An Introduction to Ocean Turbulence

This textbook provides an introduction to turbulent motion occurring naturally in the ocean on scales ranging from millimetres to hundreds of kilometres. It describes how turbulence is created and varies from one part of the ocean to another, what its properties are (particularly those relating to energy flux and the dispersal of pollutants) and how it is measured. Examples are given of real data and the instruments that are commonly used to measure turbulence. Chapters describe turbulence in the mixed boundary layers at the sea surface and seabed, turbulent motion in the density-stratified water between, and the energy sources that support and sustain ocean mixing.

   Little prior knowledge of physical oceanography is assumed and the book is written at an introductory level that avoids mathematical complexity. The text is supported by numerous figures illustrating the methods used to measure and analyse turbulence, and by more than 50 exercises, which are graded in difficulty, that will allow readers to expand and monitor their understanding and to develop analytical techniques. Detailed solutions to the exercises are available to instructors online at www.cambridge.org/9780521676809. Further reading lists give direction to additional information on the background and historical development of the subject, while suggestions for further study encourage readers to probe further into more advanced aspects.

   An Introduction to Ocean Turbulence is intended for undergraduate courses in physical oceanography, but will also form a useful guide for graduate students and researchers interested in multidisciplinary aspects of how the ocean works, from the surface to the seabed and from the shoreline to the deep abyssal plains. It complements the graduate-level text The Turbulent Ocean, also written by Professor Thorpe (Cambridge University Press, 2005).

STEVE THORPE was a Senior Scholar at Trinity College, Cambridge, where he studied mathematics and fluid mechanics, his PhD being awarded in 1966. He then spent 20 years at the UK Institute of Oceanographic Sciences, before being appointed Professor of Oceanography at Southampton University in 1986. He has carried out laboratory experiments on internal waves and turbulent mixing, and has measured and developed instrumental and analytical methods for studying waves and mixing in lakes, as well as making seagoing studies of turbulence in the boundary layers of the deep ocean and shelf seas. Professor Thorpe was awarded the Walter Munk Award by the US Office of Naval Research and the Oceanography Society, for his work using underwater acoustics, The Fridtjof Nansen Medal of the European Geophysical Society, for his fundamental experimental and theoretical contributions to the study of mixing and internal waves, and the Society’s Golden Badge for introducing a scheme to assist young scientists. He became a Fellow of the Royal Society in 1991 and is now an Emeritus Professor at the University of Southampton and an Honorary Professor at the School of Ocean Sciences, Bangor.

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An Introduction to
Ocean Turbulence

S. A. Thorpe

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CAMBRIDGE UNIVERSITY PRESS
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Cambridge University Press
The Edinburgh Building, Cambridge CB2 8RU, UK

Published in the United States of America by Cambridge University Press, New York

www.cambridge.org
Information on this title: www.cambridge.org/9780521676809

© S. A. Thorpe 2007

This publication is in copyright. Subject to statutory exception
and to the provisions of relevant collective licensing agreements,
no reproduction of any part may take place without
the written permission of Cambridge University Press.

First published 2007

Printed in the United Kingdom at the University Press, Cambridge

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ISBN 978-0-521-85948-6 hardback
ISBN 978-0-521-67680-9 paperback

Cambridge University Press has no responsibility for the persistence or accuracy of URLs 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.

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Contents

		Preface	Page ix
		Notes on the text	xi
		Acknowledgements	xiii
		Abbreviations	xv
		Standard parameters and symbols	xvi
		Units and their symbols	xviii
		SI prefixes	xix
		Approximate values of commonly used measures	xx
1		Turbulence, heat and waves	1
		1.1  Introduction	1
		1.2  Reynolds’ experiment	3
		1.3  Joule’s experiment	5
		1.4  The surf zone: waves and turbulence	8
		1.5  The nature of turbulent flow	13
		1.5.1  Stirring + diffusion = mixing	13
		1.5.2  Entrainment and detrainment	15
		1.6  Shear, convergence and strain	18
		1.7  Ocean stratification and buoyancy	19
		1.7.1  Density	19
		1.7.2  Buoyancy, and the buoyancy frequency, N	22
		1.7.3  The oceanic density profile	23
		1.8  Consequences of stratification	25
		1.8.1  Internal waves and turbulent motion	25
		1.8.2  Isopycnal and diapycnal mixing	28
		Suggested further reading	32
		Further study	32
		Problems for Chapter 1	33
2		Measurement of ocean turbulence	37
		2.1  Characteristics of turbulence	37
		2.1.1  Structure	37
		2.1.2  Stress and flux	39
		2.1.3  Dissipation	39
		2.2  Transport by eddies	39
		2.2.1  Reynolds stress	39
		2.2.2  Heat and buoyancy flux	42
		2.3  Energetics	43
		2.3.1  Turbulent dissipation, ∊, and isotropy	43
		2.3.2  The range and observed variation of ∊	45
		2.3.3  The rate of loss of temperature variance, v_T	47
		2.3.4  The Kolmogorov length scale, l_K	48
		2.3.5  The turbulence cascade and the structure of turbulence	49
		2.3.6  The Taylor hypothesis and the spectrum of turbulent energy	51
		2.4  The terms in the energy balance equation	54
		2.4.1  The rate of production of turbulent kinetic energy by the mean flow	56
		2.4.2  The turbulent potential energy	56
		2.4.3  The rate of dissipation	59
		2.5  Measurement techniques and instruments	59
		2.5.1  The first measurements of turbulence: spectra	60
		2.5.2  The air-foil probe: the measurement of ∊	60
		2.5.3  First measurements of Reynolds stress, and the related dissipation per unit area	67
		2.5.4  Estimates of Reynolds stress and ∊ using an ADCP	71
		Suggested further reading	73
		Further study	74
		Problems for Chapter 2	75
3		Turbulence in oceanic boundary layers	77
		3.1  Introduction: processes, and types of boundary layers	77
		3.2  Convection in the absence of shear	81
		3.2.1  Convection below a cooled surface or over a heated seabed	81
		3.2.2  Buoyant plumes and entrainment	83
		3.3  Stress and no convection; the law of the wall	85
		3.4  Stress and buoyancy flux	87
		3.4.1  The Monin–Obukov length scale	87
		3.4.2  Diurnal and seasonal heat cycling of the mixed layer	89
		3.4.3  Other mixing processes in the upper ocean	95
		3.4.4  The benthic (or bottom) boundary layer	100
		3.4.5  Tidal mixing and straining in shallow seas	102
		Suggested further reading	106
		Further study	107
		Problems for Chapter 3	110
4		Turbulence in the ocean pycnocline	116
		4.1  Introduction	116
		4.1.1  Processes of turbulence generation	116
		4.1.2  The first observations of turbulence in the thermocline	117
		4.2  Shear-flow instability and the transition to turbulence	119
		4.3  The Richardson number in the ocean	125
		4.4  Further turbulence parameters derived from microstructure measurements	129
		4.4.1  Estimation of v	129
		4.4.2  Estimation of eddy diffusion coefficients	131
		4.4.3  Rf and the ratio of the eddy coefficients of mass and momentum	133
		4.5  Entrainment into the surface mixed layer	135
		4.6  Observations of mid-water mixing processes	135
		4.7  The rate of diapycnal mixing	139
		4.8  Double diffusive convection	144
		Suggested further reading	149
		Further study	150
		Problems for Chapter 4	152
5		Turbulent dispersion	158
		5.1  Introduction	158
		5.1.1  The properties of dispersants	158
		5.1.2  Appropriate measures	163
		5.1.3  Effects of relative eddy and patch sizes	164
		5.2  The dispersion of particles	168
		5.2.1  Autocorrelation and integral scales	168
		5.2.2  Richardson’s four-thirds power law	170
		5.2.3  Dispersion of pairs of particles	171
		5.2.4  Effects of closed vertical circulations on buoyant particles	171
		5.3  Observations of the dispersion of floats	174
		5.3.1  Surface floats	174
		5.3.2  Subsurface floats	179
		5.4  The dispersion of solutes: methods and observations	180
		5.4.1  Dispersion (or horizontal diffusion) of a solute	180
		5.4.2  Dye releases in the surface boundary layer	180
		5.4.3  Tracer releases in the pycnocline	182
		5.4.4  Natural and anthropogenic tracers	187
		Suggested further reading	189
		Further study	190
		Problems for Chapter 5	192
6		The energetics of ocean mixing	197
		6.1  Introduction	197
		6.2  How much energy is required to mix the abyssal ocean?	199
		6.3  The tides	200
		6.3.1  The surface or barotropic tides	200
		6.3.2  The internal or baroclinic tides	201
		6.4  The atmospheric input of energy through the sea surface	204
		6.4.1  The wind stress	204
		6.4.2  Surface waves	205
		6.4.3  Buoyancy flux	207
		6.5  The mean circulation and mesoscale eddies	208
		6.6  Internal waves	209
		6.7  Dissipation produced by bottom stress	210
		6.8  Flow through and around abyssal topography	210
		6.9  Geothermal heat flux	216
		6.10  Discussion	217
		Suggested further reading	218
		Further study	219
		Problems for Chapter 6	220
		References	225
		Index	235

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Preface

My book entitled The Turbulent Ocean (referred to later as TTO) was written in 2003. It provides an account of much of the knowledge that there was then of the processes leading to turbulence in the ocean, but it was not written as a course that might be followed and used to introduce students to turbulent flow. Rather, it is a text useful for those beginning or already involved in research. It might form the basis of a number of advanced courses about ocean physics, teachers selecting material according to their needs or specialities.

I was asked to write a shorter book, an introductory course on turbulence in the ocean. Although believing that the best undergraduate and postgraduate courses are based and modelled on a teacher’s own experience and enthusiasms, and that to follow a ‘set text’ may be less enjoyable for students, I became convinced that a simplified text, more directly usable in teaching students unfamiliar with fluid motion, might be of value. Turbulence is a subject of which at least a basic understanding is essential in engineering and in many of the natural sciences, but particularly for students of oceanography. Moreover, many students, whose main interests are not in oceanography and who will not later address their talents to the study of the ocean, find interest in the sea and are motivated by aspects of their studies that are related or have application to matters of public and international concern, for example those of pollution and climate change that are at present being addressed by ocean scientists. A study of turbulent motion set in an oceanographic context can be attractive, satisfying and stimulating.

The purpose of the present book is consequently to provide a text that might be used in constructing and teaching an introductory course to students with a variety of academic abilities but who know little of ocean physics or turbulence. Much of the content has developed from a second-year course on ocean physics given in the Department of Oceanography at Southampton University, UK, in some 16 hours of lectures over a period of 4 weeks, supplemented by problems and additional reading undertaken by the students, a course attended by students whose main interests were in mathematics, physics, geology, biological oceanography or, generally, in marine science.

As in The Turbulent Ocean, the intricacy of turbulence theory is omitted. I recall, when an undergraduate, being totally mystified, if not frightened, by introductory lectures on turbulent motion that dealt with the subject in a largely statistical and analytical way, giving little or no insight into the dynamical processes of how it works. Unless students have a relatively high degree of ability in mathematics, the theoretical background is better faced after the basic concepts and ideas underlying the processes relating to turbulent motion have been absorbed and understood, and perhaps even after students have some understanding of the methods used to observe and measure turbulence. Neither is the numerical modelling of turbulence discussed here. That is best introduced to students in a separate and probably more mathematically demanding course once the processes involved in turbulence are firmly understood.

Unlike the earlier text, the material is almost entirely (but not quite!) restricted to what is well established and known, but I have also tried to explain the present limits of knowledge. I have taken the opportunity to include information that has been published since TTO was written, and to draw attention to errors that have come to my attention (specifically in footnotes 6 and 13 of Chapter 1, and footnote 13 of Chapter 6). I should be glad to be informed of any further errors found by readers in either that or this book.

S. A. Thorpe
‘Bodfryn’, Glanrafon, Llangoed, Anglesey LL58 8PH, UK

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Notes on the text

The symbol • denotes important points or summary statements.

There are six chapters with substantial cross-referencing between them. The first is intended as a general introduction, before means of quantifying and measuring turbulence are introduced in Chapter 2. Chapter 3 deals with the turbulent boundary layers near the sea surface and seabed. Chapter 4 describes the relatively weak and patchy turbulent motion that is found in the density-stratified water between these two boundary layers. Chapter 5 is about turbulent dispersion, whilst Chapter 6 is a discussion of the present (and rapidly developing) knowledge of the sources and rates of supply of turbulence energy required to support mixing in the deep ocean.

The illustrations are a very important supplement to the text. It is through pictures that information is carried most readily, and often in the most pleasurable form, to the mind and memory of a reader. ‘Cartoons’ (or sketches) conveying new ideas or concepts, photographs and data presented in graphical form are often an output of research, to which they provide a useful introduction or overview. The figure captions add substantial information that is not always included within the text.

Lists of Suggested further reading are provided at the end of each chapter. These are of literature that students might be expected to peruse, if not read in detail, in the course of their study of the contents of the chapter, e.g., to appreciate better the historical derivation of knowledge. Also listed are reference works that will provide information about basic fluid dynamics or ocean physics, should it be required.

Papers referred to under Further study are guides to encourage more extensive in-depth study of the material of the chapter, possibly leading to new research. In many cases another pathway into such further study is through the sources of figures referred to in the figure captions.

Problems are listed at the end of each chapter and are denoted at a point in the text where they might be attempted by [Pm.n], where m is a chapter number and n the problem number within the chapter. Each problem number in this list is followed by a letter that denotes the problem’s degree of difficulty: E = easy, M = mild, D = difficult and F = fiendish. The problems allow students to re-discover for themselves some of the now-accepted relationships, and provide experience in calculation and problem solving. These problems are essential elements in developing the ideas introduced in the text, and provide much additional information. They should preferably be read (if not solved) as students or readers advance through the course. The solutions to the problems are not given in this book but password-protected solutions to the problems are available online at www.cambridge.org/9780521859486. Quantitatively correct solutions are less important than the concepts introduced by the problems.

Lists of abbreviations, useful values etc. are provided on pages xv–xx for easy reference, and a map showing locations of places to which reference is made in the text is included on page xxii.

Scientific papers and books mentioned in text are all listed in the References, together with the numbers of pages on which they are mentioned.

The Index provides an entry to subjects that students may wish to locate or pursue, including ‘dimensional arguments’.

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Acknowledgements

Mrs Kate Davis has kindly helped in the preparation of figures and I am most grateful for her care and professional attention to detail. I am also grateful to Drs Larry Armi and Jim Moum for providing very helpful information about their teaching courses.

I am particularly grateful to the many individuals who have allowed me to reproduce their figures and photographs, or who have helped me to find suitable material, including M. H. Alford, L. Armi, E. D’Asaro, O. Brown, B. Brügge, J. Bryan, D. R. Caldwell, M. Carle, R. E. Davis, A. J. Elliott, R. Evans, D. M. Farmer, B. Ferron, O. Fringer, M. C. Gregg, A. J. Hall, A. D. Heathershaw, B. Hickey, H. E. Huppert, B. C. Kenney, B. King, J. Larsen, J. R. Ledwell, J. E. Lupton, W. K. Melville, J. Miles, J. N. Moum, A. Nimmo Smith, T. R. Osborn, G. Ostlund, P. E. Oswald, K. Polzin, B. S. Rarity, R. D. Ray, A. Roshko, P. L. Richardson, L. St. Laurent, R. W. Schmitt, R. Scorer, J. H. Simpson, W. D. Smyth, A. Souza, P. Stegmann, J. C. Stephens J. R. Toggweiler, C. Troy, F. Veron, H. W. Wijesekera, P. J. Wiles, J. D. Woods and V. Zhurbas, and to the publishers and organizations that have granted permission, including The American Association for the Advancement of Science, The American Geophysical Union, The American Institute of Physics, The American Meteorological Society, Blackwell Publishing Co., Cambridge University Press, CEFAS, Elsevier Ltd, The Journal of Marine Research/Yale University, MacMillian Publishers Ltd, NASA/GSFC and ORBIMAGE, The Royal Society and the Scientific Commission on Oceanic Research.

I also much appreciate the friendly help provided by the staff of Cambridge University Press in the preparation of this book and their care in its publication.

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Abbreviations

AABW	Antarctic Bottom Water
abl	atmospheric boundary layer
ADCP	acoustic Doppler current profiler
ALACE	Autonomous Lagrangian Circulation Explorer
AMP	Advanced Microstructure Profiler
AUV	autonomous underwater vehicle
bbl	benthic or bottom boundary layer
CTD	conductivity–temperature–depth probe
FLIP	Floating Instrument Platform
HAB	harmful algal bloom
HOME	Hawaiian Ocean Mixing Experiment
HRP	High Resolution Profiler
LES	large eddy simulation
lhs	left-hand side
MSP	Multi-Scale Profiler
PIV	particle image velocimetry
pd	potential difference
pdf	probability distribution function (or histogram)
rhs	right-hand side
rms	root mean square
RFZ	Romanche Fracture Zone
SOFAR	SOund Fixing And Ranging
STABLE	Sediment Transport And Boundary Layer Equipment
TTO	The Turbulent Ocean by S. A.Thorpe, Cambridge University Press, 2005
VACM	vector-averaging current meter

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Standard parameters and symbols

(with the section and, where appropriate, equation in which they are introduced)

C	Cox number (Section ; –)
CD	the drag coefficient on the seabed (Section ). (CDa is used in this text to denote the drag coefficient of the wind on the water surface, with subscript a – standing for air – to emphasize that its value is different from CD; it is defined in Section .)
I	isotropy parameter (Section ; )
KH	eddy dispersion coefficient (Section ; )
KH∞	eddy dispersion coefficient at times > > TL (Section ; )
KS	eddy diffusion coefficient of salinity (Section )
KT	eddy diffusion coefficient of heat or eddy diffusivity of heat (Section ; )
Kρ	eddy diffusion coefficient of density (Section )
Kν	eddy viscosity (Section ; )
LL	Lagrangian integral length scale (Section ; )
LMO	Monin–Obukov length scale (Section ; –)
LO	Ozmidov length scale (Section ; )
LRo	Rossby radius (of deformation) (Section )
lK	Kolmogorov length scale (Section )
N	buoyancy frequency (Section ; )
Ra	Rayleigh number (Section , footnote 4)
Re	Reynolds number (Section ; )
Rf	flux Richardson number (Section ; )
Ri	gradient Richardson number (Section ; )
RiB	bulk Richardson number (Section )
Rρ	density gradient ratio (Section 48; )
TL	Lagrangian integral time scale (Section ; )
Σ	efficiency factor (Section ; ())
ґ	rate of dissipation of turbulence kinetic energy per unit mass (Section ; –)
ΣT	sigma-T (temperature) (Section )
Σθ	sigma-theta (potential temperature) (Section , footnote 15)
xS	rate of loss of salinity variance (Section )
XT	rate of loss of temperature variance (Section ; –)

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Units and their symbols

		kg–m–s
Unit	SI symbol (name)	equivalent
Force	N (Newton)	kg m s2
Pressure (force per unit area)	Pa (Pascal, Pa = 10- 5 bar)	kg m-1 s-2
Energy	J (Joule)	kg2 s-2
Power, energy flux	W (Watt,1 W = 1 Js-1)	kg2rm s- 3
Energy dissipation rate per unit mass	W kg−1	m2s- 3
Volume flux	Sv (Sverdrup)	106 m3 s-1

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SI prefixes

Symbol	Name	Factor
E	exa	1018
P	peta	1015
T	tera	1012
G	giga	109
M	mega	106
k	kilo	103
d	deci	10-1
m	milli	10−3
▤	micro	10−6
n	nano	10−9
p	pico	10−12
f	femto	10−15

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Approximate values of commonly used measures

Radius of a sphere with the same volume as the Earth = 6371 km

Rotation rate of the Earth, γ = 7.292 10^^-5s^-1

Mean depth of the ocean = 3.795 km

Area of the ocean surface = 3.61 × 10¹⁴ m²

Mean area of sea ice 10¹³ m² in March and ¹⁰13m² 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 ≈ 5×10¹⁷ kg

Speed of sound ≈ 1500 m s⁻¹

von Kármán’s constant, k = 0.40–0.41

1 knot = 0.5148 m s⁻¹

Image not available in HTML version

The general positions of some of the locations and currents referred to in the text.

1. Agulhas Retroflection Zone

2. Antarctic Circumpolar Current

3. Baltic Sea

4. Bahamas

5. Banda Sea

6. Bay of Fundy

7. Bermuda

8. Black Sea

9. Bosphorus

10. Brazil Basin

11. Discovery Gap

12. Discovery Passage

13. East Australia Current

14. East Greenland Sea

15. East Pacific Rise

16. Florida Current

17. Gulf of Lions

18. Gulf Stream

19. Hatteras Abyssal Plain

20. Hawaiian Ridge

21. Hunter Channel

22. Irish Sea

23. Juan de Fuca Ridge

24. The Kattegat

25. Knight Inlet, British Columbia

26. Kuroshio

27. Labrador Sea

28. Mediterranean Sea

29. Monterey Bay, California

30. New England Continental Shelf

31. North Sea

32. Oregon Continental Shelf

33. Romanche Fracture Zone (RFZ)

34. Ross Sea

35. Sargasso Sea

36. Strait of Gibraltar

37. Straits of Florida

38. Vema Channel

39. Vøring Plateau

40. Weddell Sea

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