The Turbulent Ocean

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The Turbulent Ocean The subject of ocean turbulence is still in a state of discovery and development, with many intellectual challenges. This book provides a description of the principle dynamic processes that control the distribution of turbulence, its dissipation of kinetic energy and its effects on the dispersion of properties such as heat, salinity, and dissolved or suspended matter in the deep ocean, the shallow coastal and the continental shelf seas. Particular focus is given to the measurement of turbulence, and to the consequences of turbulent motion in the oceanic boundary layers at the sea surface and near the seabed, especially near sloping and rough topography, sea straits and fjords. The processes leading to turbulence and its generation by breaking internal waves are described in some detail and illustrated by examples taken from laboratory experiments and field observations. Because of its all-pervading nature, turbulence is at the heart of ocean science, and some appreciation of its effects is essential for those involved in research, not only in physical oceanography but in biological, chemical and geophysical aspects of ocean science. The Turbulent Ocean provides an excellent resource for senior undergraduate and graduate courses, as well as an introduction and general overview for researchers. It will be of interest to all those involved in the study of fluid motion, in particular geophysical fluid mechanics, meteorology and the dynamics of lakes. S t e v e T h o r p e joined the staff of the National Institute of Oceanography, later the Institute of Oceanographic Sciences, in 1962, and became Professor of Oceanography at Southampton University in 1986. He has made experiments to study the nature of turbulence and internal waves in the laboratory, at sea and in lakes, and has been involved in the development of novel sensors, instruments and methods of observation. He was awarded the Walter Munk Award by the US Office of Naval Research and the Oceanography Society for his work on underwater acoustics. He is also proud to have been elected a Fellow of the Royal Society of London, to be awarded the Fridtjof Nansen medal of the European Geophysical Society in recognition of his fundamental experimental and theoretical contributions to the study of mixing and internal waves in the oceans, and the Society’s Golden Badge for his introduction of a travel award scheme for young scientists. He is now an Emeritus Professor at the University of Southampton and an Honorary Professor at the School of Ocean Sciences, Bangor.

The Turbulent Ocean By S. A. Thorpe

   Cambridge, New York, Melbourne, Madrid, Cape Town, Singapore, São Paulo Cambridge University Press The Edinburgh Building, Cambridge  , UK Published in the United States of America by Cambridge University Press, New York www.cambridge.org Information on this title: www.cambridge.org/9780521835435 © S. A. Thorpe 2005 This publication is in copyright. Subject to statutory exception and to the provision of relevant collective licensing agreements, no reproduction of any part may take place without the written permission of Cambridge University Press. First published in print format 2005 - -

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Cambridge University Press has no responsibility for the persistence or accuracy of s for external or third-party internet websites referred to in this publication, and does not guarantee that any content on such websites is, or will remain, accurate or appropriate.

With love to Daph, my wife of 41 years, for her great support and enduring toleration.

‘For I have learned to look on nature, . . . . . . And I have felt A presence that disturbs me with the joy Of elevated thoughts, a sense sublime Of something far more deeply interfused, Whose dwelling is the light of setting suns, And the round ocean and the living air, And the blue sky, and in the mind of man.’ From ‘Lines composed a few miles from Tintern Abbey’ by William Wordsworth, 1798.

Contents Preface Structure and résumé Acknowledgements

page xi xiv xvii 1

1

Heat, buoyancy, instability and turbulence

1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9

Introduction Heat and temperature Density Instability and oscillations resulting from buoyancy forces Transfer of heat Heat capacity of the ocean Turbulence Turbulent dispersion Diapycnal heat transfer in the ocean

1 3 5 8 12 17 19 30 37

2

Neutral stability: internal waves

44

2.1 2.2 2.3 2.4 2.5 2.6 2.7 2.8

Introduction Interfacial waves Internal inertial gravity waves in continuous stratification Energy and energy flux The Garrett–Munk spectrum; the energy in the internal wave field Wave–wave interactions Generation of internal waves Internal waves and vortical mode

44 50 52 62 64 67 70 77

3

Instability and transition to turbulence in stratified shear flows

80

3.1 3.2 3.3 3.4 3.5

Introduction The onset of instability in shear flows The transition from Kelvin–Helmholtz instability to turbulence Unstratified shear flows Energy dissipation in stratified shear flows and the efficiency of mixing Holmboe instability The shape of billow patches and the length of billow crests Instability in a rotating ocean

3.6 3.7 3.8

vii

80 83 92 105 106 109 110 112

Contents

viii

4

Convective instabilities

115

4.1 4.2 4.3 4.4 4.5 4.6

Introduction The onset of convective motion Convection near surfaces of uniform buoyancy flux Convection from localized sources Convection and rotation Double diffusive convection

115 117 121 123 125 131

5

Instability and breaking of internal waves in mid-water

144

5.1 5.2 5.3 5.4 5.5 5.6 5.7 5.8 5.9 5.10

Introduction Static instability or convective overturn Self-induced shear The superposition of waves: caustics and standing waves Resonant interactions and parametric instability Breaking of internal waves in shear flows Breaking and double diffusive convection Breaking of wave groups or wave packets Three-dimensional breaking Discussion: mixing processes

144 145 146 150 153 157 165 165 168 169

6

The measurement of turbulence and mixing

172

6.1 6.2 6.3 6.4 6.5 6.6 6.7 6.8 6.9 6.10

Introduction Instrument platforms and measurement systems Estimation of ε Estimation of χ T and χ S Estimation of Kν Estimation of KT or Kρ Estimates of Isotropy Intermittency and patchiness Acoustic detection of turbulence

172 173 174 179 180 180 182 183 185 188

7

Fine-structure, transient-structures, and turbulence in the pycnocline

190

Introduction Causes of fine-structure Shear-driven turbulence in stratified regions Tracer dispersion experiments in the pycnocline Diapycnal diffusion in the abyssal ocean Turbulence from shear and double diffusion Two-dimensional turbulence

190 192 197 206 208 209 211

7.1 7.2 7.3 7.4 7.5 7.6 7.7

Contents

ix

8

The benthic boundary layer

213

8.1 8.2 8.3 8.4

Introduction The structure of the benthic boundary layer Observations in the deep ocean Nepheloid layers

213 215 220 225

9

The upper ocean boundary layer

228

9.1 9.2 9.3 9.4 9.5 9.6 9.7 9.8

Introduction Breaking surface waves Mixed layer turbulence below the wave-breaking layer Langmuir circulation Temperature ramps Horizontal dispersion in the mixed layer The base of the mixed layer and mixed layer deepening Turbulence and marine organisms

228 231 246 251 257 260 264 267

10

Shallow seas

269

10.1 10.2 10.3 10.4 10.5 10.6

Introduction Near-bed turbulence Mobile sediments and biological effects Mixing by tidal processes Dispersion in steady and periodic flows Regions of horizontal variation in temperature and salinity

269 270 278 282 287 290

11

Boundary layers on beaches and submarine slopes

291

11.1 11.2 11.3 11.4 11.5 11.6 11.7

Introduction The near-shore zone Shoaling internal waves in the thermocline Internal waves in quasi-uniform stratification Along- and upslope circulation Exchanges between the boundary layer and the interior ocean Winter cascading and turbidity currents

291 292 299 303 316 317 318

12

Topographically related turbulence

321

12.1 12.2 12.3 12.4 12.5 12.6

Introduction Headlands, promontories and curved coastlines Canyons Isolated topography Complex rough topography Straits and channels

321 321 322 323 325 326

Contents

x

12.7 12.8

Fjords Lakes

332 338

13

Large-scale waves, eddies and dispersion

340

13.1 13.2 13.3 13.4 13.5 13.6

Introduction Eddy kinetic energy Coherent structures Dispersion Rossby waves Long-term variations

340 343 348 357 365 367

14

Epilogue

368

14.1 14.2 14.3

The nature and effects of ocean turbulence Prediction and unknown energetics Conclusion

368 370 371

Appendices

373

Parameters/symbols, typical values (where these can be given) and section of first introduction or definition Units and symbols Approximate values of commonly used measures Values of typical energy levels and fluxes Acronyms used in text

373 375 376 376 378

References

380

Index of laboratory experiments

424

Subject index

426

1 2 3 4 5

Preface

The idea of writing this book developed from the need for a text suitable in teaching a course on ocean physics to undergraduates and masters course students with some knowledge of physical processes, but not necessarily in fluids. The course demanded a text describing basic processes in fluids (many of which are beautifully illustrated in Van Dyke’s (1982) book, a valuable introduction for students unfamiliar with fluid motion), and how these processes are manifest and why they are important in the ocean. In choosing as a title, The Turbulent Ocean, rather than Ocean Turbulence, I wish to convey an intention not to focus heavily on the complex and involved study of turbulence, but rather on the processes in the ocean leading to turbulence, its extraordinary and sometimes unexpected structure, and its multifaceted and profound effects in ocean dynamics. This is therefore not intended to provide the sort of basic introduction to turbulence given, for example, in the excellent book by Tenneckes and Lumley (1972). Little is said of the general theory of turbulent motion or of the problems involved in its statistical sampling and in the treatment of data, particularly the problems encountered in producing and interpreting energy spectra even though these must be faced by those measuring turbulence in the ocean. Rather I describe and focus attention on the physical processes that lead to turbulence, and on its effects in the ocean. I was encouraged by a reviewer of an early description of the book to choose a title including the word ‘mixing’, but this is only one aspect of the effect of turbulence and although arguably the most important, there are many others. There was also an expectation that ‘modelling’ would be a dominant subject in a text dealing with turbulence in the ocean. But to do justice to that subject would demand a further (and, for me and perhaps for some students too, a duller) book. The reader will find references to modelling, but not details.

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Preface

I have deliberately included much that will be beyond the scope of many undergraduate and postgraduate courses, so that the text provides both an introduction and a review of present knowledge that may be of value to those beginning or already engaged in research. My aim is to provide a text that is broad but not comprehensive, a text of which parts can be selected and used according to the requirements of the reader, whether teacher, student or researcher. I do not present an account ready-made as a teaching course – the best courses reflect the interests, strengths and expertise of the teacher. Rather the text is intended to provide material from which many different courses might be taught by the selection of appropriate sections. I have tried to assemble some illustrations of processes that may be compared, critically, with one another. Readers wishing to discover more are directed in the footnotes and references to more extensive and, sometimes, more rigorous accounts of the subject matter. Unlike a conventional textbook that generally sticks strictly to proven (or currently accepted) facts, the text includes conjectures and speculations as well as established knowledge. This is inevitable because of the state the subject is in – knowledge is incomplete with even gross energy balances yet uncertain, if not unknown and unclosed, with a frequent lack of suitable oceanographic data against which to test hypotheses. In learning about a subject, it is moreover of value to discover both what is known and what is surmised. Whilst the former is often exciting, the latter (offering the opportunity to devise a better explanation) can be stimulating. I have tried to distinguish clearly between the facts and the speculation, but have included the latter ‘in the hope of stimulating others to remedy the defect’ (Phillips, 1966). Because the subject is of great importance, of current interest and considerable activity, it is inevitable that more will soon be discovered. Whilst some of the information presented here has already stood the test of time, much is yet uncertain and hopefully, before long, will be revised and improved by those unwilling to accept the unsatisfactory status quo. I have drawn extensively in the early chapters on the findings and images of laboratory experiments to illustrate and quantify processes that occur in the ocean. These are examples of some of the rich and mutually beneficial interrelations between fluid mechanics and the study of the physics of the oceans (and lakes – for limnology too provides and gleans information from oceanography). Both fields – fluid dynamics and physical oceanography – have gained from the other, either through the stimulation to investigate new and far-reaching problems to which attention has been drawn through observations or by the explanation of previously unexplained processes active in the often very high Reynolds number flows of the natural environment, Reynolds numbers unattainable in the laboratory. It is common to find papers in physical oceanography supported by information coming from theoretical, laboratory or lake studies in fluid dynamics, and many papers in the fluid dynamics literature address problems first recognized in the context of oceanography. Addressing generic problems at the heart of fluid processes, these often go well beyond what is sometimes referred to as ‘geophysical fluid dynamics’. The control of fluid motion achievable in the laboratory experiments described in this book, from Reynolds’ and Joule’s onwards, is never obtainable in the ocean. In

Preface

xiii

consequence observations made at sea are often affected by the presence of unforeseen or apparently irrelevant variations that prevent a clear definition or identification of processes. This, and the commendable desire to obtain an ‘overall’ description, for example of energetics, have sometimes led to a predominantly numerical approach to measurement and analysis that disregards the underlying patterns of motion that provide vital clues to causality. Sometimes the opportunity to describe mechanisms and ‘structure’ has been lost by the adoption of spectral approaches to data collection and analysis. ‘Coherent structure’ is a thread leading through the text. The patterns seen in turbulent flows (of which some are illustrated in the figures), sometimes detected by conditional sampling or pattern recognition through a veil of overlying variability, are fascinating and sometimes remarkably beautiful. How they come about is still much of a mystery, but they carry information about the causes and nature of turbulence that may be of value when representing turbulence by ‘parametrization’ in predictive models. Discovering more about these features and causative processes remains a major challenge in the science of oceanography. Finally I must remark that those wishing to obtain an impression of the dynamics of large bodies of fluid such as the ocean, and who do not have the means to measure for themselves, can learn a great deal by observation or photography of the sea surface, carefully watching the movement of waves and foam, and by following the motions traced in the turbulent atmosphere by clouds or plumes of smoke. They should challenge conventional ideas, and seek quantification and explanation of even the smallest details, structures and patterns for which none is obvious, available or satisfactory. There is still much about ocean turbulence that is not understood, as the content of the book makes clear. S. A. Thorpe ‘Bodfryn’ Llangoed Anglesey LL58 8PH

Structure and résumé

I have tried to provide the reader with some information about the motivation for studying the ocean and of the excitement of being involved in discovery and innovative research. Where it seems appropriate, some account is given of the relevant theoretical studies and of related laboratory experiments. Observations made at sea are introduced at an early stage where this is possible without its being too contrived. Reference is made (sometimes in footnotes) to the way in which the subject has evolved and to those scientists who have been most prominent in its development. Although there are dangers in being carried far into the seductive and alluring subject of historical research (a pursuit that can capture and absorb the time and energy of the researcher – and too much knowledge of what is already known may deter interest and suppress novel ideas, for which there is always scope), I hope the details I have provided may tempt the interested readers into discovering more of the background to oceanographic research and how it is done. What is achieved or achievable sometimes depends more on the attitudes and social habits of individual scientists than on the facilities available to them. Chapter 1 is intended to provide an introduction to the subject, and contains some basic information that is used in later chapters. Some of the factors that are germane to ocean turbulence, heat, temperature, density and energy, oscillations and flow stability, fluxes and stress, are discussed. Scales and parameters are defined and dispersion of particles and solutes is briefly addressed. The chapter concludes with an account of the evolution of knowledge in the still unfinished study of mixing in the deep ocean and its energetics, a discussion that sets the scene for some of the later chapters, and to which further reference is made in the concluding chapter. Chapter 2 is about internal waves. This is by no means comprehensive, nor is it intended to be. But internal waves are very important in producing turbulence in

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the ocean – their breaking is described in Chapter 5 – and its purpose is to provide a description of the properties and nature of internal waves in the ocean, sufficient for the reader to appreciate their dynamics and, in later chapters, to understand their very important part in mixing the ocean. No comparable chapter is devoted to surface waves simply because whilst there are many ways in which internal waves can promote mixing, those associated with surface waves are more limited (see Chapter 9). Chapters 3 and 4 describe two ways in which turbulence is commonly generated, through shear and convection, respectively. The account is based largely on laboratory experiments, and deals with ‘transitional processes’, those that lead from a relatively quiescent flow to a turbulent one, and which need to be explained before the processes by which internal waves are known (or may be assumed) to break can be addressed in Chapter 5. Convection is most frequently connected to boundary processes, and the law of the wall is introduced in Chapter 4 in connection with Monin–Obukov scaling. Double diffusive convection is also introduced here. Although many of the concepts and scales relating to ocean turbulence have already been discussed in Chapter 1, it is in Chapter 6 that the means available for its measurement are described. One of the most significant characteristics of the density-stratified ocean is its fine-structure. The causes of fine-structure and their relation to turbulence in mid-water, the region beyond the direct influence of the solid boundary of the seabed or the air–sea interface, are discussed in Chapter 7, together with the effects of turbulence on diffusion and dispersion in this region. The following five chapters are linked in one way or another to the effects of the oceanic boundaries. The nature of turbulent flows in the benthic boundary layer of the deep ocean, beyond the influence of motions induced by surface waves, is explored in Chapter 8. Turbulence driven by waves at the upper boundary of the ocean, the air– sea interface, and through shear and convection in the underlying mixed layer where coherent structures are sometimes evident (if not properly understood), is reviewed in Chapter 9. Turbulence in the shallow seas, strongly affected by the presence of the seabed, by surface wave and tidal motions, and sometimes by biological organisms, the regions most sensitive to and degraded by anthropogenic effects, is the subject of Chapter 10. The nature of turbulence near sloping boundaries on beaches at the edge of the sea and, in deep water, over continental and seamount slopes, is described in Chapter 11, and that near more irregular topography, including headlands and canyons, and in straits, fjords and lakes, is discussed in Chapter 12. Much of the earlier account of turbulent motion is directed to the processes, generally those at small scales, which lead directly to turbulent dissipation of kinetic energy. Chapter 13 describes larger-scale motions that bear many of the characteristics of turbulence, contain sometimes-persistent coherent structures and, in particular, lead to dispersion at scales affected by the Earth’s rotation. The final chapter, Chapter 14, reviews some of the important aspects of ocean turbulence that are yet to be fully investigated or quantified, one of which is the still poorly quantified sources of energy required to support ocean mixing.

xvi


The Appendices list parameters, values of some units and measures, particularly of the (sometimes presently uncertain) levels of energy and energy flux, and acronyms. The subject spans a broad field in which there are so many references that a book could be filled by the reference list alone if all were included. Sometimes only the most recent significant papers are listed, those that refer to earlier papers, so that interested readers can trace the development of ideas to their originators. In so doing it is appreciated that injustice has been done to those who have made substantial advances, and the author can only apologise, pleading economy of space. He also apologises for the number of references made to his own publications, the inadequate excuse being that these are often, to him, the most familiar, or those from which it has been possible to reproduce figures with least difficulty.

Acknowledgements

Many have helped in the preparation of this book. I am particularly grateful to Professor Mike Gregg for reading an early draft, for kindly providing many penetrating and helpful comments, and for several suggestions that have led to the correction of mistakes or misunderstandings. Professor Larry Armi also generously read a draft and tried out parts of it on a class of students. I am most grateful for his very kind and pertinent comments that have led to several improvements in presentation. I have been fortunate to have the advice of these two experts. Their contributions to the understanding of ocean turbulence are implicitly acknowledged by my use of their figures and in my references to their work. Several other colleagues and friends provided information or gave advice. These include Professor Harry Bryden, Professor Walter Graf, Dr Peter Taylor and Dr S. A. Josey. I am indebted to them for their help. I am also grateful to others, too many to be listed, who (sometimes unknowingly) have provided information, stimulation or sympathy that has led to the development of the ideas described in the text. No blame can be ascribed for any remaining errors or inaccuracies to any who have advised or commented on the text. The fault is entirely mine and I welcome notification from readers of any errors that they may find. I thank all who gave permission to reproduce figures and particularly to those who provided copies of figures, some that required time-consuming searches through past data sets and files. These include Y. C. Agrawal, M. Alford, L. Armi, P. Atsavapranee, P. Baines, M. G. Briscoe, O. Brown, B. Brügge, J. Bryan, H. L. Bryden, M. Curé, A. G. Davies, R. E. Davis, M. A. Donelan, C. C. Eriksen, D. M. Farmer, I. Fer, A. M. Fincham, P. Flament, D. C. Fritts, A. E. Gargett, C. J. R. Garrett, M. C. Gregg, R. W. Griffiths, L. R. Haury, P. Hazel, A. D. Heathershaw, K. R. Helfrich, T. Hibiya, J. Holt, J. C. R. Hunt, H. Huppert, M. Inall, G. N. Ivey, A. T. Jessup, J. Jiménez, E. Kunze,

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Acknowledgements

J. N. Lazier, J. R. Ledwell, J. M. Lilly, M. S. Longuet-Higgins, J. E. Lupton, I. N. McCave, T. McClimans, L. Magaard, J. Malarkey, G. O. Marmorino, J. Marshall, N. Matsunaga, T. Maxworthy, W. K. Melville, J. N. Moum, K.Nadaoka, D. Nicolaou, ¨ W. A. M. Nimmo Smith, M. H. Orr, G. Ostlund, H. T. Ozkan-Haller, G. Pawlak, K. L. Polzin, B. S. H. Rarity, R. D. Ray, P. B. Rhines, P. L. Richardson, A. Roshko, T. B. Sanford, R. W. Schmitt, J. Sharples, J. H. Simpson, D. N. Slinn, J. Small, J. A. Smith, W. D. Smyth, C. Staquet, R. T. Tokmakian, J. S. Turner, F. Veron, S. I. Voropayev, J. D. Woods, and V. Zhurbas, Figures 1.8, 9.5, 9.6, 9.10, 10.10 and 13.7 are reproduced with kind permission of Nature, copyright Macmillian Publishers Ltd. The following also kindly gave permission for the reproduction of figures: the Editor of the Journal of Marine Research for Fig. 1.14, The Royal Society for Figs. 9.1, 12.6 and 13.5, The American Geophysical Union for Fig. 4.7, The American Meteorological Society for Figs. 2.4, 2.11, 3.6, 4.8, 4.9, 4.14, 5.1, 6.6, 7.7, 7.8, 7.9b, 9.11, and 13.8, Elsevier Ltd. for Figs. 4.13, 7.3, 7.5, 7.11, 8.7, 9.20, 10.6, 12.4, 13.14 and 13.11, The American Institute of Physics for Fig. 10.9, Taylor and Francis Ltd for Fig. 3.12, Blackwell Publishing for Fig. 8.3 and D. Glasscock, Publications Unit, CEFAS, Lowestoft, UK, for Fig. 10.11. To all, I am most grateful. The Librarians of the National Oceanographic Library, Southampton Oceanography Centre, and the School of Ocean Sciences, Bangor, gave valuable assistance in obtaining material, for which I thank them most sincerely. Line illustrations were expertly drawn by Mrs Kate Davis, SOES, Southampton Oceanography Centre, UK. I am deeply indebted to her for her invaluable help and elegant and careful work. Assistance of Anita Malhotra and Gwyn Roberts in the production of figures is also gratefully acknowledged. I am also grateful to the staff of the Cambridge University Press (especially Matt Lloyd, Sally Thomas, Beverley Lawrence and Jo Bottrill) for their kindness and efficiency in the preparation of this book.

Chapter 1 Heat, buoyancy, instability and turbulence

1.1

Introduction

The fluids, air and water, of the atmosphere and ocean1 that cover the solid surface of the Earth, are almost everywhere in a state referred to as ‘turbulent’. At its simplest level, turbulence involves the sort of eddying motions visible in clouds and smoke plumes, and that are felt in the gustiness of the wind or are seen in the movement of patches of foam on the surface of the sea. Some of the manifestations of turbulence in the ocean are illustrated in figures in this and subsequent chapters. Turbulence is very effective in the transfer of momentum and heat in the ocean. It disperses, stresses and strains the particles and living organisms within the ocean, and it stirs, spreads and dilutes the chemicals that are dissolved in the seawater or released into the ocean from natural and anthropogenic sources. Knowledge of ocean turbulence and its effects is crucial in understanding how the ocean works and in the construction of models to predict how the ocean will change or how its interactions with the atmosphere will be altered in the future. Although estimates of the rate of dissipation of the energy of the tides through turbulence in shallow seas were made as early as 1919, direct observations of turbulence in the ocean date back only to the measurements of near-bed turbulent stress made in the 1950s and to studies of the spectra of small-scale motions in the upper ocean in the early 1960s. In spite of ingenious developments in techniques for measuring turbulent motions, the ocean is still grossly under-sampled and, in comparison with the atmosphere, there are

1 By the ‘ocean’ is meant, here and later, the sum of the major oceans and their connected seas, including the continental shelf seas and those seas, such as the Mediterranean, Black Sea and Baltic, connected by straits to the larger ocean basins.

1


2

(a)

(b)

(c) Figure 1.1. Reynolds’ sketches of the appearance of a ribbon of dye in liquid flowing from the left through a narrow tube of circular cross-section. (a) Laminar flow at subcritical Reynolds number. (b) Transition to turbulent flow at super-critical Reynolds number. (c) Transition to turbulent flow seen when the dye is illuminated by a spark. (From Reynolds, 1883.)

few sets of data against which to test models of the ocean that include representation of its turbulent nature. Much is still to be discovered and quantified. This chapter describes some of the ideas that underlie the understanding of the part played by turbulence in the ocean. Much of this background is derived from studies of turbulence and heat transfer in laboratory experiments. The scientific study of turbulence began late in the nineteenth century. In 1883 a paper by Osborne Reynolds was published describing how a smooth flow of water down circular tubes with diameters, d, ranging from about 0.6 to 2.5 cm, breaks down when the speed of the flow, U, becomes sufficiently large. In his laboratory experiment, Reynolds introduced a thin line of dye into the water entering the tube to make the flow visible (Fig. 1.1). He describes his observations as follows. When the velocities were sufficiently low, the streak of colour extended in a beautiful straight line through the tube,

but later he makes the following comments. As the velocity was increased by small stages, at some point in the tube, always at a considerable distance from . . . the intake, the colour band would all at once mix up with the surrounding water, and fill the tube with a mass of coloured water. On viewing the tube by light of an electric spark, the mass of colour resolved itself into a mass of more or less distinct curls, showing eddies.

The experiment shows that the ‘laminar flow’, the smooth flow through the tube at low flow speeds, undergoes a transition to a random eddying ‘turbulent’ motion at

1.2 Heat and temperature

3

higher speeds when a non-dimensional parameter, U d/ν, exceeded a value of about 1.3 × 104 . Here ν is the kinematic viscosity, which for water has a value of about 10−6 m2 s−1 . The parameter is now known as the Reynolds number, usually denoted by Re.2 The eddies illustrated in Fig. 1.1c are of a size comparable to d. In geophysical contexts, the values of the speed, U, and length, d, are usually taken to be those characterizing the flow, for example the changes in mean speed of the local flow and the vertical distance over which such changes occur. The precise value of Re at which turbulence sets in depends on the geometry of the flow and the nature of disturbances to which it is exposed, but flows in which Re exceeds a critical value of about 104 , common in the ocean, are generally turbulent unless constrained by effects not represented in Reynolds’ experiment. No really precise, robust, unambiguous or clear definition of what is meant by turbulence has, however, been devised. Turbulence is generally accepted to be an energetic, rotational and eddying state of motion that transpires to produce dispersion of material and to transfer momentum, heat and solutes at rates far higher than those of molecular processes alone. Perhaps its most important property, and one that is generally used to characterize it, is that by generating relatively large gradients of velocity at small scales, typically 1 mm to 1 cm, turbulence promotes conditions in which viscous dissipation transfers its kinetic energy into heat. Reynolds’ experiment provides a first example of a transition, in this case a very rapid transition, from a steady flow to turbulence. Not all transitions from laminar to turbulent motions are so rapid. Several distinct stages have been identified within transitions found to occur in the ocean that involve buoyancy forces, and in these cases a Reynolds number is not necessarily the principal parameter that determines the onset of turbulence: examples are described in Chapters 3 and 4. 1.2

Heat and temperature

Determination of the processes and rates of transfer of energy between its kinetic and potential forms in the ocean, and the dissipation of energy caused by turbulence and its transfer to heat, are central to the science of oceanography. Heat is a form of energy contained at a molecular level within a body of fluid, and temperature is a measure of heat content. The relation between heat and temperature can be expressed in terms of a change, H , in the heat per unit mass (measured in J kg−1 ) and corresponding change, T , in temperature (measured in degrees kelvin, K, or in ◦ C) by H = ρcp T , where ρ is the fluid density (in kg m−3 ) and cp is the specific heat at constant pressure which, for seawater, has a value now known to be about 3.99 × 103 J kg−1 K−1 .3 The experimental determination of the value of cp 2 Later experiments have shown that the critical Re depends on the level of the background disturbances to the flow, values consequently ranging from about 1 × 103 for relatively substantial disturbances to about 4.5 × 104 in very carefully controlled, low disturbance, pipe flows. 3 The specific heat varies with temperature, salinity and pressure. For more precise values of cp and for values of ρ see Gill (1982, Section A3.4 and Table A3.7).

4


Figure 1.2. A sketch of Joule’s original apparatus for determining the ‘mechanical equivalent of heat’ (or cp ), showing (a) a side view and (b) a plan view of the cylinder. The cylinder, containing a little over 6 kg of water, is 0.2 m in diameter and about the same in height. (An additional experiment was made using mercury in an apparatus of about half this size.) The weights are each about 13.6 kg and descended 1.6 m in about 26 s driving the paddles within the cylinder via pulleys and the winding spindle. Heat transmission from the water up the brass paddle shaft was reduced by inserting a piece of boxwood. Experiments were made in a ‘spacious cellar’ in which the temperature was fairly uniform, and were repeated 23 times to provide reliable mean estimates of cp . (After Joule, 1850.)

provides a nice illustration of the role of fluid motion, in particular of turbulent motion, in the transfer of energy. The unit in which energy is measured is named after J. P. Joule.4 Joule’s most celebrated experiment is that to estimate of cp in the apparatus sketched in Fig. 1.2. In essence, the falling weights drive paddles, which churn water in the cylinder, leading to its heating. In the experiment the weights descended through a distance of 1.6 m at a speed of about 6.1 cm s−1 . They were repeatedly lifted, and over a period of about 35 min in which the weights descended some 20 times, the temperature of the water in the cylinder increased by about 0.31 K. The temperature change was carefully 4 James Prescott Joule (1818–1889) received private lessons in chemistry in his home city of Manchester from John Dalton (1766–1844), by inclination a meteorologist (see, for example, Oliver and Oliver, 2003) and now best known as the discoverer of the law of partial pressures of gases. As a young man, James observed the aurora borealis and sounded the depth of Lake Windermere in northwest England with his elder brother, Benjamin. The Joule family owned and managed a brewery but to what extent James was actively engaged in its running is unclear; Osborne Reynolds (see Section 1.1), a friend and biographer, asserts that James had little to do with the brewery, although he did make experiments within its premises as part of an extensive study of the relationship between different forms of energy. Cardwell’s (1989) biography of J. P. Joule provides informative details of his early years and of his contacts with other scientists of the time, including Michael Faraday and G. G. Stokes.

1.3 Density

5

measured, the accuracy attained being about 3 mK. (Temperature is now routinely measured at sea to an accuracy of 1 mK, and often in particular studies, e.g. in boundary layers where the temperature is relatively uniform, with a resolution of 0.1 mK or better.) Great care was taken to minimize heat loss during the period of the experiment, and a wooden screen was erected to avoid effects of radiant heat from the observer. Joule calculated the potential energy lost by the weights in descending and, by subtracting their kinetic energy at the end of their descent and accounting for a small heat loss from the cylinder during the experiment, was able to relate the energy imparted to the water per unit volume through the paddles, i.e. its change in heat energy, to its rise in temperature. The results were communicated to the Royal Society by no less august a person than Michael Faraday, and published in 1850. This experiment was later refined to obtain greater accuracy. But, as it is, it contains a major subtlety that involves the motion of the water within the cylinder. As Fig. 1.2 shows, there are baffles fixed to the inside of the cylinder. They are important for two reasons. The first is that without them a circulatory flow would be set up which, having kinetic energy, would need to be accounted for in the energy balance. (Alternatively, Joule could have waited until the circulation died out before measuring the temperature, but that would have required a means to ensure there was no substantial residual motion and would have taken time, during which heat would have been lost from the cylinder to the air.) The second reason is perhaps more important. The rotating paddles drive fluid past the stationary fixed baffles and promote small-scale eddying and enhanced shearing motions, characteristic of turbulence. These greatly increase the rate at which molecular viscosity dissipates the kinetic energy imparted to the water, transferring mechanical energy into heat much more rapidly than can a mean circulation gradually spun down through viscous drag at the cylinder walls. It is by increasing the mean square shear that turbulence enhances the effect of viscosity in the transfer from mechanical energy to heat, at a rate explained in Section 1.7.8. In the ocean and in lakes, the heating generated by turbulence is usually miniscule in comparison with other sources of heat5 and has little effect on the dynamics but, as will be shown later, the energy lost by turbulent motions is substantial and provides a measure of ‘mixing’. The transfer of kinetic energy by turbulence into heat in the ocean is a one-way ‘irreversible’ process, meaning that the former state cannot be recovered except by doing work on the fluid system. Turbulent dissipation represents a loss in energy that, to maintain a quasi-steady state in the ocean, must be replenished from some external source such as the atmosphere or through the tide-generating gravitational forces of the Moon and Sun.

1.3

Density

The density of fluids has important effects on their dynamics and, in particular, on the onset and nature of turbulence. Forces are derived from the variations of density in fluids 5 An exception may be in the very high dissipation region of surf zones described in Section 11.2.1.


6

lying within gravitational fields. Differences in weight lead to pressure differences that, if unopposed, drive motion. These forces are termed buoyancy forces and, often derived from temperature or salinity variations or driven by heat fluxes at the boundaries of fluids, are the cause of a wide range of turbulence-related phenomena (Turner, 1973; Simpson, 1997). These include several that involve instability, a topic introduced in the next section and described with examples from the oceans and lakes in later chapters. The majority of liquids expand when heated and contract when cooled. Their density consequently increases as their temperature decreases. Fresh water between its freezing point at 0 ◦ C and a temperature of about 4 ◦ C is a well-known exception, becoming denser as temperature increases. The salts dissolved in seawater and measured as ‘salinity’,6 modify this behaviour. The freezing point of seawater differs from that of fresh water, and decreases as salinity increases. The density of seawater with salinity greater than about 24.7 psu (and most of the ocean has a higher salinity, typically about 35 psu) behaves like that of most liquids, increasing as temperature decreases until freezing occurs (at a temperature of –1.92 ◦ C for a seawater with a salinity of 35 psu). As a consequence, the dynamical properties of the ocean and freshwater lakes may differ at low temperatures. Temperature, salinity and depth (or pressure) are now measured routinely to the full depth of the ocean basins by using lowered Conductivity–Temperature–Depth (CTD) probes (Lawson and Larson, 2001), salinity being derived from the conductivity with a temperature correction. (The mean depth of the oceans is about 3795 m, and the depth of abyssal plains is typically 5000 m.) The density of seawater is determined from the temperature, salinity and pressure by using an expression known as the equation of state. This is illustrated in Fig. 1.3, which shows several of the factors mentioned in the preceding paragraph, the density maximum of fresh water at about 4 ◦ C, the decrease in freezing point as salinity increases, and the monotonic increase in density as temperature decreases when the salinity exceeds 24.7 psu. Increased salinity increases density because of the increase in the mass of salts, and may be a very substantial component in some circumstances. For small variations in temperature, T, and salinity, S, from reference values where the density is ρ 0 , the equation of state for density, ρ, may be approximated by ρ = ρ0 (1 − αT + β S).

(1.1)

The coefficients, α and β, relate to the expansion of seawater and are specified at the reference values of T and S, but depend on depth. Seawater near the sea surface with a salinity of 35 psu has a thermal expansion coefficient, α, the increase in volume per unit volume per kelvin, of 5.26 × 10−5 K−1 at 0 ◦ C, 7.81 × 10−5 K−1 at 2 ◦ C, 1.67 × 10−4 at 10 ◦ C and 2.97 × 10−4 at 25 ◦ C. The value of α increases with depth, at 1000 m being equal to 1.84 × 10−4 K−1 in water at 10 ◦ C and 35 psu. The coefficient for the 6 The component ions contributing to salinity in the ocean are in nearly constant ratio. The main ions (and proportions) are chlorine 55.0%, sodium 30.6%, sulphate 7.7%, magnesium 3.7%, calcium 1.2%, and potassium 1.1%. The unit of salinity is the psu, approximately equal to 1000 times the mass of dissolved salts per unit mass of seawater (Unesco, 1983).

1.3 Density

7

Figure 1.3. The equation of state: the relation between density and temperature for fluids of different salinity. The salinity for each curve is shown, ranging from 0 to 38 psu. The freezing point and temperature of maximum density, both decreasing as salinity increases, are also shown.

contribution of salinity to density, β, is 0.82 psu−1 at 0 ◦ C and about 0.79 psu−1 at 10 ◦ C. An increase in temperature of T = 0.1 K in an upper layer of thickness, h, = 1000 m at a temperature of 10 ◦ C would result in a ‘steric’ rise in sea level of αhT , or about 1.76 × 10−4 × 0.1 × 1000 m = 1.76 cm. Changes in sea level are a very important consequence of climate change.7 The density of seawater is often expressed as σ T (sigma-T), the difference in density from 1 × 103 kg m−3 but with units conventionally omitted (and with qualifications explained by Gill, 1982, Appendix 2). The density of water in the upper 100 m of the sea is typically about 1.028 × 103 kg m−3 or σ T = 28. Surfaces of constant density in the ocean are known as isopycnal surfaces (or just isopycnals). Much of the attention given to ocean mixing at small scales relates to the transfers that occur across such surfaces.

7 Calculation of sea-level change in response to climatic variations is complicated by the contributions of fresh water entering the oceans from the continents as a result, for example, of the melting of glaciers, causing a ‘eustatic’ rise in sea level. Change in the salinity of the sea resulting from the melting of ice must also be taken into account (Munk, 2003).

8


Pressure-induced adiabatic heating (changes in temperature that occurs without any exchange of heat with surroundings) caused by the compression of seawater leads to a gradually increasing temperature with increasing depth in an otherwise uniform layer. The ‘adiabatic lapse rate’, a = gαϑ/cp , where ϑ is the temperature in kelvin, increases with depth and temperature in the surface waters, being about 1.16 × 10−4 K m−1 at the surface in a temperature of 10 ◦ C and about 1.31 × 10−4 K m−1 at 4000 m depth in a temperature of 4 ◦ C, both values being at a salinity of 35 psu.8 Such gradients are sometimes apparent in relatively uniform regions of the ocean, for example in the deep convectively mixed layers described in Section 4.5.2. Care must be taken to account for adiabatic changes of temperature particularly in conditions where water is transported over large vertical distances or in comparing the density of water at widely separated depths, with the use of potential temperatures and densities, where appropriate. (The potential temperature is that which a fluid parcel would have if it were moved adiabatically to a given pressure reference level. Potential density is defined similarly; see section 3.7 in Gill, 1982.) Although compressibility plays an important role in the atmosphere, it is generally small in the ocean. The first floats capable of remaining at one level and of following the slow drift of deep ocean currents were those devised in the 1950s and now called after their inventor, John Swallow (1923–1994). The main body of a Swallow float consists of a sealed aluminium tube. This depends for its stability in the water column on being less compressible than seawater. If displaced downwards from a level at which it has the same density as the sea, it will compress less than the seawater and so be less dense and buoyant, rising back towards its original position. Similarly, if moved upwards, its reduction in volume is less than that of the seawater, and so it is denser, and descends. The float is consequently ‘stable’ (a topic discussed further in the next section), tending to follow a highly damped oscillation about its mean level if perturbed from it. Tracked acoustically from a ship9 , Swallow floats provided the first evidence of the variable nature of deep ocean flows in the early 1960s. With greatly improved means of tracking and design, floats now provide a means to measure mean circulation and dispersion at large scales which is described in Chapter 13.

1.4

Instability and oscillations resulting from buoyancy forces

The density of the ocean generally increases with depth, and this is an important factor in turbulence and mixing. The effect of density is illustrated by the two idealized distributions of density shown in Fig. 1.4. In both, water is supposed incompressible and inviscid. In the first, part (a), two layers each of height h but of different densities 8 Values of α and cp from which other values of a can be derived are listed by Gill (1982, Table A3.1). 9 Acoustic tracking was no simple matter. There was no Global Positioning System (GPS) when the floats were first used, and precise position fixing was made relative to topographic features identified in careful echo-sounding surveys.

1.4 Instability and oscillations

r

9

r

r2

r1

−h −z

r1

r2

−2h r

r

−z

−H

Figure 1.4. Stable (i, on left) and unstable (ii, right) stratification, for (a, top) two layers of different density and (b, bottom) a linear density variation with depth. The horizontal undisturbed interface in (a) and undisturbed isopycnals in (b) are shown as dashed lines. Periodic disturbances, internal waves, are shown in (i) by full lines and dotted lines, whilst growing disturbances in (ii) are illustrated as full lines.

(e.g. different salinities or temperatures) are separated by a thin interface. In (i), to the left, the upper layer is of density ρ 1 and the lower is of greater density, ρ 2 . In (ii), to the right, the two layers have been reversed so that now the water of greater density, ρ 2 , is on top. As anyone who has turned a beaker of water upside down will know, in this situation the upper layer will begin to descend into the lower. But why? Case (ii) has greater potential energy, PE, than (i). The weight of the layer of density ρ 2 is gρ 2 h per unit area and its centre of gravity is a distance h higher in (ii) than in (i), so that its potential energy per unit area is gρ 2 h2 greater. Similarly, the PE of the layer of density ρ 1 is gρ 1 h2 less in (ii) than in (i). By subtraction, the PE per unit area of (ii) is therefore g(ρ 2 − ρ 1 )h2 greater than (i).10 An adjustment that converts the situation illustrated in (ii) to that in (i) releases energy, whereas energy must be provided to convert (i) into (ii). The situation in (ii), with denser water overlying less-dense water, is ‘statically unstable’, a perturbation gaining in energy and growing. In this example, the instability involves convection (as sketched), the release of potential energy through the 10 The acceleration due to gravity appears in product with the density difference. The acceleration due to gravity times the fractional density difference, denoted as g = 2g(ρ2 − ρ1 )/(ρ1 + ρ2 ), appears frequently in connection with the effects of buoyancy and is often referred to as the ‘reduced gravitational acceleration’ or simply as ‘reduced gravity’.

10


descent and growth of one or more plumes of the denser water and ascent of similar plumes of the lighter water, the potential energy of the initial state being converted into kinetic energy, KE, of the developing convective motion. The instability is called ‘Rayleigh–Taylor instability’. Lord Rayleigh (1883) studied the stability of fluids at rest with density increasing upwards, and G. I. Taylor (1950)11 recognized the essential ingredient of the instability, that the acceleration producing instability must be directed from the lighter fluid towards the denser, usefully generalizing Rayleigh’s results to cases in which the vertical acceleration varies, as in wave motions.12 Equally, any change that converts (i) into (ii) requires additional energy to be found, and such changes, without suitable sources of energy, are inhibited. In (i), the water column is ‘statically stable’ or stably stratified in density. Small disturbances, providing a rise in the centre of gravity or modest supply of KE, result in interfacial waves at the boundary between the two layers (as shown in the figure). A large energy input, tending towards the development of state (ii), may however result in instability. An example is provided in Fig. 5.5a. Generally in the ocean the density changes gradually with depth and not discontinuously. Figure 1.4b shows a fluid of height, H, with lines representing isopycnal surfaces and a density that varies linearly in depth. (Here, and later, z is taken as the upward vertical coordinate.)13 In (i), the density, ρ, decreases upwards at a rate, dρ/dz = −qρ0 , where q is a positive constant with dimensions (length)−1 , and in (ii) it increases at the same rate. The PE in (ii) is greater than that in (i) by an amount gqρ0 H 3 /6, and, as before, the illustrated situation in (ii) is unstable to disturbances that convert it into (i). Indeed any mass-conserving adjustment of the density distribution (or ‘profile’) in (ii) will lead to a state of lower PE, so that any disturbance, however small, will lead to a release of PE.14 A condition of ‘static instability’ occurs whenever denser fluid lies above less dense. In this case there is a simple alternative argument that indicates the dynamically unstable nature of (ii) and the stability of (i). Imagine that a small volume, V, of water of density ρ 0 in (i) is moved upwards by a small distance into less-dense water. By Archimedes’ Principle, there is an up-thrust or buoyancy force acting on the displaced volume equal to the weight of water it displaces. Since the displaced water is of lower 11 Sir Geoffrey Ingram Taylor (1886–1975), known to friends and colleagues as ‘GI’, was a brilliant theoretician and experimentalist whose research provided the basis of understanding many aspects of fluid mechanics, particularly turbulent flow. His work is characterized by the requirement to find experimental verification of predictions, and by the elegant simplicity of explanations of phenomena in terms of the essential physical processes – and no more. Batchelor (1996) gives an entertaining and comprehensive account of Taylor’s life and achievements. Turner (1997) provides a shorter personal description of some aspects of Taylor’s research, particularly during his later years. 12 Taylor’s results mean that a downward acceleration exceeding g is necessary at the sea surface before the air–water interface will become unstable as a consequence of the induced acceleration. A similar downward acceleration of a layer of dense fluid overlying a less-dense layer would promote stability. 13 Depth is usually measured from the mean level of the sea surface. In some figures, the vertical axis is given as pressure in pascals (Pa) or decibars (db). In seawater, 1 MPa ≈ 97 m and 1 db ≈ 0.97 m, but the conversion factor depends on the density of seawater. 14 Exchange of any two thin layers of equal thickness in the water column in (ii) will result in the denser being lowered and the lighter raised, a situation which case (a) leads to decreased PE.

1.4 Instability and oscillations

11

density, the buoyancy force is less than the weight of the displaced volume, and it consequently experiences a net downwards force and sinks towards its original position. Arriving there, it overshoots and is displaced downwards into water denser than it is. The small volume consequently experiences an upward buoyancy force that exceeds its weight, so that it decelerates and eventually again rises; any vertical displacement is opposed by a restoring force, a ‘stable’ condition which leads to oscillations. Equally in (ii), buoyancy forces tend to amplify any displacement, whether upwards or downwards. Disturbances grow and the situation is unstable. This example illustrates that, although buoyancy is usually thought of as lightness or having a tendency to rise, buoyancy forces can act upwards or downwards. The latter are often described as forces of negative buoyancy. If independent of the volume of the displaced water and of its distance from its original position, the frequency of oscillations, σ , in (i) can only depend on the acceleration due to gravity and the density gradient, represented by q. It follows, by dimensional reasoning, that σ ∝ (gq)1/2 . The constant of proportionality is not determined by the dimensional argument. It can, however, be found as follows: suppose the volume is displaced upwards by a distance η. Then, because it has passed upwards through the density gradient, dρ/dz, the density of the surrounding fluid is ρ 0 + ηdρ/dz, and consequently, by Archimedes’ Principle, the upthrust (the weight of fluid displaced) is gV (ρ 0 + ηdρ/dz), and the net downward force acting on the displaced volume, its weight minus the upthrust, is gV (ρ0 + dρ/dz) − gVρ0 = gV ηdρ/dz. But by Newton’s second law, the force on the small volume is equal to its mass, Vρ 0 , times its acceleration, d2 η/dt 2 , and so Vρ 0 d2 η/dt 2 = gV ηdρ/dz, or d2 η/dt 2 = −gqη (since dρ/dz = −qρ 0 , with q > 0). This is the equation of simple harmonic motion and so the small volume will oscillate about its original position with frequency σ = (gq)1/2 . The constant of proportionality is therefore unity. In the situation (ii) on the right, however, the density increases upwards and dρ/dz = qρ0 with q > 0. The governing equation becomes d2 η/dt 2 = gqη, which has solutions growing exponentially in time at a rate (gq)1/2 . This case is statically unstable with growing convective motions as illustrated in Fig. 1.4b(ii). (In practice, because of viscous forces, motion or ‘dynamical instability’ will not always result when the water is statically unstable, as explained in Chapter 4.) The way in which buoyancy inhibits or leads to instability and turbulence is a major topic of later chapters. The frequency, N, given by N 2 = −g(dρ/dz)/ρ 0 , is known as the buoyancy frequency15 and characterizes the oscillations or internal waves (described in Chapter 2) which the stable stratification supports. The buoyancy frequency is expressed as an angular frequency in radians per second (although usually written simply as s−1 ) and 15 Strictly, the buoyancy frequency, N, is defined by N 2 = −g(dρ/dz)/ρ, but since the density of seawater generally varies by less than about 2% from a constant value of ρ 0 of about 1.035 × 103 kg m−3 over its depth, to a good approximation the value of density, ρ, appearing in the denominator of the expression for N2 can be equated to a constant reference or mean value, ρ 0 . This approximation is consistent with the Boussinesq approximation made in the theoretical description of internal waves (Section 2.3.1) and of stratified shear flows (Section 3.2.3), as well as with the assumption of uniform density made in determining the Reynolds stress (Section 1.7.3).


12

2/N is the ‘buoyancy period’ of internal waves having the buoyancy frequency. ‘Uniformly stratified regions’ are those in which N is constant. N has real values only in stably stratified conditions.

1.5

Transfer of heat

The ocean gains and loses heat mainly through its boundaries, the air–sea interface and the seabed. Some heat (as well as dissolved chemicals) is carried into the ocean by rivers and in seepage flow through the seabed in the shelf seas and continental slopes. Several distinct processes contribute to the transport and transfer of heat both at the boundaries and within the body of the ocean, and some of these are described below. It is customary in physics texts describing the transfer of heat to include convection as one of the processes of heat transfer. Already introduced in Section 1.4, convection is a major process leading to turbulence in the ocean and is addressed in Chapter 4.

1.5.1

Eulerian and Lagrangian flow descriptions: advection and Stokes drift

It is common in the oceans to measure fluid velocity and other properties of the water at fixed locations, for example by a current meter on a mooring. The equations of motion are also usually formulated in reference to the time (t) variation of quantities at specified (x, y, z) locations. This Eulerian description of a flow contrasts with a Lagrangian description in which quantities (velocity, acceleration, temperature, etc.) are specified at virtual particles or water parcels that follow the fluid motion. Heat can be transported from one place to another by fluid motion or ‘advection’. In the absence of other heat transfer processes, the temperature at a fixed (Eulerian) point will change when a temperature gradient in the fluid is transported past it by the fluid’s motion. Figure 1.5 illustrates the transport of water at a speed, u, in the positive x-direction (to the right) past a point, A. There is a temperature gradient (also in the x-direction), within the flowing water. The temperature at A at time t = 0 is T0 . After a small time t, the water initially at position, B, at a distance, ut, to the left of A will arrive. Its temperature is equal to T0 minus the distance times the temperature gradient. If the gradient is positive, as illustrated in Fig. 1.5, the temperature at A decreases as time increases. The temperature change, T, in time, t, is therefore equal to −ut∂T/∂x. In the limit as t tends to zero, the rate of change of temperature at a point is therefore T/t = ∂T/∂t and ∂T /∂t = −u∂T /∂x.

(1.2)

Equivalently, the heat per unit volume at A will increase at a rate ∂(ρ0 cp T )/∂T = −ρ0 cp u∂T /∂x,

(1.3)

1.5 Transfer of heat

13

Figure 1.5. Temperature change at a point A caused by horizontal advection at speed, u, of fluid having a horizontal variation in temperature, T(x, t). The initial variation of temperature is shown by the curve t = 0 when the temperature at point A is T0 . After a time t, the variation of temperature with x is shown by the curve marked t = t, and fluid originally a B, a distance ut upstream of A, reaches A. Its temperature is T0 − utdT /dx.

where the reference density, ρ 0 , and cp are assumed to be almost constant, changing relatively far less than the temperature, T (see footnote 15). The difference between Eulerian and Lagrangian descriptions is vividly apparent in wave-induced motions. The Stokes drift is the mean horizontal Lagrangian translation of water, or of a neutrally buoyant marker such as dye, by wave motion relative to a mean Eulerian flow. Such a drift can exist even when the fluid motion measured at any fixed point is sinusoidal with zero mean (i.e. when the Eulerian mean flow is zero), provided that the amplitude of the oscillatory flow varies in space. The source of the drift can be illustrated in small gravity waves on the sea surface when there is no mean Eulerian flow. In this case the amplitude of the velocity fluctuations are sinusoidal in time and decrease exponentially with distance below the mean surface level. Relative to the direction of wave propagation, the motions are forward at and directly beneath the wave crests, and backward at the troughs. A particle in the wave field carried with the fluid moves forward below wave crests at a higher level than it moves backwards beneath the troughs. The forward motion, being at a smaller distance below the mean surface level, is therefore faster than the backward motion, and the particle is therefore moved forward a small distance in the wave propagation direction in every wave period. There is a consequent Lagrangian drift of fluid particles in the direction of the wave propagation, and this is found to decrease exponentially with depth with an e-folding length scale half that of the oscillating velocity field. Internal waves and tides also have an associated Stokes drift. (This is sometimes referred to as a ‘residual drift’.) Such drifts will lead to the horizontal transfer of water, and its properties, for example heat and suspended sediment. 1.5.2

Molecular conduction

Heat is conducted through the water surface into the ocean by molecular processes when warm air overlies colder water. Generally such ‘sensible heat transfer’ is


14

r

Figure 1.6. Sketch showing the conduction of heat through a stationary fluid of thickness, d. The temperatures of the upper and lower surfaces are T1 and T2 , respectively, with T2 < T1 . Stratification is statically stable with density decreasing with z.

determined not just by air–water temperature difference, but by wind speed and by the state of the sea surface, and is usually exceeded by other means of heat transfer. Heat is transferred by conduction whenever there is a gradient of temperature within the water. In the situation illustrated in Fig. 1.6, a surface, maintained at temperature, T1 , overlies a thin horizontal layer of water of thickness d, that is supported beneath by a surface or plate kept at a temperature, T2 < T1 . (T2 is supposed to be greater than that of the maximum density of the water. The fact that the bounding surfaces in this illustration are horizontal is important. Buoyancy forces could drive motion if they were not.) The water conducts heat by molecular processes acting within the water with molecular thermal diffusivity coefficient, κT , about 1.4 × 10−7 m2 s−1 , and a steady state will be reached in which the temperature of the water decreases uniformly from T1 at the upper surface to T2 at the lower surface. The density of the water in this state increases downwards and the water is statically stable. (It would not be so if T2 were greater than T1 : see Fig. 4.2.) It will remain motionless unless forces are applied. The downward flux of heat per unit area is proportional to the temperature gradient, ρcp κ T (T 1 − T 2 )/d. The flux measured in the (upward) direction, z, in which temperature is increasing, is F = −ρcp κ T (T1 − T2 )/d, so that (in the limit as d tends to zero) F = −ρcp κT dT /dz,

(1.4)

where dT/dz is the rate of change of temperature with z. In this case dT/dz is positive, implying that F is negative. The flux of heat transferred through the conducting surfaces is –ρcp κ T times the gradient of temperature normal to the surfaces. If the surfaces are non-conducting, there is no heat flux and dT /dz must be zero. The conductive heat flux depends on temperature gradient: heat transfer is enhanced when temperature gradients are intensified by dynamical processes, such as those that occur in turbulence (Section 1.7). Salinity is also transferred at a molecular scale, but with a molecular transfer coefficient, κS , about 1.5 × 10−9 m2 s−1 , much smaller than that of heat, a property that has surprising consequences in the ocean, some of them described in Section 4.6.

1.5 Transfer of heat

15

Figure 1.7. Sketch showing the vertical flux of heat, here radiation, through an ocean layer of thickness, d. If the heat flux per unit surface area, F1 , entering at the z = d, differs from that leaving, F2 , at z = 0 the temperature of the layer will change with time.

1.5.3

Radiation

The solar radiation entering through the sea surface is typically about 200 W m−2 , but considerable variations occur diurnally, and with cloud cover, latitude and season. Some 45% of the solar radiation incident on the sea surface is in the long-wavelength infrared (IR; wavelengths >700 nm). About 53% is in the visible range between 350 nm and 700 nm, and the remaining 2% is ultraviolet (UV) with wavelength 1, the waves are like surface waves, with relatively narrow crests. If, however, h1 /h2 < 1, the troughs are narrower than the crests, and solitons appear as a series of troughs as shown in Fig. 2.4 (Plate 3).6 In the case h1 = h2 waves are symmetrical, with flattened crests and troughs compared to a sinusoid. Internal hydraulic jumps and bores have properties similar to those of their counterparts on the water surface when abrupt changes in level occur in flow over confined geometry such as weirs, or in tidal bores, with characteristic properties that depend on a Froude number. The number of degrees of freedom is, however, increased in fluids with two (or more) layers and the problem of hydraulic control is more complex (see Section 12.6.1). The layers can move at different mean speeds, U1 and U2 , and the nature of the transition depends on the depth ratio, h1 /h2 . The properties of wave elevation or depression described above for solitary waves, however, generally translate into those of hydraulic jumps. The mean water surface is raised by a surface hydraulic jump or as surface bore passes and, provided U1 and U2 are almost equal, so too is the interface in a two-layer fluid when h1 /h2 > 1. The mean interface is lowered by an internal bore if h1 /h2 < 1 as illustrated in Fig. 2.5. In Loch Ness, for example, the upper mixed layer is typically 10–50 m thick, whereas the lower layer is over 100 m, and the interface is consequently lowered as a solitary wave or jump passes, 6 The waves can sometimes be very large. Stanton and Ostrovsky (1998) give an example of an internal soliton packet in which the depth of the wave troughs, the distance of the wave troughs below the pycnocline level ahead of the packet, is about 30 m, locally increasing the thickness of the mixed layer by a factor of about four.


52

Figure 2.5. The form of standing waves on an interface between pink paraffin (US kerosene), of density 780 kg m−3 , and water in a long tube rocked at frequency close to that of waves with wavelength twice the tube length. Only part of the 0.1 m deep, 4.87 m long, tube is shown. (The vertical bars at the ends of the rule are 0.45 m apart.) The hydraulic jumps and following waves travel to the right. The ratios of layer depths, h 2 / h 1 , are (a) 0.25, (b) 4.5, (c) 0.42, (d) 2.6, (e) 0.66, (f) 1.6. The interface level is raised by the advancing internal bore when h2 < h1 , decreased when h2 > h1 . The waves in Loch Ness (Fig. 2.2) correspond roughly to those with the depth ratio shown in (b). (From Thorpe, 1971a.)

leading to the abrupt temperature rises detected by sensors at fixed depths described in Section 2.1 and shown in Figs. 2.1 and 2.2. An associated phenomenon is that of gravity (or density) currents formed as a layer of greater (or less) density spreads below (or above) a existing static layer. These currents appear in a wide set of circumstances (Simpson, 1997), often with turbulence being generated by shear in the flow over the leading ‘nose’ of the intruding flow and with interfacial waves on the following interface. These and other effects of turbulence associated with internal hydraulic phenomena are discussed later (see, for example, intrusions in Section 7.2 and jumps in Section 12.5). In the absence of viscosity, interfacial waves generate a temporally varying discontinuity in velocity, u, at the interface that may result in unstability (see Section 3.2.2). Although density changes abruptly at the sea surface from that of air to that of seawater, the density of seawater is never discontinuous within the ocean because of diffusion, and nor is the velocity; interfaces are never infinitesimally thin and the model of interfacial waves is consequently of limited application.

2.3

Internal inertial gravity waves in continuous stratification

2.3.1

Wave modes

Internal waves in the ocean are modified by the effect of the Earth’s rotation. The local effect of the rotation with radian frequency, = 7.27 × 10−5 s−1 (2/1 day),

2.3 Internal inertial gravity waves

53

is represented through the Coriolis or inertial frequency, f = 2 sin ϕ, where ϕ is the latitude; f is sometimes called the Coriolis parameter. At 45◦ N, for example, f = 1.03 × 10−4 s−1 . For a general, stably stratified, inviscid fluid with buoyancy frequency, N, with no mean shear and in a frame of reference rotating steadily about the vertical axis with frequency f/2 (as on the Earth), and when the Boussinesq approximation is valid,7 the equation for the stream function, , of small perturbations from a static equilibrium state independent of the horizontal y-direction is ∂2 /∂t 2 (∇ 2 ) + N 2 ∂2 /∂x 2 + f 2 ∂2 /∂z 2 = 0,

(2.2)

where ∇ 2 ≡ (∂2 /∂x2 + ∂2 /∂z2 ). The x (horizontal) and z (upwards vertical) velocity components are u = ∂/∂z, w = −∂/∂x. Solutions for plane inertial internal gravity waves, periodic in time and sinusoidal in the propagation direction, x, can be found in the form = ϕ(z) sin(kx − σ t), with velocity components u = ϕ sin(kx − σ t), v = −( f /σ )ϕ cos(kx − σ t) and w = −kϕ cos(kx − σ t),

(2.3)

with ϕ = dϕ/dz, provided ϕ(z) satisfies (σ 2 − f 2 )d2 ϕ/dz 2 + k 2 (N 2 − σ 2 )ϕ = 0,

(2.4)

and is subject to appropriate conditions on the upper and lower boundaries. The shear, with components, (∂u/∂z, ∂v/∂z) = [ϕ sin(kx − σ t), −(f/σ )ϕ cos(kx − σ t)] at a fixed location, rotates in the horizontal (x, y) plane as the wave propagates past a fixed point. A set of analytical solutions of (2.4) exists when N 2 is constant: ϕ n = ϕ0n sin(mn z), where the coefficients, ϕ0n , are constants. These solutions satisfy (2.4) and boundary conditions w = 0 at z = (−h, 0) appropriate in an ocean of constant depth, h,8 provided the frequency is given by the dispersion relation, 2 k + m 2n , (2.5) σn2 = N 2 k 2 + f 2 m 2n and if mn = n/h, where n is a positive integer. The solutions represent a set of internal inertial gravity wave modes, the corresponding waves propagating in the x-direction with a periodic structure in the vertical. There are levels z = −ph/n, p = 0, 1, 2, . . . , n, where ϕ n = 0, and where therefore the vertical velocity component, w = −∂/∂x = −kϕ 0n sin(n/h)cos(kx − σn t) (and the density perturbations) are also always zero. 7 In the Boussinesq approximation it is supposed that the effect of a variable density in the momentum of fluid particles can be neglected, although its variation in determining gravitational or buoyancy forces is non-negligible and usually essential in determining the physical processes in stratified fluids. The approximation is usually valid because of the small density variations in the ocean. 8 The boundary conditions at the ocean surface should be one of constant pressure together with a kinematic condition expressing the compatibility of the velocity and the surface elevation, η. Generally |η| is very much less than the maximum displacements of isopycnals and can be neglected, and so for most purposes (excepting those when vertical displacements close to the surface are of interest) the surface boundary conditions can then be replaced with w = 0 at z = 0. However, as shown in Fig. 2.7, internal waves do produce a sometimes detectable, though relatively small, surface perturbation that can be used to identify their presence.

54


Figure 2.6. Wave modes. The variation with depth, z, of the horizontal velocity, ϕn = dϕn /dz, of the first four internal wave modes, n = 1−4, of the M2 internal tide normalized to a surface value of unity and calculated for a site in the western North Atlantic where the buoyancy frequency reaches a maximum of about 2.5 × 10−2 s−1 at 650 m, is between 10−2 and 10−3 s−1 from 100 to 1000 m, and f, as is usually the case in the ocean, waves of a given wavenumber, k, travel in the x-direction more slowly for larger mode number, n, and with decreasing and bounded frequencies; f < . . . < σ n < . . . < σ 2 < σ 1 < N. Their group velocity, cgxn = ∂σ n /∂k, is also in the positive x-direction. Analytical solutions of (2.4) are also known in an infinitely deep fluid with buoyancy frequency N0 sech bz. These represent internal waves on an interface of finite width, 2/b, and with density distribution, ρ =ρ0 [1 − tanh(bz)], where N02 (= gb) and ρ 0 are constant. The first mode has eigensolution ϕ = ϕ 0 sechp (bz), where ϕ 0 is constant and p is given by p2 (1 − F2 ) − F2 p − k2 /b2 = 0 with F = f/N0 , and the dispersion relation is σ 2 = N02 p/(1 + p). The second mode eigensolution is ϕ = ϕ 0 tanh(bz) sechp (bz), where ϕ 0 is again a constant, with dispersion relation, σ 2 = N02 p2 /(1 + p)(2 + p). For a general buoyancy frequency, N(z), a set of eigensolutions, ϕ n , and corresponding eigenvalues, σ n , describing waves of different ‘wave modes’ are readily found numerically using a shooting method to solve (2.4) for specified wavenumber, k. The eigensolutions can be ordered as wave modes according to the number of times they pass through zero in the water column. Figure 2.6, for example, shows the first four eigensolutions, n = 1−4, for the horizontal velocity perturbation, proportional to ϕn , for internal waves of M2 tidal frequency in the stratification at a site in the western 9 The dye bands in Fig. 5.9a shows isopycnal surfaces of the mode 1 wave.


55

Figure 2.7. Measurements of the surface elevation produced by internal tidal waves propagating away from the Hawaiian Ridge made by the TOPEX-Poseiden satellite altimeter. Filtered M2 amplitudes of order O(5 cm) are plotted along ten satellite tracks, positive surface elevations being towards the top left, where the amplitude scale is shown. The shaded area marks the shallower topography and the main axis of the Hawaiian Ridge. Only internal tides that are coherent with the barotropic M2 tides over the measurement period of 3.5 years are detected. These internal tidal waves are apparent to ranges of at least 1000 km from their source at the Ridge. (From Ray and Mitchem, 1997.)

North Atlantic where the depth is 2700 m. The shear generated by the waves is largely confined to the upper 1000 m and, as is commonly the case for internal waves, the largest horizontal currents are at the surface even though the vertical displacements there are relatively small. Most of the energy of tidal internal waves appears to reside in the lower, first to third, modes. Some general and analytical solutions can also be found for both progressive and standing internal wave modes of finite amplitude (see Thorpe, 2003). 2.3.2

Wave rays

There is growing evidence, illustrated by Fig. 2.7, of internal waves in the ocean with tidal period (‘internal tides’) and of modes 1−3, at distances between several tens of kilometres and some 1000 km, or more, from their source. Another form of solution of (2.2) proves very useful in describing internal waves in mid-water and close to the


56

sites of wave generation when N is constant or only varies in depth over distances large compared to the vertical scale of the wave: = 0 sin(kx + mz − σ t),

(2.7)

where 0 is constant and m is the vertical wavenumber, together with the dispersion relation, σ 2 = (N 2 k 2 + f 2 m 2 )/(k 2 + m 2 ).

(2.8)

This represents a wave with lines of constant phase (e.g. those joining locations of maximum isopycnal displacements) inclined at an angle tan−1 (−k/m) to the x-axis or, using the dispersion relation, at an angle β = sin−1 {[(σ 2 − f 2 )/(N 2 − f 2 )]1/2 }

(2.9)

(or β = sin−1 (σ /N), if f = 0), which depends on the wave frequency but not on the wavenumber or length of the wave. The dispersion relation can also be written as σ 2 = (N 2 sin2 β + f 2 cos2 β).

(2.10)

If, as usual in the ocean, f < N, then σ has its largest value, σ = N, when β = /2 and its smallest value, σ = f, when β = 0. Internal waves are limited in frequency to the range f ≤ σ ≤ N. The lines of constant phase move along the x- and z-axes at phase speeds, cx and cz , equal to σ /k and σ /m, respectively. The phase speed, the speed of advance of lines of constant phase in direction of the wavenumber (k, m), is c = σ /K, where K = (k2 + m2 )1/2 is the wavenumber.10 The group velocity, cg = (∂σ /∂k, ∂σ /∂m) or (cgx , cgz ) is equal to [(N2 − f 2 )mk/σ K4 ](m, −k), in a vector direction (m, −k) inclined at angle β = tan−1 (−k/m) to the x-axis and therefore not in the direction of advancing phase, but at right angles, parallel to the lines of constant phase (see Fig. 2.8b). The presence of a non-zero vertical component of group velocity implies that these waves (unlike the interfacial waves or wave modes) transport wave energy upwards or downwards through the water column. As noted above, β, which gives the direction of wave energy propagation, depends on the frequency, σ , but not on the wavenumber, K, of the waves. The magnitude of cg , can be written |cg | = sin β cos β(N 2 − f 2 )/K σ,

(2.11)

and this depends on both σ (directly, and through β) and K. The ray-like nature of the waves corresponding to (2.7) is shown in Fig. 2.8a, shadowgraph images from Mowbray and Rarity’s (1967) laboratory experiment in which a horizontal cylinder is oscillated in a non-rotating (f = 0), uniformly stratified (constant N) salt solution. The figure shows black and white bands of constant phase extending as ‘rays’ further and further away from the cylinder (here seen ‘end-on’) as time passes, each inclined at the same angle to the horizontal, sin−1 (σ /N) and moving (in a direction normal to the bands) towards the horizontal. The extension of 10 It should be noticed that c is not equal to (c2x + cz2 )1/2 .


57

(a)

(b) C

C

C

C

Figure 2.8. Ray propagation of internal waves. (a) Schlieren images showing internal wave rays generated by and radiating away from a cylinder that is oscillating horizontally in a stratified salt solution with uniform buoyancy frequency, N. The oscillation frequency, σ , is (i) 0.419N, and (ii) 0.900N. The cylinder is seen end-on and driven by thin vertical bars that appear black in the images. The light and dark bands are lines of constant phase in a St Andrew’s cross pattern caused by the distortion to the density field produced by waves spreading radially away from the cylinder at angles to the horizonta1, β = sin−1 (σ/N ), given by (2.9) with f /N = 0: about 25◦ in (i), and 64◦ in (ii). In (i), harmonics with frequency 2σ , radiate at angles, β = sin−1 (2σ/N ) (about 57◦ ) along fainter rays. Evidence of reflection from the tank sidewalls is visible at top left. In (ii), 2σ > N, and harmonic waves cannot exist. (From Mowbray and Rarity, 1967.) (b) The direction of the wavenumber vectors (marking the direction of phase advance) and the group velocities, cg , of the four internal wave rays radiating at inclination, β, to the horizontal from the oscillating cylinder in (a).


58

the bands in time corresponds to energy propagation in four directions away from the oscillating cylinder at the wave group velocity. The movement of the bands corresponds to the phase speed of the waves, directed normal to the group velocity. The dispersion relation, (2.8), is satisfied by four waves with the same frequency and wavelength, 2/K, but with different wavenumbers (k, m), (k, −m), (−k, m), (−k, −m). These travel in different directions but are each inclined at the same (absolute) angle, |β|, to the x-axis depending on frequency, not wavenumber, and this accounts for the four rays emanating from the location of the oscillating cylinder as sketched in Fig. 2.8b. Although the rays in Fig. 2.8a are dominated by waves with a wavelength of about the diameter of the cylinder, the oscillating cylinder does not actually produce waves of a single wavenumber, K (or waves of infinitesimally narrow bandwidth in wavenumber). Because, however, all waves with the same frequency (that of the cylinder) travel in the four directions whatever their wavelength, there are only four rays. When the cylinder first starts to oscillate, other frequencies are generated travelling in all directions, but these ‘transients’ soon radiate away leaving the dominant four-ray configuration. If the cylinder oscillation is stopped, the generation of the rays ceases, a growing gap appears between the cylinder and the bands which continue to propagate away, now as four ‘wave packets’. Each packet, containing waves with a range of wavenumbers, K, will grow in length because the size of the group velocity of the component waves, |cg | given by (2.11), depends on K; although (having the same frequency) the waves forming a packet travel in the same direction, they have different group velocities and therefore disperse, the length of the bands increasing in time, with longer waves (smaller K) travelling faster than shorter by (2.11).11 The velocity components of a wave in the x–z plane corresponding to (2.7) are: u = m0 cos(kx + mz − σ t), v = ( f /σ )m0 sin(kx + mz − σ t), w = −k0 cos(kx + mz − σ t),

(2.12)

an exact solution of the equations of motion for internal inertial gravity waves of finite (and not only very small) amplitude in a very deep fluid when the Boussinesq approximation is valid, and this provides a means of examining the stability of waves in Section 5.2. Although the y-component of the Reynolds stress, −ρ 0 vw , averaged over a horizontal plane, is zero, the x-component, −ρ 0 uw = −ρ 0 km 20 /2, is non-zero, but independent of z. The vertical flux of momentum is therefore constant, with no transfers occurring to the mean flow except at critical layers, as explained in Section 5.6.3. The corresponding density field is ρ(x, z, t) = ρ0 {1 − (N 2 /g)[z − a sin(kx + mz − σ t)]},

(2.13)

where ρ 0 is a constant reference density and a = k 0 /σ is the amplitude of the waveinduced displacements, η(x, z 0 , t), of isopycnal surfaces and where z = z0 is the level 11 This ignores possible interactions between waves (see Section 2.6).


59

Figure 2.9. Sketch showing the wave-induced displacements, η, of an isopycnal surface of density ρ(z 0 ) from the level, z 0 , that it has in the absence of internal waves.

of the isopycnal surfaces in the absence of waves (Fig. 2.9). If the stratification is a consequence of temperature variation, with density related to temperature by (1.1), then the temperature fluctuation corresponding to the density fluctuation in (2.13) is T = −(N2 a/gα)sin(kx + mz − σ t). The phase of T is 90◦ out of phase with w and so the mean value, wT , is zero and the internal waves described by (2.7) do not produce a vertical transfer of heat. An equation for η can be found by noting that on the curve z = z0 + η, the density is equal to that at z = z0 in the absence of waves, so ρ(x, z 0 + η, t) = ρ0 [1 − (N 2 /g)z 0 ], = ρ0 {1 − (N 2 /g)[z 0 + η − a sin(kx + m(z 0 + η) − σ t)]},

from (2.13), and so η(x, z 0 , t) = a sin[kx + m(z 0 + η) − σ t].

(2.14)

Solution is complicated because η appears on both sides of the equation. The shape of isopycnals perturbed by waves of different wave steepness, s = am, is shown in Fig. 2.10. Differentiating (2.13) with respect to z gives the vertical density gradient in the wave field: ∂ρ/∂z = −ρ0 (N 2 /g)[1 − s cos(kx + mz − σ t)],

(2.15)

and, differentiating (2.14) with respect to x, the slope of the isopycnals is ∂η/∂x = ak cos(kx + mz − σ t)/[1 − s cos(kx + mz − σ t)].

(2.16)

Equation (2.15) implies that travelling internal waves modulate the density field, producing layers of increased or reduced gradient visible as the bands in Fig. 2.8a, inclined to the horizontal at angle β and moving vertically through the stratified water at the phase speed, σ /m, of the internal waves. Internal waves generated near the sea surface and carrying energy downwards − with a positive downwards component of group velocity (e.g. the (k, m) and the (−k, m) rays in Fig. 2.8b) − have an upwards vertical phase speed, and consequently lead to variations in density gradient that move upwards. Such propagating variations in density structure have been observed in timeseries measurements with moored arrays of thermistors or repeated CTD profiles, and were first investigated by Lazier (1973; see also Lazier and Sandstrom, 1978) in Lake Bala, North Wales (Fig. 2.11). A characteristic of this wave-produced fine-structure that may help distinguish it from others is that the density gradients migrate through

60


Figure 2.10. The pattern of isopycnals caused by internal waves propagating through a fluid with N = constant and with β = /6. The waves are propagating as shown by the (k, m) vector of Fig. 2.8b. Values of the wave steepness, s, are (a) 0.5; (b) 1.0, when isopycnals just become vertical; and (c) 1.5, where there is static instability in the shaded region. (From Thorpe, 1994a.)

the density field in space as well as time. It will therefore be apparent in horizontal sections made by towing a vertical array of thermistors at speeds much greater than the horizontal phase speed of the waves, typically 1 m s−1 , in the form of layers with small vertical density gradient but along which the density changes monotonically. A possible example is given in Fig. 5.2.


61

Figure 2.11. The migration in depth and time of regions of high and low density gradient through the density field derived from measurements of temperature made using a fixed vertical array of sensors in the Lake Bala, North Wales. The temperature interval between isotherms is about 0.03 ◦ C. The lake is fresh (with negligible salinity), and so the isotherms are also isopycnals, surfaces of constant density. Letters and arrows denote regions of enhanced vertical gradient, the ‘sheets’, where isotherms come together. The thickness of layers between sheets is about 4 m. In the case shown, the transient regions of high gradient move upwards through the field of isotherms, implying upward phase, but downward group, velocity of the internal waves. Compare with the spatial pattern shown in Fig. 2.10. (From Lazier, 1973.)


62

The relation of the solution (2.7) to the wave mode solution, = ϕ 0 sin(mn z) × sin(kx − σ t), is apparent if the latter is written as = (ϕ 0 /2) {sin[kx + mz − σ (t−π/2σ )] − sin[kx − mz − σ (t − π /2σ )]}, representing the superposition of two of the wave ray solutions with stream functions having the same magnitude, ϕ 0 /2, the first travelling with group velocity in direction (m, −k) and the second in direction (m, k), the first downwards, the other upwards but with a temporal change in phase of /2. The two ray solutions combine to satisfy the boundary conditions z = 0, h. Having equal and opposite vertical group velocities, they result in energy propagation solely in the x-direction. Although (2.7) does not satisfy the boundary conditions w = 0 at z = −h and 0 adopted earlier to represent to ocean floor and sea surface, it does provide some very useful guides to the propagation of waves and wave packets in regions in which, relative to the scale of the wave, the mean density gradient is uniform, or where the buoyancy frequency, N, varies only slowly in time or space.

2.4

Energy and energy flux

In a general stable stratification, but with no mean shear, the frequency of internal waves lies between f and the largest value of N. Motions in this frequency band in the ocean appear to be dominated by internal waves, a conclusion supported by testing whether measured fluctuations in temperature and currents at mid-depth far from topographic influence are consistent with the properties of internal waves (Müller and Siedler, 1976; Müller et al., 1978). Whilst there is no corresponding theoretical bound on the wavelengths of the waves, in the ocean they appear to range in scales from about 1 m to 100 km. The energy per unit surface area of internal waves is composed of potential energy (PE) associated with the vertical movement of isopycnal surfaces by the waves, and the kinetic energy (KE) of their induced motion with velocity components (u, v, w). These may be written as averages over the horizontal wavelength, λ, of the waves as 2 2 2 ηdx dz 0 , PE = (ρ0 /2λ) N η dxdz 0 + (ρ0 /λ) N z 0 (2.17) where η is the displacement of an isopycnal surface from its reference level, z0 , and KE = (ρ0 /2λ) (u 2 + v 2 + w 2 )dxdz, (2.18) with integrals from x = 0 to λ, and over the depth of the ocean from z = −h to 0. The energies of internal waves of amplitude, a, and frequency, σ, per unit volume in an ocean of uniform depth and constant buoyancy frequency, N, are PEIW = ρ0 a 2 N 2 /4 and KEIW = ρ0 a 2 N 2 [σ 2 N 2 + f 2 (N 2 − 2σ 2 )]/[4(σ 2 − f 2 )],

(2.19)

2.4 Energy and energy flux

63

and their sum, E, the energy density, is E = ρ0 a 2 σ 2 (N 2 − f 2 )]/[2(σ 2 − f 2 )].

(2.20)

The ratio of the potential energy density to the kinetic energy density, R = PEIW /KEIW , is given by R = (σ 2 − f 2 )/[ f 2 + σ 2 (1 − 2 f 2 /N 2 )],

(2.21)

as shown by Gill (1982). R tends to one as σ tends to its largest value, N, implying a trend towards equipartition of energy between PEIW and KEIW . R, however, tends to zero as σ tends to the inertial frequency, f. The particle motions in inertial waves are nearly horizontal circles with relatively small density perturbation so that PEIW KEIW .12 Internal waves travelling through the ocean as rays from a source near the sea surface or seabed retain their frequency, σ , and horizontal wavenumber, k. From (2.8), the vertical wavenumber is given by m = ±k(N 2 − σ 2 )1/2 (σ 2 − f 2 )−1/2 ,

(2.22)

and, as waves propagate upwards into a region of increasing N in the thermocline, m also increases as (N 2 − σ 2 )1/2 , and the wavelength, λ = 2/(k2 + m2 )1/2 , of the waves is reduced. The magnitude of vertical energy flux is given by |Ecgz | = ρ0 a 2 σ (N 2 − σ 2 )1/2 (σ 2 − f 2 )1/2 /2k.

(2.23)

Bretherton and Garrett (1968) show that wave action, E/σ , rather than E is conserved following the motion of a packet of waves; provided σ remains unchanged the flux of E is conserved. The wave frequency is unchanged in propagation in the absence of mean shear so, provided dissipation is negligible, the amplitude, a, varies roughly as N−1/2 , generally diminishing as upward-propagating waves encounter the thermocline. As, however, waves approach regions where N decreases to values less than σ , m tends to infinity by (2.22), |Ecgz | tends to zero by (2.23), and waves are reflected as illustrated in Fig. 2.12. Internal wave energy may be effectively trapped within regions where N is large. When f = 0, (2.23) reduces to |Ecgz | = ρ0 a 2 σ 2 (N 2 − σ 2 )1/2 /2k,

(2.24)

which is a maximum for given a and k when σ /N = (2/3)1/2 or, using (2.9), when β = sin−1 (2/3)1/2 ≈ 54.7◦ .

12 Near a sloping boundary, R depends on the bottom slope (Thorpe and Umlauf, 2002).

(2.25)

64


Figure 2.12. The trapping of internal waves within a layer of high buoyancy frequency, N, > wave frequency, σ . (a) A Schlieren image of internal waves propagating as four rays (as in Fig. 2.8a) from a cylinder at top right that oscillates horizontally at frequency, σ = 0.7 s−1 , in a region where N varies with depth as N = N0 sech(bz) with b−1 = 16.2 cm and N0 = 1.02 s−l . The width of the image is 39 cm. At the level of the cylinder, σ < N. N decreases above this level, and wave rays become more vertical and are reflected at the level where N becomes equal to σ (as shown in (b)). Wave rays propagating downwards from the cylinder pass first through a region of increasing N, where their inclination to the horizontal, β, decreases [β sin−1 (σ/N ) from (2.9) with f /N = 0], and then through decreasing N until they reach a level where σ = N , where β = /2 and they reflect. (From Nicolaou et al., 1993.)

2.5

The Garrett–Munk spectrum; the energy in the internal wave field

In the early 1970s, Garrett and Munk (1972a, 1975) synthesized the various measurements of internal wave energy spectra then available and, making some assumptions about the nature of the waves, proposed a spectral form that, with relatively minor changes, has since proved to be a remarkably good description of internal waves within oceans and lakes. This apparently near-universal energy spectra describes the variation of wave energy in frequency, horizontal and vertical wavenumber space as shown in Fig. 2.13.13 An experiment designed to test the model spectrum, the Internal Wave Experiment (IWEX), was made over the Hatteris Abyssal Plain in 1973 with instruments set on a very stable three-leg mooring at depths between 604 and 2050 m. The observations are in good agreement with the proposed model (Müller et al., 1978). Although there is a contribution apparent from the advection past the mooring of a layered temperature fine-structure (see Section 7.2), the fluctuations in temperature and velocity are mainly a result of random linear waves (although not entirely so, as explained in Section 2.8). Averaged over long time periods, the wave 13 Problems associated with spectra, such as aliasing, are discussed by Tennekes and Lumley (1982).

2.5 The Garrett–Munk spectrum

65

field is horizontally isotropic and vertically symmetric except at tidal and near-inertial frequencies. They indicate the presence of a broadband field of waves that are interacting with one another so strongly as to effectively destroy any signal that identifies the location of their source, with the exception of tidal and near-inertial waves for which there are particularly strong sources of generation. Integration of the wave energy spectrum provides an estimate of the total energy contained in the internal wave field per unit horizontal area. Disregarding for the moment the internal tides, the energy is about 3.8 kJ m−2 averaged over depth, equivalent to about 1 J m−3 , with KE about three times greater than PE because of the dominance of near-inertial waves (Munk, 1981).14 Multiplying the vertically integrated energy by the ocean surface area, 3.61 × 1014 m2 , the total energy in the internal wave field is about 1.4 × 103 PJ (1 PJ = 1015 J).15 The energy is partly supported by a downward flux of near-inertial wave energy from the ocean surface, estimated to be about (2−4) × 10−4 W m−2 (Leaman, 1976), or globally about 0.072−0.14 TW (1 TW = 1012 W).16 If this is the only source of internal wave energy, the decay time scale of the waves required to retain a constant energy level of 1.4 × 103 PJ is about 100−200 days. The rate of decay of the internal wave field cannot be measured directly, but a decay time of about 60 days leads to a dissipation rate of 6.8 × 10−4 W m−2 or 0.2 TW globally, and is broadly consistent with the mean vertical diffusion coefficient estimated from microstructure measurements in mid-water.17 As explained in Section 1.9, Munk (1997) and Munk and Wunsch (1998) estimate the internal tidal dissipation as 0.9 TW, with surface-generated internal waves possibly providing a further 1.2 TW to support the energy dissipation of 2.1 TW required to maintain the abyssal stratification of the

14 For comparison, Gill (1982) finds that the ‘available potential energy’ of a typical mid-latitude ocean gyre is of order 105 J m−2 . The available potential energy, APE, is that which could be released by processes, usually involving an adiabatic redistribution of mass, that would allow the ocean to change to a horizontally uniform, stably stratified, static reference state in which density surfaces would be equi-geopotential surfaces. Excluding boundary currents, the mean potential energy in the large-scale oceanic circulation is about 1000 times that of the kinetic energy (Gill et al., 1974). 15 The value given in Fig. 5 of the review by Wunsch and Ferrari (2004) is incorrect. 16 Leaman (1976) derives the downward radiation of internal (mostly near-inertial) wave energy from a rotary decomposition of the currents measured between the surface and a depth of 2500 m by using an electromagnetic velocity profiler, an early version of the Advanced Velocity Profiler mentioned in Section 6.2. Clockwise polarization of the currents implies downward energy flux as can be shown from (2.12) and the formula for cgz . The data, collected during the MODE experiment referred to again in Section 13.1, are, however, limited to only a few days duration. Leaman’s estimate of 0.072−0.14 TW is rather less than the numerically derived value of about 0.6 TW for the mean global wind-induced flux of energy into inertial motions in the surface mixed layer by Watanabe and Hibiya (2002) mentioned in Section 2.7.1. 17 Gregg and Sanford (1988) measured dissipation rates in the eastern North Pacific in a region removed from fronts or strong currents, and where the water was not unstable to double diffusive processes (see Section 4.6), leaving the internal wave field as the only factor contributing to turbulent dissipation. The measured dissipation rate, ε, decreased from 2 × 10−9 W kg−1 in the seasonal thermocline to 1.8 × 10−10 W kg−1 at depths of about 900 m, although the largest dissipation rates in this depth range are about 100 times the average and the vertical diffusivity, Kρ, was about 4.3 × 10−6 m2 s−1 , with no trend in depth. These estimates are in accord with predictions made on the basis of the Garrett–Munk spectrum, and imply decay times of the internal wave field to be about 30 days at 100 m depth and 130 days at 950 m.

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Figure 2.13. The Garrett–Munk spectrum of internal waves in the ocean. The spectral energy is plotted vertically and is shown in logarithmic coordinates as a function of frequency and wavenumber. (a) The horizontal scales are frequency, ω, made nondimensional by dividing the observed frequency (σ ) by a nominal frequency, N0 = 5.24 × l0−3 s−l (or 3 cph), and horizontal wavenumber, α, made non-dimensional by multiplying the horizontal wavenumber (k) by a depth scale equal to a nominal scale of variation in buoyancy frequency of 3 km. The frequency is shown between the limits of ωt = f /N 0 and n = N /N 0 . An integral of the energy spectra in wavenumber, α,

2.6 Wave–wave interactions

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ocean, although their estimate of the surface-generated wave energy flux may be too high. The directional isotropy and apparent universality of a broadband internal wave spectrum should not be taken to imply that, everywhere in the ocean, internal waves of all periods and wavenumbers are present simultaneously, travelling in all directions.18 Indeed it is probable that, whilst a long-term, isotropic background mean ‘noise level’ exists, at any time and place the motion will often be dominated by packets or groups of waves travelling predominantly in some well-defined direction with a relatively narrow frequency band and a small range of wavelengths. Interacting slowly with the background waves or as they encounter other groups, these may have a measure of stability; they retain their group form whilst propagating over distances that depend on their chance interactions with the background waves and with other groups that they encounter. In some cases their propagation may be through the full ocean depth: internal tides propagating downward through the ocean from a generation site at the shelf break and inertial waves generated by the passage of an atmospheric front (as described in Section 2.7), are cases in point. Wave groups of other frequencies may be less evident and presently unrecognized in data sets simply because they are less energetic. The implications of the existence of such wave groups for ocean mixing are discussed in Section 5.8.

2.6

Wave–wave interactions

2.6.1

Introduction

It has been known since the work of O. M. Phillips in the 1960s (see Phillips, 1966) that, in particular conditions, a train of internal waves can interact resonantly with other internal waves or with surface waves, leading to exchanges of energy that may diminish or enhance the amplitude of the wave train. These wave–wave interactions 18 Briscoe (1999) re-analysed the IWEX data and finds that, over periods of 72 h, waves are not horizontally isotropic but show marked directionality.

← plotted onto the constant wavenumber plane, MS, corresponds to the spectra measured from a stationary moored instrument. The symbols γ = (1−f 2 /σ 2 )1/2 and α ∗ = 6(ω2 − ωt2 )1/2 . TLC and MHC stand for towed lagged coherence and moored horizontal coherence, respectively, and provide information about the coherence of measurements made by towed and moored instruments, as explained by Garrett and Munk (1975). (b) The horizontal scales are frequency, ω, and vertical wavenumber, β, made non-dimensional by multiplying the vertical wavenumber, m, by the depth scale of 3 km. The spectrum has a peak at the inertial frequency and falls off like ω−2 at higher frequencies up to the buoyancy frequency, N. The symbol β ∗ = 6n. MVC and DLC stand for moored vertical coherence and dropped lagged coherence, respectively. The spectral energy decreases as both the horizontal and the vertical wavenumbers increase. (From Garrett and Munk, 1975.)


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require at least three waves to be involved (e.g. three internal waves or one internal wave and two surface waves), and for resonance to occur their wavenumbers, ki , and frequencies, σ i , (i = 1, 2, 3, . . .), must satisfy the following conditions:19 k1 ± k2 ± k3 ± · · · = 0

(2.26)

σ1 ± σ2 ± σ3 ± · · · = 0.

(2.27)

and

Just how effective the interactions are depends on the time the waves occupy the same volume of water as they propagate, and whether their phase relationship remains steady. This process, resonant interaction, results in the possibility of an internal wave in the ocean gaining energy from surface waves or through the transfer of energy from other internal waves. The former energy exchange process implies internal wave generation or at least the growth of initially small disturbances, and the latter has a strong effect in dictating the form (e.g. the variation of energy with wave frequency shown in Fig. 2.13) of the internal wave energy spectrum. Theoretical studies have been made to examine the way in which internal waves interact with each other in the hope of investigating and explaining the apparently stable and universal form of the oceanic internal wave spectrum (see, for example, Müller et al., 1986). Idealized distortions of the spectra from the Garrett–Munk form show that it relaxes rapidly to its universal form, a form that is increasingly viewed as providing a steady cascade of energy from large to small vertical scales, where energy is dissipated through a process of wave breaking. The cascade is driven by energy transfer between waves by resonant interactions, of which McComas and Bretherton (1977) identify three kinds that are predominant in determining the shape of the oceanic internal wave spectrum. These are described in the following sections. A general theorem governs the interactions of three internal waves. A necessary condition discovered by Hasselmann (1967) for the growth of a wave characterised by frequency σ 0 through resonant interaction with two others with frequencies σ 1 and σ 2 is that |σ 0 | = |σ 1 | + |σ 2 |. The difference interaction with |σ 0 | = |σ 1 | − |σ 2 | is neutrally stable, with no change in amplitude of the ‘primary’ σ 0 wave. This is now known as Hasselmann’s theorem. 2.6.2

Elastic scattering

This type of interaction results in the back-scatter of relatively high frequency internal waves by low frequency near-inertial internal wave motions. There is little exchange 19 For internal wave interactions, the wavenumbers are the three-dimensional wavenumbers, (k, l, m), the wavenumber components in two horizontal directions and the vertical. Surface waves propagate on the water surface and have only two-dimensional wavenumbers, (k, l). For resonance, these, and the horizontal wavenumbers of the internal waves, must satisfy the resonance conditions. The strength of interaction, the rate at which energy is transferred between surface and internal waves as a result of resonant interaction, depends on the vertical structure (i.e. the mode) of the internal waves involved, and consequently on the density stratification.

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of energy in the process, but the vertical wavenumber of the high frequency waves is reversed. It contributes to an observed feature of the wave field, one of near-isotropy of the directional spectrum, masking the sources of the waves. 2.6.3

Induced diffusion

Induced diffusion characterizes the scattering of high-frequency and high-wavenumber internal waves by low-frequency and low-wavenumber internal waves. Small-scale, high-frequency wave packets may be thought of as experiencing random perturbations as they interact with the shear of larger scale waves in the fashion of a random walk, a process that leads to the effective diffusion of their wave action in wavenumber space. Caustics are formed when short waves are focussed by progressive inertial waves.20 Although strong interaction is expected in conditions in which the phase speed of an inertial wave train or wave group is equal to the group velocity of short waves, so that the short waves are refracted by the approximately steady shear of the inertial waves in their vicinity, Broutman and Young (1986) find this is relatively unimportant for realistic values of shear produced by oceanic inertial waves. Downward-propagating inertial waves tend permanently to decrease the magnitude of the vertical wavenumber of upward-propagating short waves groups. 2.6.4

Parametric instability

The periodic modulation of the local buoyancy frequency by internal waves results in instabilities that transfer energy from the waves and result in the growth of internal waves with frequencies which are harmonics or the first sub-harmonic of the primary waves. The sub-harmonic instability is the most effective of the parametric instabilities and consequently is the most thoroughly studied. (The necessary condition for wave growth of Hasselmann’s theorem is satisfied if the sub-harmonic is counted twice: σ 1 = σ 0 /2 and σ 2 = σ 0 /2.) This instability or resonant wave process tends to transfer wave energy in successive ‘halving’ stages through the wave spectrum towards the smallest frequency of waves, the inertial frequency, f, and towards large wavenumbers, increasing the shear induced by the waves. If, however, a wave has a frequency between f and 2f, sub-harmonic instability will not occur because the frequency of the sub-harmonic is less than f and therefore does not correspond to an internal wave. Such near-inertial waves may consequently be less likely to lose their energy through wave–wave interactions and may be more stable, persisting for relatively longer, than those of frequency >2f. Because of their regularity and persistence, internal tides are more prone to instabilities involving resonant interactions, in particular parametric instability, than are other, more transient, waves. The frequency 2f (= 4 sin φ) is equal to the semidiurnal M2 tidal frequency, σM2 (= 1.40 × 10−4 s−1 ), at latitude of 28.9◦ , and so 20 See Section 5.4 for further reference to caustics.

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internal M2 tides closer to the Equator (where f has smaller values, and therefore σM2 > 2f ) may be prone to parametric sub-harmonic instability, ‘psi’, and to be less stable than are those at higher latitudes. For example, the Hawaiian Ridge, a known site of M2 internal tides (see Fig. 2.7), lies between about 19◦ N and 28◦ N and the internal tides there may be subject to psi,21 whilst those generated at higher latitudes are not. This is consistent with the observation that the M2 internal tides generated at the Hawaiian Ridge propagate over shorter distances than those emanating from the Aleutian Ridge near 50◦ N and observed by using TOPEX/Poseiden satellite altimeter data (Ray and Cartwright, 1998). Evidence of the mixing caused by the psi of the M2 internal tides has been obtained by Hibiya and Nagasawa (2004). They estimate the vertical eddy diffusivity from measurements of shear between 900 and 1400 m depth in regions of the North Pacific identified as being sources of M2 internal tides of similar energies (Fig. 2.16).22 The mean vertical eddy diffusivity in the near-equatorial region south of about 30◦ N, where the instability becomes possible, is found to be greater than that to the north, where it is not, by a factor of about three. Whilst wave–wave interactions will often lead to energy transfer or ‘leakage’ from one wave frequency and wavenumber to others, effectively destabilizing a wave train, it is also possible that interactions may help to stabilize a group of waves of different frequencies by preventing their dispersion. In general, waves of different frequencies and wavenumbers travel at different group velocities (see Equation (2.11)) and in different directions, β. Interactions between pairs of waves with suitably matched frequencies and wavenumbers can, however, result in their travelling in the same direction with the same group velocity, leading to the possibility of modulated wave trains, or groups, with narrow, but finite, wavenumber and frequency bands (Thorpe, 2002a).

2.7

Generation of internal waves

Internal waves are generated in a number of ways, both within the body of the ocean and at its boundaries. Figure 2.14 is a sketch that includes some of these. 2.7.1

Processes at, or near, the sea surface

Rapid changes in wind stress on the sea surface caused by moving atmospheric fronts, and the effects of the Earth’s rotation, have been shown to result in the generation of internal or near-inertial waves with frequencies which are close to the inertial frequency, f (Pollard and Millard, 1970). These waves often dominate the oscillations 21 Evidence of fluctuations with half the frequency of the M2 tide has been observed near Hawaii (Rainville and Pinkel, 2005). Gerkema and Shira (2005) show that the effects of rotation and the Earth’s curvature (the ‘beta effect’) allows psi to act typically 1◦ –5◦ poleward of the 28.9◦ latitude. 22 Hibiya and Nagasawa (2004) estimate the vertical diffusion coefficient using the formula (7.7) derived by Gregg.

Figure 2.14. A cartoon showing some of the ways in which internal waves are generated in the deep ocean. Generation over sloping topography and at the shelf break is not represented. (Based on Thorpe, 1975.)

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of the upper pynocline, particularly in the shear field. Waves of higher frequency are also known to occur and although several mechanisms (discussed below) have been postulated for their generation, the processes that dominate their formation have not been conclusively established by observation and nor has their energy flux been measured. As mentioned in Section 2.6.1, resonant interactions occur between surface and internal waves, and these have been shown theoretically to contribute to an increase in the energy of internal waves (Watson, 1990). No measurements have yet been made to verify the transfer rates. The buffeting of the thermocline by motions within the mixed layer caused by convection or by other processes described in Chapter 9, has been known as a potentially effective mechanism of internal wave generation since the work of Townsend (1966, 1968). Radiation of internal waves from a turbulent layer is observed in laboratory experiments, for example those of Piat and Hopfinger (1981) in a stratified wind tunnel. Linden (1975) studied the deepening of a mixed layer produced by turbulence driven by a rapidly oscillating horizontal grid in a tank filled initially with a uniformly stratified salt solution. He compared the rate of increase in potential energy of the fluid in the tank with that produced by the oscillating grid, and estimated the radiation of energy from the mixed layer by internal waves propagating away into the stratified region. He found that the energy lost to mixing by wave generation may contribute to a reduction in the rate of mixed layer deepening of as much as 50%, recommended that internal wave radiation should be included in estimates of mixed layer deepening, and concluded that it may provide a significant source of internal waves in the deep ocean. Sutherland and Linden (1998) found that, in a (non-rotating) laboratory experiment, internal waves radiating into a uniformly stratified, stationary fluid from a turbulent layer moving at mean speed, U, propagate predominantly in directions β between 30◦ and 45◦ from the horizontal. If the waves have horizontal wavenumber, k, and frequency, σ , their horizontal phase speed matches that of eddies advecting in the turbulent flow when σ /k = U. Substituting this into (2.23) gives a vertical energy flux, |Ecgz | = ρ 0 a2 U(N 2 − σ 2 )1/2 (σ 2 − f 2 )1/2 /2. For fixed U, N and a, this is maximum when σ 2 = (N 2 + f 2 )/2 or, from (2.10), when β = 45◦ , at the edge of the observed range. Sutherland and Linden propose that some, yet unspecified, interactive feedback mechanism between the waves and eddies may be operating. Doron and Sutherland (2003) also found internal waves propagating downwards at angles 35◦ < β < 45◦ into a uniformly stratified region from a turbulent grid-mixed region with no mean flow, implying that the result of Sutherland and Linden may have wider application in the generation of internal waves with frequencies that are near (but less than) the buoyancy frequency, N. Although Wijesekera and Dillon (1991) observed internal waves propagating downwards through the thermocline in the equatorial Pacific after sunset, presumably generated by the impact of developing convective motions on base of the mixed layer, there is as yet insufficient observational evidence on which to base reliable estimates of the consequent downward flux of wave energy or the spectra of waves generated by

2.7 Generation of internal waves

73

Figure 2.15. The distribution of wind-induced inertial wave energy input per unit surface area averaged over periods of 3 months and determined from data for the period from 1989 to 1995. Seasonal (as well as spatial) variability is apparent, particularly in the North Pacific and Atlantic Oceans and in the Darke Passage. (From Watanabe and Hibiya, 2002.)

turbulent processes in the mixed layer under the natural range of forcing conditions. (The nature of flow at the base of the mixed layer is discussed in Section 9.7.) Travelling changes in air pressure, precipitation and convection are predicted to leave ‘internal wave wakes’ in the lee of travelling storms (Gill, 1982) but observational data to quantify accurately the generation processes are lacking. The seasonal variation of the vertically integrated energy in internal waves with frequencies between 2f and 3f generated by the wind field in the North Pacific have been estimated by Nagasawa et al. (2000) using a numerical model. The greatest energies are found in the south and west as a consequence of hurricane activity. Watanabe and Hibiya (2002) estimate the global energy flux into inertial motions in the oceanic mixed layer caused by wind to be 0.6 TW. The seasonal flux distributions are shown in Fig. 2.15. Much of this energy may be dissipated in the mixed layer or upper thermocline; only a small proportion is required to support the internal near-inertial wave flux of 0.072–0.14 TW estimated by Leaman (1976) and mentioned in Section 2.5. Although the calculation does not include the energy flux into waves of higher frequency, their contribution is likely to be relatively modest. As mentioned in Section 1.9, the total contribution of surface generated internal waves may consequently be substantially less than the value of 1.2 TW required by Munk and Wunsch (1998), together with 0.9 TW of


74

tidal dissipation, to supply the total 2.1 TW needed to support mixing in the abyssal ocean.

2.7.2

Topography

Ekman described how internal waves are generated by a moving vessel (Section 2.1). Waves are similarly generated as ‘lee waves’ if, instead of the hull of the vessel moving through the water, the stratified water moves over and past a fixed obstacle, such as a mound or small seamount on the sea bed. The same wave-generation process occurs as air flows over mountains, in suitable conditions resulting in beautiful displays of lenticular wave clouds, appearing sometimes as a ‘pile d’assiette’ (pile of plates). Bell (1975) examined the generation of waves by flow over topography, taking as an example the rough topography of the abyssal hills found on the floor of the Pacific. He recognized that, close to the seabed, the stratification may act to ‘steer’ the flow around topography so that the effects of flow over topography are diminished, a point discussed in Section 12.4. He estimated the energy extracted from the mean flow and converted to (and eventually dissipated by) internal waves to be about 10−3 W m−2 . Integrated over the area of the whole ocean, it is concluded by Baines (1982) that the barotropic tides consequently lose about 0.4 TW to the generation of internal tides by flow over abyssal topography. This is, however, an underestimate: Bell and Baines overlooked the importance of ocean ridges, as explained below. The periodic flow of the barotropic tide that carries stratified fluid over topography, either the continental slopes and shelf break or over sills, is an effective generator of internal waves in the pycnocline (Farmer and Smith, 1979). Often the stratification resembles, and can be represented as, two uniform layers. Flow in one direction causes a stationary jump or train of internal lee waves to form at the density interface in the lee of the topography and, as the tidal flow changes direction, these are carried over the topography and propagate away, as demonstrated in laboratory experiments by Maxworthy (1979). In some cases, of which examples are described in Section 12.6.1, the depressions of the thermocline or hydraulic jumps that are formed in the lee of sills are, on being advected over the topography, rapidly transformed into soliton packets of typically 3−8 internal waves travelling mainly as waves of mode number one on the pycnocline. Internal soliton packets, perhaps formed in a similar manner, are particularly evident on the continental shelf, travelling away from the shelf break towards shore. An example is shown in Fig. 2.4.23 Mixing produced by internal tides at or near the shelf break is sometimes apparent through the lower sea-surface temperatures or phytoplankton blooms in these region, the plankton benefiting from the increased levels of nutrients mixed upwards from deeper water (New, 1988; New and Pingree, 23 The internal waves generated near the sill in Knight Inlet, British Columbia, Canada, and illustrated in Figs. 12.8 and 12.9, however, appear not to be generated by the propagation of the large depression in the thermocline over the sill as the tide decreases, but as a transient response to an adjustment of hydraulic control of the stratified tidal flow (Cummins et al., 2003); see Section 12.7.1.

2.7 Generation of internal waves

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1990). The presence of colder near-surface water is sometimes taken as evidence of wave-produced mixing (see, for example, Ffield and Gordon, 1996). Internal tide generation by the interaction of barotropic tides and topography is particularly efficient when the water overlying the topography is stratified and when internal waves of tidal frequency propagate at a beam angle, β, that matches the slope of the topography, i.e. the angle of its inclination to the horizontal. The slope is then said to be ‘critical’, in a sense explained in Section 11.4.1. For typical values, N = 10−3 s−1 and f = 1.03 × 10−4 s−1 corresponding to a latitude of 45◦ , and with σ = 1.40 × 10−4 s−1 (the frequency of the 12.43 h, M2 tide), the angle, β, of the internal tidal ray given by Equation 2.9 is about 5.5◦ . The inclination of the bed of the continental slopes is typically 1−10◦ , and the internal tides are therefore often critical somewhere on a continental slope. In such critical conditions internal wave rays are generated that propagate away from the slope. (Turbulence generated in an internal tidal ray originating from the continental slope is shown in Fig. 5.4.) Whilst generation over such critical slopes may be very effective in the generation of waves of high mode number − waves that consequently produce relatively large shear which, as shown later, may be effective is promoting mixing − it is likely that the height and shape of topography (e.g. a ridge or continental slope) may be of greater importance in the generation of the low internal wave modes which appear to dominate the internal wave energy in much of the ocean. Baines (1982) estimated the total annual mean rate of barotropic tidal energy transferred to internal tides around the continental slopes of the ocean. For the M2 tide the total is 14.5 GW and for the S2 tide 2.73 GW, with possible errors of ± 50%. About half the internal tidal energy is directed towards shore and contributes to the mixing on the continental shelves. The remainder is far less than required to provide a significant fraction of the 2.1 TW needed to support mixing in the deep ocean, and smaller even than Bell’s estimate of 0.4 TW for the generation of internal tides in the abyssal oceans. Neither Bell nor Baines took into account the substantial internal tide generation that can occur at ridges where, as mentioned in Section 1.9, the barotropic tidal flow often carries water directly across the topography resulting in a more effective generation of internal tides than when, as common at the continental slopes, relatively little of the flow is across topography. Figure 2.7 shows the radiation of the internal tidal waves generated by the barotropic tides from a ridge, in this case the Hawaiian Ridge. The depth-integrated kinetic energy of the M2 internal tide in the Pacific Ocean calculated by Niwa and Hibiya (2001a, b) using a numerical model is shown in Fig. 2.16 (see Plate 4 for (b)). The largest values, exceeding 320 J m−2 , are all in regions of ocean ridges or continental slopes. The total energy transfer rate from the barotropic to the baroclinic M2 tide in the North Pacific alone is 0.27 TW. As mentioned in Section 1.9, the dissipation of internal tidal waves summed over all the ocean basins may contribute about half the power required to maintain the abyssal ocean mixing. St Laurent and Garrett (2002) have estimated the flux of energy in internal tidal waves generated by the interaction of surface tides with topography

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Figure 2.16a. The generation of internal tides in the Pacific predicted using numerical models. The depth-intergrated rate of energy flux from the barotropic to the internal M2 tide in selected boxed areas in the North Pacific. (From Niwa and Hibiya, 2001a.)

in the region of the mid-Atlantic Ridge in the South Atlantic at the edge of the Brazil Basin, where Polzin et al. (1997) observe elevated rates of dissipation. They find a vertical energy flux of 3−5 mW m−2 , with 1−2 mW m−2 contributing to local mixing and the remainder being radiated away. The former value compares favourably with Polzin et al.’s vertically integrated dissipation rates that reach 3 mW m−2 but which show a spring-neap tidal modulation. Processes that lead to the dissipation of internal wave energy by breaking, and to the generation of turbulence in mid-water, are described in Chapter 5.

2.7.3

Interior or mid-water processes

Within the stratified water column internal waves radiate away from regions of active turbulent mixing. They are also generated in the process of gravitational collapse of a mixed region of decaying turbulence (Maxworthy, 1980; Amen and Maxworthy, 1980). Both processes are secondary in the sense that the regions of turbulence are themselves likely to be wave-generated and so, like resonant interactions between waves, the processes do not contribute to the overall energy of the internal wave field, only to its dissipation or re-distribution in frequency and wavenumber. Internal waves may also be generated where the ocean adjusts towards a state of geostrophic balance in the creation or interaction of the mesoscale eddies described in Chapter 13, but the contribution of this to the overall energy of the internal wave field has yet to be fully assessed.

2.8 Internal waves and vortical mode

2.7.4

77

Lakes

Internal waves in lakes are often generated in the process of adjustment towards static equilibrium after wind stress has induced flows that transport water in the mixed layer towards the lee shore or, if the wind blows for periods comparable to the local inertial period, the side of a lake towards which near-surface water is moved by Ekman drift. The thermocline is there locally depressed, but raised on the upwind side of the lake. The subsequent tendency of the thermocline to recover a horizontal level once the wind has fallen or changed direction, leads, in relatively small or narrow lakes, to internal seiches. In long lakes such as Loch Ness that are of width less than the internal Rossby radius, LRo (defined in Section 3.8.2), these seiches may take the form of undular bores (see Fig. 2.2). In lakes of dimensions exceeding LRo , the excited oscillations often take the form of nonlinear internal Kelvin waves modified by the presence of the sloping sides of the lakes and travelling cyclonically around the lake’s circumference.24 ‘Resonant forcing’, by winds that repeatedly blow at intervals close to the period of a lake seiche, is also very effective in driving oscillations. Smaller scale waves may be radiated into the body of a lake when those of large scale, for example the Kelvin waves, pass over or around topography adjacent to the lake shore, or may be generated at the water surface as described in Section 2.7.1. Mixing generated by wind forcing and internal waves in lakes is discussed in Section 12.8.

2.8

Internal waves and vortical mode

Because of its close association with internal waves and turbulence, and in preparation for later chapters, it is appropriate here to mention the vortical mode of motion. Internal waves are not the only type of motion that can exist with a span of horizontal length scales from about 1 m to 100 km. By the late 1970s, the analysis of oceanic measurements of fluctuations in shear and temperature gradients showed that they are not entirely consistent with those produced by linear internal waves. The velocity fluctuations in some frequency ranges appeared to be too large in comparison with the temperature fluctuations, and vertical coherence scales for velocity are smaller than those of temperature (see Müller et al., 1978; Eriksen, 1978). It was recognized that there are solutions of the equations of motion in this spatial bandwidth other than those corresponding to internal waves, solutions associated with variations in potential vorticity and with frequencies very much less than the inertial frequency, f (Müller, 1984). 24 Kelvin’s theoretical description of the waves named after him assumes that the side-walls are vertical, with zero through-flow. The waves depend on the Earth’s rotation, and the presence of sloping boundaries introduce further factors that are associated with the stretching of the water column as it is moved up and down the slope and related to the conservation of vorticity. Kelvin waves diminish in amplitude offshore with e-folding length scale, LRo . In a relatively long but narrow lake, such as Loch Ness, of width much less than LRo , the amplitude of waves travelling along the lake like those illustrated in Fig. 2.2 diminishes only slightly across its width.


78

The vortical mode is defined as the class of very low-frequency motions with vertical scales less than 100 m that (unlike internal waves) have potential vorticity, q. This is defined as q = (2Ω + ∇ × v) · ∇(gρ/ρ0 ),

(2.28)

where 2Ω = (0, f cot φ, f ) at latitude φ, and v is the velocity with coordinates in the east, north and vertical directions. Ertel (1942) showed that in the absence of dissipative processes, like turbulence or breaking waves, q is conserved following fluid motion. This result is known as Ertel’s theorem. (The effect of frictional forces and dissipation in changing q have been examined by Haynes and McIntyre (1987). The rate of change of potential vorticity following its advection by the fluid is equal to −∇·[(g∇ρ/ρ 0 ) × F], where F is the frictional force. This may lead to either the production or reduction of q.) In the simplest case of horizontal motion and when density, ρ, varies only in the vertical direction, the potential vorticity reduces to ( f + ∂v/∂x − ∂u/∂y)∂ρ/∂z or (f + ωz )∂ρ/∂z, where ωz is the vertical component of vorticity (Kunze, 2001). Vortical mode motions are associated with rotation and the vertical stretching of the density field (or strain). Presently the generation mechanisms of the vortical mode are largely conjectural: some of the related processes will be described in later sections. Possible candidates in mid-water are: internal wave interactions (Lelong and Riley (1991) − internal waves can interact with vortical mode as well as with one another); the collapse and spread of a layer mixed by turbulence, particularly that produced by breaking internal waves (see Section 5.9); and processes associated with the instability of large-scale motion (e.g. baroclinic instability; see Section 3.8.2), possibly in zones of interaction of the neighbouring mesoscale eddies or breaking of Rossby waves described in Chapter 13. In regions affected by deep convection, vortices may remain as collapsed remnants of convection following the seasonal re-establishment of the mixed layer. There are also processes associated with boundary mixing, such as boundary layer convergence following the non-uniform transfer of momentum into boundary currents by breaking internal waves described in Section 11.4.4, the interaction of Meddies with seamounts (Section 13.3.5), and the separation from the bottom or sloping sides of the ocean of water mixed in near-bottom boundary layers (see, for an example, Fig. 12.10).25 A conceptual picture emerges of an ocean populated at vertical scales of about 100 m or less by internal waves and by horizontal pancake eddies of near-uniform density, sliding slowly around or over one another, the eddies or vortical mode motions possibly controlling the nature and rates of horizontal dispersion in mid-water at scales up to about 10 km (Section 13.4.1). 25 Methods of categorizing vortical motions have been devised according to the variation of Berger number, (NH/f L)2 , with the Rossby number, Ro = U/f L (Müller, 1984) or with the ratio of potential to kinetic energies (Kunze and Sanford, 1993), where H and L are the vertical and horizontal scales of the vortical structures.

2.8 Internal waves and vortical mode

79

In later chapters the peculiar properties of internal waves are described further, for example those involved in their encounter with topography (Section 11.4). The ways in which internal waves break and, losing energy, contribute to the generation of turbulence and mixing in the ocean, are discussed in Chapter 5. Before addressing wave breaking, however, it is appropriate to introduce and describe some of the processes of transition to turbulent motion within the moving stratified ocean, and these are the subjects of Chapters 3 and 4.

Chapter 3 Instability and transition to turbulence in stratified shear flows

3.1

Introduction

How turbulence develops from the instability of a smooth, laminar, stably stratified, shear flow is described in this chapter. It commences with an account of the early history of the study of the instability and how it was eventually observed in the ocean. The story nicely illustrates how the study of fundamental processes in fluid dynamics aids the investigation of naturally occurring fluids, in this case first the atmosphere and later the ocean and lakes. The shear flow instability involves several features characteristic of turbulent flow: the formation of eddies, the stretching of fluid between neighbouring eddies – intensifying gradients of fluid properties, notably density – their pairing or amalgamation and the transfer of vorticity to larger scales, and their break-down to form an energetic field of smaller eddies. Shear instability is by no means the only form of instability to occur in the stratified ocean, as is revealed in later chapters. The temperature measurements that were to form the basis of Watson’s discovery of internal seiches in Loch Ness, described in Section 2.1, and which underpinned Wedderburn’s later investigations, began in April 1903, using reversing thermometers. Measurements were made at about ten depths on a roughly daily basis.1 Wedderburn 1 The period of the internal seiche in Loch Ness is about 54 h, so once per day was barely enough to resolve it! More intense and frequent observations were made at four stations, each three times a day, in September 1904. A year earlier, in September 1903, the yacht Rhoda was anchored in about 80 m of water at the southeast end of the lake, equipped with platinum resistance thermometers that could be lowered from the yacht and connected by cable to a recorder on shore. Electrical methods of temperature measurement had been tried much earlier on the Challenger Expedition, 1872–1876. This ‘elaborate and costly apparatus did not come up to our expectations’, wrote Wedderburn (1907), but the changes observed did provide clues to mixing. Although a student in arts and law at Edinburgh University (and eventually knighted for his services to the Scottish legal profession), Wedderburn had been attracted to geophysical studies

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3.1 Introduction

81

(1907) reports the following, using the Germanic ‘Sprungschicht’ to mean a layer of changing temperature or density. The behaviour of water in the neighbourhood of a Sprungschicht is very similar to the behaviour of water near the surface. Before this was understood, the observers were often puzzled when working with mercury thermometers by obtaining readings which seemed to show the presence of very marked inverse stratification, which always disappeared in a very illusory fashion when any attempt was made to follow it up.2

It appears that Wedderburn had breaking surface waves in mind in his reference to the ‘water near the surface’. The matter is readdressed and clarified in a later paper (Wedderburn, 1911), where he refers to the temperature seiche producing . . . rapid currents both in the upper and lower layers [i.e. above and below the thermocline] the currents in the two layers being opposite in direction and therefore their relative velocity will be very large. Vortices are certain to be caused at the discontinuity where one current slips over another.

Other than the mention of the illusory density inversions, no evidence (or reference to other scientific works) is given to support the claim that shear might lead to instability and, in particular, to the generation of vortices. But evidence from theory and from a laboratory experiment already existed! The theory was that devised by Helmholtz (1868) and Kelvin (1871) to describe the stability of small (or ‘infinitesimal’), spatially periodic disturbances to an interface between two layers of differing density in relative motion, a theory developed in connection to the study of the generation of waves on a water surface by wind. The instability is now known as Kelvin–Helmholtz instability, but this term is widely used (and is here) to describe the growth of disturbances in stably stratified shear flows, however the density and velocity vary in depth, provided only that the former increases with depth. The instability of such stably stratified shear flows is considered in Section 3.2.3. The laboratory experiment had been made by Osborne Reynolds. Together with his classic account of the onset of turbulence in a pipe (Reynolds, 1883; see Section 1.1) is described an experiment in a 1.5 m long tube . . . half filled with bisulphide of carbon, and then filled up with water and both ends corked. The bisulphide was chosen as being a limpid liquid but little heavier than water and completely insoluble, the surface between the two liquids being clearly distinguishable. When the tube was placed in a horizontal direction, the weight of the bisulphide caused it to spread along the lower half of the tube, and by the professor of mathematics, George Chrystal, and helped in Chrystal’s observations of surface seiches as part of the Bathymetrical Survey of the Scottish Fresh-Water Lochs mentioned in Section 2.1. He continued his studies of the internal dynamics of lakes, making observations during summer holidays every year from 1901 to 1913, was elected as a Fellow of the Royal Society of Edinburgh in 1907 when only 23 years old. He died in 1958, aged 74. There is an obituary notice in Quart. J. R. Meteor. Soc., 84, 485 (1958). 2 Evidence of transient density inversions in the deep ocean was first described by Cooper (1967) and later, but more conclusively, at relatively shallow depths by Woods (1968).

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Instability and transition to turbulence

Figure 3.1. Reynolds’ tilted-tube experiment. Reynolds’ sketch shows four or five waves growing through shear flow instability on the interface between two immiscible layers of fluid. Surface tension has a significant effect on the development of the growing unstable disturbance in this case. (From Reynolds, 1883.)

the surface of separation of the two liquids extended along the axis of the tube. On one end of the tube being slightly raised the water would flow to the upper and the bisulphide fall to the lower, causing opposite currents along the upper and lower halves of the tube, while in the middle of the tube the level of separation remained unaltered. The particular purpose of this investigation was to ascertain whether there was a critical velocity at which waves or sinuosities would show themselves in the surface of separation. It proved a very pretty experiment and completely answered its purpose . . . When by increasing the rise [the tilt of the tube] the velocities of flow were increased, the waves kept the same length but became higher, and when the rise was sufficient, the waves would curl and break, the one fluid winding itself into the other in regular eddies.

The development of waves and their curling up is sketched by Reynolds as shown in Fig. 3.1. Reynolds makes no reference to the studies of Kelvin or Helmholtz. Had he been aware of their theoretical work and referred to it, the experiment might have been established as a major key in unravelling fluid instability. Instead Reynolds’ experiment on stratified shear flow was largely overlooked in the 80 years after his paper was published, and not until the 1960s were further laboratory studies of Kelvin–Helmholtz instability made in tilted rectangular tubes.3 In the meantime, in his Adams Prize Essay of 1915, G. I. Taylor devised a theory of instability in fluids with particular continuous variations of density and velocity in depth, but he did not published it until some 16 years later, ‘. . . hoping to be able to undertake experiments designed to verify, or otherwise, the results’ (Taylor, 1931). Even then, the work was published only with the encouragement of Goldstein (1931) who had made a similar theoretical study. Further major theoretical advances were made in the late 1950s, and in subsequent years, notably by Miles (1961) and Howard (1961). They found a criterion necessary for instability, based on the minimum Richardson number in the flow, as explained in Section 3.2.3. Laboratory experiments using other means to generate shear flows followed in the 1970s, and succeeding years, and numerical models were developed 3 G. H. Mittendorf made experiments in a tilting tube filled with two layers, water and salt solution, as part of his MSc degree at the University of Iowa in 1961, and observations of the instability are reported in his thesis, but without reference to Reynolds. (G. I. Taylor appears to have seen and had commented on the experiment during a visit to Iowa, expressing concern that the instability might be caused by the accelerations involved in tilting the tube.) Thorpe’s experiments described later were first made in ignorance of Reynolds’ work.

3.2 The onset of instability in shear flows

83

in the 1980s, eventually setting on a secure foundation the understanding of at least the early stages of the transition to turbulence. By the early 1950s ‘billows’ in stratified shear flows associated with atmospheric lee waves had been reported and their evolution described by Scorer (1951). He suggested that they might cause the turbulence sometimes encountered by aircraft flying in clear air: clear air turbulence or CAT. Over 15 years later Ludlam (1967) described visual observations of billow clouds forming in free shear layers, but it was not until the following decade when billows were observed in clear air in the troposphere and lower stratosphere by radar, and turbulence was measured simultaneously by aircraft (Browning, 1971; Browning et al., 1973), that their link to CAT was firmly established. In the late 1960s came the remarkable studies that most clearly and dramatically demonstrated the link hinted at by Wedderburn between the shear flow produced by internal waves and Kelvin–Helmholtz instability. In investigations led by Woods (1968), divers injected dye into sheets of high density gradient (Wedderburn’s ‘Sprungschichts’) within the seasonal thermocline, making visible internal, or interfacial, waves travelling along them and documenting the waves and associated features by photographs (Fig. 3.2a). Billows were observed to grow on the crests of some internal waves (where wave-induced shear is greatest), vertically spreading the dye and transferring water through the sheets (Fig. 3.2b), the first uncontroversial evidence of active mixing by the process of Kelvin–Helmholtz instability in the ocean, and supportive of the link between internal waves and mixing postulated by Munk in 1966. Before describing further observations of the manifestations of this instability in the ocean, it is important to understand its nature, and how and when it develops.

3.2

The onset of instability in shear flows

3.2.1

Unstratified flows

It is commonly the case in the ocean that the effects of molecular viscosity and heat or salinity diffusion have little effect on the instability which forms the first stage in a transition from a relatively steady shear flow to turbulence, and they are neglected in this section.4 ‘Stability’ is determined by examining whether or not small disturbances to the fluid grow. (A physical explanation of why a shear flow becomes unstable is given by Drazin and Reid, 1981, based on a vorticity argument devised by Batchelor, 1967.) For unstratified flows, Rayleigh (1880) showed the following. A necessary condition for the growth of small disturbances to a steady unidirectional shear flow varying in one coordinate direction (here taken as the vertical, z), and in which the density is uniform, is that it should contain an inflection point.

4 Diffusive effects do, however, become of critical importance in conditions of double diffusive instability as explained in Section 4.6.

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(a)

(b)

(1)

(2)

(3)

(4)

(5)

(6)

Figure 3.2. Billows on internal waves photographed in the Mediterranean thermocline near Malta. (a) In the centre of the photograph a dye sheet in the stratified interface between two uniform layers, viewed obliquely from above, is breaking up into a linear pattern of billows. The undulation in the dye sheet caused by the internal wave can be seen in the more uniform layer of dye at the top of the photograph beyond the billows. Woods’ photographs are the only source of information about the horizontal extent and coherence of oceanic billows. (b) The roll-up of a billow viewed horizontally ((1)–(6)). The distance between neighbouring billows is about 2.5 m. (From Woods, 1968.)


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Figure 3.3. Inflexion points in a shear flow. Type (a) is unstable by Fjørtoft’s theorem, whilst (b) is stable.

Tollmein (1929) demonstrated that the theorem does not provide a sufficient condition for instability. If the speed of the mean current in the x-direction in a fluid confined by horizontal boundaries in z1 ≤ z ≤ z2 with z1 < 0 < z2 is U(z) = U0 sin z, the flow does contain an inflection point at z = 0 but is found to be stable if z2 – z1 < , although unstable if z2 – z1 > . Not until the middle of the twentieth century was Rayleigh’s inflection point theorem extended by Fjørtoft (1950), who showed the following. It is necessary for instability that, somewhere in a flow, U(z), the condition (d2 U/dz 2 )(U − Us ) is less than 0, where Us is the speed of the flow at the inflection point where d2 U/dz 2 = 0.

This is satisfied in a flow in which a maximum in the shear, dU/dz, occurs at the inflexion point (see Fig. 3.3). One further important theorem is that of Squire (1933), which reduced the need for study of three-dimensional disturbances to two dimensions. To each three-dimensional disturbance (i.e. one depending on y as well as on x and z) of a unidirectional shear flow, U(z), in the x-direction there corresponds a two-dimensional one with greater growth rates.

This is now known as Squire’s theorem. It was shown by Yih (1955) that this also applies to disturbances in density-stratified flows. This theorem implies that, for unidirectional flows, it is necessary to consider only disturbances which vary in x and z, and not in the transverse y-direction. The clearest evidence of a shear instability in which stratification plays little or no part is to be seen in surface patterns formed by small waves or foam at the edge of strong tidal flows through channels or between islands, particularly as they emerge into open water. The jet-like flows produced in this way have large velocity gradients at their boundaries that result in instability and the formation of eddies, typically 5–100 m apart. An example is shown in Fig. 3.4, Plate 5, a photograph of the edge of a tidal current in the Saltfjord, Norway. In extreme conditions, such as those in the


86

Naruto Strait, Japan, or the Gulf of Corryvreckan, W. Scotland, intense eddies form whirlpools that may entrain bubbles of air and carry them to depths of 20 m or more (Thorpe et al., 1985).

3.2.2

Two-layer flows

The simplest example of a shear flow that includes stratification is that considered by Helmholtz and Kelvin. It consists of two layers each with uniform speed, U/2 and −U/2 (measured relative to the mean) and densities, ρ 1 , and ρ 2 , respectively, with ρ 2 > ρ 1 and with the former lying above the latter and separated from it by a sharp interface, at which there is surface tension, γ . In such stability problems, solutions are conventionally sought that correspond to a small spatially periodic disturbance proportional to exp (ikx + t). If is pure imaginary ( = −iσ , say, where σ is real, representing frequency), then this solution represents an internal wave that does not change its amplitude in time or is ‘neutrally stable’, propagating through the ocean without change. If, however, contains a real and positive part, then the solution grows exponentially in time, and the flow is unstable to such spatially periodic but temporally growing disturbances. In the case under consideration it is found that is given by = ikU (ρ2 − ρ1 )/[2(ρ1 + ρ2 )] ± {k 2 (U )2 ρ1 ρ2 /(ρ1 + ρ2 )2 − [γ k 3 + gk(ρ2 − ρ1 )]/(ρ1 + ρ2 )}1/2 . (3.1)

There is a positive real part of , corresponding to an exponentially growing disturbance, if the term in the curly brackets is positive, or when (U )2 > [(ρ1 + ρ2 )/ρ1 ρ2 ][yk + g(ρ2 − ρ1 )/k],

(3.2)

and the flow is then unstable. (If the term in the curly brackets is negative, is imaginary and the solution represents a neutral wave.) The right-hand side of (3.2) has a minimum value when the wavenumber, k = [g(ρ 2 −ρ 1 )/γ ]1/2 , = kc , say, so that the condition for instability becomes U > {[2(ρ1 + ρ2 )/ρ1 ρ2 ][gγ (ρ2 − ρ1 )]1/2 }1/2 ≡ (U )c , say.

(3.3)

If U is slowly increased past the critical value (U)c , the first waves to become unstable are those with wavenumbers near kc . The flow conditions in this example are appropriately replicated in Osborne Reynolds’ tilted tube experiment (Fig. 3.1). Further experiments (Thorpe, 1969) show that, provided the tube is tilted through a sufficiently small angle so that the subsequent flow acceleration along the tube is relatively small, waves with wavenumbers equal to kc (equal within the bounds of experimental uncertainty) do appear ‘spontaneously’, presumably originating from some small background ‘noise’ or vibrational disturbance in the laboratory tube, and subsequently grow, maintaining their original wavenumber, and break as reported by Reynolds.


87

Apart from at the air–sea interface (and in bubbles and spray), surface tension plays no part in ocean dynamics. Putting γ = 0, (3.2) shows that a non-zero difference in speed, U, across a fluid interface will result in unstability and lead to the growth of waves with sufficiently large wavenumbers, k, whatever the density difference, ρ2 − ρ1. It is useful here to divert for a moment from the problem of flow stability and to extend the results of Section 2.2 to include the effects of shear on the dispersion properties of internal waves. When γ = 0 and the two layers have finite depths, h1 and h2 , respectively, the equation for becomes ρ1 (U/2 − c)2 coth kh 1 + ρ2 (U/2 + c)2 coth kh 2 = g(ρ2 − ρ1 )/k,

(3.4)

where c = i/k, and tj = tanh khj , for j = 1, 2. If is pure imaginary and equal to −iσ , where σ is the wave frequency, then c is real and equal to the phase speed of the (neutral) internal wave, σ /k. (If there is no shear so U = 0, (3.4) reduces to the interfacial wave dispersion relation, (2.1), as it should.) For long waves (kh1 and kh2 tending to zero),5 Equation (3.4) becomes ρ1 (u/2 − c)2/h 1 + ρ2 (U/2 + c)2 / h 2 = g(ρ2 − ρ1 ).

(3.5)

When (ρ 2 − ρ 1 )/ρ 2 1 the two solutions of this equation are c± = U (h 2 − h 1 )/[2(h 1 + h 2 )] ± [(h 1 h 2 )1/2 /(h 1 + h 2 )][g (h 1 + h 2 ) − (U 2 )1/2 ],

(3.6)

with reduced gravity, g = 2g(ρ 2 − ρ 1 )/(ρ 2 + ρ 1 ). If U < [g (h1 + h2 )]1/2 , the two roots are real, equal to the phase speeds of neutral, non-growing, long interfacial waves. The group velocity of the interfacial waves, found as cg = ∂σ /∂k by differentiating (3.4) with respect to k after putting c = σ /k, tends to the phase speed as the wavelength of the waves tends to infinity. These results are referred to again in Section 12.6.1 in connection to hydraulic flows over or through topography.6 The two-layer model demonstrates that neutral internal waves are possible in some shear flows, whilst others may contain growing (or decaying) wave-like disturbances. 5 Equation (3.5) applies to long interfacial waves in a rectangular channel. A more general form can be derived by extending a result by Kelland (1840; or see Lamb, 1932, Section 169) for long waves in a single layer in a channel of non-rectangular shape. For long interfacial waves in a flow through a uniform channel of arbitrary cross-sectional shape, ρ 1 (U/2 − c)2 (b/A1 ) + ρ 2 (U/2 + c)2 × (b/A2 ) = g(ρ 2 −ρ 1 ), where A1 , A2 are the cross-sectional areas of the upper and lower layers respectively, and b is the breadth of the interface. The depths, h1 and h2 in (3.5) (or in 3.6) are now replaced by A1 /b and A2 /b, respectively. This may be useful in the study of flows through straits (Section 12.6.1). Effects of the sloping sides of channels on long interfacial waves (including nonlinear effects, but in the absence of shear) are described by Grimshaw (1978). 6 The shape of interfacial waves of finite amplitude (if they exist and are not unstable) is affected by the presence of shear. In the case of waves formed behind a density front, the lower layer spreading beneath the less-dense, stationary upper layer, the waves travelling at the speed of the lower layer will have exactly the same shape as surface gravity waves of finite amplitude, but inverted (Thorpe, 1974, appendix C). They therefore have relatively narrow troughs and wide crests, unlike those in a shallow lower layer in the absence of shear (see Section 2.2). For interfacial waves of wavelengths h 1 and h 2 , the crests are narrower than the troughs if U2 h1 /U1 h2 > (ρ 1 /ρ 2 )1/2 , where U1 and U2 are the speeds of the upper and lower layers, respectively, and vice versa (Thorpe, 1968b).

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Differences may occur depending on the range of wavenumbers under consideration or that are generated within the ocean, or the magnitude of the flow or, more generally, the shear and stratification. The model described above is of a flow that changes abruptly in depth at a discrete interface in density. Layers of different density within the ocean are not separated, however, by sharp interfaces, and in the next section it is shown that the finite thickness of interfacial layers has a substantial effect on the onset of stability. 3.2.3

General, statically stable, stratification

The stability of small, spatially periodic disturbances in a steady, stably stratified, shear flow for which the Boussinesq approximation is valid, is governed by the Taylor– Goldstein equation. The effects of the Earth’s rotation in so far as it affects the stability of flows (but not necessarily the nature of the flows themselves) are presently neglected; the processes leading directly to turbulence are of such small scale (generally much less that a relevant internal Rossby radius, LRo , see Section 3.8.2) that rotation is insignificant. A steady horizontal flow, taken as (U(z), V(z)), is perturbed by a small twodimensional disturbance with velocity components (∂ψ/∂z, v, −∂ψ/∂x), independent of y and spatially periodic in the x-direction. The governing equation, named the Taylor–Goldstein equation in recognition of its derivation and exploitation by these two scientists, has the form d2 ψ0 /dz 2 + {N 2 /(U − c)2 − k 2 − (d2 U/dz 2 )/(U − c)}ψ0 = 0,

(3.7)

where the stream function, ψ, is equal to the real part of ψ0 (z)exp[ik(x−ct)], N(z) is the buoyancy frequency, and the corresponding equation for v is ∂v/∂t + U ∂v/∂x = (dV /dz)K ψ0 exp[ik(x − ct)],

(3.8)

where again real parts are to be taken and with the condition that ψ0 tends to zero at horizontal boundaries or, in an unbounded flow, as z tends to ± infinity. (Equation 3.7 reduces to 2.3 if U = 0 and f = 0.) Here the phase speed, c, may be complex: c = cr + ici .7 If kci is positive, the solution is exponentially growing and the flow is unstable. Neutral solutions, neither growing nor diminishing, exist when ci = 0, which often (but not always) defines a boundary separating stable and unstable flows. Although several analytical solutions of (3.7) were known, no general conditions that could be applied to investigate the stability of the broad range of naturally occurring flows were discovered until the late 1950s. Two theorems that may apply in the ocean were then discovered for the case of a two-dimensional flow (i.e. one with V (z) = 0). The first, known as the Miles–Howard theorem (Miles, 1961; Howard, 1961), is that In a steady, inviscid, non-diffusive, parallel, stably stratified, two-dimensional, horizontal shear flow, small disturbances are stable provided the gradient Richardson number, Ri = N2 /(dU/dz)2 , is greater than 1/4 everywhere in the fluid. 7 This is equivalent to putting c = i/k in the previous section.


89

The Richardson number is named after L. F. Richardson (see footnote 32 in Chapter 1). This theorem provides a condition necessary, but not sufficient, for instability, namely that somewhere in the flow, Ri must be less than 1/4; the minimum Richardson number in the flow, Rimin , (often denoted in the literature as J) must be less than 1/4 for it to be possible that disturbances may grow. It introduces the idea of a critical Richardson number, Ric ≤ 1/4, such that, if Ri min < Ric , the flow is unstable, whilst if Ri min > Ric it is not. Some particular examples are mentioned in Section 3.3.5 including the case of a flow with uniform shear and constant N, for which Ric = 0 so that all small disturbances are stable, even though Ri min may be less than 1/4. The theorem implies that in a flow in which Ri min > 1/4, small disturbances will be stable and can propagate as internal waves. The special case when U is constant (i.e. independent of z) and Ri tends to infinity, is that described in Chapter 2. The second theorem, known as Howard’s semicircle theorem (Howard, 1961), is as follows. For unstable waves in flows that conform to the conditions of the Miles–Howard theorem,

[cr − (Umax + Umin )/2]2 + ci2 ≤ [(Umax − Umin )/2]2 .

(3.9)

This extends an earlier result of Rayleigh’s that if ci = 0, then Umin < cr < Umax ; unstable disturbances propagate at speeds within the range between the fastest flow speed, Umax , and the minimum, Umin . 3.2.4

Internal waves in shear flows

A categorization of internal waves by Banks et al. (1976) helps to explain the close relationship and distinctions between neutral internal waves and disturbances that grow in stratified shear flows. When Ri min > 1/4, the flow is stable and can support propagating neutral (ci = 0; neither growing nor decaying) internal waves. Their phase speed, cr , is either in the range Umin − D Nmax /[2 + (k D)2 ]1/2 < cr < Umin or Umax < cr < Umax + D N max [(2 + (k D)2 ]1/2 , where D is the depth of the fluid and Nmax is the maximum buoyancy frequency, outside the range of the mean flow speeds. These internal waves are ‘non-singular’ in the sense that nowhere in the flow does their horizontal phase speed equal the flow speed (the condition for the existence of a critical layer, Section 5.6.3).8 When Ri min ≤ 1/4, neutral, ‘marginally stable’ waves may also occur at the boundary of an unstable region in (k, Rimin ) space. If Ri min < 1/4 (i.e. when the flow may be unstable), neutral wave modes may exist with speeds cr outside the range Umin to Umax . Unstable, exponentially growing disturbances when Ri min < Ric are matched by ‘conjugate’, exponentially decaying disturbances, but these are 8 The properties of interfacial waves are different, as can be deduced from (3.6). For example, although c+ exceeds the maximum flow speeds, U/2, when (U)2 < g h2 , there is a range of speeds given by g h2 < ( U)2 < g (h1 + h2 ) (and consequently before instability set in) within which –U/2 < 0 < c+ < U/2, and the phase speed is therefore in the range of the flow speed. Even though the speed of long waves on a relatively thin interface may be accurately predicted by the speed of those on an interface, continuous stratification makes a key difference to their stability.

90


unlikely to be apparent in reality as, in unstable conditions, growing disturbances will dominate the field of motion. There is also a continuous spectrum of ‘singular’ wave modes with speeds within the range Umin to Umax . These modes have singularities in their vertical velocity component that are no worse than a discontinuity in the first derivatives, and are associated with an algebraic (rather than exponential) decay in amplitude (Drazin and Reid, 1981). Whilst the continuous spectrum may be important in the solution of initial value problems, it is not know whether they play a significant part in ocean waves or hydraulic phenomena (Pratt et al., 2000; see also Section 12.5). 3.2.5

Analytical solutions in particular cases

Among the analytical solutions of (3.7) that may be useful in their application to the ocean is an eigensolution discovered by Holmboe (1962) for U = U0 tanh(bz), N2 = N 20 sech2 (bz), corresponding to a vertically unlimited flow with a density profile of tanh(bz) form, i.e. with an interface of thickness 2/b at z = 0. (Analytical forms for neutral wave mode solutions are known when U0 = 0 and are given in Section 2.3.1). The Richardson number has a minimum value, Ri min = N 20 /(bU0 )2 at z = 0. The neutral, marginally stable solutions, ci = 0, are found to divide the growing from the diminishing disturbances, and lie on the ‘stability boundary’, Ri min = α(1 − α), shown in Fig. 3.5, where α is the non-dimensional wavenumber, k/b. Rimin reaches its maximum value, 1/4, at α = 1/2, and the Miles–Howard theorem is consequently satisfied. The wavenumber of the disturbance that first becomes unstable as shear increases or Ri min decreases is b/2 (i.e. a wavelength 4/b, about 2 times the interface thickness). Flows with values of Ri min < 1/4 are unstable over a band of wavenumbers that increases in extent as Ri min decreases, and with growth rates, kci , that vary with α and become greater as Ri min decreases. The fastest growing disturbances have zero real phase speed, cr , and so remain stationary within the flow, satisfying the conditions of Howard’s semicircle theorem. Many other analytical solutions of the Taylor–Goldstein equation are known (see, for example, Drazin and Howard, 1966; Drazin and Reid, 1981). Numerical solutions have been found for the eigensolutions, the stability boundaries, and the rates of growth of unstable waves (see, for example, Hazel, 1972). As mentioned above, Ri min = 1/4 is not always the critical value of Ric below which instability occurs. The uniform shear flow with constant buoyancy frequency, N, corresponding to a uniform density gradient, is stable at all positive values of Ri (Davey and Reid, 1977; Brown and Stewartson, 1980), and there are other distributions of U and N for which the critical value of Ri min lies between 0 and 1/4. There is no formal justification for accepting a minimum Richardson number of 1/4 as a universal critical value below which flow instability will occur or below which the flow will be turbulent although, as is explained in Section 7.3, naturally occurring flows in which Ri (based on the mean flow) is less than 1/4 are generally found to be turbulent or in the transitional state described in Section 3.3. Moreover, whilst it is perhaps intuitive that introducing a statically stable


91

Figure 3.5. Stability diagram of the stratified shear flow, U (z) = U 0 tanh(bz),with buoyancy frequency, N (z), given by N 2 = N02 sech2 (bz). The stability boundary is Ri min = α(1 − α), where Ri min is the minimum Richardson number and α is the non-dimensional wavenumber of disturbances with α = k/b, where k is the horizontal wavenumber of a disturbance to the flow. The maximum value of Rimin on the stability boundary is 1/4 at α = 1/2. Points above the boundary correspond to stable disturbances; all flows with Rimin > 1/4 are stable. Points below the stability boundary correspond unstable flows; all flows with Rimin < 1/4 are unstable for a range of disturbance wavenumbers surrounding α = 1/2. The line beneath the stability boundary marks the wavenumbers of disturbances having the fastest growth rate for given Rimin (those most likely to be observed, at least in the early linear stages of flow development with a uniform ‘noise’ level of disturbances), and the accompanying values are those of their non-dimensionalised growth rates, kci N0−1 . (From Hazel, 1972.)

density gradient to a previously unstratified shear flow will tend to make it more stable, this is not always the case. The presence of stratification allows additional modes of oscillation that may lead to instability; there are shear flows, stable in the absence of stratification, that become unstable when the fluid has a statically stable stratification. Holmboe (1962) gives a further analytical solution that provides some useful insight into the development of instability, particularly that occurring near the base of the oceanic near-surface, mixed layer, discussed in Section 9.7. He examined the effect of disturbances on an interface where the thickness of the velocity transition layer is thicker than that of the density. A specific example, one studied numerically by 2 2 2 Hazel (1972), √ is that of N = N 0 sech (az) and U = U0 tanh(bz), with a > b. For 1 < a/b < 2, the minimum Richardson number occurs at z = 0, and the neutral curve is the stability boundary. Stationary growing wave solutions are found below this boundary, and Ric = 1/4. At higher a/b, the minimum Ri occurs at |z| > 0, and instability takes the form of pairs of waves travelling in opposite directions in the relatively unstratified regions above and below the level of the density gradient maximum at z = 0. The finite amplitude form of the observed instability is described in Section 3.6 and shown in Fig. 3.17.

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3.2.6


Non-parallel mean flows

In some non-parallel, or three-dimensional, flows the motion can usefully be divided into two components. An example is that of internal inertial gravity waves in a steady mean flow, where the locally quasi-steady flow induced by the waves can be divided into components in and normal to the wave propagation direction. For this reason, (3.7) was deliberately posed in terms of a plane two-dimensional disturbance (with wavenumber in the x-direction) to a three-dimensional flow, (U(z), V (z)). For a small two-dimensional disturbance in the y-direction with stream function ψ = ψ0 (z)exp[il(y − ct)] having no x variation, (3.7) is transformed into an equation with V and l replacing U and k, respectively, and (3.8) is changed likewise, and for this disturbance the Richardson number becomes Ri y = N 2 /(dV /dz)2 . The Miles– Howard theorem holds, with disturbances in the y-direction being stable if Riy is everywhere greater than 1/4. In general the theorem will apply to two-dimensional disturbances orientated at an angle, φ, to the x-axis, where the flow has a velocity profile, U(z) cos φ + V (z) sin φ, in the direction of the disturbance, and a corresponding Richardson number, Riφ = N2 /[d(U cos φ + V sin φ)/dz]2 . All disturbances will be stable when Riφ > 1/4 for all φ. The direction of unstable two-dimensional disturbances will depend on how the value of the critical Richardson number (which may possibly be less than 1/4) varies with φ. An example relating to the instability of inertial internal gravity waves is given in Section 5.3.

3.3

The transition from Kelvin–Helmholtz instability to turbulence

3.3.1

The formation of billows

A number of studies of the development of Kelvin–Helmholtz instability to large amplitude have been made, based mainly on numerical, rather than analytical, methods (see Klaasson and Peltier, 1985, 1989, 1991; Scinocca, 1995; Fritts et al., 1996a; Cortesi et al., 1999; Caulfield and Peltier, 2000; Staquet, 2000; Smyth et al., 2001). These have provided considerable insight into the transitional dynamics when the instability becomes large and motion is no longer two dimensional, and have shown clearly the severe limitations of confining motion to lie in two dimensions rather allowing the development of the three-dimensional evolution that occurs in reality. As shown in the example in Fig. 3.6, Plate 6, it is now possible to follow the threedimensional development from the initial instability to the subsequent turbulent motion and its collapse. The discussion below, however, mainly follows laboratory studies, if only because in many cases they were the first to reveal the processes and stages that lead from instability to turbulence which are illustrated diagrammatically in Fig. 3.7. Stratified shear flows have been produced in several ways in laboratory conditions, for example as follows:

3.3 The transition from Kelvin–Helmholtz instability

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Figure 3.7. The stages in the transition from a laminar stratified shear flow to turbulence as described in the text. (From Thorpe, 1987a.)

(i) Use of a tilted tube as in Reynolds’ experiments (Thorpe, 1971b, 1973; Atsavapranee and Gharib, 1997). This results in an accelerating, spatially uniform, two-way shear flow, although if the tube is returned to the horizontal, the flow is subsequently steady with, in suitable conditions, a temporally developing instability. (ii) Unidirectional, two-layer flow through a channel (Koop and Browand, 1979) or wind-tunnel (Scotti and Corcos, 1972), the layers of different density being separated by a splitter plate at the upstream end of the flow section, and instability growing in space downstream of the splitter plate. (iii) An exchange flow in a channel between two large reservoirs containing fluids of different densities (Hogg and Ivey, 2003). (iv) The flow of a dense layer through a contraction or over a sill into a region containing a quiescent, less dense fluid (Pawlak and Armi, 1998, 2000).


94

(a)

t = 1.00 s

t = 3.00 s

t = 4.00 s

t = 5.00 s

t = 1.67 s

t = 3.33 s

t = 4.33 s

t = 5.33 s

t = 2.33 s

t = 3.67 s

t = 4.67 s

t = 5.67 s

5 cm

(b)

Figure 3.8. Billows formed at a thin interface between two layers uniform in density in tilted tube experiments. In (a) the lower layer is dyed. The scale is of length 0.45 m. (From Thorpe, 1971b.) In (b) a narrow light sheet normal to the billow crests makes visible fluorescent dye in the lower layer. Times after the tube is tilted are given beneath each photograph. Billows forming after 1 s are seen sharpen the interface in the braids between them (compare t = 1.00 s and t = 1.67 s), later to pair (t = 4.00s), and develop small-scale turbulence (t = 5.00 s). The Reynolds number based on interface thickness and velocity difference is 2150 and billows form at Ri = 0.012. (From Atsavapranee and Gharib, 1997.)

The latter simulates the instability of flow down the sloping side of the sill in Knight Inlet, British Columbia (Fig. 12.7) and represents a spatially growing instability on an accelerating flow, without the need to account for the boundary layers from above and below the splitter plate in the experiments of type (ii).9 In all cases studied so far, the mean flow is confined to one direction. 9 The differences of temporal and spatially growing waves and the effects of a tilted interface are discussed by Pawlak and Armi (1998).


95

Figure 3.9. A set of 4–5 billows measured by towing a chain of thermistors over a distance of about 500 m in the seasonal thermocline of the Sargasso Sea. Particularly notable are the tilted (getting deeper from left to right) braids between the central billow cores where the isotherms are almost vertical or overturning. The contours are at 40 mK interval, and the horizontal resolution is about 0.7 m. Depth is shown on the right. (From Marmorino, 1987.)

Both wind-tunnel and tilted-tube experiments have examined the instability of a thin shear layer across which there is a change in density, with the density transition having approximately a thickness between 1 and 0.6 times that of the velocity at the onset of instability. The initial flow Reynolds number, Re = dU/ν, based on the velocity layer thickness, d, and the velocity difference across the layer, U, at the onset of instability, ranges from about 120 to 16000. It is found that, at values of Ri min less than the theoretical value of Ric , growing waves appear that are stationary with respect to the flow at the centre of the layer, and with wavenumber parallel to the flow direction, i.e. normal to the vorticity vector of the mean flow. These waves have wavelengths, λ, close to those with the fastest growth rates predicted by the linear stability theory in two-dimensional flows and, provided their slope, 2a/λ, is small (where here 2a is the height of the waves), the growth rates are reasonably well predicted by the linear theory. By the time their slopes exceed about 0.06, the growing waves are visibly asymmetrical. Continuing to grow in amplitude but with no change in wavelength, the waves ‘roll up’, forming the rotating spiral ‘billow’ structure shown in Fig. 3.8, and stretching the narrowing ‘braid’ regions between billows (structures denoted as stage 1 in Fig. 3.7). The maximum slope (height/wavelength), reached by billows in separate experiments, decreases as the minimum Richardson number in the flow, Ri min , at the onset of instability increases towards the critical value, Ric , which, in these studies, is equal to 1/4: maximum slopes of about 0.5 are found for Ri min = 0.06, and about 0.1 when Ri min = 0.17. Structures similar to these laboratory billows are observed in the ocean as illustrated in Fig. 3.2b and by the data obtained by Marmorino (1987) shown in Fig. 3.9. Marmorino analysed measurements made by using a towed array of temperature sensors in a frontal zone in the Sargasso Sea, revealing coherent billow-like thermal


97

structures 5–10 m high. Acoustic observations of billows have been made in straits and near sills or banks (see Figs. 10.10, 12.2 and 12.3) and within packets of solitary waves on the continental shelf (Fig. 5.1). There seems little doubt that Kelvin–Helmholtz instability is of common occurrence in the ocean with billows 0.1–20 m or more in height, the size depending on the scale of the vertical variation of stratification and flow within which the billows develop. Billows have also been observed in Loch Ness by using thermistor arrays (Fig. 3.10), confirming Wedderburn’s speculation about the nature of mixing between stratified layers in the lake mentioned in Section 3.1. In contrast to the flows with thin, but finite, density and velocity interfaces, no signs of instability are found in tilted-tube experiments with a uniform density gradient, and consequently also with uniform shear, at least to values of Ri of about 0.015. This is in accordance with the theoretical prediction that the critical Richardson number, Ric , in such a flow is zero; the flow is stable to infinitesimal disturbances for all Ri > 0. Large-amplitude internal waves propagating in such a flow may, however, become unstable at a critical layer or when the flow is accelerating (see Sections 5.6.3 and 5.6.4, respectively); the flow is unstable to finite, but stable to infinitesimal, periodic disturbances.

3.3.2

Billow pairing

The stage of transition following the formation of billows on a thin interface depends on the minimum Richardson number, Ri min , of the mean flow in which the disturbances leading to billows are growing. At small Ri min , much less than Ric (1/4 in this case), billows often rotate round neighbours as illustrated in Fig. 3.7 at stage 2a, and shown in Fig. 3.8b, and amalgamate together in pairs, so ‘pairing’ and doubling the wavelength of

← Figure 3.10. Billows and their decay over a period of 40 min in the thermocline of Loch Ness, Scotland. The billows are in a mean flow of about 6 cm s−l moving past a vertical array of 13 thermistors spaced 0.45 m apart. The time axis restarts from zero 1200 s into the record, time increasing from the left. (a) The temperatures. The clustering of temperatures at times of about 400 s and 800 s from the left occurs in the overturning, nearly homogeneous, central region of billows. Abrupt temperature changes occur as the braids advect past the fixed sensors, near 200 s and 800 s from the left. (b) The corresponding depths of isotherms at 43 mK intervals. Marks on the depth scales (to the left and in the centre) show the depths of thermistors. The two billows visible in the first period of 1000 s are about 3 m in height and 26 m in wavelength, supposing that they are moving with the mean flow, giving a slope, 2a/λ, of 0.12. The steppy structures in the rising braids at times of about 0−400 s and 700−900 s are a consequence of imperfect contouring. Temperature inversions occur within the billows where the isotherms are shown as vertical lines. The subsequent temperature structure shows some remains of the tilted braids at 200–300 s and 800 s after the time change at 1200 s and the continuation of high-frequency or short horizontal-scale thermal activity, although of smaller amplitude than in the earlier billows. (From Thorpe, 1978a.)

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Figure 3.11. Plan view of billows observed by shadowgraph from above a tilted tube, 0.85 m wide and 0.06 m deep. The tube is tilted downwards at the bottom of the photographs so the vorticity vector of the mean flow is directed across the photographs. Billows are shown in the central 0.28 m wide part of the tube at times (b) 0.125 s, (c) 0.25 s and (d) 0.41 s, after (a). Billows A and B extend across the width of the region. Other billows join together along their 1ength in (b), (g and h into n, k and m into f ), subsequently forming turbulent knots. Billow e pairs with A, but only over a limited part of its length. The arrow in (d) indicates the first appearance of convective rolls. Also visible in (c) as near-vertical pairs of lines between neighbouring billows are vortex tubes (see also Fig. 3.13). (From Thorpe, 2002b.)

the dominant structure.10 If Ri min is much smaller than 0.09, this pairing may reoccur, with the already paired billows themselves pairing (see Brown and Roshko, 1974) before signs of coherent structure are lost in a disorganized turbulent flow. Pairing is less frequent at larger Ri min , perhaps disappearing entirely for a value of Ri min in the range 0.15 < Ri min < 0.2. Experiments in wide tubes reveal that this two-dimensional picture of evolution ignores the possibility of neighbouring billows amalgamating by pairing at some locations along their length. The growing waves may be non-uniform in the horizontal direction normal to the flow (i.e. along their crests), leading to billows of finite length, typically four times the distance between neighbouring billows, the ends of which amalgamate with neighbours in ‘knots’ illustrated at stage 2c in Fig. 3.7 and shown in the vertical view of billows in Fig. 3.11c. There is a violent tangle of vortex

10 The processes and stages involved in the pairing of vortices in unstratified surroundings, in particular the processes driving the centroids of the pair together and the diffusive stages of amalgamation, are described by Cerretelli and Williamson (2003).


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tubes in the knots and evidence of the development there of small-scale turbulence before it occurs in the other parts of the billows.11 A net displacement of billows from the original shear layer may occur in some accelerating flows. The experiments of Pawlak and Armi (1998, 2000) (Fig. 3.12) show the development of instability on a relatively thin sloping interface in a spatially accelerating two-layer flow in a contraction in a channel. Billows are formed that, in moving downstream, lift above the streamlines of the mean flow. A narrowing and stretching ‘streamer’ of fluid, denser than its surroundings, is left between billows (e.g. at d) where ‘braids’ occur in steady flows. A pairing process develops (e.g. at e) but it appears to be more complex than in steady flows, with disruption to the ‘streamers’ and ‘pinching off’ of the billows. Movement of billows away from a nearby plane rigid boundary is observed by Holt (1998) in experiments in a tilted tube with layers of different depths. He explains the evolution as a consequence of billow pairing: billows displaced downwards in the pairing process are impeded by the presence of the lower boundary and interact with its viscous boundary layer, leading to their rapid dissipation and leaving a dominant set of billows with upward displacement.

3.3.3

Three-dimensional evolution of billows

The axes of billows are not necessarily parallel to the vorticity vector of the mean flow. Such effects can result from the presence of topography. Billows that are inclined to the mean flow vorticity are predicted and observed in laboratory experiments in which a two-layer flow is confined between parallel upper and lower boundaries that are tilted from the horizontal in a direction transverse to that of the flow (Fig. 3.13); the depths of the two layers vary in a direction normal to their flow, although the vorticity vector of the mean flow is horizontal and normal to the flow direction (Thorpe and Holt, 1995; Holt, 1998). ‘Vortex tubes’ form on the braids between the ‘twisted’ billows shown in the figure and illustrated in stage 2d of Fig. 3.7, and lead to turbulence where the tubes link with the billows. It may be anticipated that billows occurring over topography that varies in a direction transverse to the flow (or parallel to the vorticity vector in the mean flow), will be twisted in a similar way. A non-parallel, three-dimensional shear flow will also affect the orientation of billows, as explained in Section 5.3.

3.3.4

Convective rolls within billows

A further stage of development towards transition occurs within the statically unstable fluid within the spiral structure of the billows. This has the form of rolls with axes aligned around the billow circumference, the ‘convective rolls’ shown in Fig. 3.14 and illustrated as stage 2b in Fig. 3.7. These rolls are found in both laboratory experiments (Thorpe, 1987a; Atsavapranee and Gharib, 1997) and numerical simulations of billows 11 Such billow amalgamation may lead to more intense turbulence than does the development of more two-dimensional billows (Thorpe, 2002b), but more research is needed to test this.

0

4

6

(b)

(c)

(d)

n flow

rtex cores

f mea

line o

Stream

Path of vo

(e)

Figure 3.12. Billows forming in a spatially accelerating flow down a sloping interface through a contraction in a laboratory channel. A composite figure showing (top) the splitting of vortex cores from the mean flow streamlines, together with (middle) photographs of the development of billows made through a technique using a light sheet and rhodamine fluorescence, and (bottom) the associated vorticity determined from particle imaging velocimetry of the flow field. The size of the images is approximately 10 cm × 7.5 cm, and the vorticity scale is in units of s−1 . (From Pawlak and Armi, 1998.)

−4

(a)


101

(a)

(b)

(c)

(d)

Figure 3.13. The development of a pair of billows, viewed from above, that are formed between two layers of depths that vary from top to bottom of the frames across the width of a tilted tube. The lower layer decreases in depth from the bottom of the frames to the top, and flows to the right. The upper layer flows to the left, deepening towards the top of the frame, so that the vorticity vector of the mean shear flow is downwards. The growing billows are twisted and are not parallel to vorticity of the mean flow. A ‘vortex tube’ (arrowed in part (c)) appears between the twisted billows. Turbulence at the top and bottom of frames is caused by viscous effects at the sidewalls of the tube. (From Holt, 1998.)

(Klaassen and Peltier, 1985, 1991). The distance between rolls divided by the billow wavelength decreases with the initial flow Reynolds number, Re, the ratio being about 0.15 when Re = 600, 0.1 at Re = 1000, and 0.05 at Re = 2000–3000, and they persist in the flow before it becomes fully turbulent for a time of the order of the buoyancy period, 2/N, at the centre of the initial interface.

102


(a)

(b)

Figure 3.14. The onset of convective rolls within a billow in a tilted tube viewed from above using shadowgraph. The frames are obtained at 0.11 s intervals and the vorticity vector of the mean flow is from right to left. The convective rolls cause the striations, vertical in the images, within the boundaries of the billow. The frame width is 20.2 cm. The initiation of turbulence near the right and left edges of the frames is a consequence of viscosity at the sidewalls of the tube. (From Thorpe, 1987a.)

(c)

(d)

(e)

The range of Reynolds numbers, Re, of the shear flows in which Kelvin–Helmholtz billows are observed in the ocean is about 3 × 102 − 1 × 105 , similar to that covered by the laboratory experiments. Whilst the separation scale of the ‘convective rolls’ that precede the onset of turbulence should be of order 0.02–0.2 m, there are presently no definitive observations of this structure within billows in the ocean. 3.3.5

The onset and ‘collapse’ of turbulence

No further stages of transition identified by evidence of the development of new coherent flow structures have been identified in laboratory experiments before the onset


103

of three-dimensional small-scale turbulence (stage 4 in Fig. 3.7) that generally occurs following the formation of the convective rolls. Other than the turbulence associated with knots, turbulent motion is first apparent within billows. These are sheared and amalgamate in the continuing mean shear flow to form a continuous layer of turbulent motion. ‘Secondary billows’, illustrated in Fig. 3.7, are sometimes seen on the braids during this stage of the transition (see also Fig. 10.10). Figure 3.15 shows the full sequence of transition from laminar flow to turbulence viewed from a direction parallel to the billow axes. The layer of turbulence formed in Fig. 3.15e spreads vertically by engulfing surrounding fluid, before eventually vertical motions decay (or ‘collapse’) to leave a laminated density structure within an overall velocity and density transition layer (Fig. 3.15m and n) which is appreciably thicker than that of the original flow. Collapse occurs at a time, τKH after the onset of turbulence given empirically by12 τKH ≈ 15U/(gρ/ρ0 ),

(3.10)

where the speed varies from −U/2 below the interface to U/2 above and the density increases from ρ 0 − ρ/2 above the interface to ρ0 + ρ/2 below. The residual structure in the salt-stratified laboratory experiments is composed of a field of small salinity layers, including some remaining small-scale density inversions of millimetre vertical scale with low characteristic Rayleigh and Reynolds numbers, and with a very low level of residual motion relative to a uniform shear. The mean gradient Richardson number in this residual layer of collapsed motion is about 0.32, a value for which a laminar flow is stable. The mean velocity and density profiles after the collapse of turbulence are generally similar to each other. However, Thorpe (1973) and Koop and Browand (1979) found a continuous mean flow and density structure across the transition layer in laboratory experiments, unbounded by higher gradients of velocity or density, whilst observations in the atmosphere by Browning and Watkins (1970) and Chapman and Browning (1997, 1999) found that the original layer is split (‘layer splitting’) after the decay of billows, to produce a layer of more uniform density bounded above and below by relatively narrow layers of large density gradient and shear. Smyth and Winters (2003) also report numerical experiments with Ri min = 0.15 that lead to a final density gradient with a two-peak structure. Scinocca’s (1995) numerical model predicts that two layers form when there is a gradient in density that extends beyond the unstable shear layer, allowing a possible loss of energy through the vertical radiation of internal waves from the turbulent layer. Whilst it should be noted that the conclusions of the laboratory, numerical and atmospheric results are based on rather few studies, the differences, if real, may be a consequence of wave radiation, different Prandtl numbers,13 leading to different final stages of mixing, or of the much higher Reynolds number of the atmospheric billows, typically order 107 , compared with the order 103 in the laboratory 12 See also footnote 15 in Chapter 6 for further reference to τ KH . 13 The Prandtl number, Pr = ν/κ, where κ is the relevant diffusivity coefficient, is about 0.73 for air, 7 for heat stratified water and about 670 for salt-stratified water.

Figure 3.15. The transition to turbulence in a stratified shear flow in a tilting tube. The frames show shadowgraphs (in negative) of flow with a narrow density and velocity interface with: (a) laminar flow; (b) and (c) the generation of turbulence through Kelvin–Helmholtz billows; (d) and (e) their amalgamation and spread; and (g)–(i) the subsequent collapse of turbulence to leave, (m) and (n), a finely layered flow. The Richardson number of the flow in (a) is about 0.06 and the tube is subsequently horizontal so the mean flow beyond the turbulent region is non-accelerating. (From Thorpe, 1971b.)

3.4 Unstratified shear flows

105

and numerical studies. The numerical experiments of Caulfield and Peltier (2000) find, however, that, whilst a more layered structure like that in the atmosphere results from a transition when the minimum initial Richardson number, Ri min , is about 0.05, the more continuous structure of the laboratory experiments results from both smaller and higher values of Ri min . Caulfield and Peltier also remark on the laminated fine-structure of the density remaining within the transition layer, concluding that ‘mixing induced by flow breakdown within a stratified shear flow leads inevitably to a layered density structure’ (see also Smyth et al., 2001). The consequences and implications of the mean and fine-structure are discussed further in Sections 3.5 and 7.2.4.

3.4

Unstratified shear flows

Although many shear flows occurring in the ocean are accompanied by stratification, some, like that in a mixed layer with little or no vertical buoyancy flux or the example shown in Fig. 3.4 (Plate 5), are not. Transition in an unstratified mixing layer appears to develop differently to a stratified layer. Laboratory experiments have been made in a water tunnel, with two layers, separated at the upstream end by a splitter plate, driven to move at different speed. Unlike the tilted-tube experiments, where growth occurs simultaneously along the length of the tube (a temporal development), in these experiments instability begins just downstream of the splitter plate and develops with distance (a spatial development). Billows grow in the first stage of instability as described above, but a second instability is found in the form of an array of streamwise vortices of alternating sign, periodic in a direction parallel to the billow crests (i.e. in the spanwise direction) and growing on the highly sheared regions (the ‘braids’) between billows. Bernal and Roshko (1986) find the separation of these vortices to be approximately equal to the thickness of the mixing layer, independent of the ratio of the velocities of the two layers.14 The streamwise vortices observed in these studies are relatively of much larger scale than the convective rolls appearing in the stratified billows at stage 2b in Fig. 3.7. The vortex tubes forming on the braids between billows in the stratified flow experiments make acute angles with the lines of the crests of the billows as in Fig. 3.13, and interact with them to form ‘knots’, like those observed in the interactions between the billows (Fig. 3.11).

14 In experiments made with helium and nitrogen in a wind tunnel, the separation is also found to be independent of the density ratio, 7, for these gases. In experiments by Lasheras and Choi (1988) the wavelength of the streamwise vortices is controlled by corrugations on the splitter plate or indentation of its edge. They identify the subsequent stages of instability as a stage of the development of waves along the axes of the billows induced by interaction with the streamwise vortices, followed by a stage of tangling of the two sets of vortices and eventually the pairing and growth of coherent billows in the, now turbulent, mixing layer unconstrained by stratification described by Brown and Roshko (1974; see Fig. 1.12).


106

3.5

Energy dissipation in stratified flows and the efficiency of mixing

Kinetic energy of the mean flow is lost in the time, τKH , given by (3.10) between the onset of turbulence in the stratified mixing layer and its decay, whilst the potential energy of the mean stratification increases. To obtain a measure of the energy involved in mixing and the differences in the final states described in Section 3.3.5, we suppose that the velocity (U) and density (ρ) profiles just before the onset of instability (subscript I) and after turbulence has decayed (subscript F) are similar in form and are given by the piecewise continuous linear functions of z illustrated in Fig. 3.16 and defined in the caption. The coordinate z in this model is measured from the centre of the velocity and density interfaces. A layer of thickness 2h (Fig. 3.16a) grows during the transition to one of overall thickness, H + d, with a uniform layer of thickness 2(H − d) in its centre (Fig. 3.16b). A parameter q is defined as q = d/H, measuring the degree of homogenisation that occurs. As shown in Fig. 3.16c, when q = 1 the final velocity and density profiles are continuous at |z| = H + d = 2H, ‘spreading’ the initial interface as found in the laboratory experiments, whilst if q = 0 the central region around z = 0 is uniform with jumps at z = ±H to the uniform exterior values, an extreme case of ‘layer splitting’. The initial Richardson number is RiI = 2gρh/[ρ 0 (U)2 ] and is taken to be less than the critical Richardson number which, for the profiles of density and velocity shown in Fig. 3.16a, is known to be 1/4. The final Richardson is RiF = 4gρd/[ρ 0 (U)2 ], and is made equal to 0.32 to conform to the laboratory experiments. There is a loss in kinetic energy, KE, calculated by comparing the the initial and final states, of KE = ρ0 (U )2 (3H + d − 2h)/12,

(3.11)

and a gain in potential energy, PE, of PE = gρ[3H 2 + d 2 − h 2 ]/6.

The ratio PE/KE can be written as a function of q, RiI and RiF :

PE/KE = Ri F2 (3 + q 2 ) − 4q 2 Ri I2 {2q[Ri F (3 + q) − 4q Ri I ]}.

(3.12)

(3.13)

This is a measure of the efficiency of the instability in mixing by transferring kinetic energy into potential energy, and is plotted in Fig. 3.17a as a function of RiI . The thickness of the homogeneous layer increases with the efficiency ratio for a given initial Richardson number. Apparently the atmospheric billows that produce a more uniform final central region are more efficient than those of the billows in the laboratory. In the absence of internal wave radiation (and vertical radiation is excluded in this model example because the upper and lower layers are unstratified15 ), the difference, 15 Localized instability might result in internal wave radiation along the stratified layer, but this is not accounted for, nor has its possible importance been quantified in the ocean.

3.5 Energy dissipation in stratified flows

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Figure 3.16. Sketch of the initial and final velocity (UI and UF ) and density (ρ I and ρ F ) profiles before and after Kelvin–Helmholtz instability. The velocity (UI and UF ) and density (ρ I and ρ F ) distributions are as follows. Initially (as shown in a): UI = −zU/2hif −h ≤ z ≤ h, and −zU/(2|z|) if |z| > h; ρI = ρ0 − zρ/2h if −h ≤ z ≤ h and ρ0 − zρ/(2|z|) if |z| > h. Finally (as shown in b): UF = 0, if |z| < H − d, and −zU/(2|z|) if |z| > H + d, UF = (H − d − z)U/4d if H − d ≤ z ≤ H + d, and −(H − d + z)U/4d if −(H + d) ≤ z ≤ −(H − d); ρF = ρ0 , if |z| < H − d, and ρ0 − (zρ/(2|z|) if |z| > H + d, ρF = ρ0 − (z − H + d)ρ/4d if H − d ≤ z ≤ H + d, and ρF = ρ0 − (z + H − d)ρ/4d if −(H + d) ≤ z ≤ −(H − d). Part (c) shows the two limiting cases: q(= d/H ) = 1, representing ‘layer spreading’, and q = 0, total ‘layer splitting’.


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(b)

3.5 3

(a) 0.5

q = 0.5 ∆PE ∆KE

0.4

2.5 96g∆rD r02(∆U )4 2

q = 0.75 0.3

q=1

1.5

0.2

1

0.1

0.5

q = 0.5

q = 0.75

0.05

0.1

0.15 RiI

0.2

0.25

q=1 0.05

0.1

0.15 RiI

0.2

0.25

Figure 3.17. The effects of layer ‘spreading’, q = 1, or ‘splitting’, q < 1, caused by Kelvin–Helmholtz instability. (a) The ratio of the change in potential to kinetic energy of the mean flow, PE/KE, as a function of the initial Richardson number, Ri I , for various values of q. (b) The non-dimensionalized loss of energy to turbulence, 96gρ D/[ρ 0 (U )2 ]2 , as a function of RiI for various values of q.

(KE−PE), is equal to the total energy dissipated in turbulence, D, during the transition from laminar flow, through instability and turbulence, to the final relaminarization as turbulence collapses. A non-dimensional form of the dissipation, 96gρD/[ρ 0 (U)2 ]2 , is shown in Fig. 3.17b.16 For given values of q, this decreases with increasing initial Richardson number, RiI , but, at a specified value of RiI , more energy is dissipated in the processes of layer splitting than in simply spreading the interface. Although unjustified assumptions are made in choosing the velocity and density profiles in Fig. 3.16 and in imposing the condition RiF = 0.32, the figure indicates that the form of the density and velocity structure following billow events may be a useful indicator of the efficiency of mixing and the related dissipation of energy by turbulence caused by Kelvin–Helmholtz instability, notably those when q = 1 that correspond most closely to the laboratory results. Laboratory values of the efficiency, PE/KE, vary from 0.212 to 0.098 (with an uncertainty of about 0.04) as RiI varies from 0.06 to 0.14 (Thorpe, 1973). The efficiency of mixing is more commonly defined as the ratio of the rate of increase of potential energy to the rate of loss of kinetic energy. Values tabulated by Fringer and Street (2003) depend on the process leading to mixing and range from 0.05 for mixing produced by an oscillating grid in a stratified fluid to 0.5 in the Rayleigh– Taylor instability described in Section 1.4. The majority of values are in the range from about 0.15 to 0.35.

16 The energy dissipated is D = {[ρ 0 (U)2 ]2 /(96gρ)}(2RiF – 8RiI − 3Ri F2 + 4 Ri I2 + 6RiF / q − 3Ri F2 /q2 ).

3.6 Holmboe instability

109

(a)

(b)

Figure 3.18. Holmboe instability on the coloured interface between two uniform layers in relative motion. In (a) the instability occurs only on the upper edge of the salt-stratified interface, the periodic disturbances ejecting thin wisps of denser fluid from the interface into the upper layer and propagating to the left with the flow in that layer. Later, in (b), this instability has resulted in mixing at the upper edge of the coloured interface, whilst Holmboe instability is just commencing on the lower edge, with periodic disturbances moving to the right past those still visible and moving to the left on the upper interface, and carrying less dense fluid downwards from the interfacial region. (From Thorpe, 1968a.)

3.6

Holmboe instability

A different type of billow structure can occur when the centres of the velocity and density interfaces are coincident but the ratio of the thickness of the velocity interface to that of the density, R, is greater than about 2, the precise value depending on the actual profiles of velocity and density. As mentioned in Section 3.2.5, Holmboe (1962) was the first to study such flows, using analytical methods. The conditions favouring this form of instability occur in the laboratory because of the more rapid diffusion of momentum through viscosity, ν, than density by heat or salinity (proportional to the relatively smaller coefficients, κ T or κ S , respectively). In this case billows are observed in the shear flows above and below the centre of the interface, moving with the flow at their level and therefore, in tilted-tube experiments, in opposite directions. Their form is shown in Fig. 3.18. The Holmboe instability has been subject to numerical modelling by Smyth and Peltier (1989, 1990) and Smyth and Winters (2003), and observed in laboratory experiments (Thorpe, 1968a; Browand and Winant, 1973; Lawrence et al., 1991; Pouliquen et al., 1994; Strang and Fernando, 2001). The initial instability appears to be closely two dimensional, but in their two-layer channel flow Lawrence et al. found that, as the billows develop further, transverse three-dimensional perturbations develop and pairing appears possible.17 Although at first sight it appears possible that the effect of the billows might be to sharpen a density interface by mixing at its edges, Smyth and Winters find no evidence 17 The effects of the presence of horizontal boundaries on Holmboe instability are studied by Haigh and Lawrence (1999). The consequences of non-coincident density and velocity interfaces are examined by Lawrence et al. (1991).

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of this. Both density and velocity interfaces are thickened by the instability in their numerical study made with Pr = 9. What is apparent is that, although the instability grows less rapidly than Kelvin–Helmholtz instability, it does appear to be almost as effective in its transfer of kinetic energy into potential energy and consequently in promoting mixing. Hogg and Ivey (2003) describe the transitions between Kelvin–Helmholtz and Holmboe instabilities as the velocity to density length scale ratio, R, varies. They define a Richardson number, Ris = g d/(U)2 , to describe the flow, where g is the reduced gravity, d is the thickness of the velocity interface and U is the difference in the velocity across the interface. They show that according to linear theory, for certain velocity and density profiles and values of R such that Holmboe instability is possible, Kelvin–Helmholtz instability grows faster than Holmboe if Ris is small, but there is a range of larger Ris in which Holmboe instability grows more rapidly than Kelvin–Helmholtz, whilst at sufficiently large Ris only Holmboe instability is possible. Their laboratory experiments in an exchange flow between two reservoirs of fluid are qualitatively consistent with the conclusions of linear theory, but there are problems – that also occur in the ocean – of suitably defining an appropriate ‘initial state’ in an evolving flow that is already affected by the growing unstable wave disturbances. Whilst conditions favourable for Holmboe instability appear likely to be common, for example at the base of the surface mixed layer described in Section 9.7, there appear to be no definitive observations of its occurrence in the open ocean, although there is evidence for its presence in highly stratified river mouths (Yoshida et al., 1998).

3.7

The shape of billow patches and the length of billow crests

Other than the photographs of billows by Woods (1968) little is known of the horizontal structure of billows in the ocean. The development of Kelvin–Helmholtz instability in the ocean is, however, very similar to that in the atmosphere (De Silva et al., 1996), where small localized patches and sometimes very extensive trains of billows are observed. It is likely that, in many cases, the size of the patches of billows depend on the scale of the processes, such as the topography or internal waves, which lead to flow acceleration and a local reduction in Ri. When the region of low Ri is very extensive, Smyth (2004) shows that a patch of billows triggered by a local disturbance will grow with a horizontally elliptical shape, the eccentricity depending on the Richardson number at the onset of billow growth, and size growing linearly in time. Patches are predicted to be nearly circular when the minimum Richardson number, Ri min , is just sub-critical and elongated in the flow direction if Ri min Ri c . It is remarkable that the billow clouds formed in the atmosphere by Kelvin– Helmholtz instability sometimes appear to be continuous along their length for distances much exceeding four times their wavelength, contrary to the conclusions drawn

3.7 The shape of billow patches

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Figure 3.19. The evolution of long-crested billows (indicated in plan form as thin lines) by a linear disturbance (heavy line) moving with velocity ud . (a) The billows have just formed behind the moving disturbance. (b) After a time, t, the disturbance has moved a distance ud t and the original billows have provided finite disturbances and sites for the evolution of the subsequent billows. The overall pattern is one in which there are ‘young’ billows near the moving disturbance and the age of the billows increases with distance along their length.

from laboratory experiments in Section 3.3.2. The theory of Kelvin–Helmholtz instability accounts for the regular spacing of billows but does not explain how very long crests might be formed.18 A mechanism that might lead to the formation of extensive coherent billows which is absent in the laboratory experiments is that of billow generation by a linear moving disturbance, for example a front, with a horizontal component of motion that reduces the minimum Richardson number, Ri min , to a sub-critical value, Ri min < Ric , as it passes. If the crests of growing billows are not parallel to the disturbance then, as this linear disturbance advances with speed, ud , it extends the region of sub-critical Ri min where billows are growing, as sketched in Fig. 3.19. The wavelength and orientation of the fastest-growing, newly developing billows will be similar 18 The reader may call to mind an image of surface waves approaching a long regular beach. These are sometimes observed to break almost simultaneously, forming long lines of breakers, reminiscent of, but quite distinct from, the long, fairly regular, billow clouds. The processes leading to long crests in the two cases are different. The waves arriving at a beach are usually swell formed in and radiated from a distant storm. Having spread in a roughly circular pattern (like ripples from a stone thrown into a tranquil pond or from raindrops on a puddle), they reach the beach, sometimes several thousand kilometres from their source (Snodgrass et al., 1966), propagating as long-crested wave groups, and are locally almost two-dimensional plane waves. If approaching a beach at right angles they arrive, and so break, almost simultaneously. In the absence of other long-period waves, the crest length of the breaking waves is then often determined by the presence of offshore sand bars and the uniformity of beach slope.


112

to those already present unless the stratification or shear changes along the path of advance of the linear disturbance. The existing billows will act, however, as a finite perturbation that fixes the location of those developing. In this way sets of coherent billows are formed with an extent that depends on the product of ud and the lifetime of billows, typically 1–10 min in the ocean. The ‘age’ of billows produced in this way will vary along their crests, being ‘young’ wave-like disturbances at the end of their crests near the moving disturbance and ‘old’, well-developed billows undergoing a transition into three-dimensional turbulence, at the end furthest from the disturbance. Measurements of, say, temperature or density, made rapidly and horizontally through such a structure, may first detect old or young billows or, if made parallel to the disturbance (e.g. an advancing front that reduces the Richardson number below Ric ), billows of similar age. In summary, any process that translates the region of Ri min < Ric through the fluid in a direction other than normal (i.e. at right angles) to billow crests may allow existing billows to act as finite perturbations in the subsequently instable region, so extending their length. The billows observed by Woods and shown in Fig. 3.2 are caused by shear generated near the crest of an interfacial wave. The billows are orientated in lines normal to the direction of propagation of the disturbance (the wave) that leads to the reduction of Ri, and so the mechanism described above for extending the billow crests does not operate. In this case the horizontal crest lengths of the billows may be limited to about four wavelengths as in the laboratory experiments, or related to the coherence of the internal wave crest at which the billows are formed and the consequent extent of the region in which Ri min < Ric . At first sight, internal inertial gravity waves appear likely candidates to produce long coherent billows because they generate a rotating shear and this may lead to growing disturbances and subsequent billows with wavenumbers that are not parallel to the direction, x, of wave propagation. However, as explained in Section 5.3, the disturbances that are most likely to grow first and become dominant have wavenumbers in the x-direction. Like the internal waves travelling along sheets in the thermocline and observed by Woods, inertial waves therefore appear unlikely to generate billows with very long crests by the proposed mechanism.

3.8

Instability in a rotating ocean

3.8.1

Shear flow

The Earth’s rotation affects the occurrence of instability. The Rayleigh criterion for the instability of an unstratified circular vortex is d/dr[(r v)2 ] < 0 (Drazin and Reid, 1981), where v(r) is the azimuthal velocity in non-rotating coordinates and r is the radial coordinate from the vortex centre. In a fluid rotating about a vertical axis with angular velocity 2f (f is the Coriolis frequency) Munk et al. (2000) show that this

3.8 Instability in a rotating ocean

113

criterion can be transformed into a condition for the instability of a shear flow, U(y) in direction x, measured relative to the rotating frame: dU/dy < − f.

(3.14)

Laboratory experiments by Bidokhti and Tritton (1992) show that the form of the instability is one of developing waves that roll-up into cyclonically rotating eddies similar in appearance to those of Kelvin–Helmholtz billows. The form of instability may account for the predominently cyclonic spiral eddies of 10 km scale photographed from space from the US Space Shuttle Challenger Mission by Scully-Power that are described by Munk et al. (2000) and shown in Fig. 13.5. 3.8.2

Baroclinic and barotropic instability

A full discussion of the effects of rotation on large-scale oceanic flows is beyond the scope of this text. However, in view of the importance of rotation in the break-up of thermal plumes and convective columns in deep convection described in Section 4.5, and in the dynamics of mesoscale eddies and Rossby waves referred to in Chapter 13, brief mention is appropriate, particularly in relation to instability. The most common form of instability at a scale in which the flow is affected by rotation is baroclinic instability, that of flow perturbed from a state of geostrophic balance, one in which the flow conforms to the thermal wind equation f dU/dz = − (g/ρ0 )dρ/dy.

(3.15)

(Here dU/dz is the vertical gradient of the flow, U, in the horizontal x-direction and dρ/dy is the density gradient in the direction at right angles, the horizontal y-direction.) Instability is found with waves and subsequent eddies of scale comparable to the internal (or baroclinic) Rossby radius, LRo . This is the horizontal scale, sometimes called the ‘internal Rossby radius of deformation’, at which rotational effects become as important as those of buoyancy (see Gill, 1982, section 7.5). In the gravitational collapse and intrusion of a mixed layer into a stratified region, it represents the horizontal scale at which spreading along isopycnals is impeded by rotational effects, and it therefore characterizes the horizontal scale of a spreading region at which adjustment takes place in the flow field tending towards the establishment of a geostrophic balance. It is defined as c/f, where c is the speed of long internal waves, determined as an eigenvalue of the equation d2 ϕ/dz 2 + (N 2 /c2 )ϕ = 0,

(3.16)

with ϕ = 0 at z = 0 and h.19 The speed of such waves is different for each wave mode and there is therefore an internal Rossby radius for each mode. However, waves of the first mode travel fastest, and it is conventional to define the internal Rossby radius in 19 This may be derived from (2.4) when k tends to zero, f σ and c = σ /k.


114

respect of this mode. It is given by Nh/ f in water of depth, h, and constant buoyancy frequency, N. Typical values of LRo are 5–10 km at 60◦ , 30–40 km at 30◦ and 200 km at 5◦ latitude (Emery et al., 1984; see also Stammer, 1997, although using a formula different from (3.16)). Barotropic instability (Gill, 1982; section 13.6) involves free-surface motions at scales comparable to the barotropic Rossby radius, (gh)1/2 /f, where (gh)1/2 is the speed of long surface waves in an ocean having the local depth, h. Typical values range from about 1000 km at 60◦ N to 20 000 km at 5◦ N. These are comparable to the dimensions of ocean basins and are also those in which the Earth’s curvature becomes important through the variation of f with latitude. The dispersion relation (relating frequency to wavenumber) of Rossby waves, referred to in Section 13.5, also depends on the Earth’s curvature. The effects of curvature are represented by a term usually symbolized as β (not to be confused with the angle of inclination of internal wave rays to the horizontal defined in (2.9)), equal to the meridional gradient of f, β = 2 cos φ/R,

(3.17)

where is the angular rate of rotation of the Earth, 2/(1 day) = 7.27 × 10−5 s−1 , ϕ is the latitude and R is the radius of the Earth, about 6370 km. The ‘β-effect’ is the variation of the Coriolis frequency, f, with latitude.

Chapter 4 Convective instabilities

4.1

Introduction

Convection occurs in the upper ocean as a consequence of the development of static instability during periods of surface heat loss or salt rejection as ice is formed on the sea surface, leaving dense salty water beneath. It also occurs in the process of geothermal heat transfer from the solid Earth through the seabed, and in processes involved in internal wave breaking. The turbulent heat flux, H, per unit area is given by H = ρ0 cp wT (Section 1.7.4), where T is the temperature variation from a mean or ambient value, w is the velocity component normal to the surface, and the average (. . . ) is taken over unit area. From the equation of state (1.1), the variation in density corresponding to T is ρ = −ρ0 αT (salinity changes being zero). The buoyancy flux, B = −(g/ρ0 )wρ , is therefore related to the heat flux by B = gα H/(ρ0 cp ).

(4.1)

A convective heat flux (or likewise salinity flux) results in buoyancy flux and consequent buoyancy forces that drive convective motion. It seems unlikely that buoyancy was the only process driving fluid motion in the laboratory experiments made by Bénard (1900) in a variety of very shallow (1 mm deep) liquid layers heated from beneath, since variation in surface tension on the free surface may also have contributed. The experiments nevertheless vividly demonstrated the development of regular flow patterns, and were instrumental in stimulating Lord Rayleigh to devise a theory (Rayleigh, 1916) which concludes that only if a certain parameter (now known as the Rayleigh number) exceeds a critical value can instability set in through convection in a thin fluid layer that is bounded above and below by plane

115

116


horizontal rigid surfaces. The theory of convection in thin layers, discussed further in Section 4.2.1, is helpful in establishing the nature of the processes that may determine and control convection in a broader context. The more common situation in which static instability is of recognized importance in driving convection in the ocean is, however, in relatively deep, mixed layers. As explained in Section 4.2.2, the nature of convection differs from that in a thin layer, being characterized by transient plumes.1 The buoyancy flux associated with the cooling of the sea surface in wintertime leads, in extreme conditions, to convection reaching many hundreds of metres below the sea surface. It is evident that convection arising from a flux of buoyancy across a surface may strongly affect the motion field in the adjoining fluid layer. The nature of turbulent flow in boundary layers is therefore introduced in this chapter before the more detailed description of ocean boundary layers in Chapters 8–10. Heat produced in the Earth’s core passes as a geothermal heat flux through the sediments and the seabed into the overlying water. Its effect is of little dynamical importance on the water column over much of the ocean bed, for example the abyssal plains. Things are, however, different near hydrothermal vents! The presence of an active circulation of seawater through the newly formed oceanic crust in the midocean spreading centres, where new sea floor is being formed, was first proposed in 1973 following earlier speculation about thermal springs, and was subsequently supported by observations of locally increased, near bottom water temperature. It was not, however, until the dives of the submersible Alvin on the Galápagos Rift in 1977 described by Corliss et al. (1979) that the nature of the hydrothermal vent fields and their extraordinary attendant biological communities became apparent. In these regions at depths of some 3000 m, groups of vents, each vent typically of order 0.5 m across, emit very hot fluid, sometimes 350 K greater in temperature than their surroundings (Fig. 4.1, Plate 7). The total rate of flux of water through oceanic vent systems is estimated to be about 12 Sv (see Huang, 1999; 1 Sv = 1 × 106 m3 s−1 ), sufficient to replace the total ocean volume of 1.36 × 1018 m3 in about 3600 years, a time short in geological terms. Discharge from individual vents may persist for months if not years. The resulting convective plumes can rise for hundreds of metres above their source and have influence tens of kilometres away.2 A quite different kind of convective instability has a widespread effect in the ocean. The presence of salt in seawater and its contribution to density through the equation of state provides the mechanism for instability. Its discovery derives from the concept of a ‘perpetual salt fountain’ proposed by Stommel et al. (1956). They imagined what would happen if a vertical, heat-conducting tube was located in a region of the ocean 1 Even though the water surface may act as an effective upper boundary, the thin ‘cold skin’ layer at the ocean surface described in Section 1.5.5, and which may occur even in conditions of a net flux of heat from the atmosphere into the ocean, has no limiting lower boundary, and the convective motion resulting from static instability in the cold layer is therefore not confined to the thickness of the layer. 2 See Section 4.5.1. General descriptions of the dispersion of fluid from hydrothermal vents are given by Lupton (1995) and by Helfrich (2001).

4.2 The onset of convective motion

117

Figure 4.2. Fluid confined between two horizontal plates separated by a small distance, d, the lower at a temperature T higher than the upper so that the density, ρ, increases upwards and the fluid is statically unstable. The form of rising or descending plumes as convection sets in when the Rayleigh number, Ra, is marginally greater than the critical value, Rac , is sketched.

in which the upper layers are saltier and warmer than those below, as often is the case. If given an upward displacement, water within the tube would become warmer through the heat conducted through the tube walls from the surrounding seawater but, retaining its lower salinity (salt being unable to pass through the tube walls), would, by the equation of state, be less dense than the surroundings. Water in the tube would therefore be at a lower hydrostatic pressure (pressure given by the acceleration due to gravity, g, times the integral of density in depth) than the surroundings. Driven by the higher external pressure at the bottom of the tube, it should continue to rise, sustaining the upward motion indefinitely, or at least until the favourable ocean temperature and salt gradients were removed. (If, however, the water in the tube were first moved downwards instead of upwards it would, by the same argument, continue to sink!) Whilst this idea proved inadequate to provide power to drive a ‘perpetual mechanical engine’, it led to the rediscovery of double diffusive instability and to the recognition of its importance in the ocean, as described in Section 4.6.

4.2

The onset of convective motion

4.2.1

Convection in thin layers and the effects of shear

Figure 4.2 is a sketch showing a fluid of molecular kinematic viscosity, ν, and molecular diffusivity, κ T , confined between two horizontal plates separated by a distance, d, the lower at temperature, T, greater than the upper, the conditions examined by Rayleigh in his study of instability. The configuration is similar to that in Fig. 1.6, but here the upper plate is at the lower temperature and the resulting stratification is statically unstable with density increasing with height, z. (The coefficient of expansion, α, in the equation of state (1.1) is supposed to be positive.) In laboratory experiments it is usual to drive convection by heating the lower plate, it being easier to steadily and uniformly heat a plate at the bottom of a tank than to cool one at the top, the opposite of the ocean surface mixed layer where it is cooling at the upper surface that drives convection. The two cases are, however, dynamically similar.


118

The non-dimensionalized equations of motion for small disturbances can be reduced to one in the vertical component of velocity, w: (∂/∂t − ∇ 2 )[(1/Pr )∂/∂t − ∇ 2 ]∇ 2 w = Ra∇12 w

(4.2)

with boundary condition w = 0 at z = 0, d, where the Laplacian operators are ∇ 2 = ∂2 /∂x2 + ∂2 /∂y2 + ∂2 /∂z2 and ∇12 = ∂2 /∂x2 + ∂2 /∂y2 , lengths are scaled with d, time with d2 /κ T , velocity components with κ T /d, and temperature fluctuations with αT, and the Prandtl number, Pr = ν/κ T .3 If the Rayleigh number, Ra ≡ gαT d 3 /νκT ,

(4.3)

is greater than a critical value, Rac , the fluid is found to be convectively unstable with small disturbances growing in time. The ‘critical Rayleigh number’, Rac , is 1708 if the horizontal boundaries are ‘non-slip’ so that the flow speed there is zero, and 657.5 if the boundaries are ‘free’, so that the fluid can move along (but not through) them. The magnitude of Ra may be regarded as a measure of the effect of buoyancy forces in driving convective motion. Fluid near the lower boundary becomes less dense when its temperature is raised by heat conduction from the lower plate, and similarly more dense at the upper boundary: the buoyancy, gαT, appears in the numerator of Ra. Motion is resisted by viscous forces and so ν appears in the denominator of Ra. At first sight it may be surprising that κ T is also in the denominator, but high heat transfer reduces the thermal gradients leading to buoyancy forces and consequently diminishes their effect.4 For values of Ra just greater than Rac , the convective motion is one of stationary cells, but as Ra is increased a variety of patterns, some oscillatory, are found, their form depending on Pr. Motion becomes chaotic and turbulent at very large, highly supercritical, values of Ra. Transient, unbounded, unstably stratified, near-horizontal layers may develop in the convectively unstable breaking internal waves described in Section 5.2 or possibly, following Kelvin–Helmholtz instability, during the process of collapse of the sheared turbulent layer described in Section 3.3.5. To illustrate the onset of instability in this situation, suppose density varies in z as ρ 0 [1 − (N2 /g)z + c sin K z], with constant mean buoyancy frequency, N, and vertical layer wavenumber, K. The constant c is chosen so that r = N2 /gKc < 1, ensuring that the density profile has layers of static instability (e.g. dρ/dz > 0 at z = 0; Fig. 4.3). An equation similar to (4.2) is found to apply in this case, but with the coefficient of w on the right-hand side depending on z. 3 Drazin and Reid (1981) give a description of the nature of the unstable flow and temperature fields, and derive (4.2); an identical equation applies for the temperature fluctuations. The equation simplifies considerably if solutions are constrained to be of the form w = w0 e pt sin kx sin ly sin mz, with growth rate, p = 0, at marginal stability. 4 The form of Ra is not apparent simply from dimensional arguments, since the right-hand side of (4.3) could be multiplied by, say, any function of the non-dimensional Prandtl number. An alternative energy approach to stability makes the role of conduction more apparent. For instability and motion to occur, the rate of production of potential energy through heating must overcome the dissipation of kinetic energy of the induced motion by viscosity and the loss of potential energy though the removal of the density gradients through conduction.

4.2 The onset of convective motion

119

Figure 4.3. The form of the density profile, ρ = ρ 0 [1 − (N2 /g)z + A sin Kz], with constant mean buoyancy frequency, N, and fluctuations with vertical wavenumber, K. The constant, A, characterizing the amplitude of the fluctuations, is such that r = N 2 /gK A < 1 so that the density profile, although with mean density decreasing upwards, contains layers of static instability characterized by the distances, d0 < /K, but which extend over vertical distances, D, that may, as shown, exceed 2/K. (From Thorpe, 1994b.)

Although the fluid contains no boundaries and has layers that are statically unstable, it is dynamically unstable only if a Rayleigh number, Ra1 , equal to gc(1 − r)/K3 νκ T , exceeds critical values that depend on r and Pr (Thorpe, 1994b), when convective motions set in. A shear flow (e.g. that remaining after Kelvin–Helmholtz instability) is found to stabilize the statically unstable layers, instability beginning only at values of Ra1 greater than those in the absence of shear, and with the growth of vortices with axes aligned in the direction of the flow. A fluid containing statically unstable layers of small thickness is not necessarily unstable to convective motion. The form of instability found in statically unstable shear flows is commonly as described above, with vortices having axes aligned in the mean flow direction (‘streamwise vortices’) and consequently transverse to the vorticity vector of the mean flow. An example is illustrated in Fig. 5.7b. The form of instability therefore differs from that usually found in Kelvin–Helmholtz instability of a stably stratified shear flow where, in the absence of the effects of boundaries, billows grow with axes aligned transverse to the mean flow and parallel to the mean flow vorticity. 4.2.2

Convection in deep layers

In the ocean, convection is most commonly associated not with thin layers, but with cooling at the sea surface in relatively deep water. Townsend (1959) made laboratory

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experiments to study the nature of convection in such conditions although, having the atmosphere in mind and because it was simpler to do it, generated convection by heating the base of a deep tank of water. (Recall that the two situations, surface cooling or bottom heating, are dynamically similar.) The results of his measurements of fluctuations in temperature, and its space and temporal derivatives, are consistent with a scaling in which the depth of the tank, provided it is sufficiently large, has no effect; unlike that in thin layers, the nature of the motions driven by convection is not dependent on the fluid depth. Near the boundary there are periods of great activity in temperature, alternating with periods of relative quiescence, the active periods being those of turbulent convection and the calmer periods typical of the turbulent motions distant from the boundary. Convection occurs after buoyant fluid formed near the warm base breaks away either as a rising blob of fluid (or ‘thermal’) or as an unsteady plume. In contrast to convection in a thin layer at moderately supercritical values of Ra, motion is unsteady and intermittent, a feature characteristic of the irregular formation of convective cumulus clouds on a sunny summer’s day. Howard (1964) devised a model that encapsulated much of the physics of this unsteady motion. Initially a column of water is of uniform temperature. At time t = 0, the temperature at the bottom is raised an amount T, resulting in the formation of a layer heated by conduction with temperature raised by T(1 – erf{z/[2(κ T t)1/2 }), where erf is the error function. The thickness, δ = (κ T t)1/2 , of the conductive layer adjoining the bottom grows in time, and so does its Rayleigh number, Ra = gαTδ 3 /νκ T , = gαT(t)3/2 κ T 1/2 /ν. Eventually Ra reaches a critical value of order 1000 when convective activity begins, rising plumes of buoyant, relatively warm water stripping away the water from the heated boundary layer, and reinstating the initial conditions at the bottom. The cycle is then repeated leading to the unsteady bursts of thermal activity observed by Townsend. The time between convective events is about 10(ν/B)1/2 , where B is the buoyancy flux through the boundary (Foster, 1971). Convection in deep layers may have a coherent structure, at least near the boundary. One situation in which there is some evidence that regular cell patterns may be produced is that of convection under a thin layer of ice covering the surface of an otherwise tranquil, fresh water lake. Because of the anomalous equation of state for fresh water, with water below 4 ◦ C becoming denser when heated, solar radiation passing through a thin floating layer of ice into an underlying layer of water, in which the temperature is uniform and near-freezing, will cause heating, increased density, and conditions of static instability. It has been suggested that the consequent circulation produces the quasi-regular patterns of ice cracks with scales of order 5 m that sometimes appear on the surface of frozen lakes. The under-ice convective motion provides an explanation of how the cells of some organisms, denser than water, remain in suspension in the water column in the otherwise low turbulence environment, shielded as it is by ice from the direct effect of wind stress (Matthews and Heaney, 1987, Matthews, 1988; Jonas et al., 2003a). In the ocean, the sheared and turbulent nature of the near-surface boundary layer may impose structure on the convecting motions (as described in Section 9.3.3, for

4.3 Convection near surfaces

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example). Rather than on a Rayleigh number, convection will depend on parameters of the turbulent motion and on the flux of buoyancy from the atmosphere to the ocean.

4.3

Convection near surfaces of uniform buoyancy flux

Neglecting the effects of surface gravity waves at the sea surface which will be described in Section 9.2, the properties of turbulence in the boundary layers just below the sea surface or above the seabed are determined mainly by the flux of heat per unit area, H (or by the equivalent buoyancy flux B: see (4.1)), across the boundary, and the stress, τ , at the boundary. The latter is imposed by the wind on the surface (τ w , as described in Sections 1.5.5 and 9.3) or by the seabed as a current flows over it (as in Section 8.2.3). Given the dimensions of the available parameters, B and τ /ρ 0 (dimensions L2 T−3 and L2 T−2 , respectively, where L is length and T is time), there is only one length scale on which turbulence can depend. Known as the Monin–Obukov length scale, it is L MO = −(τ/ρ0 )3/2 /k B

(4.4)

L MO = −u 3∗ /k B,

(4.5)

or

where u ∗ = (τ /ρ 0 )1/2 is the friction velocity in the water and k is an empirical constant, known as von Kármán’s constant, approximately equal to 0.41. With the sign convention that H is positive for an upward flux of heat, LMO is negative in destabilizing conditions, and positive in stabilizing conditions. At the seabed, the upward flux of geothermal heat tends to reduce the density and destabilize the water column; H > 0, so B > 0 (by (4.1)) and LMO < 0 (by (4.4)). At the sea surface, a cooling of the water or upward flux of heat, H > 0 giving B > 0 and LMO < 0, contributes to an increase in density in the water and so to unstable conditions. A downward heat flux (H < 0, so B < 0) at the sea surface gives LMO > 0 and stable conditions. The Monin–Obukov length scale determines the vertical extent of regions or ‘sublayers’, within which one or other of two processes, convection or the mechanical mixing induced by the stress at the boundary, is dominant. At small distances, z, from a rigid boundary, but beyond a distance at which viscous effects at the boundary, boundary texture, or small-scale roughness have important influences on the flow, the balance in the turbulent energy equation in a steady state is principally between the rate of production of turbulent kinetic energy by shear and the turbulent dissipation, ε, the terms (iii) and (iv) in Section 1.7.12. The effect of buoyancy flux (term (v)) is relatively small. The approximate equality of the dominant terms implies that ε = (τ/ρ0 )dU/dz,

(4.6)


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where U(z) is the mean flow. If, as is usually found near a boundary in near-neutral conditions of small buoyancy flux, the Reynolds stress (given as uw = −τ /ρ 0 by (1.8)) is uniform, it changes little with z. Further, if the buoyancy flux is still negligible and, because the momentum is transferred by turbulent motions and not viscous stresses, the motion is independent of kinematic viscosity, on dimensional grounds the velocity gradient must be given by dU/dz = u ∗ /kz,

(4.7)

where k is von Kármán’s constant. On integration, this leads to U (z) = (u ∗ /k)ln(z/z 0 ),

(4.8)

where z0 is the ‘roughness length’. Equations (4.6) and (4.7) give ε = u 3∗ /kz.

(4.9)

Equations (4.7)–(4.9) are the so-called ‘law of the wall’ relationships. In an unstratified and steady mean flow (e.g. in the absence of periodic currents induced by waves), these are found to provide accurate descriptions of variation U and ε with distance, z, from a horizontal seabed. Equation (4.9) appears to be valid near the sea surface, at least beyond the direct influence of breaking waves. The buoyancy flux has a small, but notable, effect: when z/|LMO | is less than about 0.3, (4.8) is modified to give U (z) ≈ (u ∗ /k)[ln(z/z 0 ) + 5z/L MO ].

(4.10)

In unstable conditions when LMO < 0, the shear flow, given by dU/dz ≈ (u ∗ /k)[1/z + 5/ LMO ], is reduced because of the enhanced vertical transfer of momentum in the convectively driven motion. The vertical momentum transfer is suppressed by the requirement to do work against the buoyancy forces in stable conditions with LMO > 0, for example in a sustained flow over sub-sea permafrost in the seabed (Osterkamp, 2001) or during daytime heating of the surface ocean in moderate winds (see Section 9.3.3). In these conditions, the stress required to maintain the same shear, dU/dz, is reduced or, equivalently, the shear produced by a given stress is increased, resulting in conditions at the sea surface sometimes referred to as a ‘slippery sea’.5 When the buoyancy flux is stabilizing (e.g. when the transfers into the sea surface contribute to a reduction of density, usually in day-time conditions of surface heating, so B < 0 and LMO > 0), the conditions at depths > 0.03|LMO | tend towards stable stratification. If, however, the buoyancy flux promotes convection (e.g. strong cooling at the sea surface, B > 0, LMO < 0), a mixed regime is found when 0.03 < d < |LMO | in which both shear and convection are important, and when d > |LMO |, there is a regime in which motions are dominated by buoyancy. In the latter, dissipation and buoyancy are related through approximate equality of the terms (iv) and (v) in 5 Similar ‘stable conditions’ affecting the relation of stress and shear may occur at the seabed when the flux of sediment eroded from the bed into a bottom boundary layer is large (Section 10.3.2).

4.4 Convection from localized sources

123

Section 1.7.12, and, when d is greater than about 2|LMO |, ε is approximately equal to B, the surface buoyancy flux. This is consistent with observations presented in Fig. 9.10 that emphasize the similarity of the oceanic and atmospheric convective boundary layers. This similarity indicates that convection in a convective surface mixed layer of the ocean will generally be unsteady, with a temporal periodicity of plumes scaling, on dimensional grounds, with LMO /u∗ .

4.4

Convection from localized sources

4.4.1

Plumes in unstratified surroundings; the entrainment assumption

An example of buoyant plumes in the ocean is shown in Fig. 4.1 (Plate 7). A plume rising from a localized flux of heat at a level, z = 0, will entrain fluid and spread as it rises and, in an unstratified medium, continue to rise indefinitely. The overall vertical heat flux, and consequently, the vertical buoyancy flux, B, in the plume is conserved and provides a quantity of dimensions L4 T−3 on which the plume must depend.6 If the plume has negligible vertical momentum and a source of very small area, and if molecular effects are insignificant in the turbulent rise of the plume, its overall properties must depend on B alone. If the radius of the plume at a height, z, above its source, is R, the mean vertical speed of motion in the plume is W and the mean non-dimensional difference in density is ρ/ρ 0 , then dimensionally R = c R z, W = cW (B/z)3/2 , and gρ/ρ0 = cρ B 2/3 z −5/3 ,

(4.11)

where cR , cW and cρ are dimensionless constants 7 . The plume consequently spreads at a constant rate or a constant spreading angle, tan−1 (R/z), found from observations to be O(5◦ ). Its vertical speed decreases with height and its density approaches that of the ambient fluid. The vertical flux of fluid in the plume, R2 W is proportional to B1/3 z5/3 , so (differentiating with respect to z) the change in a distance z is proportional to B1/3 z2/3 z. This change in flux must be supplied by an inflow or entrainment around the circumference of the plume at a mean inflow speed, v, given by 2Rvz ∝ B1/3 z2/3 z, so that v ∝ (B/z)1/3 . The inflow speed is therefore directly proportional to the rise speed, W. This is in accord with an assumption, made by Morton et al. (1956), that the local rate of entrainment of fluid into a rising plume is proportional to the speed at which the fluid in the plume is rising through the ambient fluid.8 This entrainment assumption incorporates 6 A factor of length squared contributing to the dimensions of B comes from the average flux being taken over the plume area. 7 If the source of the plume has finite radius, b, these constants must be replaced by functions of z/b. 8 This basic assumption seems to have been first introduced by G. I. Taylor in a study made during World War II of the temporary clearance of fog from airfields by using lines of oil burners (see Batchelor, 1996, p. 200).


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the notion that entrainment is through a turbulent process of fluid engulfment by eddies, and that viscosity and molecular conductivity play an insignificant role in the process, although they must be involved in the eventual homogenisation of fluid entrained. The only scale on which the local inflow of fluid into a rising plume can depend is then the difference in speeds of the plume and the ambient fluid. This has proved a successful and powerful tool in many applications in which turbulent entrainment is involved, including the prediction of the rise and spread of plumes from scales of tens of centimetres in the laboratory to heights of several kilometres in the atmosphere.9

4.4.2

Plumes in stratified surroundings

Even at abyssal depths, the ocean is stratified. The sinking of plumes from regions of surface cooling or their rise from hydrothermal vents is limited by stratification. Even though the density of a plume at its source (say at the sea surface) may apparently be sufficient for it to pass right through the ambient stratification to reach the distant boundary (in this case when its density exceeds the density of water at the seabed), it may not extend through the full ocean depth. This is because, whilst sinking or rising, plumes entrain and mix with the external fluid, so their density changes with distance. The buoyancy flux also changes with distance from source in stratified conditions. In surroundings of constant buoyancy frequency, N, a turbulent rising plume generated by a steady buoyancy flux, B0 , from a region of small area will ascend to a height, zm , that depends only on B0 and N. By dimensions, z m = cz (B0 N −3 )1/4 ,

(4.12)

where cz is a constant found empirically to be about 3.8. Plumes reaching a level where their density matches that of the ambient fluid, have vertical momentum and are consequently observed to ‘overshoot’ before spreading and intruding into the surrounding fluid at a level of about 0.8zm . Figure 4.4 shows a plume of salty water, here denser than the fluid surrounding its source, sinking into uniformly stratified water, overshooting, and spreading (or ‘intruding’) at the level of its eventual density in the surrounding fluid. The flux of fluid, Q (the volume flux), at the spreading level is found to be 1.3(B03 N −5 )1/4 . With a typical value of B0 = 10−2 m−4 s−3 for the buoyancy flux emitted by a hydrothermal vent and N = 10−3 s−1 to represent the buoyancy frequency of the ambient seawater, (4.12) gives zm = 211 m, roughly consistent with the heights to which hydrothermal plumes are found to rise in the ocean. The corresponding volume flux at the spreading level is Q = 230 m3 s−1 , compared with typical values of order 10−2 m3 s−1 at source. The considerable increase is a result of entrainment, much of it in the lower part of the rising plume. In (4.12), zm depends on B0 to the quarter power,

9 Care is needed in applications to stratified flows. For example, the rate of entrainment into a turbulent gravity current moving down a slope may additionally depend on the local Richardson number (and the slope angle). For further discussion of entrainment (and engulfment) see Section 9.7.1.

4.5 Convection and rotation

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Figure 4.4. A shadowgraph image of a dyed turbulent plume of salt solution spreading or intruding into its uniformly stratified surroundings after ‘overshooting’ to the level where the vertical velocity vanishes. The depth of the spreading region is about 0.1 m below the source of the plume. (From Turner, 1986.)

and is consequently insensitive to moderate changes in buoyancy flux. Conversely, estimates of B0 using (4.12) are very sensitive to the accuracy of estimates of zm . Equation (4.12), although providing useful estimates, can be applied only under restricted conditions. For plumes rising in a mean horizontal flow, U, it is found that zm ∝ (B0 U−1 N−2 )1/3 , decreasing as U increases.10 In the ocean, effects of rotation may also be important in the spread of fluid from hydrothermal plumes and are described in the next section.

4.5

Convection and rotation

4.5.1

Hydrothermal plumes

Although hydrothermal vent fluid rises rapidly and reaches its spreading level in about an hour, too short a time for the Earth’s rotation to have any great effect, the subsequent radial spreading is affected by Coriolis forces that cause it to turn (to the right in the Northern Hemisphere) and to form an eddy that rotates anticyclonically. The entrainment of fluid within the rising plume at lower levels results in an inflow that, being of long duration in a steady plume, is also affected by Coriolis forces and rotates cyclonically (Fig. 4.5). Helfrich and Battisti (1991) demonstrated the existence of these process in laboratory experiments in which dense salty water is released continuously at the centre of the surface of a tank rotating about a vertical axis and filled with a uniformly stratified salt solution, a reverse of the hydrothermal vents (as also in Fig. 4.4) but one in which plumes with the similar, but inverted, geometry develop. A sinking plume is formed 10 Effects of non-uniform stratification on rising plumes have been examined by Caulfield and Woods (1998).

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Figure 4.5. Sketch of a plume rising from a hydrothermal vent in a rotating fluid and spreading. The sense of rotation shown in the spreading mushroom cap or canopy is that appropriate to the Northern Hemisphere. The spreading canopy rotates anticyclonically and the underlying inflow, supplying the fluid entrained into the rising plume, rotates cyclonically.

which spreads at its equilibium level, producing an eddy. The aspect ratio (maximum thickness to radius) of the anticyclonic eddy is about 0.75f/N, where f is the Coriolis frequency, and its azimuthal velocity is of order (B0 f )1/4 , a few centimetres per second for typical ocean values. The eddy becomes unstable when its radius is about zm N/f (effectively an internal Rossby radius, though not defined conventionally as in Section 3.8.2), breaking up into two eddies that move away from the source (Fig. 4.6). The plume then again begins to spread at its equilibrium level and to form another anticyclonic eddy, the process repeating over a time of about 102 Nf −2 , a period that, in the ocean, is typically of order a few months. Plumes observed in the ocean carry substantial amounts of dissolved and particulate chemicals. The consequent light attenuation and the chemical content of the water can serve as very clear markers of the subsequent spread of plumes. Lupton and Craig (1981) find that the 3 He released into the Pacific Ocean in plumes rising some 400 m above the crest of the East Pacific Rise at 15◦ S can be traced by to a distance of about 2000 km to the west of the source (Fig. 4.7). In contrast, the geothermal plumes in the relatively high-sided median valley of the North Atlantic rise to barely the height of the surrounding ridges, and spread of vent fluid beyond the valley is restricted by the ridge topography. Dispersal of heat and solutes from the vents depends on processes leading to the flushing of the valley. The megaplumes over the southern Juan de Fuca Ridge in the northeast Pacific discovered by Baker and Massoth (1987) and Baker et al. (1989) are some 20 km across and 600 m thick and are probably caused by transient emission from multiple vents. The estimated total heat content of the plumes (based on measured temperatures exceeding the ambient) is (10 –100) PJ (1 PJ = 1015 J), corresponding to an emitted fluid volume at the source of (3−8) × 107 m3 at 350 ◦ C. The hydrothermal heat flux typical of the southern Juan de Fuca Ridge is about 7 GW,11 or a net buoyancy flux of order 3 m4 s−3 . 11 This is much smaller than the rate of energy supply of 2.1 TW required by Munk and Wunsch (1998) for deep ocean mixing.


(a)

(b)

(c)

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Figure 4.6. Vortices generated by a convective plume. Successive plan-form images of a uniform plume of dyed water descending in surroundings rotating with angular frequency, f/2, and with uniform buoyancy frequency, N, where N/f = 0.67. The image in (a) is when f t = 225, where t is the time after the plume fluid is first released. Break-up of the spreading canopy over the plume source is beginning. In (b), at f t = 299, the plume has broken into two vortex pairs. At stage (c), f t = 379, a new eddy is forming below the source. In experiments with N/f = 3.11, the canopy vortex does not split, but moves away from the source as a single baroclinic vortex. (From Helfrich and Battisti, 1991.)


128

Figure 4.7. The westward spread of the plume of 3 He from its source on the East Pacific Rise at 15◦ S. Contours of δ(3 He) are shown in a zonal section. The plume extends for over 2000 km. (From Lupton, 1995.)

4.5.2

Deep convection

In contrast to the relatively long-lived and narrow plumes of very hot water arising from localized vent sources on the seabed, deep convection resulting from cooling at the sea surface often persists for periods of only a few days but occurs over regions of horizontal extent of order 100 km. It has been found only in certain geographically limited regions of the ocean, notably the Labrador and East Greenland Seas and the Gulf of Lions in the northwest Mediterranean. Here winter cooling promotes convection in the upper boundary layer, penetrating in broad ‘chimneys’ to depths of 1–3 km and sometimes reaching the seabed. Because such deep convection affects the properties of intermediate and deep waters, its effect on the ocean’s thermohaline circulation is profound. The processes within regions of deep convection may be similar in form to those occurring during the nocturnal periods of diurnal heat cycling with the vertical transport of heat being carried by plumes (observations of which are described in Section 9.3.3). The horizontal dimensions of the regions of deep convection are however much larger than the water depth and of order 3–10 times greater than the internal Rossby radius, LRo , and their persistence exceeds the local inertial period. The Earth’s rotation is therefore likely to be important in determining the evolving structure of convection. Convection in such rotating systems can be described in terms of the Coriolis frequency, f, the fluid depth, H, and the surface buoyancy flux per unit area, B, that combine to give a deep convection scaling parameter, R* = (B/f 3 H2 )1/2 . This parameter is typically of order 0.1 in the regions where deep convection occurs. Laboratory and numerical experiments provide useful guides to the dynamics of convective regions. Maxworthy and Narimousa (1994) applied a buoyancy flux to a


129

limited area of the surface of a dish of unstratified water in solid body rotation by imposing a fine shower of brine. In the experiments, the value of R∗ ranged from 0.004 to about 1. The applied buoyancy flux results in a region of three-dimensional turbulence that gradually extends to a depth z* = (12.7 ± 1.5)R* H. Below this depth (when H > z* ) the effects of rotation subsequently become evident, resulting in quasi two-dimensional vortices with near-vertical axes of diameter (15.0 ± 1.5)R* H with maximum swirl velocities (4.0 ± 0.4) (B/f )1/2 , descending and transporting dense water downwards at speeds of about (1.0 ± 0.1) (B/f )1/2 . On reaching the bottom of the dish, the dense water carried in the vortices spreads outwards, with continuity maintained by water being carried inward towards the convective region at the water surface, the consequent vertical circulation tilting the outer edge of the region. The turbulent and convective column eventually becomes baroclinically unstable, the instability producing vortices that propagate away from the convecting region. Send and Marshall (1995) describe a numerical model with spatial resolution fine enough to resolve plumes forming at times, t, less than the Coriolis period, 2/f. At a time of about 2/f, the effect of rotation on the induced vertical circulation results in the generation of geostrophic ‘rim currents’ around the convective ‘chimney’, flowing cyclonically near the surface (inward flow being turned to the right in the Northern Hemisphere) as shown in Fig. 4.8. The circulation becomes baroclinically unstable after a time, t = tE ≈ 6.6H( f/B)1/2 (tE is an Eady time scale) with eddy diameters of about 5R*1/2 H. These cause the convective region to break up at a time of about t = 2tE , to form cones that, when the surrounding water is stratified as in the ocean, eventually slump at a level of neutral buoyancy. Observational evidence of the structure of a region of deep convection comes from measurements in winter in the Gulf of Lions by Schott and Leaman (1991) and Schott et al. (1996). They observed regions of cold surface water, some 50–100 km across, within which the water column is almost uniform to depths of 1–2 km, approaching, if not reaching, the seabed at a depth of about 2200 m. The parameter R* is about 0.16. In observations at depths between 40 and 640 m, Schott et al. (1996) found convective plumes with downward speeds sometimes exceeding 0.05 m s−1 and weaker upward motions between. The vertical extent of the plumes is at least 500 m, some passing below the depth range of instruments, and their diameters are 300–500 m. As observed in plumes developing in the atmosphere with which dynamical similarity is likely, there appears to be some increase of plume diameter with depth (i.e. with distance from the buoyancy source, the sea surface). No evidence could be found, however, that the plumes rotate. Although individual plumes could be observed for about 2 h, their persistence beyond this time could not be established because of their advection beyond the moored array of instruments. As predicted by the results of numerical experiments (Fig. 4.9, Plate 8), the convective region is bounded by a cyclonic surface ‘rim’ current with speed of about 0.1 m s−1 in a strip of width about 20 km, the same order as the internal Rossby radius, LRo , with a cyclonic surface circulation broken into cyclonic boundary eddies of 10 km scale, roughly that of LRo .

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Figure 4.8. Sketch showing the stages of development of deep convection at successive times, t, after the onset of cooling at the surface, and the influence of the Earth’s rotation. (From Send and Marshall, 1995.)

There is evidence that the location of the region of deep convection in the Gulf of Lions is partly determined by the cold offshore Mistral winds and by bathymetry that may impose geostrophic constraints limiting the local circulation and help contain the location of the convective region. The latter is supported by the observations of Swallow and Caston (1974) and a theoretical study by Hogg (1974). Convection in

4.6 Double diffusive convection

131

the Labrador Sea appears much less constrained in its location. Observations by Lilly et al. (1999; see also Lazier et al., 2001), using a Profiling Autonomous Lagrangian Circulation Explorer (PALACE) floats and moorings, find a transience of convection at a particular location that derives from the advection of large mesoscale eddies, some of which appear to originate in the East Greenland coastal current. Convection resulting from a mean surface rate of heat loss of 300 W m−2 consequently produces a near-uniform surface layer that deepens progressively, but spasmodically, during the winter, to reach depths of 1000–1750 m by March in water about 3500 m deep (Fig. 4.9, Plate 8). Complexity is added by the fact that, in summer, the surface waters are fresher than those below, and only after they have cooled to about 0.8 K below the deeper water temperature does convection set in. Acoustic Doppler observations of vertical currents are made more difficult by acoustic scattering from zooplankton that are found to migrate diurnally at speeds of about 0.04 m s−1 at 460 m depth. It is nevertheless possible to identify convective plumes which are 200–1000 m wide with vertical downward speeds reaching 0.13 m s−1 (the average is about 0.02 m s−1 ) generally embedded within horizontal motions of order 0.2 m s−1 induced by the mesoscale eddies. No evidence has yet been found for plume rotation. The arrival of convection at a particular depth level is marked by the onset of large high-frequency, but largely density-compensating, fluctuations in temperature and salinity, apparent as water is advected past fixed moorings, variations that decay only gradually with an e-folding scale of about 170 days. Rudnick and Ferrari (1999) have found that, although strong frontal regions also exist, a ubiquitous feature of the oceanic surface mixed layer is a tendency for there to be horizontal variations in temperature and salinity which compensate one another to maintain a relatively constant density, a conclusion derived earlier in a limited area by Stommel and Fedorov (1967; see footnote 18 on p. 140). Lazier et al. (2001) suggest that some of the variability observed after the arrival of deep convection in the Labrador Sea is a result of the downward transport by the plumes of the T–S variability inherent in the mixed layer. They report that some of the PALACE floats were found to penetrate below the average bottom of the mixed layer, suggesting that some plumes ‘overshoot’, possibly entraining the underlying water. Measurements of small-scale turbulence within and around deep convective plumes appear not yet to have been made. 4.6

Double diffusive convection

4.6.1

Instability

Salt is less diffusive than heat by a factor of about 100; the diffusivity of salt, κ S , ≈ 1×10−9 m2 s−1 is much less than the diffusivity of heat, κ T , ≈ 1.4 × 10−7 m2 s−1 . Stern (1960) recognized that the presence of a tube, an essential part of the perpetual salt fountain described in Section 4.1, is unnecessary to prevent a horizontal transfer of salt from a column of vertically displaced fluid – its transfer rate is always relatively slow


132

compared with that of heat.12 The instability can be described in a manner analogous to that of static instability in Section 1.4. Where salinity and temperature increase upwards, but the density is uniform or decreases upwards so that stratification is stable, a small fluid volume displaced upwards into warmer and saltier surroundings will rapidly warm through the heat conducted from the fluid around it. However, because of the much lower diffusivity of salt, its change in salinity will be relatively less. Being less salty and almost at the same temperature as its surroundings, it may become buoyant and continue to rise. A small fluid volume displaced downwards will correspondingly cool, but retain its salinity and so, being denser than its surroundings, will continue to sink. Motion is, however, opposed by viscous forces. In the conditions described above of warm, salty water lying over relatively cold and fresh water, the instability is found to take the form of long and narrow convecting cells or ‘salt fingers’. These have been extensively studied in the laboratory (see, for example, Turner, 1967, 1973, 1974, or Huppert and Turner, 1981, and Fig. 4.10). The instability is called the ‘finger regime’ of double diffusive convection. The fastest growing fingers have a predicted half-width l = [4νκT /(gαdT /dz)]1/4 ,

(4.13)

(Stern, 1960), which is typically 0.2–6 cm in the ocean. The exponential growth rate, r, of the fastest-growing fingers is found to depend on the ‘stability ratio’, Rρ = αdT/dz/(βdS/dz):13

1/2 r = (κT /ν)1/2 (gαdT /dz)1/2 1 − 1 − Rρ−1 (4.14) (see Schmitt and Evans, 1978). This increases as Rρ decreases towards unity and in the ocean implies growth periods of typically tens of minutes. In the absence of shear the plan form of the fingers appears to be generally square, but laboratory experiments by Linden (1974) show that shear results in vertical sheets of convective motion in the plane of the flow. A second instability regime is the ‘diffusive regime’ in which salinity and temperature both increase downwards, stable stratification now being produced by the increase of salinity with depth. In this the instability can take the form of growing oscillations, known as ‘overstability’. Conditions for instability in a layer of thickness, d, found by Baines and Gill (1969) are summarized in Fig. 4.11 in terms of the relative sizes of the two effective Rayleigh numbers for salinity and heat, Rs = −gβ(dS/dz)d 4 /κ T ν and Ra = −gα(dT/dz)d 4 /κ T ν (positive when temperature increases downwards), respectively. Here α and β are the 12 Whilst Stern, with others including Stommel and Turner (see Gregg, 1991), contributed to the analytical and experimental description of the instability, in fact Jevons (1857) was the first to recognize the existence of salt fingers in connection with a model of the form of clouds and, although not recognizing their possible importance in the ocean, was their original discoverer (Schmitt, 1995). Jevons’ work was subsequently overlooked for even longer than were Osborne Reynolds’ experiments on the onset of instability in stratified shear flows described in Section 3.1. 13 When fingers exist of an interface between an upper warm salty layer and a lower cold fresh layer, with temperature and salinity differences, T and S, respectively, Rρ is defined as |αT/βS|. In both finger and diffusive regimes of double diffusive convection, Rρ > 0.


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(a)

(b) Figure 4.10. ‘Salt fingers’ in the laboratory. (a) Fingers produced by adding a weak warm layer of salt solution over a layer of fresh water stratified in temperature. The fingers are made visible by adding fluorescein to the salt solution that fluoresces when lit from below through a narrow slit aligned normal to the view direction. (From Huppert and Turner, 1981.) (b) Fingers developed between layers of water with dissolved sugar (above) and salt (below). It is easier to control diffusion of salt and sugar, rather than salt and heat, because of the inevitable heat loss in the latter case at tank walls in long-duration experiments. The molecular diffusivity of sugar is about a third of that of salt, and so acts as a proxy for salt in the ocean. The upper layer, one that in the ocean is relatively warm and salty, is therefore replaced by a relatively sugary layer in this laboratory experiment. The finger interface has a thickness of about 2.5 cm and fingers are made visible by shadowgraph. Above and below the finger layer the salt and sugar fluxes drive convective motions. (From Schmitt, 2001.)

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Figure 4.11. Diagram showing the types of instability that develop through double diffusive convection in a fluid of depth, d, with uniform vertical temperature (dT/dz) and salinity (dS/dz) gradients. The horizontal axis is the ‘salinity Rayleigh number’, Rs = −gβ(dS/dz)d 4 /kT ν, and the vertical axis is Ra = −gα(dT /dz)d 4 /κT ν. The dashed line, PQ, shows the curve of neutral buoyancy with no vertical density gradient (αdT /dz = βdS/dz). Below the line the fluid is statically stable and above it is statically unstable. Regions of the finger (in the third quadrant where dT /dz and dS/dz are both >0) and the diffusive (‘unstable oscillations’, in the first quadrant below PQ, dT /dz and dS/dz < 0 mode of convection are shown. The critical value, Ra c , of the Raleigh number in the absence of salinity gradients is marked. Oscillatory modes of convection occur in statically unstable fluid when dT/dz and dS/dz are < 0, around the line XV. (From Baines and Gill, 1969.)

coefficients in the equation of state (1.1), ρ = ρ 0 (1 – αT + βS ), relating changes in density, ρ , to small variations in temperature, T , and salinity, S , from their reference values where the density is ρ 0 . The application of these conditions to the ocean is, however, constrained by the need to define a value of d in some artificial way, and a more useful criterion is based on the Turner angle, Tu, related to the stability ratio, Rρ , by Rρ = −tan(Tu − /4). The corresponding flow regimes are shown in Fig. 4.12. The water is statically unstable and subject to possible dynamical convective instability when Tu < –/2 or when Tu > /2. The finger regime is found when /4 < Tu < /2, and the diffusive regime when −/2 < Tu < −/4. ‘Doubly stable’ for angles


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Figure 4.12. The Turner angle, Tu. The horizontal and vertical axes, β∂S/∂z and α∂θ /∂z, respectively, represent the contributions of salinity, S, and potential temperature, θ respectively, to g −1 N 2 , where N is the buoyancy frequency (which increases towards the top left). The doubly diffusive regimes are labelled around the origin. (From McDougall et al., 1988.)

−/4 < Tu < /4 implies a region in which temperature increases upwards and salinity decreases upwards, both contributing in a positive way to a stable density gradient. In both regimes, the effect of the unstable motion is to transport density downwards as discussed further in Section 4.6.4, thus leading to an ever-increasing overall density gradient. The density flux is in the direction in which density increases, contrary to the flux described by (1.4) or (1.9); a corresponding diffusion coefficient would be negative. 4.6.2

Layers formed by double diffusion

Laboratory experiments in the finger regime by Stern and Turner (1969) and Shirtcliffe and Turner (1970) show that, if Rρ < 1.7, the overall temperature and salinity structure breaks down to produce a series of relatively thick layers that are fairly uniform in temperature and salinity, separated by relatively thin layers within which salt fingers persist and where the vertical mean temperature and salinity gradients are relatively large. The temperature and salinity decrease downwards in a series of steps and the structure is referred to as a ‘thermohaline staircase’.

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Figure 4.13. Thermohaline staircases. Temperature, T, and salinity, S, are plotted as functions of depth (or pressure). The layers become less evident as Rρ increases. Data are from the Tyrrhenian Sea where Rρ ≈ 1.15, (top left), the Mediterranean Outflow in the Gulf of Cadiz (Rρ ≈ 1.3, top right), and the Subtropical Underwater (Rρ ≈ 1.6, bottom left). Layers are almost entirely absent in the North Atlantic Central Water where Rρ ≈ 1.9 (bottom right. In this part the T and S records cycle repeatedly across the range as depth increases). (From Schmitt, 1981.)

Observations in the ocean confirm the existence of this thermohaline structure and provided the first firm evidence that double diffusive convection might have a substantial effect. Tait and Howe (1968, 1971) surveyed an area in a region favourable to salt fingering beneath the warm and salty Mediterranean outflow in the eastern Atlantic and discovered 10–100 m thick layers of uniform temperature and salinity, as shown in Fig. 4.13. Although at the time there were no detailed observations of salt fingers in the interfaces between the layers, the similarity of structure with that found in the laboratory and shown in Fig. 4.10b strongly suggested that convection in the finger regime was the cause of the layers. The parameter, Rρ , is about 1.3 in this area. Finger staircases were subsequently observed elsewhere where Rρ is less than about


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1.7, for example in the Tyrrhenian Sea (Rρ ≈ 1.15) and in sub-tropical regions, also shown in Fig. 4.13. Layers are now known to be relatively long lived and extensive. In the Tyrrhenian, for example, layers of temperature and salinity are found to remain virtually unchanged for many years, whilst those below the Mediterranean outflow have lateral coherence of 50–100 km. Layers are found to form in laboratory experiments made in the diffusive regime where relatively warm and salty water lies below colder and fresher water (Shirtcliffe, 1969). They are observed too in oceanic diffusive regimes, particularly in arctic regions, the first to be discovered being beneath an ice island in the Arctic (Neal et al., 1969).14 In the Weddell Sea, Foster and Carmack (1976) find layers 5–50 m thick with Rρ ranging from 1.39 to 1.03, whilst Muench et al. (1990) find two regions of layers, one in which they are 1–5 m thick with Rρ ≈ 1.52 and a second with layers about 100 m thick with Rρ ≈ 1.36. 4.6.3

Observations of salt fingers: shadowgraph images and temperature variations

The gradients in refractive index that occur within small-scale organized temperature and salinity structures in stratified regions provide an optical means to detect their shape and form. The Self-Contained Imaging Micro-Profiler (SCIMP) is a specially designed instrument that sinks freely at about 0.12 m s−1 , control and recovery to the surface being achieved by jettisoning lighter or denser fluid. A collimated horizontal light beam about 0.1 m across, a dimension that determines the largest resolved scale, is passed through water undisturbed by the profiler. The beam is refracted by gradients in refractive index in the water to produce shadowgraph images, much as in the laboratory experiments shown in Figs. 3.15 and 4.10b. Williams (1975) shows columnar patterns of salt fingers in the interfaces between the layers of the Mediterranean outflow (where the 6 mm dimension of fingers compares reasonably well to the 7.8 mm predicted by (4.13)), the Tyrrhenian Sea and the Caribbean, as well as disorganized patterns resulting from turbulence. Later examples of such images obtained by Kunze et al. (1987; including one that appears to be of a Kelvin–Helmholtz billow) are shown in Fig. 4.14. Kunze (1990) notes that fingers rarely appear to be vertical, more often being twisted or perhaps drawn into sheets by the presence of shear, in accord with Linden’s laboratory findings.15 Gargett and Schmitt (1982) were the first to identify salt fingers in temperature records. They made horizontal tows of a high-response temperature sensor at depths of 100–300 m in the sub-tropical gyre of the North Pacific. In a region favourable to salt fingers they found, in addition to patches in which the temperature spectra showed a broad band structure indicative of turbulence, regions in which the spectra have 14 Diffusive layers have also been found in arctic lakes, e.g. Lake Vanda in Antarctica (Hoare, 1966). For a theory of diffusive layering, see Linden and Shirtcliffe (1978) and Huppert and Linden (1979). 15 Further shadowgraph images of fingers are presented by Williams (1981) and Schmitt and Georgi (1982).

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Figure 4.14. Images of ocean microstructure obtained by a shadowgraph. Each image is 0.1 m across and is obtained by shining a parallel beam of light through a distance of 0.6 m. Shown are (upper left) turbulence in a stably stratified region; (lower left) a layer formed in conditions of diffusive instability; (top right) a billow-like structure; and (bottom right) shear-tilted salt fingers. (From Kunze et al., 1987.)

significant spectral peaks at wavelengths of 5–10 cm, close to the wavelength of about 5 cm found from (4.13) of the fastest growing fingers in the area. The observed scale is not expected to conform exactly to that predicted, if only because the width of the sides of fingers (with, on average, square plan form in the absence of shear) will not always be the dimension that is sampled in horizontal sections made in an arbitrary direction through an array of fingers. It appears, however, that the thin layers found in thermohaline staircases may not always contain either salt fingers or the sheets formed in shear in laboratory experiments. A major experiment to observe ocean structure in a region of salt-finger convection, ‘Caribbean Sheets and Layers Transects’ (C-SALT), was undertaken in 1985 (Schmitt et al., 1987). Records obtained by towing a streamlined body carrying


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temperature microstructure and velocity shear probes for 35 km along a thin interface in a staircase region led Fleury and Lueck (1992) to conclude that organized fingers or sheets, whether tilted or not, did not exist in the interface sampled in this study, in spite of Rρ being about 1.5, favourable to salt fingering. The reason why is not explained and the results appear to vary from others.

4.6.4

Density flux by salt fingers and the cause of layers

Analytical or numerical modelling of double diffusive intrusions does not resolve the individual fingers or diffusive instabilities, but incorporates their effect in a parametric way, accounting for their overall transports of heat and salinity by formulae derived largely from dimensional reasoning or laboratory models. Schmitt (1979), for example, shows from laboratory experiments that the ratio of the flux of density associated with heat carried by salt fingers to that of salt, called the density flux ratio and defined as γ = αFT /βFS , where FT is the vertical flux of heat and FS is the vertical flux of salt, is not a constant value (as earlier supposed). He finds that when the stability parameter, Rρ , is less than 2.5, the density flux ratio γ is about 0.7, but when 2.5 < Rρ < 4, γ ≈ 0.58, whilst when Rρ > 6, γ ≈ 0.3. The flux of density associated with the salt carried across a salt finger interface is β FS = C(gκT )1/3 (βS)4/3 ,

(4.15)

where C = 0.051 if Rρ > 3.5 and C tends to 0.1 as Rρ tends to unity. In an extensive review of double diffusive convection in the ocean, Schmitt (1994) proposes a mechanism to explain the formation of thermohaline staircases in the finger regime. Because γ < 1, fingers carry a net downward flux of density, or upward buoyancy flux, as remarked in Section 4.6.1. The flux increases with the gradients of S (see (4.15)) and of T (T is related to S by Rρ ). A perturbation to a vertically extensive finger region that increases and reduces the vertical density gradient of T and S in proportion to one another will also perturb the density and the density flux. The relatively higher downward flux in regions of high density gradient will tend to reduce the density in the lower part of overlying weaker stratification and increase the density at the top of an underlying weak density gradient, so intensifying the density perturbation. The extensive finger layer is consequently unstable.16 Radko (2003) used analytical and numerical models to argue that it is the decrease in γ with increase in Rρ that leads to the growth of perturbations of a uniform T−S gradient, and to the formation of layers. Relatively less-dense, fresher, water is released into the bottom of the uniform thicker layers from the top of the underlying thin layers of fingers. Relatively dense salty water passes from thin finger layers into underlying 16 This instability is very similar to the Phillips–Posmentier instability described in Section 7.2.1, but the density (or buoyancy) fluxes in the two cases are opposite. Here the density flux is in the direction of the mean density increase (dense fluid being carried downwards), whilst in the case proposed by Phillips and Posmentier the flux is contrary to the density gradient (dense fluid being carried upwards).

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thick layers. Both serve to drive the convection and sustain turbulent mixing in the thicker layers, thereby limiting the length of salt fingers in the thinner layers between. Other mechanisms have also been proposed to generate thermohaline staircases,17 one of which is a ‘collective instability’ resulting from the mutual interaction of several fingers. Another type of layering is described in the next section. 4.6.5

Intrusive layers: compensating temperature and salinity gradients

Nearly horizontal layers are sometimes found, one above another and separated by relatively thin, stably stratified interfaces, much as described above, but within each layer the temperature and salinity change in the horizontal whilst maintaining an approximately constant density. The first of such layers to be discovered are those documented in a remarkable paper by Stommel and Fedorov (1967) describing their analysis of data obtained near Timor and Mendinao. They came to no definite conclusion as to the cause of layers, remarking, ‘It is difficult to conjure in our minds the processes which cause the interfaces to form’.18 Since the 1960s there have been many reports of such interleaving layers or intrusions with horizontally changing properties. Layers have been found in several regions where zones of one characteristic temperature–salinity relationship adjoin another, including the Antarctic polar front in the Drake Passage and the region south of New Zealand (Toole, 1981a), the Equatorial Pacific (Toole, 1981b) – some layers even crossing the Equator and extending several hundred kilometres (Richards and Pollard, 1991; Ruddick, 2003) – and at the edges of both the Gulf Stream Rings (Schmitt et al., 1986) and Meddies described in Section 13.3. Perhaps the most extraordinary layers are those formed around icebergs (Huppert and Turner, 1978, 1980). The generation of these is similar to that of the layers illustrated in Fig. 4.15 that occur when the side of a container of fluid, stratified by sediment or dissolved salt, is cooled or heated (Thorpe et al., 1969).19 Intrusive layers are generally observed in thermohaline frontal regions (the rigid boundary of an iceberg is unnecessary) where horizontal gradients of temperature and salinity compensate in their contributions to density. The layers have been described analytically (Stern, 1967), simulated numerically (Walsh and Ruddick, 1998) and studied extensively in the laboratory (Turner, 1978; Ruddick and Turner, 1979). Their formation is illustrated in Fig. 4.16. This shows, at the right, a stably stratified patch of relatively warm and salty water that, at any level, has the same density as a colder and 17 Radko (2003) provides further references. 18 As remarked in Section 4.5.2, Stommel and Fedorov also noticed compensating variations in temperature and salinity in the surface mixed layer: ‘. . . the vertically mixed layer tends towards lateral homogeneity of density, but not towards lateral homogeneity in temperature and salinity individually’, and remarked on the substantial horizontal variations in the vertically integrated heat and fresh water storage in the layer. 19 In this case of layers formed by icebergs, however, care is needed particularly in estimating density of the low-salinity, near-freezing point, meltwater from the iceberg because of the anomalous variation of density with temperature at low temperatures (see Section 1.3).

(a)

(b)

(d)

(c)

(e)

Figure 4.15. A horizontal view of layers formed when the sidewall of a tank containing water uniformly stratified with dissolved salt is heated. Dye (which plays no part in the layer development) is added near the wall to make the pattern of flow visible. The sequence of photographs are at times after (a) of (b) 3.5 min, (c) 5.75 min, (d) 8 min and (e) 12.25 min. The horizontal bars in the sidewall are 5 cm apart. The circulation in the layers is anticlockwise, being driven by upward motions near the wall where fluid is heated by the heat flux through the wall and becomes less dense and rises, raising the isohalines near the wall. The initial condition with horizontal temperature and salinity gradients near the wall that develops in the early stage of heating is unstable to small disturbances. Cells form, setting a vertical scale for the subsequent layers and commencing the movement of the warm and relatively salty fluid away from the wall. Being warmer and saltier than fluid immediately below, this fluid is then unstable through the development of salt fingers, leading to the formation of, and mixing in, a layer that intrudes into the interior. The layer lies beneath overlying cooler and less salty water that approaches the wall in the overlying layer, conditions that are favourable to the diffusive regime of double diffusive convection. The layers eventually extend right across the 0.2 m wide tank. (From Thorpe et al., 1969.)

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Figure 4.16. The tilt of double diffusive intrusions. A stably stratified patch of warm and salty water at the right is separated horizontally from a colder and fresher patch on the left. The vertical density gradients in these layers are equal, with no horizontal density gradient. Between the two patches a perturbation forms intrusive layers. Two are illustrated. In the lower, water from the warm salty layer is moved to the left by the perturbation as indicated by a tilted arrow, between water from the colder fresher layer, with a diffusive convective regime above and finger regime below. The relative variations of the salt (FS ) and heat (FT ) fluxes are shown on the left by vertical arrows and variations of the flux of density (Fρ ) in these regimes are indicated on the right. The warmer and saltier layer moving to the left loses heat and salinity. It is, however, a region of vertically divergent density flux and therefore becomes less dense and so rises, whilst rightward moving, colder and fresher layers become denser and sink. (Adapted from Walsh and Ruddick, 1998.)

fresh region on the left. A vertically periodic horizontal perturbation transporting water between the two patches, as indicated by the curved line, results in layers favourable to salt fingers where warm salty water is carried over cold fresh, and to diffusive instability where cold fresh water is carried over warm salty water (marked ‘finger’ and ‘diffusive’ in the figure). The net downward density flux, Fρ , in the finger layers is, however, greater than in the diffusive layers (Turner, 1967) as indicated at the right by arrows, resulting in flux convergence and divergence that increases the density of the colder and fresher water moved to the right by the perturbation and reduces the density of the salty warmer layers moved towards the left. The perturbed, relatively salty water consequently tends to rise as it moves along the layers, drawing in more fluid from the salty patch at the right, whilst water in the fresher layers sinks, leading to the tilted layer structure illustrated in Fig. 4.16. This simplistic discussion does not provide an account of the initial driving forces leading to a perturbation and ignores the temporal, viscous and geostrophic effects that limit the intrusive motion field and which must be accounted for in applications (Toole and Georgi, 1981). Ruddick and Turner (1979) argue that the vertical height of layers produced by intrusions should scale with gβS/N2 , where β is the salinity expansion coefficient,


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S is the horizontal salinity difference across the region of salinity change and N is the mean buoyancy frequency. They find observations at sea where intrusive layers range in thickness from about 10–100 m to be broadly consistent with this scaling with a constant of proportionality of about 0.7. Discussion of the effect of salt fingers on the efficiency of mixing of heat and salt in turbulent regions of the ocean is continued in Sections 6.7 and 7.6.

Chapter 5 Instability and breaking of internal waves in mid-water

5.1

Introduction

Like turbulence, there is no commonly accepted or robust definition of wave breaking, although noble attempts have been made to formulate one (for example, see McIntyre, 1992, 2000; Staquet and Sommeria, 2002). Breaking is a process by which, usually in times comparable to their characteristic period, waves lose energy and generate turbulent motions that, at least initially, are commonly of scale smaller than the wavelength of the breaking waves themselves. On longer time scales, breaking may also result in the generation of residual mean motions including, for example, circulatory motions. (Examples are the ‘rotors’ generated by surface gravity waves breaking in deep water described in Section 9.2.1 and shown in Fig. 9.2.) In the case of internal waves, breaking is accompanied by distortions of isopycnals, and leads to an irreversible diapycnal transfer of water properties such as heat and salinity, or ‘mixing’. Secondary waves, shorter than the breaking waves, are sometimes involved in the process of breaking: steep, short surface gravity waves may generate shorter capillary gravity waves that lead to the rupture of the water surface or the generation of turbulence (Section 9.2.1). Breaking can be regarded as a short-lived transitional process leading from a relatively quiescent, quasi-periodic flow (but one that may already contain some residual turbulent motions, for example from earlier breakers) to a state of greater small-scale activity, although this active state may be transient, dissipating fairly rapidly, and localized in space. As defined above, breaking involves energy transfer to turbulence. Energy loss from waves (and momentum transfer through Reynolds stresses) is possible, however, without the immediate generation of turbulence or mixing, for example as waves interact with mean flows. It is furthermore possible that breaking may contribute

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5.2 Static instability or convective overturn

145

to residual motions that become large compared with the wavelength of the breaking wave or with the region in which turbulence is first generated. Some form of instability is a usual cause of breaking, but the processes leading to the instability of oceanic internal waves may be complex, involving several waves and interactions with mean flows and are, more often than not, obscure or unidentified by conventional measurements at sea. Although a stably stratified fluid is, by definition, neutrally stable to small disturbances, including those that propagate as internal waves, relatively large internal waves can lead to the onset of instability, and even waves of moderate amplitude may be unstable. Breaking is often found on slopes or over rough topography; surface waves approaching shore from deep water break in the surf zone, and internal waves break near slopes and sills, as shown in Chapters 11 and 12. Except for the observations of billows associated with internal waves, particularly those of Woods mentioned in Section 3.1, much of the evidence of internal wave breaking in locations distant from topography is circumstantial or indirect. The finger of blame for the cause of intermittent turbulent patches in mid-water (particularly where double diffusive processes can be excluded), and of the generation of the postulated vortical mode motions referred to in Section 2.8, however, points towards breaking internal waves as the prime suspect. This chapter is mainly about the mechanisms that lead to internal wave breaking and consequently to the generation of turbulence in the ocean. Much of the evidence reviewed below about the nature of breaking derives from laboratory or theoretical studies. Examples from the ocean are described in some of the relatively rare cases in which definite observations are available to define the causes of breaking and the way in which breaking occurs.

5.2

Static instability or convective overturn

An exact solution for internal waves propagating in a fluid of uniform buoyancy frequency with no shear is given in Section 2.3.3. When the wave steepness, s = am, is equal to 1, (2.15) and (2.16) show that isopycnals become vertical, as shown in Fig. 2.10b, in locations where the phase, kx + mz − σ t, is equal to 2n, where n is an integer, defining surfaces (through wave nodes) in space at a time, t. At greater values of wave steepness, the isopycnals fold over or ‘overturn’ to produce regions of static instability around these surfaces, where density increases upwards (Fig. 2.10c). The static instability will generally lead to convection and irreversible turbulent motions if the local Rayleigh number is sufficiently high (see Section 4.2.1).1

1 Sonmor and Klaassen (1997) extend Thorpe’s (1994b) study of the development of instability in statically unstable horizontal layers to include the effects accompanying tilting of the layers when static instability develops in an overturning wave ray (i.e. as in Fig. 2.10c). Although there is general agreement in the results at very small propagation angles, β, angles of modest size reduce the wavenumber of fastest growth significantly. The authors provide a classification of the types of fastest growing instabilities as a function of wave steepness, s, and β, derived from linear theory, but not including rotational effects.

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Orlanski and Bryan (1969) appear to have been the first to draw attention to this form of internal wave breaking, in connection with the step-like or staircase variation of temperature and salinity with depth occasionally observed in the ocean, suggesting that the structure is derived from breaking waves.2 A dynamic criterion for such breaking by ‘convective overturn’,3 and of the (perhaps transient) formation of a region of closed circulation, a rotor, which may be advected forward with the wave, is that the horizontal speed of the particles induced by wave motion, u, will, in the overturning region, exceed the forward phase speed, cx , of the wave. Waves approaching the thermocline from below, whilst conserving their frequency, horizontal wavenumber and vertical energy flux, are likely to become less stable for two reasons. From (2.22) and (2.23), the wave steepness, s varies as N1/2 , and so increases with N. The Richardson number, Ri, decreases as N increases,4 so that both static instability and the shear instability described in the following section are more likely as waves propagate into regions of larger N. (This, however, ignores the effects of a mean shear that are described in Section 5.6.)5

5.3

Self-induced shear

As demonstrated by Woods’ observations (Fig. 3.2), internal waves travelling on a relatively thin interface produce a shear that varies with wave phase and can result in the waves breaking through Kelvin–Helmholtz instability. When the mean shear between layers is negligible, the magnitude of the wave-induced shear is greatest at the wave crests and troughs but has opposite signs, with the result that billows forming at the wave crests rotate in a sense opposite to those at the troughs. (A small mean shear will lead to the shear being augmented at either the crests or troughs, and billows may form only where the shear is consequently greatest.) In the absence of a mean shear, when the Boussinesq approximation is valid and when the layers above and below the interface are both relatively deep so that finite amplitude waves are symmetrical (Section 2.2.2), billows may form simultaneously at both the wave crest and the wave trough. Fringer and Street (2003) describe a 2 It is now recognized that double diffusive convection is a more likely cause of the sometimes extensive thermohaline staircases in the ocean (Sections 4.6.3 and 4.6.4). 3 ‘Convective overturn’ is perhaps a misnomer since the instability is derived from advection of denser water over less dense water, or equally less dense under denser, but is in common use. 4 From (2.12) and using (2.22) and (2.23), the magnitude of self-induced shears of the component of flow in, and transverse, to the direction of propagation, |∂u/∂z| and |∂v/∂z|, are Qσ (N2 − σ 2 )3/4 /(σ 2 − f 2 )5/4 (1, σ /f ), respectively, where Q2 = (2k3 σ |Ecgv |/ρ 0 )(σ 2 − f 2 )–1 . The minimum Richardson numbers in the x- and y-directions given by Rixmin = N2 (1 − s)/(∂u/∂z)2 and Riymin = N2 (1 − s)/(∂v/∂z)2 , are then N2 {1 − Q[(N2 − σ 2 )/(σ 2 − f 2 )]1/4 }/{Q2 σ 2 × (N2 − σ 2 )3/2 /(σ 2 − f 2 )3/2 } times 1 and σ 2 /f 2 , respectively, decreasing as N increases for fixed wave frequency, σ , and wavenumber, k. 5 In the upper atmosphere, but not in the ocean, the reduction in density is a significant factor. Internal waves propagating upwards from the troposphere into the greatly reduced density of the stratosphere grow to reach large amplitudes and appear to steepen and overturn, their structure sometimes visible in the appearance of noctilucent clouds. (See also Section 5.6.4.)

5.3 Self-induced shear

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numerical model of such symmetrical conditions, with waves travelling along an interface of thickness, d, between two layers of uniform density. It reveals that the nature of breaking depends on two parameters, the non-dimensional thickness, dk, and the wave slope, ak, where k is the wavenumber and a the amplitude of the interfacial wave. On a thin interface with dk < 0.56, waves break at low values of the wave induced Richardson number by their generation of shear instability, and billows are formed with an appearance similar to those found in the studies of Kelvin–Helmholtz instability. This is in accord with Woods’ observations for which dk was about 0.06. For relatively thicker interfaces with 0.56 < dk < 2.33, breaking occurs by shear instability at a wave induced Richardson number of about 0.13 only when ak exceeds 0.85dk 1/4 . In this range of dk, the growing counter-rotating billows formed by shear at the wave crests and troughs are so long in comparison with the wavelength of the interfacial wave6 that they begin to interfere, leading to regions of overturn near the wave nodes, both in front of and behind the wave crest. The wave slope required for this to occur is, however, greater than 0.74 (with isopycnals tilted at more than 35o to the horizontal), a large value that may rarely be found in the ocean. At even greater values of dk, breaking is found when ak > 1.05 by convective overturn, occurring at the wave nodes. The instability following convective overturn, either of the wave or of the billows, is dominated by a set of vortices of alternating sign with axes aligned in the wave propagation direction. The efficiency of mixing defined in Section 3.5 is a weak function of dk, with a maximum of 0.36 ± 0.02. Observations of breaking through shear in large asymmetrical internal waves are described by Moum et al. (2003). Waves on the Oregon continental shelf travel towards the coast as a packet of internal soliton waves in each of which the thermocline level is depressed (Fig. 2.4, Plate 3). They cause bands of rippled water visible on the sea surface and were imaged acoustically within the water column, being detectable both because of the vertical displacement of layers of sound scattering organisms in the water column and because the turbulence produced by the breaking waves produces temperature microstructure that reflects sound.7 The onset of turbulence, visible in Fig. 2.4 (Plate 3) as an intensification of acoustic scattering, appears whilst the thermocline is still descending, just ahead of the wave troughs. The example in Fig. 5.1, Plate 9, shows a particular, and exceptional, case in which billows with height exceeding 5 m are produced in a wave trough. Measured values of Richardson number ahead of the wave troughs determined from estimates of velocity from an ADCP at 2 m vertical separation to derive velocity gradient, together with CTD density measurements to obtain the buoyancy frequency, do not find values less than 1/4, the necessary condition for Kelvin–Helmholtz instability in a steady flow, but are probably overestimates of the minimum gradient Richardson number because of the large separation of velocity 6 As mentioned in Section 3.2.5, billows growing on a thin interface in a uniform flow have a wavelength of roughly 2 times the interface thickness, so their wavenumber, K, is given by dk ≈ 1, and is comparable to the wavenumber of the internal waves, k. 7 The measured microstructure in the turbulence is generally sufficiently intense to explain its reflection of sound and consequent acoustic detection. This is discussed further in Section 6.10.

Figure 5.2. A 6–8 m thick layer of relatively uniform temperature in the seasonal thermocline of the Sargasso Sea measured using a towed thermistor chain. The temperature in the layer decreases with increasing distance (the isotherms drawn at 40 mK intervals migrating upwards), and in places the vertical temperature gradient is negative, suggesting the presence of overturning isopycnals although not providing conclusive evidence without information about salinity. (From Marmorino et al., 1987.)

5.3 Self-induced shear

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samples.8 The diffusivity in the thermocline, Kρ , resulting from the wave-produced turbulence is estimated to be about 5 × 10−3 m2 s−1 . A laboratory experiment by Grue et al. (2000) in which solitary waves are generated in a thin layer with uniform density gradient overlying a deep uniform density layer reproduces many of the features of breaking observed by Moum et al. Breaking is characterized by the appearance of billow vortices on the leading face of the wave trough when the fluid velocity at the water surface induced by the wave exceeds about 80% of the propagation speed. Persistent mixing caused by near-inertial waves has been subject to particular study. Generated at the sea surface by winds and atmospheric fronts (Section 2.7.1), the waves travel with downward group velocity, but upward phase speed, through the thermocline, straining the density field to create typically 5–10 m thick layers of almost uniform, or slightly unstable, density (Gregg et al., 1986). A section of a layer extending for several kilometres and followed for some 16 h in the Sargasso Sea by Marmorino et al. (1987) is shown in Fig. 5.2. It contains weak but detectable horizontal gradients of temperature in accordance with the structure of propagating internal wave rays described by Lazier (1973; see Section 2.3.3). Particularly clear measurements of inertial waves were obtained in the Banda Sea within the Indonesian Archipelago by Alford and Gregg (2001), using an ADCP and a microstructure profiler (Fig. 5.3, Plate 10). A group of at least three near-inertial waves was found, with a downward group speed of about 6 × 10−5 m s−1 (5 m/day), upward phase speeds and an observed frequency, σ , of 1.02f, where 2/f is the inertial period of about 4.4 days. Bursts of high dissipation, ε, are observed at the locations where the inertial waves produce the greatest strain, reducing the vertical density gradient by factors of at least 2. These periods of high ε dominate the turbulent contributions to the mean vertical diapycnal diffusion. Understanding of the process of breaking in near-inertial waves is not complete, partly because of the complexity introduced by the rotating currents and shear in the essentially three-dimensional waves. The onset of instability in trains of plane inertial gravity waves propagating in uniform stratification has been studied by Dunkerton (1997) and Lelong and Dunkerton (1998). They conclude that there is no preferred phase of the waves at which the instability is most likely to occur. There is, however, some doubt. Making the assumption that, for the generation of instability by shear, the local flow produced by the waves is quasi-steady and using the exact wave solutions (2.12) and (2.13), the minimum Richardson numbers of the flows in the x-direction 8 If ρ and U are density and velocity differences measured over a narrow shear and density layer of thickness δz separating relatively thick uniform layers, then the gradient Richardson number of the layer, Riδ, is −gδzρ/[ρ 0 (U)2 ]. An estimate of the Richardson number for the layer determined from values of density and velocity difference measured over a vertical separation, z, greater than δz, is Ri = −gzρ/[ρ 0 (U)2 ], which is greater than Riδ. In general, estimates, Ri , determined from discrete samples in the vertical provide upper bounds of the minimum gradient Richardson number − and it is the minimum value that is important in determining stability from the Miles–Howard theorem. MacKinnon and Gregg (2003a) do, however, observe values of Ri (measured over vertical scales of 4 m) less than 1/4 in the troughs of internal waves in soliton packets on the New England continental shelf. More is said about the dissipation of such internal waves in Section 10.4.3.

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of wave propagation and the transverse y-direction can be found. In principle, from Section 3.2.6, the orientation of the disturbances first to become unstable can be determined. The minimum Richardson number, Riymin , associated with plane disturbances with wavenumbers in the transverse y-direction is less than than Rixmin , the minimum Richardson number of disturbances with wavenumbers in the plane of wave propagation. The critical Richardson number in the y-directed flow is, however, less than 1/4 whilst that in the x-direction is equal to 1/4, with the consequence that shear instability appears most likely to set in with billow crests aligned in the y-direction and with their wavenumber in the x-direction (Thorpe, 1999a). But there is a further complication. The Richardson numbers of the near-inertial waves decrease as s increases and increase with increasing σ/f. A larger steepness, s = am, is required to induce the small Richardson numbers required for instability at larger wave frequency, σ.9 Although the x-wavenumber disturbances are the first to grow as the wave steepness increases for fixed wave frequency, because of the modulated density field, they do not grow when the ∂u/∂z component of shear is greatest and when ∂v/∂z is zero, but at a time and position that depends on the wave steepness, s.10 This means that although, for small s (and wave frequency very close to f ), the disturbances will first grow when cos(kx + mz − σ t) is small and the strain is negligible, as s tends to unity they grow closer to locations where cos(kx + mz − σ t) is unity and the strain is greatest. The conclusion is that, even in idealized conditions, the prediction of the location and orientation of billows growing as a result of shear, even in a relatively simple temporally varying and rotating flow, presents difficulties. There are unfortunately no observations of the orientation or even existence of growing billows in the available data to test the above conclusions, but the large strain values and the almost uniformity of the density found in the layers, implying that s is large, suggest that observations in waves of near (but not very near) inertial frequency are not inconsistent with the theoretical predictions. 5.4

The superposition of waves: caustics and standing waves

In the circumstances just described, in which a near-inertial wave with large horizontal scale strains the density field and enhances shear to such an extent that the local Richardson number, Ri, is reduced to a value near, but exceeding, 1/4, it is likely that the additional strain and shear produced by smaller internal waves that coexist with larger will often be sufficient, not only to reduce Ri to a value below 1/4, but to induce conditions for the growth of shear instability. In the case of near-inertial waves, therefore, mixing may first occur at a time when Riymin (which is smaller than Rixmin ) is close to or less than 1/4, even though this is not a sufficient condition for the growth 9 The minimum Ri for disturbances in the x-direction, Rix min ≈ 1/4 for s ≈ 0.4 when σ /f = 1.02, and for s ≈ 0.8 when σ /f = 1.1. √ 10 Instability first occurs when the phase, kx + mz − σ t, is equal to cos−1 {[1 − (1 − s2 )]/s}.

5.4 The superposition of waves

151

of billows induced by the presence of the inertial wave alone. If instability occurs in this way, the structure of motion leading to turbulence may not resemble billows that scale with the wavenumber of the inertial wave, but smaller-scale overturns related to the scale of the smaller internal wave. The same applies to regions in which the density gradient is substantially reduced as a wave approaches conditions of static instability or convective overturn: the presence of other waves may be sufficient to destabilize the fluid. It is also possible that relatively small-scale waves are modified by the presence of an inertially varying shear (e.g. by the development of caustics described below, or in ways to be described in Section 5.6), and consequently break in locations determined by the inertial wave but in a manner different from that in which the inertial wave itself might break. There are many cases in which the involvement of internal waves in ocean mixing is not evident from available measurements. An example in which the major cause of turbulence, however, appears to be clear is shown in Fig. 5.4, Plate 11. The turbulent region delineates the path of an internal tidal beam originating near the shelf break (Lien & Gregg, 2001). As in the case of near-inertial waves, the geometry of the apparently spatially continuous mixing identifies the likely cause, the internal tidal wave, but not precisely the process through which turbulence is generated through its presence. One mechanism leading to wave breaking in conditions of superimposed waves is the focussing of short waves by much longer waves. Broutman (1986) examines the formation of ‘caustics’ where the ray paths of short waves, traced as they propagate in a background flow that is generated by a near-inertial wave, cross one another and where their amplitude may be expected to be large. By using linear theory, he examines particular ‘test’ waves or wave packets, finding that in some cases there is indeed substantial amplification as waves approach a caustic. Broutman and Young (1986) conclude that permanent changes imposed on the small-scale, high-frequency waves are most likely to be an increase in their vertical wavenumber, together with an increase in their energy, this being extracted from the near-inertial waves (see also Section 2.6.3, and Broutman and Grimshaw, 1988). The sometimes-severe distortion of short waves by much longer waves is also examined theoretically and experimentally by Thorpe (1989), but in none of these studies is there irrefutable evidence of the interactions leading directly to wave breaking, and definitive observations are lacking. The discussion of the breaking of waves by convective overturn or self-induced shear has so far been concerned with progressive waves. In view of the generally observed isotropy in the direction of wave propagation in the ocean,11 it is likely that (as for surface waves) ‘standing waves’ resulting from the collision of a pair of waves of equal, but oppositely directed, wavenumbers (and equal frequencies) will sometimes be generated. In laboratory experiments on standing interfacial waves in a two-layer fluid, breaking occurs through overturn at the wave nodes (Thorpe, 1968c; Fig. 5.5a). Analytical (Thorpe, 2003) and numerical (Carnevale et al., 2001) studies 11 Qualifications regarding wave isotropy are expressed in Section 2.5. Isotropy is a long-term description of the wave field.

Figure 5.5. Breaking of standing internal waves in laboratory experiments. In (a) standing interfacial waves are generated at the interface between a layer of fresh water and a dyed layer of salty water by oscillating plungers in the sidewalls of the tank. Convective overturn and subsequent mixing develop at the wave node. Photographs (i)–(vi) are at intervals of 2–3 wave periods. (From Thorpe, 1968c.) In (b) a mode 1 standing wave is shown in uniformly stratified salt solution in a long tube that is rocked through a small angle about the horizontal. Arrows at the right of (i) mark the top and bottom of the 0.1 m deep tube. The sequence covers about a quarter of a wave period surrounding the time of maximum wave amplitude at (iv). Convective overturning in a dyed layer at mid-depth is visible in (ii), eventually relaxing (although with some continued mixing), as the near-horizontal wave-induced flow reverses after (iv), a time at which the flow is close to zero. (From Thorpe, 1994a.)

5.5 Resonant interactions and parametric instability

153

(b)

(i)

(ii)

(iii)

(iv)

(v)

(vi)

(vii)

Figure 5.5. (cont).

show that standing waves in a non-rotating fluid with constant buoyancy frequency also break by overturning, although not until their amplitude is such that the wave steepness, s, is of order unity. (Such breaking of a mode 1 internal wave in a laboratory experiment is shown in Fig. 5.5b.) The effect of rotation is to increase the likelihood that shear-induced instability will occur at a smaller wave steepness than does advective overturn. It is found that standing waves in uniform stratification are, however, prone to parametric instability. 5.5

Resonant interactions and parametric instability

The resonant wave interactions described in Section 2.6 are known from theoretical and laboratory studies to lead to the disintegration of a ‘primary’ wave, one that is initially

154


most energetic, through the interaction with two others. In the case of waves of mode one travelling along a thin interface between two layers of uniform density, Davis and Acrivos (1967) showed that interactions result in the exponential growth of a wave of mode two with relatively large horizontal wavenumber, provided (and depending on viscous effects) the amplitude of the primary is sufficient. In laboratory experiments the secondary waves may eventually become very steep and, in the presence of the shear in the primary, result in convective overturning and mixing causing the interface thickness to increase. The appearance of the primary and secondary waves is illustrated in Fig. 5.6. Yih (1960) appears to have been the first to recognize the possibility that internal gravity waves are prone to the parametric instability described in Section 2.6.4, the ‘parameter’ responsible for forcing the instability being the periodic variation in buoyancy frequency caused by the waves’ straining of the density field. Although parametric instability can occur at the frequency of the waves or at super-harmonics, most attention has been directed towards the sub-harmonic instability, that at which growing disturbances have a frequency half that of the primary waves and for which the growth rates of small disturbances are fastest. It is this instability that is most apparent in laboratory experiments. McEwan and Robinson (1975) first drew attention to the possibility of its importance in the ocean, having recognized the consequences of features in laboratory experiments that McEwan calls ‘traumata’, vertically thin regions of mixing that are apparent in shadowgraph or Schlieren images of standing internal waves in constant density gradients (e.g. McEwan, 1983). These ‘traumata’ are inclined at an angle of about sin−1 (σ /2N) to the horizontal, where σ is the frequency of the primary wave, and are therefore in the direction of the group velocity vector of waves of sub-harmonic frequency (see (2.9) with f = 0), and indicate their presence. Standing internal waves are excited through parametric sub-harmonic instability (psi) in a laboratory experiment devised by Benielli and Sommeria (1998). A rectangular tank filled with a uniformly stratified solution of brine is oscillated vertically at a frequency twice that of the natural mode that is to be excited. (This has the advantage that wave forcing involves no moving flaps or paddles that may drive unwanted motions.) These standing waves grow and subsequently become parametrically unstable themselves, being disturbed by waves of period twice that of the primary (so four times that of the external forcing) that subsequently develop, overturn and produce convectively unstable regions (Fig. 5.7). A layer of counter-rotating vortices with axes aligned almost horizontally in the plane of the standing wave motion is observed within the regions of static instability. The vortices are similar to those predicted in the convective overturn of progressive waves on an interfacial layer (Section 5.3).12 McEwan (1983) finds that the efficiency of mixing in standing waves subject to psi is 0.26 ± 0.06. 12 In the turbulent regime following the breaking, the wavenumber spectrum of the density fluctuations corresponding to frequencies exceeding the buoyancy frequency (and therefore arguably not internal waves which must have lower frequency) varies as k−3 . This is derived from temporal measurements and use of the Taylor hypothesis (that fluctuations in space are advected by the mean flow past a fixed point with no change).

5.5 Resonant interactions and parametric instability

155

(a)

(b)

Figure 5.6. The instability of an interfacial wave of mode 1 through a resonant interaction in which a shorter wave of mode 2 grows exponentially in time. In (a) the primary is travelling towards the right on the diffuse interface between two layers, the upper of fresh water coloured with potassium permanganate, and the lower of salt solution. The small-scale undulations mark the mode 2 waves that have developed after some 40 oscillations of the primary. The width of the image is about 0.3 m. In (b) the interface in marked by small neutrally buoyant drops of a toluene and carbon tetrachloride mixture coloured with a red oil dye. The growing second mode waves, which are initially undetectable (although they were presumably, present in the background noise), have a wavenumber about 4–5 times that of the primary, and are close to overturning. (From Davis and Acrivos, 1967.)

Although wave instability and breaking can be caused by psi in the laboratory, and whilst the instability may occur in progressive waves as well as in standing waves, growth rates are small in internal waves of moderate amplitude; waves of large steepness are required if the instability is to grow to a significant amplitude over a few wave periods. Oceanic wave trains appear rarely to be steep or more than a few waves in length, and the growth of the instability to a stage at which convective overturn occurs

156


(a)

Figure 5.7. Parametric subharmonic instability. Breaking of standing waves that have been forced in the laboratory by parametric instability through a vertically oscillating rectangular tank filled with uniformly stratified brine solution. In (a), a primary, ‘sloshing’, two-dimensional oscillation of mode 1 in the vertical with a frequency twice that of the forcing has been excited. The wave itself is unstable to a parametric sub-harmonic instability that causes the dyed isopycnal layers to overturn in a given location every second cycle of the primary (a frequency of a quarter of the externally imposed oscillation.) Image (b) is a view through the side of the tank showing the isopycnal displacements in a vertical plane normal to the plane of oscillation. Displacements are caused by an array of vortices of alternating sign that lie in the plane of the primary wave motion shown in (a). (This photograph is not taken same time as (a).) (From Benielli and Sommeria, 1998.)

(b)

may be rare. Internal tides near the sites of their generation are probable exceptions. They persist much longer than do waves of other periods, being continually generated by the barotropic tides. As mentioned in Section 2.6.4, psi can occur in M2 internal tides only at latitudes lower than 28.9o . Hibiya and Nagasawa’s (2004) observation of greater shears, and consequently larger vertical eddy diffusivities, in these low latitudes than in regions of similar internal tide generation at higher latitudes, is persuasive evidence that the psi of the internal M2 tides can contribute significantly to the enhancement of oceanic mixing.13 13 Effects of psi have been observed in the atmosphere (Klostermeyer, 1984, 1990).

5.6 Breaking of internal waves in shear flows

5.6

Breaking of internal waves in shear flows

5.6.1

Steep waves in the presence of a shear

157

The properties of oceanic internal waves, including their dispersion relation and their breaking, depend on both stratification and mean shear. As seen in Section 5.4, the shear to which waves are exposed may be generated by low-frequency waves of much greater scale. The breaking of internal waves of amplitude, a, travelling with horizontal wavenumber, k, on a thin, but finite, density and velocity interface has been studied in the laboratory (Thorpe, 1978b). When the Richardson number of the mean flow (i.e. not accounting for the velocities and density perturbations induced by the waves) has a minimum value, Rimin , at the interface that is small, but >1/4, so the mean flow is stable, breaking occurs at the wave crest or troughs of sufficiently steep waves by convective overturn as shown in Fig. 5.8.14 Breaking typically occurs at a wave slope, ak, = 0.1 when Rimin = 0.9, and at larger slopes for larger Rimin . The wave breaking is akin to that which, in a more vigorous and extreme form, occurs in spilling breakers on the sea surface; the advection of dense fluid from the crest results in the formation of a statically unstable region, within which fluid may be dynamically unstable. No study has been made of the subsequent transition to turbulence, although it may well be similar to that described in Section 5.6.4. Although wave–shear interaction must occur, for example in the shear at the base of the mixed layer, there appear to be no direct and definitive observations of this breaking process in the ocean. 5.6.2

An accelerating shear flow

Laboratory experiments and numerical models have been made to examine the evolution of wave properties and the breaking of waves in an accelerating shear flow. In the ocean this might be generated by the presence of a larger scale wave. The laboratory experiments shown in Fig. 5.9 are made in a tilting tube filled with a fluid of uniform buoyancy frequency, N. A uniform train of waves of mode 1 is generated by a wave maker at one end of the tube before it is tilted at some time, t = 0, from the horizontal through a small angle, φ. The subsequent mean flow parallel to the tube boundaries at z = −h/2 and z = h/2 is U(z, t) = N2 zt sin φ, corresponding to an accelerating Couette flow with a uniform gradient, dU/dz. The effect of acceleration is to modify the wave structure, distorting the modal structure and reducing the wave amplitude over 14 The relative orientation of three orthogonal vectors, the horizontal wave propagation velocity, c = c(1, 0, 0), the vorticity vector of the shear, Ω = (dU/dz)(0, 1, 0), and gravity, g = g( 0, 0, −1), appears to determine whether breaking occurs at the wave crest or the wave trough. When the direction of the shear is such that these form a left-handed triad (i.e. the mean flow, U, in the wave propagation direction increases upwards and c × Ω · g < 0), the interfacial waves break at their crests with denser fluid being carried forwards from the crest in the wave propagation direction at a speed which exceeds the phase speed (as in Fig. 5.8). When the triad is right handed, breaking occurs at the trough, with lighter fluid being carried from the trough in the direction of wave advance.

158


(a)

(b)

(c)

(d)

(e)

(f )

(g)

(h)

(i)

(j)

Figure 5.8. The instability of interfacial waves by convective overturn in a slowly accelerating shear flow in a tilted tube containing a wave-maker at one end. The waves are travelling to the left and the flow is to the right in the lower layer, to the left in the upper. The wave amplitude is rapidly reduced as a convective breaking occurs near the wave crests, static instability being first evident at (e) with dense dyed fluid from the lower layer carried over less-dense clear fluid. Kelvin–Helmholtz instability is visible in (j), well after the wave breaking. Photographs (a)–(c) are about 1.5 s apart and the remainder are at intervals of about 1 s. (From Thorpe, 1978b.)

160


(a)

(b)

(c)

(d)

(e)

(f)

(g)

Figure 5.9. Internal waves in an accelerating shear flow. Waves of mode 1 propagating to the left in a brine solution with constant N subjected to a uniformly accelerating shear flow, to the left at the bottom of the tank, to the right at the top. Time increases from (a) to (g). The image in (a) is just before the tube is tilted through about 4◦ , initiating the shear flow, and (g) is after 2.8 wave periods (11.3 s after tilt) when the Richardson number in the mean flow (ignoring momentum transfer from the waves) is about 0.31. The vertical variation of wave amplitude, initially sinusoidal with a maximum at mid-depth, is changed by the shear, with the position of greatest amplitude tending towards the bottom of the tank. Convective overturn of the isopycnals occurs at the location of the thickened dye lines marking isopycnals near the bottom of the tank in (g). The isopycnals become almost horizontal, indicating the rapid transfer of, at least, potential energy from the wave to the mean flow. (From Thorpe, 1978c.)

much of the depth range except close to where the mean flow in the wave propagation direction is greatest (i.e. near z = h/2 or −h/2, depending on the direction of wave propagation).15 In this location, near the bottom of the tube in Fig. 5.9g, the waves break by convective overturn. Numerical experiments (Bouruet-Aubertot and Thorpe, 1999) show that the acceleration leads to an energy transfer between the waves and the 15 The position is determined by the sign of c × Ω · g. (See footnote 14 on p. 157.)


161

mean flow through the terms (iii) in Section 1.7.12, involving the mean shear and the Reynolds stress in the wave field. Much of the energy from the wave field is transferred to the mean flow over the period of a few waves. The interaction between the flow and the waves also reduces the vertical scale of the waves and increases their steepness, s = am, until convective breaking occurs.16 Most of the transfer of energy from the waves to the mean flow, however, takes place before the waves break, and without any mixing taking place.

5.6.3

Critical level interaction

As internal waves travelling as rays propagate through the moving stratified ocean, they conserve both their horizontal wave number and their frequency, ω, relative to the ground (Bretherton, 1966). If the wave frequency measured by an observer moving with the mean flow is σ (the ‘intrinsic frequency’, the wave frequency as determined in Chapter 2 – where there is no mean flow), and now the water has a steady mean speed U(z), then the Doppler shifted frequency observed by a stationary observer is ω = kU (z) + σ.

(5.1)

Since σ is related to the wavenumber of the waves by (2.8), in the absence of rotation this gives ω = kU (z) ± N k/(k 2 + m 2 )1/2 ,

(5.2)

an equation that may be used to determine the change of vertical wavenumber, m, as the wave propagates. There is a singularity as a level, z = zc , is approached where U(zc ) = ω/k, or where the horizontal mean flow speed equals the horizontal phase speed of the wave. As z tends to zc , m tends to infinity, and so the vertical wavelength of the wave tends to zero. The properties of small amplitude waves as this ‘critical level’, z = zc , is approached were first examined by Bretherton (1966) and Booker and Bretherton (1967). Although the vertical flux of horizontal momentum in a sinusoidal train of waves in a steady non-rotating shear flow (i.e. the waves’ Reynolds stress; see Section 2.3.2) is generally constant as shown by Eliassen and Palm (1961), it is discontinuous at the critical level.17 If the Richardson number of the mean flow at the critical level, Ricrit = Ri(zc ), is greater than 1/4, the Reynolds stress is attenuated by a factor exp{−2[Ricrit − 1/4]1/2 }, and horizontal momentum is transferred to the mean flow, which consequently accelerates. The Reynolds stress is proportional to the square of the wave amplitude and, since the

16 As mentioned in Section 3.2.5, a steady uniform shear flow with constant buoyancy frequency is known to be stable to infinitesimal disturbances for all Ri > 0. This example is illustrative of the effects that finite wave amplitude and unsteady flow may have on flow stability. 17 The vertical flux of energy in a shear flow is not conserved as the waves travel through the varying flow; there are exchanges of energy between the waves and the mean shear flow.

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attenuation factor is large for Ricrit > 1, this implies that the waves are almost totally absorbed, with little transmission through the critical level.18 Numerical investigations show that when 1/4 < Ricrit < 2, a significant amount of wave energy is reflected at the critical level, whilst when Ricrit < 1/4, the reflected waves may have a greater amplitude than the incident waves, energy and momentum being drawn from the mean flow (Jones, 1968; Breeding, 1971). In this case waves are said to be ‘over-reflected’. Numerical experiments with Ricrit > 2 find good agreement with Booker and Bretherton’s analytical predictions for waves of small amplitude: there is almost complete absorption of the waves at the critical level, and momentum transfer to the mean flow. For waves of larger amplitude, the transfer of energy to the mean flow does not prevent the wave energy density near the critical level from increasing, and waves steepen to produce regions of static instability (Winters and D’Asaro, 1994). The three-dimensional model of Winters and Riley (1992) shows that, as is common in sheared statically unstable regions (e.g. as shown in Fig. 5.7b), the subsequent instability of the statically unstable regions occurs through the formation of vortices of alternating sign with axes in the direction of the horizontal mean flow. Laboratory experiments have confirmed the theoretically derived conclusion that waves do not pass through a critical layer, and have revealed what happens when their amplitude is large. Experiments have been made in a tilting tube fitted with an undulating floor and filled with a brine solution with uniform buoyancy frequency, leading to a uniform shear flow. Stationary lee waves generated by the flow over the floor of the tube propagate upwards but, as predicted, do not pass beyond the critical layer at mid-depth where the mean flow matches the waves’ phase speed, both being zero. Instead they steepen and become statically unstable as shown in Fig. 5.10.19 Koop and McGee (1986) made experiments in a flow channel using a technique developed by Odell and Kovasnay (1971) in which a shear flow is generated in a stratified fluid by two piles of differentially rotating horizontal plates. Packets of lee waves are generated as the flow passes over a sinusoidal bed or a Gaussian ridge. The experiments compare favourably with a theory devised by Koop (1981) that includes the effect of viscosity in modifying waves near a critical layer that is embedded in a layer in which the mean flow increases with height. It predicts that convective overturn of gravity waves approaching the critical layer will occur at a distance yc from the location of the critical layer if there are real solutions for yc to the equation exp[F()] = [N w/c2 k]2/3 ,

(5.3)

18 In a rotating system, (5.2) becomes ω = kU (z) ± (N 2 k 2 + f 2 m 2 )/(k 2 + m 2 )1/2 and, where m tends to infinity, kU(z) = ω ± f ; the Doppler shifted wave frequency, ω − kU(z) = ± f. Instead of the conservation of Reynolds stress, Jones (1967) shows that in a rotating system it is the vertical transport of angular momentum that is conserved in a shear flow, except at critical levels where ω − kU(z) = ± f, where it is attenuated much as described by Booker and Bretherton. 19 The regions of instability observed in these experiments are predicted quite well by an accompanying numerical model.


163

(a)

(b)

(c)

Figure 5.10. Critical layer absorption. An experiment in a tilted tube filled with salt solution having a uniform density gradient. The slowly accelerating mean shear is uniform in z with zero flow at mid-depth, flow to the left below mid-depth and to the right above. Lee waves generated by flow over a sinusoidal bottom propagate upwards (e.g. at (a), when Ri = 0.96). In (b; Ri = 0.25) and (c; Ri = 0.2) wave steepening and overturn is visible at the critical layer, where the horizontal phase speed of the waves (zero relative to the bottom) matches the speed of the mean flow (zero at mid-depth) and where the waves steepen. There is little or no wave disturbance to the dyed isopycnals above the mid-depth level until (c) when a surge arrives at the left, coming from the distant end of the tilted tube. (From Thorpe, 1981.)

where = (yc /c)dU/dz, F() = [νN3 /(9kc3 dU/dz)](−3 − 1), w is the amplitude of the wave’s vertical velocity component at the edge of the shear layer, k is the wave’s horizontal wavenumber, c is its phase speed and N is the buoyancy frequency in the shear layer containing the critical layer. The mean horizontal velocity, U, in the layer has dU/dz N, so that the Richardson number, Ri = (N/dU/dz)2 , is large. Wave breaking is suppressed by viscosity when (5.3) has no solutions for yc . In none of these laboratory experiments was it possible to confirm the predicted changes to the mean flow resulting from wave breaking.20 Numerical experiments of Dörnbrack (1998), however, reproduce the evolution of the density field in laboratory 20 Preliminary investigations in an annual channel with flow driven by a rotating lid and waves through a flexible membrane on the channel floor by Delisi and Dunkerton (1989) do, however, provide evidence that changes in the mean flow are induced by internal wave breaking near a critical layer. They also indicate that, in non-uniform mean density and velocity profiles, waves approaching a critical layer may break by shear-induced instability rather than by convective overturn.

164


experiments such as those shown in Fig. 5.10, and investigate the development of instability and changes to the mean flow following the initial convective wave overturn. Soon after the first occurrence of static instability the mean flow within the breaking zone is reduced from that originally at the critical layer in qualitative accord with the predictions of theoretical analysis. (In Fig. 5.10, the flow below the critical layer is to the left, but the waves travel to the right relative to the flow. Their transfer of momentum to the mean flow consequently reduces the speed of the mean flow towards the left.) As found in studies of waves breaking by convective overturn described in the following section, static instability is rapidly followed by span-wise, counter-rotating rolls and shear-induced billow vortices in the region just below the critical layer. (This threedimensional evolution has yet to be examined in the laboratory except in conditions of parametric instability, e.g. see Fig. 5.7b.) Dörnbrack finds the efficiency of mixing to be about 0.20. 5.6.4

Breaking by convective instability in shear

A high-resolution numerical model devised by Fritts et al. (1994, 1996b) in connection with the stratospheric internal waves that are sometimes visible in noctilucent clouds at altitudes of about 83 km, allows the development of internal wave breaking in a shear flow to be studied at high Reynolds (Re) and Rayleigh (Ra) numbers. The waves are assumed to travel in the plane of the mean flow, and break by convective overturn. The model shows the transition from static instability through several stages that are dominated by the appearance of vortical motions (a feature characteristic of transition in stratified flows, e.g. see Section 3.3.4). A set of counter-rotating vortices with axes aligned in the direction of the flow appears at an early stage. The vortices subsequently interact with the background shear and enhance the spanwise local vorticity (i.e. the vorticity in a horizontal direction normal to the flow), resulting in the generation of Kelvin–Helmholtz instability, with billow crests normal to the plane of the mean flow. These link with the original spanwise vortices to produce a complex field of intertwined vortex loops, as shown in Fig. 5.11, Plate 12 (Andreassen et al., 1998; Fritts et al., 1998). Even at this stage the small-scale flow field appears to be overturning against buoyancy forces; it might be surmised that a subsequent collapse will leave a field of eddies with predominantly vertical axes, or vortices rotating about axes that are mainly vertical, as discussed further in Section 7.2.4. There appears to be no definitive observational evidence of wave breaking by convective overturn and the subsequent generation of small-scale motions in midwater in the ocean. Theoretical studies based on the Garrett–Munk spectrum indicate that the low Richardson conditions necessary for the development of self-shear instability are most likely in near-inertial waves of high vertical wavenumber, and will usually occur before a wave reaches an amplitude sufficient for convective overturn.

5.8

5.7

Breaking of wave groups or wave packets

165

Breaking and double diffusive convection

Variations in temperature and salinity gradients, and in shear, caused by internal waves in conditions favourable to salt fingers affects their mutual development. Numerical studies by Stern et al. (2001) and Stern and Simeonov (2002) show that shear leads to the differential tilting of groups of salt fingers, changing the direction and magnitude of the convected heat and salinity fluxes, and leading to flux convergence. This modifies the distribution of buoyancy forces within the wave field at scales much larger than the fingers. In consequence, the waves may be locally amplified and driven towards conditions of static instability. The consequences of these interactions in the ocean have yet to be evaluated.

5.8

Breaking of wave groups or wave packets

In view of the wide variety of ways in which internal waves may lose energy by breaking, interaction with mean flow or through interactions with other waves, it is not surprising that the understanding has developed that internal waves in the deep ocean will not survive long, or propagate far, before their form is substantially altered. The generally transitory nature of wave generation (Section 2.7) suggests that, at least close enough to their source, internal waves will characteristically be formed in packets and have a group-like structure similar to that which is often apparent in surface gravity waves. Although there is little information about the existence of internal wave groups, observations by Sanford find that groups of downward-propagating near-inertial waves are evident in measurements of vertical velocity profiles through the thermocline (see Broutman and Young, 1986). It is usually impossible, however, to follow packets of waves as they propagate at their group velocity in three dimensions through the ocean, or to examine the changes that occur during their encounter with other waves or currents. Only for internal tides radiating many hundreds of kilometres from the shelf break or ridges has it yet been possible to follow the wave signal far with any degree of certainty. Gerkema (2002) finds that small changes in stratification near a generation site of internal tides can lead to large shifts of their phase relative to the generating barotropic tides, changes that may produce an intermittent or group-like structure in the radiating baroclinic tides. This is in accord with observations by Magaad and McKee (1973) of internal tides at a site in a depth of about 2600 m the western North Atlantic, where they appear as short ‘bursts of tidal energy’. An example of what appears to be a localized group of short waves that are close to breaking is provided by Fig. 5.12, reconstructed by Marmorino et al. (1987) from data obtained by towing an array of thermistors in the Sargasso Sea. The group is in the thermocline, and has a vertical scale of 5–8 m and appears to be at most half a wavelength long in the vertical direction. The horizontal wavelength of the waves is

166


Figure 5.12. A packet of internal waves in the seasonal thermocline of the Sargasso Sea and measured using a towed thermistor chain. Temperature contours are at 40 mK separation. (From Marmorino et at., 1987.)

uncertain because of the unknown relative direction of the section through the group, but is at most 30 m. If internal waves do propagate as groups, what are the characteristics of their breaking, for example as the waves comprising a group are amplified by local effects that promote breaking, perhaps as the group propagates through a uniform field of smaller waves?21 In a group of internal waves that travels along a ray path through a uniformly stratified ocean, breaking re-occurs with a period equal to the dominant wave period, 2/σ . Consequent ‘scars’ illustrated in Fig. 5.13, regions of decaying turbulence or microstructure left as the group propagates away,22 are inclined to the horizontal at angle ϕ = −tan−1 [( f /N )2 cot β]

(5.4)

where β, given by (2.9), is the inclination of the direction of propagation of the wave group to the horizontal, and f the Coriolis frequency (Thorpe, 1999b). The angle φ is near zero for high-frequency waves, those with frequency near N, and approaches 90o as the frequency of waves in the group tends to f in near-inertial waves. Although information about the shape of such scars could be a guide to the processes leading to their generation, there appear to be no observations presently available that can be used to test the predictions. 21 The breaking of waves travelling within a group of surface waves re-occurs with a periodicity twice that of the waves in the group. The explanation is given in Section 9.2.2. 22 Active turbulence in neighbouring but separate scars will coexist in time if the turbulent motion formed by breakers persists for more than one wave period, the time between the commencement of generation of neighbouring scars. Active or three-dimensional turbulence in the scars caused by breaking of whatever cause is unlikely to persist in the scars for a time much exceeding a few buoyancy periods (e.g. see the collapse of turbulence in Fig. 5.14 and footnote 15 in Chapter 6 referring to the collapse of turbulence generated by Kelvin–Helmholtz instability.). The time between waves beginning to break at points A and C in Fig. 5.13 is one wave period, which exceeds the buoyancy period (because σ < N). The likelihood of concurrent active turbulence in scars that are not immediate neighbours is small, but depends on wave period and is more likely for waves of higher frequency.

Figure 5.13. Breaking of internal wave groups. Scars (stippled) are left by a group of breaking internal waves propagating in the vertical (x, z) plane towards to the top right with velocity, cg . Two waves in the group are presently breaking at the location marked by the thicker full lines within the dashed region marked V, leaving regions of (stippled) decaying turbulence in their wakes. The velocity of advance of the centre of the breaking regions is given by cb . In (a) the regions of wave breaking do not overlap in space, leaving separate ‘scars’ of decaying turbulence that are tilted to the horizontal at an angle that depends on the direction of vector, cb . In (b) the group produces a continuously connected band of overlapping scars within which breaking has occurred. The band is aligned in space in direction cg . (From Thorpe, 1999b.)

168


Sutherland (2001) has made an interesting study of the stability of two-dimensional packets of internal waves dispersing in a uniformly stratified non-rotating region of buoyancy frequency, N, within which waves have initial amplitude, a, and horizontal wavenumber, k. Static instability may develop in packets of sufficiently large amplitude when their propagation angle to the horizontal, β > sin−1 (2/3)1/2 or 54.7◦ , the direction of maximum vertical energy flux for fixed N, k and a (see (2.25)). A condition for the development of static instability and for a growth rate that exceeds the wave frequency, providing conditions under which dynamic instability may have time to develop (but not necessarily accounting for the growth of convective motions), is that s = am > (1 + sin2 β)/2,

(5.5)

where s is the wave steepness, a is the wave amplitude and m = k cot β is the vertical wavenumber. Sutherland proposes a further criterion for the onset of static instability resulting from a resonant instability between the waves and the mean flow: s > cos2 β/(22 )1/2 ,

(5.6)

with, however, the caveat that numerical experiments suggest that the wave packet is stable at greater wave steepness if β ≥ /4 and if the packets have a small horizontal extent, one comparable to the horizontal wavelength, 2/k. These results may be related to the finding of Sutherland and Linden (1998), referred to in Section 2.7.1, that internal waves are predominantly generated by a turbulent layer with β between /4 and /3, i.e. with wave frequencies of (0.70–0.87)N, but further study is required.

5.9

Three-dimensional breaking

Numerical models referred to in Sections 5.3, 5.6.3 and 5.6.4 have been used to examine the three-dimensional structures developing within waves breaking either by shear or static instability. In both cases the breaking effectively occurs along an unlimited length of a line of constant phase in the advancing wave, for example the wave crest, rather than just over a short length corresponding to a localized overturn of isopycnal surfaces. Nor have laboratory experiments been devised to examine three-dimensional aspects of breaking. There appears to have been no study made of the effects of breaking occurring along a limited extent of a wave, which is presumably the situation common in the ocean, and what happens there can only be conjectured. Wave breaking will lead to the mixing of fluid, a transfer of energy into the mean potential energy and the local reduction of the density gradient in the breaking zone. The mixed patch will spread and collapse under gravity as illustrated in Fig. 1.13, and spin up to form an anticyclonic eddy over a period proportional to f −1 . But a reduction of local stratification is not the only process involved in wave breaking. The breaking of internal waves by either shear induced instability or convective instability will generally

5.10 Discussion: mixing processes

169

transfer momentum within the breaking zone from the wave to the mean flow.23 Such a transfer has already been described in Section 5.6.3 in the case of waves breaking on approach to a critical layer. A localized transfer may result in the production of vortices that are constrained by stratification to remain nearly two-dimensional, a process analogous to that observed in laboratory experiments by Voropayev et al. (1991) and Forseka et al. (1998) and shown in Fig. 5.14, in which a horizontal impulse is applied in a stratified region.24 Forseka et al. found that pairs of pancake-like eddies develop in a time of (15–25)N−1 after the impulse is applied, where N is the buoyancy frequency in the ambient fluid. (The collapse of turbulence generated by breaking may also result in a transient patch of eddying motion as described in Section 5.6.4.) It appears likely that if internal waves break in a region where 20N−1 f −1 (i.e. f/N 0.05), there will be a time period, t, following breaking lasting from t ∼ 20N−1 to t ∼ f −1 , in which eddies, possibly resembling those shown in Fig. 5.14, dominate the motion field produced by the collapse of the fluid mixed by breaking. At later times anticyclonic eddies will begin to dominate. If, however, f/N 0.05 in the region where waves break, anticyclonic eddies will be formed after a time of about f −1 . How these patterns may depend on the duration of breaking or on dimensions of the regions of breaking (e.g. the length wave crest over which breaking occurs), or what these basic characteristics of breaking internal waves are in the ocean, are unknown.

5.10

Discussion: mixing processes

Observations of turbulence or of static instability at mid-ocean depths (some to be described in Sections 7.3 and 7.5) are rarely sufficient to identify the cause beyond doubt. Nevertheless internal waves appear often to be strongly implicated in the processes leading to mixing. The conclusion by Garrett and Munk (1972b) that mixing is much more likely to be caused by shear instability than by convective instability in internal waves (except perhaps at great depths), has directed attention towards the occurrence of the former. Estimates of vertical eddy diffusivity, based on the probability of occurrence of conditions of shear-induced breaking of waves described by the Garrett–Munk internal wave spectrum, are of order 7 × 10−6 m2 s−1 , and these are generally consistent with values found from microstructure profilers in mid-water in regions where mixing by double diffusive processes is excluded.25 There is, however, little firm information about the detailed process of internal wave breaking from direct 23 The momentum transfer is related to the loss of wave energy; see Section 11.2.3 and 11.4.4. Whilst Kelvin–Helmholtz instability in tilting-tube experiments leads to a transfer of momentum normal to the isopycnal surfaces, there is no change in the integrated horizontal momentum of the water. (Neglecting the density difference between layers, the vertically integrated horizontal momentum in Fig. 3.16 is zero, both before and after instability.) 24 The eddies with near-vertical vorticity observed to develop at the interface in laboratory experiments on the breaking of standing interfacial waves appear to be driven by viscous effects at the sidewalls and not as a consequence of breaking (Thorpe, 1968c). 25 For a discussion and further references to the connection between the Garrett–Munk spectrum and mixing see Polzin et al. (1995).


170

(i)

0

5

10

15

(ii)

(iii)

(iv)

(v)

Figure 5.14. A vertical view of development of pancake eddies by the horizontal impulsive injection of fluid through a thin tube (at the left) into a uniformly stratified brine solution. The buoyancy frequency is N = 1.5 s−l (corresponding, a buoyancy period 2/N of 4.2 s) and the Reynolds number, Re = Q/νd, based on the rate of dyed fluid injection, Q (volume per second) for 5 s and tube diameter, d, is 566. The width of the frames is about 0.36 m and the images are at (i) 3 s; (ii) 5.5 s; (iii) 12 s; (iv) 22 s, by which time a dipole pair of eddies is apparent; and (v) 57 s after the introduction of the dyed fluid. (From Voropayev et al., 1991.)

5.10 Discussion: mixing processes

171

oceanic observations of both the breaking waves themselves and of the turbulence produced by their breaking. It is still much as Gargett (1989) eloquently put it: . . . the present state of our understanding of the connection between the ocean internal-wave field and the dissipation levels that are associated with it through shear instability is, to put it mildly, incomplete.

More evidence will be provided later of possible connections between breaking internal waves and vortical mode motions of sub-inertial frequency.26 In some cases internal waves are known from direct observations to coexist with an active field of motion in which frequencies exceed the buoyancy frequency. Being beyond the range of frequencies possible for non-Doppler shifted internal waves, such motions are presumably best described as turbulence. D’Asaro and Lien (2000) have tracked the movements of 28 neutrally buoyant floats over and near the sill of Knight Inlet,27 where the tidal flow is hydraulically controlled and becomes unstable with the development of active turbulence and internal waves. The data were spectrally analysed, and it is found that in the internal wave range of frequencies, σ , less than the buoyancy frequency, N, both spectra and ratios of the horizontal to vertical kinetic energies are consistent with the internal wave model devised by Garrett and Munk. At higher frequencies, σ > N, beyond the internal wave range, the spectra are isotropic and have a spectral slope consistent with an inertial subrange of turbulence. It is concluded that the field of fluctuating motions can be represented as a sum of internal waves with σ < N and of turbulent motions with σ > N, with relatively little dynamic interaction indicative of wave breaking. 26 See Sections 7.2.4 and 13.4.1 for further reference to connections between internal waves and vortical mode motions. 27 Processes near this sill are described further in Section 12.7.1.

Chapter 6 The measurement of turbulence and mixing

6.1

Introduction

Woods’ method of studying the breaking of internal waves using divers to photograph dye movements described in Section 3.1 was constrained by weather, limited to the working depth of divers, and confined to scales of order 20 m because of the rapid attenuation of underwater visibility.1 It was soon recognized that a more thorough, quantitative investigation was required to observe and record the small-scale temperature and velocity structure of the water column, whilst insulating instruments from extraneous vibrations such as those induced by waves or ships’ engines. The first documented measurements of turbulence within the thermocline were published at almost the same time as Woods’ observations. They were made by Grant et al. (1968) off the west coast of Vancouver Island using hot film anemometers2 mounted on a submarine. Turbulence was found to be continuous in the mixed layer at 15 m depth, with measured mean values of the turbulence dissipation rate, ε, of 2.5 × 10−6 W kg−1 estimated by fitting the observed one-dimensional spectra to the theoretical spectrum (1.14). The rate of loss of temperature variance, χT , was 5.6 × 10−7 K2 s−1 . In and below the seasonal thermocline, turbulence was patchy with mean values, ε = 1.5 × 10−8 W kg−1 and χT = 7.2 × 10−8 K2 s−1 , at 90 m depth. 1 In fact none of these limitations put an end to Woods’ flow visualization studies: it had achieved the results needed by the Royal Navy who had commissioned the research. No one chose to continue in spite of the potential value of obtaining further visual images of internal wave breaking or perhaps of mixing at the base of the mixed layer. 2 The principle of flow measurement by a hot film anemometer is that a film, heated by an electric current, is cooled on exposure to flowing water at a rate proportional to flow speed over the film. Cooling causes its electrical resistance to diminish and current to vary if not controlled. Variations in current (or control) provide a measure of flow speed. Problems of operation are described by Hinze (1959; Chapter 2). Generally the use of hot film anemometers in turbulence measurements is limited to regions of high turbulence (e.g. the surf zone, see Section 11.2.1).

172

6.2 Instrument platforms and measurement systems

173

Except in boundary layers, such ‘patchiness’ has proved to be a usual characteristic of oceanic turbulence. Whilst Woods was able to identify the cause of turbulence, but not determine the dissipation rates, the early instrumental measurements were not able to identify unequivocally the cause of turbulence (and rarely have those made since), but dissipation was measured, and it has since been measured over ever wider and more varied areas, and to greater depths.

6.2

Instrument platforms and measurement systems

Ship-towed turbulence measuring devices proved at first susceptible to vibrations induced by the towing vessel or of the cable connecting the device to the ship, but were eventually used successfully, particularly by Russian scientists. (For reviews, see Monin and Ozmidov, 1985; Paka et al., 1999. A towed body more recently designed for turbulence measurements, ‘MARLIN’, is described by Moum et al., 2002.) An alternative, and much-used method that removes ship- or cable-induced motions entirely, is to mount probes on a free-fall instrument that, released at the sea surface from a ship, records data on a near-vertical profile whilst sinking through the water.3 With careful design, these instruments suffer far less vibration than those that are towed but involve greater risk to the instrument package, particularly during recovery. Turbulence probes are mounted at the bottom of the instrument, exposed to undisturbed water. Recovery to the surface is either by releasing a weight or through a very light, flexible tether, usually with fibre optic cable to transmit data back in real time. A variety of such instruments have been designed, built, and used, mainly by the scientists involved in the development of each one. Examples include the Octoprobe (Oakey and Elliott, 1977), the Advanced Velocity Profiler (AVP; Sanford et al., 1985), the Advanced Microstructure Profiler (AMP; Gregg et al., 1986), the High-Resolution Profiler (HRP; Schmitt et al., 1988), and CHAMELEON (Moum et al., 1995). The Fast Light Yoyo (FLY) system (Dewey et al., 1987) is now in relatively general use for measuring turbulence in shallow water. Even expendable dissipation probes (XDPs) have been developed and used (Johnson et al., 1994). Measurements of turbulence have now been made from submarines and submersibles (Gargett et al., 1984, Osborn and Lueck, 1985; Osborn et al., 1992), Automated Underwater Vehicles (AUVs) (Levine and Lueck, 1999; Dhanak and Holappa, 1999; Thorpe et al., 2003a) and from moorings (Lueck et al., 1997). In parallel with the development and use of these instrument platforms came the necessary advances in sensors and analytical methods. Conventional sensors such as the Conductivity–Temperature–Depth probe (CTD),4 usually lowered from a research 3 Rising probes, with sensors mounted on top, have also been designed. Released from a lowered and weighted platform, or dropping a weight after reaching a specified depth, these slowly rising probes are able to sample right up to the sea surface distant from any near-surface disturbance caused by a research vessel. 4 The CTD measures electrical conductivity and temperature (from which, together, salinity can be determined) and pressure, from which depth is derived.


174

6.4 mm Stainless diameter steel tube

Hard epoxy

Electrical leads

Rubber tip

Bimorph Heat shrink beam tubing

Figure 6.1. Section through the piezoelectric air-foil shear probe designed by Dr T. R. Osborn. The probe has a diameter of 6.4 mm and is sensitive to changes in lateral forces. The probe is mounted on an instrument platform so that its tip, shown at the right, is directed into the relative flow (e.g. on a freefall profiler making measurements of micro-scale shear during its descent, the tip is directed vertically downwards.) (From Gregg, 1999.)

vessel, or acoustic or mechanical current meters deployed on moorings, do not have the time-response or spatial resolution required to measure microstructure or to make a direct determination of the rates of dissipation. Developments were successfully directed towards measurement of microscale ( 0. The points, A–O, represent the discrete measured values of density at their respective levels, z. Those between C and N are statically unstable in the sense that, because of the density inversion, there is denser fluid above or less dense below them even though the density only decreases with depth between G and J. The vertical lines and arrows show the displacements in z required to re-sort the observed density profile into the statically stable order shown in (b). The vertical lines in (b) show the depth increments that are represented by the measured density values. These set the separations of the reordered points and need not necessarily be equal. They depend on the instruments’ sampling interval in time, the instruments’ response time, and the lowering rate. The latter may not be steady, for example when data is obtained by a lowered CTD in rough conditions. The sorting conserves mass but not the potential energy.

stability (Thorpe, 1977).6 It is often assumed that temperature can be used as a surrogate to infer density, justifiably so in freshwater lakes or at temperatures > 4 ◦ C in the ocean if the relation of temperature to density is monotonic. The mean displacement scale is sometimes referred to as the Thorpe scale. The length, LT , is equal to the root mean square (rms) of the vertical displacements, d, found in resorting to convert the observed profile into one of stable stratification.7 The proportionality of LO and LT leads to the relation: ε = c1 L 2T N 3 ,

(6.3)

where c1 is a constant, equal to (LO /LT ) , and found empirically to be approximately 0.64 by Dillon (1982), 0.91 by Peters et al. (1988) and 0.90 by Ferron et al. (1998). Wesson and Gregg (1994) use measurements in the Strait of Gibraltar to show a scatter 2

6 The method of resorting fails in a convective surface boundary layer where relatively dense water sinking from the cooled water surface would be placed by the sorting algorithm at the foot of the mixed layer and not at the top from where it came. The length, LO , applies to eddies generated by turbulence in a mean stable stratification. 7 The rms value of d is found over many profiles or over vertical scales that exceed the maximum local estimates of d. Values of N in (6.3) are those derived from the average of the resorted density profiles.

6.3 Estimation of ε

177

Figure 6.3. The variation of the displacement scale, LT , with the Ozmidov scale, LO = ε 1/2 N3/2 , both measured in metres. (From Wesson and Gregg, 1994.)

of measured values about a linear relation, LO ≈ LT , with points mostly in the range 0.25LO < LT < 4LO (Fig. 6.3).8 Stansfield et al. (2001) have compared model pdfs of d with those derived from measurements in the Strait of Juan de Fuca. Even though the probability of small displacements greatly exceeds that of the large,9 the larger and more easily resolved displacements contribute most in determining the magnitude of LT , which may consequently often be accurately determined using conventional CTD data, provided the overall stratification is not very small. Lorke and Wüest (2002) find that the pdf of the displacements, d, has a universal form with the maximum displacements, dmax , divided by LT depending on ε, and argue that it is more appropriate to use dmax to derived measures of turbulence because it can be estimated from measurements with greater statistical confidence than can LT . 6.3.3

‘Indirect’ methods using acoustic Doppler

Two methods are used in coastal and shelf-sea areas to obtain remote estimates of ε. Both stem from the idea that energy produced by the largest turbulent eddies cascades down to the small scales at which it is dissipated, and that the rate of energy production of turbulence is equal to the dissipation rate. In many cases the latter is at least approximately true, although in stratified regions some energy ‘leakage’ in the form of radiating internal waves will occur. Both methods use acoustic Doppler current profilers (ADCPs), either mounted on the seabed or on a ship’s hull. These instruments measure the difference in the frequency between short pulses of emitted sound and those reflected back from particles suspended in the water. The change or ‘Doppler shift’ is proportional to the speed of the particles along the beam direction, and the speed is estimated from the measured frequency shift. The time delay in receiving the returning pulses from different distances along the acoustic beam is used to determine 8 Gargett (1999) finds a similar scatter in measurements made in the neighbourhood of Boundary Pass, between the Straits of Juan de Fuca and Georgia. 9 From observations near the head of the Monteray submarine canyon, Carter and Gregg (2002) find that the majority of measured displacements, d, the scales containing much of the turbulence, are smaller than 1 m, but there is good agreement between the Advanced Microstructure Profiler (AMP) measurements and those derived from (6.3).

178


Figure 6.4. The derivation of uw from two beams of an ADCP. Two upwardpointing, but inclined, beams are shown here for illustration. Both are inclined at an angle, θ, to the horizontal. The instrument is arranged to measure the two components of velocity, V 1 and V 2 along the directions of the two, co-planar, beams, at a height, z (or sonar range z/sin θ ). If u1 and w1 are the horizontal and vertical components of velocity in the first beam and u2 and w2 , are those in the second, then V 1 = u 1 cos θ + w 1 sin θ , V2 = w2 sin θ − u2 cos θ. If the scale of the turbulent eddies is large so that, approximately, u1 = u2 , = u, and w1 = w2 , = w, say than V12 − V22 = 2 uw sin 2θ, and so uw = (V12 − V22 ) /2 sin 2θ . The mean horizontal velocity is given by U = (V1 − V2 ) /2cos θ .

the range of the scattering volume, using the known speed of sound in water (about 1500 ms−1 ). The component of water speed along the acoustic beam can therefore be measured as a function of distance along the beam provided that the particles move in suspension at the same speed as the water. G. I. Taylor (1935) showed by dimensional reasoning that the presence of larger eddies containing energy should lead to a dissipation rate, ε = c2 u3 /l, independent of viscosity, where l characterizes their length scale and u characterizes their velocity. The empirically determined constant of proportionality, c2 , is about 0.73 ± 0.06 (Moum, 1996a). Gargett (1999) showed by comparison with airfoil probe measurements that accurate estimates of the dissipation rate can be obtained from ε = c3 q3 /d, where q is the r.m.s. vertical velocity component measured by a vertical Doppler sonar beam, and d is a vertical scale (typically a few metres in the oceanic mixed layer), characteristic of the large eddies and determined from the sonar measurements using an algorithm based on the distance between the locations where the vertical velocity crosses zero. The constant c3 is approximately unity. The second method was derived by Lohrmann et al. (1990; see also Lu and Lueck, 1999) and uses a more conventional ADCP with two pairs of beams. All the beams are inclined at approximately equal angles to the vertical (usually 20–30◦ ), the two pairs being in planes at right angles to one another. The correlation terms in the Reynolds stress, uw , with horizontal and vertical velocity fluctuations, u and w respectively, can be derived by differencing the square of the velocity components measured along the two beams of a co-planar sonar pair (e.g. one beam pointing into the flow and the other in the flow direction) as explained in the caption of Fig. 6.4. The production

6.4 Estimation of χT and χS

179

rate, approximately equal to ε when stratification is negligible, is then given by −uw dU/dz (see (iii), Section 1.7.12), where U is the mean horizontal flow at height z, determined by averaging the difference of the two horizontal velocity components estimated from the two beams. 6.3.4

Further methods

In well-mixed, non-stratified conditions the dissipation rate is sometimes obtained by fitting estimated wavenumber spectra to the theoretical spectrum in the inertial subrange, (1.14) (see Sections 6.1 and 11.2.1). Two further semi-empirical expressions used to estimate ε in stratified regions are given by (7.6) and (7.7) in Section 7.3.2. They are related to the Richardson number and the field of internal waves.

6.4

Estimation of χT and χS

The rate of loss of temperature variance, χT = 2kT (∂ T /∂ x)2 + (∂ T /∂ y)2 + (∂ T /∂z)2 ,

(1.12)

is determined from microstructure measurements of temperature usually using very fast response glass-enclosed thermistors. (Problems involved in cold films, and possible alternative sensors, are described by Gregg, 1999.) The average in χ T should be taken over all three directions, but is frequently equated to 3(dT /dz)2 derived from measurements in the z-direction obtained from free-fall instruments, assuming isotropy (often an uncertain assumption in stratified regions!) An accurate determination of χ T requires estimates of spatial gradients with resolution to the Batchelor scale. This is, however, very small, and often beyond the capability of conventional sensors because of their relatively poor spatial or temporal resolution. Interpolation is therefore made, as in finding ε in Section 6.3.1, by fitting the measured spectra (determined over a limited wavenumber range) to a universal form of the spectra given by Batchelor (1959). Even so, Gregg (1999), in reviewing the uncertainties of measurement, concludes that the ‘limitations on measuring χ T are severe and preclude estimates in most situations of interest’. Values of 3 × 10−10 K2 s−1 to 1 × 10−7 K2 s−1 are found by Gregg (1976) in the area of the Pacific Equatorial Undercurrent, but higher values, 10−6 K2 s−1 in a front, and 10−4 K2 s−1 in an active patch in the seasonal thermocline, are reported by Gargett (1978) and Oakey (1982), respectively. The ranges represented by these values are probably fairly reliable even though particular values may be uncertain by factors of order 2. Salinity is usually found from measurement of electrical conductivity. Conductivity, however, depends on both temperature and salinity, its variation being dominated by that of temperature, and so the determination of salinity microstructure demands a conversion from conductivity that requires simultaneous measurements of temperature at comparable scales. Because κ S κ T , the length scale at which salinity variations cease to be significant is less than the temperature Batchelor scale, and measurement


180

to such scales is presently impossible. Density determination by conventional CDTs usually provides salinity estimates at scales of a few centimetres. Although measurement of the microscale variations of conductivity is possible with a modification of the single electrode probe developed by Gibson and Swartz (1963), this has been used as a means of estimating the dominating temperature variation rather than salinity (Paka et al., 1999). Estimates of χS have, however, been obtained by Nash and Moum (1999) using measurements from a miniature four-point conductivity probe with spherical platinum electrodes spaced at 150–250 µm and by fitting data to a theoretical conductivity gradient spectrum. Values of χS ranging from 2 × 10−9 to 4 × 10−5 psu2 s−1 were found at depths of 15–52 m in the North-eastern North Pacific from measurement with the sensor mounted on the free-fall instrument, CHAMELEON.

6.5

Estimation of Kν

The eddy viscosity coefficient, Kν , may be derived directly from measurements of the Reynolds stress using the definition equation, K ν = −uw /(dU/dz),

(1.8)

if the stress can be measured by measuring the u and w fluctuations of the mean flow and its variation with height, dU/dz. In the unstratified boundary layer described in Section 4.3 where the stress, τ , is constant, uw = −τ /ρ 0 = − u 2∗ , and dU/dz = u ∗ /kz, so that K ν = ku ∗ z. The coefficient Kν can therefore be found from values of uw measured, for example, by the electromagnetic current meters described in Section 10.1, so giving u ∗ . Alternatively u ∗ can be determined from measurements of the mean logarithmic velocity profile by using (4.8). More generally in steady and unstratified conditions, or those in which the buoyancy flux is negligible, the Reynolds stress and ε are approximately related through the turbulent energy conservation equation (Section 1.7.12) by uw dU/dz = −ε,

and Kν can be determined as K ν = ε/(dU/dz)2 ,

(6.4)

provided methods are available to find both ε and the vertical gradient of the mean velocity. The latter may be measured, for example, by the AVP or possibly by an ADCP.

6.6

Estimation of K T or K ρ

Methods to measure the vertical eddy diffusivity of heat were needed to test Munk’s canonical value (Section 1.9). Three relationships that proved useful in the

6.6 Estimation of K T or K ρ

181

interpretation of measurements from free-fall instruments and vital in relating turbulence, microstructure and mixing were discovered in the 1970s. The first is K T = kT C,

(6.5)

where C is the Cox number, C = (∂T /∂x)2 + (∂T /∂y)2 + (∂T /∂z)2 /(dT /dz)2 ,

(6.6)

measuring the variance of the temperature gradient averaged over all three coordinate directions divided by the square of the mean vertical temperature gradient (dT /dz)2 (Osborn and Cox, 1972).10 If the temperature field is isotropic, C = 3(∂T /∂z)2 /dT /dz)2 . In principle this allows KT to be estimated from free-fall instruments measuring ocean temperature only in the z-direction, but in practice it places severe demands on the resolution of small-scale variations.11 The second method is one usually used to determine the vertical eddy diffusivity of mass or density, often called the ‘diapycnal eddy diffusivity’, in conditions of mean static stability (when N is real). It is K ρ = ε/N 2 ,

(6.7)

(Osborn, 1980), where is an ‘efficiency factor’ and N is the mean buoyancy frequency.12 The relation (6.7) is derived from the assumption that the last three terms in the turbulent energy equation (Section 1.7.12) sum to zero, supposing a steady state (sometimes an unjustified assumption) and that the first two terms are negligible. The balance becomes uw dU/dz + ε + gρ w /ρ0 = 0.

(6.8)

The ‘flux Richardson number’, R f (distinct from the gradient Richardson number, Ri), is defined as the ratio of the rate of removal of energy by buoyancy forces to the production of turbulent kinetic energy by the shear, R f = −g(ρ w /ρ0 )/(uw dU/dz).

(6.9)

Using (6.8) and (6.9) and noting that Kρ = −ρ w /(dρ /dz) is equal to gρ w /ρ 0 N2 from the definition of the buoyancy frequency, N, the diffusivity K ρ can be written in terms of R f as K ρ = [R f /(1 − R f )]ε/N 2 ,

(6.10)

or as (6.7) if = R f /(1 − R f ).

(6.11)

10 Equation (6.5) follows from (1.15); the vertical flux of heat is cp K T (dT /dz) = −cp wT = 3cp κT (∂T /∂z)2 /(dT /dz), giving (6.5) with use of (6.6). The Cox number is related to χ T by C = χT /[2κT (dT /dz)2 ]. 11 As in Section 6.4, interpolation is, however, possible to the scale lB . 12 Often ε can be measured more accurately than (∂T /∂z)2 , and (6.7) often proves the more practical than (6.5) as a means of estimating turbulent diffusion coefficients.


182

Measured values of R f (see, for example, Ellison, 1957; but see also Section 6.7) give ≈ 0.2, and this value is commonly adopted in deriving Kρ from ε using (6.7). In many cases, in particular those of mixing in stably stratified environments when the input of kinetic energy is derived from the shear rather than from convection (and R f > 0), the flux Richardson number is equal to the efficiency of mixing referred to in Section 3.5. Assuming that the eddy diffusion coefficient of heat, KT , is equal to that of density, K ρ , as should be the case when density variations are dominated by those of temperature, (6.7) relates diffusion to the turbulent dissipation rate, ε.13 In principle the two relationships for KT and K ρ , (6.5) and (6.7), respectively, provide a further way to find ε: ε = N 2 κT C/ .

(6.12)

There is, however, some debate about the appropriate value to use for , as explained in the following section.

6.7

Estimates of

From the definitions of χ T and C, and using (6.12), which supposes that KT = Kρ : = N 2 χT /[2ε(dT /dz)2 ].

(6.13)

Oakey (1982) substitutes measured values of N, χ T , dT /dz and ε into (6.13), and finds = 0.24, but with a standard deviation of about 0.14, whilst Gregg et al. (1986) report a value = 0.2. Itsweire et al. (1993) find that = 0.2 (with an uncertainty of ±20%) gives a more reliable estimate of Kρ using (6.7) when the gradient Richardson number is between 0.058 and 0.37 than do estimates based on the Cox number formulation (6.5). A study by Peters and Gregg (1988) in the Equatorial Pacific seeking to test the suggestion by Gargett (1988) that might vary with the isotropy index, ε/νN2 , that will be described in the following section, found no statistically significant variation over the range 1 < ε/νN2 < 106 . The mean value of was 0.12, corresponding to R f = 0.11, but with a scatter in estimates of more than two decades. Barry et al. (2001), however, argue that values of lower than the conventional 0.2 should be used, since laboratory experiments show that R f decreases with increasing ε/νN2 . Moum (1996b) finds values of R f are about 0.13–0.17 in the main thermocline, consistent with ∼ 0.2, but draws attention to observations by Gargett and Moum (1995) made in a tidal front in the channels connecting the Juan de Fuca Strait and the Strait of Georgia, where puzzling but significant differences in the magnitude of appear to be related to the direction of the density flux. It appears possible that physical processes contributing to flux and dissipation are present in such regions that are absent in the deep ocean. 13 Except in conditions where molecular transports are important (e.g. where double diffusive convection is possible), the coefficients KT , KS , Kρ are approximately equal (see Section 7.3.3).

6.8 Isotropy

183

It has also been argued that the value of will depend on the state of the turbulent motion, whether it is actively growing or decaying. Using a numerical model of Kelvin– Helmholtz shear instability, Smyth et al. (2001) find that first increases rapidly as billows grow, reaching values of order 1–2, and then decreases towards a value of about 0.2. The ratio, LO /LT , also varies, increases with time after the onset of turbulence in the billows from values of about 0.3 to about 1.4. Supposing that the measured oceanic turbulence results from Kelvin–Helmholtz instability, Smyth et al. propose that the empirical relation, = 0.33(LO /LT )−0.63 provides a more accurate estimate of K ρ in (6.7). Applying this to microstructure data they find an increase in the median estimates of K ρ of 50%–60%. Turbulence in which double diffusive convection is active adds a further complication. The vertical diffusivity of heat, KT , in the double diffusive salt finger regime depends on Rρ and the density flux ratio, γ (Schmitt, 1994): K T = (γ /Rρ )[(Rρ − 1)/(1 − γ )]εN −2 ,

(6.14)

or about 0.87εN−2 for values of γ = 0.7 and Rρ = 1.6 typical of the ocean. The effective value of is therefore increased.14 The corresponding value of salt diffusivity is K S = 2εN −2 .

(6.15)

These relationships imply an inequality of KT , KS , and Kρ . Salt fingers are more efficient than are other forms of turbulent transfer or may be regarded as dissipating less energy to sustain an equal diapycnal transfer of properties. The uncertainty in , and consequently the possible errors in deriving estimates of K ρ using (6.7), must be viewed in relation to the uncertainties mentioned in Section 6.3.1 that are common in determining ε, or to those of identifying the actual cause or state of development of turbulence in different regions. The accuracy demanded depends on how the estimates are used and on the sensitivity of the conclusions of, for example, ocean circulation models, to the estimated values.

6.8

Isotropy

The hypothesis that turbulent motions will be isotropic at scales of motion that are sufficiently small is based on the following concept: although the large ‘energy-containing’ eddies (for example those visible in the transient conditions of Figs. 1.12 and 3.7 or the large coherent structures in the mixed layer described in Sections 9.4 and 9.5) may be highly anisotropic, motion becomes more and more disorganised as the nonlinear process of successive instabilities and interactions transfer energy towards smaller scales, and the original anisotropy is not impressed on the smaller scales, but is lost. The turbulent motion in local regions containing eddies much smaller in scale than the energy-containing eddies should therefore be isotropic. 14 Using (6.13), Oakey (1988) finds several of values of > 1 in double diffusively unstable water in the periphery of a Meddy.

184


It is often assumed, sometimes more for pragmatic convenience than for absolute accuracy, that small-scale turbulence in the ocean is consistent with spectral forms that suppose isotropy and homogeneity of the turbulence, for example in finding ε using (6.1) or in estimating χ T from (1.11). This may be justified where turbulence is active and well developed, but not in general and, in particular, not where the effects of stratification remain or are impressed on the turbulent motion as in the final stages of collapse of Kelvin–Helmholtz billows (Fig. 3.15k–m). An extensive study of isotropy was made by Gargett et al. (1984) based on turbulence data obtained using a submersible, Pisces IV, in the highly turbulent stratified motion in the lee of the sill in Knight Inlet, British Columbia, and in the trains of internal waves that radiate from it. (These are described in Section 12.7.1; see Fig. 12.9). When the ratio of the Ozmidov to the Kolmogorov length scales, sometimes called the intermittency factor, I = (ε/N 3 )1/2 /(ν 3 /ε)1/4 = (ε/ν N 2 )3/4 ,

(6.16)

is sufficiently large, the length scale of the largest overturning eddies and the smallest are well separated, providing a broad spectral range of turbulent eddies, the smallest of which may have isotropic properties. Gargett et al. find that if the isotropy index ε/ν N 2 > 200, there does indeed appear to be isotropy.15 A further limiting value is I = 8 or ε/ν N 2 = 2.45, below which turbulence produces no significant net buoyancy flux and consequently insignificant ‘mixing’. (Stillinger et al., 1983; although Itsweire et al., 1993, suggest a rather higher value, ε/ν N 2 = 19, for negligible buoyancy flux.) Measurements by Gregg and Sanford (1988) in the eastern North Pacific, a region where internal waves are likely to dominate mixing, show that in only 0.8% of their sampled estimates of turbulence does ε exceed 200 νN2 , the threshold for dissipationscale turbulence to be isotropic, and the averaged value, ε is approximately equal to 13 νN 2 . The criterion for isotropy is applied in Fig. 6.5. Although in the mixed layer about 90% of estimated values of ε exceed 200 ν N 2 , most fail to exceed this threshold in the stratified pycnoline where a general assumption of isotropy is consequently invalid and turbulence is generally weak. (Denman and Gargett (1988) suggest that values of ε derived using the factor 15/2 in (6.1) may be overestimated in such regions by factors of order 3.) During the collapse of turbulent motion in a stratified fluid, the energy containing eddies become strongly affected by buoyancy and there is a transition, described by Gibson (1988) as ‘fossilization’. Fossil turbulence is defined by Gibson as any fluctuation of, say, temperature, salinity or vorticity, remaining when inertial forces of earlier turbulent eddies are no longer dominant. In conditions when density changes are dominated by temperature variation, Itsweire et al. (1993) found fossilization to 15 Gargett et al. show that if I ≥ 3000 (i.e. ε/ν N 2 ≥ 43 300), there is an inertial sub-range in the spectra of the cross-stream and streamwise velocity components, with the ratio, ϕ 22 /ϕ 11 , of the cross-stream to streamwise spectral densities equal to a value indicative of isotropy of 4/3. As I decreases, the inertial sub- range vanishes, but isotropy is maintained until I is less than about 53, or ε/ν N 2 ≈ 200.

6.9 Intermittency and patchiness

185

Figure 6.5. The fraction of samples of ε that are less than the instrumental noise level, 10−10 m2 s−3 , and that are less than 16 νN2 (taken as an approximate condition for significant buoyancy flux to occur) or 200 νN2 , the condition for isotropy, plotted as functions of depth (1 MPa ≈ 100 m). Data come from the California Current in the eastern North Pacific Ocean with a mean mixed layer depth of about 40 m (0.4 MPa) and with buoyancy frequency, N, that exceeds 5.2 × 10−2 s−1 in the seasonal thermocline between 40 m and 240 m, and decreases to about 2.5 × 10−2 s−1 at l000 m (10 MPa). The area of the darkest shaded region, representing the fraction of the fluid volume within which the value of ε may be high enough to ensure isotropy, is relatively small except in the mixed layer near the sea surface. Over a large proportion of the water column, the value of ε is too small to ensure either isotropy or substantial buoyancy flux. (From Gregg and Sanford, 1988.)

occur when ε is less than about 4κ T C N 2 or (substituting from (6.5) and using a Prandtl number, ν/κ T , of about 7 for the ocean) when ε/ν N 2 ≈ 0.6KT /κ T , which may be large. They concluded that the assumption of isotropy made in applying (6.1) may lead to underestimates of ε by factors of 2–4 when the Richardson number exceeds 0.37, as is frequently the case.

6.9

Intermittency and patchiness

Intermittency, characterizing of a state with rare but intense periods of energetic motion, is usual even in persistent turbulence, and the distribution of ε is often log normal

186


Figure 6.6. Microstructure measurements of temperature (left) and temperature gradient (centre), and the derived vertical displacements (right), made from a free-fall instrument in the main thermocline of the Pacific sub-tropical gyre. The lighter trace in the temperature profile shows the reconstruction of the profile as in Fig. 6.2. (From Gregg, 1980.)

(i.e. the probability distributions functions, or pdfs, of logε are Gaussian) as in Fig. 9.7, which shows measurements made in an oceanic mixed layer. Microstructure measurements soon made it evident that the characteristics of turbulence in the pycnocline (characteristics that distinguish it from turbulence near the sea surface or seabed) are that, not only is it often not very energetic, but it is commonly transient (developing or decaying) and rarely isotropic, usually being produced irregularly (if not randomly), and consequently patchy in both space and time, with near-laminar periods of motion between active periods. Figure 6.6 shows an example of a temperature profile made with a free-fall instrument in the main thermocline, together with the temperature gradient and displacement scale, LT . The irregular finestructure of layers of high and low vertical temperature gradients with vertical scales of about 0.1–5 m, respectively, noted in Section 2.1.1, is pronounced. Furthermore the variance of the local temperature gradient is very variable, being in some places large whilst elsewhere there is no measurable variance for over 1 m or more, a symptom of a patchy vertical distribution of mixing. The stratified water is often locally in a quiescent or ‘transitional’ state, with growing instabilities developing, stage-by-stage, towards turbulence as in Fig. 3.2a. Patchiness will depend on the spatial distribution and frequency of the generation of turbulence,

6.9 Intermittency and patchiness

187

on the duration of its local generation, and on how long it takes to decay. The time for which the turbulence observed by Woods (1968) continued to be active was typically about 5 min, comparable to the period of the breaking waves, but in general the duration or persistence of turbulence is not known, and nor is it known how this time is related to the processes that generate mixing in the deep ocean.16 The period of time that internal waves continue to break as they advance (distinct from the duration of turbulence once initiated by their breaking referred to in Section 5.8) is also unknown, and will depend on the conditions leading to instability, but is again likely to be of the order of a wave period. The overall effect of these factors is that turbulence in the pycnocline is often patchy in time and space (Gregg, 1980), with intermittent bursts of turbulent mixing. Although log normal in the mixed layer (Section 9.3.1), probability distribution functions (pdfs) of both ε and χ T are rarely log normal in mid-water (Gregg, 1987), and are often very variable in form.17 Peters and Gregg (1988) describe the variation of pdfs of logε measured at the Equator in the Pacific. In the upper mixed layer, well above the core of the Equatorial Undercurrent (where its speed is greatest), pdfs averaged over 24 h are generally broad with flat, and sometimes double, peaks, a consequence of the temporal variations between day and night: there is greater turbulent energy dissipation during the more active convective mixing at night.18 In the relatively high shear region at the foot of the mixed layer but above the level of the core of the Undercurrent where the mean Richardson number was ρ 1 , by a circular cylindrical barrier. When this is removed, the less-dense layer spreads over the surrounding fluid and is affected by Coriolis forces, reaching a geostrophic

196

Fine-structure, transient-structures, and turbulence

Figure 7.3. The field of eddies left behind a vertical grid moved horizontally through a uniformly stratified fluid with N = 2.3 s−1 . (a) The horizontal velocity vectors in a plane at mid-depth in the fluid obtained using a digital particle image velocimeter. (b) The vertical component of vorticity, positive being shown by full contour lines, negative by dotted lines. The zero vorticity line is omitted and the contour interval is 0.02 s−1 . The horizontal axis is the horizontal distance parallel to the plane of the grid and the vertical axis is the distance from the grid (or equivalently, increasing time after its passage at 0.5 cm s−1 through the stratified fluid.) The grid is composed of flat plates, 3.8 cm wide, spaced 15 cm apart. (From Fincham et al., 1996.)

equilibrium after moving a distance [g H/2]1/2 , where g = g(ρ 2 − ρ 1 )/ρ 2 is the reduced gravity and is the angular velocity of the tank rotation. The front between the two fluids then becomes baroclinically unstable, forming waves of length about 7[g H/2]1/2 that break, forming the eddies shown by the streak lines in Fig. 7.5. The field of vortices shown in Figs. 7.3–7.5 suggests a turbulent motion that is primarily two dimensional, quite unlike the isotropic three-dimensional turbulence considered in earlier sections. The implications of the possible existence of such a field

7.3 Shear-driven turbulence in stratified regions

197

Figure 7.4. Eddies produced by moving a vertical grid of bars back and forth (from left to right) through a 0.5 cm thick layer of dyed water overlying a deep brine layer. The figure shows the evolution of vortex dipole (or ‘hammerhead’ eddies) at times after turbulence is formed of (a) 2 s, (b) 8 s, (c) 36 s, and (d) 98 s. The width of the region shown in the image is 0.3 m and the grid is composed of 0.05 cm diameter bars separated by 1 cm. (From Voropayev and Afanasyev, 1992.)

of vortical mode motion and the evidence for its presence in the ocean are discussed further in Sections 7.7 and 13.4.1.

7.3

Shear-driven turbulence in stratified regions

7.3.1

Kelvin–Helmholtz billows

Establishing the precise process or processes that leads to a particular patch of turbulence in the stratified ocean is often difficult, if not impossible. Rarely is sufficient known of the time-history or field of temperature, salinity, density and motion surrounding a turbulent patch to be sure of its cause. There are, however, a few instances in which recognizable signatures are evident in the data sets, and of these some of the clearest are those of Kelvin–Helmholtz instability. Woods’ observations described in Section 3.1 are the only photographically documented records of billows and the only records that provide information about the threedimensional structure of the billows. The billows observed by Marmorino (1987), and shown in Fig. 3.9, have large values of the Cox number correlating with the displacement scale. In Admiralty Inlet, a salt-stratified tidal channel near Puget Sound, Seim et al. (1995) used a 200 kHz sonar to identify braids associated with very energetic,

198


Figure 7.5. Streak lines in a 7 s long exposure photograph of neutrally buoyant particles in eddies resulting from baroclinic instability in a rotating tank experiment. The photograph is 40 rotation periods after a central cylindrical barrier confining the less-dense upper fluid is removed. The streaks are images of neutrally buoyant particles in the upper layer illuminated by a horizontal light sheet. The square rotating tank is 1.1 m across. (From Griffiths and Hopfinger, 1984.)

20 m high, Kelvin–Helmholtz billows (Fig. 7.6). They used the advanced microstructure profiler (AMP) to measure shear and temperature gradient spectra. Dissipation rates within billows exceed the value 200νN2 required for isotropy by two orders of magnitude. The acoustic scattering calculated to occur from the turbulent microstructure is enough to account for the measured acoustic backscatter, without substantial contributions from the acoustic scattering from marine organisms.8 Further acoustic observations of billows have been made in straits and near sills, and are illustrated in Figs. 12.2, 12.3 and 12.7. Although in no sets of observations has the development of turbulence been fully followed or quantified, the observations point to the value of relating measurements of turbulence to larger-scale features if its likely source is to be identified. Maps in depth and time of the location of statically unstable regions, inversions in density, have been constructed from repeated profiles made from the floating instrument 8 The acoustic scattering cross section at 200 kHz within the billows has a probability distribution that is close to log normal.


199

Figure 7.6. Kelvin–Helmholtz billows in an acoustic image showing depth vs time of the acoustic scattering strength in Admiralty Inlet, Washington State, USA. The black lines marked with station numbers mark trajectories of a free-fall microstructure probe (AMP) used to sample turbulence in and around the billows. They illustrate how difficult it may be to obtain comprehensive dissipation measurements of the transient phenomenon. (From Seim et al., 1995.)

platform, FLIP, by Alford and Pinkel (2000a). An example is shown in Fig. 7.7, Plate 13. The maps suggest that the inversions occur at depths and in time periods in which the mean vertical density gradients are diminished by wave-like features, sometimes (but not always) where Ri is small (see Fig. 7.7c and the more detailed structure shown in Fig. 7.8). Internal waves are the plausible cause. With the available vertical resolution of some 2 m at about 350 m depth in these measurements, it is not yet clear in which cases the shear and in which the convective overturn of waves leads to the observed static instability of the density field. Several problems are encountered in studying the development of turbulence in the ocean and in comparing observations to theory. It is rarely possible to follow the motion of the water or of the process leading to turbulence as they propagate or advect in time. The theoretical analysis of Kelvin–Helmholtz instability described in Section 3.2 shows that the minimum Richardson number, Ri min , is a critical parameter in determining whether a given flow will become unstable. In practice both the density and velocity are measured over some short times and often averaged over periods of tens of seconds or more. Crucially, the density and velocity profiles, and hence buoyancy frequency and shear from which the Richardson number, Ri, is estimated, can be measured only over finite vertical distances, often at least 1 m or so. Such large-scale estimates can be shown to be an upper bound of the minimum Richardson number, Ri min , of a flow that is fully resolved in the vertical (see also footnote 8 of Chapter 5), and so Ri min will consequently be overestimated in data sets. There is the further complexity,

(a) 330 0.8 Richardson Number

Depth (m)

340

350

360

0.6

0.4

0.2 370 0 67.92

67.94 67.96 Time (UTC 1995)

67.98

(b) 320

1.5 Richardson Number

Depth (m)

325

330

335

1

0.5

0

340 67.71

67.72

(c)

67.73 67.74 67.75 Time (UTC 1995)

67.76

67.77

200

0.8 Richardson Number

Depth (m)

210

220

220

240

0.6

0.4

0.2

0

250 67.52

67.54

67.56 67.58 Time (UTC 1995)

67.60

67.62

Figure 7.8. Three overturning events and their relation to Richardson number, Ri. The isopycnals are shown in depth and time with Ri estimated over a vertical scale of 6.4 m. The black single dots indicate regions of overturning isopycnals where the maximum value of LT > 2 m. The almost vertical lines of small dots mark the times of CTD profies, 4 min apart. A time of 0.02 = 28.8 min. (From Alford and Pinkel, 2000a.)


201

not properly resolved in theoretical studies, that turbulence remaining from earlier instabilities may continue to be active in naturally occurring flows, although perhaps at a scale that is much smaller than that at which billows grow. This ‘pre-existing’ motion may provide initial disturbances that determine the location and possibly the scale of the growing billows, as well as contributing to dissipation. The usual approach in models is to represent residual turbulence by an eddy diffusion coefficient, effectively making the flow ‘viscous’. Care is needed in comparing models and observations where small scales are not well-resolved. There are, nevertheless, coarse tests that can be made to determine how the nature of the flow varies with Ri.

7.3.2

The variation of Ri, and the related stratification or mixing

Whether or not a local flow satisfying the requirements for the development of Kelvin– Helmholtz instability will lead to a transition to turbulence depends not only on the Richardson number, Ri, being below the critical value for instability, but for how long it remains sub-critical. There will be circumstances in the unsteady ocean environment in which wave growth may be discontinued before overturning occurs to form billows because the shear has relaxed and Ri has become greater than critical. The small value of Ri, 0.13, required in some conditions for billows to form in steep progressive interfacial waves (Fringer and Street, 2003; Section 5.3) indicates that a maintained low Ri is necessary for mixing to develop. Nevertheless statistics of Ri, and the presence and the co-existence of temperature, or density, microstructure, do suggest that a value of Ri near 1/4 identifies a transition in ocean structure. Eriksen (1978) describes measurements from a rigid array 8 m in height and 20 m in length moored 250 m off a sloping seabed in a water depth of 900 m off Bermuda. The scatter diagram, Fig. 7.9a, of N2 versus the squared shear, S2 = (dU/dz)2 , with both parameters measured by differencing measurements over some 7 m in the vertical, shows a distinct reduction in the density of points below the line, N2 = 0.25S2 where Ri = 1/4. It appears possible that the internal wave field may be ‘saturated’, limited by breaking whenever the combined contributions of the waves to the local shear and strain reduce the value of Ri to less than about 1/4; the few values of Ri less than 1/4 may be in transitional conditions of developing instability, returning Ri to a supercritical value >1/4. Observations made by Shen and Meid (1986) using a linear thermistor array spanning a depth range from the surface to about 90 m with vertical and horizontal resolution of about 0.5 m and towed at about 2.5 m s−1 in the Sargasso Sea provided interesting, if inconclusive, information about the frequency of regions of thermal inversions (interpreted as implying static instability) as a function of Richardson number, Ri. Unfortunately it was then possible to estimate Ri only on vertical and horizontal scales of 7 m and 450 m, respectively (and much better resolution could be attained


0.30 0.20 0.00

0.10

N2 (10−4 s−2)

0.40

0.50

202

0.00

0.10

0.20

0.30

0.40 2

0.50

−4 −2

S (10

s )

0.60

0.70

0.80

(a)

(b)

Figure 7.9. Variations with the square of the buoyancy frequency, N2 , and square shear, S2 = (dU/dz)2 . (a) A scatter diagram showing the reduction in the number of observed values below the line Ri (= N2 /S2 ) = 1/4. Data are from measurments at 250 m depth over the eastern slope of Bermuda in water of depth about 900 m. Shear is derived from differences in velocity over 6.3 m and N2 over 7.13 m, 6881 values are plotted derived from observations over 78 hrs. (From Eriksen, 1978.) (b) Contours of the variation of the rate of dissipation of turbulent kinetic energy ε, as a function of N2 , and S2 , measured on vertical scales of 4 m between 400 and 800 m in the eastern North Atlantic (the area of the NATRE experiment). The straight lines show values of the Richardson number, Ri. The ε contour values are (1, 2, 5, 10, 20, 50) × 10−10 W kg−l . (From Polzin, 1996.)

now), but the probability of finding regions of apparent static instability increased as Ri decreased, most rapidly when Ri = O(1), but the resolution of Ri was too poor to discriminate values of Ri between 0 and 2. Estimates of the ratio of particle speed, u, and horizontal wave phase speed, cx , in the regions of apparent static instablity were used in an attempt to distinguish between convective overturn (where u/cx > 1; see


203

Section 5.2) and shear instability (u/cx < 1) and, although both appeared to contribute to the occurrence of static instability, their relative importance could not be determined. Measurements made from below the thermocline to the surface of Loch Ness in a variety of wave and wind conditions (Thorpe, 1977) show that there is an increase in the mean fraction of the water column that is unstably stratified as the Richardson number, Ri (estimated over a vertical scale equal to the displacement scale, LT ), decreases from 1000 to 0.01. The increase in the fraction is particularly rapid (from about 0.1 to 0.38) as Ri decreases from 1 to 0.1, indicating a marked increase in the frequency of overturning and mixing in this range. The statistical studies of the occurrence of low values of Ri and related unstable stratification referred to above, although indicative of the importance of Ri in determining the small-scale nature of the flow, are not a guide to the presence of coherent structures, either of billows and their related structures resulting from shear instability or of those processes such as internal waves that lead to the region of small Ri. Since the ability of measured values Ri to define the unstable or stable nature of a flow is a question often raised in discussion, it may be useful here to attempt to summarize what is known. The Miles–Howard theorem relates to the stability of a laminar steady flow. In practise oceanic flows are not steady and rarely laminar. In conditions where the effects of viscosity and molecular diffusion are negligible, it appears that the solutions of the Taylor–Goldstein equation, a linearized equation, do provide accurate predictions of the onset of instability and the wavelength of finite amplitude billows. In some steady and unsteady flows, the critical Richardson number, Ri c , below which the flow is unstable is known to be less than 1/4. Consequently when, in the unsteady ocean, Ri is locally less than Ri c (a value determined by solution of the Taylor–Goldstein equation or possibly of an equation representing a periodic progressive wave disturbance), small disturbances will begin to grow, first as waves and subsequently as billows. Whether they will lead to turbulence will depend on whether the flow or wave motion is sustained. Although the largest of the disturbances that grow first at any time will be determined by the nature of the ‘noise’ or ‘background disturbance’ spectrum in the unstable flow, it is likely that the scale of the evolving structure will be predicted by solution of the governing equation (e.g. the Taylor–Goldstein equation). Also present may be pre-existing disturbances of smaller scale, perhaps amplified by the shear and strain of the growing billows, and causing small-scale turbulence. In the range of Richardson numbers, Ri c < Ri < 1/4, the onset of instability of a steady laminar flow is precluded. However, the large-scale billows remaining from a period when Ri was less than Ri c may be present, and pre-existing turbulence or internal waves, although perhaps small, may be sufficient in some localities to reduce Ri to a value at which turbulence is initiated. If Ri is between 1/4 and a greater value at which turbulent motion collapses (a value of about 0.32 is found in the laboratory experiments on Kelvin–Helmholtz instability), a collapsing turbulent flow may be found resulting from the previous shear instability initiated when Ri was less than Ri c . This may include still-coherent structures left from


204

billows. For larger Ri, the growth of disturbances is unlikely and coherent structure, if present, is less likely to be a remnant of earlier billows, although even here waves reaching critical layers may promote turbulence. It is apparent from this indecisive discussion that although, statistically, turbulence and small-scale regions of static instability appear to be more frequent when Ri < 1/4, the turbulent character of flow at a given location and particular time is not determined by the local value of Ri, but from the past history of shear and strain within the fluid, the rates of energy dissipation or radiation from it, and by the passing internal wave field.

7.3.3

Empirical expressions for vertical eddy coefficients

Measurements of the vertical gradients of horizontal velocity, temperature and density, and estimates of ε and χ T , were obtained by Peters et al. (1988) using the AMP free-fall probe and ADCP data in the region of shear above the core of the Equatorial Undercurrent where typically the mean Richardson number is of order 0.25–0.5. The mean turbulent dissipation rate is about 10−7 W kg−1 but, owing the effects of nocturnal convection, ε varies by a factor of 100 between its minimum values in the early afternoon to its maximum at night, with the variation being detected to depths of 90 m, well into the stratified zone below the surface mixed layer. The data are used to estimate the vertical eddy viscosity coefficient, Kν , based on (6.4), and the vertical eddy diffusivities of mass or density, Kρ , given by (6.7), and the vertical diffusivity of heat, KT , taken to be equal to χ T /[2(dT /dz)2 ] using (6.5), (6.6) and (1.11). With taken equal to 0.2, values of KT and Kρ are well correlated and of similar magnitude, appropriate in a region where density variations are dominated by those of temperature. As might be expected from the previous section, the eddy diffusion coefficients are very low at large Ri, but decrease as Ri decreases, rapidly so as Ri decreases through a value of 1/4. At small Ri (0.2 < Ri < 0.4) the eddy viscosity and diffusivity coefficients are given approximately by K ν (m2 s−1 ) = 5.6 × 10−8 Ri −8.2

(7.2)

K ρ (m2 s−1 ) = 3.0 × 10−9 Ri −9.6 ,

(7.3)

and

whilst, when Ri > 0.5, K ν (m2 s−1 ) = [5 × 10−4 /(1 + 5Ri)1.5 ] + 2 × 10−5 ,

(7.4)

K ρ (m2 s−1 ) = [5 × 10−4 /(1 + 5Ri)2.5 ] + 1 × 10−6 .

(7.5)

and


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These expressions fit and parametrize the data, giving eddy coefficients to a multiplicative factor 1, and when the stability parameter is moderate (1 < Rρ < about 2), it is found that has values between 0.4 and 1. In relatively weak shear and moderate stability, salt fingers contribute substantially to the mixing process, even though uniform thermohaline staircase layers are not produced. 12 A value of 0.2 was adopted for and the Cox number, C, was replaced by χ T using (6.5) and (1.12); see footnote 10 in Chapter 6. 13 Here εµ = g2 a2 χ T (1 +Rρ−1 )2 /(2N2 ), where a is the thermal expansion coefficient, Rρ is the ratio of the contributions to density of temperature and salinity, and is the efficiency factor, usually taken to be about 0.2, and εT = c1 L 2T N 2 (see 6.3) where c1 = 0.78 ± 0.14 and LT is the r.m.s. displacement scale (Section 6.3.2). 14 The HRP carries fine structure probes, including an acoustic current meter, sampling at a rate of 10 Hz, in addition to microstructure airfoil probes and microbead thermister, and a dual electrode microconductivity cell.

7.7 Two-dimensional turbulence

7.7

211

Two-dimensional turbulence

It is evident from the laboratory experiments described in Section 7.2.5 and others in which turbulence is generated in the wake of moving bodies (see, for example, Hopfinger, 1987; Chomaz et al., 1993), that the stage of turbulence following the suppression of vertical motions by stratification may generally be one in which motion is dominated by an irregular quasi-two-dimensional field of lens- or pancake-like vortices having small vertical to horizontal aspect ratios. Such clues to the consequences or ‘collapse’ of three-dimensional turbulence support proposals by Lilly (1983) and others, developed mainly in the context of atmospheric dynamics, for the existence of ‘two-dimensional’ turbulence. In this the vertical motion is supposed to be severely constrained by stratification, and motions are dominated by vertically shallow, but horizontally wide, pancake-like eddies of near-uniform density rotating about vertical axes as in Figs. 5.14 or 7.4, a field of vortical mode possibly combined with a still active field of internal wave motions. Given the apparent ease of generating vortices in stratified fluids in the laboratory, it might be thought that the vortical mode motions may be ubiquitous in the ocean. Observations are, however, less than decisive. Measurements so far indicate vortical mode energy levels in the ocean that are substantially less than those of internal waves. Müller et al.’s (1988) analysis of data from the IWEX experiment finds that the kinetic energy of the horizontal motions are sometimes higher than can be inferred from the vertical displacements of isopycnals on the assumption that only internal waves are present. They conclude that vortical mode motions are present with energy levels at wavelengths greater than 30 m of about one tenth those in the internal wave field. Kunze et al. (1990a) suggest, however, that this assessment of vortical mode energy may be at least 100 times too large. Furthermore, Kunze et al. examined data from a neutrally buoyant float at 180–200 m depth off southern California and concluded that the ocean structure at the small vertical scales usually associated with the vortical mode appeared to be dominated by near-inertial internal waves rather than the vortical mode. An investigation of potential vorticity made by Kunze and Sanford (1993) near the Ampere Seamount in the Northeast Atlantic proved inconclusive. It was supposed that the flow passing the seamount might result in water mixed near the sloping sides to separate and evolve into eddies, to be carried as vortical mode into mid-water. Vortical structures of horizontal scales of about 3–7 km and vertical scales of 40–400 m were found to be in geostrophic balance, but their source was not evident; neither their being a consequence of turbulent mixing nor the remnants of water formed in a deep winter mixed layer (see Section 9.3.4) could be ruled out. The balance of evidence has, however, swung back in favour of the presence of vortical mode motions. Examination of strain and shear obtained using the free-falling High-Resolution Profiler (HRP) and moored current meters, mainly in the area of the NATRE study by Polzin et al. (2003), shows that the two are not consistent with their being caused entirely by internal waves. A contribution from sub-inertial (frequencies

212


< f ) fine structure with small height-to-width aspect ratio, vortical mode interacting with the internal waves, is used to explain the observed discrepancy. The inferred vortical mode structure has an aspect ratio of about f/N (or a Berger number near unity), a dominant vertical scale of about 25 m, and an r.m.s. velocity of 0.7 cm s−1 . At vertical wavelengths greater than 30 m the vortical mode structure is inferred to be in approximate geostrophic balance, whilst at vertical wavelengths less than 10 m there appears to be an approximate energy equipartition between internal waves and vortical mode. It is speculated that the source of vortical mode in this mid-water region is the breaking of internal waves, although the mechanism of breaking and energy transfer is not described. The horizontal dispersion contributed by the inferred vortical motions appears consistent with the observed spread of tracer, as explained in Section 13.4.1. Although it is not firmly established that a field vortical mode is present throughout the ocean, the laboratory findings and the postulated generation processes referred to in Section 2.8 suggest that a patchy distribution is a likely possibility. Although energetically weak, perhaps the remains of active turbulence produced by far more energetic processes, it is possible that at horizontal scales of 100 m to about 10 km horizontal dispersion in mid-water is dominated by such horizontal eddies which, in some cases, are features described and identified as fine-structure. At still larger horizontal scales, dispersion appears to be dominated by energetic ‘mesoscale eddies’, as explained in Section 13.4.2.

Chapter 8 The benthic boundary layer

8.1

Introduction

The ‘benthic’ boundary layer (bbl) is the layer overlying the deep ocean seabed within which the mean speed of the overlying water is reduced from its ambient value distant from the boundary to zero at the seabed (or nearly to zero relative to the solid Earth if the seabed is covered by mobile and moving sediment particles). In shallow water the layer, discussed in Chapter 10, is usually referred to as the bottom boundary layer, but many (not all) of its properties are similar, if not identical, to those of the bbl. Turbulence in the bbl is sustained by shear resulting from the stress exerted on the seabed by the water flowing over it, and by flow separation over bottom roughness. The stress can equally be thought of as that exerted by the seabed on the water in reducing its speed to zero. Because of its importance in the understanding of flow over aircraft wings and of the variation in wind speed and gustiness near the ground, the nature of turbulence near rigid boundaries has been the subject of intensive investigation since the early years of the twentieth century, through both laboratory experiments and measurements in the atmospheric boundary layer over land, supplemented in more recent years by numerical simulations. Much of what is know about the structure of turbulence in such boundary layers comes from these extensive and comprehensive studies. Where comparison has been made, the properties of turbulence in the bbl appear similar to those in the laboratory and the atmosphere in equivalent parameter ranges. Unlike the conditions found in stratified flow in mid-water referred to in Section 6.9, the water in the bbl is almost invariably in a state of turbulent motion except in

213

214


a very thin viscosity-dominated sub-layer near the bed.1 Although flow structures within this quasi-laminar sub-layer may interact with the overlying field of motion, the development of turbulence from a laminar or much-less-turbulent state in the bbl is not such an important factor as it is in mid-water; transitional processes are relatively insignificant. Near the seabed, but beyond the viscous sub-layer, shear or ‘mechanical’ mixing dominates and usually the stratification of the water column has a negligible or only small effect. The term principally contributing to the maintenance of turbulent kinetic energy is the shear production term, −ρ 0 uw dU /dz ((iii) in Section 1.7.12, a product of Reynolds stress and mean velocity gradient); this and the dissipation term ((iv) in Section 1.7.12) dominate the steady state turbulent energy balance. Because the stress is a factor important in determining the properties of the layer (including the transport of sediment) and can be measured fairly easily, whereas ε cannot, it is the Reynolds stress (represented via a friction velocity) that is often used to characterize turbulence in the boundary layer, rather than ε. At the outer limits of the bbl, however, turbulence may be constrained by stratification (when the buoyancy flux term, (v) in Section 1.7.12, is significant). In shallow seas without stratification, the presence of the sea surface provides a natural upper limit to the boundary layer. This chapter describes the nature of turbulence in the bbl and its effect on the overlying water in the deep ocean in conditions where the seabed is almost flat and horizontal, typical of the vast abyssal plains. Effects of hydrothermal plumes in ocean ridge topography have already been described in Section 4.5.1. Some effects particular to sloping or rough topography are described in Chapters 11 and 12, respectively. Whilst the more general properties of boundary layers described in Section 8.2 apply equally in shallow seas, some of the effects that are important there (for example, the consequences of wave-induced motions) are left until Chapter 10. Some of the properties of boundary layers touched on in Section 4.3 are again referred to below, whilst those related to the boundary layer at the sea surface are discussed further in Chapter 9. Study of structure and turbulence in the bbl in the deep ocean began with investigations by Wimbush (1970). These were prompted by reports of surprisingly large, super-adiabatic, gradients of temperature (temperature increasing with depth at rates of 10 a to 1000 a , where a is the adiabatic lapse rate, about 0.12 mK m−1 ) extending many metres above the seabed, that had been found in measurements made in connection with studies of the geothermal heat flux within the bottom sediments. On the basis of the predicted gradients described in Sections 8.2.2 and 8.2.3, Wimbush remarks ‘It seems inconceivable that a strongly unstable layer several metres thick can persist’. As explained in Section 8.3.1, he made observations demonstrating that indeed the temperature gradients are much smaller than those reported earlier and that are now known to have been founded on erroneous measurements.

1 Wimbush and Munk (1971) provide a very thorough review of the basic theoretical results relating to the variability of currents and temperature within the bbl, including their spectral forms, based largely on theoretical and laboratory studies by Townsend (1962).

8.2 The structure of the benthic boundary layer

215

Armi and Millard (1976) later investigated the structure and nature of the mixed layer that results from the turbulence generated at the deep seabed, and of the response of the layer to variations (e.g. fronts) in the overlying water column. This benthic mixed layer is described in Section 8.3.2. There have been few investigations of smallscale turbulence in the bbl in the deep ocean, although several studies of sediment transport have been made, notably the high-energy benthic boundary layer experiment (HEBBLE) at a site about 4900 m deep on the Nova Scotia Rise in 1985–1986 (see McCave et al., 1988). These have demonstrated the variability of currents near the deep ocean floor and the episodic occurrence of what are termed ‘deep-sea storms’ by Hollister and McCave (1984), periods of relatively fast currents at abyssal depths, lasting for several days to a few weeks, with speeds generally exceeding 0.2 m s−1 and peak speeds greater than 0.3 m s−1 . These are sufficient to cause erosional features, such as ripples, to appear on a sedimentary sea floor and lead to high concentrations of sediment in the water column above the bed, the benthic nepheloid layers described in Section 8.4.

8.2

The structure of the benthic boundary layer

8.2.1

The viscous sub-layer

It is generally assumed on dimensional grounds, supported by observations, that the stress on the seabed, τ , is related to the speed, U∞ , well above the boundary (beyond the direct effects of influence of the boundary) by the equation 2 τ = ρ0 C D U ∞ ,

(8.1)

where CD is a ‘drag coefficient’ with a empirical value of about 2.5 × 10−3 . The stress √ is represented by a friction velocity, u ∗ = (τ /ρ 0 ), which, from (8.1), is given by U∗ ≈ 5 × 10−2 U∞ .

(8.2)

When the substrate is flat and immobile,2 the flow at the bed is reduced to zero by viscosity. Viscous effects dominate the flow in a layer, the viscous sub-layer, which extends to a height, z ν , of about 10ν/u ∗ from the bed and in which the mean flow is U(z) = zu 2∗ /ν. The layer thickness, z ν , is often small (e.g. if U ∞ = 0.01 m s−1 , u ∗ ≈ 5 × 10−4 m s−1 , so the height z ν ≈ 2 cm: in greater currents it will be even smaller). In many parts of the ocean it may be less than height, δ, of irregularities or roughness elements in the bed that then determine the nature of the more distant flow.

2 The definition of the location of the bottom or ‘boundary’ becomes imprecise when sediment on the bed is fully, or even partly, in motion, and the appropriate boundary conditions on the flow speed at the bottom must involve some consideration of the composition of the bed and the mechanisms by which it is set into motion by the fluid.


216

8.2.2

The viscous–conductive sub-layer

Heat may be conducted through the viscous sub-layer. The mean geothermal flux of heat, H, through the seabed, excluding the relatively high rates in the hydrothermal vent regions of mid-ocean ridges, is about 46 mW m−2 (Huang, 1999). In a viscous– conductive sub-layer very close to the seabed heat is conducted upwards by molecular conduction that, from (1.4), results in a mean vertical temperature gradient of –(H /(κ T cp ρ) + a ) or about −80 mK m−1 , which, in its magnitude, exceeds the vertical adiabatic temperature gradient of about −0.12 mK m−1 .

8.2.3

The outer boundary layer

At distances from the bed exceeding z ν and the vertical scale, δ, of the bottom roughness elements, a regime is found in which the motion is turbulent, eddies scale with distance from the boundary, and dissipation and mean flow are given the law of the wall relationships (4.9) ε = u 3∗ kz, and U (z) = (u ∗ /k)ln(z/z 0 ),

(4.8)

where k is von Kármán’s constant, about 0.41, and z 0 is the roughness length, the ‘effective level’ of z at which U approaches zero. The size of z 0 is about 0.1ν/u ∗ if δ < 3ν/ u ∗ , or δ/30 if δ > 3ν/u ∗ .3 The eddy viscosity derived from (1.8) is simply K ν = ku ∗ z, increasing linearly with height off the bottom. The mean vertical temperature gradient within this layer resulting from a geothermal heat flux, H, is dT /dz = −(H/(ku ∗ zcp ρ0 ) + a ),

(8.3)

which is generally small, dynamically passive and only slightly less than − a . (For example, if H = 46 mW m−2 and U = 0.02 m s−1 , so u ∗ ≈ 1 × 10−3 m s−1 , and k = 0.41, cp ≈ 3.9 × 103 J kg−1 K−1 and ρ 0 ≈ 1.04 × 103 kg m−3 , the gradient 1 m off the bed is about –0.15 mK m−1 , or −1.2 a .) This is the basis of the doubts expressed by Wimbush regarding the reliability of early observations. The presence of stable stratification resulting from relatively large erosion rates and high concentrations of suspended sediment may modify the variation of mean current with height off the bed as does the high buoyancy flux described in Section 4.3 (Adams and Weatherley, 1981). A consequence is that, in conditions of high sediment concentrations, values of the bed stress, τ = ρ 0 u 2∗ , obtained by fitting observed mean velocities to the logarithmic form (4.8) will be overestimates of the actual stress.

3 More about the nature of boundary layers above rough beds can be found in the review by Jiménez (2004).


8.2.4

217

Coherent structures

The viscous sub-layer at distances less than about 10zν from the boundary has been most thoroughly studied in wind and water tunnels in high Reynolds number flows approaching those found at sea, where the Reynolds number, Re = U∞ h/ν is typically about 105 or more. Here h is the thickness of the boundary layer. (A layer may be ‘capped’ by overlying stratification, a long-lasting layer will be limited by the effects of the Earth’s rotation as described in Section 8.2.6 or, in shallow water, a layer may be bounded by the presence of the sea surface or limited by tidal effects as explained in Chapter 10.) Close to the bottom two structures are found and illustrated in Fig. 8.1, Plate 16. There are arrays of sinuous ‘streaks’, parallel flow-aligned bands in which the flow is alternately greater or less than the mean. Shear, and viscous wall stress, is increased in streaks of enhanced flow. The streaks are typically 100z ν in length and separated by about 10z ν in the across-flow direction. ‘Streamwise vortices’ are also found. These are of somewhat smaller scale and more random location. They are slightly tilted away from the bottom, remaining in the region below 10z ν for distances of about 20z ν . Several are associated with each streak, typically separated by a downstream distance of order 40z ν . These coherent structures are generally regarded as being initiated by processes of nonlinear, viscous, shear instability. Outside the viscous sub-layer and beyond a distance of about 10zν from the boundary in high Reynolds number flows are found long ‘hairpin vortices’ (Fig. 8.2), tilted up from the bottom.4 Several streamwise vortices may be connected to the trailing legs of the hairpins (as to the streaks), but most appear to merge with the disorganised and unstructured field of vorticity beyond the wall layer. More important, because of their substantial, but intermittent, contribution to the uw component of Reynolds stress, are events known as ‘sweeps’ (or ‘inrushes’) with u > 0 and w < 0, and ‘bursts’ (‘eruptions’ or ‘ejections’) with w > 0 and generally u < 0, vertically moving eddy structures. Of the two classes, the former leads to the rapid transfer of relatively fast moving water towards the bottom, and the latter transfers water from the bottom into the body of the log layer and sometimes beyond (Fig. 8.3). Although vortices are probably involved, the cause of these events and their connection to the other coherent structures within the layer are not fully understood. The time between bursts observed at a fixed location appears to scale with the dimensions of the log layer, being about (3–7)h/U ∞ (Jackson, 1976).5 There is, however, debate about whether their onset may be a consequence of convergence of streaks within the viscous boundary layer, that might infer a scaling with ν, or an amalgamation of hairpin vortices.

4 Robinson (1991) gives a well-illustrated review of the history and development of ideas about coherent vortical structures in the log layer and of their ‘anatomy’. 5 For example, the time is about 30−80 s if h ≈ 5.5 m and U∞ ≈ 0.5 m s−1 , although these values are more typical of flows in shallow seas described in Chapter 10. In ‘deep-sea storms’, with currents of 0.2 m s−1 and a boundary layer thickness of, say, 20 m, the times will be longer, typically 300−700 s.

218


Figure 8.2. The form and development (A to C) of a hairpin vortex in a boundary layer at high Reynolds number. (After Robinson, 1991.)

Figure 8.3. A record of velocity fluctuations at a height of 1.5 m off the bottom showing sweeps and ejections in the boundary layer. Top: u, the variations from the mean downstream velocity component (0.49 m s−l ). Middle: w, the vertical velocity component. Bottom: the product, uw, showing the highly intermittent nature of the contributions to the Reynolds stress, −ρ 0 uw . The scale on the right shows uw/σ uw where σ uw is the standard deviation of uw. Sweeps and ejections are marked as S and E, respectively. Both make large contributions to the Reynolds stress. (These records come, not from the bbl but from a water depth of 52 m in the Irish Sea.) (From Heathershaw, 1979.)


219

Groups of hairpin vortices can occur. Laboratory experiments by Adrian et al. (2000) using particle image velocimetry (PIV) find coherent groups of 3–4 hairpin vortices aligned in the streamwise direction. The groups of vortices originate in the vicinity of the boundary and extend into the log layer, the aggregated effect of the vortices creating momentum anomalies that are more extensive than those of the individual vortices. The origin of groups and the cause of their alignment or the number of vortices occurring in groups are presently unknown. 8.2.5

Geothermal heat flux and effects of organic matter

In the deep ocean the effect of the geothermal heat flux is sometimes profound, particularly in spreading ridge systems where hydrothermal vents occur, ejecting fluid in jets at temperatures of as much as 350 ◦ C (Fig. 4.1, Plate 7). As explained in Section 4.4.2, the heated buoyant vent fluid sometimes rises to heights of several hundred metres above the seabed and may be detectable hundreds of kilometres away. Elsewhere, however, the effect of the geothermal heat flux is not significant. Excepting the ridges, the mean heat flux of about 46 mW m−2 gives a flux of buoyancy per unit area, B, of 1.88 × 10−11 m2 s−3 , from (4.1).6 Use of (8.2) to determine u ∗ gives a Monin–Obukov length scale, L MO = −u 3∗ /k B, of about −1.7 × 107 U3 . (L MO is measured in metres and U in m s−1 , and k = 0.41.) For currents of 0.05 m s−1 , typical of the deep ocean, L MO is about −2 km. A regime of forced convection driven by mechanical mixing, not by the heat flux, is therefore expected at heights above the bed less than about 0.03|LMO | ≈ 60 m (Section 4.3), whilst free convection can only occur above about |LMO |, ≈ 2 km, a height where density stratification in the body of the ocean is dominant; free convection is very unlikely to occur. Whilst the total geothermal heat flux contributes to the heating of the abyssal ocean (see, for example, Joyce et al., 1986), the mean heat flux, and consequent buoyancy flux, generates potential energy available for mixing the ocean at the relatively small rate of only 0.05 TW (Huang, 1999; values corrected by Wunsch and Ferrari, 2004); geothermal heat flux does not contribute substantially to the deep ocean mixing (requiring an estimated 2.1 TW). In some areas there is a further contribution to heat flux that requires consideration; the heat produced by the decomposition of organic matter, the sinking detritus produced by algal blooms in the upper ocean. Graf (1989) describes observations made on the Vøring Plateau off the coastal margin of Norway at a water depth of 1430 m. At the end of May pulses of faecal copepod pellets arrive at the seabed, reaching a peak of 45 mg carbon m−2 d−1 . The pellets are incorporated into the sediment at speeds exceeding 1 cm d−1 , being carried down the burrows of benthic organisms (Nephasoma sp.). The heat (per unit volume) produced by the subsequent decomposition of the organic matter measured by direct microcalorimetry decreases from about 30 W m−3 at 1–5 cm 6 The value of B is found using α = 1.71 × 10−4 K−1 (appropriate at a temperature of 2 ◦ C at a depth of 4000 m; see Gill, 1982, table A3.1), cp = 3.99 × 103 J kg−1 K−1 , and ρ 0 = 1.03 × 103 kg m−3 .


220

depth, to zero at about 10 cm, providing a net upward flux of heat (per unit area) of order 1 W m−2 from the sediments, substantially greater than the mean geothermal heat flux and possibly driving convective motions in the bbl during short and intermittent periods following surface blooms.

8.2.6

The effect of the Earth’s rotation

Boundary layers that persist for times of order of the local inertial period are affected by the Earth’s rotation. First studied by Ekman (1905), mean flows spiral in direction as the boundary is approached, a consequence of Coriolis and frictional forces. As a result, the net transport within the bbl is not in the direction of the overlying flow. The thickness of the Ekman boundary layer, hE , determined from laboratory experiments, is about 0.4u ∗ /f. (For U = 0.1 ms−1 and f = 1.03 × 10−4 s−1 at 45◦ N, u ∗ is about 5 × 10−3 m s−1 and h E is about 19 m.) When driven by a mesoscale eddy, the ‘Ekman transport’ in the layer leads to convergence or divergence depending on whether the eddy is cyclonic or anticyclonic, producing a tendency for upward or downward ‘Ekman pumping’ motions. Even in non-steady conditions, the consequences of the Earth’s rotation may be substantial, leading, for example, to the generation of diurnal jets in the ocean surface mixed layer (see Section 9.3.3).

8.3

Observations in the deep ocean

8.3.1

Turbulence near the seabed

Wimbush (1970) describes observations of temperature made at heights up to about 3.5 m off the bottom in depths of 3700 m off the coast of southern California. Using a vertical array of quartz crystal sensors that change their resonant frequency with temperature, he estimates the vertical temperature gradient in the region above the 1–4 cm thick viscous–conductive boundary layer. The sensors had a resolution of 1.7 × 10−5 K, and inter-calibration was achieved by arranging for a pair of the sensors to fall on a boom during the period of their deployment, so as to obtain comparative measurements at the same levels as other sensors. Photographs established that the bottom was very smooth. The observed vertical temperature gradient was about −1.3 a , in good agreement with the value predicted using (8.3). Further observations by Wimbush and Munk (1971) in the same area at depths of 2000–4000 km used indirectly heated thermistors7 to measure current speed to a precision of 0.1 cm s−1 . The measured variation of the mean flow with height is consistent with the law of the wall up to heights of about 1 m. Measurements of thermal gradients and spectra are also in accord with measurements in other boundary layers over rigid surfaces. 7 These work on the same principle as a hot film anemometer (footnote 2 in Chapter 6).

8.3 Observations in the deep ocean

221

Elliott (1984) made near-bed measurements with electromagnetic current meters8 at a depth exceeding 4 km in the Northeast Atlantic. The Reynolds stress at a height of 0.5 m from the seabed implies a drag coefficient of about 2 × 10−3 , in fair accord with observations in shallow water. Estimated values of the non-dimensionalized variance of the downstream, cross-stream and vertical velocity fluctuations, σ u /u ∗ , σ v /u ∗ and σ w /u ∗ , equal to 2.35 ± 0.14, 1.75 ± 0.14 and 1.37 ± 0.06, respectively, are also in approximate agreement with the respective values, 2.4, 1.9 and 1.2, found in the bottom layer of shallow seas (Soulsby, 1983). There appears to be no reason to suppose that near-bed turbulence in the abyssal ocean differs in any fundamental way from that dominated by tidal flows in shallow seas described in Section 10.2.2. 8.3.2

The structure of the benthic mixed layer; benthic fronts

The boundary layer on the 5500 m deep Hatteras Abyssal Plain in the western North Atlantic was studied in the 1970s by Armi and Millard (1976) using long-term moorings and yo-yo surveys made by raising and lowering a CDT, towed at slow speeds. Water in the bottom layer is well mixed, with potential temperature changes generally less than 1 mK between the bed and the top of the 10–60 m thick layer. Often the mixed layer is capped by a sharp change in potential temperature of order 10 mK in 10 m. The thickness of the mixed layer over a flat seabed is typically about 2.4u ∗ / f , or six times the Ekman layer thickness, h E = 0.4u ∗ / f . For comparison, the boundary structure over rough or sloping topography surrounding the Plain is shown in Fig. 8.4. It is more variable there and the layer thickness varies between hE and 6hE . Even on the Plain, there are substantial variations of boundary layer structure in time at a fixed location, as illustrated in Fig. 8.5. Although the mixed layer is sometimes uniform over horizontal distances of typically 10 km, it may exhibit horizontal gradients of up to 20 mK km−1 (Armi and D’Asaro, 1980). Temperature fronts have been found in the bbl on the 5300 m deep Madeira Abyssal Plain. Typically 300 m wide, these benthic fronts separate bottom mixed layers that differ in temperature by 2–4 mK (Fig. 8.6). They have lengths of at least 8 km, and isotherms within them are tilted at about 10◦ to the horizontal. A probable cause of the benthic fronts is the straining of the temperature field and the Ekman pumping caused by neighbouring mesoscale eddies. 8 The original electromagnetic current meters, including those used by Bowden and Fairbairn (see Section 10.1) were discus-shaped, 20 cm or more in diameter, containing a solenoid that produces a magnetic field. The flow of the conducting seawater through this field generates a potential difference (pd) as a result of the Faraday effect, and this is sensed by two orthogonal pairs of electrodes exposed to the water on the face of the discus. The component of flow normal to the line joining a pair of electrodes is proportional to the measured pd. The two pairs allow the horizontal and vertical flow components to be measured simultaneously and consequently their product, proportional to the Reynolds stress, can be obtained. The absence of moving parts gives the instrument relatively high sampling rates and avoids frictional problems found in mechanical current meters in low flows, but it has not been found possible to reduce its size so that it can resolve spatial variations over scales of order 1 cm. The sensors used by Elliott are torroidal in shape, like flattened ring doughnuts, with open centres to reduce flow obstruction, and are 15 cm in diameter.

Figure 8.4. Profiles of potential temperature over the benthic seabed showing the nature and variability of the nearbottom mixed layer. The vertical axis is height off the bottom. Data are in four groups, moving (left to right) eastward from the flat Hatteras Abyssal Plain onto the rough topography of the mid-Atlantic Ridge where profiles, and mixed layer thickness, are much more variable. The benthic mixed layer is often capped by a sharp change in potential temperature of order 10 mK in 10 m. (From Armi and Millard, 1976.)

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Figure 8.5. Variability of the benthic boundary layer. The structure of the bbl on the Hatteras Abyssal Plain measured over a period of almost a month from a fixed mooring with 7 sensors spanning 15–85 m off the bottom. (a) Contours of isotherms. None is shown below the level of the deepset sensor at 15 m off the bottom. The top of the bbl is particularly clear in the periods 18–23 June and 8–13 July. There is a benthic front where temperatures increase at all levels on 4–5 June and a (possibly associated) mixed interior layer at about 45 m, above the bbl, in the period 4–8 June. (b) Values of potential temperature. Large changes occur when the top of the benthic mixed layer rises or falls past a sensor. (From Armi and D’Asaro, 1980.)

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Figure 8.6. Four examples of benthic fronts observed on the Madeira Abyssal Plain at a depth of 5290 m, reconstructed from data collected at a fixed mooring as the fronts were advected past. Temperature contours are drawn at 0.5 mK intervals. The bottom mixed layer in the area is about 20–50 m thick. Fronts are typically 300 m wide, have lengths of at least 8 km, and are tilted at about 10◦ to the horizontal, separating bottom mixed layers with temperature differences of 2–4 mK. (From Thorpe, 1983.)

Layers of nearly constant potential temperature above the mixed layer, that Armi and Millard find are present for about 25% of the time and have horizontal extent of 2–100 km, may also be related to the fronts. The layers sometimes appear to connect to the mixed layer below, suggesting their formation by a process of the detachment of the mixed layer from the bottom by the intrusion of water of greater density. The fronts are possibly related to the ‘deep-sea storms’ referred to in Section 8.1.

8.4 Nepheloid layers

8.3.3

225

Dispersion in the boundary layer

Measurements reported by Armi and D’Asaro (1980) of the vertical distribution of radon, a naturally occurring radio-isotope with a half-life of 3.8 days that enters the bbl from the seabed, found uniform vertical profiles in the mixed layer and no excess radon above it. This indicates that the time required to mix water through the mixed layer is less than 3.8 days and that the time of exchange with the overlying water exceeds 3.8 days. Based on measurements made from an array of near-bottom current meters on the Madeira Abyssal Plain and on the tracking of clusters of neutrally buoyant floats, Saunders (1983) estimates the horizontal dispersion coefficient of particle pairs to be about 0.5 m2 s−1 at separations of 1 km, and about 200 m2 s−1 at separations of 20 km, with rather greater diffusivity in the east–west direction than north–south.9 The average time for the mean squared separation of floats to double is 10–15 days. The horizontal dispersion coefficient at a 300 m thick benthic front is estimated to be 2 × 10−2 m2 s−1 (Thorpe, 1983).10

8.4

Nepheloid layers

Nepheloid layers are regions of relatively low light transmission resulting from the presence of suspended matter, usually sediment particles that have been lifted from the seabed by currents large enough to scour sediment. Detrital pellets and ‘marine fluff’, flocculated material of biological origin in the upper ocean, may be contributory components. Nepheloid layers are most often observed adjacent to the seabed, and such layers are referred to as benthic nepheloid layers. Observations during the HEBBLE experiment found that typically about three ‘storms’ occur each year bringing sediment into suspension in a region well removed from coastal or continental shelf sources of sediment. In one ‘storm’ the sediment concentration varied from its usual ‘background’ level of less than 500 µg l−1 (about 5 × 10−4 kg m−3 ) to about 5000 µg l−1 (5 × 10−3 kg m−3 ) at a height of 2 m off the seabed. Small particles, those less than 5 µm in diameter which provide most of the light scattering recorded by deep-sea nephelometers, have long residence times (Hollister and McCave, 1984), their fall speed in tranquil water being about 4.6 × 10−6 m s−1 (McCave, 1984). Consequently, it may be several months before eroded sediment sinks back to the seabed and, during this time, the nepheloid layer may be advected far from the location in which sediment was originally carried into suspension.11 9 These values compare with estimates based on Okubo’s formula, (9.9), of 0.3 m2 s−1 and 9.1 m2 s−1 , respectively, derived from observations in the upper ocean. 10 This is a little less than 7 × 10−2 m2 s−1 found by using Okubo’s formula (9.9). 11 In a period of a month, a benthic nepheloid layer may be carried by a mean flow of about 5 cm s−1 over a distance of about 130 km. (For quick, ‘rough and ready’ estimation, it is worth recalling the relationship that 1 cm s−1 is roughly 1 km per day; actually it is 0.864 km per day.)


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Figure 8.7. Benthic and intermediate nepheloid layers formed over the Porcupine Bank, west of Ireland. (a) Five density profiles made across the slope at the edge of the Bank from (right) over the Bank to (left) deep water. The mixed benthic boundary layers are of small vertical extent. (b) The corresponding five nephelometer profiles, lower light transmission being indicated by relatively high values of nephels, showing bottom nepheloid layers (BNL) and an intermediate nepheloid layer (INL) separating from the bottom at a depth of about 700 m. Also shown is a near-surface layer of relatively low light transmission (SNL). (From Dickson and McCave, 1986.)

One effect of the return of suspended sediment to the seabed in low flow conditions following a ‘storm’ is to blanket roughness features. This smoothing is however sometimes relatively short lived: a small roughness length, z 0 , following one storm and subsequent sediment deposition was found to be increased by a factor of about 5 as a consequence of benthic fauna tracks, burrows and defecation during a period of low flow lasting for only a few days (Gross et al., 1988). Nepheloid layers are sometimes found as ‘intermediate nepheloid layers’ (INL) in mid-water, commonly as a consequence of the off-slope advection of water from an erosion site on the continental shelf (Drake, 1971; Pak et al., 1980; Hickey et al., 1986; Dickson and McCave, 1986). An example is shown in Fig. 8.7. Intermediate

8.4 Nepheloid layers

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nepheloid layers thought to be caused through the erosion of the seabed by barotropic tides, or by internal tidal waves having frequencies close to critical (as described in Section 11.4.1), have been observed adjacent to the continental slope and rise (Moum et al., 2002). One such layer adjacent to the Porcupine Bank in the Northeast Atlantic at a depth of about 2550 m apparently persisted for at least 8 months, and was found to extend along the slope for over 100 km and off-slope to a distance of about 1.4 times the internal Rossby radius, L Ro (Thorpe and White, 1988). Intermediate nepheloid layers are also common in the spread of the plumes caused by hydrothermal vents described in Section 4.4.2.

Chapter 9 The upper ocean boundary layer

9.1

Introduction

The upper ocean boundary layer is taken here to include the sea surface and its wave field, the mixed layer and the upper part of the pycnocline, the depth range of the ocean most directly affected by the atmosphere. The mixed layer has some properties that are similar to those of the benthic boundary layer described in Chapter 8, but a number that differ, notably those associated with the variable forcing of the overlying atmospheric boundary layer and the presence of surface gravity waves on the air–sea interface: the sea surface is mobile and changes shape rapidly, unlike the relatively fixed seabed.1 The atmosphere drives the upper ocean boundary layer through buoyancy fluxes derived from heating (Section 1.5 and Equation (4.1)) and precipitation, and through the stress, τ w , of the wind on the water surface. Measurement of τ w is no easy matter. 2 , to the square of the wind speed, It is usually related by the equation, τ w = ρ a CD W10 W10 , measured at a standard height of 10 m above the mean water surface, and the density of air, ρ a , about 1.2 × 10−3 ρ 0 . The drag coefficient, CD , is typically (1−2.5) × 10−3 but, for W10 greater than about 4 m s−1 , increases with W10 (see Jones and Toba, 2001; Csanady, 2001), although recent studies show a decrease in CD in hurricanes (Powell et al., 2003). (The rate at which energy is transferred to the ocean from the atmosphere is discussed in the context of large-scale processes in Section 13.2.2.) 1 Conditions within the boundary layer under extensive ice floes may more closely resemble those in the benthic boundary layer. Measurements of turbulent stress by McPhee (2002), for example, show that the roughness length, z0 , under un-deformed multiyear ice at a site in the Arctic lies in the range 0.48 cm < z0 < 0.7 cm. Complexity is however added by the wakes of isolated large ice keels (Skyllingstad et al., 2003) and by the presence of ice leads (McPhee and Stanton, 1996) within which diurnal cycles of heating can sometimes be detected.

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9.1 Introduction

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Ekman pumping resulting from large-scale horizontal divergence produced by variations in the wind-driven Ekman transport (see Gill, 1982, section 9.4) generates vertical motions of order 30 m per year in mid-latitudes to which the ocean mixed layer may respond by entraining or detraining fluid. Density fronts formed at the boundaries of different water masses, upwelling regions and mesoscale eddies, are a significant and sometimes long-lived feature of the large-scale structure of the mixed layer, one that requires careful assessment in modelling the large-scale vertical transports of heat and momentum at the sea surface. Over sufficiently large scales the mixed layer is consequently not horizontally uniform: reference has already been made in Section 4.5.2 to the density-compensating changes in temperature and salinity discovered by Stommel and Fedorov (1967) and Rudnick and Ferrari (1999). Many of the processes relating to turbulence in the upper ocean boundary layer are, however, of relatively small space and time scales in which fronts and horizontal gradients are insignificant. The existence of a mixed layer at the sea surface with little or no variation of temperature and salinity with depth has been long known but, in comparison with the atmospheric boundary layer over land, rather few studies have been made of the turbulent processes that lead to its uniformity. Irvine Langmuir, the eminent American chemist and physicist, was the first to recognize the dynamical significance of the accumulation of flotsam in ‘windrows’. During a sea crossing of the North Atlantic in 1927 he had seen narrow lines of floating sargassum aligned in the wind direction. He subsequently conducted studies in a lake that indicated that, associated with windrows, there is a pattern of circulation consisting of a set of vortices of alternating rotation and with axes directed downwind. This is now know as Langmuir circulation. The windrows form where this pattern leads to convergence at the water surface. Langmuir postulated that this circulatory motion was the main mechanism by which the mixed layer is produced. The circulation has been found to be highly variable and unsteady, with a turbulent character, and is by no means the only process active in the mixed layer that helps to maintain its uniformity. Breaking surface waves, convection and shear are other contributory processes discussed in this chapter. It is sometimes useful, following Turner (1973, section 4.3.1), to classify mechanisms that produce turbulence according to whether their energy is derived from an external or internal process. The wind stress, and the consequent breaking waves, and air–sea buoyancy flux, may be regarded as external to the upper ocean boundary layer, and the associated turbulence caused by convection, Langmuir circulation or breaking waves is externally driven, and should therefore scale with the external forcing, the heat flux and wind in the atmospheric boundary layer, as well as with distance from the sea surface.2 It is less evident that turbulence that is derived from shear within the mixed layer, possibly that leading to the temperature ramps described in Section 9.5, or that produced in the upper pycnocline by breaking internal waves, the immediate sources 2 Precipitation (e.g. rain or snow) may act as a buoyancy flux or directly generate turbulence – see Sections 9.2.5 and 9.3.3.

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of which may not be the atmosphere, will scale directly with atmospheric forcing, or at least that present at the time at which turbulence is observed; the internal processes generating turbulence may be a consequence of past external forcing events. This is the case when swell, propagating from its generation at a distant location several days earlier, affects the local wave breaking. Internal waves may similarly be of an origin distant in time, if not space. The physical presence of boundaries, the sea surface and the pycnocline, will affect the form and scale of turbulence within the mixed layer. Turbulence generated internally or, for example, generated earlier by breaking waves and impinging on the surface after breaking has ceased, will interact with and be modified by the presence of the water surface (Hunt and Graham, 1978; Brumley and Jirka, 1987, 1988). As small eddies approach the surface, their scale measured normal to the sea surface becomes less than the tangential scale. The ratio of the normal to tangential ratio may, however, be strongly affected by any surfactant contamination of the water surface which limits the freedom of tangential motion. Vertical profiles made by lowered CTD instruments or free-fall probes have provided a wealth of information about the microstructure of the upper ocean boundary layer and of the relation of small-scale turbulence to surface forcing. There are, however, as yet, few data to connect coherent structures, such as those described by Langmuir, to the small-scale turbulence that is mainly responsible for the dissipation of energy in the layer. Furthermore, the region very close to the sea surface, although most accessible, has proved to be one of the hardest to sample, mainly because of the sometimes violent motion of the waves that makes precise measurement extremely difficult. The difficulty is basically that of designing and constructing a robust, stable, non-vibrating platform from which to measure turbulence that does not, through its wake or physical presence, affect the water motion: measurements from instruments lowered from a research vessel are reliable only at depths well beyond the vessel’s draft and, because of wave orbital motions, may be in their own wake. Inverted (rising) probes do, however, provide a possible means of sampling the water column right up to the surface. The value of their near-surface measurement may be limited by whether it is possible to collect the large number of profiles necessary to make statistically robust and unbiased estimates of mean quantities in the very intermittent field of randomly breaking surface waves, the principal sites of intense turbulence. It is also hard to discriminate between turbulent motions and the orbital motions produced by waves. Only in recent years has much advance been made in investigating turbulence near the sea surface. The horizontally averaged dissipation rate, ε, is found to follow the law of the wall variation in depth in the upper part of the mixed layer. In the early 1990s it became clear that values of ε close to the sea surface are higher than those of the law of the wall relationship, turbulence generated by breaking waves being the likely cause (Agrawal et al., 1992). Breakers, discussed in the following section, disrupt the cold skin of the ocean and produce sub-surface bubble clouds, both processes that affect air–sea gas fluxes.

9.2 Breaking surface waves

9.2

Breaking surface waves

9.2.1

Types of breaker

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Three distinct types of breaker are known in deep water.3 In the first two, energy is directly lost from gravity waves without the involvement of other waves. The plunging breakers frequently observed as waves breaking at the edge of the surf zone also occur in deep water, although the processes leading to breaking differ, the effects of shoaling water being predominant at the edge of the surf zone whilst wind forcing and wave interactions are most important in deep water. Approaching breaking, the wave crest moves forward relative to the remainder of the wave, steepening the forward face of the wave until the crest eventually overhangs the free surface ahead of it and falls under gravity, impacting on the wave surface and trapping a tube of air that rapidly fragments into clouds of bubbles with accompanying creation of turbulence and spray. Spilling breakers are those in which ‘white water’, clouds of bubbles, are entrained into the water near the wave crest as it advances, a process sometimes initiated by a small plunging breaker or steep short gravity waves on the wave crest. Capillary waves with wavelengths less than about 2 cm are formed on the forward faces of steep gravity waves, typically 10–20 cm in wavelength, that occur at short fetches in strong winds or are seen on the upwind slopes of large waves or breakers. These ‘parasitic capillary waves’ are sometimes sufficiently steep to break by entraining small bubbles or, through augmenting the shear at the water surface, to create turbulence in the water without the water surface necessarily being broken. In very strong winds, 10–30 cm gravity waves form a succession of foaming bores on the upwind face of larger, often breaking waves, resulting in intense air entrainment. The presence of capillary waves and bores, as well as bubbles, introduces the additional complexity of measuring and accounting for the magnitude and variability of surface tension caused by the presence of natural and anthropogenic surface films on the sea surface which can strongly damp the capillaries and affect gas transfer from bubbles. Such microscale wave breaking without air entrainment may be relatively insignificant in the generation of turbulence within the ocean mixed layer, although it may be important in some laboratory-scale experiments in which capillary wave breaking dominates. It can, however, result in the generation of 1 cm scale coherent vortices that may affect gas transfer by disrupting the surface skin and by replacing the water at the air–sea interface, a process of ‘surface renewal’ (Siddiqui et al., 2004). Molecular processes may dominate the transfer of some properties across the air– sea interface in sufficiently low wind speeds, typically those less than about 3 m s−1 or slightly higher if the sea surface is covered with a layer of surface-active material that damps capillary ripples (i.e. in conditions in which spilling and plunging waves are infrequent). One such property is that of dissolved gases, such as oxygen. The 3 Surface wave breaking is reviewed by Melville (1996).


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cold skin of the ocean described in Section 1.5.5 is a manifestation that a quiescent layer exists. Consequently, even though the air above and the water below the interface are turbulent, the fluxes of some properties at or very close to the air–sea interface depend on the molecular coefficients. (This may appear contrary to the usual independence of the turbulent fluxes and other mean properties of a turbulent flow from the molecular coefficients, assumed for example in the dimensional argument to establish the law of the wall in Section 4.3. At low wind speeds, the sea surface may behave, over much of its area, like a viscous–conductive boundary layer above a rigid surface, described in Sections 8.2.1 and 8.2.2, a layer in which molecular properties are dominant in determining flux, and one that is largely unaffected by waves, microscale breaking, or the secondary motions described in Section 9.4.2.4 ) As wind speed increases, however, disruptive events associated with the generation of turbulence, including wave breaking, and bubble and foam production, become more frequent and widespread, and eventually control the transfer rates at sufficiently high wind speeds. Even then, however, processes (such as gas flux) that are enhanced by the dissolution of bubbles may depend on bubble-related coefficients, such as surface tension, as described in Section 9.2.4. In very high winds, the air–sea interface becomes a liquid– gas two-phase fluid, with turbulent properties and gas fluxes that are yet to be fully described.5

9.2.2

Laboratory studies and vorticity generation

Much of what is known of the process of wave breaking comes from laboratory studies. Whilst these are usually of breakers in quiescent, non-turbulent water, without the presence of shear or rotational flow that may affect the turbulence produced by breakers at sea or the character of the breaking process itself, they do provide quantitative information about processes that affect breaking and the rates of transfer. To a first approximation in wave slope, the energy density in surface gravity waves of amplitude, a, is E = gρ0 a 2 /2,

(9.1)

and the energy flux is Ecg , where cg = ∂σ /∂k is the group velocity, cg = c [1 + 2kh/ sinh(2kh)] ,

(9.2)

4 Unlike the benthic or bottom boundary layers, the velocity conditions at the water surface are approximately ‘free-slip’ but subject to the effects of surfactants. The scales of near-surface motion induced by eddies impinging on the surface from below will be modified as described in Section 9.1. 5 Brocchini and Peregrine (2001) have begun a systematic study of such violent turbulence at the sea surface. They characterize turbulence by Froude and Weber numbers defined as Fr = q(2gL)−1/2 and We = q 2 L(2γ /ρ 0 )−1 , measuring the relative effects of turbulence and of gravity or surface tension, respectively, where q is the rms turbulent velocity in patches of size, L, and γ is the surface tension, and illustrate the nature of very strong turbulence (with large Fr) by the wake of a fast-moving surface craft.


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where k is the wavenumber, σ is the wave frequency given by the dispersion relation, σ 2 = gk tanh kh, c = σ /k is the phase speed, and h is the water depth.6 In deep water (9.2) reduces to cg = c/2,

(9.3)

The rate of energy loss per unit length of a breaking wave crest, Et , is found by Duncan (1981) to be related to the phase speed of the breaking wave, cb , by E t = bρ0 cb5 /g,

(9.4)

where b is an empirical constant of order 10−3 −10−2 . Laboratory experiments by Rapp and Melville (1990) find that the breaking of twodimensional gravity waves through spilling breakers involves the loss of about 10% of the wave energy, whilst as much as 25% is lost in plunging breakers.7 Of the energy lost by the breakers, some 90% is dissipated by turbulence in the water column within four wave periods of the breaking, turbulent energy subsequently decaying at a rate proportional the reciprocal of time (t−1 ).8 Velocity fluctuations of magnitude 0.005cb , where cb is the phase speed of the breaker, remain in the water even after 50 wave periods after breaking as shown in Fig. 9.1. Dye placed on the water surface before breaking occurs is diffused vertically by breakers of height, H, to depths of about 1.2 H within one wave period after breaking, with further increase being roughly proportional to t1/4 . As much as 40% of the energy lost by breakers may go into the production of bubbles (Lamarre and Melville, 1994). Also prominent in the motion field left by breakers are coherent ‘rotors’ of diameter roughly equal to the wave height with axes parallel to the wave crests. The circulation of water in rotors shown in Fig. 9.2 leads a motion near the surface that is in the wave direction. The rotors slowly advance in the wave direction under the influence of their virtual images in the mean sea surface. Their axes gradually deepen, and their energy decays by a factor of about 10 at a time of 50 wave periods after breaking and at a rate proportional to t−1 (Melville et al., 2002). Although Hornung et al. (1995) describe the rotational flow downstream of a hydraulic jump and the mechanisms through which breakers generate vorticity in the 6 Like interfacial waves (but unlike the internal waves in uniformly stratified fluids) described in Chapter 2, the group velocity of surface waves is parallel to the direction of advance of lines of constant phase, and so normal to wave crests. 7 Three-dimensional breaking waves are commonly observed in deep water; waves have finite crest lengths and a generally broad directional spread. These are similar to breakers formed as waves recombine after passing a small island or rock, and have been produced in laboratory experiments by Wu and Nepf (2002) by spatial focusing with a controlled wave generator. Three-dimensional plunging waves lose relatively about twice as much energy as two-dimensional breakers, but there is little difference for spilling breakers. 8 Much more rapid rates of energy dissipation (∼ t−7 ) and anisotropy (possibly caused by bubble rise) are reported by Gemmrich and Farmer (personal communication) in observations made at about 0.2 m mean depth in breaking waves at sea in winds exceeding 10 m s−1 . At mean depths of 1–1.7 m with a significant wave height of about 2 m, and wave periods of 8 s, the motion is isotropic and dissipation decays as t−4.3 . An increase in dissipation begins about 1/4 of a wave period before the arrival of the crests of steep waves, possibly caused by the non-zero vorticity of the finite amplitude, nearly breaking waves.


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Figure 9.1. Decay and spread of turbulent motion after surface wave breaking in a laboratory experiment. The figure shows the turbulent r.m.s. velocities for (a) a spilling wave and (b) a plunging wave, at times of (top to bottom) 1, 4, 6, 10, 20 and 50 wave periods after breaking. The lines at times 1–20 periods show the edge of a patch of dye being diffused from the surface by the turbulent motion. The horizontal scale is the horizontal distance, x − xob , from the location of breaking multiplied by the wavenumber, kc , so that 2 ≈ 6.28 represents a distance of one wavelength. The vertical scale is depth multiplied by kc . A depth of H, the wave height, below the mean water surface is at k0 z = 0.56 in (a) and 0.70 in (b). The scale bars show r.m.s. speeds of 0.01C, where C is the phase speed of the wave. (From Rapp and Melville, 1990).

water,9 little is known of the three-dimensional structure of the vorticity field left by breakers, particularly those of limited crest length sketched in Fig. 9.3. Csanady (1994) suggests that localized breaking along a short length of a wave crest may, through its impulsive transfer of momentum to the flow, drive a pair of counter-rotating vortices that will be twisted by the Stokes drift, producing vortices with axes aligned downwind. Interacting with one another (and with their images in the free surface), the vortex pair will be carried downwards, away from the surface. It has been suggested that this vorticity input may contribute to the evolution of Langmuir circulation (Csanady, 1994; 9 A particular point of interest in Hornung et al.’s theoretical discussion and laboratory experiments is how vorticity may be generated by the breaking of an irrotational wave without violating the constraints of Kelvin’s circulation theorem.

Figure 9.2. Rotor formation. The mean velocity field at (top to bottom) 3, 10.5, 26.5, 34.5 42.5, 50 and 58 wave periods after a wave moving to the right has broken in a laboratory tank. The arrows show speed normalized with the phase speed of the breaker, and the horizontal and vertical scales (the latter being depth below the mean free surface) are normalized using the wavelength of the breaking wave. The main feature is a rotor or vortex, rotating clockwise, that moves in the wave direction by about 0.85 wavelengths in 50 wave periods. (From Melville et al., 2002.)


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Figure 9.3. Sketch showing the conjectured form of circulation produced by a three-dimensional (3D) breaker in deep water. The stippled area represents the foam left by the breaker. One, or possibly more, rotors (R) are formed with horizontal axes, overturning in a vertical plane oriented roughly in the direction of wave propagation, c, and perhaps having a banana shape as a result of the spreading of the region of breaking along the wave crest and the consequently later formation of rotors at the edges. Eddies (E) with vertical vorticity are formed at the spreading edges of the breaker and are tilted by the Stokes drift, those at the right and those at the left of the breaker possibly combining to form a pair of counter-rotating eddies with vorticity aligned in direction c and that may contribute to the formation of Langmuir circulation (Csanady, 1994).

Teixeira and Belcher, 2002). The importance of the generation of vortices by breaking waves to the processes acting within the mixed layer has, however, not been established by observations. 9.2.3

Observations at sea

Surface waves are locally generated as ‘wind waves’ by the wind prevailing in their vicinity, and radiate across ocean basins from such generation sites, particularly those of severe storms, to arrive as ‘swell’ and to be dissipated on distant beaches. Wind waves and swell will usually both be present at any particular location. Both types of waves exhibit periodicity in their amplitude, a tendency to travel in ‘groups’ of some 4–9 waves. (The periodic arrival of higher waves is particularly evident on beaches.) It is not known over what distances the groups retain their form but, in strong or increasing winds, the highest waves, often those near the centre of wave groups, break. The breaking is visible on the sea surface as the repetitive appearance of transient patches of white foam in a slowly advancing location, a feature that can often be detected either from shore or aircraft. The repetition was first described and explained by Donelan et al. (1972). Individual waves advance at a phase speed, c, that in deep water is about twice the speed of wave groups, cg (by 9.3), and so the speed of a wave relative to the group within which it propagates is c − cg = c/2. The time taken for a wave to advance one wavelength, λ, through the group to the location in the group where its predecessor had broken (i.e. to the location in the group where waves are large enough to break) is therefore λ/(c/2) = 2λ/c, or 2T, where T is the wave period, since the phase speed, c = λ/T. The interval of time between the successive breaking of waves travelling in groups is therefore approximately equal to twice the wave period.


237

This is shown graphically in Fig. 9.4. The duration of breaking and the subsequent persistence of foam or turbulence determine the properties of their distribution in space and time as explained in the figure caption.10 Observations by Gemmrich and Farmer (1999) show that the phase speed of the wind-generated breaking waves, cb , has a wide range, from 0.1 to 1.3 times the speed of the dominant waves,11 and so the local energy input to turbulence, depending on the fifth power of cb (see Equation (9.4)), is highly variable. A significant theoretical study by Phillips (1985) demonstrated the importance, particularly in relation to energy loss, of the variation of the average length of breaking crests with their speed of advance. This prompted air-borne photographic measurements of breakers in deep water by Melville and Matusov (2002) to discover the distribution of the lengths of breakers (found to be proportional to the cube of the wind speed) and how the length of a breaker is related to its speed. As is apparent even to the casual observer, although wave crests are sometimes visible for relatively large distances in deep water, the length of the breaker regions themselves is rather short as shown in Fig. 9.5 (and sketched in Fig. 9.3): the length of a wave crest that generates turbulence by breaking is of the order of, or less than, the wavelength of the breaking waves. This is in contrast to the edge of the surf zone on an extensive and smooth sandy beach, where waves approaching the beach from a direction normal to the shoreline break almost simultaneously and continuously along a long length of wave crest. Melville and Matusov conclude that the fraction of the ocean surface mixed by breakers is dominated by relatively slow-moving and short breaking waves. As shown in Fig. 9.6, Agrawal et al. (1992) found that turbulent dissipation exceeds that predicted by the law of the wall in a region below the mean sea surface of thickness about 105 u 2∗ g−1 , (or about Hs, the significant wave height, equal to the mean height of the highest 1/3 of the waves), suggesting that the cause is turbulence generated by breaking waves.12 Gemmrich and Farmer’s (2004) finding that high dissipation rates, four orders of magnitude greater than the background rates at 1 m depth, are linked to wave breaking provides convincing evidence that breaking waves are the cause of the enhancement of mean values of ε near the sea surface. The thickness of the layer in which turbulence is enhanced by breakers is, however, not known precisely. Based on measurements of turbulence from an electromagnetic sensor attached to the bow of a ship operating in the tropical and equatorial ocean, Soloviev and Lukas (2003) conclude that most of the wave-breaking energy is dissipated within a layer that is of thickness 0.25Hs.

10 The results for breaking surface wave groups can be extended to other classes of waves, e.g. internal waves modes or rays (see Section 5.8). 11 Since, from the dispersion relation in deep water, c2 (= g/k) = gλ/2, the wavelength, λ, of breaking waves in given conditions varies over a wider range, but one biased towards relatively short waves. 12 The friction velocity in the water, u ∗ , is usually estimated as u ∗ = (τ w /ρ 0 )1/2 . Implicit in this formulation of u ∗ is the assumption that the stress measured within the water is equal to the wind stress, τ w , on the water surface. It will not be valid if there is a substantial transport of momentum from the wind into the surface wave field.


238

(a)

2

(b)

2

Figure 9.4. Idealized representation of the breaking and generation of turbulence by a group of waves in deep water. The vertical axis is the distance, x, in which waves are propagating, and the horizontal is time, t. The diagonal band represents a group of waves travelling in deep water. Its tilt indicates the speed of advance of the group, cg = c/2 where c is the phase speed. Individual waves of period, T, are marked with lines passing through the group with greater speed, c, therefore being represented by steeper lines. They break for periods τ in locations marked by thicker lines enclosed between the dashed lines, the breaking zone near the centre of the group, leaving foam (or turbulence) in the water lasting for a period Tf in stippled locations. (The effects of wind drift and Stokes drift are ignored and so the foam does not move downwind; the stippled regions do not vary in x as t increases.) In (a) τ < T. The period between wave breaking is 2T. (Waves begin to break at A and D. The first (A) ceases breaking at E. ABC is a line at constant location, x, with C corresponding to the time at which the


239

Terray et al. (1996) have proposed a parametrization of the energy loss by breakers, or equivalently of the production of energy within the upper ocean, using the wave age (the ratio of the phase speed of the dominant waves to the friction velocity, u ∗ ), but careful observations of breaker frequency and speed by Gemmrich and Farmer (1999) suggest that formulation may need to be more sophisticated, depending on the local rate of input of energy from the wind to the wave field.13 9.2.4

Bubbles and foam

Clouds of small (20–400 µm radii) bubbles produced by breakers are detectable by side-scan sonar operating at frequencies of about 250 kHz. Since the clouds remain detectable for sometimes hundreds of seconds after a wave has broken, they offer a marker to track the turbulence from its source in the breakers (Thorpe et al., 2003a). Two ways in which breakers produce sub-surface bubbles are identified by Deane and Stokes (1998, 2002) using a novel optical system capable of recording bubbles of radii >200 µm. The air cavity trapped by an overturning wave disintegrates into bubbles in a time of about 90 ms, deduced from hydrophone measurements of the noise produced as bubbles fragment and oscillate. A plume of bubbles is created by the entrainment of air as the falling jet of a plunging breaker impacts on the water surface. The violent turbulence at and just behind the toe of a spilling breaker, the impact of rain or spray droplets and, as mentioned earlier, capillary ripples forming ahead of the crest of short gravity waves, can also entrain air and cause bubbles. Bubbles of radius, a, less than about 1 mm that are formed in deep water breakers appear to be stabilized 13 Swell arriving from distant sources may also affect breaking rates and energy input to the mixed layer.

← second wave (D) begins to break and B being the time at which this wave passes the location, x. Lines BD and AE have slopes, c/2 and c, corresponding to the group and phase speeds, respectively, and so DC = (c/2) · AC = c · BC, so BC = AC/2. Therefore AB = BC = AC/2 and since the wave period is AB = T, the time between the onset of breaking is AC = 2T.) The x coordinate of E, xE , when the first wave ceases to break, is less than that of D, xD , when the second begins. There are locations, xE < x < xD , when no waves break (a spatially patchy input of turbulence), and foam patches cannot overlap in space, no matter how long they persist (at least when the drifts are ignored). No more than one wave breaks at any time t and there are periods between E and D when no waves are breaking. If the foam persists for times, Tf > 2T−τ , there will always be foam at some location. In (b) the duration of breaking is longer: T < τ < 2T. There are no locations at which breaking does not occur (or foam is not produced) as the breaking group passes, but still times at which there are no waves breaking (there is always wave breaking somewhere in the group if τ > 2T), and at most only one wave breaks at any given time. There are times at which no foam is present (and the group would not be detectable from the presence of its generated foam) if Tf < 2T − τ , the same condition as in (a). The most common situation in the ocean is that illustrated in (a) with τ < T . Foam often persists for several wave periods (see Fig. 9.8) and turbulence for even longer (Fig. 9.1).

240


Figure 9.5. Aerial view of surface breaking waves. Airborne camera photographs taken from a height of about 1500 m of waves breaking in deep water in the Gulf of Tehuantepec off the Pacific coast of Mexico. The width of the image is about 950 m and the wind is blowing at 20–25 m s−l towards the top right. Similar images showing the patchy and short-crested nature of breakers may be found in Melville and Matusov (2002.) (Photograph kindly provided by Professor W. K. Melville.)

by surface tension, but those of greater radius are fragmented by differential pressures in the turbulent motion that accompanies their formation. Large bubbles have a bubble size distribution (the number of bubbles of radii between (a − 0.5) µm and (a + 0.5) µm, per m3 ) that is proportional to a−10/3 , whilst the distribution of those of smaller radii is proportional to a−3/2 , the transition occurring at a bubble radius equal to 2−8/5 ε−2/5 (γ Wec /ρ 0 )3/5 (the Hinze scale). Here Wec is a critical value, about 4.7, of a Weber number defined as We = 2(ρ 0 /γ )q2 a, γ is the surface tension, and q is the turbulent velocity on the scale of the bubbles of radius, a, given by q2 = 2ε 2/3 a2/3 . Many of the gaseous components of air within very small bubbles pass rapidly into solution (the pressure inside bubbles being greater than that outside by a factor 2γ a−1 , and increasing as a decreases), bubbles eventually dissolving completely or their radius decreasing until further gas transfer is inhibited by the increasing concentration of particles or surfactants carried on their decreasing surface area. When a is less than


241

Figure 9.6. The variation of the turbulent kinetic energy dissipation rate, ε, with depth, z, below the sea surface. Dissipation is nondimensionalized with u 3∗ /kz and depth with u 2∗ /g, and the law of the wall variation, ε = u 3∗ /kz, is shown by the vertical line, where u ∗ is the friction velocity in the water. (From Agrawal et al., 1992.)

about 30 µm, the bubble size distribution usually decreases with decreasing bubble radius. Bubbles of 30 µm radius rise at only 3–4 mm s−1 and, in winds of 6 m s−1 or more, persist within the water column for periods of order 100 s following their generation in breakers, during which time they are often advected into the linear bands associated with Langmuir circulation described in Section 9.4. Turbulence is higher than average in the near-surface bubble clouds as shown in Fig. 9.7, indicating the association of turbulence with the generators of bubbles, the breaking waves. The effect of turbulence is generally to impose lift and drag forces on bubbles that reduce the rise speed of small bubbles by factors that are sensitive to the wavenumber spectrum of the turbulence and which depend on the ratio, q = σ w /wb , where σ w is the root mean square turbulent vertical velocity fluctuation and w b is the rise speed of a bubble in still water. In homogeneous and isotropic turbulence and when 0 < q < 1, Spelt and Biesheuvel (1997) show that the rise speed decreases as q increases, and may be reduced by a factor of 2–3 at q = 1. The reduction depends critically, however, on the chosen representation of turbulence. The numerical results suggest that at q ∼ 1 the reduction in rise speeds is sometimes associated with the transient capture of bubbles in turbulent vortices, although this requires further investigation.

Figure 9.7. The variation of ε near the sea surface in winds of about 11 m s−1 . (Left) The pdfs of (4 + log10 ε) with ε in W kg−1 at depths of (top) 2.22 m and (bottom) 4.16 m. Values measured in bubble clouds with void fraction >2 × 10−8 are shown in black. Both distributions are close to Gaussian. (Right) The ratio of values in bubble clouds to the total samples. The majority of high values of ε at the shallower depth are in bubble clouds. (From Thorpe et al., 2003a.)


243

In the ocean, σ w is of order u ∗ ,14 so the parameter q is of order u ∗ /w b . In winds of 10–12 m s−1 this is approximately unity for bubble radii of 70–90 µm, which is near the peak of the volume size distributions of oceanic bubble clouds (The relatively small rise speeds contribute to the formation of a peak.) In the upper ocean, however, and particularly in bubble clouds formed by breaking waves, turbulence may be neither isotropic nor homogeneous. Whilst the size of q suggests that turbulence is likely to reduce the bubble rise speeds, particularly since higher dissipation rates accompany bubbles, the knowledge of the nature of turbulent motion and of its spectral form in bubble clouds are presently insufficient to make any assessment of the magnitude of the reduction.15 The thin sheet of floating bubbles or ‘foam’ left on the water surface after a wave has broken is soon disrupted. Its disintegration is accompanied by the formation of some repeated and coherent patterns within the foam layer.16 Holes are shown in Fig. 9.8. These appear to be caused by clouds of large rising bubbles that transfer momentum (through drag) to the water, causing upward flows that, on meeting the surface after the faster rising bubbles, diverge and spread the foam layer. At a later stage of its disintegration, a foam patch is drawn into a series of wind-aligned streaks, as shown in Fig. 9.8e and f and, at higher wind speeds, in Fig. 9.9. These appear similar to patterns formed in the early stages of development of Langmuir circulation studied by Veron and Melville (2001; see also Section 9.4.2). Foam has, however, a complex rheological behaviour that depends on the surface-active nature of the surfaces of its component floating bubbles (see, for example, Kraynik, 1988), and care must be taken in relating foam patterns to water motion. 9.2.5

Effects of rain on wave breaking

It is seafaring lore that ‘Rain knocks down the sea’ or that heavy rainfall reduces wave breaking. It is not yet known precisely why this is so. A direct effect seems unlikely. Laboratory experiments, simulating the fall of rain onto waves in windless conditions (Tsimplis, 1992; Poon et al., 1992), show that rain can substantially damp short gravity waves with wavelengths of 5–30 cm, apparently because of its enhancement of turbulence and mixing within the top 0.1 m or so of the water column (Green and Houk, 1979). This is in accord with the suggestion of Reynolds (1900) that vortices generated by raindrops may dissipate waves. The breaking waves that affect ships and are consequently of concern to mariners are, however, far larger in scale than the radiating ripples produced by drop impact on the sea surface or of the short waves known to be directly damped by rain. It is possible 14 See Section 9.3.1 for values of σ w in the upper ocean. Measurement of σ w is complicated because of the need to distinguish between turbulent dispersive motions and the largely advective and periodic wave-induced motions, and to remove the effect of the latter. In practice, a total and unambiguous separation may be impossible, for example in breaking waves where wave motions and turbulence are interrelated. 15 The effects of this, and other processes, on the transfer of gases from the atmosphere to the ocean by bubbles are reviewed by Thorpe et al. (2003b). 16 See Section 11.2.1 for reference to the patterns formed by the break-up of foam in the surf zone.

244


Figure 9.8. Holes and streaks in a layer of foam formed by a breaking wave in deep water; a sequence of photographs taken from the Floating Instrument Platform, FLIP. Frames are every 3 s. Waves with a period of about 1.9 s travel from top left to bottom right, roughly in the wind direction. The wind speed is 7.5 m s−1 . The frames cover a width of 6.5 m (at bottom) to 16.3 m (at top) and are about 17.5 m in height. Foam at top right in (a) is an ‘old’ patch left from a wave that broke before the start of the sequence. Almost circular holes of O(1m) diameter are visible in the foam sheet at centre frame in (c) and (d), first forming at about 2–3 wave periods after wave breaking. Streaks of foam, some 0.1 m apart and aligned in the wind direction, develop some 15–20 s after foam formation and are visible in the ‘old’ foam patch at (c)–(e). (From Thorpe et al., 1999b.)

that the rain damping of short waves, smoothing the sea surface rather in the same way as does an oil film, reduces the wind stress on the larger waves and consequently diminishes the energy transfer from the wind into these waves, and so reduces their breaking. But even the damping of the shorter waves is in doubt. The laboratory experiments do not replicate the effects of a real sea or the turbulent atmospheric boundary layer over waves. Rain increases the horizontal stress on the sea surface by an amount ρ r V R, where ρ r is the density of the rainwater, R is the rainfall rate in units of volume per second per unit surface area (i.e. m s−1 ), and V is the horizontal component of the velocity of the rain at the water surface. (V is not necessarily equal to the wind speed at the conventional measurement height of 10 m above the sea surface, or higher. Rain drops will decelerate in falling through the atmospheric boundary layer above the sea surface; see Caldwell and Elliott, 1971.) Rain may enhance the stress by 7%–25% (Poon et al., 1992). The theoretical analysis of Le Méhauté and Khangaonkar (1990) shows that short waves in deep water with phase speed, c, may grow as a consequence


245

Figure 9.9. The appearance of the sea surface in a wind of about 26 m s−l (force 10, gusting to 11) showing streaks of foam aligned downwind. As well as larger breakers (one is visible at top left) there are many small breaking waves visible. (Photograph taken by Mr J. Bryan from RRS Charles Darwin off Northwest Spain on 2 January 1986, and reproduced with his kind permission.)

of the stress resulting from the vertical transport of the raindrops’ horizontal momentum if c < V /3.

(9.5)

If, for example, V = 10 m s−1 , then since c(m s−1 ) ≈ 1.56T(s), where T is the period of surface gravity waves, (9.5) implies that waves of period less than 2.1 s, or wavelength less than 6.8 m, may be amplified, contrary to expectations based on the laboratory findings.17 There are other effects of rainfall that may affect waves and the turbulence they generate, such as the formation of a layer of relatively low density, a ‘freshwater pool’, on the sea surface, that may impede vertical mixing (Green and Houk, 1979).18

17 The dispersion relation of small amplitude surface gravity waves of wavenumber, k, in deep water is c2 = g/k or (λ/T )2 = λg/2, where λ = 2/k is the wavelength, and T the period. Rearranging, c = (λ/T) = (g/2)T or, since g ≈ 9.81 m s−2 , c(m s−1 ) ≈ 1.56T(s), a relation that is useful to remember. 18 Methods have been devised to detect rainfall acoustically. Rain produces an audible ‘hiss’ that can be detected by hydrophones. The mechanism of sound generation is not the impact of raindrops with the sea surface, but their production of sub-surface bubbles that oscillate at acoustic frequencies of about 14 kHz as they are formed (Franz, 1959; Prosperetti et al., 1989). These oscillations persist for only a few milliseconds after the formation by individual bubbles and are likely to have little effect on the turbulent motion in the surrounding water.


246

9.3

Mixed layer turbulence below the wave-breaking layer

9.3.1

Near-neutral conditions

In water which is so deep that the mixed layer is isolated by an underlying pycnocline from any source of turbulence at the seabed, and in the absence of double diffusive effects, the energy for turbulence in the layer must be derived externally from the sea surface (the wind stress or the surface buoyancy flux), and internally from the shear within the mixed layer or, at the base of the layer, from shear or breaking internal waves. Close to the surface are the thin viscous–conductive sub-layer, found in low wind conditions and described in Section 9.2.1 and, when wave breaking is established at higher wind speeds, a region extending to a depth roughly equal to the significant wave height, Hs, in which the turbulence produced by breakers is important, as explained in Section 9.2.3. Below these, turbulence in the upper part of the mixed layer follows the law of the wall, properties scaling with depth and with the friction velocity in the water, u ∗ = (τ w /ρ 0 )1/2 , where τ w is the mean wind stress on the sea surface. The pdf of the turbulent energy dissipation rate, ε, is approximately log normal (Oakey, 1985; Thorpe et al., 2003a).19 However D’Asaro (2001) finds that, in near-neutral conditions (negligible buoyancy flux or when the modulus of the Monin– Obukov scale, LMO , is much greater than the mixed layer depth), the root mean square vertical velocity, σ w (excluding motions induced directly by surface waves), exceeds that normally found in the flow over fixed boundaries (see Section 8.3.1), being in the range (1.22–1.73)u ∗ , rather than about 1.2u ∗ . The reason is not apparent, but may be a consequence of Langmuir circulation.

9.3.2

The effects of a surface buoyancy flux: severe and prolonged cooling

Shay and Gregg (1984) reported observations made using a free-fall instrument in a warm-core Gulf Stream Ring during a period of severe surface cooling resulting from a cold air outbreak. In a period of just over a day during which the air temperature was near 0 ◦ C, some 13 K below the surface water temperature, the mixed layer deepened from less than 50 m to more than 150 m. Mean values of ε in the convective region between 2|LMO | and the bottom of the mixed layer at a depth, D, shown and compared with atmospheric observations in Fig. 9.10, are equal to 0.72B, where B is the (upward) surface flux of buoyancy resulting largely from evaporative cooling which reached 3.2 × 10−7 W kg−1 . Values of ε gradually decrease through the mixed layer, but by no more than a factor of 3, but decrease abruptly, over distances of 2–10 m, by factors of 10–100 at the base of the convectively mixed layer. 19 Rates of dissipation measured by Oakey (1985) at depths of 4–20 m below the sea surface using the free-fall Octoprobe have a cubic relationship with the wind speed in accord with the law of the wall and, to within the uncertainty imposed by poor signal to noise, have a log normal distribution.

9.3 Mixed layer turbulence

247

e/B 10−3

10−2

10−1

100

101

102

1.4 1.2

OCEAN

z /D

1.0 0.8 0.6 0.4 0.2 0.0 1.4 ATMOSPHERE

1.2

z /D

1.0

Figure 9.10. The variation of ε with depth in convective conditions. (Top) Values measured using a free-fall instrument in the ocean during a period of severe surface cooling. Depth, z, divided by the mixed layer depth, D, is plotted upwards and ε is divided by the surface buoyancy f1ux, B. (Bottom) Comparative values in the atmospheric boundary layer determined by Caughey and Palmer (1979), with height, z, measured upwards. Over much of the layer, ε ∝ B, as expected from Section 4.3. (From Shay and Gregg, 1984.)

0.8 0.6 0.4 0.2 0.0

10−3

10−2

10−1

100

101

102

e/B

9.3.3

The effects of a surface buoyancy flux: diurnal heating

A negative surface buoyancy flux, B, caused by surface heating, or by freshwater introduced by rainfall (see, for example, Smyth et al., 1997, and Section 9.2.5), may quite rapidly promote stratification near the sea surface and a decay of turbulence in the underlying mixed layer. A pronounced diurnal cycle in heat flux can lead to daytime heating that produces near-surface stratification, capping what, at night, is a relatively deep mixed layer. Shay and Gregg (1986) observed the effects of this transient forcing on turbulent dissipation and temperature fluctuations in measurements made in the Bahamas. The diurnal variation results in the mixed layer depth reaching 100 m at night, followed by daytime restratification and reduction in ε as shown in Fig. 9.11, parts 2 and 3. Recommencement of cooling at night leads to erosion of the stratification and a deepening turbulent layer (parts 4–6).20 From vertical profiles of temperature or density 20 Skyllingstad et al. (1999) also give examples of the time–depth plots of ε and of χ T in a mixed layer and underlying thermocline during the transition from daytime heating to nocturnal cooling.

248


Figure 9.11. The diurnal cycle of temperature (thin lines) and dissipation, ε (shaded) vs depth, illustrated by observations at six times over a period of 30 h. (1 MPa ≈ 100 m.) The surface flux of buoyancy from the sea to the air, B, is shown below, stable heating conditions being when B < 0. Periods of surface cooling are shaded and the times of the six profiles are marked and labelled. The maximum value of B corresponds to a Monin–Obukov length, LMO , of about −17 m. Conditions are initially convective. Daytime heating of the surface water and restratification at time 3, results in a lowering of the mean dissipation rates at depths of 10–100 m by 2–3 orders of magnitude from those of the earlier convective conditions (times 1 and 2). Convection has set in again by time 4, deepening the region of high dissipation and gradually, over a period of 6 h (times 4–6), eroding the stratification through the mixed layer down to the depth of the seasonal thermocline at about 100 m. (From Shay and Gregg, 1986.)

alone it is sometimes almost impossible to establish the true depth to which surfacegenerated turbulence directly mixes the boundary layer (see Fig. 9.11, profile 5). During the convective periods ε /B was approximately equal to 0.61.21 21 Anis and Moum (1992) report values of ε /B ranging from 0.69 to 0.87 (mean value 0.81) in their observations in convective surface layers in the Pacific. A further example of diurnal mixing and the variation of dissipation is given by Moum and Smyth (2001; Fig. 1).

9.3 Mixed layer turbulence

249

These estimates of ε /B in the ocean mixed layer and those recorded in the previous section are in general accord with the conclusions of Section 4.3, and are comparable to the value of 0.64 found in convective atmospheric boundary layers (Caughey and Palmer, 1979; see Fig. 9.10), suggesting dynamical similarity. Whilst, however, the ratio D/|L MO | is typically in the range 120–240 in the highly convective atmospheric boundary layer, values of 3–25 are typical in the ocean mixed layers, suggesting that convective plumes may be less well developed in the ocean. The diurnal cycling of the thickness of the region of relatively intense turbulence at the surface has a substantial effect on the mean horizontal velocity in the mixed layer (Price et al., 1986). A nearly constant wind stress during the cycle of heating and cooling is found to lead to the generation of a near-surface ‘diurnal jet’.22 During the daytime, the stabilization of the upper part of the water column by solar heating reduces the thickness of the region of high ε and consequently the depth to which momentum is transferred from the surface. The wind stress drives a downwind current (or jet) that, during the morning, accelerates and reaches speeds of order 0.1 m s−1 by mid afternoon. The current is, however, rotated cum sole (to the right in the Northern Hemisphere) by the effect of Coriolis forces, and by sunset is almost at right angles to the direction of the surface wind stress. Overnight, convection deepens the region of active mixing and momentum transfer, and further rotation by the Coriolis force causes the current to turn into the wind, so that by the following morning the jet-like flow is often completely erased. The mean velocity profiles have a structure resembling that of a classical Ekman spiral, but (perhaps contrary to first expectation) an increase of wind stress causes enhanced mixing and a reduction in the mean vertical shear. Although there are observations of large plumes in regions of deep convection where the depth |L MO | (Section 4.5.2), knowledge of the structure of convective plumes in relatively shallow mixed layers or near the surface is fragmentary. Figure 9.12 shows the conditionally sampled structure of convective plumes measured from an array of 11 thermistors sampling at 4 Hz and hanging beneath a catamaran being towed directly into a 10 m s−1 wind in convective conditions south of Iceland. The tilted plumes are about 4 m wide near the surface, growing to 10 m wide at a depth of 8.5 m. Observations made with a 2.5 m vertical array of temperature sensors mounted on a submarine at a depth of 10 m in Lake Geneva in strongly convective conditions, detected plumes some 10 µK cooler than their surroundings of horizontal scale of about 5–10 m (Thorpe et al., 1999a). This scale is consistent with that determined by Jonas et al. (2003b) from observations of vertical velocities in plumes formed in similar convective conditions in the Soppensee, a relatively small Swiss lake, but in a mixed layer where ε is relatively low, only about 20% of the surface buoyancy flux, B. The downward speeds in plumes, typically 3–6 mm s−1 , exceeds the upward speeds in their surroundings, leading to a skewed pdf of vertical velocity. The persistence, horizontal shape, and turbulent structure of such plumes have yet to be examined.

22 This is akin to the atmospheric ‘nocturnal jet’ modelled by Thorpe and Guymer (1977).

Figure 9.12. The mean structure of cold plumes. Conditionally averaged structure of plumes measured from array of 11 thermistors on a spar hanging from a towed catamaran (Thorpe and Hall, 1987) towed directly into the wind direction in conditions of air temperature = 0.5 ◦ C, sea surface temperature = 6.92 ◦ C and wind = 10 m s−1 . Data are identified from records collected at 4 Hz by seeking times, tc , at which the temperature at 3.5 m depth is more than two standard deviations below the mean at that level, and then averaging temperatures obtained at each thermistor at sequential sampling times before and after these conditionally selected times, tc . The resulting set of averaged temperatures are contoured at 1 mK intervals. The 6.919 ◦ C contour is shown in bold to emphasize the down-wind tilted shape of the plume. (Kindly provided by Dr M. Curé.)

9.4 Langmuir circulation

9.3.4

251

The effects of a surface buoyancy flux: seasonality

In springtime, the convection that forms a relatively deep mixed layer in winter, ceases, and heating creates a shallow seasonal thermocline that overcaps the water in the lower part of the winter mixed layer. This water is removed from direct contact with the atmosphere, at least until convection in the following winter again deepens the mixed layer. During the late spring and summer, the water will retain some of the properties and dynamical features, particularly the vortical structure or potential vorticity, imposed by wind-forcing or other air–sea interaction during its contact with the atmosphere in winter. If, during the spring and summer, the water is carried by the ocean circulation or propagates in eddy form23 to a region (perhaps at a lower latitude) in which convection in the following winter is shallower and does not penetrate to its depth, or if a following winter is relatively mild with convection reaching only to small depths, the imposed structure within the water will persist for much longer periods and may become a quasi-permanent feature of the pycnocline.

9.4

Langmuir circulation

9.4.1

General description

Although in 1927 he first recognized the dynamical significance of the accumulation of flotsam in ‘windrows’, bands of floating material aligned parallel to the wind, it was not until 1938 that Langmuir published his conclusions. These were supported by experiments in Lake George, New York State, designed to test his deduction that a regular pattern of circulation must exist within the water column (Langmuir, 1938).24 Numerous observational studies have since been made of Langmuir circulation in the ocean, notably those of Weller et al. (1985), Weller and Price (1988), Plueddemann et al. (1996) and Smith (1998). These have substantiated the main features of the circulation described by Langmuir. The bands of floating material, often consisting of foam produced by breaking waves, are typically 2–300 m apart and some ten times greater in length. The circulation, at least in its conceptually simplified state, consists of a set of downwind-directed vortical motions (Fig. 9.13a). The windrows, illustrated in Fig. 9.13b and c, Plate 17, form on the water surface in convergence regions between the counter-rotating vortices. Beneath them the downward vertical water speed increases with depth, reaching a maximum of about 1–20 cm s−1 at a depth of 0.2–0.5 times the mixed layer depth, before decreasing towards the base of the mixed layer. The surface water is replenished by a broader weaker upward flow between the convergence lines. The mean flow in the downwind direction is increased, typically by a few centimetres per second, in the vicinity of the windrows. Buoyant algae and bubbles accumulate in the downward flow, partly as a consequence of the Stommel effect described in 23 See, for example, Fig. 13.9 and Section 13.3.6. 24 For reviews of Langmuir circulation, see Leibovich (1983) and Thorpe (2004).

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Figure 9.13. Langmuir circulation. A sketch showing the circulation pattern with enhanced downwind flow in the surface convergence regions and including a ‘Y junction’, where neighbouring windrows join together.

Figure 9.14. Observations of the components of surface currents directed across the wind direction and produced by Langmuir circulation. The near-surface flow speed is measured by Doppler sonar pointing approximately across wind. The wind increased rapidly at time = 0730 in (a), leading to the development of a pattern of cells with gradually increasing scales (given by the mean distance in range between the bands) which slowly drift towards the sonar, so leading to the tilt in the appearance of the bands of high and low relative surface velocity component. Some 19 h later (b) the patterns show stronger flows and larger cell sizes moving away from the sonar. The near-vertical bands are caused by velocity fluctuations induced by surface waves. (From J. A. Smith, 1996.)


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Figure 9.15. The unsteadiness and amalgamation of bubble bands. A sonograph image, range vs time, from a sidescan sonar directed across Loch Ness. The wind is directed along the loch in a direction normal to the sonar beam with a speed of about 15 m s−1 . The dark bands are associated with lines of bubble clouds formed in the surface convergence zone of Langmuir circulation and are being advected through the sonar beam. Bands amalgamate at, for example, a range of 40 m and time of 55 min.

Section 1.8.5 in which the tendency of particles or bubbles to rise is countered by the downward flow. The consequent bands of sub-surface bubbles can be detected acoustically (Thorpe, 1984; Zedel and Farmer, 1991; Farmer and Li, 1995) as shown by the images in Figs. 9.14, 9.15 and 9.16b, and this provides a useful means of detection. Although attention had been drawn earlier to its effect on dispersion by Csanady (1973), until the early 1990s it was usual to regard Langmuir circulation as being rather regular and steady, apparently constraining lateral dispersion by carrying material into narrow bands and so inhibiting, rather than enhancing, its dispersion. This concept has now changed. Based on ideas stemming from observations made using sidescan sonar, Thorpe (1992) showed that a uniform array of vortices may be unstable to threedimensional perturbations, whilst Csanady (1994) demonstrated the log-normality of pdfs of the distance between bands, suggesting its turbulent nature. The concept of ‘Langmuir turbulence’ is derived largely from developments in numerical simulation, particularly Large Eddy Simulation (LES) models, and from observations that have allowed continuous monitoring of the pattern of circulation and of its variability, using side-scan Doppler sonar (Fig. 9.14; Smith et al., 1987; J. A. Smith, 1996) and freely drifting instruments (Farmer and Li, 1995). Whilst the bands and vortices produced by Langmuir circulation are generally orientated downwind, they are rarely steady, linear or regularly spaced as previously imagined, but are more often twisted and subject to amalgamation one with another (Fig. 9.15). They meander and vacillate in strength, with a hierarchy of simultaneously occurring separation scales. Langmuir circulation


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Figure 9.16. The onset of Langmuir circulation. (a) Infrared images showing the rapid development of bands of warmer water on the surface of a pond and the transition to a more turbulent field after an increase of wind from about 1.7 to 2.3 m s−1 . The images, ordered along rows from top left to bottom right, are at 1 s intervals and are 53.4 cm across. The wind direction is shown by the arrow. (From Veron and Melville, 2001.)


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is now regarded as one of the several turbulent processes that operate in the upper boundary layers of the ocean and lakes, contributing substantially to the dispersion of dissolved or floating material, such as oil. It complements, interacts with, and often dominates, other turbulent processes that transport momentum and heat in the upper ocean. 9.4.2

Development of the circulation

It has been known for some time that, shortly after the onset of wind, material floating on a water surface, such as foam or small dust particles, often becomes aligned in bands some 2–10 cm apart lying in the direction of the wind (Faller and Caponi, 1978; Kenney, 1993), but only recently have detailed laboratory and field studies been made to study the phenomenon (Veron and Melville, 2001; Handler et al., 2001). The onset of cellular circulation in a previously quiescent fluid is observed to occur at Re = 530 ± 20 and after the first appearance of surface waves at Re = 370 ± 10, where Re = U0 (t/ν)1/2 , U0 is the surface water speed, ν is the kinematic viscosity and t is the time since the onset of wind. The across-wind wavelength of the circulation pattern is of order 150ν/u, where u2 = U0 (ν/t)1/2 at its onset. The circulation appears to be the form of an instability that affects the structure, stability and transports across the cool skin of the ocean in low winds (Section 1.5.5) and is visible in infrared images of surface temperature (Fig. 9.16a). Numerical studies by Handler et al. that exclude both buoyancy and surface waves appear to show that this instability is caused by viscous shear at the free surface and, unlike the instability and circulation described below, not by waves or thermal convection. Veron and Melville (2001) describe the onset and transition to turbulence of the cellular motion in observations made in a large pond in Mission Bay, San Diego, and the consequent increase in ε at a depth of 5–8 cm from about 3 × 10−6 to 8 × 10−4 W kg−1 as the wind increases from 2 to 5 m s−1 . Much of the theoretical study of Langmuir circulation has been directed to predicting the onset of circulation at much larger scales. Instability leading to circulation can be generated through the presence of a vortex force associated with the wave-induced Stokes drift, the so-called ‘Craik–Leibovich 2’ (or CL2) mechanism (Leibovich, 1983). The instability may be explained physically by supposing that in a uniform downwind Stokes drift, us , and a wind-induced shear flow, there is a small perturbation that increases the downwind flow. This perturbation necessarily involves the presence of ← (b) A sonograph image, range vs time, from a sidescan sonar directed across Loch Ness during a period in which the wind speed along the loch increased gradually from 2 to 10 m s−1 . The dark bands are associated with bubble clouds. As time increases, their intensity increases as more bubbles are generated by breaking waves, and their mean separation becomes greater (as in Fig. 9.14), being about 2 m at the start and 3.6 m after 30 min, by which time the pattern is drifting slowly to the right of the wind direction, consistent with the development of Ekman drift.


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vorticity, ω, with a vertical component, and consequently introduces a horizontal vortex force, us × ω, directed towards the downwind-orientated vertical plane of the maximum flow perturbation. There is therefore an acceleration of fluid towards this plane where (neglecting downwind variation) continuity implies that the flow must sink, leading to a circulation with downwind vorticity. Kinematically, the vertical gradient of Stokes drift tilts the vertical vortex lines resulting from the perturbation, creating streamwise vorticity. So far this disregards the frictional effects of small-scale turbulence that may damp or assist the motion. Downwind flow in the convergence region, enhancing the original perturbation and providing ‘feed-back’, is a consequence of acceleration caused by the stress of the wind on water particles as they move at the surface between the upwelling and the convergence regions, perhaps affected by changes in turbulent drag near the downwelling site (Li and Garrett, 1993). The generation process depends critically on the Stokes drift induced by surface waves, and consequently has no immediate analogue in the atmosphere. The process of instability may be formulated in equations of motion averaged over the wave field. The wind stress, τ w , is characterized by the friction velocity in the water, u ∗ . The waves include a Stokes drift of magnitude us = 2S0 e2βz , decreasing in depth, −z, below the surface. Small-scale turbulence represented by an eddy viscosity, Kν , is usually taken to be independent of depth. The onset of instability is found in terms of a Langmuir number, La, defined as La = (K ν β/u ∗ )3/2 (S0 /u ∗ )−1/2 ,

(9.6)

that expresses a balance between the rates of downward diffusion of downwind vorticity and its production by vortex tilting and stretching by the effect of Stokes drift. The flow is found, analytically, to be unstable to the development of downwind vortices if La is less than a critical value of about 0.5. Whilst such analysis is valuable in establishing the cause of the circulation, the problem of choosing Kν precludes the predictive use of this analytical theory: values estimated at sea already include some contribution of unknown magnitude from the existing circulation and, because of their mutual interactions (for example that described in Section 9.4.3), the contributions of small-scale turbulence and the turbulent circulation cannot be separated even in conditions in which the two can be distinguished. In the presence of the circulation, the eddy viscosity is a property that should be derived as part of the solution, and not one imposed as part of an ‘initial condition’ of the unperturbed flow. This problem is avoided in LES models devised by Skyllingstad and Denbo (1995) and McWilliams et al. (1997) in which a ‘turbulent Langmuir number’, La turb , is defined in terms of the forcing by Laturb = (u ∗ /2S0 )1/2 .

(9.7)

Values of La turb are typically about 0.3 in the ocean. The LES models successfully describe the initial development of vortices and their subsequent breakdown, with three-dimensional interactions between vortices. They reproduce many of the observed

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features of Langmuir circulation, particularly its unsteadiness and the tendency for vortices to have a non-uniform distribution in size, to amalgamate together in so-called ‘Y junctions’ pointing downwind (as sketched in Fig. 9.13a and observed by Farmer and Li, 1995) and to draw floating particles into linear bands. The LES models predict some features that are still to be confirmed by observations and have yet to incorporate realistically the effects of intermittent turbulence produced by breaking waves. 9.4.3

Associated small-scale turbulence in the mixed layer

The locations of acoustically detected bubble bands marking the convergence zones have been used to provide the means of conditionally sampling ε and temperature data collected by an AUV moving at constant speed and at constant depth, these quantities being averaged according to their location relative to the bubble bands (Thorpe et al., 2003a). Turbulent dissipation rates measured at depths of about 2Hs to 12Hs within 5–10 m wide zones of downwelling flow in the convergence regions beneath the windrows are found to be from 22% to 130% greater than the average at the given depth, the highest values being near the surface (Fig. 9.17). This ‘signal’ of enhanced dissipation may result from the advection by the converging flow of turbulence generated by breakers rather than the local generation of turbulence by the circulation (e.g. by vortex stretching). The mean water temperature in the downward moving water differs from the ambient by a few millidegrees, the difference being positive or negative in accordance with the direction of the heat flux at the sea surface. The effect of Langmuir circulation on the vertical distribution of subsurface bubbles increases in conditions of high wind speed when bubbles are injected by breakers to a depth at which the speed of their rise is less than the (depth-increasing) speed of downward motion in the Langmuir circulation. Large-scale Langmuir cells may subduct smaller cells, rotors or other vortices generated by breakers, enhancing subsurface turbulence and dispersion, although this effect is presently unquantified.

9.5

Temperature ramps

Although the water in the mixed layer is strongly stirred by turbulence, there are residual temperature variations of a few mK, a consequence of the vertical heat flux. Some of the variability is associated with Langmuir circulation as mentioned in Section 9.4.3, but not all. Observations made with towed or moored temperature sensors reveal the presence of abrupt shifts in temperature of the same sign (i.e. falls or rises) as shown in Fig. 9.18, leading to a mean skewness, S, in the space or time derivative of the temperature. These 0.05–0.2 m thick ‘temperature ramps’ (referred to as ‘microfronts’ in the atmospheric physics parlance) are coherent in a vertical plane in the flow direction along lines that are tilted downwind at about 45◦ to the horizontal, as illustrated in Fig. 9.19. The distance between ramps is typically about 1.5 times the mixed layer depth.

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Figure 9.17. Conditional plots showing the mean variation of ε in Langmuir circulation. Data obtained from an AUV at (a) 2.07 m and (b) 4.04 m. Data are averaged together about the centre of Langmuir bubble bands (placed at time t = 0) identified by sidescan sonar mounted on the AUV. The wind speed is 11.6 m s−1 and bands are crossed at about 45◦ to the wind direction. The width of the plots, 40 s, corresponds to a distance of about 50 m. Shown are (top to bottom) ε, V 8 (a measure of acoustic scattering from bubbles, greatest in the centre of the bubble bands), T (water temperature, near the centre of the bands being typically 1–10 mK less than average) and d (the depth of the AUV from which ε is measured). The averaged depth, d, varies by about 0.1 m; the AUV is affected by the vertical water motions as it passes through the flow induced by Langmuir circulation. (From Thorpe et al., 2003a.)

Data obtained from towed thermistors show that S varies approximately sinusoidally with the angle, θ , of the tow direction to the wind direction, as shown in Fig. 9.20, with an amplitude of order unity and sign that depends on the sign of the buoyancy flux at the sea surface (Thorpe, 1985; Thorpe et al., 1991; Wijesekera et al., 2004). Non-zero values of S at θ = ±/2 are consistent with a rotation of eddies and ramps in the Ekman layer. Conditional sampling, this time relative to the location of high temperature gradients, shows there to be a greater concentration of bubbles and higher ε on the upper side of a ramp than on the lower, as shown in Fig. 9.19. The greater concentration of bubbles is consistent with a downward advection of water from near the surface. Downwelling flow also carries higher than average turbulence from near the sea surface. As in Langmuir circulation, the enhanced dissipation on the upper side of the ramps is possibly related mainly to advective effects, rather than to local generation.

9.5 Temperature ramps

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Figure 9.18. The variation of temperature with time measured by a fixed vertical array of sensors spanning depths of 4.55–7.05 m in the mixed layer of Loch Ness in conditions of surface heating. Rapid rises occur as the ‘temperature ramps’ are advected past the array at about 8 cm s−1 . (From Thorpe and Hall, 1980.)

Figure 9.19. Sketch showing the two-dimensional structure of a temperature ramp. Bubbles (stippled) and high dissipation are found on the upper side of ramps and imply that downward motions occur there. The relatively warm and cold water corresponds to conditions of surface heating to accord with Fig. 9.18. (From Thorpe et al., 2003a.)

The ramps appear similar to the braids formed between billows in Kelvin–Helmholtz instability, narrow tilted regions of high shear and density gradient. The cause or precise topology of the structures is unknown. In a case in which the Taylor–Goldstein equation was solved numerically by using the mean flow and density structure measured within a mixed layer containing ramps, the conditions were found to be ‘marginal’, close to those of at which Kelvin–Helmholtz instability could develop in an inviscid flow with a horizontal length scale similar to that of the observed ramps (Thorpe and Hall, 1977). Whilst this suggested shear-flow instability as a cause, precisely what three-dimensional form the temperature ramps have is unknown. They may possibly

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Figure 9.20. The skewness, S, of the temperature time derivative measured in tows of temperature sensors at constant depth but at different angles, θ, to the wind direction (see inset) during conditions of surface heating. Circular points are from observations at sea and squares from Loch Ness. (From Thorpe, 1985.)

resemble the hairpin vortices in the bottom boundary layer (see Section 8.2.4) or their origin may, in some cases, be linked to the rotors produced by large breaking waves. Other than the rather meagre clues offered by the linear foam lines formed on the water surface, the acoustic evidence of underlying bubble bands, the circulation patterns, and the presence of temperature ramps and diurnal jets, rather little is known from observations of the large-scale, coherent and vortical structures of the mixed layer and their interaction. How temperature ramps may interact with Langmuir circulation, for example, is yet to be discovered.

9.6

Horizontal dispersion in the mixed layer

9.6.1

Dispersion of dye or neutral solutes

Measurements of the horizontal spread of patches of dye released into the surface layer in 20 different studies in the North Sea, off Cape Kennedy and Southern California, in New York Bight and in the Banana and Manokin Rivers, are reviewed by Okubo (1971) and used to estimate horizontal dispersion at scales ranging from 30 m to 100 km over times of 2 h to 1 month. The measured area, A(C), enclosed by a horizontal curve of constant dye concentration, C, is set equal to the area, R 2 , of an equivalent circle (although as Okubo remarks, ‘The real diffusion pattern never shows radial symmetry’), and the variance, σ r , is found from 1/2 2 σr (t) = C(R)2RdR , (9.8) R C(R)2RdR i.e. the second moment of the radial distribution of dye. Okubo estimates the horizontal dispersion coefficient, K H , at time, t, from (9.9) K H = σr2 4t.

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Plots of logσ 2r vs logt, indicate a linear relationship with an approximate fit, σr2 = 1.08 × 10−6 t 2.34 ,

(9.10)

with σ r measured in metres and t in seconds. The variation of diffusivity with length scale l, taken to be equal to 3σ r (t),25 is found to be given by K H = 1.03 × 10−4l 1.15 ,

(9.11)

with K H in m2 s−1 and l in metres.26 Although having a different power law, Okubo’s values are consistent with the dispersion coefficients determined from particle tracking by Stommel (1949a) and by Kirwan et al. (1978; see also Section 13.4.2) at scales of about 100 m.27 As Okubo points out, closer fits to the data can be found by fitting K H with a l4/3 power law over segments with 30 m < l < 1 km and 1 km < l < 100 km with the break at 1 km corresponding to a time of about 10 h (Fig. 9.21). It appears possible that K H might be equal to the dimensionally correct c1 ε1/3 l4/3 , with a constant c1 , the segments corresponding to two ranges of different dissipation rates, ε (or rates of transfer of energy, or energy cascade, through the energy spectrum), the relationship between K H and l changing at scales that correspond to those at which there is substantial energy input, as envisaged in a generalized theory of turbulence by Ozmidov (1965). Obuko’s relationships appear not to have been subjected to thorough testing. There is as yet insufficient knowledge of how horizontal dispersion responds to environmental factors such as wind speed and gustiness, or to variations in the Monin–Obukov scale, L MO , to waves and sea state, and to changes in mixed layer depth. Fragmentary evidence that dispersion in Langmuir circulation is greater down than across wind indicates that, at least at the scales of 2–300 m of Langmuir circulation, K H is not isotropic. Perhaps the break in the two K H vs l segments represents the scale at which other processes begin to dominate over Langmuir circulation? As Obuko suggests, ‘A further study should include these environmental factors as parameters in the description of the diffusion regime’. This still remains an important goal,28 particularly in view of the desirability to improve the prediction of the spread of harmful algal blooms or spills of toxic chemicals at scales of about 10 m–100 km.

9.6.2

Dispersion of floating films or particles

The spread of fluids that are buoyant and immiscible with seawater or particles that float on the sea surface differs from that of dye or other neutral solutes because, remaining on the surface, the spread of the former is effectively two-dimensional and subject to 25 If the concentration distribution is radially symmetric and Gaussian, 95% of dye is within 3σ r . 26 Actually, use of (9.9) and (9.10) and the definition of l, gives KH = 2.00 × 10−4 l1.15 , but the form (9.11) appears to fit the scatter of data points. 27 Davis (1991b) remarks that it is ‘remarkable that such agreement is found between different ocean conditions and by such diverse methods’. 28 See also Section 10.5.


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l, cm Figure 9.21. Diagram showing the variation of the horizontal dispersion coefficient, K H , with horizontal scale, l, of dye patches. Full and dashed lines show relations K H ∝ l 4/3 . (From Okubo, 1971.)

the direct effects of wind and waves. The mixed layer depth is no longer relevant as a physical constraint to dispersion as it may be for dissolved material. In the case of film-forming substances such as oil, their presence will, however, affect the air–sea interface and hence the waves (especially capillary waves) and wind stress. Little is known about how the variety of factors affecting floating particles contributes to promote dispersion, and there is consequently considerable uncertainty over the appropriate dispersion coefficients to use in predictive models. A model has been devised to describe the dispersion of floating markers caused by the presence of a uniform current and by Langmuir circulation that is driven by a wind aligned in a direction different from the current. The markers, continuously released from a fixed location, are drawn into a set of lines as shown and compared to

9.6 Horizontal dispersion in the mixed layer

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Figure 9.22. The movement of floating particles into Langmuir bands. (a) A model simulation of floating particles released from the origin into an array of Langmuir cells aligned in the wind direction at angle, α, to the x-axis with surface velocity varying with amplitude, v 0 , and advected by a mean current, V, in the x-direction. The particle distribution is shown at two times separated by half the time required to advect one Langmuir cell past the source. The pattern meanders near the source. Particles accumulate in the jets directed along the surface convergence lines of the Langmuir circulation shown as tilted full lines. Because of the consequent correlation of particle location and high downwind surface flow in the circulation, the mean particles’ track is not parallel to the direction of the mean surface flow unless α = 0, in which case particles travel on average faster than the spatial mean x-directed current. The direction of the mean surface particle flow (marked ‘mean flow’) is shown. (b) Aerial photograph showing the break-up of a plume of oil released from a moored boat at the left in a tidal current with non-parallel wind (arrowed). (From Thorpe, 1995.)

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observations in Fig. 9.22.29 Later stages of dispersion depend on the unstable nature of Langmuir circulation. Csanady (1973) pointed out that in conditions in which Langmuir circulation is active, cross-wind dispersion of floating material captured in windrows is determined largely by their duration, TLc , before amalgamating with neighbouring windrows or otherwise dispersing their captured material, typically about −1 , where 2 min to 1 h. The cross-wind dispersion coefficient, K y , is then of order L 2Lc TLc L Lc is the distance between windrows. Values of K y using Csanady’s estimate in winds of 5–10 m s−1 are about 4.5 × 10−1 m2 s−1 , whereas a more sophisticated model of Faller and Auer (1988) gives 1.7 × 10−2 to 2.6 × 10−2 m2 s−1 . The dispersion of surface floats at the scale of mesoscale eddies and estimates of the large-time dispersion coefficient, K H∞ (see Equation (1.21)), are described in Section 13.4.2.

9.7

The base of the mixed layer and mixed layer deepening

9.7.1

Engulfment and entrainment

Processes of ‘engulfment’ or ‘entrainment’ at the base of the mixed layer, the region of transition between the relatively uniform near-surface layer and the stratified pycnoline or thermocline, control much of the deepening of the layer30 and the transfer of heat and momentum derived from the atmosphere into the body of the ocean. There is no precise commonly accepted definition of the two processes; they are not necessarily independent or distinct. Both carry relatively dense water from the pycnocline into the mixed layer. Turner (1973) describes ‘entrainment’ in the context of mixing of outside or ‘ambient’ fluid into a buoyant plume. There is ‘a sharp boundary separating nearly uniform turbulent buoyant fluid from the surroundings. This boundary is indented by large eddies and the mixing process takes place in two stages, the engulfment of external fluid by the large eddies, followed by a rapid smaller scale mixing across the central core’ of the rising plume. The, often transient, transition between the mixed layer and the pycnocline observed in vertical profiles is frequently marked by a sharp local increase in density with depth, and a rapid decrease in ε. Mixing processes contribute to the transience and to the high gradients observed. ‘Engulfment’ is envisaged as a process involving the movement of water from the turbulent mixed layer into and around denser stratified water below the foot of the mixed layer, perhaps by convective plumes. This is a process of invasion, like that of the overshoot shown in the descending plume in Fig. 4.4, the consequence of which is the folding of isopycnal surfaces, the subsequent upward transport of denser water, and its mixing to smaller scales over much of the vertical extent of the mixing layer. ‘Entrainment’ involves the action of transient motions in the lower part of the mixed layer (caused, for example, by Langmuir circulation or eddies associated with 29 As explained in the caption to Fig. 9.22, this provides an example of how the drift of particles may differ from that of the mean flow, in this case because particles accumulate in regions of the flow field in which the local flow differs from the mean. 30 Ekman pumping is a further process: see Section 8.2.6.

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temperature ramps) that locally enhance the mean shear at the top of the pycnocline, generating instability and billows. These result in a local vertical mixing, spreading the interface between the mixed layer and the pycnocline, so transferring mixed water, now of some intermediate density, up into the mixed layer, from where it is swept even higher, eventually throughout the mixed layer, by the large mixed layer eddies.31 Engulfment may be evident, for example, when, after daytime heating, a stratified layer produced just below the sea surface is ‘overturned’ by the rapidly growing Langmuir circulation at the onset of wind or by the onset of convective plumes after nightfall. Subsequent mixing across an established, relatively sharp, density gradient at the foot of the resulting mixed layer may mainly be through an entrainment process. Definitive observations of the processes of engulfment and entrainment at the foot of the mixed layer are lacking.32 Shear caused by mean flow and large eddies at the foot of the mixed layer is thought to generate instabilities akin to those of Kelvin–Helmholtz instability or Holmboe instability. The associated process of entrainment has been studied extensively in the laboratory, although at relatively low Reynolds number (order 104 , based on the mixed layer depth). Strang and Fernando (2001) found that Kelvin–Helmholtz instabilities dominate when Ri B < 3.2, and Holmboe instabilities dominate when Ri B > 5.8, where Ri B = gρh/ρ 0 (u)2 is a ‘bulk’ (or layer) Richardson number based on the depth, h, of the mixed layer and the density, ρ, and velocity, u, differences across the underlying pycnocline. A mixed regime exists when 3.2 < RiB < 5.8. The consequence of these instabilities is to diffuse the interface, allowing larger eddies in the mixed layer to engulf the less well-stratified water.33 A corresponding process of detrainment in which turbulent fluid is expelled from a boundary layer is described in Sections 10.2.3 and 11.6. 9.7.2

Effects of Langmuir circulation in deepening the mixed layer

There is some doubt about the role of Langmuir circulation on the deepening of the mixed layer. Although Li and Garrett (1997) and Li et al. (1995) argue from theoretical grounds for its importance as a process of engulfment and shear enhancement, Weller and Price (1988) concluded from their observations that there is no evidence that it has a direct effect in mixing near the base of a 40–60 m deep mixed layer. In the regions of downward flow in Langmuir circulation in the depth range 1.5 < z/Hs < 12, Thorpe et al. (2003a) failed to find any signs of an enhanced presence of ‘temperature ramps’, as might be expected if they are shear-generated and if Langmuir circulation increases the shear substantially. Skyllingstad et al. (1999) compare LES simulations with observations made during a 24 h period in fairly steady winds of about 12 m s−1 31 See Turner (1986) for further discussion. 32 Experiments to determine the mean vertical diffusivity at the base of the mixed layer using SF6 (Section 7.4) are described by Law et al. (2001, 2003) in connection with the transfer of nutrients into the layer from the seasonal thermocline. 33 Langmuir circulation is not present in the laboratory experiment and how its effect should be represented in the parametrization of mixed layer deepening is not clear.


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and that includes nocturnal cooling with |LMO | exceeding 20 m. Except in the upper 10 m, where observations are not available because of the effect of the ship’s wake, the comparison of model and measured values of ε and χ T in the mixed layer is satisfactory, although the measured ε increases more rapidly towards the surface than the model estimates. The model results imply that Langmuir circulation is the dominant process controlling both ε and χ T in the upper 10 m, but at greater depths in the nocturnal mixed layer, extending to 40–60 m, eddies generated by shear and buoyancy flux dominate. Regularity of the structures of Langmuir cells is lost at depths of about half the mixed layer depth in the LES model of McWilliams et al. (1997). In an LES study of the generation of inertial waves by a resonant wind field, Skyllingstad et al. (2000) concluded that the effects of the circulation are mostly confined to the initial stages of mixed layer growth. It is likely that the dominance of Langmuir circulation over other turbulent processes in mixed layer deepening is limited to the early stages of mixed layer development and, thereafter, its effects in maintaining the homogeneity of the layer are mainly in a limited depth range near the surface. 9.7.3

Numerical models

Numerical models of the changes of the properties of the mixed layer, such as its thickness, temperature or density, that are used to forecast or study the general circulation of the atmosphere and part, if not all, of the underlying ocean, do not have the capacity or resolution to represent the intricate turbulent structures that have been described above. The Krauss–Turner (1967) model of the mixed layer is essentially one in which it is regarded as a uniform slab of water, responding to heat flux, with no shear or advection. In the model it is assumed that the rate of mechanical working by the wind is constant, and that a small, and constant fraction of the kinetic energy generated by convection is used in transferring the denser water from the thermocline into the mixed layer. The Pollard–Rhines–Thompson (1973) model revises the Krauss– Turner model to include a representation of the processes of dynamical erosion of the thermocline using a bulk Richardson number derived from wind stress and heat flux. Substantial developments of both models have been made (Niiler and Krauss, 1977; Price et al., 1986). These, and models that include some semi-empirical parametric representation of turbulent diffusion or that represent some aspects of turbulence by applying a ‘closure’ condition at second or higher order, are described by Kantha and Clayson (2000, section 2.10). Although, as mentioned above, large eddy simulation (LES) models that include Langmuir circulation have been devised and compared favourably to observations (Skyllingstad et al., 1999), they are probably too intricate for general use in large-scale modelling. 9.7.4

Internal wave radiation

Convective plumes or eddies that perturb the interface between the mixed layer and the pycnocline may generate internal waves that radiate energy from the mixed layer

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267

and promote mixing within the underlying pycnocline, as mentioned in Section 2.7.1. Evidence of such mixing activity is found at the Equator, where the diurnal variation in surface heat flux is pronounced and convection results in the night-time enhancement of the variance of the vertical velocity to depths of 20–40 m below the base of the surface mixed layer, with oscillations at frequencies near the buoyancy frequency and increases in ε and in χ T (Lien et al., 2002). In other regions internal wave radiation from the base of the mixed layer and breaking within the pycnocline may reduce the stability and ‘precondition’ the pycnocline, so as to enhance the rate of mixed layer deepening. Evidence for a connection between Langmuir circulation in the mixed layer and internal waves in the underlying pycnocline is, however, much weaker. Perhaps the strongest comes from acoustic observations of bubble clouds accompanying highresolution measurements of temperatures in a storm, with winds of about 14 m s−1 , by Farmer et al. (2001). The times at which relatively cold water appears in the mixed layer (water accompanying the deeper-going bubble clouds, those that are related to downwelling in Langmuir circulation) are linked in phase to temperature fluctuations within the thermocline. Further study is warranted.

9.8

Turbulence and marine organisms

Because of their demand for light, many marine organisms such as phytoplankton develop through their larval and adult stages in the upper ocean boundary layer. Turbulence there has both constructive and detrimental effects.34 Algal blooms are commonly limited by the availability of nutrients that are provided by upward turbulent transfers from deeper water. In addition to this control in the provision of nutrients, turbulent eddies may determine the net amount of light to which organisms are exposed by physically transporting them irregularly up and down through the mixed layer, and increase predator–prey contacts. Langmuir circulation affects the distributions of positively and negatively buoyant algae. The Stommel effect described in Section 1.8.5 not only contributes to an accumulation of bubbles but also causes increased concentrations of buoyant algae to form in the presence of Langmuir circulation (see Bainbridge, 1957; Smayda and Reynolds, 2001). Buoyant algae, and motile organisms swimming upwards to reach higher light levels, tend (like bubbles) to be found in higher concentrations in the downwelling waters beneath windrows where ε is enhanced (Fig. 9.17), whilst negatively buoyant algae concentrations are greatest in upward-moving water roughly halfway between windrows. Turbulence causes damage to flagellates. The impact depends on the organisms involved. For example, the growth of the red-tide dinoflagellate, Genyaulex polyedra Stein, is found to be inhibited when the turbulent shear, (ε/ν)1/2 , is greater than about 34 For review of the effect of turbulence on algae, see Estrada and Berdalet (1998).

268


3 s−1 (Thomas and Gibson, 1990), although a much lower threshold may apply when turbulence is intermittent as is usual in the ocean (Gibson and Thomas, 1995).35 The measured extreme values of ε at depth, z = 2.2Hs in Langmuir cells formed in wind speeds of about 11 m s−1 are found to exceed these thresholds, although the mean values do not, suggesting that some limitation of plankton growth may occur at least for those buoyant algae that are found in high concentrations within the downwelling regions of Langmuir cells (Thorpe et al., 2003b). The non-uniform distributions of turbulence and algae, but possible correlation of the two, need to be considered in the construction of models of algal dynamics. Intense fish shoals or the wakes of large marine mammals may result in enhanced, localized and transient levels of turbulence (Farmer et al., 1987), but are not regarded as major sources of ocean turbulence. At a very much smaller scale, periodic or irregular water motions may be caused by organisms in the processes of filtering material from the water or in the capture of prey but, except locally, have little substantial effect in oceanic dispersion or dissipation.36 35 Turbulent shear has a negative effect on nitrogenase activity (NA) and CO2 fixation by Nodularia strains of cyanobacteria at values of (ε/ν)1/2 > 2.2 s−1 (Moisander et al., 2002). 36 The sound produced by the snapping shrimp, for example, is known not to be produced by the halves of its large or ‘giant’ claw coming together. Instead, the closure of the claw produces a thin water jet that is so fast that a cavitation bubble is formed (Versluis et al., 2000). The sound is that made as the bubble collapses. The associated shock wave is enough to stun, if not kill, a shrimp’s prey, leaving it to be picked up by a second claw and eaten. Fluid motions may occur in extraordinary ways!

Chapter 10 Shallow seas

10.1

Introduction

Many of the processes described in the preceding chapter also occur in the upper layers of shallow seas. Tidal flows are, however, generally more prominent, especially in the shelf seas,1 and the oscillatory motions induced by surface waves will, in some regions, affect the near-bed flow within the bottom boundary layer. (Waves are particularly effective in generating turbulence in shoaling water at the edge of the shallow seas, a topic deferred to Chapter 11.) The bottom boundary layer in shallow seas shares with the benthic boundary layer (bbl) the physical characteristics described in Sections 8.2.1 and 8.2.3, but often has a substantial effect on the whole water column. It is far more accessible to observation than the bbl of the deep ocean described in Chapter 8 and, because of its importance, for example in the transport of sediment and the movement of sandbanks and channels, its study is one of considerable practical application. Some 2.6 TW of tidal energy are dissipated in the bottom boundary layers of shallow seas. The first estimation of the importance of boundary dissipation, and indeed of ε in the ocean, was by Taylor (1919). He made two independent determinations of tidal dissipation during Spring tides in the Irish Sea. The first was made by finding the averaged product of the bottom stress, ρ 0 C D U 2 , and the tidal current, U, integrating over the area of the Sea and making use of the value, 2 × 10−3 , of the drag coefficient, C D , that he had derived from measurements of the atmospheric boundary layer on Salisbury Plain, UK (Taylor, 1916).2 The second was obtained by calculating the net mean flux 1 But not all shallow seas; see, for example, observations by Stips et al. (1998) in the Baltic. 2 The calculation takes no account of the ‘form drag’ defined in Section 10.2.4 that, as in a case described in Section 12.4, may greatly exceed the bottom stress.

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of tidal energy into the Irish Sea through the North and South Channels that connect it to the North Atlantic, correcting the estimate for the tidal generation within the Irish Sea itself.3 Remarkably, given the approximations involved, the two estimates were in fair agreement, the first giving 1.04–1.3 W m−2 , and the second giving 1.53 W m−2 . These lead to a mean value of ε during Spring tides of about 2 × 10−5 W kg−1 . It was not until the development of electromagnetic current meters,4 able to measure rapid fluctuations in two components of motion simultaneously on small, O(0.1 m), length scales, that measurements of turbulent motions and turbulent stress close to the seabed became possible. The first recordings were made by Bowden and Fairbairn (1956) over the sandy bed of Red Wharf Bay in Anglesey (also in the Irish Sea), and these heralded the possibility of linking the study of turbulence to that of sediment transport. Turbulence near the seabed in shallow water is strongly modulated by oscillatory flows induced by surface waves when the water depth is less than about 0.6λ or 0.1gT 2 , where λ and T are the wavelengths and periods of the dominant waves. Sediments are often mobile, resulting in bedforms that may change during a tidal cycle and that affect the stress of the flow on the bed and the turbulence in the overlying boundary layer. Some of the consequences of sediment mobility and the shape of the seabed are described in following sections.

10.2

Near-bed turbulence

10.2.1

The viscous sub-layer

Caldwell and Chriss (1979) made measurements from 19 cm above to 2 cm below a relatively flat sediment–water interface at a depth of 200 m on the Oregon Shelf with a heated thermistor from which flow speed can be found. In a region in which the roughness length estimated in the overlying log layer is z 0 = 5.5 × 10−6 m, the speed decreases linearly from about 8 cm s−1 at a height of 0.6 cm off the bottom to near-zero at the boundary. Estimates of the turbulent stress in the log layer derived from (4.8) are in good accord with the viscous stress in the sub-layer, demonstrating that the stress in the log layer is supported by viscosity at the smooth boundary.

10.2.2

Turbulence above the viscous sub-layer: quasi-steady flows over flat topography

Although the velocity fluctuations produced by surface waves often have their effect, observations by Lien and Sanford (2000) show that, where water is sufficiently deep and wave action small so that flows are quasi-steady, the structure of the boundary 3 Taylor has overestimated the flux values by omitting a term representing the flux of ‘equilibrium energy’ in and out of the Irish Sea (Munk, 1977). 4 See footnote 8 in Chapter 8.

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layer is similar to that in the atmospheric boundary layer. The log relationship, (4.8), extends to distances of typically a few metres from the bed. In tidally varying flows, Lueck and Lu (1997) find the log layer reaches to heights, h, of about (0.03 –0.04) u∗ /ω, where ω is the angular frequency of the tide (i.e. 2/(12.43 h) for the M2 tide) and u ∗ is a mean friction velocity. This means that if the tidal current well above the bed, U∞ , is 0.5 m s−1 , then u ∗ ≈ 0.025 m s−1 (from (8.2)), and the mean thickness of the log layer is about 5.5 m. 10.2.3

Flow over rippled beds: form drag and detrainment

There are two distinct contributors to drag in flow over rough topography: the shear stress derived from viscous forces that reduce the flow on an immobile boundary to zero, and the ‘form drag’ resulting from the horizontal forces caused by differences in pressure from the upstream side of roughness elements to the downstream. The latter may be associated with flow separation over the roughness elements (e.g. ripples or sand waves) or, in stratified flow, with internal lee wave generation. The example below explains the relation of the boundary drag resulting from flow separation (the form drag) to turbulent dissipation. Oscillatory and steady flow over fine sediment can lead to the formation of bed ripples. In purely oscillatory flows, vortices form on the lee slopes of steep ripples every half cycle of the flow as a result of flow separation. These vortices are subsequently ejected from the shelter of the ripples as the flow reverses. In oscillatory flow with orbital amplitude a, over ripples of wavelength λ, and height h, vortices usually dominate the near-bed dynamics if the amplitude of the flow is such that 0.5 < a/λ < 2 and the ripple steepness is in the range 0.15 ≤ h/λ ≤ 0.2. In general, flow separation does not occur if either a/λ or h/λ are below these ranges, whilst if a/λ exceeds about 2, the flow loses eddy structure, appearing similar in character to that of a turbulent flow over a randomly rough bottom. An analytical theory, supported by numerical experiments, that encapsulates much of the physics of oscillatory flow around sharp boundary projections was devised by Longuet-Higgins (1981). He considers an inviscid oscillatory flow over a twodimensional periodic train of symmetrical sharp-crested ripples, neglects the effect of the suspended sediment on the flow or changes to ripple shape that occur during a wave cycle, and supposes that flow separation occurs at the wave crests with shed vorticity being represented as a succession of discrete point vortices as illustrated in Fig. 10.1. Since the flow is inviscid, the drag on the bed is confined to form drag: there is no viscous stress. Vortices shed during one phase of the flow are advected back over the ripple crest in the following return phase of the oscillatory flow as vorticity of the opposite sign is shed in the separated flow. After one complete wave cycle vortex pairs move under their mutual interaction away from the boundary. Assuming that the vortex or form drag on the boundary (which can be calculated from the analytical expression for the pressure on the boundary) is given by τ = ρ 0 C D U |U |, where U is the instantaneous velocity and CD is a drag coefficient (possibly a function of time),

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then the mean rate of energy lost from the mean flow and transferred into the field of shed vortices (or turbulent eddies) over a wave cycle is D = ρ 0 C D U 2 |U | . In an equilibrium situation this is just the vertically integrated rate of energy supplied to eddies and is equal, if the eddies dissipate producing a steady state, to the mean rate of dissipation of eddy energy per unit horizontal area. If written as D = ρ 0 CD U2 |U| , where CD is a mean drag coefficient, and if U = U0 sin σ t, then CD = (3/4)D /U03 , which is found to be about 0.1, in good agreement with observed values. Alternatively, an estimate of the mean rate of energy lost from the mean flow per unit surface area and per unit fluid mass is (4/3)CD U03 or about 0.042U03 . The rate of energy dissipation is conventionally written as (2/3) f e U03 , where f e = 2C D is called the dissipation or wave friction factor. A related and ingenious laboratory experiment to measure the energy dissipation in flow over rippled beds was made by Sleath (1985; see also Fig. 10.2) following an earlier experiment by Bagnold (1946). Sleath shows that there is considerable variation in fe as flow separation occurs and eddies are formed by the flow over the ripples.5 For a given sinusoidal bed shape, fe reaches a maximum at a wavelength λ ≈ 130δ, where δ = (ν/σ )1/2 is the Stokes thickness of an oscillatory boundary layer and σ is the wave frequency. At large values of the Reynolds number, Re = aU 0 /ν, the dissipation factor fe ∝ Re−1/6 . The latter experimental results are consistent with numerical models devised by Malarkey and Davies (2002; see Fig. 10.3). Not unexpectedly, the models show that a sharpening of the ripple crests causes fe , and hence the dissipation rate, to increase. The movement of vortices away from the rippled wall under their mutual interaction mentioned in connection with the shed vortices illustrated in Fig. 10.1 is a means whereby turbulent fluid from a wall layer may be transported away from the wall, or ‘detrained’ from a turbulent sub-layer. Any process that leads to the development of coherent vortical structures, such as vortex rings or vortex pairs of approximately equal strength but opposite sign (to which hairpin vortices are related), will, through 5 For more recent work measurements of the stress caused by oscillatory flow over rippled beds, see Rankin and Hires (2000).

← Figure 10.1. The shedding of vortices from ripples in an inviscid periodic flow. Successive positions of point vortices released from the ripple crests are shown at specified values of σ t, where the mean flow speed to the right is proportional to sinσ t, starting from rest. The strength of vortices is shown by the area of the semicircles that mark the vortices’ locations. The vortex strength is determined by the vorticity in the flow near the ripple crests at the time of their generation and shedding. The flow increases to the right from σ t = 0 to σ t = /2 ≈ 1.57, forming clockwise rotating vortices shown as reversed ‘C’ shapes, and then diminishes to zero at σ t = ≈ 3.14, by which time the interaction of vortices has resulted in eddies forming to the right of the ripple crests. The flow then reverses, increasing to the left until σ t = 3/2 ≈ 4.71, forming anticlockwise vortices marked by the ‘C’ semicircles, before decreasing again to zero after one complete cycle, at σ t = 2. The region of anticlockwise vortices and that of the clockwise vortices swept over ripple crests by the reversed flow produce a vortex pair, and its self interaction and consequent propagation results in vorticity being diffused upwards. (From Longuet-Higgins, 1981.)

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Figure 10.2. A sketch of the apparatus used by Sleath (1985) to determine the dissipation of energy in flow over periodic ripples. The apparatus consists of a vertical plate on which there are modelled ripples with their crest lines parallel and also vertical. The plate is made to oscillate horizontally with simple harmonic motion in a tank of water, so simulating an oscillatory flow over the ripples. The plate hangs from rollers free to move along a horizontal rail, and motion is induced by a lever connected at one end to the rollers supporting the plate, and at the other to a bar fixed to a cylindrical drum free to rotate about a horizontal axis. The rotation of the drum causes the plate and ripples to oscillate horizontally. The drum is made to rotate by winding around it a thread that leads over a pulley to a weight. As the weight descends, the thread causes the drum to rotate, and the plate to oscillate. Frictional effects are first assessed by running the system without water in the tank. The mean rate of energy dissipation per unit mass, E, is then determined as E = MgCf/Aρ, where C is the circumference of the drum, f its frequency in Hz, A the area of ripples on the vertical, g is the acceleration due to gravity and M is the effective mass of the weight driving the motion (that in water minus that in the air tests at the same frequency and stroke).

their self (for rings) or mutual (for vortex pairs) interaction, propagate and carry fluid with them. Smoke rings are an example of such transport. Some coherent vortical structures are however unstable and, contrary to a general transfer of vorticity to larger scales, their disintegration may contribute to the production of small-scale turbulence.6 The log layer in quasi-steady flow is sometimes found to be composed of two layers, in each of which the mean speed varies logarithmically with z, the lower determined by the local boundary structure and the upper by the drag on larger structures, sand waves 6 A discussion of the motion of vortex pairs and vortex rings can be found in Batchelor (1967, sections 7.3 and 7.2, respectively) or Lamb (1932, sections 155 and 163, respectively). The stability of vortex rings has been investigated by, for example, Widnall and Sullivan (1973) and Widnall and Tsai (1977).


275

Figure 10.3. A numerical simulation of the vorticity field in periodic flow over ripples. Almost half a wave cycle is shown, from zero mean flow at zero phase (phase, σ t, is given in degrees below the wave ripple crest) to a maximum flow to the right at σ t = 90◦ , reducing towards zero at 180◦ (not shown, but similar to that at 0◦ , but with left–right transposition). Contours are of the non-dimensionalized vorticity, λω/U0 (anticlockwise positive), where U0 is the amplitude of the free-stream oscillatory flow of frequency, σ , and ω is the vorticity and λ is the ripple wavelength. The parameter a/λ = 0.8, where a is the orbital amplitude of the oscillatory motion. The eddy E is newly ejected at σ t = 0◦ and has almost vanished by σ t = 120◦ . G is an eddy forming in the left-to-right flow over the ripple crest. In this model diffusion is represented by a random walk of decaying discrete vortices shed from the ripples. (From Malarkey and Davies, 2002.)

or ripples, surrounding the measurement site. In measurements in the tidal Pickering Passage, Washington, USA, described by Sanford and Lien (1999), the lower log layer extends up to about 3 m from the bottom. It is shown in Fig. 10.4 as a region in which U(z) varies linearly with logz. The U vs log z relation in the upper layer between 5 and 12 m has a different slope corresponding to a different u ∗ in (4.8) or a different stress, stress here deriving from the form drag on surrounding undulations in the bed, observed to be 0.3 m high and 16 m in wavelength. Chriss and Caldwell (1982) find similar layers, the stress in the upper layer (dominated by form drag) being some four times that of the drag in the lower layer, where the stress is equal to that in the

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276

(a)

25

surface

25

(b)

surface

20 10

1.0

logz 0.5

3

(m)

15 5 4

10

2 5 1

0 0.5

0.6 U (m s−1)

0.7

0.8

0 0.5

0.6

U (m s−1)

0.7

0.8

Figure 10.4. Logarithmic fits to the mean flow speed measured in the 1 km wide Pickering Passage, Washington, USA, during ebb tide. The bottom is covered in undulations in the bed with height 0.3 m and wavelength 16 m with 100 m long crests orientated across the flow. (a) Speed as a function of log (height), and (b) as a function of height, z, together with the logarithmic fits (full lines). The mean speed is plotted as circles as a function of logz. The shading indicates 95% confidence limits and the thick (thin) lines are fits to the data in the lower (upper) log layers. The shading at the top represents the range of surface elevations during the observations made over 2–3 h. (From Sanford and Lien, 1999.)

underlying viscous sub-layer. Precisely how the two log layers coexist, or turbulence in them interacts, has yet to be discovered. 10.2.4

Turbulence above the viscous sub-layer: oscillatory flows with zero mean flow

Considerable attention has been given in laboratory conditions to the nature of oscillatory flow near a rough or rippled bed. Although two-dimensional vortices form behind steep ripples, study of the onset of turbulence in oscillatory flow over two-dimensional periodic ripples shows that the evolving motion is not necessarily confined to two dimensions. The recognition that there are three-dimensional sediment patterns as well as two-dimensional ripples has stimulated the investigation of the effects of hairpin vortices and centrifugal instability in the flow (see, for example, Hara and Mei, 1990).7 A Görtler-type instability over concave parts of a train of ripples produces streamwise vortices, vortices with axes aligned in the flow direction, and its onset and scale depend largely on the size of two parameters, the ratio a/λ, and the Taylor number, Ta = (a/λ)2 (h/δ), a measure of the centrifugal forces to the viscous stress on the ripples. Various modes of instability occur at sufficiently large values of Ta. 7 Centrifugal and Görtler instabilities are described by Drazin and Reid (1981, chapter 3).


277

Numerical studies by Scandura et al. (2000) of flow over steep ripples find an evolution of complex three-dimensional vortex structures and mushroom vortices, indicative of a stage of transition to turbulence. Sleath (1987) provides references to earlier study of the nature of turbulent oscillatory flow over flat and rough beds. His laboratory experiments show that near the bed the maximum turbulent intensity and maximum Reynolds stress occur not at the maximum flow, but as the flow decelerates. The location of the maximum turbulent intensity propagates outwards from the bed at approximately the same speed for all bed compositions, leading to a difference between the time of maximum turbulent intensity near the bed and that at some height above it. In typical conditions, a 90◦ phase difference (i.e. occurring at zero rather than maximum free stream velocity) in peak intensity is observed at a height of between 0.4h and 0.8h, where h is the height at which the amplitude of the local flow in the boundary layer reaches 95% of that in the free stream. Close to the bed, the generation and detachment from the bottom of vortices formed in the lee of roughness elements leads to an irregular, non-monotonic height variation of the Reynolds stress. Jensen et al. (1989) describe laboratory experiments at Reynolds numbers, Re = aU0 /ν, up to 6 × 106 (where a is the amplitude of the oscillatory motion of the free stream with maximum current, U0 ), over smooth beds or beds with roughness elements of height h = 0.35–1.5 mm (a/h of order 2000). Flow over a smooth bed undergoes a transition to turbulence when Re ∼ 103 at a time just prior to the reversal of the nearbed flow. At higher Re, the fraction of a wave period for which the flow is turbulent increases, but even when Re ∼ 1.6 × 106 there are still times during the wave cycle at which the turbulence is relatively weak or negligible. Near the bed even very small bed roughness increases the kinetic energy density of turbulence and the Reynolds stress, typically by 50%, but at distances beyond 0.3a there is little difference found between smooth and rough beds. Logarithmic velocity profiles are found at times during the wave cycle above both smooth and rough beds, the log form appearing soon after the flow reverses direction: the extent of the part of the profile that is logarithmic increases as the flow develops.8 10.2.5

Turbulence above the viscous sub-layer: oscillatory flows with non-zero mean flow

Novel devices have been constructed in recent years to successfully make measurements in the bottom boundary layer.9 The present state-of-the-art instrument for quantifying the three-dimensional eddy structure of the turbulence and ε within the bottom boundary layer uses particle image velocimetry (PIV). This measures and maps the motion of tiny particles suspended in the water in both space and time (Doron et al., 2001; Nimmo Smith et al., 2002, 2005). Pairs of digital camera images of the 8 Numerical models of oscillatory flow over smooth and rough beds have been developed and are described by Sleath (1990) and Guizien et al. (2003). 9 An example is the vorticity sensor described by Sanford et al. (1999).

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location of particles are obtained by generating double pulses of light illuminating a sheet normal to the camera view direction at a frequency of several times per second. Particle displacements are used to produce a velocity map within the plane of the light sheet.10 Dissipation is derived by a spectral method. The device has been deployed on a bottom-mounted rig that can be remotely raised or lowered to sample at heights up to about 10 m from the seabed. Being demanding in power and of high data rates, the rig is connected to an anchored vessel by an umbilical cable containing an optical fibre through which light is carried to the submerged optical probe. Figure 10.5, Plate 18, shows two instantaneous vector maps of currents separated in time by 0.6 s obtained using the PIV system, with vorticity superimposed in colour. The images span a height above the sea floor from about 0.4 to 0.7 m. Various structures in the vorticity field, including an eddy at a height of about 60 cm from the bed and a thin, elongated, vortical layer, some 2 cm thick and 10–15 cm in length at 50–55 cm off the bed, are advected between the two images by a mean flow to the right. Much of the Reynolds stress is associated with the presence of the larger vortices. Measurements have been made at heights of 0.3–2.7 m above the bottom in mean currents of about 0.005–0.4 m s−1 , in waves producing near-bed currents of about 0.1 ms−1 , and in water depths of 12 and 21 m (Nimmo Smith et al., 2005). There is relatively little variation in ε with wave phase. Turbulence in the boundary layer is found to be anisotropic, with the ratio, ϕ 33 /ϕ 11 , of the vertical to streamwise spectral densities less than the isotropic value of 4/3 (see also Section 6.8). The anisotropy appears to arise from the formation of thin, horizontally elongated, vortical layers.

10.3

Mobile sediments and biological effects

10.3.1

The threshold of sediment motion and sediment erosion

A description of the complex subject of sediment transport is beyond the scope of this book, but the interaction of sediment and turbulence is of such importance that some mention is appropriate. The transport of sediment induced by currents is often divided into two parts, the movement of grains along or close to the seabed, the bedload transport, and the transport of sediment in suspension in the water column, the suspended load. Their relative importance depends on sediment type and size. In spite of the enormous effort put into its study, the basic mechanisms leading to the movements of particles in the turbulent bottom boundary layer are not yet well understood, even those beyond the viscous layer that, because of its small thickness, has been less amenable to study. Two major problems, both involving turbulent motion, are central to sediment transport and are yet to be fully solved: the processes of erosion of 10 In 2002 the system was capable of measuring up to about 127 × 127 velocity vectors spanning an area of 50 cm × 50 cm at a frequency of 4 Hz.

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particles from the seabed, and the interaction of suspended particles, often of irregular shape and a range of sizes, with the turbulence surrounding them. The threshold of motion of a non-cohesive sediment lying on a horizontal bed under a uniform turbulent flow is usually determined by a critical value of a Shields parameter, ρ0 u 2∗ /[gd(ρs − ρ0 )], a ratio of the bottom stress to a scale related to the mean sediment diameter, d, and to its density, ρs . The Shields relation (see, for example, Graf, 1998) expresses the critical value as a function of d[g(ρs − ρ0 )/(ρ0 ν 2 )]1/3 , where ν is the kinematic viscosity of the fluid. Fine sandy sediment typically becomes mobile at flow speeds of 0.2–0.4 m s−1 . Although the rate of erosion of sediment from a flat seabed is known to be related to the local shear stress and to depend critically on the nature of the sediment (e.g. whether it is cohesive or not), the precise mechanisms of erosion in a turbulent flow are not fully understood (see Redondo et al., 2001). There appear to be at least three mechanisms through which sediment is carried into suspension. At values of the Shields parameter exceeding critical, non-cohesive sediment particles may begin to move by the process of rolling, but at different speeds depending on the size or exposure to the flow or as they are driven by eddies inducing different near-bed speeds. Collisions will therefore occur, with some particles gaining vertical momentum on collision and moving upwards into suspension, a process like that of the change in direction of a billiard ball on impact with another. Once detached from the bed, particles become exposed to vertical components of the turbulent motion and are further dispersed vertically. A second mechanism through which sediment particles can be ejected from the bed is when the upward pressure component induced by turbulence in the liquid of the sediment bed becomes so great that the resulting upward force is sufficient to cause particles to rise and detach, effectively sucked, from the bed. Both these processes may act on a level bed. On non-level beds, particles being rolled along the bottom over a region where the bed slope changes abruptly, for example at a sediment crest, may be carried off the bed by the separating mean (or turbulent) flows illustrated in Figs. 10.1 and 10.3. Processes that lead to motions transverse to the mean flow, and to ripples with crests aligned downstream,11 may also contribute substantially to the suspension of sediment. 10.3.2

Suspended sediment

The intermittent nature of turbulence in boundary layers described in Section 8.2.4, and the consequent variations in stress and pressure on the bed that contribute to the process of sediment being carried into suspension, cause the suspended sediment to have a patchy structure. Heathershaw and Thorne (1985) seem to have been the first to make detailed simultaneous observations of sediment movement and fluid turbulence in a tidal flow over a sandy bed. They use electro-magnetic current meters to measure 11 An example is Görtler instability in the strongly curved topography of scour holes in the lee of rocks or, in river flows, in the lee of sluice gates (Hopfinger et al., 2004).

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the stress and detect sediment motion by recording the noise arising from particle collisions or from the mobile bed, and report a correlation between the occurrence of turbulent bursts and high sediment suspension. They find that form drag appears to be of greater importance than the local bed stress in the transport of sediment. Over a sandy bed, ‘ejections’ are the dominant mechanism for the sediment suspension whereas ‘sweeps’ dominate the process of bedload transport (Soulsby et al., 1994). Once in suspension, particles are exposed to the turbulent motions but, being denser than their surroundings, tend to sink back towards the bed. If particle inertia is ignored, small rigid spheres with density greater than water settle, on average, in a turbulent flow at the same rate as in still water. Particle inertia, however, produces a bias in the trajectories of particles that carries them towards regions of flow convergence, where the strain rate is high or the vorticity is low (Maxey, 1987). This tends to increase the mean settling velocity.12 Numerical studies by Wang and Maxey (1993) find that, for particles in homogeneous isotropic turbulence (which, as mentioned in Section 10.2.5, may not faithfully represent the conditions near the seabed), the settling velocity may be increased by 50% above that in still water. An effect that may have implications for the aggregation of sediment or algae is that the particles tend to collect in elongated sheets on the peripheries of local vortical structures within the field of turbulent motion. Turbulence can increase the rates at which particles come into contact and results in larger, more rapidly sinking particles. Field experiments by Kawanishi and Yokosi (1997), however, suggest that the aggregation of muddy sediments in suspension and the formation of flocs may be of greater influence on the mean settling velocity of some sediment types than are the interactions of particles with turbulent motions, such as those described by Maxey. Measurements by Rose and Thorne (2001) over a sandy rippled bed in the River Taw, SW England, indicate that the ratio of the eddy diffusion coefficient for suspended sediment to the eddy viscosity lies between 1 and 2. That the eddy diffusion coefficient of particles exceeds that of momentum is a result that is also commonly found in the atmosphere, for example in snow blown by the wind over a snowfield. It also appears likely to be the case for bubbles produced by breaking waves in the upper ocean boundary layer, although there as a consequence of transport by the large-scale coherent structure of Langmuir circulation (Thorpe et al., 2003b). Other effects may enhance or limit sediment transport. A vertical gradient of suspended sediment near the seabed results in a mean stabilizing density gradient and tends to suppress turbulent momentum transfers, therefore reducing the stresses on the bed contributing to its erosion, a negative feed-back leading to control of sediment suspension (Adams and Weatherley, 1981). The presence of waves effects turbulence and consequently sediment transport. The variable amplitude of the bottom flow induced by wave groups influences the vertical distribution of suspended sediment. Villard et al. (2000), for example, measured the sediment concentrations over a sandy bed with a ripple wavelength, λ, of about 0.2 m, in flows caused by groups of surface waves 12 A similar but opposite effect is found for bubbles as described in Section 9.2.4. The settling speed of spherical sediment particles in quiescent fluid is given in Chapter 1, footnote 35.

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producing maximum horizontal motions of about 1 m. The greatest concentrations at heights less than about 0.1 m from the bed are found below the highest waves of the groups. Maximum concentrations at greater heights are however lagged in phase by up to three wave periods. Such time-lags, in this case in sediment concentration, are reminiscent of those of turbulent intensity described in Section 10.2.4, and are characteristic of periodic flows. 10.3.3

Bed ripples and oscillatory flow

Turbulence directly affects the bed topography; the equilibrium topography of the surface of a mobile sediment changes as the energy of the overlying wave-induced motions increases. There is a progression of small-scale bed forms with increasing horizontal wave-induced motion, from irregular features, through a regime of ripples, with crossing and linear ripple patterns, and finally to a flat bed in the most energetic conditions.13 Smyth and Hay (2003) made measurements of turbulence with a coherent acoustic Doppler current profiler up to distances of 1 m from a sedimentary layer with a typical grain diameter of 170 µm in a nearshore region with mean large-scale slope of about 0.5◦ . Over rippled beds the vertical r.m.s. velocity has a maximum at a height of about 1 cm from the bed. The dissipation factor, fe , has its maximum value of about 0.2 in the low energy conditions when the bed is rippled, and lowest value for the flat bed, high energy conditions. This is consistent with the observations of Trowbridge and Edgar (2001) of a lower drag coefficient under breakers than under unbroken waves in the outer part of the surf zone. 10.3.4

Biological effects on seabed erosion and turbulence

Reference has already been made in Section 8.4 to the tracks and mounds left by bottom-living organisms that alter the small-scale topography of the seabed and affect the motion field above it, making the bed rougher and, in some cases, substantially increasing the drag coefficient. (Grant and Madsen (1986) find a change by a factor of two.) In some cases, for example in mussel beds, organisms completely change the character of the seabed. Living organisms may have a considerable effect on sediment transport in both shallow and deep waters by contributing, through burrowing or bioturbation, to sediment movement within the seabed and to the disruption of the sediment surface, by producing the (sometimes annual) deposition of detritus onto the bed (see Section 8.2.5) or by making the surface sediment less prone to erosion by currents, for example by the secretion of mucous material. Marine organisms also affect the flocculation of marine particles by producing polymers that increase the 13 An example of this process given in Section 11.2.2. Sedimentary features have a great diversity of shapes. The generation of those of much larger scale than ripples, such as sand banks and ridges, are known to be associated with mean or tidal currents, or with eddies. Turbulence at both small and larger scale is involved in their formation. Description of the processes, and of the need for further study of the evolution and stability of sedimentary topography, is beyond the scope of this book, but for a review see Dyer and Huntley (1999).

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strength of flocs and consequently the maximum diameter that they may attain before disruption by turbulent shear.14 A similarity between the oceanic and atmospheric boundary layers has already been noted. The effects of vegetation, forests or fields of crops, on the atmospheric boundary layer are reviewed by Finnigan (2000). Depending on the plants’ density and characteristics, the flow conditions near the solid boundary are changed and the overlying boundary layer structure is modified. Ghisalberti and Nepf (2002) have made laboratory experiments on the flow of water over a simulated field of seagrass. The mean flow contains an inflexion point of the type shown in Fig. 3.3a and, like that over forest canopies, is consequently susceptible to shear flow instability leading to propagating billows. These, other eddies in the turbulent boundary layer, and motions induced by surface waves, cause the flexible (and sometimes buoyant) aquatic vegetation to oscillate in coherent moving patterns similar to those seen travelling across fields of corn or barley in windy conditions.

10.4

Mixing by tidal processes

10.4.1

Tidal mixing fronts

Stratification is seasonally and often spatially variable in the shelf seas. Whilst the southern North Sea is unstratified throughout the year, the northern part becomes strongly stratified in summer. The two regions are separated by a front, typically 1–10 km wide. The presence of such fronts led to a very important discovery by Simpson and Hunter (1974; see also Simpson, 1981) that revealed the connection between bottom-generated turbulence and larger-scale features, and had an impact on the understanding of biological processes, for example that of larval dispersion which is restricted by the presence of fronts in shelf seas. To a first approximation and neglecting relatively short-term effects such as those of storms, the mean location of fronts between well-mixed (unstratified) and densitystratified waters is determined by a competition between the promotion of stratification by surface heating with buoyancy flux, B, and the destruction of stratification through the mixing caused by seabed-generated turbulence. For a mean tidal flow, U, in water of depth, H (both U and H depending on position), there is then only one non-dimensional parameter on which the location of the transitional front can depend, HB/U 3 . In shallow regions of relatively fast tidal flows where HB/U 3 is relatively small, turbulence generated by shear stress on the bottom is able to reach the surface and results in mixing throughout the water column, sustaining unstratified conditions. Where turbulence is relatively weak or water depth large, so that HB/U 3 is large, the input of solar heat in summer produces stratification that prevents complete mixing and here a thermocline is maintained. Fronts form at a critical value of HB/U 3 . 14 The processes leading to the contact or coagulation of suspended particles are reviewed by Pruppacher and Klett (1978) and some of the bio-physical processes of floc break-up in the deep ocean are described by McCave (1984).

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283

A value of log10 [H (m)/U (m s−1 )3 ] = 2.7

(10.1)

provides a satisfactory prediction of the location of summertime ‘tidal mixing fronts’ on the European Continental Shelf, some of which are shown in the infrared image of Fig. 10.6, Plate 19. The simple ‘HB/U 3 = constant’ formulation has been successfully tested in a number of shelf sea regions around the World. Simpson and Bowers (1981) examined the effects of seasonal heating, the spring–neap tidal cycle, and the competing input of turbulent energy from the wind, concluding that the latter is relatively unimportant on the European Shelf because the tidal flows are so large. There are smaller horizontal changes in the location of fronts during spring–neap tidal cycles than are predicted from the changes in H/U 3 that result from variation of the tidal flow, U, because energy is expended in removing existing stratification. Large gradients in the intensity and scale of turbulence are known to occur in the tidal mixing fronts between the stratified and unstratified regions (Simpson, 1981). There are, however, few direct measurements to test the predictions of frontal structure by models that incorporate turbulence closure schemes (Simpson and Sharples, 1994).15 10.4.2

The effect of seabed-generated turbulence on the overlying water

Measurements of turbulence using a FLY free-fall instrument in a variety of differing conditions, including mixed and stratified regions, have defined and more precisely quantified the processes involved in tidal mixing. Because turbulence is generated by the shear produced by the tidal currents over the seabed with usually two maximum values of flow speed per tidal cycle, the dissipation rate, ε, has a periodicity of half that of the tide (i.e. about 6.2 h for the M2 tide). Turbulent kinetic energy diffuses upwards from its source in the bottom boundary layer, and consequently the time at which the maximum dissipation occurs is later at greater heights off the seabed. In a tidal current of order 1 m s−1 in unstratified conditions in water 60 m deep, the dissipation rates decrease by three orders of magnitude from a position close to the seabed to one 5 m below the sea surface, and the phase lag between the two depths is 1–1.5 h (Fig. 10.7, Plate 20). Greater time lags, some 4 h, are observed when stratification limits the turbulence to a region of thickness of 40 m adjacent to the seabed (Simpson et al., 1996). In the tidally mixed region of the 45 m deep southern North Sea where tidal flows reach 1 m s−1 , Nimmo Smith et al. (1999) used acoustic and visual observations to show that turbulent eddies generated near the bottom in strong tidal flows can reach the water surface to form boils (Fig. 10.8) like those often seen in rivers when they are in flood. The patterns formed on the surface appear to be similar to those caused by ‘upwellings’ in open channel laboratory experiments made by Pan and Banerjee 15 Oakey (1990) and Loder et al. (1994) give examples of the cross-frontal profiles of ε.

Figure 10.8. Surface manifestations of ‘boils’ in a shallow tidal sea. (a) An acoustic sidescan image of the sea surface taken from a seabed-mounted sonar. The boils are the dark (high-scattering) semi-circular features made visible in the acoustic image by enhanced wave breaking and bubble production. The near vertical bands at the bottom are caused by surface waves. Bands (see arrows) in the dark scattering region of bands are roughly aligned with the wind and probably caused by Langmuir circulation. (b) A composite video image of an oil slick (seen, for example, at A and B) made from an aircraft, but not coincident with the acoustic image (a). Boils make holes in the oil film or bring clouds of sediment to the surface (C). The scale bar in 50 m long, and the water depth is 45 m. (From Nimmo Smith et al., 1999.)

10.4 Mixing by tidal processes

285

Figure 10.9. The formation of boils. Streak patterns of particles on the water surface disturbed by the arrival of a turbulent burst generated in flow over the bottom of a 2.5 cm deep laboratory tank. The images follow the mean flow of about 0.11 m s−1 from left to right. In (a) the dark patch is caused by divergent flow and removal of particles around the surface-impinging burst. The image in (b) shows the spreading of the dark, particle-free, patch transverse to the flow direction and the formation of a dipole vortex in its lee. By (c) and (d) the divergent flow is largely replaced by the flow around the dipole vortex. The frames show an area 6.7 cm × 4.5 cm. (From Kumar et al., 1998.)

(1995) and Kumar et al. (1998), although these experiments are at much smaller Reynolds numbers (Re = UH/ν of order 9 × 103 ) than in the sea (about 5 × 107 ). Upwellings are related to bursts originating in the bottom boundary layer and appear to generate dipole eddies after impact with the free surface as shown in Fig. 10.9. The tide-generated boils, sometimes containing sediment derived from the seabed (a clue to their likely origin), are of diameter comparable to the water depth. Whether or not the coherent structures observed on the sea surface are associated with ‘bursts’ in the bottom boundary layer has, however, yet to be demonstrated but, if so, they may provide a means of monitoring turbulent processes below the sea surface by aircraft or satellite remote sensing. Their effect on the spread of a floating oil plume

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286

23.0

23.4

23.8

στ

24.2

24.6

25.0

23.0

στ 23.4

23.8

DEPTH, m

0

15

30

45 0

5

10

TIME, min

Figure 10.10. Billows in Massachusetts Bay. A 200 kHz echosounder image of billows in the Bay in a water depth of 80 m made shortly after the passage of a depression in the thermocline but before the arrival of internal waves that are formed by the tidal flow over the Stellwagen Bank. The image is remarkable because billows of two scales are apparent: the larger are 5–15 m high and the smaller, on the braids between the larger billows, are 1–2 m high. Superimposed on the images are two CTD density profiles showing regions of static instability within the billows. Acoustic reflections from the CTD cause the diagonal lines in the acoustic image. (From Haury et al., 1979.)

shown in Fig. 10.8 demonstrates that they supplement the dispersion caused at the sea surface by other processes such as Langmuir circulation.16 10.4.3

Mixing produced by internal waves

Attention has been drawn earlier to the breaking on the continental shelf of soliton packets of internal waves generated, for example, by tidal flows at the shelf edge. Figure 10.10 is an acoustic image of billows obtained in Massachusetts Bay. Internal waves are generated as lee waves in the ebb tide over the Stellwagen Bank in the outer part of the Bay and, on the flood tide, cross the Bank and propagate across the Bay towards the shore, much as described by Maxworthy (1979). The billows shown in the figure formed shortly after the thermocline was depressed by some 20 m and before the appearance of 10 m amplitude internal waves, suggesting that here mixing occurs during the process of development of internal solitons. The image shows the presence of small 1–2 m high billows on the braid between the larger billows.17 The loss of energy from internal waves may not be predominantly in mid-water. Inall et al. (2000) made measurements of turbulent dissipation at a location 5 km inshore 16 At horizontal scales greater than the water depth, turbulence is likely to be anisotropic, as demonstrated in laboratory experiments of shallow flow through a turbulence-producing grid by Uijttewaal and Jirka (2003). 17 Such secondary billows are also observed in some tilted tube laboratory studies of Kelvin–Helmholtz instability (see, for example, Atsavapranee and Gharib, 1997, figs. 7k and 8l).

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287

of the shelf break on the Malin Shelf, west of Scotland. Much of the dissipation over a tidal cycle is associated with soliton internal waves but, both in neaps and spring tides, a majority of dissipation, about 70%, is in the boundary layer at the seabed, the remainder being in mid-water presumably much as described by the observations of Moum et al. (2003) illustrated in Fig. 2.4 (Plate 3). The depth averaged rates of turbulent energy dissipation are 1.1 × 10−2 W m−2 at neap tides and 4.0 × 10−2 W m−2 at springs,18 and lead to values of Kρ of 5 × 10−4 m2 s−1 at neaps and 12 × 10−4 m2 s−1 at springs.

10.5

Dispersion in steady and periodic flows

10.5.1

Shear dispersion in shallow water

Measurements of near-surface dispersion, some from shallow seas, have already been mentioned in Section 9.6. On dimensional ground the horizontal dispersion coefficient in a layer of shallow water of depth, H, with a mean current, U, uniform in depth (an approximation to the flow in a shallow tidal sea with no significant wind or buoyancy forcing), must be K H = cUH,

(10.2)

where c is a non-dimensional constant. Data reviewed by Fischer (1973) give a value of c = (1.03 ± 0.44) × 10−2 . Bowden (1965) pointed to a further factor that may be particularly important for diffusion in shallow tidal seas, now known as ‘shear dispersion’. Dispersion in shear flow was first investigated by Taylor (1953) in connection with the spread of dye injected into a steady flow through a tube with a circular section of radius, a. When the Reynolds number is so low that the flow through the tube is laminar (Section 1.1), the flow speed at radius, r, is u = 2U(1 − r 2 /a2 ), where U is the mean flow speed. The spread of a patch of dye is such that its concentration is centred on a point that moves along the tube with the mean speed, U, and (in spite of the radial variation of the flow) is symmetrical about this point but, because of shear, spreads in time with an along-tube diffusivity, κ H , given by κH = a 2 U 2 /48κd ,

(10.3)

inversely proportional to the molecular diffusion coefficient of the dye, κd .19 Taylor shows that the relation, applied to experimental measurements of the diffusion of dye, provide an accurate estimate of κd . In a subsequent paper, Taylor (1954) showed that 18 Further measurements of turbulent dissipation in internal soliton packets are described by MacKinnon and Gregg (2003b). 19 Taylor’s results are valid when 4L/a Ua/κd 6.9, where L is the length of tube over which there are appreciable changes in concentration. It is well-worth reading Taylor’s original paper, if only to appreciate his comments about earlier observations by a Dr Griffiths: ‘The only parts of his statement that do not seem clear are those which follow the words “It can easily be shown that . . .” and “it is obvious that . . .” ’.

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288

in a turbulent flow through the tube, the longitudinal diffusion coefficient is 10.1au∗ , a form similar to (10.2) since U and the friction velocity, u∗ , are linearly related. (Here u * = (τ /ρ 0 )1/2 , where τ is the stress on the tube walls.) Shear dispersion of a passive tracer in a horizontal mean current, U (z), varying in depth, is a consequence of the tracer at different levels in the flow being carried in a downstream direction at different speeds by the mean shear flow, whilst being diffused vertically by turbulent motion. The horizontal diffusivity, KH , in the downstream direction is found to be inversely proportional to the tracer’s vertical diffusivity, KV , (as in (10.3)),20 and may therefore depend on the factors that affect vertical flux, such as tidal currents, surface buoyancy flux, water depth and wind stress. The effect of shear is to greatly increase the horizontal downstream diffusivity. The effect of these processes on the estimates of dispersion synthesized by Okubo (Section 9.6.1) is unquantified.

10.5.2

Dispersion by waves and tides

Dispersion in tidal flow over and around irregular bottom topography (e.g. sand banks) that leads to the formation of eddies and residual flows has been studied by Ridderinkhof and Zimmerman (1992) and applied to the 20–30 km wide Dutch Wadden Sea. A purely periodic Eulerian tidal motion everywhere can lead to chaotic Lagrangian particle paths in some (but not necessarily all) regions, with particularly rapid dispersion along the axis of the channel leading from the North Sea into the Wadden Sea. Thorpe et al. (1994) used sidescan sonar observations of bubble bands formed by Langmuir circulation in the southern North Sea to simulate the paths of floating particles released into a corresponding surface flow field (one with convergence towards the observed bands). They find values of across-wind dispersion coefficient, Ky , in the range 5 × 10−3 to 0.5 m2 s−1 in wind speeds of 5–10 m s−1 . These are comparable with values in Langmuir circulation mentioned in Section 9.6.2, but it appears likely that bottom-generated boils may have contributed to the break-up of the bubble bands, and so to dispersion.

10.5.3

Dispersion in the thermocline

Although several of the dispersion experiments included in Okubo’s overview of dispersion summarized in Equation (9.11) are from the mixed layer of shallow seas, none is in the stratified shelf-sea thermocline. Five experiments each lasting for 2.5–5 days in 20 Bowden (1965) finds, in a steady mean flow in which U = u f (ζ ) and K V = K g(ζ ), with ζ = z/H , and where f and g are arbitrary non-dimensional functions of ζ , that K H = −(u 2 H 2 / ζ K )G, where G = f 1 (ζ )F(ζ ) , with f 1 (ζ ) = f (ζ ) − f (ζ ) , and F(ζ ) = 0 {[1/g(ζ2 )] × ζ 0 f 1 (ζ1 )dζ1 }dζ2 . The symbols . . . imply that mean values are taken over the depth, H, or from ζ = 0 to 1. If, for example, U = uz/H = uζ (f = ζ , a uniform shear) and KV = 4K(z/H) × (1 − z/H) = 4Kζ (1 − ζ )(g = 4ζ (1 − ζ ) or a vertical diffusivity that is zero at the sea bed, z = 0, and at the sea surface, z = H, with maximum value, K, at mid-depth, z = H/2), then KH = u2 H2 /96K, whilst if KV is constant, KH = u2 H2 /120KV . Bowden also examined dispersion in an oscillatory current flow.

10.5 Dispersion in steady and periodic flows

289

Figure 10.11. Dispersion around the UK. The distribution of caesium-137 on the UK Continental Shelf in 1983 resulting from a continual release into the Irish Sea from the Sellafield nuclear plant which is marked by a star. The units are Bq kg−1 . The discharge of caesium-137 in 1983 totalled 1.2 PBq. (1 PBq = 1015 Bq, Note that 2 PBq were released in 1982.) Much of the caesium is carried northward, through the North Channel of the Irish Sea and around the north-western coast of Scotland. Compare C British Crown copyright, 1985, with Fig. 13.15, Plate 33. (From Hunt, 1985; reproduced by permission of CEFAS, Lowestoft.)

which rhodamine or fluorescein dye is released into the summer or autumn thermocline in water about 70 m deep some 100 km south of Martha’s Vineyard, Massachusetts, are described by Sundermeyer and Ledwell (2001). Dye introduced in a line about 1 km in length on a chosen isopycnal surface had, by the end of an experiment, spread to scales of about 10 km. The horizontal diffusivity, KH , estimated from a Fickian model of the dye distribution, ranges from 0.3 to 4.9 m2 s−1 , to which estimates of 0.29– 4.1 m2 s−1 at scales of 1–10 km using (9.11) compare favourably. Estimates of dispersion driven by shear dispersion or of the interleaving of dyed patches, however, fall short of the measured values, and Sundermeyer and Ledwell suggest that the presence of vortical mode derived from the collapse of intense diapycnal mixing events,

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290

the spread and spin-up of mixed fluid,21 may supplement dispersion to the extent of contributing 0.2–4.6 m2 s−1 to KH . Direct evidence for this process of dispersion is lacking. 10.5.4

Anthropogenic markers of mean flow and dispersion

The pattern of dispersion over larger scales around the coastline of the United Kingdom, largely dominated by advection in the mean flow, has become apparent from surveys of caesium-137 released into the Irish Sea from the Sellafield Nuclear Plant (Fig. 10.11). Caesium has a half-life of 30.17 years and the figure illustrated the distribution after a release of about 10 years.

10.6

Regions of horizontal variation in temperature and salinity

In regions of large horizontal density gradient, tidal straining acts to produce periodic stratification that modulates turbulence generated at the seabed. This is particularly so where fresh water is carried from land, in rivers mouths and estuaries, called ‘regions of freshwater influence’ (ROFI). Study of such regions is often of particular importance because of the associated anthropogenic influence on the content and composition of the fresher water, and the need to discover and quantify the distribution, effect and fate of pollutants in coastal waters. During the ebb flow of the tide, less dense, fresh water, its speed enhanced by its run-off, flows over the underlying denser salty water that moves relatively slowly because of the bottom stress. The mean isopycnal surfaces become more horizontal and the stratification of the water column is consequently increased. Observations by Rippeth et al. (2001) show that in this period strong turbulence is mainly confined by stratification to the lower near-bed layer of the water column. In flood, however, stratification is decreased as denser off-shore water moves towards shore; bottom friction again reduces the speed of the deeper water but now leads to a steepening of the mean isopycnal surfaces and reduction in vertical density gradient. Mixing may now extend from the bottom throughout the water column as shown in Fig. 10.12, Plate 21. Advection and bottom friction may even result in the denser off-shore water being carried over the fresher inshore water, resulting in conditions of static instability and convectively forced turbulence described by Nepf and Geyer (1996) as ‘overstrained’. Definitive observations of turbulent dissipation rates or turbulent structure in such conditions are not yet available. 21 Polzin and Ferrari (2004) have a similar explanation to explain the width of streaks of the SF6 tracer in NATRE (see Section 13.4.1, where the Fickian model used to estimate dispersion is described).

Chapter 11 Boundary layers on beaches and submarine slopes

11.1

Introduction

This chapter describes the nature of turbulent flow induced by waves and other processes in two contrasting regions where the presence of a sloping boundary has profound effects. Much of the energy of surface waves that have sometimes crossed the full width of ocean basins (Snodgrass et al., 1966) is eventually dissipated by turbulence in the near-shore zone. Perhaps the most familiar, violent and extensive turbulence anywhere in the ocean is that produced in long lines by plunging breakers formed as waves of approximately symmetrical shape in deep water encounter a gently shelving and uniform beach and enter the surf zone, the gradually shoaling region between the location of the first breaking of the shoreward-propagating waves and their final demise in the swath zone at the shoreline itself. Waves travel and arrive in groups; as those who have paddled at the edge of a sandy beach know, about every seventh – but with substantial variation – wave is higher than the others. Breaking is a transitional process. After breaking, the waves continue to advance shoreward, no longer symmetrical but commonly in the form of a hydraulic jump or turbulent bore, leaving a floating trail of foam.1 Patterns visible in foam indicate a degree of order and structure both within the breakers themselves and within the turbulence left as bores advance through the surf zone. Much of what is known of the turbulence in the surf zone is to be found in the coastal engineering literature and derives from laboratory studies, and many of the observations in the ocean surf zone have been directed towards improving the prediction of erosion or deposition of sediment. 1 Hydraulic jumps are described in Section 12.6.1.

291


292

Relatively little is known about turbulence within the corresponding sub-surface zone where internal waves encounter the sloping boundaries surrounding the deep ocean. Because of the peculiar properties of internal waves, this ‘internal surf zone’ differs in a number of ways from its surface wave equivalent. It is a region identified by Munk (1966) as possibly supporting a substantial part of the diapycnal mixing in the abyssal ocean. Armi (1978) made the interesting suggestion that a combination of turbulent mixing within the near-bed boundary layers on the sloping sides of topographic features in the deep ocean, and the subsequent along-isopycnal advection of mixed water from the mixing region and away from the ocean boundary, might provide much of the mixing required to account for Munk’s canonical value of KT . By the mid 1990s, evidence of enhanced mixing near the lateral boundaries of the oceans had been discovered; Ledwell and collaborators (see Ledwell and Bratkovich, 1995; Ledwell and Hickey, 1995) showed that the vertical diffusion of tracers released in mid-water within the Santa Cruz and Santa Monica deep-water ocean basins off the Californian coast is much increased when, after spreading along isopycnal surfaces, the tracers come into contact with the sloping sides of the basins. A number of analytical and observational studies, some described below, confirm Munk’s conjecture that the sloping boundaries of the ocean are indeed sites of active mixing. They now appear to contribute substantially to the vertical diffusion of heat and dissolved material in the deep ocean, notably in ‘hot spots’ where internal tides are large.

11.2

The near-shore zone

11.2.1

The surf zone

The flux of energy carried by a surface wave that, in deep water, has an amplitude, a, and a phase speed, c, is, from (9.1) and (9.3), Ecg = (1/4)ρ 0 ga 2 c per unit crest length. Much of this energy will be dissipated by turbulence as the wave breaks at the edge of the surf zone and runs through it as a hydraulic jump or smaller breaker towards shore. Relatively little is reflected except on steep beaches, seawalls or cliff faces. Ignoring energy lost from the waves approaching a beach through their reflection, or in maintaining a field of subsurface bubbles, producing spray, moving sediment and generating mean or transient currents, the mean rate of dissipation per unit mass by turbulence within the surf zone, ε , is (1/4)ga 2 c/A, where A is the area of a vertical section of the water within the surf zone taken in a direction normal to the shoreline. To obtain a rough measure of the mean dissipation rate in the surf zone, suppose that a = 0.3 m, c = 10 m s−1 (corresponding to a wave with period of about 6.5 s), and that the surf zone is 40 m wide and extends from shore with a uniform slope to a depth of 1 m at its outer edge: then ε ≈ 0.11 W kg−1 , a relatively large value. If no heat is lost from the water within the surf zone either through the sea surface and the seabed, or by advection into deep water (perhaps by rip currents), the wave energy dissipated

11.2 The near-shore zone

293

by turbulence will result in an increase in heat per unit mass at a rate, cp dT / dt, where dT/dt is the rate of rise in temperature. With the adopted values, and taking cp = 3.99 × 103 J kg−1 K−1 , this implies dT /dt ≈ 2.2 × 10−5 K s−1 , or about 80 mK h−1 . Although the surf zone is often warmer than the deeper water beyond, the dominant cause of its higher temperature is likely to be the incoming solar radiation rather than the dissipation of wave energy. Heat transferred across the surface in shallow water by solar radiation leads to a rate of increase in temperature greater than that caused by the same radiant heating distributed over a thicker layer in deeper water. It appears that, as in breakers in deep water (Section 9.2), only a small proportion, typically 2%–6%, of the energy lost by a wave in breaking at the outer edge of the surf zone is dissipated at depths greater than the wave height. This is supported by measurements made by Trowbridge and Elgar (2001) in the outer surf zone at a height of 1 m above the bottom in a mean depth of 4.5 m. The average shear production of turbulence in this location well below the wave troughs is of the same magnitude as the mean dissipation rate, ε, but both are some two orders of magnitude less than the depth average rate at which shoaling waves lose energy; generally turbulence does not penetrate far below the breaking wave. In general the time taken for turbulence to dissipate is much longer than the period of the breakers, so that waves break into water that is already in a state of decaying, if not still very active, turbulence. A degree of structure is evident in the foam left by breakers. Even though bubbles are generated within and all along the tube of air trapped by a plunging breaker, foam is not produced in a smooth continuous sheet along the length of a breaking wave crest, but in a series of ribbons (Fig. 11.1), presumably indicating lines of convergence possibly in motions akin to surface Taylor vortices (Wang and Domoto, 1994). Turbulence is formed at the air-entraining ‘toe’ of a spilling breaker advancing towards shore. Behind is a turbulent bubbly, wake-like flow (Peregrine, 1983). Bands of foam, orientated parallel to the wave crests, are often visible in the surf zone (Fig. 11.2). These are caused by an array of vortices rotating with near-surface motion in the direction of wave advance, the ‘horizontal eddies’ or shed vortices shown in Fig. 11.3 and described by Nadaoka et al. (1989), each similar to the rotors in deep water examined by Melville et al. (2002). They are possibly caused by repeated breaking of waves, the first plunge being followed by bounce up and subsequent plunging (Miller, 1976; Peregrine, 1983). Deane (1997) describes acoustic measurements of sound created by bubbles formed by breakers in the surf zone and shows photographs of the surface and sub-surface appearance of bubble clouds and plumes. The linear bands appear to be broken up into deeper penetrating plumes of diameter roughly equal to the breaker height. Bubble-carrying ‘oblique eddies’ with axes inclined at some 45◦ to the vertical penetrate below the horizontal eddies. Both horizontal and oblique eddies are significant in their contribution to Reynolds stress. Other structures become visible in the surface foam layer a few metres behind a hydraulic jump advancing through the surf zone, presumably as new processes become dominant, including those involving the arrival at the surface of coherent structures, for example those generated by the shear flow over the seabed. Almost circular holes

294


Figure 11.1. Ribbons of foam produced by breaking waves in the surf zone. The crest of the breaker is at the right and is travelling to the right encountering foam patches left by earlier breakers. (From Thorpe et al., 1999b.)


295

Figure 11.2. Bands of foam behind breakers approaching shore. The triangular regions of foam are formed as the crest length, over which breaking occurs, increases as the waves advance into shallower water. A sharp-crested wave is just about to break at the edge of the surf zone. Bands of foam lying roughly parallel to the breaking crests are visible within the triangular regions. (From Thorpe et al., 1999b.)

Horizontal eddies

Obliquely descending eddies

Figure 11.3. Sketch showing the formation of large eddies in the wake of a breaking wave travelling towards the top right up a sloping beach. This may be compared to the sketch of wave breaking in deep water shown as Fig. 9.3. (From Nadaoka et al., 1989.)

appear within the wispy foam layer as shown in Fig. 11.4, perhaps caused by divergent flows induced by rising plumes of greater bubble content (as in deep water, Fig. 9.8) or by bed-generated ‘boils’ (as illustrated in Fig. 10.9a). Estimates of turbulent dissipation rates within the surf zone have been made using hot film anemometers sampling at 0.125 Hz in conditions when the significant wave

296


(a)

(b)

Figure 11.4. Holes in sheets of foam produced by breaking waves travelling as hydraulic jumps into shoaling water. (a) Holes within the foam layer behind a breaker moving through the surf zone over a relatively smooth bed. The hydraulic jump is at the right of the photograph and is travelling to the right. No ribbons of foam behind the crest like those in Fig. 11.1 are visible at this stage of breaking. The diameter of holes is about 1 m. (b) Holes in a broad foam sheet in a rocky surf zone. (From Thorpe et al., 1999b.)


297

height, H 0 , off-shore is between 0.5 and 1.2 m (George et al., 1994).2 Values of ε increase with distance off the seabed, and are in the range (10−6 to 7 × 10−4 )(g3 h)1/2 , where h is the local mean water depth in the surf zone. Depth averaged dissipation rates, ε , are in accord with a model of a train of propagating periodic bores proposed by Thornton and Guza (1983); ε = (1/8)g3 hσ b3 ,

(11.1)

where = H 0 /h (0.25–2.5 in the observations), σ is the wave frequency and b is a ‘breaker-intensity’ coefficient found to range from 0.4 to 0.9 in the observations. If, for example, = 1.2, h = 0.5 m, σ = 2/(6.5 s) and b = 0.65, Equation (11.1) gives ε = 0.09 W kg−1 . The mean kinetic energy of the turbulent motion (TKE) in the surf zone has a generally weak variation with depth (Svendsen, 1987).

11.2.2

The swath zone

At the landward boundary of the surf zone lies the swath zone. This is the region at the edge of a sloping beach that is periodically covered by pulses of shallow water (the remains of waves largely dissipated in the surf zone), which run up the slope, lose their kinetic energy, and then drain back, again exposing the beach. Severe turbulence is caused by the collapse of bores at the seaward edge of the swath zone as they meet the down-slope draining flow of a preceding wave, and in the high Reynolds number flow of water up and down the beach. Dissipation rates of 0.1 W kg−1 are reported by Raubenheimer et al. (2004), compared to values of 0.04 W kg−1 just beyond the swath zone. As the swath zone is slowly carried down a sloping beach by an ebbing tide, it moves over sediment previously covered in the surf zone. Sedimentary bedforms (e.g. ripples) generated in the surf zone may be smoothed or obliterated by sediment movement in the highly turbulent swath zone, leaving a smooth, exposed and drying, beach at low water (Clark and Werner, 2004; see also Section 10.3.3).

11.2.3

Along-shore flows

Longuet-Higgins (1970) showed that the energy flux of surface waves, Ecg , and their ‘radiation stress’ or momentum flux are directly related. The flux of along-shore momentum per unit length along the shoreline carried by a wave train approaching a beach from an incident angle, θ (θ is zero for waves approaching along a direction normal to the shoreline), is given by Sx y = (Ecg /c) sin θ cos θ,

(11.2)

2 The dissipation rate, ε, is estimated by translating the measured frequency (σ ) spectrum of the velocity fluctuation to a wavenumber (k) spectrum using the Taylor hypothesis, k = σ /U, where U is the mean velocity, and by fitting the resulting spectrum to the theoretical wavenumber spectrum in the inertial sub-range (1.14): (k) = α 2 ε 2/3 k−5/3 , where α 2 is a constant, approximately 0.5.

298


Figure 11.6. An aerial photograph of the surf zone showing evenly spaced jet-like rip currents carrying material from the surf zone and producing vortex pairs. Waves are approaching the beach obliquely from the top left. (From Inman et al., 1971).

where c is the phase speed of the waves in deep water. Consequently the loss of energy through the breaking of waves that approach shore from a non-normal direction (so that θ = 0, for example as in Fig. 11.6) involves a transfer of momentum from waves that drives a mean alongshore flow in the surf zone, positive in the direction of the alongshore component of the wave’s phase speed.3 There is evidence that mean currents in the surf zone with well-developed along-shore drift have a law of the wall variation with height above the bed at depths well below wave troughs (Cox et al., 1996; Garcez Faria et al., 1998). Such along-shore flows are, however, known to be unstable (Oltman-Shay et al., 1989; Bowen and Holman, 1989; Noyes et al., 2004), developing periodic disturbances, ‘shear waves’ or ‘vorticity waves’ (Shrira et al., 1997), with observed alongshore wavelengths of order 50–300 m and periods of 45–1000 s. Numerical models 3 The onshore component of radiation stress is balanced by pressure gradients resulting from changes in mean sea level in the near-shore zone, or wave ‘set-up’.

11.3 Shoaling internal waves in the thermocline

299

¨ (Allen et al., 1996; Slinn et al., 1998; Ozkan-Haller and Kirby, 1999) show that the waves grow to produce eddies or vortices that propagate offshore as illustrated in Fig. 11.5, Plate 22.4

11.2.4

The outer edge of the surf zone

Vortex pairs form at the head of rip currents (Fig. 11.6), narrow jets flowing offshore to distances of order 100 m from the surf zone, perhaps linked to the instability of the alongshore flow. Measurements of rip currents made from the pier at the Scripps Institution of Oceanography using an acoustic Doppler sonar are described by Smith and Largier (1995). The currents transport warm and sediment-rich water from the surf zone at speeds reaching 0.7 m s−1 , and are consequently a hazard to bathers. A further process leading to organised mixing outside the surf zone is that discovered by Matsunaga et al. (1994; see also Li and Dalrymple, 1998) in laboratory experiments in which waves propagate towards a slope. Vortices, with horizontal axes, are found in shoaling water before the waves break, as shown in Fig. 11.7. The vortices rotate with an offshore flow near the surface, and therefore opposite in sense to the horizontal eddies shown in Fig. 11.3. In water of depth, h, the distance between vortices, dv , in this ‘offshore vortex train’ is given empirically by dv = 6 × 10−3 (tan α)−1/3 (λh/a),

(11.3)

where a and λ are the local wave amplitude and wavelength, respectively, and α is the bottom slope. The vortices drift slowly offshore, sometimes ‘pairing’, one joining with a neighbour, and decay as they reach deeper water. They appear to be generated in the shear between drift currents directed onshore and offshore. No direct measurements of turbulence within the vortices have been made, but acoustic observations of persistent bands of bubbles outside the surf zone, possibly trapped within the vortices, are described by Nimmo Smith (2000) and are shown in Fig. 11.8.

11.3

Shoaling internal waves in the thermocline

Laboratory studies by Wallace and Wilkinson (1988) of the approach of an interfacial wave to a uniform slope show that the wave steepens and evolves into a structure resembling a density (or gravity) current or bore, carrying deeper and denser water 4 Guidance on the motion of such eddies is provided by the use of image systems: vortices in the wedge-like near-shore region can be represented as segments of complete vortex rings, the segments of the ring outside the wedge forming the image (Peregrine, 1998). Behaviour of multiple eddies in a shoaling region is dynamically similar to that of smoke rings. This assumes, however, that the bottom slope is small and constant and neglects frictional effects. In such restrictive conditions, an along-shore flow in the surf zone can be represented as a segment of an axisymmetric jet, again forming an image system, so that the instabilities of along-shore flows and axisymmetric jets are also dynamically similar.

Figure 11.7. An ‘offshore vortex train’ in a laboratory experiment made visible by the addition of dye at the water surface. The vertical bars, E, D, C and B are 0.5 m apart and at positions of increasing depth ranging from 0.08 m at E to 0.14 m at B. Waves approach a smooth 1.5◦ slope from the right and break about 0.68 m to the left of the vertical bar E. Image (a), nearest the shoreline, shows the initiation of vortices. They become larger with increasing offshore distance ((b)–(c)), before dissipating (as shown in (d)). (From Matsunaga et al., 1994.)

11.3 Shoaling internal waves in the thermocline

301

Figure 11.8. Two examples of acoustic images obtained using a 250 kHz sidescan sonar pointing towards shore up a smooth sandy beach with a bottom slope of about 1◦ . A scale bar of 50 m is shown. The vertical axis points towards shore, and range is derived from the time delay of acoustic pulses and the known speed of sound. The horizontal scale, originally time of acoustic pulse emission, has been adjusted to along-shore distance using the known along-shore current. The shoreline is well beyond the top (furthest range) of the image. The irregular dark bands near the top of the images mark the location of an acoustic wall, possibly a region of intense sound absorption by bubbles formed within the surf zone, which is even further up the image. The images therefore show mainly reflections from acoustic scatterers outside the surf zone. The relatively dark and generally continuous acoustic scattering bands are possibly regions where bubbles are temporally captured by vortices in an ‘offshore vortex train’. The conditions of the observations differ from the laboratory (Fig. 11.7) because there are along-slope flows and θ = 0. (From Nimmo Smith, 2000.)

upslope but diminishing in size as it does so, as shown in Fig. 11.9. A front is formed with large density gradients on the forward face of the density current. Water carried upslope by a preceding wave and draining down-slope ahead of its successor, is lifted off the bottom over the rotating nose of the density current following the front (Fig. 11.9a, iii). There is often evidence of billows formed by shear instability over, and immediately behind, the front. Water is entrained at the rear of the density current nose leading to a circulation with, near the bed, a forward motion comparable in speed to that of the bore’s speed and, over the nose, a flow contrary to the bore’s propagation direction. (This circulation has a sense of rotation opposite to that found within a plunging breaker at the sea surface.) The down-slope drainage, and the lifting over the nose of the bore, of water that has been mixed during the run-up phase and is consequently of intermediate density, results in an intrusive layer slowly spreading along the interface away from the sloping bottom and into deeper water (Fig. 11.9b). The layer may contain a relatively high concentration of suspended sediment if the near-bed currents are sufficiently enhanced by run-up to cause sediment erosion from the bed.


302

(a)

(i)

(ii)

(iii)

(iv)

(v)

Figure 11.9. Internal waves on a thin interface between two uniform layers approaching a uniformly sloping beach. (a) Interface shapes near the slope at successive times over one wave period. (b) A 1aboratory experiment showing the lifting of dye placed on the slope ahead of the approaching waves and its entrainment into the rear of the up-slope running front of the wave at about the stage (iii) shown in (a). The arrow at left shows the interface location before the arrival of the waves and the lines are 1 cm apart. The interface to the left of the entrainment region is marked by dye that has been lifted from the slope and spread along the interface by earlier waves.

11.4 Internal waves in quasi-uniform stratification

303

Similar effects are found as a solitary interfacial wave of depression between two layers with upper layer depth, h 1 , less than the lower, h 2 , that impinge on a slope, but with the bore forming as the rear face as the wave reaches the slope (Michallet and Ivey, 1999).5 The efficiency of mixing, the ratio of the increase in potential energy resulting from mixing to the energy lost by the wave, is less than 0.25. Hosegood and van Haren (personal communication) detected the presence of such bores, and an associated increase in sediment concentration, on the southern flank of the Faeroe– Shetland Channel.6 The breaking of the large-amplitude solitary waves on the Oregon continental shelf (Fig. 2.4, Plate 3, and Fig. 5.1, Plate 9) may be a consequence of wave amplification as the water depth decreases. Most of the evidence in this case points to shear as the cause of breaking. The maximum horizontal component, u, of currents in the water above the wave troughs in the direction of wave propagation (at speed, c) are, however, close to the condition (u = c) that must be exceeded if breaking is to occur by convective overturning, with the possible subsequent formation of rotors, cores of re-circulating fluid over the wave troughs that are carried forward with the wave.7

11.4

Internal waves in quasi-uniform stratification

11.4.1

Wave reflection and critical slopes

The presence of sloping topography leads to a variety of processes involving internal waves, some of which are illustrated in the cartoon shown in Fig. 11.10 and described below. Phillips (1966) was the first to describe the changes that occur in the reflection of an internal wave train from a uniform sloping boundary when the waves are propagating as a ray in a fluid of uniform buoyancy frequency, N. Wave frequency is conserved and, as shown in Fig. 11.11, the reflected wave energy must therefore travel at the same inclination, β, to the horizontal as the incident wave, given by (2.9). The wavenumber of the waves is changed on reflection and therefore, by (2.11), so is the group speed. Eriksen (1982) recognized that, whatever the azimuthal angle of approach of the incident wave train to the slope,8 singular effects occurs when β is equal to the angle of inclination, α, of the bottom to the horizontal. As the wave frequency is changed so that β approaches α, the direction of the reflected wave energy tends towards the direction of the line of greatest slope, the wavenumber of the reflected wave tends to 5 Vlasenko and Hutter (2002) have made a related numerical study of waves breaking on slopes. 6 Observations using seismic reflection profiling have also detected internal wave run-up, on slopes west of Norway (Dr W. S. Holbrook, personal communication). 7 Lamb (2003) examines the conditions under which rotors may develop as solitons run into shallower water. His study includes the effects of a mean current with near-surface shear of the same sign as that caused by the soliton waves. 8 The azimuthal angle is that between the vertical planes in the directions (i) up the line of steepest ascent on the slope, and (ii) of the group velocity of the wave. It determines the horizontal direction of approach of the waves to the plane. For surface waves it would be identical to the incident angle, θ (Section 11.2.3).

cg

E

E

(k, l, mR)

E

L

cgR

I

cgI

cgI

R

I

1

2σ ≈ (N 2 sin2 α + f 2 cos2 α) 2

1

σ ≈ (N 2 sin2 α + f 2 cos2 α) 2

Figure 11.10. Some of the effects of the presence of sloping topography including, at top left to right: lee waves, wave generation by oscillatory flow, resonant interactions between incident and reflected waves with generation of harmonics, wave breaking at critical frequencies; and at second row, left to right: wave steepening and formation of fronts, the generation of Eulerian (VE ) and Lagrangian (uL ) mean flows by reflecting waves, and mixing by reflecting sub-critical waves when the first harmonic is near critical. The final illustration represents the ejection of fluid from the boundary resulting from the convergence of flows produced by the irregular generation of Lagrangian flows by reflecting wave packets. (From Müller and Briscoe, 2000.)

cg

1

σ ≈ (N 2 sin2 α + f 2 cos2 α) 2


305

Figure 11.11. The reflection of internal waves at a slope. The symbols cgI and cgR indicate the group velocity of waves incident and reflected from the slope, and the lines indicate surfaces of constant phase (e.g. crest or troughs). Parts (a) and (b) show the super-critical case with α < β, and phase lines travelling upslope or downslope, respectively, at velocity, cb , whilst (c) and (d) show subcritical conditions, α > β, with downward and upward incident waves (and phase lines moving upslope or downslope, respectively). The distance between lines of constant phase, proportional to the wavelength of the waves (see Fig. 11.12), is reduced on reflection in (a) and (c), increased in (b) and (d). Singular critical conditions are approached as β tends to α in (a) or (c) when the reflected wavelength tends to zero and the reflected ray becomes parallel to the slope. (From Thorpe and Umlauf, 2002.)

infinity, and its group velocity tends to zero. The condition, α = β, can be regarded either as defining a critical slope, αc , for waves of a given frequency, σ , given by αc = sin−1 {[(σ 2 − f 2 )/(N 2 − f 2 )]1/2 },

(11.4a)

(from (2.9)), or as defining a critical frequency, σ c , of waves on a slope of inclination, α, given by σc = (N 2 sin2 α + f 2 cos2 α)1/2 ,

(11.4b)

(from (2.10)). Waves with σ > σ c are called supercritical waves. In practice, both α and N vary as depth increases down, say, a continental slope. The right-hand side of (11.4b) is a function of depth, and the equation determines the critical frequency on the slope as a function of depth, or alternatively the depth at which waves of a given frequency are critical.

306



307

Eriksen (1985) showed that the Garrett–Munk spectrum of internal waves in the ocean should be severely modified by wave reflection from a sloping boundary and that, in accordance with this prediction, spectra are distorted in such regions, energy near the critical wave frequencies exceeding the Garrett–Munk spectral levels to distances of several hundred metres off the bottom, as shown in Fig. 11.12. Eriksen concludes that the dissipation of only a small proportion of the excess of energy flux in the internal wave field within such regions is required to account for Munk’s basin-wide vertical diffusivity of about 1 × 10−4 m2 s−1 . Dissipation where energetic internal waves reflect from near-critical slopes is indeed likely. Conservation of wave action on reflection implies that the amplitude of the reflected wave must increase indefinitely as critical conditions are approached (see Fig. 11.13), and so too do the wave steepness and the shear. The Richardson number decreases towards zero. The likelihood of wave breaking by the development of static instability, or through shear-induced instability in internal waves of near-critical frequency, is consequently enhanced by reflection, a conclusion consistent with the occurrence of mixing in laboratory experiments when near-critical waves reflect from a plane slope of inclination, α (Ivey and Noakes, 1989; Taylor, 1993), as illustrated in Fig. 11.14. Laboratory experiments by Ivey et al. (2000) and a numerical model of Legg and Adcroft (2003) are in general agreement in their estimates of the thickness of the mixing layer on the sloping boundary, h m , produced by an internal wave of frequency, σ , approaching the sloping boundary in a plane normal to it: h m = (1 − r )qλI / cos(α + β),

(11.5)

and of the resulting dissipation rates in the layer: ε = U02 σ sin[2(α + β)]/[8 q cos2 β],

(11.6)

where r is the reflection coefficient, the ratio of the wave energy flux reflected to that incident on the slope (zero if all the incident energy flux is dissipated), λI is the incident wavelength, U0 is the amplitude of the oscillating upslope velocity component of the incident wave and q is a constant, found empirically to be 0.1–0.15. The agreement of the laboratory and numerical experiments gives some confidence in the two estimates. It should be remembered, however, when applying the results or in comparing with ← Figure 11.12. The spectra of the upslope component of velocity, showing a peak near the critical frequency, σ c , and its variation with increasing distance from a sloping boundary. Data were obtained from a mooring set at a depth of 1455 m on the southwest flank of the Fieberling Guyot in the North Pacific at 32◦ 25 N, 127◦ 47 W. The depths of sensors are marked and each spectrum is offset from that at depths below by one decade. The Q1 , P1 , K1 and M2 tidal frequencies are shown by arrows, so too are the critical and buoyancy frequencies. Thin lines show the Garrett–Munk spectra. The substantial enhancement in energy near the critical frequency, to levels well above those of the Garrett–Munk spectra, can be detected up to heights of about 300 m off the bottom (i.e. to sensor F308). (From Eriksen, 1998.)


308

cgR cgI

b a

Figure 11.13. Sketch showing internal waves reflecting from a slope. The slope is of inclination α. The wave propagates from a direction lying in a plane normal to the slope at angle β to the horizontal and reflects at the same angle. The tilted lines represent surfaces of constant phase separated by 2 in the incident and reflected waves. Comparing the incident and reflected wavelength, AB and CD, respectively, the ratio of the reflected and incident wavenumber, KR /KI = AB/CD = sin(α + β)/ sin(β − α ). If the wave action is conserved then, since the frequency is conserved on reflection, the energy flux is conserved. Equating the incident flux across AB with that reflected across CD gives EI cgI (2/K I ) = ER cgR (2/K R ), where E I , E R , are the energy densities of the incident and reflected waves and cgI , cgR , are their respective group velocities. Use of (2.20) and (2.11) gives the wave amplitude ratio aR /aI = KR /KI = sin(α + β)/sin(β − α), which tends to infinity as β tends to the slope angle, α. The steepness of the reflected wave is proportional to aR KR , which also tends to infinity.

observations in the ocean, that the incident waves of the laboratory and numerical models are at a zero azimuthal angle and that ε/ν N 2 is O(10) in the laboratory and numerical studies, so that mixing is relatively weak and anisotropic. Direct evidence of the enhancement of mixing at critical slopes was obtained by Moum et al. (2002). They used a towed instrument, MARLIN, carrying turbulence sensors to investigate the horizontal structure of turbulence along the Oregon continental slope. Turbulence is continuous over distances of several kilometres within a few hundred metres of the bottom, but is most intense near slopes at which the M2 internal tide is critical. Values of dissipation rate, ε, and eddy diffusivity, K ρ , increase towards the slope, with mean values of K ρ equal to about 4 × 10−4 m2 s−1 near the slope. It is evident from these observations (and, for example, those shown in Fig. 11.15) that internal waves of tidal period are important in mixing near the continental slope, but where do the internal tidal waves arriving at continental slopes originate? Until the observations illustrated in Fig. 2.7, it was supposed that internal tides do not propagate far from regions of topographic forcing. Although no geographical surveys or robust models have yet been made of their presence and characteristics, it is becoming commonly accepted that internal tides can propagate as low order modes for distances


309

(a)

(b)

Figure 11.14. Schlieren images of laboratory experiments in which mixing is produced on an inclined slope by incident internal waves of critical frequency, σ /N = sin−1 α, travelling in a vertical plane normal to the slope. The waves of mode 1 propagate from the right and N = 0.6 s−1 , wave frequency, σ = 0.3s−1 , and slope angle α = 30◦ . The vertical lines are 0.1 m apart. (a) A bore running upslope, with large but short-lived clockwise eddies forming behind it, shortly after the first arrival at the slope of the incident internal waves. (b) The developed turbulent boundary layer, with evidence of entraining eddies at the upper edge. (From Ivey and Nokes, 1989.)

310


of at least 1000 km from their source. Generated at ridges or other topography within ocean basins, they may be incident on many of the surrounding continental slopes and contribute to the local mixing. (Enhanced turbulence observed by Nash et al. (2004) on the near-critical continental slope off Virginia with K ρ ∼ 10−3 m2 s−1 is attributed to such an incident M2 tidal wave.) An alternative is that internal tidal waves meeting a continental slope or rise at some depth well below that of the continental shelf are generated by interaction of the barotropic tide and the topography at some other location on the same continental slope, perhaps at the shelf break or possibly at critical slopes, before propagating as a ray (see Fig. 5.4, Plate 11) that is refracted back to the continental slope. Except close to their generation site, it is rarely possible to identify the source of internal tides unambiguously. The effect of the shape of the continental slope on mixing produced by reflection has been studied analytically and numerically. Gilbert and Garrett (1989) examine the consequences of non-uniform, but idealized, shapes of boundaries, predicting greater energy enhancement and possibly dissipation near the critical frequency above locally convex bottom shapes than over concave. Müller and Liu (2000a, b) discuss the scattering of incident, two-dimensional wave modes from various shapes of topography, including a continental shelf and a ridge. They conclude that a convex sloping boundary is more effective in scattering the incident waves to high wavenumbers than is a concave boundary. Numerical model experiments by Legg and Adcroft (2003), however, find no significant reduction in mixing for concave slopes. For both convex and concave slopes they find that the mean stratification is eroded in a layer above the sloping bottom that is bounded at its upper edge by a line lying parallel to an internal wave ray path with inclination corresponding to the frequency of the incident wave mode. Reflection of a wave ray approaching sinusoidal sloping topography from an azimuthal direction, γ, results in a ‘primary’ reflected wave with properties similar to that of a wave reflected from a smooth slope. There are, in addition, ‘scattered’ secondary waves. These have wavenumbers in the plane of the slope equal to the sum and difference of the incident waves’ wavenumber component in the plane and the wavenumber of the sinusoidal topography. The distribution of energy flux between the primary and scattered waves is a complex function of α, β, and γ , and depends on the orientation of the sinusoidal topography on the slope and the ratio of the wavelengths of the topography and the incident internal wave. Legg (2003) uses a numerical model including the effects of rotation to show that the interaction of an internal tidal ray generated at the shelf break with rough topography on the continental slope can generate a high vertical wavenumber structure in the velocity field and may lead to mixing. The main effect of rotation is to add a velocity component normal to the plane of the incident internal wave propagation direction, and this contributes to driving fluid up and over the topography, so leading to greater secondary wave generation. As well as the mixing on continental slopes, active mixing is found in the boundary layer over the sloping sides of seamounts in the eastern North Pacific. Observations to depths of 300 m around the shallow Cobb seamount by Lueck and Mudge (1997)


311

using a free-fall FLY instrument, reveal enhanced mixing, with values of ε exceeding 10−7 W kg−1 extending horizontally to distances of order 6 km. At heights of 20–40 m off the sloping side of the Fieberling Guyot, regions of static instability extend to vertical scales exceeding 10 m (i.e. displacement scales exceed 10 m) for 12% of the time, and values of Richardson number calculated from differences of velocity and density over 10 m in the vertical are less than 0.25 for about 23% of the time, indicating a very dynamic regime (Eriksen, 1998).9 Even so, Toole et al. (1997), interpolating microstructure measurements near the seamount to the seamounts of the whole North Pacific, concluded that the present limited data set does not support the idea that boundary mixing sustains an effective basin-average diffusivity of 1 × 10−4 m2 s−1 at mid-depth in the Pacific.

(Nor, as mentioned in Section 2.7.2, can internal tides generated on continental slopes provide sufficient energy, but tidal flow over ridges is a more effective cause of internal tides and potentially of mixing.) One further effect deserves mention. Internal waves are generated by the flow of the barotropic tide over critical slopes. A surface (an indentation of a continental slope) that has a uniform inclination to the horizontal, α,10 but which is concave (like the interior surface of a cone with its vertex pointing downwards), may act as a generator and may also contribute to the focussing of the generated waves. Such topography may also focus waves reflected from it. Canyons may be regions of enhanced internal wave energy (as found in Section 12.3) because of such generation and reflection processes, involving the three-dimensional shape of the reflecting boundary. 11.4.2

Resonant interactions

Internal waves in a train approaching a slope interact with those reflected. If α < β, so that the wave frequency is supercritical (>σ c ), and if α is sufficiently small ( 1 only if the signs of the long wave speeds are the same, that is if both travel and carry energy in the same direction. In supercritical flow, wave energy can be transmitted in only one direction and with speeds, c+ and c− , both greater than −U or both less than −U . In sub-critical flow, waves will travel in opposite directions. The concept of upstream or downstream, possible in surface hydraulics, is lost in two-way, two-layer flows, but the concept of a region on one side of the location where G = 1 having influence on the other, remains valid. The flow at a given location is characterized by whether interfacial waves can travel in both, or only one, direction and so transmit information to adjust conditions elsewhere. In view of the analogies with single-layer 5 Tidal bores propagate upstream, and in these cases the relevant flow speeds, U, are those measured relative to the bores. In this frame of reference the flow upsteam of the bore is super-critical and that downstream sub-critical. In the case of hydraulic jumps formed in flow over a weir, or more generally over a shallow ridge, the conditions upstream of the obstacle are usually sub-critical (Fr < 1), but with decreasing depth and increasing flow speed, the Froude number becomes equal to and exceeds unity over the weir or ridge crest, before reaching the location of the jump on the downstream side of the obstacle where the flow undergoes a transition to return to subcritical conditions.

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flow (and of the analysis of two-layer flows), interfacial hydraulic jumps are expected in straits, and they are indeed found there. Only in recent years has it become technically possible to measure mixing in straits. An extensive set of observations of the dynamics of the two layer flow through the Strait of Gibraltar in the 1980s, together with important developments in the theoretical understanding of flow through constrictions, defined the principal processes involved and the nature of the control on the flow imposed by the Tarifa Narrows, some 9 km wide in the central Strait, and by the two sills to the west (Armi and Farmer, 1988). The westernmost is the Spartel Sill at the Atlantic entrance to the Gulf of Cadiz. It has a depth of about 300 m and is 30 km wide, but the more substantial, influential and shallower sill is the 280 m deep Camarinal Sill, about 12 km wide. Although the flow in the Strait is in general accord with predictions based on two-layer hydraulic control and the magnitude of G, the flow is strongly affected by the M2 tide and this results in considerable variability, transient mixing and the production of propagating internal bores. Observations by Wesson and Gregg (1988, 1994) have provided substantial evidence of mixing in the shear flow over the sills and in the propagating internal bores that are generated in the Strait. Stationary internal hydraulic jumps form in the flow over the sills, particularly to the west of the Camarinal Sill when the tidal flow is to the west, augmenting the westward flow in the undercurrent. During neap tides, Wesson and Gregg (1994) observed that the deeper layer flows down the western slope of the Carmarinal Sill, forming billows 30–75 m in height, creating intense turbulence with ε > 10−2 W kg−1 (Fig. 12.2, Plate 24) and thickening the transitional interface between the two flowing layers, from about 30–50 m east of the sill to about 130–150 m at a distance of 1–2 km to the west. The mean dissipation is estimated to be about 340 MW, much greater than both the estimated dissipation at the floor of the sill (about 6 MW) in the same area and that usually associated with the Garrett–Munk internal wave spectrum (about 72 kW integrated over the same region). Acoustically detected, 50 m high, Kelvin–Helmholtz billows over the Carmarinal Sill are illustrated in Fig. 12.3, Plate 25. As the westward tidal flow decreases, the jump on the west side of the sill is carried eastwards over its crest, becoming detached and propagating eastward as an internal bore through the Strait, much as described by Maxworthy (1979; see Section 2.7.2). Figure 12.4 shows the dissipation rates measured during the passage of one large internal bore travelling eastwards through the Tarifa Narrows to the east of the Camarinal Sill. The black line is the 28.0 potential density contour, which lies approximately at the interface between the upper and lower layers. The bore height is about 100 m and the vertical acceleration at the leading edge is almost equal to g . Flows are weak before the arrival of the bore, but increase sharply to about 0.6 m s−1 in an easterly direction in the upper layer and to 0.4 m s−1 westward in the lower layer as the bore arrives, leading to a velocity shear of about 2.4 × 10−2 s−1 . The largest dissipation, about 10−4 W kg−1 , is in the trough immediately following the bore. Further to the east, the bores break up to become packets of internal solitary waves that radiate from the Strait into the Alboran Sea

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Figure 12.4. Dissipation during the passage of an internal bore through the Tarifa Narrows east of the Carmarinal Sill in the Strait of Gibraltar. The vertical scale is pressure (1 MPa ≈ 100 m) and the horizontal is time. The jump is travelling eastwards past the sampling location. Values of log10 ε are shown with ε in W kg−l . The thick line marks the depth of the 28.0 isopycnal (σθ ). The vertically integrated dissipation rates are enhanced for a period of about 1.5 h as the bore passes. (From Wesson and Gregg, 1988.)

and which are detectable in satellite images (see, for example, Farmer and Armi, 1986). The rate of energy dissipation by the eastward-going bore appears to be less than that in the hydraulic jump west of the Camarinal Sill, and Armi and Farmer (1988) find that its contribution is a relatively minor contribution to the total rate of dissipation of the tides in the Strait. They argue that frictional effects, including those of the wind stress on the surface, have small or negligible effect on the supercritical section of the Strait where G > 1, and consequently the flow may be accurately predicted on the basis of inviscid internal hydraulics, with hydraulic control where G = 1. In contrast to the Strait of Gibraltar, the Bosphorus is almost non-tidal and the exchange flow is quasi-steady. The channel is 0.75–3 km wide and 28–100 m deep, with layer speeds of typically 0.5 m s−1 and g ∼ 0.16 m s−2 . It is bounded by sills of depth about 40 and 60 m with the shallower at the southern, Sea of Marmara, end, between which there is a relatively narrow channel, but one having sharp bends. The flow is far from satisfying the hydraulic conditions for a simple two-layered channel flow. Estimated values of G2 in the channel are less than 0.25 and there is no evidence of a control region within the strait itself (Gregg et al., 1999), although there is control ¨ outside the limits of the strait (Gregg and Ozsoy, 2002). Observations are in general accord with a numerical frictional two-layer model of flow through a contraction by

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Winters and Seim (2000) that finds flows with lower Froude numbers, larger transports, and wider regions of sub-critical flow, than in the frictionless case. A full explanation of the flow and its variability probably requires better understanding of the effects of the very complex bathymetry of the 30 km long strait, especially its sharp bends (that may produce flow separation), as well as the effects of the finite thickness of the ¨ interface, never less than 28% of the water depth (Gregg and Ozsoy, 2002). 12.6.2

Sea straits: finite interface thickness

Some progress in understanding how finite interface thickness may affect hydraulic control have been made by Pratt et al. (2000) in a study of the conditions for critical flow in the Strait of Bab el Mandab at the mouth of the Red Sea. They find longwave solutions of the Taylor–Goldstein equation, (3.7), using the mean density and velocity distributions that are observed and allowing the channel to have sloping sides to simulate the real topography.6 If the minimum Richardson number of the mean flow, Rimin , exceeds 1/4, and the flow is therefore stable, the speed of the stable waves is nearly always outside the range, Umin to Umax , of the flow speeds (always so if the side walls are vertical, see Section 3.2.4), so that C− < Umin < Umax < C+ . This implies that, unless the flows in the two layers are in the same direction, there are generally no solutions with zero phase speeds that gave G = 1, the critical condition in the two-layer case. Only subcritical conditions are possible if Umin < 0 < Umax .7 Pratt et al. conclude that hydraulic control (with accompanying ‘maximal flow’) will be an intermittent feature influenced greatly by tide-induced fluctuations. These fluctuations are also important because the estimated phase speeds of the long waves are quite close to the bounds, Umin and Umax , of the mean flow. A relatively small variation in mean flow will carry the disturbances into the range of the mean flow where shear instability is possible. These conclusions raise questions, yet to be fully addressed, about the validity of two-layer analysis and the earlier interpretation of the observed flows and hydraulic effects in straits. A numerical simulation of a quasi-two-layer flow through a laterally contracting channel, a stratified flow with an interface of finite velocity and density thickness resulting from frictional and diffusive effects, has been made by Hogg and his colleagues. Taking a turbulent Prandtl number, Kν /Kρ , to be unity, Hogg et al. (2001a) find the flow is characterized by a parameter g H 5 /L 2 K ν2 , where H is the total depth and L the length of the contraction, with values near 107 when control is purely hydraulic, and 102 when the flow is at a viscous–diffusive–advective limit. Although there is considerable uncertainty about the appropriate values of Kν , the Strait of Gibraltar appears to be near the hydraulic limit and the Bosphorus near the viscous–diffusive–advective 6 See also footnote 5 in Chapter 3. 7 These conclusions rely on the speed of energy propagation of long waves being the same as the phase speeds, as in two-layer flows. Although the group velocity of internal waves in shear flows may be greater or less than the phase speeds, and consequently the group speed of waves may lie within the range Umin to Umax , the group speeds do tend to the phase speeds as the length of the waves tends to infinity (Baines, 1995).

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limit. The flow model is used to examine the propagation of waves through the contraction (Hogg et al., 2001b), seeking critical conditions for no wave propagation. The form and propagation of the waves are determined by using an equation similar to the Taylor–Goldstein equation, but including frictional and diffusive effects. Two types of wave mode are identified according to whether they represent disturbances effectively trapped in one or other layer (‘vorticity modes’ with eigenfunction maxima above or below the interface, similar in form to Holmboe waves), or at the interface (‘density modes’ with eigenfunction maximum at the interface, like growing Kelvin– Helmholtz billows). The propagation of waves and the changes in their form depend on an assumption of quasi-uniform conditions and ignore any mean tilt of the interface, but may be related to the location of control points in the flow. Further study is required to establish the relation of these studies to observations. 12.6.3

Sea straits: effects of rotation

A further factor in the control of stratified flow through straits is the Earth’s rotation. This will have an effect in straits of width comparable to or greater than the internal Rossby radius, L Ro = c/f, where c is the speed of long internal waves in the observed mean stratification. The width of the Strait of Gibraltar is about 15 km, comparable to L Ro (about 12 km), and CTD and acoustic transects across the Strait show that the interface between the two layers is typically 50 m deeper on the northern side than on the southern (Armi and Farmer, 1988). This is likely to affect the internal hydraulics and the nature and development of hydraulic jumps and bores. The Bosphorus is relatively narrow, 0.75–3 km, in comparison to a radius, LRo , of about 17 km and there the flow is unlikely to be strongly affected by the effects of the Earth’s rotation. The Straits of Florida, through which passes water from the Caribbean to form the Gulf Stream, are of much larger scale, 760 m deep, 75 km wide and with LRo ≈ 25 km, and are not subjected to any evident hydraulic control. The measurements of turbulence by Winkel et al. (2002) identify several different mixing regimes characterized by the level of the Richardson number and the proximity of topography. Only near the bottom and sloping sides do values of Kρ exceed 10−4 m2 s−1 . Overall, the level of turbulence is moderate and the authors conclude that this is not a ‘hot spot’ for ocean mixing. In contrast, Oakey and Elliott (1980) find Kρ is about 2.5 × 10−3 m2 s−1 at depths between 233 and 671 m in the geostrophically controlled flow on the western side of Denmark Strait. The Strait has a sill depth of about 600 m and a total width of about 300 km, greatly exceeding LRo . In view of the doubts raised over the application of the two-layer approximation to the flow that has largely been used in the interpretation of data, the still uncertain importance of rotation and mixing, and the effects of flow separation in complex channels like the Bosphorus, the understanding of flow through straits cannot be regarded as complete. The subject of flow over sills is returned to in Section 12.7 in relation to fjord dynamics. Recent observations in fjords indicate that a purely two-dimensional approach can disregard important factors.


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12.6.4

Deep water passages and channels

Several major passages, such as the Faeroe–Shetland channel linking the Norwegian and Iceland basins of the North Atlantic, form connections allowing water to pass through or around topographic barriers separating neighbouring ocean basins. A remarkable study of mixing in deep water has been made by Polzin et al. (1996) and Ferron et al. (1998) using fine and microstructure instruments in the Antarctic Bottom Water (AABW) as it flows eastward in its route towards the Northern Hemisphere through the confines of the Romanche Fracture Zone in the mid-Atlantic Ridge in the Equatorial Atlantic. Intense turbulence occurs in the cold bottom water downstream of the main sill at a depth of about 4300 m, where mean values of Kρ reach 0.1 m2 s−1 . Although the Romanche and nearby Chain Fracture Zones represent only 0.4% of the area covered by the adjoining Sierra Leone and Guinea Abyssal Plains, the diffusive heat fluxes in the fracture zones amount to about half those of the plains. Bryden and Nurser (2003) argue that, even with such large values of Kρ , the mixing within these fracture zones is underestimated and that much of the deep mixing in the Atlantic occurs in the passage of AABW through deep passages, the Vema Channel, the Romanche Fracture Zone and the Discovery Gap. It appears possible that some of the enhanced diffusivity observed over the rough topography of the western Brazil Basin (Fig. 1.15, Plate 2) may be caused by flows across or through gaps in the east–west trending fracture ridges.

12.7

Fjords

12.7.1

Sill dynamics

Fjords are generally deep, glacially carved estuaries, usually with a shallow sill produced by past glacial debris at their connection with the open sea. Dense water is commonly trapped at some depth below that of the sill, so that the fjords are often stratified.8 Although (as in lakes) freshwater carried by rivers, winds, atmospheric heating or cooling, and ice formation all have their effect, the circulation and exchange of water within fjords are often controlled by the conditions near the sill, particularly by the amplitude of the tidal flow and the processes of its conversion to waves and eddies or dissipation. It is perhaps pedantic to separate sills in fjords from those in straits. Mixing in the flow over sills is often very similar to that in straits, and consequently there are parallels to be found between this section and Sections 12.6.1 and 12.6.3. In both cases the local dynamics affect those of the adjoining basins, the fjords or inland seas. Apart from the differences in scale (straits sometimes being longer than the regions surrounding fjord sills and containing more than one sill), the separation is made largely for convenience of presentation. 8 A wide-ranging and thorough introduction to mixing in fjords is provided by the review by Farmer and Freeland (1983).

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The tidal energy entering fjords is lost mainly in four ways, by: (a) reflection of energy from the fjord system back to the open sea as barotropic tidal waves; (b) loss of energy in working against the bottom shear stress, particularly as the tidal flow crosses the sill; (c) the generation and maintenance of eddies at the sill; and (d) the generation of internal waves that carry energy into the fjord. Both (c) and (d) are associated with the form drag derived from the pressure difference on the two sides of the sill. The internal waves may be seiches of the fjord or transient lee waves, stationary in the incoming flow but ‘leaking’ energy into the fjord because their group velocity is less than their phase speed, just as in the case of the hydraulic jumps described in Section 12.6.1. Energy is ultimately dissipated from the eddies and the waves by turbulent mixing in the water column or in the boundary layers around the basin sides; dissipation process (b) may be linked to (c) and (d). Two expressions are available to estimate the mean rate of energy lost from the barotropic tide in crossing the sill (Farmer and Freeland, 1983) and hence to calculate the energy available for mixing within the fjord: E 1 = (ρg Aσ a 2 /4) sin(2ϕ1 ),

(12.3)

where A is the horizontal surface area of the landlocked fjord basin, σ is the tidal frequency, a the tidal amplitude outside the fjord near the sill and ϕ 1 is the phase lag of the barotropic tide across the sill, and E 2 = (ρgU Ba/2) cos ϕ2 ,

(12.4)

where ϕ 2 is the phase difference between the barotropic flow over the sill and the surface elevation, U is the amplitude of the barotropic tidal current at the sill and B is the vertical cross-sectional area normal to the flow direction of the sill. Stigebrandt and Aure (1989) identify two different classes of dynamic response of stratified fjords to tidal forcing at the sill. When the maximum barotropic flow, U, into the fjord over the sill exceeds the speed of long internal waves in the fjord, c, the incoming flow forms a jet that advances on each tide into the fjord at shallow depths with cross-section dimensions similar to that of the sill. An example is shown in Fig. 12.5. The jet may be accompanied by internal hydraulic jumps and internal waves, and eddies with vertical components of vorticity are generated within and around the jet, but only some 1% of the energy flux from the surface tide is used in its production. If U < c, however, internal waves radiate from the sill into the fjord; barotropic tidal energy is converted into baroclinic (internal) wave energy. This is subsequently dissipated largely at the boundaries of the fjord, although the internal waves may nevertheless be more effective in mixing the deeper waters of the fjord than are the phenomena associated with the jet flow, produced by higher inflow speeds at the sill. It is estimated that typically about 5.6% of the energy flux from the surface tide is used for work against the buoyancy forces within a fjord basin. Although little is yet known of the locations or processes of turbulent dissipation by direct observations within fjords, more is being discovered about the processes that occur at sills. The flow over the sill in Knight Inlet, British Columbia, has been particularly well studied (Farmer and Smith, 1979, 1980; Gargett, 1979; Farmer and

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Figure 12.5. A jet-like intrusion of water over the sill into a fjord, upper Loch Etive, western Scotland. Horizontal currents are measured by an ADCP mounted on a ship. The sill is to the left and the tidal jet is seen in the upper 30 m of the water column. The scale is speed to the right in cm s−l . Part (a) is at a time of about 86 min before (b). Positions of stagnant flow and the separation point from the sloping bottom are marked. (From Inall et al., 2004.)

Armi, 1999a; Armi and Farmer, 2002). The Inlet is 120 km in length, and some 3 km wide. The fjord narrows abruptly to about 2 km at the sill where the water depth is some 65 m. The sill is about 1 km long and separates the 540 m deep landward basin on its eastern side from a 250 m deep seaward basin to the west. Armi and Farmer have made observations of velocity and density along the central axis of the sill in an east–west direction using ADCP and CTD, whilst obtaining acoustic imagery of subsurface scattering layers (Farmer and Armi, 1999a; Armi and Farmer; 2002). During the seaward-flowing ebb tide, a wedge of partly mixed water develops on the seaward side of the sill, the apex of the wedge being over the sill and pointing into the flow. The flow bifurcates over the sill, a weak flow passing over the mixed water in the wedge and a stronger flow moving rapidly down the sloping seaward side of the sill beneath the wedge, before rising rapidly in a hydraulic jump as shown in Fig. 12.6. The flow plunging down the sill slope (Fig. 12.7) shows marked evidence of a shear instability with form and evolution similar to that found in the laboratory experiments of Pawlak and Armi (1998, 2000; see Fig. 3.11). The observed flow in the wedge appears weak but, unexpectedly, to be contrary to the seaward tidal flow. As tidal currents over the sill increase, the bifurcation is forced downstream, and internal waves are observed to radiate upstream, contrary to the subcritical flow

Figure 12.6. A hydraulic jump formed in flow over the sill in Knight Inlet, British Columbia, Canada. An acoustic image with Doppler-measured velocity vectors is superimposed, coded (as shown in the inset) for magnitude, and with isopycnals shown as full lines. East is to the left. The jump is apparent in the sharply rising σ T = 24 isopycnal about 250 m west of the sill crest. (From Farmer and Armi, 1999a.)

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Figure 12.7. Flow plunging down the side of a sill. An echo-sounder image obtained in a transect over the sill in Knight Inlet, British Columbia, Canada. The tidal flow is from left to right. The depth at the right of the image is about 60 m and the horizontal width is about 0.5 km. Billows are similar in form to those of Fig. 3.11. (From Pawlak and Armi, 1998.)

approaching the sill as illustrated in Fig. 12.8 (Plate 26) and Fig. 12.9 (Farmer and Armi, 1999b; Cummins et al., 2003). In the case shown, an undular bore or train of internal waves forms upstream (relative to the tidal flow) to the east of the top of the sill. This formation occurs as the ebbing tidal flow accelerates and as the flow over the sill rapidly becomes uncontrolled, with the bifurcation moving to the lee of the sill crest (i.e. to its western side; Fig. 12.8, Plate 26). In these conditions, a quasisteady analysis of the bore generation appears inadequate, the bore being formed as a transient response to the adjustment in the location of the bifurcation. After its formation, the bore remains almost stationary for about an hour before, as the tidal flow weakens, propagating eastwards away from the sill to form the train of waves shown in Fig. 12.9.9 Although aerial photographs of the patterns of surface ripples caused by the waves in the undular bore show that they are long crested and with relatively little curvature, evidence that the flow in the lee of the sill is sometimes very three-dimensional comes from ADCP sections made in a north–south direction along the top of the sill on the lee side in both ebb and flood tides by Klymak and Gregg (2001). Figures 12.10a (Plate 27) and 12.10b show a north–south section just east of the sill during flood tide. An hourglass-shaped jet of water flows into the eastern basin with contrary flows on either side. These counter flows, of magnitude typically half that of those in the jet, are interpreted as being caused by re-circulating eddies formed in the sill constriction as sketched in Fig. 12.10b. Their vertical component of vorticity is a consequence of flow separation, and their effect is to transport water from the boundary layers on the side 9 This formation of internal waves differs from that proposed by Maxworthy (1979) and described in Section 2.7.2.

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Figure 12.9. An internal wave train radiating eastward from the sill in the Knight Inlet, British Columbia, Canada. The record is obtained from a 200 kHz echo-sounder and the leading edge of the wave packet is to the right, denoted by cg . Measurements to test for isotropy referred to in Section 6.8 were made through a similar packet of internal waves in Knight Inlet by the submersible, Pisces IV. (From Gargett et al., 1984.) Figure 12.10b. The circulation in the lee of the sill in Knight Inlet, British Columbia, Canada. A sketch indicating the likely location of eddies and (dashed) the location of the section shown in (a) (Plate 27). (From Klymak and Gregg, 2001.)

of the channel into its central part. Klymak and Gregg also provide evidence that the hydraulic jump in the lee of the sill is bowed, with a three-dimensional structure.10 Estimates of dissipation in the flow between the crest of the sill and the jump are about 8 × 10−5 W kg−1 . These compare with slightly lower values, about 10−5 W kg−1 , derived from estimates of LT and the use of (6.3) by D’Asaro and Lien (2000; see Section 5.10). The vertical diffusivity is very large, about 10−2 m2 s−1 . 12.7.2

Basin response

The energy lost and changes in density caused by mixing as flow crosses the sill determines the tidal energy left to drive waves and mixing within the fjord, and also whether or not the inflowing seawater can reach and ventilate its deeper levels. In 10 This structure is perhaps analogous to the cow-horn eddy found by Brighton (1978) in the very different ‘obstacle’ topography described in Section 12.4.


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some fjords ventilation is a rare event, occurring only every few years. It may result when wind-induced upwelling on the seaward side of the sill carries water of density exceeding that in the deep fjord basin to the level of the sill so that it overflows into the fjord.11 In general the baroclinic wave response of a fjord depends on its stratification and dimensions. If these are such that the frequency of internal seiches are at or near harmonics of the tidal forcing, the wave response is correspondingly greater. In large fjords of width greater than the internal Rossby radius, the circulation and waves are affected by the Earth’s rotation with a tendency for cyclonic motions to develop as they do in large lakes, described in the next section.

12.8

Lakes

Unlike the open ocean (but not all semi-landlocked seas), tides are insignificant in lakes, and the energy available for mixing derives from atmospheric forcing (wind, heat flux, precipitation and ice formation) or river inflow. A rough measure of the response of a lake with a relatively shallow upper layer of mean thickness, h1 , to forcing by wind blowing along its length (but not of forcing by convection) is given by the size of the Wedderburn number, We = c2 h1 /u 2∗ L (Spigel et al., 1986; Stevens and Imberger, 1996). Here c is the phase speed of long internal waves, approximately equal to (g h1 )1/2 where g is the reduced gravitational acceleration, u ∗ is the friction velocity in the air and L is the length of the lake. When, in relatively light winds and short lakes, We 1, mixed layer deepening occurs largely through processes at the base of the layer discussed in Section 9.7. In relatively strong winds and long lakes where We 1, the thermocline tilts as described in Section 2.7.4, with deep water being raised towards the surface at the up-wind end of the lake. This exposes the thermocline more directly to the effect of wind-induced mixing. If brought right up to the surface, the originally deep and colder, relatively dense water below the thermocline may be driven downwind over the less-dense surface water, leading to convective mixing.12 The tendency of a thermocline, tilted by the effect of wind, to recover a horizontal equilibrium level, results in the generation of internal seiches and, in large lakes, Kelvin-like internal waves propagating cyclonically around the lake’s boundary, as explained in Section 2.7.4. The pre-existing ‘ambient’ mean shear, the shear caused by lake circulation and small-scale waves, is augmented by the seiches and Kelvin waves, and this increases the likelihood of shear-induced turbulence. In Lake Geneva, with dimension exceeding LRo , the magnitude of the estimated dissipation, ε = O(6 × 10−5 W kg−1 ), as Kelvin-like waves propagate past a fixed location, leads to the conclusion that much of the energy of a wave will be lost in some 90 h, time for it to 11 Coastal upwelling is produced by winds that, blowing parallel to the coast, produce an Ekman drift away from the coast allowing surface water to be replaced by deeper denser water. It is necessary for ventilation that the upwelled water is denser than that within the fjord, but this is not a sufficient condition – see footnote 15 on p. 339. 12 The latter process of mixing is similar to that in ROFI conditions described in Section 10.6.

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propagate little more than once around the approximately 120 km long periphery of the lake.13 Much of the kinetic energy in lakes generated by wind action is contained in the internal wave field, rather than in the mean circulation or mean available potential energy, and is dissipated in the boundary layers around their sides (Wüest et al., 1994; Imberger, 1998). In this sense, internal waves are relatively more energetic in lakes than in the ocean, and the lateral boundaries of lakes and fjords (often being within an internal Rossby radius of much of the basin’s volume) may be relatively more significant dissipaters of energy than are the boundaries of the ocean. Estimates of density in lakes are often simplified because of the absence, or low levels, of salts; density may be derived directly from temperature measurements. Double diffusive effects are generally negligible, although complication is introduced by the density maximum of freshwater near 4 ◦ C. This may affect convection under ice as mentioned in Section 4.2.2, and can also lead to the formation of ‘thermal bars’. These occur in freshwater lakes that are frozen over in winter. In spring, after the ice has broken up, the shallower regions around the edges of a lake receive about the same heat from solar radiation as does the water further offshore. Consequently, since the heat is distributed over a smaller depth, the near-shore temperature rises more rapidly, and the near-surface temperature decreases with distance from shore. As heating continues, density of the near-surface water increases until the temperature of maximum density, about 4 ◦ C, is reached. Further temperature increase in the shallow water region leads to a decrease in its density. There is then a region between the near-shore and the off-shore surface water where the surface temperature is about 4 ◦ C and the density is greatest. Here water sinks, forming two circulating cells, one in the shallow nearshore region and the other connected to the deeper, off-shore water. The convergence of surface water towards the 4 ◦ C surface isotherm leads to the accumulation of floating material or a slick that ‘bars’ offshore movement of buoyant material, whilst the closed near-shore circulation inhibits dispersion of solutes originating from shore.14 The replenishment and ventilation of water at the bottom of some deep lakes is a rare event occurring only during the coldest winters when the surface is cooled sufficiently to cause major convection or ‘lake overturn’ or when water entering a lake from inflowing rivers becomes sufficiently dense to reach its deepest parts.15 The winter cascading from the cold shallows down the sloping sides of lakes described in Section 11.6 can be a significant seasonal contributor to lake ventilation and cooling. 13 The calculation, by Thorpe and Jiang (1998) using Equation (7.6), neglects the effects of bottom friction and of the radiation of wave energy into the centre of the lake by the scattering of relatively high-frequency internal waves as the quasi-Kelvin waves encounter rough topography or headlands and lose energy through working against form drag (helping to sustain the internal wave field in the body of the lake). The decay time may therefore be significantly less than 90 h. In practise the waves can often be detected for at least 70 h, about the time to travel once around the lake (Dr Ulrich Lemmin, personal communication). 14 Zilitnikevich et al. (1992) describe a model of the formation of a thermal bar and provide references to earlier observations, laboratory studies and models. 15 As explained in Section 4.4.2, it is not sufficient that water at the surface is denser than that in the deepest part, because of mixing and consequent density reduction of the surface water as it sinks.

Chapter 13 Large-scale waves, eddies and dispersion

13.1

Introduction

Except for the partly ice-covered polar oceans – the Southern Ocean that circumscribes the Antarctic continent, and the Arctic Ocean – the major oceans of this planet, the Atlantic, Pacific and Indian Oceans, are confined in the meridional direction by the presence of the continental landmasses to basins of scales of between 0.8 and 2.3 Earth radii. These boundaries, the wind fields and the presence of the Equator, result in a mean circulation pattern of the upper levels of the subtropical oceans with the form of anticyclonic ‘gyres’ in which the currents are intensified on their western boundaries, forming major currents, for example the Gulf Stream in the North Atlantic and the Kuroshio in the North Pacific. Until the early 1960s the circulating flows, particularly in the major gyres, were believed to be subject to relatively little variability.1 Although some changes in surface currents in response to variations in wind forcing were known to occur, the deeper motions were supposed to be much slower and almost steady. This view was proved wrong by the measurements of Swallow and Crease in the deep Atlantic. Using the newly invented, neutrally buoyant floats, the ‘Swallow floats’ described in Section 1.3, it was discovered that the currents at a depth of 4000m are sometimes as strong as those near the surface, of order 0.1 m s−1 , and that they vary at a particular location over time scales of a few weeks (Crease, 1962; Swallow, 1971). Attempts were made to explain the variability in terms of planetary or Rossby waves, large-scale waves that

1 In World War II, the pilots of US aircraft flying over the Pacific Ocean were issued with handkerchiefs imprinted with a map of the major flow patterns so that, if forced down at sea, they would know which way to paddle or swim for land, taking advantage of the currents.

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13.1 Introduction

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Figure 13.1. Spaghetti diagrams; tracks of surface floats released in the North Atlantic. From Brügge (1995) showing tracks of floats all drogued at 100 m.

depend on not just the local Coriolis frequency, but on its variation with latitude, β (see Longuet-Higgins, 1965). By the early 1970s, improvements in mooring technology allowed the deployment of arrays of moorings carrying current meters recording for periods of many months. Several major experiments were made, the Mid-Ocean Dynamics Experiment (MODE), the Soviet Polygon-70, and subsequently Poly-mode, and these led to the discovery and description of eddies with horizontal dimensions of 30 km to 150 km, small in comparison with the size of the gyres. These are now described as mesoscale eddies. The mooring arrays were themselves a major international initiative, and led to novel ideas: synoptic maps were constructed in which potential vorticity conservation was examined and demonstrated by McWilliams (1976), and the first attempts were made to use data collected at sea for ocean forecasting and prediction. Accompanying the advances in mooring technology were developments in SOFAR (SOund Fixing And Ranging) float tracking by Rossby and Webb (1970) making use of the long ranges in acoustic propagation obtainable in the sound channel at the level of minimum sound speed. The channel is at about 800 m depth in midlatitudes but close to the surface in the polar oceans. Analysis of the ‘spaghetti diagram’ (see Fig. 13.1) tracks of floats at 1500 m in the North Atlantic by Freeland et al. (1975) revealed a westward propagation of streamline patterns at about 0.05 m s−1 ,

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significantly greater than the mean flow of about 0.01 m s−1 . Float tracks loop in both cyclonic and anticyclonic senses (Richardson, 1993).2 With improvements in satellite technology to locate position and especially the introduction of the ARGOS position fixing system, surface floats came into common use in the late 1970s, followed later by Autonomous Lagrangian Circulation Explorer (ALACE) floats that spend most of their time drifting at a set depth, only occasionally returning to the surface to report location and to transmit data. Satellite images also became widely available in the late 1970s and showed that the ocean surface is rich in mesoscale patterns made visible by their advection of phytoplankton blooms (Fig. 13.2, Plate 28) or through their infrared-detected thermal structure. Among the eddy-like features are many with similar coherent spatial structure, including single eddies and dipole structures such as vortex pairs or ‘hammer-head eddies’. Even at the large scale of such mesoscale eddies, the ocean appears to have a turbulent character. These eddy-like features have horizontal scales much greater than the Ozmidov scale, LO (defined in (6.2), the vertical scale of the largest overturning eddies in stratified flow) and are comparable to or greater than the internal Rossby radius, LRo . Motion within eddies is restricted to be largely horizontal by the constraints of stratification, buoyancy forces and water depth. Whether they should be described as being in a state of ‘horizontal turbulence’ (a designation favoured at the time to describe the state of motion within the atmosphere; Lilly, 1983) was subject to debate. But by 1981, Wunsch was able to conclude that an eddy-like field is universal in the ocean, and that there had been a shift of perspective, led by the new measurements, from one in which the mesoscale variability was thought to be caused by Rossby waves to the other extreme position: that linear dynamics cannot be adequate and that the ocean is completely ‘turbulent’ (Wunsch, 1981). In the 1980s a novel method for ocean observation was introduced: acoustic tomography. The brainchild of Munk and Wunsch, this relies on the accurate measurement of the relative time taken for pulses of low frequency sound to travel between emitters and receivers separated horizontally in the ocean by hundreds of kilometres. The speed of sound relative to the Earth depends on temperature and pressure, and to a lesser extent salinity, as well as currents, and with an array containing a sufficient number of receivers and emitters, the times of arrival of pulses, refracted by the water structure to travel along different paths through the ocean, can be inverted to produce maps of the changing temperatures and currents caused by mesoscale eddies, as well as by deep convection and tides, in the area covered by the acoustic paths within the array (Munk et al., 1995). Sound frequencies of around 250 Hz have been used, with lower frequencies at greater ranges to overcome attenuation.3 By the 1990s quantitative information was also becoming increasingly available from satellites. The altimeter data from the 2 Lagrangian, float measurements in the ocean are reviewed by Rossby et al. (1983) and Davis (1991b). 3 An experiment made with transmission at a frequency of 57 Hz from Heard Island in the South Indian Ocean proved that sound can be detected at ranges up to 15 000 km, making it possible to monitor the average temperature changes in the ocean or ‘ocean climate’ by measuring the variation of acoustic travel time. An average warming of 4 mK over 15 000 km would lead to a reduction in the travel time (about 10 000 s) of about 0.13 s, which is well within detection limits.

13.2 Eddy kinetic energy

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TOPEX/POSEIDON mission provided the means to determine the variability of sea surface height and, from the surface slope, of the surface currents4 over periods of several years. These data have proved to be a powerful tool in the study of the mesoscale eddy field. Complementary to these observational developments have been advances in analytical and numerical models of so-called ‘geostrophic turbulence’ (see Rhines, 1979), a field of eddying motion that is affected by the Earth’s rotation and curvature, and close to being in geostrophic balance. Represented in the models are motions driven by wind that are subject to baroclinic instability leading to the generation of eddies with vigorous transports of momentum, mass and heat (see McWilliams and Chow, 1981). Eddies tend to interact and grow in scale as described in Section 7.2.5, but at very large scales the effects of the Earth’s curvature become a significant factor in limiting their size. At a wavenumber, kβ = (β/2U)1/2 , the upward scale evolution of closely packed eddies with r.m.s. current, U, on a β plane ‘nearly stops’ (Rhines, 1975). The limiting scale of such eddies in the ocean is about 220 km. (The equivalent scale in the atmosphere is approximately 3000 km.) The purpose of this chapter is not to provide a comprehensive account of large-scale ocean circulation or of its variability, but to offer a brief description of some of the processes and coherent structures that occur at large length and time scales, and which affect dispersion.

13.2

Eddy kinetic energy

13.2.1

Distribution

By the early 1980s, a general picture of the distribution of the energy of the mesoscale eddies had emerged. A long time series of temperature measurements in the main thermocline at Bermuda by Wunsch (1972) showed that most of the potential energy resides in a frequency band surrounding that of the advecting mesoscale eddies, periods of order 100 days.5 Analysis of global data from ship drift, the flow statistics from long term moorings and drifters by Wyrtki et al. (1976), Dickson (1983), Schmitz et al. (1983) and Richardson (1983a),6 showed that the kinetic energy in the eddy field (the eddy kinetic energy per unit mass, EKE) at the surface of much of the ocean exceeds 4 The horizontal pressure gradient in the water near the sea surface in the x-direction along the altimeter track is approximately related to the slope of the sea surface, s, by ∂p/∂x = gρo s. The horizontal currents in the direction of the altimeter track, u, and the transverse component, v, are consequently related through the linearized x-equation of motion by ∂u/∂t − f v = −gs. If u and v are of similar magnitude and, as in the mesoscale eddies, the timescales of changes are much longer than f, |∂u/∂t| | fv|, and this reduces to the geostrophic balance relation between v and s: v = gs/f . 5 A 100 km eddy being advected unchanged with a westward propagation speed of 0.01 m s−1 takes 116 days to pass a fixed location. 6 There was some uncertainty in the data from drifters because, although the majority were drogued to provide a fairly reliable indication of water motion, in most cases the drogue became detached at some, usually unknown, time of the order of a few months after release, and thereafter wind and waves may have had some unquantified and uncertain effect. The general conclusions of the analysis were, however, subsequently proved correct.

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Figure 13.3. The eddy kinetic energy, EKE, at the surface of the North Atlantic derived by Richardson (1983a). Contours with values marked in cm2 s−2 according to the scale at bottom right are those added by Brügge after further analysis of float tracks. Dots show locations at which there are sufficient observations in 2◦ × 2◦ boxes to make reliable estimates. (From Brügge, 1995.)

that of the mean flow; only in the strongest currents (e.g. the Gulf Stream) are the mean and eddy kinetic energies comparable.7 The distribution of EKE at the surface of the North Atlantic is shown in Fig. 13.3. The surface EKE is largest, typically 0.1 m2 s−2 , near strong flows, pointing to their being a source of eddies; indeed Wunsch (1983) concluded that the Gulf Stream is the only strong source of eddy energy in the North Atlantic. (This is not quite true; see Section 13.3.5.) On the eastern side of each ocean basin both north and south of the Equator are large pools of low EKE, with values of about 0.02 m2 s−2 . Less is known of the sub-surface eddy field, although the eddy to mean ratio there is also large (see Imawaki and Takano, 1982). The EKE beneath the Gulf Stream at a depth of 700 m is about 0.03 m2 s−2 , whilst at 2000 m it falls to about 0.008 m2 s−2 , about a third of the surface value. More recent study of the distribution of EKE in the surface North Atlantic determined from tracked drogued floats confirms much of the earlier conclusions (Brügge, 1995; Reverdin et al., 2003). Although heterogeneous in distribution, the EKE is generally isotropic with a basin average ratio of EKE to the mean of 0.74. Examination 7 So much for the use of circulation maps on handkerchiefs (see footnote 1 on p. 340)!


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of the Reynolds stress shows that there is an energy transfer from the eddy field to the mean circulation in some regions, for example in the vicinity of the North Atlantic Current. There are striking differences between northward (e.g. the West Greenland Current) and southward flowing currents (e.g. the Labrador Current) in the sub-polar gyre, the former having much higher EKE. 13.2.2

The global energy balance

The EKE should be seen in the context of a global energy balance, although this is only poorly know at present. Gill et al. (1974) considered the difference between the KE of the mean flow and that of mesoscale eddies. They showed on dynamical grounds that the available potential energy (APE) in the mean circulation, about 105 J m−2 , must exceed its kinetic energy by a factor of about 1000.8 The ratio is equal to the square of the gyre scale divided by the internal Rossby radius, LRo . (Estimates reported by Wunsch and Ferrari (2004), however, suggest that the total energy in mesoscale eddies is about 1.3 × 104 PJ or, since the area of the ocean is about 3.61 × 1014 m2 , a depth integrated energy per unit surface area of 3.6 × 104 J m−2 , whilst Oort et al. (1994) estimate the total available potential energy in the general circulation of the ocean to be about 1.5 × 105 PJ.) If the eddies are generated by baroclinic instability,9 their energy being derived from the potential energy of the mean flow, their EKE may exceed that of the mean flow, but nevertheless be limited (in a steady-state ocean) by the rate of flow of energy into the mean circulation from the wind, a rate of about 10−3 W m−2 , and by the time in which energy is lost from the eddies. The depth-integrated EKE of about 3.6 × 104 J m−2 therefore implies a dissipation time scale of the order of 3.6 × 107 s, or about 1.1 year, not inconsistent with the time span of individual eddies that are described in Section 13.3. However, the energy flux from the wind into the mean circulation is, at best, a rough estimate. Summed over the area of the ocean, the flux of 10−3 W m−2 into the mean circulation implies a total flux about 0.36 TW. (Wunsch estimates, however, that about 1 TW of wind energy goes into driving the global circulation, but this includes internal wave generation; see Munk and Wunsch, 1998). The energy flux to the water surface 2 ), is τ w Us per unit area, where Us is the surface by a wind stress, τ w (= CD ρ a W10 drift current. Taking values of 1.3 × 10−3 for the drag coefficient (CD ), 1.25 kg m−3 for the air density (ρ a ), and supposing that the surface drift current, Us , is equal to 0.04W10 (Wu, 1975), then the rate of energy flux from the wind field into the water is 3 and, for a global mean wind speed, W10 , of 7.5 m s−1 is of order proportional to W10 −2 −2 2.7 × 10 W m . Integrated over the ocean area, this is about 10 TW. (Values are, however, very approximate: the mean wind cubed rather than the cube of the mean wind should be used, and Lueck and Reid (1984) estimate a value between 7 TW and 8 See footnote 14 of Chapter 2 for an explanation of APE. 9 Stammer and Wunsch (1999) conclude that both barotropic and baroclinic instability may lead to the generation of mesoscale eddies – see later.


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36 TW. Wunsch and Ferrari (2004) take an intermediate value of 20 TW.) Much of this energy passes into surface waves and, in their breaking, is dissipated by turbulence in the mixed layer (Chapter 9) or in the surf zone (Section 11.2). These values compare with the 3.5 TW of tidal energy dissipation in the ocean (Section 1.9) or the estimated 0.6 TW of energy passing from the wind field into the near-inertial internal wave field (Watanabe and Hibiya, 2002). 13.2.3

Seasonality

Dickson et al. (1982) found that the level of kinetic energy at a depth of 4000 m in the eastern North Atlantic has a seasonal cycle and suggested that, except in boundary currents, the wind is responsible for most of the observed eddy variability. This suggestion is not supported by results of a comprehensive study of the variability in eddy energy in the World Oceans undertaken by Stammer and Wunsch (1999) using TOPEX/POSEIDON sea-surface slope measurements and current meter data. The study builds on earlier analysis by Wunsch (1997) in which the surface currents determined from the altimeter slope data are compared with the surface currents derived from vertical interpolation of currents measured by moored instruments.10 Although the variations in the height of the sea surface determined from altimeters are strongly affected by barotropic eddies, in most regions the fluctuations in the altimeter slope records appear to be dominated by surface currents driven by the first baroclinic mode, and may therefore provide information about the vertical motion of the subsurface thermocline. Stammer and Wunsch conclude that barotropic and baroclinic instability of the mean oceanic flow field appears to be the major source of eddy energy. Most seasonal changes are related to changes in the strength and stability properties of these currents or of their interaction with topography. Over most of the subtropical oceans and in the locations of the major fronts, seasonal variations of eddy energy are negligible. There are some regions in which annual cycles of EKE do occur, although with variations of less than 25%: the eastern North Pacific, the eastern and sub-polar North Atlantic, and the tropical oceans. In two areas, the north-eastern North Pacific and the eastern and sub-polar North Atlantic, variations in EKE over seasonal and inter-annual periods show a significant correlation with variation in the wind stress. The impact of wind variations, however, appears to be limited to high latitudes, and even there only a fraction of the observed eddy energy variation is directly related to wind stress, contrary to the suggestion of Dickson et al. (1982). 13.2.4

Spectra

The spectra of the large-scale variations provide quantitative information, useful in providing insight into the dynamics of the ocean and, by making comparison with models, helping to determine how the mesoscale eddy field originates. 10 Assuming isotropy of horizontal currents, the mean horizontal eddy kinetic energy per unit mass is given by (g/ f )2 s2 where s2 is the mean square surface slope.


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Wunsch and Stammer (1995) analysed the first two years of the TOPEX/ POSEIDON altimeter data. Neglecting the tides, about 50% of the variance in sea surface height is at periods greater than 150 days. Short-period fluctuations ( 30 km can be represented by the conventional advection–diffusion equation. Only for a field in which the time changes following a particle path, τ , are of the order of 10 days or less would a history correction to the conventional advective–diffusive equation and flux balance be important. Davis (1991a,b) also addresses the problems of inferring Eulerian statistics from the Lagrangian measurements made by floats. The Eulerian measures of currents involve a combination of space and time averaging complicated by the uncertainty in the observed float locations. Rarely do two floats pass through the same location in space, and therefore determining exactly the time variation of current at a particular point is impossible; some choice must be made of an area through which floats pass and that characterizes a ‘point’ (and possibly of a time interval within which the floats are in the area). The averaging leads to an uncertainty, δU, of a measured value, U, that depends on spatial averaging scale. The consequent uncertainty in the estimates of Eulerian diffusivity, KH , is found to increase with time, even after KH has reached KH∞ . Zhurbas and Oh (2003) apply Davis’ (1987, 1991a) results to their estimation of surface diffusivity in the Pacific from drifters, removing the effects of apparent dispersion caused by mean shear and finding single values of the length and time scales, and of a resulting ‘lateral dispersion’ coefficient, rather than the more common zonal and meridional values. The diffusivities and Lagrangian length and timescales generally lie within the ranges of earlier estimates, with lateral diffusivity shown in Fig. 13.16 ranging from about 2 × 103 m2 s−1 at 50◦ N to about 3 × 104 m2 s−1 in the eastern Equatorial Pacific. There appears to be a relation between the lateral diffusivity, KH∞ , and the internal Rossby radius, LRo : K H∞ = ce V L Ro ,

(13.4)

where V is the Lagrangian velocity scale and ce is an empirical constant determined from the Pacific Ocean data as 1.02.23 23 V is not equal to the Lagrangian scale, u , defined under (1.16) in Section 1.8.1, but is defined as the square root of the minor principle component for the Lagrangian covariance matrix at a particular location. Interested readers are advised to consult Zhurbas and Oh (2003). Equating K H∞ in (1.21) and (13.4), gives LL ≈ 1.02VLRo /u . Zhurbas and Oh find L L = 0.96LRo − 0.42 km, suggesting that LL ≈ LRo , approximately equal to the size of mesoscale eddies. (Compare also with the relation (13.1).) Later calculation described by Zhurbas and Oh (2004) based on Atlantic floats shows a much less convincing relation between L L and L Ro .

13.5 Rossby waves

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Further refinements and study of dispersion in other regions of the ocean will be made as the duration, depth range, and number of tracked floats increases yet further.

13.5

Rossby waves

As well as mesoscale eddies, propagating Rossby waves are evident in Geosat and TOPEX/POSEIDON satellite altimetry measurements (Fig. 13.17) and appear to be ubiquitous in the World Ocean (see Tokmakian and Challenor, 1993; Hughes, 1995; Chelton and Schlax, 1996). The barotropic mode is not well resolved by orbiting satellites; the waves cross ocean basins in about a week and travel too fast for their motion to be well resolved by the relatively long repetition times of satellite tracks. The first baroclinic mode is dominant in the observations and leads to westwardpropagating fluctuations in sea level of amplitude 0.1 m, with periods of 0.5 to 2 years and wavelengths of 500 km to thousands of kilometres. Typical propagation speeds are a few centimetres per second and consistent with the higher than mean flow propagation of streamline patterns found by Freeland et al. (1975) mentioned in Section 13.1. Waves appear to be larger to the west of the mid-ocean ridges, perhaps a consequence of the interactions of barotropic and baroclinic modes and topography. Chelton and Schlax (1996) ascribe 10% of the r.m.s. variation in sea level at latitudes

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Figure 13.17. Rossby waves. The sea surface height variations in metres measured by the Geosat altimeter in the eastern North Atlantic and plotted as a Hovmöller diagram with contours in longitude vs. time (unit 17 days) at latitudes of (a) 30◦ N, (b) 35◦ N, and (c) 40◦ N. The tilt of the features to the left (to greater longitude as time increase) indicates propagation to the west, and increased tilt with increasing latitude implies slower propagation. (From Tokmakian and Challenor, 1993.)

time (units of 17 days)

13.6 Long-term variations

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ϕ > 30◦ to Rossby waves, and 30% at ϕ < 15◦ . These figures are not dissimilar to the 5%–15% variance contribution by barotropic Rossby waves found in Gaspar and Wunsch’s (1989) analysis of Geosat data in the NW Atlantic. The energy of Rossby waves appears to be smaller than that of mesoscale eddies, except perhaps in some local areas. It is known, however, that eddies interact with Rossby waves (Sutyrin, 1994), and eddies with eddies, Rossby waves with Rossby waves. At these large scales, just as has been found at the smaller scales of internal waves, there appears to be a synergy between waves and vortical motions. A sharp distinction may be worth drawing only when it is helpful in defining the sources and transfer of energy, when it impacts on the ability to forecast the motion field, or when it helps distinguish between those processes that are the cause of dispersion, or of heat, mass or solute transfer, and those which simply move fluid back and forth.

13.6

Long-term variations

There are known to be variations of water properties and flow over time scales exceeding the life of mesoscale eddies or the period of baroclinic Rossby waves, and that have an effect at length scales comparable to the dimensions of the great ocean basins or the Southern Ocean. These range from those lasting from one or two years, such as the El Niˇno phenomenon in the Pacific Ocean (Trenberth, 2001), to decadal changes (e.g. the ‘great salinity anomalies’ in the North Atlantic described by Dickson et al., 1988, and Belkin et al., 1998), or the indirectly measured changes in sea level over interglacial periods (see Church and Gregory, 2001) and to the sometimes ‘abrupt’ changes in climate or ocean circulation occurring within shorter time scales (see Rahmstorf, 2001). Although still apparently containing some of the random or chaotic properties of turbulent motions, particularly intermittency and the sometimes relatively sudden transition between different modes (or climatic states), and important because of their relation to climate change, its mitigation or even future control, these effects are beyond the scope of the present text.

Chapter 14 Epilogue

14.1

The nature and effects of ocean turbulence

In the nineteenth and early twentieth centuries, considerable advances were made in understanding the transitional processes that lead from laminar flow towards turbulence. These studies, largely of the onset of flow instability in quiescent flows, were supported by laboratory experiments, sometimes devised and made well after the development of a relevant theory, as explained in the early chapters of this book. Turbulence itself proved a far more difficult subject. Whilst there are the shining examples of Osborne Reynolds’ papers in 1883 and 1895, and the insightful use of dimensional reasoning, advances in the understanding of turbulence have been severely constrained and delayed by the analytical complexity of the subject. Only in more recent years have computational methods been devised, and computers of sufficient power developed, to begin at last to replicate the small-scale, high Reynolds number processes that accompany turbulent flow in the ocean. Technological advances in the last 40 years of the twentieth century, including novel instrumentation, vastly greater capacity of data recorders, faster recording rates, and more reliable and smaller sensors, have formed the basis for relatively rapid advances in the measurement and understanding of turbulence in the ocean. Some of the most important technical developments have been the design and construction of the piezoelectric shear probe to measure velocity shear, thermistors to measure temperature at sub-centimetre scales, and free-fall instruments to carry sensors and measure from near the sea surface to the depths of the abyssal plains, supplemented recently by autonomous vehicles. It has become possible to address problems of dispersion through the development of the means to detect passive tracers at minute concentrations, the methodology needed to release and follow a patch of tracer in the

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14.1 The nature and effects of ocean turbulence

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ocean, neutrally buoyant floats that can follow water motion in both two and three dimensions, and the global positioning systems and remote sensing that have allowed survey of ocean variability from the scales of internal waves to those of mesoscale eddies. These remarkable advances have provided knowledge that has, in many cases, entirely changed the previously existing perception of how the ocean works. Future advances will provide a more comprehensive and statistically robust quantification of ocean turbulence, and extend the presently limited understanding of turbulence and its interaction with particles and live organisms. In some regions, and in relation to some processes, knowledge of turbulence is as patchy as is turbulence itself in the stratified thermocline. There is too little known to establish precisely how, on average, internal waves ‘break’, contribute to dispersion and lose their energy, or to quantify the associated energy dissipation rates near sloping realistically rough boundaries. The dynamical consequences of the three-dimensional breaking of short-crested internal waves are largely unknown, even though their breaking may dominate in the process of generation of turbulence in the deep ocean. Little is known of turbulent dissipation when motions caused by surface waves are large, for example in the vicinity of the air–sea interface in hurricane conditions, or near a shallow sea floor that consists of mobile sediment or is covered by aquatic plants. There is still a lack of knowledge about the larger-scale processes within the mixed layer and how these interact with those in the pycnocline. It is usually assumed that at length scales approaching the Kolmogorov scale, turbulence is isotropic and has a universal form. It is, however, uncertain when or whether this is really so, when it is appropriate to think of the structure as consisting of small interacting spherical or ellipsoidal eddies or whether the flow field is really fibrous, with thin tubes or sheets of relatively high vorticity. Isotropy and homogeneity are often assumed in theoretical models but, as shown in earlier chapters, the ocean is rarely like that. For many applications, the structure is of little consequence, and only the overall effect on, say, momentum transport is important. For others, the structure of the motion may be highly significant. In laboratory experiments in which the uptake of nutrients or the growth, predation and mortality rates of algae are being measured, it is vital to know whether the turbulence produced in the laboratory tank replicates that encountered by such organisms in the ocean. In calculations or experiments to examine the formation or break-up of sediment flocs or the disruption of algal assemblies, or to establish the rise speed of bubbles in turbulent flow, it is also essential to be able to simulate the pertinent properties of ocean turbulence in relevant conditions. As seen in Sections 9.2.3 and 10.3.2, the rise speed of bubbles and the fall speed of sediment particles, respectively – and their distributions in the turbulent flow – depend on the (presently unknown) structure of turbulent flow at small scales. This lack of knowledge of ocean turbulence imposes a limitation on the accuracy of numerical models of sediment transfer and gas flux from bubbles. Some of the ideas coming from theoretical and laboratory studies of turbulence, for example the effect of the very high accelerations to which small particles or organisms may be subjected in intense near-surface turbulence (La Porta et al., 2001), have yet to have their impact on the

Epilogue

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understanding of ocean turbulence. At larger scales, improved means to describe and parameterize mixing produced by turbulent flow over and around rough topography in stratified and unstratified conditions will be a major advance.

14.2

Prediction and unknown energetics

Within the directly useful or applicable areas of the subject of ocean turbulence, the scientific study of dispersion is perhaps the most important and the most neglected. Attention has been drawn in Section 9.6.1 to the lack of understanding of environmental effects (notably atmospheric forcing and mixed layer structure) on the dispersion of dye released at or near the sea surface – a reservation expressed by Okubo in presenting his empirical formulae for dispersion over 30 years ago.1 Tracer release experiments, such as those described in Sections 7.4 and 13.4.1, are greatly aiding the understanding of three-dimensional dispersion within the ocean. Many more are needed before confident predictions are possible. Numerical models in which the processes of turbulence are represented in some parametric fashion (e.g. by an eddy diffusion coefficient), have made useful progress towards prediction. The validity of the parametrization and, for example, the truncation of sets of terms representing turbulence or of the limitation of resonant wave interactions to those that occur between only three or four waves, still require careful testing against data: model ‘verification’ must be critical and severe. Particular problems are encountered when models of climate change are to be applied beyond the range of their tested validity. This is especially so in the present conditions of rapidly changing atmospheric (notably green-house gas) composition, when some processes may have greater or less effect than in the, perhaps, more benign conditions of the past. It is now firmly established that deep convection occurs only in limited regions, and that consequently the regions where changes in atmospheric conditions can be communicated rapidly into the deep ocean are restricted. The principle sites of diapycnal mixing in the deep ocean are also in restricted areas: ‘the hot spots’, notably where internal tides are generated by barotropic tidal flow over ridges, where there is mixing resulting from flow through gaps connecting neighbouring abyssal basins, and regions where internal waves are strongly forced by atmospheric disturbances such as storms and hurricanes. It is yet to be determined how this affects the deep ocean circulation.2 This is in part because the geographical positions of the sites of intense and sustained mixing have yet to be identified and the dissipation still needs to be quantified through observation. 1 Remarkably, when comparison is made between estimates based on Okubo’s empirical formula (9.11) – that strictly applies only to dispersion in the near-surface waters from which it was deduced – and observations, some far from the sea surface (for example in Sections 8.3.3, 13.3.3 and 13.4.1), the agreement is not bad. 2 A model of the circulation of water through the ocean, the ‘global ocean conveyer belt’, has been devised by Broecker (1991) and provides a basis for more recent ideas and models. Zenk (2001) gives a comprehensive account of the development of knowledge and of observations of the general circulation of the deep ocean.

14.3 Conclusion

371

A related matter of concern, one raised in Section 1.9, is that the equation of energy balance, the equation from which the energy that is available for mixing in the ocean might be determined, has not been closed; the magnitude of energy available for ocean mixing is unknown and many of the terms are not known with great certainty (Wunsch and Ferrari, 2004). Munk and Wunsch (1998) estimate that some 2.1 TW are needed to support the observed deep ocean mixing, but this is possibly an overestimate. Webb and Suginohara (2001) point out that numerical models indicate that more deep water than was previously thought may be brought to the surface in the Southern Ocean by Ekman suction. It is subsequently driven north in the surface Ekman layer and reduced in density by surface freshening. They conclude that it is therefore not necessary for bottom water to be mixed through the full water depth, but only to the level of deep water that is brought to the surface in the Southern Ocean. If this is correct, Munk’s canonical estimate of K T may be reduced from 1 × 10−4 m2 s−1 to about 3 × 10−5 m2 s−1 , and the power required for mixing is not 2.1 TW, but only about 0.6 TW, an amount that appears to be readily provided from the identified possible sources.3 But there is uncertainty even of these. About 0.6 TW appears to be available from the surface-generated, near-inertial internal waves (Watanabe and Hibiya, 2002) and, at most, there is 0.9 TW from the internal tides (Munk and Wunsch, 1998) although much of the energy of both these sources may be dissipated in the upper ocean or near sites of barotropic to baroclinic tide conversion, and not radiated far into the ocean (Lozovotsky et al., 2003) or to great depth. This possibly leaves some 0.5 TW unaccounted for. Other potential sources, such as flow through deep water channels and passages, and the effects of mesoscale eddies and Rossby waves, are presently unquantified. Very basic information is still being discovered about the spatial and temporal variations in the energy available to generate turbulence (see Rudnick et al., 2003) and about the latitudinal variations in dissipation and mixing (Gregg et al., 2003; Hibiya and Nagasawa, 2004). The uncertainty in estimates is being reduced, but at present it cannot be said that the sources or the location of ocean mixing are known with the certainty required to proceed with confidence to predict the consequences of mixing on long-term circulation or to develop a predictive model of the response of deep ocean circulation to changes in climatic forcing.

14.3

Conclusion

Whilst preparing this book I came upon the words in Evelyn Waugh’s quintessentially English novel, Brideshead Revisited, which recall our illusory perception of life, and that use the ocean deeps in simile. They are from Chapter 1 of Book 2, as Charles, the narrator, is leaving his friend Sebastian’s ancestoral home, Brideshead:

3 Munk and Wunsch (1998) acknowledged that their estimate of 2.1 TW could be too high by a factor of about 2.

Epilogue

372

I had come to the surface, into the light of common day and the fresh sea-air, after long captivity in the sunless coral palaces and waving forests of the ocean bed. “I have left behind illusion,” I said to myself. “Henceforth I live in a world of three dimensions – with the aid of my five senses.”

He goes on: I have since learned that there is no such world, but then, as the car turned out of sight of the house, I thought it took no finding, but lay all about me at the end of the avenue.

Any who uses the latter, finely expressed and evocative sentence, as a guide to ocean dynamics will be sadly misled! The working of the ‘real ocean’ is known to be threedimensional, even if the details of its functioning are not yet fully understood. Obstinate and blinkered affection for models that represent an effectively sluggish or passive ocean, or for the concept of an ocean that is simply a two-dimensional shell of water covering 70% of the Earth’s surface with little effect on climate, is misplaced. Such concepts are now plainly absurd and have been largely abandoned to history.4 The ocean is, however, still sometimes regarded as being dominated by stratification-dominated, rotational turbulence and, in modelling the circulation, it has often, inappropriately, been supposed that diapycnal mixing is unimportant. Disregard of diapycnal motions can provide only an illusionary realisation of the real ocean; three-dimensional processes and turbulence at both small and large scales are known to play an essential part in its workings, as examples provided in this text have demonstrated. 4 Ideas of ocean activity have changed greatly in the past 40 years. Heezen and Hollister (1971) wrote ‘. . . the warm water sphere’, meaning the warm upper layers of the ocean, ‘protects and isolates the abyss from the more familiar forces of weather, leaving as the only significant dynamic factors in the nearly closed abyssal system the thermohaline currents which spread and mix from pole to pole, . . . and the tides of the moon and sun which gently, but incessantly, caress the abyssal waters.’ This vision of a quiescent abyssal ocean, largely derived from photographic observations of the nature of the deep seabed, was later contradicted, partly by Hollister himself (Hollister and McCave, 1984), in the light of observations of deep-sea storms (Section 8.4). No longer is the body of the ocean regarded as a rather tranquil, steady and passive fluid mass.

Appendices

1

Parameters/symbols, typical values (where these can be given) and section of first introduction or definition

1.1

Non-dimensional parameters

C cE Fr I La Laturb Pr

Cox number (Section 6.5) Dalton number, about 1.5 × 10−3 (Section 1.5.5) Froude number (Section 12.2) Intermittency factor (Section 6.7) Langmuir number (Section 9.4.2) Turbulent Langmuir number, typically about 0.3 (Section 9.4.2) Prandtl number = ν/κT , about 7 in the ocean (Section 4.2.1) (the turbulent Prandtl number is Kν /KT ; Section 1.7.5) Rayleigh number (Section 4.2.1) Reynolds number (Section 1.1) (the turbulent Reynolds number is Ret ; Section 1.7.5) Flux Richardson number (Section 6.5) Richardson number (Section 3.2.3) Critical Richardson number (Section 3.2.3) Richardson number of the mean flow at a critical layer (Section 5.6.3) Minimum Richardson number (Section 3.2.3) (the symbol J is often used to denote the minimum Ri) Rossby number (Section 13.3.1)

Ra Re Rf Ri Ric Ricrit Rimin Ro

373

Appendices

374

R* Rρ S Sc Ta Tu We We γ

Deep convection scaling parameter, (B/ f 3 H 2 )1/2 (Section 4.5.2) Stability ratio = |αT/βS| = 2.3 (average value) (Section 4.6.1) Berger number (Section 12.4) Schmidt number = ν/κ S , about 670 in the ocean (Section 1.7.10) Taylor number (Section 10.2.4) Turner angle (Section 4.6.1) Wedderburn number (Section 12.8) Weber number (Sections 9.2.2 and 9.2.3) Efficiency factor, typically 0.2 (Section 6.5; see also Section 6.6) Density flux ratio αFT / βFs (Section 4.6.4) or surface tension (Section 9.2.4)

1.2

Derived dimensional parameters

kβ

Rhines wavenumber, (β/2U )1/2 , typically about 1/(70 km) (Section 13.1) Dispersion coefficient (Section 1.8.1) Vertical (or diapycnal) eddy diffusion coefficient, or diffusivity, of salinity (Section 6.6) Vertical (or diapycnal) eddy diffusion coefficient, or diffusivity, of heat (Section 1.7.4) Vertical (or diapycnal) eddy diffusion coefficient, or diffusivity, of mass (Section 6.5) Eddy viscosity (Section 1.7.3) Batchelor scale = 2 × 10−6 to 4 × 10−4 m (Section 1.7.10) Kolmogorov length scale = 6 × 10−5 to 10−2 m (Section 1.7.9) Lagrangian integral length scale (Section 1.8.1) |Monin–Obukov length scale|, typically 1 to 102 m (Section 4.3) Ozmidov length scale = 1–10 m (Section 6.3.2) Internal Rossby Radius, = 10 km (polar), 30 km (typical), 200 km (equatorial) (Section 3.8.2) Displacement (or Thorpe) scale (Section 6.3.2) Buoyancy frequency = 10−2 − 10−4 s−1 (Section 1.4) The Eady time scale ≈ 6.6H ( f /B)1/2 (Section 4.5.2) Lagrangian integral time scale (Section 1.8.1) The β parameter, = 2 cos φ/R (Section 3.8.2) Rate of loss of kinetic energy per unit mass of the turbulent motion (the dissipation rate), ranges from about 10−10 to 10−1 m2 s−3 (Section 1.7.8) Rate of loss of salinity variance (Section 1.7.8) Rate of loss of temperature variance, ranges from about 7 × 10−10 to about 10−4 K2 s−1 (Section 1.7.8)

KH KS KT Kρ Kν lB lK LL LMO LO LRo LT N tE TL β ε

χS χT

2 Units and symbols

375

1.3

Symbols for dimensional measures of seawater and its dynamics

cp

Le

Specific heat of seawater at constant pressure ≈ 3.99 × 103 J kg−1 K−1 (Section 1.2) Thermal expansion coefficient ≈ (5 – 30) × 10−5 K−1 (Section 1.3) Coefficient of salinity contribution to density, the salinity expansion coefficient ≈ 0.8 psu−1 (Section 1.3) (the distinction between this coefficient and the β parameter is made clear in their use in text) Adiabatic lapse rate ≈ 1.2 × 10−4 K m−1 (Section 1.3) Molecular thermal diffusivity coefficient ≈ 1.4 × 10−7 m2 s−1 (Section 1.5.2) Molecular transfer coefficient for salinity, ≈ 1.5 × 10−9 m2 s−1 (Section 1.5.2) Latent heat of evaporation ≈ 2.5 × 106 J kg−1 (Section 1.5.4)

1.4

Other commonly used symbols

CD

ν

Drag coefficient (in benthic boundary layer, Section 8.2; at the air–sea interface, Section 9.1) Coriolis frequency (Section 2.3.1) Acceleration due to gravity ≈ 9.81 m s−2 (Section 1.4) Significant wave height (Section 9.2.2) von Kármán’s constant, about 0.41 (Section 4.3) Friction velocity in the water (Section 4.3) Friction velocity in the air = (τ w /ρ a )1/2 , where, ρ a is the density of air (Section 1.5.5) A stress, for example wind stress on the sea surface (τ w , Sections 1.5.5 and 9.1), bottom stress (Section 8.2.3) or Reynolds stress (Section 1.7.3) Molecular kinematic viscosity ≈ 1 × 10−6 m2 s−1 (Section 1.1) Earth’s rotational frequency = 7.27 × 10−5 s−1 (Section 2.3.1)

2

Units and symbols

α β

a κT κS

f g Hs k u∗ u ∗a τ

unit

SI symbol (name)

kg–m–s equivalent

Force Pressure (force per unit area) Energy Power, energy flux Energy dissipation rate per unit mass Volume flux

N (Newton) Pa (Pascal, 1 Pa = 10−5 bar) J (Joule) W (Watt, 1 W = 1 J s−1 ) W kg−1 Sv (Sverdrup)

kg m s−2 kg m−1 s−2 kg m2 s−2 kg m2 s−3 m2 s−3 106 m3 s−1

Appendices

376

symbol

name

factor

P T G M k m

peta tera giga mega kilo milli micro nano pico femto

1015 1012 109 106 103 10−3 10−6 10−9 10−12 10−15

µ

n p f

3

Approximate values of commonly used measures Radius of a sphere with the same volume as the Earth = 6371 km Mean depth of the ocean = 3.795 km Area of the ocean surface = 3.61 × 1014 m2 Area of ice sheets and glaciers = 1.62 × 1013 m2 Area of sea ice = 1.75 × 1013 m2 in March and 2.84 × 1013 m2 in September Volume of the ocean = 1.37 × 1018 m3 Mass of the atmosphere = 5.3 × 1018 kg Mass of the ocean = 1.4 × 1021 kg Mass of water in lakes and rivers = 5.0 × 1017 kg Speed of sound ≈ 1500 m s−1 (see Section 7.1) von Kármán’s constant, k = 0.40–0.41 1 knot = 0.5148 m s−1

4

Values of typical energy levels and fluxes

(A summary of values mentioned in text. These are in many cases very approximate values subject to future revision, and the reader should refer to the sections mentioned for more detail.) 4.1

Turbulent dissipation Rate of supply of energy to support ocean mixing 2.1 TW (Section 1.9) (This figure may be an overestimate by a factor of about 2 – see Section 14.2.)

4 Values of typical energy levels and fluxes

4.2

377

The tides Tidal energy dissipation rate: 3.7 TW (Section 1.9) (This value is known with some certainty. About 0.2 TW of tidal energy goes into driving the tides of the solid Earth and 0.02 TW into atmospheric tides, leaving about 3.5 TW to be dissipated in the ocean.) M2 tidal energy dissipation rate: 2.5 ± 0.05 TW (Section 1.9) Tidal dissipation rate in shallow seas: 2.6 TW (Section 1.9) Total internal tidal energy dissipation rate in canyons: 58 GW (Section 12.3) Rate of energy transfer from barotropic to baroclinic tides on the continental slopes: 14.5 GW (M2) 2.73 GW (S2) (Section 2.7.2) Rate of energy transfer from barotropic to baroclinic tides in the N. Pacific: 0.27 TW (Section 2.7.2)

4.3

Mesoscale eddies Net global eddy kinetic energy (EKE): 13 × 103 PJ (Section 13.2.2) Rate of loss of energy from Gulf Stream Rings: about 0.03 W m−2 (Section 13.2.1) (For comparison, the total available energy in the general circulation of the ocean is estimated by Oort et al., 1994, as 1.5 × 1020 J or 1.5 × 105 PJ.)

4.4

The wind stress Net rate of transfer of energy from wind to water: 7–36 TW (Section 13.2.2) Rate of energy used to drive the mean ocean circulation: 1 TW (Section 13.2.2) Rate of energy generation of near-inertial oscillations in the mixed layer: 0.6 TW (Section 2.7.1) Energy flux radiated as internal near-inertial waves: 0.072–0.14 TW (Section 2.5) Rate of generation of APE by Ekman pumping: 0.4 TW (Gill, 1982, p. 569)

4.5

Internal waves Net energy (not including tides): 1.4 × 103 PJ (Section 2.5)

Appendices

378

4.6

Geothermal heat flux Mean rate of increase of potential energy from the geothermal heat flux into the ocean 0.05 TW (Section 8.2.5) Mean geothermal heat flux (except in ridge systems of hydrothermal activity) 46 mW m−2 (Section 8.2.5) Heat flux from the Juan de Fuca Ridge hydrothermal vents: about 7 GW (Section 4.5.1)

4.7

Straits Dissipation over the Carmarinal Sill in the Straits of Gibraltar 340 MW (Section 12.6.1)

5

Acronyms used in text

AABW ADCP ALACE AMP APE AUV AVHRR AVP BNL BT CAT CTD EKE FLIP FLY GPS HEBBLE HRP INL IR IWEX KE LES MODE NA

Antarctic Bottom Water Acoustic Doppler Current Profiler Autonomous Lagrangian Circulation Explorer Advanced Microstructure Profiler Available potential energy Automated Underwater Vehicle Advanced Very High Resolution Radiometer Advanced Velocity Profiler Benthic nepheloid layer Bathythermograph Clear air turbulence Conductivity Temperature and Depth probe Eddy Kinetic Energy per unit mass Floating instrument platform Fast Light Yo-yo Global positioning system High Energy Benthic Boundary Layer Experiment High-Resolution Profiler Intermediate nepheloid layer infrared Internal Wave Experiment kinetic energy Large Eddy Simulation Mid-Ocean Dynamics Experiment nitrogenase activity

5 Acronyms used in text

NATRE pd pdf PE PIV psi psu rhs r.m.s. ROFI SCIMP SNL SOFAR T–S TKE UV XDP

North Atlantic Tracer Release Experiment potential difference probability distribution function potential energy particle image velocimetry parametric subharmonic instability practical salinity units right-hand side (of an equation) root mean square Regions of freshwater influence Self-Contained Imaging Micro-profiler Surface nepheloid layer Sound fixing and ranging Temperature–salinity Turbulent Kinetic Energy ultraviolet expendable dissipation probe

379

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Some of the laboratory experiments referred to in the text are presented here. laboratory experiments on along-slope currents (Dunkerton et al., 1998) 314 baroclinic instability and eddies (Griffiths and Hopfinger, 1984) 195 boils (Pan & Bannerjee, 1995; Kumar et al., 1998) 283 breaking internal waves in shear flow (Thorpe, 1978b) 157 breaking surface waves (Rapp and Melville, 1990) 233 breaking surface waves produced by hydrofoil (Duncan, 1981) 233 coherent structure in shear (Brown and Roshko, 1974) 24, 98 convection (Bénard, 1900) 115 convection with rotation (Maxworthy and Narimousa, 1994) 128 critical layers (Koop and McGee, 1986) 162 critical layers (Thorpe, 1981) 162 exchange flows in straits (Marsigli, 1681) 326 flow over bed of seagrass (Ghisalberti and Nepf, 2002) 282 groups of hairpin vortices (Adrian et al., 2000) 219 impulsive generation of vortices (Voropayev et al., 1991) 169 interfacial waves on a slope (Wallace and Wilkinson, 1988) 299

424

internal wave generation by oscillatory flow over topography (Maxworthy, 1979) 74 internal wave generation by turbulence (Sutherland and Linden, 1998) 72 internal waves in (accelerating) shear (Thorpe, 1978c) 157 internal waves on critical slopes (Ivey and Nokes, 1989) 307, 312 internal wave resonant interactions (Davis and Acrivos, 1967) 154 internal wave rays (Mowbray and Rarity, 1967) 56–58 Kelvin–Helmholtz instability in different depths (Holt, 1998) 99 Kelvin–Helmholtz instability in downslope flow (Pawlak and Armi, 1998) 99 Kelvin–Helmholtz instability in rotating flow (Bidokhti and Tritton, 1992) 113 Kelvin–Helmholtz instability in tilting tube (Reynolds, 1883) 81–82, 86 Kelvin–Helmholtz instability in tilting tube (Thorpe, 1971b, 1973) 86, 93, 95 Kelvin–Helmholtz instability in wind tunnel (Scotti and Corcos, 1972) 93, 95 mechanical equivalent of heat (Joule, 1850) 4–5 mixed layer deepening and wave radiation (Linden, 1975) 72


mixed layer deepening (Strang and Fernando, 2001) 265 mixing with vertical grid (Fincham et al., 1996; Liu et al., 1987) 193 mixing with vertical grid (Yap and van Atta, 1993) 194 offshore vortex train (Matsunaga et al., 1994) 299 onset of Langmuir circulation (Handler, 2001; Veron and Melville, 2001) 255 onset of turbulence in pipe flow (Reynolds, 1883) 2–3, 37, 132 oscillating grid (Ivey and Corcos, 1982; Park et al., 1994; Thorpe, 1982) 193 oscillatory flow over rough bed (Jensen et al., 1989) 277 oscillatory flow over rippled bed (Sleath, 1985) 273 oscillating grid (Voropayev and Afanasyev, 1992) 195 parametric sub-harmonic instability (McEwan, 1983) 154

425

rain and mixing (Green and Houk, 1979) 243 rain and wave damping (Tsimplis, 1992; Poon et al., 1992) 243 rotors (Melville et al., 2002) 233 salt-finger convection (Huppert and Turner, 1980, 1981; Stern & Turner, 1969; Turner) 132, 135 salt fingers and shear (Linden, 1974) 132 ship-generated internal waves (Ekman, 1904) 46 slope-mixing (Salmun and Phillips, 1992) 318 solitary interfacial wave on a beach (Michallet and Ivey, 1999) 303 solitary internal waves (Grue et al., 2000) 149 standing internal waves (Benielli and Sommeria, 1998) 154 standing internal waves (Thorpe, 1968c, 1994a) 151 wakes in stratified flow (Hunt and Snyder, 1981) 323 wakes in stratified oscillatory flow (Boyer and Zhang, 1990) 325

Subject index

abyssal ocean 37, 208–209, 219 abyssal plain 116, 208, 214, 221, 225, 319, 332 canyons 208 accelerating flow 94 accelerometer 175 acoustic detection of billows 197, 198 turbulence 188–189 acoustic Doppler current profiler (ADCP) 131, 149, 177–179, 281, 336, 337 acoustic reflection 188, 334 acoustic scintillation thermography 188 acoustic tomography 188, 342 acronyms (see Appendix 5) across-wind dispersion coefficient (Ky ) 288 adiabatic heating 8 lapse rate (a ) 8, 214, 220 Admiralty Inlet 197 Advanced Microstructure Profiler (AMP) 173, 177, 191, 198 Advanced Velocity Profiler (AVP) 65, 173 advection 12, 38 advective–diffusive balance 39 aggregates 36, 37 Agulas Current 347 air-foil shear probes 174–175, 368 air–sea gas exchange 230, 232, 240, 298 Alborian Sea 328

426

Aleutian Ridge 70 algae 37, 280, 369 algal bloom 219, 267 Alfin (submersible) 116 along-shore flow 297–299 instability of 298, 299 along-slope flows 314, 316–317, 318 Ampere Seamount 211 angle of repose 319 anisotropy (see isotropy) Antarctic 14, 39, 47 Bottom Water (AABW) 38, 209, 332 anticyclonic vortices (see eddies) anthropogenic markers 290 Archimedes Principle 10, 11 Arctic Ocean 137 ARGOS satellite navigation system 342 Atlantic (see North Altantic) atmospheric boundary layer 123, 213, 228, 244, 249, 269, 271, 282, 348 autocorrelation function 31, 361 automated underwater vehicle (AUV) 173 available potential energy 65, 339, 345 Bahamas 247, 352 Baltic 269, 290 Banda Sea 149 baroclinic (see also internal) 41 instability 78, 129, 193, 195, 343, 345, 346, 347

Subject index

tide (see internal tide) wave (see internal wave) barotropic 41 eddies 346 instability 346 Rossby wave 365, 367 Tide (see also tidal flow) 42, 314, 333 Batchelor length scale 26, 179 Bathymetrical Survey of Scottish freshwater lochs 46, 81 bathythermograph (BT) 190 Bay of Biscay beach (see also surf zone) waves approaching 111 Beaufort Gyre 321 bed roughness (see roughness elements) benthic boundary layer 45, 213–225 boundary layer thickness 221 fronts 221–224, 225 nephloid layer 215 Berger number 78, 212, 325, 361 Bermuda 47, 201, 343 beta (β) effect 114 billows (see also numerical models) 83, 92–102, 137, 197–201, 286, 301, 328 convective rolls in 99–102, 105 length of crests of 98, 102–103, 110–112 pairing 97–99 shape of patches of 110 Woods’ observations of 83, 110, 112, 146, 172, 197 bioturbation 281 Black Sea 326 boils 189, 283–286, 288 bore (see hydraulic jump) on sloping boundary 299, 312–314 Bosphorus 326, 329–330, 331 bottom boundary layer (see also benthic boundary layer) 188, 213, 269, 270–282 boundary layer (see also sub-layer; mixed layer) atmospheric 123, 213, 228 benthic 45, 213–225 bottom 188, 213, 269, 270–282 on beaches 291–299 on submarine slopes 299–320 under ice 228 upper ocean 228–268 Boussinesq approximation 11, 53, 88, 146 braid 95, 197, 259 Brazil Basin 41, 76, 208, 325, 332, 362

427

breakers microscale 231 plunging 231, 233, 291, 293 spilling 231, 233, 293 breaking internal waves 39, 41, 144–171, 361, 369 breaking near-inertial waves 149–150 breaking waves (see breakers; breaking internal waves) bubble 239–243 advection by Langmuir circulation 257 bands 253, 257 break-up 37 cloud 189, 231, 239–243, 293 energy in producing 233 entrainment 86 generation 239–240 oscillations 245 rise speed 241–243, 369 size distribution 240–241, 243 buoyancy flux 115, 121, 122–123, 184, 228, 246–251, 258 force 6, 10–11, 53 frequency (N) 11, 53 frequency modulation 69 buoyant algae 251, 267, 268 bursts 217, 280, 285 caesium 34–37, 290 canyons 177, 311, 322–323, 352 dissipation of internal tidal energy in 323 Cape Kennedy 260 Cape St Vincent 352 capillary waves 231 Caribbean 137 Carabean Sheets and Layers Transects (C-SALT) program 138 caustics 69, 151 cells (see also Langmuir circulation) 118 centrifugal instability 276 Chain Fracture Zone 332 Challenger spacecraft 348 CHAMELEON 173, 180 channels 332 in deep water 332 chaotic particle paths 288 chimney 129 clear air turbulence (CAT) 83, 188, 325 climate change 6, 7, 367, 370 clouds of bubbles 189, 231, 239–243, 293 of sediment 188

428

CO2 fixation 268 Cobb seamount 310 coherent structures 23, 24, 203, 217–219, 230, 285, 348–357 cold skin 17, 116, 232 collapse and spreading 34, 78, 164, 168 of turbulence 102–105, 184, 194, 205, 211 of Kelvin–Helmholtz billows 103, 184 compressibility 8 concave continental slope 310, 311 conditional sampling 257, 258 Conductivity–Temperature–Depth probe (CTD) 6, 59, 173, 191 continental slope 41, 52, 303, 308, 310, 315 continuous spectrum of wave modes 90 convection (see also double diffusive convection) 115–131 in deep layers 119–121 in thin layers 117–119 under ice 120 convective instability (see also convective overturn; static instability) 115–143 overturn 145–146, 199, 202, 303 plume 116, 120, 123–125, 129, 131, 249, 265 surface boundary layer 176, 205–206 convergence 19–20, 22–23 convex continental slope 310 Coriolis force 124, 249 frequency ( f ) 53, 112 parameter 53 correlation (see autocorrelation function) cow-horn eddy 324, 337 Cox number 181 critical frequency 305, 307, 312 Rayleigh number 118 Reynolds number Richardson number 89, 90, 97, 150, 201, 203 slope 305 critical layer absorption 161–162 interaction 161–164 viscous effects 162 C-SALT 138 Dalton, John 4 Dalton number 16 dead water 45 decomposition of organic material 219 deep convection 38, 78, 128–131, 370

Subject index

deep-sea storms 215, 217 Denmark Strait 331 density 5–8 compensation 131, 140, 229 current 299 flux, by salt fingers 139 flux ratio (γ ) 139–140 inversions (see convective overturn) maximum 6, 339 profile 10, 45, 50 detrainment 22, 265, 273, 317, 318 diapycnal mixing 23, 34, 37, 38, 144 diffusion (see eddy diffusion; molecular diffusion) diapycnal (see also diapycnal mixing) 23, 37–43 Fickian 33, 289, 290, 360 horizontal (see diffusion, isopycnal) in tube flow 287–288 isopycnal 23 vertical (see diffusion, diapycnal) 23 diffusive regime 132 diffusivity (see also diffusion) of heat 14, 39, 47, 131–135 of salinity 14, 131 Discovery Gap 209 Discovery Passage 28 dispersion 30–34 coefficient 225 horizontal 31–33, 34, 78, 212, 225, 260–264, 287 meriodional 361 of passive solutes 33–34 of floating films 261–264 vertical 34, 288 zonal 361 dispersion coefficient (K H ) 32, 33 across wind component 264, 288 at the surface in the North Atlantic 363 at the surface in the Pacific Ocean 362, 364 estimated by Okubo 225, 260–261, 362, 370 estimated by Stommel 33 in Brazil Basin 362 in filament fronts 350 in Langmuir circulation 261, 262–264 in NATRE 360, 361 in the thermocline 289–290 of floats or floating material 262–264, 362, 363 on the Madeira Abyssal Plain 225 Richardson’s 33 dispersion in the benthic boundary layer 225 Brazil Basin 362 pycnocline 207, 288–290, 357–362 North Altantic 357–362, 363 Pacific Ocean 362, 364

Subject index

shallow seas 287–288 upper ocean boundary layer 260–264 dispersion relation of interfacial waves 50 interfacial waves in shear flow 87 internal wave modes in ‘tanh’ density profile 54, 90 internal wave modes in uniform density gradient 53 internal wave rays 56 internal wave rays in shear flow 161, 162 surface gravity waves 233, 245 displacement scale 176 dissipation factor 273, 281 dissipation rate of turbulent kinetic energy per unit mass (ε) (see rate of loss of turbulent kinetic energy per unit mass (ε)) diurnal heating 247–249, 265 jet 249 migration 39, 188 variations 267 divergence 19–20, 22–23 Doppler shift 161, 162, 171 double diffusive convection 31–33, 34, 39, 131–143, 165, 183, 209–210, 352, 353 instability 83, 117, 131 layers 135–137, 139–143, 192 drag coefficient over solid bottom boundary 215, 221, 271, 281 of wind on sea surface 228, 345 Drake Passage 41, 140 drift current (see also Stokes drift) 345 Eady time scale 129 East Greenland Sea 128 East Pacific Rise 126 Earth’s rotation (effects of) 52, 125, 128, 220, 317, 325, 331 eddies (see also vortices) anticyclonic 34, 125, 126, 169, 321, 355 cyclonic 113, 125, 129 intra-thermocline 348 mesoscale 42, 76, 229, 341–345, 346 pairing of 355–357 sub-surface 352–355 eddy diffusion 19–20, 22–23 eddy diffusion coefficient of dissolved solutes 23 eddy diffusion coefficient of heat (K T ) (see also eddy diffusion coefficient of mass (K ρ )) 23, 39, 180–181

429

in the abyssal ocean 39, 40 in the Brazil Basin 41, 208 in the North Pacific 40, 70, 311 Munk’s canonical value in the deep ocean 40, 41, 42, 207, 208, 292 near seamounts 40 over abyssal plains 41, 209 over rough topography 41 eddy diffusion coefficient of mass (K ρ ; see also semi-empirical equation for) 23, 180, 181–182 above the Equatorial Undercurrent 204 at the edge of the Scotia Sea and Drake Passage 41 at the Porcupine Bank 312 at the Stonewall Bank 323 in internal solitons 149, 287 in internal waves 169 in Knight Inlet, BC 337 in Monteray canyon 323 in NATRE 207 in the abyssal ocean 208 in the Denmark Strait 331 in the Emerald Basin, Scotian Shelf 314 in the Romanche Fracture Zone 332 in the Straits of Florida 331 on the Oregon continental slope 308 on the Scotian Shelf 314 on the Virginia continental slope 310 over seamounts 311 eddy diffusion coefficient of salinity (K S ) 23 eddy diffusion coefficient of sediment particles 280 eddy kinetic energy (EKE) 343–345, 347 eddy shedding (see also flow separation) 321–322 eddy viscosity 22, 216, 280 eddy viscosity coefficient (K ν ; see also semi-empirical equation for) 22, 180, 256 above the Equatorial Undercurrent 204 in the benthic boundary layer 216 efficiency factor () 181, 209, 210 of mixing 106, 154, 164, 182, 303 ejection from sloping boundary layer 315, 317 ejections 217, 280 Ekman boundary layer 220, 258, 317 drift 77 pumping 220, 221, 228 spiral 220 transport 220, 229 elastic scattering 68–69

430

electromagnetic current meter 180, 181–182, 221, 237, 270, 279 velocity profiler 65 El Niˇno 367 Emerald Basin, Scotia Shelf 314 energetics (see also Appendix 4) 40–42, 65–67, 73–74, 293, 370, 371 energy available potential (APE) 65 cascade 68 -containing eddies 27, 183 density 63 flux (see also Appendix 4) 63, 73, 232 kinetic (KE) 10, 62, 65 internal wave 62, 63 potential (PE) 5, 9, 62, 65 ratio 63 spectrum (see also spectrum) 27, 65 vertical flux of 63, 65, 76 energy loss by breakers 233 from internal wave field 65 engulfment 264–265 entrainment 20, 30, 264–265, 301 assumption 123–124 equation of rate of change of temperature variance 29–30 of motion 118 of rate of change of turbulent energy 28 of state 6, 117, 120, 339 Taylor–Goldstein 88, 90, 203, 259, 330 Equatorial Pacific 72, 187, 364 Undercurrent 187, 204 Ertel’s theorem 78 Eulerian flow 13, 314 length scale 32 time scale 32 European Continental Shelf 283 evaporation 16, 326 rate 16 expendable dissipation probes (XDPs) 320 external processes (see also internal processes) 229–230 Faeroe–Shetland Channel 303, 332 fall speed of particles 35, 280, 369 sediment 213 Fast Light Yo-yo (FLY) 173, 283, 311

Subject index

Fickian diffusion 33, 289, 290, 360 Fieberling Guyot 311 filaments 350 fine-structure 59, 189, 191 contamination (of spectra by) finger regime 132, 135 finite amplitude effects on interfacial waves 51 fish shoals 268 fjords 332–338 Fjørtoft’s theorem 85 flagllates 267 Floating Instrument Platform (FLIP) 198, 210 floats Autonomous Lagrangian Circulation Explorer (ALACE) 342, 363 neutrally buoyant 8, 31, 171, 225 Profiling Autonomous Lagrangian Circulation Explorer (PALACE) 131 Swallow 8, 340 flocs 280, 282, 369 flow over isolated topography 323–325 rough topography 325 flow separation 317, 322, 330 flux Richardson number (Rf ) 181 foam 243, 293–295 form drag 271–276, 280, 322, 323, 333 fossilization 184 four-thirds power law 32–33, 261 Fram 45 Franklin, Benjamin 45 free-fall profilers 181, 368 Advanced Microstructure Profiler (AMP) 173, 177, 191, 198 Advanced Velocity Profiler (AVP) 65, 173 CAMELEON 173 Fast Light Yo-yo (FLY) 173, 283, 311 High-Resolution Profiler (HRP) 173, 211, 361 Octoprobe 173, 246 freezing point 6 freshwater flux 7, 16 pool 245 friction velocity 121–122, 215, 233, 237 front benthic 221–224 on slope 312–314, 315 surface 229 tidal mixing 282–283 Froude number 232, 322, 326–328 composite 316, 327, 329, 330

Subject index

Galápagos Rift 116 Garrett–Munk spectrum 64–70, 164, 169, 171, 205, 307 estimated eddy diffusivity in 169 isotropy of internal waves in 67, 151 general circulation of the ocean 38 geostrophic balance 76, 113, 114, 195 current 47 geothermal heat flux 115, 116, 214, 216, 219 Global Positioning System (GPS) 8, 369 golf balls 33, 154 gravity current (see density current) Great Salinity Anomaly 367 groups of hairpin vortices 219 of internal waves 67, 165–168, 315 of surface waves 166, 236–237 group velocity 51, 314, 330 of internal wave modes 54 of internal wave rays 56 of surface waves 232, 236 Gulf of Cadiz 320, 352 Corryvreckan, W. Scotland 86 Lions 128, 129, 130 Gulf Stream 331, 344, 347, 348 ring 140, 246, 348 gyres 340 hairpin vortices 217, 219, 260, 276 Hasselmann’s theorem 28, 68, 69 Hatteras Abyssal Plain 64, 221 Hawaiian Ridge 42, 70, 75, 312 headland 321–322 Heard Island 342 heat 3–4 capacity 17–19 flux 14, 23, 115 generated by turbulence 5, 293 produced by decomposition of organic matter 219–220 transfer 12–16, 38 heat flux 14, 23, 115 geothermal 115, 116, 214, 216, 219 heat transfer (see also radiation) 12–16, 38 by internal waves 59 diapycnal 37–43 High Energy Benthic Boundary Layer Experiment (HEBBLE) 215 High-Resolution Profiler (HRP) 173, 211, 361 Hinze scale 240

431

histograms (see pdfs) Holmboe instability 91, 109–110, 265 homogeneity 24, 352, 363 hot film anemometer 28, 172, 174, 295 Howard’s semicircle theorem 54, 89, 90 hydraulic control 74 hydraulic jump internal 328–329, 333–337 in two-layer flow 327–328 surface 291, 292, 326–327 hydrothermal vent 116, 124, 192, 219 ice convection under 120 leads 228 icebergs 140 Indian Ocean 325 inertial (see also internal inertial gravity waves) sub-range 27, 297, 314 infrared (IR) 15 infrared image 283 sensors 17 instability (see also convection) 8–12 baroclinic 113, 129, 193, 195, 343, 345 barotropic 113, 114 centrifugal 276 collective 140 convective double diffusive 83, 117, 131 Görtler 276, 279 Holmboe 91, 109–110, 265 Kelvin–Helmholtz 81–92, 147, 183, 199, 265 parametric subharmonic 42, 69–70, 156, 206 Phillips–Posmentier 139, 150 Rayleigh–Taylor 10 static 9, 10, 115, 116, 145–146, 168, 204, 290, 315 instability in double diffusive diffusion regime 132 in double diffusive finger regime 132 in the near-surface mixed layer 259 in thin layers 118–119 leading to Langmuir circulation 255–257 of along-shore flow 298, 299 of continuously stratified static fluid 10–11 of internal tides 69–70 of non-parallel flows 92 of statically unstable shear flow 119 of stratified shear flow 80–112 of unstratified shear flow 83–86 of two-layer flows 86–88

432

instability (cont.) of two static layers 8–10 of turbulent flow 139, 150 instruments (see measurements of turbulence; acoustic Doppler current profiler (ADCP); bathythermograph; Floating Instrument Platform; free-fall profilers; sidescan sonar; single electrode probe) interfacial hydraulic jumps 51, 327–328 interfacial waves 83 with shear 87–88, 89, 327–328, 330 without shear 10, 50–52 intermittency 23, 185 factor 184 internal dissipation 65, 75 gravity waves 44–79 inertial gravity waves (see also near-inertial waves) 53, 187 Kelvin wave 77 roll waves 318 seiche 46–47, 77, 80, 338 surf zone 292 internal processes (see also external processes) 229–230 internal Rossby radius 77, 88, 113–114, 128, 129, 193, 227, 317, 331, 339, 342, 345, 347, 364 internal tide 41, 42, 55, 156, 165, 206, 208, 227, 292, 308–310, 314, 325 dissipation 65, 75 instability 69–70 generation 75–77 internal wave (see also interfacial waves; internal Kelvin wave; internal tide; near-inertial intenal waves; soliton packets) 44–79 breaking 39, 41, 144–171, 361, 369 generation of 41, 42, 70–77, 333, 336 groups 165–168 in lakes 77 in stratosphere 146, 164 Kelvin 77 modes 52–55 rays 62, 166, 303 radiation 72–73, 106, 184, 266–267 shape 58–59 slope 59, 157 steepness 59, 161, 168, 355–357 strain tidal vertical propagation 63 Internal Wave Experiment (IWEX) 64, 67 intra-thermocline eddies 348

Subject index

intrinsic frequency 161 intrusions 140–143, 192–193, 301, 317, 318, 352, 357 intrusive layers (see intrusions) inversions 81, 198, 201 Irish Sea 37, 269 isopycnal dispersion 34 surfaces 7, 10, 37, 38, 54 isopycnal shape produced by internal waves 58–59 isotropy 24, 175, 179, 181, 183–185, 278, 308, 346, 369 index 182, 184 isotropic turbulence 24, 174 Juan de Fuca Ridge 126 jet (in fjord) 333 Joule, James Prescott 4 Joule’s experiment to measure c p 4–5 Kelvin–Helmholtz billows (see billows) instability 81–92, 147, 183, 199, 265 transition to turbulence 92–108 Kelvin waves 77, 338 kinematic viscosity 3 kinetic energy (KE; see also eddy kinetic energy; turbulent kinetic energy) 62, 65, 211 Knight Inlet, British Columbia 74, 94, 171, 184, 197, 333–337 knots 98 Kolmogorov 26 length scale 26, 175, 184, 369 minus five thirds law 27 Krauss–Turner model 266 Labrador Sea 128, 131, 348 laboratory experiments 424–425 Lagrangian description of flow 31 drift 13, 314 integral length scale (L L ) 31, 363 integral time scale (TL ) 31, 363 velocity scale 31 Lake Geneva 249, 338 George, New York State 251 Longemer, Vosges, France 46 Vanda, Antarctica 137 laminar flow 2, 37 Langmuir, Irvine 229, 251

Subject index

Langmuir cells 35 circulation 229, 243, 246, 251–257, 265–266, 286, 348 circulation and internal waves 267 number (La) 256 number (turbulent, Laturb ) 256 turbulence 253 Langmuir circulation and buoyant algae 267, 268 large eddy simulation (LES) model 253, 265, 266 latent heat of evaporation 16 law of the wall 121–122, 216, 220, 230, 246, 298 layer generation by double diffusion 135–137, 139–143, 192 internal waves 59, 192 intrusions 192–193 Kelvin–Helmholtz instability 103, 192 oscillating vertical grids 193–194 Phillips–Postmentier instability 193 layer splitting 103, 108 layers 59, 186, 191, 192 convection in 118–119, 145 lee waves (wakes) 46, 73, 74, 83, 324, 325 Loch Ness 46, 77, 80, 97, 203 logarithmic velocity profiles 122, 216, 220, 230, 271, 274–276, 298 lognormality 187, 246 Madeira Abyssal Plain 225 marine organisms 188, 198, 267–268, 281–282 MARLIN 308 Marsigli 326 measurement of turbulence measurements of turbulence using acoustic Doppler 177–179 AUV 173 electromagnetic current meters 221, 237 expendable probes 173 free-fall profilers 173, 177, 181, 246, 283 ship mounted instruments 28, 177, 237 towed body 173, 308 measurements of turbulence from moorings 173 submarine 172, 173 submersibles 173, 184 the seabed 177 Meddies 78, 183, 192, 352–355 Mediterranean eddies (see Meddies) outflow 136, 137 undercurrent 326, 352 megaplumes 126

433

mesoscale eddies 42, 76, 229, 341–345, 346 microfronts (see also temperature ramps) 257 microstructure 39, 40, 174, 188, 201 mid-Atlantic Ridge 41, 76, 208, 323, 332 Mid-Ocean Dynamics Experiment (MODE) 65, 341 Miles–Howard theorem 88, 203 Mission Bay, San Diego 255 mixed layer (see also upper ocean boundary layer) 44, 46, 246–251 convection 246–251 deepening (see also numerical models of) 264–266 turbulence 228–268 mixing 20, 30 at the shelf break 73, 74 in the abyssal ocean 38, 65, 74, 75 length 32 near boundaries 40 modes barotropic (of Rossby waves) 365 baroclinic (of Rossby waves) 365 internal wave 53 molecular conduction 13, 216 diffusion 20, 287 diffusivity 14, 39, 47 transfer coefficient of salinity 14 viscosity 3 momentum transfer to mean flow (see also along-slope flows) 161, 169, 210 Monin–Obuko length scale (L MO ) 121, 219, 246 Monteray submarine canyon 177, 322 mucous Munk, Walter 38, 47 Munk’s canonical value of K T 40, 41, 42, 207, 208, 292 estimation of K T in the abyssal ocean 39 Murray, Sir John 46 Nansen, Fridtjof 45–46 Naruto Strait, Japan 86 Nasmyth universal spectrum 175 natural tracers 357 NATRE 40, 205, 206–207, 211, 357–362 near-inertial internal waves 42, 65, 68, 69, 70, 149, 164, 188 nepheloid layer 227 benthic 215, 226 intermediate 226–227, 318

434

neutral stability 86, 145 neutrally buoyant float 8, 31, 171, 183, 206, 211, 340, 352, 369 New York Bight 260 nitrogenase activity (NA) 268 noctilucent clouds 146, 164 North Atlantic 206, 341, 344, 346, 362 Deep Water 38 Tracer Release Experiment (NATRE) 40, 205, 206–207, 211, 357–362 North Pacific 40, 70, 73, 184, 346, 362 North Sea 260 Nova Scotia Rise 215 numerical models of mixed layer deepening Krauss–Turner 266 LES 265, 266 Pollard–Rhines–Thompson 266 numerical models of along-shore flow 298 billows 92 billows produced by internal waves 147 deep convection 129 flow over a sloping ridge 322 flow through a constriction, with dissipation and stratification 330 internal waves and salt fingers 165 internal waves at a critical level 162, 163 internal waves in shear flow 160, 164 Kelvin–Helmholtz instability 183, 266 Langmuir circulation 253, 256–257 mixed layer (see also numerical models of mixed layer deepening) 266 near surface flow and instability 255 wave reflection from sloping boundaries 307, 310, 315 wave reflection from sinusoidal sloping boundaries 310 ocean 1, 340 Octoprobe 173, 246 Odell and Kovasznay water channel 162 offshore vortex train 299 oil film 244, 261–264 Okubo’s formula for dispersion 225, 260–261, 362, 370 Oregon continental slope 303, 308 oscillating grid 193–194 oscillatory flow 271–274, 276–277, 322 over reflection 162 overstrained 290 overstability 132

Subject index

Ozmidov length scale (L O ) 175, 184, 194, 342 theory of turbulence 261 Pacific Ocean 38, 42, 126, 325 Equatorial 140, 182, 187 pairing of billows 97–99 eddies 355–357 vortices 98 pancake eddies 78 parameters (see Appendix 1) parametric instability 42, 69–70, 156, 206 parasitic capillary waves 231 parsnips 33 particle image velocimeter (PIV) 277 patches of billows 110 patchiness 173, 186–188 pdfs (probability distribution functions) of acoustic scattering cross section 198 ε 187 χT 187 vertical velocity in presence of plumes 249 pendulum day 355, 357 perpetual salt fountain 116–117, 131 persistence of turbulent motion 103, 187 phase speed 51, 56, 59 of internal waves in shear flow 89 Phillips, O. M. 67 Phillips–Postmentier instability 139, 150, 193 phytoplankton blooms 342 Pisces IV submersible 184 platinum resistance thermometer 312 plumes bubble 239, 293 convective 116, 120, 123–125, 129, 131, 249, 265 hydrothermal 124, 125–126, 188 in stratified surroundings 124–125 mega- 126 Pollard–Rhines–Thompson model 266 Poly-mode 341 Porcupine Bank 227 Portland Bill 322 potential energy (PE) 5, 9, 62, 65, 343 available 65, 345 density 8, 328 potential temperature 8, 221, 224 potential vorticity 77–78, 211, 341, 355 Prantdl number (Pr) 26, 103 preditor–prey interaction 37 probability distribution function (see pdfs)

Subject index

Profiling Autonomous Lagrangian Explorer (PALACE) float 131 Puget Sound 197, 322 pycnocline 44, 228 quartz crystals 220 radar radiation infrared (IR) 15 ultraviolet (UV) 15 radiation stress 297, 314 radon 225 rain 243–245 rate of change of turbulent kinetic energy 28 rate of loss of salinity variance (χS ) 26, 179–180 range of values 180 rate of loss of temperature variance (χT ) 25, 29, 172, 179, 210 range of values 26 rate of loss of turbulent kinetic energy per unit mass (ε) (see also semi-empirical equation for) 25 above the Equatorial undercurrent 204 during a diurnal heating cycle 247 during severe winter cooling 246 in breaking waves 233 in internal Kelvin waves 338 in Knight Inlet 336, 337 in Langmuir circulation 257 in Monteray Bay canyon 322 in near-bottom oscillatory flow 278, 283 in shallow seas 283 in internal solitons 287 in temperate ramps 258 in the eastern North Pacific 65 in the Irish Sea 270 in the Mediterranean outflow 320 in the seasonal thermocline 28, 172 in the Strait of Gibraltar 328, 329 in the surf zone 292–293, 295–297 in the swath zone 297 in the upper ocean boundary layer 172 latitudinal variation 206 methods to estimate 174, 205–206 near the sea surface 237, 255–256 over Cobb Seamount 311 over Stonewall Bank 323 range of values 25 Rayleigh, Lord 10 Rayleigh number (Ra) 115, 118 criteron for instability of circular vortex 112

435

Rayleigh’s inflection point theorem 83 Rayleigh–Taylor instability 10 reduced gravitational acceleration (see also reduced gravity) 9, 327, 338 reduced gravity 9, 196 Red Wharf Bay, Anglesey, UK 270 reflection coefficient 307 regions of freshwater influence (ROFI) 290 resonant wave interactions 67, 72, 153–156, 311 Reynolds, Osborne 2, 4, 22, 81–82, 243, 368 Reynolds experiment on onset of turbulence in tube flow 2–3, 37 number (Re) 3, 14–28, 37 tilted tube experiment 81–82 Reynolds stress 22, 29, 161, 178, 180–181, 214, 217, 277, 278 conservation in internal waves 58, 161, 162, 163 Rhines wavenumber 343, 347 Richardson, Lewis Fry 33 Richardson number after collapse of turbulence 103, 106, 203 bulk (Ri B ) 326 critical 89, 90, 97, 150, 201, 203 flux (R f ) 181 gradient (Ri) 82, 89, 149, 150, 199–204, 210, 307, 311, 330 measured values in the ocean 147 Richardson’s four-thirds power law 32–33, 261 ridges (see also mid-Atlantic Ridge) 41, 42, 280, 311 rip current 292, 299 rippled bed 271–277, 280, 281 rise speed (of bubbles) 241–243, 369 roll waves 318 Romanche Fracture Zone 332 Rossby number 78, 325, 348, 352 Rossby radius of deformation external (barotropic) 114 internal (baroclinic) 77, 88, 113–114, 128, 193, 227, 317, 331, 339, 342, 345, 347, 364 Rossby waves 78, 340, 365–367 rotation, of the Earth 114 rotor in surface wave 144, 233, 293 rough topography 41, 42 roughness elements 215, 216, 271, 277 roughness length 216 effect of organisms on 226, 281 effect of sediment on 226 effect of swath zone on 297

436

salinity 6, 7 variance (see rate of loss of salinity variance (χS )) Salinity–Temperature–Depth probe (STD) 191 salt fingers 132, 137–140, 165, 183 growth rate 132 regime 132 width 132, 138 Saltfjord, Norway 85 Sargasso Sea 165, 201, 348 sargassum 229 scars (caused by breaking internal wave groups) 166 Schlieren 24 Schmidt number 26 Scotia Sea 41 Scotian Shelf 308 seagrass 282 sea level change in 7 eustatic rise 7 Sea of Marmara 326 seamount 40, 78, 310–311, 325, 353 Ampere 211 Cobb 310 Fieberling Guyot 311 seasonal energy flux 73 seasonality 251, 283, 346, 363 sediment concentration 281 erosion 122, 278–279, 291, 301, 303 particles 213 slumping 319 transport 215, 278–279 sediment–water interface (see also roughness elements; roughness length) 215, 219, 270, 279, 281, 319 seiche 46–47, 77, 80, 338 seismic reflection profiling 189 Self-Contained Imaging Micro-profiler (SCIMP) 137 semi-empirical equation for ε 205 K ν and K ρ 204–205 sensible heat 13 shadowgraph 24, 137, 210 Shallow Meddies (Smeddies) 352 Shallow subtropical subducting westward-propagating eddy (Swesty) 355 Shallow water ocean eddies (Swoddies) 351 shear 19 dispersion 273 instability 80

Subject index

production term (−ρ0 < uw > dU/dz) 29, 121, 178, 181 vertical 19 wave-induced 146–150 shelf seas 269–290 Shields parameter 279 shrimp 268 sidescan sonar 253 sigma-T (σT ) 7 significant wave height 246 single electrode probe 180 skewness of temperature derivatives 257–258, 312 slippery sea 122 sloping boundaries 291–320 concave 310 convex 310 sinusoidal 310, 322 Smeddy 352 snapping shrimp 268 solar radiation 3–15 solitary interfacial wave 51–52, 303 soliton packets 46, 47, 51, 74, 149, 303, 328, 334 sonar 190 sorting algorithm 175 sound channel 341 fixing and ranging (SOFAR) 341 speed 190 South China Sea 319 spaghetti diagram 341 specific heat at constant pressure (cp ) 3 spectrum (see also continuous spectrum of wave modes) internal wave energy (see also Garrett–Munk spectrum) 39, 64–70, 164, 169, 171, 191, 205, 307 mesoscale eddy 346–347 Nasmyth universal 175 turbulent energy 27, 297 speed of sound 190 spiral patterns 95, 113, 348–349 spray 16 Sprungschicht 81, 191 Squire’s theorem 85 stability boundary 90 stability diagram double diffusive convection 132, 134 Kelvin–Helmholtz instability 90 Turner angle 134 stability parameter (Rρ ) 139, 210 stably stratified 12 shear flows 80

Subject index

standing waves 151–153 static instability 9, 10, 115, 116, 145–146, 168, 204, 290, 315 stirring 20, 30 Stokes 4 Stokes drift 13, 19, 237, 255–256, 314 Stokes Law 36 stress (see Reynolds stress; viscous stress) Stommel, Henry Stommel effect (trapping mechanism) 34–37, 251 Stonewall Bank, Oregon 323 strain 19 Strait Denmark 331 of Gibraltar 176, 320, 326, 328–329, 330, 331 Naruto 86 straits 326–331 Straits of Bab el Mandab 330 Florida 331 Georgia 177, 182 Juan de Fuca 177, 182 stratification 44–45 streaks 30, 217, 243, 290, 357–361 homogenization of 361 streamwise vortices 105, 119, 164, 217, 276 sub-layer Ekman 220 logarithmic 216, 271, 274–276, 277 viscous 214, 215–216, 217, 270 viscous–conductive 216, 220, 232, 246 submersible, Pisces IV 184 successive wave breaking (see groups of internal waves; groups of surface waves) sulphur hexafluoride (SF6 ) 206, 208–209 Sulu Sea 319 super-adiabatic temperature gradient 214 supercritical waves 305, 307, 311 superposition of waves 150–153 surface buoyancy flux 246–251 drift current 345 drifters 362–363 renewal 231 tension 86, 87, 232, 240 waves 111 surfactant 230 surf zone 231, 291, 292–297 suspended sediment 225, 279–281, 301 Swallow, John 8 Swallow floats 8, 340

437

swath zone 291, 297 sweeps 217, 219, 280 swell 230, 236 Swesty 355 Swoddies 351 symbols (see Appendix 1) Taylor, G. I. 10, 39–287 Taylor dispersion theorem 32 hypothesis 27, 297 number 276 vortices 293 Taylor–Goldstein equation 88, 90, 203, 259, 330 temperature 3 ramps 257–260, 265 variance (see rate of loss of temperature variance (χT )) thermal bar 339 diffusivity coefficient (K T ) 14, 39, 47 expansion coefficient 6 thermal wind 27, 314 equation 113 thermistor array 59–60, 97, 165, 201 chain 46, 258 heated 270 thermocline main 44, 182 seasonal 44, 67, 72, 77, 172, 187, 191, 265 thermohaline staircase 135, 138 thermometer (see also thermistor) mercury 81 platinum resistance 80 reversing 46, 80 thickness of boundary layer over the sea bed 221 on slope 307, 314 Thorpe scale (L T ) 175–177 tides 41 barotropic 42, 314, 333 baroclinic (see internal tide) Spring–Neap cycle 41 tidal bore 327 energy 269 energy dissipation 40, 41 flow 41, 74, 208, 269, 281, 282–283, 286, 288, 290, 311, 334–336 mixing front 282–283 time lag 283

438

Tollmein’s modification of Rayleigh’s theorem 85 TOPEX/POSEIDON satellite missions 343, 347, 365 topography (effects of) 74–76, 299, 338, 340 tracers 357–362 tritium 357 transition between Kelvin–Helmholtz and Holmboe instability 110 transition to turbulence 186 in flow over rippled bed 277 in Kelvin–Helmholtz instability 92–108 in Lagmuir circulation 255 traumata (caused by psi) 154 turbidity currents 319–320 turbulence 1, 2–3, 19, 369–370 definition 3 fossil 184 geostrophic 343, 347 horizontal (see also two-dimensional) 342 Langmuir 253 onset, in stratified shear flow 92–108 persistent 103, 187 two-dimensional 196, 211–212, 342 turbulence in benthic boundary layer 213–227 boundary layers over slopes 299–320 complex topography 325 canyons 322–323 deep water passages 281, 332 fjords 332–337 lakes 338–339 near-shore zone 291–299 pycnocline 190–212 shallow seas 269–290 straits and channels 326–331 upper ocean boundary layer 228–268 turbulence near headlands 321–322 seamounts or submarine hills 315, 323 turbulent energy equation 28 kinetic energy (TKE) 23, 27, 29 Langmuir number (Laturb ) 256 Prandtl number 23, 330 Reynolds number 23 Turner angle (Tu) 134 Tyrrhenian Sea 137 ultraviolet (UV) 15 undular bore 77

Subject index

undercurrent Equatorial 187, 204 Mediterranean 326, 352 uniformly stratified regions 12 units (see Appendix 2) upper ocean boundary layer (see also mixed layer) 228–268 upslope flow 316–317 upwelling 338 upwellings 283 US Space Shuttle Challenger 113 Vema Channel 209 ventilation 38, 339, 355 vertical collapse 34 divergence 20 velocity autocorrelation function 31 viscous dissipation 26 stress (see also sub-layer, viscous) 215, 217, 270, 276 von Kármán’s constant 121, 216 vortex force 256 loops 164 pairs 169, 271, 299, 342 rings 299 street 324 tube 98, 99 vortical mode 77–78, 86, 145, 197, 211–212, 361 vortices (see also eddies) 217 generated by breaking surface waves 233–236 generated by breaking internal waves 168–169 hairpin 217, 219, 260, 276 quasi-horizontal 169 spanwise 164 streamwise 105, 119, 164 vorticity cascade 194 vorticity waves 298 vortex stretching 28, 257 Vøring Plateau, Norway 219 wakes (see also lee waves) 323–325 wave (see also groups; interfacial waves; internal tide; internal wave; surface waves; tides) action 63, 307 age 239 breaking (see breakers; breaking internal waves) friction factor 273 wavelength of internal waves 62

Subject index

wavenumber 50, 58 vertical 56, 63, 69 wave–wave interaction 67–70 elastic scattering 68–69 induced diffusion 69 parametric instability 42 Weber number (We ) 232, 240 Weddell Sea 137

439

Wedderburn, E. M. 80–81 Wedderburn Number (We) 338 wind stress 228 windrows 229, 251 winter cascading 318, 339 Yih’s extension to Squire’s theorem 85 Y-junctions 257

Figure 1.8, Plate 1. The disruption of the thermal skin of the ocean by a breaking wave. A sequence of simultaneous video visual (left) and infrared (right) images of a breaking wave taken from the floating instrument platfom, FLIP, in the open ocean spanning a period of 1.6 s. The image size is approximately 5 m × 10 m, and the breaking wave propagates to the left. The whitecap is the warmest region and leaves a roughly circular patch of warmer water. Wind speed 7.7 m s−1 . The skin reforms some 10 s after such disruption. (From Jessup et al., 1997.)

Figure 1.15, Plate 2. The variation of vertical diffusivity of heat, KT , in an east–west section across the Brazil Basin in the South Atlantic Ocean. To the right is the rough, faulted terrain of the edge of the mid-Atlantic Ridge (the sea bed is shown in black), above which values of KT exceed Munk’s canonical value of 1 × 10−4 m2 s−1 . Values of KT lower than the canonical value are found over the relatively smooth topography in the centre and east of the Basin. (From Polzin et al., 1997.)

Figure 2.4, Plate 3. Four waves in a packet of solitary internal waves travelling shoreward (to the right) in a depth of about 110 m on the Oregon continental shelf, observed with a downward-pointing 120 kHz sonar. The packet appears as a series of waves of depression on the thermocline, about 600 m in wavelength, 25 m in depth and with troughs about 200 m wide. The colours indicate the intensity of acoustic scatter, red being relatively high and blue being low. This intensity increases just ahead of the wave troughs, indicating the generation of microstructure by wave breaking, and persists after a wave trough has passed, almost until the location of the next trough. (From Moum et al., 2003.)

longitude

(J m−2)

Figure 2.16b, Plate 4. The vertically integrated kinetic energy in the M2 internal tide in the Pacific Ocean. The tide often radiates away from ridge topography along linear, horizontally narrow, ray-like paths. (From Niwa and Hibiya, 2001b.)

latitude

Figure 3.4, Plate 5. A photograph of eddies about 10 m across formed at the edge of a tidal current passing through a narrow channel in the Saltfjord, Norway, and photographed from a bridge that spans the channel by Dr T. A. McClimans.

Figure 3.6, Plate 6. Billows, and their development, simulated in a direct numerical simulation model of a temperature stratified shear flow with an initial narrow shear and density layer with minimum Richardson number of 0.08. The Reynolds number, Re = 1354 and the Prandtl number = 7. Billows are formed (a), which pair (b), and eventually collapse (d), (e) and (f) show the rate of dissipation of turbulent kinetic energy, ε (W kg−1 ), on a log scale at the same times as (c) and (d), and (g) and (h) show the corresponding rates of dissipation of temperature variance, χT (K2 s − 1 ) again on a log scale. Particularly evident in (f) and (h) is the tendency for a layered structure to develop in the layer, thickened by mixing, between the uniform upper and lower layers. (From Smyth et al., 2001.)

Figure 4.1, Plate 7. Plumes rising from a group of hydrothermal black smoker vents, so called because of the blackening of the vented plume by tiny metal-sulphide crystals. The vent fluid, emitted at a speed of some 5 m s−1 , has a density of about 0.7 times that of the surrounding water and so is very buoyant and continues to rise rapidly. Eddies in the boundaries of the plumes engulf surrounding fluid. The plume on the right is about 1m wide at the top of the image. Those on the left originate from a number of vents in a column or black smoker chimney. (From Huppert, 2000.)

°C

0 3

500

2.95

Depth, m

2.9 2.85

1000

2.8 2.75

1500

2.7 2.65

2000

2.6 2.55

2500

2.5

J

J

A

S

O

N

D

J

F

M

A

M

Figure 4.9, Plate 8. Annual cycle of temperature in the Labrador Sea measured from instruments between 110–2510 m depth in a fixed mooring. White lines show the estimated depth of each instrument, occasionally increasing as the mooring is bent over by strong currents. Letters on the horizontal axis mark the first day of each month, June, July, . . . , May, through the year. A general warming occurs at the upper levels until January or at deep levels until March–April when the deepening cooled, and eventually convective, surface layer (cold and coloured deep blue) penetrates to about 1500 m depth. Heating resumes in May–June. (From Lilly et al., 1999.)

Figure 5.1, Plate 9. Billows in the trough of an internal solitary wave, detected using 120 kHz downward pointing sonar. (The wave is one of a train of internal solitary waves observed on the Oregon shelf. Such a train is shown in Fig. 2.4, Plate 3.) The wave is travelling to the right (phase speed marked as cw ) and the horizontal scale of the 10 m high billows appears to be about 70 m. Arrows near the wave trough show two other layers of high acoustic backscatter that possibly contain billows of smaller scale. (From Moum et al., 2003.)

100 uz /s−1

(a)

2

50 0

100 150 100 vz /s−1

(b)

−2 2

50 0

100 150

Depth / m

50

(c)

Strain

−2 1

100 N(z) 150

0

200 250 300 50

log10 ε/W kg −1

(d)

100

−1 −7 −8

150 200

−9

250 300 50

log10 K ρ /m2 s−1

(e)

100

−10 −4 −5

150 200

−6

250 300

296

298

300 302 Yearday 1998

304

306

−7 308

Figure 5.3, Plate 10. Mixing by inertial oscillations in the Banda Sea. Time vs depth plots over a period of 14 days of the 3 h averaged (top to bottom) (a) zonal shear, (b) meridional shear and (c) strain of the density field. Isopycnals are shown as black lines in (a)–(c) and the variation of buoyancy frequency is shown in (c). Upward propagation of the observed structures corresponds to a downward propagation of the inertial wave energy (as in Fig. 2.11). The next frame (d) shows the dissipation, ε, measured by microstructure probes, and the bottom frame (e) is the vertical diffusivity, Kρ , showing its periodicity in the pycnocline. The black lines in (d) and (e) at depths 2 m. Black lines are isopycnals at 20 m mean vertical spacing. The measurements were made about 30 km off the Californian coast in depths of about 800–900 m. The relative currents are about 0.1 m s−1 and the mean buoyancy frequency decreases from about 8.7 × 10−3 s−1 at 100 m to 3.5 × 10−3 s−1 at 400 m depth. (From Alford and Pinkel, 2000a.)

Pressure (dbar)

3500

4000

4500

5000

21

20

19

18

16 17 Longitude, °W

15

14

13

21

20

19

18

17 16 Longitude, °W

15

14

13

Pressure (dbar)

3500

4000

4500

5000

0

2

4

6

8

10

12

14

16

SF6 concentration, (fM)

Figure 7.11, Plate 14. The vertical spread of tracer, SF6 , released at a depth of about 4000 m in the Brazil Basin. The mid-Atlantic Ridge bounding the Basin to the east is labelled MAR and the injection site is marked INJ. The east–west sections of tracer concentration are at (top) 14 months and (bottom) 26 months after release. The white dots show the locations at which samples were taken and the white lines are isopycnal surfaces with potential density anomaly referenced to the 4000 dbar pressure level shown in units of kg m−3 . (From Schmitt and Ledwell, 2001.)

Figure 7.13, Plate 15. The variation of the efficiency parameter, , in (a) regimes stable to double diffusive convection, and (b) regimes favourable to salt fingering. The parameter, , is shown as a function of Richardson number, Ri, and of Rρ . Relatively high values are found where Ri is large (indicating stability to shear) but where (Rρ − 1) is small and positive (indicating instability to salt fingers). (After St Laurent and Schmitt, 1999.)

Figure 8.1, Plate 16. Streamwise velocity streaks and quasi-streamwise vortices in the viscous boundary layer close to a rigid flat seabed simulated in a numerical model by A. Pinelli and M. Uhlmann. The plan view shows (i) ∂u/∂z at the boundary in shades of grey. The dark regions are those of relatively low shear and therefore mark the streaks of relatively low downstream (direction x) velocity, u, just above the seabed. The light regions are streaks of relatively high downstream flow. Also shown are (ii) coloured objects, which are isosurfaces of streamwise vorticity within a distance of about 8z v of the boundary, purple being positive and red negative. The downstream, x, and cross-stream, y, axes are in units of v/u ∗ . (From Jiménez, 2000.)

(b)

(c)

Figure 9.13b,c, Plate 17. Langmuir circulation. Photographs showing windrows (marked by arrows) aligned with the wind on the water surface of a lake. The separation of windrows in (b) is about 10 m, and (c) is taken from the surface of the lake.

Figure 10.5, Plate 18. PIV image showing two maps of current vectors (arrows), with the mean current of 0.11 m s−1 to the right subtracted, and vorticity (coloured with positive, blue, being clockwise), at 0.6 s intervals in a vertical section in the plane of the mean flow. The images span from about 0.4 to 0.7 m above the seabed in a depth of 21 m off the coast of New Jersey. Large eddies and a string of smaller eddies are advected through the image sequence. The wave period is about 10 s. (After Nimmo Smith et al., 2002.) Figure 10.6, Plate 19. A satellite Advanced Very High Resolution Radiometer (AVHHR) infrared image showing the location of tidal fronts formed around the southwest of the UK. Violet–blue represents a temperature of 13–14 ◦ C and shows regions of the shelf that are unstratified. Red represents 18–19 ◦ C and indicates the surface temperature of strongly stratified waters, whilst green–yellow represents 16–17 ◦ C and marks regions of weak stratification and tidal mixing fronts. Three of the latter are apparent at A (the Ushant front), B (the Celtic Sea front) and C (the Western Irish Sea front). Clouds are shown in black at bottom left and in a north–south band across the English Channel. (From Sharples and Simpson, 2001.)

log(e; W kg −1) −1.50

100

−2.00

Height above bed (m)

80

−2.50 −3.00

60

−3.50 −4.00

40

−4.50 −5.00

20

−5.50 0 87.6

87.5

87.7

87.8

87.9

88

88.1

88.2

88.3

88.4

Decimal days

Figure 10.7, Plate 20. The lag of bottom-generated turbulence with increasing height above the seabed in well-mixed unstratified shelf sea waters of the southern Irish Sea. The record obtained from repeated FLY microstructure profiles covers two M2 tidal cycles and shows the half-M2 tidal period (6.2 h) variation of ε. Greatest values of ε are found near the bottom and there is a time lag in ε as height above the bottom increases. The tidally varying surface elevation is show at the top. (From Simpson et al., 2000.)

−1.50

30

−2.00

1024.3

4.1

1024.5

−2.50

25

2 10

−3.50 .3

.5

4 102

−4.00

.7

−4.50

1024.7

10

4.5

1024

4.3

102

102

15

−3.00

1024

20

.5 1024

Height above bed (m)

log(e; W kg −1) 35

−5.00

5 −5.50 0 186.7

186.8

186.9

187

187.1

187.2

187.3

187.4

187.5

187.6

187.7

Decimal days

Figure 10.12, Plate 21. Turbulence in a region of freshwater influence (ROFI): the variation of density (lines, in units of kg m−3 ) and log ε (colours) made by a FLY microstructure instrument in Liverpool Bay, eastern Irish Sea. The upper red line shows the position of the sea surface. High values of dissipation are confined to the bottom 15 m during ebb (water depth shown at top decreasing), but extend throughout the water column during flood. (From Simpson et al., 2002.)

M=0

M = 0.25

M = 0.5 0.04

2500

0.03 2000 0.02

0.01

y (m)

1500 0

1000

−0.01

−0.02 500 −0.03

−0.04

0 0

500

x (m)

0

500

x (m)

0

500

x (m)

Figure 11.5, Plate 22. Eddies developing in the wave-induced along-shore flow of a numerical model. The vertical component of vorticity, q (scale at right in s−1 , anticlockwise eddies positive), is shown at three different values, 0, 0.25 and 0.5, of an eddy viscosity coefficient, M, where K ν = Md(εb /ρ)1/3 , d is the local water depth and ε b is the rate of energy dissipation by wave breaking per unit area. The x-axis is distance off-shore and the y-axis along-shore, with the shoreline at x = 0. The induced mean along-slope flow in the −y direction has a maximum of about 0.8 m s−1 at a distance, x, off-shore of 75 m. The shape of the bottom and the wave conditions in the model replicate those in the area of the Superduck experiment made in Duck, North Carolina, USA, in Autumn 1986, where the overall slope is about 0.8◦ . The off- and along-shore components of bottom stress are taken as (2.23 × 10−3 u0 / d)(u, v), where u0 is the amplitude of the horizontal component of the orbital velocity of waves approaching the beach. The number of eddies is reduced as the coefficient M increases. Most of the isolated eddies at x > 200 m are anticlockwise. Vortex pairs propagating offshore can be seen, e.g. at x ≈ 300 m, y ≈ 1900 m when M = 0.25 and at x ≈ 350 m, ¨ y ≈ 900 m when M = 0.5. (From Ozkan-Haller and Kirby, 1999.)

t = t0

(a)

t = t0 + 11.5T

(b)

t = t0 + 18.5T

(c)

r 10.0 9.6

2

9.2 z

8.8 8.4

1

8.0 7.6 7.2

0

(d)

(e)

v/ c y

(f )

1.00 0.77 2 0.54 z

0.31 0.09

1

−0.14 −0.37 −0.60

0

(g)

(h)

e / vN 2 10.00

(i)

8.61 2

7.21 5.82

z

4.43 1 3.04 1.64 00

0.5 1.0 1.5 x

0

0.5 1.0 1.5 x

0

0.5 1.0 1.5 x

0.25

Figure 11.16, Plate 23. Along-slope flow and dissipation rates generated by internal waves reflecting from a slope at critical frequency. The numerical calculations for waves approaching at angle γ = 71.6◦ show that wave breaking produces a growing near-bottom boundary layer. The sections in a plane normal to the bottom slope are: (a)–(c), isopycnal surfaces; (d)–(f) along-slope currents divided by the along-slope phase speed of the internal waves; and (g)–(i) ε/ν N 2 (but only values >0.25 are shown). (The model values are much smaller than those likely to be found in the ocean.) The second and third columns are at times 11.5 and 18.5 wave periods, respectively, after the first. The along-slope flow develops over some 12 wave cycles with evidence of mixing occurring some distance off the bottom, but relatively little homogenisation of the boundary layer. (From Zikanov and Slinn, 2001.)

Figure 12.2, Plate 24. Acoustic image showing mixing on the western side of the Camarinal Sill in the Strait of Gibraltar. The image is from a section made over a period of 45 min (time scale at bottom) and is digitised with red colouring indicating the greatest scattering. The vertical scale is pressure (1 MPa ≈ 100 m). The vertical exaggeration is 2.9:1. The flow of the lower layer, the Mediterranean water, is westward (to the left). The braid-like features of billows are first detectable at 3.1 km on the horizontal scale. The upper part of the scattering layer rises abruptly at 2.95 km. Large, 70 m high, billows are detectable at 2.3–2.7 km. (From Wesson and Gregg, 1994.)

Figure 12.3, Plate 25. Billows over the Carmarinal Sill in the Strait of Gibraltar. The digital image is from a section made over a period of 11 min (time scale at bottom) and is coloured red where scattering is greatest. The vertical scale is pressure (1 MPa ≈ 100 m). The vertical exaggeration is 1.6:1. The flow of the lower layer, the Mediterranean water, is westward (to the left). The lower of the three distinct scattering layers visible at the left is not entrained into the billows, visible at 1 MPa and between 2.9 and 3.1 km, until about 2.95 km on the horizontal scale. The upper scattering layer rises between 2.90 and 2.95 km. (From Wesson and Gregg, 1994.)

Figure 12.8, Plate 26. An acoustic image of an internal wave train or undular bore (top, left) and flow bifurcation during ebb tide over the sill in Knight Inlet, British Columbia, Canada. East is to the left. The lower plot shows a (1:1) aspect ratio image of the bifurcation region where the west-going surface flow plunges below a slow eastward going flow. Doppler velocity vectors are coded for magnitude (as shown in the inlet) and the easternmost (blue) and western (red) density structure is shown at the bottom left. (From Armi and Farmer, 2002.)

0

depth, m

50

100

150 −1000

−500

0

500

1000

south–north/m

Figure 12.10a, Plate 27. The circulation in the lee of the sill in Knight Inlet, British Columbia, Canada. The vertical south–north section across the crest of the sill shows an hourglass-shaped region of eastward flow (red and yellow) in the centre of the strait and westward flow on either side.

Figure 13.2, Plate 28. Eddies visible on the sea surface in a coccolithaphore bloom south west of Iceland, visible at top right. The width of the image is about 400 km.

Warm Core Rings

Cold Core Rings

Figure 13.4, Plate 29. Gulf Stream meanders and rings. This is a false colour image of temperature in the Western North Atlantic based on data from the NOAA-7 Advanced Very High Resolution Radiometer (AVHRR) infrared observations. The image roughly 3000 km square, was obtained in June 1984 and shows the Gulf Stream extending northward as a bright red band of relatively warm water off the eastern seaboard of the United States from the coast of Florida. Further to the northwest the Gulf Stream beings to meander, forming Cold Core (green) and Warm Core (yellow) Rings as indicated. (The image was produced and kindly provided by O. Brown, R. Evans and M. Carle at the Rosenstiel School of Marine and Atmospheric Science, Miami, Florida.)

(a)

(b)

(c)

Figure 13.10, Plate 30. Eddy pairing. The path of a float that has passed south of Hawaii and caught in the circulation of an eddy that is advected westward in the North Equatorial Current. (a) The float track, rotating anticyclonically with increasing amplitude. The path of the centre of the vortex is shown by the dashed line. (b) The orbital period of rotation, which successively doubles as the float is advected westwards (to the left). (c) Sea surface height in cm determined from the altimeter tracks dotted in (a). The eddy lies in a region of mean surface elevation. (From Flament et al., 2001.)

Figure 13.11, Plate 31. The spread (or intrusion) of the tongue of Mediterranean Water entering the North Atlantic through the Strait of Gibraltar (arrowed). Red dots mark the locations of identified Meddies. Contours show the difference (or anomaly) of the mean salinity (in psu) from 35.01 psu near a depth of 1100 m. The track of the decaying Meddy in Fig. 13.7 is not shown, but is southwards from about 32.1◦ N, 22.9◦ W to 22.0◦ N, 22.2◦ W. (From Richardson et al., 2000.)

latitude, °S latitude, °S

longitude, °W

depth, m Figure 13.14, Plate 32. The horizontal spread of tracer, SF6 , released at a depth of 4000 m in the Brazil Basin. The vertically integrated concentrations in nmol m−2 at (a) l4 months and (b) 26 months after release. The underlying topography is shown in colour. (From Schmitt and Ledwell, 2001.)

Figure 13.15, Plate 33. The effect of seasonality and topography on the tracks of drifters released at the same location over a continental slope in (a) winter and (b) summer. In both seasons the drifters are followed using the ARGOS tracking system for 240 days after release. The surface drifters are drogued at 50 m and released almost simultaneously in three clusters of seven drifters each along a 20 km line across the upper part of Hebrides continental slope in the location that is arrowed. The tracks of drifters originating from the three clusters are shown in different colours. The water depths at the release positions range from about 200 m (the tracks coloured blue) to about 1200 m (the red tracks). The green tracks are of floats released in a cluster between the other two. The 1000 m isobath is shown as a continuous black line and the 200 m isobath as a dashed black line. The general drift follows the slope current along the slope to the northeast. There is however much greater advection in the winter release. Most of the floats in the 200 m cluster (blue) remain in shallow water, whilst many of the floats released in the deepest water (red) move off-slope. Diversions to the west can be seen in some tracks, notably towards the Anton Dohrn Seamount near 57◦ 20 N, 11◦ W, and along the Wyville–Thompson Ridge at 60◦ N. Many of the floats from the deep cluster (red) remain for a long time in deep water, several of those released in summer moving southwards into an eddying region west of Northern Ireland. A few floats pass around the north of Scotland into the North Sea. (From Burrows and Thorpe, 1999.)

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