The Dynamics of the Upper Ocean

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The Dynamics of the Upper Ocean

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NUNC COCNOSCO EX PARTE

TRENT UNIVERSITY LIBRARY

Digitized by the Internet Archive in 2019 with funding from Kahle/Austin Foundation

https://archive.org/details/dynamicsofupperoOOOOphil

CAMBRIDGE MONOGRAPHS ON MECHANICS AND APPLIED MATHEMATICS GENERAL EDITORS

G. K.

Batchelor,

Ph.D., F.R.S.

Professor of Applied Mathematics at the University of Cambridge

J. W.

Miles,

Ph.D.

Professor of Engineering, University of California, Los Angeles

THE DYNAMICS OF THE UPPER OCEAN

THE DYNAMICS OF THE UPPER OCEAN BY

O. M. PHILLIPS Professor of Geophysical Mechanics, The Johns Hopkins University, Baltimore

CAMBRIDGE AT THE UNIVERSITY PRESS 1966

Published by the Syndics of the Cambridge University Press Bentley Plouse, 200 Euston Road, London N.W. 1 American Branch: 32 East 57th Street, New York, N.Y. 10022 Library of Congress Catalogue Card Number: 66-17054

Printed in Great Britain at the University Printing House, Cambridge (Brooke Crutchley, University Printer)

V

CONTENTS page vii

Preface

I. INTRODUCTION 1.1

The ocean environment

i

1.2

The development of the subject

3

2. THE EQUATIONS OF MOTION 2.1

Specification of the motion

7

2.2

The equations of motion

8

2.3

The mechanical energy equation

13

2.4

The Boussinesq approximation

14

2.5

The Reynolds stresses

19

3. THE DYNAMICS OF SURFACE WAVES 3.1

The governing equations

22

3.2

Infinitesimal waves

25

3.3

Particle motions in irrotational flow

31

3.4

The influence of molecular viscosity

33

3.5

The ‘ conservation of waves ’

43

3.6

The dynamical conservation equations

44

3.7

Some applications

52

3.8

Wave interactions

63

4. OCEAN SURFACE WAVES 4.1

The specification of a wave field

72

4.2

The generation of waves by wind

79

4.3

The coupling between wind and waves

87

4.4

Wave interactions

107

4.5

The equilibrium range

109

4.6

The development of the spectrum

119

4.7

Ripples and short gravity waves

131

110990

CONTENTS

VI

page 139

4.8

The air flow over the sea

4.9

Wave propagation and attenuation

145

4.10 The probability structure of the surface

154

5.

INTERNAL WAVES

5.1

Introduction

158

5-2

Infinitesimal waves

161

5-3

The lowest internal mode

166

5-4

Small scale internal waves

173

5-5

The influence of a weak mean shear

178

5-6

The degradation of the lowest internal mode

185

5-7

Waves of tidal period

190

6.

OCEANIC TURBULENCE

6.1

The occurrence of turbulence

198

6.2

The energy equation for the turbulence

200

6.3

The spectrum of turbulence

202

6.4

Local similarity theory

205

6-5

The maintenance of Reynolds stresses

217

6.6

Turbulence in the surface layer

222

6.7

Entrainment

235

6.8

Thermocline erosion

239

Bibliography Index

244 257

Vll

PREFACE A proper understanding of the dynamical processes that occur in the upper ocean is crucial to much of physical oceanography. During the last ten years, close contact between theory on the one hand and observation and experiment on the other has resulted in many important advances in this area. Some of these are described in papers scattered among the various journals and their connexions are not immediately apparent; others have not hitherto been published. The time seems particularly opportune, then, to attempt to provide a coherent account of the recent developments in this subject in the hope of making them accessible to a greater number of oceanographers and, at the same time, of stimulating the interest of other workers in this rewarding and fertile field of geophysical fluid mechanics. In some aspects, our understanding is already fairly detailed, but in others we are handicapped by the lack of either exploratory or definitive experi¬ ments, so that theoretical studies are necessarily rather tentative and subject to revision. Where this is so, I have tried not to conceal the fact, in the hope of stimulating others to remedy the deficit. In any event, the success or failure of this monograph will, I suppose, be measured by the degree to which it provides the basis for future developments or, possibly, dissent. An earlier draft was awarded the Adams Prize for the years 1963-64 by the University of Cambridge. To acknowledge all those who, directly or indirectly, have contributed to this book would be to acknowledge all my friends who work in this area. Nevertheless, I am particularly indebted to Dr M. S. Longuet-Higgins and to my colleagues at Cambridge and at The Johns Hopkins University for the stimulation of many discussions, to Dr M. N. Brearley for his critical reading of the manuscript and the discovery of many errors, to the British Admiralty and the U.S. Office of Naval Research for their support, to my wife for her patience and, for many things, to Professor G. K. Batchelor, to whom, with respect, this book is dedicated. Baltimore September ig6$

O. M. P.

I

CHAPTER I

INTRODUCTION 1.1. The ocean environment From the beginnings of history, the sea has excited our fascina¬ tion and respect. Whether Phoenician or Polynesian, Arab, Viking or Medieval European, the lore of the sailor and his achievements have contributed to the streams of our culture; they have provided recurrent themes in painting, in literature and in music. Today, as in time past, the sea continues to attract or repel those who know her; few are neutral. Many generations of navigators had wondered about the ocean depths beneath them, and their ignorance bore apprehension. By the middle of the nineteenth century, the exploration of the ocean surface was virtually complete, but the depths remained unknown. One of the earliest of the great oceanographic expeditions was the circumnavigation of the earth by the Challenger in the eighteenseventies. For the first time there was a systematic collection of data and specimens from not only the surface but the ocean depths as well. In later expeditions, ships with evocative names such as Discovery, Meteor and Atlantis made enduring contributions to the description of the world’s oceans. These explorations continue— it is, for example, barely ten years since Cromwell, Montgomery and Stroup established the existence of the Pacific equatorial undercurrent, by any account one of the largest oceanic current systems (Montgomery and Stroup, 1962). As a result of many such painstaking studies, our present knowledge has accumulated of the oceans and of the basins wherein they are contained. This provides the environment for the various kinds of motion described in this book, and for which we must account. In geographical terms, the sea bed can be divided into two or three quite distinct regions. Surrounding the continental shore¬ lines, for a distance of the order of 100 or 200 km, the water is relatively shallow, averaging some 200 m in depth. The sea floor of this continental shelf usually undulates gently, rising occasionally to the surface in off-shore banks or islands. This is the region of I

PD U

2

THE DYNAMICS OF THE UPPER OCEAN

most concern to the returning navigator and is the most accessible with his instruments; though the least homogeneous, it is the best known. The water above the continental shelf is moved by the tides and by esturine currents; it can be stirred throughout its depth by the wind stress at the surface. Beyond this region, the sea floor falls away to the ocean depths beyond. The transition is remarkably abrupt, the overall slope being generally of the order o-1, but frequently corrugated by submarine canyons where locally the slope is much larger. At the foot of the continental slope lies the deep ocean bed whose depth is of the order 4000 m and which lies beneath the great majority of the ocean surface area. This again is relatively flat, the abyssal plain being broken by an occa¬ sional sea-mount or trough or by a long mid-ocean ridge. Little is known about the motions that occur at these great depths, though recent scattered observations (see, for example, Crease, 1962) have indicated the presence of surprisingly rapid and rather irregular movements. In the open ocean above the deep sea bed, the vertical structure of the water mass is characterized by the presence of one or more thermoclines, regions of large temperature gradient within the upper few hundred metres. The thermocline is strongest in equatorial waters, where the contrast between the warmed surface layer and the cold deeper water is greatest; at increasing latitudes it weakens, even vanishing in certain polar waters. The main¬ tenance and structure of the thermocline are central questions in an understanding of the general oceanic circulation. In the upper ocean, from the thermocline region to the surface, are found the variety of inter-related motions, internal waves, turbulence and surface waves with whose dynamics we shall be concerned. The motions in the upper ocean provide the means for the exchange of matter, momentum and energy between the atmosphere and the underlying deep ocean. On a global scale, these exchanges produce in the oceans the general circulation pattern and at the same time constitute one of the most important factors in the world-wide distribution of climate. A proper understanding of the behaviour of the upper ocean is at the heart of these larger questions, and to this ultimate end are our efforts directed.

INTRODUCTION

3

1.2. The development of the subject The threads to be woven into the fabric of this subject lead far back in time. They are of three kinds. There is first the observation of the phenomena of the upper ocean, the identification, descrip¬ tion and measurement of the various modes of motion and dynamical processes to be found under natural, uncontrolled (but hopefully, measured) conditions. Secondly, there is the experi¬ mentation in the laboratory, where it may be possible to isolate one or two of these phenomena, and to study their properties and mutual interactions. Finally, there is the analysis of these motions and their inter-relations, the development of a coherent theory. It is only by drawing these threads together in a consistent way that the pattern of the subject can be discerned and its texture be felt. In each of the three major topics involved in the dynamics of the upper ocean this process can be followed. Casual observations on surface waves and their relation to the wind have been made for time immemorial, so that it was not unnatural that the pioneers of theoretical fluid mechanics, Lagrange, Airy, Stokes and Rayleigh sought to account for the elementary properties of surface waves in terms of perfect fluid theory. Even simple experiments allowed the frequency-wavelength relations to be compared with this theory, and the results must have been encouraging. Nevertheless, surface waves in the ocean were less simple and provided a constant reproach to the elementary theory. Rayleigh wrote: ‘ The basic law of the seaway is the apparent lack of any law.’ In the last century, the irregularity of ocean waves defied description—this had to await the developments in probability theory made during the first forty years of this century. The problem of relating the rate of wave growth to the wind was recognized by Kelvin, but no real progress was made. By 1850, Stevenson had made observations on surface waves in a number of lakes and derived an empirical rela¬ tionship between the ‘ greatest wave height ’ and the fetch (distance from the windward shore).

Seventy-five years later, Jeffreys

attempted to model the generation of waves by wind in a laboratory experiment. But as recently as 1956, Ursell could reasonably write: ‘Wind blowing over a water surface generates waves in the

4

THE DYNAMICS OF THE UPPER OCEAN

water by physical processes which cannot be regarded as known. There was no even remotely adequate theory; the results of oceanic observation and laboratory experiment seemed flatly in¬ consistent. The threads were either missing or disjoint. Ursell’s pessimism provoked new efforts to develop a sound theory and to make well-documented experiments and observa¬ tions. These attempts are described in Chapter 4 herein; whatever their shortcomings it appears that theory and observation may at last be becoming mutually relevant. Internal waves, on the other hand, are not the creatures of our common experience that surface waves are. The simplest solutions for waves on an internal interface were given by Stokes in 1847; their properties could again be illustrated by simple experiments. Ekman, writing in the record of Nansen’s North Polar Expedition of 1893-6 (vol. v, 1904, 562), described some laboratory observa¬ tions on the generation of internal waves by a slowly moving ship model. There have been numerous rather cursory indications of the presence of internal waves in the ocean (some of which are mentioned later), but it is both difficult and expensive to make a detailed, systematic study of these motions. Those that do exist are correspondingly rare and valuable. In the laboratory the greatest contributions have undoubtedly been made by Long (1953a, b, 1954, 1955) and have been concerned with the general problem of the excitation of internal waves by the flow over irregularities in the bed. Nevertheless, since our acquaintance with internal waves, both in the ocean and in the laboratory, is still quite limited, we can hardly conceive all the subtleties of their dynamical behaviour. Until the last few years, theoretical studies were mainly confined (with some notable exceptions) to the finding of eigen solutions for particular density distributions and were still far from giving an understanding of the physical processes that might occur in the ocean. Even now, the strands have not drawn together as one would wish. In the absence of extensive observa¬ tional experience, the theoretical endeavours described later are sometimes rather speculative, not (it is hoped) in their internal correctness, but possibly in their immediate relevance to what actually occurs in the sea. This is a situation to be remedied only by time and patient endeavour.

INTRODUCTION

5

The application of our ideas about turbulence to specifically oceanographical contexts has taken place only fairly recently. In 1863, Osborne Reynolds published his celebrated account of laboratory observations on turbulent (‘sinuous’) flow, pointing out the statistical nature of the problem and the need for describing the motion in terms of its average properties. Later, under the stimulus given by the growth of aerodynamics, the mixing length concepts were developed, in which the motion of eddies was likened to that of discrete entities such as the molecules in a gas. A close collaboration between theory and experiment during this time revealed both the success (limited though it was) of these concepts, and more important, their shortcomings. Gradually, it became evident that before any real insight into the dynamics of turbulence could be gained, a more fundamental approach would be needed. This was ultimately given by G. I. Taylor in 1935 in a series of five papers to the Royal Society; from these has developed the modern theory of turbulence. The particular case of homo¬ geneous turbulence allows some analytical simplifications, and at the same time is approximately realizable in laboratory wind tunnels. The penetration of this part of the subject by Batchelor, Kolmogorov, Kraichnan, Townsend and others has consequently been substantial, though the central problem—the statistical mechanics of the non-linear interactions—cannot yet be con¬ sidered solved. None the less, a considerable insight was gained that is relevant and valuable in a consideration of other turbulent motions. At the same time, detailed experimental studies of turbulent shear flows were being made, particularly by Townsend, Laufer and Corrsin. These gave, if nothing else, a greater apprecia¬ tion of the power and limitations of similarity methods in describing such flows. As early as 1915, G. I. Taylor had made astute observations on the structure of atmospheric turbulence, but thirty years were to elapse before suitable instruments were available to allow a detailed and systematic investigation. Since the war, this has been a fertile field for the application and extension of our under¬ standing of turbulence, notable contributions having been made by groups in the Soviet Union and in Australia. Oceanic turbu¬ lence, on the other hand, is less accessible and (with some shining

6

THE DYNAMICS OF THE UPPER OCEAN

exceptions) virtually unexplored. But from a combination of our background experience with homogeneous and with atmospheric turbulence, the use of similarity reasoning together with an appre¬ ciation of the phenomena specific to the ocean (such as the presence of breaking waves) it is possible (in Chapter 6) to make a number of inferences about the structure of oceanic turbulent motions. Some of these can already be compared with direct observations; others, not yet. These three modes of motion do not, of course, occur in isolation; their mutual interactions are of interest. When these are correctly understood, there is hope that we may have some appreciation of the role of the upper ocean in the dynamics of the ocean as a whole. In the following pages, each of these modes is considered in turn. In Chapter 3, the elementary properties of surface waves, their conservation laws and interaction characteristics are derived as simply as possible. These are basic to an understanding of the sometimes complex problems posed by a random field of ocean waves, its generation and decay. The structure of internal waves is then described, the emphasis being not with the analytical details of the mode structure under various particular conditions but rather with the kind of physical processes that may be associated with such motions. Finally, we turn to the mechanisms and phenomena involved in turbulence in the upper ocean and to the interactions between this and the other types of motion. But first, it is necessary to state briefly the relevant governing equations in the forms to be used throughout the book, and to this is the next chapter devoted.

7

CHAPTER 2

THE EQUATIONS OF MOTION 2.1. Specification of the motion A fluid motion is usually specified in one of two ways. The descriptions are of course equivalent, the choice being a matter not of principle but of convenience. In an Eulerian description of the motion, physical quantities such as the velocity u, pressure p and density p are regarded as functions of position x and time t. Thus u = u(x, t) and p = p(x, t) represent the velocity and density of the fluid at prescribed points in spacetime. The partial derivative with respect to x or t represents the gradient in the field at a given instant or the rate of change at a given point. Alternatively, the fluid elements can be identified by their position a at some initial instant t0, and the motion specified by the subse¬ quent position and velocity of these fluid elements. This is usually known as a Lagrangian specification of the motion, the independent variables being the initial co-ordinates a and the elapsed time t —10. ^US

x = x(a ,t — t0);

x(a,o) = a.

(2.1.1)

The velocity of a fluid element is the time derivative of its position: u(a ,t-t0) = jx(a,t-t0).

(2.1.2)

f u(a>

(2.1-3)

so that

t — t0) dt.

JU

The fluid acceleration is the time derivative du/d£ of the velocity of a fluid element as it moves in space. The total time derivative, or the derivative ‘following the motion’, can be expressed in Eulerian terms as

,

^

d, = %+(uV)’

(2.1.4)

the sum of the time rate of change at a fixed point and a convective rate of change. Most of the dynamical problems discussed in this book are posed and solved more simply with an Eulerian specification. Many

THE DYNAMICS OF THE UPPER OCEAN

8

instruments, measuring fluid properties at a fixed point, provide Eulerian information directly. On the other hand, in questions of diffusion or mass transport, the motion of the fluid elements is of interest and a Lagrangian specification of the problem may be more natural. In observation, the marking of fluid elements by dye or other tracer gives Lagrangian information. It is sometimes difficult to relate the two specifications, particularly when the flow is un¬ steady; the general problem of transformation from one to the other in an arbitrary motion has, in fact, not been solved. However, in some particular motions, such as the surface waves described in Chapter 3, the transformation is fairly simple and the Lagrangian characteristics can be inferred readily from the Eulerian solution. In the upper ocean, many of the dynamical phenomena have length scales much smaller than the radius of the earth. Lor these, it is sufficient to use a system of orthogonal Cartesian co-ordinates (x,y,z) fixed with respect to the earth; the ^-axis will be taken vertically upwards. When the notation of Cartesian tensors is used, the co-ordinates xi = (xly x2, x3) will be used interchangeably, with x3 corresponding to z. The velocity components are accordingly («, v, w) or (zq, zz2, u3). Sometimes, it is convenient to consider the horizontal velocity components separately from the vertical one; the notation u = (qa, w) being then used, with the Greek suffix a taking values 1, 2 so that q = (u, v) = (zq, zz2).

..

2 2

The equations of motion

The motion of a fluid is governed by the conservation laws of mass and momentum, by the equation of state and the laws of thermodynamics. The first of these is the conservation of mass, (2.2.1) In virtue of (2.1.4) this can be expressed alternatively as (2.2.2) If the density of a fluid element does not change (though it may differ for different elements), (2.2.1) simplifies to

V .u = o.

(2.2.3)

THE EQUATIONS OF MOTION

9

The momentum equation, referred to axes at rest relative to the rotating earth, takes the form du

_

_

p d7+Pfixu + v/)-Pg = f-

(2.2.4)

The first term represents the mass-acceleration and the second the Coriolis force, in which £2 is the rotation vector, or twice the earth’s angular velocity. Its magnitude £2 = |£2J = 27r/i2h~1 = 1-46 x io~4sec_1, is for the purposes of this book, constant. In the gravitational term, g = (o, o, — g) represents the apparent gravitational acceleration, or the true (central) gravitational acceleration modified by the small (‘centrifugal’) contribution normal to the axis of the earth’s rotation. The direction of g defines the local vertical; its magnitude varies throughout the ocean from its mean value of approximately 981 cm sec-2 by less than 0-3 %, and for dynamical purposes it can be considered constant. The term f on the right of equation (2.2.4) represents the resultant of all other forces acting on unit volume of the fluid. The most important of these arises from the molecular viscosity. In almost all oceanic circumstances where viscous effects are important, the water can be regarded as an isotropic, incompressible Newtonian fluid, and the stress tensor Pv = -p80. Conventionally, p0 is taken as one standard atmosphere. In the upper ocean, the differences between the potential temperature and the actual in situ temperature are small, but not always insignificant. The potential density is defined similarly

ppo,=p~S1,M),sdp' rP = p—\ dp/c2.

(2.2.20)

J Po

Oceanographic density data are often presented in terms of the iuantity

a = io»(p-i),

where p is the relative density of the sea water. The symbol crt represents the value of a at atmospheric pressure and at the in situ temperature. Since this differs only slightly from the potential temperature, the difference between crt and the value of cr found from the potential temperature is very small. 2.3. The mechanical energy equation An equation that describes the balance of mechanical energy is found by forming the scalar product of u with the respective terms of (2.2.9). d Pjt{\u2) + u.SJp-pu.g = M.{. (2.3.1) Now if £ measures the vertical displacement of a fluid element (measured upwards), then -pu.g= pgw = pgd^jdt and with the use of the continuity condition (2.2.1), this equation can be expressed as 0 {\pu2+pgQ + V .{u(p + \pu2+pgQ)-pV. u = u.f. dt

(2.3.2)

THE DYNAMICS OF THE UPPER OCEAN

The rate of change of kinetic and potential energy per unit volume is specified in terms of the divergence of the energy flux F = u (p + ^pu2+pgQ, together with the rates of working in compressing the fluid and against frictional forces. If the fluid is incompressible, V. u = o; if it can be supposed inviscid, f = o. It is noteworthy that the Coriolis forces do no work, since their direction is always normal to the velocity u. They can, however, influence the energy flux indirectly (as in inertial waves) by contributing to the pressure variations in the fluid. In an incompressible Newtonian fluid, f is given by (2.2.7), so that the rate of working against viscous forces is u ji = 2/iut de^/dxj

(2-3-3) where

6

=

ev =

(2-3-4)

since etj is a symmetric tensor. The first term on the right of (2.3.3) can be interpreted as a viscous energy flux. The quantity e is essentially positive; it represents the rate of energy dissipation per unit volume by molecular viscosity. 2.4. The Boussinesq approximation In oceanography, it is convenient to compare the actual oceanic state with a reference state in which the entropy and salinity are constant and the fluid is at rest relative to the rotating earth. In this reference state, the potential temperature and potential density are, of course, constant. The momentum equation (2.2.4) reduces to a simple balance between the vertical pressure gradient and the gravitational force:

0p -^+Prg = 0>

(2-4-0

where the suffices r indicate the properties of the reference state. Since in this state the fluid is isentropic, dpr = c2 dpr, and (2.4.1) gives for the density distribution (2.4.2)

THE EQUATIONS OF MOTION

15

where p0 is the reference density at the free surface z = o and can be taken as the mean oceanic surface density. The density field (2.4.2) is characterized by the local scale height H = c2jg\ in the ocean, i/~ 200 km, considerably greater than the water depth which is at most of the order 5 km. If, in the actual ocean, the variations in pressure on a fluid element are predominantly the result of variations in its depth, then

&P — — Pg dsr. Although the ocean is not isentropic, the entropy of a given fluid element varies only as a result of molecular diffusion and (very near the free surface) of radiative transfer; if, for a first approximation, these are neglected, dp dt From (2.2.1), then

1 dp =

PS.

-S TT = -At w

(2-4-3)

=4 w.

(2.4.4)

7 c\t

V.u

In this equation, the ratio of the term on the right to the term dw/dz on the left is of the order IJH, where lz is the differential length scale of the vertical motion and is necessarily of the order of the oceanic depth at most. This ratio is always negligibly small, so that (2.4.4) can be approximated by the incompressibility condition V.u = o.

(2.4.5)

In almost all oceanic motions, (2.4.5) is an adequate approxima¬ tion to the continuity equation. Even if the pressure gradient is partly balanced by the fluid acceleration, its order of magnitude is usually no larger than the hydrostatic gradient, and this approxima¬ tion can still be made. The one important example to the contrary is afforded by the propagation of sound waves in the ocean, where the pressure fluctuations on a fluid element are associated intimately with its dilatation. Although sound waves in the sea are involved in a number of important measurement techniques, they play little role in the overall dynamics of the upper ocean, and so will not be considered in this book. The hydrostatic pressure gradient of the reference state can be subtracted from the momentum equation (2.2.4), giving />^ + P^xu + v^-(p-Pr)g = f>

(2.4.6)

l6

THE DYNAMICS OF THE UPPER OCEAN

where p — p -pr. Now, in the ocean, the difference between the actual state and the reference state is the result primarily of surface heating or cooling, whose influence is distributed by the processes of convection and mixing. The difference is very small, in the sense that (P~Pr)lpr< i, being of the order io~3 at most. In the inertia terms of (2.4.6), then, it is possible to approximate p by pr. An even greater simplification is in fact possible in the oceans since IJH (2.4.6) reduces to ~ + J2xu + ~V^—^°g = pV2u, at p0 p0

(2.4.7)

where p now represents the difference between the actual and the hydrostatic pressure in an ocean at rest with constant density p0, and v = p/p0 is the kinematic viscosity. The set (2.4.3, 5> 7) are the equations of motion to this, the Boussinesq approximation. The primary conditions for its validity, namely that the actual density distribution differ only slightly from the reference state (2.4.2), that the vertical scale of the motion be small compared with the scale height and that the Mach number of the flow be very small, are all well satisfied in the oceanic motions of interest. In the Boussinesq approximation, variations in the fluid density are neglected in so far as they influence the inertia; variations in the weight (or buoyancy) of the fluid may not be negligible. It is important to recognize that in these equations of motion, the gravitational acceleration and the variations in density occur only in the combination b =

(2-4-8) H0

which describes the buoyancy per unit volume. Equation (2.4.7) can be written , du _ 1 _ { „ -j- + £2xin—Vp — bm = vV2u, (2.4.9)

at

p0

where m is a unit vector vertically upwards, while from (2.4.3)

17

THE EQUATIONS OF MOTION

If the density field be regarded as the sum of a mean density p(z) together with the fluctuation p'(x, t) about this mean, and similarly if b = B(z) + b(x, t), db

then

dt

w.

3z g dp

(A)dz =

+

zv.

-N2w,

say.

(2.4.11)

The Brunt-Vaisala frequency

1

g dp

£2)*

(

g 3ppot)

1

Po dz

c2)

[

Po

dz

1

(2.4.12)

is shown in Chapter 5 to be the natural frequency of oscillation of a vertical column of fluid given a small displacement from its equi¬ librium position. The fluid is statically stable when iV is real, that is, when dpp0t/dz < o. The distribution of N is one of the most important dynamical characteristics of the ocean. The corresponding period 2n/N varies usually from a few minutes in the thermocline to many hoursf in the deep oceans where the water is nearly neutrally stable. In the upper ocean, in situations where the stratification is important at all, g_dp

g2

Po dz so that (2.4.12) can be approximated by

dB)i Po dz and (2.4.3) by

dz I

dp/dt = o.

(2.4.13) (2.4.14)

An alternative form of the momentum equation (2.4.7), corre¬ sponding to (2.2.9) is ~ + (Si + to) x u + V{(p/p0) + iu2} -bm = pV2u.

(2.4.15)

f It is very difficult to measure with any accuracy such large values. 2

PD U

THE DYNAMICS OF THE UPPER OCEAN

18

The curl of (2.4.15) gives for the rate of change of vorticity: ^ = (£2 + to).Vu + V x (6m) + pV2io.

(2.4.16)

The terms on the right describe the generation of vorticity by the stretching of the lines of total vorticity (£2 + to) and by the horizontal variations in buoyancy, together with the diffusion of vorticity by molecular viscosity. When the time scale of a motion is small com¬ pared with both 277-/O (12 h) and 2n/N, the vorticity equation reduces to

, ^ = to.Vu + pV2to, d£

(2.4.17)

the form appropriate to a homogeneous incompressible fluid in an inertial frame of reference. The corresponding momentum equation in terms of the total pressure is ^ + 10 xu + S7(p/p0 + ^ii2+gz) = pV2u.

(2.4.18)

It was indicated in §2.2 that in many oceanic motions, the influence of the viscous term in the momentum equation is quite negligible. In this event, (2.4.17) shows that if, at some initial instant the vorticity of a fluid element is zero, then dco/dt = o and it remains zero. If the vorticity vanishes everywhere in the field of flow, the motion is irrotational, and, in the absence of viscous or buoyancy effects, remains so. In such a flow, since 10 = V x u = o, it follows that u can be represented as the gradient of a scalar function, the velocity potential = o, V2u = o and (2.4.18) becomes ,

P

THE EQUATIONS OF MOTION

19

which can be integrated immediately to give

~+^ + W+g* =f(t),

(2.4.21)

where f(t) is an arbitrary function of time determined by the pressures imposed at the boundaries of the motion. Another form of Bernoulli’s equation can be found for steady frictionless flow, which may be stratified and rotational. It can be shown from (2.4.15) that in these circumstances

P + ml li/2 . Po

Po

g*

is constant along streamlines.

2.5. The Reynolds stresses As in most branches of geophysical fluid mechanics, we are continually confronted in the ocean with motions that vary in a random manner in both space and time. They may be turbulent, or they may be associated with an irregular wave field of one kind or another, but it is the randomness that is their characteristic property. Even under carefully controlled laboratory conditions, it is found that in such motions, the detailed velocity field and its development with time are not reproducible from one experiment to the next, though the experimental conditions are unchanged. Only the average or statistical properties of the motion can be reproduced, so that only these properties can be regarded as physically significant or predictable theoretically. The specification of the averaging process is a matter of some importance. In this book, averages will usually be interpreted as ensemble averages, taken over a large number of independent experiments or observations in which the macroscopic (or observ¬ able) initial and boundary conditions are always the same. With averages defined in this way, it is evident that the operations of averaging and those of addition, differentiation and integration commute. For some purposes, various types of space or time averages will be required; these are noted specifically in the appro¬ priate contexts.

20

THE DYNAMICS OF THE UPPER OCEAN

Though a full discussion is outside the scope of this book, it should be appreciated that the problems involved in making observational approximations to these averages are by no means trivial. To find an ensemble average, the observer must repeat his measurement a large number of times under independent but macroscopically identical conditions. In geophysical measurements, control of the macroscopic environment is rarely possible, and the observer is faced with many problems involving the choice of a meaningful ensemble. These are particularly acute when largescale phenomena are involved, as Stommel (1958) notes in his discussion of the Gulf Stream, but are present to some degree in all field measurements. Kinsman’s (i960) study of the generation of waves on a branch of the Chesapeake Bay is typical of those in which such problems had to be faced. However the averaging process (denoted by an overbar) is defined, it allows a separation of the motion into mean and fluctuating parts. The velocity field can be expressed as U + u, where ti = o. When ensemble means are taken, both U and u may be functions of x, y, z and t. The incompressibility condition is V.(U + u) = o;

(2.5-1)

if this equation is averaged, there results V.U = o for the mean motion and by subtraction of this from (2.5.1), V.u = o for the fluctuating velocity field also. With the same substitution, and with the neglect of molecular viscosity, the momentum equation (2.4.9) can be written

g (U + u) + (U + u). V(U + u) + £2 x (U + u) + V(P+p') -(B + b) m = o,

(2.5.2)

where the total derivative is written out in full and the pressure and buoyancy fields are similarly expressed as the sum of mean and fluctuating parts. The constant density factor p0 has been incor¬ porated into the pressure term:

PlPo = P +P' • The mean of equation (2.5.2) reduces to 0U -^ + U.VU + S2xU + VP-Pm = O,

(2-5-3)

21

THE EQUATIONS OF MOTION

where, in tensor notation, ■u.

du^

3 dxj’

~bx^

=

The ‘Reynolds stress’

(3-1-6)

where y is the ratio of surface tension to water density, Rx and R2 are the principal radii of curvature of the free surface and^>a the atmo¬ spheric pressure. If pa is assigned, these two equations together constitute the second free-surface condition on the motion. An alternative form of (3.1.5) can be found that involves explicitly only p and its derivatives, the fluid velocity. The total derivative d/dt of the Bernoulli equation (2.4.21) is dp

d

+g dz

dt

d2p ^ dt2

u-v!t!(u^ u.v(u*)

o,

which holds in particular at the free surface z — £. Consequently 'd.

P

d t PI J

+

d^p

dj

dt2

g dz j?

+

0U2 dt

+ i[u.Vu2]e = o,

(3-1-?)

an equation used by Longuet-Higgins (1962). If the atmospheric pressure is constant and the influence of surface tension negligible, the first term vanishes. With a prescribed atmospheric pressure and given initial condi¬ tions, these equations suffice to determine the subsequent motion.

THE DYNAMICS OF THE UPPER OCEAN

24

The upper boundary conditions are non-linear, but more seriously, they are specified on the free surface £ which is unknown a priori and which is in general a function of x, y and time t. Very few exact solutions are known, and these are for very particular circumstances, so that some approximation technique is usually required. One method, now widely used in problems where the water depth is not small compared with a typical wavelength, was suggested by Stokes (1847) and involves the use of expansions about the mean water level £ = o. The convergence in the case of steady wave motion was proved by Levi Civita (1925). Convergence in general has not yet been established completely, but it is not unreasonable to anticipate that the expansions will generally con¬ verge provided the expansion parameter (the wave slope) is sufficiently small. The free-surface conditions are expressed as Taylor series expansions about z = o. From (3.1.7) d

(Pa

d t\p

i+4+«2 +

_a2 dz2

0^

+ 0U21

1

dz

dt Jo

dtp

d

+g*z-v*.m dz dt

+ {|u.Vu2+...}0 = o.

(3.1.8)

The success of this expansion procedure depends on the property (which becomes evident a posteriori) that the ratio of a typical non¬ linear term, for example [du2/dt]0 or g£[82£>/8.sr2]0 to a linear term, say [d2p/dt2]0 is in deep water of the order of the wave slope at most, which (for reasons involved with the stability of the free surface) is necessarily rather small. Consequently, the non-linear terms can be regarded as imposing a regular perturbation on the linear solu¬ tion, and an expansion of successive approximations based on the linear problem would be expected to converge towards the solution. If the wave slopes are infinitesimal, the perturbations are negligible, but if they are not very small, higher-order approximations may be necessary. When the waves are about to become unstable and break, the wave slopes approach the order unity, and such an expansion is impractical. So, let us expand (p in powers of an ordering parameter e which subsequently will be found to be of the order of the wave slope. tp = etp1 + e2tp2 + e3tp3+...,

(3.1.9)

THE DYNAMICS OF SURFACE WAVES

25

with corresponding expansions for u and £. The governing equations for each of the

(3-i-i3)

(3-i-i4)

both to be applied at z = o. The next and higher approximations, which contain rather more terms, can be obtained in a similar way.

3.2. Infinitesimal waves In this section, first-order solutions are given for the propagation of waves in water of uniform depth, d. Certain mean properties of the motion such as the energy and momentum density, both of second order, can be found very simply from the first-order solu¬ tions. A sinusoidal disturbance is considered, but this involves no loss of generality since an arbitrary disturbance can, by Fourier’s theorem, be considered as a superposition of elementary waves each of which to the first order propagates independently.

26

THE DYNAMICS OF THE UPPER OCEAN

The motion is specified by (3.1.10) to (3.1.12) with the lower boundary condition d/dz = o at z = —d. If the surface displacement^S

£ = acos(k.x — nt)

(3.2.1)

(the suffices 1 having been dropped), the associated velocity potential is readily found to be na cosh k(z + d)

0

k sinh kd

(3.2.2)

sin (k.x — nt),

where k — |k|. The frequency n is determined by the dynamical free-surface condition (3.1.12). With the atmospheric pressure pa constant, it is found that n2 = gk( 1 + yk2/g)tanhkd = cr2(k), The phase velocity of the wave c = (cr/k) 1 where

say.

1

(3.2.3)

= k/k is a unit

vector in the direction of the wave-number. The group velocity cg = V,.a(k), where Vfc = (d/dkv d/dk2) is the gradient operator in wave-number space. Note that the symbol cr is used to denote the particular frequency associated with wave-number k; n refers to frequency in general. When kd > 1, the waves are said to be in ‘ deep water ’.f The phase velocity then has a minimum when k — (g/y)^, or at a wavelength A = ■zn{yjg)^. In waves shorter than this, the restoring force per unit displacement is predominantly the result of surface tension, and these are usually called capillary waves. Gravity waves are those for which A > 2ir(yjg)^. In clean water at 20 °C, the minimum phase velocity of surface waves is 23 cm sec-1 at a wavelength of 1-7 cm. Most of the energy of ocean waves is found in deep water gravity waves for which (3.2.3) reduces to cr2 = gk. Only the very shortest components are influenced by capillarity but these may contribute significantly to the mean square surface slope (§4.7). In tsunamis and tidal waves, kd may not be large even in the deep oceans and for these waves the water is, in effect, shallow everywhere. In deep water (3.2.2) reduces to (f) =k~1 era ekz sin (k.x —crt), f In practice, this condition is unnecessarily severe. 0-99 < tanhkd < i.

(3.2.4) When

kd >

THE DYNAMICS OF SURFACE WAVES

27

where cr2 = gk(i + yk2/g); it is readily confirmed from (3.1.13) that the quadratic terms in the full surface boundary condition are of the order (ak) times a typical linear term. The mean energy per unit area of the wave motion can be found readily from these expressions. The mean kinetic energy T = \p

u2 d^

exactly,

-d

= ip\° u= ds-

J

correct to second order,

-d

= '°a^coth kd,

(3-2-5)

from (3.2.2). Now, in any conservative dynamical system under¬ going small oscillations, the mean potential and kinetic energies are equal. Consequently, the total energy density is E — {2k)-1 pcr2a2 coth.kd. In gravity waves, such that yk2jg

(3-2.6)

1, this reduces to

E = \pg(P = Pg£2,

(3.2.7)

while in capillary waves, where yk2/g > 1, E = \pyk2a2 - /Oy(V£)2

(3.2.8)

These expressions can be found in a number of alternative ways. In gravity waves, the potential energy density V = ^pgC2 exactly, and this, added to (3.2.5) with cr2 = gkta.nh.kd gives (3.2.7). Also in capillary waves the potential energy arises from the stretching of the surface against the restoring force of surface tension; V = pySA, where 8A is the mean increase in area of surface per unit projected area. So SA = {1 + (VQ2}d — 1 — |(V£)2

to second order.

Hence, V = ipy(VQ2 and the total energy density is twice this. An exact expression for the kinetic energy density that involves only quantities measured at the free surface £ was given by Phillips (1961 a). It is

T = ip(l>^, =

(3-2.9)

correct to the second order,

whence (3.2.5) follows immediately from (3.2.1) and (3.2.2).

28

THE DYNAMICS OF THE UPPER OCEAN

The mean momentum per unit area M is another second-order quantity that can be found simply from the first-order solutions. M=pP qdz = p[ J —d J

Vft0ds

(3.2.10)

d

in irrotational flow. Now, consider the identity

vT ^d» = (v»osst+r v^d*. J

J

—d

d

the two terms on the right arising because £ is a function of hori¬ zontal position. The term on the left is the gradient of an oscillating quantity, whose mean vanishes, while the mean value of the last term is evidently Mjp. Thus M = -/o0cVfc£ = —p d/dx. The vorticity equation simplifies further to da)

d2co

dt

V dz2 ’

(3-4-3)

0w' ^ du'

where

dx

dz'

This equation can be solved readily, subject to the boundary condition (3.4.2) and the requirement that far from the bottom, the first-order motion approaches the irrotational solution and u' -» o. It is found that the horizontal component of the rotational part of the velocity is u' = — aacosech.kdt~Pz' cos(x +

(3-4-4)

where z' = z + d is the distance above the bottom and

y? = (o-/2P)i

(3.4.5)

The flow in the boundary layer is thus oscillatory, and the thickness 8 of the layer is

The ratio 8/ A = (tt

where Rw = cr/vk2

is the wave Reynolds number. The horizontal non-uniformity of the boundary-layer flow induces small compensating vertical velocity fluctuations in the boundary layer which can be found from the continuity condition. w'

=

2“2 craRcosech kd (e_^'[cos (x + (3z') + sin (x + fiz')] - sinx —cos(3.4.6)

THE DYNAMICS OF SURFACE WAVES

35

I he vorticity in the boundary layer is u = (3cr a cosech kd e_^*'{cos (% + /fe') + sin (% + (3z')},

(3.4.7)

neglecting terms of order R~3 relative to this. These solutions will be used later to infer the nature of the second-order mean motion near the bottom. The free-surface layer must be considered in a rather different manner. Since the boundary-layer thickness may be small compared with the wave amplitude, it is hardly satisfactory to apply the surface-boundary conditions at the mean water level z — o. A better alternative is to choose a reference frame moving with the wave and to use a system of orthogonal curvilinear co-ordinates in which the free surface is a co-ordinate line. In this frame, the motion is steady apart from the slow attenuation. Therefore, let r a cosh k(z + d) . 7 5 = v-. , \ j—- sin kx, sinh kd (3-4-8)

a sinh k(z 4- d) v — z-. , \ cos kx. sinh kd

To the first order, the line y = o corresponds to the free surface 2- = o. cos kx, while at the bottom, 7] = ~d. The Jacobian of the transformation is

7 .

V) d(x, z)

2 ak cosh k(jj + d) sinh kd

cos &£,

(3-4-9)

correct to the first order. Since the motion is two-dimensional, a stream function T can be defined, and the irrotational wave motion is given simply by T = —a/, where c is the wave speed. In the surface layer, then, let

T = —cr/ + i/r, where

is the first-order vortical contribution to the

stream function, whose distribution is to be found from the dynamical equations. The vorticity equation in curvilinear co¬ ordinates is given in a number of text-books such as McConnell (1947) and by Benjamin (1959); its form is generally rather com¬ plicated. With the co-ordinates (3.4.8), however, when second-order terms are neglected, it reduces to the simple diffusion equation d

Cd£ = vdrf’

(3.4.10) 3-2

36

THE DYNAMICS OF THE UPPER OCEAN

when d/dr/^> 3/0£ and the vorticity 0) = —

+ ^vv) — —

to

the first order. The nature of the surface layer depends very strongly on the velocity or shear stress condition to be imposed at 7] = o. At one extreme is the situation when the surface is free of any wind stress and is not contaminated by a surface film. The shear stress at the free surface then vanishes: rs = p{(TF;),-(7T^} = o

at

7i = o,

where, as before, the subscripts £ and rj denote differentiation. To the first order, from (3.4.9), this requires that o) = —f/7

= —2ak(rcoskt;

at

7) = o.

The solution to (3.4.10), satisfying this boundary condition and vanishing as rj ->• — 00, is = — 2akerc^ cos(A£ — fiy), where again J3 = (cr/2v)% = (%RW)^ k.

(3.4.11)

This free-surface layer is

evidently very weak; the vorticity is restricted to a layer of thickness /?-1 and in this region the magnitude of the vorticity is no larger than the irrotational rate of strain in the wave motion. Its influence on the overall wave dynamics is insignificant. If, however, the surface is covered by a slick—a thin film of oil or a layer of adsorbed material—the surface layer may be much more intense. Slicks are usually quite flexible, but their reactions to tangential surface strain vary widely. At an extreme, a densely packed layer of adsorbed material or a film of very viscous oil is practically incompressible to the tangential stresses that can be set up by the wave motion. The surface condition at

= o is then

^y = ^y + ^vVy = -c,

which reduces to

\Jr,t = — akc coth kd cos k£,

correct to the first order. The solution of (3.4.10) with this boundary condition leads to a tangential velocity perturbation in the boundary layer of

u'= —era coth. kdePy cos (kg —fir/),

(3.4.12)

and a vorticity field whose magnitude is greater than (3.4.11) by a factor of the order R^. The surface layer is then very similar to the bottom boundary layer in shallow water, though the velocity and vorticity fields are more intense by the factor cosh kd.

THE DYNAMICS OF SURFACE WAVES

37

Wave attenuation In a real fluid, the energy of the motion is gradually dissipated by viscosity. The rate of energy dissipation per unit area is given from (2.3.4) as

v J

E=—

pr

n

/0«.

du\2

W

(3.4.0)

Contributions to this integral arise from the surface and bottom layers and from the straining in the irrotational motion, so that the attenuation rate depends on the value of kd and on whether or not the surface is contaminated. The simplest case is that of deep water (kd^> 1) and a clean surface. The surface layer makes a negligible contribution to (3-4-13) since the rates of strain there are no larger than in the irrotational flow and the layer thickness is small. The energy losses arise almost entirely from the straining of the irrotational motion where u = V0, so that

E = — 2/if

(d2^>/dxidxj)2dz,

correct to the second order. With

0 this reduces to

sin A3

E = — 2fi(T2a2k.

(3-4-H)

Since, from (3.2.6), E =(2k)~xpcr2a2 in deep water, the attenuation coefficient j,, „ 70 yv = — E/2E = 2vk2, (34-I5) where v = ju,/p is the kinematic viscosity of the water. The energy density of the wave field decreases as exp (— 2y„ t) and the ampli¬ tude as exp( — yvt). If there is a surface film, or if the water is sufficiently shallow that sinh2& yv. Short waves incident upon a slick are quite rapidly attenuated, and since the apparent smoothness of the sea surface is very much dependent on the small-scale components, this attenuation evidently accounts for the smooth appearance of oil slicks in the ocean. In shallow water with a clean surface, the energy dissipation occurs mainly at the lower boundary, and the attenuation coefficient is readily shown to be ys = F/?Acosech2 kd. (3.4.19) Induced streaming The existence of a non-vanishing viscosity of the water results in the development of streaming motions—second-order mean velocity fields—which are germane in questions of the mass transport in waves. Several of these effects were discovered by Longuet-Higgins (1953) in a detailed study of the second-order dynamics of waves in a viscous fluid; they are rather analogous to the classical phenomenon of acoustic streaming. In the first place the wave attenuation results in the generation of a mean vorticity field near the free surface. Since the wave momen¬ tum M = E/c, the dissipation of wave energy is accompanied by a decrease in M. It was shown in § 3.2 that in an Eulerian description of the motion, the wave momentum is contained in the region of space above the wave troughs. In the absence of wind, a decrease in the mean momentum of this region must be accompanied by a mean stress S across horizontal planes below the free surface, given

THE DYNAMICS OF SURFACE WAVES

39

Attenuation rate yja

Wavelength A (cm)

X

Fig. 3.1. Viscous attenuation rates of surface waves in deep water with either a clean surface (yv) or in an in extensible surface film (yf). The frequency relation is also given. Capillary waves are to the right of the break, gravity waves to the left.

by S = M = E/c, which is clearly of the second order. But if there is no surface contamination, the velocity potential (3.2.2) is correct to the second order and the first-order vortical flow vanishes beyond the thin surface layer; neither can provide a second-order Reynolds stress. The only possibility is that a second-order mean viscous stress S — — ju,dU/dij is set up to balance the momentum loss. Below the free-surface layer, then, there must be a mean second-order vorticity

W u>(-o) =

drj

E_ lie'

In deep water with a clean surface dUjdTj = &>( — o) = 2 cra2k2,

(3.4.20)

40

THE DYNAMICS OF THE UPPER OCEAN

from (3.4.14). As time goes by, the second-order mean vorticity field diffuses deeper and deeper into the water. A notable aspect of this result is that hi( — o) is independent of the magnitude of v, although the existence of this mean vorticity depends crucially on v being non-zero, however small. Besides exemplifying the singular nature of the viscous perturbation to the equations of motion, it has an interesting physical consequence. A column of marked fluid is distorted both by the mean (Eulerian) velocity gradient (3.4.20) and by the gradient of the drift velocity (3.3.5). Equation (3.3.3) shows that, to this order, the effects are simply additive, and the vertical gradient of the velocity of the marked column is •j—(u + Uj) = 4 (3-4-22)

correct to the second order, where z' = z + d. The mean momentum equation in this region reduces to a balance between viscous and Reynolds stresses:

a _ d2U ^—uw = Voz 8z 2



dU _ _ v — = uw — uw„ oz

so that

(3-4-23)

where —puwm represents the Reynolds stress just outside the boundary layer, and is given by the limiting value of (3.4.22) as fiz' ->

00,

namely

^2/j2£> 4/? sinh2 kd'

(3-4-24)

An integration of (3.4.23) with (3.4.22) gives the mean velocity profile near the bed. U(z)

crcdk 4 sinh2 kd

{3 — 2(/?#' + 2) Z~Pz' COS fiz' — 2 (j3zr — 1) er^smflz' + e-2^2'}.

(3.4.25)

Thus outside the boundary layer the (Eulerian) mean velocity Um in the direction of wave propagation is of the second order, and again is independent of the magnitude of v provided only that it is non-zero.

9,

u-~:fiffljH-

(3'4-26)

42

THE DYNAMICS OF THE UPPER OCEAN

The Lagrangian mean velocity, or the mass transport velocity is (correct to the second order) the sum of Ux and the irrotational expression (3.3.5). Consequently $cra2k ur = 4sinh2 kd'

(34-27)

or f times the inviscid value. This result was established first by Longuet-Higgins (1953).

Fig. 3.2. The mass transport velocity outside the lower boundary layer com¬ pared with measurements by Collins. When W;A > io3 cm3 sec-1 in these experiments, the flow appeared to be intermittently turbulent.

The existence of this forward streaming near the bottom is clearly relevent to questions involving the movement of sediment and sand by wave action. It has been studied in a number of experi¬ mental investigations, notably those of Vincent and Ruellan (1957), Russell and Osorio (1958) and Collins (1963). Fig. 3.2 shows some of Collins’s results. The straight line represents equation (3.4.27) written in terms of the wavelength A and wave period T:

",A

= 57,2 (vriShw)2'

Collins’s measurements agree well with this expression provided a/(Ti sinhkd) < 4-6 in water. If we construct a boundary layer

THE DYNAMICS OF SURFACE WAVES

43

Reynolds number Rs based on the orbital velocity us just outside the layer, this corresponds to Rs = us8/v — 160. Above this value the flow in the boundary layer is turbulent, and although a qualitatively similar streaming phenomenon occurs, the velocities appear to be rather less than those given by (3.4.27). Russell and Osorio found that if the waves broke, the direction of drift near the bottom tended to be reversed. Although this effect is not properly understood, it appears to provide a mechanism for the maintenance of an off-shore sand bar, since opposite drifts are set up, meeting at the top where the waves break.

3.5. The ‘conservation of waves’ The propagation of a wave component is specified by the equation K> — a e1*, where a(x, t) is the local amplitude and y the phase func¬ tion. The propagation of phase points is given by ^(x, t) = const. The wave-number and frequency, which may be slowly varying functions of x and t, can be defined in terms of the phase function: k = Vx,

n = -(3.5.1)

From the first of these, it follows immediately that V x k = o; the wave-number vector in space is irrotational. It also follows that 0k dt

+ Vn

o.

(3-5-2)

This is a kinematical conservation equation for the wave-number. In a single wave train, it can be interpreted as the conservation of the number of waves per unit length, the rate of change balancing the convergence of n, the flux of the number of waves. In a confused sea containing many wave components, (3.5.2) holds for each component, but this simple interpretation is not possible since individual wave crests and troughs are not conserved. The frequency of surface waves is known as a function of k and, in any particular case, of the location x (determining the local depth d). If there is also a mean current U, n — cr(k, x) + k.U,

44

THE DYNAMICS OF THE UPPER OCEAN

which can be regarded as a known function. From (3.5.2) then

l+(c«+u)'vk = _F’

since dkjdxj

(3-5-3)

dkj/dxi and where

3cr

dUj

Fi ~ dXi+kj

3Xi

, . ’

da

(c^“av

(3-5-4)

cg is the group velocity of the wave motion. Equation (3.5.3) states that the rate of change of wave-number k following a point moving with the combined group and convection velocity is equal to — F; if a and U are independent of x then k is constant following such points. These geometrical equations hold for any kind of wave motion; they were given in this context by Ursell (1960) and Whitham (1960). With given initial and boundary values, (3.5.3) specifies the distribu¬ tion of the irrotational vector k, and so the form of the wave pattern at subsequent times. In this chapter we are concerned with gravity waves in regions where d and U vary slowly over distances of the order of a wavelength, so that cr2 = gk tanh kd. The phase velocity is c = l[(g/k)ianhkd]? and the group velocity

~dkt~ H1 + sinh 2kd) ’

^3'5'5^

The contribution to F arising from depth variations is conveniently expressed as Vcr = ^(^2-1) W = icrAfcoth^ —tanh^JW.

(3.5.6)

Some applications of these expressions are given in §3.7 below.

3.6. The dynamical conservation equations In this section, simple expressions are derived for the conserva¬ tion of mass, momentum and energy per unit area when a wave motion is superimposed on a variable current. The energy equation was first given in an approximate form (correct to the second order) by Longuet-Higgins and Stewart (i960, 1961) and the three approximate conservation equations by Whitham (1962).

THE DYNAMICS OF SURFACE WAVES

45

Any geopotential can conveniently be taken as the reference level for potential energy. The presence of currents and waves may well disturb the mean water level, so that £ = h cannot be taken as zero and must be regarded as unknown. The velocity field u can be considered as the sum of its mean and fluctuating parts U + u' where, for algebraic simplicity, U is supposed independent of depth. U may, however, vary horizontally with a length scale large compared with a typical wavelength. An elementary theorem is used repeatedly in this section. If D is a differential operator, then Dp fdz = p Dfdz+^DC+fdBd, J -d

(3.6.1)

J -d

where the suffix £ or d indicates evaluation at the free surface z = £ or the bottom (or an undisturbed streamline) z = —d. Conservation of total mass A vertical integration of the continuity equation (2.2.2) from — d to the free surface gives

dvftTqd^+pK-^] = °>

(3-6-2)

where q = (Ua + uf), (a =1,2) represents the horizontal vectorial component of the velocity field and w the vertical component. The theorem (3.6.1) enables the differential operators to be placed outside the integrals:

T, I dz=IS £ dp

-ddt

P

I

Pdz~P^ = JtP^+d)~pL

V^.pqdxr = VA.

J—d

P

pqdz-pq^.V^-pqd.Vd.

J —d

We also have kinematic conditions at the free surface and at z = - d, Wr

^ + qs.

Wd =

-q a-^d-

(3-6-3)

These expressions can be substituted into (3.6.2) and the resulting equation averaged. This gives for the conservation of total mass per unit area

2 0 ^ ^{p(d+h)} + w{Ma + MJ = o,

(3-6-4)

46

THE DYNAMICS OF THE UPPER OCEAN

where ct, = 1,2. The mass flux per unit width of the mean flow is Ma = f pUadz = p UJfl+A), J -d

(3-6-5)

while the mass transport of the wave motion is

The sum of the two in (3.6.4) gives the total mass flux. Conservation of total momentum The horizontal momentum equation can be expressed as

0

0

0

Jt (PUa) + 0^ (PUa Ufi

^ (PUa «0 =

where a, fl = 1,2. A vertical integration, followed as before by the use of (3.6.1) and (3.6.3), yields

=

-irf)Jpu^+fS^iz+p^a-

(3-6-7)

A

The mean value of the first integral is again Ma + MaJ which is here interpreted as the total horizontal momentum per unit area, the sum of the current momentum and the wave momentum. Since from (3.2.16) pd = pg(d + h), the mean value of the last term is = l-{ipg(d+hy}-pg(d+h)if.

(3.6.8)

The second integral is, in mean, with ua = Ua + u'a, pUa Ufld+h)+ UaMp+ UpMa+ f (pu^u^+pSa/5)dz. J —d

(3.6.9)

This can be expressed conveniently in terms of the momentum flux of a steady stream having the same mass flux Ma = Ma + Ma and mean transport velocity 0„

= XtMd+h) =

U.+MJp(d+h)

(3.6.10)

as the actual flow together with an excess momentum flux arising from the superimposed wave motion. Since pua U,(d+h)+U.Mt+ UfMa = 0a fflp — Ma Mp\p{d + h),

THE DYNAMICS OF SURFACE WAVES

47

the mean value of (3.6.7) can be written

+

= T«’

(3-6.Il)

where s*fi = j_Jpu* u'fi +

dz ~

+ hf 8*p ~

^

(3-6.12)

represents the excess momentum flux that results from the unsteady motion and

T* = -PZ(d + h)faT

(3-6-I3)

is the net horizontal force per unit area arising from the slope of the free surface. The balance of total momentum per unit area of the motion is expressed by equation (3.6.11). Conservation of total energy An equation for the balance of mean total energy can likewise be found by integration of (2.3.2) over the depth of the water. With the use of (3.6.1) and (3.6.3) again, this becomes 0

/»g

0

/» £

gj ^(^l+^ck+g^J _ua{fu\+pgz+p)dz = o.

(3.6.14)

The substitution ut = (Ua + u'a, w') in the first integral leads to

r/

p(\u\+gz)&z = \pU>(d+h)+U„Ma + hMl?-d*) + E

J -d

= iOaM^ + ipg(h*-d>)-

Mi + E, 2 p{d + h)

(3-6.15) from (3.6.10), where E is the energy density of the unsteady motion. The second integral can be treated similarly and there results for its mean value Ua {\pv%i+h)+UfMf + E+ ipgth* -d2)+jfldzj

where

+ \U}Ma+uA^ pu'au'p-+ J_J£

dz + A, {J Jdz- ipg(d+hf + ter2},

(3-6.23) correct to the second order in the wave slope (ak), and where

T*/? = - f

J —d

P((uda K)/?>

(3-6-24)

is the integrated turbulent Reynolds stress. The mean partp of the pressure field is given by (3.2.17), whence —per2 a2

hPg{d + hf

Ch

2 sinh2 k(d + h) J

sinh.2k(z — h + d)dz,

-d

8k sinh2 k(d+h) ^Smh 2k

Xr=K-X~nrt,

r—± 1

the mean square slopes k2raraf of the components are necessarily small for reasons concerned with the stability of the water surface. Consequently, the magnitude of the non-linear terms in an equation such as (3.1.8) is always small compared with the dominant linear terms (being relatively of the order of the root mean square slope). Their influence on a first-order motion is therefore in the nature of a small perturbation. The response of the motion will be small also, unless some phenomenon such as dynamical resonance occurs. To the second order, equation (3.1.8) reduces to

9V dt2

a*

at

dz \ dt2

z — o.

(3-8.i)

In this approximation, it is sufficient to consider the interaction of two primary wave trains whose first-order solutions can be sub¬ stituted on the right of (3.8.1). So let 0 = | 2

e*r0expi(kr.x- oy£)j + 0x(x, z, t),

where 04 = (gkr)i and 0X is the product of the interaction, initially small. From (3.8.1), Cj2i 7\/4\ -0Jr1 +£ -|^ = (M2 - ki • k2) A b2 exp [ifai + X2)] + V* exp [i(A4-;&)]}, where Xr =

(3-8-2)

. x — 04 t and the condition that 0 be real, br = b.*

has been used. Clearly, the dependence of 0X upon spatial position is as exp {i(kx ± k2). x}, and since Laplace’s equation must be satisfied, 0i — 0(0 efc°3 e^o -x, k0 = kj + k 2 >

and from (3.8.2) gj* +gk0 0 = (&i k2-k1. k2) {Zq b2 exp [-i(o4 + cr2) t] + blb'i exp [-i(oq-cr,,) f]}.

65

THE DYNAMICS OF SURFACE WAVES

1 his is the equation of a linear oscillator excited by the second-order forcing term on the right. The response

is also of second order and

bounded unless resonance occurs; that is, unless the conditions k0 ± kx + k2 = o,

or0 ± 01 + 0=

JkJn

dA(k,«) exp {i(k.x-»*)},

(4.1.10)

(see, for example, Cramer’s (1962) book), where the integration is over all wave-number, frequency space. In terms of these FourierStieltjes coefficients, it can be shownf that (x, t0)p(x + r,t0 +1) e_lk-r dr, _ drzr(k, f0)dw#(kT0 + £) dk±dk2

/ ’

N

(4'2'2)

where dkx dk2 = dk represents an element of area in the wavenumber plane. If the wind field is statistically steady, fl(k,

t)

is

independent of the time origin t0. The wave-number, frequency spectrum is the Fourier transform of this with respect to t. II(k,«) = (27r)_1 f J

fi(k, t) tint dt.

(4-2-3)

—00

If the turbulent pressure pattern were rigidly convected with some velocity Uc, the only frequencies present in the wave-number, frequency spectrum (4.2.3) would be those for which n = ± k.Uc, representing two planes in (k, n) space. Since, however, the pattern is evolving as it is being convected, spectral contributions arise from a certain region surrounding this surface; for each k there is a range of frequencies contributing to II (k, n). Suppose, however, that n0 is the frequency at which the spectral density If(k, n) is greatest for a particular wave-number k. The convection velocity Uc(k) can now be defined as the velocity (in the direction of the mean wind) for which n0 = +k.Uc(k). It should be noted that, in this more 6

PDU

82

THE DYNAMICS OF THE UPPER OCEAN

general definition, the convection velocity may be a function of the wave-number concerned. It might be anticipated that Uc is a decreasing function of k, since the larger wave-numbers in the surface pressure distribution are generated by small eddies, close to the surface, which are convected more slowly.

Fig. 4.2. The frequency spectrum of surface pressure fluctuations in the turbulent flow over a flat plate, as measured by Willmarth and Wooldridge (1962). The frequency n is measured in radians per second, t0 is the surface stress, 8 the boundary-layer thickness and Ux the free stream velocity. For the points O, Uao = 156 ft/sec; for •, U^ = 206 ft/sec.

An important property of the spectrum II(k, n) is the integral time scale 0(k) of the pressure components with wave-number k, measured in a frame of reference moving with the convection velocity Uc(k). The wave-number spectrum with time delay, as observed in this frame, is given by (4.2.2) as Q(k, t) exp (ik. Uc f},

83

OCEAN SURFACE WAVES

on putting r = r1 + Uci on the right-hand side. Consequently, the integral time scale in this frame of reference is given by f* CO

0(k, i)

t) exp {i(k.Uc t)}&t =

27rII(k,n0),

from (4.2.3),

— CO

= Q(k,o)0(k),

say,

(4.2.4)

where n0 = k.Uc. In physical terms, 0(k) is a measure of the time interval over which the pressure components with wave-number k remain correlated. Alternatively, from (4.2.4), /*00

2JT 0(k)

D(k.c) n(k,n0)

J_„n(k'")d" n(k,n0) '

, , (4'2'5)

so that 27r/0(k) can be interpreted geometrically as a measure of the spread in the frequency direction of the spectral density II(k, n) about the surface n = n0 = k. Uc. It has already been pointed out that there may arise random pressure fluctuations at the wave-number k from the air flow over wave components with wave-numbers other than k. For example, wave-numbers kx andk2 in the surface configuration may, through inertial effects, jointly induce pressure variations at wave-numbers

kx + k2, 2kj_ + k2, etc. It seems beyond our present wit to calculate the magnitude of the pressure variations induced in this rather indirect way. Fortunately, however, it appears that most of these indirectly induced pressure components are dynamically insigni¬ ficant, and result only in a slight distortion of the wave profiles. The response of the water surface to such pressure components poses a mathematical problem that is closely akin to the one involved in the very weak interactions of gravity waves, described in §3.8, where the only combinations of components that can result in appreciable wave growth are those satisfying resonance conditions (3.8.6). It is not clear in detail how these specific combinations could arise in the air flow over the sea so that tentatively, they will be neglected. The total component of the pressure field can then be taken as the sum of the directly induced part given by (4.2.1) and the random contribution arising from the atmospheric turbulence. The turbu¬ lent pressures provide an energy input over a wide spectral range, 6-2

84

the dynamics of the upper ocean

while the directly induced pressure provides a selective ‘feedback’ that will be found to augment greatly the rate of growth of certain components. In a consideration of wave generation, the weak non¬ linear effects and the influence of vorticity in the water can be neglected, and the motion supposed to be irrotational with linearized free surface conditions. The motion in the water is consequently governed by Laplace’s equation V20 = o together with

PlPw+g£-yVl£+ = o,

]

>

at

z — o.

(4-2.6)

£ = Wd. In deep water, p -> o as z -> — 00. If the surface displacement is specified by

r

£(x,*) =

dA(k, t) e^-*, J k

the velocity potential

M ^ 1 almost invariably,! and the complex frequency is given to sufficient accuracy by N = cr(i+ii/i), (4.2.10) t Though when the surface shear is extremely large, the magnitude of v (which is negative) may become such that gk + yk3 + vc2k2 is negative for a certain range of k, whereupon Kelvin-Helmholtz instability results (see §4.3).

OCEAN SURFACE WAVES

85

where cr — (gk + yk3)b is the frequency of free surface waves with wave-number k. To this approximation, the wave frequency (the real part of N) is unchanged by the coupling with the wind; the imaginary part of N, proportional to the part of the induced pressure m phase with the wave slope, is very relevant to the rate of wave growth. In dynamical terms, this is the part of the induced pressure that supplies energy and momentum to a moving wave. The solution to (4.2.9), subject to the initial conditions cL4(k, t) = 0(k) and at > 1 (though the magnitude of peat > o is not restricted), the wave spectrum is given by =2Pb(Sl^L/)Jln(k,T)C0S that is, when the dominant frequency of the convected pressure fluctuations is equal to the frequency of a free surface wave with the same wave-number. This mechanism was described by Phillips (1957) and clearly involves a type of resonance between the free surface waves and the exciting turbulent pressure fluctuations. If the condition gert < 1 is satisfied by all the wave components at a given frequency cr (and so at a given scalar wave-number k) then at this wave-number one would expect a directional maximum in the polar wave spectrum T(k) = T(A, a) at the angles for which the condition cr — k.Uc = kUccosocm, say, is fulfilled, since at these angles, II(k, cr) is a maximum. For gravity waves am = cos-1 (cj Ue) = cos-1 (g/k u*)i.

(4.2.15)

The magnitude of the wave spectrum at these wave-numbers k = (k,ocm) is, from (4.2.4) and (4.2.14), vm( k, 0

II(k)Q(k)f 2/4, C2

where



(4.2.16)

II(k) = £2(k, o) is the wave-number spectrum of the simul¬ @(k) is the convected

taneous surface pressure fluctuations and

integral time scale. At the scalar wave-number k = g/U%, ocm = o and the wave energy density is greatest for components travelling in the direction of the wind. When k > g/U%, ccm > o; for these shorter waves, the directional distribution becomes bi-modal with the maxima (4.2.16) at angles ± ctm to the wind.

OCEAN SURFACE WAVES

87

But as /icrt approaches and exceeds unity (as will be found to happen quite rapidly for short waves) the induced pressure on the growing wave becomes important; the system develops a ‘feed¬ back . The pressure component in phase with the wave slope increases as the slope does, and continually supplies energy to the wave. The rate of growth becomes more rapid and approaches the exponential form characteristic of an instability: (4-2-I7) This transition from a forced to an unstable mode of growth has important consequences on the spectral shape and the directional distribution of the wave components. The time T = (/^cr)-1 can be taken to mark this transition in a motion starting from rest. Those components for which T is small compared with the wind duration grow much more rapidly then they would under the influence of the atmospheric pressure fluctuations alone. Their directional distribu¬ tion is consequent not only on the properties of II (k, n) but on the directional dependence of ju. It is evident that this parameter is of cardinal importance, and the next section is devoted to its determination.

4.3. The coupling between wind and waves Consider the air flow over a water surface disturbed by waves. Our object is to determine the induced surface pressure and the consequent rate of energy transfer to the moving waves. In 1924, Jeffreys was forced to postulate an expression for this (incorrectly, as it now appears); thirty three more years were to elapse before Miles (1957) would make the first calculation of the induced pressure component in phase with the wave slope. In a later series of papers (1959a,b, 1962a) Miles extended his theory, further contributions being made by Brooke Benjamin (1959, i960) and Lighthill (1962). These developments constituted a great advance but, in all of them, a rather idealized version of the problem was considered. The air flow was regarded as quasi-laminar, atmospheric turbulence being neglected except in so far as it determined the basic mean velocity profile. This allows the problem to be posed and solved analytically, but the neglect of the diffusive effects of the

88

THE DYNAMICS OF THE UPPER OCEAN

turbulence is hardly realistic in this context.f It will in fact appear that the turbulence can have a profound influence on the induced pressure distribution, particularly if the wave speed is comparable with the wind speed. The difficulties introduced by the presence of the atmospheric turbulence are not trivial, but the importance of the problem demands that they be faced if a realistic and reliable theory is to be constructed. The motion will be supposed to be statistically steady in time and homogeneous in the horizontal space variables, at least over intervals large compared with a typical wave period or wavelength. The wavy surface may contain many Fourier components, but we will be concerned with the pressure induced by the flow over one particular component, with wave-number k, say. This can be selected, as in §4.2, by averaging in a frame of reference moving with the velocity c of this component. The v-axis can be chosen in the direction of c, and in this frame the mean surface displacement (0 can be taken as z = {0 = a cos kx, where the operator { ) denotes an average overy, in the direction of the crests of this component. The position of the actual free surface will, of course, fluctuate about (0 because of the presence of the other wave components. Several important points should be mentioned at this stage. First, the overall mean velocity, averaged over both x andy, of the air at points that are always above the free surface is U(z) — c, where U(*) is the mean velocity as measured by an instrument at a fixed spatial point as the waves go by. Near the moving free surface, however, averages at a point cannot be defined (or measured!) so readily since a given Eulerian space point may be sometimes in the air and sometimes under water. Note, however, that if any function is integrated vertically upwards from the free surface, the resultant integral is a function of x, y and t, existing for all values, and so can be averaged in the usual way. Finally, since a periodic wave component has been selected out, the ( >-average values may be periodic functions of x. We must therefore distinguish between overall mean values, averaged over both x and y (or t), functions of z and indicated by an overbar, and the y or < )-averages, which can be functions of both x and z. t Bryant (1966) was the first to consider this question at all seriously, and succeeded in formulating some aspects of the problem.

OCEAN SURFACE WAVES

89

The horizontal components of the momentum equation can be written as ~ „ oua 3 9 ! dp

1i+^u^+rzu‘w

= ~pd^+^u‘’

(«•■)

where a, /? = 1,2 denote components in the horizontal plane. At points well above the water surface, the overall average can be taken, and it follows that the mean stress

(To )a =-pt&o + pdUJdz, (where eIk-x. k

(4-3-6)

90

THE DYNAMICS OF THE UPPER OCEAN

Clearly, rj = o corresponds to the instantaneous free surface z — and as rj -> 00, variables rj and z approach coincidence. Motzfeld’s and Benjamin’s results show that the overall mean properties in the flow at a given value of rj approximate closely to the corresponding properties in the flow past a flat surface with the same mean stress r0. This is not to say that the < )-averaged quantities are the same as in the flow over a flat surface—indeed, it is just the difference that is responsible for the induced pressure field—but it does assure us that such averages are the same in the actual flow in the convected frame (with all the other wave components) as in the flow over the isolated component of interest, in which the others are absent. Our subsequent analysis can then realistically concentrate on this equivalent (and simpler) situation.

Kinematics of the flow As a result of the previous discussion, the free surface (Jf) — a coskx can be regarded as fixed, and above this, the total velocity field can be represented as

where

u = U(T) -c + °U{x, z) + u'{x,y, z, t),

(4-3-7)

= U(*)-c + °ll{x,z),

(4.3.8)

the jy-average of the velocity field, and » = U(*) —c,

(4.3.9)

the overall average. °i/ = (fU, K, W) then represents the waveinduced perturbation and u' = (u,v,w) the random velocity fluctu¬ ations of the atmospheric turbulence. The incompressibility condition is V.u = o; on taking the jy-average of this equation, there results ~{U(z)c°sa~c + ^} + ~^ = 0,

(4-3-io)

where cc is the angle between the mean wind and the direction of wave propagation. This is sufficient to ensure the existence of a stream function for the ()-mean motion such that U cos a — c + °U = dW/dz, 1K=

-dW/dx.

(4-3-11)

OCEAN SURFACE WAVES

91

Since the wave-induced perturbations °ti, iV are periodic in x, the streamlines of the mean motion can be represented as the real part of ^ =

{^(£)cosa — c}d£ + cp(2)eiAa: = const.,

(4.3.12)

where the arbitrary lower limit of the integral is taken as the height zm where U(zm) cos a — c = o, or where the wind speed is just matched by the wave speed. Away from zm, the mean streamlines are merely smooth undula¬ tions on a uniform stream. If

Re{„