are higher. b Not all of these conditions are required, nor should failure of one condition on its own require the use of a more severe limit.

Table 14.5.

Limits for overtopping for vehicles. Mean discharge q (l/s per m)

Max volumea Vmax (l/m)

Driving at low speed, overtopping by pulsating flows at low flow depths, no falling jets, vehicle not immersed

10–50a

100–1000

Driving at moderate or high speed, impulsive overtopping giving falling or high velocity jets

0.01–0.05b

5–50b at high level or velocity

Hazard type and reason

a These

limits probably relate to overtopping defined at highway. limits relate to overtopping defined at the defense, but assumes the highway to be immediately behind the defense. b These

steadily as the recipient’s ability or willingness to anticipate or receive the hazard reduces. A further precautionary limit of q = 0.03 l/s per m might apply for unusual conditions where pedestrians have no clear view of incoming waves; may be easily upset or frightened or are not dressed to get wet; may be on a narrow walkway or in close proximity to a trip or fall hazard. Research studies have, however, shown that this limit is only applicable for the conditions identified, and should NOT be used as the general limit for which q = 0.1 l/s per m in Table 14.4 is appropriate. For vehicles, the suggested limits are rather more widely spaced as two very different situations are considered. The higher overtopping limit in Table 14.5 applies where wave overtopping generates pulsating flows at roadway level, akin to driving through slowly varying fluvial flow across the road. The lower overtopping limit in Table 14.5 is, however, derived from considering more impulsive flows, overtopping at some height above the roadway, with overtopping volumes being projected at

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speed and with some suddenness. These lower limits are however based on few site data or tests, and may therefore be relatively pessimistic. Rather fewer data are available on the effects of overtopping on structures, buildings, and property. Site-specific studies suggest that pressures on buildings by overtopping flows will vary significantly with the form of wave overtopping, and with the use of sea defense elements intended to disrupt overtopping momentum (not necessarily reducing discharges). Guidance derived from the CLASH research project and previous work suggests limits in Table 14.6 for damage to buildings, equipment, or vessels behind defenses. A set of limits for structures in Table 14.7 has been derived from early work by Goda7 and others in Japan. These give a first indication of the need for specific protection to resist heavy overtopping flows. It is assumed that any structure close to the sea will already be detailed to resist the erosive power of heavy rainfall and/or spray. Two situations are considered: • embankment seawall or dike, elevated above the defended area, and so overtopping flows pass over the crest and down the rear face; • promenade defense in which overtopping flows remain on or behind the seawall crest before returning seaward.

Table 14.6.

Limits for overtopping for property behind the defense.

Hazard type and reason

Mean discharge q (l/s per m)

Max volumea Vmax (l/m)

50

5,000–50,000

10a 1b 0.4a

1,000–10,000 ∼ ∼

Significant damage or sinking of larger yachts Sinking small boats set 5–10 m from wall. Damage to larger yachts Building structure elements Damage to equipment set back 5–10 m a These b This

limits relate to overtopping defined at the defense. limit relates to the effective overtopping defined at the building.

Table 14.7.

Limits for overtopping for damage to the defense crest or rear slope.

Hazard type and reason Embankment seawalls/sea dikes No damage if crest and rear slope are well protected No damage to crest and rear face of grass covered embankment of clay No damage to crest and rear face of embankment if not protected Promenade or revetment seawalls Damage to paved or armoured promenade behind seawall Damage to grassed or lightly protected promenade or reclamation cover

Mean discharge q (l/s per m) 50–200 1–10 0.1 200 50

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Fig. 14.29.

Principle of the wave overtopping simulator.

The limits for the latter category cannot be applied where the overtopping flows can fall from the defense crest where the nature of the flow may be more impulsive. The limits in Table 14.7 are precautionary and are generally based on old data. In order to clarify the erosion resistance of grass protection under wave overtopping, tests were performed using the overtopping simulator (Figs. 14.29 and 14.30) in 2007 on a sea dike in the Netherlands. The grass dike had a 1:3 inner slope of fairly good clay, sand content smaller than 30%. The overtopping simulator was used to test the erosion resistance of this inner slope for a simulated 6 h storm for each overtopping condition. These started with overtopping equivalent to a mean discharge of 0.1 l/s per m and increased to 1, 10, 20, 30, and finally even 50 l/s per m. After all these simulated storms, the slope was still in good condition and showed little erosion. The erosion resistance of this dike was very high. Another test was performed on bare clay by removing the grass sod over the full inner slope to a depth of 0.2 m. Overtopping conditions of 0.1 l/s per m, 1, 5, and finally, 10 l/s per m were performed, again for 6 h each. Erosion damage started for the first condition (two erosion holes) and increased during the other overtopping conditions. After 6 h at a mean discharge of 10 l/s per m (see Figs. 14.30 and 14.31), there were two large erosion holes, about 1 m deep, 1 m wide, and 4 m long. This situation was considered as “not too far from initial breaching.” The overall conclusion of this first overtopping test on a real dike is that clay with grass can be highly erosion-resistant. Even without grass the good quality clay also survived extensive overtopping. The conclusions may not yet be generalized to all dikes as

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Fig. 14.31.

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Overtopping simulator discharging a large volume on the inner slope of a dike.

Wave overtopping test on bare clay; result after 6 h with 10 l/s per m width.

clay quality and type of grass cover still may play a role and, therefore, more testing is required to come to general conclusions. One remark, however, should be made on the strength of the inner slopes of dikes by wave overtopping. Direct erosion of the slope is one possible failure mechanism. A major failure mechanism, especially in the past, was slip failure of the (rear) slope. Slip failures may directly lead to a breach, and such slip failures often occur

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mainly for steep inner slopes like 1:1.5 or 1:2. For this reason, most dike designs in the Netherlands in the past 50 years have used a 1:3 inner slope, where it is unlikely that slip failures will occur due to overtopping. This mechanism might, however, occur for steep inner slopes, and so it should be taken into account in safety analysis. 14.6.3. Tolerable maximum volumes and velocities 14.6.3.1. Overtopping volumes and velocities Guidance on suggested limits for maximum individual overtopping volumes have been given in Tables 14.4–14.7 where data are available. Research studies with volunteers at full scale or field observations suggest that danger to people or vehicles might be related to peak overtopping volumes, with “safe” limits for people covering: Vmax = 1000 − 2000 l/m for trained and safety-equipped staff in pulsating flows on a wide-crested dike; Vmax = 750 l/m for untrained people in pulsating flows along a promenade; Vmax = 100 l/m for overtopping at a vertical wall; Vmax = 50 l/m where overtopping could unbalance an individual by striking their upper body without warning. Few data are available on overtopping velocities and their contribution to hazards. Chapter 15 gives guidance on overtopping flow velocities and depths at crest and inner slope for simple sloping embankments as well. Velocities of 5–8 m/s are possible for maximum overtopping waves during overtopping discharges of 10– 30 l/s per m width. Studies of hazards under steady flows suggest that limits on horizontal velocities for people and vehicles will probably need to be set below vx < 2.5–5 m/s. On vertical and battered walls, upward projected velocities (vz ) have been related to inshore wave celerity (see Chap. 16). Relative velocities, vz /ci , have been found to be roughly constant at vz /ci ≈ 2.5 for pulsating and slightly impulsive conditions, but increase significantly for impulsive conditions, reaching vz /ci ≈ 3–7. 14.6.3.2. Overtopping loads Post-overtopping wave loads have seldom been measured on defense structures, buildings behind sea defenses, or on people; so little generic guidance is available. If loadings from overtopping flows could be important, they should be quantified by interpretation of appropriate field data or by site-specific model studies. An example model study during the CLASH research project indicates how important these effects might be. A simple 1 m high vertical secondary wall was set in a horizontal promenade about 7 m back from the primary seawall, itself a concrete recurve fronted by a steep beach and short rock armour slope. Pulsating wave pressures were measured on the secondary wall against the effective overtopping discharge arriving at the secondary wall. (This discharge was deduced by applying Eq. (14.7) to the overtopping measured at the primary wall, 7 m in front.) Whilst

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strongly site-specific, these results suggest that quite low discharges (0.1–1.0 l/s per m) may lead to loadings up to 5 kPa.

14.6.4. Effects of debris and sediment in overtopping flows There are virtually no data on the effect of debris on hazards caused by overtopping, although anecdotal comments suggest that damage can be substantially increased for a given overtopping discharge or volume if “hard” objects such as rocks, shingle, or timber are included in overtopping. It is known that impact damage can be particularly noticeable for seawalls and promenades where shingle may form the “debris” in heavy or frequent overtopping flows.

14.7. Model Effects, Scale Effects, and Uncertainties 14.7.1. Model and scale effects This section deals with the types of model and scale effects that result from the use of hydraulic models to quantify wave overtopping. Firstly, scale and model effects are defined. Secondly, a methodology based on the current knowledge is introduced on how to account for these effects. Model or laboratory effects originate from the incorrect reproduction of the geometry or materials of the prototype structure, or of the waves and currents, or due to the boundary conditions of a wave flume (sidewalls, wave paddle, etc.). Modeling techniques have developed significantly, but model effects may still influence test results. One noticeable feature is that very few model tests include wind, whilst wind effects may be significant for some forms of overtopping, particularly splash or spray. Scale effects result from incorrect (or distorted) reproduction of a prototype water-structure interaction in the model. The ideal model requires that both Froude’s and Reynolds’ laws are met simultaneously. This is not possible without changing the test fluid; so, scale effects cannot be avoided when performing scaled model tests. They can, however, be minimized for the main processes, and/or corrections can be applied where the distortion is understood. Gravity, pressure, and inertial forces are the relevant forces for wave motion; so, physical models of seawalls/breakwaters are scaled according to Froude’s law. Viscosity forces are governed by Reynolds’ law, elasticity by Cauchy’s law, and surface tension forces by Weber’s law, and these forces are generally neglected for most models. Distortions or errors resulting from ignoring these forces are called scale effects, and are generally unquantified. Measurement effects result from distortions to the process by the use of measurement equipment and/or data sampling methods. These distortions may significantly influence the comparison of results between prototype and model, or between two models. It is therefore essential to quantify the effects and the uncertainty related to the different techniques available.

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Whilst these definitions are reasonably clear, in reality, it is sometimes difficult to assign all the causes of differences between model and prototype data. During CLASH, the major contributions to model effects were found to be wind since this is seldom included in hydraulic models. Additional differences were also found and were ascribed to the model effects in addition to those by the wind. Examining repeatability of example tests on armoured slopes showed that wave parameters (Hm0 , Tp , Tm−1,0 ) have a coefficient of variation CoV ∼ 3%. Differences between overtopping measured in two wave flumes were CoV ∼ 13% and CoV ∼ 10%. Different time windows for wave analysis and different types of wave generation methods had little or no influence on the estimated wave parameters (CoV ∼ 3%). The number of waves in a test influences overtopping, and the use of 200 waves compared to 1000 waves shows differences in the mean overtopping rates up to 20%. The position of the overtopping tray at the side of the flume also showed differences in the overtopping rates (CoV ∼ 20%) from the results where the tray was located at the center of the crest. This could be because of the different arrangements of the armour units in front of the overtopping tray or due to the influence of the sidewalls of the flume. Scale effects have been investigated by various authors, and this has led to some generic rules that should be observed for physical model studies. Generally, water depths in the model should be much larger than h = 2.0 cm, wave periods larger than T = 0.35 s, and wave heights larger than Hs = 5.0 cm to avoid the effects of surface tension. For rubble mound structures, the Reynolds number for the stability of the armour layer should exceed Re = 3 × 104 ; for overtopping of coastal dikes Re > 1 × 103 ; and stone sizes in underlayers and core of rubble mounds should be scaled according to the velocities in the core rather than the stone dimensions, especially for small models. This leads to the use of larger material in the core than that demanded by simple Froude scaling. Critical limits for the influence of viscosity and surface tension are given in Table 14.8. From the observations in the prototype and scaled models, a methodology was derived to account for these differences without specifically defining the contribution from the model, scale, or measurement effects. These recommendations are given Table 14.8.

Scale effects and critical limits.

Process

Relevant forces

Similitude law

Critical limits

Wave propagation

Gravity force Friction forces Surface tension Gravity force Friction forces Surface tension Gravity force Friction forces Surface tension Gravity force Friction forces Surface tension

F rW , ReW , We F rW , ReW , We F rA , F rq Req , We F rA , F rq , Req , We

ReW > ReW,crit = 1 × 104 T > 0.35 s; h > 2.0 cm

Wave breaking

Wave runup

Wave overtopping

ReW > ReW,crit = 1 × 104 T > 0.35 s; h > 2.0 cm Req > Req,crit = 103 W e > W ecrit = 10 Req > Req,crit = 103 W e > W ecrit = 10

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in Chaps. 15 and 16 for dikes, for rubble mound slopes, and for vertical walls, respectively.

14.7.2. Uncertainties in predictions Sections 14.2–14.5 have proposed various models or ways to predict wave overtopping of coastal structures. These models will now be discussed with regard to their uncertainties.

14.7.2.1. Empirical models Unless the empirical method has been configured to give an “upper bound” estimate, model uncertainty can usually be described by using a mean factor of 1.0 and a Gaussian distribution around the mean prediction. The standard deviation is derived by comparing model data and the prediction. This has two implications for design. Probabilistic design values for all empirical models used in this manual describe the mean approach for all underlying data points. This means that, for normally distributed variables, about 50% of the data points exceed the prediction by the model, and 50% are below the predicted values. This value should be used if probabilistic design methods are used. Deterministic design values may be given as the mean value plus one standard deviation, which in general gives a safer approach, and takes into account that model uncertainty for wave overtopping is always significant.

14.7.2.2. Neural network When running the NN, the user is provided with overtopping rates based on the CLASH database and the NN prediction. Together with these results the user will also obtain the uncertainties of the prediction through the 5% and 95% confidence intervals. Assuming a normal distribution will allow the standard deviation of overtopping to be estimated from those confidence intervals, and (if required) the whole Gaussian distribution.

14.7.2.3. CLASH database The CLASH database was described earlier. It provides a large data set of model data on wave overtopping of coastal structures. As these are model data, it is noted that corrections for model/scale effects discussed in Sec. 14.7.1 above have not been applied to the database. The user will therefore need to decide to apply any scale/model correction procedure whenever these data are used for prototype predictions. With respect to uncertainties, all model results show variations in measured overtopping. Most of these variations will result from the measurement and model effects as discussed earlier, and will therefore apply to the database.

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14.8. Guidance on Use of Methods The EurOtop Overtopping Manual5 is accompanied by an overall Calculation Tool, which includes the following elements: • Empirical Calculator programmed with the main empirical overtopping equations in this chapter and the next two (limited to those that can be described explicitly, that is without iteration). • PC-Overtopping, which codes all the prediction methods for mean overtopping discharge and other parameters, for (generally shallow sloped) sea dikes, see Sec. 14.3. • Neural Network tool developed in the CLASH research project to calculate mean overtopping for many types of structures (see Sec. 14.4). • CLASH database, a listing of input parameters and mean overtopping discharge from each of approximately 10,000 physical model tests on both idealized (research) test structures, and site-specific designs. These data can be sifted to identify test results that may apply for configurations close to the reader’s (see Sec. 14.5). None of these methods give the universally “best” results, and indeed there may still be a need for site-specific model tests for some defenses. The most reliable method to be used will depend on the type and complexity of the structures, and the closeness with which it conforms to simplifying assumptions used in the previous model testing (on which all of the methods above are inherently based). In selecting which method to use, or which set of results to prefer when using more than one method, the user will need to take account of the origins of each method. It may also be important in some circumstances to use an alternative method to give a check on a particular set of calculations. To assist these judgments, a set of simple rules of thumb are given here, but as ever, these should not be treated as universal truths. • For simple vertical, composite, or battered walls which conform closely to the idealizations in Chap. 16 on vertical walls, the results of the Empirical Calculator are likely to be more reliable than the other methods as test data for these structure types do not feature strongly in the Database or NN, and pcovertopping is not applicable. • For simple sloped dikes with a single roughness, many test data have been used to develop the formulae in the Empirical Calculator; so, this may be the most reliable, and simplest to use/check. For dikes with multiple slopes or roughness, pc-overtopping is likely to be the most reliable, and easiest to use, although independent checking may be more complicated. The Database or NN methods may become more reliable where the structure starts to include further elements. • For armoured slopes and mounds, open mound structures that most closely conform to the simplifying models may best be described by the formulae in the Empirical Calculator. Structures of lower permeability may be modeled using pcovertopping. Mounds and slopes with crown walls may be best represented by the application of the Database or NN methods.

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• For unusual or complex structures with multiple elements, mean overtopping discharge may be most reliably predicted by pc-overtopping (if applicable) or by the Database or NN methods. • For structures that require use of the NN method, it is possible that the use of many data for other configurations to develop a single NN method may introduce some averaging. It may therefore be appropriate to check in the database to see whether there are already test data close to the configuration being considered. This procedure may require some familiarity with manipulating these data. In almost all instances, the use of any of these methods will involve some degree of simplification of the true situation. The further the structure or design (analysis) conditions depart from the idealized configurations tested to generate the methods/ tools discussed, the wider will be the uncertainties. Where the importance of the assets being defended is high, and/or the uncertainties in using these methods are large, then the design solution may require use of site-specific physical model tests.

14.9. Conclusions and Outlook It is clear that increased attention to flood risk reduction, and to wave overtopping in particular, have increased interest and research in this area. The EurOtop Overtopping Manual5 is therefore not expected to be the “last word” on the subject; indeed even whilst preparing the first version of the manual, the author team expected that there will be later revisions. The reader of this handbook is therefore advised to check whether an improved version of the EurOtop Overtopping Manual has been released. Beyond that manual, it is probable that there will be significant improvements in numerical modeling, although it should be acknowledged that improved numerical models will require substantial measurement data to validate them before their results can be relied upon in detailed analysis or design.

Acknowledgments This chapter, and also Chaps. 15 and 16, are based on the EurOtop Overtopping Manual,5 which was funded in the the United Kingdom by the Environmental Agency, in Germany by the German Coastal Engineering Research Council (KFKI), in the Netherlands by Rijkswaterstaat and Netherlands Expertise Network (ENW) on Flood Protection. The Project Team for the creation, editing, and support of the manual was formed by Prof William Allsop (HRW) Dr Tom Bruce (UoE) Dr Andreas Kortenhaus (LWI) Dr Jentsje van der Meer (Infram) Dr Tim Pullen (HRW) Dr Holger Sch¨ uttrumpf (BAW)

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The project was guided and supervised by a Project Steering Group: Dr Chrissy Mitchell (EA) Dr Daniel Schade (IM Gmbh) Dick Thomas (Faber Maunsell) Hans van der Sande (Waterboard Zeeuwse Eilanden) Mark Klein Breteler (Delft Hydraulics) Dr Mervyn Bramley (EA Advisor) Michael Owen (EA Advisor) The EurOtop authors are grateful to a wide group of contributors including (but not limited) John Alderson, Phillip Besley,2 Laurence Banyard, Karl-Friedrich Daemrich, Leopoldo Franco, Daan Heineke, Hocine Oumeraci, Jon Pearson, Thorsten Piontkowitz, and Piebe van den Berg. References References in this chapter have been kept to a real minimum. An extensive list of relevant references, however, can be found in the EurOtop Overtopping Manual.5 1. N. W. H. Allsop, P. Besley and L. Madurini, Overtopping performance of vertical and composite breakwaters, seawalls and low reflection alternatives, Paper 4.7 in MCS Project Final Report, University of Hannover (1995). 2. P. Besley, Overtopping of Seawalls — Design and Assessment Manual, R & D Technical Report W 178, Environment Agency, Bristol, ISBN 1 85705 069 X (1999). 3. CLASH, Crest level assessment of coastal structures by full scale monitoring, neural network prediction and hazard analysis on permissible wave overtopping, Fifth Framework Program of the EU, Contract no. EVK3-CT-2001-00058. www.clash-eu.org. 4. DELOS, Environmental design of low crested coastal defence structures, Fifth Framework Program of the EU, Contract no. EVK3-CT-00041. www.delos.unibo.it. 5. EurOtop Overtopping Manual, Wave Overtopping of Sea Defences and Related Structures — Assessment Manual, eds. T. Pullen, N. W. H. Allsop, T. Bruce, A. Kortenhaus, H. Sch¨ uttrumpf and J. W. van der Meer, www.overtopping-manual.com. 6. L. Franco, M. de Gerloni and J. W. van der Meer, Wave overtopping on vertical and composite breakwaters, Proc. 24th Int. Conf. Coastal Eng. Kobe (1994), pp. 1030–1044. 7. Y. Goda, Random Seas and Design of Maritime Structures, 2nd edn. (World Scientific Publishing, Singapore, 2000), ISBN 981-02-3256-X. 8. TAW, Technical Report — Wave run-up and wave overtopping at dikes, J. W. van der Meer (author) Technical Advisory Committee for Flood Defence in the Netherlands (TAW), Delft (2002). 9. H. Verhaeghe, Neural network prediction of wave overtopping at coastal structures, PhD. thesis, University Gent, Belgium, ISBN 90-8578-018-7 (2005).

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Chapter 15

Wave Run-Up and Wave Overtopping at Armored Rubble Slopes and Mounds Holger Sch¨ uttrumpf Institute of Hydraulic Engineering and Water Resources Management RWTH-Aachen University, 52065 Aachen, Germany [email protected] Jentsje van der Meer Van der Meer Consulting P. O. Box 423, 8440, AK Heerenveen, The Netherlands [email protected] Andreas Kortenhaus Leichtweiss-Institute for Hydraulics Technical University of Braunschweig Beethovenstr 51a, 38106, Braunschweig, Germany [email protected] Tom Bruce School of Engineering, University of Edinburgh King’s Buildings, Edinburgh, EH9 3JL, UK [email protected] Leopoldo Franco Department of Civil Engineering, University of Roma Tre Via V. Volterra 62, 00146 Roma, Italy [email protected]

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Wave overtopping and to a lesser extent wave run-up for armored rubble slopes and mounds have been subject to a number of investigations in the past. The objective of the present chapter is to summarize existing information to be present as a closed guidance on the use of wave run-up and wave overtopping formulae for a wide range of possible applications in practice. Therefore, guidance is given first on the use of wave run-up and wave overtopping formulae for simple slopes, excluding the effects of composite slopes, direction of wave attack, roughness, wave walls, etc. Then, formulae are presented to include these parameters in the calculation procedure. Guidance is also given on wave overtopping volumes, overtopping velocities, and the spatial distribution as well as for wave overtopping for shingle beaches. Finally, the effect of model and scale effects on the calculation of average overtopping rates are discussed. This chapter has mainly been composed from Chap. 6 of the EurOtop Overtopping Manual (2007), with some additions from Chap. 5. The present chapter is related to the previous Chap. 14 and the next Chap. 16 of this manual.

15.1. Introduction Armored rubble slopes and mounds are characterized by a mound with some porosity or permeability, covered by a sloping porous armor layer consisting of large rock or concrete units. In contrast to dikes and embankment seawalls, the porosity of the structure and armor layer plays a role in wave run-up and overtopping. The cross section of a rubble mound slope, however, may have great similarities with an embankment seawall and may consist of various slopes. As an example for armored slopes and mounds, a rock-armored embankment is given in Fig. 15.1.

Fig. 15.1.

1:4 rock-armored embankment.

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15.2. Wave Run-Up and Run-Down Levels, Number of Overtopping Waves 15.2.1. Introduction to wave run-up The wave run-up height is defined as the vertical difference between the highest point of wave run-up and the still water level (SWL) (Fig. 15.2). Due to the stochastic nature of the incoming waves, each wave will give a different run-up level. In many design formulae, the wave run-up height Ru2% is used as a characteristic parameter to describe wave run-up. This is the wave run-up height, which is exceeded by 2% of the number of incoming waves at the toe of the structure. The idea behind this was that if only 2% of the waves reach the crest of a coastal structure, the crest and inner slope do not need specific protection measures. It is for this reason that much research in the past has been focused on the 2%-wave run-up height. In the past decade the design or safety assessment has been changed to allowable overtopping instead of wave run-up. Wave run-up has always been less important for rock-armored slopes and rubble mound structures, and the crest height of these types of structures has mostly been based on allowable overtopping, or even on allowable transmission (low-crested structures). Still an estimation or prediction of wave run-up is valuable as it gives a prediction of the number or percentage of waves which will reach the crest of the structure and eventually give wave overtopping. And this number is needed for a good prediction of individual overtopping volumes per wave, overtopping velocities, and flow depths. The general formula that can be applied for the 2% mean wave run-up height is given by Eq. (15.1): The relative wave run-up height Ru,2% /Hm0 in Eq. (15.1) is related to the breaker parameter ξm−1,0 .

Ru2% = wave run-up height RC = freeboard Hm0 = wave height at the toe of the structure h = water depth at the toe of the structure α = seaward slope steepness Ru2%

RC

Hm0

SWL

α h

Fig. 15.2.

Definition of the wave run-up height Ru2% on a smooth impermeable slope.

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Ru2% = 1.65 · γb · γf · γβ · ξm−1,0 Hm0
 , = 1.00 · γb · γf · γβ 4.0 − √ 1.5 with a maximum of RHu2% m0

(15.1)

ξm−1,0

where Ru2% Hm0 γb γf γβ

= = = = =

wave run-up height exceeded by 2% of the incoming waves [m]; significant wave at the toe of the structure [m]; influence factor for a berm [see 15.3.4(b)] [–]; influence factor for roughness on the slope [–]; influence factor for oblique wave attack (see  15.3.3) [–]; √ 2 ξm−1,0 = breaker parameter = tan α/ sm−1,0 = tan α 2πHm0 /(gTm−1,0 ) [–]; tanα = average slope angle (see Fig. 15.2) [–]; Tm−1,0 = spectral moment at the toe of the structure, based on m−1 and m0 [s]. The breaker or surf similarity parameter ξm−1,0 relates the slope steepness tan α 2 (or 1/n) to the fictitious wave steepness sm−1,0 = 2πHm0 /(gTm−1,0 ) and is often used to distinguish different breaker types. The combination of structure slope and wave steepness gives a certain type of wave breaking (Fig. 15.3). For ξm−1,0 > 2–3 waves are considered not to be breaking (surging waves), although there may still be some breaking, and for ξm−1,0 < 2–3 waves are breaking. Waves on a gentle foreshore break as spilling waves and more than one breaker line can be found on such a foreshore. Plunging waves break with steep and overhanging fronts and the wave tongue will hit the structure or back washing water. The transition between plunging waves and surging waves is known as collapsing. The wavefront becomes almost vertical, and the water excursion on the slope (wave run-up + run-down) is often largest for this kind of breaking. Values are given for the majority of the larger waves in a sea state. Individual waves may still surge for generally plunging conditions or plunge for generally surging conditions. The relative wave run-up height increases linearly with increasing ξm−1,0 in the range of breaking waves and small breaker parameters less than about 2. For nonbreaking waves and higher breaker parameters, the increase is less and

Fig. 15.3.

Type of breaking on a slope.

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becomes more or less horizontal. The relative wave run-up height Ru,2% /Hm0 is also influenced by the geometry of the structure, the effect of wind, and the properties of the incoming waves.

15.2.2. Wave run-up on rock-armored slopes Figure 15.4 gives 2% wave run-up heights for various rocks slopes with cot α = 1.5, 2, 3, and 4, and for an impermeable and permeable core of the rubble mound. These run-up measurements were performed during the stability tests on rock slopes of van der Meer.11 First of all, the graph gives values for a large range of the breaker parameter ξm−1,0 , due to the fact that various slope angles were tested, but also with long wave periods (giving large ξm−1,0 -values). Most breakwaters have steep slopes 1:1.5 or 1:2 only and then the range of breaker parameters is often limited to ξm−1,0 = 2–4. The graph gives rock slope information outside this range, which may be useful also for slopes with concrete armor units. The highest curve in Fig. 15.4 gives the prediction for smooth straight slopes (γf = 1). A rubble mound slope dissipates significantly more wave energy than an equivalent smooth and impermeable slope. Not only both the roughness and porosity of the armor layer cause this effect, but also the permeability of the under-layer and core contribute to it. Figure 15.4 shows the data for an impermeable core (geotextile on sand or clay underneath a thin under-layer) and for a permeable core (such as most breakwaters). The difference is most significant for large breaker parameters. Equation (15.2) includes the influence factor for roughness γf . For two layers of rock on an impermeable core, γf = 0.55. This reduces to γf = 0.40 for two layers of rock on a permeable core. This influence factor is used in the linear part of the runup formula, say, for ξm−1,0 ≤ 1.8. From ξm−1,0 = 1.8, the roughness factor increases linearly up to 1 for ξm−1,0 = 10, and it remains 1 for larger values.

4.0 3.5

Ru2%/Hm0

3.0 2.5 2.0 1.5 1.0

imp. cota=2 imp. cota=3 imp. cota=4

0.5

perm. cota=1.5 perm. cota=2 perm. cota=3

0.0 0

1

2

3

4

5

6

7

8

9

10

Spectral breaker parameter ξ m−1, 0 Fig. 15.4. Relative run-up on straight rock slopes with permeable and impermeable core, compared to smooth impermeable slopes.

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The prediction for the 2% mean wave run-up value for rock or rough slopes can be described by Ru2% = 1.65 · γb · γf · γβ · ξm−1,0 , Hm0 with a maximum of

Ru2% Hm0


= 1.00 · γb · γf surging · γβ 4.0 − √

(15.2) 1.5 ξm−1,0



.

From ξm−1,0 = 1.8, the roughness factor γf surging increases linearly up to 1 for ξm−1,0 = 10, which can be described by γf surging = γf + (ξm−1,0 − 1.8) · (1 − γf )/8.2 γf surging = 1.0 for ξm−1,0 > 10. For a permeable core, however, a maximum is reached for Ru2% /Hm0 = 1.97. The physical explanation for this is that if the slope becomes very steep (large ξm−1,0 -value) and the core is impermeable, the surging waves slowly run up and down the slope, and all the water stays in the armor layer, leading to a fairly high run-up. The surging wave actually does not “feel” the roughness anymore and acts as a wave on a very steep smooth slope. For a permeable core, however, the water can penetrate into the core which decreases the actual run-up to a constant maximum (the horizontal line in Fig. 15.4). Equation (15.2) may also give a good prediction for run-up on slopes armored with concrete armor units, if the right roughness factor is applied. Deterministic design or safety assessment. For design or a safety assessment of the crest height, it is advised not to follow the average trend, but to include the uncertainty of the prediction. As the basic equation is similar for a smooth and a rough slope, the method to include uncertainty is also the same. This means that for a deterministic design or safety assessment, Eq. (15.3) should be used: Ru2% = 1.75 · γb · γf · γβ · ξm−1,0 , Hm0 with a maximum of From ξm−1,0 for ξm−1,0 γf surging γf surging

= = = =

Ru2% Hm0


= 1.00 · γb · γf surging · γβ 4.3 − √

(15.3) 1.6 ξm−1,0



.

1.8, the roughness factor γf surging increases linearly up to 1 10, which can be described by γf + (ξm−1,0 − 1.8) · (1 − γf )/8.2 1.0 for ξm−1,0 > 10.

For a permeable core a maximum is reached for Ru2% /Hm0 = 2.11. Probabilistic design. For probabilistic calculations, Eq. (15.2) should be used together with a normal distribution and variation coefficient of CoV = 0.07. For prediction or comparison of measurements, the same Eq. (15.2) is used, but now for instance with the 5% lower and upper exceedance lines.

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15.2.3. Number of overtopping waves or overtopping percentage Till now only the 2% run-up value has been described. It might be that one is interested in another percentage, for example, on the design of breakwaters where the crest height may be determined by an allowable percentage of overtopping waves, say, 10–15%. A few ways exist to calculate run-up heights for other percentages, or to calculate the number of overtopping waves for a given crest height. van der Meer and Stam10 give two methods. One is an equation like Eq. (15.2) with a table of coefficients for the 0.1%, 1%, 2%, 5%, 10%, and 50% (median). Interpolation is needed for other percentages. The second method gives a formula for the run-up distribution as a function of wave conditions, slope angle, and permeability of the structure. The distribution is a two-parameter Weibull distribution. With this method, the run-up can be calculated for every percentage required. Both methods apply to straight rock slopes only and will not be described here. The given references, however, give all the details. The easiest way to calculate run-up (or overtopping percentage) different from 2% is to take the 2%-value and assume a Rayleigh distribution. The probability of overtopping Pov = Now /Nw (the percentage is simply 100 times larger) can be calculated by   2  √ Rc − ln 0.02 , (15.4) Pov = Now /Nw = exp − Ru,2% where Pov Now Nw Rc

= = = =

probability of overtopping [–]; number of overtopping waves in a sea state [–]; number of waves in a sea state [–]; crest freeboard [m].

Equation (15.4) can be used to calculate the probability of overtopping, given a crest freeboard Rc or to calculate the required crest freeboard, given an allowable probability or percentage of overtopping waves. One warning should be given in applying Eqs. (15.2)–(15.4). The equations give the run-up level in percentage or height on a straight (rock-armored) slope. This is not the same as the number of overtopping waves or overtopping percentage. Figure 15.5 gives the difference. The run-up is always a point on a straight slope, where for a rock-armored slope or mound the overtopping is measured some distance away from the seaward slope and on the crest, often behind a crown wall. This means that Eqs. (15.2)–(15.4) always give an overestimation of the number of overtopping waves. Figure 15.6 shows measured data for rubble mound breakwaters armored with Tetrapods, AccropodeTM , or a single layer of cubes. All tests were performed at Delft Hydraulics. The test setup was more or less similar to Fig. 15.4 with a crown wall height Rc a little lower than the armor freeboard Ac . CLASH-data on specific overtopping tests for various rock and concrete armored slopes were added to Fig. 15.6. This figure gives only the percentage of overtopping waves passing the crown wall.

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run-up level (eq. 2 and 3) calculated here

CREST Gc

1.5 Hm0toe

Ac

overtopping measured behind wall

Rc

swl 1.5 Hm0toe

Fig. 15.5.

Run-up level and location for overtopping differ.

Percentage of overtopping waves (%)

100 90

tetrapod DH

80

accropode DH 1 layer cube DH

70

CLASH 60 50 40 30 20 10 0 -0.2

0

0.2

0.4

0.6

0.8

1

2

Dimensionless crest height Ac*Dn/Hm0

Fig. 15.6. Percentage of overtopping waves for rubble mound breakwaters as a function of relative (armor) crest height and armor size (Rc ≤ Ac ).

Analysis showed that the size of the armor unit relative to the wave height had 2 influence, which gave a combined parameter Ac · Dn /Hm0 , where Dn is the nominal diameter of the armor unit. The figure covers the whole range of overtopping percentages, from complete overtopping with the crest at or lower than SWL to no overtopping at all. The CLASH-data give maximum overtopping percentages of about 30%. Larger

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percentages mean that overtopping is so large that it can hardly be measured and that wave transmission starts to play a role. Taking 100% overtopping for zero freeboard (the actual data are only a little lower), a Weibull curve can be fitted through the data. Equation (15.5) can be used to predict the number or percentage of overtopping waves or to establish the armor crest level for an allowable percentage of overtopping waves. Pov

  1.4  Ac Dn . = Now /Nw = exp − 2 0.19Hm0

(15.5)

It is clear that Eqs. (15.2)–(15.4) will come to more overtopping waves than Eq. (15.5). But both estimations together give a designer enough information to establish the required crest height of a structure given an allowable overtopping percentage. 15.2.4. Wave run-down on rock-armored slopes When a wave on a structure has reached its highest point, it will run down on the slope till the next wave meets this water and run-up starts again. The lowest point to where the water retreats, measured vertically to SWL, is called the run-down level. Run-down often is less or not important compared to wave run-up, but both together they may give an idea of the total water excursion on the slope. Therefore, only a first estimate of run-down on straight rock slopes is given here, based on the same tests of van der Meer,11 but re-analyzed with respect to the use of the spectral wave period Tm−1,0 . Figure 15.7 gives an overall view. Breaker parameter ξ m- 1,0 0

1

2

3

4

5

6

7

8

9

Relative run-down R d2%/Hm0

0

0.5

1

1.5

2

imp; cota=2 perm; cota=1.5 hom; cota=2

imp; cota=3 perm; cota=2 imp; Deltaflume

imp; cota=4 perm; cota=3 perm; Deltaflume

2.5

Fig. 15.7. Relative 2% run-down on straight rock slopes with impermeable core (imp), permeable core (perm), and homogeneous structure (hom).

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The graph shows clearly the influence of the permeability of the structure as the solid data points (impermeable core) generally show larger run-down than the open data symbols of the permeable core. Furthermore, the breaker parameter ξm−1,0 gives a fairly clear trend of run-down for various slope angles and wave periods. Figure 15.7 can be used directly for design purposes, as it also gives a good idea of the scatter.

15.3. Overtopping Discharges 15.3.1. Simple armored slopes The mean overtopping discharge is often used to judge allowable overtopping. It is easy to measure, and an extensive database on mean overtopping discharge has been gathered in CLASH. This mean discharge does not of course describe the real behavior of wave overtopping, where only large waves will reach the top of the structure and give overtopping. Random individual wave overtopping is random in time, and each wave gives a different overtopping volume. But the description of individual overtopping is based on the mean overtopping, as the duration of overtopping multiplied with this mean overtopping discharge gives the total volume of water overtopped by a certain number of overtopping waves. The mean overtopping discharge has been described in this section. The individual overtopping volumes is the subject in Sec. 15.4.1. Wave overtopping occurs if the crest level of the coastal structures is lower than the highest wave run-up level. In that case, the freeboard Rc defined as the vertical difference between the SWL and the crest height becomes important (Fig. 15.2). Wave overtopping depends on the freeboard Rc and increases for decreasing freeboard height Rc . Usually, wave overtopping for rubble slopes and mounds is described by an average wave overtopping discharge q, which is given in m3 /s per m width, or in l/s per m width. An average overtopping discharge q can only be calculated for quasi-stationary wave and water level conditions, a so-called sea state. If the amount of water overtopping for a structure during a storm is required, the average overtopping discharge has to be calculated for each more or less constant storm water level and constant wave conditions. Many model studies were performed to investigate the average overtopping discharge for specific dike geometries or wave conditions. For practical purposes, empirical formulae were fitted through experimental model data which obey often one of the following expressions: Q∗ = Q0 (1 − R∗ )b

or Q∗ = Q0 exp (−b · R∗ ) .

(15.6)

Q∗ is a dimensionless overtopping discharge, R∗ is a dimensionless freeboard height, Q0 describes wave overtopping for zero freeboard, and b is a coefficient which describes the specific behavior of wave overtopping for a certain structure. Sch¨ uttrumpf 8 summarized expressions for the dimensionless overtopping discharge ∗ Q and the dimensionless freeboard height R∗ .

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3 The dimensionless overtopping discharge Q∗ = q/ gHm0 is a function of the wave height, originally derived from the Weir formula. Deterministic design or safety assessment. The equation, including a standard deviation of safety, should be used for deterministic design or safety assessment: 0.067 q γb · ξm−1,0 =√ 3 tan α g · Hm0   Rc · exp −4.3 ξm−1,0 · Hm0 · γb · γf · γβ · γv with a maximum of √

q 3 g·Hm0

(15.7)


 = 0.2 · exp −2.3 Hm0R·γcf ·γβ ,

where γv = the influence of a small wall on top of the embankment. Probabilistic design. The mean prediction should be used for probabilistic design, or prediction of or comparison with measurements. This equation is given by

q 3 g · Hm0

with a maximum of √

0.067 γb · ξm−1,0 = √ tan α   Rc · exp −4.75 ξm−1,0 · Hm0 · γb · γf · γβ · γv

q 3 g·Hm0

(15.8)


= 0.2 · exp −2.6 Hm0R·γcf ·γβ .

The reliability of Eq. (15.8) is described by taking the coefficients 4.75 and 2.6 as normally distributed stochastic parameters with means of 4.75 and 2.6 and standard deviations σ = 0.5 and 0.35, respectively. For probabilistic calculations, Eq. (15.8) should be taken together with these stochastic coefficients. For predictions of measurements or comparison with measurements also Eq. (15.8) should be taken with, for instance, 5% upper and lower exceedance curves. It has to be mentioned that the first part of Eqs. (15.7) and (15.8) is valid for mostly breaking waves. Considering the steep slopes of armored rubble slope and mounds this part has less importance in practice than the second part of the equation, describing the maximum of overtopping. In that case, the relative freeboard does not depend on the breaker parameter ξm−1,0 for nonbreaking waves (Fig. 15.8), as the line is horizontal. This means that a composite slope and even a, not too long, berm leads to the same overtopping discharge as for a simple straight rubble mound slope. Only when the average slope becomes so gentle that the maximum part in Eqs. (15.7) and (15.8) do not apply anymore, then a berm and a composite slope will have effect on the overtopping discharge. Generally, average slopes around 1:2 or steeper do not show influence of the slope angle, or only to a limited extent, and the maximum part in Eqs. (15.7) and (15.8) are valid.

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.

relative overtopping discharge q/(gHm0 )

3 1/2

[-]

0,016 1:3 slope

0,014

1:4 slope 1:6 slope

0,012 0,01 0,008 0,006 0,004 0,002 0 0

0,01

0,02

0,03

0,04

0,05

0,06

2

wave steepness 2πHm0/(gTm-1,0 ) [-]

Fig. 15.8. Wave overtopping as a function of the fictitious wave steepness sm−1,0 = 2πHm0 / (gTm−1,0 2 ) and a smooth slope.

As part of the EU research program CLASH2 tests were undertaken to derive roughness factors for rock-armored slopes and different armor units on sloping permeable structures. Overtopping was measured for a 1:1.5 sloping permeable structure at a reference point 3Dn50 from the crest edge, where Dn50 is the nominal diameter. The wave wall had the same height as the armor crest, so Rc = Ac . As discussed in Sec. 15.2 and Fig. 15.5, the point to where run-up can be measured and the location of overtopping may differ. Normally, a rubble mound structure has a crest width of at least 3Dn50 . Waves rushing up the slope reach the crest with an upward velocity. For this reason, it is assumed that overtopping waves reaching the crest will also reach the location 3Dn50 further. Results of the CLASH-work are shown in Fig. 15.9 and Table 15.1. Figure 15.9 gives all data together in one graph. Two lines are given, one for a smooth slope, Eq. (15.8) with γf = 1.0, and one for rubble mound 1:1.5 slopes, with the same equation, but with γf = 0.45. The lower line only gives a kind of average, but shows clearly the very large influence of roughness and permeability on wave overtopping. The required crest height for a steep rubble mound structure is at least half of that for a steep smooth structure, for similar overtopping discharge. It is also for this reason that smooth slopes are often more gentle in order to reduce the crest heights. TM R R , Xbloc , and In Fig. 15.9, one-layer systems, like Accropode , CORE-LOC one layer of cubes, have solid symbols. Two-layer systems have been given by open symbols. There is a slight tendency that one-layer systems give a little more overtopping than two-layer systems, which is also clear from Table 15.1. Equation (15.8) can be used with the roughness factors in Table 15.1 for the prediction of mean overtopping discharges for rubble mound breakwaters. Values in italics in Table 15.1 have been estimated/extrapolated, based on the CLASH results.

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Wave overtopping q/(gHm0 )

3 0.5

1.E+00 Smooth Cube rough Anitfer Tetrapod Accropde Xbloc rough gf=0.45

1.E-01

1.E-02

Rock Cube flat Haro Cube-1 layer Coreloc smooth gf=1.0

1.E-03

Eq. 8

f

= 1.0

1.E-04

1.E-05

Eq. 8

f

= 0.45

1.E-06

1.E-07

0

0.5

1

1.5

2

2.5

3

3.5

4

Relative crest height Rc/Hm 0

Fig. 15.9.

Mean overtopping discharge for 1:1.5 smooth and rubble mound slopes. Table 15.1. Values for roughness factors γf for permeable rubble mound structures with slope of 1:1.5. Values in italics are estimated/extrapolated. Type of armor layer

γf

Smooth impermeable surface Rocks (one layer, impermeable core) Rocks (one layer, permeable core) Rocks (two layers, impermeable core) Rocks (two layers, permeable core) Cubes (one layer, random positioning) Cubes (two layers, random positioning) Antifers HARO’s AccropodeTM R Xbloc R CORE-LOC Tetrapods Dolosse

1.00 0.60 0.45 0.55 0.40 0.50 0.47 0.47 0.47 0.46 0.45 0.44 0.38 0.43

15.3.2. Effect of armored crest berm Wave overtopping on simple straight slope include an armored crest berm up to about three nominal diameters. It is clear, however, that wide crests will certainly decrease the overtopping as much more energy will be dissipated in a wider crest. The crest width can be described by Gc (see Fig. 15.5). The EA-Manual (Besley1 ) describes in a simple and effective way the influence of a wide crest. First, the wave overtopping discharge should be calculated for a simple slope, with a crest width up to 3Dn50 . Then, the following reduction factor on the overtopping discharge can

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be applied: Cr = 3.06 exp(−1.5Gc /Hm0 ),

with maximum Cr = 1.

(15.9)

Equation (15.9) gives no reduction for a crest width smaller than about 0.75 Hm0 . This is fairly close to about 3Dn50 and is, therefore, consistent. A crest width of 1Hm0 reduces the overtopping discharge to 68%, a crest width of 2Hm0 gives a reduction to 15%, and for a wide crest of 3Hm0 , the overtopping reduces to only 3.4%. In all cases, the crest wall has the same height as the armor crest: Rc = Ac . Equation (15.9) was determined for a rock-armored slope and can be considered as conservative, as for a slope with Accropode, more reduction was found. 15.3.3. Effect of oblique waves In the CLASH-project, specific tests on a rubble mound breakwater were performed with a slope of 1:2 and armored with rock or cubes7 to investigate the effect of oblique waves on wave overtopping. The structure was tested both with long-crested and short-crested waves, but only the results by short-crested waves are given. Results for the effect of oblique waves on smooth slopes, dikes, or embankments are given in the EurOtop Overtopping Manual,4 and in the TAW-report.9 Here, only the results for armored rubble mound slopes will be discussed. For oblique waves, the angle of wave attack β (deg.) is defined as the angle between the direction of propagation of waves and the axis perpendicular to the structure (for perpendicular wave attack, β = 0◦ ). And the direction of wave attack is the angle after any change of direction of the waves on the foreshore due to refraction. Just like for smooth slopes, the influence of the angle of wave attack is described by the influence factor γβ . Just as for smooth slopes, there is a linear relationship between the influence factor and the angle of wave attack, but the reduction in overtopping for rock slopes is faster with increasing angle: γβ = 1 − 0.0063|β|

for 0◦ ≤ |β| ≤ 80◦ for |β| > 80◦ , the result of β = 80◦ can be applied.

(15.10)

The wave height and period are linearly reduced to 0 for 80◦ ≤ |β| ≤ 110◦. For |β| > 110◦ , the wave overtopping is set to q = 0. 15.3.4. Composite slopes and berms, including berm breakwaters In every formula where a cot α or breaker parameter ξm−1,0 is present, a procedure has to be described how a composite slope has to be taken into account. Hardly any specific research exists for rubble mound structures, and, therefore, the procedure for composite slopes at sloping impermeable structures like dikes and sloping seawalls is assumed to be applicable. (a) Average slopes. A characteristic slope is required to be used in the breaker parameter ξm−1,0 for composite profiles or bermed profiles to calculate wave runup or wave overtopping. Theoretically, the run-up process is influenced by a change

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of slope from the breaking point to the maximum wave run-up height. Therefore, often it has been recommended to calculate the characteristic slope from the point of wave breaking to the maximum wave run-up height. This approach needs some calculation effort, because of the iterative solution since the wave run-up height Ru2% is unknown. For the breaking limit, a point on the slope can be chosen which is 1.5Hm0 below the still water line. It is recommended to use also a point on the slope 1.5Hm0 above water as a first estimate to calculate the characteristic slope and to exclude a berm (Fig. 15.10). First estimate: tan α =

3 · Hm0 . LSlope − B

(15.11)

As a second estimate, the wave run-up height from the first estimate is used to calculate the average slope [LSlope has to be adapted (see Fig. 15.11)]: 

1.5 · Hm0 + Ru2%(from 1st estimate) Second estimate: tan α = . (15.12) LSlope − B 1st estimate:

tan α =

3H m0 LSlope − B

B Average slope

SWL

dB 1.5 HS

1.5 HS

B

LSlope

Fig. 15.10.

Determination of the average slope (first estimate).

2nd estimate:

tan α =

Ru2% (from 1st estimate)

1.5 H m 0 + Ru 2% ( from 1st estimate) LSlope − B Average slope

SWL

dB 1.5 HS

B

LSlope Fig. 15.11.

Determination of the average slope (second estimate).

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If the run-up height or 1.5Hm0 comes above the crest level, then the crest level must be taken as the characteristic point above SWL. Also, the influence of a berm in a sloping profile has been adapted from smooth sloping sea dikes for rubble mound structures. There is, however, often a difference in the effect of composite slopes or berms for rubble mound and smooth gentle slopes. On gentle slopes, the breaker parameter ξm−1,0 has large influence on wave overtopping (see Eqs. (15.2) and (15.3) as the breaker parameter will be quite small). Rubble mound structures often have a steep slope, leading to the formula for “nonbreaking” waves, the maximum in Eqs. (15.7) and (15.8). In these equations, no slope angle or breaker parameter is present, and the effect of a small berm will be very small and probably negligible. (b) Influence of berms. A berm is a part of a dike or breakwater profile in which the slope varies between horizontal and 1:15. A berm is defined by the width of the berm, B, and by the vertical difference dB between the middle of the berm and the SWL (Fig. 15.12). The width of the berm, B, may not be greater than 0.25 · Lm−1,0 . If the berm is horizontal, the berm width B is calculated according to Fig. 15.12. The lower and the upper slope are extended to draw a horizontal berm without changing the berm height dB . The horizontal berm width is therefore shorter than the angled berm width. dB is 0 if the berm lies on the still water line. The characteristic parameters of a berm are defined in Fig. 15.12.

(a) Calculation of width B and height db of berm Bhorizontal

SWL

db Bangled

(b) Calculation of berm length Lberm

1.0 H S SWL

db

1.0 HS B LBerm

Fig. 15.12.

Determination of the characteristic berm length LBerm .

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A berm reduces wave run-up or wave overtopping. The influence factor γb for a berm consists of two parts. γb = 1 − rB (1 − rdb )

for 0.6 ≤ γb ≤ 1.0.

(15.13)

The first part (rB ) stands for the width of the berm LBerm and becomes 0 if no berm is present. rB =

B LBerm

.

(15.14)

The second part (rdb ) stands for the vertical difference dB between the SWL and the middle of the berm and becomes 0 if the berm lies on the still water line. The reduction of wave run-up or wave overtopping is maximum for a berm on the still water line and decreases with increasing dB . Thus, a berm lying on the still water line is most effective. A berm lying below 2 · Hm0 or above Ru2% has no influence on wave run-up and wave overtopping. Different expressions are used for rdB in Europe. Here, an expression using a cosine-function for rdb (Fig. 15.13) is recommended which is also used in pcovertopping (see Chap. 14).   db rdb = 0.5 − 0.5 cos π for a berm above SWL , Ru2%   db for a berm below SWL , (15.15) rdb = 0.5 − 0.5 cos π 2 · Hm0 rdb = 1 for berms lying outside the area of influence . Berm lying at Ru,2%

Berm lying at SWL Berm lying above SWL

Berm lying . at 2 Hm0 below SWL Berm lying below SWL

1 0.9 0.8 0.7 rdb [-]

0.6 0.5 0.4 0.3 0.2 0.1 0 -1

-0.5 db/Ru2% [-]

Fig. 15.13.

0

0.5 . db/(2 Hm0) [-]

Influence of the berm depth on factor rdh .

1

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The maximum influence of a berm is actually always limited to γB = 0.6. This corresponds to an optimal berm width B on the SWL of B = 0.4 · LBerm . The definition of a berm is made for a slope smoother than 1:15, while the definition of a slope is made for slopes steeper than 1:8. If a slope or a part of the slope lies in between 1:8 and 1:15, it is required to interpolate between a bermed profile and a straight profile. For wave run-up, this interpolation is written by

 (1/8 − tan α) Ru2% = Ru2%(1:8slope) + Ru2%(Berm) − Ru2%(1:8slope) · . (1/8 − 1/15)

(15.16)

A similar interpolation procedure should be followed for wave overtopping. 15.3.5. Wave overtopping on a berm breakwater A specific type of rubble mound structure is the berm breakwater. The original idea behind the berm breakwater is that a large berm, consisting of fairly large rock, is constructed into the sea with a steep seaward face. The berm height is higher than the minimum required for construction with land-based equipment. Due to the steep seaward face the first storms will reshape the berm and finally a structure will be present with a fully reshaped S-profile. Such a profile has then a gentle 1:4 or 1:5 slope just below the water level and steep upper and lower slopes (see Fig. 15.14). The idea of the reshaping berm breakwater has evolved in Iceland to a more or less nonreshaping berm breakwater (Fig. 15.15). The main difference is that during rock production from the quarry, care is taken to gather a few percent of really big rocks. Only a few percent is required to strengthen the corner of the berm and part of the down slope and upper layer of the berm in such a way that reshaping will hardly occur. An example with various rock classes (class I being the largest) is given in Fig. 15.16. Therefore, distinction has been made between conventional reshaping berm breakwaters and the nonreshaping Icelandic type berm breakwater. In order to calculate wave overtopping on reshaped berm breakwaters, the reshaped profile should be known. The basic method of profile reshaping is given in van der Meer,11 and the program breakwat (WL | Delft Hydraulics) is able to calculate the profile. The first method described here to calculate wave overtopping at reshaping berm breakwaters is by using Eqs. (15.7) or (15.8) which have been developed for smooth slopes. Equations (15.7) and (15.8) include the effect of an average slope with the roughness factors given in Table 15.1 of γf = 0.40 for

Fig. 15.14.

Conventional reshaping berm breakwater.

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Fig. 15.15.

Fig. 15.16.

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Icelandic berm breakwater.

Nonreshaping Icelandic berm breakwater with various classes of big rocks.

reshaping berm breakwaters and γf = 0.35 for nonreshaping Icelandic berm breakwaters. The method of composite slopes and berms should be applied as described above. The second method is to use the CLASH neural network. As overtopping research at that time on berm breakwaters was limited, this method also gives quite some scatter, but a little less than the first method described above. Recent information on berm breakwaters has been described by Andersen.5 Only part of his research was included in the CLASH database and consequently in the Neural Network prediction method. He performed about 600 tests on reshaping berm breakwaters and some 60 on nonreshaping berm breakwaters (fixing the steep slopes by a steel net). The true nonreshaping Icelandic type of berm breakwaters with large rock classes, has not been tested and, therefore, his results might lead to an overestimation. One comment should be made on the application of the results of Andersen.5 3 0.5 The maximum overtopping discharge measured was only q/(gHm0 ) = 10−3. In practical situations with wave heights around 5 m, the overtopping discharge will then be limited to only a few l/s per m width. For berm breakwaters and also for conventional rubble slopes and mounds, allowable overtopping may be much higher than this value.

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The final result of the work of Andersen5 is a quite complicated formula, based on multi-parameter fitting. The advantage of such a fitting is that by using a large number of parameters, the data set used will be quite well described by the formula. The disadvantage is that physical understanding of the working of the formula, certainly outside the ranges tested, is limited. But, due to the fact that so many structures were tested, this effect may be neglected. The formula is valid for berm breakwaters with no superstructure and gives the overtopping discharge at the back of the crest (Ac = Rc ). In order to overcome the problem that one encounters when calculating the reshaped profile before any overtopping calculation can be done, the formula is based on the “as built” profile, before reshaping. Instead of calculating the profile, a part of the formula predicts the influence of waves on recession of the berm. The parameter used is called fHo , which is an indicative measure of the reshaping, and can be defined as a “factor accounting for the influence of stability numbers.” Note that fHo is a dimensionless factor and not the direct measure of recession and that Ho and To are also dimensionless parameters (see below). ∗ fHo = 19.8s−0.5 om exp(−7.08/Ho ) for To ≥ To ,

fHo = 0.05Ho To + 10.5 for To < To∗ ,

(15.17)

where Ho = Hm0 /∆Dn50 , To = (g/Dn50 )0.5 Tm0,1 , and To∗ = {19.8s−0.5 exp(−7.08/Ho) − 10.5}/(0.05Ho). om The berm level dh is also taken into account as an influence factor, d∗h . Note that the berm depth is positive if the berm level is below SWL, and therefore, for berm breakwaters often negative. Note also that this influence factor is different from that for a bermed slope. This influence factor is described by d∗h = (3Hm0 − dh )/(3Hm0 + Rc )

for dh < 3Hm0 ,

d∗h = 0 for dh ≥ 3Hm0 .

(15.18)

The final overtopping formula then takes into account the influence factor on recession, fHo , the influence factor of the berm level, d∗h , the geometrical parameters Rc , B, and Gc , and the wave conditions Hm0 and the mean period Tm0,1 . It means that the wave overtopping is described by a spectral mean period, and not by Tm−1,0 . 3 1.34 q/(gHm0 )0.5 = 1.79 · 10−5 (fHo + 9.22)s−2.52 ∗, op 1.39 . exp[−5.63(Rc/Hm0 )0.92 − 0.61(Gc /Hm0 )1.39 − 0.55h1.48 b∗ (B/Hm0 )

(15.19)

Equation (15.19) is only valid for a down slope of 1:1.25 and an upper slope of 1:1.25. For other slopes, one has to reshape the slope to a slope of 1:1.25, keeping the volume of material the same and adjusting the berm width B and for the upper slope also the crest width Gc . Note also that in Eq. (15.19), the peak wave period

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Tp has to be used to calculate sop , where the mean period Tm0,1 has to be used in Eq. (15.17). Although no tests were performed on the nonreshaping Icelandic berm breakwaters (see Fig. 15.16), a number of tests were performed on nonreshaping structures by keeping the material in place with a steel net. The difference may be that Icelandic berm breakwaters show a little less overtopping, due to the presence of larger rocks and, therefore, more permeability. The tests showed that Eq. (15.19) is also valid for nonreshaping berm breakwaters, if the reshaping factor fHo = 0. 15.3.6. Effect of wave walls Most breakwaters have a wave wall, capping wall, or crest unit on the crest, simply to end the armor layer in a good way and to create access to the breakwater. For design, it is advised not to design a wave wall much higher than the armor crest, for the simple reason that wave forces on the wall will increase drastically if directly attacked by waves and not hidden behind the armor crest. For rubble mound slopes as a seawall, design waves might be a little lower than for breakwaters and a wave wall might be one of the solutions to reduce wave overtopping. Nevertheless, one should realize the increase in wave forces in designing a wave wall significantly above the armor crest. Equations (15.7) and (15.8) for a simple rubble mound slope include a berm of 3Dn50 wide and a wave wall at the same level as the armor crest: Ac = Rc . A little lower wave wall will hardly give larger overtopping, but no wave wall at all would certainly increase overtopping. Part of the overtopping waves will then penetrate through the crest armor. No formulae are present to cope with such a situation, unless the use of the Neural Network prediction method. Various researchers have investigated wave walls higher than the armor crest. None of them compared their results with a graph like Fig. 15.9 for simple rubble mound slopes. During the writing of the EurOtop Overtopping Manual, 2007, some of the published equations were plotted in Fig. 15.7 and most curves fell within the scatter of the data. Data with a wider crest gave significantly lower overtopping, but that was due to the wider crest, not the higher wave wall. In essence, the message is: use the height of the wave wall Rc and not the height of the armored crest Ac in Eqs. (15.7) and (15.8) if the wall is higher than the crest. For a wave wall lower than the crest armor the height of this crest armor should be used. The Neural Network prediction might be able to give more precise predictions. 15.3.7. Scale and model effect corrections Results of the recent CLASH-project suggested significant differences between field and model results on wave overtopping. This has been verified for different sloping rubble structures. Results of the comparisons in this project have led to a scaling procedure which is mainly dependent on the roughness of the structure γf [–]; the seaward slope m of the structure [–]; the mean overtopping discharge, upscaled to prototype, qss [m3 /s/m]; and whether wind is considered or not. Data from the field are naturally scarce, and hence the method can only be regarded as tentative. Furthermore, it is only relevant if mean overtopping rates are

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Zeebrugge (1:1.4) Ostia (1:4)

30,0

Eq. (20), f_q,max = 7.9 Eq. (20), f_q,max = 30

fq [-]

20,0

10,0

0,0 1,E-06

1,E-05

1,E-04

1,E-03

1,E-02

3

qss [m /s/m]

Fig. 15.17. Proposed adjustment factor applied to data from two field sites (Zeebrugge 1:1.4 rubble mound breakwater, and Ostia 1:4 rubble slope).

lower than 1.0 l/s/m but may include significant adjustment factors below these rates. Due to the inherent uncertainties, the proposed approach tries to be conservative. It has, however, been applied to pilot cases in CLASH and has proved good corrections with these model data. The adjustment factor fq for the model and scale effects can be determined as follows (Fig. 15.17): 1.0 for qss ≥ 10−3m3 /s/m , (15.20) fq = min{(− log qss − 2)3 ; fq,max } for qss ≥ 10−3m3 /s/m , where fq,max is an upper bound to the adjustment factor fq and can be calculated as follows:  for γf ≤ 0.7 ,  fq,r (15.21) fq,max = 5 · γf · (1 − fq,r ) + 4.5 · (fq,r − 1) + 1 for 0.7 < γf ≤ 0.9 ,   1.0 for γf > 0.9 . In Eq. (15.21), fq,r is the adjustment factor for rough slopes which is mainly dependent on the slope of the structure and whether wind needs to be included or not:  for m ≤ 0.6 ,  1.0 fq,r = fw · (8.5 · m − 4.0) for 0.6 < m ≤ 4.0 , (15.22)   fw · 30 for m > 4.0 , in which m = cot α (slope of structure); fw accounts for the presence of wind and is set to fw = 1.0 if there is wind and fw = 0.67 if there is no wind. This set of equations include the case of smooth dikes which will, due to γf = 0.9 in this case, always lead to an adjustment factor of fq = 1.0. In case of a very rough

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1:4 slope with wind fq,max = fqr = 30.0, which is the maximum, the factor can get to (but only if the mean overtopping rates get below qss = 10−5 m3 /s per m). The latter case and a steep rough slope is illustrated in Fig. 15.14.

15.4. Individual Overtopping Waves 15.4.1. Overtopping volumes per wave The following section is a summary of Sec. 14.2.2 in Chap. 14 of this handbook. Parts of that section are repeated in the following with a focus on rubble mound structures. Wave overtopping is a dynamic and irregular process and the mean overtopping discharge, q, does not cover this aspect. But by knowing the storm duration, t, and the number of overtopping waves in that period, Now , it is easy to describe this irregular and dynamic overtopping, if the overtopping discharge, q, is known. Each overtopping wave gives a certain overtopping volume of water, V , with dimension m3 per m width or l per m width. As many equations in this chapter, the two-parameter Weibull distribution describes the behavior quite well. This equation has a shape parameter, b, and a scale parameter, a. The shape parameter gives a lot of information on the type of distribution. The exceedance probability, PV , of an overtopping volume per wave is similar to Eqs. (15.23) and (15.24).     0.75 V , (15.23) PV = P (V ≤ V ) = 1 − exp − a with a = 0.84 · Tm ·

q = 0.84 · Tm · q · Nw /Now = 0.84 · q · t/Now . Pov

(15.24)

Equation (15.24) shows that the scale parameter depends on the overtopping discharge, but also on the mean period (not the spectral period Tm−1,0 !) and probability of overtopping, or which is similar, on the storm duration and the actual number of overtopping waves. The probability of wave overtopping for rubble mound structures has been described in Sec. 15.2 and Eq. (15.4). Equations for calculating the overtopping volume per wave for a given probability of exceedance is given by Eq. (15.25): V = a · [− ln (1 − PV )]4/3 .

(15.25)

The maximum overtopping during a certain event is fairly uncertain, as most maxima, but depends on the duration of the event. In a 6-h period, one may expect a larger maximum than only during 15 min. The maximum during an event can be calculated by Eq. (15.26): 4/3

Vmax = a [ln (Now )]

.

(15.26)

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15.4.2. Overtopping velocities and spatial distribution The hydraulic behavior of waves on rubble mound slopes and on smooth slopes like dikes, is generally based on similar formulae, as clearly shown in this chapter. This is different, however, for overtopping velocities and spatial distribution of the overtopping water. A dike or sloping impermeable seawall generally has an impermeable and more or less horizontal crest. Up-rushing and overtopping waves flow over the crest, and each overtopping wave can be described by a maximum velocity and flow depth. These velocities and flow depths form the description of the hydraulic loads on crest and inner slope and are part of the failure mechanism “failure or erosion of inner slopes by wave overtopping.” This is different for rubble mound slopes or breakwaters where wave energy is dissipated in the rough and permeable crest and where often overtopping water falls over a crest wall onto a crest road or even on the rear slope of a breakwater. A lot of overtopping water travels over the crest and through the air before it hits something else. Only recently in CLASH and a few other projects at Aalborg University, attention has been paid to the spatial distribution of overtopping water at breakwaters with a crest wall.6 The spatial distribution was measured by various trays behind the crest wall. Figure 15.18 gives different cross sections with a setup of three arrays. Up to six arrays have been used. The spatial distribution depends on the level with respect to the rear side of the crest wall and the distance from this rear wall (Fig. 15.19). The coordinate system (x, y) starts at the rear side and at the top of the crest wall, with the positive y-axis downward. The exceedance probability F of the travel distance is defined as the volume of overtopping water passing a given x- and y-coordinate, divided by the total overtopping volume. The probability, therefore, lies between 0 and 1, with 1 at the crest wall. The spatial distribution can be described with the following equations, which have slightly been rewritten and modified with respect to the original formulae by Andersen and Burcharth.6 The probability F at a certain location can be

Fig. 15.18.

Definition of y for various cross sections.

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Definition of x- and y-coordinate for spatial distribution.

described by F (x, y) = exp[−1.3/Hm0 · {max(x/ cos β − 2.7ys0.15 op ; 0)}] .

(15.27)

Equation (15.27) can be rewritten to calculate directly the travel distance x (at a certain level y) by rewriting the above equation: x/ cos β = −0.77Hm0 ln(F ) + 2.7ys0.15 op .

(15.28)

Suppose cos β = 0, then we get F = 1 with x = 0; F = 0.1 with x = 1.77Hs; and F = 0.01 with x = 3.55Hs . It means that 10% of the volume of water travels almost two wave heights through the air and 1% of the volume travels more than 3.5 times the wave height. These percentages will be higher if y = 0, which is often the case with a crest unit. The validity of Eqs. (15.27) and (15.28) is for rubble mound slopes of approximately 1:2 and for angles of wave attack between 0◦ ≤ |β| < 45◦ . It should be noted that the equation is valid for the spatial distribution of the water through the air behind the crest wall. All water falling on the basement of the crest unit will, of course, travel on and will fall into the water behind and/or on the slope behind.

15.5. Overtopping of Shingle Beaches Shingle beaches differ from the armored slopes principally in the size of the beach material, and hence its mobility. The typical stone size is sufficiently small to permit significant changes of beach profile, even under relatively low levels of wave attack. A shingle beach may be expected to adjust its profile to the incident wave conditions, provided that sufficient beach material is available. Run-up or overtopping levels on a shingle beach are therefore calculated without reference to any initial slope. The equilibrium profile of shingle beaches under (temporary constant) wave conditions is described by van der Meer.11 The most important profile parameter for run-up and overtopping is the crest height above SWL, hc . For shingle with Dn50 < 0.1 m, this crest height is only a function of the wave height and wave

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steepness. Note that the mean wave period is used, not the spectral wave period Tm−1,0 . hc /Hm0 = 0.3s−0.5 om .

(15.29)

Only the highest waves will overtop the beach crest and most of this water will percolate through the material behind the beach crest. Equation (15.29) gives a run-up or overtopping level which is more or less close to Ru2% . Acknowledgments This chapter was based on the EurOtop Overtopping Manual,4 which was funded in the United Kingdom by the Environmental Agency, in Germany by the German Coastal Engineering Research Council (KFKI), in the Netherlands by Rijkswaterstaat and Netherlands Expertise Network (ENW) on Flood Protection. The Project Team for the creation, editing, and support of the manual; the Project Steering Group for guidance and supervision; and a number of individual persons, have been listed and acknowledged in Chap. 14. References References in this chapter have been kept to a real minimum. An extensive list of relevant references, however, can be found in the EurOtop Overtopping Manual (2007). 1. P. Besley, Overtopping of seawalls — Design and assessment manual, R & D Technical Report W 178, Environment Agency, Bristol (1999), ISBN 1 85705 069 X. 2. T. Bruce, J. W. van der Meer, L. Franco and J. M. Pearson, Overtopping performance of different armour units for rubble mound breakwaters, Coastal Eng. 56(2), 166–179 (2009). 3. CLASH, Crest level assessment of coastal structures by full scale monitoring, neural network prediction and hazard analysis on permissible wave overtopping, Fifth Framework Programme of the EU, Contract no. EVK3-CT-2001-00058, www.clash-eu.org. 4. EurOtop Overtopping Manual, Wave Overtopping of Sea Defences and Related Structures — Assessment Manual, eds. T. Pullen, N. W. H. Allsop, T. Bruce, A. Kortenhaus, H. Sch¨ uttrumpf and J. W. van der Meer (2007), www.overtoppingmanual.com. 5. T. L. Andersen, Hydraulic response of rubble mound breakwaters. Scale effects — Berm breakwaters, PhD. Thesis, Aalborg University, Denmark (2006), ISSN 0909 4296 Series Paper No. 27. 6. T. L. Andersen and H. F. Burcharth, Landward distribution of wave overtopping for rubble mound breakwaters, Proc. First Int. Conf. Application of Physical Modelling to Port and Coastal Protection (2006). 7. T. L. Andersen and H. F. Burcharth, Overtopping and rear slope stability of reshaping and non-reshaping berm breakwaters, Proc. 29th Int. Conf. Coastal Engineering, Lisbon (2004).

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8. H. Sch¨ uttrumpf, Wellen¨ uberlaufstr¨ omung bei Seedeichen — Experimentelle und Theoretische Untersuchungen, PhD. thesis (2001), http://www.biblio.tu-bs.de/ediss/ data/20010703a/20010703a.html. 9. J. W. van der Meer, Technical Report — Wave run-up and wave overtopping at dikes, Technical Advisory Committee for Flood Defence in the Netherlands (TAW), Delft (2002). 10. J. W. van der Meer and C. J. M. Stam, Wave runup on smooth and rock slopes, ASCE J. WPC OE 188(5), 534–550 (1992); also, Delft Hydraulics Publication No. 454. 11. J. W. van der Meer, Rock slopes and gravel beaches under wave attack, PhD. thesis, Delft University of Technology (1988); also, Delft Hydraulics Publication No. 396.

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Chapter 16

Wave Overtopping at Vertical and Steep Structures Tom Bruce School of Engineering, University of Edinburgh King’s Buildings, Edinburgh, EH9 3JL, UK [email protected] Jentsje van der Meer Van der Meer Consulting P. O. Box 423, 8440 AK Heerenveen, The Netherlands [email protected] Tim Pullen HR Wallingford Howbery Park, Wallingford, Oxon, OX10 8BA, UK [email protected] William Allsop HR Wallingford Howbery Park, Wallingford, Oxon, OX10 8BA, UK [email protected] Wave overtopping prediction at vertical structures in earlier days was mainly based on caisson-type structures in relatively deep water. Recent research in many EU-projects has been concentrated on shallower water with waves breaking onto the structure as well. It has led to the definition of two situations: nonimpulsive and the most severe impulsive condition. This chapter relies on the EurOtop Overtopping Manual, as well as the two previous chapters, 14 and 15, in this handbook. It first describes the mean overtopping discharges for many configurations of vertical and composite vertical structures. Later sections quantify influences such as oblique wave attack, wind effects, model, scale effects, etc.

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Individual overtopping volumes are then described. Finally, post-overtopping processes and parameters — landward distribution of discharge; velocities and downfall pressures — are described.

16.1. Introduction This chapter presents guidance for the assessment of overtopping and postovertopping processes at vertical and steep-fronted coastal structures such as caisson and blockwork breakwaters and vertical seawalls (see Figs. 16.1 and 16.2). Also included are composite vertical wall structures (where the emergent part of the structure is vertical, fronted by a modest berm) and vertical structures which include a recurve/bull-nose/parapet/wave return wall as the upper part of the defense. Large vertical breakwaters (Fig. 16.1) are almost universally formed of sand-filled concrete caissons usually resting on a small rock mound. Such caisson breakwaters may reach depths greater than 100 m, under which conditions with no wave breaking at all at the wall would be expected. Conversely, older breakwaters may, out of necessity, have been constructed in shallower water or indeed, built directly on natural rock “skerries.” As such, these structures may find themselves exposed to breaking wave, or “impulsive” conditions when the water depth in front of them is

Fig. 16.1. Examples of vertical breakwaters: (left) modern concrete caisson and (right) older structure constructed from concrete blocks.

Fig. 16.2. Examples of vertical seawalls: (left) modern concrete wall and (right) older stone blockwork wall.

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sufficiently low. Urban seawalls (e.g., Fig. 16.2) are almost universally fronted by shallow water, and are likely to be exposed to breaking or broken wave conditions, especially in areas of significant tidal range. There are three principal sources of guidance on this topic preceding this chapter; in the United Kingdom, the Environment Agency’s Overtopping of Seawalls: Design and Assessment Manual 2 ; in the United States, the US Army Corps of Engineers’ Coastal Engineering Manual 6 ; in Japan, Goda’s design charts.11 The guidance presented in this chapter builds upon that of Besley,2 with adjustments to many formulae based upon further testing since 1999. For those familiar with Besley,2 the principal changes/additions are: • minor adjustments to recommended approach for distinguishing impulsive/ non-impulsive conditions (Sec. 16.2); • all formulae are now given in terms of wave period Tm−1,0 resulting in an adjusted definition of the h∗ , d∗ , and Vbar parameters (Secs. 16.2.2, 16.2.3, and 16.4.2, respectively) in order to maintain comparability with the earlier work; • in line with convergence on the Tm−1,0 measure, formulae using fictitious wave steepness sop have been adjusted to use the new preferred measure sm−1,0 (Secs. 16.3.1 and 16.4.2); • new guidance on mean overtopping under conditions when all waves break before reaching the wall (part of Sec. 16.3.1); • new guidance on overtopping at vertical walls at zero freeboard (Sec. 16.3.1); • minor adjustments to formulae for mean overtopping under impulsive conditions due to the availability of additional data, from e.g., the CLASH database7 (Sec. 16.3.1); • extension of method for mean overtopping to account for steep (but not vertical) “battered” walls (Sec. 16.3.2); • new guidance on prediction of mean and wave-by-wave overtopping for oblique wave attack under impulsive conditions (Sec. 16.3.4); • new guidance on reduction in mean overtopping discharge due to wave return walls/parapets/recurves (Sec. 16.3.5); • new guidance on the effect of wind on mean overtopping discharge (Sec. 16.3.6); • inclusion of summary of new evidence on scale effects for laboratory study of overtopping at vertical and steep walls (Sec. 16.3.7); • new guidance on “post-overtopping” processes, specifically; velocity of “throw”; landward spatial extent of overtopping (Sec. 16.5); • all formulae are now given explicitly in terms of basic wave and structural parameters without recourse to intermediate definitions of dimensionless overtopping discharge and freeboard parameters specific to impulsive conditions. The qualitative form of the physical processes occurring when the waves reach the wall are described in Sec. 16.2. Distinctions drawn between different wave/structure “regimes” are reflected in the guidance for assessment of mean overtopping discharges given in Sec. 16.3. The basic assessment tools are presented for plain vertical walls (Sec. 16.3.1), followed by subsections giving advice on how these basic tools should be adjusted to account for other commonly-occurring configurations, such as battered walls (Sec. 16.3.2); vertically composite walls (Sec. 16.3.3);

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the effect of oblique wave attack (Sec. 16.3.4); the effect of recurve/wave-return walls (Sec. 16.3.5). The effect of wind is described in Sec. 16.3.6, with scale and model effects reviewed in Sec. 16.3.7. Methods to assess individual “wave-by-wave” overtopping volumes are presented in Sec. 16.4. The current knowledge and advice on post-overtopping processes including velocities, spatial distributions, and postovertopping loadings are reviewed in Sec. 16.5.

16.2. Wave Processes at Walls 16.2.1. Overall view For assessment of overtopping at steep-fronted and vertical structures the regime of the wave/structure interaction must be identified first, with quite distinct overtopping responses expected for each regime. On steep walls (vertical, battered, or composite), “non-impulsive” or “pulsating” conditions occur when waves are relatively small in relation to the local water depth, and of lower wave steepnesses. These waves are not critically influenced by the structure toe or approach slope. Overtopping waves run up and over the wall giving rise to (fairly) smoothly-varying loads and “green water” overtopping (Fig. 16.3). In contrast, “impulsive” conditions (Fig. 16.4) occur on vertical or steep walls when waves are larger in relation to local water depths, perhaps shoaling up over the approach bathymetry or structure toe itself. Under these conditions, some waves will break violently against the wall with (short-duration) forces reaching 10–40 times greater than for non-impulsive conditions. Overtopping discharge under these conditions is characterized by a “violent” uprushing jet of (probably highly aerated) water.

Fig. 16.3. A non-impulsive (pulsating) wave condition at a vertical wall, resulting in a nonimpulsive (or “green water”) overtopping.

Fig. 16.4. An impulsive (breaking) wave at a vertical wall, resulting in an impulsive (violent) overtopping condition.

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Broken wave at a vertical wall, resulting in a broken wave overtopping condition.

Lying in a narrow band between non-impulsive and impulsive conditions are “near-breaking” conditions where the overtopping is characterized by suddenness and a high-speed, near-vertical up-rushing jet (like impulsive conditions), but where the wave has not quite broken onto the structure and so has not entrained the amount of air associated with fully impulsive conditions. This “near-breaking” condition is also known as the “flip through” condition. This condition gives overtopping in line with impulsive (breaking) conditions and is thus not treated separately. Many seawalls are constructed at the back of a beach such that breaking waves never reach the seawall, at least not during frequent events where overtopping is of primary importance. For these conditions, particularly for typical shallow beach slopes of less than (say) 1:30, design wave conditions may be given by waves which start breaking (possibly quite some distance) seaward of the wall. These “broken waves” arrive at the wall as a highly-aerated mass of water (Fig. 16.5), giving rise to loadings which show the sort of short-duration peak seen under impulsive conditions (as the leading edge of the mass of water arrives at the wall), but smaller in magnitude due to the high level of aeration. For cases where the depth at the wall hs > 0, overtopping can be assessed using the method for impulsive conditions. For conditions where the toe of the wall is emergent (hs ≤ 0), these methods can no longer be applied and an alternative is required. In order to proceed with the assessment of overtopping, it is therefore necessary first to determine which is the dominant overtopping regime (impulsive or nonimpulsive) for a given structure and design sea state. No single method gives a discriminator which is 100% reliable. The suggested procedure for plain and composite vertical structures includes a transition zone in which there is significant uncertainty in the prediction of dominant overtopping regime, and thus a “worst-case” is taken.

16.2.2. Overtopping regime discrimination — plain vertical walls A method will be described to distinguish between impulsive and non-impulsive conditions at a vertical wall where the toe of the wall is submerged (hs > 0; Fig. 16.6). When the toe of the wall is emergent (hs < 0), only broken waves reach the wall. For submerged toes (hs > 0), a wave breaking or “impulsiveness” parameter, h∗ is defined based on the depth at the toe of the wall, hs , and incident wave conditions

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Fig. 16.6.

Definition sketch for assessment of overtopping at plain vertical walls.

inshore: h∗ = 1.3

hs 2π hs 2 Hm0 g Tm−1,0

or h∗ =

hs 2π hs , 2 Hm0 g Tm

(16.1)

where Hm0 = the significant wave height at the toe of the structure and Tm−1,0 = the spectral wave period based on spectral moments m−1 and m0 , at the same location. Non-impulsive (pulsating) conditions dominate at the wall when h∗ > 0.3, and impulsive conditions occur when h∗ < 0.2. The transition between conditions for which the overtopping response is dominated by breaking and nonbreaking waves lies over 0.2 < h∗ < 0.3. In this region, overtopping should be predicted for both non-impulsive and impulsive conditions, and the larger value assumed. It should be noted that, for certain long-period waves, h∗ < 0.2 can be found under conditions for which no breaking at the vertical wall would be expected. Although the term “impulsive overtopping” may not be accurate in these conditions, the overtopping responses nevertheless follow those described by the “impulsive” formulae.

16.2.3. Overtopping regime discrimination — composite vertical walls For vertical composite walls where a berm or significant toe is present in front of the wall, an adjusted version of the method for plain vertical walls should be used (see Fig. 16.7).

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Definition sketch for assessment of overtopping at composite vertical walls.

A modified “impulsiveness” parameter, d∗ , is defined in a similar manner to the h∗ parameter (for plain vertical walls, Sec. 16.2.2): d∗ = 1.3

d 2π hs 2 Hm0 g Tm−1,0

or d∗ =

d 2π hs , 2 Hm0 g Tm

(16.2)

with parameters defined according to Fig. 16.7. Non-impulsive conditions dominate at the wall when d∗ > 0.3, and impulsive conditions occur when d∗ < 0.2. The transition between conditions for which the overtopping response is dominated by breaking and nonbreaking waves lies over 0.2 < d∗ < 0.3. In this region, overtopping should be predicted for both nonimpulsive and impulsive conditions, and the larger value assumed.

16.3. Mean Overtopping Discharges for Vertical and Battered Walls 16.3.1. Plain vertical walls This section will give design equations for wave overtopping. The use of these equations may be different, depending on the design process followed. First, an equation is given, which describes the mean of the prediction and the associated uncertainty. This mean prediction should be used for probabilistic design, or for prediction of, or comparison with, measurements. In the first case, the associated uncertainty as given should be considered, and in the latter case, for instance, 5% upper and lower exceedance curves. As prediction of wave overtopping is fairly uncertain, the mean prediction should not be used for deterministic design or safety assessment. In that case a safety margin should be included in order to account for the uncertainty. Often, this will

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be one standard deviation. Each equation will be given as a mean prediction with uncertainty, followed by the application in a deterministic way. For simple vertical breakwaters, the following equations should be used. 16.3.1.1. Non-impulsive conditions (h∗ > 0.3) Equation (16.3) gives the mean prediction for non-impulsive conditions and should be used for probabilistic design, or for comparison with measurements. The coefficient of 2.6 for the mean prediction has an associated normal distribution with a standard deviation of σ = 0.4. This standard deviation can be used to calculate, for instance, 5% exceedance curves. Equation (16.3) has the same shape as equations for smooth and rubble mound slopes (see Chaps. 14 and 15 in this handbook):   q Rc
valid for 0.1 < Rc /Hm0 < 3.5. = 0.04 exp −2.6 (16.3) 3 Hm0 gHm0 Note that a steep (smooth or rubble mound) slope has the same coefficient of 2.6, which means that curves in a log-linear graph will be parallel. For steep slopes, however, the first coefficient is 0.2 and not 0.04, which gives more overtopping. (See also Fig. 16.1 in Chap. 14 of this handbook.) A well-known formula for vertical walls is given by Allsop et al.1 It is similar to Eq. (16.3) with coefficients 0.05 and 2.78. These coefficients are fairly close to those of Eq. (16.3) with 0.04 and 2.6, and, therefore, this known formula can also be used for simple vertical walls and non-impulsive conditions. For deterministic design or safety assessment, Eq. (16.3) should be used with a coefficient of 2.2 instead of 2.6. Figure 16.8 shows measurements taken from the CLASH database and Eq. (16.3) with a mean prediction, the curve for deterministic design and 5% exceedance curves. Zero freeboard: For a vertical wall under non-impulsive conditions only one investigation is available,18 with zero freeboard (Rc = 0 or Rc /Hm0 = 0). Figure 16.9 gives
the measurements as a function of the fictitious wave steepness. 3 The mean was q/ gHm0 = 0.062 with a standard deviation of σ = 0.006. The value of 0.062 is slightly larger than the coefficient 0.04 in Eq. (16.3), and even closer to the coefficient of 0.05 of Allsop et al.1 Figure 16.9 shows that the value does not depend on the wave steepness, which is according to Eq. (16.3).
For probabilistic 3 = 0.062 for design or comparison with measurements, one can use q/ gHm0 plain vertical walls with zero freeboard. For deterministic design or safety assessment, it is recommended to increase the average overtopping discharge by one standard deviation and use the value of 0.068. No data are available for impulsive overtopping at zero freeboard at vertical walls. 16.3.1.2. Impulsive conditions (h∗ < 0.2) The mean prediction for impulsive conditions is given by Eq. (16.4) (Fig. 16.10). The reliability of this equation is described by considering the scatter in the logarithm of

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1.E+00

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0.5

CLASH database set 028 CLASH database set 106 CLASH database set 224

1.E-01

CLASH database set 225 CLASH database set 351 CLASH database set 402

1.E-02

CLASH database set 502 probabilistic = mean deterministic

1.E-03

5%

1.E-04 5% 1.E-05

1.E-06 0.0

0.5

1.0

1.5

2.0

2.5

3.0

3.5

4.0

dimensionless freeboard Rc/Hm0

Fig. 16.8.

Mean overtopping at a plain vertical wall under non-impulsive conditions [Eq. (16.3)].

RC=0

0.09

3

dimensionless discharge q/(gHm0 )0.5

0.10

0.08

3 0.5

q/(g Hm0 )

= 0.062 ± 0.006

0.07 0.06 0.05 0.04 0.03 0.02 0.01 0.00 0.00

0.01

0.02

0.03

0.04

0.05

(fictitious) wave steepness sm-1,0 Fig. 16.9.

Dimensionless overtopping discharge for zero freeboard.18

0.06

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dimensionless discharge q / h *2(ghs3)

0.5

1.E+03 1.E+02 1.E+01

CLASH database set 028

CLASH database set 224

CLASH database set 225

CLASH database set 351

CLASH database set 502

CLASH database set 802

probabilistic

deterministic

1.E+00 1.E-01 1.E-02 1.E-03

5%

1.E-04 5% 1.E-05 1.E-06 0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

(impulsive) dimensionless freeboard h* Rc/Hm0

Fig. 16.10.

Mean overtopping at a plain vertical wall under impulsive conditions [Eq. (16.4)].

the data about the mean prediction: log10 (qmeasured )−log10 (qpredicted ) is taken as a normally distributed stochastic parameter with a mean of 0 and a standard deviation σ = 0.37 (i.e., 68% of predictions lie within the range of ×/÷ 2.3). For probabilistic calculations, Eq. (16.4) should be taken together with these stochastic coefficients. For predictions of measurements or comparison with measurements also Eq. (16.4) should be taken with, for instance, 5% upper and lower exceedance curves. For deterministic design or safety assessment, a coefficient of 2.8 × 10−4 should be used in Eq. (16.4), instead of 1.5 × 10−4 :  −3.1 q Rc Rc −4
h∗ = 1.5 × 10 valid over 0.03 < h∗ < 1.0 . (16.4) 2 3 H H h∗ ghs m0 m0 For h∗ Rc /Hm0 < 0.02 arising from hs reducing to very small depths (as opposed to from small relative freeboards), there is evidence supporting an adjustment downward of the predictions of the impulsive formulae due to the observation that only broken waves arrive at the wall.4 The mean prediction is then described by Eq. (16.5). The reliability of this equation is described by considering the scatter in the logarithm of the data about the mean prediction: log10 (qmeasured ) − log10 (qpredicted ) is taken as a normally distributed stochastic parameter with a mean of 0 and a standard deviation σ = 0.15 (i.e., 68% of predictions lie within the range of ×/ ÷ 1.4):  −2.7 q Rc Rc
= 2.7 × 10−4 h∗ valid for h∗ < 0.02, broken waves. Hm0 Hm0 h2∗ gh3s (16.5)

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1.E+04

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0.5

Bruce et al (2003) data broken waves only- probabilistic (Eq. 5) broken waves only - deterministic (Eq. 5)

1.E+03

breaking waves - probabilistic (Eq. 4)

1.E+02 5%

1.E+01 5%

1.E+00 0.000

0.005

0.010

0.015

0.020

0.025

0.030

(impulsive) dimensionless freeboard h* Rc / Hm0

Fig. 16.11. Mean overtopping discharge for lowest h∗ Rc /Hm0 (for broken waves only arriving at wall) with submerged toe (hs > 0). For 0.02 < h∗ Rc /Hm0 < 0.03, overtopping response is illdefined — lines for both impulsive conditions (extrapolated to lower h∗ Rc /Hm0 ) and broken wave only conditions (extrapolated to higher h∗ Rc /Hm0 ) are shown as dashed lines over this region.

For probabilistic calculations or predictions of measurements or comparison with measurements, Eq. (16.5) should be taken together with these stochastic coefficients or, for instance, 5% upper and lower exceedance curves. For deterministic design or safety assessment, a coefficient of 3.8 × 10−4 should be used in Eq. (16.5), instead of 2.7 × 10−4 . For 0.02 < h∗ Rc /Hm0 < 0.03, there appears to be a transition between Eq. (16.4) (for “normal” impulsive conditions) and Eq. (16.5) (for conditions with only broken waves). There is, however, insufficient data upon which to base a firm recommendation in this range. It is suggested that Eq. (16.4) is used down to h∗ Rc /Hm0 = 0.02 unless it is clear that only broken waves will arrive at the wall, in which case Eq. (16.5) could be used. Formulae for these low h∗ Rc /Hm0 conditions are shown in Fig. 16.11. 16.3.1.3. Impulsive conditions, toe emergent (hs ≤ 0) Data for configurations where the toe of the wall is emergent (i.e., at or above still water level, hs ≤ 0) is limited. The only available study suggests an adaptation of a prediction equation for plunging waves on a smooth slope may be used, but particular caution should be exercised in any extrapolation beyond the parameter ranges of the study, which only used a relatively steep (m = 10) foreshore slope. This adaptation is given in Eq. (16.6). The reliability of Eq. (16.6) is described by taking the coefficient 2.16 as a normally distributed stochastic parameter with a

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1.E-04

dim'less discharge q/(gHm03)

0.5

.(sm-1,0 x m)

0.5

1.E-03

1.E-05

5%

Bruce et al (2003) data

1.E-06

emergent toe - probabilistic (Eq. 6)

5%

emergent toe - deterministic (Eq. 6)

1.E-07 0.0

1.0

2.0

3.0

dimensionless freeboard Rc/Hm0 x

Fig. 16.12.

4.0 0.33 sm-1,0

5.0

6.0

xm

Mean overtopping discharge with emergent toe (hs < 0).

mean of 2.16 and a standard deviation σ = 0.21:   q Rc √  · m sm−1,0 = 0.043 exp −2.16 m s0.33 , m−1,0 Hm0,deep 3 gHm0,deep valid for 2.0 < m s0.33 m−1,0

Rc < 5.0; Hm0,deep

0.55 ≤ Rc /Hm0,deep ≤ 1.6;

sm−1,0 ≥ 0.025 . (16.6)

Note: Data only available for m = 10 (i.e., 1:10 foreshore slope). For deterministic design or safety assessment, Eq. (16.6) should be used with a coefficient 1.95 instead of 2.16. Equation (16.6) for overtopping under emergent toe conditions is illustrated in Fig. 16.12. It is emphasized that this formula is based upon a limited dataset of small-scale tests with 1:10 foreshore only and should not be extrapolated outwith the given limits. 16.3.2. Battered walls Near-vertical walls with 10:1 and 5:1 batters are found commonly for older UK seawalls and breakwaters (e.g., Fig. 16.13). Mean overtopping discharges for battered walls under impulsive conditions are slightly in excess of those for a vertical wall over a wide range of dimensionless

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Fig. 16.13. Battered walls: typical cross-section (left), and Admiralty Breakwater, Alderney Channel Islands (right, courtesy G. M¨ uller).

dimensionless discharge q / h *2(ghs3)0.5

1.E+01

plain vertical (Eq. 3)

1.E+00

10:1 batter (Eq. 7a) 5:1 batter (Eq. 7b)

1.E-01

1.E-02

1.E-03

1.E-04 0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

(impulsive) dimensionless freeboard, h* Rc/Hm0 Fig. 16.14.

Overtopping for a 10:1 and 5:1 battered walls.

freeboards. Multiplying factors are given in Eqs. (16.17a) and (16.7b) (plotted in Fig. 16.14). 10:1 battered wall:

q10:1batter = qvertical × 1.3 ,

(16.7a)

5:1 battered wall:

q5:1batter = qvertical × 1.9 ,

(16.7b)

where qvertical is arrived at from Eq. (16.4). The uncertainty in the final estimated overtopping discharge can be estimated as per the plain vertical cases. No dataset is available to indicate an appropriate adjustment under nonimpulsive conditions. An alternative method, however, may be to calculate

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overtopping for a vertical structure and for a steep smooth slope 1:1. By using the cot α of the battered wall as the parameter (between the limits 0 and 1), a fair guess will be reached (see also Fig. 14.2 in Chap. 14 of this handbook). 16.3.3. Composite vertical walls It is well established that a relatively small toe berm can change wave-breaking characteristics, thus substantially altering the type and magnitude of wave loadings. Many vertical seawalls may be fronted by rock mounds with the intention of protecting the toe of the wall from scour. The toe configuration can vary considerably, potentially modifying the overtopping behavior of the structure. Three types of mounds can be identified: (i) Small toe mounds which have an insignificant effect on the waves approaching the wall — here, the toe may be ignored and calculations proceed as for simple vertical (or battered) walls. (ii) Moderate mounds, which significantly affect wave-breaking conditions, but are still below water level. Here, a modified approach is required. (iii) Emergent mounds in which the crest of the armor protrudes above still water level. Prediction methods for these structures may be adapted from those for crown walls on a rubble mound (Chap. 15 of this handbook). For assessment of mean overtopping discharge at a composite vertical seawall or breakwater, the overtopping regime (impulsive/non-impulsive) must be determined (see Sec. 16.2.3). When non-impulsive conditions prevail, overtopping can be predicted by the standard method given previously for non-impulsive conditions at plain vertical structures [Eq. (16.3)]. For conditions determined to be impulsive, a modified version of the impulsive prediction method for plain vertical walls is recommended, accounting for the presence of the mound by use of d and d∗ . 16.3.3.1. Impulsive conditions (d∗ ≤ 0.2) The mean prediction for impulsive conditions at a composite vertical structure is given by Eq. (16.8), and Fig. 16.15. The reliability of this equation is described by considering the scatter in the logarithm of the data about the mean prediction: log10 (qmeasured) − log10 (qpredicted ) is taken as a normally distributed stochastic parameter with a mean of 0 and a standard deviation σ = 0.28 (i.e., 68% of predictions lie within the range of ×/ ÷ 1.9). For probabilistic calculations, Eq. (16.8) should be taken together with these stochastic coefficients:

d2∗

q


gd3

= 4.1 × 10

valid for 0.05 < d∗

−4

 −2.9 Rc , d∗ Hm0

Rc < 1.0 and h∗ < 0.3 . Hm0

(16.8)

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dimensionless discharge q / d *2(gd3)0.5

1.E+03 VOWS data

1.E+02 composite vertical - probabilistic (Eq. 8)

1.E+01

composite vertical - deterministic (Eq. 8)

1.E+00

1.E-01

1.E-02 5% 1.E-03 5% 1.E-04 0.0

0.2

0.4

0.6

0.8

1.0

1.2

(impulsive, composite) dimensionless freeboard d* Rc/Hm0

Fig. 16.15.

Overtopping for composite vertical walls.

For deterministic design or safety assessment, Eq. (16.8) should be used with a coefficient of 7.8 × 10−4, instead of 4.1 × 10−4 , and an exponent of −2.6 instead of −2.9. 16.3.4. Effect of oblique waves Seawalls and breakwaters seldom align perfectly with incoming waves. The assessment methods presented thus far are only valid for shore-normal wave attack. In this subsection, advice is given on how the methods for shore-normal wave attack (obliquity β = 0◦ ) should be adjusted for oblique wave attack. As for zero obliquity, overtopping response depends critically upon the physical form (or “regime”) of the wave/wall interaction — non-impulsive, impulsive, or broken. As such, the first step is to use the methods given in Sec. 16.2 to determine the form of overtopping for shore-normal (zero obliquity). Based upon the outcome of this, guidance under “non-impulsive conditions” or “impulsive conditions” should be followed. For non-impulsive conditions, an adjusted version of Eq. (16.3) should be used:   2.6 Rc q
= 0.04 exp − , (16.9) 3 γβ Hm0 gHm0 where γβ is the reduction factor for angle of attack and is given by γβ = 1 − 0.0062β

for 00 < β < 45◦ ,

γβ = 0.72 for β ≥ 45◦ , and β is the angle of attack relative to the normal, in degrees.

(16.10)

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dimensionless discharge q / h *2(ghs3)0.5

1.E+01 beta = 0 degrees (Eq. 3) beta = 15 degrees (Eq. 11b)

1.E+00

beta = 30 degrees (Eq. 11c)

1.E-01

1.E-02

1.E-03

1.E-04 0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

(impulsive) dimensionless freeboard, h* Rc/Hm0 Fig. 16.16.

Overtopping of vertical walls under oblique wave attack.

For conditions that would be identified as impulsive for normal (β = 0◦ ) wave attack, a more complex picture emerges.13 Diminished incidence of impulsive overtopping is observed with increasing obliquity (angle β) of wave attack (Fig. 16.16). This results not only in reductions in mean discharge with increasing β, but also, for β ≥ 60◦ , a switch back over to the functional form observed for non-impulsive conditions [i.e., a move away from a power-law decay such as Eq. (16.4) to an exponential one such as Eq. (16.3)]. The mean predictions are given by Eqs. (16.11a)–(16.11d). Data only exist for the discrete values of obliquity listed: for β = 15◦ , h∗

Rc ≥ 0.2, Hm0

 −3.7 q Rc
= 5.8 × 10−5 h∗ , Hm0 h2∗ gh3s

(16.11a)

for β = 15◦ , h∗

Rc < 0.2, Hm0

as per impulsive β = 0◦ (Eq. 16.4) ,

(16.11b)

for β = 30◦ , h∗

Rc ≥ 0.07, Hm0

 −4.2 q Rc
= 8.0 × 10−6 h∗ , Hm0 h2∗ gh3s

(16.11c)

for β = 60◦ , h∗

Rc ≥ 0.07, Hm0

as per non-impulsive β = 60◦ (Eq. 16.10) .

(16.11d)

Significant spatial variability of overtopping volumes along the seawall under oblique wave attack are observed/measured in physical model studies. For deterministic design, Eqs. (16.12a)–(16.12c) should be used, as these give estimates of

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the “worst case” conditions at locations along the wall where the discharge is the greatest: for β = 150 , h∗

Rc ≥ 0.2, Hm0

for β = 300 , h∗

Rc ≥ 0.07, Hm0

as per impulsive β = 15◦ (Eq. 16.11b),

(16.12b)

for β = 600 , h∗

Rc ≥ 0.07, Hm0

as per non-impulsive β = 0◦ (Eq. 16.3).

(16.12c)

as per impulsive β = 0◦ (Eq. 16.4),

(16.12a)

16.3.5. Effect of bullnose and recurve walls Designers of vertical seawalls and breakwaters have often included some form of seaward overhang (recurve/parapet/wave return wall/bullnose) as a part of the structure with the design motivation of reducing wave overtopping by deflecting back seaward uprushing water (e.g., Figs. 16.17 and 16.18). The mechanisms determining the effectiveness of a recurve are complex and not yet fully described. The guidance presented here is based upon physical model studies.12,15 Parameters for the assessment of overtopping at structures with bullnose/recurve walls are shown in Fig. 16.19. Two conditions are distinguished: • the familiar case of the parapet/bullnose/recurve overhanging seaward (α < 90◦ ), and

Fig. 16.17. An example of a modern, large vertical breakwater with wave return wall (left) and cross-section of an older seawall with recurve (right).

Fig. 16.18. A sequence showing the function of a parapet/wave return wall in reducing overtopping by redirecting the uprushing water seaward (back to right).

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Fig. 16.19. Parameter definitions for assessment of overtopping at structures with parapet/wave return wall.

• the case where a wall is chamfered backward at the crest [normally admitting greater overtopping (α > 90◦ )]. For the latter, chamfered wall case, influence factors γ should be applied to Franco et al.’s equation10 for non-impulsive mean discharge [Eq. (16.13)] with the value of γ selected as shown8 :   4.3 Rc q = 0.2 exp − , (16.13) 0.5 3 γ Hm0 (gHm0 ) γ = 1.01 for α = 120◦ , γ = 1.13 for α = 135◦ , γ = 1.07 for α = 150◦ . For the familiar case of overhanging parapet/recurve/bullnose, the effectiveness of the recurve/parapet in reducing overtopping is quantified by a factor k defined as k=

qwith recurve . qwithout recurve

(16.14)

The decision chart in Fig. 16.20 can then be used to arrive at a value of k, which in turn can be applied by multiplication to the mean discharge predicted by the most appropriate method for the plain vertical wall (with the same Rc , hs , etc.). The decision chart shows three levels of decision: • whether the parapet is angled seaward or landward; • if seaward (α < 90◦ ), whether conditions are in the small (left box), intermediate (middle box), or large (right box) reduction regimes;

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Fig. 16.20. “Decision chart” summarizing methodology for tentative guidance. Note that symbols R∗0 , k23 , m, and m∗ used (only) at intermediate stages of the procedure are defined in the lowest boxes in the figure. Please refer to text for further explanation.

• if in the regime of largest reductions (greatest parapet effectiveness; Rc /Hm0 ≥ R0∗ + m∗ ), which of the three further subregimes (for different Rc /hs ) is appropriate. Given the level of scatter in the original data and the observation that the methodology is not securely founded on the detailed physical mechanisms/processes, it is suggested that it is impractical to design for k < 0.05, i.e., reductions in mean discharges by factors greater than 20 cannot be predicted with confidence. If such large (or larger) reductions are required, a detailed physical model study should be considered. 16.3.6. Effect of wind Wind may affect overtopping processes and thus discharges by (1) changing the shape of the incident wave crest at the structure resulting in a possible modification of the dominant regime of wave interaction with the wall;

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(2) blowing up-rushing water over the crest of the structure (for an onshore wind, with the reverse effect for an offshore wind) resulting in possible modification of mean overtopping discharge and wave-by-wave overtopping volumes; (3) modifying the physical form of the overtopping volume or jet, especially in terms of its aeration and breakup resulting in possible modification to postovertopping characteristics such as throw speed, landward distribution of discharge, and any resulting post-overtopping loadings (e.g., downfall pressures). The modeling of any of these effects in small-scale laboratory tests presents very great difficulties owing to fundamental barriers to the simultaneous scaling of the wave-structure and water–air interaction processes. Very little information is available to offer guidance on effect (1) — the reshaping of the incident waves. Comparisons of laboratory and field data (both with and without wind) have enabled some upper (conservative) bounds to be placed upon effect (2) — the intuitive wind-assistance in “pushing” of up-rushing water landward across the crest. These are discussed immediately below. Discussion of effect (3) — modification to “postovertopping” processes — is reserved for Secs. 16.5.3 and 16.5.4 (on distributions and downfalling pressures, respectively). Several investigations on vertical structures have suggested different adjustment factors fwind ranging from 30% to 40% to up to 300% (Fig. 16.21) either using a paddle wheel or large fans to transport uprushing water over the wall. When these tests were revisited, a simple adjustment factor was proposed for the mean discharge based upon small-scale tests qss , which is already scaled up by

10,0 De Waal et al. (1996) Davey (2004) Pullen and Allsop (2004)

f Wind [-]

Eq. (15)

5,0

0,0 1,0E-07

1,0E-06

1,0E-05

1,0E-04

1,0E-03

qss [m3/s/m]

Fig. 16.21. datasets.

Wind adjustment factor fwind plotted over mean overtopping rates qss for three

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appropriate scaling to full-scale (see also Ref. 9):  for qss ≤ 10−5 m3 /s/m  4.0 fwind = 1.0 + 3 · (− log qss − 4) for 10−5 < qss < 10−4 m3 /s/m   1.0 for qss ≥ 10−4 m3 /s/m .

(16.15)

From Eq. (16.15), it becomes clear that the influence of wind only gets important for very low overtopping rates below qss = 0.1 l/s/m. Hence, in many practical cases, the influence of wind may be disregarded. The mean overtopping discharge including wind becomes qwith wind = fwind × qss .

(16.16)

16.3.7. Scale and model effect corrections Tests in a large-scale wave channel (Fig. 16.22) and field measurements (Fig. 16.23) have demonstrated that with the exception of wind effect (Sec. 16.3.6), results of overtopping measurements in small-scale laboratory studies may be securely scaled to full-scale under non-impulsive and impulsive overtopping conditions.16,17 No information is yet available on the scaling of small-scale data under conditions where broken wave attack dominates. Comparison of measurements of wave loadings on vertical structures under broken wave attack at small scale and in the field suggests that prototype loadings will be over-estimated by small scale tests in the presence of highly-aerated broken waves. Thus, although the methods presented for

dimensionless discharge q / h *2(ghs3)

0.5

1.E+01 Large-scale data (Pearson et al, 2002)

1.E+00

Eq. 7a (10:1 batter)

1.E-01 1.E-02 1.E-03 1.E-04 1.E-05 1.E-06 0.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

(impulsive) dimensionless freeboard, h* Rc/Hm0

Fig. 16.22. Large-scale laboratory measurements of mean discharge at 10:1 battered wall under impulsive conditions showing agreement with prediction line based upon small-scale tests [Eq. (16.7a)].

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dimensionless discharge q / d *2(gd3)0.5

1.E+04 field data (Pullen et al, 2005)

1.E+03

composite vertical - probabilistic (Eq. 8)

1.E+02 composite vertical - deterministic (Eq. 8)

1.E+01 1.E+00 1.E-01 1.E-02 5% 1.E-03

5%

1.E-04 0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

(impulsive, composite) dimensionless freeboard d* Rc/Hm0 Fig. 16.23. Results from field measurements of mean discharge at Samphire Hoe, UK, plotted together with Eq. (16.8).

the assessment of overtopping discharges under broken wave conditions given in Sec. 16.3.1 have not been verified at large scale or in the field, any scale correction is expected to give a reduction in the predicted discharge.

16.4. Overtopping Volumes 16.4.1. Introduction While the prediction of mean discharge (Sec. 16.3) offers the information required to assess whether overtopping is slight, moderate, or severe, and make a link to any possible flooding that might result, the prediction of the volumes associated with individual wave events can offer an alternative (and often more appropriate) measure for the assessment of tolerable overtopping levels and possible direct hazard. First, a method is given for the prediction of maximum overtopping volumes expected associated with individual wave events for plain vertical structures under perpendicular wave attack (Sec. 16.4.2). This method is then extended to composite (bermed) structures (Sec. 16.4.3) and to conditions of oblique wave attack (Sec. 16.4.4). Finally, a short section on scale effects is included (Sec. 16.4.5). The methods given for perpendicular wave attack are the same as those given previously in UK guidance,2 but now using the Tm−1,0 period measure (with Weibull parameters, steepness values, etc., adjusted accordingly). The extension to oblique wave attack is new.

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16.4.2. Overtopping volumes at plain vertical walls The first step in the estimation of a maximum expected individual wave overtopping volume is to estimate the number of waves overtopping (Now ) in a sequence of Nw incident waves. For non-impulsive conditions, this was found to be well described by Franco et al.,10 as follows:  2

Rc (16.17) (for h∗ > 0.3) . Now = Nw exp −1.21 Hm0 Under impulsive conditions, Now is better described by2 Now = 0.031 Nw ×

Hm0 h∗ Rc

(for h∗ < 0.3) .

(16.18)

The distribution of individual overtopping volumes in a sequence is generally well described by a two-parameter Weibull distribution (see also Chap. 14, Sec. 14.2.2 of this handbook):  

b V , (16.19) PV = 1 − exp − a where PV is the probability that an individual event volume will not exceed V ; a and b are Weibull “shape” and “scale” parameters, respectively. Thus, to estimate the largest event in a wave sequence predicted to include (e.g.) Now = 200 overtopping events, Vmax would be found by taking PV = 1/200 = 0.005. Equation (16.19) can then be rearranged to give Vmax = a (ln Now )

1/b

.

(16.20)

The Weibull shape parameter a depends upon the average volume per overtopping wave Vbar , where Vbar = 0.8

qTm−1,0 Nw Now

or

Vbar =

qTm Nw . Now

(16.21)

For non-impulsive conditions, there is a weak steepness-dependency for the scale and shape parameters a and b: 0.66 for sm−1,0 = 0.024 0.74Vbar b= (16.22) a= for h∗ > 0.3 . 0.82 for sm−1,0 = 0.048 0.90Vbar (Note that the fictitious steepness values sm−1,0 = 0.024 and 0.048 in Eq. (16.22) correspond to the values of sop = 0.02 and 0.04 quoted by Besley2 .) For impulsive conditions2,16 : a = 0.92Vbar ,

b = 0.85

for h∗ < 0.3 .

(16.23)

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measured max. individual overtopping 3 volume, Vmax [m /m]

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1.E-01

1.E-02

1.E-03

1.E-04 1.E-04

1.E-03

1.E-02

1.E-01

1.E+00 3

predicted max. individual overtopping volume, Vmax [m /m] Fig. 16.24. Predicted and measured maximum individual overtopping volume — small- and largescale tests.16

The effectiveness of the predictor under impulsive conditions can be gauged from Fig. 16.24. Note that all the a values are fairly similar, leading to similar steep distributions (see also Sec. 14.2.2 in Chap. 14 of this handbook). A further step in development could be to choose a fixed value for a and modify b accordingly. 16.4.3. Overtopping volumes at composite (bermed) structures There is very little information available specifically addressing wave-by-wave overtopping volumes at composite structures. The guidance offered by Besley2 remains the best available. No new formulae or Weibull a, b values are known; so, for the purposes of maximum overtopping volume prediction, the methods for plain vertical walls (Sec. 16.4.2) are used. The key discriminator is that composite structures whose mound is sufficiently small to play little role in the overtopping process are treated as plain vertical, non-impulsive, whereas those with large mounds are treated as plain vertical, impulsive. For this purpose, the significance of the mound is assessed using the “impulsiveness” parameter for composite structures, d∗ [Eq. (16.2)]. “Small mound” is defined as d∗ > 0.3, with d∗ < 0.3 being “large mound.” 16.4.4. Overtopping volumes at plain vertical walls under oblique wave attack For non-impulsive conditions, an adjusted form of Eq. (16.17) is suggested10 :  2

Rc 1 for h∗ > 0.3 . (16.24) Now = Nw exp − 2 C Hm0

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Table 16.1. Summary of prediction formulae for individual overtopping volumes under oblique wave attack. Oblique cases valid for 0.2 < h∗ Rc /Hm0 < 0.65. For 0.07 < h∗ Rc /Hm0 < 0.2, the β = 00 formulae should be used for all β. β = 15◦ Now = 0.01 Nw × a = 1.06 Vbar b = 1.18

“

β = 30◦ Hm0 h∗ Rc

”−1.6

Now = 0.01 Nw × a = 1.04 Vbar b = 1.27

“

β = 60◦ Hm0 h∗ Rc

”−1.4

Treat as non-impulsive Treat as non-impulsive Treat as non-impulsive

C is given by

 0.91 C = 0.91 − 0.00425β  0.74

for β = 0◦ for 0◦ < β < 40◦ for β ≥ 40◦

for h∗ > 0.3 .

(16.25)

For impulsive conditions (as determined for perpendicular i.e., β = 0◦ wave attack), the procedure is the same as for perpendicular (β = 0◦ ) wave attack, but different formulae should be used for estimating the number of overtopping waves (Now ) and Weibull shape and scale parameters (Table 16.1).13 16.4.5. Scale effects for individual overtopping volumes Measurements from large-scale laboratory tests indicate that formulae for overtopping volumes, based largely upon small-scale physical model studies, scale well (Fig. 16.24).16 No data from the field is available to support “scale-ability” from large-scale laboratory scales to prototype conditions. 16.5. Overtopping Velocities, Distributions, and Down-Fall Pressures 16.5.1. Introduction to post-overtopping processes There are many design issues for which knowledge of just the mean and/or waveby-wave overtopping discharges/volumes are not sufficient, e.g., • assessment of direct hazard to people, vehicles, and buildings in the zone immediately landward of the seawall; • assessment of potential for damage to elements of the structure itself (e.g., crown wall, crown deck, secondary defenses). The appreciation of the importance of being able to predict more than overtopping discharges and volumes has led to significant advances in the description and quantification of what can be termed “post-overtopping” processes, specifically, the current state of prediction tools for: • the speed of an overtopping jet (or “throw velocity”), • the spatial extent reached by (impulsive) overtopping volumes, and

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max. dim'less vertical throw speed, u z / ci

7.0

6.0

5.0

4.0

3.0

2.0

1.0

0.0 0.0

0.1

0.2

0.3

0.4

0.5

h* parameter

Fig. 16.25. Speed of upward projection of overtopping jet past structure crest plotted with h∗ parameter (after Bruce et al.5 ).

• the pressures that may arise due to the downfalling overtopped jet impacting on the structure’s crown deck. 16.5.2. Overtopping throw speeds Studies at small scale, based upon video footage suggest that the vertical speed with which the overtopping jet leaves the crest of the structure (uz ) may be estimated as (Fig. 16.25): uz ≈ where ci =

√

2 to 2.5 × ci

for non-impulsive conditions ,

5 to 7 × ci

for impulsive conditions ,

(16.26)

ghs is the inshore wave celerity.5

16.5.3. Spatial extent of overtopped discharge The spatial distribution of overtopped discharge may be of interest in determining zones affected by direct wave overtopping hazard (to people, vehicles, buildings close behind the structure crest, or to elements of the structure itself). Under green water (non-impulsive) conditions, the distribution of overtopped water will depend principally on the form of the area immediately landward of the structure’s crest (slopes, drainage, obstructions, etc.) and no generic guidance can be offered (though, see Sec. 16.5.2 for information on speeds of overtopping jets).

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1 0.9 0.8

c. 95% of discharge falls within 0.25 × wavelength

0.7 0.6 0.5

c. 90% of discharge falls within 0.2 × wavelength

0.4 0.3 0.2

approx. lower (worst case) envolope

0.1 0 0.00

0.05

0.10

0.15

0.20

0.25

0.30

0.35

dimensionless landward distance x/L0 Fig. 16.26. Landward distribution of overtopping discharge under impulsive conditions. Curves show proportion of total overtopping discharge which has landed within a particular distance shoreward of seaward crest.

Under violent (impulsive) overtopping conditions, the idea of spatial extent and distribution has a greater physical meaning — where does the airborne overtopping jet come back to the level of the pavement behind the crest? The answer to this question, however, will (in general) depend strongly upon the local wind conditions. Despite the difficulty of directly linking a laboratory wind speed to its prototype equivalent (see Sec. 16.3.6), laboratory tests have been used to place an upper bound on the possible wind-driven spatial distribution of the “fall back to ground” footprint of the violently overtopped volumes.3,17 Tests used large fans to blow air at gale-force speeds (up to 28 m/s) in the laboratory. The lower (conservative) envelope of the data (Fig. 16.26) gives the approximate guidance that 95% of the violently-overtopped discharge will land within a distance of 0.25 × Lo , where Lo is the offshore (deepwater) wavelength. No data is available for the case of a shallow foreshore with no clear definition of Lo . 16.5.4. Pressures resulting from downfalling water mass Wave impact pressures on the crown deck of a breakwater have been measured in small- and large-scale tests.19,20 These impacts are the result of an impacting wave at the front wall of the breakwater generating an upward jet which in turn falls back onto the crown deck of the structure. Small-scale tests suggest that local impact pressure maxima on the crown deck are smaller than but of the same order of magnitude as wave impact pressures on the front face. For high-crested structures (Rc /Hm0 > 0.5), pressure maxima were observed to occur within a distance of ∼ 1.5×Hm0 behind the seaward crest. For lower-crested structures (Rc /Hm0 < 0.5),

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this distance was observed to increase to ∼ 2 × Hm0 . Over all small-scale tests, pressure maxima were measured over the range 2<

p1/250 < 17 with a mean value of 8. ρ gHm0

(16.27)

The largest downfall impact pressure measured in large-scale tests was 220 kPa (with a duration of 0.5 ms). The largest downfall pressures were observed to result from overtopping jets thrown upward by very-nearly breaking waves (the “flip through” condition). Although it might be expected that scaling small-scale impact pressure data would overestimate pressure maxima at large scale, approximate comparisons between small- and large-scale test data suggest that the agreement is good. Acknowledgments This chapter was based on the EurOtop Overtopping Manual,14 which was funded in the United Kingdom by the Environmental Agency, in Germany by the German Coastal Engineering Research Council (KFKI), and in the Netherlands by Rijkswaterstaat and Netherlands Expertise Network (ENW) on Flood Protection. The Project Team for the creation, editing, and support of the manual; the Project Steering Group for guidance and supervision; and a number of individual persons, have been listed and acknowledged in Chap. 14.

References References have been kept to an absolute minimum. An extensive bibliography can be found in the EurOtop Overtopping Manual (2007). 1. N. W. H. Allsop, P. Besley and L. Madurini, Overtopping performance of vertical and composite breakwaters, seawalls and low reflection alternatives, Paper 4.7 in MCS Project Final Report, University of Hannover (1995). 2. P. Besley, Overtopping of seawalls — Design and assessment manual, R&D Technical Report W 178, Environment Agency, Bristol (1999). 3. T. Bruce, T. Pullen, W. Allsop and J. Pearson, How far back from a seawall is safe? Spatial distributions of wave overtopping, Proc. Int. Conf. Coastlines, Structures and Breakwaters 2005, ICE London, Thomas Telford (2005), pp. 166–176. 4. T. Bruce, J. Pearson and N. W. H. Allsop, Violent wave overtopping — Extension of prediction method to broken waves, Proc. Coastal Structures 2003, ASCE, Reston, Virginia (2003), pp. 619–630. 5. T. Bruce, N. W. H. Allsop and J. Pearson, Hazards at coast and harbour seawalls — Velocities and trajectories of violent overtopping jets, Proc. 28th Int. Conf. Coastal Eng., Cardiff (2002), pp 2216–2226. 6. CEM/H. F. Burcharth and S. A. Hughes, Fundamentals of design, Coastal Engineering Manual, eds. L. Vincent and Z. Demirbilek, Part VI, Design of Coastal Project Elements. Chapter VI-5-2, Engineer Manual 1110-2-1100 (US Army Corps of Engineers, Washington, DC, 2002).

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7. CLASH, Crest level assessment of coastal structures by full scale monitoring, neural network prediction and hazard analysis on permissible wave overtopping, Fifth Framework Programme of the EU, Contract N. EVK3-CT-2001-00058, www. clash-eu.org. 8. A. Cornett, Y. Li and A. Budvietas, Wave overtopping at chamfered and overhanging vertical structures, Proc. Int. Workshop on Natural Disasters by Storm Waves and their Reproduction in Experimental Basins, Kyoto, Japan (1999). 9. J. De Rouck, J. Geeraerts, P. Troch, A. Kortenhaus, T. Pullen and L. Franco, New results on scale effects for wave overtopping at coastal structures, Proc. Coastlines, Structures and Breakwaters 2005, ICE London, Thomas Telford (2005), pp. 29–43. 10. L. Franco, M. de Gerloni and J. W. van der Meer, Wave overtopping on vertical and composite breakwaters, Proc. 24th Int. Conf. Coastal Eng., Kobe (1994), pp. 1030–1044. 11. Y. Goda, Random Seas and Design of Maritime Structures, 2nd edn. (World Scientific Publishing, Singapore, 2000). 12. A. Kortenhaus, J. Pearson, T. Bruce, N. W. H. Allsop and J. W. van der Meer, Influence of parapets and recurves on wave overtopping and wave loading of complex vertical walls, Proc. Coastal Structures 2003, ASCE, Reston, Virginia (2003), pp. 369–381. 13. N. Napp, T. Bruce, J. Pearson and N. W. H. Allsop, Violent overtopping of vertical seawalls under oblique wave conditions, Proc. 29th Int. Conf. Coastal Eng., Cardiff (2002), pp. 4482–4493. 14. Overtopping Manual, Wave Overtopping of Sea Defences and Related Structures — Assessment Manual, eds. T. Pullen, N. W. H. Allsop, T. Bruce, A. Kortenhaus, H. Sch¨ uttrumpf and J. W. van der Meer (2007), www.overtopping-manual.com. 15. J. Pearson, T. Bruce, N. W. H. Allsop, A. Kortenhaus and J. W. van der Meer, Effectiveness of recurve wave walls in reducing wave overtopping on seawalls and breakwaters, Proc. 29th Int. Conf. Coastal Eng., Lisbon (2004), pp. 4404–4416. 16. J. Pearson, T. Bruce, N. W. H. Allsop and X. Gironella, Violent wave overtopping — Measurements at large and small scale, Proc. 28th Int. Conf. Coastal Eng., Cardiff (2002), pp. 2227–2238. 17. T. Pullen, N. W. H. Allsop, T. Bruce, J. Pearson and J. Geeraerts, Violent wave overtopping at Samphire Hoe: Field and laboratory measurements, Proc. 29th Int. Conf. Coastal Eng., Lisbon (2004), pp. 4379–4390. 18. R. Smid, Untersuchungen zur Ermittlung der mittleren Wellen¨ uberlaufrate an einer senkrechten Wand und einer 1:1,5 geneigten B¨ oschung f¨ ur Versuche mit und ohne Freibord, Student study at Leichtweiss-Institute for Hydraulics. Braunschweig (2001) (in German). 19. T. Bruce, L. Franco, P. Alberti, J. Pearson and N. W. H. Allsop, Violent wave overtopping: Discharge throw velocities, trajectories and resulting crown deck loading, Proc. Ocean Wave Measurement and Analysis (“Waves 2001”), ASCE (1999), pp. 1783–1796. 20. G. Wolters, G. M¨ uller, T. Bruce and C. Obhrai, Large scale experiments on wave downfall pressures on vertical and steep coastal structures, Proc. ICE Maritime Engineering 158, 137–145 (2005).

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Chapter 17

Surf Parameters for the Design of Coastal Structures Dong Hoon Yoo Department of Civil Engineering, Ajou University Suwon, 443-749, Korea [email protected] Iribarren number is widely used by coastal engineers for classifying the type of breaking waves and for estimating armor weight, run-up height, and even for estimating wave overtopping. Iribarren number is the ratio of breakwater slope or beach slope to the square root of wave steepness. Another existing surf parameter is the ratio of breakwater slope to the wave steepness. In the present chapter new surf parameters, which are called “first order wave action slope” and “second order wave action slope,” are introduced for representing the local wave conditions in shallow waters by employing local values of wavelength as well as wave height. The use of linear wave theory on a flat bed of the depth at the front of breakwater might be considered far better than the simple adoption of the deepwater wavelength for characterizing surfing waves at a shoaling depth. The wave action slopes are formed by the product of the breakwater slope and the celerity ratio to the wave height. The run-up height is related to the first order wave action slope, and the optimum or minimum weight of armor unit is related to the second order wave action slope.

17.1. Introduction When waves approach a coast, wave particles transform from elliptic movement to horizontal movement and energy transport velocity (group velocity) continues to reduce to near zero. Then wave heights increase up to a certain limit, that is, until they lose their stability of wave formation. The type of breaking waves is one of the major factors for the determination of wave forces affecting beaches and coastal structures. Iribarren1 suggested a wave breaking parameter to determine the type of breaking waves and the Iribarren number is in recent years widely employed for the estimation of run-up height and unit weight of armor block. Iribarren number is the ratio of beach slope to the square root of wave steepness with introducing 441

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Fig. 17.1.

Wave deformation on a beach slope.

deepwater wavelength. The use of deepwater wavelength is reckoned to be largely due to the difficulty of measuring or estimating the wavelength at a local point. In practice, the definition of wavelength does not seem to be clear particularly when waves approach a beach slope or sloping breakwater, as the water depth varies from deep to shallow as shown in Fig. 17.1. The wavelength of offshore direction from the breaking point is certainly longer than the wavelength of onshore direction on a sloping beach. It is, therefore, often practice to use a deepwater wavelength instead of a local value, although a local value of wave height is normally employed in order to represent partly the local condition at the breaking point. It may possibly be very difficult to estimate or measure or even define the wavelength at a sloping bed in shallow waters. But the wavelength of the local point water depth given by any wave theory, assuming the sloping bed is flat, may be far better than the deepwater wavelength for the estimation of wave steepness at a local point. Iribarren adopted the square root of wave steepness for his nondimensional physical number. Direct use of wave steepness also results in nondimensional physical number, which yields new form of Iribarren number. On the other hand, new surf parameters are formulated by introducing Froude number which is the ratio of wave celerity to group velocity of wave height. On the result four sets of surf parameters are suggested, and it is found that one set is related to the other set. All of them are employed for developing empirical equations of run-up height and unit weight of armor block. 17.2. Iribarren Numbers Using local values of wavelength and wave height, the first order Iribarren number I is represented by1 : S I= √ , M

(17.1)

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where M = H/L (wave steepness) and S = tan θ (beach slope or breakwater slope). Offshore Iribarren number Io uses the values of wave height Ho and wavelength Lo at deepwater for the estimation of wave steepness. Inshore Iribarren number Ii uses the local value of wave height but still uses the value of wavelength at deepwater condition. On the other hand, the local Iribarren number I takes the local values of both wave height and wavelength for the estimation of wave steepness. The second order Iribarren number J is also, using local values of wavelength and wave height, represented by: J=

S . M

(17.2)

The second order Iribarren number is directly related to the wave steepness, while the first order Iribarren number is related to the square root of the wave steepness. The former is associated with the square of wave celerity, and the latter with the wave celerity with no power. Similarly, offshore second order Iribarren number Jo uses the values of wave height Ho and wavelength Lo at deepwater for the estimation of wave steepness. Inshore second order Iribarren number Ji uses the local value of wave height but still uses the value of wavelength at deepwater condition. On the other hand, the local second order Iribarren number J takes the local values of both wave height and wavelength for the estimation of wave steepness. 17.3. Wave Action Slopes Wave action slope is the product of the beach slope S and the Froude number associated with wave celerity C and wave height H as suggested by Yoo and Na.2 Two types of wave action slopes are suggested; first order and second order. The first order wave action slope SX is represented by: SX = FH S, (17.3) √ where FH = C/ gH. Similarly, the offshore first order wave action slope SXo uses the values of wave height Ho and wave celerity Co at deepwater. And the inshore first order wave action slope SXi uses the local value of wave height but still uses the value of wave celerity at deepwater condition. On the other hand, the local first order wave action slope takes the local values of both wave height and wave celerity. The second order wave action slope SY (Ref. 3) is represented by: 2 S. SY = FH

(17.4)

Offshore second order wave action slope SY o uses the values of wave height Ho and wave celerity Co at deepwater. And inshore second order wave action slope SY i uses the local value of wave height but still uses the value of wave celerity at deepwater condition. On the other hand, the local second order wave action slope takes the local values of both wave height and wave celerity. Using dispersion relation as follows: σ 2 = gk tanh kh,

(17.5)

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Relationship of wave action slopes and Iribarren numbers. SXo

SXi

Definition

√Co S gHo

√Co S gH

Relationship

√1 Io 2π

√1 Ii 2π

SX q

SY o

√C S gH

2 Co gHo

tanh kh I 2π

1 J 2π o

S

SY i 2 Co

gH

S

1 J 2π i

SY C2 S gH tanh kh J 2π

where σ is the angular frequency (σ = 2π/T ) and k is the wave number (k = 2π/L), the local first order wave action slope SX is related with the local first order Iribarren number I as follows:
tanh kh C SX = √ . (17.6) S=I 2π gH √ For√the offshore condition tanh kh ≈ 1, and hence SXo = Io / 2π and SXi = Ii / 2π. That is, the first order wave action slopes are found simply proportional to the first order Iribarren numbers. Similarly, the local second order wave action slope SY is related with the local second order Iribarren number J as follows: SY =

C2 tanh kh S=J . gH 2π

(17.7)

For the offshore condition tanh kh ≈ 1, and hence SY o = Jo /2π and SY i = Ji /2π. That is, the second order wave action slopes are found simply proportional to the second order Iribarren numbers. The first order wave action slope is closely related to the first order Iribarren number and the second order wave action slope is closely related to the second order Iribarren number. It is, however, found that the relations do not have simple linear proportionality. The wave action slopes are related to the Iribarren numbers as presented in Table 17.1. As summarized in the table, the offshore numbers and the inshore numbers are linearly proportional to each other, but the local numbers are not directly proportional to each other. Further effect of depth dispersion is reflected in the wave action slopes through dispersion relation.

17.4. Run-up Height The run-up height is the vertical length from the mean surface to the top of runup reach, and the run-up height ratio is defined by the ratio of run-up height to wave height. Saville4 conducted laboratory experiments on a concrete slope with various conditions of beach slope and wave steepness for the monochromatic waves, and presented two separate graphs of run-up height ratio to the slope with various inshore wave steepness (Mi = H/L0 ) as shown in Fig. 17.2(a). It is found that the run-up height ratio is almost proportional to the beach slope up to a certain limit, from where reflection might be a significant factor on the wave particle movement on the slope. Depending on the water depth condition, the run-up height ratio is

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Fig. 17.2.

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Run-up height ratio ηR versus (a) S or (b) SXi using the laboratory data of Saville.4

related to the Iribarren number or first order wave action slope as follows: ηR = 2.64SXi ,

(SXi < 0.427Mi−0.25, 1 < h/H < 3),

(17.8a)

ηR = 2.39SXi ,

(SXi < 0.674Mi−0.16, 3 < h/H),

(17.8b)

where ηR = hR /H (run-up height ratio), hR is the run-up height, H is the wave height. As shown in Fig. 17.2(b), the effect of wave period is found well implied in the expression of inshore wave action slope SXi . Further simplification can be made by the use of local wave action slope, and the result is given by: ηR = 2.80SX (SX < 0.394M −0.25).

(17.9)

When SX < 0.394M −0.25, the influence of reflection is considered very small. Otherwise, the effect of reflection becomes strong and then the run-up height becomes almost irrespective of slope or the wave action slope. Mase5 conducted laboratory experiments of run-up on smooth concrete slope (S = 0.033 − 0.25) for irregular waves, and suggested empirical equation of run-up height related to the offshore Iribarren number as follows: ηR−n = αIoβ ,

(17.10)

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Table 17.2. Coefficients α, β, and δ for the estimation of run-up height ratio.

hR−n hR−max hR−2 hR−1/10 hR−1/3 hR−avr

Eq. (17.10)

Eq. (17.11)

Eq. (17.12)

α 2.32 1.86 1.70 1.38 0.88

α — 0.37 0.34 0.27 0.17

α 0.41 0.44 0.38 0.30 0.20

β 0.77 0.71 0.71 0.70 0.69

β — 1.38 1.27 1.04 0.66

β 2.2 1.64 1.52 1.25 0.79

Eq. (17.13) δ 1 0.742 0.688 0.568 0.359

where ηR = hR /HS , HS is the significant wave height, and the subscript n indicates the reaches of different probability. The values of constants α and β for different probability are presented in Table 17.2. Hedges and Mase6 re-analyzed the laboratory data of run-up height of irregular waves of Mase.4 They deducted the set-up height from the total run-up height, and related the pure run-up height to the inshore Iribarren number as follows: ηR−n = α + βIi ,

(17.11)

where α and β are constants, but the values vary depending on the probability of run-up reaches as presented in Table 17.2. For the case of irregular waves, the run-up height ratio is determined by using significant wave height HS , that is, ηR = hR /HS . For the case of 2% reach ηR−2 they suggested α = 0.37 and β = 1.38. New empirical equation of run-up height ratio is developed using the local Iribarren number as follows: ηR−n = α + βI.

(17.12)

Comparison of the equations (17.10), (17.11), and (17.12) are shown in Fig. 17.3 against Mase’s laboratory data.5 Up to the value of Ii = 2, Eqs. (17.10) and (17.11) show very good correlation with the laboratory data. But when the inshore first order Iribarren number exceeds approximately 2, the results of Eqs. (17.10) and (17.11) deviate considerably from the laboratory data. On the other hand, when using the local Iribarren number as expressed by Eq. (17.12), good correlations are found over the whole region of I. Another empirical equation of run-up height ratio is developed using the local wave action slope SX instead of the inshore Iribarren number Ii or the local Iribarren number I. Hedges and Mase6 used different separate values of constants of Eq. (17.10) or (17.11) for different reaches of probability in order to keep the accuracy. For the engineering use the run-up height of different reaches of probability ηR−n can be simply related to the maximum run-up height as follows: ηR−n = δηR−max ,

(17.13)

where δ is the reduction factor of probability. The maximum run-up height ratio of irregular waves ηR−max is expressed by: ηR−max = 0.32 + 6.9SX

(17.14)

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Fig. 17.3.

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Run-up height ratio versus Io , Ii , I, and SX using the laboratory data of Mase.5

or ηR−max = 8.1SX .

(17.15)

The degree of accuracy of Eq. (17.11) or (17.12) is slightly better than that of Eq. (17.13), but Eq. (17.13) is also acceptable for an engineering use with less than 5% endurance. The reduction factor of probability δ = 1 for the maximum reach, δ = 0.742 for the 2% reach, δ = 0.688 for the 10% reach, δ = 0.568 for the 1/3 reach, and δ = 0.359 for the average reach as summarized in Table 17.2. The computation results are presented in Fig. 17.4. As shown in the figure, the correlation of Eq. (17.13) is generally found very good simply by introducing the

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448

Fig. 17.4.

Various run-up height ratios versus SX using the laboratory data of Mase.5

reduction factor. The correlation coefficients are estimated over 0.97 for all cases, and the determination coefficients are about 0.93 for most cases. Comparing both constants of Eqs. (17.9) and (17.15) each other, the maximum run-up reach of irregular waves is found almost tripple times bigger than the run-up reach of monochromatic waves. The reduction factor of average reach of irregular waves is about 0.36, and this is almost the same as the ratio of the two constants of Eqs. (17.9) and (17.15). The run-up height can be significantly reduced for the rough slope of breakwaters from the values given for the smooth concrete slope. Ahrens7 suggested to use another reduction factor for the rough wall. The run-up height is, hence, finally computed by: ηR−n = ξδηR−max ,

(17.16)

where ξ is the reduction of run-up height for rough wall. ηR−max is the maximum run-up height ratio on smooth concrete slope. The values of ξ are given in Table 17.3. 17.5. Armor Weight The equation of Hudson8 is still world-widely employed for the estimation of minimum or optimum weight of armor block. Hudson equation is expressed by: ηW =

S , (s − 1)3 KD

(17.17)

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449

Reduction factor of wall roughness for the estimation of run-up height.

Quality of a slope

Deferment system

Roughness coefficient

— Fitted Fitted Fitted — — Fitted Random Random Random Random

1.00 0.90 0.85∼0.90 0.85∼0.90 0.85∼0.90 0.80 0.75∼0.80 0.60∼0.65 0.60∼0.65 0.50∼0.55 0.45∼0.50

Smooth and impermeable slope Concrete block Basalt block Gobi block Grass One layer of quarry stone Quarry stone Rounded quarry stone Three layer of quarry stone Quarry stone Concrete slope of 50% percolation

where ηW = W/γH 3 , W is the armor weight, γ is the specific weight of armor block, s is the specific gravity of armor block, and KD is the empirical constant which is primarily associated with the type of armor block. Hudson equation indicates that the weight ratio is solely dependent on the breakwater slope, but various workers found that the optimum weight of armor block is also significantly influenced by wave period, percolation of slope, and the formation of irregular waves. van der Meer9 employed the inshore first order Iribarren number, and suggested a set of two equations depending on the condition determined by the Iribarren number. The equation of van der Meer9 is represented by: ηW =

0.0042NS Ii1.5 (s − 1)3 P 0.54

(0.4 < Ii < 2.5),

(17.18a)

NS P 0.39 S 1.5 (2.5 < Ii ), (17.18b) (s − 1)3 Ii3P √ where P is the percolation, NS = ( N/D)0.6 , N is the number of incident waves, and D is the degree of damage which is defined by the ratio of damage area to the projected area of armor unit. The number of incident waves for a field is normally between 1000 and 3000, and the degree of damage is between 1 and 3. The armor weights measured in the laboratory by van der Meer9 are related to the Iribarren number Ii and the second order wave action slope SY . As shown in Fig. 17.5, the use of the second order wave action slope gives the best correlation with the laboratory data. As found by van der Meer,9 the variation of ηW becomes complicated when Ii is bigger than 2.5. New empirical equation has been developed using the second order wave action slope as follows: ηW =

ηW =

0.178KY SY , (s − 1)3

(17.19)

where KY is a constant depending on percolation, degree of damage, the condition of reflection, but primarily depending on the type of armor block. The constant 0.178 is determined by adopting the values of N = 3000, D = 3, and P = 0.1. The empirical constant KY = 1 for the armored slope of three-layered round stones.

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Armor block weight ratio versus Ii and SY (laboratory data of van der Meer9 ).

Sealock is one of the most well-devised concrete armor blocks. It is widely used in Japan and recently introduced to Korea. The stability of the block is well tested in the laboratory experiments. The experiments were conducted in the channel of 1.5 m deep, 1.0 m wide, and 30 m long against the breakwater of slope S = 3/4. The number of incident waves was 500. Waves are generated with four different wave periods 1.3, 1.5, 1.8, 2.1, and 2.4 s, and six or seven different wave heights for each wave period between 0.15 m and 0.24 m. The degree of damage (DS ) is defined by the percentage of failure units, and it varies from 0.0 to the maximum value 4.9. For the zero damage level Hudson parameter of Sealock was estimated 12.0 for T = 2.4 s, 14.7 for T = 2.1 s, 21.3 for T = 1.8 s, 22.0 for T = 1.5 s, and 17.8 and 28.5 for T = 1.3 s as shown in Fig. 17.6(b). The values of Hudson parameter are found to become small for longer wave period, that is, longer waves are found more damageable to the stability of armored breakwaters. The parameter KD of the Hudson equation is found strongly dependent on the wave period. From the results Hudson parameter KD is almost proportional to the degree of damage but strongly dependent on the wave period as follows: KD = 3.874DS + 23.31 (T = 1.3 ∼ 1.8 s),

(17.20a)

KD = 3.529DS + 13.51 (T = 2.1 ∼ 2.4 s).

(17.20b)

For the safety with a confidence Samsung Hydraulic Engineering suggested that KD equals to 10.0 for the armor block Sealock.10 The set of the data is re-analyzed by relating them to the local second order wave action slope SY . The results are demonstrated in Figs. 17.6(c) and 17.6(d). The figure demonstrates that even the local second order wave action slope does not seem to include sufficiently well the effect of wave period. There still exists some deviation between the regression lines of different wave period. It may be further refined by considering the effect of set-down and set-up for the computation

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Fig. 17.6.

451

Weight ratio of armor block Sealock versus S with KD and SY with KY .

of the local wavelength or wave celerity. The variation of the parameter KY is not ignorable as shown in Fig. 17.6(d). Nevertheless, the variation of the parameter KY becomes almost irrespective of wave period and it is given by: KY = −0.010DS + 0.140.

(17.21)

When the degree of damage defined by Samsung Hydraulic Engineering is zero, KY for the armor unit Sealock is approximately 0.14. The values of the parameter KY for various armor blocks are suggested in Table 17.4. The values of three-layered rounded quarry stones and Sealocks are determined by analyzing the laboratory data, and other values are deduced by relating it with the Hudson parameter KD . More confident values should be determined by testing each relevant set of laboratory data.

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452 Table 17.4.

KD and KY of various armor blocks. KD

Two-layer rounded quarry stones Three-layer rounded quarry stones Sharp edged quarry stones TTP Dolos T-bar Sealock

KY

Nonbreaking

Breaking

(Nonbreaking)

2.4 3.2 4.0 8.0 31.8 10.0 10.0

1.2 1.6 2.0 7.0 15.8 7.0 10.0

1.33 1.0 0.8 0.4 0.1 0.32 0.14

17.6. Conclusions In the present chapter, several new surf parameters are presented primarily using the local values of wavelength as well as wave height. The first one has the similar form to the Iribarren number, and it is called “second order Iribarren number.” The second order Iribarren number is related to the wave steepness, while the original (first order) Iribarren number is related to the square root of wave steepness. The second group is called “wave action slope.” √ It is the product of the beach slope S and the Froude number defined by FH = C/ gH, where C is the wave celerity and H is the wave height. The first order wave action slope is FH S(= SX ), the second order 2 wave action slope is FH S(= SY ). The first order wave action slope is closely related to the first order(original) Iribarren number and the second order wave action slope is closely related to the second order Iribarren number. It is, however, found that the relations do not have simple linear proportionality. Run-up height is related to the first order wave action slope, and unit weight of armor block is related to the second order wave action slope. Both relations have simple linear forms, and excellent agreements are found in comparison with laboratory data. The effects of wave period or wavelength are found to be well included in the new parameters. The use of the new parameters are considered to be suitable particularly when the wave period is relatively big and the water depth is relatively deep. Acknowledgment This research was supported by Heyin ENC. References 1. R. C. Iribarren, Generalization of the formula for calculation of rock fill dikes and verification of its coefficients (Revista de Obras Pulicas, Madrid, 1950 WES Translation N.51-4). 2. D. H. Yoo and D. Y. Na, Hydraulic condition of breaking waves, J. Korean Soc. Civil Eng. 25(2B), 159–163 (2005) (in Korean).

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3. D. H. Yoo, S. K. Koo and I. H. Kim, Minimum weight of breakwater armor unit, Proc. 1st Asian and Pacific Coastal Engineering Conf., Dalian, China (2001), pp. 605–612. 4. T. J. P. Saville, Wave run-up on shore structures, Proc. ASCE, Paper No. 925 82(WW2), 359–373 (1956). 5. H. Mase, Random wave run-up height on gentle slope, J. Waterway, Port, Coastal, and Ocean Eng. 115(5), 649–661 (1989). 6. T. S. Hedges and H. Mase, Modified Hunt’s equation incorporating wave setup, J. Waterway, Port, Coastal and Ocean Eng. ASCE 130(3), 109–113 (2004). 7. J. P. Ahrens, Approximate Upper Limit of Irregular Waves Run-Up on Riprap, Department of the US Army Corps of Engineers (1988). 8. R. V. Hudson, Laboratory investigation of rubble-mound breakwater, ASCE, Trans., Paper No. 3213, 126 (1961). 9. J. W. van der Meer, Rock slopes and gravel beaches under wave attack, Delft Hydraulics Communication No. 336 (1988). 10. Samsung Hydraulic Engineering, Hydraulic experiments of Sealock, Report No. 11 (1978) (in Japanese).

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Chapter 18

Development of Caisson Breakwater Design Based on Failure Experiences Shigeo Takahashi Port and Airport Research Institute 3-1-1, Nagase, Yokosuka 239-0826, Japan takahashi [email protected] The experiences of caisson breakwater failures are explained and the current design of caisson breakwaters is described to prevent such failures. The design method of conventional caisson against wave forces is especially explained in this chapter. The design method of caisson breakwaters is still under development including a new design method of performance design. The future direction will be discussed in the final section.

18.1. Introduction Composite type breakwater is one of major types of breakwaters in the world, consisting of vertical wall installed on a rubble mound foundation. To resist the enormous power of storm waves, concrete caissons are usually employed as the vertical wall.1 The composite breakwater with caissons (caisson breakwater) requires a smaller body width and quantities of material compared with conventional rubble mound breakwaters. This is one of the biggest advantages of using a caisson breakwater, which makes the breakwater construction more economical, especially in deep water. The design and construction methods of the caisson breakwater have developed rapidly in the latter half of the 20th century through many bitter experiences of failures in Japan. Especially, the design technique for the caisson breakwaters is nearly established including the calculation method for determining the wave forces acting on the breakwater. In addition, the improvement of the conventional caisson breakwaters has been encouraged. The horizontally composite caisson breakwater is an improved version of the conventional composite caisson breakwater. Such breakwaters are not new, however, since vertical wall breakwaters suffering damage to the vertical walls were often strengthened by placing large stones or concrete blocks in front of them so as 455

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to dissipate the wave energy and reduce the wave force, and is now frequently being constructed especially in shallow surf zone in Japan. Various new caisson breakwaters have been invented and commercialized in order to mitigate the drawbacks associated with conventional composite breakwaters. One of typical new caisson breakwater is the perforated wall caisson breakwater invented by Jarlan2 in 1961. The caisson dissipates wave energy by front perforated wall and wave chamber. Many breakwaters of this type were subsequently constructed throughout the world with further improvement. In this chapter, the experiences of caisson breakwater failures are explained and the current design of caisson breakwaters is described to prevent such failures. The design method of conventional caisson breakwaters against wave forces is especially described in this chapter. The design method of caisson breakwaters is still developing including an integrated method of performance design. The future direction will be discussed in the final section.

18.2. Failures of Caisson Breakwaters It is very important to learn from the actual failures. The design methods of the breakwaters have been developed significantly from such experiences, and the development of the design methods has prevented them from suffering failure. It should be noted that the total length of breakwaters in Japan is more than 800 km and the failures of the breakwaters are now very limited compared with the total amount of caissons in the country. Before explaining each failure, it should be noted that the most important cause of failures of caisson breakwaters is the caisson sliding and breakage due to impulsive wave pressures. It should also be noted that the rubble mound/rubble foundation of composite breakwaters is vital to prevent the failure of the upright section by scouring, as well as stabilizing the foundation against the wave force and caisson weight.3

18.2.1. Meandering failures Photo 18.1 shows an offshore breakwater at Sendai Port after Typhoon 9119 hit causing several caissons to slide. The length of the breakwater is 700 m. Although the attacking waves were estimated to be about 20% higher than the design wave height, only caissons at particular locations suffered sliding. This is called “Meandering Sliding.” The breakwater consisted of caissons of 11.8 m wide on 6 m thick rubble foundation. The water depth is 21 m, and the estimated incident wave was H1/3 = 6.8 m and T1/3 = 12 s, with an incident wave angle 65◦ . The waves attacking the caisson were not breaking waves but nonbreaking waves. This meandering sliding is a typical sliding phenomenon due to nonbreaking waves. This is caused by diffracted waves from breakwater heads in an oblique wave condition as discussed in Sec. 18.3.3.

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Photo 18.1.

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Meandering failure of caisson.

Photo 18.2.

Impulsive wave pressure.

18.2.2. Impulsive wave pressures Photo 18.2 shows a wave hitting the offshore side of a caisson at Minamino-hama Port. The breakwater forms a jetty type breakwater designed to protect small ferryboats, with its rear side to be used as a quay wall. Big splash in the photo is typical, when an impulsive breaking wave force act on the vertical wall. During a typhoon, waves equivalent to the design wave or larger attacked the breakwater head caisson from the breakwater alignment direction. Plunging breakers almost completely destroyed the caisson at the breakwater head. Caisson damage started when the sidewall of the caisson began breaking, then progressed to the whole caisson. Such caisson breakage was caused by impulsive wave pressures acting on a caisson installed on a steep seabed slope. Actually, the breakwater was under construction and the damaged caisson was going to be protected

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by another caisson which was designed to resist such severe wave pressures, but unfortunately not in place before the storm. Similar failure due to impulsive wave pressure occurs when breaking waves acting on a caisson installed on a high/wide rubble foundation. Impulsive wave pressures occur when the vertical wall is attacked by an almost vertical wave front, and therefore larger vertical wave front due to plunging or surging breakers gives larger impact pressures. Such caisson failures due to impulsive pressures had occurred frequently in history but have recently been greatly reduced using accumulated knowledge about impulsive wave pressures including impulsive pressure coefficient as discussed in Sec. 18.3.2. 18.2.3. Scattering of armor for rubble foundation Photo 18.3 shows a typical caisson failure at a breakwater head, where the caisson moved toward the harbor side. It should be noted that the caisson was not moved by wave force, rather by scouring of the rubble foundation. Photo 18.4 shows scouring of the rubble foundation at the breakwater head in a model experiment. It is known that very strong wave-induced current occurs at the corner of the breakwater head caisson. Scattering of armor stones occurs when the weight of armor stones was insufficient against very high water particle velocity around the breakwater head. In such cases scour of the rubble mound and the sandbed under the rubble mound occur. This type of failure can particularly occur during the construction period, although the design method against such high water particle velocity is well established as explained in Sec. 18.3.4. 18.2.4. Scouring of rubble stones and seabed sand due to oblique waves Figure 18.1 shows the inclined caisson in a relatively calm harbor. This is due to the scouring of rubble mound stones and the sandbed under the rubble mound.

Photo 18.3.

Breakwater head.

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Scouring

Photo 18.4.

Fig. 18.1.

Scouring of rubble mound around head.

Scouring of rubble mound along caisson.

Oblique waves caused a strong wave-induced current along the breakwater caissons, although the wave height is not large. An estimation method of the wave-induced current in front of the caisson due to normal as well as oblique waves was already proposed and was included in the current design.4 18.2.5. Erosion of front seabed (scouring of mound toe) Figure 18.2 shows a cross section of a large composite breakwater which suffered severe erosion of the mound toe area. Due to high waves, which exceeded 7 m in significant wave height, the front area was scoured more than 3 ms. This phenomenon is not seldom when such high waves attack a breakwater. The front erosion of breakwaters comprises two phenomena; large-scale sea bottom change and local front scouring. It is really difficult to protect a breakwater from such front erosion although Irie et al.5 described its fundamental mechanism. Only empirical countermeasures such as a gravel mat or asphalt mat are usually adopted to reduce such scouring.

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Fig. 18.2.

Photo 18.5.

Erosion at rubble mound toe.

Settlement of caisson due to through-wash.

It should be noted that protecting the caisson is essential even though some part of the rubble foundation is scoured. The rubble foundation is usually designed with an enough length considering its deformation due to front erosion.

18.2.6. Seabed through-wash Photo 18.5 shows settlement of a breakwater due to through-wash of the sandbed under it.6 The relatively fine sand under the rubble mound was washed away by severe wave actions. This type of damage is normally prevented by placing a geotextile sheet under the rubble foundation as discussed in Sec. 18.3.4. If, however, high waves hit the breakwater during construction, this may lead to improper placement of the geotextile sheet, which results in settlement as shown here.

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18.2.7. Rubble foundation failure Breakwater failure due to foundation failure is seldom seen since the current design method seems to be a little conservative which evaluates the bearing capacity of rubble mound and seabed foundation using the Bishop method.7 Figure 18.3 shows a special case of that due to rubble mound failure. An asphalt mat was placed under the caisson to increase the friction coefficient between the caisson and the rubble foundation. Due to high waves exceeding the design wave, sliding took place, which did not occur at the caisson bottom, rather in the foundation. It was thought that this sliding occurred between the rubble mound and sandbed.

Fig. 18.3.

Rubble mound failure.

Fig. 18.4.

Failure path diagram.

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Use of an asphalt mat is very effective in reducing the necessary width of the caisson, and has been widely applied in recent designs. Care must be taken to check foundation stability, however. 18.2.8. Failure path diagram Figure 18.4 is a failure path diagram of composite breakwaters. It shows the weaker parts of breakwaters and major causes of failures that were already explained above. For example, due to wave-induced strong current around the breakwater head, scattering of armor blocks around the head occurs, which results in settlement of mound and therefore the settlement of the caisson. Then caisson can easily slide. It should be noted that damage to breakwaters seldom occurs even when storm waves exceed the design wave. Typically, only a part of the breakwater weaker than other parts suffers damage due to storm wave heights less than the design wave height. The failure can be reduced by more careful design especially against armor layer scattering and seabed scouring. It will be also effective to reduce failures to include the wave height increase along the breakwater alignment. Most of the design methods against such failures are already established but a new integrated design method should be developed further to increase the reliability and to reduce the total cost of breakwaters.

18.3. Design of Conventional Caisson Breakwaters 18.3.1. Example of a caisson breakwater Figure 18.5 shows a typical cross section of a conventional caisson breakwater. The upright section is a 21.5 m × 27 m caisson installed on a 3.5-m-thick rubble foundation. The depth of the caisson h is 12.5 m and the height of the crown hc is 6.1 m at LWL. The caisson is divided into 5 × 6 chambers by 20-cm-thick inner walls and 40-cm-thick outer walls. The chambers are filled with sand, capped by concrete, and a concrete superstructure is placed on the caisson.

Fig. 18.5.

Typical cross section of caisson breakwater, Noshiro Port.8

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Foot protection blocks are placed to prevent through-wash of the rubble foundation and the sand below, while shape-designed concrete blocks are installed to act as the armor layer of the rubble foundation. The water depth d above the rubble mound including the armor layer is 10 m at LWL, and the berm width BM of the rubble mound is 12.8 m. For scour protection, additional gravel is placed, being called a “gravel mat.” A vinyl sheet is also used to prevent scouring of the sand under the rubble foundation. This breakwater is designed to withstand a wave of HD = 11.6 m (significant wave height H1/3 = 6.66 m), significant wave period T1/3 = 13 s, and wave angle θ of 22◦ at a water depth h of 21 m. The design significant wave in deepwater H0 is 12.2 m at a return period of 50 years. The design wave at the breakwater site was evaluated using a wave transformation calculation, with wave pressures on the caisson being evaluated by the extended Goda pressure formula. 18.3.2. Wave forces on vertical walls 18.3.2.1. Extended Goda formula (Goda–Takahashi formula) In 1973, Goda9 used his own theoretical and laboratory studies to establish a comprehensive formula to calculate the design wave forces. After a later modification to account for the effect of oblique wave incidence, this formula was successfully applied to the design of vertical breakwaters built in Japan. The original Goda formula10,11 has many advantageous features, with the main ones being as follows: (1) It can be employed for all wave conditions, i.e., both for standing and breaking waves. (2) The formula’s design wave is the maximum wave height and can be evaluated by given diagrams and/or equations. (3) It is partially based on nonlinear wave theory and can represent wave pressure characteristics by considering two pressure components: the breaking and slowly varying pressure components. Consequently, it is relatively easy to extend the Goda formula in order to apply it to other vertical wall-type structures. (4) The Goda formula clarifies the concept of uplift pressure on the caisson bottom, since the buoyancy of the upright section in still water and its uplift pressure due to the wave action are defined separately. The distribution of the uplift pressure has a triangular shape. The Goda formula was subsequently extended to include the following parameters: (1) The incident wave direction,12 (2) The modification factors applicable to other types of vertical walls, (3) The impulsive pressure coefficient.13 In the extended Goda formula (named here the Goda–Takahashi formula), the wave pressure acting along a vertical wall is assumed to have a trapezoidal distribution both above and below the still water level, while the uplift pressure acting

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Fig. 18.6.

Goda pressure formula.

on the bottom of the upright section is assumed to have a triangular distribution as shown in Fig. 18.6. The buoyancy is calculated using the displacement volume of the upright section in still water at the design water level. As indicated, h denotes the water depth in front of the breakwater; d, the depth above the armor layer of the rubble mound foundation; h , the distance from the design water level to the bottom of the upright section; and hc , the elevation of the breakwater above the design water level. The elevation at which the wave pressure is exerted, η ∗ , and the representative wave pressure intensities p1 , p3 , p4 , and pu can be written in a generalized form as η ∗ = 0.75(1 + cos θ)λ1 HD , p1 = 0.5(1 + cos θ)(λ1 α1 + λ2 α∗ cos2 θ)w0 HD , p 3 = α3 p 1 , p 4 = α4 p 1 , pu = 0.5(1 + cos θ)λ3 α1 α3 w0 HD , in which


α1 = 0.6 + 0.5

(4πh/LD ) sinh(4πh/LD )

h∗c , η∗

h∗c = min{η ∗ , hc },

(18.2) (18.3)

2 ,

α∗ = max{α2 , αI },   
hb − d (HD /d)2 2d , , α2 = min hb 3 HD 
 1 h 1− , α3 = 1 − h cosh(2πh/LD ) α4 = 1 −

(18.1)

(18.4) (18.5) (18.6) (18.7) (18.8) (18.9)

where θ is the angle between the direction of wave approach and a line normal to the breakwater;

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λ1 , λ2 , λ3 are the modification factors dependent on the structural type; HD , LD are the wave height and wave length applied to calculate design wave forces; αI is the impulsive pressure coefficient; w0 is the specific weight of sea water (= ρo g); hb is the offshore water depth at a distance five times the significant wave height H1/3 ; min{a, b}, the minimum of a and b; and max{a, b}, the maximum of a and b. 18.3.2.2. Pressure component and pressure coefficients (α1 , α2 , and αI ) Figure 18.7 shows the transition of wave pressure from nonbreaking to impulsive pressure, where the pressure component is indicated by coefficients α1 , α2 , and αI . α1 represents the slowly varying pressure component and α2 the breaking pressure component, while αI represents the impulsive pressure component, which includes the dynamic response effect of the caisson sliding. α1 increases from 0 to 1.1 as the relative depth decreases, and α2 increases as d/hb decreases, though it peaks and then decreases as d/hb decreases, ranging 0–1.0. The value of αI is described next in more detail. 18.3.2.3. Impulsive pressure coefficient αI Takahashi et al.13 obtained the impulsive pressure coefficient αI by reanalyzing the results of comprehensive sliding tests, being a nondimensional value representing the impulsive pressure component, which should be regarded as an additional effect to the slowly varying pressure component. The effect of the dynamic (impulsive) pressure indicated by α2 in Goda’s formula does not under all conditions accurately estimate the effective pressure (equivalent static pressure) due to impulsive pressure, and therefore, αI was introduced. Figure 18.8 shows a calculation diagram for αI , in which it is expressed by the product of αI0 and αI1 , where αI0 represents the effect of wave height on the

Fig. 18.7.

Transition of wave pressure.

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Fig. 18.8.

Calculation diagram of impulsive pressure coefficient.13

mound, i.e., αI = αI0 αI1 ,  H/d when H ≤ 2d, αI0 = 2.0 when H > 2d,

(18.10) (18.11)

and αI1 represents the effect of the mound shape (shown by the contour lines). This term can be evaluated using   cos δ2 when δ2 ≤ 0, (18.12) αI1 = cosh δ1  0.5 −1 when δ2 > 0, [(cosh δ1 )(cosh δ2 ) ]  20δ11 when δ11 ≤ 0, (18.13) δ1 = 15δ11 when δ11 > 0,  4.9δ22 δ2 = 3δ22

when δ22 ≤ 0, when δ22 > 0,


 h−d BM − 0.12 + 0.36 − 0.6 , = 0.93 L h 

h−d BM − 0.12 + 0.93 − 0.6 . = −0.36 L h

(18.14)




δ11 δ22

(18.15)

The value of αI reaches a maximum of 2 at BM /L = 0.12, d/h = 0.4, and H/d > 2. When d/h > 0.7, αI is always close to zero and is less than α2 . It should be noted that the impulsive pressure significantly decreases when the angle of incidence θ is oblique.

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18.3.2.4. Modification factors (λ1 , λ2 , and λ3 ) For the ordinary vertical breakwater, λ1 , λ2 , and λ3 are taken as unity since the Goda formula was originally proposed to describe this type of breakwater. The modification factor λ1 represents the reduction or increase of the wave’s slowly varying pressure component, λ2 represents changes in the breaking pressure component (dynamic pressure component or impulsive pressure component), while λ3 represents changes in the uplift pressure. These modification factors are introduced to express the pressures on other types of caisson breakwaters. 18.3.2.5. Design wave height The wave height and length applied to calculate the design wave forces are those of the highest wave in the design sea state. The height of this wave is taken as HD = Hmax = H1/250 = 1.8H1/3 seaward of the surf zone, or within the surf zone as the largest wave height Hb of random breaking waves at the water depth hb . The term H1/250 is the average height of the highest one-two hundred fiftieth waves. 18.3.2.6. Wave direction The wave angle θ is the angle between the direction of wave approach and a line normal to the breakwater alignment. Starting from its principal direction, the wave direction should be rotated toward the line normal to the breakwater alignment by up to 15◦ . This adjustment is made to compensate for both the uncertainty in estimating the wave direction and the waves’ directional spreading. 18.3.3. Other problems related to wave forces 18.3.3.1. Concave section Since the breakwater alignment is usually somewhat complex and a vertical breakwater is so reflective, the reflection and diffraction from it should be taken into account in wave force calculations. Figure 18.9 shows the calculated distribution of the ratio of the wave height to the incident wave, kd , in front of a vertical breakwater which has an alignment forming two lines having a concaved shape with respect to the incident waves. Due to the reflection from one of the lines of the breakwater, the wave height along the vertical wall is not simply 2 for standing waves. This effect should be taken into account by considering the amplification factor of the incident wave to be equal to kd /2. It should be noted that the amplification factor is limited by standing wave breaking and that its maximum value is recommended to be 1.4 based on carrying out a series of experiments. 18.3.3.2. Meandering effect Another problem related to the plane shape of the breakwater is the so-called meandering effect.15 Figure 18.10 shows the calculated value of kd for a singleline detached breakwater with length LB = 200 m and wavelength L1/3 = 92.3 m,

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Fig. 18.9.

Fig. 18.10.

Wave height distribution along concaved breakwater alignment.14

Wave height distribution along detached one-line breakwater.14

where the wave height fluctuates along the breakwater alignment and significantly increases near the breakwater head. If waves exceeding the design wave height attack the breakwater, and if the breakwater caissons slide, the shape of these caissons will subsequently form a meandering shape. 18.3.4. Design of rubble mound foundation 18.3.4.1. Armor for rubble foundation 1. Wave force on armors The rubble mound foundation under a vertical wall should be protected to prevent it from scattering due to wave actions. This is accomplished by covering it with armor

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Fig. 18.11.

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Scattering of armor stones of rubble foundation.16

stones or concrete blocks. Figure 18.11 illustrates the movement of the armor stones in observed sections, where in the upper figure, the stones are scattered from the slope of the mound when incident waves approach perpendicular to the breakwater alignment. Note the occurrence of heavy scattering around the breakwater head. In the middle figure where θ = 45◦ , the armor stones not only scatter away from the slope, but in the flat berm near the caisson as well. The stones also scatter around the breakwater head on its downstream side and especially at the caisson edge. In the lower figure with θ = 60◦ , the stones on the slope do not move, though they significantly scatter in the flat berm, especially near the caisson. It is postulated that the stability of armor stones or blocks is mainly threatened by the water particle velocity induced by the waves, i.e., by the drag and uplift forces produced by the water particle velocity. In Fig. 18.11, the water particle velocity is high where the stones moved, which supports this hypothesis. Figure 18.12 shows a hodograph of the water particle velocity at the breakwater trunk. When θ = 0, the water particle only moves perpendicular to the breakwater alignment and the velocity is almost zero near the vertical walls, being largest at the node of the clapotis (standing wave). As θ increases, the water velocity component parallel to the breakwater alignment increases, with the velocity near the vertical wall also significantly increasing. Figure 18.13 shows the distribution of the peak water particle velocity around the breakwater head, where it is revealed that a high velocity occurs around the edge of the upright section.

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Fig. 18.12.

Fig. 18.13.

Water particle velocity.

Water particle velocity distribution around the breakwater head.

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To withstand velocity-induced forces, the armor stones or blocks should have enough weight, which can be evaluated using Isbash’s equation for stones embedded in the bottom of a sloped channel, i.e., W =

πγr U 6 , 48g 3 y 6 (Sr − 1)3 (cos α − sin α)3

(18.16)

where W is the necessary weight of armor stone; γr , the specific weight of armor stone (γr = ρr g); y, the Ishbash number (1.2 for embedded stones and 0.86 for stones placed on a flat bottom); Sr , the specific gravity of stone; U , the water particle velocity on the stone; g, the acceleration of gravity; and α, the bottom slope. 2. Necessary weight of armor Ishbash’s equation relates the stable weight of stones to the water particle velocity. Brebner and Donnelly,16 however, proposed a method to directly determine the necessary weight from the wave height. In their method, the stable weight of armor units W can be expressed as W =

3 γr H1/3

Ns3 (Sr − 1)3

,

(18.17)

where γr denotes the specific weight of the armor unit; H1/3 , the design significant wave height; and Ns , the stability coefficient. This is a kind of Hudson’s equation, which uses Ns instead of KD cot α, such that H1/3 {(Sr − 1)(W/γr )1/3 } = Ns .

(18.18)

Ns depends on variables such as the shape of the armor unit, their manner of placement, the shape of the rubble mound foundation, and wave conditions (height, period, and direction). Tanimoto et al.17 proposed a formula to calculate the stability coefficient for two layers of quarry stones, being based on analytical considerations and the results of random wave experiments. Takahashi et al.18 modified Tanimoto’s formula so that it can be applied to obliquely incident waves, i.e., 

 

 

1−κ h (1 − κ)2 h + 1.8 exp −1.5 Ns = max 1.8, 1.3 H1/3 H1/3 κ1/3 κ1/3 (18.19) in which κ = κ1 (κ2 )B , 2kh , κ1 = sinh 2kh (κ2 )B = max{αs sin2 θ cos2 (kBM cos θ), cos2 θ sin2 (kBM cos θ)},

(18.20) (18.21) (18.22)

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Fig. 18.14.

The calculated necessary weight of armor stones for rubble foundation.18

where max {a, b} denotes the maximum of a and b; h , the water depth above the rubble mound foundation; L , the wavelength corresponding to the significant wave period at the depth h ; k, the wave number (= 2πL ); and BM is the berm width as shown in Fig. 18.14. αs is a correction factor obtained using wave tank experiments and is 0.45. Equation (18.19) is also extended to include the stability of the rubble mound armor layer in the breakwater head. In the breakwater head, the term (κ2 )T is used instead of (κ2 )B to represent the water particle velocity at the breakwater head, where κ = κ1 (κ2 )T ,

(18.23)

1 (αs τ 2 ), 4

(18.24)

(κ2 )T =

where τ expresses the ratio of the water particle velocity at the breakwater head to that of the incident wave and is determined to be 1.4 for the wave angle less than 45◦ . Figure 18.14 shows sample calculations to determine the necessary weight of armor stones. The shape of the breakwater is indicated and the necessary weight

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for different wave conditions and water depths in front of the vertical wall are calculated. When θ = 60◦ , the weight is the largest, whereas that for θ = 30◦ is the smallest. It is obvious to see that the weight increases as the wave period increases or the water depth in front of the wall decreases. When the weight is large, armor concrete blocks must be used to cover the rubble mound. 18.3.4.2. Foot protection of the upright wall Foot-protection concrete blocks are usually placed in front of the upright section. Figure 18.15 shows the failure modes related to such foot-protection, whereas the foot-protection blocks are removed, erosion of rubble mound takes place near the foot of the upright section. Also, through-wash (rapid current through the rubble mound) will cause scouring of the sand under the rubble mound; thus, the foot-protection concrete blocks must prevent the direct intrusion of wave pressure into the rubble mound and also the subsequent pressure-induced current in the mound. These blocks work as a filter and also provide weight for the rubble mound. The critical force on a foot-protection block is that due to the pressure difference between the upper and lower faces of the block. The absolute value of the wave pressure under the block was experimentally found to be 5–40% less than that on the upper side. This pressure difference can be reduced and the stability increased by making holes in the blocks, although if the holes are too large, the filtering effect is reduced. An opening ratio of 10% is therefore recommended by Tanimoto et al.15 Several ways exist to empirically determine the necessary weight. Figure 18.16 shows a diagram to determine the thickness t of concrete blocks having a 10% opening, with the dimensions of the blocks being subsequently determined and summarized in Table 18.1. Figure 18.17 shows a concrete block with t = 1.2 m. It should be noted that the foot-protection blocks also act as armor blocks for the rubble mound as discussed, especially in oblique seas and at the breakwater head. Therefore, they should be stable against the velocity-induced force. Additionally, care should be taken to prevent the occurrence of scouring underneath the rubble foundation. Too thin a rubble mound may cause this type of scouring to occur due to severe wave actions, and consequently, a vinyl sheet is sometimes placed on the sand bed to prevent it.

Fig. 18.15.

Failure due to damage to toe protection blocks.

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Fig. 18.16.

Thickness of foot-protection blocks.19

Table 18.1. blocks.

Specification

t
(m)

l(m) × b(m) × t
(m)

W (t)

∼0.8 ∼1.0 ∼1.2 ∼1.4 ∼1.6 ∼1.8 ∼2.0 ∼2.2

2.5 × 1.5 × 0.8 3.0 × 2.5 × 1.0 4.0 × 2.5 × 1.2 5.0 × 2.5 × 1.4 5.0 × 2.5 × 1.6 5.0 × 2.5 × 1.8 5.0 × 2.5 × 2.0 5.0 × 2.5 × 2.2

6.2 15.6 24.8 37.0 42.3 47.6 52.9 58.2

Fig. 18.17.

of

foot-protection

Foot-protection block.

18.3.4.3. Toe protection against scouring Figure 18.18 shows scouring damage at the toe of the rubble mound for a composite breakwater situated on a sandy seabed.20 Due to the weight of the caisson, the entire breakwater slightly settles and the toe area is significantly scoured, which deepens the toe area by about 2 or 3 m. Even though this toe erosion occurs, the

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Fig. 18.18.

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Scour around rubble mound toe.20

caisson remains intact and the foot-protection blocks and armor blocks of the rubble mound still function as designed. In actuality, one of the primary roles of the rubble mound and the foot-protection blocks is to protect the caisson from such scouring. The cause of toe area scouring is probably due to the strong wave-induced velocity. These velocities are especially increased by reflected waves from the vertical wall. However, the scouring mechanism is very complicated and has not yet been properly explained. Generally, two types of toe scouring exist: local scouring and large-scale sand movement. Local scouring was investigated by Xie,21 Irie et al.,5 Oumeraci,22 and others. Irie found the occurrence of N- and L-type scours, in which the former type scour is due to suspension of relatively fine sand which causes accretion at the node, while the later type is due to “bed load” of relatively coarse sand that causes erosion at the node and accretion at the loop. L-type scour appears to be predominant in proto type seas where erosion at the node is usually found. Scour is usually inevitable for vertical breakwaters built on a sandy sea bottom. However, scour is not a fatal problem due to the protection features provided by the rubble mound. Nevertheless, scour protection should be included in breakwaters in which severe scouring is expected. There are several scour protection methods, e.g., the use of gravel, geotextile, or asphalt mats. These methods can prevent scouring to some extent, though no fully sufficient method has yet been realized.

18.4. Recent Development of Conventional Caisson Breakwater Design Over the last 15 years improved awareness of wave impact induced failures3,23,24 has focused attention on the need to account for the dynamic response of maritime structures to wave impact loads. Also, the attention was focused on the probabilistic consideration in the design of breakwaters.25–28 The most relevant collaborative research projects carried out in the recent past in the European Union on wave loads on caisson breakwaters and seawalls include: (1) PRObabilistic design tools for VERtical BreakwaterS, see Oumeraci et al.29 and references therein;

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(2) Violent Overtopping of Waves at Seawalls, see Cuomo30 and references therein; (3) Breaking Wave IMpacts on step fronted COastal Structures, see Bullock et al.31 and references therein. Recently, a new innovative design method is being developed for the design of breakwater caissons which is called the performance design method. Performance design can be considered as a design process that systematically and clearly defines performance requirements and respective performance evaluation methods. In other words, performance design allows the performance of a structure to be explicitly and concretely described.32 A new performance design for the stability of breakwater caissons was being developed which is called deformation-based reliability design.33–37 Sliding distance is selected as the performance evaluation item and the probabilistic nature is fully considered. Performance design requires a reliable performance evaluation method. Thus, in deformation-based reliability design of a breakwater caisson, a calculation method to determine the sliding distance due to wave actions is used, employing Monte Carlo simulation to include the probabilistic nature of waves and response of the breakwater caisson. Since breakwaters are designed to withstand wave heights having a particular return period, such as 50 years, a high probability exists that higher waves than the design wave will attack them and there still remain some failures by heavy storms. Recently, it becomes very important to evaluate the extent of damage/deformation as well as the consequence including restoration and the total economic losses. The performance design is just the design process to consider these items for more reasonable and economical design of breakwaters. Also, the performance design is becoming increasingly important for the design of coastal defenses against storm surges and tsunami.38

References 1. S. Takahashi, Design of vertical breakwaters, Reference Document No. 34, Port and Harbour Research Institute (1996) 85 pp. 2. G. E. Jarlan, A perforated vertical breakwater, The Dock and Harbour Authority 41(488), 394–398 (1961). 3. S. Takahashi, K. Shimosako, K. Kimura and K. Suzuki, Typical failures of composite breakwaters in Japan, Proc. 27th Int. Conf. Coastal Eng., ASCE (2000), pp. 1899–1910. 4. K. Kimura, S. Takahashi and K. Tanimoto, Stability of rubble mound foundations of caisson breakwater under oblique wave attack, Proc. 24th Int. Conf. Coastal Eng. Kobe, ASCE (1994). 5. I. Irie, Y. Kuriyama and H. Asakawa, Study on scour in front of breakwaters by standing waves and protection method, Rep. Port. Harbour Res. Inst. 25(1), 3–86 (1986) (in Japanese). 6. K. Suzuki, S. Takahashi and Y. H. Kang, Experimental analysis of wave-induced liquefaction in a fine sand bed, Proc. 26th Int. Conf. Coastal Eng. (1998), pp. 3643–3654.

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7. M. Kobayashi, M. Terashi and K. Takahashi, Bearing capacity of a rubble mound supporting a gravity structure, Rept. Port and Harbour Res. Inst. 26(5), 215–252 (1987). 8. S. Kataoka and S. Saida, Compilation of breakwater structures, Tech. Note of Port and Harbour Res. Inst., No. 556, 150 pp. (1986) (in Japanese). 9. Y. Goda, Laboratory investigatation of wave pressures exerted upon vertical and composite walls, Rept. Port and Harbour Res. Inst. 11(2), 3–45 (1972) (in Japanese) or Y. Goda, Experiments on the transition from nonbreaking to post breaking wave pressures, Coastal Eng. 15, 81–90 (1972) (in Japanese). 10. Y. Goda, A new method of wave pressure calculation for the design of composite breakwater, Rept. Port and Harbour Res. Inst. 12(3), 31–70 (1973) (in Japanese) or Proc. 14th Conf. Coastal Eng., ASCE, Copenhagen (1974), pp. 1702–1720. 11. Y. Goda, Random Seas and Design of Maritime Structures (Univ. Tokyo Press, Tokyo, 1985). 12. K. Tanimoto, K. Moto, S. Ishizuka and Y. Goda, An investigation on design wave force formulae of composite-type breakwaters, Proc. 23rd Japanese Conf. Coastal Eng. (1976), pp. 11–16 (in Japanese). 13. S. Takahashi, K. Tanimoto and K. Shimosako, A proposal of impulsive pressure coefficient for design of composite breakwaters, Proc. Int. Conf. Hydro-technical Eng. for Port and Harbor Construction, Port and Harbour Res. Inst. (1994). 14. K. Kobune and M. Osato, A study of wave height distribution along a breakwater with a corner, Rept. Port and Harbour Res. Inst. 15(2) (1976) (in Japanese). 15. Y. Ito and K. Tanimoto, Meandering damages of composite type breakwaters, Tech. Note of Port and Harbour Res. Inst. 112 (1972) (in Japanese). 16. A. Brebner and D. Donnelly, Laboratory study of rubble foundations for vertical breakwaters, Proc 8th Int. Conf. Coastal Engineering, New Mexico City, ASCE (1962), pp. 406–429. 17. K. Tanimoto, T. Yagyu and Y. Goda, Irregular wave tests for composite breakwater foundation, Proc. 18th Conf. Coastal Eng., Capetown (1982), pp. 2144–2163. 18. S. Takahashi, K. Kimura and K. Tanimoto, Stability of armor units of composite breakwater mound against oblique waves, Rept. Port and Harbour Res. Inst. 29(2) (1990) (in Japanese). 19. R. Ushijima, R. Mizuno and T. Imoto, Laboratory stability test of foot-protection blocks for upright section of composite breakwaters, Rept. of Civil Engineering Research Institute, Hokkaido Development Bureau, No. 424 (1988), pp. 1–14 (in Japanese). 20. H. Funakoshi, Survey of long-term deformation of composite breakwaters along the Japan sea, Proc. Int. Workshop on Wave Barriers in Deepwaters, PHRI, Yokosuka, Japan, PHRI (1994), pp. 239–266. 21. S. L. Xie, Scouring patterns in front of vertical breakwaters and their influence on the stability of foundation of breakwaters, Rept. of Department of Civil Engineering, Delft University of Technolgy (1981), 61 pp. 22. H. Oumerachi, Scour in front of vertical breakwaters — Review of problems, Proc. Int. Workshop on Wave Barriers in Deepwaters, PHRI (1994), pp. 281–317. 23. H. Oumeraci, Review and analysis of vertical breakwater failures — Lessons learned, Special issue on vertical breakwaters, Coastal Engineering 22, 3–29 (1994). 24. L. Franco, Vertical breakwaters: The Italian experience. Special Issue on vertical breakwaters, Coastal Engineering 22, 31–55 (1994). 25. H. F. Burcharth, Development of a partial safety factor system for the design of rubble mound breakwaters, PIANC PTII working group 12, subgroup F, Final Report (PIANC, Brussels, 1993).

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26. M. A. Losada and J. Llorca, Breakwaters under the action of sea oscillations. Spanish recommendations ROM 1.1, Coastal Structures (2007). 27. J. W. van der Meer, Deterministic and probabilistic design of breakwater armor layers, Proc. American Society of Civil Engineers, J. Waterways, Coastal Ocean Eng. 114(1), 66–80 (1988). 28. J. K. Vrijling, H. G. Vorrtman, H. F. Burcharth and J. D. Soren-sen, Design philosophy for a vertical breakwa-ter, Proc. Coastal Structures ’99 (1999), pp. 631–635. 29. H. Oumeraci, A. Kortenhaus, N. W. H. Allsop, M. B. De Groot, R. S. Crouch, J. K. Vrijling and H. G. Voortman, Probabilistic Design Tools for Vertical Breakwaters (Balkema, Rotterdam, 2001), 392 pp. 30. G. Cuomo, Dynamics of wave induced loads and their effects on coastal structures, PhD. dissertation, University of Roma TRE, Italy (2005). 31. G. N. Bullock, D. H. Peregrine, H. Bredmose, C. Obhrai, G. Walter and G. Muller, Characteristics and design implication of breaking wave impacts, Proc. 29th Int. Conf. Coastal Engineering (ASCE, 2004), pp. 3966–3978. 32. SEACO, Vision 2000-Performance-based seismic engineering of bridges (1995). 33. K. Shimosako and S. Takahashi, Application of deformation-based reliability design for coastal structures, American Society of Civil Enginners (ASCE), Coastal Structures ’99 (1999), pp. 363–371. 34. T. Takayama, S. Ikesue and K. Shimosako, Effect of directional occurrence distribution of extreme waves on composite breakwater reliability in sliding failure, Proc. 27th Int. Conf. Coastal Engineering (ASCE, 2000), pp. 1738–1750. 35. Y. Goda and H. Takagi, A reliability design method for caisson breakwaters with optimal wave heights, Coastal Engineering 42(4), 357–387 (2000). 36. M. Hanzawa, N. Yamagata, T. Nishihara, Y. Umezawa, T. Takayama and S. Takahashi, Performance of seawall against wave overtopping, American Society of Civil Engineers, Proc. 29th Int. Conf. Coastal Eng. (2004), pp. 4314–4325. 37. J. A. Melby and N. Kobayasi, Damage progression on breakwaters, Proc. 26th Int. Conf. Coastal Eng. (1998), pp. 1884–1897. 38. S. Takahashi, H. Kawai, T. Tomita and T. Takayama, Performance design concept for storm surge defenses, American Society of Civil Engineers, Proc. 29th Int. Conf. Coastal Eng. (2004), pp. 3074–3086.

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Chapter 19

Design of Alternative Revetments Krystian Pilarczyk (Former) Hydraulic Engineering Institute Public Works Department, Delft, The Netherlands HYDROpil Consultancy, 23 Nesciohove, 2726BJ Zoetermeer, The Netherlands [email protected] Revetments are most common structures in coastal engineering. The design of revetments is a complex process and needs proper understanding of loads and structural interactions. The basic information on composition and dimensioning of various types of revetments under wave and current attack are provided. Special attention is given to filter structures using geotextiles and transitions into splash areas and toe protection. Reference to actual developments and manuals is also provided.

19.1. Introduction There has been an increasing need in recent years for reliable information on design methodology and stability criteria of revetments exposed to wave and current action. The use of revetments, such as riprap, blocks and block mats, various mattresses, and asphalt in civil engineering practice is very common. The granular filters, and more recently the geotextiles, are more or less standard components of the revetment structure.6,14,20,22,23,26,28−30 However, the proper design rules are not always available. Within the scope of the research on the stability of open slope revetments, much knowledge has been developed about the stability of rock under wave and current load and stability of placed (pitched) stone revetments under wave load.5,7,8 Until recently, no or unsatisfactory design tools were available for a number of other (open) types of revetment and for other stability aspects. This is why the design methodology for placed block revetments has recently been extended in applicability by means of a number of desk-studies for other (open) revetments: • interlock systems and block mats; • concrete mattresses; • gabions; 479

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• geosystems, such as sandbags and sand mattresses; and • other stability aspects, such as flow-load stability, soil-mechanical stability, and residual strength. This chapter aims at giving a summary of the increased knowledge, especially that concerning the design tools that have been made available. The details behind it can be found in Refs. 9, 16, and 26.

19.2. Basics of Hydraulic Loading and Structural Response 19.2.1. Wave-load approach Waves attack on revetments, especially those with low permeability of cover layer, will lead to a complex flow over and through the revetment structure (filter and cover layer). During wave run-up the resulting forces by the waves will be directed opposite to the gravity forces. Therefore, the run-up is less hazardous than the wave run-down. The wave run-down will lead to two important mechanisms: • The downward flowing water will exert a drag force on the cover layer and the decreasing phreatic level will coincide with a downward flow gradient in the filter (or in a gabion). The first mechanism can be schematized by a free flow in the filter or gabion with a typical gradient equaling the slope angle. It may result in sliding. • During maximum wave run-down there will be an incoming wave that a moment later will cause a wave impact. Just before the impact, there is a “wall” of water giving a high pressure under the point of maximum run-down. Above the rundown point the surface of the revetment is almost dry and therefore there is a low pressure on the structure. The high pressure front will lead to an upward flow in the filter or a gabion. This flow will meet the downward flow in the rundown region. The result is an outward flow and uplift pressure near the point of maximum wave run-down (Fig. 19.1). The schematized situation can be quantified on the basis of the Laplace equation for linear flow. After complicated calculations, the uplift pressure in the filter or a gabion can be derived.9,26 The uplift pressure is dependent on the steepness and height of the pressure front on the cover layer (which is dependent on the wave height, period, and slope angle), the thickness of the cover layer, and the level of the phreatic line in the filter or a gabion. In case of riprap or gabions, it is not dependent on the permeability of the cover layer, if the permeability is much larger than the sublayer and/or subsoil. For semi-permeable cover layers the equilibrium of uplift forces and gravity forces (defined by components of a revetment) leads to the following (approximate) design formula26 :
0.67 D Hscr =f ∆D Λξop

 with Λ =

bDk k


(19.1a)

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Fig. 19.1.

or

bk Dk


and

or Hscr −0.67 = F ξop ∆D

481

Pressure development in a revetment structure.

 Λ/D =

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0.33 D k
Hscr −0.67 =f ξop ∆D b k


with maximum

Hs ∆D

(19.1b)

 cr

= 8.0 and ctgα ≥ 2,

(19.1c)

where √ Hscr = significant wave height at which blocks will be lifted out (m); ξop = tan α/ (Hs /(1.56Tp2)) = breaker parameter; Tp = wave period at the peak of the spectrum (s); Λ = leakage length (m), ∆ = (ρs − ρw )/ρw = relative volumetric mass of cover layer, with: ρs = density of the protection material and ρw = density of water (kg/m3 ); b = thickness of a sublayer/filter (m), D = thickness of a top (cover) layer (m), k = permeability of a sublayer (m/s), k
= permeability of a top layer (m/s), f = stability coefficient, mainly dependent on structure type, tan α and friction; F = total (black-box) stability factor. The leakage length (Λ) and stability coefficient (F ) are explained more in detail for specific applications in the next sections. There are two practical design methods available: the analytical model based on Eq. (19.1a) and the black-box model based on Eq. (19.1c). In both cases, the final form of the design method can be presented as a critical relation of the load (H) compared to strength (∆D), depending on the type of wave attack. More general form (also applicable for riprap and ctgα ≥ 1.5) is provided by Pilarczyk24,26 as: 
F cos α tan α Hs = , where ξop =  (19.2) b ∆D cr ξop Hs /Lop in which F = revetment (stability) factor, Hs = (local) significant wave height (m), ∆ = relative density, D = thickness of the top layer (m), ξop = breaker parameter

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(–), Lop = gTp2 /(2π), deep-water wavelength at the peak period (m), and Tp = wave period at the peak of the spectrum (s) and b = exponent; 0.5 ≤ b ≤ 1.0. For porous top layers, such as sand mattresses and gabions, the relative density of the top layer must be determined, including the water-filled pores: ∆=

ρs − ρw ρw

and ∆t = (1 − n) · ∆

(19.3)

in which ∆t = relative density including pores and n = porosity of the top layer material. D and ∆ are defined for specific systems such as: • for rock: D = Dn = (M50 /ρs )1/3 (= nominal diameter) and ∆t = ∆ = (ρs − ρw )/ρw , • for blocks: D = thickness of block and ∆t = ∆, • for mattresses: D = d = average thickness of mattress and ∆t = (1 − n)∆, where n = bulk porosity of fill material and ∆ = relative density of fill material. For sand and common quarry stone (1 − n) ∆ ∼ 1. The approximate values of stability factor F are: F = 2.25 for riprap, F = 2.5 for pitched stone of irregular shape and placed geobags, F = 3.0–3.5 for pitched basalt, F = 4.0 for cement geomattresses, 3.5 ≤ F ≤ 5.5 for block revetments (4.5 as an average/usual value), 4.0 ≤ F ≤ 6.0 for block mats (higher value for cabled systems), 6.0 ≤ F ≤ 8.0 for gabions, and 6.0 ≤ F ≤ 10 for (asphalt or concrete) slabs. Exponent b refers to the type of wave–slope interaction and its value is influenced by the roughness and the porosity of a revetment. The following values of exponent b are recommended: b = 0.5 for permeable cover layers (i.e., riprap, gabions, pattern grouted riprap, open block mats, and geobags), b = 2/3 for semipermeable cover layers (i.e., pitched stone and placed blocks, block mats, concreteor sand-filled geomattresses, and 2/3 < b ≤ 1.0 for slabs). The advantage of this black-box design formula is its simplicity. The disadvantage, however, is that the value of F is known only very roughly for many types of structures. The analytical model is based on the theory for placed stone revetments on a granular filter (pitched blocks). In this calculation model, a large number of physical aspects are taken into account. In short, in the analytical model, nearly all physical parameters that √ are relevant to the stability have been incorporated in the “leakage length”: Λ = (bDk/k
). The final result of the analytical model may, for that matter, again be presented as a relation such as Eq. (19.1) where F = f (Λ). A system without a filter layer is (directly on sand or clay and geotextile) not the permeability of the filter layer, but the permeability of the subsoil (eventually with gullies/surface channels) is filled in. For the thickness of the filter layer, it is examined to which depth changes at the surface affect the subsoil. One can fill in 0.5 m for sand and 0.05 m for clay. The values for D and ∆ depend on the type of revetment. In the case of a geotextile situated directly under the cover layer, the permeability of the cover layer decreases drastically. Since the geotextile is pressed against the cover layer by the outflowing water, it should be treated as a part of the cover layer. The water flow trough the cover layer is concentrated at the joints between the

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blocks, reaching very high flow velocities and resulting in a large pressure head over the geotextile. The presence of a geotextile may reduce k
by a factor 10 or more. To be able to apply the design method for placed stone revetments under wave load to other systems, the following items may be adapted: the the the the the

revetment parameter F ; (representative) strength parameters ∆ and D; design wave height Hs ; (representative) leakage length Λ; increase factor Γ (friction/interlocking between blocks) on the strength.

Only such-like adaptations are presented in this summarizing review. The basic formulae of the analytical model are not repeated here. For these, the reader is referred to Refs. 9 and 26. The wave attack on a slope can be roughly transformed into the maximum velocity component on a slope during run-up and run-down, Umax , by using the following formula:  (19.4) Umax = p gHs ξop where p ≈ 1.5 can be assumed as a first approximation. 19.2.2. Flow-load stability There are two possible approaches for determining the stability of revetment material under flow attack. The most suitable approach depends on the type of load: • flow velocity: “horizontal” flow, flow parallel to dike; • discharge: downward flow at slopes steeper than 1:10, overflow without waves; stable inner slope. The general velocity criterion can be expressed as5,8,25 : U2 = Λh Ψ, 2g∆D

(19.5)

where U = depth-average (mean) velocity, Ψ = Shields parameter, ∆ = relative density, g = acceleration of gravity (g = 9.81 m/s2 ), Λh = depth related flow resistance function; i.e., Λh = C 2 /2g for Chezy approach, C = 18 log(12h/ks ), Λh = 33(h/ks )1/3 for Strickler–Manning, Λh = 33(h/ks )0.2 for Pilarczyk25 (based on Neill’s data19 ). When the flow velocity is known, or can be calculated reasonably accurately, Pilarczyk’s relation is applicable24−26 : ∆D = 0.035

Φ KT Kh u2cr Ψ Ks 2g

(19.6)

in which D = characteristic thickness (m): for riprap D = Dn = nominal diameter as defined previously, Ucr = critical vertically averaged flow velocity (m/s), Φ = stability parameter, Ψ = Ψcr = critical Shields parameter KT = turbulence factor,

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Kh = depth parameter: Kh = 33/Λh, and Ks = slope parameter. These parameters are explained below. Stability parameter Φ The stability parameter Φ depends on the application. Some guide values are: Revetment type

Riprap and placed blocks Sand-filled units Block mats, gabions, washed-in blocks, Concrete-filled geobags, and geomattresses

Continuous toplayer

Edges and transitions

1.0

1.5

0.75

1.1

Shields parameter Ψcr With the critical Shields parameter Ψ, the type of material can be taken into account: • riprap, small regular bags Ψ ≈ 0.035 • placed blocks, geobags Ψ ≈ 0.05 • blockmats Ψ ≈ 0.05−0.07 (if washed-in) • gabions Ψ ≈ 0.07 • geomattresses Ψ ≈ 0.07 Turbulence factor K T The degree of turbulence can be taken into account with the turbulence factor KT . Some guide values for KT are: • Normal turbulence of rivers: KT ≈ 1.0 • Increased turbulence: river bends: KT ≈ 1.5 downstream of stilling basins: KT ≈ 1.5 • Heavy turbulence hydraulic jumps: KT ≈ 2.0 strong local disturbances: KT ≈ 2.0 sharp bends: KT ≈ 2.0 to 2.5 • Load due to water (screw) jet: KT ≈ 3.0 to 4.0 Depth parameter Kh With the depth parameter Kh , the water depth is taken into account, which is necessary to translate the depth-averaged flow velocity into the flow velocity just above the revetment. The depth parameter also depends on the development of the flow profile and the roughness of the revetment. The fully developed profile can be found in natural rivers. In case of local civil engineering works as bottom protection or slope protection, the nondeveloped profile is usually present. The following formulae

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Fig. 19.2.

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Kh factor as a function of relative roughness h/ks .

are recommended (see also Fig. 19.2): 2 • fully developed velocity profile: Kh =   2 log 12h ks −0.2
h • nondeveloped profile: Kh = 1 + ks 
h < 5 : Kh = 1.0 • very rough flow ks

(19.7a)

(19.7b) (19.7c)

in which h = water depth (m) and ks = equivalent roughness according to Nikuradse (m). In the case of dimensioning the revetment on a slope, the water level at the toe of the slope must be used for h. The equivalent roughness depends on the type of revetment/geosystem. For riprap, ks is equal usually to one or twice the nominal diameter of the stones; for bags, it is approximately equal to the thickness (d); for mattresses, it depends on the type of mattress: ks of about 0.05 m for smooth types and about the height of the rib for articulating mats. Note: Usually 12h/ks is applied, however, by using (1 + 12h/ks ) the discontinuity at small values of h can be avoided; the same adjustment is also applied for other velocity distributions. The effect of this additional (imaginary) depth practically vanishes for h/ks > 2.

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Slope parameter Ks The stability of revetment elements also depends on the slope gradient under which the revetment is applied, in relation to the angle of internal friction of the revetment. This effect on the stability is taken into account with the slope parameter Ks , which is defined as follows:   2 2

sin α tan α Ks = 1 − = cos α 1 − (19.8a) sin θ tan θ or Ks = cos αb ,

(19.8b)

where θ = angle of internal friction of the revetment material, α = transversal slope of the bank (◦ ), and αb = slope angle of river bottom (parallel along flow direction) (◦ ). The following values of θ can be assumed as a first approximation: 40◦ for riprap, 30◦ –40◦ for sand-filled systems, and 90◦ for stiff and anchored mortar-filled mattresses and (cabled) block mats (Ks = cos α). However, for flexible nonanchored mattresses and block mats (units without contact with the neighboring units) this value is much lower, usually about 3/4 of the friction angle of the sublayer. In case of geotextile mattress and block mats connected to geotextile lying on a geotextile filter, θ is about 15◦ –20◦. Pilarczyk’s formula was verified using Neill’s data (see Fig. 19.3). The advantage of this general design formula is that it can be applied in numerous situations. The disadvantage is that the scatter in results, as a result of the large margin in parameters, can be rather wide. With a downward flow along a steep slope and small h/ks values, it is difficult to determine or predict the flow velocity because the flow is very irregular. In such case formulae based on the discharge are developed.17,26

Fig. 19.3.

Comparison of Pilarczyk’s equation with the experimental data collected by Neill.19

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It should be noted that Pilarczyk formula can be transformed into similar form as Maynord formula, which is more or less a standard formula in the USA. Maynord formula4,18 : 


2.5 2.5 1/2 γw U U  ⇒ CM · h  D30 = Sf Cs Cv CT h γs − γw (Ks gh) (Ks g∆h) (19.9) where D30 = characteristic riprap size of which 30% is finer by weight, Sf = safety factor (minimum = 1.1), Cs = stability coefficient for incipient failure: = 0.30 for angular rock and = 0.375 for rounded rock, Cv = velocity distribution coefficient: = 1.0 for straight channels, 1 < Cv < 1.28 for river bends = 1.25 downstream of concrete channels, CT = blanket thickness coefficient (CT = 1 for blanket thickness t = 1.5D50 ), Ks = side slope correction factor, and h = local depth (use depth at 20% upslope from toe for side slopes). D30 can be translated (approximately) into D50 using gradation ratio of rock (D85 /D15 ) as: D50 = D30 (D85 /D15 )1/3 . At the first approximation one may assume: D30 = 0.75D50 ≈ 0.90Dn50. The minimum value of the total stability coefficent CM is about 0.3. Transformation Pilarczyk formula: 2.5 2.5

1.25


0.035ΦKT U U h  = CP · h  . Dn50 = 2Ψcr (Ks g∆h) (Ks g∆h) (19.10) The original minimum value of CP is about 0.3; however, for practical applications CP = 0.4 is recommended. Equation (19.10) has the same structure as Eq. (19.9) by Maynord. Comparison of the total stability coefficients CP and CM provides a similar result when using comparable input data and taking into account the difference between Dn50 and D30 . 19.2.3. Soil-mechanical stability The water motion on a revetment structure can also affect the subsoil, especially when this consists of sand. Geotechnical stability is dependent on the permeability and stiffness of the grain skeleton and the compressibility of the pore water (the mixture of water and air in the pores of the grain skeleton). Wave pressures on the top layer are passed on delayed and damped to the subsoil under the revetment structure and to deeper layers (as seen perpendicular to the slope) of the subsoil. This phenomenon takes place over a larger distance or depth as the grain skeleton and the pore water are stiffer. If the subsoil is soft or the pore water more compressible (because of the presence of small air bubbles), the compressibility of the system increases and large damping of the water pressures over a short distance may occur. Because of this, alternately, water under-tension and over-tension may develop in the subsoil and corresponding to this an increasing and decreasing grain pressure occurs. It can lead to sliding or slip circle failure; see Fig. 19.4.

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Fig. 19.4.

Schematized development of S-profile and possible local sliding in sand.

Fig. 19.5. Geotechnical stability; design diagram for block mats and wave steepness Hs / Lop = 0.03.

The design method with regard to geotechnical instability is presented in the form of design diagrams. An example is given in Fig. 19.5 (more diagrams and details can be found in Refs. 9, 26, and 27). The maximum admissible wave height is a function of equivalent thickness of cover layer defined as: Deq = D + b/∆t , where D is the cover layer thickness, b is the filter thickness, and ∆t is the relative system density of the cover layer (inclusive open area of blocks).

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19.2.4. Filters Granular and/or geotextile filters can protect structures subjected to soil erosion when used in conjunction with revetment armor such as riprap, blocks and block mats, gabions and mattresses, asphalt or concrete slabs, or any other conventional armor material used for erosion control.11,14,21,22 However, there is still a misunderstanding about the function of geotextiles in the total design of these structures, especially in comparison with the granular filters. In this section, the general principles of designing revetments incorporating granular or geotextiles are reviewed. Attention is paid to the replacing of a granular filter by a geotextile, which may often lead to geotechnical instability. Furthermore, it appears that a thicker granular filter gives larger geotechnical stability, but lower cover layer stability (uplift of blocks). The conclusion is therefore that the wave loads must be distributed (balanced) adequately over the sand (shear stress) and the cover layer (uplift pressure). Filters have two functions: erosion prevention and drainage. Traditional design criteria for filters are that they should be “geometrically tight” and that the filter permeability should be larger than the base (soil) permeability. However, it results in a large number of layers which are often unnecessary, uneconomical, and difficult to realize. In several cases, a more economical filter design can be realized using the concept of “geometrically open filters” (e.g., when the hydraulic loads/gradients are too small to initiate erosion). Recently, some criteria for “geometrically open” filters including geotextiles were developed (and are still under further development). However, the application of these criteria requires the knowledge/prediction of the hydraulic loads. In the cases when the erosion exceeds an acceptable level, a filter construction is a proper measure for solving this problem. In revetment structures, geotextiles are mostly used to protect the subsoil from washing away by the hydraulic loads, such as waves and currents. Here, the geotextile replaces a granular filter. Unfortunately, the mere replacing of a granular filter by a geotextile can endanger the stability of other components in the bank protection structure. The present section shows that designing a structure is more than just a proper choice of geotextile. Filter structures can be realized by using granular materials (i.e., crushed stone), bonded materials (i.e., sand asphalt, sand cement), and geotextiles, or a combination of these materials. The choice between the granular filter, a bonded filter, or geotextile depends on a number of factors. In general, a geotextile is applied because of easier placement and relatively lower cost. For example, the placement of granular filter underwater is usually a serious problem; the quality control is very difficult, especially when placement of thin layers is required. When designing with geotextiles in filtration applications, the basic concepts are essentially the same as when designing with granular filters. The geotextile must allow the free passage of water (permeability function) whilst preventing the erosion and migration of soil particles into the armor or drainage system (retention function). In principle, the geotextile must always remain more permeable than the base soil and must have pore sizes small enough to prevent the migration of the larger particles of the base soil. Moreover, concerning the permeability, not only

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the opening size but also the number of openings per unit area (Percent Open Area) is of importance.27 It has to be stressed that geotextiles cannot always replace the granular filter completely (see Sec. 19.2.3). A granular layer can often be needed to reduce (damp) the hydraulic loadings (internal gradients) to an acceptable level at the soil interface. After that, a geotextile can be applied to fulfill the filtration function. In respect to the filters for erosion control (granular or geotextile), the distinction can be made between: • geometrically tight filters, • geometrically open filters, and • transport filters (when a limited settlement is allowed). Only geometrically tight filters are discussed. For other type of filters, the reader is guided to Refs. 6, 26, and 27. 19.2.4.1. Design criteria for geometrically tight granular filters In this case, there will be no transport of soil particles from the base, independent of the level of hydraulic loading. That means that the openings in the granular filter or geotextile are so small that the soil particles are physically not able to pass the opening. This principle is illustrated in Fig. 19.6 for granular filters. The main design rules (criteria) for geometrically tight (closed) granular filters are summarized below. • Interface stability (also called “piping” criterion): Df 15 ≤ 4 to 5, Db85

(19.11)

where Df 15 is the grain size of the filter layer (or cover layer) which is exceeded by 15% of the material by weight in m; Db85 is the grain size of the base material (soil) which is exceeded by 85% of the material by weight in m.

Fig. 19.6.

Principles of geometrically tight filters.

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Sometimes, a similar equation is defined as: Df 50 < 6 to 10. Db50

(19.12)

However, Eq. (19.12) is generally less than Eq. (19.11) and can be used for “narrow” gradation only. Therefore, Eq. (19.11) is recommended for general use. • Internal stability In the case of very “wide” gradation, the situation requires an additional check with respect to the internal migration. In this respect, an important parameter is the so-called “uniformity coefficient” Cu , defined by Eq. (19.13) and the shape of the sieve curve: Cu =

Db60 , Db10

(19.13)

where Cu is the coefficient of uniformity. Usually, if this ratio is smaller than 6 (to 10), the soil or filter structure is considered internally stable. However, internal stability can be judged more carefully by the following rules [Eq. (19.14)]: D10 < 4D5 , D20 < 4D10 , D30 < 4D15 ,

(19.14b) (19.14c)

D40 < 4D20 .

(19.14d)

Df 15 > 5. Db15

(19.15)

(19.14a)

• Permeability criterion

19.2.4.2. Summary of design rules for geotextiles Current definitions for geotextile openings There are a large number of definitions of the characteristic of geotextile openings. Moreover, there are also different test (sieve) methods for the determination of these openings (dry, wet, hydrodynamic, etc.) which depend on national standards. These all make the comparison of test results very difficult or even impossible. That also explains the necessity of international standardization in this field. Some of the current definitions are listed below: O90 corresponds with the average sand diameter of the fraction of which 90% of the weight remains on or in the geotextile (or 10% passes the geotextile) after 5 min of sieving (method: dry sieving with sand). O98 corresponds with the average sand diameter of the fraction of which 98% of the weight remains on or in the geotextile after 5 min of sieving. O98 gives a practical approximation of the maximum filter opening and therefore plays an important role

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in the sand tightness criterion for a geotextile in strong cyclic loading situations. O98 is also referred to as Omax . Of filtration opening size (FOS). Of is comparable with O95 (hydrodynamic sieve method). AOS is the apparent opening size (acc. to ASTM method), also called EOS (effective opening size). The AOS is determined by sieving spherical glass particles of known size through a geotextile. The AOS, also frequently referred to as O95 (dry sieve method), is defined as a standard sieve size, x, mm, for which 5% or less of the glass particles pass through the geotextile after a specified period of sieving. Dw effective opening size which corresponds to the sand diameter of the fraction of which 10%, determined by the wet sieve method, passes through the geotextile. Dw is comparable with O95 . The transport of soil particles within a grain structure is possible when there is enough space and a driving force (groundwater pressure, hydraulic gradients within the soil). In most cases, it is the intention to prevent the transport of small-sized soil particles in the subsoil and therefore the term soil tightness is used and not the term space for transport or pore volume (in the case of the transport of water the terms pore volume and water permeability are used). The relation between pore magnitude and grain diameter can be characterized by: pore diameter ≈ 20% of the grain diameter. Just as for the characterization of the performance of a grain structure with regard to the transport of soil particles, for geosynthetics, too, the term soil tightness is used. As was mentioned before (Fig. 19.6), in a theoretical case when the soil is composed of spheres of one-size diameter, all spheres can be retained if all apertures in the geosynthetic are smaller than the diameter of the spheres. Usually, the soil consists of particles with different diameters and shapes, which is reflected in the particle-size distribution curves. Smaller particles can disappear straight across the geosynthetic by groundwater current. In this case, the retained soil structure can

Fig. 19.7.

Schematic representation of a natural filter with a soil-retaining layer.

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function as a natural filter; see Fig. 19.7. The better the soil particles are distributed, the better the soil tightness of the soil structure is effected. Smaller soil particles get stuck into the spaces between larger ones and the soil structure prevents the flow of fine particles. When certain particle-size fractions are lacking, the soil structure is not stacked very well and cavities develop through which erosion can occur. The displacement of soil particles not only depends on the soil tightness but also on the hydraulic gradient in the soil structure. Moreover, the dynamic effects due to heavy wave loading may not allow the forming of a natural filter, and the process of washing-out may continue. According to some researchers the forming of a natural filter is only possible for stationary flow.6 However, for small values of the hydraulic gradients, this is also possible for nonstationary flow. For heavy wave attack (i.e., exposed breakwaters, coastal revetments), this is usually not the case. In extreme situations, soil liquefaction is even possible. In such situations the soil particles can still reach the surface of a geotextile and be washed out. In order to judge the risk of wash-out of soil particles through the geosynthetics, some aspects have to be considered. An important factor is the internal stability of the soil structure. In the case of a loose particle stacking of the soil, many small soil particles may pass through the geosynthetic before a stable soil structure is developed near the geosynthetic. Also, a proper compaction of soil is very important for the internal stability of soil. The internal stability is defined by the uniformity coefficient Cu [see Eq. (19.13)]. In case of vibration, for instance, caused by waves or by traffic, the stable soil structures can be disturbed. To avoid such situations, the subsoil has to be compacted in advance and a good junction between geosynthetic and subsoil has to be guaranteed and possibly, a smaller opening of geotextile must be chosen. The shape of the sieve curve also influences the forming of a natural filter. Especially, when Cu > 6, the shape of the base gradation curve and its internal stability must be taken into account.27 For a self-filtering linearly graded soil, the representative size corresponds to the average grain size, Db50 . For a self-filtering gap graded soil, this size is equal to the lower size of the gap. For internally unstable soils, this size would be equivalent to Db30 in order to optimize the functioning of the filter system. It is assumed that the involved bridging process would not retrogress beyond some limited distance from the interface. Soil tightness With respect to the soil tightness of geotextiles, many criteria for geometric soil tightness have been developed and published in the past.2,6,10,12,15,20,27 An example of such design criteria, based on Dutch experience, is presented in Table 19.1. An additional requirement is that the soil should be internally stable. The internal stability of a grain structure is expressed in the ratio between Db60 and Db10 . As a rule, this value has to be smaller than 10 to guarantee sufficient stability. However, in many situations additional requirements will be necessary, depending on the local situation. Therefore, for design of geometrically tight geotextiles the method applied in Germany can be recommended.2,13,20,27 In this method a distinction is made between the so-called stable and unstable soils.

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Design requirements for geosynthetics with a filter and separation function.

Description

Filter function/soil tightness

• Stationary loading • Cyclic loading with natural filter (stable soil structure) • Cyclic loading without a natural filter (unstable soil structure) • When wash-out effects acceptable • When wash-out effects not acceptable

O90 ≤ 2Db90 O98 ≤ 1(to 2) Db85

Table 19.2.

O98 ≤ 1.5Db15 O98 ≤ Db15

Design criteria for geometrically soil-tight geotextiles. Soil type Db40 < 60µm

Stationary loading Dynamic loading

Db40 < 60µm

Stable soil

Unstable soil

Stable soil

Unstable soil

O90 < 10Db50 and O90 < 2Db90

O90 < 10Db50 and O90 < Db90 O90 < Db90 and O90 < 0.3 mm (300 µm)

O90 < 5Db10 Cu and O90 < 2Db90 O90 < Db90

1/2

O90 and O90 O90 and O90 O90

1/2

< 5Db10 Cu

< Db90 1/2 < 1.5Db10 Cu < Db50 < 0.5 mm

O90 is determined by wet sieve method.

Soils are defined as unstable (susceptible to down-slope migration) when the following specifications are fulfilled: • • • •

a proportion of particles must be smaller than 0.06 mm; fine soil with a plasticity index (Ip ) smaller than 0.15 (it is not a cohesive soil); 50% (by weight) of the grains will lie in the range 0.02 < Db < 0.1 mm; clay or silty soil with Cu < 15.

If Ip is unknown at the preliminary design stage, then the soil may be regarded as a problem soil if the clay size fraction is less than 50% of the silt size fraction. The design criteria are presented in Table 19.2.27 In the case of fine sand or silty subsoils, however, it can be very difficult to meet these requirements. A more advanced requirement is based on hydrodynamic sand tightness, viz. that the flow is not capable of washing out the subsoil material, because of the minor hydrodynamical forces exerted (although the apertures of the geotextile are much larger than the subsoil grains). Requirements concerning water permeability To prevent the formation of water pressure (uplift) in the structure, causing loss of stability, the geotextile has to be water permeable. One has to strive for the increase of water permeability of a construction in the direction of the water current. In

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the case of a riverbank protection, it means that the permeability of the geotextile has to be larger than the permeability of the soil on which the geotextile has to be applied. In the case of a dike slope or dike foundation, the geosynthetic is often applied on an impermeable layer of clay. Proper permeability of geotextiles is very important in respect to the stability of relatively less permeable cover layers as, for example, block and block mats. When a geotextile lies directly under the cover layer it considerably reduces the open area of the cover layer, and as a result the uplift forces increase (see example in Sec. 19.3.2). The water permeability of woven fabrics and nonwovens may decrease in the course of time owing to the fact that fine soil particles, which are transported by the groundwater flow from the subsoil, block the openings in the geotextile, or migrate into the pores of the geosynthetic (clogging). To prevent mineral clogging, the pore size of the geotextiles has to be chosen as large as possible; but, of course, this pore size has still to meet the requirements for soil tightness. The danger of clogging increases when the soil contains more than 20% of silt or in the case of gap-grading of a soil. On the other hand, there usually is no danger of clogging when the total hydraulic gradient (over the subsoil and geotextile together) is less than 3, or when the subsoil is well graded. In all situations it holds that the soil must be internally stable. For less critical situations no clogging can be expected if: • for Cu > 3: O95 /Db15 > 3; • for Cu < 3: criterion of internal stability of soil should be satisfied or geotextile with maximum opening size from soil-tightness criteria should be specified. In respect to the water permeability of geosynthetics/geotextiles, a distinction should be made between “normal to the interface” and “parallel to the interface.” For geotextile filters the permeability parallel to the interface, is of importance, while for drainage structures the permeability normal to the interface is of most importance. As a general design criterion for flow normal to the interface one can hold that the water permeability of a geosynthetic/geotextile has to be greater than that of the soil at the side from where the water flow comes. As a rule one can keep to: kgeotextile (filter) = ksoil × factor, where ks and kg are usually (basically) defined as permeability for laminar conditions. For normal (stationary) conditions and applications and clean sands a factor of 2 is sufficient to compensate the effect of blocking. If a geotextile is permeable with a factor of 10 more than the (noncohesive) subsoil, overpressure will usually occur, neither below the geosynthetic, nor in the case of reduced permeability caused by clogging or blocking. However, for special applications (i.e., for dam-clay cores with danger of clogging) this factor can be 50 or more.27,32

19.3. Stability Criteria for Placed Blocks and Block Mats 19.3.1. System description Placed block revetments (or stone/block pitching) are a form of protection lying between revetments comprised of elements which are disconnected, such as rubble, and monolithic revetments, such as asphalt/concrete slabs. Individual elements of

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Fig. 19.8.

Examples of block mats.

a pitched block revetment are placed tightly together in a smooth pattern. This ensures that external forces such as waves and currents can exert little drag on the blocks and also that blocks support each other without any loss of flexibility when there are local subsoil irregularities or settlement. A (concrete) block mat is a slope revetment made of (concrete) blocks that are joined together to form a “mat”; see Fig. 19.8. The interconnection may consist of cables from block to block, of hooks connecting the blocks, or of a geotextile on which the blocks are attached with pins, glue, or other means. The spaces between the blocks are usually filled with rubble, gravel, or slag. The major advantage of block mats is that they can to be laid quickly and efficiently and partly underwater. Block mats are more stable than a setting of loose blocks, because a single stone cannot be moved in the direction perpendicular to the slope without moving other nearby stones. It is essential to demand that already with a small movement of an individual stone a significant interactive force with the surrounding stones is mobilized. Large movements of individual blocks are not acceptable, because transport of filter material may occur. After some time, this leads to a serious deformation of the surface of the slope. The block mats are vulnerable at edges and corners. If two adjacent mats are not joined together, then the stability is hardly larger than that of pitched loose stones. 19.3.2. Design rules with regard to wave load The usual requirement that the permeability of the cover layer should be larger than that of the underlayers cannot be met in the case of a closed block revetment and other systems with low permeable cover layer. The low permeable cover layer introduces uplift pressures during wave attack. In this case, the permeability ratio of the cover layer and the filter, represented in the leakage length, is found to be the most important structural parameter, determining the uplift pressure. This is also the base of analytical model. The analytical model is based on the theory for placed stone revetments on a granular filter.9 In this calculation model, a large number of physical aspects are taken into account (see Fig. 19.1). In short, in the analytical model, nearly all physical parameters that are relevant to the stability have been incorporated in the “leakage length” factor. The final result of the analytical model may, for that matter, again be presented as a relation such as Eq. (19.1) or (19.2) where F = f (Λ). For

FA

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systems on a filter layer, the leakage length Λ is given by   bf Dk bf k or Λ/D = , Λ= k
Dk


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(19.16a)

where Λ = leakage length (m), bf = thickness of the filter layer (m), kf = permeability of the filter layer or subsoil (m/s), and k
= permeability of the cover layer (m/s). With a system without a filter layer (directly on sand or clay, without gullies being formed under the top layer) not the permeability of the filter layer, but the permeability of the subsoil (eventually with gullies/surface channels) is filled in. For the thickness of the filter layer, it is examined to which depth changes at the surface affect the subsoil. One can fill in 0.5 m for sand and 0.05 m for clay. The values for D and ∆ depend on the type of revetment. When schematically representing a block on a geotextile on a gully in sand, the block should be regarded as the top layer and the combination of the geotextile and the small gully as the filter layer (Fig. 19.9). The leakage length can be calculated using:  (kf dg + kg Tg )D Λ= , (19.16b) k
where kf = permeability of the filter layer (gully) (m/s), dg = gully depth (m), kg = permeability of the geotextile (m/s), Tg = thickness of the geotextile (m), D = thickness of the top layer (m), and k
= permeability of the top layer (m/s). In the case of a geotextile situated directly under the cover layer, the permeability of the cover layer decreases drastically. Since the geotextile is pressed against the cover layer by the outflowing water, it should be treated as a part of the cover layer. The water flow through the cover layer is concentrated at the joints between the blocks, reaching very high flow velocities and resulting in a large pressure head over the geotextile. The presence of a geotextile may reduce k
by a factor 10 or more (see Fig. 19.10). The leakage length clearly takes into account the relationship between kf and k
and also the thickness of the cover layer and the filter layer. For the theory behind this relationship, reference should be made to literature.9,16 The pressure head difference which develops on the cover layer is larger with a large leakage length

Fig. 19.9.

Schematization of a revetment with gully (cavity).

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Fig. 19.10.

Combined flow resistance determining the permeability of a system.

than with a small leakage length (effect of kf /k
in the leakage length formula). The effect of the leakage length on the dimensions of the critical wave for semi-permeable revetments is apparent from the following equations: 0.67

0.33 Hscr Hscr D D k
Hscr −0.67 −0.67 =f =f = F ξop ξop = , ∆D Λξop ∆D b k ∆D

(19.17)

where Hscr = √ significant wave height at which blocks will be lifted out (m); ξop = tan α/ (Hs /(1.56Tp2)) = breaker parameter; Tp = wave period (s); ∆ = relative volumetric mass of cover layer = (ρs − ρ)/ρ; f = stability coefficient mainly dependent on structure type and with minor influence of ∆, tan α, and friction, and F = total (black-box) stability factor. These equations indicate the general trends and have been used together with measured data to set up the general calculation model.9,26 This method works properly for placed/pitched block revetments and block mats within the following range: 0.01 < k
kf < 1 and 0.1 < D/bf < 10. Moreover, when D/Λ > 1 use D/Λ = 1, and when D/Λ < 0.01 use D/Λ = 0.01. The range of the stability coefficient is: 5 < f < 15; the higher values refer to the presence of high friction among blocks or interlocking systems. The following values are recommended for block revetments: f = 5 for static stability of loose blocks (no friction between the blocks), f = 7.5 for static stability of a system (with friction between the units), f = 10 for tolerable/acceptable movement of a system at design conditions. From these equations, neglecting the minor variations of “f
,” it appears that: • An increase in the volumetric mass, ∆, produces a proportional increase in the critical wave height. If ρb is increased from 2300 to 2600 kg/m3 , Hscr is increased by about 23%;

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• If the slope angle is reduced from 1:3 to 1:4 (tan α from 0.33 to 0.25), Hscr is increased by about 20% (due to the breaker parameter, ξop ); • An increase of 20% in the thickness of the cover layer, D, Hscr increases by about 27%; • A 30% reduction in the leakage length, Λ, Hscr increases by about 20%. This can generally be achieved by halving the thickness of the filter layer or by doubling the k
/kf value. The latter can be achieved by approximation by: — — — —

reducing the grain size of the filter by about 50%, or by doubling the number of holes in (between) the blocks, or by making hole sizes 1.5 times larger, or by doubling joint width between blocks.

Example: In 1983, the Armorflex mat on a slope 1:3 was tested on prototype scale at the Oregon State University: closed blocks with thickness D = 0.12 m and open area 10% on two types of geotextiles and very wide-graded subsoil (d15 = 0.27 mm, d85 = 7 mm). In the case of a sand-tight geotextile, the critical wave height (instability of mat) was only Hscr = 0.30 m. In the case of an open net geotextile (opening size about 1 mm), the critical wave height was more than 0.75 m (maximum capacity of the wave flume). The second geotextile was 20 times more permeable than the first one. This means that the stability increased by factor 200.33 = 2.7. In most cases, the permeability of the cover layer and sublayer(s) are not exactly known. However, based on the physical principles as described above, the practical “black-box” method has been established where parameter Λ and coefficient “f ” are combined to one stability factor “F ”. F depends on the type of structure, characterized by the product of the ratios of k
/kf and D/bf . With the permeability formulae9 it is concluded that the parameter (k
/kf ) × (D/bf ) ranges between 0.01 and 10, leading to a subdivision into three ranges of one decade each. Therefore, the following types are defined: (a) Low stability: (k
/kf )(D/bf ) < 0.05–0.1, (b) Normal stability: 0.05–0.1 < (k
/kf )(D/bf ) < 0.5–1, (c) High stability: (k
/kf )(D/bf ) > 0.5–1. For a cover layer lying on a geotextile on sand or clay, without a granular filter, the leakage length cannot be determined because the size of bf and k cannot be calculated. The physical description of the flow is different for this type of structure. For these structures there is no such a theory as for the blocks on a granular filter. However, it has been experimentally proved that Eq. (19.1) or (19.17) are also valid for these structures. It can be concluded that the theory has led to a simple stability formula [Eq. (19.17)] and a subdivision into four types of (block) revetment structures: • • • •

(a1) (a2) (a3) (a4)

cover cover cover cover

layer layer layer layer

on on on on

granular filter possibly including geotextile, low stability; granular filter possibly including geotextile, normal stability; geotextile on sand; clay or on geotextile on clay.

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Lower and upper value for F .

Type

Description

a1.1

Pitched irregular natural stones on granular filter Loose closed blocks/basalt on granular filter, low stability Loose blocks on granular filter, normal stability Loose blocks on geotextile on compacted sand/clay Linked/interlocked blocks on geotextile on good clay or on fine granular filter

a1.2 a2 a3 a4

Low F

High F

Usual F

2.0

3.0

2.5

3.0

5.0

3.5

3.5

6.0

4.5

4.0

7.0

5.0

5.0

8.0

6.0

The coefficient, F , is quantified for each structure type by way of fitting Eq. (19.1c) or (19.17) to the large collection of results of model studies from all over the world. Usually, only large-scale studies are used because both the waves and the wave induced flow in the filter should be well represented in the model. In the classification of structures according to the value of (k
D/kf bf ), the upper limit of (k
D/kf bf ) is 10 times the lower limit. Therefore, the upper limit of F of each structure type (besides a1.1) is assumed 100.33 = 2.14 times the lower limit, since F = f (k
D/kf bf )0.33 . A second curve is drawn with this value of F . In Table 19.3, all available tests are summarized and for each type of structure a lower and upper boundary for the value of F is given (see also an example in Fig. 19.11). The lower boundary gives with [Eq. (19.1c)] a stability curve below which stability is guaranteed. Between the upper and lower boundaries, the stability is uncertain. It depends on various unpredictable influences whether the structure will be stable or not. The upper boundary gives a curve above which instability is (almost) certain. The results for structure type a3 (blocks on geotextile on sand) may only be applied if the wave load is small [Hs < 1 or 1.5 m (max.)], or to structures with a subsoil of coarse sand (D50 > 0.3 mm) and a gentle slope (tan α < 0.25), because geotechnical failure is assumed to be the dominant failure mechanism (instead of uplift of blocks). A good compaction of sand is essential to avoid sliding or even liquefaction. For loads higher than H = 1.2 m, a well-graded layer of stone on a geotextile is recommended (e.g., layer 0.3–0.5 m for 1.2 m < H < 2.5 m). The results for structure type a4 can be applied on the condition that clay of high quality and with a smooth surface is used. A geotextile is recommended to prevent erosion during (long duration) wave loading. The general design criteria for geotextiles on cohesive soils are given by Pilarczyk.27 In the case of loose blocks, an individual block can be lifted out of the revetment with a force exceeding its own weight and friction. It is not possible with the cover layers with linked or interlocking blocks. Examples of the second type are: block mattresses, ship-lap blocks, and cable mats. However, in this case, high forces will be exerted on the connections between the blocks and/or geotextile. In the case of blocks connected to geotextiles (i.e., by pins), the stability should be treated as

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Fig. 19.11. Example of stability function for type a1.2 (loose closed blocks on granular filter); low stability. Table 19.4. Recommended values for the revetment parameter F for blockmats (the lower values refer to blocks connected to geotextile while the higher ones refer to cabled blocks). Type of revetment

F (−)

Linked blocks on geotextile on compacted sand or good clay For uncertain conditions/mediocre clay use Linked blocks on a granular filter:

5–6 4–5

— Favorable construction — Normal construction — Unfavorable construction

5–6 4–5 3–4

for loose blocks in order to avoid the mechanical abrasion of geotextiles by moving blocks. The lower boundary of stability of cabled mats can be increased by a factor of 1.25 (or 1.5, if additionally washed-in with granular material) in comparison with loose blocks. Such an increase of stability is only allowable when special measures are taken with respect to the proper connection between the mats. The upper boundary of stability (F = 8) remains the same for all systems. Application of this higher stability requires optimization of design (including application of geometrically open but stable filters and geotextiles).6,9 To be able to apply the design method for placed stone revetments under wave load to other semi-permeable systems, the following items may be adapted: the revetment parameter F , the (representative) strength parameters ∆ and D, the

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design wave height Hs , and the (representative) leakage length Λ. Table 19.4 gives an overview of usable values for the revetment stability parameter F in the blackbox model for linked blocks (block mats). The terms “favorable,” “normal,” and “unfavorable” refer to the composition of the granular filter and the permeability ratio of the top layer and the filter layer.9 In the case of fine granular filter and relatively permeable top layer, the total composition can be defined as “favorable.” In a case of very coarse granular layer and less permeable top layer, the composition can be defined as “unfavorable.” In a case of blocks connected to a geotextile and concrete-filled mattresses on a filter layer, the construction can be usually defined as between “unfavorable” and “normal,” and the stability factor F = 3.0–3.5 (max. 4.0) can be applied. For blockmats and permeable mattresses on sand F = 5 (max. 6.0) can be applied. The higher values can also be used in cases that the extreme design loading is not very frequent or when the system is (repeatedly) washed in by coarse material providing additional interlocking. This wide range of recommended values for F only gives a first indication of a suitable choice. Furthermore, it is essential to check the geotechnical stability with the design diagrams (see Fig. 19.5 and for a full set of diagrams see Refs. 26 and 27). 19.4. Stability Criteria for Concrete-Filled Mattresses 19.4.1. Concrete mattresses Characteristics of concrete mattresses are the two geotextiles with concrete or cement between them. The geotextiles can be connected to each other in many patterns, which results in a variety of mattress systems, each having its own appearance and properties. Some examples are given in Fig. 19.12.

Fig. 19.12.

Examples of concrete-filled mattresses.

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Fig. 19.13.

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Principles of permeability of Filter Point mattress.

The permeability of the mattress is one of the factors that determine the stability. It is found that the permeability given by the suppliers is often the permeability of the geotextile, or of the so-called Filter Points (Fig. 19.13). In both cases, the permeability of the whole mattress is much smaller. A high permeability of the mattress ensures that any possible pressure buildup under the mattress can flow away, as a result of which the uplift pressures across the mattress remain smaller. In general, with a subsoil of clay and silty sand, the permeability of the mattress will be higher than the permeability of the subsoil. Therefore, the water under the mattress can usually be discharged without excessive lifting pressures on the mattress. The permeability of the mattress will be lower than the permeability of the subsoil or sublayers if a granular filter is applied, or with sand or clay subsoil having an irregular surface (gullies/cavities between the soil and the mattress). This will result in excessive lifting pressures on the mattress during wave attack. 19.4.2. Design rules with regard to wave load The failure mechanism of the concrete mattress is probably as follows: • First, cavities will form under the mattress as a result of uneven subsidence of the subsoil. The mattress is rigid and spans the cavities.

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• With large spans, wave impacts may cause the concrete to crack and the spans to collapse. This results in a mattress consisting of concrete slabs which are coupled by means of the geotextile. • With sufficiently high waves, an upward pressure difference over the mattress will occur during wave run-down, which lifts the mattress (Fig. 19.1). • The pumping action of these movements will cause the subsoil to migrate, as a result of which an S-profile will form and the revetment will collapse completely. It is assumed that local settlement of the subsoil will lead to free spans of the concrete mattress. Then, the wave impact can cause the breaking of these spans, if the ratio of Hs /D is too large for a certain span length. A calculation method is derived on the basis of an empirical formula for the maximum wave impact pressure and the theory of simply supported beams. The collapsing of small spans (less than 1 or 2 m) is not acceptable, since these will lead to too many cracks. The empirical formula for the wave impact is16 : Fimpact = 7.2Hs2 tan α, ρg

(19.18)

where Fimpact = impact force per m revetment (N). Calculation has resulted in an average distance between cracks of only 10–20 cm for a 10-cm thick mattress and wave height of 2 m. This means that at such a ratio of Hs /D the wave impacts will chop the mattress to pieces. For a mattress of 15 cm thick and a wave height of 1.5 m, the crack distance will be in the order of 1 m. Apart from the cracks due to wave impacts, the mattress should also withstand the uplift pressures due to wave attack. These uplift pressures are calculated in the same way as for block revetments. For this damage mechanism the leakage length is important. In most cases, the damage mechanism by uplift pressures is more important than the damage mechanism by impact. The calculated representative values of the leakage length for various mattresses are presented in Table 19.5. The results of calculated stability for various values of leakage length (permeability) are presented in Fig. 19.14.

Table 19.5.

Estimated leakage length for concrete mattresses.

Mattress Leakage length Λ(m) Standard — FP FPM Slab Articulated (Crib) a Good

On sanda

On sandb

On filter

1.5 1.0 3.0 0.5

3.9 3.9 9.0 1.0

2.3 2.0 4.7 0.5

contact of mattress with sublayer (no gullies/cavities underneath). assumption: poor compaction of subsoil and presence of cavities under the mattress. b Pessimistic

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Fig. 19.14. Calculation results for concrete mattresses (Hs /∆D < 4 because of acceptable crack distance due to impacts on spans).

Taking into consideration the above failure mechanisms, the following design (stability) formula has been derived for the mattresses [similar to Eq. (19.1c)]:

Hs F Hs = 2/3 with = 4, (19.19) ∆D ∆D max ξop where D ∆ ρs F

= = = =

mass per m2 /ρs (which can be called Deffective or Daverage ), relative volumetric mass of the mattress (−) = (ρs − ρw )/ρw , volumetric mass of concrete (kg/m3 ), stability factor (see below).

For an exact determination of the leakage length, one is referred to the analytical model.9,16 However, besides the mattresses of a type as, for example, the tube mat (Crib) with relative large permeable areas, the other types are not very sensitive to the exact value of the leakage length. It can be recommended to use the following values of F in design calculations: F = 2.5 or (≤ 3) — for low-permeable mattresses on (fine) granular filter, F = 3.5 or (≤ 4) — for low-permeable mattress on compacted sand, F = 4.0 or (≤ 5) — for permeable mattress on sand or fine filter (Df 15 < 2 mm). The higher values can be applied for temporary applications or when the soil is more resistant to erosion (i.e., clay), and the mattresses are properly anchored.

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19.5. Stability of Gabions and Stone-Mattresses 19.5.1. Introduction Gabions are made of rectangular baskets of wire mesh, which are filled with stones. The idea of the protection system is to hold the rather small stones together with the wire mesh. Waves and currents would have easily washed away the small stones, but the wire mesh prevents this. A typical length of gabions is 3–4 m, a width of 1–3 m, and a thickness of 0.3–1 m. The gabions with small thickness (less then 0.5 m) and large length and width are usually called Reno-mattresses or stone-mattresses. An important problem of this protection system is the durability. Frequent wave or current attack can lead to a failure of the wire mesh because of the continuously moving grains along the wires, finally cutting through. Another problem is the corrosion of the mesh. Therefore, meshes with plastic coating or corrosion resistant steel are used. On the other hand, the system is less suitable where waves and currents frequently lead to grain motion. 19.5.2. Hydraulic loading and damage mechanisms Wave attack on gabions will lead to a complex flow over the gabions and through the gabions. During wave run-up the resulting forces by the waves will be directed opposite to the gravity forces. Therefore, the run-up is less hazardous than the wave run-down. Wave run-down, as it was already mentioned in Sec. 19.2, will lead to the following mechanisms: The downward flowing water will exert a drag force on top of the gabions and the decreasing phreatic level will coincide with a downward flow gradient in the gabions. During maximum wave run-down, there will be an incoming wave that a moment later will cause a wave impact. Just before the impact, there is a “wall” of water giving a high pressure under the point of maximum run-down. Above the run-down point the surface of the gabions is almost dry and therefore there is a low pressure on the gabions. The interaction of high pressure and low pressure is shown in Fig. 19.1. A simple equilibrium of forces leads to the conclusion that the section from the run-down point to the phreatic line in the filter will slide down: • if there is insufficient support from gabions below this section, • if the downward forces exceed the friction forces: (roughly) f < 2 tan α with: f = friction of gabion on subsoil and α = slope angle. From this criterion, we see that a steep slope will easily lead to the exceeding of the friction forces, and furthermore, a steep slope is shorter than a gentle slope and will give less support to the section that tends to slide down. Hydrodynamic forces, such as wave attack and current, can lead to various damage mechanisms. The damage mechanisms fall into three categories: (1) Instability of the gabions (a) The gabions can slide downwards, compressing the down slope mattresses. (b) The gabions can slide downwards, leading to upward buckling of the down slope mattresses.

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(c) All gabions can slide downwards. (d) Individual gabions can be lifted out due to uplift pressures. (2) Instability of the subsoil (a) A local slip circle can occur, resulting in an S-profile. (b) The subsoil can wash away through the gabions. (3) Durability problems (a) Moving stones can cut through the mesh. (b) Corrosion of the mesh. (c) Rupture of the mesh by mechanical forces (vandalism, stranding of ship, etc.).

19.5.3. Stability of gabions under wave attack An analytical approach of the development of the uplift pressure in the gabions can be obtained by applying the formulae for the uplift pressure under an ordinary pitched block revetment, with leakage length: Λ = 0.77D. With this relation the stability relations according to the analytical model are also applicable to gabions. Substitution of values, which are reasonable for gabions, in the stability relations provides stability function which indeed match the line through the measured points.16 After complicated calculations the uplift pressure in the gabions can be derived.16 The uplift pressure is dependent on the steepness and height of the pressure front on the gabions (which is dependent on the wave height, period, and slope angle), the thickness of the gabions, and the level of the phreatic line in the gabions. It is not dependent on the permeability of the gabions, if the permeability is larger than the subsoil. The equilibrium of uplift forces and gravity forces leads to the following (approximate) design formula: Hs −2/3 = F · ξop ∆D

with 6 < F < 9 for slope of 1:3 (tan α = 0.33)

(19.20a)

or, using Pilarczyk’s equation (19.2) with b = 2/3 and F = 9 as an upper-limit (Fig. 19.15):


Hs ∆D

 = cr

F cos α 9 cos α = 2/3 , b ξop ξop

(19.20b)

where Hs ∆ D F ξop Tp

= = = = = =

significant wave height of incoming waves at the toe of the structure (m), relative density of the gabions (usually: ∆ = ∆t ≈ 1), thickness of the gabion (m), stability factor, √ breaker parameter = tan α/ (Hs /(1.56Tp2) wave period at the peak of the spectrum (s).

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Fig. 19.15.

Summary of test results (Ashe1 ) and (Brown3 ) and upper-limit design curve.

For practical applications, F ≤ 7 is recommended in (19.20b) or F = 6 and b = 0.5 in Eq. (19.2) or (19.20b). It is not expected that instability will occur at once if the uplift pressure exceeds the gravity forces. On the other hand, the above result turns out to be in good agreement with the experimental results. The experimental verification of stability of gabions is rather limited. Small scale model tests have been performed by Ashe1 and Brown3 ; see Fig. 19.15.

19.5.4. Motion of filling material It is important to know if the filling material will start to move during frequent environmental conditions, because it can lead to rupture of the wire mesh. Furthermore, the integrity of the system will be affected if large quantities of filling material are moved. During wave attack the motion of the filling material usually only occurs if ξop < 3 (plunging waves). Based on the van der Meer’s formula for the stability of loose rock5,7,8 and the assumption that the filling of the gabion will be more stable than loose rock, the following criterion is derived (van der Meer formula with permeability factor: 0.1 < P < 0.2; number of waves: N < 5000; and damage level: 3 < S < 6): Hs F = , ∆f D f ξop

(19.21)

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where: ∆f = relative density of the stones in the gabions (usually: ∆f ≈ 1.65), Df = diameter of stones in the gabion (m), and F = stability factor, 2 < F < 5. In all situations the stone size must be larger than the size of the wire mesh in the basket; this defines the minimum size.

19.6. Scour and Toe Protection Toe protection consists of the armoring of the beach or bottom surface in front of a structure which prevents it from scouring and undercutting by waves and currents. Factors that affect the severity of toe scour include wave breaking (when near the toe), wave run-up and backwash, wave reflection, and grain size distribution of the beach or bottom materials. Toe stability is essential because failure of the toe will generally lead to failure throughout the entire structure. Toe scour is a complex process. Specific (generally valid) guidance for scour prediction and toe design based on either prototype or model results have not been developed as yet, but some general (indicative) guidelines for designing toe protection are given in Refs. 4, 5, 22, and 31. The maximum scour force occurs where wave downrush on the structure face extends to the toe and/or the wave breaks near the toe (i.e., shallow water structure). These conditions may take place when the water depth at the toe is less than twice the height of the maximum expected unbroken wave that can exist at that water depth. The width of the apron for shallow water structures with a high reflection coefficient, which is generally true for slopes steeper than about 1 on 3, can be planned based on the structure slope and the expected scour depth. The maximum depth of a scour trough due to wave action below the natural bed is about equal to the maximum expected unbroken wave at the site. To protect the stability of the face, the toe soil must be kept in place beneath a surface defined by an extension of the face surface into the bottom to the maximum depth of scour. This can be accomplished by burying the toe, when construction conditions permit, thereby extending the face into an excavated trench the depth of the expected scour. Where an apron must be placed on the existing bottom, or can only be partially buried, its width should not be less than twice the wave height. If the reflection coefficient is low (slopes milder than 1 on 3), and/or the water depth is more than twice the wave height, much of the wave force will be dissipated on the structure face and a smaller apron width may be adequate, but it must be at least equal to the wave height (minimum requirement). Since scour aprons generally are placed on very flat slopes, quarrystone of the size (diameter) equal to 1/2 of the primary cover layer probably will be sufficient unless the apron is exposed above the water surface during wave action. Quarrystone of primary cover layer size may be extended over the toe apron if the stone will be exposed in the troughs of waves, especially breaking waves. The minimum thickness of cover layer over the toe apron should be two quarrystones. Quarrystone is the most favorable material for toe protection because of its flexibility. If a geotextile is used as a secondary layer it should be folded back at the end, and then buried in cover stone and sand to form a Dutch toe. It is recommended to provide an additional flexible

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Fig. 19.16.

Alternative toe protections.

edge (at least 1 m) consisting of loose material which may easily follow the scour at the toe. Some alternative designs of toe protection are shown in Fig. 19.16. The size of toe protection against waves can be roughly estimated by using the common formulae on slope protection and schematizing the toe by mild slopes (i.e., 1 on 10) or by using formulae developed for breakwaters.

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Toe protection against currents may require smaller protective stone, but wider aprons. The necessary design data can be estimated from site hydrography and/or model studies. Special attention must be given to sections of the structure where scour is intensified; i.e., to the head, the areas of a section change in alignment, the channel sides of jetties, and the downdrift sides of groynes. Where waves and reasonable currents (>1 m/s) occur together, it is recommended to increase the cover size at least by a factor of 1.3. Note that the conservatism of the apron design (width and size of cover units) depends on the accuracy of the methods used to predict the waves and current action and to predict the maximum depth of scour. For specific projects, a detailed study of scour of the natural bottom and at nearby similar existing structures should be conducted at a planned site, and/or model studies should be considered before determining a final design. In all cases, experience and sound engineering judgment play an important role in applying these design rules.

19.7. Protection Against Overtopping If a structure (revetment) is overtopped, even by minor splash, the stability can be affected. Overtopping can: (a) erode the area above or behind the revetment, negating the structure’s purpose; (b) remove soil supporting the top of the revetment, leading to the unraveling of the structure from the top down; and (c) increase the volume of water in the soil beneath the structure, contributing to drainage problems. The effects of overtopping can be limited by choosing a higher crest level or by armoring the bank above or behind the revetment with a splash apron. For a small amount of overtopping, a grassmat on clay can be adequate. The splash apron can be a filter blanket covered by a bedding layer and, if necessary, by riprap, concrete units, or asphalt. No definite method for designing against overtopping is known due to the lack of the proper method on estimating the hydraulic loading. Pilarczyk (1990) proposed the following indicative way of design of the thickness of protection of the splash area (Fig. 19.17): Hs 1.5 cos αi  , = Rc ∆Dn ΦT ξ 2b 1 − R n

(19.22)

where Hs = significant wave height, ξ = breaker index; ξ = tan α(Hs /Lo )−0.5 , α = slope angle, αi = angle of crest or inner slope, L0 = wavelength, b = coefficient equal to 0.5 for smooth slopes and 0.25 for riprap, Rc = crest height above still water level, Rn = wave run-up on virtual slope with the same geometry (see Fig. 19.17),

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Fig. 19.17.

Definition of splash area.

D = thickness of protective unit (D = Dn for rock), and ΦT = total stability factor equal to 1.0 for rock, 0.5 for placed blocks, and 0.4 for block mats. The length of protection in the splash area, which is related to the energy decay, depends on the permeability of the splash area. However, it can be roughly assumed as equal to: Ls =

ψ  T g(Rn − Rc ) ≥ Lmin , 5

(19.23)

with a practical minimum (Lmin ) equal at least to the total thickness of the revetment (including sublayers) as used on the slope. ψ is an engineering-judgment factor related to the local conditions (importance of structure), ψ ≥ 1, usually 1 < ψ < 2. Stability of rockfill protection of the crest and rear slope of an overtopped or overflowed dam or dike can also be approached with the Knauss formula.17 The advantage of this approach is that the overtopping discharge, q, can be used directly as an input parameter for calculation. Knauss analyzed steep shute flow hydraulics (highly aerated/turbulent) for the assessment of stone stability in overflow rockfill dams (impervious barrages with a rockfill spillway arrangement). This kind of flow seems to be rather similar to that during high overtopping. His (simplified) stability relationship can be re-written to the following form: √ q = 0.625 g(∆Dn )1.5 (1.9 + 0.8Φp − 3 sin αi ),

(19.24)

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where q = maximum admissible discharge (m3 /s/m), g = gravitational acceleration (9.81 m/s2 ), Dn = equivalent stone diameter, Dn = (M50 /ρs )1/3 , ∆ = relative density; ∆ = (ρs − ρw )/ρw , αi = inner slope angle, and Φp = stone arrangement packing factor, ranging from 0.6 for natural dumped rockfill to 1.1 for optimal manually placed rock; it seems to be reasonable to assume Φp = 1.25 for placed blocks. Note: When using the Knauss formula, the calculated critical (admissible) discharge should be identified with a momentary overtopping discharge per overtopping fraction of a characteristic wave, i.e., volume of water per characteristic wave divided by overtopping time per wave, roughly (0.3–0.4)T (T = wave period), and not with the time-averaged discharge (q).

19.8. Joints and Transitions Despite a well-designed protective system, the construction is only as strong as the weakest section. Therefore, special care is required when designing transitions. In general, slope protection of dike or seawall consists of a number of structural parts such as: toe protection, main protection in the area of heavy wave and current attack, upper slope protection (very often grass mat), berm for run-up reduction, or as maintenance road. Different materials and different execution principles are usually applied for these specific parts. Very often a new slope protection has to be connected to an already existing protective construction which involves another protective system. To obtain a homogeneous strong protection, all parts of protective structures have to be taken under consideration. Experience shows that erosion or damage often starts at joints and transitions. Therefore, important aspects of revetment constructions, which require special attention are the joints and the transitions; joints onto the same material and onto other revetment materials, and transitions onto other structures or revetment parts. A general design guideline is that transitions should be avoided as much as possible, especially in the area with maximum wave attack. If they are inevitable, the discontinuities introduced should be minimized. This holds for differences in elastic and plastic behavior and in the permeability or the sand tightness. Proper design and execution are essential in order to obtain satisfactory joints and transitions. When these guidelines are not followed, the joints or transitions may influence loads in terms of forces due to differences in stiffness or settlement, migration of subsoil from one part to another (erosion), or strong pressure gradients due to a concentrated groundwater flow. However, it is difficult to formulate more detailed principles and/or solutions for joints and transitions. The best way is to combine the lessons from practice with some physical understanding of systems involved. Examples to illustrate the problem of transitions are given in Fig. 19.18.

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Fig. 19.18.

Transitions in revetments.

As a general principle, one can state that the transition should be of strength equal to or greater than the adjoining systems. Very often it needs reinforcement in one of the following ways: (a) increase the thickness of the cover layer at the transition, (b) grout riprap or block cover layers with bitumen, and (c) use concrete edge strips or boards to prevent damage progressing along the structure. Top edge and flank protection are needed to limit the vulnerability of the revetment to erosion continuing around its ends. Extension of the revetment beyond the point of active erosion should be considered but is often not feasible. Care should therefore be taken that the discontinuity between the protected and unprotected areas is as small as possible (use a transition roughness) so as to prevent undermining. In some cases, open cell blocks or open block mats (eventually vegetated) can be used as transition (i.e., from hard protection into grass mat). The flank protection between the protected and unprotected areas usually needs a thickened

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Fig. 19.19. Construction aspects of revetments; examples of toe protection and placing of block mats (mattresses), and some methods of anchoring.

or grouted cover layer, or a concrete edge strip with some flexible transition, i.e., riprap. 19.9. General Construction (Execution) Aspects Revetments are constructed in a number of phases, for example: — construction of the bank/dike body, — placement of toe structure,

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— placement of revetment sublayers (clay and/or filter layers), — laying the blocks or mattress, — anchoring the mattress and, possibly, applying the joint filler. A well-compacted slope is important in order to produce a smooth surface and thus ensure that there is a good connection between the mattress and the subsurface. When laying mattresses on banks, it is strongly recommended that they are laid on undisturbed ground and that areas excavated too deeply are carefully refilled. Before using a geotextile, the slope must be carefully inspected for any projections which could puncture the material. When laying a mattress on a geotextile, care must be taken to ensure that extra pressures are not applied and that the geotextile is not pushed out of place. Geotextile sheets must be overlapped and/or stitched together with an overlap of at least 0.5–1.0 m to prevent subsoil being washed out. This is particularly important if the mattress is laid directly on sand or clay. Block mattresses are laid using a crane and a balancing beam. The mattress must be in the correct position before it is uncoupled because it is difficult to pick up again and also time-consuming. Provided that part of the mattress can be laid above the water line, it can generally be laid very precisely and joints between adjacent mattresses can be limited to 1–2 cm. Laying a mattress completely under water is much more difficult. The spacing between the blocks of adjacent mattresses, nonetheless, should never be more than 3 cm. Once in place, mattresses should be joined so that the edges cannot be lifted/turned up under the action of waves. Loose corners are particularly vulnerable. In addition, the top and bottom edges of the revetment should be anchored, as shown in Fig. 19.19. In such a case, a toe structure is not needed to stop mattresses sliding. More information on execution aspects of revetments can be found in Refs. 4, 5, 7, 8, 9, and 31

19.10. Conclusions The newly derived design methods and stability criteria will be of help in preparing the preliminary alternative designs with various revetment systems. However, there are still many uncertainties in these design methods. Therefore, experimental verification and further improvement of design methods are necessary. Also, more practical experience at various loading conditions is still needed.

References 1. G. W. T. Ashe, Beach erosion study, gabion shore protection, Hydraulics Laboratory, Ottawa, Canada (1975). 2. BAW, Code of practice: Use of geotextile filters on waterway, Bundesanstalt f¨ ur Wasserbau, Karlsruhe, Germany (1993). 3. C. Brown, Some factors affecting the use of Maccaferi gabions, Water Research Lab. Australia, Report 156 (1979). 4. CEM, Coastal Engineering Manual (US Army Corps of Engineers, Vicksburg, 2006).

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5. CIRIA-CUR-CETMEF, The Rock Manual (CIRIA, London, 2007). 6. CUR, Filters in Hydraulic Engineering (Civil Engineering Research and Codes (CUR), Gouda, the Netherlands, 1993) (in Dutch). 7. CUR/CIRIA, Manual on Use of Rock in Coastal Engineering, CUR/CIRIA Report 154 Gouda, the Netherlands (1991). 8. CUR/RWS, Manual on Use of Rock in Hydraulic Engineering, CUR Report 169, Gouda, the Netherlands (1995). 9. CUR/TAW, Design Manual for Pitched Slope Protection, CUR Report 155 (A.A. Balkema, Rotterdam, 1995). 10. DVWK, Guidelines for water management no. 306: Application of geotextile in hydraulic engineering, German Association for Water Resources and Land Improvement (DVWK), Bonn, Germany (1993). 11. Flexible Armoured Revetments, Proceedings of the International Conference, Thomas Telford Ltd., London (1984). 12. FHWA, Geosynthetics Design and Construction Guidelines, Federal Highway Administration, FHWA-HI-95-038, Washington, DC (1995). 13. G. Heerten, Dimensioning the filtration properties of geotextiles considering long-term conditions, Proc. 2nd Int. Conf. on Geotextiles, Las Vegas (1982). 14. G. Heerten, Geotextiles in coastal engineering, 25 years experience, Geotex. Geomemb. 1(2), 119–141 (1984). 15. R. D. Holtz, B. R. Christopher and R. R. Berg, Geosynthetic Engineering (BiTech Publishers Ltb., Richmond, Canada, 1997). 16. M. K. Breteler, K. W. Pilarczyk and T. Stoutjesdijk, Design of alternative revetments, Proc. 26th Int. Conf. on Coastal Engineering, Copenhagen (1998); http://www. wldelft.nl/rnd/publ/search.html (insert for author: Breteler or Pilarczyk). 17. J. Knauss, Computation of maximum discharge at overflow rock-fill dams, 13th Congress des Grand Barrages (ICOLD), New Delhi, Q50, R.9 (1979). 18. S. T. Maynord, Corps riprap design guidance for channel protection, in River, Coastal and Shoreline Protection, eds. R. Thorne Colin et al. (John Wiley & Sons, Chichester, UK, 1995). 19. C. R. Neill, Stability of coarse bed material in open channel flow, Edmonton (also, IAHR Congress, Fort Collins) (1967). 20. PIANC, Guidelines for the design and construction of flexible revetments incorporating geotextiles for inland waterways, Report WG 4, PTC I, Supplement to Bulletin No. 57, Brussels, Belgium (1987). 21. PIANC Bulletin, Special issue on propeller jet action, erosion and stability criteria near the harbour quays, Pianc Bulletin no. 58, Brussels, Belgium (1987). 22. PIANC, Guidelines for the design and construction of flexible revetments incorporating geotextiles in marine environment, Report WG 21, PTC II, Supplement to Bulletin No. 78/79, Brussels, Belgium (1992). 23. PIANC, Guidelines for the design of armoured slopes under piled quay walls, Supplement to Bulletin No. 96, Brussels, Belgium (1997). 24. K. W. Pilarczyk (ed.), Coastal Protection (A.A. Balkema, Rotterdam, 1990), www.enwinfo.nl (select English, downloads). 25. K. W. Pilarczyk, Simplified unification of stability formulae for revetments under current and wave attack, in River, Coastal and Shoreline Protection, eds. R. Thorne Colin et al. (John Wiley & Sons, Chichester, UK, 1995). 26. K. W. Pilarczyk (ed.), Dikes and Revetments (A.A Balkema, Rotterdam, 1998). 27. K. W. Pilarczyk, Geosynthetics and Geosystems in Hydraulic and Coastal Engineering (A.A. Balkema, Rotterdam, 2000), www.balkema.nl.

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28. K. Pilarczyk, International perspectives on coastal structures uses, in Advances in Coastal Structure Design, eds. K. Mohan Ram et al. (ASCE, 2003). 29. EAU, Recommendations of the committee for waterfront structures, German Society for Harbour Engineering (Ernst & Sohn, Berlin, 2000). 30. RWS, The Closure of Tidal Basins, Rijkswaterstaat (The Delft University Press, Delft, 1987). 31. SPM, Shore Protection Manual, U.S. Army Corps of Engineers, Vicksburg (1984). 32. G. Van Santvoort (ed.), Geotextiles and Geomembranes in Civil Engineering, revised edition (A.A. Balkema, Rotterdam, 1994).

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Appendix A. Comparative Stability of Revetments (Pilarczyk, 1990, Coastal Protection) Limits φ(rock) = 2.25 ctgα ≥ 2

Criterion Hs cos α cos α = ψu · φ b = ψu 2.25 b ∆m D ξp ξp

0.5 ≤ b ≤ 1.0 Sublayer

Riprap (2 layers) Riprap (tolerable damage)

Granular Granular

Pitched Stone

1.00 1.33 1.50

Poor quality (irregular-)stone Good quality (regular-)stone Natural basalt

Granular Granular Granular

Loose closed blocks; Hs < 1.5 m Loose (closed-)blocks Blocks connected to geotextile Loose closed blocks Cabled blocks/Open blocks (>10%) Grouted (cabled-) blocks/Interlocked blocks adequately designed Surface grouting (30% of voids) Pattern grouting (60% of voids)

Geotextile on sand Granular Granular Geotextile on clay Granular Granular

Blocks/Blockmats

Grout

1.50 1.50 1.50 2.00 2.00 ≥2.50 1.05 1.50

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Appendix A. (Continued ) Limits φ(rock) = 2.25 ctgα ≥ 2

Gabions Fabric Containers Grass

2.00 2.50 2 + 3.0 2 + 2.5 1.00 1.50 2.00

Open stone asphalt; Up ≤ 6 m/s Open stone asphalt; Hs < 4 m

Geotexile on clay Sand asphalt

Gabion/mattress as a unit, Hs < 1.5 Stone-fill in a basket, dmin = 1.8 Dn Pm  1 less permeable mattress Pm ≈ 1 (Pm = ratio permeab. top/sublayer Pm
2 Permeable mattress of special design Grass-mat on poor clay; Up < 2 m/s Grass-mat on proper clay; Up < 3 m/s/s

Geotexile on sand or Geotextile on clay Sand or Clay and ev. Geotextile

Clay (Up = permiss. velocity)

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Chapter 20

Remarks on Coastal Stabilization and Alternative Solutions Krystian Pilarczyk (Former) Hydraulic Engineering Institute Public Works Department, Delft, The Netherlands HYDROpil Consultancy, 23 Nesciohove 2726BJ Zoetermeer, The Netherlands [email protected] Erosion control and coastal stabilization are common problems in coastal engineering. A brief overview of some available alternative systems for shore stabilization and beach erosion control is presented. Special attention is paid to artificial reefs and geosystems. Geosystems (geotubes, geocontainers, etc.) have gained popularity in recent years because of their simplicity in placement, cost effectiveness, and environmental aspects. However, all these systems have some advantages and disadvantages, which have to be recognized before application.

20.1. Introduction Some coastal environments may be regarded as rather stable (rock and reef coasts) while others are more vulnerable (sand and mud coasts, soft cliffs). In this last case, coastal users and managers all over the world are frequently faced with serious erosion of their sandy coasts. Possible causes of erosion include natural processes (i.e., action of waves, tides, currents, sea level rise, etc.) and sediment deficit due to human impact (i.e., river regulations, sand mining, and coastal engineering works). Countermeasures for beach erosion control function depend on local conditions of shore and beach, coastal climate, and sediment transport. Continuous maintenance and improvement of the coastlines, together with monitoring and studies of coastal processes have yielded considerable experience on various coastal protection measures all over the world. This contribution presents an overview of the various available methods for shore stabilization and beach erosion control, with special emphasis on the novel/alternative systems in various design implementations. More detailed information on other coastal protection systems and measures applied nowadays throughout the world can be found in extensive list of references.15,16,51,52,54,80 521

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20.2. General Approach 20.2.1. Types and functions of coastal structures In general, a coastal structure is planned as a practical measure to solve an identified problem. Starting with identification of the problem (e.g., shoreline erosion), a number of stages can be distinguished in the design process for a structure: definition of functions, determination of boundary conditions, creating alternatives, geometrical design, and the final choice of functional solution. After the choice of functional solution has been made the structural design starts including creating structural alternatives (i.e., using different materials and various execution methods). The final choice will be made after verification of various structural solutions in respect to the functional, environmental, and economic criteria. Coastal management, in its broadest sense, must take into account all factors, which bearing on the future of the coastal zone. Politics, policy making, planning, economy, and a multitude of economic, and noneconomic users (e.g., natural reserve areas), environment protection and, by and large, sustainable development, all play significant roles and provide both motivation and background for coastal management. The factors of coastal management may well entail many scientific and engineering disciplines other than coastal engineering and at sites and locations far removed from the costal zone. Some of these factors interact with one another; others are incompatible. The extent to which this applies in any particular region, area, or specific site needs careful evaluation and compromise solutions. This is one of the major roles of coastal management. The primary objectives of a typical coastal management study are to formulate long-term engineering planning, including financial strategies for the future usage, development, and conservation of the coastal zone. In this process, priorities should be defined both for new works and essential maintenance, with estimates for contingency items to cover emergency situations, which inevitably occur (see Fig. 20.1). The key element in any coastal management study is thorough understanding of coastal processes by which is meant the interaction between the hydraulic environment of winds, waves, tides, surges, and currents with the geological conditions in the coastal zone. To be effective, this may require a very broad view to be taken on a regional basis in the first instance. A regional cell could then be subdivided into smaller cells once the basic coastal processes had been established, and so on, with decreasing cell sizes until the cell in question becomes the specific one of the project itself. It is only in this way that the impact of new works in the coastal zone can be evaluated satisfactorily or long-term planning undertaken. The basic tools of the coastal engineering are still fairly limited and comprise cross-shore structures (such as groins, jetties spurs, etc.) shore-parallel structures (offshore breakwaters, seawalls, revetments generally close to shoreline), and dikes, headland structures, and artificial beach nourishment (see Fig. 20.2). • Groins generate considerable changes in wave and circulation patterns but their basic function — to slow down the rate of littoral drift — is sometimes overlooked. In the absence of beach nourishment, groins can redistribute the existing supply and, in a continuous littoral system, may be expected to create a deficiency at

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Analysis of coastal strategies.

the downdrift end where the uncontrolled drift rate is re-established. Without an adequate supply of beach material, groins are of limited value. In addition to controlling the rate of drift, groins are also used extensively to control the distribution of material along a frontage and to limit the temporary effects of drift reversal. There are unfortunately many examples where either bad design or failure to provide for the downdrift consequences has resulted in an adverse effect

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Fig. 20.2.

Schematic presentation of various shore protection measures.

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on the coastline. In other instances, failure to maintain groin systems might be more worse than having no groins at all. Offshore breakwaters are usually provided either to reduce wave energy at shoreline structures or to modify the wave climate and enhance sediment transport patterns so as to improve beach levels and create desirable beach features, such as salients.17,22 Offshore breakwaters can be shore-connected or detached, submerged or emerging, alongshore or oblique. Perched beach is a system consisting of a submerged breakwater (“sill” or “reef”), usually located not far away from shoreline, and artificial beach nourishment providing sand to the area extending between sub-aerial beach and the sill crest. Seawall (wording sometimes used interchangeably with bulkhead) is either a retaining wall intended to hold or prevent sliding of the soil behind it or a massive structure whose primary purpose is to protect the backshore from heavy wave action. Sometimes one speaks of “beach wall” or “shore wall.”15,16,52,53 Revetment is placed on a slope to shelter the adjacent uplands from erosion. Wave reflection, a serious disadvantage of vertical-wall bulkheads (seawalls), does not accelerate toe erosion as strongly at revetments as it does at seawalls.51 Dikes are generally intended as means of flood prevention. The crest of a dike is elevated high enough to counteract or confine overtopping in rare storm surge events.53,55,56 Beach nourishment or fill (or recharge), consists of importation of granular material to beach from an external source.17,53 It is not new, and has been used in some countries for decades, but is now being applied to an increasing extent and in a greater variety of ways. The resulting beach provides some protection to the area behind it and also serves as a valuable recreational resource. The beach fill functions as an eroding buffer zone, and its useful life will depend on how quickly it erodes. One must be prepared to periodically renourish (add more fill) if erosion continues. Headland control has been devised by analogy to the Nature’s efforts to keep in equilibrium a certain crenulate shape of erosion bays sculptured for thousands or so years. The crenulate shaped bays can be kept in equilibrium by use of a system of headlands. The headland system is claimed to be in feedback with coast and to combine the advantages of groins and detached breakwaters (shore-parallel or oblique).64

All forms of shore protection (i.e., groins, breakwaters, seawalls, revetments, bulkheads, beach-fill, etc.) have certain advantages and disadvantages. A shoreparallel (or oblique) breakwater, placed near the shoreline or offshore and designed either to intercept a portion of moving sediment or to protect a placed beach-fill, has the potential to perform close to the ideal for many types of coastal environments. A number of innovative solutions are given by Silvester and Hsu.64 Recent experience with design of beach stabilization structures is reviewed by Bodge.12 Various low-cost, environment-friendly, emergency and temporary measures, and their combinations provide alternatives to the principal measures. These systems are often appropriate for application only in sheltered waters. Inherent in the concept of environmental friendliness and low cost is the assumption on the equal importance

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of physics, engineering, ecology, and economy. The approach in this contribution is to provide a background for some of these alternatives. 20.2.2. Functional requirements and design It is common knowledge that the realm of coastal processes and the interactions of offshore breakwaters and beaches can be broadly classified as far-field and nearfield phenomena. The scales are somehow arbitrary but can be roughly identified as those greater and smaller, respectively, than characteristic dimensions of a structure or coastal feature. The design procedures for coastal structures should embody functional/ geometrical design and structural design reflecting, respectively, the far-field and near-field requirements imposed on structures. This corresponds to the division of design procedures into two basic groups concentrating on the overall layout and configuration of a structure as a whole, and its interaction with the coastal environment to produce desirable sedimentation patterns and coastal management effects stability and reliability of the structure and its components, hence dimensioning of structural constituents, associated with possible unavoidable and undesirable hazards due to the loadings exerted by the coastal environment. In other words, the first group involves design parameters producing the best environmental effectiveness of a structure in “ideal” conditions, i.e., upon negligence of possible “harmful by-effects,” such as different modes of failures and instabilities, both overall and internal. The second group is concerned about these “by-effects” and provides the tools, which secure the integrity and proper operation of the structure and its components. As in many other engineering activities, the design of coastal structures should encompass the following considerations and stages (see also Fig. 20.3): (a) (b) (c) (d) (e) (f) (g) (h) (i) (j)

specification of the structure’s function(s); description of the physical environment (boundary conditions); selection of envisaged construction technologies; inclusion of design of the structure’s operation and maintenance; conceptual design; preliminary design and selection of alternatives; geometrical dimensioning basing on far-field considerations; detailed design basing on near-field factors, including structural design; inclusion of possible construction constraints affecting the design; inclusion of some design flexibility allowing for redesign basing on monitoring of the operation and effectiveness of the structure after construction.

Functional requirements and design outline are depicted in Fig. 20.3. It is seen that the design in various stages is verified through the use of simulation models at different levels of complexity. Boundary conditions (bottom) constitute input to both design considerations and the models employed, while the functional requirements (top) ensure evaluation of the suitability of the design and provide design objectives at the same time.

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Design process.

The starting point in the design process consists of the identification of the beach erosion problem, followed by the selection of the type of protection measure; the final design can incorporate the risk analysis. Attention is drawn to the proper choice of the shore protection measure. The selection is usually affected by the cost. For example, beach nourishment can be cost-effective for low sediment deficit but might be comparable with offshore breakwaters for higher erosion rates. Aside from the cost, many other aspects must also be taken into account upon selection of the shore protection measure. Not shown in the drawing are legal restrictions, regional constraints and priorities, construction, operation and maintenance aspects, etc. Coastal structures are constructed to protect life and property against storm surges, to combat erosion and/or to create (often artificial) beaches for recreational purposes, and to preserve the natural environment. However, the absolute safety of an area or structure is nearly impossible to achieve. Therefore, it is much better to speak about the probability of failure (or safety) of a certain protection system. To implement this concept, all possible causes and outcomes of failure have to be analyzed. This concept is actually being developed for breakwaters52 and the dike and dune design, mostly in the Netherlands (see, www.enwinfo.nl).18,53,55 The “fault tree” is a handy tool for this aim. In the fault tree all possible modes of failure of elements, which can eventually lead to the failure of a structure section and to inundation are included. They can also badly affect the behavior of the structure, even if the latter is properly designed on the whole. Although all categories of

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events which may cause the inundation of a land or damage of structure are equally important for the overall safety, the engineer’s responsibility is mainly limited to the technical and structural aspects. In the case of coastal structures the following major events can be distinguished: • • • • • •

overflow or overtopping of the structure (i.e., instability of the superstructure); erosion or instability of slopes; instability of inner sections leading to progressive failure; scour and instability of toe-protection; instability of the foundation and internal erosion (i.e., piping); instability of the whole structure.

20.3. Alternative Systems for Coastal Protections Various coastal structures, as already discussed, can be applied to solve, or at least, to reduce erosion problems. They can provide direct protection (seawalls, dikes, revetments) or indirect protection (groins and offshore breakwaters of various designs), thus reducing the hydraulic load on the coast (Fig. 20.4). Rock and concrete are usually the construction materials. However, there is a growing interest both in developed and in developing countries in low cost or novel methods of shoreline protection particularly as the capital cost of defense works and their maintenance continues to rise. The shortage of natural rock in certain geographical regions can also be a reason for looking to other materials and systems. Despite this interest there is little published and documented information about the performance of low cost or patented structures especially at more exposed wave climate. Novel systems as geosystems (geotubes, geocontainers, geocurtains) and some other (often patented) systems (Reef Balls, Aquareef, prefabricated units, beach drainage, etc.) have gained popularity in recent years because of (often but not always) their simplicity in placement and constructability, cost effectiveness, and their minimum impact on the environment. These new systems were applied successfully in number of countries and they deserve to be applied on a larger scale. Because of the lower price and easier execution, these systems can be a good alternative for traditional coastal protection/ structures. The main obstacle in their application is, however, the lack of proper design criteria. An overview is given on application and performance of some existing novel systems and reference is made to the actual design criteria. Additional information on these systems can be in references and on websites. 20.3.1. Low-crested structures Low-crested and submerged structures (LCS) as detached breakwaters and artificial reefs are becoming very common coastal protection measures (used alone or in combination with artificial sand nourishment).5,9,17,22,57 As an example, a number of systems and typical applications of shore-control structures is shown in Figs. 20.4–20.7.

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Fig. 20.4.

Fig. 20.5.

Examples of shore-control and low-crested structures.

Definitions and objectives of low-crested/reef structures.

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Fig. 20.6.

Fig. 20.7.

Example of Aquareef.33

Example of Reef Balls units.31

The purpose of LCS structures or reefs is to reduce the hydraulic loading to a required level allowing for a dynamic equilibrium of the shoreline. To obtain this goal, they are designed to allow the transmission of a certain amount of wave energy over the structure in terms of overtopping and transmission through the porous structure (emerged breakwaters) or wave breaking and energy dissipation on shallow crest (submerged structures). Due to aesthetical requirements low freeboards are usually preferred (freeboard around SWL or below). However, in tidal environment and frequent storm surges they become less effective when design as a narrow-crested structures. That is also the reason that broad-crested submerged breakwaters (also called, artificial reefs) became popular, especially in Japan (Figs. 20.5, 20.6 and 20.10).5,33,40,41,48,60−62 They can also be an important measure in combating the effect of sea-level rise; the water level will gradually rise but the wave heights and thus, coastal erosion, run-up and overtopping can be reasonable reduced.

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However, broad-crested structures are much more expensive and their use should be supported by a proper cost-benefit studies. On the other hand, the development in alternative materials and systems, for example, the use of sand-filled geotubes as a core of such structures, can reduce effectively the cost.24,45,56 The upgrading of (integrated/multidisciplinary) design criteria for LCS structures recently took place in the scope of European project DELOS22 ; (see also: www.delos.unibo.it). The relatively new innovative coastal solution is to use artificial reef structures called “Reef Balls” as submerged breakwaters, providing both wave attenuation for shoreline erosion abatement, and artificial reef structures for habitat enhancement.7 An example of this technology using patented Reef BallTM is shown in Fig. 20.7. Reef Balls are mound-shaped concrete artificial reef modules that mimic natural coral heads. The modules have holes of many different sizes in them to provide habitat for many types of marine life. They are engineered to be simple to make and deploy and are unique in that they can be floated to their drop site behind any boat by utilizing an internal, inflatable bladder. Worldwide a large number of projects have already been executed by using this system. The first applications were based purely on experience from previous smaller projects. Since recently, more well-documented design criteria are available. Stability criteria for these units were determined based on analytical and experimental studies. For high energetic wave sites the units can be hydraulically anchored with cables to the sea bed. Wave transmission was studied in Canada.4 Technical design aspects are treated by Harris.31 20.3.2. Prefabricated systems There exist a number of other novel and/or low-cost materials and methods for shore protection (gabions and stone mattresses, open stone asphalt, used tire pile breakwaters, sheet pile structures, standing concrete pipes filled with granular materials, concrete Z-wall (zigzag) as breakwater, geotextiles curtains (screens), natural and mechanical drainage of beaches, and various floating breakwaters, etc.)54,56 Most of them are extensively evaluated and documented. However, more recently, a new family of prefabricated concrete elements as SURGEBREAKER offshore reef system, BEACHSAVER reef, WAVEblock, T-sill elements, and others have been developed and applied.54 The details on these systems can be found in references and on the websites. However, because of very narrow crest these prefabricated breakwaters are only efficient during mild wave conditions and their effects usually disappear during storm conditions, and because of scour and/or settlement, even losing their stability. The recent evaluation on performance of prefabricated, narrowcrested breakwaters can be found on the US Army website and publications.66 Some of these breakwaters are applied for comparison with other systems in recent US National Shoreline Erosion Control Development and Demonstration Program (Section 227), and more reliable information on the effectiveness of these systems can be expected within a few years (http://chl.erdc.usace.army.mil/CHL.aspx?p = s&a=PROGRAMS;3). The website provides details on sites/systems, and also documentation, if available.

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532

Fig. 20.8.

Principle of distorted ripple mat and application.50

20.3.3. Some other systems 20.3.3.1. Distorted ripple mat A new concept for creating shore accretion is actually developed and applied in Japan. A distorted (precast concrete blocks) ripple mat (DRIM) laid in the surf zone induces a landward bottom current providing accretion of a shore; see Fig. 20.8.50 The strong asymmetry of (artificial) ripple profile generates current near the bottom to one direction and thus sediment movement, whose concentration is high near the bottom, can be controlled with only very little environmental impact. The hydraulic condition on which the distorted ripple mat can control the sediment transport most effectively is studied experimentally and numerically and its capability to retain beach sand is tested through laboratory experiments and field installation. The definite onshore sediment movement by the control of DRIM is expected if the relative wave height H/h is less than 0.5, where H is the wave height and h is the water depth. The optimum condition for the efficient performance of DRIM is that d0 /λ > 1.7, where d0 is the orbital diameter of water particle and λ is the pitch length of DRIM, and this condition coincides with the condition in which natural sand ripples grow steadily. 20.3.3.2. Beach drainage (dewatering) systems Beach watertable drainage is thought to enhance sand deposition on wave uprush while diminishing erosion on wave backwash (Fig. 20.9). The net result is an increase in subaerial beach volume in the area of the drain. The larger prototype drainage by pumping installations used in Denmark and Florida suggest that beach aggradation

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may be artificially induced by beach watertable drainage. The state of the art of this technique is presented by Vesterby.77 An extensive evaluation of drainage systems is provided by Curtis and Davis18 and actual information can be found on the website: http://shoregro.com/. It is concluded that the drainage system has, in general, a positive effect on diminishing the beach erosion; however, its effectiveness is still difficult to control. The idea to achieve lowering of the water table without pumps by enhancing the beach’s own drainage capacity or hydraulic conductivity through the use of strip drains has been applied in Australia and in Japan.39

20.4. Some Remarks on Wave Transmission and Coastal Response 20.4.1. Wave transmission For shoreline control the final morphological response will result from the timeaveraged (i.e., annual average) transmissivity of the applied systems. However, to simulate this in the designing process, for example, in numerical simulation, it is necessary to know the variation in the transmission coefficient for various submergence conditions. Usually, when there is need for reduction in wave attack on structures and properties the wave reduction during extreme conditions (storm surges) is of interest (reduction of wave pressure, run-up and/or overtopping). In both cases the effectiveness of the measures taken will depend on their capability to reduce the waves; the submergence ratio and crest width are important factors (see Fig. 20.10).14,19,21,22,26,57,63,75,76,81 The transmission coefficient, Kt , defined as the ratio of the wave height directly shoreward of the breakwater to the height directly seaward of the breakwater, has the range 0 < K < 1, for which a value of 0 implies no transmission (high, impermeable), and a value of 1 implies complete transmission (no breakwater). Factors that control wave transmission include crest height and width, structure slope, core and armor material (permeability and roughness), tidal and design level, wave height, and period. As wave transmission increases, diffraction effects decrease, thus decreasing the size of sand accumulation by the transmitted waves and weakening the diffractioncurrent moving sediment into the shadow zone.30 It is obvious that the design rules

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Fig. 20.10.

Prototype measurements for Yugawara reef, Japan.3,49

for submerged structures should include a transmission coefficient as an essential governing parameter. Example of the transmission over the submerged structures (Aquareef) is shown in Fig. 20.11. More detailed descriptions of the functional and technical design of these reefs can be found in Hirose et al.33 The construction of detached breakwaters and, especially, artificial reefs (= submerged breakwaters with broad crest) is very popular and advanced in Japan. Their application had already started in the 1970s, supported by extensive model studies. The design techniques were gradually improved by using the results of a large number of prototype measurements and by monitoring completed projects (Fig. 20.10). Examples of prototype measurements in Japan and the Japanese design procedure can be found in Refs. 25, 27, 49, 68, and 82. 20.4.2. Layout and morphological response Most commonly an offshore obstruction, such as a reef or island, will cause the shoreline in its lee to protrude in a smooth fashion, forming a salient or a tombolo. This occurs because the reef reduces the wave height in its lee and thereby reduces the capacity of the waves to transport sand. Consequently, sediment moved by longshore currents and waves builds up in the lee of the reef.9–11,35,36,46 The level of protection is governed by the size and offshore position of the reef, so the size

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Fig. 20.11. General transmission characteristic for Aquareef33 (Ht is the transmitted wave height recorded on the landward side, H1/3 and L1/3 are the significant wave height and wavelength at the toe of the rubble mound, B is the crown width, and R is the submergence of the crown).

of the salient or tombolo varies in accordance with reef dimensions. Of course, one can expect this kind of morphological change only if the sediment is available (from natural sources or as sand nourishment). The examples of simple geometrical empirical criteria for the layout and shoreline response of the detached, exposed (emerged) breakwaters can be found in Refs. 20, 32, and 53. To include the effect of submergence (transmission) Pilarczyk57 proposes, at least as a first approximation, adding the factor (1− Kt) to the existing rules. Then the rules for low-crested breakwaters can be modified to (for example): • Tombolo: Ls /X > (1.0–1.5)/(1 − Kt ) or X/Ls < (2/3 to 1)(1 − Kt ),

or

X/(1 − Kt ) < (2/3 to 1)Ls . • Salient: Ls /X < 1/(1 − Kt ) or X/Ls > (1 − Kt ) or X/(1 − Kt ) > Ls . For salients where there are multiple breakwaters: GX/L2s > 0.5(1 − Kt ), where Ls is the length of a breakwater and X is the distance to the shore, G is the gap width, and the transmission coefficient Kt is defined for annual wave conditions. The gap width is usually L ≤ G ≤ 0.8Ls , where L is the wavelength at the structure defined as: L = T (gh)0.5 ; T = wave period, h = local depth at the breakwater. As first approximation, Kt = 0 for emerged breakwaters and Kt = 0.5 for submerged breakwaters can be assumed for average annual effects. These criteria can only be used as preliminary design criteria for distinguishing shoreline response to a single, transmissive detached breakwater. However, the range of verification data is too small to permit the validity of this approach to be assessed for submerged breakwaters.

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In general, one may conclude that these simple geometrical design rules, both for emerged as well as submerged breakwaters, are of limited value for design calculations because they do not include the effect of the rate of sediment transport, which can be very different for a specific coast. It is supported by the studies on effect of offshore breakwaters along UK coast.6,67 On the other side, it can be stated that numerical models (i.e., Genesis, Delft 2D-3D, Mike 21, etc.) can already be treated as useful design tools for the simulation of morphological shore response to the presence of offshore structures. Examples can be found in Refs. 13, 17, 28–30, 42, and 70. As mentioned above, while considerable research has been done on shoreline response to exposed offshore breakwaters, very little qualitative work has been done on the effect of submerged offshore reefs, particularly outside the laboratory. Thus, within the Artificial Reefs Program9 (www.asrltd.co.nz), Andrews2 examined aerial photographs seeking cases of shoreline adjustment to offshore reefs and islands. All relevant shoreline features in New Zealand and eastern Australia were scanned and digitized, providing 123 different cases. A range of other statistics, particularly reef and island geometry, was also obtained. Some of these results are repeated below. To examine the effects of wave transmission on limiting parameters, data for reefs and islands were considered separately. The data indicated that tombolo formation behind islands occurs with Ls /X ratio of 0.65, and higher and salients form when Ls /X is less than 1.0. Therefore, for islands the Ls /X ratios determining the division between salients and tombolos are similar to those from previously presented breakwater research. Similarly, data resulting from offshore reefs indicate that tombolo formation occurs at Ls /X ratios of 0.6 and higher, and salients most commonly form when Ls /X is less than 2. The data suggests that variation in wave transmission (from zero for offshore islands through to variable transmission for offshore reefs) allows a broader range of tombolo and salient limiting parameters. Thus, a reef that allows a large proportion of wave energy to pass over the obstacle can be (or must be) positioned closer to the shoreline than an emergent feature. Thus, from natural reefs and islands the following general limiting parameters were identified: • Islands: Tombolos form when Ls /X > 0.65; Salients form when Ls /X < 1.0; • Reefs: Tombolos form when Ls /X > 0.60; Salients form when Ls /X < 2.0; • Nondepositional conditions for both shoreline formations occur at Ls /X < ≈0.1. The choice of the layout of submerged breakwaters can also be affected by the current patterns around the breakwaters. The Japanese Manual (1988) provides (indicative) information on various current patterns for submerged reefs.82 However, for real applications it is recommended to simulate the specific situation by numerical or hydraulic modeling.

20.5. Geosystems in Coastal Applications 20.5.1. General overview Geotextile systems utilize a high-strength synthetic fabric as a form for casting large units by filling with sand or mortar. Within these geotextile systems a distinction

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Examples of application of geosystems: (a) Geotube as a breakwater, (b) BEROSIN

can be made between: bags, mattresses, tubes, containers, and inclined curtains.56 All of which can be filled with sand or mortar. Some examples are shown in Fig. 20.12. Bags are suitable for slope protection and retaining walls or toe protection but the main application is the construction of groynes, perched beaches, and offshore breakwaters. The tubes and containers are mainly applicable for construction of groynes, perched beaches, and (offshore) breakwaters, and as bunds for reclamation works (Fig. 20.13). Geotubes can form an individual structure in accordance with some functional requirements for the project but they can also be used complementary to the artificial beach nourishment to increase its lifetime. Especially for creating the perched beaches, the sand tubes can be an ideal, low-cost solution for constructing the submerged sill.34,45,56 Geotubes and geocontainers can also be used to store and isolate

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Example of reef structure composed with geotubes24 ; http://geotecassociates.com/.

contaminated materials from harbor dredging, and/or to use these units as bunds for reclamation works. An interesting application for shore erosion control is the geocurtain known under the name BEROSINTM,71 [Fig. 20.12(b)]. The BEROSIN curtain is a flexible structure made of various woven geotextiles, which after placing by divers near the shore and anchoring to the bed catches the sand transported by currents and waves providing accretion on a shore and preventing the erosion. The horizontal curtain (sheet) can be easily spread by a small workboat and two divers. The upper (shoreside) edge, equipped with some depth-compensated floaters, should be properly anchored at the projected line. The sea-side edge is kept in position by the workboat. By ballasting some of the outside pockets at the lower edge with sand or other materials and with the help of divers, the lower edge is sinking to the required position. The proper choice of permeability of geotextile creates the proper conditions for sedimentation of suspended sediment in front/or under the curtain and at the same time allowing the water to flow out without creating too high forces on the curtain and thus, on the anchors. In case of Pilot project at the coast of Vlieland (NL), some of the horizontal curtains placed in the intertidal zone have provided a growth of a beach/foreshore of 0.5–1.0 m within a week while others within a few weeks.56 These geocurtains can also be applied for the construction of submerged sills and reefs. In the past, the design of geotextile systems for various coastal applications was based mostly on vague experience rather than on the general valid calculation methods. However, the increased demand in recent years for reliable design methods for protective structures have led to the application of new materials and systems (including geotextile systems) and to research concerning the design of these systems. Contrary to research on rock and concrete units, there has been no systematic research on the design and stability of geotextile systems. However, past and recent research in the Netherlands, USA, Germany, and in some other countries on a number of selected geotextile products has provided some useful results which can be of use in preparing a set of preliminary design guidelines for the geotextile systems under current and wave attack.31,56,58 The recent, large scale tests, with large geobags, can be found on the website: http://sun1.rrzn.unihannover.de/fzk/e5/projects/dune prot 0.html.

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When installing geosystems, one should see to it that this does not take place on a rough foundation. Sharp elements may easily damage the casing of the element. Geosystems must usually not be filled completely. With a fill ratio of approximately 75% an optimum stability of the elements is reached. A sound soil protection is necessary if gravel (sand) sausages are used in circumstances where they are under attack of flow or waves. Stability criteria are discussed by Pilarczyk.56 The main (large) fill-containing geosystems (geobags, geotubes, and geocontainers filled with sand or mortar) and their design aspects are briefly discussed below. For more detailed information, the reader will be guided to the relevant manuals and publications (see references and websites). 20.5.2. Geobags Geobags can be filled with sand or gravel (or cement, perhaps). The bags may have different shapes and sizes, varying from the well-known sandbags for emergency dikes to large flat shapes or elongated “sausages” (see Fig. 20.14). The most common use for sandbags in hydraulic engineering is for temporary structures. Uses for sandor cement-filled bags are, among other things: • • • •

repair works; revetments of relatively gentle slopes and toe constructions; temporary or permanent groynes and offshore breakwaters; temporary dikes surrounding dredged material containment areas.

Fig. 20.14.

Application of geobags (sand or cement-filled).

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Because this material is easy to use and cheap, it is extremely suitable for temporary structures. A training groyne is a good example. The working of a groyne is difficult to predict in advance. That is why it is a good procedure to make such a construction using a relatively cheap product first, to see how one thing and another works out, and subsequently either make improvements or, after some time, a permanent structure. However, it is recommended not to use sand-filled geobags for a wave height above 1 m and/or a flow velocity above 2 m/s. The geosystems filled with sand cannot be used at these more extreme conditions because the sand in the systems is no longer internally stable. Sandbags can be placed as follows: (1) As a blanket: One or two layers of bags placed directly on the slope. An “interlocking” problem arises if the bags are filled completely. The bags are then too round. A solution is not to fill the bags completely, so that the sides flatten out somewhat, as a result of which the contact area becomes larger. (2) As a stack: Bags stacked up in the shape of a pyramid. The bags lie halfoverlapping with (usually) the long side parallel to the shoreline. The stacking of geotubes increases their stability due to interlocking with the neighboring bags. More recently, prototype experience and large scale tests with large geobags, are described by Restall,58 and more detailed information can be found on the website: http://sun1.rrzn.uni-hannover.de/fzk/e5/projects/dune prot 0.html. 20.5.3. Tube system Geotube is a sand/dredged material filled geotextile tube made of permeable but soil-tight geotextile.56 The desired diameter and length are project specific and only limited by installation possibilities and site conditions. The tube is delivered to the site rolled up on a steel pipe. Inlets and outlets are regularly spaced along the length of the tube. The tube is filled with dredged material pumped as a water–soil mixture (commonly a slurry of 1 on 4) using a suction dredge delivery line (Fig. 20.15). The choice of geotextile depends on characteristics of fill material. The tube will achieve

Fig. 20.15.

Filling procedure of geotube.

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its desired shape when filled up to about 70–80% (or a height equal to about a half of the flat width of the tube); a higher filling grade is possible but it diminishes the friction resistance between the tubes. Tube can be filled on land (e.g., as dikes for land reclamation, bunds, toe protection, or groyns) or in water (e.g., offshore breakwaters, sills of perched beaches, dikes for artificial islands, etc.). The tube is rolled out along the intended alignment with inlets/outlets centered on top. When a tube is to be placed in water, the effects of buoyancy on the tube geotextile prior to filling as well as on the dredged material’s settling characteristics must be considered. In order to maximize inlet/outlet spacing, an outlet distant from the inlet may be used to enhance the discharge of dredged slurry and thereby encourage and regulate the flow of fill material through the tube so that sufficient fill will flow to distant points. Commonly, the filter geotextile on both sides of a tube (against scour) and flat main tube are fully deployed by floating and holding them in position prior to beginning the filling operation. The geotextile filter is often furnished with small tubes at the edges which when filled with sand holds the filter apron in place. The required length of apron is usually two times the local wave height.

20.5.3.1. Shape and mechanical strength of geotubes For the selection of the strength of the geotextile and calculation of a required number of tubes for a given height of structure, knowledge of the real shape of the tube after filling and placing is necessary. The change of the cross-section of the tube depends on the static head of the (sand) slurry. Depending on this static head, the laying method, and the behavior of the fill-material inside the tube, it is possible that the cross-sectional shape of the filled tube will vary from a very flat hump to a nearly fully circular cross-section. More recently, Silvester64 and Leshchinsky43,44 prepared some analytical or numerical solutions and graphs allowing the determination of the shape of sand- or mortar-filled tubes based on some experiments with water. The Leshchinsky’s method combines all the previous developments and can be treated as a design tool. The design of the shape of the geotube is an iterative process. To obtain a proper stability of the geotube and to fulfill the functional requirements (i.e., required reduction of incoming waves/proper transmission coefficient, the width, and the height of the tube must be calculated). If the obtained shape of geotube does not fulfill these requirements, a new (larger) size of a geotube must be taken into account or a double-line of tubes can be used.

20.5.4. Container system Geocontainer is a mechanically filled geotextile and “box” or “pillow” shaped unit made of a soil-tight geotextile.56 The containers are partially prefabricated by sawing mill widths of the appropriate length together and at the ends to form an elongated “box.” The “box” is then closed in the field, after filling, using a sewing machine and specially designed seams (Fig. 20.16). Barge placement of

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Fig. 20.16.

Example of split barge and filling and placing of geocontainer.

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the site-fabricated containers is accomplished using a specially configured bargemounted crane or by bottom dump hoppers scows, or split barges. The containers are filled and fabricated on the barge and placed when securely moored in the desired position. Positioning of barge for consistent placement — a critical element of constructing “stacked” underwater structures — is accomplished with the assistance of modern surveying technology. The maximum volume of applied geocontainers was up to now 1000 m3 . The advantages of these large barge-placed geocontainers include: • Containers can be filled with locally available soil, which may be available from simultaneous dredging activities. • Containers can be relatively accurately placed regardless of weather conditions, current velocities, tides, or water depths. • Contained material is not subject to erosion after placing. • Containers can provide a relatively quick system buildup. • Containers are very cost competitive (for larger works). 20.5.4.1. General design considerations When applying geobags, geotubes, and geocontainers, the major design considerations/problems are related to the integrity of the units during filling, release and placement impact (impact resistance, seam strength, burst, abrasion, durability), the accuracy of placement on the bottom (especially at large depths), and the stability under current and wave attack. The geotextile fabric used to construct the tubes is designed to: • • • • •

contain sufficient permeability to relieve excess water pressure, retained the fill-material, resist the pressures of filling and the active loads without seams or fabric rapture, resist erosive forces during filling operations, resist puncture, tearing, and ultraviolet light.

The following design aspects are particularly of importance for the design of containers: (a) change of shape of units in function of perimeter of unit, fill-grade, and opening of split barge, (b) fall-velocity/equilibrium velocity, velocity at bottom impact, (c) description of dumping process and impact forces, (d) stresses in geotextile during impact and reshaping, (e) resulting structural and executional requirements, and (f) hydraulic stability of structure. Some of these aspects are briefly discussed hereafter.

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Fig. 20.17.

Development of forces during dumping of geocontainers.

20.5.4.2. Dumping process of containers and practical uncertainties A summary of various forces during the dumping and placement process is given in Fig. 20.17.56 (1) The required perimeter of geotextile sheet must be sufficient enough to release geocontainer through the given split width b0 for a required cross-sectional area of material in the bin of barge Af (or filling-ratio of fill-material in respect to the max. theoretical cross-section). The derivation of the required minimum length of perimeter of geotextile sheet is given by Pilarczyk.56 After opening of the split of a barge the geocontainer is pulled out by the weight of soil but at the same time the friction forces along the bin side are retarding this process. Due to these forces the tension in geotextile is developing at lower part and both sides of the geocontainer. The upper part is free of tension till the moment of complete releasing of geocontainer. The question is how far we are able to model a friction and the release process of geocontainer. (2) Geocontainer will always contain a certain amount of air in the pores of soil and between the soil and the top of (surplus) geotextile providing additional buoyancy during sinking. The amount and location of air pockets depends on soil consistency (dry, saturated) and uniformity of dumping. The air pockets will exert certain forces on geotextile and will influence the way of sinking. The question is how to model in a proper way the influence of soil consistency and air content on shape and stresses in geotubes/geocontainers. (3) The forces due to the impact with the bottom will be influenced by a number of factors: • consistency of soil inside the geocontainer (dry, semi-dry, saturated, cohesion, etc.) and its physical characteristics (i.e., internal friction);

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• amount of air; • permeability/air tightness of geotextile; • strength characteristics of geotextile (elasticity/elongation versus stresses, etc.); • fall-velocity (influenced by consistency of soil; saturated soil diminish amount of air but increases fall speed); • shape and catching surface of geocontainer at impact including effect of not horizontal sinking (i.e., catching of bottom with one end); • type of bottom (sand, clay, soft soil, rock, soil covered with rockfill mattress, etc.) and/or type of sublayer (i.e., layer of previous placed containers). During the impact the cross-sectional shape of geocontainer will be undergoing a continuous reshaping; from cone shape, first probably into a transitional cylindrical shape, and through a certain relaxation, into a semi-oval shape or flat triangular/rectangular shape dictated by soil type, perimeter, and elongation characteristics of geotextile. The question is how far we are able to model this impact phenomena and resulting forces/stresses in geotextile. The impact forces with the bottom are a function of the fall velocity (dump velocity) of a geocontainer. (4) In final situation the geocontainers will perform as a core material of various protective structures or as independent structure exposed to loading by currents and waves, and other loadings (ice, debris, ship collision, vandalism, etc.). In most cases the geocontainers will be filled by fine (loosely packed) soils. The question is how these structures will behave in practice under various types of external and internal loadings. Practical note: The prototype experience indicates that geocontainers with volume up to 200 m3 and dumped in water depth exceeding 10 m have been frequently damaged (collapse of seams) using geotextile with tensile strength lower than 75 kN/m, while nearly no damage was observed when using the geotextile with tensile strength equal to or more than 150 kN/m. This information can be of use for the first selection of geocontainers for a specific project. The placing accuracy for depths larger than 10 m is still a problem.8 20.5.5. Durability of geotextiles Durability of geotextiles is a frequently asked question especially concerning the applications where a long life-span is required. Geotextile is a relatively new product. The first applications are from 1960s. Recently, some 30-year-old geotextiles used as a filter in revetment structures in the Netherlands have been evaluated. In general, these geotextiles were still in a good condition. The technology of geotextiles is improved to such an extent that the durability tests under laboratory condition indicate the life-time of geotextiles at least of 100 years (when not exposed to UV radiation). There is no problem with durability of the geosystems when they are submerged or covered by armor layers. However, in case of exposed geosystems the UV radiation and vandalism are the factors, which must be considered during the design.

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All synthetics are vulnerable to UV. The speed of UV degradation, resulting in the loss of strength, depends on the polymer used and type of additives. Polyesters (PET) are by nature more light stable than, for example, polyamide (PA) and polypropylene (PP). The long-term prototype tests in the Netherlands indicated that the geosynthetics under water and in the intertidal zone show very little degradation in strength in comparison with geosynthetics placed on land; in the inter-tidal zone the geosynthetics are covered very soon by algae which provide very good UV protection. To avoid the problem with light degradation the fabrics must be properly selected (i.e., polyester) and UV stabilized. As the period in which the fabric is exposed is short (in terms of months), no serious problems are to be expected. In case of more or less permanent applications under exposed conditions the fabric must be protected against direct sunlight. There are a number of methods of surface protection for geosystems. To provide additional UV and abrasion protection to the exposed sections of tubes, a coating of elastomeric polyurethane is often used. This coating, however, has a tendency to peel after about a number of months and therefore, has to be re-applied. The permanent surface protection by riprap or block mats is a rather expensive solution and it will normally be applied only when it is dictated by necessity due to a high wave loading or danger of vandalism or other mechanical damage, i.e., boating, anchoring, etc. In other cases it will be probably a cheaper solution to apply a temporary protection of geotextile tubes by an additional layer of a strong geotextile provided with special UV-protection layer. This extra protection can be realized by adding the highly UV-stabilized nonwoven fleece needled onto the main fabric. The function of this felt layer is also to trap the sediment particles and algae, which give again extra UV protection.

20.6. Remarks on Stability Aspects Structural design aspects of low-crested structures are relatively well described in a number of publications.1,15,16,22,23,37,51,52,65,68,69,74,78,79,82 Some useful information on the design of breakwaters on reefs in shallow water can be found in Jensen et al.38 Usually for submerged structures, the stability at the water level close to the crest level will be most critical. Assuming depth limited conditions (Hs = 0.5 h, where h = local depth), the (rule of thumb) stability criterion becomes: Hs /∆Dn50 = 2 or, Dn50 = Hs /3, or Dn50 = h/6, where Dn50 = (M50 /ρs )1/3 ; Dn50 = nominal stone diameter and M50 and ρs = average mass and density of stone. The upgraded stability formulae for LCS structures, including head effect and scour, can be found in Delos report22 (www.delos.unibo.it). It should be noted that some of useful calculation programs (including formula by van der Meer)72,73 are incorporated in a simple expert system CRESS, which is accessible in the public domain (http://ikm.nl/rwscress/ and http://www.ihe.nl/ we/dicea). Useful information on functional design and the preliminary structural design of low crested-structures, including cost effectiveness, can be found in CUR report.17 Alternative solutions, using geotubes (or geotubes as a core of breakwaters), are treated by Pilarczyk.56 An example of this application can be found in Refs. 21, 24, and 57.

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20.7. Conclusions The author does not intend to provide the new design rules for alternative coastal structures. However, it is hoped that this information will be of some aid to designers looking for new sources, which are considering these kinds of structure and improving their designs. Offshore breakwaters and reefs can be permanently submerged, permanently exposed or inter-tidal. In each case, the depth of the structure, its size, and its position relative to the shoreline determine the coastal protection level provided by the structure. To reduce the cost some alternative solutions using geosystems can be considered. The actual understanding of the functional design of these structures needs further improvement but may be just adequate for these structures to be considered as serious alternatives for coastal protection. Continued research, especially on submerged breakwaters, should further explore improved techniques predict shore response and methods to optimize breakwater design. A good step (unfortunately, limited) in this direction was made in a collective research project in the Netherlands.17 Research and practical design in this field is also the focus of the “Artificial Reefs Program” in New Zealand (www.asrltd.co.nz), the International Society for Reef Studies (ISRS) (www.artificialreefs.org), the European Project DELOS (Environmental Design of Low Crested Coastal Defence Structures; http://www.delos.unibo.it), and recent US National Shoreline Erosion Control Development and Demonstration Program (Section 227). Some useful documents can also be found on the website: http://www.citg.tudelft.nl/live/pagina.jsp?id=4de0d195-5207-4e6784bb-455c5403ae47&lang=en; www.hydraulicengineering.tudelft.nl. Also, the past and recent research in the Netherlands, USA, Germany, and in some other countries on a number of selected geotextile products (geosystems) has provided some valuable results, which can be of use in preparing a set of preliminary design guidelines for the geotextile systems under current and wave attack. The following conclusions can be drawn on application of geosystems based on the actual developments and experience. • Geosystems offer the advantages of simplicity in placement and constructability, cost effectiveness, and minimal impact on the environment. • When applying this technology the manufacturer’s specifications should be followed. The installation needs an experienced contractor. • When applying geotubes and geocontainers the major design considerations/problems are related to the integrity of the units during release and impact (impact resistance, seam strength, burst, abrasion, durability, etc.), the accuracy of placement on the bottom (especially at large depths), and the stability. • The geotextile systems can be a good and mostly cheaper alternative for more traditional materials/systems. These new systems deserve to be applied on a larger scale. However, there are still many uncertainties in the existing design methods. Therefore, further improvement of design methods and more practical experience under various loading conditions are still needed. • The state of the art of the actual knowledge on the geosystems in hydraulic and coastal engineering can be found in Refs. 8, 34, 45, 56, and 58.

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These new efforts will bring future designers closer to more efficient application and design of these promising coastal solutions. The more intensive monitoring of the existing structures will also help in the verification of new design rules. The intention of this literature search is to uncover the technical information on these systems and make them available for the potential users. It will help to make a proper choice for specific problems/projects and it will stimulate the further developments in this field. International cooperation in this field should be further stimulated.

References 1. J. Ahrens, Characteristics of Reef Breakwaters, USAE CERC TR 87-17, Vicksburg (1987). 2. C. J. Andrews, Sandy shoreline response to submerged and emerged breakwaters, reefs or islands, unpublished thesis, University of Waikato, New Zealand (1997) (www.asrltd.co.nz). 3. T. Aono and E. C. Cruz, Fundamental characteristics of wave transformation around artificial reefs, 25th Coastal Eng., Orlando, USA (1996). 4. H. D. Armono and K. R. Hall, Wave transmission on submerged breakwaters made of hollow hemispherical shape artificial reefs, Canadian Coastal Conf. (2003). 5. T. Asakawa and N. Hamaguchi, Recent developments on shore protection in Japan, Coastal Structures and Breakwaters’91, London (1991). 6. P. Axe, S. Illic and A. Chadwick, Evaluation of beach modelling techniques behind detached breakwaters, ASCE, Proc. 25th ICCE, Orlando, USA (1996). 7. T. Barber, What are Reef Balls, Southwest Florida Fishing News (2000). 8. A. Bezuijen et al., Placing accuracy and stability of geocontainers, 3rd EuroGeo, Munich, Germany (2004), http://www.wldelft.nl/rnd/publ/search.html (insert for Author: Bezuijen). 9. K. Black and S. Mead, Submerged structures for coastal protection, ASR, Marine and Freshwater Consultants, New Zealand (1999) (www.asrltd.co.nz). 10. K. Black and C. J. Andrews, Sandy shoreline response to offshore obstacles, Part I: Salient and tombolo geometry and shape, J. Coastal Res. Special Issue on Surfing (2001). 11. K. Black, Artificial surfing reefs for erosion control and amenity: Theory and application, J. Coastal Res. 1–7 (2001). 12. K. R. Bodge, Beach fill stabilization with tuned structures; experience in the Southeastern USA and Caribbean, Coastlines, Structures and Breakwaters’98, London (1998). 13. K. J. Bos, J. A. Roelvink and M. W. Dingemans, Modelling the impact of detached breakwaters on the coast, 25th ICCE, Orlando, USA (1996). 14. M. Buccino and M. Calabrese, Conceptual approach for prediction of wave transmission at low-crested breakwaters, J. Waterways, Port, Coastal Ocean Eng. (May/June) (2007). 15. CEM, Coastal Engineering Manual, US Army Corps of Engineers, Vicksburg (2006). 16. CIRIA-CUR-CETMEF, The Rock Manual, CIRIA, London (2007) (also CUR/ CIRIA, Manual on use of rock in coastal engineering, CUR/CIRIA report 154, Gouda, the Netherlands, 1991). 17. CUR, Beach nourishments and shore parallel structures, R97-2, Centre for Civil Engineering Research and Codes (CUR), P.O.Box 420, Gouda, the Netherlands (1997).

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18. W. R. Curtis and J. E. Davis, Field evaluation/demonstration of a multisegmented dewatering system for accreting beach sand in a high-wave-energy environment, WES/CPAR-CHL-98-1 (1998). 19. K. d’Angremond, J. W. van der Meer and R. J. de Jong, Wave transmission at lowcrested structures, 25th Int. Conf. on Coastal Eng., Orlando, Florida (1996). 20. W. R. Dally and J. Pope, Detached breakwaters for shore protection, Technical report CERC-86-1, U.S. Army Engineer WES, Vicksburg, MS (1986). 21. Delft Hydraulics, AmWaj Island development, Bahrain; physical modelling of submerged breakwaters, Report H4087 (2002). 22. DELOS, Environmental design of low crested coastal defence structures; D 59 DESIGN GUIDELINES, EU 5th Framework Programme 1998–2002, Pitagora Editrice Bologna (2005) (www.delos.unibo.it). 23. Ch. Fleming and B. Hamer, Successful implementation of an offshore reef scheme, 27th Coastal Engineering, Sydney (2000). 24. J. Fowler, T. Stephens, M. Santiago and P. De Bruin, Amwaj Islands constructed with geotubes, Bahrain, CEDA Conf., Denver, USA (2002), http://geotecassociates. com/. 25. H. Funakoshi, T. Siozawa, A. Tadokoro and S. Tsuda, Drifting characteristics of littoral sand around submerged breakwater, Hydro-Port’94, Yokosuka, Japan (1994). 26. Y. Goda, Wave damping characteristics of longitudinal reef system, Advances in Coastal Structures and Breakwaters’95, London (1995); M. D. Groenewoud, J. van de Graaff, E. W. M. Claessen and S. C. van der Biezen (eds.), Effect of submerged breakwater on profile development, 25th ICCE, Orlando, USA (1996). 27. T. Hamaguchi, T. Uda, Ch. Inoue and A. Igarashi, Field experiment on wavedissipating effect of artificial reefs on the Niigata Coast, Coastal Engineering in Japan, JSCE 34(1) (1991). 28. H. Hanson and N. C. Kraus, GENESIS: Generalised model for simulating shoreline change. Report 1: Technical Reference, Tech. Rep. CERC-89-19, US Army Eng., WES (1989). 29. H. Hanson and N. C. Kraus, Shoreline response to a single transmissive detached breakwater, Proc. 22nd Coastal Eng. Conf., ASCE, The Hague (1990). 30. H. Hanson and N. C. Kraus, Numerical simulation of shoreline change at Lorain, Ohio, J. Waterways, Port, Coastal Ocean Eng. 117(1) (1991). 31. L. E. Harris, www.artificialreefs.org/ScientificReports/research.htm; Submerged Reef Structures for Habitat Enhancement and Shoreline Erosion Abatement; FIT Wave Tank and Stability Analysis of Reef Balls, http://www.advancedcoastaltechnology.com/science/DrHarrisWavereduction.htm. 32. M. M. Harris and J. B. Herbich, Effects of breakwater spacing on sand entrapment, J. Hydraulic Res. 24(5) (1986). 33. N. Hirose, A. Watanuki and M. Saito, New type units for artificial reef development of eco-friendly artificial reefs and the effectiveness thereof, PIANC Congress, Sydney (2002) [see also 28th ICCE, Cardiff (2002)]. 34. G. Heerten, Geotextiles in coastal engineering, 25 years experience, Geotex. Geomem. 1(2) (1982). 35. J. R. C. Hsu and R. Silvester, Accretion behind single offshore breakwater, J. Waterway Port Coastal Ocean Eng. 116, 362–381 (1990). 36. J. A. Jimenez and A. Sanchez-Arcilla, Preliminary analysis of shoreline evolution in the leeward of low-crested breakwaters using one-line models, EVK3-2000-0041, EU DELOS workshop, Barcelona, 17–19 January 2002.

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37. K. Itoh, T. Toue and D. Katsui, Numerical simulation of submerged breakwater deformation by DEM and VOF, Advanced Design of Maritime Structures in the 21st Century, Yokosuka, Japan (2001). 38. Th. Jensen, P. Sloth and V. Jacobsen, Wave dynamics and revetment design on a natural reef, 26th Coastal Engineering, Copenhagen (1998). 39. K. Katoh, S. Yanagishima, S. Nakamura and M. Fukuta, Stabilization of beach in integrated shore protection system, Hydro-Port’94, Yokosuka, Japan (1994). 40. T. Kono and S. Tsukayama, Wave transformation on reef and some consideration on its application to field, Coastal Eng. Jpn. 23 (1980). 41. Y. Kuriyama, K. Katoh and Y. Ozaki, Stability of beaches protected with detached breakwaters, Hydro-Port’94, Yokosuka, Japan (1994). 42. M. Larson, N. C. Kraus and H. Hanson, Analytical solutions of the one-line model of shoreline change near coastal structures, J. Waterway Port Coastal Ocean Eng., ASCE, 123(4) (1997). 43. D. Leshchinsky and O. Leshchinsky, Geosynthetic confined pressurized slurry (GeoCops): Supplement notes for Version 1.0, May (1995) (Nicolon/US Corps). 44. D. Leshchinsky, O. Leschinsky, H. I. Ling and P. A. Gilbert, Geosynthetic tubes for confining pressurized slurry: Some design aspects, J. Geotech. Eng., ASCE, 122(8) (1996). 45. C. R. Lawson, Geotextile containment: International perspectives, Proc. Seventeenth GRI Conf., Geosynthetic Institute, Philadelphia, USA, December (2003), pp. 198–221. 46. D. Ming and Y.-M. Chiew, Shoreline changes behind detached breakwater, J. Waterway Port Coastal Ocean Eng. 126(2) (2000). 47. A. Nakayama, N. Horikosi and H. Kobayashi, The planning and design of multipurpose artificial barrier reefs, Coastal Zone’93, Coastline of Japan II, New Orleans (1993). 48. A. Nakayama, M. Yamamoto, J. Yamamoto and A. Moriguchi, Development of waterintake works with submerged mound (WWSM), Hydro Port’94, Yokosuka, Japan (1994). 49. S. Ohnaka and T. Yoshizwa, Field observation on wave dissipation and reflection by an artificial reef with varying crown width; Hydro-Port’94, Yokosuka, Japan (1994). 50. N. Ono, J. Irie and H. Yamaguchi, Preserving system of nourished beach by distorted ripple mat, Coastal Engineering, Lisbon (2004), http://www.civil.kyushu-u.ac.jp/ engan/drim.html. 51. PIANC, Guidelines for the design and construction of flexible revetments incorporating geotextiles in marine environment, Brussels, Belgium (1992). 52. PIANC, Analysis of rubble mound breakwaters, PIANC, P.T.C. II/WG. 12, Suppl. To Bulletin 78/79, Brussels (1992). 53. K. W. Pilarczyk (ed.), Coastal Protection (A.A. Balkema, Rotterdam, 1990). 54. K. W. Pilarczyk and R. B. Zeidler, Offshore Breakwaters and Shore Evolution Control (A.A. Balkema, Rotterdam, 1996), www.balkema.nl. 55. K. W. Pilarczyk, Dikes and Revetments (A.A. Balkema, Rotterdam, 1998). 56. K. W. Pilarczyk, Geosynthetics and Geosystems in Hydraulic and Coastal Engineering (A.A. Balkema, Rotterdam, 2000), www.balkema.nl (select Pilarczyk). 57. K. W. Pilarczyk, Design of low-crested (submerged) structures: An overview, 6th COPEDEC, Sri Lanka (2003), www.enwinfo.nl (select English, downloads). 58. S. Restall et al., Australian and German experiences with geotextile containers for coastal protection, 3rd EuroGeo, Munich, Germany (2004). 59. A. Sanchez-Arcilla, F. Rivero, X. Gironella, D. Verges and M. Tome, Vertical circulation induced by a submerged breakwater, 26th Coastal Engineering, Copenhagen (1998).

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60. T. Sawaragi, I. Deguchi and S. K. Park, Reduction of wave overtopping rate by the use of artificial reef, Proc. 21st Int. Conf. Coastal Eng., ASCE (1988). 61. T. Sawaragi, Detached breakwaters; short course on design and reliability of coastal structures, Venice, 1–3 October 1992. 62. T. Sawaragi, Coastal Engineering-Waves, Beaches, Wave–Structure Interactions (Elsevier, 1995). 63. S. R. Seabrook and K. R. Hall, Wave transmission at submerged rubble mound breakwaters, 26th Int. Conf. Coastal Eng., Copenhagen (1998). 64. R. Silvester and J. R. Hsu, Coastal Stabilization, (Prentice Hall Inc., Englewood Cliffs, 1993), http://www.worldscibooks.com/engineering/3475.html. 65. D. Smit et al., Submerged-crest breakwater design, Advances in Coastal Structures and Breakwaters’95, London (1995). 66. D. K. Stauble and J. R. Tabar, The use of submerged narrow-crested breakwaters for shoreline erosion control, J. Coastal Res. 19(3), 352 (2003), http://chl.erdc.usace. army.mil/CHL.aspx?p=m&a=MEDIA. 67. F. Thomalla and Ch. Vincent, Designing offshore breakwaters using empirical relationships: a case study from Norfolk, UK, J. Coastal Res. 20(4) (2004). 68. T. Uda, Function and design methods of artificial reef (in Japanese), Ministry of Construction, Japan (1998) (see also, Coastal Zone’93). 69. US Army Corps, Engineering design guidance for detached breakwaters as shoreline stabilization structures, WES, Technical Report CERC–93–19, December 1993. 70. S. C. Van der Biezen, J. A. Roelvink, J. Van de Graaff, J. Schaap and L. Torrini, 2DH morphological modelling of submerged breakwaters, 26th ICCE, Copenhagen (1998). 71. van der Hidde, BEROSIN, Bureau van der Hidde, Harlingen, P.O.B. 299, the Netherlands (1995). 72. J. W. van der Meer, Stability of breakwater armour layers, Coastal Eng. 11 (1987). 73. J. W. van der Meer, Rock slopes and gravel beaches under wave attack, Doctoral thesis, Delft University of Technology (1988) (Also: Delft Hydraulics Communication No. 396). 74. J. W. van der Meer, Low-crested and reef breakwaters, Delft Hydraulics Rep. H 986 (1990). 75. J. W. van der Meer, Data on wave transmission due to overtopping, Delft Hydraulics (1990). 76. J. W. van der Meer and K. d’Angremond, Wave transmission at low-crested structures, Coastal Structures and Breakwaters (Thomas Telford, London, UK, 1991), pp. 25–40. 77. H. Vesterby, Beach drainage — state of the art — Seminar on Shoreline Management Techniques, Wallingford, 18 April 1996. 78. C. Vidal, M. A. Losada, R. Medina, E. P. D. Mansard and G. Gomez-Pina, A universal analysis for the stability of both low-crested and submerged breakwaters, 23rd Coastal Engineering, Venice (1992). 79. C. Vidal, I. J. Losada and F. L. Martin, Stability of near-bed rubble-mound structures, 26th Coastal Engineering, Copenhagen (1998). 80. N. Von Lieberman and S. Mai, Analysis of an optimal foreland design, 27th Coastal Engineering 2000, Sydney (2000). 81. T. Wamsley and J. Ahrens, Computation of wave transmission coefficients at detached breakwaters for shoreline response modelling, Coastal Structures’03, Portland, USA (2003). 82. K. Yoshioka, T. Kawakami, S. Tanaka, M. Koarai and T. Uda, Design manual for artificial reefs, in Coastlines of Japan II, Coastal Zone’93 (ASCE, 1993).

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Chapter 21

Geotextile Sand Containers for Shore Protection Hocine Oumeraci∗ and Juan Recio Leichtweiss-Institute for Hydraulic Engineering and Water Resources, Technical University Braunschweig, Beethovenstr. 51a, 38106 Braunschweig, Germany ∗ [email protected] This chapter aims at providing a brief overview on geotextile sand containers (GSCs) applied in coastal engineering for shore protection. First, the engineering properties required for the geotextile used for sand containers as well as the durability and the lifetime prediction issue are discussed. Second, some example applications are provided to illustrate the versatility of GSCs as an appropriate soft shore prediction alternative to conventional hard coastal structures made of rock and concrete units. However, the major part of the chapter is aimed to address the hydraulic stability of the containers constituting a shore protection structure subject to wave attack. For this purpose, simple formulae are first proposed for the stability of the slope and crest containers. The processes which may affect the hydraulic stability are then discussed to highlight the necessity of developing more process-based stability formulae. New stability formulae are finally proposed which can also account explicitly for the effect of deformation of the containers. Finally, a discussion is provided on the comparative analysis of the stability of the slope and crest containers with and without consideration of the deformation effect.

21.1. Introduction In view of the increasing storminess associated with climate change and its effect on coastal flood and erosion, more versatile materials and solutions are required for the design of new, cost-effective shore protection structures as well as for the reinforcement of existing threatened coastal barriers and structures, including dune reinforcement and scour protection. In search for low-cost, soft, and reversible solutions, the concept of geotextile sand containers (GSCs) as “soft rock” for coastal defense structures was introduced for the first time in the 1950s in the form of “sand bags” for the construction of a dike to close the inlet “Pluimpot” in the Netherlands.53 553

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At the same time some test groins were constructed by using “sand bags” at the German North Sea coast.10,57 Later, GSCs have mainly been used as temporary shore protection measures due to the difficulties associated with the assessment of the long-term performance. Meanwhile, significant advances have been made with respect to the following issues: • improvement of the long-term performance of geotextiles (additives and stabilizers against UV-radiation, coating against abrasion, etc.), • assessment of the durability and lifetime prediction (accelerated testing, standards, etc.),3,14,46 • survey of GSC-built structures and analysis of past experience with respect to the degradation mechanisms, and • understanding of the mechanisms of failure, including hydraulic instability under severe wave action.32,34 These advances, together with numerous advantages of GSCs as “soft rock,” have contributed to extend the use of GSCs to permanent coastal defenses, including a wide range of types of structures such as seawalls, revetments, groins, artificial reefs, offshore breakwaters, perched beaches, dune reinforcement, core of rubble mound structures, scour protection, etc. (Fig. 21.1). Generally, in coastal engineering any containment of sand encapsulated in geotextile to build flexible and erosion-resistant gravity structures is called “geotextile sand container.” A variety of size and shapes of GSCs have been used, including “geotubes,” “geocontainers,” and “geobags.” The latter have a volume of about 0.05–5 m3 and are generally filled offsite. Geocontainers are much larger (sausage shape up to 700 m3 ) and generally filled in split bottom barges. Geotubes have a diameter up to 5.5 m and are filled directly at the location where they are built (see Fig. 21.2 and Table 21.2). In most cases, it is advantageous to use smaller volume containers because: (i) they are more versatile and can be adapted to build any type of structures; (ii) they can better fulfill any requirements with respect to structure slope and geometry (better tolerance); (iii) maintenance and remedial work are much easier in case of vandalism or failure by wave action; (iv) less tensile strength is required and less change of shape will be experienced, thus resulting in longer lifetime; (v) higher density of the sand fill can be achieved; (vi) less risk of liquefaction of the sand fill is expected and thus less GSCdeformations; (vii) simpler mobilization of the required equipment; and (viii) the smaller the containers, the larger the self-healing capacity of the structure. However, larger containers may be required, for instance, in the case of higher waves forces and for temporary structures. A brief illustration of the container sizes used in coastal engineering is given in Fig. 21.2.

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Fig. 21.1.

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Coastal engineering applications of geotextile sand containers.

In the following, some basic information on geotextiles, including a discussion on the durability issue are first provided. Second, the versatility of the use of GSCs as a soft shore protection alternative to conventional hard structures made of rock and concrete units is illustrated by some example applications. The major part of the chapter will, however, focus on the hydraulic stability of the GSCs under wave action. For this purpose, simple formulae for the stability of slope and crest GSCs will first be proposed which do not take explicitly into account the effect of GSC-deformation and friction between containers. A detailed description of the processes, which may lead to failure under wave action, is then provided to illustrate the necessity of developing more process-based stability formulae. Finally, the new detailed stability formulae and the simple formulae are comparatively analyzed to stress the effect of GSC-deformation on the hydraulic stability.

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Fig. 21.2.

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Range of nonwoven geotextile sand container sizes applied in coastal engineering.

21.2. Geotextile Properties and Durability of Geotextile Sand Containers 21.2.1. Required properties of the geotextile for sand containers Geotextiles and geomembranes, including their related products, are called geosynthetics;19,50 i.e., fabrics that are specially manufactured for civil and environmental engineering applications. While geomembranes are impermeable to water, geotextiles are permeable. The most widely used polymer for geotextiles is polypropylene (PP > 90%), followed by polyester (PET ≈ 5%) and polyethylene (≈ 2%). Based on the manufacturing process two major categories of geotextiles may be distinguished: nonwoven (≈ 60%) and woven geotextiles (≈ 40%). Nonwoven geotextiles are composed of directionally or randomly oriented fibers which are mechanically (needle punching), chemically, or thermally bonded into a loose web. Woven geotextiles are obtained by interlacing two or more sets of yarns (one or several fibers), using conventional weaving processes with a weaving loom. The yarns can be mono-filament, slit film, fibrillated or multi-filament. Besides these two main groups there are also knitted and stitched geotextiles. The main engineering properties of nonwoven and woven geotextiles are comparatively summarized in Table 21.1.

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Properties of nonwoven and woven geotextiles (After Lawson and Kemplin, 1995)∗ .

Types of geosynthetics Nonwovens Heat-bonded Needle-punched Resin-bonded Wovens Mono-filament Multi-filament Flat tape Knitted Weft Warp Stitch-bonded

Tensile strength (kN/m)

Extension at max. load (%)

Apparent opening size (mm)

Water flow rate (volume permeabiltiy) (liters/m2 /s)

Mass per unit area (g/m2 )

3–25 7–90 5–30

20–60 30–80 25–50

0.02–0.35 0.03–0.20 0.01–0.25

10–200 30–300 20–100

60–350 100–3000 130–800

20–80 40–1200 8–90

20–35 10–30 15–25

0.07–4.0 0.05–0.90 0.10–0.30

80–2000 20–80 5–25

150–300 250–1500 90–250

2–5 20–800 30–1000

300–600 12–30 10–30

0.20–2.0 0.40–1.5 0.07–0.50

60–2000 80–300 50–100

150–300 250–1000 250–1000

∗ C. R. Lawson and G. T. Kempton, Geosynthetics and Their Use in Reinforced Soils (Terram Ltd., UIC, 1995).

As shown in Table 21.1, nonwoven and woven fabrics have significantly different properties which can be exploited to produce the best solution for each specific need, including composite fabrics to combine the advantages of both types. The lower tensile strength of nonwoven geotextiles as compared to woven geotextiles might represent a drawback, if the containers have to accommodate very large stresses during installation without failure. Generally, however, the stresses on the containers are much lower in service than during installation. The higher capacity for elongation of nonwoven geotextiles can to some extent compensate the disadvantage of lower tensile strength in the sense that it allows to accommodate large strains without failure. This is particularly important when the container is required to reshape during installation and in service (adaptation to scour development and settlement, self-healing effect). Project experiences have shown that the final dimensions of sand-filled containers made of nonwoven geotextile are comparable to those made of woven geotextile. Based on tests with both materials — woven and nonwoven — the final height of hydraulically filled geotextile containers (tubes) was approved to be 80% of the theoretical diameter.17 The hydraulic permeability of geotextile used for sand containers is particularly important when subject to cyclic wetting and drying (e.g., in tidal regime). Water should be drained from the sand container fast enough to avoid excess pressure build up and to ensure overall stability. Therefore, it is generally required that the geotextile should have a much higher permeability than the sand fill without losing the finer fractions. Alternatively, the geotextile can be selected to fulfill the commonly used filter criteria. Nonwoven geotextiles have a higher permeability and a higher fine retention capability than their woven counterparts, but it should be stressed that the permeability is a function of the fabric thickness, and thus depends on its compressibility under normal stresses. The abrasion resistance is particularly important in the surf environment where coarse angular sand, shell fragments, or coral debris are present. The larger thickness of nonwoven geotextiles and the ability of their structure to retain sand material

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make them more resistant. For testing abrasion resistance of different geotextile products, a special test setup has been introduced by the Federal Waterways Engineering and Research Institute (BAW, Germany) already about 30 years ago.47 For geotextiles being used in the surf environment abrasion tests, standard test setup (as described in BAW Guidelines for Testing Geotextiles for Navigable Waterways4,47) are recommended. The puncture resistance is important in the case of vandalism, drift ice, drift wood, or dropped rock material during construction. The catalogue for geotextile testing in waterway engineering established by the German Institute for Waterway Engineering (BAW) also includes a puncture test setup which allows a 1:1 test according to the actual stone drop energy expected for the specific job conditions.23,47 Due to their high elongation capacity and to the retained sand within the fabric structure, nonwoven geotextiles are more susceptible to limit damage from puncture, including vandalism. A higher friction angle between sand containers is desirable to enhance the hydraulic stability against wave and current actions. Due to their structure, nonwoven geotextiles provide a higher friction (see also Sec. 21.4.25). The resistance against UV-radiation and oxidation still represents one of the most critical issues which can limit the service time of exposed GSCs. The resins, fibers, and yarns of the used woven and nonwoven fabrics need special stabilization packages to avoid deterioration already during production and to guarantee longterm stability for more than 50 years. As they are reasonable cost factors, the special stabilization packages need to be described in the specifications or material certificates with, e.g., confidential information to the certification institute. In general, increasing the thickness of fibers and yarns and increasing weight of the fabric improve UV resistance and long-term stability.49 Dug-up operations from woven and nonwoven geotextiles carried out in the seventies have already shown up to 25 years long-term stability with no indication of further concern to the upcoming years. This is also linked to the fact that the design parameters for surviving handling and installation are giving much reserve for the container simply lying on the sea bed after successful installation. Well-designed and well-installed woven and nonwoven geotextiles for coastal engineering applications have indeed shown that they can be resistant over a long term.9 Despite the significant progress in the use of UV-stabilizers, coating or/and armoring of the exposed geotextile containers still remain the sole alternative to achieve a satisfactory lifetime without damage. Simultaneously, the coating/armoring will also protect the geotextile against abrasion, vandalism, drift ice, and drift wood. The resistance against chemical and microbiological attack of polymers (e.g., polypropylene) is very high, so that no significant loss of strength is expected during the design lifetime. The ability of geotextiles to enhance marine growth and to attract/support diverse invertebrate communities also becomes an important issue, if maximizing the biodiversity of the recruiting communities is relevant for the choice of the type of geotextile. First, results of investigations comparing woven and nonwoven geotextiles in Australia have shown that the latter are more favorable in this respect.5 The marine growth may represent an enhancement of the resistance against

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UV-radiation and abrasion. However, it is still not clear whether it may cause serious changes or detrimental effects on the mechanical properties of the geotextile over lifetime. 21.2.2. Durability and lifetime prediction Unlike conventional construction materials such as rock and concrete, synthetic geotextiles are, even after 50 years experience, for most coastal engineers relatively new products. Therefore, their degradation and long-term performance are still not well understood. Instead of trying to answer the most frequently asked question: “How long will a geotextile structure last?”, it is more practical from the engineering point of view to ask, how long must a geotextile structure last. The expected lifetime when designing “permanent” shore protection structures is typically 20–100 years. Modern geotextiles are designed to be resistant to degradation from UV-radiation, chemical/biological attack, abrasion, and hydraulic loading. Generally, a lifetime in the order of 20–25 years can be expected if damages during construction and through vandalism are avoided. However, not all applications in coastal engineering require such a level of lifetime, for instance, in the case of temporary protection measures. Although lifetimes up to 100 years have been suggested based on accelerated testing and extrapolation, the following question still remains unanswered: “How to predict/achieve a 100 years or more lifetime for geotextile structures applied for shore protection?” Since much is known about the degradation mechanisms and degradation rates of polymer materials, and how these can be reduced or prevented (Fig. 21.3), it seems reasonable to use this knowledge:

Fig. 21.3.

Degradation mechanisms and reduction procedures.

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• to develop further “index” tests similar to those proposed in Annex B of the European Standards EN 13249–13257 and EN 13265 which are believed to ensure a minimum durability of 25 years and • as a basis for planning and interpreting site monitoring (see ISO 13437), including the development of further procedures and techniques to reduce the degradation rates. Ideally, a degradation curve over the design lifetime for each relevant property of the geotextiles such as tensile strength, specific mass in g/m2 , elongation capacity, and hydraulic permeability should be determined, together with the associated acceptable degradation limits at which the geotextile cannot perform its primary function (Fig. 21.4). The most authoritative evidence for long-term durability is generally obtained from monitoring of the degradation under real service conditions. In fact, the empirical evidence of long-term durability from retrieved (nonexposed) geotextiles has shown that the reduction of tensile strength and other important engineering properties strongly depends on the prevalent service conditions and therefore significantly differs from one site to another. Samples of PP (polypropylene) and PET (polyester) nonwoven geotextiles retrieved from 25 sites in France lost up to 30% of their tensile strength after 10–15 years service time as filters, separators, and drains, while no chemical/biological effects were identified.51 A further interesting case was reported by Lefaive,21 showing that the reduction in tensile form strength after 17 years of the same PET straps embedded in a

Fig. 21.4.

Degradation curves (principle sketch).

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concrete facing wall and in the backfill was completely different: 2% in the backfill with pH = 8.5 and up to 40% at the transition between the wall and the backfill where pH values of 13–14 and temperature up to 30◦ C prevailed. This case well illustrates the contribution of alkaline surface attack (25%), internal hydrolysis (5–10%), and mechanical damage to degradation of PET fibers or yarns. The effect of UV-radiation is illustrated by a case reported by Troots et al.,52 where woven PET samples were retrieved after 13 years from an earth embankment: while along the section of the slope covered by vegetation and bitumen to provide protection against UV-attack no significant changes of the geotextile properties occurred; a reduction up to 50% of the tensile strength was identified in the nonprotected part. These and further numerous examples from the literature show that the results/ data from retrieved (essentially nonexposed) fabrics, although very valuable, have serious limitations when intended to be used for the prediction of long-term durability and lifetime of permanent structures. Moreover, the obtained data are often incomplete and relate to conditions that are generally far from those for which the prediction/assessment is being made (Fig. 21.5). In order to make the best use of field data some brief recommendation for future site monitoring are given in Plate 21.1. These limitations and the urgent necessity for both users and manufacturers to predict lifetime of geotextiles have led to the development of accelerated tests which also have serious limitations (Fig. 21.5).

Fig. 21.5. Limitations of present approach to predict/assess lifetime of permanent geotextile structures for shore protection.

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Plate 21.1.

Fig. 21.6.

Recommendations for future site monitoring.

Principle and limitations of accelerated tests.

The principle of accelerated testing is briefly summarized in Fig. 21.6, showing that (i) generally only one dominant degradation mechanism can be considered, thus ignoring the interaction with other mechanisms and (ii) the approach cannot be applied to all degradation mechanisms.

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Moreover, if two degradation mechanisms occur sequentially (e.g., UV-radiation followed by mechanical degradation), then the two mechanisms are analyzed separately and the predicted lifetimes simply added. Further difficulties arise when extrapolating the short lifetime obtained from accelerated tests at increased load frequency/intensity and increased temperature to predict longer lifetime at service conditions. Power laws are often used for extrapolation. For more details, refer to Greenwood and Friday.7 Ideally, the results of both site monitoring and accelerated tests in combination are expected to provide the best basis for the prediction of long-term durability and lifetime. However, a consistent methodology to combine both approaches still needs to be developed (Fig. 21.5). In summary, it can be stated that geotextile applications, although using previous weaker versions of polymer material, performed relatively well over many decades when not exposed to UV-radiation.56 Most failures observed were rather due to faulty design, incorrect choice of material, and poor quality of installation. The knowledge available on the degradation mechanisms, although still limited, allows to predict rationally lifetime up to about 25 years. Since rational prediction cannot foresee problems for which there is no empirical evidence or scientific basis, the primary goal of future research toward the assessment of durability and lifetime is to improve the understanding of all relevant degradation mechanisms — separately as well as in combination — by making use of both site monitoring and laboratory testing. Future significant improvement of UV-resistance and aesthetical aspects as well as the improvement of the long-term performance of GSCs against large wave loads represent further R&D challenges toward avoiding to cover or armor geotextile structures.

21.3. Example Applications of Geotextile Sand Containers for Shore Protection The types of GSCs used in coastal engineering as temporary or permanent structures are generally referred to in the literature as geotubes, geocontainers, or geobags (see Table 21.2 and Sec. 21.1). For permanent coastal structures, small volume containers offer more advantages and are therefore often preferred (see Sec. 21.1). In particular, they are more versatile in application and are used for different class of structures, including dune reinforcement, seawalls and revetments, detached breakwaters and artificial reefs, groins, etc. (see Fig. 21.5). Comprehensive largescale model investigations on GSCs used for the scour protection of monopile foundation of offshore wind turbines have also been conducted by Gr¨ une et al.8 The results are described in more detail in a final report by Oumeraci et al.29 Reviews of example applications related to geotextiles in general but, also including geocontainers, can be found in Heerten,11,12 van Santvoort et al.,53 and Pilarczyk.31 Comprehensive reviews on the applications of geosynthetics in hydraulic engineering and for the protection of land fill (including coastal areas) which may also provide valuable information and inspiration for the application

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564 Table 21.2. Type 1. Geotubes

2. Geocontainers

3. Geobags

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Volume (m3 ) Sand fill Generally >700 m3

On site

Generally 100–700 m3

Split bottom barge Offsite

0.05–5 m3

Shape

Applications

• • • Cylindrical/pillow • (D < 5 m) • Cylindrical (D = 1–55 m)

Pillow, box, mattress

Groins Containment dikes Nonpermanent structures Reef structures (surf zone) Defense structures against tsunami As soft rock units to build any type of coastal structures. Also for scour protection and dune reinforcement

in coastal engineering are given by Heibaum et al.15 and Kavazanjian et al.,18 respectively. More specific reviews related to geocontainers are provided by Fowler and Trainer,6 Lenze et al.,22 Lawson,20 Saathoff et al.,48 and Jackson et al.16 Rather than trying to duplicate the examples from the aforementioned reviews and to provide a further comprehensive review, it is attempted in the following to select only few examples from three classes of structures and applications: (a) long-shore barriers in the form of seawalls, revetments and dune reinforcement, (b) cross-shore barriers in the form of sea groins, and (c) a new possible application of GSCs as a core of rubble mound structures. The latter type of application may also become particularly important when armoring the GSC-structure is required due to too severe wave attack, abrasion, UV-radiation, and vandalism. 21.3.1. Revetments, seawalls, and reinforcement of beach–dune system Most of the applications of geotextile containment in coastal engineering belong to this type of shore protection; i.e., the containment is built directly along the shoreline to prevent erosion and to stabilize a beach–dune system during storm surge (Fig. 21.7). For this purpose, different types of containments have been applied, very often as a last defense line in combination with beach nourishment. An impressive example of the performance of such a last defense line behind a beach nourishment is the wrapped sand containment needle-punched composite geotextile (woven PP slit film and nonwoven PET) to reinforce a dune on the island of Sylt (North Sea, Germany) is shown in Fig. 21.8. The stability of this stepped barrier was successfully tested in the Large Wave Flume (GWK) of Hannover. Since 1990, it survived several storm surges with water levels of about 2.5 m above normal and wave heights up to 5 m. Only the sand cover was removed, confirming that the nickname “Bulletproof Vest” commonly given to this type of construction is appropriate. More details on the design and construction of this shore protection are given by Nickels and Heerten26 and Lenze et al.22 Similar geotextile sand containments have also been used successfully at many other sites. Some of them are well documented by ACT.1

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Fig. 21.7. Seawall made of geotextile sand containers (principle sketches): (a) plan view and (b) cross-section.

Fig. 21.8. Geotextile containment for dune reinforcement, Sylt/Germany (extended and modified from Ref. 26).

More flexible and much simpler in both engineering design and installation, but equally efficient are smaller volume containers (Fig. 21.9). Moreover, smaller containers have many advantages over larger containers and tubes (see Sec. 21.1 and Ref. 2). A comparative analysis of containers with V = 0.75 m3 and containers with a 30 times larger volume for dune reinforcement which confronted the pros and cons of both methods from the client and contractor view point clearly resulted in the selection of the smaller containers.2

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(a)

(b)

(c) Fig. 21.9. Beach and dune reinforcement with geotextile sand containers. (a) Beach reinforcement in Australia, (b) Stockton Beach Revetment,48 and (c) Dune Reinforcement in Wangerooge, North Sea/Germany.54

Using geotextile containment for a long-shore barrier in the form of seawalls, revetment or beach-dune reinforcement has several advantages over hard barrier such as rock structures (see Sec. 21.1), but there are also some drawbacks. The most important drawback is that the sand cover has to be fully or partially rebuilt after each important storm surge because a natural recovery is not always possible. Due to their lower permeability and larger slope steepness, GSC-structures have generally higher reflection coefficients and higher wave overtopping rates than rubble mound structures (Fig. 21.10).

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(a)

(b) Fig. 21.10. Wave reflection and overtopping performance of GSC-seawalls.28 (a) Overtopping performance and (b) reflection performance.

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Fig. 21.11.

Down drift erosion induced by seawalls.

Moreover, it should be kept in mind that a long-shore GSC-barrier also induces a down drift erosion similar to those induced by hard structures (Fig. 21.11), including erosion of the foreshore. 21.3.2. Sea groins Geotextile sand containers, including geotubes, have often been used for emerged and submerged groins (Fig. 21.12). Generally, the containers are uncovered and directly exposed to wave impact, abrasion, and UV-radiation. Therefore, the fabric should be heavily treated for UV-stability and should consist of an inner layer for strength and an outer layer for robustness, durability, and abrasion resistance (geocomposite). Sometimes, the

Fig. 21.12.

Sea groins (principle sketches).

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fabric is made in a color that blends naturally with the beach environment.24 Heavy-duty UV stabilized nonwoven needle-punched geotextiles with high tenacity polyester thread in all seams (min. 80% of basic fabric strength) have also been successfully used.44 Despite large displacement that occurred during storms, the groin continued to provide protection and even withstood abrasion and UV attack over 10 years.48 To reduce the costs in the case of larger groins and larger projects, the groin core can be made of smaller container with lower requirements while strengthening the outer protective layer.24 Further example applications are provided by Saathoff et al.48 and Restall et al.44,45 21.3.3. Core of rubble mound structures There are several reasons which might lead the engineer and other decision-makers in practice to use sand instead of conventional quarry run for the core of rubble mound breakwaters and structures, including among others: (i) Nonavailability of rock material in sufficient quantities and at affordable costs. (ii) Avoiding sediment infiltration through rubble mound structures which may result in the shoaling of navigation channels and harbor basins, and thus in higher maintenance dredging costs. (iii) Reduction of wave transmission through the structure which might particularly be crucial in the case of long waves. On the other hand, the use of sand as a quasi-impervious core instead of quarry stone would result in an increase of (i) wave setup and runup at the structure, (ii) wave overtopping, (iii) wave reflection, which might be detrimental to the stability of the structure, to the operation on and behind the breakwater (due to excessive overtopping) as well as to navigation and seabed stability. Moreover, serious difficulties arise in practice when trying to design and construct the filter to protect the sand core against wash out by wave action. Applying geometrically closed filter criteria would result in very complex, multiple, and relative thin filter layers which will not only be very costly and very difficult to build in larger water depths, but also might certainly fail due to the almost unpredictable very complex loading conditions of the sand core under cyclic pulsations by waves and entrained air at the interface with the last filter layer. Such failures have indeed been observed under both laboratory and field conditions in the past. Laboratory evidence has also shown that introducing the so-called “geometrically open filter” criteria to design a “hydraulic sand-tight filter” may reduce the number of filter layers. However, the main practical difficulties mentioned above will remain, including those associated with the long-term stability of the sand-core due to the high complexity of the loading and its uncontrollability during the entire storm duration and over the life cycle of the structure.

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Geotextile filters might present themselves as an alternative to the very complex, costly, and uncertain filter made of multiple layers of granular material. However, geotextile mats are not only difficult to install under waves and currents, but also may introduce a shear surface which might be detrimental to the stability of the armor layer. Therefore, geotextile filters need careful installation by experienced contractors with special equipment, and the design should address the shear plane between the geotextile filter and the layers below and above. Also, “sandmats” as composite products of nonwoven geotextiles and sand needle-punched together may be a suitable solution.13,14 A more feasible alternative is to use a core made of GSCs. This will not only allow to overcome the aforementioned core stability problems, but also to provide (i) a better erosion stability of the core and (ii) an increased stability against seismic loads as compared to a core simply made of loose sand. However, many of the drawbacks mentioned above remain with respect to wave setup, runup, overtopping, reflection, and armor stability in comparison to a conventional core. Therefore, an extensive research program has been initiated at LeichtweissInstitute to study both hydraulic performance and armor stability, including the processes involved and the development of prediction formulae for the design of a class of rubble mound structures with a core made of geotextile sand containers (Fig. 21.13). The first phase of this research program which is concerned with hydraulic model tests to study in the twin-wave flumes of LWI the hydraulic performance and the armor stability of a rubble mound breakwater made of GSCs as compared to its conventional counterpart with a core made of quarry stones, has been completed

Fig. 21.13.

Class of geocore structures in comparison to conventional rubble mound structures.

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and the results are published by Oumeraci et al.30 Prediction formulae for wave reflection, wave runup and overtopping, wave transmission, and armor stability have been determined for the nonconventional breakwater in comparison with the conventional rubble mound breakwater.30 21.4. Hydraulic Stability of Geotextile Sand Containers Depending on the prevailing wave loads and degradation mechanisms, GSCs may experience different types of failure modes: (i) hydraulic failure modes, (ii) geotechnical failure modes, and (iii) failure modes related to the geotextile itself (Fig. 21.14). In the following, only the first type which is related to hydraulic stability under wave loads will be addressed. First, simple stability formulae without explicit account of the effect of deformation will be proposed separately for the containers on the slope and the containers on the crest of the structure (called hereafter “slope containers” and “crest containers”). Second, the necessity of a better understanding of the processes responsible for the deformation of the sand containers under wave loads as well as their effect on the hydraulic stability is illustrated by some selected results from recent research. Finally, more detailed stability formulae are proposed which can also explicitly account for the deformation effects, including a comparison with the simple stability formulae.

Fig. 21.14.

Failure modes for geotextile sand containers.

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21.4.1. Simple stability formulae Due to the different wave loads and boundary conditions which prevails on the slope and on the crest of a coastal structure, a different stability behavior and thus different stability formulae are expected for the containers on the slope and those on the crest. The following results are extracted from the research report27 of two comprehensive laboratory studies: small-scale model tests performed in the wave flume of Leichtweiss-Institute, using 1 liter sand containers subject to random waves up to 20 cm height and large-scale model tests in the large wave flume of the Joint Coastal Research Centre (FZK) of both Universities of Hannover and Braunschweig, using 150 liter sand containers subject to random waves up to 1.6 m height.27,28 21.4.1.1. Stability of slope containers The sand containers on the slope which are located around the still water level are repeatedly moved up and down by the waves rushing up and down the slope, leading to an incremental seaward displacement of the containers. This dislodgement/pull out effect is illustrated by Fig. 21.15 as observed in the wave flume and in the field. Based on the HUDSON-formula for the hydraulic stability of rock armor units (nondeformable) and similarly to WOUTERS,56 a stability number Ns is formulated and postulated to be a function of the surf similarity parameter ξ0 , which includes both the slope steepness tan α as well as the significant wave height Hs and the wavelength Lop (Plate 21.2): Ns =


Cw =√ . ξ0 −1 ·D Hs

ρE ρW



(21.1)

 With the surf similarity parameter ξ0 = tan α/ Hs /Lop expressed in terms of the deepwater length Lop = gTp2 /2π (Tp = peak period of wave spectrum) the following stability formula is obtained in terms of the characteristic size D of the container: 3/4

D=

Hs

1/2

· Tp · (tan α)1/2 .  2π 1/4
ρE Cw · d ρW − 1

(21.2)

Defining the characteristic size D as D = lc sin α according to Wouters56 and according to the principle sketch in Plate 21.2, Eq. (21.2) can be reformulated in terms of the length lc of the slope containers as: 3/4

lc = Cw ·




2π g

Hs · 1/4


 Tp

ρE ρW

.  sin 2α −1 2

(21.3)

The empirical parameter Cw was determined by stability tests in a large and a small wave flume to Cw = 2.75. In Fig. 21.16 only the results of the large-scale model tests are plotted to illustrate that the threshold curve between stable and unstable containers is obtained for Cw = 2.75.

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(a)

(b)

(c)

Fig. 21.15. Hydraulic failure modes of slope containers. (a) Wave uprush and downrush on slope containers, (b) pull-out effect in the FZK-large wave flume, and (c) pull-out effect in a dune reinforcement (Courtesy of Heerten).

For Cw = 2.75 and g = 9.81 m/s2 , Eq. (21.3) reduces to:  Tp   − 1 · sin(2α)

3/4

lc =

1.74 ·




with Hs = Tp = α= ρE =

significant wave height (m), peak period of waves (s), slope angle of structure (◦ ), bulk density of GSC (kg/m3 ),

Hs ρE ρW

·

(21.4)

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Plate 21.2.

Fig. 21.16.

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Stability of slope containers based on Hudson formula.

Stability of slope containers from large-scale model tests (modified from Ref. 28).

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ρW = ρE = n= ρS =

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density of water (kg/m3 ), (1 − n) · ρs + ρW · n (with ρE ≈ 1800 kg/m3 for sand), porosity of fill material (–), density of grain material (kg/m3 ).

The stability formula in Eq. (21.4) expressed in terms of the required container length lc is plotted in Fig. 21.17 for a structure slope angle α = 45◦ in order to illustrate the sensitivity to the significant wave height Hs and to the peak period Tp . To compare the results with those of the Hudson-formula for rock armor, the volume of the tested slope containers (V = lc ·0.46lc ·0.14lc = 0.065lc3) is considered. For Hs = 1.5 m and Tp = 4 s, about the same required weight for slope containers with ρE = 18,000 kg/m3 is obtained as for a rock armor with ρE = 2700 kg/m3 and Kd = 2.0. Using a 10 times higher Kd -value (Kd = 20) and applying Hudsonformula in this specific case for slope containers would indeed provide the same result as the proposed formula. However, such an approach is not applicable as the stability of slope container is very sensitive to the wave period and the Kd value is expected to be a function of both wave height and wave period, due to the deformation effect caused by wave action. In fact, if for the same wave height Hs = 1.5 m the wave period is increased from Tp = 4 s to Tp = 6 s the required length lc will increase by more than 20% (Fig. 21.17). Moreover, due to the effect of deformation of the slope containers on the long-term stability, it is not advisable to use unprotected slope containers for design wave heights of Hs > 1.5–2.0 m.

Fig. 21.17.

Required length of slope containers for hydraulic stability.

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Fig. 21.18.

Hydraulic failure modes of crest containers.

21.4.1.2. Stability of crest containers The sand containers on the crest of the structure may fail due to two possible mechanisms (Fig. 21.18): (i) uplifting during the wave uprush process and shoreward displacement by the wave overtopping flow, (ii) dislodgement and pull-out effect similar to the mechanism observed for the slope containers. Due to the boundary conditions of the crest containers which are more critical than those of the slope containers (no overburden from upper layers), it is expected that the stability of the crest containers will be more critical than that of the slope containers if the crest level of the structure is not high enough. The relative freeboard Rc /Hs , therefore, represents the most important influencing parameter. In fact, it was difficult to identify a noticeable effect of the surf similarity parameter

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Fig. 21.19.

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Stability of crest containers form large-scale model tests.28

ξ0 on the stability number Ns as clearly observed for slope containers (Fig. 21.19). However, plotting at the top left of Fig. 21.19, the stability number Ns against the relative freeboard Rc /Hs shows that the stability number of the crest containers increases with increasing relative freeboard Rc /Hs according to the following linear relationship (Fig. 21.19): Ns =


Rc  = 0.79 + 0.09 , Hs −1 ·D Hs

ρE ρW

(21.5)

(see Plate 21.2) with D = lc sin α substituted in Eq. (21.5), a stability formula for crest containers is obtained in terms of the required container length lc : lc =


ρE ρW

−1




Hs

 . Rc 0.79 + 0.09 H sin α s

(21.6)

In a similar way as in Fig. 21.17 for slope containers, the required container length lc in Eq. (21.6) is plotted against the design wave height Hs for different relative freeboard Rc /Hs = 0 − 2.0 (Fig. 21.20). Considering exemplarily a typical value Rc /Hs = 1.2 and Hs = 1.5 m, a container length lc = 3.15 m and a container weight W = 36.6 kN are obtained which are much larger than those obtained for the slope containers in Fig. 21.17.

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Fig. 21.20.

Required length of crest containers for the hydraulic stability.

21.4.2. Processes affecting the hydraulic stability 21.4.2.1. Position of the problem and need to improve process understanding As observed in the experiments and shown in Fig. 21.21, the dislodgment and pull out of the slope containers by wave action, including the sliding and overturning of crest containers are strongly affected by the deformation of the sand containers. Simple stability formulae like those proposed in Sec. 21.4.1 cannot explicitly account for the deformation and other mechanisms affecting the hydraulic stability. An improved understanding of the processes and mechanisms is needed in order to (i) possibly avoid failure (engineering judgment), (ii) develop more process-based stability formulae (see Sec. 21.4.3). For this purpose it is necessary to address the following aspects: (i) hydraulic permeability of GSC-structures and its effect on the stability, (ii) wave loads and identification of the most critical loading case and location of the containers, (iii) internal movement of sand fill and its effect on the stability, (iv) effect of the friction angle between geotextile containers on the stability, and (v) effect of the container deformations on the stability. The content of this section and next section represents a brief summary of selected key results which have essentially been obtained from comprehensive laboratory studies, including several types of experiments in combination with numerical

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Fig. 21.21.

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Observed failure mechanisms under wave loads.

studies using a CFD-code (RANS-VOF model COBRAS developed at Cornell University) partially coupled with a CSD-code (FEM-DEM code “UDEC” developed by Itasca). These studies were performed in the framework of the PhD-research work by Recio37 and the results are described in more detail in several research reports33,35,36,38–40 and in a PhD thesis.37 21.4.2.2. Hydraulic permeability of GSC-structures Surprisingly, no information on the assessment of the hydraulic permeability of structures made of GSCs could be found in the published literature, although permeability represents an important parameter for both hydraulic stability and hydraulic performance (wave transmission, reflection runup, and overtopping). Moreover, reliable permeability values of GSC-structures are also needed for numerical simulations. Comprehensive laboratory experiments supplemented by numerical simulations using the COBRAS-code were performed for the first time to investigate the hydraulic permeability of several types of GSC-structures for different modes of placement and sizes of the containers, and based on the achieved results then to develop a conceptual model for the assessment of the hydraulic permeability.37,38,41 The key results may be summarized as follows: (i) The permeability of a GSC-structure is essentially governed by the size of the gaps between the containers. The flow through the sand fill can therefore be neglected in the computation. (ii) The hydraulic permeability of a GSC-structure is generally more than 10 times higher and 10 times lower than the permeability of sand (k ≈ 10−3 m/s) and gravel (k = 10−1 m/s), respectively (see Fig. 21.22).

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Fig. 21.22. Hydraulic geocontainers.38,41

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permeability

coefficients

for

different

mode

of

placement

of

(iii) The mode of placement of GSCs generally affects the permeability of the GSC-structure (Fig. 21.22). However, randomly placed GSCs and longitudinally placed GSCs have approximately the same permeability coefficient: k = 2.4 cm/s and k = 2.3 cm/s, respectively. To achieve a lower permeability an “interlaid” placement (gaps of the longitudinally placed containers are blocked by transversally placed containers) is required, allowing to reduce the permeability up to k = 1 cm/s and even to k = 0.5 cm/s. (iv) A conceptual model and a systematic procedure to predict the hydraulic permeability is proposed by Recio and Oumeraci.38,41 Validation by experimental data show that uncertainties less than 30% would be expected when applied to prototype containers. 21.4.2.3. Wave-induced loads and critical location of slope containers Based on the pressure recorded by load sensors around an instrumented container placed at different locations along the slope of the GSC-structure subject to both breaking and nonbreaking waves, the horizontal and vertical components of the total wave force on the instrumented containers is obtained for each time during a wave cycle. A typical result of the measurement is exemplarily provided in Fig. 21.23

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Fig. 21.23.

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Wave-induced force: Identification of critical container location.38– 40,42

to illustrate that the highest horizontal force in seaward direction and the highest vertical force in upward direction occur on the container located just below still water level. Using pressure measurements within the gaps between slope containers for both breaking wave impacts and nonbreaking waves, it is found that the former are less critical for the hydraulic stability than the nonbreaking waves rushing up and down the slope and causing more damage. In fact, the impact pressure induced by breaking waves is of much shorter duration. Moreover, it is strongly damped when propagating inside the gap (Fig. 21.24). 21.4.2.4. Movement and redistribution of sand inside the containers induced by wave action The analysis of the video records of the movement of colored sand inside a transparent permeable container built in the slope of the GSC-structure (Fig. 21.25) and subject to wave action has shown that (Fig. 21.26): (i) Noticeable sand movements occur only for larger waves that are capable to substantially move the front part of the container up and downward during the wave runup and rundown process [Fig. 21.26(b)]. After about 30 wave cycles the internal sand movement decreases significantly due to the accumulated sand at the seaward front of the container [Fig. 21.26(c)]. (ii) Due to the sand fill redistribution at the seaward front of the container, the latter deforms, thus offering a larger impact surface area to the mobilizing wave forces and reducing the contact area with neighboring containers [Fig. 21.26(c)].

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Fig. 21.24.

Fig. 21.25.

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Impact pressure propagation inside a gap between geotextile sand containers.42

Permeable transparent container for the investigation of internal movement of sand.38

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Fig. 21.26.

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Internal movement and redistribution of sand fill in the transparent container.42

With the increased mobilizing forces and the decreased resisting forces, an incremental lateral seaward displacement of the deformed container occurs (pull-out effect) which causes again the start of the internal sand movement in a similar way as during the first wave cycles [Fig. 21.26(a)]. These results have considerable practical implications as the internal sand movement is responsible for the deformations of the container which affect • the hydraulic stability by reducing the contact area between GSCs and by increasing the drag and lift forces due to the increased exposed areas, • the crest level of a GSC-structure. Even in the laboratory it is observed that the height of sand containers is reduced by 4% after placement under water and by further 6% due to wave action. As a result, a total reduction of the height of the GSC-structure of about 10% was observed.38,41 Since these effects are strongly dependent on the adopted sand fill ratio, future research and design guidance should be directed toward the definition of an optimal sand fill ratio by accounting for the deformation properties of the geotextile and

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Table 21.3.

Results of friction tests in a shear box25 (30 cm × 30 cm).25 Measured peak friction stress τ (kN/m2 ) at normal stress σ (kN/m2 ) σ = 50 kN/m2

σ = 100 kN/m2

Nonwoven versus nonwoven geotextiles

27 29

48 47

Woven versus woven geotextiles

16 14

29 26

Interface

by balancing the advantages and drawbacks of high sand fill ratio. Moreover, this issue should also be explicitly addressed in future standards and guidelines due to the considerable effect of the sand fill ratio on stability and long-term performance. 21.4.2.5. Friction between geotextile sand containers Friction angles between woven geotextiles vary from about 12◦ (e.g., MirafiGT500) and 20◦ (e.g., Geolon PP120S), while for nonwoven geotextiles values of 20–26◦ (mechanical bound) or even 20–30◦ (thermal bound) are more common. Generally, the friction behavior between geotextiles increases with the surface roughness of the geotextiles. The higher the transmittable shear stress τ in relation to the induced normal σ stress in an interface (τ /σ), the higher is the friction behavior. Table 21.3 shows shear box test results for different interfaces at normal stresses of 50 kN/m2 and 100 kN/m2 .25 The results of numerical simulations using the partially coupled CFD and CSD codes (COBRAS and UDEC) previously validated by laboratory data, as shown in Fig. 21.27, highlight the significant effect of friction between the GSCs on the hydraulic stability. The effect of the friction angle on the stability is particularly important within the range of the practically relevant values (15–30◦), implying that the friction between the containers should be explicitly considered in future stability formulae. 21.4.2.6. Effect of container deformations on the stability It was shown in Sec. 21.4.2.4 that the sand fill is redistributed, resulting in deformations of the containers (Fig. 21.26). These deformations affect the stability of the containers in the following manner: (i) Reduction of the stability against sliding caused (a) by the increase of the drag and uplift forces as a result of the increased exposed areas AS and AT (Fig. 21.28) as well as (b) by the decrease of the resisting force as a result of the decreased friction area between the containers (Figs. 21.28 and 21.29). (ii) Reduction of the stability against overturning caused by the increase of the mobilizing drag and uplift forces as mentioned above, but also by the seaward

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Fig. 21.27.

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Effect of friction between GSCs on hydraulic stabilization.42

Fig. 21.28. Increase of exposed areas of drag and uplift forces and decrease of friction areas due to container deformation.

shift of the center of gravity of the deformed container leading to a reduction of the resisting moment (Fig. 21.30). A closer experimental and numerical examination of the variation of the “effective” contact areas between containers during wave action (i.e., those areas

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Fig. 21.29.

Fig. 21.30.

Effect of container deformation on sliding stability.38,42

Effect of container deformation on overturning stability.42

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Fig. 21.31.

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Reduction of effective contact areas between containers during wave action.37,42

which contribute to the resistance against dislodgment by friction) has shown that the “effective” contact areas (i) decrease due to the upward movement of the front part of the containers (Fig. 21.31). (ii) increase with increasing slope angle of the GSC-structure. 21.4.3. Process-based stability formulae The insight in the processes affecting the hydraulic stability as described in Sec. 21.4.2 has clearly highlighted the necessity of an explicit account of these processes in future stability formulae, at least those processes which mostly affect the stability such as the effect of deformations of the container and the friction between the containers. In order to examine more closely the effect of deformation on the stability, new process-based stability formulae are proposed: first, without any account of

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Fig. 21.32.

Sliding stability formulae without deformation effect.

the deformation effect, and then correction factors are introduced to account for the deformation effect. Finally, a comparative analysis between the results of the stability formulae with and without deformation effect is provided for both slope and crest containers. It should be underlined that the new formulae proposed below were derived analytically for the geometry of the containers commonly used in coastal structures, i.e., with a container length lc which is twice the container width (lc /2) and five times the container height (lc /5) resulting in the following relationships for the volume of the container V and the application area of the drag and uplift forces AS and AT , respectively [Fig. 21.32(a)]. V = 0.1lc3 , AS = 0.1lc2 ,

(21.7)

AT = 0.5lc2 . These relationships provide the geometrical parameters that govern the resisting forces (weight) and the mobilizing forces (drag, inertia, and uplift forces), thus allowing to express the stability formulae in terms of the container length lc (see also Sec. 21.4.1). If, however, other container geometries, and thus other relationships which differ from those in Eq. (21.7) are adopted, the stability formulae can be modified accordingly. Further indications on how to proceed with such modifications and on the limitations of the proposed stability formulae will also be given below.

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21.4.3.1. Stability formulae without deformation effect (a) Stability against sliding: A sand container is stable as long as the resisting force FR = µ(FGSC − FL ) generated by the resulting normal force (FGSC − FL ) due to friction remains larger than the drag force FD and the inertia force FM (Fig. 21.32): µ (FGSC − FL ) ≥ (FD + FM ) .

(21.8)

And with the relative density parameter ∆ = (ρE /ρW ) − 1 (see Plate 21.2): 

µ ∆gV − 0.5CL AT · u

2



∂u 2 . ≥ 0.5CD AS u + CM V ∂t

(21.9)

Given the considered container geometry, relationships similar to those in Eq. (21.7) which provide V , AT , and AS as a function of the container length lc can be obtained and substituted in Eq. (21.9) which is then solved to obtain either the required length lc or the required mass WGSC of the container. Using, for instance, the geometry described by Eq. (21.7), the stability formulae are given in Fig. 21.32 in terms of the required length lc or mass WGSC of the container. (b) Stability against overturning: A sand container is stable as long as the stabilizing moment induced by the weight of the container under buoyancy FGSC remains larger than the mobilizing moment induced by the drag, inertia, and uplift forces FC , FM , and FL (Fig. 21.33): FGSC · rs ≥ FD · rh + FM · rh + Fc · rs

(21.10)

(ρE − ρw )gV · rs ≥ 0.5ρw CD u2 As · rh + ρw CM

∂u V · rh + 0.5ρW CL u2 AT · rs . ∂t

(21.11)

Given the considered container geometry, relationships similar to those in Eq. (21.7) which provide V , As , and AT , but also the lever arms rs and rh as a function of the container length lc can be obtained and substituted in Eq. (21.11). The latter is then solved to obtain either the required length lc or the required mass WGSC of the container. Using, for instance, the geometry described by Eq. (21.7), the overturning stability formulae are obtained in terms of the required length lc or mass WGSC of the container (Fig. 21.33). More details on the force coefficients CD , CM , and CL as well as on further input parameters required in the stability formulae summarized in Figs. 21.32 and 21.33 will be given in Sec. 21.4.3.2. 21.4.3.2. Stability formulae including deformation effect The effect of the container deformations on the stability is explicitly accounted for by introducing analytically derived deformation factors into the formulae for the drag force, lift force, inertia force, and resisting forces for both hydraulic failure

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Fig. 21.33.

Overturning stability formulae without deformation effect.

modes37,43 : sliding and overturning. The deformation factors are obtained as correction factors, describing the changes of (Plate 21.3): The resisting force FR against sliding. The correction factor KSR is obtained as the ratio of the effective weight contribution to the frictional force FR with and without deformation. The resisting moment against overturning. The correction factor KOR is obtained as the ratio of lever arm rs of the container weight FGSC under buoyancy with and without deformation effect. The mobilizing drag, lift, and inertia forces contributing to sliding. The correction factors of the drag force FD and lift force FL (KSCD and KSCL ) are obtained as the ratios of the areas AS and AT with and without deformation effect. The correction factor KSCM is assumed to be 1.0 since the container volume V remains constant. The mobilizing moment induced by the drag, lift, and inertia forces. The correction factor for the moments induced by the drag and lift forces (KOCD and KOCL ) area is obtained as the ratio of describing the changes of both surface areas (AS and AT ) and lever arms (rm and rs ) of the drag and lift forces FD and FL . The correction factor KOCM is obtained as the ratio of lever arms rsn of the inertia force with and without deformation effect. The values of the correction factors suggested in Table 21.4 are derived on the basis of a number of simplifying assumptions (see Ref. 37 for more details).

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Plate 21.3.

Stability formulae including the effect of deformation.

Table 21.4.

Deformation factors and force coefficients.

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Plate 21.4.

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Defining of parameters and typical values to be used in the stability formulae.

Among the most important assumptions the following are noteworthy: sand fill ratio of 80% and a slope angle of the GSC-structure of 45◦ . Indications to account for other slope angles and recommendations on further research to overcome most of the simplifying assumptions are given by Recio.37 Moreover, the force coefficients CD , CL , and CM are also given as a function of the Reynolds number for different locations and boundary conditions which may represent different practical applications (scour protection on the sea bed, artificial reef, slope containers, and crest container of a surface piercing structure such as revetments, seawalls, groins, etc.). The proposed values of CD , CM , and CL have been determined on the basis of systematic laboratory experiments.35 In Plate 21.4 the parameters used in the stability formulae described in Figs. 21.32–21.35 are defined and typical values are also given, including some remarks on the limitations of the suggested values. 21.4.3.3. Comparative analysis of stability formulae with and without including deformation effect In order to illustrate the effect of deformation the results of the new more processbased stability formulae with and without consideration of the deformations of the container as proposed in Secs. 21.4.3.2 and 21.4.3.1 are compared in Figs. 21.35 and 21.34 for a sloping revetment with an angle of 45◦ , a water depth d = 4 m at the structure, and a peak period Tp = 6 s of the waves. Moreover, the simple stability formulae for slope containers (Fig. 21.34) and crest containers (Fig. 21.35) proposed

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Fig. 21.34.

Stability formulae for slope containers: Comparative analysis.

Fig. 21.35.

Stability formulae for crest containers: Comparative analysis.

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in Sec. 21.4.1 are also considered to illustrate the difference with the results from the more process-based formulae. Depending on the range of the design wave height Hs , the following remarks may be drawn from the comparison of the formulae for the slope containers (Fig. 21.34): (i) For smaller design waves (Hs ≤ 1.5 m): the simple stability formulae proposed in Sec. 21.4.2 are too conservative and the deformation effect on the stability obtained from the comparison of the new formulae proposed on Sec. 21.4.3 are relatively small. (ii) For larger design waves (Hs ≥ 2.5 m): the simple stability formulae become more unsafe with increasing wave height. The effect of deformation also increases with the increase of the design wave height. Depending on the range of the design wave height Hs the following remarks may be drawn from the comparison of the stability formulae for the crest containers (Fig. 21.35): (i) For smaller design waves (Hs ≤ 1.5 m): the simple stability formula proposed in Sec. 21.4.2 is slightly conservative and becomes unsafe as soon as the significant wave height exceed 1.5 m. It can therefore be used instead of the more processbased formula only for Hs ≤ 1.5 m. However, the effect of deformation is higher than in the case of slope containers. (ii) For larger design waves (Hs ≥ 2.5 m): the simple stability formula by Oumeraci et al. (2003) becomes more unsafe with increasing wave height Hs . The effect of deformation on the stability also increases with increasing wave height. Comparing Figs. 21.34 and 21.35 also confirms that for commonly used relative freeboards Rc /Hs in the order of 1.2, much larger containers are required for the crest than for the slope of the structure. Moreover, it also shows that the effect of deformation on the stability dramatically increases with increasing design wave height and is much more pronounced for the crest containers than for the slope containers.

21.5. Concluding Remarks After about 50 years of successful experience of geotextile applications in coastal engineering, applications for shore protection are well established. Most failures which have yet been experienced are rather due to bad design, bad choice of material, and/or bad installation. Geotextile sand containers (GSCs) represent nowadays a soft and low cost alternative to conventional hard structures made of rock and concrete. Moreover, GSCmade structures are environmentally more appropriate and more easily reversible as they need essentially sand as construction material which is generally available at any coastal site. As “soft rock” GSCs can be manufactured at any size and used to build any type of shore protection structure, including scour protection, dune reinforcement, and repair of undermined structures.

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However, several problems still need to be solved in order to make use of the full potential of GSCs, particularly including (i) the long-term durability and lifetime prediction and (ii) the hydraulic stability under severe wave action. The facts, limitations and research needs related to the durability and lifetime issue may be summarized as follows: (i) Although field evidence (mostly nonexposed geotextile) is available over about 50 years, useful information extracted from nonretrieved samples is often very incomplete or very limited as the results can hardly be transferred to other sites, to present geotextile products and to time durations and conditions beyond the experienced service time/conditions. (ii) Results of accelerated tests — even in combination with those from site monitoring and retrieved samples — are still very limited when trying to predict lifetime of more than about 25 years. In fact, a systematic methodology to combine both laboratory and field monitoring approaches for this purpose is still missing. (iii) Recommendations for future research in the mid-term and long-term should focus on two directions: (a) improvement of the understanding of the degradation mechanisms, including physical, biological, and chemical processes, both isolated and in combination, and (b) development of a consistent framework for the assessment of long-term durability and lifetime (up to 100 years and more) based on the results of the above and including site monitoring, laboratory testing, and theoretical/numerical modeling. (iv) Meanwhile, in order to contribute to solve the present problems in practice, the following two recommendations might be helpful. (a) Apply engineering judgment based on the present knowledge of degradation mechanisms rather than relying on “extrapolation approaches” to predict lifetime and (b) apply where feasible and necessary well-established measures to enhance long-term performance, including, for instance, appropriate stabilizers and additives (e.g., against UV-radiation), more robust geotextiles (e.g., multi-layer), and geotextile coating (e.g., against abrasion and vandalism), sand covering (e.g., against UV-radiation and vandalism and to enhance aesthetical aspect), rock covering (e.g., against ice loads, debris, very high waves, UV, and vandalism) and setup of a consistent maintenance plan. Moreover, the collaboration with experienced companies with regard to geotextile production, manufacturing of sand containers, filling, handling, and placing is also strongly recommended. Regarding the hydraulic stability under severe wave action, the present state of knowledge, the limitations, and the needs for future research may be summarized as follows: (i) The large experience and stability formulae available for rock and concrete armor units cannot simply be transferred to GSCs, essentially due to the deformation of the GSCs under very severe wave attack. (ii) The deformations of the GSCs are essentially induced by the internal movement of the sand fill of the containers.

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(iii) The effect of the deformations of the GSCs on the hydraulic stability rapidly increases with the severity of wave attack, depending on the size and sand fill ratio of the containers as well as on the degree of exceedance of the wave loads required for the inception of the internal movement of the sand fill. (iv) Besides, the effect on the hydraulic stability, the internal sand movement in submerged geotextile containers may lead to a substantial reduction of the height of GSC-structures (up to about 10%) when subject to severe wave attack. (v) The deformation of GSCs affects the hydraulic stability, essentially due to two mechanisms: (a) reduction of the contact areas between GSCs caused by the uplift of the containers by wave action, thus decreasing the stabilizing forces/moments and (b) increase of the surface areas exposed to drag and lift forces which represent the main destabilizing forces/moments. Therefore, the deformation effect should be accounted for explicitly in the stability formulae. (vi) Friction between GSCs affects the hydraulic stability much more than commonly assumed in past and present design practice. Therefore, and due to the possible changes in service life, friction should be incorporated explicitly in stability formulae. (vii) The hydraulic permeability of structures made of GSCs is not only important for the prediction of the hydraulic performance (e.g., wave transmission, runup, overtopping, etc.), but also slightly affects the hydraulic stability. However, no clear correlation could be found between stability and permeability for the range of practical permeability coefficients of GSC-structures in the order of k = 1–3 cm/s. The permeability of a GSC-structure is essentially determined by the gaps between the containers, so that the flow through the sand fill itself can be neglected. Therefore, the hydraulic permeability essentially depends on the mode of placement of the containers. For randomly placed containers and longitudinally (in wave direction) placed containers, the permeability coefficient is in the order of k = 2.5 cm/s. (viii) The effect of breaking wave impact on sliding and overturning stability of slope containers has been found much less than expected, due to the potential of the GSCs to effectively damp impact pressure propagation inside the gaps. More efficient to destabilize the slope containers are the uprush and downrush of the longer nonbreaking waves and partially breaking waves. (ix) The proposed simple stability formulae derived in Sec. 21.4.1 on the basis of the Hudson-formula take additionally into account the effect of the wave period for the slope containers and that of the relative freeboard (Rc /Hs ) for the crest containers. In both cases, the effect of container deformations is taken into account implicitly through the empirical parameters determined from laboratory experiments. These formulae are conservative for waves up to about Hs = 1.5 m and can thus be used for design wave heights not larger than about 2 m. For higher waves (Hs > 2 m) the effect of deformation on the stability becomes more important and must therefore be considered more explicitly in order to ensure long-term stability performance.

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(x) The more detailed stability formulae proposed in Sec. 21.4.3 do not only take into account the most relevant processes, but also allow us to better quantify the effect of deformation on the required size of the container as a function of the incident wave height for both slope containers (Fig. 21.34) and crest containers (Fig. 21.35). In fact, these new process-based formulae clearly highlight the effect of container deformation which dramatically increases with the design wave height and is much more pronounced for the stability of the crest containers than for that of the slope containers. Moreover, they also show that even for the commonly used relative freeboard of about Rc /Hs ≈ 1.2 much larger containers are required for the crest than for the slope of a GSCstructure to ensure hydraulic stability. (xi) Although significant advancement has been achieved in the understanding of the processes and mechanisms responsible for the hydraulic failure of GSCs, more systematic research is further needed to investigate and better control the friction between GSCs, the effect of the sand fill ratio, the effect of the slope steepness of the GSC-structure, the internal movement of the sand fill, the container deformations and their more explicit consideration in both stability formulae and numerical simulations. A fully coupled CFD and CSD model system well-validated by experimental data will be needed as a necessary tool in combination with new laboratory experiments to achieve these goals.

Acknowledgments The financial support by StAUN Rostock for the experimental investigations described in Sec. 21.4.1 and by NAUE GmbH & Co. KG for those described in Sec. 21.4.3.2 are gratefully acknowledged. Sections 21.4.2 and 21.4.3 are essentially the part of the PhD-Thesis of the second author which has been supported by the Deutsche Akademische Austausch Dienst (DAAD). The first author would like to thank Prof. Heerten, Mr. Pilarczyk, and Mrs. Werth for their valuable comments and for reading the manuscript. Thank is also due to Mrs. Werth for providing the first author with valuable information and references on the properties and durability of geotextiles. References 1. ACT, Innovative Technology for Coastal Erosion Control — Subsurface Dune Stabilization, Advanced Coastal Technologies, LLC (2006), http://bcs.dep.state.fl.us/ innovative/report/appedic c/06 Subsurface Dune Protect.pdf. 2. J. Buckley and W. Hornsey, Woorin beach protection–chasing the tide sand fill tubes versus sand filled containers, Proc. Int. Conf. Geosynthetics, Yokohama, Japan (2006), pp. 761–764. 3. CEN, Guide to Durability of Geotextiles and Geotextile Related Products (European Normalization Committee (CEN/TC189/WGS/N210/Paris), 1998). 4. EAG-CON, German recommendations on geotextile containers in hydraulic engineering, WG-UG5 “Geotextile Containers of AK 5.1” Geosynthetics, Geotechnics and Hydraulic Engineering (German Geotechnic. Society (DGGT), 2008).

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5. R. A. Edward and S. D. A. Smith, Optimising biodiversity of macro invertebrates on artificial reefs, Poster presentation AMSA National Conf., Brisbane, Australia (2003). 6. J. Fowler and E. Trainer, Overview of geocontainers projects in the United States, Proc. Western, Dredging Conf., USA (1998). 7. H. H. Greenwood and A. Friday, How to predict hundred year lifetimes for geosynthetics, Proc. Int. Conf. Geosynthetics, Yokohama, Japan (2006), pp. 1539–1542. 8. J. Gr¨ une, U. Sparboom, R. Schmidt-Koppenhagen, Z. Wang and H. Oumeraci, Stability test of geotextile sand containers for monopile scour protection, ASCE Proc. Int. Conf. Coastal Eng., San Diego, USA (2006), pp. 5093–5105. 9. G. Heerten, Long-term experience with the use of synthetic filter fabrics in coastal engineering, Proc. 17th Int. Coastal Engineering Conf., Vol. 3, Sydney, Australia (1980), pp. 2174–2193. 10. G. Heerten and F. F. Zitscher, 25 Jahre Geotextilien im K¨ ustenschutz — Ein Erfahrungsbericht. 1. Nationales Symposium Geotextilien im Erd-und Grundbau, Mainz, Germany, Forschungsgesellschaft f¨ ur Straßen- und Verkehrswesen (Hrsg.), K¨ oln, Germany (1984), pp. 7–15 (in German). 11. G. Heerten, Geotextiles in coastal engineering — 25 years experience, Geotextiles and Geomembranes, Vol. 1 (Elsevier, 1984), pp. 119–141. 12. G. Heerten, A. Jackson, S. J. Restall and F. Saathoff, New development with mega sand containers of non-woven needle-punched geotextiles for the construction of coastal structures, ASCE Int. Conf. Coastal. Eng., Sydney, Australia (2000). 13. G. Heerten, The challenge for the use of geosynthetic construction materials in environmental, coastal and offshore engineering applications, Offshore Arabia 2006, Conf. and Exhibition, Dubai, United Arab Emirates (2006). 14. M. Heibaum, Geosynthetic containers — A new field of application with nearly no limits, Proc. 7th Int. Conf. Geosynthetics, Nice, France (2002). 15. M. Heibaum, A. Fourie, H. Girard, G. B. Karunararne, J. Lafleur and E. M. Palmeira, Hydraulic applications of geotextiles, Proc. Int. Conf. Geosynthetics, Yokohama, Japan, Millpress, Rotterdam (2006), pp. 79–120. 16. L. A. Jackson, R. Tomlinson, I. Turner, B. Corbett, M. D’Agatha and J. McGrath, Narrow artificial reef — Results of 4 years of monitoring and modifications, Proc. 4th Int. Surfing Reef Symposium, Manhattan Beach, California, USA (2005). 17. Th. Jansen, Erprobung und Verwendung von sandgef¨ ullten Kunststoffschl¨ auchen aus Gewebe und Vlies beim Seedeichbau in der Leybucht, Ostfriesland, 1. Kongress “Kunststoffe in der Geotechnik” K-Geo, deutsche Gesellschaft f¨ ur Erd- und Grundbau e.V., Hamburg (1988) (in German). 18. E. Kavazanjian, Jr., N. Dixion, T. Katsumi, A. Kortegast, P. Legg and H. Zanziger Geosynthetic barriers for environmental protection of landfill, Proc. Int. Conf. Geosynthetics, Yokohama, Japan (2006), pp. 121–152. 19. R. M. Koerner, Designing with Geosynthetics, 5th edn. (Pearson Prentice Hall, Ltd., London, 2005), 796 pp. 20. C. R. Lawson, Geotextile containment for hydraulic and environmental engineering, Proc. Geosynthetics Int. Conf. (Mitchpress, Rotterdam, 2006), pp. 1–48. 21. E. Lefaive, Durability of geotextiles: The French experience, Geotextiles and Geomembranes 7, 553–558 (1988). 22. B. Lenze, G. Heerten, F. Saathoff and K. Stelljes, Geotextile sand containers — Successful solutions against beach erosion at sandy coasts and scour problems under hydrodynamic loads, Proc. EUROCOAST, Littoral 2, Porto, Portugal (2002), pp. 375–381.

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23. MAG, Code of Practice — Use of Geotextile Filters on Waterways (Federal Waterways Engineering and Research Institute (BAW), Karlsruhe, 1993), www.baw.de. 24. A. McClarty, J. Cross, L. Gilbert and O. M. James, Design and construction of coastal erosion protection groyne using geocontainers, Langebaan, South Africa. Proc. Int. Conf. Geosynthetics, Yokohama, Japan (2006). 25. NAUE, Scherversuche im Labor der Firma NAUE f¨ ur Soil Filters Australia, NAUE Fasertechnik (2002). 26. H. Nickels and G. Heerten, Objektschutz Haus Kliffende, HANSA, Band 137(3), 72–75 (2000) (in German). 27. H. Oumeraci, M. Bleck, M. Hinz and S. K¨ ubler, Large-scale model test for hydraulic stability of geotextile sand containers under wave attack, Leichtweiss-Institute for Hydraulic Engineering and Water Resources, LWI Report No. 878 (2002) 62 and Annexes. 28. H. Oumeraci, M. Hinz, M. Bleck and A. Kortenhaus, Sand-filled geotextile containers for shore protection, Proc. COPEDEC, Colombo, Sri Lanka (2003). 29. H. Oumeraci, J. Gr¨ une, H. Sparboom, R. Schmidt-Koppenhagen and Z. Wang, Investigations on scour and scour protection for monopile foundation of wind offshore turbines, Forschungszentrum K¨ uste (FZK), Res. Report (2007) 79 and Annexes (in German). 30. H. Oumeraci, A. Kortenhaus and K. Werth, Hydraulic performance and armour stability of rubble mound breakwaters with core made of geotextile sand containers — Comparison with conventional breakwaters, Proc. Int. Conf. Coastal Structures, Venice, ASCE (2007). 31. K. W. Pilarczyk, Geosynthetics and Geosystems in Hydraulic and Coastal Engineering (A.A. Balkema, Rotterdam, The Netherlands, 2000). 32. J. Recio and H. Oumeraci, Effect of the deformation on the hydraulic stability of revetments made of geotextile sand containers, Proc. Int. Symp. “Tsunami Reconstruction with Geosynthetics”, Bangkok, Thailand (2005), pp. 53–68. 33. J. Recio and H. Oumeraci, A numerical study on the hydraulic processes associated with the instability of GSC-structures using a VOF-RANS model, LeichtweissInstitute for Hydraulic Engineering and Water Resources, LWI Report No. 941 (2006). 34. J. Recio and H. Oumeraci, Processes affecting the stability of revetments made with geotextile sand containers, Proc. Int. Conf. Coastal Engineering, ICCE 2006, San Diego, USA (2006). 35. J. Recio and H. Oumeraci, Geotextile sand containers for coastal structures, hydraulic stability formulae and tests for drag, inertia and lift coefficients, Leichtweiss-Institute for Hydraulic Engineering and Water Resources, LWI Report No. 936 (2006). 36. J. Recio and H. Oumeraci, Preliminary experiments and numerical simulations of solitary wave acting on a submerged-filter-reef, Leichtweiss-Institute for Hydraulic Engineering and Water Resources, LWI Report No. 945 (2006). 37. J. Recio, Hydraulic stability of geotextile sand containers for coastal structures — Effect of deformations and stability formulae, PhD-Thesis, Leichtweiss-Institute for Hydraulic Eng., Technical University Braunschweig (2007). 38. J. Recio and H. Oumeraci, Permeability of GSC-structures, model tests and analyses, Leichtweiss-Institute for Hydraulic Engineering and Water Resources, LWI Report No. 943 (2007). 39. J. Recio and H. Oumeraci, “Coupled” numerical simulations on the stability of coastal structures made of geotextile sand containers, Leichtweiss-Institute for Hydraulic Engineering and Water Resources, LWI Report No. 942 (2007).

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40. J. Recio and H. Oumeraci, Effect of the deformations on hydraulic stability of coastal structures made of geotextile sand containers, Geotex. Geomemb. J. 25, 278–292 (2007). 41. J. Recio and H. Oumeraci, Hydraulic permeability of GSC-structures: Laboratory results and conceptual model, Geotex. Geomemb. 26, 473–487 (2008). 42. J. Recio and H. Oumeraci, Processes affecting the hydraulic stability of GSC structures: Experimental and numerical investigations, Coastal Eng. 56, 632–658 (2009). 43. J. Recio and H. Oumeraci, New stability formulae for coastal structures made of geotextile sand containers, Geotex. Geomemb. 56, 260–284 (2009). 44. S. J. Restall, L. A. Jackson, G. Heerten and W. P. Hornsey, Case studies showing the growth and development of geotextile sand containers — An Australian perspective, Geotex. Geomem. 20(5), 321–342 (2002). 45. S. J. Restall, W. P. Hornsey, H. Oumeraci, M. Hinz, F. Saathoff and K. Werth, Australian and German experiences with geotextile containers for coastal protection, Proc. Euro-Geo 3, Munich, Germany (2005), pp. 141–146. 46. A. L. Rollin, Long term performance of geotextiles, 57th Canadian Geotechnical Conf., Session 4D (2004), pp. 15–20. 47. RPG, Guidelines for Testing Geotextiles for Navigable Waterways (Federal Waterways Engineering and Research Institute (BAW), Karlsruhe, 1994), www.baw.de. 48. F. Saathoff, H. Oumeraci and S. Restall, Australian and German experiences on the use of geotextile containers, Geotex. Geomem 25, 251–263 (2007). 49. H. F. Schroeder, H. Bahr, P. Herrmann, G. Kneip, E. Lorenz and I. Schmuecking, Durability of polyfine geosynthetics under elevated oxygen pressure in aqueous liquids, Proc. Second European Geosynthetics Conf. EUROGEO, Bologna, Italy (2000). 50. S. K. Shukla and J. H. Yin, Fundamentals of Geosynthetics Engineering (Routledge, 2006). 51. M. Solton, B. Leclerq, J. L. Paute and D. Fayoux, Some answer’s components on durability problems of geotextiles, Proc. 2nd Conf. Geotechnics 2, 553–558 (1982). 52. G. H. Troost, G. Den Hoedt, P. Risseeuw, W. Voskamp and H. M. Schmidt, Durability of a 13 year old embankment reinforced with polyester woven fabric, 5th Int. Conf. Geotextiles, Geomembranes and Related Products, Singapore, IGS (1994) pp. 1185–1190. 53. G. van Santvoort (ed.), Geotextiles and Geomembranes in Civil Engineering (A.A. Balkema, Rotterdam, The Netherlands, 1994). 54. J. Vohlken, H. Lind and J. Witte, Dune reinforcement with geotextile sand containers, HANSA 4, 60–62 (2003) (in German). 55. J. D. M. Wisse, C. J. M. Broos and W. H. Boels, Evaluation of the life expectancy of polypropylene geotextiles used in bottom protection structures around the Ooster Schelde storm surge barrier — A case study, Proc. 4th Conf. Geosynthetics (1990), pp. 697–702. 56. J. Wouters, Slope Revetment — Stability of Geosystems, Delft Hydraulics Report H. 1939 (1998) Annex 7 (in Dutch). 57. F. F. Zitscher, Kunststoffe f¨ ur den Wasserbau, Bauingenieur-Praxis, Heft 125, Verlag von Wilhelm Ernst & Sohn, Berlin, M¨ unchen, D¨ usseldorf (1971).

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Chapter 22

Low Crested Breakwaters Alberto Lamberti∗ and Barbara Zanuttigh Department of Civil Engineering (DISTART), University of Bologna Viale Risorgimento, 2; 40136 Bologna, Italy ∗ [email protected] The chapter describes typical features of low crested breakwaters, their hydraulic stability, effects on waves, induced circulation, erosion, and problems related to construction and maintenance, providing basic tools for design.

22.1. Introduction As low crested breakwaters (LCBs) we mean breakwaters that are frequently overtopped or weakly submerged. They are in any case breakwaters, i.e., the height over the bed of these structures is sufficient to cause systematic wave breaking on the crest (submergence less than one significant wave height), but they are also low, i.e., not so high to make overtopping rare (freeboard less than one significant wave height). In this sense they are distinguished from artificial reefs (deployed mainly to enhance fisheries) or bed protections on one hand, that are deeply submerged and do not cause breaking, and by usual harbor breakwater on the other hand that are high enough not to cause significant overtopping. LCBs mimic in some manner the functioning of coral reefs, that however have usually a much wider crest and a narrower range of submergence/emergence due to their generation mechanism. As a consequence they are sometimes named also “reefs.” LCBs are usually built with natural rock for the defence of beaches (Fig. 22.1), even if some examples of LCBs protecting the main breakwater from highest waves or protecting an external harbor from excessive agitation and currents (Fig. 22.2) can also be found. Beach defence breakwaters are usually built in a few meter water depth within the surf zone at design conditions. The freeboard (positive or negative) is usually of the same order of magnitude as the tidal range, i.e., in contrast with most harbor breakwaters, freeboard variability in lifetime is substantial. 601

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Fig. 22.1. Detached LCBs protecting the beach of Skagen (DK, 57◦ 44 07 N 10◦ 37 50 E; coordinates are given to let the reader obtaining a plan view from Google-Earth).

Fig. 22.2. Jetties protecting Ijmuiden entrance (NL, 52◦ 27 55 N 4◦ 32 18 E) during a storm surge event. From http://www.vandermeerconsulting.nl/.

As it will be shown later, the height of the structure in this case is not greater than a few armor stones, posing serious problems in building the traditional filter structure separating the armor from the sandy bed. LCBs are usually located in the surf zone where design wave height is depth limited and, therefore, lasts for long periods. It is therefore unsafe to accept any,

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even small, permanent damaging rate in design conditions, since the damage will cumulate in a few years at a significant level. For all the reasons sketched above, LCBs cannot be treated as a small scale version of rubble mound harbor breakwaters, even if they share some evident common features. Due to their wide use in beach defence, however, they deserve some special attention. This contribution is dedicated to highlight differences between high and low breakwaters. We summarize the experience accumulated in the last two decades on the subject, making use of several results focused within DELOS research project, that we had the opportunity to coordinate; we acknowledge at the same time the contribution of project partners, evident to everyone will read one at least of the two major deliverables of the project: the special issue of Coastal Engineering8,24,28,29,35,46,57,68 and the Design Guidelines.5 The sections are arranged as follows: Section 22.2 — typical features, Section 22.3 — hydraulic stability, Section 22.4 — wave reflection and transmission, Section 22.5 — overtopping, Section 22.6 — induced circulation, Section 22.7 — scour and erosion, Section 22.8 — construction and maintenance problems.

22.2. Typical Layout and Sections LCBs were used since long time as detached emergent breakwaters for the defence of beaches. The low crest was motivated by economy reason in conditions where some agitation in the protected area could be accepted and sometimes required in order to avoid silt sedimentation. The problem of visual impact was not posed at that time. Later during the last two decades of the 20th century, in Europe particularly along the Mediterranean coasts subject to intensive tourism exploitation, the aim of reducing visual impact of defence works promoted the use of submerged breakwaters. Figure 22.3 shows one of the most deeply submerged breakwater constructed in Pellestrina (near Venice, IT) and completed in 1997.29 It is a 9 km long defence work, where a 4.5 × 106 m3 capital nourishment of 0.2 mm sand was protected by a LCB system. The offshore part of the LCB is a submerged breakwater defending the beach against offshore sand losses. The groins limit the long-shore displacement of sand and are characterized by an emerged part in the aerial beach zone and a submerged continuation to reach the submerged longitudinal breakwater. Average water depth at structure is 4.5 m. Design submergence is −1.5 m a.m.s.l. Spring tidal range near Venice is 1.0 m, but high tide combined with storm surge caused by Scirocco wind may reach 1.5 m a.m.s.l. Incident significant wave height with return period 10 years is 3.70 m, therefore, extreme waves at the structure are depth limited. Since the construction of Venice outlet jetties (end of the 19th century), the beach, deprived of any natural sand nourishment, suffered a slow but continuous erosion. The beach, after the construction of the defence system, showed to be rather stable with low southerly directed littoral drift.

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Fig. 22.3. Pellestrina (IT, 45◦ 15 57 N 12◦ 18 10 E) offshore submerged breakwater and groins. Courtesy of Consorzio Venezia Nuova.

Fig. 22.4.

Lønstrup defense section (DK, 57◦ 28 37 N 9◦ 47 44 E). From Burcharth et al.6

Figure 22.4 shows the section of a Danish beach protection example at Lønstrup in northern Jutland.29 Mean tidal range at the site is 0.3 m; exceptional high tide may reach 1.5 m; wave are usually depth limited. Longshore net transport is great, approx. 600 × 103 m3 /year. Beach sand size is 0.2 mm. A 1.1 km long segmented breakwater was constructed to protect the cliff and the village from an ongoing 1.5 m/year erosion. To maintain the shoreline position, a 20–30 × 103 m3 /year nourishment was required even after the breakwater construction.

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Fig. 22.5. Sea Palling defence section (UK, 52◦ 47 32 N 1◦ 36 11 E). Adapted from Halcrow;21 see also Sea Palling site at http://argus-data.wldelft.nl/index.html, reefs located north of the video station.

Another example of a typical emerged breakwater for coastal defence is represented by detached breakwater built near Sea Palling in North Norfolk, UK, Thomalla and Vincent.48 The defence system was motivated by severe flooding in 1953 (causing seven casualties in the area) and consisted of a continuous seawall protecting dune foot, requiring for its stability an adequate beach at the toe. In order to reduce wave intensity on the wall and required nourishment volumes in the most critical part of the defence, nine breakwater (or reefs) were constructed since 1993. Figure 22.5 shows the section of the first built reefs 5–8.21 The site is characterized by 3.0 m spring tidal range (from −1.3 to +1.7 m over datum -ODN-); extreme level reach 3.7–4.0 m ODN with return period 50–100 years. Offshore significant wave height reaches 7.9 m with return period 100 years, therefore, extreme waves are depth limited at the structure. Longshore tidal currents are relevant with a peak south-easterly flood of 0.8 ms−1 and a peak spring ebb to the northwest of 0.6 ms−1 . The net longshore sediment drift has been estimated to be 100–150×103 m3 per year in a southerly direction. The sandy beach shows to be highly volatile, with temporary spatial average erosion under severe storms reaching up to 70 m3 /m. Finally, an example of low crested harbor breakwater is worth to be presented. The outer breakwater prolongation in Leix˜ oes harbor (near Porto, PT) is selected, detailed, described in Vera-Cruz and Reis de Carvalho.58 The two long jetties enclosing the 100 ha outer harbor were built in 1884–1892. The outer harbor was not designed to provide load/unload capacities, but the limited extension of the initial dock area forced ships to stay anchored or moored in it. A storm in January 1917 caused severe damages to several of these ships. The construction of an outer vertical wall breakwater was therefore initiated in 1934, but during the first winter the last 120 m of the 400 m long built breakwater were destroyed. Based on economy considerations, a submerged breakwater was proposed. The design was verified with a hydraulic model carried out in Lausanne (CH), showing that the structure safety and harbor tranquility resulted satisfactory. The main environmental conditions are: water depth at roundhead −15.0 m, tidal excursion 0.0–4.0 m, significant wave height 9.5 m, wave period 13–19 s. The construction of the submerged extension was carried out in years 1937–1942 and its behavior was satisfactory, requiring only minor maintenance works. In 1966, in order to provide berthing facility to 100,000 t oil-carriers, the construction of berth A along the outer breakwater and of a wave wall with crest elevation 15 m a.s.l. was decided. The designed wall, apparently tested with regular waves, was protected by a steep (3:4) 40 t tetrapod slope resting on the crest of the 90 t cube submerged

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breakwater. In 1972, two years after the construction ended, a replacement of more than one-third of the placed tetrapods was necessary, and a similar replacement was required in 1974. In 1979, the last 90 m of the wave wall were destroyed. Breakage of tetrapods was observed in prototype that, together with the insufficient characterization of design waves (regular rather than irregular waves with equal significant wave height), were recognized as the main causes of the failure. The construction of some protection works of the roundhead was decided and, after comparison of several alternative solutions in a hydraulic model, the construction of a submerged breakwater with section similar to the previous one but shifted 52 m offshore was selected, aiming to reduce through breaking the wave height incident on the wave wall. The construction of the selected design (see Fig. 22.6) ended in 1982; the crest was constructed 1 m higher than shown in the drawing anticipating a 2 m settlement. The behavior of the submerged breakwater was evaluated satisfactory in 1993 either regarding its stability under wave action or regarding the protective effect on tetrapod stability and wave wall overtopping. The prototype verification appears significant since during the 11 year monitoring period a 9.0 m peak significant wave height event occurred (January 1985), reaching almost design intensity. PERFIL 6 - 6 0

L = 65,0 m - Solução 1 L = 40,0 m - Solução 2 L = 52,0 m - Solução 3 (ADOPTADA)

25

+ 11,5↓

MACIÇO DE PROTECÇÃO

+ 4,0 (PMAV)↓

+ 1,0↓

Z.H.↓ 2

7,0

1

1

1

PARALELEPÍPEDOS DE BETÃO

ENR.

- 12,0 ↓

t. 0t S4 DE . PO 3t t Á > . TR NR TE

OBRA A PROTEGER

L

12,0 4

+ 15,0↓

50 m

4

2

~ 90 tt.

1

1

2

↓+7,25

4

E

PARALELEPÍPEDOS DE BETÃO

~ 90 tt.

↓+4,0

↓+3,5 T. O T. T. O T. SELECIONADO

EN

EN R . R. > 3 > tt

T. O T.

ENR.

1 - 5 tt.

6 −1

3

0, 5 . tt.

Z.H. 3

2 5

< 1tt.

4

− 15

- 105,0

−1

6

4

( - 12,0) − 13

− 12

L

(0,0)

(0,0)

(+ 5,0) (+ 7

,5) (+ 15,0)

(+ 0,5)

(+ 6,8)

(+ 7,25)

(- 5,0)

PLANTA

6 0

50

100 m

POSTO A

Fig. 22.6. Terminal section and layout of the Leix˜ oes harbor breakwater (PT, 41◦ 10 22 N 8◦ 42 30 W). From Vera-Cruz and Reis De Carvalho.58

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22.3. Armor Stability in Shallow Water and Significant Tide Excursion A low breakwater compared to a high one is less exposed to breaker impact and to wave downrush, since the breaker falls on the crest often covered with water and the impact finds vent shorewards in the absence of a wave wall. This section is dedicated to describe this effect and to provide proper tools for armor design. Systematic laboratory experiments regarding rock armor stability in low crested breakwaters were carried out by several authors. Givler and Sørensen20 provided results of 2D experiments with regular waves, van der Meer51 and Burger7 with irregular waves. Vidal et al.59,60 Kramer and Burcharth,27 and Kramer et al.28 performed experiments with long crested and short crested irregular waves in a wave basin observing also damages at roundheads. A synthetic presentation of these results with a critical analysis is presented in Burcharth et al.6 on which this section is essentially based. Experiments are described in detail in the referred papers. All experiments are carried out on a fixed bed, where no settlement or erosion can take place, and therefore they do not provide information on these failure modes. Results refer to where the damage takes place and how severe the damage is. Assuming equal armor stone size at trunk and roundhead, all authors agree on where first and how damage occurs, see Fig. 22.7. • For submerged or zero freeboard breakwaters (Rc ≤ 0), hydraulic damage occurs first at trunk crest and results in a shorewards displacement of stones; roundhead in not more critical than the trunk and damage is distributed all around the head. • For slightly emerged breakwaters (0 < Rc < Dn50 ), the crest remains the most critical part of the trunk with some stones thrown shorewards and other dragged Seaward and middle head Leeward head

Stability number Hs/∆Dn50

5

4

3

X D

Trunk seaward slope Trunk crest

º

Trunk leeward slope

2

1

Least stable section, Eq. (22.1) 0 -3

-2

-1

0

1

Normalized freeboard Rc/Dn50 Fig. 22.7.

Partial and global start of damage curve, from Burcharth et al.5

2

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down the offshore slope; leeward roundhead becomes the first damaged part of a breakwater. • For emerged LCBs (Dn50 < Rc ), the seaward slope is the most critical part of the trunk with most stones dragged down the slope and the leeward roundhead is the first damaged part of a breakwater. • The crest width, built with stones of the same size as the armor slope, has no evident effect on hydraulic stability. • In all the mentioned tests, slope was 2:3 or 1:2 and no significant effect of the slope was observed. Burger7 showed that gradation and stone shape within reasonable limits (D85 /D15 < 2.5, L/D < 3) have not an evident effect on stability. Based on all the available information Burcharth et al.6 proposed the following formula representing the initiation of damage in some part of the breakwater
2 Hs Rc Rc = 0.06 − 0.23 + 1.36 (22.1) ∆Dn50 Dn50 Dn50 for relative crest freeboard less than 1.9. Above this value the stability number remains constant and equals to 1.14, the minimum of Eq. (22.1). This value is significantly lower than the one that can be derived from Hudson formula (∼ =1.5). This is due partially to the fact that no damage is assumed in this formula against a moderate damage (5%) accounted for in Hudson formula and partially to the deliberate use of Hs as the characteristic wave height (Hudson formula is used sometimes with H1/10 and some other with Hmax ). Most LCBs are placed in the littoral zone and in design conditions are subjected to depth breaking waves. For many of these the mean water level varies in a relatively wide range, regularly due to tide and occasionally due to storm surge. If waves are depth limited and the breaking limit is represented as a limit ratio γ between significant wave height and the local water depth, among the incident wave height Hi , the water depth at structure toe hs , the structure height hc , and structure freeboard Rc the relation holds Hi = γ · hs = γ · (hc − Rc ),

(22.2)

where the breaker index γ depends on seabed slope, wave steepness, and the characteristic wave height in use (significant, extreme . . . ). Figure 22.8 shows a typical situation at failure boundary. The shape of the two domains (failure domain and possible sea conditions) is such that stability is assured if breaking limit does not cross the start of damage curve or is tangent to it as a limit. This leads to the hydraulic stability condition under variable water level and depth limited waves characterized by the significant wave height: γs /∆ Dn50 ≥ . hc 1.36 − (γs /∆ − 0.23)2 /(4 · 0.06)

(22.3)

If the tangency point does not lie within the possible sea level range or breaking conditions are unlikely, the condition contains an implicit safety margin.

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Fig. 22.8.

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Failure representation under variable water level, from Burcharth et al.6

Table 22.1. Stability condition and critical degree of submergence for different foreshore slope and breaker index. Wave solicitation Foreshore slope 1/∞ 1/200 1/100 1/40 1/20 — —

Tangency point

Hsi /hs

Rc /hc

Dn50 /hc

Hsi /∆Dn50

0.40 0.45 0.50 0.55 0.60 0.65 0.70

−0.02 −0.08 −0.16 −0.25 −0.36 −0.48 −0.64

0.18 0.21 0.23 0.26 0.29 0.33 0.37

1.39 1.46 1.5 1.6 1.7 1.8 1.9

Results for the minimum stone size according to Eq. (22.3) for relative stone density ∆ = 1.6 and different values for significant breaker index γs are reported in Table 22.1. The breaker index values are evaluated according to van der Meer52 with sop = 0.03 and hs /Lop = 0.05. For a gentle foreshore slope the following simple and cautious rule of thumb is found Dn50 ≥ 0.3 · hc .

(22.4)

The ratio Dn50 /hc imposes serious restrictions to the structure design. A double armor layer with Dn50 /hc > 0.4 require that at least part of the filter is placed below the natural bed level; this could be beneficial in case a bed erosion might occur. A value Dn50 /hc > 0.3 implies that there is no sufficient space for a core and the two layer armor stones rest directly on a coarse filter. Only when Dn50 /hc < 0.2 there is space for a conventional core and filter layer.

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It should be noted that Eq. (22.1) is based on model tests with sea-side slope 1:2 (some 1:1.5). For milder slopes, smaller armor stones can presumably be stable. According to Hudson formula, for a slope 1:5, an increase of the stability number for emerged breakwaters by a factor 1.36 is expected; a smaller increase in stability is argued in the case of submerged crest because the crest (including edges) is the most damaged part in this case. Also, some laboratory test cases support this conclusion (Ostia IT, Ferrante et al.18 ; Sea Palling UK, Leasowe Bay UK, Barber and Davies1 ). Offshore and inshore the armor slopes, a toe berm is normally constructed with the function of supporting armor and preventing the propagation of damage resulting from scour or settlement. Regarding berm stability no specific analysis was carried out on low breakwaters. Design rules suggested by van der Meer et al.56 were evaluated as reasonably safe for the offshore berm of low breakwaters in Burcharth et al.6 For a sufficiently wide crest, the breaker will fall on the crest and the inshore toe may be designed with the same material as the offshore one. For narrow and submerged breakwaters, the breaker caused by the offshore slope can fall beyond the crest causing a severe solicitation of the inshore toe berm; care should be paid in this case. It is worth noting that the above formulae are derived from fixed bed models, and do not represent any effect of bed mobility and/or erosion, whereas in most prototype conditions the fine sand seabed is subjected to intense mobilization and transport. Indeed, the observed damage of prototype rubble mound (as described by Lamberti et al.29 ) is often the consequence of geotechnical or morphodynamic instabilities. First of all, when proper bed protection is not used, dumping of stones onto bed cause their rapid sinking already during construction followed by a slower settlement. Moreover, breaking waves and related strong currents produce local scour (Sumer et al.46 ) which affects toe stability either directly (stone sliding in scour hole) or indirectly (increase of water depth and breaker height). To avoid sinking of the rubble stone material into a sandy bed it is necessary to separate the two materials with an appropriate filter: widely dispersed small stone layers, geotextiles, or mattresses. If sinking of stones stops within construction period, the rubble mound volume used for construction must be oversized and that is all; if sinking last longer, the structure must be designed and initially built higher (see, e.g., the Leixoes example). The overheight will correspond to the anticipated settlement, and depends on seabed characteristics, construction method, and structure height. For instance, a large rubble mound built on muddy seabed by means of floating equipment demands a large overheight, whereas a low mound structure placed on a coarse sandy seabed by landbased equipment running on already placed material demands much less overheight (but great overvolume) as settlement will be almost completed during construction. Burcharth et al.6 describes in detail some settlement cases. Statistics from a much wider set of cases from Japan is presented in Uda.49 Depending on the nature of the bed material the statistics in Table 22.2 was observed. The fraction of settling breakwater increases from 10% on rock (probably degradation of armor units), up to 43% on gravel, 63% on sand and 100% on sylt. The settlement height is highly variable in the range 0–2.80 m, with average and modal value around 1.00–1.20 m.

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Table 22.2. Settlement distribution of detached breakwater as function of bed material. Settlement Bed material

Exist

Nil

Unknown

Total

Rock Gravel Sand Silt

10%(19) 43%(139) 64%(678) 100%(1)

73%(141) 56%(183) 33%(347)

17%(34) 1%(5) 3%(42)

100%(194) 100%(327) 100%(1067) 100%(1)

Total

53%(837)

42%(671)

5%(81)

100%(1589)

Results are also given for scattering (displacement, hydraulic damage) of armor units. Globally scattering was observed in 14.2% of the 1,552 cases, but the frequency varies significantly between breakwaters showing evident settlement, for which the scattering frequency was 17.9%, and those for which no settlement was observed, for which scattering frequency was 9.4%. The frequency difference is highly significant, proving that scattering and settlement are related. In our experience hydraulic damage, i.e., the removal of stones from the breakwater crest, is able to justify only a minor part of crest settlement. On the other side, bed erosion around the structure and stone sinking preferably along the structure perimeter are the cause of block dilation on the slopes and therefore of reduced interlocking and instability. The mutual influence of settlement and block scattering is therefore bilateral and can explain the evident correlation observed.

22.4. Effect on Waves: Transmission and Reflection The main effect of a LCB compared to a high one is that a significant fraction of incident energy can pass over its crest and generate waves behind the structure. The increase of wave transmission leads, on one hand, to lower beach protection from erosion or to less calm areas for swimming purposes inshore the LCBs. On the other hand, it guarantees a high degree of water recirculation and thus a water quality improvement within the protected cell. Transmission, as well as reflection, can be characterized by a global intensity factor, normally the transmission and reflection coefficients, and by a description of energy distribution within the energy containing frequency range. The EU-projects DELOS (www.delos.unibo.it) and CLASH (www.clash-eu.org) collected and generated wide datasets from tests on all kind of structures, where the transmission and reflection coefficients were available in almost all cases. An extensive and homogeneous database on wave transmission (van der Meer et al.57 ) and on wave reflection (Zanuttigh and van der Meer64 ) were prepared, whose analyses produced for LCBs the major outcomes synthesized below. For rubble mound LCBs, to be able to estimate the transmission coefficient Kt also in presence of submerged structures with very wide crests, the following two formulae are given for design conditions (Briganti et al.3 ; van der Meer et al.57 ).

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Equation (22.5a), which is the formula by d’Angremond et al.,12 is used for Bc /Hsi < 8; Eq. (22.5b) for Bc /Hsi > 12: Rc Kt = −0.40 + 0.64 Hsi




Bc Hsi

−0.31

(1 − e−0.50ξop )

(22.5a)

with imposed lower and upper limit values Ktmin = 0.075, Ktmax = 0.8 and Kt = −0.35

Rc + 0.51 Hsi




Bc Hsi

−0.65

(1 − e−0.41ξop )

(22.5b)

with Ktmin = 0.05, Ktmax = 0.93 − 0.006(Bc /Hsi ). A linear interpolation between the two is performed in the range 8 < Bc / Hsi < 12. In Eqs. (22.5) Rc is the structure freeboard (negative for submerged structures), Bc is the crest width, and ξop is the surf similarity parameter based on structure offshore slope and peak wave period. Equations (22.5) were applied in the range −5 ≤ Rc /Hsi ≤ 5. Not all LCBs are of the rubble mound type. Sometimes smooth and impermeable structures exist, for example, LCBs covered with asphalt or armored with a block revetment. Often the slope angles of these structures are gentler (1:3 or 1:4) than for rubble mound structures, mainly for construction reasons. Wave transmission over smooth LCBs is completely different from rubble mound structures. First of all, the wave transmission is larger for the same crest height, simply because there is less energy dissipation by friction and structure permeability. Furthermore, the crest width has less or even no influence on transmission, as also on the crest there is lower energy dissipation. Only for very wide (submerged) structures there could be some influence on the crest width, but this is not a case that will often be present in reality as asphalt and block revetments are mainly constructed in the dry and not under water. The presence of tide makes it possible to construct these structures above water. For smooth LCBs the prediction formula by van der Meer et al.57 is   Rc −0.5ξop + 0.75(1 − e ) (22.6) Kt = −0.3 Hsi with imposed lower and upper limit values Ktmin = 0.075, Ktmax = 0.8, and limitations: 1 < ξop < 3, 1 < Bc /Hsi < 4. Transmitted spectra are often different from incident spectra. Waves breaking over a LCB may generate two or more transmitted crests on the lee side per incident crest. The effect is that more energy is present at higher frequencies than in the incident spectrum. Usually the peak period is quite close to the incident one, but the mean period may decrease considerably. If the reduction of wave energy is mainly led by the dissipations due to the flow through the armor layer, however, higher frequencies may be cut. A first analysis on this topic can be found in van der Meer et al.,53 who developed a simple model where in average the 60% of the transmitted energy is present in the range f < 1.5fp and the other 40% of the energy is evenly distributed between 1.5fp and 3.5fp. This scheme was analyzed little further by

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Briganti et al.3 and by van der Meer et al.,57 concluding that the redistribution is applicable only to submerged rubble mound structures. Particular care to the representation of the shape of the transmitted wave spectra was paid by Lamberti et al.30 The transmitted wave is reconstructed from the incident wave as the sum of the overtopped and filtered components. Incident waves transform seawards the structure into overtopping events and then regenerate, leeward the structure, due to perturbations caused by impulsive overtopping volumes. The overtopping and filtration discharges are added and converted into displacements of an ideal wave maker placed leeward the structure. A sample of the reconstructed transmitted wave spectrum with comparison of an experimental one is given in Fig. 22.9. The approach describes fairly the spectrum modification. Related to spectral change is the effect on obliquity described below. Similarly, for traditional structures several formulae exist that predict the reflection coefficient Kr as function of the Iribarren–Battjes surf similarity parameter ξ and are empirically determined for different types of armor units, of tested conditions and layouts. The only formula that can be applied, with varying coefficients (a, b) for different kinds of structures is the one by Seelig and Ahrens42 Kr =

a1 · ξ 2 . ξ 2 + b1

(22.7)

The coefficients (a1 , b1 ) are, respectively, equal to (0.6, 6.6) for rubble mound structures, and to (1.0, 6.2) for smooth slopes.

Fig. 22.9. Measured transmitted spectra (solid line with +), transmitted spectra components due to filtration (dot line) and to overtopping (dash line), in case of abundant overtopping. From Lamberti et al.30

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A more recent formula by Zanuttigh and van der Meer64 exists that • can be applied to all kinds of structures or revetments in design conditions, provided only that the roughness factor γf is known, • represents physical bounds, • was validated against a wide database. This formula reads: Kr = tanh(a · ξob ),

(22.8)

where a = 0.167 · [1 − exp(−3.2 · γf )]; b = 1.49 · (γf − 0.38)2 + 0.86 and ξo is the Iribarren–Battjes parameter calculated with the mean (−1, 0) spectral period at the structure toe. Based on CLASH results, the parameter ξo is used in this formula since it allows to represent also bi-modal spectra or shallow water with a flat spectrum where a peak period is not well defined. For the most typical cases, the coefficients in Eq. (22.8) are: γf = 0.40, a = 0.12, b = 0.87 for rock slopes; γf = 0.55, a = 0.14, b = 0.90 for rock slopes with impermeable core; γf = 1.0, a = 0.16, b = 1.43 for smooth slopes. Based on the DELOS database, van der Meer et al.,57 proved that as expected low crest breakwaters have smaller reflection than high ones, due to the fact that more energy passes over the structure. Moreover, the key parameter to describe the reduction of Kr is the relative crest height Rc /Hsi . Based on these observations, van der Meer et al.57 provided a first rough reduction factor for Kr as function of Rc /Hsi to be generally applied to formulae for well-emerged structures. The dependence of Kr on Rc /Hsi can be appreciated for permeable structures in Fig. 22.10, where the measured value of Kr is a dimensionalized by the value 1.20

1.00

K rm/K rc

0.80

0.60

UPC UCA UPD UFI Seabrook

0.40

0.20

0.00 -2.0

-1.5

-1.0

-0.5

0.0

0.5

1.0

1.5

2.0

2.5

Rc/Hsi Fig. 22.10. Measured to computed values based on Eq. (22.10) of the reflection coefficient with varying degree of overtopping.

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obtained from Eq. (22.6). Similarly to van der Meer et al.,57 Zanuttigh and van der Meer64 proposed an extension of their formula to rubble mound LCBs by including a simple linear regression on Rc /Hsi . Reflection is not influenced if the relative crest height is > 0.5 and decreases to 0 when submergence is twice the wave height. The formulae (22.5), (22.6), and (22.7) are given for perpendicular waves only. Based on a limited set of data, van der Meer et al.57 and Wang et al.61 analyzed the effects of oblique wave attack on wave transmission. With increasing incident wave angle βi , Kt tends to decrease and the obliquity of the transmitted wave βt is a little smaller than the incident βi . The influence of wave obliquity on transmission is more evident for smooth structures, where it results proportional to cos2/3 β, and hardly observable for rubble LCBs. The deviation of transmitted waves (obliquity reduced to about 80%) can be interpreted as refraction of the average wave due to spectral change and reduced nonlinearity of transmitted waves. Explanation of effects is provided in Wang et al.63 Analysis of the effect due to wave obliquity on Kr can be found in Wang et al.,62 who derived for smooth LCBs a Kr reduction proportional to cos2/3 β.

22.5. Wave Overtopping Overtopping refers to the process of water passing over the structure crest. If the structure is not continuously submerged the process can be naturally interpreted and made up of several events strictly related to the passage of wave crests and characterized by a certain mass and momentum (or volume and velocity). The consequences of overtopping are dependent on specific statistics of the process related to the considered effect: for instance, the extreme momentum of one overtopping crest may be responsible for damaging exposed elements, whereas the mean volume per unit time (or mean overtopping discharge) is responsible for flooding. Nevertheless, the last is the standard parameter for characterizing overtopping intensity, probably because it is the easiest to measure. The total elapsed time and the total volume of water passing over the structure are simply the sum of elapsed time and passed volume within each wave. This trivial observation is the basis of the relation among mean overtopping discharge qot , overtopping probability Pot , mean overtopping volume ·Vot , and mean period of well formed waves Tm : qot

 Volumes Vot  Not · Vot  = waves = Pot , = Periods N · T Tm w m waves

(22.9)

where the overtopping probability Pot is the ratio between the number of overtopping waves Not and the number of waves Nw . Since every wave whose runup exceeds the structure crest (and only these waves) cause overtopping, the overtopping probability at the offshore edge of the crest is equal to the probability that a single wave runup is higher than the crest. For irregular waves, van der Meer and Stam54 suggested for runup at rubble mound structures a Weibull probability distribution, with parameters k1 and k2 related to

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incident waves and slope characteristics, from which the overtopping probability can be derived: Prob(Ru ≤ z) = 1 − exp(−(z/k1 )k2 ) ⇔ Pot = exp(−(Rc /k1 )k2 ),

(22.10)

where k1 = 0.4Hsi s−0.25 cot gα−0.2 , som is the mean wave steepness and α the om −0.75 structure offshore for plunging waves (ξm < 2.5) and k2 = √ slope; k2 = 3.0ξm −0.3 P 0.52·P ξm cot α for surging waves (ξm > 2.5), ξm is the Iribarren number based on mean wave period and P is the notional structure permeability. Later, van der Meer and Janssen55 provided a simpler expression for the runup distribution at dikes: the shape parameter is constant k2 = 2, i.e., the distribution is Rayleighian, and the location parameter is proportional to incident wave height k1 = 0.81 · γh · γf · γβ · min(γb · ξop , 2) · Hsi . The Rayleigh distribution fits in any case rather well the distribution of runup. In Eq. (22.9) Pot is the factor controlling the order of magnitude of mean overtopping discharge and is the expression of the main effect of relative crest elevation Rc /Hsi , as it can easily vary by orders of magnitude as a consequence of modest variations of structure crest elevation, see the CLASH database (Steendam et al.44 ; wwwclash-eu.org). The data obtained in wave basin by Zanuttigh and Lamberti67 for perpendicular and oblique layouts of LCBs show that the pattern and values of Pot are in good agreement with predictions from Eq. (22.10) and that the overtopping volumes can be well approximated by a Weibull distribution, as in van der Meer and Janssen.55 The mean value of such distribution Vot  shows a more or less parabolic relation with the runup Ruot , that is the median potential runup of waves causing overtopping (Fig. 22.11). Ruot is derived from Eq. (22.10) as the value giving exceeding probability just half the overtopping probability. As suggested by Pilarczyk,38 when overtopping is rare, Vot  is almost proportional to (Ruot − Rc )2 ; this implies a fixed shape of the overtopping crests. The scaled volumes, however, increase significantly as soon as Pot exceeds 0.4–0.5. The location parameter k1 of the volume distribution (typical overtopping volume) can be obtained from the mean value as k1 = Vot /Γ(1 + k2−1 ). The shape parameter k2 of the distribution increases with increasing Pot (Fig. 22.12). Its values range from something below 1 up to 3; the lowest values refer to the case of “rare” overtopping and the highest values to very frequent overtopping. If due attention is paid to difference in overtopping frequency and parameter uncertainties, the lowest figures are not substantially different from the value 3/4 suggested by van der Meer and Janssen.55 In conclusion, for LCBs that are overtopped by most waves, not just by the highest (Pot > 0.4 or Rc < k1 ), the crest level is so low in the approximate Rayleigh distribution of runup that the shape of all overtopping characteristics distributions become more symmetric and less variable (the shape parameter in a Weibull distribution becomes greater) and the overtopping events become longer. The combined model of a Weibull (or Rayleigh) distribution of runup, providing the overtopping probability, and a shape model for overtopping crests, as the one suggested by Pilarczyk,38 represents properly the process for high and low freeboards if the shape parameters are assumed variable with relative freeboard or overtopping probability.

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8 F=0, Layout 1, Narrow berm F=0, Layout 1, Wide berm F=3, Layout 1, Narrow berm F=3, Layout 1, Wide berm F=0, Layout 2, Narrow berm F=3, Layout 2, Narrow berm F=0, Layout 2, Wide berm F=3, Layout 2, Wide berm

7

/(Ruot-Rc)2

6 5

F

Ru

4 3 2 1 0 0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

Pot Fig. 22.11. Mean overtopping volumes Vot  versus overtopping probability Pot , emergent and eres et al.8 zero-freeboard structures. In the plot, F = Rc . From Cac´

3.0

F=0, Layout 1, Narrow berm

F=0, Layout 1, Wide berm 2.5

F=3, Layout 1, Narrow berm

F=3, Layout 1, Wide berm Rayleigh 2.0

F=0, Layout 2, Narrow berm

F=3, Layout 2, Narrow berm

k2

F=0, Layout 2, Wide berm F=3, Layout 2, Wide berm 1.5

Exponential 1.0

0.5 0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

Pot Fig. 22.12. Shape parameter k2 of the Weibull volume distribution versus overtopping probability Pot , emergent and zero-freeboard structures. From Cac´eres et al.8

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The extrapolation below the aforementioned limits of formulae adapted to high structures has two contrasting effects: the intrinsic limitation Pot < 1 is not recognized and the factor is overestimated; the increasing duration of overtopping events is not recognized and the mean volume is underestimated. The total result on the discharge is weak and may be confused with formulae uncertainty. No specific formula exists so far to estimate the overtopping discharge at LCBs. The widely adopted formula for surging waves by van der Meer 
qot Rc  , (22.11) = 0.2 · exp −2.6 3 Hsi γf γβ gHsi where the symbol γ denotes the runup reduction factors accounting for slope roughness (γf ) and wave obliquity (γβ ), whose values and expressions can be found in TAW,47 tends to overestimate the overtopping discharge for LCBs, even with the indeed small correction by Kofoed and Burcharth.26 Equation (22.11) can be thus used only when overtopping is rare, i.e., well-emerged LCBs or weak wave attack with respect to the crest freeboard. 22.6. Induced Setup and Planimetric Circulation Unlike high barriers, the flow rate over LCBs is abundant and related to the rear piling-up. Overtopped water accumulates behind the structure, establishing a higher mean water level, or piling-up, which forces return flows along different paths: water may return offshore through gaps, originating the so-called rip currents, and, if the crest is submerged, also over the structure itself. In this latter case, the flux over the crest during the wave cycle is alternately directed inshore and offshore, driven by waves and piling-up. Since LCBs are typically made of permeable rubble mound, return flow occurs also through the structure, due to the unbalance between the steady hydraulic gradient induced by piling-up and the breaking wave thrust. Gaps are normally shorter but proportionate to breakwater length (∼ = 1/3), but breakwaters, even when permeable, are far less permeable than the open gap, therefore, when the breakwater is emerged the overtopping discharge returns to the sea almost only through the gaps. The flux through gaps qG globally equals the net flux across the structure qIS , which is the algebraic sum of the inshore directed flux over the structure crest qD and the return flow across the structure qU , see Fig. 22.13. The return flow through gaps qG is frequently named also recirculation flow, since it is usually simulated in a wave flume by a recirculating system. The process can be schematized in analogy to a pump system (Lamberti et al.31 ; Zanuttigh et al.66 ). The head losses associated to return flow through gaps are represented by a characteristic curve and the relation between piling-up and net mass flux across the structure is similarly described by a barrier pumping curve. The system operational point at equilibrium is the intersection between the two curves and is the solution of the equation. The pumping curve was experimentally investigated in wave flumes equipped with a recirculation system and it was found to be approximately linear for emerged conditions (Ruol et al.40,41 ; Cappietti et al.10 ). Piling-up reaches its maximum p0 in

FA

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qIS

zO

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qD

zI

qU qG

p ≡ zI − zo qG ⋅ LG LB

Fig. 22.13. At the left-hand side scheme of planimetric circulation; at the right-hand side, scheme of the model approach. From Lamberti et al.31

absence of recirculation and decreases to zero when qIS is recirculated without any flow resistance. For given wave conditions, p in confined conditions is quite greater than for a structure with gaps: indeed, the overall return flow resistance decreases with increasing the gap to structure length ratio LG /LB , and