Nonlinear Waves

NONLINEAR WAVES

Edited by

LOKENATH DEBNATH Professor and Chairman, Department of Mathematics University of Central Florida, Orlando, Florida and Professor of Mathematics and Adjunct Professor of Physics East Carolina University, Greenville, North Carolina

CAMBRIDGE UNIVERSITY PRESS CAMBRIDGE LONDON NEW YORK NEWROCHELLE MELBOURNE SYDNEY

CAMBRIDGE UNIVERSITY PRESS Cambridge, New York, Melbourne, Madrid, Cape Town, Singapore, Sao Paulo, Delhi 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/9780521254687 © Cambridge University Press 1983 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 1983 This digitally printed version 2008 A catalogue record for this publication is available from the British Library Library of Congress Cataloguing in Publication data LOKENATH DEBNATH NONLINEAR WAVES (Cambridge monographs on mechanics and applied mathematics) Includes index. 1. Nonlinear waves. 2. Fluid dynamics. 3. Plasma (Ionized gases) 4. Inverse scattering transform. 5. Evolution equations. I. Debnath, Lokenath. QA927.N665 1983 531'.1133 83-15102 ISBN 978-0-521-25468-7 hardback ISBN 978-0-521-09304-0 paperback

CONTENTS PREFACE

PART I NONLINEAR WAVES IN FLUIDS 1. Towards the Analytic Description of Overturning Waves MICHAEL S. LONGUET-HIGGINS

1

2.

The Korteweg-de Vries Equation and Related Problems in Water Wave Theory R. S. JOHNSON

25

3.

Solitary Waves in Slowly Varying Environments: Long Nonlinear Waves R. GRIMSHAW

44

4.

Nonlinear Waves in a Channel M.C. SHEN

69

5.

Soliton Behaviour in Models of Baroclinic Instability IRENE M. MOROZ and JOHN BRINDLEY

84

6.

Waves and Wave Groups in Deep Water PETER J. BRYANT

100

7.

Two-and Three-Wave Resonance ALEX D. D. CRAIK

116

PART II NONLINEAR WAVES IN PLASMAS 8.

Nonlinear Electromagnetic Waves in Flowing Plasma N.E. ANDREEV, V. P. SILIN and P. V. SILIN

133

9.

Superluminous Waves in Plasmas P.C. CLEMMOW

162

10.

Electrostatic Ion Cyclotron Waves and Ion Heating in a Magnetic Field H.OKUDA

177

11.

Solitons in Plasma Physics P.K. SHUKLA

197

12.

A Theory for the Propagation of Slowly Varying Nonlinear Waves in a Non-Uniform Plasma R. J. GRIBBEN

221

PART HI SOLITONS, INVERSE SCATTERING TRANSFORM, AND NONLINEAR WAVES IN PHYSICS 13.

On the Inverse Scattering Transform in Two Spatial and One Temporal Dimensions A. S. FOKAS

245

14.

Linear Evolution Equations Associated with Isospectral Evolutions of Differential Operators ANTONIO DEGASPERIS

268

15.

Inverse Scattering for the Matrix Schrbdinger Equation with Non-Hermitian Potential PETER SCHUUR

285

16.

A General nth Order Spectral Transform P. J. CAUDREY

298

17.

Recurrence Phenomena and the Number of Effective Degrees of Freedom in Nonlinear Wave Motions A. THYAGARAJA

308

18.

Action-Angle Variables in the Statistical Mechanics of the Sine-Gordon Field R. K. BULLOUGH, D.J. PILLING and J. TIMONEN

326

INDEX

356

PREFACE The last two decades have produced major advances in the mathematical theory of nonlinear wave phenomena and their applications.

In an effort to

acquaint researchers in applied mathematics, physics, and engineering and to stimulate further research, an NSF-CBMS regional research conference on Nonlinear Waves and Integrable Systems was convened at East Carolina University in June, 1982.

Many distinguished applied mathematicians and sci-

entists from all over the world participated in the conference, and provided a digest of recent developments, open questions, and unsolved problems in this rapidly growing and important field. As a follow-up project, this book has developed from manuscripts submitted by renowned applied mathematicians and scientists who have made important contributions to the subject of nonlinear waves.

This publication

brings together current developments in the theory and applications of nonlinear waves and solitons that are likely to determine fruitful directions for future advanced study and research. The book has been divided into three parts. Waves in Fluids, consists of seven chapters.

Part I, entitled Nonlinear

Nonlinear Waves in Plasmas are

the contents of Part II, which has five chapters.

Part III contains six

chapters on current results and extensions of the inverse scattering transform and of evolution equations.

Included also is recent progress on statis-

tical mechanics of the sine-Gordon field. The opening chapter, by M.S. Longuet-Higgins, is devoted to recent progress in the analytical representation of overturning waves.

Among the forms

suggested for the fluid flow are, for the tip of the jet, a rotating Dirichlet hyperbola, and, for the tube, a "/T-ellipse" or a parametric cubic. these have been expressed in a semi-Lagrangian form.

All

The semi-Lagrangian

form for the rotating hyperbola is derived by a new and simpler method, and certain integral invariants are obtained which have the dimensions of mass, angular momentum and energy.

The relation of these to the previously known

constants of integration is discussed, and directions for further generalizations are indicated.

Also, a new class of polynomial solutions of the semi-

Lagrangian boundary conditions is derived.

These, or their generalizations,

may be of use when combining different solutions so as to form a complete description of the overturning wave.

In Chapter 2, R.S. Johnson describes

how the classical problem of inviscid water waves is used as the vehicle for introducing various forms of the Korteweg-deVries (KdV) and nonlinear Schrodinger (NLS) equations.

The appropriate equations in one and two

PREFACE dimensions are given with some discussion on the effect of shear and variable depth.

It is shown that KdV and NLS equations match in a suitable limit

of parameter space, and the various KdV solutions-notably similarity-are themselves matched to corresponding near fields.

Some other equations based

on more complicated physics are mentioned together with a brief comment on two-dimensional problems with shear or variable depth.

In Chapter 3, R.

Grimshaw discusses canonical equations for the evolution of long nonlinear solitary waves in slowly varying environments. type and include the effects of dissipation.

These equations are of KdV

The slowly varying solitary

wave is constructed as an asymptotic solution of these equations by a multiscale perturbation expansion, and is shown to consist of a solitary wave with slowly varying amplitude and trailing shelf.

The specific case of a solitary

wave decaying due to dissipation is described in detail.

Chapter 4, by

M.C. Shen, is concerned with some approximate equations for the study of nonlinear water waves in a channel of variable cross section.

He gives a sys-

tem of shallow water equations for finite amplitude waves, and a KdV equation with variable coefficients for small amplitude waves. more study are mentioned in this chapter.

Some problems deserving

Chapter 5, by I.M. Moroz and J.

Brindley, is concerned with the derivation of a system of evolution equations for slowly varying amplitude of a baroclinic wave packet.

The self-induced

transparency, sine-Gordon and nonlinear Schrbdinger equations, all of which possess soliton solutions, each arise for different inviscid limits.

The pre-

sence of viscosity, however, alters the form of the evolution equations and changes

the character of the solutions from highly predictable soliton solu-

tions to unpredictable chaotic solutions.

When viscosity is weak, equations

related to the Lorenz attractor equations obtain, while for strong viscosity the Ginzburg-Landau equation obtains.

P.J. Bryant, in Chapter 6, discusses

specific wave geometries which occur in deep water and are calculated by a numerical method based on Fourier transforms.

Examples are presented of

permanent waves and wave groups of permanent envelope in two and three dimensions without restriction on wave height.

Although the method is ap-

plied here only to gravity waves in deep water, it may be generalized to further forms of nonlinear wave motion.

Chapter 7, by Alex Craik, deals with

linear, or direct, resonance of two waves, and weakly nonlinear three-wave resonance.

Special attention is given to non-conservative three-wave sys-

tems, for which the mathematical theory is least developed.

In addition,

subharmonic resonance and further complications involving quadratic interaction of more than three waves are discussed.

PREFACE In Chapter 8, N.E. Andreev, V.P. Silin, and P.V. Silin discuss various aspects of the stationary theory of the interaction of an electromagnetic field with moving plasmas, with special attention to the field self-restriction phenomena in supersonic plasma.

The authors also suggest a direction

for further research and study on the theory.

In Chapter 9, P.C. Clemmow

discusses finite-amplitude plane waves travelling with uniform speed through a cold homogeneous plasma in a Lorentz frame of reference.

This problem can

be reduced to solving a single nonlinear ordinary vector differential equation.

Periodic solutions of this equation are investigated.

It is found

that some new results for propagation in a direction perpendicular to the ambient magnetostatic field go some way towards elucidating the conditions under which various types of wave can exist.

H. Okuda presents the results

of analytic theory as well as of numerical simulations on electrostatic ion cyclotron (EIC) waves in Chapter 10.

In Chapter 11, P.K. Shukla presents

an evaluative review on theories of solitons in plasma physics along with a discussion on some open questions and unsolved problems.

Chapter 12, by

R.J. Gribben, is concerned with uniformly-valid perturbations of uniform, monochromatic nonlinear, periodic wave solutions of the Vlasov and Poisson equations in one space dimension in the absence of a magnetic field.

Also,

a theory for the propagation of slowly varying nonlinear waves in a non-uniform plasma is presented.

Appropriate basic uniform wave solutions are re-

viewed, some general consequences of the theory given, and current work described, including solutions obtained for particular cases, and directions in which further study might proceed. In Chapter 13, A.S. Fokas describes some recent results and developments on the extension of the inverse scattering transform to solve nonlinear evolution equations in one time and two space dimensions.

Based on the

SchrOdinger partial differential operator as a simple mathematical model, A. Degasperis studies linear evolution equations associated with isospectral evolutions of differential operators in Chapter 14.

He also discusses how

to solve the corresponding initial value problem using the spectral properties of the Schrbdinger operator.

Then the scattering operator expression is

divided in the case of a linear evolution equation associated with a pure many-soliton solution. results are pointed out.

Some natural extensions and generalizations of these In Chapter 15, Peter Schuur develops an inverse

scattering formalism for the NxN matrix SchrOdinger equation with arbitrary, in general non-Hermitian potential matrix, decaying sufficiently rapidly for | x | -v oo . A general nth order spectral transform and a technique for inverting this transform are developed by P.J. Caudrey in Chapter 16.

The

PREFACE use of the whole procedure is illustrated by the solution of a system of nonlinear Klein-Gordon equations.

In Chapter 17, A. Thyagaraja gives an

elaborate account of recurrence phenomena and the number of effective degrees of freedom in nonlinear wave motion.

The relationships between re-

currence phenomena and different motions of stability due to Lagrange, Poisson, and Lyapunov are described.

The chapter concludes with a brief

discussion of some unsolved problems relevant to applications.

The final

chapter, by R.K. Bullough, D.J. Pilling, and J. Timonen is devoted to the statistical mechanics of the sine-Gordon (s-G) field.

Functional integrals

for the classical and quantum partition functions Z for the s-G field <J>(x,t) are calculated in different ways including methods which exploit the complete integrability of the classical s-G and its canonical transformation to a Hamiltonian involving action variables alone. poses no problems.

The free Klein-Gordon field

But discrepant results for the s-G kinks and anti-kinks

are explained by the observation that the functional integrals for Z are defined best by discretization to a lattice of spacing a on finite support L. The s-G problem then becomes that of a sequence of problems involving a finite number of degrees of freedom; and for L -> °° and a-> 0 kinks and antikinks are dressed by coupled K-G modes.

These dressings are calculated in

different ways both quantally and in classical limit, and connections established with kinks and anti-kinks are largely resolved, but quantum WKB methods, for example, pose problems of their own. I am grateful to the authors for their cooperation and contributions, and hope that this monograph brings together all of the most important, recent developments in the mathematical theory and physical applications of nonlinear waves and solitons in fluids and plasmas, besides describing all major current research on the inverse scattering transform.

I want the reader to share in

the excitement of present day research in this rapidly growing subject and to become stimulated to explore nonlinear phenomena. I express my grateful thanks to Dr. Carroll A. Webber for his help in improving the readability of several papers.

I am thankful to my wife for

her constant encouragement during the preparation of the book.

In conclu-

sion, I wish to express my sincere thanks to the Cambridge University Press for publishing the monograph. LOKENATH DEBNATH

CHAPTER 1 TOWARDS THE ANALYTIC DESCRIPTION OF OVERTURNING WAVES MICHAEL S. LONGUET-HIGGINS Department of Applied Mathematics and Theoretical Physics, Silver Street, Cambridge, England and Institute of Oceanographic Sciences, Wormley, Godalming, Surrey

1.

INTRODUCTION. Till recently, one notable hiatus in the theory of surface waves was

the absence of any satisfactory analysis to describe an overturning wave. In this category we include both the well-known "plunging breaker" and also any standing or partially reflected wave which produces a symmetric or an asymmetric jet, with particle velocities sometimes much exceeding the linear phase-speed. A first attempt to describe the jet from a two-dimensional standing wave was made by Longuet-Higgins (1972), who introduced the "Dirichlet hyperbola", a flow in which any cross-section of the free surface takes the form of a hyperbola with varying angle between the asymptotes.

Numer-

ical experiments by Mclver and Peregrine (1981) have shown this solution to fit their calculations quite well.

The solution was further analysed

in a second paper (Longuet-Higgins, 1976) where a limiting form, the "Dirichlet

parabola", was shown to be a member of a wider class of self-

similar flows in two and three dimensions.

Using a formalism introduced

by John (1953) for irrotational flows in two dimensions, the author also showed the Dirichlet parabola to be one of a more general class of selfsimilar flows having a time-dependent free surface. All the above flows were gravity-free, that is to say they did not involve g explicitly; they are essentially descriptions of a rapidly evolving flow seen in a frame of reference which itself is in free-fall. A useful advance came with the development of a numerical time-stepping technique for unsteady gravity waves by Longuet-Higgins and Cokelet (1976, 1978).

As later refined and modified by Vinje and Brevig (1981), Mclver

and Peregrine (1981) and others, this has

given accurate and reproducible

results for overturning waves, with which analytic expressions can be compared. A further advance on the analytic front came with the introduction

M.S. LONGUET-HIGGINS

t=O

Figure 1 (after Longuet-Higgins 1980b).

Example of the free surface in a

rotating hyperbolic flow wheniSf • 0.30 (see equation (6.15)). 0 is in a free-fall trajectory.

The origin

ANALYTIC DESCRIPTION OF OVERTURNING WAVES

3

by Longuet-Higgins (1980a) of a general technique for describing free-surface flows, that is flows satisfying the two boundary conditions p = 0 at a free surface.

and

Dp/Dt = 0

(1.1)

Particular attention was paid to the parametric repre-

sentation of the flow in a form X - X(tf,t)

,

z = Z(uS,t)

(1.2)

where both the complex coordinate z = x + iy and the velocity potential x are expressed as functions of the intermediate complex variable w and the time t.

This was a generalisation of the formalism of F. John (1953), in

which a) was, however, assumed to be Lagrangian at the free surface, though not elsewhere in the fluid.

For this reason John's formalism was called

"semi-Lagrangian". The more general formalism was put to immediate use in a second paper (Longuet-Higgins 1980b) in which the "Dirichlet hyperbola" of previous papers was generalised to include "rotating hyperbolic flow".

Besides the

time-variation of the asymptotes, the principal axes were allowed to rotate, as shown in Figure 1.

This solution, in which the velocity potential x

was given in closed form, allowed for the first time a convincing possible description of the later stages of a plunging jet. of the jet, however, was not included.

The initial evolution

In a third paper (Longuet-Higgins

1981a) the author made use of the more general (non-Johnian) formalism to derive a plausible analytic description of the development of the flow, up to about the instant when the free surface first becomes vertical.

This

description introduced the approximate potential X = j ig tf3 + Uto2 + 2Aw

(1.3)

where U is a constant, A is a linear function of the time and z = o>2

.

(1.4)

The first term on the right of (1.3) by itself represents Stokesfs 120° corner-flow.

The third term represents a finite-amplitude perturbation of

the Stokes flow.

The expression (1.3) gives a rather convincing represen-

tation of the initial development of the breaking wave (see Figure 2 ) . However, the task of matching this flow to the later stages, including the time-dependent jet, remains still to be accomplished. In another direction New (1981) found empirically that in some of his

M.S. LONGUET-HIGGINS

Figure 2 (from Longuet-Higgins 1981).

Initial development of the over-

turning flow as given by equations (1.3) and (1.4) when g - 1, U • (-1,0.5), and A(t) is chosen so as to minimise /(Dp/Dt)2ds on p = 0. is in uniform motion; the solution includes gravity.

The origin 0

ANALYTIC DESCRIPTION OF OVERTURNING WAVES

5

numerical calculations of breaking waves the forward face, or "tube", of the breaker was remarkably well fitted by part of the circumference of an ellipse, with axes in the ratio /3:1.

Whereupon Longuet-Higgins (1981b,

1982) pointed out that the free surface was even better fitted (see Figure 3) by the cubic curve: z = itw3 + 3t2a)2 - it3w - j t ^

(1.5)

which is a limiting case of one of the self-similar flows found previously (Longuet-Higgins 1976).

Moreover the flow (1.5) contains another surface

p = 0 which comes close to the rear surface of the wave, though the second boundary condition Dp/Dt = 0 is not satisfied on it.

Nevertheless there

was perhaps some possibility that by suitably perturbing the flow (1.5) and by matching it to a rotating hyperbolic flow near the tip of the jet a complete solution might be found.

Since (1.5) is expressed in semi-

Lagrangian form a next step would be to express the rotating hyperbolic flow in semi-Lagrangian form also. This has in fact been done in a very recent paper (Longuet-Higgins 1983) where the rotating hyperbolic flow is shown to be expressible in the form z = F(t)cosho) + G(t)sinha)

(1.6)

the functions F and G being related to the solutions of a Kiccati equation. The corresponding particle trajectories have also been computed. Meanwhile in still unpublished work New (1983) has succeeded in finding a flow, in semi-Lagrangian representation, which is outside his elliptical free surface, and he has shown that the velocity field resembles that in numerically calculated waves, over about half the circumference of the ellipse.

Unlike the cubic (1.5), the elliptical model cannot of course de-

scribe the velocity discontinuity which must occur when the jet meets the forward face of the wave.

Greenhow (1983) has made further progress in

deriving a semi-Lagrangian expression, polynomial in a), which for large values of t approximates the hyperbolic jet on the one hand and New's ellipse on the other.

His expression also provides a "rear face" to the wave,

but is still gravity-free. The purpose of the present paper is twofold:

first, to derive the

semi-Lagrangian representation for the rotating hyperbolic flow in an alternative, and perhaps simpler, way than in Longuet-Higgins (1983).

The

present method has the advantage that it brings to light naturally some

M.S.

LONGUET-HTGGINS

101

(b)

Figure 3 (after Longuet-Higgins 1982).

Profile of the surfaces p = 0 in

the cubic flow (1.4)(a) when t = 1.0; (b) when t = 0.5. both p and Dp/Dt vanish.

On II only p vanishes.

indicates a possible perturbation.

On the curves I,

The broken curve III

ANALYTIC DESCRIPTION OF OVERTURNING WAVES

7

integral invariants K, y, and v which in turn provide constraints on the functions F and G.

It is shown how K, y, and v are related to the constants

of integration found in earlier papers.

Moreover the method suggests some

possible generalizations. A second purpose is to give some exact polynomial solutions to one of the problems investigated by Greenhow (1983).

The same methods may, in

turn, be employed in other, more general, problems occuring in the same context. 2.

SEMI-LAGRANGIAN COORDINATES. In the semi-Lagrangian representation of irrotational, free-surface

flows in two dimensions, the coordinate z - x + iy is expressed as an analytic function of a complex parameter w and the time t: z • z(o),t)

(2.1)

such that on the free surface w is real (w = w*) and Lagrangian (Dw/Dt • 0) . The condition that the pressure be constant along this surface can then be expressed as ztt - g = i « u

(2.2)

where g denotes gravity (the x-axis being vertically downwards) and r(w,t) is some function that must be real when a) is real.

If gravity is negli-

gible, or if the motion is viewed in a free-fall reference frame, then (2.2) reduces to z

tt -

±r

(2

\

-3)

In the interior of the fluid, the coordinate w is generally not Lagrangian, and the velocity is given by z (a>*,t), which of course equals z (o),t) on the boundary.

The vanishing of the derivative z

singularity in the flow field, unless at the same point [z vanishes also, hence z* (w,t) = 0.

implies a (to*,t)]*

In other words

OJt

z

- 0

U)

implies

z*_ = 0

(2.4)

U)t

everywhere in the interior. When equations (2.3) and (2.4) are satisfied we can, if necessary, find a velocity potential x(w>fc) throughout the fluid by calculating X(o),t) - / z*(w,t) zaj(o),t)do) for then

(2.5)

M.S. LONGUET-HIGGINS

Z

(2

t

'6)

as required. 3.

ROTATING HYPERBOLIC FLOW. As a very simple form of solution suppose that z - au> - bto

(3.1)

where a(t) and b(t) are some functions of the time, to be determined.

This

will satisfy equation (2.3) with r = o)R(t) provided a and R is real.

a) — b

(A)

— iR(aoo + bo) ) •

(3.2)

Also from equation (2.4) the two equations a + bw

=0

j

, * -2 a* + b*a)

=0

l

(3 3)

'

J

are to be satisified simultaneously, if the corresponding point z(ca,t) is to lie within the fluid. Starting from equations (3.2) and (3.3) we shall deduce a chain of results leading eventually to a differential equation for the unknown function R(t). On equating coefficients of GO and OJ

in equation (3.2) we have

iRa (3.4) -iRb where R is not necessarily real.

On eliminating R from these two equations

we get ab t t + a tt b - 0 .

(3.5)

Again, on eliminating a) from equations (3.3) we have ab* - a* b = 0 .

(3.6)

From (3.6) and its complex conjugate there follows (ab* - a*b) t - 0 ,

(3.7)

hence ab* - a*b =

constant

= IK

(3.8)

ANALYTIC DESCRIPTION OF OVERTURNING WAVES

9

say, where K is real. Next, if we differentiate (3.6) with respect to t and use equations (3.4), we get (atb* - a*bfc) + iR* (ab* + a*b) = 0

*

i(a b

(3.9)

it

- a b )

ab

+ a b

* Clearly R = R , so R is real, as required. t - 2a t b t

(4.8)

so by (4.1) (ab) t t = 2A* 2 (ab)*

.

(4.9)

Comparing (4.7) and (4.9) we see that

2ab = -v

1— (AA ) 2

.

(4.10)

(ab* - a*b) 2 » -K 2

(4.11)

Now on squaring each side of (3.8) we have

and so 4 a b a V - (ab* + a*b)2 = K2 .

(4.12)

ANALYTIC DESCRIPTION OF OVERTURNING WAVES

11

Using (4.3) and (4.10) we find * A A

.

— ^ (AAV say.

2

1;—— (AA )

2

=P v

(4.13)

2

Hence

AfcA* - (AA*)2 = P ( A A V .

(4.14)

This equation involves both the amplitude and phase of A, whereas R(t) by (4.5) involves only the amplitude.

To eliminate the phase (or rather, its

rate of change with time), we note that by (3.10) and (4.1)

say.

A a b - A*a*b* = iy

(4.15)

(i/v)(A*a*b* - Aab) = y/v = X

(4.16)

From (4.10) this implies that * A A

* - AA

a (^*)2

=x

(4 17

->

that is i (A*Afc - AA*) = X(AA*) 2 .

(4.18)

A = aeia

(4.19)

Now write

where a and a are real.

Then (4.14) and (4.18) yield respectively (a2 + a 2 a 2 ) - ak = Pa 8

(4.20)

a2afc = \ak .

(4.21)

and

Eliminating a

from these two equations we obtain a 2 = 0^(1 - X 2 a 2 + Pa 4 )

an equation for a alone.

(4.22)

Since by (4.5) R = (R(t) as in Section 3

and write (8.1) then (2.3) becomes

ANALYTIC DESCRIPTION OF OVERTURNING WAVES

19

then (2.3) becomes ztt = i r '

(8.2)

V

where rf - ^ R

.

(8.3)

Substituting for R from equation (6.17) we find in the limit as 73* + 0

5

(8.4)

If we choose the time origin so that t- = 0 and scale t so as to make A - 1, hence T = t, then for large values of t equation (8.4) becomes 1

(8.5)

(1 + t2 very nearly.

As pointed out by Greenhow (1983) this is essentially the

same function r(oa,t) as occurs in the representation of New's ellipse (New 1983).

Accordingly, any simple solutions of the boundary problem

tt are of particular interest.

(1 +

t2)2

(8.6)

0)

For, by combining such solutions linearly we

may be able to find a flow which incorporates the desirable features both of the rotating hyperbolic flow (for the tip of the jet) and New's ellipse (for the forward face of the wave). We shall now derive some simple solutions of equation (8.6). (a)

Circular functions of a)

Writing z

= eik(Vt)

(8.7)

in (8.6) we have

*tt

=

k

(i + t 2 ) 2

•

(8.8)

of which a solution is • - (1 + t 2 )

i Y arctan

(8.9)

provided Y2 = 1 - k

.

Hence if k ^ 1 we have solutions of the form

(8.10)

20

M.S. LONGUET-HIGGINS

]

(8.11)

where M and N are arbitrary constants and 0 = arctan t

.

(8.12)

In particular when k = 0, equation (8.11) becomes z = (1 + t2)4

(Me 10 + Ne" 1 9 )

(8.13)

and when k = -1 z = e- iw (l +

t

2 ) % (Me i / 2 Q + N e - i / 2 0 ) .

(8.14)

As k -»• 1 we find in the limit z = e ^ d + t 2 ) ^ (Mf + iN f 0)

(8.15)

where M1 and Nf are other constants; and when k - -2 z = e" 2lu (l +

t

2 ) % (Me l / 3 e + N e - 1 / 3 0 ) .

(8.16)

Similarly for values of k > 1 we have in general

[

^ 1 ^ + Ne"*-1^ |

(8.17)

and in particular when k = 2 z =

e

2ia)

(l +

t

2 ) % (Me0 + Ne" 9 ) .

(8.18)

New's ellipse corresponds to (8.11) with

(b)

Exponential functions of m

Writing k - -im

(8,20)

in (8.11) we have

[ Me i(1 + im)^9 + Ne- 1(1+ lm)%Q ] in which the free surface (w real) tends to infinity as mw -> °°.

(8.21) By adding

two such solutions, one with a positive value of m and one with a negative value, we get a solution with the boundary tending to infinity in two directions.

21

ANALYTIC DESCRIPTION OF OVERNTURNING WAVES (c)

Polynomials in o), t and 0

Writing (8.11) in the form z = eika)(l

N

.] <s.

22)

where M and N are arbitrary constants (not the same as before), let us differentiate with respect to the parameter ik and then set k = 0, remembering that (1 + t 2 ) ^ cos0= 1,

(1 + t 2 ) ^ sin0=* t.

(8.23)

Then we get a new solution z

- M(a) - %Lt0) + N(u)t + %i0).

(8.24)

Similarly if we differentiate (8.11) twice and set k = 0 we get z o = M[o)2 - it0u) - ^(t0 - 0 2 )]

N[ta)2 + lew + h(0 + t0 2 )]

(8.25)

and in general NQn(t,0,aj)

(8.26)

where _ P

n

n . n—1 , , . = a a ) + a nw + ... + a, a) + a n n-1 1 o n

(8.27)

11 1

Q = b o) + b -w " + ... + b-o) + b n n n-1 1 c

and

n!

R (t) (8.28)

n! n

with

"" m ) •

22

M.S.

R

o

LONGUET-HIGGINS

= 1

\

» te R0

= - -H02 - te)

L

2. • (8.29)

30 2 - 3t0)

R = (G>I+

4 A

" 6tG)3 "" 15G>2 + 15t0)

~ 45t03 " 105°2 + 1O5t0) etc., the (I + l)th coefficient in the series for R being m . 2~l

(m + I - 1 ) ! (m - I - 1) !

(8.30)

In the series for S (t), we simply replace 0 P by t0 P and t0 P by - 0 P in the series for R (t). m Finally we note that there exist solutions to (8.6) obtainable from (8.11) by differentiation with respect to k, in which k is not set equal to zero after differentiation.

These will in general have a spiral form.

To obtain physically valid flows, the above solutions must be combined in such a way as to satisfy equation (2.4), at least for large t. Such a procedure has been carried out by Greenhow (1983) in the case (8.31)

r = -

(namely the first two terms in the expansion of (8.5)) and it would be possible in principle to carry this process further. 9.

FURTHER SUGGESTIONS. We have shown how previous solutions, each representing part of the

flow in an overturning wave, can be represented very simply in semiLagrangian form, in particular the rotating hyperbola (for the tip of the wave) and the /3-ellipse (for the tube).

In the first instance we have

also found three integral invariants of the motion. By combining the methods of Sections 3 to 7 with that of Section 8 it may be possible to further generalize these solutions so as to obtain a rather complete representation of the flow, in semi-Lagrangian form. On the other hand it must be emphasised that all solutions to the homogeneous boundary condition (2.3) that do not contain g are essentially

ANALYTIC DESCRIPTION OF OVERTURNING WAVES

23

gravity-free, and can give only a local picture of the flow over a limited duration of time.

So far, the most promising solution incorporating

gravity has been that given by equations (1.3) and (1.4) above (see Figure 2) . This solution, however, is not in John's semi-Lagrangian form (the free surface does not correspond to u> real) but is in the more general parametric form suggested by Longuet-Higgins (1980a).

Quite probably it will prove

necessary to make use of the greater flexibility of the general formulation to represent the complete wave. One feature which the solution of equations (L.3) and (1.4) has in common with the semi-Lagrangian flow (1.5) is the existence of a branch-point, given by z

= 0 , outside the fluid domain.

It is this feature which can account

for the discontinuity in fluid velocity where the jet meets the forward face of the wave.

The reader will take note that the former solution (1.3),

(1.4) though approximate, is in some ways simpler that the cubic (1.5), since the velocity potential x corresponding to (1.5) is actually of the sixth degree in o>, as can be seen from (2.5).

Hence the further general-

isation of (1.3) and (1.4) may yet prove to be the most promising line for future investigation. REFERENCES JOHN, R. (1953) Two-Dimensional Potential Flows with a Free Boundary, Comm. Pure App. Math. £, 497-503. GREENHOW, M. (1983) Free Surface Flows Related to Breaking Waves, J. Fluid Mech. (in press). LONGUET-HIGGINS, M.S. (1972) A Class of Exact, Time-Dependent, Free-Surface Flows, J. Fluid Mech. 55, 529-543. LONGUET-HIGGINS, M.S. (1976) Self-Similar, Time-Dependent Flows with a Free Surface, J. Fluid Mech. 21» 603-620. LONGUET-HIGGINS, M.S. (1980a) A Technique for Time-Dependent, Free-Surface Flows, Proc. R. Soc. Lond. A.371, 441-451. LONGUET-HIGGINS, M.S. (1980b) On the Forming of Sharp Corners at a Free Surface, Proc. R. Soc. Lond. A.371, 453-478. LONGUET-HIGGINS, M.S. (1981a) On the Overturning of Gravity Waves, Proc. R. Soc. Lond. A.376, 377-400. LONGUET-HIGGINS, M.S. (1981b) A Parametric Flow for Breaking Waves, Proc. Symp. on Hydrodynamics in Ocean Eng., Trondheim, Norway, August 1981, pp. 121-135. LONGUET-HIGGINS, M.S. (1982) Parametric Solutions for Breaking Waves, J. Fluid Mech. 121, 403-424. LONGUET-HIGGINS, M.S. (1983) Rotating Hyperbolic Flow: Particle Trajectories and Parametric Representation, Quart. J. Mech. Appl. Math. 36, 247-270.

24

M.S. LONGUET-HIGGINS

LONGUET-HIGGINS, M.S. and COKELET, E.D. (1976) The Deformation of Steep Surface Waves on Water. I. A Numerical Method of Computation, Proc. R. Soc. Lond. A.350, 1-26. LONGUET-HIGGINS, M.S. and COKELET, E.D. (1978) The Deformation of Steep Surface Waves on Water. II. Growth of Normal-Mode Instabilities. Proc. R. Soc. Lond. A.364, 1-28. McIVER, P. and PEREGRINE, D.H. (1981) Comparisions of Numerical and Analytical Results for Waves that are Starting to Break, Proc. Symp. on Hydrodynamics in Ocean Eng., Trondheim, Norway, August 1981, pp.203215. NEW, A.L. (1981) Breaking Waves in Water of Finite Depth. Proc. Brit. Theor. Mech. Colloq., Univ. of Bradford, England, April 1981 (Abstract only). NEW, A.L. (1983) An Elliptic Class of Free Surface Flows, J. Fluid Mech. (in press). VINJE, T. and BREVIG, P. (1981) Breaking Waves on Finite Water Depths: a Numerical Study, Ship. Res. Inst., Trondheim, Norway, Rep. R-111.81; see also Adv. Water Resources 4, 77-82.

CHAPTER 2 THE KORTEWEG-DE VRIES EQUATION AND RELATED PROBLEMS IN WATER WAVE THEORY R.S. JOHNSON School of Mathematics The University, Newcastle upon Tyne NE1 7RU, United Kingdom

1.

INTRODUCTION. The last decade and more has produced an altogether unlooked-for im-

petus in the study of certain partial differential equations by use of the inverse scattering transform.

Two of the (now) standard equations which

are susceptible to this technique arise quite naturally in the study of water waves:

the Korteweg-de Vries (KdV) equation and the nonlinear Schro-

dinger (NLS) equation.

This suggests the possibility that there exist

other equations of a similar character which are also relevant in water wave theory.

The similarity may merely be that the equations are generali-

sations (more terms, variable coefficients, etc.) which convert the problem into a non-integrable one.

On the other hand a conceivable result is that

we generate other integrable equations which are extensions of the classical equations to different - possibly higher dimensional - co-ordinate systems.

The overall picture is that of a number of diverse equations which

describe various aspects of the same underlying problem.

This has the

virtue that we can more readily compare and contrast the equations, and in some cases specifically relate one to another. In this paper we shall collect together many of the varied forms of KdV (and to a lesser extent NLS) equations which arise in water wave theory. To emphasise the connecting themes the same variables and parameters will be used throughout.

We shall start from an appropriate set of basic equa-

tions and thence develop both KdV and NLS equations for one spatial dimension.

Since both equations describe alternative aspects of the same problem

(by employing different limits in parameter space), it should be possible to match these two equations:

this is readily demonstrated.

We then turn

to the two-dimensional problems which correspond to both the KdV and NLS equations.

Some properties of the relevant similarity solutions in one and

two dimensions are mentioned, together with matching to the near-field, i.e.

to initial data.

Finally, we briefly comment on a few other more

26

R.S. JOHNSON

involved equations of KdV-type which describe the role of other physical processes (such as viscous dissipation)• 2.

BASIC EQUATIONS. The fluid is assumed to be both incompressible and inviscid (although

not necessarily irrotational), bounded above by a free surface and below by a rigid surface.

The fluid is supposed to extend to infinity in all

horizontal directions so that, for example, the effects of beaches are ignored.

The free surface is characterised as a surface of constant pressure.

The appropriate equations are then non-dimensionalised by using an ambient depth (d), a typical wave length (A) and the acceleration of gravity.

The

non-dimensional equations and boundary conditions for an irrotational flow, with a flat horizontal surface below, are 4

+ 62V^(j) = 0; (j) - 0

on

z = 0;

6)22]] - 0, \ (V6)

V±±n)] n)] 62[nt +a(v±) • (V

I

, ))

(2.1)

on z - 1 + an.

The vertical coordinate measured up from the bottom of the fluid is z, and the free surface is at z = 1 + ar\ where a is an amplitude parameter.

The

gradient operator perpendicular to z is represented by V , and the ratio of vertical to horizontal length scales is given by 6 = d/X. In addition to equations (2.1), we shall also require the equations (in one horizontal dimension only) which describe the perturbation of a main shear flow and which include variable depth.

If the flow has a velo-

city distribution u - U(z), moving in the x-direction, with a bottom topography given by z = b(x), then = 0 ; + (U + au)u + w(Uf + au ) + p t X Z X 2 6 [ w + (U + a u ) w + a w w ] + p = 0 ; x t X Z Z u + w = 0 ; with w/u = b f (x) on z = b(x),

u

and

P - ri, w = r| + ( U + au)r|

on

(2.2)

z = 1 + ari»

The prime denotes a derivative, p is the pressure and the velocity components are (U + au, aw). Equations (2.2) are therefore essentially the rotational counterparts of (2.1), being based on Euler's equation rather than Laplace's equation. Now, by suitable choice of the limits of the parameters and corresponding scaled variables, we may construct asymptotic expansions which

KdV EQUATION AND PROBLEMS IN WATER WAVE THEORY yield the KdV or NLS equations to leading order.

27

Other phenomena can be

incorporated by adjusting the size of the new parameter with respect to a and/or 6. 3.

ONE DIMENSIONAL PROBLEMS:

KdV.

The simplest theory for the KdV equation, which harks back to the 2 derivation of Korteweg and de Vries (1895), is obtained by choosing 6 =0(a) as a •> 0. §6). V

(More general scalings, valid for arbitrary 6, will be given in

For one-dimensional irrotational flow we use equations (2.1) with

= (3/3x,O); the wave is taken to be propagating in the positive x-direc-

tion only.

If we introduce £ = x - t,

x - at,

(3.1)

and expand both $ and r) in powers of a, then to leading order we obtain 2

i

?

= o.

(3.2)

This is the classical KdV equation, the solution of which exhibits the well-known soliton behavior. Corresponding equations can also be derived if the flow incorporates a basic shear (in the x-direction), or there is an appropriate slow variation in the ambient depth.

For both these analyses it is either necessary

or convenient to use equations (2.2).

In the case of the propagation over

a shear flow we take b = 0, i.e. the bottom is flat and horizontal.

The

characteristic and long time variables take the same form as in (3.1), and so we set £ = x - ct,

T

where c is a speed to be determined.

f

1

to leading order. speeds c:

(a -»• 0)

The expansion used abbve now yields

-2 [U(z) - c]

J

- at

Z

dz = 1,

0

This is the Burns integral (Burns, 1953) defining the

it is assumed here that U(z) ^ c for 0 £ z £ 1, and hence there

is no critical layer in the fluid.

At the next order we obtain the KdV

equation which now takes the form -2I 3 n T + 3I 4 nru + (62/a)J r W r - 0 where I n

-

(3.3)

f1 J

0

[W(z)]

-n

dz; J =

W 2 (y) y — dx dy dz; W - U(z) - c, J J J 0 z 0 for(z)fcr(x) f

(see Freeman and Johnson, 1970).

1

(3.4)

^

9

The special case of a solitary wave mov-

ing over a shear was given by Benjamin (1962).

The effect of a shear - at

28

R.S. JOHNSON

least in the absence of a critical layer - is merely to alter the numerical coefficients in the KdV equation.

It is clear that if we set U(z) = 0,

then c = ±1 and we recover equation (3.2) (with c = 1 ) . The case that arises when a critical layer is present can also be discussed.

Velthuizen and van Wijngaarden (1969) argue that the Burns inte-

gral (3.3), must be interpreted as a Cauchy principal value (with a corresponding contribution to the imaginary part of c ) . Using their approach it is then fairly straightforward to obtain (3.4) again, provided all the integrals are taken to mean principal values.

(A discussion of large ampli-

tude waves over a shear flow in the presence of a critical layer is given in Varley et al. 1977.) The problem of wave propagation over variable depth can be examined in terms of aKdV theory if the bottom varies slowly, specifically on the scale a.

If we note that the characteristic variable is to imply a varia-

tion in propagation speed, then we write 5 * g(x;a) - t,

X - ax

with b - b(X).

(3.5)

The leading order then serves to define g, g = I [D^x)]" 1 * dx,

D - 1 - b(X),

and the next order yields the dominant equation for r),

v

D?

^

^ = 0,

where the prime denotes a derivative with respect to X.

(3.6)

Equation (3.6) is

a variable coefficient KdV equation (Kakutani 1971, Johnson 1973a), which is often re-written under the transformation n

2H

2 = D Z ( X )H(x,S),

f x

=

J

0

V D 2 dX, D 2 dX,

+ 3HH r + 4(6 /a)H r r r - - f ~

H.

(3.7)

(3.8)

This equation predicts the soliton fission that occurs as a solitary wave moves into a shelving region (Madsen and Mei 1969, Tappert and Zabusky 1971; Johnson 1972b); see Figure 1.

The equation has also been used as the basis

for a discussion of the effects of a perturbation to the KdV equation, i.e. D'/D small.

This can be accomplished either by direct methods (Johnson,

1973b) or more satisfactorily via the inverse scattering transform (Kaup and Newell 1978; Karpman and Maslov 1978; Candler and Johnson 1981).

In

particular the phenomenon of the shelf that appears behind the solitary wave is now well-understood (Knickerbocker and Newell 1980).

29

KdV EQUATION AND PROBLEMS IN WATER WAVE THEORY

i

O-OI

D 71 ;. 1 1 1 1

r\ : / \:

10-

/ v / :'\ / * \ •

/ .•vyo

• 1 f

'•; ft / / /

I \

/ I // \\

1 • /

0-

-40 A « Figure 1. The geometry of the shelf (which changes between X=0, 0.01) and the two-soliton formation (do = 0.614) as predicted by equation (3.6). Initial solitary wave( ) ; solution at X=0.075 (••••); solution at X=0.25 ( ) ; solution at X = 0 . 5 ( — ) . Before we leave the classical KdV problem in one dimension, it is of interest to relate it to the so-called Boussinesq equation.

This single

equation, correct at 0(a), accommodates waves moving in both directions. From equations (2.1), with V

0(a), a direct expansion

= (9/3x,O) and 6

in powers of a yields a -* 0; n

n,

- n-,

lxx where 0 ot

+ 2n

ltt

2

+ n

2

0;

oxx - n ott

+ n n

- 20

n

+

1 2 T(5

/ct)n

fxn 0,

where c.c. denotes the complex conjugate and 5 - x - cpt,

C = a(x - c g t),

T

= a 2 t.

(4.1)

The expansion for r\ (and <j>) is so constructed that it be periodic (to all orders) in £, whence higher order terms must contain higher harmonics generated by the nonlinear coupling.

The carrier wave moves at the phase speed

(c ) and the amplitude modulation moves at the corresponding group speed (c ) , although the specific forms of c ,c O

C

respectively.

are not assumed a priori.

From

CO

equations (2.1), with V 2 p

= (3/8x,O), the leading and next order give

tanh 6k • — I k " '

1 ,. 26k °g " 2 C p ( 1 + sinh 26k >'

„ ON '2)

(4

At the very next order, which incorporates cubic nonlinear-

ity, we obtain -2ikc

AT + q A

+ r A|A|2 = 0

(4.3)

where q and r are involved functions of 6k (see Hasimoto and Ono 1972). Equation (4.3) is the NLS equation for water waves, but it has appeared earlier in a more general formulation given by Benney and Newell in 1967. The coefficient r(6k) has a zero at 6k = a

~ 1.363, and it is well-known

that the Stokes wave is then unstable if 6k > a (Benjamin and Feir 1967). o This suggests that the nature of the Benjamin-Feir instability could be examined via a suitable generalisation of (4.3) valid near r = 0.

It is

clear that new scales are now required since we can anticipate the appearance of quintic nonlinearity. Thus we set C = a 2 (x - c t), x = a t p and obtain the equation

with

6k = a o

KdV EQUATION AND PROBLEMS IN WATER WAVE THEORY 2iAT + a j A ^ + a2|A|2A^ + a^ A ( | A | 2 ) ^ + a4A|A|4 + where i|; - |A|

(see Johnson 1977).

&5

31

A* T = 0

(4.4)

Equations (4.3) and (4.4) can be com-

bined to yield an equation which is uniformly valid for all 6k, but with coefficients a

(i = 2,...,5) known only on 6k = o .

The inclusion of a background shear, U(z), in the x-direction follows rather similar lines to the corresponding problem for the KdV equation, However, the new coefficients are immensely more complicated and involve, in particular, the solution to a second order ordinary differential equation.

Let V(z) be the solution of (V f /W 2 ) f - (6k) 2 (V/W2) = 0

(W - U - c ) P

(4.5)

with V(l) = 1; V 1 (0) = 0, then using exactly the same expansion and variables as for the standard NLS equation, we obtain rl 2 (WI1) dz - 1

dz - 1,

c = c -I—

=

1

(4.6)

to the leading and next order, respectively, where I(z) -

VW

Jo

dz

Equations (4.6) serve to define c that c

= d(kc )/dk.

and

W

1

= U(l) - c .

P

and c , and it is comforting to find

The NLS equation is recovered at the next order as

8

2ik W (1 + W 1

f1 1 Jo

IIfWfdz)A

T

+ qA

where q,r are given in Johnson (1976).

^

+ rA|A|

2

- 0,

(4.7)

The condition for the onset of

Benjamin-Feir instability is still r = 0, although this is now a functional of U(z). 5.

MATCHING OF KdV AND NLS. Both the KdV and NLS equations are derived under the assumption that

a + 0; however, 6 is held fixed for the NLS equation whereas 6 -> 0 for the KdV equation.

This suggests that it should be possible to match these two

equations in the limit:

NLS as 6 -»• 0 with KdV as 6 •> °°.

We shall demon-

strate that this is indeed the case and, further, we shall apply the method to the equations valid for arbitrary shear, (3.4) and (4.7).

In the case

of the KdV equation, we must employ an appropriate expansion which allows a modulated carrier wave as the solution.

XAx, and write

Z - (5 + UAT)/A,

Let us therefore introduce

T = T/A,

A = 62/a

32

R.S. JOHNSON

11

~Z^Z 00

in equation (3.4), and

OO

+ c.c.

At successive orders we obtain JkZ __ 3Jk2 y _ A - - — , \i - - -fi—

(I ) 2 21 I 3 A ol . + 3kJA Q i e $ - | - A - - A o l |A o l | 2 = 0.

(5.1)

The NLS equation, (4.7), is to be approximated for 6 -• 0 which involves estimating the coefficients in this limit.

So, for example, we use

V(z) ~ 1 - (6k) 2 f f W2(x)w"2(y)dydx; c - c - -(6k) 2 y- , whence

eventually we obtain 21 6 2 I 3 A T + 36 4 kJA^ - | — £ j - A|A|2 = 0.

It is clear that (5.1) and (5.2) match, i.e. k

(5.2) 2

= A, since T=6 f, C=62£.

For more details of the matching procedure, see Freeman and Davey (1975) and Johnson (1976). 6.

TWO DIMENSIONAL PROBLEMS: KdV. The various equations which describe propagation on the whole surface,

rather than on the line, will be developed for irrotational flow over a flat horizontal surface. Thus we deal here only with equations (2.1), although the special cases of nearly plane waves over a variable depth and ring waves over a shear will be mentioned briefly in §10. The KdV equations which apply for nearly plane waves, concentric waves and nearly concentric waves will be given in terms of general scalings valid for arbitrary 6, as a •> 0. The nearly plane (or 2-D) KdV requires V

= O/8x,8/9y) with

? - (o*/6)(x - t); Y = (a/6)y; T - (a3/2/6)t; $ = (ofV6)cf> and then, as a -»• 0, we obtain to leading order

(2n + 3nn +

3-n

+ n Y Y - 0.

(6.1)

This reduces to equation (3.2), upon one integration, if r| is independent 2 of Y and we re-define £> T with 6 = 0(a). The equation was first suggested by Kadomtsev and Petviashvili (1970) and its inverse scattering representation was given by Dryuma (1974).

The oblique solitary wave interaction

solution was given by Satsuma (1976), and Miles (1977a) has shown in detail how the equation becomes relevant as oblique waves become nearly parallel. The equation also exhibits resonant soliton solutions (see Miles 1977b, Anker and Freeman 1978).

KdV EQUATION AND PROBLEMS IN WATER WAVE THEORY

33

The problem of concentric waves on the surface is a little more involved since the amplitude decays as the radius increases.

I f V = (8/8r,O)

then we define £ « (a/6)2(r - t); R = (a6/64)r; $ = <j>/a; H = (62/a3)T?

(6.2)

4 2 and the equations (2.1) now contain the single parameter A = a /6 • As A -> 0, the leading order problem for H yields 2H R + | H + 3HH ? + j H ^ =

0.

(6.3)

This is the so-called cylindrical KdV equation which was first derived in another context by Maxon and Viecelli (1974).

The inverse scattering trans-

form for (6.3) involves a linearly increasing potential which generates eigenfunctions based on the Airy function (see Calogero and Degasperis 1978).

A discussion of the solutions of both the nearly plane and concen-

tric KdV equations is to be found in Freeman (1980). As the classical KdV equation has a two-dimensional counterpart (in the nearly plane equation), so does the concentric equation. V

We introduce

E O/^r, r^a/aS) and use the variables (6.2) together with 6 = Q/&^

whence, as A -> 0, the dominant equation defining H(£,R,G) becomes iH + 3HH ? + iH^) 5 + i I H e e =0. R

(6.4)

It is clear that as (6.1) corresponds to (3.2), so (6.4) corresponds to (6.3).

There is, at present, no inverse scattering transform for this

equation when written in all three independent variables (£,R,O), although some special transformations will be mentioned below which reduce (6.4) to a standard problem but in one few variables.

One simple interpretation of

this new equation is that it applies to a small angular sector (of 0(A 2 )) over which the wave may vary i n 0 :

since the wave is nearly straight in a

small angular region it is nearly plane, and in this sense it is analogous to (6.1). The three equations developed above take differing forms depending on whether cartesian or polar coordinates are employed.

It is therefore worth

investigating the possibility that transformations exist between the equations.

Thus, for example, we have that 2 r - t ~ x(l + j ^ ) ~ t x 2 h

= (6/a ) (5 + £ 2 _ )

as y/x + 0

34

R.S. JOHNSON

1 2 which suggests that if n = n(T,£ + j Y / T ) in (6.1) then we would obtain (6.3) (after one integration).

This is easily confirmed (if we interpret

T as R ) . Similarly, if we set H = H(T,£ - ~ 10 ) in (6.4), then we recover the classical KdV equation, (3.2) (again, after one integration).

The

transformation from the nearly plane (2-D) equation to the cylindrical equation can also be used in a generalised form to construct the inverse scattering transform for this latter equation (see Johnson 1979).

Since we

have shown that the nearly concentric equation can be transformed into the KdV equation, (3.2), it follows that a limited class of solutions is now available to equation (6.4). 7.

TWO DIMENSIONAL PROBLEMS:

NLS.

The nearly plane KdV equation was obtained by allowing the variation in the y-direction to be slow (or weak); the same approach can be adopted for the NLS equation.

From equations (2.1), using V

= (3/8x, 8/9y), with

variables (4.1) and Y = ay, we now let A = A ( £ , Y , T ) (see §4). The expressions for c and c are unaltered (see (4.2)), and for A we obtain the P g coupled system -2ikcp A T + q A ? c - c ^ ^

= r A|A|2

o

(1- V * C C

+

*YY

where q,r,s,$ are involved functions of 6k (see Davey and Stewartson 1974; also Benney and Roskes 1969).

Equations (7.1) recover the NLS equation

(4.3) upon the assumption that there is no dependence on Y.

Solutions of

(7.1) in the long wave length limit (6 -*• 0) are discussed by Anker and Freeman (1978), and the deep water limit discussed by Lake et al. (1977) corresponds to 6 •> °° (in (7.1) and (4.3)). 8.

SIMILARITY SOLUTIONS:

KdV.

In more recent years the studies in inverse scattering theory have brought to prominence the role of Painleve equations generated by seeking similarity solutions of the evolution equations.

In particular both the

classical and concentric KdV equations possess similarity solutions of the 2 For the plane KdV equation (3.2), with 6 = a (or from

appropriate form.

(6.1) in the absence of any dependence on Y) we set

n = i(2/x) 2/3 F(X), x = c/(2x) 1/3 whence V" - XV - V 3 = 0, where

/ T F '- F

« V and V + 0 (exponentially).

(8.1) Equation (8.1) is a

35

KdV EQUATION AND PROBLEMS IN WATER WAVE THEORY

Painleve of the second kind (P-2) and its solutions are discussed at some length by Berezin and Karpman (1964) and Rosales (1978). (8.1) which decays exponentially as X •+• +°° grows like |x|

The solution of as X -* -°° (see

Figure 2 ) .

-10

Figure 2. Similarity solutions, F(X). For the concentric KdV equation: A - - 8 ( — ) ; A - 0 with F(O)=1 (•••). For the plane KdV equation with F(0)=l( ). The concentric KdV equation (6.3) gives rise to a far more acceptable similarity solution.

First we introduce

H - (2/(3R 2 )) 1/3 F(X),

X = £/(2R) 1 / 3

and so FF" - j F | 2 + 2(F 3 - XF 2 ) = where A is an arbitrary constant.

(8.2)

If, as above, we seek the solution for

which F -* 0 (exponentially), then A = 0 and we can write v" - Xv + v 3 = 0,

F - v2.

(8.3)

This is another P-2 equation, simply related to (8.1) by an elementary (although non-real) transformation (i.e. V = iv). Equation (8.3), which has been discussed by Ablowitz and Segur (1977), Miles (1978b), Rosales (1978), has a solution which decays like |X| ^ as X •*• -°° and might therefore describe a realizable phenomenon.

A particularly simple exact solution

to equation (8.2) can be obtained by working from the nearly plane KdV equation and then using the transformation between (6.1) and (6.3) (for details, see Johnson and Thompson 1978, Johnson 1979); this yeilds

36

R.S. JOHNSON F(X)

2 r°° 2 i y [£a( A 2 dX) ], dX^ Jx 1

which corresponds to A = -8 (see (8.2)).

(8.4)

Solution (8.4) decays algebraical-

ly as X •> +°°; both (8.4) and the solution of (8.3) are depicted in Figure 2. This special closed-form solution is directly related to one of the set of similarity solutions found by Airault (1979) for the plane KdV equation. 9.

MATCHING TO A NEAR-FIELD:

KdV.

The various KdV equations introduced here, together with the few solutions mentioned above, are all to be interpreted in terms of water waves. Specifically, we can examine the appropriate near-field of these equations and thereby derive the general form of the initial value problem for each KdV equation.

In the case of the similarity solutions it is also possible

to find the precise nature of a near-field which will match the far-field, although a neighbourhood of the origin must be excluded. The nearly plane (or plane) KdV equation can be examined as T -»• 0 by introducing the near-field variables X = (a^/6)x,

T = (a**/6)t,

Y = (a/6)y,

whence

n T T - n ^ = o, to leading order.

Thus for right-travelling waves, we have

n ~ f(x - T,Y), where f is an arbitrary function, and matching from the far-field is therefore possible if n + f (£,Y)

as

t+0.

(The plane wave corresponds to excluding the dependence on Y.) The concentric KdV equation can be discussed in a similar vein, where we now use the near-field variables ft = (a/6)2r,

T = (a/6)2t,

h = (6/a)n,

which gives rise to RR

R

R

to leading order as A -> 0 (see §6). The outwards moving wave is described by h ~ R

2

f (R - T)

as

R •* °°,

KdV EQUATION AND PROBLEMS IN WATER WAVE THEORY

where f is arbitrary.

37

Again, matching is possible if H ~ R-"1 f (£)

as

R -> 0.

The corresponding problem for the nearly concentric equation is less straightforward, and since it requires rather special initial data (which turn out to be barely realistic) we shall not pursue the analysis here: for more information, see Johnson (1980). Of rather more interest is the manner in which the similarity solutions can be matched to an appropriate near-field.

In particular, for both

A = 0,-8 in equation (8.2) (from the concentric KdV), we are able to match through a linear dispersive wave region to a linear concentric wave and finally recover the full equations (2.1) in a neighborhood of the origin (in r - t space).

The similar problem with A = 0, but for small amplitude

solutions of (8.3), has been considered by Miles (1978a).

The dispersive

wave region is most conveniently expressed in terms of a multiple-scale representation with

p = A- 3n R, R

(i X




anc

*

are

obtained

In the case of A ~ -8, we obtain

2 2 ** f°° -± 1 3 H ~ ±j ( — ) k 2 cos(kx + 7 p k - 7T/4)dk, 77

A + 0,

JO

(x > 0) ,

'

with the same ordering as above, and this solution matches to the similarity solution described by (8.4). The above solutions can be matched, in turn, to a linear concentric wave solution.

It we introduce R = A" n R,

T = A~ n T

then we obtain | -j I sin(kT) JQ(kR)dk

(A = 0; R - T > 0)

and H -

~3 ( sin(kT) Y (kR)dk (A - -8;R - T 0), * '0 A close correspondence between our two similarity solutions is now very evident:

Jn

°

they each make use of one of the available solutions of Bessel's

38

R.S. JOHNSON

equation which is generated by the concentric wave equation. tion to — < n < —

The restric-

is necessary to obtain the correct ordering of the terms

as A •* 0; if n = -r , then the scalings used in defining R,T recover the full water-wave equations with all terms of comparable size.

The solution

of this latter set can not be determined and so the complete matching to an initial value problem (at T = 0) is not possible. If we adopt the same prescription for the similarity solution of the plane KdV equation (see (8.1)), then it is fairly easy to see that there is no regime where a linear problem is relevant.

It turns out that any

allowable scaling always recovers the original KdV equation or, when n = y as above, the full governing equations given in (2.1) but devoid of any parameters.

This full problem applies in an 0(a 2 ) neighbourhood of the

origin (in x - t space). 10.

SOME OTHER EQUATIONS OF KdV TYPE. To conclude this discussion of KdV equations, in the context of water

waves, we shall mention a few other versions which can be obtained by either incorporating new physical effects or extending still further some of our earlier work.

We start with the important property of dissipation by

the viscous stresses in the fluid.

For a relatively thin layer, where the

whole flow can be regarded as totally immersed in a boundary layer, the dominant gradients occur across the wave front giving rise to an equation of the form

This is the Korteweg-de Vries-Burgers equation, with properties reminiscent of those for the two underlying classical equations.

Equation (10.1) also

pertains for thicker layers when the surface is subjected to a wind-loading represented by Jeffreys sheltering model (Jeffreys 1925). An alternative approach to the role of the viscous terms is to assume that a thin boundary layer is formed below the wave-front and extending behind it.

If the basic flow is a fully developed viscous profile (as it

must be to avoid any inconsistences) then in a suitable parameter range, we obtain -2I Q n

3 T+

31, nnr + Jr\rcc = -o

4 5

5SS

Jo n5r (5

+ C O

V %r T

.

(10.2)

Here, as in (10.1), 0 represents the appropriate non-dimensional viscous coefficient.

In (10.2), we have used the notation introduced for equation

(3.4), but now the integrals are to be evaluated only for the Poiseuille

KdV EQUATION AND PROBLEMS IN WATER WAVE THEORY

profile.

39

Both equations (10.1) and (10.2) have very similar looking solu-

tions which adequately model the undular bore, the crucial point being that both include a dissipative term and it does not really matter the form that this takes.

More details are given in Byatt-Smith (1971a), Johnson

(1972a), Pfirsch and Sudan (1971). Higher order generalisations of the KdV equation can also be derived. For example, if surface tension effects are included then an equation of the fourth order is possible (10 3)

2nT + 3nn5 + 3 n K ? - ^^£>¥ in a special parameter range.

'

The constant W is a function of both the

surface tension coefficient and kinematic viscosity (see Byatt-Smith, 1971b), This same equation also describes the propagation of a wave on a surface which is composed of a material with density proportional to W.

(Such a

model is particularly relevant in cylindrical geometries for waves propagating through a liquid-filled flexible tube; equation (10.3) is again obtained.)

We can note that the inclusion of surface tension in the classi-

cal KdV problem merely alters the coefficient of the third order term: the original work of Korteweg and de Vries (1895).

see

Of course, by judicious

choice of the relative sizes of the various parameters, we can also produce equations wich combine the additional term in (10.3) with those in (10.1) or (10.2). Finally, we turn to two problems which are extensions of the work presented in §§3,6.

First we describe the nearly plane (2-D) KdV equation

over variable depth: from equations (2.1), with V = (3/8x, d/3y), we introduce x = (a*/5)x,

y - (a/6)y,

T - (a*/6)t,

* - (a*/6).

The bottom topography (introduced in equations (2.2)) is now written as z - b(X,Y);

X - C K , Y = ay,

whence with 5 = a

v

g(X,Y) - T;

g(X,Y) =

r D"

J

we obtain

2

dX,

D(X,Y) = 1 - b,

0 2

to leading order as a -> 0.

This can be expressed more conveniently as

(2D^ n x + \ v~h Dxn + 3D" 1 nn£ + \ D ng££)£ + D n ^ = o, (io

40

R.S. JOHNSON

where

y = y +f\

1 J

D^(

o

J

o

D"J/Z D y dX) dX . Y

Equation (10.4) may be compared with (3.6) and (6.1); the variables used in (10.4) agree with those required for general wave front propagation as given by Varley and Cumberbatch (1965). Our last example addresses the problem of how a shear flow distorts a ring wave:

this is therefore a combination of the ideas embodied in equa-

tions (3.4), (6.3) and (6.4).

If we use cylindrical polar coordinates, and

define C= (a/6)2[h(6)r - t], R = (a6/64)r, then the leading order (as a /6

2

2

[h + (h') ] f J

with 6 = 0(1),

-> 0) implies that

[(U - c){h cos 6 - h 1 sin 6} - I ] " 2 dz • 1,

(10.5)

0

where U(z) is the shear profile.

Here, c is a free parameter (by virtue of

the Galilean invariance) and the integral condition, which is a generalisation of Burns1 (1953) result, can be interpreted in terms of wave-front propagation in an inhomogeneous medium.

The solution, h(9), of (10.5),

describes the dominant distortion of the wave-front from circular (which itself corresponds to h E 1, U E c ) . The next order yields the equation for the surface profile which takes the form

a

+

A

T

H+

T He + a4 HHc + a5 HCCC = °'

(10 6)

'

where a. = a.(0) are complicated functions related to (10.5) (c.f. equations (3.3) and (3.4)).

The properties of (10.5) and (10.6) are currently under

investigation, but some novel features are already evident.

For example, if

U(z) > 1, then equation (10.5) predicts a critical layer but only for some 6 in a neighbourhood of the 'back1 of the ring wave.

The critical layer

moves off the bottom surface, rises to a maximum height and then drops down again, as 9 varies.

Furthermore, the usual singularity found in the trans-

verse velocity component for a wave moving obliquely over a shear is removed by the curvature effects.

The presence of the critical layer, for certain

6, is transmitted to the surface wave through the functions a.(0).

We may

note, however, that in the context of inverse scattering theory, there would seem to be very little immediate possibility of solving equation (10.6).

KdV EQUATION AND PROBLEMS IN WATER WAVE THEORY

41

REFERENCES ABLOWITZ, M.J. and SEGUR, H. (1977) Asymptotic solutions of the K dV equation, Stud. Appl. Math. 57_, 13-44. AIRAULT, H. (1979) Rational solutions of Painleve" equations, Stud. Appl. Math. 6^, 31-53. ANKER, D.A. and FREEMAN, N.C. (1978) Interpretation of three-soliton interactions in terms of resonant triads, J. Fluid Mech. 87, 17-31. BENJAMIN, T.B. (1962) The solitary wave on a stream with an arbitrary distribution of vorticity, J. Fluid Mech. 12, 97-116. BENJAMIN, T.B. and FEIR, J.E. (1967) The disintegration of wave trains on deep water, Part I: Theory, Jo Fluid Mech. 27, 417-430. BENNEY, D.J. and NEWELL, A.C. (1967) The propagation of nonlinear wave envelopes, J. Math. Phys. 46, 133-139. BENNEY, D.J. and ROSKES, G.J. (1969) Wave instabilities, Stud. Appl. Math. 48, 377-385. BEREZIN, Y.A. and KARPMAN, V.I. (1964) Theory of nonstationary finite-amplitude waves in a low-density plasma, Sov. Phys. JETP 19, 1265-1271. BURNS, J.C. (1953) Long waves in running water, Proc. Camb. Phil. Soc. 49, 695-706. BYATT-SMITH, J.G.B. (1971a) The effect of laminar viscosity on the solution of the undular bore, J. Fluid Mech. 48, 33-40. BYATT-SMITH, J.G.B. (1971b) Waves on a thin film of viscous liquid, AICHE J. 17, 557-561. CALOGERO, F. and DEGASPERIS, A. (1978) Solution by the spectral-transform method of a nonlinear evolution equation (including the cylindrical K dV equation), Lett. Nuovo Cim. 23, 150-153. CANDLER, S. and JOHNSON, R.S. (1981) On the asymptotic solution of the perturbed K dV equation, Phys. Lett. A86, 337-340. DAVEY, A. and STEWARTSON, K. (1974) On three-dimensional packets of surface waves, Proc. R. Soc. Lond. A338, 101-110. DRYUMA, V.S. (1974) Analytic solution of the two-dimensional K dV equation, Sov. Phys. JETP Lett. 1£, 387-388. FREEMAN, N.C. (1980) Soliton interactions in two dimensions, Adv. Appl. Mech. 2£, 1-37. FREEMAN, N.C. and DAVEY, A. (1975) On the evolution of packets of long surface waves, Proc. R. Soc. Lond. A344, 427-433. FREEMAN, N.C. and JOHNSON, R.S. (1970) Shallow water waves on shear flows, J. Fluid Mech. 42. 401-409.

42

R.S. JOHNSON

HASIMOTO, H. and ONO, H. (1972) Nonlinear modulation of gravity waves, J. Phys. Soc. Japan 33, 805-811. HIROTA, R. (1973) Exact N-soliton solutions of the wave equation of long waves in shallow water, J. Math. Phys. 14, 810-814. JEFFREYS, H. (1925) On the formation of water waves by wind, Proc. Roy. Soc. Lond. A107, 189-205. JOHNSON, R.S. (1972a) Shallow water waves on a viscous fluid - the undular bore, Phys. Fluids 11, 1693-1699. JOHNSON, R.S. (1972b) Some numerical solutions of a variable-coefficient Korteweg-de Vries equation, J. Fluid Mech. 54, 81-91. JOHNSON, R.S. (1973a) On the development of a solitary wave moving over an uneven bottom, Proc. Camb. Phil. Soc. 73, 183-203. JOHNSON, R.S. (1973b) On an asymptotic solution of the Korteweg-de Vries equation with slowly varying coefficients, J. Fluid Mech. 60, 813-824. JOHNSON, R.S. (1976) On the modulation of water waves on shear flows, Proc. R. Soc. Lond. A347, 537-546. JOHNSON, R.S. (1977) On the modulation of water waves in the neighborhood of kh « 1.363, Proc. R. Soc. Lond. A357, 131-141. JOHNSON, R.S. (1979) On the inverse scattering transform, the cylindrical Korteweg-de Vries equation and similarity solutions, Phys. Lett. A72, 197-199. JOHNSON, R.S. (1980) Water waves and Korteweg-de Vries equations, J. Fluid Mech. £7, 701-719. JOHNSON, R.S. and THOMPSON, S. (1978) A solution of the inverse scattering problem for the K-P equation, Phys. Lett. A66, 279-281. KADOMTSEV, B.B. and PETVIASHVILI, V.I. (1970) The stability of solitary waves in weakly dispersing media, Soviet Phys. Dokl. 15, 539-541. KAKUTANI, T. (1971) Effect of an uneven bottom on gravity waves, J. Phys. Soc. Japan 30, 272-276. KARPMAN, V.I. and MASLOV, E.M. (1978) Structure of tails produced under the action of perturbations on solitons, Sbv. Phys. JETP 48, 252-259. KAUP, D.J. and NEWELL, A.C. (1978) Solitons as particles, oscillators, and in slowly changing media: a singular perturbation theory, Proc. Roy. Soc. Lond. A361, 413-446. KNICKERBOCKER, C.J. and NEWELL, A.C. (1980) Shelves and the K dV equation, J. Fluid Mech. 9£, 803-818. K0RTEWEG, D.J. and de VRIES, G. (1895) On the change of form of long waves advancing in a rectangular canal, Phil. Mag. 39, 422-443.

KdV EQUATION AND PROBLEMS IN WATER WAVE THEORY

43

LAKE, B.M., YUEN, H.C., RUNDGALDIER, H. and FERGUSON, W.E. (1977) Nonlinear deep-water waves: theory and experiment, Part 2, J. Fluid Mech. 83, 49-74. MADSEN, O.S. and MEI, C.C. (1969) The transformation of a solitary wave over an uneven bottom, J. Fluid Mech. 39, 781-791. MAXON, S. and VIECELLI, J. (1974) Cylindrical solitons, Phys. Fluids 17, 1614-1616. MILES, J.W. (1977a) Obliquely interacting solitary waves, J. Fluid Mech. 79_, 157-170. MILES, J.W. (1977b) Resonantly interacting solitary waves, J. Fluid Mech. 79, 171-180. MILES, J.W. (1978a) An axisymmetric Boussinesq wave, J. Fluid Mech. 84, 181-192. MILES, J.W. (1978b) On the second Painleve transcendent, Proc. Roy. Soc. Lond. A361, 277-291. PFIRSCH, D. and SUDAN, R.N. (1971) Conditions for the existence of shocklike solutions of the K dV equation with dissipation, Phys. Fluids 14, 1033-1035. ROSALES, R. (1978) The similarity solution of the K dV equation and the related Painleve transcendent, Proc. Roy. Soc. Lond. A361, 265-275. SATSUMA, J. (1976) N-soliton solution of the two-dimensional K dV equation, J. Phys. Soc. Japan 40, 286-290. TAPPERT, F. and ZABUSKY, N.J. (1971) Gradient-induced fission of solitons, Phys. Rev. Lett. 27., 1774-1776. URSELL, F. (1953) The long-wave paradox in the theory of gravity waves, Proc. Camb. Phil. Soc. ^9, 685-694. VARLEY, E. and CUMBERBATCH, E. (1965) Non-linear theory of wave-front propagation, J. Irist. Math. AppljLc. 1_, 101-112. VARLEY, E., KAZAKIA, J.Y. and BLYTHE, P.A. (1977) The interaction of large amplitude barotropic waves with an ambient shear: critical flows, Phil, Trans. 287, 189-236. VELTHUIZEN, H.G.M. and VAN WUNGAARDEN, L. (1969) Gravity waves over a non~uniform flow, J. Fluid ttech. 39, 817-829.

CHAPTER 3 SOLITARY WAVES IN SLOWLY VARYING ENVIRONMENTS: LONG NONLINEAR WAVES R. GRIMSHAW Department of Mathematics University of Melbourne, Parkville Victoria 3052, AUSTRALIA

1.

INTRODUCTION. It is well known that for many different physical systems weakly non-

linear long waves are described by the Korteweg-de Vries equation when the wave propagation is uni-directional and the background medium is homogeneous (e.g. WhJtham, 1974, Chapter 17). In some circumstances, the canonical evolution equation is modified from the K-dV, either in the nonlinear term, or more commonly, in the dispersive term; an example of the latter case is the deep fluid equation (BDA) derived by Benjamin (1967) and Davis and Acrivos (1967)•

Most existing theories discuss these equations for

the uni-directional homogeneous case, when it is well known that they possess N-soliton solutions, and are exactly integrable through the inverse scattering transform technique (e.g. Ablowitz and Segur, 1981, Chapter 4 ) . On heuristic grounds we claim that the canonical equation to describe weakly nonlinear uni-directional long waves in an inhomogeneous medium is the variable-coefficient K-dV equation

Here the coefficients y and X are functions of T alone.

In (1.1) if £

2

is

a small parameter measuring the amplitude of the wave, then T is £ x, and r/*x -1 E, is the convected coordinate £i/ n c~

dx - t[, where x is distance, t is

time and cQ is the linear long wave phase speed; the medium is assumed to be inhomogeneous in the x-direction on a scale £ 3 ion of £ x.

, and so c n is a funct2

The wave amplitude A is chosen so that A

wave action flux in the x-direction.

is a measure of the

Equations of the type (1.1) were

first derived by Ostrovsky and Pelinovsky (1970) for the case of a surface gravity wave travelling over variable depth (see also Johnson (1973) and Shuto (1974)); in other physical contexts, equations of the type (1.1) arise in plasma physics (Nishikawa and Kaw (1975)) and for internal gravity waves (Grimshaw (1981a)).

SOLITARY WAVES IN SLOWLY VARYING ENVIRONMENTS

45

In Section 2 we present a generalization of (1.1) which allows for more general kinds of inhomogeneity, includes transverse variations and dissipative effects, and also replaces the dispersive term in (1.1) (i.e. the term whose coefficient is A) with a general linear operator.

It will

be shown how the linear long wave dispersion relation based on the linear long wave phase speed cft allows the introduction of a set of rays, which in turn provide the natural coordinates for the evolution equation.

In

the Appendix, we outline a general method for the derivation of the canonical evolution equation.

Our starting point here is a general system of

partial differential equations and boundary conditions, with the property that their linear part represents a slowly varying waveguide. In the remainder of this article we consider that class of asymptotic solutions of the canonical evolution equation which can be described as slowly varying solitary waves.

Unlike the case when the coefficients fi

and A. in (1.1) are constants, the variable-coefficient equation does not, in general, have exact solutions.

Hence instead we assume that the coef-

ficients A and // vary slowly with respect to a locally defined solitary wave.

The asymptotic solution is constructed using a multi-scale pertur-

bation method applied directly to (2.1), a technique first used in this context for the particular case of a surface solitary wave propagating over variable depth by Grimshaw (1970) and Johnson (1973).

A more general

account of this method is given by Grimshaw (1979) (see also Gorshkov and Ostrovsky (1981), or Kodama and Ablowitz (1981)).

We shall show that this

asymptotic solution consists of a solitary wave with a slowly varying amplitude and a trailing shelf.

An alternative procedure which leads to the

same result involves an adaptation of the inverse scattering transform technique (Karpman and Maslov (1977), or Kaup and Newell (1978)). In Sections 3 and 4, we consider the slowly varying solitary wave solution of the variable-coefficient K-dV equation (1.1), supplemented by the inclusion of a small dissipative term (see (3.1)). sists of two parts:

The expansion con-

an inner expansion, discussed in Section 3, for the

structure of the solitary wave, and an outer expansion, discussed in Section 4, for the trailing shelf.

Then in Section 5, the results of Sections

3 and 4 are applied and extended to discuss the decay of a solitary wave due to dissipation.

Finally in Section 6, we outline the analogous theory

for the BDA equation. 2.

CANONICAL FORM FOR THE EVOLUTION EQUATION. We shall consider waves propagating horizontally in a waveguide.

In

46

R. GRIMSHAW

general a waveguide can support an infinite set of vertical modes; here we select just one of these modes, whose linear long wave phase speed is c n , and seek an equation for the evolution of its amplitude.

Long waves of

small amplitude are characterized by two small parameters; one, a, is a measure of nonlinearity and is typically the ratio of the wave amplitude to the vertical scale of the waveguide; the other, £, is a measure of dispersion and is typically the ratio of the vertical scale of the waveguide to the horizontal wavelength. The evolution equation for the wave amplitude

A(T , £, rj) is

| p + l J A | | + A X ( | | ) + VT(A) + B = 0 2

iS. = 6 9 A 8?

an 2 '

(2.1a)

(2 ib)

Here T is a slow time variable, £ is a phase variable which describes slow spatial modulations in the wave direction relative to a frame which moves with the linear long wave phase speed c , and r) describes modulations transverse to the wave direction. X(A) is a linear operator describing dispersion, given by

w

= - i [ f(k)exp(ik£) ?(A)dk,

f

(2.2a)

exp(-ik£)Ad£.

(2.2b)

J-00

For instance, when f(k) = k 2 , X (A) - d2A/dE,2 and (2.1a) is an equation of K-dV type; the balance between nonlinearity and dispersion requires that 2 a = e . These equations typically occur when the waveguide has limited vertical extent, and there is an analogy with shallow water theory. By contrast, when the waveguide is abutted by a deep passive region, it is typically found that f(k) = |k|, and (2.1a) is a generalization of an equation derived by Benjamin (1967) and Davis and Acrivos (1967) for internal waves in deep fluids. requires that a = £.

The balance between nonlinearity and dispersion now T(A) is a linear operator describing dissipation,

given by T(A) = ^

I

g(k)exp(ikS)2(A)dk.

(2.3)

J 00

For instance, when g(k) - -k 2 , T(A) is d2A/d£2 and (1.1a) is a K-dV-Burgers equation.

When dissipation is principally due to frictional drag at a ri-

gid boundary, g(k) = (-ik) '

(Grimshaw, 1981).

pose that g(k) is equal to (-ik)

In general, we shall sup-

where m > 0; for a dissipative process,

we require Re{vg(k)} to be non-negative, and the sign of V i s chosen

SOLITARY WAVES IN SLOWLY VARYING ENVIRONMENTS

accordingly. here.

47

More general forms can be chosen, but will not be considered

Finally the coefficients y,A,v

and 8 are known functions of T, de-

termined from the structure of the vertical mode in the waveguide.

Equa-

tions of the type (2.1a,b) occur in a variety of physical situations; for internal waves in stratified shear flows a general derivation is described by Grimshaw (1981a).

An abstract rendering of this derivation is given in

the Appendix. In a homogeneous and horizontally isotropic medium x = £Ott, £=e(x-cot) where x is a coordinate in the wave direction and t is the time coordinate, while rj = ea

y where y is a coordinate normal to the wave direction.

coefficients y, A,V

The

and 6 are constants in this instance; in particular

note that the frequency u) for linear waves of wave number k in the x-direction, and I in the y-direction is given by 2

a) - kc Q + -5£- - Akf(k) - ivae 1-m g(k).

(2.4)

This dispersion relation provides an immediate interpretation of the coefficients A, V and 6.

Further it may be shown that 6 = y c Q .

Note that in

the dissipative term V is a non-dimensional measure of dissipation, and its relation to a dimensional quantity is such that the latter is 0(a£ m ) . In general, the coefficient y of the nonlinear term has no such simple interpretation. In an inhomogeneous medium the background environment varies on length and time scales of order $~ .

If the effect of this slow variation is to

be comparable with the effects of weak nonlinearity and dispersion, then 3 2 we must choose 3 = £(X (i.e. £ case).

for the K-dV case, or £

for the deep fluid

The main consequence of this is that the linear long wave phase

speed c_ will now vary on these long length and time scales, thus defining a set of rays whose trajectories in time and space define the wave direction and phase.

We introduce slow time and space variables T - 8t, X ± - 3x±,

(2.5)

where i = 1,2 corresponding to the number of horizontal space variables in the waveguide.

Then we introduce the phase variable £ - | S(T, X ± )

(2.6a)

and

0) - - |f, K ± - ||- .

(2.6b)

Now the linear long wave phase speed c^ will be a function of T, X. and if the medium is not isotropic, it will also be a function of the wave direct-

48

R. GRIMSHAW

ion £. which is the unit vector in the wave direction (£. = K . / K where K

2

i

= KiKi).

ii

The phase n then satisfies the dispersion relation 0) - Kc Q (T, X ± , £ ± ) - W(T, X ± , K ± ) .

Note that W is homogeneous in K with degree one.

(2.7)

Equation (2.7) is a par-

tial differential equation for the phase, whose solution is obtained by the method of characteristics or rays. long a ray.

Let T be a time-like parameter a-

Then the ray equations are dT

dX

-

do> _ 8W dT " dT '

i

.. dK

dz

8W

,

i _ 8W " " ax ± •

.

(2#8b)

Here V. is the long wave group velocity, and is related to c n by the expression

rv

(2 9)

'i-^+s^tj-vV-

-

Note that V.K. = c~ and so the component of group velocity in the wave direction is equal to the phase speed; in an isotropic medium the group velocity is equal to the phase velocity.

The ray equations (2.8a,b) are

solved subject to the initial conditions T, X ± = T Q (T i ),

X^C^),

H = H 0 (T ± ),

on T = 0,

(2.10a)

on T = 0.

(2.10b)

These equations determine an initial manifold on which T

are interior co-

ordinates; (2.10b) determines the initial phase on this manifold, and the for 0), K are are determined from (2.10b) and the dispersion initial values for relation (2.7).

The solution of (2.8a,b) is X

I

=

X (T

I J>>

K

=

I

K



I< T J>>

where we are adopting the convention that X Q = T, T Q = T, K = -a) and a capital letter index takes the values 0, 1, 2. The first set of equations in (2.11) determine T (i.e. T , T ) as functions of X , and if J is the J

Jacobian of this mapping, then

1

1 3J

i.

8V

i

,9 1 9 v

(2 12a)

j-fr = w:> where

-

. J - det[ -^

].

Next we observe that as a consequence of (2.8a) and (2.9),

(2.12b)

SOLITARY WAVES IN SLOWLY VARYING ENVIRONMENTS

^ -

49

-co + K ± V ± = 0,

(2.13)

and so H = - n ( T j) is an interior coordinate on the initial manifold. out loss of generality we may now identify!

With-

with H, and define r|=ot

T?.

It may then be shown that the evolution equation is (2.1a,b) provided that 2 the amplitude A is chosen so that A is a measure of the wave action flux along a ray tube (i.e. the quantity which is conserved in linear long wave theory in the absence of dissipation), and is proportional to J (wave action density).

The coefficients y, A, v and 6 generally differ from their

counterparts in a homogeneous medium by geometrical factors, and are functions of T ; however, the dependence on T-, T j

i

is passive, and they may be

z.

effectively regarded as functions of T only.

An outline sketch of the de-

rivation of (2.1a,b) is given in Appendix A (see also Grimshaw 1981a). As an illustration of the foregoing theory, suppose that the environment varies in just one space direction, and the wave is propagating in this direction.

Then we may assume that c~ = cn(X ) , and the solution of

the ray equation is

rxi « •

E=

T

" T> T 2 =2 V

2

(2 14)

'

The Jacobian J = c , the coefficient 6 = — c Q , and the coefficient \ differs from its counterpart in a homogeneous medium (i.e. when defined by 3 2 (2.4)) by the factor 1/c case.

in the K-dV case, and l/cQ

in the deep fluid

In the absence of transverse variations and dissipative effects, it

can be seen that for this uni-directional case,(2.1a,b) reduces to (1.1). 3.

SLOWLY VARYING SOLITARY WAVE (K-dV):

INNER EXPANSION.

Now let us consider the one-dimensional K-dV equation in an inhomogenous and dissipative medium.

Thus in (2.1a,b) we shall assume that A is in2 dependent of r\ and so B is zero; for the K-dV case, f(k) = k and (2.1a) becomes

-~+yA|f+X-^4 + av T(A) = 0. 3r

dE,

(3.1)

3?2

Here we shall further assume that the coefficients ]i, X and V are slowly varying functions of T , and so are functions of the slow variable s = ax.

(3.2)

Consistently with this hypothesis we have introduced a factor O into the dissipative term. variation.

Here a is a small parameter characterizing the slow

Because of the variable coefficients (3.1) does not in general

possess exact solutions analogous to the N-soliton solutions of the constant coefficient K-dV equation.

Instead we shall construct the slowly

50

R. GRIMSHAW

varying solitary wave asymptotic solution, using a multi-scale perturbation method applied directly to (3.1) (see Grimshaw (1979), Gorshkov and Ostrovsky (1981), or Kodama and Ablowitz (1981)).

An alternative procedure which

leads to the same result involves an adaptation of the inverse scattering transform technique (Karpman and Maslov (1977), or Kaup and Newell (1978)). To be specific we shall suppose that y and A are constant and v is zero for s < 0, so that in s < 0 equation (3.1) has a uniform solitary wave solution, Our aim is to determine the continuation of this solution into s > 0. We put A - A Q (0, s) + OA 1 (0, s) + ... ,

(3.3a)

1 fs 0 = £ - ± b(s f )ds f ,

(3.3b)

where

and b Q + ab

+ ... .

(3.3c)

Here 0 is a rapidly varying phase and b can be interpreted as the "speed" of the wave.

For the case when there is also a slow variable a£, see

Grimshaw (1979).

At leading order, we obtain the solitary wave solution

where

A Q - a seen2 36,

(3.4a)

ya = 3b Q - 12A$ 2 .

(3.4b)

Here a is the solitary wave amplitude, depends on s and is the principal quantity we wish to determine.

Note that it is permissible to center the

solitary wave at 0 = 0 , as the term in 0 arising from b- is equivalent to a phase shift. At the next order, we find that

-bo 9^ +

v

h

(A

iV

+ x

7 3 Ai = N r

(3 5a)

-

00 where

o. A

N

l

=

dA

~~os

~. b

tfA

l~a¥~ m

We note here that T(A ) with g(k) = (-ik)

V T(A

0)#

(3#5b)

can be expressed in either of

the alternative forms

_a3m r(m+l)sin™ f° sechV+u) ^ J 0 u a or,

m

a(- TTfr )

±f a i g

^

m

±nteger>

(3#6a)

2 sech 30,

if m is an integer,

(3.6b)

SOLITARY WAVES IN SLOWLY VARYING ENVIRONMENTS

T(An) = a3 0

r — cos(k36 - ±-TTm)dk. 2 Josinh(^)

51

(3.6c)

In (3,6a) the integral is divergent, and must be interpreted in the Hadamard sense of finite parts.

Integrating once, and using (3.4a,b), we find

that

2

2

^ Al

1

*

bn{-A- + (3 sech ee)A. + - ~ - — T T > - N- , U l X 43 36 -1N 1 = cQ + | N ^ e .

(3.7a) (3.7b)

Two independent solutions of the homogeneous equation (3.7a) are f - sech236 tanh 36, g = $6f + -~ cosh236 + ~ - sech236.

(3.8)

Note that f is proportional to 3A0/96. The method of variation of parameters then gives the general solution of (3.7a) as xf

+ C2g + ~ j Nxg d6 - I j Nxf d6,

(3.9a)

where 3W = - 3 j b Q .

(3.9b)

The general solution (3.7a) thus contains three undetermined functions of s, namely C , C , and C«. The solution (3.9a) contains terms which are exponentially large, proportional to exp(±236) as 6 -*• ±°°. These terms must be removed, and hence,

wc2

f N-fdS - - [ J0

L

J-oo

N.f d6, 1

(3.10a)

and so N f d6 = 0.

(3.10b)

Recalling that f is proportional to 3A0/36, (3.10b) may be integrated by parts, and hence we obtain

|-( f f Ajrae) + v f A T(A )d = o. J —00

(3.11)

J —00

Here we have used (3.5b) for N . Recalling (3.4a) we see that (3.12a) and I J _oo

where

A Q T(A Q )d6 = m o a 2 3 m " 1 ,

(3.12b)

52

R. GRIMSHAW

•r The constant m

T(sech2(())d(().

(3.12c)

can be evaluated using Parseval's theorem for Fourier trans-

forms; for instance, when m = y, we find that m~ = 0.72, and for m = 2, m n = - •=-=• .

Since k is determined in terms of a by (3.4b), (3.11) is the 1 2 Noting that r A is the wave

equation which determines the amplitude a.

action flux along a ray tube, we can identify (3.11) as the equation for wave action.

Next, it can be shown from (3.9a) that A± -> A*

as

where

6 + ±«, /•zoo

b A

0 l " ~C0 " J

Since C

(3.13a)

N d9#

l

(3.13b)

is still free, we now choose it so that A

is zero, and conditions

ahead of the solitary wave are undisturbed (Johnson, (1973).

It then fol-

lows that, using (3.5b),

b Q A~ = [ N d 8

(3.14a)

J —oo

b A

O l

=

" "55" (J

A d6)

O

*

(3.14b)

This latter equation shows that the solitary wave loses mass to a shelf trailing behind the wave whose amplitude at the rear of the solitary wave is A .

r

From (3.4a) we see that

J — oc

Proceeding to the second order, we find that

- b o "fe + v l e

+x

^ A2 " N2>

(3 16a

- >

where 8A N

2

Equation (3.16a) for A

3A

§1" *i-§T

3A + b

i-§e-

vT(A

i>-

(3 16b)

'

can be solved in a manner similar to that for A ;

the removal of exponentially large terms leads to the condition

[ J —oo

N 2 A Q d9 = 0.

(3.17)

On substitution of (3.16b), and some rearrangement using (3.5a) we obtain

|j ([ A^dS) + V [ {A2 T(A ) + AQ T(A1)}d6 + \ bQ(A~)2 = 0. « _OO

J ^00

(3.18)

SOLITARY WAVES IN SLOWLY VARYING ENVIRONMENTS

53

This can be identified as the second order wave action equation, recalling that (3.11) is the first order such equation.

Assuming, for simplicity,

that v is zero, (3.18) becomes ~

a b

aA~

?

and so serves to determine the phase speed correction b . zero, an analogous equation again determines b .

When v is not

At this point we have

determined the amplitude a by (3.11), the phase speed correction b (3.18) and the coefficients C remaining coefficient C

by

and C 2 in the expression (3.9a) for A .

The

and the second order phase speed correction b« are

determined by proceeding to the third order, although we note that it is permissible to put C- equal to zero as its effect is equivalent to the phase shift provided by b_.

Also note that

Ao ^ r - ~ ^ + 2 b Q 9s

A* 2

as 0 + ±°°.

(3.20)

Assuming that conditions ahead of the solitary wave are undisturbed, we have already put A- equal to zero, and may likewise put A_ equal to zero. However, A

is then determined (3.14b) and similarly we can find an expres-

sion for A~. 4.

SLOWLY VARYING SOLITARY WAVES (K-dV):

OUTER EXPANSION.

At this point, we observe that although the calculation of the first two terms in the expansion (3.3a,b,c) has been completed, certain non-uniformities have arisen.

The most severe of these are those due to the terms

A" and A~ (see (3.13a) and (3.20)), although there are also non-uniformities due to terms with algebraic decay as 0 •> -°° (arising from the dissipa2 tive term), and terms proportional to 0 exp(±2$0), and 0exp(±2$0) as 0-*±«> in A .

Our procedure for dealing with these is a generalization of that

used by Johnson (1973).

Since we have chosen A

and A_ equal to zero, let

us first consider the simpler case when 0 -> +°° (i.e. ahead of the solitary wave).

As 0 ->• °°, it may be shown from (3.4a) and (3.9a) that A ^ 4a exp(-230)(l + a(H*3 2 0 2 + H*30 + . . . ) + 0(a 2 )},

(4.1a)

where H* = C2bo3)~1( i -|^ ) , and H + = (2b 3)" 1 (- - — + — — 1 0 a 9s 28 9s

(4.1b)

54

R. GRIMSHAW

Thus we regard (3.3a,b,c) as an inner expansion, and seek an outer expansion of the form A ^ 4a exp{- ~ f(r,s; a )} + 0{exp(- ^j- )}, r - a£.

(4.2a)

To match with (4,1a), we require that f ^ 23© - H*3 2 © 2 + a(-H*3© + . . . ) + 0(a 2 ),

as 0 -> 0+,

(4.3a)

where 0 = ae «

b(s f )ds f .

-

r

(4.3b)

Jo The outer expansion is to hold in the region 0 > 0.

Since A is now a

function of r and s alone, equation (3.1) becomes |A + ^ |A + O2X d^ + Q m ds 9r 8r3 Substitution of (4.2a) then gives

If >3>+ ^H+ ^ f fi+ ^ < If >m 9r

Next we put

f = fQ + Of1 + ... ,

(4.6)

Substitution into (4.5) gives 8f

3f-

3fn

2 3f.

8f

0 .

0

3 9fn 3 2 f n

a

3fn

_

The initial conditions are obtained from (4.3a,b) and are f 0 - 0,

-97=2)8

on

0Q = r - j

f x - -23 [ b 1 (s t )ds t

b()(sl)dst=0,

on 0 Q = 0.

(4.8a) (4.8b)

The solution is obtained by the method of characteristics, and is f

3

= 163J(s )

fS s

X ds,

(4.9a)

0

where

fS0 2 fS U f f - I b o (s )ds = 123 (sQ) j X ds.

(4.9b)

SOLITARY WAVES IN SLOWLY VARYING ENVIRONMENTS

55

Here S-. is an initial condition on 0. = 0, and is related to s and r by (4.9b) which is the equation for the characteristics.

The solution remains

valid provided the characteristics do not intersect, which is the case provided that fs

3ft

3 | | (s0) j

s

Ads * X(so)B(so).

This is certainly satisfied initially (0 is negative.

(4.10)

0 = 0 ) , and is always true it 33/3s

However, if 33/3s is positive, (4.10) is violated at s = s.

say, and the solution cannot be continued beyond this point.

When the

characteristics do not intersect, f~ develops into a similarity solution as r,s •»• °°,

f

9

r

o^I

\IO

{3

f

x ds}

— 1 /9

0. (3.14b) and (3.15).

The shelf A, is found from

3vm

A" = - ^ 3m""1 .

(5.2)

o/ Hence as s •> °°, the amplitude decays as s" cays as s - m / m .

and the shelf amplitude de-

Since m < 3 the amplitude a decays faster than the shelf;

indeed, for m < 1, the shelf amplitude increases as s •* °°.

The solitary

wave position is given by 0 Q - 0 (4.8a); using (3.4b) we find that

1 +V

m

° ^

C

2 r

- [1 + T vmmn3nsj *

, if m * 2,

(5.3a)

SOLITARY WAVES IN SLOWLY VARYING ENVIRONMENTS

1r

59

?

f- = log{l + vmn3^s} if m = 2.

(5.3b)

Note that for 2 ^ m < 3, r -* °° as s -• °° and so the solitary wave travels to infinity (albeit with a decreasing speed).

However, for 0 < m < 2

r -*• r^ (a finite constant) as s •+ °°, and so the solitary wave stops in this frame of reference (i.e. travels with the linear long wave phase speed c ). The shelf is given by (4.18); using (5.2) and (5.3a,b), we find that

B

e

O-V r 3vm

{ i Vm+

3Vm BQ =

r }y = ~

vm n r -—- } eexpix { P

- 33 Q

Thus for 0 < m < 1, the shelf B decreases with r.

-v- c (m-2)

if

.»> 2

^

m = 2.

(5.4b)

increases with r, but for 1 < m < 3, B Q

Since V is a constant in s > 0, and by hypothesis is

zero in s < 0, the similarity solution (4.23a) is needed at r

0.

It is apparent from these expressions that the shelf amplitude, given by (5.2) at the rear of the solitary wave and by (5.4a,b) elsewhere will surpass the solitary wave amplitude (5.1).

This indicates a breakdown of

the asymptotic solution described in Sections 3 and 4. introduction of a longer scale for T than a then on a scale for s of 0(a

.

The remedy is the

First, suppose that K m < 3 ;

) , the solitary wave amplitude is compar-

able with the maximum shelf amplitude of 0(a) which occurs at r = 0. we put

Hence

m+2 g=

o

m/2

s = a

2

T,

(5.5)

and seek a solution of the form (compare (3.3a,b,c)) A = GA(8, s ) , where

1-m /\

e=

and

a 1 / 2 e = aj

a

f J

b = ob.

(5.6a) (5.6b)

0 (5.6c)

On substituting into (3.1), we find that

3s m-1 2 Using an expansion parameter a , the analysis of Section 3 may now be 39

86

dQ

repeated and we again obtain a slowly varying solitary wave with a trailing

60

R. GRIMSHAW

shelf.

Indeed the solution is precisely that obtained in Section 2 when it

is rewritten in the new variables s and 6. For the outer expansion corresponding to (5.6a,b,c) we first consider 2 2, but that when r is 0 ( a T ~ ) (i.e. r is 0(1)) the nonlinear effect increases to the extent that the solution (5.10) develops a singularity, when as is 0(1), which occurs first at r = 0.

When m = 2,

r = r and A is 0(a ' ) as r + °° (strictly when r is 0(|log a|)), but there is again a singularity which develops first at r = 0.

We resolve this

singularity by invoking a similarity solution in the vicinity of r = 0. Hence we put and

s

n

6s,

where 6 = on9

p = r(3a 2 As)" 1 / 3 ,

A = (a6) 2/3 A(s , p ) .

(5.11a) (5.11b)

n Here p is the similarity variable (4.23b), and for p of 0(1) r is O(cr

6

) when the evolution scale is given by s . n to (4.4), we find that

On substitution in-

SOLITARY WAVES IN SLOWLY VARYING ENVIRONMENTS

3s

n

3s

n

op

^

/o

A \ 1 / 3 op

(3s A)

*

3s

^ 3

r/o

n 3p

61

•vxni/3

6 (3s A)

2/3 1/3 When 6 = 1 , the boundary layer has width r of 0(0 ) , A is 0(a ) since A is 0(a), and the linear similarity solution (4.23a) pertains. However, 1/2 when 6 = 0 , A is 0(1) as A is 0(a), and the similarity solution is now nonlinear; it will have the structure F(p,s .•* r ) is a slow variable. power series in (3s

wnere

r = a

p(As . y

As s .„ -> 0, F may be constructed in the form of a

whose leading term is (4.23a). But as s- ,--*» ' -2/3 G p>s F is constructed in the form (3s-,/2^) ^ l / ? ; r^ w h e r e G i s a P o w e r series in (3s . A ) . As the scale for evolution is increased further to 1/3 6 = a, A is 0(1), but A is 0(a ' ) ; the boundary layer width is now r of 0(a ) . This solution is to be matched to (5.10) as p •> 3(A)dk,

(6.8a)

J —00

2

2 1/? - in ) x / ^ > 0, 2 2 1/2 Xk Im (k - in r > 0.

where either

Re(k

or

(6.8b)

The Fourier components with |k| < m correspond to radiating waves.

Assum-

ing that m is small it may be shown that (3.11) becomes (Grimshaw 1981b)

fc T 0, g(x) » 1 and as x ->• °°, g(x) ~ 3x~2.

Thus when v is zero and X

and y are constants, (6.10a) gives a linear rate of decay for large amplitudes and an algebraic decay for small amplitudes. APPENDIX:

DERIVATION OF THE EVOLUTION EQUATION.

Suppose'first that the coordinate system consists of a vertical coordinate z which varies across the waveguide, and horizontal coordinates x-,

SOLITARY WAVES IN SLOWLY VARYING ENVIRONMENTS

x 2 which vary along the waveguide, while t is the time.

65

In order to de^

scribe long waves, we re-scale the horizontal coordinates and the time, and we put xj. = £x I , where x~ = t, etc.

(A.I)

Next we propose that the physical system is described

by L ( p r -j| ; X I , z; a, e)v + M = 0,

(A.2a)

where

Pl Here we recall that X

= -£r .

(A.2b)

= eax T (2.5), and describe the slow variation due

to the background environment.

In (A.2a), L is a linear operator, consist-

ing of a set of differential equations and boundary conditions, operating on the vector v, while M is a nonlinear term.

In (A.2a) the dependence on

a arises due to the slow variation of the environment, while the dependence on £ describes higher order dispersion. We introduce another set of slow 1/2 coordinates x" = £a x and note that the next member of this sequence is xf'f= X .

Then we seek a solution of the form v = v(?; x1^, Xz; z; a ) .

(A.3)

The equation to be satisfied is then

3L + a{

82L

8 V

3?: ~$t

+ eL1Qv + where

L = L(K

o

+

e2

3K

3 V

2 9pT3pT 9XT 3f} + L2ov +

M

+

•••

aL

=

01 v °>

i If 'fe; x r z;0>0)-

(A.4a)

(A 4b)

-

Here L ni is the first term in the expansion of L with respect to a (i.e. dL -5— ) , and L ., L o n are the first and second terms in the expansion of L

dot

1U ZU

with respect to e; note that in the K-dV case L..-. is zero. At the leading order, the solution is v = av Q = ar(K]. ||- ; Xj, z)A(5; x1^, XJ , where

L Q ( p r - ~ ; Xv

z)r( Pl ; X-j., z) = 0.

(A.5a) (A.5b)

66

R. GRIMSHAW

Thus r is the right null vector for L Q and (A.5b) is satisfied only when the dispersion relation p Q + W(X l5 is satisfied (compare (2,7)),

P±)

= 0,

Note that L

(A.6)

is an operator in z alone de-

scribing linear long waves and this is reflected in the homogeneity of the dispersion relation (A.6).

Next we introduce an inner product |w(-p x ), v(p t )},

(A.7)

which typically takes the form of an integral with respect to z (i.e. acrossthe waveguide), together with some boundary terms.

We shall assume that L^

is self-adjoint with respect to this inner product

|Lou, v } - { u , L o v(.

(A.8)

Consider now the inhomogeneous equation L Q v + f - 0.

(A.9)

A necessary and sufficient condition for a solution is

{r, f } - 0.

(A.10)

For all p_, we define D( P l ;

Xj.) - Jr, L Q r|.

(A.ll)

Clearly the dispersion relation (A.6) is equivalent to D = 0. Next, we put v - a(vQ + a 1 ' v± + av 2 + . . . ) , and substitute into (A.4a).

(A.12)

At the next order, we obtain

8L 8 v

o o

4

This is an equation of the form (A.9) and the compatibility equation (A. 10)

gives ^-r^T

-0

on D - 0,

(A.14a)

=

.

(A.14b)

or ^

^

0

Here we recall that V. is the long wave group velocity (2.8a), and (A.14b) shows that to leading order the amplitude A propagates with the group velocity.

At the next order, we obtain an equation for v« of the form (A.9)

and it may be shown that the compatibility condition (A.10) leads to (see Grimshaw, 1981a)

SOLITARY WAVES IN SLOWLY VARYING ENVIRONMENTS

3D

3A

,1

3

, 3D

57

3D_ + {r , M} - T(A)] • ;A -t- v-r— ) T^A;J + tr, n/ 2

x

2

2 3 P l 8 P j 3 x^3x Here n = 1, except in the K-dV case (L

J

0.

is zero) when n = 2.

The dissipa-

tive operator is given by

v |2- T(A) = {r, LQ1r}A - \ {r, -J- ( -^ )r}A.

(A.16)

At this point, we introduce the ray coordinates T , £ and r\9 where we recall 1/2 that T- = aE, and T = a n- !t may then be shown that the first group of terms in (A.15) is

Throughout these equations, p_ is the operator K T -r-p , and in particular 3D 3D -r— is an operator in £. However, since D is homogeneous in p , -r— is a dpQ

I

dpQ

homogeneous operator in £, and by a combination of adjusting the d e f i n i t i o n of A and integrating (A,15) with respect to £, we may replace (A.17) with dD d\

118

,

8D .

" ^ 3 7 " 2 J af( J ^ ) A "

dD

^

T.

T(A)

.

.

(A 18)

'

.

'

From this expression, we can identify |j T ~ | A as the wave action flux, and defining a new amplitude A as the square root of this expression, we see that (A.18) reduces to the first and fourth terms in (2.1a).

Next the

nonlinear term {r, M} will generally take the form of the second term in (2.1a), while the dispersion term {r, L rtr}A will become the third term in nU (2.1a); the details of the calculations involving these two terms requires a knowledge of the specific physical system being considered (for an application to stratified shear flows, see Grimshaw , 1981a). be shown that the last term in (A.15) takes the form 3D _ , 3B 1 32W 92A ^ - - B , where — = - ^ _ ^ - ^ . -1/2 Introducing the transverse coordinates ri = a T_(X ) , £f = a

T (X,) = a

Finally, it may ,A 1 Q N (A.19)

?, and noting that the homogeneity of W implies that

K^d W/Bi^dK. is zero, it follows that in (A.19) A contains only derivatives with respect to r). Hence 95

3n 2

68 where

R. GRIMSHAW . ~ 0 <S - I 9 W 2 2 6 " 2 8K i 9K j 9X ± 3Xj '

(A.20b)

and so the terms in (A. 19) correspond to the last term in (2.1a) and (2.1b). REFERENCES ABLOWITZ, M.J. and SEGUR, H. (1981) transform, SIAM, 440 pp.

Solitons and the inverse scattering

BENJAMIN, T.B. (1967) Internal waves of permanent form in fluids of great depth, J. Fluid Mech. 29^, 559-592. DAVIS, R. and ACRIVOS, A. (1967) J. Fluid Mech. 29^ 593-607.

Solitary internal waves in deep water,

GORSHKOV, K.A. and OSTROVSKY, L.A. (1981) Interaction of solitons in nonintegrable systems: direct perturbation method and applications, Physica 3D 112, 428-438. GRIMSHAW, R. (1979) Slowly varying solitary wave. equation, Proc. Roy. Soc. A368, 359-375.

I Korteweg-de Vries

GRIMSHAW, R. (1981a) Evolution equations for long, nonlinear internal waves in stratified shear flows, Studies Applied Math. 65, 159-188. GRIMSHAW, R. (1981b) Slowly varying solitary waves in deep fluids, Proc. Roy. Soc. A376, 319-332. GRIMSHAW, R. (1970) The solitary wave in water of variable depth, J_. Fluid Mech. 42^, 639-656. JOHNSON, R. (1973) On the asymptotic solution of the Korteweg-de Vries equation with slowly varying coefficient, J. Fluid Mech. 60, 313-324. KARPMAN, V.I. and MASLOV, E.M. (1977) Perturbation theory for solitons, Zh. Exsp. Teor. Fiz. 73, 537-559. KAUP, D.J. and NEWELL, A.C. (1978) and in slowly changing media: Roy. Soc. A361, 413-446.

Solitons as particles, oscillators a singular perturbation theory, Proc.

KODAMA, Y. and ABLOWITZ, M.J. (1981) Perturbations of solitons and solitary waves, Studies Applied Math. 64, 225-245. NISHIKAWA, K. and KAW, P.K. (1975) Propagation of solitary ion acoustic waves in inhomogeneous plasmas, Phys. Lett. 50A, 455-456. OSTROVSKY, L.A. and PELINOVSKY, E.N. (1970) Transformation of surface waves in fluids of variable depth, Izv. Atmospheric arid Oceanic Physics 6, 934-939. SHUTO, N. (1974) Nonlinear waves in a channel of variable section, Coastal Eng. in Japan 17, 1-12. WHITHAM, G.B. (1974) 636 pp.

Linear and Nonlinear Waves, Academic Press, New York,

CHAPTER 4 NONLINEAR WATER WAVES IN A CHANNEL M. C SHEN Department of Mathematics and Mathematics Research Center University of Wisconsin Madison, Wisconsin 53706

1.

INTRODUCTION. In this paper, we consider some approximate equations for the study

of nonlinear water waves in a channel of variable cross section.

For fi-

nite amplitude waves, a system of shallow water equations are given; for small amplitude waves, we present a K-dV equation with variable coefficients.

Some of their applications are discussed.

Some problems deserving

more study are mentioned at the ends of the following two paragraphs and in the conclusions to Sections 3 and 4. One of the interesting problems of water waves in a sloping channel concerns the breaking of a wave moving toward a shoreline, the development of a bore, and the movement of the shoreline after the bore reaches it. For the two dimensional case corresponding to a rectangular channel of variable depth, the bore run-up problem was studied by Keller et al. (1960), Ho and Meyer (1962), and Shen and Meyer (1963a,b) on the basis of shallow water equations (Stoker, 1957).

Later Gurtin (1975) derived a criterion

for the breaking of an acceleration wave in a two-dimensional channel, and his result was extended by Jeffrey and Mvungi (1980) to the case of a rectangular channel of variable width and depth.

We generalize Gurtin1s re-

sult to predict the breaking point of an acceleration wave in a channel of variable cross section and review some existent results regarding the bore run-up problem for a rectangular channel with a uniformly sloping bottom. Needless to say, the use of shallow water equations for the study of bore propagation may be open to criticism.

The issue would be settled if we

knew the precise conditions for the validity of shallow water equations. Up to date, the shallow water equations for a two-dimensional channel with analytical initial data have been justified by Kano and Nishida (1979), and for the three-dimensional case with a priori assumptions on the free surface by Berger (1976).

At present we may accept shallow water equations

as model equations, and the bore run-up problem for a general channel certainly deserves further investigation.

M. C. SHEN

70

Another application of our results deals with the development of a solitary wave in a channel of variable cross section.

Recently there have

been discussions on the so-called infinite mass dilemma, which arises from the formation of a shelf behind the solitary wave.

If the shelf were ex-

tended to infinity, then infinite mass would be created or annulled by a perturbation on the solitary wave.

A study of this problem may be found

in Miles (1979) and Knickerbocker and Newell (1980), based upon the K-dV equation for a rectangular channel of variable depth or width (Kakutani, 1971; Johnson, 1973; Shuto, 1974).

We shall establish a global existence

theorem for the solution of the K-dV equation for a general channel as a consequence of the existence results due to Kato (1975, 1980).

It follows

that the shelf, if formed behind the solitary wave in a general channel, can only be finite.

A rigorous justification of the validity of the K-dV

equation here should be an important contribution to the theory of water waves.

Work in this direction has been done by Berger (1974) and Nishida

(private communication). 2.

SHALLOW WATER EQUATIONS AND THE BREAKING OF A WAVE. We consider the irrotational motion of an inviscid, incompressible

fluid of constant density under gravity in a channel with a boundary defined by h*(x*, y*, z*) = 0, where z* is positive upward and x* is in the longitudinal direction (Figure 1 ) .

Fig. I.

ACROSS

SECTION OF THE CHANNEL

71

NONLINEAR WATER WAVES IN A CHANNEL The governing equations are V* • q* = 0,

(2.1)

V* xq* - 0,

(2.2)

p(q* + q* • Vq*) = - V*p* + g t* subject to the boundary conditions

(2.3)

n* + q* • V*£* = 0 t*

(2.4) at

C* = -£* + n*(t*,x*,y*) = 0,

p* = 0

(2.5)

q* . y*h* = 0

at

h* = 0.

(2.6)

Here V* - (3/3x*, 3/3y*, 3/3z*), q* » (u*, v*, w*) is the velocity, t* is the time, g = (0,0,-g) is the constant gravitational acceleration, p is the constant density, p* is the pressure, and z* = r\* is the equation of the free surface. assumptions.

To derive the shallow water equations, we make the following The channel boundary is convex, sufficiently smooth, and var-

ies slowly in the longitudinal direction; the magnitude of the transverse velocities is much smaller than that of the longitudinal velocity. As suggested by Friedrichs (1948), we introduce non-dimensional variables t = e~ 1 / 2 t*/(h/g) 1 / 2 ,

(x,y,z) = (3" 1/2 x*/H, y*/H, z*/H),

n = n*/H,

(u,v,w) = (u*/(gH) 1 / 2 , 3**v*/(gH)1/2, 3 * V ( g H ) 1 / 2 ) ,

h = h*/H ,

where 3^ = L/H and L and H are respectively the horizontal and transverse length scales.

In terms of them, (2.1) to (2.6) become u x

+ v + w =0, y z '

3 u = v , y x* u

(2.7)

u = w , v = w , z x' z y

t

+ uu + vu + wu + p = 0, x y z *x

t

+ uv + w + wv + 3p = 0 , x y z *y '

v

(2.9) v

w + u w + v w + w w + 3(p + 1) = 0, t x y z z at

z= n

P - 0, uh x

+ vh + wh = 0, y z

(2.8)

(2.10)

v

y

(2.11)

(2.12) (2.13)

at h = 0.

(2.14)

72

M. C. SHEN Assume that u,v,w,p and 3 possess an asymptotic expansion of the form

and substitute (2.15) in (2.7) to (2.14).

The equations for the zeroth

approximation are u

+ v

0x

0y

+ W

(2

0z = °»

V = u0z " °> u

0t

+ u u

0 0x + P0x + v0Uoy

V

(2 17

' >

+ W U

(2

0 0z = °'

p

0z " -1'

" °'

'18) (2 19)

'

n

0t + V o x + V o y "w 0 = °'

(2 20)

'

at

z-

Q

P o = 0,

U h +

+

0x W

'16>

(2.21)

at

V z " °'

h=0

'

(2 22)

-

As seen from (2.17), (2.19) and (2.21), u Q is a function of t,x only, and P 0 = -z + n Q .

(2.23)

This implies n is also a function of t,x only.

It follows from (2.17),

(2.18), and (2.23) that

V

+

Vox + %x - °'

(2 24)

'

Now we integrate (2.16) over a cross section D of the channel, apply the divergence theorem and make use of (2.20) and (2.22) to obtain

II

(v

0y

+ w

oz)dydz

=

-u0xA(t'x) =

" U 0 L h x (h y

+ h

z >

ds +

(n

0t

By rearranging the terms, we have n ot +u o n Ox +u Ox A(t,x)/B(t,x)-[u o

f

j

9

9-1/9

n o t +u o n O x +u O x A(t,x)/B(t,x)-[u o /B(t,x)]j h x (h y +h z Z )

where A(t,x) is the area.

i/Z

ds = 0,

(2.25)

B(t,x) is the width and L is the wetted bound-

ary of the cross section D (Figure 1 ) .

(2.24) and (2.25) form a system of

nonlinear equations, which may be used to model bore formation and its subsequent development in a channel of variable cross section. In the following we extend Gurtin's method to the case of a general channel.

The assumptions made are the following:

NONLINEAR WATER WAVES IN A CHANNEL

(1)

(2)

73

U Q J ^ Q are continuous

the first and second derivatives of u. and ru possess at most jump discontinuities,

(3)

u~ = ru = 0 ahead of the wave.

Denote the value of a function f immediately behind the wave front by f • Hereafter we also drop the subscript 0.

From assumptions (1), (2), we have

u" = r)~ = 0.

(2.26)

By total differentiation, u~ = -cu~ ,

n~ = -cn~ ,

where c is the speed of the wave front.

(2.27)

From (2.24), (2.25), and (2.26),

it follows that u" + n" = 0, t

vT + U~A"7B"~ = 0.

X

t

(2.28)

X

Comparing (2.27) and (2.28), we have -

-1 -

ut = c \

-1/9

,

c - (A /B )

± / Z

.

(2.29)

Now we differentiate (2.24) with respect to t and (2.25) with respect to x, and evaluate the equations behind the wave front. n

Then we eliminate

and make use of the expression 2 — — 2 — c u x t - ufct = c d(u x )/dx - cdu t /dx,

to obtain -2c d d ^ r V d x + ( D " ) " 1 [c f - r~(B~«] + 3c

x

= o,

where 2 2 —1/2 i v(h + h ) ds. x y z '

Hence, aQc

±/Z

[(3/2)aQ

c •'x o

D/Z

exp

T (2A ) •'x o rx -

L

dx1 dx + 1]

- _i

r (2A ) L dx,

exp

x

(2.30)

J

where a^ is the initial value of r) at x = x Q ,

We call x = £ a shoreline

if A"(£) = 0, but B~(£) ± 0, and let I(x) = (3/2)

fX

-5/2 fxl - -1 c D / Z exp r (2A ) dx1 dx.

X O

'X O

74

M. C. SHEN

Suppose a < 0. o

If I(&) • °°, then r\ = °° and the wave breaks before it x

reaches the shoreline.

If I(i) ^ °°, then either the wave breaks before it

reaches the shoreline or it breaks at the shoreline. If I(&) ^ °°, then the wave breaks at the shoreline.

Next suppose a

> 0,

Otherwise if I(&) = °°,

we evaluate the limit of r\~ given by (2.30) as x •+ I and obtain lim n x - (2/3)[-(d")f/4 + r"/2B"] x=: ^

Hence the wave will never break if (d~) f is finite at

where d~ - A"/B~. x = I.

(2.31)

However, for channels of variable cross section the equilibrium

water surface may converge to a point and this case is also of interest. Assume again a

> 0, I(£) - «.

If B""(Jl) - d~ (I) - 0 and (d~)' is finite

at x - i9 we assume h(x,y,z) - -z + g(x,y) = 0, and F

-

h (h

+ h

ds :

)

where y = -b, , b« are the endpoints of the width B (x). It follows from (2.31) that lim rf - (2/3)[-(d~)74 + g x /2] x=sjl , and the wave will never break. 3.

RUN-UP PROBLEM. We consider a bore propagating toward a shoreline in a rectangular

channel with a uniformly sloping bottom.

On the basis of the shallow water

equations, we can find a fairly complete solution of the bore run-up problem, which appears to be the only one available to date.

The bore path

from the point of breaking to the shoreline may be approximately determined by Whitham's rule (Whitham 1958) of which a justification was given by Ho and Meyer (1962).

Here we shall only consider the movement of the shore-

line after the bore reaches the shore. The shallow water equations for a rectangular channel of variable depth are obtained from (2.24) and (2.25) as u

+ uu

t n

+ri

=

0,

+ [u(n + d )] A.

(3.1)

xx O

- 0,

(3.2)

X

where we also drop the subscripts for u and ri and d

= -yx, y > 0.

sume t = 0 when the bore reaches the shoreline x = 0.

We as-

It was shown by Ho

and Meyer (1962) that u tends to a positive limit u° as the bore approaches the shore.

Let

NONLINEAR WATER WAVES IN A CHANNEL

75

c 2 = n + dQ, a = 2c + u + yt - u°,

(3.3)

3 = 2c - u - yt + u°.

(3.4)

In terms of a and 3, (3.1) and (3.2) can be expressed as x a = (u - c)ta,

Xg = (u + c)tg.

(3.5)

By cross differentiation of (3.5) and making use of (3.4), we have taf3 + 3(ta + t3)/[2(a + 3)] = 0.

(3.6)

If we introduce the canonical variables, a = (a + 3) 3 / 2 t a ,

b = (a + 3) 3 / 2 t 3 ,

(3.7)

(3.6) yields a system of equations (a + 3)a 3 = -3b/2,

(a + 3)ba = -3a/2.

(3.8)

Z = a - b.

(3.9)

Let Y = a + b, It follows from (3.8) that Y a 3 = 15Y/[4(a + 3) 2 ],

Z a 3 = 3Z/[4(a + g) 2 ].

(3.10)

In the a,3-plane, we prescribe sufficiently smooth data on 0c:a=0, 0 < e < 3 ^ 3*, and on CD:3 = 3*, 0 < a < a* (Figure 2 ) . However, the precise nature of the data is immaterial. to(0,3) < 0, and that as 3 + 0

We require only t (a,3*) > 0,

along a = 0,

p

lim a = a°

>

0,

lim x = lim t = 0,

lim y

=

u° > 0,

b(0,3) = 0 ( 3

where the existence of the positive limit a

9/2

(3.11)

),

and the behavior of b(0,3) for

small 3 were established by Ho and Meyer (1962).

Our discussion will be

based upon (3.10) instead of (3.6) so that the derivations will be simpler, but we have to assume smoother data on the boundary. Let G* be the region {(a,3), 0 < e < 3 ^ 3 * , 0 < a < show that the mapping from G be single-valued.

a*}. We first

in the a,3-plane to the x,t-plane ceases to

Since the Jacobian of transformation from the a,3 coor-

dinates to the x,t coordinates is 9(s,t)/9(a,3) = -2c"2ab, where c > 0 in G , we look for the lines a = 0 or b = 0 in G .

The Riemann

functions of (3.10) for Y and Z respectively are (Courant and Hilbert, 1962)

R(a,3; a',31) = F(-3/2, 5/2; 1, z) = F ^ z ) , F(-l/2, 3/2; 1, z) = F 2 (z),

(3.12)

M. C. SHEN

76

c N.

X

D(a?/3*) Ge

0 \

Fig.2.

Go IN THE

E

a , £ - PLANE

where z

= >(a' - a)(3f - 3)/(af + 3f)(a + 3),

and the Riemann representations for Y and Z in G

are found as

* ± (a,B) - * i (0,B*)F ± [a(e* - B)/(B*(ui + B) ]

J

a1 a '

i

t _ a)(3* - 3)/(a1 + 3*)(a + 3)]daf

* B ,(0,B l )F i [a(B l where $ 1 = Y, $

= Z,

To show that a and b change signs in G , we need

the following two lemmas. LEMMA 1.

(3.13)

As 3 + o+ along a • k3, 0 < k < K Q for arbitrary K Q , lim (a + b) - a°F1(k/(l + k)), llm (a - b) = a°F2(k/(l + k)) ,

uniformly for 0 < k < K .

NONLINEAR WATER WAVES IN A CHANNEL

77

LEMMA 2. lim (a°/2)[F (k/(l + k)) + F. (k/ (1 + k)) ] = -2a°/37r, lim (a°/2)[F (k/(l + k)) - F (k/(1 + k)) ] = a%JT. 1

Z

As seen from Lemma 1,2, a and b indeed change signs in their limiting values along a = k , and we have THEOREM 1.

Let ~G be the limiting region of G

as e -> 0.

Then there

exists both lines a = 0 and b = 0 in G, which terminate at the origin with finite slopes.

Along a line 3 = constant or a = constant, starting from

the boundary 0C or CD, the line a = 0 is encountered first.

The proof of

the Lemmas and Theorem 1 may be found in (Shen and Meyer, 1963). We choose a* so small that the line a = 0 starts from the boundary a = a*.

Denote by G

the line a = 0. x,t-plane.

the region delimited by a = o, 3 - 3*,

Our next step is to examine how G

a = a

* and

is mapped back to the

By taking limits along various directions towards the origin

in G , we find that the origin a = 3 = 0 is mapped to the parabola x = -Yt2/2 + u°t along which the water depth is zero.

t > 0,

(3.14)

As observed from (3.14), the shore-

line starts to move upwards after the bore reaches it.

At t = u /y, x

reaches the maximum value (u ) /2y and starts to move downwards and continue to recede.

However, since the line a = 0 approaches the origin with

a non-zero slope, its image in the x,t-plane can be shown to approach the shoreline eventually, and forms an upper bound of the physical admissible region.

Hence only an advancing bore which forms a part of the boundary

of the physically admissible region can catch up the shoreline.

We may

summarize our results as THEOREM 2.

The shoreline advances and recedes following the parabola x = -yt /2 + u°t,

t £ 0.

The solution breaks down again in the process of shoreline recession, and an advancing bore appears. From the above discussion, a rather complete solution for the bore run-up problem has been obtained for a two-dimensional sloping channel and one may marvel at the applicability of such simple equations to so complicated a problem.

It should be of great interest if the results could be

extended to general channels.

78 4.

M. C. SHEN K-dV EQUATION AND THE DEVELOPMENT OF A SOLITARY WAVE. We only sketch the derivation of the K-dV equation for a channel of

variable cross section; the details may be found in Shen and Zhong (1981). We introduce the non-dimensional variables -1/9

t - 3

J/

1 /9

-1/9

t*/(H/g) ± / Z ,

(x,y,z) - (3

r|, h, p and (u,v,w) are the same as before.

x*/H, y*/H, z*/H), The method used here is a

specialization of the procedure developed by Shen and Keller (1973).

We

assume that u,v,w,p and rj also depend explicitly upon a new variable 5 = 3S(t,x) where S is a function of t and x only, will be called a phase function. Then we assume that they possess an asymptotic expansion of the form

2 + ... And we assume that the zeroth approximation is given by

(u

=

o> V V

°' Po= "z« % = °'

The equation for the first approximation determines a Hamilton-Jacobi equation for S.

Let k = S , U) = ~St« 0) = kG(x),

Then

G(x) = ±[a(x)/b(x)] 1/Z ,

(4.1)

where a(x) is the area of a cross section D , and b(x) is the width of D , of water at rest (Figure 1 ) .

(4.1) may be solved by the method of charac-

teristics and the corresponding characteristic equations are dt/da = y,

dx/da = yG(x),

dk/da * -kyG'(x),

,. 2\

do)/da - dS/da - 0. where p is a proportionality factor.

We choose y = 1 so that O = t.

The

solutions of (4.2) determine a one-parameter family of bicharacteristics called rays, x = x(t,a 1 ), where O

is constant along a ray.

The equations for the second approxima-

tion determine a K-dV equation with variable coefficients

Vlt

+

Vlx

( 4 )

Here m Q = 2b(x),

(4.4)

NONLINEAR WATER WAVES IN A CHANNEL

n^ = 2 a ( x ) / G ( x ) ,

m9 = -[G(x)] 2

(4.5) 9

f

JLQ

9—1/9 1/Z

h (h yZ + h zZ ) x y z

—9

ds - G Z ( x ) G f ( x ) a ( x ) ,

(4.6)

m 3 = 3k[G(x)]"1b(x) - u T ^ y O : , x, y^ 0) - (|)y(t, x, y±, 0)],

(4.7)

m, — co-1

(4.8)

4

L

-1 L

79

ff JJ

2 dy dz, (V6)

Do

is the wetted boundary of D ; y = y , y are the endpoints of the width

of D ; and $ is the solution of the following Neumann problem V2 = k 2

in

2 = co

d)h+d)h =0 y y z z

D, o

at at

z -0 L. o

Since from (4.2) d/da = 3/9t + G(x) 9/9x,

dx/da - G(x),

along a ray, we may express (4.3) in terms of a and £

Vio

+ m n +

2 i -a V i e + V I K C

= 0)

or in terms of x and ^

To be definite, we choose G(x) S - -t +

fx J

0

-1 [G(z)] ^dz,

which i s a solution of ( 4 . 2 ) , and i t follows that OJ - 1,

k - G" 1 (x).

Note that other choices of S are also possible.

For rectangular and tri-

angular channels, the coefficients given in (4.4) to (4.8) can be explicitly evaluated (Shen and Zhong, 1981). 1)

Rectangular Channel Let d(x) and b(x) be the variable depth and width respectively.

m Q = 2b(x),

m± = 2b(x)d 1 / 2 (x),

m 3 = 3d~ 1 (x)b(x),

m,, = b f (x)d 1 / 2 (x) + d" 1 / 2 (x)d' (x)b(x)/2,

m 4 = (l/3)b(x)d(x).

80

2)

M. C. SHEN

Triangular Channel Let the two sides of a cross section D be defined by z = y, (x)y - d(x), o 1

z = -y2(x)y - d(x), where y ± (x) = d(x)/b i (x), i = 1,2. m o = 2[b1(x) + b 2 (x)] = 2b(x),

m1 =

2 d 1 / 2 ( x )b(x),

m2 = 2 d~1/2(x)[b'(x)d(x) + df(x)b(x)/2]/2, m 3 = 5d" 1 (x)b(x), 1 2 m 4 = [d" (x)/4][b(x)d (x) + (b^(x) +b 2 (x))/3]. In both cases, m- > 0, m

> 0, m, > 0 if d(x) > 0, b(x) > 0.

cases, as seen from (4,5), (4,8), m but the sign of m

For general

> 0, m, > 0 where we choose G(x) > 0,

given by (4.7) is not obvious.

We shall assume that

m_ ^ 0 is the case. Let T = [ m (xf)m " 1 (x f )dx f ,

(4.10)

A = m 3 (x)m 4 " 1 (x)n 1 .

(4.11)

In terms of T and A, (4.9) becomes

A

+

AA

•+-A

= H^T^ A

T > 0

—oo < £" < oo

(A 1 9^

subject to A(O,Q = A o ( Q ,

-«> < g < oo,

(4

.13)

where H(T) = -m o (x)m. z4

(x) - m1 (x)m, 14J

(x)m o (x) [m, (x)/m o (x) ] f . 4 3

A global existence theorem for (4.12) and (4.13) can be easily obtained by extending the existence result for the K-dV equation with constant coeffic c i e n t s due to Kato (1975,1980). Assume H(T) i s continuous and l e t H (-00,00) 2 denote the Sobolev space of order S of the L - t y p e . Since H(T)A s a t i s f i e s the conditions ( f 1 ) , (f2) in Kato (1975), we have the following

local

existence r e s u l t (Kato, 1980). THEOREM 3.

(4.12) subject to (4.13) with A e H S , S > 2 , has a unique

solution A e C[O,T f ; HS] n (^[O.T 1 ; H S ~ 3 ] ,

A(0,O -

AQ(O,

for some T* > 0; and A(t) depends upon A

continuously in the H -norm.

NONLINEAR WATER WAVES IN A CHANNEL

Hereafter, we shall denote an H -norm by || • || || • || .

81

and an L -norm by

To show that there exists a global solution in [0,T] for any T > 0 ,

we need a regularity result and an a priori estimate for | | A | L , which may be obtained by means of the first three conservation laws for the K-dV equation with constant coefficients (Lions, 1969). If A € C[0,T; H S ] is a solution to (4.12) with S > 2 and

THEOREM 4. if A(0) A

£ H

S

with S 1 > S, then A e C[0,T; H S ] with the same T.

This theorem is a simple extension of Katofs result (1980). o

To derive the H estimate for A ( T ) , T in [0,T], where T > 0, we first 4 2 assume A Q , A e H . Then we multiply (4.12) successively by A, A + 2A~£ and A + 3A^ + 6AA,. + (18/5)A^^^.^ and integrate to obtain estimates for

||A||, llAl^ and where $ T ( # ) depending upon T is a monotone increasing function with 1 2 $ (0) = 0.

For A , A e H , we may approximate A ,A by sequences of smooth-

er functions, make use of the results of Theorems 3 and 4 to complete the derivation. On the basis of the H

2

estimate and Theorem 4, we have the global

existence theorem: THEOREM 5.

For any T > 0, (4.12) possesses a unique solution

S

A c C[0,T; H ] n C^OjT; H S " 3 ] satisfying A(0,Q = A pends upon A

e H S , S > 2.

A de-

continuously in H -norm.

It is evident from the above theorem that if we prescribe A =a sech 3S at x = 0, then lim A = 0 for any T > 0.

If there is a shelf created behind

the solitary wave, it can never be extended to infinity for finite T.

We

also remark in passing that (4.3) has been used to study the fission of solitons in a channel of variable cross section (Johnson, 1973; Zhong and Shen, 1983) and a justification of the asymptotic method used should also be of interest. ACKNOWLEDGEMENTS.

The work reported here was supported in part by the

National Science Foundation under Grant MCS-800-1960 and the United States Army under Contract No. DAAG29-80-C-0041. REFERENCES BERGER, N. (1974) Estimates for the derivatives of the velocity and pressure in shallow water flow and approximate shallow water equations, SIAM J. Appl. Math. 27^ 256-280.

82

M.C. SHEN

BERGER, N. (1976) Derivation of approximate long wave equations in a nearby uniform channel of approximately rectangular cross section, SIAM J. Appl. Math. 31, 438-448. COURANT, R. and HUBERT, D. (1962) Methods of Mathematical Physics Vol. II, Interscience, New York. FRIEDRICHS, K.O. (1948) On the derivation of the shallow water theory, Comm. Pure Appl. Math. 1^, 81-85. GURTIN, M.E. (1975) On the breaking of water waves on a sloping beach of arbitrary shape, Quant. Appl. Math. 33, 187-189. HO, D.V. and MEYER, R.E. (1962) Climb of a base on a beach, Part 1, J_. Fluid Mech. 14, 305-318. JEFFREY, A. and MVUNGI, J. (1980) On the breaking of water waves in a channel of arbitrarily varying depth and width, J. Appl. Math. Phys. ZAMP 31, 758-761. JOHNSON, R.S. (1973) On the development of a solitary wave moving over an uneven bottom, Proc. Camb. Phil. Soc. 73, 183-203. KAKUTANI, T. (1971) Effects of an uneven bottom on gravity waves, J. Phys. Soc. Japan 30, 271-276. KANO, T. and NISHIDA, T. (1979) Sur les Ondes de Surface de l'Eau avec une Justification Mathematique des Equations des Ondes en Eau Peu Profonde, J. Math. Kyoto Univ. 1£, 335-370. KATO, T. (1975) Quasi-linear equations of evolution, with applications to partial differential equations, Lecture Notes in Mathematics 448, Springer-Verlag, New York, 25-70. KATO, T. (1980) The Cauchy Problem for the Korteweg-de Vries Equation, Research Notes in Mathematics 53, Pitnam, New York, 293-307. KELLER, H.G., LEVINE, D.A., and WHITHAM, G.B. (1960) Motion of a bore over a sloping beach, J. Fluid Mech. ]_, 302-316. KNICKERBOCKER, C.J. and NEWELL, A.C. (1980) Shelves and the Korteweg-de Vries Equation, J. Fluid Mech. 9£, 803-817. LIONS, J.L. (1969) Quelques Methodes de Resolution des Problems aux Limites Non Lineaires, Dunod, Paris. MILES, J.W. (1979) On the Korteweg-de Vries Equation for a gradually varying channel, J. Fluid Mech. 91^, 181-190. SHEN, M.C. and MEYER, R.E. (1963a) Climb of a bore on a beach, Part 2, J. Fluid Mech. 1(3, 108-112. SHEN, M.C. and ZHONG, X.C. (1981) Derivation of K-dV equations for water waves in a channel with variable cross section, J. de Mec. 20, 789-801. SHUTO, N. (1974) Nonlinear waves in a channel of variable cross section, Coastal Eng. in Japan 17, 1-12.

NONLINEAR WATER WAVES IN A CHANNEL

83

WHITHAM, G.B. (1958) On the propagation of shock waves through regions of non-uniform area of flow, J. Fluid Mech. 4_, 337-360. ZHONG, X.C. and SHEN, M.C. (1983) Fission of solitons in a symmetric triangular channel with variable cross section, Wave Motion, to appear.

CHAPTER 5 SOLITON BEHAVIOUR IN MODELS OF BAROCLINIC INSTABILITY IRENE M. MOROZ and JOHN BRINDLEY School of Mathematics The University Leeds LS2 9JT W.Yorks. England

1.

INTRODUCTION One of the major concerns of the meteorologist is the degree of

predictability of atmospheric motions.

The classic remarks made by

Lorenz (1963), in the now-celebrated paper in which deterministic equations were first shown to exhibit aperiodic and consequently unpredictable behaviour, that it may be impossible to predict the weather accurately beyond a few days, only too truly reflect the current state of affairs.

Although the availability of very fast computers and greater

accuracy of initial data have brought about some improvement in weather forecasts, there is still a disappointing limit on the length of time for which a weather prediction can be considered to be accurate.

Nevertheless

certain features of atmospheric motion are observed to persist for considerable lengths of time, usually associated with what are known as blocking situations (Berggren et al 1949); a notable persistent factor in another planetary atmosphere is Jupiter's Red Spot.

Among the models

proposed for these phenomena are modons (Fleierl et al 1981) and solitons (Maxworthy and Redekopp, 1976). We shall not be concerned in this article with direct modelling of atmospheric predictability; instead we shall concentrate on phenomena occurring in simple models exposing the essential physical behaviour, and demonstrate that under certain conditions, coherent persistent behaviour is possible. Cyclones, anticyclones and their associated frontal systems are a prominent feature of the mid-latitude westerlies of the Earth's lower atmosphere.

Their importance as weather-bearing systems and more gener-

ally their role in the general circulation of the atmosphere is wellknown if not yet well understood.

Their occurrence and rapidly changing

behaviour is strongly influenced by the existence of larger scale "longwaves" which are remarkable for their persistence and coherence over

SOLITON BEHAVIOUR IN MODELS OF BAROCLINIC INSTABILITY longer periods of time.

85

Both phenomena owe their existence to the avail-

ability of potential energy associated with the baroclinicity of the fluid, i.e. the non-coincidence of surfaces of constant gravitational potential and constant density, which is a possible equilibrium in a rotating system. Such an equilibrium is unstable and wave-like perturbations can develop at the expense of the potential energy if the trajectories of fluid particles are contained within the geopotentials and isopynals.

This

process is known as sloping convection or baroclinic instability, and the consequent waves as baroclinic waves. Laboratory studies of baroclinic instability have contributed much to our understanding of the properties of baroclinic waves.

Experimental

systems usually comprise two concentric cylinders containing the working fluid in the annular region between them.

The Earth's rotation is

simulated by co-rotating the cylinders and stable density stratification is achieved either by differentially heating the fluid (see Hide and Mason, 1975) or by imposing a differential shear between two immiscible fluids of different density (see Hart, 1979). An analysis of the thermally driven system reveals the existence of sixteen non-dimensional parameters which specify the behaviour of the system (Fowlis and Hide, 1965), although in practice only a small subset is of significance.

The behaviour of the flow is conventionally

summarised in a regime diagram of Thermal Rossby Number (a stratification parameter) against Taylor number (a viscosity parameter).

For a given

temperature contrast a variety of flows is observed ranging from axisymmetric zonal flow at low rotation speeds, through bands of periodic and modulated waves at moderate rotation speeds to aperiodic flow at high rotation speeds.

The review by Hide and Mason (1975) contains an

excellent account of the thermally driven annulus system;

similar

behaviours are encountered in the mechanically driven two layer model (see Hart, 1979 for a review). The development of mathematical models has proceeded in parallel with the development of laboratory models.

Mathematical models are

almost invariably infinite channel models (with a spatial periodicity imposed

to mimic an annular geometry);

the simplest are the

heterogeneous model due to Eady (1949), and the two layer model due to Phillips (1954).

The basic state for the Eady model is one of linear

vertical shear;

for the two layer model the zonal velocity is constant

(but different) in each layer.

Over the last decade a number of authors

86

I.M. MOROZ and J. BRINDLEY

have studied various aspects of the weakly nonlinear behaviour of wavelike perturbations to the basic states of both models, when the amplitude is permitted to vary slowly in both time and/or space.

Significant

contributions in this area have been made by Drazin (1970,1972) for the continuously stratified model and Pedlosky (1970,1971,1972) for the twolayer model. Depending on the orders of magnitude of various parameters which enter the formulation of the problem, it is possible to obtain a wide variety of nonlinear evolution equations, some of which are completely integrable equations with soliton solutions when friction is ignored, others admit aperiodic solutions and have strange attractors is present.

when friction

In this article was shall concentrate on the integrable

equations although it is of interest to indicate how friction modifies the form of the equations and the character of the solutions.

A fuller

treatment can be found in Moroz and Brindley (1982) . The motivation for considering baroclinic wave-packet behaviour lay partly in the experiments of Hide, et al (1977) and other experimenters who had long observed a modulation of the baroclinic wave and a recurrence property of data.

They interpreted the modulation in terms of a triad

interaction between the dominant wave, its sidebands, and the long wave. Subsequent numerical integrations performed by F a m e 11 and James (1977) failed to support this conjecture and a wave-packet model was thought to be an alternative and possibly better way Pedlosky (1972)

of viewing long-wave modulations.

had derived a coupled pair of equations describing

the evolution of a wave-packet in an inviscid two-layer model of baroclinic instability,

and Gibbon et al (1979)

were able to transform Pedlosky1s

wave-packet equations into the self-induced transparency (S.I.T.) and sineGordon equations, both of which admit soliton solutions.

Subsequently

Moroz (1981) and Moroz and Brindl'ey (1981) were able to show that the same equations arise in the three-layer model (and more generally the N-layer model) and the continuously stratified nonlinear Eady model. The material of this article draws heavily from the main results appearing in Gibbon et al (1979), Moroz (1981) and Moroz and Brindley (1981) .

§2 contains descriptions of both the continuously stratified

and the two-layer models as well as the linear stability theory for both. In §3 we indicate the nonlinear theory and show how the completely integrable equations arise.

§4 is a comparison of theoretical and

experimental results and finally in §5 we show how viscosity enters the problem.

88

I.M. MOROZ and J. BRINDLEY

y

= o

x lim _1_ f x-x» 2x J -x

and

at

y

yt

dx1

on y = + -

(2.2)

on y = + - 2

(2.3)

(2.4)

V'i'x

"(x,y)

where V2

=

B (_

3z 2 J

= x (f/9) f gr - f g , and (x,y) " xy y^x field. In the following we assume

is the non-dimensionalised pressure

,

s

1

= -s2 *

For a two

" l a Y e r model we

have ) + BY|

(x,y,

=

O

i = 1,2 (2.1)

subject to on

ix

y = 0,1

(2.2) '

-x where we have introduced the non-dimensional parameters B

Burger Number

(Stratification Parameter) (Dispersion Parameter)

, and F

Non-dimensional Boundary Slopes * Internal Rotational Froude Number

3

3-effect

(Dispersion Parameter)

s.

(i = 1,2)

(Stratification Parameter)

We seek wave-like perturbations to a simple zonal flow, writing ¥

=

if; =

-yz + \\)

for the continuous model

(Pcosh 2qz + Qsinh 2qz) e -U.y + ij/.

=

,1. ik(x-ct) (y) e

ft (-x—r )

for two-layer

. sin imry

model

(2.5) sin m7r(y+J5)

SOLITON BEHAVIOUR IN MODELS OF BAROCLINIC INSTABILITY

89

and linearise the resulting equations, we obtain characteristic relations: \, E

4q 2 c 2 + 2soq(tanhq + cothq) c + s 2 - (1 + q 2 ) Z £

=

+ q(tanhq. + cothq) 2 2 2 2 4q + B(k + m ^ ) j

for the continuous model where \

(2.6) and

a2 (a (a22 + 2P) c 2 - c [a 2 (a2 + 2 F ) (U. + U o ) - 26 (a2 + F)J

= Li

+ a 2 (a2 + 2F) u U_ + B 2 + F a 2 U 2 - $ (n 12 c

+ U j (a2 + F) 1 2 (2.6) '

for the two-layer model, where

a

= k

+ m T"

The dispersion relations are found by solving ^ The condition for marginal stability,

kc, = O ,

= O

for

c .

yields surfaces which

separate stable and unstable regions of parameter space.

These surfaces

often have a local form dominated by one particular parameter and can be conveniently represented in a two-dimensional plot of that (stability) parameter against zonal wave number stratified model,

s

(Fig.l) .

For the continuously

is the relevant stability parameter and the

condition for marginal stability is

s

2

_ -

4lq(tanhq + cothq) - (1 + q2)J

\z.n

(tanhq - cothq) for the two-layer model it is 9 XT

=

U

4B2F2 a^(4F 2 a' + )

=

U

- U

and

.

(2.7) •

The validity of a wave packet analysis rests on the existence of suitable behaviour in the linear problem.

This must yield a neutral

curve which, at least locally, takes the form shown schematically in figure 1. A ,

For small departures of order

A

of the stability parameter,

from critically, we see that a band of wave numbers of width

unstable, centred on the critical wave number, (1981) have shown that the stationary point for

k

.

A

is

Moroz and Brindley

in the "Eady" model occurs

s^ = 1 and at that point we have a coalescence of two modes with

90

IoM. MOROZ and Jo BRINDLEY

Figure 1 - Typical Neutral Stability Curve

SOLITON BEHAVIOUR IN MODELS OF BAROCLINIC INSTABILTY

identical phase speeds,

(c = -h) but different group speeds.

else on the neutral curve the phase speeds differ.

91

Everywhere

A similar result is

true for the two-layer model; Pedlosky (1972b) has shown that the stationary point is given by

U

= $/F

and at that point

c

c = U (twice). ^L

We shall see later that this is why we expect to obtain an evolution equation for the wave amplitude which is second order in space and time. The shape of the neutral curve near the stationary point therefore determines the bandwidth of excited waves;

the growth rate near the

marginal stability curve can also be computed. we have

For inviscid problems

kc. = 0(A ) .

3. NONLINEAR THEORY The results from linear theory suggest that we introduce new ables variables

X

and

T

resp , scaled respectively on the bandwidth and growth

rate of the most unstable wave, i.e. (3.1) We also require the variables amplitude

A(X.,T.)

and time.

(X ,T ) = |A|(x,t)

and assume that the

of the wave is a slowly varying function of space

The solution may then be developed as an expansion in a small

parameter, related to the departure from neutral stability.

This method

has been formalised by Newell (1972), Weissman (1979) and Gibbon and McGuinness (1981), and the result is a system of evolution equations which must be satisfied by the wave amplitude at each order of the expansion.

The coefficients of the linear terms in these equations are

identifiable as derivatives of the characteristic function, T respect to its various arguments;

, with

the nonlinear terms are specified by

the particular problem under consideration. For the inviscid baroclinic wave models we have •*A

+

=

0

(3.2)

\

-T\T(A)

(3.4)

92

I.M. MOROZ and J. BRINDLEY

A number of special cases arise as follows: (i) Marginally unstable wave packets with 3 or s For an inviscid model, ^ ^

= 0

X

and T

= 0

a

of 0(1)

everywhere on the neutral curve but

only at the critical point (Weissman, 1979).

The slow scales

are not required and the amplitude evolves according to the

second order equations

( +

° $( +C A> A 5'ANAB

where 52

is the square of the linear growth rate

N

is a positive constant

B(XJT )

is a second order correction to the basic state

and

_»1 T

g

ak

i

ft 2 _%

±

Qk Voo

l,2

1 }*2 oo

kk

are the group speeds. Gibbon et al (1979) have shown that the transformation S

=

± 1 - NB/|a| 2

R = /2 A

(3.6)

and the change of variable », ( x " c g i T )

i-i2

-"»

(X C

-