MARINE PHYSICS
FURTHER T l T L E S IN THIS S E R I E S 1 J.L. MERO THE MINERAL RESOURCES OF THE SEA 2 L.M. FOMIN THE ...
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MARINE PHYSICS
FURTHER T l T L E S IN THIS S E R I E S 1 J.L. MERO THE MINERAL RESOURCES OF THE SEA 2 L.M. FOMIN THE DYNAMIC METHOD IN OCEANOGRAPHY 3 E. I. F. WOOD MICROBIOLOGY OF OCEANS AND ESTUARIES 4 G. NEUMANN OCEAN CURRENTS 5 N.G. JERLOV OPTICAL OCEANOGRAPHY 6 V. VACQUIER GEOMAGNETISM IN MARINE GEOLOGY 7 W.J. WALLACE THE DEVELOPMENTS OF THE CHLORINITY!SALINITY CONCEPT IN OCEANOGRAPHY 8 E. LlSlTZlN SEA-LEVEL CHANGES 9 R.H. PARKER THE STUDY OF BENTHIC COMMUNITIES J.C. I. NIHOUL (Editor) 10 MODELLING OF MARINE SYSTEMS 11 0. I. MAMAYEV TEMPERATURE-SALINITY ANALYSIS OF WORLD OCEAN WATERS 12 E.J. FERGUSON WOOD and R. E. JOHANNES (Editors) TROPICAL MARINE POLLUTION 13 E. STEEMANN NIELSEN MARINE PHOTOSYNTHESIS 14 N.G. JERLOV MARINE OTPICS 15 G.P. GLASBY MARINE MANGANESE DEPOSITS 16 V. M. KAMENKOVICH FUNDAMENTALS OF OCEAN DYNAMICS 17 R.A. GEYER (Editor) SUBMERSIBLES AND THEIR USE IN OCEANOGRAPHY AND OCEAN ENGINE:ERINIG 18 J.W. CARUTHERS FUNDAMENTALS OF MARINE ACOUSTICS j. C. I. NIHOUL (Editor) 19 BOTTOM TURBULENCE 20 P. H. LEBLOND and L. A. MYSAK WAVES IN THE OCEAN C. C. VON DER BORCH (Editor) 21 SYNTHESIS OF DEEP-SEA DRILLING RESULTS IN THE INDIAN OCEAN 22 P. DEHLINGER MARINE GRAVITY 23 J. C.J. NIHOUL (Editor) HYDRODYNAMICS OF ESTUARIES AND FJORDS 24 F. T. BANNER, M. 8. COLLINS and K. S. MASSIE (Editors) THE NORTH-WEST EUROPEAN SHELF SEAS: THE SEA BED and THE SEA IN M O l'ION 25 J.C. J. NIHOUL (Editor) MARINE FORECASTING 26 H.G. RAMMING and Z. KOWALIK NUMERICAL MODELLING MARINE HYDRODYNAMICS 27 R. A. GEYER (Editor) MARINE ENVIRONMENTAL POLLUTION 28 J. C. 1. NIHOUL (Editor) MARINE TURBULENCE 29 M. WALDICHUK. G. 6. KULLENBERG and M. 1. ORREN (Editors) MARINE POLLUTION TRANSFER PROCESSES 30 A. VOlPlO (Editor) THE BALTIC SEA 31 E. K. DUURSMA and R. DAWSON (Editors) MARINE ORGANIC CHEMISTRY 32 J.C. J. NIHOUL (Editor) ECOHYDRODYNAMICS 33 R. HEKlNlAN PETROLOGY OF THE OCEAN FLOOR 34 J. C.J. NIHOUL (Editor) HYDRODYNAMICS OF SEMI-ENCLOSED SEAS 35 8. JOHNS (Editor) PHYSICAL OCEANOGRAPHY OF COASTAL AND SHELF SEAS 36 1. C. 1. NIHOUL (Editor) HYDRODYNAMICS OF THE EQUATORIAL OCEAN 37 W. LANGERAAR SURVEYING AND CHARTING OF THE SEAS 38 1. C. 1. NIHOUL (Editor) REMOTE SENSING OF SHELF SEA HYDRODYNAMICS (coutinued on p. 516)
Elsevier Oceanograghy Series,
53
MARINE PHYSICS JERZY DERA Institute of Oceanology
Polish Acudemy of Sciences, Sopof, Poland
ELSEVIER Amsterdam-Oxford-New
York-Tokyo
PWN - POLISH SCIENTIFIC PUBLISHERS Warszawa
1992
Tranlated by Peter Senn from the revised Polish edition Fizyka morza, published in 1983 by Palistwowe Wydawnictwo Naukowe, Warszawa Distribution of this book is being handled by the following publishers:
For the USA and Canada ELSEVIER SCIENCE PUBLISHING CO., INC. 655 Avenue of the Americas, New York, NY 10010 For Albania, Bulgaria, Cuba, Czecho-Slovakia, Hungary, Korean People’s Democratic Republic, Mongolia, People’s Republic of China, Poland, Romania, the USSR, Vietnam and Yugoslavia PWN -Polish Scientific Publishers Miodowa 10, 00-251 Warszawa, Poland For all remaining areas ELSEVIER SCIENCE PUBLISHERS B.V. Sara Burgerhartstraat 25 P.O. Box 21 1, loo0 AE Amsterdam, The Netherlands
Library of Congress Cataloging-in-Publication Data Dera, Jerzy. [Fizyka morza. English] Marine physics / Jerzy Dera. (Elsevier oceanography series ; 5 3) cm. P. Rev. and updated translation of: Fizyka morza. Includes bibliographical references and index. ISBN 0-444-98716-9 1. Oceanography. I. Title. 11. Series. GC150.5.D4713 1991 551.46--dC20
--
90-48486 CIP
0-444-98716-9 (vo~.53) 0-444-41623-4 (series) Copyright
0by PWN-Polish
Scientific Publishers-Warszawa
1992
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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 copyright owner Printed in Poland by D.N.T.
PREFACE
Contemporary science and technology is equipping us with increasingly sophisticated methods of studying the sea. Hand in hand with these goes the advance in techniques that render accessible the ocean’s vast reserves of space, raw materials and energy, whose utilization has become a key factor in the solution of modern civilization’s problems. The ocean’s paramount influence on climate formation and living conditions on the Earth has also been recognized, and the world-wide threat to the environment by the wasteful exploitation of the seas is perfectly plain. As a result, the science of the sea, traditionally known as oceanography, has undergone rapid development. In trying to discover as much as possible about the nature of the marine environment as a pre-condition of its rational use, oceanography has evolved so far, that today it embraces a number of separate disciplines such as marine physics, marine chemistry, marine biology and marine geology, which are often lumped together under the general heading of oceanology. Researching the nature of the oceanic environment in all its complexity, however, requires even greater specialization, such as we are witnessing in the fundamental sciences. An illustration of this trend is the large number of highly specialized monographs on various aspects of the marine cnvironment. So too in marine physics there are numerous monographs separately dealing with wave-action, ocean currents, turbulence in the sea, hydrodynamic models of the ocean’s upper layer, the optical properties of the sea, the solar light field in the sea and the propagation of sound waves in the ocean. My intention is to introduce the reader to these complex problems of marine physics, to explain the mechanisms of the principal physical processes in the sea and their inter-relationships, and to enable him/her to embark on a more detailed study of the subject with the aid of the quoted source literature. In attempting to produce a synthesis of the results of the latest research in marine physics, and to keep this volume to a manageable size, 1 have limited the text to a discussion of the physical properties of sea water, some thermodynamic processes in the sea, the transfer and inter-relationships of sunlight and sound waves in the sea, the molecular and turbulent exchange of mass, heat and momentum in the sea, and air-sea interaction. I have given all those classic problems of marine dynamics like waves, tides and currents only cursory treatment as they are comprehensively dealt with in numerous text-books.
VI
PREFACE
I hope that my book will be of service to a wide range of readers interested in the marine environment, from students, teachers and scientists to engineers and management personnel involved in all aspects of the maritime economy. While writing this book I received generous assistance and support from my colleagues at the Institute of Oceanology, Polish Academy of Sciences, in Sopot, in particular Professor Czeslaw Druet, Docent Andrzej Zielinski, Dr Miroslaw Jonasz and Mrs Janina Jackowska. Professor Antoni Sliwinski of the University of Gdansk also made many valuable comments. To all these people, and to many others not mentioned here, who contributed to the creation of this book, I extend my warmest gratitude. JERZY DERA
CONTENTS
PREFACE
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
V
CHAPTER 1 A GENERAL PICTURE OF PHYSICAL PROCESSES IN THE OCEAN 1 .1 The Earth as a Thermodynamic System . . . . . . . . . . . . . . . . . . 1.2 Forces Inducing the Motion of Water Masses in the Ocean . . . . . . . . . . The Force of Gravity and Its Components . . . . . . . . . . . . . . . . . Geopoten tial . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Pressure. Buoyancy . . . . . . . . . . . . . . . . . . . . . . . . . . . . Vertical Stability . . . . . . . . . . . . . . . . . . . . . . . . . . . . The Vaisala-Brunt Oscillation Frequency . . . . . . . . . . . . . . . . . The Inclination of Isobaric Surfaces . . . . . . . . . . . . . . . . . . . . The Action of the Coriolis Force . . . . . . . . . . . . . . . . . . . . . Frictional Forces . . . . . . . . . . . . . . . . . . . . . . . . . . . . The Equation of Motion of Waters . . . . . . . . . . . . . . . . . . . .
CHAPTER 2 SEAWATER AS A PHYSICAL MEDIUM . . . . . . . . . . . . . 2.1 The Structure of the Water Molecule . . . . . . . . . . . . . . . . . . . 2.2 The Association of Water Molecules. The Structure of an Ice Crystal and Liquid Water . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3 Ion Hydrates in Seawater . . . . . . . . . . . . . . . . . . . . . . . . 2.4 The Principal Chemical Constituents and the Salinity of Seawater . . . . . . 2.5 Electrical Conductivity as an Indicator of Seawater Salinity . . . . . . . . . Measurement of Salinity on the Practical Scale . . . . . . . . . . . . . . . 2.6 Yellow Substances in Seawater . . . . . . . . . . . . . . . . . . . . . . 2.7 Suspended Particles in Seawater, Their Concentration and Dimensions . . . . 2.8 Gas Bubbles in Seawater . . . . . . . . . . . . . . . . . . . . . . . .
1 1 13 14 20 21 23 29 31 33 37 43 49 51
56 64 68 73 79 85 89 101
CHAPTER 3 THE THERMODYNAMICS OF SEAWATER . . . . . . . . . . . 3.1 Seawater State Parameters and the Equation of State . . . . . . . . . . . 3.2 The Thermal Expansion of Seawater . . . . . . . . . . . . . . . . . . . . 3.3 The Compressibility of Seawater . Potential Temperature and Potential Density in the Sea . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.4 The Salinity Effect on the Specific Volume of Seawater . . . . . . . . . . . . 3.5 The Empirical Equations of State for Seawater . . . . . . . . . . . . . . .
122 128 130
CHAPTER 4 THE INTERACTION OF LIGHT AND OTHER ELECTROMAGNETIC RADIATION WITH SEAWATER. THE INHERENT OPTICAL PROPERTIES OF THE SEA . . . . . . . . . . . . . . . . . . . . . . . . 4.1 Radiance and Other Basic Photometric Quantities in Hydrooptics . . . . . . 4.2 Light Absorption in Seawater . . . . . . . . . . . . . . . . . . . . . . .
141 145 155
.
107 113 116
VIIl
CONTENTS
Light Absorption by Water Molecules . . . . . . . . . . . . . . . . . . . Light Absorption by Seawater Constituents . . . . . . . . . . . . . . . . . 4.3 Light Scattering in Seawater . . . . . . . . . . . . . . . . . . . . . . . Rayleigh's Theory of Scattering . The Volume Scattering Function and the Total Scattering Coefficient . . . . . . . . . . . . . . . . . . . . . . . . . . Molecular Scattering According t o the Smoluchowski-Einstein Fluctuation Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Scattering by Marine Suspended Particles . Principles of the Mie Theory . . . . . A Matrix Description of Scattering Stokes Parameters . . . . . . . . . . . . 4.4 The Transparency of Seawater t o Light and Other Electromagnetic Waves. The Radiant Energy Transfer Equation in the Sea . . . . . . . . . . . . . . . .
.
CHAPTER 5 SOLAR RADIATION INFLOW AND THE NATURAL IRRADIANCE I N THE SEA. THE APPARENT OPTICAL PROPERTIES OF THE SEA . . . The Solar Constant . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.1 The Influx of Solar Radiation t o the Sea Surface . . . . . . . . . . . . . . The Optical Thickness of the Atmosphere . . . . . . . . . . . . . . . . . The Single Scattering Model . . . . . . . . . . . . . . . . . . . . . . . The Transmittance of a Real Atmosphere . . . . . . . . . . . . . . . . . 5.2 Reflection and Transmittance of Sunlight at the Sea Surface. The Albedo of the Sea Reflectance Functions . Albedo . . . . . . . . . . . . . . . . . . . . . . Reflection from a Roughened Sea Surface . . . . . . . . . . . . . . . . . 5.3 The Penetration of Natural Light into the Sea Depths . The Optical Classification of Waters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Vector Irradiance . . . . . . . . . . . . . . . . . . . . . . . . . . . . The Diversity of Irradiance and the Optical Classification of Waters . . . . . . Fluctuations in the Underwater Irradiance . . . . . . . . . . . . . . . . . 5.4 The Apparent Optical Properties of the Sea and Their Relationships with the Inherent Properties in an Underwater Light Field . . . . . . . . . . . . . . Definitions of Apparent Optical Properties . . . . . . . . . . . . . . . . . The Interrelationships among the Optical Properties of the Sea . . . . . . . . The Asymptotic Light Field . . . . . . . . . . . . . . . . . . . . . . . CHAPTER 6 THE TRANSFER OF MASS. HEAT AND MOMENTUM I N THE MARINE ENVIRONMENT . . . . . . . . . . . . . . . . . . . . . . 6.1 Molecular Transport of Mass. Heat and Momentum in Seawater . . . . . . . The Equation of Diffusion . . . . . . . . . . . . . . . . . . . . . . . . The Thermal Conductivity Equation . . . . . . . . . . . . . . . . . . . . The Navier-Stokes Equation of Motion . . . . . . . . . . . . . . . . . . 6.2 The Turbulent Exchange of Mass. Heat and Momentum in the Sea . . . . . . Conditions for Turbulent Motion . . . . . . . . . . . . . . . . . . . . . Average and Fluctuating Component of Velocity . . . . . . . . . . . . . . Averaging of the Navier-Stokes Equation of Motion . . . . . . . . . . . . The Turbulent Exchange of Momentum . . . . . . . . . . . . . . . . . . The Turbulent Exchange of Mass and Heat . . . . . . . . . . . . . . . .
156 165 174
177 187 192 209 213
227 227 232 234 236 243 248 250 257 268 271 274 281 292 293 297 308
315 317 324 329 332 341 344 348 351 354 361
CONTENTS
IX
CHAPTER 7 SMALL-SCALE AIR-SEA INTERACTION AND ITS INFLUENCE ON THE STRUCTURE O F WATER MASSES IN THE SEA . . . . . . . . 7.1 Fluxes of Momentum. Mass and Heat Across the Sea Surface . . . . . . . . . The Solar Radiation Flux Qb . . . . . . . . . . . . . . . . . . . . . . . . The Long-wave Radiation Flux Qb . . . . . . . . . . . . . . . . . . . . The Surface Boundary Layer. Momentum Flux . . . . . . . . . . . . . . . The Fluxes of Sensible Heat Qn and Latent Heat Q.. and of the Mass of Water Vapour Me . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The Laminar Surface Layer . . . . . . . . . . . . . . . . . . . . . . . FluxesofWaterDroplets. SaltParticlesandElectricCharge . . . . . . . . . . 7.2 The Energy Budget of a Basin and Its Influence on the Structure of Water Masses The Equation of the Turbulent Energy Budget . . . . . . . . . . . . . . . The Horizontally Stratified Sea Model . . . . . . . . . . . . . . . . . . . The Mixed Layer Model . . . . . . . . . . . . . . . . . . . . . . . . . The Heat Budget of a Sea Basin . . . . . . . . . . . . . . . . . . . . .
387 391 393 397 398 403 411 421
CHAPTER 8 THE ACOUSTIC PROPERTIES OF THE SEA . . . . . . . . . . . The Wave Equation for Unattenuated Waves . . . . . . . . . . . . . . . . The Energy and Intensity of Sound . . . . . . . . . . . . . . . . . . . . 8.1 The Velocity of Sound in the Sea . . . . . . . . . . . . . . . . . . . . . Sound Velocity Distributions in the Sea . . . . . . . . . . . . . . . . . . 8.2 The Absorption and Scattering of Sound in the Sea . . . . . . . . . . . . . Relaxation Processes in Seawater . . . . . . . . . . . . . . . . . . . . . The Absorption of Sound Energy in the Sea . . . . . . . . . . . . . . . . Sound Scattering Functions . . . . . . . . . . . . . . . . . . . . . . . Sound Scatter at Small Scattering Centres and Bubbles . . . . . . . . . . . 8.3 Introduction to the Ray Theory of Sound Propagation in the Sea . . . . . . . The Eikonal Equation . . . . . . . . . . . . . . . . . . . . . . . . . . Trajectories of Sound Rays in the Sea . . . . . . . . . . . . . . . . . . .
425 428 431 435 440 443 446 451 454 460 465 466 470
BIBLIOGRAPHY . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
479
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
507
INDEX
369 370 371 373 376
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CHAPTER 1
A GENERAL PICTURE OF PHYSICAL PROCESSES IN THE OCEAN
1.1 THE EARTH AS A THERMODYNAMIC SYSTEM
Imagine the Earth’s globe as a sphere suspended in the cosmic vacuum and unceasingly heated by solar radiation. As when any body is irradiated, the incident radiation is partially reflected from the Earth and partially transmitted to the atmosphere, land and sea where it is scattered and absorbed. That part of the solar radiation energy AQn which in a finite time (a century, a year) is absorbed by the Earth increases its internal energy by AU, in accordance with the first law of thermodynamics
AQ, = AU+ W
(1.1 .l)
where the work of the Earth W done against external forces can be assumed to be equal to zero. The increase in the Earth‘s internal energy in a given time is manifested above all by an increase in the temperature of its constituents, i.e. by an increase in the kinetic energy of particles of air, water, soil, etc. (sensible heat). But also contributing to this overall increase in energy are the rise in the potential energy of water molecules after these have been pulled away from the influence of hydrogen bonds when ice melts or water evaporates (latent heat-see Chapter 2), and, furthermore, the increase in the kinetic and potential energy of the mobile macrocomponents of the Earth, that is, the mechanical energy of the masses of air, water vapour and water in the atmosphere, oceans and rivers (winds, clouds, currents). Lastly in this global energy increase due to the absorption of solar radiation we have to include the increase in chemical energy accumulated as the result of the photosynthesis of organic matter (carbohydrates, proteins, fats and their conversion products such as wood, coal, oil, natural gas). The other forms of the Earth’s internal energy mentioned are in various ways almost all converted to heat as well. This is what happens when heat is produced during the oxidation of organic substances, by the friction between moving masses, during the condensation of water vapour and freezing of liquid water, and in other processes. Thus heat is continuously supplied to and evolved by the Earth. However, the Earth is prevented from heating up by emitting its excess heat into
1 A GENERAL PICTURE OF PHYSICAL PROCESSES IN THE OCEAN
2
space in the form of electromagnetic waves. These are infra-red (IR) waves emitted by the land, sea and atmosphere in accordance with the rule that every body whose absolute temperature is T > 0 emits electromagnetic radiation into its surroundings. Figure 1.1.1 is a simplified illustration of the basic mechanism of this global Solar radiation
Fig. 1.1.1. The Earth rotating with an angular velocity w as a thermodynamic system illuminated by solar radiation, whose absorbed radiation Q. [J] is approximately equal to the energy emitted Q,,[J in the infra-red part of the radiation spectrum (for details, see Chapters 5 and 7).
process. We can assume that the Earth behaves like a radiating black body having an average (effective) temperature of T z 255 K, whereas the Sun is a radiating black body whose temperature is T z 6000 K. The characteristics of such radiation are described by the weli-known Stefan-Boitzmann and Wien laws. The first of these states that the total emissivity of a perfectly black body E ~ i.e., , the flux of energy [W/m2] emitted by a unit area of that body, is proportional to the fourth power of the absolute temperature of that body ET
=
(1.1.2)
aT4
where @ = 5.6687 x Wm-2K-4 is the Stefan-Boltzmann constant. The second of these laws, Wien’s law, describes the spectral distribution of the radiant energy of a black body and can be written down as follows: EA,T
=
A-’f(A
T)
(1.1.3)
where = dsT/dA is the spectral density of the emissivity, that is, the emissive power of wavelengths contained in an infinitesimally small interval dA surrounding a given wavelength R at temperature T,and f is a certain function of the product
1.1 THE EARTH AS A THERMODYNAMIC SYSTEM
3
AT. From this law we can conclude that the maximum emissivity (el, T)max at an absolute temperature T falls at the wavelength
a,
b T
=-
(1.1.4)
where b = 2.898 x m . K is Wien’s constant (see e.g., Szczeniowski, 1971). This law is called Wien’s displacement law since it describes the displacement of the maximum of the radiation spectrum. The upshot of this is that the Earth, with its average temperature of T x 255 K (including that of the atmosphere) most strongly emits electromagnetic radiation of wavelength I , x 11 pm (IR), whereas the Sun, whose average surface temperature T x 6000 K, most strongly emits radiation of wavelength I , x 0.48 pm (blue-green light). The principal difference in the total electromagnetic radiation spectra of the Earth and Sun,
1
I/:
i
I
,\
03 0.2 05 1 Wavelength 1 Ivml (b)
-
Earth’s radiation
Wavelength A
bml
Fig. 1.1.2. A comparison of the radiation spectra of the Earth and Sun. (a) Normalized spectra of the Sun’s (T c 6000 K) and the Earth’s (T c 255 K)radiation illustrating the differences in bands and radiation maxima of both these bodies; (b) the approximate absolute spectra of solar and terrestrial radiation passing in opposite directions through the surface of the Earth.
4
1 A GENERAL PICTURE OF PHYSICAL PROCESSES I N THE OCEAN
compared in Fig. 1.1.2, results from the temperature difference between these two bodies. That the overall balance of the energy absorbed Qa and radiated Qb by the Earth's mass is zero is confirmed by the relatively stable temperature of the Earth over hundreds of years. But we know that on the geological time scale this thermodynamic equilibrium shifts in one direction or the other, bringing about glaciation or the reverse. So, for example, some 17.000 years ago, in the Ice Age, the average temperatures of the surface waters of the Atlantic were much lower than they are now. This is illustrated by Fig. 1.1.3 in which the isotherms from 17000 years ago have been reproduced on the basis of information derived
Longitude E
Fig. 1.1.3. Mean winter temperatures ["C] at the surface of the Atlantic Ocean: (a) now, and (b) 17 000 years ago. The isotherms on the upper map were plotted from contemporary oceanographic data, those on the lower map from past temperatures estimated from fossilised Foraminifera prbtozoans present in about 90 deep-water cores of Atlantic Ocean bottom sediments examined during the CLIMAP programme (Climate Long-Range Investigation, Mapping and Prediction Study; from NSF Report, 1973).
1.1 THE EARTH AS A THERMODYNAMIC SYSTEM
5
from fossilised marine organisms found in cores of sea-bed sediments (investigations made during the international CLIMAP study-see NSF, 1973). These shifts in the Earth’s thermodynamic equilibrium may be due both to changes on the Earth itself (amount of carbon dioxide or volcanic dusts, hence increased absorption by the atmosphere and seas, also different distribution of land) and to possible changes in the solar radiation to the Earth as a result of changes in the Sun’s temperature or its distance from the Earth (see Robock, 1978). The exchange of the kinetic energy of particles between the Earth and outer space, also the radiation of heat from the Earth’s interior, and energy from nuclear reactors and other sources are of minor jmportance in the global energy budget in comparison with the intense flux of solar radiation (see Chapter 5). The large-scale combustion of fossil fuels (oil, coal, natural gas) at the present could be of some significance in this budget. Their chemical energy, produced by plants from solar energy over millions of years, is being released during a single century. Changes in the absorption properties of the atmosphere and seawaters have been brought about by this combustion and the development of transport and industry (see Chapter 4). During recent decades, the mean balance of energy influx and radiation within the terrestrial system has nevertheless been close to zero, so that despite certain oscillations (Angel1 and Korshover, 1978), a constant mean temperature has been maintained on the Earth. This is the temperature at which the waters of the oceans, seas and rivers can exist in the liquid state and promote the development of diverse forms of life. This simplified thermodynamic mechanism becomes much more complicated as we inspect ever smaller elements of the terrestrial system. The main factor complicating the thermodynamic macroprocesses in this system is the spatially and temporally unequal supply of solar energy to its various elements. We could assume that, apart from slight fluctuations due to changes in the Sun’s activity and in the Earth’s orbit, the total solar energy flux reaching the Earth is constant in time (see Chapter 5). But there are two major reasons why the energy supplied to the Earth is not evenly distributed. Firstly, the Sun’s rays illuminate only about half the Earth’s surface at any one moment and moreover do so very unevenly because the Earth’s spherical shape means that they impinge on the surface at different angles of incidence. Secondly, the movement of the Earth, together with the spatially differentiated optical and thermodynamic properties of its surface (including the atmosphere and oceans), causes the ratio of energy absorbed to that reflected and radiated to change constantly. The directly visible consequences of the superimposition of these two factors are not only the diurnal and seasonal differences in temperature and the existence of climatic zones, but
6
1 A GENERAL PICTURE OF PHYSICAL PROCESSES IN THE OCEAN
also the distinct latitudinal variations in climate (continental, maritime and other climates). The classic example of the powerful influence exerted by the reflective properties of an insolated surface is the fact that the polar ice-caps could be melted if they were covered with soot. During the polar day (summer) (see Fig. 1.1.4),
Time (month)
Fig. 1.1.4. Differences in the insolation of the Earth a t various latitudes during the year. The isolines and the figures describing them express, in calories, the amount of solar radiation energy incident in 24h on 1 cm2 of a horizontal surface at the upper boundary of the Earth's atmosphere [lo2 cal cm-2 . d-'1 (after Fritz, 1951, with permission of the American Meteorological Society).
the ice-caps receive a very large amount of solar energy, but a considerable part of this is reflected from their white surfaces and lost to the Earth for ever. The very clean atmosphere, whose thickness is about half that at the equator, assists this escape of radiation from polar regions.
1 . I THE EARTH AS A THERMODYNAMIC SYSTEM
7
The quantity of energy locally radiated into space by a given area of the Earth also depends strongly on the nature of the surface covering, its heat capacity and absorption capabilities (Jucewicz, 1970), since the Earth’s surface is not, in fact, a black body. Emission and absorption depend particularly on the occurrence of clouds in the atmosphere (Herman, Wu and Johnson, 1980) and on the water vapour, carbon dioxide and other IR-absorbing compounds contained in it (Zuev, 1970). Also, because of the considerable difference between the heat capacities of water and that of soil and rocks, the surface of the sea can absorb far more heat than the land, and heated to a temperature T, can in turn radiate far more heat than a land surface heated to the same temperature. For these same reasons, at any given instant there are subareas in the surface layers of the Earth’s system (atmosphere, ocean, land surfaces) differing in the amount of sensible and latent heat that they have accumulated, because some of them receive more radiation from the Sun than they give out, and vice versa. How the radiation surplus varies with latitude is illustrated in Fig. 1.1.5.
Latitude
LO1
Fig. 1.1.5. The radiation surplus at different lattitudes (after Budyko, 1956).
The differences in the quantities of heat accumulated in contiguous subarear of the same thermodynamic system must obviously lead to its immediate transfes to areas of lower temperature by all possible means, i.e., by radiation, molecular conductivity, and by the small- and large-scale turbulent exchange associated with the movement of masses of water, air and water vapour over large distances (ocean currents and winds). This transfer of heat (see Chapter 6 ) means that areas suffering a continual loss of radiation do not cool indefinitely, neither do areas
8
1 A GENERAL PICTURE OF PHYSICAL PROCESSES IN THE OCEAN
having a net gain of heat warm up ad infiniturn as might be expected from the local balance of radiation absorption and emission. Heat transfer between different parts of the Earth is therefore a basic natural process moderating the Earth’s climate and much reducing the annual fluctuations of temperature in climatic zones (Izrael and Sedunov, 1979; Berger, 1979). Notice now that it is only the mobile masses of atmospheric air and oceanic water which chiefly transfer the tremendous quantities of heat over long distances, from the strongly insolated areas of the hot zone (summer hemisphere) towards the cold areas (winter hemisphere and poles), because only in such a way can heat transfer over great distances be effected (see Chapter 6). In the process balancing out radiation gains and losses between warm and cold areas, masses of air and water transfer gigantic amounts of heat, of the order of 1015 J/s, for distances of thousands of kilometres (Bennett, 1978; Bryden and Hall, 1980; Hasternrath, 1977, 1980), so this transfer in the atmosphere can often be extremely violent, taking the form of hurricanes and typhoons (Vetroumov, 1979; ROSS, 1979). The very process of heat transfer is a highly complicated one, composed as it is of multi-stage and time-variable atmospheric circulation systems (Lau Ngar Cheung, 1978; Marchuk et al., 1979) and ocean currents (see next section; Magnier et al., 1973; Gurgul, 1981). These are large-scale vortices of air and water masses (of the order of 1000 km) which gradually scatter their energy by inducing ever smaller (meso-scale, of the order of 100 km) vortices, right down to tiny vortices and molecular movements (see turbulence, Chapter 6). The essence of the principal mechanism involved in the formation of atmospheric circulations can be explained on the basis of convection in a liquid medium, non-uniformly warmed up from below. If solar radiation is intense and the atmosphere transparent, atmospheric air is heated mainly from below, because the absorption of solar radiation and the emission of heat are greatest in the surface layer of water and at the surface of the land. Heated from below, air expands and increases its specific volume, and hence its uplift pressure: it therefore rises. The loss of air mass at the bottom is immediately compensated for by air flowing in from the sides. After some time, a closed circulation must be set up, equalising the distribution of air masses in space. The shape and range of this circulation depend on the different kinds of substrates and on the influx of energy. The simplest and most typical examples of such circulations are monsoon winds, which blow over adjacent areas of ocean and land. Since land has a much lower heat capacity than water, the former becomes much warmer is summer than the latter. So the warmer air over the land rises, and cooler air, saturated with water vapour, flows in from over the sea to
1.1 THE EARTH AS A THERMODYNAMIC SYSTEM
9
replace it: a humid, cool wind blows in low from over the sea. As these moist masses of air are raised up over the land they undergo further cooling as a result of adiabatic expansion in the upper regions of the atmosphere where the pressure is lower (see Chapter 3). This cooling causes the water vapour in this saturated air to condense and intense rainfall is the result. In the upper regions of the atmosphere, the air-flow cycle is closed by a wind blowing in the opposite direction (Fig, 1.1.6). In winter the reverse takes place: the land is cooler than the ocean
Fig. 1.1.6. The monsoon circulation of air and water (a) in summer, when the land is warmer than the ocean, and @) in winter, when the ldnd is cooler than the ocean.
which, thanks to its large heat capacity, has accumulated a lot of heat and cools much more slowly. Hence the air warmed by the water rises over the ocean, and a cool, dry wind blows over the land out to sea, while aloft, the cycle is closed by a wind blowing in the opposite direction. This cool dry air blowing out to sea from large land areas removes heat from the sea (temperature difference
10
1 A GENERAL PICTURE OF PHYSICAL PROCESSES I N THE OCEAN
between water and air-conductivity, see Chapters 6 and 7) and increases evaporation of oceanic water as it is dry (far from being saturated) and can absorb a lot of water vapour. The salinity and density of the surface layer of water increase at the same time due to evaporation. On rising into the atmosphere, the wind also takes up water droplets, and with these, many sea salt constituents (see Chapters 2 and 7). The monsoon-type circulation takes place on a large scale and is especially clearly defined in India where in summer (April-July) the moist monsoon wind from the sea gives rise to exceptionally intense rainfall in the coastal zone, amounting to as much as several metres of water per month. The long-term friction between the relatively stable monsoon wind and the water surface brings a warm surface current into existence which, after having flowed for some distance and caused water to pile up (increasing hydrostatic pressure), induces a cold reverse current to flow back deeper down. Therefore, a circulation of ocean water comes about together with heat transfer, which in the Indian Ocean changes seasonallyjust as the monsoon winds do (see Hastenrath and Lamb, 1980). A circulation similar to this but on a much smaller scale can be observed along sea coasts. This results from the diurnal changes of insolation and cooling of the land and sea, giving rise to sea breezes, that is, fairly stable winds blowing inland off the sea during the day when the land is warmer and the air above it is rising, and out to sea from the land, which has cooled more quickly, at night. These breezes are accompanied by water circulation, and obviously, bring about wave action at the sea surface and cause water droplets containing salt to be taken up into the atmosphere. In the early morning and evening, before they change direction, these winds cease, and the sea surface becomes calmer; fishermen usually take advantage of this to sail through the breaker zone. These examples show how solar energy is converted into the heat and mechanical energy of the masses of air and water and how it is transferred to neighbouring regions of the environment. Similar circulations arise as a result of the varying insolation which different parts of the ocean receive. Furthermore, the cooling of surface water due to a radiation deficit leads to an increase in its density (see Chapter 3) and may disturb the vertical hydrostatic equilibrium. In this case, the surface water sinks, and this again brings about a circulation compensating the distribution of the water masses; in other words, ocean currents come into existence. So far, we have not yet said anything about how these circulations are complicated as a result of the Earth’s rotation, the inertia of the air and water masses, and the mutual thermodynamic interactions between parts of these masses.
1.1 THE EARTH AS A THERMODYNAMIC SYSTEM
I1
A global model of the mean atmospheric circulation (see e.g., Schelsinger and Gates, 1980) is shown diagrammatically in Fig. 1.1.7. Here we see in section the major three-stage circulation from the Equator to each of the poles, the main highs and lows of atmospheric pressure which form with the inflow of air, and the east-west lateral movement of the air masses caused by their inertia over the rotating Earth (see Section 1.2). Examples of oceanic circulation will be described in the next section. Water (see Chapter 2), and hence the seas and oceans, plays an especially important role in the transfer of energy around the globe. In the mass of the ocean, water, together with everything it contains, absorbs solar radiation almost like a black body (see Chapters 4 and 5). As a result of evaporation, it not only releases into the atmosphere gigantic quantities of latent heat contained in water vapour (high latent heat of evaporation), but also continuously supplies the land with moisture essential to life. On average, something of the order of l O I 4 t of water vapour evaporates each year from the surface of the oceans, that is, a layer of water about one metre thick is removed from the world’s oceans and their adjacent seas. a total surface area of some 361 million km’.
Fig. 1.1.7. A model of the global circulation of atmospheric air masses (based on the ideas given in Von Am, 1962 and Thurman, 1978). Regions of low and high atmospheric pressure and the polar fronts are indicated.
Precipitation, the run-off of water into the sea via rivers, cold deep-sea currents, cold northerly winds and the partial melting of polar ice complete this global circulation of water and transfer of heat in Nature which determine the Earth’s
12
1 A GENERAL PICTURE OF PHYSICAL PROCESSES IN THE OCEAN
climates. Water also carries dissolved salts (from the sea, soil and rocks) essential to life and plant growth. The ocean-atmosphere system of the Earth, unevenly supplied with radiant energy from the Sun, is therefore a kind of enormous heat engine in which, on a global scale, the heat source is the surplus of solar radiation in the hot zones, the cooler the insufficiently warmed polar regions, and the working substance the masses of air and water in the atmosphere and oceans. Ignoring for the moment how complex this engine is and to what extent it is thermodynamically ideal, we can say that its power is gigantic, and that its action is comparable with a reversible Carnot cycle; Fig. 1.1.8 illustrates this. Heat source
Working substance
Cooler
(a) Work done
Solar radiation Masses + of atmosphere ---+ surplus in hot zones and ocean
Radiation-deficient cold areas
Potential and kinetic energy of air and water masses
Fig. 1.1.8. A qualitative comparison of the mechanisms of water and heat circulation in nature (b) with a reversible Carnot cycle in a heat engine (a).
This simplified diagram helps us imagine how the chief mechanism of Nature works, in which all movement of the atmospheric and oceanic masses, and with it the circulation of water and the transfer of heat and chemical substances, results from the conversion of solar radiation energy into mechanical energy. This movement is usually accompanied by friction (already mentioned), which calls into existence turbulent vortices of various dimensions passing on their energy to the adjacent medium, and leads to the irreversible conversion of mechanical energy into heat. This movement could not therefore exist for long without the inflow of solar energy. In fact, without this inflow, most existing sea currents would expire within about three years.
1.2 FORCES INDUCING THE MOTION OF WATER MASSES IN THE OCEAN
13
Likewise, all movement and life processes of people and animals, which take place at the expense of chemical energy, could not continue for long if it were not for the replenishment of the stocks of this energy during the photosynthesis of organic matter, about half of which takes place in the oceans, in the cells of marine phytoplankton (see e.g., Steemann Nielsen, 1975). A by-product of photosynthesis is the regeneration (release) of oxygen in Nature (see Barber, 1977), consumed in all oxidation processes, including respiration, the large-scale combustion of chemical energy sources and the natural oxidation of huge amounts of organic waste matter. That the ocean absorbs carbon dioxide from the atmosphere and supplies it with free oxygen is as much a factor determining the conditions of life on Earth as the effect of the ocean on the Earth’s climate. So no matter how far from the sea we live, the ocean, with its global influence on the natural environment, also determines the living conditions in our local environments. Since this is so, pollution of the ocean is particularly dangerous, especially activities which lead to permanent, uncontrolled changes in the natural properties of the ocean and atmosphere. Such activities include the indirect or direct dumping of excessive quantities of wastes and sewage effluent from rivers or the atmosphere. This may limit the transparency of surface waters to light and their permeability to carbon dioxide and oxygen, restrict evaporation from the sea’s surface because of contamination by petroleum products, poison marine plankton that produce organic matter and free oxygen, and bring about the excessive consumption of oxygen (see Johnston, 1976; Goldberg, 1976; American Institute ..., 1978). The lack of free oxygen in seawater means that organic compounds oxidised by bacteria take the oxygen they require from sulphates (a component of sea saltsee Chapter 2). The product of such a reaction is hydrogen sulphide, which accumulates in the lower water layers and poisons all forms of life there. Such a situation already exists, e.g., in the Black Sea, particularly below a depth of 200 metres. In conclusion, the reader’s attention should be drawn to the complexity of the natural phenomena occurring in the sea and hence to the inter-disciplinary character of oceanology. Marine physics is just a part of this complex science of the marine environment and cannot develop properly in isolation from its other branches. 1.2 FORCES INDUCING THE MOTION OF WATER MASSES IN THE OCEAN
Let us now examine the globe from the point of view of mechanics, as a rotating solid almost spherical in shape. The surface of this sphere is largely covered with
14
1 A GENERAL PICTURE OF PHYSICAL PROCESSES IN THE OCEAN
a fluid mass of water having an average thickness of about 3.85 km. Occupying only some 29% of the total surface area of the Earth, the continents protrude like great islands above the enormous, roughly 360x lo6 km2 area of the world ocean. The fluid atmosphere that envelops the planet also has many physical features in common with the ocean (see e.g., Davydov et al., 1979; Eagleson, 1978). A multiplicity of forces acts ceaselessly on the constituents of this fluid mass of air and water to keep them in motion. The movement of water masses is therefore the predominant physical process in the ocean. The forces acting on the masses of oceanic water can generally be divided into external or primary forces, which initially set these masses in motion, and internal or secondary forces, which come into play once motion has been induced by the primary forces. These primary forces include gravitational forces (weight, tidal forces), wind stress at the sea surface, and forces due to the atmospheric pressure gradient. Among the secondary forces we have friction between masses of moving water, and the Coriolis force.
The Force of Gravity and its Components Every element of a mass of oceanic water m is first and foremost uplifted together with the Earth in its rotation around a circle of radius vector r, = RCOSQ), where ~1 is the geographical latitude, and R is the distance of the mass element
Fig. 1.2.1. The Earth as a solid rotating with an angular velocity o. The centripetal force F c acting ~ on mass rn in circular motion arises as a component of the gravitational force G. The other component is the apparent force of gravity (thrust) Ge.
1.2 FORCES INDUCING THE MOTION OF WATER MASSES IN THE OCEAN
15
from the centre of the Earth (see Fig. 1.2.1). The linear velocity vL of this rotatory motion is always one of the components of the resultant velocity of the motion of this water mass with respect to a motionless (inertial, extraterrestrial) system of coordinates xyz. The angular velocity of the Earth’s rotation is a vector w pointing north along the axis of rotation. The absolute value of the angular velocity of the Earth’s rotation is co = 2n/T = 7.29211 x rad s-l, where T ( z 24 h) is the period of the Earth’s rotation about its own axis. When the linear velocity of the circular motion of an element of mass is vL, then in accordance with known relationships, its centripetal acceleration is expressed by the vector product a, = w x vL = o x (oxrJ
(1.2.1)
while the centripetal force acting on that element of mass in its circular motion together with the rotating Earth is equal to
Fcp = ma,,, its value being F,, = mw2r = mco2Rcospl.
(1.2.2a) (1.2.2b)
This centripetal force Fcp,directed perpendicularly to the Earth’s axis of rotation, arises as one of the components of the Earth’s gravitational force G, and it is this force alone that causes the 1.45 x 1OI8 t of the fluid mass of the world ocean to remain in circular motion together with the Earth. The second component of the Earth’s gravitational pull is a force acting at right angles to the planet’s surface G, which can be called the apparent gravity force or the thrust due to gravity onto a substrate (see Fig. 1.2.2a). This resolution of the gravitational force into components is due to the slight flattening of the Earth’s sphere (see Dehlinger, 1978). The gravitational force acting on a mass m is thus resolved into the sum of vectors G = F,,+G,. The centripetal force F,, is, however, very small in comparison with the thrust force G,, i.e., F,, < G,. This is why the Earth’s force of gravity G, or at least its direction, is frequently identified with the thrust force G , % G, although this is not strictly accurate. We shall return to this distinction later in this chapter, when we shall look at the apparent acceleration due to gravity G,/m = g,-this differs marginally from the real acceleration due to gravity Gjm = g, which depends on geographical latitude and sea depth. In the mobile reference system connected with the Earth, the centrifugal force acts on a mass m on the Earth itself. This force is in fact the reaction of an
1 A GENERAL PICTURE OF PHYSICAL PROCESSES IN THE OCEAN
.T.w
-Ge
Buoyancy
Fig. 1.2.2. The components of forces acting on a mass m in the sea, in a system of reference coordinates rotating together with the Earth. (a) The force due to gravity 8 is balanced by the buoyancy -Go and the centrifugal force +FCJ = -Fcp; (b) the centrifugal force FCJ can be resolved into components: a normal one Fn and a component Ft tangential to the Earth's surface. Their description by the given equations that include the geographical latitude p is an approximate one because. the small angle between the directions of the forces G and G , has been neglected.
inert mass to the centripetal force F, = -Fcp. Also acting on this mass is the thrust of the substrate, which in the sea is the buoyancy - G,. These two forces acting on mass m are balanced out by the gravitational force G (Fig. 1.2.2a). The centrifugal force, equalised in this way, can be resolved into a horizontal component, tangential to the Earth's surface Ft z Fcfsinq, and a vertical one, normal to the Earth's surface, P,, z Fcfcosq; this is illustrated in Fig. 1.2.2b. Approximate equality signs are used in these equations because the angle q is the latitude only if the slight difference in direction between the forces G and G , is neglected. A force F divided by unit mass m expresses acceleration Flm [N/kg] 3 [m/s2]
1.2 FORCES INDUCING THE MOTION OF WATER MASSES IN THE OCEAN
17
where m is a mass. With respect to a continuous medium of ocean water, the forces acting on this mass are usually related to a unit volume of water F / V at a given point (x, y , z ) in the water body (where V is a volume). A force thus defined acting on a unit volume of water is expressed in N/m3 and is called a speciJic force (e.g. specific gravity) or a volume force. Where a gravitational force acting on a mass m is concerned, we can state that G = mg = eVg, where e = m/V is the density of that mass (specific mass). The volume force of gravity, in other words, the specific gravity f, = G/V, is given by the equation f, = eg. (1.2.3) As we can see, this force is proportional to the density of the water in the sea, and can therefore change within the body of the ocean and in time, just as the water density in the sea changes (see also Fig. 1.2.6 and Chapter 3). The gravitational forces of the Moon and the Sun act simultaneously on the masses of water in the ocean. Although the strength of these forces is barely a fraction of one per cent of that of the Earth's gravity, they are none the less real forces which cause ocean water to pile up and flow; these flows of seawater are known as tides (see Dehlinger, 1978). The tidal forces arising from the interaction of the Earth and the Moon only are illustrated diagrammatically in Fig. 1.2.3. They are the result of the gravitational attraction of the Moon and the centrifugal reaction of the mass of water induced by the Earth's rotation about the centre of mass of the Earth and the Moon. This centre of mass is situated within the Earth about 4600 km from its centre. The tidal forces induced by the Moon are modified by a similar though weaker action of the more distant Sun. Local variations and complexities in the whole system of tidal forces are due to the complicated motion of these bodies with respect to each other. The force of gravitational attraction F, of two masses M and m, whose centres are separated by a distance r = Iri, is defined by the law of gravitation
Mm (1.2.4a) r2 . the gravitational constant. The vector where y = 6.6720 x lo-" m3 kg-l s - ~is of this force is located on the line joining the centres of the two masses and points in the direction of mass M which is attracting mass m. We can therefore write
Fg = y-
F,
=
Mm y3-r. r
(1.2.4b)
However, every element of the water mass m is acted upon by the sum of gravitational forces derived from many elements of mass distributed in space, most
18
1 A GENERAL PICTURE OF PHYSICAL PROCESSES IN THE OCEAN
(a)
Direction of Mooil's
Earth
motion
T Moon
I Direction of Earth's motion
Earth I t )
/'
,$!:
. %
'./'"'
Direction
.. Direction
I of Earth's
..motion _--.-L---.-~
t
of Moon's motion
Moon ( f I '6
'
\
i, / /
\
(.\--/'
Earth
(t')
Fig. 1.2.3. How tides arise. (a) The tidal forces P are the result of the Moon's gravitational attraction Goand the centrifugal reaction CII?of the elements of mass rn in the relative rotation of the Earth about the centre ofmasses. The actual dimensions and distances are not shown t o scale. because the distance from the Moon to the Earth is roughly equal to 60 Earth radii. T h e components PI of tidal forces P, tangential to the Earth's surface, induce a horizontal flow of water m a s m and are greatest where a vertical line (the z axis) forms an angie of 45" with the line joining the centres of the Earth and Moon. The vectors of the forces are shown in some places on the outline of the Earth's section. The origin of tidal forces due to the action of the Sun can be explained in the same way; (b) two successive positions of the Earth and Moon at times f and t' > f during their motion around the centre of masses Cm, explaining why the centrifugal reaction CFr (see (a)) is identical at every point on the Earth. As it rotates about its axis, the Earth maintains its angular momentum, which means that its axis maintains a constant direction in space. As a result, every point on the Earth (ea. A, B, S) during its rotation about the centre of masses Cmdescribes a circle of radius equal to the distance of the centre of the Earth S from the centre of masses Cm Hence the centrifugal force acting on a mass m in the reference system connected with the Earth-Moon system is the same at every point on the Earth, and therefore also the same as at the centre of the Earth S where it is equal to Go (the force with which the Moon attracts a mass m placed at S). An analogous anticlockwisemovement of the hand placed flat on a table with the fingers always pointing in the same direction helps us to understand this rotary motion of the Earth.
.
1.2 FORCES INDUCING THE MOTION OF WATER MASSES IN THE OCEAN
19
of them being proximate elements of the Earth‘s mass. This resultant sum of gravitational forces acting on a unit of free water mass imparts to it an acceleration which we call gravitational acceleration. We can assume to a good approximation that every element of water mass m on the Earth’s surface is situated at a distance of radius R from the centre of the Earth. Then, according to the law of gravity (1.2.4), the resultant force of attraction of this element m by the mass of the MI?? Earth M is equal to Fg = y R and so the acceleration due to gravity on the R3 Earth FJm is equal to I _
M g = Y--R
R3
(1.2.5a)
and its value is M
s=rF.
(1.2.5b)
We have assumed here that the Earth is a sphere of radius R whose vector is taken from the mass m to the centre of the Earth. By substituting in this equation the value of the gravitational constant given earlier, the mean radius of the Earth R z 6.371 x lo6 m and the mass of the Earth M = 5.98 x kg, we get the mean value of the acceleration due to the Earth‘s gravity at the Earth’s surface g = 9.83 m/s2. In more accurate calculations, the normal component of the centrifugal acceleration Fll z
m
02Rcos2y,
(1.2.6)
dependent on the latitude rp, is subtracted from this acceleration, as can be seen in Fig. 1.2.2b. We have neglected the insignificant deviation in the direction of the force F, from that of the radius R, that is, in the directions of forces G and G,. The value of the apparent acceleration due to gravity g, at the Earth‘s surface is thus the difference g, = g-dRcoS2p
(1.2.7)
and its direction is practically the same as the direction of the Earth’s radius. At a depth z in the ocean, the distance of an element of water mass m from the centre of the Earth is R -z. The apparent mass of the Earth attracting this element towards the centre is thus diminished by a certain value dM, since part of this mass remains on the outside, nearer the Earth’s surface than the given element.
1 A GENERAL PICTURE OF PHYSICAL PROCESSES IN THE OCEAN
20
Using (1.2.5a) we can state that the acceleration due to the Earth's gravity at depth z in the ocean is approximately equal to (1.2.8)
The apparent acceleration g,(z) at depth z will again be lower than g(z) by the value of the normal component of the centrifugal acceleration at depth z equal to w2(R-z)cos2p As z << R, we can in practice assume that the latter is independent of z. The apparent acceleration due to gravity g,(z) at depth z in the ocean is therefore also described by (1.2.7) after having taken the value of g(z> from (1.2.8) into consideration. The difference g-g, is small and we shall neglect it in later chapters, because in the processes discussed there it is of no significance. Geopotential
The apparent acceleration due to gravity g,(z) is usually linked with the concept of geopotential and geopotential surfaces in the sea, denoted by horizontal directions. Let us imagine an enormous volume of ocean several kilometres deep and the elements of the water mass within it. If such an element of mass is to rise to the surface of the sea, work must be done to counteract the force of gravity. The geopotential @,(z) is just such a measure of the work that has to be done against the apparent force of gravity in order to transfer a unit mass of water from depth z to the mean (equalized) surface of the sea. Moving the unit mass through an infinitesimally small distance dz against the apparent force of gravity mg, (where m = 1 kg) requires that dQa = -g,dz
(1.2.9)
joules per kilogram [J/kg] of work be done. The z axis here points vertically downwards, as does g,, and the minus sign indicates that movement is taking place in the direction opposite to that of the z axis. The work described by this equation is equal to an increase in the potential energy of the unit mass in the apparent terrestrial gravitational field as a result of shifting this mass vertically by dz. The movement of unit mass from depth z to the surface of the sea (z = 0) thus corresponds to a change in its potential energy which will be the integral of the increments (1.2.9) across the depth interval from 0 to z, that is (1.210)
1.2 FORCES INDUCING THE MOTION OF WATER MASSES IN THE OCEAN
21
This function @,(z) is a measure of the potential energy of unit elements of mass at depth z relative to the calm surface of the sea and is called thepotential of the apparent terrestrialJield of gravity, or in short, the geopotential. We assume that the geopotential on the free, equalized surface of the sea is zero, i.e. Qa(2 =
(1.2.10a)
0) = 0.
Surfaces defined in the sea on which the value of the geopotential @a is the same at all points are known as geopotential surfaces @,, = const. These geopotential surfaces are locally referred to as horizontal surfaces. On a global scale they are curved like the surface of the Earth but are always perpendicular to the vector g,, and as the value of this alters, so do the distances between them, in accordance with (1.2. lo), (1.2.7) and (1.2.8). Hence the geopotential surfaces in the vicinity of the poles lie closer to one another than near the Equator. This is because in high latitudes, where g, is greater (1.2.7), a smaller vertical shift of a mass is enough to produce the same change in its potential energy. Pressure, Buoyancy Every element of mass or volume of water in the sea is subjected to many forces of a nature different from that of gravitation. These include the pressure forces of adjacent masses resulting from atmospheric and hydrostatic pressure (their source is also gravity acting on the surrounding elements of mass), the forces of pressure due to the inertia of moving masses (dynamic pressure), the frictional forces of moving masses which are contiguous with a given element
zl
'
ap F ( z i d ~ l = [ / A ~ ) +dzIdx dy
az
Fig. 1.2.4. A geometrical sketch explaining how a force results from a pressure gradient.
1 A GENERAL PICTURE OF PHYSICAL PROCESSES IN THE OCEAN
22
of mass m (wind, sea current), and the Coriolis inertial force on the rotating Earth. The resultant force due to external pressure (atmospheric, hydrostatic and others) acting on an element of volume of a fluid comes into existence as a result of the gradient of this pressure across the space occupied by that element. This is illustrated diagrammatically by Fig. 1.2.4, in which one of the walls of the volume element of water dxdydz = dV, lying, say, in the z plane, is acted upon by an external pressure p(z), and another wall, lying in the zi-dz plane, is acted upon by a slightly greater pressure p(z) (Jp/az)dz. The resultant force acting on that element of volume dV parallel to the z axis is therefore equal to
+
8 ’ --dV.
az
(1.2.11) Since the body of water in the sea is a continuous medium, we again refer this force to unit volume of water dFJdV = ,fi, so fz=
- -aP az
(1.2.12)
The expression applaz is, in general one of the three components of the pressure gradient along the axes of the coordinate system. The resultant of all three components of this pressure gradient is thus equal to their vector sum, which gives an expression of the resultant force [N/m3] acting on a unit volume of the water mass ( 1.2.13)
where V is the nabla operator (of the gradient), i, j, k, are the unit vectors directed along the x, y, z axes respectively; the Cartesian system of coordinates is assumed to be dextrorotatory, with the z axis pointing vertically downwards. The pressure p(z) which exists in the sea at every depth z comprises mainly the sum of the atmospheric pressure p a acting on the surface of the sea, and the hydrostatic pressure. This, when a liquid is at rest in equilibrium, is equal to the ratio of the weight of the water column, measured from the actual surface of the sea to depth z, to the cross-sectional area of the column. The weight of a water
1.2 FORCES INDUCING THE MOTION OF WATER MASSES IN THE OCEAN
column of cross-sectional area dxdy and depth-dependent density is equal to
23
e = e(z)
z
(1.2.14)
G, = dxdySeg,dz. 0
The pressure p(z) [N/m2] can therefore be expressed by the equation (1.2.15 ) The volume force fi [N/m3] induced by this pressure is, according to (1.2.13) and (1.2.15), equal to fi
=
--aP
az
= -eg
(1’
(1.2.16)
Obviously, this is the buoyancy, since it is the upthrust to which each element of unit volume of water is subjected and, in accordance with Archimedes’principle, is equal to the weight of water displaced by that element. Vertical Stability
The density of a given volume element of water e‘ at depth z does not always have to equal that of the surrounding water e(z) in equilibrium at the same depth. When these densities are not equal, the weight of the given element differs from the buoyancy: e’g,-eg, # 0. The difference, that is, the resultant of these two oppositely directed forces, gives our element of water a vertical acceleration. This element is then not in hydrostatic equilibrium in the water column at depth z. The difference between densities e’ and e(z) in the sea may have several causes, e.g., a local cooling of surface waters (see thermal expansion in Chapter 3), or an intrusion of water as a result of other forces. It is therefore essential to establish the state of stability of the mass elements in the water column as this gives some indication of the equilibrium of the distribution of water masses in the column, or of their vertical mixing together with resources of heat and chemical substances. In the sea it is the vertical distribution of water density e(z) which determines the hydrostatic equilibrium and its degree of stability. The specific buoyancy and the specific gravity are products of the density e and the acceleration due to gravity g,; over a short distance, the latter can be regarded as constant.
1 A GENERAL PICTURE OF PHYSICAL PROCESSES IN THE OCEAN
24
Most often, the water density e(z) increases with depth in the sea, because the stable equilibrium of the water column (minimum potential energy) requires just this. Such an arrangement of the water masses is usually favoured by the chief factors influencing density, i.e. the temperature of warmed waters, almost always at the surface, and the pressure tending to compress the water at great depths (see the thermal expansion and the compressibility of seawater in Chapter 3). Temperature has the greatest effect on the density of sea water, but other influential factors are salinity and pressure. In low and (in summer) middle latitudes, the temperature of the warmed surface waters is 295-305 K. The usual vertical temperature distributions in oceanic water in these latitudes are typified by a very small temperature drop over the first 100 metres or so resulting from the turbulent mixing of the waters by wind and wave action at the surface (Fig. 1.2.5a). Below this upper layer, often assumed to be isothermal in marine models (T(z) = const, aT/az = 0), the temperature falls quite sharply with depth (see Chapter 7; Kraus, 1977). The water layer in which aT(z)/az < 0 is known as the thermocline. Deeper down, below 800-1000 metres in the tropical zone but nearer the surface in higher latitudes, the water temperature is once again practically constant, though low, close to the temperature at which the seawater density is at its greatest (see Chapter 3). As we approach the cold zones, the thermocline gradually shrinks and finally disappears as the
Temperature 7
(b)
["Cl
Salinity S
34.0
3&-5 350
355
360
I%ol
,365
370
100 L
200 -
300 -
400 c
-1I 500
E 500-
I
r
603-
% 700-
0
900
1x01-
1
Fig. 1.2.5. Typical vertical distributions of temperature and salinity in the ocean at low latitudes, with a visible thermocline where aT/az < 0 (a) and halocline where as/& < 0 (b). These graphs illustrate the results of measurements in successive stages of research. The data have been selected from a large number of measurements made from the research ship "Lomonosov" in the tropical Atlantic as part of the GARP programme-from Diiing ef al., 1980.
1.2 FORCES INDUCING THE MOTION OF WATER MASSES IN THE OCEAN
25
surface waters become cooled, so that in polar seas, the water temperature from the surface to the bottom varies only slightly around the 273 K (0°C) mark. This, however, is only a very general outline of temperature distributions in the ocean; they are subject to many slight spatial and temporal modifications as a result of the thermodynamic processes taking place in the sea (see Chapters 3 and 7). The salt concentration, which also affects the density of sea water, varies roughly from 33 to 37 g per kg of seawater. In much of the world ocean, the salinity is around 35 g/kg, that is, 35 units in the water salinity scale; it is a little lower in polar seas (salinity is defined in Chapter 2). The vertical distribution of the salinity S is also complicated. In the tropical zone (Fig. 1.2.5b), the salinity of the upper layer is higher due to intensive evaporation from these surface waters, but below this layer it falls rapidly aS/dz < 0-this region is called the halocline. In actual fact, however, this vertical distribution of salinity S(z) is dependent on a number of processes such as water mixing, geochemical and biological processes. As it is mainly temperature, besides salinity and pressure, that determines the density of water in the sea, the density e(z) at depths approximating to those of the thermocline-the pycnocline-also increases sharply. The vertical distribution of density in the tropical Atlantic and its variation over a period of 20 days is shown in Fig. 1.2.6. The density “jump”, i.e., the pycnocline, in which dp(z)/dz 9 0, is where the isopycnals (lines of equal density) are bunched
-
h
--27.0
-
\
-
--------------
1000 Day31 /
2
1 ‘
5
3 4 “
\
-b.12
-21.4-
“
7
6 ‘
8 ~
9 10 11 12 13 14 15 16 17 18 I9 ~
‘
‘
‘
~
i
~
‘
~
1
September
Fig. 1.2.6. The vertical distribution of the abbreviated density uI of water in the tropical Atlantic during 20 days in September (from measurements by the ship “Musson” during the GARP programme-see Diiing et al., 1980). The abbreviated density UT is defined by equation (3.5.16) in Chapter 3. The bunching of pycnocline, in which de/dz > 0.
UT
values indicates the
’
~
1 A GENERAL PICTURE OF PHYSICAL PROCESSES IN THE OCEAN
26
together, from depths of c. 50-100 m. The vertical distribution of water density in the sea is thus a complex function of the depth e(z), and its vertical differentiation (stratification) is characterised by the vertical density gradient de/dz which, as we shall see later on, determines the hydrostatic equilibrium of the water column. If a volume element of water is in equlibrium with its surroundings at depth z, both this element and its surroundings must have the same density e(z), and its apparent specific gravity g,e(z) is equal to the buoyancy -g,e(z) acting on it in the opposite direction. Let us assume that this element has been displaced from its equilibrium position at depth z to a new position, say, a distance dz below z, where the hydrostatic iressure is greater and the density of the surrounding medium is higher. In this new position, its specific gravity will have risen somewhat as a result of adiabatic compression caused by the increased pressure
[
equal to g, e(z)+
(t).1. dz
~
(31, (%),,
and is therefore
At the same time, the buoyancy acting on its
unit volume has increased, since in a medium of density force is equal to -g,
[
21.
e(z)+-dz
The algebraic sum of these forces gives
~
us a resultant force acting on our element equal to g, and on dividing it by the mass of our unit volume of water, i.e. by the density e(z), we obtain the acceleration along the z axis, that is, d2z/dt2 imparted to the
element by this force (1.2.17) As can be seen from this equation, the sign, and therefore the direction of this acceleration, depsnds on whether the vertical density gradient of the water de/dz is greater or smaller than the “adiabatic compression gradient” (dp/dz), . When de/dz > (dp/dz),, the acceleration always acts in the direction which will restore the original stratification at depth z. A displacement dz > 0 implies a downward deviation, so, according to (1.2.17), the acceleration has a minus sign; in other words, it is directed upwards. If, on the other hand, the displacement from equilibrium is in the upward direction, dz < 0, and the acceleration d2z/dt2 takes
1.2 FORCES INDUCING THE MOTION OF WATER MASSES IN THE OCEAN
27
a plus sign, i.e. is directed downwards. The stratification is therefore stable, since any imbalance is immediately cancelled out by forces restoring the equilibrium. When de/dz = (de/dz),, we see from (1.2.17) that the acceleration d2z/dt2 is zero, hence the equilibrium is neutral, as it exists both at depth z and at depth z+dz. A similar comparison of buoyancy and weight when de/dz < (de/dz), leads us to conclude that every fresh displacement will still further deepen the imbalance of forces and cause convectional mixing of the waters; the water column is therefore unstable. The conditions for vertical hydrostatic stability in the water column are thus defined by the relationships (1.2.1 Sa)
neutral
$
=
($)4,
unstable dz <
($)a.
(1.2.18b) (1.2.18~)
It should be remebered that each of these conditions can occur only within a certain depth interval and not throughout the whole water column. The condition for stable stratification (1.2.18a) is usually fulfilled in the thermocline, but the other two conditions may obtain near the surface, especially in winter or in polar seas. In (1.2.17) it is convenient to use the symbol E, to represent the expression (1.2.19)
which, like the conditions (1.2.18), defines the vertical stability [m-'1 of the water column (the minus sign is due to the chosen direction of the z-axis, here oriented downwards). Since e is always greater than zero, the stratifications (1.2.18) defined by this function E, will be E, > 0,
(1.2.2Oa)
neutral E, = 0,
(1.2.20b)
unstable E, < 0.
(1.2.20c)
stable
This is why the function E,(z) is called the function of the hydrostatic vertical stability of the water column and is universally applied in oceanology to define the state and degree of stability of water masses at various depths of a given region. However, it is written down in such a way that one can apply density-
1 A GENERAL PICTURE OF PHYSICAL PROCESSES IN THE OCEAN
28
dependent parameters like temperature T, salinity S and pressure p of the water, which are directly measurable in the sea. Assuming that when a volume element of water is adiabatically expanded or compressed, only its temperature and pressure change (its salinity hardly at all), we can write (1.2.21) On the other hand, the density gradient in the column is simultaneously dependent on all three parameters (see Chapter 3)) that is, ~
de dz
ae dp i3p dz
= .---+--+
dT i3T dz
ae dS
as
dz
(1.2.22) ’
Substituting these expressions (Doronin, 1978) in (1.2.19) and performing a simple rearrangement, we obtain the vertical stability function in the form (1.2.23) which in oceanology is known as the Hesselberg-Sverdrup function (1915). The vertical distribution of the magnitude of this function in the sea E,,(z) shows a sharp peak Ee > 0 in the depth interval where the density gradient de/dz % (defdz),. This occurs in the pycnocline where, in accordance with condition (1.2.18a), the vertical hydrostatic stratification is extremely stable. The pycnocline therefore prevents vertical mixing and separates layers of water of different density. Apart from the main layers of water separated by the pycnocline shown on Fig. 1.2.6, the water column in the sea is often composed of a large number of thinner layers about 1 metre thick. These make up the fine structure of oceanic water masses which, when examined in detail, are found to reflect the vertical distribution of the Hesselberg-Sverdrup function EJz) (see Fedorov, 1976 ; Druet and Siwecki, 1980). Enhanced stability E,(z) at the boundaries of water layers is prevented by convection and by the vertical, turbulent8exchange of salts and heat in the sea (see Chapters 6 and 7). Woods (1971, 1977), and Woods and Wiley (1972) have suggested a model of the formation of the fine-layer structure of ocean waters. There is also a distinct bilamellarity and density “jump” in the water of river estuaries (Nihoul, 1978) where low-density and often warm river water flows over the top of the much denser, cold, salt water of the sea. An example of the stability function E,(z) graph for a river estuary is given in Fig. 1.2.7; the sharp
1.2 FORCES INDUCING THE MOTION OF WATER MASSES
Temperature T
Salinity
OC
1
IN THE OCEAN
29
Stability function €, I10-5m?I
s “?60l
Fig. 1.2.7. The vertical distributions of temperature T, salinity S and vertical stability function
E9 near the mouth of the river Vistula. The maximum value of EQdenotes the boundary between waters of differing densities-between river and seawater (after Dera, 1965).
peak of this function at the river-seawater boundary is quite plain. In depth intervals where EJz) < 0, intensive vertical exchange of waters should be expected (see turbulent exchange, Chapter 6). The Vaisala-Brunt Oscillation Frequency When the stratification is stable, the acceleration described by (1.2.17) rises along with the upward or downward displacement of an element of the water mass from its equilibrium position, but diminishes to zero at the equilibrium position. The force restoring that element to the equilibrium position thus increases with the displacement, and it may be compared to the directing force in oscillating motion, for example, in the harmonic motion of a sphere suspended from a helical spring. Like this sphere, an element of water mass displaced from the equilibrium position attains its greatest momentum at the instants it returns to this position; however, it does not stop there but oscillates about it. The continuity of the aqueous medium means that horizontal components of motion are also induced, so that the movements of elements of the medium are in fact orbital (elliptical, circular). This complex oscillatory motion of water volume elements about an equilibrium position is, in the sea, the basis of the mechanism of internal waves (and also of surafce waves) originating at the boundary between water layers of different density (see Le Blond and Mysak, 1978). This is particularly common in the region of the pycnocline, where dpldz $ (de/dz), . The frequency of these oscillations must emerge from (1.2.17), though not absolutely accurately, because this equation takes no account of the damping of the oscillation by the water.
1 A GENERAL PICTURE OF PHYSICAL PROCESSES IN THE OCEAN
30
If we substitute the function E,, expressed by (1.2.19) or (1.2.23), in (1.2.171, we get a simple form of the equation for the acceleration of a volume element of water displaced from its equilibrium position (1.2.24) This is the equation for harmonic vibratory motion in which the acceleration is proportional to the displacement dz. Its solution is thus a harmonic function describing the oscillation of a given volume element of the water. It will be seen from this solution that the circular frequency of these vibrations is equal to (1.2.25) This frequency is an important parameter of the stratification of water in the sea and is known as the VuisuZafreguency(Vaisala, 1925; Eckart, 1960) or the VaisalaBrunt frequency (Druet and Kowalik, 1970; Le Blond and Mysak, 1978). By transferring Ee from (1.2.19) to (1.2.25) when the stratification is stable (E, > 0), we obtain the Vaisala frequency in the form (1.2.26) To rewrite this equation we can use the equation for pressure (1.2.15), and (8.0.14) (from Chapter 8) which links adiabatic changes in pressure and density with the velocity of sound c in a given medium. If we assume that the adiabatic change in temperature in the oscillating region is negligibly small (see Chapter 3), we get
Ha 2 @ ’
__
from (1.2.21). On the basis of (1.2.15) and (8.0.14),
this new expression is moreover equal to
1 C
which, after substitution in
(1.2.26), allows us to write the Vaisala frequency in a practically useful form (1.2.27) However, depending on the approximation we have applied to the density changes in the medium, the definitions of E, and N, are subject to fine differences (see e.g., Le Blond and Mysak, 1978 ; Monin, 1978-the Boussinesq approximation). The above approximation takes account of the compressibility of water (see Chapter 3).
1.2 FORCES INDUCING THE MOTION OF WATER MASSES IN THE OCEAN
31
The density gradient in the sea is a function of the depth, therefore the Vaisala frequency Np also changes with depth, and in accordance with (1.2.27) reaches a maximum in the pycnocline where the gradient dp/dz is the greatest. In summer in the ocean this maximum is of the order of N,,,, % lo-' s-l, which corresponds to an oscillation period T = 2x/N, of about 10 minutes. The lowest Vaisala frequencies that one comes across in the ocean are of the order of 10-3-10-4 s-l, which are equivalent to oscillation periods of from 1.7 to 17 hours (Monin, 1978). These values give us some idea of the period of internal waves in the sea. When the oscillation frequency exceeds the Vaisala frequency, the oscillatory (wav?) motion becomes turbulent motion (see Chapter 6 , Section 2). The reader w31 find a detailed theoretical description of internal and surface waves in the sea in Cherkesov (1976) and Le Blond and Mysak (1978).
The Inclination of Isobaric Surfaces
So far we have been quietly assuming that density and pressure are merely functions of the depth z, and that in the horizontal x y planes they are constant. Such a simplifying assumption (also applied to other functions describing the properties of the sea) is an approximation of the description of the sea called the horizontal layer model or one-dimensional model of the sea. Such an approximation considerably simplifies the equations describing various phenomena in the sea, because the functions in these equations, generally dependent on three variable coordinates, simplify to the function of one variable: p(x, y , z ) = &), p ( x , y , z) = p(z), etc. Hence, the horizontal components of the gradients of these functions are assumed to be zero: &/ax = 2 ~ / a y= 0 and ap/ax = appl2y = 0. This approximation is quite widely applied and is close to the actual state of affairs over not too large an area of the sea. Assuming this with regard to pressure, isobaric surfaces are horizontal, and so parallel to geopotential surfaces which determine the levels. The pressure gradient is then perpendicular to the geopotential surfaces. In reality, however, pressure in the sea is spatially differentiated; it is the functionp = p(x, y , 2). This springs from the differences in atmospheric pressure above various areas of the sea, from Iocal piling up of water above the mean sea level by wind or tidal forces, and from local differences in the temperature and salinity of the water which, in turn, affect its specific gravity (see Chapter 3). As we can see, then, there are many reasons why the combined weight of a water column and the atmospheric column above it exerting a pressure on the medium at depth z is different in differentregions of the sea. The pressure gradient therefore
1 A GENERAL PICTURE OF PHYSICAL PROCESSES IN THE OCEAN
32
has all three components different from zero and, being a vector, is not directed vertically downwards but along the resultant of these three components iaplax, jap/ay,kap/&. The isobaric surfaces are thus inclined at an angle CI (and not parallel) to the geopotential surfaces (Fig. 1.2.8). In such a real situation, we can
Isobaric surface,
4
const ' "
;<(xl,Ol
Geopotential surface
u:
*qj,(Xl,Z) Olez=const
6X
vz
Oe0:const
i p,=const
k 2 dx
'i
Fig. 1.2.8. The inclination of isobaric surfaces p = const at an angle a to the geopotential sur-
faces
@a
=
const. This inclination gives rise to resultant horizontal forces
= - @gatanax; f, =
8P
- --
= -@gotancc,,
fx
= -
aP ~
ax
where a, and a,, are the angles between these sur-
aY
faces in the xz and y z planes, respectively.
see from Fig. 1.2.8 that the pressure at depth z, e.g., at point (xo,z ) , is less than the pressure at the same depth but at point (xl,2). Being a scalar, the pressure acts equally in all directions, so in this case it also exerts a horizontal thrust on the water masses which is different at these two points. The horizontal component of the volume force acting in the x-direction is proportional to the component of the pressure gradient in this direction, as emerges from (1.2.13): (1.2.28)
This force can also be simply described with the aid of the angle at which the isobaric surface is inclined to the geopotential surface in the xz plane. Using (1.2.15) we can express the pressure increment by the equation Sp = pga6z and by dividing this by the increment ax, corresponding to a pressure increment of 8p in the horizontal, we get the expression 8p/6x = Qga6z/Bx= egotanax. On differentiating we get
aP -_ ax
-
-pegatanax
= fx.
(1.2.29)
1.2 FORCES INDUCING THE MOTION OF WATER MASSES I N THE OCEAN
33
Remembering that the pressure is a function of the other two coordinates x and y as well, we can write analogous relationships for the directions of these coordinates, e.g. in the y-direction the component of the volume force is (1.2.30) where cly is the angle which the isobaric surfaces make with the geopotential surfaces in the yz plane. Forces ,fx and ,& give the water masses horizontal accelerations not counteracted by the Earth’s gravity, which acts exactly along the z-axis. These masses are therefore induced to flow as gradient marine currents.
The ‘4ction of the Coriolis Force
A mass of water induced to flow across the rotating Earth is at once subject to the so-called Coriolis effect, which deflects the current from its original direction. Regardless of this current, the volume elements of the flowing mass of water have a component linear velocity taking them towards the east together with the Earth‘s rotation equal to vL = o r , (where co is the angular velocity of the Earth‘s rotation and rp,is the radius vector of the given mass in its original position at latitude qj and depth z in the sea). If now a certain mass initially rotating with the Earth at position r+,, shifts with the current to a point rp,, < r,, nearer the Earth’s axis of rotation (towards a pole or deeper into the sea), its retained original eastward velocity vLl = ow,, becomes greater than the eastward velocity vL2 = of the elements (waters, islands) surrounding it in its new position. Moving eastwards, this mass therefore overtakes the elements of its new environment, in other words, the current is deflected to the east with respect to, say, continents and islands, which are fixed to the Earth. In the opposite, westward direction, that mass of water will be deflected which is moving away from the Earth’s axis with an eastward velocity less than that of the new surroundings (see Fig. 1.2.9). This change in direction of the flow velocity vector v with respect to the Earth during the flow defines the new, apparent acceleration a, = ( a v / d t ) , which the mass is subjected to as a result of inertia. This is the Coriolis acceleration which is actually experienced in the Earth‘s coordinate system. A special case of the Coriolis acceleration arises when a mass m flows along some line of latitude with a velocity v y with respect to the Earth. In this case, it is not the radius vector r,’ which changes but the resultant angular velocity of the rotation of this mass. Instead of being u) = vL/rv, this velocity is co’
34
1 A GENERAL PICTURE OF PHYSICAL PROCESSES IN THE OCEAN
= (vL+zlY)/rp, where the plus sign is used with eastward motion (increase in angular velocity) and the minus sign with westward motion. This is why the centrifugal reaction of this mass changes by a value of mr,(d2 -02)and so the balance between centrifugal reaction, the reaction of the substrate to the buoyancy
Fig. 1.2.9. The Coriolis effect on the rotating Earth: deflection of a flow to the right in the northern hemisphere and to the left in the southern hemisphere (see text). (a) Deflection of a flow initiated in a longitudinal direction; @) deflection of a flow initiated in a latitudinal direction (c) vector components of the angular velocity of the Earth's rotation-tangent o. and normal o . with oy = 0.
1.2 FORCES INDUCING THE MOTION OF WATER MASSES IN THE OCEAN
35
effect and the weight is disturbed; this is shown in Fig. 1.2.2a. This unequalized addition to the centrifugal reaction (or loss with westward flow) imparts an acceleration, the Coriolis acceleration, to the flowing mass which can be resolved into components: a horizontal one, causing a latitudinal flow to be deflected towards the equator (if flow is eastward), and a vertical one which is added to the acceleration due to gravity and which deflects the flow along the z-axis closer to the surface or deeper into the sea (if flow is eastward) (see Fig. 1.2.2b). Deflection towards the Equator and the surface of the sea thus occurs when the flow velocity is to the east, i.e., when the centrifugal reaction of the flowing mass of water increases. Poleward and bottomward deflection will take place when the flow starts off in a westerly direction. The velocity with which water flows in the sea with respect to the Earth is in general a vector in three-dimensional space v(x, y, 2). Using this vector and the angular velocity vector of the Earth’s rotation w, the Coriolis acceleration can be expressed by the vector product (1.2.3 1) the reasoning behind which can be found in physics textbooks (see e.g., Kittel et al., 1973; Piekara, 1961; Le MChautC, 1976). By multiplying this acceleration by the mass m we obviously get the force acting on this mass in the Earth’s coordinate system, whereas multiplying it by the density of the water e yields the volume force [N/m3]. The properties of the vector product (1.2.31) enable us to deduce that the Coriolis acceleration is always directed perpendicularly to the flow velocity vector v and the angular velocity vector of the Earth’s rotation w; in other words, it acts perpendicularly to the plane delimited by these two vectors. It is also clear that only a flow whose velocity is directed parallel to the Earth‘s axis of rotation, that is, to the direction of vector w, is not subject to the Coriolis effect, since the value of la,/ = 2mvsina = 0, when the angle a between these two vectors is zero. It is convenient to resolve the flow velocity vector v with respect to the Earth into component flow velocities in the directions OF the coordinate axes and to express the components of the Coriolis acceleration, or the components of the Coriolis volume force, acting in these three directions. So by pointing the x-axis of the Earth’s system of coordinates to the north, the y-axis to the east and the z-axis vertically downwards along the Earth’s radius R, we obtain a dextrorotatory system of coordinates (Fig. 1.2.9~). The components of the velocity v will be v, = U, v, = v, v, = w. Likewise, we resolve the angular velocity w into compo-
1 A GENERAL PICTURE OF PHYSICAL PROCESSES IN THE OCEAN
36
nents oz= ocosp), o, = osiny, where p) is the latitude, and oy= 0 since vector w lies in the xz plane (see Fig. 1.2.9~).If vector v has components (u, v, w), and vector w has components (cox, 0, uZ), the respective components of the vector product (1.2.31) expressing the components of the Coriolis acceleration along the x,y, z axes are (1.2.32) Expanding these determinants and multiplying them by the density of water we get the Coriolis volume forces acting in the three component directions of motion
fCx = -2gwvsinp), fEY
=
-2p(wco~p)-usin~),
fc. = -2Qwvcosy.
(1.2.33a) (1.2.33b) (1.2.33~)
Hence the components of the Coriolis force acting in the x and z directions (along a meridian and along the vertical axis) are dependent only on the E-W components of the flow velocity v. They are moreover proportional to this velocity v which, as we have said, changes the centrifugal reaction of the moving mass. The force component along a meridianf,, increases with latitude, i.e., it is greatest at the poles, whereas the force component acting along the verticalf,, decreases with increasing latitude. The Coriolis force component acting in the y-direction (to the east and west) depends on the two components of the flow velocity u and w in the x and z directions. If we ignore the often very small vertical velocity w z 0, we see that the Coriolis volume force acting to the east or west (along the y-axis) is f,, z 2~wusinp).
(1.2.34)
It is thus proportional to the flow velocity u along a meridian and is greatest at the poles. The expression 2osinp is often called the Coriolis parameter. The direction in which the initiated motion is deflected can be analysed on the basis of the sign of each of the components of the Coriolis force (1.2.33). This sign tells us whether it is acting in accordance with (+) or in opposition to (-) the sense of the appropriate axis of the coordinate system. In analysing this sign we must take into consideration the signs of the components u, v, w ("-" when they are directed oppositely to the coordinate axes) and also the fact that, unlike the latitude in the northern hemisphere, the latitude p in the southern hemisphere must have a minus sign; on the other hand, o is always positive.
1.2 FORCES INDUCING THE MOTION OF WATER MASSES IN THE OCEAN
37
Frictional Forces Any motion of a mass within a body of water always gives rise to yet another force, namely, friction. It counteracts that motion and often balances out the forces causing it so that flow becomes approximately uniform. Friction arises between neighbouring masses of a medium whose velocities are different. In the sea, friction, usually composed of molecular friction (viscosity) and so-called turbulent friction (“turbulent viscosity”), results from the exchange of momentum between masses of water flowing past one another and induces more and more of thein to change their velocity. Finally, as a result of the exchange and scattering of kinetic energy, it is converted into heat. Molecular friction arises out of the mutual transfer of the flow momentum of single water molecules resulting from their thermal motion. On the other hand, turbulent friction arises out of the mutual transfer of the flow momentum of larger volumes of water resulting from pulsations of their flow velocities (vortices). These complex and extremely important natural processes are examined in detail in Chapter 6. A frictional force acts along two contiguous layers of medium moving relative to each other. The force is parallel to the flow velocity vector v of a layer and acts in the opposite direction to it. To a first approximation, this force is proportional to the difference between the flow velocities of neighbouring layers. However, in a continuous medium it is difficult to distinguish between layers flowing past one another with different velocities. The difference in flow velocities of the touching masses is therefore described by the gradient of this velocity dv/dn, where dn denotes an infinitesimally small vector of shift perpendicular to the flow velocity v. Friction acts on rubbing surfaces, so it is expressed in units of force per area [N/m2]. The reaction to friction is the stress arising at the friction surface directed against the frictional force and is called the tangential tension z. To a first approximation, the tangential tension in molecular friction is proportional to the flow velocity gradient, i.e. z = q-
dv
dn’
(1.2.35)
where q is the coeficient of dynamic molecular viscosity. This relationship is called Newton’s LW (see Section 6.1). By analogy with molecular friction, the same proportionality is hypothetically assumed in the case of turbulent friction. But instead of the coefficient q, we use another, much larger coefficient, Kc”), called the coeficient of turbulent viscosity. In general, however, this latter coefficient
1 A GENERAL PICTURE OF PHYSICAL PROCESSES IN THE OCEAN
38
depends on direction in the water body, and hence is different for each component of the velocity of motion. In a fixed system of coordinates xyz it forms the tensor of coefficients of turbulent viscosity K:;).The stresses induced in various directions by turbulent friction also form a tensor of stresses t i j ,called the Reynolds stresses (see Section 6.2, equation (6.2.31)). For turbulent friction, we have to write down the assumed proportionality, analogous to (1.2.35), for the tension components, which in generalized notation can be written down as follows: (1.2.36) where t i jrepresents the components of the Reynolds stresses, Ui the components of the mean flow velocity, and xj the coordinates; i = 1,2, 3, j = 1,2, 3 (see Chapter 6). The components of a frictional force related to unit volume of medium flowing with a spatially variable mean velocity v(x, y , z), having components along the coordinates U i , are, as will be further demonstrated in Chapter 6, the following : (I .2.37)
For molecular friction, in which the coefficient of viscosity q is constant, this expression simplifies and can be written in vector form f,, = qV2v [N/m3], (1.2.38) as will be shown in Section 6.1. The volume force of viscosity, or specific viscosity, thus depends on the spatial differentiation of the flow velocity, which is inherent in the formula Vzv (see equation (6.1.37)) or can be deduced indirectly from (1.2.35). In the case of turbulent friction, the components of a turbulent frictional force related to unit volume of medium are, in accordance with equation (1.2.37), equal to
a (-@u:U;), fi = ax, __
(1.2.39)
where eu;u; are the mean values of the product of the medium’s density and components of the pulsation parts of the flow velocity, and express the components of the Reynolds stresses zij (see equations (6.2.30) and (6.2.31)). This complex turbulent frictional force, therefore starts to appear only when the flow becomes turbulent, with velocity pulsation components u:, u;, and not during laminar flow. However, flow in the sea usually is turbulent (see the criterion for such flowequation (6.2.3)).
1.2 FORCES INDUCING THE MOTION OF WATER MASSES IN THE OCEAN
39
A special and very important case of friction in nature is that between the wind and the sea surface. This friction brings about the Iarge-scale movement of water masses in the ocean and mixing of ocean waters together with their resources of heat and chemical compounds. It also potentiates the exchange of energy and masses of substances between the sea and the atmosphere, thus fundamentally affecting the Earth's climate (see Chapter 7). Accidental pulsations of air pressure on the sea surface at first cause the elements of this surface to oscillate about their equilibrium positions. Thereby tiny surface waves called capillary waves are generated. In this way, the surface becomes rough and changes the flow of air just above the water from almost laminar to turbulent, although only a thin layer of air "stuck" to the surface of the water remains laminar (see Chapter 7). So the friction of the wind on the sea surface rapidly becomes turbulent friction (see Chapter 6). This induces a stress which is tangential to the surface, and is dependent on the surface roughness and the wind velocity. This stress is far greater than that which would exist if the sea surface were smooth. Supposing, to simplify the description of this case, we point the x axis of the coordinate system in the direction of the wind. Using equations (1.2.37) and (1.2.39), the tangential stress due to the wind will then be ___
(1.2.40)
z,(h) = Q ~ U ' W ' ,
where h is the height above the water surface, ea is the density of air, and u', w' are the pulsation components of the wind velocity at height h (see Section 6.2). But to define these magnitudes at the very surface of the sea is problematical (see Chapter 7). The tangential wind stress at the sea surface zo is therefore usually expressed by an empirical equation containing the mean wind velocity at a certain standard height h (see Roll, 1965; Zilitinkevich, 1970; Zilitinkevich et al., 1978; Wu, 1980; Smith, 1980; and others; see also Chapter 7). This stress can, to a certain approximation, be given by the equation =5
eachvh",
(1.2.41a)
where uh is the mean wind velocity at height h in the lower layer of the atmosphere, and ch is a dimensionlesscoefficientknown as the drag coeficient linking the wind stress at the sea surface with the wind velocity at height h. The wind stress at the sea surface zo is related to the mean wind velocity U , , measured at a standard height of 10 metres above sea level by the following equation :
.
70
= eat,, G o .
(1.2.41b)
1 A GENERAL PICTURE O F PHYSICAL PROCESSES IN THE OCEAN Wind direction
fq/
~
Direction of surface water flow
Fig. 1.2.10. The flow of water due to wind friction. (a) The equilibrium of lateral forces of wind thrust f T and Coriolis force fc, which establishes the direction of flow of surface waters at an angle of 45" to the wind direction (or a smaller angle if conditions are not ideal-see text); (b) the Ekman spiral, illustrating the changes in direction of a drift current with depth in a uniform, deep sea as a result of friction between water layers and the Coriolis force, D-the Ekman depth.
1.2 FORCES INDUCING THE MOTION OF WATER MASSES IN THE OCEAN
41
The drag coefficient C1,depends on the roughness of the sea surface and on the stability of the atmosphere (see Smith, 1980; Wu, 1980; Roll, 1965; see also Chapter 7). The tangential wind stress zo [N/m2] gives the surface layer of water an acceleration in the direction of the wind. The mass of water, stimulated into motion by the fixed wind, is immediately affected by the Coriolis force, which in the northern hemisphere deflects the water to the right of the wind's direction. The deflection of the direction of flow from the direction of the wind can, however, take place only until equilibrium between the components of lateral forces acting perpendicularly to the direction of movement has been established. This is illustrated in Fig. 1.2.10a in whichf, represents the Coriolis force and ft is the differencebetween the friction of the wind and the friction of the substrate of the surface layer. Because of friction, the surface layer of flowing water induces a tangential stress in the layer beneath. This stress causes this lower layer to move, though in a direction already deflected by 45" from the direction of the wind. But again, the Coriolis force and friction on the substrate affect the motion of this layer. So its flow is once more deflected to the right of the flow of the superficial layer until a direction is reached in which the lateral forces (perpendicular to the direction of motion), i.e., the components of friction and the Coriolis force, are balanced. The lower layer, thus flowing in a direction deflected even further from that of the wind, induces tangential stress due to friction in the next water layer below it, and so on, and so forth. In this way, the process whereby motion is initiated as a result of friction and its direction changed owing to the Coriolis effect is transferred into the depths of the sea. The result of an idealized model of this phenomenon is the Ekman spirai (1902). It describes the vertical differentiation of the water flow velocity vector under the influence of wind in a homogeneous and infinitely deep sea (Fig. 1.2. lob). As can be seen from the diagram, the end of this vector describes a spiral, such that the rate of flow at depth z is (1.2.42) where /v(d$')/ is the surface flow velocity, D is the depth of the lower boundary of flow at which the rate is e-R times smaller than at the surface, that is, barely about 1/23 vg;?. The direction of flow, that is, the angle adriftwhich the flow velocity vector makes with the wind direction, changes linearly with depth
42
1 A GENERAL PICTURE OF PHYSICAL PROCESSES
IN THE OCEAN (1.2.43)
The justification of these equations is to be found in many textbooks (see e.g., McLellan, 1965; Neumann and Pierson, 1966). They are derived from an equation of motion taking into account only friction and the Coriolis force in equilibrium, that is when, according to Newton’s 2nd law of motion, their sum is equal to zero and the velocity of their motion is constant (stationary flow). A real sea is neither homogeneous nor infintely deep as the Ekman model assumes, and flows are never purely drift currents but usually result from a combination of all the forces mentioned. The complexity of these flows therefore requires much more complicated hydrodynamic modelling (see e.g., Druet, 1978; Druet and Kowalik, 1970; Monin, 1978; Felzenbaum, 1960). Instances do exist, however, where the movement of ocean water masses is influenced primarily by wind friction and the Coriolis force (see Defant, 1961;Monin, 1978). Of global importance, such phenomena are particularly common along the western coast of North America where they bring about the westward flow of surafce water away from the coast. This flow in turn causes water to pile up in adjacent areas of the ocean and the isobaric surfaces to slope towards the shore. This induces currents of cold bottom waters to rise towards the shelf, and from there on up
Ekrnan iayer
of deep water
Fig. 1.2.11.The transport of water masses near land due to wind friction and the Coriolis force. The current raising deep water to the surface near the coasts is called upwelling.
to the surface of the ocean. Cold, deep waters are thus raised to the surface, closing the cycle initiated by the wind. Such rising currents, called “upwelling” (see Fig. 1.2.11), fertilize the surface waters of the ocean and are the reason why fish are so abundant on the shelves off the west coasts of continents. These
1.2 FORCES INDUCING THE MOTION OF WATER MASSES IN THE OCEAN
43
currents bring nutrient salts and biogenic substances from the depths of the ocean to the insolated surface layers where the plankton producing organic matter utilises them.
The Equation of Motion of Waters The considerable variety of forces acting on the mass of water in the sea suggests that it is very unlikely that they will balance out; more probably they will remain unbalanced and thus stimulate this mass into motion. So too in reality we see in the ocean a ceaseless, complicated movement of water with a certain velocity v which is a random figure. The equations describing this motion are generally known as equations of motion, and there are two principal methods of describing it. (1) The Lagrange method, in which the equations of motion describe the changes in position of a given element of mass in time. The coordinates of this element (x,y , z) are then dependent variables, that is, functions of time t and position (x,, y o , zo) at the starting instant to; this can be written as x = f(x,y,z, , t- to), with analogous expressions for the other two coordinates. Though rarely used to describe marine currents, this method also allows properties of the water mass, such as density @, salinity S, temperature T and pressure p , to be described for an element of mass in its successive positions during motion, (2) The Euler method, in which the equations of motion describe the state of motion of the water mass (velocity v and its changes in time t ) at fixed points in space (x,y , z), irrespective of the fact that ever different “parcels” of water are flowing past this fixed point (x,y , z). In the Euler method, both the coordinates x,y , z and the time t are independent variables, because we can choose any point and research period that we like. But the ffow velocity of the water mass at a selected point and time is dependent on these independent variables, and so is a vector function v(x, y , z, t). In the same place and time, regardless of the exchange of flowing water, we can define the density of the medium e ( x , y , z, t), its temperature T ( x ,y , z, t ) , salinity S(x, y , z , t ) , ambient pressure p(x, y , z , t ) , etc. The values of these functions in a given time at all points (x, y, z) of a given space of sea define the physical fields in this space. These fields can be divided into vector fields such as fields of force, flow velocity, heat flux, etc. and scalar fields such as temperature, pressure and density fields. Of course, it is more complicated to describe vector than scalar fields. For a vector, we have to define not only its value but also its direction at every point in the studied space by giving the magnitudes of its three components along the x, y , z coordinates.
44
1 A GENERAL PICTURE OF PHYSICAL PROCESSES IN THE OCEAN
The vector velocity field v(x, y , z ) in the sea and its changes with time are determined theoretically from the Navier-Stokes equation of motion for a viscous fluid or its modification for an ideal fluid. This equation always takes some form of Newton's 2nd law, which tells us that the mass multiplied by the sum of its accelerations is equal to the sum of the forces acting on that mass. This product of mass and sum of accelerations can also be called the sum of the forces of inertia of that mass. For the mass of a unit volume of water they will aV
be the following: e __-local
inertia, Q(W) v-advectional
at
inertia resulting from
an influx of kinetic energy (see Chapter 6), 2 p x v-the Coriolis inertia resulting from the Earth's rotation. The chief forces acting on the mass, which we discussed earlier, are -Vp-the pressure gradient, eg-the weight, acting vertically downwards only, qVZv-the viscosity (molecular friction), and F,-other forces acting on the fluid. By writing Newton's 2nd Law in such a way as to compare the above-mentioned active and inertial forces, we get an equation of motion for an incompressible viscous fluid called the Navier-Stokes equation for an incompressible fluid
@[(2)
I
+(vV)v+2w x v = -Vp+qV2v+eg+Fi.
(1.2.44)
Notice that this equation includes forces dependent on the density e ( x , y , z , t ) , salinity S(x,y , z , t), pressure p ( x , y , z , t ) and temperature T(x, y , z , t ) ; these in turn are dependent on, among other things, the absorption of sunlight a(x, y , z , t ) . We therefore have to find a way of delineating the fields or spatial distributions of these scalar functions as well in order to solve the equations of motion. We do this by using other equations besides the equations of motion, so that the number of unknowns is the same as the number of equations-only then can we solve the set of equations unequivocally. So we make use of the Law of Conservation of Mass-the continuity equation, the equation of state, and others; they are discussed in Chapter 6. In spite of these various combinations, the analytical solution of the general equation of motion of the type given in (1.2.44), i.e., finding an analytical expression of the vector velocity field v(x, y , z , t ) is only feasible for a few simplified cases. The same applies to the Reynolds equation (Section 6.2, equation (6.2.30)), which reflects more faithfully what really happens in the sea. We therefore usually look for simplified solutions, e.g., for motion in the x-direction only, in a homogeneous medium of constant density (neglecting certain forces of secondary importance) and so on; in other
1.2 FORCES INDUCING THE MOTION OF WATER MASSES IN THE OCEAN
45
words, solutions using simplified models of flows and other physical processes in the sea (see e.g., Felzenbaum, 1960; Stern, 1975; Kraus, 1977; Zilitinkevich et al., 1979; Ramming and Kowalik, 1980). At the same time, detailed measurements of physical and chemical fields in the sea are being made-this will enable 20 N
10
0
10
S 20 60 W
40
50
20
30
0
10
Fig. 1.2.12. Mean surface currents in the tropical Atlantic in February and August (after Diiing et a!., 1980). Abbreviations: NEC-North Equatorial Current; Gy-Guyana Current; NECC-North Equatorial Counter Current; Gu-Guinea Current; S E C S o u t h Equatorial Current; Br-Brazil Current. The degree of steadiness and flow speeds are as follows: STEADINESS SPEED (knots) SPEED(knots1 ~9Very
Steady
-
--+ Steady
__j_
025-05
--+
__ ->
05-075
Unsteady
-On-’
.
1-15
15-2
7e
,z
. I
46
'
1 A GENERAL PICTURE OF PHYSICAL PROCESSES IN THE OCEAN
the results of calculations to be verified (see e.g., Duing et al., 1980; Kolesnikov and Capurro, 1976; Monin and Shifrin, 1974). Consider too the part played by the boundaries of seas and oceans, which alter the system of forces and further complicate the movement of water masses. The reaction of a substrate in the sea to pressure is buoyancy, which diminishes practically to zero at the surface of the sea where weight is the predominant force. At the bottom of the sea, the weight and inertia of the water mass are balanced by the reaction of the bottom to the pressure and by the friction of the water on the bottom. The equilibrium of the system of forces at these boundaries is thus completely changed. Like the bottom, the coastline equalises the pressure of the mass of water, including the pressure due to its inertia in motion. As further flow of water in a direction perpendicular to the coastline is thus retarded, water flowing in towards the shore becomes piled up, the hydrostatic pressure rises and the direction of flow changes in accordance with the pressure gradient. Besides the many random flows, we can distinguish certain mean, more or less stationary flows of water masses, These are the marine currents, known by such names as the Gulf Stream, Cromwell Current, North and South Equatorial Currents, Antarctic Circumpolar Current, and many others (see e.g., Monin, 1978; Duing et al., 1980; and atlases of the ocean which give most of the fundamental mean hydrological characteristics of the ocean, e.g., Gorshkov et al., 1974 and 1977). Figure 1.2.12, which has been compiled from recent data, is an example of how complex just the surface currents in the tropical Atlantic are. It shows the great masses of ocean waters flowing with spatially variable velocities and the quite distinci effects of the Coriolis force and continental coastlines on them. These and other more or less stationary, mean flows in the sea obviously undergo considerable fluctuations in time and make up part of the large-scale turbulence discussed in Chapter 6 (see e.g., Townsend, 1977; Ozmidov, 1968; Richardson, 1980). Of the many forms of complicated seawater movement, we have so far mentioned marine currents-gradient currents induced by a pressure gradient-and drift currents, due to wind friction on the sea surface. We have also seen that a drift current can initiate a gradient current. If we can assume that a given current is the result of the interaction of a pressure gradient, gravitational forces and the Coriolis inertial force only, we call it a geostrophic current. Each of these terms encompasses a certain approximation of the various components of the complex motion of waters which facilitates the description of this motion. A separate class of water motion is the periodic oscillatory motion of the medium which is described by means of wave equations (Cherkesov, 1976).
1.2 FORCES INDUCING THE MOTION OF WATER MASSES IN THE OCEAN
47
In particular, these are the oscillations of the free surface of the sea called surface waves (see Le Blond and Mysak, 1978). The complex and continuous spectra of these oscillations can be divided into oscillation intervals having various periods T. Thus we conventionally distinguish capillary waves (T x 0,1 s), wind-generated gravity waves (ultragravity T z 1 s, ordinary gravity T x 10 s and infragravity T z lo2 s), and long-period waves (T x 103-104 s) generated by storms and earthquakes, and finally waves with very long periods (T z 12 hordinary tidal waves, T z 24 h and longer-transtidal waves) induced by tidal forces. Gravity waves with a period of about 10 s usually amass the greatest potential and kinetic wave energy. The height of these waves is on average about 1 m, though it may exceed 13 m in exceedingly strong winds (see Ross, 1979). A special case of long gravity waves are the free oscillations of an entire body of water called seiches. They are comparable to the free oscillations of the water in a bowl or bath. The complex forms of water motion, the dominant physical process in the ocean, have been examined in detail in the textbooks and monographs on marine dynamics mentioned previously. The present book contains no more of the analysis of such motion than is necessary to explain the other marine physical phenomena discussed here. On the other hand, the reader will find in it a discussion of the principal problems of the thermodynamics of sea water, energy transfer in the marine environment, and marine optics and acoustics.
This Page Intentionally Left Blank
CHAPTER 2
SEAWATER AS A PHYSICAL MEDIUM
Water itself makes up over 96% of seawater, and so as a chemical compound it determines the chief physical properties thereof. Some of these properties, however, are very different from those of pure water. This is because seawater contains, among other things, salts, dissolved organic substances, and mineral and organic suspended matter. The differences between pure and seawater are especially great with respect to, say, the absorption and scattering of light, and electrical conductivity. The properties of water are highly unusual in comparison with those of other substances which we come across in nature. In the first place, it is surprising that under normal conditions water is a liquid at all and can fill the ocean and stimulate life processes on Earth. If we look at the melting and boiling points of compounds of hydrogen with the elements of Group VI in the Periodic Table-oxygen is one of them-the unusual behaviour of water is immediately striking. By analogy with the other compounds of this group, water ought to boil, that is, be entirely TABLE 2.0.1 The melting and boiling points of compounds of hydrogen and Group VI eIements a t normal pressure Compound HzTe HzSe
HS
Melting point
Boiling point
_____
"C
K
-53 -65 - 83 0
220 208
190 273
"C
K
-5
268 228
-45 - 63
+ 100
210
373
converted to steam, not on heating to +100"C (373 K)", but already at about -80°C (193 K) (Table 2.0.1). Under such conditions, the ocean could not exist at the temperatures normally occurring on Earth. The heat capacity and latent heat of evaporation are exceptionally large. Moreover, water has an extraordinary
* We shall be using temperature expressed in ["C] rather than [K] and some other traditional units instead of SI units whenever original empirical formulas, definitions or data require this.
2 SEAWATER AS A PHYSICAL MEDIUM
50
capacity for dissolving salts, whereas its thermal expansion and many features essential to life are anomalous. These unusual properties of water can only be explained by its molecular structure, and in particular by the electronic configuration within the water molecule. We can describe this configuration using atomic and molecular orbitals (Coulson, 1969). An orbital is the wave function y(x, y , z) describing the space in which an electron in a stationary state bound to an atom may be found with a density probability of ( y / * .The whole space in which the density probability of the electron is high is, as it were, “filled with the electric charge” of the rapidly moving electron. We say that this “filled space” is “an electron charge cloud” which can in its entirety interact electrostatically with its surroundings. We can imagine this to be like the “mass cloud’’ of a whirling aeroplane propeller. The characteristic shapes of “charge clouds”, corresponding to the various known types of atomic orbitals, have been determined from quantum mechanics. Different types of atomic orbitals have different shapes and corresponding designations: Is, 24 2p, 3s, 3d, etc. The numeral in the designation expresses the value of the principal quantum number n = 1,2, 3 ..., while the letters s, p , d,f denote the value of the secondary quantum number I = 0 , l ,2, ... ...,n - 1. Non-spherical orbitals, e.g., p orbitals, may be variously oriented with respect to the x, y , z coordinates, so we distinguish between p x,p y and p z orbitals differing by the magnetic quantum number m,which takes values from - I to + I N
2s orbital
X
( i = 2 , f=O>
z
I
z
2p orbital (n=2, /=I)
X
2 Px
2P,
2Pz
Fig. 2.0.1. The types of electronic orbitals occurring in atoms of hydrogen and oxygen.
2.1 THE STRUCTURE OF THE WATER MOLECULE
51
Regardless of how the orbital is described, every electron in an atom must obey the Pauli exclusion principle: it must differ from all the other electrons by at least one of the four quantum numbers. The same kind of orbital can therefore contain at most two electrons differing in their fourth or spin quantum number, which takes the value + 1/ 2 or - 1/ 2 (Fig. 2.0.1). In other words, only one pair of electrons, whose spins are different, can fill an orbital of the same type, but not every orbital must contain two electrons. Furthermore, there is no stationary distribution of charges in atoms between their distribution in an s orbital and a p obrital. The water molecule, consisting of hydrogen and oxygen atoms, must therefore also have its charges spatially distributed in a certain way. We shall now proceed to discuss the structure of the water molecule and its effects on the physical properties of water. In so doing, we shall be using the term “orbital” and the concept of “charge cloud”. 2.1 THE STRUCTURE OF THE WATER MOLECULE
The configuration of electrons in hydrogen and oxygen atoms and in the water molecule (Horne, 1969) can be characterized as follows: (a) there is a single 1s electron in the orbital of a hydrogen atom in the ground state; (b) in the oxygen atom there are eight electrons distributed in the (2~)~, ( 2 ~ 3 ,(2p,), (2p,) orbitals, where the superscript figure 2 denotes the number of electrons in an orbital; (c) there must therefore be 10 electrons in the water molecule; OH bonds are formed when the incomplete Is orbitals in the two hydrogen atoms pair up with the 2p, and 2p, orbitals of the oxygen. Thus the atoms can approach one another and become bound by the attractive forces between the charges of the electrons and those of the nuclei (Fig. 2.1.1). The illustration shows that the combination of two atoms of hydrogen with one of oxygen has brought about hybridisation, that is, the s and p orbitals have mixed, thereby substantially altering the original configuration of the electrons. These changes are attributable to the varying interaction of the electrostatic forces between the combined electron charge clouds and the nuclear charges of these atoms. The effect of these changes is to considerably enlarge the p x and p,, electron clouds of the oxygen atom, now the OH bonds, with a double charge as a second electron, from a hydrogen atom, has been added. Further, these clouds are shifted away from the oxygen nucleus towards the hydrogen nucleus, and the
2 SEAWATER AS A PHYSICAL MEDIUM
52
J X
0
7 H
H/o\H
s H
Fig. 2.1.1. The configuration of hydrogen and oxygen in the water molecule. The orbitals not important to the bonding are not shown. (a) Atoms of hydrogen approachingoxygen atoms; (b) and (c) front and side views respectively of a model of the water molecules; underneath, the structure in symbolic chemical notation (see Coulson, 1963; Horne, 1969).
angle between them has been widened from 90" to about 105" owing to electrostatic repulsion. A second important change is the formation of two strongly negative electron clouds (twice times two electrons) shifted away from the molecule centre (see Fig. 2.1.1),one of which has been formed from the (2pJZorbital and the other from the ( 2 ~ orbital )~ of the oxygen atom. The electron charge clouds surrounding the oxygen and hydrogen nuclei (or rather the electrons moving about in these regions) at the same time prevent the nuclei from moving apart; in other words, they form OH bcnds. Electrostatic repulsive forces prevent these singly-charged nuclei from approaching any closer. The distribution of the charge clouds of electrons and nuclei in the inolecule thus corresponds to the minimum potential energy of the molecule (i.e., the charge distribution is in permanent equilibrium) resulting from the electrostatic attractive and repulsive forces between the electron and nuclear charges in the molecule (Fig. 2.1.2). The bonded nuclei are not, of course, at rest, but oscillate about an equilibrium position, and there is also an important connection between the energy of these oscillations and certain physical properties of water, such as the absorption of electromagnetic waves in the infra-red band. The most significant feature
2.1 THE STRUCTURE OF THE WATER MOLECULE
Distance bepveen nuclei
I
53
1
Fig. 2.1.2. The potential energy of the O-H bond in a water molecule as a function of the distance between the oxygen and hydrogen nuclei (calculated from the formula for the LenardJones potential). mean characteristic bond length; Ep(ro) z -8.19 x 10-19 J-characteristic bond energy; AED z -7.93 x 10-19 J+nergy of dissociation; Ar-amplitude of oscillation of a hydrogen atom relative to an oxygen atom; AE,-energy of oscillation (see Fig. 4.2.1).
ro z 0.96 A-the
of a water molecule of this shape is the considerable shift of the positive charge centres away from the negative ones. A permanent dipole is thereby formed, and at close range, the charges on the molecule thus distributed also act on other atoms or molecules like an electric quandrupole, that is, two positive poles around the hydrogen nuclei and two strongly negative ones in the region of the negative electron clouds protruding from the other side of the molecule. This means that the water molecule has a high, permanent electric dipole moment and moreover, because of its electrical poles, it can combine with other molecules into groups in an ordered fashion (Fig. 2.1.3). The dipole moment of the water molecule is pe = q . r, where q is the resultant positive or negative charge, and r is the vector of the distance the charge centres have been shifted from each other, directed from the negative to the positive charge. This moment is as much as 1.89 x lo-’* electrostaticunits (g’/2 cm5” s-I), i.e., 1.89 units called debyes (the symbol is D-from the name of the Dutch physicist Debye). In SI units this is 6.29 x C * m. A glance at a table of dipole moments will tell us immediately that the moments of some chemical compounds are much lower, e.g., ethanol 1.7 D, ammonia 1.46 D, chloroform 1.2 D, while
2 SEAWATER AS A PHYSICAL MEDIUM
54
Fig. 2.1.3. A water molecule as a dipole in a non-uniform electrical field E, (e.g., of other molecules) in which it is affected by the moment of fixing forces M = P, x E, and the resultant pulling force equal t o - qEel+qECZz. F. = (pe V)E,.
-
many other compounds, such as benzene, acetone and carbon disulphide, are non-polar, that is, they have a zero dipole moment. Such a high dipole moment enables water molecules to bring fairly strong electrostatic forces to bear on other molecules. This is the main reason why water dissolves salts and many other compounds so well. The water molecules break the molecular bonds in these substances, and even bonds between the atoms in these molecules, thereby causing salt crystals to dissolve and salt molecules to dissociate into ions. The polarity of water molecules, that is, the fine structure of the charge distribution within them, coupled with the small size of the hydrogen atom, makes it possible for them to combine into groups of molecules called associations. As we shall see, this phenomenon, and also the spatial distribution of nuclei in the water molecule, explain many of the anomalous properties of water, its exceptionally high specific heat and its high boiling point, to mention only two. A partial confirmation of the spatial structure of the water molecule, shown in Fig. 2.1.1 is offered by the agreement between the theoretical calculations of the specific heat of water vapour and the actual results obtained (Shuleykin, 1968). We know from Maxwell and Boltzmann’s molecular theory of gases that any molecule of a gas (denoted by i = 1,2, 3, ...) at an absolute temperature T has for every degree of freedom of movement ( j = 1 , 2 , 3 , ...), a statistically averaged mechanical energy of
Wi,,= 3kT,
(2.1.1)
2.1 THE STRUCTURE OF THE WATER MOLECULE
55
where k = 1.3806 x J . K-’ is Boltzmann’s constant. The number of degrees of freedom of movement of a molecule is equal to the number of coordinates required to describe completely the molecule’s position in space. A water molecule, looking like the model shown in Fig. 2.1.1., therefore has 6 degrees of freedom, since it can perform translatory motion whose components are along the three axes of a system of rectangular coordinates ( x , y, z) and circular motion at angles 8, @, x to the three axes of symmetry of the molecule’s mass distribution (Fig. 2.1.4).
X
Fig. 2.1.4. The six degrees of freedom of movement of a single water molecule.
In view of (2.1.1), the total kinetic energy of a free water molecule is 6
Wr:j
=
= 3kT.
(2.1.2)
j= 1
The kinetic energy W of all the molecules contained in 1 mole of water vapour will therefore be equal to the sum of the energies of the individual molecules of which there are N = 6.02217 x mol-l (the Avogadro number), i.e. W = 3kT. N .
(2.1.3)
-
But kN = R,, where R, = 8.3142 J mol-’ K-I is the gas constant, so the kinetic energy of the molecules in 1 mole of water vapour at an absolute temperature T is
w = 3Ro T .
(2.1.4)
The specific heat of a gas at constant volume C, is given by the ratio of the increase in kinetic energy of the molecules to the temperature increase of the gas, i.e., (2.1.5)
56
2 SEAWATER AS A PHYSICAL MEDIUM
Measurements of the specific heat of water vapour made at a pressure of 1 atm and temperature of 473 K give the value C, = 1968 J kg-IK-l z 35 J * mol-lK-l.
The ratio of specific heats in this case in C p / C , E 1.3, and so for unassociated water molecules in water vapour (at 473 K) we get C, E Cp/1.3 E 27 J x x mol-I K-l. Such a result is a satisfactory confirmation of the theoretical calculations, which produce a value of Cy N” 25 J . mo1-lK-l. The somewhat high experimental result in comparison with the one expected from the theory can be explained by the fact that when water vapour is heated, not all the heat energy is consumed in increasing the translational and circular kinetic energy of the molecules; some of the energy may be used up in exciting the atoms in the molecules to a higher energy level, and some may be dissipated in breaking the bonds between some of the associated molecules. For this latter reasons, the theoretical calculations have been done for rarefied water vapour, because in dense steam, and the more so in liquid water, the bond energy of associated molecules is considerable, as we shall see.
2.2THE ASSOCIATION OF WATER MOLECULES. THE STRUCTURE OF A N ICE CRYSTAL AND LIQUID WATER
The hydrogen atoms bound in a water molecule are positively polarized because the electron charge centre of the OH bond has shifted towards the oxygen nucleus. By means of attractive electrostatic forces, such a polarized hydrogen atom can join up with the negative “free” electron clouds of the oxygen atom in another water molecule. In this way two water molecules become bonded, or rather, two oxygen atoms become bonded via one of the polarized hydrogen atoms. On the one hand we have an ordinary QH bond in which the common electron cloud surrounds the nuclei of both atoms, while on the other, the same hydrogen is bonded to the oxygen of another water molecule (Fig. 2.2.la,b). This kind of bond between two atoms (e.g., of oxygen) of neighbouring molecules via one of the polarized hydrogen atoms is called a hydrogen bond. The hydrogen atom is capable of forming such a bond because of its small size (radius c. 0.3 A), and the small electron cloud surrounding the nucleus (a single electron in the atom). The proton H+ can therefore get sufficientlyclose to the negative electron cloud of the oxygen. As in the OH bond described earlier, the forces of attraction between charges of different valency and the repulsive forces of charges of equal valency in the two molecules are in equilibrium at a distance between the oxygen
2.2 THE ASSOCIATION OF WATER MOLECULES
57
Fig. 2.2.1. The combination of two water molecules by means of hydrogen bonds. (a) Model of the “translinear”-typebond; (b) structural formula of the ‘‘translinear’’-typedimer (H,O), (after Richards, 1979); (c) structural formula of the less stable ring dimcr (H,O)* (after Rao, 1972).
nuclei of c. 2.98 A (Richards, 1979). If the nuclei come closer to each other, there is a net repulsive force; if they move away from each other, attractive forces prevail, so that the distribution of the potential energy of two oxygen atoms bonded in this way is similar in shape to the distribution shown for an OH bond in Fig. 2.1.2. Both water molecules, joined together by a hydrogen bond therefore oscillate about the position of equilibrium between attractive and repulsive forces (as if held by a spring). They form a pair of molecules called a dimer, which contains a total of 20 electrons and a corresponding number of positive charges in the atomic nuclei. These two molecules are so placed with respect
58
2 SEAWATER AS A PHYSICAL MEDIUM
to one another (see Fig. 2.2.1) in the field of inutually interacting forces that their potential energy is at a minimum. There is also a hypothesis that in a network of many combined water molecules (H,O),, i.e., in polymers, the proton binding two molecules is continually jumping backwards and forwards from one oxygen atom to the other, that is, from the electron cloud of the OH bond to the free electron cloud of the oxygen in the neighbouring molecule, so that to all intents and purposes it belongs in equal measure to both molecules (see Horne, 1969). Moreover, the quadrupole distribution of ‘‘electric poles” in water molecules makes it possible for them to be joined by hydrogen bonds in different ways (see Rao, 1972), as, for instance, in Fig. 2.2.1~. The ability of water molecules to combine in various ways by means of hydrogen bonds to form dimers and polymers is thus a consequence of their structure, and particularly of their electronic configuration. This ability to form polymers or the regular crystalline lattice of ice is responsible for the anomalous properties of water, such as the high melting and boiling points, the exceptionally high specific heat, and the maximum density at about 4°C (Franks, 1972, 1979). This is easily explained if we consider the energy consumption when ice or water are heated not just to increase the kinetic energy of the molecules by a suitable amount, i.e., to make the water perceptibly warm, but also to break down the hydrogen bonds by which molecules are bound in an ice crystal lattice or by which many of them are bound in liquid water. Molecules of compounds which do not form hydrogen bonds can, under favourable conditions, combine by means of weak van der Waals forces whose bond energy is barely c. 2500 J mol-l. On the other hand, hydrogen bonds of the kind formed by water molecules have a bond energy of c. 18.840 J * mol-l, and this, moreover, depends on the way in which these bonds are formed. A considerable quantity of the heat supplied to water in order to warm it up is therefore consumed in breaking hydrogen bonds. That is why the specific heat of water of 4187 J kg-lK-l is so high, some 5-10 times higher than that of other liquids, except for the few, like ammonia NH3, which also form hydrogen bonds. This is also partly the reason for the relatively high melting point of ice”, at which the kinetic energy of the molecules must be sufficient to break down the lattice of hydrogen bonds in the ice. A further consequence is the high boiling point of water. Water molecules combine into ice crystals at temperatures below O’C, when
-
* The very much higher melting points of, say, metals is due to the fact that their crystalline lattices consists of ionic bonds, whose energy is roughly 10 times greater than that of hydrogen bonds.
2.2 THE ASSOCIATION OF WATER MOLECULES
59
the mean thermal kinetic energy of the moIecules is not high enough to break the hydrogen bonds that are forming. Attracted by electrostatic forces, the molecules can therefore become combined into a permanent ice crystal lattice below the freezing point. The spatial ordering of the hydrogen bonds is determined by the geometry of the charge distribution in each of the molecules. The spatial distribution of this charge (see Fig. 2.1.1) allows the molecules to combine in a number of different arrays depending on the external pressure during freezing; if this is high, it “forces” the molecules into a denser lattice. It is likely that the two “free” electron clouds of the oxygen can separately or jointly participate in the bonds in various ways. The spatiaI structure of the lattices, determined by the distances between oxygen nuclei and the angles which the planes of neighbouring water molecules make with each other, depends on these various ways in which the molecules can combine. Different types of ice lattices are known: ordinary ice is called ice I, while others, generally forming at pressures above 2000 atm, are called ice 11, ice 111, and so on to ice VII. There exist two further kinds of ice besides these forms (Kavanau, 1968; Horne, 1969; Franks, 1972). In the crystal lattice of ordinary ice I, tbe oxygen atoms are joined in layers so that each one forms a network of twisted hexagonal rings (Fig. 2.2.2). In each
Fig. 2.2.2. A model of hydrogen-bonded water molecules in the crystal Iattice of ice I after Barnes, 1929 (based on Nemethy and Sheraga, 1962, with permission of the American Institute of Physics). Thickened lines, and spheres delimiting the volume of the molecules, have been used to show how the oxygen atom in one molecule is hydrogen-bonded to four other oxygen atoms in neighbowing molecules.
layer, the oxygen atoms lie alternately higher and lower, and each layer is the mirror image of the layer adjacent to it. Each oxygen atom is linked by hydrogen bonds to four other oxygen atoms to form a tetrahedron, the distance between the
2 SEAWATER AS A PHYSICAL MEDIUM
60
oxygen atoms being 2.76 A (Fig. 2.2.2). It is interesting that between the water molecules combined in such a way there are empty spaces each surrounded by six molecules and slightly larger than a single water molecule. The density of such ice is c. 0.924 g cm-3, and is therefore less than the density of water at 4°C. Ice XI is formed from other ice structures when these are subjected to pressure and lower temperatures. Its density is as high as 1.17 g cm-3 and its crystalline structure differs from that of ice I (see Kavanau, 1968). Now, a block of ice composed entirely of a regular crystalline lattice is called an ice monocrystal. In nature, however, we usually come across polycrystals-ice which is a conglomerate of tiny monocrystals. Disturbances of the regular crystal lattice structure of large blocks of ice may be due to foreign molecules, tensions which arose during freezing, and other factors. In the crystal lattice of ice, the thermal motion of the molecules is basically the vibration of each molecule about its equilibrium position in the field of interacting forces set up by neighbouring molecules. If ice is heated, that is, if energy is supplied to it at temperatures below freezing point, the hydrogen bonds between the molecules are not yet broken. That is why the specific heat of ice is roughly half that of water. But as the temperature of the ice rises, so does the vibrational energy of the molecules in the crystal lattice, and the average distance between them increases. The physical manifestation of this is the normal expansion of ice as its temperature rises. When the temperature of 0°C is reached, at normal pressure, the average oscillational energy of the molecules is sufficient to break the hydrogen bonds. Ice changes into liquid water-and here we come to a gap in our knowledge: we still do not know whether the whole lattice structure of hydrogen-bonded molecules breaks down when ice melts, and if this is the case, what the structure of liquid water is. At the melting point the average kinetic energy of the molecules W j is in fact comparable with the energy of the hydrogen bonds W,, so that the latter can be both broken and reformed. But the fluidity of water suggests that the molecules have a freedom of movement which they did not have in the ice crystal lattice; the hypothesis that the hydrogen bonds between the molecules are broken when ice melts would therefore seem to be a reasonable one. We know from experimental evidence that the latent heat of melting of ice W, Z 335 J g-' z 6030 J * mo1-I. We know too that the energy needed to break all the hydrogen bonds is about W, = 18.840 J . mo1-l. On comparing these two values we may conclude that the heat needed to melt ice is sufficient to break only about 30% of the hydrogen bonds, even if it were all used to this end. It seems very probable, however, that about half of this heat is used up in 9
2.2 THE ASSOCIATION OF WATER MOLECULES
61
increasing the kinetic energy of the molecules, so the conclusion may now be that in water at 0°C formed from melted ice some 85% of the molecules are still hydrogen-bonded and that only 15% of the hydrogen bonds need be broken for water to be a liquid. Various authors have found that, depending on the experimental procedure used, there may be from 3 to 72% free (unbound) molecules in water at 0°C from melted ice. There are quite a number of theories on the structure of liquid water (Horne 1969; Franks, 1972-79; Clementi, 1976). Each explains some of the physical properties of water, but is invalidated by other experimentally proven properties. A11 these theories set out from the assumption that ice is a regular crystalline lattice which on melting either disintegrates partially or is converted in whole or in part to other structures. Undoubtedly, not all the hydrogen bonds are broken during melting, so in liquid water there must exist crystalline structures or at least associations of molecules such as (H,O), , (H20)4, (H,O), ,etc., having some structure or other. A further conclusion is that there will be more bound molecules at lower temperatures than at higher ones, at which the average kinetic energy of the molecules far exceeds the energy of the hydrogen bonds. Lastly, we may infer that at lower temperatures there will be a greater percentage of large groups of molecules or even quasi-crystalline structures than at higher temperatures at which the higher average kinetic energy of the molecules would make it difficult
Q
Q
Fig. 2.2.3. Diagrammatic comparison of the structures of ice, water and steam. W is the kinetic energy of the molecules; WHis the hydrogen bond energy, Q denotes heat supply.
for hydrogen bonds to exist (Fig. 2.2.3). We may also suppose that the composition of groups of linked molecules is dependent on the experimental procedure used with a given sample of water, its storage time, the temperature changes it was subjected to before the experiment, etc. Separate samples of water may there-
62
2 SEAWATER AS A PHYSICAL MEDIUM
fore differ considerably from one another and in very accurately controlled experiments may demonstrate different physical properties. A convincing model of liquid water structure which explains most of its physical properties is the cluster model put forward by H. S . Frank and W. Y. Wen in 1957-1963 (see Horne, 1969; Kavanau, 1968). According to this model, the existence of a hydrogen bond between a pair of oxygen atoms causes a shift of charge such that each of these atoms tends to become hydrogen-bonded to its nearest-neighbour atom, and a whole cluster of molecules is formed (Fig. 2.2.4). Again, if one link in such a cluster is broken, the whole cluster will tend to disintegrate. The lifetime of such a cluster is estimated to be 10-lo-lO-ll s; this is, however, from 100 to 1000 times longer than the period of vibration of a water molecule (see Fig. 4.2.1 in Chapter 4), so the existence of clusters can be demonstrated during the interaction of electromagnetic waves or acoustic waves with water, or when it is heated or compressed. In other words, the existence of clusters strongly influences the physical properties of water.
Clusters of molecules
Fig. 2.2.4. The Frank-Wen cluster model of liquid water (adaptation taken from Nemethy an Scheraga, 1962, with permission of the hmerican Institute of Physics; see also Horne, 1969)
The forming and disintegrating clusters of molecules are hydrogen-bonded, with unbonded single molecules being retained near these groups by Van der Waals forces. No particular structure is assigned to these clusters; the only requirement is that they contain as many molecules as possible and are as compact as possible,
2.2 THE ASSOCIATION OF WATER MOLECULES
63
so that the system of bonding forces remains stable. Special attention must be drawn to the fact that these clusters are “loosely packed” groups of molecules and therefore occupy a larger volume than the same molecules would occupy were they unbound. So these “loosely packed” groups of molecules can be compressed or broken down on being heated, and, as we shall see later, this is of fundamental importance as regards the anomalous changes in the density of water with temperature and pressure. Cluster formation or disintegration is initiated by a local fall or rise in t& kinetic energy of the molecules in their random thermal motion. A local fall in kinetic energy is equivalent to a local fall in temperature which may decrease sufficientlyfor the molecules there to “freeze” or become bound in a “piece of ice”. Conversely, a local rise in the kinetic energy of the molecules causes the clusters to “melt”. It is estimated that under favourable thermal conditions, the clusters forming in this way contain an average of 57 water molecules each. 70% of all water molecules are in clusters; 23% are within the body of the cluster, each molecule being hydrogen-bonded to four others just as in the ice lattice, whereas the other 47% are attached by one, two or three hydrogen bonds to the outside of the cluster (see Kavanau, 1968). A much-criticized aspect of the ordered structure model of water was the reference to experiments on water super-cooled below the freezing point. If ice crystals already exist in liquid water as “seeds”, it should be easier for water to freeze at the freezing point. But we know that if pure water is slowly and carefully cooled it can be retained in the liquid state down to temperatures of even -40°C.On the basis of his cluster model, Frank explains the difficulty of joining clusters into a crystal of ice by suggesting that charges are variously distributed over the surfaces of the different “walls” of the clusters-the poles of the molecules happen to be so oriented that the ‘‘walls’’ of the network do not fit onto one another and so cannot be combined into a regular lattice. The cluster must first break down; only then can a new regular ice lattice be formed in a larger space than was occupied by the cluster. So supercooled water just needs to be shaken to turn it into ice. Studies of water structure have also revealed that certain structural changes take place not only at 0°C during the phase change from solid to liquid, but at several higher temperatures, e.g., + 30°C or + 45°C (Drost-Hansen, 1956). These changes may well have a significant influence on life processes in living cells and could be connected with the body temperatures of humans and animals. The forces due to hydrogen bonds are also those forces which create a high surface tension at the air-water interface and attract molecules which happen
64
2 SEAWATER AS A PHYSICAL MEDIUM
to be on the surface. This makes it difficult to break through the surface membrane of bound molecules and for single molecules to escape during evaporationhence inter alia the high latent heat of evaporation of water. This surface tension is of very great importance in the mechanism of molecular exchange of mass and heat between the sea and the atmosphere. It is probable that the thickness of the surface membrane, i.e., the range of the molecular surface tension forces, is roughly the size of a molecular cluster, i.e., of the order of cm. Lastly, it has to be said that the structure of liquid water, though chemically much simpler than that of many complex organic compounds whose properties are now well known, still remains to be finally elucidated (Richards, 1979). Its study is an extremely laborious affair, despite the availability of optical, atomic and nuclear spectroscopes, not to mention other modern equipment for investigating the structure of chemical compounds. 2.3 ION HYDRATES I N SEAWATER
The molecular structure of pure water described in the previous section explains most of the physical properties of pure water and also to a large extent those of seawater. We see that with an average concentration of 35 g of salts per kilogram of ocean water, there are only 3-4 molecules of salts to every 100 molecules of water. None the less, their presence fundamentally affects many natural processes in the sea, even such processes as the water mass circulation, freezing of the sea surface, the ability of the ocean to conduct an electric current, and the interaction between seawater and living marine organisms. On adding crystals of a salt to pure water, their lattices break down to give ions; in other words, we have solution and dissociation. The polarity of water molecules is responsible for this. Positive and negative salt ions can, moreover, exist in water separately, but only because they are kept apart by water molecules, which set up a kind of electrostatic screen between them (Fig. 2.3.1). Otherwise, the ions would have to recombine as a result of the interaction between charges of opposite sign. The appearance of salt ions in water is most clearly manifested by an increase in its electrical conductivity. The specific electrical conductivity of seawater is of the order of 10-1 C2-I cm-l which, in comparison with that of pure water (lod8f2-I cm-I), indicates that a salt in seawater is almost completely dissociated. The presence of salt molecules and then ions in water brings about three important changes in its microstructure: (a) the structure of pure water is disrupted, i.e., a certain number of “loosely packed” clusters of water molecules break down;
2.3 ION HYDRATES IN SEAWATER
65
Fig. 2.3.1. The dissociation of a salt ion in water (a) and the formation of ion aggregates (b) and (c).
(b) a new, much more stable structure is formed, with the water molecules clustering around the ions because the ion-water bond is much stronger than the water-water hydrogen bonds; (c) the molecules are concentrated within a smaller volume as a result of the powerful ionic attraction (electrostriction), thereby raising the density of the medium.
66
2 SEAWATER AS A PHYSICAL MEDIUM
The cluster of water molecules around a salt ion is called an ion hydrate (or ion aggregate) and, according to the Frank-Evens-Wen (1957) model, consists of three distinctly different zones surrounding the ion (Fig. 2.3.2). The ion is immediately surrounded by a zone of water molecules strongly bound to it and oriented such that the appropriate end of the water dipole is nearest the ion. This is the zone A of powerful electrostriction (concentration due to electrical forces), so because of the close packing of the molecules, the density here is high and the compressibility low. This first zone is surrounded by a second zone B
Fig. 2.3.2. The Frank-Evens-Wen model of a hydrate atmosphere (1957) (adaptation taken from Home, 1968, by permission of Academic Press Inc.).
2.3 ION HYDRATES IN SEAWATER
67
of water molecules in a new structure. The ordering of these molecules is to some degree still dictated by the electrical field of the ion, but as this is much weaker than in zone A, water molecules are also linked together by hydrogen bonds. Beyond this zone is a third one C, comprising so-called free water. But even here some induced polarisation of the water molecules is possible. The density of the zones decreases in the order A, B, C. The total number of water molecules contained in zones A and B is estimated to be around 52 at 5"C, 34 at 20°C and 21 at 50°C, so clearly, this number is closely dependent on temperature-the higher the kinetic energy of the water molecules, the fewer of these molecules an ion can surround itself with in a hydrate. The number of molecules around the ion is also determined by the charge-to-mass ratio of the ion q/m (where m is the mass of the ion), which establishes the intensity of the electrical field around the ion. Finally, since there is a greater concentration of charge at the negative end of the water dipole, the number of molecules bound by positive ions is usually greater than the number bound by negative ions. Accurate experiments have shown that the number of water molecules bound to an ion cannot be determined precisely because it depends on many other factors such as the concentration of ions in the water, the pressure, and changes in these parameters before the sample is tested (see Whitfield, 1975). Coulomb's Law can be invoked to explain the occurrence side by side of the positive and negative ions of a salt in water:
(2.3.1) where ql, q2 are the interacting charges, e0 is the permittivity of a vacuum, E is the permittivity of the medium under investigation, and r is the distance between the electric charges. The large dipole moment of the water molecule means that the dielectricconstant ofwater is also exceptionally large: E z 81. Hence, in accordance with (2.3.1), the force of mutual attraction between ions of opposite charge decreases markedly, in fact by 81 times with respect to a vacuum, when such ions are separated by water molecules. The dynamic equilibrium of the dissociation of salt molecules and the recombination of some of them in consequence of thermal collisions, as e.g. in NaCl + Na+ + C1-, is shifted strongly in the direction of dissociation. The number of ions ni is thus proportional to the number of molecules of the dissolved salt n, n, = 01,12,, (2.3.2) where a, is the degree of dissociation.
2 SEAWATER AS A PHYSICAL MEDIUM
68
In dilute solutions, and seawater can be regarded as such, the number of salt molecules is very small as compared with the number of water molecules. The ions are therefore separated by such a large number of water molecules that the probability of their recombination is extremely small. Hence the degree of dissociation a, is close to unity, which means almost complete dissociation.
2.4 THE PRINCIPAL CHEMICAL CONSTITUENTS AND THE SALINITY OF SEAWATER
The great complexity of seawater as a chemical substance is due to the fact that, in theory at least, every naturally occurring chemical element and many organic compounds besides are dissolved in it (Goldberg, 1974; Riley and Skirrow, 1975; Trzosiliska, 1977; Pempkowiak, 1977). Also present in the water are dissolved gases and numerous complex suspensions, among them the active cells of living organisms (Parsons, 1963; Zalewski, 1977). However, among all these many constituents, there is a group of ions which make up the major part, 99% in fact, of the mass of dissolved inorganic salts in seawater (Fig. 2.4.1) (Culkin, 1965; Wilson, 1975). They include Na+, K+, Mg2+,C1-, Br-, SO:- and HCO;,
Seawater
Sea salt
Fig. 2.4.1. The proportions of seawater constituents. (a) the mass of sea salt compared with the mass of pure water; (b) the concentrationsof various ions in sea salt expresscd as equivalent percentages.
2.4 THE PRlNCIPAL CHEMICAL CONSTITUENTS AND THE SALINITY
69
and are derived from the weathering and solution of rocks by circulating natural waters. To a good approximation, the proportions between the mass concentrations of the principal constituents of sea salt in open waters are constant. But accurate analyses have indicated that their proportions do alter slightly in various waters from one season to another owing to the participation of many ions in biological and geochemical processes in the sea. So in areas of abundant plant growth there is a slight lowering of the amounts of potassium and calcium because of the greater consumption of these elements during the growing season. Elsewhere, salts are precipitated when river water mixes with seawater, bottom waters may TABLE 2.4.1 The average chemical composition of ocean water (from Oceanograpkicul Tables, 1975) Weight %
Element Oxygen Hydrogen Chlorine Sodium Magnesium Sulphur Calcium Potassium Bromine Carbon Nitrogen Strontium Boron Silicon Fluorine Argon Rubidium Lithium Phosphorus Iodine Barium Arsenic Zinc Aluminium Iron Copper Lead
0 H Cl Na Mg S Ca
K Br C N Sr B Si F Ar Rb Li P J Ba As Zn Al Fe Cu Pb
85.94 10.80 1.898 1.056 1.272 x lo-' 8.84 x lo-' 4.00 x lo-' 3.80 x lo-' 6.5 x 10- 3 3 x 10-3 1.7~ 10-3 1.33 x 10- 3 4.6x 10-4 2~ 10-4 1.3 x 10-4 6.1 x 1 0 - 5 2~ 10-5 1 x 10-5 1X1O-S 5x 5 x 10-6 1.5~ 1 x 10-6 1x 10-6 1 x 10-6 6x 4~ 10-7
Element Manganese Mn Selenium Se Tin Sn Caesium Cs Uranium U Titanium Ti Germanium Ge Molybdenum Mo Gallium Ga Thorium Th Nickel Ni Vanadium V Cerium Ce Yttrium Y Lanthanum La Krypton Kr Bismuth Bi Ne Neon Cobalt co Silver Ag Xenon Xe Scandium Sc Mercury Hg Helium He Gold Au Radium Ra
Weight % 4~ 10-7 4 x 10-7 3 x 10-7 2 x 10-7 1.5 x 10-7 1 x 10-7 1 x 10-7 5x 5 x 10-8 5 x lo-* 3 x 10-8 3 x 10-8 3 x 10-8 3 x 10-8 3 x lo-" 2.8 x 2 x 10-8 1.1x lo-" 1 x 10-8 1x 10-8 9 . 4 ~10-9 4~ 10-9 3 x 10-9 5.5 x 10-10 5 x 10-10 0.2-3 x lo-''
2 SEAWATER AS A PHYSICAL MEDIUM
70
be enriched with calcium from the sediments, or salts may be selectively precipitated when seawater freezes at various temperatures. Seen against the high concentrations of the major sea salt constituents, these differences in the principal ionic content of various seawaters are minimal and can be revealed only by careful analysis. On the other hand, the distribution of certain other constituents of seawater, present in incomparably smaller concentrations (of the order of mg per kg of water and less, see Table 2.4.1) but often significantly affecting the properties of seawater and the natural processes occurring therein, show considerable spatial and temporal differentiation. They include the biogenic compounds of phosphorus, nitrogen and silicon, which are nutrient salts essential to the maintenance of life in marine organisms. A large number of polyvalent metal ions such as Fe3+, Mg2+, Ni2+, C o 2 + , Cu2+, Ti4+, Th4+ and others, which are important as constituents of many enzymes in marine organisms, are present in concentrations of micrograms or submicrograms per kg of seawater. The role of these trace elements, as they are called, is very significant in the formation of bottom sediments, which include useful minerals like the well-known iron-manganese nodules. Another important group of substances in seawater are certain organic compounds called “yellow substances’’ (Gelbstog). Among these are melanoids and humic substances which are strong absorbers of the short-wave portion of the solar radiation in the sea and are the reason why the optical properties of seawaters vary in space and time (see Sections 2.6 and 4.2). The relatively constant composition of the main constituents of sea salt has made it possible for a single parameter defining the salt concentration in seawater to be introduced. This is known as the salinity and is denoted by S. Salinity was defined by Knudsen in 1901 in the following terms: the salinity is the weight of inorganic salts contained in 1kG of seawater after having converted bromides and iodides to an equivalent quantity of chlorides, and carbonates to an equivalent quantity of oxides. In order to calculate the salinity in line with this definition, the salts were weighed after the water had been dried for 72 hours at a temperature of 480°C,the weights being those obtained in a vacuum. The salinity so defined was compared with the chemically determined chloride content, and this led Knudsen to formulate his now well-known empirical formula for determining salinity from the chloride content of the water. This formula is valid on the assumption that the other components of sea salt occur in constant proportions with respect to the chloride:
s%,= 1.805CE%,f0.030.
(2.4.1)
2.4 THE PRINCIPAL CHEMICAL CONSTITUENTSAND THE SALINITY
71
This formula was accepted as a standard by the International Commission of 1902 and, until recently, this definition of salinity was never revised. The Cl in this formula, i.e., the chlorinity, was not very precisely defined and it was not until 1940 that Jacobsen and Knudsen restated this in the words of the following operational definition: the chlorinity is a value expressed in parts per thousand numerically equal to the mass (in grams) of chemically pure silver required to precipitate halides from 0.3285234 kg of seawater. Evidently, how the term salinity is defined is a matter of convention (Wallace, 1974), but as we shall see, it very frequently appears in equations for calculating other oceanographicparameters. Recent research has also shown that the Knudsen formula is not strictly correct either for the waters of semi-enclosed seas like the Black Sea and the Baltic, which to a large extent receive river waters of different ionic composition, or for certain areas of the oceans where the expected ionic compositionhas been altered by the biological, physical and geochemical processes mentioned earlier. According to Trzosiriska’s observations (1977), the free term in the Knudsen formula (2.4.1) for Baltic water takes the average value of 0.082, so for the Baltic the formula reads
+
S%, = 1.805C1%, 0.082.
(2.4.2)
It has been found that these operational definitions of salinity can be replaced by one which permits us to use the widely applied physical methods based on the electrical conductivity of water to give quick and accurate measurements of salinity. So the detailed investigations carried out by the group of UNESCO experts on tables and standards (see UNESCO, 1976,1981) of the interdependence between the electrical conductivity and the chlorinity of seawater have resulted in the introduction of an operational definition of salinity. New standards have been fixed and tabies compiled which link the salinity S with the electrical conductivity of seawater (see Section 2.5), and in the following way with the chlorinity
I%]: S%,
=
1.80655 Clx,.
(2.4.3)
Similar reiationships have also been obtained for other parameters indicating salt concentration,e.g., the refractive index. This, however, is only weakly dependent on changes in salinity, and to be useful must be measured to seven decimal places. The salinity has to be measured with a high accuracy (minimum of +O.Ol%, S), because it is so often used in various calculations and because of the far-reaching
72
2 SEAWATER AS A PHYSICAL MEDIUM
hydrodynamic, biological and other consequences of even slight changes in its value in the enormous mass of ocean water. An appropriate measurement precision of the refractive index, and hence indirectly of salinity, can only be guaranteed by high-technology optical interferometers. It would seem then that the definition of salinity is losing its chemical character and no longer corresponds to the actual salt ion content of the water. Despite appearances, though, the differences between the salinities determined by the definitions based on the various methods mentioned are slight and are revealed only in detailed investigations; the differences in the ionic composition of waters are similarly inconsiderable. The new operational definitions, particularly the one connecting salinity with electrical conductivity, allow salinity to be measured much more quickly and accurately than was possible with the chemical method, and, more importantly, permit remote-controlled measurement in situ. As a result of all this, the Practical Salinity Scale came into force on 1 January 1982 (see equation (2.5.8)). The salinity of ocean water is close to 35%,. Regional differences are usually within the range 33-38%, and are brought about by intensive evaporation in the tropical zone or ice formation near the Poles; both processes cause water to become more saline. On the other hand, water flowing into the sea from rivers and glaciers and the melting of icebergs reduce the salinity. So because of melting icebergs and the lower rate of evaporation, the salinity in polar regions is rather less, around 33-34%,. In semi-enclosed seas where evaporation exceeds precipitation, the salinity may rise to figures not normally found in the ocean; 45%, has been recorded in the Red Sea, and even higher figures elsewhere. Conversely, in cold, semi-enclosed seas like the Baltic, inflowing river waters have a decisive influence on the salinity. The salinity of the surface waters of the Baltic is barely 7-8%,, although in the bottom layer it is about 18%,. The average salinity of the Baltic increases as one approaches the Danish Straits, through which saline waters flow in from the North Sea; as these are denser, they flow along the bottom and aerate the oxygen-deficient waters of the Baltic deeps. As we are primarily interested here in the physical aspects of the natural processes occurring in the sea, we shall not go any further into the complexities Of seawater chemistry, neither shall we be dealing with the complicated chemical reactions that are continually taking place in the sea (Goldberg, 1974). In later sections of this chapter, we shall, however, be looking at the organic substances, which absorb sunlight, and the gas bubbles and suspended particles which are also an integral part of the marine environment. They have a striking effect on the transference and accumulation of chemical substances, the absorption of solar
2.5 ELECTRICAL CONDUCTIVITY AS AN INDICATOR OF SEAWATER SALINITY 73
energy, suspension currents, the propagation of sound and the life of marine organisms. Before that, however, we shall examine the electrical conductivity of seawater, a basic indicator of its salinity.
2.5 ELECTRICAL CONDUCTIVITY AS A N INDICATOR OF SEAWATER SALINITY
As in other electrolytes, the carriers of electrical charges in seawater are ions which have a mass of their own but also bear the mass of the hydration layers associated with them. At the same time, a naturai phenomenon in water and other fluids is the random thermal motion of the molecules and ions. On placing a sample of water in an electrical field of intensity E,, the randomly moving ions carrying a charge of + q or - q are acted upon by an additional force F, = If:qE,. This force does not eliminate the thermal motion of the ions, but merely gives them a certain additional acceleration in the direction in which the electrical force field is acting. Assuming that the charge carriers are spheres of radius r (e.g. zone A of an ion hydrate), we can compare the internal frictional force acting on such a carrier during the time of motion Fq = 6mqu (Stokes’ formula) with the force with which an electrical field acts F, = qE, (where q is the coefficient of molecular viscosity of the solution). The balance of these two forces causes ions to “drift” in the solution along the lines of force of the field with a mean velocity of v
=
PEe,
(2.5.1)
where is the ionic mobility (for water ,u is of the order of 10-4[cm2.V-l -s-’ 1) which has a significant bearing on the electrical conductivity of the solution. Having compared these two forces and substituted the relationship in (2.5. I), we obtain an expression for the ionic mobility: p = q/(6xr?7).We see from this simplified case that the average ionic mobility is proportional to the charge q, but inversely proportional to both the dimensions of the entire charge carrier (radius r ) and the coefficient of viscosity of the solution 7. In fine detail, this phenomenon is of course much more complicated, because in order to move, an ion must find empty spaces between the other molecules and ions into which it can “jump”. Sometimes, the resultant momentum an ion receives from collisions with other molecules can even make it move in a direction opposite to that of the lines of force of the electrical field E, (Fig. 2.5.1). But, when acted upon by a field of forces Fq, ions are subject to a resultant “drift” of average velocity z‘, which creates an electric current in the water of intensity i = dq/dt.
2 SEAWATER AS A PHYSICAL MEDIUM
74
Fig. 2.5.1. The motion of an ion in water. (a) the path of an ion in thermal motion in the absence of an external electrical field; (b) the path of an ion after applying an external field: (c) an ion “jumping” into a vacant space as a result of an external electrical field being applied.
The ability of seawater to conduct electricity i s given by its specific conductivity Y e , defined by Ohm’s Law written in the form j = YeEe,
(2.5.2)
where j = i / A i s the average density of electric current flowing across a crosssectional area A , and E, is the intensity of the electrical field acting on this area. The relationship between the specific conductivity y e on the one hand, and the
2.5 ELECTRICAL CONDUCTIVITYAS AN INDICATOR OF SEAWATER SALINITY 75
concentration of the solution (salinity) and ionic mobility on the other, can be expressed by a simple formula only for single-component solutions (e.g. of NaCI) which contain one kind of cation (e.g. Na') of mobility ,u+ and one kind of anion (e.g., Cl-) of mobility p- : ye
= FaeC(P++P-),
(2.5.3)
where Fis Faraday's constant (= 96 486.7 C mol-I), a, is the dissociation constant, and C is the normality of the solution. In seawater, a multicomponent dilute electrolyte, the participations of each component (charge carrier) are additive; in other words, its conductivity is the sum of expressions analogous to (2.5.3) for all the dissolved salts. We see from (2.5.3) that the higher the concentration of ions C, the proportionally greater is the solution's specific conductivity y e ; this gives some indication of the concentration of the solution, and hence too of the salinity in seawater. However, the conductivityis also directly dependent on the sum of ionic mobilities, and, as is clear from (2.5.1), the mobility is an expression of the ionic drift velocity in a field E, of unit intensity. In the light of the mechanism of ionic drift outlined earlier, this velocity must in turn be strongly dependent both on the specific charge of charge carriers q/m, i.e., on the size of the ion hydrates, and on the internal friction generated when they move, i.e., on the viscosity of the solution (see Chapter 6). Many physical properties of the electrolyte, principally its ionic composition, temperature and pressure, affect both these factors. How complex the effects of these physical factors on the conductivity of a simple electrolyte are is illustrated by the Debye, Huckel and Onsager equation. This expresses the dependence of the equivalent conductivity "/R =
V yex
(2.5.4)
on these factors
(2.5.5)
where V is the molar volume, N is the Avogadro number (= 6.022169~ x loz3 moF1), e is the charge carried by an electron (= 1.6021917x lo-'' C), zl,z2 are the respective valencies of the anions and cations, E is the permittivity of the medium; k is Boltzmann's constant (= 1.3806 x J K-l), T is the
76
2 SEAWATER AS A PHYSICAL MEDIUM
absolute temperature, ’7 is the coefficient of molecular viscosity of the medium, and E is given by
in which q1 and q2 are the respective charges on cations and anions, p l and pz are the respective ionic mobilities of cations and anions; and C is the concentration of the electrolyte. Equation (2.5.5) demonstrates how involved is the relationship between the electrical conductivity of a solution and its concentration when the other parameters of the medium are variable. This relationship is even more intricate in seawater, whose ionic composition is so complex. Every ion in seawater is strongly bound to water molecules in a hydrate and at the same time borders on clusters of similarly bound molecules. To move the ionic charge in the direction of the field E, therefore requires the large mass of all or part of a hydrate to be moved as well, and this, moreover, has to be “squeezed” in between other clusters of molecules. The flow of a current of electric charges is therefore accompanied by the flow of a large mass, different for different ions, This mass reduces the mobility of the charge carriers, and both this mass (the size of the hydrates) and internal friction (the resistance offered by large hydrates when moving them in the medium) increase as the temperature falls. A temperature decrease thus causes the ionic mobility and the viscosity of the medium to change in such a way that the electrical conductivity of seawater falls when the temperature does likewise, so much so that, when the salinity is 35%,, its conductivity at 213 K (OOC) is almost half its value at 298 K (25°C). We have also already mentioned the fact that the clusters of “loosely packed” water molecules, of which there are many in water at low temperatures and low pressures, may break down when compressed. At the temperatures and hydrostatic pressures normally obtaining in the sea, an increase in the pressure causes the viscosity of the water to fall somewhat (Franks, 1972; see Table 6.1.2 in Chapter 6) and, therefore, the ionic mobility and electrical conductivity to rise. For pressures rising to lo7 Pa, there is a corresponding gradual increase in the electrical conductivity of about 10% with respect to its value at atmospheric pressure. The effect of pressure on these changes is thus greater at lower temperatures and pressures, that is, when more “loosely packed clusters of molecules” still remain to be squeezed together.
2.5 ELECTRICAL CONDUCTIVITY AS A N INDICATOR OF SEAWATER SALINITY 77
6
298 K 293 K
288 K 283 K 278 K
273 K (O°C)
0 0
10
20 Salinity
-\ 6-' v)
2 . >
:5 -
40
30
s [%.I
-c297.91 ._.-.-.-*-.---*-!
r
3
.
U
-,o L-./L-.*-*-o 0
(278.11 K
c
a- ..- -
.' 0
3-m-----*
L273.18. K
Fig. 2.5.2. The electrical conductivity of seawater, expressed in siemens per metre (S
= C2-l)
(a) with respect to salinity at various temperatures at atmospheric pressure; (b) with respect to pressure for m a n water, S = 3 5 O / 0 0 (from the data of Popov et al., 1979).
2 SEAWATER AS A PHYSICAL MEDIUM
78
The relationship between the electrical conductivity of seawater and the salinity, temperature and pressure is so complex that no theoretical equation yet exists to express it. All that what is available are precise measurements of conductivity, some of which are presented in Table 2.5.1 and Fig. 2.5.2. The empirical formula describing the dependence of the specific conductivity of seawater on its salinity S and temperature T at atmospheric pressure (after Bogorodski et al., 1978) is ye = aT+bS+cTS+d, (2.5.6) where y e is the specific electrical conductivity in Q-lcm-', T is the temperature in "C, S is the salinity in and the constant coefficients a, by c, d determined for the intervals 7°C < T < 30°C and 24x0 < S < 38x0 are: a = 4 . 0 ~ Q-lcm-' (OC)-l, b = 7.9 x Q-lcm-l(%o)-l, c = 2 . 2 lov3 ~ Q;2-1cm-1("C)-1(%o)-1, d = 3.0 x Q-'cm-l. This dependence allows the electrical conductivity of ocean water to be calculated at different temperatures and salinities, or conversely, the salinity of the water can be evaluated from its specific electrical conductivity y e at a given temperature T.
x0,
TABLE 2.5.1 The electrical conductivity of seawater of different salinity and at different temperatures at atmospheric pressure (from Oceunogruphical Tables, 1975) y e ( W 1cm-' x W 5 I ) ye(S m-I x
1 TT % '' 0 5 10 15 20 25
6
8
10
18
574 752 924 1585 664 869 1067 1829 759 993 1219 2086 858 1123 1378 2356 961 1258 1542 2635 1067 1396 1712 2921
20
30
32
34
36
38
40
1747 2015 2298 2594 2899 3215
2527 2912 3317 3739 4175 4624
2679 3086 3514 3961 4422 4896
2830 3261 3713 4183 4669 5168
2979 3432 3907 4400 4910 5435
3128 3603 4101 4619 5152 5703
3276 3772 4292 4833 5392 5966
Another empirical formula for ocean water (Weyl, 1964) links the electrical conductivity ye at various temperatures with the chlorinity logye = 0.57627+0.892 log Cl-
- 1Od4~[88.3+0 . 5 5 +0.0107tZ ~ - CI(0.145 -0.002t+ 0.0002~~)]
(2.5.7)
2.5 ELECTRICAL CONDUCTIVITY AS AN INDICATOR OF SEAWATER SALINITY 79
where z = (25- T)"C; the value of Cl is given in [%,I, and ye in [Q-lcm-l x x 10-31. Although it is somewhat involved,we can use this formula to calculate the absolute specific electrical conductivity of seawater for the range 17%, < CI < 20%,, i.e. a salinity range of 31X0 < S < 36%, at temperatures 0 < T < 25"C, at atmospheric pressure, with an error of no more than 0.1%.This equivalent to an accuracy of 0.03%, in calculating the salinity S. Measurement of Salinity on the Practical Scale It is hard to measure absolute values of the electrical conductivity of seawater with the accuracy required in oceanology for calculating its salinity. Measurement errors are far fewer if we determine the relative electrical conductivity, that is, if we compare the measurement for a particular sample of water with that of a standard solution under the same conditions. For this reason, an international group of experts on tables and standards from UNESCO-IOC*, SCOR* and IAPSO" have arrived at a new operational definition of salinity on the so-called practical scale. It is based on the ratio of the electrical conductivity of a test sample of seawater and a standard solution of potassium chloride (see UNESCO, 1981). The use of this scale has been obligatory in oceanography since 1 January 1982. Thus the practical salinity of a sample of seawater, denoted by the letter S, is defined below by equation (2.5.8) by means of the conductivity ratio K15, that is, the ratio of the electrical conductivity of a given sample of seawater at a temperature of 15°C and a pressure of one standard atmosphere (101 325 Pa) to the electrical conductivity (under identical conditions) of a standard solution of potassium chloride (KCI) of mass concentration 32.4356 g KCl per kg of The equation defisolution, i.e., having a mass concentration of 32.4356 x ning the practical salinity is S= ao+alK~~+a2K,5+a3K~~2+a,K~5+a,K~~,
(2.5.8)
is the ratio of the electrical conductivities of the ( 1 seawater sample and the standard solution at a temperature of 15°C and a pressure
where K15
=
Yewatersample
ye KCl standard
of one standard atmosphere. The constant coefficients in equation (2.5.8) are: = 0.0080, u3 = 14.0941,
a,
* IOC-In
a, = -0.1692, a4 = - 7.0261,
a2
=
25.3851
a5 = 2.7081.
tergovemmental Oceanographic Commission; SCOR-Scientific Committee Association for the Physical Sciences of the
on Oceanic Research; IAPSO-International OW.
2 SEAWATER AS A PHYSICAL MEDIUM
80
The sum of these coefficients c u j = 35.000, which means that when the conductivity ratio K15for a given seawater sample is equal to exactly 1, the practical salinity of that sample is S = 35. In other words, the standard solution of potassium chloride of mass concentration of 32.4356 x at 15°C and one standard atmosphere has exactly the same specific electrical conductivity as seawater whose practical salinity is 35.0000. Equation (2.5.8) is applicable to a range of salinities in the practical scale of 2 < S < 42 which, on the previously used scale corresponds to a salinity range of 2%,-42%,. Notice that the “units” of the salinity S on the practical scale are 1000 times larger than those of the formerly used scale: a salinity of 34.15%,, for example, denotes a mass concentration of salt equal to 34.15 x and on the practical scale this is roughly equal to 34.15. Various seawaters, having identical Kl conductivity ratios, are also of identical salinity S on the practical scale; however, this does not have to imply identical chlorinity CI. This latter value should be treated as a separate parameter, which has no straightforward connection with the practical salinity. So according to the definition, the practical salinity of seawater is based on a comparison of measurements of the electrical conductivities of a seawater sample and a standard solution of KCl. As a secondary standard for calibrating measuring instruments, we can also use seawater which at 15°C and atmospheric pressure has the same electrical conductivity as the KCl standard, i.e., which has a K15 conductivity ratio of exactly 1 and therefore a practical salinity of exactly 35. Such standard water can be obtained from ocean water of slightly lower salinity by evaporation. Standard samples of such water are produced in specialist laboratories such as the National Oceanographic Institute of Great Britain. The ratio of the electrical conductivity of the tested water to that of the standard seawater sample of practical salinity S = 35 at 15°C and 1 standard atmosphere is denoted by R I 5 and takes the same value as K15. If a secondary seawater standard of salinity S = 35.0000 is used, the term K15 is replaced by R15in the equation (2.5.8) for calculating practical salinity. In practice, the conductivity ratio is frequently measured at any convenient temperature, and not necessarily at 15°C. We then get a conductivity ratio of
RT =
( 7-)
at temperature T, which is nearly the same as the ratio
Y e sample
estandard wafer
T
R I S ,and can be used directly in (2.5.8) instead of K15 to calculate the practical salinity. In very accurate calculations, though, a small correction A S must be added. If we make a direct substitution of RT for K15 in (2.5.8), we obtain the uncorrected salinity at temperature T, whereas the corrected value can be evaluated from the equation
2.5 ELECTRICAL CONDUCTIVITY AS AN INDICATOR OF SEAWATER SALINITY 81
S = a,
+ a, R;I2 +a2RT+ a3R$l2+a, R; +a, R;/’
+AS,
(2.5.9)
where the values of the constant coefficients a,, a, etc. are the same as those in (2.5.Q while the correction AS, which has to be inserted when measurements are made at temperatures other than 15”C, is described by the equation: ( T - 15) AS = __ (2.5.10) (b, b, R+/’ +b, R , b3R;/2+ b, R; + b , I?:/’), 1+ k ( T - 15)
+
+
in which the constant coefficients are: b, = 0.0005, b , = -0.0056, b, = -0.0066, b, = 0.0636, b, = -0.0144, k = 0.0162.
b, = -0.0375,
The sum of the bj coefficients, i.e. c b j = 0. Equations (2.5.9) and (2.5.10) are valid over the temperature range from - 2°C to + 35°C and over the range of practical salinities from 2 to 42. The conductivity ratio RT of the tested water and standard seawater, introduced into (2.5.9) is, as we have said, justified by the fact that a comparison of the conductivities of two samples measured on any conductometer gives results of far greater accuracy than attempts to measure the absolute specific electrical conductivity of the tested water. Table 2.5.2 shows part of the oceanographic tables which supply values of the practical salinity S calculated from (2.5.9) TABLE 2.5.2 A section of the oceanographic tables linking the practical salinity of seawater S with the electrical conductivity ratio RT (Intern. Oceanogr. Tubl., 1981) RT
0.990 1 2 3 4 5 6
I 8 9
1.ooo 1 2 3
0
1
2
3
4
5
6
34.609 648 687 726 765 804 843 883 922 961 35.000 039 078 118
613 652 691 730 769 808 847 886 926 965 004 043 082 121
617 656 695 734 773 812 851 890 930 969 008 047 086 125
620 660 699 738 777 816 855 894 933 973 012 051 090 129
624 663 703 742 781 820 859 898 937 977 016 055 094 133
628 667 706 746 785 824 863 902 941 980 020 059 098 137
632 671 710 750 789 828 867 906 945 984 023 063 102 141
7
8
636 640 675 679 714 718 753 757 793 796 832 836 871 875 910 914 949 953 988 992 027 031 067 070 106 110 145 149 -
9 644 683 722 761 800 840 879 918 957 996 035 074 114 153
2 SEAWATER AS A PHYSICAL MEDIUM
82
for various values of RT. From these tables we can read off the uncorrected salinity S for various values of RT (column 1, and the top line for the last digit); or else if we have data RT = R I 5, we can read off the exact value of S calculated from equations (2.5.9) and (2.5.10). So, for instance, when RT = R15 = 0.9978, the practical salinity S = 34.914. These tables came into use on 1 January 1982 and replaced those published earlier. A further complication in connection with measuring the conductivity ratio and determining S from it arise with in situ measurements of salinity. We now have to compare the electrical conductivity of seawater at some temperature T and some hydrostatic pressure p greater than a standard atmosphere, with the conductivity of S = 35 standard water at a temperature of 15°C and hydrostatic pressure p = 0 (i.e., 1 standard atmosphere in the laboratory). This conductivity ratio is generally denoted by R and is expressed as the product of three values: R
=
(2.5.11)
R,rTRT,
where R, is the ratio of the conductivity of the water in situ at hydrostatic pressure p to the conductivity of the same water at the same temperature and at hydrostatic pressurep = 0, i.e., at 1 standard atmosphere; rT is the ratio of the conductivity of a sample of standard ocean water of practical salinity S = 35 at temperature T to the conductivity of the same water at 15°C; RT is the conductivity ratio discussed above. By rearranging (2.5.11) we can find the value of R T , and we can use this in (2.5.9) for salinity or in the appropriate oceanographic tables. R is obtained from measurements, whereas R , and rT are calculated from the following empirical equations (UNESCO, 1981): R,=l+----
&l+ ezP + e3P2) 1 +dl T+d, T2+(d3+d4 T ) R *
(2.5.12)
The temperature T (in "C)and the hydrostatic pressurep (in bars; I bar = los Pa) are also obtained from in situ measurements; the coefficients ej and dj take the following constant values :
e, = - 6 . 3 7 0 ~ d2 = 4 . 4 6 4 ~
2.070~ dl = 3 . 4 2 6 ~lo-,, d4 = - 3 . 1 0 7 ~
el =
e3 = 3.989 x d3 = 4 . 2 1 5 ~lo-',
Similarly,
+
+
+
rT = c0 c1 T+ c2 T2 c3 T 3 c4 T4,
(2.5.13)
2.5 ELECTRICAL CONDUCTIVITY AS AN INDICATOR OF SEAWATER SALINITY 83
where
c0 =
0.6766097, c3 = -6.9698 x lo-',
lo-', c4 = 1.0031 x lo-'.
c1 = 2.00564 x
c2 = 1.104259x
The above algorithm makes it possible to determine the practical salinity of seawater from the ratio of its electrical conductivity to that of a standard, both in the laboratory, where the sample temperature T is controlled, and in situ, where the temperature T (from -2°C to f35"C and the hydrostatic pressure p (from 0 to lo8 Pa) are measured at the same time as the electrical conductivity of the tested water. The origins and principles of this algorithm are discussed in detail in UNESCO Technical Publication No. 37 (1981). The reader is invited to use the values in the table below to verify the accuracy of the equations given earlier. R 1 1.2 0.65
T P C ] p [bar]
15 20 5
0 200 150
RP 1.OoOo0Oo
1.016 942 9 1.020486 4
rT
1.o0OOoOo 1.116 492 7 0.779 565 85
RT
S
1.o0OoO00 1.056 887 5 0.817058 85
35.000000 37.245 628 27.995 341
Notice that, taking into account the inaccuracies outlined earlier, the ratio of the mass of salt to the mass of seawater is defined as if it were the absolute salinity; thus it is now denoted by S,. It differs from the practical salinity S by a factor of lo00 plus a tiny difference resulting from the imprecision of the connection between the concentration of the constituents of any sea salt and its electrical conductivity. We can therefore say that S is only approximately equal to S, x lo3, and therefore an absolute salinity of ocean water S, = 0.03456, i.e. 34.56%, is approximately equivalent to a practical salinity of S = 34.56. The method of accurately recalculating old-scale salinities to give values on the practical scale is described by Lewis and Perkin (1981). Salinity, understood today as an absolute value, and impossible to determine exactly, is represented by the symbol S in many earlier theoretjcal and empirical equations in oceanology. We shall leave it as such, expressed in %,, in the equations yet to be discussed in this book, but we should also bear in mind the subtleties of the new definition. Salinometers (accurate to +0.003%,) are used to determine salinity from electrical conductivity. Figure 2.5.3 shows how the sensor in ordinary salinometers works. This one is an induction salinometer as opposed to a contact salinometer (conductometer-see Lopatin, 1966). In order to determine the electrical conductivity of a water sample (and then of the standard under the same conditions),
2 SEAWATER AS A PHYSICAL MEDIUM
4 Water inflow
Rw
,!,
Water outflow
Fig. 2.5.3. Mode of action (a) and construction (b) of the transducer (sensor) of an induction salinometer (explanations in the text).
we measure the resistance of a loop of water R,-one coil of the transformer with which the water (e.g., in a glass tube) is tested. The transformer winding U T , is supplied with an alternating voltage U 1whose frequency is c. 3 kHz. By means of a ferromagnetic core, the alternating current in this winding induces an alternating current in the water sample which here acts as the secondary “winding”. Now the induced current in the water (the primary coil with respect to transformer UT,) induces, through a second ferromagnetic core a voltage in the winding of transformer UT,. The ratio of the output voltage in winding UT2 to the input voltage in winding UT, is highly dependent on the electrical resistance of the loop of tested water (&). This is easily understood in extreme cases: if the resistance of the water loop is infintely great (conductivity equal to zero), no electric current can flow in it and so there cannot be any electromagnetic induction in winding UT,; the converse holds if the resistance of the water is very small-a large current will flow and the induction is high. Quantitatively, this depends on the transformation ratio, the amount of electrical energy lost as heat in the cores, and other factors resulting from the construction of the equipment. An accurate electronic system, working in tandem with the transducer described must also have a temperature detector, and for in situ measurements
2.6 YELLOW SUBSTANCES IN SEAWATER
85
using STD probes (Salinity, Temperature, Depth), the temperature and pressure must be very carefully controlled (or compensated for) over a much wider range than is necessary with laboratory salinometers (see e.g., Brown, 1968). Therefore a set of sensors is used to measure simultaneously the conductivity, temperature and pressure, and from these the salinity is calculated, often automatically with the algorithm mentioned above and an electronic calculator coupled to the measuring instrument. The induction sensor for measuring electrical conductivity in situ must, of course, be constructed differently, so that the windings and cores are enclosed in watertight housings made of some insulating material, while water flows freely around and through the area surrounded by the transformer cores (see Fig. 2.5.3b). Before concluding this section, we should mention that the use of contact salinometers, i.e., those that measure the resistance of the water between a pair of immersed electrodes, has been abandoned. This is because it was found that the electrolytic polarization of the electrodes, their oxidation and other chemical and physical changes occurring at the water-electrode interface, introduced errors making it impossible to measure the salinity with the precision required today in oceanography. Such salinometers may still be of some use, however, where great accuracy is not needed, for example, in brackish waters with a widely varying salinity such as in river estuaries, or perhaps in the Baltic. The induction salinometer is one of the contactless devices (i.e., no contact between electrode and water) and is devoid of the sources of error just mentioned, as also are the hitherto rarely used salinometers with a four-electrode system or with capacity “contact” of the electrodes with the water through a glass wall (Lopatin, 1966). Besides temperature and pressure, the salinity of seawater is a fundamental parameter of the state of the water, and is usually measured in other investigations as a basic reference parameter.
2.6 YELLOW SUBSTANCES IN SEAWATER
By yellow substances, physicists mean a rather indeterminate mixture of dissolved organic substances which have an important effect on the absorption of light in the sea. It was K. Kalle (1966) who initiated and later continued research into the optical properties of these substances. Since they strongly absorb violet, these substances appear yellow in daylight, and hence their name. Their presence in water in large concentrations also makes the water yellow in daylight, and this colouration is particularly apparent in bays and river estuaries (Kirk, 1976). But
2 SEAWATER AS A PHYSICAL MEDIUM
86
it is by no means a simple matter to distinguish between the dissolved phase of these substances and their suspended phase, because of the size of many of their molecules and their ability to flocculate, that is, to come out of solution and form colloids, fine suspensions or organic clusters (Riley, 1963). It is thought that the molecular mass of organic compounds in seawater usually varies from 500 to 15 000, a very high figure in comparison with the molecular mass of water (18) or that of common salt NaCl (58). It has also been found that in coastal waters about 20% of organic matter particles are greater than 0.005 pm in diameter, some 10% are larger than 0.05 pm and about 3% are larger than 1 pm-the last mentioned quite clearly belong to the suspension category (Ogura, 1974; Sharp, 1975; Wheeler, 1976). Although the nature of yellow substances in seawater has not yet been finally elucidated, we do know that they are complex organic compounds which form as a result of the metabolism and breakdown of organisms living in the marine environment or carried into it. According to Kalle (1966), some of the more important and commonly found yellow substances are melanoids. These are readily formed in an environment where free hydrocarbons and amino acids are present. In the sea, organisms in the fullness of life exist side by side with dead organisms, and this situation gives rise to an extremely complex set of chemical compounds, among which sugars or their anaerobic conversion products are important substrates in the melanin reaction with proteins or their decomposition products. We know in a general way that in the melanin reaction, basic nitrogen compounds (amines) become attached to carbonyl compounds such as aldehydes and ketones. The reaction is initiated as follows:
I
I
I
R
H
R
H O
I
R
I
I
H
H,
/
OH -NH-R
I
R
The compounds forming during the first phase of this reaction then undergo further complicated polycondensation reactions which produce undefined polymolecular substances. The characteristic absorption bands of these substances in both the short-wave and visible spectra are due to the presence of conjugated
2.6 YELLOW SUBSTANCES IN SEAWATER
87
double bonds (see the melanin formula) whose absorption bands correspond to the relatively low electron transition energy between permitted energy levels (Simons, 1976). Their good solubility in water, on the other hand, is a consequence of the hydrophilic groups like -COOH, -OH, and -NH, which they contain. The structural formula of melanin* shown below is an illustration of a molecule containing a considerable number of double bonds.
/ \0
0
Besides yellow substances of the melanin type, many other complex organic compounds have been found in seawater, but their structures are for the moment somewhathypothetical. It is highly probable that compounds such as the following may be present (Stuermer, 1975; Gagosian and Stuermer, 1977):
* In the structural formulae of complex organic molecules, C atoms are not drawn, though they are, of course, present at all the angles in the formula.
88
2 SEAWATER AS A PHYSICAL MEDIUM
Hydrocarbon fragments, amino acids, amino sugars, organic acids and aromatic rings are all recognisable here. They contain polar structural groups with double C = O bonds such as carboxyl, amide and ketone, but have far fewer aromatic rings as compared with melanin. The polarity of the structural groups (molecular fragments) increases the ability of these molecules to join up with other molecules to form clusters, to become sorbed on suspensions, etc. Typical humus substances have also been found, not only in river estuaries (and in the rivers themselves), but in the open sea as well. Humus usually arises from the natural decomposition of dead plants. It is chiefly composed of lignins, which condense to give low-molecular-weight substances which are initially water-soluble (Flaig, 1960). Later in the condensation process, their molecular weight rises and their solubility declines to yield the typical large particles of humus. Hence, a good quantity of typical humus substances occurs in water in the colloidal state, whereas the melanoids and other yellow substances mentioned earlier are to a large extent in solution. Many organic substances are carried into the sea by rivers, both in soluble form and in suspension. In estuarine areas, the heavy suspensions fall to the bottom close to the shore and are there subjected to further chemical reactions. Because sea and river waters have different ionic compositions, the substances dissolved in the estuarine waters of a river are flocculated to a high degree on contact with seawater to form colloids and fine suspensions. These are fairly light particles and are carried far out to sea. To take the example of the river Vistula, which flows into the Gulf of Gdansk, more than 80% of the dissolved organic substances carried by this river undergo flocculation (Pempkowiak, 1977). This has a significant effect on the absorption properties of the waters and of the suspensions themselves in the Gulf of Gdahsk and the adjacent waters of the Baltic (Dera et al., 1978). The concentrations of organic substances in seawater range from 0.05 to 0.5 mg/dm3 of water in clean, open ocean basins to about 5 mg/dm3 in enclosed seas and estuarine areas. In rivers their concentrations are of the order of 10 mg/dm3. It has been estimated, moreover, that about 50% of the total dry mass of organic substances is composed of carbon-organic carbon incorporated in the molecules of organic matter. While it has been possible to develop accurate and fairly easily applicable analytical methods for determining organic carbon in seawater, it is extremely difficult to determine individual organic compounds therein. The usual analytical procedures for determining organic carbon are mostly based on the mineralization of the organic compounds. In the dry method, the sediment is roasted in a quartz tube after having been dried, and in the wet
2.7 SUSPENDED PARTICLES IN SEAWATER
89
method, potassium persulphate is added to the previously acidulated water. After either method, the quantity of carbon dioxide thereby released is measured. This can be done in several ways, the best-known ones being (1) absorption in a solution of a barium salt followed by titration with a base, (2) IR spectrophotometry and (3) gas chromatography. The reader will find more information on the topics just discussed in the monographs by Williams (1975), Parsons (1975), Riley et al. (1975), and Bordovskii and Ivanenko (1979). A special group of organic light-absorbing substances in sea water are the pigments contained in phytoplankton cells. They include chlorophyll a, b, c and d, carotenes, xanthophylls, phicobilins and other pigments essential to the photosynthesis of organic matter in the sea (Kamen, 1963). Outside the cell, these pigments are unstable in seawater, but inside them, during the season of rapid phytoplankton growth, they exert a significant effect on the absorptive properties of seawater.
2.7 SUSPENDED PARTICLES IN SEAWATER, THEIR CONCENTRATIONS AND DIMENSIONS
Suspensions of solid particles are present in large quantities in the waters of all seas and oceans (Parsons, 1963; Zalewski, 1977; Jonasz, 1978). Their average mass concentrations in various waters investigated by different authors are given in Table 2.7.1. TABLE 2.7.1 The mass concentrations of organic and inorganic suspensions in the waters of different parts of the ocean
Study area
Littoral zone of Pacific Pacific NE Pacific North Sea Wadden Sea (Holland) Bering Sea Average for all oceans Baltic
Total dry mass of suspended Of matter in mg/dm3 Organic suspensions of water 10.5 3.8 0.45-1 6.0 18.0 2-4 0.8-2.5 0.2-12
62 29
27 14 20-60 -
Authors
D.L. Fox et al., 1953 D.L. Fox et al., 1953 L. A. Hobson, 1967 H. Postma, 1954 H. Postma, 1954 A. P. Lisitzin, 1959 A. P. Lisitzin, 1959 M. Jonasz, 1980
90
2 SEAWATER AS A PHYSICAL MEDIUM
The suspended matter comprises organic particles of biological origin and inorganic particles of mineralogical origin. The first group includes bacteria, fungi, phytoplankton, zooplankton and detritus-the decomposition products and remains of marine organisms. The inorganic suspensions contain various kinds of mineral particles derived from rock crushing, rock debris transported by rivers into the sea, and atmospheric, volcanic and cosmic dusts. It has been estimated that every day the rivers of the world alone deposit 400 thousand million tons of solid matter in the seas and oceans (Romanowski, 1966). The amount of suspended organic matter produced by photosynthesis in all the world’s seas is roughly 50 thousand million tons dry mass per annum. Apart from the biological production of bacteria and plankton in the water, many other processes take place therein which alter the concentration and qualitative composition of marine suspensions: some of the suspended particles dissolve in seawater, other substances dissolved in the sea are sorbed on their surface. The flocculation of dissolved substances and other processes like aggregation or coagulation, brought about by electrostatic forces are also known to occur, particularly where different masses of water mix (Riley, 1963). Suspended particles also form when dissolved organic substances are deposited on the surface of air bubbles in the water. Even after the bubble has dissolved, the substance that was earlier deposited on its surface remains in suspension (Johnson, 1976). It is merely a matter of convention where we set the smallest size for suspended particles, as the degree of mechanical comminution of particles is theoretically infinite, though in water this does depend on the solubility of the particle. Bacteria and colloids are among the smallest suspended particles in the sea and may be less than 1 micron in size. So when talking about the numbers of suspended particles in water we have to set certain limits to their dimensions to which we shall adhere. To do this we require some parameter which describes the size of the suspended particle fairly accurately, because marine particles are irregular in shape and one cannot speak of their diameters in the literal sense. We therefore use what are called substitute diameters; these are variously defined, depending on the measurement technique applied (Zalewski, 1977). When we examine particles under the microscope, we usually see the projection of an accidentally positioned particle on to a plane. Then we can take the diameter to be the distance between the two tangents to the outline of the particle’s projection which are perpendicular to the baseline of the microscopic picture-this is called the Feret diameter (Fig. 2.7. la). Alternatively, we can measure the length of the straight line running parallel to the picture baseline which cuts the area of the particle’s projection at half the height-this is the Martin diameter (Fig. 2.7.la). In modern
2.7 SUSPENDED PARTICLES IN SEAWATER
91
studies of particle size distribution, we use an electric conduction technique with a Coulter counter (description follows), and we take the substitute diameter of the particle to be the diameter of a sphere of the same volume as the investigated particle (Fig. 2.7.lb).
Martin diameter
TzGzr Fig. 2.7.1. How the diameters of irregularly-shaped suspended particles are measured in seawater (a) outline of the microscopic picture ofa particle with Martin and Feret diameters shown; (b) equivalent diameter D used in the Coulter technique, i.e., the diameterof a sphere whose volume is the same as that of the investigated particle.
It is these volume-related diameters which are what is meant by the dimensions or diameters of suspended particles. We usually have to set a lower boundary to the size of these particles, because the number of suspended particles increases rapidly as their dimensions decrease and, as we have already said, this lower boundary can only be a conventional one. The upper boundary of particle size is not always of any importance, because the number of large particles quickly decreases as they become larger in size, so that suspended particles larger than, say, 50ym make up only a tiny fraction of one per cent of the total number of suspended particles. In the cleanest ocean waters, the iiumber of suspended particles greater than 1 ym in size is generally of the order of 1000 per cm3 of water, and a similarly low (but not lower) concentration can be obtained in a carefully filtered sample of distilled water, In an average sample of seawater the number of suspended particles larger than 1 ym is around 104-105 per cm3 of water, but in polluted seas, bays, river estuaries, etc., the concentration often rises to lo7 particles per cm3 of water (Zalewski, 1977; Jonasz, 1980). A better way of characterising the concentration and dimensions of marine suspensions, one that is in general use, is to estimate the suspended particle size distribution. It specifies the numerical concentration of suspended particles of
2 SEAWATER AS A PHYSICAL MEDIUM
92
various diameters over as wide as possible a range of diameters. Integrated size distributions N, = f ( D ) are used, where N, is the accumulated number of particles, i.e. the number of particles per unit volume of water with diameters larger than and equal to diameter D. Differential distributions dN/dD = f ( 0 ) are also used,
'"
1
10
Equivalent diarneier of particles D
[ p":
10G
Fig. 2.7.2. Typical integrated size distributions N,(D) of suspended particles in the surface waters of the Atlantic and Baltic, selected from the results of Zalewski (1977) and Jonasz (1980). The lines drawn through the experimental points are determined by the least-square method and illustrate the course of the function described by equation (2.7.2).
2.7 SUSPENDED PARTICLES IN SEAWATER
93
in which dN is the number of particles in a given set (in a unit volume of water) whose diameters lie within the interval from D to D+dD. A typical way of presenting graphically the integrated size distributions of marine suspended particles is a plot of the accumulated number of particles N , = f(0)against the diameter D in a logarithmic scale. Figure 2.7.2 gives examples of such plots for Atlantic and Baltic waters. It shows what is usually observed in seas: a rapid rise in suspended particle numbers as diameters decrease. This is the main reason why such size distributions are plotted on a log-log scale; a certain regularity inherent in them is also revealed by this technique. We see that, to a first approximation, they consist of one or two sections of a straight line. Each section can therefore be described mathematically by the equation for a straight line y = ax+b, which on the log-log scale gives
logN,(D) = -rnlogD+logk.
(2.7.1)
Here, the coefficients m and k are distribution parameters; the first describes the slope of the particle-size distribution curve, the second their total numerical concentration for D > 1, because it emerges from equation (2.7.1) that for D = 1, k = N , . The parameter m can therefore be called the slope coefficient of the particle size distribution and k the numerical concentration coefficient of the suspended matter. Equation (2.7.1) written in another way (without the logarithms) is obviously the equation for a hyperbola
N,(D)
=
kD-".
(2.7.2)
We call this a hyperbolic distribution, and it is also generally known as the Junge distribution (1969). Many workers on marine suspensions have found that the Junge distribution (2.7.2) is an excellent approximate description of the particlesize distribution that one comes across in clean ocean waters and in many peripheral seas, especially where there is a high percentage of mineral suspensions, that is, comminuted products of rock crushing (Bader, 1970; Gordon and Brown, 1975; Zalewski, 1977; Jonasz, 1980). According to Zalewski (1977), about 80% of the suspended particle-size distributions which he studied in the Baltic comprise two straight-line sections (on the log-log scale), as in Fig. 2.7.2, well described by the Junge distribution (2.7.2) with different values of the coefficients k and m for diameter intervals larger and smaller than about 7 pm. For diameters D > 7 pm, the statistically most probable value of m is about 2.8 (in the Baltic) and the most likely value of k is approximately 2 x 10". For diameters D < 7 pm, the most probable values of m and k are m z 4 and k z 6 x lo". The numerical
2 SEAWATER AS PHYSICAL MEDIUM
94
concentration coefficient k changes very considerably for distributions investigated at different times and locations in the Baltic, from lo4 to los in fact, whereas the slope coefficient m is far more stable, as is the case in other seas. Why the plots of many suspended particle-size distributions should have kinks in them is not yet entirely clear. It is thought that in some cases, some irregularity or other in the distribution is due to the superposition of two sets of suspensions of different nature and origin, e.g., organic and inorganic (Jonasz, 1980). Hyperbolic particle-size distributions are observed not only in marine suspensions, but also in many other natural aggregations of particles such as atmospheric dusts; even the size distribution of the craters on Mars can be so described (Baldwin, 1965). The Junge distribution is not the only mathematical approximation of the experimentally measured particle-size distributions of marine suspensions, A number of other functions describing these distributions are used which sometimes give better agreement with the measurements obtained in some seas. Zaneveld and Pak (1973), for example, use a distribution of the type Nc(D) = Ae-BD,
(2.7.3)
where A and B are constant distribution parameters defined on the basis of the measurements. Jonasz (1980) showed good agreement with many experiments with a distribution of the type N,(D) = AD-Be-CD,
(2.7.4)
where A, B, C are constant parameters of the distribution. Lastly, Jonasz (1975) and Kitchen (197 5) described statistically the experimental suspended particle size distributions measured in a sea basin with the aid of the characteristic vectors (eigenvectors) y,(D) of a covariance matrix of the set of distributions NJD) n
(2.7.5) -__
where Nc(D) is the mean size distribution in the basin, and ai represents the constant coefficients of the distribution. Because the number of suspended particles increases rapidly with decreasing diameter, the differential size distributions of marine suspensions differ little in their course on the plot from the integrated ones. The complexity of these distributions is better revealed on the latter plots: for instance, the intermediate maxima due to the presence in the water of a large number of phytoplankton cells of similar dimensions are better shown up (Fig. 2.7.3).
2.7 SUSPENDED PARTICLES IN SEAWATER
95
Fig. 2.7.3. Comparison of the differential m l d D and integrated N, distributions of suspended particle sizes in the same sample of Baltic Sea water, which contains a large number of phytoplankton cells whose differential distribution reveals a characteristic maximum for a diameter of c. 6 pm (after Zalewski and Jonasz, 1977, 1980).
If we take the approximate mathematical description of the particle-size distribution in the form of equations (2.7.2) or (2.7.3-2.7.5), we can describe seawater as regards the suspended matter contained in it by using a small number of distribution parameters instead of whole ranges of numbers of particles with given diameters. A complete description of the suspended-particlesize distributions in the sea, however, requires much statistical data because the diversity of suspensions is considerable, especially in peripheral seas. Biological growth, which is seasonally variable, can bring about considerable deviations from the typical distributions (see Fig. 2.7.3).
2 SEAWATER AS A PHYSICAL MEDIUM
96
The erosion of the sea bottom and shores by currents, storms, and in spring also by influxes of river or glacier water, can affect the variability of suspendedparticle concentration to no small extent. But as yet no statistical data are available on the particle size distributions for most areas of the world’s seas and oceans, though such data was gathered over a period of several years in the Baltic Sea and the Gulf of Gdarisk by Zalewski and Jonasz (1977, 1980). Knowing the suspended particle size distributions, we can calculate the total volume occupied by these particles in a given volume of seawater. For suspended particles within a selected diameter interval from Dminto D,,,, this volume is: V
=
1 x 6
Dr
D3dN(D)
(2.7.6)
and in the sea it is of the order of 1 mm3/dm3 of water. From the particle size distributions we can also estimate the minimum total surface area of the suspended matter contained in a unit volume of seawater. These irregularly-shaped particles must have a surface area equal at least to that of sphzres of the same volume. This lowest possible total surface area is Dmax
A =n
)
D2diV(D)
(2.7.7)
D min
which is the result of summing the surface areas of spheres of diameter D. This minimum surface area of marine suspended particles is as high as 10 cmz/dm3 water: this is the upshot of the substantial comminution of this low total volume. The accurate estimation of this surface area in a given situation is important as regards determining the extent of sorption and the transference of chemical substances from the sea water to the surface of the suspended particles. The velocity of fall of suspended particles through the water column depends on their density and dimensions.To estimate this velocity of fall, we can apply Stokes’ formula, which gives the frictional force to which a spherical particle of diameter D, moving with a velocity w through a liquid, is subjected:
Fq = 3nDyw,
(2.7.8)
where y is the coefficient of dynamic viscosity. The free-fall velocity of a single suspended particle w is constant once the force of gravity Fg has been equalised by the sum of the buoyancy Fb and friction Fq which increases along with the velocity. This can be expressed by the equation: F,
=
Fb fFq
(2.7.9)
97
2.7 SUSPENDED PARTICLES IN SEAWATER
that is
(2.7.10) and hence the rate of fall of the particle is (2.7.1 1)
where 10’ and Q are the respective densities of the particle and the surrounding water. With the aid of this formula we shall now estimate the approximate velocity of free fall of suspended organic particles whose density is close to that of water, e.g., 1.1 g/cm3, and of suspended mineral particles (e.g., silica, whose density is roughly 2.1 g/cm3). We shall assume that the density of water is approximately 1 g/cm3 and that the molecular coefficient of dynamic viscosity of ocean water at a mean temperature of 10°C is 7
=
1.4 x lO-’g cm-’s-l.
Substitutingthese values in (2.7.1 1) gives us the approximate velocities of free fall of suspended particles in the sea; the results are set out in Table 2.7.2. TABLE 2.7.2 The approximate velocity of free fall of suspended particles of different sizes in seawater as calculated from equation (2.7.1 1) _-_____
Particle diameter D
-
_____I
Velocity-of fall of particles organic particles of density e’ = 1.1 g/cm3
Pm
cm/s
1 10 30 100
3.9x 10-6 3 . 9 ~10-4 3 . 5 ~10-3 3.9x 10-2
--
______
m/year 1.23 123 1105 12 307
____
silica particles of density e’ = 2.1 g/cm3 cm/s
m/year
4.3 x 4.3 x 10- 3 3.9x 10-2 4.3 x 10-1
14 1357 12 307
________
As we can see from the table, fall velocities of suspended matter in seawater are very low; for example, it would take particles 10 pm in diameter from 4 to 40 years to reach the bottom of the Atlantic at a depth of 5000 m. Even very weak horizontal marine currents, with speeds of around 1 cm/s, would in this time carry suspended particles 1200-12000 km from their starting point. But there are a number of easily identified suspended particles, e.g. quartz dust blown out
98
2 SEAWATER AS A PHYSICAL MEDIUM
to sea from the desert, which are found in the bottom sediments of the ocean almost immediately below the point at which they entered the water (Spencer et al., 1978). There must therefore be some factor accelerating the removal of suspended matter from the surface of the sea to the bottom. Estimates of the velocity of fall of suspended particles using Stokes’ formula may well be very inaccurate with respect to organo-mineral aggregates (Chase, 1979), whose real velocity of fall could be a whole order of magnitude higher than the Stokes velocity. But even this correction does not explain the observed similarity between the distributions of suspended matter at the ocean surface and on its bottom at horizontal distances of less than 100 km from the examined surface area. A highly probable factor accelerating the vertical transport of suspensions in the sea is the clustering of suspensions by zooplankton (Spencer et al., 1978; Lerman et al., 1977). Zooplankton organisms filter seawater in order to obtain nourishment which is comprised of phytoplankton and organic particles smaller than themselves. The remains of these suspensions and undigested mineral particles make up the faecal particles excreted by zooplankton, and these are around 100 pm in size. Moreover, their density is such that they are capable of reaching the ocean floor within 10-15 days, that is, in a time interval allowing the formation of bottom sediments almost immediately under the spot where the suspended particles entered the ocean or were produced. A statistical model of the free fall of particles through successive layers of water is described by Lerman et al. (1974) and La1 and Lerman (1973). The theoretical vertical distributions of particles in the sea are discussed by Jerlov (1959). According to his model, such a distribution in the stationary state can be described by the following equation: (2.7.12) where C is the concentration of suspended particles, z is the depth (the z-axis pointing downwards, positively), R, are local temporal changes (positive and negative sources) within the concentration of suspensions, w is the velocity of fall of the suspended matter, K, is the coefficient of vertical turbulent diffusion (see Chapter 6 ) ; C, w and K, are functions of depth and take positive values. R, is controlled mainly by biological processes and may be either positive or negative. To determine the precise values of w, R, and K, as a function of depth in the sea is generally very difficult if not impossible. This is why equation (2.7.12) can be applied successfully only to highly simplified particular cases. Significant is the model description of the concentration maxima of suspended matter observed
2.7 SUSPENDED PARTICLES IN SEAWATER
99
at particular depths in the sea which are called optical scattering layers (Jerlov, 1959). For phytoplankton particles, assuming a constant velocity of fall w = const and coefficient of vertical turbulent diffusion Kz = const (see Chapter 6), equation (2.7.12), describing the vertical distribution of particle concentration C in the water column, can be written as follows:
(2.7.13) where y denotes the rate coefficient of biological growth of these particles. We assume here a two-layer model of the sea, that is, an upper euphotic layer in which y = const > 0, and a lower unproductive layer, in which the particles decompose, i.e. y = const < 0. The solution of (2.7.13) under these assumptions (Riley et al., 1949) indicates a maximum concentration of suspended matter in the lower part of the euphotic layer. This means that at a certain depth aC/az = 0, and a2C/az2 < 0. At this point it would be worth mentioning the part played by suspensions in the creation of “suspension currents” in the sea (see e.g., King, 1975). Suspensions which have arisen near the bottom as the result of bottom disturbances (e.g., seismic activity) sometimes form such a dense mixture with water that the average total density of this mixture is far greater than the density of seawater alone. If this happens on a steep slope of the bottom, this dense mass flows downhill to the bottom of the slope. Once such a flow has been initiated, further sediments are raised up from the bottom and the flow along the bottom is maintained; this is accompanied by a compensatory flow. In this way, large benthic rivers come into existence in the ocean, which flow very fast and are capable of damaging such objects as underwater cable installations. The importance of suspended particles in the sea is manifested in their effects on the scattering and attenuation of light-this will be discussed in Chapter 4. Statistical studies of the size distributions of suspended particles in the sea can now be performed very accurately with the aid of rapid electronic counting techniques. The Coulter counter, a schematic representation of which is illustrated in Fig. 2.7.4, is used for this (Coulter, 1973; Sheldon, 1972; Zalewski, 1977). In this counter, seawater with its suspended matter is pumped through a special aperture about 100 pm in diameter (or some other, as need be), bored in the wall of the measuring glass tube. At the same time, a continuous electric current is passed through this aperture in the tested water between platinum electrodes placed on either side of the aperture. Whenever a suspended particle passes
2 SEAWATER AS A PHYSICAL MEDIUM
100 To suction pump
Seawater tested
I
passing through Mixer the measurementaperture
Fig. 2.7.4. Schematic representation of the Coulter counter for measuring size distributions of suspended patricles in seawater.
through the aperture, it sets up an additional resistance to the electric current; in an appropriate meter circuit, this resistance induces a voltage pulse which is registered by the counter. In this way, the number of suspended particles passing through the aperture in the measuring tube is measured. A very important property of this device is the experimentally proven proportionality between the volume (!) of the particle passing through the aperture and the magnitude of the voltage pulse. The analyser connected to the Coulter counter can therefore automatically measure the volume of the tested particles from the magnitude of the pulses. Note that this whole procedure must be carried out in a liquid that can conduct electricity, such as seawater (blood cells have also been counted by this method in medical laboratories). The proportionality between the voltage pulse height in the counter and the particle volume holds for all suspensions regardless of their chemical composition. This is thought to be due to the presence of a double layer of electrical charge on the surface of the particles in seawater which prevents the electrical properties of the suspended particles themselves from interacting with the external electrical field. From the measured volume of the irregularly-shaped particle we can calculate its substitute diameter (discussed earlier) as the diameter of a sphere of the same volume. The number of particles whose individual diameters lie within the interval D 6D is counted in a volume of water (0.5 or 1 cm3) which is measured auto-
+
2.8 GAS BUBBLES IN SEAWATER
101
matically in the Coulter counter. A special mercury manometer with electrical contacts is used for this purpose: the mercury level in the U-tube follows the sucked in water, activating successively the “start” and “stop” contacts; that is, it starts the counter and stops it after the requisite volume of water has passed through the nozzle. Depending on the type of counter and to what extent it is automated, counting may take place simultaneously in many electronic channels corresponding to different pulse magnitudes (particle volumes) or in one channel which can be adjusted manually. A high-precision part in this instrument is the accurately calibrated orifice made in a special plate which is inserted in the wall of the measurement tube. The application of the Coulter counter to the measurement of suspended particles is limited mainly by the diameter of the orifice which cannot be too wide when small particles are counted, but should be wide enough to allow as many particles as possible, the larger ones too, to pass through. For seawater, nozzles c. 100 pm wide are usually used; this enables particles from c. 2 pm to c. 40 prn in diameter to be counted. Problems do arise when the nozzle becomes blocked by larger particles, and errors may occur when two particles pass through the nozzle simultaneously (Zalewski, 1977). These difficulties can be overcome and the hundreds of thousands of particles in 1 cm3 of water can be analysed in this way in a few minutes. 2.8 GAS BUBBLES IN SEAWATER
Bubbles filled with gases dissolved in water and with water vapour should also be regarded as particles suspended in the sea. They exert a powerful influence on the scattering and attenuation of sound waves in the aquatic medium (see Chapter 8). Their numbers and size distributions depend on many local factors, such as the processes contributing to bubble formation, forces acting to change their volume and position in the sea, and the properties of the gases themselves, i.e., their solubility in water, saturated vapour pressure under given conditions, etc. The chief source of gas bubbles in the topmost layer of water are the waves breaking upon the surface of the sea which gather up atmospheric air and tsap it under the surface. Apart from the visible effect of this which is the appearance of foam on the surface of the sea, some of the air bubbles (< 100 pm in diameter), get pulled into the turbulent motion of the water mass and are carried under the sea surface, sometimes down to depths of 20 metres. Bubbles are also formed when raindrops hit the sea surface.
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2 SEAWATER AS A PHYSICAL MEDIUM
Yet other sources of bubbles include biochemical reactions in the water or on the sea floor, the products of which are gases evolved in quantities greater than those needed to saturate the surrounding water. Examples are the evolution of oxygen during photosynthesis in the sea, and the biochemical breakdown of organic substances under anaerobic conditions near the bottom producing methane with admixtures of other gases (“marsh gas”). Conditions are usually right for other gases evolving from biochemical reactions, such as hydrogen sulphide or ammonia, to be dissolved in the water. This is why they are a rather unlikely source of gas bubbles in the sea; but the processes by which bubbles are formed during chemical reactions in the sea have so far received little study. Marine animals, particularly large concentrations of them, also make their contribution to bubble formation in the water. Bubbles of gases dissolved in the water can be formed when the water temperature rises, since the solubility of gases in water decreases rapidly as the temperature goes up. Ocean water of salinity 35 is saturated with oxygen when it contains 8.05 cm3/dm3 at 0°C but only 4.73 cm3/dm3 at 25°C. An artificial source of large numbers of bubbles filled with gas and water vapour is the acoustic wave set up by an explosion, or other powerful mechanical disturbance with a high pressure amplitude such as that which occurs at the surface of vibrating bodies like ships’ propellers. When there is a rapid local fall in pressure during the rarefaction phase (the acoustic wave pressure is at its minimum), the continuity of the fluid may be broken and dissolved gases and water vapour from the surrounding water immediately fill the regions of vacuum that are thereby formed. It is easy to imagine what happens when the pressure of the acoustic wave in the water just below the surface attains a value less than the atmospheric pressure during the rarefaction phase. Deep down in the water, the amplitude of pressure changes required to break up the continuity of the medium must be even higher in order to compensate the hydrostatic pressure. Perfectly pure water at normal temperatures can withstand a very great stretching tensionover 250 kG/cm2-before continuity is broken. In seawater, however, there are always unhomogeneities in the form of microbubbles, for example, on the surface of inadequately wetted solid suspended particles. They reduce the ability of water to transfer stretching tensions and so become centres initiating the breakdown of the continuity of flow of the disturbed medium. This process is known as cavitation, and the micro-unhomgeneities mentioned are called cavitation centres (Flynn, 1964; Clay and Medwin, 1977). If microbubbles-cavitation centres-are present, the pressure in the rarefaction phases of the acoustic wave may be not much lower than the saturated vapour
2.8 GAS BUBBLES IN SEAWATER
103
pressure under the given conditions, so that the microbubble containing vapour loses its stability within the medium and begins to expand rapidly to a large size (see Birkhoff and Zarantonello, 1957). If cavitation occurs in a microbubble of radius r o , the critical pressure at which the bubble starts to expand is Pcr
= Pu- -__
3J3
2y [ l + ( p , - p u ) ~2Y ]-i’2, ro
(2.8.1)
where pu is the saturated vapour pressure under the given conditions, p , is the pressure of the undisturbed medium, and y is the surface tension of the water. As the condensation phase of the wave passes, the bubble is now compressed, and there is partial liquefaction of the vapour inside it and a partial escape of gas back into the water; but there is also a rapid rise in the temperature of the bubble because of the adiabatic compression effect. It is also possible that the bubble may break up into many finer ones. At the same time, chemical reactions and even luminescence effects can occur when the bubbles are compressed, if the changes in pressure and temperature are great enough. The whole process also depends on the frequency of the acoustic wave and usually takes place at wave frequencies of < 10 kHz; for higher frequencies, much greater pressure wave amplitudes are necessary (Flynn, 1964). Once bubbles have formed in one of the ways described, mostly during wave action at the surface of the sea, their further fate in the water depends on the forces acting on each bubble and on the physical and chemical properties of the medium. The bubble cannot attain static equilibrium in the water because of the considerable buoyancy force resulting from the difference in density between it and the surrounding water. When the bubble is in motion, the pressure p inside it changes dynamically; this pressure is equal to the sum of the atmospheric
s
z
pressurep,,, the hydrostatic pressue pgdz, and the pressure due to the surface 0
tension forces of the water at the bubble surface. The total pressurep can therefore be expressed as the sum (2.8.2)
in which the significant parameter is the coefficientof surface tension y. Because water molecules are hydrogen-bondedythe surface tension of water is 2-3 times greater than that of other liquids; that of ocean water at 15°C is 74.25 x N/m. The dynamic changes of pressure in the bubble bring about dynamic changes in its volume and diffusion of gas through the walls of the bubble. Small bubbles
2 SEAWATER AS A PHYSICAL MEDIUM
104
in water not saturated with gases may collapse altogether, but larger ones, on moving up to the surface, in the direction of decreasing hydrostatic pressures, may expand. In one way or another, the presence and size distribution of bubbles in seawater are determined by the dynamic equilibrium between their formation and collapse or escape from the water. The “lifetimes” of individual bubbles are thus short-in the surface layer no more than some minutes, except for those which can maintain a hold on suspended solid particles thanks to the cohesive forces between the water molecules and suspended particles. The survival of these bubbles then largely depends on the rate of gravity to buoyancy ratio of such a cluster. The largest numbers of bubbles come into being in the top 10 metres or so of the ocean. Not much has been done in the way of investigating their size distributions because of the difficulties in performing measurements (Blanchard and Woodcock, 1957; Clay and Medwin, 1977). The gas bubbles get into resonant vibration with the vibrations of acoustic waves of suitable frequencies which are dependent on the sizes of the bubbles (for frequencies of free vibrations of
Fig. 2.8.1. Bubble population in various media. From Clay and Medwin, 1977; by permission of John Wiley and Sons, Inc. Symbols: ro-bubble radius; N(ro)dro-the number of bubbles in 1 m3 of water of radii in the interval from ro to rO+dro (where dro = 1 rrm); (0. XI-in degassed water (Messino ef ul., 1963); GI,..., Gqand hatched area-in tap water after 25 min., lh, Zh, 3h, 5h, and the estimated terminal value (Gavrilov. 1969); M (broken lines)-in the coastal zone at depth 3.3 m (Medwin, 1970). The expressions rF3. rC3’’, etc. describe the respective slopes of the plots and denote the proportionality N(r0) rF3 or something similar.
-
2.8 GAS BUBBLES IN SEAWATER
105
bubbles-see Chapter 8). This is why they strongly absorb and scatter the energy of waves of resonant frequencies. This phenomenon makes it possible for bubbles to be detected and counted with the aid of acoustic waves even under such dynamic conditions as exist under the rough surface of a sea. The size distributions of bubbles in water are therefore a topic of investigation in hydroacoustics and most of the information on this subject is to be found in hydroacoustics monographs (Brekhovskikh, 1974; Clay and Medwin, 1977). The results of the few, difficult studies that have been undertaken on the size distributions of bubbles in the sea probably contain serious errors. But these studies do show that there is wide differentiation in the concentrations and sizes of bubbles in various natural environments (Clay and Medwin, 1977). According to Medwin (1970), the diameters of most bubbles lie within the range from a few to several tens of micrometres; he obtained his results from studies in the coastal zone at depths of about 3 metres. Apart from the maximum, the size distributions on a log-log scale are, as is the case with marine suspensions, straight lines inclined steeply to the size axis, i.e. the number of large bubbles decreases very rapidly-from c. 20 000 bubbles in 1 m3 of water of diameter 40 pm (more precisely within the diameter interval 39 pm < D < 41 pm) to c. 500 bubbles per m3with diameters close to 100 pm. The differential plots of the size distributions of bubbles investigated by a number of authors under diverse conditions in situ are illustrated, after Clay and Medwin, in Fig. 2.8.1. For comparison, the size distributions of bubbles are also given for degassed water, tap water and the above-mentioned coastal water. Notice that bubble radii and not diameters are given on the size scale, and also that the inclinations of the size distribution plots are, according to various data, equal to the reciprocal of the radius raised to the third or fourth power (e.g. ro4). Somewhat different data on the size distributions of bubbles in seawater can be found in the monograph by Brekhovskikh (1974) and in Blanchard and Woodcock’s paper (1957).
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CHAPTER 3
THE THERMODYNAMICS OF SEAWATER
We shall start this chapter by summarizing the fundamental tenets of thermodynamics, which when applied to seawater will enable us to describe its thermodynamic properties. We consider any set of water molecules together with other substances mixed with them in a certain volume V to be a thermodynamic system. Each of these N molecules has a certain kinetic energy, potential energy of interaction and bond formation, energy of oscillation, etc. Such a system, as an entity, can exist in various energetic states which are the consequence of the spatial distributions of the kinetic energies of the molecules. But when isolated from its surroundings, a system will tend towards a state of equilibrium, that is, towards such a distribution of the molecular energy which under the given cixumstances is the most probable, e.g., the even filling of a volume V by the molecules. This state of thermodynamic equilibrium is reached by an isolated system when it has existed in an unaltered state for a sufficient length of time called the relaxation time. This state is described by certain kxed macroscopic parameters such as volume V, temperature T, pressure p and total internal energy
U. As the quantization of the energy of atomic orbitals or of hydrogen bonds shows, not every energy state of a system of atoms and molecules is permissible. It emerges from the laws of statistical physics (see e.g., Reif, 1975) that the number of permitted states 2 of a macroscopic system increases rapidly as the energy of the system rises. This number describes a function of the state of the system called the entropy" S , which, according to the statistical definition is S, = klnZ,
(3.0.1)
where k is Boltzmann's constant. In macroscopic systems whose energy U comprises the energy of a large number of molecules, the number of permitted energy states 2 is very high, and the entropy can be treated as a function of the macroscopic parameters (V, V,N , xi), where xi = x1,x2 ,xg,x4... is a generalized coordinate.
* &--by placing an asterisk after the S, we distinguish the usual thermodynamic symbol for entropy from the symbol S used in oceanography to denote salinity.
3 THE THERMODYNAMICS OF SEAWATER
108
If an adiabatically isolated macroscopic system is to be stable, its entropy, as the function S,(U, V,N , xi), must take the maximum value, i.e. S, = max, SS, = 0 , 82S, < 0 , (3.0.2) where SS, is the variance of this function. The condition for equilibrium (3.0.2) is the direct consequence of the second law of thermodynamics, which states that for every infinitesimally small change in a system’s equilibrium due to an amount of heat SQ (at absolute temperature T ) absorbed or lost, the entropy of the system changes by an infinitesimally small amount SQ . ss, = T
(3.0.3)
When the change in equilibrium is reversible, the variance 8Q is a total differential. The macroscopic state of the equilibrium of a system can also be described by means of other thermodynamic functions. These include internal energy U(&, V , N , x i ) , free energy F(T, V , N , x i ) , Gibbs potential G ( T ,p , N , xi), enthalpy H(S* P,N7 xi), potential P(T7 V7 4, where ,u is the chemical potential defined by equation (3.0.12); for the j-th component of a system, pi describes the amount of energy by which any of the thermodynamic potentials increases on adding one molecule of that compound to the system. These thermodynamic functions act as thermodynamic potentials for different types of change. By characterizing the macroscopic state of a system’s equilibrium by means of the internal energy U,we can state the 1st law of thermodynamics clearly: when a system is isolated, its internal energy U = const, but if it interacts with its surroundings, passing from one equilibrium state to another, the change in its internal energy is A U = AQ+AW, (3.0.4) where AQ is transferred heat energy which is not produced by changes in external parameters such as pressure and volume; A W on the other band is the work done on or by the system as a result of changes in these external parameters. For an infinitesimally small, quasi-static deviation of the system from equilibrium we can write d U = SQi-BW. (3.0.5) 7
~2
3 THE THERMODYNAMICS OF SEAWATER
109
Any (but not infinitesimally small) reversible changes in the system’s equilibrium, described by (3.0.4) can be regarded as a series of consecutive, infinitesimally small changes taking place in accordance with (3.0.5),i.e. a series of consecutive quasi-static passages of the system through consecutive states of equilibrium. It is evident from (3.0.5) that a change in the internal energy of the system can be caused by only two things: an exchange of heat or of work with the surroundings. The work SW is in general equal to the sum of three components: (1) the work pdV done by the forces acting at the “wall” of the system when its volume changes; (2) the work done by forces (generalized X i ) acting directly on the individual molecules of the system (e.g., the work done by the forces of a gravitational or electrical field) equal to Xidxi; (3) changes in the chemical energy of the system due to the change N j , equal to c p j d N j , in the number
c
i
of molecules of every j-th component of the system which may take place as a result of a chemical reaction, dissociation, etc., or of the influx of molecules from the surroundings by diffusion. pj is the chemical potential of the j-th component of the system (e.g., of salt ions). Assuming that the work done is positive when the system gains energy and vice versa, we can write (3.0.6) Substituting in (3.0.5) the expression for SQ from (3.0.3) and that for SW from (3.0.6), we obtain the equation
dU
= TdS,-pdV-
xXidx,+ xpjdNj, i
(3.0.7)
i
which is a basic thermodynamical statement and for d U = 0 describes the condition of thermodynamic equilibrium in an isolated system. Notice that a simple rearrangement of (3.0.7) yields its identity, which expresses the entropy increment dS, (3.0.8) similar identities can be obtained for the other thermodynamic potentials. From (3.0.7), (3.0.8) and similar thermodynamic identities, we can get useful relationships between the macroscopic parameters of a system and its thermodynamic potentials. Consider equation (3.0.7) first and notice that for a quasistatic change (one parameter changes slightly, the others are regarded as constant), we can derive from it
3 THE THERMODYNAMICS OF SEAWATER
110
for y = const, xi = const, Ni= const
( z) =T, as*
(3.0.9)
V,x,N
for S , = const, xi = const, N i = const
(g)
-
-P,
(3.0.10)
S*,X.N
for S, = const, V = const, N = const (3.0.11) for S, = const, V
(””)a 4
=
const, xi = const
= pj.
(3.0.12)
s,,v.x,
Likewise, from (3.0.8) we can also obtain useful relationships linking the system parameters with its entropy for processes in which three of the four parameters are constant (%)V,~,N
(%-)u,x,N
- _1
(3.0.13)
-_ P
(3.0.14)
- T’
-
T’
(3.0.15)
(3.0.16) In the case of seawater, the thermodynamic potentials and macroscopic parameters of the system usually refer to units of the water mass. The following parameters are therefore introduced: V = tc specific volume,
(3.0.17)
_s*_ _- 6 specific entropy, m
(3.0.18)
-
M
U
- = E specific internal energy. m
(3.0.19)
3 THE THERMODYNAMICS OF SEAWATER
111
At the same time, instead of the number of molecules of the j-th component N j and chemical potentials pj ,we use here the salinity of seawater S and its chemical potential p (see Doronin et al., 1978). This is an approximation resulting from the assumption that the proportions of the components in sea salt are constant, and that a proportionality exists between the concentration of a given component and the total salinity of seawater. The chemical potential of the components of seawater can be divided into the chemical potential of pure water p; and the chemical potential of the sum total of sea-salt constituents ,us.If we denote the number of molecules of the j-th constituent of salt in 1 kg of seawater by NjCs) and the number of molecules of water in it by Nw, we can write the last term in (3.0.6) (for 1 kg of seawater) as the following sum: (3.0.20)
We now use the proportionality between the number of molecules Njc,, of the given sea-salt component and the salinity of the water S NjCd
(3.0.21)
= CAS)SY
where cj(,) are coefficients of proportionality. In view of the definition of salinity, a similar proportionality exists for the number of water molecules
N,
= const(1- S ) .
(3.0.22)
From this we get
dNjc,) = C , ( ~ , ~ and S
dN, = -constdS.
(3.0.23)
Substituting these expressions for dNjcS,and dN, in (3.0.20), we get (3.0.24a) and if we replace iW
cj(s)by ps and const p; by pw,we can rewrite (3.0.24a)
as follows:
where p = ps-pu, is the (specific) chemical potential of seawater. We can now express the increment in the specific entropy of seawater 6 on the basis of (3.0.8) using the relationships (3.0.17), (3.0.18), (3.0.19) and (3.0.24). In so doing, we
3 THE THERMODYNAMICS OF SEAWATER
112
assume that the work done by forces Xi is negligible. So for a unit mass of seawater we have 1 dE+ --P da- P dS. d6 = T T T
(3.0.25)
This fundamental thermodynamic equation is also called the Gibbs equation. Likewise, we can obtain an expression for the increment in the specific internal energy of seawater E from (3.0.7): d&= Td6-pdai-PdS. (3.0.26) On the other hand, the total differential of the specific internal energy, as a function of the set of three independent parameters &(a,S , T), can be expressed in the following way:
(3.0.27) If we equate the right sides of (3.0.26) and (3.0.27), and divide this equality by dT at constant volume (ct = const) and constant salinity ( S = const), we obtain a definition of the specific heat of water at constant volume and salinity given by the equation (3.0.28) Taking (3.0.5) into consideration we can express the same specific heat for a reversible reaction as follows:
(3.0.29) In order to express the specific heat of warmed water at constant pressure (p = const) and constant salinity ( S = const) when some of the heat is consumed in warming the water while another portion does work pdct, we equate (3.0.26) and (3.0.27) to obtain
(3.0.30) On dividing this equation by dT, rearranging, and including (3.0.28), we can write an expression which defines the specific heat at constant pressure and salinity
(3.0.31)
3.1 SEAWATER STATE PARAMETERS
113
This shows that Cs,p is generally larger than or equal to C, , s, but the temperaturedependent difference Cs,p- Cm,sis usually less than 40 J kg-'K-' for seawater. The dependence of the specific heat of seawater on temperature at different salinities and depths is illustrated in Fig. 3.0.1. This figure demonstrates how
s=20%0,z=Om
m
097
s=f,O%o,z=Om 0.94093-/
S=3485%o,~=1000m
0.90-/S=3f,.85%0, z-5000m 0 89-
Fig. 3.0.1. The dependence of the specific heat of seawater on temperature at different salinities S and depths z (graph based on data from Oceanographical Tables, 1975). Notice that the specific heat increases distinctly as conditions favouring the formation of clusters of hydrogen-bonded water molecules arise.
complex the effect of salinity and pressure on the specific heat of seawater isit can be explained in terms of the changes in the molecular structure of the water that are due to changes in these parameters (see Sections 3.2 and 3.3).
3.1 SEAWATER STATE PARAMETERS AND THE EQUATION OF STATE
The principal seawater state parameter is the density e = dm/dV [g/cm3] or [kg/m3], which depends in a complicated way on the salinity, temperature and pressure. The spatial distribution of the density of the water masses in the sea basically determines their state of hydrostatic equilibrium or motion, and so this parameter occurs in a number of hydrodynamical equations describing the flowof water, the transfer of mass, heat and momentum in the sea (see Chapters
3 THE THERMODYNAMICS OF SEAWATER
114
1, 6 and 7), the propagation of acoustic waves (see Chapter 8), and many other processes. It is paradoxical that the density of seawater has to be determined by very indirect methods, because, unlike the many direct means of measuring salinity, temperature and pressure that are in general use, no practically useful method of measuring the density of water in the sea in situ has yet been invented. This is the main reason why much attention is being paid in oceanography to the precise elucidation of the relationships between the density of seawater and its salinity, temperature and pressure. Frequently, in order to simplify thermodynamic equations, instead of the density of seawater Q we use its specijic volume ct = l/e = V/m,i.e., the volume of a unit mass of water. The connection that usually exists between the density (or specific volume), absolute temperature T and pressurep of a substance is called the equation of state of that substance. This equation is fairly simple only for ideal gases, and is known as the Clapeyron equation pV = nRT, (3.1.1) where V is the volume of the gas, R is the universal gas constant, and n is the number of moles of the gas. For real gases and vapours, in which we have to take into account the finite dimensions of the molecules and the forces of their interaction, the equation of state becomes more complicated, and takes the form known in physics as the Van der Waals equation
(
p+
__ :z2)
(V-nb) = nRT,
(3.1.2)
where a and b are Van der Waals coefficients. Such an interdependence between the specific volume, pressure and temperature as applied to seawater would not really reflect the actual state of affairs, although if the Van der Waals coefficients were carefully selected, it could stand as a rough approximation of the equation of state for seawater of constant salinity. A change in the composition of the substance-in this case the variability in the concentration of the salts dissolved in seawater-introduces into the equation of state further independent parameters: one, if the percentage proportions between the sea-salt components remain constant, or more, if they vary. The assumption that these proportions are to a good approximation constant means that we can use one single parameter to define the total salt concentration, i.e., the salinity, which was discussed in Chapter 2. Under this assumption, the equation of state of seawater is described by four parameters, i.e., the salinity S, temperature T, pressure p and specific volume M (or density l/ct).
3.1 SEAWATER STATE PARAMETERS
115
These four seawater state parameters are linked to the specific internal energy of the system E = U / M (the internal energy U per unit of unalterable mass of the system M ) by the following expressions which ensue from the relationships (3.0.9)-(3.0.12), or directly from equation (3.0.26) : (3.1.3)
where 8 is the specific entropy (d8 =
[-$I - [%I),
and ,u is the specific
chemical potential of seawater. On the other hand, the equation of state of seawater defines the relationship between the variables a,S, T,p , which is most often formulated as the dependence of the specific volume on the other three variables: a@, T,p). However, this relationship is an extremely complex one, and we can express it as a simple analytical function only in special cases, e.g., when the hydrostatic pressure p = 0, S = const. In general cases, we usually use empirical data and formulae or other approximations (Mamaev, 1975), of which more in Section 3.5. We shall now consider the nature of the dependence a(S, T , p ) on the basis of empirical data, bearing in mind the anomalous properties of water which are the consequence of its structure. We can express a change in the specific volume a(S, T , p ) with the aid of a total differential of this function: (3.1.4) If, when rearranging this equation, we use the definitions of the coefficients of thermal volumetric expansion kT, compressibility k p , and volume change due to salinity (contraction) k s , we can write it down as an equation of state in the form
(3.1.5) where g o is the initial specific volume before its increase due to changes in the variables S, T,p, (3.1.6)
116
3 THE THERMODYNAMICS OF SEAWATER
is the coefficient of thermal expansion at constant salinity and pressure (isohaline, isobaric), (3.1.7)
is the coefficient of contraction (volume change due to salinity) at constant temperature and pressure (isothermal, isobaric), and (3.1.8)
is the compressibility coefficient at constant salinity and temperature (isohaline, isothermal)*. We see from equation (3.1.5) that a relative change in the specific volume dalao of seawater depends on the temperature through the coefficient of thermal expansion k T , on the salinity through the coefficient of volume change due to salinity k s , and on the pressure through the compressibility coefficient k,. Closer examination of these individual dependences will facilitate our understanding of the reasons for the considerable complexity of the relationship between the specific volume (or density) of the water in the sea and the set of simultaneously changing environmental variables.
3.2 THE THERMAL EXPANSION OF SEAWATER
The thermal expansion of seawater is the thermodynamic process whereby changes in its specific volume are induced by temperature changes. Quantitatively, a given substance is thus characterized by the coefficient of thermal expansion kT, defined by equation (3.1.6). The process in which a given mass of water expands or contracts owing to changes in average intermolecular distances depends on the kinetic energy of the molecules and their clusters. From the molecular theory of the structure of water, we know that this thermal motion of molecules consists mainly of the rotation and oscillation of molecules about their equilibrium position and of molecular jumps from one equilibrium position to another. In order to escape from the field of bonding forces (forces
* Note that simple isothermal or isobaric processes involving substances whose composition is constant now becomecomplicated owing to the appearance of a fourth parameter, the salinity S. This is why we need a two-index definition in the name of the coefficient in order to describe the process accurately.
3.2 THE THERMAL EXPANSION OF SEAWATER
117
due to hydrogen bonding and dipole interaction) in one equilibrium position and jump across to another (crossing the potential barrier), a molecule needs additional energy-the activation energy E. It receives this energy in random fashion from adjacent molecules which are also in thermal motion. The average time Z that an oscillating molecule spends in one equilibrium position and in the jump across to another, is highly dependent on the activation energy and the absolute temperature T 7 = toeE/kT,
(3.2.1)
where zo is the mean period of vibration of a molecule about its equilibrium position, and k is the Boltzmann constant. If the average shift of a molecule in time Z is denoted by & the average velocity of the molecule's translation at temperature T is, in view of (3.2.1), the following:
(3.2.2) We may assume that the average shift 3 is of the order of the diameter of the free space (hole) between the molecules, that is, of the order of the average,:distance between the molecules. The order of magnitude of this distance, dependent on the specific volume, can be expressed by the equation
(3.2.3) where no is the number of molecules in a unit volume, M is the molar mass of water (equal to 18 kg/kmol), a is the specific volume of water (equal to c. m3/kg), and N is the Avogadro number. For water, 8 is therefore equal to 3.1 x cm. Equations (3.2.2) and (3.2.3) also indicate indirectly the close and complex dependence of the specific volume a on temperature. The pitching motion of the water molecules and the distances between them are, as we know, highly dependent on the bonds which group molecules into clusters. This is very significant as regards the anomalous thermal expansion of water which, at normal pressure, occupies the smallest volume at a temperature of c. 277.16 K ( z 4°C). The reasons for this anomaly are readily explained by Frank and Wen's cluster model (1 957). When the temperature of water is raised, two competing processes take place. The first is the usual increase in the kinetic energy of the molecules, and the consequent increase in the intermolecular distance 6 , in other words, the usual
3 THE THERMODYNAMICS OF SEAWATER
118
expansion of a liquid which is heated. In the other processes, more and more molecular clusters break down as the temperature, and therefore the molecular kinetic energy, rises. The molecular cluster is “loosely packed”, and when it breaks down, the same molecules occupy a rather smaller volume even though the temperature has risen (Fig. 3.2.1). The upshot of these two competing processes
(4 ----__
(b)
(c)
/
\
\
,’/
‘\ Tw277U
T=277K
273ucn 2 7 7 ~
Fig. 3.2.1. The mechanism of the anomalous thermal expansion of pure water (a) at a high temperature, the distances between free molecules are large, and there are few “loosely packed” clusters of molecules; (b) at c. 277 K, the specific volume reaches its lowest value because the distances between free molecules are small and the number of “loosely packed” clusters is not yet very great; (c) at temperatures between 273 K and 277 K the low kinetic energy of the molecules permits large numbers of moleculesto be hydrogen-bonded into clustersthere are now relatively few free molecules in the nearest neighbourhood of the clusters. It is these “loosely packed” clusters which bring about the increase in specific volume of water despite the low temperature.
(change in S) leads to the non-linear relationship a(T) shown in Fig. 3.2.2. When the temperature falls below 277.16K, so many “loosely packed” molecular clusters are formed in pure water that the specific volume now tends to increase rather than decrease owing to a fall in the kinetic energy of the molecules. This is why from 277.16K to freezing point, the coefficient of thermal expansion of pure water is negative: the specific volume increases with a fall in temperature. Above 277.16 K the volume increase, caused by the increased kinetic energy of the molecules on being heated, is decidedly greater than the volume decrease due to the breakdown of clusters, so above 277.16K (4°C) the coefficient of thermal expansion is positive. These two processes are in equilibrium in pure water at atmospheric pressure and a temperature of 277.14K (3.98”C) (Franks, 1972). At this temperature water has the greatest specific volume, that is, the highest density, and its coefficient of thermal expansion is then zero.
3.2 THE THERMAL EXPANSION OF SEAWATER
119
As discussed in Chapter 2, the appearance of salt-ions in water causes ion hydrates to form which, owing to electrostriction, occupy a very small volume in comparison with that occupied by the same number of molecules unassociated with ions. The number of molecules “densely packed” in hydrates rises rapidly as the temperature falls. So when the temperature of salt water changes, a third process-electrostriction-affects the specific volume: this always decreases when the water temperature falls. As the salinity rises, the electrostriction effect becomes stronger, and the temperature T(um3,,)at which the specific volume of seawater reaches a minimum, i.e., at which its density is at a maximum, falls. At salinities greater than c. 24.7%,, the specific volume of seawater decreases with temperature right down to freezing point (Tsurikov, 1976) without showing any other minimum (Fig. 3.2.2). Ocean water does not therefore demonstrate anomalous thermaI expansion in the sense that, unlike pure or weakly saline 1.005
1.000
n
To
0.99F:
W
E,
r_l
I 0
5
0.99c
0
0 .-
5al 0.98: a
v)
0.98C
0.9F 0.971
Fig. 3.2.2. The specificvolume of pure and sea water as a function of temperature at atmospheric pressure. The coefficient of thermal expansion kT changes sign from plus on the right of the broken line joining highest-density temperatures to minus on the left of this line (compiled from tables in Popov et al., 1979).
3 THE THERMODYNAMICS OF SEAWATER
120
water, it does not have a negative coefficient of thermal expansion (see Table 3.2.1). Because of the effect of the three processes described, the k,(T) dependence for ocean water is still non-linear. In simple liquids (e.g., mercury), where there are neither hydrogen bonds linking molecules into clusters nor ion aggregates, TABLE 3.2.1 The coefficients of thermal expansion at atmospheric pressure of seawater of different salinity and different temperatures (after Neumann and Pierson, 1966) k T - lo6 [K-'1
s [%,I 0 10 20 25 30
40
T [Kl -_____271 -105 -65 -27
273
278
283
288
293
298
303
-67 -30
17 47 75 88 101 126
88 113 135 146 157 177
151 170 188 197 206 222
207 222 237 244 251 263
257 270 281 287 292 301
303 314 324 329 332 331
5
- 10 7 38
21 36 65
it is principally the kinetic energy of the molecules and hence the intermolecular distances which change with temperature. The k,( T ) relationship, at constant pressure, in sucb a liquid is then practically linear; in other words, the specific
35 %D
Pressure p t107 pal
Fig. 3.2.3. The coefficient of thermal expansion of ocean water as a function of pressure at different temperatures (compiled from data in Popov et al., 1979).
3.2 THE THERMAL EXPANSION OF SEAWATER
121
volume of such a liquid rises in proportion to the temperature increase. This is the reason why the scale on a mercury thermometer is linear. The coefficient of thermal expansion will, of course, change as the pressure does (Fig. 3.2.3). Keeping in mind the picture of the molecular structure of water, we should remember that when subjected to a large external pressure, e.g. the pressure of the water column at great depths or the pressure of high-amplitude acoustic waves, the “loosely packed” clusters of water molecules will become squashed and will cease to play their usual role when changing temperature affects the water volume. The changes in the thermal expansion coefficient under the influence of pressure will be greatest in pure water where there are “loosely packed” clusters of molecules but no dense ion hydrates. The coefficient of thermal expansion ought thus to change the most with pressure in non-saline water; but in any water the effect of pressure on thermal expansion should be greatest at low temperatures (near freezing) at which there exist the largest numbers of “loosely packed” clusters that become crushed under pressure. This is confirmed by Table 3.2.2, in which we see that the greatest increases A in the coefficient TABLE 3.2.2
A comparison of coefficients of thermal expansion kT
at atmospheric pressure
=
(lo5N/m2) and at a pressure of lo7 N/m2 (Pa) for seawater of different salinity (based on data in Neumann and Pierson, 1966; and Dietrich, 1952) k T . lo6 [K-’1 T tK1
0
______
273
-
105
107
-67
-30
5
40
65
A = 28
__
107
105
107
105
113
207
222
-
303
107
A = 15
303 311 A=8
37
135 156 A = 21
237 250 A = 13
324 331 A = 7
93
177 195 A = 18
263 274 A = 11
337 343 A = 6
A = 32
88
___
293
A = 25
A = 37 20
105
283
of thermal expansion due to pressure are at low temperature (273 K) and salinity (Ox,). The higher this pressure is, the greater the thermal expansion at a constant applied pressure will be since a higher pressure at the same temperature means,
3 THE THERMODYNAMICS OF SEAWATER
122
in unit volume, a larger number of molecules jostling for space when heated (outside the low-temperature anomaly). As the values of kT on the plots and in the tables show, the empirical data completely confirm the qualitative conclusionsdrawn from the presented molecular model of the structure of seawater. Notice finally, that such a complex coefficient of thermal expansion as the function kT(S,T , p ) describes only the first term in the equation of state (3.1.5) expressing the relative changes in the specific volume of seawater. 3.3 THE COMPRESSIBILITY OF SEAWATER. POTENTIAL TEMPERATURE AND POTENTIAL DENSITY IN THE SEA
The compressibility of seawater is the thermodynamic process in which its specific volume changes as a result of pressure being applied to it. The process is described quantitatively by the coefficient of compressibility k, defined by (3.1.8). Its mechanism can also be readily explained in terms of the molecular model of the structure of seawater. As the pressure rises, the “loosely packed” clusters of molecules become squeezed until their structure breaks down, after which the average intermolecular distance diminishes. The mechanism of the reduction in the volume of water under pressure is therefore a complex one and the relationship between pressure and volume is non-linear. At first, when the pressures are still low, smaller pressure increases are needed to break down “loosely packed” clusters and to push the molecules closer together than when the molecules are already fairly tightly packed. The compressibility of water must therefore fall with a rise in pressure. Moreover, the compressibility of fresh or weakly saline water must be greater than that of salt water, since in the former there are more “loosely packed” clusters which can be squeezed together, while TABLE 3.3.1 The isothermal, isohaline compressibilitycoeffiicientk , =
-
of seawater under various extreme conditions
273 273 303 303
P IN/m21
S [%,I
k , . 10’’ [m*/Nl
0 0 0
0 35 35 35
5.25 4.67 4.17 3.28
1os
3.3 THE COMPRESSIBILITY OF SEAWATER
123
in the latter there are more incompressible hydrates crowded together (see Fig. 3.3.1). So the compressibility of seawater must also fall with increasing salinity: Fig. 3.3.2 clearly demonstrates this. Finally, as we know, a temperature rise breaks down the clusters, so there are less of them or they are smaller at a higher
Fig. 3.3.1. The isothermal, isohaline compressibility of distilled water and ocean water of 35Xv salinity as a function of pressure at various temperatures (based on Wilson and Bradley’s data, quoted in Horne, 1969).
0
10
20 Salinity
30 S l%ol
LO
Fig. 3.3.2. The isothermal, isobaric compressibility of seawater as a function of salinity at different temperatures and atmospheric pressure (based on data from Lepple and Millero, 1971).
3 THE THERMODYNAMICS OF SEAWATEL
124
temperature, and hence the compressibility of water also decreases as it gets warmer. This decrease in the compressibility coefficient when all three variables (S, T , p ) increase is obvious from Figs. 3.3.1 and 3.3.2, and its values under extreme conditions are given in Table 3.3.1. In many natural phenomena, for example, the periodic compression of a medium by the pressure of acoustic waves, compression is an adiabatic process. The pressure rises so rapidly that the medium, though warming up as it is compressed, has no time to exchange heat with its surroundings. When a medium is compressed or rarefied without any exchange of heat taking place with the surroundings, we describe the process with the aid of the coefficient of adiabatic compressibility
(3.3.1) The simple relationship k, = xk,, connects the isothermal compressibility coefficient with the adiabatic one, and x = C p ,s/C,, is the ratio of the specific heats of seawater of salinity S (for seawater 1.00 < x < 1.02, see equations (3.0.28) and (3.0.31)). A similar adiabatic compression or rarefaction of water occurs during changes in its vertical position in the body of the water, as for example, during convectional circulation, or when a sample of seawater is being hauled up to the surface from a great depth. While a large mass of water is being raised, it becomes more Sea surface
T---
+ Fig. 3.3.3. The adiabatic rarefaction of a water mass raised during circulation.
3.3 THE COMPRESSlBILITY OF SEAWATER
I25
rarified, and therefore cools as its pressure decreases on approaching the surface; heat from the surroundings usually manages to penetrate only as far as the thin external layer of the moving water mass. Such decompression of water, even if raised slowly, is accompanied by hardly any exchange of heat with the surroundings, and is therefore an adiabatic process (Fig. 3.3.3). This adiabatic cooling can be explained quantitatively on the basis of the thermodynamic principle given in equation (3.0.5) written in the form: da+pdcc = dQ. For an adiabatic process, dQ = 0, hence work during decompression p d a is done at the expense of a fall in the internal energy ds of the given water sample, in accordance with the equation dE =
-pda,
(3.3.2)
which implies a change in the temperature of this mass of water. The change in water temperature 6T during an adiabatic change in the pressure 8p exerted on the water mass is described by Kelvin’s formula of 1857 (3.3.3) where kT is the coefficient of thermal expansion (discussed above), and C p , is the specific heat of seawater (see equation (3.0.31)). We see from this equation that the ratio of temperature rise to pressure rise is proportional to the coefficient of thermal expansion (STIGp), kT. Remember, however, that the thermal expansion of seawater behaves anomalously, i.e., it can even take a negative value, at low salinities and temperatures around 273 K (see Table 3.2.1). When kT < 0, we must also have (ST/Sp), < 0, i.e., during the decompression (Sp < 0) which takes place on raising a mass of water whose coefficient kr is negative, its temperature rises (6T > 0); the converse takes place when the water is compressed. In seas where the salinity exceeds 24.1%,, the thermal expansion kT is always positive (as can be roughly worked out from Table 3.2.1), so deep ocean water cools as it is raised to the surface. The hydrostatic pressure, i.e., the pressure of a water column at depth z in the sea can be written in the form N
p ( z ) = i:di, 0
(3.3.4)
3 THE THERMODYNAMICS OF SEAWATER
126
therefore a small increase in pressure will be equal to g sp = -82. cc
(3.3.5)
Substituting this last expression in Kelvin's formula (3.3.3) we get (3.3.6)
The ratio (ST/&), is called the adiabatic temperature gradient and generally expresses the change in temperature of water raised adiabatically in the sea through a vertical distance of 1000 m. So, for ocean water at 278 K the adiabatic temperature gradient according to Kelvin's formula is (ST/6z), = 0.1 1 K/1000 m (see the component (dT/dz)a of the hydrostatic stability function (1.2.23) in Chapter 1). For various combinations of S, T , p , Ekman (1914) compiled tables of (sT/Bz),, later verified by Cox and Smith (1959). In consequence of this adiabatic decompression and cooling of sea water, the temperature I9 of a raised water sample measured at the surface will be different from the actual in situ temperature T at the depth where the sample was taken. This difference is clearly 6T = T - 8 . (3.3.7) Thus we distinguish the potential temperature 13 of water (with respect to the surface of the sea or any other position in the water column) from its temperature in situ T. So 35%, saline water at 273.16 K (OOC) at a depth of 10000 m has a potential temperature relative to the surface of 8 = 272.003 K (- 1.157"C). The adiabatic changes in water temperature under various other conditions can be found in tables (Oceanographical Tables, 1975). The adiabatic changes in temperature and pressure on raising a water mass clearly elicit corresponding changes in the specific volume or density as well. This is why the density at a given spot in the sea (in situ) differs from the potential density which this same water will have as the potential temperature at an adiabatically changed pressure in a new position in the water column. We shall now return to the compressibility of seawater, which is the reason why it cools when raised adiabatically to the surface. From what we said earlier and from empirical data, we know that the compressibility coefficient of water decreases inter a2ia with an increase in pressure. But as our water mass rises from depth z to the surface, it is subjected to a considerable pressure change, and so the values of the compressibility coefficient vary widely along the way up. It was
3.3 THE COMPRESSIBILITY OF SEAWATER
I21
therefore found necessary to introduce an average coefficient of compressibility for pressures from p ( z ) to p(O), i.e. for the water column from the surface down to depth z, and to find a connection between this average coefficient and its real values. Ekman did this in 1908 by introducing the average coefficient of isothermal compressibility
k-
= -
1 AdSY TYP) 7 “S, T Y 0) AP
where Ap = p(z)-p(0) hence
(3.3.8)
= p(z), A a ( S , T , p ) = a(S,
T , p ) - a ( S , T , 0),
(3.3.9)
(3.3.10)
If we insert expression (3.3.10)for a(S, T ,p ) into the equation for the compressibility kpywe obtain the connection between the real and average coefficient of compressibility (for the water column from 0 to z metres)
(3.3.11) Ekman worked out an empirical formula for the average coefficient of isothermal compressibility of water k, (often denoted by p in other publications), This was for many years a unique formula, used to calculate the density of seawater in situ (see (3.3.10))where below a depth of a few hundred metres we can assume that T z const. The Ekman formula, fairly complicated in appearance but straightforward in computer calculations, reads -
k , 109 ~ =
4886 - -227 28.33T-0.551 T Z 0.W4T3) 1 0.0000183p + p x 10-4(105.5+ 9.50T- 0.158T2)- 1.5p2Tx lo-*o0- 28 - -r147.3-2.72T+0.04TZ -p x 10-4(33.4-0.87T+ lo
+
+
uo-28 + 0.02T2)]-t (T)
+
+
2
x r4.5-0.1 T - p x 10-4(1.8 -O.O6T)],
(3.3.12)
3 THE THERMODYNAMICS OF SEAWATER
128
where the average compressibility k, is expressed in reciprocal decibars (dbar-l), the pressure in the sea p in decibars*, and the temperature in "C. The symbol q, in this formula denotes the so-called abbreviated specific gravity of seawater defined by (3.5.14) in Section 3.5. The average coefficients of compressibility calculated according to this formula are contained in oceanographical tables, although experts believe (UNESCO, 1976; Mamaev, 1975) that in the light of the latest research, this formula introduces certain errors with respect to the values actually measured. It is therefore proposed to introduce a new and more accurate formula to replace the Ekman formula for calculating the specific volume of seawater, which will be in line with the general equation of state in the form (3.3.10) (see Section 3.5).
3.4 THE SALINITY EFFECT ON THE SPECIFIC VOLUME OF SEAWATER
The volume contraction process whereby the specific volume of water changes with its salinity is particularly important in estuarine regions where waters of differing salinities mix. It is also manifest in areas where there is intense evaporation, where the water surface freezes, where icebergs and smaller calves of ice melt, and also in regions subject to bottom influxes of salt waters as, for example, in the Baltic. The molecular masses of the components of sea salt are for the most part far greater than that of water. The complexity of the process of volume change with salinity, however, results from the complexity of the seawater structure, in which ion aggregates of various sizes may form, electrostrictive effects due to these aggregates may take place and clusters of water molecules may fall apart. These processes, in which new structures affecting the specific volume of salt water are formed because of the presence of salts, must therefore also be temperature- and pressure-dependent. When the temperature T and pressure p are constant, an increase in the number of salt ions causes a proportionate increase in the number of ion aggregates. Therefore, a relative change in the specific volume of seawater Aala, is to a first approximation, proportional to a change in salinity 4S ($)T,p
=
-ks4S. -
(3.4.1)
The coefficient of proportionality k, E k, is the average value of the contraction coefficient (for intervals of AS) described by definition (3.1.7) and appearing
* 1 dbar = 104 Pa (the pressure of a 1 metre-tall water column).
3.4 THE SALINITY EFFECT ON THE SPECIFIC VOLUME O F SEAWATER
129
in the equation of state of seawater (3.1.5). The relationships between the specific volume a and the salinity S at various temperatures are illustrated in Fig. 3.4.1. These relationships have been calculated from Knudsen's formula which describes the dependence of the abbreviated density on the salinity, of which more in the next section.
0.93'
0
I
I
I
I
10
20 Salinity
s[%J
Fig. 3.4.1. The specific volume of seawater as a function of salinity at different temperatures, at atmospheric pressure and at a pressure of c. 1.3 x lo8 Pa (based on data Wilson and Bradley, and Newton and Kennedy, quoted in Popov et al., 1979).
130
3 THE THERMODYNAMICS OF SEAWATER
3.5 THE EMPIRICAL EQUATIONS OF STATE FOR SEAWATER
The dependence of the density e ( S , T , p ) or the specific volume a(S, T,p) on the salinity S, temperature T and pressure p is called the equation of state for seawafer.In a general differential form, this equation has already been formulated inrelationship (3.1.4) or in an often used equivalent (3.1.5). Because of the complexity of the functions k,(S, T , p ) , k s ( S , T ,p ) and k,(S, T ,p ) appearing in this equation, other ways of expressing the dependence of u(S, T , p), like equations (3.3.10) together with (3.3.12), are being sought. Historically speaking, such research was aimed at finding various special cases of this dependence using empirical methods. Many textbooks mention the names of scientists such as Knudsen, Forch and Ekman, all well known in the oceanographical literature, who worked out empirical formulae at the beginning of the 20th century. Their empirical formulae have lost nothing of their usefulness and continue to be used to compile oceanographical tables of seawater densities (Oceanographical Tables, 1975). Modern research, capable as it is of providing far more accurate data, has helped to make these formulae more precise and has modified them for the purposes of numerical modelling of water flows in the sea (Bryan and Cox, 1972; Friedrich and Levitus, 1972; and others). The monograph by Mamaev (1975) contains an excellent review of these various forms of the equation of state for seawater, while the latest practical thinking, equations and numerical data are presented in reports by the UNESCO group of experts on tables and standards (UNESCO, 1976, 1981) and in the papers quoted therein (see equations (3.5.23) to (3.5.32) below). The difficulties in finding an analytical form of u(S, T ,p ) have led to a search €or approximations as a polynomial whose degree depends on the degree of accuracy required by the description of this function. In theory we can get such a polynomial by expanding the function u(S, T , p ) into a Taylor series in the neighbourhood of the point (S, T , p ) corresponding to the assumed standard values of these parameters for pure or ocean water. The densities or specific volumes of natural waters do not differ much from, say, those of ocean waters under standard conditions, ie., those having a salinity of 35%, at a temperature T = 273 K (OOC) at atmospheric pressure, i.e. at a hydrostatic pressure of p = 0. These differences (3.5.la) 6a = a(S, T , p ) - cr(35,0,0) appear only in the second decimal place (in cm3/g) and can be treated as corrections to the deviations from standard values (3.5.1b) a(S, T , p ) = (x(35,O70)+8a.
131
3.5 THE EMPIRICAL EQUATIONS OF STATE FOR SEAWATER
The initial terms in the Taylor expansion, the functions of the three variables a(S, T,p) in the neighbourhood of the point (35,070) yield the following ex-
pression
(3.5.2)
from which we see that the sum of the consecutive terms with the derivatives of the function CI corresponds to the correction 6a in expression (3.5.lb). On expanding (3.5.2) we get a(S, T, p) = ~r(35,0,0)
+
aZa
asap dSdp+
as aT a3a
+2-
+3-
aZa
a2a
a2a
+-dT2+-dp2+2--dSdT+2aTz aP2
a3a
amp
+
a3a
a3a
dT2dp 3 ___ dp2dS+ 3 ap2
as
~
a p w
dp2dT+
asmap dSdTdp) + ...
(3.5.3)
a3a
+
This series is a convergent one, so further derivatives of higher orders contribute less and less to the total sum expressing the value of a(S, T, p). Breaking off the series after the third term generally furnishes a satisfactory approximation of this function. Notice, after Mamaev (1975), that if we rearrange the terms in the series (3.5.3) we obtain first adjacent series of expressions dependent on only one parameter, then mixed series containing terms with two parameters, and finally series with all three parameters: a(S, T,p) = a(35,0,0)+
...)+
3 THE THERMODYNAMICS OF SEAWATER
132
...
a2
1 a3a 1 a3a dSdp+ - __ dS2dp+ -__ dp2dSi2 ap2as 2 aszap
...)+
dT2dp+ 1 ___ a3a dp2dT+ 2 ap2aT
...)+
1 __ a3cc
2 aT2ap
+( aTasap a3u
dSdTdp+ ...).
(3.5.4)
The consectutive terms of this equation, contained within separate sets of brackets, ,~ can be denoted by the symbols 8us, 8uT, 8aP, ~ c L ~ 8, a~ s, , p ,~ u T , ~~ C, L S . Trespectively. The expansion (3.5.4) of the function u(S,T,p) can then be written briefly as a(S, T, p) = a ( 3 5 , 0 , 0 )
+ Sa, + Su, +8ap+8 ~ sT.+
CIS,
+B
~ T+. Sas, ~ T
, ~ .
(3.5.5) Thus, the consecutive components of the rearranged expansion of a@, T ,p ) can be treated as constituents of the correction 8u, expressing separately the effects of temperature, salinity and pressure on the specific volume and the interdependences of these effects. Hence we get
+ + +
8a = 8uS 8aT
8ap
8ES, T
+ Gas, + p
8c(T, p
f 8aS, T , p
(3.5.6)
Notice further that (3.5.5) allows us to write clearly special cases of changes in the specific volume : when the temperature and salinity are constant (standard) and only the pressure changes, that is, when the process is isothermal and isohaline
~ ( 3 5 , 0p) ,
=
a(35,0,0)
+ Sap
(3.5.7)
or, when the salinity and pressure are standard and only the temperature changes, that is, when the process is isohaline and isobaric
a(35, T , 0) = a(35, 0 , o)+6aT
(3.5.8)
or finally, when the temperature and pressure are standard and only the salinity changes, that is, when the process is isothermal and isobaric a(S, 0,O) = a(35,0,0)+8cr,.
(3.5.9)
3.5 THE EMPIRICAL EQUATIONS OF STATE FOR SEAWATER
133
On the other hand, we denoted the respective terms in (3.5.4) by 6as, 8aT and 8% (3.5.10)
so that the correction 6a, is also an approximation in the form of the new series (3.5.10), like the other corrections Bas and 6 a T . Taking into consideration two parameters in (3.5.4) or (3.5.5) already gives us more complicated formulae, e.g. the changes in specific volume due to changes in temperature and salinity for a sample of water at the surface of the sea ( p = 0). a(S, T , 0) = cz(35,0, O)+Sczs+BaT+Sas,T.
(3.5.1 1)
Finally, if we take into account all three parameters S, T, p simultaneously, we come back to the complete equation (3.5.5) which formally describes the specific volume of seawater in situ. The problem is now reduced to the accurate determination of the correction 6a as a function of S, T, p , which is extremely difficult to do. These questions are examined in detail in the monograph by Mamaev (1975) and in the papers by Bjerknes-Sandstrom et al. quoted therein. Notice too, that there is an obvious connection between the density e(S, T , p ) and the specific volume a(S, T ,p )
and that by differentiating this equation we get an expression for the increase in density (3.5.13)
By similarly expanding the function e(S, T , p ) into a Taylor series we obtain, as for the specific volume, corrections for the deviations of the density from the standard. The connection between these two expansions emerges from (3.5.12). In order to illustrate certain analogies between the equations we have just derived and the empirical formulae hitherto applied in oceanography for the dependence of a on S, T or p , we have to introduce the traditional concepts of abbreviated specijic gravity a. and abbreviated density aTused in these formulae (Oceanographical Tables, 1975). They were fixed by Knudsen in 1901 so that the density of seawater could be referred to that of a distilled water standard,
3 THE THERMODYNAMICS OF SEAWATER
134
and so as to facilitate a more convenient recording of multidigit values of this parameter in oceanographical tables. The abbreviated specific gravity of seawater of salinity S, at temperature T and atmospheric pressure is the name given by Knudsen to the following expression, known in oceanography as uo: (3.5.14)
where e,(S, 0,O)is the specific gravity of seawater at temperature T = 0°C at atmospheric pressure, i.e. at a hydrostatic pressure of p = 0; e,(O, 4,O) i s the specific gravity of distilled water at T = 4°C and atmospheric pressure. Notice that by subtracting one from the relative specific gravity (pg/egStandal,,) and multiplying the result by 1000 we can simplify the notation of such a value; i.e., instead of writing pg/egstandard = 1.01985, we simply write cro = 19.85. For the abbreviated specific gravity a. defined in this way, there is Knudsen’s well-known empirical formula which connects this value with the chlorinity CE
[%,I 00
= -0.069
+ 1.4708Cl- 0.0O1570Cl2+0.0000398C13,
(3.5.15)
and, in the case of formula (2.4.1), with the salinity S I%0]. The abbreviated density of seawater oT (sigma-T) was defined by Knudsen as follows: (3.5.16)
The ratio of the density of seawater of salinity S at temperature T and atmospheric pressure to that of distilled water at 4°C and the same pressure is contained within the brackets. The abbreviated specific volume of seawater is defined in the same way (3.5.17)
and these values have been used up to now in oceanography and included in oceanographical tables (see Oceanographical Tables, 1975; Diiing et al., 1980). Based on laboratory investigations, Knudsen’s empirical formula for determining the sigma-T value of seawater at atmospheric pressure is the following ‘T
=
2;.f (go +0,1324)[1 -A T 4BT(00 -0.1 3291,
(3.5.18)
3.5 THE EMPIRTCAL EQUATIONS OF STATE FOR SEAWATER
135
where
ZT = - (T- 3.98)'
T+ 283°C 503.570 T+ 67.26"C' AT = T(4.7867-0.098185T+0.0010843T2) x B T = T(18.030-0.8164T+0.01667T2) x
This formula expresses the dependence of oT on the salinity by means of co, and on the temperature T ["C] by means of the expressions for ZT,AT and 3,. Knudsen also wrote this dependence in the following form OT =
(3.5.19)
oO-D,
where
D
= (TO - OT =
-L'T-O.
1324
+ +0.1324)[AT-BT(oo -0.1 3291. (GO
(3.5.20)
ClearIy there is an analogy with expression (3.5.8). Employing data contained in oceanographical tables, Mamaev (1975) worked out, by the least square method a new expression for cT in the form of a polynomial OT
= 28.152-0.07357'-0.00469T2+
(0.802-0.002T)(S- 3 9 ,
(3.5.21)
which he assessed as being useful for ocean waters of salinities 0 < S < 40%, and temperature 0 4 T < 30°C and yielding resuits agreeing with the Knudsen formula with an error of less than lod4 g/cm3. For typical ocean water of salinity 32%, 6 S < 37%,, this error is less than 3 x g/cm3. For calculating the density of seawater at in situ pressures, empirical equations of state must take into account the pressure, that is, the compressibility effect of the water, besides the influence of temperature and salinity. The Knudsen formula (3.5.18) for IT^ can be used for this, together with the Ekman formula (3.3.12) for the compressibility, and relationships (3.3.10) and (3.5.12). This procedure leads to the method of calculating the in situ density of seawater using the Knudsen-Ekman formulae, still basic in oceanography. Bearing in mind the degree of accuracy required of present-day determinations of seawater density, the above formulae have a number of weak points. Two of the most important ones are the ambiguity of the definition of standard distilled water and the imprecision of the concept of salinity which they use. Distilled water prepared as an ordinary water standard from any source may have a varying isotopic composition (i.e., different percentages of '*O and D isotopes), one which is usually different from that of ocean water. The extent to which it is saturated with atmospheric gases can also vary, and this will affect the density of such a standard.
3 THE THERMODYNAMICS OF SEAWATER
136
The new, accurate standard pure water is now made from ocean water. It is called Standard Mean Ocean Water (SMOW) and is of a standard isotope content and is free from atmospheric gases (Craig, 1961). The maximum absolute density of such a standard distilled water at a pressure of 101325 pascals (1 standard atmosphere) and a tzmperature of 277 K (4°C) is em(SMOW) = 999.9750 kg/m3 (see Table 3.5.1). Such water is prepared by the International Atomic Energy Agency which distributes it in small quantities on demand to scientific institutions. The density of SMOW at various temperatures from 0 to 40°C at a pressure of 101 325 Pa is calculated in kg/m3 from the following empirical formula (M. Menach&-UNESCO, 1976) pSMOW= ao+a, T + a z T 2 + a 3 T 3 + a 4 T 4 + a 5 T 5 , (3.5.22) where a, = 999.842594, a, = 6.793952x 10-20C-1,a2 = -9.095290 x oC-2, a3 = 1.001685x a, = - 1.120083 x a5 = 6.536332~ With the aid of this equation, a table of the absolute density of SMOW has been compiled, part of which is shown in Table 3.5.1. The determination of the absolute density of other water samples requires their isotopic composition to be determined on a mass spectrometer which will enable the appropriate correction to be applied (UNESCO, 1976). O C 4 ,
O C 3 , O
C
5
.
TABLE 3.5.1 The absolute density of Standard Mean Ocean Water (SMOW) e tkg/m31, free from atmospheric gases, at a pressure of 101 325 Pa and at temperatures T "Caccording to the International Practical Scale of Temperatures of 1968 (values at selected temperatures from the UNESCO tables, 1976)
0.0 4.0 10.0 15.0 20.0 25.0 30.0 35.0 40.0
999.8426 999.9750 (max) 999.7021 999.1016 998.2063 997.0480 995.6511 994.0359 992.2204
3.5 THE EMPIRICAL EQUATIONS OF STATE FOR SEAWATER
137
Recent precision experimentation (Cox et al., 1970; Krembling, 1972; Millero et al., 1975) shows that there exists a systematic difference in specific volume
between the values obtained from the Knudsen formulae and the measured values, which are usually higher. As regards the specific volume of 35%,, salinity ocean water at atmospheric pressure and at temperatures naturally occurring in the sea, this difference is on average (8.7F 1.0) x cm3/g, while the greatest cm3/g, occurs underrating of data calculated by Knudsen’s formula, 1 x within the temperature interval from 10 to 15°C. In the light of these comparisons, the accuracy of the data calculated for ocean water using Knudsen’s formula is of the order of 1 x cm3/g. Results which also take into consideration the pressure in the sea according to the Ekman formula of 1908 (see equations (3.3.12) and (3.3.10)), i.e. for the in sifu specific volume are lower still in comparison with the results of the latest research (Chen and Millero, 1975; Fine et al., 1974): for ocean water of 35%,, salinity, this underrating ranges from 18 x low6cm3/g at a pressure of 1 x lo7 Pa to 89 x cm3/g at a pressure of I x lo8 Pa. Because of the nature of the pressure effect on thermal expansion, the accuracy of measurements obtained using the Knudsen-Ekman formulae for ocean water is estimated cm3/g at 10’ Pa to be from 3 x cm3/g at atmospheric pressure to 5 x (UNESCO, 1976). Thus one can gauge the accuracy of the universally applied oceanographical tables in which these formulae have been used and which, of course, will long serve their purpose whenever a precision greater than the above is not required. Because of the systematic deviations of the values obtained using the KnudsenEkman formulae from the new high-accuracy results using pure water standards, various revised versions of the empirical equation of state for seawater have been proposed. These latter are based infer aha on the results of detailed studies by Chen and Millero (1976), Millero, Gonzales and Ward (1975), and others. And these new versions of the equation of state have in their turn been disputed and revised (Gebhart and Mollendorf, 1978). The upshot of all this is that the afore-mentioned international group of experts on tables and standards (see UNESCO, 1981), besides introducing a practical scale of salinity, has also verified and established a new empirical equation of state for seawater (1980). This equation makes use of the practical salinity S of seawater, the temperature T expressed in centigrade and the hydrostatic pressure p in bars (1 bar = lo5 Pa). In these units, the equation for the density of seawater e [kg m-3] as a function of S, T, p is expressed as follows:
e(X T ,PI
= e(S, T , O)/[l
- P / W ,T , P)l,
(3.5.23)
3 THE THERMODYNAMICS OF SEAWATER
138
where K(S, T , p ) here is the secant bulk modulus, further described by the expression in (3.5.26). On the other hand, the specific volume CI = l/e is given by a(S, T ,P) = a(S,T , 0)r1-PIK(S, TY Pll*
(3.5.24)
Q(S,T , 0) in (3.5.23), i.e. the density of seawater at a pressure of one standard atmosphere ( p = 0), can be derived from the following dependence: ~ ( 5 ' 7
T , 0) = ew+(bo+b,T+b2T2+b3T3+bqT4)S+ (co c, T+ c2 T 2 )S312 do S 2 ,
+ +
+
(3.5.25)
where b, = 8.24493 x lo-', b3 = -8.2467 x lo-', co = - 5 . 7 2 4 6 6 ~ do = 4 . 8 3 1 4 ~
b, = -4.0899 x b4 = 5.3875 x c, = 1.0227~
b,
=
7.6438 x lo-',
c2 = - 1 . 6 5 4 6 ~
whereas ew = &Mow is the density of pure SMOW given by (3.5.22). The secanf bulk modulus K of seawater is given by the expression
+
K(S, T ,p ) = K(S, T , 0)+Ap Bp2, where, when the pressure p = 0, K(S, T , 0) = K,
+ (fo +fi T+f2T 2
+f3
(3.5.26) T3)S+(go+ g , T+ g2 T2)S3'2, (3.5.27)
whereas the coefficients in these expressions are equal to f o = 54.6746, fi = -0.603459, f 2 = 1.09987 x f 3 = - 6 . 1 6 7 0 ~lo-', go = 7.944 x lo-', g , = 1.6483 x g, = -5.3009 x A = A,f(io+i, T+i2T2)SfjoS3'2, (3.5.28) where io = 2 . 2 8 3 8 ~ i, = - 1.0981 x j o = 1.91075x B = 3,+(m , +m1 T+m2 T2 ) S ,
i, = - 1.6078 x
(3.5.29)
where
m,
= -9.9348 x lo-',
m 1 = 2.0816 x lo-*,
m2 = 9.1697 x lo-".
The values K , = KsMow,A , 5 ASMOW,B," I BsMowfor pure standard mean ocean water are given by polynomials
K,
= eoSe, T+e2 T 2+e3 T3+e, T4,
(3.5.30)
3.5 THE EMPIRICAL EQUATIONS OF STATE FOR SEAWATER
139
where eo = 19 652.21, el = 148.4206, e2 = -2.327105, e, = -5.155288 x
A, = ho+h, T f h , T 2 + h 3T 3 , where h, = 3.239908, h, = 1 . 4 3 7 1 3 ~ h, h3 = -5.77905 x B,,, = ko+k,T+kzTZ, where ko = 8.50935~loW5, k , = - 6 . 1 2 2 9 3 ~
e3 = 1,360477~lo-,,
(3.5.31) =
1.16092~ (3.5.32) k2
=
5 . 2 7 8 7 ~lob8.
This whole algorithm, which goes by the overall name of the International Equation of State for Sea Water, is valid over intervals of practical salinity S from 0 to 42, of temperature T from -2 to 40°C and of pressure p from 0 to lo00 bars. The conversion of salinity from the former units to practical scale units is discussed by Lewis and Perkin (1981). The following data are given so that the reader can check for himself the correctness of the calculations from the formulae given above, the respective units being Q [kg m-3], T [“C],p [bar], K [bar] (from UNESCO publication No. 36, 1981): -
S
0 0 0 0 35 35 35 35
T [“Cl
P bar1
5 5 25 25 5 5 25 25
0 1000 0 1000 0 1000 0 lo00
K V ,T ,P) ___
999.966 75 1044.128 02 997.047 96 1037.90204 1027.675 47 1069.489 14 1023.34306 1062.538 17
[bar] 20 337.803 75 23 643.525 99 22 100.721 06 25 405.097 17 22 185.933 58 25 577.498 19 23 726.349 49 27 108.94504
Very slight errors may creep in when these formulae are used with respect to atypical seawaters whose chemical compositions deviates from the usual (see Millero et al., 1976, 1978; Poisson et al., 1980, 1981; Lewis, Perkin et al., 1978; see UNESCO, 1981). Other papers dealing with the equation of state of seawater include that by Fofonoff and Bryden (1975), while Mamaev (1975) reviews earlier work in his monograph.
140
3 THE THERMODYNAMICS OF SEAWATER
A final comment to this chapter: notice how much importance is attached in oceanography to the exact determination of seawater density. This is justified by the many laws of physics and hydrodynamics in which density appears as a sensitive parameter in a number of processes, e.g. the hydrostatic stability of waters (see Chapter l), turbulence in the sea (Chapters 6 and 7), sound propagation (Chapter 8), sediment deposition (Chapter 2), etc. Although the mass of water in the oceans is gigantic, even minimal differences in the density of the water in this fluid environment can provoke colossal effects in flows of water, and the transfer of heat and chemical substances.
CHAPTER 4
THE INTERACTION OF LIGHT AND OTHER ELECTROMAGNETIC RADIATION WITH SEAWATER. THE IN HERENT OPTICAL PROPERTIES OF THE SEA
Electromagnetic waves carry energy whose electrical field acts on the molecules of the medium through which it passes. Such interaction excites the electrons of atoms and molecules, sets up oscillations and rotational vibration of molecules, ionises atoms, breaks chemical bonds (causing molecules to dissociate), and gives rise to a number of other reactions. Some of these excited or ionised atoms or molecules react chemically with other molecules to yield products which are often of great importance in nature. On the other hand, the particles of matter tend to prevent the free propagation of electromagnetic radiation within the medium: they scatter it in all directions and absorb its energy, in the long run converting it by these processes into heat, or in certain cases, into chemical energy. This is manifested by a temperature rise or a chemical change in the irradiated substance. The mechanisms and effects of the interaction between electromagnetic radiation and matter depend primarily on the frequency of the waves and on the nature of the constituents of the irradiated substance. This interaction is one of the fundamental problems of physics and far exceeds the confines of this book. But we shall just recapitulate briefly the principal phenomena which explain the macroscopic picture of radiant energy transfer and conversion in the sea. Electromagnetic radiation in the sea is basically sunlight, which is the primary source of energy in all natural processes taking place in the marine environment. A huge quantity of solar radiation is continuously pouring into the sea where, as a result of atomic and molecular excitations, it is converted into heat and other kinds of energy. This, a key process in nature, takes place on a gigantic scale, but none the less always as a result of an energy transfer to the molecules of the medium by elementary wave trains, called photons, during their individual “collisions” with these molecules. Light rays, including solar rays, are not a continuum of electromagnetic waves but comprise thousands of millions of finite wave trains emitted by the single atoms or molecules of the light source. So, for example, the influx and absorption of 1 J of red light energy of wavelength 1 = 700 nm by the medium is tantamount to the influx and absorption of over 3 . 5 x 10” photons in as many collisions with molecules. This figure emerges
4 THE INTERACTION OF LIGHT WITH SEAWATER
142
from the well-known quantum mechanical relationship between the energy of a photon E,,, the frequency of the vibrations v and Planck's constant h = 6.62517 x x J s, i.e.
E,
=
(4.0.1)
hv = h--, CO
R
where co = 3 x los m/s is the velocity of light in a vacuum. Since the motion of light is uniform, we can express this velocity as a simple relationship of distance travelled to time, most conveniently as the distance of one wavelength A to the TABLE 4.0.1 The relationship between wavelength A, frequency v, photon energy E, and the colour of light ~
Wavelength [nml
100 150 200 250 300 350 400 450 500 550 600 650 700 750 800 1000 2000 3000 5000 10 000 20 OOO 50 OOO
Frequency
3.001 x 1015 2.000 x 1015 1.500~ 1015 1 . 1 9 9 ~1015 9.993 x 1014 8.566 x 1014 7.495 x 1014 6 . 6 6 2 ~1014 5.996 x 1014 5.451 x loL4 4.997 x 1014 4.612 x 1014 4.283 x 1014 3 . 9 9 7 ~1014 3.748 x 1014 2.998 x loz4 1 . 4 9 9 ~1014 9.993 x 1013 6.038 x lOI3 3.019 x 1013 1 . 5 0 9 ~1013 6 . 0 3 2 ~10l2
Photon energy E,[10-19 J]
19.88 13.25 9.93 7.95 6.62 5.68 4.97 4.41 3.97 3.61 3.31 3.06 2.84 2.65 2.48 1.99 0.99 0.66 0.40 0.20 0.10 0.04
Photon energy E,[eV]
12.41 8.27 6.20 4.96 3.87 3.54 3.10 2.76 2.48 2.25 2.07 1.91 1.77 1.65 1.55 1.24 0.62 0.41 0.25 0.13 0.06 0.03
._
Number of photons per joule of energy
0.503 0.756 1.006 1.258 1.510 1.761 2.012 2.268 2.519 2.770 3.021 3.268 3.521 3.774 4.032 5.025 10.10 15.15 25.00 50.00 100.00 250.00
Colour of light (kind of waves)
ultra-violet
violet blue green greenish-yellow orange red
infra-red
far infra-red
4 THE INTERACTION OF LIGHT WITH SEAWATER
143
time of the wave period T, during which light travels that distance. In other words, co = A/T = Av, where the wave frequency v = IIT. From (4.0.1) we see that the energy of a single photon of wavelength il = 700 nm is c. 2.8 x J, and the frequency of this wave is of the order of l O I 4 Hz. Obviously, the energy of a single photon increases in proportion to the frequency Y, but decreases as the wavelength increases (see Table 4.0.1). The effects on a medium of light of different wavelengths must therefore be various. Photons of ultraviolet radiation carry far more energy than those of infrared radiation. On passing through a medium, the former can, for example, ionize certain atoms or excite their electronic energy levels, or break chemical bonds, whereas the energy of the latter is generally barely sufficient to intensity molecular oscillation or rotation. In this process, the energy of the photon is transferred to the molecule, whose energy level is now raised: this means that, the photon is absorbed by the molecule. As we know, the energy levels of atoms, molecules, and arrays of molecules in matter are quantized. The excitation of atoms or molecules to free vibration is therefore most likely when they collide with a photon of a particular resonance frequency Y, , and hence of a strictly defined energy hv, , The molecules of the various constituents of a medium, having as they do various excitation energies (definite quantum conditions), will therefore absorb photons selectively. This means that light of different wavelengths is absorbed by the medium to different extents, depending on the composition and nature of the medium’s constituents. The dependence of the extent to which a medium absorbs light energy on the wavelength of that light is called the absorption spectrum of light in the given medium. The differences between absorption spectra in different seas therefore characterize the differences in the nature of the water in those seas. The waters in different seas vary because of the biological, geochemical and thermodynamical processes taking place in them, and these are indirectly detectable with the aid of absorption spectra. We shall discuss the absorption of light in the sea in Section 4.2. Apart from absorption, an extremely important phenomenon accompanying the transfer of light through a medium is scattering. This also happens when the electrical field of a wave interacts with the charges of the medium’s molecules. According to one of the accepted mechanisms of scattering (Rayleigh, 1871), it is assumed that under the influence of an electromagnetic wave, the molecules of a medium become vibrating electrical dipoles which radiate secondary waves in various directions, that is, they scatter part of the energy derived from the primary waves. In effect, rather than being absorbed, some of the numerous photons change their direction of motion. For this very reason, these photons
144
4 THE INTERACTION OF LIGHT WITH SEAWATER
of sunlight, originally travelling along direct sun rays, change their directions in the atmosphere and sea many times even, so that diffused light, visible at every point above the horizon and within the sea itself, is produced. Since light is scattered, the path taken by photons through a given layer of a medium is greatly extended. Along this extended path there is a much greater probability of a photon colliding with a suitable molecule which can absorb it in accordance with the quantum rules. The fact that light is scattered thus has an important, though indirect, influence on the efficiency with which its energy is absorbed by a medium containing absorbant molecules. We shall deal with light scattering in greater detail in Section 4.3. The sum total effect of the scattering and absorption of light as it passes through a medium is called attenuation. The total attenuation is the process determining the transmittance of light-beam energy of various wavelengths through a medium and is a characteristic optical property of various seawaters. It will be discussed in Section 4.4. Electromagnetic waves other than light differ from the latter only in their wavelength 1 and hence in the energy of their photons in accordance with formula (4.0.1). Their passage through a medium is therefore accompanied by the same effects as those involving light, i.e. absorption, scattering and the resultant attenuation of the radiation transferred in a certain direction. The absorption efficiency in this case depends in just the same way on the presence of molecules in the medium capable of absorbing energy quanta hv. As we shall see, the scattering of electromagnetic waves depends on unhomogeneities in the medium, known as scattering centres (e.g. suspended particles), and on the dimensions of these centres with respect to the wavelength of the radiation. We recall the well-known fact that the non-uniform, scratched and even rusty surface of an ordinary sheet of metal is an excellent reflector of radio waves because these are very long in relation to the dimensions of the unhomogeneities on the metal surface. On the other hand, light waves, whose wavelengths are short as compared with the dimensions of the surface unhomogeneities, will be strongly scattered. The dimensions of the unhomogeneities of an aqueous medium with respect to wavelength exert a similar effect on the scattering of electromagnetic radiation passing through such a medium. As Section 2.7 on the dimensions of suspended particles in the sea has shown, these have a significant effect on the scattering of light waves but hardly any on that of long-wave electromagnetic radiation like radio waves. The scattering and absorption of electromagnetic waves in the sea, outlined here in brief, will be discussed in detail in the following sections of this chapter.
4.1 RADIANCE AND OTHER BASIC PHOTOMETRIC QUANTITIES IN...
145
4.1 RADIANCE AND OTHER BASIC PHOTOMETRIC QUANTITIES IN HYDROOPTICS
A uniform set of optical quantities applicable in hydrooptics and their symbols was compiled by an international group of experts drawn from the International Association for the Physical Science of the Ocean (IAPSO). The symposia and other scientific activities of this association have had a decisive influence on the development of physical oceanology, including hydrooptics, and so in this book we shall be employing the IAPSO-recommended terminology and symbols of the optical quantities (see Jerlov, 1976). The terminology and definitions of these quantities are largely based on the work of Preisendorfer (1961) who proposed a uniform set of operational definitions of the optical properties of the sea and the two-flow analysis of the light field in the sea (upward and downward flux in the horizontal homogeneity model). The origins of his reasoning and his successful attempt at unifying the description of radiation propagation in the sea go back to atmospheric optics, which had developed far earlier, and to Chandraselchar’s theory of radiant energy transfer (1950). In the accepted set of photometric quantities, apart from light intensity and irradiance, a basic parameter is the radiance. In order to clarify the description of the interaction and transfer of light energy in the sea, the appropriate quantities will now be defined. Notice that the wave surfaces of an electromagnetic wave determine the planes of a uniform phase oscillation of the electric vector E, of this wave. The lines perpendicular to the wave surfaces, whose direction at any point in space is given by the wave vector k, are called Zight rays, where [k[ = 2x11. The light rays thus determine the directions in which electromagnetic waves are propagated, that is, the directions in which radiant energy is propagated (Fig. 4.1.1a). Light rays emitted from a point source in all directions along the geometrical radii of the sphere enclosing the source represent a spherical wave; parallel rays on the other hand represent a plane wave, in which the wave vectors at any point are identical and parallel to one another. Let the quantity of radiant energy be denoted by Q; it is usually expressed in joules [J], but may also be given in calories [call or electronvolts [ev]. It is the overall quantity of energy transferred by electromagnetic radiation in an indeterminate direction and time. By means of this radiant energy quantity Q, we can define the radiant flux F as (4.1.1)
146
4 THE INTERACTION OF LIGHT WITH SEAWATER
which is the quantity of energy transferred by radiation in unit time, but usually we have to state the direction of this energy transfer. Since the rays emitted by a source propagate in all directions, we can exactly define the direction of radiation transfer (Fig. 4.1.lb) using the unit vector or the direction angles of this vector (0, @), but only when we take into consideration a portion d F of the radiant flux that is propagated through a very small solid angle around that direction In this way, we can define the direction of the flux dF(g) contained within the solid angle dl2g). The angular densitj~of the radiantflux thus defined in a given direction is described by
a@.
(4.1.2)
sin 6 dt? d@
4.1 RADIANCE AND OTHER BASIC PHOTOMETRIC QUANTITIES
147
_‘&I= Fig. 4.1.1. Geometrical sketches complementing the definitions of radiant flux F and radiant
intensity I. (a) Wave surfaces and light rays emitted from a point source; (b) a solid angle and the direction of radiation; (c) maintaining the direction of light of intensity ICE) = -with the aid of a convergent lens (‘‘pencil of radiation”).
AWE)
27C
vector, [kl = --, AQ-element of a solid angle A whosevalue is finite; dQ-inlinitesimally small element of a solid angle: Ee-electrical intensity vector of the electromagnetic wave; Qquantity of radiant energy; F(Q-radiant flux in direction 5; %-unit vector determinii direction in space; 0, @-angular coordinates; dQ(n-element of a solid angle around direction 6 or direction (0, 0 ) ; dWE) E dQ(0, 0); r(@-iintensityof radiation in direction 5.
Symbols: So-point source of waves; A-wavelength;
k-wave
and is called the intensity ojradiation or light intensity (of a source in a given direction). The intensity of radiation therefore expresses the radiant flux emitted by a source in an infinitesimal cone containing the given direction, divided by a solid angle of the cone. The light intensity actually measured is generally the mean radiant intensity AF/AQ through a small, but not infinitesimally small solid angle AQ so that in long path any lateral deviation from direction ,E can be neglected. The rays can be kept parallel artificially by placing a convergent lens perpendicular to the direction 5 on the axis of the flux AFE) emitted by a point source of light, at the focal distance from the source (Fig. 4.1.1~). So from a divergent beam we get a parallel beam of intensity I. In such a beam, in which ideally the rays are parallel, the light waves are plane waves, which are particularly useful in optical experiments. Of course, the lens refracts different wavelengths of light through different angles, so, assuming the lens is free from defects, such an ideally parallel beam is obtainable only from a point source of monochromatic light. In practice, we mostly have to deal with continuous sources, not point sources of light, that is, with surfaces or volumes of bodies excited to light emission or which reflect or scatter light. The concept of radiance is introduced for light from non-point sources. The individual points on the surface of a source are point sources, and so the flux emitted by a non-point source and the radiant intensity of the source in a given direction can be referred to a given point in the source. This point, which is exclusively a non-dimensional geometrical creation,
148
4 THE INTERACTION OF LIGHT WITH SEAWATER
fc)
Fig. 4.1.2. Geometrical sketches accompanying (a) the definition of the radiance of light emitted by an element of the source surface dA; (b) the definition of the radiance of light corning from direction ,E to a point in space surrounded by the element of area d A : ; (c)the definition of vector irradiance E and scalar irradiance Eo . n-a unit Vector normal to the element of area dA (flat horizontal or spherical respectively) surrounding P i n t ( x, Y , 2) in space.
4.1 RADIANCE AND OTHER BASIC PHOTOMETRIC QUANTITIES
149
has, however, to be described physically, even as an infinitesimally small element of the surface of the light-emitting source dA containing that point. This is how we consider the light intensity emitted by an infinitesimally small element of the source area dA through an infinitesimally small element of the solid angle Urn).The light Aux thus defined, flowing in direction 5 (4.1.3)
is called the radiance (Fig. .4.1.2a). Here dA, = dAcos0 is the projection of the surfacearea of the source dA onto a plane normal to the direction of the light flux under consideration (Fig. 4.1.2). Taking into account the intensity (4.1.2), the definition of radiance (4.1.3) is equivalent to the expression (4.1.4) The radiance is therefore the flux of radiant energy per unit solid angle per unit projected area of a surface. We shall return to the concept of radiance a little later in order to distinguish between the radiance emitted by an element of the source area and that incident on an element of an irradiated surface. The light flux being propagated in space at any point passes through a geometrical area (e.g. the cross-sectional area of a beam of light at any point) or is incident on a real surface of a body, which it illuminates. We must then know the irradiance on the illuminated surface, that is, the radiant flux incident on a unit area of that surface. We can generally express this by the ratio (4.1.5)
In classical electrodynamics and its applications to the description of light scattering, the concept of light intensity expressed as the ratio (4.1.6)
is commonly used; the flux d F transfers a directed beam of light perpendicularly through an element of its cross-section dA,. We shall be using the intensity I ' when we describe the theories of light scattering, so as not to complicate this description with additional rearrangements of classic equations. None the less, we have to remember that the intensity I' is a special case of irradiance, i.e., it is the irradiance of the cross-section of the beam of light En.It is easily demonstrated that irradiance by a divergent beam emanating from a point source diminishes
150
4 THE INTERACTION OF LIGHT WITH SEAWATER
with the square of the distance r from the source (in other words, when there is no attenuation of the light in the medium, I'(r) = I'(r = O)/r2). This results from the geometry of divergent rays representing a given radiant flux d F which as one moves away from the source, is incident on an ever larger spherical area. Regardless of this rule, in a real medium light can, of course, be additionally attenuated or referated by the medium itself. In the sea, we are often interested in the sum of the fluxes dFG) of light arriving (owing to scattering) from all directions of space at a given point (x, y , 2). Here too, as for a point source, we have to take an infinitesimally small but nevertheless real area dA containing the point in question, in which we could measure the irradiance of the incoming radiation. This area may be an element of a horizontal surface dA containing the given point at depth z in the sea, or it may be an infinitesimally small, closed, spherical area surrounding the given point at depth 2. Depending on whether we refer the sum of incident radiation to an element of a plane or to an elemental sphere surrounding a given point, we obtain two different irradiances at our point in space, called respectively the normal or vector irradiance E (because of the given direction of the plane) and the scalar irradiance Eo at the point (x, y, z) in space. We shall further introduce detailed definitions of these irradiances with the aid of the radiance of the light reaching a given point in space (Fig. 4.1.2~).The definition of this radiance does, however, require some further elucidation, from which it emerges that the radiance of light incident in a solid angle, whose apex is at the illuminated surface, is described in exactly the same way as the radiance of light emitted through such a solid angle whose apex is at the point source (Fig. 4.1.2b). Let us pause for a moment at Fig. 4.1.2b. Let us assume that, as we can see from the drawing, the element of the area of the source dA, emits in a normal direction a flux of radiation dF through a solid angle Ctn, which illuminates entirely a normal area dAA a distance r from that source. According to definition (4.1.4), the radiance of this flux is equal to dF L= (4.1.7a) dOdA, Again, the radiance describing the radiant intensity incident on an element of area dAi can be written in analogous fashion (4.1.7b) It is a simple matter (Jerlov, 1976) to show that so long as there is no attenuation or any other source of light along the distance r, both these radiances are equal
4.1 RADIANCE AND OTHER BASIC PHOTOMETRIC QUANTITIES
151
to each other L = L'. The radiance L can therefore describe both the radiant flux emitted in a given direction by an element of a source surface through a unit solid angle, and the radiant flux incident upon an element of area of a receptor from an element of a solid angle around a given direction. From the general definition of irradiance (4.1.5) and radiance (4.1.4), it now emerges that the irradiance of an element of area perpendicular to the direction of the illuminating flux can be expressed as dEo = L(g)dQ(Q.
(4.1.8)
So the irradiance of an infinitesimally small spherical surface surrounding a given point (x, y , z) by the radiance of light coming from all directions 5 of a sphere Q to that point-the scalar irradiance-will be (4.1.9a) Considering that when integrating over the sphere we sum the solid angles dQ(5) = sin%d%d@, we can express this same scalar irradiance using the direction angles 2n x
Eo =
1 SL(6, @)sinOdOd@.
(4.1.9b)
0 0
The irradiance on a normal plane (e.g. a horizontal plane at depth z in the sea) by an irradiating element of the light flux from direction 5 will be
dE = In. 51-mdQ(5),
(4.1.10)
where n is a unit vector normal to the irradiated surface, In * 51 E IcosOl, whereas angle 8 here is the angle between these vectors, i.e. between the normal to the irradiated surface and the direction of the radiance of the incoming light. The irradiance on an element of a plane z by the radiance from all directions of a sphere i2 is therefore equal to (4.1.1 la) which when written using direction angles, reads 2z x
E=
5 IcosOIL(6, @)sinOdOdd,.
0 0
(4.1.1 lb)
4 THE INTERACTION OF LIGHT WITH SEAWATER
152
In the two-flow analysis of the light field in the sea, that is, of the flux from the upper and lower hemispheres, with respect to the horizontal plane z under consideration, we can distinguish between the downward irradiance E , and the upward irradiance E , , which can be both scalar and vector irradiances. In equations (4.1.9) or (4.1.11) we integrate respectively over the upper hemisphere of the angles, i.e. 0 6 8 < in or over the lower hemisphere +r 6 0 6 z, adding up the radiances of the light arriving from all directions of the upper or lower hemisphere (we assume 0 to be the angle at which the given direction deviates from the zenith, @ is the azimuth). At depth z in the sea, we therefore get the scalar downward irradiance 2 X x/2
(4.1.12) 0 0
and the scalar upward irradiance 2X
Eo,(z) =
1
X
L(z, 8 , @)sinBd0d@.
(4.1.13)
0 x/2
From the above definitions and (4.1.9b) we can write
EOG) = Eo 1. (4+Eo t (4*
(4.1.14)
The corresponding relationships for the vector irradiances of a horizonta1 plane are (4.1.15) (4.1.16) The applications and characteristics of these irradiances in the sea will be discussed in Chapter 5. Besides the notation E , , E , , E,, , Eor, the subscripts u and d for upward and downward are commonly used with irradiances-E,,, Ed, Eo,, E0dand also with other optical quantities applied in the two-flow analysis of the light field in the sea. Notice that all the irradiances defined above can be determined from the direction distribution of the radiance L(0, @), hence the radiance, as we have said, is one of the basic photometric quantities in hydrooptics. For in situ measurements of the radiance L in the sea we use suitably sensitive photometers known as radiance meters. An important part of a radiance meter
4.1 RADIANCE AND OTHER BASIC PHOTOMETRIC QUANTITIES
153
is the precision optical collimator which coIlects radiation from only a very small solid angle surrounding the desired direction (e.g., an aperture A0 = 0.5"). To ensure the adequate sensitivity of such a meter, this collimator is constructed (a) Cable
I
Watertight window
-f-------i (c)
(b)
o
Flat collector
-L
Spherical collector
_1
Watertight window
Field
~
I
Interference filter
Screen
Lens Screens
I Detector
,Watertight housing
Cable
Fig. 4.1.3. The mode of action of light meters in the sea: (a) radiance meter; (b) upward or downward irradiance meter (flat light collector); (c) upward or downward scalar irradiance meter (spherical light collector with screen). Meters (b) and (c) differ only in the type of removable, flat or spherical light collector they employ and which can be placed over the window of the radiance meter.
4 THE INTERACTION OF LIGHT WITH SEAWATER
154
from a system of lenses and screens (see Fig. 4.1.3). Thus it lets into the light detector (photomultiplier) a relatively broad beam of rays travelling almost parallel, i.e., from a given direction at an angle no wider than the one assumed, The divergence of such a beam is given by the ratio of diameter of the apertured to the focal length f of the system of convex lenses. The light wavelengths to be analysed are selected by an interference filter, or in more sophisticated equipment, by an optical monochromator. A difficult technical problem in hydrooptics is the measurement of the directional distribution of radiance deep down in the sea, Apart from being watertight and sea-proof, a radiance meter must also be selfpropelling and remotely controllable. The point is that the positioning of such a meter in the required directions, whilst suspended at the end of a cable, and the reception of the requisite signals on board the research ship must be controllable therefrom (Sasaki, 1964; Lundgren, 1971; see also Kalinowski and Dera, 1968). The practical application of the definition of irradiances for calculating them from the radiance distribution measured in the sea requires a numerical method with discrete values of the radiance Li in the particular solid angles AQi of a spherical lattice. Tyler et al. (1959) describe a method of such numerical integration. Because of the great difficulties, already hinted at, of measuring the directional distribution of radiance in the sea, irradiance is in practice usually measured directly in situ. Irradiance meters equipped with appropriate light collectors automatically integrating the radiant flux from all directions according to one of the deEnitions (4.1.12), (4.1.13), (4.1.15) or (4.1.16) are used for this reason, A basic type of collector is the Lambert cosine collector (Smith, 1969). This is a flat disc of milk glass having a very small refractive index with respect to water and capable of strongly scattering light. It collects and transmits to the detector the component fluxes of radiation incident from direction 8 in proportion to dE, = dL(8)cosO.
(4.1.17)
So when light falls on it from various directions, a flux of light proportional to integral (4.1.15) or (4.1.16) (depending on how the instrument is oriented in space) is transmitted to the detector (Fig. 4.1.3b). The collector is placed on the window of the radiance meter, which in effect creates an upward or downward irradiance meter if the collector has a solid angle of observation of 2 x of the upper or lower hemisphere in the water. Like a radiance meter, an irradiance meter also requires the wavelength of the measured light to be set by a suitable optical filter or monochromator placed over the photosensitive element of the detector. The upward or downward scalar irradiance can also be measured in situ
4.2 LIGHT ABSORPTION IN SEAWATER
155
using a meter with a milk-glass sphere for an optical input, every element of whose wall is a Lambert cosine collector (Fig. 4.1.3~).On the meter side, this sphere must be masked by a large, flat, black (non-reflecting) screen which cuts off the light incident from one hemisphere (upper or lower) from the field of view. The detector “sees” this sphere through a small opening in the screen, but does not itself block the field of view as it is in the covered hemisphere. Some details of the mode of action of these meters are explained by Fig. 4.1.3, and others can be found in the cited literature (hanov, 1975; JerIov, 1976; Shifrin, 1972 and 1981).
4.2 LIGHT ABSORPTION I N SEAWATER
The energy of photons absorbed in the sea is determined by the permitted energy transitions of the atoms and molecules of seawater constituents, and it also takes account of the influence of combinations and interactions between them. The different concentrations of these constituents and the different probability of exciting given energy states in them further qualifies the extent to which various wavelengths of light can be absorbed, i.e., they further restrict the absorption spectrum of seawater. Analysis of the interaction between light and each individual component of seawater is practically impossible, because all the naturally occurring chemical elements are present in it, and so are a large number of chemical compounds. As we saw in Chapter 2, the number of elements and compounds present in large quantities is also high. This situation compels us to consider principally those seawater constituents for which we have experimental evidence that they strongly affect its absorption spectrum. The most important of these constituents is the water itself, both because of the huge number of water molecules in seawater, and because they strongly absorb solar radiation, infra-red in particular. Second in order of importance are the organic yellow substances, which were discussed in Section 2.6. Also responsible for absorbing much light in the sea are the suspended particles, especially organic ones. The major constituents of sea salt, on the other hand, do not affect the absorption of light energy in the sea to any great extent. There are practically no differences between the absorption spectra of clean, salt water and distilled water, from the violet through to the infra-red. Careful examination of these spectra does, however, show up some slight differences between them, chiefly in the ultra-violet part of the spectrum; they are discussed by Jerlov (1968) who cites some papers on the subject.
4 THE INTERACTION OF LIGHT WITH SEAWATER
156
Light Absorption by Water Molecules
As we said in Chapter 2, the water molecule’s structure is complex enough for fragments of it (e.g., OH bonds) to be subject to several kinds of normal oscillation, each with its own frequency, just as in coupled mechanical oscillators. The maintenance of each of these kinds of oscillation in the molecule requires energy, and this can be drwan from the photons during their absorption. The mechanism of this phenomenon is such that the periodically variable electric field of the photon wave interacts with the electric charges on the molecule, exciting them to periodic oscillations. If the frequency of the wave oscillations are the same as that of certain free oscillations of the molecule, the excitation to oscillation-resonance-is particularly strong, and takes place at the expense of the energy of the photon, which is thus absorbed. There are three known normal oscillational modes in the water molecule, and they are shown diagrammatically (with the aid of arrows) in Fig. 4.2.1. These oscillations are called valence oscillations, as they are due to changes in the bond lengths between atoms and in the angles between the bonds. The same diagram
t
L
=
1.1268~10’4Hz
,= 2 . 6 6 , ~ ~ ~
1.0949 2.74pm
Hz
0.4637
Hz
6.47~”
Fig. 4.2.1. The normal oscillations of the water molecule and their corresponding energy transitions; the light transmittance spectrum is also shown (adapted from Barrow, 1969).
4.2 LIGHT ABSORPTION IN SEAWATER
157
shows the corresponding energy transitions in the molecule and gives the photon’s parameters-energy, frequency, wavelength which are absorbed by the water molecule during these transitions, that is, which are use dup in exciting the oscillations shown in the diagram. The figure also shows the light absorption spectrum characterizing the process by which these three kinds of oscillation are maintained. There are three, somewhat broadened absorption lines (resonance lines) in this spectrum corresponding to wavelengths of 2.66 pm, 2.74 pm and 6.47 pm. We see that these lines lie in the infra-red (see Table 3.0.1), because, as we know, the visible part of the spectrum lies within the wavelength range from 0.4 pm to 0.8 pm, i.e., from 400 nm to 800 nm. In fact, it turns out that water molecules also absorb light of many other wavelengths. The absorption of shorter waves, i.e. of higher-frequency oscillations, corresponds to combinations of normal oscillations and excited higher harmonic oscillations (“overtones”). The absorption bands maintaining these oscillations therefore correspond to shorter waveseven in the visible spectrum within the range from 543 to 847 nm with a maximum between 750 and 760 nm. Many longer waves are absorbed to provide energy for the rotation of the molecules, and this rotational energy is also quantized. The complex rotational energy spectrum of the molecules is superimposed on the oscillational energy spectrum. Both kinds of oscillation are dependent on each other and form combined oscillation-rotation spectra. That this is so is revealed in the absorption by the water molecules of photons having a large number of different frequencies. The light absorption spectrum of water molecules is thus composed of very many absorption lines lying very close to each other or even overlapping. These are called absorption bands of the oscillation-rotation spectrum which has a complicated fine structure. A fragment of such a spectrum is shown in Fig. 4.2.2. The absorption bands of light wavelengths consumed in 13356 A
I
---,- Y
Fig. 4.2.2. Fragments of the absorption spectrum of water molecules in water vapour illustrating the complex, fine structure of this spectrum (compiled by Zuev (1970) from Plyler et a/.).
158
4 THE INTERACTION OF LIGHT WITH SEAWATER
maintaining the rotation of the molecules usually lie in the far infrared as the energies of rotation are much lower than those of molecular oscillations. So far, we have been talking about the oscillation-rotation spectrum of single water molecules, i.e., molecules “not reacting” to the forces of interaction of adjacent molecules. This is approximately the state of affairs in the Earth‘s atmosphere, which usually contains a small quantity of water vapour. Notice, by the way, that naturally occurring molecules of water containing the isotope deuterium (HDO) or other isotopes (3H, 1 7 0 , l 8 0 ) have a completely different spectrum of oscillations and hence different absorption spectra. In the liquid water of the sea, the oscillation-rotation spectra of molecules are further complicated by the action of the intermolecular forces which we discussed in Chapter 2. The ways in which the molecules oscillate and rotate are highly dependent on the hydrogen-bonding forces between the molecules, on the forces bonding the ionic clusters, etc. As we mentioned in Chapter 3, the number of permitted energy states in the complex macroscopic system that water is, is extremely high. Many new absorption bands therefore appear in the spectrum, that is to say, there are a lot of new reasons why light of suitable wavelengths should be absorbed in liquid water as opposed to rarified water vapour. The reader’s attention is drawn in particular to the possibility that hydrogen bonds in clusters can be broken by photons of appropriate energy. The energy of a single hydrogen bond between water molecules in liquid water is also dependent on how the molecules are joined, and on the complex molecular structure of such water, that differs under different conditions. Assuming, as we did in Chapter 2, that the hydrogen bond energy is 4.5 kcal/mol, we can, by dividing this value by the Avogadro number and converting calories to joules or electronvolts, obtain the energy of a single hydrogen bond, equal to 3.13 x 10-’O J or 0.195 eV. The photon of such an energy hv needed to break a hydrogen bond in water has a frequency v = 4.72 x Hz, which is equivalent to a wavelength in the infra-red of about 6.35 pm. In actual fact, the photolysis of hydrogen bonds is a more complicated process, also dependent on the temperature of the water and causing new absorption bands to appear in the spectrum of the light passing through the water. We have not said anything here about electronic excitations in the oxygen and hydrogen atoms or about other possible mechanisms by which light can be absorbed in pure water (e.g., by ionization), because they are of marginal importance in the process of solar energy absorption in the sea. They generally require high-energy photons (in the ultra-violet) which, as we shall see later, are far more efficientlyabsorbed by other components of seawater than by the water molecules
4.2 LIGHT ABSORPTION IN SEAWATER
159
themselves. A theoretical description of the combined interaction of all these phenomena in the hydrosphere is not used, as its excessive complexity would make it impossible for the absorption spectrum of water to be determined theoretically. These few explanations should have made it clear why the infrared portion of solar radiation is so strongly absorbed in water. In this region of the wave spectrum there are so many absorption bands for the various reasons just mentioned, that water is practically opaque to infra-red radiation. This is an exceptionally important physical property of ocean water. It means that about 50% of the solar radiation energy comprising IR at sea level is absorbed in the topmost layer of the sea, a few centimetres thick. This energy, converted to heat (molecular motion), is most efficiently used in this layer to evaporate ocean water and it thereby contributes to the basic natural process of water circulation. At the same time, it prevents the deeper layers of ocean water from overheating on sunny days, so lending stability to the living conditions of thousands of species of marine organisms. The absorption spectrum can be expressed quantitatively by the volume absorption coeficient, whose value differs with wavelength. The absorption coefficient of water, denoted by a, is usually determined experimentally on the equation for the radiant energy transfer in the medium or from other laws of basis of the optics, of which more later. In the simplest case, when the absorption of light within the analysed range of wavelengths is so great that scattering can be neglected, we can apply Lambert’s well-known absorption law. Using the radiance, we can write it as follows
dL
(4.2.1) -aL. dr This means that a change in the radiance of dL as a result of absorption over an --=
infinitesimally short distance in the medium dr is proportional to the value of that radiance L, while the proportionality coefficient a is the absorption coefficient in the medium in question. The value of the absorption coefficient depends on the nature of the medium and usually differs widely for various wavelengths of light for the reasons stated earlier. Lambert’s law (4.2.1) is at the same time the simplest differential equation, the solution of which gives an operational definition of the absorption coefficient: LJL0 = e-ar, (4.2.2) hence
(4.2.3)
4 THE INTERACTION OF LIGHT WITH SEAWATER
160
where Lo and L,are the respective values of the radiance at the beginning of the layer of medium and after the passage of the light (in a given direction) perpzndicularly through a layer of thickness r. We see from (4.2.3) that the units of the absorption coefficient a are m-l. In practice, when we wish to use this formula to measure the absorption coefficient of water, we pass a parallel beam of monochromatic light from an artificial source through a layer of the medium TABLE 4.2.la Selected values of the attenuation coefficient cw and absorption coefficient a, of light in distilled Kind of water
Source Of mation
distilled
distilled
Clarke, Clarke, James, 1939; James, 1939 Le Grand 1939*
distilled
distilled
Hale, Queny, 1973
Shuleykin, 1959
clean natural waters
Smoluchowski, 1908
Sargasso Sea
Ivanov, 1975
coefficient measured a,
W'I __ 375 390
0.045
0.0383
0.117
400
0.043
0.0379
0.056
0.033
0.0291
0.038
0.019
0.0159
0.028
0.018
0.0155
0.025
410 425 430 450 470 475 490 494 500 510 522 525 530 550 558
a
a,
h-'I
b-'I _____ 0.038
0.041
0.037
0.034
0.036 0.037 0.039
0.025 0.016 0.014
0.042
0.018
0.054
0.023
0.062 0.074
0.039 0.050
0.002 0.036
0.0340
0.025 0.002
0.041
0.0394
0.032
0.069
0.0676
0.045
-
0.038 ______-
*
Values calculated as the diffrenct uw = c,-b,,
where cw is the attenuation coefficient of light in pure WateI
4.2 LIGHT ABSORPTION IN SEAWATER
161
of thickness Y and measure the loss of energy of the flux (change in radiance) along this distance. However, for good results, three conditions must be fulfilled: (I)the law can only be applied to that region of the spectrum where the scattering effect in comparison with the absorption of the sample really is negligible (the formula does not hold for turbid media); (2) one must be able to guarantee a measurement geometry (a practically parallel beam) in which there is no energy
water, and the absorption coefficients of light u in clean natural waters
Kind of water
Source Of mation
distilled
distilled
Clarke, Clarke, James, 1939 James, 1939 Le Grand, 1939*
distilled
distilled
Hale, Querry. 1973
Shuleykin, 1959
clean natural waters
Sargasso Sea
Smoluchowski, 1908
Ivanov, 1975
a,
U
U
b-'I
W'I
fm-7
0.094
0.067
0.16
0.14
0.26
0.24
0.38
0.33
0.54
0.52
coefficient measured
570 575 590 600 602 607 610 612 617 622 625 643 650 658 675 690 700 750 800
0.091
0.0898
0.079
0.186
0.1850
0.23
0.089
-
0.173 0.200 0.233 0.234 0.239 0.228
0.2272
0.28
0.288
0.2873
0.32
0.291 0.320 0.367
0.3664
0.415
0.500 2.400 2.050
0.4995
0.60
(from Clark and James, 1939), and bw is the scattering coefficient of light in pure water (from Le Grand, 1939).
4 THE INTERACTION OF LIGHT WITH SEAWATER
162
loss due to beam divergence, i.e., all the flux must pass along the path Y and fall within the angle of observation of the radiance meter; (3) the disturbing effect of light from other possible sources and reflections of part of the light flux from the meter windows (c. 3% from each one) and possibly also on the walls of the measurement cell must be taken into account. The measurement of the absorption coefficient as an indication of the concentration of a given substance in an aqueous solution is standard practice in chemical TABLE 4.2.lb The absorption coefficients of light a(A) for pure water in the infrared (selected from data by Hale and Querry, 1973) -
A [wl 1.4 1.6 1.8 2.0 2.2 2.4 2.6 2.65 2.70 2.75 2.80 2.85 2.90 2.95 3.00 3.05 3.10 3.15 3.20 3.25 3.30 3.35 3.40 3.45 3.50 3.60 3.70 3.80 3.90
a [m-I x 1.23 0.067 0.802 6.91 16.5 50.05 15.32 31.77 88.4 269.6 516 815 1161 1269 1139 988 778 538 363 236 140 98 72 48 34 18 12 11 12
I [~ml 4.00 4.10 4.20 4.30 4.40 4.50 4.60 4.70 4.80 4.90 5.00 5.10 5.20 5.30 5.40 5.50 5.60 5.70 5.80 5.90 6.00 6.10 6.20 6.30 6.40 6.50 6.60 6.70 6.80
a [m-lx
14 17 21 25 29 37 40 42 39 35 31 28 24 23 24 27 32 45
71 132 224 270 178 114 88 60 68 63 60
4.2 LIGHT ABSORPTION IN SEAWATER
163
analyses of such solutions. This is what factory-made, precision spectrophotometers are for. In the case of water too, this measurement would seem to be a simple matter. But in spite of the large number of experiments performed up till now, we are still not sure which values of the absorption coefficient obtained from measurements are in fact accurate. Even the most meticulously conducted measurements of this coefficient by different workers have not yielded concurring results (Hale and Querry, 1973). Especially inaccurate are the figures from the visible and ultra-violet regions of the spectrum, where absorption is very low in comparison with scattering and these two effects cannot be completely separated. Another serious problem in optical studies of water is the impossibility of obtaining absolutely pure water. Even in the purest water, filtered many times over in a closed cycle, there always remain from 500 to 1000 suspended particles greater than 1 pm in diameter in 1 cm3 of water. Lastly, for the aqueous solutions to be tested, pure water is the standard with which the photometer is calibrated; but there is no standard for pure water. For these reasons, instead of the absorption coefficient of pure water a(A), the attenuation coefficient c(1) (defined later by equation (4.4.2)) is given-this expresses the total attenuation due to the absorption and scattering of light. There is no big difference between these two coefficients (b)
U I - - - . _ _ L _ _ L _ _ _ i i 200 300 400 500 600 700 Wavelength
L [nml
10-21,
0.2
,
,
,
, 1
, , , ,
,
,
.
,
2 Waveiength 1 [urn]
Fig. 4.2.3. The attenuation spectrum of light in pure water from data collated by Morel (1974) from various papers (by permission of the Academic Press, Inc. and the author): (a) in the visible and near-ultra-violet regions: Lenoble and Saint Guily (1955) (the optical path length used in the tested water: 400 cm); Hulburt (1934, 1945) (optical path length 364 cm); Sullivan (1963) (optical path length 97 cm): Clarke and James (1939) (optical path length 97 cm, cell walls covered with ceresine); James and Birge (1938) (optical path length 97 cm, cell walls covered with silver): (b) over a wide spectral range (from 0.2 to 2.8 urn): 1. Barret and Mansell (1960); 2. Lenoble and Saint Guily (1955); 3. Curico and Petty (1951); 4. Collins (1925).
164
4 THE INTERACTION OF LIGHT WITH SEAWATER
for pure water; in the infra-red, scattering in pure water can be neglected in comparison with absorption, while only in the short-wave region of the spectrum, especially in the ultra-violet, is scattering of importance. Absorption in this latter region has not been studied thoroughly. The attenuation spectrum of light in pure water and the other optical properties of water over a wide range of wavelengths are reproduced from a large number of measurements made by various workers in different narrow spectral ranges (Morel, 1974). The values of the attenuation coefficient for pure water that are best known in the literature and which are still widely applied in various calculations are those published by Clarke and James in 1939. These values, alongside the figures obtained by other workers on the absorption of light in pure water for selected wavelengths in the visible region, are given in Table 4.2.1. The attenuation spectrum of light in pure water over a wide range of wavelengths, plotted from data collected by Morel (1974) from the work of many different workers, is illustrated in Fig. 4.2.3. The attenuation coefficient plotted in this figure is practically equivalent to the absorption spectrum for waves longer than c. 0.6 pm, the scattering of which in pure water can be disregarded in relation to its absorption. As Fig. 4.2.3b shows, the spectrum is a continuous one, generally observed in fluids (unlike the line spectrum in rarefied gases). There is an abrupt increase in absorption (note that the graph has been plotted on a log scale) as the wavelengths increase in the infra-red, and there is also a clear minimum-the only “window” in the absorption spectrumin the visible region. The discrepancies in the results of experiments on this minimum done by various authors, evident from Fig. 4.2.3a, are the upshot of the difficulties in obtaining and testing pure water which were mentioned earlier. Many of the extreme values in different wavelength ranges of this spectrum are due to the superimposition of absorption bands corresponding to the excitation energies of complex molecular oscillations and rotations and to other effects mentioned previously. Longer waves, moreover, bring about other effects in the medium, e.g., changes in the orientation of molecules and clusters of them as electric dipoles, cluster deformation, etc. These are electrodynamic effects arising out of the interaction between the alternating electric field of the electromagnetic wave and the electric charges of the medium’s molecules. These are relaxation processes, which cause the absorption of wave energy in those frequency bands in which the period of oscillation of the wave is close to the relaxation time (change of position) of the molecules in a given field of forces. The relaxation times in particular processes depend in turn on the microstructure of the medium and the intermolecular forces, which themselves are pressure- and temperaturedependent. The excitation and maintenance of relaxation oscillations, strongly
4.2. LIGHT ABSORPTION IN SEAWATER
165
damped by the surrounding molecules of the medium, requires the absorption of electromagnetic wave energy. In pure water, and the more so in seawater, containing as it does permanent dipoles and diverse kinds of ions, there are many reasons why the wave energy of the far infra-red, microwaves and radiowaves is strongly absorbed. This energy is used up in moving these electrically nonneutral molecules against “frictional forces”. This is the reason why radio communicationis practically impossible in an ocean-water medium. The energy of very short electromagnetic waves, X and gamma rays, is usually sufficient to ionize atoms. For this reason, these very short waves (high-energy photons) are also strongly absorbed in water owing to ionization. But the photon energies of the so-called hard gamma rays in cosmic radiation are so great (of the order of many megaelectronvolts) that there is enough to ionize a large number of atoms, with only a small fraction being lost during a single act of ionization. This radiation can, therefore, penetrate the water down to a depth of almost 100 m, whereas infra-red radiation is, as we have said, absorbed almost entirely in the top few centimetres of water. The attenuation spectrum in seawater of electromagnetic waves within the wide range of frequency from lo2 to 1020 Hz will be illustrated further in Section 4.4 (Fig. 4.4.4). It remains to be noted at this point that water is the most transparent to those wavelength ranges of electromagnetic radiation which are most visible to human and animal eyes.
Light Absorption by Seawater Constituents In Section 2.6 we discussed the yellow substances dissolved in seawater and stated that they were indeterminate organic compounds derived mostly from the remains and metabolic products of marine plants and animals. The most important of them are melanoids, humic substances and other complex clusters of organic molecules probably containing fragments of hydrocarbons, amino acids, aromatic rings and other compounds. These substances play a highly significant role in the hydrosphere, not only in multifarious biological and biochemical processes, but also in the process whereby light is absorbed in the sea. Organic compounds can markedly affect the optical properties of seawaters, in some, preventing ultra-violet radiation from reaching the deeper layers of water, thereby inhibiting many photochemical reactions. The question arises why these organic substances, present barely in milligrams per litre of seawater, should have a far greater influence on the absorption of light and the differentiation of seas from the optical point of view than sea saIt, whose concentration is thousands of times greater. Clearly,
166
4 THE INTERACTION OF LIGHT WITH SEAWATER
it is the molecular structure of these compounds which is responsible for their absorption spectra. To excite oscillation and rotation in all the molecules in an aqueous solution normally requires the absorption of low-energy (infra-red) photons. In this region of the spectrum it is the huge mass of water molecules in the sea which absorbs the light. Seen against this strong absorption of IR radiation by the water, absorption by other compounds, present in incomparably lower concentrations, is scarcely noticeable. The absorption spectrum of seawater in the IR is therefore almost identical in all seas. On the other hand, the electronic excitation (change of orbitals) of sea salt ions, water molecules and other simple chemical compounds in the water usually requires high energies, which can only be provided by photons in the far ultra-violet. So, for example, the lowest electronic excitation energy of the OH group in water corresponds to a wavelength of 166 nm, while C1- ions demonstrate maximum absorption in the 181 nm waveband, Other ionic components of sea salt also have absorption-band maxima in the UV region, usually shorter than 250 nm. Thus a mixture of the various components of sea salts brings about only a slight monotonic increase in the absorption of ever-shortening waves (Armstrong and Boalch, 1961). Despite the dissimilar salinities of seawaters, their absorption spectra do not differ significantly from one another. In complex organic compounds, including those occurring in the sea, the molecules contain inter alia electrons weakly bound to particular atoms but occupying orbitals loosely localized within the whole cluster of neighbouring atoms in the molecule. This is why many low-energy electronic transitions are possible in them, forming absorption bands in the UV and also visible regions of the spectrum. These low-energy levels of electronic excitations are characteristic of unsaturated organic compounds, and of carbonyl groups in particular. Some of these compounds in the sea fluoresce on being excited by UV radiation (Duursma, 1974). Apart from this, there are many other reasons why light is absorbed in such complex organic compounds dissolved in water. A large number of photochemical reactions takes place in them. Electronic excitation usually initiates these reactions: the breaking of chemical bonds or the synthesis of a new compound are possible only when the molecule is in an excited state (appropriate shift of charges). A series of possible excitations and photochemical reactions in organic compounds similar to the above is described in detail by Simons (1976). One of them is predissociation, which leads to the breakdown of chemical bonds of the P * ~ type - ~
4.2 LIGHT ABSORPTION IN SEAWATER
167
Others involve bond breakage with a hydrogen atom shifting to another position, and yet others bring about intramolecular rearrangement. The electronic excitations and the various photochemical reactions undergone by organic compounds in water cause broad-band absorption of visible and UV light. So water containing a mixture of these compounds has a characteristic continuous absorption spectrum, rising rapidly on the short-wave side (Fig. 4.2.4). Intensive absorption of violet from the sunlight spectrum makes this
200 Wavelength 1 [nml
250
350
300
Wavelength A
[rim]
Fig. 4.2.4. Absorption spectra of water containing large quantities of yellow substances, compared with the spectrum for clean ocean water from the Sargasso Sea (a) in the visible region (fromKopelevich et ol.. 1974). 1. Baltic, Gulf of Riga; 2. Baltic, Gotland Deep; 3. clean ocean water from the Sargasso Sea. (b) in the ultraviolet: 1. Vistula estuary; 2, 3, 4, Gulf of Gdruisk (Dera, 1967).
water appear yellowish; when the concentration of organic compounds is high, for instance in river estuaries, the water is distinctly yellow and even brown where it contains large amounts of humic acids, phenolhumus compounds and other similar compounds originating from the land. The steep increase of absorption on the short-wave side of the light spectrum in waters containing a lot of organic substances considerably limits the influx of violet light to the deeper water layers. This has an undoubted but little-studied
4 THE INTERAmION OF LIGHT WITH SEAWATER
168
influence on the sunlight-initiated photochemical reactions occurring there. The most important of these reactions is the photosynthesis of organic matter, the first link in the food chain of marine organisms. Other important photochemical reactions in the sea include the formation of pigments essential to the well-being of marine plant and animal cells, and the photodegradation and oxidation of organic substances contaminating the aquatic environment, that is, the selfpurification of waters. We shall say more about the attenuation of light in the sea by organic substances in Sections 4.5 and 4.6. In Chapter 2 we also mentioned pigments such as chlorophyll and carotenes contained in the cells of phytoplankton, or from the point of view of physics, in marine suspended particles. The absorption spectra of these pigments (Kamen, 1963; Godnev, 1963) display distinct maxima in the visible region and also affect the absorption properties of seawater. The Red Sea takes its name from the fact that while its waters commonly appear blue in daylight, they take on a reddish tinge at certain times as a result of various optical effects, one of which is the absorption of light by the pigments in the cells of Trichodesmium erythraeum, a red Cyanophyte living there in huge numbers (Demel, 1974). Most common in the upper layer of ocean waters is phytoplankton containing chlorophyll a, b and c, and carotenes in such proportions that a mixture of these
-
0.16
0
0.12
-
0.08
E
0.OL c
.-5
0
5
0.08
8
0.04
.$
0
U
a
2 9 1
0.08
0.04 0
0.04 400
5Xl
Ilnml
n 400
500
IInml
Fig. 4.2.5. Absorption spectra i n seawater at various depths (a) in the Gulf of Riga (Baltic), and (b) in the Pacific Ocean (station 399) (from Kopelevich eta!., 1974; by permission of Nauka Publishers). 1-unfiltered samples; 2-filtered samples; 3-absorption by suspended matter; 4-absorption by dissolved substances.
4.2 LIGHT ABSORPTION IN SEAWATER
169
pigments shows maximum absorption in the 430-440 nm range. There is a second maximum, also in the red, e.g., for chlorophyll a in the 660-690 nm range (depending on the kind of cells). So, as long as the phytoplankton concentration in the water is relatively high, cell pigments give rise to greater or smaller maxima in the absorption spectra of seawater (Kopelevich et al., 1974). This is illustrated by the absorption spectra in Fig. 4.2.5, which their authors consider to be typical of waters in the large areas of the Atlantic and Pacific which they have studied (see Table 4.2.2). The absorption due to suspensions and that due to dissolved yellow substances, which are usually present with the suspensions, has been separated on these spectra. Two types of absorption spectra have been found to occur: (a) those with a monotonic increase of absorption towards the short-wave end of the spectrum, attributed to the dominant influence of dissolved yellow substances (Fig. 4.2.5a), and (b) those with a local maximum in the 41C440 nm region, attributed to the dominance of the pigments contained in the plankton cells (Fig. 4.2.5b). In clean ocean waters, the percentage fall in the activity of phytoplankton pigments with depth, and the increase in concentration and changes in the composition of yellow substances dissolved in the water, products of cell degradation at greater depths, is quite evident. The decomposition of both these groups of constituents with depth (suspensions of cells and organic compounds) also brings about a general decrease in absorption by seawater at greater depths. In surface waters, and in turbid waters like those of the Baltic Sea, the absorption spectra usually rise gradually towards the short-wave end because of the dominant role played by yellow substances in this absorption. The absorption properties of waters abundant in phytoplankton (and hence also in zooplankton and fish), i.e., fertile waters, are used in aerial and satellite surveys of the biological productivity of ocean basins. These surveys also assist in the compilation of methods of searching for potential shoals of fish which concentrate where there is plankton for them to feed on. Attempts are therefore being made at aerial and satellite measurements of the concentration of chlorophyll and other substances in the sea as indicators of the biological productivity of waters. They are based on the remote analysis of the spectra of scattered light or fluorescence in the sea, filtered by the afore-mentioned absorbents in se-awater before returning to the atmosphere. However, the spectra of such light, received a t considerable heights up in the atmosphere, are affected by many factors besides absorption and scattering in the sea. This light is influenced principally by spectrally selective absorption and scattering in the atmosphere (Gordon, 1978b)
4 THE INTERACTION OF LIGHT WJTH SEAWATER
170
and by wave action at the sea surface (Olszewski, 1979). We shall be dealing with such interactions in later chapters. Owing to this interaction and its complexity, satellite and aerial surveys of the biological productivity of seas are still at the stage of experiments and theoretical analyses (Galazii et al., 1979; Gordon, 1978, 1979). The International Council for the Exploration of the Seas-ICES-is one of the organisations dealing with their development (see also Sherman, 1979). Let us return to the absorption of light in seawater. Besides phytoplankton cells, an important absorber of light in the sea is also non-living suspended matter. Suspended organic particles contain many organic compounds with absorption properties similar to those of the dissolved yellow substances. We may therefore predict that the observed increase in absorption towards the short-wave end of the spectrum is also due in part to absorption by suspended particles of various kinds (Burt, 1958; Morel, 1970). The role of inorganic suspensions (e.g., of silicate minerals), in light absorption is thought to be inconsiderable. It is, however, difficult to determine only the absorption of light by marine suspensions when at the same time they scatter light strongly. This is the reason why few results are available on work done in this field (see Jerlov, 1976; Kopelevich et al., 1974; Dera et al., 1978). These experiments do allow us to conclude, however, that in the short-wave regions of the visible spectrum, there is from 20 to 30% attenuation of incident light due to absorption by suspended particles. Other natural constituents of seawater, not discussed here, do not significantly affectthe absorption of sunlight in the sea. Their effect on the absorption spectrum is extremely small in comparison with the effects of the substances mentioned. Assuming that the absorption of light by seawater constituents is additive means that its absorption coefficient in water can be resolved into its component parts expressing the absorption of the various groups of constituents. So denoting the absorption coefficient of pure water by a,, that of yellow substances by ay, that of suspended particles by u p ,that of sea salt by a,, and that of other admixtures such as artificial contaminants by ad, we can write the absorption coefficient of seawater as the sum = a,v+a,+a,+a,+a,.
(4.2.4)
Each of these partial coefficients is, of course, a function of the wavelength, and therefore has its own spectrum characteristic of a given group of constituents in the water. The absolute values of these coefficients are furthermore dependent on the concentration of these constituents in the water. The absorption spectrum of seawater a(A) is thus the superimposition of these partial spectra. Little attention
4.2 LIGHT ABSORPTION IN SEAWATER
171
has been paid to the shapes and relative proportions of these partial spectra in seawaters, but it is possible to reproduce them approximately and obtain the absorption spectrum of a given type of seawater as the superimposition of partial spectra. This is illustrated in Fig. 4.2.6 for waters in which there are large concentrations of yellow substances and suspended particles, such as Baltic waters.
Fig. 4.2.6. The absorption spectrum of seawater with a high content of suspended particles and yellow substances (Baltic type) obtained by superimposing the absorption spectra of these constituents and pure water. a-pure
water; uy-yellow
substances; a,-suspended
particles; a-seawater.
If we examine Fig. 4.2.6, we shall see that, depending on the concentration and diversity of the mixtures of constituents in a given water, the appearance of the partial absorption spectra and the proportions between them can vary greatly. Superimposing them will therefore yield different results, and so the absorption spectra of waters from different regions of an ocean will also differ. It is moreover quite obvious that even if we superimpose a number of such absorption bands, detailed analysis of their spectra must reveal many extremes in various regions of the spectrum-evident examples are the pigment maxima. Incidental irregularities may therefore appear on the basically monotonic absorption spectrum of seawater, and such irregularities are in fact shown up by precise analysis (Morel and Prieur, 1975). These not very regular plots of absorption spectra a(1) cannot be described by any relatively simple mathematical formula. Notice the shift of the absorption minimum in polluted seawaters towards the long waves in relation to the clean water minimum. In the cleanest ocean waters, like the Sargasso Sea, the absorption minimum lies within the wavelength
TABLE 4.2.2 Typical values of the absorption coefficients from the Pacific Ocean Atlantic Ocean and Baltic Sea a [m-'1 (selected from Kopelevich (a) et al., 1974*). (a) Pacific Ocean Study area Northern zone of subtropical convergence North Equatorial Current Gulf of Panama Galapagos Islands region
Depth [ml
390
410
430
450
470
85 60 200 10 20 200 30 10 10
0.085 0.12 0.14 0.12 0.053 0.067 0.099 0.14 0.13
0.071 0.15 0.14 0.14 0.090 0.062 0.16 0.099 0.16
0.048 0.15 0.11 0.13 0.12 0.044 0.19 0.92 0.16
0.037 0.13 0.083 0.11 0.090 0.034 0.16 0.074 0.094
0.023 0.087 0.062 0.080 0.060 0.025 0.099 0.053 0.055
Wavelength of light [nm] 490 510 530 550
0.014 0.064 0.055 0.060 0.039 0.014 0.062 0.039 0.032
0.14 0.060 0.051 0.044 0.028 0.014 0.034 0.032 0.021
570
590
610
650
690
0.029 0.039 0.063 0.060 0.067 0.085 0.062 0.071 0.085 0.048 0.051 0.062 0.032 0.037 0.058 0.028 0.032 0.055 0.032 0.034 0.055 0.032 0.037 0.060 0.028 0.032 0.055
0.14 0.17 0.15 0.14 0.14 0.14 0.14 0.14 0.14
0.24 0.26 0.25 0.24 0.24 0.24 0.24 0.24 0.24
0.33 0.36 0.34 0.33 0.33 0.33 0.33 0.33 0.33
0.52 0.56 0.54 0.52 0.52 0.52 0.52 0.52 0.52
0.14 0.14
0.24 0.24
0.33 0.33
0.52 0.52
0.055 0.14 0.067 0.14 0.060 0.14 0.055 0.14 0.055 0.14 0.25 0.28
0.24 0.24 0.24 0.24 0.24 0.37
0.33 0.33 0.33 0.33 0.33 0.41
0.52 0.52 0.52 0.52 0.52 0.62
South Equatorial Current (3 different stations along the current from east to west) Cromwell Current 150 0.13 0.12 0.10 0.078 0.053 0.039 0.032 0.037 0.041 0.062 Tonga Trench loo00 0.058 0.037 0.023 0.012 0.009 0.009 0.016 0.025 0.032 0.055 (b) Atlantic Ocean and Baltic Sea
North Equatorial Current Sargasso Sea Gulf Stream Caribbean Sea Baltic Sea, Gotland Deep Baltic Sea, Gulf of Riga
0 0 10 0 0 0
0.032 0.041 0.041 0.20 0.71 2.7
0.034 0.034 0.044 0.090 0.48 1.9
0.021 0.018 0.014 0.025 0.016 0.014 0.044 0.041 0.030 0.034 0.023 0.016 0.34 0.23 0.15 1.2 0.90 0.83
0.012 0,018 0.030 0.014 0.090 0.62
0.018 0.023 0.025 0.014 0.064 0.44
0.030 0.039 0.039 0.023 0.064 0.39
0.034 0.050 0.046 0.032 0.032 0.30
* The data for the oceans come from the fifth expedition of the Soviet research ship "Dimitrij Mendeleev" from 20.01-12.05.1971 ; the data for the Baltic are taken from Soviet studies carried out in the smmner of 1970. with permission of Nauka Publishers.
4.2 LIGHT ABSORPTION I N SEAWATER
173
range 460-480 nm, just as it does for pure water, the absorption coefficient in this band taking the minimum value of aminz 0.014 m-l. Such waters are extremely poor in yellow substances and suspensions, and hence also in plankton, and are referred to as marine deserts. In very clean ocean waters the absorption minimum lies in the range 470-490 nm, but in more fertile areas of ocean water this minimum shifts to the 510 nm band. Lastly, in bays, river estuaries, and semienclosed seas like the Baltic, the seawater absorption minimum falls in the greenyellow band, i.e., 460-600 nm. The experimentally determined absolute values of the absorption coefficients of light of different wavelengths in various seawaters also show that they are strongly differentiated in the short-wave region of the spectrum. The chief reason for this differentiation is the absorption of this part of the spectrum by suspended particles and yellow substances. Table 4.2.2, compiled from the results of research conducted by the Soviet research ship “Dimitrij Mendeleev” (Kopelevich et al., 1974), illustrates such regional differences in the absorption coefficients in different waters. The table shows that the absorption coefficients a for violet light in the cleanest ocean waters, like the Sargasso Sea, are lower by a factor of several tens than those obtained in the Baltic Sea. This gives us some idea of the different depths to which violet and UV light are capable of penetrating in various seas (see Morel and Prieur, 1975). Anticipating what follows in Chapter 5, we can state at this juncture that the product of the coefficient of absorption a (for a given wave band) and scalar irradiance Eo at depth z in the sea expresses the radiant power P absorbed in a unit volume V of water at that depth. This emerges from the law of conservation of radiant energy in the sea and can be briefly written thus: dP ~- -aEo,
dV
(4.2.5)
This expression is substantiated in Section 5.4. Equation (4.2.5) shows clearly that the absorption coefficient, although unequivocally characterizing the absorption properties of a given water, does not fully describe the process of light absorption in the sea. This absorption is equally dependent on the scalar irradiance, i.e., on the angular distribution of the radiance in the sea at any instant. This distribution, with a certain irradiance of the sea, is highly dependent on the process of light scattering, which will be discussed in the next section. Equation (4.2.5), written in a somewhat different form (see equations (5.4.22)(5.4.28))is one of the few operational equations useful in determining the absorption coefficient in situ from measurements of the appropriate irradiances, because
174
4 THE INTERACTION OF LIGHT WITH SEAWATER
of the fundamental difficulty of measuring the absorption of light in a medium which simultaneously scatters light strongly. This difficulty results from the fact that a beam of light travelling along a path r in a certain direction loses energy through being absorbed and scattered in other directions. Besides the aforementioned method of calculating the absorption coefficient of light in the sea from irradiance measurements, there are two other well-known methods of determining this parameter. In principle, their mode of action involves the detection of both the light passing through a sample of water and that scattered in it, so that the measured effect of the light beam attenuation is due almost entirely to absorption. In one of these methods, a detector having a large area (e.g., a large flat irradiance meter) is used to collect scattered light when measuring its transmittance (Gilbert et al., 1969; Bauer et al., 1971). In the other method, the measurement cell is placed inside an integrating sphere, whose walls reflect diffusionally and scatter within it all the light passing through or scattered by the sample. The part proportional to the intensity of this light is recorded by the detector and compared with a similar signal for a standard sample (see Rvachev, 1966). The methods of determining the absorption coefficient and the other optical properties of seawaters will be discussed one by one in later sections, after the functions and equations describing the phenomena used in these methods have been introduced. In particular, we have to consider the phenomenon of light scattering and the methematical description of its influence on the transfer of radiant energy.
4.3 LIGHT SCATTERING IN SEAWATER
The scattering of light is a physical phenomenon which takes place when randomly distributed optical unhomogeneities in the medium cause random changes in the directions of light rays. As we know, light is refracted at the interface between two optically different media. We can take two such media to be, for example, a suspended particle and the aquatic medium surrounding it, or even a particle consisting of clusters of water molecules and its nearest neighbourhood which has a different density. These are unhomogeneities of the medium through which light travels with different (phase) velocities. The refractive index of light at the phase boundaries, in this particular case at the interface between the unhomogeneities, is determined by the light-velocity ratio. The ratio of the velocity of light in a vacuum co to that in a particle of matter v p is the absolute refractive index of light co/vp = n,
4.3 LIGHT SCATTERING IN SEAWATER
175
for the material of the particle, and similarly, the ratio of the light velocity in a vacuum to that in the water surrounding the particle co/u,+ = H, is the absolute refractive index of that water. Since these refractive indices n p and n, are not equal, a particle in water has a refractive index relative to water which differs from unity and therefore refracts (and also partially reflects) light incident on it. When water contains a large number of such unhomogeneities, called lightscattering centres, the combined effect of all the refractions leads to scattering in all directions. An optically unhomogeneous medium like this is called a dispersion medium, and seawater is without doubt such a medium. D
'f;
1
Water
Fig. 4.3.1. A model of light scattering due to reflection, refraction and diffraction by large particles suspended in water. PI -P4--incident rays; Pi - Pi", P:- P:"--rays scattered owing to diffraction at the particle's edges; Pi -P:"-rays scattered owing to refraction and reflection.
Light scattering centres in seawater are marine suspensions (see Section 2.7)these include gas bubbles (see Section 2.8), and molecular scattering centres, which are fluctuations in the density (or packing of molecules) of the medium, first discovered by Smoluchowski (1908). Using a geometrical model of light-refracting particles, scattering is easily explained when the scattering centres are large in relation to the wavelength of the scattered light. Such a model is illustrated in Fig. 4.3.1. This shows not only the effects of refraction, but also those of multiple internal reflection, surface reflection and diffraction. Ray PI passes straight through the medium without encountering any unhomogeneity. On passing through the medium, rays P2 and P4 are principally diffracted at the edges of the particle, the end result of this being that the ray is bent and scattered in various directions. Ray P3 hits the particle head on; some of the ray's energy is reflected at the particle surface, while the remainder enters the irregularly-shaped particle. At the water-particle interface
4 THE INTERACTION OF LIGHT WITH SEAWATER
116
and at unhomogeneities within the particle, the ray is subjected to multiple internal reflection and refraction (P:-P:'). Reflection and refraction take place in accordance with Fresnel's and Snell's laws (see Section 5.2), which define how much radiant energy is reflected and how much is refracted in each act of reflection and refraction. The ratio of these two energies depends on the relative refractive index of light at the water-particle interface. The diagram shows that in consequence of all these effects, the originally parallel rays P1-P4have been scattered in all directions, though most of them forward at small acute angles to the original direction. The intensity of the refracted rays emerging from the particle must also be highly dependent on the extent to which they were absorbed within the particle. Besides the absorption coefficient of the particle, its dimensions, shape and refractive index determine how much light it absorbs, as the path length of the ray within the particle depends on these parameters. It is for these reasons that the so-called complex refractive index m was introduced. It characterises the main features of the suspended particles affecting the intensity and angular distribution of scattered light of a given wavelength A. By definition, the complex refractive index is equal to
m
=
n-in'.
(4.3.I)
The virtual part of the refractive index in' = iaA/4x contains the absorption coefficient a of the particle matter and the wavelength of the light A, on which absorption and refraction, and hence also scattering, are very much dependent (Van de Hulst, 1957). This simple, geometrical interpretation of light scattering does not fully explain the nature of this phenomenon and is unsatisfactory for the case when the dimensions of the scattering centres are comparable with or smaller than the light wavelength. Here we have to replace the macroscopic picture of refracted light rays with a microscopic picture of the electromagnetic interaction between a single light wave and the electric charges of the particles acting as scattering centres. The vast majority of these molecular scattering centres in the sea are the fluctuations in the density of the water-molecule clusters which arise as a result of their thermal motion. Scattering centres in the sea which are small with respect to the wavelength of light in the sea also include ion aggregates and clusters of other substances, colloids and the tiniest marine suspended particles with diameters less than c. 0.1 pm. The theoretical description of light scattering by small scattering centres, and also by larger suspended particles, requires that the molecular mechanism
4.3 LIGHT SCATTERING IN SEAWATER
177
of this effect, that is, the interaction of the electrical field of a single light wave with the electric charges of the scattering particles, be taken into consideration. Maxwell’s well-known equations describe this process and enable us to determine the distribution of the electromagnetic field intensities of a wave travelling through a region of unhomogeneities such as a scattering centre (Deirmendjian, 1969; Van de Hulst, 1957). To do this, however we need to know the physical properties of the material of which the particle is made, and its dimensions and shape. This last condition, a boundary condition as regards the solution of Maxwell’s equations, limits their practical application here to regularly-shaped particles (e.g., spherical ones). The solution becomes even simpler for homogeneous, isotropic particles, i.e., those whose electrical polarizability is identical in all directions. The solutions for such simplified cases of isotropic spherical particles obtained by Mie (1908) are very well known in hydrooptics and are still used, albeit with modifications and improvements by Shifrin (1951), Van de Hulst (1957), Shifrin and Salganik (1973), Morel (1973) and others. The physical principles of the theory of light scattering were explained by Rayleigh in 1871. He assumed that light scattering in pure media was caused by the discontinuous, molecular structure of matter, in other words, that this contained scattering particles. Today we know that among these particles in water are the clusters of many single molecules (see Chapter 2). Rayleigh made the assumption from the laws of electrodynamics that a scattering particle in the electrical field of a light wave became an induced, oscillating electric dipole which emitted its own (secondary) electromagnetic waves in different directions, visible as scattered light. A statistical theory of scattering in optically pure media, which assumed that the “scattering particles” were fluctuations in the medium’s density, was worked out by Smoluchowski (1908) and Einstein (1910). Scattering at such density fluctuations is called molecular scattering, and the description of its mechanism explains the scattering properties of pure water and very clean ocean water observed in nature. We shall now outline the main aspects of these theories of light scattering and discuss their application to hydrooptics. Rayleigh’s Theory of Scattering. The Volume Scattering Function and the Total Scattering Coeficient
While Rayleigh’s classic theory of light scattering cannot be applied directly to condensed media such as seawater, it does provide a good description of
178
4 THE INTERACTION OF LIGHT WITH SEAWATER
scattering in atmospheric air and a basis for the explanation and mathematical description of this phenomenon. It assumes that any given particle of matter, much smaller than the wavelength of light, becomes an induced electrical dipole when influenced by the electrical field of the light wave (Fig. 4.3.2). As long as the
Ec 4
Panicle of dimensions <
Panicie-oot dimensions
> A
Fig. 4.3.2. Dimensions of scattering centres compared with the electrical field distribution of the electromagnetic wave. (a) In Rayleigh scattering from a particle far smaller than the wavelength 1, an elementary dipole is formed all of which oscillates in time with the oscillations of this wave, emitting a secondary wave; (b) on being irradiated. a large scattering particle (a suspended particle) becomes a set of induced dipoles oscillating in various phases shifted with respect to each other by a constant value. Each of them sends out secondary waves, which interfere to different degrees in various directions. This is why particles comparable in size with and greater than the wavelength do not satisfy the assumptions of the Rayleigh theory.
velocity of the light in the material the particle consists of is not too small (n % l), at any moment within the entire volume of this small particle, the electrical field Ee due to the wave is identical (homogeneous). This particle in its entirety can therefore form a single electrical dipole with an induced dipole moment of pe = CleEe. The coefficient a, is generally referred to as the electrical polarizability of the particle’s material and in isotropic particles (about which we are for the time being concerned) it is a scalar. A single, monochromatic light wave is a harmonic wave, so its electrical field intensity at any given point in space changes with time as the harmonic function E, = Eeocoscrzt, where cu = 2x/T is the circular frequency, and T is this period of the wave’s oscillations. The electrical dipole p e induced by such a wave is therefore a dipole oscillating in time with the wave, so p e =p,,cosot. Notice further that the charges of the induced dipole produce their own electrical field in the surrounding space, and any change
4.3 LIGHT SCATTERING I N SEAWATER
179
in the position of these charges must alter the field as well. Such a change in the electrical field induces in turn a magnetic field in accordance with the law of electromagnetic induction. An electromagnetic disturbance is thus created in the space around the dipole which does not penetrate to points further and further away straight after the change in position of the charges, but does so gradually, with the speed of light. The periodic oscillations of the dipole charges induced by the incident light’s electrical field thus produce periodically changing electromagnetic disturbances which are propagated through space as an electromagnetic wave at the speed of light. So a scattering particle, as an oscillating electrical dipole, emits its own electromagnetic wave (as a dipole radio aerial does), in this way scattering part of the energy of the primary wave which induced the dipole to oscillate. The frequency of the secondary wave oscillation is therefore the same as that of the oscillations of the inducing wave, but the directions of secondary wave propagation around the dipole differ from the primary direction of the incident plane wave. The electromagnetic radiation of an oscillating dipole can be calculated from Maxwell’s wave equations (see e.g., Hippel, 1963). These calculationsshow that in a homogeneous medium, an oscilIating point dipole emits electromagnetic radiation in all directions with an intensity proportional to the fourth power of the frequency m. If the particle as a dipole is induced to oscillate by a plane wave propagating in a certain direction &,, the radiant intensity emitted at various angles Or to that direction varies. The radiant power of light incident on a unit area perpendicular to the rays (in a given direction) is called the light intensity I‘ in the classical sense (see equations (4.1.6) and (4.1.2)). Rayleigh demonstrated that if light of intensity I’ was incident on a particle in the form of a plane wave (a parallel beam), the intensity of the scattered light I; at a distance r from the particle would be given by the expression I ’ k4a,2 (4.3.2a) I ; (e,) = 2-- (1 + cos2er). r2 2 The subscript “1” represents scattering at a single particle, such a one as we have been discussing so far. Taking into account the fact that in hydrooptics we define the light intensity as I = dF/dSZ (4.1.2) and the irradiance of a surface normal to the rays as Ep = dF/dA, = I; (4.1.6), and also including the relationships dQ = dA,/r2 and k = 2x14 equation (4.3.2a) can be rewritten (4.3.2b)
This shows that the intensity of light scattered at a particle much smaller than the wavelength is inversely proportional to the fourth power of the wavelength. So at
180
4 THE INTERACTION OF LIGHT WITH SEAWATER
scattering centres much smaller than the wavelength, it is short waves like those of violet and ultra-violet light which are very strongly scattered (!). This conclusion concurs with the fact that a cloudless sky and very clean deep ocean waters are navy blue-we see these colours thanks to the strong molecular scattering of short waves from the spectrum of white sunlight. The second important conclusion from Rayleigh's law (4.3.2) is the symmetrical distribution of light intensities scattered forward through angles (Oo < 8, < 90") and backward through angles (90" < 8, < 180"), that is, the symmetry of intensity relative to a plane passing through the scattering centre perpendicular to the direction of the incident beam (Fig. 4.3.3). Since I1(Or)is independent of the second angular coordinate, there (a)
Scattering volume d V
Fig. 4.3.3. The geometry of light scattering. (a) The path of incident rays scattered through an angle 0,; b) plot o f the Rayleigh scattering function and the components o f the scattered light intensity function: il-polarized perpendicularly, i,-polarized parallel to the plane of observation (adapted from Van de Hulst, 1961).
is total symmetry of the intensity distribution of light scattered around the incident beam of unpolarized light, i.e., with respect to the axis of the beam. The electrical fields of the wave trains of particular photons in a ray of unpolarized light oscillate perpendicularly to the ray but in various planes along it. The large number of these photons following upon one another in a short time therefore excites the
4.3 LIGHT SCATTERING IN SEAWATER
181
particle to electrical oscillations in all directions of the plane perpendicular to the beam of incident light. From this emerges the symmetry of radiation around the axis of the incident beam of light through angles in front of and behind the plane of the particle's electrical oscillations. The third significant conclusion from (4.3.2) is the fact that the intensity of scattered light is lowest at right angles to the incident rays (i.e., Z1(900) = minimum) and greatest at angles approaching 0" and 180". This results from the extreme values which the term cos20r takes for these angles in (4.3.2). The Rayleigh formula (4.3.2) for the intensity Zl of light scattered by a single scattering particle much smaller than 1 can under certain assumptions be applied to the scattering of light by a unit volume V of a medium containing N such particles. We have to assume that the scattering by these particles is additive, that is, the scattered light intensity I per unit scattering volume of N scattering particles is equal to NZl . This assumption excludes the possibility of interactions between the individual particles, e.g., in which the light emitted by one of them excites another to oscillate (multiple scattering). Such an assumption is obviously an approximation which is inapplicable to high particle concentration media containing large numbers of scattering particles close to one another. If, however, we do assume that the scattering at N particles is additive in a given volume of medium, we can write in accordance with the Rayleigh formula (4.3.2b), (4.3.3) Under the assumptions given, equation (4.3.3) retains the same relationships for a set of scattering centres in a volume of medium as for a single particle. Among them is the relationship I 1/lL4. Of particular importance in nature, it is satisfied to a good approximation in a clean atmosphere and in clean seas wherever the scattering particles are much smaller than the wavelength of light. When analysing the scattering properties of a medium, it is convenient to rid their mathematical description of the incident light intensity, which may take any value and is independent of the investigated medium. To this end, Rayleigh introduced the ratio of the intensity of the light scattered by a unit volume scattering at 90" to the irradiance Ep of this volume by a parallel beam of light (more precisely, to the irradiance of the area of this volume projected onto the plane of the cross-section of the incident light beam). This ratio is
-
(4.3.4)
4 TXCE INTERACTION OF LIGHT WITH SEAWATER
182
and it corresponds to the volume scattering function at an angle of go", where N [m-3] is the number of scattering particles in a unit volume. In general, the volume scattering function ,!?(Or)is defined (Jerlov, 1976) as the ratio of the intensity dZ(8,) of light scattered at an angle 8, by a volume element of a scattering medium dV to the irradiance Ep of the volume, and to dV, i.e.,
(4.3.5) It has units of [m-l sr-"] and expresses the relative distribution of the intensity of light scattered by a given volume element of the medium at various angles O,, regardless of the incident light intensity. The function therefore exclusively characterizes the scattering properties of the medium, in the same way as the absorption coefficient characterizes its absorption properties. Light scattering in accordance with the Rayleigh theory is often called Rayleigh scattering. In view of definition (4.3.4) for isotropic particles much smaller than A, the volume Rayleigh scattering function at angles of 8, = 90" after including (4.3.2) will be equal to 8x4
#8,(90) = N -
A4
2
a,.
(4.3.6)
For any angle 6 , we obtain the function BR(f3,) corresponding to Rayleigh scattering after replacing Z,(90) with Zl(8,) in (4.3.4) and using (4.3.2) (4.3.7) By substituting the function j3,(90) given by (4.3.6) in (4.3.7) we get a convenient form of the Rayleigh scattering function @RR(er)= @R(90)(1fcosZer)* (4.3.8) Clearly, the intensity of light scattered forward (6, = 0") or backward (0, = 180") is twice as great as at 90°,where, as we have said, it is a minimum (see Fig. 4.3.3). The picture of Rayleigh scattering becomes complicated when we take into consideration the polarization of the scattered light (see Van de Hulst, 1957). The geometry of the incident wave trains shows that an unpolarized, parallel beam of natural light induces electrical dipoles in the scattering medium which oscillate in all directions of a plane perpendicular to this beam. So the oscillations of the scattered light source are to a certain extent ordered as they lie in a given plane. Scattered light is therefore partially polarized, to an extent depending
4.3 LIGHT SCATTERING IN SEAWATER
183
on the angle between the plane of dipole oscillations (perpendicular to the incident rays) and the direction of the scattered light in question. This relationship could be described equally well by the scattering angle Or, that is, tha angle between the direction of the incident beam and the direction of observation of the scattered rays. The plane of observation determined by the direction of the incident beam and by the direction of observation of the scattered rays is taken to be the reference plane when analysing the polarization of the scattered light. With respect to this plane, we can distinguish in the observed ray of scattered light two components of the electric field intensity vector of the light wave and, corresponding to them, two components of the scattered light intensity-the first polarized perpendicularly and the second parallel to this plane. As the oscillation phases of the wavetrain's electric vectors are random, the dominant intensity of one of these two components means that there exists a preferred direction of oscillation of these vectors; in other words, the resultant scattered light is polarized to a certain extent. These intensity components are described by the non-dimensional intensity functions il = i, and i2 = i,,, which are connected with the total intensity of the light scattered by a particle at angle 6, by the relationship 11
(6,) = 2k2 ED lil(6r) _ I
+ i~(@r)l
(4.3.9a)
further substantiated in the traditional notation (see equation (4.3.42)) as Il(6r) =
"
[ i l ( e r ) -5i2 (or11
7
(4.3.9b)
where I$= E p . By comparing equation (4.3.9) with the Rayleigh formula (4.3.2) for the total intensity of the light scattered by a single particle, we get il = k6CrZ,,
i2 = k6cczcos26,.
(4.3.10)
Analogous relationships can be obtained for N particles per unit volume of medium if we assume scattering to be additive. Hence, the intensity component of light polarized perpendicularly to the plane of observation (il) is constant and independent of the scattering angle. The intensity component of scattered light polarized parallel to the plane of observation (iz) is, however, dependent on the scattering angle 8 , . On combining these two components, we see that light scattered at an angle 6, = 90" is, according to the Rayleigh theory, completely polarized, since i2 = 0. Light scattered at angles of 0" and 180" (along the incident beam) is on the other hand, unpolarized, since il = iz, which with the oscillation phases
4 THE INTERACTION OF LIGHT WITH SEAWATER
184
of these components being random, leads to total depolarization (see Fig. 4.3.3). A depolarization factor (depolarization ratio) 6 = il7
(4.3.11)
12
is now introduced, while the degree of polarization of the scattered light is given by (4.3.12) As we can see from (4.3.2), the intensity of Rayleigh-scattered light depends on the polarizability u, of the scattering particles. Ihe polarizability is a scalar in the equations given here, i.e., it is assumed to be identical in all directions: the scattering particles are isotropic. If we assume further that they are spherical particles of radius yo (consisting of a large number of molecules), the polarizability a, of the material of these particles is connected with their absolute refractive index n (or the complex refractive index m for absorbent particles-see Van de Hulst, 1957) in the Lorentz-Lorenz formula:
(4.3.13) Obviously, the polarizability u, in the formula has units of m3. From the theory of the interactions of electromagnetic waves with matter there also emerges a relationship between the velocity v of the propagation of these waves in the medium on the one hand, and the dielectric permittivity E , the magnetic permeability ,u of the medium, and the velocity with which electromagnetic waves are propagated in a vacuum co on the other: v=-
-. J &P CO
(4.3.14)
Materials which are not ferromagnetic have a magnetic permeability of p
k
1.
so 0,"-
CO
(4.3.15)
./1 and since the absolute refractive index is given by n = co/v, we can write
n2 z
E.
(4.3.16)
4.3 LIGHT SCATTERING IN SEAWATER
185
Applying these values to the material of the scattering particles, and after substituting (4.3.16) in the expression for their polarizability (4.3.13), we also obtain the fallowing formula: (4.3.17)
We must nevertheless remember that this last relationship does not hold for ferromagnetic substances, and that both the refractive index n and the dielectric permittivity E are functions of the wave frequency v. So, for example, the permittivity (dielectric constant) of water, which is as high as 1.89 D (see Section 2.1), refers to the action of a constant electrical field E,, whereas when this field changes with a frequency of v = lOI4 Hz in a light wave (see Table 4.0.1), it is much lower. We can use expressions (4.3.13) or (4.3.17) for the polarizability of spherical isotropic particles in the Rayleigh formula (4.3.6) for the function ,4,(90) (see Morel, 1974). We then obtain the expressions: (4.3.18)
or (4.3.19)
Together with (4.3.8), these expressions enable us to evaluate the volume scattering function BR(0,) for N spherical isotropic particles, much smaller than the light wavelength, having a refractive index n or dielectric permittivity E, contained in a unit volume of the medium. The overall effect of light scattering by a volume element of a medium in all directions is given by the total volume scattering coeficient b, which has units of m-l. By definition, this coefficient is the integral of the scattering function fl(0,) through all directions (through elementary solid angles dQ(4)) of a sphere surroundingthe scattering volume of the medium. Ass cattering is symmetrical, we can write this definition as follows X
b
p(0,)Cm
= 4x
=
2n ,4(8,)sin8,d€Jr.
(4.3.20)
0
Integrating the function p,(0,) described by the Rayleigh formula (4.3.8) gives us the following expression for the total Rayleigh scattering coefficient (4.3.21)
186
4 THE INTERACTION OF LIGHT WITH SEAWATER
After substituting (4.3.18) for p,(90) in this we obtain (4.3.22)
or, if we substitute (4.3.19), we get
(4.3.23) It is evident from these equations how rapidly the scattering coefficient increases with increasing scattering particle size (rz !), e.g., for particles of radius ro = 40 A the coefficientis 64 times greater than its value for particles of 20 A radius. The reader's attention is once again drawn to the spectral characteristics of the Rayleigh scattering coefficient, which rises abruptly with decrease in wavelength as 1/A4. This means that violet light of 400 nm wavelength is scattered nearly 10 times more intensively than red light of 700 nm wavelength, and the scattering of IR is scarcely noticeable in comparison with that of UV (Fig. 4.3.4).
Wavelength
I. [nml
Fig. 4.3.4. The spectrum of molecular scattering according to the Rayleigh theory, illustrating the very strong scattering of short waves (-l/A4).
The results of many detailed measurements of the scattering function j3 and total scattering coefficient b have revealed certain deviations of their values from those calculated from the Rayleigh formulae for isotropic particles-even for gases, in which one could ignore the mutual interactions of the scattering particles. In particular, it was found that light scattered through a right angle is not totally
4.3 LIGHT SCATTERING IN SEAWATER
181
polarised as is required by formulae (4.3.8) and (4.3.9) for an angle 8, = 90". Rayleigh explained this discrepancy by the fact that the particles are anisotropic : the polarizability of the particles is a function linking the dependence of vector pe on vector E, in different directions, and therefore a tensor. He therefore introduced the components of a polarizability tensor along the coordinate axes (see Van de Hulst, 1957). It was also found experimentally that the intensity of scattered light is usually greater than that given by the Rayleigh formulae. Cabannes (1920) therefore introduced appropriate corrections for particle anisotropy to the Rayleigh scattering function for isotropic particles /3,(90) given by (4.3.6) (4.3.24)
in which the fraction on the right side of this expression is known as the Cabannes factor. Thus 0,(90) is the scattering function for isotropic particles and BC(90) that for the anisotropic particles which are usually found in nature; these particles must, of course be far smaller than the waveIength and must not interact with one another, i.e., they must scatter light additively. Expressions (4.3.8) for scattering at any angle and (4.3.21) for the scattering coefficient have also been suitably modified (see Morel, 1974): (4.3.25) (4.3.26)
The values of the depolarization factor 6 or the degree of polarization are linked with the afore-mentioned components of the polarized light intensity in (4.3.1 1) and (4.3.12). Molecular Scattering According to the SmoluchoM..ski-Einstein Fluctuation Theory
Light scattering is observed even in very pure gases and fluids; it would appear from their homogeneous chemical structure and the lack of suspended particles that there are no light scattering centres whatever. Such scattering also displays many features similar to those of Raylejgh scattering like scattering symmetry, polarisation of scattered light and obeisance of the law. A11 this shows that in a fluid there exist unhomogeneities whose dimensions are smaller than the wavelengths of visibfe light.
4 THE INTERACTION OF LIGHT WITH SEAWATER
188
When describing this phenomenon, Smoluchowski (1908) and Einstein (1910) assumed that the “scattering particles” in pure media were fluctuations in the density of elements of the medium. At any instant, the number of molecules which happen to be in sufficiently small units of volume of the medium SV as a result of thermal motion differs. The density of these elements of the medium fluctuates e = @+a@,and therefore their polarizability CI, = Cr,+Scr,, or the dielectric permittivity E = E+ SE and refractive index n = Z + 6n associated with them also fluctuate (see equations (4.3.13)-(4.3.17)). The instantaneous value of each of these variables around a given point (x, y , z) in the medium is therefore the sum of its mean values in time (& E or %) and the momentary deviation from that mean, is., the fluctuation (Se, Sc or 6n). Hence the density fluctuations are 6p = e-p, the fluctuations in dielectric permittivity are BE = E - 2 , etc. The thermal motion of molecules, and hence also that of these fluctuations too, depend on the absolute temperature of the medium T. We know today that in liquid water this phenomenon is connected with the formation and breakdown of Frank-Wen clusters of molecules and with the differentiation in the density of the “packing” of the water molecules (see Section 2.2). They are therefore unhomogeneities of the medium whose lifetimes are sufficiently long in comparison with the frequency of the light waves to be able to act as light scattering centres. A concise outline of the Smoluchowski-Einstein statistical theory combined with the results of experiments on molecular scattering in pure water and clean seawater, is given by Morel (1974). Here we shall just present the main points of the argument, which allow us to derive a formula for the molecular scattering function which agrees well with the results of empirical work on very pure water. Assuming density fluctuations, and hence also fluctuations in the dielectric permittivity of the medium, Smoluchowski and Einstein established that the relative intensity of scattering, described by the scattering function 0,(90) was propor__ tional to the mean square of the fluctuation in the dielectric permittivity (6~)‘ in a volume SV, and to this volume itself (4.3.27) The scattering function j3,(90) thus fixed corresponds to the Rayleigh scattering function with small isotropic particles, because it is assumed that the volume 8V is far smaller than the wavelength. At the same time, however, this volume must be sufficiently large for the number of molecules of the medium contained therein to obey the laws of statistical thermodynamics. The Rayleigh spectral dependence of scattering, DR l/A4, still obtains when these conditions are satisfied because N
4.3 LIGHT SCATTERING IN SEAWATER
189
the same mechanism of electricaI osciIlati ons of particles and secondary wave emission is in operation; only its physical manifestations, such as polarizability, dielectric permittivity and refractive index are determined by density fluctuations in the medium. Although we do not yet know the nature of the dependence of the dielectric permittivity on the density of the medium, the fluctuations of this constant a&(?) can formally be linked with density fluctuations Se thus: SE = (de/de)6e (4.3.28) where ( 8 ~ )is ~the mean square of the density fluctuation. On substituting this last expression for ( 8 ~ in ) ~(4.3.27), the latter equation can be rewritten (4.3.29) The problem is now one of establishing in this formula the thermodynamic dependence of the product 6 V ( 8 ~ on ) ~ the temperature and the derivative deldg or, since E = n2 and the assumptions for (4.3.15) hold, its connection with the refractive index n. It has been found from the laws of thermodynamics that for pure liquids, this formula takes the form (4.3.30a) or 2x2
(4.3.30b)
BR(90) = __ kTk, 14
where k is Boltzmann's constant and k , is the isothermal compressibility coefficient (see Section 3.3). Using one of the known dependences of n(e), e.g., the Lorentz-Lorenz formula 1. ) ( n ~ ,+ we 2 ) can ~n 2 - 1 - to find the derivative d s = --1 - ( ~ ~ ~ - 1-~ = const 3 n2+2 Q de e derive the Einstein expression for the function ,9,(90) from (4.3.30a) ~
(4.3.3 1) This expression can be inserted in the general form of the function /?,(Or) obtained by Rayleigh for isotropic particles (equation 4.3.8) to yield the molecular scattering function at any angle BR(er) =
z2
kTk,(n2 - 1)2((n2+2)2(1+ C O S ~ ~ , ) .
(4.3.32)
4 THE INTERACTION OF LIGHT WITH SEAWATER
190
This formula does not, however, agree sufficientlywell with experimental results on molecular scattering in liquids. The discrepancies noted are attributed, among other factors, to the existence of a dependence of d&/d@or dn/de on the temperature T and pressure p for which the equation using the Lorentz-Lorenz formula does not account. For this reason, the experimentally determined and tabulated dependences of the refractive index of a liquid n (here water) on temperature and pressure, i.e., experimentaldata (an/aT)por (anlap),, hence (ae/aT), or ( d ~ / a p ) ~ , have been used instead of the derivative de/de. The Cabannes correction for molecular anisotropy has also been included and this has yielded an expression for the molecular scattering function in liquids which stands in agreement with experimental evidence on pure liquids (Morel, 1974; Jerlov, 1976). 2x2 1 an 6+66 1-6 (4.3.33) &(Or) = __ kTn2 __ ___ I + 24 k, ( 8 ~ ) : 6-76 If8 Notice that this function still obeys the A-4 law and is analogous to the general form of the function /lc(0,) given by (4.3.25); scattering is therefore symmetrical. Integrating the function fi,(O,) according to definition (4.3.20) produces the molecular scattering coefficient. By analogy with formula (4.3.26) it takes the form (Jerlov, 1976) ~
(
(4.3.34)
In shape, the plot of function fim(O,) is similar to the Rayleigh scattering function, and retains an analogous symmetry. Its values for pure water as calculated by Morel for light of wavelength il = 475 nm (Jerlov, 1976) are as follows: Scattering angle 0, 0"and 180" 10"and 170" 20" and 160" 30" and 150" 45" and 135" 60" and 120" 75" and 105" 90"
Value of function &,(O,) 3.15 x rn-l 3.11 x rn-I 2.98 x m-' 2.78 x rn-l 2.43 x m-l 2.09~ 1.85~ rn-l 1.73 x low4rn-'
(by permission of Elsevier Scientific Publishing Company and the author)
For solutions of electrolytes (including seawater), detailed calculations require other thermodynamic dependences for {~SE)~,dependent on the concentrations
4.3 LIGHT SCATTERING IN SEAWATER
191
and chemical potentials of the solution's constituents, to be taken into consideration in formula (4.3.27), and hence also in (4.3.31) too (Stockmayer, 1950; Morel, 1974). The spectrum of the molecular scattering function at 90" for pure water and clean ocean water is illustrated in Fig. 4.3.5. It shows the rapid increase in short-
2
300
400
500
Wavelength
600
L Inml
Fig. 4.3.5. The spectrum of the molecular volume scatteringfunction Frn (90") for pure water (1) and clean ocean water of salinity S = 35-39%, (Z), plotted from data calculated by Morel (1974).
wave scattering and the effect of sea salt in the water on the increase of molecular scattering across the whole spectrum. Morel (1974) calculated the total molecular scattering coefficient b, (A = 350 nm) to be 103.5 x lob4m-l for pure water and 134.5~ m-l for ocean water. As the light waves lengthen, the coefficient decreases to bm(A= 600 nm) = 1 0 . 9 ~ m-l for pure water and 14.1 x x m-l for ocean water. This very strong scattering of short waves as opposed to long waves explains the dark blue colour of very dean, deep seas observed in daylight. In the water of such seas it is molecular scattering which is dominant : violet light is particularly strongly scattered and some of it emerges from the sea LO reach the observer. The appearance in the water of large concentrations of suspended particles and absorbers of violet light such as organic substances immediately changes this dark blue colour. One comes across such visibly clean
192
4 THE INTERACTION OF LIGHT WITH SEAWATER
waters in the upper layers of the ocean mainly in low latitudes, in the Atlantic, in the Sargasso Sea, and in the central Pacific. They are, as it were, “sunburnt” marine deserts: in these very pure waters there is very little nourishment to sustain living organisms, either plants or animals. The causes of such a situation are complex, but the heating of the upper layer of the sea is significant. The warmed waters are less dense and therefore remain at the surface and preclude the upwelling of waters containing nutrients. Deeper down, on the other hand, where there are plenty of nutrients owing to the continuous rain of animal and plant remains and their decomposition products, there is not enough light for photosynthesis to take place. The fertile areas of the sea are therefore those which contain a high concentration of suspended particles and organic compounds. Suspended particles in the sea are much larger in mass than the wavelengths of light A and the theory of molecular scattering is inapplicable to them. The grey colour of seas containing much suspended matter, like the fertile waters of the Antarctic, is evidence of this. Mie’s (1908) theory of scattering and its modifications (Van de Hulst, 1957) are generally applied to the description of light scattering by marine suspensions. This theory is based on the solution of Maxwell’s wave equations for a plane electromagnetic wave passing through a homogeneous sphere immersed in a uniform medium of different refractive index. Scattering by Marine Suspended Particles. Principles of the Mie Theory
As we saw in Section 2.7, a considerable portion of marine suspensions and air bubbles are greater in size than the wavelengths of visible light. So scattering by suspended particles cannot be described by the Rayleigh theory, or by the Smoluchowski-Einstein fluctuation theory which assumes that the electrical field of a wave within a small particle is uniform. Particles whose dimensions are comparable to or greater than the wavelength occupy an area in which the wave’s electrical field is not uniform, is of varying intensity and whose phase oscillations constantly shift to different sites in the area (Fig. 4.3.2b). Individual fragments of a large particle therefore oscillate as separately induced dipoles and emit phase-shifted secondary waves. These waves interfere to different extents in different directions, so that the actual angular distribution of scattered light is different from what might be expected from the afore-mentioned theories for small particles. So for suspended particles some other method of calculating the scattering function has to be found: Mie (1908) worked out such a method for the simplified case of spherical particles.
4.3 LIGHT SCATTERING IN SEAWATER
193
The essence of the Mie theory is the solution of the Maxwell equations for the case where they describe the electromagnetic field arising when a light wave passes through a non-homogeneous area of the medium forming a sphere which has a refractive index different from that of the surroundings. The complexity of the calculations requires that the theory be made applicable to the simple case of a plane, monochromatic wave passing through a uniform, isotropic medium containing a uniform, isotropic sphere of given radius ro . To simplify the derivation further, it is assumed that this sphere is placed in a vacuum, that is, the complex refractive index of the medium m 2 = 1, whereas the sphere itself has a refractive index of m with respect to its surroundings. A spherical system of coordinates (r, 8, 40) with the origin in the centre of the scattering particle is introduced. In a spherical system like this, the Maxwell equations, along with the boundary conditions at the surface of the sphere (field continuity conditions), become a system of ordinary differential equations whose solution is sought in the form of infinite series. The details of this complicated mathematical derivation, a development of the Mie theory, will be €ound in the monographs by Van de Hulst (1961) and Born and Wolf (1973). Here we shall merely outline the main aspects and conclusions of this derivation which, as we shall see, have a practical application not only to sets of uniform spherical particles randomly distributed in a uniform medium, but also to many natural dispersive systems such as aerosols and marine suspensions (see also Asano and Yamamoto, 1975; Wiscombe, 1980). The plane monochromatic wave, propagating along the z axis and incident on the spherical particle in question, can be described by giving the vectors of its electrical E, and magnetic He field intensities as functions of the z coordinate and time t. This is a single harmonic wave, so that these vectors are described by a harmonic function of the cosy type, where the angle 'p will be equal to (ot-kz). In these calculations it is more convenient to write this function as __ a complex number, using the Euler equation eip = cosg, f isin pl, where i = 11- 1 . The real part of this complex number is an ordinary harmonic function: Re(eiv) = cosy. Vectors E, and He can therefore be written in complex number form:
E,
=
EeoeX(Ut-kz)
He = Heoei(of-k')
(4.3.35)
where Eeoand He, are the oscillation amplitude vectors directed perpendicularly to each other along the x and y axes respectively, and assumed here to be unit vectors. Note also that k = 2x11 = w/co is the wave number, cct = 2 x /T the circular frequency of the wave, and Tits period. Finding the values and direction of the vectors E, and He for waves (4.3.35) scattered over the surface of our
4 THE INTERACTION OF LIGHT WITH SEAWATER
194
sphere is a problem based on the solution of the Maxwell equations for periodically changing fields in a medium having a complex relative refractive index m curlH, = ikm2E,, curlE, = -ikH,. (4.3.36) The solution of these equations is a fairly simple matter if we represent each of the fields E, and He as the sum of two linearly dependent electromagnetic fields "E, "Hand "'E, mH;this is possible on condition that in a system of spherical coordinates (r, 0, @) with the scattering particle at its centre, the radial component of the electrical vector of one of these field components and the radial component of the magnetic vector of the other are equal to zero, i.e., "E,. = 0 and 'H, = 0, hence "E, = E, and "H, = H , . This condition means that the field components with the superscripts e and m are transverse waves-devoid of electrical or magnetic translational oscillations respectively. The Maxwell vector equations (4.3.36) for such field components can be replaced by wave equations of the type
A n t - k2m21T = 0
(4.3.37)
for the scalar potentials "17and "Ll,which are linked with the component fields "E and "H by the equations
whereas the other two components YE and "H are linked by equations resulting from those above ones and from the'Maxwell equations (4.3.36) (Born and Wolf, 1973). The symbol A in equation (4.3.37) denotes a Laplace operator, and the solution of this equation in the spherical coordinates (r, 0, @) can be written as the following series
(4.3.38) 00
Here, the indices I are whole numbers while the other factors in the summation are Al ,&-constant coefficients, known as Mie coefficients, which are dependent on the size parameter xo = 2xr0/iz of the scattering particle and on its relative refractive index m; [$')(kr)=
Hfi\,2(kr), where Hj:\,2 is a Hankel half-
4.3 LIGHT SCATTERING IN SEAWATER
195
order function of the first type, given in mathematical tables (Jahnke et al., 1960); P{')(cosO)-an 1-th order Legendre polynomial of the first type, given in tables as above. In the spherical system of coordinates, the components of the vectors E, and He of the wave are E,, Eo, EDand H,, He, H,, while the wave scattered at the surface of the sphere is a secondary wave of the same frequency produced by the scattering sphere. Using the found form of the potentials "17and "Z7 (4.3.38), the afore-mentioned connections of these potentials with the wave field components and the boundary conditions (field continuity at the surface of the sphere), we can, after complicated rearrangements of the appropriate equations (Born and Wolf, 1973; Van de Hulst, 1961) obtain the components of the wave scattered at some distance r $ 2 from the scattering sphere, that is, where the functions [I1) can be replaced by their asymptotic form
(4.3.39)
Since the radial components of the scattered wave Er and H, diminish rapidly with distance r from the scattering sphere, they are not taken into consideration 1 i in the solution. Obviously, the factors -- cos@S,(8) and - - sin @S1(8) kr kr in (4.3.39) act as the amplitudes of the oscillation (see (4.3.35)). The functions S1 and S, are thus the amplitude functions of the angle 8, of the scattered light components, dependent on the size parameter of the particle xo = 2 x r 0 / l and on the refractive index m. As the components of the scattered wave (4.3.39) and the potentials in the form given in (4.3.38) are connected, the amplitude functions S1and S2 are expressed as infinite series which, after denoting cose = p, take the following form:
(4.3.40)
196
4 THE INTERACTION O F LIGHT WITH SEAWATER
The functions Pj1)(p)= P!')(cosO) are the Legendre polynomials mentioned earlier. The solution (4.3.39) and (4.3.40) describes the spherical wave propagating around the sphere, whose amplitude and state of polarization (angle @) at any point denoted by the coordinates (Y, 8, @) depend on the direction (8, @). Knowing the amplitude of the wave, we can define its intensity. In electrodynamics, we usually use the concept of wave intensity
I' = dF/dA, = I / r 2 . It emerges from the divergence of the stream of a spherical wave about a point source that such an intensity, as in the case of surface irradiance, descreases with the square of the distance from the source. On the basis of the well-known relationship between the amplitude of the wave (with the wave vector k directed along path Y ) and its intensity I' = /Ee0IZ, we can write (4.3.41) for the wave components (4.3.39) of unit amplitude. These are the intensities of two wave components polarised at right angles to each other. Hence the total intensity of light scattered through the angle 8 = 8,, i.e., Z'(8,) 5 Z'(8,, @) by our sphere, irradiated by a plane wave of intensity I; (of any amplitude EoO not only the unit amplitude), is found to be the sum of the component intensities 1;+I; averaged over all possible angles of polarization (4.3.42) where il = ISl(Or)!2, i2 = IS2(8,)l2.The functions i l , i 2 , introduced in (4.3.9), are called the dimensionless intensity functions of the wave components polarized at right angles to each other. The one with the subscript 1 is polarized perpendicular to the plane of observation of the scattered light (the plane delimited by the scattering angle Or). The intensity functions il (Or), i2(8,) calculated from (4.3.40) and (4.3.42) for spherical particles having different refractive indices m and different size parameters xo can be determined from tables of scattering functions (Shifrin and Salganik, 1973). The above considerations are readily applicable to the general case of a sphere placed within a medium of refractive index m 2 ,not necessarily in a vacuum, where m2 is made equal to 1 to simplify the caclulations. In order to use (4.3.40) we
4.3 LIGHT SCATTERING IN SEAWATER
197
need to introduce only two physical quantities for a given scattering particle in a given medium, i.e., the refractive index of the particle with respect to the medium m = m l / m 2 and its size parameter xo = 2nro/1 = 2nro/1,,, which contains the dependence of the scattering function on the wavelength. We stated at the beginning of this section that the results of the Mie theory also hold for disperse systems containing a large number of particles randomly distributed in the medium at distances far greater from one another than 1, that is, in concentrations which allow mutual interactions between particles to be ignored. For N identical spherical particles in a unit volume of a medium, the intensity of the light scattered by a unit volume of such a disperse medium will, as in the Rayleigh theory, simply be N times greater than that which (4.3.42) gives for one particle. So assuming that scattering is additive, we can use (4.3.42) to find the volume scattering function for N particles in unit volume p, = N I’(0 ) r2, which, when
-:IP
the irradiance is due to a plane wave, corresponds to definitions (4.3.4) or (4.3.5) (4.3.43) The subscript p (from “particles”) denotes scattering by suspended particles rather than molecular scattering (subscript m) or total scattering (p without a subscript). According to definition (4.3.20), the total scattering coefficient for N identical spherical particles in a unit volume is an integral of the function 3/, for these particles through the whole solid angle. This reads as follows c
(4.3.44a) 0
or, after expanding the wave number k = 2x11 and rearranging, v
[il(e,)+iz(6,)]sinB,dB,.
(4.3.44b)
0
We must remember at this juncture that this total scattering coefficient and the scattering function are dependent on the wavelength 1 and on the particle dimensions, this dependence being partially concealed in the intensity functions il and i 2 . This dependence is quite obvious on the plots in Figs. 4.3.6, 4.3.7 and 4.3.9.
4 THE INTERACTION OF LIGHT WITH SEAWATER
198
Fig. 4.3.6. The relative scattering cross-section Q, as a function of the particle size parameter xo = 2 x r 0 / l = kro for various values of the complex refractive index m = n-n'i (from Jonasz's
calculations). Particles not absorbing lisht of wavelength 1 take values of n' = 0.
It turns out that the total scattering coefficient b, for a set of identical particles (monodisperse system) is proportional to the summer geometrical cross-section of the scattering particles in a unit volume of the medium, i.e., XD2
b, = Q,Nxr: = Q r N -
4
'
(4.3.45)
where D =: 2v0 is the particle diameter, while N [ ~ n -can ~ ]be called the numerical concentration of pavtides. The dimensionless coefficient of this proportionality Qr is called the relative scattering cross-section of the coefficient of scattering efficiency of a given disperse system, as it characterises the efficiency of scattering by different sets of particles of the same size. Taking into account the relationships (4.3.44b), (4.3.45) and the definition of the size parameter xo = 2xr0/A = kro, we can express Qr for a system containing N identical particles thus: x
Q,
=
1
xo
5
il (6,)
+iz(0,) sin 6, do,.
(4.3.46)
0
Its values can be found from this formula for particles of given sizes and given light wavelengths (parameter xo) from the functions il(6,) f i2(0,) calculated
4.3 LIGHT SCATTERING IN SEAWATER
199
or found from tables (Shifrin and Salganik, 1973) or from the direct evaluation of these functions from equations (4.3.40) and (4.3.42). Examples of plots showing the dependences of Qr(xo)are given in Fig. 4.3.6 for different refractive indices m = n-in'. They show that as the particle radius yo increases or the wavelength shortens, Qr(2nr0/A) at first rises sharply. For values of the parameter x,, 3: 0.1, this is an increase according to Rayleigh's A-4 law that is almost invisible on the scale of this graph. Having reached the first maximum, the value of Qr(xo)for particles not absorbing light of wavelength oscillates around a boundary value of 2 and these oscillations gradually die down as xo increases. We can therefore assume roughly that the relative scattering cross-section of very large particles not absorbing light Qr (xo -+ 00) is equal to 2. The graphs in Fig. 4.3.6 were plotted for particles having refractive indices close to those of marine suspensions. They illustrate the influence of the virtual part of the refractive coefficient of light scattered by particles on the value of their relative scattering cross-section Q r .The increase in the virtual part of this coefficient makes the oscillations of Qr values subside more rapidly and reduces its boundary value, and therefore diminishes the scattering coefficient b, of large particles. For N particles (in unit a volume of medium) of different sizes and size distribution N(D) (see Section 2.7) but having an identical (or perhaps averaged) refractive index m with respect to the medium, the scattering function can be expressed as an integral (4.3.47a) where /I1(D,Or)is the scattering function for a single particle of diameter D = 2ro, as was assumed at the start of this discussion, and Dmin and D,,, delimit the boundaries of particle diameters occurring in the set in question. The function can be found straight away from the relationship (4.3.43) for a single particle ( N = 1). Hence the scattering function for a set of particles with a size distribution N(D), taking (4.3.47a) into account, can be expressed thus: Dmax
Dmin
We must remember at the same time that this is the scattering function for a certain wavelength of light A scattered by particles having a given refractive index m with respect to the medium in which they occur. Taking into consideration the
4 THE INTERACTION OF LIGHT WITH SEAWATER
200
definition of the total scattering coefficient (4.3.20), we can obtain its value from the function b,(0,) for the particles just described:
(4.3.48)
+
The intensity functions il (0, ,0) iZ(0,, 0)can be found, as before, from the Mie theory or from tables of these functions worked out from this theory for given angles O,, given particle diameters D and their refractive index m. The relative scattering cross-section will also vary in the case of particles of different diameters; it is a function of the diameter D.By analogy with expression (4.3.45), we can write the following expression for the coefficient b, for a set of N particles with a size distribution N(D) and relative scattering cross-section QJD)
s
Dmax
b,
=-
2
Q,(D)N(D)D2dD.
(4.3.49)
Dmin
By means of the Mie theory for particles not absorbing light (n’ = 0) and the Rayleigh theory for small particles, Burt (1956) worked out a practicable nomogram enabling one to determine the relative scattering cross-section Qr with respect to particle radius, wavelength and relative refractive index of the particle. This nomogram is shown in Fig. 4.3.7. On the lower graph we can find, for any desired wavelength, the required particle radius ro at the point where the horizontal line (1= const) cuts the appropriate line of the graph (ro = const). The vertical section is taken from this point to the upper graph, to a height corresponding to the value of the desired refractive index. The end of this section cuts or at least approaches one of the curves plotted on the upper graph Qr = const from which we can read off Q, on the upper scale of the graph (or possibly between the curves, by interpolation). So, for example, for a wavelength of 1 = 0.5 pm, particle radius ro % 0.07 pm, and relative refractive index m = 1.20, Qr % 0.07. On the basis of this nomogram for non-absorbing particles, notice that as the particles become bigger, Qr approaches 2, fluctuating about that value. Notice further that Q r is strongly dependent on 1, in other words, there exists a strong spectral scattering selectivity for particles of radius ro < ii and a very weak dependence of Qr on A for large particles whose radii ro % 1. If we have determined the relative scattering cross-sections and the particle size distributions, we can calculate the total scattering coefficient of sets of particles in accordance with equation (4.3.49).
4.3 LIGHT SCATTERING IN SEAWATER
m
Particle radius c0 [ p n
]
Fig. 4.3.7. Nomogram for determining the relative scattering cross-section Q , for particles not
absorbing light with respect to the wavelength of the light, the relative refractive index and the particle radius. Drawn by Burt (1956) from calculations based on the Mie and Rayleigh theories.
These results of the Mie theory refer to calculations for homogeneous spherical particles. In actual fact, they describe the electromagnetic wave field due to reflection, refraction and diffraction of light waves incident on homogeneous spherical particles in a homogeneous medium, i.e., in a monodisperse system. Seawater, however, is a polydisperse medium, containing suspended particles of multifarious shapes and diverse optical properties. It so happens, however, that sets of suspended particles, very irregularly-shaped but randomly distributed in the sea, have scattering properties approximately the same as those which emerge from the Mie theory for sets of isotropic sphericaI particles (Beardsley, 1968; Jonasz, 1975). Serious deviations from the results of theory are observed only with respect to backward scattering (Holland and Cagne, 1970), where, as we shall see, large suspensions scatter only a minute fraction to the total
4 THE INTERACTION OF LIGHT WITH SEAWATER
202
energy scattered. The above equations from the Mie theory are thus successfully applied to the description of light scattering by suspended particles with only their substitute diameters (see Section 2.7) and refractive index with respect to seawater being taken into account. The accurate determination of this refractive (a)
b,, (0,)
Suspended particles, Baltic
---.
0.1
1
10
100 180 Scattering angle 0,
["I
Fig. 4.3.8. Scattering functions p(0,) in pure water and on marine suspended particles from the Baltic, measured and calculated from the Mie theory for a wavelength of 3, = 633 nm (from Jonasz, 1980). (a) In polar coordinates (the scale for pure water has been enlarged lo4 times with respect to the scale for suspended particles because of the much weaker scattering by pure water); (b) in rectangular coordinates (in the same logarithmic scale): continuous Iine-calculated results, dots-measurements.
4.3 LIGHT SCATTERING IN SEAWATER
203
index for given sets of marine suspensions still remains a problem. In practice, however, we use various values of these indices from the range given in Figs. 4.3.6 and 4.3.7. The calculated values of the function P,(O,) are compared with its measured values in order to confirm the correct choice of refractive index for the suspended matter of a given body of water (Kullenberg, 1974). Graphs of these functions, calculated from theory and actually measured in the sea are shown in Figs. 4.3.8 and 4.3.10. These and many other light-scattering functions investigated show certain regularities. First of all, the volume functions of molecular scattering, similar in shape to the graph in Fig. 4.3.3, are symmetrical with respect to the plane perpendicular to the incident rays which pass through the centre of the scattering volume element. The volume scattering function of marine suspended particles are, on the other hand, strongly extended forward on a similar graph with polar coordinates (Fig. 4.3.8a). Apart from the characteristic extension of these functions in polar coordinates (which means strong forward scattering), their precise shapes are complicated somewhat because of interference between the waves
I 300
Loo
500
600
Wavelength in air
700
2
'
800 [ nm]
Fig. 4.3.9. Spectral distributions of the scattering coefficient standardized to unity for A = 600 nm, on scattering particles with a relative refractive index of m = 1.15 (assuming it is independent of A) and various radii in the range 0.1 p m f ro f 1.6 pm. The dashed line refers to Rayleigh scattering (from Burt, 1955).
204
4 THE INTERACTION OF LIGHT WITH SEAWATER
emitted by fragments of particles acting as dipoles. The principal feature of these functions is however their forward extension, i.e., the rapid increase in their values for small scattering angles 8, in comparison with those for wide angles 8,. This means that suspended particles which are large in comparison with the wavelength, scatter light forward most strongly through small angles 8, with respect to the direction of the rays falling upon the scattering centres. At the same time, the minimum of the volume scattering function @(Or) shifts from an angle of Or = 90" in clean waters or at small centres to wider angles, up to about 130°, in waters containing a large quantity of suspended matter, such as the Baltic (Fig, 4.3.8). Lastly, as the scattering centres become larger, the spectral selectivity of scattering becomes smaller and smaller, unlike the strongly selective molecular scattering assumed by the Smoluchowski-Einstein theory (Fig. 4.3.9). It is usual in every type of seawater that molecular scattering and scattering by suspended particles take place simultaneously, so that the measured scattering function /I(&) reflects the resultant effect of these two processes. In very clean waters, where the influence of the few suspended particles present is small in comparison with the molecular interaction of the scattering centres, molecularly scattered light prevails; the sea is strongly coloured violet in daylight and the shape of the scattering function is roughly the same as that of the molecular scattering function. Conversely, in seas containing a large quantity of suspended matter, scattering by these particles is the dominant process. It has been found that the scattering function and the relative scattering cross-section increase steeply as the scattering centres become larger within the range of sizes of particulate matter found in the sea. Thus the appearance of average concentrations of suspended particles in the sea (Section 2.7) leads to a big increase in light scattering, and so more light is scattered by suspended matter than molecularly. As can be seen in Fig. 4.3.9, the light scattered by suspensions is spectrally much less selective, so the colour of the water in daylight becomes much less violet and more bluish or even greenish. The simultaneously increasing absorption lends a greyish tinge to the water. These phenomena are of fundamental importance in the transfer and release of radiant solar energy in the sea, and that is why so much attention is given to the scattering of light and its characteristics in hydrooptics. A set of scattering functions measured in various waters is illustrated in Fig. 4.3.10. They are mostly similar in shape, while the differences between them are the upshot of differences in the size distribution and the refractive indices of the particulate matter contained. A simple analytical form of the scattering functions observed in the sea is not
4.3 LIGHT SCATTERING IN SEAWATER
205
t Pure water 10.’
1
0
1
30
1
1
a
60
1
.
’
1
-
I
3
8
120 150 180 Scattering angle 8, 1°1
90
Fig. 4.3.10. Light scattering functions measured in various waters. Gulfof Gdahsk, Baltic % a 4 (from Kullenberg, 1969).
=
633 nm (from Prandke, see Gohs et ul., 1978; Sargasso Sea, pure water-A
= 655 nm
known. They are therefore most often described by a series of Legendre polynomials (Chandrasekhar, 1950; Ivanov, 1975)
(4.3.50) where the denotation ,u = cos19, has been introduced, P l ( p )are 1-th order Legendre polynomials (first type), and x, is an expansion coefficient dependent on the shape of the scattering function. This expansion is used to solve the radiant energy transfer equation discussed in Section 4.4. and in Chapter 5 (e.g., the square formulae method presented inter alia in Chandrasekhar’s monograph, 1950). An exact description of the scattering function of seawater using this series (4.3.50) is, however, extremely complicated, since a good approximation requires a large number of terms in this theoretically infinite series to be taken into consideration. But knowing only the coefficient x1 in the first term of this expansion
4 THE INTERACTION OF LIGHT WITH SEAWATER
206
often sufficesto give us a satisfactory idea of the shape of the scattering function. For if we avail ourselves of the orthogonality of Legendre polynomials, we can easily demonstrate the existence of the following relationship
(4.3.51)
Bearing in mind the definition of the scattering coefficient (4.3.20), we can see that the numerator in (4.3.51) is the scattering coefficient b, whereas the parameter x , is proportional to the weighted average cosine of the scattering angle (weighted mean over the scattering function p(p)), i.e., to xl =
(4.3.52)
3ji = 3coser
(cf. the averaging method, e.g., in (5.2.5) or (5.4.18) in Chapter 5). The parameter x1 which, as we see, can take values 0 6 x1 < 3, can be called the elongation parameter of the scattering function B(0,). The spectral distributions of the values
z
I' 500 Wavelength
I 600
I
700
A [nm]
Fig. 4.3.11. Spectra of the extension parameter xl(A) of the scattering function for various seawaters-from very clean to highly polluted: A-Sargasso Sea; 'B-average ocean waters; C-Baltic Sea; D-Gulf of Gdarisk (from W o i n i , 1977).
of this parameter for very clean ocean waters, fairly clean seawaters and heavily polluted coastal waters are illustrated in Fig. 4.3.11. The xl(A) spectra of other seawaters can almost always be found on the graph between the spectra shown in the figure. A value of x1 = 0 would correspond to purely molecular or isotropic
4.3 LIGHT SCATTERING IN SEAWATER
207
scattering, whereas the value xi = 3 would describe exclusively forward scattering through a scattering angle of 8, = 0. In natural waters, the values of this parameter lie between the limits 2 < x1 < 2.9 (see Fig. 4.3.11). It also appears that the ratio of the scattering function through a chosen constant angle /3(Or) to the total scattering coefficient is approximately constant. Kullenberg's analysis (1974) of data shows that for the large majority of investigated instances in different seas, the ratio 8(45")/b varies from 3.2 x to 3.5 x x Obvious discrepancies also occur: e.g., the value of this ratio for Baltic Establishing the ratio P(Or = const)/b in given surface waters is c. 2.1 x waters can make the techniques for measuring the scattering coefficient much easier, because measuring /?(Or) for a lot of angles and integrating it through those angles is very troublesome. Particularly difficult is the accurate measurement of @(Or) for very small angles 8,(0-3"), which is where the most light is scattered in average sea water (as can be seen from Fig. 4.3.10). The principle of measuring p(Or) has already been illustrated to some extent by Fig. 4.3.3a, where a parallel beam of light (a plane wave) irradiating the volume element is obtained from a source fitted with an optical collimator (the lens on Fig. 4.1.1c), and the scattered light intensity, or preferably the radiance L, is measured with a radiance meter placed at a suitable angle (see Fig. 4.1.3). If the beam of light incident on a volume element of the medium in a solid angle AOo is described by the radiance L o , and the beam of light scattered through an angle 8, in a small solid angle A52 is described by an increment of the radiance hL,(rS,), the operational definition of the scattering function p(O,), equivalent to definitions (4.3.4) or (4.3.5), can be written in the form (4.3.53)
So by measuring the radiance of the incident light and that of the light scattered at angle Or, we can determine the scattering function. But this measurement is a very difficult and delicate operation, and the technical problems are exacerbated because measurement usually takes place in situ. Removing a water sample brings about considerable changes in its optical properties (the composition of particulate matter changes). The techniques of measuring light scattering in the sea are a separate, complex question with which we shall not concern ourselves here (see Tyler, 1963; Kullenberg, 1968; Tvanov, 1975; Jerlov, 1976). As a supplement to the various graphs of the scattering function and coefficient, Table 4.3.1 gives values of this coefficient measured in different waters. These
4 THE INTERACTION OF LIGHT WITH SEAWATER
208
figures give some idea of the optical diversity of waters caused by marine suspensions. Suspended particles in the sea are an important element in the marine environment which not only strongly influence the transfer of radiant energy but also the transfer and conversion of many chemical substances. This is why we are seeking TABLE 4.3.1 Light scattering coefficients in various waters Type of water, study area
1 [nml
Pure water
Scattering coefficient m-ll
Source of data
350 400 450 500 600
1.035 0.581 0.349 0.222 0.109
Morel, 1974
350 400 450 500 600
1.345 0.175 0.454 0.288 0.141
Morel, 1974
465 465 465 465 465 625 625 625 625 625
5.4 2.5 4.7 2.8 1.3 5.1 2.2 4.3 2.8 1.1
Jerlov, 1963 (see Jerlov, 1976)
North Atlantic Mediterranean Sea Black Sea North Sea
520 520 520 520
4.6-41 6.9-30 2.3-46 27-55
Kopelevich et. al., 1975 (see Ivanov, 1975)
Southern Baltic
380 655 380 655
Clean ocean water
Central Pacific (00"02'N, 152"07'W)
Om 147 m 295 m 787 m 2385 m
Om 147 m 295 rn 787 m 2385 m
Gulf of Bothnia
21 20 31 28
Jerlov, 1955 (see Jerlov, 1976)
4.3 LIGHT SCATTERING IN SEAWATER
209
possible applications of optical measurements of the study of other physical properties of suspended particles such as their size distribution, concentration, density and refractive index. The possibility of performing rapid, remote-controlled, precision optical measurements in the sea suggests that such hydrooptical methods should be applied. But then the difficult “reverse problem” has to be solved, namely, the problem of describing the physical properties of marine suspended particles from measurements of their light-scattering characteristics. Jonasz (1980) has made considerable progress in this line of research. A Matrix Description of Scattering. Stokes Parameters
From the mechanism of scattering previously described, we can see that scattered light is always partially polarized, both when the scattering centres are irradiated with ordinary-non-polarized-light, and when they are illuminated with polarized light, which is partially depolarized on being scattered. As we know, an ordinary light ray consists of a large number of independent, consecutive wave trains (photons) polarized linearly in various planes and oscillating in various phases, and propagated in a given direction. The resultant intensity of the electrical field E, at the investigated point in space through which the light is passing is therefore subjected to oscillations, not along one line but in a different direction at every instant. With a velocity comparable to that of light, the resultant vector E, thus changes its orientation in succession in all the directions in the plane perpendicular to the ray: this is the propagation of linearly unpolarized light. But this does not mean that the amplitudes of the oscillations of this resultant vector E, must be identical in all directions. Often, and especially in scattered light, there exists an angular distribution of these amplitudinal oscillations (in the plane perpendicular to the ray) such that there is a distinct maximum in one direction and a gradual transition towards a minimum in the direction perpendicular to the first one, which is what “a certain degree of polarization” implies. This results from the superimposition of wave trains whose phases and directions of polarization are not entirely random and independent of one another, but which are derived from partially-ordered or interdependent sources such as dipoles oscillating at right angles to the direction of the plane excitation wave. Much research on the polarization of scattered light in seawater has been done by Ivanoff et al. (1974), Timofeeva (1962) and others (Jerlov, 1976). The electrical vector E, in a light ray can be resolved into two component vectors along the perpendicular x and y axes at right angles to the ray. This is tantamount to resolving it into components, that is, projections onto perpendicular
210
4 THE INTERACTION OF LIGHT WITH SEAWATER
planes respectively delimited by the angles 8, and Q,, which is what we did for a single scattered wave in (4.3.39). The angle 8, denotes the plane of observation of the scattered waves, whereas the angle Q, denotes the plane at right angles to the first. We then have a vector Ee expressed by two component vectors Ee = Ee,l and ED E Eel, the former oscillating in a plane parallel to the plane of observation and the latter in a plane perpendicular to the plane of observation. These vectors completely specify the light waves in question, which the intensity or radiance of the scattered light by themselves do not do. In order to accommodate all this information in a form convenient for calculations, Stokes, in 1852, introduced four parameters associated with the components of vector E, in the following way
(4.3.54)
These parameters have the units [Wm-2] and are now called Stokes parameters or, in sum, the Stokes vector. The brackets indicate averaging over a time At -+ 03, and the asterisks by the symbols E; denote the conjugate values of Ee as a complex number". As we know the product EeEZ of a complex number and a complex conjugate number obviates the need to square the real number. In equation (4.3.41) this was denoted by the abbreviated symbol /El2. The first of the four Stokes parameters expresses the light intensity in [W/m2] (I' = dF/dA,, which can replace the radiance L). It enables us to distinguish clearly the two intensity components, polarized parallel and perpendicularly to the plane of observation, i.e., Z' = Zi+Zl, where I; = EellEZl, Z l = EeIE,*I. These components were denoted earlier by Ze ill = i2 and Z@ iL = il respectively (see equations (4.3.41), (4.3.42), (4.3.9) and others). The other three Stokes parameters define the shape of the polarization ellipse which the end of the electrical vector Ee in general describes and the inclination of the axis of this ellipse to the plane of observation in question (Fig. 4.3.11). The Stokes parameters satisfy the relationship ZJ2 2 Q 2 + U 2 + V 2 (4.3.55) N
-
and taken together, enable us to specify unequivocally the energy and state of polarization of a light beam, and at the same time, to pick out beams which
* A number conjugated to (a+ ib) is the number (a-ib).
4.3 LIGHT SCATTERING IN SEAWATER
21 1
differ from the rest by even one parameter (e.g., a beam incident on some system from a reflected, scattered, attenuated beam or one whose polarization is merely altered, etc.). The Stokes parameters for a simple scattered wave whose components are given by (4.3.39) are readily described. IF we denote generally the amplitude of these components by aL and a,, and take into account possible phase differences of the angle p, between them, we obtain the Stokes parameters expressed by these quantities in the following way: I’ = ai+a:,
Q
= ai-af,
U = 2a,,aLcospl, V = 2a,,a,sintp,
(4.3.56)
the first of which gives us the equation for the intensity of the scattered light wave (4.3.42). Instead of using the analytical form of the wave of the type in (4.3.39), expressed with the aid of amplitudes and phases, we can write the geometrical form of such a wave by using the parameters of the polarization ellipse and its inclination to the plane of observation. By comparing the analytical and geometrical parameters (Van de Hulst, 1961), we get the Stokes parameters in the form I’ = a2,
Q = a2cos2ycos2~, U = a2cos2ysin2~, V = a2sin2y, (4.3.57) Y
Fig. 4.3.12. Geometrical sketch explaining the Stokes parameters of elliptically polarized light. Ee-the position of the wave-intensity vector and its comoonents E e l and Eel j at any instant.
212
4 THE INTERACTION OF LIGHT WITH SEAWATER
where a is the amplitude of the resultant wave, whereas the meanings of angles y and x are explained by Fig. 4.3.12. Besides the full description of light waves given by the Stokes vector, and not just by their intensity, we also need a fuller description of the transformation of waves by a scattering system (or some other) than the one given by scalar scattering functions P(0,). Assuming that the very phenomenon of light modification is a linear transformation, we can introduce into the description of this transformation a table of coefficients (Chandrasekhar, 1950) called the Miiller matrix. If the Stokes parameters for incident light are I;, Q p , U p , Vp and those of scattered light are T,Q, U,V, we have (4.3.58) (Z'QUV) = const. F(ZL, Q p , U p ,V p ) . The Miiller matrix F is in fact an operator of the light-flux transformation by an optical system and contains all the information on the properties of that system that can be obtained by optical means. When it describes the transformation of a light flux by a scattering system, this operator is generally represented by a 4 x 4 element matrix called an intensity scattering matrix (as opposed to a 2 x 2 element phase matrix-Jonasz, 1974). The elements of this scattering matrix Fij are functions of the scattering angles 8, ,@, and of the physical parameters of the scattering particles. They can all be determined experimentally, although measuring the radiance and state of polarization of light scattered in all directions is complicated. The transformation of the Stokes vector of light unpolarized as a result of scattering is as follows:
(4.3.59a)
since the first parameter of the Stokes vector for unpolarized light is equal to 1; and the others are equal to 0. The above expression leads to
(4.3.59b) from which, bearing in mind the definition of the scattering function ,8(8,), we can find a connection between it and the first term of the intensity scattering matrix :
(4.3.60)
4.4 THE TRANSPARENCY OF SEAWATER TO LIGHT
213
The scattering matrix is much simplified when the scattering particles are symmetrical in shape, preferably spherical, and isotropic, that is, equally polarizable in all directions, as the Mie theory assumes. In this case, the polarisation vector of the particle is identical and always in agreement with the intensity vector of the electrical field of the incident light wave. For this reason, some of the elements of the scattering matrix are reduced to zero while the others depend only on the scattering angle 0, and not on the angle Qi,. The intensity scattering matrix for spherical, isotropic particles thus takes the form
(4.3.61)
The following equalities exist: Flz= Fzl,F34= F43,Fll= F22,F33= F44. The first term of this matrix for a homogeneous scattering sphere is 4, = a(sls:+&S3, (4.3.62) which in combination with expression (4.3.60) gives a formula for the scattering function ,b obtained earlier from the Mie theory. For Rayleigh scattering S1 = 1, S, = cose,, and the scattering matrix takes the form:
(4.3.63)
Using scattering matrices to study the properties of seawater containing natural suspended matter is extremely difficult and has so far been done only rarely (Beardsley, 1968; Kadyshevich et al., 1976). This is because the connections between all 16 elements of the operator F and the physical and chemical properties of marine suspensions are unknown. So we usually restrict ourselves to the nonpolarized light scattering function p(0,) which is linked with the first term of the intensity scattering matrix given by equation (4.3.60).
4.4 THE TRANSPARENCY OF SEAWATER TO LIGHT AND OTHER ELECTROMAGNETIC WAVES. THE RADIANT ENERGY TRANSFER EQUATION IN THE SEA
As light passes through water, its energy flux is gradually attenuated as a result of the combined action of absorption and scattering (Fig. 4.4.1a). If a pIane wave
Fig. 4.4.1. Diagrams showing the attenuation of light passing through a layer of medium dr, and the role of the path function L* and source function L in the radiant energy transfer equation. (a) The simplest case of a single beam of light rays ABCD, part of which is absorbed (ray B ) and part scattered (ray C), which results in an overall attenuation of the beam. The radiance Lo decreases by dL over a distance dr to give a value of LI;(b) the situation when the energy of the beam transferred and attenuated across distance dr is amplified by the accidental scattering in this direction of part of the light (ray C,) emitted from other directions. Although the radiance Lo decreases by dl, across dr, it is simultaneously augmented (- La)by the scattered ray C. (path function Lt); (c) the situation when the radiance transferred and attenuated across dr is amplified as a result of the luminescence from internal sourccs of light, e.g., the bioluminescence of marine microrganisms (source function Lq).
4.4 THE TRANSPARENCY OF SEAWATER TO LIGHT
21 5
of light is incident normal to the surface of a flat layer of water of thickness Sr -+ 0, the relative attenuation of the energy flux of the light in this layer is proportional to Br and is equal to car, where c is the volume attenuation goeficient of light. If we use the radiance L to describe the energy flux of the incident light, we can write the flux attenuation relative t o its initial value across the layer 8r as follo ws: SL L
___ =
-cSr,
(4.4.1)
where SL is the decrease in the initial value of the radiance L along the infinitesimally small path Sr. The same expression can be written more precisely as a differential
-dL- - -cL, dr
(4.4.2)
which expresses the law of attenuation of light rays passing through an infinitesimally thin layer of medium dr, perpendicular to its walls. In practice, this is the law of attenuation of a parallel beam of light, which is the differential form of the radiant energy transfer equation for the simplest case when only one, thin, parallel beam of radiation passes through the medium. The attenuation coefficient of a given wavelength (A) of light, i.e., c(L), is the sum of the coefficients (functions) of absorption a(/?) and scattering b(il):
c(L)
=
+
a(L) b(L).
(4.4.3)
It has units of [m-l] and is a function of the wavelength L. In general, it is also a function of the coordinates in the water space and can vary with time. The coefficients of absorption a, scattering 6 , attenuation c and the scattering function p(0,) make up a set of functions defining the inherent optical properties of the sea, as opposed to the apparent optical properties described in the next chapter. The chief characteristic of the inherent optical properties of the sea is that they are dependent not on the external irradiance of the seawater in question but only on its chemical composition and physical structure. The inherent optical properties of the sea thus characterize seawater from the optical standpoint, they enable us to distinguish between water masses having different physico-chemical properties and, as a function of the coordinates of water space, allow us to draw conclusions about phenomena other than optical ones taking place in the sea, e.g., about the concentrations of suspended matter, biological productivity of waters, layering
4 THE INTERACTION OF LIGTH WITH SEAWATER
216
of water masses, and so on (Jerlov, 1976; Neuimin et al., 1964, 1966, 1976; Dera, 1965, 1971). The passage and attenuation of light in a given direction in the sea is usually accompanied by scattered light coming from all directions (the beam A* B,C, in Figs. 4.4.lb and c). Some of this background scattered light amplifies the rays travelling in the direction of observation, i.e., in the direction of the energy transfer in question (Fig. 4.4.lb, ray C,). The attenuated radiance in this direction is augmented in every unit section of the path dr by a value of L, called the path function (units: radiance x m-l). The energy transfer equation (4.4.2) must therefore be extended by a term expressing the increase in radiance due to the added scattered light in the given direction. If we denote this direction by the unit vector go, we can write L -= L(go)and L, --= L,(go), and the transfer equation is, in short: dL dr
-cL+L*.
-=
(4.4.4)
The path function L, can be determined by means of the scattering function @(Or), where the angle 8, is, as before, the angle between the directions of the incident and scattered radiances. The background radiance incident on a volume element of the medium lying in the path of the transferred rays under consideration can have any direction (denoted by a unit vector) in space, whose directional angles are denoted by (O’, @I). The radiance scattered in the direction in question 5, obviously has the same directional angles as the direction of radiation transfer; they are denoted by (O,, Go). These angles are usually worked out from the zenith (0, or 6’) and geographical north (@, or @’). The scattering angle 0, is the angle between the vectors 5, and y, and the scattering function @(Or) = @(g‘,&) = p(%‘, 6 , ,@), expresses the degree of scattering in direction (%,, di,) of light travelling from any direction (O’, P ) . In the case of a single “interfering” beam of background radiation, denoted ~ path function is simply the product by the rays A*B,C, in Fig. 4 . 4 . 1 ~the B(O’, @‘, B0, @,)L(B‘, @‘)ha’(see equation (4.3.9, where AQ‘ -+ 0 is a small solid angle around the direction of the incident light in question. In actual fact, however, such “interfering” light fluxes can reach our volume element from any direction, and so these products have to be integrated through all directions O’, @‘ in order to obtain the full value of the path function @I,
2x x
(4.4.5a)
4.4 THE TRANSPARENCY OF SEAWATER TO LIGHT
217
- here
sine’ do’ d@’ = dQ(ff’, @’) expresses the solid angle around a given direction (Of, ds‘) defined in such a manner as to ensure correct integration through a sphere (see Fig. 4.1.1). The same expression (Preisendorfer, 1961) can be written more simply using direction vectors because, according to our earlier definitions, the unit vector g‘ corresponds to the direction e’, @’, and the unit vector go to the direction (0, , L*(50)
=
\ N E ’ Y 5o)LG’)dQ(S’),
(4.4.5b)
47t
c’,
where d 0 g ’ ) is the elementary solid angle around direction and 4 i is~ the full solid angle. The product under the integral thus expresses that part of the radiance L(e’) incident at the solid angle dQ(g‘) which was scattered in the direction of radiant transfer under consideration 5,-conventionally on a unit section of the path r in direction go. Integrating through the whole sphere of angles sums these products as contributions to the increase in radiance in the direction of transfer g o , emanating from the light present within the water, which is usually scattered light and arrives at the point under consideration from all directions 5‘. It is in this way that the path function (4.4.5) supplements the transfer equation (4.4.2), giving it the form (4.4.4). A further reason for the amplification of the light transferred in direction g o , especially at great depths or at night, could be the light sources contained within the water itself (Fig. 4.4.lc, function L,), such as certain microorganisms capable of emitting light, e.g., zooplankton of the genus Metridia, and others (see Boden and Kampa, 1964; Dera and Wggleriska, 1980). This phenomenon is known as bioluminescence. Other internal radiation sources include the IR radiation of the heated mass of water or the fluorescence due to the solar excitation of organic substances in the water. When luminous microorganisms or other sources of radiation find themselves in the path of the transferred light flux, part of their light also amplifies the light travelling in the given direction go by a radiance L, per unit path in this direction. Lq is called the source function and like the path function it has the units of [radiance x m-l]. The radiant energy transfer equation in the direction go thus has to be expanded to include a further term, the source function: dL -cL+L*+L,. (4.4.6) dr We must remember that the radiance functions in this equations are functions of direction and space coordinates and are dependent on the wavelength, i.e., L = ~ ( xy , z , 8, @, A); L, = L , ( X , y , z , e, G,a); L, = L~(x,y , z, e, Q, 2). __ =
4 THE INTORACTION OF LIGHT WITH SEAWATER
21 8
Equation (4.4.6) is the general form of the timeless radiant energy transfer equation; it describes the spatial changes of the radiance in the sea with respect to the inherent optical properties of seawater, i.e., its attenuation coefficient c = a t b and scattering function B(O,), in the presence of internal light sources (polarization effectsare neglected). We assume that the refractive index of seawater throughout the entire volume of water studied is identical and time-invariable (i.e., n = const) and that the radiance L does not change with time (i.e., the light flux is quasistationary (dL/dt = 0)). Without these assumptions we would have to include in the equation time derivatives of these parameters (see Chandrasekhar, 1950). The differential operator (4.4.7) called a Lagrange derivative, is used because the direction of radiation transfer along path r is not usually the same as the direction of the axis of the coordinate system (Fig. 4.4.2). Descriptions of the optical properties of the sea often make use of the model of horizontally homogeneous sea layers, i.e., a model of a sea in which the properties studied in the horizontal are identical at every depth z and are dependent only on that depth z (they are not dependent on x or y). The attenuation coefficient in a horizontally homogeneous sea would therefore be c(x, y , z) = c(z). Likewise the radiance Lg,x , y , z) = L(g, z), and the path function L*(g, z) and source function L,(g, z ) also depend only on z. The change in transmitted radiance is then (4.4.8) Zenith
Z
Nadir
Fig. 4.4.2. Geometrical sketch accompanying the radiant energy transfer equation in a horizontally homogeneous sea.
4.4 THE TRANSPARENCY OF SEAWATER TO LIGHT
219
since aL/ax = aL/ay = 0. Figure 4.4.2 shows that dz/dr = cos0, hence, taking equation (4.4.8) into consideration, the transfer equation (4.4.6) in a horizontally homogeneous sea takes the form cos 0 dUz7 8 , @) = -c(z)L(z, 0 , @ ) + L * ( z ,8 , @)+L,(z, 8 , @). dz
(4.4.9)
Neither the transfer equation (4.4.6) not its alternative form (4.4.9) can be solved analytically for such a general case. On the other hand, the simplified form (4.4.2) of this equation is easy to solve and is most frequently used as an operational definition of the attenuation coefficient of light in seawater c and serves to determine it from radiance attenuation measurements. If we apply a suitably powerful source of artificial radiance Lo(go) such that cLo(go) % L 9 e 0 )+ &&), that is, we can assume in the general transfer equation (4.4.6) that L , z 0 and L z 0, we can reduce this equation to the simplest case (4.4.2) and find its solution straight away in the well-known form
(4.4.10) where Lr(E,o)/Lo($o)= T, is the ordinarily measured radiance transmittance along path r in direction go in the medium when Logo)is the radiance of light entering the water layer and L,eo) is the radiance after the light has travelled a distance r through this layer. The value of the radiance transmittance in seawater along a distance of r = 1 metre is called the transparency of the water in the physical sense and expresses the general meaning of this term more precisely. The attenuation coefficient c should be regarded as the principal inherent optical property of seawater. The spectrum of this coefficient c(A) characterizes and distinguishes water masses from different seas. Moreover, c is a significant coefficient in the transfer equation, determining the effectiveness of radiance transfer as an optical signal transmitted in water in a given direction. It thus defines the underwater visibility conditions in the sea (Olszewski, 1973; Sokolov, 1974). The values of the attenuation coefficient in seawater, which are the sum of the absorption coefficient and the scattering coefficient (equation (4.4.3)), result from the physical phenomena described in detail in Sections 4.2 and 4.3. It is measured in the sea using transparency meters in all oceanographic studies and provides useful information about the non-optical properties of seawater. Typical spectra of the attenuation coefficient c(A) in the visible waveband measured in different seas are shown in Fig. 4.4.3. This shows quite plainly that in different
220
4 THE INTERACTION OF LIGHT WITH SEAWATER
Wavelength 1 [nrnl
Fig. 4.4.3. Light attenuation coefficient spectra of different seas. Pure water (like clean ocean waters)-from Clarke and James, 1939; Coastal waters-Chesapeake Bay-from Hulburt, 1945; Baltic, Gulf of Gdansk-from Gohs et a[., 1978.
waters these spectra are variously located on the graph, which implies a wide variety of water transparency over the entire visible waveband. The main cause of this variety are marine suspensions which scatter light strongly though not very selectively. Furthermore, the slope of these spectra in the short-wave band vary, and this is attributed largely to the varying absorption of light by yellow substances (this was discussed in Section 4.2). Thus in waters containing large quantities of organic substances, the minimum of the attenuation coefficient c(A), i.e., the maximum transparency, is shifted towards the longer waves-from violet in clean ocean waters to yellow and even orange in river estuaries. The slope and shape of the attenuation spectrum in its long-wave range alter very little, because the same strong absorption by water molecules always prevails in this region, onto which the poorly selective scattering on suspended matter is superimposed. The full spectrum of the attenuation coefficient for wavelengths of from hundreds of kilometres to fractions of an Angstrom unit is shown (after Ivanov, 1975) in Fig. 4.4.4. This shows the only deep window in the transmittance spectrum of seawater in the visible region of the electromagnetic spectrum (a minimum
4.4. THE TRANSPARENCY OF SEAWATER TO LIGHT
ep
0
8"
L .
Ikm
Im
221
,_---, , .
I _ _ -
Imm liJm
IA
Wavelength 1
Fig. 4.4.4. General outline (broken line) and measurements (circles) of the attenuation spec a of electromagnetic waves in ocean water over a wide range of wavelengths (after Ivanov, 1975, by permission of Nauka i Tekhnika and the author). The continuous line refers to the attenuation of sound waves-for
comparison.
value of c(A)) and a considerable attenuation of waves of all other lengths which decrease monotonically towards the edges of the spectrum. Notice the very long logarithmic scale of c(A) on this diagram. This means that only the very long waves, of the order of I = 100 km, and very short waves (gamma rays), of the order of I = 0.01 A, are attenuated in seawater no more than visible light waves are. Neither the very long nor the very short waves seem to have any practical importance in the sea, and only visible light of the remaining electromagnetic radiation travels any significant distance through seawater. This is easily checked by calculating, with the aid of equation (4.4. lo), the percentage radiance LJL0 which travels a distance of 1 metre in seawater, i.e., by working out the transparency of seawater to various wavelengths corresponding to different values of c taken from the graph in Fig. 4.4.4, When c = m-l, 99% of the radiant energy travels across a 1 m layer of the path r, but if c = lo2 m-l, which is the average for non-visible radiation, the 1 m section of path P is covered by a pracof the energy of such waves, tically undetectable fraction (of the order of so that seawater is opaque to them for the reasons discussed in the previous two sections. The c(i2) spectrum shown on such a large scale is of course only an outline
222
4 THE INTERACTION OF LIGHT WITH SEAWATER
of the shape of this dependence; in detail it displays many local irregularities, just like the absorption spectra mentioned earlier. Like the absorption coefficient, the light attenuation coefficient can be resolved into its constituents: attenuation due to the water alone c,, attenuation due to suspended particles c,, attenuation due to dissolved yellow substances c,, z a, etc. We can therefore write c = c,+cp+c,+c,+cd
w a,+b,+a,+ap+bp+as+cd
(4.4.1 1)
in which the subscripts denote the same as in (4.2.4). We assume that these components of attenuation are additive, so their percentage participation in the total attenuation of light in given waters can be studied (Dera et al., 1978). Such studies show, indeed, that the proportion of suspended matter, yellow substances and the water itself in light attenuation is so great that in ordinary ocean water the effect of sea-salt ions on visible light attenuation can be neglected. In more polluted seas, such as the Baltic, we can ignore the effect of molecular scattering on this attenuation. For example, in the Gulf of Gdansk, for the 525 nm waveband, the percentage attenuation due to suspended particles is on average 88% and there is a linear dependence between the attenuation coefficient due to suspended particles cp and the total attenuation coefficient c (Dera et al., 1978). In order to characterise the optical properties of seawater, e oftwen introduce the dimensionless parameter b
0 0
= --- =
c
b a+b’
I _
(4.4.12)
which is called the scattering-attenuation ratio or scattering albedo. Notice that according to this definition, 0 < too < 1. In an ideal, non-absorbing environment (a = 0), coo would be equal to 1, that is, the probability of a photon not being absorbed in such an environment would be 1 (i.e., 100%). Conversely, when the absorption a is very large, i.e., c >> b, coo < I (i.e., almost zero). Then the scattering-attenuation ratio is very small, that is, the probability of the photon being absorbed by the medium is high. The parameter m0 is called also the probability of photon survival. It is of course, a function of the wavelength A, just like the coefficientsa, b and c ; its values coo(;l) in ocean waters can be determined from the values of these coefficients for a given wavelength a(A) and b(il). We shall now return to the method of measuring the transmittance L,ILo, and hence of determining the light-attenuation coefficient in seawater, paying special attention to the errors in measurement which may arise when using the simplified transfer equation (4.4.2) without adequate fulfilment of the conditions
4.4 THE TRANSPARENCY OF SEAWATER TO LIGHT
223
permitting us to simplify (4.4.6). We shall start by drawing attention to the fact that accurate measurements of c(A), as of a(2) or b(A) and other optical properties of seawater, are made in situ, and at worst are done on fresh samples of water just as soon as they have been hauled on board ship. This is because we want to avoid disturbances of the natural properties of the tested water; some sedimentation of suspended particles, photoreactions, and microbiological changes occur anyway when water samples are removed from the sea, and even more so when they are stored and transported. For measuring the transmittance L,/Lo in the sea, we use a beam-transmittance meter (lowered into the sea from the ship), whose mode of action (according to (4.4.10)) is explained by Fig. 4.4.5. The
Interference filter
diaphragm
Field diaphragm
Fig. 4.4.5.Beam transittance meter used for measuring the light attenuation coefficient in situ.
principal parts of a beam transmittance meter are an artificial radiaticn source and a radiance meter, placed in the optical axis within some rigid structure, coupled to an electronic system for providing current and recording results, whose underwater and above-water parts are connected by a suitable cable. Measurements are made using this cable at the desired depths after having let the apparatus down into the sea. Depending on the extent of automation, one can (or cannot) remotely change the wavelength of the light used and thus produce a transmittance spectrum. This device must have absolutely parallel optical windows placed at right angles to the optical axis because the tested water between these windows forms a very thick “flat-parallel plate”. Any inclination of the walls of this “plate” towards the optical axis after immersion makes the test beam of light shift to one
4 THE INTERACTION OF LIGHT WITH SEAWATER
224
side so that part of it fails to pass through the diaphragm aperture. Thus the detected signal is weaker, not because it has been attenuated by its passage through seawater, but because the beam has shifted. The meter must be calibrated in very pure water, because in air, where over a distance of 1 metre we could assume L,/Lo % 1, the light reflection at the input optical window is different from that in water. The source function L,, in the transfer equation can generally be neglected as long as the light beam used is sufficiently powerful, because light from bioluminescent sources is many orders of magnitude weaker than daylight, for example (see Dera and Weglariska, 1980). The path function L, can also be neglected on condition that no scattered light enters the detector-and this includes light scattered forward through small angles from the examined light beam. In practice then the inequality Lo 9 L, must hold, and this is hard to achieve in the daytime at shallow depths in the sea without the aid of an optical shield to counteract the strong field of natural light in the water. We do without a shield, however, so as to ensure the free flow of water when immersing the meter. The radiance meter used must also have the smallest angle of observation 8, possible, as it always picks up some of the rays strongly scattered forward from the examined beam transmitted through the device. If the radiance meter of the beam transmittance meter “sees” rays scattered through angle Or, the attenuation coefficient, determined as already described, will be lower by approximately 8.
2x ,5(8,)sinO,dO,, for this part of the scattered light will not be rejected from 0
the recorded signal L,. So instead of the actual value of c, we get a depressed value or
c* = a-tb-2n
5 /3(Or)sin0,dOr.
(4.4.13)
0
From the value of the scattering function from particulate matter in the sea through a solid angle encompassing forward scattering angles 8, from 0 to 5”, some 25% of the scattered light energy is scattered, and so errors in measurements of c can be considerable. Reducing the angle of observation of the radiance meter in order to reduce this error is equivalent to reducing the ratio of the diameter of the aperture in the diaphragm P to the focal length of the lens focusing the beam onto the meter. In fact, an angle 8, c 1” is assured. Notice too, that the optical path length of r = 1 m used with this meter renders it useful only for average seawaters. Keeping in mind the law of exponential attenuation (4.4.8), L,/Lo in very clean ocean waters will be so close to unity
4.4 THE TRANSPARENCY OF SEAWATER TO LIGHT
225
that any differences between signals L, and Lo will fall within the bounds of experimental error. On the other hand, values of LJL, for extremely turbid waters will be so small that they also cannot be measured with precision. Assuming at the start the error which a measurement of the coefficient must not transgress (e.g., 1%), it is quite easy to work out from (4.4.10) the optimal optical path length in water, approximate c(A) values of which can be predicted. In very clean ocean waters, the optical path length of the beam transmittance meter should be c. 10 metres. In order to improve the accuracy of the measurements, we use differential (double-beam) systems with a broken (hence lengthened) light beam, while at the same time keeping the size of the apparatus reasonably compact (see Ivanoff, 1975, p. 111).
This Page Intentionally Left Blank
CHAPTER 5
SOLAR RADIATION INFLOW AND THE NATURAL IRRADIANCE IN THE SEA. THEAPPARENT OPTICAL PROPERTIES OF THE SEA
Radiant energy and its spectral composition are mainly dependent on the radiation of the Sun-the source of this energy-and on the conditions under which this radiation is transferred and attenuated in the Earth’s atmosphere. So before continuing our discussion of processes in the sea, we must explain, at least in brief, the most important characteristics of solar radiation and the principal optical properties of the Earth‘s atmosphere. The scientific literature dealing with these basic problems of nature is extensive and goes a long way back in time (e.g., Chandrasekhar, 1950; Kondratiev, 1954; Rozenberg, 1963; Zuev, 1966, 1970; Malkevich, 1973; Feygelson and Krasnokutska, 1978; and others). Some 99.9% of the Solar System’s mass (i.e., 1.99 x 1030 kg) is concentrated within the Sun itself and is heated to a very high temperature. The gigantic, hot mass of the Sun thus emits into space a huge quantity of energy as electromagnetic J s-l, besides expelling a considerable quantity waves at a rate of (3.86k0.3) x of elementary particles. The effective absolute temperature of the Sun’s surface T, is 5785 K, calculated on the basis of the Stefan-Boltzmann Law ( E = oT,4) from a measurement of the emissive power E of the Sun. The electromagneticradiation of the Sun is conventionally divided into gamma and X-rays (wavelength 1< 0.01 pm), light waves (0.01 pm < A < 500 pm) and radio waves (1> 500 pm) (Glagolev, 1970). Light waves from the Sun play a fundamental role in the Earth’s energy budget. More than 99% of solar radiation energy consists of wavelengths from 0.28 pm to 6.0 pm; less than 0.5% is shortwave radiation and less than 0.4% is long-wave radiation. The Solar Constant
The average annual surface density of the solar light flux incident on a surface normal to the rays at the upper boundary of the atmosphere is called the solar constant FsQ and is approximately 2 cal/cm2 min = 1395 W/m2. This value was established by Jonson in 1954, but has since been determined more accurately (Thekaekara and Drummond, 1971), the average value from these latter measurements being Fso = 1.940 cal/cm2 min = 1353 W/m2.
5 SOLAR RADIATION INFLOW AND THE NATURAL IRRADIANCE
228
Because the Earth's orbit around the Sun is elliptical, the instantaneous values of the solar radiation flux reaching the upper boundary of the atmosphere changes slightly during the year. The values rise gradually to a maximum of 3.4% above the solar constant during the perihelion (at the beginning of January, when the Earth is 1.471 x lo8 km from the Sun) and then decreases until they are 3.4% lower than the solar constant during the aphelion (at the beginning of July, when a distance of 1.521 x lo8 km separates the Earth from the Sun).
c
2
c
1.8
C In
:1.7
-m0
16 .
"
1.4
01
2
1.3
1.1
400
600 Wavelength
500
700
1 lnml __
Fig. 5.0.1. Spectral density spectrum of the solar constant Fs(A) = AFsa/AA averaged over intervals of A 1 = 25 nm (based on data from Glagolev, 1970).
The solar constant FsQ is an expression of the total spectrum of the radiant power reaching the Earth's atmosphere from the Sun (per square metre of perpendicular surface). As we know, this spectrum is strongly differentiated in different wavelength intervals. Therefore, we must also define the value of the solar constant per unit of wavelength A in the different wavelength interwals. Every such value for a wavelength il equal to F,(il) = aFsQ/aA can be called the spectral density of the solar constant, or in short, the spectral solar constant measured in W/m2 per nm of wavelength. The plot of the relationships of FJA) for 0 < 1 < o is the full solar constant spectrum, that is, the spectrum of solar radiation reaching
5 SOLAR RADIATION INFLOW AND THE NATURAL IRRADIANCE
229
the Earth’s atmosphere. Because of the wide range of the solar radiation spectrum, one often gives the radiant power per given longer wavelength interval from Izl to A2. Table 5.0.1. shows how much solar radiation flux is present in each of the spectral intervals of the solar constant. The sum of the values AFs(AA) given in the table is slightly higher than the solar constant FsQ given, although we should find that AFs(AA) = FsQ,or more precisely, rn
FSQ
=
j
Fs(A>dA.
(5.0.1)
0
These slight discrepancies result from necessary approximations used in the calculations, and also from inaccuracies and discrepancies in various measurements of Fs(A). We can determine the average spectral densities of the solar constant in the different wavelength intervals from Table 5.0.1, by dividing values TABLE 5.0.1 Light fluxes of various wavelengths in the solar constant spectrum (from Glagolev, 1970)
AF,(AA) [W.m-2]
A, t &
[nm]
[nm]
AF,(AA) [w.m-2]
0-225 225-250 250-275 275-300 300-325 325-350
0.41 1.40 4.20 11.17 19.10 28.32
525-550 550-575 575-600 600-650 650-700 700-750
49.15 47.91 47.44 86.49 78.78 71.02
1100-1200 1200-1300 1300-1400 1400-1500 1500-1600 1600-1700
52.92 42.29 34.06 27.68 22.65 18.70
350-375 375-400 4OCL425 425450 450-475 475-500 500-525
30.87 30.54 46.93 48.00 54.12 51.77 48.50
750-800 800-850 850-900 900-950 950-1000 1000-1050 1050-1100
63.56 56.65 50.36 44.72 39.71 35.07 31.63
1700-1800 1800-1900 1900-2000 2000-2500 2500-3000 3000-3500 3500-4000
15.55 13.02 10.98 35.07 17.45 9.62 5.68
A, + A2
11
+A2
[nm]
AFB(AA) [W-m-2]
A1 t 1 2
[nm] 40004500 4500-5000 5000-6000 6000-7000 ~OOO-CO 0-co
AFs(AA) [W.m-’] 3.72 2.28 2.79 1.47 2.65 1396.4
of AF,(AA) by the width of the interval AA. The spectrum of solar constant spectral densities plotted from more detailed data is shown in Fig. 5.0.1. The values in [W/m2 nm] given in this figure denote practically the maximum spectral densities of the light flux of various wavelengths which would reach the surface of the sea through an ideally transparent atmosphere.
-
5 SOLAR RADIATION INFLOW AND THE NATURAL IRRADIANCE
230
Once the solar radiation has entered the real atmosphere it is attenuated as a result of complex interactions with the components of atmospheric air, i.e., a mixture of atmospheric gases, water vapour and all kinds of chemical admixtures, not to mention water droplets, dusts and smokes which come under the collective heading of aerosols. During such interactions, part of this solar energy is absorbed by the constituents of the air and converted into heat (molecular motion), chemical energy (photochemical reactions), or is consumed in ionizing atoms. Another part of this energy is back-scattered, and so is reflected and escapes irreversibly back into space. As in the sea (Section 4.3), we can distinguish between molecular scattering and scattering at suspended particles (aerosols, water droplets in fog and clouds) in the atmosphere. These processes directly retard the flow of solar radiation through the atmosphere, part of it being diffusionally back-scattered At the same time, scattering indirectly increases light absorption in the atmosphere because the path the photons take is longer and more intricate, so the probability of their colliding with and being absorbed by absorbent molecules is enhanced. The processes by which solar radiation is absorbed in the atmosphere before it reaches the sea are no less complex than in the sea (Zuev, 1966; 1970; Paltridge and Platt, 1976; Feigelson and Krasnokutska, 1978). Their physical mechanism and mathematical description are analogous to those of seawater (see Chapter 4), while the differences are due mainly to the different chemical composition and the absorption-scattering properties of the atmosphere. The medium here consists of the rarified atmospheric gases (given in Table 5.0.2) and, in the real atmosphere, TABLE 5.0.2 Average percentage volume concentrations of atmospheric gases in dry tropospheric air (from Gudi, 1966; Zuev, 1970) Kind of gas Nz 0 2
Ar COZ Ne He CH*
% concentration 78.084 20.946 9.34 x10-I 3.3 x10-2 1.818 x 5.24 ~ 1 0 - 4 1.6 x 10-4
Kind of gas
Kr
% concentration
co
1 . 1 4 ~10-4 5~ 10-5 3 . 5 ~10-5 7x
O3 NO,+NO
0-2x
H Z
N20
considerable quantities of water vapour and aerosols. Locally, other light-absorbing substances may occur. The distribution of these atmospheric mixtures, and hence their interactions, is very uneven. Their percentage composition up to an
5 SOLAR RADIATION INFLOW AND THE NATURAL IRRADIANCE
23 1
altitude of 80 km is roughly the same, except for the concentrations of water vapour, ozone and aerosols. Of the total mass of the atmosphere (c. 5 . 1 5 ~ x loz1 g), some 90% is in the 16-km-thick layer adjacent to the Earth‘s surface, whereas only about 0.0001% of this mass is present above 100 km. The average g cm-3 at sea level, to c. density of atmospheric air changes from 1.23 x 1.66 x g at an altitude of 16 km, and c. 5 x g ~ m - 100 ~ , km above sea level. Since the concentrations of most of these atmospheric constituents are small and the coefficients of absorption of some of them are very small, three components, namely, water vapour HzO, carbon dioxide CO,, and ozone 0, absorb the largest quantities of visible and IR light energy arriving from the Sun. All three have complex electron-oscillational-rotational absorption spectra (Zuev, 1966). The chief constituent of air-nitrogen N,-has an absorption band in the far UV, like oxygen 0,and the many other components playing an important part in the photochemical processes of the upper layers of the atmosphere. The very short-wave radiation, the far UV, is strongly absorbed by atmospheric gases already at altitudes exceeding 80 km above sea level, thereby both exciting and ionizing atoms-hence the name “ionosphere”. From 20 to 80km, UV is primar0, -+ ily absorbed by oxygen molecules 0, which dissociate as a result
-
P.-&~w.+
-+ 202-. The close proximity of oxygen molecules and 0,- ions facilitates the formation of ozone 0 3 ,which is an excellent absorber of solar radiation, especially of short wavelengths. This is strong band absorption (the Hartley band), as a result of which radiation shorter than 2900 A practically does not reach the surface of the sea. Absorption due to ozone is of less importance in the visible and IR wavebands; water vapour and carbon dioxide molecules now come into their own. Ozone is concentrated mainly at altitudes from 10 to 40 km (Green, 1964), water vapour is found largely in the first 10 km above sea level, while aerosols lie even closer to the Earth’s surface. At high altitudes, the percentage composition of the atmosphere undergoes distinct changes owing to the separation by diffusion of the atmospheric gases, and also because of photochemical reactions (Levy, 1972; Sklarenko, 1978), and molecular dissociation and ionization caused by the powerful selective NO 0, action of solar and cosmic radiation (e.g., CH4 ?+ CH3 + H, NO, and others). These changes are reflected by the fall in the average molecular mass of air with increasing altitude, from 28.97 at sea level to approximately 28.8 at 100 km, after which it decreases rapidly: 23.7 at 300 km, 15.7 at 600 km and c. 4 at over 1500 km (see Standard Models of the Atmosphere CIRA 1961 and USSR
+
5 SOLAR RADIATION INFLOW A N D THE NATURAL IRRADIANa
232
1964 in Glagolev, 1970). All these phenomena significantly affect the spectral and directional distributions of the radiation arriving at the sea surface. We shall briefly discuss the effects of such influences in Section 5.1.
5.1 THE INFLUX OF SOLAR RADIATION TO THE SEA SURFACE
When the Sun is not hidden behind clouds, the sea surface is illuminated by direct sunrays attenuated somewhat in the atmosphere. As sunlight is partially scattered in the atmosphere, a substantial proportion of scattered light diffuses to the sea surface from all parts of the sky. A radiation field thus comes into being which is made up of two parts: direct sunlight, and scattered light whose presence is manifested by the lighted sky. This field is relatively uncomplicated when the sky is clear, but becomes more complex when clouds appear. An exact description of such a light field (Chandrasekhar, 1950; Rozenberg, 1963) is an extremely involved affair, owing to the complexity and variability of the atmosphere as an optical medium (Zuev, 1966,1970; Feigelsonand Krasnokutska, 1978). So in order to describe it we apply various approximations using simplified versions of the transfer equation (4.4.4), and also a number of semi-empirical methods based on measurements of light transmittance through the real atmosphere under varying cloud cover. It is fairly easy to calculate from the solar constant the approximate energy of the flux of direct solar rays reaching the sea surface out of a cloudless sky. To do so, we use the simplest form of the transfer equation (4.4.2), ignoring the source function (e.g., IR emission from clouds) and the path function, and applying the equation with certain values of the absorption and scattering coefficients which give us the total attenuation coefficient c established for such an atmosphere. We can assume that on a cloudless day, the attenuation of sunlight will be due mainly to molecular scattering (here well described by the Rayleigh theory) (bR),aerosol scattering (b,) and absorption (a) by the gases and all the other constituents of the atmosphere. The attenuation coefficient will then comprise the following sum c =
b,+b,+a.
(5. I. 1)
All these coefficients describing the inherent optical properties of the atmosphere (like those of the sea-see Chapter 4) are, of course, functions of the wavelength A, but are also highly variable functions of the space and time coordinates. In view of this variability, defining them as functions of altitude in the atmosphere
5.1 THE INFLUX OF SOLAR RADIATION TO THE SEA SURFACE
233
can refer only to the state obtaining at a given position and time or to certain simplified or idealised states of the real atmosphere. The most probable values of the scattering coefficients, e.g., for light of wavelength il = 514 nm at two very different air turbidities (horizontal sea-level visibilities) are given in Table 5.1.1. TABLE 5.1.1 Coefficients of Rayleigh scattering bR and aerosoI scatering 6. in different Iayers of the atmosphere for light of wavelength il = 514 nm and different air turbidities defined by the horizontal visbility S,,, at sea level (after Feigelson and Krasnokutska, 1978, from McClatchey) 6. [km-I]
h
bR
[kml
[km-ll
S,,, = 25 km
S,,, = 5 km
0 0-1 1-2 2-3 3-4 4-5 5-6 6-7 7-8 8-9 9-10 10-1 1 11-12 12-13 13-14 14-15
0.015 0.0143 0.0129 0.0116 0.0105 0.00951 0.00856 0.00768 0.00689 0.00616 0.0055 0.0049 0.00435 0.00385 0.00334 0.00285
0.168 0.112 0.0486 0.0207 0.00976 0.00616 0.00449 0.00364 0.00356 0.00354 0.00342 0.00327 0.00324 0.00319 0.00304 0.00291
0.82 0.496 0.181 0.0663 0.0242 0.00884 0.00443 0.00364 0.00356 0.00354 0.00342 0.00327 0.00324 0.00319 0.00304 0.00291
The first thing to notice about this table is that these coefficients are whole orders of magnitude lower than the values given earlier for seawater. This is why they are expressed in [km-l] rather than in [m-l]. In real situations, the values of the scattering and absorption coefficients in the atmosphere as functions of altitude depend on the current state of the atmosphere. As we see, aerosol scattering is more intense than molecular scattering only up to an altitude of about 4 km. Above 6 km, aerosol scattering is practically constant, regardless of changes in visibility (turbidity) near the lower boundary of the atmosphere. It should be noted that in a clear, dry atmosphere the total absorption coefficient a of visible light is extremely small in comparison with the scattering coefficient and can usually be omitted from calculations (Malkevich, 1973 ; Gelberg, 1970).
5 SOLAR RADIATION INFLOW A N D THE NATURAL IRRADIANCE
234
The Optical Thickness of the Atmosphere
Instead of using the scattering and absorption coefficients in the different layers of the atmosphere in our simplified calculations of the light influx to the sea surface based on the solar constant, it is enough to apply the optical thickness of the atmosphere z along the path r of the transmitted rays, which takes into consideration the total attenuation of these rays in the atmosphere r=O
(5.1.2)
in which we take rm to mean the upper (conventional) boundary of the atmosphere, and r = 0 its lower boundary, i.e., the sea surface. The least thickness, i.e., the Zenilh Sun
/
m
Sun
Fig. 5.1.1. The concept of the optical thickness of the atmosphere (a) and the single scattering model (b). iv---direction angle, i.e., the deviation of a given direction from the zenith: 8,-the direction angle of direct sunlight, i s . the zenith distance of the Sun. Note: in the atmosphere we shall denote the direction angles 4, (p and the Sun’s direction ivS, pS by small letters to distinguish them from 0, @ in the sea. Ls-the radiance of direct sunlight; LD-the radiance of once-scattered light incident at angle S to the sea surface; r-the path ofalight ray through the atmosphere; h-altitude; t-optical thickness ofthe atmosphere along path r (in direction 19); rh-the zenith optical thickness of the atmosphere; CI, c l , ...,cWlight attenuation coefficients in different strata in the atmosphere.
5.1 THE INFLUX OF SOLAR RADIATION TO THE SEA SURFACE
235
zenith optical thickness of the atmosphere, is measured from its upper boundary at the zenith h , to the sea surface h = 0, h=O
(5.1.3)
Its values can be determined for a given set of atmospheric conditions, e.g., in a standard atmosphere, or in a real one of low, medium or high turbidity (Glagolev, 1970). If we ignore the curvature of the atmosphere's boundary, and a certain small refraction of sunlight within its layers, and if we make the approximation c(x, y , z ) = c(z), i.e., assume a horizontally stratified atmosphere, its optical thickness for rays incident at any angle 6 can be evaluated with the aid of the zenith optical thickness as t=
z,sec6.
(5.1.4)
This is explained in Fig. 5.1.1. If we do allow for the curvature of the atmosphere and the refraction of the rays therein, the angle 6 changes with increasing altitude (the ray path r becomes a curve) and in the equation (5.1.4) for the optical thickness of the atmosphere z we have to replace sec6 with its integrated average value r=O
__
m = sec6 =
5
rm
c(r)sec@(r)dr (5.1.5)
r=O
f
m d r
' a 3
So, because of (5.1.4), z = mzh.
The value of m depends on the mass of atmospheric constituents along the ray path and is called the optical mass of the atmosphere. It can be calculated from Rozenberg's (1963) approximately empirical formula
m = (cos6+0.025e-11'oso)-1. Obviously, for angles 6 < 75", m N" see#, and €or 6 > 75", m < sec6. Like the attenuation coefficient c, we can resolve the optical thickness of the atmosphere z into components representing molecular scattering, aerosol scattering, and absorption. So for a clear sky, in view of (5.1.1), we can write -0
z=
1 b,(r)dr+
rrn
r=O
r=O
b,(r)dr+ rW
1 a(r)dr,
roo
(5.1.6)
236
5 SOLAR RADIATION INFLOW AND THE NATURAL IRRADIANCE
which in brief is r = z b R + z b , + zn. These last values, reduced to the zenith thickness according to (5.1.6), can be worked out more realistically from experiment or theory by using a standard, model or other atmospheric composition, and taking into account the pressure, humidity and other physical properties of the atmosphere. A prominent place is given to the spectral values z(A) for a given wavelength, and also (e.g. in the energy budget) the averaged value tQfor the entire solar energy spectrum. The average spectral values of the zenith optical thickness of the atmosphere th(jl), calculated by Allen (1955) for a clear sky are given in Table 5.1.2. The third line of this same table gives values of the zenith optical thickness due only to Rayleigh scattering ~~,~(il), selected for a few wavelenghts ilfrom Pendorf's (1957) data; we can assume that Rayleigh scattering plays the major role in attenuating light from a cloudless sky, i.e., zh z thR. TABLE 5.1.2 Average spectral values of the zenith optical thickness of the atmosphere sky, and values of t h , R ( A ) due only to molecular scattering
.,,(A) for a clear
1 [tLml
0.34
0.36
0.38
0.40
0.45
0.50
0.55
rh(A)
0.84
0.67
0.55
0.46 0.3630
0.31 0.2231
0.23 0.1447
0.19 0.09805
TiI,R(A)
A~ m l
0.60
0.65
0.70
0.80
0.90
1.o
th(n)
0.17
0.127
0.093
0.063
0.049
0.040
Zh,R(4
0.06880
0.004971
0.03682
The Single Scattering Model If we know the optical thickness of the atmosphere tQor z(il), we can readily calculate from the solar constant Fsa or the spectral density of the solar constant Fs(A) = aFsQ/ail, respectively, the total or spectral values of the direct sunlight energy arriving at the sea surface out of a clear sky. Using a modification of the transfer equation (4.4.2) in its simplest form affords the radiance (51.7)
5.1 THE INFLUX OF SOLAR RADIATION TO THE SEA SURFACE
237
where L"(t,6,,ps)is the radiance of a beam of direct sunrays (directed sunrayssubscript s), propagated in direction a,, rps at the solid angle AQ, z 6.8 x sr at which the Sun's disk is seen from the Earth. The solution of this simple differential equation, the exponential version of which is well known, gives the magnitude of the radiance of the attenuated direct sunlight reaching the sea surface
(5.1.8) From this, according to the definition of downward radiance (4.1.15), the irradiance of the sea surface by direct sunrays incident at an angle 8, is
E ; = L S ( z ,gS,
?,)a
AQ,c0s8~= Fscos6,e-m'h.
(5.1.9)
Now we must also take into consideration the ever-present additional irradiance of the sea surface by radiation scattered in the atmosphere EY (diffuse lightsuperscript 0).In a cloudless sky, the percentage of incident light due to diffuse irradiance is quite large when the Sun is low over the horizon (see Fig. 5.1.3). As the Sun approaches the horizon, the path length r of direct sunrays reaching the point of observation is much extended and so the light is increasingly attenuated. At the same time, light travelling higher in the atmosphere is scattered, contributing a comparably large amount of diffuse light arriving at the observation point from all directions. As the Sun approaches the horizon, the spectral proportions of the radiance from various directions change markedly. The sizeable contribution made by strongly selective Rayleigh scattering to the attenuation of direct sunrays means that relatively more long wave radiation reaches the sea-hence the red coloration of the rising or setting Sun. Again, because of Rayleigh scattering in the atmosphere, the light arriving from the sky comprises mostly shortwave radiation. That is why the sky is blue. Of course, if dust, fog or clouds are present in the air, these phenomena are further complicated. We see then, that the inclusion of diffuse light (scattered in the atmosphere)-a significant addition to the total sunlight flux reaching the sea-is essential, even when the sky is clear. This diffuseirradiance is harder to evaluate than the direct sunlight flux calculated from equations (5.1.7-5.1.9). The transfer of diffuse light from a given direction (6, p) towards the sea surface is also described by the transfer equation (4.4.6) or its modified and simplifien, versions. In describing this transfer across a cloudless atmosphere along path r7 we can ignore the source function L, but not the path function L, . This path is cut along its entire length by direct (and also multiply-scattered) rays of sunlight, which are scattered inter alia in the observed direction of transfer and are the main
238
5 SOLAR RADIATION INFLOW A N D THE NATURAL IRRADIANCE
source of radiance in this direction. The radiance of light scattered in direction (8, y ) is transmitted from sections of the path r farther away from the sea and amplified by light scattered on the remaining section of this path, which in this equation is expressed by the path function L, . Ignoring other possible sources. of radiation in the atmosphere and making use of the optical thickness of the atmosphere z (see formula (5.1.2)), we can describe the radiance of diffuse light L D ( t 6, , y ) from direction (6, y ) by means of the transfer equation dLD(t9 6 7 y, = -LD(z, 6 , y ) + L * ( z , 6 , p), dz
(5.1.10)
where L* = IlcL, is the path radiance equal to the path function L , divided by the attenuation coefficient c (see equation (4.4.4)). This path function emerges from the multiple scattering of light in the atmosphere. Because of the general form of the path function (equation (4.4.5a)), equation (5.1.10) is an involved integral-differential equation a solution of which in an analytical form is not known. Simplified versions of it are therefore being sought. One of the more important of these describes the effect of a single scattering of sunlight in a given direction (6,y ) in the atmosphere and omits the effects of secondary and further degrees of scattering of light already scattered on the assumption that they are negligibly small. In order to apply such a single scattering model, we must first divide the path radiance L* in (5.1.10) into one part due to single scattering only and another part representing the contribution made by further scattering. This can be written as follows:
277 x
+ 4x 1 s p ( t , 6,p', 6, y)L"(t, 6',p1)sin6'd8'dy'.
(5.1.11)
0 0
The first term on the right side of this equation describes the single scattering of sunlight from direction a,, y s to the observed direction (6, y), whereas the second term describes the effect of further degrees of scattering from all directions (d', p') to the same direction (6,p). The scattering function p ( z , 6,,ps,6,q) is the scattering phase function now introduced (5.1.12)
5.1 THE INFLUX OF SOLAR RADIATION TO THE SEA SURFACE
239
TABLE 5.1.3. Spectral distributions of the sea surface irradiance from a clear sky, calculated from the single scattering model for various solar altitudes a and air turbidities. Air turbidity is indicated by the zenith optical thickness % , S O 0 for light of wavelength il = 500 nm A E l ( l , s l , ) [low3cal min-' ~ m - ~ ]
a, -+A~ tnml
dch
~-
lo"
30"
50"
60"
460-480 480-500 500-520 520-540 540-560 560-580 580-600 600-620 620-640 640-660 660-680 680-700
1.14 1.28 1.71 1.90 1.98 2.03 2.21 2.34 2.42 2.54 2.52 2.52 2.54 2.55 2.50
3.53 3.96 5.25 5.81 5.99 6.05 6.42 6.65 6.71 6.84 6.60 6.49 6.37 6.32 6.06
18.5 19.2 23.9 25.0 24.5 23.8 24.4 24.5 24.3 24.4 23.1 22.4 21.8 21.2 20.3
33.4 33.9 41.3 42.4 41.0 39.3 40.0 40.1 39.5 39.4 37.3 35.9 34.8 33.9 32.3
38.8 39.1 47.4 48.7 46.9 44.9 45.6 45.7 45.0 44.8 42.4 40.8 39.5 38.5 36.5
400-700
32.18
89.05
341.3
564.4
644.6
400-420 420-440
600-620 620-640 640-660 660-680 680-700
0.89 0.98 1.27 1.41 1.45 1.47 1.59 1.68 1.74 1.83 1.81 1.83 1.83 1.84 1.80
2.54 2.94 4.00 4.50 4.70 4.82 5.17 5.41 5.50 5.66 5.48 5.41 5.33 5.31 5.12
16.0 16.7 20.9 22.1 21.8 21.3 22.0 22.3 22.2 22.4 21.4 20.8 20.3 19.9 19.0
30.3 30.8 37.7 39.1 38.1 36.9 37.9 38.3 37.9 38.0 36.1 34.8 33.7 32.9 31.2
35.7 36.0 43.8 45.3 44.0 42.5 43.5 43.9 43.4 43.4 41.2 39.7 38.5 37.5 35.6
400-700
23.42
71.89
309.1
533.7
614.0
400-420 420-440
440-460
440-460 460-480 480-500 500-520 520-540 540-560 560-580 580-600
5"
= 90O-8,
~~
t Th,500
Remarks Low air turbidity
0.2
Medium air turbidity
0.3
5 SOLAR RADIATION INFLOW AND THE NATURAL IRRADIANCE
240
TABLE 5.1.3 (cont.)
al +a2
ah
= 90"-8,
T
bml
5"
lo"
30"
50"
60"
40M20 420-440 44M60 460-480 480-500 500-520 520-540 540-560 560-580 580-600 600-620 620-640 640-660 660-680 680-700
0.53 0.60 0.80 0.89 0.92 0.94 1.01 1.07 1.11 1.16 1.14 1.14 1.14 1.14 1.12
1.45 1.65 2.22 2.52 2.66 2.77 3.05 3.29 3.47 3.69 3.69 3.73 3.76 3.80 3.71
11.6 12.5 16.1 17.3 17.4 17.3 18.1 18.6 18.7 19.0 18.3 17.9 17.5 17.2 16.5
24.2 24.6 30.4 31.9 31.6 31.1 32.6 33.4 33.5 33.9 32.3 31.4 30.6 30.0 28.7
29.5 29.8 36.5 38.1 37.4 36.7 38.2 39.0 39.0 39.3 37.5 36.3 35.3 34.6 33.0
400-700
14.71
45.46
254.0
460.2
540.2
th,500
Remarks
0.5
High air turbidity
which is the scattering function p(0) normalized to 4.n, such that 2%n
1p ( z , 8,,y s , 8, y)sin@d@dq,
=
4n
1.
(5.1.13)
0 0
This normalized scattering function p is known as the scattering phase function. The parameter uo is the scattering-attenuation ratio introduced in the previous chapter (i.e., uo(A)= b(A)/c(A)).We are thereby assuming that the ratio of the scattering and attenuation functions b(il)/c(A) in the atmosphere is constant, which is clearly just another approximation simplifying the description. We can now insert the expression for the path radiance (5.1.1 1) in the transfer equation (5.1.10), and reject the second term in (5.1.11). Because of the relatively small optical thickness of the atmosphere, the single scattering model yields LD radiance values very similar to those actually measured. Equation (5.1.8) is substituted for Ls in the path radiance (5.1.11) simplified to the single scattering term. This in turn is substituted in the transfer equation (5.1.10) which becomes (5.1.14)
5.1 THE INFLUX OF SOLAR RADIATION TO THE SEA SURFACE
241
where Fsis the spectral solar constant (see Fig. 5.0.1), m = m(6)is the optical mass of the atmosphere, and zhis the zenith optical thickness of the atmosphere (see equation (5.1.3)). Equation (5.1.14) is now an ordinary differential equation the solution of which is
Now the downward irradiance of the sea surface due to this hgnt scattered from the whole sky is obtained by integrating the scattered light radiance L D ( z ,8, 9) over all directions of the upper hemisphere, i.e., 2x nlZ
EY
5 1 L D ( z ,8,p)cos6sin6d6dply
=
0
(5.1.16 )
0
where LD is given by (5.1.15) which assumes a single scattering. The total downward irradiance of the sea surface El is the sum of the irradiances due to direct sunrays E: (5.1.9) and to diffuse light EY (5.1.16), so
El
=
E: +Ef.
(5.1.17)
Typical spectral distributions of the sea surface irradiance are illustrated in Fig. 5.1.2a and b: they show the absorption bands of water vapour and other atmospheric gases. On the plots, the total radiance flux irradiating the sea surface over the whole spectral range is given by the area under the curve and is equal to m
EQl
=
1 EJWA.
(5.1.18)
0
More exact values of sea surface irradiance are given in Table 5.1.3 for a clearsky and for various degrees of air turbidity. These values were calculated for different solar altitudes and visible light wavebands according to the single scattering model (see equations (5.1.8), (5.1.1 5) and (5.1.17)). A certain most probable shape of the scattering phase function (5.1.12) expanded into a series of Legendre polynomials p(6,) = 1 1 .4P1cos(8,) 1.3P2cos(8,)+0.5P3cos(8,) had to be included in these computations, where PI , P, , ... are Legendre polynomials of successive degrees, and 8, is the scattering angle (Wensierski, 1980). A percentage indicator of the portion of the irradiance due to scattered (diffuse) light in the total irradiance has been found advantageous in the description of sea-surface irradiance conditions, i.e.,
+
+
(5.1.19)
5 SOLAR RADIATION INFLOW A N D THE NATURAL IRRADIANCE
242
‘g
r
s
2.5 2.0
“i t‘ 1.5 C 0
$
-e
1.0 0.5
n 0
0.6
0.8
1.2
2.0
1.6
2.8
2.1
I
Wavelength.
I
3.2
1 Iyml
I
0.2
0.4
0.6
0.8
1.2
12 .
1.4
1.6
1.8
Wavelength
2.0
2.2
1 [urn)
Fig. 5.1.2. Spectral density distributions of the downward irradiance at the sea surface (a) of light ,?&(A) from direct Sun rays when the Sun is at the zenith (curve 3) compared with the spectral density distribution of the solar constant FdA) (curve 2) and the spectral density distribution of black-body radiation at 6000 K (curve 1). Hatched areas illustrate absorption of light by atmospheric gases (from Glagolev, 1970); (b) of daylight, EJA) at different aur turbidities (from Kondratev, 1954).
5.1 THE INFLUX OF SOLAR RADIATION TO THE SEA SURFACE
243
It is called the dzjiiseness of irradiance. The spectrum of the irradiance diffuseness dE(17)at different solar elevations OI,, = 90"- 6,, and the dependence of dE on the Sun's altitude is shown in Fig. 5.1.3. The shapes of the plots in this figure illustrate what was said earlier about the changes in the solar spectrum reaching the sea from various directions as the Sun approaches the horizon.
Lil
B
g
1.0
0.8
.-m 0.6
;
e
0.L
m
f 0.2 5 o n 433
500
600 Wavelength
700
I lnml
Fig. 5.1.3. (a) Spectral distributions of the irradiance diffuseness dE at the sea surface under a cloudless sky for two positions of the Sun's altitude ah = 10" and 50°, and (b) the dependence of dEon the Sun's altitude for two wavelengths of light i, = 500 nm and 580 nm (from Wensierski, 1980) (a) the continuous lines were plotted from empirical data taken from Sauherer and Ruttner (1941) for A = 535 nm, the dashed lines were calculated from the single scattering model at low air turbidity (horizontal visibility Smax = 15 km);(b) the circles denote empiricaldata for A = 540 nm collected at Sopot on the Gulf of Gdahsk; the dashed lines were calculated from the single scattering model for the two wavelengths given. Air turbidities: 1-low (S,,, = 5 km). = 15 km),medium (Smax = 10 km) and high
The Transmittance of a Real Atmosphere Notice that we derived the relatively simple method of calculating the sea surface irradiance (given above) under the reasonable assumption that the sky was clear. The problem becomes far more complex when there are clouds in the sky (see e.g., Feigelson and Krasnokutska, 1978)-the diffuseness of irradiance
244
5 SOLAR RADIATION INFLOW AND THE NATURAL IRRADIANCE
then increases fast and is equal to 1 when the sky is completely overcast. Yet more complications arise out of the non-stationarity of the conditions, which is a perfectly natural state of affairs during the day. The cloud cover, air humidity and turbidity are universally complex, time-variable states, so to calculate the influx of solar energy into the sea, we often use semi-empirical techniques drawing from statistical data of this energy measured at actinometric stations on shore and elsewhere (Czyszek et al., 1979; KreZel, 1980). A new quantity is introduced in such calculations, namely, the transmittance of the atmosphere T,, defined as the ratio of the downward irradiance (onto a horizontal surface) at sea level to that at the upper boundary of the atmosphere. In view of earlier definitions of the total sea-surface irradiance E,, ((5.1 .IS), (5.1.17)) and the solar constant FsQ,the transmittance of the atmosphere over the entire solar spectral range can be given by (5.1.20) from which it is plain that it is also a function of solar elevation and the radiance distribution of diffuse light in the atmosphere. Analogously, we can introduce the spectral density of the atmospheric transmittance Ta(A) by replacing E,, with E,(A) and FsQwith FJd) in equation (5.1.20). The instantaneous values of the atmospheric transmittance can vary considerably-in theory from 0 to 1, but in practice from 0.22 to 0.99-and they are dependent mainly on the Sun’s position and the cloud cover. A layer of cloud (or fog) intercepting sunlight attenuates the solar energy incomparably more than all the other constituents of the atmosphere under normal conditions. Haurwitz’s empirical formulae (1948) are helpful in estimating the flow of light energy through layers of the typical cloud types. They describe the transmittance of downward irradiance through a layer of clouds over the whole solar spectrum as a function of the Sun’s deviation from the zenith @s and of the atmospheric pressure pa given here in hectopascals (millibars). If we denote this transmittance by
EQl(below the clouds) Tcl*Q = EQ, (above the clouds)
’
(5.1.21)
according to Haurwitz, it will take the following values for various cloud types covering the Sun:
5.1 THE INFLUX OF SOLAR RADIATION TO THE SEA SORFACE Cloud type Fog Stratus Stratocumulus Cumulus
Altostratus Cirrus
245
Td,Q 0.1626+0,0054S 0.2684- 0.0101S 0.3658-0.01496 0.3656-0.01496 0.4130-0.00146 0.871 7- 0.01796
where 6 = p,/l013cos6,. We can often assume here that since light is so strongly attenuated by clouds, attenuation in other parts of the atmosphere both above and below the clouds is negligibly small, and when the sky is completely overcast, Tcl,Q is practically the same as the irradiance transmittance through the whole thickness of the atmosphere, i.e., Tcl,Q z Ta,a. When the Sun is visible, though not too high in the sky (ah c 507, and the cloud cover is scanty (1-3 on a 10-point scale), it can happen that the instantaneous transmittance of the atmosphere is paradoxically high, close to or even greater than 3 . This is due to the diffusional reflection of some sunlight towards the observation point from suitably positioned single clouds. Studies have also shown that light attenuation in clouds is only slightly dependent on the wavelength A. The theoretically calculated light attenuation coefficients in clouds ccl(A) for the most probable cloud microstructures (typical droplet dimensions and concentrations) are given in Zuev's monograph (1970). This shows that for a numerical concentration of droplets of 28 ~ m - c,,(310 ~ , nm) = 19.5 km-l, cc1(710nm) = 19.7 km-l and cc,(1150 nm) = 19.9 km-l, while values for intermediate wavelengths are intermediate between those given, so that across the whole spectrum, from the violet to the near infra-red, they differ only by tenth parts of km-'. It is for this reason that the relative spectral distribution of the sea surface irradiance from an overcast sky is not much different from the value recorded on a cloudless day (see Fig. 5.1.2). The grey colour of clouds is a further indication of this. When the sky is overcast, however, the absolute value of the total sea-surface irradiance EQ.L changes considerably according to (5.1.21). The sea-surface irradiance in clear, dry air is often taken to be a standard (i.e., the maximum possible), so instead of the absolute transmittance of the atmosphere we use the ratio of actual to standard sea surface irradiance. This transmittance is equal to the ratio of the actual atmospheric transmittance to the standard atmospheric transmittance, and can be called the relative transmittance of the atmosphere. Values of this for various solar altitudes and cloud covers, and other
5 SOLAR RADIATION INFLOW AND THE NATURAL IRRADIANCE
246
quantities associated with the flow of light into the sea can be found in oceanographical tables (see Oceanographical Tables, 1975, Chapter 5) and elsewhere. Usually, however, we use certain values averaged or summed over a period of time in view of the 24-hour cycle and the well-known fact that there are considerable fluctuations in irradiance during the day e.g., since the arrangement of clouds in the sky changes rapidly, the instantaneous values of the sea-surface irradiance can change by a factor of 2 + 3 or even 5 in a matter of hours or minutes. This is why we introduce the average irradiance per given time interval (e.g., per hour) and the sum of radiant energy incident on a unit area per given time: f Z
(5.1.22) where in the daily sum tl and t, denote the appropriate times of sunrise and sunset at the point of observation, but in the monthly sum, t2 - tl is the total duration of daylight in a given month. Such real daily (or monthly) sums of sea-surface irradiances for a particular day (or month) in a given place (e.g., during studies of marine photosynthesis at some location) are determined experimentally by making diurnal measurements of the irradiance and summing its energy. These can be called daily (or monthly) totals of solar energy. These are actinometric measurements which are routinely made at many stations, both on shore and on islands. From these measurements, the results of which are dependent on all real states of the atmosphere, we calculate the real many-years average of the daily Qd or monthly Qm sum of solar radiation energy reaching the sea at a particular observation point. After substituting in equation (5.1.22) the irradiance at the upper boundary of the atmosphere E l c ( t ) = FSQcos[8,(t)],we can integrate this irradiance over the requisite period of time so as to calculate the daily Q! or monthly Qg sum of energy at the upper boundary of the atmosphere. Just as for the instantaneous values (5.1.20), the ratio _.
- (Ta , a ) m
Qm/Qm F -
(5.1.23)
denotes the (absolute) average monthly atmospheric transmittances (or the average diurnal values-subscript d ) which, to a good approximation can be treated as statistical data for an area surrounding the observation station. Such detailed calculations of actual average sums of solar radiant energy and average atmospheric transmittances (e.g., for the Baltic) can be found in Czyszek ef al. (1979) and Kreiel (1980). Actual average monthly sums of radiant energy reaching the southern Baltic calculated from actinometric data collected by Polish shore stations over many years, are given in Table 5.1.4. This shows that there are
5.1 THE INFLUX OF SOLAR RADIATION TO THE SEA SURFACE
247
TABLE 5.1.4 Many-years average monthly sum of sea surface irradiance energy Qm in the southern Baltic at latitudes q j = 54", 57" and 60" N, in [cal/cm2], from Czysezk et al. (1979) Month
I1
In
IV
V
VI
1422 1027 667
2591 2168 1759
6306 5720 5135
9749 9314 8849
13214 12936 12658
15761 15612 15485
VII
VIII
IX
X
XI
XI1
14591 14406 14222
11903 11518 11113
8016 7453 6891
4410 3839 3286
1610 1267 913
1057 711 408
Latitude 54" N 57" N 60" N
Month Latitude 54' N 57" N 60" N
great differences in the monthly sums of surface solar radiation energy in the southern Baltic during the year. To some extent they are due to changes in the average states of the atmosphere, mainly in cloud cover, but chiefly responsible for them are the changes in solar altitude and the number of hours of daylight TABLE 5.1.5 Actual average atmospheric transmittances ( ~ o Q ) m of monthly sums of solar radiation energy over many years in the southern Baltic (from Czyszek et al., 1979)
I
I1
I11
Iv
V
VI
(T.Q)rn
0.31
0.32
0.43
0.44
0.44
0.55
Standard deviation from the mean
0.07
0.03
0.07
0.07
0.06
0.05
Month
VII
VIII
IX
X
XI
XI1
(T,,),
0.45
0.45
0.42
0.37
0.27
0.28
Standard deviation from the mean
0.05
0.06
0.04
0.04
0.04
0.07
Month Transmittance
Transmittance
248
5 SOLAR RADIATION INFLOW A N D THE NATURAL IRRADIANCE
during particular months. Astronomical data defining the position of the Sun (a,, p,) with respect to time and geographical coordinates can be determined from the usual spherical trigonometry formulae or the nomograms printed in the astronomical calendar for the 20th century (e.g., Janiczek, 1962). Detailed calculations require additional correactions for such factors as changes in the Earth-Sun distance. The average diurnal sum of surface solar radiation energy in any month is obtained by dividing the monthly sum by the number of days in that month. Czyszek et a2. (1979) give a method of calculating the energy sums at any latitude. Values of the actual average atmospheric transmittances over many years in the southern Baltic (reduced here to absolute transmittances in accordance with definition (5.1.23)) come from the same paper and are set out in Table 5.1.5. It is clear therefrom that a substantial proportion of solar radiation energy is held up in the real Baltic atmosphere and does not reach the surface of the sea. Obviously, these values will be different for the different average cloud cover, humidity and air turbidity that may occur over different sea areas lying at the same latitude (Egorov and Kiryllova, 1973; Girdyuk et al., 1973; Braslau and Dave, 1973; Dave and Braslau, 1975). The atmospheric transrnittances over the sea also seem to be somewhat higher than those calculated on the basis of coastal-station observations.
5.2 REFLECTION A N D TRANSMITTANCE OF SUNLIGHT AT THE SEA SURFACE. THE ALBEDO OF THE SEA
The reflection and refraction of light at the interface between two media are described by the phenomenological and molecular theories of reflection from which Fresnel's and Snell's laws of geometrical optics are derived (Kizel, 1973). These laws are readily applicable to the radiance of natural light falling on a smooth sea surface from one preferred direction (6, 9).This radiance of light incident at a given angle (e.g., the angle of incidence of direct sunrays 6 = 6, or some other fixed angle 6), is described by (5.1.8) for direct sunrays and by (5.1.15) for diffuse light. The angle of refraction 8 of the rays of this light penetrating below the water surface is given by Snell's law sin6 (5.2.1) = n. sin 19 Every refracted ray lies in the plane determined by the incident ray and the normal to the surface.
5.2 REFLECTION SURFACE OF SUNLIGHT AT THE SEA
249
16 C
c)
:.
1.4
0
L
r
P
1.3 1.2 1.1
1.0
1
0
1
5
,
10
I
15 Wavelength 1
*
20
b‘d
Fig. 5.2.1. The refractive index n of pure water as a function of wavelength (a) in the short-wave region of the spectrum, plotted according to Hale and Querry (1973), with permission of the Optical Society of America; (b) in the wide-range spectrum plotted according to Mullamaa (1964), from Centeno’s experimental data.
The relative refractive index n (its real part) at the air-water interface is almost the same as the absolute refractive index (i.e., that with respect to a vacuum). It is dependent on the wavelength of light and changes slightly with the salinity and temperature of the water. Fig. 5.2.1 illustrates typical plots of the real part of the refractive index n against the wavelength for pure water. This shows clearly that n differs widely for radiations of various wavelengths, which means a considerable dispersion of the propagation velocities of these waves in water. In the visible region of the spectrum, n is roughly 1.33 or 4/3, and alters little with the wavelength A. The effect of salinity and temperature on the refractive index of water n is also slight, as Table 5.2.1 shows. When analysing visible and near-IR light transmittance through the sea surface, it is sufficient to assume an average value of n of 4/3 (Jerlov, 1978). This implies an average velocity of light in water of 2.25 x x lo8 m/s. An important natural phenomenon emerges from Snell’s law, namely, that the cone of rays entering the water becomes narrower: this is depicted in Fig. 52.213. This shows that light rays of refractive index n = 413 incident in air on
5 SOLAR RADIATION INFLOW AND THE NATURAL IRRADIANCE
250
a smooth water surface from any direction in the upper hemisphere (from an arc of 180") all enter the water within a cone whose apex is about 2 x 48.5" = 97'. From under the water, the entire sky (180") is thus visible within a cone of 97'. Likewise, light rays from the water emerge through its smooth surface into the atmosphere only from such a cone. If the angle of incidence at the undersurface of the water is greater than 48.5", the rays are totally internally reflected and return to the water. This can be inferred directly from Snell's law, since the rays incident on the water at the critical angle 8, = y l I z = 48.5" are refracted in air through 90" in accordance with the reciprocal of the relationship (5.2.1), i.e., sin48.5"/sin9O0 z 3/4. The total internal reflection of part of the light scattered in the sea means that nearly half of the energy scattered upwards is retained within the water. TABLE 5.2.1 The refractive index of seawaters of different salinity and temperature for light of wavelength 589.3 nm (from Sager, 1974) Temperature ["C]
Salinity
0 10 20 35
0
10
20
30
1.33400 1.33597 1.33793 1.34088
1.33369 1.33557 1.33746 1.34028
1.33298 1.33482 1.33665 1.33940
1.33194 1.33374 1.33554 1.33824
RefEectanceFunctions, Albedo The energy ratios, that is, the ratios of the radiance of light reflected in air at an angle 8 (equal to the angle of incidence) to the radiance of the light incident on a smooth sea surface are given by the Fresnel equations. For rays directed at an angle of incidence of as and refracted at an angle 0,, these equations can be written thus: (5.2.2a)
R.1 =
sin2(fis-0s) sinZ(as+ 0,) '
(5.2.2b)
5.2 REFLECTION SURFACE OF SUNLIGHT AT THE SEA
25 1
Sun Zenith
,/
Fig. 5.2.2. The refraction of light rays at a smooth sea surface. (a) Geometrical sketch explaining the meanings of the symbols in Snell’s law as applied to a smooth sea surface: (b) cone through which rays enter the water (angle at which the sky is viewed under water y = 979,yt,,,-critical angle equal to 48.S0, O-observer; (c) cone through which rays emerge from the water (S-point source).
5 SOLAR RADIATION INFLOW AND THE NATURAL IRRADIANCE
252
where Rslland R,, are the reflectance functions of the component light waves from a given direction polarized* parallel and perpendicularly to the plane of incidence respectively. This shows that light polarized parallel to the plane of incidence is reflected in accordance with (5.2.2a), whereas light polarized perpendicularly to the plane of incidence is reflected according to (5.2.2b). For incident light polarized parallel to the plane of incidence, the reflectance function R,,, according to the first relationship drops to zero if the angle of incidence 6,, is such that 6,B+t3s= 90". Then, according to Brewster's law, tan6,, = n, and angle 8.sB is c, 53.1O-this is the Brewster angle (Fig. 5.2.3). The total reflectance function of direct sunlight will be the arithmetic mean of the functions R,,, and R,. This is because direct sunlight in a given direction, composed as it is of numerous randomly polarized wave trains, is thus unpolarized radiation. We can then write
+-
(5.2.3)
cl'ol'
I
I
'
I
cc" 0.8
I
.-0 Y
0.6 c
u al c m
5 0.4
.+0)
Ix 0.2
C. 0
30
90
60
Angle of incidence
I?
["I
Fig. 5.2.3. The reflectance function at an air-water interface R, versus the angle of incidence plotted on the basis of the Fresnel equations (5.2.2) and (5.2.3). R,,for the light component polarized perpendicularly to the plane of incidence, Rsl )-for the light component polarized parallel to the plane of incidence, &-for unpolarized light; B-the Brewster angle 53.1'.
* The plane of polarization is here identified with the plane of oscillation of the light wave's eiectric vector.
5.2 REFLECTION SURFACE OF SUNLIGHT AT THE SEA
253
Equation (5.2.3) connects the reflectance function R, of direct sunlight from a smooth sea surface with the angles of incidence 8, and refraction 8, of this light. It therefore refers to a particular direction in which the waves are incident, i.e., to a plane wave and, strictly speaking, also a monochromatic one, since the angle of refraction (refractive index) is a function of 1. As we have already said, this last relationship can usually be disregarded in the case of visible light. The reflectance function plotted against the angle of incidence 6, at the air-water interface in accordance with the Fresnel equations (5.2.2) and (5.2.3) is illustrated in Fig. 5.2.3. This shows the reflection of the two polarized light components, the reflection of unpolarized light, the Brewster angle and the abrupt rise in the value of the reflectance function for large angles of incidence. It also shows that for angles of incidence 6, less than 40°, R, is almost constant, close to 0.02. For rays falling perpendicularly onto the surface (6, = 0), the reflectance functions R,, R , , , , and R,, are equal to one another, and according to the Fresnel equations take the value R, = [(n- l)/(n+ 1)12, i.e., they are equal to 0.02 when n = 4/3. When analysing the flow of radiant energy through the sea surface, it is essential to consider the reflectance function of the total irradiance energy due to the light radiance reaching the sea from every part of the sky, and not just from one direction. So the energy reflected from the surface in particular directions in accordance with the Fresnel equations have to be integrated over a11 directions ( 6 , ~ )But . to do this we need to know or assume a certain directional radiance distribution of the incident light, that is, the radiances incident on the sea surface from all directions L(6, 9).In such a distribution L (8 , p), it is convenient to separate, as before, the radiance due to direct sunrays L"(@,,vs)and that arriving as diffuse light from all other directions LD(6,9).For directions (8,cp) # (G, cp,), the former is equal to zero (strictly speaking, beyond angle AQ, in which all the energy of this direct solar radiation is contained), so its integration over angles (6, p) in order to determine the incident irradiance leads to the formula (5.1.9) derived earlier. The coefficient of reflection from a smooth sea surface R, of unpolarized solar irradiance from one direction (6, , pl,) is of course given directly by the Fresnel equation (5.2.3). The ratio of the radiance reflected from a smooth sea surface to that incident through angle 6, is equal to the ratio of the irradiances of a horizontal surface by the same incident and reflected light (5.2.4) where EsCref1) is the upward irradiance due to reflected rays just above the water surface, E; is as before, the downward irradiance of the sea surface due to direct sunrays.
5 SOLAR RADIATION INFLOW AND THE NATURAL IRRADIANCE
254
The reflectance function of the diffuse irradiance RD, i.e. of the radiance of diffuse light from all directions, is harder to define, because in order to use the FresneI equations we have to know the radiance distribution LD(8,v) and to include or neglect a certain degree of polarization of this scattered light, For a cloudless sky we can assume the directional radiance distribution LD(8,v) that is yielded by the single scattering model and described by equation (5.1.15). This equation enables us to find the radiance LD for every direction (6, p), after which we can use the FresneI equation (5.2.3) to describe the reflectance function of light from this direction. Integration of these results over all directions determines the reflectance function of the diffuse irradiance. In practice, this is done by suitably averaging the directional reflectance function R,(6 , p) over all directions (6,p) # (6,, p,), which gives the resultant reflectance function of diffuse irradiance in the following form 2n n / 2 0
RD
=
5
Rs(6,
0
p)L(6Y ~ ) c o s 6 s i n 8 d 8 d ~
s s L ( 8 , p)cos6sin8d6dp
2x71/2
ED(ref1)
-~t
(5.2.5)
E:
0 0
where E?cref1) is the upward irradiance due to reflected diffuse light just above the water surface, while Ey is as before, the downward irradiance of the sea surface resulting from diffuse light scattered in the atmosphere. Such a method of averaging a function is known here, the function R s ( 6 ,p) being averaged with the weight on the angular distribution of the incident radiance (see equation (5.1.5) of sec6 for comparison). The physical implication of the ratio of the integral expressions in (5.2.5) can be seen here as the ratio of reflected to incident irradiance at the sea surface, although our assumption was that we would take only diffuse irradiance into account. The integration in this formula is done over the upper hemispherehence the limits. From the geometry of integration over the sphere of solid angles dQ = sin6d6dpy we have, as usual, the factor sin6 (the segment of the sphere becomes narrower towards the top like the segment of an orange), whereas cos6 denotes the component radiances perpendicular to the sea surface in accordance with the definition of downward irradiance E, (see (4.1.15)). The function RD is thus the reflectance function of diffuse irradiance from a smooth sea surface for a given distribution of incident light radiance L(6,q). AS we have said, this distribution of radiance may result from a single scattering of sunlight in the atmosphere, but the applicability of such a distribution is then restricted to a clear sky. It is also obvious from (5.1.15) that, in the general case, this is an extremely complicated directional radiance distribution, dependent
5.2 REFLECTION OF SUNLIGHT AT THE SEA SURFACE
255
on atmospheric variables like the scattering phase function p , the optical thickness z and the optical mass m of the atmosphere, all of which are difficult to determine precisely. In (5.2.5) we also neglect the influence of scattered light polarization and take the directional reflectance function R, which emerges from the Fresnel equation (5.2.3) for unpolarized light. When the Sun is high and shining out of a cloudless sky, direct sunlight is generally prevalent. Its reflection is described by (5.2.3) and (5.2.4). Under real conditions, when the sky is partially clouded over (see Ross et al., 1978; Dubov, 1973), the radiance distribution L ( 8 , q) is very complicated and cannot be described by an analytical formula, unless we again use time-averaged values obtained for a given degree of cloud cover (Avastye and Waynikko, 1974). The distribution becomes simpler again only when the sky is completely covered with a thick layer of cloud. Then, as empirical studies have shown, the directional radiance distribution is only slightly dependent on solar altitude, it is axially symmetrical with respect to the vertical, that is, isotropic around the zenith (independent of p), and one can describe it approximately using the cardioidal function in the form
L(8) = A(l+Bcos8), (5.2.6) where A is the radiance from the horizon L(n/2) dependent on the optical thickness of the clouds and does not affect the relative shape of the radiance distribution; B is a variable associated with the scattering phase function by the water droplets in the clouds, and usually lies within the interval 2 < B < 10. Different workers have accepted different experimentally determined average values for these functions, e.g., Moon and Spencer (1942) (see Jerlov, 1976) took B = 2, whereas Rozenberg et al. (1964) took B = 3. Assuming such a cardioidal radiance distribution and an overcast sky, we can use (5.2.5) to obtain a value of the diffuse reflectance function which, according to Preisendorfcr's (1957) calculations, comes to RD,card= 0.052 with B = 2. This indicates that when the sky is overcast, not much over 5% of the daylight energy is reflected from a smooth sea surface, while the other 95% enters the water regardless of the Sun's position in the sky. Without delving any deeper into the complexities of daylight radiance distributions, we can define the reflectance function of the total irradiance at the sea surface R, as being the ratio of the total upward irradiance just above the water (h = 0) due to reflection from the surface EY" to the total downward irradiance of this surface E , Erefl
R =t-.
EL
(5.2.7)
5 SOLAR RADIATION INFLOW A N D THE NATURAL IRRADIANCE
256
This reflestion of irradianc: from the sea surfacz could be called the albedo of the sza surFacac=,but wz must remember that in the literature this name isfrequzntly applied to a different concept which we shall here call the albedo of the sea A,,. This latter albedo describes the combined flux of light reflected from the surface E;'" and that emerging from under the surface as a result of back-scattering in the water itself E;" (for results of studies on A b , see e.g., Payne, 1972; Cogley, 1979). By the albedo of the sea we shall thus mean the ratio of the upward irradiance close to the sea surface E , ( h = 0) to the downward irradiance at the sea surface E i ( h = 0), remembering that the former is the sum E t = EY"+ +E',", so that (5.2.8) or written down clearly for the entire range of the irradiance spectrum (see (5.1.18)), (5.2.8a) Apart from this, there is an albedo of the whole sea-atmosphere system observed at great altitudes, but we shall not go into this further here. Let us now consider the definitions of the components of the reflectance functions described by (5.2.4) and (5.2.5), which are (5.2.9) regardless of how the irradiances are determined. The irradiance diffuseness dE, described in Section 5.1., is defined by (5.1.19) and the total sea-surface irradiance El is the sum E , = Ef E f . By combining these relationships, we get the following expression for the irradiance reflectance function from the sea surface
+
R,
=
Rs(l-dE)+RDdE,
(5.2.10)
which describes the ratio of the irradiance energy reflected from the surface to that incident on it. That part of the incident light energy not reflected from the surface enters the water. The ratio of the irradiance entering the water to the surface irradiance is called the surface transmittance of the sea: p a n s
T = _-?--. EL
(5.2.1 1)
5.2 REFLECTION OF SUNLIGHT AT THE SEA SURFACE
257
Expressed using the reflectance function, this transmittance is T,
=
1 - R,.
(5.2.12)
Both the transmittance T p and the reflectance R , are, of course, functions of the wavelength, and we can speak of their values for particular wavelengths Tp(A), R,(jl) or about their averaged values over all or part of some spectral interval in the same way as was done in Section 5.1. with regard to atmospheric transmittance. The concept of albedo is most often applied to the whole solar energy spzctrum, although even here we can speak of an albedo in a given spectral interval, especially in remote assessments of the physical characteristics of the environment (see Section 4.2.). Refection from a Roughened Sea Surface When we used the Fresnel equations in this simple way, we were assuming a smooth sea surface, which actually occurs only extremely rarely. So in order to get a real picture of light reflection and transmittance through the sea surface, we must consider the effect of wave action on these phenomena. For this we need to know or assume a certain geometrical profile of the roughened sea surface so that we can define the angle of incidence of the light rays. As the oscillations of the free sea surface are complex, the only practicable solution is to define statistically the probability distribution P ( 8 , ~ n )of the slopes on a given element of roughened surface in a given time, or (and this we can take to be the same thing), to define statistically the probability distributions of the slopes of the considered individual element of a roughened surface at a given instant. Experiments have shown that in the open sea, such a distribution differs but little from the normal Gaussian one. Much detailed experimental work has been done on such statistical distributions, results obtained by Cox and Munk (1956) being well known and very frequently applied in marine optics. They analysed photographs of the sea surface in the central Pacific glittering in the Sun and roughened by winds of various velocities. These distributions (Fig. 5.2.4; equation (5.2.13)) are associated with the wind velocities v over the sea surface, although this is only an approximation. Wave action and winds are rarely fixed in time and the connection between them is ambiguous (see also the distributions given by Pelevin and Burtsev, 1975). However, the principal part in the interaction of light on a wind-roughened sea is played by the steeper, short-period components of the wave spectrum (Snyder and Dera, 1970; Dera and Druet, 1975; Dera and 01szewski, 1978). The correlation of these small waves with instantaneous wind
258
5 SOLAR RADIATION INFLOW AND THE NATURAL IRRADIANCE
Surface
Zenith
c
Fig. 5.2.4. Probability density distributions of the slopes of a roughened sea surface in the direction of the wind t o windward qn = 0" and to leeward qn = 180" (b) and a t right angles to the wind pn = 90" and pn = 270" (c) at various wind velocities zi (angles .9., v,,are direction angles normal to the surface element (a): @.-from the vertical, q,,-from the wind direction). These distributions were plotted from Cox and Munk's data contained in Mullamaa's atlas (1964). The curves in (b) and (c) correspond to various winds whose velocities are given in the same order as in the figure for F" = 0.
5.2 REFLECTION OF SUNLIGHT AT THE SEA SURFACE
259
velocities is better than that of long gravity waves which are strongly dependent on the period and extent of wind action. The dependence of the surface slope distribution on the wind velocity (5.2.13) studied by Cox and Munk also speaks in favour of a dependence of this distribution on the wind velocity. But this takes no account of the breaking of wave crests and the formation of foam on the sea surface (Blanchard, 1963; Monahan, 1971; Samoylenko et al., 1974; Gordon and Jacobs, 1977), which strongly scatters and reflects incident diffuse light (Olszewski, 1979b). This process is intensive with an increasing, unsteady wind, but also with a strong steady wind. As the quoted papers indicate, the percentage of the sea surface covered with foam is inconsiderable only when the wind velocity is below 5 m/s; it may be as much as 4% when the wind velocity is c. 10 m/s, and even 10-20% with a velocity of 15 m/s. The role of wave-breaking in nature is of exceptional importance as regards the dissipation of wave energy and the formation of air bubbles in the water which, when they burst at the surface, send a mass of water, salt and electric charge into the atmosphere (Blanchard, 1963); see Fig. 7.1.3. According to Cox and Munk (1956), the form of the function describing the empirical distribution of surface slopes is
1
6 r 2 f 3 ) + ...
(5.2.13)
where f = z:/oX, v = z;/o,,, Z: = az/ax = sin(p,-qU)tan6,,, z; = az/ay = cos(q, - vu)tan6, ,6, ,pl, are the directional angles normal to the water surface, 6, is the deviation from the zenith (vertical), p, is the azimuth, pv is the wind’s azimuth, G , oy are standard deviations, 0: = 0.003+ (1.92 x lOP3w), 0,’= 0.000f + ( 3 . 1 6 ~1 0 - 3 ~ ) , cZ1 = 0.01-0.0086~, co3 = 0.04-0.033~, c40 = 0.40, ~ 2 = 0.12, co4 = 0.23, and v is the wind velocity in [m/s]. This distribution is deemed sufficiently accurate for slopes C < 2.5 and 7 < 2.5, which is equivalent to wind velocities of up to c. 15 m/s. The graph of the probability of roughened surface slopes at various wind velocities w is shown in Fig. 5.2.4 for directions parallel (b) and at right angles (c) to the wind direction. Indices of the slope of a surface element (a) are the directional angles 6,, pn normal to this surface element; 6, is the angle measured from the vertical and cp, from the
2
5 SOLAR RADIATION INFLOW AND THE NATURAL IRRADIANCE
260
direction of the wind, whose azimuth is denoted by po.Fig. 5.2.4b clearly shows the difference in the distribution of surface slopes between the windward (cp, = 0') and leeward (9,= 180') sides. When the wind is stronger, this difference widens and more steep angles of slope appear on the leeward side. Slopes transverse to the wind direction (Fig. 5.2.4~)are symeterical on both sides. Notice too, that as the wind velocity falls, the plots of these distributions become steeper, that
Deviation of sun from zenith
8$ I"]
(b) 0.07
I
I
1
2
1
0.03
I
,
51
0
.
0
0
5
10
Wind velocity v
[Ink.]
IS
Fig. 5.2.5. Graphs showing the reflectance functions from a roughened sea surface (from MulIamaa, 1964). (a) For direct sunrays Rs in the visible spectrum with respect to the Sun's deviation from the zenith 8, for different wind velocities w [m/s]; (b) for diffuselight (scattered in clouds) RD from an overcast sky with a cardioidal radiance 1/3+cos@(curve 1) and with an isotropic distribution (curve 2) with respect to wind speedw; distribution L ( 8 , p) (c) for Rayleigh-scattered diffuse light RD in very clean air and a cloudless sky with the optical thickness of the atmosphere being t = 0.1 with respect to the wind speed for different solar elevations.
-
5.2 REFLECTION OF SUNLIGHT AT THE SEA SURFACE
261
is, there is a lower percentage of large angles of slope, and when it drops to zero, we ought to get a single vertical line 6 = 0 on our plot, indicating a probability of 1 that the surface is horizontal. Calculating the reflectance functions using equations (5.2.2) and (5.2.3) for a wind-roughened sea surface thus requires surface slope distributions P(8,, , p,J to be introduced into these equations. This leads to complicated expressions resulting from the spatial geometry of the relative positions of light rays incident from the direction (6, p) and the vectors normal to the surface elements. Mullamaa (1964) derived these particular expressions and used them to produce a special atlas which gives a large number of results of directional distributions of radiance both reflected and penetrating beneath the surface, and of reflectance functions. Notice that light incident on a roughened surface from a certain direction (6, p) is reflected not in one direction, as from a smooth surface, but within a certain cone which can be described by the directional distribution of the reflected radiance. So radiance reflected from a roughened surface in a given direction should be regarded as an averaged radiance, and is indeed taken to be such when calculating the irradiance E?". The reflectance functions of downward irradiance onto a roughened sea surface calculated by Mullamaa (1964) for Cox and Munk's surface slope distributions (equation (5.2.13)) are illustrated by the plots in Fig. 5.2.5. Fig. 5.2.5a shows that the reflectance function of downward irradiance due to direct sunrays R, for the visible spectrum and solar zenith distances 6, < 40" is almost constant (when n % 4/3), and is practically independent of the wind velocity. R, then is not much greater than 0.02, the value for a smooth surface according to Fresnel's law. R, increases as the Sun approaches the horizon, at first gradually when the deviations from the zenith 6, > 40", and then more quickly when the deviations 6, exceed 70". This rapid increase in R, is fastest when the sea surface is smooth: according to Fresnel's equation (5.2.3), when 6, = 90", reflection is total, i.e. R, = 1. A wind velocity increasing from 0 to 15 m/s (increase in surface roughness) seems to affect R, only when the deviation of the Sun from the zenith 6, is greater than 70". So when the Sun is low in the sky, a rising wind considerably depresses values of R,; this can be explained by the decrease in the average angle of incidence. R, is considerably decreased with respect to the smooth-surface value even when the wind is still weak (up to 5 m/s), but any increases in wind velocity have little further effect. This also demonstrates the prevalence of small waves (short-period components) on the reflection and transmittance of light through a roughened surface. Almost 98% of direct sunlight incident on the sea is transferred under the surface at high solar altitudes (almost irrespective of wave action). At low
262
5 SOLAR RADIATION INFLOW AND THE NATURAL IRRADIANCE
solar elevations, this penetration is much less, but is considerably facilitated by a roughened surface-even by the small waves which occur when only slight breezes are blowing. Reflection from a roughened surafce of diffuse irradiance RD scattered in the cloud layer with an assumed cardioidal (or isotropic) radiance distribution (Shifrin and Pyatovska, 1959) is illustrated in Fig. 5.2.5b. The directional distribution of the radiance here is, as we have said, almost independent of solar altitude, so the reflectance function RD is also independent of 6,,and when the surface is smooth, it is equal to about 0.05-0.06. The appearance of slight waves (at wind speeds of < 5 m/s, as before for Rs)reduces the value of this function to 0.035. This value remains about constant as the wind speed rises from 5 to 15 m/s. Thus diffuse light with a cardioidal radiance distribution is transferred under the sea surface to an extent that is practically independent of solar altitude. The transmittance function (1 -RD),i.e., the transmittance of its energy through the surface is c. 95% when the sea surface is smooth and increases along with the appearance of waves to a constant value of 96.5% when the wind is stronger than c. 5 m/s. The reflection of diffuse light in atmospheric situations other than a completely overcast sky is of course more complicated. The reflectance function of diffuse light RD for a roughened surface when the sky is clear and only Rayleigh scattering is assumed (a “Rayleigh atmosphere”) is illustrated in Fig. 5.2.5~. Here the dependence of RD on wind speed at different solar altitudes 6, is visible. The values of this diffuse reflectance in a “Rayleigh atmosphere” are, on average, higher than those discussed earlier and decrease as the optical thickness of the atmosphere increases (Mullamaa, 1964). As can be gleaned from the figure, they also decrease as the Sun approaches the zenith and decrease in a fairly complex manner as the wind speed rises. Here again, the reflectance is substantially reduced as soon as small waves appear (low wind speeds up to 5 mls), but increasing wave action has a much lesser effect, the reflectance rising slightly at wind velocities of 9-12 m/s but falling again as the wind gets stronger. Olszewski (1979) conceived a detailed model of the reflectance function of the radiance in various directions from a roughened surface having a Gaussian slope distribution. These calculations are useful in remote aerial and satellite surveys of the sea which utilise optical telemetry methods. The total reflectance function of the erradiance from a real wind-roughened surface R, can be estimated from the plots in Fig. 5.2.5, so long as we take into account the summation in (5.2.10) and the irradiance diffuseness de under the various conditions discussed in Section 5.1. Now the transmittance of downward irradiance through the sea surface is
5.2 REFLECTION OF SUNLIGHT AT THE SEA SURFACE
263
1 - R, according to (5.2.12), so it can be derived from the same calculations as the reflectance function R,. Fig. 5.2.6 illustrates the results of such calculations of the irradiance transmittance through a roughened sea surface for extreme wavelengths of visible light in clear air and slight air turbidity at various solar elevations and wind velocities (Wensierski, 1977). They were obtained by assuming Cox and Munk's distribution of roughened surface slopes (5.2.13) and the distribution of skylight radiance emerging from the single scattering model. Also taken into consideration are Rayleigh scattering (1/A4), scattering from aerosols proportional (by assumption) to l / A , and the scattering phase function (from Section 5.1) written as an expansion of Legendre polynomials approximated to the first three terms. The graphs show that the irradiance transmittance T p rises distinctly with increasing wind velocity only when the wind is slight (< 5 m/s) and when the Sun is low over the horizon ah = 90"-6,. T, is only weakly dependent on an
::;I
t ____----
0.8
, ,,/--
0.6
0.51 0
'''2
=700nm
Uh
t
I
I
5
10
-15 vlm/s J
Wind velocity
.
I 0.5 0
5
=10
15
10
Wind velocity v [m/S]
(b)
0.7
0.6 0.5 Solar altitude ah
["]
Solar altitude ah
1
Fig. 5.2.6. Irradiance transmittance through a roughened sea surface Tp from a clear sky for two extreme wavelengths of visible light (from Wensierski 1977) (a) as a function of wind speed w at different solar altitudes ah in cairn (0= 0) and windy (w = 15 m/s) weather.
= 5',
lo" and 50"; (b) as a function of solar altitude
5 SOLAR RADIATION INFLOW AND THE NATURAL IRRADZANCE
264
increase in the wind velocity when the wind is already blowing more strongly. Likewise, the transmittance Tpdepends strongly on the solar altitude a,, only when this is less than c. 30". Clearly, Tyof longer wavelengths (700 nm) is more sensitive to changes in these variables (LX,,and v) than that of shorter waves (400 nm). From what we have said so far, it appears that both the sea surface irradiance and its transmittance through the surface are subject to considerable changes in time. When analysing the actual solar radiant energy entering a given area of sea, we must use certain time-averaged values, e.g., diurnal or monthly ones. The values of the actual monthly or diurnal sum of radiant energy transmitted through the sea surface will therefore not be the same as the instantaneous values of the irradiance transmittance, because the Sun's position, and the states of the atmosphere and sea surface change with time. The time-averaged transmittance Tp from tl to t Z , with the weight on the irradiance distribution in time, can be calculated in the same way as the diffuse reflectance function (cf. the form ofequation (5.2.5))
(5.2.14)
The numerator in this equation expresses the sum of the radiant energy entering the water in a time At, while the denominator gives the sum of irradiant energy incident on the sea surface. We can thus distinguish an averaged transmittance through the sea surface of a sum of light energy (a) when the sky is completely clouded over (or nearly so) Tp,c,which is scarcely dependent on solar altitude and the degree of surface roughness, i.e., (Tp,
dnt
0.95.
(5.2.15)
(?i,s)At
(b) when the sky is clear by substituting instantaneous values of TPJEi) and instantaneous irradiances El ( t ) in (5.2.14), (c) during intermediate situations (T,),, (partial cloud cover), when the transmittance should take intermediate values and requires empirical determination. In these, the most frequently encountered real conditions, the average diurnal (d) or monthly (m)transmittance of the sum of the total solar radiation energy through the sea surface can be expressed approximately using the following general formula: (K)d,rn
=
(1 -nc)(Ta,s)d,m+0.95n,,
(5.2.16)
5.2 REFLECTION OF SUNLIGHT AT THE SEA SURFACE
265
where (Tp,s)d,m is the averaged transmittance through the sea surface of the diurnal or monthly sum of radiant energy for a clear sky and average roughness (calculated for such conditions from equation (5.2. I4)), and n, is the average daily cloud cover per month expressed on a 0-1 scale. The value of n, will also be equal to 1 - tJt, where t , is the number of hours of sunshine per day or month, and t is the total number of hours of daytime. The logic of the constituents of this relationship is readily appreciated when we substitute the extreme values of the average cloud cover, n,, equal to 1 and 0 in (5.2.16), from which we obtain results for the situations included in points (a) and (b), respectively. This averaging by means of (5.2.16) is logically justified for longer periods of time (e.g., one month). Let us now return to the albedo of the sea (Ab) the general definition of which was given by (5.2.8). We recall that it describes the total energy of light reflected from the sea surface and of light emerging from under the surface as a result of daylight having been scattered within the water. The dependence of the albedo on atmospheric conditions, wave action and solar elevation with respect to the reflectance function R, is further complicated by the environmental complexities of the light emerging from under the sea surface. This emergent beam is affected directly not only by the absorbent and scattering properties of the seawater, but also by the refraction and reflection at the roughened surface of the light incident on this surface from below. Also of significance here is the directional radiance distribution of the light scattered within the water and incident on the water-air interface from below the surface. It is often assumed that the radiance distribution of this completely scattered diffuse light is isotropic, that is, that the radiance from all directions of the lower hemisphere is identical. The reflectance and transmittance of such light at the roughened water-air interface can be calculated in the same way as for diffuse light incident from above. The spectral distribution of light emerging from the sea is usually restricted to the visible and near ultra-violet, with greenish-blue light dominant in reasonably clear water and violet prevalent in very clean waters (molecular scattering)these are the colours of the sea in daylight. Jn waters containing large quantities of organic yellow substances the spectrum is narrowed even farther to the greenyellow or orange band. The spectrum of emergent light is narrowed because daylight is filtered through absorbents contained in the water: IR and red light are absorbed by water molecules, UV and some violet by organic substances (see Section 4.2). Within this narrow spectral region, 1-2x of the solar light energy entering the sea leaves it again; if the water is extremely clean, this figure rises to c. 5% (see e.g., Clarke and Ewing, 1974). So the percentage of this energy,
5 SOLAR RADIATION INFLOW AND THE NATURAL IRRADIANCE
266
multiplied by the transmittance of the downward flux (1 -R,), is added to the reflectance coefficient R,, the upshot of this being an albedo of the sea with slightly altered properties in comparison with the properties of the reflectance function. Fig. 5.2.7 illustrates the results of precision experimental work on the albedo done by Payne (1972) from a permenent rig anchored in the Atlantic off the coast of Massachusetts (USA). This is the albedo A , for practically the entire solar radiation spectrum measured just above the sea surface with a piranometer sensitive to radiation in the 280-2800 nm waveband. These results depict the dependence of the albedo A , on solar altitude a,, both in a clear sky (Fig. 5.2.7a), when the air transmittance lay within the interval 0.60 Q TQ < 0.65, and on a completely overcast day (Fig. 5.2.7b), when the air transmittance was 0 < To 6 0.1. These figures show that, like the reflectance function R,,the albedo of the sea depends on solar altitude on a cloudless day. With the Sun high in
0
0.05
21)
t
0
40 60 Solar altitude
80 ah
["I
O'
20
40
60
ah ["I Fig. 5.2.7. The albedo of the sea AQ = E Q ~ / E Q as+a function of solar altitude (from Payne, 1972; with permission of the American Meteorological Society). (a) Clear sky and atmospheric transmittance 0.60 6 TQ < 0.65 (dots) and for TQ = 0.625 (line); (b) completely overcast sky and atmospheric transmittance 0 < TQ < 0.1 (dots). The continuous line represents the average value,
Solar altitude
the dashed lines the standard deviation at wind speeds from 3 to 15 knots.
5.2 REFLECTION OF SUNLIGHT AT THE SEA SURFACE
261
the sky, the albedo’s value is 0.040: this is due to direct sunrays reflected from the roughened sea surface, light scattered in the atmosphere and reflected from this surface, and light emerging from the sea into the air. The transmittance of the last-mentioned light here (in clean water) contributes on average 12.5% (and a maximum of 15%) of the total magnitude of the albedo. The slightly greater spread of experimental points at low solar altitudes is due to changes in the wind velocity which at such altitudes, as we have already said, affects the reflection and refraction of light to a greater extent. Wind velocities here usually range from c. 1.5 to 8 m/s. The average value of the albedo when the sky is extremely cloudy does not depend on solar elevation (as can be seen in Fig. 5.2.7b) and with an average wind velocity of c. 4 m/s, is roughly 0.061 (the continuous line in Fig. 5.2.7b). The spread of measurements (dots) here too was caused by changes in wind velocity, and for a range of velocities 1.5 < u < 8 m/s the standard deviation (dashed lines in Fig. 5.2.7b) from the average value of the albedo is +0.005. Payne states further that at wind velocities less than c. 15 m/s the effect of foam on the albedo can be disregarded. He also says that the directional distribution of the radiance from a very cloudy sky turns out to be closer to the isotropic than to the cardioidal distribution described by (5.2.6). Investigations of the sea’s albedo are of great significance in climatology as it enables us to assess the radiation budget of the sea and atmosphere (Budyko, 1963, 1964; Payne, 1972). The albedo and the reflection of electromagnetic waves are also important optical indicators of many processes taking place in the sea which can be studied indirectly by means of aerial or satellite photography of the sea or other optical remote sensing techniques (Galazi et al., 1979; Sherman, 1979). But it is a difficult problem to solve, both as regards the theoretical description and the practical applications, because, among other factors, of the additional influence of light-scattering in the atmosphere seen from great altitudes (Galazi et al., 1979; Gordon, 1978, 1980; Gordon and Jacobs, 1977). The radiance of light scattered and reflected in the real atmosphere at altitudes of 100-200 km may be several times greater than the radiance reaching that height from the sea surface. So a satellite will in fact record a composite albedo of the whole atmospheric and marine system, in which the information on the properties of the sea will scarcely be perceivable. Nevertheless, the analysis of light emerging from the sea, in particular close to the surface (measured from a helicopter or an aeroplane) allows us to assess the colour of the sea (Plass et al., 1978) and the intensity of the emergent light. From this we can draw conclusions about such factors as wave action, currents and tides, the water’s constituents, chlorophyll
268
5. SOLAR RADIATION INFLOW AND THE NATURAL IRRADIANCE
concentration, fertility of waters, and lack of suspended matter (marine deserts) (see e.g., Gordon, 1978, 1979; Clarke and Ewing, 1974). The emergent light is, ofcourse, totally scattered (diffuse), hence the ratio of the radiance of this emergent light to that of light reflected from the surface in different directions may vary greatly. The direction of observation of the sea surface is therefore also very important in remote measurements of the sea’s properties using optical methods. The extent of polarization of light reflected from the sea surface as compared with that of the emergent light is also different in different directions of observations over the water and can serve as an indicator of certain properties of the marine environment (see Ivanoff, 1974). An analysis of the transmittance of daylight into deep water, which will be dealt with in the next section, will enable us better to understand the nature of the light emerging from the sea. Much information on the reflection of light from artificial sources at the roughened sea surface and on optical techniques of studying the properties of the ocean, wave action, surface temperature and marine pollution will be found in the collective work edited by Galazi et aE. (1979). The use of visible light, IR radiation and microwaves in satellite studies of the ocean are discussed by Sherman (1979) and Novogrodsky et al. (1978). The reflection of light and fluorescence excited by a laser beam enables the thickness of an oil slick polluting the sea surface to be estimated (Sato et al,, 1978; Visser, 1979).
5.3 THE PENETRATION OF NATURAL LIGHT INTO THE SEA DEPTHS. THE OPTICAL CLASSIFICATION OF WATERS
The input of sunlight energy into the sea and its spread within the water mass is initiated by the light transmitted through the roughened sea surface. The directional distribution of the radiance of this light is more complicated than that which we noted at the sea surface as a result of air-sunlight interaction. This further complication of the radiance distribution just below the sea surface L(z = 0,8,@) is due to the roughness of the surface and to refraction and reflection at such a surface. This directional radiance distribution significantly affects the efficiency with which light energy is transmitted into the depths of the sea. This is clear from the very definition of the downward irradiance of a horizontal plane (4.1.15) in which the factor is the cosine of the angle of incidence of the radiance falling onto this plane from various directions of the upper hemisphere. Note that the radiance of direct sunlight refracted at a smooth sea surface has only one direction, defined by the law of refraction. The instantaneous radiance
-
Wind direction
c
c
Angle of devlalion from the vertical 0 ["I
Fig. 5.3.1. Angular distributions of radiance in the sea in the plane of incidence of direct sunrays. (a) Radiance distribution of direct sunrays transmitted under the roughened sea surface from various directions 9 (from Mullamaa, 1964). Plot 1-wind velocity 15 m/s, plot 2-wind velocity 2 m/s; the dashed line denotes the direction of a ray refracted at a smooth surface; (b) changes in angular distribution of radiance with depth as a result of light scattering and absorption in clean waters (left) and turbid waters (right): (c) angular distributions of radiance measured at various depths in the Sargasso Sea plotted in rectangular coordinates (from Lundgren and Hdjerslev, 1971; with permission of the authors and the Institute of Physical Oceanography of Copenhagen University).
5 SOLAR RADIATION INFLOW A N D THE NATURAL IRRADIANCE
270
of direct sunrays passing through an infinitesimally small element of a roughened surface also has one direction of refracted rays. But this direction is subject to rapid temporal change, in time with the complex oscillations of the wave slopes. The velocity of light is incomparably large in comparison with the speeds with which the surface slopes change (frequency of the order of 1-10 Hz), so even the smallest and shortest-lived change in the surface slope is, as it were, a static state with respect to the incident light which is refracted always according to Snell's and Fresnel's laws, though with a time-variable angle of incidence 8. Within the period of a few seconds, then, the radiance of light refracted at a roughened surface can take many possible directions and many appropriate values resulting from Fresnel's law for different angles of incidence 8. When these have been averaged in time we obtain a statistical directional distribution of the radiance of rays refracted under the roughened surface of the sea; this is illustrated by Fig. 5.3.la. This shows that just under the surface we find the main direction of the radiance of the refracted rays, but that this is accompanied by a whole range of other directions of radiance with some smaller averaged values. In windy weather, this main direction can differ from the direction of refraction at a smooth surface by up to 20". At the same time, changes occur in both the average angle of refraction and in the plane of refraction (Mullamaa, 1964). On top of this radiance distribution of refracted direct sunrays, a more complex radiance distribution of diffuse light entering the sea from all directions is superimposed. The resultant radiance distribution L(z, 0 , @) changes again as the light is transmitted deeper into the water. It should be noted that the value of the radiance transmitted beneath the water surface changes owing to the narrowing of the cone (solid angle) of the refracted rays. As the light flux crosses the interface, the principle of energy conservation requires that
E& = 0)
=
(14JEJ,
(5.3.1)
where E , (z = 0) is the downward irradiance transmitted through the surface, while E , , as before, is the downward irradiance of the sea surface. The depth is denoted by z, assuming a coordinate axis directed vertically downwards. For an elementary beam of light, i.e., a radiance L(6, rp) incident on the sea surface from a direction (8, rp), the relationship in (5.3.1) describing the conservation of energy when light passes through the surface becomes, in view of the definition of irradiance,
L(z
=
0,0, @)cos0dQW= [1-RS(6)]L(8,cp)coS6dQ,.
(5.3.2)
5.3 THE PENETRATION OF NATURAL LIGHT INTO THE SEA DEPTHS
27 1
where L(z = 0 , 8 , @) is the radiance of the refracted light beam just beneath the surface of the water, dQa is this beam's solid angle in the atmosphere and dQw its solid angle in the water, and 8 is the angle of refraction. The solid angle dQ, in water is reduced as a result of refraction, so the radiance passing through the surface into the water, expressed as the energy flux incident on a unit solid angle (and on a unit perpendicular area-equation (4.1.4)), increases. This increase is readily determined from Snell's law (5.2.1) and the equations derived from it. The solid angles in water and air are respectively do, = sinOdOd@ and dn, = sin6d6dv. (5.3.3) But as the plane of incidence and the plane of refraction are identical, d@ = dv. Hence,
(5.3.4) According to Snell's law (5.2.1), the connection between the angle of incidence in air 6 and the angle of refraction 8 can be written B = arcsin(nsinO), (5.3.5) and so, after rearranging, the derivative d@/d8 will be d 6 - cos 8 _ - PI--dI3 cos8 .
(5.3.6)
Substituting (5.3.6) in (5.3.4), and the resulting expression for the ratio dQa/dQw in (5.3.2) and rearranging, we get the transmittance of radiance through the sea surface L(z = 0,8,@) = n2 [I - R,(6)]. (5.3.7) L@, q?) The radiance of light under the water surface is thus increased by a factor of n2 with respect to that transmitted under the water; this has already been demonstrated by Gershun (1939). Vector Irradiance The further transmittance of radiance L(z = 0, 8 , @) down into the sea is weakened by absorption and scattering, which here is usually much stronger than in the atmosphere because the components of seawater are condensed into a much smaller space than are those of the atmosphere. Light is scattered in all directions, so it flows through every point in the water in all directions with an intensity differing with depth (Fig. 5.3.lb). The resultant flux of this energy is
5 SOLAR RADIATION INFLOW AND THE NATURAL IRRADIANCE
272
described by the irradiance vector (Gershun, 1958 ; Preisendorfer, 1961) which, using the radiance L ( x , y , z, can be briefly written thus:
x),
(5.3.8a)
E
=
{LdQ.
(5.3.8b)
R
As before, 5 here is the unit vector denoting the direction of the radiance in the solid angle dQ around that direction, i.e., dQ(9. The vector dQ = EJL? = csinOdOd@. The subintegral expression in (5.3.8) for a fixed direction is a vector of direction 5. and absolute value equal to the radiant energy flux dF passing through an element of surface dA, positioned perpendicular to 5 (see the definition of radiance (4.1.4)). The irradiance vector E is a vector sum, that is, the resultant vector of all elementary fluxes from all directions, i.e., their exact integral over the full solid angle 52 = 4x. This resultant flux of energy carried by electromagnetic waves is described elsewhere by the time-averaged (time t , much greater than the period of light waves) Poynting vector Pe = E, x He of the resultant electromagnetic field of the light waves at a given point of the medium: E
= --
te I S 0
Pe(t)dt.
(5.3.9)
As a function of the spatial coordinates, this vector describes the vector light field E(x,y,z), that is, it describes the direction of the resultant momentum of the photons at every point in space ( x , y , z), or, to put it another way, the resultant direction of flow and the magnitude of the radiant energy. Both the direction and the value of the irradiance vector E change with depth (Olszewski, 1976), in a similar way to the radiance L ( z , 0, @) of light coming in from different directions (Fig. 5.3.lb; Tyler, 1977). Let us now imagine some spherical or other space-like surface A enclosing a volume V of the water medium. If the sum of all the light fluxes crossing (in both directions) each element dA of this closed surface area is other than zero, then, assuming there to be no internal sources of light, the law of conservation of energy requires that this difference be equal to the light energy absorbed P within volume Vin unit time. This can be done by integrating the irradiance vector E over the closed area A and simultaneously using the well-known Gauss-Ostrogradski theorem :
P = $ E d A = SVEdV, A
Y
(5.3.10)
5.3 THE PENETRATION OF NATURAL LIGHT INTO THE SEA DEPTHS
273
where dA = n dA, and vector n here is a unit vector normal to an element of the surface dA directed out of the volume V. The magnitude of P given by expressions (5.3.10) is thus the absorbed radiant power, i.e., the power released in the volume of medium V and is equal to the integral over V of the divergence of the irradiance vector E. The flow of energy and the field of irradiances in the sea are most often described using the approximation of horizontal homogeneity and the two-flow analysis of the light field, i.e., of the downward and upward fluxes (Preisendorfer, 1961; Jerlov, 1976). The irradiance vector in a horizontally homogeneous sea is E(x, y , z) = E(z). The projection of this vector onto the direction normal to the horizontal surface, i.e., its component along the z-axis is: (5.3.1 1) E,(z) = n * E(z), where n is the unit vector normal to the horizontal plane at depth z and directed upwards. E, can be referred to as the vector irradiance of the horizontal plane at depth z. It expresses the resultant radiant energy flux flowing vertically downwards into the sea through a unit area at depth z. In view of the definition of the irradiance vector (5.3.8), its component in the z direction can be expressed with the help of the radiance L(z, g) = L ( z , 8 ,@) (5.3.12a) or
1 s cosOL(z, 8, @)sinOdOd@,,
2x x
E,(z)
=
(5.3.12b)
0 0
where n 5 = cose; angle 8 is the deviation from the vertical, d, is the azimuth. This vector irradiance of the horizontal plane E, can be resolved into component irradiances-downward E , and upward E , -by separately integrating in (5.3.12) over angles 6 of the upper hemisphere (from 0 to ~ T C for ) the downward irradiance and over the angles of the lower hemisphere (from in to n) for the upward irradiance. Thence we obtain the definitions of downward and upward irradiances given by (4.1.15) and (4.1.16). The upward irradiance emerges here with a minus sign, so the total E,, the algebraic sum of the downward and upward irradiances, is E&) = E,(z)-E,(z). (5.3.13) 1
The large number of measurements made in various seas shows that the upward irradiance E,(z) due to light scattered upwards is, on average, 50 to 100 times
214
5 SOLAR RADIATION INFLOW AND THE NATURAL IRRADIANCE
weaker than the downward irradiance E,(z) (Tyler and Smith, 1967; Morel and Caloumenos, 1974). For instance, in the Baltic Sea, the upward irradiance is usually less than 2% and is very frequently less than 1% of the downward irradiance (Dera et al., 1974; Gohs et al., 1978). This is why studies of light energy in the sea usually concentrate on the downward irradiance E , (2). This describes the radiant energy flux penetrating down into the sea and, since E l $ E , ,this process is practically irreversible. The Diversity of Irradiance and the Optical Classification of Waters The transmittance of downward irradiance into various seas differs as a result of the diverse absorption-scattering properties of their waters (see Chapter 4). Oceanographers have thus come to speak of the optical classification of seawaters, a term first proposed by Jerlov (1964, 1977, 1978). This classification refers to the upper layer of waters which differ in their irradiance transmittances E,(z)/E,(z = 0) at different depths in the sea, The percentage decrease in the downward irradiance El ( z ) with depth with respect to the irradiance just below the surface El ( z = 0) is depicted for various waters in Fig. 5.3.2. The appropriate typical plots of E l ( z ) in different waters for a clear sky, high solar altitude and calm sea have been numbered by Jerlov and characterise ocean waters of types I, IA, IB, I1 and 111, and coastal waters of types 1 to 9. Note that these are the characteristics of the penetration and attenuation of the entire light flux directed and scattered from the upper hemisphere, which is often called the diffuse attenuation of irradiance penetrating the sea or the diffusion of light energy into the sea. One should clearly distinguish between this irradiance attenuation and the attenuation of a directed beam of light, which was discussed in Chapter 4. The curves shown in Fig. 5.3.2a, plotted by Jerlov (1978) from measurements and calculations, refer to the wavelength il = 475 nm which lies in the band of greatest irradiance transmittance in ocean waters. This figure shows that in very clean ocean waters (type I) the irradiance is attenuated to 1% of the surface value at c. 140 m, whereas in type I11 waters this takes place already at 40 m, and in type 5 coastal waters at around 10 m. The cleanest surface waters of the ocean, type I according to Jerlov’s classification, are to be found in tropical areas of the Pacific, in the Sargasso Sea in the Atlantic, and at the eastern part of the Mediterranean Sea (see Jerlov, 1976). Baltic water falls between type I11 and type 5, depending on season and region (Woiniak et al., 1977b). Obviously, the irradiance attenuation will vary for light wavelengths different from those shown in Fig. 5.3.2a. The further away in either direction one goes from
5.3 THE PENETRATION OF NATURAL LIGHT INTO THE SEA DEPTHS
215
the 450-500 nm band, the stronger, attenuation becomes. The spectral diversity of this attenuation is illustrated by the transmittance spectra of irradiance through a 1 m layer of various types of water (Fig. 5.3.2b) which show the increasing (a) Percentage of surface irradiance
5
2
1
10
9) for A =475nm E.+ (z-o), 20
50
5
3
100 0 10 20
1
30 40 50
60 70
II
z;
80 90 100 110 120
IB IA ,
-
I
I
/
/
I
Wavelength
I
I
,
1
J
I
130 iJ140
I [nm]
Fig. 5.3.2. Jerlov’s optical classification of seawaters in the euphotic zone (1978). (a) Depth profiles of downward irradiance distribution as a percentage of surface irradiance for 475 nm Wavehgh for oman water types I-III and coastal water types 1-5 according to Jerlov’s optical classification (1978); with permission of the author and the Institute of Physical Oceanography of Copenhagen University; @) spectral diversity Of inadiance transmittance through a 1 m layer of water of various types (from Jerlov, 1976). with permission Of the Elsevier Scientific Publishing Company and the author.
216
5 SOLAR RADIATION INFLOW AND THE NATURAL IRRADIANCE
influence of yellow substance absorbing more and more violet in successive types of water. The maximum of the irradiance transmittance spectrum therefore shifts gradually in the direction of the longer waves. The roughly parallel shift of the attenuation spectra in different waters (visible in the red) is due chiefly to the not very selective scattering of light by particulate matter (see Section 4.3). Thus the irradiance of light of wavelength A = 600 nm is attenuated to 1% of its surface value at 20 m in type I waters and at 18 m in type I11 waters. On the violet side, the optical diversity of water types is greater as it is highly dependent on the yellow substance content of these waters. So the surface irradiance of light of wavelength 400 nm is reduced to 1% in clean waters of type I at c. 115 m, but in type 111waters, containing much yellow substance, already at 25 m. Detailed attenuation spectra are given by Jerlov (1978) for all types of water. However, not all waters can be so rigidly classified, if we take their spectral properties into consideration. Coastal polar waters, for example, demonstrate substantial deviations from the “norm” as they contain large quantities of mineral particles but disproportionately little yellow substance (Dera, 1979). (b)
Wavelength 1 lnrnl
Wavelength 1 Inn1
Fig. 5.3.3. Downward irradiance spectra at various depths in the sea (a) in the Ocean (from Jerlov, 1976; with permission o f the Elsevier Scientific Publishing Company and the author; (b) in Ezcurra Inlet (Antarctica) on a January afternoon-from measurements by Woiniak et al. during the 2nd Antarctic Expediation of the Polish Academy of Sciences (Dera, 1979).
5.3 THE PENETRATION OF NATURAL LIGHT INTO THE SEA DEPTHS
277
IR irradiance is strongly attenuated in all waters because this light is strongly absorbed by water molecules (see Chapter 4, Fig. 4.2.3). So the layers of water filter daylight, and are most transparent to the 400-500 nm waveband. At depths of around 100 m, even in the cleanest waters, the irradiance spectrum is thus practically restricted to this band. In waters containing large amounts of yellow substance (types I11 and 1,2,3, ...), this spectrum is even narrower and is shifted to the 500-550 nm band. Figure 5.3.3 presents the changes in the downward irradiance spectrum E, (A) with depth under average conditions and high solar altitude (see also Tyler and Smith, 1967; Morel and Caloumenos, 1974; Okami et al., 1978). The cells of marine phytoplankton producing organic matter and free oxygen during photosynthesis are adapted to the irradiance spectra in the sea. These cells are capable of energetic and spectral photoadaptation, amongst other things, as they sink to greater depths (Dera, 1971; Halldal, 1974; Steeman-Nielsen, 1974). This adaptation is based on changes in the composition of enzymes and photoactive pigments in the cells. In the complex process of multistage absorption and energy transfer in cells, even tiny admixtures of light from some wavebands improve the efficiency of photosynthesis powered by the energy of other wavebands (e.g., the Emerson effect; see Govindjee, 1975). Thus it is that phytoplankton cells can make use of the whole irradiance spectrum in the sea for photosynthesis, even though chlorophyll a, directly involved in the synthesis, has absorption bands in the red and in the violet (Kamen, 1963). So the attenuation characteristics with depth of all the irradiance in the photosynthetically useful wavebands are important in determining the energetic conditions of photosynthesis. It has been found that this useful waveband is that from 350 to 700 nm (SCOR, 1974) and for this waveband one defines an irradiance E,(350-700 nm) or a suitable sum of light quanta called the quantum irradiance q (350-700 nm) as being useful for photosynthesis (Hajerslev, 1978). This irradiance is defined as follows: 700
E,(350-700)
=
S E,(A)dl,
(5.3. 4)
350 700
5 n
q,(350-700)
=
A
E,(A)--dA, hco 350
(5.3. 5)
where the fraction under the integral is the reciprocal of the energy of a single photon, h = 6.626 x J . s is Planck’s constant, and co z 3 x lo8 m/s is the velocity of light in a vacuum. Plots of attenuation versus depth over the whole
278
5 SOLAR RADIATION INFLOW AND THE NATURAL IRRADIANCE
waveband suitable for photosynthesis ("Photosynthetically Active Radiation"PAR) for seawater of different types according to Jerlov's classification (1978) are shown in Fig. 5.3.4. As far as the suitability of light for photosynthesis is concerned, the ends of the useful waveband are not irreversibly fixed in the (a) Percentage of surface irradiance
1
2
5
10
€+ ( 2
1
for 1
20
=(350+700mrn) 50 100
0
5 3
10
2030- E
rn1
"2 50 5
il
60
70
80 90
18 IA I
100 110 120
(b) Percentage of surface number of light quanta
J
10
- 20 - 3oY
-:04
-5og -60 0"
- 70
-80 -90 - 100 Fig. 5.3.4.Vertical distribution of downward irradiance in the euphotic zone of different types of water in the Photosynthetically Active Radiation waveband 350-700 nm (from Jerlov, 1978, with permission of the author and the Institute of Physical Oceanography of Copenhagen University) (a) energetic irradiance, defined by equation (5.3.14); (b) quantum irradiance, defined by equation (5.3.15).
5.3 THE PENETRATION OF NATURAL LIGHT INTO THE SEA DEPTHS
219
literature, as the very influence of light on photosynthesis is complicated (Govindjee, 1975). Sometimes, the useful range of light is simply taken to be the visible waveband (400-700 nm). Since violet is strongly absorbed in fertile waters, the energy differences in the water depths ensuing from such shifts in the endpoints of the waveband concerned are small (around 1%). The light energy consumed by marine photosynthesis is estimated at around 0.5%, and in extreme cases 2-3% of the total daylight energy entering the sea (Koblents-Mishke et al., 1975). The quantum efficiency of this process in the sea is not really known. We do know that to assimilate one atom of carbon and produce one molecule of oxygen, a minimum of 8 photons of usable light is required (Govindjee, 1975). Experiments have shown, however, that a considerable quantity of the energy absorbed by cell pigments is lost to other processes, e.g., luminescence or conversion to heat, so that from eight to possibly several hundred photons must be absorbed by a cell so that a single atom of carbon can be assimilated in the sea (Koblents-Mishke et al., 1975). Spatial and temporal fluctuations of this efficiency depend on many environmental factors and they are being intensively studied and modelled (Bannister, 1974; Woiniak et al., 1977). Marine biology often delimits a euphotic zone in the sea, i.e., the upper layer of waters irradiated with enough daylight to make photosynthesis possible. The lower boundary of this zone is determined by an average level of diurnal irradiance at a given depth such that the amount of oxygen produced during photosynthesis falls to a level comparable with the quantity consumed by the same cells during respiration. This compensation depth, somewhat fluid and not very precisely defined, is roughly that at which the surface irradiance of photosynthetically active radiation falls to 1%. The characteristics of the decrease in irradiance with depth give a rough idea of the depth of the euphotic zone in a given sea. The further attenuation of downward irradiance below the euphotic zone usually becomes more and more exactly exponential, i.e., the gradient dE, /dz approaches a constant asymptotic value, so long as the water medium is homogeneous in a given area. The plot lnE, = f(z) thus becomes a straight line, and the light is completely diffuse (scattered). More precisely, this happens at great depths (1000 m and more, depending on the cleanliness of the water), where the directional radiance distribution L(z, 8 , @) of this diffuse light becomes symmetrical around the vertical regardless of the external irradiance conditions of the sea and the state of its surface. Such an abyssal light field, whose radiance distribution is symmetrical around the vertical, is called an asymptotic or boundary field. The mathematical description of such a field is very much simpler than that which we find nearer the surface on a sunny day (see Section 5.4.).
5 SOLAR RADIATION INFLOW AND THE NATURAL IRRADLANCE
280
A radiance field something like an asymptotic one actually occurs much closer to the surface, at depths of around 200-400 m for visible light, depending on the zenith optical thickness of the water t,(A) (defined in the same way as for the atmosphere tltin equation (5.1.3)). The directional radiance distribution in such a field is not yet symmetrical around the vertical, but the light is completely scattered and the decrease in irradiance with depth is practically exponential. Many physical characteristics of such a field are simplified thanks to their simpler dependences on the inherent optical properties of seawater (Ivanov, 1975). The physical variables describing the light field, together with some of the criteria defining the asymptotic field, will be presented in the next section. The level of daylight irradiance penetrating down to great depths, close to the asymptotic field, is now very low-roughly six to ten orders of magnitude lower than the sub-surface irradiance. Its spectrum is extremely narrow, usually reduced to the blue region. This weak irradiance (almost total darkness during the day) 500-1000 metres down in the ocean is often comparable with the bioluminescent light also obtaining there (Clarke and Kelly, 1965), produced by many marine organisms, e.g., Crustacea, Hydrozoa and many others (Boden and Kampa, 1964, 1974; Gutielzon, 1977; Dera and Weglenska, 1981). The total lack of red in daylight at great depths in the sea has contributed to the red colouration of many abyssal animals. This colour is invisible in light Percentage of surface irradiance 1
2
5
10
M
€*I2 ) I
E,[z=O
so
100
0
LO
70
Fig. 5.3.5. The vertical distribution of downward irradiance in percent of surface irradiance for 1 = 550 nm in various seas (based on measurements of Dera et al., see Dera, 1971).
5.3 THE PENETRATION OF NATURAL LIGHT INTO THE SEA DEPTHS
28 1
devoid of red, and so acts protectively. Likewise, the black skin of many creatures of the abyss camouflages them in the murky twilight existing there. Nearer the surface, fish, for example, are afforded protection by their dark dorsal surfaces, which make them difficult to see from above against the background of deep water, and by their light-coloured undersides, which render them almost invisible against the bright sea surface. Fluctuations in the Underwater Irradiunce
Since the Sun’s position, the state of the atmosphere and the roughness of the sea surface are always changing, the irradiance in the sea is never constant. Besides the obvious changes in the diurnal and annual cycles caused by the Earth’s movement with respect to the Sun, the instantaneous values of the irradiance are also subject to rapid fluctuations in time. These fluctuations are due chiefly to clouds cutting off direct sunlight as they are blown across the. Sun’s disk (Dera et al., 1975) and to the refraction of sunlight at the roughened sea surface (Schenck, 1957; Snyder and Dera, 1970; Severnev, 1973; Dera and Olszewski, 1978). The underwater irradiance fluctuations which result from the superimposition of these causes are particularly great in the upper subsurface layer of the sea. An example of typical changes of irradiance with time E , ( t ) is shown in Fig. 5.3.6. Here we see slower fluctuations due to cloud movement, and superimposed on them, short-period ones caused by the refraction of sunlight on the roughened sea surface-the lens effect or focusing eflect. Much research is being done on these short-period fluctuations (Fig. 5.3.6b) because in nature they exist only under the roughened surface of water bodies and they probably affect photochemical reactions and other biological phenomena in the sea (Dera et aZ., 1975). Both theoretical and experimental work on these underwater irradiance fluctuations has shown that short-period fluctuations are induced mainly by the short-period components of the sea surface wave spectrum and result from changes in the slope and curvature of the surface, changes in the height of the water column and other phenomena affecting the reflection, refraction and attenuation of light rays over the study area within the water (Snyder and Dera, 1970). Under favourable conditions, in sunny weather and in clean waters, the focusing effect prevails in the top few metres of the sea (Schenck, 1957). Statistical analysis of short-period irradiance fluctuations under the waves enables us to obtain fluctuation power spectra. Examples of such spectra are compared with corresponding wave-oscillation power spectra in Fig. 5.3.7.
282
5 SOLAR RADIATION INFLOW AND THE NATURAL IRRADIANCE
In spite of the already considerable number of fluctuation spectra, analytical models (e.g., Schenck, 1957; Snyder and Dera, 1970) and statistical models (e.g., Nikolayev et al., 1972; Shevernev, 1973) obtained by experiment, the phenomenon of short-period irradiance fluctuations in the sea is far from being n +.-
sun
I
11.42 Local time
Sun
Cloud
I
I
11.LO
I
Cloud
I
11.38
11.36
5.3 THE PENETRATION OF NATURAL LIGHT INTO THE SEA DEPTHS
283
unravelled. None the less, we can draw several general conclusions about it from Fig. 5.3.7 and other data contained in the works cited: (1) the principal maximum in the irradiance fluctuation power spectrum at shallow depths (1-5 m below the surface) lies in the 1.5-2 Hz frequency range. Thus, it is shifted quite a long way towards the higher-frequency end in comparison with the frequencies of the principal maximum of the power spectrum of the wind-induced waves prevailing at the time of the experiment; (2) as the observation depth increases, the maximum of the fluctuation power spectrum gradually moves forward towards the lower frequencies and approaches the wave-spectrum maximum. This means that longer waves affect irradiance fluctuations at greater depths (Snyder and Dera, 1970; Khulapov and Nikolaev, 1976), although these fluctuations are already weak there. Below 15 m their amplitude does not normally exceed 15% of the average irradiance; (3) the principal spectral maximum of these strongest irradiance fluctuations, occurring at shallow depths, increases abruptly and shifts towards the Lhigher
E 0
c
0
1
2
3
4
6
5 Time t
7
[sl
Fig. 5.3.6. Examples of natural irradiance fluctuations under a roughened sea surface (wavelength of tested irradiance A = 530 nm). (a) Composite fluctuations over a period of several minutes at two depths (3 and 6 m), due to the combined effect ofwind-blownclouds covering the Sun’s disk and of the focusing by the waves; (b) short-period fluctuations, depicted clearly over a period of several seconds at 1.5 m depth, due to the focusing effect of the waves o n direct sunlight @era et al., 1974); (c) changes ofirradiance with time E&+the diagram explains the parameters denoting the instantaneous concentrations of sunlight flux under the waves. E ( t ) E E,(t)-changes in downward irradiance with time at a selected depth; 1,2, ...- successive events in the increase of irradiance E above the value 1.5Eat the observation point-called Bashes; N ( E 3 lSE)-average flash frequeucy;N ( E S, 4FI-average frequency of appearance of flashes of intensity greater than 4E (for example); At(l.5E)duration of flash.
5 SOLAR RADIATION INFLOW A N D THE NATURAL IRRADIANCE
284
Y L
; 0.8 .-.a C
0
m
._
2
0.6
0.4
02
0 Oscillation frequency
/ [Hzl
Fig. 5.3.7. Power spectra of underwater downward irradiance fluctuations in the 530 nm wave band (a) compared with power spectra of the surface waves at the time of the experiment (b); the results were obtained in the Baltic coastal zone at Zingst (GDR) during the Zingst (1973) experiment (Dera and Olszewski, 1978).
frequencies when the energy of the short-period wave components in the wave spectrum rises; (4)the irradiance fluctuation spectra at shallow depths show many local maxima, which reflect the complexity of the wave spectra, and are particularly marked when the Sun is high in the sky (Dera and Olszewski, 1978). The focusing of sunrays by the crests of fine waves induces especially rapid instantaneous increases in the irradiance just under the surface which exceed the average irradiance by 2, 3 or even 5 times. These are the greatest instantaneous concentrations of solar radiation energy found in nature. They are called underwaterjlashes (Dera, 1977) if they exceed the average downward irradiance at the point of observation by 50% or more. One can analyse the frequency with which s uch flashes occur, statistical distributions of their intensity, and distributions
5.3 THE PENETRATION OF NATURAL LIGHT INTO THE SEA DEPTHS
285
of their durations under various environmental conditions. Typical flash frequencies under favourable conditions are illustrated in Fig. 5.3.8 for four different depths in clean water in the Atlantic and Mediterranean, and at 1 m depth at various wind speeds in the turbid waters of Ezcurra Inlet. It is clear from this figure (and from many other similar results) that the frequency of flashes diminishes with their increasing intensity to a good approximation according to the exponential law N
=
N,e
IA E ~ ,
Fig. 5.3.8. Typical empirical relationships between the frequency of flashes and their intensity (a) at several depths in slight wind in the Altantic (continuous lines-white points) and in the Mediterranean (dashed lines-black points) (Dera. 1977); (b) at 1 m depth in wind of various velocities in Ezcurra Inlet (King George Island. Antarctica) (Dera, 1979) - diameter of the downward irradiance collector 2.5 mm, - wavelength of tested light 525 nm.
28 6
5 SOLAR RADIATIAN lNFLOW AND THE NATURAL IKRADIANCE
which has a physical sense for E , > E,, and has been verified experimentally for E , > 1.5E,.The constants No and A depend on the environmental variables obtaining at the time of observation, such as water transparency, state of the atmosphere, and above all, considering the result shown in Fig. 5.3.8b, the wind velocity. This is shown by the gradients of the plots in Fig. 5.3.8b which correspond to conditions with different wind velocities. The graph shows moreover that the mean frequency of flashes stronger than 5E,under the most favourable
Fig. 5.3.9. The empirical dependence of flash-frequency 1 m below the waves on the velocity of a fixed wind at a standard height of c. 10 m above sea level. The data in particular wind-velocity ranges are the averages ofa number of results from separate 20-minute recording series. The vertical segments indicate standard deviations. The results for wind velocities greater than 10 m/S We* obtained only from the semi-enclosed Ezcurra Inlet in Antarctica; the others also include data from the Baltic and Atlantic.
5.3 THE PENETRATION OF NATURAL LIGHT INTO THE SEA DEPTHS
287
conditions is of the order of one every ten minutes. These strongest flashes may be the result of rays focused by two waves becoming superimposed, perhaps at the mode of two waves crossing each other's path, or of other cases where light from two or more surface sources has become superimposed. The conditions most propitious for fluctuations seem to occur at wind speeds of 2-5 m/s; this emerges from the statistical dependence of the flash frequency N on the wind velocity o, shown in Fig. 5.3.9. The large standard deviations on this figure are mainly due to the fact that with too few data available as yet, the influence on this phenomenon of no factors other than the wind has been taken into account. Further experimental work will have to discover the effect of such environmental factors as the irradiance diffuseness of the sea surface (which determines the efficacy of the focusing effect of the waves), the light scattering and attenuation coefficients in the tested water, and solar elevation; in addition, wave-action parameters should be used instead of wind velocity.
Average wind velocity
v
[rn/sl
Fig. 5.3.10. The empirical dependence on the wind velocity of the coefficient A characterizing the diminishing frequency of occurrence 1 m below the disturbed water surface of light flashes N(E) whose intensities E are increasing according to the exponential law N = Nee-*". The continuous line is the regression curve described by equation (5.3.17) in which e, is the average wind velocity at a standard height of 10 m above sea level.
5 SOLAR RADIATION INFLOW AND THE NATURAL TRRADIANCE
288
The search for some empirical dependence of the coefficient A from (5.3.16) on wind velocity has so far yielded the result shown in Fig. 5.3.10. The regression curve is described by the equation
A
=
( 0 . 1 0 2 ~ 0 . 0 3 ) ~ ~ - ( 0 . 8 7 ~ 0 . 4 0 ) ~ + ( 4 1.24), .71~
(5.3.17)
where A is expressed in units of E y l , and the wind velocity v in m/s. According to this formula, A will be at a minimum value when the wind velocity is c. 4.2 m/s. This clear, though flat minimum of A indicates that the best conditions for the occurrence of intense flashes exist when the wind speed is from about 3 to 5 m/s. Obviously this only happens when direct sunlight reaches the sea,
0 3 0 6 0 9 0 a o Flash duration E*15& ~dmsl
Fig. 5.3.1 1. Typical empirical probability density distribution of the durations of underwater light flashes (above an irradiance level of E+(z)= l.SE,(z) (a) recorded at various depths in the Atlantic, wind speed 1-2 m/s (Dera, 1977); (b) recorded in Ezcurra Inlet in slight wind (1.5 m/s) and strong wind (1 1 m/s).
5.3 ‘THE PENETRATION OF NATURAL LIGHT INTO THE SEA DEPTHS
289
since only such light and not diffuse light (scattered by clouds) can be efficiently focused by the wave crests. In calm weather, when the sea surface is smooth, no flashing occurs; likewise, when the wind is stronger than 9 m/s, the rough sea surface scatters light rather than focuses it. The duration of an irradiance value momentarily exceeding 1.5E4 can be called the flash duration. These flash durations are directly linked with the oscillation dynamics of the sea-wave slopes, but this connection has not as yet been studied. On the other hand, many statistical probability distributions of fIash durations have been studied experimentally (Dera and Olszewski, 1978; Dera, 1979b), typical examples of which are illustrated in Fig. 5.3.11. This shows that ordinary flashes with durations of from 15 to 50 ms are the most likely. These distributions are steeper and less complicated when only slight winds are blowing, than when they are stronger. Because of the fluctuations, the slowly changing oscillations and the diurnal irradiance cycle, it is useful to define the sum of radiant energy falling onto the sea surface or into the sea during a time interval from tl to t, (for a given range of light wavelengths): 12
E,(t)dt.
QAt(t> =
(5.3.18)
11
This may be the actual sum of radiant energy evaluated in a given time (e.g., a day) or the diurnal or monthly statistical average. These latter sums are calculated from the surface sum of radiation (see e.g. Table 5.1.4) and the transmittance of this radiation energy sum through the sea surface (see equation 5.2.16), and from the transmittance TE(z)down into the sea which will be explained in the next section (Czyszek et al., 1979). The actual average monthly sums of solar radiation energy measured over a period of many years [Q
TABLE 5.3.1 The average monthly sums of solar energy over the entire spectrum = 54", 57" and 60" N [cal/cm2]; from Czyszek et al., 1979 Z
0
p7 =
54"
0 0.2 0.5
rp = 57"
1 2 3 5 10 15 20
particular depths in the southern Baltic at latitudes
I
n
I11
IV
V
VI
VII
VIII
IX
x
XI
1314 795 660 535 396 327 223 66 26 11
2417 1463 1215 986 725 587 387 109 41 15
5902 3 559 2940 2367 1706 1352 862 230 77 24
9184 5473 4454 3472 2397 1828 1102 239 73 18
12487 7 367 5856 4383 2822 2048 1186 225 50 13
14894 8 698 6806 5124 3 306 2427 1355 223 45 7
13 788 8 052 6 301 4 660 2 840 2 040 1 029 152 28 6
11 225 6 566 5 186 3 828 2 458 1818 954 146 34 6
7 519 4 399 3 481 2 579 1681 1 263 744 120 30 8
4 110 2 413 1911 1561 933 707 407 70 21 4
1539 921 748 588 397 326 225 49 15 6
977 592 492 400 301 260 183 60 25 12
945 572 474 384 284 235 161 47 19 8
2016 1220 1014 823 605 490 323 91 34 12
5337 3218 2658 2 140 1542 1222 779 203 69 21
8 755 5 218 4 246 3 309 2 285 1742 1 050 282 70 18
12 212 7 205 5 727 4 286 2 760 2 003 1160 220 49 12
14 738 8 607 6 735 5 070 3 272 2 402 1341 221 44 7
13 599 7 942 6 215 4 596 2 801 2 012 1 006 150 27 5
10 850 6 347 5 013 3 700 2 365 1758 922 141 32
6 976 4 081 3 230 2 393 1563 1172 656 112 28 7
3 563 2 091 1657 1354 809 613 353 61 17 3
1177 704 572 450 304 250 172 38 12
653 396 329 268 202 173 122 40 17 8
Iml
0.2 0.5 1 2 3 5 10 15 20
em@) reaching
5
5
XI1
q~ = 60"
0 0.2 0.5 1 2 3 5 10 15 20
610 369 306 248 183 152 104 30 12 5
1629 985 819 665 489 396 261 73 28 10
4776 2880 2378 1915 1380 1094 697 181 62 19
8 291 4 942 4 021 3 134 2 164 1650 995 216 66 17
11 936 7 042 5 598 4 189 2 697 1956 1134 215 48 12
14 602 8 528 6 673 5 023 3 241 2 380 1 329 219
44 7
13 411 7 832 6 128 4 538 2 663 1985 992 148 27 5
10 457 6 118 4 831 3 566 2 280 1 694 889 136 31 5
6 429 3 761 2 977 2 206 1 440 1080 604 102 26 6
3 036 1782 1412 1154 689 522 300 52 15 3
845 505 41 1 323 218 179 123 27 8 3
373 228 188 153 115 98 70 23 10 5
5 SOLAR RADIATION INFLOW A N D THE NATURAL IRRADIANCE
292
changes in plankton growth and water turbidity. The influx of solar radiation energy in other seas will of course differ, depending on the geophysical parameters of the Earth’s motion with respect to the Sun, the state of the atmosphere and the type of seawater occurring in a given region. These real sums of energy penetrating into the waters of different parts of the ocean have not yet been adequately studied. They are, nevertheless, estimated by various means, principally in order to assess the energy budget of the sea (see Chapter 6) and its effect on climate and weather (Budyko, 1963; Payne, 1972). Measurements of irradiance and its attenuation with depth are one of the most important aspects of oceanographic research. These measurements are usually made on board a research vessel. Many conditions have to be satisfied so that this can be done properly: the diurnal irradiance must be stationary during the measurement, the irradiance collector must have a suitable geometry and properties (see Section 4.1) and the spectral sensitivity characteristics of the measuring device must be known (Tyler and Smith, 1966; Neuymin et al., 1966; Smith, 1969; Jerlov and Nygard. 1969; Prieur, 1970; Dera et al., 1972).
5.4 THE APPARENT OPTICAL PROPERTIES OF THE SEA AND THEIR RELATIONSHIPS WITH THE INHERENT PROPERTIES IN A N UNDERWATER LIGHT FIELD
The plots showing the decrease of daylight irradiance with increasing depth in the sea, illustrated in the previous section, are approximately straight lines on a semi-logarithmic scale and over long depth ranges. The reduction in irradiance in different waters can therefore be described by the slope of these lines, expressed as the following ratio: -AlnEl(z) AZ
=
K,(Az) .
(5.4.la)
zl
(Az), describes the average attenuation of the irradiance This ratio, denoted by E , (more exactly, the natural logarithm of the irradiance) per unit thickness of a layer Az through which the downward light flux diffuses. The precise value of this ratio at depth z has to be referred to an infinitesimally thin layer of medium dz at that depth. Equation (5.4.la) thus becomes the derivative (5.4.1 b)
5.4 THE APPARENT OPTICAL PROPERTIES OF THE SEA
293
Definitions of Apparent Optical Properties If we denote the derivative (5.4.lb) by K, (z), we define the diffuse attenuation coeficient of the downward irradiance at depth z in the sea as (5.4.2) and likewise the diffuse attenuation coefficient of the upward irradiance as (5.4.3) and also the difuse attenuation coefficient of the total vector irradiance, i.e., the algebraic sum E, = El - E , as K (z)
= -
1 ~. dEZ(z) Ez(z) dz ’
(5.4.4)
In the same way, we can obtain the diffuse attenuation coefficieat of scalar irradiances for - the downward scalar irradiance (5.4.5)
- the upward scalar irradiance (5.4.6) -
and the total scalar irradiance, i.e., the sum E,
=
Eo
+Eo, (5.4.7)
In like manner too, we can define the difluse attenuation coefficient of the radiance from any direction (0, @) with depth (5.4.8) This set of coefficients, as coordinate functions in the water space, characterizes the conditions for the vertical transfer of light energy within the sea and makes up the group of so-called apparent optical properties, that is, those which occur under a given set of marine environmental conditions. The apparent optical properties of the sea obviously depend mainly on the inherent optical properties of sea water, which we discussed in Chapter 4. They
5 SOLAR RADIATION INFLOW AND THE NATURAL IRRADIANCE
294
are also dependent to a certain extent on the irradiance conditions existing at any instant in the sea, but chiefly on the directional distribution of the light flux entering it from the atmosphere. A number of other optical functions, widely used in hydrooptics, are also included among the apparent optical properties of the sea (Preisendorfer, 1961): the dzfuse reflectancefunction of the downward irradiance R , and of the upward irradiance R , ,also known as the irradiance ratio at depth z in the sea (5.4.9) (5.4.10)
and the directional distribution functions of the light flux at depth z (5.4.1 1)
We also distinguish between the directional distribution function of the downward flux (5.4.12)
and that of the upward flux (5.4.1 3)
The physical sense of these latter functions becomes apparent when we use them in the definitions of the appropriate irradiances (see Sections 4.1. and 5.3). SO after taking into account the definitions of the downward scalar irradiance (4.1.12) and downward vector irradiance (4. 1.15), the directional distribution function of the downward light flux (5.4.12) is 2n x / 2
L(z, 8, @)sinedOddi DL(4 =
0 0 2xx/2
Y
(5.4.14)
cosOL(z, 8, @)sin@dBd@ 0
0
hence (5.4.15)
5.4 THE APPARENT OPTICAL PROPERTIES OF THE SEA
295
The directional distribution function of the downward flux D , (2) is thus the reciprocalof the average cosine of the angle delimiting the average direction of the radiance from the upper hemisphere at depth z in the sea. If the downward irradiance at depth z resulted from an isotropically distributed radiance L(8, @) = L(8) = L($n)= const, then according to (5.4.14), the average cosine of such a distribution would be equal to f , and the function D s ( z ) = 2. The magnitude of this function just below the sea surface D + ( z = 0) -= D(0) serves to define the external conditions of light penetration into the sea (the boundary condition in calculations). Hydrooptics is particularly concerned with the downward irradiance El (z), but also with the diffuse attenuation coefficient of the downward irradiance K , (z). For if (5.4.2) is formally treated as a simple differential equation, a well-known form of the solution to this equation is the downward irradiance E , ( z ) as a function of depth E + ( z )= E,(z
= O)e
-
z
Kd-W
(5.4.16a)
O
or written another way E,(z) = E l ( z = O)e-K1z.
(5.4.16b)
It is frequently found in practice that as a function of depth, the coefficient K , (z) alters little with depth. It is then accorded an average value in a given layer of water (e.g., in the euphotic zone), so that K,(z) = K,(z) = const. Using the coefficient KL(z) we can express directly the transmittance of the downward irradiance in water TwEpassing from the sea surface down to depth z z
(5.4.17) Like all the optical functions so far discussed in this section, the irradiance transmittance into the sea TwE(O+- z ) refers to a particular wavelength of light. Both the downward irradiance E , (see Fig. 5.3.3) and its attenuation coefficient K , (see Fig. 5.4.1) are functions of the wavelength A. But we are often interested in the irradiance transmittance of a wider range of wavelengths from 1, to A, (see Fig. 5.3.4) or even of the whole solar radiation energy spectrum 0 < 1 < co (see equations (5.3.14) or (5.1.18)). Equation (5.4.17) can be applied to such a transmittance [ E ,(z)/E,(0)Ial as well, by replacing the spectral attenuation coefficientK , for a given wavelength with the apparent coefficient of diffuse attenuation (of the downward irradiance) K,,(z), i.e. averaged, with the weight on the spectral irradiance distribution
296
5 SOLAR RADIATION INFLOW AND THE NATURAL IRRADIANCE
450
500
550 Wavelength
600
650
'
1 [nm]
Fig. 5.4.1. Spectral dependences of the diffuse attenuation coefficient of the downward irradiance Kl(rl) in various natural waters, at high solar altitudes. Curve 1-Pacific: t = 106'22'W; p = 20"21'N (Islas Tres Marias). Measured 28.11.1968, depth 12.5 m (Tyler and Smith, 1970). Curve 2-Atlantic, Antarctic Basin: 1 = 66'57'W; p = 64"08'S (South Shetlands). Measured 29.01.1976, depth 10 m (Wensierski-1st Polish Antarctic Expedition 1975/76, r/v "Profesor Siedlecki"). Curve 3-Pacific: A = 113"ll'W; p = 29"14" (Gulf of California). Measured 9.12.1968, depth 12.5 m (Tyler and Smith, 1970). Curve &Atlantic, Antarcti cBasin: I = 56'29'W; p = 6193's (Bransfield Strait). Measured 6.02.1976, depth 10 m. (Wensierski-as for curve 2). Curve 5-Ckntral Baltic. Measured 18.06.1973, depth 4.5 m (Hapter et al., 1974). Curve 6-Baltic, Gulf of Gdansk, Measured 3.09.1973, depth 4.5 m (Hapter et a1.-as for curve 5). Curve 7-Atlantic, Ezcurra Inlet (South Shetlands). Average values for January 1978 in the euphotic zone (Woiniak ef al. ;PAN Antarctic expedition, see Dera, 1979).
5.4 THE APPARENT OPTICAL PROPERTIES OF THE SEA
291
(5.4.18)
Equations analogous to (5.4.17) and (5.4.18) and useful in describing the underwater light field, also exist for the other kinds of irradiance, for sums of radiant energy (equation (5.3.18)) and for radiance from any direction (8, @). Examples of experimentally determined spectral relationships of the diffuse attenuation coefficientEl, averaged over the depths of the euphotic zone in different seas, are shown in Fig. 5.4.1. In relation to (5.4.17),the plots on this figure characterize, according to optical criteria, the average conditions of solar radiation energy penetration and the modifications of the daylight spectrum in different waters. It also shows the greatest differentiation of this penetration in the short-wave region, i.e., violet and UV. We shall now proceed to discuss the dependence of the function K,(A) on the external irradiance conditions of the sea, and its connections with its inherent optical properties. As apparent optical property of the sea, the function K , (A) depends to some degree on the directional distribution of the light flux entering the sea (Wensierski, 1980). Values of this function for different seas are thus compared for similar external conditions, i.e., high solar altitude, sunny weather and slightly roughened sea surface. Under different conditions, in the euphotic zone, K,(A) can differ from the usual value by 10-30%. This depends on the directional distribution of the downward light flux D r ( z ) , and thus on the Sun’s position, the state of the atmosphere, the roughness of the sea surface and the scattering function of light in the water. This dependence will be partially elucidated by the interrelationships between the various optical properties of the sea, to be discussed below, and by the plots in Fig. 5.4.5. The Interrelationships among the Optical Properties o j the Sea
We can obtain an exact description of all the optical properties of the sea under stationary conditions if we solve the radiative transfer equation (4.4.6) for a given boundary condition of the sea area in question. The solution would be an analytical expression of the radiance function at all points of the water space L ( x , y , z , 8 , @) or at all depths in a horizontally homogeneous sea L ( z , 8, @). It is with the aid of the radiance that we can define the inherent optical properties of seawater (Chapter 4) and all the different types of irradiance; from these we can then define the apparent optical properties of the sea. Unfortunately, however,
298
5 SOLAR RADIATION INFLOW AND THE NATURAL IRRADIANCE
such a general analytical form of the solution of the radiative transfer equation has not yet been worked out. Intensive research is in progress to find various hydrooptically useful approximations of this equation. The simplest of these, based on the single scattering model, were discussed in Section 5.1, when we were analysing the transmittance of sunlight through the atmosphere. When applied to such a condensed medium as seawater, this model is approximately valid only for the upper layers of the sea (Jerlov, 1976). In such a medium, secondary and multiple scattering of light soon takes place to a significant degree. Applications are being sought of other, necessarily very complex methods of calculating the radiance L(z, 0 , @) from the transfer equation which would account for multiple scattering. These methods are numerical methods, which include the successive approximations method or iteration (Sugimori and Hasemi, 1971 ; Raschke, 1972, 1975; Woiniak, 1980), Monte Carlo simulations (Funk, 1973, Gordon et al., 1975) and a numerical quadrature (Chandrasekhar, 1950: Schellenberger, 1964; Herman and Lenoble, 1968). The results of these complicated computations enable us to determine the spatial and directional radiance distributions L(z, 0, @) for given inherent properties (the scattering function and absorption coefficient) and a given distribution of the radiance entering the examined area of the sea. As the mathematical equations thereby obtained are so involved, they are of real significance mainly in theoretical considerations and in the theoretical analysis of the properties of the light field in the sea. In practice, we do not always have to know the directional distribution of the radiance in the sea. It is often sufficient to know its integral defining the irradiances which describe the essential characteristics of light influx into the depths. We have already demonstrated this by means of examples, and we shall now substantiate this further with equations that link the irradiances with the inherent optical properties of the sea. Many practical conclusions about the nature of the light field in the sea and about the relationships between irradiances and the inherent and apparent optical properties of the sea are reached by carrying out some simple mathematical operations on the radiative transfer equation. Suppose we consider equations (4.4.6) and (4.4.7), but ignore the source function L, we can then write the transfer equation briefly as gvL= -cL+L,. (5.4.19) Integrating the terms of this equation (Preisendorfer 1961) over all directions of the sphere SZ, we get
(5.4.20)
5.4 THE APPARENT OPTICAL PROPERTIES OF THE SEA
299
From this expression, after making use of the scattering function (4.3.2) and the definitions of the irradiance vector (5.3.8a), scalar irradiance (4.1.9a) and path function (4.4.5b), we obtain the following extremely important relationship
VE = -cEo+bEo.
(5.4.21a)
Since absorption and scattering are additive in light attenuation, c = a+b, (5.4.21a) is equivalent to
VE = -aEo. (5.4.21b) In this way we have derived a very important equation, the Gershun equation (Gershun, 1958), which enables us to determine absolute values of the light absorption coefficient in the sea. To do this, it is sufficient to establish empirically the divergence of the irradiance vector E and the scalar irradiance Eo. Taking (5.3.10) into consideration, the divergence of the irradiance vector E is equal to the radiant power absorbed in a unit volume of water dP/dV. From (5.4.21b) and (5.3.10) we can thus derive an important expression describing the radiant power d P absorbed by a given volume element of the medium dV anywhere within the sea dP ~- -aEo W/m3]. (5.4.22) dV In order to calculate the energy absorbed by the water mass, it is obviously sufficient to know the scalar irradiance Eo and the light absorption coefficient a for the waveband in question. Equations (5.4.21b) and (5.4.22) therefore become simple operational expressions, enabling one to determine the absorption of light energy in the sea. In a horizontally homogeneous sea, the irradiance vector E(x, y , z) = E(z) and the scalar irradiance E&, y , z) E Eo(z). Equation (5.4.21) then simplifies to -= dEz(z)
-a(z)Eo(z). (5.4.23) dz By using the symbol a(z) for the light absorption coefficient, we intend to indicate clearly that this coefficient depends on the depth z in the sea, where light usually comes into contact with a varying content of absorbent substances. If we divide (5.4.23) by the vector irradiance E&), we incidentally obtain an important connection between the light absorption coefficient a(z) and the total directional distribution function of the light Aux at depth z, D(z) (5.4.24a)
5 SOLAR RADIATION INFLOW A N D THE NATURAL IRRADIANCE
300
from which it is obvious that K,(z) = a ( z > W ) (5.4.24b) (see definitions of K, (5.4.4) and D (5.4.11)). Equation (5.3.13) shows that the total vector irradiance E, is the algebraic sum of the downward and upward irradiances. Equation (5.4.23) can therefore be written using the downward E , (z) and upward E , (z) irradiances: (5.4.25)
This is the basic operational equation which allows us to determine the absorption coefficient of light in seawater a(z) in situ. So four fundamental irradiances are measured in the sea: E , (z),E , (z), and Eo (z),Eo (z), remembering that the sum of the latter two is the total scalar irradiance Eo(z). Bearing in mind the law (5.4.22), the same equation may also serve to determine the radiant power absorbed by a unit volume of the water mass at given depths in the sea (Czyszek et al., 1979). Taking into account the definition of the diffuse reflectance function of the downward irradiance R,(z), (5.4.25) can also be written in the form
,
d ---{E,(z)[l dz -R,(z)l) = - a ( z ) ~ o ( z ) .
(5.4.26)
The values of the reflectance function R,(z) in seas usually lie within the range 0.01 < R , (z) < 0.1 ; most often they are close to 0.02, but in turbid seas like the Baltic, they are around 0.01. Thus equation (5.4.26), simplified by the assumption that E , (z) E , (z) or that R , (z) "N 0, can be used with a certain accuracy. So from (5.4.25) or (5.4.26) for very turbid waters we get the simplified form of this basic equation dEJz) dz
z -a(z)E,(z).
(5.4.27)
If we apply here the definition of the diffuse attenuation function of the downward irradiance (5.4.2), we can rewrite this last simplified equation as
K , (4E , (4 z5 a ( 4Eo (4
(5.4.28)
This simplification obviates the need to establish the upward irradiance E , (2) which is difficult to measure with precision in the sea. The difficulty lies in the errors made when integrating the radiance incident at large angles from the lower hemisphere on the integrating irradiance collector,
5.4 THE APPARENT OPTICAL PROPERTIES OF THE SEA
301
The relationships obtained above point to the fundamental parts played by the vertical measurements of the distributions of the four principal irradiances in the sea, i.e., E,(z), E,(z), E,,(z) and E,,(z). Without having to know the radiance distribution L(z ,0, @), we can use them to determine the basic apparent optical properties of the sea, find and utilise interrelationships between them and the inherent optical properties, and define the radiant power absorbed by the medium in different layers of the sea. Some vertical distributions of the four principal irradiances taken from the results of experimental work done in the Baltic (Dera et al., 1974; Gohs et al., 1978) are given in Fig. 5.4.2. They show that E,(z) B E,(z) and likewise that Eor(z) % E, ( z ) ; also, of course, that according to the definitions of these irradiances, EoL(z) > E , (z) and E, (z) > E , (z), i.e., the scalar irradiances are greater than the vector ones at the same depths. Comparison of the plots for 1 = 425 nm
,
t
lrradiance E [relative units] 161
10-2
lo-&
lo5 I
11.03.76
1 525 nm 15 *O
-
25 30
t
1
11.03.76
\
M 25
1~425nrn 30
lo-~
0
103
10'
lo5
15
20
30 -
4
1 z 525 nm
V
Fig. 5.4.2. Typical results of measurements of the four principal irradiances in the Baltic for two different wavelengths. (Data from Polish-GDR research expeditions in the Baltic (Gohs et al., 1978).
5 SOLAR RADIATION INFLOW AND THE NATURAL IRRADIANCE
302
with those for il = 525 nm shows the considerably greater attenuation of short waves due to the relatively high concentrations of yellow substance in Baltic waters. Examples of values of the functions introduced above to describe the apparent optical properties of the sea are compared with the absorption coefficient a(z) in Table 5.4.1. TABLE 5.4.1 Example of values of the apparent optical properties of the sea and of the absorption coefficient calculated from equation (5.4.25) (data selected from Preisendorfer, 1961)
4.24 10.42 16.58 28.96 41.30 53.71
1.247 1.288 1.291 1.313 1.315 1.307
2.704 2.727 2.778 2.781 2.751 2.763
0.153 0.174 0.169 0.165 0.158
0.150 0.172 0.169 0.165 0.158
0.0215 0.0183 0.0204 0.0227 0.0235 0.0234
0.115 0.118 0.117 0.117 0.112
Preisendorfer (1961) describes many interrelationships between the apparent and inherent optical properties of the sea; he has introduced the concept of hybrid optical properties. These are chiefly products comprising the inherent coefficients a(z), b(z) and c(z) multiplied by the flux distribution function D,,(z) for the downward flux (arrow pointing down) or for the upward flux (arrow pointing up) respectively. These products make up the hybrid volume absorption function for the downward or upward flux
a,t(z) = 4 4 D ,t ( 4 the volume scattering function for the downward or upward flux
(5.4.29)
b,t(z) = a(z)D,,(z) (5.4.30) and the volume attenuation function for the downward or upward flux Crt(4 =
c(z)D,t(z).
(5.4.31)
Moreover, Preisendorfer has introduced the hybrid volume scattering coefficient for the downward (1) or upward (f) light flux in the forward bf or backward bb hemisphere, depending on the direction of the flux marked -1 or f. They are defined as follows
(5.4.32)
5.4 THE APPARENT OPTICAL PROPERTIES OF THE SEA
303
(5.4.33) Here L?, means the upper hemisphere and 52, the lower one; as we can see, e.g., to get the backward scattering of the flux from the upper hemisphere bbJ we integrate over the lower hemisphere 52, of angles dn(Q which is thus implied by the first of the two arrows in the subscripts. The integrals inside the square brackets of these equations express the path function L, in direction 5 which has arisen from the radiance from all directions 5' of the upper or lower hemisphere being scattered at depth z (see equation 4.4.5b where 5 = go). Integration of these path functions (the integral in front of the bracket) over all directions 5 of the upper (or lower) hemisphere thus describes all the light scattered in the hemisphere in the same direction as (bf) or in the opposite direction (b,) to that of the irradiance incident on plane z. We thus have the following relationships for the hybrid optical properties of the sea so defined: b,,(z)
=
bf,,(4+bb,,(Z)
(5.4.34) (5.4.35)
Using these hybrid optical properties, we can derive transfer equations for the downward and upward irradiances which are analogous to the transfer equation for the radiance. So integrating the terms of the transfer equation (5.4.19) over the upper hemisphere we get
v1 Q,
5 - w 7
5)dQ(5) =
s
Q,
C(ZW(Z7
ndJw3+
L*(z7 5M-4517
(5.4.36)
Q ,
hence the left side of this equation is dE,(z)/dz. The first term of the right side is c(z)Eo (z), which, after multiplying top and bottom by El (z) and using the definition of the downward flux distribution function 0,(5.4.12), is equal to c(z)D, (z)E, (z) = c 4(z)E, (z), where the right side of this equality emerges from the definition of the hybrid downward flux attenuation function (5.4.31). Finally, in view of the path function definition (4.4.5b), the second term of the right side of (5.4.36) is
5 SOLAR RADIATION INFLOW A N D THE NATURAL IRRADIANCE
304
What is more, in view of the definitions of the hybrid scattering functions (5.4.32) and (5.4.33), this is equal to b,, (z)El ( 2 )+bbt ( z ) E ,( 2 ) . Substituting these expresions for the various terms in equation (5.4.36) and using (5.4.34) and (5.4.35), we obtain after Preisendorfer (1961), the timeless transfer equations for the downward and upward irradiance we have been seeking (5.4.37)
--_dEt(z) dz
-
- [at (2)+bbt (z)]E+(z)+bbc (')El
(z).
(5.4.38)
Equations (5.4.37) and (5.4.38) allows us to derive useful relationships between the optical properties of the sea. So, for instance, by dividing (5.4.37) by E,(z) and using the definitions of K,(z) and R,(z), we get a relationship for the downward flux K $ ( z >= aL(z)+.bb((')-bbf(z)Rt(z),
(5.4.39a)
which shows how the diffuse attenuation function of the downward irradiance K,(z) is linked to the absorption and scattering coefficients in the sea in a way which is dependent on the flux distribution D(z). A similar relationship can be obtained from (5.4.38) for the upward irradiance attenuation function
-K T (1 '
= at
(z)+bb t (z)-bb 1 ( Z ) R $ (z).
(5.4.39b)
Rearrangements of these relationships reveal further interdependences between the optical functions introduced here. With the aid of the definition of K,(z), we can write (5.4.37) in the form (5.4.40) (see (5.4.36)). Hence it is easily deduced that K,(z) must always be less than the hybrid attenuation function of the downward flux c,(z), or at most, be equal to it, i.e., K , (4
CJ
(z)
(5.4.41a)
at any depth z in the sea. Moreover, since c , ( z ) = c(z)DL(z),the following relationship exists between the inherent coefficient of light attenuation c(z) and the downward irradiance attenuation function K , (z) and the downward flux distribution function D,( z ) K&(Z> c(z)D,(z).
(5.4.41 b)
5.4 THE APPARENT OPTICAL PROPERTIES OF THE SEA
305
From (5.4.39), it is also evident that there is a connection between the inherent coefficientof absorption a(z) and these functions :
K&)
4W,(z).
(5.4.42)
Combining these last two inequalities, we reach the important conclusion that (5.4.43)
A frequent and more universal way of describing the apparent optical properties of the sea is to link the irradiance attenuation with the optical depth
s
z
tZ(4 =
c(z)dz,
(5.4.44)
0
rather than with the actual depth z in the sea. The coefficients of the diffuse attenuation of irradiance with optical depth are defined in the same way as those coefficients introduced earlier in equations (5.4.2)-(5.4.7), z being replaced by z,. We denote them by y with the appropriate indices and by analogy with definition (5.4.2), we can write the diffuse attenuation function of the downward irradiance with optical depth as (5.4.45) From this it is immediately evident that K, and y , are interrelated: (5.4.46)
In the same manner we can obtain all the other attenuation coefficients of irradiances and radiance with respect to the optical depth 7,. By analogy with the inequality (5.4.43), if we divide this by the light attenuation coefficient c, we obtain a universal inequality for attenuation with respect to optical depth (5.4.47a) Hence, from the scattering-attenuation ratio w0 = b / c (see equation (4.4.12)), we gain another useful form of this inequality (5.4.47b)
306 (a) theoretical relationships calculated from the single scattering model with corrections for multiple scattering I-for typical turbid seas, when the scattering function elongation parameter xi = 2.87, 2-for typical clean seas when x1 = 2.17, 3-for isotropically scattering media (xl = 0). The (straight) dotted line y4( rz)/DJ(0)= 1-coo represents this dependence common to all waters for an optical depth rZ = 0; the continuous lines (curves) correspond to an optical depth of rz = 0.2, the dashed lines to an optical depth rz = 4. the Baltic and Gulf of Gdarisk for various visible light wavelengths (b) Experimental relationships within the optical depths range 0 < zz < 5. Symbols: +-in (from Woiniak. 1974: 0-in the vicinity of the Canary Islands in the Atlantic for L = 425 nm and L = 525 nm (from Woiniak, 1974); &-in the Mediterranean for violet and red light (based on H+jerslev, 1973).
5 SOLAR RADIATION INFLOW A N D THE NATURAL IRRADIANCE
Fig. 5.4.3. The ratio of the diffuse attenuation coefficient of the downward irradiance to the directional distribution of the downward light flux y~ ( ~ , ) / D(0) J as a function of the parameter w o = b / c for different seas (from Woiniak, 1977).
5.4 THE APPARENT OPTICAL PROPERTIES OF THE SEA
307
The inequalities in (5.4.43) and (5.4.47) both specify only the range of variability of the diffuse attenuation function of irradiance with actual depth z or optical depth z, in the sea. They do not, however, specify definite dependences of this attenuation coefficient on the inherent optical properties of seawater for a given distribution of the downward flux denoted by D , . Such strict dependences between these magnitudes are yielded by these inequalities only for the extreme cases where we can replace the inequality sign by an equals sign. This is possible (1) in an ideal medium in which there is no scattering, so that K , /Dl= a = c, or yJ/D, = 1, which is almost true for IR in water, where b c$ a, and (2) in any real sea just below the surface, where it is easy to demonstrate that K , (z = O)/Dl ( z = 0) = a(z = 0) or y,(z, = O)/Dl(zz = 0) = l-uzo(zz = 0). In all other cases-for real media and different depths-the dependences between these opticalparameterscannot be definedprecisely on the basis of such simple mathematical operations on the transfer equation (5.4.19); this equation has to be solved. In order to do this, we use the various more or less complicated models referred to earlier. Woiniak (1977) performed such calculations using the single scattering model with corrections for multiple scattering; Gordon et al. (1975) used Monte Carlo methods, while Raschke (1972) and Ziege et al. (see Ivanov, 1975) applied the iteration method which takes account of multiple scattering. These calculations furnish the radiance L ( t , , 0, @) (see (5.1.15)), and hence the relationships between
1
.
0
1
~
"
0.1 0.2
"
"
0.3
" " 0.L
0.5
0.6
0.7
Parameter
0.8
0.9
10
w0
Fig. 5.4.4.Theoretical examples of the dependence of the downward light flux distribution function D&(zz)on the parameter wo = b/c in differentseas and at different depths when direct sunlight falls on a smooth sea surface irradiating the horizontal underwater surface at an angle of incidence of 30" (from Woiniak, 1977). I-for typical turbid seas (xi = 2.87); 2-for typical clean seas (xl = 2.17); 3-fOr isotropically scatterins media = 0). The (straight) dotted l i e represents these dependencesfor an optical depth of zz = 0, common to all waters: the continuous lines (curves) correspond to an optical depth of zz = 0.2. and the dashed lines to an optic& depth of r, = 4. (XI
308
5 SOLAR RADIATION INFLOW AND THE NATURAL IRRADIANCE
the inherent and apparent optical properties of the sea ;(in view of the definitions of the latter) for a given flux distribution D , (z = 0) = D , (0) of the light entering the sea. Examples of these dependences, calculated theoretically and measured in various seas are illustrated in Figs. 5.4.3 and 5.4.4. The first of these shows the relationship between the ratio of functions y r ( z Z ) / D c ( 0 )and the ratio of inherent coefficients b/c = coo. The parameters of these graphs are the optical depth and the elongation parameter x1 of the scattering function which was defined by equation (4.3.52). Fig. 5.4.3b compares these plots with the results of actual measurements in the sea. Characteristic of the diffuse attenuation coefficient of the downward irradiance K , or y L (where the external flux distribution D1(0) is the same) is that it descreases as the parameter oo increases, in all seas and at all depths. The slope of the plot, however, does vary in different seas. This differentiation results from the fact that the diffuse attenuation coefficient of the downward irradiance decreases as one passes from clean seas (where x1 is small) to turbid ones (where x1 is large) for any value of coo greater than 0. Figure 5.4.4 shows theoretical examples of the dependence of the downward flux directional distribution function D , ( t z ) on parameters coo and xl. Typical of these dependences are the increase in D , ( T = )as scattering plays an ever greater part in attenuation oo = b/c, and the fall in value of this function D,(zz) as the elongation parameter x1 of the scattering function increases. It is important to note that under fixed external conditions ( D , (0) = const) the pair of variables ooand x1 unequivocally determine the values of the pair of variables y and D,. Woiniak (1 977) provides detailed nomograms of these interdependences for various depths and external conditions. The Asymptotic Light Field
All the relationships between the inherent and apparent optical properties of the sea become considerably simpler at great optical depths. Then we are dealing with an asymptotic or boundary light field (mentioned in Section 5.3) in which, as Preisendorfer (1961, 1964) and others have shown, the angular radiance distribution L(z, 6 , @) is symmetrical around the vertical (see Fig. 5.3.1) and does not change its shape with depth. In such an asymptotic light field, all the attenuation coefficients of radiances and irradiances are independent of the external irradiance conditions and all take the same boundary value, also independent of depth, so long as the medium itself is homogeneous (in the sense that the inherent optical properties are not invariable). These conditions describing the asymptotic field which below some critical depth z,, can beas written follows:
5.4 THE APPARENT OPTICAL PROPERTIES OF THE SEA
309
(5.4.48a) (5.4.48b) where zzcris the asymptotic depth, ,,z, is the asymptoticoptical depth, and K, and y m are the asymptotic attenuation coefficients of radiances and irradiances, i.e., limK, = limK, = limK, = limKt = limk,, = limkOt = K, (5.4.49a) -:+
Z4W
m
and similarly limy, = limy, rz-+m
z4m
Z’W
Tz-+ m
=
limy,
=
2-
00
:-+
a3
... - Y m
(5.4.49b)
T.7’00
where Ym
=
Km -*
(5.4.50)
C
The depth-dependent irradiance attenuation coefficients thus tend towards a constant asymptotic value K,; this means that above the asymptotic depth Attenuation coefficient y (; T;) 1.6
1.8
2.0
Fig. 5.4.5. Vertical changes in the diffuse attenuation coefficient of the downward irradiance with optical depth y r ( t l ) in the sea, calculated by Wensierski (1980). The relationships between t: and mh and the inherent values of t: and wo are elucidated by equations (5.4.56). 1-D(o)
= 1;
2-D(O)
=
I , 2; 3--0(0)
= I , 4;
4-D(0)
=
2 (idealized case); 5-asymptotic
value y ; .
5 SOLAR RADIATION INFLOW AND THE NATURAL IRRADIANCE
310
they vary to a greater or lesser extent, even in a homogeneous medium. Figure 5.4.5 illustrates the vertical changes and the tendency towards an asymptotic value of y c (tZ) with optical depth t, in different environmental conditions. These changes were computed using a transfer approximation (see equation (5.4.54)) and the gaussian numerical quadrature (Wensierski, 1980). Comparison of these plots for two widely differing values of o& shows that values of y 1 reach the asymptotic value y m much faster when the scattering-absorption ratio is high than when it is low. The figure also shows that when co; is higher, the asymptotic value of the coefficients y; is lower. In an asymptotic field, which satisfies conditions (5.4.48)-(5.4.50), the radiative transfer equation (4.4.9), on being included in (5.4.48) for z 2 z,, becomes x
(c-~mcose)L(z,e) = 2rS ~ ( ze’)p(e‘, ,
P,e, @)sinB’de’
(5.4.51a)
0
or for
t,
2 z, +1
-1
where p = c o d andp is the scattering phase function defined by equation (5.1.12). The right sides of equations (5.4.51) express the path function, the left sides result from the other two terms of the transfer equation (4.4.9); the source function Lll is ignored, but the derivative of the radiance in a form appropriate to the asymptotic field (5.4.48), is included. This form of the transfer equation in the asymptotic field is an integral (Volterra) equation whose solution is far simpler than that of the general transfer equation. Many investigators have attempted to solve this equation for an asymptotic field in the sea (Lenoble, 1961 ; Prieur and Morel, 1971 ;Zaneveld, 1974; and others; see also Ivanov’s monograph, 1975). Particularly easy is the solution for an ideal medium which assumes an isotropic form of the scattering function, i.e., ,8(0’, W , 0 , @) = b/4n = const o r p ( p ’ , p) = 1. Then the right-hand sides of equations (5.4.51) are constant and independent of direction 8, and these equations reduce to the forms
L ( ~e),
=
const c - K , cos 8
(5.4.52a)
for z 2 zcr,or (5.4.52b)
5.4 THE APPARENT OPTICAL PROPERTIES OF THE SEA
311
for t, 2 zCr.When scattering in the asymptotic field is isotropic, the directional radiance distribution is ellipsoidal. Substituting solution (5.4.52) back into the transfer equation (5.4.51) yields the characteristic equation (Chandrasekhar, 1950; Ambarcumian, 1952)
b l+K,/c ____ = I In 2Km l-Km/C
(5.4.53a)
or (5.4.53b) which describes the dependence of the apparent optical properties of the sea on the inherent properties in the asymptotic field when scattering is isotropic. In comparison with (5.4.48), this is at the same time the solution to the transfer equation for the idealized asymptotic field case and the isotropic light scattering function. It serves largely as an aid to the approximate solution of this equation for real conditions using the transfer approximation. (see (5.4.54) In a real sea, which contains particulate matter, the light scattering function is, as we remember (Section 4.3), strongly elongated forward through small angles, and is therefore far from isotropic. We can apply the above-mentioned transfer approximation to such a real function p ( p ’ , p), which requires this function to be resolved into two components : (1 -piso) expressing only forward scattering (through small angles, and more exactly, through a scattering angle 8, = 0), and piso expressing isotropic scattering through the whole sphere. This can be written as follows P@’, = Piso+(l -Piso)d, (5.4.54) where the function d(8,) = S@-1) is the delta function, one which is equal to unity only when ,u = 1, i.e., when the scattering angle 8 = 0, and which is equal to 0 for all other scattering angles. Furthermore, the following property of the B function emerges from the normalising condition for the scattering phase function (5.1.13): B(p- 1)dQ = 4x. That a simple relationship exists a
between pisothus defined and the elongation parameter of the scattering function x1 is readily demonstrated. If we substitute in equation (4.3.51) the expression for the scattering function (5.4.54) introduced here and take into account the relationship between and p (5.1.12), we obtain, after integrating and rearranging, the following equation: piso =
X1 1- T.
(5.4.5 5)
312
5 SOLAR RADIATlON INFLOW AND THE NATURAL IRRADIANCE
The physical sense of the transfer approximation is this. Since the scattering function for seawater is strongly elongated forward, it is assumed that the major portion of the scattered light energy is borne in the same direction as the incident light (forward through small angles 8, -+ 0) and that this portion is not distinguished from unscattered light passing through the medium; in other words it is not included in the scatterance. The remaining scattered light, which makes only an insignificant contribution to the energy transfer, is described by the approximate isotropic scattering function piso. We have already obtained a fairly straightforward solution for this isotropic function in the asymptotic field (5.4.53) which we can use here. This scattering function piso(describing only part of the scattered light), however, leads to scattering and attenuation coefficients b' and c', the parameter co; and the optical depth z: which are underestimated, and to overestimated irradiance attenuation coeficients y'. Nevertheless, we have the relationship (5.4.55) between the idealized function piso and the elongation parameter x1 of the real scattering functions for real media. In view of this relationship and of the definitions of the coefficients given below, we can readily specify a relationship between these changed (dashed) coefficients and the real ones:
(5.4.56)
Only the value of the real absorption coefficient a does not change with this approximation, but the proportion of absorption in the total attenuation 1--@A instead of 1 -I& does seem to rise. With the approximations (5.4.56) expressed by only two parameters (xl and coo) at our disposal, we can now readily apply to real media the solutions of the transfer equation which we obtained earlier for isotropic scattering (equations (5.4.53a) and (5.4.53b)). To this end, we substitute in (5.4.53) the dashed factors given in (5.4.56) instead of the real ones, after which we include expressions (5.4.56) of these factors by the actual parameters XI and coo for real media. This substitution brings us straight away to the relationship between the asymptotic value of the irradiance and radiance attenuation coefficients ym and the real parameters x1 and coo. These dependences, computed for seas with different inherent optical properties, are illustrated in Fig. 5.4.6. They also show that the asymptotic values of the attenuation coefficients fall as the properties
5.4 THE APPARENT 0PTICAL PROPERTIES OF THE SEA
Fig. 5.4.6. The asymptotic attenuation coefficient of radiances and irradiances y m as a function of the parameter wo in different seas (from Woiniak, 1977). (a) Theoretical dependences for seas with different scattering function elongations characterised by parameter x1 (see equation (4.3.51)). 1 -XI = 2.87; 2-Xi = = 2.76; 3-XI = 2.36;4--rl = 2.27; 5-xl = 2.17;6--x1 = 0 (isotropic scattering). The straight dashed line expresses the dependence 7 = I - % . (b) Empirical relationships at great optical depths (zZ c 10 and more) observed for various light wavelengths. Symbols: 0-in the Baltic (from Woiniak); -in the Baltic (from Jerlov and Nygard); n-in the clear Lake Pend Oreille (from Tyler); A-in the Mediterranean (estimated) (based on data from Hnjerslev). The lines on this figure are plots 1 and 5 from (a).
313
314
5 SOLAR RADIATION INFLOW AND THE NATURAL IRRADIANCE
of scattering in the total light attenuation wo rises. The asymptotic values also fall when the parameter x1 increases, i.e., when the water turbidity increases. It should be noted at the end of this section that besides the daylight field described, there often exists an abyssal light field of biological origin (bioluminescence), of which mention has already been made. So as to describe the influence of this light on the resultant radiation field, we must now include in the radiative transfer equation (5.4.19) the hitherto neglected source function Lq (see (4.4.9)). But the form of this function L, (z, 0, @) has been little studied and that is why this particular problem still awaits a solution. It should be added that it is experimentally difficult to distinguish between the daylight field and the bioluminescence field at great depths.
CHAPTER 6
THE TRANSFER OF MASS, HEAT AND MOMENTUM IN THE MARINE ENVIRONMENT
The transfer of electromagnetic radiation, described in the last chapter, is just one way of transferring energy in nature. The effectiveness of this transfer is principally due to the fact that this energy covers enormous distances at the speed of light and, moreover, does not require any intermediary substance. Thus, Percentage of energy absorbed in a layer of water from 0 to z ,
0
20
40
60
AQ,/Q, [Yo1
80
100
0
5 3-
.o Y)
3 0)
r
4 Clean ocean water
01
;5 -
(Sarcjasso Zca)
Y 0
.-
8
6-
m
u1
5
4
n
8' 9-
,cL
J p I - _ 2 -
I
~
\' I
I
Fig. 6.0.1. The percentage of solar radiation energy AQ.(0+z)/Qs absorbed in a layer of water from the surface z = 0 to depth Z.
es
Ear(h = O)-EQt(h = 0) is the total solar radiation flux absorbed by the sea over the whole spectrum U/m2 sl (see Chapter 7); AQ, (OT.?) is the solar radiaaon flux absorbed by d layer of water from the sed surface z = 0 to a de-
pth z over the whole spectrum [J/mZ s].
316
6 THE TRANSFER OF MASS, HEAT AND MOMENTUM
through the vacuum of space the Sun regularly supplies the various regions of the rotating Earth with energy. This energy is released, i.e., most is converted into heat, and the rest into chemical and other kinds of energy, when photons collide with the molecules of absorbent substances in the atmosphere, in the sea and on the land. The heat energy thus acquired by these molecules is conducted throughout the entire medium and is converted into various kinds of mechanical work, energy of flows, etc., so that by a process of gradual scattering, it is reconverted into heat and radiated back into space. The radiant power released in a unit volume of a medium, that is, the volume density of (mainly) heat energy sources Qss in the sea (and in the atmosphere too), is described by equation (5.4.22) discussed in Chapter 5. The extent to which this energy is released in the various layers of the atmosphere and sea is strongly differentiated, and this differentiation in turn derives from the absorption-scattering properties of the medium, discussed in Chapters 4 and 5. The absorption properties of water are responsible for the fact that most of the solar energy is absorbed in the upper layer of the sea. From the equations given in Chapter 5 and the absorption coefficient @(A) of clean ocean water, we know that as much as c. 37% of the solar radiant energy entering the sea is absorbed in the top 10 cm of water. About 57% is absorbed in the first metre, even in clear ocean water, and about SO% will have been absorbed by the time a depth of 10 metres is reached. This is shown diagrammatically in Fig. 6.0.1, a plot of this absorption for clear ocean waters and turbid Baltic Sea waters. Since water is only slightly transparent to the total spectrum of the solar radiation energy flux, radiation plays an ever-diminishing part in transferring this energy deeper into the sea. The dominant role is therefore taken over by the mechanical processes of its molecular and turbulent exchange among the various elements of the medium; it is this which we shall be discussing in this chapter. These processes are constantly accompanied by mass motion, and therefore by the transfer of masses of water and the substances they contain. We have thus to consider the transfer of this energy in the sea in association with the transfer of mass, that is, with diffusion, and the flow of elements of the medium, and the f.' . between them. This is an extensive branch of science, mostly dealt with llction by hydrodynamics, and since there already exists a sufficient number of monographs on marine hydrodynamics, we shall say no more about it in this book. Nevertheless, we shall proceed to examine the basic physical aspects of the transfer of mass, heat and momentum in the sea, and to provide a mathematical description of them.
6.1 MOLECULAR TRANSPORT OF MASS, HEAT AND MOMENTUM
317
6.1 MOLECULAR TRANSPORT OF MASS, HEAT AND MOMENTUM IN SEAWATER
The random thermal motion of molecules gives rise to their exchange as elements of mass between adjacent zones of the medium. When molecules collide with one another, this same thermal motion also promotes the transfer of their momentum and thus the transfer of the energy of their molecular motion, that is, heat. This molecular exchange always proceeds in the direction tending to restore thermodynamic equilibrium between neighbouring volume elements of the medium and is an irreversible process.
a #0 az Fig. 6.1.1. The irreversible transfer of mass, heat and momentum due to the random thermal
motion of molecules (a) the mechanism of diffusion-a greater number of molecules passes from a high-concentration to a low-concentration layer; (h) the mechanism of thermal conductivity-the greater (average) momentum of molecules is transferred during collisions to a lower-temperature layer; (c) the mechanism of molecular exchange of flow momentum causing friction (viscosity)-the greater flow momentum (transportation) of particles m u l transferred during collisions to a layer with a lower flow velocity u l ; (d) a stratified medium with concentration gradients of diffusing substance aCl8.Z temper&tureaT/& and flow velocity (in the x direction) au/az.
318
6 THE TRANSFER OF MASS, HEAT AND MOMENTUM
Imagine first, a horizontally stratified medium in which adjacent layers differ in temperature T(x,y , z) = T(z), i.e., in their average molecular kinetic energy, in the concentration C ( x , y , z ) = C(z) of some dissolved, passive substance (like the ion of sea salt), and in their laminar flow velocity v(x, y , z ) = v(z) (Fig. 6.1.1). During their thermal motion in all directions, some molecules will pass from one layer to another or vice versa. There will, however, be more molecules of a component passing from a layer containing more of them (higher concentration) to an adjacent layer where their concentration is lower. This resultant flow of molecules from a zone where their concentration is high to one where it is lower is called molecular dzyusion.This is a fundamental means by which mass is transferred and differences in the concentrations of constituents in a medium are equalized. Because molecules collide, the temperature difference, that is, the difference in the average momentum of molecules in their thermal motion in adjacent layers of the medium causes the energy of their random thermal energy to be transferred towards a zone of lower temperature. This elementary way of transferring heat in a medium is called the molecular conductivity of heat and causes temperature differences to balance out. When neighbouring layers of liquid are moving relative to each other (they do so with different velocities vx = u), every molecule has not only a randomly directed thermal velocity, but also an additional transfer velocity u in the direction of flow. On colliding with one another, molecules carried at differing speeds in the x direction transfer both their thermal momentum and their kinetic momentum at the interface between the layers. During these collisions, the large number of such momentum components mju due to the motion of single molecules of mass mi gives the molecules in the adjacent layer an impulse Fiti in the x direction according to Newton’s 2nd law of motion (where ti is the collision time, and Fi is the force acting in the x-direction during the collision). Thus is created the frictional force Fi tangential to the direction in which the layers are moving. This force accelerates the motion of the layer having the lower velocity u, while retarding the motion of the adjacent layer which has the greater velocity u (the force of reaction - Fi), i.e., it promotes the exchange of flow momentum. This is an elementary process of momentum exchange (flow energy) at right angles to the direction of fluid flow and is called the molecular viscosity. The direct exchange of mass, heat and momentum among single molecules takes place at a distance of the mean free path of molecules 6 in their thermal motion in the medium. The differences in the state of a liquid over infinitesimally short distances
6.1 MOLECULAR TRANSPORT OF MASS, HEAT A N D MOMENTUM
319
are described by the gradients of its macroscopic state parameters, whose components along the z-axis for the concentration of a passive substance C, temperature T and transportation velocity u in the x direction are aC/az, aT/az, and au/&, respectively. Numerous experiments have shown that during molecular exchange, fluxes of molecular mass, heat and momentum are directed in the respective directions of the gradients given (here perpendicularly to the z plane) and are to a good approximation proportional to these gradients. As regards the horizontally homogeneous medium assumed here, these laws can be written as follows: (a) for the diffusion of a component of a solution (6.1.1)
where qm(z)is the surface density on the z plane of a mass flux of a diffusing substance [kg/s mZ], i.e., the resultant quantity of mass passing across unit area z in unit time; C(z) is the concentration of this substance expressed in kilograms per cubic metre of solution [kg/m3] or in dimensionless units, i.e. the ratio of the mass of constituent to the mass of solution in a given volume; D is the coe@cient of molecular diffusion of a substance in a solution expressed in [mZ/s] if the concentration is in kilograms per cubic metre of solution, or in [kg/ms] if the concentration is given as a dimensionless ratio of masses, like, e.g., the salinity; (b) for the thermal conductivity
4*(4 = -Y-3
aT
(6.1.2)
aZ
where %(z) is the surface density on the z plane of a heat flux conducted across this plane, in [J/smZ], i.e., the resultant quantity of heat passing across a unit area z in unit time; T(z)is the temperature in kelvins [K], and y is the coeficient of molecular thermal conductivity expressed in [W/m.K] ; (c) for the viscosity (6.1.3)
where q&) is the surface density on the z plane of a momentum flux transferred
across this plane in
m2 s (or impulse) in the x direction passing across a unit area z in unit time; u(z)
320
6 THE TRANSFER OF MASS, HEAT AND MOMENTUM
is the flow velocity in the x direction in [m/s], and q is the coeflcient of dynamic molecular viscosity expressed in [N * s/m2] = [Pa * s]. Also used is yu = q/e, the coeficient of kinematic molecular viscosity [m2/s], where Q is the density of the medium. The differential laws (6.1. l), (6.1.2) and (6.1.3) are called respectively Fick‘s law of diffusion, Fourier’s law of thermal conductivity and Newton’s law of viscosity. At the same time, we see that the momentum flux density qu is a measure of the force per unit area, i.e., the.frictionalforce per unit area, or the tangential stress, shearing off layers of liquid parallel to their boundary. Ordinarily, stress is denoted by z, but here this symbol is reserved for the Reynolds stresses occurring during the turbulent exchange of momentum (see Section 6.2). The density of momentum flux just introduced is thus equal to the tangential stress acting on the shearing boundary surfaces of the adjacent layers of liquid, tending to deform the layers in the x direction. The density of each of the fluxes q is, in general, a vector directed in the same direction as the concentration, temperature or flow velocity gradients respectively, whereas the stress acts in the same direction as the flow velocity. The analogous form of all three laws of molecular transfer (6.1.1)-(6.1.3) emerges from the fact that the same thermal motion of molecules (or their clusters) transfers elementary masses and momenta in all three cases. The minus sign in these equations indicates that the flux is flowing in the direction of decreasing concentrations. The linear relationships (6.1.1)-(6.1.3) are, however, approximate relationships, valid only for the relatively small concentration, temperature or flow velocity gradients encountered in nature. When the gradients in the equations describing the flux density of molecular exchange are very steep, further terms of their expansion, i.e,, higher-order derivatives and products of derivatives, appear (see equation 3.5.3). The molecular-kinetic theory of fluids and gases defines the limits of applicability of these linear equations describing the processes of molecular exchange (see e.g., Szczeniowski, 1971). The coefficients of molecular diffusion, thermal conductivity and viscosity are charcteristic (inherent) physical properties of every substance, including seawater. They are constant when the salinity S, temperature T and pressure p are also constant, but generally they are slowly-variable functions of these quantities. The complexity of these functions is due to the complexity of the water structure, and in particular to the changes in the numbers and sizes of molecular and ionic clusters resulting from changes in temperature, salinity and pressure (see Chapter 2). We can state generally, however, that when conditions favour
6.1 MOLECULAR TRANSPORT OF MASS. HEAT A N D MOMENTUM
321
the existence of the large, less mobile molecular clusters, when their mean free path is reduced, the coefficients of molecular exchanges of mass and heat decrease, whereas the coefficient of molecular exchange of flow velocity (internal friction) increases. The coefficient of molecular diffusion of salt ions in sea water, under the
Ions Ca2+
K+ Na+
c1-
so:-
j
1
1I
at 5°C
at 23.7"C
5.0
7.5 17.9 13.4 18.6 9.8
11.4 8.o 11.5 5.8
Clearly, the coefficient of diffusion increases with a rise in temperature, which causes the large molecular clusters in the water to break down. The few available values of the molecular coefficient of thermal conductivity y in seawater given by different workers do not always agree (Popov et ul., 1979; Oceunogruphical Tables, 1975). At the salinities, temperatures and pressures usual in the sea, they fall within the range 0.55 < y < 0.64 W m-lK-l. At higher temperatures (313K) and pressures (lo8 Pa), y approaches the upper boundary of this range, but increasing salinity causes y to fall slightly in value. This behaviour of the coefficient of molecular thermal conductivity in sea water is equally readily explained on the basis of an analysis of the changes in its molecular structure. The dependence of the coefficient 9 of molecular dynamic viscosity of pure and seawater on temperature is illustrated in Fig. 6.1.2 and Table 6.1.2.This information shows that 7 decreases rapidIy and non-linearly with temperature rise. Again, one can explain this as being due to the breakdown of the large and not very mobile water molecule clusters whose presence at low temperatures effectively inhibits the free movement of the liquid (see Section 2.2). Increasing
322
6 THE TRANSFER OF MASS, HEAT AND MOMENTUM
Temperature
[ K]
Fig. 6.1.2. The coefficient of molecular dynamic viscosity of pure and ocean water as a function of temperature at atmospheric pressure (compiled from tables in Popov et al., 1979).
salinity raises the viscosity of water slightly because of the rise in the number of ion clusters which are also less mobile than single molecules or their dimers. Finally, a pressure rise at low temperatures slightly reduces the viscosity because the large molecular clusters are squashed; but this reduction in viscosity is only slight as the average distances between the molecules decrease under pressure and consequently, the intermolecular forces acting between them increase. The relationship between viscosity and pressure is thus a more complex one, highly dependent on the temperature and salinity.
6.1 MOLECULAR TRANSPORT OF MASS, HEAT AND MOMENTUM
323
TABLE 6.1.1 The coefficients of dynamic viscosity of seawater (selected from tables in Popov et comprising data by various authors): (a) at atmospheric pressure lo-^ Pa. s]
.*.
(1979)
Salinity S [%,I
Temoerature
, I
a!.
["CI
0
5
10
20
36
0 4
1.79 1.57 1.31 1.11 1.01 0.91 0.80
1.80 1.58 1.32 1.12 1.02 0.93 0.81
1.82 1.59 1.33 1.13 1.03 0.94 0.82
1.84 1.62 1.36 1.16 1.05 0.96 0.83
1.88 1.66 1.40 1.20 1.09 0.99 0.87
10 16 20 24 30
(b) at higher pressures in ocean water of salinity S = 35.29%, (selected from Stanley and Batten) Pressure
Relative increment qP/qarat temperature
p [lo4 Pa]
-0.024"c
10.013"C
1 720 5 170 10 360 12 060
0.983 0.962 0.953 0.953
0.992 0.984 0.987 0.990
20.013"C 0.998 0.998 1.007 1.011
___ * Because of the
discrepancies in the data from various sources, values have been rounded off to two decimal places in table (a) and three decimal places in table 0).
The laws of molecular exchange (6.1.1)-(6.1.3) can be used to derive important equations specifying spatial and temporal changes in the concentration C(x, y , z, t ) , temperature T(x,y , z , t) and flow velocity v(x, y , z , t ) due to molecular exchange. These are the equations of dzfhsion, thermal conductivity and molecular exchange of momentum (viscosity). In seeking these equations, we make use of the laws of conservation of mass, heat and momentum, and of their budget as they flow through a given space, e.g., an elementary cube dxdydz or between planes z and z -t- dz in a water column of unit cross-section. Moreover, we can assume that in seawater, molecular exchange is isotropic, that is to say, the coefficients of this exchange D, y, are identical in whichever direction the process in the medium takes place.
6 THE TRANSFER OF MASS, HEAT AND MOMENTUM
324
The Equation of Dz#iision We shall first find the equation of diffusion in a horizontally homogeneous medium for which we write the laws (6.1.1)-(6.1.3). The concentration gradient of the diffusing substance is then directed vertically downwards, along the z-axis. x ,
1
4 Fig. 6.1.3. Geometrical sketch to the equation of diffusion. q,(z)-flux
density of the diffusing mass of passive substance [kg/m* s].
The mass m,(z) which diffuses across area dxdy (the upper one in Fig. 6.1.3) of the z plane during a time interval from t to t+dt is equal to m l ( 4 = qm(Z)dxdydt, (6.1.4) where, as we recall, qm(z)is the flux density of the diffusing mass on the z plane (see Fig. 6.1.3), so we obtain the flux crossing the area dxdy by multiplying qm by dxdy, and the mass diffusing across in time dt by multiplying the flux by dt. The mass m,(z+dz) which diffuses further downwards across the same area dxdy of plane z dz in the same time interval from t to t dt is
+
+
[
1
mz(z+dz) = qm(Z)+-dZ dxdydt. (6.1.5) aqm az Allowing for the law of conservation of mass of the diffusing substance (conservative by assumption), its budget requires that the difference in mass which has diffused through the upper and then the lower wall of the cube dxdydz, i.e., dm = m, -m, ,be equal to the mass increment dm within this cube in the same time dt. We have assumed the medium to be horizontally homogeneous, so exchange will occur only in the z direction. The equation of this budget in time dt for a volume dzdxdy can be written thus dxdydt = - __ +lm dxdydzdt, az
(6.1.6a)
6.1 MOLECULAR TRANSPORT OF MASS, HEAT A N D MOMENTUM
325
so the mass increment dm in a unit time is then equal to
am - -*dxdydz. ~-
az
at
(6.1.6b)
The mass m [kg] needed in a volume dV = dxdydz [m3] to increase its concentration by AC = C- Co [kg/m3) is equal to
m
=
ACdV,
(6.1.7)
provided that the solution does not become supersaturated and that the diffusing substance is neither evaporated nor precipitated. Combining this last expression with the law of diffusion (6.1.1) describing the flux density qm(z), we can rewrite equation (6.1.6b) as follows:
ac
a
__ = at
az
(Dg),
(6.1.8)
where C is the average concentration of the substance diffusing through an infinitesimally thin layer between planes z and z fd z , i.e., C + C(z). This is the equation of :diffusion in a horizontally homogeneous medium, where aC/& = aC/ay = 0, assuming that there are no sources of the diffusing substance within the space under considerations. In the general case, the medium need not be horizontally homogeneous; then, the concentration C = C(x,y , z) and gradC(x, y , z ) can take any direction n with respect to the coordinate axes. The resultant flux of the diffusing mass will also flow in this direction. The flux density is thus a vector qm(x,y , z ) and, like the concentration C, is time-variable. The flux density vector q,(x, y , z) can therefore be resolved into three components along the x, y and z axes and, in view of equation (6.1.1), can be written in the form
(6.1.9)
ac aZ
qmz= -Dz-.
Here, O x , D,, D, are the coefficients of diffusion in given directions within the medium which does not have to be isotropic. Obtaining the budget equation of the mass diffusing through area A closing the volume of the elementary cube dV = dxdydz now requires the flux density of the mass diffusing through all
6 THE TRANSFER OF MASS, HEAT AND MOMENTUM
326
the walls of this cube to be taken into consideration. The algebraic sum of these fluxes flowing from time t to t+dt is equal to the mass increment dm within the volume of the cube d V in this time. This sum can be expressed as the sum of component flux density increments between opposite walls of the cube dxdy dz multiplied by the area of the walls, e.g., in the x direction,
(-
%ax)
dydz
(see (6.1.6)). By comparing such a sum of component flux increments with the diffusing mass increments within the cube in a unit time, we obtain the following expression for the mass budget in a unit time: (6.1.10)
Now, using the expressions for the mass in (6.1.7) and component flux densities in (6.1.9), we get an equation of diffusion which in anisotropic medium, where D, = D , = D, = D , for any spatial distribution of the concentration of the diffusing substance C(x, y , z), takes the form
ac a ac a ac a ac at = ax ( D a , ) +ay( D a y ) -z(%I-
(6.1.11)
+
The same diffusion equation can be derived from the Gauss-Ostrogradski theorem, taking into consideration the flux density vector of the diffusing mass
4
q&,
s
Y , z)dA = Vq&,
A
Y , z)dV,
(6.1.12)
V
where A is an area of any shape closing volume V, vector dA = ndA, and n is a unit vector (directed outwards) normal to element dA of the area closing the volume V. This equation implies that the total mass flux diffusing in a certain time across the closed area A enclosing the volume V, is equal to the mass increment within volume V during this time. In view of (6.1.7), this mass increment per unit time is equal to CdV. When this is made equal to the right-hand V
side of (6.1.12) (within a minus sign to show that the mass increment corresponds to a loss of this flux at the sides of volume V), we obtain the equation
a at
CdV = V
[ Vq,(x,
y , z)dV.
(6.1.13)
V
Volume V is indefinite, so that (6.1.1 3) is satisfied when the subintegral functions are equal to each other. We can then write: (6.1.14)
6.1 MOLECULAR TRANSPORT OF MASS, HEAT AND MOMENTUM
327
Substitution of the expressions for the components of the flux density vector of the diffusing mass (6.1.9) in (6.1.14) immediately yields the equation of diffusion written in the form (6.1.11) for an isotropic medium. In the special case when the differences in concentration C and temperature T in the volume under consideration are not too great, we can assume that the coefficient of diffusion D (slightly dependent on concentration) is roughly spatially constant, so that D ( x , y , z ) = D = const. From (6.1.11) we thus get Fick's 2nd law in the form (6.1.15)
where A = Vz is here a Laplace operator (dz/ax2+ az/dy2+ a2/az2). In certain cases, diffusion may be stationary. This is so when, even though a mass is diffusing through the volume under scrutiny, its inflow at every point in this volume is equal to its outflow, so that the concentration C(x, y , z, t ) = C(x, y , z ) is constant in time. The derivative X / a t is then equal to 0 and we obtain the Laplace equation for diffusion from (6.1.15) : AC = 0.
(6.1.16)
So far, in our considerations leading to the three-dimensional equation of diffusion (6.1.11) or (6.1.15), we have been tacitly assuming that no externally induced flows of medium have occurred in volume V, i.e., there has been no advection of diffusing mass. When at the start of this section we spoke of layers moving with respect to one another, we were assuming a horizontally homogeneous medium, where the concentration C(x,y , z) = C(z). In such an environment, horizontal laminar flow of the medium at any spot ( x , y , z ) does not alter the concentration, i.e., it does not cause advection of the diffusing substance. In the more general case, however, we must accept the fact that the medium may flow in any direction with a velocity v. In the general case, this velocity has three components v, = u , z,t = v , v, = w , differing from zero in all three dimensions. So, for instance, the velocity along the z-axis is w,that is, the medium moves with this velocity along the vertical, transporting a mass of diffusing substance through the space element dxdydz under consideration. If at the same time, the concentration of this substance C(x, y , z) differs along the vertical, i.e., there exists a component of the gradient X / a z # 0, at certain fixed points in space (the Euler method), the concentration will change with time as a result of flow, irrespective of any local changes due to diffusion. This change in concentration with time resulting from flow in the z direction is proportional to
6 THE TRANSFER OF MASS, HEAT AND MOMENTUM
328
the flow velocity in this direction w = dz/dt and to the magnitude of the gradient aC/az. We can express this as
(aC/at), = (aC/az)(dz/dt) = (aC/az) w . The same conclusion can be drawn from analogous reasoning applied to the flow velocity components and concentration gradients in the x and y directions, 1.e.
ac
dx - -acu ax dt ax
and
ac
(&
ac
dy ay dt
ac V . ay
--
Alternatively, we can say that the concentration is the function C ( x , y , z , t ) and that the coordinates of the element of medium considered in motion are functions of time (the Lagrange method). Then the total change of concentration with time is a Lagrange derivative d c - ac + at dt
ac~ ax
dx dt
ac-
dy+ ay dt
ac ~ az
dz ~ dt
+ ac ~ ac~ at ax
=ac ~ ay
ac+ az
~ *
Both these methods of reasoning lead to the conclusion that the total change in concentration dC/dt is the sum of local changes aC/at due to diffusion, described by equation (6.1.15), and the change of concentration with time as a result of the inflow (advection) of the investigated substance. Therefore, on the left side of (6.1.15) we have to add terms describing the advection of the diffusing substance
At the same time, we have added the function Ms to the right side which allows for possible sources of diffusing substance within the investigated space itself. The source density function M,(x, y , z) expresses the mass of diffusing substance which is created (or lost) in a unit volume of the medium (e.g. by formation or decomposition during a chemical reaction) around a point (x, y , z ) in unit time (units kg m-3 s-l). The same equation of diffusion with advection, written in brief with the aid of operators is
ac +vVC = DAC+M,. at
__
(6.1.17b)
Here we have also assumed that the coefficient of diffusion is constant in space D ( x , y, z) = D = const, which in practice is valid only for gentle gradients of concentration C, temperature T and pressure p , since this coefficient is in
~
6.1 MOLECULAR TRANSPORT OF MASS, HEAT A N D MOMENTUM
329
general the function D(C, T , p ) . We must also assume that the liquid under study is imcompressible, so that the concentration C does not change as a result of the medium being compressed by external forces. The reader will have noticed that since the coefficients of diffusion of various substances in water are small, the flow described by the advection component vVC in the diffusion equation (6.1.17) can often be the main reason for changes in the concentration C of an investigated substance in a given place in the water. A special case of molecular exchange of mass is osmosis. This is the diffusion of a substance through a porous barrier, selectively allowing the molecules of various constituents of a solution to pass through. Such permeable barriers include the cellular membranes of marine phytoplankton or parts of the skin of animals living in the sea. Water readily passes through such membranes in either direction, but the diffusion of ions is limited and selective. As water molecules diffuse into a cell, the excess of salt ions remaining outside gives rise to an additional external pressure called the osmotic pressure that “bombards” it during the thermal motion of salt ions. Osmosis therefore has a tremendous influence on the living conditions of marine organisms and their adaptation to life in sea water of a given salinity. The Thermal Conductivity Equation
We shall now briefly discuss the thermal conductivity equation, whose derivation and form are analogous to those of the equation of diffusion. When dealing with the conductance (diffusion) of heat, we replace the diffusing mass m in the equation of diffusion by the quantity of heat Q [J], and the concentration of diffusing substance C by the temperature of the medium T. We make use of the approximately linear laws of thermal conductivity (6.1.12) resulting from temperature differences and apply them to the three components of the flux density vector of the conducted heat qQ
aT
q Q y = -yp.-,
(6.1.18)
3Y
qQz = - ,7-
aT
We have assumed that the coefficient of molecular thermal conductivity is independent of direction in space, i.e., that the medium is isotropic. Bearing in mind
6 THE TRANSFER OF MASS, HEAT AND MOMENTUM
330
our elementary cube dxdydz containing a motionless medium (see Fig. 6.1.3), we can calculate the heat increment within it in a unit time aQ/& resulting from the various amounts of heat entering and leaving it as a result of heat diffusion (conduction through its walls). So, for example, in the x direction, the difference in the heat flux entering through the left wall and leaving through the right wall in a unit time is qQx(x)+ axdx
1)
dydz
=
-2Edxdydz.
ax
(6.1.19)
The sum of such differences in the heat flowing through all three pairs of walls must be equal to the heat increment in a unit time within the cube, i.e., (6.1.20) We now ecxlude the occurrence of phase changes-evaporation or freezing, or structural changes in the volume of medium considered. Then the amount of heat Q needed to raise the temperature of an element of medium dV = dxdydz of mass m by A T = T- To at constant pressure and salinity is equal to (6.1.21) Q = C,mAT = pC,ATdV, where C, 2 C,, is the specific heat at constant pressure and constant salinity (see (3.0.31)), and p is the density of the medium. Applying this last expression together with the law of thermal conductivity (6.1.18) to each of the components of the heat flux density vector qQin (6.1.20) yields the thermal conductivity equation in an isotropic medium having any spatial temperature distribution T(x, y , 2). (6.1.22) The same equation is easily derived from the Gauss-Ostrogradski theorem applied to the flux density vector of the conducted heat (#
A
qQ(x, Y , z)dA
vqQ(x, Y , z)dV,
=
(6.1.23)
V
where dA = ndA, and n is a unit vector (directed outwards) normal to the element of area dA of any shape and enclosing the volume V. This equation means that the total heat flux passing in a given time through a closed area A limiting volume V is equal to the heat increment within this volume in the same time. In the light of (6.1.21), this heat increment in volume V in unit time is, equal
6.1 MOLECULAR TRANSPORT OF MASS, HEAT A N D MOMENTUM
331
to aJat$ QC, TdV. We can therefore equate this heat increment with the right V
side of (6.1.23) (with a minus sign to denote loss of flux), which yields
[
S
(6.1.24) eC,TdV = - VqQ(X, y , Z)dV. V at v Since the volume V can take any value, this equation holds, as long as the subintegral functions are equal to each other. Hence we get (6.1.25)
If we substitute in this equation the expressions for the components of the flux density vector of the conducted heat (6. I. 18), we immediately get the thermal conductivity equation (6.1.22). Here again, in particular cases in seawater, where temperature, pressure and salinity differences are spatially insignificant, we can take the thermal conductivity coefficient to be approximately constant y ( x , y , z ) = y = const, since, as we mentioned earlier, it is only weakly dependent on these variables. Equation (6.1.22) then becomes (6.1.26)
Here we are also assuming that the medium is motionless and that there are no heat sources or sinks, i.e., no heat is converted into other kinds of energy, and none is developed within the volume considered, for instance, as a result of insolation or friction. Since the density of the medium e and the specific heat C, of seawater change only slightly, we often assume eC, = const. Equation (6.1.26) then becomes (6.1.27)
where y/eC, is sometimes denoted by a single symbol, e.g., yT, and is known as the coeficient of temperature conductivity. In a special case, thermal conductivity may be a motionless process, i.e., one in which the quantities of heat entering and leaving any given volume element are identical, despite the existence of a heat flux. Then the temperature T ( x ,y ,z ,t ) = T ( x , y , z) is constant in time, hence the derivative aT/at = 0 and the thermal conductivity equation (6.1.26) take the form of a Laplace equation A T = 0.
(6.1.28)
6 THE TRANSFER OF MASS, HEAT AND MOMENTUM
332
If in (6.1.26) we allow for the fact that the medium flows with a velocity Y, that is, for a temporal change of temperature resulting from heat advection, we arrive (in the same way as for diffusion) at a general thermal conductivity equation; or to put it another way, at an equation for the heat budget for a unit volume of medium in a unit time.
____ a(ecpT) +vV(&,T) = yV2T+QSsat
(6.1.29)
Here we have added the advection component vV(eC,T) and also the source density function QsS(x,y , z ) expressing the amount of heat released from sources in a unit volume per unit time [J/m3s]. In the sea, this source of heat is the solar energy absorbed in a unit volume of the medium in a quantity given by the m
equation
Qss
z
s n(il)E,(A)dl and this is usually nearly all converted into heat
0
(a(il) is the coefficient of light absorption, Eo(A) is the scalar irradiance; see (5.4.22)).
The Navier-Stokes Equation of Motion We still have to discuss the equation of molecular momentum exchange, which gives rise to friction between volume elements of a liquid that are moving with respect to one another. This friction, known as viscosity or internal friction comes into play while the medium is in motion. So the end result of the discussion that follows ought to be the equation of motion of a viscous liquid, generally known as the Navier-Stokes equation. The same reasoning that provided as with the equations of mass diffusion and of thermal conductivity will also lead us to the equation of the molecular exchange of momentum. This reasoning is, however, more complicated than before, because transferred momentum is a vector whereas a diffusing mass or heat are scalars. Consider a space containing a flowing liquid in which the flow velocity field is v(x, y , z , t ) . As the mass elements of this liquid do not all move at the same velocity, friction arises between them, that is, flow momentum is transferred. At each wall of an elementary cube dxdydz isolated in space, the velocity v generally has three components u, v,w, so the flow momentum of molecules of mass m itransferred perpendicular to the velocity vector v as a result of friction has three components miu+miv+m,w at each wall. The decrease (or increase) in the flux density of the total momentum between any pair of opposite walls is now the sum of the decreases of its three components in the directions of the
6.1 MOLECULAR TRANSPORT OF MASS, HEAT AND MOMENTUM
333
Fig. 6.1.4. Geometrical sketch explaining the equation of molecular viscosity and the tensor of stresses. u, v, w-flow
components
z] [z]
velocity components [mls] along the x, y and z axes respectively, qu, qv. qk-momentum flux density
[
transferred along the z-axis as a result of friction arising from the motion of the medium
~
with a velocity having the components density components
3
-
u, v, w respectively, qxr. qpr.etc.-set
of values of all the momentum flux
N/m21 _= pal, known as molecular stresses and making up the tensor of mol-
ecular stresses (see equation 6.1.36).
velocities u, v, w (Fig. 6.1.4). So, for example, the decrease along the z-axis of the total momentum flux between walls z and z+dz has there component decreases of the momentum flux in each of the three directions of velocities u, v, w.
(6.1.30)
334
6 THE TRANSFER OF MASS, HEAT AND MOMENTUM
Likewise, between walls x and x+dx, i.e., along the x-axis, the decreases in the fluxes of the momentum along the three coordinate axes will be
(6.1.31)
and between walls y and y
+dy, these decreases are
(6.1.32)
In view of the principle of conservation of momentum, we can now assume that the sum of the decreases of the component momentum fluxes in the x direction between all three pairs of walls of the cube dxdydz is equal to the increment along the x-axis of the momentum of the mass of medium within the cube in unit time. We express this mass as the product of its density and its volume, i.e., m = edxdydz. So we can write down the momentum increment of the mass of the cube in the x direction in unit time as [a(~u)/at]dxdydzand equate it with the sum of the momentum flux increments in the x direction between all three pairs of walls given in the first lines of expressions (6.1.30), (6.1.31) and (6.1.32).
(6.1.33) Likewise the increment per unit time in the momentum component of the mass of cube dxdydz along the y-axis, due to the molecular exchange of momentum, is equal to
(6.1.34) and the momentum increment along the z direction is equal to
(6.1.35)
6.1 MOLECULAR TRANSPORT OF MASS, HEAT AND MOMENTUM
335
We recall from our interpretation of law (6.1.3) that the flux density of momentum during its molecular exchange is a measure of the tangential stress at the surfaces of the adjacent volume elements of liquid given in units of force per unit area. The set of all the stress components q acting on the walls of our elementary cube dxdydz can be seen in equations (6.1.33)-(6.1.35). This set of stress components can be written as a table representing the tensor of molecular stresses qij. Double subscripts are used to distinguish them from one another: one subscript i equal to x, y or z for stresses due to momentum components of velocity u, v or w respectively, and a second subscript j equal to x,y or z indicating the direction in which the stress is transferred (direction of momentum exchange). Thus qu in the components of (6.1.33) is respectively denoted by q x y , qxy and qx2, qv in the components of equation (6.1.34) by qYx,qYYand qyz, and qw in the components of equation (6.1.35) by qzx, qzY and qz2. From this (and 6.1.3) we obtain a set of expressions representing the tensor of molecular stresses
(6.1.36)
At the same time, the subscripts in these expressions indicate the direction in which the medium is distorted by particular stress components. We can easily imagine such distortions if we take our cube dxdydz (see Fig. 6.1.4): its successive layers can be sheared off in the component directions of the momentum, i.e., in planes parallel to the walls, and compressed in the component directions of the momentum perpendicular to the walls. Thus, for example, qxx indicates the compressive or stretching stress due to the momentum component in the x direction acting perpendicularly on the wall lying in the x plane, i.e., it is the dynamic pressure acting on this wall as a result of flow. On the other hand, qxY is the shearing stress due to the momentum component in the x direction acting in this direction on the wall lying in the y plane; in other words, it is the frictional force acting in the x direction on unit area in th e y plane. The action of the other stress components can be deduced in a similar manner. Furthermore, it is not at all difficult to see that if an element of liquid-the cube dxdydz-does not rotate, the appropriate moments of the forces acting on walls lying perpendicular to each other must be equal to each other, hence qxY= qYx,qxz = qzx and qYz = qZy.Returning once more to our concept of the flux of molecular momentum exchange during friction, we can now apply the linear approximate law governing the exchange (6.1.3) to each of the three flux components qu, q v , qw in
6 THE TRANSFER OF MASS, HEAT AND MOMENTUM
336
equations (6.1.33)-(6.1.35). In this way we get the three components of the equation for the molecular exchange of momentum
(6.1.37)
We have assumed here, of course, that the medium is isotropic, i.e., that the coefficient of dynamic viscosityq does not depend on the direction of the molecular exchange of momentum. If we further assume that the coefficient q(x, y , z) = q = const, i.e., that it is practically independent of the diversity of the medium in the space in question, we can remove it from in front of the differential signs in equations (6.1.37). Then the set of equations (6.1.37), written briefly in vector form with the aid of a Laplace operator A, reduce to (6.1.38) This vector equation expresses Newton’s 2nd law of motion as applied to the inertial motion of a liquid subject only to internal frictional forces, i.e., viscosity. Thus the change of momentum ev of unit volume of this liquid with time, i.e., a(p)/at, is equal to the frictional force qAv = fq acting on this unit volume in the direction opposite to that of the motion. We have now obtained an expression for the volume density of the viscous force measured in [N/m3] which has to be taken into consideration in the equation of liquid motion. But (6.1.38) does not account for the total change of momentum with time d(pv)/dt at point (x, y , 2); this also includes the inflow to that point (x, y, z) of a medium having a differentiated momentum in space, i.e., “the momentum advection” caused by forces Fi other than just the friction. We must therefore add to (6.1.38) an advection term (vV) (ev) describing the change in momentum with time at any point in space (x, y, z) as a result of the inflow to that point of a liquid with velocity v and having a different momentum. So if the derivative a(ev)/at determines the local change in momentum due to viscosity, the total change of this momentum will be
_ dm_ -~ - a(ev) dt
at
+ (VV)(@V)
(6.1.39)
6.1 MOLECULAR TRANSPORT OF MASS, HEAT AND MOMENTUM
337
(see (6.1.17b) and (6.1.29) for diffusion and thermal conductivity). Hence the full equation for the molecular exchange of momentum (viscosity), based on (6.1.38) and (6.1.39), takes the form
a(ev)
.~
at
+ (vV)(p)
=
~ A v +Fi,
(6.1.40a)
where Fi-forces other than viscosity (explained below). Expanding this equation to three equations for the component momentum increments (in a unit time per a unit volume) gives
At the end of our discussions on the diffusion and thermal conductivity equations we also allowed for internal sources of diffusing substance or heat. The sources of momentum within a medium are forces Fi other than friction (acting on a unit volume) which give rise to momentum changes in time within the investigated volume that we have called “advection”. Such “volume sources” are usually volume forces due to spatial differences in the pressure p ( x , y , z), volume forces due to gravity, the Coriolis inertia force, etc. If we take into account, for instance, only the pressure gradients (on the walls of element dV) and the gravitational forces acting on every element of the medium dV, we can replace the “source function” Fi in (6.1.40) by the forces mentioned which take the forms -Vp [N/m3] and pg [N/m3]. Thus view of (6.1.40) and the simplifying assumption that e = const, we obtain (6.1.41) Thisis known as the Navier-Stokes equation of motion of a viscous (incompressible) liquid; it disregards the Coriolis force, possibly other forces acting in special cases, and also the compressibility of the medium. The role of viscous forces in this equation is evident: it causes kinetic energy to be dissipated, i.e., scattered and converted into heat (see Chapter 1). The transfer of momentum associated with the viscosity of seawater takes place only during the movement of water masses with respect to one another.
6 THE TRANSFER OF MASS, HEAT A N D MOMENTUM
338
This is why our analysis of the viscosity of water led us to the Navier-Stokes equation of motion of a viscous liquid (6.1.41). This equation is traditionally derived from Newton's 2nd law, which states that the change in momentum in a time dt of an element of mass d(mv)/dt is equal to the sum of the forces acting on that element. The Euler method can be implemented to write this for every point in space (x, y , z) with reference to the mass of medium per unit volume dxdydz surrounding that point dv @+= -Vp+eg+f '7' (6.1.42) at
in which the sum of forces acting includes the pressure gradient -Vp, the force of gravity eg, and the previously found viscosity force of the medium f, = +qAv acting on a unit volume of the medium. We have ignored the Coriolis inertial force and other possible forces as they are irrelevant to this argument. The total rate of change of momentum edv/dt at a point in space (x, y , z ) comprises a local change in momentum due to molecular friction @v/& and a change in momentum due to advection, i.e., due to the inflow of medium having a different momentum. This total change is given by (6.1.39) which, when equated with (6.1.42), leads directly to the Navier-Stokes equation (6.1.41) (where e = const). The Navier-Stokes equation contains at least 5 unknown variables (u,Y, w,p , e), but consists only of three scalar equations of the type (6.1.40b). So in order to solve it we need at least two more equations. One which is often used is the equation of state of the medium (see Chapter 2), but as the new variables of temperature T and salinity S appear in it, we still need the thermal conductivity and diffusion equations. The second equation required in order to close the set of equations so as to be able to solve them is usually the continuity equation of the medium. This latter is derived directly from the mass budget of the medium flowing through a volume element dxdydz in a time from t to t+dt. For such a budget we denote the flux density vector of the mass of medium [kg/m2s] flowing through a closed area A by (6.1.43) q,(x, Y, 2 ) = e(x7 Y , z)v(x, Y, z). In view of the law of conservation of mass, we can apply the Gauss-Ostrogradski theorem to this vector:
I%dA A
= SVq,dV,
(6.1.44)
V
where A is an area enclosing the volume Y ; dA = ndA is an element of this area multiplied by the unit-normal vector directed to the exterior. As before, the
6.1 MOLECULAR TRANSPORT OF MASS, HEAT AND MOMENTUM
339
right side of this equation describes the mass increment SedV within volume V
Vin a given time, e.g., from t to t+dt. So, per unit time, this increment is equal to
(6.1.45)
After substituting (6.1.43) in (6.1.45) and rearranging, we get
1
[g+V(p)]dV
=
0.
(6.1.46)
V
The function under the integral must therefore be equal to zero which, as written below, is the continuity equation of the medium we have been seeking
ae -+V(p) at
=
0.
(6.1.47)
Equations of motion and flows in the sea are discussed in detail in textbooks and monographs on hydrodynamics (e.g. Defant, 1961 ; Druet and Kowalik, 1970; Monin, 1978; Druet, 1978). Here we have restricted ourselves to the most essential aspects of the description of flows which we use in our discussion of the molecular and turbulent exchange of mass, heat and momentum in the sea. In order to perform various mathematical operations and seek solutions, the equation of motion (6.1.41) written in vector form must be expanded into the three following component equations :
(6.1.48)
The right side of the more general form of these equations includes an additional term taking into consideration the compressibility of the medium and the volume viscosity on compression, which are essential for the propagation of acoustic waves in the sea (see Section 8.2). Scalar versions of the vector continuity equation (6.1.47) can also be derived (6.1.49)
6 THE TRANSFER OF MASS, HEAT A N D MOMENTUM
340
So as to simplify these equations, we use a system of generalized notation, i.e., coordinates xi = (x, y , z), where i = 1, 2, 3, velocity components ui= (u, v , w ) where i = 1, 2, 3, etc. With such symbols, all three scalar versions of the NavierStokes equation of motion for an incompressible liquid can be written briefly as
(6.1.50) where fi generally means the components of the external volume forces acting on the liquid, e.g., the force of gravity (acting only in the z direction). The significance of the indices i or j will emerge from a comparison of expressions (6.1.50) and (6.1.48). So indices (i or j or any others expressed by Latin minuscules) occurring once in a given term in the equation are fixed and take the values 1 , or 2, or 3. But indices occurring twice in a term conventionally indicate the summation of this expression with respect to these indices which here take au . axj
the consecutive values of 1, 2, 3. The expression u j - - I conventionally implies
2 2. uj
j=l 3
aui
.
where i is fixed; likewise, the conventional expression __ will mean axi
The explanation of this and other more complicated cases of such
i=l
a conventional notation associated with tensor calculus can be found in Synge and Schild (1964) or Korn and Korn (1970), and with direct reference to hydromechanics, in Monin and Yaglom (1965). In the generalized notation, the continuity equation (6.1.49) now reads
(6.1.51) We can write the diffusion equation (6.1.17) in the same way
(6.1.52) and likewise the thermal conductivity equation (6.1.29)
(6.1.53) These equations as written in this form (6.1SO)-(6. 1.53) are often used in oceanology (Monin, 1978). At the end of this section, it behoves us to draw attention to the thermody-
6.2 THE TURBULENT EXCHANGE OF MASS, HEAT AND MOMENTUM
341
namic nature of the exchanges of heat, mass and momentum and their direct connection with the change in internal energy (equations (3.0.7), (3.0.26)) or entropy (equations (3.0.8), (3.0.25)) of the elements of the medium as thermodynamic systems (see e.g., Doronin, 1978). The exchange (equalization) of mass, heat or momentum can take place across the surface of any selected volume element. This indicates that the element in question is not in thermodynamic equilibrium with its surroundings. This deviation is due to the differences in concentration of a given component of the solution C, temperature T, pressure p , etc. Such a deviation from thermodynamic equilibrium can be characterised by the rate of changes in the specific entropy of the system d8/dt, which we can define from the thermodynamic equations given in Chapter 3. The variables of the state of the medium (e.g., temperature, salinity) occurring in these equations are, on the other hand, described by the equations of molecular exchange given above.
6.2 THE TURBULENT EXCHANGE OF MASS, HEAT AND MOMENTUM IN THE SEA
The processes of molecular exchange of mass, heat and momentum are of major importance in the numerous cases of this exchange in the sea. In particular, the molecular exchange of momentum, i.e., viscosity in the sea, regardless of the degree of turbulence and the complexity of the flows within the medium, is always the final stage in the dissipation of flow energy and its conversion into the random thermal motion of molecules. But the molecular thermal conductivity and diffusion of masses of passive substances are the last stages in the thermal and chemical equalization of unhomogeneities in the medium, which restore thermodynamic equilibrium to the system (this is the tendency for a system to achieve minimum entropy). The viscosity and thermal conductivity of seawater also play a significant part is attenuating acoustic waves in the sea. The surface skin layer of the ocean is of especial importance as regards molecular exchange (Bortkovskii et al., 1974)-it is here that the fluxes of heat and masses of substances exchanged between air and sea are determined by molecular exchanges when winds are slight. Molecular exchange also has a significant part to play in the vertical flow of mass and heat in a strongly stratified medium (in which density increases rapidly with depth), where the high degree of hydrostatic stability effectively counteracts convection and, as we shall see later, damps turbulent exchange. Finally, molecular exchange is often of importance in the ocean abyss in areas of stagnant water, and especially at the interfaces between
342
6 THE TRANSFER OF MASS, HEAT AND MOMENTUM
layers of water of different densities, where there is insufficient energy to induce and maintain the turbulent flows described below. When transferring mass, heat and momentum over distances incomparably larger than the mean free path of molecules, the dominant role is taken over by flows of the medium and their associated advection of mass, heat and momentum. Of varying dimensions and caused by diverse time-variable forces, these flows usually give rise to complex, closed cycles which on a large scale are referred to as circulations and on a small scale as eddies. The continuity law of the aquatic medium and the condition that the water mass tend towards hydrostatic equilibrium, i.e., attain minimum potential energy in the gravitational field, requires a closed-cycle flow in a basin. The scale of circulations or eddies in the ocean depends on the power of their energy sources and can range from tiny eddies to intercontinetal ocean currents. The actual dimensions of eddies in the ocean are of the order of from 0.01 to lo6 m, and because of their irregular, random nature, they go by the name of turbulent motion. In view of such a wide range of dimensions of flows within the same environment, main. tained by :energy from many different time-variable sources, the considerable variability and randomness of the flow velocity vector v(x, y , z, t ) in time and space is hardly surprising. At any point within the water we can distinguish certain flow components whose overall velocity make up the resultant vector of the instantaneous velocity v(x, y , z , t ) . The large-scale flow is particularly conspicuous and because of its huge size and great inertia, varies very slowly in time. The velocity of this flow over not too great a time interval can be locally regarded as constant and can be described by the usual equation of motion, At the same time, however, local changes in the flow velocity as a result of local kinetic energy supply or dissipation cause this velocity to fluctuate. These fluctuations can be separated into wave motion, i.e., more or less regular oscillations, periodic in simpler cases, and extremely irregular eddies which can only be desscribed by statistical methods. The randomness of turbulent motion recalls the random thermal motion of molecules. In the turbulent motion of mass, however, the transported mass, its momentum and the heat it contains, are incomparably greater than that carried by the single molecules involved in molecular exchange. Moreover, the mixing distance in a single vortex is incomparably greater than the free path between colliding molecules in molecular motion. The velocity vector of an eddy usually has a large component perpendicular to the principal flow, and so mass and the energy of the elements of the medium are transported a considerable distance perpendicular to the direction of flow. This is why turbulence in the ocean is
6.2 THE TURBULENT EXCHANGE OF MASS, HEAT AND MOMENTUM
343
of such great importance. In this process, energy and mass are transferred in a cascade-like manner from larger eddies to the ever decreasing ones generated by them. In the end, these lose all their mechanical energy, converted into heat as a result of molecular friction (Fig. 6.2.1). In spite of the different mechanisms and the quantitative differences involved, there are none the less many analogies between the turbulent and molecular transfer of mass, heat and momentum
6 Time t
Fig. 6.2.1. The turbulent cascade exchange of momentum which leads to the dissipatioli of the principal flow energy. (a) Lines parallel to the flow velocity (stream lines); 01)the dependence of the velocity component (e.g., wX = u ) on time at a selected point in the water during a statistically stationary flow (Z = const).
6 THE TRANSFER OF MASS, HEAT A N D MOMENTUM
344
in a medium. Thus in the early days of research into turbulence, Boussinesq (1 877) assumed that an analogy existed between the molecular and turbulent exchange of momentum and consequently introduced the concept of turbulent stresses and a coefficient of turbulent viscosity in a horizontally homogeneous turbulent flow. Schmidt (1925) extended these concepts to include the exchange of heat and mass. Prandtl (1925) coined the concept of average mixing length I' in turbulent motion by analogy to the mean free path of molecules 6 in molecular motion. The fundamental condition enabling turbulence, and thus the turbulent exchange of mass, heat and momentum to be maintained, is a supply of energy from an external source compensating eddy energy lost in countering frictional and buoyancy forces, and finally dissipated as heat. A local, volume source of eddy energy is usually the energy of a basic (larger scale) flow which, as a result of an instantaneous imbalance of the moments of the forces acting on the elements of the medium, generates smaller eddies. Thus the laminar motion of the medium becomes turbulent. Karman (1930) worked out a theory of turbulence based on the assumption of kinematic similarity of eddies, i.e., velocity pulsations at any point of a horizontally homogeneous flow which differ only in their dimensions and velocity. This has made it possible to simplify the statistical description of turbulence with the aid of the functions of dimensionless variables, identical for all the points in the space where turbulence exists. Ertel (1937) extended the Boussinesq and Prandtl theories to include three-dimensional flow with a non-isotropic exchange of momentum, which he called the tensor of turbulent stresses.
Conditions for Turbulent Motion
It was Reynolds (1894) who defined the condition under which laminar flow becomes turbulent. He demonstrated that laminar flow destablises and changes into turbulent flow when the inertial forces of the elements of a liquid exceed the viscous forces acting on those elements during motion. The ratio of these forces is expressed by a dimensionless parameter, known nowadays as the Reynolds number Re
=
V L -,
(6.2.1)
rlu
where v is the velocity of the principal flow, L its characteristic scale (e.g., the
6.2 THE TURBULENT EXCHANGE OF MASS. HEAT AND MOMENTUM
345
diameter of the tube through which the liquid is flowing), 9)" is the coefficient of kinematic molecular viscosity 7" = T/Q, where 7 is the coefficient of dynamic viscosity, and e is the density of the medium. The critical value of this number, Re = Re,, above which laminar flow becomes turbulent is thus easily established when the characteristic scale of turbulence L is known. So, for instance, it has been found experimentally that for a flow of water in a pipe of diameter L with reasonably smooth walls, motion will be laminar so long as the Reynolds number Re does not exceed c. 2000. But, depending on the smoothness of the inner walls of the pipe, the rate of change of the forces inducing flow, and other factors, this critical value may reach 40000. In the ocean it is hard to fix the characteristic scale of the basic flow, seeing that the ocean is so large and that the movement of water masses is so complex. If we took L for the ocean to be its depth or width, the Reynolds number would always exceed the critical value, even with weak flows, of the order of 1 cm/s, and so unceasing turbulent motion would occur everywhere in the ocean. Turbulent motion in the ocean is indeed a frequent occurrence, but mostly after faster flows have been locally induced by strong winds whipping up the sea surface, by the wave motion of the surface-water layers, by the motion of internal waves occurring near the boundary of water layers of different densities, and other locally-induced strong flows. At great depths, and also over large areas of the upper ocean layers in calm weather, flows are often approximately laminar. Evidence for this is the rate with which chemical or radioactive indicators diffuse through the water (Kagan and Ryabchenko, 1978) or the maintenance of the thermal and density microstructure of the water masses (Gregg, 1975). So we cannot take the size of the whole basin to be the characteristic scale L of the turbulence, only the size of the flow in the area where the energy of this flow is sufficient to induce turbulence (Ozmidov, 1966). In practice, this is the width of the zone upon which a given wind is acting, the size of the area disturbed by internal wave-action (wave length), etc. Notice that the original source of energy initiating all these movements is the radiant energy of the Sun which heats the atmosphere and ocean unevenly. A separate source of principal flows in the ocean is the work done by the gravitational forces of the Moon and Sun that give rise to the tides which in straits and over shallows are such a significant source of turbulent flow energy (see Chapter 1). On the basis of the energy budget of eddies, it is possible to evaluate the local conditions under which turbulence is maintained in the sea (Doronin, 1978). Let us assume that in a flow vortices arise having a characteristic scale I, velocity ol and duration t. The kinetic energy (mvf/2)consumed for this per unit mass of medium m and in unit time t I/vi is thus proportional to v;/l, or
-
6 THE TRANSFER OF MASS, HEAT AND MOMENTUM
346
v: & N -
(6.2.2)
I'
This eddy energy is used up in doing work E,, against internal frictional forces and work ee against buoyancy forces-per unit mass of medium and in unit time, respectively. Eddy turbulence is thus not damped only when the energy E taken from outside in a unit time to set eddies in motion is greater than the energy eq+eP lost in unit time. So the condition under which turbulence is maintained is that the following inequality be satisfied & > &,,+EQ. (6.2.3) The eddy energy consumed in unit time to counteract internal friction is (6.2.4)
This proportionality can be inferred from dimension analysis (see Cole, 1964). Energy is used up in doing work against buoyancy forces ce when a permanent vertical equilibrium exists in the fluid, i.e., when its density rises with depth and the buoyancy and the gravitational force acting on each element of the liquid are balanced. The vertical transport of liquid elements associated with eddies requires work to be done against the excess of buoyancy over gravity if the element is moving downwards, or against the excess of gravity over buoyancy if movement is upwards. This work is proportional to the potential energy increment of the liquid element shifted from its equilibrium position at a depth where the density (the potential with respect to the new environment) is el:to a depth where the density of the surrounding water is e. It is thus proportional to the buoyancy increment per unit mass
-
L* 1 ____ @* - @ g
and to the magnitude of the
vertical displacement AZ I. Hence the energy of unit mass of an eddy consumed in a unit time t llv, against the buoyancy is given by the proportionality N
(6.2.5) where g is the acceleration due to gravity. We shall now examine the turbulence condition (6.2.3) in two extreme cases: (1) when c,, 9 E, and (2) when E,, < E ~ The . first case arises in a practically uniform medium, when ex-@ -+ 0; then the work done by the eddy in overcoming the buoyancy can be disregarded in comparison with the work done in overcoming internal friction. The turbulence condition (6.2.3) then simplifies to the inequality & > Etl (6.2.6)
6.2 THE TURBULENT EXCHANGE OF MASS, HEAT AND MOMENTUM
347
which in view of (6.2.2) and (6.2.4) can be written (6.2.7a) hence (6.2.7b) where Rel is the local Reynolds number, because it includes the characteristic scale of local eddies I and their flow velocity vl. Where marked density stratification occurs (particularly along steep density gradients at water layer boundaries), e.g., in the thermocline, the stability of the elements of a liquid in equilibrium is high and moving them vertically requires a considerable amount of work to be expended against buoyancy forces. We can then assume that E, % e V , so the turbulence condition (6.2.3) simplifies to E
>
EQ
(6.2.8)
which, in the light of (6.2.2) and (6.2.5), can be written (6.2.9a) Hence (6.2.9b) where Ri, is the local Richardson number. The generally known form of the (dimensionless) Richardson number for horizontal laminar flow with a velocity gradient duldz and potential density gradient de,/dz is
(6.2.10)
It expresses the ratio of the buoyancy gradient to the inertia gradient and, in the sea, is a measure of the flow stability of a horizontally homogeneous medium. At the same time, as can be deduced from (6.2.8), it is the ratio of the kinetic energy of turbulent eddies lost in doing work against the buoyancy to the energy drawn from outside to produce turbulence. It is clear from (6.2.10) that an appropriately small Richardson number, critical as regards the inititation of turbulence in the sea, can exist both as a result
6 THE TRANSFER OF MASS, HEAT AND MOMENTUM
348
of a fall in the vertical density gradient (e.g., when the sea surface cools), and when the vertical flow gradient rises (e.g., when the wind gets stronger and there is increasing friction between it and the surface). The overall condition whereby turbulence is maintained is, as we can see, a suitably small value of the Richardson number and a suitably high Reynolds number, which physically implies that the inertial forces in a flow prevail over the buoyancy and viscosity forces. A special case is the not infrequent situation where the vertical distribution of the water mass is hydrostatically unstable. This means that along certain depth intervals the density of the water decreases with depth. In this case, the work done by buoyancy forces is added to the energy of the turbulent vortices. The turbulence condition for eddies having a characteristic scale I then takes the form of an inequality E+Ee
> ev
(6.2.1 1)
which, in view of (6.2.2), (6.2.4) and (6.2.5), becomes
(6.2.12) The theory of turbulence in the sea, although still based on hypotheses and empirical equations, has already had a number of monographs devoted to it (Monin and Yaglom, 1965, 1967; Ozmidov, 1968; Phillips, 1969; Monin, 1973, 1978; and others). The equations of turbulent exchange of mass, heat and momentum in the sea and in the adjacent layer of the atmosphere are the principal elements in the analysis and modelling of hydrophysical fields in the ocean (Kraus 1977; Bortkowski et al., 1974; Zilitinkevich et al., 1978, 1979). This is the reason why we present here only these basic equations describing turbulent exchange in the sea, without delving into the hydrodynamic aspects of turbulence which are the subject of discussion in many textbooks and monographs on hydrodynamics (Monin and Ozmidov, 1981; Monin, 1978; Defant, 1961). Average and Fluctuating Component of Velocity As we have already seen, the separation of large circulations from turbulent eddies in the ocean is a matter of agreement, as it depends on the scale according to which we average out the flow velocity in space and time. If the real instantaneous velocity of the medium at a particular spot is v, we can assume that it is the sum of the average velocity V, i.e., the velocity of the basic flow, and of
6.2 THE TURBULENT EXCHANGE OF MASS, HEAT AND MOMENTUM
random fluctuations of the velocity vector v’ and reflect turbulent motion
v
349
which satisfy statistical laws
= v-7
(6.2.13)
= VfV’.
The average value in a time t, of this vector of the instantaneous velocity is by definition equal to 1.
v
=
1
vdt,
-
(6.2.14)
ta 0
where t, is the time interval over which averaging is performed. There is moreover an averaging condition which states that a second averaging of this vector (or any other function) must not change the result of the first averaging. This condition can be expressed as - _ _ j = v-v’ =
(6.2.15)
v- ,
so the average value of the velocity fluctuation interval fa must be equal to zero v
- 1
=-s
V’
during the averaging time
1.
1
fdt
=
0.
(6.2.16)
ta 0
Obviously then, the fulfilment of this condition depends on the averaging time, which must be long enough for the “number” of random velocity fluctuations to obey statistical laws but sufficiently short for the average velocity of the primary flow to remain unchanged during it. Strictly speaking, f should equal the boundary value of (6.2.14) when to + cx), but then theoretically, over an infinitely long period of time, the average flow velocity must be constant, so turbulent motion must be stationary. In practice in the sea, this is usually quasi-stationary motion, i.e., motion fixed in a certain finite averaging time, though generally not fixed over a much longer period of time (Fig. 6.2.2). Separating the components of the average 1 and fluctuating part v‘ of the velocity therefore depends on the averaging scale applied, and hence on the scale of eddies which in the sea we regard as turbulence. The choice of averaging time is made easier by the empirically investigated scale of oceanic circulations, and the occurrence on this scale of three evident scales in which one observes in the ocean energy influx maxima (mainly from the atmosphere) maintaining the turbulence of such scales (Ozmidov, 1966, 1968) (see Fig. 6.2.3). The turbulent motion of ocean waters thus falls into three distinct categories : (1) large-scale (global) circulations having
6 THE TRANSFER OF MASS, HEAT AND MOMENTUM
350
5
At
Time t
At
Time t
-
Fig. 6.2.2. The dependence of flow velocity on time u(t) in stationary turbulent flow (a) and non-stationary turbulent flow (assumed quasi-stationary, i.e., almost stationary during the averaging time t,, = A t ) (b). 7
in
m
E
I
v)
Turbulence scale I lml
Fig. 6.2.3. The spectral density S of the kinetic energy of turbulent oceanic circulations as a function of the characteristic scale I of turbulence (from Omidov, 1968, with permission of the author). q,az. rr3--intervals
of energy supply to maintain turbulence.
6.2 THE TURBULENT EXCHANGE OF MASS, HEAT AND MOMENTUM
351
characteristic scales of the order of lo6 m (the size of an ocean), which are sometimes referred to as macroturbulence, and which can be both quasi-stationary circulations of ocean waters and non-stationary reaction to synoptic processes in the atmosphere (e.g., drift currents); (2) medium-scale circulations with characteristic scales of 1 O4 m, sometimes called mesoturbulence, associated with the energy of gravitational oscillations (tides, seiches); and (3) fine-scale circulations with characteristicscales of around 10 m, usually associated with the energy of local wind and wave action. The smallest sometimes go by the name of microturbulence in the sea and it is from their scale that we usually get the averaging time when studying local turbulent exchange, and the dissipation of kinetic energy and its conversion into heat. The characteristic scale of fine-scale turbulence is from c. 10 cm to c. 10 m for vertical turbulence, and 10-1000 m for horizontal turbulence; characteristic eddy survival times range from fractions of a second to tenths of a minute. The marked differences between the scales of vertical and horizontal turbulence are due to condition (6.2.8) which shows that vertical turbulence in the sea, that is the eddy components in the vertical plane and the vertical exchange of their energies, is highly dependent on the vertical distribution of water density and usually limited by buoyancy forces. Vertical turbulence may often take place locally in horizontal water layers or “lenses” of the same density without effectively penetrating the boundaries of layers where the density gradients are large. The turbulent exchange of mass, heat and momentum is inextricably linked with the turbulent motion of the elements of the medium, just as molecular exchange is linked with the molecular (thermal) motion of molecules of the medium. It does not always happen, however, that molecular motion is simply the motion of single molecules (see Chapter 2). The boundary between these classes of motion may, in extreme cases, become blurred and as a result of momentum exchange may pass from one to the other (conversion of kinetic energy into heat). Averaging of the Navier-Stokes Equation of Motion
The condition for the turbulent exchange of mass, heat and momentum is turbulent flow within the medium. We shall therefore deal with the turbulent exchange of flow momentum first. To this end, we use the Navier-Stokes equation of motion of a viscous fluid as applicable to an incompressible liquid. The z-axis is directed downwards, i.e., the reverse of its usual direction in hydrodynamics. We write the scalar components of the Navier-Stokes equation (6.1.48) for the
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352
instantaneous velocity Y by adding (to simplify the derivation) the continuity equation (6.1.49) multiplied by u, ZI or w as appropriate. Adding this sum of the expressions of the continuity equation which is equal to zero does not alter the equation of motion (Neumann and Pierson, 1966). Hence, for the first component we ob!ain the expression
=
aP yvzu- -.
(6.2.17)
ax
After rearranging the terms of this expression in the form of derivatives of products, we can write them as (6.2.18a)
In the same way, we get expressions for the other two components of the NavierStokes equation (6.1.48) (after having added the continuity equation multiplied by v and w respectively): (6.2.18b)
Using generalized notation, we can write this briefly : (6.2.19) where ai, is the Kronecker delta (4,= 1 for i = 3; d i 3 = 0 for i # 3). We now average this equation in time, assuming that the instantaneous velocity of the motion is the sum of the average velocity and fluctuating velocity of turbulent motion according to equation (6.2.13), i.e., u =
U+u’,
ZI =
V+d,
w
=
Gfw’
(6.2.20a)
or in generalized notation ui = ai+u;. (6.2.20b) We can also assume that in an area of turbulence, the instanantaneous temperature of the medium T is the sum of the average T and fluctuation part T’, i.e.,
T = T+T’.
(6.2.21)
6.2 THE TURBULENT EXCHANGE OF MASS, HEAT AND MOMENTUM
353
The same applies to the concentration of somz passive substance (or the salinity)
s = S+S‘,
(6.2.22)
the pressure, p = P+p‘ the density of the medium,
(6.2.23)
e = @+e’,
(6.2.24)
etc. But to simplify things when averaging the Navier-Stokes equation, we assume that z e andp z p , so that p‘z 0 and p‘ z 0. (6.2.25) This assumption is a justifiable approximation, since fluctuations of pressure p‘ and density e‘ are usually small in the sea in comparison with their average values j and e. Assumption (6.2.25) considerably simplifies the averaged equation (6.2.18), which after applying the averaging rules reads (6.2.26)
and likewise for the other two component equations (6.2.18). Here we have availed ourselves of the condition that the averages of the derivatives of the functions in this equation are equal to the derivatives of the average values of these functions. In view of the complex velocity (6.2.20) and the averaging rules (6.2.14)-(6.2.16), the averaged velocity products in (6.2.26) are _ _ _ ~ _
- --
uu = (5+ u’)(ii +24’)
=
-_ uzr =
=
(U + u’)(%+v‘)
uw = (ii+U’)(W+W‘)
- __
uu + UU’ +U’U+u’u‘,
- - _ _ uv +U d +245’ +u’d, - _ _ _ - _ _
(6.2.27a)
= uW+uw‘+u‘W+u‘w’.
But according to the averaging conditions, U,V , W are constant, i.e., they are not functions of time, while 2, 6’ and W‘ are equal to zero, hence ilu’ = UU‘ = 0, and likewise vv’ = 0 and G7 = 0. Expressions (6.2.27a) appearing in (6.2.26) thus reduce to __ __ __ uu = Uii+u’u’, uv = uzI+u’v’, uw = Uw-tu‘w’. (6.2.27b) Equation (6.2.26) thus now reads
(6.2.28)
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354
and similar equations are obtained for the other two averaged components of the Navier-Stokes equation. Using generalized notation, we can write them all in one equation
(6.2.29)
If the density e is fixed, the left side of this equation can be expanded @-
aiii at
+u, ae + c i a o +euj- au, ~
at
axj
ax,
a axj
__
+--(@u+;),
In the light of the continuity law (6.1.51) which, as is easily demonstrated, also holds for the average turbulent flow velocity, the sum of the second and third terms in this expression equals zero and can be ignored (in other words, we subtract the continuity equation multiplied by the velocity we had added at the start). This finally yields the equation of turbulent motion which, for an assumed constant density, reads (6.2.30a) and is called the Reynoldsequation. Notice that this differs from the Navier-Stokes equation of motion only in the additional “advection” term describing the addition of “momentum advection”
a-
@ -- (uf u;)
ax,
brought about by the fluctuation
component v‘ of the flow velocity vector. The Turbulent Exchange of Momentum Equation (6.2.30a) can be rewritten in such a way that this “momentum advection addition” is placed on the right side of the equation, beside other volume forces acting on the liquid in motion (6.2.30b) This addition can be treated as the volume force of turbulent friction, equal to the sum of component increments of the momentum of the mass of an elementary cube dxdydz (in unit time) resulting from turbulent exchange-by analogy with molecular exchange (see expressions (6.1.33)-(6.1.35)). The products guzrepresent the flux density of the transferred momentum (units:
m
I
momentum per m2 per second, i.e., [kgs/m2s]
= [N/m2]], and
expresses
6.2 THE TURBULENT EXCHANGE OF MASS, HEAT AND MOMENTUM
355
stresses, just as in molecular exchange (see the tensor of molecular stresses (6.1.36)). These products are therefore called Reynolds turbulent stresses, and after expanding all the components usually represented by the symbol t i j ,we get the Reynolds stress tensor ~ ~ _ _ ,,z zxy ,z, eu‘u’ eu‘d eu’w‘ _ _ ~ _ _ __(6.2.31) zij E zy, z , t y z= p‘u‘ ev‘o‘ ev’w’ = eui uj. ~ _ _ _ _ ,z, zzy z, gw’u’ @ W ’ d ew‘w’
QS
,
-
I
As in the case of molecular stresses (6.1.36), the components ,,z, zyu, z,, here express compressive stresses normal to the walls of the cube dxdydz, that is, they represent the dynamic pressure due to turbulent motion; the other components with mixed indices express shearing stresses. The very phenomenon whereby these stresses arise is called turbulent friction, by analogy to molecular friction. The following equalities also exist: ,,z, = z y x ,z,, = ,z, and zyz= zzy for the same reasons that applied to molecular stresses (equilibrium of moments of forces). This is immediately evident from the products of the velocity fluctuations in (6.2.31), where the order of multiplication cannot alter the result, i.e., -~ eu'v' = ev'u', etc. Reynolds stresses can be calculated by measuring the fluctuation of the flow velocities of the medium. To do this in the sea, current meters with a very low inertia of recording of changes in currents are required; moreover, they have to be highly sensitive and of such a construction as not to disturb the often weak natural current fluctuations in the study area. In the wind field above the sea surface, Doppler shift flowmeters using ultrasonic waves of frequencies of c. 10 MHz or laser beam are employed (see Nielsen and Jacobsen, 1980). As a result of the Doppler effect, changes take place in the frequency of the waves scattered by or reflected from elements of the moving water. These changes in frequency, compared interferometrically with the frequency of waves emitted by the source, are an indicator of current fluctuations (Squier, 1968). A source system producing a thin beam of waves and three wave detectors placed at right angles to one another allow the components of the flow velocity vector, and hence its instantaneous values and directions, to be measured. The volume of water under investigation is "affected" merely by a wave beam and is not directly disturbed by the bulk of the measuring device. The mode of action of a Doppler shift flowmeter is shown in Fig. 6.2.4. More often, however, fiowmeters acting on other principles are used in the sea. The most widely employed ones include electromagnetic and thermoelectric detectors. The former measure the electromagnetic induction induced by the
356
analyser of relative changes
6 THE TRANSFER OF MASS, HEAT AND MOMENTUM
waves
of frequency v =
yo-
1
I*_ w
Fig. 6.2.4. The mode of action of a Doppler shift flowmeter. According to the geometry assumed in the diagram, the detector reacts to changes in the velocity component w = @+d.
movement of saline water. The latter make use of the thermoresistant properties of electrical resistors. The resistor takes the form of a thin film applied to the tiny head of the sensor (c. 1 mm in size) and heated by a high-frequency electric current-hence the name hot film sensor. The changes in the velocity with which a water current flows around such a hot film affects its rate of cooling. The temperature changes thus elicited indirectly induce changes in the resistance of the sensor which indicate changes in the flow velocity. Such a device records slight fluctuations in the flow velocity while it is in uniform motion within the water-usually while it is being lowered through the water column (see Gibson et al., 1975; Belaev et al, 1976). Trials using piezoelectric detectors are being carried out; such a device registers pressure fluctuations induced by the flow, thus allowing the fluctuations in flow velocity to be measured (Siddon, 1969, 1974; Obsorn and Crawford, 1977). Eesides flow velocity fluctuations, oscillations of temperature and salinity or salt concentration are also recorded. The results of such measurements also enable turbulence to be studied and the turbulent mass and heat fluxes in the water to be determined directly (see (6.2.44) and (6.2.50)). However, technical difficulties are encountered here, because of the excessive inertia of reaction of the generally available detectors of these parameters. For investigating micro-
6.2 THE TURBULENT EXCHANGE OF MASS, HEAT AND MOMENTUM
3 57
turbulence in the sea, sensors reacting to temperature changes of <0.01 K with a frequency of a few hundred Hertz are needed, whereas their spatial resolution should be around 1 mm. Sensors for measuring fluctuations in salinity or the concentration of other passive substances in the sea should be made to similarly exacting specifications. Special devices calied marine turbulimeters have therefore been constructed for the study of turbulence in the sea. Their construction and the techniques used in studying turbulence at sea, e.g., by towing suitable sensors through an undisturbed medium, are described amongst others, in works by Ozmidov (1973, 1980), and Belaev et al. (1975, 1976), and thc literature cited therein. In view of the many difficulties involved in measuring flow velocity fluctuations in the sea, the Reynolds stresses, tantamount to the flux density of the turbulent momentum exchange, are often expressed by macroscopic quantities called the coeflcients of turbulent momentum exchange or turbulent viscosity, which were introduced long before present-day developments in research techniques. By analogy to molecular exchange, these coefficients are introduced on the basis of the hypothesis that the dependence between the turbulent momentum flux and the gradients of the appropriate components of the averaged turbulent flow velocities is a linear one. The beginnings of this approach are found in Boussinesq's theory (1877) in which he derived the coefficient of the turbulent exchange of momentum K(")in a horizontally homogeneous turbulent flow in the x direction. The components of the instantaneous velocity of such a current are u = i i f u ' , e: = n' and M: = M', and since the flow is horizontally homogeneous, the mean velocity is only a function of depth, i.e., U = U(z). By analogy to molecular exchange (see Newton's equation 6.1.3), Boussinesq defined the coefficient of the turbulent (vertical) exchange of momentum (turbulent viscosity) K(") by an equation based on the hypothesis that a proportionality exists between the flux density of the average flow momentum in the x direction, transferred to the z direction during turbulent exchange (i.e., perpendicular to U), and the gradient of the average velocity modulus (6.2.32) Here the flux density of the momentum QU"= zxz clearly represents only one component of the Reynolds stress tensor (6.2.31), i.e., the stress resulting from the "turbulent friction", tangential to the z-plane in the turbulent flow with a mean velocity directed along the x-axis. The coefficient of turbulent viscosity K@'),both in this, the simplest case, and in others, has units of [m2/s], i.e., the
6 THE TRANSFER OF MASS, HEAT AND MOMENTUM
358
same ones as the coefficient of kinematic molecular viscosity. Its value is not, however, a physical property of a given medium, but a characteristic of a given turbulent flow dependent on the scale of the turbulent eddies and on the flow velocity field. In this turbulent mixing length theory, derived by analogy to the mean free path of molecules in molecular motion, Prandtl (1925) attempted to connect this coefficient with the structure of turbulence. This not very precisely defined average mixing length 1’, having components I:, I;, I:, is supposed to represent the average distance which elements of medium in single random eddies move and over which they transfer their average flow momentum during turbulent motion from the point where the eddy came into existence to that where all its energy has been dissipated. In this way, along with a single eddy, the momentum of an average flow of medium elements is transported a distance I’. So then, if a non-zero average velocity gradient diildz exists in a horizontally homogeneous turbulent flow, an element of medium is transferred with the eddy a distance 1: from the depth where the mean velocity is u(z) to one where the mean velocity of the environment is U(z+IL). It is along this mixing length in the z direction that an element of mass thus transfers its average momentum, du corresponding to a difference in velocity of U(z)-U(z+Ii) = I: -. The indz
crement in the average velocity expressed in this way is, according to Prandtl’s assumptions, proportional to the velocity fluctuation in the z direction, i.e., du wt “N I,---. (6.2.33) dz I
If we assume that turbulence is roughly isotropic, i.e., Jut/z Iw’[(which is not strictly true when buoyancy forces act along the vertical), in view of (6.2.33) we can write
(6.2.34) The square of duldz has been replaced by the product with the modulus of the value of one of the terms in order to retain the physical sense of the agreement of the sign of the momentum exchange with the sign of the average velocity modulus gradient diildz. Comparing Boussinesq’s hypothesis (6.2.32) with (6.2.34) derived from Prandtl’s theory, we can express the coefficient of turbulent viscosity as the product
(6.2.35)
6.2 THE TURBULENT EXCHANGE OF MASS, HEAT AND MOMENTUM
359
Taylor (1931) worked out a somewhat different theory but which leads to the same conclusions. Comparison of the results of this discussion with the Reynolds stress tensor shows that the turbulent coefficient of momentum exchange K(") introduced earlier describes only the simplest case of isotropic or one-dimensional exchange. Combining expressions (6.2.32) and (6.2.33) shows that this coefficient is equal to K(") = ,!:u'.
(6.2.36)
Ertel (1937) generalized this conclusion by introducing analogous coefficients of turbulent exchange of momentum in all directions in a three-dimensional, non-isotropic flow, which overall make up a second-order tensor (6.2.37)
This tensor of exchange coefficients, already complicated enough and dependent on a rather vague mixing length, does not yet describe the most general case of exchange. The principal hypothesis of a semi-empirical theory of turbulence, which assumes a proportionality between the momentum exchange Rux and the velocity modulus gradient, in the most general case connects the linear interdependence of the Reynolds stress tensor zij with the tensor aui/axj, called the deformation rate tensor of the averaged velocity field. This connecting linear function contains an exchange coefficient tensor which in the most general case of non-isotropic turbulence, where geometrical symmetry with respect to the coordinate axes may be lacking, is a 4th degree tensor K$'$ (Monin, 1978). But since it is not possible to determine all its components, many practicable simplifications are implemented in the sea (Saint-Guily, 1956; Fofonoff, 1962; Kamenkovich, 1967; and others). Use is made of the frequent prevalence of horizontal, and locally, horizontally stratified flows in the sea. One considerable simplification (a reduction in the number of components of the exchange coefficients) is based on the assumption that the exchange coefficient tensor K;& in the ocean is symmetrical with respect to the vertical axis (Kamenkovich, 1967). An even simpler though less easily defended approximation is to assume only two different components of the coefficients: KP) for vertical exchange, and K f ) for horizontal exchange. Assuming horizontal exchange to be isotropic, the components expressing the volume forces of turbulent friction in the Rey-
360
6 THE TRANSFER OF MASS, HEAT A N D MOMENTUM
nolds equation (6.2.30b) take the following form, often used in oceanology (Monin, 1978)
They are simplified even further if we assume additionally that the coefficient of horizontal exchange Kh is constant and independent of the coordinates. Then we have
(6.2.39) (6.2.40) Much experimental work has been done on determining the coefficients of turbulent exchange in the sea (see Sverdrup, 1937; Defant, 1961; Ozmidov and Popov, 1966; Monin, 1978). This usually involved observing the average-velocity field of flows in the sea using current meters, or more often, some sort of indicator floating on the water. These latter included both special mechanical floats and chemical, radioactive or fluorescent compounds or dyes. The monograph by Timofeev and Panov (1962), to mention but one, reviews the methods and results of many such experiments. Kagan and Ryabchenko (1975) on the other hand discuss the natural and man-made indicators (chemical and radioactive tracers) of various processes in the ocean and their dispersal therein. In particular case of different turbulent flows, the coefficients of turbulent momentum exchange (turbulent viscosity) values characteristic of a given flow, can take very different values. So the opportunities of practically applying even these simplified equations (6.2.39, 6.2.40) to ocean currents is limited. The values of the exchange coefficients usually rise with the scale of turbulence, which is in fact already evident from (6.2.36) or (6.2.37). As far as horizontal exchange @)is concerned, the orders of magnitude of these coefficients are estimated to be lo4 m2/s for macroturbulence (large-scale currents), lo-' m2/s, for mesoscale turbulence and mz/s for fine-scale turbulence. For vertical exchange KP), the corresponding orders of magnitude of the exchange coefficients for the same decreasing scales of turbulence are estimated to be lo-'lo-' mz/s, 10-2-10-3 mz/s and m2/s. They are therefore mostly smaller
6.2 THE TURBULENT EXCHANGE OF MASS, HEAT AND MOMENTUM
361
than the horizontal exchange coefficients because of the buoyancy forces which usually counteract vertical turbulent mixing (Doronin, 1978). These approximate values of the coefficients of turbulent viscosity do not exhaust the much more complex set of results of various experimental studies. The coefficients of turbulent viscosity are usually whole orders of magnitude greater than those of kinematic molecular viscosity qv mZ/s (see the explanation to equation (6.1.3) and Table 6.1.1). The coefficients of turbulent exchange in the sea, including that of momentum exchange, are subject to certain regularities which make their value dependent on the scale of turbulence. We shall be discussing this regularity in the next section, which will deal with the turbulent exchange of heat and mass of substances contained in the water. The reader should be aware of the fact that it is only recently that the technical difficulties of measuring turbulent viscosity in the sea have been overcome (see Belaev et al., 1975; 1976; Ozmidov, 1980); so far, the coefficients of the turbulent exchange of mass and heat have been much more accurately determined. N
The Turbulent Exchange of Mass and Heat Turbulent motion in the sea is always accompanied by an additional kind
of exchange of mass and heat, which by analogy to diffusion and molecular thermal conductivity is called the turbulent dzfusion of substances contained in the water and the turbulent exchange of heat. The turbulent diffusion of mass and heat is a kind of advection of mass and heat from the randomly variable velocity fluctuation u; of a turbulent flow. We obtain the equation accounting for the effect of this turbulent motion of the water on the average concentration distribution C of a passive substance or the heat Q by averaging the equations of molecular exchange in the flowing water ((6.1.52) and (6.1.53)), i.e., by averag-
ac in
ing the equations of diffusion with an advection component uj--
ax,
which
u, is treated as the instantaneous velocity of the turbulent motion equal to Uj+ uJ. Thus, in order to simplify the derivation, we multiply equation (6.1.52) by the density of the medium e and add to its left side the sum of the terms of the continuity equation (6.1.51) (equal to zero) multiplied by the concentration C of the passive substance (admixture) in question present in the water. If we ignore the source function Ms we have e-
ac + C -a@ ac - +eu.-at
at
J
axj
a(Qu.1 + C --’-ax,
=
eDV2C,
(6.2.41)
3 62
6 THE TRANSFER OF MASS, HEAT AND MOMENTUM
which after rearranging yields (6.2.42)
This expression, still only a modification of the diffusion equation (6.1.52) is now averaged on the assumption that the medium is in turbulent motion. The instantaneous velocity is thus uj = uj+ui and the instantaneous concentration C = +C', but the density fluctuations are negligibly small in comparison with e.g., After averagthe fluctuations in concentration C', so that we can assume e z ing (in the same way as (6.2.19) was averaged, and taking into account the fact that the sum of components making up the continuity equation is equal to zero), we get the following equation of diffusion which describes the changes in the average concentration of an investigated substance in space and time (in an area devoid of sources)
ct
e.
(6.2.43)
Here, the fluctuation component of the advection of the substance in question with concentration C, i.e., a / a x j ( - u ~ )has been transferred to the right side of the equation and treated as a component describing the turbulent diffusion beside the component describing the molecular diffusion D V 2 c . If the concentration C_ _of the substance in the water is expressed in [kg/m3], the averaged ' units of [kg/s m2] and expresses the surface density of the flux product u ~ Chas of a mass of this substance transported during turbulent diffusion towards the fluctuation component ui of the turbulent flow velocity. Since the value of this turbulent diffusion flux is large in comparison with the molecular diffusion flux, the molecular component D V 2 c in (6.2.43) can often be ignored. By analogy to molecular diffusion (see (6.1.I)), we usually set up the hypothesis that the turbulent diffusion flux of a mass of a given passive substance (admixture in water) is proportional to the gradient of its averaged concentration, and the coefficient of proportionality K("'), known as the coeficient of turbulent dzfluxion, is thus introduced (Schmidt, 1925). In the general case of three-dimensional turbulence flow, with spatially non-uniform and non-isotropic turbulence, the coefficients of turbulent diffusion in the various component directions form a second-degree tensor K$) linking by a hypothetical linear dependence __ the vector components of the turbulent diffusion flux ujC' (per unit area) with -
ac
the vector components of the average concentration gradient _ -. 3x1
6.2 THE TURBULENT EXCHANGE OF MASS, HEAT AND MOMENTUM
3 63
(6.2.44) where the index 1 = 1, 2, 3 respectively. The unit of the coefficients of turbulent diffusion Ki(7) are [m'/s]; they are thus the same as those of the coefficient of molecular diffusion. By introducing the hypothesis (6.2.44) into (6.2.43) and neglecting the component describing molecular diffusion, we can rewrite (6.2.43) as
(6.2.45)
We must remember that on expanding this equation with terms in which the indices j and I repeat themselves, we must sum these terms over j = 1, 2, 3 and over I = 1, 2, 3 (see explanations to equation (6.1.50)). Equation (6.2.45) is a general one describing the changes in the average concentration of a given
(
substance in space and time c ( x , y , z, t ) resulting from its average influx component of average advection
a
Uj--
axj
1
and turbulent diffusion, i.e., the advection
( K $ ) Z ) , described with the aid of exchange axj coefficients Kj'J). In particular cases of weak turbulence, this equation will contain an additional term allowing for molecular diffusion DV'C (see 6.2.43), and if volume sources of the investigated substance are present in a given area, a source function M sdescribing these sources has to be included as well. Practical oceanological simplifications of the turbulent diffusion equation (6.2.45) emerge from the assumption of a given symmetry of turbulent currents in the sea, e.g., a horizontally homogeneous flow with a plane-parallel field of average velocity. Then, when pointing the x-axis in the direction of the average flow, we have velocity U, = U(z), Uz = 5 = 0,U3 = w = 0, and the coefficients of turbulent diffusion KJ'J)(x,y , z ) = K,'Y)(z) are merely functions of the depth z. Equation (6.2.45) then takes the form (Monin, 1978) fluctuation component
__
(6.2.46)
6 THE TRANSFER OF MASS, HEAT A N D MOMENTUM
364
Further simplification of this equation in oceanology, determining its solution under fixed boundary and initial conditions, is possible by making use of scarcely justifiable assumptions simplifying the exchange coefficients K:;) = Kj;) = Kim)= const, and KiT) = Kim),while K::) = K$ = 0; but the chief problem here is that we do not know their values. Such assumptions lead to the following equation of turbulent diffusion in a uniform, plane-parallel, horizontal turbulent flow, often applied in oceanology:
With the aid of these simplified equations of turbulent diffusion (6.2.46) and (6.2.47), we can explain the experimentally determined elongation of slicks of passive substances in the sea (Monin, 1978). If, moreover, we assume a planeparallel stratification of the water with respect to the average concentration of the substance in question C ( x , y , z ) = C(z), the simple equation of vertical turbulent exchange, derived from (6.2.47), is (6.2.48) Similar considerations lead to the turbulent heat exchange equation. We average the equation of molecular thermal conductivity in the flowing water (6.1.53) assuming that the density Q z 5 = const, the specific heat C, = const, and that the instantaneous flow velocity and the instantaneous temperature of the water are sums of the average and fluctuation component of turbulent motion uj = Ujt +u:. and T = T? T'.Hence we get the equation for heat oxchange in turbulent flow* (6.2.49) We have neglected the source function QEs,and have assumed that there are no internal sources of heat in the area under investigation. On the other hand, we have transferred the fluctuation component of heat advection a/ax,(&,u;r) to the right-hand side of the equation (hence the minus sign), treating it as a com-
* In heat transfer equations in real compressible media one should use the potential temperature 8 rather than the absolute temperature T. The rise in the potential temperature of a volume element of the medium (in any changed position) reflects the real heat influx and omits the gain (or loss) of heat due only to a conversion of work p d a to heat (or vice versa in adiabatic compression or rarefaction) of that element, during its motion in a non-uniform pressure field.
6.7 THE TURBULENT EXCHANGE OF MASS, HEAT AND MOMENTUM
365
ponent describing the turbulent conductivity of heat, beside the component __ yV2T describing the molecular conductivity. The product (pCpu; T')has units of [J/m2s] and expresses the surface density of the heat flux transferred during turbulent exchange. One also assumes the hypothesis, similar__ (to 6.2.44), that there exists a linear dependence betwen thz hsat flux (eC,uJ") (per a unit area) transferred during turbulent diffusion, and the appropriate components of the average temperature gradient (6.2.50) where Kj?) is the tensor qf turbulent heat exchange coeficients, measured in [m2/sl as well. After having applied this hypothesis to the heat exchange equation (6.2.49) and omitting the molecular exchange component yV2T (this has a relatively small value), we get an equation for the turbulent exchange of heat analogous to the equation for turbulent diffusion (6.2.45) and simplifications of this equation analogous to the diffusion equations (6.2.46)-(6.2.48). Unlike the coefficients of molecular exchange, the coefficients of turbulent diffusion KjZ) and turbulent exchange of heat Kj?) are not real physical properties of the water but, just like the coefficients of the turbulent exchange of momentum, are characteristic properties of a given turbulent flow, dependent on the scale 1 of the turbulent eddies. Like the coefficients of turbulent viscosity, they can thus take values which differ from one another by whole orders of to lo4 m2/s), depending on the scale of turbulence. magnitude (from This dependence on the scale 1 of turbulent eddies was established by Richardson (1926) on the basis of numerous experiments (in the atmosphere); he described an equation which expresses the order of magnitude of a turbulent diffusion coefficient KP) = k1-W. (6.2.51) Nowadays, this formula is known as the "$our-thirds" law. Here k is a constant coefficient. This law was later proven by Obukhov (1941) by the methods of dimensional analysis for the scale ranges of turbulence in which this occurs in an inertial flow (not supplied with energy) and is dependent only on the rate of dissipation of turbulent energy. Richardson's law (6.2.51), which does not always agree with experiment in cases of turbulence of any scale, was generalized by Ozmidov (1957, 1959) in the equation (6.2.52)
6 THE TRANSFER OF MASS, HEAT AND MOMENTUM
366
where the function f ( l / L )is a certain universal function, whereas L is the characteristic scale of the principal turbulent flow (the largest one in cascade-like eddies) and corresponds to the spatial scale on which the examined turbulent exchange is supplied with energy. The relationship between the coefficient of horizontal turbulent diffusion and the scale of turbulent eddies 1 established by Ozmidov (1968) and studied experimentally on the basis of the dispersal of dye stains in the sea, is shown in Fig. 6.2.5 (Okubo and Ozmidov, 1970). This illustrates the action of the four-
-3-01
106 104 lo2 Scale of turbulence I Irnl
Fig. 6.2.5. The coefficient of horizontal turbulent diffusion K, = K,”(Z) as a function of the scale of turbulence I, established on the basis of experimental investigations of the diffusion of dye stains in the ocean (according to Okubo and Ozmidov, 1970, with permission of the authors). Symbols: d o t s d a t a from 1962; circles-data from 1968; straight-line sections obey the law K I
-
f4f3.
thirds law over a wide range of turbulence, but with a different constant of proportionality on the sections of the plot on either side of the characteristic kink. This kink corresponds to the turbulence scale which in the sea corresponds to the scale of maximum inflow of energy from the exterior directly maintaining turbulence of this scale. This was shown in Fig. 6.2.3. In the fairly narrow scale range where the kink in the plot lies, the turbulent flow is clearly supplied with energy, that is, it has no inertia and does not obey the four-thirds law. Ozmidov (1968) extended this dependence of the coefficient of horizontal turbulent exchange
6.2 THE TURBULENT EXCHANGE OF MASS, HEAT AND MOMENTUM
3 67
(the four-thirds law) to include the entire range of turbulence outside the section where turbulence has energy supplies to it (see Fig. 6.2.3); i.e., the four-thirds law holds in turbulence ranges where the turbulent flow is inertial and loses energy only through dissipation. The coefficients of horizontal turbulent exchange in the sea are thus usually far greater than those of vertical exchange. As we have said, this is because the vertical component of turbulence is damped by buoyancy forces in the (mostly) stratified marine environment (see condition (6.2.3)). The mechanism of turbulent transport of mass and heat differs from that of momentum transfer. The turbulent transfer of momentum depends on the transfer of only the kinetic energy of the eddies between elements of the medium (i.e., on a certain type of collision between elements of medium and on pressure fluctuations). The turbulent diffusion of mass and heat on the other hand, requires the mutual mixing of volume elements of different temperatures of substance concentrations. This is why the coefficients of turbulent diffusion K(m)and turbulent heat exchange KtQ) are usually lower than the coefficients or turbulent momentum exchange K(”)by a factor of several tens. Thus the ratios of the last-mentioned coefficient to the two former ones are 20 or 30 times greater than unity
(6.2.53)
and are called the Prandtl and Schmidt numbers, respectively. Further details characterizing turbulent motion and exchange in the sea, empirical interdependences of turbulence parameters, methods of studying turbulence and statistical descriptions of physical fields in the sea, not to mention a key to the world literature on this topic, can be found in Druet (1980), Ozmidov (1980), Zeidler (1980), Massel (1980), Belaev (1980), Monin (1978), and Monin and Ozmidov (1981). To end this section, the reader’s attention is drawn to the energy of turbulent motion contained within a given area of water. The turbulent motion of the medium fills the space with the kinetic energy of an average flow whose density is feG,and with the energy of the fluctuation components of the flow, i.e., the turbulent energy, whose density is (units [J/m3]). A basin in which there is turbulent motion thus contains an often huge quantity of turbulent kinetic
368
6 THE TRANSFER OF MASS, HEAT AND MOMENTUM
energy besides the thermal energy it already contains. It could be called scattered kinetic energy, by a certain analogy to the energy of scattered light. This is because both the photons of scattered light and turbulent eddies are of a random nature and are usually the result of the energy of a suitable primary flow being scattered ; moreover, both these forms of scattered energy are usually converted into heat. A certain analogy also exists between the absorption by the basin of electromagnetic radiant energy (sunlight) reaching the sea surface and the absorption of the kinetic energy of air masses (winds, gales, etc.) reaching the sea surface (by friction and pressure fluctuations), So the scattering of the energy of a flow as a result of the formation of turbulent eddies and the absorption of this energy by molecular friction is similar in a sense to the scattering of rays of sunlight and the absorption of the photons by the water. In both cases, scattering increases the probability of energy being absorbed and converted into heat. The action of both these processes thus leads to the dissipation of the energy fluxes entering the sea and their conversion, mainly to heat. However, the absorption of energy by the water mass takes place only up to the moment that thermodynamic equilibrium with the surroundings is achieved, after which the direction of the flow of energy, both heat and mechanical, is reversed. We shall continue to discuss turbulent energy in a basin in the next chapter.
CHAPTER 7
SMALL-SCALE AIR-SEA INTERACTION AND ITS INFLUENCE O N THE STRUCTURE OF WATER MASSES I N THE SEA
Because the Earth’s surface is unevenly heated by the Sun, the thermodynamic state of sea basins is usually instantaneous and variable in time, just like the state of the atmosphere. The termodynamic equilibrium of water elements with their surroundings is very unlikely to be lasting, so it is instantaneous and no less random than transitional states of imbalance. Typical of these transition states are temperature, salinity, pressure and density gradients (different from zero) which cause energy and mass to flow in such a direction as to restore equilibrium. The chief means by which this is achieved were discussed in previous chapters, notably, quasi-stationary fluid flows (sea currents) giving rise to the advection of energy and mass, turbulent exchange, molecular exchange (see Chapter 6 ) and electromagnetic radiation (see Chapter 5). In the mass of water, energy is also transported by mechanical waves (Le Blond and Mysak, 1978), i.e., surface, internal and acoustic waves (see Chapter 8). The dominant role of some means of transferring mass and energy and the marginal role of others depends on the currently available sources of energy, on the temporal-spatial scale of the process in question and on the environmental conditions obtaining. An element of the analysed sea basin is often reduced to a water column delimited by lateral geometrical planes. The natural upper boundary of such a water column is a section of the sea surface, the lower boundary a section of the sea bottom. Various forms of energy and various components of mass flow across all the walls of such an enclosed volume of water in such directions as to restore thermodynamic equilibrium. The normal situation is that while exchange is intensive across the sea surface but only slight across the bottom, the intensity of exchange across the sides of the water column largely depends on the horizontal gradients and the horizontal flow velocity of the water. That the exchange of energy and mass across the air-sea interface is intensive is usually due to the fact that the sea’s greatest sources of energy, i.e., solar radiation and wind friction at the water surface, supply this energy across this surface. The complex set of processes of energy and mass exchange across the
370
7 SMALL-SCALE AIR-SEA INTERACTION
surface together with its direct effects in the sea and air is generally known as air-sea interaction and is one of the basic problems of modern oceanology (Faver and Hasselmann, 1978: Monin, 1978; Zilitinkevich et al., 1978; Dubov, 1973, 1974; Bortkovskii et al., 1974; Kraus, 1972; Kitaigorodskii, 1970; Zilitinkevich, 1970; Roll, 1965). On the one hand, this exchange determines the terrestrial climate, the circulations and variability of the atmospheric states (Schlesinger and Gates, 1980; Eagleson, 1978; Zverev, 1977; Dzerdzeevskii, 1975; Reshetov, 1973, and others), on the other, it determines the circulation and thermohaline structure of water masses in the sea (Zilitinkevich et al., 1979; Faver and Hasselmann, 1978; Nihoul, 1978; Kraus, 1977; Fedorov, 1976; Monin et al., 1974, 1978; Shtokman, 1970; Druet and Kowalik, 1970; Philips, 1969), their chemical composition (Bordovskii and Ivanenkov, 1979; Goldberg, 1974; Horne, 1969), and in the long run the conditions of life of countless marine organisms (Vinogradov, 1977), the production of free oxygen and the absorption of carbon dioxide by the ocean (Steeman Nielsen, 1975; Johnston, 1976; Goldberg, 1976). In this chapter we shall be characterizing briefly the main fluxes of mass and energy between the sea and the atmosphere, we shall be discussing their effects on the energy budget of a basin, and their influence on the structure of water masses in the sea. 7.1 FLUXES OF MOMENTUM, MASS AND HEAT ACROSS THE SEA SURFACE
The principal fluxes of energy and mass across the sea surface are illustrated diagrammatically in Fig. 7.1.1. It shows the fluxes of solar radiation absorbed by the sea Qs, the effective IR radiation of the sea surface Q b , the sensible heat conducted across the air-sea interface Qh, the latent heat escaping into the air with water vapour Qe, and the heat transferred in either direction in water droplets Qw. We also have the mechanical energy flux E, due to the work done by wind friction on the sea surface, etc. All these fluxes are usually expressed per unit average (horizontal) sea surface, so strictly speaking, they are the surface densities of these fluxes measured in [J/s m2]. The mass fluxes include the water vapour flux M e , the water droplet flux M,, the flux of salt and other solid substances M,, the flux of gases Mo2, MCq, and others. They are also expressed per unit sea surface, and are therefore the surface densities of these fluxes given in [kg/s m2]. A slightly different kind of flux is the electrical charge flux qe [ C / s m2],which takes place as ions transported into the atmosphere become separated. We shall now examine these fluxes in greater detail.
7.1 FLUXES OF MOMENTUM, MASS AND HEAT ACROSS THE SEA SURFACE
371
c
Transport in the atmosphere (winds)
Atmosphere
Fig. 7.1.1. The main fluxes of energy, mass and electrical charge across the sea surface. &-solar radiation flux absorbed by the sea; Q,--effective IR flux radiated by the sea surface (difference between air-to-sea and sea-to-air radiation; Qh-sensible heat flux penetrating by conductivity from sea to air or vice versa; Q,-latent heat passing from sea to air (or vice versa) together with the mass of water vapour; Q,-heat flux transported into the atmosphere (or vice versa) with the mass of water droplets (spray, precipitation); E,-e5ective mechanical energy flux passing from air t o sea (mainly as a result of the work done by turbulent wind friction on the sea surface); Me-effective mass flu of water vapour transported from sea to air; M,,, effective flux of water mass transported as droplets from air to sea as rain or from sea to air; M,-effective flux of salt (or other solid substances) transported from sea to air with water droplets and from air to sea with aerosol fallout and atmospheric precipitation; Mo,,co,--effective flux of oxygen, carbon dioxide and other gases exchanged between sea and air; q,--electric charge flux due (mainly) to the separation of ions when seawater droplets are transported from the sea Surface.
The Solar Radiaion Flux Qs The solar radiation flux absorbed by every square metre of the sea surface Q, [J/s m2]is the main source supplying a sea basin with energy under average conditions during the daytime. About 99% of this energy lies within the shortwave interval 0.3 < il < 3 pm. It is equal to the difference between the incoming and outgoing solar radiation fluxes at the sea surface (h = 0). The incoming flux is the downward irradiance transmitted through the atmosphere FQ (h = 0), and the outgoing flux E, (12 = 0) that is the upward irradiance due to the reflection of part of the sunlight within the sea and from its surface (see Chapter 5). We can therefore write that (7.1 -1) Q, = E,,(h = O)-EQt(h = 0).
By multiplying the top and bottom of the right side of this equation by EQl(h = 0), and bringing into play the definition of the albedo A , given by (5.2.8a), we obtain
7 SMALL-SCALE AIR-SEA INTERACTION
372
Q6 = E Q , ( h = O ) [ ~ - A Q ] .
(7.1.2)
When investigating the flux Qs in detail, we thus have to measure precisely, or define in some other way, the instantaneous downward irradiance Ear and the albedo of the sea A , just above the sea surface. A measurement made at a height of h = 5-10 m will be a good approximation because of the small absorption coefficient of light in air. Piranometers are used to measure irradiance over the entire spectral range E,, , EQ, (in practice from c. 300 to 3000 nm). Widely used are piranometers with a flat irradiance collector consisting of a set of plates coated matt black and matt white and arranged in a chequer-board pattern. Made from special metal alloys, these plates are in contact with a set of thermoelements forming a thermopile (thermoelectric battery). The black plates absorb incident light of all wavelengths, the white ones reflect it. The black plates therefore become warmer than the white ones and the temperature difference between them generates an electromotive force in the thermoelectric battery. The EMF, which is proportional to the irradiance measured, is then recorded on a millivoltmeter as a measure of this irradiance. If we do not measure the irradiance, we can obtain an approximate value of Q, by using the value of the solar constant Fso = 1353 J/s m2 given in the introduction to Chapter 5, atmospheric transmittance data TaQ under the given conditions (see Section 5.1, (5.1.20)) and data on the albedo of the sea A , (see Fig. 5.2.7). Applying (5.1.20) we can express the sea surface irradiance Eo,(h = 0) in (7.1.2) by means of the solar constant FsQand the atmospheric transmittance TaQ Qs = TaQ F ~ Q ( l CoS 6 s (7.1.3) 9
where 6, is the deflection of the Sun from the zenith. So, for example, if the transmittance of a real though very clear, cloudless atmosphere is TaQ= 0.9, with the Sun at the zenith and the sea's albedo A , = 0.04, we obtain a value of Qs= 1169 [J/s m2]. On expressing this flux per square kilometre of sea surface, we end up with a power of over 1 1 0 0 MW, which is equivalent to the power of a generating station supplying a large city with electricity. On an overcast or foggy day, assuming a very low atmospheric transmittance of TaQ w 0.1, an albedo of A, % 0.06 and a solar altitude of 6s= 40", (7.1.3) gives Qs2 N" 97 J/s m2 or 97 MW/km2. In actual fact, this flux takes intermediate values with different solar positions and atmospheric transmittances during the day, and is zero during the night. There are many empirical formulae describing the atmospheric transmittance or the average downward irradiance E,,(h = 0) for differing cloud cover N
7.1 FLUXES OF MOMENTUM, MASS AND HEAT ACROSS THE SEA SURFACE
373
(measured on a 0-1 scale) with respect to the irradiance E,o,(h = 0) from the Sun at the same altitude and shining out of a cloudless sky (i.e., the relative atmospheric transmittance; see Chapter 5; see Ivanoff, 1972, 1977). One of the latest of these formulae is the one conceived by Matsuike et al. (1970) E Q= ~ I?: (1 -0.52N1.3),
(7.1.4)
which can be applied to estimate Qsfrom (7.1.2). The value of EZo,for a cIoudIess sky can be evaluated from the solar constant in accordance with the equations given in Chapter 5. When the sky is completely overcast, it can be assumed in (7.1.3) that the atmospheric transmittance is roughly equal to the cloud transmittance described by the Hurwitz equation (5.1.21) (see also Dave and Braslau, 1975; Braslau and Dave, 1973). It is perfectly obvious that estimates of Qsusing various empirical formulae (see also Eagleson, 1978) are rarely identical. This flux is strongly influenced by the atmospheric transmittance, and is dependent in a complicated manner on a number of environmental factors, such as cloud cover, fog, humidity, aerosol concentration (see Chapter 5), which cannot always be accurately determined. That is why precise, direct measurements of Qs and of the other air-sea fluxes are required in detailed studies of small-scale air-sea interactions (see “Kamchiya 1979”). Global or averaged values of this flux can be estimated from compilations and geophysical tables (see e.g., Hurtwitz, 1948; Egorov and Grillova, 1973; Timofeev, 1970; Czyszek et al., 1979; Kondratev, 1954; Smithsonian Tables, 1951; Oceanographical Tables, 1975). Finally, the reader is remainded that about 57% of the solar energy flux Qs is absorbed in the top metre of the sea, and c. 80% is absorbed in the top 10 metres (the upper sea layer) (see Fig. 6.0.1). The strong absorption in the subsurface layer of the sea is due to the fact that over 50% of the energy of this flux (depending on the solar altitude and state of the atmosphere) is near IR radiation (0.7 < )3 < 3 pm) which is strongly absorbed by water molecules (see Chapters 4 and 5 ; also Ivanoff, 1977; Czyszek et al., 1979).
The Long-wave Radiation Flux Qb This flux is the difference between the heat flux radiated by the sea surface into the atmosphere E,? and the heat radiated by the atmosphere down to the sea E , ~ . So, as for sunlight, we can write Qb
= E,&(z =
0)-
=
0).
(7.1.5)
374
7 SMALL-SCALE AIR-SEA INTERACTION
The basic laws governing thermal radiation-the Stefan-Boltzmann law and Wen’s law for an ideally black body are given in Chapter 1 (equations (1.12) and (1.1.4)). But neither the sea nor the atmosphere are ideally black bodies, so the description of their radiation is more complicated. The IR radiation emitted by the atmosphere and passing from the atmosphere to the sea is almost entirely long-wave radiation, quite different from the waveband encompassing the solar radiation arriving at the sea surface. As we stated in Chapter 5 , water vapour, ozone, carbon dioxide and also water droplets in fog and clouds all strongly absorb long-wave radiation (see Zuev, 1966, 1970; see Fig. 7.1.2). The efficiency with which IR radiation from the sea and Sun is
“1,
i i i i
5
6
i .iB i o i i Wavelength i
12iiii
[~~rnl
Fig. 7.1.2. An outline of the atmospheric transmittance spectrum from 0 to 14pm (after Holter and Legault, 1965). The visible transmittance maxima (unshaded peaks) are called “windows” in the atmospheric transmittance spectrum. Beyond these windows, the atmosphere is opaque to radiation, i.e.. strongly absorbs it.
absorbed by the atmosphere thus depends on the vertical concentration distribution of these atmospheric constituents existing at any given instant. So the heating of the atmosphere due to this absorption and its own secondary TR radiation towards the sea also depend on these constituents and, of course, on the Sun’s position in the sky. Finally, the IR radiation of the upper layers of the atmosphere (e.g., strong cloud radiation) is absorbed by the lower layers intercepting the radiation on its way down to the sea. So although clouds radiate in something like the same way as black bodies at an absolute temperature T, the flux E , ~entering the sea differs from this emittance, as it is strongly modified by the other layers of the atmosphere and by reflection of part of the IR radiation from the sea surface (2-4%). As this process is a complex one, the atmospheric
7.1 FLUXES OF MOMENTUM, MASS AND HEAT ACROSS THE SEA SURFACE
375
radiation flux entering the sea E , ~is given by approximate empirical formulae. For a cloudless sky, this formula reads (7.1.6) a C ( a + b v/e>~ r 7 where cr = 5 . 6 6 8 7 ~lo-* W m-2 K-" is the Stefan-Boltzmann constant, T, is the absolute temperature of the surface boundary layer of the atmosphere over the sea (difficult to determine accurately), e is the vapour pressure in this layer expressed in millibars, 6, = 0.96-0.98 is the transmittance from the atmosphere to the sea (l-albedo) estimated for JR, a and b are here empirical coefficients varying within the ranges 0.34 < a < 0.66 and 0.03 < b < 0.09 (see Doronin, 1978), but usually taking the values a = 0.56 or 0.61 and b = 0.08 or 0.06 (see Ivanoff, 1972 and the papers cited therein). The atmospheric radiation flux (7.1.6) entering the sea is much greater when the sky is overcast. Clouds have a higher absorption capacity of solar energy and thus a higher emissivity than air. This increased atmospheric radiation entering the sea is also given by the empirical relationship Ear = (&,ifcloudlesstl+CN), (7.1.7) (&al)cloudless
=
where (&,$)cloudleasis the flux given by (7.1.6), N is the cloud cover expressed on a O+ 1 scale, and c is an empirically established coefficient which unfortunately varies over a wide range-from 0.05 to 0.4, depending on atmospheric conditions (see Doronin, 1978). The second part of the effective long-wave radiation flux Qb is the sea surface radiation into the atmosphere when the surface temperature is T,; this radiation is taken into consideration in (7.1.5). A deep, clean sea radiates almost like an ideally black body, though strictly speaking like a grey body whose total emissivity has been reduced with respect to a black body by an amount proportional to (l-albedo) in the IR. We can therefore write ewt
z 6,aT$,
(7.1.8)
where T, is the absolute temperature of the water surface; CI is the Stefan-BoItzmann constant; S , as in (7.1.5) is estimated at 0.96-0.98: the higher value should better reflect the actual situation for a very clear, deep sea. From this last formula we can readily calculate that the surface radiation flux for a warm sea ( T, = 295K) is considerable (E,+ z 421 J/s m2) and does not cease at night. We can also apply Wien's shift law (see 1.1.4) to sea surface radiation. This law tells us that the spectral maximum of the surface radiation at temperature T = 295 K is A, z 9.8 pm, while at T = 275 K, A,,, z 10.5 pm. The spectrum
7 SMALL-SCALE AIR-SEA INTERACTION
376
of this radiation stretches a long way towards the longer waves (see Fig. 1.1.2). But its maximum lies in that part of the spectrum where there is a wide “window” in the atmosphere transmittance spectrum (Fig. 7.1.2), and so much of the sea’s radiant energy escapes high up into the atmosphere. This radiation can therefore be made use of to make remote measurements of the sea surface temperature with IR detectors from aeroplanes or satellites (Keyes, 1977; Holter and Legual, 1965; Hackforth, 1964). Knowing the long-wave radiation fluxes from air to sea E , , (7.1.7 and 7.1.6) and from sea to air E , , , ~ (7.1.8), it is possible using equation (7.1.5) to find the resultant long-wave flux across the sea surface
Q, = 6,0T4 [(a+b dF)(l + c N ) - I ]
(7.1.9)
in which we have assumed that T, = T,,, = T, i.e., that the air temperature T, and the surface water temperature T, are the same. The other symbols were explained when we derived the intitial equations. The coefficients denoted here by a, 6, c have nothing to do with the inherent optical properties of seawater denoted by the same letters in Chapter 4. Neither should the temperature be mistaken for the light transmittance; both phenomena are represented by the letter T. As we can see, the empirical equations for calculating Qbcontain many inaccuracies in the coefficients a, by c and also in the determination of the temperature T of the sea surface and the atmosphere. Because of radiation and evaporation, the temperature of the surface skin layer of the ocean is frequently 0.3-0.5 K or even I K cooler than that of water a few centimetres lower down (Dubov, 1974; Monin, 1978). Measurements of this flux are also extremely difficult to make; its accurate determination is thus still an open research problem.
The Surface Boundary Layer. Momentum Flux The exchange of other energy fluxes and of mass is completely different from radiation transfer and consists of the mechanical processes of molecular and turbulent exchange, which can be described using the appropriate equations from Chapter 6. These processes are highly dependent on the wind velocity gradient and the temperature and salt concentration gradients of the adjacent lower layer of the atmosphere and the top sea layer. It is extremely difficult to evaluate precisely the actual fluxes of this mechanical transfer for any wind, wave-action and any state of the vertical hydrostatic equilibrium. The analytical description of exchange in this most general case requires the analytical solutions of the most general equations of momentum, mass and heat transfer; these are
7.1 FLUXES OF MOMENTUM, MASS AND HEAT ACROSS THE SEA SURFACE
377
still not known. A wide range of theories has thus been developed, describing air-sea interaction based on simplified models of the structure of the air-sea surface boundary layers and on approximate methods of solving the momentum, heat and mass transfer equations (Roll, 1965; Kitaigordskii, 1970; Zilitinkevich, 1970; Zilitinkevich et al., 1978. 1979; Mellor and Furbin, 1975; Kraus, 1972, 1977; Faver and Hasselmann, 1978). Experimental data and formulae are used in order to simplify and close the system of equations. The various methods of determining the flux of mechanical energy transfer have been reviewed by Busch (1977) and Kowalik (1979), among others. It is usually assumed that as far as small-scale exchanges are concerned, the atmospheric surface boundary layer (SBL), which is from several to several tens of metres thick, is horizontally homogeneous. It is further assumed that the average wind velocity is also horizontal only, while turbulent motion is statistically homogeneous in the horizontal plane and, moreover, time-invariable (stationary). This means that the average wind velocity U ( x , y , z ) = U(z), and the average values of the other meteorological parameters of the SBL-density $(x,y,z) = ~ ( z )pressure , ~ ( xy ,, z ) = ~ ( z )air , temperature T ( x , y , z ) = T(z) and air humidity 4(x, y , z ) = q(z) are only functions of altitude and are independent of time. With these assumptions, the Reynolds momentum transfer equation (6.2.30b), the mass diffusion equation (6.2.43) and the thermal conductivity equation (6.2.49) become much simpler. As we shall see, these assumptions allow us to conclude that the fluxes of vertical momentum, heat and mass transfer in the SBL are almost constant, and so are equal to the exchange fluxes across the sea surface, a place where they are impossible to determine directly. The horizontal homogeneity assumed for flowing air (i.e., the assumption that the average horizontal gradients of the atmospheric parameters are zero, e.g., a6/ax = aU/dy = 0, ae/ax = a@/dy = 0) is often justified in practice over not too large areas of the open sea well away from shores. This assumption is, however, not valid for areas close to shores, in fiords, at ocean current boundaries, etc., where there is strong horizontal heterogeneity and the atmosphere is vertically unstable owing to the diversity of the substrate. The assumption that the wind is statically stationary act)= const, a u / a t = 0 in the SBL holds in practice in stable weather (though not when atmospheric fronts are passing) and only during periods shorter than the periods when the effectsof diurnal changes in atmospheric parameters make themselves felt. In conditions in which the above assumptions are tenable, we can speak of certain fixed vertical distributions or profiles of meteorological parameters, whose gradients are vertical and define the vertical turbulent and molecular exchange
7 SMALL-SCALE AIR-SEA INTERACTION
378
of energy and mass right down to the sea surface. The intensity of the vertical turbulent exchange of energy and mass is strongly dependent on the turbulent movements of the air during which heat, water vapour, gases, etc. are trans. ferred. Wind friction on the sea surface causes the horizontal air flow to lose momen. turn. The wind velocity thus decreases as the surface is approached. This gives rise to a vertical gradient of horizontal air-flow velocity which induces a down. ward turbulent flux of horizontal momentum transfer. This turbulent momentum flux acts on the sea surface, bringing the top layer of water into horizontal turbulent flow. The water surface thus becomes mobile and rough, which potentiates its "feed-back" effect on the air flow-by forces of reaction on friction and pressure fluctuations. The complexity of this process is further complicated by the Coriolis force which, strictly speaking, ought to be taken into consideration in the Reynolds equation (6.2.30b). As a result of the Coriolis force, the vectors of the mean wind velocity and the induced flow of water have different directions (see Chapter 1, equation (1.2.43), Ekman spiral). The boundary condition on the sea surface is assumed to be the continuity of a vertical flux of mechanical energy from air to water (Schmitz, 1962; Roll, 1965). Consequently, the velocity components of turbulent flow, normal to the surface must be equal in the water and in the air at every instant for as long as the process lasts-this is the kinematic boundary condition. At the sea surface, moreover, the tangential stress vectors to (see Chapter 6, equations (6.2.34), (6.2.31)) of the air acting on the water and of the water acting on the air must balance out-this is the dynamic boundary condition (Schmitz, 1962). Of course, this does not have to mean that the vectors of the average flow velocities on either side of the surface are equal. There is usually some skidding of the air over the more slowly flowing water. But every excess of work done by the average air flow against the turbulent friction forces of the water surface which has not been converted into the energy of the average water flow must be converted into the energy of its turbulent motion. If vertical gradients of heat, gases, vapour, concentration etc. exist at all, their fluxes accompany the vertical turbulent (or molecular) momentum flux (see Chapter 6). In order to describe these fluxes we apply approximations based on assumptions derived earlier. Let us take a stationary and horizontally homogeneous wind whose velocity has components u1 = U,(x,)+u;,
(7.1.10)
7.1 FLUXES OF MOMENTUM, MASS AND HEAT ACROSS THE SEA SURFACE
379
where x3 = z ; the generalized notation is used so that the Navier-Stokes equation (6.1.50) or the Reynolds equation (6.2.30b) can be employed. We then apply the so-called Botlssinesq approximation : we assume that the atmosphere as a medium is statistically stratified, i.e., its density e = e(x3), temperature T = T(x3), pressure p =p(x3)
(7.1.1 1)
are only functions of height above sea level, and that they differ only very slightly from certain standard (fixed) values of p a , T,, pa for the SBL where the forces of gravity and buoyancy balance out (7.1.12)
so that the hydrostatic equation (see Section 1.2) is satisfied. Hence the density in the SBL is almost constant e z e,, so to a good approximation the continuity equation of the medium (6.1.51) simplifies to (7.1.1 3)
i.e., the divergence of the flow velocity of the air is zero. Deviations from the standard values (7.1.14a) s p = p’ = p - Pa
are very small /@‘I < ea, so that although their effect on buoyancy is noticeable, their influence on friction and other forces can be disregarded. When averaging the Navier-Stokes equation of motion (cf. (6.2.19)), we thus have to take into consideration the effect of changes in these parameters in the term (eg - a p / i 3 ~ , ) 8 ~where , ~ , Bi,3 is the Kroneker delta, whereas in the other terms (using the Boussinesq approximation) we can assume a practically constant air density equal to e . Since, furthermore, the deviations e’, p’, T‘ from the standard values e., p a , Ta are very small, these slight changes in density can be assumed linear @ =
@,+p:
p = pa-t-p:
T
=
TafT’,
(7.1.14b)
i.e., we can put only the first (linear) term in the expansion of these functions into a power series; see e.g., (3.5.2). It is convenient here to use the same dashes
7 SMALL-SCALE AIR-SEA INTERACTION
380
as for ordinary turbulent fluctuations, though strictly speaking, they are not deviations from the average but from the standard values, so that the mean values of these deviations differ from zero -
-
p’#O, T ’ Z O , YfO. It is evident from (7.1.14b) that the algebraic sum of the gravity and buoyancy forces which appears in the third component of the equation of motion is
(7.1.15)
so in view of (7.1.12), (7.1.16) which holds when e’/e < 1, i.e., in practice in the SBL, and indeed, everywhere in the sea. When p B p’, we also have aplax, = @‘/axi. The assumption that the velocity divergence is equal to zero (7.1.13), the replacement of the left side of (7.1.16) by its right side in the equation of motion, and the replacement of the density e by its standard value en in the other terms of the equation of motion together constitute the main essence of the Boussinesq approximation. On the basis of (6.2.30b) and also taking the Coriolis forcef, into account, the averaged equation of motion in the Boussinesq approximation can be generally written
where is the coefficient of dynamic molecular viscosity, expressed in [N s/mz]. On the left side of this equation we recognise the mass of unit volume (density iiU.
ea) multiplied by its average local acceleration 2 and its average advectional at aii . while on the right side we have the volume forces of molacceleration U 2 ,
ax,
ecular viscosity qV2Ui and of turbulent viscosity
~
a (e,u%), axj
the difference between
1
aj: the average excesses of gravity and the pressure gradient ergsi, - -__ ,and the
(
axi
components of the Coriolis force (f&, all expressed in [N/m3]. The assumption that the average flow of air takes place only in a horizontal direction (see 7.1 .lo) and that in the horizontal it is statistically uniform reduces
7.1 FLUXES OF MOMENTUM, MASS A N D HEAT ACROSS THE SEA SURFACE
volume force of molecular viscosity in another way qVzUi =
381
a
__
2Xj
Equation (7.1.17) will then read (7.1. IS)
aiii In the first term on the right in the bracket we see the sum of molecular q __
axj (6.1.36) and turbulent stress tensors (-pas) (6.2.31). The second term in the bracket represents the resultant of the unbalanced forces of gravity and buoyancy. If we assume a horizontal, statistically uniform wind, only the derivatives from x3 E z, that is, the vertical gradients of the horizontal stress components, from among the stress tensor derivatives a/axj , differ from zero. In other words, only the vertical gradients of the vertical fluxes transferring horizontal momentum differ from zero. If the hydrostatic equilibrium of the atmosphere is neutral, we have to take into consideration only the following two averaged components of the equation of motion when we expand (7.1.18)
(7.1.19a) (7.1.19b)
A further wind stationarity assumption means that G ( t ) = const and G ( t ) = const, the derivatives over time of these wind velocity components (left sides of equations (7.1.19)) are equal to zero. Hence SO
(7.1.20a)
(7. I .20b)
ie., when a fixed stratisfied wind is blowing, the internal friction and Coriolis force are equal. This was mentioned in Chapter 1. Now we must assess the value of the Coriolis force; this will show that in the SBL it is small enough to be ignored. At intermediate latitudes, the Coriolis force is roughly equal to the product of the density of the medium ea, the wind
7 SMALL-SCALE AIR-SEA INTERACTION
382
velocity U and the angular velocity of the Earth's rotation w (see (1.2.33)).The density of the air in the SBL is ea = 1.23 x g/cm3, i.e., about 1 kg/m3; we can assume the wind velocity U to be c. 1 m/s, while the angular velocity of the Earth is w = 7.29211 x lods s-l, i.e., of the order to per second, N/m3. The force fc ea U w is therefore around At the same time, it is clear from equations (7.1.20) that the total vertical lfiomentum flux emerges from the vector sum of wind stresses N
(7.1.21)
7 = 7,+TY,
where (7.1.22a) av
z y = q---e,v az
__ , I
w
.
(7.1.22b)
The value of this flux is t=
jt:+z;.
(7.1.23)
The thickness h, of the SBL is defined as being a layer in which the presumably constant flux z may vary with height by 10% at most. In view of equations (7.1.20), the constancy of this flux implies that Cf& z Uc),z 0, that is, we disregard the Coriolis force, and hence the derivatives of the flux at,/az z az,/az z 0 and t xz const. If we wish to ignore the force cf,), in the h, layer, the friction, e.g.,
__ a2
z
_ I
, must be at least one order of magnitude
hs
greater than the Coriolis force, i.e.,
__
z lOCf,), z 10fc. If we take
tx
h S
0.1 N/m2, a typical value at the ocean surface, a 10%change in z, corresponds A% 0.01 N/m2 = 10m. to the value A t , = 0.01 N/m2 and hence h, z ___ 10fc lox N/m3 The thickness hs of the SBL thus estimated may, however, change by one order of magnitude (from 1 m to 100 m, depending on the wind velocity and other factors), so it is often assumed to be some 60 m thick (see Zilitinkevich et a]., 1978). Therefore, in a roughly 10 m thick SBL we can assume that the total vertical momentum flux and other fluxes are constant and equal to the corresponding fluxes at the sea surface. The assumption of statistical stationarity and homogeneity of a horizontal wind in a stratified SBL thus leads to this layer being "N
7.1 FLUXES OF MOMENTUM, MASS AND HEAT ACROSS THE SEA SURFACE
383
defined as a constant flux layer. As we have seen, this statement is only approximately valid and is insufficient to describe the atmosphere, although it is enough to describe the fluxes close to the sea surface and their flow across this surface. Under the above assumptions, when in equation (7.1.20) we can disregard the Coriolis force as being small in comparison with the friction, the derivative with respect to z of the sum of vertical components of the molecular and turbulent momentum fluxes is equal to zero, which proves that these fluxes are constant in the vertical dU
ty =
__
-.pau‘w’ z const,
(7.1.24a)
q - - - ~ ~ v ‘ w ‘ z const.
(7.1.24b)
z, = q-
a2
az
~
3Z
The conditions which we have assumed in order to justify the constancy of the vertical momentum flux are idealized, but even so they frequently approach reality in the few-metres thick layer of the atmosphere above the sea surface (see Zilitinkevich et aZ., 1978). If the stress in the SBL T = const, we can introduce a new set of coordinates whose x-axis points in the same direction as the stress T (the sum of the former and TJ, assuming that this is also the direction of the wind velocity which has components u = U+u‘, v = O+o’, w = O+ w’. Usually, we can also neglect the molecular flux in the atmosphere rau/az as being small in comparison with the turbulent exchange Then expressions (7.1.24), in the new system of coordinates, with the x-axis pointing in the direction of the wind stress T = const = T ~ reduce , to the simple expression (if we disregard the molecular flux)
u,
earn.
~
t
z eau’w’z zo z const,
(7.1.25)
where zo is the shearing stress of the wind on the sea surface, directed along the x-axis in the new set of coordinates (this is a measure of the momentum flux across the surface). We assumed earlier that the density pa of the SBL was almost constant; more precisely, we now assume that the gradient of this density is equal to the adiabatic gradient (see Chapters 1 and 3), i.e., that the hydrostatic eqilibrium of the atmosphere is neutral. Under such conditions, turbulence is isotropic, induced only by the mechanical energy of the wind (see Section 6.2), and satisfies Prandtl’s postulate (6.2.34) which can be written briefly as
(7.1.26)
7 SMALL-SCALE AIR-SEA INTERACTION
384
where I = :I is the Prandtl turbulent mixing length (see (6.2.33)). After this equation has been rearranged and t/ea denoted by u:, it reads (7.1.27) This defines an important auxiliary parameter known as the local friction velocity u* for a wind at any height within the SBL. The local friction velocity at the sea surface is usually denoted by the same symbol with an additional subscript 0 (i.e. u * ~ just ) as the wind stress at the surface is to.I n general, the shearing stress can thus be expressed in terms of the local friction velocity: it is propor. tional to the square of this velocity t = u*Qa, 2 (7.1.28a) whereas at the sea surface To =
(7.1.28b)
u2,oe..
For the SBL in a coordinate system with the x-axis pointing in the direction of the stress t and the z axis pointing vertically downwards __ t = c0a u’w’ = eau: =
Qau:o =
to.
(7.1.29)
We can therefore determine the momentum flux t across the sea surface and hence the exchange of the wind’s kinetic energy E, from measurements of the velocity fluctuations of the wind in the SBL. When estimating this flux one can also make use of the local friction velocity of the wind, a typical value of which (Busch, 1977) corresponds to the order of magnitude of the typical shearing stress at the ocean surface t o,N 0.1 N/m2a and is equal to u* ,N 0.28 m/s according to equation (7.1.29). Of course, these are only rough values, which can vary by a factor of 10 depending on the wind. Implementing the coefficient of turbulent exchange of momentum in this simplest case of a horizontal wind directed along the x-axis, the momentum flux can also be expressed by equation (6.2.32); this will be done in the ensuing discussion. The experimentally determined mixing length I appears in the definition of the local friction velocity u* (7.1.27). Experiments performed when the SBL was in neutral equilibrium showed that I is a linear function of the distance z from the boundary surface (Roll, 1965). For a flow over a roughened sea surface it can be written as 1=
X(ZfZo),
(7.1.30)
7.1 FLUXES OF MOMENTUM, MASS AND HEAT ACROSS THE SEA SURFACE
385
where the coefficient ~t= 0.4 is called the Karman constant (i.e., a universal dimensionless constant), whereas zo is the surface roughness parameter measured in metres and is dependent on the shapes and sizes of the surface waves. From the empirical equation (7.1.30) we may conclude that the mixing length at the sea surface itself is finite and is equal to I, = xz,. In the “constant flux” layer u* = const, so from (7.1.27) and (7.1.30) we obtain the simple differential equation
(7.1.3 1) whose solution leads to the well-known logarithmic vertical distribution of the average wind velocity in the SBL -
u*
U(z) = -In 31
z+zo -__
(7.1.32a)
ZO
At heights z much greater than the heights of the waves, i.e., when z 9 z o , this distribution simplifies to
(7.1.32b) This vertical distribution of the average wind velocity c(z) with the wind stress t(z) = zo z const, provides a good description of reality when the SBL is in neutral hydrostatic equilibrium, and frequently also holds to a good approximation in somewhat more complicated real conditions (see e.g., Smith, 1980; Wu, 1980). From this we can determine the vertical gradient of the wind velocity from a measurement of it at one standard height h = - z (e.g., 10 m) and describe the vertical momentum flux with the aid of the coefficient of momentum exchange (see (6.2.32) or (6.2.39)) or efse using the drag coefficient C, of a windroughened sea surface €or a wind flowing around it at height h (see equation (L2.41a)). So by substituting (7.1.30) and (7.1.31) in (6.2.35) (where we made I F I;), we obtain the coefficient of vertical turbulent momentum exchange in the SBL as
K(”)= .(Z
+20) u* .
(7.1.33)
On the other hand, we can express the momentum Aux across the sea surfice per square metre of surface on the basis of (6.2.32) and (7.1.29) thus:
(7.1.34)
7 SMALL-SCALE AIR-SEA INTERACTION
386
By combining these last two equations and also by using other expressions given above, we can determine the momentum flux -co in a variety of ways. If we compare equations (1.2.41a) and (7.1.27), it is immediately apparent that the drag coefficient of the wind (or the coefficient of flow around a rough surface) C,,, defined from the average wind velocity v
t o = calk: =
e a c h u2(h).
(7.1.35b)
This last relationship can also serve to determine the exchange of momentum using, e.g., (7.1.32). But the problem here is to define the roughness parameter of the sea zo and the local friction velocity u*. Experimental work has led to a number of approximate links between these values. One which is particularly well known and which has been confirmed by many workers (Wu, 1980; Garrat, 1977) is the Charnock formula (1955) zo
=
a -4 , g
(7.1.36)
in which a is a dimensionless proportionality constant (the Charnock constant), and g is the acceleration due to gravity. This equation suggests that the roughness parameter z,, is proportional to the square of the friction velocity, i.e., proportional to the wind stress at the sea surface (we have assumed u, “N u*~).The Charnock constant takes a wide variety of values in changing conditions and has to be determined experimentally (Roll, 1965; Kraus, 1972; Smith, 1980). This description of momentum exchange (and a similar one of heat and mass exchange, about to be presented) becomes rather more complicated when the atmosphere is in non-neutral hydrostatic equilibrium (see condition (6.2.3)). A more universal description of the profile of the wind and other environmental characteristics resulting from turbulent exchange on any scale requires the introduction of the “stability length” or the Monin-Obukhov scale of Zengtli L = - u:/x/?w’e’, where 8‘ is the fluctuation component of the potential temperature, and /3 = g / T is the buoyancyparameter for the atmosphere. A scale_. of velocities which is the friction velocity u, and a temperature scale T, = -w‘W/xu* are also needed. Using these scale “units” we get the dimensionless characteristics
7.1 FLUXES OF MOMENTUM, MASS AND HEAT ACROSS THE SEA SURFACE
387
of the environment (e.g., the dimensionless, universal wind profile-see Zilitinkevich el al., 1978) as universal functions of the dimensionless value 6 = z/L. This value is very small when the atmosphere is in neutral hydrostatic equilibrium, where the effect of buoyancy on the vertical movement of the medium can be ignored. The Monin-Obukhov theory of similarity (Monin and Obukhov, 1953, 1954) is applied to the description of the SBL using the above-derived values. The literature is extensive: Zilitinkevich et a/. (1978); Smith (1980); Wu (1980); Coulman (1979); Kitaigorodskii et al. (1973); Kraus (1972). The Fluxes of Sensible Heat Qh and Latent Heat Qe, and of the Mass of Water Vapour Me
The description of the vertical exchange of heat and mass is again reduced to the simplest case in which all the previous assumptions are valid. These lead to the concept of the SBL as a layer of constant fluxes. We shall first deal with the heat exchange equation (6.2.49) in which we replace the absolute temperature T by the potential temperature 8. This is because the absolute temperature can change even as a result of air being adiabatically compressed during its vertical motion, and such changes do not reflect any inflow of heat energy. We therefore assume the process to be stationary, that is, the vertical heat flux in the SBL is constant in time, hence ae/at = 0; we assume further the horizontal thermal homogeneity of the air, so that l#/ax = ae/ay = 0. As a result, when we have horizontal wind flow U3 E il; = 0, and in (6.2.49) the term uj(a6/axj)also equals 0, so that the entire left side of this equation is equal to zero. On rewriting the term describing the molecular thermal conduc-
-
tivity in air yaV28 =
a
,the averaged thermal conductivity equation
(6.2.49) under these assumptions reads
(7.1.37)
where Qs is a source function, added here to draw attention to the possibility Of heat being evolved as a result of radiation, condensation of water vapour etc. [J/m3s]; 8 and 8’ are the average and fluctuation components of the potential temperature respectively [K], ya is the coefficient of the molecular thermal conductivity in atmospheric air [J/m K s]; CP,@is the specific heat of atmospheric air at constant pressure [J/kg K]; pa is the density of air, as above [kg/m3].
7 SMALL-SCALE AIR-SEA INTERACTION
388
We recognize the vertical molecular and turbulent heat fluxes per unit area inside the bracket of (7.1.37). If now we again assume that there are no heat sources in the atmosphere (or that they are relatively small QsFZ 0), the derivative with respect to z of the sum of these fluxes is equal to zero, that is, their sum is constant in the vertical and equal to the sensible heat flux Qh [J/s mz] transported vertically from the sea surface into the atmosphere __
Q, = ~,C,,,w'13'-y,
ae az
= const.
(7.1.38)
Note that we have altered the signs so that the heat flux supplied to the sea is positive (the z axis points vertically downwards). The heat flux Q, can reverse its direction, i.e., instead of passing from the sea to the air it may travel the other way (inversion). Analogous reasoning with respect to the water vapour concentration in the -
8C
SBL using the diffusion equation (6.2.43) allows us to state that __ = 0 and
-
that
ac Uj-__
at
= 0, where C
ax,
=q
is the concentration of water vapour in the air,
i.e., the absolute humidity of the air. After rearranging the molecular diffusion term D,V 2-q -
"1 a (D"3q
- axj
this equation again takes on a simplified form (7.1.39)
where M, is a source function intended to draw attention to the possibility of water vapour being evolved (e.g., from water droplets in the air) or condensed [kg/m3s]; 4, q' are the average and fluctuation components of the air humidity [kg/m3]; D, is the coefficient of molecular diffusion of water vapour in air expressed in [mz/s] (see (6.1.1)). When conditions are favourable, we can again disregard internal water vapour sources and sinks in the SBL as they are small in comparison with the quantities of water vapour rising vertically from the seainto the air, so that Ms z 0. The derivative with respect to z of the sum of the molecular and turbulent water vapour flux (in (7.1.39)) is thus nearly zero, so the sum of these fluxes is roughly constant
ay. z const. Me = w'q'- D, -~
az
(7.1.40)
We have altered the signs so that the mass flux lost by the sea is negative.
7.1 FLUXES OF MOMENTUM, MASS AND HEAT ACROSS THE SEA SURFACE
389
This water vapour flux Me [kg/m2s] also raises the latent heat of evaporation from the sea surface into the air. Water molecules tear themselves away from the hydrogen bonds and get into the atmosphere at the expense of heat energy drawn from the sea. They take this energy up into the air in the form of latent heat Q e . The heat removed from the sea by 1 kg of water vapour is equal to the heat of evaporation of the water given by L. The latent heat of evaporation L [J/kg] multiplied by the flux density of the raised water vapour mass Me [kg/s m2] thus yields the flux density of the latent heat Qe [J/s m2] removed together with the water vapour from sea to air. (7. I .41) LM,. By substituting the water vapour mass flux given by (7.1.40) in this last equation we get the flux density of the latent heat in the form Q,
=
aq z const. LD, __
az
(7.1.42)
With the exception of a thin layer of air adjacent to the sea surface, of which more below, the molecular fluxes in the air are usually very small in comparison with the turbulent ones
This is why molecular fluxes are often omitted from expressions (7.1.37) to (7.1.42), which are therefore greatly simplified. The constant fluxes in the SBL thus defined, i.e., equal to the fluxes crossing the sea surface, can be studied directly by measuring the fluctuation of the wind velocity w‘, the potential temperature 8‘ and the humidity q’ of the air. In Chapter 6 mention was made of methods of measuring the fluctuations of velocity and temperature in the sea (Ozmidov and Yampolski, 1965; Squier, 1968; Siddon, 1969, 1974; Ozmidov et al., 1974; Gibson et al., 1975; Belaev, 1976; Osborn and Crawford, 1977). Many of these methods are equally suitable for the air. Measuring fluctuations in air humidity q’ is more difficult, but it can be done with optical sensors (UV or IR absorption) or with electrochemical sensors. In the atmosphere, fluctuations of these parameters are either measured in one fixed spot with their correlation moments being averaged in time, or else they are measured in space (from an aircraft, assuming stationary turbulence) with their correlation moments also being measured in space (see Smith, 1974, and the papers cited by Busch, 1977 and Zilitinkevich et al., 1978).
7 SMALL-SCALE AIR-SEA INTERACTION
390
Since molecular fluxes can be disregarded in equations (7.1.37) to (7.1.42)they are small in comparison with turbulent fluxes-these latter can also be determined on the basis of the hypothesis that they are proportional to the respective temperature or humidity gradients (see equations (6.2.32) for momentum, (6.2.44) for mass and (6.2.50) for heat). The fluxes so described can be expressed by the appropriate coefficients of turbulent exchange and gradients of average potential temperature or average air humidity, respectively. Obviously, in this simplest case, in the constant flux layer, they can be written as (7.1.43) (7.1.44) (7.1.45) and K(") are respectively the coefficients of vertical turbulent exchange where PQ) of heat and mass in this simplified SBL structure; and 4 are the average values of the potential temperature and air humidity, respectively. Like the momentum coefficient KC"),the turbulent exchange coefficients of heat K(Q)and mass K(") are not inherent properties of the medium but are properties of a given turbulent flow (the wind flow in this case). Therefore, they are not fixed, but different for flows having a different characteristic scale of turbulence; this was explained in Chapter 6. Consequently, they are more readily studied with the aid of equations (7. l .43)-(7. l .45) by directly measuring the fluctuations and mean gradients of the appropriate physical quantity than used to determine fluxes. When determining the turbulent exchange coefficients we can also make use of their connections with the Prandtl or Schmidt numbers (6.2.53) and the momentum exchange coefficient in the form (7.1.33), and indirectly with the local friction velocity u* and the surface roughness parameter zo . If we ignore the molecular fluxes and assume a simple case where the turbulent heat and vapour fluxes in the SBL are constant, the gradients of the average
ae and the average air humidity potential temperature of the air -
aZ
~
a. az
can be taken
to be roughly constant in the vertical, regardless of the height h = therefore approximate them by means of finite equations -
84 __
aZ
N
-
~
ae az
N"
-2.
Go-$, ~
-2
We can
and
40--4z , where the subscript 0 denotes a given quantity just above the sea
-___ -2
7.1 FLUXES OF MOMENTUM, MASS A N D HEAT ACROSS THE SEA SURFACE
391
surface and the subscript z denotes the height h = -2. So by measuring the difference in average potential temperature and humidity on two levels, we can make an estimate of the coefficients of turbulent exchange in equations (7.1.43)-(7.1.45) and determine the exchange fluxes. But the exchange coefficients depend in general on the characteristic scale of the turbulence which is affected by the wind field, sea-surface roughness and the hydrostatic equilibrium of the atmosphere, here assumed to be neutral. These coefficients must therefore be determined for each individual situation, unless one uses a different method based on the Monin-Obukhov similarity theory (Monin and Obukhov, 1953, 1954; Zilitinkevich and Monin, 1974; Busch, 1977; Zilitinkevich et al., 1978). The ratio of fluxes of the sensible heat to the latent heat in water vapour Qh/Qe is often called the Bowen ratio and is an aid in the' estimation of these fluxes. For the large scale of the ocean, this ratio (averaged from a large number of experiments (Jacobs, 1951)) is estimated at between 0.1 and 0.2. This implies that the averaged latent heat flux removed from the sea to the air along with water vapour is 5 to 10 times greater than the sensible heat flux transmitted by the sea surface into the atmosphere. Among other things, this is due to the high latent heat of evaporation of water L,which is 2.494 x lo6 J/kg at 273 K and 2.448 x lo6 J/kg at 293 K. Bearing in mind the fact that on average the surface of the world ocean loses a roughly one-metre thick layer of water to the atmosphere every year, we can obtain a rough value for the mean flux density of the vapour mass Me z 3 x x ~ O - kg/m2 ~ s. Hence, from (7.1.41), the averaged latent heat flux Qe z 74 J/m2s, while Qh is around 10 J/m2s.
The Laminar Surface Layer
In spite of the mainly turbulent transport of momentum, heat and mass in the atmosphere, the mechanical exchange of heat across the air-water interface is a molecular process, described in Section 6.1. Experiments on the vertical gradients of wind velocity and air temperature or humidity over the water have shown that these gradients are quite steep just above the air-water interface (see Businger et al., 1971; Foken et al., 1975). This is caused by, among other things, the adhesion of the air to the water surface and the friction of the wind against the surface, abruptly lowering its average velocity at the surface itself and reducing the Reynolds number Re to below its critical value Re, (see equation (6.2.1)). As a result, there exists a roughly 1 mm thick laminar layer of air above and a similar laminar layer of water just below the air-water interface. These surface
392
7 SMALL-SCALE AIR-SEA INTERACTION
layers are not subject to turbulent mixing until they are torn apart by breaking wave crests or by bursting air bubbles. Thermal conductivity across these thin layers must therefore be molecular. But this molecular flux may be very considerable, seeing that, as we stated in Section 6.1, it is approximately proportional to the temperature gradient, which at this point is very steep (Foken et al., 1975). For instance, if the average temperature difference across the air-water interface over a distance of 1 mm is c. 0.5 K, this would correspond to a vertical temperature gradient of 500 K/m. However, in the air up to a height of c. 60 m above sea level (in the SBL), the average vertical temperature gradients as a a result of turbulent mixing of the air are scarcely of the order of 0.01 K/m. They are thus four orders of magnitude smaller than in the laminar surface layer. This ratio of temperature gradients is this the inverse of the ratio of the orders of magnitude of the molecular and fine-scale turbulent coefficients of the vertical heat exchange. This ensures that the molecular heat exchange across the sea surface and the turbulent heat exchange in the air column are about equal in intensity. The thickness of the laminar surface layer of air 6, depends on the distance z = 6, from the water surface at which the Reynolds number Re = Uz/q,,,. (see (6,2.1)) takes its critical value Re,, essential for the initiation of turbulence ( c i s the air flow velocity, qU,, is the coefficient of kinematic viscosity of air in this layer). In this viscous layer, assuming a smooth water surface, the coefficient of momentum exchange becomes the coefficient of molecular viscosity K(")-+ vu, and in view of (7.1.34) we can write (7.1.46)
On integrating this expression, we have (7.1.47)
Using this last expression for z = 6, and the expression for the critical Reynolds number Re, = ~(da)6,/qu,,, we can eliminate Cf(S,) and obtain the thickness of the surface laminar layer of air over a smooth sea surface in the form (Kraus, 1972) (7.1.48)
Assuming further that at the boundary of the laminar layer with a smooth surface the coefficient of molecular exchange becomes equal to the coefficient of
7.1 FLUXES OF MOMENTUM, MASS AND HEAT ACROSS THE SEA SURFACE
393
turbulent exchange qv,, s K$“ and implementing-(7.1.33) for z+z, = do, we obtain the expression
(7.1.49) from which we can calculate that with the kinematic viscosity of air qU., being around 1 . 4 ~ m2/s and the friction velocity u* being, say, 0.1 m/s, the thickness of the laminar layer of air 8, is about 0.55 mm. Under real conditions, where the surface is wind-roughened, this thickness is given by a more general equation
(7.1.50) where k is a new dimensionless parameter dependent on the surface roughness parameter zo that takes values of from 5 to 10, usually around 7 (see Foken et a/., 1975; Kitaigorodskii et al., 1973). Using the values of q”,, and u* given above, the thickness 6, z 1 mm. Krauss (1972) defines the ratio of the thickness of the laminar surface layer of air 6, to the laminar surface layer of water 6, (under the surface) as
(7.1.51) where qu,oand quVware the respective coefficients of kinematic viscosity of air and water, and 8, and 6, are the respective densities of air and water. Hence molecular processes under the water surface determine exchange within a roughly 1-2 mm thick layer. We have already mentioned the cooling of this laminar surface layer of the ocean due to evaporation-often by 0.3-0.5 K and more-as compared to the temperature of deeper water, which can be measured with a thermometer (see Dubov, 1974; Fedorov et al., 1979). In studies of this layer, it is therefore important to determine the surface temperature of the sea from measurements of its IR radiation (Aumann and Chahine, 1975; Hornam, 1976; Galazi et al., 1979; Lintz and Simonett, 1976). Fluxes of Water Droplets, Salt Particles and Electric Charge
Water droplets expelled into the atmosphere by bursting air bubbles play a special part in the exchange of water and sea salt. Produced mainly by windinduced waves at the surface (see Section 2.8), the bubbles are forced some way down into the sea along with the eddies. That they are usually small (see Fig.
7 SMALL-SCALE AIR-SEA INTERACTION
394
2.8.1) is due to the hydrostatic pressure and the solubility of gases in water, Buoyancy forces raise the bubbles to the surface at a velocity of around 10 cm/s. On rising, they expand because the pressure decreases as the surface is approached (see equation (2.8.2)): a bubble whose radiusis, say, 20 pm a few metres below the surface has a radius of about 1 mm at the surface itself. Having reached the surface, the bubble is overlain by a skin of water which is due to surface tension forces (hydrogen bonds-see Chapter 2). The size of such a water skin at the surface is estimated to be 3 x lo4 pmZ. It contains some 3 x l0lz water molecules which are ejected as droplets into the atmosphere when the skin breaks (see Fig. 7.1.3; McIntyre, 1965-see Horne, 1969). The mechanism by which
A
8
C
D
E
Fig. 7.1.3. Consecutive phases in the formation of drops by air bubbles bursting at the water surface according to the McIntyre model (1965) (adapted from Horne’s drawing (1969),with permission of John Wiley and Sons, Inc). A-air bubble rising and expanding in water; B-bubble held up at the water surface; formation of a water skin 2 ~ 3 1 thick containing c. 3 X 1012water molecules; C and D-the skin breaks and up to 20 water droplets 20 vm in diameter are ejected into the air; E-expulsion of 1-4 jet droplets from the bottom o f the bubble, containing sea salt ions (with positive ions prevailing).
seawater droplets are expelled into the atmosphere is a complicated one (Kientzler et al., 1954; Blanchard, 1963; Roll, 1965). There are two distinct phases: (1) the bubble’s skin breaks and ejects, to some considerable height in the air a group of up to 20 pure water droplets 1-20 pm in diameter from this skin, and (2) 1-4 large saline jet drops ca. 100 pm in diameter are ejected from the bottom of the bubble when this bursts. If the air bubbles are about 2 mm in di-
7.1 FLUXES OF MOMENTUM, MASS AND HEAT ACROSS THE SEA SURFACE
395
ameter, these water drops are thrown 15-20 cm into the air with a velocity of some 10 m/s. These “jet drops” are particularly important because they take seawater containing the ions of salts and other chemicals into the atmosphere. According to McIntyre, whose results are presented in Horne (1969), such a drop takes into the atmosphere about 3 x lo1’ molecules of water (about 4 nanolitre) and 30 nanograms of sea salt. Moreover, it raises into the air about 200 elementary positive electrical charges as salt ions become separated at the water surface and transfer a kinetic energy of about 4 erg (i.e., 5 x lo-* J). Jf we take into consideration the pressure in the bubble, given by equation (2.8.2), we shall see how significantly pollution of the sea surface can affect this process, as it reduces the surface tension ip. The drops thus lifted into the SBL are carried farther away by the turbulent motion of the air and the air-sea turbulent water, salt and charge fluxes. Blanchard (1963) found that with winds blowing at 10-30 knots, the jet drop fluxes was 300-1000 drops/m2s. In the air, these drops are wholly or partially evaporated, leaving salt crystals which also varry an electrical charge and can, if conditions are right, become condensation nuclei for water vapour. Mass fluxes moving in the opposite direction are atmospheric precipitation and aerosol fall-out into the sea (see e.g., Eagleson, 1970). The search for a method that would describe this aerosol exchange of matter between the sea and the atmosphere has been undertaken by many (see Blanchard and Woodcock, 1957; Toba, 1965; Garbalewski, 1977, 1980). Since the physicochemical processes involved in this exchange at the sea surface are so complex, Garbalewski (1977) has proposed and justified a description of the flux of the i-th component of the mass (of drops, salt, etc.) raised into the air with the air of a dimensionless emission coefficient Cit . The rising salt mass flux Mst[kg/m2s]can be described by the equation
Mst = CstS@,u,Re
for
z = zo,
(7.1.52)
where C, is the dimensionless coefficient of salt emission from the sea in drops. This postulates the proportionality of a given mass component taken from the sea to its concentration in the sea ( S Qfor ~ salt if the salinity is S ) and to the friction velocity of the wind u* and the Reynolds number Re = u*zo/rl,,, at level z = zo. The analogous equation with the immission coefficient of this component from the atmosphere into the sea C,. describe the downward flux of its mass M, .I,and the difference between these fluxes determines the effective mass flux participating in aerosol exchange and its direction (upwards or down-
396
7 SMALL-SCALE AIR-SEA INTERACITON
wards). Unfortunately, the coefficients of this aerosol exchange are variable and are being studied under diverse environmental conditions (Garbalewski, 1980). It has been estimated that the effective flux of sea-salt particles of radii less than 20 pm, lifted up into the air, is of the order of 1012-1013 kg per annum (Blanchard, 1963), i.e., the average flux density is around lo-* kg/mZs andis chiefly the result of bubbles bursting. This is from c. 40% to SO% of the total mass of aerosol particles emitted into the atmosphere from all sources (see Robinson and Robbins, 1971). Once the water has been evaporated from jet drops, the salt ions taken up into the atmosphere there form condensation nuclei for water vapour in the air. Most importantly, it is mainly positively charged ions that are lifted into the atmosphere. In this selection process, ions, and hence electrical charges are separated and carried up into the air (Blanchard and Woodcock, 1957; Blanchard, 1963; Piotrowicz, 1977; Piotrowicz et al., 1979). The atmosphere is thereby enriched in positive metal ions, including those of the heavy metals present in trace amounts in the sea (see also Duce et al., 1975, 1976). A model
Fig. 7.1.4. The model of a mechanism by which the atmosphere becomes charged with positive ions by jet drops that form when air bubbles burst. (a) The inflow of positive ions into the jet drop as a result of the formation of a double charge layer at the water surface and the stronger bonding of negative ions with the surface water skin (from Blanchard. 1963, by permission of Pereamon Press PLC); (b) theprobable distribution of water molecules and salt ions just under the water surface.
7.2 THE ENERGY BUDGET OF A BASIN
397
showing how positive ions come to prevail in the atmosphere has been demonstrated by Blanchard (1963). This model assumes that water molecules are arrayed in an orderly manner at the sea surface because of the hydrogen bonding (see Chapter 2). As water molecules lie with their oxygen atoms uppermost, the polarity of these molecules means that negative ions will tend to accumulate under a monomolecular layer of water molecules. These negative ions are attracted by the positively polarized hydrogen atoms in the water molecules. Lower still, for the same reason, a layer containing largely positive ions will form (see Fig. 7.1.4). These sub-surface positive ions enter the drop the moment the jet drop is formed, because the subsurface layer of medium is more fluid than the surface skin of linked water molecules. In this way, when the air bubble bursts, the jet drop becomes filled with seawater containing a greater concentration of positive than negative ions. As this mechanism of ion separation at the sea surface is highly probable, many workers have been inclined to accept it (see Blanchard, 1963 and the literature cited therein). When waves are whipped up by the wind, it is to be expected that inert water drops will rise from the crests of breaking waves. This direct transport of water and salts into the atmosphere is probably dominant as wave action develops (as waves attain dynamic equilibrium) in a strong wind. However, this process should not give rise to the separation of sea-salt ions. Significant changes in the exchange of mass between sea and air are to be expected in basins whose surface has been polluted and whose physical properties have thus been altered.
7.2 THE ENERGY BUDGET OF A BASIN AND ITS INFLUENCE ON THE STRUCTURE OF WATER MASSES
The fluxes of energy and mass crosssing the sea surface directly affect the thermodynamic state of a basin. The values of these fluxes define the boundary conditions at the free sea surface when describing energy and mass transport in the sea and the associated evolution of the thermohaline structure of water masses in the sea. Here, we need the transfer equations, derived in Chapters 5 and 6, which are a restatement of the laws of conservation of energy or mass in a unit volume of the medium in unit time. For radiant energy in [J/m3ss],the transfer equation is (5.4.21), for heat energy in [J/m3s], it is (6.2.49) (written for the potentiaI temperature B), and for the diffusing mass of a passive substance in [kg/m3sJ it is (6.2.43) Likewise, the momentum budget per unit
7 SMALL-SCALE AIR-SEA INTERACTION
398
volume of medium in a unit time in [kg . m.s-l/m3s] 5 [N/m3] which is the budget of volume forces, is described by the Navier-Stokes equation of motion (6.1.50) (instantaneous momentum) and by the Reynolds equation of motion (6.2.30) (the momentum of an averaged flow). In order to close the set of equations of motion (so that their number equals the number of unknowns), further equations are needed. These are made up from the continuity equation of the medium (6.1.51), the equation of state of seawater (3.1.5), and the Reynolds stress tensor-a special case of which is the equation of turbulent kinetic energy (to be described)-and other empirical and semiempirical equations. How this is done depends on the approximation methods used in the necessarily idealised description of the structure of water masses in the sea (see Gill and Turner, 1976; Niiler and Kraus, 1977; O’Brien et al., 1977; Zilitinkevich et al., 1978; Ramming and Kowalik, 1980). The equation for the flow kinetic energy budget can apply to the kinetic energy of an instantaneous flow 3
(7.2.1)
E,, = +puiui
(7.2.2)
Ek =Z! @uiui the kinetic energy of an averaged flow
---
and to the difference between these two quantaties, i.e., the kinetic energy of turbulent flow fluctuations-in short, the turbulent energy (7.2.3) At the same time, the kinetic energy (specific) of an instantaneous flow [J/m3] is the sum
Ek
=
E,+Et.
(7.2.4)
The Equation of the Turbulent Energy Budget The averaged Navier-Stokes equation of motion in the Boussinesq approximation can be written as (7.2.5) Remember that the generalized notation, discussed in connection with (6.I SO), is used here. The physical sense of the terms in this equation was discussed earlier when (7.1.17) was described. Notice here that the sum of the terms inside
7.2 THE ENERGY BUDGET OF A BASIN
399
the brackets expresses the total flux density of the momentum (the sum of in-
ternal stresses) of the average turbulent flow in volume Coriolis force and other possible volume forces involved (see Section 1.2). The conversion of (7.1.17) into (7.2.5) emerges from the following relationships :
a
_ _
aiii
-
Po u . __ , if the density p x Po = const and the velocity axj axi divergence aiij/axj = 0 (see the continuity equation (6.1.51));
(1)
__
(eouiu j )
=
J
a (2) --p
axj for i # j ;
(3)
,
6ij
aj-
= __
au
axi
4.. =7 2= $q aJ axj
since the unit tensor d i j
=
1 for i
=j
and 8,,
=
0
aiii (=+ ) ; , -auj is the molecular stress tensor (6.1.36) (see
also 6.1.37), whereas the right-hand equality of this expression results from the assumed symmetry of this tensor. The coefficient 17 [N s/m2J is the coefficient of molecular dynamic viscosity (see Section 6.1). The reference density ell, introduced for the atmosphere in the previous section, is denoted here by go for seawater and will continue to be treated as the reference density (the density for neutral hydrostatic equilibrium of the medium), with respect to which the deviations el = e - eo are counted. Multiplying equation (7.2.5) by the average flow velocity Ui leads us directly to the equation for the kinetic energy budget of an averaged turbulent flow E,, i.e., to a connection between a local change in this energy in unit volume of the medium (specific energy) in unit time aEa/at, and its inflow, dissipation and conversion into other kinds of energy on the one hand, and its formation due to the action of local sources (work done by pressure and other forces) on the other. So multiplying (7.2.5) by Ui and arranging the terms in accordance with (7.2.2) yields
(7.2.6)
The sum of the terms inside the brackets on the left side expresses the total flux density of the kinetic energy of an averaged turbulent flow [J/mzs] across the
400
7 SMALL-SCALE AIR-SEA INTERACTION
“walls” of the volume element dxdy dz of the medium under consideration, The individual terms of this equation, denoted A , B, C,etc., can be presented together with their conversion from the straightforward multiplication of the terms in (7.2.5) by U i :
(4Ui
a(eoui) =
at
a (+eouiiii) = 2. This term describesthe local (resultant) at
changes in specific kinetic energy (or energy density) of an average flow in a unit time [J/m3s]. These local changes are due to various causes described by the sum of the remaining terms.
term describes the local changes in the specific kinetic energy of an average flow in a unit time as a result of its advection, that is, the difference between the inflow and outflow of energy across the walls of the volume element considered.
nents of this product express respectively:
--
a
(BD) __ (Po ui uju,) is the contribution to the local changes in specific kinetic
3x1 energy of the average flow in unit time as a result of its turbulent exchange across the walls of the volume element considered (the turbulent exchange budget), and __
aui
( J ) - Po ul u; a.j are the local losses of the specific kinetic energy of an average
flow in unit time as result of its conversion into the turbulent energy (vorticity) of the flow (minus sign), or conversely, the gain in kinetic energy of an average flow as a result of the energy of turbulent fluctuations being converted into average flow energy. This last important term in (7.2.6) is present at the end of its right side. (BE) U j ___ a’di = ___ since the unit tensor djj = 1 only ifi = j , andis ax, ax, aiij 6 i j = 0 if i # j . As before __ = 0. This expression describes the specific kinetic
ax,
energy increment of an average flow in a unit time as a result of the work done by external pressure forces.
7.2 THE ENERGY BUDGET OF A BASIN
401
algebraic sum is at the end of the expression on the left side of (7.2.6) (BF) and describes the local change in the specific kinetic energy of an average energy Bow in unit time due to the exchange of this energy with the surroundings resulting from molecular friction. As before, qij is the molecular stress tensor (see 6.1.36). The second term in this sum ( I ) reads (7.2.7)
where 9 is the coefficient of molecular dynamic viscosity [N s/m*]. This expression describes the local losses of the specific kinetic energy of an average flow in unit t i e as a result of its viscous dissipation, i.e., conversion to heat (conversion to internal energy) due to molecular friction. The introduced quantity E , is frequently applied to abbreviate the notation of the equations and can be called the viscous dissipation rate of the kinetic energy-in this case of the energy of an average flow (subscript a) (see Zilitinkevich et al., 1978). As it has units of [mz/s3], it has to be multiplied by Po in order to express the specific kinetic energy lost by an average flow in a unit time [J/m3s] owing to viscous dissipation. (G)iijFgdi3 describes a local change in the specific kinetic energy of an average flow as a result of work done by buoyancy, which force causes the medium to flow in a vertical direction (ai, = 1 for i = 3, and di3 = 0 for i # 3). (H)UiE describes the local changes in the specific kinetic energy of an average flow in unit time [J/m3s] owing to the work done by other volume forces, including the Coriolis force. In the same way that we wrote the averaged equation (7.2.5), we can write the Navier-Stokes equation of motion for the instantaneous flow velocity (6.1 S O ) . On multiplying this by the instantaneous velocity ui and taking (7.2.1) into consideration, we obtain the equation for the specific kinetic energy budget of an instantaneous flow in unit time. This reads
The significance of the terms in this equation is the same as in (7.2.6), i.e., they describe the changes in the specific kinetic energy (kinetic energy density) of instantaneous motion in unit time, for the same reasons as before. Turbulent fluctuations are not given separately this time, so the terms D and J from (7.2.6) are absent. The final term in (7.2.8), describing the viscous dissipation of energy can again be denoted by Q ~ E where ~ , &k can be called the viscous dissipation rate of the kinetic energy of an instantaneous flow.
402
7 SMALL-SCALE AIR-SEA INTERACTION
In order to produce an equation for the kinetic energy budget of exclusively turbulent fluctuations of the flow velocity (the average energy of the turbulence itself), it is sufficient to subtract (7.2.6) from (7.2.8) and average out. If we take, into account equations (7.2.1) to (7.2.4), we get
(7.2.9) As in (7.2.6), the terms of (7.2.9) describe the changes in the specific kinetic energy (energy density) in unit time [J/m3s]-though in this case only the energy of the
turbulent fluctuations of the flow velocity, i.e., the energy of turbulence separated from the energy of an average turbulent flow. The sum of the terms in the square brackets expresses the density of the total kinetic energy flux of turbulent motion E, [J/m2s] crossing the walls of a given volume element of the medium. The terms in (7.2.9) describe the changes in the turbulent energy: ( A ) the local aE,/at which are the result of changes due to (BQ-the advection of turbulent energy, (BD)-its turbulent exchange, (BF)-its molecular exchange, (BE)-the work done by external pressure forces, and on the right side of the equation, (G) the conversion of the work done by buoyancy forces into turbulent energy (or the reverse), (H)the conversion of the work done by other volume forces into turbulent energy (or the reverse), (Z) he viscous dissipation of turbulent energy and its conversion into heat
and ( J ) the conversion of the kinetic energy of an average flow into turbulent energy or (sometimes) vice versa. This latter process is the conversion of the work done by turbulent friction forces (the work done by the turbulent stresses of an average flow) into the kinetic energy of the turbulent fluctuations of the
i I:-.
flow (or sometimes the other way round). In the sea this term - poujuj-is usually positive, which indicates that the kinetic energy of an average flow is being converted into energy of turbulence. Within a particular region of the sea, the kinetic energy of an average flow is very often the main source of the energy of turbulence. The reverse case, when the average flow receives energy at the expense of the energy of the turbulent fluctuations of the flow, is less
7.2 THE ENERGY BUDGET OF A BASIN
403
probable, although it does occur in the ocean, for instance, in the Gulf Stream (see e.g., Richardson 1980; Ozmidov et al., 1974). The values of the terms in the turbulent-energy budget equation (7.2.9) depend on the scale of turbulence and are the object of hydrodynamic research studies in the sea (see e.g., Monin, 1978; Launder and Spalding, 1972). Note also that the appearance in (7.2.9) of the third degree correlation moments __ U;uiu;, which are new unknowns, means that the set of equations of motion again cannot be closed. In model descriptions of energy exchange processes in the sea, simplifications reducing certain terms in these equations are made in order to solve them; they are the terms expressing stresses (second degree correlation moments) with the aid of coefficients of turbulent exchange and average flow gradients etc. (see Kraus, 1977). Numerical methods of solving these equations are also frequently applied (see Ramming and Kowalik, 1980). The Horizontally Stratijied Sea Model
The main fluxes of thermal and kinetic energy enter the ocean from the atmosphere by crossing the former’s free surface. The direct range over which this energy is transported into the sea is limited to the top layer of the ocean, called the active layer. Its thickness, usually from 100 to 300 mydepends on the amplitude of the annual variations in the fluxes of the vertical exchange of mass and energy with the atmosphere. This is why this thickness differs in various parts of the Earth, and is much greater in polar regions than in low latitudes. Energy thus flows from the atmosphere into the sea-mainly solar radiation (heat) energy, turbulent energy due to wind friction, and potential buoyancy energy as the surface evaporates and cools-while a largely turbulent flux of this energy flows down into the depths of the sea. This flux weakens with increasing depth and is periodically inverted when the surface sources of energy tail off. Energy afrom the atmosphere thus enters the upper layer of the ocean, raises its temperature, and after several conversions, returns to the atmosphere in the form of the radiant flux Q b , the heat of evaporation Qe and the other fluxes shown in Fig. 7.1.1. It is for this reason that the temperature of the waters in the upper layer of the ocean is usually higher than that of those deeper down, where heat from the sea surface hardly ever penetrates. The vertical flow of the greatest fluxes of energy and mass thus gives rise to the thermohaline structure of water masses in the ocean such that it becomes statistically almost homogeneous in the horizontal. So the average vertical gradients of temperature, salinity, pressure and density in the top layer of the ocean are several orders of magnitude
7 SMALL-SCALE AIR-SEA INTERACTION
404
higher than the average horizontal gradients of these parameters. For example, the temperature in the active layer of the ocean changes by as much as 10 K over a vertical 100 m, but by only 10 K over lo00 km horizontally. So the average vertical temperature gradient is greater than the horizontal one by a factor of 4 (see e.g., Figs. 1.1.3 and 1.2.5). On the basis of this, we can assume to a good approximation that the active layer of the ocean in the horizontal is statistically homogeneous, in other words, its horizontally stratified structure (a)
Temperature T 1-
275
__
I I
IKl
290
Winter I
J, -
I
Active
70
layer of the ocean
_ _ _ _ _ - - - - <
f
-E
i
r
,-
I
‘Seasonal
thermoctine
100
h-
f P n ‘Permanent
(principal) thermocline
Permanent thermocline layer
1000 Isothermal
deep layer
1
10000 ---_______--
(b)
Latitude
-
[* ]
0
E 100
u
5
2 200 300 Fig. 7.2.1. The vertical thermal macrostructure of ocean waters. (a) Diagrams showing the plainly variable seasonal thermocline and permanent thermocline separating the active layer of the ocean from deep waters whose temperature is fixed; @) examples of vertical temperature distribution at various latitudes (Wolff, 1979), illustrating the diverse thermal macrostructure o f the active layer of the ocean.
(a) 0.0
0
2.0
4.0
6.0
Temperature T [ K ] 8.0 10.0 12.0 14.0
(b)
T e m p e r a t u r e g r a d i e n t dT/dz[ K/m]
~ l ' i ' i ' l ' l ' l ' l ' l ' l ' l ' l ' ~ ' l ' l '
n
30
Fig. 7.2.2. The fine thermal structure of the upper layer of Baltic waters (from Druet and Siwecki, 1980). (a) The vertical distribution of instantaneous temperature with visible thermal microstratification of waters (Baltic 13 July 80, 17"00'E, 55"13'N); (b) the vertical distribution of instantaneous temperature gradients corresponding to the temperatures in (a) (time and place of measurements as in (a)). P VI 0
406
7 SMALL-SCALE ATR-SEA INTERACTION
is taken to be a model. Obviously, this is reasonable only in a model of a small or meso-scale sea area and cannot apply on a global scale. The assumption that the sea is statistically homogeneous in the horizontal and that the average flow velocity of the waters is along the horizontal as we$ enables us to reduce all the budget equations to equations of the one-dimensional function of the coordinate xj = z, as we did with the SBL (Section 7.1). Sim. plifying the description of the active layer of the ocean in such a manner provides one-dimensional models of it, and these are the most widely used in oceanology (see Niiler and Kraus, 1977; Zilitinkevich et al., 1978, 1979). We have already availed ourselves of the horizontally homogeneous sea in Chapters 4 and 5 where in describing the radiant flux, whose range in the sea is small, the as. sumption of a horizontally homogeneous sea is almost the rule. Now, while the water masses in the ocean are quasi-horizontally stratified, the transition from the warmer upper layer to the deep layers is a more or less complex one. Characteristic are the many thermally quasi-homogeneous sub. layers (Lineikin, 1974) and the sharp drops in temperature between the m which form thermoclines. It is possible to distinguish a diurnal thermocline near the surface, which exists in a calm sea and may reach depths of several metres; further down there is the seasonal thermocline which reaches a depth of around 100 m, and finally the permanent or principal thermocline which goes down to some 1000 m (see Fig. 7.2.1). In addition, careful measurements of the vertical temperature distributions reveal the fine structure of ocean water masses, i.e., the occurrence in the depths of the ocean of a large number of fine-scale unhomogeneities. There are numerous jumps and steps in the vertical temperature distributions, particularly within the seasonal or permanent thermoclines (see Fig. 7.2.2). We find quasi-homogeneous layers up to a few metres thick and differing but slightly in their physical properties-e.g., their temperatures may differ by scarcely tenth parts of one degree. These layers are, however, clearly separated by a steep temperature gradient. They are formed and transformed as a result of turbulence, the vertical diffusion of heat and salts in opposite directions, the inflow of water intrusions and other hydrodynamic processes; their study is the key to learning the mechanisms of these processes (see Fedorov, 1976; Woods, 1971, 1977; Osborn and Bilodeau, 1980; Druet and Siwecki, 1980; Monin and Ozmidov, 1981). The active layer of the ocean extends down to the lower boundary of the seasonal thermocline and is conventionally taken to be a layer going from the surface down to the depth at which the vertical temperature gradient below the seasonal thermocline falls to 0.1 Kjm (Filushkin, 1968). Below this lower
407
72THE ENERGY BUDGET OF A BASIN
boundary the water is practically thermally homogeneous and is not subject to any seasonal variations. On the basis of this criterion, the active layer of the ocean atlow and medium latitudes is barely 100 to a few hundred metres thick (Filushkin, 1968; Zilitinkevich et al., 1978; see also Figs. 7.2.1 and 7.2.2). It is thus a thin surface layer lying over the several-kilometres-deep ocean (average depth =e. 4 km). In polar seas where the great cooling of the sea surface favours the deep turbulent and convectional vertical exchange of waters (little work done against buoyancy forces; see condition (6.2.3) in Chapter 6), the active layer may go down much deeper, sometimes right down to the bottom. This is an important feature of polar seas which, together with the horizontal circulation of ocean waters around the globe, exerts an important effect on the renewal (refreshing) of the deep ocean waters all over the world. For a horizontally homogeneous sea with a homogeneous horizontal turbulent flow, the equation for the turbulent energy budget (7.2.9) can be considerably simplified. Advection and other components of turbulent kinetic energy transport in the horizontal direction can be ignored since in the horizontal, the density of this energy is constant by assumption. All these components of turbulent energy transport are described in (7.2.9) by the derivative-
a
()
ax,
with the complex sum of terms inside the bracket. The sum of these terms C + t D t E + F is the total flux density of the turbulent energy (7.2.10)
which is thus only a function of the depth z, and __ aE‘ = __ aE‘ - 0. Moreover, ax ay the vertical component of the average flow velocity W in a stratified sea is asm e d to be zero, so that u = U+u’, v = ?5+v‘, w = O+ w’.The term describing 2 vertical advection, i.e., the convection of turbulence energy -(E,W) = 0, az
also falls away from equation (7.2.9). Under such assumptions, the equation for the turbulence energy budget (7.2.9) now reads
--a
az
[$@O(U’ZW’+B’2W’+
W ’ W ) +p‘w’]-@o E .
(7.2.1 1)
7 SMALL-SCALE AIR-SEA INTERACTION
408
In this equation we have also disregarded the Aux of the molecular exchange of the turbulence energy, as it is small in comparison with that of the turbulent exchange of this energy, and we have neglected the action of the vertical fluctuations of other volume forces E 0), e.g., the vertical component of the Coriolis force which, in the opinion of Niiler and Kraus (1977), has no significant effect here. As before, the symbol E denotes the viscous dissipation rate of turbulent energy [m2/s3],so that
(pi
(7.2.12)
expresses this dissipation in [J/m3s]. The first two terms on the right side of equation (7.2.11) usually describe the work done by turbulent friction forces, i.e., the work of the shearing stresses z, = and sy = of an average (horizontal) flow, which is converted into kinetic turbulence energy (turbulent fluctuations of flow) [J/m3s]. With the turbulence energy being generated in this manner, the energy budget must be conserved, i.e., the fall in the kinetic energy of an average flow must be equal to the increase in the energy of turbulent fluctuations in the flow. The differential a [ 1/82 in (7.2.11) describes the increase in the specific kinetic energy of turbulence in a layer dz in a unit time due to the difference between the inflow and outflow from this layer of the vertical turbulent flux of this energy (i.e., as a result of losses in the vertical flux of energy in this layer). Equation (7.2.1 1) can be included in the set of equations of motion, although as we have said, it introduces new unknowns such as third degree correlation __ moments of the type ( ~ ’ ~ w ’and ) , in doing so does not allow the closure of this set of equations without further assumptions being made. One of these is to disregard local temporal changes of the turbulent energy caused by losses from its vertical Aux in layers dz (the differential a[ ] / a 2 is ignored), as they are small in comparison with changes resulting from the generation and dissipation of this energy in the presence of a strong wind (described by the other terms in (7.2.1 1)). In the one-dimensional model of the upper ocean layer, the equations of motion are also simplified (terms in (7.1.17) or (7.2.5)) depending on the assumptions made, e.g., as to the form given by (7.1,19); other equations discussed in Chapter 6 are also simplified. Thus the equation for the turbulent exchange of heat (6.2.49) becomes
porn
eom
(7.2.13)
7.2 THE ENERGY BUDGET OF A BASIN
409
although we have disregarded here the molecular heat exchange flux, and have taken into consideration internal sources of heat at depth z in the layer dz (as before 0 is the potential temperature, and C, is the specific heat of seawater at constant pressure and salinity). Equation (7.2.13) is a kind of heat budget in a unit volume of medium and unit time, assuming that any local change in the quantity of heat with time (in layer dz) at depth z in the sea is due solely to the turbulent exchange of heat and to the evolution of heat from internal sources Qs(z). The product poC,expresses the density of the vertical turbulent heat flux in [J/m2s], whereas the derivative
a
--
aZ
(Po C,Tf?) [J/m3s] is the quan-
tity of heat lost or gained by this flux in layer dz in a unit volume of the medium in unit time. The derivative aQ,(z)/az is the energy of internal sources evolving [J/m3s] in layer dz at depth z. In practice in the upper ocean layer down to a depth of c. 100 m, QJz) is mainly the solar radiation energy flux reaching down to depth z and converted to heat as a result of partial absorption by seawater in a layer dz at that depth (see Fig. 6.1.1). It is found from the general law (5.4.17) of the diffuse attenuation of the solar flux entering the sea Qs = Qs(0) with the effective coefficient of attenuation (5.4.18). Near the surface, other internal sources of heat (e.g., the viscous dissipation of turbulence energy and its conversion into heat) can usually be neglected in view of the powerful radiant flux from the Sun. The equation for the turbulent exchange of mass (6.2.43), e.g., of sea salt in water of average salinity 3; ie., an average concentration = in the one-dimensional model takes the following form if molecular diffusion is ignored
c
-
as @(I ~
at
+ aza ~
--
(7.2.14)
(@* W’S’) = 0.
Again, this is a kind of budget (conservation principle) of the mass of salt in a unit volume of medium in a unit time, assuming that there are no internal sources of salt in the sea, and local changes in the quantity of salt in a unit volume of __
medium in a unit time
@
O -as 27
[kg/m3s] (or changes in the salinity 3)are solely the
upshot of the verticaI turbulent exchange of salt. Here
a ~
az
(eo w’S’)is the salt
increment in a layer of medium dz at depth z as a result of the evolution in this layer (or removal from this layer) __ of part of the salt [kg/m3s] carried by its turbulent vertical flux of density w’S’ [kg/m2s].
7 SMALL-SCALE AIR-SEA INTERACTION
410
An analogous equation for the budget of buoyancy forces is also applied to the set of vertical exchange equations. To this end, a buoyancy pavameterb (see Kraus, 1972, 1977) is introduced. It has units of acceleration and is de. fined as b= -g
g [ k ~ ( e 00) - - ks(S- So) - k&
eO-co
eo
-Po)],
(7.2.15)
where B,, So and p o are respectively the reference values of the potential tern. perature, salinity and pressure in the sea (values at neutral hydrostatic equi. librium), from which deviations (fluctuations) are measured; k T , ks,k , are the coefficients of thermal expansion, saline contraction and compression of sea. water, respectively (see Chapter 3). The left side of this expression is a term in the equation of motion which we can see in the Roussinesq approximation of this equation (7.1.18) as the product Q’gSi, divided by ea (here equal to eo). From (1.2.26) there is also a connection between the buoyancy b and the Vaisala frequency N (7.2.16) The approximate equality on the right side of (7.2.15) emerges from the linear approximation of the equation of state for seawater written down as (3.1.5) and discussed in Chapter 3. This linear approximation is permissible if we assume very slight deviations e-eo = Q‘ < e, 0-0, = 0’ < 8, S - S o = S’ < S and p - p o = p‘ < p . When the deviations are small, there exists a simple relationship between the relative changes of the specific volume of the medium a, occurring in the equation of state (3.1.5), and the relative changes in the density of this a-a,
-___ . This relationship is further justified in Chapter eo 8 when we describe the transition from (8.0.6) to (8.0.7). Because the compressibility of seawater is low in the upper layer of the ocean, the last term in (7.1.15) is usually neglected. The equation for the volume buoyancy budget, which includes the buoyancy parameter b, (as in (7.2.13)) reads
medium:
-
N N
@-@’
a0
(7.2.17) where b’ is the fluctuation component of the buoyancy b. We gather from this equation that a local change in the average volume buoyancy in unit time
aZ [N/m3s] is equal eo at
to the algebraic sum of its change
7.2 THE ENERGY BUDGET OF A BASIN
a -
-(p,,w‘b’) az
41 1
due to the turbulent influx (turbulent convection) of a flux of this
force (eow’b’) [N/m2s] and its change _____ gkT aQs(z) due to a local conversion
c,
a2
of heat from internal sources Q&) into potential energy. This last term in the equation results from a local heating of the medium as a result of heat being evolved from the solar radiation
aQ (4 in a layer of water dz at depth z in the az
sea. The thermal expansion of the water, and hence the increase in buoyancy, are responsible for the fact that this heat is partially converted into potential energy of the medium. The vertical turbulent buoyancy flux mentioned earlier is associated with the vertical mixing of waters of different buoyancy. The fluxes of mass, heat and momentum exchange in this set of equations should, when they are solved analytically, be expressed by means of turbulent exchange coefficients or in some other way, as shown in Chapter 6, or in Section 7.1 for the atmosphere (see Busch, 1977; Zilitinkevich et al., 1978).
The Mixed Layer Model For the upper layers of the ocean, a particularly simple and widely used model of a horizontally stratified sea is the quasi-homogeneous mixed layer model (Kraus and Turner, 1967; Niiler, 1975; Gill and Turner, 1976; Niiler and Kraus, 1977; Zilitinkevich et al., 1978). It is endorsed by experimental work at sea and in the laboratory (see Kato and Phillips, 1969) which demonstrates that almost everywhere in the ocean, the subsurface layer of water, going right down to the seasonal thermocline, is pretty well mixed. Strictly speaking, this refers to an upper ocean layer whose surface has been well stirred by the wind. Intensive turbulent mixing of this upper layer is induced by the energy of the wind, drift currents and the work done by the buoyancy (see Mellor and Durbin, 1975). The following assumptions are made in the mixed layer model: (1) the temperature, salinity and horizontal flow velocity in the whole layer are quasi-homogeneous (the last assumption is not really valid for the sea); (2) the distribution of the parameters has the form of quasi-discontinuous “jumps” at the sea surface and at the lower mixed layer boundary (their values just above the surface and just below the layer which is practically the seasonal thermocline are different); (3) local changes in the turbulence energy due to vertical transfer within the
7 SMALL-SCALE AIR-SEA INTERACTION
41 2
mixed layer (the differential
a
--
a2
[ ] in equation (7.2.1 1)) are small in comparison
with the generation and viscous dissipation of turbulence energy by an average flow and buoyancy forces (see the remaining terms in (7.2.11)); (4) small temperature changes in the mixed layer resulting from the dissipation (conversion to heat) of turbulence energy and from changes in the che mica1 potential of the water (salinity) can be neglected. It is further assumed that in this latter 19 x T x T o ,i.e., that the potential temperature is practically the same as the absolute temperature, and this is the same throughout the layer, and therefore equal to the surface temperature of the sea To. Likewise, the salinity is equal to that a t the surface S = So,which is justified by the assumed mixing of the waters. In spite of the fact that the water is fairly well mixed and assumed to be quasi-homogeneous, there are still many minute temperature jumps in this layer and thus too, a slight average vertical temperature gradient. Since water has a high specific heat C , and a high density, this slight density gradient may inducea vertical heat flux ( P o C,w'Ti = Q,, C,-K(Q)aT/az,see (7.1.43)) of the sam eorder as that in the SBL. Comparison of the appropriate air and water parameters shows that a temperature difference about 100 times smaller in water than in air will induce the same heat flux. So a generally disregarded temperature difference of 0.01 K per 10 m of the water column is sufficient to induce the same heat flux as in air when the temperature difference is 1 K per 10 m (Zilitinkevich et al., 1974). The mixed layer can also be regarded as a good conductor of heat, turbulently conducting heat from the atmosphere to the layer below the thermocline, between which a considerable difference in temperature exists. The mixed-layer model was adopted for the ocean (from the atmospheric model) by Kraus and Turner (1967), and it is this model, subsequently developed by other workers, which is widely applied in the subject literature (Niiler, 1975; Niiler and Kraus, 1977; Zilitinkevich et al., 1978). We shall now outline Niiler and Kraus's version of it (1977). In order to describe the evolution of the thickness of the mixed layer h,, its temperature To, salinity So and other parameters, we make use of the set of equations discussed earlier which were applied to the horizontally homogeneous model, i.e., the equations of motion of the type (7.1.19), (molecular fluxes of momentum are neglected as they are small), the equations of turbulent exchange of heat (7.2.13), and salt (7.2.14) and buoyancy (7.2.17) and the equations of the budget of turbulence kinetic energy (7.2. l l) for the stationary case, As we have assumed the mixed layer h, to be uniform (see Fig. 7.2.3, p. 418) and P
7.2 THE ENERGY BUDGET OF A BASIN
41 3
hence that the parameters of the medium’s state are independent of z, the equations are simply integrated from z = 0 to z = h,. The varying thickness of the mixed layer h, is described with the aid of the rate w e (inflow due to turbulence) at which water is entrained into this layer through its bottom, i.e., from, below the seasonal thermocline. This is tantamount to the thermocline moving downwards. An inflow in the reverse direction is physically unreasonable if we have assumed that the mixed layer has a boundary. This condition is written thus we
= ~-d h z
for
dt
we = 0
for
dt --
dt
> 0, (7.2.18)
6 0,
i.e., when dh,/dt > 0, colder water enters the mixed layer from below, increasing its thickness and reducing its temperature. If, conversely, dh,/dt 6 0, Kraus and Turner (1967) assume that exchange across the lower boundary of the mixed layer ceases altogether. This means that the turbulence energy in this layer is then not sufficient to overcome the work done by buoyancy in the thermocline and no mixing between these adjacent layers takes place. The mixed layer is then heated or cooled only as a result of a flow of heat across the sea surface, if we disregard the flux of molecular conductivity through the bottom of the layer. The rate of entrainment we of the thermocline is an additional unknown which is determined by means of the turbulent energy equation (7.2.11) integrated from 0 to h, for the stationary case; this closes the system of equations mentioned earlier. Obviously, we have to know the boundary conditions, i.e., the exchange fluxes across the boundaries of the mixed layer-at the free sea surface (for z = 0) and at the interface with the thermocline (for z = hz). The surface fluxes were discussed in Section 7.1. We shall now avail ourselves of that discussion and the symbols introduced there to make a brief presentation of the conditions at the surface and at the lower boundary of the mixed layer in line with Niiler and Kraus’s model (1977). We shall start with the boundary conditions at the free sea surface for the total flux of heat, salt, buoyancy, momentum and turbulent energy, corresponding to the equations introduced earlier: (7.2.13), (7.2.14), (7.2.17), for the momentum (7.1.19) (Section 7.1, equations applied to the sea) and for the kinetic energy (7.2.11).
7 SMALL-SCALE AIR-SEA INTERACTION
414
(1) The condition for a vertical heat flux at the sea surface z = 0 is a kind of heat budget for unit surface area in unit time: __ (7.2.19) e o Cpw’T’Iz=o = Qs- Q ~ LQb- Qh- Qei
where Qsl denotes that part of the solar radiation flux Qs which is not evolved as heat, conventionally at the surface-strictly, not the IR which is absorbed just below the surface but the remaining part which penetrates down into the sea (conventionally QSLrepresents visible light, i.e., about 45% of the flux energy Q,-see Fig. 6.0.1, and Chapter 5). The other heat fluxes (Qb-sea surface radiation, Q,-sensible heat exchange, Qe-latent heat removed with water vapour) were explained in Section 7.1. The algebraic sum of all these fluxes describes the resultant density of the heat flux at the sea surface [J/m2s] given by the left side of equation (7.2.19). (2) The condition for the vertical sea salt flux at the surface z = 0 will be __ @O w’s’lz=O = SO (7.2.20) 7
where Po is the flux density of the mass of atmospheric fallout, Me is the flux density of the mass of water vapour [kg/mzs], and So is the salinity at the surface (and in the mixed layer) This is a simplified budget of the salt flux at the surface. (3) The condition for the buoyancy flux at the surface is _. _ _(7.2.21) eo w’glz=o = -gee [(krw’T’),=o - (ksw‘S‘)Z,ol Bo where b, k r , ks were explained in connection with equation (7.2.15), and this flux appears in (7.2.17). Condition (7.2.21) is the budget of buoyancy fluxes at the sea surface [N/m2s] z [J/m3s]. It describes the ratio at which potential energy escapes from or flows into the water column across the surface as a result of changes in the temperature and salinity of the water (e.g., when the surface cools, the buoyancy of the water decreases, but its potential energy with respect to the surface increases-see Section 1.2). In this equation we have ignored the compressibility of the water as the pressure changes in this thin upper sea layer are negligible and because we are using equations of motion__ that are applicable to incompressible liquids. The correlation moments w‘T’ and w’sl in this equation can be expressed using the heat and salt fluxes at the surface in line with the boundary conditions for the heat (7.2.19) and salt fluxes (7.2.20), respectively. (4) The condition for the vertical momentum flux z at the sea surface emerges from the assumption that it is equal to the shearing stress of the wind zo at the surface, given by (7.1.35). Hence 9
l4,=0
-
=
To =
euu:0 =
@uc,u2(h),
(7.2.22)
7.2 THE ENERGY BUDGET OF A BASIN
41 5
where T is the shearing stress of the wind given by the general equations (7.1.21) and (7.1.22), so that if we neglect the molecular fluxes we obtain
This expression becomes much simpler if the direction in which the stress T is acting is the same as that of one of the coordinate axes; en is the air density; u , ~is the friction velocity of the wind; C, is the surface drag coefficient of the wind; u(h) is the average wind velocity at height h. (5) For a vertical flux of turbulent energy E, (i.e., for (7.2.10), which appears in (7.2.1 1) for a stratified sea), the boundary condition at the sea surface emerges from the assumption that this flux is equal to the work done by the wind fiction on this surface. It would thus emerge from the product of the shearing stress of the wind and the wind speed at the sea surface. But it is difficult to assess thisspeed over a wind-roughened surface, so the authors of this model have formulated this condition with the aid of an additional parameter ml and the friction velocity of the wind uzk0. [--)Qo(u’2w’+v’2w’+w’2w’)+p72]z=o =
rn, u&.
(7.2.23)
Now the boundary conditions at the bottom of the mixed layer, i.e., the exchange fluxes for z = h, have been defined as __
eOCpw’T’Iz=h, = eoCpweAT, __
e o w’S’I,=h, __ eoW’b’lz=h,
eo w
JS,
= e o weAb,
(7.2.24a) (7.2.24b) (7.2.24~)
where AT, AS and Ab are the respective differences in temperature, salinity and buoyancy between the mixed layer and the layer beneath its lower boundary. Thus, when the temperature of the mixed layer is T o ,that of the water entraining from below is AT lower, and likewise for the other parameters. The signs of these fluxes have been chosen such that on entering the mixed layer they are positive (the z-axis in this discussion points vertically downwards; Niiier and Kraus had it pointing in the opposite direction). The first of these equations describes the heat flux across the lower mixed layer boundary. The downwardflowing turbulent heat flux moves the thermocline down at a rate we, which is equivalent to water colder than To by AT entraining into the mixed layer, whose temperature is thereby reduced. As we said earlier, when the turbulence energy in the mixed layer is too small to overcome the buoyancy of the thermocline, turbulent exchange between these layers ceases. The thermocline does not then
7 SMALL-SCALE AIR-SEA INTERACTION
416
move down, we = 0 in accordance with condition (7.2.18), and the vertical turbulent heat flux across the thermocline (7.2.24a) is zero. Equations (7.2.24b) and (7.2.24~) describe similar conditions for the salt and buoyancy fluxes at the lower mixed boundary, below which the salinity differs by A S and the buoyancy by Ab from So and b, in the quasi-uniform mixed layer. The condition for vertical exchange of momentum across the lower mixed layer boundary is, as in the other cases, more complicated and requires the introduction of an additional parameter-the drag coefficient C, . Then t/z=h,-@O
=
___
@O
c,t;l’/
(7.2.25)
7
where jvl = Juz + v z is the horizontal flow velocity of waters in the mixed layer, which is written in this way, so long as none of the coordinate axes (e.g., the x-axis) coincides with the direction of this flow (see Section 7.1). Lastly, the boundary condition for the vertical flux of turbulence energy will be _
_
-
[ ~ @ o ( U ’ z ~ w ’ + ~ ’ z ~ ’ + w ’ z ~ ’ ) - ~ ‘ ~ ’= ] ,W = he E , f.
(7.2.26)
This is the density of the vertical turbulent energy flux [J/mzs] at the lower mixed layer boundary, and is equal to the product of this energy Et and the rate we of its downward movement together with the thermocline. Having formulated the boundary conditions, we can now integrate the equations of motion, and the heat, salt, buoyancy and kinetic energy budgets within the mixed layer (from z = 0 to z = hz). This yields explicit expressions describing the changes of temperature, salinity, etc. in it. Integrating (7.2.13) over the boundaries of the mixed layer, i.e., from z = 0 to z = h,, where the variables are independent of z and 0 z T, yields (7.2.27) After applying the boundary conditions (7.2.19) and (7.2.24a) and rearranging the terms, we obtain an expression describing the changes in time of the temperature To of the mixed layer: (7.2.28) This equation gives the rate of change of the mixed layer temperature owing t o : (a) the entrainment of water cooler by AT from below the thermocline (the
first term on the right side), and (b) the exchange of heat fluxes with the atmos-
7.2 THE ENERGY BUDGET OF A BASIN
417
phere (the second term on the right side). The flux Q,le-K..lhz describes the escape of unabsorbed solar radiation across the lower boundary of the mixed layer. We recall that Qsl indicates that portion of the solar radiation flux Qs which is not converted into heat at the sea surface but penetrates deeper (conventionally visible light), and which is attenuated with an effective attenuation coefficientKer (see equations (5.4.16)-(5.4.18)). Similar integration of the salt exchange equation (7.2.14) from z = 0 to z = h, with the inclusion of the boundary conditions (7.2.20) and (7.2.24b), yields the expression describing the temporal changes of the salinity So in the mixed layer. (7.2.29)
It describes the rate of change of salinity in the mixed layer as a result of the entrainment of water whose salinity differs by A S across its lower boundary, and owing to evaporation Me and precipitation Po at the surface. Similar equations can be obtained for the momentum flux and the buoyancy. The everpresent additional unknown velocity we with which the thermocline shifts is determined from the turbulent energy equation (7.2.11) integrated over the layer from 0 to h, for the stationary case which closes this sef ot equations. After integrating (7.2.15) with the boundary conditions (7.2.21) and (7.2.24c), we get the buoyancy changes (7.2.30)
This equation describes the rate of change of buoyancy in the mixed layer as a result of its exchange at the boundaries of this layer, assuming that the internal source of heat is Qs(z) = Q,le-Kebz. Now we have to eliminate the derivative dbo/dt from this equation in order to determine the turbulent buoyancy flux pow". So we again integrate (7.2.17), but this time from any depth z within the mixed layer to its surface z = 0. This gives an equation similar to (7.2.30) which when subtracted from (7.2.30) yields
(7.2.31)
7 SMALL-SCALE AIR-SEA INTERACTION
418
This expression for the turbulent buoyancy flux isneeded in order to integrate, / the turbulent energy equation (7.2.1 l), in which gw’e’ - eow‘b’ . For the stationary case, the turbuIent energy equation (7.2. I I) can therefore be written in the form
(7.2.32) This integration requires a certain amount of finesse and a further assumption about the “bottom” of the mixed layer. We have to assume that the lower boundary of this layer is not a geometricalplane at which the parameters of the state of the medium change sharply, but that it is a thin transition layer h = hi-h, rthick (see Fig. 7.2.3). This is assumed because the terms describing the gener2ii av ation of turbulence energy by an average flow P o ?? __ and p o r n -must 2Z
a2
be equal to zero at the lower boundary, if this boundary is to shift together with Temperature T buoyancy b
0 N
Fig. 7.2.3. The mixed layer model according to Niiler and Kraus (1977, with permission of Pergamon Press PLC). O-b,--quasi-uniform mixed layer; h,-b,-thin transition layer (hi -A, 0); To-temperature of the mixed layer; bo-buoyancy of the mixed layer; Bo-buoyancy flux across the sea surface (potential energy flow, see equation (7.2.21)).
any inflow of energy, whereas at a sharp boundary, where h:-h, -+ 0, the gradient 2iipz 4 co. We must allow for the possibility that internal waves are generated at the boundary z = h, at the expense of turbulence energy. If this is so, we integrate (7.2.32) over the mixed layer from z = 0 to z = h,
7.2 THE ENERGY BUDGET OF A BASIN
I
2
419
3
3
h,
a + -[+Q~(u'~w'+v'~w'+w'~w')+~'WI]~Z+ eOEdZ= 0. az
S
(7.2.33)
0
0
4
5
According to this notation, we have denoted the consecutive terms of this equation by the numbers I , 2 , 3 , ...Integrals I and 2 can be combined in vector form
@oi hz
w'v'
a+ ~
a2
d z and solved with the boundary conditions for momentum (7.2.22)
and (7.2.25), where T hz
aT eo W'faZdz
=
=
_-
Po w'v'. Hence
1 1 1 --~owejV12+3~oC,/V~3-3p.~*o. 2
3
(7.2.34)
0
Integral number 3 in (7.2.33) is solved by substituting in it (7.2.31) and integrating from z = 0 to z = h,. This gives h:
eomdz = 0
1
1
-QO h, Bo + 7Poh, w,Ab 2 (7.2.35)
Integral number 4 in (7.2.33), with boundary conditions (7.2.23) and (7.2.26) gives the following solution (7.2.36) Integral 5, which describes the dissipation of the energy of turbulence, is left as it is. By substituting in (7.2.33) the solutions of integrals I , 2,3, ... and rearranging, we obtain a final expression describing the rate of change of the mixed layer thickness we = ah,/& > 0 as a result of a flow of turbulent energy across its boundaries. It is convenient to write this as expressing the budget of the turbulentenergy flux
7 SMALL-SCALE AIR-SEA INTERACTION
420
(7.2.37) This is the equation of turbulence energy in the mixed layer integrated over depths z. Its terms have units of flux density of energy [J/m2s] and express the various fluxes of this energy: we E,-the turbulence energy flux propagated downwards which lowers the thermocline (lost by the layer within its initial boundaries); w,+po h,Ab-the turbulence-energy flux converted into the work done in raising denser water and mixing it throughout the whole layer. The product h,Ab = c: is, according to the model's authors, the square of the velocity of the long internal waves arising at the boundary between layers at z = h,; - - w e ~ ~ o ~ v ~ 2 - tflux h e energy of an average flow lost in doing the work of turbulent mixing in the layer (a gain of turbulence energy at the expense of the average flow); ~ o r n , u ~ o + ~ , o a u ~ o flux - t h eof turbulence energy gained at the expense of the work done by wind friction. The expression -$eau:o can be disregarded as it is small owing to the low density of air; +poh,Bo-the turbulence energy flux gained as a result of potential energy flowing in across the sea surface; - 4eo C,jVl 3-the turbulence energy lost during interaction with internal waves; gkT h -t-- (1 - e-KeA)-the turbulence-energy flux gained from the __ Qsl CP Kel L ) influx of solar energy radiating within the mixed layer and converted into potential energy. In turbid waters, the effective downward irradiance attenuation coefficient Ker is large and the quantity of radiant energy escaping across the lower mixed layer boundary is in practice close to zero. This means that e - K J ' z z 0;
(2
'r
eoedz-the
turbulence energy flux lost to viscous dissipation within the
0
whole mixed layer. In the hope that this short description has explained the physical implication of these commonly applied models, the reader is recommended to seek further detailed information in the specialist literature on the subject. It is not the pur-
7.2 THE ENERGY BUDGET OF A BASIN
421
pose of this book to delve deeper into the problems of the complex and highly specialized branch of knowledge that is the mathematical and numerical modellingof the structure and dynamics of ocean waters (Kraus, 1977;Zilitinkevich, Monin and Chelikov, 1978; Ramming and Kowalik, 1980). The Heat Budget of a Sea Basin We know that the two most powerful sources supplying energy to the ocean are the solar radiation flux and the kinetic energy flux arising out of the friction of the wind on the sea surface. Both these forms of energy are, in the long run converted into heat and as such transmitted back into the atmosphere in several ways. The dominant process is the absorption of solar radiation and its conversion into heat. The heat evolved from the viscous dissipation of kinetic energy is usually small in comparison with the solar energy flux, although it may seem to be the greatest source at the sea surface when we see the crests of surface waves breaking upon the shore. The principle of conservation of thermal energy as applied to a basin having acertain volume is called the heat budget of that basin. This budget us the algebraic sum of all the heat gained and lost by this volume of sea in a given time At = t 2 - t l (e.g., 24h or a year). The resultant gain or loss of heat in this time is manifested by a change in the average temperature of this basin in the given time. Instead of the heat budgets for a unit volume of fluid in unit time (differential budgets) which we have been discussing up till now, let us now examine the heat budget of a basin of volume V formed by a water column from the free sea surface to the bottom underneath an area of free surface equal to A . [mZ](integrated budget). Let A [m2] be the total surface area delimiting the sea basin in question, i.e., the area of all the real sides (free surface and bottom) and of the assumed sides (areas of the walls of the water column). Then, without making any assumptions about the column, its heat budget in time At = t2- tl will be the following integral : ti
QW) =
1 (Qs-Q~-Qt,-Qe)dAodt+
I1 A ,
iQ.dAdt+ fl
I
r2
tz
A
1
rl
11
QidVdt.
(7.2.38)
V
III
We recognize in this equation the symbols of all the main fluxes of heat Aowing across the free sea surface (Section 7.1). Apart from these, Qu = uj a(ecp dxj expresses the thermal advection ffux [J/m2s] flowing across 3x1
7 SMALL-SCALE AIR-SEA INTERACTION
422
the walls of the column (in practice only across the side walls), whereas Qi is the rate of evolution of heat from internal sources in a unit volume of fluid [J/m3s]. The radiation flux Q3 was accounted for in (7.2.38) in toto as the heat flux entering the water across the free surface, and therefore the internal source of heat (besides radiation) is chiefly the viscous dissipation of kinetic energy of flows Qi z5 Q&k (see (7.2.6)-(7.2.9)). This last component is frequently ignored as it makes a negligibly small contribution to the heat budget; for the same reasons, one disregards heat from chemical or nuclear reactions, heat flowing from the centre of the Earth through the sea bottom (of the order of [J/m%]) and other possible sources. According to equation (7.2.38), the resultant gain or loss of heat in the water column Q(At) [J] in time At = t2--tl consists of the gains resulting from (a) the inflow of heat across the sea surface (integral I ) , (b) the advection of heat owing to the flow of water (integral ZZ), and (c) the evolution of heat by internal sources (integral ZIZ) in this time; heat emanating from other, weak sources is neglected. This resultant gain or loss of heat in the column [J] in the time from to t, is reflected in a change in its average temperature of T(tJ-T(t,) - t,T2-TI Q(At) = eVCp(E- G),
(7.2.39)
where 8 is the average density of the water in the column, Y is its total volume and is its average specific heat. In view of the difficulties involved in determining the advection of heat and the heat-exchange fluxes between sea and air, the implementation of (7.2.38) for longer periods of time (e.g., one year) is problematical. This equation is, however, helpful when studying the average heat fluxes entering the column. For instance, it may serve to estimate the advection of heat from measurements or calculations of the other terms in the equation. If we disregard internal sources Qi we can, on the basis of (7.2.38) write
c,
t2
e v ~ p < ~ ~S -S T(Qs-Qb-Qh-Qe)dAodt ~)tl
A,
tz
S SQudAdt.
(7.2.40)
ti A
So from temperature measurements and determinations of heat fluxes across the sea surface, we can approximately define the advection of heat into the column during a period of 24 h. If the advection of heat were equal to zero, as is assumed in a horizontally homogeneous sea, the sum of heat fluxes Q s - Q b - Qh-Qe flowing across the sea surface when the average column temperature N” 0) would also be equal to zero. The is constant over long periods ( E -
7.2 THE ENERGY BUDGET OF A BASIN
423
estimated average annual heat fluxes at various latitudes (Defant, 1961) are illustrated in Fig. 7.2.4. This shows that the sum of these fluxes is not equal to zero. There is an obvious gain of heat by the column due to the sum of these fluxes at low latitudes and a loss of heat at high ones. Oceanographic observations show, however, that the average annual temperatures of sea basins are relatively constant. The conclusion to be drawn from this is that the heat advection (integral ZZ in (7.2.38), which is not equal to zero) and the viscous dissipa-
-0.5O
,Net heat gain Z
*
0 10 20 30 40 50 60 70 80 90 Latitude
["I
Fig. 7.2.4. Latitudinal variation in the average annual heat fluxes at the ocean surface (from Defant's data 1961, after McLellan, 1968, with permission of Pergamon Press PLC).
tion of kinetic energy (integral ZZZ in (7.2.38) balance out these differences in heat flow across the sea surface in various regions of the world ocean. Over large areas, therefore, we cannot assume that the sea is horizontally homogeneous, or that there is no horizontal advection of heat, neither can we disregard in our budget the heat arising out of the dissipation of kinetic energy or from other sources. The great turbulent circulations of the ocean (and hurricane-force winds in the atmosphere), which we mentioned in Chapter 1, are connected with heat advection. In a small scale water column, 1 km in length, say of the kind in which many
424
7 SMALL-SCALE AIR-SEA INTERACTION
oceanographical experiments are carried out, the surface unhomogeneity of the average water temperature can usually be neglected, and likewise, the advection of heat can be disregarded in comparison with the heat fluxes flowing across the free sea surface. Also during a single day, the heat flux arriving with the solar radiation is sufficiently large that by comparison, the heat from the dissipation of kinetic energy, in average weather and at low and medium latitudes can be ignored.
CHAPTER 8
THE ACOUSTIC PROPERTIES OF THE SEA
There are numerous analogies between the mathematical descriptions of the propagation of sound waves and of electromagnetic waves in the sea. The mathematica: notation used to describe these waves is similar, and so are the wave equations; sound and light absorption, scattering and attenuation coefficients are similarly defined, and the laws of geometrical optics are applied to the description of sound reflection and refraction. The nature of electromagnetic waves is totally different from that of sound waves: the former are due to oscillations of an electromagnetic field, whereas the latter arise out of the mechanical elastic vibrations of the particles of the medium. Electromagnetic waves can be propagated in a vacuum, but sound waves can exist only in a material medium, Electromagnetic waves in the sea are transverse waves, because the vector E, oscillates at right angles to their direction of propagation; on the other hand sound waves in the sea are longitudinal waves, since a medium which is only elastic in volume and not in form can transfer only elastic vibrations oscillating in the direction in which these disturbances are propagated. The importance of hydrooptical processes in the sea is the result chiefly of the inflow and interaction in the sea of solar radiation energy. The significance of hydroacoustical processes in the sea, however, lies in their many applications in the hydrolocation of the bottom, shoals of fish, and other objects, and in studying the structure of the marine environment. Both light and sound affect the marine biosphere in various ways, many of which are not yet fully understood. Hydroacoustics therefore deals with the detailed mechanism by which acoustic signals emitted from given sources are propagated in the sea, and also with the noise field in it. The complex mathematical description of these processes in the sea has been the subject of a number of monographs, e.g., De Santo (1979), Clay and Medwin (I977), Brekhovskikh (1974), Albers (1 972). The theoretical foundations of hydroacoustics are set out in the monographs by Brekhovskikh (1973), Tolstoy and Clay (1966) and others, and in an extensive chapter in Vol. I1 of Monin’s Oceanology (1978). The books by Urick (1975) and Bergmann and Yaspan (1968) include a richly illustrated discussion of the laws of hydroacoustics. These works are but a fraction of what has been published so far in this field.
426
8 THE ACOUSTIC PROPERTIES OF THE SEA
In this chapter, therefore, we shall merely give the reader an overview of the subject and discuss briefly the more important acoustic properties of the marine environment. Consider an infinitesimally small cubic element of medium dxdydz = dV (as in Fig. 1.2.4), which we shall call an acoustic particle. If a pressure difference should appear across opposite walls of this particle, a force will act on that particle in the direction of decreasing pressure. If pressure p(x) acts on one of the aP walls in the x-plane, and a higher pressure p(x)+ __ dx acts on the wall in the ax xfdx-plane, the resultant force acting on the particle due to this pressure difference along the x-axis will be
Similar equations can be written for the component forces along the y and z axes (see (1.2.13)). So the force acting on a unit volume of medium along any one of the coordinate axes can be written as (8.0.1)
We shall be assuming in this chapter that the pressure p is only the acoustic pressure, that is, the pressure of an acoustic wave, which is the excess over the ambient pressure in the medium when this is at rest: p = presultant-prest. This latter pressure in the sea in generally the hydrostatic pressure which will be denoted by P(z) in this chapter (see equation 1.2.15). Volume forces acting in a compressible medium condense its molecules : their density increases, their specific volume a falls, and so the molecules are displaced with respect to their equilibrium position. This tighter packing of particles in an elastic medium will induce the elastic forces. acting in the opposite drection, to react at once. So particles, and hence the pressure and density of the medium, will oscillate about their equilibrium position. The rate at which the particles oscillate due to acoustic pressure is called the acoustic velocity Y, The energy of this oscillatory motion is transferred through the medium when momentum is exchanged during elastic collisions between neighbouring particles; in other words, these pass on the pressure to particles located further and further away from the source of the disturbance. The movement of such pressure oscillations through the medium is called an acoustic wave, and is characterised by such parameters as wavelength A, frequency v and amplitude of the pressure oscillations pa.
8.1 THE VELOCITY OF SOUND IN THE SEA
427
The acoustic disturbance of the medium is described by the relative acoustic compression due to the wave pressure p (8.0.2)
by the relative acoustic deformation of volume, a-a.
- 6cc
a0
a0
(8.0.3)
the acoustic displacement E, and the acoustic velocity of the particle v during any phase of its oscillatory motion. Here Q and CI express the local density and specific volume at the point of disturbance, whereas eo and clo denote the same parameters in elements of medium undisturbed by an acoustic wave. The space filled by the energy of the acoustic oscillations of molecules is generally referred to as the acoustic field. Strictly, though, we should speak about an acoustic pressure field p(x, y , z , t ) , an acoustic compression field Be(x, y , z , t ) , an acoustic deformation field Sa(x, y , z , t), or about a vector field of acoustic velocities v(x, y , z , t ) or acoustic displacements g ( x , y , z, t). Instead of using the velocity vector v in theoretical descriptions of an acoustic field, we often apply a scalar function called the acoustic velocity potential y ( x , y , z , t ) defined as v = -vy
(8.0.4a)
or (8.0.4b) and sometimes the similarly defined potential of acoustic displacements (Tolstoy and Clay, 1966). In practical applications of hydroacoustics to the description of an acoustic field, we usually use the acoustic pressure p ( x , y , z, t ) as a function of time and space coordinates. This pressure expresses the pressure increment induced by an acoustic wave at any given point in the medium. Knowing the acoustic pressure as a function of the time and space coordinates is equivalent to describing the acoustic field; this function is derived from the solution of a fundamental differential equation-the wave equation-and which will be introduced in the next section for the simple case of an unattenuated wave. To this end we shall use the equation of motion, the law of elasticity, and the continuity equation of the medium.
8 THE ACOUSTIC PROPERTIES OF THE SEA
428
TJie Wave Equation for Unattenuated Waves
The oscillations of molecules in a real medium can be generally described by the Navier-Stokes equation (6.1.50). In the ideal case, of unattenuated oscillations, we can reduce this equation to its simplest form, taking into consideration only the action of the volume forces given by (8.0.1) and resulting from the gradients of the acoustic pressure p . On the other hand, we shall assume that there is no advection of momentum induced by external forces; moreover, we shall initially assume that there is no friction between the molecules in their motion; the coefficient of viscosity 7 is thus 0. Consider, therefore, wave motion in an ideal medium, in which oscillations are not damped, i.e., wave energy is not converted to heat by friction or other possible processes. Such an assumption often holds to a good approximation in water when the amplitude of the waves is small (small pressure changes) and the frequency is also low (low acouttic velocity). The compression and rarefaction of the medium’s particles can then be treated as an adiabatic process (no heat exchanged with the surroundings), and the oscillatory motion of the particles as being frictionless. When the density e varies with the acoustic pressure, the equation of motion (6.1.50) in this simplified case reads (8.0.5a)
where ui are the component velocities of the acoustic particles’ oscillations. Since Q = Po + Be, the left side of this equation can be expanded to N
aui
aui
a[(e~+Q>~il
at
.
Po __,disregarding a(seui) as it is far smaller than eo __ Then the equa-
at at at tion of motion of an acoustic particle (8.0.5a) takes on the approximate form
(8.0.5b) which is the version of Newton’s second law as applied to acoustics. We now use the simplified form of the equation of state: the connection between the specific volume of the fluid c1 (or its density ,g) and the acoustic pressure p in an adiabatic process. This connection emerges from the equation of state of seawater (3.1.5) which, if we assume small changes in the acoustic pressure p and constant temperature and salinity of the water, can be approximated to the following linear relationship, expressing the law of elasticity (8.0.6)
8 THE ACOUSTIC PROTERTIES OF THE SEA
429
where k,.o is the coefficient of adiabatic compressibility. In order to obtain similar laws of elasticity which would describe the connection between density and pressure, we see that at small pressure wave amplitudes 8c1/ao < 1 and likewise 6a Fp/eo 4 1, hence p z Po, and eo/e z 1. We can therefore state that __ c1
-
1ie-1leo . - ___ eo(Q0-e)
F -
eeo
ueo
e se e eo
2
-
N
N
-- . Using this last approximation eo
in equation (8.0.6) gives us another form of the approximate law of elasticity (8.0.7)
The sign of the volume acoustic deformation Galao or acoustic compression
6e/eo may be either positive or negative, depending on the sign of the acoustic pressure (on the compression or rarefaction phase in the fluid). By differentiating equation (8.0.7) over the coordinates, we obtain a new expression (8.0.8)
which we can substitute in the equation of motion (8.0.5b). This furnishes a new version of the equation of motion (8.0.9) which on differentiating over the coordinates yields (8.0.10)
Now we can apply the continuity equation (6.1.51) so that
which, if we omit the expression a(6eui) as it is relatively small in value, gives axr
the equation
aui We) _ __ - --&-.
ax,
at
If this is now substituted in (8.0.10), we get
J2(W
-___ at2
+
1 eokp.0
a2(w ax,%
=
0.
(8.0.1 1)
8 THE ACOUSTIC PROPERTIES OF THE SEA
430
A simple rearrangement of this equation followed by substitution of (8.0.7) yields the wave equation in the form: (8.0.12)
so on substituting this in (8.0.12), we obtain yet another form of the wave equation ____ a2p
axi axi
=
a2(Q)
_ I _
at2
(8.0.13)
*
The compressibility of the medium k,,o and its density eo directly determine the velocity c with which sound waves are propagated in this medium. These parameters are combined in the basic equation of Newton (1687) and Laplace (1816) (8.0.14)
where k , is the isothermal compressibility coefficient of the medium, x is the ratio of specific heats (see equation 3.0.31), and (ap/a& means this differential at constant entropy 6. If we substitute this last relationship in the wave equation (8.0.12) and replace the generalized notation by a nabla operator, we obtain the usual form of the wave equation a2P at 2
__ =
c’V2p.
(8.0.15)
This equation of an undamped wave is valid not only for the acoustic pressure but also for other parameters characterising the acoustic field, such as the acoustic compression and the acoustic velocity potential. Applying the definition of the velocity potential in the equation of motion (8.0.5) and integrating this over the coordinates, the existence of a relationship between the acoustic veloc ity potential y and the acoustic pressure p is readily demonstrated. For a wave of small amplitude this is
-
P = Qo-. at
(8.0.16)
8 THE ACOUSTIC PROPERTIES OF THE SEA
43 1
In hydroacoustics, equation (8.0.15) is the basic equation for describing lowfrequency and low-amplitude waves. As we shall see later, waves like this can in practice be regarded as undamped in an acoustic environment. But a complete solution to the wave equation for the acoustic pressure or potential in the sea exists only for simple or idealized cases. One usually assumes a point source of sound and a horizontally homogeneous sea in which the velocity of sound c(x,y, z ) = c(z). An analytical solution exists only for certain classes of the function c(z) (Brekhovskikh, 1973; De Santo, 1979). For the sake of simplicity, the sea basin is considered to be of constant depth, and the reflective properties of the surface and bottom are defined in a simple manner (Albert, 1972; Brekhovskikh 1974; Clay and Medwin, 1977). So because of this restriction in our analysis of sound propagation in the sea, we make use not only of wave acoustics, but also of a geometrical acoustic approximation based on the concept of an acoustic ray introduced in the same way as a light ray (see Fig. 4.4.la). The principal equation of geometrical acoustics is the eikonal equation, which we shall introduce later, and from which Snell's law of refraction can be derived. The Energy and Intensity of Sound The energy transferred by an acoustic wave in water consists of the kinetic energy of the oscillating elements of medium and free energy which, in an ideal medium, is reduced to the potential energy of molecules deflected from their equilibrium positions (compressed or rarefied) in a field of mutually interacting forces. The kinetic energy of the wave motion contained at a given instant in a unit volume of medium, i.e., the instantaneous density of the acoustic kinetic energy at a given point in the acoustic field, is Ek = +pZ, where v is the velocity of oscillation of the acoustic particles. On the other hand, the potential energy is equal to the work done against elastic forces during the compression or rarefaction of the medium, that is, the work that needs to be done to change the volume from cl0 to CI.A relative change in the volume Sa/a, in view of the relationships leading to (8.0.7), is equal to a compression of the medium with the opposite sign - 6p/eo. An infinitesimally small change in the volume 6a is there-
(3 ( 3
fore (in view of 8.0.3) equal to aod - z - trod
~
, whereas the work
involved in producing this infinitesimally small change in volume is (8.0.17)
8 THE ACOUSTIC PROPERTIES OF THE SEA
432
where the final equality is derived from (8.0.7). The work done to effect the total change in volume from a. to a , that is from zero relative compression to 6e/eo, is thus the integral of (8.0.17) (8.0.18)
in which the final equality is again derived from (8.0.7). This work done on unit volume of medium is equal to the instantaneous potential energy density at any given point in the acoustic field E, =
s,QP2 2
.
(8.0.19a)
After applying the Newton-Laplace equation (8.0.14) it can be rewritten as (8.0.19b) The instantaneous density of the total acoustic-wave energy (units [J/m3]) as the sum Epf Ek at a given point ( x , y ,z ) is thus equal to (8.0.20)
assuming that the amplitude of the waves is small, so that e + Po. In the light of (8.0.4) and (8.0.16), the instantaneous density of the acoustic energy (8.0.20) can also be expressed by means of the acoustic velocity potential E,,,,
= * [2( ' 7 y ) 2 + - $ ( - $ ) z ] .
(8.0.21)
When the medium in which we are investigating an acoustic field is an ideal one (ideally compressible), the potential energy of the waves at a given point in the fluid is alternately converted entirely into kinetic energy and vice versa. So the time-averaged potential energy density is equal to the time-averaged kinetic energy density E, = E k and we can assume that the time-averaged density of the total acoustic energy is equal to 2Ep or 2 E k . In view of (8.0.20), we can therefore write that (8.0.22)
8 THE ACOUSTIC PROPERTIES OF THE SEA
433
The average value of the acoustic pressure p is easily found for the simple case of a single plane harmonic wave propagated along the x-axis, for which the changes in time and space are described by the harmonic function
p = p,cos(wt-kx),
(8.0.23)
where p a is the pressure amplitude of the wave, OJ = 2 n / T is the circular frequency of the oscillations, T is the period of the oscillations, and k = 2x11 is the wave number. Averaging this pressure over the time of one or more oscillation periods and substituting it in (8.0.22) gives us the following expression for the average energy density of a plane harmonic wave
(8.0.24) Pa where pelf = __ jl4
*
The surface density of an energy flux transferred by acoustic waves is denoted by the vector qwave.This expresses the quantity of energy transferred by acoustic waves in unit time across unit area of a surface perpendicular to the direction of propagation of the waves (units: [J/m2s] = m/m2]). The instantaneous density of this energy flux is called the instantaneous wave intensity I,. This flux fluctuates in harmony with the wave pressure fluctuations, so its average value in a given time interval z is usually of practical significance. This average value of the flux density of the transferred energy is called the wave intensity or sound intensity (8.0.25)
for a periodic wave, it is sufficient to average one period z = T over time. Obviously, the time-averaged energy flux transferred (with velocity c) across a unit surface area, called the effective intensity of the wave, is (8.0.26)
where the second equality is derived from equation (8.0.22).
8 THE ACOUSTIC PROPERTIES OF THE SEA
434
The effective intensity of a sine wave, in view of its average energy density given by (8.0.24), is (8.0.27)
and is thus proportional to the square of the acoustic pressure wave amplitudep,. The energy from the source of a disturbance (e.g., a piezoelectric plate elec. trically induced to oscillate) is propagated in all directions through the medium, filling an ever-increasing space with energy. From a point source in a homogeneous medium, the energy radiates in all directions as a spherical wave. As the distance r from the source increases, the energy flux emitted from the source in a cone of constant solid angle AQ, thus cuts off a section of the surface area of this sphere which increases in proportion to rZ. Hence, the flux per unit area, that is, the surface density of the flux or the sound intensity is, for geometrical reasons, inversely proportional to the square of the distance I l/rz. There thus exist very considerable spatial differences, possibly of many orders of magnitude, in the sound intensity. Likewise, the scale of sound intensities audible to the human ear from the threshold of audibility to the pain threshold covers twelve orders of magnitude. Therefore, when we are comparing sound intensities it is convenient to use a logarithmic scale. Again, instead of intensities, we can compare acoustic pressure, because, as we have just seen, I pz. The logarithms of the ratio of two intensities, e.g., the investigated one I and the standard one Io, logl/Io = 2 log pipo, is called the relative sound intensity or the sound-intensity ZeveZ and is expressed in bels [B]. But these units are too large and impractical, so we use units which are ten times smaller-decibels [dB]. The sound intensity level (or acoustic pressure level) expressed in decibels is therefore
-
N
J
=
I = 20 log ~. P 10 log -I0
Po
(8.0.28)
Clearly, among other things, this formula can describe the transmission of sound through a medium I/Io(or through an amplifier), if I. is taken to be the intensity at the beginning of the sound path. In practice, lois often conventionally defined as the standard or “zero” intensity-different for different purposes-it may be the audibility threshold or the intensity of natural noise, or it could be the intensity at 1 m from a sound source. In hydroacoustics, the “zero” pressure is currently taken to be p o = 1 pPa = N/mZ. In the old literature, the most usual reference pressure was p o = 1 pbar = lo5 pPa. So in a particular case,
8.1 THE VELOCITY OF SOUND I N THE SEA
435
the acoustic pressure measured in the sea ( p = lo-* Pa) would, according to (8.0.28), correspond to a sound intensity level of 80 dB. Two sound intensities may also be compared on the natural logarithmic scale rather than on the (log,,) scale. Then the units in which the acoustic pressure level or sound intensity level are measured, derived from an equation analogous to (8.0.28) (8.0.29)
are called nepers [Np], 1 Np being equal to 8.686 dB. It should be noticed here too that only in the simplest case are the oscillations of the acoustic pressure p in a given place in the medium described by a simple harmonic function (8.0.23). Usually the measured acoustic pressure p ( x , y ,z , t ) is complex and comprises a Iarge number of superimposed harmonic oscillations of differing frequencies and amplitudes, that is to say, it has a complex oscillation spectrum. 8.1 THE VELOCITY OF SOUND IN THE SEA
It is merely a coincidence that in hydroacoustics the miniscule c means the velocity of sound, whereas in hydrooptics it stands for the light attenuation coefficient. We shall see that although the role of these two parameters is different, the light-attenuation coefficient primarily characterizes the diversity of conditions under which light is propagated in the sea, while the velocity of sound c acts in the same capacity as regards sound propagation in the sea. A basic equation in physics which defines the velocity at which elastic waves (such as sound waves) are propagated is (8.0.14), which can be briefly restated as c=
1
,
Ik~.Q@O
(8.1.1)
where k p . Qis the adiabatic compression coefficient of the fluid, and Q, is the density of the fluid undisturbed by waves. But this basic equation is not applicable in practice to sound velocity measurement in the sea. Both the density of sea water @,, and its compressibility k,,Q are complex functions of the salinity, temperature and pressure, as will have been seen in Chapter 3. But of major importance as regards the propagation of sound in the sea is the vertical distribution of its velocity c(z). If we assume a horizontally homogeneous sea, we get layers of medium in which sound travels
436
8 THE ACOUSTIC PROPERTIES OF THE SEA
at different velocities c(z,), c(z,), c( z 3) , etc. with the sound being refracted at the boundaries between them. The relative coefficient of refraction of sound is defined in the same way as for light (8.1.2)
and its values significantly affect the path taken by sound in the sea. Equation (8.1.1) can be successfully applied to the qualitative interpretation of changes and the vertical sound velocity distributions measured in the sea. This equation demonstrates that increasing compressibility and density cause sound to travel more slowly. The density of the sea usually increases non-linearly with depth because of the rise in pressure and fall in temperature. At the same time, however, the compressibility coefficient k p , usually decreases with depth as a result of the same pressure rise, but increases as the temperature falls (see Table 3.3.1). The whole relationship is further complicated by the diversified vertical distribution of the salinity. As we recall from Chapter 3, increasing salinity increases the density but decreases the compressibility of sea water. It is therefore not easy to predict the relationships of c(z) without first making detailed measurements and experimentally establishing the relationships between the sound velocity and the temperature, salinity and pressure in the sea c(S, T , P). It has been found that the velocity of sound in the ocean varies from 14301540 mjs near the surface to 1580 m/s at great depths. Temperature has the greatest effect on this velocity; in typical oceanic conditions, temperature decreases with increasing depth in the upper layers of the ocean are accompanied by a decrease in sound velocity of 3.5 m/s per 1"C, mainly as a result of the increasing density of the water. An increase in salinity with depth corresponds to an increase in sound velocity of about 1.3 m/s per I%,, salinity; the effect of increasing salinity is greater on the decrease in compressibility than on the increase in density (see 8.1.1). A rise in the hydrostatic pressure with depth increases the sound velocity by some 1.8 m/s per 100 m water column; this is because, as we can see from (8.1. I), the pressure affects the decrease in compressibility more than the increase in water density. In particular cases, changes in the sound velocity c caused separately by changing salinity, pressure or temperature, are highly dependent on the absolute values of all these three parameters simultaneously. A theoretical solution to this complicated relationship c ( S , T , P) has not yet been found in the form of an analytical function. Experimental work has, however, provided a number of formulae which establish this relationship with
8.1 THE VELOCITY OF SOUND IN THE SEA
437
good accuracy. The most accurate and most widely used of these formulae is Wilson’s empirical formula (1960) which lays down a relationship between the velocity of sound in seawater c [m/s] and the temperature T [“C],the salinity S KO]and the pressure P [kG/cm2]. The formula, in the form of a polynomial, reads (8.1-3) c(S, T, P) = 1 4 4 9 . 1 4 + A ~ ~ + A ~ ~ + A c ~T,P, +Ac~, where 1449.14 m/s = c(35,0,0) is the velocity of sound under standard conditions assumed for ocean water of 35%, salinity at temperature 0°C and atmospheric pressure. The other terms is this formula are corrections for other conditions deviating from the standard ones and have been fixed as follows:
+
Acs = 1.3980(S- 35) 1.692 x 10-3(S- 35)2, AcT = 4.5721 T-4.4532 x 10-2TZ- 2.6045 x 10-4T3+7.985 x 10-6T4, Acp = 1.60272x lO-lP+ 1.0268 x 10-5P2+3.5216 x 10-9P3- 3.3603 x 10-”P4
A C ~ .=~ (S-35)(-1.1244~ , ~ 10-2T+7.7711 x 10-7T2+ +7.7016x lO-’P-1.2943 x 10-7Pz+3.1580xlO-*PT+ f 1 . 5 7 9 0 ~10-9PT2)+P(- 1.8607~10-4T+7.4812x 10-6TZ+ +4.5283x 10-*T3)+P2(-2.5294x 10-7T+1.8563 x 10-9T2)t +P3(-1.9646x 10-lO)T. To distinguish it from the acoustic pressure p , the combined atmospheric and hydrostatic pressure in the sea is denoted by a capital P and is expressed in Wilson’s formula in [kg/cm2]. This formula is widely employed in hydroacoustics, and its accuracy is estimated to be i 0 . 3 mjs for salinities 0 < S < 37%,, temperatures -4°C < T < 30°C and pressures 1 kG/cm2 < P < 1000kG/cm2 (see Tolstoy and Clay, 1966). Its practical application to almost all bodies of natural waters is obvious, while its precision is of the same order as Ithat attainable by direct measurement of the velocity of sound in the sea using modern equipment (Mackenzie, 1971). One drawback of Wilson’s formula is that it takes no account of differences in the salt composition in water bodies, or of the content of gases and the extent to which waters are saturated by them, neither does it allow for the effect of gas bubbles, or organic substances, or for the fact that the velocity of sound is slightly dependent on the frequency of the oscillations, i.e., the sound dispersion (Del Grosso and Veissler, 1951). All these factors taken together slightly alter the velocity of sound, by roughly 3% under natural conditions, regardless of the
8 THE ACOUSTIC PROPERTIES OF THE SEA
438
salinity, temperature or pressure of the water. In this situation, less precise empirical formulae for the relationship c(S, T , P) (Leroy, 1969; Medwin, 1975; Frye and Pugh, 1971) have the same practical significance as Wilson’s equation. Medwin (1975) for example, the author of many papers and monographs on hydroacoustics (see Clay and Medwin, 1977), gives the following relationship
+
+
c = 1449.2 4.6T- 0.055T2-I-0.00029T3
+(1.34-0.010T)(S-35)+
1.58 x 10-6P,
(8.1.4)
in which c is the velocity of sound in seawater [m/s], T is the water temperature TC],S is the salinity p4,,], P A is the hydrostatic pressure [N/m2]. If we ignore the compressibility of water, the pressure PA at depth z in the sea can, according to Medwin, be simply calculated from the equation Pa = &&z, where en 2 (1 + S x 10-3)103 kg/m3, g = 9.8 m/s2, z is the depth in metres. From such an evaluation, the last term in (8.1.4) can be replaced by 0.0162. Hence, the vertical velocity gradient due solely to the hydrostatic pressure, i.e., in isothermal and isohaline waters-and polar waters come the closest to these-is dc/dz z 0.016 s - l . This implies an increase in the velocity of sound with depth of c. 1.6 m/s per 100 m water column. According to other calculations (Doronin, 1978), the sound velocity increments at various depths induced by hydrostatic pressure are the following: depth z [m]
m’s
]
0
10
100
1000
5000
0.165
0.165
0.165
0.168
0.1826
and induced by temperature changes in various temperature intervals : T Wl
m/s
ACT
[y]
1-10
10-20
20-30
30-40
4.446-3.635
3.635-2.734
2.734-2.059
2.059-1.804
These tables show that at greater depths, the increments due to increasing hydrostatic pressure are rather larger than nearer the surface. The temperature-induced changes in c are much smaller at higher temperatures. This can again be explained qualitatively using (8.1.1). Sound Velocity Distributions in the Sea
As the relationships of c ( S , T , P) are also complex, the vertical sound vel-
8.1 THE VELOCITY OF SOUND IN THE SEA
439
ocity distributions c(z) in the sea are highly differentiated in time and space, although a number of characteristic cases are distinguishable : these are illustrated in Fig. 8.1.1. The most typical of them, resulting from typicai vertical distributions of temperature and salinity at low and medium latitudes in the ocean, are the patterns shown in Fig. 8.1.la. Here a fall in temperature from over 20°C near the surface to around 4°C below 1000 m, which is within the principal thermocline, brings about a rapid decrease in the sound velocity c(z) with depth (a)
(b)
Sound velocity c Irn/sl
Sound velocity c [rnfsl
440
8 THE ACOUSTIC PROPERTIES OF THE SEA
since the water density is increasing. Below 1000 m temperature changes are minimal, so that in fact we can talk about an isothermal abyssal layer. Here temperature changes no longer affect the velocity of sound, but the increasing hydrostatic pressure does reduce the compressibility of the water which produces a perceptible increase in the sound velocity. As long as the changes in salinity are not too great, the minimum speed of sound c(z) is fixed within the depth range 400-1200 mydepending on the climatic zone, sea currents and other hydro(c)
Sound velocity c
5 a n
i( ?
?;/* 1_ i\ c
lili
Fig. 8.1.1. Vertical sound velocity distributions in the sea.
(a) Typical distributions in the ocean at low and medium latitudes (examples from the Atlantic, selected from Brekh. ovskikh, 1974, with permision of Nauka Publishers); (b) typical distributions in the ocean presented diagrammatically: 1- low and medium latitudes with a heated sea surface; 2- medium latitudes with a cooled surface; 3high latitudes where the T and Sgradients are inconsiderable; the principal thermocline lies at depths of 0.8-1 km; (c) other possible or idealized distributions in shallow seas.
8.1 THE VELOCITY OF SOUND IN THE SEA
441
dynamic factors affecting the vertical temperature and salinity distributions. This minimum is usually located sufficiently deeply for the effects due to seasonal changes in the surface temperature to be avoided; it is therefore fixed, and is subject only to certain fluctuations brought about by internal waves, turbulence, etc. As we shall see, this minimum sound velocity in the sea determines the axis of a certain kind of waveguide created by the medium around this minimum and is ideal for propagating low-frequency (unattenuated) sound waves over long distances (c. 1000 km). These waveguides are known as SOFAR channels (Sound Fixing And Ranging). Various types of vertical distributions of c(z) are illustrated by the diagrams in Figs. 8.1.lb and c. Graph 1 in Fig. 8.1.lb is a diagram of the plots in Fig. 8.1.la, typical of waters in low latitudes where the surface layer is heated; the sound channel axis (minimum c(z)) lies about 1000 m below the surface. Plot 2 in Fig. 8.1.1b shows a similar situation in medium latitudes in winter, that is, when the surface layer is cooled; a second seasonal minimum of c(u) appears at the surface, forming a second acoustic channel there. Plot 3 in Fig. 8.1.lb shows the situation in polar seas, where the vertical differentiation in the water temperature is minimal (low surface temperature, weak vertical stability, ease of turbulent mixing), but the salinity falls already at the surface because of the inftow of freshwater from glaciers. In these practically isothermal and isohaline basins, it is mainly the hydrostatic pressure that affects the vertical distribution of sound velocity below the thin surface layer: the velocity increases monotonically with depth. Fig. 8.1.1~depicts other possible distributions of c(z) which are encountered in the particular conditions obtaining in shallow seas, and distributions idealized in order to simplify the analysis of sound propagation in the sea-such as the ideally uniform distribution (plot 5), the ideal linear plots 6 and 7, or the obviously stratified distribution in plots 8 and 9. The real sound velocity distributions in a given situation in the sea can be fairly accurately determined using such empirical formulae as (8.1.3) or (8.1.4) (and also from tables: Leroy, 1969; Frye and Pugh, 1971) derived from the measured vertical distributions of temperature and salinity in the waters of a given basin. Results for the Baltic Sea, for instance, can be found in Tymariski (1973) and Klusek and Wawryniuk (1980); they show that the assumption of a horizontal homogeneous sea for acoustic waves of particularly low frequencies and propagating over great distances, is only a general approximation of the true picture (Fig. 8.1.2). The function c(z) is very variable in the horizontal at different locations in the sea, which means that the velocity of sound is in fact a complex function of the spatial coordinates in the sea and of time c(x, y , z , t).
8 THE ACOUSTIC PROPERTIES OF THE SEA
442
Fig. 8.1.2. The mean distribution of sound velocity levels [m/s] in August in the Baltic, along the vertical transect Karlshamn-Ustka (see map) calculated using Wilson’s formula (8.1.3) from 10-year measurements of water temperature and salinity (from Tymahski, 1974).
1550j-
> a,
1350 Temperature T [ K l
Fig. 8.1.3. Sound velocity in seawater (a) as a function of temperature in water of varying salinity at atmospheric pessure, (b) as a function of pEsSUrC h ocean water at different temperatures (from data in tables published by Popov ef ol., 1979).
The velocity of sound in seawater is plotted as a function of temperature, salinity and pressure in Fig. 8.1.3. The monotonic increase in the velocity c when S, Tor P increases is evident. In atypical waters which, besides sea salt, contain large quantities of other dissolved substances or gas bubbles (under the wind-roughened surface), errors
8.2 THE ABSOPTION AND SCATTERING OF SOUND IN THE SEA
443
in calculations of the velocity of sound based on salinity and temperature may be serious, from a few to several tens of metres per second on an absolute scale. For this and other reasons, the techniques for measuring the velocity of sound in the sea directly are of great importance (Mackenzie, 1971). The motion of sound waves is uniform, so the principle of such measurements is simple, usually entailing the measurement of the time taken by a signal to travel along a fixed path from transmitter to receiver. This path must, however, be short, about 10 cm, so as to ensure “point” measurements in a non-homogeneous environment, and to avoid sound refraction and other undesirable effects. With a velocity of c. 1500 m/s, considerable technical difficulties have to be surmounted in order to ensure adequate accuracy of measurement (to c. 0.1 m/s) over such a short distance; this involves the electronic measurement of the time taken for a specially-shaped acoustic signal to travel along the path once or many times. Experiments and achievements in this field are discussed by Mackenzie (1971), Wehr (1972) and Jagodzidski (1974).
8.2 THE ABSORPTION AND SCATTERING OF SOUND IN THE SEA
Apart from the spatial distributions of velocity, the absorption of sound by seawater (Schulkin and Marsh, 1962) and the absorption and scattering due to unhomogeneities in th emedium-such as gas bubbles in the water and in marine organisms-(Clay and Medwin, 1977) are of importance as regards the propagation of sound in the sea. In order to discuss these processes, we must at the outset define the coefficient of absorption of wave energy, which various workers have understood somewhat differently (Brekhovskikh, 1974; Urick, 1975; Clay and Medwin, 1977). If we consider a plane sound wave travelling along a path r in an absorbing and scattering medium, the attenuation of the sound intensity along r ensues only as a result of this energy being absorbed and scattered by the medium. A wave of any other shape additionally requires changes in the spatial intensity to be considered for purely geometrical reasons, i.e., if the flux is divergent or convergent, since, unlike the new definition of light intensity (4.1.2), the definition of sound intensity (8.0.25) does not take account of them. In conditions in which scattering can be disregarded, the attenuation of a plane wave or acoustic pressure is described by the simplest transfer equation, which has already been derived for light (4.2.1). By analogy, we can express an infinitesimally small change in the intensity (energy flux density) of a wave d I over an infinitesimally short path dr as
8 THE ACOUSTIC PROPERTIES OF THE SEA
444
~
dZ = -a,I, dr
(8.2.1)
where the coefficient ar [m-l] may be called the energetic (or intensity) coejicient of sound absorption in the medium, as it characterises the ability of the medium to absorb wave energy. Integrating (8.2.1) over the path from r = ro = 0 to r leads straight away to the exponential law of absorption I = Joe-a.r (8.2.2) which, after taking logarithms and substituting (8.0.29) now reads (8.2.3) and expresses the attenuation due to absorption of the sound intensity level or acoustic pressure in nepers [Np]. This equation includes the" amplitude" coefficient of absorption a, = a1/2 which expresses the attenuation of the acoustic pressure level (pressure amplitudes) along a unit path in [Np/m]. The subscript e in cc, indicates that we are using natural rather than base 10 logarithms, and hence nepers as the unit in which we are expressing the attenuation of the acoustic pressure level. However, this is usually expressed in decibels. In order to do this, we must use (8.0.28) and the relationship between base 10 logarithms and natural ones: logx = lnxloge. Using the right-hand equality in (8.2.3), we can write the attenuation of the acoustic pressure level in decibels as Po 2010g -~P - = -201n-loge Po
P
=
4.343aIr
=
8.686zer = ar.
(8.2.4)
Thus we can obtain connections between differently defined sound absorption coefficients = 201, = 0.23026~ (8.2.5) in which aI is expressed in [m-'1, a, in [Np/m] and a in [dB/m]. The absorption of sound and its conversion to heat were well described already in the 19th century by the classic theories of Stokes and Kirchhoff (Mikhailov et al., 1964; Malecki, 1964; Elpiner, 1968; Kwiek et al., 1971). Stokes' theory describes absorption brought about by molecular viscosity in a medium. It is possible, however, to distinguish between viscosity conceived as friction between particles of medium as they oscillate and the volume viscosity, seen as the friction in a compressible liquid when this is compressed, with old structures breaking up and the molecules rearranging to form new ones. This is what happens in water when acoustic pressure forces elements of the medium to change their specific
8.2 THE ABSORPTION A N D SCATTERING OF SOUND IN THE SEA
445
volume (e.g., when clusters of water molecules are crushed in a period shorter than their lifetime-see Chapters 2 and 3). This second viscosity appears only when the elements of the medium undergo rapid deformation as an acoustic pressure wave passes. Part of the energy of an acoustic wave does work against both these kinds of friction and is thereby converted into heat. Molecular regrouping depends, moreover, on the ratio of the relaxation time of given structures to the period during which the pressure of the acoustic wave changes, i.e., it is dependent on the frequency of the sound wave. This process is called molecular relaxation and we shall discuss its importance in seawater in the next section. The coefficient of absorption due only to molecular friction was obtained theoretically by Stokes at the end of the 19th century (8.2.6) where p is the density of the medium, c is the velocity of sound in the medium, and 7 is the molecular dynamic viscosity (Chapter 6). Notice the particularly important proportionality of the coefficient to the square of the acoustic wave frequency m2 = 4x2v2 which means that high-frequency waves are strongly absorbed owing to the viscosity of the medium, so that their range in water is short. Kirchhoff's theory (Mikhailov, 1964) explains the second reason why sound is absorbed : this is due to molecular thermal conductivity between compressed and rarefied volume elements of the medium. To say that the compression and rarefaction of volume elements caused by the sound wave is approximately adiabatic (as we assumed when deriving the equation for a non-attenuated wave (8.0.15) is not strictly true. This is because elements of medium warm up when compressed and release part of this heat to neighbouring elements owing to the temperature difference between regions of acoustic compression and rarefraction. So according to the law of conservation of energy, the work done by the wave pressure during the compression phase is not returned in its entirety during the rarefaction phase as work done by the elasticity of the medium because part of this work has been given up to the surroundings in the form of heat. How much of this energy is lost as heat therefore depends on the molecular thermal conductivity coefficient y of the medium. The theoretically obtained coefficientof sound wave absorption (Landau and Lifshits, 1958) will therefore be (a,), =
y QC"C3
x-1 x
__ co2
(8.2.7)
446
8 THE ACOUSTIC PROPERTIES OF THE SEA
where x is the ratio of specific heats Cp/Cv. Once again, we see the characteristic proportionality of this coefficient to the square of the wave frequency, although in the case of seawater it is negligibly small and energy losses sustained therefrom are barely 1/1000 of those resulting from the viscosity of the medium. Relaxation Processes in Seawater
Let us now return to the absorption of sound energy due to the two types of viscosity, whose mechanism must take molecular relaxation processes into consideration. The wave energy losses in these processes are the upshot of changes in the molecular structure of the elements of medium induced by the varying pressure of the wave. So once the pressure has changed byp, there follows a more or less successful regrouping of molecules into a new structure, and a new density, differing by Be from the previous one, is established. The regrouping may just be a different packing with the molecules combined into clusters in a different way (structural relaxation), or it may involve an alteration to the chemical structure of the molecules, e.g., dissociation and ion hydration (chemical relaxation). After the pressure has changed, the molecules of medium take up new positions corresponding to the minimum potential energy, i.e., the relaxation position given in the new ambient pressure. The time required for the molecules to regroup themselves is finite and is known as the relaxation time z,. A more precise definition of this relaxation time will emerge from equation (8.2.11); it is characteristic of a given set of chemical or structural changes within a component of the medium and depends fairly closely on the temperature and pressure. So the ratio of this time and the period of the wave pressure changes T is usually an incidental one. The result of all this is that waves of different frequencies v = 1/T will induce relaxation processes in different components of the medium with varying success, depending on the ratio z,/T. When T % z, (wave frequency much lower than the relaxation frequency), the molecules are easily able to regroup in phase with the changes in the wave pressure. If on the other hand, T < z, (wave frequency much greater than the relaxation frequency), the regrouping molecules of medium cannot keep up with the pressure changes, so their participation in relaxation is less effective. Energy is lost by a wave during relaxation when part of the energy consumed in the compression phase to produce the new structures is not returned to the wave during the rarefaction phase as work done by the elastic forces in the medium. When T > z,, struc-
8.2 THE ABSORPTION AND SCATTERING OF SOUND IN THE SEA
441
tural breakdown keeps up with rarefaction, so a large proportion of the energy used up in the compression phase to produce the new structure is returned during the rarefaction phase and wave energy losses are small. As the period of the wave oscillations is gradually shortened when T is approaching z, but is still greater than z,, energy losses begin to increase since less and less of the energy of the new structures is given back to the wave during the rarefaction phase. When the oscillation period gradually becomes shorter than the relaxation time, the structural regrouping process becomes less efficient, but there is a greater delay in the energy return with respect to the oscilIation phase of the wave. The highest energy losses per cycle occur when there is resonance (i.e., T = r,.), but the number of cycles per second increases with the wave frequency. It seems then that when wave periods are short (T Q z,-high wave frequencies), a certain equilibrium is established between these processes, so that the absorption coefficient in a given relaxation process is constant for T < z, which in seawater is independent of the frequency; this is illustrated in Fig. 8.2.1. But when T 9 z, this coefficient increases with the square of the circular frequency of the wave w until a maximum is reached at resonance T = z, (Brekhovskikh, 1974; Clay and Medwin, 1977). A quantitative description of these relationships in seawater now follows. Relaxation in seawater, which has an obvious significance as regards sound absorption, has so far been discovered only for three molecular processes: (1) structural changes in molecular clusters in pure water with a relaxation time T , , ~z s; (2) structural changes in magnesium sulphate MgSO, (a weak electrolyte in water) involving bonding or dissociation and ion hydration of this seasalt component, with relaxation times for the first two stages in this process of zrMg,z s and z ~ , % ~2 x~lo-* , ~ s (Eigen and Tamm, 1962); (3) structural changes of boric acid, B(OH), , in sea water with a relaxation time T , . ~z lo-, s (Yeager et al., 1973; Fisher and Levison, 1973; Fisher and Simmons, 1975, 1977; Mellen et a]., 1979, 1980). It seems that other components of sea salt present in large quantities-even sodium chloride, NaCI--have no detectable effect on sound absorption in seawater. The relaxation changes in the structure of magnesium sulphate (see Eigen and Tamm, 1962), depending on the amplitude of the pressure wave inducing them, are thought to occur in a number of stages: MgSO,
+ 0H'
+
MgS0,O /H + O/" OH ' Mg\H \H \H \H (Mg2+)nH20 (SOi-)nH20.
+
+ SO20/H
o/H \H \Hr
8 THE ACOUSTIC PROPERTIES OF THE SEA
448
So the MgSO, molecule is thought to become polarized and hydrated by degrees until it is completely dissociated and both ions are hydrated (see Chapter 2; Horne, 1969). However, these processes have not yet been studied sufficiently thoroughly. Studies on the effect of B(OH)3 on sound absorption were begun not so long ago (Yeager et al., 1973), and were part of a priority programme to discover the reasons why the absorption of low-frequency waves (< 10 kHz) was several times greater than was to be expected from the relaxation changes in magnesium sulphate (Clay and Medwin, 1977). The relaxation times given above indicate that a sound wave is absorbed owing to relaxation processes in seawater around and above the frequency v, = l/zr,,, = 10’ MHz as a result of interaction among water molecule clusters, Y ~ = l~ / ~, , ,z ~~100 ~ .kHz ~ and v ~ = I~/ - c , ~ , , , M~ 200 MHz due to the interaction of MgSO, and vB = 1 /T,, % 1 kHz due to the interaction of B(OH), , In various hydroacoustical applications, one usually uses sound waves of frequencies from around 0.1 Hz to several hundred kHz, but no higher because of the strong absorption of higher frequencies brought about by the viscosity of water ((aI)l wz). It is only for very special purposes, such as obtaining ultrasonic pictures of objects in turbid water (see e.g., Muehlner, 1967) or measuring flow fluctuations using the Doppler effect (Squier, 1968) that frequencies of around 10 MHz are employed. The practical significance of the relaxation absorption of waves with frequencies of 200 MHz and higher is therefore small. The relationship between the absorption of sound waves during relaxation and the circular frequency of the wave o = 2xv and the relaxation time z, will now be outlined, after Clay and Medwin (1977). By combining (8.0.2), (8.0.7) and (8.0.14), we obtain the following relationship p = c26p (8.2.8) which is one way of expressing the law of elasticity. Since changes in the density e are dependent on the tinie t during relaxation, we shall add a term taking these density changes into account N
(8.2.9) where b is a constant coefficient. Let us now assume that at a certain instant t = 0 the pressure p acting hitherto vanishes, so that from instant t = 0, p = 0, while the resulting change in density by takes place gradually in time, depending on the relaxation time z,. Then on the basis of (8.2.9) we can write: 1
-2
?- dt. b
(8.2.10)
8.2 THE ABSORPTION AND SCATTERING OF SOUND IN THE SEA
Integrating this equation from t
=
0 to t and denoting 6~ =
(Be),
449
for the instant
t = 0 yields
(8.2.1 1) which defines the relaxation time as z, = b/c2.This is therefore the time during which (as we can see from (8.2,ll)) the density increment Se falls to l/e of its initial value (Be)o once the pressure has disappeared. In theory, only after an infinitely long time t will this increment, induced by the previously acting pressure, fall to zero. The absolute value of this density increment during the relaxation time depends on how large a proportion of the molecules of a given medium actively participates in the relaxation process with a relaxation time t, at a given temperature and pressure. Assuming at the start, for the sake of simplicity, that all the molecules in the medium take part in this process, Clay and Medwin (1977) derived a general expression for the coefficient of absorption due to molecular relaxation. Their derivation is based on the solution of the wave equation (8.0.13) for the pressure of a plane harmonic wave travelling along the x-axis. By substituting the expression for the acoustic pressure (8.2.9)a form of the law of elasticity-in (8.0.13) and assuming that a/& = d/dt, we get the following equation for a plane wave p = p ( x )
(8.2.12) where the relaxation time t, = b/c2,and b is the coefficient from (8.2.9). If the wave is a harmonic one, 6~ can be written as
Be = exeiwt,
(8.2.13)
where the amplitude ex depends only on spatial position (on the x coordinate) and not on time. Equation (8.2.13) is thus readily differentiated with respect to the stipulations of (8.2.12), and when the results of differentiation have been inserted in this latter equation we get
(8.2.14a) This equation is now rearranged by dividing it by the expression in brackets and by multiplying the second term top and bottom by ( 1 - i ~ ~ ~This ) . gives (8.2.14b)
8 THE ACOUSTIC PROPERTIES OF THE SEA
450
If we now assume an exponential fall in the amplitide of the waves ex along the path x with the attenuation coefficient of this amplitude ctk due to absorption of this wave energy, the following solution of this equation is suggested
ex = const e-(ik+eLP,
(8.2.15)
where k is the wave number. Substituting this expression and its second derivative in (8.2.14b) yields (8.2.16)
Two equations emerge directly from this, since the sum of the real terms and the sum of the imaginary terms (in square brackets) must each be zero for (8.2.16) to be satisfied. By combining these two equations, we get an expression for the absorption coefficient a: (amplitude attenuation in Np//m) in the relaxation process in which, as we have assumed, all the molecules are taking part (8.2.17) where cp s cu/k is the phase velocity of wave propagation, i.e., the speed of the wave crest in the direction of its propagation. This phase velocity depends on the wave frequency o and on the relaxation time 7, in a way that emerges from (8.2.16) discussed earlier c, = ( c v 2 ) (1 +o"z:)""(1--02z:)1'~+
1]-l/2.
(8.2.18)
Thus there is a certain diversity in the phase propagation velocity of sound waves of different frequency in the medium (sound velocity dispersal). In seawater, this diversity is very small, the phase velocity c p differing from the velocity of sound in a non-dispersed medium c by less than 1%. If we therefore assume that c p z c, (8.2.17) can be simplified (8.2.19) If not all the molecules but only a certain number of them take part in a relaxation process, we have to multiply (8.2.19) by a constant coefficient which defines this number of molecules. In fact, relaxation is a multicomponent process and may be a multistage process for each constituent (e.g., MgSO,). A fraction of fluid molecules A; thus participates in the next j-th stage of the relaxation process with a relaxation time of tj.Then, in view of (8.2.19) and taking A;/c = A j , we obtain the sound absorption coefficient during relaxation
8.2 THE ABSORPTION AND SCATTERING OF SOUND IN THE SEA
45 I
(8.2.20)
where zj in seawater may correspond to the relaxation times of these processes rl = z,,~, z2 = z , , ~ ~ z3 , ~=, z , , ~ ~t4, = ~ ,z, etc. It should be remembered that relaxation times in sea water depend on pressure, temperature and salinity and will therefore change with depth (Bezdek, 1973; Eigen and Tamm, 1962). The Absorption of Sound Energy in the Sea
The absorption of sound waves in seawater is chiefly the result of the relaxation processes described generally by (8.2.20), and more precisely, of its terms for the water itself, and for the magnesium sulphate and boric acidcontained therein. The relaxation time in (8.2.20) or (8.2.19), which can be expressed for water clusters by the two kinds of viscosity qw and 7: requires more precise definition. So in order to take into consideration the absorption of sound due to molecular friction in the water itself, we must use not the simplified equation of motion (8.0.5b) but the full Navier-Stokes equation (6.1.41), omitting forces such as gravitation which do not affect sound propagation, but including a term describing the volume viscosity in a compressible medium (Landau-Lifshits 1958). @o [$+(vV)v]=
-Vp+qV2v+
(q’+-q:
i
(8.2.21)
V(Vv).
By employing the following relationship between field functions
v x v x v = V(VV)-V2V,
(8.2.22)
equation (8.2.21) can be reduced to the form eo
[g +
(VV)V] =
- vp +
(fq + ) V(Vv) +q(V qr
x
v x v) .
(8.2.23)
For both kinds of viscosities q and q’ we assume Newton’s approximately linear law of momentum exchange (stress proportional to Vv, see (6.1.3)), thus we can speak of a Newtonian fluid. In a plane wave propagating along the x-axis, the particles oscillate in the direction of this axis (a longitudinal wave), hence their oscillation velocity v, = u , vy = v, = 0 and au/ay = au1a.z = 0. We also assume that &/at S (vV)v and this latter term is omitted from the equation. With these simplifications for a plane wave, the last term on the right side of (8.2.23) is equal to zero and the equation becomes
8 THE ACOUSTIC PROPERTIES OF THE SEA
452
(8.2.24) Differentiating this equation over x and using the approximate continuity equation of the medium eo&/ax = - 2 ( s ~ > / a t= - & / a t (see the transformation from (8.0.10) to (8.0.11)), yields
(8.2.25) Implementing (8.2.8) and remembering that d(Q) = de, we obtain the wave equation for an attenuated plane wave
(8.2.26a) which, after dividing by cz and rearranging can be written as
(8.2.26b) In this way we have obtained an equation analogous to (8.2.12) for a plane wave, but from comparing the two equations, it is clear that the relaxation time is z, = 44/3 f 41'
(8.2.27) eoc2 The appropriate values for pure water at 287K given by Clay and Medwin (1977) are qw z 1 . 1 7 ~ N s/m2, qk "N 2 . 8 ~ ~ ~ ew = lo3 kg/m3, c, G 1 . 4 8 ~lo3 m/s,
-
so that the relaxation time in pure water, according to (8.2.27) is z, z 2.1 x 10-12 s ; in practice it is much shorter than the period of sound oscillations in the sea tr < T. A11 the water molecules can therefore take part in the relaxation process attenuating sound waves; the attenuation coefficient for water alone results from a combination of equations (8.2.19) and (8.2.27). Since for water z, < T and 0~z.5< 1, we can therefore assume that for water the expression in the denominator of (8.2.19) is (l+02tS) 2 1 . With this simplification we obtain the attenuation coefficient of the wave amplitude by water alone (from (8.2.19) and (8.2.27)) in [Np/m]
(8.2.28)
8.2 THE ABSORPTION AND SCATTERING OF SOUND I N THE SEA
453
which allows for the effect of both viscosities of water qw and 11; on this process and also indicates that for z, < T, the absorption of sound waves in this process increases with the square of the wave frequency. Clay and Medwin (1977) also give empirical equations for the relaxation absorption of the other two important terms of the general equation (8.2.20), i.e., the absorption relaxations due to MgSO, and B(OH)3. Denoting (after Clay and Medwin, 1977) the resonance frequency by 1 (8.2.29) 2xz, the sum of the three main components of the sound absorption coefficients xS in seawater of salinity S(%,) can be given in [dB/m] as v, =
as =
-,
1.71 x *
+ A”v,, Y2 + V;,B BY’
(1 - 1.23x 10-3P,)+
(8.2.30)
in which the first term describes the absorption by water from the equation (8.2.28) derived earlier (after converting nepers to decibels and giving the frequency in kHz), the second term describes the absorption by magnesium sulphate and the third one by boric acid; the last two terms emerge from the general form of the attenuation coefficient in the relaxation process with empirically determined coefficients A’ and A”. Here a, denotes the absorption coefficient of seawater, v the wave frequency in kHz, v , . , the ~ ~ resonance frequency for magnesium-sulphate relaxation (in kHz), v,, the resonance frequency for boricacid relaxation (in kHz; defined by (8.2.29)), P, the hydrostatic pressure in the sea (in atmospheres), S the salinity [7g0], while the coefficients A’ = 2.03 x x dB/kHzm. The total absorption of dB/kHz %,,m and A” = 1.2 x sound in seawater and the various components of this absorption, in accordance with (8.2.30), as a function of the wave frequency, is illustrated in Fig. 8.2.1. The plots on this figure indicate principally the rapid increase in the sound absorption coefficient in pure water with frequency (-02), so that the range of higher frequency waves in water descreases at a high rate. At the same time in seawater we see the rapid rise in absorption due to the relaxation processes of magnesium sulphate and boric acid with increasing frequency (-w2), until resonance frequencies are reached, which differ widely for these two components. Above the resonance frequencies, absorption becomes constant, so that the relaxation absorption coefficient is now independent of the frequency.
454
8 THE ACOUSTIC PROPERTIES OF THE SEA
Wave frequency v
I kHz]
Fig. 8.2.1. The absorption coefficient of sound in pure water and ocean water of salinity S = 35%, as a function of sound wavefrequency, plotted according to equation (8.2.30)for a temperature of 287 K at the sea surface (P,w 0). The broken lines illustrate the separate participation in sound attenuation resulting from the relaxation of magnesium sulphate and boric acid (a compilation from Clay and Medwin, 1977). With permission of John Wiley and Sons Inc.
Sound Scattering Functions As with the propagation of light (see Chapter 4), sound propagation in the sea is attended not only by absorption in the medium but also by scattering at unhomogeneities within it called scattering centres. As far as light-electromagnetic waves-is concerned, these centres are particles having different electromagnetic properties from their surroundings, with different polarizability and magnetizability in an electromagnetic wave field. For sound, on the other hand, these centres are particles whose mechanical properties differ from those of the surroundings ;they have a different compressibility and a different density, hence a different inertia.
8.2 THE ABSORPTION AND SCATTERING OF SOUND IN THE SEA
455
The principal groups of sound scattering centres in the sea are marine organisms and gas bubbles. The latter include free air bubbles created largely beneath the sea surface by sea waves (see Section 2.7), bubbles attached to suspended particles, and a special group of swim bladders within the bodies of many marine organisms (see e.g., Barham, 1963; Lovik and Hovem, 1979). Thanks to these latter, echosounders easily receive sound signals backscattered from concentrations of these organisms, which makes it much easier to discover and locate them in the water. The elastic integuments of many fish and plankton species fulfil a similar role (Lovik and Hovem, 1979 Hargreaves, 1976). The quantitative description of sound scattering from various objects in the sea requires the introduction of several functions characterizing it. The set of functions used in hydroacoustics is different from that applied in hydrooptics. We shall first introduce a few definitions of the functions employed in Clay and Medwin’s (1977) excellent monograph and compare them with the definitions of the corresponding functions universally applied in hydrooptics (Jerlov, 1976; see Chapter 4). Above all, we must remember that the sound intensity from equation (8.0.25) expresses the power of the waves incident on a unit surface area perpendicular to the direction of their propagation, that is, it corresponds to the classical and not the modem definition of light intensity, which in Chapter 4 we denoted by I’ or Ep. Such a sound intensity, denoted here simply by I, for a spherical wave emitted by a point scattering centre in a homogeneous medium (not distorting the spherical wave) is inversely proportional to the square of the distance R from that centre
(8.2.31) for purely geometrical reasons. This is because at any distance R from the source, the same total power emitted by the point source of a spherical wave, that is the total wave energy flux F [w] is distributed over the whole surface area 4xR2 of the sphere of radius R surrounding the source. The wave intensity I [W/m2], i.e., the energy flux per unit area of this sphere, is therefore equal to
I=-
F N-
4nR2
1 R2
(8.2.32) ¶
in other words, it decreases with the square of the distance.
In hydroacoustics, the scattering function y(&) is defined for a single scattering centre, which at a sufficiently great distance R can be regarded as the point source of a spherical scattering wave. This function expresses the ratio of the
8 THE ACOUSTIC PROPERTIES OF THE SEA
456
intensity of a wave I, scattered by a given object through a scattering angle of 0, to the intensity I,, of a plane wave incident on that centre, but as converted to unit cross-sectional area of the centre A,, normal to the direction of the incident wave (a kind of shield for the wave) (8.2.33) Multiplying the ratio Ir/Ip by R2 renders the function p so defined independent of the distance R in view of what we have just said, so the function p(Or) is merely a physical feature (an inherent acoustic property) of the scattering centre. Notice too that the ratio of a sector AA of the sphere to the square of its radius is by definition the solid angle AQ (in steradians) over which this sector of the surface is visible from the centre. So for a spherical particle R 2 / A , 2 1/AQ is approximately the reciprocal of the solid angle over which the scattering particle can be seen from a distance R, i.e., in reality the function ~ ( 0 , )is the ratio of the intensity of a wave scattered in a given direction 0, over unit solid angle Ir(O,)/AQ to the intensity of the incident wave I,, and is therefore given in [sr-'1. As in hydrooptics, we can define the volumetric sound scattering function po(O,) which here will express the ratio of sound wave intensity I,, o , scattered over angle 0, by N particles contained in a unit volume of medium, to the intensity of a plane wave Zp incident on that volume-as calculated per unit area of the total cross-section A n , of those N particles (8.2.34)
The linear dimensions of the scattering centre must also be much smaller than the observation distance R, and the medium must be homogeneous for the 1/R2to hold. Assuming that scattering at the N individual relationship particles contained in a unit volume of medium is additive, I,, = I,, whereas N
N
A , , and for N identical particles I,, = NI,. From (8.2.34), we can
A,,, = N
write for this last case I, R2 To(&) = N - - - - , Zp A n , o
(8.2.35)
from which it is obvious that the volume sound scattering function yO(Sr)has units of [m-3 sr-l], i.e., the same units as N -
R2 An, O
. By comparing this with the
8.2 THE ABSORPTION AND SCATTERING OF SOUND IN THE SEA
457
volume scattering function B(0,) used in hydrooptics (see Chapter 4, equation
(4.3.5)), there emerges a simple connection between these functions if they are to be applied to the same waves (acoustic) (8.2.36) where A n , Ois, as we have said, the total cross-sectional area of N particles contained in a unit volume of medium-the cross-section being perpendicular to the direction of the incident wave. The geometrical cross-section A , of a scattering centre in the path of a wave (at right angles to the direction in which this wave is propagating) acts as a kind of shield, retaining part of the wave’s energy which is then scattered or absorbed. Depending on the physical properties of the centre, including the range within which it can interact with the wave, the incident wave flux so “removed” from its original direction may be considerably larger or smaller than that which is incident on the geometrical cross-section of the object, i.e. A , I p = F p . For this reason, the concept of the active (absolute) scattering cross-section of the particle (centre) is introduced and defined in the following way Fr a, =-, I, where F, =
(8.2.37) I,(6,)RzdQ. This gives the relationship between the total wave 4X
flux F, [W] scattered in all directions (over a full solid angle 47t) and the intensity Ip IW/m2] of a wave incident on a given scattering centre. Hence a, is expressed in units of area [m’] and is equal to the geometrical cross-section of the centre A,, only when F, = Fp = A,IP. The absolute active absorption cross-section of a centre is defined anaIogously (r (I
=-.Fa I,
(8.2.3 8 )
where Fu is the energy flux absorbed by that centre. The total power absorbed and scattered by the centre makes up the attenuation of the wave, so we can also introduce the active attenuation (extinction) cross-section of the centre (re
= 0,
+ Gr
(8.2.39a)
or (8.2.39b)
8 THE ACOUSTIC PROPERTIES OF THE SEA
458
The same parameters have applications in optics, but there was no need to introduce them in Chapter 4. The relative active scattering cross-section Qr is commonly used in hydrooptics (Jerlov, 1976; Kullenberg, 1974) and, it is readily demonstrated that it is the ratio of the absolute active scattering cross-section or to the geometrical cross-section A , of the scattering centre Qr =
-.0;
(8.2.40)
An
Since or is defined as the ratio of the total wave power scattered over a full solid angle, i.e., F, = Zr(8,.)R2dSZ, to the intensity of the wave I p incident on the
s
4K
scattering centre, from definitions (8.2.37) and (8.2.33) we can write
- "
0, = --
=
1
a R 2 d Q
4x
ZP
=
S
p(8,>AndQ.
(8.2.41)
452
Likewise, for unit volume of the medium in which N particles additively scatter a sound wave with a summed intensity of Zr,o(O) = NIr(8,.), we can introduce the active (absolute) scattering cross-section for this volume or, = No, and on the basis of (8.2.41) and (4.3.20) write
(8.2.42) Hence the absolute scattering cross-section through a volumeelement of the medium containing scattering particles is simply the volumetric scattering coefficient which was defined for light waves in Chapter 4. It is also applied in hydroacoustics, but mainly as an average statistical value (or,o>in view of the fluctuations in the active cross-sections of scattering centres such as fish that alter their position and orientation in a given volume of water. Besides this, in hydroacoustics, the sound source and receiver of an echosounder operate in water right by the ship's hull whereas the scattering centres, such as fish, are many tens or hundreds of metres away. Under these circumstances, the only practical thing to do is to record only backscattered sounds originally emitted from the ship. So especially important is the backscattering function pl,.(n)= vr,band the corresponding active backscattering cross-section (over unit solid angle) @r,b which according to definitions (8.2.41) and (8.2.33) is written as (8.2.43)
8.2 THE ABSORPTION AND SCATTERING OF SOUND IN THE SEA
459
where I,,* = Zr(x) is the intensity of the backscattered wave (over angle 8, = x). This active backscattering cross-section determines the relative intensity of the signal reflected from the given object (or objects in the water volume (No,, b ) ) , i.e., it determines the efficiency with which this signal is recorded by the acoustic echosounders generally used in marine hydrolocation. When scattering is isotropic (Zr(Or) = I,,, = const and hence pr(&) = piso= const), the integration of the expression for the total active cross-section (8.2.41) becomes simpler o,,i.o= ~ l i ~ , A , = 4 ~4 ~ -ZrR 2 . (8.2.44) 1, Then the scattering over a unit solid angle is identical in all directions, so the active cross-section due to backscattering over the unit solid angle is
(8.2.45a) The converse is also true: if we assume that the centre scatters sound isotropically in all directions, we can determine the total active scattering cross-section from the magnitude of the recorded signal I,. backscattered over the unit solid angle:
or,iso
=
4x0,.
b,
= 4~ _-17 is0
R2.
(8.2.45b)
1,
The active backscattering cross-section G,, b divided by the unit area A = 1 m2 or by the square of unit distance R2 = 1 m2 (to get rid of the units) is called the target strength (TS) and is usually expressed in decibels according to the definition
TS
=
10 log
a..b[dB]. A1
(8.2.46)
Using the connection between the intensity Z and the pressure p of the wave Z = p 2 / e c , we can rewrite (8.2.43) as
(8.2.47) from which, in view of (8.2.46), we obtain the target strength expressed in terms of the acoustic pressure 2
TS
=
lolog '(x)An A,
=
lolog($$-)
[dB].
(8.2.48)
460
8 THE ACOUSTIC PROPERTIES OF THE SEA
The pressure of an incident wave can also be expressed by means of the pressure of a wave emitted from a source p s at a distance R, from that source. Assuming that the law (8.2.32) holds for the emitted wave, we can assume that pg(R)/p,2(R) = R,2/Rg. In the sea, the acoustic targets are often fish, whose spatial distribution and orientation require the target strength to be statistically averaged (see Foote, 1980). All these parameters are, of course, functions of the frequency of the acoustic wave. Sound Scatter at Small Scattering Centres and Bubbles
After these few definitions and digressions, we now turn to a short description of the nature of sound propagation in the sea. The mechanism of this scattering at particles small in comparison with the wavelength is to some extent similar to Rayleight scattering of light, always allowing, of course, for the different nature of the vibrations of the scattering particles. This mechanism was also described by Rayleigh in 1896. Under the influence of a pressure wave, a spherical particle of radius ro < R may be subjected to two kinds of forced oscillations: (1) elastic deformation (compression and rarefaction, i.e., fluctuations of volume), and (2) oscillations within an elastic medium (periodic displacement in the medium). The first type of forced oscillations, the volume fluctuations of the particle, are determined by the ratio of the compressibility of the medium surrounding the particle to the compressibility of the particle itself, and can be given by the ratio of the adiabatic compressibility coefficient (kp,Q)waler/(kp, and denoted by the letter e (elasticity ratio). The second type of oscillations, that is the oscillations of a particle with respect to the surrounding medium, is determined by the ratio of the density of the particle to that of the surrounding water Q ~ ~ ~ which ~ ~ is denoted ~ ~ by~ d (density / Q ratio). ~ ~ The different densities of the scattering particle and water cause a difference in the inertia of the oscillatory motion fo these two elements of medium, i.e., a difference between the motion of the particle with respect to the water. Both these modes of particle oscillation, I and I1 (volume fluctuations -an oscillating monopole, and oscillations within the medium-an oscillating dipole), are sources of new waves of pressures which disperse within the medium like scattered waves, because they have their own directional characteristics and draw their energy from the primary wave. Thus a scattering particle much smaller than the wavelenth is the source of a new (scattered) wave composed in general
~
~
~
8.2 THE ABSORlTION AND SCATTERING OF SOUND IN THE SEA
461
of two constituent waves: one due to the volume fluctuation of the particle, and the other due to the oscillation of the particle within the medium. The condition that the particle or scattering centre be far smaller than the wavelength (i.e., that ro << 1, or kr, < 1) springs, as in Rayleigh scattering of light, from the assumption that the entire particle is at any instant enclosed in the uniform pressure field of the wave and that all its elements oscillate with the wave in the same phase, the whole making up a single oscillating centre (see Chapter 4, Fig. 4.3.2). The sound waves applied in hydroacoustics usually satisfy this condition with respect to suspended matter and air bubbles contained in sea water. On the other hand, it does not always hold with respect to the swim bladders and bodies of the larger marine animals; the sound waves used in this case may have frequencies of at least 100 kHz. For centres which are small in comparison with the wavelength, Rayleigh obtained an expression for the sound-scattering function ?(Or) which for scattering centres in water can be written thus:
(8.2.49) where ro is the radius of the scattering centre, e = (kp,Q)water/(kp,Q)partic,e is the elasticity ratio, d = Q ~ ~ ~ ~is the ~ ~density , ~ ratio, / Q k~ =~2x11 , ~is ~the wave number in water, and 8, is the scattering angle, i.e., the angle between the direction of the incident wave and the direction of observation of the scattered wave (see Chapter 4, Fig. 4.3.3a, and remembering that the drawn rays-of sound also-are perpendicular to the wave surfaces). Assuming that scattering at N identical particles in a unit volume of medium is additive, we get the Rayleigh volume scattering function simply by multiplying pRby N ( y R N = The active Rayleigh scattering cross-section for sound on the other hand is obtained by integrating over a full solid angle 4x the function y R multiplied by A,, = nr: according to definition (8.2.41). This yields
(8.2.50) Expressions (8.2.49) or (8.2.50) refer to small spherical particles whose resonance frequencies differ from the frequencies of the sound waves used. They include the volume pulsation and oscillation wave components which comprise the resultant wave emitted into the medium by a given particle. These two wave components may differ for different scattering centres, so the shape of the scattering function may change. In the extreme case, when the matter of which the
8 THE ACOUSTIC PROPERTIES OF THE SEA
462
particle is composed has the same density as water but differs from the latter only in compressibility, d = 1 and pR(Or) = const in (8.2.49), that is, the scattering is derived only from the fluctuation volume of the centre and is isotropic (independent of the scattering angle). The relationship between the scattering intensity and the wavelength of the sound is of significance here too. It is analogous to the situation in Rayleigh light scattering (see Chapter 4, (4.3.7)), hence the intensity of sound scattered at small particles is inversely proportional to the 4th power of the sound wavelength (pR IR ,- l/14).This is quite clear from (8.2.49) in which k4 = 16x4/A4, but it does mean that short waves are very strongly scattered and that their range in water containing a large number of small particles will be very restricted. Just as white light that has been scattered at small particles contains much violet, so too scattered sound which originally contained various wavelengths, contains a high proportion of short-wave energy. Unlike violet light, however, this is very strongly absorbed even by pure water, as we have seen. Because this is so, the noise which can be heard in the sea contains far more low-frequency energy (see Wenz, 1962; Perrone, 1969; Klusek, 1974). One particular set of sound-scattering centres in the sea are gas bubbles (see Section 2.8; Medwin, 1970, 1977). Above all, their density, sound velocity, and compressibility are very different from those of water, and so their so-called acoustic resistance (ec)ai, < (ec),ater differs strongly. With the hydrostatic pressure near the sea surface being small, e < 1 and d << 1 for bubbles, so that according to the Rayleigh equation (8.2.49), they are scattered very widely and almost isotropically. However, a pressure wave may alter the physical properties of both bubbles and the surrounding water as gas and water vapour diffuse from the bubbles into the water and vice versa. Finally, since the size distribution of the bubbles in the water is complex, a large number of them may begin to resonate with the sound-wave oscillations as waves in a given frequency interval pass (see e.g., Nayfeh and Mook, 1979). At resonance, the amplitude of the bubbles’ volume pulsations (mode I and other modes, e.g., III-of quadrupolar oscillations) increases greatly so that sound scattering by resonating bubbles rises sharply. Obviously then, sound scattering at gas bubbles is actually much more complicated than the Rayleigh theory for non-resonating particles with fixed physical properties would give us to understand (see Nayfeh and Mook, 1979; Hsieh, 1979; Drumheller, 1980). Sound scattering at bubbles at resonance frequencies is particularly important in the sea, because the intensity of the scattered sound is then at its greatest. At the same time, however, since the volume pulsation amplitude of the bubbles
-
8.2 THE ABSORPTION AND SCATTERING OF SOUND IN THE SEA
463
in resonance is large, the intensity of processes leading to the absorption of part of the wave energy by the bubbles and its conversion into heat also rises rapidly. Thus sound absorption in seawater containing a large number of resonating bubbles is several times greater than in seawater alone, and sound waves are then very strongly attenuated. In a rough sea, there are great numbers of air bubbles in the upper layers of the water, and sound waves travelling up from the sea bottom often do not reach the surface because they have been totally scattered and absorbed by the bubbles. Sound propagation in a shallow sea is thereby considerably worse; in a calm sea, however, sound waves are reflected from the surface and bottom like light waves from a mirror (see Section 8.3). As bubbles resonate at certain sound-wave frequencies (Uberalch et al., 1979), bubbles sizes in water can be measured: at resonance, there is a distinct, very great increase in the scattered signal-this is illustrated in Fig. 8.2.2. This drawing also shows that as the scattering particles increase in size, their active scat-
'oo
Size parameter of particle xo=kro
Fig. 8.2.2. Variation of ratio of total scattering to geometrical cross section a,/xr; with kr, for a fixed rigid sphere and a bubble. The height of the bubble resonance peak was calculated for the sea level resonance 52 kHz. The low frequency tail of the bubble resonance curve may also be calculated from (8.2.50) for kro Q 1 and e < 1, d < 1. From Clay and Medwin (1977), with permission of John Wiley and Sons, Inc.
464
8 THE ACOUSTIC PROPERTIES OF THE SEA
tering cross-scetion tends to become the same as their geometrical cross-section (a,/xr; -+ 1). The considerable complexity of sound scattering at gas bubbles in water has been well described by Clay and Medwin in their monograph (1977). For bubbles much smaller than the wavelength of sound (kr, 4 l), they obtained the following expression for the active scattering cross-section (8.2.51) where p r and p p are the amplitudes of the scattered and incident waves respectively; ro is the bubble radius; v is the frequency; Y , is the resonance frequency; and S is the damping coefficient, dependent on the frequency and the physical properties of the bubble in a given medium. Clearly, when a bubble is resonating with the sound wave Y, = v, its active scattering cross-section will be (8.2.52) Sound scattering by small bubbles in water is practically isotropic, because its mechanism is dominated by radial pulsations of the volume-that is to say, monopolar sources prevail, although weak dipolar oscillations and other, perhaps quadrupolar oscillations distorting the bubbles can occur as well. When scattering is isotropic, the backscattering function vr,b = yis0= q(&J,). So applying (8.2.45) and (8.2.51) we can express the backscattering function for bubbles as (8.2.53) while the target strength for a bubble from (8.2.48) and (8.2.51) is (8.2.54) Suspended matter and gas bubbles in the sea usually satisfy the condition that ro < I for commonly used sound waves. From the standpoint of an analysis of the reasons why sound is attenuated in seawater, we shall therefore not discuss sound scattering at particles larger than the wavelength, which is also complicated to describe. Such scattering is nevertheless of great significance in nature, and especially in hydroacoustical applications, since it is applied to such scattering centres as fish, cuttlefish, unhomogeneities of the water masses in the sea, ships, submarines, also irregularities in the sea bottom and the wind-
8.3 INTRODUCTION TO THE RAY THEORY OF SOUND PROPAGATION
465
roughened surface (see Brekhovskikh, 1974).The functions of sound scattering at large centres are highly dependent on their shapes, the materials of which they are constructed, and hence on the mode of their forced oscillations and their resonance frequencies. Often, these functions are extended forward on a polar plot in the same way as for light scattered at particles greater than the light wavelength (see Section 4.3). A description of this scattering and plots of the function vr(e,)for cylindrical and spherical centres can be found in Faran (1951), Clay and Medwin (1977-on p. 192 of this monograph after Stenzel, 1938) and elsewhere. The total effect of absorption and scattering makes up the attenuation of sound in seawater, and can be given by the sum of the absorption coefficient and the volume scattering coefficient (as for light see (4.4.3)), and described by a transfer equation (for sound intensities) similar to the equation (4.4.6) for light. There exists, however, a fundamental difference between the conditions under which light and sound are propagated in a given direction in the sea. Although light energy is much more strongly attenuated that the energy of an average sound in the sea, it is practically not refracted at unhomogeneities in the water mass. It travels in straight lines, whereas sound is strongly refracted at these unhomogeneities and its path through the water is much extended. We shall discuss this phenomenon with respect to sound in the next section. It is due to the relatively large differences in the coefficient of sound refraction (sound velocity c) in layers of water of different temperatures, salinities and hydrostatic pressures (see Section 8.1, equation (8.1.2) and Fig. 8.1. l), whereas the coefficient of refraction of light for different water layers usually differs only in the fourth place after the decimal point.
8.3 INTRODUCTION TO THE RAY THEORY OF SOUND PROPAGATION IN THE SEA
A sound field emitted from a source within the sea and its changes with time can be described mathematically by solving the wave equation. To do this, we must establish the characteristics of the source initiating the acoustic disturbance (the angular distribution of the emitted power, frequency of oscillations) and the conditions of transferring these disturbances in the sea and at its boundaries (velocity of propagation, attenuation coefficient, reflection coefficient). TWObasic ways of solving the wave equation are known and applied to sound waves in the sea. The first is called the normal mode theory (i.e., normal oscil-
8 THE ACOUSTIC PROPERTIES OF THE SEA
466
lations or normal waves), and the second, the ray theory or geometrical acoustics approximation (see Brekhovskikh, 1974). In the normal mode theory, an acoustic field is described with the aid of characteristic functions called normal modes, each of which defines a particular way in which oscillations are propagated in a basin or layer of medium, and each is a solution of the wave equation (e.g., the unattenuated wave equation (8.0.15)). The algebraic combination of modes (assuming that they are additive), selected so as to correspond to boundary and initial conditions, describes the resultant spatial distribution of sound energy and its changes with time. For the natural conditions obtaining in the sea, these are usually extremely complicated calculations, applicable only in special cases, e.g., in a basin shallow in comparison with the wavelength, with a flat, reflecting bottom, smooth free surface, and horizontally homogeneous as regards sound velocity distribution c ( ~ ) . More commonly used and of greater practical application in the sea is the ray theory or geometrical acoustics approximation, elements of which will be discussed briefly. It uses the concepts of wave surfaces, wave fronts and sound rays, analogous to those of geometrical optics. The conversion of the wave equation into the eikonal equation leads to the ray theory; Snell’s law of light refraction can be derived from the eikonal equation. This law makes it possible to determine the path of a small section of an acoustic wave front in an unhomogeneous environment, since it defines the direction of the path for a given velocity. Recently a mixed solution to the propagation of sound in a non-homogeneous medium has been worked out using elements of the mode and ray theories (Weinberg and Burridge, 1974). This complex approach extends the possibilities of applying the wave equation to a description of sound propagation in an unhomogeneous medium in all three dimensions of a sea basin of varying depth.
The Eikonal Equation The wave equation (8.0.15) for the acoustic velocity potential (8.0.4) is 821,” __at2
- c2v22y.
(8.3.la)
After rearranging and expanding the nabla operator, it reads (8.3.lb)
8.3 INTRODUCTION TO THE RAY THEORY OF SOUND PROPAGATION
467
To carry out this conversion, we write a “characteristic” equation of this wave equation, replacing in the latter the indices showing the degree of differentiation by power exponents (8.3.2)
The new equation enabling us to describe the wave field is obtained by substituting the general form of the solution of the wave equation (8.3.1) in the characteristic equation (8.3.2). The general solution of the wave equation (8.3.1) is the potential (the acoustic disturbance) as a certain function of spatial and temporal coordinates. In this function we can separate the dependence of the potential on the coordinates W(x,y, z ) from the dependence of the potential on the time V(t) = c o t , where co is a constant (the speed of sound in a given area of medium). This general solution can be symbolically represented as Y(X,
Y , z, t ) = f [ W ( x ,Y , z)+ c o t ] .
(8.3.3)
For a single harmonic wave of circular frequency m, this potential has the wellknown form (8.3.4)
where the amplitude of the wave A ( x , y , z ) may be variable in space in the general case; in the ray theory, we assume A = 1. Substituting solution (8.3.4) or its general form (8.3.3) in the characteristic equation (8.3.2) and making certain assumptions about the function f , yields a new, time-independent equation
( ~ ) 2 aw+ ( +(%) ~) 2
=$=n2
(8.3.5)
which defines the condition which the function W(x, y , z) must satisfy. This is the basic equation of the ray theory and is called the eikonal equation. The coefficientn = co/c is the coefficient of sound refraction in the medium with respect to the same medium in a reference area where the velocity of sound is co. It is obvious from the form of the potential (8.3.4) for a harmonic wave that the function W(x,y , z ) described by the eikonal equation (8.3.5) in some way expresses the phase distribution of the acoustic disturbance in space in a fixed time.
468
8 THE ACOUSTIC PROPERTIES OF THE SEA
The solution of the eikonal equation (8.3.5) is a good approximation of the solution of the wave equation (8.3.1), so long as certain conditions are fulfilled. It can be demonstrated that for a harmonic wave (8.3.4) this approximation is a good one if the increment in the sound propagation velocity gradient on the path of one wavelength is small in comparison with the average gradient of this velocity (see Officer, 1958). This means in practice, that as far as acoustic waves are concerned, this approximation cannot be applied at the sharp boundaries between media in which the speed of sound is very different or in areas of the same medium where over the distance of one wavelength the sound velocity changes abruptly. We also say nothing about the amplitude of the wave which is assumed not to be subject to attenuation, or else must be determined in some other way. The eikonal equation is a partial 1st order differential equation whose solution describes surfaces in three-dimensional space W ( x ,y , 2 ) = const. (8.3.6) With reference to a field of acoustic disturbances they are wave surfaces, i.e., surfaces of identical oscillation phases, among them the surface of a wave front. The vector dr perpendicular to the wave front determines the direction of propagation of the section of the wave front surrounding it, while its successive positions in time delineate the path of this section of the wave front-this is called the trajectory of the sound ray, or simply the sound ray (Fig. 8.3.1). Also normal to the wave surface W(x,y , z ) = const is the wave vector k (see Fig. 4.1.1a), in other words, the sound rays are in accord with the directions of propagation of the wave energy. To determine these directions, we write the equations of a line normal to the surface W ( x ,y , z ) = const at point x,yozo (8.3.74 where awlax, awjay, awl& are the directional coefficients of this normal, and x - xo, y - yo, z- zo are its components along the coordinate axes. Being normal to the wave front, an infinitesimally small increment of the path of the wave front dr of components dx, dy, dz can be described by analogous equations (8.3.7b)
The directional cosines of this increment in the ray dr emerge from simple trigonometrical relationships
8.3 INTRODUCTION TO THE RAY THEORY OF SOUND PROPAGATION
dY cos?9., = -, dr
dx
cos?9x = --, dr
dz cos19z = __ dr
469
(8.3.8)
and the sum of their squares is equal to one.
gq+(gY+(g) 2
(8.3.9)
= 1.
We also know that the directional cosines of the normal are proportional to its directional coefficients, which can be written down using a constant proportionality coefficient c,
aw -dx dr- - cr __ ax ’
(8.3.IOa)
-dY- - G aaw y ’ dr
(8.3.10b)
dz aw cr(8.3.1Oc) dr aZ Substituting these expressions in (8.3.9) and using the eikonal equation (8.3.5) it is easy to find that the coefficient of proportionality c, = I/n, so (8.3.10) takes the form __ =
-
(8.3.1 1 a) (8.3.1 1b) (8.3.11~) Differentiating these equations over the ray increments dr yields the general form of Snell’s law (see Officer, 1958). Differentiating the first of them (8.3.11a), gives d
-&(n$)
=
aW
a
F(x)=-(-ax
i3W dx ax dr
aW dy dr
+-- ay
*)
+z (8.3.12) dr az
which after applying (8.3.1 l), becomes (8.3.13) If, further, we take (8.3.9) into consideration in this equation and do likewise with (8.3.11b) and (8.3.11c), we shall obtain the general form of Snell’s law,
8 THE ACOUSTIC PROPERTIES OF THE SEA
470
which describes the changes in direction of a ray's path depending on the coefficient of refraction of sound n (8.3.14a) (8.3.14b) (8.3.14~) From these equations we can state that the increments in the products of the coefficient and the directional cosine (see (8.3.8)) along the ray's path are equal to the corresponding components of the gradient of the refraction coefficient n(x, y , z ) in the medium. Trajectories of Sound Rays in the Sea Let us now assume a horizontally homogeneous sea with the sound velocity distributions c(x, y , z ) = c(z) discussed in Section 8.1. Then the coefficient of refraction n(x, y , z ) = n(z) and (8.3.14) reduce to dx dr
=
const
or
ncos6x = const,
(8.3.15a)
n-du dr
=
const
or
ncosi?,, = const,
(8.3.15b)
=
2
n-
-$(n$)
or
d dr
dn dz -'
~(ncos6,) = -'
(8.3.15~)
The first two conditions show that the ratio cos8x/cos~,= const, which means that a sound ray in a horizontally homogeneous environment travels in a single, vertical plane (up or down, but not laterally). We can therefore make the x-axis run along the plane, so that the description of the ray's path in space is simplified to its description in the xz plane. Then C O S ~ . , = cos90" = 0, and condition (8.3.15) is reduced to equations (8.3.15a) and (8.3.15~).If the angle of incidence of a sound ray onto a horizontal plane is denoted by 8 3 @, and the angle which this ray makes with the horizontal plnae, the shear angle, by 6 = 8x = 90-0, we can write (8.3.15) as ncos6 = const
(8.3.16a)
8.3 INTRODUCTION TO THE RAY THEORY O F SOUND PROPAGATION
471
or
nsine = const and d -ncos8 dr
dn dz
= --
(8.3.16b)
(8.3.17)
Notice further that n = co/c (as denoted in the eikonal equation (8.3.5)), and that co is constant. We can assume that co is the velocity of a sound in the immediate vicinity of the source of that sound at depth zo in the sea, while the ray is emitted from that source at an angle 6, to the horizontal. Further away from the source, the velocity of the sound along the path of the ray may take the values c(zl), c(z,), c(zg), ... ,c(z), and the angle at which this ray cuts the consecutive horizontal planes then takes the values @,, @, G3,...,@.Since co = c(zo) is a constant, condition (8.3.16a) can be rewritten in the practically useful form of Snell’s law (8.3.18) where a, can be called the Snell’s law constant for a given ray, or the rayparameter. A geometrical illustration of these relationships and symbols is shown in Fig. 8.3.1, which depicts a certain vertical sound-velocity distribution c(z) (Fig. 8.3.la) and the corresponding path of the sound ray emitted from the source at a shear angle Go in the xz plane (Fig. 8.3.lb). The figure shows that in a horizontally homogeneous medium with a continuous though gentle velocity distribution c(z), there are no reflected rays (waves), and the refracted ray is a continuous extension of the incident ray, the two together creating a single ray whose direction changes. This refraction of a sound ray (sound waves) at a continuous unhomogeneity in the medium is called the refraction of sound in a medium. They are, as it were, successive refractions of the ray at the boundaries of uniform but infinitesimally thin layers of medium, in which the sound velocity varies but an idnitesimal fraction from that in adjacent layers. To calculate the ray’s path and the time taken to cover this distance by an element of the wave front requires sections of the ray du and the time intervals dt for the wave front to cover them to be integrated along the ray path. Thus on the basis of the trigonometrical relationships shown in Fig. 8.3.1, we can write that an element of the ray path is equal to dr
dz sin8
= __
(8.3.19)
8 THE ACOUSTIC PROPERTIES OF THE SEA
472
On the other hand, the definition of the velocity of sound propagation drldt = c indicates that the time element dt in which a ray (wave front) covers the section of the path du at depth z in the sea can be given by dr dz _ _ ~ (8.3.20) c(z) c(z)sin6 ' where the right-hand equality in this expression comes from (8.3.19). The constituent element of the horizontal path of the ray (the horizontal distance covered by the disturbance) can also be defined from the geometrical relationships in Fig. 8.3.1 thus: dt
= __ = _
dz dx = --. tan 6
(8.3.21)
Before integrating these equations, we apply Snell's law (8.3.18) so that the trigonometrical functions in them can be replaced by algebraic ones whose only variable is the speed of sound c(z). This operation yields (8.3.22a) cos8 = a,&), sin6 = (1 - C O S ~ ~ )=' )[' t/ -~a , Z ~ ~ ( z ) ] ~ ~ ~ , (8.3.22b) (8.3.22~) Substituting these new expressions in (8.3.19) and (8.3.20) yields these equations in another form, more convenient for integration, as long as the form of the function c(z) and the initial depth of the ray z, and its initial shear angle 6, are known (Snell's law constant a, = cos80/c(zo)-equation (8.3.18)). The horizontal distance travelled by the acoustic disturbance can be worked out from (8.3.21) and (8.3.22~)thus: X
x-xo = i d " = xo
z
a,c(z) dz [-[l-U,Z,C2(Z)]1'2
(8.3.23) *
20
Likewise, the passage time of the disturbance, from (8.3.20) and (8.3.22b) is f
(8.3.24) It is clear from Fig. 8.3.1 that a sound ray in a horizontally homogeneous medium changes direction in accordance with Snell's law. As long as such a ray does not encounter an obstacle like the boundary of the medium, at some place
8.3 INTRODUCTION TO THE RAY THEORY OF SOUND PROPAGATION (a)
Velocity of sound c ( z )
473
(b) Horizontat distance x
7 zo---
EZ,-
------
--_-_~-
Fig. 8.3.1. Geometrical sketch of the ray theory of sound propagation in the sea. (a) Vertical velocity distribution of sound propagation in the sea with an obvious minimum c(z) at a certain depth; (b) the trajectory of a sound ray corresponding to this distribution c(z) (the path taken by a section of the wave front) in the .rz plane, emitted from the source at an angle e0 at depth 20. 81,Q1, .,., 4 are the shear angles of the ray, dr = d z / s h 8 = dx/cos8 is an element of the ray's path, B0 = 90°--80 is the angle of incidence.
along its path it reaches a turning point: a downward course turns upwards, or vice versa; in other words, the ray is totally internally reflected within the medium. The integrations in (8.3.23) and (8.3.24) must therefore be done separately for the sections of the ray path between successive turning points. At a turning point, the ray is horizontal (i.e., dz/dr = 0; cos6 = 1). In view of Distance [km]
0
5
10
15
20
25
30
35
40
45
Fig. 8.3.2. The trajectories of a number of sound rays in an abyssal sound channel with the vertical velocity distribution c ( x , y , Z) = c(z) shown in the left-hand drawing. The sound source is placed on the axis of the channel (in the c(z) minimum region), and the analysed rays are emitted at various angles Bo to the channel axis from -7" (extreme ray sent upwards) to +16" (extreme ray emitted downwards). This result was computed from equation (8.3.29) by Klusek and Szczucka (1981).
8 THE ACOUSTIC PROPERTIES OF THE SEA
474
Snell's law (8.3.18), this condition is satisfied at a depth z where cosSo/c(zo) = I/+), i.e., the turning point of the ray at depth z is given by (8.3.25)
As a result, depending on the vertical distribution of the sound propagation velocity in the basin c(z), the angle at which the ray leaves the source and the depth of the source zo, the sound ray path accordingly becomes complicated and can have any number of turning points, particularly if it does not arrive at the boundary of the medium (see Fig. 8.3.2). A characteristic feature of sound-ray trajectories is their continual change of direction such that they always turn towards the direction of lower velocity c(z). This is the same as changing the shape of the wave front, whose element in that part of the medium where the sound velocity is c(zJ leads other elements of the wave which are in the slower speed region c(zl) < c(z2) (see Fig. 8.3.lb). Equation (8.3.23) actually expresses the function x(z), which describes in an involved way the path taken by a wave-front element, i.e., the trajectory of a sound ray. The function in this form is not convenient for application to numerical calculations of ray trajectories. So we have to introduce another form of the ray trajectory equation derived from the rearrangement and the differentiation of (8.3.21) with respect to z using Snell's law d2z __ dx2
1
cos26(z)
dW) 1 d6(z) dz -_ dx cos26(z) dz dx
___
From Snell's law we have at the same time cos6(z) = U,C(Z)
(8.3.26)
(8.3.27)
which on differentiating with respect to z and rearranging yields dG(z) . a, dC(Z> dz
-sin6(z)
dz
(8.3.28)
*
Combining these last three equations, and taking into account the trigonometric relationship dz/dx = sin6/cos6, we obtain a practically useful differential equation that describes the trajectory of a sound ray d2z_ - -1 dc(z) _ dx2 u , " c ~ ( z ) dz
(8.3.29) *
This equation is a form of the ray-trajectory equation which can be implemented in computer calculations without the need to divide the trajectory into sections
8.3 INTRODUCTION TO THE RAY THEORY OF SOUND PROPAGATION
475
delimited by turning points (Moler and Solomon, 1970). Yet another way of describing this, applicable also to the three-dimensional distribution of the sound velocity c(x, v , z), is presented by Eliseevnin (1964). Some characteristic sound ray trajectories in the ocean, computed from (8.3.29), are shown in Fig. 8.3.2. There are many turning points concentrated (a)
c ( z ) [rn/s] f422 1457
Dis?ance [ m ] 0
1000
2000
3000
60
80
Fig. 8.3.3. Sound ray trajectories in a sea with a subsurface sound channel (after Klusek and Szczucka, 1981). (a) In the subsurface channel created by the velocity c(z) profile shown on the left side of the drawing, Sound rays are emitted from a source located at a depth of 25 m in a cone of aperture 5", directed diagonally downwards. They are deflectedand reflected many times from the sea surface; (b) in a seasonal subsurface channel in the ocean, where some of the rays emitted at a large angle above the channel axis also enter the abyssal channel. Notice the points at which sound energy is concentrated in the water, the areas of silence avoided by the emitted sound, and where energy is transferred from one channel to the other.
476
8 THE ACOUSTIC PROPERTIES OF THE SEA
around the depth at which the velocity C(Z) is at a minimum. This depth delimits the axis of the sound channel in the ocean, formed by a suitable profile of velocities c(z) for the reasons given in Section 8.1. Recalling that each ray describes the path of a selected element of the sound-wave front, it is apparent from the plots of these few rays that the original geometrical profile of the wave undergoes considerable distortion along its path. Because the wave energy is concentrated in sound channels, low-frequency waves (ie., weakly absorbed, see Section 8.2.) can be propagated across the ocean for thousands of kilometres. This is possible, however, with the sound source in a suitable position in the water, or more precisely, with an appropriate angle of shear 6,, at which the ray enters the sound channel. Depending on the vertical distrituion of the sound velocity c(z) and on the position and directional characteristics of the sound source, sound-ray trajectories become variously complicated in the sea. When the surface layers of the ocean are cooled during the winter at medium and high latitudes, the vertical velocity distribution c(7) allows for a second minimum just below the surface, thus producing a second, subsurface sound channel (Fig. 8.3.3). In this subsurface channel (seasonal or typical of cold seas), sound rays are bent up towards the surface and are reflected from it many times. The effective concentration of rays in certain regions in the water shown in Fig. 8.3.3a indicates that sound energy is strongly concentrated there. This geometrical effect of ray concentration, and also the attenuation of sound energy in accordance with the laws discussed in Section 8.2. must be allowed for when calculating sound-intensity field distributions in the sea-the ray theory says nothing about them (Matveenko and Tarasiuk, 1976). Certain areas of concentrations of sound rays emitted by sources into a given cone of angles, as we can see on Fig. 8.3.3a, can be enclosed in an envelope which delimits the acoustic area. In theory, no sound energy emitted from that source will penetrate beyond this area. Away from the sound source in the sea there are zones where the sound can be heard very well, and others in which there is only silence. The efficiency with which sound is transferred along the subsurface sound channel depends on the efficiency with which the sound is reflected off the undersurface (see e.g., Dera et al., 1974). When the surface is smooth, reflection is very efficient, as the coefficient of reflection is then almost unity. When the surface is rough, however, sound is scattered at the surface elements, there is scattering and attenuation at air bubbles in the water, the Doppler effect makes itself felt as a result of the motion of the surface, not to mention other phe-
8.3 INTRODUCTION TO THE RAY THEORY OF SOUND PROPAGATION
477
nomena impairing the efficiency of the subsurface channel. Disturbances in the sound-ray path also occur because of density fluctuations and other unhomogeneities in the medium (e.g., internal waves or turbulence) deep in the water, also in the abyssal sound channel (Brekhovskikh, 1974). As a result of unhomogeneities and fluctuations in the state of the medium, part of the energy of a sound signal emitted from a source is scattered within the water, while the remainder reaches the receiver along various ray trajectories and is received at the same point in portions after different times. A received sound signal in the sea is thus usually very much distorted. So, for example, the energy of a single sound impulse emitted from a source is divided into a number of elements travelling along diverse paths and reach the receiver after varying periods of time as a sequence of impulses of variable and gradually decreasing amplitude (reverberation). This process of sound propagation in the sea is rendered even more complicated when the bottom participates in the reflection of rays. The bottom may have very different properties of reflecting and scattering sound (see e.g., Brzozowska, 1977)-it may be hard and rocky, soft and multi-layered, or of some other kind, and moreover it may not lie parallel to the surface. The sea bottom usually comes into play in sound propagation when the sea is shallow in comparison with the wavelength of the emitted sound. The surface and bottom of such a “shallow” basin delimit the boundaries of the sound channel (waveDistance
Fig. 8.3.4. Sound ray trajectories in a shallow sea-with surface and bottom (after Klusek and Szczucka, 1981).
[ml
multiple reflection of rays from the
478
8 THE ACOUSTIC PROPERTIES OF THE SEA
guide). The conditions under which sound can be propagated in such a channel will depend on the depth of the basin and on the acoustic properties of its surface and bottom. An example of sound trajectories in such a channel is shown in Fig. 8.3.4. For details on this extensive branch of hydroacoustics, the reader is referred to the many monographs already cited, and in particular to the book by Brekhovskikh (1974).
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INDEX
abbreviated density 25, 134-135 absorption of light 8, 141, 155-174, 231233, 299, 315-316, 414 - _ _ by pure water 156-165 - _ _ by seawater 165-174, 221-222 - - _ in the sea 173, 276, 277, 299, 300, 315-316, 414, 46-417 of sound 443-454, 451, 463 - - - by pure water 445-446, 451-453, 454 - _ _ by seawater 446-451, 454 - - - in the sea 451-454, 457, 463 active scattering cross section 457460 active layer of the ocean 403407 acoustic compression 427, 429 - pressure 426, 428, 448 - relative deformation 427, 428, 429 - resistance 462 waveenergy 432433 - velocity 426 - _ potential 427 additive attenuation of light 222 adiabatic compression 26, 124, 125, 387, 428-429, 432, 435 - density gradient 126, 346 -- temperature gradient 26, 387 advection 328, 336, 354 - ofheat 332, 364, 421-423 - of mass 328, 362-363 - ofmomentum 336, 354 - of turbulent energy 400, 402, 407 aerosols 233, 395-397 air bubbles 101-105, 393, 394, 464 air-sea interaction 8-10, 39-43, 369-424 composition 230-231 - _ interface 391-397, 413-415 albedo 256, 266, 267
-
-
-
angular distribution of radiance 255, 268-271 - - of scattered light 180, 181, 190, 202, 203, 204, 205 Antarctic (polar) waters 192,276, 285, 296 apparent optical properties 293 - - - of the sea 296, 302, 306, 307, 308-31 3 association of water molecules 56-64 asymptotic depth 309 - light field 279, 280, 308-314 Atlantic Ocean 4, 25, 45-46, 172, 191-192, 208, 266, 285, 296, 306, 438-439 atmospheric air circuIation 8-12 - transmittance 243-241, 374 attenuation of light 214-215, 219-222 - of electromagnetic waves 220-221 averaging of the Navier-Stokes equation 351-354 Baltic Sea properties 72, 89, 92, 93, 94, 95, 96,167,171,172,173,202, 205, 207, 208, 220, 222, 246-248, 280, 284, 289-291 296,301,306,313,315,405, 441-442 bar 137 beam of light 147 - transmittance meter 223 bioluminescence 280, 314 black body emissivity 2, 3 _ _ _ approximation for sea surface 375-376 bonds in water molecules 52, 53, 57, 59-64 boric acid (sound absorption) 447, 448, 453, 454 boundary conditions at sea surface 371-373, 376,378, 384-385, 390, 395, 413415 Boussinesq hypothesis 357 - approximation 379, 380 Bowen ratio 391
508
Brewster angle 252 breezes 10 bubbles 101-105, 393-394, 464 -, bursting at sea surface 393-397 -, resonance of 40, 464 -, size of 104 -, target strength of 464 buoyancyforce 23, 26, 346, 380, 410 - flux 414 - parameter 410 Cabannes correction 187 cardioidal radiance distribution 255 cavitation 102-103 centripetal force 14-16 Charnock constant 386 - equation 386 chemical potential of seawater 111 chlorinity of seawater 71 classification of ocean waters (optical) 274 -277 cloud transmittance of irradiance 244-245 cluster of water molecules 62-63 coefficient oflight absorption 159, 160, 162, 167, 168, 169, 172 - - contraction (due to salt in seawater) 128 - - irradiance attenuation 292, 293, 294, 296, 305, 309 - _ light attenuation 160, 163, 215, 220, 221, 222 - - _ scattering 185, 186, 203, 208 - - molecular diffusion 319, 321 - - - thermal conductivity 319, 321 - - - Viscosity 320, 322, 323 - - seawater compressibility 122-124, 126-128, 435 _ _ sound absorption 444, 445, 453, 454 - - temperature conductivity 331 - - thermal expansion 115, 116-1 22 - - - diffusion 362-376, 390 - - turbulent heat exchange 365, 367, 390 - - - viscosity 357-361, 367 colour of animals 280-281 - of the sea 85, 191-192, 204
INDEX compensation depth 279 complex refractive index 176, 193 concentration of bubbles 104-105 - - atmospheric gases in the air 230-232 _ _ ions in seawater 68-70 _ _ organic substances in seawater 88 _ _ suspended particles 89-90, 91-95, 98-99 continuity equation 339, 340 constant flux layer 382-386,' 390 Coriolis effect 34, 40-41 - force 33-36, 40-42, 44, 380-383 - parameter 36 Coulter counter 99-101 current meter 355-357 - velocity fluctuation 348 350, 352 currents in tropical Atlantic 45 daylight at great depth 269, 280, 308, 309 deep water layer 280, 406, 440 deformation rate tensor 359 density 133-1 37 -, acoustic compression 427, 429 -, determination of 134-139 - fluctuation 62-63, 66-67, 116-118, 175, 188-189, 353, 379-380 - in the sea 25 - of ice 60 - of seawater 66-67, 137, 139 - of Standard Mean Ocean Water 136 - potential 126, 346 diameter of irregular particles 90-91 diffuse attenu ationcoefficient 293, 305 _ _ of clouds 244, 245 _ _ of irradiance 293, 295-297, 304-307 _ _ of radiance 293 - reflectance function 294, 302 diffuseness of irradiance 241, 243 diffusion 317,318, 320-321,324329,361-367 dissipation of turbulent energy 343, 368,399, 401, 402, 408, 419 distribution function of light flux 294, 295 dipole moment of water molecule 53, 54 directional distribution function D 294-295, 297, 302, 304-308
INDEX diurnal thermocline 406 Doppler flowmeter 355-356 downward irradiance 152, 242, 273-280, 281, 284, 295, 300, 301 drag coefficient 39, 416 dynamic viscosity coefficient 320, 322, 323 Earth 1-3, 11 - gravity 15-20 - isolation 6 - radiation 3 - rotation 14-16 eikonal equation 467 Ekman spiral 40-42 - formula 127 elasticity 428, 429 electrical conductivity of seawater 64, 73, 77-79 - - ratio 80 electrostriction of seawater 65-67 energy budget of eddies 345-348 energy of light absorbed in the sea 299, 315-316 - - light underwater 276, 290-291 - - photons 142 -, solar radiation 227-229, 242, 276, 280, 290-291, 315-316 -, sound 431-433, 454 -, turbulent flow 345-348, 350, 398-403 -, viscous dissipation of 401, 402 entropy 107-110, 112 equation, acoustic waves 427-431, 452 - characteristics of an asymptotic light field 311 -, continuity 339, 340 -, diffusion 324, 327, 328, 340, 361-365 -, eikonal 467 -, Fick molecular diffusion 327 -, Gershunlight 229 -, Gibbs thermodynamics 112 -. Laplace diffusion 327 -, Maxwell electromagnetic field 194 -, Navier-Stokes 44, 337, 340, 351, 398, 45 1
509
-, heat budget 421422 - of motion 43-44, 337, 340, 354, 398, 407 -, radiant energy transfer 213-219, 236-241, 304, 310 - of state of seawater 115, 134-135, 136-140 -, thermal conductivity 331, 332, 340, 364, 387 -, turbulent diffusion 362-364 -, - energy 399-403, 418, 420 -, - heat exchange 364-365 -, Reynolds turbulent motion 354 Ertel exchange coefficient tensor 359 Euler description of motion 43 euphotic zone 279 evaporation 11, 64, 371, 387-391 Feret particle diameter 91 fine structure of water layers 28, 405, 406 fluctuation of underwater irradiance 281-284 - of water density 62-63, 66-67, 116-118, 175, 188, 353, 379-380 flux, aerosol 395-396 -, buoyancy 414 -, diffusing mass 319, 325, 327, 361, 362 -, latent heat 389-391 -, light energy 145-146, 227, 357, 371-376 -, momentum 319, 332-335, 354-355, 376, 382-386 -, sea surface radiation 373-376 -, sensible heat 387-388, 390 -, solarradiation 2-3, 6, 371-373 -, sound wave energy 431-433 -, turbulent energy 399, 402, 407, 416, 420 -, vapour 387-390 -, water droplet 393-397 fluxes across sea surface 370-371 foam at sea surface 259, 267 focusing effect 281-289 force, buoyancy 23, 26, 346, 280, 410 -, centripetal 14-16 -, Coriolis 33-37, 40-42, 44, 380-383 - of gravity 14-20
510
-, pressure gradient
22, 31-33, 44, 380, 428 -, turbulent friction 38, 354, 355, 380 -, viscous molecular friction 37-38, 44, 96, 320, 335, 336, 381 -, volume (specific) 17 -, wind friction 39-40, 382-386, 415 formula, Borogodskii et al. electrical conductivitv 78 Clay and Medwin sound velocity 438 Cox and Munk surface slope distribution 259 Ekman compressibility 127 Hurwitz clound transmittance 244-245 Kelvin adiabatic temperature gradient 126 Knudsen salinity 70 - abbreviated density (r 134-135 - specific gravity 134 Mamaev abbreviated density 135 Menache standard water density 136 Matsuike et a/. solar irradiance 373 Newton and Laplace sound velocity 430, 435 Rayleigh scattering 179, 461 Richardson-Ozmidov turbulent diffusion 365 -, Stokes sound absorption 445 -, - frictional force 96 -, UNESCO seawater density 137-140 - _ - salinity 79-83 -, Wilson sound velocity 437 -, Weyl electrical conductivity 78-79 Fresnel reflectance 250-254 frequency, -, light wave 142 -, relaxation 446-448 -, sound wave 454, 461 -, underwater flash 285, 286 -, Vaisala-Brunta oscillation 31 friction 37, 38, 318, 320, 354, 444, 445, 451 friction velocity 384-386 of particle 96 - of wind 39-42, 378-386, 414-415
-
INDEX frictional force 37, 38, 39, 44, 320, 335, 336, 354-355, 380, 381 geopotential 20-21, 32 global atmospheric circulation 11 heat budget 1-3, 7, 421-423 - isolation 6, 227-230 Gershun equation 229 Gibbs equation 112 gravity, force of 14-20, 379 - waves 41
-
halocline 24, 25 heat budget 421-424 - capacity (specific heat) 8, 58, 60, 112-113, 330, 331, 446 - transfer on Earth's surface 7-13 - _ equation 331-332, 340, 364, 408 - - through sea surface 370, 387-390, 414 horizontal homogenity assumption 218-219, 235, 273, 319, 360 horizontal homogenity assumption 218-219, 235, 273, 319, 360, 363, 317-379, 304-401, 470-477 hot film sensor 356 hurricanes 8 hybrid optical properties 302-303 hydrogen bonds 56-64 hydrostatic stability (balance) 23, 26-28, 379 pressure 22-23
-
ice crystal 58-60 inherent optical properties 215, 302 inorganic (mineral) suspensions 89-90, 93, 97, 98 ion hydrates 64-68 mobility 73-76 - separation 396-397 ions of polyvalent metals 70 - separation 396-397 ions of polyvalent metals 70 isobaric surfaces 31-33 irradiance 151-1 52
-
INDEX
- a t sea surface 237-240, 242, 247 - attenuation 274-281, 292-293, 301,
309 -, downward 152, 273, 276, 278-280, 284 fluctuation 281-289 - in various waters 274-281 - ratio (diffuse reflectance function) 294, 300, 302 -, scalar 151, 152 - spectra 239-240, 242, 276, 277 - transmittance 295 -, upward 1522, 73, 274, 294, 301 -, vector 272, 273 isobaric surfaces 32 isothermal deep layer 404
-
Jerlov’s optical classification of waters 274-278 Junge particle size distribution 93-94 Karman constant (coefficient) 385 Kelvin’s formula 125 kinematic viscosity coefficient 320 KirchhoE sound absorption 445-446 Knudsen salinity definition 70 Knudsen’s formulae 70-71, 134-135 Lagrange method of describing motion 43 218 laminar surface layer 391-393 latent heat 60, 64, 389, 390, 391, 141, 423 layers of stratified sea 403-407 light absorption in atmosphere 230-232, 233, 236, 374 - - in pure water 156-1 65 _ _ in seawater 155-174, 315 - _ in water column 299, 300, 315 attenuation 21 3-225 -, colour of 142 emerging from the sea 251, 256, 265) 267, 268 - energy 141-143 - - absorbed in a volume 173, 272, 273, 299 - flashes under sea surface 283-289 - reflection 248-257
- derivative
-
511
- scattering
174-213 182, 185-191, 197-199 209-21 3 142, 227 local Reynolds number 347
- _ function - _ matrix - wavelength
magnesium sulphate sound absorption 447-448, 453, 454 mass transfer (diffusion) 315, 317, 361, 371, 390, 395, 414 Martin particle diameter 90-91 Mie amplitude functions 195 - intensity functions 196, 200 - theory of light scattering 192 mixed layer model 411 mixing length 358 model of atmospheric air circulation 11 - of Blanchard ion separation 396-397 - of Ekman spiral 40-42 - of free fall of particles 96-98 - of horizontally homogeneous sea 44-45 218, 273, 299, 325,364, 4 0 3 4 0 6 , 411, 470 - of liquid water structure 62-63 - of McIntyre drop emission 393-395 -, monsoon circulation 8-9 -, Niiler and Kraus mixed layer 413-420 -, single scattering of light 238-241 -, surface boundary layer 377 modes of particle oscillation 460 molecular diffusion 318, 321 - exchange through sea surface 341, 391-393 -- stress tensor 335 - thermal conductivity 318, 321, 392 - viscosity 318, 322, 323, 329-330, 444-445, 446, 451, 452 Monin-Obukhov scale 386 Morel molecular scattering data 190, 191 Navier-Stokes equation, 44, 337, 340, 35 I , 398, 451 nomogram for light scattering 201 number, Prandtl 367 -, Reynolds 344, 347, 348, 395
INDEX
512
-, Richardson 347, 348 -, Schmidt 367 -, of suspended particles 91-95, 163
-
ocean, absorption of solar energy in 7-8, 165-169, 276, 315-316 -, active layer of 403-407 -, average depth of 14 -, circulation (currents) in 10, 40-43, 45-46, 99, 342, 345, 348-351 -, evaporation from 11, 389, 391 -, pollution of 13 -, salinity of 24, 25, 69, 72 -, salt emission from 395, 396 -, sound channel in 441, 473, 476 -, surface area of 14 -, - fluxes of 371, 423 -, temperature of 4,24-25,376,393,403,404 -, tidal forces in 18 optical classification of seawater 274-277 - mass of atmosphere 235 - thickness (depth) 234-236, 305 osmosis 329 Ozmidov turbulent scale 365-367 path function (radiance) 216,217,238, 303 particle concentration 89, 90-92, 163 - counter 99-101 - diameter 90-91 - free-fall 96-99 - scattering of light 175-177,192,197-204 - size distribution 92-96 - surface area 96 particles suspended in seawater 89-101, 192, 222, 461, 462 - volume 91, 96, 100 periods of surface waves 47 - of Vaisala oscillations 31 permanent (principal) thermocline 404,406, 440 permittivity (dielectric constant) 67, 1 84, 185, 188, 189 photoadaptation of phytoplankton 277 photochemical reaction 166-168, 231 photon energy 142
photosynthetically active radiation (PAR) 277-279 pigments 89, 168-169 polar (Antarctic, Arctic) waters 72, 192, 276, 438-439 polarization of light 180, 183-184, 168 potential density 346 - energy 20, 53, 432 - temperature 126, 387, 388, 391 practical salinity scale 79-83 Prandtl assumption 358 - mixing length 358, 359, 384 - number 367 pressure gradient force 22, 31-33, 44, 380, 428 probability distribution of sea surface slopes 258-261 pycnocline 25 quantity of radiant energy 145 quantum irradiance 277, 278 radiance
147-149, 151
-, cardioidal distribution of 255 -, diffuse attenuation coefficient of 293
- distribution in atmosphere 255, 267 _ - a t the water-air interface 265,269-270 - - in the sea 268-271, 279, 280
- transmittance throught
sea surface 27& -271 radiant flux 145 - intensity 146-147 radiation of the Sun 2, 3, 227-229 - of the Earth 2, 3 - of the sea surface 373-376 radiative transfer approximation 311, 312 - - equation 213-219, 236241,298 304, 310 ratio of scattering function 207 ray theory of sound propagation 465-478 Rayleigh scattering 177-186, 188-190, 199, 232, 233, 236, 460-462 reflectance of light 250-254, 257, 260-262 - function 294, 300 refraction of light 248-250, 251, 268-271
INDEX - of sound 467478 refractive index 176, 198, 199,200,201, 203, 249, 250, 467 relaxation processes 446-451 - time 446, 447, 448, 449, 451453 relative scattering cross-section 198-201 remote sensing 267, 268 reverberation 477 Reynolds equation 354 - number 344-345, 437-348, 392, 395 - stress 38, 39, 355, 357 Richardson number 347-348 - turbulent diffusion coefficient 365 roughness parameter 385, 386
salinity 70-72 - measurements 70-71, 79, 83-85 -, practical scale of 79-83 salinometer 83-85 Sargasso Sea 167, 171, 191-192, 205, 206, 269, 274, 315 scalar irradiance 151, 152, 173, 299-300, 301 scattering albedo 222 - coefficient 185, 186, 187, 190,191, 197, 198, 200, 203, 208 458, 465 - cross-section 198-201, 457-460 - function 182, 185-191, 197-199, 202, 205, 455-457 - matrix 212-213 -, Mie theory of 192-201 - of light 174-213 - of sound 454-465 -, Rayleigh theory of 177-1 82, 460-462 -, Smoluchowski-Einstein theory of 187-190 -, spectral distribution of 186, 191, 201, 203, 206, 463 sea surface aerosol exchange 395-396 - - albedo 256 - - boundary conditions 371-373, 376 378, 384-385, 390, 395, 413-415 - -, bubbles bursting at 394-396 - - downward irradiance 242, 247, 276, 281-282
513
drop emission
I -
394-396
- - electrical charge emission 396-397 - -, evaporation from 11, 388-391 - - foam 259, 267
- - heat budget 421423 - -, laminar (skin) Iayer of 341, 376, 391-393, 395 oscillations (waves) 39, 47, 258, 284, 378 - - radiance distribution 253, 255, 268-27 I radiation 271, 373-376 roughness parameter 385, 386 - - slope distribution 258-260 - - temperature 4, 24-25, 376, 393, 403-404 , transmittance of light through the 256-257, 263-265, 270-271 - wind stress 39, 382-386, 414-415 seawater 64 absorption coefficient 159, 167-173, 444, 445, 453, 454 -, chemical composition of 68-70, 85-89 - compressibility 122-124, 127-128 - density 130, 134-135, 137-140 --, molecular structure of 64-68 - salinity 70-72, 79 -, specific heat of 112, 113 - viscosity 320, 321-323 seasonal thermocline 404, 406 shearing stress 335, 355, 382-385 single scattering model 236-241 size distribution of particles 92-96 - parameter 195 Snell’s law 248, 249-251, 271, 471 SOFAR 441 solar constant 227 solar radiation 2, 3, 6, 7, 227-229 - - distribution on the Earth 6, 7, 423 energy at the sea surface 239-240, 242, 247 - - spectrum 3, 227-229 in the sea 274-276, 280, 290, 291, 315, 371-373 sound absorption 443454, 457, 463
--
--
--
-
-
---
514 - channel 441, 473-478 - ray 466, 468 - refraction 436, 467-478 - trajectories 468, 473478 - velocity 435-443, 476 source function 217, 328, 332, 340, 364, 417 specific volume 110, 115, 117-119, 128-133, 137, 421-428 - heat 58 - - ratio 124 stability, vertical, of water column 26-29, 347 STD probes 8685 Standard Mean Ocean Water 136 Stefan-Boltzmann constant 2, 375 law 2, 374-375 Stokes’ parameters 209-212 - sound absorption 445 substitue diameter 90-91 surface boundary layers SBL 376-391, 395 - laminar layer 391-393, 395 - roughness parameter 385-386 - waves 47, 258-259 suspended particles 89-101, 208 suspension currents 99
-
tangential shearing stress 39, 41, 320, 335, 354-355 target strength 459, 464 temperature conductivity coefficient 331 - gradient 24, 319, 365, 392, 4003405, 412 - vertical distribution 24, 29, 404, 405, 418 tensor of deformation rate 359 -, molecular stress 335 -, Reynolds turbulent stress 355 - - turbulent diffusion coefficients 362-363 - _ _ momentum exchange coefficients 359 - - _ heat exchange coefficients 365 thermal conductivity coefficients 319, 321 - - equation 331, 332, 340 thermocline 24, 404, 40G thermodynamic equilibrium condition 108, 109
INDEX tidal force 17, 18 trace elements in seawater 70 trajectories, sound ray 468, 470-478 transmittance of irradiance 253, 295 - _ through the atmosphere 243-247,314 - - through the sea surface 256, 251, 262-267 - - through a water layer 274-280 transparency (definition) 219 turbulent diffusion 361-364 - energy 350, 398-403, 407, 408 - friction 354-355 - mixing length 358-359, 384, 395 - motion 341-351, 354, 398-403, 41 1 - scale 342, 350, 351, 365-367 underwater flash irradiance 283-289 upward irradiance 152, 273, 294, 301 upwelling 42 Vaisala-Brunt frequency 29-31, 410 velocity components 332, 333 - - in generalized notation 340, 352, 378 - - in turbulent flow 343, 348-350, 378 - - measurements 355-357 velocity of sound 435-443 vertical distribution of density 23, 24, 25-26 - _ of irradiance 215, 276, 278, 280, 301 - - of salinity 25, 29 of sound velocity 438-441 _ - of suspended particles 99 - _ of temperature 24-25, 29, 404-406, 418 of wind velocity 385 - stability 26-29, 347 viscosity 317-320, 322, 323, 392, 444-445, 45 1 viscosity 317-320, 322, 323, 392, 444-445, 45 1 -, dynamic coefficient of 320, 322, 323 -, kinematic coefficient of 320 -, turbulent 357-361 viscous dissipation of energy 341,401, 405, 420, 344-445 _I
--
INDEX water, absolute density of 136 51-55 -, molecular structure of 56-64 - molecule cluster 62-63 - standard 136 wave equation (acoustic) 427, 430 - number 193, 461 - periods 47
- molecule
515
- vector
145, 146 wind drag coe5cient 39, 386, 415 - friction 40-41, 42, 257-259, 378, 420 - - velocity 384 - stress 39, 382, 383, 384, 385, 414-415 - velocity vertical distribution of 385
yellow substances
85-88, 169, 170, 171
516 F U R T H E R T I T L E S IN T H E SERlES (cont. from p. /I) 39 T. ICHIYE (Editor) OCEAN HYDRODYNAMICS OF THE JAPAN AND EAST CHINA SEAS 40 1. C. J. NIHOUL (Editor) COUPLED OCEAN-ATMOSPHERE MODELS 41 H. KUNZENDORF (Editor) MARINE MINERAL EXPLORATION 42 J. C. J. NIHOUL (Editor) MARINE INTERFACES ECOHYDRODYNAMICS 43 P. LASSERRE and J. M. MARTIN (Editors) BIOGEOCHEMICAL PROCESSES AT THE LAND-SEA BOUNDARY 44 I.P. MARTINI (Editor) CANADIAN INLAND SEAS 45 J. C. 1. NIHOUL and 6. M. JAMART (Editors) THREE-DIMENSIONAL MODELS OF MARINE AND ESTUARINE DYNAMICS 46 J. C. I. NIHOUL and 6. M. JAMART (Editors) SMALL-SCALE TURBULENCE AND MIXING I N THE OCEAN 47 M. R. LANDRY and 6. M. HICKEY (Editors) COASTAL OCEANOGRAPHY OF WASHINGTON AND OREGON 48 S.R. MASSEL HYDRODYNAMICS OF COASTAL ZONES 49 V. C. LAKHAN and A. S. TRENHAILE (Editors) APPLICATIONS IN COASTAL MODELING 50 J. C. J. NIHOUL and B. M. JAMART (Editors) MESOSCALE/SYNOPTIC COHERENT STRUCTURES I N GEOPHYSICAL TURBULENCt 51 G.P. GLASBY (Editor) ANTARCTIC SECTOR OF THE PACIFIC 52 P. GLYNN GLOBAL ECOLOGICAL CONSEQUENCESOF THE 1 9 8 2 4 3 EL NI~~O-SOUTHERN OSCILLATION 53 J. DERA MARINE PHYSICS
