An introduction to the mathematical theory of geophysical fluid dynamics

AN INTRODUCTION TO THE MATHEMATICAL THEORY OF GEOPHYSICAL FLUID DYNAMICS

This Page Intentionally Left Blank

NORTH-HOLLAND MATHEMATlCS STUDIES

41

Notasde Matematica (70) Editor: Leopoldo Nachbin Universidade Federal do Rio de Janeiro and University of Rochester

An Introduction to the Mathematical Theory of Geophysical Fluid Dynamics SUSAN FRIEDUNDER Department of Mathematics University of Illinois at Chicago Circle Chicago, Illinois, U.S.A.

1980

NORTH-HOLLAND PUBLISHING COMPANY - AMSTERDAM

NEW YORK

OXFORD

Q North-Holland Publishing Company,

1980

All rights reserved. No part of this publication may be reproduced, stored in a retrieval system or transmitted, in any form or by any means, electronic, mechanical, photocopying, recording or otherwise, without the prior permission of the copyright owner.

ISBN 0 444 86032 0

Publishers NORTH-HOLLAND PUBLISHING COMPANY

AMSTERDAM*NEWYORK*OXFORD Sole distributors for the U S A . and Canada: ELSEVIER NORTH-HOLLAND. INC. 52 VANDERBILT AVENUE. NEW YORK. N.Y. 10017

Library of Congress Cataloging in Publication Data Friedlander, Susan, 1946Introduction to the mathematical theory of geophysical fluid dynamics. (Notas de matem6tica ; 70) (North-Holland mathematics studies ;41) Bibliography: p. Includes index. 1. Fluid dynamics. 2. Geophysics. I. Title. 11. Series. QAl.N86 no. 70 [QC809.F5] 510s [532 '.05] 80-16811 ISBN 0-444-86032-0

PRINTED IN THE NETHERLANDS

To E r i c

This Page Intentionally Left Blank

PREFACE

This work i s based on a s e r i e s of l e c t u r e s given t o graduate students both a t the University of I l l i n o i s , Chicago Circle i n t h e United S t a t e s , and a t Oxford University i n England.

It i s intended t o provide a framework f o r such a

course given primarily t o graduate students i n applied mathematics, a s well a s t o be a u s e f u l supplementary t e x t f o r students of oceanography, meteorology and engineering.

A

f a m i l i a r i t y with b a s i c f l u i d dynamics i s assumed and some knowledge of asymptotic techniques would be h e l p f u l .

Since

the material presented i s confined t o a s i n g l e course, the t o p i c s covered r e f l e c t t o a c e r t a i n extent personal preference.

A number of important t o p i c s , p a r t i c u l a r l y those con-

cerning aspects of non-linear r o t a t i n g f l u i d dynamics a r e omitted here and await treatment elsewhere.

However, the

fundamentals of t h e o r e t i c a l geophysical f l u i d dynamics a r e given from f i r s t p r i n c i p l e s i n order t h a t they may be e a s i l y a c c e s s i b l e t o a motivated reader. The authors wishes t o thank Professor L. Nachbin f o r h i s p a t i e n t supervision of t h i s monograph and t o acttnowledge t h e very constructive c r i t i c i s m given by Professors V . Barcilon, F. Busse, E . Isaacson, N. Lebovitz and W . Siegmann.

The author

i s most g r a t e f u l t o the Mathematical I n s t i t u t e a t Oxford f o r Y ii

viii

Preface

t h e i r h o s p i t a l i t y and support. supported by N.S.F.

This work was a l s o p a r t i a l l y

Grants MCS 78-01167 and MCS 79-01718.

F i n a l l y , t h e author extends thanks t o Ms. S h i r l e y Roper

f o r h e r e x c e l l e n t typing of t h i s book.

Susan Friedlande r

Chicago, I l l i n o i s January, 1980

TABLE O F CONTENTS Page INTRODUCTION CHAPTER 1: CHAPTER 2:

CHAPTER 3 : CHAPTER 4:

1

EQUATIONS OF MOTION POTENTIA L VORTIC ITY Problems NON-DIMENSIONAL PARAMETERS Problems GEOSTROHIC FLOW Taylor-Proudman Theorem Taylor Column Application t o Geophysical Motion f3 Plane Approximation Problems

-

CHAPTER 5:

CHAPTER 6:

CHAPTER 7:

CHAPTER 8:

CHAPTER 9:

THE EKMAN LAYER EKmn Layer Equations Example of C y l i n d r i c a l Flow Ekmn Layer S p i r a l Mass Transport i n t h e EkLcman Layer Spin-up Time S c a l e Tea-cup Experiment Problems THE GEOSTROPHIC MODES The Geostrophic Mode i n a Sphere Geostrophically Free, Guided, and Blocked Regions Circulation Problems INERTIAL MODES X Real and 1x1 < 2 Orthogonality Mean C i r c u l a t i o n Theorem I n i t i a l Value Problem I n e r t i a l Modes i n a Cylinder Plane Wave S o l u t i o n Problems ROSSBY WAVES S l i c e d Cylinder @-Plane Problem Plane Wave S o l u t i o n Problems VERTICAL SHEAR LAYERS E Laye r E1’4-Layer S l i c e d Cylinder An Ocean Model: Sverdrup’s R e l a t i o n Problems

’-

ix

5 11

15 17

20

21 21

23

26 28 33 35 39 43 46 47 48 52 54

62 63 65 67 68 70 71 72 74 77 80 85 86 89 95 97 99 100

102 110

114 120

X

Table of Contents Page

CHAPTER 10:

ANALOGIES BETWEEN ROTATION AND STRATIFICATION Problems

CHAPTER 11:

THE NORMAL MODE PROBLEM FOR ROTATING STRATIFIED FLOW The Steady Flow Potential Vorticity Problems ROSSBY WAVES I N A ROTATING STRATIFIED FLUID The P o t e n t i a l V o r t i c i t y Equation Rossby Waves f o r a S t r a t i f i e d Fluid Roasby Radius of Deformation Problems

CHAPTER 12:

CHAPTER 13:

CHAPTER 14:

CHAPTER 15:

CHAPTER 16:

APPENDIX BIBLIOGRAPHY INDEX

INTERNAL WAVES I N A ROTATING STRATIFIED FWID Plane Wave S o l u t i o n Waves i n Bounded Geometry Variable N ( z) Oceanographic Results Problems BOUNDARY LAYERS I N A ROTATING STRATIFIED FLUID The S t r a t i f i e d Ekman Layer The Side-wall Layers Problems SPIN-DOWN I N A ROTATING STRATIFIED FLUID Spin-down i n a Cylinder S e c u l a r Growth The Steady S o l u t i o n The Decaying Modes Further Comments Problems BAROCLINIC INSTABILITY The Eady Model The S t a b i l i t y C r i t e r i o n Experiments : Laboratory Models Problems BOUNDARY LAYER METHODS

123 131 133 137 141 147 151

151 153

156 159 161

163 166 176 187 189 191 193 196 206

'Log 2 12 2 19 220 222 'L 26 228 231 232 2 36 243 247 249 263 269

INTRODUCTION

For many c e n t u r i e s man has attempted t o g a i n some understanding of the behavior of the ocean and the atmosphere, with the impetus f o r such work coming from the need t o p r e d i c t the motion of the water and a i r t h a t surround us.

I n ancient times

knowledge came almost e n t i r e l y from records of p r a c t i c a l observation, but the l a s t century has seen g r e a t advances i n t h e t h e o r e t i c a l , numerical and experimental techniques which a r e used t o study t h i s important branch of science. Geophysical f l u i d dynamics, in i t s broadest sense, is the study of f l u i d motions in t h e e a r t h .

The purpose of t h i s book

is t o give a mathematical d e s c r i p t i o n of a c e r t a i n c l a s s of such phenomena.

We w i l l be concerned with those problems f o r

which the length s c a l e i s s u f f i c i e n t l y l a r g e t h a t t h e e a r t h ' s r o t a t i o n has a s i g n i f i c a n t e f f e c t on the dynamics of the f l u i d . Hence we w i l l exclude many i n t e r e s t i n g small s c a l e problems, f o r example, those connected with s u r f a c e tension, but we w i l l discuss the mathematics t h a t describes b a s i c models f o r the motion of the ocean and t h e atmosphere.

Besides the relevance

t o geophysics, the s u b j e c t i s a n appealing one t o a mathemat i c i a n because the p a r t i a l d i f f e r e n t i a l equations which a r i s e frequently d i s p l a y i n t e r e s t i n g and r a t h e r unusual p r o p e r t i e s . We consider the t h e o r e t i c a l aspects of geophysical f l u i d

I n t roduc t i o n

2

dynamics by g i v i n g a n i n t r o d u c t i o n t o t h e mathematical theory of r o t a t i n g f l u i d motion.

I n Keeping with t h e theme of r e l e -

vance t o geophysical problems, t h e l a t e r s e c t i o n s of t h e booic i n c l u d e a f u r t h e r c h a r a c t e r i s t i c f e a t u r e of t h e ocean o r t h e atmosphere, namely t h a t t h e motion i s influenced by t h e e f f e c t s of g r a v i t y on a f l u i d of non-uniform d e n s i t y .

Thus t h e f i r s t

h a l f of t h e book concerns a r o t a t i n g homogeneous f l u i d , and t h e second h a l f c o n s i d e r s a f l u i d s u b j e c t t o t h e f o r c e s of both r o t a t i o n and s t r a t i f i c a t i o n . We develop t h e mathematical a n a l y s e s i n a n ordered f a s h i o n , s t u d y i n g f i r s t t h e equations t h a t d e s c r i b e t h e s i m p l e s t physics, namely small p e r t u r b a t i o n s from t h e e q u i l i b r i u m of a homogeneous i n v i s c i d r o t a t i n g f l u i d .

We t h e n proceed from

t h i s base t o add l a y e r upon l a y e r of mathematical complexity a s f u r t h e r r e l e v a n t p h y s i c a l f a c t o r s a r e included i n t h e model. Where i t i s a p p r o p r i a t e , we w i l l d e s c r i b e simple l a b o r a t o r y experiments t h a t i l l u s t r a t e phenomena c h a r a c t e r i s t i c of a rotating fluid.

Given t h e scope of t h i s book i t i s not p o s s i -

b l e t o provide d e t a i l s about t h e many a p p l i c a t i o n s o f mathematics t o geophysics.

We w i l l however mention s e v e r a l r e l e -

vant problems and g i v e a more e x t e n s i v e d i s c u s s i o n o f t h r e e r e p r e s e n t a t i v e examples.

I n t h e oceanic example we u s e

boundary l a y e r theory t o e x p l a i n t h e e x i s t e n c e of t h e Gulf Stream on t h e western s i d e of t h e A t l a n t i c .

I n t h e metero-

l o g i c a l c o n t e x t we show t h a t a n i n s t a b i l i t y a s s o c i a t e d w i t h t h e l a t i t u d i n a l v a r i a t i o n of s o l a r h e a t i n g of t h e atmosphere i s c r u c i a l t o t h e formation of cyclone waves.

We a l s o g i v e a n

i n t e r e s t i n g a s t r o p h y s i c a l a p p l i c a t i o n , d e s c r i b i n g i n some

Introduction

3

d e t a i l , the formulation of a well posed mathematical problem i n terms of a s i n g l e p a r t i a l d i f f e r e n t i a l equation w i t h approp r i a t e boundary conditions, whose s o l u t i o n sheds l i g h t on the s o l a r spin-down controversy. The basic construction of each mathematical model t r e a t e d i n t h i s book is given i n d e t a i l i n order t o provide s u f f i c i e n t information t o communicate the essence of the material t o an u n i n i t i a t e d reader.

However, a f a i r l y extensive l i s t of r e f e r -

ences and sources i s provided f o r those who wish t o pursue a p a r t i c u l a r topic i n g r e a t e r depth.

Included i n the references

a r e basic t e x t s , e a r l y seminal papers and recent surveys of r e s u l t s , a s well a s c u r r e n t advances i n c e r t a i n a r e a s .

We

o f f e r the following b r i e f s e l e c t i o n of b a s i c t e x t s t h a t a student should find p a r t i c u l a r l y valuable i n the study of geophysical f l u i d dynamics. Fluid dynamics :

Batchelor Lamb

[5]

Rotating f l u i d s :

Greenspan Carrier Howard

[ 271

Stratified fluids:

Yih

[ 741

Geophysical f l u i d dynamics:

PedlosKy

[51], [ 5 2 ]

Oceanography :

Kamemovich [ 371 Krauss [393 Phillips [55]

[ 401

[81 [ 331

I n the appendix we give a b r i e f i n t r o d u c t i o n t o boundary layer techniques a s they a r e used t o study s i n g u l a r perturbat i o n problems.

For a more d e t a i l e d exposition of t h i s branch

of asymptotic a n a l y s i s the reader is referred t o S c h l i c h t i n g [60] Van DyKe [67].

This Page Intentionally Left Blank

CHAPTER 1

EQUATIONS O F MOTION

The problems t h a t we w i l l consider a r e characterized by the importance of r o t a t i o n .

To r e a l i z e i t s s i g n i f i c a n c e i n the

geophysical context we observe t h a t

-

6 x 108 em

R -

n

( r a d i u s of the e a r t h )

7.5 x 10-5 s e c - l (angular v e l o c i t y )

hence the v e l o c i t y i n equational l a t i t u d e s , r e l a t i v e t o the a x i s of r o t a t i o n , i s of the order

4 x lo4 c d s e c .

Such a

v e l o c i t y i s very l a r g e compared with t y p i c a l winds i n t h e atmosphere ( f o r example, a hurricane wind i s

0(104) cm/sec).

Also, the v o r t i c i t y ( t h e physical concept t h a t measures the llswirl'lo r " c u r l " i n a f l u i d motion) due t o the e a r t h ' s rotat i o n i s very l a r g e compared w i t h the v o r t i c i t y of t y p i c a l motions t h a t occur on a large s c a l e i n the ocean o r atmosphere. Thus, when t.he h o r i z o n t a l length s c a l e i s comparable t o the radius of the e a r t h , i t i s always necessary t o take i n t o a c c m n t the e f f e c t of the e a r t h ' s r o t a t i o n . It i s frequently convenient i n the study of r o t a t i n g f l u i d

motion t o w r i t e the equations of motion i n the r o t a t i n g coordinate system.

Let us b r i e f l y review r o t a t i o n i n 2-

dimensional motion. angular v e l o c i t y

n

Consider a plane r o t a t i n g with constant about the

k axis. 5

Let

(i,j,k) . ) A

denote

6

Rotating co-ordinate System

Cartesian u n i t vectors i n the i n e r t i a l frame of reference and

(1' ,3'

,&I

)

denote Cartesian u n i t s vectors i n the r o t a t i n g

frame of reference.

I

V 'i

The i n e r t i a l and r o t a t i n g co-ordinates FIGURE 1 A t a time

Let

&

t

the u n i t vectors s a t i s f y t h e r e l a t i o n s

It

=

t

COB

3'

=

-t

sin

nt + 3 nt + 3

sin cos

nt nt.

(1.1)

(1.2)

denote d i f f e r e n t i a t i o n following a p a r t i c l e .

Let

be a vector which can be w r i t t e n as

+ A23 + A ~ +Z A~k j l +

A = All

=

A3k

( i n e r t i a l frame) ( r o t a t i n g frame).

9

Equations of motion Then

2 ?' + 2 1' + 2 3' -

d4 = dA'

dA'

dA'

dA'

dt

I n p a r t i c u l a r , i f we taKe

+

A:

&+A!& dt

L dt

-

AiP )

+ n(Al,j'

4

=

7

from ( l . l ) - ( l . Z ) .

the radius vector measured

f,

from the common o r i g i n of the co-ordinate s y s t e m , we o b t a i n

-d,r- -dtI

Where dz dtR

=

[Note:

ds dtI

&'

=

gI,

dr dtR

+ nk

x

4.

v e l o c i t y measured i n the i n e r t i a l frame and

v e l o c i t y measured by an observer i n the r o t a t i n g frame

.

i f the p a r t i c l e s a r e i n r i g i d body r o t a t i o n

91

=

nR

I f we now s e t

x

4 = gI

qR = 01.

and

we o b t a i n the r e l a t i o n s h i p between the

a c c e l e r a t i o n i n the two frames of reference, x

[gR + nii x 41

nf x 5). I n t h i s equation the f i r s t term i s a c c e l e r a t i o n i n the r o t a t i n g co-ordinate system, the second term i s c a l l e d C o r i o l i s ' a c c e l e r a t i o n , and the t h i r d term is c a l l e d c e n t r i fugal acceleration.

With a l i t t l e more work a general formula

corresponding t o (1.5) can be derived i n terms of general c u r v i l i n e a r co-ordinates

.

We r e c a l l the Navier-Stokes equations which govern the motion of a viscous f l u i d .

I n an i n e r t i a l frame they a r e

8

Nav i e r-S t okes equations

given by the following two vector equations:

T h i s i s the equation of conservation of mass which mathematlc-

a l l y describes the f a c t t h a t , i n the absence of 8ources o r sinks, each f l u i d p a r t i c l e may move around but the t o t a l mass remains constant dtI

.

= -VP

+

pot3

+ po 2CJ* + Lfp

V'cJI.

T h i s i s the equation of conservation of momentum which is the

a p p l i c a t i o n of Newton's law of motion [Force = mass x acceler a t i o n ] t o a f l u i d system. Thus rewriting the Navier-Stokes equations I n terms of a uniformly r o t a t i n g co-ordinate system gives

dQ p[
b e

t h e d e p t h averaged v e l o c i t y of a n i n e r t i a l

I n i t i a l v a l u e problem

where

z = z

container.

d e f i n e s t h e t o p and

T

z = z

B

t h e bottom of t h e

I n t e g r a t i n g e q u a t i o n ( 6 . 4 ) with r e s p e c t t o

from t o p t o bottom gives ihm>

d i s s i p a t e d by viscaus a c t i o n . waves with

2 g2

2

> u1

2

K ~ a , re

those most s t r o n g l y

Hence energy t h a t a r r i v e s i n

a t a western boundary has a tendency t o

be r e f l e c t e d i n waves t h a t a r e d i s s i p a t e d by v i s c o s i t y . Therefore not a l l of t h e a r r i v i n g energy i s returned t o t h e i n t e r i o r , but some of t h e energy i s used t o b u i l d up a viscous boundary l a y e r .

This formation of a western boundary l a y e r

by a mechanism of "trapping energy" i s of g r e a t importance i n t h e dynamics of oceans s i n c e i t gives r i s e t o s t r o n g western boundary c u r r e n t s of which t h e A t l a n t i c Gulf Stream and t h e Kuroshio c u r r e n t off t h e c o a s t of Japan a r e two examples.

8.1)

(a)

Problems

97

Discuss the Rossby waves t h a t e x i s t i n a cylinder with a h o r i z o n t a l top a t and a parabolic bottom

z

= L

z = a ( x2+y 2 ) .

Obtain the s o l u t i o n t o t h e Rossby wave equation. (b) Is t h e r e a geostrophic mode f o r t h i s

container? (c)

What modes c a r r y the i n i t i a l c i r c u l a t i o n ?

z = L

I I

I I I

1.

k’

/

/

/

2

= cL(x+Y

’’

FIGURE 17

2

1

->

98

Problems

8.2)

(a)

Consider the Bout-

p-plane problem

hemisphere.

i n the

Discuss the d i r e c t i o n

of propagation of the phase v e l o c i t y and the group v e l o c i t y of Rossby waves i n a southern ocean basin. (b)

Does the boundary layer form on the western o r e a s t e r n boundary?

(c)

I n t e r p r e t your answer i n terms of the strong ocean boundary currents i n the southern hemisphere.

8.3)"

( a ) What i s the flow i n the annular region

bounded by i n f i n i t e l y long concentric cylinders of radius with angular v e l o c i t y respectively.

.

rl

and

nl

r2

and

rotating

n2

[The f l u i d i s incompressible

and v i s c ou6 ] (b)

Assuming t h a t t h i s flow i s not t o o much a f f e c t e d by h o r i z o n t a l end walls, can the v a r i a t i o n of the zonal v e l o c i t y be used t o model the

p-plane e f f e c t ?

[ I n o t h e r words,

can we use d i f f e r e n t i a l r o t a t i o n of a v e r t i c a l wall of an annulus mounted on a t u r n t a b l e t o simulate the

@-effect.]

CHAPTER

9

VERTICAL SHEAR LAYERS

I n problems concerning flow i n a r o t a t i n g cylinder t h a t we considered i n Section 5 J we have shown t h a t i t i s possible t o determine the

0(1)

i n t e r i o r motion by considering t h e e f f e c t s

of Ekman l a y e r s u c t i o n .

We r e c a l l the example i l l u s t r a t e d i n

Figure 7 where the angular v e l o c i t y o f the bottom of the cylinder i s increased.

I n t h i s c a s e J Ekmn l a y e r sunction

induces a negative v e r t i c a l mass f l u x of

O(E1’2).

In a

general problem we again expect a small i n t e r i o r v e r t i c a l flux.

I n order t o r e t u r n t h i s mass f l u x i t i s c l e a r l y

necessary t o i n v e s t i g a t e the v e r t i c a l shear l a y e r s supported by a r o t a t i n g f l u i d . We w i l l f i r s t consider the case of a right c i r c u l a r cylinder.

We r e c a l l the steady viscous equations of motion I

2k x

3

=

0.9 =

-VP

+

2

Ev j

(9-1) (9.2)

0.

The manipulation of these equations given i n S e c t i o n 5 gives the pressure equation

99

100

El/3-w We consider the s t r u c t u r e of a s i d e wall layer by writing X E = ~

r

- a,

thus

The dominant terms i n equation (9.3) become

Hence, balance between these two terms requires

1.e

., t h e r e

thickness

e x i s t s a v e r t i c a l shear l a y e r of dimensionless O(Ev3).

We now seek an asymptotic expansion f o r denotes a n E113-layer

quantity].

6

and

Since we require t h i s

boundary layer t o r e t u r n a v e r t i c a l mass f l u x of the v e r t i c a l component of v e l o c i t y

[(-)

= O(Ea)

O(Ev2)

where

The problems we a r e considering is axisymmetric, hence we w i l l a8sume

= 0.

The divergence equation then gives

Thus a balance of t e r m requires

V e r t i c a l shear l a y e r s

6

=

O(E 1/2 )

101

.

The components of the momentum equation give: -23 =

-E-’I3Fx

+

-

E 1/3-uxx

E1/3;

2u =

-

0 = -P

+

z

E

xx

1/3wXx*

Hence bala ce of terms i n (9.7) and (9.8) requires

5

=

and

O(E

i

=

O(E 1/2 )

.

Thus the v e l o c i t y components and pressure a r e given by an asymptotic expansion i n powers of

-

E1’2G3

u =

-

v = E 1/67

+

i

+

=

-P =

+

...

+

...

......

...... EU2F

And the highest order equations a r e : i l

3X

+w

lz

=

-P

2 3 =

7

-23,

= o

3x

o=-F

IXX

3z

+ i

lxx

as:

102

B1/4-Lave We have found a v e r t i c a l boundary l a y e r of

O ( E’’l

)

where

t h e o r d e r of t h e v e l o c i t y components i s such t h a t t h e v e r t i c a l

mass flux i s of t h e o r d e r of t h a t of

t h e i n t e r i o r , namely

0(E1I2). However we observe t h a t t h e t a n g e n t i a l v e l o c i t y component

G

is

Hence i t i s

O(E1’6).

t h i s boundary l a y e r t o match t h e

possible t o use velocity

O(1)

v

i n t e r i o r with a g e n e r a l s i d e w a l l boundary c o n d i t i o n . that the i n t e r i o r velocity

v

i n the [Recall

i s determined by t h e Ekman

l a y e r s u c t i o n c o n d i t i o n and w i l l not n e c e s s a r i l y s a t i s f y t h e s i d e wall boundary c o n d i t i o n ] .

It i s t h e r e f o r e necessary t o

s e e k a f u r t h e r v e r t i c a l s h e a r l a y e r i n which t h e v e l o c i t y component

v

i s prescrib.ed t o be

O(1).

I n t h e second l a y e r we w r i t e qEB = r

-

a , where

p

4 l/j.

Hence

Let

(*)

denote a boundary l a y e r q u a n t i t y .

The dominant

terms i n t h e p r e s s u r e e q u a t i o n ( 9 . 3 ) g i v e

For v a l u e s of

p

>

1/3

t h i s e q u a t i o n reduces t o

V e r t i c a l shear layers

103

However t h e r e i s no n o n - t r i v i a l s o l u t i o n t o an equation of t h i s form t h a t could s a t i s f y the boundary conditions on t h e

v e l o c i t y component

#u

v

and a l s o ensure t h a t lim

-

P = 0.

-03

Hence

must be l e s s than

f3

1/3

and the equation (9.13)

reduces t o

9 = 0. az 2-

The components of the momentum equation a r e

and the symmetric divergence equation i s

Recall

m

v = 0(1), hence from (9.16) we require, 1-28 ) = O(E and from (9.18), m w = O ( E 1-23). m


,

giving A,

=

2

K 1

for to

n n

=

0,1,2,.....

= 0

.

+

2

K2

+

2 2 2 ’

fon

(12.17)

N2h2

We note t h a t t h e mode corresponding

has eigenvalue

Baroclinic Rossby waves

Ao=

and eigenfunction

=

2 K1

+

155

(12.18)

2

K2

constant.

T h i s mode c l e a r l y corresponds t o the homogeneous Rossby wave.

It i s c a l l e d the barotropic mode and i s unaffected by s t r a t i fication.

Because t h i s mode i s independent of

a r y conditions imply t h a t

w

must be

z, t h e bound-

zero everywhere.

Hence

the motion i s purely h o r i z o n t a l and a f l u i d p a r t i c l e i s not required t o cross the density gradient and experience the e f f e c t s of s t r a t i f i c a t i o n . The v e r t i c a l s t r u c t u r e of s t r a t i f i e d Rossby waves appears i n the higher modes

n = 1,2,.

.... .

These a r e c a l l e d t h e

b a r o c l i n i c modes and the pressure i s given by

From the geostrophic balance Implied by equation (11.5) the h o r i z o n t a l v e l o c i t y components a r e :

And from ( 1 2 . 1 2 ) and (12.13) the v e r t i c a l v e l o c i t y i s given by

Thus the higher modes have a s t r o n g degree of v e r t i c a l s t r u c -

156

Rossby radius of deformation

t u r e with the v e l o c i t y components having s i n u s o l d a l dependence.

z-

We a l s o note t h a t unlike t h e homogeneous problem,

wn

the v e r t i c a l v e l o c i t y I n f a c t , the values of

m = 0, il, a 2 , ...)

i s not zero f o r a l l values of a t which

z

wn

i s zero

(z

=

z.

mh

n ' a r e p r e c i s e l y those values a t which the

magnitude of t h e h o r i z o n t a l v e l o c i t y i s g r e a t e s t . i t-

0

on

0

Let us consider the dispersion r e l a t i o n (12.16) f o r the n-th

Baroclinic mode.

It can be r e w r i t t e n as

where the e f f e c t s of s t r a t i f i c a t i o n a r e r e f l e c t e d i n t h e term

0 f2n n2 . 1 N2h2 Clearly the eigenvalue

Xn

[K: + K:]

only d i f f e r e e a-gnificantly from

t h a t of the barotropic ( o r homogeneous) mode

Lo,

given by

( 1 2 . 1 8 ) , when

Hence the e f f e c t s of s t r a t i f i c a t i o n a r e only s i g n i f i c a n t , f o r each mode, when the following c r i t e r i o n i s s a t i s f i e d , namely

15 7

Baroclinic Rossby waves

(12.19)

Now

2 [U1

+

is t h e square of the h o r i z o n t a l wave number,

Ui]

hence o(gl 2

where

L

+ u2)2

=

o(n2n2L-2),

i s the h o r i z o n t a l length s c a l e .

Thus t h e s t r a t i f i -

c a t i o n c r i t e r i o n (12.19) can be r e w r i t t e n i n t h e form

The value

* fO

L > = . fO =

$

(12.20)

i s c a l l e d the Rossby radius of deformation.

We conclude t h a t , f o r a given s t r e n g t h of s t r a t i f i c a t i o n and given depth

N

h, Rossby waves a r e only influenced by s t r a -

t i f i c a t i o n i f the h o r i z o n t a l length s c a l e exceeds the Rossby radius of deformation.

We remark t h a t t y p i c a l values of

i n the atmosphere and ocean a r e lOOOkm and 60km, r e s p e c t i v e l y . These a r e length s c a l e s t h a t a r e frequently encountered i n t h e study of motions I n the atmosphere and ocean. We have given here a n i n t r o d u c t i o n t o t h e theory of Rossby waves i n a geophysical context.

Since the i n i t i a l study of

the t o p i c by Rossby [ 5 7 ] , i n 1939, the complexity of the subj e c t has increased considerably.

A s i g n i f i c a n t body of work

has developed which shows the importance of Rossby waves i n understanding the movement of l a r g e s c a l e disturbances i n the oceans and the atmosphere.

The e s s e n t i a l f e a t u r e s , t o d a t e ,

a r e given i n a n a r t i c l e by Dickinson [ 141.

V a r i a b l e depth We p a r t i c u l a r l y mention s e v e r a l e x t e n s i o n s of t h e work d e s c r i b e d i n t h i s c h a p t e r , t h a t a r e discussed i n d e t a i l by DicKinson.

F i r s t , i n geophysical problems, t h e depth

t h e l a y e r of f l u i d is not g e n e r a l l y c o n s t a n t .

h

of

The i n t r o -

d u c t i o n of a v a r i a b l e depth l e a d s t o a f u r t h e r term i n t h e p o t e n t i a l v o r t i c i t y e q u a t i o n ( s e e Chapter 8 and e q u a t i o n (8.21)).

It is t h e n a p p r o p r i a t e t o g e n e r a l i z e

mean p o t e n t i a l v o r t i c i t y g r a d i e n t .

'N

=

-Pa az

4

t o the

Second, t h e parameter

is not c o n s t a n t , i n f a c t , i n t h e ocean i t has a

f a i r l y high degree of v e r t i c a l s t r u c t u r e which d e f i n e s t h e thermoclines.

It can be shown t h a t t h e f i r s t b a r o c l i n i c mode

( n = l ) i s s t r o n g l y dependent on t h e s t r u c t u r e of .'N

Hence

a r e a l i s t i c study of ocean Rossby waves r e q u i r e s working w i t h

a p o t e n t i a l v o r t i c i t y e q u a t i o n with non-constant c o e f f i c i e n t s .

Prob lens Chapter 12 Problems 12.1)

I n a s t r a t i f i e d f l u i d t h e equation f o r Rossby waves i s given by

Obtain t h e s o l u t i o n f o r waves i n a closed cylinder by SeeKing a s o l u t i o n of t h e form P = A(x,y,z)e

- A

i x ,ipxt

t h a t s a t i s f i e s the above equation, together with t h e boundary conditions

az

=

o

at

z = 0,1

and

u = o at

ae

12.2)

r = a .

How do the Rossby waves i n a s t r a t i f i e d f l u i d obtained i n problem ( 1 2 . 1 ) d i f f e r from t h e barotropic Rossby waves described i n Chapter 8?

12.3)

I n oceanic models the presence of a f r e e s u r f a c e modifies the boundary condition.

~

We t h e r e f o r e consider the problem

159

Problems

160

w 1t h t h e boundary c ondi t i o n s w = g z at

Z = O

w=o

z = -h.

and

a)

Obtain t h e plane wave s o l u t i o n of t h e form P = r(z)e

b)

at

i(KIX

+

K2Y)

How does t h e s o l u t i o n d i f f e r from t h e Rossby wave s o l u t i o n i n a c o n t a i n e r w i t h a r i g i d lid?

CHAPTER 13 INTERNAL WAVES I N A ROTATING STRATIFIED FLUID

S t r a t i f i c a t i o n provides a r e s t o r i n g force and hence allows the existence of i n t e r n a l waves ( s e e Problem ( 1 0 . 2 ) .

I n view

of the analogy between r o t a t i o n and s t r a t i f i c a t i o n (Chapter lo), we would expect i n t e r n a l g r a v i t y waves t o have similar

p r o p e r t i e s t o t h e i n e r t i a l waves supported by r o t a t i o n :

this

i s i n f a c t the case ( Y i h [ 7 6 ] ) . We w i l l now consider the int e r n a l waves t h a t e x i s t when a f l u i d i s both r o t a t i n g and stratified. We seek wave-like s o l u t i o n s t o t h e i n v i s c i d l i n e a r i z e d equations of motion given by

a9

at +

28 x

9

0.9

= -0P

+

Tk

= 0

We have assumed t h a t t h e Boussinesq approximation is v a l i d , 6

t h a t t h e equilibrium temperature f i e l d s a t i s f i e s

vTo = K,

and t h a t t h e l i n e a r i z e d equation of s t a t e i s

-uT (Chapter

10).

We s u b s t i t u t e

( q , P , T ) = eiAt($,,l,s)

tions t o obtain

16 1

p =

i n t o t h e s e equa-

Pressure equation

162

(13.2) (13.3) We manipulate these vector equations to obtain the equation for the pressure field 5 . computing ko(l3.1) gives

-

+ s,

2(wii-$1

=

ixw =

(13.4)

and (13.3) and (13.4) give

Now

k

(13.1) gives

X

il(L€J)+

-ic

x Ol

.

(13.6)

Hence v (13.6) implies that

-

0

Uv-(kxQ)

We substitute for

k

x

GJ

+

(13.7)

from (13.5) g i v e s the equation for

the pressure as

+

% = 0.

from (13.1) and (13.3) to obtain

Thus, substitution for w

V21

2

2 N

q az

= 0.

(13.9)

We note that in the case of no stratification, i.e., N2 = 0,

163

I n t e r n a l waves

t h i s equation reduces t o Poincard‘s equation (7.5) f o r iner-

t i a l waves. e Wave S o l u t i m We consider a plane wave s o l u t i o n , i n an unbounded f l u i d ,

for equation (13.9).

4.;

= K1x

+

KZy

+

Writing

# = # 0e i ( b ’ z ) , where

K3z, and s u b s t i t u t i n g t h i s form i n t o equa-

t i o n (13.9) gives

Hence t h e d i s p e r s i o n r e l a t i o n i s

( 13.10)

This, of course, reduces t o expression (7.24) f o r homogeneous i n e r t i a l waves when f l u i d where

N

‘2

f 0

N2

= 0.

We note t h a t for a s t r a t i f i e d

t h e frequency depends not only on t h e

d i r e c t i o n of the wave vector, but a l s o on i t s magnitude. The phase v e l o c i t y

Sp =

fi il

is given by

Again the system i s d i s p e r s i v e with long waves t r a v e l i n g fastest. The group v e l o c i t y that

Cg = v K A .

A l i t t l e manipulation shows

Plane wave s o l u t i o n

164

We remark t h a t f o r a l l

t h e product

N2

= 0.

Thus, t h e introduction of s t r a t i f i c a t i o n , however s t r o n g , does not change a basic property of i n t e r n a l i n e r t i a l waves, namely t h a t energy i s transported a t r i g h t angles t o the phase velocity.

A s Garrett and Munk [24] observe, t h i s r e s u l t implies

t h a t a packet of waves would appear t o s l i d e sideways along the c o a s t s .

This property is i l l u s t r a t e d i n laboratory ex-

periments of Mowbray and Rarity [45]. We a l s o note t h a t s u b s t i t u t i o n of a v e l o c i t y vector of plane wave form

i n t o the divergence equation

Thus the p a r t i c l e v e l o c i t y

Sg

0'9 =

go,

0

gives

a s well a s the group velocity

i s perpendicular t o the wave number vector

1.

The dispersion r e l a t i o n (15.10) can be rewritten i n the form

X2

=

4

sin27

+

N2 cos2Y

Internal waves

\

\

, 2 4 0 /

Illustration of

I

Y

5 , 90

FIGURE 21

and

Eg'

166

Waves i n bounded geometry

where

y

is the angle given by

i s close t o 2 ( t h e i n e r t i a l frequency),

When t h e frequency the angle

y

is almost a r i g h t angle; when A

( t h e buoyancy frequency), the angle

i s close t o

i s almost zero.

y

Figure 21 i l l u s t r a t e s the perpendicular properties of

jo i n these two cases.

and

9,

N

sgy

braves i n Bounded Ge one t ry I n t h e previous s e c t i o n we described t h e dispersion r e l a t i o n f o r a plane wave s o l u t i o n t o equation (13.9).

A n y spe-

c i f i c physical problem requires the study of i n t e r n a l raves in a bounded region of f l u i d .

For example, an ocean b a s i n has

h o r i z o n t a l boundaries a t the coast l i n e of the adjacent land

mass:

we could crudely approximate the geometry of the ocean

by a rectangular box, o r a cylinder.

The atmosphere can be

modeled by a region bounded by a s p h e r i c a l annulus.

A labora-

tory experiment t o study i n t e r n a l waves would n e c e s s a r i l y be performed i n a bounded geometry. The mathematical model t h a t we have constructed f o r i n t e r n a l waves i n a r o t a t i n g s t r a t i f i e d f l u i d neglects t h e e f f e c t s of viscous and thermal d i f f u s i o n .

The appropriate boundary

condition f o r equations ( 13. 1)

(13.3) i s t h e r e f o r e the

-.

condition t h a t the normal v e l o c i t y i s zero on t h e boundary, i.e.,

9.;

= 0

on the boundary

C.

I n t e r n a l waves

167

A l i t t l e manipulation of the equations enables

us t o w r i t e

t h i s boundary condition i n terms of the pressure f i e l d

We r e c a l l t h a t when t h e s t r a t i f i c a t i o n parameter zero we proved t h a t t h e frequency

X

satisfied

as

4

N2

is

1x1 < 2 and

hence the equation f o r the pressure was always hyperbolic. We w i l l now o b t a i n the c o n s t r a i n t s on zero. where

when

We construct the energy i n t e g r a l by taking

CJ*

CJ,

i s the complex conjugate of

=

-s

2

- $J

s*.v*dv

V

v - 9*

that

= 0

J

N2

i s non

CJ* - ( 1 3 . 1 ) ¶

and i n t e g r a t e over

T h i s procedure gives

the volume of the container.

Since

JAJ

2

( 13.15 1

I w I dv.

V and

n-$ = 0

$.v#dv

on

Gauss' theorem implies

= 0.

V Thus equation (13.15) becomes 1$I2dv

-A2

+

N's

V We w r i t e

V

V i n component form a s

3

=

* CJ .k

Iwl2dv = i2X

uz

+ v j + wic

x J€ d v .

(13.16)

168

Bounds on t h e frequency

u

where

and

can be w r i t t e n i n r e a l and imaginary p a r t s

v

as

u = u Hence

+

R

-

* .k % 9

Q

v =

iU1y

VR

-2i(uRvI

=

+

iVI.

- vRU I

) y

and e q u a t i o n (13.16) becomes

IQ

A2

'dv

+

4~

J

(uIvR

-

uRvI)dv

-

N2

f

lwI2dv = 0. (13.17)

V

Equation (13.17) g i v e s a q u a d r a t i c e q u a t i o n f o r

A

with d i s -

A

are real

crimlnan

which is never n e g a t i v e .

Hence t h e eigenvalues

and t h e s o l u t i o n s a r e p u r e l y wave l i k e with no e x p o n e n t i a l growth

.

(AI

To o b t a i n bounds on

we r e w r i t e e q u a t i o n (13.17) t o

give (luI2

k2

+

lvI2)dv

+

V

+

(A2

-

N2)

21

s

V

This e q u a t i o n has t h e form

J

2(uIvR

-

V

2

I w I dv

= 0.

uRvI)dv

I n t e r n a l waves

where

P

2

Case a )

and

0

N




However a s o l u t i o n

2:

Again (13.19) can not have a s o l u t i o n

X

t h i s would imply both terms a r e p o s i t i v e .

x2


4 s i n c e

However a s o l u t i o n

is possible.

Thus we observe t h a t

1x1 i s bounded from above

l a r g e r of the two dimensionless frequencies ever

2

N

and

1x1 is & bounded from below by Min(N,2)

by the

How-

2.

and we can

expect t o find s o l u t i o n s t o the eigenvalue problem f o r a l l values of A

such t h a t 0

< x2
4

and

the curves

N2




0

and

r

>

Hence a l l t h e w e l l known r e s u l t s of

0.

Sturm-Liouville t h e o r y [Morse and Feshbach [ 4611 can be a p p l i e d t o o b t a i n information about t h e v e r t i c a l s t r u c t u r e of i n t e r n a l waves.

S i n c e ( 1 3 . 4 4 ) and (13.45) a r e of t h e same

form and d i f f e r only a s t o t h e c o e f f i c i e n t s , i t i s s u f f i c i e n t t o analyze t h e problem f o r for

G(z)

and t h e analogous r e s u l t s

GK(z) follow immediately.

We may f i r t h e r remark t h a t both (13.44) and (13.45) a r e

i n f a c t forms of t h e one-dimensional Schrodinger e q u a t i o n

&2 ?dz2

+

[E

- V(z)]u =

0.

T h i s o b s e r v a t i o n was f i r s t made by Eckart [ 1 6 ] f o r i n t e r n a l

waves i n a n o n - r o t a t i n g f l u i d .

We s e e now t h a t i t a l s o

a p p l i e s t o both c l a s s e s of i n t e r n a l waves i n a f l u i d t h a t i s s t r a t i f i e d and r o t a t i n g .

The known r e s u l t s of c l a s s i c a l

quantum mechanics can t h e r e f o r e b e used t o study our p r e s e n t problem. Consider problem (13.44).

From Sturrn-Liouville t h e o r y we

can conclude t h e following r e s u l t s . 1) There e x i s t s a d i s c r e t e i n f i n i t e spectrum of eigen-

values

where and

wmkn

with the property t h a t

Nmax

=

Max N2(z)

181

I n t e r n a l waves Recalling t h a t the frequencies

2)

-

= 1/(A2

(1j2

'

4)

we conclude t h a t

satisfy

Xmkn

There i s a complete s e t of eigenfunctions

GmKn

which a r e mutually orthogonal w i t h respect t o the weight function

(N2(z)

-

4)

S i m i l a r r e s u l t s hold f o r the problem (13.45) f o r

frequency spectrum of

Nmax

>

X

X2 Kko

'

KKn

>

and the eigenfunctions

KKn

Kkn

>

N2 z )

The frequencies

A-

< 2

4

.... > x 2K k n

-

0,

( 19-47)

form a complete s e t and a r e

orthogonal with weight function The case

The

satisfies

X2

G

(GK,u).

N2(z).

can be t r e a t e d i n the same manner. satisfy

0

The problem f o r N

2

(2)

crosses

(GK,AK)

i s unchanged.

I n the case where

4 ( t h i s may be the case i n the deep ocean) a

s l i g h t l y more s u b t l e treatment i s required s i n c e problem (13.44) i s no longer of the standard Sturm-Liouville form. The frequency spectrum i n t h i s case i s i l l u s t r a t e d i n Figure 24.

Frequency spectrum

182

I

I

N2

4

I

I

Pi*

I

I

I

I

I

I

I

I

I

I

I

I

,

I

I

I

I

I

1

I

1

I

I

I

I I

N2 'pax

I

Frequency spectrum f o r

I

I

2

m,

XniKn, 2

%in


x.

In the

neighborhood of t h e turning p o i n t , equation (13.44) can be approximated by an equation i n which t h e c o e f f i c i e n t of is l i n e a r i n

G

This has a s o l u t i o n i n terms of A i r y func-

z.

t i o n s which must be matched t o the o s c i l l a t o r y s o l u t i o n (13.49).

Details of these refinements of

(sometimes known as

methods

W.K.B.

methods) can be found i n the

W.K.B.J.

book by Murray [ 4 7 ] . The problem f o r i n t e r n a l Kelvin waves given by (13.45) can be t r e a t e d i n the same manner t o give

GK

Because

\AK\




v, i . e . ,

O(N2cr)




O(E-')

we r e t u r n t o equation

The dominant terms a r e now

6 9 + VEP = 0 ag a = 9 .

(14.16)

196

S i d e wall layers

Equation (14.16) i s analogous t o equation (9.4) describes the

EU3-boundary

J

which

l a y e r a t t h e s i d e wall of a

homogeneous r o t a t i n g f l u i d , however the r o l e s of the v e r t i c a l and h o r i z o n t a l a r e interchanged.

I n f a c t , when t h e s t r a t i -

N 20

f i c a t i o n is s u f f i c i e n t l y s t r o n g t h a t

>

O(E'l),

the

boundary layer s t r u c t L r e is dominated by s t r a t i f i c a t i o n . The h o r i z o n t a l boundary l a y e r s a r e then analogous t o the v e r t i c a l Stewartson boundary layers a s we mentioned I n Chapter 10.

-3 We examine the boundary l a y e r s on side-walls p a r a l l e l t o

the a x i s of r o t a t i o n by w r i t i n g

From equation (14.9) we observe t h a t the highest order terms i n the boundary l a y e r equation a r e

6

E2-68 bp ax6

+

2 Ea-2B hp 2

ax

+

4Pzz

= 0.

(14-17)

The boundary layer s t r u c t u r e is once again dependent on the s i z e of

a.

L.&e A:

a

>

2/3

The dominant terms a r e

4+ ax

and

4Pzz = 0

8 = l/3.

Hence when the s t r a t i f i c a t i o n is small enough s o t h a t

(14.la)

S t r a t i f i e d boundary l a y e r s

N2a




, Springer-

P h i l l i p s , N.A. , Geostrophic motion, Rev. Geophys. (1963) 9 125-176.

A

P h i l l i p s , O.M., Energy t r a n s f e r i n r o t a t i n g f l u i d s by r e f l e c t i o n of i n e r t i a l waves , Phys. Fluids 5 (1963) , 513-520.

1551 P h i l l i p s , O.M.,

m e dynamics Of t h e UDDer o c w , 2nd ed. Cambridge University Press , Cambridge (1977).

[561

Proudman, J . . On t h e motion of s o l i d s i n l i a u i d s possessing v o r t i c i t y , Proc. ROY. SOC. A 9i: (1916), 408-424.

Bibliography

267

C .G. , Relation between v a r i a t i o n s i n t h e i n t e n s i t y of the zonal c i r c u l a t i o n of t h e atmosphere and the displacements of the semi-permanent c e n t e r s of the a c t i o n . J . Mar. Res. 2 (1939), 38-155.

E 571 Rossby,

1581 Sakurai, T., Spin-down of a r o t a t i n g s t r a t i f i e d f l u i d i n thermally i n s u l a t e d c i r c u l a r c y l i n d e r s , J . Fluid Mech. (1969), 689-699. [591

Sakurai, T., ClarK, A . and Clark, P . , Spin-down of a Boussinesq f l u i d of small Prandtl number i n a c i r c u l a r c y l i n d e r , J . Fluid Mech. 3 (1971), 753-773. S c h l i c h t i n g , M. , j3oundar.v l a v e r the orv. New York (1955)

.

McGraw-Hill,

Spiegel, E.A. and Veronis, G., On the Boussinesq approximation f o r a compressible f l u i d , Astrophys. J. (1960) , 442-447. Stewartson, K., J . Fluid Mech. Stommel, M., Press , BerkelT-

On almost r i g i d r o t a t i o n s ; Part 2 ,

6 (1966) , 131-144.

, University

of C a l i f o r n i a

S u t c l i f f e , R . C . , The quasi-geostrophic advective wave i n a b a r o c l i n i c zonal c u r r e n t , Q u a r t . J . R . Met. SOC. 12 (1951), 226-234. Taylor, G . I . , Experiments on t h e motion of s o l i d bodies i n r o t a t i n g f l u i d s . Proc. ROY. Soc. A U (1923), 213-218. [66]

Thorpe, S . A . , The e x c i t a t i o n , d i s s i p a t i o n and i n t e r a c t i o n Of i n t e r n a l waves i n the deep ocean, J. Geophys. Res &2 (1975), 328-338.

[67]

VanDyke, M., J'ertu r b a t i o n methods i n f l u i d m e c h u , Academic Press, New York (1964).

[68]

Veronis, G., The analogy between r o t a t i n g and s t r a t i f i e d f l u i d s , Annual Rev. of Fluid Mech. 2 (1970) , 37-66.

[69]

Walin, G., Some aspects of time dependent motion of a s t r a t i f i e d r o t a t i n g f l u i d , J. Fluid Mech. & (1969), 289-307.

[TO]

Williams, G.P., Thermal convection i n a r o t a t i n g annulus 1: The b a s i c a x i s y m e t r i c flow, J . Atmospheric S c i . (196'0, 144-161.

.

a [ 711

Williams, G.P., Thermal convection i n a r o t a t i n g annulus 2: Classes of a x i s y m e t r i c flow, J . Atmospheric S c i . &j (1967), 162-174.

268

Bibliography

[ 721

Williams, G.P., Thermal convection i n a r o t a t i n g annulus 3: Suppression of the f r i c t i o n a l c o n s t r a i n t on l a t e r a l boundaries , J. Atmospheric S c i . 15 (1968), 1034-1045.

[ 731

Williams, G.P.9 Numerical i n t e g r a t i o n of the threedirnensi o n a l Navie r-St okes equations f o r incompressible flow. J . Fluid Mech. 32 (1969), 727-750.

[ 741

Wunsch, C , Geographical v a r i a b i l i t y of the i n t e r n a l wave f i e l d : a search f o r sources and sinks. J . Phys. Oceanogr. h (1976), 471-485.

.

[751 Y i h , C.S., Unarnics of n o n - h o r n w o u s f l u i u , MacMillan Co., New York (1965). [76]

Yih, C.S.,

S t r a t i f i e d Flows, Annual Rev, of Fluid Mech.

1 (1969)s 73-110.

INDEX

Annulus models

2 43

p -plane approximation Baroclinic i n s t a b i l i t y Bgnard convection Blocking Boussinesq approximation Brunt -Vais a l a frequency Buoyancy l a y e r

28 231 2 32 129 126 12 7 200

Centrifugal force Circulation C o r i o l i s force

125, 7

63 7

Depth averaged v e l o c i t y

71

Eady model Ekman l a y e r mass t r a n s p o r t spiral s u c t i o n condition Ekman number Energy equation Equation of s t a t e Equilibrium s t a t e E r t e l ' s theorem

2 32

F r i e d r i c h ' s example

253

38 47 46 42

18 134 9 125 12

Geostrophic balance Geost rophic contours Geostrophic mode Geostrophically f r e e , guided and blocked

21

61 57 62

269

2 70

Index

Gravitational potential Group v e l o c i t y Gulf stream

9 77 32

Heat equation Helmholtz equation Hydrostatic l a y e r

10

202

I n e r t i a l modes eigenvalues f o r i n a cylinder plane wave s o l u t i o n r e f l e c t i o n of I n i t i a l value problem Inner and outer expansions I n t e r n a l g r a v i t y waves i n a container Kelvin waves v a r i a b l e N( z ) Inviscid modes

68 74 77 79 72, 136 256 162 166 170 176 57

Matching p r i n c i p l e Mean c i r c u l a t i o n theorem Metamorphosis of s ide-wall layers Navier-St okes equations Oceanographic r e s u l t s Orthogonality Phase v e l o c i t y P o i n c a r e " ~equation Potential v o r t i c i t y Prandtl number Pressure equation Regular perturbat ion Rossby number Rossby radius of deformation Ros s by waves p-plane generated by topography plane wave s o l u t i o n s l i c e d cylinder

94

259 71 198 8 187 90 78 67 11

126 35 251

18 156, 240 88 89 90 95 86, 94

Index Rossby waves i n a stratified fluid Rotating co-ordina t e system Rotating s t r a t i f i e d flow geos trophic contours normal mode problem potential vorticity steady mode Schrodingerls equation Singular p e r t u r b a t i o n Solar spin-down Spin-up time s c a l e Stability criterion Stew r t s o n l a y e r s EL7 3-layer E114- l a y e r i n a s l i c e d cylinder i n a stratified fluid Stommel's model S t r a t i f i e d Ekman l a y e r S t r a t i f i e d spin-down Stretched co-ordinate Sverdrup s r e l a t i o n

271 15 3

6 139 139 141, 1 5 1 137 180 25 1 2 10

48 2 36

100 102 110

196 118 193

2 11 25 4

114

Taylor-Proudman theorem Taylor column Tea-cup experiment Thin-shell approximation Trapping of energy

96

Viscous d i f f u s i v e time s c a l e Vortex l i n e s t r e t c h i n g

51 51

Western boundary l a y e r

96, 112

21

23 52 26

2 72 R

-.

K

L i s t of Symbols

Angular v e l o c i t y v e c t o r . Unit v e c t o r i n t h e d i r e c t i o n of t h e a x i s of r o t a t i o n . Position vector Cartes i a n c o-ordinat e s S p h e r i c a l p o l a r co-ordinates Velocity vector V e l o c i t y components Time Pressure Temper a t u r e Gravitational potential Density Length s c a l e C o e f f i c i e n t of v i s c o s i t y C o e f f i c i e n t of Kinematic v i s c o s i t y C o e f f i c i e n t of thermal diff’usion Vorticity vector Potential vortic i t y Ekman number P r a n d t l number C o r i o l i s parameter Brunt-Vaisala frequency Reynolds number Rayleigh number Stress vector Wave number v e c t o r Pressure eigenfunc t i o n Frequency