Introduction to Coastal Dynamics and Shoreline Protection

Introduction to Coastal Dynamics and Shoreline Protection

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Introduction to Coastal Dynamics and Shoreline Protection G. Benassai University of Naples Parthenope, Italy

Introduction to Coastal Dynamics and Shoreline Protection G. Benassai University of Naples Parthenope, Italy

Published by WIT Press Ashurst Lodge, Ashurst, Southampton, SO40 7AA, UK Tel: 44 (0) 238 029 3223; Fax: 44 (0) 238 029 2853 E-Mail: [email protected] http://www.witpress.com For USA, Canada and Mexico WIT Press 25 Bridge Street, Billerica, MA 01821, USA Tel: 978 667 5841; Fax: 978 667 7582 E-Mail: [email protected] http://www.witpress.com British Library Cataloguing-in-Publication Data A Catalogue record for this book is available from the British Library ISBN: 1-84564-054-3 No responsibility is assumed by the Publisher, the Editors and Authors for any injury and/or damage to persons or property as a matter of products liability, negligence or otherwise, or from any use or operation of any methods, products, instructions or ideas contained in the material herein. © WIT Press 2006 Printed in Great Britain by Lightning Source UK Ltd., Milton Keynes All rights reserved. No part of this publication may be reproduced, stored in a retrieval system, or transmitted in any form or by any means, electronic, mechanical, photocopying, recording, or otherwise, without the prior written permission of the Publisher.

Contents

Preface

xi

CHAPTER 1 Integrated approach to coastal dynamics 1. Coastal dynamics basic approach 2. Coastal erosion and remediation study 2.1 Data acquisition 2.2 Critical erosion evaluation 2.3 Planning analysis 3. Causes of coastal erosion 4. Space and time scales 5. Meteomarine factors 5.1 Wind 5.2 Waves 5.3 Currents 5.4 Sea level variations 6. Sediment transport and coastal structures 6.1 Modes of sediment movement and their appearance region 6.2 Coastal structures and sediment transport 7. Elements of coastal management

1 1 3 4 6 7 7 8 9 10 10 14 16 17 17 19 21

CHAPTER 2 Beach morphology and sediment analysis 1. Introduction 2. Beach classification 3. Beach morphology and sediment transport 4. Seasonal profiles, bars and berms 5. Equilibrium beach profile 6. Sediment analysis 6.1 Bathymetric and geophysical surveys 6.2 Physical and chemical analysis

27 27 27 30 33 33 35 35 37

6.3 Sediment size classification 7. Case study

39 42

CHAPTER 3 Linear wave analysis 1. Introduction to linear wave theory 1.1 Governing equations 1.2 Boundary conditions 1.3 Linearized boundary conditions 2. Results of the linear theory 2.1 Wave profile, length and celerity 2.2 Group celerity 2.3 Velocity components 2.4 Particle displacements 2.5 Wave pressure 3. Case study

45 45 45 49 50 51 51 54 56 58 60 62

CHAPTER 4 Sea level variability 1. Introduction 2. Astronomical tide 3. Long waves (tsunami and seiches) 4. Wave set-up and set-down 4.1 Radiation stress 4.2 Water level fluctuations due to radiation stress 5. Storm surge 6. Case study

67 67 67 71 73 74 76 77 85

CHAPTER 5 Random wave measurement and analysis 1. Wave measurements 1.1 Ultrasonic and pressure gauges 1.2 Wave buoys 1.3 Italian Sea Wave measurement Network 1.4 Satellite remote sensing 1.4.1 Radar altimeter 1.4.2 Synthetic Aperture Radars 2. Statistical properties of random waves 2.1 Data sampling 2.2 Data processing 2.2.1 Time domain analysis 2.2.2 Directional wave spectra 2.2.3 Pierson – Moskowitz and Jonswap spectrum 3. Statistical representation of wave climate 4. Case study

87 87 87 88 89 90 90 92 93 93 95 95 99 100 102 103

CHAPTER 6 Short term wave prediction 1. Introduction 2. Elements of wind measurement analysis 2.1 Wind information needed for wave hindcasting 2.2 Geostrophic and low-height winds 3. Wave prediction on deep water 3.1 Fetch and duration limited growth 3.2 Significant wave (SMB) model 3.2.1 Case study 3.3 Spectral wave models 3.3.1 First, second and third generation models 3.3.2 Third generation models 3.3.3 WaveWatch III 3.3.4 Case study (WWIII application for the Gulf of Naples)

107 107 108 109 110 111 112 114 116 117 118 119 122 124

CHAPTER 7 Long term wave statistics 1. Introduction 1.1 Wave data 1.2 Data selection 1.3 Extreme value probability distribution 2. Data fitting to the probability distribution 2.1 Normal Probability Distribution 2.2 Log-Normal Distribution 2.3 Gumbel distribution 2.4 Weibull distribution 3. Parameter calculation 3.1 Statistical tests of fit 3.2 Confidence intervals 3.3 Statistics of offshore extreme waves 3.4 Wave height persistence 3.5 Case study

127 127 128 129 129 129 130 131 131 132 133 133 134 135 137 138

CHAPTER 8 Wave transformation in the coastal zone 1. Wave energy and energy flux 1.1 Potential Energy 1.2 Kinetic Energy 1.3 Energy Flux 2. Refraction and shoaling 2.4 Discussion on Kr and Ks 3. Total reflection 4. Wave diffraction 5. Numerical models for wave propagation 5.1 Phase-averaged model 6. Finite depth spectral wave models 6.1 Other finite depth spectral wave models

143 143 143 144 145 146 150 152 155 156 158 159 161

6.2 Phase-resolving models 6.2.1 Boundary Integral Models 6.2.1.1 Mild Slope Equation Models 6.2.1.2 Boussinesq equation model 6.2.2 Lagrangian models – Ray method for wave transformation 6.2.3 Eulerian models 6.3 Grid models 7. Wave breaking

161 163 163 165 166 169 169 172

CHAPTER 7 Sediment transport 1. Introduction 2. Basic concepts of sediment transport 2.1 Critical bed shear stress 2.2 The Shields parameter and modified Shields diagram 2.3 Sediment fall velocity 2.4 Bed load and suspended load 2.4.1 Bed-load and shear stress 2.4.2 Steady bed load in sheet flow transport 2.4.3 Basics of suspended load transport formulation 2.5 The bottom boundary layer and the bed roughness 2.6 Bed load and suspended load: a simple parametrical model 2.7 Case study 3. Basic shore processes 3.1 Nearshore circulation 3.2 Wave run-up in the swash zone 3.3 Bar formation by cross-zone flow mechanisms

175 175 176 176 177 179 180 181 182 183 185 187 189 194 194 198 199

CHAPTER 10 Beach profile modeling 1. Cross-shore transport 2. Cross-shore sediment transport and equilibrium beach profile 3. Dean’s model for equilibrium beach profile 3.1 Equilibrium parameter A 4. Processes of accretion and erosion 4.1 Surf zone 4.2 Swash zone 5. Erosion/accretion parameters 5.1 Case study 6. Analytical profile modelling 6.1 Case study 7. Numerical beach profile modeling 7.1 Example of numerical model: SBEACH

201 201 203 204 207 208 208 208 209 213 214 219 220 222

CHAPTER 11 Shoreline modeling 1. Introduction

227 227

2. Longshore transport 2.1 Case study 3. Numerical shoreline modeling 3.1 GENESIS 3.1.1 Governing equations 3.1.2 Model parameters 3.1.3 Model implementation 3.1.4 Model calibration

228 230 231 234 234 236 239 241

CHAPTER 12 Comparison and choice among alternative protection systems 1. Introduction 2. Insertion of protection systems on the coastline 3. Shoreline protection systems 4. Hard measures 4.1 Detached emerged breakwaters 4.2 Detached submerged breakwaters 4.3 Emerged or semi-submerged groins 4.4 “T”-shaped emerged of semi-submerged groins 4.5 Adherent breakwaters 4.6 Seawalls 5. Soft measures 5.1 Artificial nourishment 5.2 Dune restoration 6. Schematic indications for the choice 7. Mechanisms of protection 7.1 Efficiency 7.2 Induced efforts

245 245 245 246 248 248 249 250 252 252 254 254 254 256 258 260 260 261

CHAPTER 13 Hydraulic design 1. Dimensional analysis 2. Wave run-up Ru and run-down Rd 3. Overtopping discharge 4. Transmission coefficient 5. Reflections 6. Case study

263 263 265 268 270 271 273

CHAPTER 14 Structural design 1. Introduction 2. Structural stability 2.1 Hudson formulation 2.2 Van der Meer Formulation 2.3 Comparison of Hudson and new formulae 3. Armour layers with concrete units

275 275 277 279 280 283 283

4. Low-crested structures 5. Reef breakwaters 6. Statically stable low-crested breakwater 7. Submerged breakwaters 8. Filter and core characteristics 9. Toe stability and protection 10. Breakwater head stability 11. Fundamentals of probabilistic design 12. Deterministic design – case study

285 286 286 287 287 288 289 289 291

CHAPTER 15 Beach fills 1. Introduction 2. Beach fill profile 3. Volume computation 4. Beach planform evolution 5. Longevity of beach fills 6. Effect of fill length and of wave climate 6.1 Case study 7. Compatibility of the borrow material 7.1 Case study 8. Sediment sources 9. Monitoring

293 293 296 298 298 301 303 305 306 308 309 310

References

313

Preface This book was developed from lecture notes for a course on Coastal Dynamics and Shoreline Protection addressed to students of Environmental Sciences. This is the reason why it is organized to introduce the reader to the fundamental principles of the topics treated in each chapter. It can be used as a training aid both for students and for practicing engineers, as almost every topic is developed with case studies. The book, which deals primarily with sandy coastlines, is divided into three parts. In the first part, which is limited to Chapter 1 – Integrated approach to coastal dynamics, the reader is introduced to the approach of a coastal erosion and remediation study and to coastal management. In the second part, the meteomarine factors cited in Chapter 1 are dealt with in some detail, together with the mechanisms of sediment transport. The topics addressed are, amongst others, linear and higher order waves, random waves and spectra, wave transformation in the coastal zone, water levels, short-term and longterm wave prediction, sediment transport, shoreline and beach profile modeling. The third part deals with the choice between various protection systems and tries to give the reader some basic elements of hydraulic and structural design for both rigid structures and beach fills. Acknowledgements have to be addressed to all the people who inspired me, in particular my father Edoardo for the first approach to the maritime structures, Giulio Scarsi and Laura Rebaudengo of Genoa University for their rigorous analytical approach to the maritime hydraulics. Many thanks have to be addressed to all the students to whom I had the privilege of teaching the concepts reported in this book for more than 14 years, and to the University Parthenope of Naples where I currently teach the course on coastal dynamics and shoreline protection. Additional thanks have to be addressed to the people who typed the manuscript for their patience and their contributions, especially Dr I. Ascione and Dr. E. Chianese. Finally, particular thanks is due to my wife Maria Pia and my sons Edoardo Maria and Rossella who have given me total support and encouragement, and to whom this book is dedicated. G. Benassai Naples, 2006.

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Chapter 1 Integrated approach to coastal dynamics 1 Coastal dynamics basic approach The littoral area is a system in unstable dynamic equilibrium: storms create short term erosional events, natural recovery after the storm and seasonal fluctuations may not be in balance to produce long term erosion. Shore protection projects moderate the long term average erosion rate and reduce damage caused by flooding and wave attack. The coastal environment constitutes a fragile, little known and complex ecosystem that is an important resource for most nations. The high economic interest relative to environmental resources and man’s extensive and growing use of the coastal zone drives the need for the optimization and the mitigation actions. In order to reach a satisfactory design, we must understand each element and its interactions, the inputs and the outputs, and how they affect the system and neighboring systems. In fact, any kind of protection scheme interfering with the littoral transport (groynes, breakwaters etc.) modifies this equilibrium. The relative consequences cannot be estimated without the analysis of the physical phenomena and the simultaneous consideration of the local process and their interaction with adjacent systems. In the past, mistakes were made by not considering the proper system boundaries or by not considering a super-system, when necessary. This restricted vision produced huge damages for the neighboring beaches, with catastrophic environmental, economic and social consequences. For this reason, an integrated approach to the coastal zone has to be done with a preliminary identification of all systems that influence the design and that are affected by it. The monitoring of the systems must be done on both a short and long time scale and has to be extended to the entire “physiographic unit”. This is characterized by the circumstance that the coastal sediments move inside it without external influences. The identification can be based on the study of the

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INTRODUCTION TO COASTAL DYNAMICS AND SHORELINE PROTECTION

natural factors (wind, currents, human impact), or on the consequences of these factors: erosion and accretion. The identification requires the knowledge of the dynamic conditions and of the interaction with the littoral evolution. The limits of the area involved can be variable in time in relation to the events (natural or anthropic), which modify the coastal dynamics. The analysis of the wave conditions and in particular the interaction between the waves and the sediment plays a relevant role for understanding the physical processes. This analysis, then, has to be focused on the monitoring of the wave climate and on the statistical analysis of its spatial and temporal characteristics (wave height, direction and period). A statistically valid analysis can be obtained through the monitoring of the waves for a period of 10 years or more. When the waves approach the shoreline, they are modified by the seabed through processes such as refraction, shoaling, bottom friction and diffraction (if waves interact with structures). The waves, then, change in height and direction. This feature plays a relevant role in the littoral transport process. A numerical model is then required in order to consider the role of the bottom in changing the wave characteristics, due to lack of field local data regarding the littoral transport. This interaction between the waves and the bottom is due primarily to second order shallow water phenomena, such as the wave drift (which is responsible for mass transport) and the radiation stress, which is associated with the longshore current for obliquely incoming waves. The wave drift is directed normal to the shoreline, transporting the water particles near the surface in the direction of wave propagation. The shore-parallel component of the radiation stress generates the longshore current, which carries sediments along the shoreline (the so-called littoral drift). Other significant interactions can occur between waves and currents. Wavecurrent interaction may affect the development of rip currents. In fact the weak currents generated by a gentle alongshore variation in the wave field can cause significant refractive effect on the waves so as to change the structure of the forcing which drives the currents. This can cause the development of instabilities of the cellular circulation and can influence the rip channels spacing and depth. From the literature it is assumed that the longshore sediment transport is more relevant than the cross-shore one. The latter process, although significant during storms, gives little final influence on the sediment dispersion along the coastline. The littoral transport on a long time scale is directly connected with the longshore component of the energy transmitted during the breaking phenomena. This parameter can be computed on the basis of both the offshore wave climate (frequency of events classified by period, direction and wave height) and the propagation features of the waves. So a statistical analysis has to be carried out through all the possible combinations of wave height, period and direction, statistically computed from field measurements. The sum of these contributions, related to the breaking wave energy flux, allows to compute the

INTEGRATED APPROACH TO COASTAL DYNAMICS

3

longshore transport on several time scales. The applicability of the results depends on the analysis of the effective sediment transport, which means to put in the numerical model more information, such as grain size in relation to the depth and the breaking wave type (the most important parameters during the sediment suspension). The results obtained have to correspond with an independent assessment about the beach morphology and the influence of existing structures. The complexity of environmental phenomena makes the design process extremely complex. Even if the more sophisticated methods are used (backanalysis of a finite number of real cases, or the results of the close reproduction of phenomena), it is possible to make large errors in relation to the techniques and materials used and the boundary conditions imposed. This is the reason why the design is done by trial and error. This approach is obviously unacceptable in the prototype, but it is possible, however, in numerical models. These numerical models are used to verify the impact of structure in several environmental conditions, optimizing the localization, typology and the dimension of protection systems. In this phase a quantitative evaluation is carried out by numerical simulation of the coastline evolution in order to identify the potential environmental impact of the project. This prognostic step of the design process requires the definition of boundaries and the identification of the coastal dynamics through the analysis of available data. The impact assessment of the sediment budget supplied by the rivers, the analysis of available maps and aerial photographs and the study of the impact of existing structures define the boundaries of the physiographic unit. The numerical simulation in the final phase allows the quantification of the specific environmental impact, the time scale and the significant trend of the coastline modifications resulting from the interaction between the new structure and the mean wave climate.

2 Coastal erosion and remediation study The fundamental criteria of a coastal dynamics study are based on an accurate evaluation of the environmental, economic and social aspects of the area of interest on a spatial scale of the physiographic unit. In fact, the key concept in coastal design is the integration, that is the awareness that nothing should be done within a littoral cell without thinking about how it affects the rest of the cell, which by definition is part of the same system. The main goals of the coastal dynamics study are the assimilation and collection of a huge quantity of experiences, actually distributed among several private companies and government agencies, in order to reach a complete vision of the existing knowledge and in order to plan the coastal protection in the best way. In fact, a deep knowledge of the integrated physical processes can avoid unwise and dangerous actions and plays a substantial role in minimizing possible local negative consequences.

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INTRODUCTION TO COASTAL DYNAMICS AND SHORELINE PROTECTION

With these premises, the main technical steps of a coastal dynamics study are the following: • • •

Data acquisition Critical erosion evaluation Planning analysis

A typical scheme of a coastal dynamics study is given in figure 1.1. Offshore Waves and Winds

Tides and Currents

Offshore Wave Climate

Hydrodynamics

DATA ACQUISITION

Coastal Morphology

PLANNING ANALYSIS

CRITICAL EROSION EVALUATION

Bathimetry

Coastal Dynamics Analysis

Inshore Wave Climate

Sediment Transport Analysis

Littoral Transport Analysis

Sediment Transport Evolution Satellite and Aerial Maps Bathimetry Dispersion of Sediments Effects of Structures

Numerical Model Shoreline Evolution

Wave-Current Interactions

Sediment Budget Physiographic Unit Significant Parameters

Trial-Error Design

Shoreline Simulations

Figure 1.1- The main steps of an integrated approach to coastal design. 2.1 Data acquisition The first phase of the study will involve the identification and acquisition of available data. Coastal data are usually acquired through field observation and may be divided into the following categories:

INTEGRATED APPROACH TO COASTAL DYNAMICS

5

• •

Field data (bathymetric and topographic surveys, sedimentology); Office data provided by: interpretation of historical maps, aerial photographs, etc; numerical simulations of the coastline, laboratory and office data analyses. Data are provided by various sources, such as: • Literature sources (University Departments, Journals and conference proceedings, reports of research projects); • Local sources, which provide detailed and sometimes unique data pertinent to the site (reports of local newspapers, interviews with fishermen, etc.); • Government agencies, which provide data archives and studies with relevant coastal information; • Private companies, such as oil, gas and construction companies. Some of these data are in the public domain and include also environmental impact reports containing extensive coastal process data; • Computerized literature databases containing information that may be acquired by key terms, subjects, titles and author names, available to major Universities and Government agency libraries. Data requested to characterize the significant coastal processes and to analyze the characteristics of severe storms include the following: Wave data, which include wave height, period, steepness and direction and breaker type. These data may be available through various techniques: recorded wave data, obtained using buoys and other recording instruments located offshore (piers or other coastal structures near the study site); hindcast data obtained from numerical prediction models and used to estimate wave statistics. Water level data are provided using tide gauges deployed near the study site. The water level is commonly measured through three types of instruments: pressure transducers, floating gauges and staff gauges. Typically, water level measurements are related to a given datum; daily records are usually published in reports, while predicted water levels and tidal current information for each day can be obtained from the annual Tide Tables. Geologic and sediment data include geologic maps and data collected during bathymetric surveys. These data are important to characterize the response of the coastline to severe storm events. Aerial photographs are useful for studies over long time scales to understand shoreline change assessments and to get information about coastal landforms and materials, behavior of engineering structures, location of rip currents, storm effects on the coastline, etc. Satellite data, like aerial photographs, are useful for assessing large-scale changes of the coastal zone. Finally, remote sensing allow to define spatial patterns of suspended sediments in shallow waters and freshwater discharges near estuaries.

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2.2 Critical erosion evaluation The most part of defense actions in the past were done in emergency conditions and economic constraints. Usually the actions were focused on the restoration of the previous conditions without paying attention to the causes of erosion. Besides, the actions were not focused on the efficiency, on the past experiences and, above all, on the need to reduce the environmental impact. For this reason, the first step of the critical erosion evaluation is the analysis of the existent structures, which means: • Impact analysis of the structures on the coastline evolution and on the local bathymetry; • Localization, time and costs evaluation for dredging operations and maintenance of ports and waterways; • Evaluation of the amount of volume for artificial nourishment; • Sustainability of the project location at all the technical and administrative levels. The basic critical shoreline changes can be summarized in three types: • Natural shoreline accretion • Shoreline erosion • Interactions between coastal structures and littoral processes The fundamental criteria to detect these critical situations are the following: • Physiographic unit • Littoral transport intensity • Coastline evolution • Sediment loss • Human impact Coastal erosion studies involve part of the following phenomena: • Erosion processes caused by natural or human impact • Sediment supply by rivers or dunes • Conflicting demands on the coastal area After the data acquisition and analysis, a hypothesis of critical area localization on provincial or regional scale has to be defined in order to focus the areas with the greater erosion problems. All the coastal area characterized by different erosion levels has to be detected and classified, in order to identify the urgency level and the correct strategy. The classification of the critical areas starts from the following identification of the damaged areas: • Areas with high erosion level: they need huge restoring activity of the littoral transport and/or beach nourishment • Areas with medium erosion level: they need a beach stabilization but specific restoring actions are not required. • Areas with low erosion level: they only need maintenance of existing equilibrium.

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2.3 Planning analysis The planning analysis requires: • • • • •

Analysis of the environmental local risks (erosion processes; loss of sediment; human impact, etc); Analysis of the existing negative responses; Final setting of physiographic unit limits; Classification of critical zones according to priority criteria; Checklist of possible alternatives to shore protection (retreat, do nothing);

In relation to the erosion level found out in the physiographic unit, the following options can be chosen: • • •

Passive hard structures: revetments, bulkheads, sea walls; Active hard structures: groins, detached breakwaters (emerged or submerged); Soft structures: artificial nourishment alone or protected with submerged structures, sand bypass.

The choice among alternative solutions is based on extensive coastal knowledge, which is the combination of theory and experience. Some of these solutions will be illustrated in chapter 12.

3 Causes of coastal erosion The main causes of coastal erosion can be divided into natural and human. Nevertheless, it has to be observed that the latter causes are indirectly related to the most part of the “natural” erosion processes. Natural causes The main natural cause of erosion is the imbalance between the loss of sediment due to offshore transport and the decrease of the longshore transport, which results in a negative sediment budget. The global climatic change emphasizes this problem and affect the sediment transport; the decrease of run off is also responsible for loss of littoral transport. The forces acting on littoral sediments are related to wave conditions. The strong dissipative processes associated to breaking events generate alongshore currents responsible for the littoral sediment transport between the breaker zone and the swash zone, which are responsible for spatial change. During severe meteomarine events, the sediment near the coastline is transported offshore and deposited in a bar far away from the shoreline by the steeper waves. During the following calm periods, the smaller waves tend to move the sand from the bar location toward the beach, even if part of the sediment is lost in the open sea. Other factors affecting the sediment transport are density currents and tidal currents which may change the wave circulation pattern. Another natural cause

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INTRODUCTION TO COASTAL DYNAMICS AND SHORELINE PROTECTION

of erosion is the mean sea level and the direct sediment transport by the wind, which results in the dunes formation. Human impact Human causes depend on the interaction between coastal structures and littoral transport, because they can modify the natural equilibrium through the imbalance of longshore sediment budget. Dams and other flood prevention structures reduces sedimentation in rivers and accelerates erosion processes. Besides, urbanization of coastal areas decreases the longshore sediment transport causing the erosion of some areas and the silting up of others. Localized structures like emerged and submerged breakwaters, groins and their combinations reduce the damages, although they need a focus study on sediment dynamics in the interested area and on the total extension of physiographic unit. This analysis has to be carried out to avoid additional damage both on the shoreline evolution and on the quality of the coastal environment.

4 Space and time scales Before the examination of the physical phenomena directly responsible for coastal processes, a first fundamental issue concerns the spatial and temporal scale on which processes take place. In this context the processes developing in relatively shallow water will be considered, that is between the coast line and the breaker zone where the interaction between the water forces and the bottom is particularly relevant. The usual approach when hydrodynamics (turbulence, waves, tides, stormsurges, currents, etc.) and morphodynamics (ripple formation, bar formation, cross-shore transport, longshore transport, etc.) are involved is the division of processes in relation to the spatial and temporal scale of observation (see figure 1.2). The processes involved can be distinguished in relation to the spatial/temporal scale of observation: • • •

Small scale (0.1mm – 10m, 0.1sec – 1 day) Intermediate scale (1m – 10km, 1sec – 1 year) Large scale (1 km – 100km, 1month – ten-years periods)

Different processes are connected to each other: in fact, the large scale processes modify boundary conditions for the processes on the intermediate scale, which, in turn, influence the processes on smaller scales. Some interesting questions about the influence of resolution on the correct description of the phenomena may be raised: the relationships between spatial resolution and predictability show that while increasing resolution provides more descriptive information about the patterns in data, it also increases the difficulty of accurate modeling.

INTEGRATED APPROACH TO COASTAL DYNAMICS

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Figure 1.2 - Space-time scales relationships, (modified from De Vriend, H.J., 1991). Processes on small scale Small scale processes involve interactions between bottom currents and sediments, which generate the sediments transport. Several bed forms are shaped by wave action (ripples and dunes). The bottom morphology and the bottom boundary layer dynamics are crucial at this scale to define the processes involved in the sediment transport. Processes on intermediate scale Intermediate scale processes involve the wave dynamics, the littoral wave circulation and the evolution of the bottom morphology due to sediment transport. Processes of wave dynamics include refraction and shoaling (caused by the depth decrease) diffraction, reflection (in presence of sheltering structures), breaking and dissipation. The circulation in the surf zone has to be evaluated in order to compute the sediment transport. The next step is to consider bathymetry evolution, which, in turn, modifies the circulation. The processes go on till an equilibrium is reached. Processes on large scale Large scale processes involve the behavior of wide coastal areas on long times, which is known in literature as LSCB (Large Scale Coastal Behavior). Large scale models are developed integrating the ones on small and intermediate scale. The starting input is a known morphology, then the forcing (waves, currents, river inlets, structures, etc) have to be imposed to find out a new configuration. The large scale observation is important from an economic and social point of view, and it represents an efficient tool for the correct coastal zone management.

5 Meteomarine factors Meteomarine factors are directly associated with climate, such as winds and atmospheric pressure variation, waves, currents, tides and storm surge, which affect coastal morphology.

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5.1 Wind Wind is caused by pressure gradients between adjacent areas. Pressure variations are generally the result of temperature differences due to disomogeneous insulation of adjacent zones reached by solar radiation (e.g. between lands and oceans). Wind fields vary in a wide range of spatial and temporal scales, from large-scale (permanent winds) to local (short winds). Hydrodynamics is strongly influenced by wind action, which is responsible for wave generation, wind setup, surge and surface currents. Wind forces directly modify coastal morphology through transport and deposition of sediments on the beach and the dunes, while they indirectly modify coastal morphology generating waves responsible for sediment transport. Wind action has a substantial role in some phenomena such as sea breezes, water level fluctuations and seiches. Sea breezes occur in a coastal area, especially in summer, and are generated by pressure gradients between water and coast: during the day, the warm air over the land becomes lighter and rises, thus forming an area of low pressure, resulting in a landward-directed breeze. During the night, the cooling of land is faster than water, which results in a seawarddirected breeze. Breezes don’t play a significant role in shaping the coastal morphology and give little contribution in sediment transport. Rapid changes in atmospheric pressure and directional shifts of strong winds on small seas and confined water bodies, cause periodic water level oscillations called seiches. 5.2 Waves Waves are the main factor in determining the shape and morphology of beaches and significantly influence the planning and design of harbours and coastal structures. The wave crest is the high water level reached by the free surface, whereas the wave trough is the low water level. The wave height (H) is defined as the vertical distance between the crest and the trough. The wave length (L) is the distance over which the wave pattern repeats itself. The wave speed (C) is the velocity of a single wave, while the wave period T is the time required for a wave to pass a particular location. The inverse of the period is the wave frequency f. When waves become large or travel toward shore into shallow water, higherorder wave theories are often required to describe wave phenomena. These theories represent nonlinear waves, while the linear theory is valid when waves are small and travel on deep water (but it still provides some insight for finiteamplitude periodic waves). However, the linear theory cannot account for the fact that wave crests are higher above the mean water line than the troughs are below the mean water line (U.S. Army Corps of Engineers - Coastal Engineering Research Center, 2001). Waves are strongly related to wind action on surface water, which provides a source of energy for wave formation. Waves propagate energy supplied by the

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wind at air-sea interface, contribute to longshore transport and move bottom materials onshore and offshore. The mechanical characteristics of waves are important factors in planning and design of coastal structures because they interact with obstacles (breakwaters and coastal protection works), changing their characteristics until breaking occurs. Waves are divided in two main groups: short waves (period lower than 20s) and long waves, also called long-period oscillations (period between 30s and 40min). Water-level oscillations with a period longer than 1 hour (called astronomical currents and storms surge) are related to water-level variations. Short waves include wind waves and swell, whereas long waves can be divided into surf-beats, harbour resonance, seiche and tsunamis. Wave Classification A typical classification based on the frequency representation of all oceanic waves is given in Kinsman, B., 1965 (figure 1.3). A wave field is formed by many individual wave components, each characterized by a wave height, a wave period and a direction. Wave fields with many different wave periods and heights are called irregular, whereas wave fields with many wave directions are called directional. A wave field can be more or less irregular and more or less directional. Gravity waves are one of the most important factors acting on coastal morphology. Wave conditions vary extensively from one site to another, and depend on the wind climate and on the fetch extension, that is the water distance covered by wind without obstructions. Short waves are divided in two groups: •

•

Wind waves (also called sea waves), are generated by the local wind field. They are generally steep (high and short), irregular and directional waves. Their directionality makes difficult to recognize defined wave fronts, so they are also named short-crested waves. Swell are waves that traveled long distances away from their generation area and are more regular and unidirectional than wind waves. A portion of wave energy is lost during the propagation in deep water due to dissipation phenomena, which results in long period waves.

Wave Generation Wind waves are generated as a result of the action of the winds on the sea surface. Wave properties (height, period, direction) at a site depend on the following factors: • The wind field (speed, direction and duration); • The fetch of the wind field (meteorological fetch) or of the water area (geographical fetch); • The water depth in the wave generation area.

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INTRODUCTION TO COASTAL DYNAMICS AND SHORELINE PROTECTION

Figure 1.3-Waves classification by frequency (after Kinsman, B.,1965). Wave height and period are closely related to wind conditions, from which they are estimated. The analysis of historical measured wind records is useful to reconstruct wave climate at a site. Such a computation is known as wave hindcasting. Forecast wind conditions are used to perform the wave forecasting. Wave transformation When waves approach a shoreline, they are affected by the seabed through refraction, shoaling, bottom friction and wave-breaking. Wave-breaking can also occur at deep water when the waves are getting too steep. The following transformations may occur: • Shoaling is the wave deformation, starting when water depth becomes less than about half the wavelength. This process causes a reduction in the wave propagation velocity and a shortening of the waves. • Refraction is caused by the fact that waves propagate more slowly in shallow than in deep water. When the wave front travels at an angle with the bathymetry, a change in direction of wave propagation occurs. An example of refraction phenomenon due to the bottom interaction is given in figure 1.4. • Diffraction is caused by sheltering structures such as breakwaters. This is the process by which the waves propagate into the lee zone behind the structures by energy transmittance laterally along the wave crests. • Bottom friction causes energy dissipation and thereby wave height reduction as the water depth becomes shallower. Friction is of special importance over large shallow water areas.

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13

Figure 1.4 – Waves usually do not approach the shoreline parallel to the coast, but interact with bottom, resulting in wave refraction (after Thurman, H.V., 1985). • Depth-induced wave breaking happens when the wave height becomes greater than a certain fraction of the water depth. The wave height of an individual wave at breaking is often said to be around 80% of the water depth, but this is an approximate number, depending on breaking type and beach slope. • Wave-current interactions occur in the presence of a large scale current field (like a river mouth) and in the presence of rip currents. They influence the associated sediment transport and the subsequent shoreline evolution. • Wave-wave interactions result from nonlinear coupling of wave components and result in energy transfer from some wave component to other ones. Statistical description of wave parameters A statistical description of waves is necessary in order to make a correct analysis of the wave climate and of the littoral transport. This statistical description is based on the directional distribution of the following wave characteristics: • The significant wave height Hs, which is defined as the mean of the highest third of the waves in a time-series of waves representing a certain sea state (Hs computed on the basis of a spectrum is referred to as Hm0). • The mean wave period Tm, which is defined as the mean of all wave periods in a time-series representing a certain sea state. • The peak wave period Tp, which is the wave period with the highest energy. The analysis of the distribution of the wave energy as a function of wave frequency for a time-series of individual waves is referred to as a spectral analysis. The peak wave period is extracted from the spectra. • The mean wave direction θm, which is defined as the mean of all the individual wave directions in a time-series representing a certain sea state.

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INTRODUCTION TO COASTAL DYNAMICS AND SHORELINE PROTECTION

These parameters are often calculated from a time-series of the surface elevations once every three hours. The time-series is thereafter statistically analyzed to arrive at a synthetic description of the wave conditions in order to reach two main goals: •

•

Identify the occurrence probability of the extreme wave events for the design of maritime structures; this is often done through the percentage of exceedance of the wave height Hs; Identify the wave climate for the computation of coastal dynamics processes; this is often done through a directional distribution of the wave heights, often presented in the form of a wave rose or scatter diagram of Tp, versus Hs.

5.3 Currents Currents occur within the deep ocean, over the continental shelf and within the offshore and nearshore zones. They may be quasi-steady and persist for several hours to several weeks (ocean and shelf currents), or they may be oscillatory with periods of seconds (currents under waves). Currents may be limited to the surface or to the seabed, or they may extend over the full depth of water. Surface currents can have a different direction to those at the seabed. A partial currents classification based on the water depth divides the currents into the ocean, shelf and nearshore currents. The largest currents are those of the open ocean (ocean currents), which are driven by global scale interactions between the atmosphere and the sea. The continental shelf can extend between 1 and several kilometers from the coast. Continental shelf currents, are a complex mix of several components, among which there are internal waves, coastal trapped waves, tides and local wind induced currents. Shelf and ocean currents are generally of little significance within the shallower waters of the nearshore zone. This area is the preserve of wave induced currents which include: • • • •

oscillatory currents at the seabed prior to wavebreaking; mass transport of water shoreward as waves break; rip currents; longshore currents.

Currents are also an important aspect of many designs, particularly those involving the environment, water quality and habitat. Ocean currents are driven by the circulation of winds above the surface water, but coastal design needs the analysis of the wave-induced currents rather than the ocean currents. Wave-induced currents Nearshore currents are mainly caused by surface waves breaking on a beach. In fact, when surface waves break on a beach, wave energy is lost to turbulence generated in the process of breaking, and wave momentum is transferred into the

INTEGRATED APPROACH TO COASTAL DYNAMICS

15

water column generating nearshore currents. There are two current systems whose flow structures are predominantly horizontal, alongshore currents caused by obliquely incident waves and cell-like circulations, which can occur when waves are nearly at normal incidence. The scheme of a nearshore current system is indicated in figure 1.5. Cross shore currents It is known that the propagation of surface waves produces a mass transport in the direction of wave propagation, which is a second-order correction to the linear wave theory. This mass transport is called the wave drift and is directed toward the coast. When the waves are breaking, water is also transported in surface rollers towards the coast. These two contributions are concentrated near the surface. As the net flow must be zero, they are compensated by a return flow in the offshore direction, which is concentrated near the bed (undertow). Long shore currents The longshore current is generated by the shore-parallel component of the stress associated with the breaking process for obliquely incoming waves, the socalled radiation stresses. This current, which is parallel to the shoreline, carries the sediments alongshore and it is approximately proportional to the square root of the wave height and to sin(2αb), where 2αb is the wave incidence angle at breaking.

UNIT CELL

RIP HEAD

MASS TRANSPORT

WAVE BREAKING

RIP CURRENT

LONGSHORE CURRENT

SURF ZONE

SHORELINE

Figure 1.5 – Nearshore current system. Rip currents Rip currents have been defined as currents in the offshoreward direction which return the sea water transported shoreward by wave action. These currents,

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INTRODUCTION TO COASTAL DYNAMICS AND SHORELINE PROTECTION

which are concentrated within the surface layer, are part of the cellular circulations, depicted in figure 1.5, ‘‘fed’’ by the converging alongshore flows close to the shoreline. Rip currents cause a seaward transport of beach sand, hence have direct impacts on beach morphology. Recent studies (Yu, J. & D.N. Slinn, 2003) show that the offshore directed rip currents interact with the incident waves to produce a negative feedback on the wave forcing, reducing the strength and offshore extent of the currents. 5.4 Sea level variations Water-level variations are important because they lead to identification of the swash zone and the fluctuations of the coastline. The extreme water levels are associated to flooding events. Water-level variations can be divided in two types: • Regularly oscillating variations with periods from half day up to one year (astronomical tides); • Non regular variations with recurrence periods from days up to several years, mainly caused by meteorological conditions. Astronomical tide The astronomical tide consists of the periodic rise and fall of water level generated by gravitational interaction among the earth, moon and sun. These factors determine the tide at a given location, mainly generated in the deep oceans from which travels into the coastal waters. The tidal wave height in deep water is normally less than 0.5m, whereas in shallow water it is modified by shoaling and friction; at specific locations the tide can be up to 15m high. The tidal conditions at a specific location vary according to semi-diurnal and diurnal tidal constituents and are published in Tidal Tables. If the semi-diurnal constituents determine the tide at a given location, the tide is called semi-diurnal, and if the diurnal constituents dominate it is called diurnal tide. A semi-diurnal tide has two high waters and two low waters every day, whereas diurnal tides have only one of each every day. In addition to these diurnal and semidiurnal variations the contribution of the fortnightly variations must be added, that cause the tide to be higher than normal at full moon and at new moon (spring tide), and lower than the normal at the quarters (neap tide). Meteorological water-level variations The water level also varies as a function of the wind impact and atmospheric pressure variations on the water surface: • Wind action drives onshore and offshore the surface waters and is responsible for the water-level rise (wind set-up) in restricted areas subject to wind stress: when wind drives water offshore, deep waters move onshore, and vice versa. • Weather disturbances cause water level variations called barometric surge and storm surge. High onshore wind over the shallow water surface associated with low barometric pressure causes a temporary

INTEGRATED APPROACH TO COASTAL DYNAMICS

17

sea level rise, which results in flooding and coastal damage. The storm surge is the result of the combined impact of the wind stress on the water surface, and the atmospheric pressure reduction. The storm surge does not include the effect of the astronomical tide. The combined effect of astronomical and meteorological surges is often referred to as a tidal-wave. Barometric surges are often associated to storm surge and are generated by the inverse relationship between sea level and barometric pressure: water surface level rises as atmospheric pressure decreases of about 0.1 m for each kPa of pressure difference. In figure 1.6 an example of barometric surge superimposed on astronomical tide is given.

6 Sediment transport and coastal structures 6.1 Modes of sediment movement and their appearance region

Mean sea level (m)

The inshore sediment movement is usually divided into bed load, suspended load and sheet flow, as shown in the scheme of Figure 1.7. On deep water, waves have only a minor influence on sediment transport. As a wave enters shallow water, it begins to “feel” the bottom, its profile deviating from the linear one. In

Time (Days) Figure 1.6 - Example of barometric surge superimposed on astronomical tide (G. Benassai, 2002). fact, linear wave theory predicts that the crest and trough heights of the wave are equal (see Figure 1.8), that is, the wave is evenly distributed about the still water level. In contrast, finite amplitude wave theories predict waves with peaked crests and flat troughs. The crests are further above the still water level than the troughs are below this level (Young, I.R., 1999). When a resisting force of the

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INTRODUCTION TO COASTAL DYNAMICS AND SHORELINE PROTECTION

sand particle on the bottom becomes smaller than a wave force on it, the sand begins to move. The depth of this point on the beach is called a critical depth for sediment movement and the critical velocity is defined as the water particle velocity at this depth. When a shearing force larger than the critical condition for sediment movement acts on the sea bed a sediment movement of traction mode (bed load transport) takes place. As the shear stress increases further, sand ripples are formed on the bed. Suspended sediment occurs by the advection of the trapped sediment in the vortex formed periodically around the sand ripple (Sawaragi, T., 1995). When the bottom shear stress increases beyond a certain limit, ripples disappear and the so-called sheet flow in which sediment in high concentration is transported within a thin layer appears. Because a great deal of the sand particles are transported in the thin layer in this range, fluid turbulence near the bottom is repressed and any systematic turbulence (such as vortex on the ripple), does not occur there. In the shallow water region where incident waves break, a great deal of sediment is brought into suspension by the turbulence caused by breaking waves. Breaking waves of the plunging type suspend more sediment than breaking waves of the spilling type. In fact in the plunging breakers the crest becomes faster than the trough, curls and violently collapses. swash zone

breaker zone

breaking point

Disapperance of ripples Initial motion Generation of ripples

Bed load transport

Figure 1.7- Sediment movement and mode of sediment transport on the beach. A considerable amount of energy is released into a downwardly directed mass of water and the turbulence reaches the area beyond the breaking zone. Sediment materials are suspended and transported offshore by undertow currents. A large trough is formed and becomes deeper until the energy is completely exhausted. Offshore a bar is formed, localized between the breaking point and the deepest zone reached by vorticity. In the swash zone, fluid motions become entirely different from that in the shallow water region and sand movement also differs from the other part of the

INTEGRATED APPROACH TO COASTAL DYNAMICS

19

beach. In fact, the swash zone is characterized by a fluctuation of coastline associated with sediment transport which occurs when the energy of the waves advancing up the beach is not completely exhausted. Swash dynamics depend on the wave frequency oscillations: at low frequencies, if the beach is permeable and the sand is not saturated, the water percolates through the substrate and the backrush decreases. An amount of sediment is accumulated on the beach in the swash zone and the beach slope increases. The result is a convex beach profile. At high frequencies (storm waves), if the substrate is saturated, the water cannot percolate through the sand and the backrush current cannot decrease. In this case the sediment transport associated with backrush is greater than uprush, which results in a concave beach profile. 6.2 Coastal structures and sediment transport Coastal structures interfering with littoral transport are the most common causes of coastline modification. The principal structures that may cause erosion are: • • • •

Groynes or similar shore normal structures Ports Inlet jetties at river mouths Detached breakwaters (a)

(b)

(c)

Figure 1.8 – Wave profiles: sinusoidal (a), cnoidal (b) and solitary (c). The accumulation and erosion pattern adjacent to coastal structures depend on the type of coastline, (the wave orientation of the shoreline) and on the extent and the shape of coastal structures. The typical impact of the coastal processes and the related erosion problems for different types of structures will be briefly discussed in the following.

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INTRODUCTION TO COASTAL DYNAMICS AND SHORELINE PROTECTION

Groynes Groynes are normally built perpendicular to the shoreline with the purpose of protecting a section of the shoreline by blocking (part) of the littoral transport, whereby sand is accumulated on the upstream side of the groyne. However, the trapping of the sand causes a deficit in the littoral drift budget, and the coastal protection is always associated with a corresponding erosion on the lee side of the structure. In other words, a groyne just shifts the erosion problem to the downstream area. This is the reason why groynes are often built in long series along the shoreline, a so-called groyne field. Ports The primary purpose of a port is to provide safe mooring and navigation for vessels but when built on the shoreline it interferes with littoral drift budget and the results are sedimentation and shoreline impact. Like a groyne, the port acts as a blockage of the littoral transport, whereby it causes trapping of the sand on the upstream side in the form of an accumulating sand filet, and the possible bypass causes sedimentation in the entrance. The sedimentation requires maintenance dredging and deposition of the dredged sand. The result is a deficit in the littoral drift budget which causes lee side erosion on the adjacent shoreline. A port must consequently minimize sedimentation and coastal impact. These requirements have not always been given adequate attention. The result is that many ports trap large amounts of sand and suffer from severe sedimentation. Inlet jetties at river mouths River mouths are often by nature shallow and variable in location, which makes them unsuitable for navigation. In order to improve navigation conditions (and to some extent flushing conditions), many river inlets have regulated mouths. The regulation may consist of jetties, possibly combined with maintenance dredging programs. For the above reasons, regulated inlets are normally obstructions to the littoral transport which means upstream sand accumulation along the upstream jetty, loss of sand due to sedimentation in the deepened channel and the associated maintenance dredging. Detached breakwaters Detached breakwaters are used as shore and coast protection measures. In general terms, a detached breakwater is a coast-parallel structure located inside or close to the surf-zone. Breakwater schemes have many variables, depending on the distance from shoreline and location relative to the surf-zone, length and orientation crest height, (emerged or submerged). The breakwater shelters the coast partly from the waves, however as the waves diffract into the sheltered area, a complete shelter cannot be obtained. Submerged breakwaters have a lower visual impact, but provide less shelter. The longshore current is partially blocked by the circulation currents which cause some of the longshore current to be diverted outside the breakwater. So the littoral transport in lee of the

INTEGRATED APPROACH TO COASTAL DYNAMICS

21

breakwater is decreased due the attenuated wave and longshore currents in the area sheltered by the breakwater. This causes trapping of sand behind the breakwater dependent on the length and shoreline distance, until a tombolo is eventually formed. For shorter or submerged breakwaters, only a salient in the shoreline will develop. In case of trapping of sand, a tombolo will be developed, which causes lee side erosion downstream of the breakwater very similar to what is developed for groynes.

7 Elements of coastal management The coastal system is comprised of a complex, dynamic web of interrelationships among human activities, societal demands, natural resources, and external natural and human inputs. The system is driven by human activities in terms of societal demands for use of the natural resources of the coastal area to produce desired products and services, e.g., seafood, marine transportation, recreation etc. Societal demands for outputs from a coastal area usually exceed the capacity of the area to meet all the demands simultaneously. Coastal resources, e.g., fish and recreations, are often “common property resources” with “open” or “free” access to users. Free access often leads to excessive use of the resource, e.g. degradation or exhaustion of the resource, coastal pollution and habitat degradation. The Integrated Coastal Management (ICM) is the process used to decide what mix of outputs will be produced. Integrated coastal management can be defined as “a continuous and dynamic process by which decisions are taken for the sustainable use, development, and protection of coastal and marine areas and resources. ICM acknowledges the interrelationships that exist among coastal and ocean uses and the environments potentially affected, and is designed to overcome the fragmentation inherent in the sectorial management approach. ICM is multi-purpose oriented, it analyzes and addresses implications of development, conflicting uses, and interrelationships between physical processes and human activities, and it promotes linkages and harmonization among sectorial coastal and ocean activities” (Cicin-Sain, B. & R.W. Knecht, 1998; Pernetta, J. & D. Elder, 1993). Pressures The coastal areas, which accommodate one of the more fragile and precious habitats of the world, are under pressure from economic causes, because recent large migrations have resulted in coastal stress and overloads (Kamphuis, J.W., 2000). The following pressures on the coastal zone are indicated: •

Population density: According to the 1994 distribution of population in relation to the distance from the nearest coastline, 20.6 percent of the world’s population lives within 30 km of the coast, and 37 percent within 100 km (Gommes R., J. Du Guerny, F. Nachtergaele & R. Brinkman, 1997). The coastal regions of

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INTRODUCTION TO COASTAL DYNAMICS AND SHORELINE PROTECTION

•

•

•

European countries are subject to constant pressures due to population density: half of European population lives in a narrow strip about 50 kilometres far away from the sea. The resources of the coastal zones produce a great part of the economic wealth. Population density has a substantial influence: on the equilibrium of natural ecosystems, as a result of human activities; on the quality and the amount of the natural resources (forests, grounds, waters, fishing zones, shores, etc.) as result of activity and population concentration. In turn, these factors increase the demand for use and exploitation of resources and require waste disposal; on the natural landscape, modified as a consequence of the human activities, the dimension and the scale of the relative structures and associated impact; Migration: as a result of migration to coastal areas, the coastal population is growing at a faster rate than the world population. Probably within the next 20 to 30 years, the coastal population will almost double; Tourism generates large fluxes of people who take vacations in far away coastal areas. The Mediterranean is one of the main tourist destinations of the world, giving accommodation to 30% of the international tourists: a third part of introits comes from the international tourism. Tourism usually has a seasonal frequency and increases in years. Probably the pressures on the littoral zones will continue to grow in future, with a doubling of tourism in the Mediterranean in the next 20 years. Tourism interacts with coastal environment in terms of land use and water resources consumption; Erosion caused by natural and human factors increase resources exhaustion, pollution and habitat degradation: these factors emphasize the fragility of coastal areas and push priority on protecting and upgrading the coastal system. Erosion of the habitat takes place mainly for the competitive use of the coastal zone. The analysis of erosion data (Table 1.1) shows that in the European coastal zones 1500 km are artificial coasts (Balearic islands, Gulf of the Lion, Sardinia, Adriatic, Ionian and Aegean), occupied by harbours (1250 km) (Corine, 1998). According to the data collected by Corine, approximately 25% of the Italian Adriatic coast and 7.4% of the Aegean coast shows an evolutionary tendency to erosion, whereas approximately 50% of the Euro-Mediterranean coastal zone is considered stable.

Conflicts The analysis of conflicts takes into account the interactions among natural resources and different economic, political and environmental factors. Conflicts among user groups and current social, economic and environmental conditions

INTEGRATED APPROACH TO COASTAL DYNAMICS

23

represent a problem that often requires an immediate solution. The challenge in this context is to combine the following interests: • •

Public interests: shore protection, resource preservation, development of infrastructure and public utilities, etc.; Private interests: development of projects and coastal protection, industrial development, navigation, etc.

There are often inherent conflicts of interests in such projects, both with respect to the objectives and with respect to costs sharing. Resolving these planning matters is often as difficult as it is to find a suitable technical solution. However, the dissemination of the planning concepts, of the principles of sustainable development and environmental protection as well as of the physical mechanisms in the coastal zone, are all important for the success of the entire planning and design process. Table 1.1 - Evolutionary trends of some coasts of the European part of the Mediterranean Sea for both rocky coasts and beaches as % of coasts (Corine, 1998). MARITIME REGION Balearic Islands Gulf of Lion Sardinia

Adriatic Sea Ionian Sea Aegean Sea

Total coastline (km)

Stability

Erosion

Sedimentation

No information

Not applicable

2861

68.8%

19.6%

2.4%

0.5%

8.7%

1366

46.0%

14.4%

7.8%

4.1%

27.8%

5521

57.0%

18.4%

3.6%

16.0%

5.0%

970

51.7%

25.6%

7.6%

3.9%

11.1%

3890

52.3%

22.5%

1.2%

19.7%

4.3%

3408

49.5%

7.4%

2.9%

37.5%

2.6%

Coastal management In the European coastal regions local administrations have the best knowledge of the real problems. They interact with citizens, private companies and non governing organizations. Regional agencies give the guidelines and principles to coordinate the local initiatives, whereas political and national programs facilitate the participations on regional and local level. In many cases the cooperation between States is also necessary: as an example, it would be better that countries

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INTRODUCTION TO COASTAL DYNAMICS AND SHORELINE PROTECTION

joining the same sea try to coordinate their own initiatives, rather than propose different and contrasting political actions. The strategy of coastal zone integrated management (figure 1.9) encourages this transnational approach in the countries joining the “regional seas” like the Mediterranean or the Baltic Sea. Another aim is to prevent political strategies that damage the coast. In the case of the agricultural pollution, the integrated coastal management will give to the PAC (Common Agricultural Policy) the measures to decrease the impact of fertilizers on coastal waters.

Figure 1.9 -Coastal management strategy (after Townend, I.H., 1994). Coastal management Objectives and Activities Chapter 17 of Agenda 21 so describes the objectives and processes of Integrated Coastal Management programs: Coastal States commit themselves to integrated management and sustainable development of coastal areas and the marine environment under their national jurisdiction. To this end, it is necessary to, inter alia: • • • •

Provide for an integrated policy and decision-making process, including all involved sectors, to promote compatibility and a balance of uses; Identify existing and projected uses of coastal areas and their interactions; Concentrate on well-defined issues concerning coastal management; Apply preventive and precautionary approaches in project planning and implementation, including prior assessment and systematic observation of the impacts of major projects;

INTEGRATED APPROACH TO COASTAL DYNAMICS

•

•

25

Promote the application of methods, such as environmental accounting, that reflect changes in value resulting from uses of coastal and marine areas, including pollution, marine erosion, loss of resources and habitat destruction; Provide access, as far as possible, for concerned individuals, groups and organizations to relevant information and opportunities for consultation and participation in planning and decision-making at appropriate levels.

The activities associated with the coastal management concern: •

•

•

Coordinating mechanisms (such as a high-level policy planning body) for integrated management and sustainable development of coastal and marine areas and their resources, at both the local and national levels. Such mechanisms should include consultation, as appropriate, with the academic and private sectors, nongovernmental organizations, resource user groups, and local communities; Data and Information: Coastal States, where necessary, should improve their capacity to collect, analyze, assess and use information for sustainable use of resources, including environmental impacts of activities affecting the coastal and marine areas. Information for management purposes should receive priority support in view of the intensity and magnitude of the changes occurring in the coastal and marine areas; International and regional cooperation and coordination: The role of international cooperation and coordination on a bilateral basis and, where applicable, within a sub-regional, interregional, regional or global framework, is to support and supplement national efforts of coastal States to promote integrated management and sustainable development of coastal and marine areas.

Planning and management of the coastal zones must necessarily be founded on the so-called “principle of precaution”: responsible subjects must try to preview the possible damages for the coastal zone and identify the more appropriate interventions. According to this principle, if they are not sure that some participation can take negative repercussions for a coastal zone, they must address their action with maximum caution. This approach assumes particular importance in vulnerable zones that could have negative consequences as a result of urbanization or tourism. In the past, many attempts to preserve the coastal regions failed, in spite of the good intentions, because they were addressed to sectorial aspects. As an example, the tourism in the coastal zones cannot be treated without paying attention to other concurrent factors, such as water supply, regional planning, impact of tourism on the existing natural habitats. The Integrated Coastal

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INTRODUCTION TO COASTAL DYNAMICS AND SHORELINE PROTECTION

Management has the objective to facilitate contacts between local, regional and national administrations in order to allow the responsible subjects to obtain a precise picture of the real necessities involved. The interested local subjects and organisms must be involved in planning and management of the coastal zones: without the contribution of companies and citizens who live and work in the coastal zones, the ICM will not be able to work successfully.

Chapter 2 Beach morphology and sediment analysis 1 Introduction A beach is an accumulation of loose material around the limit of wave action. According to King (1972) the beach may be taken to extend from the extreme upper limit of wave action to the zone where the waves, approaching from deep water, first cause appreciable movement of beach material. According to the nature of the beach material, we can define gravely, sandy and silty beaches. Beaches have variable planar shapes (eg. tongues, tombolos, cuspids) and their classification depends on sediment characteristics (mineral and granular properties) obtained by topographic and bathymetric surveys, as described in the following sections. The beach profiles show a typical morphology extending between the low tide level and the first major change in topography (e.g. a dune), and covering also the submerged profile. The coastline is subject to cyclical changes on different temporal scales (a few seconds to geological ages). This variability provides great interest for researchers in the field of geology, sedimentology and coastal engineering.

2 Beach classification Geologists describe beach classification using different criteria: according to grain size criteria we distinguish gravely, sandy and silty beaches. Some criteria are based on lithology: marine processes modify chemical and physical properties of a rock (erosion and dissolution); strong temperature changes, crystallisation and weathering processes cause physical alteration of rocks (mechanical disintegration); chemical processes like oxidation, reduction and biochemical reactions cause decomposition of rocks. Coasts consisting of firm material are defined as consolidated coasts, whereas unconsolidated coasts are subject to vigorous erosive processes. Volcanic coasts are residues of calderas originated by volcanic explosion and are characterised by circular physiognomy (convex or concave) and deposits of volcanic material.

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INTRODUCTION TO COASTAL DYNAMICS AND SHORELINE PROTECTION

Early classifications, based on a genetic approach, distinguish coast affected by rising sea level (submergence), coasts affected by falling sea level (emergence), or compound coasts, affected by both phenomena (Davis, R.A., Jr., & Hayes, M. O. 1984; Johnson, D. 1919). Later classifications are based on onshore and shoreline morphology as those of Cotton (1952) and Shepard (1937). Inman and Nordstrom (1971) classification includes conditions of the offshore bottom. The most widely used coastal classification was introduced by Shepard in 1937, who distinguishes primary coasts - formed mostly by non-marine agents and secondary coasts – shaped primarily by marine processes or by marine organisms. As described in Shepard (1973), primary and secondary coasts include many groups and subgroups, so that Shepard’s classification is more detailed than others and includes all the existing world’s coasts. According to Shepard’s classification, primary coasts include: 1.

Land erosion coasts - shaped by subaerial erosion and partly drowned by postglacial rise of sea level (with or without crustal sinking) or inundated by melting of an ice mass from a coastal valley;

2.

Subaerial deposition coasts – for example, coasts originated by rivers, glacial and wind deposition;

3.

Volcanic coasts – originated by lava flows, fragmental volcanic products, volcanic collapse or explosion;

4.

Coasts shaped by diastrophic movements;

5.

Ice coasts – formed by various types of glaciers.

Primary coasts morphology reflects either tectonic or terrestrial or land-based processes. Generally they are coarse and irregular, because they have not been straightened out by currents and waves. Secondary coasts include: 1. 2. 3.

Wave erosion coasts – coasts eroded by wave action Marine deposition coasts - prograded by waves and currents Coasts built by organisms - formed by the growth of animals and/or plants.

Major features of secondary coasts are sea cliffs and wave-cut platforms; secondary coasts are affected by shoreline straightening over time. frequently they are characterised by the formation of beaches, sand spits, bars, and barrier islands. Longshore drift (the movement of sand along the shoreline in the direction of the waves) is one of the most important processes affecting secondary coasts. Energy-based classification Tides and waves are dominant forces driving littoral processes on open coasts because they sort the sediment and move it alongshore. Davies (1964) applied an energy-based classification to coastal morphology by dividing the world’s shores

BEACH MORPHOLOGY AND SEDIMENT ANALYSIS

29

according to tide range. Hayes (1979) expanded this classification, defining five tidal coastlines categories: 1. 2. 3. 4. 5.

microtidal <1m low-mesotidal, 1-2 m high-mesotidal, 2-3.5 m low macrotidal, 3.5-5 m Macrotidal >5 m

The Hayes (1979) classification (see Figure 2.1) was primarily based on shores with low to moderate wave power and was intended to be applied to trailing edge, depositional coasts. In the attempt to incorporate wave energy as a significant factor modifying shoreline morphology, five shoreline categories were identified based on the relative influence of tide range versus mean wave height (Hayes, M. O. 1979; Nummedal, D. & Fischer, I. A. 1978). 1. 2. 3. 4. 5.

Tide-dominated (high) Tide-dominated (low) Mixed-energy (tide-dominated) Mixed energy (wave-dominated) Wave-dominated.

The relative effects of these processes are significant, not the absolute values. Also, at the lower end of the energy scales, there is a delicate balance between the forces;

Mean tidal range

Tide dominated

Wave dominated Mean wave height Figure 2.1 - Energy-based classification of shorelines.

tide-dominated, wave-dominated, or mixed-energy morphologies may develop with very little difference in wave or tide parameters (U.S. Army Corps of Engineers - Coastal Engineering Research Center, 1995).

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INTRODUCTION TO COASTAL DYNAMICS AND SHORELINE PROTECTION

3 Beach morphology and sediment transport In Italy the total development of shoreline amounts approximately to 7500 km, of which 55% are rocky coasts and 45% are beaches (most of them are sandy beaches). In northern Adriatic there are mild sloped sandy beaches, while steep shores are typical of Liguria, Calabria, Sicily and Sardinia, where little beaches (pocket beaches) are confined between adjacent capes. In Italy there are few silty beaches: actually they have an ecological importance because they are related to the so-called humid areas (areas covered by vegetation used by migratory birds). 32% of the Italian beach is subject to erosion, only 5% is in advance and 63% is stable; protection systems defend 15% (500 km) of beaches A general compendium (approximately 2000 km) of the morphology of Italian beaches is described in 60 maps (scale 1:100000) of CNR Italian Beach Atlas (Fierro G., AA. VV., 1999). Coastal zone definition is not simple because temporal and spatial variations modify boundaries and features. Besides, different definitions are used by researchers to describe elements and characteristics of a coast. Because of this, the following basic classification (U.S. Army Corps of Engineers - Coastal Engineering Research Center, 1995) presents a general definition of coastal zone, (based on geological criteria) which defines four subzones: • • • •

Coast Shore Shoreface Continental Shelf

A coast is a transitional zone between land and water extending from the maximum reach of storm waves to the first major change in topography. The coastline is defined by the presence of cliff formed by wave erosion, dune or permanent vegetation. In some areas tide action obscures the landward limit of the coast (large deltas like Mississippi). The shore zone (Figure 2.2) is a sloping area where unconsolidated sediments are subject to wave action. The term ‘backshore’ is used for the zone above the limit of the swash of normal high spring tide, and is, therefore, only exceptionally under the direct influence of the waves. On a rocky coast it includes the cliffs, while a low coast may consist of sand dunes or mature salt marsh. The ‘foreshore’ zone is a sloping portion of the beach including all that part of the beach which is regularly covered and uncovered by the tide. On a tideless beach this zone will be narrow, only covering the distance between the limit of the swash and backwash of the larger waves. The foreshore zone includes the surf zone (a zone traversed by breaking waves extending between the wave breakpoint and the maximum run-up of the swash) and a swash zone, where wave swash (uprush of water) and backwash (back rush of water) foreshore.

BEACH MORPHOLOGY AND SEDIMENT ANALYSIS

31

The ‘berm’ is a terrace formed in the backshore zone above the limit of the swash at high tide to form a flat terrace, or a ridge with a reverse slope. Below low water the positive features on the sandy floor in the offshore zone are called ‘submarine bars’. The hollows found on the landward side of the submarine bars of sandy beaches are termed ‘troughs’. Another feature of coastal zones is the shoreface, a narrow sloping zone between the continental shelf and the low water limit, where unconsolidated sediments are subject to vigorous transport. Continental shelf is a slightly sloping zone of submerged continental margin extending from the offshore limit of shoreface to the slope of shelf break. Researchers divide continental shelf in a inner part, a middle part and a outer part.

Figure 2.2 Beach profile Sediment budget in the littoral zone is related to movements of materials between offshore and onshore zones. Sediments are mainly supplied by rivers, channels, beaches and dunes erosion. Secondary sources are artificial replenishment, industrial refuse materials, civil works, etc. In shallow water bottom materials are influenced by wave action, whereas in deep water currents action on bottom material is the main cause of sediment transport. In the littoral zone, sediment patterns and composition depend on the beach slope, characteristics of the area, external and internal sources of materials. Analysis of sediment characteristics, illustrated in the following subsection, plays a significant role in the study of littoral transport and sources of bottom materials. Offshore zone In deep water, waves have only a little influence on sediment transport. As a wave enters in shallow water, it begins to drag the bottom; wave length and speed increase, whereas the period remains constant. The wave energy increases with the wave height. In this phenomena, wave-bottom interactions produce some bed forms depending on the bottom contours and wave strength:

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INTRODUCTION TO COASTAL DYNAMICS AND SHORELINE PROTECTION

• • • •

Moderate waves produce vortices of suspended particles, and generate some bed forms such as ripples. Strong wave action produces flat bottom and sediments are transported by sheet flows. When swell waves interact with bottom materials, ripples dimension depend on fluid particles excursion. In case of high wave speed, even if sheet flows occur, flat bottoms are not observed. Sea waves produce small ripples. Near the breaking zone, fluid motion becomes turbulent and many particles are suspended.

Surf zone and breaking zone: In this area breaking processes cause energy dissipation and produce vortices and turbulence phenomena. Breaking mechanisms are usually divided into two groups: spilling and plunging. •

•

Spilling occurs when steep waves advance up a gentle beach: when the wave breaks, the energy is gradually released over a considerable distance, and wave deformations are smaller than in plunging breakers. Suspended sediments are transported offshore by undertow currents or superficial currents associated with breaking. Breaking mechanisms continue their action with less intensity in the part of the surf zone nearest to the offshore zone, where the waves which generate the breaking process are recomposed. Plunging usually occurs on slightly steeper beaches: the crest becomes faster than the wave base, curls and violently descends into the wave trough. A considerable amount of energy is released into a downwardly directed mass of water and the turbulence reaches the area beyond the breaking zone. Bottom materials are suspended and transported offshore by undertow currents. A large bore is formed and becomes deeper and deeper until the energy is completely exausted. Ahead of the bore, a sediment accumulation is formed, localized between the breaking point and the deepest zone reached by vorticity.

Swash zone The swash is a fluctuation of coastline associated with sediment transport which occurs when the energy of the waves advancing up the beach is not completely exhausted. Swash dynamics depend on the frequency oscillations. At low frequencies, if the beach is permeable and the sand is not saturated, the water percolates through the substrate and the backrush decreases. An amount of sediment is accumulated on the beach in the swash zone and the beach slope increases. The result is a convex beach profile; At high frequencies (storm waves), if the substrate is saturated, the water cannot percolate through the sand and the backrush current does not decrease. In this case the sediment transport associated with backrush is greater than uprush, which results in a concave beach profile.

BEACH MORPHOLOGY AND SEDIMENT ANALYSIS

33

4 Seasonal profiles, bars and berms The beach slope is related to grain size: larger grain sizes generate steeper beaches as shown, for example in CERC (1984). During the winter storm waves move sand offshore, while the summer waves move sand onto the beach. Kamphuis (2000) shows that beach slopes are a function of the ratio (H/D) which represents the ratio of disturbing wave forces to restoring particle forces. On this basis, several qualitative accretion/erosion criteria were conceived (see Chapter 9). The interface where the sea surface meets the shoreline moves up and down the beach in a small area called swash zone. Even in case of low flow velocity, the small flow depth makes the shear stress extremely large. As a result, even coarse material can be mobilized by slow-moving swash. During the action of larger waves the beach material is redistributed offshore to become a longshore bar or sandbar, generally visible at low tide. The bars are an accumulation of material near the point of breaking waves and are considered offshore features of a beach (Kamphuis, J.W., 2000). Bars are formed near the breaking point (sometimes seaward of a trough) because the breaking waves set up a shoreward current with a compensating counter-current along the bottom. Sand carried by the offshore moving bottom current is deposited where the current reaches the wave break (Kamphuis, J.W., 2000). Berms form as a result of preferential shoreward transport within the swash. They are generally preserved (left behind) after a highwater event (stormy high tide, etc.). The berm is periodically overtopped during storms or extreme high tides. On sandy and shingle beaches berms build seaward through the multiple accretion of bars to the beach face. Vertical accretion to the berm is accomplished by swash, which is influenced by wave height. A beach may have more than one berm or none at all (e.g. an eroded beach backed by a seawall). Some beaches, particularly mesotidal gravel beaches, may exhibit multiple berms (e.g. LHT berm, HHT berm, summer berm, winter berm, storm berm, etc). High-water berms are formed during storm surges or spring tides (HHT berm).

5 Equilibrium beach profile The equilibrium beach profile results from steady wave forcing during the seasonal cycle. As described above, summer wave conditions move sand onto the beach, while winter storm waves move sand offshore. Beach profiles fluctuate with the seasonal cycle of wave energy: storms cause larger and more energetic waves in winter than in summer. As a result, the winter profile is characterized by the presence of a bar or by a berm on the foreshore (Figure 2.3). Dean (1983), conceived an equilibrium profile on the basis of constant energy dissipation.

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INTRODUCTION TO COASTAL DYNAMICS AND SHORELINE PROTECTION

Summer shoreline

Summer profile

Winter shoreline

berm

m.s.l.

cliff trough

bar Winter profile

bar

Figure 2.3 - Winter and summer beach profiles. The equilibrium profile is expressed in equation 2.1.

1 dF = De h dx

(2.1)

where: h is the water depth at distance x from the coast line F is the wave energy flux in shallow water; De is the dissipation coefficient of energy. Wave energy flux in shallow water is expressed as follows (see chapter 3): ρgH b2 ( gd )1 / 2 8 Hb is the breaking wave height, given by: F=

H b = d bγ

(2.2)

(2.3)

where γ is the breaker index (0.78 ↔1) and hb is the breaking depth. We define the equilibrium parameter A as: d = Ax 2 3

(2.4)

substituting this definition in (2.2) we obtain: F = (1 / 8) ρg 3 / 2γd 5 / 2 = (1 / 8) ρg 3 / 2γ 2 A5 / 2 x 5 / 3

(2.5)

With a little algebra we have: De =

1 dF 1 ρg 3 / 2γ 2 5 5 / 2 2 / 3 A x = d dx d 8 3

(2.6)

BEACH MORPHOLOGY AND SEDIMENT ANALYSIS

35

Finally, A can be expressed as a function of De according to the following expression:  24 1   A =  De 3/ 2 2  ρ g γ  5 

2/3

(2.7)

Dissipation of energy due to breaking waves destabilize sediment particles. When destabilizing forces are balanced by restoring forces, dynamical equilibrium occurs. The profile parameter A depends on grain size and fall velocity of particles. According to Moore (1982) and Dean (1983), it is possible to define relationships over ranges of grain sizes. For example: A = ( 1.04 + 0.086 ln( D )) 2

for 0.1×10 −3 ≤ D ≤ 1.0 ×10 −3 m

for 0.1×10−3 m ≤ D ≤ 0.2 ×10−3 m

A = 20 ⋅ D0.63

(2.8) (2.9)

Where D is the grain diameter. Dean (1983) also proposed a simple relationship between A and fall velocity: A = 0.50 ⋅w f 0.44

(2.10)

where wf is the fall velocity in m/s.

6 Sediment analysis 6.1 Bathymetric and geophysical surveys Temporal and spatial scales are important to define the zone of shoreline object of surveys as the time interval between two successive samplings. With regard to the spatial scale, it is necessary to survey the whole physiographic unit. From the engineer’s point of view, a physiographic unit covers the zone where any coastal change in plan or profile influences the adjacent coastline. In other words, it is a confined area where sediments move and the exchanges with the adjacent beaches are limited or negligible. The definition of a physiographic unit is important to define the limits of the area (generally variable) where the studies of the effects of coastal dynamics are necessary. With regard to the temporal scale, the study of the short term coastal dynamics requires a short time interval (analysis of the daily phenomena covering the observed period). In order to study long term phenomena, a long time interval is needed (seasonal phenomena require at least a time step of ten years). A preliminary characterization of physiographic units is carried out by analysis of coastal morphology. The first step is the comparison between historical charts and the definition of the area limits; the following step is the

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INTRODUCTION TO COASTAL DYNAMICS AND SHORELINE PROTECTION

analysis of photographs and maps obtained by aero-photographic and topographic surveys. Then, the inspection of the coastline variations allows to examine the following characteristics: • • • •

the coastline cyclical fluctuations; evolution of river mouth as flèches (one-direction oriented flows) or cuspids (divergence of the flow); concave coastline (to which center sediment flow gradually converges); deposits and erosions of buildings near coastal structures.

Topographic surveys The topographic surveys are integrated with bathymetric surveys and coastal profiles. The analysis of the maps concurs to estimate the type of coastal morphology (emerged and submerged coast) and allows to characterize the elements of short-long term equilibrium. The topographic surveys are then related with the sediment characteristics, which pattern gives an alternate characterization of the physiographic unit. The topographic characteristics of the emerged and submerged beach are needed to verify the morphological evolution and they should be measured in the same period of the year, in order to avoid influence of the seasonal variability of the equilibrium profile. The ‘dry’ topographic survey is performed along transects (50 meters width or more) or from the coastline until buildings or any human infrastructure is reached; Bathymetric surveys are normally carried out using transects spaced by 50-100 meters or more until the 10-meters bathymetric line is reached. When steep zones are found, a more fine spacing and orthogonal sections are then needed. Bathymetric surveys Bathymetric surveys can be carried out until 1.5 meters depth, using an accurate echo sounder with an emission cone not wider than 10°. The depth measured has to be corrected in order to account for the tide excursions. The submerge profile for depths lower than 1.5 meters is generally measured with a graduated stick. Finally, the survey is plotted on scales 1:1000 or higher, interpolating the bathymetric lines every 0.5 or 1 meter. In transparent and shallow waters (10 meters depth) the surveys can also be integrated by aero photographic survey techniques. Sampling sediments The distribution of sediments and their selection can be studied through an accurate sampling strategy. The sampling site must be selected according to beach morphology and site variability: for example, trough, bar and foreshore regions are more variable than the nearshore and dune zones. Typically, grain size distributions are better sorted in the summer than in the winter. Finally, shoreline seasonal changes and engineering structures should be considered in selecting sampling points. Number of samples collected depends on research objectives:

BEACH MORPHOLOGY AND SEDIMENT ANALYSIS

• •

•

37

If the intent is to grossly characterize the sediments of a beach, usually only a few samples are needed; If the objective is to characterize the site as a whole, a set of samples needs to be collected. According to Hobson (1977), combining samples from different transects across the beach can reduce the high variability in spatial grain size distribution; If the intent of the study is to analyze the differences among the portions of the same beach, many more samples are needed.

In the third case, the first step of the survey is the development of a sampling project to define all the sampling locations. In the cross shore, according to Stauble and Hoel (1986), sampling points along the profile have to be located at all major changes in morphology: dune base, mid berm, mean high water, mid-tide, mid berm, mean high water and then at 3-m intervals until the depth of closure. In the longshore direction, sediment sampling should coincide with survey profile lines so that the samples can be spatially located and related to morphology and hydrodynamic zones (U.S. Army Corps of Engineers - Coastal Engineering Research Center, 1995). 6.2 Physical and chemical analysis The most important sediment characteristic is the particle grain size (the measure of grain dimensions and their statistical distribution). Other interesting parameters are colour, texture, surface morphology (aspect and structure), shape and degree of rounding. The definition of sediment morphology and texture gives some indications of their origin and evolution; the analysis of petrography is useful to define the origin of sediment matrix. Grain size can be defined by direct measurement of particle diameters or, indirectly, by determination of their ‘hydraulic equivalents’ based on settling velocities of quartz spheres. The analysis is carried out using samples of 50 grams collected on the beach. The containers are opportunely labelled using a resistant plastic support where are signed place, date and hour of sampling. The oldest method of grain size determination uses a set of nested sieves with different mesh size. Each sieve is formed by woven stainless or brass steel wires. An amount of sediments pass through a set of nested sieves in which the size is gradually smaller down the stack. Grains are trapped on a sieve if their size is smaller than mesh openings. The sieves are agitated by hand (or mechanically) to make the selection more efficient (without forcing grains through the mesh). The weight of each size class is expressed as a percent of the total sample weight. The sieve analysis is a widely accepted method of grain-size determination because it is relatively easy and quick, but it can be a deadly process if the number of samples is too high. The amount of grains can affect the accuracy of the analysis: when the sample is too little, small errors can be more significant; if too much sand is used, one of the sieves in the stuck may become overloaded. Besides, grains can stick together or tend to get stuck in the mesh.

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INTRODUCTION TO COASTAL DYNAMICS AND SHORELINE PROTECTION

Figure 2.4 - Sampling points along the beach profile (U.S. Army Corps of Engineers - Coastal Engineering Research Center, 1995). Analysis of muddy sediments is commonly carried out by pipette analysis. The sample of sediments is put in a one litre graduated cylinder containing distilled water. An amount of dispersing agent is added to avoid particle flocculation. After agitation, 20 ml aliquots of mixture are taken using a pipette. The water in each aliquot is evaporated and the sediment amount is determined measuring weights of containers with and without sediments. In some cases, the presence of contaminants can have a significant influence on the accuracy of measurements. Pipette analysis is often a tedious process and requires great accuracy. Rapid Sediment Analyzer (RSA) is a method faster than the ones described above. RSA is a long cylinder containing distilled water, where an amount of sediment is introduced and allowed to settle. Particles are collected on a pan connected with a balance recording sediments weight. Some RSA have a computerized system to record weight data over time. The method is based on the principle that the falling velocity of grains in water varies with the diameter, shape and specific weight of particles. The system measures indirectly the grain size, on the base of settling and hydraulic behavior of particles. Measured velocities are compared with known settling rates and the distribution of particles is expressed as “equivalent diameters”. Some disadvantages are that RSA are calibrated with quartz spheres, whereas particles analyzed can have various shape and nature; finally, walls of the container can interfere with natural falling down of sediments.

BEACH MORPHOLOGY AND SEDIMENT ANALYSIS

39

6.3 Sediment size classification Sediments occur in a wide range of sizes, between micrometers and centimetres. The range of grain sizes of practical interest to coastal engineers is enormous, covering about seven orders of magnitude, from clay particles to large breakwater armour stone blocks. Diameter is the most important property; morphological parameters are defined by several geometric indices of rounding. Sediment size classification is usually performed with the assumption that particles are roughly circular and the grain size can be expressed as a projected cross section. Most common classifications are based on Wentworth scale (1922) and Krumbein scale (1936). Wentworth divides grains into four size classes based on particle diameter: mud (less than 0.06 mm) sand (between 0.05 and 2 mm) gravel (between 2 and 64 mm), cobble/boulder (larger than 64 mm). Statistical analysis of sediments demonstrates that size distribution of particles has a logarithmic distribution, so Krumbein (1936) proposes an alternate system called ‘phi scale’, in which grain diameter is expressed as:

ϕ = − log 2 ( D)

(2.11)

where d is the particle diameter in mm. Phi diameters are indicated by writing ϕ after the numerical value. For example, a 2.0 ϕ sand grain has a diameter of 0.25 mm. To convert from phi units to millimeters, the inverse equation is used: D = 2 −ϕ

(2.12)

The benefits of the phi unit include: 1) it has whole numbers at the limits of sediment classes in the Wentworth scale 2) it allows comparison of different size distributions because it is dimensionless. Disadvantages of this phi unit are: (a) As grain size get coarser, phi size gets smaller, which is both counterintuitive and ambiguous; (b) it is difficult to physically interpret size in phi units without considerable experience; (c) phi doesn’t represent a unit of length because it is a dimensionless unit (U.S. Army Corps of Engineers - Coastal Engineering Research Center, 1995). After the sieve analysis, the weights of the sediment trapped on each sieve are tabulated and divided by the total weight of the sample in order to calculate weight percent. The cumulative frequency for each size is calculated summing up the weights for sediment trapped on that sieve and all those higher in the stack. Sediment character is described by a histogram that graphically shows the relative abundance of each size class. Poorly sorted samples can have multiple modes, whereas better-sorted samples will have a single mode (a single peak). Table 2.1 compares Wentworth and Krumbein scales. Method of moments Mathematical determination of grain size utilizes a statistical measure called the “method of moments”.

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INTRODUCTION TO COASTAL DYNAMICS AND SHORELINE PROTECTION

Table 2.1 - Wentworth and Krumbein scales

Gravel

Sand

Mud

Size categories

Millimetres

phi

Cobble/Boulder

>64

<-6.0

Pebble

64↔4

-6↔–2

Granule

4↔2

-2↔–1

Coarse

2↔0.5

-1↔+1

Medium

0.5↔0.25

+1↔+2

Fine

0.25↔0.06

+2↔+4

Silt

0.06↔0.002

+4↔+9

Clay

<0.002

>+9

The statistical moment is the moment of a given size class with respect to the arbitrary point (i.e., to the origin of the curve). The statistical moment of the distribution is the moment per unit frequency. It is determined by finding the moments of each size class (frequency percentage in each size class times the distance of each from the origin), adding them up, and dividing the sum by 100. The first moment is the mean calculated using the frequency (f) in percent for each size class and the midpoint (m) of each class in phi values. In the first moment the distance term (m) appears in the first power. Successively higher moments are defined by raising the distance term to progressively higher powers. In the higher moments the distance term is represented with respect to the mean, so the higher moments are moments about the mean, whereas the first moment is the moment with respect to an arbitrary point. The second moment is the square of the standard deviation. The third moment is the skewness, and the fourth moment is the kurtosis (see Table 2.2). Graphic statistics method The simplest method to calculate grain statistic distribution is based on few points from the plot representing grain size (on an arithmetic axis) versus cumulative frequency (on a probability ordinate). The resulting curve shows three important points: − − −

the 50th percentile ϕ50 (the sample median) the 16th percentile ϕ16 the 84th percentile ϕ84

Other points are the 5th and 95th percentiles (indicated as ϕ5 and ϕ95), used to calculate standard deviation. The graphic mean (Mϕ) is computed by the average

BEACH MORPHOLOGY AND SEDIMENT ANALYSIS

41

between the median (ϕ50) and the values one standard deviation unit to either side: Mϕ =

(ϕ16 + ϕ50 + ϕ84 )

(2.13)

3

Table 2.2 Formulas for calculating statistics using the method of moments. f= frequency in percent for each size class ; m= midpoint of each class in phi values;n=total number in sample. When f is expressed in percent, n=100 Mean (first moment)

∑fm 1 σ = f (m − x ) 100 ∑ 1 Sk = f (m − x ) 100σ ∑ xϕ =

Standard deviation (second moment)

1 n

2

ϕ

Skewness (third moment)

ϕ

3

ϕ

ϕ

3

ϕ

Kurtosis (fourth moment)

Kϕ =

1 100σ ϕ4

∑ f (m − x )

4

ϕ

Standard deviation can be computed using Folk (1974) formulation:

σϕ =

(ϕ84 − ϕ16 ) + (ϕ 95 − ϕ 5 ) 4

6

(2.14)

If the sample is distributed such that values for the ϕ5 and ϕ95 cannot be derived from the graph, then a less accurate determination (graphic standard deviation) can be made using:

σϕ =

(ϕ 84 − ϕ16 ) 2

(2.15)

The degree by which the distribution departs from symmetry is measured by the phi coefficient of skewness defined in Folk 1974 as:

αϕ =

(ϕ16 + ϕ 84 − 2ϕ 50 ) + (ϕ 5 + ϕ 95 − 2ϕ 50 ) 2(ϕ 84 − ϕ16 ) 2(ϕ 95 − ϕ 5 )

(2.16)

A perfectly symmetric distribution indicates that the skewness is zero. If the skewness is positive there is an excess of fine material and the curve has a tail to the right. Conversely, a negative value of skewness indicates excess of coarse material (a tail to the left). The phi coefficient of kurtosis βϕ is a measure of the peakedness of the distribution; that is, the proportion of the sediment in the middle of the distribution relative to the amount in both tails. Following Folk (1974), it is defined as:

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INTRODUCTION TO COASTAL DYNAMICS AND SHORELINE PROTECTION

βϕ =

(ϕ 95 − ϕ 5 ) 2.44(ϕ 75 − ϕ 25 )

(2.17)

Frequency

Frequency

Mode Mean Mode Median

coarse

Particle size A

fine

Median Mean

coarse

Normal

Particle size B

fine

Positively skewed

Frequency

Mode Median Mean

Particle size

coarse C

fine

Negatively skewed

Figure 2.5 – Frequency curves illustrating the mode, median and the difference between normal frequency curves and asymmetrical (skewed) curves.

7 Case study Find the statistics of the sediment size distribution shown in Figure 2.6. Given the needed phi values are: ϕ05 = 2.224, ϕ16 = 2.498, ϕ25 = 2.727, ϕ50 = 3.506, ϕ75 = 4.211, ϕ84 = 4.506, ϕ95 = 4.718. In phi units, the median grain size is given as:

ϕ50 = MDϕ = 3.506 ϕ From equation (2.12) the median grain size in millimeters is found as: MD = 2-3.506 = 0.088mm The mean grain size is found in phi units as (equation 2.13): Mϕ = (ϕ16+ϕ50+ϕ84) / 3 =3.503ϕ From equation (2.12) this is converted to millimeters as: D = 2-3.503 = 0.088 mm

BEACH MORPHOLOGY AND SEDIMENT ANALYSIS

From equation (2.14) the standard deviation is found as:

σϕ = (4.506-2.498)/4 + (4.718-2.224)/6 = 0.918ϕ

Figure 2.6 - Sediment size distribution. From equation (2.16) the coefficient of skewness is found as:

αϕ =

2.498 + 4.506 − 2(3.506) 2.224 + 4.718 − 2(3.506) + = −0.016 2( 4.506 − 2.498) 2( 4.718 − 2.224)

From equation (2.17) the coefficient of kurtosis is found as:

βϕ =

4.718 − 2.224 = 0.689 2.44 ( 4.211 − 2.727 )

43

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INTRODUCTION TO COASTAL DYNAMICS AND SHORELINE PROTECTION

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Chapter 3 Linear wave analysis 1 Introduction to linear wave theory The problem of describing wave motion is generally approached through the hypotheses of a frictionless and incompressible fluid in irrotational motion. The equation of mass and momentum conservation written for this ideal fluid must be closed by appropriate boundary conditions. Two serious difficulties arise in the attempt to obtain an exact solution. The fact is that the free surface boundary conditions are nonlinear, and the second is that these conditions are prescribed at the free surface z=η which is initially unknown. The simplest and most fundamental approach is to seek a linear solution of the problem by taking the wave height H to be very much smaller than both the wave length L and the still water depth d: that is H<
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INTRODUCTION TO COASTAL DYNAMICS AND SHORELINE PROTECTION

H/L H = d/L d

(3.1)

which is termed the relative wave height. Like the wave steepness, large values of the relative wave height indicate that the small-amplitude assumption may not be valid. A fourth dimensionless parameter often used to assess the relevance of various wave theories is the Ursell number, given by: 2

2 ⎛L⎞ H L H UR = ⎜ ⎟ = 3 d ⎝d ⎠ d

(3.2)

The value of the Ursell number is often used to select a wave theory to describe a wave with given L and H (or T and H) in a given water depth d. High values of UR indicate large, finite amplitude, long waves in shallow water that may necessitate the use of nonlinear wave theory. In fact, the Stokes expansion method is valid only if d/L >1/8 or kd>0.78 or UR<79 which restrict the applicability of Stokes theories only to deep waters. In shallow waters, the cnoidal wave theory is often used, for which reference is made to Sarpkaya & Isaacson, 1981. Le Méhauté (1976) presented a convenient plot showing the approximate limits of validity of various wave theories, as shown in Figure 3.1.

Figure 3.1 - The applicability of various wave theories. after Le Méhauté (Le Méhauté B., 1976).

LINEAR WAVE ANALYSIS

1.1

47

Governing equations

In developing the linear wave theory the following assumptions are made: • • • • • • • •

Fluid homogeneous; Density ρ constant; Surface tension neglected, so that pressure at free surface can be considered uniform and constant; Coriolis effect neglected ; Inviscid fluid (lack of internal friction); Irrotational flow (shearing forces negligible); Horizontal, fixed, impermeable bed, which provides the boundary condition that vertical velocity at the bed is null; Long-crested Waves (two-dimensional motion).

With reference to figure 3.2, we consider a right-hand Cartesian system of coordinates axes (x,y,z) with its origin on the still water level. Let x-axis be oriented in the positive direction of wave propagation, positive z-axis upward, yaxis orthogonal to x and z. C Direction of Propagation L

Crest

η(θ) 0

a

H

π/2

3π/2

π

SWL

-a

2π

θ=kx z=0

Trough w

u

Bottom z=-d

Figure 3.2 - Sinusoidal progressive wave With the assumptions of irrotational motion and incompressible fluid, a velocity potential exists which should satisfy the continuity equation: div v = 0

(3.3)

div(∇φ ) = 0

(3.4)

or: the divergence of a gradient leads to the Laplace equation:

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INTRODUCTION TO COASTAL DYNAMICS AND SHORELINE PROTECTION

∂ 2Φ ∂ 2Φ ∂ 2Φ + + =0 ∂x 2 ∂y 2 ∂z 2

(3.5)

which states that the Laplacian of potential function Φ must be null. ∇ 2Φ = 0

(3.6)

∂ 2Φ ∂ 2Φ + =0 ∂x 2 ∂z 2

(3.7)

In a bidimensional case:

Equation (3.7) is the continuity equation for an ideal fluid (unviscid and uncompressible) in irrotational motion. The governing equations of motion in the fluid for the x-z plane are the Euler equations: ∂u ∂u ∂u 1 ∂p +u +w =− ∂t ∂x ∂z ρ ∂x ∂w ∂w ∂w 1 ∂p +u +w =− −g ∂t ∂x ∂z ρ ∂z

Substituting in the two-dimensional irrotationality condition ( ∇ × v = 0 ) ∂u ∂w = ∂z ∂x the equations can be rewritten as:

(3.8) (3.9)

(3.10)

∂u ∂ (u 2 / 2) ∂ ( w 2 / 2) 1 ∂p + + =− ∂t ∂x ∂x ρ ∂x

(3.11)

∂w ∂ (u 2 / 2) ∂ ( w 2 / 2) 1 ∂p + + =− −g ∂t ∂z ∂z ρ ∂z

(3.12)

Now, since a velocity potential exists for the fluid, we have u=−

∂φ ∂φ , w=− ∂x ∂z

(3.13)

Therefore, if we substitute these definitions into eqs. (3.9) and (3.12), we get ∂ ⎡ ∂φ 1 2 p⎤ + (u + w2 ) + ⎥ = 0 ⎢− ∂x ⎣ ∂t 2 ρ⎦

(3.14)

p⎤ ∂ ⎡ ∂φ 1 2 + (u + w 2 ) + ⎥ = − g − ⎢ ρ⎦ ∂z ⎣ ∂t 2

(3.15)

where it has been assumed that the density is uniform throughout the fluid. Integrating equation (3.14) with respect to x yields: p ∂Φ 1 2 (3.16) − + − ( u + w 2 ) = F1 ( z ,t ) ρ ∂t 2

LINEAR WAVE ANALYSIS

49

where the constant of integration varies only with z and time t. Integrating zequation yields: p ∂Φ 1 2 (3.17) − gz − + − (u + w 2 ) = F2 ( x, t ) ρ ∂t 2 Subtracting equation (3.16) from equation (3.17) gives gz = F2 ( x, t ) − F1 ( z , t )

(3.18)

Since g is not a function of x, it follows that F2 does not depend on x and is a function only of time and the same for F1=F2 − gz. Thus equations (3.16) and (3.17) can be reduced to the single equation − gz −

p

ρ

+

∂Φ 1 2 − (u + w 2 ) = F2 (t ) ∂t 2

(3.19)

The latter represents the “generalized Bernoulli theorem of Hydraulics” ∂Φ which must hold throughout the fluid. For steady flows, = 0 and ∂t F2=constant and equation assumes the expression: gz +

p

ρ

+

1 2 (u + w 2 ) = const 2

(3.20)

Which is the usual form of the steady-state Bernoulli equation.

1.2

Boundary conditions

In the hypothesis of irrotational and ideal fluid, the differential equations governing wave motion are Laplace equation and (Bernoulli) motion equation − gz −

p

ρ

+

∂Φ 1 2 − ( u + w 2 ) = const Motion equation (3.21) ∂t 2

∂2Φ ∂ x2

+

∂2Φ ∂ z2

=0

Continuity equation (3.22)

which must be valid throughout the fluid, that is in the region –d
50

INTRODUCTION TO COASTAL DYNAMICS AND SHORELINE PROTECTION

(x,t) must remain the same during the motion and this implies that fluid parcel speed in the normal direction to free surface must match the velocity of free surface in the same direction: d [z − η ( x, t )] = 0 on the surface z = η dt

(3.23)

where η is the displacement of free surface about the horizontal plane z=0.

∂η dz ∂η − −u =0 ∂x dt ∂t

(3.24)

dz ∂Φ =w=− dt ∂z

(3.25)

Considering that

and that u=−

∂Φ ∂x

(3.26)

equation (3.25) can be written −

∂Φ ∂η ∂Φ ∂η − + ⋅ =0 ∂z ∂t ∂ìx ∂x

∂Φ ∂η ∂Φ ∂η + − ⋅ =0 ∂z ∂t ∂x ∂x

(3.27) (3.28)

Equation (3.28) kinematic boundary condition on the surface. On the bottom the non-penetration condition imposes that vertical component of velocity be null z = − d . The kinematic condition on the bottom becomes: w=−

∂Φ =0 ∂z

(3.29)

A further boundary condition is obtained by considering that fixed surfaces, such as the air-sea interface, cannot support variation of pressure across the interface and hence must respond in order to maintain the pressure as uniform. Therefore dynamic boundary condition on the surface requires that pressure be invariant along the wave form (i.e. z = η ) and be equal to atmospheric pressure. So, Bernoulli equation with pressure pη=constant=pa (usually taken as 0) is to be applied to free surface z=η(x,t)

∂Φ 1 ⎡⎛ ∂Φ ⎞ ⎛ ∂Φ ⎞ + ⎢⎜ ⎟ ⎟ +⎜ ∂t 2 ⎢⎣⎝ ∂x ⎠ ⎝ ∂z ⎠ 2

gη −

2

⎤ ⎥ = constant ⎥⎦

(3.30)

LINEAR WAVE ANALYSIS

51

1.3 Linearized boundary conditions The difficulty in obtaining a general analytical solution of continuity equation and motion equation arises from the fact that boundary conditions on free surface are non-linear, and that free surface is itself unknown in the problem. So a linear solution to the problem is looked for by hypothesizing that wave height H is much smaller than wave length L and than depth d:

H << L

H << d

(3.31)

So the following assumptions can be made:

• • •

free surface displacement η is very small, so that O(η)=0 wave motion is slow (i.e. u small) and hence the square of particle velocity is negligible: O(u2)<
A further assumption is made by treating Bernoulli constant as to be included into the velocity potential Φ(x,z,t). In this way non-linear terms, which involve products of terms with the order of H, are negligible with respect to linear ones having the same order as H; furthermore boundary conditions on the surface are applied in correspondence of still level (z=0). With these assumptions, the kinematic boundary condition on the surface (3.28) is linearized as:

∂η ∂Φ ∂η ⎡ ∂Φ ⎤ + =0→ = −⎢ ⎥ ∂z ∂t ∂t ⎣ ∂z ⎦ z = 0

(3.32)

Dynamic boundary condition on the surface becomes: gη −

1 ⎡ ∂Φ ⎤ ∂Φ = cost. = 0 → η = ⎢ ⎥ ∂t g ⎣ ∂t ⎦ z = 0

(3.33)

By deriving (3.33) with respect to t and substituting in (3.30), we have: 1 ∂ 2 Φ ∂Φ = g ∂t 2 ∂z

(3.34)

2 Results of the linear theory Integrating equation (3.34) through separation of variables, the solution yields potential velocity Φ(x, z, t), given by: Φ(x, z, t) =

ag cosh(d + z) ⋅ ⋅ cos(kx − σt) σ cosh(kd)

(3.35)

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INTRODUCTION TO COASTAL DYNAMICS AND SHORELINE PROTECTION

2.1 Wave profile, length and celerity In linear theory, solution of equations are harmonic waves and vertical displacement η describing wave profile can be expressed through a regular progressive wave.

η=

1 g

⎡ ∂Φ ⎤ ⎢ ∂t ⎥ = a sin(kx − σt ) ⎣ ⎦ z =0

(3.36)

which is periodic in time and space.

η = a sin(kx − σt ) = a sin k ( x − ct )

(3.37)

where a=H/2 is the wave amplitude, k=2π/L is the wave number, σ=2π/T is the angular frequency, H is the wave height, L the wave length, T the wave period and C=σ/k=L/T is the wave celerity. The latter represents the speed at which a wave form propagates. ∂ 2Φ ∂Φ and we obtain the From equation (3.34) by explicating the terms ∂t 2 ∂z dispersion relation for the general case: σ 2 = gk ⋅ tgh(kd)

(3.38)

which can be otherwise written as: L=

gT 2 ⎛ 2 πd ⎞ ⋅ tgh⎜ ⎟ 2π ⎝ L ⎠

(3.39)

or through wave celerity

C=

gT ⎛ 2 πd ⎞ tgh⎜ ⎟ 2π ⎝ L ⎠

(3.40)

Equations (3.38), (3.39) and (3.40) state the relationship between wave lengths and wave periods; in particular equation (3.40) evidently shows that waves with different period travel with different speed. Equation (3.39) is noticeably implicit, and is to be solved numerically. In the resolution of practical problems it is usual to implement tabulated values of d/L0 and d/L in order to obtain corrisponding values of tanh (kd) and then to simplify the solution of equation (3.40) (see table 3.1). It may be convenient to distinguish the type of wave on the basis of the ratio between d and L called “relative depth” because the dispersion relation specializes according to the range which the ratio d/L lies in. d 1 > L 2

(deep water)

(3.41)

LINEAR WAVE ANALYSIS

53

⎛ 2 πd ⎞ tgh⎜ ⎟≅1 ⎝ L ⎠

(3.42)

gT 2 2π

(3.43)

gL0 L gT = 0 = 2π 2π T

(3.44)

L = L0 = C0 =

Because of the behavior of hyperbolic functions if relative depth d/L is less than 1/20, then the depth is small compared with wavelength and tgh(kd)≈kd. Equation (3.39) specializes into “shallow water” form (or “long wave”), d 1 < (shallow water) L 20

⎛ 2 πd ⎞ ⎛ 2 πd ⎞ tgh⎜ ⎟≅⎜ ⎟ ⎝ L ⎠ ⎝ L ⎠ L=

gT 2 2 πd ⋅ 2π L

L =T

gd

C = gd

(3.45) (3.46)

(3.47) (3.48) (3.49)

For a wave traveling in shallow water, wave celerity depends on the depth alone. If d/L is greater than 1/2 the waves are “short” and dispersion relation specializes into “deep water” form, because the hyperbolic fuction tanh(kd) approaches unity (see figure 3.3).

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INTRODUCTION TO COASTAL DYNAMICS AND SHORELINE PROTECTION

Figure 3.3 – Relative depth and asymptotes to hyperbolic functions The case of 1/20
2.2 Group celerity When a group of waves propagates through a medium, the speed of the wave train (or of the wave group) is generally not the same as the speed with which individual waves within the group travel. Let us consider the superposition of two regular waves moving in the same direction and for simplicity let us suppose they have the same amplitude. Since the linear theory is applied, superposition of solutions is admissible and the two components can be summed.

η = η1 + η 2 = a cos(k1 x − σ 1t ) + a cos(k 2 x − σ 2 t )

(3.50)

LINEAR WAVE ANALYSIS

55

where, in the hypothesis of small ∆σ and ∆k in order that ki and σi satisfy dispersion relation k1 = k −

∆k 2

σ1 = σ −

∆σ 2

(3.51)

k2 = k +

∆k 2

σ2 =σ +

∆σ 2

(3.52)

In terms of wavelengths and periods, we can write

η=

⎛ 2πx 2πt ⎞ H ⎛ 2πx 2πt ⎞ H ⎟⎟ + cos⎜⎜ ⎟ − − cos⎜⎜ 2 T1 ⎠ 2 T2 ⎟⎠ ⎝ L1 ⎝ L2

(3.53)

Because wave lengths L1 and L2 are supposed different, for some values of x at a given time t, the two harmonics will be in phase enhancing wave amplitude to 2H, for some other x they will be out of phase and will “destroy” each other. Using trigonometric identities, the two sinusoidal waves may be combined: ⎡1 ⎣2

⎤

⎡1 ⎣2

⎤

η = H cos ⎢ [(k1 + k 2 ) x − (σ 1 + σ 2 )t ]⎥ cos ⎢ [(k1 − k 2 ) x − (σ 1 − σ 2 )t ]⎥ = ⎦

⎦

(3.54) ⎡1 ⎛ ∆σ ⎞ ⎤ = H cos(kx − σt ) cos ⎢ ∆k ⎜ x − t ⎟⎥ ∆k ⎠⎦ ⎣2 ⎝

The resulting equation represents a wave propagating with velocity C=σ/k which is modulated by a wave traveling with speed ∆σ /∆k. Figure 3.4 shows the characteristics of a group formed by sinusoidal waves. The last is an enveloping wave and the speed is called “group velocity”. Cg =

∆σ ∆k

(3.55)

The group velocity constitutes the speed at which wave energy propagates. In fact, by letting ∆k to tend to 0, we obtain the group velocity of a wave train of infinite wave length Lg=2π / ∆k. C g = lim

∆k → 0

∆σ dσ = dk ∆k

(3.56)

Using the dispersion relation, the derivative can be evaluated: σ 2 = gk tanh(kd) 2σ

dσ = g tanh( kd ) + gkd sech 2 ( kd ) dk

(3.57) (3.58)

Table 3.1 - Wave table (Kamphuis, 2000) d/L 0,0000 0,0179 0,0253 0,0311 0,0360 0,0403 0,0496 0,0576 0,0648 0,0713 0,0775 0,0833 0,0888 0,0942 0,0993 0,104 0,109 0,114 0,119 0,123 0,128 0,132 0,137 0,141 0,150 0,158 0,167 0,175 0,183 0,192 0,200 0,208 0,225 0,234

kd 0,000 0,112 0,159 0,195 0,226 0,253 0,312 0,362 0,407 0,448 0,487 0,523 0,558 0,592 0,624 0,655 0,686 0,716 0,745 0,774 0,803 0,831 0,858 0,886 0,940 0,994 1,05 1,10 1,15 1,20 1,26 1,31 1,41 1,47

sinh(kd) 0,000 0,112 0,160 0,196 0,228 0,256 0,317 0,370 0,418 0,463 0,506 0,547 0,587 0,627 0,665 0,703 0,741 0,779 0,816 0,854 0,892 0,930 0,967 1,007 1,085 1,166 1,254 1,336 1,421 1,509 1,621 1,718 1,926 2,060

cosh(kd) 1,00 1,01 1,01 1,02 1,03 1,03 1,05 1,07 1,08 1,10 1,12 1,14 1,16 1,18 1,20 1,22 1,24 1,27 1,29 1,31 1,34 1,37 1,39 1,42 1,48 1,54 1,60 1,67 1,74 1,81 1,90 1,99 2,17 2,29

d/L0 0,22 0,23 0,24 0,25 0,26 0,27 0,28 0,29 0,30 0,31 0,32 0,33 0,34 0,35 0,36 0,37 0,38 0,39 0,40 0,41 0,42 0,43 0,44 0,45 0,46 0,47 0,48 0,49 0,50 0,75 1,00

tanh(kd) 0,909 0,917 0,926 0,933 0,940 0,946 0,952 0,956 0,961 0,965 0,969 0,972 0,975 0,978 0,980 0,983 0,985 0,986 0,988 0,989 0,990 0,992 0,992 0,993 0,994 0,995 0,995 0,996 0,996 1,000 1,000

d/L 0,242 0,251 0,259 0,268 0,277 0,285 0,294 0,303 0,312 0,321 0,330 0,339 0,349 0,358 0,367 0,377 0,386 0,395 0,405 0,415 0,424 0,434 0,443 0,453 0,463 0,472 0,482 0,492 0,502 0,746 0,981

kd 1,52 1,57 1,63 1,68 1,74 1,79 1,85 1,90 1,96 2,02 2,08 2,13 2,19 2,25 2,31 2,37 2,43 2,48 2,54 2,60 2,66 2,73 2,79 2,85 2,91 2,97 3,03 3,09 3,15 4,69 6,16

sinh(kd) 2,18 2,30 2,45 2,59 2,76 2,91 3,10 3,27 3,48 3,70 3,94 4,15 4,41 4,69 4,99 5,30 5,64 5,93 6,30 6,69 7,11 7,63 8,11 8,61 9,15 9,72 10,3 11,0 11,6 54,4 236,7

cosh(kd) 2,40 2,51 2,65 2,78 2,94 3,08 3,26 3,42 3,62 3,84 4,06 4,27 4,52 4,80 5,09 5,40 5,72 6,01 6,38 6,77 7,18 7,70 8,17 8,67 9,21 9,77 10,4 11,0 11,7 54,4 236,7

INTRODUCTION TO COASTAL DYNAMICS AND SHORELINE PROTECTION

tanh(kd) 0,000 0,112 0,158 0,193 0,222 0,248 0,302 0,347 0,386 0,420 0,452 0,480 0,506 0,531 0,554 0,575 0,595 0,614 0,632 0,649 0,666 0,681 0,695 0,709 0,735 0,759 0,782 0,800 0,818 0,834 0,851 0,864 0,887 0,900

56

d/L0 0,000 0,002 0,004 0,006 0,008 0,010 0,015 0,020 0,025 0,030 0,035 0,040 0,045 0,050 0,055 0,060 0,065 0,070 0,075 0,080 0,085 0,090 0,095 0,10 0,11 0,12 0,13 0,14 0,15 0,16 0,17 0,18 0,20 0,21

LINEAR WAVE ANALYSIS

Cg =

dσ g tanh( kd ) + gkd sech 2 ( kd )σ = = dk 2gktanh ( kd )

57

(3.59)

2kd ⎞ C⎛ ⎟ = nC = ⎜⎜1 + 2⎝ sinh (2kd ) ⎟⎠

where the factor n=

1⎛ 2kd ⎞ ⎜⎜1 + ⎟ specializes for deep and shallow water as 2 ⎝ sinh (2kd ) ⎟⎠

follows:

•

•

Deep water:

n=1/2

and

n=1

and

Cg =

C 2

(3.60)

Shallow water: Cg = C

(3.61)

Cg = energy propagation speed. The energy travels at half the speed of the wave profile; in shallow water energy and wave profile move at the same velocity. 2.3

Velocity components

By the definition of the velocity potential, from equation (3.41) velocity components u,w can be obtained by deriving potential function Φ(x, z, t) with respect to x and z, respectively: u=−

w=

∂ Φ agk coshk ( d + z ) = ⋅ sin (kx − σt ) ∂x cosh ( kd ) σ

∂ Φ agk sinhk (d + z ) = ⋅ sin(kx − σt ) ∂z cosh(kd ) σ

(3.62)

(3.63)

The two velocity components u and w, both harmonic in space and time, are π/2 out of phase, and the comparison with equation (3.36) shows that horizontal component u is in phase with the profile, showing its maximum absolute value in correspondence for the crest and the trough, that is when: kx − σt =

π 2

+ nπ (n = 0,1,.........)

(3.64)

and is null in the nodes. kx − σt = nπ (n = 0,1,.........)

(3.65)

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INTRODUCTION TO COASTAL DYNAMICS AND SHORELINE PROTECTION

η Cg 2α

η envelope

η=η1+ η2

C α 2H

-α

-2α

Lg

Figure 3.4 - Characteristic of a group formed by sinusoidal waves Vice versa the absolute value of vertical component w of velocity is maximum in the nodes and shows null value in correspondence of the crest. The horizontal component in crest is: uc =

agk coshk ⋅ (d + z) ⋅ σ cosh(kd)

(3.66)

Hence for z=0 and for z=–d (that is on the surface and on the bottom), the horizontal component is respectively: u c = u max =

agk

uc = u f =

σ

agk 1 ⋅ σ cosh(kd)

(3.67)

For what concerns w behavior in the node, we have: wn (z) = −

agk sinh k ⋅ (d + z) ⋅ σ cosh(kd)

(3.68)

Therefore for z=0 and z=–d, we have respectively: wn = wmax = −

agk

σ

tgh(kd )

wn = 0

(3.69)

The latter equation satisfies the cinematic boundary condition at the bottom.

LINEAR WAVE ANALYSIS

59

2.4 Particle displacements With reference to Figure 3.5 , let ζ be the horizontal displacement of surface parcels, and ξ be the vertical displacement. Velocity components are the time derivatives of the displacement components: dζ =u dt

dξ =w dt

(3.70)

Integrating with respect to time, we have:

ζ = ∫ udt = ∫

agk coshk (d + z ) ⋅ sin(kx − σt )dt = coshkd σ (3.71) agk coshk (d + z ) = 2 ⋅ ⋅ cos(kx − σt ) coshkd σ

ξ = ∫ wdt = ∫ −

agk sinh k(d + z ) cos(kx − σt )dt = ⋅ coshkd σ (3.72) agk sinhk(d + z) = 2 ⋅ ⋅ sin(kx − σt ) coshkd σ

The equations (3.71) and (3.72) can be simplified by using the dispersion relation (3.38). ζ=a

coshk (d + z ) ⋅ cos(kx-σx) coshkd

(3.73)

ξ=a

sinhk (d + z ) ⋅ sin (kx − σt ) coshkd

(3.74)

It is possible to demonstrate that particle paths are in general elliptical in shape. In fact equations (3.73) and (3.74) can be jointed (using the fundamental trigonometric relation): ζ

2

A2

+

ξ2 B2

=1

(3.75)

Being: coshk (d + z ) sinhk (d + z ) (3.76) B=a sinhkd sinhkd Equation (3.75) represents an ellipse with a major horizontal semi-axis A and a minor vertical semi-axis B. Equations (3.76) specialize depending on depth and trajectories assumes specific forms which can be easily determined by examining the values of A and B. A=a

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INTRODUCTION TO COASTAL DYNAMICS AND SHORELINE PROTECTION

u As

w

Bs

Ab

Figure 3.5 - Water particle orbits and relative displacement. shallow water: sinh k(d + z) ≅ sinh kd ≅ kd ; cosh k(d + z) ≅ cosh k d ≅ 1 A=

a kd

B=a

d+ z d

(3.77) (3.78)

Therefore the amplitude of horizontal semi-axis does not depend on vertical coordinate, whereas the amplitude of vertical semi-axis decreases with increasing depth until it becomes null on the bottom (z=–d), reducing the ellipse to a segment. sinhkd ≅ coshkd ≅

A = a⋅

k (d + z) e 2 kd e 2

kd e2

=

;

kz a⋅ e 2

sinhk (h + z ) ≅ coshk (d + z ) ≅

B = a⋅

k (d + z ) e 2 kd e2

kd e2

(3.79)

kz

= a⋅ e 2

(3.80)

Semi-axes being equal, the elliptical paths become circular, with the circumference radius exponentially decreasing with depth. For example in z=– L/2 we have A=B=0.042a. Below this depth, as already mentioned, wave perturbation is negligible. The described behavior of particle paths is depicted in Figure 3.6.

LINEAR WAVE ANALYSIS

d/L < 1/20

1/20 < d/L < 1/2

d/L > 1/2

Shallow water

Intermediate water

Deep water

61

Figure 3.6 - Water particle orbits for shallow, intermediate and deep water 2.5

Wave pressure

The generalized Bernoulli equation for an ideal fluid is: p

ρ

+ gz +

u 2 + w 2 ∂Φ − = f (t ) 2 ∂t

(3.81)

where the quadratic term is neglected (small amplitude waves). By matching in the same instant this relation to any depth z and to the surface z=η where the relative pressure is null, we have: ⎛p ∂Φ ⎞ ⎡ ∂Φ ⎤ ⎜⎜ + gz − ⎟⎟ = gη − ⎢ ⎥ ∂t ⎠ z ⎣ ∂t ⎦ z =η ≅ 0 ⎝ρ

(3.82)

The wave profile is obtained by equation:

η=

1 ⎡ ∂Φ ⎤ g ⎢⎣ ∂t ⎥⎦ z = 0

(3.83)

and hence: p

ρ Being:

⎡ ∂Φ ⎤ = − gz + ⎢ ⎥ ⎣ ∂t ⎦ z

(3.84)

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INTRODUCTION TO COASTAL DYNAMICS AND SHORELINE PROTECTION

cosh k(d + z ) ⎡ ∂Φ ⎤ ⎢ ∂t ⎥ = a g ⋅ cosh kd sinh(kx− σt) ⎣ ⎦z

(3.85)

We have: p = − ρgz + ρga⋅

cosh k(d + z) sinh(kx − σt) = − ρgz + ρgη⋅ k p (z) cosh kd

(3.86)

The quantity k p (z) =

cosh k(d + z) cosh kd

(3.87)

is called the response factor and is always less than unit below the mean water level. In equation (3.86) the first term represents hydrostatic pressure, which would exist in absence of wave motion, the second term represents dynamic pressure due to wave motion. The dynamic pressure results from two contributions: hydrostatic pressure surcharge associated with free surface displacement and pressure (surcharge or discharge) produced by vertical accelerations accompanying the orbital motion. If the response factor Kp(z) is unitary, the pressure contribution from the free surface displacement is purely hydrostatic; indeed the response factor is equal to unity from the surface to the mean watel level (z=η≅0). Vertical acceleration is 180° out of phase with free surface displacement and its contribution modifies pure hydrostatic pressure by enhancing or decreasing its magnitude depending on wave phase. The behavior of pressure in correspondence of crest and trough is shown in figure 3.7. +η -η

η(1-Kp)

-η(1-Kp) -z

Actual pressure head

d

kx-σt

Hydrostatic p/γ=η-z

Hydrostatic p/γ=η-z

Figure 3.7 - Pressure diagram within the wave profile.

LINEAR WAVE ANALYSIS

63

At the crest, pressure is hydrostatic from z=η to z=0, below z=0 pressure is higher than hydrostatic pressure referred to mean water level, but lower than hydrostatic pressure relative to the crest level. At the trough, pressure is higher than hydrostatic pressure relative to the trough level, but lower than hydrostatic pressure corresponding to mean water level. Anyway, with increasing depth, pressure tends to the hydrostatic value corresponding to the mean water level.

3 Case study A calibrated pressure gauge located on the sea bed at a depth of z=13m, measures a value of maximum pressure ∆p=0.146 Kg/cm2 and a frequency f=0.11 Hz. Evaluate wave height and length and the relative wave celerity. Calculate the values of horizontal and vertical components of velocity for phase angles θ=kx-σt=90° e θ=180° at the surface, at half depth and on the bottom. Calculate also the value of semi-axes of the particle paths at the crest, at the surface, at half depth and on the sea bed. Given data: ∆p=0.146 Kg/cm2=1460 Kg/m2 z=13m f=0.11 Hz θ=kx-σt=90° θ=180°

•

Wave length, height and celerity

Consider the wave pressure equation:

p = − ρgz + ρgηK p ( z ) where Kp(z) is the pressure response factor given by Kp( z ) =

cosh(d + z) cosh(kd)

Because z=–d, the pressure equation can be written as: p = − ρg d + ρgη⋅

1 cosh(kd)

If the pressure gauge is calibrated, it gives only the dynamic pressure alone, and therefore we have: H ∆pmax ηmax = a = = ⋅ cosh(kd) 2 ρg Because k=2π/L the knowledge of wave length L is required for the calculus of wave number k and the determination of wave height. The value of L is determined from the period T and the depth d, with reference to the dispersion relation, which is implicit.

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INTRODUCTION TO COASTAL DYNAMICS AND SHORELINE PROTECTION

f=0.11 Hz

T=1/f = 1/ 0.11 s=9.09 s

Deepwater wave length may be calculated from wave period: L0 =

gT 2 = 1.56 ⋅ T 2 = 1.56 ⋅ 9.09 2 m = 129 m 2π

Entering in the table 3.1 with the ratio

d 13 = = 0.1 , the value of tanh(kd) is L0 129

given as 0.709. From the formula d d = tanh(k d) L0 L

wave length corresponding to the depth of 13m can be determined as:

L = L0 tanh(kd) = 129 ⋅ 0.709 m = 91,46 m Now the wave number is given by: k=2π/L =2⋅ 3.14/91.46 m-1 =0.06869 m-1

•

Maximum elevation ηmax

∆p=0.146 Kg/cm2=1460 Kg/m2 ρg=γ=1020 Kg/m3

ρ=104 Kg s2/m4

g=9.8 m/s2

d=13m cosh(kd)= 1.4239 1460 ⋅ 1.4259m = 2.04m 1020 For the small amplitude waves, wave height is obtained as: H = 2η max = 2 ⋅ 2.04 = 4.08m Wave celerity is given by: L 91.46 = 10.1 m / s C= = 9.09 T

η max = a =

•

Horizontal and vertical components of velocity

Consider the equations (3.62) and (3.63) for velocity components: u=

agk coshk ( z + d ) sin (kx − σt ) σ cosh (kd )

w=−

agk sinhk ( z + d ) cos(kx − σt ) σ cosh (kd )

LINEAR WAVE ANALYSIS

65

Horizontal velocity u θ=kx-σt=90° and θ=180° z=0 , z=-d/2 , z=-d k=2π/L=0.06869 m-1 σ=2π/T=0.691 d=13 m When θ=180° (i.e. in the node) horizontal component is null. For θ=90°, that is on the crest, we have: agk

σ z=0 z=-d/2 z=-d

=

2.04 ⋅ 9.81 ⋅ 0.0686 m / s = 1.98m / s 0.691

cosh k(d + z) =1 cosh(kd) coshk(d + z) = 0.774 cosh(kd)

⇒ uc = ⇒ uc =

coshk(d + z ) 1 = = 0.703 cosh(kd) cosh(kd)

agk = 1.98m / s σ

agk ⋅ 0.774 = 1.53m / s σ

⇒ uc =

agk ⋅ 0.703 = 1.39m / s σ

Vertical velocity component w: θ=kx-σt=90° and θ=180° z=0 , z=-d/2 , z=-d k=2π/L=0.06869m-1 σ=2π/T=0.691 d=13 m When θ=90° (on the crest and in the trough) vertical component is null. In the nodes, θ=180° we have:

z=0 z=-d/2 z=-d

sinhk(d + z) = tanh(kd) = 0.711 cosh(kd) sinhk(d + z) = 0.323 cosh(kd) sinhk(d + z) =0 cosh(kd)

⇒ wn = −1.41m / s

⇒ wn = −0.64m / s ⇒ wn = 0

The last result is in agreement with the kinematic boundary condition on the bottom (vertical velocity component null).

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INTRODUCTION TO COASTAL DYNAMICS AND SHORELINE PROTECTION

•

Semi-axes of particle paths

Major and minor semi-axes of particle paths are given respectively by: cosh k(d + z) sinh kd sinh k(d + z) B=a sinh k d A=a

Let us consider separately the behavior of the terms at the three depths: 1 ⎧ coshk(d + z) ⎪ sinh(kd) = tanh(kd) = 1.406 ⇒ A = a× 1.406 = 2.04 × 1.406 = 2.87 m z=0 ⎪⎨ ⎪ sinhk(d + z) = 1 ⇒ B = a = 2.04m ⎪⎩ sinh(kd)

z=-d/2

z=-d

⎧ cosh k(d + z) ⎪ sinh(kd) = 1.237 ⎪ ⎨ ⎪ sinh k(d + z) = 0.454 ⎪⎩ sinh(kd)

⇒ A = a× 1.237 = 2.52m ⇒ B = a× 0.454 = 0.93m

1 ⎧ cosh k(d + z) ⎪ sinh(kd) = sinh(kd) = 0.989 ⎪ ⎨ ⎪ sinh k(d + z) = 0 ⇒ B = 0 ⎪⎩ sinh(kd)

⇒ A = a× 0.989 = 2.02m

Horizontal semiaxis decreases with depth much more slowly than the vertical semiaxis, which becomes null for z=-d (on the bottom).

Chapter 4 Sea level variability 1 Introduction The mean sea level is referred to as the average level on a time interval of several minutes, in order to exclude both high frequencies, due to gravity surface waves, and sea level variations, which period is much longer than those of oceanographic interest. In fact coastal water levels are influenced by a variety of astronomical, meteo-oceanographic and tectonic factors, which often interact in a complex way to increase water levels significantly above normal tide level. Storms, which develop low atmospheric pressure, strong onshore winds and large waves, are the most common cause of abnormally high water levels. These increases of water levels are of concern because they intensify beach erosion and increase damage to coastal structures because they allow larger waves to cross offshore bars and break closer to the beach. Sea level fluctuations are classified according to time scales in: • astronomical tides (caused by the attraction of sun and moon on water masses); • long waves (tsunami and seiches); • wave setup (due to wave breaking); • storm surge (mean sea-level variations due to atmospheric causes). These categories of sea level fluctuations show periods ranging from a few minutes to some days.

2

Astronomical tide

The sum of forces caused by gravitational interaction among the Earth, Moon and Sun generates a periodic variation of big water masses. The rising and depression of water masses due to these forces is called astronomical tide, whereas the consequent horizontal motion is called tidal current. The most important tidal component is caused by Moon attraction and is called M2, with maximum values 12h and 25min intervals; consequently

68

INTRODUCTION TO COASTAL DYNAMICS AND SHORELINE PROTECTION

maximum and minimum fluctuations of mean-sea-level are delayed 50min per day, in perfect agreement with the Moon day which lasts 24h and 50 min. The second component of the tide, called S2, is caused by Sun attraction, with a period of 12h, which in turn enhances or damps the semi-diurnal effects of the Moon. When the two attracting forces caused by Moon and Sun are in conjunction or in opposition of phase (full-Moon or new-Moon) the tide at a site shows the maximum level and is called spring tide; when the two forces are onequarter period out of phase (Moon in first or third half) the tide assumes the minimum value and is called neap tide. The tide fluctuations depend on the relative positions of Earth, Moon and Sun. The elements contributing to tidal forces are almost 100 on long time scales, 160 diurnal, 115 semi-diurnal and 14 one-third diurnal. But in practice only few components are accounted for and they are listed in Table 4.1. Examples of diurnal, semidiurnal and mixed tides are given in Figure 4.1. Table 4.1 - Tidal constituents and arguments (U.S. Army Corps of Engineers Coastal Engineering Research Center, 2002). TIDAL CONSTITUENTS AND ARGUMENTS Symbol M2 S2 N2 K1 M4 O1 M6 (MK)3 S4 (MN)4 ν2 S6 µ2 (2N)2 (OO)1 λ2 S1 M1 J1

Speed (deg/hr) 28.984 30.000 28.439 15.041 57.968 13.943 86.952 44.025 60.000 57.423 28.512 90.000 27.968 27.895 16.139 29.455 15.000 14.496 15.585

Period (hr) 12.421 12.000 12.659 23.935 6.2103 25.819 4.140 8.177 6.000 6.269 12.626 4.000 12.872 12.906 22.306 12.222 24.000 24.834 23.099

Symbol Mm Ssa Sa Msf Mf ρ1 Q1 T2 R2 (2Q)1 P1 (2SM)2 M3 L2 (2MK)3 K2 M8 (MS)4

Speed (deg/hr) 0.544 0.082 0.041 1.015 1.098 13.471 13.398 29.958 30.041 12.854 14.958 31.015 43.476 29.528 42.927 30.082 115.936 58.984

Period (hr) 661.765 4390.244 8780.488 354.680 327.869 26.724 26.870 12.017 11.984 28.007 24.067 11.607 8.280 12.192 8.386 11.967 3.105 6.103

To classify different sites of the Earth according to the tidal type, the following ratio is used: F = (K1 + O1)/(M2 + S2)

(4.1)

SEA LEVEL VARIABILITY

69

if: (1) F = 0÷0.25 gives a semi-diurnal tide: two daily maxima and two minima of almost the same height; ‘spring tide’ = 2(M2 + S2). (2) F =0.25 ÷ 1.5 gives a mainly semi-diurnal tide with irregular variation of amplitude and frequency between the minimum and maximum. (3) F =1.5 ÷ 3 gives a mainly diurnal tide, with one maximum a day; spring tide = 2(K1 + O1). (4) F > 3.0 gives a diurnal tide; spring tide like 3). Diurnal tide

Semi-Diurnal tide

Mixed Tide

M

6

12 First day

18

M

6

12 Second day

18

M

Figure 4.1 - Examples of diurnal, semidiurnal and mixed tides. Tides are waves several hundreds kilometres long. Being the dimensions of wave length often comparable with the geometric dimensions of the gulf or bay, it is important to study the tidal wave propagation in shallow water conditions, because rising of sea level in a site does not only depend on the magnitude of tidal forces, but also on resonance between tidal period and free oscillation period of the site. The shallow water motion equations for a constant depth h are: ∂U ∂U τ bx ∂U ∂ζ +U +V = fV − g − ∂t ∂x ∂y ∂x ρ (ζ + d )

(4.2)

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INTRODUCTION TO COASTAL DYNAMICS AND SHORELINE PROTECTION

∂V ∂V τ by ∂V ∂ζ +U +V = − fU − g − ∂t ∂x ∂y ∂y ρ (ζ + d )

(4.3)

where: U,V = are respectively x and y components of mean velocity averaged over the water column. t = time f = 2ωsinϕ = Coriolis parameter, with ϕ latitude (in degree) ω = angular velocity of Earth (7.29 × 10-5 rad/s) g = gravity acceleration ζ = free surface elevation relative to the mean-sea-level τbx , τby = x and y components of bottom friction ρ = water density The mass conservation equation is: ∂ζ ∂ ∂ + [(ζ + d )U ] + [(ζ + d )V ] = 0 ∂t ∂x ∂y

(4.4)

The simplified equations can be obtained neglecting bottom friction, hypothesizing ζ + d ≅ d and considering only small acceleration terms, such that U ∂U ∂x and V ∂V ∂y can be neglected. The linearized equations obtained by (4.2), (4.3) e (4.4) are: ∂U ∂ζ = fV − g ∂t ∂x

(4.5)

∂V ∂ζ = − fU − g ∂t ∂y

(4.6)

 ∂U ∂V  ∂ζ  = 0 + d  + ∂t  ∂x ∂y 

(4.7)

In order to determine the tidal wave celerity and mean-sea level variation, let us consider a progressive tidal wave, travelling in a finite length channel (x direction). Equations become: ∂U ∂ζ = −g ∂t ∂x

(4.8)

∂ζ ∂y

(4.9)

fU = − g

∂ζ ∂U +d =0 ∂t ∂x

Elimination of U gives:

(4.10)

SEA LEVEL VARIABILITY

∂ 2ζ ∂t 2

= gd

∂ 2ζ ∂x 2

71

(4.11)

while the elimination of ζ gives a similar equation for U: ∂ 2U ∂t 2

= gd

∂ 2U ∂x 2

(4.12)

A simple harmonic solution of a progressive wave can be expressed in the form: ζ = Ae − my cos(kx − ωt )

(4.13)

U = U 1e − my cos(kx − ωt )

(4.14)

where: U1 =

g and m = f gd dA

where A is wave amplitude. Equations (4.13) and (4.14) represent a Kelvin wave propagating in x direction. Due to Coriolis force, wave amplitude exponentially decreases moving away from the right coast of the Northern hemisphere.

3

Long waves (tsunami and seiches)

Tsunamis are long-period water waves generated by undersea shallow focus earthquakes or by undersea crustal displacements (subduction of tectonic plates), landslides, or volcanic activity. Tsunamis can travel great distances, undetected in deep water, but shoaling rapidly in coastal waters and producing a series of large waves capable of destroying harbour facilities, shore protection structures, and upland buildings. There is no simple answer to how large an event needs to be to create a tsunami. Generally, it is thought that a quake (the most common cause of tsunami) needs to have a magnitude of 6.5 (Bryant, E., 1991) to 7.5 to produce a dangerous one. Tsunami are hard to detect in deep waters. Their wavelengths are very long (hundred of miles), but only a few feet high in the open ocean. Their speed is a function of the water depth: the deeper the water, the faster the tsunami. If they are created in the deeper parts of the Pacific, their speeds can be hundreds of miles per hour. Offshore tsunamis would strike the adjacent shorelines within minutes and also cross the ocean at speeds as great as 600 miles per hour to strike distant shores. As the ocean floor rises, the front of the wave slows, causing the rest of the wave to push up behind it, increasing the height.

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INTRODUCTION TO COASTAL DYNAMICS AND SHORELINE PROTECTION

Not all tsunamis break when they reach land. Some just rush ashore as a huge mass of water, like a sudden massive tide. Others break far from land and come ashore as a turbulent cascading mass called a bore. The size of the tsunami, its speed, as well as the coastal area’s form and depth are the factors that affect their shape. The power of a tsunami comes from the huge amount of water behind the leading edge of the wave. Normal waves have a small volume, so they dissipate quickly when they strike the shore. Tsunamis do not. Their huge volume pushes the water far inland. This phenomenon is called ‘run up’ and its size is what often determines a tsunami’s destructiveness (Myles, Douglas, 1985). Tsunamis rarely crash ashore in one huge wave. They are often preceded by coastal flooding, followed by a recession of the water, and then, finally, a series of waves. This effect is dangerous since many people assume the trouble is over after the first wave breaks. Unaware of the looming danger they venture too close to shore and are swept away by subsequent waves. The first wave to strike Crescent City, California, caused by the Alaska earthquake in Prince William Sound in 1964, was 9 feet above the tide level; the second, 29 minutes later, was 6 feet above tide; the third was about 11 feet above the tide level, and the fourth, most damaging wave, was more than 16 feet above the tide level. When an earthquake that might generate a Pacific Coast tsunami is detected, the Alaska Tsunami Warning Center calculates the danger to the northeast Pacific Coast and notifies the communities at risk. Those warnings may give people a few hours to prepare and evacuate (depending on the distance to the earthquake). If the earthquake occurs off our coast, however, there may be no time to send out hazard warnings. The first waves could arrive within minutes of the earthquake. The only tsunami warning might be the earthquake itself. Seiches are long waves with different causes. They develop when a enclosed body of water is shaken. The water literally sloshes around in its ‘container’ like water in a cup. Seiches are free oscillations with a natural frequency depending on the geometric dimension of the basin. Resonance effects can develop when the long wave period matches the frequency of oscillation of the body of water. Causes providing free surface oscillations, mainly in channels, are: • • • •

Pressure oscillations; Wind direction variations; River discharge in fluvial deltas and estuaries; Seismic oscillations during earthquakes.

The main cause in seiches generation has a meteorological feature. By supposing the force acting only at the initial process of generation of basin oscillations, seiches can be dealt with by using equations elaborated in the previous section. In particular, the example of a rectangular, narrow, long and constant-deep basin, similar to the one used in the previous section, but without open boundaries, will be re-considered here.

SEA LEVEL VARIABILITY

73

Solving the linearized shallow water equations, a solution for seiches of a steady wave is: ζ =

A0 cos kx cos ωt 2

(4.15)

with A0= amplitude of free surface oscillation Supposing that the cross-shore component of the velocity is null at the boundary of the basin; this condition implies that at the two ends of the basin, two anti-node are present, that is using equation (4.8) one obtains: senkx = 0

for x=0,l

(4.16)

This implies kl=nπ where n is the oscillation mode of the basin. Substituting k one obtains: L = 2l / n Figure 4.2 shows the basin wave length for three values of n. Basing on the characteristics of forces, seiches with several kinds of typical modes it is important to underline that energy is predominantly associated to the first modes of oscillation. Because: L = gd T the oscillation period of the basin is determined by: C=

T=

2l n gd

(4.17)

(4.18)

Wilson, 1972 has demonstrated that in the case of a rectangular open basin, equation (4.18) becomes: T=

4l n gd

(4.19)

that is: 1 n gd T 4 where n can only assume odd values. l=

4

(4.20)

Wave set-up and set-down

Wave set-up and Wave set-down are defined as the elevation and depression of the mean sea level due to breaking phenomenon. These variations of the mean sea level phenomena are associated with conversion of kinetic energy of breaking waves into quasi-steady potential energy (increased water level). When waves enter shallow water over a shoaling bottom, the characteristic vertical scale of wave motion is comparable with the local water depth. The

74

INTRODUCTION TO COASTAL DYNAMICS AND SHORELINE PROTECTION

actual water depth in the arbitrary point of the coastal zone depends on many factors such as the geometric water depth, the mean water level influenced by the wave motion, tidal oscillations, seiching and storm surges. Waves approaching a sloping bottom are affected by some modifications of their parameters (wave height H, wave length L, crest angle α, depth d) resulting from shoaling, refraction, diffraction, reflection, and breaking processes (Massel, S.R., 1989).

Figure 4.2 – Seiches wave length for three values of n. The forward flow of the water particles in the breaking waves usually sets up longshore currents (see chapter 9) and “pumps” water across the breaking zone, increasing the water level there. The onshore momentum of the waves holds some of this water close to shore, causing an elevated water level near the shoreline (wave set-up). This phenomenon can be explained by the concept of radiation stress, introduced by Longuet-Higgins and Stewart (Longuet-Higgins, M. S., & R.W. Stewart, 1964) described in the following sub-section. 4.1 Radiation stress Let us first consider an off-shore steady condition, by assuming that a wave train beats against a coastline for a time long enough to establish an equilibrium condition. With reference to a system of Cartesian axes moving with the waves and oriented with x-axis in the direction of propagation, and y-axis orthogonal to it, the mean value of the total longitudinal momentum flux integrated over the vertical, deprived of the hydrostatic pressure per unit length, is equal to:

SEA LEVEL VARIABILITY ζ

S XX =

∫ (p + ρu )dz − ∫ 2

−d

0

p0 dz

75

(4.21)

−d

Where SXX is the normal component of radiation stress; d = bottom depth; ζ = mean-sea-level setup; p = pressure; p0 = hydrostatic pressure; u = horizontal wave velocity component; ρ = water density; The transversal component of radiation stress (SYY ) can be expressed as: ζ

SYY =

2 ∫ (p + ρv )dz −d

SYY =

∫

0

ρw2 dz +

−d

−

∫

0

p0 dz

(4.22)

−d

1 ρg ζ 2 2

(4.23)

The shearing stress component is denoted by SXY and results null because xcomponent of speed is zero as y-axis is orthogonal to the direction of wave propagation: S XY =

∫

ζ

ρuvdz

(4.24)

−d

In order to carry out an analytic treatment, radiation stress components are expressed in a system of coordinate with x-axis cross-shore direction, and y-axis long-shore direction. By transforming the old system of coordinates (X,Y) into the new system coordinates (x,y) one obtains: S yy =

1 (S XX − S YY ) − 1 (S XX − S YY ) cos 2α 2 2

(4.25)

S xx =

1 (S XX + S YY ) + 1 (S XX − S YY ) cos 2α 2 2

(4.26)

S xy =

1 (S XX − SYY )sen2α 2

(4.27)

where α is the angle between the wave fronts and the coast. Sxy is the shearing stress component with respect to long-shore direction caused by the difference between transversal and longitudinal momentum fluxes of an incident wave with an oblique front. Substituting orbital velocities u and w and surface elevation ζ and adopting linear wave theory, the following expressions are obtained: 1  S XX = E  2m −  2 

(4.28)

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INTRODUCTION TO COASTAL DYNAMICS AND SHORELINE PROTECTION

1  SYY = E  m −  2 

(4.29)

and so: 1 1 3 S yy = E  m −  − Em cos 2α 2 2 2 S xy = Emsenα cos α

(4.31) (4.32)

where E is the total energy and m is the following ratio: m=

Cg C

=

1 2kd  1 +  2  senh2kd 

(4.33)

where Cg = group velocity and C = phase velocity. 4.2 Water level fluctuations due to radiation stress Waves approaching a sloping bottom are modified by shoaling, refraction, diffraction, reflection, and breaking processes. Therefore, the radiation stress components are modified and a new time-averaged equilibrium for the timeaveraged water level is estabilished. Longuet-Higgins and Stewart (1964) from the conservation of momentum flux in an incident wave train, derived the following momentum balance between the radiation stress variation and the changes in sea level: dS xx dζ + ρg (d + ζ ) =0 dx dx

(4.34)

If Sxx is known, the equation (4.34) can be integrated to yield a mean waterlevel set-down outside the breaker line and a mean water level set-up from the surf zone, until the shoreline. For spilling type breakers on dissipative beaches, the assumption commonly employed is that the breaker index γb (ratio of breaking wave height to mean depth at breaking) remains a fixed ratio throughout the entire surf zone :

γb =

Hb H ≈  (d + ζ )  d b

(4.35)

The breaking index can be calculated using various expressions, as described in chapter 8. If linear wave theory is applicable to compute the radiation stress component, Sxx=(3/2)E, where E is the total wave energy, so: 3 ρgγ b2 (d + ζ ) 2 (4.36) 16 Using the equation (4.36) in the momentum equation balance (4.34), yields: S xx =

SEA LEVEL VARIABILITY

 3γ b2  d ζ 3γ 2 dd  1 + =− b  8  dx 8 dx 

77

(4.37)

or:  8 dζ = −1 + 2  dx  3γ b

−1

  tan β  

(4.38)

Therefore, the mean water surface slope is proportional to the beach slope. The total rise of the mean water level in the surf zone can be calculated by integrating the equation (4.37):

ζ =−

3γ b2 8

−1

 3γ b2  1 +  d ( x) + C  8  

(4.39)

Evaluating the constant C at x=xb, the breaker line, where ζ = ζ , gives: b

ζ ( x) = ζ b +

3γ b2 8

−1

 3γ b2  1 +  [d b − d ( x)]  8  

(4.40)

Using shallow water theory, ζ becomes: b

ζb =−

1 γ bHb 16

(4.41)

For the maximum set-up at the shoreline, the equation (4.40) shows that: −1

2  2   3γ b2  1 + ζ max = ζ b + 3γ b 1 + 3γ b  d b  8  8  8  

(4.42)

or:

ζ max = ζ b +

3γ b2 db 8

(4.43)

Substituting (4.41) in (4.43) we get: ζ max =

5 γ bHb 16

(4.44)

Equation (4.44) shows that the mean water level at the shoreline is about 25% of breaker wave height due to wave set-up (Massel, S.R., 1989). Figure 4.3 gives a schematic representation of wave set-up and set-down.

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INTRODUCTION TO COASTAL DYNAMICS AND SHORELINE PROTECTION

5

Storm surge

The term “storm surge” includes all the mean-sea-level fluctuations due to the passage of an atmospheric perturbation, which are due to both wind-wave interaction and sea response to pressure variations. Wind action gives a double contribution to sea-level variation: first, according to Miles’ theory, wind generates waves heights and so a wave setup is experienced; on the other hand, wind action on sea surface generates coastal currents, which in conditions of slightly sloping bottom cause an increase of mean-sea-level.

breaker line

Wave set-up

ζ

Wave set-down

ζs ζ max

ζb

d

Horizontal level ζ max = maximum wave set-up

ζ s = set-up at the still water level ζ b = wave set-down at the breaker line db

= breaking depth

Figure 4.3 – Wave set up and set-down. The last mechanism is called wind setup and depends on the magnitude, duration, and wind incidence angle with respect to the coast. Sea response to atmospheric fluctuations (storm surge) is not simultaneous to the transit of a cyclonic vortex, but requires the permanence of storm for a few hours. In the presence of a storm surge, equations describing coastal marine dynamics, that are mass conservation equation and motion equations, must account for the interaction between atmospheric effects and sea surface. In the following, the motion equations in the presence of a storm surge will be treated with some simplifications, that is they are vertically integrated and all terms are expressed with respect to current mean velocity, and to mean momentum flux over the water column. The following simplifications are established:

SEA LEVEL VARIABILITY

(1) (2) (3) (4)

79

vertical accelerations neglected; wave effects (wave setup) neglected; friction not considered; fixed and impermeable bottom.

A system of Cartesian axes is established with x and y oriented in horizontal plane with the origin on s.w.l., x axis cross-shore and y axis long-shore; θ is the anti-clockwise angle between wind direction and x-axis; W is wind speed. Motion equations are the linearized shallow water equations used at the beginning of this chapter: ∂U ∂U τ bx ∂ζ ∂U − = fV − g +V +U ∂x ρ (ζ + d ) ∂y ∂x ∂t

(4.45)

∂V ∂V τ by ∂ζ ∂V − = − fU − g +V +U ∂y ρ (ζ + d ) ∂y ∂x ∂t

(4.46)

and mass conservation equation is: ∂ ∂ ∂ζ + [(ζ + d )U ] + [(ζ + d )V ] = 0 ∂t ∂x ∂y

(4.47)

Symbols are also the same as those used at the beginning of this chapter. Solutions of this system of equations which describes marine dynamics can be obtained through numerical integration (e.g. finite difference method). The highest value of mean-sea-level variation during the transit of a storm, neglecting astronomical tide effects, is called maximum surge, while the absolute maximum over all values collected in all sites affected by the storm is called peak surge. In order to calculate mean-sea-level rise during a storm surge, a separate calculation of inverted barometer and wind setup is used. Sea-level variation caused by inverted barometer can be studied by considering sea-level fluctuations as a phenomenon of long wave propagation generated by atmospheric pressure fluctuation. Thus let us consider a negative pressure variation which propagates with speed U in x direction: p0 = f1 (Ut − x )

where x axis is cross-shore with entering shore direction. The linearized motion equations in x direction is: dζ d dp0 du = − gd − d dx ρ dx dt

(4.48)

(4.49)

Continuity equation can be expressed by considering a system of coordinates moving together with the long wave caused by pressure fluctuations, which makes the system stationary and the horizontal velocity equal to: u–U.

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INTRODUCTION TO COASTAL DYNAMICS AND SHORELINE PROTECTION

By considering a constant transport Q, time and space-invariant, and by supposing that particle speed is proportional to free surface displacement: Q = (u − U )(d + ζ ) = −Ud

(4.50)

and thus linearizing: u =U

ζ

ζ

≅U

d +ζ

d

(4.51)

by assuming the form: ζ = f 2 (Ut − x )

(4.52)

mass conservation equation as a function of ζ dζ dζ = −U dx dt

(4.53)

)

(4.54)

Combining all equations: −

(

d dp0 dζ 2 U − gd = − ρ dx dx

which is an exact differential and can be integrated between a point of null pressure variation (and so of mean-sea-level) and a point of interest, obtaining:

ζ d

=

p0 ρ U − gd

(

2

)

(4.55)

In static conditions, that is in absence of propagation: ζS = −

p0

ρg

(4.56)

If the speed of translation U assumes a value close to gd = C , the denominator tends to null and mean-sea-level variation should become unlimited. In practice, friction effects damp this amplification. Figure 4.4 shows the absolute ratio between the sea-level variation caused by a moving depression and the same variation caused by inverted barometer in steady state. Also, the comparison between the cases with and without friction is given.

SEA LEVEL VARIABILITY

81

5 4.5 4

|ζ |

ζS

Frictionless case

3.5 3 2.5

A/(kdC)=0.5

2 1.5 1 0.5 0

0

1

2

3

4

5

U /C

Figure 4.4 - Dynamic response of translating pressure disturbance, with and without friction. For U << C, i.e. on deep bottoms, mean-sea-level variation is similar to the steady case, and assumes the value: ζ =−

p0

ρg

(4.57)

where ζ is mean-sea-level variation expressed in centimetres, p0 is atmospheric pressure variation expressed in millibar. Therefore to each negative variation of pressure of 1 millibar, sea responds with 1 centimeter of mean-level increase. In order to calculate storm surge due to wind setup the Cartesian plane previously discussed is adopted. X-component of wind stress is equal to τ θ = τ cosθ and let us suppose that the horizontal scale system is so small that Coriolis force effects are negligible. Linearized equations become: Sx

Sx

τ Sxθ τ bxθ ∂U dζ = −g + − ∂t dx ρ (d + ζ ) ρ (d + ζ )

(4.58)

after a long time period, supposing wind action remains constant, flux becomes stationary because of the presence of the coast, thus a dynamical equilibrium is established between wind stress which is balanced by pressure gradient coastward and by bottom friction. Bottom stress is not expressible in terms of flux U (being it null), so it is useful to introduce a scale parameter n defined as:

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INTRODUCTION TO COASTAL DYNAMICS AND SHORELINE PROTECTION

nτ WX = τ Sxθ − τ bxθ

(4.59)

nτ WX = τ Sxθ − τ bxθ

(4.60)

that is:

τbxθ is negative as it is oriented in the opposite direction with respect to x, so -τbxθ is positive and n parameter is always greater that 1. Typical values of n vary between 1.15 and 1.30 (CERC, 1984); in the following it is set to the value 1.20. Equation (4.58) becomes: nτ wx dξ = dx ρg (d + ζ )

(4.61)

Wind stress is given by the following formula: τ S = CD ρ a Vw VW

(4.62)

Vw is wind speed measured in a fixed height (generally 10m);

ρa is air density

CD is the drag coefficient caused by wind action on free sea surface and it’s normally calculated at 10m above sea-mean-level (thus called C10); Equation (4.62) can be written as: τ S = C10 ρ a V10 V10

and the air density being ρa =0.001

(4.63)

kgs 2 m4

For a strong wind condition, the coefficient may be set to: nτ Wx = 1.2 × 0.001 × 2.5 × 10− 3 = 3 × 10− 6 C102

(4.64)

To calculate wind setup, let us consider a constant wind, being l continental shelf length and in this first case suppose a constant bottom depth d0

(d 0 + ζ ) dζ

dx

=

nτ Wx ρg

(4.65)

as d0 is not a function of x: 2 nτ 1 d (d 0 + ζ ) = Wx 2 dx ρg

(4.66)

by solving:

(d 0 + ζ )2 = 2nτ Wx x + C ρg

(4.67)

SEA LEVEL VARIABILITY

83

to determine integration constant C let us consider a null mean sea level variation in x=0 (at the beginning of continental shelf) located far enough from the coast.

(d 0 + ζ )2 = 2nτ Wx x + d 02

(4.68)

(d 0 + ζ )2 = 2nτ Wx x + d 02

(4.69)

ρg

and so:

ρg

ζ ( x) d0

 2nτ Wx xl   − 1 = 1 + 2 Ax − 1 = + 1 + 2  l ρgd 0 l  

(4.70)

with A parameter given by: A=

nτ Wx ρgd 02

(4.71)

With the purpose of determining wind setup on a variable bottom with constant slope, that is for linearly variable depth: d = d0 ( 1 −

x l

)

(4.72)

equation (4.61) can be written as

(d + ζ ) d (d + ζ ) − (d + ζ ) dd dx

where

dx

=

nτ WX ρg

(4.73)

d dd = − 0 = constant dx l

By dividing the two variables: −

(h + ζ )d (d + ζ ) = dx

d 02 l

  d +ζ  − A    d0

(4.74)

by solving:  d +ζ x + C = l  A − d0 

and by determining C as before:

   d +ζ  − A ln − A     d0

(4.75)

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INTRODUCTION TO COASTAL DYNAMICS AND SHORELINE PROTECTION

  d +ζ x = l 1 −  d0  

 d +ζ  − A    d   − A ln 0   1 − A      

(4.76)

or in non-dimensional form  x  d + ζ = 1 − l  d0  

 d +ζ  − A    d   − A ln 0   1 A −     

(4.77)

Equations (4.70) and (4.77) solutions are reported in Figure 4.5 to show slope bottom effects on mean-sea-level variation. It is clear that for the same distance from the coast and same depth of continental shelf limit, a variable depth bottom produces a wind setup greater than the one pertaining to a constant depth. This characteristic may be explained by considering that mean-sea-level variation is increased approaching the coast and with bottom depth decreasing. Figure 4.6 shows the value of mean-sea-level variation for x/l=1 and h=0, that is on the coast, as a function of A parameter. This figure is used in this study because, sea-level rise is determined on the coast line by calculating parameter A. In order to obtain a value of free sea surface level in a site not coincident with the coast line, it is preferable to solve iteratively equation (4.77).

ζ0 d0

Figure 4.5 –Storm surge elevation versus shoreward distance for different wind shear stress and sloping bottoms.

SEA LEVEL VARIABILITY

85

(d + ζ ) d0

Figure 4.6 - Storm tide for x/l =1.0 for a sloping shelf. For the case of no Coriolis force, the ordinate is equal to ζ/d0, the storm surge at the coast, as d=0 at x/l=1.

6

Case study Example 1 Calculate wind setup caused by an atmospheric disturbance. Reference parameters values: c10 = 2.5.×10-3; ρair = 0.001; V10 = 29 m/s; l = 4000 m; =10m;

d0

The calculation of wind setup can be dealt with by using formula (4.71) which allows to calculate parameter A. With the help of Figure 4.5 and Figure 4.6 mean-sea-level variation can be estimated. By applying (4.71): A=

(3 × 10 × (29) × 4000) ≅ 0.01 −6

2

9.8 × (10 )

2

where A=0.01 and by supposing that the bottom between the coast and off-shore zone is smooth slope decreasing, from table in Figure 4.5, on coast line, that is x/l=1, one obtains that:

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INTRODUCTION TO COASTAL DYNAMICS AND SHORELINE PROTECTION

ζ d0

= 0.03 ⇒ ζ = 0.03 × 10 ⇒ ζ = 0.3meters

mean-sea-level set-up in the in-shore, in calm conditions, is 30 cm. Example 2 Calculate wind setup across the surf zone (beach slope tanβ = 0.01) for a normally incident wave with breaker height Hb=2.7 m. The breaking index is γb=0.84. Reference parameters values: Acceleration due to gravity g = 9.81 m/s2 The breaking depth db is obtained by equation (4.35): db=Hb/ γb =2.7/0.84 = 3.2 m Set-down at the breaking is given by equation (4.41): ζb =−

1 γ b H b =1/16(0.842)·3.2= -0.14 m 16

The maximum set-up is given by equation (4.44): ζ max = ζ b +

3γ b2 3 ⋅ 0.84 2 d b = −0.14 + 3 .2 = 0 .7 m 8 8

Chapter 5 Random wave measurement and analysis 1

Wave measurements

Information on sea state severity can be obtained by wind hindcasting or forecasting, by visual estimation and by direct measurement of sea surface elevation through single point or remote sensing measurements. Generally wave measurement concerns only a few quantities such as height, period and direction, while other characteristics like wavelength and wave celerity are calculated by the previous data. In field wave experiments, three groups of instruments are used: instruments which need a reference height (wave staffs, acoustic and pressure sensors) and floating instruments, like wave buoys and remote sensing instruments. • Wave staffs can be resistive or capacitive, while acoustic sensors directly measure vertical displacement of free surface elevation. Pressure sensors indirectly measure the vertical displacement through a pressure measurement. Acoustic and pressure sensors usually adopt the bottom as the reference level, wave staffs generally refer to the height of a platform or pier. • Wave buoys are floating instruments which measure vertical acceleration by which vertical displacement is obtained through a double time integration. They are used in deeper water, where the supporting structures for wave staffs are not available and the water depth is too high to use a pressure gauge. • Conventional buoy systems are difficult to run and rather expensive in deep oceans, which comprise the vast majority of the globe. Satellite remote sensing systems have the capability of imaging wave conditions day and night through all weather conditions. 1.1 Ultrasonic and pressure gauges Wave gauges can be mounted on platforms offshore or on the sea floor in shallow water. Many different types of sensors are used to measure the surface elevation or wave pressure which is related to the former. Sound, infrared beams,

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and radio waves can be used to measure the distance from the sensor to the sea surface provided the sensor can be mounted on a stable platform that does not interfere with the waves. Bottom-mounted pressure gauges measure water level changes by sensing pressure variations with the passage of each wave. The gauges are either selfrecording or are connected to onshore recording device by cables. The pressure distribution below a gravity wave is given by: p = − ρgz + ρgηk p (z )

(5.1)

where: ρ is the sea water density; z is the vertical distance (positive upward) with the origin at the mean sea level; η is the surface elevation referred to the meansea level; kp(z) is the pressure response factor, given by: k p ( z) =

cosh(d + z ) ≤1 cosh(kd )

(5.2)

The first term in eqn. (5.1) ps=-ρgz represents the hydrostatic pressure which would occur in absence of wave motion, and the second term p=ρgηkp(z) is the dynamic pressure induced by wave motion. The pressure sensor, mounted on the bottom, is beforehand calibrated so as to record only dynamic pressure pd. In this case, vertical displacement is given by:

η=

pd ρgk p (− d )

(5.3)

Being: k p (− d ) =

1 cosh(kd )

(5.4)

kp(−d) is a function through k of the angular frequency σ by means of the dispersion relation. Short-period waves present very small values of kp(−d), whereas long waves show values of kp(−d) close to unity; this implies that very short waves are filtered by the pressure sensor. The depth range of pressure sensors goes from 10m to 20m. Lower depths may produce the breaking of the higher waves, which therefore are not correctly measured, while higher depths involve an attenuation of the response for short waves which becomes unacceptable. Pressure gauges can also be coupled to electromagnetic current meters, in order to obtain directional measurements of wave motion. 1.2 Wave buoys In deep water, where acoustic sensors and pressure gauges cannot be used, wave buoys are employed. They have generally a spherical or discus shape, with a

RANDOM WAVE MEASUREMENT AND ANALYSIS

89

diameter from 0.5 to 2.0 meters and weight from some tenth of kilos to some tons. The vertical acceleration is integrated twice with respect to time, to produce a measure of the instantaneous surface profile. In order to avoid measurement of unwanted accelerations due to roll and pitch of buoy, the sensitive axis of the accelerometer is mounted on a stabilized platform. Once the vertical acceleration measure is obtained, the buoy transmits the signal to a remote station for further analysis. The most common type of remote transmission is via radio; it is cheaper but requires, for a good reception, that the buoy is set within 30 miles; the second is via satellite but, due to the limitation of data exchange, it requires a microprocessor on board which allows the calculation of wave spectra and of selected parameters, so that the amount of transmitted data is smaller than that of recorded data. Thus the disadvantage of satellite transmission is that raw data are not generally available for further analysis. The transfer function between acceleration and displacement should be constant. For the most popular wave rider buoy manufactured by Datawell, the transfer function shows that the buoy response is essentially constant for wave periods between 5 and 10 s, while some attenuation is observed for periods between 10 and 25 s. A comprehensive comparison of various types of wave buoy is given by Allender et al. (1989). One of the most used buoys, the Datawell directional Waverider, has the following specifications: Hull Weight Range of wave height Resolution Accuracy Range of wave period Directional range Resolution Buoy heading accuracy Power supply

0.9m diameter 260kg ±20m 0.01m 3% FS 1.6-20s (when moored) 0-360° 1.5° 0.5-2° 25 VDC (internal batteries, 12 month capacity)

During the WADIC Project (Wave Direction Measurement Calibration Project) which was held at or in the vicinity of the Edda platform in the Ekofisk field in the North Sea during winter 1985-1986, several wave buoys, platforms and wave staffs were intercalibrated. The authors compared the performance of a particular buoy against the best estimated data set obtained from the platform based measurements. Although differences in accuracy have been found, comparisons for the most important engineering parameters, significant wave height, mean wave period and wave direction at the spectral peak are satisfactory. 1.3 Italian Sea WAve measurement Network In 1989 the Italian Ministry of Public Works started up the national Sea WAve measurement Network (SWAN). Since the beginning of 1996 SWAN is managed by the National Hydrological and Marine Survey (NHMS) which

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provides measurements and analysis of wave data systematically, water levels and other meteo-oceanographical parameters recorded in the Italian seas. SWAN was originally formed by 8 Datawell WAVEC buoys placed offshore La Spezia, Alghero, Ortona, Ponza, Monopoli, Crotone, Catania and Mazara del Vallo (see map in figure 5.1). Between 1999 and 2001 the RON was significantly upgraded with respect to instruments, management and data recording. The most significant change since March 1999 was the addition of two translation type WAVERIDER Datawell directional buoys at Cetraro and Ancona, in areas that were not covered by the first eight stations. In 2002 the network was completely restructured and upgraded. Four new stations were installed at Punta della Maestra near Chioggia, Civitavecchia, Capo Comino and Palermo, and the whole network was planned to work in real-time. Data were measured every thirty minutes and sent by radio link to local onshore stations, where they were processed and classified (APAT). Fourteen Triaxis (by Canadian Axis) translation type remote-controlled buoys are currently employed. Their main features are: • Spherical shape with diameter 0.9 m (weight = 200 Kg); • Solar panel and buffer battery power system; • Solid sensors; • Data sampling frequency of 4 Hz; • Onboard spectral and zero-crossing data analysis; • Radio link data transmission with assigned frequency of 44.8 MHz; • Onshore transmission of wave height and direction raw data time series, resampled as ∆t=0.78125 s; • Satellite tracing of buoy position by GPS and Inmarsat D+. These features allow more accurate analyses of wave parameters in Italian seas and minimize permanent data loss risks. The network operativity in real-time has enabled the realisation of automatic consultation services both on Internet, on RAI Televideo and transmission of measured data to the main meteorological centres through the World Meteorological Organization Global Telecommunication System. 1.4 Satellite remote sensing There are three standard active microwave instruments of principal interest for surface wave detection; namely, the altimeter, the Synthetic Aperture Radar (SAR) and scatterometer. These instruments can provide global, all weather, day and night data coverage. An overview of all satellites deploying instruments detecting the ocean surface is given by Komen et al., (1994). 1.4.1 Radar altimeter The most recent Radar altimeters missions include: • •

GEOSAT: November 1986 – January 1990 TOPEX: September 1992 – present

RANDOM WAVE MEASUREMENT AND ANALYSIS

• •

91

ERS1: August 1991 – present ERS2: May 1995 – present

GEOSAT transmitted 1000 pulses per second towards the Earth’s surface. The returned energy from these pulses was averaged to produce data values at a frequency of 1 per second. The ground speed of the satellite was approximately 6.5 km/sec and hence one data value was obtained every 6.5 km along the ground track. The pulses radiate away from the satellite altimeter antenna as a spherical shell. The pulse reflects from the water surface and the return pulse is sampled by the satellite. Rather than the return pulse being continuously sampled, it is measured in discrete intervals called “range gates”. Hence it is convenient to consider the pulse during each time increment corresponding to each range gate (Dobson and Monaldo, 1996). During the first range gate, the pulse illuminates a small circular region directly below the satellite. In subsequent time increments, the surface is illuminated by a series of expanding annular rings. The area of each of these annular rings is constant. The orbital characteristics depend on the use to which the data are to be put. For wave climate purposes, exact repeat orbits are the most valuable with the satellite returning to exactly the same ground track with time intervals of the order of 1 week to 1 month. During the exact repeat period the satellite’s ground track forms a web over the global oceans limited north and south by the satellite’s inclination. If the ocean surface was perfectly flat, the return pulse would exhibit a rapid rise, corresponding to the return from the inner circular region. This would be followed by a flat section resulting from each of the constant area annular rings. If the ocean surface is roughened by the presence of waves, the leading edge of the transmitted pulse will interact with the crests of the waves a small time before interacting with the troughs. As a result, the leading edge of the return will be broadened in comparison to the flat ocean case. As the wave height increases, this broadening of the leading edge of the return pulse increases. Therefore the slope of the leading edge of the return pulse can be used as a measure of the wave height.

Figure 5.1 - Location of the Italian directional wave buoy network (Corsini S. et al, 2002).

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INTRODUCTION TO COASTAL DYNAMICS AND SHORELINE PROTECTION

3 1

2

Figure 5.2 - TRIAXYS (1) WAVEC (2) and WAVERIDER (3) buoys. 1.4.2 Synthetic Aperture Radars Synthetic aperture radar (SAR) is basically a method of ground mapping that uses computer processing to make radar work better. It first appeared in the early 1950s but did not reach a high state of development for almost 30 more years with the introduction of digital processing and other advances. SAR yields high-resolution, photograph-quality images because it combines radar images made many miles apart. SAR uses antennas positioned on moving bodies, such as satellites, and then mathematically combines the separate signals transmitted as the antenna moves, simulating or mimicking the transmission of radar from a source with a larger “aperture,” or a larger opening. For instance, a European Space Agency satellite with a 10-meter SAR antenna mimics the performance of a 4-kilometer-long stationary antenna. The motion of the satellite, combined with the wide beam of the radar, covers a swath along the ground, allowing a large area to be searched quickly typically 256 square kilometers. If a few dozen of these swaths are collected, an area of several thousand square kilometers will be covered. SAR is currently the only satellite-borne instrument which can measure directional characteristics of the ocean wave field independent of the weather conditions. However, from the wave analysis point of view, the actual image of the surface is not used directly, and a SAR wave analysis starts by a spectral analysis of the image. Although sometimes denoted the wave spectrum, the SAR image spectrum has turned out very far from being so. A rather complicated

RANDOM WAVE MEASUREMENT AND ANALYSIS

93

post-processing of the SAR image spectrum is therefore necessary for extracting quantitative wave information. The core of the methodology is the Hasselman nonlinear ocean-SAR spectral transform developed power the last few years (Hasselmann et al., 1991).

Figure 5.3 - SAR: example of ground tracks. Even if the methodology for extracting wave spectra from SAR data is available, there is still some way to go before this is becoming routine. The modulation mechanisms are incompletely understood and appear to be strongly dependent on centimeter waves generated by the local wind. The current resolution of space borne SAR also limits the observations to waves longer than about 40m. The SAR image spectrum is from the outset symmetric and thus suffers from a 180 degrees ambiguity in direction. This is the main reason for introducing the a priori spectrum in the inversion functional. However, in most cases, for example in coastal areas, the choice between the two possible wave directions is obvious.

2 Statistical properties of random waves 2.1 Data sampling The wave recorder will provide a continuous record of the surface elevation η(t) and should be used over a reasonably long period (a few years) so as to obtain data representative of the various conditions occurring. In order to avoid the

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INTRODUCTION TO COASTAL DYNAMICS AND SHORELINE PROTECTION

accumulation of too much data, waves are recorded intermittently as indicated in figure 5.4. The recording interval is the time that lapses between successive starts of the recording instrument and could, for example, be 3 or 6 hours. The recording period is the duration of a single continuous recording and may be say 10 or 20 minutes. The sampling interval between two digitized ∆t is included between 1/10 and 1/20 of the significant period, generally goes from 0.25 to 1.0 second, more often equal to 0.39 or 0.5 seconds. Once ∆t is chosen, the highest frequency at which the spectrum can be estimated is automatically determined by: fc =

1 2∆t

(5.5)

denoted as Nyquist frequency. Wave component with this frequency is sampled with two point per wave, and the energy contained in frequencies higher than fc is added to the constant energy in the range of 0
η

recorder on

recorder on

t recording period

recording period

recording interval

Figure 5.4 - Recording interval and recording period for a random signal η(t).

RANDOM WAVE MEASUREMENT AND ANALYSIS

2.2

95

Data processing

Data processing is preceded by a preliminary filtering in order to identify and eliminate instrumental errors; the most frequent is the comparison of one or more peaks (spikes) occurring due to interference of radio signal. Starting from the vertical displacement data, the characteristics of the significant wave are determined through two different ways: the first (time domain) passes through the zero crossing method (up- or down-crossing) in order to obtain wave heights and periods of random waves, and then the characteristic waves (e.g. significant wave); the second (frequency domain) requires the determination of spectral energy density of vertical displacements; from which spectral moments are obtained which allow the estimation of a spectral significant wave height and a significant wave period (see figure 5.5).

Automatic vertical displacements

Quality control

Zero up-crossing analysis Random waves

{H i } {Ti }

Spectral analysis

Energy Spectrum

S(f) Spectral moments

Characteristic waves

{H1 / c }

{T } H 1/ c

∞

m0 = ∫ S ( f )df 0

Figure 5.5 - Diagram for the estimation of a spectral significant wave height and significant wave period. 2.2.1 Time domain analysis The random wave is defined as that part of registration included between two subsequent crossings of the zero line (i.e. MSL) with a positive derivative of surface elevation η(t) (zero up-crossing method) or negative derivative (zero down-crossing method). The single wave is characterized by a waveheight

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INTRODUCTION TO COASTAL DYNAMICS AND SHORELINE PROTECTION

intended as the difference of the maximum and minimum values recorded, and a period defined as the difference of temporal coordinates of two zero crossing. Zero up-crossing height of random waves is defined as the vertical distance between the two points of maximum and minimum height (that) surface elevation assumes between two subsequent SML crosses. H u = η max − η min

(5.6)

whereas zero up-crossing wave period Tu is the time interval between the mentioned subsequent crossings. To obtain the significant wave, which by definition is a regular wave derived from a third of highest waves, heights of n random waves are sorted in increasing order yielding a continuous succession {Hi} such that i=1,n (where i=1 represents the minimum wave, i=n the highest wave). Significant wave height Hs comprises the n/3 highest waves Hj where j=(1-1/3)+1. Height and period of this significant wave are respectively given by:

Hs =

3 n ∑ Hi n i= j

Ts =

3 n ∑TH n i= j i

(5.7)

Random waves are arranged ascending order with relative periods so that the first wave is the highest of the wave record. On the total of m waves of a record, the ratio c=m/n and the mean height of the c highest waves H1/n, and relative periods T1/n, is calculated; usual values of n are 3,10,20 and in particular H1/3 and T1/3 are known as significant height and period. In the hypothesis that random waves are characterized by a narrow band spectrum, Longuet-Higgins (1952) proposed the Rayleigh distribution for the significant wave height Hs, which probability distribution and cumulative probability are respectively expressed by: p( H ) =

 − H 2  exp  2  4σ η2  8σ η  H

 H  P( H ) = 1 − exp − 2   8σ η 

(5.8)

(5.9)

~ Introducing the non dimensional variable H = H / σ η and reminding the relation ~ ~ p ( H ) dH = p ( H ) dH , equations (5.8) and (5.9) become: ~ ~  H2 H ~ p( H ) = exp−  4  8  ~  H2 ~ P ( H ) = 1 − exp −   8 

(5.10) (5.11)

RANDOM WAVE MEASUREMENT AND ANALYSIS

97

The Forristall (1978) distribution, is given by: ~ ~α   H ~ H (α −1)  p( H ) = α exp− (5.12)  β  β  ~α ~  H  p ( H ) = 1 − exp − (5.13)   β  where α=2.126 and β=8.42. The following quantities can be obtained through the time domain analysis:

• • • • •

Hm = mean height [m] obtained from the average of n zero up crossing waves Hs = significant wave height [m] Ts = significant wave period [m] Hmax = maximum wave height [m] obtained as maximum value of height of a record, after a crescent sorting of wave heights. Tmax = period of the maximum wave [s]

Having significant example, H1/ 3 = H S

adopted a theoretical distribution, it is possible to estimate the wave height for the surface displacement characteristics. For using Rayleigh distribution, Longuet–Higgins obtained = 4σ . H 1 / 10 = 2.03H

H 1 / 10 = 1.27 H 1 / 3

H 1 / 3 = 1.60 H

(5.14)

Values obtained by field data respectively give the following results: H 1 / 10 = 1.26 H 1 / 3

H 1 / 10 = 2.00 H

H 1 / 3 = 1.58 H

(5.15)

in perfect agreement with (5.8). Analogous to heights, characteristic wave periods are mutually correlated. Goda (1985) suggests the following ranges of variability. TH max = (0.6 ÷ 1.3)TH 1 / 3

(5.16a)

TH1 / 10 = (0.9 ÷ 1.1)TH1 / 3

(5.16b)

TH1 / 3 = (0.9 ÷ 1.4)TH1 / 1

(5.16c)

The average value of these variations may be given as: TH max ≅ TH 1 / 10 ≅ TH 1 / 3 ≅ (1.1 − 1.3)TH

(5.17)

from the analysis of University Parthenope field data we obtained: TH1 / 10 = 1.02 TH1 / 3 = 1.18 TH1 / 1

(5.18)

For what concerns the maximum wave height recorded, it is a random variable, because different records which show the same value of H1/3 may show different

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values of Hmax. Nevertheless it is possible to define a probability density for the ratio Hmax/H1/3 which depends on the number N of wave records. The distribution mode, that is the most probable value, is given by: ( H max / H1/ 3 ) mode ≅ 0.706 ln N

(5.19)

Thus the prediction of the maximum wave height in a wave record falls approximately in the range: H max = (1.6 ÷ 2.0) H 1/3

(5.20)

For example, Goda assumes Hmax=1.8 H1/3 as design wave height. In the hypothesis that the temporal series η(t) is stationary, information on the characteristics of sea state may be obtained in frequency domain by a spectral analysis, that is by applying Fourier transform to the autocorrelation function. The auto-correlation function of vertical displacement correlates the value of η at time t to its value at time t+τ and therefore gives an information of the correlation of the signal with itself for different temporal shifts τ. 1 R (τ ) =< η (t ) ⋅η (t + τ ) >= lim  T →∞  T 

T /2



−T / 2



∫ η (t ) ⋅η (t + τ )dt 

(5.21)

In practice the period T used to obtain the temporal mean is sufficiently large to ensure that its value does not influence the estimation of R(τ). The autocorrelation function R(τ) expressed in nondimensional form yields to the autocorrelation coefficient Ψ(τ) given by: Ψ (τ ) =

R(τ )

σ η2

=

< η (t ) ⋅ η (t + τ ) >

σ η2

(5.22)

Obviously the signal is perfectly correlated with itself for a null temporal shift, whereas it is completely uncorrelated with an infinite time shift. τ = 0 ⇒ Ψ (τ ) = 1 τ = ∞ ⇒ Ψ (τ ) = 0

(5.23)

The autocorrelation function is directly connected to the spectral energy density because R(τ) and S(f) are given by the Fourier transformation, that is ∞

S ( f ) = ∫ 4 R(τ ) cos(2πfτ )dτ

(5.24)

0

∞

R (τ ) = ∫ S ( f ) cos(2πfτ )df

(5.25)

0

Spectral moments are given by the following relation: ∞

m n = ( 2π ) n ∫ f n S ( f )df 0

(n=0,1,2,…)

(5.26)

RANDOM WAVE MEASUREMENT AND ANALYSIS

99

In particular, moment of zero order, equal to the variance of the recorded data, is given by: ∞

m0 = ∫ S ( f )df

(5.27)

0

For each spectral moment, a corresponding frequency exists: fn =

1 2π

 mn  m  0

   

(n=0,1,2,….)

(5.28)

In particular f2=(m2/m0)1/2 is the zero up-crossing frequency of the stationary time series {η(t)}, and the relative zero up-crossing period (mean period) is given by Tz=T0,2=1/f2. Considering a Rayleigh distribution for random wave heights, the significant wave height obtained by a spectral analysis is equal to Hm0=4m01/2, while the root mean square wave height Hrms results equal to H rms = 2 2 m0 . From spectral analysis, also peak frequency fp may be obtained, that is the frequency corresponding to the maximum value of energy spectral density, which inverse is the peak period Tp. Through a spectral analysis the following quantities are obtained: • • • • •

Hrms Hm0 Tz Tp Dm

= root mean square height [m] = significant height [m] = mean period [s] = peak period [s] = mean direction [°N]

The theoretical spectrum most used in practical applications is the JONSWAP spectrum (Hasselmann et al., 1973), given by (see § 2.2.2):

S J = αg 2 2π −4 f

−5

{

}

exp − 1.25( f / f p−4 ) γ

[

exp 0.5( f / f p −1) 2 / ω 2

]

(5.29)

For γ=1, The spectrum is that of Pierson-Moskowitz, from which JONSWAP spectrum is derived, the mean value of γ obtained in North Sea is 3.3, the mean value obtained in Italian coasts is about 2.0. 2.2.2 Directional wave spectra

In time domain directional spectra are treated using the same method as non directional ones while for the definition of spectral forms generally used is the auxiliary function G(j∆F,θ) which models the spectral shape defined above: S ( j∆F ,θ ) = S ( j∆F )G ( j∆F ,θ )

(5.30)

where G satisfies the relation: π

∫ G( j∆F , θ )dθ = 1

−π

(5.31)

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allows to make the present definition of wave spectrum, coherent with the nondirectional ones and to develop a directional analysis for each frequency value. The function G(j∆F,θ) is given as discrete directions in some cases (Endeco) and as mean value, directional spreading, asymmetry coefficients and exceed parameters in other (Datawell). In the time domain directional spectra may be obtained considering that the quantity S(f,θ)dfdθ represents the contribution to the variance ση2 due to waves with frequency between f and f+df and direction between θ and θ+dθ, that is: π

∞

σ η2 = ∫

∫ S ( f , θ )dfdθ

(5.32)

0 −π

Directional spectra are often proposed in the form of a product of a monodimensional spectrum S(f) and a function of directional dispersion called “directional spreading function” D(f,θ): S ( f , θ ) = S ( f ) D( f , θ )

(5.33)

Several semi empirical expressions have been proposed for D(f,θ). The first exemplification is that to consider D(f,θ)=D(θ) independent of frequency. Longuet-Higgins, Cartwright (1961) proposed a power cosine form: 1  D(θ ) = C ( s ) cos 2 s  (θ − θ ) 2  

(5.34)

being θ the propagation direction of the whole wave motion sea state, and: C ( s) =

Γ( s + 1) Γ 2 π ( s + 0.5) 1

(5.35)

The value of s describes the degree of dispersion around the direction θ , a high value of s corresponds to a more concentrated spectrum around θ . In the particular case of random waves propagating only in direction θ , D(θ) tends to the Dirac function. 2.2.3 Pierson – Moskowitz and Jonswap spectrum

The Pierson – Moskowitz (1974) is given by: 2

S ( f ) = αg (2π )

−4

f

−5

−4   f   5   exp −   4  fp      

(5.36)

where α=8.1×10-3 (Phillips constant). This spectrum depends only on the wind speed U and so refers to fully developed conditions. The JONSWAP spectrum constitutes a modification of the Pierson - Moskowitz spectrum to account for the effect of fetch restrictions and provides a much more sharply peaked spectrum. It is given by:

RANDOM WAVE MEASUREMENT AND ANALYSIS

S( f ) =



−4 

5  f   a αg 2 γ exp −  5 4  4  fp   f (2π )   

(

σ a = 0.07 σ = σ b = 0.09

(5.37)



)2 / 2σ 2 f p2 ]

a = exp[− f − f p

101

for

f ≤ fp

for

f > fp

(5.38) (5.39)

Here fp is the peak frequency at which S(f) is a maximum and is found to be related to the fetch parameter by fp=2.84(gF/U2)-0.33; σa and σb relate respectively to the widths of the left and right sides of the spectral peak; α is equivalent to the Phillips’ constant but is now taken to depend on the fetch parameter α=0.066(gF/U2)-0.22; and finally γ is the ratio of the maximum spectral density to that of the corresponding Pierson Moskowitz spectrum. 2.00 fp

α=(0.65 f p )

∫ (2π )

4

g 2 f 5 S ( f ) df

(5.40)

1.35 fp

The peak enhancement parameter γ can be calculated as the ratio between the maximum spectral energy and the maximum energy of the Pierson-Moskowitz spectrum with the same value of α:

γ = S(fp)(2π)4 fp -5exp(5/4)(αg2)-1

(5.41)

where α is obtained from equation (5.50). The nondimensional energy, peak frequency and fetch are expressed by the following equations:

ε = σ2g2/ Ua4

(5.42)

ν = S(fp) Ua/g

(5.43)

χ = g x/ Ua

(5.44)

2

are related with each other the following regression lines (Hasselmann et al.,1973):

ε =7.13×10 –6 ν -3.33 ν = 3.5χ -0.33

(5.45) (5.46)

A relevant number of regression lines was obtained by other studies, all of them valid for fetch limited conditions: Kahma (1981) obtained a dependence of ν with χ comparable to the JONSWAP one but with growth rates for ε greater than JONSWAP of a factor about 2. So Kahma achieved the following growth curves:

ε = 3.60×10 -7χ

(5.47)

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INTRODUCTION TO COASTAL DYNAMICS AND SHORELINE PROTECTION

ν = 3.18χ - 0.33

(5.48)

Benassai (1994) obtained the following relationships between the spectral parameters:

ε =2.71×10 -6 ν -3.76 α = 0.027ν

0.51

ν = 2.6χ -0.29 ε = 4.1×10 χ -8

(5.49) (5.70) (5.71)

1.16

α = 0.044χ - 0.15

(5.72) (5.73)

3 Statistical representation of wave climate Wave climate indicates the statistical characteristics of synthetic wave parameters recorded for a number of years averaged on a yearly basis (but also on a monthly or seasonal basis). Reference is normally made to their occurrence with reference to a single or a couple of wave parameters. Below, the attention is focused on two different probabilistic distribution: • •

joint-occurrence frequency of wave height/direction mean exceedance or persistence of significant wave height above assigned thresholds.

The first information is very useful in the analysis of coastal morphodynamics and in coastal protection as well as in harbour design, while the second one gives useful indications for maritime and navigation purposes. Occurrence frequencies can easily be calculated by dividing the value interval of all measurements taken into different classes. The frequency, for each class, is the ratio between the number of events falling within the considered class and the total number of data. Similarly, exceedance frequencies can easily be calculated by dividing the interval of all measured values into different classes. The frequency, for each class, is the ratio between the number of events above the minimum threshold and the total number of measured events. In order to calculate the joint-occurrence frequency of wave height/direction, sea states with significant wave heights below 0.50 m may be defined as calm and may not considered further. The interval of measured wave heights is divided into classes, while each direction sector covers an angle of 10°, 22.5° or 30°. The information is graphically represented with polar diagrams, similar to those also used for winds. They enable an immediate visual identification of the most frequent wave direction (prevailing seas) and the direction of sea states with the most frequent highest waves (dominant sea). Their eventual coincidence marks a predominant sea. In order to calculate the wave height over threshold persistence, the interval of measured values can be divided into classes. The information is graphically

RANDOM WAVE MEASUREMENT AND ANALYSIS

103

represented with histograms which allow an immediate calculation of the mean annual duration of sea states above a given wave height. Finally, the joint-occurrence of wave/direction can be calculated also on a quarterly/seasonal basis as follows: Winter = January, February, March; Spring = April, May, June; Summer = July, August, September; Autumn = October, November, December. This information, represented with frequency polar diagrams, may be useful to design temporary installation or sea activities (e.g. summer navigation lines, seasonal moorings, etc.). The main elements of wave climate are parameters relative to wave height, period and wave direction. Generally one refers to significant wave and significant period obtained by the analysis in time domain and to the peak period obtained by spectral analysis, and mean direction. Data relative to wave climate must cover a time period long enough, one year at least in order to have a sufficiently reliable description, ten years minimum if it is necessary to abstract information about extreme waves. In fact Goda showed that mean annual value of significant wave height obtained in a single year may deviate by 15% with respect to the value obtained on long time intervals. In any case, the representation of wave climate in a determined site may be performed in different ways, by reporting for example: • Mean annual, seasonal or monthly value, of significant wave height, complete or not with standard deviation and with the maximum value assumed in the considered time interval. • Polar diagram of appearance frequency of significant wave height that is histogram of significant wave height which fall in assigned provenience directions. • Frequency tables of the joint distribution significant wave height- mean period (or significant period) distinguished or not basing on a assigned directional class (the first hypothesis is preferable). The results of the analysis for the Italian Seas have outlined the existence of two different groups of directional wave climate, which can be related to different geographical areas of the Italian Peninsula. Namely, the Western group (Alghero, La Spezia, Ponza and Mazara) and the Eastern group (Pescara, Monopoli, Crotone and Catania). The first cluster has typically an unimodal annual direction climates, characterized by high waves mainly originating from the West, while the second one, instead, often exhibits a bimodal wave climate tending towards a North-South axis. Furthermore, the frequencies of waves coming from the western sectors are low in the Western group and High in the Eastern one. Mean seasonal climate shows that stations in the Western group have a lower seasonal variability with respect to stations in the Eastern ones. Finally, the persistence of significant wave heights over threshold shows that Western seas are more stormy than the Eastern ones.

4 Case study Calculate H1/3, H1/10, T1/3 and T1/10 from table 5.1 , the distribution mode and compare the values reported in eqn. (5.20a) and (5.20b) with empirical values of Hmax / H1/3 and H1/10 / H1/3

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Figure 5.6 –Sea level variation recordings. The values of Hu and Tu from figure 5.6, are reported in table 5.1. The following values of H1/3, H1/10, T1/3 and T1/10 can be obtained from table 5.1: H1/3=(1.94+1.92+1.71+1.56+1.55+1.52+1.50+1.49+1.40+1.31)/10=1.59 m H1/10=(1.94+1.92+1.71)/3=1.86 m T1/3-=(5.990+8.955+6.240+7.633+7.727+7.054+8.190+8.975+6.408+7.987)/10 =7.515 s T 1/10 =(5.990+8.955+6.240)/3=7.0617 s Hmax / H1/3 = 1.2

H1/10 / H1/3 = 1.16

According to the theoretical expressions (5.20) Hmax/H1/3 = 1.6÷2.0 m According the equation (5.19) the distribution mode is given by: ( H max / H1 / 3 ) mode ≅ 0.706 ln N =0.706× [ln(31)]1/2=1.308

RANDOM WAVE MEASUREMENT AND ANALYSIS

Table 5.1 Hu

Tu

1.920 1.550 0.690 1.400 0.650 1.060 1.310 1.710 1.940 0.970 0.240 0.360 0.650 0.910 0.810 1.520 0.760 0.100 0.340 0.550 0.510 1.560 0.340 0.180 1.160 0.920 1.490 1.500 0.850 0.720 0.240

8.955 7.727 6.310 6.408 4.425 3.276 7.987 6.240 5.990 4.856 2.284 2.750 4.088 3.882 4.476 7.054 3.511 1.011 3.747 3.006 2.867 7.633 3.042 4.657 6.157 7.126 8.975 8.190 7.702 4.143 2.795

Wave Number 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31

Order Number 30 27 12 23 11 20 22 29 31 19 4 7 10 17 15 26 14 1 5 9 8 28 6 2 21 18 24 25 16 13 3

105

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Chapter 6 Short term wave prediction 1

Introduction

The study of ocean wave forecasting dates back centuries. During the second world war, Sverdrup and Munk (1947) introduced a forecasting method to predict the so called ‘significant wave’, defined as ‘the wave which describes the sea state’ for the invasion of Normandy in 1945, which was later published. They proposed a parametrical description of the sea state and used empirical wind and swell laws. The method was later improved by Bretschneider, 1958 and was called SMB (1958). An important advance in the sea state prediction was in the fifties, with the introduction of the wave energy spectrum, based on the assumption that the sea surface can be represented as a Fourier series of superimposed waves with different wave lengths and with statistically random phases. Each of the Fourier modes was associated to the mean wave energy of the spectrum. The first dynamical equation describing the evolution of spectrum was given by Gelci et al. (1957) with the formulation of the spectral balance equation. The lack of adequate theories forced Gelci to use empirical expressions to define the spectral evolution. The Phillips (1957) and Miles (1957) studies introduced a deeper insight on wave generation. An experimental milestone was the JONSWAP field study in the North Sea (published in 1973) in which the spectral evolution was formulated as a function of fetch length, through a parametrical approach, giving rise to the JONSWAP spectrum, which is still widely used. Another important concept was the introduction of the self-similarity shape, which was later developed for the spectral transfer on shallow water. After Hasselmann et al., (1985) who derived the source function for the nonlinear interactions it was possible to write down a general expression for the source function (Hasselmann et al., 1985), consisting of three terms representing the input from the wind, the nonlinear transfer and the dissipation due to whitecapping. The spectral wave energy balance in its complete form is actually the basis for all numerical wave prediction models.

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The first models, developed in the early 1960s, only took into account the wave energy growth and dissipation. Later these models became known as “first generation” wave models. From their results it became clear that interactions between waves of different frequencies were important in determining the wave spectral evolution with the fetch length. These non-linear interactions were computationally expensive for the time (1970’s), so parameterizations were developed to account for this effect, the most popular of which was the JONSWAP one. Wave prediction models using a parameterization of the nonlinear interactions are known as “second generation”. Further development led to an approximation for the non-linear transfers, which considers the four most important interacting waves at each frequency out of the infinite number of interactions theoretically possible. This approximation is more expensive to compute than the parameterizations of a second-generation model, but it improves significantly the model, because the wave spectrum is not forced to take a particular form for growing wind-sea, which is free to evolve according to the physical equations. Such a wave model is called “third generation”, and now is routinely run by several meteorological offices

2

Elements of wind measurement and analysis

The wind is an air flow caused by pressure gradients between adjacent areas. Horizontal pressure gradients arise in the atmosphere primarily because of density differences, which in turn are generated primarily by temperature differences. Wind results from nature’s effort to eliminate the pressure gradient, but is modified by many other factors. The altitude divides the boundary layer into three sub-layers (Figure 6.1): in the geostrophic layer (up to 1000 m height) the wind has a constant speed and experiences the Coriolis forces to the right (in the northern hemisphere). The pressure gradient is balanced by the Coriolis force and is perpendicular to the wind direction. Below the geostrophic region, the boundary layer may be divided into two sections: a constant stress layer (<100 m) called hydrodynamic layer and above which an Ekman layer (100 to 1000 meters in height). A detailed description of the boundary layer mechanics is given in Resio and Vincent (1977). The pressure gradient is nearly always in approximate equilibrium with the acceleration produced by the rotation of the earth. The geostrophic wind (defined by assuming that exact equilibrium exists) varies proportionally with the pressure gradient and is a function of the Coriolis parameter (f=2Ωsinϕ; Ω=7.29⋅10-5 rad/s); its velocity depends on the air densityρa (Kg/m3), the distance between two adjacent isobars ∆n (in meters) the pressure gradient ∆p (Pa) through the equation: U =

1 ∆p

ρ a f ∆n

(6.1)

SHORT TERM WAVE PREDICTION

109

The equation (6.1) can be explicitated in terms of the latitude ϕ (degrees) as: U =

Z ≅ 1000 m

4.82 ∆p sin(ϕ ) ∆ϕ

(6.2)

Geostrophic layer

Ekman layer

Z ≅ 100 m Constant shear layer

Figure 6.1 - Atmospheric boundary layer over waves. The geostrophic wind blows parallel to the isobars leaving the low pressure to the left, when looking in the direction toward which the wind is blowing, in the Northern hemisphere. Geostrophic wind is usually the best estimate of the true wind in free atmosphere. When the trajectories of air particles are curved, equilibrium wind speed is called gradient wind. Gradient wind is stronger than geostrophic wind for flow around a high pressure area, and weaker than the geostrophic wind for flow around low pressure. The magnitude of the difference between geostrophic and gradient winds is determined by the curvature of the trajectories. If the pressure patterns do not change with time and friction is neglected, trajectories are parallel with the isobars. 2.1 Wind information needed for wave hindcasting Wave hindcasting from meteorological data must be preceeded by the following activities: a. Estimate the mean surface wind speed and direction; b. Delineate a fetch distance over which the wind is reasonably constant in speed and direction, and measure the fetch length; c. Estimate wind duration over the fetch. These calculations can be made in different ways depending on the location and the type of meteorological data available. For restricted bodies of water, like

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INTRODUCTION TO COASTAL DYNAMICS AND SHORELINE PROTECTION

the Mediterranean Sea, the fetch length is the distance from the forecasting point to the opposite shore measured along the wind direction. The most common form of meteorological data used are depicted on synoptic surface charts, which report lines of equal atmospheric pressure called isobars. In Figure 6.3 the distance between adjacent isobars is 3 mb. (1 bar=105 Pa=105 N/m2; 1 mb=105/1000 Pa=105/1000 N/m2). H and L indicate high and low pressure areas, respectively. Triangular markers identify the presence of an atmospheric front. The distance between the flex points of two adjacent isobars is the wind fetch. The wind speed and direction are represented by the symbol : each mark value is 5 m/s and a half mark is 2.5 m/s, so the wind speed is 22.5 m/s at 35°N 130°E. Winds can be divided into three groups: high frequency winds, high speed winds and winds with both characteristics, called “prevailing winds”. Wind frequency, speed and direction are usually plotted on polar diagrams, in which different wind speeds are grouped into classes (e.g. 0-5 m/s, 5-10 m/s, 10-15 m/s) represented by segments of different width departing from the centre of the diagram. The segments length is proportional to the wind frequency. In Figure 6.2 winds are divided into six speed classes: the first (<1 m/s), the second (1-2 m/s), and so on. It is evident that the prevailing winds come from Eastern directions. Wind speed (m/s)

Figure 6.3 - Polar diagram showing directional occurrence of wind speed. 2.2 Geostrophic and low – height winds The wind speed in the geostrophic region is used to calculate the wind speed at 10 meters height : 1/ 7

 10  U (10) = U ( z )   z 

(6.3)

z is the height above the mean water level in the geostrophic layer (z=1000 m), e.g. if U(z)=22.8 m/s ⇒U(10)=11.8 m/s.

SHORT TERM WAVE PREDICTION

111

In the aerodynamic layer, it is possible to write an equation for the wind speed profile: U ( z) =

U*   z   z  ln  − Ψ   0.4   z0   L 

(6.4)

U*= friction velocity (the shear stress is given by ρU ) k= Von Karman constant z0= surface roughness Ψ= function which accounts for the effects of stability of the air column on the wind velocity L =a length scale associated with the mixing process, dependent upon the airsea temperature difference. Wind speed in the aerodynamic layer is measured by anemometers. The measured wind speed is corrected to obtain the corresponding value at 10 m height above the mean sea level. In order to take account of the different resistance encountered by wind on land and on sea, another correction factor is needed : 2

*

(6.5) U(10)= RL ⋅U’(10) RL is a function of wind speed at 10 m height above the sea level, as shown in Figure 6.4. We assume that RL=1 when the anemometer is at 10 m above the mean water level. The Figure 6.4 shows that RL >1 at low wind speed because the shear stress on land is higher on sea. At high wind speed has a higher friction sea surface with waves than land surfaces (RL <1).

3

Wave prediction on deep water

Short term wave prediction (at time scales of 1 hour) is carried out in the usual practice by the calculation of the characteristics of the significant wave. This is a regular wave which describes the sea state, and is characterized by two parameters: the significant wave height (Hs) and the corresponding wave period (Ts). The significant wave height (Hs) is defined as the average height of the waves which form the highest 33% of waves in a given sample time interval (typically 20 to 30 minutes). The significant wave period (Ts) is the average of the periods of the highest one-third of wave heights. In order to characterize the wave characteristics, two methods are frequently used: - Significant wave model (SMB) - Spectral wave models SMB model, introduced by Sverdrup and Munk (1947) and refined by Bretschneider (1958), is the simplest and the most frequently used method. The model is based on empirical formulations of wave height and period as functions of wind speed (U), fetch (x) and time duration (t) of wind events. Spectral models are based on the analytical formulation of spectral wave action and take

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INTRODUCTION TO COASTAL DYNAMICS AND SHORELINE PROTECTION

Figure 6.3 -Surface synoptic chart of Pacific Ocean – 27 October 1950 – ( Shore Protection Manual, 1984).

account of the interaction between different frequency components. The spectral models solve the spectral energy balance as a function of a 2-D spectrum, which frequency and direction are elements of a 2D matrix.

3.1 Fetch and duration limited growth The development of a wave field on deep water is governed by wind speed duration and direction variations, shape and position of the coastline. The combination of these variables leads to a vast number of situations which cannot be simply characterized. Two idealized cases are, however, commonly examined: fetch limited growth and duration limited growth. Although highly idealized, these cases provide a valuable insight into many of the physical processes responsible for wind wave evolution and provide estimates of wave conditions which could be expected at a site. Fetch limited growth occurs when a wind of constant magnitude and direction blows perpendicular to a long and straight coastline. The water is assumed deep and the wind blows for a

SHORT TERM WAVE PREDICTION

113

sufficiently long time that the wave field reaches a steady state (independent of time). Hence, for the given wind speed, the wave field becomes a function of the only fetch, x. 2.0

1.5 RL

1.0

0

0.5

10

U’(10) (m/s)

20

30

Figure 6.4 – RL is a function of wind speed at 10 m height above the sea level. A related problem to fetch limited growth is duration limited growth. This case considers the development of the wave field from an initially calm sea. All land boundaries are assumed sufficiently distant that there is no fetch limitation to growth. The wind field is of constant speed and direction and spatially homogeneous. The water is also assumed to be deep. For a given wind speed, if it blows for a limited time, the resulting wave field will be spatially homogeneous and only a function of the duration or time that the wind has been blowing, t. Sverdrup and Munk (1947), and Kitaigorodskii (1962, 1970) considered the variables which may be active in fetch and duration limited growth. They assumed that the most significant quantities were the variance of the surface elevation, the wind speed at a reference height, the fetch, the gravity acceleration, the wind duration, the frequency of the spectral peak. Dimensional analysis indicates that there should be four non-dimensional groups of parameters:

ε= ν=

σ 2 g 2 the non dimensional energy

(6.6)

f pU a

(6.7)

U a4 g

the non dimensional peak frequency

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INTRODUCTION TO COASTAL DYNAMICS AND SHORELINE PROTECTION

χ=

gx the non dimensional fetch U a2

(6.8)

ς=

gt the non dimensional duration Ua

(6.9)

Therefore for fetch limited conditions:

ε = f1 ( χ ) ; ν = f 2 ( χ ) (6.10) and for duration limited conditions:

ε = f 3 (ς );

ν = f 4 (ς )

(6.11)

where f1, f2, f3 and f4 are functions to be determined. 3.2 Significant wave (SMB) model The wave energy balance in the generation area without currents is given by:

∂E + ∇ ⋅ (C g E ) = Qin + Q ds ∂t

(6.12)

E is the wave energy, Cg is the group speed (the energy propagation speed). The right side of the equation (6.12) is the net source of energy per unit time in the generation area obtained summing up the wind energy (Qin) and the energy dissipated by whitecapping (Qds). A first part of the source term is a supply of wave energy and a second part is transferred away from the generating area. The wave energy is a function of the sea density (ρ), gravity (g), time (t), space (x) and wind speed (U) as shown by: (6.13) E = f ( ρ , g , t , x, U ) The equation (6.13) can be rewritten in a nondimensional expression:

gE  gt gx  = f , 2  4 ρU U U 

(6.14)

where gt/U is the time duration parameter and gx/U2 is the fetch parameter. The equation (6.12) can be applied to many wave generation events: if the wind does not blow for enough time to reach the stationary case, the wave energy is not transferred out of the local generating area (transient case), so the wave energy increases locally:

∂E = Qin + Q ds ∂t

(6.15)

SHORT TERM WAVE PREDICTION

gE  gt  = f  4 ρU U 

115

(6.16)

If the wind blows for enough time, the local generating area is saturated, so wave energy can be transferred to adjacent areas (stationary case). The wave energy is only a function of the fetch parameter:

∇ ⋅ (C g E ) = Qin + Qds

(6.17)

 gx  (6.18) = f 2  U  In this case, the duration parameter is greater than a threshold value tm. If both the fetch parameter and the duration parameter are higher than a minimum value the maximum energy input is reached with the associated wind input, so that any further energy increase is dissipated (fully developed sea), this means that the wave energy content is constant for the given wind speed: gE

ρU 4

Qin + Qds = const

gE

ρU 4

= const

(6.19) (6.20)

The SMB model calculates the significant wave height Hs and period Ts on deep water as functions of wind speed (U), fetch extension (x), gravity (g), and time duration (t):

H S = f 1 (U , x, t , g ) ,

TS = f 2 (U , x, t , g )

(6.21)

At a given wind speed, the fetch and the time duration are given by:

gH s  gt gx  gTs  gt gx  (6.22) = f1  , 2  ; = f2  , 2  2 ρU  U U  ρU U U  In the transient case, the significant wave height and period depend on the time duration parameter gt/U (duration limited sea): gH s  gt  gTs  gt  (6.23) = f1   ; = f2   2 ρU  U  ρU U  In the stationary case the significant wave height and period are expressed as functions of the fetch parameter (fetch limited sea): gH s

ρU 2

 gF  gTs  gF  = f1  2  ; = f2  2  U ρ U   U 

(6.24)

The fully-developed sea occurs when the fetch no longer controls the development of the waves and there is no more net transport of energy from the wind to the waves. In this case gx/U2> gxm/U2 and the significant wave height and period are constant and only functions of the wind speed:

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gTs gH s = const = const ; 2 ρU ρU

(6.25)

According to Bretschneider (1958, 1973) the dependence of the significant wave height and period by the fetch and duration parameters can be formulated as: 0.42  gH s  gx   = ⋅ 0 . 283 tanh 0 . 0125     2 ρU 2  U   

(6.26)

0.25  gTs  gx   = 1.2 ⋅ tanh 0.077 2   ρU  U   

(6.27)

The equations (6.26) and (6.27) are valid only for the stationary case, (t>tm) with tm given by:

{[

]

0.5 gt m = 6.5882 ⋅ exp 0.0161λ2 − 0.3692λ + 2.2024 + 0.8798λ ρU

}

(6.28)

where λ = ln( gx / U 2 ) . If gx/U2 exceeds a critical value, the fully developed sea occurs:

gH s = 0.283 ρU 2 gTs

ρU

= 1. 2

(6.29) (6.30)

Equations from (6.26) to (6.30) are depicted in figure 6.5. The lines plotted in Figure 6.5 are used to calculate Hs and Ts for given values of U, t and x. The wave condition is identified by the intersection between the values of U and x. The dotted line t=tm is the minimum time duration needed for the stationary case. If t
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Figure 6.5 - Significant wave height and period forecasting curves (Shore protection manual 1977 )

(a) In figure 6.5 the intersection between U=35 Kn and x=60 n.m. gives:

6htm ⇒ stationary case (fetch limited sea). The intersection between U=35 Kn and x=60 n.m. gives Hs=11 ft ≅3.3 m and Ts =7.3 s. (b) In figure 6.5 the intersection between U=35 Kn and x=200 n.m. gives tm=18 h⇒ t
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of terms considered, models are divided into three groups: first, second and third generation models. A key concept in the description of random waves is the two dimensional spectral distribution function of surface elevation variance. It contains the information required for a dynamic and probabilistic description of the local wavefield. To lowest order of wave steepness, surface elevation variance and wave energy density are equivalent, apart from a constant factor. Energy divided by the intrinsic frequency is called wave action, and presents a very useful concept because it is conserved in case of wave-current interactions (in absence of energy sources or sinks). The spectral distribution function of variance, energy or wave action is the principal prognostic quantity in the phase-averaged modelling of random waves. Its slow evolution in time and space can be calculated on the basis of an energy or action balance. This approach was initiated by Gelci et al. (1956, 1957) (using energy) and has been followed ever since, with numerous and continuing improvements and extensions (given by Komen et al. (1994), Battjes(1994). Phase averaging models predict average or integral properties of the wave field. These may be quantities such as significant wave height and peak period. Phase averaged models solve a single equation, the energy (or action) balance equation:

∂E ∂ [(C g ⋅ ∇θ ) E ] = S + ∇(C c E ) − ∂t ∂θ

(6.31)

where E(x, t, f, θ) is the directional wave spectrum, x is the position in space, Cg is the group velocity. S is a source function which includes a description of the physical processes representing wave growth, interaction and decay. This equation allows swell to be represented in addition to local wind generated waves. (6.32) Stot = Sin + S nl + S ds where Sin is the atmospheric input, Snl represents quadruplet non linear interactions between spectral components and Sds is the dissipation by whitecapping breaking. 3.3.1 First, second and third generation models The first generation models, developed in the early 1960s, took only account of wave energy growth and dissipation (Q=Qin+Qds). The dissipation source Qds was obtained assuming that the wave components suddenly stopped growing as soon as they reached a universal saturation level (Phillips, 1957). The saturation spectrum, represented by Phillips’ one dimensional f−5 frequency spectrum and empirical equilibrium directional distribution, was prescribed.

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The source energy transferred by the wind was expressed as the sum of the Phillips (1957) increase during the initial wave motion development and the Miles (1957) exponential growth of energy. The relative importance of the nonlinear transfer Qnl on the spectral evolution become more evident after extensive wave growth experiments (Hasselman et al., 1973) and direct measurements of the wind input to the waves (Snyder et al., 1981). Non-linear interactions between different wave components are the main cause of spectral growth with fetch extension; low frequency wave components causing the spectrum to quickly increase. This led to the development of second generation wave models, which attempted to simulate the nonlinear interaction in the source balance. A major impediment to the inclusion of Qnl was the significant computational time required for its evaluation. In order to overcome this problem a class of wave models which utilized approximations to this term were developed (Barnett, 1968; Ewing, 1971; Sobey and Young, 1986; Young, 1988). The significant advancement with this class of models was the inclusion of a parametric representation for the nonlinear interaction term Qnl . In almost all cases these approximate forms represented the model spectrum in terms of a small number (3 to 5) parameters. The JONSWAP spectral parameters were a common choice (Barnett, 1968; Ewing, 1971; Young 1988). Of course, problems occurred when the model spectrum did not conform to any of those for which Qnl had been precomputed. A further sub-classes of models, termed hybrid models, took this process one step further and predicted the evolution of the parameters rather than the discrete values of the two dimensional directional spectrum. The Sea Wave Modeling Project, SWAMP (SWAMP Group, 1985) demonstrated that it was necessary to overcome the limitations in the parameterization of Qnl of the second generation wave models. Some of the SWAMP (1985) participants developed a third generation wave model (WAM, 1988) which applieed no restrictions on the spectral shape, gave an exact Qnl parametrization (Hasselmann et al., 1985) and calculated Qds from field measurements of Snyder et al., (1981). The WAM model is a third generation wave model which solves the wave transport equation explicitly without any presumptions on the spectral shape. Third generation models utilize the discrete interaction approximation (DIA) to the Qnl term (Hasselmann et al., 1985). Although the DIA is still an approximation, it is very different in its formulation to the simple parametric forms used in second generation models. The DIA retains the basis physics of the nonlinear interaction process, but considers a very small sub-set of all the possible interactions. In contrast to second generation parameterization for Qnl, the DIA has many degrees of freedom as the values in discretely specified directional spectrum.

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3.3.2 Third generation models Third generation models solve the energy balance equation using a finite difference method based on a gridded representation of the bathymetry allowing refraction and shoaling to be modelled. The spectral balance equation can take many different forms, dependent on: • • • • • • •

Time variation (stationary or nonstationary) geometry of propagation space (plane or spherical) space coordinates (Cartesian, polar, spherical) spectral domain (wavenumber vector, frequency/direction) spectral density (energy or action) medium (deep water; finite depth, constant or variable; currents) source functions (processes and representation)

Wave Spectral Density Sη(f)

Envelope of spectra

Frequency

Figure 6.9 - Evolution of one-dimensional frequency spectrum. To illustrate the basic structure, a formulation in terms of action density in wave number space is summarized in the following. Dispersion equation 1

σ = ( gk tanh kd ) 2 = ω − kα U α

(α = 1,2)

(6.33)

G σ =intrinsic frequency; ω = absolute frequency; k = wave number; k = k

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121

Propagation equation

xα = c gα + U α =

∂σ + Uα ∂kα

∂U β ∂k ∂k ∂σ ∂d − kβ kα = α + x β α = − ∂t ∂x β ∂d ∂t ∂xα ω =

Action density

∂U α ∂ω ∂σ ∂d ∂ω + kα = + xα ∂t ∂xα ∂d ∂t ∂t

G G G G N (k ; x , t ) = E (k ; x , t ) / σ

(6.34) (6.35)

(6.36)

(6.37)

in which E is energy density Action balance equation

∂N ∂N  ∂N + xα + kα = ∑ S Qi ∂t ∂xα ∂kα i

(6.38)

Qi is the source term (i=1,2, …) The propagation formulation applies to each of the three domains considered here (oceans, shelf sea and nearshore) but distinguishing the source terms. Ocean models: The spectral evolution of wind-driven ocean waves, in the absence of currents, is mainly determined by wind input, whitecapping and nonlinear resonant quadruplet interactions; a suitable parameterisation for these source functions has allowed the development of the so-called third generation wave model, in which the spectrum is calculated without a priori constraints. Shelf sea models: the additional effects of restricted depth and relatively stronger currents have to be taken into account. The major change in energy budget is increased dissipation due to bottom friction (Weber, 1988) and, under extreme conditions, depth-induced breaking. Refraction due to varying depth and currents plays a minor role. Consider the individual source terms: • •

Wind input and whitecapping are taken to be the same as in deep water, with the finite depth influence on wavenumber and phase speed taken into account (WAMDI, 1988). The deep-water parameterisation of the quadruplet interactions is scaled up with a multiplying factor dependent on k d , where k is the spectrally-averaged wavenumber. This approximation is said to be valid for k d > 1 (WAMDI, 1988).

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•

•

In shelf seas, bed friction is important. Significant advances have been made in the understanding of the hydrodynamics and the modelling of the oscillatory near-bottom boundary layer. However, prediction remains problematic, particularly because the development of flow– dependent roughness on sedimentary bottoms and the role of wavecurrent interactions in the bottom boundary layer are insufficiently known. Depth dependent breaking does not need to be taken into account in shelf sea models under normal conditions. However a number of numerical studies (e.g. extreme waves in Northern Sea using Tolman’s model by Holthuijsen et al. 1994) have shown that inclusion of this process is necessary.

Near-shore models: If shelf sea models are made to include even depthinduced breaking among the finite-depth effects, one can wonder whether there is a need for so-called near-shore models and why such shelf sea models are not simply used all the way to the shore. The reason is that the very small depth and the more rapid depth variation typical for coastal areas would require a high spatial resolution, which for time dependent models also translates into small time increments of the numerical integration. This makes application of shelf sea models in near-shore regions unattractive and causes the need to use separate models to cover a rather restricted area from the coastline out to seaward boundary which is usually taken at some chosen depth contour, say 10m or 20m below MSL. 3.3.3

WaveWatch III

WaveWatch III is a third generation wave model developed at NOAA/NCEP after the WAM wave model, as a further development of WaveWatch I, (Delft University of Technology) and WaveWatch II (NASA, Goddard Space Flight Center). The governing physical equations, the physical parameterizations and the numerical methods reflect some modifications of previous models. The solution of the governing equations is based on a first and a third order accurate numerical scheme. The breaking waves physics are not modeled, hence the applicability of this model is outside of the surf zone and on large scale. Outputs from the model include significant wave height gridded fields with the associated wave directions and periods, spectral information about wave energy at the different wavelengths. The governing equations simulate variations in time and space of wave growth and decay produced by the surface wind, dissipation (e.g. due to whitecapping), and the bottom friction effects. For irregular wind waves, the random variance of the sea surface is described using variance density spectra (usually denoted as energy spectra). The variance spectrum F depends on all independent phase parameters, i.e., F(k,σ,ω), and furthermore varies in space x

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123

and time t, e.g., F(k,σ,ω,x,t), where k,σ and ω are the wave number vector, the intrinsic frequency and the absolute frequency respectively.

σ 2 = gk tanh kd

(6.39)

ω = σ + k ⋅U

(6.40)

where U is the current speed averaged in time and space and d is the water depth. If the individual spectral components satisfy the linear wave theory (locally), the dispersion relation and Doppler type equation interrelate the phase parameters; only two independent phase parameters exist, and the local and instantaneous spectrum becomes two-dimensional. Within WWATCH the basic spectrum is the wavenumber-direction spectrum F(k,θ), which has been selected because of its invariance characteristics with respect to physics of wave growth and decay for variable water depths. The output of WWATCH, however, consists of the more traditional frequency-direction spectrum F(fr,θ). The different spectra can be calculated from F(k,θ) using straightforward Jacobian transformations. Without currents, the variance (energy) of a wave packet is a conserved quantity. With the addition of currents the energy or variance of a spectral component is no longer conserved, due to the work done by current on the mean momentum transfer of waves (Longuet-Higgins et al., 1961). In a general sense, however, wave action A≡E/σ is conserved (Whitham,1965; Bretherthon and Garrett, 1968). This makes the wave action density spectrum N(k, θ)≡F(k, θ)/σ the spectrum of choice within the model. Wave propagation then is described by:

DN Q = Dt σ

(6.41)

where D/Dt represents the total derivative (moving with a wave component) and S represents the net effect of sources and sinks for the spectrum F. In a numerical model, a Eulerian form of the balance equation (6.41) is needed. The balance equation for the spectrum N (k,θ,x,t) in a spherical grid as used in WWATCH is given as (for convenience of notation, the spectrum is henceforth denoted simply as N):

∂  ∂  Q ∂N kN + + ∇ x ⋅ xN + θN = ∂k ∂θ σ ∂t

(6.42)

x = c g + U

(6.43)

∂σ ∂d ∂U k = − −k⋅ ∂s ∂d ∂s

(6.44)

∂U  1  ∂σ ∂d −k⋅ θ = − − k  ∂d ∂m ∂m 

(6.45)

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where cg is the group celerity and θ is the wave direction, s is a coordinate in the direction θ and m is a coordinate perpendicular to s. The above equations are valid for a Cartesian Grid. The net source term is generally given by summing up a wind-wave interaction term Qin, a nonlinear wave-wave interactions term Qnl and a dissipation (whitecapping) term Qd .In shallow water additional processes have to be considered, most notably wave-bottom interactions Qbot (e.g., Shemdin et al., 1978). The general source terms used in WWATCH as:

Q = Qin + Qnl + Qds + Qbot

(6.46)

Nonlinear interactions are optionally modelled using the Discrete Interaction approximation (DIA, Hasselmann et al., 1985), or the Webb-Resio-Tracy method (WRT). The model includes two source term options: the first one is based on WAM model cycles 1 through 3 (WAMDIG 1988); the second one is based on Tolman and Chalikov (1996) formulation. Input to WWIII can consist of wind, current, water level, temperature and ice concentration fields on the spatial wave model grid. WWIII model gives various types of output, such as fields of mean wave parameters on the spatial grid and input fields driving the model (wave height, maximum wave height, primary and secondary wave direction, primary and secondary wave period, sea height, swell height, sea period, swell period, sea direction, swell direction, and whitecap probability). 3.3.4

Case study (WWIII application for the Gulf of Naples)

The spectral third-generation ocean wind-wave model WAVEWATCH III (WW3), operational since January 2005 at the Department of Applied Sciences of the University of Naples “Parthenope” (Italy), was adopted for simulating wave propagation in the Gulf of Naples. The model was coupled with PSU/NCAR mesoscale model (MM5), which gives wind forcing at 1-h intervals. The model was implemented using a four-nested grid configuration covering the Mediterranean Sea until the Gulf of Naples, the inner mesh with higher resolution (1 km x 1 km). The simulated directional spectral waves were compared with APAT storm wave data recorded in year 2000 offshore the Gulf of Naples and with wind and wave data collected by Servizio Idrografico e Mareografico offshore the mouth of river Sele in the Gulf of Salerno. The MM5/WW3 model domains cover four areas (from regional ocean to small scale): • DOMAIN 1 (Mediterranean sea) • DOMAIN 2 (Seas around Italy) • DOMAIN 3 (Tyrrhenian sea) • DOMAIN 4 (Gulf of Naples) Information about the spatial dimension of four domains is summarized in table 6.1.

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Table 6.1 – Spatial information about the four domains of WWIII Latitude range

Longitude range

Latitude Increments

Longitude Increments

DOMAIN 1

30.02 47.84

-5.53 41.83.

0.24

0.24

DOMAIN 2 DOMAIN 3 DOMAIN 4

36.11 48.31 39.80 41.67 40.41 41.08

3.76 22.41 12.50 16.47 13.72 14.69

0.08 0.03 0.01

0.08 0.03 0.01

Domain 3

Domain 4

Fi

Figure 6.10 – Example of significant wave height, domains 3 and 4. An example of nested grids is in figure 6.10 (domains 3 and 4). The comparison between simulated and observed waves is given for two locations close to the Gulf of Naples during the November 2000 storm: the first one for deep water (Ponza 40°52’00’’N e 12°57’00’’E) and the second one for intermediate depth (Sele mouth 40°29’06”N, 14°55’30”E). Time histories of simulated and recorded wave storms gives more insight into the physical aspects of the simulation: the wave simulations November 2000 storm (characterized directions spread from 200°N to 230°N) are in good agreement with the data (fig. 6.11 and 6.12). Figure 6.14 gives map of WWIII 2004 storm in the Gulf of Naples.

INTRODUCTION TO COASTAL DYNAMICS AND SHORELINE PROTECTION

DDw [°N]

126

Hs [m

Figure 6.11 – Measured and simulated wave directions- Ponza and Sele mouth stations - November 2000. M=measured; S=simulated.

Figure 6.12 – Measured and simulated significant wave height Ponza and Sele mouth stations - November 2000. M=measured; S=simulated

Figure 6.14 – WWIII results for the 2004 storm in the Gulf of Naples

Chapter 7 Long term wave statistics 1 Introduction The characterization of wave climate in a coastal zone for a long period deals with the estimation, through an extrapolation from data directly collected or hindcasted, of the wave parameters besides the period of observation, in order to obtain an extreme wave, also called “design wave”, which by definition is the highest wave which may occur in a given return period. The procedure for the evaluation of a design wave is based on the analysis of recorded or estimated data and passes through the following steps: • • • •

Wave height data selection relative to a period of observation; Choice of a known probability distribution of extreme values; Fitting of wave height data to the chosen distribution; Data extrapolation through the probability distribution chosen to identify the extreme wave height in a given return period.

The sample independence is somewhat guaranteed by fixing a threshold value for the time interval between two subsequent sea storms. The size of this interval can be calculated on the basis of the auto-correlation function of the observed time series. By imposing a weak sample independence, a long enough time lag implies that the two subsequent sea storms are independent, which makes the auto-correlation function value sufficiently small (the value of 0.4 is normally recommended in literature). Generally, this time interval is equal to 48 hours. In order to assure the independence of two consecutive events, the combination of different criteria is used (e.g. a time interval of 48 h and a deviation of 60° by the mean wave direction). The choice of threshold wave height is also a delicate operation, because it has a considerable influence on wave height estimates with a given return period. In fact, if the threshold value is too low, it produces too many sample elements, most of which do not really represent extreme conditions. These circumstances will determine stable but over-estimated predictions of the wave height with a

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given return period. On the contrary, if the threshold value is too high, it produces very few sample elements, even if they do not represent extreme conditions. These circumstances determine strongly variable wave height predictions according to the number of samples collected. The analysis of SWAN (Sea Wave Italian Network) data shows that H0=2.0 m is a value acceptable for all the Italian sites, with a number of storms variable between 10 and 30. 1.1 Wave data Long term wave prediction can be performed on the basis of three categories of data: 1. 2. 3.

Visual data; Data obtained by hindcasting model based on wind data; Instrumental data collected in situ or by remote sensing.

Visual data, reported by ships during their normal activity, are less reliable than those obtained by hindcasting. In the past visual observations of waves and wind were the only basic source of ocean wave statistics. Although other types of wave data are now available, visual observations are still the source which covers most of the ocean areas. In the early stages of observations, the sea state was represented using the Beaufort scale, which gave the gradation of wind speed associated with some description of the sea state. However, there was not a straightforward relationship between wind speed and wave height. Only in 1947 did the World Meteorological Organization introduce the international code for the observation of wind and waves. There are two main sources of wave visual data, i.e. data from merchant ships and observations by weather ships (Ocean Weather Stations — OWS). Merchant ships are generally expected to avoid bad weather, thus being subjected to less severe wave conditions than weather ships. The stations cover most of the ship routes between Europe and North America. The Pacific Ocean is not so well documented and only a few stations operate in the northern part of the Pacific. Major sources of visual data are the compilations made by Hogben and Humb (1967) and Hogben (1988). Wave hindcasting are used at locations where no information on the wave climate is directly available. Several wave hindcasting models starting from the wind field have been used to generate large data set, eventually calibrated with instrumental data. Instrumental data collected in situ by wave gauges or through remote sensing are the most reliable source of data. Nevertheless their use is reduced due to the limited availability (few subsequent years of observation) and because of the high cost of acquisition. A wave recorder at a site will provide a continuous record of the surface elevation η(t) and should be used on a reasonably long period (ten years or longer) so as to obtain a data archive representative of the various conditions occurring.

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129

1.2 Data selection The first step to calculate the design wave height is the selection of the set of wave heights and periods in the site of interest. The time duration of the sample is limited by the data availability. In the case of measured data, the sample length may be five or ten years, in the case of hindcasting the length may be longer (from 20 to 50 years). Selection of wave height values is done in different ways, for example the maximum significant wave height occurring during a storm can be considered, or the maximum annual significant wave height (if the length of data sample is a few tens of years). Data used for the extreme wave analysis should be taken only from significant events in the recorded time history and each value should be from a different event to ensure stochastic independence. The preferred approach to data selection is to take the maximum value from each wave storm to create a series of extreme values. Thus the extreme analysis can focus on a smaller series representing truly significant events. A set of declustered events can be identified by local maxima with a sufficient separation in time such that independence is a reasonable assumption, as already stated before. 1.3 Extreme value probability distribution There is no theoretical basis for selecting any particular probability distribution to represent the observed data. Therefore generally a best-fitting distribution function is chosen among the candidates (log-normal, Gumbel, Frechèt, Weibull). All these distributions have a theoretical base, but in long term statistics they are used as empirical data fitting. To describe the different distributions, it is useful to introduce the following parameters: • α - Shape parameter: determines the basic shape of a particular distribution • θ - Scale parameter: controls the degree of spread along the abscissa • ε - Location parameter: locates the position of the density function along the abscissa.

2 Data fitting to the probability distribution The probability that a wave height H ' is less than a specified wave height H is defined as: P = P( H ' < H )

(7.1)

The probability that H ' is greater than a specified wave height H can be defined as: Q = Q( H ' > H ) = 1 − P

(7.2)

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The first is called Cumulative Distribution Function, the second Probability of Exceedence. In the following some probability functions are listed: they are all possible candidates as Extreme Value Probability Distributions. 2.1 Normal Probability Distribution We begin with this common distribution, because it forms the basis for the LogNormal distribution described below. The equation for the CDF, based on a Normal distribution with sample mean H and sample standard deviation σH is: P=

1

σ H 2π

H

∫e

1 H −H −  2 σ H

   

2

−∞

H −H = Φ  σH

  = Φ ( Z ) 

(7.3)

where H is wave height, H mean waveheight, σH standard deviation of waveheight and Z the standard normal variate H −H Z =   σH

 H 1  = H− σ σ H H 

(7.4)

Standard Normal Probability Tables are used to relate P to Z. However because we are interested in extreme values, the common version of Normal Probability tables do not cover a large enough range. Table (7.1) presents an extended range, in which only negative values of Z are shown (only the left hand of the normal distribution is represented here), If Z is known, such a table will yield P=Φ(z) as defined by equation (7.4). Since the standard normal probability distribution is symmetrical, the positive counterpart of the probability distribution (i.e. the values P for positive values of Z), is obtained as Ppos =1-Pneg. The tables can also be used in reverse: Z can be obtained from P. We define that inverse operation symbolically as:

Z = Φ −1 ( P)

(7.5)

Table 7.1 - Probability table: P=Φ(z) Z 0 -0.1 -0.2 -0.3 -0.4 -0.5 -0.6 -0.7 -0.8 -0.9

0 0.5000 0.4602 0.4207 0.3821 0.3446 0.3085 0.2743 0.2420 0.2119 0.1841

-1 0.1587 0.1357 0.1151 0.0968 0.0808 0.0668 0.0548 0.0446 0.0359 0.0287

-2 0.0228 0.0179 0.0139 0.0107 0.0082 0.0062 0.0047 0.0035 0.0026 0.0019

-3 1.350E-03 9.677E-04 6.872 E-04 4.835 E-04 3.370 E-04 2.327 E-04 1.591 E-04 1.078 E-04 7.237 E-05 4.812 E-05

-4 3.169 E-05 2.067 E-05 1.335 E-05 8.456 E-06 5.417 E-06 3.401 E-06 2.115 E-06 1.302 E-06 7.944 E-07 4.799 E-07

-5 2.871 E-07 1.701 E-07 9.983 E-08 5.802 E-08 3.340 E-08 1.904 E-08 1.075 E-08 6.008 E-09 3.326 E-09 1.824 E-09

Equation (7.4) shows that Z is actually a linear function of H and therefore, the appropriate transformation of axes for a normally distributed CDF would be:

LONG TERM WAVE STATISTICS

Y = Z = Φ −1 ( P );

131 (7.6)

X =H

The slope and the intercept of the straight line, according to eq. (7.4), would be 1 ;

A=

B=−

σH

H

(7.7)

σH

This transformation can be tested if a series of points is normally distributed. The test for normality is: Are the points on a straight line? The Y-axis transformation comes from eqn. (7.6). When reduced variate is plotted against H, points are much closer to a straight line than they would be if the probability value were used. 2.2 Log-Normal Distribution Using variables Y=Φ-1(P) and lnH as X, Y and X should produce almost a straight line. The CDF of P versus lnH is given by: Y = Z = Φ −1 ( P ) =

ln H − ln H

σ ln H

=

1

σ ln H

ln H −

ln H

σ ln H

(7.8)

or:

A=

Y = Φ −1 ( P )

(7.9)

X = ln H

(7.10)

1 ;

σ ln H

B=−

ln H

σ ln H

(7.11)

The individual points do not lie exactly on a straight line, but their relationship may be approximated by the straight line. The equation of the straight line of best fit is obtained by linear regression analysis. 2.3 Gumbel distribution In addition to log-normal probability distribution, it is possible to use distributions developed specifically for analysis of extreme values. These models were originally derived for a limited number of “ordered statistics” such as a set of maximum annual floods sorted in descending order. The Gumbel distribution is given by:   H −γ P = exp − exp − β  

   

(7.12)

This equation can be linearized by taking the logs by both sides.  H −γ ln P = − exp − β  and taking logs again

  

(7.13)

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ln(− ln( P)) = −

H −γ

(7.14)

β

or 1 γ  1  H −γ − ln  ln  = = H− β β β  P

(7.15)

The reduced variate Y will be called G. The resulting transformation is: 1  1 Y = − ln ln  = G ; X = H ; A = ; β  P

B=−

γ β

(7.16)

2.4 Weibull distribution The above distributions all have two parameters. A more versatile extreme value distribution is the three-parameter Weibull distribution: α

  H −γ P = 1 − exp −    β 

   

  

(7.17)

which may also be expressed as:   H −γ Q = exp −    β 

α

  

   

(7.18)

Linear transformation may be accomplished by taking the logs of both sides:  H −γ − ln Q =   β

α

  

(7.19)

H −γ

(7.20)

which results in: 1/α

 1  ln   Q

=

β

1/α

 1 Y =  ln   Q

= W; X = H; A =

1

β

;B=−

γ β

(7.21)

The Weibull distribution has three parameters (α, β and γ). Linear regression provides only two constants (A and B) and if we want to continue to use linear regression analysis, the determination of the third coefficient (α) will require some trial and error. Assuming a different value of α the curvature of the points will change. The parameter γ in the Weibull and Gumbel distributions has physical meaning. It is the lowest limit of H (when H=γ, Q=1 or P=0). Thus γ is

LONG TERM WAVE STATISTICS

133

theoretically equal to the threshold value in a Peak over Threshold data set. This can be used as a check.

3 Parameter calculation Once the probability distribution has been chosen, the selected wave heights are reduced to a set of points to be plotted on a probabilistic paper. With this aim, data are sorted in descending order with reference to an index m such that m=1 corresponds to the highest wave, and m=N to the smallest one and a value of exceedence (or non-exceedence) probability must be assigned to each extreme data value. A simple estimation of the exceedence Q(Hm) and of the total probability P is given respectively by equations: Q( H m ) =

m ; N +1

P ( H m ) = 1 − PS ( H m )

(7.22)

So it is possible to individuate an interpolating line which represents the best fitting of data to the chosen distribution. Generally for this purpose two different approaches may be used: graphical or computational. Graphical approach: any set of measured data is plotted such that the selected distribution lies on a straight line; in such a way the goodness of fit and the extrapolation procedure are visually evaluated. Thus, once the particular probability paper is selected, abscissa and ordinate scales must be constructed in order to make the distribution appear as a straight line. The linear ordinate scale y is related to the cumulative probability P, and the linear abscissa x is related to the variate H according to the relationship relative to the chosen probability distribution. Computational approach: the Method of Moments is often used, but also Least Squares Method and Maximum Likelihood Method may be employed. Method of Moments consists in equating the first two or three moments of the distribution to the estimated moments calculated by data set. Regardless to the method used, it is preferable to plot the computed distribution and data together to ensure that the fit is consistent with good engineering judgement. 3.1 Statistical tests of fit When several candidate distribution functions are under consideration, one is selected as a best fit to the data. The selection criteria can range from visual inspection of plotted results and simple statistics such as the correlation between data and model. A possible method for testing the goodness of probability distribution is the chi-square test. The general procedure involves the use of a statistic with an appropriate chisquare distribution as a measure of discrepancy between an observed probability density function and the theoretical probability density function which is

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expected for the theory in exam. A hypothesis of equivalence is then tested by studying the sampling distribution of this statistic. To be more specific, consider a sample of N independent observation from a random variable x with a probability density function p(x). Let the N observations be grouped into K intervals, called class interval, which together form a frequency histogram. The number of observations falling within the i-th class interval is called observed frequency in the i-th class and will be denoted by fi. The number of observation that would be expected to fall within the class interval if the true probability density function of x were p0(x) is called expected frequency in the i-th class interval and will be denoted by Fi. Now, the discrepancy between the observed frequency and the expected frequency within each class interval is given by fi−Fi. To measure the total discrepancy for all class interval, the squares of the discrepancies in each interval are normalized by the associated expected frequencies and summed to obtain the sample statistic K

( f i − Fi )2

i =1

fi

χ2 =∑

(7.23)

The goodness of the chosen probability distribution is related to the tendency of chi-square to approach to zero. In the ideal case that the chosen probability distribution is coincident with the observed one, chi-square matches zero. For each value i, the χ2-test calculates the squared weighted mean of differences between analytical and empirical frequencies. For χ2=0 the sample and the distribution agree; the difference between the two distributions increases with χ2. The ratio (fi−Fi)2/fi represents the relative difference between the analytical and empirical frequency, for a given value n. The χ2-test is carried out thought the following steps: • • • •

Divide the sample into k classes with the same amplitude; Select a confidence threshold P; Individuate the value χ2 by the table 7.2, for a given number of degrees of freedom (ν=k−1). Compare the empirical and the analytical values χ2 .

3.2 Confidence intervals Confidence intervals associated with the chosen distribution function should also be estimated, to appraise the closeness of fit of the data points to the fitted distribution on either sides of the fitted line. The scatter of data is best described in terms of confidence limits individuated by a pair of curves drawn on either sides of the best-fit line; they provide the confidence bands within which data are expected to lie with corresponding probability, namely a 50% or 90% confidence bands border a zone within which data are expected to lie with a 50% or 90% probability.

LONG TERM WAVE STATISTICS

135

For given sample size N and chosen confidence probability level, a pair of height limits for the particular statistics is provided. This method assumes that the residual are normally distributed and is only valid for the range of data used in the fit; thus it cannot be extrapolated beyond the observed data and therefore cannot be used for the prediction. An alternative method of confidence bands determination involves the Monte Carlo simulation to generate random data sets deriving from the best-fit distribution that have been obtained. The spread of simulated data can be used to estimate the confidence or the uncertainty attached to any chosen value. Differently from the previous, this approach is advantageous as it can be used on either observed and predicted values of wave height. Once data have been fitted to the probability distribution, the latter must be extrapolated in order to evaluate design wave height corresponding to a fixed return period. Return period is defined as the average time interval running between two successive events of the design wave height being equalled or exceeded, and is correlated to the exceedance probability QS by the relation: TR 1 1 = = r Q S ( H TR ) 1 − Q ( H TR )

(7.24)

where r is the interval of record relative to each wave height measurement. From equation (7.24) the value of cumulative probability associated to a fixed return period is derived and the corresponding wave height may be determined from the best-fit line that has been plotted. This design value is defined as the individual data points used in the analysis (significant height, maximum height, etc. depending on adopted data) and should be equalled or exceeded on average once over the duration of any return period. In the project contest the encounter probability E may also be used, which is the probability that the design wave height is once equalled or exceeded during the structure lifetime L. The relation between the encounter probability and the return period is the following:  r E = 1 −  TR

  

L/r

(7.25)

3.3 Statistics of offshore extreme waves For the determination of the so called “design wave” needed for the functional and structural design it is necessary to perform a statistical elaboration of the highest wave heights, recorded during the peak of independent sea storms relative to the time series deduced by wave data. When wave height data are available, the statistical analysis of incomplete data series is applied. In this method, the significant wave heights for different directional sectors are selected; each height selected is greater than a known threshold value and identifies an independent storm peak. The incomplete series method (POT) is generally preferred to the annual maxima method (because of

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the limited number of data) and the complete data series method (because some wave events may be not independent). The POT method defines the temporal characteristics of storms by the selection of the events associated to wave heights greater than a threshold wave height value H0. In the simplest case of the local maxima method the random and independent variables are equally distributed and governed by a random counter. The random counter is the expected number of storms exceeding the threshold wave height H0 during a year. If the storms with height H>H0 have a Poisson distribution, the following equation is valid: P ( Na = n) = λn exp − λ / n!

(7.26)

Table 7.2 - χ2 table for given values of degrees of freedom ν.

Where λ = λˆ = n / M is the mean number of storms with H>H0 during a year, n is the total number of storms with H>H0. The implicit relationship between Hs,T and T is:

LONG TERM WAVE STATISTICS

F ( H s ,T ≥ H 0 ) = 1 − 1 / λˆT

137

(7.27)

for a given value T, Hs is determined through the equation (7.27). The equation (7.27) is independent by the process and is generally valid if the storms with H>H0 have a statistic distribution and if λˆ represents how many times the threshold value H0 is exceeded per time unit. Let us introduce the exponential distribution F: F ( H s | H s ≥ H 0 ) = 1 − exp[( H s − H 0 ) / β ]

(7.28)

we obtain: H s = H 0 + βˆ ln λˆ + β ln T

(7.29)

The eqn. (7.29) relates the wave height and the return period, and βˆ , λˆ are empirical parameter. For a given H0, β is the estimation of the maximum probability and can be calculated also through the method of moments. The variance associated to the estimation of Hs,T, is given by: Var ( Hs, T ) =

β2 N

[1 + (ln λˆ + ln T ) ] 2

(7.30)

Through the equation (7.30) the standard error σ = Var ( H s,T ) and the confidence interval can be obtained. If the probability is 95%, the “true” value of Hs,T is included between Hs±2σ, where Hs,T is given by the equation (7.29) and σ is given by equation (7.30). The data fitting is carried out by selecting the wave height values, plotted on a probability map. Data are taken in decreasing order and are represented by the following expression: C H m = {H m }mN =1

(7.31)

where the m=1 and m=N are the highest and the smallest height indexes, respectively. The coefficients of the fitting curve are identified through the method of moments, which is based on the identity between the analytically determined mean µH , and variance σ2H and the statistical mean µHm and variance σ2Hm . After the data fitting, the wave height associated with the return period can be determined. The return period Tr is the mean time interval between two consecutive wave events with H≥H0. Tr depends on exceeding the probability as follows: Tr 1 1 (7.32) = = r 1 − P ( H Tr ) Q ( H Tr ) where r is the mean time interval at which the statistic data are referred.

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3.4 Wave height persistence Wave persistence traditionally refers to duration of conditions in which wave heights are above or below a certain threshold value. This information is needed in connection with the operations of many marine structures (for example, most pipeline dredges cannot work in wave heights greater than 1m). A simple parametrical model is described in the Italian Wave Atlas (APAT, 2004) to evaluate the main persistence. Another model commonly used is the one of Kuwashima & Hogben (1986), which is based on the mean persistence τ ( H ) obtained by the Weibull distribution which parameters can be determined by the Least Squares Method: τ = A[ − ln( Q )] − β

(7.33)

where Q is the exceedence probability of the threshold H ' given by the Weibull distribution:

{

Q( H ' ) = exp − [(H '−ε ) / ϑ ]α

}

(7.34)

and the exceedence probability of the nondimensional duration x (τ normalized with the mean duration τ ) is given by:

Q ( x ) = exp − ( x / ϑ s ) α s

(7.35)

where αs and ϑs are given by the lower-limited Weibull distribution:

α s = 0.267γ ( H ' / H ) 0.4

(7.36)

ϑ s = 1 /[ Γ( 1 + 1 / α s )] α s

(7.37)

being H’ the threshold value height and H the mean value given by: H = ϑ Γ( 1 + 1 / α ) + ε

(7.38)

γ = α + 1.8ε /( H − ε )

(7.39)

and:

ε being the lower limit of the Weibull distribution. 3.5 Case study

In order to obtain long term statistics of wave data, we refer to significant wave heights measured by the APAT wave buoy located offshore the coastline of Monopoli, in the Adriatic Sea, which refers to a period of observation r=9 years (1989-1997). The values of 29 peak-over-threshold significant wave heights for the direction DD=350°N have been reported in table 7.3 together with their exceedence probability Q(Hm). The mean and variance of data sample are given by:

LONG TERM WAVE STATISTICS

µ= σ2 =

1 N

N

1 N

139

∑ Hm

(7.40)

∑ (H m − µ) 2

(7.41)

m =1

N

m =1

The Probability Distribution Function is given by:   H −γ P = exp − exp − β  

    

where γ and β are obtained through the method of moments: µˆ = γ + kβ = γ + 0.577 β the variance σ2 is given by:

σ = 2

π

2

6 So the method of moments gives:

µˆ =

β ≅ 1.645β

1 N ∑ H m =2.2690 m N m =1

2

and

2

σˆ 2 =

1 N

N

2

∑ ( H m − µˆ ) 2 =1.1429 m

m =1

the above values give:

σˆ = 1.645β ⇒ β = 2

σˆ

2

2

1.645

=0.8335 m-1

µˆ = γ + 0.577 β ⇒ γ = µˆ − 0.577 β =1.7880 m The PDF is P(H) =exp(−exp[−1.1997 (H−1.7880)]) The transformed variable is : W=-ln{-ln[P(H)]}= 1.1997 (H-1.7880). The significant wave heights are represented in figure 7.1. On the y-axis the transformed variable W and the PDF P is reported. The application of the χ2 test requires the definition of the following four significant wave height classes: 1) Hs<1.5 m 2) 1.5 ≤Hs<2.5 m 3) 2.5 ≤ Hs <3.5 m 4) 3.5≤ Hs <5.0 m The significant wave heights are obtained as: H=1.7880−{ln[−lnP(H)]}/1.1997 2

In order to perform χ test, in table 7.4 the analytical and empirical significant wave heights are given, with a confidence level P=95%. The analytical value of χ2, reported in table 7.2 with reference to the confidence level of 95% and the number of degrees of freedom, is greater than

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the empirical value. The hypothesis that the sample follows the Gumbel distribution is so verified with a confidence value P=95%. The mean time interval between the peak values of the significant wave height in the storms is r=9years/29=0.30 years. The degrees of freedom are given by ν=k-1 where k are the four classes. The number of linearly independent conditions is 3; one of these conditions identifies the fourth class, the other two conditions are associated with the parameters calculated from the sample. The empirical value of χ2 is calculated as: χ2=(9-8)2/9+(10-10)2/10+(5-5)2/5+(5-6)2/5=0.31 Tr [years] 6.00

W

0.9975

5.00

P(Hs) 0.9932

4.00

0.9818

3.00

0.9514

2.00

0.8734

1.00

0.6922

0.00

0.3678

-1.00

0.0659

-2.00

0.0006 0.00 1.00 2.00 3.00 4.00 5.00 6.00 7.00 8.00 9.00 10.00

Tr=100 yr Tr=50 yr Tr=20 yr Tr=10 yr

Hs [m]

Figure 7.1 – Gumbel distribution fitted with data. The cumulate probability can be calculated from the definition of return period, and the “design” wave height can be obtained by knowing the empirical parameters above calculated: TR 1 = r 1 − P ( H TR )

P(HTR)=1−r/TR =1−0.30/50 = 0.994 W=5.113=1.1997 (H-1.7880)

LONG TERM WAVE STATISTICS

141

HTR =(5.113/1.1997) + 1.7880 =6.05m The wave period can be given by (Boccotti, 1997): 1/ 2 TH TR = 0.95 ⋅ 9π {H Tr / 4 g } =10.55s

The calculated value of the design wave height is 6.05m (TR=50 years) and its wave period is 10.55s. Table 7.3 - Ordered peak significant wave heights and associated Q(Hm).

m

Hm

Q(Hm)

1

4.4

0.9808

2

3.9

0.9464

3

3.8

0.9121

4

3.7

0.8777

5

3.6

0.8434

6

3.5

0.8091

7

3.4

0.7747

8

3.1

0.7404

9

3

0.7060

10

2.8

0.6717

11

2.7

0.6374

12

2.4

0.6030

13

2.3

0.5687

14

2.2

0.5343

15

2.1

0.5000

16

2

0.4657

17

1.9

0.4313

18

1.8

0.3970

19

1.7

0.3626

20

1.6

0.3283

21

1.5

0.2940

22

1.4

0.2596

23

1.3

0.2253

24

1.2

0.1909

25

1.1

0.1566

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INTRODUCTION TO COASTAL DYNAMICS AND SHORELINE PROTECTION

26

1

0.1223

27

0.9

0.0879

28

0.8

0.0536

29

0.7

0.0192

Table 7.4 - Analytical and empirical significant wave heights Hm[m]

class

Hmt[m]

class

4.4

4 4 4 4 4 4 3 3 3 3 3 2 2 2 2 2 2 2 2 2 2 1 1 1 1 1 1 1 1

4.8

4 4 4 4 4 3 3 3 3 3 2 2 2 2 2 2 2 2 2 2 1 1 1 1 1 1 1 1 1

3.9 3.8 3.7 3.6 3.5 3.4 3.1 3 2.8 2.7 2.4 2.3 2.2 2.1 2 1.9 1.8 1.7 1.6 1.5 1.4 1.3 1.2 1.1 1 0.9 0.8 0.7

3.9 3.5 3.5 3.5 2.8 2.6 2.6 2.5 2.5 2.2 2.1 2.0 1.9 1.8 1.7 1.6 1.6 1.5 1.5 1.3 1.2 1.2 1.1 1.0 0.9 0.7 0.6 0.3

Chapter 8 Wave transformation in the coastal zone 1 Wave energy and energy flux When a wave train propagates through a fluid, part of its of energy is transmitted throughout the fluid; the rate of energy transfer matches the group celerity Cg which is the propagation velocity of the wave packet obtained by the envelope of the group of random waves. Energy consists of two contributions: potential energy resulting from the displacement of the free surface and kinetic energy due to the motion of water particles throughout the fluid. The total energy and its transmission is of extreme importance in determining how wave properties change when a wave approaches to a shore or/and interacts with a structure. 1.1 Potential Energy During the propagation of waves, water particles are displaced against gravity from a position of equilibrium (e.g. calm-level z=0) to a higher (or lower) position (z= η) resulting in a change of potential energy. It can be shown that when the water is at rest with a uniform free surface elevation, the potential energy assumes its minimum value. The displacement of free surface due to the movement of an aggregate of water particles, requires that work must be done on the system against gravitational field and this causes an increase of potential energy. With reference to figure 8.1, mean potential energy per unit surface area is given as the difference of potential energy occurring at the transit of the wave and potential energy without the wave, that is relative to the mean sea level. E p = (E p )d +η − (E p )d

(8.1)

Assuming the bottom as a reference level, the elementary potential energy of a small water column with mass dm in figure 8.1, is given by: (dEp )d +η = dmgz

(8.2)

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INTRODUCTION TO COASTAL DYNAMICS AND SHORELINE PROTECTION

where dm is the elementary mass per unit width given by dm = ρ(d + η) dx and z = (d + η) / 2 is the height of the gravity centre of the water column.

The potential energy (E p )d + η averaged over one wave length for a progressive wave of height H is:

(E p )d + η = ρg = 2L

x+L

∫( x

1 L

x+L

∫

dE p =

x

1 L

x+L

∫

ρg

x

(d + η) 2 dx = 2

)

d2 H2 d + 2 dη+ η dx = ρg + ρg 2 16 2

2

(8.3)

In absence of wave, potential energy is related to the weight of the water column and is given by:

(E p ) = ρg

d2 2

(8.4)

Thus the wave potential energy per unit area depends only on the wave height:

E p = (E p )d + η − (E p )d = ρg

H2 16

(8.5)

1.2 Kinetic Energy During the transit of the wave, water particles move within the fluid describing elliptical paths. Mean kinetic energy associated to a water parcel with mass dm is proportional to the square of particle velocity and is given by

dE c = dm

u 2 + w2 u 2 + w2 = gdxdz 2 2

(8.6)

Mean kinetic energy averaged over one wave length per unit width is: A 

  η 2 1 u + w2 Ec = dx g dz L x 2 −d x +L

∫

∫

(8.7)

Substituting the known solutions for velocity components u and w: agk cosh k ( d + z ) sin( kx − σt ) cosh kd σ agk sinh k (d + z ) w=− cos( kx − σt ) cosh kd σ

u=−

the integral A can be written as:

(8.8)

WAVE TRANSFORMATION IN THE COASTAL ZONE

145

2

A=

g ⎛ H gk 1 ⎞ ⎜ ⎟ ⋅ 2 ⎝ 2 σ cosh kd ⎠

(8.9)

η

∫ cosh

2

2

2

2

k(d + z)sin (kx − σt) + sinh k(d + z)cos (kx − σt) dz

−d

Using the trigonometric identities: cosh 2 (d + z) =

1 [1 + cosh 2 k(d + z)] ; sinh 2 k(d + z) = − 1 [1 − cosh 2 k(d + z)] 2 2

(8.10)

After some algebra one obtains: 2

Ec =

ρ ⎛ H gk 1 ⎞ 1 sinh2 kd ⎜ ⎟ 4 ⎝ 2 σ cosh kd ⎠ 2 k

(8.11)

Further, making use of the identity sinh2 k d = 2sinh k d cosh k d , kinetic energy may be written as: E c = ρg

H2 16

(8.12)

where the dispersion relation has been used. Thus the total wave energy per unit length will be determined by the sum of mean potential and kinetic energy averaged over the same wave length: E t = E p + E c = ρg

H2 H2 H2 + ρg = ρg 16 16 8

(8.13)

It is interesting to underline how in linear theory neither mean potential energy E p nor mean kinetic energy Ec do depend on depth and on wave length, but they

are both functions of wave height only. 1.3 Energy Flux In small amplitude theory, waves do not transport mass during the propagation since particle trajectories are closed circles whose ray decrease with depth and water paths are not affected by bottom. Yet waves transmit energy, with a speed denoted as group velocity. The rate at which energy is transferred has the dimension of a power and is called energy flux Ef which represents a useful concept in several applications, particularly when dealing with a number of wave phenomena that involve energy transfer. In fig. 8.1 consider the section separating the fluid into two parts. Energy flux represents the rate at which work is performed by the fluid on the left of section AA' over the fluid to the right of the section; or in other words energy flux is the average rate of kinetic and potential energy transfer across an elemental plane area of unit width.

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INTRODUCTION TO COASTAL DYNAMICS AND SHORELINE PROTECTION

The total work is given by the integral over depth of dynamic pressure multiplied by horizontal velocity. Thus energy flux is η

Ef =

⋅ u ⋅ dz

(8.14)

coshk(d + z) cosh kd

(8.15)

∫p

d

−d

p d = ρgη

after some algebra, reminding that c=σ/k, energy flux is given by: E f = Et

2k d ⎤ c⎡ 1+ = Et c g ⎢ 2 ⎣ sinh 2 kd ⎥⎦

(8.16)

where cg is the group celerity, given by: cg =

c⎡ 2k d ⎤ 1+ 2 ⎢⎣ sinh 2 k d ⎥⎦

(8.17)

In deep water sinh 2 kd → ∞ ⇒ c g =

c 2

(8.18)

In shallow water sinh 2 k d → 2 k d ⇒ c g = c

(8.19)

2 Refraction and shoaling When a regular wave train approaches the shore coming from deep water, wave properties undergo several changes due to the presence of obstructions in the flow or to variations of water depth. One common case for wave structure interaction is a wave train hitting a vertical wall (a seawall or a breakwater). The boundary condition of the wall imposes that the wave energy is totally reflected backwards (energy is not actually completely reflected since a portion is dissipated and another part is transmitted). The resultant wave field may be described as a superposition of an incident and a reflected wave components. The interaction of a regular wave train with bottom variations is developed by considering two different cases depending on the angle of approach of the wave. In fact if wave fronts are parallel to seabed contours (that is propagation in crossshore direction) the interaction is due only to the change of wave group celerity (the so-called shoaling). If the wave train approaches obliquely to the bathymetry, wave fronts are imposed to rotate by bottom condition so as to become more parallel to the shore, (the so-called refraction).

WAVE TRANSFORMATION IN THE COASTAL ZONE

A

147

η(x,t) x C. d

P

h

z

dx

A

Figure 8.1 - Definition sketch for potential energy and energy flux. The most common assumption for wave refraction and shoaling is to consider a steady-state, monochromatic (and thereby long-crested) wave propagating across a region of a straight shoreline with all contours evenly spaced and parallel to the shoreline. In addition, no current is present.

Shoaling In the hypotheses of a gently sloping bottom, (that is negligible wave reflection) and that energy is neither supplied (by the wind) nor dissipated by wave breaking or by bottom friction, the rate of energy transfer in the direction of propagation can be assumed constant. Furthermore, the motion is supposed to be twodimensional and the period constant. Under these assumptions the rate of energy transfer, (that is energy flux), is constant so its expression for deep water can be matched to the one for intermediate depth. On deep water:

E0 C g 0 = EC g

(8.20) 1

H ⎛ C go ⎞ 2 1 1 =⎜ ρ gH 02C g 0 = ρ gH 2C g → ⎟ H 0 ⎜⎝ C g ⎟⎠ 8 8 C g0 =

C 0 gT 0 = 2 4π

(8.21)

(8.22)

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INTRODUCTION TO COASTAL DYNAMICS AND SHORELINE PROTECTION

On intermediate depth:

Cg =

C 2

2kd ⎤ gT 2kd ⎤ ⎡ ⎡ ⎢1 + sinh2kd ⎥ = 4π tanhkd ⎢1 + sinh2kd ⎥ ⎣ ⎦ ⎣ ⎦

(8.23)

Assuming the period to be constant with depth T=T0=cost., one can write Cg0

−1

⎧ 2kd ⎤ ⎫ ⎡ = ⎨tanh kd ⎢1 + ⎥⎬ = Cg ⎩ ⎣ sinh 2kd ⎦ ⎭

(8.24) cosh kd sinh 2kd cosh kd 2 sinh kd cosh kd = = sinh kd 2 kd + sinh 2kd sinh kd 2kd + sinh 2kd 2 cosh 2 kd = 2kd + sinh 2kd The ratio of wave height in intermediate depth H to the deep water wave height H0 defines the shoaling coefficient: =

1

H ⎛⎜ 2cosh 2 kd ⎞⎟ 2 = = KS H o ⎜⎝ 2 kd + sinh 2 kd ⎟⎠

(8.25)

Refraction If a wave crest initially has some angle of approach with respect to the bed contours, a portion of the wave will travel in shallower water than another longcrest point, so in accordance with dispersion relation its velocity will be lower than the portion propagating on deeper water. Consequently differential speed along the wave crest causes the crest to turn in order to become more parallel to shore. Figure 8.2 shows this process for a time interval dt, where the wave celerities are obtained by simply applying Snells’ law l s sin α 2 l s sin α1 (8.26) C2 = 2 = C1 = 1 = dt dt dt dt where α is the angle that wave crest makes with seabed contours. The ratio: C1 sin α 1 (8.27) = C 2 sin α 2 may be applied to whatever depth. If reflection is negligible and if energy is not dissipated during the propagation, the rate of energy transfer between two orthogonals remains constant. It must be noted that the distance b between the two orthogonals is not constant such as the shoaling case effects, and therefore the equation of energy transfer may be properly adjusted as: 1

1

H ⎛ b0 ⎞ 2 ⎛⎜ C go ⎞⎟ 2 =⎜ ⎟ E 0 C g 0 b0 = EC g b = cost ; → H 0 ⎜⎝ b ⎟⎠ ⎜⎝ C g ⎟⎠

(8.28)

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149

The shore-parallel component of the distance s between the wave orthogonals is independent on the position on the shore (that is on the depth) (fig 8.2) b0 b b cosα 0 = = cost. → 0 = b cosα 0 cosα cosα t

(8.29)

t+δt WAVE FRONTS l1

α1

1

SEABED CONTOUR

α2

2 l2

S WAVE ORTHOGONALS

Figure 8.2 –The refraction of wave fronts and orthogonals in a time interval dt. 1

Remembering that cos α = (1 − sin 2 α ) 2 the ratio may be written as: ⎛ b0 ⎜ ⎜ cos α 0 ⎝

1

⎞ 2 ⎡1 − sin 2 α ⎤ ⎟ =⎢ ⎥ ⎟ 2 ⎢⎣ cos α 0 ⎦⎥ ⎠

−

1 4

sin α C L = = = tanh k d → sin α = sin α0 tanh k d C0 sin α0 L0

(8.30) (8.31)

The ratio of the distance between two adjacent wave rays on deep water to the one on the shore defines the refraction coefficient. 1

2 ⎛ b0 ⎞ 2 ⎡1 − sin α0 ⎤ ⎜ ⎟ =⎢ ⎥ 2 ⎝b⎠ ⎢⎣ cos α0 ⎦⎥

−

1 4

= Kr

(8.32)

Definitely wave height at a generic depth may be determined from deep water wave height through the following equation:

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1

H ⎛ b0 ⎞ 2 ⎛⎜ C g0 =⎜ ⎟ H 0 ⎝ b ⎠ ⎜⎝ C g

1

⎞2 ⎟ = ⎟ ⎠

⎡1 − sin 2 α 0 tanh 2 kd ⎤ =⎢ ⎥ cos 2 α 0 ⎦ ⎣ H = H0 Kr Ks

−

1 4

1

⎡ 2 cosh 2 kd ⎤ 2 ⎢ ⎥ = Kr Ks ⎣ 2kd + sinh 2kd ⎦

(8.33)

2.1 Discussion on Kr and Ks In the case of rectilinear and parallel seabed contours, expression of shoaling coefficient are given by: 1

1

⎛ Cg Ks = ⎜ 0 ⎜C ⎝ g

⎞ 2 ⎡ 2 cos 2 kd ⎤ 2 ⎟ =⎢ ⎥ ⎟ ⎣ 2kd + sinh 2kd ⎦ ⎠

1

−

2 2 ⎛ b ⎞ 2 ⎡1 − sin α0 tanh kd ⎤ Kr = ⎜ 0 ⎟ = ⎢ ⎥ cos 2α 0 ⎝b⎠ ⎢⎣ ⎦⎥

1 4

(8.34)

(8.35)

Shoaling accounts for group celerity variation with depth and is therefore given by the ratio of wave height at the generic depth d and deep water wave height in absence of orthogonal deformation. Refraction coefficient is related to the variation of the distance between orthogonals and may be seen as the ratio of generic incident wave height at the depth d and the hypothetical wave height which would occur in the same depth if wave incidence is normal, that is wave height due to solely shoaling effect. Eq. (8.34) shows that KS depends on depth and on wave number k, that is on d and L. The analysis of the dependence of KS on the ratio d/Lo, sketched in fig. 8.3 shows that is possible to distinguish three different behaviors according to the values of d/Lo.

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151

Figure 8.3 - Estimation of shoaling coefficient (Goda, 1985). In the range d/Lo>0.16, KS decreases with the ratio d/Lo until it reaches a minimum value for d/Lo=0.16. In the range 0.06
(8.36)

In table 8.1 values of Kr and α are reported for decreasing depth for a deep water wave of period T0 =10 s, whose wave fronts are 40° tilted with respect to offshore bathymetry.

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Figure 8.4 - Kr curves reported with dashed line as a function of the ratio d/gT02 for constant deep water incidence angle of approach α0 . Table 8.1 – Kr values for decreasing depths. d(m)

d/gT2

α(°N)

Kr

100

0.10

40

1

80

0.08

40

1

50

0.05

38

1

20

0.02

30

0.94

10

0.01

22

0.91

Figure 8.5 shows with dashed lines the curves of the product KS Kr as a function of the ratio d/gT2 for constant values of deep water angle α0. In table 8.2 the values of KS Kr are reported, that is the ratio H/H0 whose fronts approach the coastline with a 40° angle; from the comparison between Kr KS and Kr it is possible to obtain KS values. It is noticeable that with decreasing depth, the product Kr KS first decreases, then increases and this behaviour is due to the shoaling coefficient KS . In fact the KS value almost equal to 1 is verified in correspondence of d/L0=0.06 (see also fig. 8.3).

WAVE TRANSFORMATION IN THE COASTAL ZONE

153

Table 8.2 - Kr Ks values for decreasing depths d(m)

d/gT 2

d/L o

α (°N)

KrKs

Kr

100

0.10

0.64

40

1

1

80

0.08

0.51

40

0.99

0.99

50

0.05

0.32

38

0.94

0.94

20

0.02

0.13

30

0.86

0.91

10

0.01

0.06

22

0.80

0.89

.

3 Total reflection When a regular wave train hits an obstacle, a part of the incident energy is reflected, a part is transmitted beyond the obstacle, the remaining is dissipated within the obstacle. In the case of a vertical, hard structure, the fraction of reflected wave energy can be large, while for permeable structures or gentle slopes, the reflection will be much less. In the ideal case of total reflection of a vertical wall, incident energy is completely reflected (that is no energy is transmitted or dissipated) and surface elevation η must satisfy the boundary condition imposed by the wall that velocity is null u=0 in the abscissa of vertical wall. For small amplitude waves, reflection condition is satisfied in terms of the component incident progressive wave train by superimposing a second reflected progressive wave train travelling in the opposite direction with a wave height matching the incident wave height. It is largely demonstrated that the velocity potential function and the profile of the retrograde profile of the retrograde wave are respectively given by:

φr = −

ag cosh k ( d + z ) cos(kx + σt ) cosh kd σ

ηr = a sin (kx − σt)

(8.37) (8.38)

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Figure 8.5 –Values of the product Kr Ks as a function of relative depth. Therefore the reflection condition is expressed by superimposing two surface elevations η1 progressive and η2 retrograde given by: η1 = a sin(kx − σt + ϑ 1 )

η 2 = a sin(kx − σt + ϑ 2 )

(8.39)

being σ1 and σ2 the phases of incident and reflected waves. Operating on the position of the origin, one may impose that σ1=0 and thus σ2 = σ will be the relative phase between the two waves. Each component motion separately satisfies the governing equation and since they are linear because of the small amplitude wave theory, the resulting surface elevation is determined by superposing the two components so that the combined motion still is a solution of wave equation.

η t = a sin(kx − σt ) + a sin(kx − σt + ϑ ) = = a sin(kx − σt ) + a sin(kx − σt ) cos ϑ + a cos(kx − σt ) sin ϑ

(8.40)

In order to estimate σ, the reflection condition for horizontal velocity may be written in the abscissa x=B corresponding to the vertical wall position.

⎡ ∂φ ⎤ ut = −⎢ t ⎥ =0 ⎣ ∂x ⎦ x =B

(8.41)

WAVE TRANSFORMATION IN THE COASTAL ZONE

155

Like the surface elevation component, the wave potential is given by the sum of the incident wave and reflected wave potential. φT = φ 1 + φ 2 =

ag cosh k(d + z) [cos (kx − σt) − cos (kx + σt + ϑ)] σ cosh kd

(8.42)

from which velocity components may be determined by deriving potential function with respect to x and z: uT =

∂φt agk cosh k(d + z) [sin (kx − σt) − sin (kx + σt + ϑ)] = σ cosh kd ∂x

(8.43)

after some algebra one obtains:

η t = 2a sin(kB − σt ) cos(kx − kB)

(8.44)

The resulting surface elevation represents a standing wave in which the height is twice that of the incident wave. From equation (8.44) it is evident that for some value of x, ηt=0 at any instant t. These points which are called nodes represent points of continuously null surface elevation and are such that:

π →x= 2 π L = (2 n − 1) + B = (2 n− 1) + B 2k 4

cos(kx− kB) = 0 → kx − kB = (2 n− 1)

(8.45)

Points in which elevation assumes maximum value are called antinodes and correspond to: (8.46) cos(kx− kB) = 1 → kx− kB Vertical and horizontal components of velocity may be deduced from equation (8.42), with σ = π − 2kB and again by deriving potential velocity function with respect to x and to z: 2ag cosh k (d + z ) [cos(kB − σt ) cos(kx − kB)] φt = σ cosh kd (8.47) ∂φ t 2agk cosh k ( d + z ) ut = − = [cos(kB − σt ) sin(kx − kB)] ∂x σ cosh kd Therefore vertical velocity wt is null in the nodes and maximum in the antinodes as well as surface elevation, whereas horizontal velocity is null in the antinodes and maximum in the nodes. In most practical cases wave reflection is not complete and the amplitude of the reflected wave is consequently less than that of the incident wave. It is often used a reflection coefficient Kr defined as the ratio of reflected wave height to incident wave height so that total surface elevation can be written as:

ηt =

1 H [cos(kx − σt ) + K r cos(kx + σt )] 2

(8.48)

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x

Figure 8.6 – Sketch of a fully reflected wave.

4 Wave diffraction When an incoming wave encounters a vertical barrier such as a breakwater or an island, wave fronts tend to curve around the barrier and penetrate into the sheltered area; so wave disturbance is transmitted into the “geometric shadow zone”. A qualitative understanding of this phenomenon is important in investigating the wave action behind breakwaters and around small islands or large off-shore structures. It is a process by which energy spreads laterally perpendicular to the main direction of wave propagation. Diffractive phenomena may be also of importance in the case of wave propagation across long distances; in such cases classical wave refraction effects considered alone would indicate zones of wave convergence and an extremely high concentration of wave energy. As the energy accumulates between two converging wave orthogonals, some of this energy will spread towards regions of lower energy density, leading to a redistribution of wave energy across the wave rays. Refraction and diffraction obviously take place simultaneously, but a separation of the two strictly related processes constitutes an academic manner to investigate the problem of wave transformation. It must be specified that refraction is concerned with gently changing depth causing (besides the wave shoaling) wave crest and wave rays to bend, whereas diffraction is concerned with zero depth changes and solves for sudden changes in wave conditions such as the presence of obstacles that cause the wave energy to be forced across wave rays. General methods of treatment for diffractive phenomenon assume the fluid incompressible and the motion irrotational so that the velocity potential satisfies the Laplace equation. Furthermore the theory is based on linear wave assumptions so that wave height is considered to be small.

5 Numerical models for wave propagation A computational model is a means of representing the most relevant processes in a suitable schematic form and solving the mathematical equations describing the series of physical processes involved.

WAVE TRANSFORMATION IN THE COASTAL ZONE

157

Practical wave prediction applications involve different processes, such as local wind generated sea and swell, variable winds and irregular bathymetry and coastline. Wave prediction in these demanding conditions requires a model which represents the physical processes occurring in nature. The processes to be considered are propagation (diffraction and refraction), conservative wave-wave interaction (between triads and between quadruplets) and energy sources and sinks (whitecapping, shallow water breaking and bed friction). In deriving the mathematical equations simplifications and assumption are made about the nature of physical process depending on the degree of approximation and resolution for the intended class of application, because it determines which processes are dominant. A comprehensive model which incorporates all processes and is applicable in all situations would be computationally very expensive. Instead a variety of models have been proposed for application in particular situations. For a broad classification, we shall distinguish four domains of application: • • • •

oceans: bottom influence can be neglected shelf sea: area between the deep ocean and the shoaling zone nearshore area: area where shoaling becomes significant harbours: representative for various kinds of problems of wave-structure interaction.

Models can be classified according to different criteria. Wave generation models are those derived on the basis that sea-bed effects are negligible compared with those of the wind. Wave transformation models can be divided into two general classes according to a criterion based on the rate of spatial evolution of wave field, which in turn is determined by the strength of the processes causing the evolution. From this viewpoint, one can distinguish between rapidly varying waves and slowly varying waves. In rapidly varying waves, the phase averaged local properties vary strongly within the distances of the order of a wavelength or less. They require phaseresolving models. In slowly varying waves, the variation of these properties on the scale of a wavelength are weak, the existence of only slow variations implies length scale (wavelength and slow variation) allowing the treatment of the wave field as quasi-uniform. Slowly varying waves can be described with phase averaged models. Table 8.3 - Negl-negligible; Signif-significant; Domin-dominant; Min Imp: minor importance. Physical processes Diffraction Depth Refraction/Shoaling Current Refraction

Deep Oceans Negl

Negl

Nearshore Zone less import.

Negl

Signif

Domin

Signif

Negl

Min Imp

Signif

Negl

Shelf Sea

Harbours Domin

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Quadruplet Interaction Triad Interaction Atmospheric Input White-Capping Depth Breaking Bottom Friction

Domin

Domin

Min Imp

Negl

Negl

Min Imp

Signif

Min Imp

Domin Domin Negl Negl

Domin Domin Min Imp Domin

Min Imp Min Imp Domin Signif

Negl Negl Negl Negl

Phase-resolving models comprise equations describing the instantaneous state of motion, either in the time domain or in the frequency domain (with amplitudes and phases). Phase-averaged models deal with averaged properties, using kinematic propagation equations expressed in terms of wavenumber (magnitude and direction), group velocity, wave rays and so on, and an energy balance in one form or another. The following stages may be needed to be passed through in selecting a model: • • •

What are likely to be the dominant physical processes- this requires engineering judgement. What models are available to represent these processes- this may result in several models or a combination of models being identified. What are time/ financial/ quality constraints on the modelling – for example a highly sophisticated model may be not needed for a preliminary feasibility study, it may be sufficient to use a simplified model and accept that not all processes are modelled.

A model should be made or chosen such that it will yield the required output at minimum effort. It is important to remember that however efficient the solution method, the accuracy of representation of the physical processes is only as good as the assumptions made in deriving the equations. Wind-generated waves are essentially random phases, whose specific values are ultimately irrelevant in application (and impossible to predict because they cannot be related to causative factors). The spectral wave energy distribution is considered to contain the necessary and sufficient information to determine the most important dynamic and probabilistic properties of the wave field. This rests on the assumption of independent and uniformly distributed random phases in lowest order. However it is important that higher-order properties including bound harmonics and their effects such as skewness can be derived from the spectrum, if necessary, under the additional (mild) assumption of a (quasi-) homogeneous wave field. The energy spectrum is a phase-averaged quantity; it is most economical to compute such quantity with a phase-averaged model, if at all possible. Phaseresolving models are computationally much more demanding (per unit area of computational domain) so they should be used only where they are strictly needed.

WAVE TRANSFORMATION IN THE COASTAL ZONE

159

5.1 Phase-averaged models

Phase-averaged models are for slowly varying waves in regions of slowly varying depth and/or currents. In their derivation, the vertical coordinate is separated from the horizontal ones. The vertical coordinate functions as crossspace, containing the modal structure, and the horizontal plane is the propagation space, containing the phase structure. The modal structure is determined first, independent of the horizontal variation, and expressed analytically so that depth integration can be carried out. The results are used to derive transport equations describing the horizontal propagation, which are usually integrated numerically. These can describe instantaneous values (phase-resolving models), but for slowly varying waves they are phase-averaged. The small cross-shore extent of the computational region of near-shore models implies a relatively short travel time of the waves from the seaward boundary to the coast, short compared to the timescale of variation in incoming wave field and of tidally-induced changes in depth and currents. This can be exploited to make the model stationary and thereby more computationally efficient. Source terms The small cross-shore extent of the computational region of nearshore models also has implications for the relative importance of source terms, in that it makes local generation, whitecapping and quadruplet interactions relatively unimportant (which does not necessarily imply “negligible”). The wave field is mainly determined by propagation (assuming exposure to open water) and depth-induced breaking; in very shallow water, triad interactions become important. • •

•

•

Wind input and whitecapping, if included, are commonly taken as in shelf seas, i.e. as in deep water with local values of wavenumber and phasespeed. The quadruplets interactions could be taken as in deep water, with the depth-dependent upscaling as in WAM , but this is nominally valid for k d > 1 . This condition is not fulfilled in the very shallow-water nearshore region. Rather than having to cope with a divergent source term, it may be better to omit it altogether. Its contribution is relatively weak anyway, in comparison with the effects of shoaling and breaking. In very shallow-water, triad interactions become near-resonant, leading to significant cross-spectral energy transfer and harmonic generation. Abreu et al. (1992) first derived a Phase-averaged formulation for it, suitable as a source term in a spectral energy balance. However their derivation is restricted to nondispersive shallow water waves ( C = (gd ) 0.5 ). This is a serious restriction, which implies a continued, resonant, one-way transfer to the higher harmonics, in contrast to the case of weakly dispersive waves for which the transfer is non-resonant and back-and-forth, because of the mismatch in phase speed. For bottom friction, the same comment applies as for shelf sea models, except that its effect is relatively small because of the shorter propagation distances and the stronger effect of breaking.

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•

•

Depth induced breaking is the dominant source term in nearshore, shallow water areas. The derivation of models for the average dissipation rate due to this process in random waves has made it possible to make fairly realistic predictions of wave heights in nearshore regions, with typical relative errors less than 10% (Battjes and Jansen, 1978; Thornthon and Guza, 1983); Battjes and Stive, 1985; Roelvink, 1993). In this connection, it is worthwhile to point out that a good representation of the dissipation is not only important for the prediction of wave field itself but also for that of wind-driven currents, since the wave-induced force driving the currents is proportional to the local dissipation rate (Longuet-Higgins, 1973). Models with a good representation of the energy loss due to depth-dependent breaking are essential for simulation of coastal morphological processes.

The breaking-wave models referred to above predict only the total dissipation rate, but not its spectral distribution, which is required in spectral energy balance models. A possible approach is to distribute the total in proportion to the local spectral density, on the assumption that the breaking does not alter the spectral shape.

6 Finite depth spectral wave models The TMA model, proposed by Bouws et al., (1985) gives in stationary case the saturation spectrum S(f,d) on finite depth d for a given JONSWAP spectrum S(f,∞)=S0(f) on deep water. The TMA model, tested using field data series (TEXEL, MARSEN, ARSLOE), is based on the self-similarity hypothesis formulated by Kitaigorodski et al. (1975), which implies the maintenance of the analytical formulation of spectrum in the deep water wavenumber space for whatever depth. If the wavenumber k0 is replaced by the finite depth wave number k, the spectrum F0(k0) is replaced by the corresponding spectrum F(k,d) , being k0 and k related by the dispersion equation. For given values of F0(k0) and F(k,d) the associated frequency spectra S0(k0) and S(k,d) are given by:

S 0 (f) = F0 (k0 )

(8.49)

S(f, d) = F(k, d) dk/df

(8.50)

where the derivative can be obtained through the dispersion relation which can be expressed both on deep and shallow water, respectively:

(2 πf)2 = gk0

(8.52)

( 2ππ ) = gk tanh kd

(8.53)

2

where g is the acceleration due to gravity. The TMA frequency spectrum S(f,d) = S TMA (k,d) is given by:

S TMA (f,d) = S PH 0 (f)φ PM 0 (f,f p0 )φ J 0 (f,f p 0 ,γγ,ωφ k (f,d)

(8.54)

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161

In the above equations, α0 is the equilibrium parameter; γ0 is the amplification factor; ω0 is the peak amplitude parameter; φPM0 and φJ0 are the shape functions suggested by Pierson and Moskowitz (1964) and Hasselmann et al., (1973); φk is the depth function given by Kitaigorodskii et al (1975). The peak frequency fp0 and the dimensionless quantities α0 , γ0 and ω0 are given by: fp0=3.5 ~ x -0.33 g/U0

α0=0.076 ~ x

(8.55)

-0.22

(8.56)

γ0 = 3.3 if f ≤ f p and 0

ω 0 = 0.07

(8.57) ω 0 = 0.09

~ x = gx/U 02

if f ≥ f p 0

(8.58) (8.59)

The equation (8.59) is the dimensionless fetch parameter, being x the fetch and U0 the wind speed at 10 meters above the mean sea level. The dimensionless functions φPM0 , φJ0 and φk are given by:

φ PM = exp { − 1.25(f/f 0

p0

)−4 }

φJ = exp{ ln γ0 exp[ − 0.5( f f p − 1)2 /ω02 ]} 0

0

φk = χ −2{1 + 2σ d2 χ/ sinh 2σ d2 χ}−1

(8.60) (8.61) (8.62)

where σd is the dimensionless depth given by:

σ d = 2ππf(d/g ) 1 / 2

(8.63)

The dimensionless function χ = χ(f,d) is given by: χ tanh(σd2χ) =1

(8.64)

and depends on the wave number: χ = 1/ tanhkd

(8.65)

which is given by the dispersion relation. k can be readily calculated through the Fenton (1990) equation: k = ( 2ππ ) 2 /{ f ( tanhσ 3d / 2 ) 2 / 3 }

(8.66)

which allows to obtain the equation (8.65) as: χ = 1 / tanh ξ

; ξ = σ d2 /(tanh(σ d2 )3 / 2 ) 2 / 3

(8.67)

By considering above equations, the saturation spectrum STMA(f,d) can be correctly written as:

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STMA (f,d) = α0 g 2 (2 π ) −4 f −5 exp{−1.25( f f p0 ) −4 } ⋅ ⋅ exp{ln γ0 exp[−0.5( f/f p 0 − 1) /ω02 ]} ⋅

(8.68)

⋅ { χ −2 [1 + 2 σ d2 χ sinh2 σ d2 χ ]−1} The last equation is correctly associated with the mean JONSWAP spectrum, being φk=1 on deep water. 6.1 Other finite depth spectral wave models

As shown in Scarsi et al. (1992), the TMA model gives the following inconsistency: the 0-order momentum m0 obtained on deep water using the TMA model is smaller than the wavenumber frequency spectrum. This inconvenience can be avoided through a full application of the self-similarity hypothesis. TMAR spectra at fp0 =0.10 Hz and U~0 =0.2 for six values of depth d in figure 8.7 are depicted. Each curve gives the TMAR spectrum at a different depth d, which decreases from 120 to 12 m. d=120 m is associated with infinite depth case. The comparison between the table 8.4 and the figure 8.7 shows that the saturation spectrum S T (f,d) reaches a maximum value at frequency nearest fp0 , even if S T (f,d) is a function of χ and χp. The frequency fp0 maintains all the characteristics to the peak frequency also at finite depth, so we can conclude that fp = fp0 and Tp = Tp0 . Spectral waves associated with the above spectra are generally stable in a given field of σhp valid under linear assumptions. In fact, the σdp field depends on the parameter U~0 : the smallest depth d in figure 8.8 and in table 8.4 (d=12 m) gives a σdp value contained in the prescript field at U~0 =0.20. 6.2 Phase-resolving models

Phase-resolving models are required for rapidly varying waves and can be divided in a few basically distinct classes, depending on: 1- the nonlinearity (ak or a/d), 2-relative depth (kd) and 3-bottom slope (α), where a is wave amplitude, d is the depth, and k the wavenumber (characteristic values). If no assumption are made about the magnitude of these parameters except that they must correspond to physically realizable conditions, then a class of the so-called “exact” models are needed (the term “exact” refers to the absence of approximations within the domain of irrotational gravity waves on the surface of an incompressible , non viscous liquid). In the exact models, the vertical structure in the waves is determined simultaneously with and in an equal manner as the horizontal variations, which makes them very computing-intensive. (In contrast the assumption of mild bottom slope allows the separation of the vertical coordinate from the horizontal one(s), which leads to the following , more efficient depth-integrated models.)

WAVE TRANSFORMATION IN THE COASTAL ZONE

163

ST (f, d)

m2/Hz

d =120 m 60 m 45 m 30 m 20 m 12 m

Figure 8.7– The TMAR spectrum ( ST (f, d) ) for different d (Scarsi et al.,1995) Table 8.4 - Spectral wave height H(m0)T obtained by the TMAR model and the corresponding breaking wave height H(m0)f calculated through the Goda (1975) formulation on horizontal bottom. d

H(m0)T

H(m0)f

(m)

(m)

(m)

120

5.83

22.80

60

5.76

19.60

45

5.62

17.40

30

5.24

13.95

20

4.72

10.60

7.10 4.01 12 In the hypothesis of arbitrary depth is (kd=O(1)), the bottom slope is small (α<
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The leading-order nonlinear term in these equations is quadratic in the amplitude. If only forward propagation is considered (i.e. no reflection) the Boussinesq equation reduces to the Korteweg-de Vries equations For α≤ kd<<1, a/d=O(1), the classical shallow water equations are obtained. 6.2.1 Boundary Integral Models

This class of models solves the Laplace equation: ∂ 2φ ∂x 2

+

∂ 2φ ∂y 2

+

∂ 2φ ∂z 2

=0

(8.69)

where φ(x,y,z) is the velocity potential. Such models are computationally very expensive as both the horizontal and the vertical structure of wave field is estimated simultaneously. Although these models are excellent tools for the investigation of detailed properties of irrotational wave, they are limited by the implicit assumption in Laplace equation development, so that real fluid effects such as viscosity are neglected. Furthermore, such models neglect the existence of the bottom boundary layer and bottom friction. 6.2.1.1 Mild Slope Equation Models

The linear mild-slope equation (MSE) in its elementary form was derived by Berkoff (1972). The equation describes the effects of shoaling, refraction and diffraction (including reflection) of periodic gravity waves with energy conservation, in the absence of currents. The MSE in its complete form is elliptic, requiring boundary conditions along the entire curve enclosing the computational domain, and the simultaneous solution of the wave field in all the interior points. This becomes computationally very demanding even for regions of moderate size, comprising only a few wavelengths. This problem can be overcome if amplitude variations along the wave rays can be neglected compared to those along the crests: absence of reflection from downwave obstacles is usually a sufficient condition for this approximation. This allows a parabolization, in which only lateral diffraction effects are taken into account, so that no downwave conditions are required and the computation can proceed from the seaward side into the domain. Berkoff development results the most widely accepted description of linear water wave propagation equation: ⎛ Cg ∇(CC g Φ ) + ω 2 ⎜⎜ ⎝ C

where Φ(x,y) is the velocity potential

⎞ ⎟Φ = 0 ⎟ ⎠

(8.70)

WAVE TRANSFORMATION IN THE COASTAL ZONE

φ ( x,y,z ) = Φ

cosh k (d + z ) cosh kd

⎛ ∂ ∂ ∇ = ⎜⎜ , ∂ ∂ x ⎝ i yi

⎞ ⎟ ⎟ ⎠

165

(8.71) (8.72)

C is the phase velocity, Cg is the group velocity ω the radian frequency Mild slope equation provides a solution φ for amplitude and phase of the waves in the horizontal plane. To obtain the equation, Berkdoff assumed that the bottom slope was mild (no abrupt steps, shoals, or trenches). Often slopes of interest violate this assumption, but the models based on the mild-slope equation perform better than the ray approach. Many approaches have been taken to computationally solve this equation. Berkoff’s approach solves the velocity potential of the wave in the horizontal, which can require 5-10 computational grid points per wave length. This is impractical for many cases. Another approach, is to use a parabolic approximation, which is far more computationally efficient (but subsequently adds more limitations). The equation describes the propagation of linear periodic small amplitude surface gravity waves over a sea bed of mild slope and will represent the combined effects of refraction, shoaling, reflection and diffraction. The potential Φ is directly related to the surface elevation, so by solving the equation the variation of wave height over the modelled area can be calculated. The equation can also be modified to include currents effects. The mild slope equation can be solved using either a finite difference or finite element techniques. A requirement of the formulation is that the spatial mesh over which the equations are solved must be selected so that the wavelength is accurately resolved. Therefore, when modelling short period waves over typical coastal areas, the number of grid points required to achieve accurate solutions may exceed several hundreds in both horizontal coordinates. This constraint, combined with the elliptical form of the equation, means that standard finite difference or finite element techniques can be computationally very expensive. and time consuming to implement for realistically sized regions. A direct consequence of this has been to simplify the equation to a form which is more amenable to rapid computation. This has resulted in the so called “parabolic” form of the mild slope equation, first derived by Radder (1979). The form which is more commonly solved is that due to Booji (1981). The derivation assumes that the reflected wave field is negligible small so that only forward travelling waves are considered. This leads to the equation i ∂ 2Φ ⎧ 1 ∂k ⎫ ∂Φ = + ⎨ik − ⎬Φ 2k ∂x ⎭ ∂x 2k ∂y 2 ⎩

(8.73)

where Φ is the velocity potential, k is the wave number, x is the main direction of wave propagation and y the transverse direction. The parabolic approximation will allow refraction, shoaling and seabed diffraction to be modelled. The approximation works where the important effects occur in the direction of wave

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propagation, as transverse effect are only included in a weak sense. Reflections are not modelled in this equation. The equation can be solved numerically using the evolutionary finite difference technique; this type of method only requires storage of one or two adjacent rows of solution points and, as a consequence, is considerably less expensive in terms of cost and storage than the equivalent numerical solution to an elliptic equation. Thus, the main advantage of the parabolic equation is that it permits a more rapid and straightforward method of solution than would be possible for the elliptic equation. In many coastal situations the use of the parabolic form of the equations does not represent a significant limitation as reflected wave activity will be negligible. 6.2.1.2 Boussinesq equation model

The family of Boussinesq models is for shallow water waves with weak nonlinearity and weak frequency dispersion of equal order. The nonlinearity is retained to second order in amplitude, and the frequency dispersion to second order in relative depth, which corresponds to third derivatives in the equations. An approach for wave propagation problems close to the coast and in harbours is the use of vertically integrated shallow-water equations in which a Boussinesq approximation has been made. The equations are: ∂ η ∂(ud) ∂(vd) + + =0 ∂t ∂x ∂y ∂u ∂u ∂u ∂ η d0 ⎛⎜ ∂ 3 (u d0 ) ∂ 3 (v d 0 ) ⎞⎟ +u +v +g − + + ∂t ∂x ∂y ∂ x 2 ⎜⎝ ∂ x 2 ∂ t ∂ x ∂ y ∂ t ⎟⎠ d 2 ⎛ ∂3 u ∂ 3 v ⎞⎟ + 0 ⎜⎜ 2 + =0 6 ⎝ ∂ x ∂ t ∂ x ∂ y ∂ t ⎟⎠ ∂v ∂v ∂v ∂ η d 0 ⎛⎜ ∂ 3 (ud0 ) ∂ 3 (vd0 ) ⎞⎟ +u +v +g − + + ∂t ∂x ∂y ∂ y 2 ⎜⎝ ∂ x ∂ y ∂ t ∂ y 2 ∂ t ⎟⎠ d 2 ⎛ ∂3 u ∂ 3 v ⎞⎟ + 0 ⎜⎜ + =0 6 ⎝ ∂ x ∂ y ∂ t ∂ y 2 ∂ t ⎟⎠

(8.74)

(8.75)

(8.76)

where η is the surface elevation, d0 is the still water depth, d=d0+η the total water depth, and u and v the depth averaged velocities in x and y directions respectively. The Boussinesq equations include implicitly diffraction (including reflection) and refraction, as well as wave-current interaction. Dissipation due to the depthinduced breaking has also been incorporated in Boussinesq models. Such models express the effect of breaking in terms of horizontal velocity gradients, as opposed to vertical gradients, which is not realistic; moreover, they give no indication about the onset of breaking.

WAVE TRANSFORMATION IN THE COASTAL ZONE

167

A more realistic formulation is based on the roller concept (Svendsen, 1974), with a criterion of a critical surface slope to trigger the onset of breaking. Their model provides remarkably accurate representations of wave profile evolution and decay due to breaking on a slope, and the wave-induced mean water level variations (set-up) including the so-called transition region immediately following the onset of breaking. Models based on these equations can represent both primary and group bounded long waves and account for refraction, shoaling, reflection and diffraction. They provide a sophisticated description of wave propagation and use either finite difference and finite element solution methods. For similar reasons to those discussed for the mild slope equations they can be time consuming and expensive in terms of computational time. However, in a situation where a comprehensive description of the physics of the problem is required, their use should certainly be considered. 6.2.2 Lagrangian models - Ray method for wave transformation

In computing a solution of wave refraction, it is convenient to work with a set of rays equations describing the paths of orthogonals. Basically, the (x,y) coordinate system is transformed into (s,n) coordinates where s is the along-ray coordinate and n is a coordinate orthogonal to it. For straight and parallel contours, it is easy to show that the Snell’s law holds for wave direction. Given the wave direction, in deep water θ1, Snell’s law gives the direction in shallower water, θ2 . sin θ1 sin θ 2 = C1 C2

(8.77)

This relationship is a mean to estimate wave direction in shallow water given the deep water direction by simply computing wave celerities. If the bottom contours are not straight and parallel, then a ray tracing is needed. By using the Snell’s law at each contour crossing, assuming straight and parallel contours locally, a wave ray can be drawn offshore to onshore and in reverse. For irregular bathymetry, where the assumption of locally straight and parallel contours may lead to erroneous results, the irrotationality of wave number has been used to generalize the Snell’s law results. G k = ∇S ( x , y , t ) (8.87)

σ =−

∂S ∂t

(8.79)

G Identically ∇ × k = 0 . This can be expanded to: ∂k sin θ ∂k cos θ − =0 ∂x ∂y

(8.80)

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INTRODUCTION TO COASTAL DYNAMICS AND SHORELINE PROTECTION

For realistic bathymetry this equation can be resolved for θ in a number of ways. If we state that rays always follow the shortest transmission path between two points, refraction may be described by the Fermat’s Principle. ∂θ 1 ∂C = ∂s C ∂n

(8.81)

where s,n are coordinates along and normal to the ray. If b0 is the original spacing between two adiacent wave rays and b is the local spacing of the rays, and defining β as b/b0, Munk and Arthur derived the following second order differential equation for β. ∂2β ∂s

2

+p

∂β + qβ = 0 ∂s

(8.82)

where:

cosθ ∂C sin θ ∂C − C ∂x C ∂y

(8.83)

sin 2 θ ∂ 2C sin 2θ ∂C cos 2 θ ∂ 2C + − C ∂x 2 C ∂x∂y C ∂y 2

(8.84)

p0 −

q=

The local wave height is found from H = H0

C0 2C g

1

β

(8.85)

where Cg is the local group velocity. Noda (1974) solved the refraction/shoaling problem by simultaneously solving the set of four first order differential equations with a four order Runge-Kutta scheme. ∂x = cos θ ∂s ∂y = sin θ ∂s ∂θ ∂C ∂C 1 = [sin θ − cos θ ] ∂s C ∂x ∂y

(8.86) (8.87) (8.88)

∂β =r ∂s

(8.89)

∂r = − pr − qβ ∂s

(8.90)

WAVE TRANSFORMATION IN THE COASTAL ZONE

169

Considering first the processes of refraction and shoaling, a good method for representing the combined effects of refraction and shoaling is to use a ray technique. In a ray method these effects are predicted by assuming that wave energy is conversed between neighbouring pairs of rays. (A wave ray is a line drawn orthogonal to a wave crest in the direction of propagation.) Such an assumption is valid provided the local variation in water depth is small and the currents are negligible. y WAVE FRONTS

b+db WAVE RAYS α+dα b

s n

α x

Figure 8.9 - Wave front and wave rays In rays models, the ray tracking is carried out automatically over a realistic bathymetry, and the results are processed to calculate in-shore wave height, period and direction from the off-shore parameters which are provided. In a forward tracking model, rays at a specified separation are tracked over a grid, which represents the bathymetry, from a line in deep water. As each ray crosses a grid square its path and energy are modified to take account of the depth changes and model the refraction and shoaling behavior. On completion of the ray tracking process the model calculates the energy in a grid square and converts this to a wave height. The wave direction is obtained directly from the wave ray direction. Each model runs for a single period and direction component. Therefore to represent refraction and shoaling of a wave spectrum several runs for different frequency and direction components would need to be made and the results combined. In a back tracking model a large number of rays are tracked seawards on a grid representing the bathymetry from an in-shore point at which conditions are required. This process gives information on how energy is transformed between

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INTRODUCTION TO COASTAL DYNAMICS AND SHORELINE PROTECTION

the seaward edge of the area of interest and the inshore point. The changes in energy and direction for each ray each frequency and each direction are stored as a series of matrices (transfer functions). Ray models can be extended to include current refraction effects. This requires a modification to the computational procedure for calculating ray paths and the subsequent analysis of results. Inclusion of currents in the wave refraction process will also mean that a gridded representation of the current field will be needed by the model. As the current field will vary through the tidal cycle this may mean that the refraction model needs to be run at several discrete states of the tide, and the results from these recombined to give the wave climate at the site. This process also requires a good definition of the current field at the depth grid points which defines the bathymetry of the area. 6.2.3 Eulerian models Finite difference and finite element methods for wave transformation. Ray models work well in areas where refraction and shoaling are the dominant physical processes. There are however situations in which diffraction by the bed or obstacles may be important, or where the propagation of a random wave train with a non linear wave-wave interaction will have to be modelled. In both these cases the governing equations are more complex than those used in a ray model, and so they are solved by other means. This usually involves either a finite difference or finite element solution over a gridded representation of the bathymetry. 6.3 Grid Models

More recently, refraction calculations have been carried out by solving the irrotationality condition (8.39) on a rectangular grid (Perlin and Dean, 1983) The gridded results can then be used for input to other models of interest (waveinduced circulation, for example). These models divide the offshore region into a grid, say, x=m∆x, m=1,2,…,M and y= n∆y, n=1,2,…,N. Equation (8.39) must be solved in finite difference form. Dalrymple rewrites the equation as: ∂A ∂B + =0 ∂x ∂y

(8.91)

where A=kcosθ and B= ksinθ=(k2-A2)1/2 Mathematical model for wave refraction The model is based on the principle of conservation of wave action A(x,y,t) defined as: A( x, y, t ) = E ( x, y, t ) / σ =

1 ρgH 2 ( x, y, t ) / σ 8

and on the principle of irrotationality of wave number k.

(8.92)

WAVE TRANSFORMATION IN THE COASTAL ZONE

171

In eq. (8.37) σ represents angular frequency of propagation.

σ = ω − k ⋅V

(8.93)

where ω=2π/T is wave radial frequency, k is the vector wave number, whose module is given by: k =2π/L and V is the current velocity vector (in case it is present, its components are U=Vcosθ, V=Vcosθ. Using these terms, the equation of conservation may be expressed as:

∂A ∂ ∂ + (C x A) + (C y A) = T ∂t ∂x ∂y

(8.94)

where T is a dissipative term. In steady conditions and with the lack of this term, the equation becomes:

∂ ∂ (C x A) + (C y A) = 0 ∂x ∂y

(8.95)

in which Cx and Cy are the propagation celerity components given by: C x = U + C g cos θ

(8.96)

C y = V + C g sin θ

(8.97)

here Cg is the group celerity given by: Cg =

C 2

2kd ⎤ ⎡ ⎢⎣1 + sinh 2kd ⎥⎦

(8.97)

Definitely (after all) eq. (8.40) becomes:

∂ ⎡E ⎤ ∂ ⎡E ⎤ (U + C g cos θ ) ⎥ + ⎢ (V + C g sin θ ) ⎥ = 0 ⎢ ∂x ⎣ σ ⎦ ∂y ⎣ σ ⎦

(8.98)

This latter is coupled with the following non linear dispersion relation:

σ 2 = gk tanh[kd + f 2 (ka)][1 + f 1 (ka) 2 D]

(8.99)

being D a parameter of non linearity given by: D=

cosh 4kd + 8 − 2 tanh 2 kd 8 sinh 4 kd

(8.100)

and: f1 = tanh 5 k d

⎡ kd ⎤ f2 = ⎢ ⎥ ⎣ sinh(kd) ⎦

(8.101) 4

(8.102)

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INTRODUCTION TO COASTAL DYNAMICS AND SHORELINE PROTECTION

For what concerns the equation of refraction: ∇×k = 0

(8.103)

∂ (k cosθ ) ∂ (k sin θ ) − =0 ∂x ∂y

(8.104)

∂A ∂B − =0 ∂x ∂y

(8.105)

B ≡ k cos θ = k 2 − A 2

(8.106)

this can be written as:

fixed A=k senθ one obtains:

where B is defined as:

Eqn. (8.105) is a differential equation with partial derivatives to the first order, which may be solved with initial condition x=0, corresponding to the direction of the incident wave. This equation, taking account of the expression of B, results to be a nonlinear equation in A which may be solved through an iterative process. Assuming a coordinate system with x-direction oriented cross-shore, and split in segments x=i∆x, for 1=0, 1,2,….n, and with y-component long-shore oriented and divided analogously into segments, y=j∆y, with j=0,1,2,….,n, eq. (8.105) may be solved through the finite difference method for A(x,y)→A i, j . Solution are calculated by starting from off-shore gridlines in the direction of increasing I, considering that the value of θ is known on the external boundary of the grid. (i=0). The wave height is determined by :

∂ ⎡ E (U + C g cos θ ) ⎤ ∂ ⎡ E (V + C g sin θ ) ⎤ ⎢ ⎥+ ⎢ ⎥=0 ∂x ⎣ σ σ ⎦ ∂y ⎣ ⎦

(8.107)

By starting from eq. (8.107) and introducing expressions for wave action flux in x and y direction respectively:

A= B=

E

σ E

σ

(U + C g cos θ )

(8.108)

(V + C g sin θ )

(8.109)

The following equation is obtained:

∂A ∂B + =0 ∂x ∂y

(8.110)

This equation differs from eqn. (8.105) solely for the minus sign, and it is therefore possible to resolve by using the aforementioned method. The value of significant wave height is given as a function of wave action flux assumes the following form

WAVE TRANSFORMATION IN THE COASTAL ZONE

H i, j

⎡ ⎤ 8 Ai , jσ =⎢ ⎥ ⎣⎢ σ g (U i , j + C gi , j cosθ i , j )⎦⎥

173

1/ 2

(8.111)

7 Wave breaking As the depth of the bottom on which a wave is propagating decreases, wave steepness (ratio of height H to length L) increases until it assumes a limit value in correspondence of which a wave becomes unstable and breaks. It is therefore interesting to state a value of steepness, (that is of height), for a wave on a given bottom. Stokes (1880) states that the instability condition, that is the critical steepness, is reached when horizontal velocity of a particle on the crest matches wave celerity. In such condition the crest contains an angle of 120°. (fig.3.7) Using Stokes’ criterion, a formulation for the critical steepness on infinite depth is:

H0 1 = 0.142 ≅ L0 7

(8.112)

From eq. (8.112) it is evident that in deep water wave height is limited by wave length. Indeed the critical value of steepness reported by eq. (8.112) is almost never reached for ex. T=10s, Lo = 156m, Ho = 22m. On shallow water, wave height is no longer limited by wave length but it is limited by depth. By applying Stokes’ solitary wave theory, McCowan (1891) proposed for wave height the critical value:

H = k = 0.78 d

(8.113)

Weggel (1972) makes the value of k depend on the bottom slope m.

H H = k ( m) = b ( m ) − a ( m) 2 d gT

(8.114)

being a ( m) = 43.8(1 - e -19 m ) b( m) = 1. 56(1+ e -19.5m )-1. The value of K(m) tends to 0.78 for m→0. For intermediate depth, Miche (1944) proposed for the critical steepness the expression: H = 0.142tanh k d L

(8.115)

coherent with Mitchell expression for infinite depth. Nevertheless, eq. (8.115) overestimates wave height on shallow water, resulting H/d=0.89. For what concerns the shape of breaking waves, they are classified by Galvin (1968) in breaking wave types spilling, plunging, collapsing and surging with increasing bottom slope. With reference to fig. 3.8, spilling waves, which are present in deep water or on mild slope beaches, are characterized by a localized breaking in a stripe

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INTRODUCTION TO COASTAL DYNAMICS AND SHORELINE PROTECTION

along-crest, plunging waves, (which occur on a more sloping bottom), are characterized by a greater downward concavity so that water mass falls ahead; surging waves happen on very steep beaches, on which reflection is considerable, they present foam in the lower part of the crest. The depth at which wave breaking happens may be determined through the expression of shoaling coefficient and refraction coefficient by knowing the offshore wave characteristic.

⎛ Cg H = H 0 ⎜⎜ 0 ⎝ Cg

1

1

⎞ 2 ⎛ cos α 0 ⎞ 2 ⎟⎟ ⎜ ⎟ ⎠ ⎝ cos α ⎠

(8.116)

On shallow water it becomes:

⎛ Cg 0 H = H0 ⎜ ⎜ 2 gd ⎝

1

1

⎞ 2 ⎛ cos α ⎞ 2 0 ⎟ ⎜ ⎟ ⎟ ⎝ 1 ⎠ ⎠

(8.117)

Using McCowan theory : 1

2

⎡ C0 ⎤2 1 ⎛ H 2C cos α0 ⎞ 5 ⎟⎟ cosα 0 ⎥ → db = 1 4 ⎜⎜ 0 0 kdb = H 0 ⎢ 2 ⎠ 5 5 ⎝ ⎣⎢ 2 gdb ⎦⎥ g k

(8.118)

If the seabed slope m is constant so that db = mxb , the distance xb of breaking line from the shoreline may be obtained by dividing db by m. 2

d xb = b = m

⎛ H 02C0 cos α0 ⎞ 5 ⎜ ⎟⎟ 1 4 ⎜ 2 ⎠ 5 5 ⎝ mg k 1

(8.119)

Finally breaking wave height is gained by multiplying k by breaking depth df, that is by mxb: 1

2

⎛ k ⎞ 5 ⎛ H 2C cos α0 ⎞ 5 ⎟⎟ H b = kmxb = ⎜⎜ ⎟⎟ ⎜⎜ 0 0 2 ⎝g⎠ ⎝ ⎠

(8.120)

WAVE TRANSFORMATION IN THE COASTAL ZONE

SPILLING

PLUNGING

COLLAPSING

SURGING

Figure 8.10

175

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Chapter 9 Sediment transport 1 Introduction The Airy (Airy G. B., 1845) small amplitude theory (also named linear wave theory), described in chapter 3, provides a useful first approximation to the wave kinematics and is often extended to describe also processes related with nonlinear phenomena. However, waves are usually not small in amplitude, and larger waves produce the largest forces and greatest sediment movement, so nonlinear waves have to be studied for the analysis of some coastal processes. Sea level variations (wave setup and setdown) can be explained using some concepts of the linear theory; but the observed decrease and increase in the mean water level usually involve the wave height to the second power, which results in nonlinear quantities. These processes are explained with the radiation stress concept, that is the mean value of horizontal momentum across unit area of a vertical plane with respect to time, minus the mean flux in the absence of waves. Gradients in this quantity therefore correspond to a net addition or loss of momentum to a water column, i.e. a net force, arising from the processes of wave shoaling and breaking. Another nonlinear quantity is the mean transport of water toward the shoreline, the mass transport, which is not predicted by the linear Airy theory, which assumes that each water particle under a waveform is travelling in a closed elliptical orbit. We define the mass transport as:

M=

1 t2 η ∫ ∫ ρu ( x, z )dzdt t 2 − t1 t1 − d

(9.1)

where the time interval between t1 and t2 is a sufficiently long (many wave periods for irregular waves; one wave period for periodic waves). If we integrate over the depth from the bottom to the mean water surface, z=0, rather than the instantaneous water surface η, we obtain M equal to zero, as predicted by linear theory. If we continue the integration up to η, the mass transport becomes:

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AN INTRODUCTION TO COASTAL DYNAMICS AND SHORELINE PROTECTION

M =

E C

(9.2)

which shows that there is a nonlinear transport of water in the wave direction due to the larger forward transport of water under the wave crest because the total depth is greater when compared with the backward transport under the trough. From this formula, it follows that the mass transport is larger for more energetic waves. This mass transport has momentum associated with it, which means that forces will be generated whenever this momentum changes magnitude or direction by Newton’s second law. To determine this momentum, we integrate the momentum flux from the bottom to the surface as follows:

Mf =

1 t2 η ∫ ∫ ( ρu )udzdt t 2 − t1 t1 − d

(9.3)

This quantity has a first approximation Μf=MCg=En, which indicates that the flux of momentum is described by the mass transport times the group velocity. The sediment transport is usually divided into bed load, suspended load and swash load, as shown in the scheme of Figure 1.7. The bed load transport is either in sheet flow or rolled along the bottom, the suspended load is carried up within the fluid column and moved by currents, the swash load is moved on the beach face by the swash. The bed load transport is initiated when the resisting force of the sand particle on the bottom becomes smaller than a wave force on it. The depth of this point on the beach is called a critical depth for sediment movement and the critical velocity is defined as the water particle velocity at this depth. In the shallow water region where incident waves break, suspended load is generated by a great deal of sediment brought into suspension by the turbulence caused by breaking waves. Sediment materials are suspended and transported offshore by undertow currents. A large trough is formed and becomes deeper until the energy is completely exhausted. Offshore a bar is formed, localized between the breaking point and the deepest zone reached by vorticity. The swash load is generated in the swash zone, when fluid motion depends on the fluctuation of coastline associated with the frequency of oscillations: at low frequencies, if the beach is permeable and the sand is not saturated, the water percolates through the substrate and the backrush decreases, with accumulation of sediment. At high frequencies (storm waves), if the substrate is saturated, the water cannot percolate through the sand and the backrush current cannot decrease, with erosion.

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179

2 Basic concepts of sediment transport 2.1 Critical bed shear stress The sediment on the sea bed is transported when it is exposed to large enough forces, or shear stresses, by the water movements. These movements can be caused by the current or by the wave orbital velocities or by a combination of both. In fact, while on deep water there is no orbital motion of particles on the bottom, when the water depth roughly reaches one half of the wave length, the wave particle orbits begin to interact with the bottom, causing bed shear. If a steady flow over a bed composed of cohesionless grains is considered, these grains will not move at very small flow velocities, but when the flow velocity becomes large enough, the driving forces on the sediment particles will exceed the stabilizing forces. This flow velocity is called the critical flow velocity. A now classical solution to the problem based on dimensional analysis was offered by Shields (1936). The threshold of particle motion is supposed to be attained for a given ratio between driving and stabilizing forces. 2.2 The Shields parameter and modified Shields diagram The driving force acting on the bottom grains is a function of the second power of sediment diameter D, given by F = τ b D 2 . On the other hand, the motion is contrasted by the individual grains tendency to stay on the bottom due to the friction caused by their submerged weight and to the presence of neighboring grains. For a non cohesive sediment, the particles submerged weight is defined as follows: W = ( ρ s − ρ ) gD 3

(9.4)

where ρs is the sediment density, ρ is the fluid density, g is the gravity acceleration and D is the particle diameter. The ratio F/W is defined as Shields parameter θ (Shields, 1936):

θ=

τb τb u*2 = = ( ρs − ρ ) gD (s − 1) ρgD (s − 1) gD

(9.5)

where s (=ρs/ρ) is the ratio between sediment density (ρs) and fluid density (ρ) and u* is the friction velocity, defined as u*2 = τ / ρ . The parameter determining the characteristics of the near-bottom flow and hence the mobilizing force acting on individual sediment grains is the dimensionless boundary Reynolds number: R e* =

u* D

ν

(9.6)

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AN INTRODUCTION TO COASTAL DYNAMICS AND SHORELINE PROTECTION

where:

ν=kinematic fluid viscosity=µ/ρ µ =(absolute) dynamic fluid viscosity Physical studies demonstrated that the condition for motion inception may be expressed as a critical Shields parameter ψc which value is a function of the Reynolds number f(Re*) :

θc =

τc τc u*2c = = = f ( Re* ) ( ρ s − ρ ) gD ( s − 1) ρgD ( s − 1) gD

(9.7)

So the critical Shields parameter θc is the effective Shields parameter (θ′) at which sediment movement starts. The empirical diagram of θc versus Re* is given in figure 9.1. Typical θc values for sand in water are of the order 0.05.

Figure 9.1 – Shields diagram for initiation of motion in steady turbulent flow (after Raudkivi, 1976). The Modified Shields diagram Madsen and Grant (1976) introduced the sediment-fluid parameter S*, which gives the relation between the critical Shields parameter and the sediment.

SEDIMENT TRANSPORT

181

Figure 9.2 - Modified Shield diagram (after Madsen and Grant, 1976). From the definition of θc we obtain:

u*c = ( s − 1) gD θ c

(9.8)

which can be introduced in the definition of Re* to obtain the S* expression: S* =

R D ( s − 1) gD = e* 4v 4 θc

(9.9)

The sediment fluid parameter is often used to give a modified representation of the traditional Shields diagram, in which the values of S* are represented on the abscissa instead of the Reynolds parameter values. 2.3 Sediment fall velocity

In order to describe suspended sediment transport, it is important to understand the behavior of suspended sand grains in different type of flow. The simplest case is when the fluid accelerations are negligible compared to the acceleration of gravity. In this case the relative velocity between sand and water is everywhere equal to the settling velocity wf. The rate at which a particle settles depends on grain and fluid properties (i.e. grain size, shape and density, water density, flow viscosity rate and turbulence). Assuming a spherical sediment grain, the force balance of submerged weight and fluid drag on a grain falling through an otherwise quiescent fluid gives: ⎛π ⎞ 1 ⎛π ⎞ ( ρ s − ρ ) g ⎜ D3 ⎟ = ρCD ⎜ D 2 ⎟ w2f ⎠ 2 ⎝4 ⎠ ⎝6

(9.10)

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AN INTRODUCTION TO COASTAL DYNAMICS AND SHORELINE PROTECTION

wf ( s − 1) gD

=

4 3C D

(9.11)

Where: wf= sediment fall velocity; D = grain diameter; ρs= density of the particle; ρ= density of the fluid; s=ρs/ρ. The drag coefficient CD is a function of the Reynolds number ReD =Dwf/v which is a function of S*, the sediment-fluid parameter defined by equation (9.9). From the empirical relationship of CD versus ReD , CD is obtained for a specified value of ReD. C D = 1.4 + 36 / ReD

(9.12)

With this value of CD the dimensionless fall velocity is obtained, and the value is used with the specified value of ReD to obtain the corresponding value of S*. In this manner (Madsen and Grant 1976), the graph of nondimensional fall velocity as a function of the sediment fluid parameter, shown in figure 9.3 is obtained. The sediment fall velocity can be calculated as a function of the sediment fluid parameter: wf ( s − 1) gD

= 1.82 for S* >300

(9.13)

For small values of S*, e.g. for quartz grains of D<0.1 mm in seawater, the sediment fall velocity is calculated from the Stokes law: ( s − 1) gD 2 18v

wf =

(9.14)

or, using the S* parameter: wf ( s − 1) gD

=

2 S * for S* <0.8 9

(9.15)

The fall velocity can also be calculated as a function of the median grain size D50 as follows:

wf =

⎛ 36ν ⎜ ⎜D ⎝ 50

2

⎞ 36ν ⎟ + 7.5( s − 1) D50 − ⎟ D50 ⎠ 2.8

(9.16)

where v is the kinematic viscosity N/m2; D50 is the median grain size; s is the ratio between the sediment density and the water density (s=ρs/ρ). 2.4 Bed load and suspended load

Of the total sediment load a distinction between two categories is usually made, on the basis that two different mechanisms are effective during the transport.

SEDIMENT TRANSPORT

183

− Bed load − Suspended load

Figure 9.3 - Nondimensional fall velocity for spherical particles versus the sediment fluid parameter (Madsen and Grant, 1976). The basic idea of this distinction is that the bed load is defined as the part of the total load that is in more or less continuous contact with the bed during the transport, so it involves that part of total load which is supported by intergranular forces (Bagnold, 1966). Thus the bed load must be determined almost exclusively by the effective bed shear acting directly on the sand surface. The suspended load is the part of the total load that is moving without continuous contact with the bed as a result of the agitation of fluid turbulence. 2.4.1 Bed-load and shear stress Making use of the Bagnold’s definition of bed-load, it is fairly easy to estimate the weight of material which will be moved as bed-load under a certain effective stress τ’. The bed-load must, due to this immersed weight, deliver an effective normal stress σl (M/LT2) onto the top most of the immobile bed: ∞

σ l = ρ ( s − 1) g ∫ c B ( z )dz

(9.17)

0

where cB is the volumetric concentration of bed-load in vol/vol, ρ and s are the sediment density and porosity, respectively, and g is the acceleration due to gravity. Assuming that the yield criterion for the top layer of immobile grains is

τ max = τ c + σ l tan ϕ s the amount of bed-load which is in equilibrium with τ′ is given by:

(9.18)

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AN INTRODUCTION TO COASTAL DYNAMICS AND SHORELINE PROTECTION ∞

τ '−τ c

∫ c B ( z )dz = ρ ( s − 1) g tan ϕ 0

(9.19) s

Here it is convenient to introduce the maximum concentration cmax, which is the volumetric concentration of solid sediment in the immobile bed. In terms of cmax, the vertical scale of the bed-load distribution is then defined by LB =

1

c max

∞

∫ c B ( z )dz

(9.20)

0

Introducing this expression for LB into equation (9.18), we see that the vertical distribution scale measured in grain diameters is:

θ '−θ C LB = d c max tan ϕ s

(9.21)

Bagnold (1966) gave tan ϕs =0.63 as a typical value for fairly rounded grains corresponding to a maximum concentration of the same value, i.e. cmax=0.63 [vol/vol], and he noted that the product cmax tan ϕs =0.63 is fairly constant at about 0.4 for different grain shapes. Hence, as rules of thumb we have: LB = 2.5(θ '−θ c )d

(9.22)

c max L B = 2.5(θ '−θ c )dc max

(9.23)

LB is the equivalent thickness at rest of the bed-load, and cmaxLB is the corresponding solids volume per unit area of the bed. 2.4.2 Steady bed load in sheet flow transport The bed load transport rate can be expressed as : ∞

qb = ∫ c B ( z )u s ( z )dz = c max L BU B

(9.24)

0

us is the sediment velocity distribution and cmax is the maximum concentration of solid sediment in the immobile bed. We can predict qb values empirically with reasonable confidence for steady flow because it was measured directly in a large number of experiments. One of the first theoretical approaches to the problem of predicting the rate of bed load transport was presented by Einstein (1950). One of the most important innovations in his analysis was the application of the theory of probability to account for the statistical variation of the agitating forces on bed particles caused by turbulence. Based on experimental observations, Einstein assumed that the mean distance traveled by a sand particle between erosion and subsequent deposition, is simply proportional to the grain diameter and independent of the hydraulic conditions and the amount of sediment in motion.

SEDIMENT TRANSPORT

185

The principle in Einstein’s analysis is as follows: the number of particles, deposited in a unit area, depends on the number of particles in motion and on the probability that the dynamical forces permit the particles deposit the number of particles eroded from the same unit area depends on the number of particles within the area and on the probability that the hydrodynamic forces on these grains are sufficiently strong to move them. For equilibrium conditions the number of grains deposited must equal the number of particles eroded. In this way, a functional relation (bed load function) is derived between the two nondimensional quantities. Φ =

qB ( s − 1) gd 2

(9.25)

where qB is the rate of bed load transport in volume of material per unit time and width. The general trend of Φ can be also expressed by the Meyer-Peter and Muller (1948): (9.26) Φ = 8(θ '−θ c )1.5 Another expression is given in Nielsen (1992) Φ = 8( θ ' −θ c ) θ '

(9.27)

The representation of the Φ functions is given in figure 9.4. 2.4.3 Basics of suspended load transport formulation Under the assumption of an uniform flow, the relation between the average velocity (v), the water level slope (i) the water depth (h) and the bed shear friction coefficient (C) is given by the Chezy formula:

v = C di

i=

v2 C 2d

(9.28)

In that case, the shear stress is given by:

τ c = ρgv 2 / C 2

(9.29)

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AN INTRODUCTION TO COASTAL DYNAMICS AND SHORELINE PROTECTION

θ‫׳‬ Figure 9.4 - Representation of the Φ functions for such flow the vertical velocity gradient dv( z ) dz can be written as: dv( z ) dz = τ ( z ) ρε f

(9.30)

where εf is the diffusion coefficient and is the shear stress at height z from the bed. The diffusion coefficient is given by the mixing length theory:

ε f = l2

dv( z ) dz

(9.31)

where l is the mixing length, given by: l=kz

(near the bed)

l = kz 1− z / d (for the entire water column)

(9.32) (9.33)

so the shear stress varies linearly with the height above the bed: 2

⎛ dv( z ) ⎞ ⎟ = τ c (1 − z / d ) ⎝ dz ⎠

τ ( z ) = ρ (kz ) 2 (1 − z / d )⎜

(9.34)

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187

the vertical gradient dv( z ) dz is: dv( z ) dz =

(τ c / ρ )

kz

(9.35)

The solution of the above differential equation is : v( z ) =

τC 1 u z z ln = * ln k z0 ρ k z0

(9.36)

where u* is the shear stress velocity, which is the velocity occurring at a certain elevation above the bed, assuming a logarithmic velocity profile. A physically important quantity is the velocity which marks the change of the turbulent flow of the logarithmic velocity profile to a much less turbulent or even laminar sublayer close to the bed: the velocity distribution near the bed is usually assumed be linear and tangent to the logarithmic distribution at a height zt above the bed. From dvt / dz = vt / z t follows that zt=ez0. The velocity at the height zt is found : vt =

g v* = v kC k

(9.37)

with equation (9.35) a new formulation for the bed shear stress τc is found:

τ = ρk v 2

c

2 t

(9.38)

This expression relates the bed shear stress to the velocity near the bed for the combination of the two fundamentally different velocity profiles of a uniform flow and the orbital velocity due to the waves. The value of z0 is related to the apparent bottom roughness . Experimentally, Nikuradse found: z0/r=1/33

(9.39)

The above given information is required to calculate the concentration of the material in suspension. This material is kept into suspension by the exchange of upward and downward transport as result of the turbulent diffusion. This upward diffusion coefficient for the sediment is related to the turbulent fluid diffusion coefficient. Thus, the upward transport due to turbulent diffusion is, in the equilibrium situation, equal to the downward motion of the sediment due to the fall velocity:

wc( z ) + ε s ( z )

dc( z ) dz

(9.40)

where z is the height above the bed; w is the fall velocity of the sediment particles in still water; c(z) is the average concentration at height z above the bed; εs(z) is the diffusion coefficient for the sediment at height z.

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AN INTRODUCTION TO COASTAL DYNAMICS AND SHORELINE PROTECTION

ε s ( z ) = 4ε s max

z ⎡d − z ⎤ d ⎢⎣ d ⎥⎦

(9.41)

this results in a concentration distribution given by :

⎡d − z a ⎤ c ( z ) = c0 ⎢ ⎥ ⎣ z d − a⎦

Z0

(9.42)

where z* = w / kv* = dw / 4ε s max ; c0 is the reference concentration at level z=a above the bed. Einstein calculated the value of c0 at a height a of only some grain diameters from the bed (Einstein, 1950). For a rippled or ondulated bed this assumption is not realistic. Bijker assumed therefore that a would be equal to the bed roughness r. The concentration ca at the top of this layer is calculated under the assumption that the bed load is transported in this layer by the average velocity and that the concentration is constant (in the same layer). The suspended load can be calculated as: qb = rvbedlayerca

(9.43)

Over the height z0 the velocity distribution is linear, from z0 to r the velocity distribution is logarithmic. This results in the following formula for the average velocity v∗ in the bottom layer:

vbedlayer =

1 ⎡ 1 v* z ⎤ r v z 0 + ∫z0 * ln dz ⎥ ⎢ r ⎣2 k k z0 ⎦

(9.44)

vbedlayer = 6.34v*

(9.45)

or :

2.5 The bottom boundary layer and the bed roughness

The bottom boundary layer is the zone in the immediate vicinity of the bed where the fluid motion is significantly influenced by the frictional resistance of the bottom. The velocity in the boundary layer grades from zero at the bed to the velocity of the free stream at some distance above the bed. The thickness of the wave boundary layer (δ) depends on the wave period and is approximately (Madsen and Grant, 1976):

δ≈

ku*,w

ω

(9.46)

where k is the von Karman’s constant (=0.4), u*,w is the wave-induced shear velocity and ω is the wave radian frequency (2π/T). The thickness of the wave boundary layer is typically in the order of a few centimeters. While an oscillatory boundary layer more or less breaks down and reforms twice every wave cycle, a current boundary layer is significantly thicker, as this layer has ample time to

SEDIMENT TRANSPORT

189

develop. The most important physical aspect of the boundary layer is the shear stress (τb) , i.e. the force which the fluid motion exerts on the bed. The shear stress can be defined as τ b = ρu*2 and determinates the velocity gradient close to the bed and the mobilising forces applied to the sediment grains on the bottom. This quantity is of fundamental importance to sediment entrainment and transport. Under unidirectional currents, the bed shear stress can be computed from the velocity profile. In the lower 1-2 m of the water this profile is generally well described by the von Karman-Prandtl equation: uz =

u* ln( z / z 0 ) k

(9.47)

where u* is the mean velocity at elevation z above the bed, u* is the shear velocity associated with the current and z0 is the zero intercept of the velocity profile. However, sediment entrainment and concentration in surf zone are mainly determined by the shear stress due to waves. This is because the wave boundary layer is much thinner than the current boundary layer. Therefore, velocity gradients (the shear velocity) due to the waves are significantly larger than for the mean current. An appropriate way of determining shear velocity and bed shear stress is through the use of the definition of bed shear stress under waves:

τ b, w = ρu*,2 w =

1 ρf w u 2 2

(9.48)

where ρ is the density of the water, u is the free stream wave orbital velocity and fw is a wave friction factor, i.e. a coefficient of proportionality describing the relation between shear velocity and the free stream velocity. A commonly accepted expression for the non-dimensional wave-friction factor is due to Swart (1974): f w = exp(5.213(k s / A) 0.194 − 5.977)

k s / A < 0.63

(9.49)

where ks is the bed roughness (or hydraulic roughness) and A is the water particle semi-excursion (i.e. orbital amplitude, A=umT/2π) . For ks/A>0.63, fw=0.30. The bed roughness is usually defined as: (Nielsen, 1992): ks=30z0

(9.50)

Hence, the rougher the bed the steeper the velocity gradients and the more stress exerted by the fluid motion. However, in sediment transport modeling, the specification of ks remains a problem. When waves and currents coexist, the bed roughness is mainly linked to the wave motion. Total bed roughness is composed of the grain roughness (roughness due to the individual particles on the bed kd), roughness exerted by bedforms (form drag roughness, kr) and due to sediment movement (km). The latter two terms are called moveable bed roughness. Thus: ks=kd+kr+km

(9.51)

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AN INTRODUCTION TO COASTAL DYNAMICS AND SHORELINE PROTECTION

For a flat fixed bed, ks=kd=2.5D50 (where D50 is the mean grain size). However, in the surf zone the bed is rarely immobile and for a moveable bed under waves, Nielsen (1992) proposed ks =

8η r2

λr

+ 170 D50 (θ '−0.05)1 / 2

(9.52)

where ηr is the bedform height, λr is the bedform wavelength and θ′ is the skin friction Shields parameter. 2.6 Bed load and suspended load: a simple parametrical model

Bed load Bagnold (1966) pointed out one of the shortcomings in Einstein’s formulation by stating the following paradox. Consider the ideal case of fluid flow over a bed of uniform, perfectly piled spheres in a plane bed, so that all particles are equally exposed. Statistical variations due to turbulence are neglected. When the tractive stress exceeds the critical value, all particles in the upper layer are peeled off simultaneously and are dispersed. Hence the next layer of particles is exposed to the flow and should consequently also be peeled off. The result is that all the subsequent underlying layers are also eroded, so that a stable bed could not exist at all when the shear stress exceeds the critical value. Bagnold explained the paradox by assuming that in a water-sediment mixture the total shear stress τ′ would be separated in two parts:

τ′ =τF+τG

(9.53) where τF is the shear stress transmitted by the intergranular fluid, while τG is the shear stress transmitted because of the interchange of momentum caused by the encounters of solid particles, i.e. a tangential dispersive stress. The existence of such dispersive stresses was confirmed by his experiments. Bagnold argues that when a layer of spheres is peeled off, some of the spheres may go into suspension, while others will be transported as bed load. Thus a dispersive pressure on the next layer of spheres will develop and act as a stabilizing agency. Hence, a certain part of the total bed shear stress is transmitted as a grain shear stress τG, and a correspondingly minor part as fluid stress τ′ =τF+τG. Continuing this argument, it is understood that exactly so many layers of spheres will be eroded that the residual fluid stress τF on the first immovable layer is equal to (or smaller than) the critical tractive stress τG . The mechanism in transmission of a tractive shear stress τ greater than the critical is then the following: τc is transferred directly from the fluid to the immovable bed, while the residual stress τ′ -τc is transferred to the moving particles and further from these to the fixed bed as a dispersive stress. The effective bottom shear stress ( τ c' ) is given by:

τ c' =

⎞ 2 1 ⎛ 0.06 ⎟U ρ⎜ 2 ⎜⎝ (log(12d 2.5 D50 )) ⎟⎠

(9.54)

SEDIMENT TRANSPORT

191

Bijker (1986) presented a method for the calculation of sediment transport in combined wave and current motion. The mean bed shear stress ( τ wc ) by Bijker (1986) in this situation is given by:

1 (9.55) 2 τc is the bed shear stress by current alone, and τw,max is the maximum bed shear stress by wave alone:

τ wc = τ c + τ w, max

τc =

⎞ 2 1 1 ⎛ 0.06 ⎟U ρU 2 = ρ ⎜⎜ 2 2 ⎝ (log(12d k s )) ⎟⎠

τ w, max =

1 ρf cU m2 2

⎛ ⎛ k ⎞0.2 ⎞ f w = exp⎜ 5.5⎜⎜ s ⎟⎟ − 6.3 ⎟ ⎜ ⎝ Aw ⎠ ⎟ ⎝ ⎠

(9.56) (9.57) (9.58)

where: d=water depth [m]; ks=bed roughness [m]; U=average velocity of current [m/s]; Aw=amplitude of the water particle on the bottom [m]; Um=maximum horizontal velocity of the water particle on the bottom [m/s]. The bed load transport is given by:

⎛ − 0.27( s − 1) D50 ρg ⎞ τ ⎟⎟ qB = 2 D50 c exp⎜⎜ ρ µrτ wc ⎝  

  ⎠ transporting

(9.59)

stirring up

3

where: qB= bed load transport [m /m×s]; D50=median sediment grain size [m]; g=acceleration of gravity [m/s2]; µr=ripple factor=τ’c/τwc [-]; 0.27=experimental coefficient. Suspended load The suspended load is defined as the part of the total load which is moving without continuous contact with the bed as the result of the agitation of the fluid turbulence. The appearance of ripples will increase the bed shear stress (flow resistance). On the other hand, more grains will be suspended due to the flow separation on the lee side of the ripples, thus the suspended load is related to the total bed shear stress. Einstein-Bijker formula for suspended sediment transport is: ⎛ ⎞ ⎛ d ⎞ ⎟⎟ + I 2 ⎟ qS = 1.83qB ⎜⎜ I1 ln⎜⎜ ⎟ ⎝ 0.033k s ⎠ ⎝ ⎠

where I1 and I2 are the Einsten integrals given by:

(9.60)

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AN INTRODUCTION TO COASTAL DYNAMICS AND SHORELINE PROTECTION

I1 = 0.216

I 2 = 0.216

A(z* −1)

1⎛ 1 −

(1 − A) z*

∫

A(z* −1)

1⎛ 1− B ⎞

(1 − A) z*

z

B⎞ * ⎜ ⎟ dB A⎝ B ⎠

∫A ⎜⎝

⎟ B ⎠

(9.61)

z*

lnBdB

(9.62)

where A=ks /d; B=z/d; z*=wf /(κu*,C); u *,c = τ c / ρ = friction velocity. κ is the Von Karman constant (κ=0.40, dimensionless) and qB is the bed load transport under combined wave and current [m3/m×s]. The values of Integrals I1 and I2 can be used to calculate the Einstein Total Integral Q as follows:

Q = [ I1 ln(d / 0.033k s ) + I 2 ]

(9.63)

For given values ks, d and z* , the Einstein Total Integral Q can be also calculated using the table 9.1 or figure 9.5, which gives the representation of 1.83Q (= qS /qB) versus A(=ks//h). The suspended load qS is a function of the bed load transport and the Einstein Total Integral: q S = 1.83Qq B

(9.64)

This indicates that the suspended load transport is directly and linearly proportional to the bed load. The total transport QT can now be written as: QT = q B + q S = q B (1 + 1.83Q)

(9.65)

2.7 Case study Example 1

Calculate the sediment fall velocity in sea water (ρ=1,025 Kg/m3 ; ν=10-6 m2/s) for a quartz sediment sand (ρs =2,650 Kg/m3) with diameter D=0.15 mm. Solution: The sediment fall velocity can be calculated using the sediment fluid-parameter S* : s=ρs/ρ=2.59 g=acceleration due to gravity=9.8 m/s; D=0.15mm=1.5⋅10-4m. D 1.5 ⋅ 10 −4 ( s − 1) gD = ( 2.59 − 1) ⋅ 9.8 ⋅ 1.5 ⋅ 10 − 4 = 1.81 4v 4 ⋅ 10 − 6 The figure 9.5 in correspondence of S*=1.81 the dimensionless fall velocity can be calculated by the following expression: S* =

SEDIMENT TRANSPORT

wf ( s − 1) gD

= S * ≈ 0.3

so we have: w f = S * ( s − 1) gD = 0.0145m / s = 1.45cm / s

Example 2

In a coastal region, (sea water density ρ=1025 kg/m3; kinematic viscosity v=10-6 m2/s) the flow speed is U=1 m/s at 2 m water depth; Wave parameters are: H=0.5 m and T=8 s (L=35 m); the sediment density is 2650 kg/m3 and D50=0.15 mm. The bed roughness is ks=2 cm. Calculate the sediment transport under current and under combined wave and current. Solution: The effective bottom shear stress is given by:

1 ⎛

0.06

⎞

⎟U 2 = 1.33N / m 2 τ c' = ρ ⎜⎜ 2 ⎝ (log(12d 2.5 D50 ))2 ⎟⎠ The total bottom shear stress is:

τc =

⎞ 2 1 ⎛⎜ 0.06 ⎟U = 3.24 N / m 2 ρ⎜ 2 ⎟ 2 ⎝ (log(12d k s )) ⎠

The ripple factor is:

µr =

τ c' 0.41 = τc

The bed load transport is: q B = 2 D50

⎛ − 0.27( s − 1) D50 ρg ⎞ τc m3 ⎟⎟ = 1.0395×10-5 exp⎜⎜ ρ µ rτ wc m× s ⎝ ⎠

The relative density of the sediment is: s = ρ s / ρ = 2.59

The fall velocity is:

wf =

⎛ 36ν ⎜ ⎜D ⎝ 50

The friction velocity is:

2

⎞ 36ν ⎟ + 7.5( s − 1) D50 − ⎟ D50 ⎠ = 0.012 m/s 2.8

193

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AN INTRODUCTION TO COASTAL DYNAMICS AND SHORELINE PROTECTION

u *,c = τ c / ρ = 0.056 m/s the Q values can be calculated by numerical integration or using figure 9.5 or using table 9.1. By numerical integration: A ( z* −1) I 1 = 0.216 (1 − A) z* I 2 = 0.216

A ( z* −1) (1 − A) z*

z*

⎛1− B ⎞ ∫A ⎜⎝ B ⎟⎠ dB = 2.62 1

z*

⎛1− B ⎞ ∫A ⎜⎝ B ⎟⎠ ln BdB = -4.83 1

The suspended sediment transport is: ⎛ ⎛ h q S = 1.83q B Q = 1.83q B ⎜ I 1 ln⎜⎜ ⎜ 0 . 033 ks ⎝ ⎝

⎞ ⎞ m3 ⎟⎟ + I 2 ⎟ = 3.0755×10-4 ⎟ m× s ⎠ ⎠

Figure 9.5 - Suspended sediment transport parameters.

SEDIMENT TRANSPORT

195

The total sediment transport is: QT = q B + q S = 3.1795×10-4

m3 m× s

If we consider the combined wave and current, by linear wave theory the amplitude of water particle on the bottom is: Aw =

H 1 = 0.68 2 sinh(2πh / L)

the maximum horizontal velocity of water particle on the bottom is: 2π = 0.53 m/s T

U m = Aw ω = Aw

the wave friction coefficient is: ⎞ − 6.3 ⎟ = 0.028 ⎟ ⎠ the maximum bottom shear stress by wave is: ⎛ ⎛k f w = exp⎜ 5.5⎜⎜ s ⎜ ⎝ Aw ⎝

⎞ ⎟⎟ ⎠

0 .2

1 ρf cU m2 =4.07 N/m2 2 The mean bottom stress under combined wave and current is:

τ w, max =

1 2

τ wc = τ c + τ w, max =5.28 N/m2 the bed load transport is: q B = 2 D50

⎛ − 0.27( s − 1) D50 ρg ⎞ τc m3 ⎟⎟ = 1.2532×10-5 exp⎜⎜ ρ µ rτ wc m× s ⎝ ⎠

the friction velocity is: u *,WC =

τ Wc =0.072 m/s ρ

The suspended sediment transport parameters are: A=ks /d =0.01 z*=wf /(κu*,wC)=0.42 the Q values can be calculated by numerical integration or using figure 9.5 or using table 9.1 . By numerical integration: I 1 = 0.216

A ( z* −1) (1 − A) z*

z*

⎛1− B ⎞ ∫A ⎜⎝ B ⎟⎠ dB = 3.8 1

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AN INTRODUCTION TO COASTAL DYNAMICS AND SHORELINE PROTECTION

I 2 = 0.216

A ( z* −1) (1 − A) z*

z*

⎛1− B ⎞ ∫A ⎜⎝ B ⎟⎠ ln BdB = -6.3 1

. ks/h

z*=0.00

Table 9.1- Einstein Integral factor Q.

z* =3.00 0.432 0.432 0.432 0.432 0.431 0.431 0.430 0.428 0.424 0.417 0.404 0.374 0.339 0.317

z* =4.00 0.276 0.276 0.276 0.276 0.275 0.275 0.275 0.274 0.273 0.270 0.264 0.249 0.236

SEDIMENT TRANSPORT

0.00001 303000 0.00002 144000 0.00005 53600 0.0001 25300 0.0002 11900 0.0005 4360 0.001 2030 0.002 940 0.005 336 0.01 153 0.02 68.9 0.05 23.2 0.1 9.8 0.2 3.9 0.5 0.8 1 0

z* =0.20 32800 17900 7980 4320 2330 1020 545 289 123 63.9 32.8 13.1 6.3 2.8 0.7 0

EINSTEIN INTEGRAL FACTOR Q z* 0.40 z* z* z* z* z* =0.60 =0.80 =1.00 =1.50 =2.00 3880 527 88 20.0 2.33 0.973 2430 377 71.6 17.9 2.31 0.973 1300 239 53.6 14.4 2.28 0.967 803 169 42.7 13.6 2.25 0.967 496 119 33.9 11.9 2.21 0.967 260 74.3 24.6 9.8 2.13 0.962 158 51.2 19.1 8.4 2.05 0.951 95.6 35.1 14.6 7.0 1.96 0.940 48.5 20.8 10.0 5.4 1.78 0.907 28.6 13.8 7.3 4.3 1.62 0.869 16.5 8.9 5.2 3.3 1.42 0.809 7.7 4.8 3.1 2.2 1.10 0.694 4.1 2.8 2.0 1.5 0.84 0.568 2.0 1.5 1.2 0.9 0.55 0.414 0.6 0.5 0.4 0.3 0.17 0 0 0 0 0

197

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The suspended sediment transport is: 3 ⎛ ⎞ 5.6481×10-4 m ⎛ d ⎞ ⎟⎟ + I 2 ⎟ = qS = 1.83qB Q = 1.83qB ⎜⎜ I1 ln⎜⎜ m× s ⎟ ⎝ 0.033k s ⎠ ⎝ ⎠

The total sediment transport is: QT = q B + q S = 5.7734×10-4

m3 m× s

3 Basic shore processes When waves approach a sloping beach and break, nearshore currents are generated, which action depends on the beach characteristics and the wave conditions. Beach morphology is strongly controlled by nearshore currents because of sediment movements; water fluxes between the coast and the offshore zone contribute to renew the coastal waters. Nearshore current patterns are a combination of longshore currents, rip currents and undertow. For large incident wave angle, alongshore momentum generated by the wave breaking process sets up strong longshore currents. The forward flow of the water particles in the breaking process sets up longshore currents. Smaller incident wave angle generate weaker longshore currents. The forward flow of the water particles in the breaking waves also “pumps” water across the breaking zone, increasing the water level there. The onshore momentum of the waves holds some of this water close to shore, causing a shoreward elevated water level (wave set-up). This phenomenon can be explained by the concept of radiation stress, introduced by Longuet-Higgins and Stewart (1964) and described in chapter 3. 3.1 Nearshore circulation

The explaination for the generation of the cell circulation was developed following the introduction of the concept of radiation stress by Longuet-Higgins and Stewart (1964), defined as the excess of flow of momentum due to the presence of waves. The shoreward component of the radiation stress produces a set-down immediately offshore of the breakers and a set-up within the surf zone. In a two-dimensional case, the wave crests are always parallel to the shoreline. Averaging over one wave period, continuity of mass must be satisfied at every cross section. This necessitates a vertical distribution of mass transport velocity: forward flow at the surface and near the bottom, return flow near middepth (Ippen, 1966). The forward flow at the surface transports the water in surface rollers toward the coast,and the wave drift is also directed toward the coast. These contributions are concentrated near the surface. As the net flow must be zero, they are compensated by a return flow in the offshore direction, which is concentrated near the bed (undertow).

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In a three-dimensional case, a cellular circulation takes place, which is constituted by longshore currents and rip currents.

Figure 9.6-Nearshore circulation pattern. Two dimensional case (after Ippen, 1966). The longshore current is generated by the shore-parallel component of the radiation stress associated with the breaking process for obliquely incoming waves. This current, which is parallel to the shoreline, carries the sediments alongshore and it is approximately proportional to the square root of the wave height and to sin(2αb), where αb is the wave incidence angle at breaking. The movement of beach sediment along the coast is referred to as littoral transport or longshore sediment transport, whereas the actual volumes of sand involved in the transport are termed the littoral drift. This longshore movement of beach sediments is of particular importance because the transport can either be interrupted by the construction of jetties and breakwaters (structures which block all or a portion of the longshore sediment transport), or can be captured by inlets and submarine canyons. In the case of a jetty, the result is a buildup of the beach along the updrift side of the structure and an erosion of the beach downdrift of the structure (CEM, 2001). The rip currents are part of cellular circulations fed by longshore currents within the surf zone that increase from zero at a point between two neighboring rips, reaching a maximum just before turning seaward to form the rip (see figure 9.6). The longshore currents are in turn fed by the slow shoreward transport of water into the surf zone from breaking waves. A nearshore circulation cell thus consists of longshore currents feeding the rips, the seaward flowing rip currents that extend through the breaking zone and spread out into rip heads, and a return onshore flow to replace the water moving offshore through rips. (Shepard and Inman, 1950). The cell circulation results from alongshore variation in wave heights, which in turn produce a longshore variation in set-up elevations. The set-up results from the Sxx onshore component of the radiation stress being balanced by the pressure gradient of the seaward-sloping water surface in the nearshore. Balancing those forces yields:

⎛ ∂η 8 ⎞ ∂d = −⎜⎜1 + 2 ⎟⎟ ∂x 3 γ ⎝ ⎠ ∂x

(9.66)

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For the cross-shore slope of the set-up denoted by η . There is a direct proportionality with the beach slope S 0 = ∂d / ∂x , but not a direct dependence on the

Figure 9.7 - The nearshore cell circulation consists of (1) feeder longshore currents, (2) seaward-flowing rip currents, and (3) a return flow of water from the offshore zone into the surf zone (after Komar, 1988). wave height. However, the larger waves break in deeper water than smaller waves, and the set up therefore begins farther seaward at longshore locations where the larger waves occur. Inside the surf zone, the mean water level is higher shoreward of the larger breakers than it is shoreward of the small waves. A longshore pressure gradient therefore exists, which will drive a longshore current from positions of high waves and set-up to adjacent position of low-waves. In addition of the Sxx component of the radiation stress, there is a Syy component, a moment flux acting parallel to the wave crest, in this case parallel to the shoreline. This component is given by:

⎡ ⎤ kd 1 S yy = ρgH 2 ⎢ ⎥ 8 ⎣ sinh(2kd ) ⎦

(9.67)

since the wave height varies alongshore, Syy will similarly vary and there will exist a longshore gradient:

∂S yy ∂y

=

⎡ ⎤ ∂H 1 kd ρgH ⎢ ⎥ 4 ⎣ sinh(2kd ) ⎦ ∂y

(9.68)

This longshore gradient produces a flow of water away from the regions of high waves and toward position of low waves. The flow then turns seaward as a rip current where the waves and the set-up are lowest and the longshore currents converge.

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The cell circulation therefore depends on the existence of variations in wave heights along the shore. The most obvious way to produce this variation is by wave refraction, which can concentrate the wave rays in one area of a beach, causing high waves, and at the same time spread the rays in the adjacent area of the beach and then produce low waves. The position of rip currents and the overall cell circulation will then be governed by wave reflection and hence by the offshore topography. Headlands, breakwaters and jetties can affect the incident waves by the partial sheltering of the shore and thereby produce significant longshore variations in wave height and set-up. Wave reflection and diffraction produce alongshore gradients with lower waves and set-up in the lee of the headland or breakwater, which in turn generate longshore currents flowing inward toward the sheltered region. In some situations this process can account for the development of strong rip currents adjacent to jetties and breakwaters. An example of rip current acting on bottom topography is the phenomenon of rip channels. On a barred profile the wave breaking on the bar will induce a wave set-up, causing an increase of water level inshore of the bar. A bar will be in many cases interrupted by holes (rip channels) found at more or less regular intervals. The wave breaking is less intensive in the rip-channels due to the larger depth and because the wave refraction may concentrate the wave energy on the bars at the sides of the channel. Wave-current interaction may affect the development of rip-currents. In fact, the weak currents generated by a gentle alongshore variation of the wave field can cause significant refractive effect on the waves as to change the structure of the forcing which drives the currents and the instability of the cellular circulation. When currents are weak compared to the wave group velocity, their effects on waves are small, but such effects are sometimes not negligible. This is the case of rip currents produced by alongshore topographic variations on otherwise alongshore uniform beaches. These alongshore variations in the topography, like gentle rip channels, produce longshore variations in the radiation stress and provides the source of vorticity and of horizontal circulations, which interact with the waves, so wave radiation stress will be modified. Such changes of course are small relative to the effect due to wave breaking, but can be comparable to the variations caused by the topography. When this is the case, the circulations of interest can be significantly affected by the wave-current interaction. The interaction of the narrow offshore directed rip currents and incident waves produces a forcing effect opposite to that due to topography, hence it reduces the strength of the currents and restricts their offshore extent. The two physical processes due to refraction by currents, behind the wave rays and changes in the wave energy, both contribute to this negative feedback, on the wave forcing (Yu and Slinn, 2003).

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Figure 9.8 - The circulation current generated by normally incident waves on a barred coast with rip-channels (after Fredsøe and Deigaard, 1992). 3.2 Wave run-up in the swash zone

The wave run-up is defined as the maximum elevation of the wave uprush above the mean sea level. The uprush is given by the sum of two components: the positive elevation of the mean water level caused by wave action and the fluctuations above the mean water level (swash). The concept of wave run-up is frequently used to describe the beach profile processes. The wave run-up parameter calculation depends on the processes of wave transformation, such as the wave reflection, the interaction between the bottom and the waves and the sediments properties (e.g. porosity and permeability). Actually, some formulations of wave run-up are based on the empirical studies carried out by Hunt (1979) for regular waves and for irregular waves. Regular waves For regular breaking waves, the run-up is a function of the beach slope, incident wave height and wave steepness. According the Hunt (1979) formulation: R = ξ 0 for 0.1<ξ0 <2.3 H0

(9.69)

ξ0 is the so-called surf-similarity parameter: ⎛ H0 ⎞ ⎟ ⎟ ⎝ L0 ⎠

ξ 0 = tan β ⎜⎜

−1 / 2

(9.70)

where β is the beach slope, H0 and L0 are the incident wave height and the wave length on deep waters, respectively, and the ratio H0 / L0 is the wave steepness. Irregular waves For irregular waves the wave run-up is a function of the similarity parameter and depends on the interaction between individual runup bores (CEM, 2001).

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203

The maximum run-up is an important parameter to describe the active portion of the beach profile. Mase (1988) describes the expressions (Table 9.2) for the maximum run-up (Rmax) and other run-up parameters valid for flat and impermeable beaches and for 1 / 30 ≤ tan β ≤ 1 / 5 , H 0 L0 ≥ 0.007 . 3.3 Bar formation by cross-shore flow mechanisms

The classic response of a planar beach to storm waves consists of erosion of the beachface and inner surf zone and deposition around the wave breakpoint resulting in the formation of a storm bar. The first explanation of bar formation was given as early as 1863 by Hagan (in Komar, 1976) who explained a bar formation by ‘a seaward going undertow meeting the shoreward moving waves’. This intuitive idea has now been formalised where the ‘undertow’ is represented by the bed return flow and the intuitive expression ‘shoreward moving waves’ refers to the shoreward flow asymmetry associated with the highly non-linear shoaling waves. Thus, offshore of the breakpoint, the residual flow close to the bed is shoreward, whereas onshore of the breakpoint, the near-bed flow is seaward. This results in a region of flow convergence and hence sediment accumulation close to the breakpoint and the formation of a bar (Dyhr-Nielsen and Sorensen, 1970). For bars formed according to the cross-shore flow mechanism, the distance from the shoreline to the bar crest (xbar) is described by (Holman and Sallenger, 1993)

xbar =

db γH b = tan β tan β

(9.71)

where db is the depth at the break point, tan(β) is the nearshore gradient, γ is the breaker criterion and Hb is the breaker height. The cross-shore flow mechanism of bar formation does not account for the formation of multiple bars because only one breakpoint or breaker zone will occur. However, several explanations can be given for the formation of multiple bars according to the cross-shore flow mechanism. The first involves the presence of multiple breakpoints. Once waves break on an outer bar of a gently sloping beach they may recover and reform as they travel across the deeper shoreward trough. These reformed waves may break for a second or perhaps a third time before they eventually reach the shoreline, resulting in multiple bar morphology. The presence of several distinct wave climates, each one responsible for a single bar, may result in multiple bar morphology (Evans, 1939). Another mechanism may involve varying water levels and breaker position due to tides (Komar, 1976).

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Table 9.2 - Mase (1988) expressions for run-up parameters. Symbol Maximum Run-up

Rmax

Formulation Rmax = 2.32ξ 00.77 H0

Run-up exceeded by 2% of the R2% crests

R 2% = 1.86ξ 00.71 H0

Average of the highest 1/10 of the R1/10 run-ups

R1 / 10 = 1.70ξ 00.71 H0

Average of the highest 1/3 of the R1/3 run-ups

R1 / 3 = 1.38ξ 00.70 H0

Mean run-up

R = 0.88ξ 00.69 H0

R

Chapter 10 Beach profile modeling 1 Cross-shore transport Cross-shore sediment transport encompasses both offshore transport, which occurs during storms, and onshore transport, which dominates during mild wave activity. Transport in these two direction appears to occur in significantly distinct modes and with markedly different time scales. Offshore transport is of greater relevance due to the potential for damage to structures, and tends to occur with greater speed, so it is of greater concern (CEM, 2001). When the coastal profile is exposed to high waves and storm surge, the sediments near the shoreline will be transported offshore and typically be deposited in a bar resulting in a overall flattening of the slope of the shoreface. However, the inner part of the shoreface as well as the foreshore will get steeper in this process, and the shoreline will recede. During following periods of smaller waves, swell and normal-water level conditions, the bar will travel very slowly towards the coastline again, practically rebuilding the original coastal profile (Mangor, 2001). Cross-shore sediment transport takes place when an existing beach profile changes. If the beach profile is close to its equilibrium with the existing environmental conditions, little cross-shore sediment motion will take place. Should the environmental conditions change, however, substantial cross-shore sediment transport must be expected in order to come to the new equilibrium that accompanies the new conditions. The rate of cross-shore transport is normally assumed to be proportional to the difference between the existing beach profile and the equilibrium profile that matches the new environmental conditions (Bakker, 1968). This means that cross-shore sediment movement is large, immediately after a change in environmental conditions and subsequently slows down. As a result, shoreline change also begins rapidly and then slows down in time. The annual beach profile changes from a summer berm profile to a winter bar profile (figure 10.1). The summer berm is more or less in equilibrium with the small summer waves and when the higher and steeper winter waves come, the beach seeks a new equilibrium. Similarly, at the beginning of summer, the winter equilibrium beach must respond to the more gentle summer wave

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climate and the beach adjusts itself again. In both cases, cross-shore motion is the main sediment transport vehicle. In the same way, temporarily higher water levels such as storm surge permit larger waves to come closer into shore, and the beach will respond suddenly and dramatically through extensive offshore transport (Kamphuis, 2000).

Figure 10.1 - Annual change in beach profile (after Kamphuis, 2000). Dean (1973) noted that suspended sediment can move either onshore (constructive) or offshore (destructive), depending on how high a sand grain is suspended off the bottom. Under the wave crest, if the sediment particle is suspended a distance above the bottom proportional to the wave height H, and if the particle has a fall velocity wf, then the time required for the grain to fall back to the bottom would be proportional to H/wf. If this fall time is less than one-half of the wave period, then the particle should experience net onshore motion, whereas the particle should move offshore if the fall time is greater than one-half the wave period. So it has been shown that a net onshore or offshore sediment transport can be correlated with the so-called fall time parameter H/wfT. The condition H0/wfT≅1 approximates a critical threshold. If the ratio exceeds 1, sediment moves offshore (tends to produce a bar profile); if it is less than 1, sediment moves offshore to produce a berm. Further information may be found in Kraus et al., (1991). On the basis of this concept, some erosion/accretion criteria were conceived, which will be discussed in this chapter. The long-term and short-term limits of cross-shore sediment transport are important in engineering considerations of profile response. During short-term erosional events, elevated water levels and high waves are usually present and the seaward limit of interest that is to which significant quantities of sand-sized sediments are transported and deposited (CEM, 2001). The seaward limit of effective profile fluctuation over long-term (seasonal or multi-year) time scales is a useful engineering concept and is referred to as the “closure depth”, denoted by dc. Based on laboratory and field data, Hallermeier (1978, 1981) developed the first rational approach to the determination of closure depth. Hallermeier defined a condition for sediment motion resulting from wave conditions that are relatively rare. Effective significant wave height He and

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effective wave period Te were based on conditions exceeded only 12 hr per year; i.e., 0.14 percent of the time. The resulting approximate equation for the depth of closure was determined to be:

 H2 d c = 2.28H e − 68.5 e2  gTe

  

(10.1)

in which He can be determined from the annual mean significant wave height H and the standard deviation of significant wave height σH as (CEM, 2001):

H e = H + 5.6σ H

(10.2)

Erosion of the beach (decrease in volume of beach material) will cause recession of the beach profile (movement landward). One classic example of beach recession results from sea level rise. Higher water levels allow larger waves to come closer into shore, resulting in erosion of the top portion of the profile to adjust to the more severe wave conditions. It is possible to estimate the net beach recession accompanying sea level rise by assuming that the wave climate remains the same and the beach profile retains its shape. This beach profile eventually must rise with the water level and the volumes of sand required to raise the profile in the foreshore must come from a landward movement of the profile. This results in Bruun’s Rule.

R=

xc d (d d + d c )

(10.3)

where R is the recession, d is the water level rise, dc is the closure depth and dd is the dune height. On the basis of the Bruun’s rule, a number of formulas for predicting the shoreline recession were conceived, which will be discussed in the next chapter.

2 Cross-shore sediment transport and equilibrium beach profile The shape of a beach (its water depth as a function of distance from offshore) is called the beach profile (Kamphuis, 2000). Natural beaches are modified by wave action, which transport sediments and generate variations in geomorphologic features (cusps, tombolos, longshore bars, features caused by the presence of coastal structures). The complexity of topographic changes is usually schematised into cross-shore and longshore-processes; regarding crossshore processes, the beach profile usually shows seasonal changes, such as coastline progress and regression, erosion and berm formation in the foreshore zone, and the appearance/disappearance of a longshore bar in the surf-zone. The cyclical changes of the beach profile are the result of constructive and destructive alternating forces, which action produces deposition and erosion processes. Destructive storm waves erode dune and beach, then transport the sediment offshore. The combination of the calm waves and storm waves causes a beach

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cycle. Varying wave conditions result in varying onshore and offshore transports over the coastal profile. These transports are to some extent reversible and therefore irrelevant in terms of longshore littoral drift. When the coastal profile is exposed to high waves and storm surge, the sediments near the shoreline will be transported offshore and typically be deposited in a bar resulting in an overall slope flattening of the shoreface. However, the inner part of the shore face as well as the foreshore will get steeper in this process, and the shoreline will recede. During following periods of smaller waves, swell and normal water-level conditions, the bar will travel very slowly towards the coastline again, practically rebuilding the original coastal profile. During such a sequence of profile erosion and rebuilding, certain parts of the coastal profile may experience temporary erosion. This may not be recorded in profile surveys, because they are possible after the storm when some rebuilding has already taken place. It is important to take such temporary profile fluctuations into account when designing structures in the coastal zone. It is particularly important to have a sufficiently wide beach so that the temporary beach erosion will not cause erosion of the coast (Mangor, 2001). The beach profile responds to wave action between two limits: the limit on the landward side where the wave run-up ends, and the other limit in deeper waters where the waves can no longer produce a measurable change in depth. The latter is the so-called depth of closure, that is not the location where sediment ceases to move, but that location of minimum depth where profile surveys before and after a period of wave action, a storm perhaps, lie on top of one another. The onshore and offshore transport is closely related to the form of the coastal profile. Several investigations have revealed that a coastal profile possesses an average, characteristic form which is referred to as the theoretical equilibrium profile. The equilibrium profile has been defined as “a statistical average profile, which maintains its form apart from small fluctuations, including seasonal fluctuations.” The depth d [meters] in the equilibrium profile increases exponentially with the distance x from the shoreline according to the equation (Dean, 1977):

d = Ax n

(10.4)

where A is the sediment scale parameter (depending on the mean grain size D50), x is the distance from the shoreline and d is the water depth. According to field studies carried out by Bruun (1954) and theoretical studies conducted by Dean (1977), the value of n is 2/3 when wave energy dissipation per unit water volume is the dominant force. In figure 10.2 equilibrium profiles for various values of D50 are given.

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3 Dean’s model for equilibrium beach profile A model for the equilibrium beach profile expected under a given wave condition was developed by Dean (1977). The model considers the high level of turbulence in the surf zone, due to the energy dissipation of the breaking and broken waves. The hypothesis is that the bed on a beach profile can only withstand a certain rate of energy dissipation (and thus production of turbulence). If this level is exceeded, it will lead to a reshaping of the beach profile to give a wider surf zone with a lower intensity of the wave energy loss. Dean considers the energy loss per unit bed area, as well as per unit water volume. The latter has shown to be the most successful and is therefore treated in the following. Dean (1977) introduces a number of simplifications in order to obtain simple analytical expressions: the waves are described as linear shallow water waves and the wave height (H) in the surf zone is taken to be proportional to the water depth (d):

H = Kd

(10.5)

The analysis is based on the energy conservation equation, which for steady conditions reads:

dF = De dx

(10.6)

where D e is the depth integrated energy dissipation. The wave energy flux F can be expressed as:

F=

ρgH 2 8

( gd )1 / 2

(10.7)

the ratio between the depth integrated energy dissipation ( D e ) and the water depth (d) gives the mean energy dissipation per unit volume ε : 2  De 1 d  ρgH ( gd )1/ 2  = =  d d dx  8  2 1 d  ρgK 1/ 2 5 / 2  5 d (d 3 / 2 )  ρgK 2 g g d  = = d dx  8 dx  24

ε=

(10.8)

Following Dean’s hypothesis, that ε is constant for a given bed material on an equilibrium beach profile, leads to the following shape of the beach profile:

d 3 / 2= or:

24 ε ( D) x 5 ρgK 2 g

(10.9)

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d ∝ x2/3

(10.10)

where ε(D) signifies that the equilibrium energy dissipation is a function of the grain size D of the bed material. The relation (10.10) has been supported by many field measurements. A similar relationship was proposed by Bruun (1954) as an empirical result based on analyses of beach profiles under very different conditions. It should be noted that (10.9) describes a monotonic profile, with the water depth always increasing with the distance from the shore. In the case of barred profile it can, at most, only be expected to describe a part of the profile (Fredsøe and Deigaard, 1992). It should be noted that the concept of equilibrium profile is only valid for the breaker zone (or the surf-zone), i.e. out to the closure depth dc.

 H2  d c = 2.28 H e − 68.5 e2   gTe 

(10.11)

as defined by Hallermeier (see chapter 9). It is seen that the equilibrium profile does not depend on the wave height because the water depth limits the wave height inside the surf-zone. The main influence of the wave height is the increasing wave heights implying that the breaking line is moved further offshore, or in other words, the surf zone is widened. The width of the surf-zone and the average slope of the surf-zone thus depend on the mean grain size as well as on the wave condition as shown in figure 10.3.

Figure 10.2- Equilibrium profiles for various values of D50 In the real beach profiles there is often a sorting of the sediments: the mean grain size decreases with increasing distance from the shoreline. If this variation is introduced in the equilibrium profile considerations, one gets a more accurate

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representation of the equilibrium profile at a specific site. The concept of the equilibrium profile is a rather practical “tool” for the analysis of coastal conditions and, as already mentioned, for preliminary design considerations (Mangor, 2001).

. Figure10.3 - Width of the surf zone as a function of D50 for various wave climates represented by Hs, 12 h/y. 3.1 Equilibrium parameter A The sediment scale parameter A [m-3] depends on the dissipation of energy according to the following formulation:  24 1 A =  De 3/ 2 2  5 ρ g γ 

   

2/3

(10.12)

Where De is the equilibrium wave energy dissipation per unit volume, ρ is the water density, g is the acceleration due to gravity and γ is the ratio between wave height and local depth within the surf zone. Dean (1983) proposes a simple relationship between A and the fall velocity of sediment grains: A = 0.50 ⋅ w 0f .44

(10.13)

where wf is the fall velocity in m/s. If the water temperature is 20°C and grain dimension is similar to the sandy beach sediments (which fall velocity is between 1 and 10 cm/sec), we get:

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 w 2f A = 2.25  g 

   

1/ 3

(10.14)

Moore (1982) calculates the profile parameter as a function of the median grain size. The fitting of field data gives the formulations in table 10.1. Table 10.1 - Profile parameter A as a function of the grain size. A

D50 (mm)

0.41( D50 ) 0.94

D50 < 0.4

0.23( D50 ) 0.32

0.4 ≤ D50 < 10

0.23( D50 ) 0.28

10 ≤ D50 < 40

0.46( D50 ) 0.11

D50 ≥ 40

4 Processes of accretion and erosion 4.1 Surf zone The beach profile is the result primarily of the cross-shore sediment transport process. Typical examples of this wave induced sediment movement are the summer (swell profile) and winter (storm profile) beach profiles. In summer the mean wave height decreases, the mean wave period usually increases and beach recovery begins. Sand is transported onshore and a new berm is created by the run up. Because of the onshore sediment transport, the slope in the surf zone becomes steeper. In winter the mean wave conditions are more intense than in summer. Material is moved offshore and the beach slope in the surf zone becomes gentler. As a result one or more offshore bars may form. In other words, erosion processes are related to storms that produce shorter period waves and greater wave heights and steepness (H/L is high). Deposition occurs by vertical accretion onto the berm when wave steepness (H/L) is low and wave period (T) is long, and by welding of bars to the beach face. Deposition increases the berm width and steepens the beach face. The rate of bar migration depends on the wave height, the beach slope, the grain size. Bars migrate faster in larger swell conditions and on steeper beaches. Fine grain size contribute to slower bar migration because the beach is flatter. Erosion and accretion processes depend on the beach face permeability (grain size, sorting, degree of saturation), sediment characteristics (critical wave steepness values required for erosion are typically greater for coarse beaches); settling velocity of sediment (largely determined by grain size); velocity asymmetry in shallow water waves, wave height (which influences the height of suspension), and particle settling velocity relative to wave period.

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4.2 Swash zone The study of swash dynamics has both practical and research applications to coastal engineering. The local elevation of the water table and its influence on groundwater infiltration and exfiltration within the swash zone may contribute to accretionary or erosional trends at the shoreline. The vertical flow through the bed has several important applications to sediment transport in the swash zone. These include: uprush-backwash flow (figure 10.4) asymmetry resulting from the loss/addition of swash volume, the altered effective weight of superficial sediment due to vertical fluid drag (Nielsen, 1992), and modified shear stress exerted on the bed due to boundary layer “thinning” or “stretching”. Some studies shows that the “beach dewatering” technology (Turner and Leatherman, 1997) is a viable tool for shoreline stabilization and a practical endpoint of more basic research questions of groundwater interactions at the beach face. In fact, if the beach is permeable and the sand is not saturated, the water percolates through the substrate and the backrush decreases. An amount of sediment is accumulated on the beach in the swash zone and the beach slope increases. The result is a convex beach profile. At high frequencies (storm waves), if the substrate is saturated, the water cannot percolate through the sand and the backrush current cannot decrease. In this case the sediment transport associated with backrush is greater than uprush, which results in a concave beach profile.

5 Erosion/accretion parameters Wave steepness is an important factor for the modification of the beach profile. Steep winter storm waves tend to remove material from the beach face and deposit it offshore as a bar, whereas summer swell and swell generated during the decay of a storm tend to build the berm and widen the beach. In fact, most erosion and accretion predictors are based on the wave steepness, and many of these criteria are summarized in Kraus et al. (1991).

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Figure 10.4 - Uprush- backwash flows in storm and swell conditions (after Nielsen, 1992). The subject of beach erosion and accretion was greatly stimulated by Dean (1973), who derived a predictive criterion from a simple quantitative model of sand particle motion produced by a breaking wave. Suppose that under a breaking wave of height Hb, a sand particle is lifted to some elevation z in the water column that is a fraction of the wave height:

z = bH b

(10.15)

where b is less than (but in the order of) unity. The time t required for the particle to settle to the bottom depends on the fall speed wf, as:

t = z / wf

(10.16)

Dean assumes that under a breaking wave crest, the wave particle velocities are directed onshore, and the resulting net displacement is onshore for half of a wave period. Therefore, if the fall time t is less than half the wave period, T/2, the net onshore water motion under the wave crest would carry the sand particle onshore, whereas if t>T/2 (but less than T), the particle will move offshore under the wave trough. In this situation, the following equations are valid: 2bH b <1 wf T

onshore motion

(10.17)

2bH b > 1 offshore motion wf T

(10.18)

Dean also expressed these conditions as:

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H b πw f < L0 bgT

onshore motion

(10.19)

H b πw f > L0 bgT

offshore motion

(10.20)

The condition expressing onshore motion signifies a summer or accretionary beach profile, and the condition expressing offshore motion signifies a winter or erosional beach profile. By replacing Hb by H0 and examining small-scale largetank data of Saville, Dean (1973) obtained the criterion:

πw f H0 = 1.7 L0 gT

(small scale)

(10.21)

From Dean’s results, it is clear that the sand size (through the fall velocity) plays a major role in the formation of the different seasonal profiles as does the wave period. Beaches with finer sand (and smaller fall velocities) require a smaller value of wave steepness for the formation of a storm profile. Some studies demonstrated that criteria based on two parameters were more predictive than criteria based on a single parameter. Sunamura and Horikawa (1974) introduced also the dependence on the beach slope and the sand diameter : H 0 L0

C* = tan( β )

− 0.27 

D  L  0

   

0.67

(10.22)

where H0 and L0 are the offshore wave height and wave length, D is the sand grain diameter, and β is the beach slope. • • •

C*>8 C*<4 4< C*<8 (10.23)

destructive (erosion) constructive (accretion) the wave action is intermediate

Hattori and Kawamata (1980) modified the Dean (1973) equation as: C* =

H 0 gT tan( β ) L0 w f D

(10.24)

H0 and L0 are the wave height and the wave length on deep waters, T is the wave period, wf and D are the sediment fall velocity and the grain diameter, respectively. • •

C*>0.5 C*<0.5

destructive (erosion) constructive (accretion)

(10.25)

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Kriebel, Dally and Dean (1986) criterion is based on indicators H0/L0 and

πw f gT :

( L0 > 5(πw f

) gT )

•

H 0 L0 < 4 πw f gT

summer conditions

•

H0

winter conditions

(10.26)

A modified formulation of the Sunamura and Horikawa (1974) equation uses different interval limits: • •

C*<3.5 C*>9

constructive (accretion) destructive (erosion)

(10.27)

for intermediate values the wave action is transitional. In the case of random waves, the authors suggest that the mean wave should be taken into account. If the significant wave is used, the limits of intervals are modified as: 3.5
C * < 4.8 ⋅ 10 8 ( H 0 / D50 ) −3.05

constructive (accretion)

(10.28)

•

C ≥ 4.8 ⋅ 10 ( H 0 / D50 )

destructive (erosion)

(10.29)

*

8

−3.05

Larson and Kraus (1988) modified the criterion originally developed by Dean (1973) which is based on erosion and accretion produced in small wave tanks. The resulting criterion is expressed as:

πw f H0 = 1.7 L0 gT

(small scale)

(10.30)

πw f H0 = 5.5 L0 gT

(prototype)

(10.30)

 πw f H0 = 115 L0  gT

   

1.5

(10.31)

Equation (10.32) is valid in situation of better separation between accretionary and erosional cases. In figure 10.5, line A expresses the Dean (1973) criterion; line B lies parallel to line A with a coefficient evaluated for prototype scale data. The change in coefficient value represents a scale effect, since the transport processes are different under small and large waves, even if the steepness is the same. Although line B gives an improvement over line A, the rotated line C provides a better separation of all erosional and accretionary cases.

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Figure 10.5 – Criterion for distinguishing bar and berm profiles by use of deepwater wave steepness and Dean’s parameter (Dean, R.G. & Dalrymple, R. A , 2002). Kraus, Larson, and Kriebel (1991) examined large wave tank data and proposed the following criterion:  H H0 ≤ 0.0007 0  wf T L0 

3

   (erosion) H 0 > 0.0007 H 0   wf T L0  

3

  (accretion) (10.32)  

5.1 Case study Two examples, distinguished by different median grain size, are given to illustrate some of the above criteria for predicting beach erosion and accretion caused by cross-shore transport. The following criteria are considered: Criterion 1: Kraus, Larson, and Kriebel (1991) eq. (10.32) Criterion 2: Larson and Kraus (1988) eq. (10.29) Criterion 3: Larson and Kraus (1988) eq. (10.28 -10.29) Given : [A] Quartz sand D50=0.2 mm H0=1 m T=10 sec

[B]

Quartz sand D50=0.4 mm H0=1 m T=10 sec

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Determine, using the criteria presented, whether the beach will experience erosion and accretion. Required constants: density of quartz sand ρs=2.65 g/cm3; density of the sea water at 20°C, ρ=1.025 g/cm3; and kinematic viscosity of seawater at 20°C, v=0.01 cm2/sec. Solution: (a) calculate L0 [metric units]: L0 =1.56⋅T2=1.56(10)2=156 m (b) calculate wf as: 1.1 w f = [( ρ s / ρ − 1) g ] 0.7 D50 /[6v 0.4 ] :

[A] wf=[(1.59)⋅981]0.7 (0.02)1.1/ [6⋅(0.01)0.4]=2.4 cm/sec (=0.024 m/sec) [B] wf=[(1.59)⋅981]0.7 (0.04)1.1/ [6⋅(0.01)0.4]=5.2 cm/sec (=0.052 m/sec) (c) evaluate relationships for each criterion: H0/L0=1/156=0.006 Criterion 1: [A] 0.00070⋅(H0 /wfT)3=0.00070⋅(1 / [0.024⋅10])3=0.051 0.006 < 0.051 Æ erosion [B] 0.00070⋅(H0 /wfT)3=0.00070⋅(1 / [0.052⋅10])3=0.005 0.006 > 0.005 Æ accretion

Criterion 2: [A] 4.8⋅108(H0/D50)-3.05=4.8⋅108(1/0.0002) -3.05=0.002 0.006 > 0.002 Æ erosion [B] 4.8⋅108(H0/D50)-3.05=4.8⋅108(1/0.0004) -3.05=0.021 0.006 < 0.021 Æ accretion Criterion 3: [A] 115⋅(πwf /gT)1.5 =115⋅([3.14⋅0.024]/[9.81⋅10])1.5=0.002 0.006 > 0.002 Æ erosion [B] 115⋅(πwf /gT)1.5 =115⋅([3.14⋅0.052]/[9.81⋅10])1.5=0.008 0.006 < 0.008 Æ accretion

6 Analytical profile modeling Analytical models are closed-form mathematical solutions of a simplified version of the equation for shoreline and profile change, respectively. By developing analytical solutions originating from mathematical models that describe the basic physics involved, essential features of beach response may be derived, isolated, and more readily comprehended. Also, with an analytical solution as a starting point, direct estimates can be made of characteristic

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219

parameters associated with a phenomenon, such as the time elapsed before bypassing of a groin occurs, percentage of volume lost from a beach fill, and growth of a salient behind a detached breakwater. Thus, analytical solutions serve mainly as a means to identify characteristic trends in beach change through time and to investigate basic dependencies of the change on the incident waves and water levels as well as the initial and boundary conditions. As a result, analytical models will typically have a longer time perspective than their numerical counterparts. Bruun’s model The first model used to calculate the response of coastline to the sea water level is the Bruun’s model (1968). Bruun expressed the sediment volume variation on the active portion of beach profile in the x-direction as:

∆V = R∞ (d c + B)

(10.33)

where R∞ is the regression of the coastline: for y<0, (dc+B) is the vertical extension of the profile, as a summation of the berm height B and the closure depth dc. The volume needed to maintain the profile in equilibrium is: ∆V = SWb

(10.34)

Where S is the increase of the sea level wave set-up and Wb is the amplitude of the equilibrium profile. Bruun combined the “generated” sand volume variation with the “required” sand volume variation to obtain the following formulation: Wb (10.35) R∞ = S dc + B defined as Bruun’s rule. The Bruun’s rule is independent of the profile shape. The last equation was tested by laboratory experiments and field studies, and the shoreward and landward limits of the equilibrium profile were calculated. The Bruun’s rule is illustrated in figure 10.6. The Bruun’s equation is useful for its simplicity, but it overestimates beach regression. Edelman (1972) model Edelman (1972) uses a modified version of the Bruun’s equation to represent the large sea level variations due to storm waves. By considering the profile variation due to the positive variation of the sea level the following equation can be formulated:  dR dS  Wb =   dt dt  hb + B (t ) 

(10.36)

where S is the increase of the sea level and [B(t)=B-S(t)] is the total instantaneous height of the active profile above the sea level, B is the berm elevation above the still water level, db is the breaking depth and Wb is the amplitude of the breaking area given by: Wb = (d b / A)3 / 2 , where A is the scale parameter. The integration of the last equation gives:

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  db R (t ) = Wb ln    d b + B − S (t ) 

(10.37)

the last equation is similar to the Bruun’s rule. Dean’s (1991) model Dean (1991) derived similar solutions for storm-induced berm retreat based on theoretical pre- and post-storm profile forms given by the equilibrium profile in equation (10.1). Because pre- and post-storm profiles were defined by an analytical form, the equilibrium beach response could be obtained by integrating the areas (volume per unit length) between the initial and final equilibrium profiles and by equating the resulting eroded and deposited areas. Solutions were obtained for both the case of a uniform water level rise and for the case where breaking waves create a distribution of wave setup across the surf zone. One interesting result is the case where water levels are elevated by both a storm surge and by breaking-induced wave setup. For this case, an approximate solution for the steady-state erosion is given as:

R∞ = (S + 0.068H b )

Wb B + db

(10.38)

where S is the sea level rise, B is the berm height, Hb is the breaking wave height, Wb is the amplitude of the breaking area and db is the breaking depth. Dean’s model is very similar to Bruun’s rule. Kriebel and Dean (1993) model A simple analytical model for predicting dynamic profile response during storms is the so-called Convolution Method of Kriebel and Dean (1993). This method is based on the observation that beaches tend to respond toward a new equilibrium exponentially over time. For laboratory conditions, where a beach is suddenly subjected to steady wave action, the time-dependent shoreline response R(t) may be approximated by the form: t  −  R (t ) = R∞ 1 − e Ts  

    

(10.39)

where R∞ is the equilibrium beach response given by equation (10.39) and Ts is the characteristic time-scale of the system. A more general result for the dynamic erosion response may be obtained by noting that the last equation suggests that the rate of profile response is proportional to the difference between the instantaneous profile form and the ultimate equilibrium form.

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221

Figure 10.6 – a) Sand volume balance due to sea level rise and associated profile retreat according to Bruun’s rule (CEM, 2001). An approximate differential equation governing the profile response to time dependent variations in water level may be assumed in the form:

dR(t ) 1 = [ R∞ f (t ) − R(t )] dt Ts

(10.40)

where f(t) represents a unit-amplitude function of time that describes the storm surge hydrograph, while R∞ represents the equilibrium beach response for the peak water level. The general solution to this system may be expressed as: R R (t ) = ∞ Ts

t

∫ f (τ )e

−

t −τ Ts

dτ

(10.41)

0

As a result, several important characteristics of dynamic beach profile response are evident. First, a beach has a certain “memory,” so that the beach response at any one time is dependent on the forcing conditions applied over some preceding time period. As a result, the beach response will lag behind the erosion forcing. In addition, because of the exponential response characteristics of the beach system, the beach response will be damped so that the actual maximum response will be less than the erosion potential of the system (CEM, 2001).

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Figure 10.6 – b) and c) Sand volume balance due to sea level rise and associated profile retreat according to Bruun’s rule (CEM, 2001). A useful application of the convolution method is to analyze the erosion associated with an idealized storm surge hydrograph. Consider the case where the storm surge is approximated by the function:

S (t ) = S sin 2 (σt ) = Sf (t )

(10.42)

with σ= π/TD and where TD is the total storm surge duration. The maximum storm surge level S would be used to determine the maximum potential erosion R∞ :

db m0 R∞ = S S B + db − 2 Wb −

(10.43)

where B is the berm height , m0 is the beach slope, db is the breaking height and Wb is the breaking zone width, calculated as:

d  Wb = y0 +  b   A

3/ 2

(10.44)

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223

A is the equilibrium parameter and y0 is a small offset area between the beach inclination and the imaginary origin of the equilibrium profile, given by: y0 =

4 A3

(10.45)

27 m03

As shown by Kriebel and Dean (1993), solution of the convolution integral in Equation (10.32), with the unit-amplitude forcing term f(t) equal to the sinesquared function, gives the following time-dependent erosion response:   2σt  R(t ) 1  β2 1  − = 1 − exp − [cos(2σt ) + β sin( 2σt )] (10.46) 2 2  1 + β 2 β R∞   1+ β 

where β is the ratio of the erosion time scale to the storm duration, which is given as β=2πTS/TD. The maximum value of Rmax R∞ is the maximum reached by equation (10.43) for given values of β and TD . The value of Ts is given by :

H 3/ 2  d mW  Ts = 320 1/ b2 3 1 + b + 0 b  g A  B db 

−1

(10.47)

In figure 10.7 the relative response R(t)/R∞ for various values of β is illustrated.

Figure 10.7 - Example of profile response to idealized sine-squared storm surge for different values of β. 6.1 Case study Given TD=10 hr, m0=25%, B=1.40 m, m=5%. Calculate R∞ , Rmax / R∞ in two different cases: (a) D50=2 mm, S= 1.50 m, db=5.73 m

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INTRODUCTION TO COASTAL DYNAMICS AND SHORELINE PROTECTION

(b) D50=5 mm, S= 1.50 m, db=5.73 m Solution: (a) According to the formulation in table 10.1: A= 0.23( D50 ) 0.32 =0.23(2)0.32=0.29m-3 The breaking wave height is calculated as (Kamphuis, 2000):

H b = d b 0.56 exp(3.5 ⋅ m) =0.56exp(3.5⋅0.05)⋅5.73=3.82 m y0 =

4 A3 27m 03

d  Wb = y0 +  b   A

3/ 2

=

4 ⋅ 0.29 3 27 ⋅ 0.25 3

= 0.22 m

 5.73  = 0.22 +    0.29 

3/ 2

=89.38 m

H 3/ 2  d mW Ts = 320 1 / b2 3 1 + b + 0 b g A  B hb = 320

3.82 3 / 2

5.73 0.25 ⋅ 89.38   + 1 +  5.73 9.81 / 2 0.29 3  1.40 

−1

  = 

−1

= 3587.40 sec

db 5.73 89.38 − m0 0.25 = 15.62 m = 1.50 ⋅ R∞ = S S 1.50 B + db − 1.40 + 5.73 − 2 2 Wb −

β=2πTS /TD =2⋅3.14⋅( 3587.40/(10⋅3600))=0.63 The ratio Rmax /R∞ can be derived from figure 10.8 (a): Rmax /R∞ ≅0.92 R max =

R max ⋅ R∞ = 0.92 ⋅ 15.62 = 14.42 m R∞

(b) According to the formulation in Table 10.1: A= 0.23( D50 ) 0.32 =0.23(5)0.32=0.38m-3 The breaking wave height, calculated before, is Hb=3.82 m y0 =

4 A3 27m 03

d  Wb = y0 +  b   A

=

3/ 2

4 ⋅ 0.38 3 27 ⋅ 0.25 3

= 0.54 m

 5.73  = 0.54 +    0.38 

3/ 2

=57.97

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225

−1

Ts = 320

= 320

H b3 / 2  d b m0Wb   = 1 + + g 1 / 2 A3  B d b 

3.84 3 / 2

5.73 0.25 ⋅ 57.97   + 1 +  5.73 9.81 / 2 0.38 3  1.40 

−1

= 1756.15 sec

β=2πTS /TD =2⋅3.14⋅(1756.15/(10⋅3600))=0.30

db 5.73 57.97 − m0 0.25 = 8.24 m R∞ = S = 1.50 ⋅ 1.50 S 1.40 + 5.73 − B + db − 2 2 Wb −

The ratio Rmax /R∞ can be derived from figure 10.8 (b): Rmax /R∞ ≅0.98

Rmax =

Rmax ⋅ R∞ = 0.84 ⋅ 23.87 = 8.06 m R∞

7 Numerical beach profile modeling Some models, referred as “closed models”, restrict the profile response to a predetermined shape known as the equilibrium beach profile proposed by Dean (1977). Other models let the profile respond to the input forcing function without a pre-determined shape and are called “open models”. Open models are based on detailed hydrodynamics and incorporate weighing functions, spread functions, residual angles, or soil stability criteria to modify the profile shapes. These models usually provide a more realistic representation of bottom features, such as berm, bars and are often based on numerical techniques to avoid numerical instability around the shoreline and bar. Houston (1996) developed an open model in which the effects of wave breaking, wave nonlinearity, and beach slope on sediment transport were considered. In other models, the sediment transport is not divided into cross-shore and longshore processes, but it is given by current and wave transport terms. A review of open models can be founded in Roelvink and Broker (1993). Numerical models can also be classified on the basis of their capability in one-line models, N-line models, 2D-models and 3D models. The last two types of models require more computational time and are used for short term prediction, while one-line models and N-line models are used for long term prediction Dally (1990) cited the expected capability of the numerical model as follows: •

The ability to generate profiles of both normal and storm types depending on the wave conditions and sediment characteristics;

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INTRODUCTION TO COASTAL DYNAMICS AND SHORELINE PROTECTION

•

R(t)/Rinf

• •

The ability to predict the proper shape of these profiles i.e., (a) the normal should be monotonic and concave upwards, and (b) the bar(s) of the storm profile should have the proper spacing and shape; The ability to accurately predict the rate of profile evolution; The ability to respond to changes in water level due to tides, storm surges, or long term fluctuations; 1 0,9 0,8 0,7 0,6 0,5 0,4 0,3 0,2 0,1 0 0

2

4

6

8

10

Tim e [hour]

(a) 1

R(t)/Rinf

0,8 0,6 0,4 0,2 0 0

2

4

6

8

10

Tim e [hour]

(b) Figure 10.8 – Relative response as a function of the time for the examples described in case study a) and b). The stability of the model is judged by its tendency to approach a stable profile asymptotically, if all the relevant parameters are held constant. An asymptotically stable profile is a profile approaching equilibrium.

BEACH PROFILE MODELING

227

Conservation equations Most beach profile models are based on cross-shore transport equations and the beach profile evolution is usually simulated by solving the mass conservation equation for bed material. Two types of grid spacing are used in numerical models: in the first approach, the cells are finite increments of the distance variable y. The distance is the independent variable, whereas the depth varies with the time. In the second approach, the cells of the computational grid are formed by finite increments of the depth d. In this case, y and d are the dependent variable and the independent variable, respectively (y is a function of the time and varies with the d values). When the grid cells used are the first type, the conservation equation is given by (CEM, 2001):

∂d ∂q y = ∂t ∂y

(10.48)

If y and t are independent variables, the conservation equation is:

∂q y ∂y ( d ) (10.49) =− ∂t ∂d in which d is the depth and y is the associated distance from the coastline. qy is given by the so called “transport relationship”, which formulation differs in open and closed models. In closed models the Kriebel and Dean (1985) transport relationship is used: (10.50) q y = K ' ( D − De ) Where D is the energy dissipation per unit water volume. K’ is a parameter which values can be tuned to calibrate the model by correlating the excess of energy dissipation to the sediment transport rate. In order to take account of the gravitational effects, the transport relationship can be modified adding the beach face slope, ∂d (10.51) q y = K ' ' ( D − De ) + ε ∂y where ε is a parameter added by Larson and Kraus (1988). In open models, the transport relationships depend on the detailed hydrodynamics and attempt to incorporate the actual processes more faithfully than the closed loop variety. Usually both bed load and suspended load transport components are represented based on the hydrodynamic properties averaged over a wave period (CEM, 2001). 7.1 Example of numerical model: SBEACH The numerical model by Larson and Kraus (1990), SBEACH, is conceptually similar to the model of Kriebel and Dean (1985) but contains a more detailed description of breaking wave transformation and sediment transport across the beach profile, especially near the breakpoint. This model approximates the equation for conservation of sand in finite difference form. Vertical changes in

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water depth are determined by horizontal gradients in sediment transport rate. In contrast to the Kriebel and Dean model, this allows simulation of breakpoint bar formation and evolution. The model SBEACH was developed using data obtained from experiments performed in large wave tanks with monochromatic waves of prototype-scale heights and periods. Cross-shore movement of sediment is governed by properties of the velocity field and sediment concentration. Sediment concentration in the surf zone is closely related to the generation of turbulent motion, which depends on wave breaking. Thus, in the model, prediction of the cross-shore velocity field is avoided by expressing the cross-shore sand transport rate directly in terms of wave and profile properties. The direction of transport is predicted from an empirical criterion derived from the large wave tank data and verified with field data, and the magnitude of transport in the main portion of the surf zone is related to the wave energy dissipation per unit water volume. This simplified approach implies a transport rate distribution directed either onshore or offshore along the entire profile at a specific instant in time. Cross shore transport direction In the previous sections are described some criteria for predicting whether a beach will erode or accrete through cross-shore sand transport processes are described. In the SBEACH model, the deepwater wave steepness H0/L0 and the parameter Ho/wfT , called the dimensionless fall speed or fall speed parameter, are used to give the distinction between profiles exhibiting mainly bar and berm formation. In the model, the separation of eroded and accreted (bar and berm) profiles is achieved by:  H H0 = M 0  wfT L0 

   

3

(10.52)

in which M = 0.00070 is an empirically determined coefficient. If the left side of the last equation is less (greater) than the right side, the profile is predicted to erode (accrete). Wave steepness describes the wave asymmetry, whereas wave height and period appearing in the fall speed parameter account for the absolute magnitudes of those quantities. The fall speed accounts for the settling characteristics of sand particles. Cross shore transport distribution Cross-shore sand transport rate distributions are calculated in SBEACH from consecutive profiles in time by integrating the mass conservation equation. By calculating the cross-shore transport rate distribution using initial and final profiles of a run, termed equilibrium distributions, the overall profile response is obtained.

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229

Equilibrium transport rate distributions are classified as three main types: • • •

Erosional (Type E): transport directed offshore along the entire profile. Accretionary (Type A): transport directed onshore along the entire profile. Mixed accretionary and erosional (Type AE): transport directed onshore along the seaward part of the profile and transport directed offshore along the shoreward part of the profile.

Transport regions and transport rates The SBEACH model identify four regions in the nearshore (see figure 10.9) with different wave characteristics, where the used transport equations are different: • • • •

Zone I: From the seaward depth of effective sand transport to the break point (prebreaking zone). Zone II: From the break point to the plunge point (breaker transition zone). Zone III: From the plunge point to the point of wave reformation or to the swash zone (broken wave zone). Zone IV: From the shoreward boundary of the surf zone to the shoreward limit of run-up (swash zone).

The transport rate relationships, based on physical considerations and analysis of the large wave tank data, are summarized in table 10.2:

Zone I

Q = Qb e

− λ 1 ( x − x0 )

Zone II

Q = Qpe

− λ2 ( x − x p )

Zone III

Zone IV

  ε dd   K  D − Deq + K dx     Q= 0 

ε dd   D >  Deq −  K dx   ε dd   D ≤  Deq −  K dx  

 x − xr Q = Qz   x z − xr

  

xb <x

x p < x ≤ xb xz ≤ x ≤ x p

xr < x ≤ x z

Table 10.2 – The SBEACH model transport rate relationships (Larson et al.,1990). The symbols in table 10.2 are explained as follows:

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Q = net cross-shore sand transport rate, m3/m ⋅sec

λ1,2 = spatial decay coefficients in Zones I and II, m-1 x = cross-shore coordinate directed positive offshore, m K = sand transport rate coefficient, m4/N D = sand transport rate coefficient, m4/N Deq = equilibrium wave energy dissipation per unit water volume, N⋅m/m3⋅sec

ε = slope-related sand transport rate coefficient, m2/sec d

= still-water depth, m The subscripts b, p, z , and r stand for quantities evaluated at the break point, plunge point, end of the surf zone, and run-up limit, respectively. Different spatial decay coefficients are used in Zones I and II, denoted by the subscripts 1 and 2, to describe the decrease in sand transport rate with distance. In fully broken waves zones, the cross-shore sediment transport rate is related to wave energy dissipation per unit volume. The transport relationships in the other zones are empirical and based directly on the data from the wave tank experiments. In the swash zone the transport rate decreases linearly from the end of the surf zone to the run-up limit.

Figure 10.9 – Principal zones of cross shore transport in SBEACH model (Larson et al., 1990).

Chapter 11 Shoreline modeling 1

Introduction

Incident wave angle at breaking is the most important element in determining sediment movements, since it determines longshore sediment transport rates. Longshore transport is often described under the assumption that the shoreline is nearly straight with parallel depth contours. This assumption is valid for not too long sections of the shore and for a gradual transition between such sections. When breaking waves approach the shoreline at an angle, the parallel component of wave motion modifies bottom materials in the same direction of the coastline and generates longshore sediment transport (also called littoral transport). The longshore sediment transport is essentially due to two different simultaneous processes: • •

High speed breaking waves generates turbulence which causes bottom sediments to be brought into suspension. Marine currents originating by breaking waves suspended sediments along the shoreline.

During the transport processes two important currents are generated: The first one is localised between the shoreline and the breaking zone. This current generates sediments down, normally to the beach. This current moves sediments accumulations called “beach drifting”. The second one (longshore current) is generated at breaking and moves parallel to the coast. This current determines a cell circulation between nearshore and offshore. When waves approach the coastline at an angle, the incoming water masses are not balanced by an offshore directed current. Littoral transport is influenced by many factors: • •

Some processes such as rifraction and diffraction may contribute to sediment transport because of the lateral expansion of sediment; The magnitude of the littoral transport depends on the wave height, the grain size and the wave incident angle. In fact, the littoral transport increases with the third power of the wave height and

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INTRODUCTION TO COASTAL DYNAMICS AND SHORELINE PROTECTION

•

with the third power of the grain size and is approximately proportional to sin2.5(2α), where α is the wave incident angle; Finally, other factors influencing the littoral transport are the sediment distribution along the beach profile, complex fluid motions over bottom irregularities and sediment-sediment interactions.

2 Longshore transport Waves approaching the shore at an angle will move sediment along the shore in the wave propagation direction. There are two mechanisms (figure 11.1): beach drifting in the swash zone and transport in the breaking zone. Beach drifting is due to the circumstance that the wave action pushes sand up the beach in the wave direction. When the wave retreats, the water and sediment particles are accelerated by gravity and travel down the steepest incline, perpendicular to the beach. Transport in the breaking zone is due to the circumstance that the turbulence in the breaking zone stirs the material into suspension and it is carried by an alongshore current generated by the momentum of the breaking waves. The same turbulence and current also transport sand as bedload along the bottom.

Figure 11.1 - Longshore sediment transport (after Kamphuis, 2000). Sediment transport along the shore is related to the wave-generated momentum or energy gradient. The energy flux or wave power between wave rays is given by:

P = nCE b

(11.1)

and the average wave power per unit length is: P=

nCEb = nCE cos α b / cos α

(11.2)

The longshore component of this power (a misnomer since P is a scalar) is

Pl = ( nCE cos α ) sin α =

1 nCE sin 2α 2

(11.3)

SHORELINE MODELING

in the breaking zone, n → 1 C b → gd b and Eb= 1/8 ρg Hb2

233

11.4)

and Plb =

1 ρg 3 / 2 H b2 d b1 / 2 sin 2α b 16

(11.5)

Using the breaker index γ b = H b d b we have: Plb =

1 ρg 3 / 2 5 / 2 H b sin 2α b 16 γ b1 / 2

(11.6)

for irregular waves, Hsb and γsb are used to define the component of wave power (in S.I. units) as (Kamphuis, 2000):

Plsb =

1 ρg 3 / 2 5 / 2 H sb sin 2α b 1/ 2 16 γ sb

(11.7)

Another representation of the longshore sediment transport rate is an immersed weight transport rate Il related to the volume rate by: I l = ( ρ s − ρ ) g (1 − n)Ql

(11.8)

Il ( ρ s − ρ ) g (1 − n)

(11.9)

or: Ql =

ρs , ρ=mass density of the sediment and of the water g= acceleration due to gravity n= in-place sediment porosity (n ≈ 0.4) The parameter n is a pore-space factor such that (1- n)/Ql is the volume transport of solids alone. One advantage of using Il is that this immersed weight transport rate incorporates effects of the density of the sediment grains. The factor (ρs -ρ) accounts for the buoyancy of the particles in water. The term “potential” sediment transport rate is used, because calculations of the quantity imply that sediment is available in sufficient quantity for transport, and that obstructions (such as groyns, jetties, breakwaters, submarine canyons, etc.) do not slow or stop transport of sediment alongshore (CEM, 2001). The immersed weight transport rate Il has the same unit as Pl (i.e., N/sec or lbf/sec) so the relationship: I l = KPl

(11.10)

is homogeneous, that is, the empirical proportionality coefficient K is dimensionless. The best known expression for bulk sediment transport rate is found in CERC (1984):

I l = 0.39 Plsb

(11.11)

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where Il is the underwater weight of sediment transported. Assuming a dense sand with ρs=1800 kg/m3 and porosity n=0.32, the equation (11.11) can be converted in m3/yr as: Ql = 2.2 ⋅ 106

H sb5 / 2

γ

3 sin 2α b (m /yr)

1/ 2 sb

(11.12)

for a flat beach, (mÆ0), γsb =0.56 and hence: 3 Ql = 2.9 ⋅ 106 H sb5 / 2 sin 2α b (m /yr)

(11.13)

or: Ql = 330H sb5 / 2 sin 2α b

(m3/yr)

(11.14)

where Hsb is in meters and Ql is a function of H and α only. Kamphuis (1991) derives an expression that includes the effects of wave period (or wave steepness), beach slope and grain size: 1.25

H  Ql = 1.3 ⋅ 10− 3  sb  3 L  ρ H sb Top  op 

0.75 b

m

 H sb     D50 

0.25

sin 0.6 2α b

(11.15)

This reduces to:

Ql = 2.27 H sb2 Tp1.5 mb0.75 D −0.25 sin 0.6 2α b

(11.16)

where Ql is in Kg/s underwater. This may be converted to: 3 Ql = 6.4 ⋅ 104 H sb2 Tp1.5 mb0.75 D −0.25 sin 0.6 2α b (m /yr)

(11.17)

The above sediment transport expressions assume that there are infinite amounts of sand along the shoreline. They imply infinitely long beaches with sandy profiles that extend far offshore. At most locations, the assumption of an infinitely long beach with unlimited amounts of sand is not valid. It is necessary to distinguish between potential alongshore sediment transport rate and actual rate (the amount of sand actual moving along the shore). The actual alongshore sediment transport rate is calculated by examining the various inflows, outflows, sources and sinks of sand, such a calculation is known as a sediment budget. Sand sources are the supplies of sand provided by rivers erosion products from dune or bluff erosion and the lowering of the foreshore that accompanies shoreline recession. Common sediment sinks are offshore losses into deeper water, onshore losses when wind blows the sand inland so it can no longer be reached by the waves and man-made losses resulting from construction, dredging and sand mining (Kamphuis, 2000). 2.1.1 Case study Example 1 Calculate the longshore sediment transport for: (a) Hsb=0.1 m, m=0.1, αb=4 (deg), D=0.1 mm T=1 and (b) Hsb=1 m, m=0.02, αb=4.4 (deg), D=0.22. T=8

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235

Solution: According to the CERC formula: (a) Qc = 2.9 ⋅ 10 6 H sb5 / 2 sin 2α b =2.9⋅106⋅0.15/2sin(2⋅4) ≈1.28⋅103 m3/yr (b) Qc = 2.9 ⋅ 10 6 H sb5 / 2 sin 2α b =2.9⋅106⋅15/2sin(2⋅4.4) ≈4.4⋅105 m3/yr According to the Kamphuis formula: (a) Ql = 6.4 ⋅ 104 H sb2 Tp1.5mb0.75 D −0.25 sin 0.6 2α b =64000⋅0.12⋅11.5⋅0.10.75⋅(0.1/1000)0.25

⋅sin(2⋅4)0.6=357.12≈0.36⋅103m3/yr

(b) Ql = 6.4 ⋅ 104 H sb2 Tp1.5mb0.75 D −0.25 sin 0.6 2α b =64000⋅12⋅81.5⋅0.020.75⋅(0.22/1000)0.25

⋅sin(2⋅4.4)0.6≈2.0⋅105 m3/yr The two expressions produce significant differences, so it must be clear that some uncertainties exist and that the results must be carefully analysed.

3 Numerical shoreline modeling Within coastal zone management, prediction of coastal evolution with numerical models has proven to be a powerful technique to assist in the understanding of the processes involved and, in the case of necessary interventions, selection of the most appropriate project design. Models provide a framework for organizing the collection and analysis of data and for evaluating alternative future scenarios of coastal evolution. In situations where engineering activities are involved, models are preferably used in developing problem formulation and solution statements and, importantly, for efficiently evaluating alternative designs and optimizing the selected design (Hanson et al., 1989). Models can be classified according to various criteria: Time-space classifications Recent classifications distinguish between Short Term and Small Area Models, Medium Term and Medium Area Models and Long Term and Large Area Models. The first ones covers prototype duration of hours (or less), and areas of 1 to 100 m2 . The second ones typically cover prototype areas of several Km2 and duration of years. Coastal applications are models of shore sections (littoral cells), harbours, inlets, estuaries, or portions of estuaries, and shore protection with offshore structures. The third category includes models which covers areas greater than 100 km2 and time scales over centuries or even millennia. These models simulate the slow coastal erosion processes and sediment exchanges between the coastal zone, the continental shelf and the backshore. Classification by purpose (Kamphuis, 2000) Kamphuis (1996, 2000) distinguish two types of physical models: design and processes models. The design models simulate actual complex prototype situations in order to provide specific information that can be used directly in design or in retrospective study of failures. The model is as close as possible to a small area

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replica of an actual prototype situation. Models to determine the effects of proposed constructions such as dams and navigation channels on estuarine flows, salinity and sedimentation; models of accretion and erosion near harbour entrances; outfall design models simulating effluent plumes, perhaps in stratified tidal flow and models of breakwater stability are all examples of design models. These models simulate specific prototypes with a defined geometry and boundary conditions. The process models study a physical process; they do not model a specific prototype. Examples are: how does stratification affect sedimentation in an estuary, how are bedform ripples related to vortices, how do vortices move sediments up into the water column, or how do wind waves cause mixing and influence dispersion. The processes to be modeled and the boundary conditions and the scales can be defined to minimize the laboratory and scale effects (Kamphuis, 2000). Models can use an analytical or a numerical approach. Analytical models are closed-form mathematical solutions of a simplified version of the equation for shoreline and profile change, respectively.

Model classification

Short Term and Small Area Time and space scales

Medium Term and Medium Area Long Term and Large Area Design

Purpose Process

Table 11.1 – Models classification by time and space scales and by purpose. As shown in chapter 10, numerical models can also be classified on the basis of their capability in one-line models, N-line models, 2D-models and 3D models. The last two types of models require more computational time and are used for short term prediction, while one-line models and N-line models are used for long term prediction. The shoreline change models developed from bulk transport models are often computed using one-line models. This class of models calculates shoreline position changes that occur over a period of years to decades. The spatial extent of one-line models varies from the single project scale of hundreds of meters to the regional scale of tens of kilometres. One-line models assume that the profile is displaced parallel to itself in the cross-shore direction and its shape always remains the same during accretion and erosion, even if the profile includes some other features, such as bars. Thus, the controlling equations may be solved for one contour line only (usually taken as the shoreline). All contour lines have similar shapes and move landward and seaward together, hence the name one-

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237

line models. The standard assumptions of the shoreline change modelling are summarized in table 11.2. Many shoreline change models have been developed and applied, (e.g. Komar 1973; Le Méhauté and Soldate 1977; Walton, Liu and Hands 1988 and many others). An example of a one-line model is the computer model GENESIS (GENEralised model for Simulating Shoreline change) (Hanson and Kraus, 1989), discussed below. SHORELINE CHANGE MODELLING ASSUMPTIONS 1

The beach profile shape is constant

2

The shoreward and seaward limits of the profile are constant

3

Sand is transported alongshore by the action of breaking waves

4

Detailed structure of the nearshore circulation can be ignored

5

There is a long-term trend in shoreline evolution Table 11.2 - Standard assumptions of the shoreline change modelling.

The one line models solve two simple simultaneous equations: the conservation equation of sediment in a control volume or shoreline reach, and the sediment transport equation. The first one is also called morphology equation; the second one is a simple function of the relevant wave climate and beach characteristics. It is assumed that the constant profile shape moves in the cross-shore direction between the closure depth dC and the berm crest elevation B. The total depth of the profile is given by (B+dc). Sediment transport is uniformly distributed over the active portion of the profile. The conservation of sediment volume is expressed as: ∂y 1  ∂Q  =− − q  ∂t (B + d c )  ∂x 

(11.18)

where Q is the longshore transport rate, q is the line source or sink of sediment along the reach, and t is the time. The last equation states that the longshore variation in the sand transport rate is balanced by changes in the shoreline position. In order to solve the last equation, it is necessary to specify an expression for the longshore sand transport rate. A general expression for this rate is given by (equation described in chapter 9):     ρ g K  H b5 / 2 sin( 2α b ) Q= P = K 1 ( ρ s − ρ ) g (1 − n)    16κ 2 ( ρ − ρ )(1 − n)  s  

(11.19)

where ρ is the water density; ρs is the mass density of sediment grains, αb is the wave breaker angle relative to the shoreline; Hb is the wave height at breaking; g is the acceleration due to gravity; n is the sediment porosity; K is an empirical

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proportionality coefficient; κ is the breaker index, given by Hb/db. db is the water depth and P is the potential longshore sediment transport rate. The use of this equation requires the knowledge of the breaking wave angle relative to the beach. The local wave angle relative to the beach is the difference between the wave angle relative to the model baseline αbg and the shoreline angle relative to the model baselineαsb:

α b = α bg − α sb = α bg −

1  dy  tan   dx 

(11.20)

where x is the distance alongshore; y is the distance offshore. If the angle of the shoreline is small with respect to the x axis and simple relationships describe the waves, analytical solutions for shoreline change may be developed. 3.1

GENESIS

The numerical model GENESIS (GENEralised model for Simulating Shoreline change) (Hanson 1987; Hanson and Kraus 1989; Gravens, Kraus, and Hanson 1991) is an example of a one-line shoreline change model used for spatial scales between 1 and 100 km and for time scales between 1 and 100 months. It calculates shoreline changes produced by spatial and temporal differences in longshore sand transport due to breaking waves. The evolution of the shoreline is based on the one-line theory, which assumes that beach profile shape remains the same and the shoreline variations are expressed by changes of a single point on the profile. The model calculates also shoaling, refraction and diffraction through a wave sub module; sand bypassing and transmission near coastal structures, such as jetties and groins; wave transmission at detached breakwaters. GENESIS is used for the analysis and evaluation of coastal projects with regard to the longterm fate of beach fills, feeder beaches, re-nourishment cycles, and coastal structures designed to enhance the longevity of placed beach fill material. The first GENESIS assumption is (the same as other shoreline change models), is that the beach profile moves landward and seaward while retaining the same shape (see figure 11.2). Thus, any point of the profile is sufficient to specific the horizontal location of the profile with respect to a baseline, and one contour line can be used to describe change in the beach plan shape and volume as the beach erodes and accretes. The second assumption is that sediments are transported parallel to the coastline between two limits: the first (shoreward) limit is located at the top of the active berm, and the second (seaward) limit is the so-called closure depth, out of which significant depth changes do not occur. 3.1.1 Governing equations The model solves two simple equations: the equation for the rate of change of shoreline position and the equation for the longshore sediment transport rate.

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239

Shoreline position The equation for the evolution of shoreline position has the same form as equation (11.18). To solve this equation, the initial shoreline position over the full reach to be modeled, the boundary conditions on each end of the beach, and values for Q, q, B, and dc are required. Longshore sand transport The empirical predictive formula for the longshore sediment transport rate in GENESIS is:

dH sb   Ql = H sb2 C gb  a1 sin 2α b − a2 cos α b  dx  

(11.21)

Figure 11.2 – Definition sketch for shoreline change calculation (after Gravens et al., 1991). The first term in the last equation corresponds to the equation for calculating volume transport rate (see chapter 9), and accounts for longshore sand transport produced by obliquely incident breaking waves. The second term is used to describe the effect of another generating mechanism for longshore sand transport, the longshore gradient in breaking wave height (CEM, 2001). Hsb is the significant wave height at breaking; ρ is the water density; ρs is the mass density of sediment grains. αb is the wave breaker angle relative to the shoreline and Cgb is the wave group speed at the breaker line. a1 and a2 are dimensionless parameters given by:

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INTRODUCTION TO COASTAL DYNAMICS AND SHORELINE PROTECTION

a1 =

K1

(11.22)

5

ρ  16 s − 1(1 − n)(1.416) 2  ρ 

and a2 =

K2 7

ρ  8 s − 1(1 − n) tan( β )(1.416) 2 ρ  

(11.23)

K1 and K2 are empirical coefficients used for calibration, and tan(β) is the average bottom slope from the shoreline to the depth of active longshore sediment transport:  A3 tan (β ) =   D LT0

  

1/ 2

(11.24)

A is the equilibrium parameter expressed as a function of the median grain size D50 (see chapter 10). D LT0 is the maximum depth of longshore transport described in the next section. 3.1.2 Model parameters Depth of longshore transport The sand bypassing calculation in GENESIS requires a depth of longshore transport DLT , which is directly related to the width of the surf zone under the assumption that the profile of distance offshore is a monotonically increasing function of the distance offshore. The depth of longshore transport DLT is calculated as:

DLT =

1.27

γ

(11.25)

H sb

where γ is the breaker index (ratio between the significant wave height (Hsb) and the water depth (db) at breaking). The maximum depth of longshore transport The maximum depth of longshore transport (closure depth) is calculated using the following formulation: D LT0 = (2.28 − 10.9 H 0 )

H0 L0

(11.26)

H0/L0 is the wave steepness in deep water; H0 is the significant wave height in deep water (in meters), L0 is the wavelength in deep water, calculated from the

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241

linear wave theory as L0=gT2/2π (in which g is the acceleration due to gravity and T is the wave period). The value of the maximum depth of longshore transport is used to compute the average beach slope tan(β). The same formulation is used GENESIS to calculate the maximum depth of closure. Average profile shape and slope The shoreline change equation does not require specification of the bottom profile shape since it is assumed that the profile moves parallel to itself. In order to determine the location of breaking waves alongshore and depth at the tips of structures that extend offshore, and to calculate the average nearshore bottom slope used in the longshore transport equation, a profile shape must be specified. The average equilibrium profile shape is empirically obtained according to Bruun (1954) and Dean (1977) formulations (see chapter 10). The average profile slope is also described in chapter 10. Depth of closure The depth of closure, the seaward limit beyond which the profile does not exhibit significant change in depth, is a difficult parameter to quantify. Empirically the location of profile closure cannot be identified with confidence, as small bathymetric change in deeper water is extremely difficult to measure. This situation usually results in a depth of closure located within a wide range of values, requiring judgment to be exercised to specify a single value. Breaking waves Wave transformation from deepwater reference depth or the nearshore reference line is initially done without accounting for diffraction from structures or landmasses located in the model reach. The solution strategy is to obtain a first approximation without including diffraction and then modify the result by accounting for changes to the wave field by each diffraction source. Omitting diffraction, there are three unknowns in the breaking wave calculation: the wave height, wave angle and depth at breaking. To obtain the latter quantities, some equations are needed. The height of breaking waves that have been transformed by refraction and shoaling is calculated as:

H 2 = K r K s H ref

(11.27)

H2 is the breaking wave height at an arbitrary point alongshore; Kr is the refraction coefficient; Ks is the shoaling coefficient and Href is the wave height at the offshore reference depth. The refraction coefficient Kr is a function of the starting angle of the ray and the angle of arrival:  cos(ϑ1 )   K r =   cos(ϑ 2 ) 

1/ 2

(11.28)

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INTRODUCTION TO COASTAL DYNAMICS AND SHORELINE PROTECTION

where ϑ2 is the breaking angle. The shoaling coefficient is expressed as:    

 c g1 Ks =   cg2 

1/ 2

(11.29)

where cg1 and cg2 are the wave group speeds. The group speed is defined as cg=cn, where c is the phase speed, given by the ratio between the wave length (L) and the wave period (T); n = 0.5[1 + (2πd / L) / sinh(2πd / L)] . The wavelength is calculated from the following relationship (dispersion relation):

 2πd  L = L0 tanh    L 

(11.30)

The equation for depth-limited wave breaking is given by:

H b = γd b

(11.31)

where db is the breaking depth and γ is the breaking index:

γ =b−a

H0 L0

(11.32)

where a and b are given by: a = [5.001-e-43tan(β)] and b = 1.12/[1+e-60tan(β)] The wave angle at breaking is calculated by the Snell’s law: sin(ϑb ) sin(ϑ1 ) = Lb L1

(11.33)

where ϑb and Lb are the angle and the wavelength at breaking, while ϑ1 and L1 are the corresponding values on deep waters. The values of Hb, db and ϑb are obtained as functions of the wave period and of the wave height and angle at the reference depth. The breaking wave angle required to calculate the longshore sediment transport is given by:

ϑbs = ϑb − ϑ s

(11.34)

where ϑ s is the angle between the shoreline and the x-axis. If ϑ s =0, the GENESIS model considers that waves move normally to the coast. If there are no structures to produce diffraction, the undiffracted wave characteristics are used as input to the sediment transport relation (eq.11.21). If such obstacles are present, breaking wave heights and directions are recalculated. If waves are affected by structures, such as breakwaters, jetties and groins extending seaward, a circular wave pattern will be generated, and the response of the shoreline will be influenced. Sand typically accumulates in the diffraction shadow of a structure and is carried by waves in the circular pattern. Waves entering the shadow region are transformed and decrease in wave height. In

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243

order to obtain realistic predictions of shoreline, an accurate calculation of wave transformations under combined diffraction and refraction is required. When detached breakwaters are present, wave transmission in the modelling system is described using a value of the transmission coefficient Kt for each detached breakwater. The transmission coefficient, defined as the ratio between the incident waves directly shoreward of the breakwater and the height directly seaward of the breakwater, generally varies between 0 (no transmission) and 1 (complete transmission). Sand bypassing In GENESIS two types of sand movement past a structure are simulated. One type of movement is around the seaward end of the structure, called bypassing, and the other is trough and over the structure, called sand transmission. Bypassing is assumed to take place if the water depth at the tip of the structure DG is less than the depth of active longshore transport DLT . Since the shape of the bottom profile is known, DG is determined from knowledge of the distance between the tip of the structure and the location of the shoreline. To represent sand bypassing, a bypassing factor BYP is introduced and defined as: BYP = 1 −

DG D LT

( DG ≤ D LT )

(11.35)

implying a uniform cross-shore distribution of the longshore sand transport rate. If DG ≥ D LT , BYP=0 Sand transmission A permeability factor PERM is introduced to describe sand transmission over, through, and landward of a shore-connected structure such as a groin. A high (in relation to the mean water level), structurally tight groin that extends far landward so as to prevent landward sand bypassing is assigned PERM=0, whereas a completely “transparent” structure is assigned the value PERM=1. Values of PERM vary between 0 and 1 and must be specified by the user through the judgment of the modeller based upon, for example, the structural characteristics of the groin (jetty, breakwater), its elevation, and the tidal range at the site. 3.1.3 Model implementation

The operations of the GENESIS model include the preparation of up to seven input data files. These input files contain information about boundary conditions, spatial and temporal ranges, model calibration parameters, beach fills configurations, presence of coastal structures, etc. The preparation of input files requires most of time spent on a GENESIS project. In figure 11.3 input-output GENESIS files are shown. The four input files shown in grey rectangles are required for all model simulations, whereas the files ‘NSWAV.EXT’ and ‘DEPTH.EXT’ are used only if a seawall is simulated. Input and output file names consist of five letters with a three-letter extension. The extension “.ext” in

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INTRODUCTION TO COASTAL DYNAMICS AND SHORELINE PROTECTION

figure 11.3 is left for user specifications. All input file extensions must be the same. At the end of each run, three output files are generated with the same input files extension.

Figure 11.3 – The GENESIS input-output files scheme. Input data required by GENESIS include the spatial and temporal ranges of the simulation, structure and beach fill configurations (if any), values of model calibration parameters, and simulated times when output is desired. Initial and measured (if available) shoreline positions as referenced to a baseline established for the simulation are also required. Offshore and nearshore (if available) wave information and associated reference depths are used to calculate longshore sand transport rates. Output files provided by GENESIS include intermediate and final calculated shoreline positions, and net and gross longshore sand transport rates. GENESIS reads the input files and calculates the shoreline changes according the instructions and data contained in the input files. The files containing the input and output data are summarized in table 11.3. The above listed files are generated by some programs included in the GENESIS package: •

• • • •

Genedit is a program designed to facilitate the generation and modification of the primary model configuration input data file. The program performs the control check of input data and generates the file Start.ext; Wtwaves generates offshore waves time series (files Waves.ext); Shorlrot is a generalized program that will automatically perform the coordinate system rotation and origin translation according to user-provided input specifications. Cuintp is used to obtain shoreline positions (Y-coordinates) at uniform spaced distances along the project reach (X-coordinates) using a cubic spline interpolation algorithm; Wtsho accept output generated by CUINTP with additional user specifications to create a shoreline position data file in the

SHORELINE MODELING

• •

3.1.4

245

appropriate format for input to genesis. Generates the files Shorm.ext and Shorl.ext; Genesis generates Setup.ext, Output.ext and Shorc.ext; Gengraf allows the user to produce typically required plots depicting the results of a GENESIS simulation using the standard GENESIS input and output files as input.

Model calibration

Before GENESIS can be used to quantitatively estimate shoreline change or predict the performance of a proposed shore-protection project at a project reach, the capability of the model to provide reliable estimates of shoreline evolution at the site must be demonstrated. The calibration is a process of determining values of tuneable coefficients that allow the model to reproduce changes in shoreline position measured over a certain time interval. The adjusted coefficients are applied in order to verify if the model predictions depend on the simulation interval (verification procedure). When all input data requested by the model are available, GENESIS calibration and verification can be performed by determining the values of coefficients K1 and K2. As an example, if the sediment supply is almost exhausted when compared with the calibration period, the transport coefficient K1 should be adjusted to reach a more appropriate value. During long term simulations, long wave data series are necessary to reproduce long term wave climate. However, available wave data often do not cover the entire calibration and verification interval. Lack of complete physical data sets in the modelling process introduces additional unknowns and reduces the reliability of model forecasts. In such cases, a standard procedure is to use short data sets and to repeat them over time. This solution does not allow to reproduce the effective variability of wave conditions: as an example, long term weather cycles may cause different wave climates during the calibration and verification interval. In this case, the lack of data should be estimated by the modeller and a series of parameters should be modified. The task of modeller is to use his experience in coastal studies and numerical modelling to optimize the calibration. Some GENESIS parameters can be adjusted to modify wave angles and height and obtain good agreement between measured and calculated volumetric change as well as shoreline position over time. Some of these parameters are HCNGF (Wave height change factor), ZCNGF (Wave angle change factor) and ZCNGA (Wave angle change amount). The standard strategy is to change only one parameter at a time in order to identify its effect and its relation with other parameters. This strategy allows to recognize the main parameters controlling known quantities. In the successive phase of calibration procedure, parameters with less influence on model results are determined. Some of the main GENESIS parameters are summarized in table 11.4. To provide an objective measure of goodness of fit, GENESIS calculates a single number called the “Calibration/Verification Error” that is the average absolute difference between calculated and measured shoreline positions at each grid point.

INTRODUCTION TO COASTAL DYNAMICS AND SHORELINE PROTECTION

GENESIS MODEL

246

Input files

Information required

Start.ext

Number and size of grid cells Time step and duration of simulation Grain diameter Berm height Empirical parameters for calibration K1 and K2 Other information on existing coastal structures, artificial nourishment, etc.

Shorl.ext

Initial configuration of coastline

Shorm.ext

Coastline shape measured after a user defined number of years.

Waves.ext

Offshore waves information

Seawl.ext

Indications about cells near walls

Output files

Information required

Setup.ext

Coastline position after each year simulation Simulation errors Calibration parameters Sediment budget in the physiographic unit

Output.ext

Coastline position after each year simulation Sediment budget in each grid cell Wave mean direction Sediment budget in the physiographic unit

Shorc.ext

Coastline position at the end of the simulation

Table 11.3 – GENESIS input and output files. If some beach portions show greater positive or negative bias than others, the criterion used to minimize the difference between calculated and measured shoreline position should be based on mathematical averages. Sometimes, also volumetric change calculations are used in shoreline modelling as a tool in calibration and verification procedures to optimize model parameters.

SHORELINE MODELING

NAME

FUNCTION

247

PRIMARY CONTROL

K1

Primary calibration coefficient

Magnitude of longshore sand transport rate

K2

Secondary calibration coefficient

Distribution of sand within calculation area.

ISMOOTH

Size of offshore smoothing window

Time scale of shoreline equilibrium shape of shore.

HCNGF

Wave height change factor

Breaking wave height and location.

ZCNGA

Wave angle change amount

Amount and direction of sand transport.

ZCNGF

Wave angle change factor

Directional variability of waves.

IX-

Grid cell number of structure tip

Shape and location of shoreline change.

Y-

Distance of structure tip to x-axis

Shape and location of shoreline change.

D-

Depth at structure tip

Wave height and direction at diffracting tip; shape and location of shoreline change.

SLOPE2

Bottom slope near groins

Groin bypassing; shoreline change near groins.

PERM

Groin permeability

Amount of sand passing through groins; shoreline change near groins.

YG

Distance from shoreline outside grid to groin tip

Amount of sand entering the calculation

Transmission coefficient detached breakwater

Amount of wave energy passing through and over detached breakwater; shape of shoreline.

TRANDB

for

response

and

area.

Table 11.4 – Some of the main GENESIS parameters and their functions.

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Chapter 12 Comparison and choice among alternative protection systems 1 Introduction Breakwaters or groins usually serve the purpose of protecting land from erosion and/or enabling safe navigation into harbour and marinas. Offshore breakwaters/ reefs can be permanently submerged partly (at low tide) or permanently exposed. In each case, the depth, size and position of the structure relative to the shoreline determines the coastal protection level provided by the structure. Offshore breakwaters are mainly built to protect the shoreline from wave action, to prevent beach erosion, and to replenish beach sand by interrupting long shore and wave-generated currents. Offshore breakwaters dissipate incident wave energy through wave reflection and diffraction. They act as a countermeasure against beach erosion and provide a sheltered area, which serves as a littoral reservoir for sediments brought in by diffracted waves. In the last decades the number of built offshore breakwaters has worldwide increased in a higher proportion than groin-type structures. This shows a strong trend towards the use of offshore breakwaters over groins as means of beach stabilization and protection. Submerged rubble mound breakwaters are being considered more often in coastal engineering design applications, especially where more natural and environmentally friendly solutions to shoreline protection problems. In this chapter an analysis of advantages and disadvantages of the various protection systems in connection with technical and economical reasons is examined.

2 Insertion of protection systems on the coastline Any breakwater inserted along a coastline constitutes an obstacle to wave propagation and to sediment transport, giving rise to alterations of the shoreline, not only in the proximity of the coastal structure, but in the whole physiographic unit.

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Therefore it is necessary to consider a single breakwater not as isolated, but as an integrated part of a defence system, which concurs to assure the integrity of the coastal area. In fact, the beach is a dynamical system in continuous evolution under the action of wave motion. The fundamental base of a natural process of beach formation and conservation is the continuous movement of sediments. For a beach in equilibrium, the outgoing sediments are immediately replaced with analogous incoming sediments. In other words, the “conservation’’ of a beach requires human intervention when a strong imbalance in the budget exists between incoming and outgoing sediments. Other observations about the human impact is that artificial interventions often show their effects after long time intervals and are in contrast with natural phenomena, so that they often do not constitute convenient solutions for coastal protection, causing imbalance in the movement between contiguous areas. Therefore the choice of the optimal intervention must be achieved through gradual and flexible planning operations in order to reach the best solutions. This study of the coastline before and after the realization of a coastal structure implies the determination of the morphologic evolution and represents an essential condition in determining the effect of the protection system on the shore. This study is performed by widely used numerical models which represent a valid help in coastal planning. Before the realization of human interventions a detailed analysis of the following points is needed: • •

•

The study of offshore waves and their propagation in the coastal area, i.e. next to the breaking point. The study of the cross-shore transport, considering the possible configurations of the beach profile until the equilibrium is reached; for the most important interventions the analysis of the changes of the bottom profile (through auxiliary models like SBEACH or similar) may be needed. The individuation of the shoreline position is important to determine the littoral transport on the basis of the directional flow of incident wave energy and the calculations of the coastline evolution (using, for example, one-line models such as GENESIS or similar).

3 Shoreline protection systems The most frequently used protection systems can be included in the following categories: • • •

Longitudinal protection systems (parallel to the shoreline); Cross-shore protection systems (normal to the shoreline); Artificial nourishment.

The combination of the three different types of protection systems is often used.

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251

A recent classification divides protection systems into passive and active defenses. The first ones assure a simple protection of the coastline by the separation of the causes (waves) and the object of the protection; the second ones also produce a localized beach sediment increment due to the reduction of the causes (waves), often by the breaking process. Protection systems can also be divided in hard and soft defenses on the basis of the deformability to the action of hydraulic factors (sea level variations, wave action etc.). The category of the passive-hard protection systems includes the so-called adherent breakwaters (longitudinal protection systems parallel to the shore), disposed parallel to the shoreline as a reinforcement of a part of the coastal profile. These structures are used in emergency situations, are economic and rapidly executable and give an immediate protection effect. Active-hard protection systems comprise detached breakwaters, and groins (narrow structures, usually normal to the shoreline). Wave action is reduced through a combination of reflection and dissipation of incoming wave energy. Breakwaters can be built as a series of little emerged or completely submerged structures (submerged breakwaters). The depth at the toe (in general -3 ÷ -4 m) coincides with the natural bar depth (if present). These structures have a variable width depending on the energy to be dissipated. The detached breakwaters modify the nearshore wave pattern, which is strongly influenced by diffraction at the heads of the structures, and will cause salients and sometimes tombolos to be formed, thus making the coastline similar to a series of pocket beaches. Once formed, the pockets will cause wave refraction, which helps to stabilize the pocket-shaped coastline. Moreover, the emergent barriers can cause hygienic-environmental problems connected with stagnation of water, in particular during the summer season. Such disadvantage is reduced or resolved in the case of submerged barriers for which the water exchange is assured.

Figure 12.1- Shoreline protection systems.

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Groins are used to reduce the effects of the longshore transport. The effect of a single groin is accretion of beach sediment on the updrift side and erosion on the downdrift side; both effects extend some distance from the structure. They are usually placed at a distance 2÷3 times their own length and, in order to intercept the sediment movement, they can extend from backshore until the seaward limit of the surf zone (long groins). Both parallel and normal breakwaters are usually rubble-mound structures armoured with rock or concrete armour units, but there are also examples of structures made by alternate piles or (submerged) groins made by sand bags, etc. Soft protection systems include the artificial nourishments, constituted by sediments placed on the eroded part of the beach to compensate the lack of natural supply of beach sediment. Sediments are extracted from sea bottom deposits or from the land and are chosen with opportune size. The grain diameter - where possible – should be equal or greater than in situ sediments, in order to assure the maximum width increase with respect to the fill volume. Otherwise, frequent and expensive maintenance interventions are needed. In order to limit the number of periodic maintenance interventions, artificial beaches can be protected by breakwaters, which limit sediment loss, (generally submerged and parallel to the coastline breakwaters) and control the distribution of beach sediment along the coastline.

4 Hard measures 4.1 Detached emerged breakwaters This breakwater system can be adopted to protect the beach when the wave front approach the shore parallel to the coastline. Such a system is constituted by a series of detached continuous elements, positioned at one wavelength distance from each other. Each breakwater reflects and dissipates some of the incoming wave energy, thus reducing wave heights in the lee of the structure and reducing shore erosion. Beach sediment transported along the beach moves into the sheltered area behind the breakwater where it is deposited in the lower wave energy region. Characteristics These structures can have a crest height between +0,5 and +1,5 m s.l.m. and a variable width depending on the energy dissipated. On the basis of the wave action intensity, they can be armoured with rock or concrete armour units. The depth at the toe varies between 3 and 5 m and generally coincides with the natural bar depth (if present). Advantages • Effective protection of the shoreline in relation to the reduction of the wave action in extreme and moderate erosive attack;

COMPARISON AND CHOICE AMONG ALTERNATIVE PROTECTION SYSTEMS

•

253

Beach sediments are transported along the beach, moved into the sheltered area behind the breakwater and deposited in the lower wave energy region.

Disadvantages • Diffraction at the heads of the structures and interruption of the solid longitudinal transport will cause salients and sometimes tombolos, thus making the coastline similar to a series of pocket beaches. • Erosion of the breakwater base (in particular in correspondence of the heads of the structures); • Stagnation of water in the sheltered area behind the breakwater, which causes hygienic-environmental problems, in particular during the summer season; • High environmental impact in relation to the negative effects for the landscape. gap

Tombolo

Currents

Salient

Original shoreline

Figure 12.2 – Tombolo formation in presence of detached breakwaters. Management and maintenance requirements: Low maintenance cost, which is limited to the replacement of some elements of the armour layer and the breakwater base, after extreme wave events. 4.2 Detached submerged breakwaters This protection system is less effective but at the same time limits the environmental impact. In fact, breakwaters act as an obstacle to the direct higher wave action and causes the waves to break and therefore dissipate the greatest part of the energy on the structure; however, the lower waves reach the beach almost undisturbed, so the tombolo formation is prevented and the coastal circulation is more similar to the case of absence of structures. Characteristics Submerged breakwaters, considered as low impact structures, are located along the shoreline and can be built with natural rocks or concrete armour units.

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Advantages: • Effective protection of the shoreline in relation to the reduction of the waves for extreme events and in presence of reduced tidal excursion; • Limited loss of sediments due to the effect on the cross shore transport directed offshore; • Preservation of water exchange in the sheltered zones behind the breakwater, which allows the beach fruition. • Reduced environmental impact in relation to the scarce visibility on the coastal landscape. Disadvantages: • Scarce effects on the wave action in presence of significant tide oscillations; • Collision risk for the coastal navigation and danger for swimmers, because of limited visibility. The position of the breakwater should be signalled.

Figure 12.3 – Example of submerged breakwater. Management and maintenance requirements: Low cost maintenance requirements (they are more stable than the emerged ones), which are limited to the replacement of the armour layer after extreme wave events. 4.3 Emerged or semi-submerged groins This type of protection system is usually adopted on shorelines subject to a definite net longshore transport. The function of a groin (more or less permeable) is the interception of a part or the totality of the longshore transport, therefore determining the accretion of the beach on the updrift side and erosion on the downdrift side. Groin systems are often used to protect the shoreline and at the same time to reduce the erosion in the downdrift side. This type of defense can be associated with artificial nourishment to contain longitudinal transport and to limit the losses of sediments. Emerged groins can have a seaward submerged end, in order to limit the interruption of the longitudinal sediment transport and to avoid localised erosion processes.

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255

Characteristics In figure 12.4 the shoreline changes due to the presence of a groin are depicted. A series of groins divides the shoreline into short sections which orientation varies with the incoming wave direction. The position of the seaward end is determined such that the groin retains the longshore sediment transport in severe wave conditions. This means that the groin must protrude some distance into the zone of littoral transport, the extent of which is largely determined by surf zone width. Groins can be classified as either long or short, depending on how far across the surf zone they extend. Such protection systems can assume various shapes, and can be also built with various sediments: natural or artificial rocks, wood palisades, etc. In the case of multiple groins systems, it is important to establish the distance between two consecutive groins: for impermeable groins, this distance is generally assumed to be 2-3 times the groin length; for permeable groins or groins with submerged portions, this distance is generally assumed to be equal to the groin length. The phenomenon of breaking waves near the groin-landward end (flanking) produces deep scour trenches and compromises groin stability; for this reason groins length, spacing and position are determined such that this inconvenient is avoided. When the transport rate across the groins is greater than the transport rate outside, the erosion-accretion process continues until the sand bypassing arrives at the updrift side. In the long period, extensive damages may be caused on the downdrift side. In order to prevent these damages, artificial nourishment can be combined with the groin construction. Cross-shore sediment transport causes the erosion inside the groin field: strong variations in sea water level (e.g. due to storm surge) remove sediments from the updrift side. If the erosion is severe, the landward end of the groin and the adjacent beach will be damaged. In this situation, artificial nourishment may not be a sufficient countermeasure and other types of structures may be needed (T-shaped groins, for example). accretion

Original shoreline

erosion groin Figure 12.4 - Shoreline changes due to the presence of a groin series. Advantages: • Accretion of the beach on the updrift side if sediments are available; • Good water exchange inside the areas comprised between consecutive groins;

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INTRODUCTION TO COASTAL DYNAMICS AND SHORELINE PROTECTION

Reduced environmental impact in relation to the landscape (if submerged), and conservation of the beach in good conditions for fruition.

Disadvantages: • Reduced protection of the coastal area in extreme wave conditions; • Erosion of the coastline at the downdrift side, which requires adequate countermeasures. Management and maintenance requirements: • Groin length depends on the effective angle between the coast and the breaking wave direction; • In order to minimize the downdrift erosion groins should be not too long; • The groin system should be designed without discontinuity, in order to minimize damages due to the adjacent downdrift areas; Low cost management requirements, such as the replacement of the more external layers and the groin base, after extreme wave events. 4.4 “T”- Shaped emerged or semi-submerged groins These structures are usually applied in most exposed shores in the presence of intense erosive conditions. The combination of the groins and longitudinal breakwaters determines the formation of cells along the shoreline where the sediments tend to be “trapped’’ and therefore both the cross-shore and longshore transport are reduced. Characteristics Emerged groins can have submerged portions (e.g. submerged seaward ends) and can also be continuous or with interruptions. Advantages: • Effective protection of the coast for extreme storm events; • Accretion of the upward side of groins if sediments are available; • Accretion of the beach in the cells; Disadvantages: • Erosion in the downdrift side: it is necessary to estimate with caution the closure of the interventions; • Reduced water exchange inside the cells and risk of water stagnation; • High environmental impact in relation to the visibility on the landscape. Management and maintenance requirements: Low cost maintenance requirements, such as the replacement of some elements of the armour layer and the groin base, after extreme wave events. 4.5 Adherent Breakwaters These breakwaters belong to the category of “passive-hard” systems, disposed parallel to the shore in the emerged part of the beach (often in correspondence of

COMPARISON AND CHOICE AMONG ALTERNATIVE PROTECTION SYSTEMS

257

the dune). The mechanism of coastal protection produces the direct protection of the shore left behind against the storm surge action. Characteristics Breakwaters are built to reduce wave action in the lee of the structure through a combination of reflection and dissipation of incoming wave energy. They are generally rubble-mound structures armoured with rock or concrete armour units. Vertical-front structures are in most cases constructed of either sandfilled concrete caissons or stacked massive concrete blocks placed on a rubble stone basement. They can have massive gravity concrete walls and vertical parapets, eventually with concave walls or terraces for a better dissipation of the wave energy.

Figure 12.5 - Adherent breakwaters. It is important to limit the use of adherent breakwaters to emergency situations, in order to assure the shore protection (for example with stones on the base of the wall) and limit the erosion of the toe of the breakwater. Advantages: • They are usually low-cost structures (if compared with detached breakwaters), which can be built in a short period of time and therefore they allow the immediate protection of the shoreline in emergency situations. Disadvantages: • This type of structure is used in emergency situations and it does not concur to resolve the problems of long term erosion processes; • These breakwaters do not supply any protection to the beach, which indeed is often damaged by the increased reflection at the toe of the structure; • Environmental impacts are not negligible in relation to the negative effects on the coastal landscape.

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Management and maintenance requirements: If the protection of the toe is not adequate, the maintenance cost could become of greater importance in the sense of an eventual reconstruction of the breakwater as a result of damage. 4.6 Seawalls A seawall is a wall built along the coastline, used to protect infrastructures or buildings which can be damaged by the waves. Seawalls range from vertical face structures such as massive gravity concrete walls to sloping structures with typical surfaces being reinforced with concrete slabs, concrete armour units, or stone rubble. Seawalls have only a little influence on the longshore sediment transport, but can cause disturbance in water flows: in fact, seawalls are exposed to direct wave actions, which moves sand offshore and alongshore away from the structure. The reflected waves off the seawall cause scour at the toe of the wall. The scour may excavate the supporting sand under the wall, compromising the structure stability. Water level variations influence the seawall design: high water levels will cause overtopping by waves and erosion at the back of the structure. Water trapped behind the structure may cause drainage problems, resulting in structure erosion and instability. Unfortunately, seawalls are often built as a last resort, without an adequate design project, and the most frequent consequence after a few years is instability and the damage of the structure under strong wave action.

5 Soft measures 5.1 Artificial nourishment Beach nourishment is a soft passive measure used for prevention of shoreline erosion, usually with sediment larger than the natural beach sediment. The fill is placed on the eroded part of the beach to compensate for the lack of natural supply of beach sediment. The beachfill should protect not only the beach where it is placed, but also downdrift stretches by providing an updrift point source of sand. Characteristics The artificial nourishment consists of a sand deposit extracted from the sea bottom or from the land with granulometry selected. In some cases large deposits of sand can be found offshore; otherwise, the sediment may be dredged by adjacent ports (if the environmental quality is acceptable). With the aim of limiting the periodic maintenance, the beac fill can be used in combination with both normal and parallel breakwaters (protected nourishments), generating semi-enclosed cells where the sediments are confined. The artificially placed sediment has a profile different from the stable profile and has a limited length along the shoreline. Because of diffusion processes,

COMPARISON AND CHOICE AMONG ALTERNATIVE PROTECTION SYSTEMS

259

sediments will spread out and tend toward a more gently curved shoreline. At the same time, in the cross-shore direction, the nourishment will tend toward a stable profile shape. The placement of sediment depends on the equipment used. Sediment are placed on the shoreface, in the breaking zone or seaward of the breaker bars. In the shoreface the sand moves seaward, forming an offshore sediment reef that protects the shore and prevents beach erosion. The sediments, placed onshore with a steep slope, are modified by wave action; the smaller grains will be winnowed out of the mass of the sand and may be lost to deep water, until the cross-shore profile of the nourishment becomes similar to the equilibrium one. James (CERC, 1984) noted that if the average grain size of the nourished sediments is smaller than the native one, more sediment needs to be placed than can be expected to stay. If the sand grain size is coarser than the native one, the sand will armour the beach, but, also in this case, a larger volume of fill is needed to reproduce the native gradation. The result is that less uniform or smaller fill sediments need larger nourishment volumes and frequent rehandling of the sand. Dean and Yoo (1993) formulated the relationship between the fill volume needed and the grain size. If the native sediments are used for nourishment, the beach profile is simply shifted seaward. The equilibrium parameter A increases with the diameter D, and the beach slope increases with the sediment grain size: coarser sediments produce a steeper profile that intersects the existing profile; finer sediments produce a flatter profile that does not intersect the existing profile. The grain size of the nourishment sediment is an important parameter: the diameter must be equal or greater than the native sediment and with small assortment to preview losses of the finer sediments in storm conditions. The nourishment is often realized using sediment of different granulometry disposed in two layers: the upper one is constitued of finer sediment (in order to assure the fruition by tourists) and the lower one is constituted of larger size sediments, in order to better resist to wave attack. In figure 12.7 an example of two – layer artificial nourishment is given. Advantages: • Immediate accretion of the shoreline without consequences on the neighbouring areas; • The efficiency of the protection system is reached when the longshore transport is of low entity and the the beach is not subject to significant erosion conditions (otherwise the maintenance cost increase); • Positive effects in terms of the socio-economical activities and tourism fruition; Disadvantages: • Requirement of huge quantities of low cost and high quality sediments (in terms of grain size, chemical and microbiological characteristics); • Environmental impact, although in some cases temporary, in relation to the local increase of the turbidity of coastal waters and to the reduction of the benthic population.

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Management and maintenance requirements: Systematic maintenance interventions are needed, in particular after a storm surge, when a restoration of the eroded beach can be necessary.

W

Figure 12.6 - Intersecting and non intersecting profiles.

Figure 12.7 - Example of protected artificial nourishment. 5.2 Dune Restoration A particular type of soft protection system is realized by building a new dune or by consolidating an existing dune. The mechanism of coastal protection is analogous to the breakwaters and is addressed to assure the direct protection of the coast from the action of the storm surges.

COMPARISON AND CHOICE AMONG ALTERNATIVE PROTECTION SYSTEMS

261

Figure 12.8 - Example of protected nourishment. Characteristics This type of intervention is associated with the increasing attention to the environmental changes due to interactions between coastal structures and littoral transport. The construction of new dunes or the restoration of the existing ones is done by deposition of sediments of adequate characteristic (grain size and mineralogy) and allows the accretion of the high part of the beach. An essential component of dune reconstruction is planting of dune vegetation and placement of netting or snow fencing to help retain wind-blown sand normally trapped by mature dune vegetation. A storm acting on a fan may be a viable source of sediment for dune construction. The anti-erosive action of the psammophylous vegetation produces the advance of dunes towards the shoreline. For this reason, the positioning of vegetation on and behind the dune requires the selection of autochthonous species or plants well adapted to the environmental conditions. Advantages: • Positive effects on socio-economical activities and beach fruition by tourists; • Positive effects for environment and landscape, because of the restored shoreline morphology; • The dune restoration has greater effects when the longshore transport is weak and the beach is not interest by strong erosive conditions (otherwise the maintenance requirements increase). Disadvantages: These coastal structures can be realized only for adequate width beaches and when environmental conditions are preserved; the dune restoration can be

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defined a “dynamical” defense system in continuous change until the equilibrium is reached. Management and maintenance requirements: • In order to control the dune evolution, systematic maintenance is needed (e.g., the restoration of a dune eroded by a storm); • The dune management involves the control and maintenance of vegetation in combination with the constant effort to assure the public fruition and to contrast vandalism and degradation.

6 Schematic indications for the choice The starting point for the choice of the type of intervention is the identification of the degree of shoreline erosion. This identification involves the study of the morphologic processes interesting the area and the analysis of the available data and the prediction of the future developments (preliminary study carried out by data extrapolation and mathematical models). Another aspect of the preliminary study is the individuation of the interests involved in the intervention: the conservation of the atmosphere, the landscape, the socio-economic influence of the operation. The comparison between these interests and the requirement to reduce the rate of beach erosion are important to produce the final decision. The determining factors involved in this comparison can be summed as follows: •

The purpose of the protection system in relation to urgency, efficiency and beach fruition. • The morphology of the coastal area; • The regime of sediment transport; • The sea level variations and meteomarine factors. Table 12.1, originally produced by Kobayashi (1987), can be useful to give a first indication of the degree of suitability for each type of intervention. The most significant factors are the following: •

Urgency: (a) Emergency. The risk for people and property is so high that the intervention must be immediately executed (within 15 days), without previous planning; (b) Urgency. The environmental conditions allow to take time to establish a program and execute surveys needed for correct planning; (c) Timeliness. There is available time to execute all necessary surveys, without risking to arrive too late.

•

Littoral transport patterns: (d) Longshore transport less than the cross-shore sediment transport;

COMPARISON AND CHOICE AMONG ALTERNATIVE PROTECTION SYSTEMS

(e) (f) (g) (h)

263

Littoral drift (net transport) less than the gross transports; Little but well defined littoral drift; Significant littoral transport but with not well defined littoral drift; Significant littoral transport and well defined drift.

•

Tide: (i) Limited tide excursion (j) High tide excursion In the Tyrrenian sea the tide excursion is limited; for this reason Tyrrenian sea is enclosed in class ‘i’. • Morphology of the coastal area: The morphology of the coastal area is connected to the dimensions of the main beach features, the height of the active beach and the erodibility of the coast, so that the following classes have to be distinguished: (k) Insignificant instability (for example cliffs); (l) Modest instability (for example thin beach of great extension and the pocket beach); (m) Important instability (for example river mouth cuspids and small physiographic units with strong transport). •

The degree of suitability is indicated in the following way: ƒ 3 advisable solution; ƒ 2 suitable solution; ƒ 1 acceptable solution; ƒ + suitable or acceptable solution for some coastal characteristics, inefficient for others; ƒ 0 inefficient solution; ƒ * unadvisable solution. Hence, the scheme gives the following general indications: • Groins are usually employed where the littoral drift is well defined, in order to redistribute the longshore transport along the shoreline, in different degrees depending on the wave regime and on the configuration of the shoreline (for example for the stabilization of river mouth systems). They must be well rooted to land and employed with precaution in shorelines with a labile equilibrium. • Detached breakwaters can be used where sea level variations and littoral transport are not significant. • Beach revetments and walls have generally not been used because they modify the stability of the beach (with the exception of emergency conditions). • Parallel protection systems, such as reef breakwaters, have not been used in cases of significant, well defined longshore transport. • Artificial nourishment can be used also in emergency conditions as an alternative to beach revetment where the longshore transport is not significant.

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Table 12.1- Scheme for the choice of the beach protection system. Meteomarine and socioeconomic factors Protection systems Groynes Det. Breakwaters Beach revetment Nourishments

Urgency

Longshore transport

a * * 2 3

d 0 2 1 3

b 1 0 + 3

c 2 2 * 2

e 1 3 0 2

f 3 + 0 2

Tide g 2 2 * 1

h 2 + * 1

i 1 2 1 2

Morphologic instability j 2 1 2 2

k 2 2 2 1

l 1 1 1 2

m * 0 * 3

Where the littoral transport is significant, the nourishment can be used together with submerged breakwaters in order to reduce the maintenance cost. For shore protection, various solutions are available, which will be further discussed. (A) Nourishment (B1) Submerged breakwaters (B2) Seawalls, emerged breakwaters (B3) Offshore structures

(C2) Short emerged groins (C3) Long emerged groins (D1) Cells with semi-submerged T shaped groins (D2) Cells with emerged T shaped groins

7 Mechanisms of protection In order to compare different solutions the following aspects have been considered: • Mechanism of shoreline protection; • Meteomarine factors with greater influence on protection systems; • Environmental impact of protection systems. The identification of the mechanisms of shoreline protection is necessary in order to detect the elements, which influence the efficiency of the intervention and, moreover, allows to consider as alternative solutions those based on the same principle. 7.1 Efficiency The efficiency for each of the considered protection systems is a condition of the elements that characterise the shoreline. In table 12.2 protection systems are classified using signs (+ or -) which indicate the more or less efficiency with the aim of reconstituting an emerged beach. The elements considered are the following:

COMPARISON AND CHOICE AMONG ALTERNATIVE PROTECTION SYSTEMS

265

Entity of the longitudinal solid transport The difficulty to quantify the sediment transport have been considered on two levels: the first level is associated with a longshore transport which determines erosion in a time interval of 1 year; the second one is associated with a longshore transport which determines erosion in a time interval of 5-10 years. Inclination of wave resulting direction • Wave resulting direction is inclinated respect to the normal to the coastline at an angle around 10° (threshold value); • Significant sea level variations with oscillations of 1 m and extreme values of 2 m; Yearly monitoring activity for protection systems and maintenance intervention every 3-5 years. Table 12.1- Protection mechanisms of the shoreline (ICCE 1992). TYPE

PROTECTION MECHANISM A Direct

Wave dissipation by breaking

In direct

Normal to the coastline Beach erosion reduction

B2

B3 B4 C1

x

o

o

Positive contribution

Widening of the beach

B1

o

x

x

o

o

o

x

x

x

Submerged beach

Whole beach

D1 D2 x

o

Emerged beach Parallel to the coastline

C2 C3

x

o

x

x

x

x Primariy Mechanism o Secondary mechanism

7.2 Induced effects In order to estimate the best type to adopt for the shoreline protection it is necessary to consider, beyond the efficiency of the protection, the effects on the structures placed on the back of the shoreline, the effects on the shoreline and the visibility of the breakwaters. They have therefore been considered the effects placed back of the shoreline, the effects on the shoreline and the visibility of the breakwaters.

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Table 12.2- Elements conditioning the efficiency of structures Types

ELEMENTS CONDITIONING STRUCTURE EFFICIENCY

A B1 B2 B3 B4 C1 C2 C3 D1 D2 High

-- -

-

-

-

Longshore transport Low

+

High

-- -

Low

++ ++ +

+ ++ -- + -- -

Wave energy vector incidence

High sea level variation

-

Frequent maintenance

++ ++ -

-

-

+

+ Weakly positively conditioned ++ Strongly positively conditioned

- Weakly negatively conditioned -- Strongly negatively conditioned

Table 12.3- Effects induced from the interventions. EFFECTS DUE TO THE REALIZATION OF EACH SHORELINE PROTECTION On backshore structures

A B1 B2 B3 B4 C1 C2 C3 D1 D2

Direct protection

High x

Indirect protection

High

Accretion On the shoreline

TYPES

x

Low Low

X x x

x

x

x

x x

x

x

x

x

x

X x

x

X x

x

X x

High x Low

Longshore High transport interruption Low High Structure visibility Low

x

x

x

x

x x

x

x

x

x

x x

x

x

Chapter 13 Hydraulic Design 1 Dimensional analysis The physical phenomena occurring when a train of regular wave interact with a rock slope, are breaking, run-up/run-down, reflection and transmission (see fig. 13.1). These phenomena depend upon the following parameters (Bruun, 1985): (a) Physical environment - depth at the toe of the dike d - bottom slope β - specific weight of water γw - acceleration of gravity g - kinematics viscosity ν=µ/ρ - wave height H and period T - angle of wave approach ϑ (b) Structural - slope angle α - characteristic width λ - roughness k - permeability p The last two characteristics depend on: -

the thickness of the armour layer (which can be characterized in terms of l = 3 W / γ r , W being the armour unit weight and γr its specific

-

weight); the way in which the armour units are placed, which can be characterized by the number of units per unit area of slope n.

So the flow characteristic Z on a rough, undefined slope under the action of a regular wave train (associated with the run-up Ru and run-down Rd, reflection and transmission) can be described as a function of the following variables: Z= f(H, T, ϑ, µ, ρ, g, α, l, n) Considering perpendicular angle of incidence, ϑ=0.

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By dimensional analysis we obtain five non dimensional groups of variables:   υT gT 2 l , , n ,α  Π z =ψ  2 ,  H H H  

(13.1)

When the wave height is considerably greater than the equivalent diameter l (implicitly we assume a smooth slope), the flow characteristics are independent of l and of the number n. Besides, assuming large scale, the Reynolds’ number H 2 υT is of minor importance. So we obtain:   gT 2 ,α  Π z =ψ    H  

(13.2)

This function can be reduced to a simpler expression with a single variable only, which is the Irribarren number ξ = tgα

s 0 = tgα

H 0 L0 :

Π z =ψ (ξ )

(13.3)

where α = slope angle s0 = deepwater wave steepness (=H0/L0), with H0 =deepwater wave height and L0 = deepwater wavelength (=gT2/2π) T = wave period g = acceleration due to gravity.

Figure 13.1 – Wave run-up and overtopping. This chapter presents the basic calculation of the main hydraulic response parameters such as run-up and run-down levels, overtopping discharges, wave transmission and wave reflection. The prediction methods are generally suitable to calculate the hydraulic response for only a few simplified cases, because tests

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269

have been conducted for a limited range of wave and structure conditions. It is therefore necessary to estimate the reliability of the formulas for related but nonsimilar structure configurations. Run-up, Ru, and run-down, Rd are defined as the maximum and minimum water-surface elevation measured vertically from the still-water level (SWL) (see fig. 13.1). Wave run-up and run-down on a structure depend on the type of wave breaking, identified by the Irribarren number ξ. When we remove the assumption of a smooth slope, Ru and Rd depend also on the surface roughness and the permeability and porosity of the slope. The maximum flow velocity and so the maximum values of Ru and Rd for a given sea state and slope angle are reached on smooth impermeable slopes.

2 Wave run-up Ru and run-down Rd Run-up and run-down levels are defined in relation to still water level (see fig. 13.1). All run-down levels are assumed positive if below swl, and all run-up levels are also assued as positive if above swl. Much of the field data available on wave run-up and run-down apply to gentle and smooth slopes. Some laboratory measurements have been made on steeper smooth slopes, and on porous armoured slopes. Prediction methods for smooth slopes can be used directly for armoured slopes that are made with concrete or asphalt, and can also be used for rough non-porous slopes with an appropriate reduction factor. When we deal with irregular waves, the Irribarren number ξ defined previously can be related with the significant wave Hs, the mean period Tm or the peak period Tp, respectively named ξm and ξp: ξm =

ta nα s 0m

or

ξp =

ta nα s 0p

(13.4)

where s om =

Hs 2πH s = L0 m gTm2

(13.5)

s op =

2πH s Hs = L0 p gT p2

(13.6)

The behavior of waves on rough porous (rubble mound) slopes is very different from that on non-porous slopes. In fact a rubble mound slope will dissipate significantly more wave energy than the equivalent smooth or nonporous slope in most cases. Run-up levels will therefore generally be reduced. This reduction is influenced by the permeability of the armour, filter and underlayers, and by the steepness and period of the waves. The difference between smooth and rock slopes is illustrated in Fig. 13.2, where 2% relative run-up, Ru2% /Hs is plotted for both smooth and rock slopes.

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The greatest divergence between the performance of the different slope types is seen for 1< ξp<5. For higher ξp values the run-up performance of smooth and porous slopes tends to be very similar. In this case the waves are surging up and down the slope without breaking and the roughness and porosity of the structure are less important. Measurements of wave run-up on smooth slopes have been analyzed by Ahrens (1981), Delft Hydraulics (1989), and by Allsop et al (1985). In each instance the test results are scattered (see fig. 13.2), but simple prediction lines have been fitted to the data. Figure 13.2 shows the data of Ahrens (1981) for slopes between 1:1 and 1:4, of Van Oorschot and d’Angremond (1968) for slopes 1:4 and 1:3 and Allsop et al. (1985) for slopes between 1:1.33 and 1:2. All mentioned data points are for smooth slopes, while the other points in fig. 13.2 are for rock slopes (Delft Hydraulics,1989).

Fig. 13.2 - Comparison of relative 2% run-up for smooth and rubble slopes (Van der Meer, 1992). Analysis of test data from measurements by Van der Meer (1988) has given prediction formulae for rock slopes with an impermeable core, described by a notional permeability factor P = 0.1, and porous mounds of relatively high permeability given by P = 0.4-0.6 (Delft Hydraulics, 1989). The empirically derived formulae for run-up on rock slopes give the run-up as a function of the self-similarity parameter ξm (Van der Meer, 1988): Rux / H s = aξ m for ξm < 1.5

(13.7)

Rux / H s = bξ mc

(13.8)

for ξm > 1.5

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271

The run-up for permeable structures (P > 0.4) is limited to a maximum: Rux / H s = d

(13.9)

where x is the percentage of highest levels. The values of coefficients a, b, c and d have been determined for Ru levels of 0.1%, 1%, 2%, 5%, 10% and are shown in table 13.1. Table 13.1 - Values of coefficients a, b, c and d for different Ru levels (%). level (%) 0.1 1 2 5 10 sign. mean

a 1.12 1.01 0.96 0.86 0.77 0.72 0.47

b 1.34 1.24 1.17 1.05 0.94 0.88 0.60

c 0.55 0.48 0.46 0.44 0.42 0.41 0.34

d 2.58 2.15 1.97 1.68 1.45 1.35 0.82

A general run-up formula based on large scale tests for the 2% run-up was given by (Van der Meer,1988): (13.10) Ru 2% / H s = 1.5γξ op

with a maximum of 3.0γ, in which: γ = total reduction factor for various influences; ξop = surf similarity parameter based on the peak period and the deepwater wave

length. The influence of berm width, roughness, shallow water wave distribution and oblique wave attack on wave run-up can be taken into account with reduction factors γb, γf , γh , γβ respectively. The total reduction factor becomes then:

γ = γ b γf γd γβ (13.11) The influence of slope roughness on run-up is given in table 13.2 with reduction factors γf  for various rough slopes. The influence of depth limited waves on run-up can be described with a coefficient calculated by γH =H2%/1.4Hs. For a Rayleigh distribution of the wave heights γH becomes 1. Table 13.2 - Reduction factors γf for run-up on slopes including roughness. Covering

Reduction factor γf

Smooth, concrete, asphalt Impermeable smooth block revetment Grass 1 layer of rock 2 layers of rock

1.0 1.0 0.90 – 1.0 0.55 – 0.60 0.50 – 0.55

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According to De Waal and Van der Meer (1992) the effect of a berm can be taken into account by a berm influence factor γb defined as:

γb =

ξ eq ξ op

= 1 − rB ( 1 − rdB ) 0.6 ≤ γ b ≤ 1.0 rB = 1 −

where:

d rdB = 0.5 B  Hs

ta nα eq ta nα    

(13.12)

(13.13)

2

0 ≤ rdB ≤ 1

(13.14)

αeq is the equivalent slope angle, α is the average slope angle (see fig. 13.3).

Figure 13.3 - Berm influence on wave run-up. The equivalent slope angle αeq is a straight line between points on the slope 1.5Hs below and above the slope (see fig. 13.3). The reduction factor for oblique waves γβ which takes into account the oblique waves is calculated differently for short-crested and long-crested waves. Short-crested waves:

γβ =1-0.0033β

(13.15)

Long-crested waves:

1.0  γ β = co s 2 ( β − 10°) 0.6 

for 0° ≤ β ≤ 10° for 10° ≤ β ≤ 50°

(13.16)

for β > 50°

3 Overtopping discharge In the design of many sea walls and breakwaters, the controlling hydraulic response is often the wave overtopping discharge, which varies greatly from wave to wave. For most cases it is sufficient to use the mean discharge, Q,

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273

usually expressed as a discharge per meter. A nondimensional discharge parameter can be defined as: Q* =

Q gH s3

s / 2π

(13.17)

The calculation of overtopping discharge for a particular structure geometry, water level and wave condition is based on empirical equations fitted to hydraulic model test results. The data available on overtopping discharge is restricted to a few structural geometries. Laboratory data were obtained by De Waal and Van der Meer (1992). The Technical Advisory Committee for Water Defenses (TAW) in The Netherlands reported limiting values of Q for different design cases which are summarized in table 13.4. This incorporates recommended limiting values of the mean discharge for the stability of crest and rear armour to types of sea walls, and for the safety of vehicles and people. The most simple dimensionless description of overtopping Q / gH 3 is given in fig. 13.4 which gives all available data, both Owen (1980) and Van der Meer (1988) data. The horizontal axis gives the “shortage in run-up height” (Ru2%. – Rc)/Hs. When this parameter is null, the crest height equals to the 2% run-up height. For negative values the crest height is even higher and overtopping will be (very) small. For the maximum value of 1.5 the crest level is 1.5 lower than the 2% run-up height and overtopping will obviously be large. The vertical axis gives the logarithmic of the mean nondimensional overtopping discharge. The formula that best describes the average of the data is given by an exponential function (according to Owen 1980):

µ (Q) = 8.10 −5 gH S3 exp[3.1( Ru 2% − Rc ) / H s ] Table 13.4- Admissible overtopping discharges (TAW).

(13.18)

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INTRODUCTION TO COASTAL DYNAMICS AND SHORELINE PROTECTION

Figure 13.4 - Final results on wave overtopping of slopes. The reliability of Eq. (13.18) can be given by assuming that log( Q ) has a normal distribution with a variation coefficient V=σ/m=0.11. Reliability bands can then be calculated for different values of mean overtopping discharges. The 90% reliability bands for some overtopping discharges are reported in table 13.5. Mean discharge 0.1 l/s per m 0.2 1.0 1/s per m 0.3 10 1/s per m

90% reliability bands 0.02 to 0.5 1/s per m 0.3 to 3.5 1/s per m 4.4 to 23 1/s per m

Table 13.5 - 90% reliability bands for some overtopping discharges.

4 Transmission coefficient Structures such as low crested breakwaters will transmit some wave energy into the area behind the breakwater. The transmission performance of low-crested breakwaters depends upon the structure geometry, mainly the crest freeboard, crest width and water depth, but also on the wave conditions, mainly the wave height and period. Van der Meer (1990) formulated a single prediction method which relates the trasmission coefficient Kt to the relative crest freeboard, Rc /Hs. The data used are plotted in Fig. 13.5. The prediction equations describing the data may be summarized in table 13.6. These equations give a very simplistic description of the data available, but will often be sufficient for a preliminary estimate of trasmission. The upper and lower bounds of the data considered are given by lines 0.15 higher, or lower, than the mean lines given above. This corresponds with the 90% confidence bands (the standard deviation was 0.09).

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275

Figure 13.5 - Wave transmission over and through low-crested structures. In order to take into account the berm width, the equations listed in table 13.7 are considered, and they are also reported in fig. 13.6: Table 13.6 - Prediction equations for Kt . Range of validity -2.00
Equation Kt=0.80 Kt=0.46 – 0.3 Rc/Hs Kt=0.10

5 Reflections Waves will reflect from nearly all coastal or shoreline structures. For structures with non-porous and steep faces, approximately 100% of the wave energy incident upon the structure will reflect. Rubble slopes are often used in harbor and coastal engineering to absorb wave action. Such slopes will generally reflect significantly less wave energy than the equivalent non-porous or smooth slope. Although some of the flow processes are different, it has been found convenient to calculate the reflection performance given by C using an equation of the same form as for nonporous slopes, but with different values of the empirical coefficients to match the alternative construction. Data for random waves is available for smooth and armoured slopes at angles between 1:1.5 and 1:2.5 (smooth) and 1:1.5 and 1:6 (rock). Battjes (1974) gives for smooth impermeable slopes:

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C r = 0.1ξ 2

(13.19)

C r = aξ p 2 /( b + ξ p 2 )

(13.20)

Seelig and Ahrens (1981) give

with: a=1.0; b= 5.5 for smooth slopes a=0.6, b= 6.6 for a conservative estimate of rough permeable slopes Table 13.7 - Wave transmission coefficients for various types of waves.   R H K t =  0.031 s − 0.24  c + b D n50   D n50

maximum Kt=0.75; minimum Kt=0.075 (conventional structure) maximum Kt=0.60; minimum Kt=0.15 (reef type structure) where: 1.84   B  H − 5.42sop + 0.0323 s − 0.0017  + 0.51 D   Dn50  n50  b= Hs  − 2.6sop + 0.05 D + 0.85 n 50 

Conventional

Reef type structure structure Hs = significant wave height of incident wave Dn50 = median of nominal diameter of rocks for design conditions Rc = freeboard, negative for submerged breakwater B = width of crest sop = deepwater wave steepness corresponding to peak period

Figure 13.6 - Wave transmission coefficients Kt given by formulas in tab. 13.7 (CEM,2001).

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277

Eqs. 13.19 and 13.20 are shown in Fig. 13.7 together with the reflection data of Van der Meer (1988) for rock slopes. The two curves for smooth slopes are close. The curve of Seelig and Ahrens for permeable slopes does not give a conservative estimate, but even underestimates the reflection for ξp large values.

Figure 13.7 - Comparison of data rock slopes of Van der Meer (1988) with other formulae.

6 Case study Calculate Ru2% and Kt for the following wave and structural parameters: Hs=5.0 m; Tp=11.0 s; tgα=0.5; sop=2πHs/gTp2=0.0265; Run-up height: ξop=tgα/sop1/2=3.1 being ξop >2 Æ Ru2%/Hs=3.0γ γβ=1 γd=1 γf =0.5 Ru2%=3.0×0.5=1.5ÆRu2%=1.5×5.0=7.5m Mean overtopping discharge: (Ru2%-Rc)/Hs=0.2 Æ Q

gH s3 =0.000955Æ Q =0.0334 m3/s ml

From table 13.6 the prediction equation is: Kt=0.46-0.3 Rc/Hs Æ Kt=0.25 Taking into account the berm width and the rock diameter, if Dn50=1.6 m and B=8 m by using table 13.7 the first term of the trasmission coefficient is given by:

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INTRODUCTION TO COASTAL DYNAMICS AND SHORELINE PROTECTION

  R H  0.031 s − 0.24  c = (0.031 × 3.125 – 0.24)×2.1875= −0.313  D D n50   n50

while the second term of tab. 13.7 is given by: 1.84

b = −5.42sop + 0.0323

 B  Hs  − 0.0017  Dn50  Dn50 

+ 0.51 =

−0.1436+0.1−0.032+0.51= 0.4336 Kt= 0.12 So the formula reported in table 13.7 gives for this type of structure a lower value of Kt .

Chapter 14 Structural Design 1 Introduction The main function of a rubble mound breakwater is to protect a coastal area from wave action. Incident wave energy is dissipated primarily through turbulent runup within and over the armour layer. The bulk of the cross-section comprises a relatively dense rockfill core, which is armoured with one or two layers of rock or one of the numerous types of precast concrete armour units (see fig.14.1). The basic deterministic design procedure for breakwaters is based on a significant wave height associated to a given return period, as calculated in chapter 7, a related period T, and a water level. These sea state parameters enter into most of the design formulae, which will be reported in the following. However, because of the uncertainty on the design sea state, such deterministic design procedure will lead to different safety levels for alternative structures because of differences in their sensitivity to variations in the sea state. Therefore the actual uncertainties related both to the waves and to the structural response should be taken into account by a probabilistic design method (which will be only cited in the last paragraph). The response of the structure under hydraulic loads will be introduced in this chapter and some basic design tools will be given. The first point to be underlined is that each design rule which has been obtained through laboratory tests with prescribed ranges of parameters, has its limitations for this reason. So, for important design projects, it is advised to perform physical model studies. Some general basic rules for the geometric design of the cross-section will be given here. These are the following: • The minimum crest width; • The thickness of armour layers; • The number of units or rocks per surface area; • The bottom elevation of the armour layer. The crest width is often determined by the construction methods used (access on the core by trucks or crane) or by functional requirements (road/crown wall on the top). In case, the width of the crest can be as small as the required minimum width prescribed, for example, by CERC (1984):

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Armour layer

Concrete blocks

Core Under layer

Figure 14.1 - Conventional multilayer rubble-mound breakwater. Bmin = (3-4)Dn50

(14.1) 2

The thickness of layers and the number of units per m are given by ta = tu = tf =n kt Dn50 , where ta , tu , tf = thickness of armour, underlayer or filter; n= number of layers; kt= layer thickness coefficients; The number of units per m2 is given by: Na= n kt (1-nv) Dn502

(14.2)

where nv=volumetric porosity. Values of kt and nv are given in CERC (1984) and CEM (2001). The number of units in a rock layer depends on the grading of the rock. The values of nv and kt that are given in table 14.1. For riprap and even wider graded material the number of stones cannot easily be estimated. In that case the volume of the rock on the structure can be used. kt

nv

Smooth rock, n=2

1.02

0.38

Rough rock, n=2

1.00

0.37

Rough rock, n >3

1.00

0.40

-

0.37

Cubes

1.10

0.47

Tetrapods

1.04

0.50

Dolosse

0.94

0.56

Graded rock

Table 14.1 - Values of kt and nv , CERC (1984). The bottom elevation of the armour layer is generally obtained by hydraulic design considerations. The armour layer should be extended downslope to an

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281

elevation below the minimum SWL of at least one (significant) wave height, if the wave height is not limited by the water depth. Under depth limited conditions the armour layer should be extended to the bottom .

2 Structural stability A number of formulas for the prediction of rock size of armour units subject to wave attack have been proposed in the last half century. Those treated in some detail here are the Hudson formula as used in CERC (1984) and the Van der Meer formula (1988). In order to understand these formulations, a preliminary description of the equilibrium forces acting on a block is needed. A breaking wave interacts with a rock structure in the following way. If the slope of the armour layer is lower than a critical value, the water flows over the structure, some blocks are moved and, when the flow speed decreases, a berm will be formed above the mean sea level. When the slope of the armour layer is greater than a critical value, the blocks are moved when the water flows out. In the first case the hydrodynamic forces associated to these flow movements are directed upward and inside the structure; in the second case the forces are directed downward and outside the structure. These hydrodynamic forces are opposite to the gravity and to the interaction forces between the blocks. On the basis of equilibrium principles, a formulation for the stability of the single blocks is reported in the following. In a first hypothesis, let us consider a wave breaking on a bottom with slope α greater than a critical value (α>αc). The wave is supposed to attack frontally on the block revetment. When the wave comes back, it generates forces F directed from the inside towards the sea.

F R

T α Ws

Figure 14.2 - Hydrodynamic forces acting on a block. In figure 14.2 the hydrodynamic forces acting on a block are: • • •

the weight of the submerged block Ws; the weight component parallel to the revetment T; the friction R.

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In order to equilibrate the forces the following condition must be satisfied: R=T

(14.3)

Let us define: • • • • •

the friction coefficient f ; the block length scale l ; the surface of the block where the wave acts S=k2 l2 ; the wave height H ; γs, γ0 the specific weight of the blocks and of the water, respectively. Ws = k1 l3 (γs - γ0)

(14.4)

T= Ws sinα

(14.5)

F = k k2 l2 γ0 H

(14.6)

R=f(Ws cosα - F)

(14.7)

k, k1, k2, are nondimensional parameters. k1 ⋅l3 (γs - γ0) ⋅(f cosα-sin α)=f kk2 l2γ0 H l=

f ⋅ k ⋅ k2 ⋅ γ 0 ⋅ H

k1 (γ s − γ 0 ) ⋅ ( f ⋅ cos α − sin α )

(14.8) (14.9)

let us define: ∆=

γs γ0

(14.10)

we obtain: l=

f ⋅ k ⋅ k2 ⋅ γ 0 ⋅ H

k1 (∆ − 1) ⋅ ( f ⋅ cos α − sin α )

(14.11)

by elevating to the third power: l3 =

(k ⋅ k 2 )3 k13

f 3 ⋅H3

(∆ − 1) ⋅ ( f ⋅ cos α − sin α )3 3

(14.12)

Ws can be expressed as k1 l3 γs , so : k' =

W s = H 3 ⋅ k '⋅

(k ⋅ k 2 )3 k12 f 3 ⋅γ s

( f ⋅ cos α − sin α )3 ⋅ (∆ − 1)3

(14.13)

(14.14)

the expression in terms of W50 is the Hudson formula, when all the constant values have been put in KD:

STRUCTURAL DESIGN

W50 =

γs ⋅H3 γ  K D ⋅  s − 1 γ0 

3

⋅

1 cot gα

283

(14.15)

2.1 Hudson formulation The Hudson equation is valid when the slope of the armour layer is greater than or equal to a critical value. The main advantages of the Hudson formula are its simplicity, and the wide range of armour units and configurations for which values of KD have been derived (see table 14.2). Block type

n

Natural blocks Rounded 2.0 Not rounded 3.0 Not rounded 1.0 Not rounded Not rounded 2.0

Placement

Structure trunk Structure head KD KD Breaking Non Breaking Non waves Breaking waves Breaking

cot α

Random Random Random Random

Not rounded Not rounded Artificial Parallelepiped Tetrapods and Quadripod

3.0 2.0

Random Special

2.0

Special Random

2.0

Tribar

2.0

Dolos

2.0

Random

Esapods “Toskane” Tribar

2.0 2.0 1.0

Random Random Random Uniform

2.0

4.0

7.0-20.0

8.5-24.0

7.0

8.0

9.0

10.0

14.8

31.8

Random

1.9 1.6 1.3

3.2 2.8 2.3

1.5 2.0 3

5.0 4.5 3.5 8.3 7.8 6.0 8.0 7.0

6.0 5.5 4.0 9.0 8.5 6.5 16.0 14.0

1.5 2.0 3.0 1.5 2.0 3.0 2.0 3.0

Table 14.2 - Recommended KD values for use in Hudson formula.. KD is a stability coefficient taking into account all other variables. KD values suggested for design correspond to a “no damage” condition where up to 5% of the armour units may be displaced. The Hudson formula has many limitations. Briefly they include potential scale effects due to the small scales in which most of the tests were conducted, and the following:

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• • • •

The use of regular waves only; No account taken in the formula of wave period or storm duration; No description of the damage level; The use of non-overtopped and permeable core structures only.

The use of KD does not always best describe the effect of the slope angle. It may therefore be convenient to define a single stability number with KD cotα . Further, it may often be more helpful to work in terms of a linear armour size, such as a typical or nominal diameter. The Hudson formula can be rearranged as: Hs = N s = 3 K D cot α ∆D n50

(14.16)

The last equation shows that the Hudson formula can be written in terms of the structural parameter Hs/∆n50 (stability number). 2.2 Van der Meer Formulation An extensive series of model tests on stability of rock slopes was conducted at Delft Hydraulics (Van der Meer, 1988). The test included structures with a wide range of core/underlayer permeabilities and a wider range of wave conditions. The results of the experiments were examined by Van Der Meer (1988), who derived a series of formulations depending on the type of structure and employed blocks. Two new formulae were derived for plunging and surging waves respectively. The relationship between slope angle θ, wave height H and offshore wave length Lo can be expressed in terms of the surf similarity parameter (ξm) or Irribarren number given by:

ξm

H = tgα  0  L0

   

−

1 2

= tgα ( s m )

−

1 2

(14.17)

where sm is the wave steepness on deep water sm=H0/L0 . The usefulness of ξm has been demonstrated in the widely referenced work of Battjes (1974). He gave a physical interpretation of ξm, showing that for waves that break on the slope, it is approximately proportional to the ratio of α to the local steepness of the breaking wave. Resonance of uprush and downrush with wave period occurs when ξm is within the range 2 to 3 and hence breakers are of the plunging or collapsing types (Bruun 1985). The formulas for plunging breaking waves can be written as:

(

Hs = 6.2 P 0.18 S ∆Dn50

N

)

0.2

ξ m − 0.5

(14.18)

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285

and for surging waves:

(

Hs = 1.0 P 0.13 S ∆Dn50

N

)

0.2

cot α ξ mP

(14.19)

where: P = permeability of structure; S = degree of damage, defined in the following; N = number of waves in a storm. The transition from plunging to surging waves can be calculated using a critical value of ξm .

[

ξ mc = 6.2 P 0.31 tan α

]

1 /( P + 0.5)

(14.20)

Van der Meer (1988) introduced a new formulation which took into account the parameters not yet considered by Hudson, such as the permeability of the structure p; the storm duration (the number of waves N); the degree of damage S; the wave period of the incident wave (s = Ho/Lo). The permeability is 0.10 ≤ P ≤ 0.60 and its value is determined on the basis of the size of the blocks of the armour layer, the core and the thickness of the structure layers. The degree of damage S is defined as the ratio between the surface of the damaged area and the second power of the nominal diameter D50. Design values for the damage level S are shown in table 14.3. The level “start” of damage, S=23, is equal to the definition of “no damage” in the Hudson formula. The storm duration is defined on the basis of the number of waves N. Some experiments were carried out by varying the value of N between 1000 and 5000. The maximum number of waves N which should be used in Eqs. (14.17) to (14.20) is 7500. After this number of waves the structure more or less has reached an equilibrium. The structure slope generally varies between 1.5 and 5. The wave steepness sm varies between 0.005<sm<0.05 and the specific weight between 2.0 and 3.1 t/mc. Table 14.3 - Design values for the damage level S. Slope

Initial damage

Intermediate damage

Failure

1:1.5

2

3–5

8

1:2

2

4–6

8

1:3

2

6–9

12

1:4

3

8 – 12

17

1:6

3

8 – 12

17

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The reliability of the formulae depends on the differences due to random behavior of rock slopes, accuracy of measuring damage and curve fitting of the test results. The reliability of the formulae (14.18) and (14.20) can be expressed by giving the coefficients 6.2 and 1.0 in the equations a formal distribution with a certain standard deviation. The coefficient 6.2 can be described by a standard deviation of 0.8 (variation coefficient 6.5%) and the coefficient 1.0 by a standard deviation of 0.08 (8%). These values are significantly lower than that for the Hudson formula at 18% for K (with mean of 4.5).

Figure 14.3 - Comparison between Hudson and Van der Meer formulas (Van der Meer, 1988)

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287

Equations 14.18 – 14.20 are more complex than the Hudson formula 14.3. They include also the effect of the wave period, the storm duration, the permeability of the structure and a clearly defined damage level. This may cause differences between the Hudson formula and Eqq. 14.18 – 14.20. 2.3

Comparison of Hudson and Van der Meer formulae

The Hs/∆Dn50 in the Hudson formula is only related to the slope angle cotα. Therefore a plot of Hs/∆Dn50 or Ns versus cotα. as shows one curve for the Hudson formula, while formulae (14.18-14.20) take into account the wave period (or steepness), the permeability of the structure and the storm duration. The comparison between the two formulas was given by Van der Meer (1988) in fig.14.3. The upper graph shows the curves for a permeable structure after a storm duration of 1000 waves (a little more than the number used by Hudson). The lower graph gives the stability of an impermeable revetment after wave attack of 5000 waves (equivalent to 5-10 hours in nature). Curves are shown for various wave steepnesses.

3 Armour layers with concrete units The Hudson formula (14.16) is valid also for concrete units, with appropriate KD values. The Shore Protection Manual gives a table with values of KD for a large number of concrete armour units. The most important ones are: KD = 6.5 and 7.5 for cubes, KD = 7.0 and 8.0 for tetrapods and KD = 14.8 and 31.8 for Dolosse. For other units the reader is referred to the CERC (1984). Extended research by Van der Meer (1988) on breakwaters with concrete armour units was based on the governing variables found for rock stability. The research was limited to only one cross-section (slope angle and permeability) for each armour unit. Therefore the slope angle, cotα, and the surf similarity parameter ξm, is not present in the stability formula developed on the results of the research. The same yields for the notional permeability factor, P, which was fixed at P = 0.4. Breakwaters with armour layers of interlocking units are generally built with steep slopes in the order of 1:1.5. Therefore this slope angle was chosen for tests on cubes and tetrapods. A uniform 1:30 foreshore was applied for all tests. Only for the highest wave heights which were generated, some waves broke due to depth limited conditions. Damage to concrete units can be described by the damage number Nod which is the number of displaced units within a strip width (along the longitudinal axis of the breakwater) of one nominal diameter Dn , Nod and Nor . As only one slope angle was investigated, the influence of the wave period should not be given in formulae including ξm; as this parameter includes both wave period (steepness) and slope angle. The influence of wave period, therefore, will be given by the wave steepness s. Final formulae for stability of concrete units include the relative damage level Nod the number of waves N, and the wave steepness, sm.

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The formula for cubes is given by: Hs/∆Dn = (6.7 Nod0.4/N0.3 + 1.0)sm-0.1

(14.21)

For tetrapods: Hs/∆Dn = (3.75 Nod0.5/N0.25 + 0.85)sm-0.2

(14.22)

Eqs. (14.21) and (14.22) are valid for the following conditions: • • •

Non-depth limited wave conditions, irregular head-on waves; Two-layer units on 1:1.5 slope; Surf – similarity parameter range 3<ξm<6 for tetrapods. Hs/∆Dn = sm-0.1 Hs/∆Dn = 0.85 sm-0.2

(Nod =0 cubes)

(14.23)

(Nod =0 tetrapods) (14.24)

The storm duration and wave period showed no influence on the stability of accropode and the ‘no damage’ and ‘failure’ criteria were very close. The stability, therefore, can be described by two simple formulas: Start of damage, Nod =0

Hs/∆Dn = 3.7

(14.25)

Failure, Nod > 0.5:

Hs/∆Dn = 4.1

(14.26)

The standard deviation of the formulas (14.21), (14.22), (14.23) and (14.24) is equal to 0.1, while for eqq. (14.25) and (14. 26) is equal to 0.2. Eqs. 14.17 and 14.18 and 14.21 and 14.25 describe the stability of rock, cubes, tetrapods and accropode. A comparison of stability is made in Fig.14.4 where for all units curves are shown for two damage levels: “start of damage” (S = 2 for rock and Nod = 0 for concrete units) and “failure” (S = 8 for rock, Nod = 2 for Cubes, Nod =1.5 for tetrapods and Nod > 0.5 for accropode). These curves are drawn for N=3000 and are given as Hs/∆Dn versus the wave steepness, s. From fig.14.4 the following conclusions can be drawn: •

Start of damage for rock and cubes is almost the same. This is partly due to a more stringent definition of “no damage” for Cubes (N = 0). The damage level S=2 for rock means that a little displacement is allowed (according to Hudson’s criterion of “no damage”, however).

•

The initial stability of tetrapods is higher than for rock and cubes and the initial stability of accropode is much higher. As start of damage and failure are very close for accropode, a safety coefficient should be esse-I for design (for example a factor 1.5 on the Hs /∆Dn value).

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289

Failure of the slope is reached first for rocks, tetrapods and accropode. The stability failure (in terms of Hs/∆Dn values) is closer for tetrapods and accropode than at the initial damage stage computation.

Fig 14.4 - Comparison of stability number for some concrete units.

4 Low-crested structures As long as structures are high enough to prevent overtopping, the armour on the crest and rear can be (much) smaller than on the front face. The dimensions of the rock in that case will be determined by practical matters as available rock, etc. Most structures, however, are designed to have some or even severe overtopping under design conditions. Other structures are so low that also under normal conditions the structure is overtopped. Structures with the crest level around swl and sometimes far below swl will always have overtopping and transmission. It is obvious that when the crest level of a structure is low, wave energy can pass over the structure. This has two effects. First the armour on the front side can be smaller than on a non-overtopped structure, due to the fact that energy is lost on the front side. The second effect is that the crest and rear should be armoured with rock,which can withstand the attack by overtopping waves. For rock structures the same armour on front face, crest and rear is often applied. The methods to establish the armour size for these structures will be given here. They may not yield for structures with an armour layer of concrete units. For those structures physical model investigations may give an acceptable solution.

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5 Reef breakwaters The analyses of stability by Ahrens (1987) and Van der Meer (1990) was concentrated on the change in crest height due to wave attack. Ahrens defined a number of dimensionless parameters which described the behavior of the structure. The main one is the relative crest height reduction factor h/h’. The crest height reduction factor h/h' is the ratio of the crest height at the completion of a test to the height at the beginning of the test. The natural limiting values of h /h' are 1.0 and 0.0 respectively. Ahrens found for the reef breakwater that a longer wave period gave more displacement of material than a shorter period. Therefore he introduced the spectral (or modified) stability number, N*s, defined by: N*s=Nsp-1/3=Hs/∆Dn50 sp-1/3

(14.27)

The relative crest height, according to Van der Meer (1990) is: hc = At / exp(aN s* )

with

a = −0.028 + 0.04C '+0.034hc' / h − 6 ⋅ 10 −9 Bn2

(14.28)

(14.29)

and hc=hc’ if hc in eq. (14.29) > hc’ Eq. (14.28) was derived by Van der Meer (1990), including all Ahrens (1987) tests. The parameters are given by At = area of structure cross section, h= water depth at structure toe C’=At/hc’ (response slope) Bn = At D

2 n 50

(bulk number)

(14.30) (14.31)

The lowering of the crest height of reef type structures can be calculated with equations (14.28) and (14.29). It is possible to draw design curves from these equations which give the crest height as a function of N*s or even Hs. reliability of Eq. (14.28) can be described by giving 90% confidence bands. The 90% confidence bands are given by hc ± 10%.

6 Statically stable low-crested breakwaters The stability of a low-crested breakwater (overtopped, Rc >0) can be related to the stability of a non-overtopped structure. Stability formulae as (14.18) and (14.19) can be used for example. The required stone diameter for an overtopping breakwater can then be determined by a reduction factor for the mass of the armour, compared to the mass for a non-overtopped structure. The derived equations are based on Van der Meer (1990). The reduction factor for Dn50=1/(1.25-4.8R*p) for 0< R*p<0.052 where: R *p = Rc H s s op / 2π

(14.32)

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The R*p parameter is a combination of relative crest height, Rc/Hs and wave steepness sop. Design curves are shown in Figure 14.5.

Figure 14.5- Design curves for low-crested breakwaters

7 Submerged breakwaters The stability of submerged breakwaters depends on the relative crest height, the damage level and the spectral stability number. The given formulae are based on a re-analysis of the tests by Van der Meer (1990). The stability is described by: * hc' / hc =(2.1+0.1S)exp(-0.14 N s)

(14.33)

S is the damage level, N*s is the spectral stability number given by eq. (14.27).

8 Filter and core characteristics Rubble mound breakwaters are built up like filters consisting of layers of stone. The center core of the breakwater is made up of quarry run rock of the most economically available size. The outside layer consists of large armour units, that can be either rock or specially designed concrete units. This primary armour layer is intended to be statistically stable with respect to the environmental condition imposed on it (the waves and currents do not move the armour stones under design conditions). It is therefore necessary to construct the breakwater as a filter of three or four layers so that the material from any layer is not removed through the layer above it. A typical filter relationship is:

D15 (upper layer ) < 5D85 (lower layer )

(14.34)

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where D is the nominal diameter and D85 means that the nominal diameter of the 85% of the sample is less than D85. When a breakwater is built on erodible material, the toe filter is of particular interest. It is located where the largest stone (the armour) and the breakwater core (often fine material such as tout-venant) are adjacent to each other. To prevent the removal of the material, this toe filter also needs to be built up of several layers, but the layers must be compact so that the total depth of the filter remains small. The toe filter is crucial to the operation of the breakwater. If it fails, the core material will be moved and the lowest armour stones will drop down into the resulting cavity and endanger the stability of the whole armour layer. If the breakwater is located in shallow water under breaking waves, the toe filter will be exposed to extreme wave action. In breaking wave conditions, the toe filter is protected by the armour layer. Core permeability affects wave runup and armour stability, with low permeability causing higher runup and lower stability. Although a dense core reduces wave transmission, a minimum of tout-venant should be used so as to avoid internal erosion. A graded filter prevents removal percentage of finer material from the core but does not prevent the internal redistribution of material which may cause differential settlement. The selection of core material generally uses empirical guidelines based on past experience.

9 Toe stability and protection The function of a toe berm is to support the main armour layer to prevent damage resulting from scour. Armour units displaced from the armour layer may come to rest on the toe berm, thus increasing toe berm stability. The following formula for the stability of the toe berm was obtained by Van der Meer, d’Angremond and Gaurding (1995), and reported in CEM (2001):

Ns =

  h Hs =  0.24 b + 1.6  N od0.15 Dn50 ∆Dn50  

(14.35)

hb is the water depth at top of toe berm, Nod is the number of units displaced out of the armour layer within a strip width of Dn50.

N od

0.5  = 2 4 

No damage Acceptable damage Severe damage

The formula gives the Nod value for assigned values of Hs, hb and Dn50. Nod should not overcome the value of 2, otherwise the damage would be unacceptable.

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10 Breakwater head stability The seaward end of a shore connected breakwater is termed the “head” and is circular in plan. Under similar wave conditions the round head of a rubblemound structure normally sustains more extensive and more frequent damage than the structure trunk. One reason is the reduced support from neighboring units in the direction of wave overflow on the lee side of the structure. One method to increase the stability of the breakwater head is to reduce KD values of Hudson formula. A different approach is adopted by Carver and Heimburg (1989), who obtained the following formula, reported in CEM (2001).

H = Aξ 2 + Bξ + C C ∆Dn 50

(14.36)

where: ξ = tanα/(H/L)1/2 , H is the characteristic wave height; L is the local wave length at structure toe; α is the structure armour slope; A, B, Cc are empirical coefficients given in table 14.4 for rock and dolos. A wave trough on the leeward side coincident with maximum runup on the seaward side may create a head for internal flow, dislodging leeward armour. Vidal et al. (1991) provide guidance on required armour weights at breakwater heads. Jensen (1984) described the hydraulic loads as follows: “When a wave is forced to break over the roundhead it leads to large velocities and wave forces. Armor Type

A

B

Cc

Range of ξ

Stone

0.272

-1.749

4.179

2.1 – 4.1

Stone

0.1983

-1.234

3.289

1.8 – 3.4

Dolos

0.406

-2.800

6.881

2.2 – 4.4

Dolos

0.840

-4.466

8.244

1.7 – 3.2

Table 14.4 – Empirical coefficients for rock and dolos used in equation 14.36. For a specific wave direction only a limited area of the head is highly exposed. It is an area around the still water level where the wave orthogonal is tangent to the surface and on the lee side of this point. It is therefore general procedure in design of heads to increase the weight of the armour to obtain the same stability as for the trunk section. Alternatively, the slope of the roundhead can be made less steep, or a combination of both”.

11 Fundamentals of probabilistic design If we have only two independent parameters involved, say a load parameter (surcharge), S, a resistance parameter R, and the failure probability as: R − S ≤ 0 ,

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then the failure probability can be calculated on the basis of known density functions (see fig 14.7). A relatively simple approach is the First Order Reliability Method, (FORM), which requires preliminary information on the distribution function of all the variables in the formulae plus the uncertainty of the formulae.

Figure 14.7 – Probability of failure for a non-correlated two-parameter system. We take as an example the Hudson armour stability equation:

ρ s Dn 3 = M =

Dn =

ρs H s3

(14.37)

K D ∆3 cot gα

Hs KD

1/ 3

(14.38)

∆(cot gα )1 / 3

Introducing the failure function:  < 0 failure  G = R − S =  = 0 li mit state  > 0 safe 

where R (resistance) and S (load or surcharge) are functions of many stochastic variables x1 , x 2 , … , x n , we obtain: G = K D1 / 3

ADn ∆(cot gα )1 / 3 R ( x1 , x2 , x3 , x4 )

−

Hs S ( x5 )

where: KD = deterministic damage coefficient, A = normal distributed variable with mean µA = 1 (if no bias) and a standard deviation, (σA = 0.18 for Hudson’s formula) signifying the uncertainty of the formulae.

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Dn, ∆ and cotgα are assumed normal distributed with variational coefficients estimated to be in the order of 6%, 4% and 5%, respectively. Hs is given by a long–term extreme value distribution (e.g. a Weibull or Gumbel distribution). FORM analysis makes it possible to calculate the relative influence of the uncertain parameters on the failure probability, and to quantify the benefit by reducing the uncertainties of the design formulae.

12 Deterministic design - case study Calculate the design armour weight for a structure built in natural rocks elements using the following input parameters: N=3000; S=2; p=0.4; Hs=3 m; cotgα=2; γs=2600 kg/m3; γ0=1030 kg/m3; Tm=8 sec; KD =2 (structure trunk); KD=1.6 (structure head). Hudson formula:

γ s 2600(kg / m 3 ) = − 1 = 1.52 γ 0 1030(kg / m 3 )

∆=

W50 =

γs ⋅ H 3

1 =4997 kg =5.0 ton ⋅ K D ⋅ (∆ − 1)3 cot g α

Calculation of ξm and ξmc leads to: sm =

2πH s gT

2

=

2π ⋅ 3 9.81 ⋅ 8

2

= 0.03

[

ξm =

ξ mc = 6.2 p 0.31 tan α

tgα

]

sm

1 /( p + 0.5)

=

0.5 0.03

= 2.89

=3.72

ξm<ξmc so eq. 14.18 has to be adopted Van der Meer formula:

Hs  S  − 0.5 = 6.2 P 0.18  ξ m ∆Dn50  N Ns =

Hs = 1.60 ∆Dn50

Dn50 =

Hs = 1.24 ∆N s

W50 = 2600 ⋅ 1.24 3 = 4957 kg = 5ton

Stability of elements of the structure head is calculated by using the Hudson formula with KD=1.6:

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W50 =

γ s H s3 2600 ⋅ 33 = = 6246 kg =6.25 ton K D cot gα∆3 1.6 ⋅ 2 ⋅ 1.52 3

By using eq. (14.36) ξ has to be calculated with the local depth wave length. We assume d=10 m. A= 0.198, B= -1.234, Cc=3.289 L0=gT2/2π = 99.92 m d/L0=0.10. From the wave table (see chapter 3) we obtain d/L=0.141 so L=70.92 m. H/L=0.042 tanα /(H/L)0.5=2.44=ξm Ns=Hs/∆Dn50 = 1.355 = 0.198×2.442−1.234×2.44 + 3.289 = 1.457

∆Dn50 = Hs/ ∆Dn50 = 1.355 W50 = 2600×1.3553 = 6500 Kg = 6.5 ton So the armour weight at the head is increased approximately by 20% with Hudson (1958) formula, by 30% with Carver and Heimburg (1989) formula.

Chapter 15 Beach fills 1 Introduction Many shorelines of the world are experiencing a trend of beach recession, due to both natural causes and human interaction. The primary cause of natural shoreline retreat is the relative rise in sea level, which is in the order of 15 to 30 cm/century. Human interactions with the shoreline can also cause beach erosion and include: (1) the modifìcation of inlets to improve navigation, (2) the reduction in supply of sand to the shoreline through the construction of river dams, (3) the construction of shoreline structures updrift of a coastal area, and (4) the removal (mining) of sediment from the shoreline. In order to advance the beach seaward, a project of beach nourishment can be performed. Beach nourishment consists in placement of large quantities of good quality sand on the beach to advance it. Usually these projects are initially carried out along the beach, where a moderate erosion trend exists and thus the beach has eroded to a degree that homes and/or infrastructure may be jeopardized by major storms or have suffered damage. Subsequent to the initial nourishment, re-nourishment may be carried out on a somewhat periodic basis. After a project has “seasoned”, which may require one or several years, the additional dry beach width is usually maintained at width of 20 to 30 m. Beach nourishment provides two distinctly different benefìts to a shoreline. The enhancement of the recreational features of the beach is obvious and may contribute significantly to the income generated by tourism. Additionally, a wider beach provides a useful (but generally unquantified) degree of protection to upland structures against large waves occurring as a result of extreme storms. The rationale for beach nourishment can be viewed as follows. The recession trend is due to the above mentioned sea-level rise or to some additional human interaction erosion stresses. The comparatively short-term erosional events followed by a slower recovery are due to steep storm waves and the mild wave conditions which follow. The idealized effects of a well-designed beach nourishment project were shown as the dotted line. The nourishment project displaces the beach seaward by a given distance, in effect setting the system “back in time”, assuming that the nourished beach behaves as the natural beach

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did at the earlier time. It will be shown later that nourished beaches may erode more rapidly than the natural beach due to losses at the ends of the fill segment or to losses of the fine fraction of the fill material. Usually the sediment is placed at slopes steeper than equilibrium, which also creates perturbations in the shoreline alignment. Nature responds by inducing sediment flows that tend to reduce this disequilibrium. The planform disequilibrium occurs in the longshore direction as longshore sediment transport and results in sediment being transported from the project area to the project adjacent areas where it is deposited and widens the beaches there. The profile disequilibrium induces seaward sediment transport that results in a profile of milder slope and a reduced dry beach width (see fig. 15.1). The process of subsequent shoreline change can be considered in three stages (see fig. 15.2): 1. 2. 3.

The profile equilibration, which generally results in a cross-shore transfer of sand from the upper to lower portions of the profile and, thus, a shoreline recession, but not a transfer of sand out of the profile; A transfer of sand along the beach from a “spreading out” of sand resulting from the planform anomaly created by the placed sand; Background shoreline erosion due to ongoing processes before the project was emplaced.

Original shoreline

Sand moves offshore to equilibrate profile

Nourished shoreline

Losses due to spreading out

Figure 15.1 - Plan view showing sand moving offshore to equilibrate profile. These three components of change are operative simultaneously; however, they usually have somewhat different time scales. Profile equilibration typically dominates on time scales in the order of years, whereas, as we will see for the longer projects, longshore diffusional losses general occur in the order of

BEACH FILLS

299

decades. Usually it is assumed that the background erosion losses continue at the same rate as before the project; thus, their effect is the same for each year. Figure 15.2 qualitatively illustrates the shoreline changes for each of the three effects noted and their sum for two background erosion rates.

Figure 15.2 – Qualitative illustration of three components of shoreline recession following a beach nourishment project shown for two background erosion rates and initial nourished width of 75m. (Dean, 2002) Adequate beach nourishment design requires an understanding of these evolutionary processes and the ability to predict these time scales with reasonable confidence. Although guidelines are available for selecting some design parameters, improved procedures are required to enhance the capability to: (1) realistically estimate the design life of the nourishment project, (2) make quantitative trade-offs between alternate borrow source materials, (3) establish optimal allocation of available funding between sand and stabilization structures, and (4) predict the slope adjustment that will occur due to placement of nourishment material on a slope steeper than that of the equilibrium profìle. Sediment transport processes in beach nourishment projects Sediment is lost from the original location of a beach nourishment project by several mechanisms and pathways. An understanding of these mechanisms is important to the design and maintenance of a beach nourishment project. Consider a nourished beach resulting in a seaward extension of the shoreline and an increase in the sand elevation relative to the original contours. Sand can be transported from the nourished area by several mechanisms. If the placed sand contains a significant fraction of fine material, then this material will be placed into suspension and eventually be carried seaward to a location where the wave agitation level is suffìciently low for the sand stability. Many early beach nourishment projects used sediments from inside bays which contained a

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substantial percentage of very fine material and the performance of the projects was understandably poor. As a rule of thumb, if the fall time of suspended material is greater than the wave period, T, then the sediment material will probably be carried seaward beyond the surf zone. Considering the material to be suspended to some fraction β of the depth d this criterion would yield: w<βd/T, in which w is the fall velocity. For a suspension height βd of 10 cm and a wave period of 10 sec, a sediment with a fall velocity of less than 1 cm/sec would be lost from the surf zone. This corresponds to a diameter of 0.1 mm which is usually the approximate lower size limit of material found in the surf zone. A second pathway whereby sand can be transported seaward from the nourished area is simply by a profile adjustment which occurs following the placement of the nourished material on a slope which is steeper than the equilibrium. Although the profile adjustment results in a shoreline recession, the material is still present offshore of the placed zone. Sand is also lost from the ends of the project area.

2 Beach fill profile The model for equilibrium beach profile expected under given wave conditions was developed by Dean (1977), and was developed by Dean (1977), and was reported in § 10.3. The monotonic profile obtained was: d(x) = Ax2/3

(15.1)

in which x is the distance offshore to a water depth d, and A is the equilibrium parameter which depends primarily on sediment size, and m is a shape parameter. In § 10.3, the calculation of the equilibrium parameter A was done both as a function of diameter D (Moore, 1982), and has a function of the sedimentation velocity. It was noted that the equilibrium profile becomes increasingly steeper with increasing grain size. It is evident from these correlations between grain size and equilibrium profile that it is very important in beach nourishment to use materials equally coarse or coarser than the native material. Otherwise the nourished sand will immediately be transported offshore in natures attempt to form the new and flatter equilibrium profile which fits the finer sand. Although this is not a universally valid form, it serves to capture many of the important characteristics of equilibrated beach profiles. The best-fit value of m was found to be 2/3 and this was shown to be compatible with and was interpreted as due to uniform wave energy dissipation by spilling breakers across the surf zone. The rationale for this interpretation is that the turbulence due to wave breaking is a mobilizing (destabilizing) agent and that a sediment of particular characteristics will be stable in the presence of a certain level of mobilizing forces. Moore analyzed a number of published beach profiles comprising sediments over a wide range of sizes and found the variation of A vs. D as presented in Figure 15.3. It is noted that the dimension of the scale parameter. A, is (length)1/3 and that Figure 15.3 presents A in metric dimensions.

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301

Sediment parameter A (m 1/3 )

1.0

0.1

0.01 0.01

0.1

1.0

10.0

100.0

Sediment size D (mm)

Figure 15.3 - Beach profile factor as a function of the sediment parameter A. The closure depth is the depth at which most of the profile disturbance will occur in an average year. Hallermeier (1978, 1981) developed the first rational approach to the determination of closure depth. Hallermeier defined a condition for sediment motion resulting from wave conditions that are relatively rare. Effective significant wave height He and effective wave period Te were based on conditions exceeded only 12 hr per year; i.e., 0.14 percent of the time. The resulting approximate equation for the depth of closure was determined to be:

⎛ H2 ⎞ d * = 2.28H e − 68.5⎜⎜ e2 ⎟⎟ ⎝ gTe ⎠

(15.2)

Beach fill profiles It can be shown that three types of nourished profiles are possible, depending on the volumes added and on whether the nourishment is coarser or finer than the material originally present on the beach. These profiles are termed “intersecting”, “nonintersecting” and submerged, respectively. It can be shown that an intersecting profile requires the added sand to be coarser than the native sand, although this condition does not guarantee intersecting profiles, since the intersection may be at a depth in excess of the depth of closure. An advantage of such a profile is that the nourished profile “toes into” the native profile thereby negating the need for material to extend out of the closure depth. Nonintersecting profile occurs when the sediment is of the same diameter or finer than the native sand. This profile always extends out to the closure depth. A third type is the submerged profile, which requires the nourished material to be finer than the native sediment. It can be shown that if only a small volume of sediment is used than all the material will be mobilized by the breaking waves and moved offshore to form a small portion of the equilibrium profile associated with this grain size. With increasing amounts of fill material, the intersection

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between the nourished and the original profile moves landward until the intersection point is at the water line. For greater volumes of sediment, there will be an increase of the dry beach width, ∆y, resulting in a profile of the second type described.

3 Volume computation The intersecting profile requires that the sediment parameter of the nourished materials AF will be greater than the native AN . In this case, the volume placed per unit shoreline length V associated with a shoreline advancement ∆y was found by Dean (2002):

∆y 3 d * V = + BW* W* 5 B

⎛ ∆y ⎞ ⎜⎜ ⎟⎟ ⎝ W* ⎠

5/3

1 ⎡ ⎛A ⎢1 − ⎜⎜ N ⎢⎣ ⎝ AF

(15.3)

⎞ ⎟⎟ ⎠

3/ 2

⎤ ⎥ ⎥⎦

2/3

where B is the berm height, W* is the reference offshore distance associated with the breaking depth and d* is the closure depth, such that W*=(d*/AN)3/2. The closure depth increases indefinitely with the distance from the coastline. That is unrealistic. A beach profile has a practical seaward limiting depth, where the wave conditions can no longer change the profile. Sediment will still move back and forth, but there is no perceptible change in depth. Hallermeier (1981) discusses this critical depth and (CUR, 1990) approximates it as d*=1.6 HS,12 , where HS,12 is the significant wave height which occurs 12 hrs/yr on average. For non-intersecting profiles, the volume V is calculated as:

∆y 3 h* V = + BW* W* 5 B

3/ 2 5/3 3/ 2 ⎫ ⎧⎡ ⎛ AN ⎞ ⎪ ⎪ ∆y ⎛ AN ⎞ ⎤ ⎜ ⎟ ⎜ ⎟ ⎢ ⎥ + − ⎨ ⎜ ⎟ ⎜A ⎟ ⎬ ⎝ F⎠ ⎪ ⎪⎩⎢⎣W* ⎝ AF ⎠ ⎥⎦ ⎭

(15.4)

It can be shown that the critical value (∆y/W*)c for intersecting/non-intersecting profiles is given by:

⎛A ∆y = 1 − ⎜⎜ N W* ⎝ AF

⎞ ⎟⎟ ⎠

3/ 2

(15.5)

with the intersection occurring if ∆y/W* is positive.

4 Beach planform evolution Beach nourishment projects represent a planform anomaly or “bulge” in the shoreline. In addition, sand is placed at slopes that are steeper than the "natural" or equilibrium profile. The steeper profile and the “bulge” in the waterline relative to the pre-nourished beach system induce sediment transport in both the

BEACH FILLS

303

longshore and cross-shore directions. Under normal wave conditions, transport in the longshore direction causes sand to be moved from the project area to adjacent beaches where sediment deposition occurs. Similarly, transport of sand in the onshore or offshore direction (cross-shore transport) results in an adjustment toward an equilibrium profile. High waves and water levels during storms result in accelerated and modified longshore and cross-shore sediment transport processes. The linearized equations for beach planform evolution were first formulated and applied by Pelnard Consideré, as a result of the transport and continuity equations. W* ∆y

(a)

B

d*

W* ∆y B

(b)

d*

W* ∆y <0

(c) Virtual Origin of the Nourished profile

yI B

d*

Figure 15.4 – Three generic types of nourished profiles (a) Intersecting; (b) Non intersecting ; (c) Submerged (Dean, 1991).

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INTRODUCTION TO COASTAL DYNAMICS AND SHORELINE PROTECTION

The transport equation, obtained using the spilling breaker assumption, and the equation for longshore sediment transport, is given by:

Q=

5/ 2 g / κ sin 2θ b K Hb 8 (1 − p )( s − 1) 2

(15.6)

In which p is the sediment porosity (generally 0.35-0.40), K is a constant of proportionality and s is the sediment specific gravity (2.65 g/cm3). Equation (15.7) will later be linearized by considering the deviation of the azimuth β of an outward normal to the shoreline and the azimuth αb from the breaking wave direction: Q=

5/2 g / κ sin 2(β − α b ) K Hb 8 (1 − p )( s − 1) 2

(15.7)

Where β=µ - π/2 – tan-1(∂y/∂x). The one-dimensional equation of the sediment conservation is: ∂y ∂Q 1 + =0 ∂t (d* + B) ∂x

(15.8)

The equation of longshore sediment transport (15.8) can be differentiated in order to obtain: 5/ 2 g /κ ∂Q K H b ∂β = cos 2(β − α b ) ∂x 8 (1 − p)(s − 1) ∂x

(15.9)

Recalling the definition of β and linearizing, we obtain:

β =µ−

π 2

π ∂y ⎛ ∂y ⎞ − tan −1 ⎜ ⎟ ≈ µ − − ∂ x 2 ∂x ⎝ ⎠

(15.10)

and considering the wave approaching the shoreline at an angle (β-αb) to be small such that cos 2(β-αb)=1, the final result is: 5/ 2 g /κ ∂2 y K Hb ∂Q =− ∂x 8 (1 − p)( s − 1) ∂x 2

(15.11)

Combining equation (15.9) and (15.12), a single equation describing the planform evolution for a shoreline which is initially out of equilibrium is obtained as ∂y ∂y 2 (15.12) =G 2 ∂x ∂t where: G=

H b5 / 2 g / κ K 8 (1 − p )( s − 1)(d * + B )

(15.13)

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305

in which x is the longshore distance, t is the time, G is the longshore diffusivity which depends on the wave height mobilizing the sediment, and eq. (15.13) is known as the “heat condution equation”. The parameter G may be considered as a “shoreline diffusivity” with dimensions of (length)2/time. Field studies have documented the variation of K with the sediment size, D. A detailed evaluation demonstrates that a more appropriate expression for G can be developed and expressed in terms of deep water conditions, where the subscript 0 indicates deep water conditions and C* is the wave celerity in water depth, d* .

5 Longevity of beach fills The longevity of the beach fill depends on the geometry of the project and the nature of fill material. In addition, longevity depends on the wave climate to which the project will be exposed during its lifetime. In order to calculate the longevity, the solution of the differential equation (15.13) is evaluated in a simple case of a narrow strip of sand. A narrow strip of sand extending into the ocean Nourishment on a long straight beach is the simplest possible scheme and has the greatest history of construction and monitoring results. The planform of nourishment on a long straight beach represents a protuberance or “bulge” that the waves tend to straighten out, resulting in sediment flows that cause shoreline advancements on both sides of the project. It is seen that beach nourishment on a long straight beach acts as a “feeder beach” to the adjacent shorelines, and diminishes the need for shoreline stabilization at those locations. Beach nourishment profiles are usually placed at slopes steeper than natural, resulting in profile adjustment that causes a narrowing of the dry beach width that may require several years to occur. Let us consider the case of a narrow strip of sand of width dx at a distance Y from the shoreline, such that the total area of the sand is m=Ydx . The solution of the differential equation (15.13) is given by:

⎛ x ⎞ (15.14) exp⎜ − ⎟ 4πGt ⎝ 4Gt ⎠ The equation (15.15) represents a normal distribution with increasing standard deviation as a function of the time. G is a parameter proportional to the wave height to the 5/2 power which provides some insight into the significance of wave height in remodelling the beach planforms which are initially out of the equilibrium. During its evolution, the planform remains symmetric and centered around the point of initial shoreline perturbation, even if waves approach the coastline obliquely. We expect that the beach should accrete on the updrift side and erode the downdrift side of the perturbation. The value of the constant G is given by: y( x ,t ) =

G=

m

2

H b5 / 2 g/κ K co s 2( β − α b ) 8 (1 − p )( s − 1)(d * + B )

(15.15)

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INTRODUCTION TO COASTAL DYNAMICS AND SHORELINE PROTECTION

If the difference between the wave direction and the shoreline orientation is greater than 45°, G will be negative, which means that the perturbation will grow until the final planform will be a very narrow distribution, exactly as was the initial planform. In this case, the factor cos(2(β-αb)) is less than 0 and the shoreline is considered unstable. Rectangular distribution of sand Consider the initial planform with a longshore length l at a distance Y from the shoreline. The planform can be schematized as the summation of a series of narrow strips similar to the previous example. The analytic solution for the initial planform is given by: y( x ,t ) =

⎡ l ⎛ 2 x ⎞⎤ ⎫ Y ⎧ ⎡ l ⎛ 2 x ⎞⎤ − 1⎟ ⎥ ⎬ + 1⎟⎥ − erf ⎢ ⎜ ⎜ ⎨erf ⎢ 2 ⎩ ⎣ 4 Gt ⎝ l ⎠⎦ ⎭ ⎠⎦ ⎣ 4 Gt ⎝ l

(15.16)

with: erf ( z ) =

2

π

z

∫e

−u 2

(15.17)

du

0

and u is a dummy variable of integration. The two ends of the planform will spread out and move toward the center, so that the planform distribution will become like a normal distribution. Beach longevity calculation It is possible to obtain (Dean, 2002) an analytical expression for the proportion of sand M(T) remaining at the placement site, where: M(t ) =

1 ∆y 0 l

l 2

∫ y( x , t )dy

(15.19)

−l 2

or M(t ) =

4Gt l π

( e −( l

4Gt )2

− 1 ) + erf ( l

4Gt )

(15.20)

which is plotted in figure 15.5 along with the asymptote: M( t ) ≅ 1−

2

π

Gt l

(15.21)

which appears to fit reasonably well for Gt0.5 / l <0.5. A useful approximation for estimating the half life time of a project t50 (in years) is: t 50 = 0.21

another approximation for t50 is given by:

l2 G

(15.22)

BEACH FILLS

t 50 = 0.172

l2 H b5 / 2

307

(15.23)

where t50 is in years, Hb is in meters and l is in km.

Figure 15.5 – Proportion of material remaining in fill area versus dimensionless time (Dean, 2002).

6 Effects of fill length and of wave climate An unstabilized beach nourishment project will experience losses from the ends of the project that are due to alongshore sediment transport to the neighboring beaches. In effect, the nourished area acts as a “feeder beach” to the adjacent shoreline segments. It is possible to evaluate the effect of both representative wave height, H, and beach fill length l, on the time tp, for p percent of the original volume to be transported out of the nourished region. Pelnard-Consideré showed that the equation (15.13) could be used to approximate the planform response of a beach system. From the above equation (15.17), it can be seen that the time required, tp, for a percentage, p, of the original volume to be lost is:

(t p ) 2 = (t p )1 =

l 22 G1 l12 G2

(15.24)

in which the subscripts (1) and (2) refer to conditions for two particular projects. The effect of fill length on its life is obtained as a square law relationship from Equation (15.24). Thus, for example, if a beach fill of a particular length looses one half of the volume from the nourished area in 10 years, a fill of twice the length and the same width under the same wave action would lose one half the volume from the filled region in 40 years. The explanation for the square law relationship is as follows. Examination of the derivation of Equation (15.13) will demonstrate that the net transport rate of sand from the nourished region is due to the slope (in planview), ∂y/∂x, of the planform. Doubling the fill length reduces the slope by a factor of one half. Secondly, the double-length fill simply has twice the volume which accounts for the other factor of one half. The

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INTRODUCTION TO COASTAL DYNAMICS AND SHORELINE PROTECTION

relationship of effective wave height to time required for a percentage, p, of the fill volume to be transported from the nourished area is:

(t p ) 2 (t p )1

=

H 15 / 2 H 25 / 2

(15.25)

Thus, the effect of doubling the wave height results in a longevity which is 1/5 of the previous one. Effects of Structures Terminal structures and groins — that is structures placed near the ends of a beach nourishment project — can be effective in reducing losses of nourished sediment. The simplest case which will be examined here, is for an initially filled groin compartment of length, l, and groin length, w, subjected to the action of unidirectional waves arriving at an angle to the shoreline. It can be shown that, neglecting wave diffraction, the solution of Equation (15.13) with the appropriate boundary conditions, is: 2 tan α ⎛ x⎞ y ( x, t ) = w − l ⎜1 − ⎟ tan α + l ⎝ l⎠ 2

∞

∑

n =0

⎧⎪ ⎡ ⎤ (2n + 1) π t ⎫⎪ 2l ⎡ (2n + 1)πx ⎤ ⎬ cos ⎢ ⎥ exp⎨− G ⎢ ⎥ 2 + n 2l π ( 2 1 ) ⎪ ⎪ l 4 ⎣ ⎦ ⎣ ⎦ ⎩ ⎭ 2

2

(15.26)

Examination of Equation (15.26) shows that the shoreline in the vicinity of the structures approaches a straight line parallel to the incoming wave crests with the downdrift end of the shoreline in the groin compartment coincident with the downdrift groin. Ultimately, the approximate percentage loss of sand from the groin compartment is founded to be: %loss =

1 l tan α 2w

(15.27)

where the above does not account for losses due to onshore/offshore transport. From Equation (15.27) it is evident that the expected percentage losses would be reduced substantially if a groin field were installed. A practical matter to be considered near the ends of a stabilized fill is the possible perturbation caused to adjacent shoreline segments. In order to alleviate such effect, it is desirable to either taper a stabilized fill including, of course, the length of the stabilizing structures and/or to place sand outside the limits of the nourished region. The perched beach The basic concept of the perched beach is to reproduce the existing profile to some convenient seward point and to intersect this profile with a submerged toe structure to retain the beach in a perched position, see Figure 15.6.

BEACH FILLS

309

x1 x2 Perched Beach B d2

d1

Toe Structure

Figure 15.6 - Definition sketch for perched beach. d2 is the depth to which the nourished profile would be configured through the use of a perched beach, d1 is the depth at the toe of the structure, B is the berm elevation and ∆x=x1-x2 is the desired seaward advancement. The sand volume reduction can be evidenced by the following case study. 6.1 Case study

Calculate the volume of sand required for given: DN= 0.3 mm, AN=0.11 m-3, DF= 0.2 mm, AF=0.09 m-3, B=2 m, d1=4 m, ∆x=30 m The beach profile is described by equation (15.2). The distance from the coastline corresponding to the closure depth is: W*= x* = (d*/A)3/2=(6/0.09)3/2 =544 m The volume of sand required to advance the shoreline seaward by 30 m without the structure present is:

V1 = B∆x +

[

x* + ∆x

∫ 0

x*

AN x 2 / 3 dx − ∫ AF x 2 / 3 dx = 0

]

3 5/3 B∆x + AN ( x* + ∆x) 5 / 3 − AF x* = 719m 3 / m 5 in presence of the structure: x1 = (d1 / AN )3 / 2 = (4 / 0.11)3 / 2 =219 m

the depth h2 is given by:

d 2 = AF ( x1 − ∆x) 2 / 3 = 0.09(219 − 30)2 / 3 = 2.96 m the height of the structure will be: d1-d2=4-2.96 =1.04 ≈1 The volume required in presence of the structure is:

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INTRODUCTION TO COASTAL DYNAMICS AND SHORELINE PROTECTION x1

x2

0

0

V2 = B∆x + ∫ AN x 2 / 3 dx − ∫ AF x 2 / 3 dx = B∆x +

[

]

3 5/3 5/3 AN x1 − AF x 2 = 249m 3 / m 5

this value is 35% of that required without the structure. If the fill length L is 1000 m, the required volume Vr is: without structure: Vr=V1×L=719×1000=719000 m3 for a perched beach: Vr=V2×L=249×1000=249000 m3 If the cost of sand is 15 € / m3 Cost of beach fill for a perched beach: 3.735.000 € Cost of beach fill without structure: 10.785.000 € The significant cost reduction justifies the cost of the structure for the beach fill protection

7 Compatibility of the borrow material Generally there will be a fraction of the borrow area material that will be significantly lower in size than the native and will be within the silt and clay size range, that is, less than 0.0625 mm. It is experienced that when the silt and clay size component of the beach nourishment is exposed to wave action, it will be ineffective in terms of contributing to the volume in the project area or to the adjacent beach volumes. Additionally, the silt and clay fraction of the material placed is generally considered environmentally undesiderable since it may adversely affect the biota in the area. Although there is no accepted proportion of fine materials that may be included within a borrow area, generally more than 5% to 10% for waters that are usually clear is considered to be unacceptable. In areas where the waters are normally more turbid, up to 15% to 20 % could be acceptable. The portion of borrow material finer than the native sediment grain size distribution is assumed to be lost offshore. James (1975) developed this concept into a method to calculate an overfill factor, RA. Conceptually, the overfill RA factor is the volume of borrow material required to produce a stable unit of usable fill material with the same grain size characteristics as the native beach sand. The overfill factor attempts to consider the distribution of grain sizes. Therefore, it does provide an additional piece of information on the amount of borrow material that might be needed to construct a beach nourishment project in more difficult design cases where the grain size characteristics of the borrow material differ significantly from those of the native beach material, especially when the borrow sediments are finer than the native sediments. The overfill factor, RA , is determined by comparing mean sediment diameter and sorting values of the native beach and nourished materials (in φ units). The phi, scale of sediment was described in chapter 2. The overfill factor is computed using the following relationships between the borrow nourished and native sediment:

BEACH FILLS

σ φb σ φn

⎡ (φ84 ⎢ ⎣ = ⎡ (φ 84 ⎢ ⎣

− φ16 ) ⎤ ⎥ 2 ⎦b − φ16 ) ⎤ ⎥ 2 ⎦n

311

(15.28)

where σφb= standard deviation or measure of sorting for nourished material σφn= standard deviation or measure of sorting for native material The values of the overfill factor RA can be obtained by interpolating between the values represented by the isolines in figure 15.7. An example of calculation of RA in the next section is given. James (1975) provide an approach to the planning and design of nourishment projects which relate to the long-term maintenance of a project, that is how often re-nourishment will be required if a particular borrow source is selected that is texturally different from the native beach sand. With this approach, different sediment sizes will have different residence times within the dynamic beach system. Coarse particles will generally pass more slowly through the system than finer sizes. This approach also requires accurate estimates of native and borrow sediment textures.

Figure 15.7 - Isolines of the adjusted overfill ratio (RA) for values of ϕ mean difference and ϕ sorting ratio (CERC, 1984). In order to determine periodic re-nourishment requirements, James (1975) defines a re-nourishment factor, RJ, which is the ratio of the rate at which borrow material will erode to the rate at which natural beach material is eroding. The renourishment factor is given as: ⎡ ⎛ M − M φn ⎞ ∆2 ⎟⎟ − RJ = exp ⎢∆⎜⎜ φb σn ⎢⎣ ⎝ ⎠ 2

⎛ σ φ2b ⎞⎤ ⎜ 2 − 1⎟⎥ ⎜σ ⎟ ⎝ φn ⎠⎥⎦

(15.29)

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INTRODUCTION TO COASTAL DYNAMICS AND SHORELINE PROTECTION

where ∆ is a winnowing function. The ∆ parameter is dimensionless and represents the scaled difference between the phi means of non-eroding and actively eroding native beach sediments. James (1975) estimates values of ∆ ranging between 0.5 and 1.5 for a few cases where appropriate textural data were available and recommends ∆=1 for the common situation where the textural properties of non-eroding native sediments are unknown. Equation (15.29) is plotted in Figure 15.8 for ∆=1. The re-nourishment and fill factors are not mathematically related to one another. Each relationship results from unique models of predicted beach fill behavior which are computationally dissimilar although both use the comparison of native and borrow sand texture as input. Nevertheless, the models address the different problems in determining nourishment requirements when fill that is dissimilar to native sediments is to be used (fill factor) and in predicting how quickly a particular fill will erode (for design purposes, the fill factor, RA, or its equivalent, should be applied to adjust both initial and re-nourishment volumes (see table 15.1). Both coefficients are simple descriptions of complex beach relationships, and there will be cases where the RA and RJ values calculated for a particular borrow material suggest quite different responses from that material. One example is where the RJ coefficient suggests that the borrow sand will erode much slower than the native beach sediments. This situation could arise with coarser and poorly sorted borrow sand where early winnowing would remove the average volume of unstable finer sizes and leave a sediment coarser than the native that erodes more slowly. For cases like this and in all cases where these coefficients are applied, engineering judgment and experience must accompany design application. Table 15.1 – Relationships of phi means and phi standard deviations of native and borrow material. Category Case Quadrant in figure 15.7 I 1

Relationship of phi means

II

Borrow material is coarser than the native material

native material

2

III IV

7.1

3 4

MΦb> MΦn

Relationship of phi standard deviations σΦb> σΦn

Borrow material is finer than the native material

Borrow material is more

MΦb < MΦn

poorly

sorted

than

the

MΦb< MΦn

σΦb< σΦn

Borrow material is coarser than the native material

Borrow material is better

MΦb> MΦn

sorted

Borrow material is finer than the native material

material

than

Case study

Calculate the overfill factor RA and the renourishment factor RJ for given: Native sediment:φ 16=1.41(0.38 mm) φ84= 2.47 (0.18 mm) Nourished sediment: φ16= (0.31 mm) φ84= 3.41 (0.09 mm)

the

native

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313

As shown in chapter 2, the mean sediment diameter in phi units is given by: Mφn=(φ84+φ16)/2=( 2.47+1.41 )/2= 1.94 (0.26 mm) Mφb=(φ84+φ16)/2=( 3.41+1.67 )/2= 2.54 (0.17 mm)

σφn=(φ84 -φ16)/2=( 2.47-1.41 )/2=0.53 σφb=(φ84 -φ16)/2=( 3.41-1.67 )/2=0.87 the sorting ratio is:

σφb/σφn=0.87 /0.53= 1.64 The phi mean differences ratio is: (Mφb - Mφn)/ σφn=( 2.54-1.94)/ 0.53=1.13 From figure 15.7 we obtain RA=2.25. That is the fill volume obtained by numerical or geometrical calculations multiplied by 2.25. From fig. 15.8 we obtain RJ =3/2. That is the time interval between two successive renourishments has to be divided by 3/2.

Figure 15.8 - Isolines of the renourishment factor (RJ) for values of φ mean difference and φ sorting ratio, ∆=1.0 (James, 1975).

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INTRODUCTION TO COASTAL DYNAMICS AND SHORELINE PROTECTION

8 Sediment sources Borrow sources for beach fill can be divided into three general categories: terrestrial, offshore, and navigation channels. Each category has favorable and unfavorable aspects; however, selection of an optimum borrow source depends more on individual site characteristics relative to project requirements than type of source. The single most important borrow material characteristic is the sediment grain size (CEM, 2001). Terrestrial sources of material suitable for beach fill can be found in many fluvial and marine terrace and channel deposits. Use of terrestrial borrow sites usually involves lower costs for mobilization-demobilization operations and plant rental, and less weather-related downtime than the use of a submerged borrow source. However, the production capacity of terrestrial borrow operations is comparatively low, and haul distances may be relatively long. Thus, costs per unit volume of placed material may exceed those from alternative submerged sites. Navigation channels and maintenance dredging of existing navigation projects often involves the excavation and disposal of large volumes of sediment. In some cases, where the dredged sediment is of suitable quality, it can be used as fill on nearby beaches rather than placing it in offshore, upland, or contained disposal sites. Operations of this type are economically attractive because dual benefits are realized at considerably less cost than possible if both operations were carried out separately. Maintenance dredging of sediment in navigation projects in low energy environments is least likely to produce suitable beach-fill material. In such areas, the dredged material often consists of material in the clay, silt, and very fine sand size range. However, when dredging new harbors, channels or waterways, or deepening of existing channels in low energy areas, the dredge may cut into previously undisturbed material of suitable characteristics. Obviously, sediment compatibility tests should be performed to determine its suitability for use as beach-fill material. Offshore deposits can be excavated by dredges designed to operate in open sea conditions. When borrow material is obtained by dredges it is typically pumped directly to the beach via pipelines. Offshore borrow sources have several favorable features. Suitable deposits can often be located close to the project area. Offshore deposits usually contain large volumes of sediment with uniform characteristics and little or no silt or clay size material. Large dredges with high production rates can be used. Environmental effects can be kept at acceptable levels with proper planning. The evaluation of the sediment source is therefore a crucial point in the design of a beach fill. The reader is referred to Dean (2002) for further details.

9 Monitoring A beach fill project monitoring can include quantification of the project response or the monitoring of response and forcing, i.e. waves, winds, currents and tides. There can be a variety of objectives of a monitoring program, ranging from

BEACH FILLS

315

documentation of nourishment performance to development of a database to assist, for example, in the improvement of cross-shore sediment transport mode. In some projects, there may be cases where stabilization structures are included and questions exist as to whether these structures will adversely affect the adjacent beaches. In other words, there may be pre-established “thresholds” or “triggers” which, when exceeded, will require certain remedial action, either renourishment or modification of the stabilizing structures. The monitoring plans involve both spatial and temporal dimensions, each of which should be commensurate with the project characteristics and anticipated evolution. In fact short projects evolve more rapidly than longer projects and all projects are generally expected to evolve more rapidly in their early stages especially near the project ends. Additionally, project evolution is more rapid in energetic wave climates. Prior to initiating a monitoring plan, it is necessary to establish a baseline which can be used for physical, biological and economic monitoring. Although the need for a physical baseline has decreased with the availability of Global Positioning System (GPS) capabilities, it is still regarded as an overall valuable reference basis for the project. The surveys of beach and offshore profiles can be accomplished by a number of approaches. The spacing of profile lines and frequency of surveys should be commensurate with the expected spatial and temporal evolution of the project. In addition to surveys conducted in the project area, in order to document the complete effects of the project, surveys should be usually conducted on project adjacent beaches. The spaces are usually the same as the general spacing over the first few lines and then perhaps at double the spacing over the next several lines. The spacing of the lines both within and adjacent to project areas can be assisted by calculations of the anticipated project evolution and consideration of resources available for monitoring. As a rule of thumb (Dean, 2002), some projects include five to ten lines on either end and outside of the project placement area. The frequency of profile surveys should be commensurate with the anticipated temporal project evolution. Generally, in addition to the pre- and post-nourishment surveys, reasonable frequencies are at one-half year, one year and then two years unless the earlier surveys identify unexpected behavior. Surveys intended to quantify the effects of major storms are also usually conducted, where the justification is based on visual observations or cursory measurements of changes in beach width. If an objective of the surveys is to quantify the volume remaining in the project area, the profiles should extend to depths exceeding the closure depth. It may be worthwhile to extend the profiles to either a depth 25% greater than the closure depth. In order to carry out the design process, sand samples are taken to quantify characteristics of the native beach and the borrow area prior to nourishment. The number and locations of the samples to quantify the native beach are flexible and depend on variability, but could include, for each profile, sampling the dune, beach berm, beach face and 3 feet (or one meter) depth intervals out to a depth or distance exceeding that associated with the closure depth by 25%.

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References ABBOT, M.B., PETERSON, H.M., & SKOVGAARD, O., 1978. On the Numerical Modelling of Short Waves in Shallow Water. Journal Hydraulic Research, Vol. 16, No. 3, pp 173-203. AHRENS, J.P., 1981. Design of Riprap Revetments for Protection Against Wave Attack. Technical Paper TP-81-5. U.S. Army Engineer Waterways Experiment Station, Coastal Engineering Research Center, Vicksburg, MS. AHRENS, J.P., 1987. Characteristics of Reef Breakwaters. Technical Report CERC-87-17, U.S, Army Engineer Waterways Experiment Station, Vicksburg, MS. AIRY, G.B., 1845. Tides and waves. Encyclopedia Metropolitana, Vol. 192. ALLENDER, J.H, AUDUNSON T., BARSTOW S.F., BJERKEN V., KROGSTAD H.E., STEINBAKKE P., VARDEL L., BORGMAN L.E. & GRAHAM C., 1989. The WADIC project: a comprehensive field evaluation of directional wave instrumentation. Ocean Eng. 16, 505-536. ALLSOP, N.W., HAWKES, P.I., JACKSON, F.A., & FRANCO, L., 1985. Wave RunUp on Steep Slopes – Model Tests Under Random Waves. Report No. SR2, Hydraulics Research Station, Wallingford, England. APAT, 2004. Italian Wave Atlas. Environmental Protection and Technical Services Agency, Rome. BAGNOLD, R.A., 1966. An Approach to the Sediment Transport Problem from General Physics. U.S. Geological Survey Professional Paper 422-I, U. S. Dept of Interior. BAKKER, W.T., 1968. The Dynamics of a Coast with a Groin System. Proceedings of the 11th Coastal Engineering Conference, American Society of Civil Engineers, pp 492-517. BARNETT, T.P., 1968. On the generation, dissipation and prediction of ocean wind-waves. J. Geophys. Res. 73: 513-529. BARSTOW, S.F., & KROGSTAD, H.E., 1995. Wave-Climate Assessment by Satellite Remote Sensing. Sea Technology. October 1995, pp 31-38.

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Environmental Problems Coastal Engineering VII Modelling, Measurements, Engiin Coastal Regions including Oil and Chemical neering and Management of Seas and Coastal Regions Spill Studies Edited by: C. A. BREBBIA, Wessex Edited by: C. A. BREBBIA, Wessex Institute of Technology, UK

This volume deals with problems related to monitoring, analysis and modelling of coastal regions, including sea, land and air phenomena. Bringing together papers presented at the Sixth International Conference on Environmental Problems in Coastal Regions, the book focuses on ecological and environmental problems and the issues of water quality. The book will be essential to researchers, engineers and professionals involved in the field of Coastal Environmental quality and the related challenges to monitoring and controlling Oil Spills. Topics of interest include: Remote Sensing; Ecology and the Coastal Environment; Water Quality Issues; Wetlands; Sediment Problems; Coastal Restoration; Atmospheric Aspects; Sea States Forecasting; Modelling of Trajectory and Fate of Spills; Bioremediation; Detection, Prevention and Clean-up Measures; Erosion Problems; Management of Risk; Preservation of Pristine Coastal Areas; Estuarial Problems; Floods; Climate Change and the Coastal Environment. WIT Transactions on Ecology and the Environment Volume 88 ISBN: 1-84564-167-1 2006 apx 400pp apx £145.00/US$265.00/€217.50

WIT Press Ashurst Lodge, Ashurst, Southampton, SO40 7AA, UK. Tel: 44 (0) 238 029 3223 Fax: 44 (0) 238 029 2853 E-Mail: [email protected]

Institute of Technology, UK, M. DA CONCEICAO CUNHA, ISEC, Portugal Recent disasters have confirmed the importance of adequate protection of coastal areas. The cumulative effect of development in these regions needs to be better understood in order to ensure that they are managed in a sustainable fashion. Given the dynamics of the processes taking place on shorelines and the multitude of aspects that need to be considered the problems involved are extremely challenging. Bringing together contributions from researchers and professionals engaged in the development of modern computational and experimental tools, this book addresses many subjects relevant to the successful management of such areas and offers new insights. The papers featured were originally presented at the Seventh International Conference on Modelling, Measurements, Engineering and Management of Seas and Coastal Regions. Topics and techniques covered include: Shallow Water Studies; Estuarine Problems; Pollutant Transport and Dispersion; Water Quality Issues; Oil Spills; Coastal Ecosystems and their Sustainability; Coastal Erosion and Sedimentation; Coastal Geomorphology; Sea Defence and Protection Systems; and Case Studies. WIT Transactions on The Built Environment, Volume 78 ISBN: 1-84564-009-8 2005 368pp £140.00/US$224.00/€210.00

Marine Technology V

Coastal Environment V

Editor: WESSEX INSTITUTE OF TECHNOLOGY, UK

Incorporating Oil Spill Studies

This book contains most of the papers presented at the Fifth International Conference on Marine Technology. Developments in the following areas are highlighted: Design and Fabrication in Shipbuilding; Shipbuilding and Design; Hydrodynamics; Navigation, Ship Operation and Multimode Transport; Inland Water Transportation; and Reliability and Safety in Marine Technology. WIT Transactions on The Built Environment, Volume 68 ISBN: 1-85312-973-9 2003 340pp £115.00/US$184.00/€172.50

Coastal Engineering VI Computer Modelling and Experimental Measurements of Seas and Coastal Regions Editors: C.A. BREBBIA, Wessex Institute of Technology, UK, and D. ALMORZA and F. LÓPEZ-AGUAYO, University of Cadiz, Spain The proceedings of the Sixth International Conference on Computer Modelling and Experimental Measurements of Seas and Coastal Regions. Contributions are grouped into sections including: Shallow Water Studies; Pollutant Transport and Dispersion; and Coastal Erosion and Sedimentation. WIT Transactions on The Built Environment, Volume 70 ISBN: 1-85312-977-1 2003 532pp £159.00/US$254.00/€238.50 Find us at http://www.witpress.com Save 10% when you order from our encrypted ordering service on the web using your credit card.

Editors: C.A. BREBBIA, Wessex Institute of Technology, UK, and J.M. SAVAL PEREZ, L. GARCIA ANDION and Y. VILLACAMPA, University of Alicante, Spain Highlighting monitoring, analysis, modelling and control, this volume contains papers presented at the Fifth International Conference on Environmental Problems in Coastal Regions and the associated Fourth International Seminar on Hydrocarbon Spills. Over 40 contributions are included and these focus on areas such as: COASTAL ENVIRONMENT: Ecology and the Environment; Water Quality Issues; Wetlands; Sediment Problems; Coastal Restoration; Atmospheric Aspects; Sea States Forecasting. OIL SPILL STUDIES: Modelling of Trajectory and Fate of Spills. WIT Transactions on Ecology and the Environment, Volume 68 ISBN: 1-85312-710-8 2004 484pp £172.00/US$275.00/€258.00

WIT eLibrary Home of the Transactions of the Wessex Institute, the WIT electronic-library provides the international scientific community with immediate and permanent access to individual papers presented at WIT conferences. Visitors to the WIT eLibrary can freely browse and search abstracts of all papers in the collection before progressing to download their full text. Visit the WIT eLibrary at http://library.witpress.com