An introduction to the mathematical theory of geophysical fluid dynamics, Volume 41 (North-Holland Mathematics Studies)

AN INTRODUCTION TO THE MATHEMATICAL THEORY OF GEOPHYSICAL FLUID DYNAMICS

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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

NEW YORK NORTH-HOLLAND PUBLISHING COMPANY - AMSTERDAM

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

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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].

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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[<

+

2n

2 2

X

qR]

=

- vP + p v ( G - n;?r)

[We remind the reader t h a t it i s only the time d e r i v a t i v e of a vector quantity that changes from an i n e r t i a l t o a r o t a t i n g frame of reference:

s c a l a r s and s p a t i a l d e r i v a t i v e s such a8

gradients and divergences a r e unaffected . ] The symbols represent the following:

Equations of motion

9

p = density

P = pressure G = gravitational potential =

c o e f f i c i e n t of v i s c o s i t y .

We note t h a t c e n t r i f u g a l force can be w r i t t e n a s the gradient of a s c a l a r and hence i t s r o l e i s only t o modify the e f f e c t i v e gravitational potential.

C o r i o l i s f o r c e , however, can not be

t r e a t e d i n t h i s manner and we w i l l show t h a t t h i s f o r c e plays a much more s i g n i f i c a n t r o l e i n the behavior of t h e equations. For n o t a t i o n a l s i m p l i c i t y we drop the s u b s c r i p t

R, however

throughout t h e following chapters we w i l l be working with t h e system of equations given by (1.4) and (1.5) which describe t h e motion of a viscous f l u i d i n a co-ordinate system r o t a t i n g with uniform v e l o c i t y

3.

TO complete the system we must add t o (1.4) and (1.5)

the

equation of s t a t e p = p(P,T), where

together with an equation f o r of thermodynamics.

T T

i s temperature, derived from the f i r s t law

For the f i r s t p a r t of t h e book we w i l l

consider the simpler case of a homogeneous f l u i d , i . e . , T

p

and

a r e constants, which w i l l of course r e s u l t i n some s i m p l i -

f i c a t i o n of ( 1 . 4 ) and (1.5).

I n t h e l a t e r s e c t i o n s we w i l l

include t h e e f f e c t s of s t r a t i f i c a t i o n which produces s i g n i f i cant modifications t o some geophysical problems.

However i n

l i g h t of the type of problems t h a t we a r e studying we w i l l not consider general equations of s t a t e and thermodynamics, but r a t h e r model the f l u i d , f o r example, water o r a i r , by the simplest reasonable r e l a t i o n s .

We assume t h a t the d e n s i t y i s

Heat equation

10

l i n e a r l y proportional t o temperatuse.

P = p0

-

We a l s o assume t h a t the i n t e r n a l energy temperature; e = cT.

(1.6)

p 0 u(T-T~).

e

is proportional t o

Physical arguments then imply t h a t

T

s a t i s f i e s the equation e = where

K

2=f

O'T

+

non-linear terms

i s the c o e f f i c i e n t of thermal d i f f u s i o n .

there a r e no i n t e r n a l heat sources).

(1.7) (We assume

We note t h a t i n using

the simple equations (1.6) and (1.7) we a r e ignoring c e r t a i n physical properties of a f l u i d , however we have retained the c h a r a c t e r i s t i c s of s t r a t i f i c a t i o n t h a t a r e s i g n i f i c a n t f o r most geophysical problems.

CHAPTER 2 POTENTIAL VORTICITY

Since we have already s t a t e d t h a t we consider r o t a t i o n t o be a dominant f e a t u r e of geophysical dynamics, i t is not s u r p r i s i n g t h a t the v o r t i c i t y f i e l d measure of the behavior of the f l u i d .

8

is a very s i g n i f i c a n t

[Recall

9

= v x

Taking the c u r l of (1.5) gives the v o r t i c i t y equation

= -0

x

=+pox-

P

v29 P *

Hence ( 2 . 1 ) becomes

11

41.

E r t e l ' s theorem

12

From (1.4) v .g =

-

gn

P dt

.

Hence we obtain the equation

Consider the case where f l u i d i s i n v i s c i d , i . e . , p = 0 . A

2 = 0,

be any s c a l a r quantity such t h a t

conserved a s a p a r t i c l e moves w i t h the f l u i d . s c a l a r product of

VA

i.e.,

X

Let

is

Take t h e

with ( 2 . 2 ) t o obtain

VX

Since gives

$$ =

+

9.VA

= 0,

a l i t t l e manipulation of ( 2 . 3 )

=

vx

*y.

(2.4)

P

Equation ( 2 . 4 ) i s known as E r t e l ' s equation. U t e l ' a meorem

i.e.,

the quantity

If we choose

TT =

* P

(€J

A

+

such t h a t

20)

Known as the p o t e n t i a l

v o r t i c i t y , is conserved as a p a r t i c l e moves with the f l u i d . For an i n v i s c i d , non-rotating, homogeneous f l u i d i t i s well Known t h a t the v o r t i c i t y i s a conserved quantity. now obtained the analogous expression

TI

We have

t h a t i s conserved

i n the more general case of an inviscid but r o t a t i n g s t r a t i f i e d fluid.

We note t h a t the d e r i v a t i o n of ( 2 . 4 ) d i d not require

P o t e n t i a l Vort i c i t y

13

t h a t the conservation of mass equation be reduced t o

v.3

= 0,

hence E r t e l ' s theorem holds even f o r a compressible f l u i d . However, i n many geophysical contexts t h e compressibility of the ocean and atmosphere can be neglected, and i t can be assumed t h a t

= 0.

Since the d e n s i t y a l s o s a t i s f i e s vp x V P

vp

the density f i e l d

I

0,

i n the case of an incompressible f l u i d

p

i s i t s e l f a s u i t a b l e candidate f o r t h e s c a l a r

X

and E r t e l ' s

theorem b ec omes

A second p a r t i c u l a r case i s t h a t of a barotropic f l u i d

where the density i s a function only of t h e pressure f i e l d , i.e.,

p = p(P)

.

I n t h i s case

v p x V P i s i d e n t i c a l l y zero,

hence E r t e l ' s theorem holds for any conserved s c a l a r X . We note t h a t i f

p

i s a constant ( i . e . ,

the f l u i d i s

and equation (2.2), i n t h e i n v i s -

homogeneous), then v p = 0 c i d case, reduces t o

Recalling t h a t

3

inertial

=

,g + 9 x 5,

we see t h a t

4+

22

i s , of course, the v o r t i c i t y i n the i n e r t i a l co-ordinate

system.

Hence equation (2.6) gives the well known r e s u l t t h a t

f o r an i n v i s c i d homogeneous f l u i d , i f the v o r t i c i t y i s i n i t i a l l y zero, the v o r t i c i t y w i l l remain zero.

14

E r t e l ' s Theorem I n l a t e r work we w i l l make use of E r t e l ' s theorem i n

s e v e r a l d i f f e r e n t contexts.

For example, we w i l l show t h a t

f o r a homogeneous f l u i d of t h i n l a y e r depth

h ( x , y ) , the

express ion =

2

h(x,y)

(a

+

2q)

is conserved.

15

Problems t e r 2 Problemg 2.1)

The r o t a t i o n of the e a r t h i s almost, but not q u i t e , constant; t h e r e is a s l i g h t “wobble”. What w i l l the e f f e c t s be on the p o t e n t i a l v o r t i c i t y equation when

n i s allowed t o be a function of time?

Derive a modified p o t e n t i a l v o r t i c i t y equation f o r an i n v i s c i d , incompressible f l u i d r o t a t i n g w i t h angular v e l o c i t y

n 2.2)

=

no + a ( t ) ,E

<<

1.

Determine the p o t e n t i a l v o r t i c i t y equation f o r a homogeneous r o t a t i n g viscous f l u i d .

Explain the

r o l e v i s c o s i t y plays i n the generation of v o r t i c i t y .

This Page Intentionally Left Blank

CHAPTER 3

NON DIMENSIONAL PARAMETERS

We have s t r e s s e d t h a t i n t h i s t e x t we w i l l consider f l u i d motion i n which the e f f e c t s of r o t a t i o n a r e important.

It

i s t h e r e f o r e u s e f u l t o rewrite t h e equations so t h a t we can a s s e s s the regions of physical parameter space i n which C o r i o l i s force i s s i g n i f i c a n t r e l a t i v e t o the o t h e r f o r c e s i n the momentum equation.

For s i m p l i c i t y we w i l l i n i t i a l l y d i s -

cuss the s c a l i n g f o r a r o t a t i n g homogeneous f l u i d :

in later

chapters we w i l l include the e f f e c t s of d e n s i t y s t r a t i f i c a -

t ion. Let

n-l, U, c h a r a c t e r i z e t y p i c a l length, time and r e l a -

L,

t i v e velocity:

we then introduce dimensionless co-ordinates

by w r i t i n g

r =

I)

LS*, t

=

t*J 3 =

uy,

I n t h e case of constant d e n s i t y

p,

P = pnuLp.

equations (1.4) and (1.5)

reduce t 0

0.3 a

~ where

(G

P

+

c

p 2Ql

x +a =

- LPv P + ! v

2

3,

i s now the reduced pressure incorporating

- @).

gives

~

= 0

S u b s t i t u t i o n of the dimensionless co-ordinates

Ekman number and Rossby number

18

=

(k

-

nu v * p ,

.5 v* 2g* L

+

i s the u n i t v e c t o r i n the d i r e c t i o n of the a x i s of rota-

tion.)

- v*)j*

+ ~(3*

Hence

=

( v = k/p

- oP*

+

2k x

g*

+ m*23*.

(3.3)

i s the Kinematic v i s c o s i t y ) .

I n most geophysical problems, both

E

and

E

a r e very small.

A small Rossby number i n d i c a t e s t h a t the deviation from r i g i d

r o t a t i o n i s small and t h a t C o r i o l i s force dominates the nonl i n e a r terms ( i n most regions).

A small Ekman number implies

t h a t Coriolis force i s much l a r g e r than the viscous force which w i l l be generally neglected almost everywhere.

In a

simple experiment t h a t we w i l l describe s h o r t l y , where water

i s s t i r r e d i n a tea-cupr L = 5 cm, n hence, E = 2 x 10-4

.

= 2

sec -1, v = 0.01 cm2 /sec,

I n a n oceanographic context i t i s a l i t t l e more d i f f i c u l t t o estimate the v i s c o s i t y since i t i s probably more reasonable t o consider some s o r t of turbulent v i s c o s i t y r a t h e r than the molecular v i s c o s i t y .

However the length s c a l e s a r e

Non-dimensional parameters s u f f i c i e n t l y l a r g e t o ensure t h a t

E

remains small.

20

3.1)

a)

Problems

Show t h a t the Rossby number measures the r a t i o of the convective force t o Coriolis force.

b)

Show t h a t t h e Ekman number measures the r a t i o of the viscous f o r c e t o Coriolis f o r c e .

3.2)

Assume t h a t a c h a r a c t e r i s t i c surface s t r e s s of magnitude fluid.

S

generates motion i n a r o t a t i n g

Introduce appropriate non-dimensional

co-ordinates and determine the relevant nondimensional parameters.

CHAPTER 4

GEOSTROPHIC FLOW

I f we neglect the terms multiplied by equation (3.3)

a

(-a t

= 0)

E

and

E

in

and we assume t h a t t h e flow is steady

, we o b t a i n 2; x

g

= -vP

(4.1)

( f o r convenience we now drop the s t a r s ) , and (3.1) remains

v.3

(4.2)

= 0.

Hence the momentum equation has reduced t o a balance between C o r i o l i s force and the pressure gradient. peostrou

balance

of r o t a t i n g flow.

T h i s is known as

and i t i s a fundamental c h a r a c t e r i z a t i o n

O f course equation (4.1) was derived a f t e r

major s i m p l i f i c a t i o n s of the physics, however it gives a remarkably good d e s c r i p t i o n of t h e b a s i c behavior of a r o t a t i n g f l u i d and we w i l l use geostrophic flow as the fundamental s t a t e t o which we w i l l add physical complexities. =lor-Prwdman

Theorem

Assume the flow is i n geostrophic balance, hence described by equations (4.1) and ( 4 . 2 ) .

TaKing t h e c u r l of ( 4 . 1 ) and

using (4.2) gives

(k*v)g 21

f

0.

(4.3)

Taylor-Proudman theorem

22

I n terms of Cartesian co-ordinates (XaYaZ), where

z

is the

co-ordinate p a r a l l e l t o the a x i s of r o t a t i o n , (4.3) can be w r i t t e n as

t h e r e i s no v a r i a t i o n i n the v e l o c i t y f i e l d with respect

i.e., to

z.

This property of geostrophic flow is known a s the

Taylor-Proudman theorem because of the foundational work of two distinguished f l u i d dynamicists, G.I. Taylor [65] and J . Proudman [56], who contributed many important r e s u l t s t o

the subject i n t h e e a r l i e r p a r t of the century.

Writing t h e vector equation (4.1) i n component form ( i n Cartesian co-ordinates with

3

= u1

2v =

u

-2u =

u

o =

u

+

+

vj

wk)

gives

ax

ay

(4.5)

az

I n vector form, equation (4.4) can be w r i t t e n a s J~

=

$x

VP, where

jH uz 3

+

v3.

Thus, as we have already observed, the v e l o c i t y f i e l d is independent of t h e v e r t i c a l co-ordinate velocity

Z,

and the h o r i z o n t a l

qH i s given as d e r i v a t i e s of the pressure

II

P(x,y)

However, the geostrophic equations alone will not uniquely determine the flow, since any function geostrophic balance.

P(x,y)

w i l l satisfy

It w i l l then be necessary t o i n t r o -

.

Geos t rophic flow

23

duce more physics ( s e e Chapter 5 ) t o determine which twodimensional pressure

P(x ,y)

describes a p a r t i c u l a r problem:

once t h i s is done the geostrophic r e l a t i o n s ( 4 . 4 ) immediately determine the h o r i z o n t a l v e l o c i t y .

-3 An i n t e r e s t i n g and f a i r l y e a s i l y observed phenomenon t h a t can be explained using t h e Taylor-Proudman theorem is t h e Taylor Column.

Consider the following experiment :

Take a right c i r c u l a r cylinder w i t h r i g i d top and bottom.

Fill the cylinder full of water i n which f i n e p a r t i c l e s (aluminum f o r example), a r e suspended. b a l l a t the bottom of the c y l i n d e r .

Place a small metal

Set the cylinder on a

uniformly r o t a t i n g t u r n t a b l e and allow enough time f o r t h e f l u i d t o achieve r i g i d body r o t a t i o n .

Without d i s t u r b i n g

the f l u i d , move the small b a l l slowly with v e l o c i t y

1

r e l a t i v e t o the r o t a t i n g f l u i d ( t h i s could possibly be done with a magnet if the b a l l were magnetized). physical parameters

c),v,U,L

such t h a t

so t h a t the flow i s geostrophic.

E

Choose a l l the

<<

1 and

E

<<

1

By shining a l i g h t through

t h e cylinder t h e alignment of the f i n e p a r t i c l e s w i l l make

i t possible t o observe t h a t t h e e n t i r e column of f l u i d above the b a l l moves as though i t were r i g i d l y attached t o the b a l l . This e f f e c t is known a s a Taylor column.

24

A Taylor Column

A Taylor Column FIGURE 2

Geostrophic flow

25

The explanation of t h i s phenomenon i s the following: cylinder has a r i g i d top perpendicular t o w

component of v e l o c i t y

is

0

k,

a t t h e top.

the

hence the normal I f the f l u i d

above the b a l l were not t o move w i t h t h e b a l l t h e r e must be flow over t h e b a l l , and such a flow requires a non-zero component of v e l o c i t y

w.

However, the Taylor-Proudman

theorem t e l l s us t h a t t h e r e can be no v a r i a t i o n i n respect t o

z = L

z , hence it i s impossible t h a t

and nonzero a t

z = z ball.

w

w

with

be zero a t

Thus t h e r e can be no flow

over the b a l l , and the same argument a p p l i e s a t any l e v e l of

z.

Hence t h e column of f l u i d above the b a l l moves as though

i t were r i g i d l y attached.

As an i n t e r e s t i n g v a r i a t i o n of t h e experiment, consider moving t h e b a l l impulsively along t h e bottom through t h e fluid.

( a ) I f t h e f l u i d i s not r o t a t i n g (n = 0 ) t h e b a l l

moves i n a straight l i n e . slowly, s o t h a t

E

( b ) I f the f l u i d i s r o t a t i n g very

i s not small and geostrophic balances does

not hold, then t h e t r a j e c t o r y of t h e b a l l w i l l be d e f l e c t e d by t h e r o t a t i o n .

(c) I f

n i s increased so t h a t

E

<<

1,

the motion is e s s e n t i a l l y 2-dimensional and the b a l l again moves i n a s t r a i g h t l i n e , but now i t c a r r i e s a column of f l u i d above i t .

We note t h a t i f the b a l l i s not kept i n

motion by a n e x t e r n a l f o r c e i t would rapidly l o s e v e l o c i t y because of transference of energy t o t h e column of f l u i d t h a t

must move with the b a l l v e l o c i t y . A Taylor column i s t h e r e f o r e a s t r i k i n g i l l u s t r a t i o n of the

2-dimensional nature of r o t a t i n g flow.

Despite t h e s i m p l i c i t y

of the geostrophic equations, they can be used t o understand an

Application t o geophysical motion

26

i n t e r e s t i n g and a t f i r s t glance, unexpected f l u i d motion. Taylor columns have even been suggested ( s e e Hide [ 2 9 1 ) a s a possible explanation f o r the "Red Spot" on J u p i t e r .

We should,

however, note t h a t t h i s explanation i s somewhat c o n t r o v e r s i a l . Alternative explanations have been proposed; Maxworthy suggests t h a t J u p i t e r ' s ''red spot", "white oval" and other c h a r a c t e r i s t i c markings a r e the r e s u l t of the v o r t i c i t y d i s t r i b u t i o n of non-linear d i f f u s i v e waves i n the presence of strong v e r t i c a l shears.

The equations ( 4 . 4 ) f o r geostrophic balance were derived by considering a f l u i d on a horizontal plane r o t a t i n g about the perpendicular a x i s .

For geophysical problems the more r e a l -

i s t i c geometry i s t h a t of Plow i n a s p h e r i c a l s h e l l ( o r a portion of a s p h e r i c a l s h e l l i n the case of a n ocean).

Let

us t h e r e f o r e consider equation ( 4 . 1 ) i n component form, i n s p h e r i c a l co-ordinates

k

=

i. cose

where

e

s i n e , q = w:

+

(r,e,+)

-6

i s co-latitude, vz

+

ui.

We w i l l a l s o e x p l i c i t l y include g r a v i t y , w r i t i n g

VG =

-g1.

The component equations a r e

e

=

-a -g ar

2v cos 6

+

2w s i n

-2u s i n

e

(4.6)

=

-

1

r sin 6

2

a+

(4.8)

Geost rophic flow

Co-ordinates i n a s p h e r i c a l s h e l l FIGURE 3

27

8-Plane approximation

28

Now i n oceanographic o r meteorological problem i t i s reasonable t o assume t h a t t h e f l u i d is constrained t o move i n a narrow s h e l l where

$, <<

1, [see Figure 31.

<<

equation then implies t h a t

can be neglected with respect t o

i n the e q u a t o r i a l regions where

The divergence

1. Hence i n (4.8)

v cos 8

cos 8

everywhere

-, 0. The narrow

s h e l l assumption a l s o enables us t o approximate radius

%

of the sphere.

w sin 8

r

by the

Thus t h e equations reduce t o

(4.10) I n most problems t h e h o r i z o n t a l v e l o c i t i e s a r e small compared with those produced by g r a v i t y , hence (4.6) approximates t o

.a2

ar

Equations (4.9),

=

-g.

(4.11)

( 4 .lo), (4.11) a r e the geostrophic approxima-

t i o n f o r a spherical s h e l l .

The equations f o r geostrophic balance i n a s p h e r i c a l s h e l l a r e of course s i m i l a r t o the Cartesian equations ( 4 . 4 ) .

They

do i n f a c t correspond exactly i f we maKe Use of the so-called "@-Plane Approximation."

Consider an ocean, o r s e c t i o n of the

atmosphere centered a t a c o - l a t i t u d e

8*:

we then assume t h a t

the l a t i t u d i n a l s c a l e of the problem is small enough t h a t we can neglect the curvature of the e a r t h and approximate the

Geos t rophic flow geometry by a tangent plane centered a t

E

The tangent plane FIGURE 4

29

8,.

[See Figure

[4]]

30

@-Plane approximation

Define l o c a l co-ordinates

eo

a(eo-e) = y, a(eO-+) s i n Let

2 cos 8 = f

= x, r

-

a = z.

( c a l l e d Coriolis parameter).

We w r i t e

f

i n a Taylor s e r i e s as f

,.

eo

‘L cos

-z

eo[e-e,l

sin

+

....

For small v a r i a t i o n s i n l a t i t u d e we approximate f

where

f o = 2 cos O o

-

and

fo +

p =

2

f

by

BY,

sin e a

The geos trophic equations then become (4.12)

fv =

-Q

iE

2 ax (4.14)

az

We have therefore approximated the problem of flow i n an ocean basin by flow on a plane r o t a t i n g about an axis ponding t o

^r a t point

(610,$o)), with

A

kt

an angular v e l o c i t y

which i s no longer constant, but v a r i e s l i n e a r l y with [Note increasing on the plane].

y

(corresy.

corresponds t o the northward d i r e c t i o n

I n l a t e r chapters we w i l l discuss i n d e t a i l

the very i n t e r e s t i n g e f f e c t t h a t t h i s l a t i t u d i n a l v a r i a t i o n has on the behavior of a r o t a t i n g f l u i d , and show how i t can be used t o explain observed oceanographic phenomena.

Geos t rophic flow

31

Clearly, equations (4.12), (4.13), (4.14) again imply t h a t u

v

and

a r e independent of

v e r t i c a l co-ordinate

.

z , where

z

i s now the l o c a l

However, combining t h i s system with

the divergence equation, V'J

=

0,

implies

Thus

hf 3Y i s no longer zero

Because

conclude t h a t

w e do

a z = 0. I n o t h e r words, t h e Taylor-Proudman theorem does not hold on

a

@-plane, where the l a t i t u d i n a l v a r i a t i o n of C o r i o l i s para-

meter

is

f

O(f3)

i s approximated by

f

-

fo

+ .@,

I n t h i s case

%

and given by

(4.16) As a f i n a l remark i n t h i s s e c t i o n we note t h a t equations ( 4 . 1 2 ) and (4.13) can be combined t o give

where

gH i s the h o r i z o n t a l v e l o c i t y .

I n o t h e r words, the

flow l i n e s of h o r i z o n t a l v e l o c i t y a r e p a r a l l e l t o l i n e s of constant pressure.

I f we were t o assume t h a t the atmosphere

i s i n geostrophic balance, which i s o f t e n a reasonable approximation, w e conclude t h a t t h e isobars on a weather map correspond t o the stream l i n e s of the flow.

Geostrophic motion

I n summary, geostrophic motion occurs when the h o r i z o n t a l components of the pressure gradient and Coriolis force a r e in approximate balance and f r i c t i o n a l e f f e c t s a r e confined t o t h i n boundary l a y e r s .

For a study of geostrophic motion as

applied t o large s c a l e motions of the oceans and the atmosphere, we r e f e r t h e reader t o the important a r t i c l e on the subject by N. P h i l l i p s [ 5 3 ] . He considers s e p a r a t e l y two types of geostrophic motion:

type 1 where the h o r i z o n t a l

length s c a l e

L

i s smaller than t h e radius of the e a r t h , and

type 2 where

L

i s comparable t o the radius of the e a r t h .

Type 1 motions a r e successfully applied t o describe such problems as wave s t a b i l i t y , v e r t i c a l propagation of energy, and Gulf Stream meanders.

Motions of type 2 have q u i t e

d i f f e r e n t properties and a r e used t o analyze the very slow motions of the i n t e r i o r of t h e ocean.

Problems

33

Dter 4 Problems I n the t e x t we discussed the two dimensional nature of strongly r o t a t i n g flow.

We remarked t h a t a b a l l

moving across a strongly r o t a t i n g flow could move i n a s t r a i g h t l i n e , but i n a weakly r o t a t i n g flow the b a l l would be d e f l e c t e d by r o t a t i o n .

Illustrate

t h i s statement by considering t h e following mathematical problem.

Consider a Taylor Column bounded

between h o r i z o n t a l p l a t e s a t

z

= 0

and

Neglect t h e e f f e c t s of boundaries i n the y

z = 1. x

and

directions.

Show t h a t the v e l o c i t y of the f l u i d i n the Taylor Column moving i n a s t r a i g h t l i n e s a t i s f i e s the l i n e a r geostrophic equations. Determine a n

O(E)

c o r r e c t i o n term t o the v e l o c i t y

t h a t s a t i s f i e s an Oseen approximation t o t h e nonl i n e a r momentum equation. Consides the f u l l Navier-StoKes equations f o r the motions of a r o t a t i n g f l u i d .

What physical assump-

t i o n s must be made i n order t h a t t h e s e equations can be used t o explain the phenomenon of a Taylor Column? Show t h a t under these assumptions

34

Problems

4.3)

Consider 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 a t i s f i e d by t h e motion of a homogeneous r o t a t i n g f l u i d i n

a

p-plane,

-&(z.k + Assume t h a t

B

f ) = 0.

i s small and c o n s t r u c t a l i n e a r i z a -

t i o n of t h i s e q u a t i o n t o g i v e -d9

at

2p

+ p g = o .

CHAPTER 5

THE EKMAN LAYER

I n t h e previous s e c t i o n we i n v e s t i g a t e d the fundamental s t a t e of a r o t a t i n g f l u i d , namely geostrophic balance. equations ( 4 . l ) and ( 4 . 2 )

,

From

i t i s immediately demonstrated

t h a t t h e v e l o c i t y and pressure f i e l d s i n a geostrophic flow a r e independent of

t h e v e r t i c a l co-ordinate, and t h a t t h e qH = 21-K x v P . However, a s we have

z,

horizontal velocity

previously observed, t h e p r e s s u r e f i e l d i t s e l f can not be uniquely determined from these equations (any f u n c t i o n P(x,y)

w i l l s a t i s f y ( 4 . 1 ) and ( 4 . 2 ) ) .

Hence, t o determine

the flow we must r e t u r n t o the f u l l momentum equation given (Again f o r convenience we w i l l drop t h e s t a r s ) .

by ( 3 . 5 ) .

Let us consider t h e case where the p e r t u r b a t i o n from r i g i d r o t a t i o n i s very small s o t h a t we can l i n e a r i z e t h e equation by s e t t i n g

E = 0.

The equations of motion a r e t h e n

aa9t +

-

2K x

9

= -0P

v.9 =

+

2

Eo 9.

0.

(5- 2 )

We w i l l now perform a l i t t l e v e c t o r manipulation of these equations t o reduce t h e system t o a s i n g l e equation f o r the pressure

P.

Consider

which implies

ka(5.1)

55

The pressure equation

36 Because

0.3

= 0

the expression

i . [ v x (5.1)] becomes

and (5.3) and ( 5 . 4 ) give

From the divergence of (5.1) we o b t a i n L

2K.V

X

9

2

= V P,

Hence (5.5) and (5.6) combine to give

($ -

,,+4q

=o.

2

az

Thus the general linearized equation s a t i s f i e d by t h e pressure rotating fluid f i e l d f o r the motion a homogeneous, VISCOUS,

i s a s i x t h order, time dependent, p a r t i a l d i f f e r e n t i a l equation. We note t h a t i f , instead of completely neglecting the non-linear term, we were t o use an Oseen approximation ( i - e - , writin@; Eg-vq

=I

E ~ ~ Q C J ,where

1

the v e l o c i t y

is assumed

known), the equation f o r the pressure becomes

[% +

EU.V

-m ]

v2P+ 4

2

az

9

= 0.

I n t h e case of the geostrophic approximation E

= 0,

and

z.

= 0,

E = 0, (5.7) reduces t o

which is consistent w i t h t h e r e s u l t t h a t of

5

P

i s independent

The reduced i n v i s c i d equation i s n a t u r a l l y of lower

37

The Ekman l a y e r

order and hence can not be used t o s a t i s f y boundary conditions on t h e t a n g e n t i a l , as well a s , the normal components of velocity.

A s i s customarily the case i n a problem i n f l u i d dyna-

mics, f r i c t i o n a l e f f e c t s must be included i f t a n g e n t i a l veloc i t y conditions a r e t o be s a t i s f i e d a t t h e boundary.

We w i l l

achieve t h i s by assuming t h a t t h e viscous parameter ( i n t h i s case the Ekman number E ) i s small but non-zero.

We w i l l then

follow t h e standard techniques of asymptotic a n a l y s i s : we w i l l o b t a i n an approximate s o l u t i o n i n terms of a matched asymptotic expansion which i s composed of a n expansion v a l i d i n the int e r i o r region t h a t i s i n v i s c i d t o the f i r s t order, and an expansion t h a t i s v a l i d i n a viscous boundary l a y e r . To i n v e s t i g a t e t h e s i z e of the h o r i z o n t a l boundary l a y e r

l e t us f i r s t consider t h e steady form of equation (5.7), E V P

+

4

5 = 0. az 2

namely

(5-8)

We note t h a t t h e f i r s t term represents t h e viscous f o r c e and t h e second term C o r i o l i s f o r c e .

T’

The Ekman l a y e r co-ordinates FIGURE 5

38

The boundary l a y e r thiclzness

I n most s p a t i a l regions

2 6

E v P

is very small since

E

<<

1,

however i n a narrow region c l o s e t o the boundary, the derivat i v e s could be l a r g e . layer by s c a l i n g

z:

We examine the h o r i z o n t a l boundary write

z

=:

EaS.

Hence

Equation (5.8) becomes

Retaining only the dominant terms gives

If the boundary layer i s t o be n o n t r i v i a l , there must be

balance between t h e two terms i n equation ( 5 . 9 ) , requiring

Hence t h e horizontal boundary l a y e r which is called the EKman laver i s of non-dimensional thickness

O(E1’2)

and the equa-

t i o n f o r the boundary layer pressure is

T h i s i s a 6-th order P.D.E.,

and thus the boundary value

problem is well posed i f boundary conditions determining t h e t h r e e v e l o c i t y components a r e prescribed a t

5 =

00.

f

= 0, and

We note t h a t the balance i n (5.10) between viscous

force and Coriolis force is c h a r a c t e r i s t i c of the Ekman l a y e r . We a l s o note t h a t we would o b t a i n t h e same boundary Layer

The EKman l a y e r

39

equation f o r any boundary l a y e r i n a r o t a t i n g f l u i d on a boundary t h a t i s

co-planar with t h e a x i s of r o t a t i o n

k.

Consider a boundary surface w i t h outward u n i t normal v e c t o r

A

and l e t

fi-v

= E

-‘I2 A

becomes

?I*;

provided t h a t

The boundary l a y e r equation then

as

We will d i s c u s s i n a l a t e r s e c t i o n

0.

the s t r u c t u r e of a boundary l a y e r a t a surface where i.e.,

5.f

= 0,

p a r a l l e l t o t h e a x i s of r o t a t i o n .

As we remarked, we w i l l use the method of matched asymp-

t o t i c expansions with t h e convention t h a t 9Interior

+

gboundary l a y e r - prescribed condition atboundary =

and as

sboundary l a y e r - r O That i s , the condition a t

5

5 - a ~ . = 0

on the boundary l a y e r solu-

t i o n c o r r e c t s f o r the behavior of the i n t e r i o r s o l u t i o n a t t h e wall.

Let us denote Ekman boundary l a y e r q u a n t i t i e s by

Since the Ekman l a y e r i s expansion of

*

P

and *

O(E1/‘)

+ G 1E l l 2 + + vlE lI2+ w 0 i &E l 1’2 + B, + PlE”2 + a

-

w

=

k

=

we consider an asymptotic

i n powers of

u = u0 a v = vo 1

a

(”).

...

... ... ...

40

The boundary l a y e r equations

*

Consider t h e steady v e r s i o n of equation (5.1) i n t h e EKman l a y e r where

EV

=

+

S u b s t i t u t i o n of t h e expan-

O(E1’2).

s i o n i n t o t h i s equation gives t o f i r s t order

Now

Po

-

O = - E - 1/2

0

as

5

-

co

Po

Po

hence

.

s

= 0.

The divergence equation (5.2) gives

wo cI

hence

Q

= 0,

f; = 0.

Thus t h e Ekman l a y e r equations a r e -ZG0

=

2Go =

G

Go

uoc 5

+ G OX

Q5

o y + 9 13

= O .

Equations (5.11) and (5.12) combine t o give

(5 14)

The Elrman l a y e r since

(Go

lim

5-a

+

16,)

41

= 0.

Let us consider a p a r t i c u l a r example where t h e r e i s a r i g i d boundary a t

z = 0

with imposed v e l o c i t y ( i n t h e

r o t a t i n g co-ordinate system)

sH = Q,I 3

Let t h e i n t e r i o r v e l o c i t y Then

+

vo

+ ?o

=

n

uo = Q1 = Q~

90

so +

E 1/’2 41 +

at

z=O,

5-0

at

z=o,

5=0

... .

s a t i s f i e s t h e geostrophic equation

‘zk

x go = -vPo

v*jo Hence

Q.3.

wo = 0 .

and

Also

uo

+

uo, vo

and

wo

= 0.

a r e independent of

z

and t h e

boundary conditions t h e r e f o r e imply n

uo = Q1

Go

=

Q2

wo = 0

- Uo(X,Y)

- V,(XJY)

at

5=0

everywhere.

S u b s t i t u t i n g (5.16) i n t o expressions, (5 .l5) gives

From equation (5.13),

can be determined t o g i v e

The Ekman l a y e r S u c t i o n c o n d i t i o n

42

Now

wo = 0, hence t h e divergence e q u a t i o n i m p l i e s uox

+ v

- 0 .

OY

And f o r c o m p a t i b i l i t y , t h e imposed h o r i z o n t a l v e l o c i t y

must a l s o s a t i s f y

Q1

+

X

Since

Qz Y

= 0.

Hence (5.18) g i v e s

wo = 0, t h e h i g h e s t o r d e r boundary c o n d i t i o n on

w

becomes i;,

+

w1 = o a t

S=O,

Z=O.

Hence (5.19) g i v e s w1 = which c a n be w r i t t e n i n v e c t o r form a s

Expression (5.20) i s known a6 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 i t i s a very important f e a t u r e of r o t a t i n g flow. We observe t h a t somewhat remarKably a p r e s c r i b e d h o r i z o n t a l velocity

CJH

induces a small v e r t i c a l v e l o c i t y

t h i s v e l o c i t y is a n o r d e r

smaller t h a n C&).

w1

(remember We a l s o

n o t e t h a t although i t was n e c e s s a r y t o s o l v e t h e EKman l a y e r equations t o o b t a i n (5.20) , t h e e x p r e s s i o n i t s e l f i n v o l v e s

The Etrman l a y e r only t h e i n t e r i o r v e l o c i t y .

43

We t h e r e f o r e c o n s i d e r t h e Ekman

l a y e r a n a c t i v e boundary l a y e r t h a t p l a y s a c r u c i a l r o l e i n determining t h e i n t e r i o r s o l u t i o n .

From t h e i n t e r i o r momen-

turn e q u a t i o n s we observe t h a t t h e v i s c o u s term i s the

O(E1’2)

O ( E ) , hence

v e l o c i t y components of t h e i n t e r i o r flow a l s o

s a t i s f y t h e e q u a t i o n s of g e o s t r o p h i c balance implying t h a t wl,

as well as

u1

and

v l,

a r e independent of

z.

Thus

t h e Etrman l a y e r s u c t i o n c o n d i t i o n , ( 5 . 2 0 ) which was d e r i v e d

a s a boundary c o n d i t i o n h o l d i n g a t f o r a l l v a l u e s of and

z=O,

must i n f a c t hold

z, s i n c e t h e q u a n t i t i e s involved, wl, go

€&, a r e a l l independent of

z.

We w i l l g i v e a n example

t o i l l u s t r a t e t h e power of t h i s r e s u l t i n d e t e r m i n i n g t h e s t e a d y flow i n a r o t a t i n g c y l i n d e r . FxamDle of C v l i n d r i c a l Flow

z = 1

z = o

FIGURE 6

A c y l i n d r i c a l example

44

Consider a c y l i n d e r of f l u i d i n r i g i d body r o t a t i o n with angular velocity bottom p l a t e t o

n:

i n c r e a s e t h e a n g u l a r v e l o c i t y of t h e

n ( 1 + ~ ) I.n

terms of t h e dimensionless para-

meters, and in t h e r o t a t i n g c o - o r d i n a t e system, t h e boundary conditions a r e

We write

9

4

=

9

=

re^ a t 0

at

Z=O

z=1.

i n c y l i n d r i c a l co-ordinates

9

=

u;

+

ve"

(r,e,z)

as

+ wi;.

Applying t h e Ekrnan l a y e r s u c t i o n c o n d i t i o n (5.20) a t

z=O

gives

To analyze t h e EKman l a y e r a t

and

a az -

-E-

''' a t

.

z=1 we w r i t e

1 - z = .gE1/2

Hence we o b t a i n a s i m i l a r e x p r e s s i o n

f o r t h e s u c t i o n c o n d i t i o n , but with a s i g n change, namely

where

€ &

is t h e imposed v e l o c i t y of t h e t o p s u r f a c e .

Applying t h i s c o n d i t i o n t o t h e problem g i v e s

Since t h e problem i s axisymmetric, we will assume t h a t t h e s o l u t i o n is a l s o axisymmetric, i . e . , A = 0. (5.21) and ( 5 . 2 2 ) are independent o f

all

z.

ae

Expressions

z and hence hold f o r

Equating t h e s e two e x p r e s s i o n s g i v e s

The Ekman l a y e r

therefore

45

v = r/Z.

S u b s t i t u t i o n of t h i s value f o r

v

w1 =

-

i n t o ( 5 . 2 2 ) gives -1 2 -

The divergence equation f o r axisymmetric geostrophic flow i s hence

ru i s a constant, which w i l l be equal t o zero i f the

s i d e walls a r e r i g i d .

Thus t h e f i r s t order steady flow i s i m -

mediately obtained from the Ekmn l a y e r conditions a t t h e t o p and bottom boundaries.

A s one might i n t u i t i v e l y expect, we

f i n d t h a t t h e i n t e r i o r angular v e l o c i t y increases t o b e t h e average of t h e top and bottom boundary v e l o c i t i e s , and t h a t t h i s i s accompanied by a small constant v e r t i c a l v e l o c i t y .

z = 1 O(E1’*)

r/

z = o

V e r t i c a l v e l o c i t y component driven by Elcman l a y e r suction. FIGURE 7

46

The Elcman l a y e r s p i r a l

-

Of course t o complete the problem i t i s necessary t o examine the s i d e wall layers which must e x i s t t o r e t u r n t h e v e r t i c a l

mass f l u x .

We w i l l consider t h i s problem i n a l a t e r s e c t i o n .

I n c y l i n d r i c a l co-ordinates the Ekman l a y e r equations a r e

again

I f we consider the case where the boundary condition i s

2

=

rO

at

5=0

and assume t h a t t h e i n t e r i o r flow is un-

a f f e c t e d , then t h e s o l u t i o n I n the Ekman l a y e r i s

5 + iG

$ = e-5

thus

-

= i r e ( l+i 1t ,

sin 5,

$ = e-5

Hence a hodograph plane diagram i n which n

against

V

5

cos 5 .

is plotted

gives the c h a r a c t e r i s t i c Ekman l a y e r s p i r a l ,

which is sketched i n Figure 8.

The Ekman l a y e r

The EKman l a y e r s p i r a l FIGURE 8

s TransDo r t I n t h e E K m a n

The

O(1) =

. I . .

ui

v e l o c i t y i n the EKman l a y e r i s

+

-1

vj

( r e t u r n i n g t o Cartesian co-ordinates).

Hence t h e mass t r a n s p o r t

i s given by

Now Hence s u b s t i t u t i n g f o r

and

i

from (5.17) gives

47

The spin-up time s c a l e

48

=

+ i; x

$[(cpJo)

I f we again assume t h a t transport is a t

z0

is zero, we observe t h a t the mass

45O t o the imposed v e l o c i t y 9.

$Din-ur, Time SI n t h e previous

WOrK

we have discussed the steady flow i n

the i n t e r i o r and the Ekman l a y e r .

We w i l l now consider the

time s c a l e on which t h i s i s achieved. The time-dependent form of the EKman l a y e r equations (5.11) and (5.12) give

aii 2 - Go

=

iio

55

The s o l u t i o n of t h i s system, together with appropriate bounda r y conditions, can be obtained using Laplace transforms ( s e e Greenspan [ 2 6 ] ) , and i t i s found t h a t t h e time dependence goes a s

erf(2it)

and

erfc((l+i)t”‘).

Hence, a s we

would expect from t h i s system of equations where each term i s 0(1), the time dependence decays t o give a steady s o l u t i o n on

a time s c a l e of

O ( 1 ) . I n other words, the steady Ekman layer

has developed a t t h e boundary a f t e r a couple of revolutions. The steady Exman l a y e r w i l l now modify t h e i n t e r i o r v e l o c i t y by means of the c i r c u l a t i o n induced by E K M n layer suction.

49

The Ekman l a y e r

This process i s known a s spin-up ( o r spin-down i f t h e boundary v e l o c i t y i s decreased r e l a t i v e t o

n), and t h e time taken f o r

the i n t e r i o r v e l o c i t y t o reach a new steady s t a t e i s c a l l e d t h e spin-up time s c a l e . Let us consider t h e time dependent equations s a t i s f i e d by the

and

O(1)

i n t e r i o r flow:

O(E112)

v.3 Thus

= 0.

x (5.25) gives

k.V

Now the v e r t i c a l v e l o c i t y boundaries is

0(E1”).

w

induced a t t h e top and bottom

I f t h e angular v e l o c i t y of both t h e

t o p and t h e bottom is increased, t h e induced v e l o c i t y a t

z

and

= 0

z = 1 w i l l be of opposite s i g n s ( r e c a l l t h e EKman

layer analysis).

in w

Hence t h e order of

per Unit change i n h e i g h t , i s

g, which

i s t h e change

O(ElL2).

We a r e assum-

ing i n i t i a l l y t h e f l u i d i s i n r i g i d body r o t a t i o n with

k-2 =

0:

i n t h e f i n a l steady s t a t e t h e angular v e l o c i t y of

t h e f l u i d is increased t o t h a t of t h e boundary, hence i n our scaling,

k-9 =

O(1).

Thus we observe from (5.25) t h a t such a

balance of terms r e q u i r e s t h a t t h e time s c a l e on which becomes

O ( 1) must be

spin-up time s c a l e i s

O(E-1’2)

:

o(E-~/~).

i .e.,

k-5

t h e non-dimensional

Spin-up c i r c u l a t i o n

50

/

/

/

/

/

/

/

1

/ \ \ \

\

\

\

\ I

\

\

\

\

\

\

? Tr I

I

I

I

/

/

\

I

\

I

I

\

\

\ \

\

\

\ \

\

/ \

\

\

\

\

\

\

/ /

\ \

I

I

I

I

I

Spin-up c i r c u l a t i o n

FIGURE 9 We can describe spin-up i n physical terms i n the example of a cylinder where the boundary v e l o c i t y i s increased from

to

n(l+c).

n

After a couple of revolutions Ekman l a y e r s form

on the upper and lower boundaries.

The f l u i d i s spun rapidly

outwards and f r i c t i o n a l forces allow the angular v e l o c i t y t o increase. ior.

A small v e r t i c a l v e l o c i t y i s induced i n the i n t e r -

We can consider the Ekman layer a s "sucking in" f l u i d

51

The Ekman l a y e r

from the i n t e r i o r t o replace the f l u i d t h a t i s spun r a d i a l l y outward.

This s u c t i o n gives r i s e t o a s t r e t c h i n g of the

vortex l i n e s and hence an increase of the i n t e r i o r v o r t i c i t y (another very common example of t h i s process i s the s t r e t c h i n g of vortex l i n e s by g r a v i t y a s water drops down the d r a i n i n a bathtub:

t h i s spins up the f l u i d i n t o a v o r t e x ) .

Equation

(5.25) gives a mathematical d e s c r i p t i o n of the process: v e r t i c a l gradient i n vorticity.

w

a

gives r i s e t o a n i n c r e a s e i n the

When t h e f l u i d is spun-up t o the increased angu-

l a r v e l o c i t y of

n(l+E)

the v o r t i c i t y ceases t o change and

equation (5.25) reduces t o the steady form

which i s a now f a m i l i a r r e s u l t contained i n the TaylorProudman theorem. The mechanism and the time s c a l e of spin-up i s of g r e a t importance t o geophysical f l u i d dynamics because i t provides a very rapid way f o r a boundary condition t o be communicated t o the i n t e r i o r flow.

Consider a non-rotating f l u i d where

viscous d i f f u s i o n i s the only means of communication.

When

the boundary v e l o c i t y i s perturbed, the time s c a l e on which the i n t e r i o r steady s t a t e i s achieved i s the viscous d i f f u s i v e s c a l e of

O($).

I n the r o t a t i n g problem, the spin-up

time s c a l e , i n dimensional terms, i s

Since

v

tional to

i s small, t h e spin-up time s c a l e , which i s propor-

v-~",

is m c h f a s t e r than t h e d i f f u s i v e s c a l e

52

Tea-cup experiment

which is proportional t o

v-’.

Sea-CuD ExDeriment We suggest performing the following simple experiment t o i l l u s t r a t e spin-down.

Take a transparent Pyrex d i s h , prefer-

ably w i t h s t r a i g h t s i d e s ; h a l f f i l l the d i s h w i t h water and add cooked t e a leaves which should sink t o the bottom.

t e a leaves tend t o f l o a t ] .

[Fresh

S t i r the water uniformly u n t i l

the water is i n r i g i d body r o t a t i o n .

Cease t o s t i r and ob-

serve and time the spin-down process.

It is very easy t o

follow the motion of the t e a leaves i f t h e d i s h is placed on a n overhead p r o j e c t o r and the experiment is projected onto a screen.

It w i l l be seen t h a t the tea-leaves s p i r a l i n t o the

c e n t e r of the d i s h w i t h reduced angular v e l o c i t y ; they ceaee t o move a f t e r a time s c a l e

’’‘

o(L/(n

T h i s is, of

vl/”)).

course, the predicted motion in the Ekman l a y e r when the angular v e l o c i t y of the boundary is decreased r e l a t i v e t o the fluid

.

Example:

Note: i.e.,

L = 5cm, 0 = Z/sec,

2 v = 0.01 cm /sec.,

spin-down time s c a l e

a

Diffusive time s c a l e

=

% = 35 s e c s . r;4

hence the

= 2500 secs.

if the process were purely d i f f u s i v e , the time f o r

motion t o s t o p would be very much longer. This experiment is only an approximation t o t h e o r e t i c a l spin-down s i n c e the perturbation of the boundary v e l o c i t y is not small, i n f a c t , t h e Rossby number

E =

1, hence the non-

53

The EKman l a y e r l i n e a r terms can not s t r i c t l y be neglected.

However, the

experiment does produce a spin-down time t h a t i s remarkably c l o s e t o the predicted value. We note t h a t t h e phenomena of spin-up o r spin-down of a f l u i d t h a t i s i n i t i a l l y i n r o t a t i o n a r e equivalent ( i n t h e l i n e a r theory).

However, the problem of spin-up of a body of

f l u i d t h a t is i n i t i a l l y a t r e s t i s fundamentally d i f f e r e n t s i n c e i n t h i s case i t i s not i n i t i a l l y possible t o induce v o r t i c i t y by t h e mechanism of vortex l i n e s t r e t c h i n g .

Viewed

i n terms of the equations, i t is c l e a r l y the case t h a t i f t h e f l u i d i s i n i t i a l l y a t r e s t , the C o r i o l i s term i s missing from equation (5.23).

Hence the s t r e t c h i n g term

e x i s t s i n the v o r t i c i t y equation (5.25),

no longer

and a c r u c i a l mechan-

i s m by which v o r t i c i t y can be changed i n a r o t a t i n g f l u i d i s absent.

Problems

54 Chapter 5 P r o b l e w

Derive the f-rst order steady

3w

i n a cy Lnd e r

with the following boundary conditions.

3

9= C J

-r8

at

z=O

+r^e

at

z=1

r5

at

z=1.

=

=

Legend has it t h a t the oceanographer Ekman was led t o i n v e s t i g a t e the viscous boundary l a y e r a t the surface of the ocean a f t e r observing an iceberg move a t r i g h t angles t o the w i n d .

Explain t h i s

phenomenon by showing t h a t t h e mass t r a n s p o r t i n the Ekman layer i s a t wind s t r e s s

9 0 '

t o a n imposed surface

z.

Consider a r o t a t i n g homogeneous viscous f l u i d bounded by a surface with outward u n i t normal

n.

vector

.)

Show t h a t t h e surface boundary l a y e r has the form

of an EKman layer everywhere except i n the regions where

n - b = 0. A

*

What i s the thickness of the Ekman layer? Why does the Ekman l a y e r a n a l y s i s break down when

= 01

55

Problems

5.4)

Determine t h e steady flow d r i v e n by t h e Ekman l a y e r i n a sphere when t h e lower hemisphere i s r o t a t i n g with angular velocity

n(l-c)

and t h e upper

hemisphere i s r o t a t i n g with angular v e l o c i t y n(i+c).

This Page Intentionally Left Blank

CHAPTER 6 THE GEOSTROPHIC MODE

We have seen t h a t i n r o t a t i n g f l u i d motion t h e r e a r e t h r e e important time s c a l e s ; the l a y e r develops; the and the

O(E’l)

residual effects.

O(E-u2)

O(1)

s c a l e on which t h e Ekman

s c a l e on which spin-up occurs;

s c a l e on which t h e r e is viscous decay of We will now d i s c u s s t h e t r a n s i e n t motion

t h a t occurs before the steady s t a t e i s achieved by t h e mechan-

ism of spin-up.

To do t h i s we w i l l assume t h a t t h e i n t e r i o r

s o l u t i o n can be w r i t t e n a s a superposition of a l l the I n v i s c i d modes, namely

so

i s the mode t h a t corresponds t o eigenvalue zero.

This

mode c l e a r l y s a t i s f i e s the equations of geostrophic balance and

i t is c a l l e d the geostrophic mode.

Since we have Observed t h a t

t h e steady geostrophic mode describes the flow on t h e time s c a l e we could reasonably expect

57

-

O(E 1/2 )

The gesotrophic mode i n a sphere

58

The time-dependent l i n e a r i z e d i n v i s c i d equations a r e

2+

zi; x

q = -VP L)

v.3 Hence the equations f o r

so zir x

= 0.

and

CJm

so =

are (6.2)

-VB0

and

I n t h i s s e c t i o n we w i l l i n v e s t i g a t e the solutions of (6.2) and (6.3) f o r the geostrophic mode i n a closed container.

I n the

following s e c t i o n we w i l l study the wave-like s o l u t i o n s t o (6.4) and (6.5) which a r e c a l l e d i n e r t i a l modes. GeostroDhic Mode i n a S ~ h a Let us consider the example of flow i n a sphere. boundary condition on (6.2) and (6.3) i s s p h e r i c a l geometry becomes,

9.;

=I

0, where

r a d i a l vector i n s p h e r i c a l co-ordinates.

9.;

= 0,

The which i n

? i s the u n i t [we note t h a t a

t a n g e n t i a l boundary condition can not be imposed upon the i n v i s c i d equations]

.

59

The geostrophic mode

I l l u s t r a t i o n of s p h e r i c a l and c y l i n d r i c a l co-ordinates FIGURE 10

Let

be s p h e r i c a l co-ordinates and

(r,e,4)

c y l i n d r i c a l co-ordinates.

Hence

;.so

=

(k

cos

e +

TaKing

k. ( 6 . 2 )

sin 0)

1 [T k x

Thus t h e boundary c o n d i t i o n

‘-so =

A

0

(R,+,z)

gives

vio

implies

+ wkl.

be

60

and hence

w

-

=

Geostrophic contours x ofo

- $ % fi.k

+w

e

cos

x vOo, on

= 0,

r = a.

Now t h e geostrophic equations i m p l y t h a t w

value of

must be the same a t

and a t

0

(6 * 7 ) Hence the

= 0.

(n-0).

From

(6.7) we observe t h a t t h i s can only occur i f

Writing

sin 0

we observe t h a t condition (6.8) re-

as

quires the p o t e n t i a l

)o t o satisfy

The above condition, together with the f a c t t h a t t h e geo-

ar 2

s t r o p h i c mode s a t i s f i e s

= 0,

implies t h a t the geostrophic

p o t e n t i a l i n s p h e r i c a l geometry must be a function only of the c y l i n d r i c a l radius, i . e . ,

fo

= o ~ ( R ) . Thus from

-’^o 8oo aR

- 2

(6.6)

(6.9)

(R).

Thus i n geostrophic flow the f l u i d moves around each c i r c l e of l a t i t u d e of radius

ar

2 bR

(R).

R

w i t h a constant v e l o c i t s of

I n a sphere t h e r e e x i s t s two c i r c l e s of l a t i t u d e

corresponding t o a given value of

R

(one i n the upper hemi-

sphere and a symmetric one i n the lower hemisphere). height

h

The

of a v e r t i c a l column bounded by these two c i r c l e s

i s a function of the radius

R.

Clearly the v e l o c i t y given

The geostrophic mode by expression (6.9) implies t h a t each COluttUl of f l u i d of height

h(R)

moves a s a u n i t , maintaining constant length,

around the boundary c i r c l e s which a r e c a l l e d geostrophic contours.

(See Figure 11).

I

Geostrophic contours i n a sphere FIGURE If

61

Free, guided and blocKed regions

62

BostroD-lv

Free. Guided. and

I n a general geometry the existence of such closed geos t r o p h i c contours bounding a column of constant height allows geostrophic motion t o occur.

h,

It i s possible t o extend

the a n a l y s i s t h a t we have given f o r the sphere t o any closed container where the boundary can be covered by geostrophic contours.

A formulation of t h e general theory, together with

some supporting experimental r e s u l t s is given by Greenspan and Howard [ 2 5 ] .

I n an a r b i t r a r y geometry it i s u s e f u l t o

c l a s s i f y the boundary i n t o t h r e e t y p e s of regions ( s e e Howard

Geostrophically f r e e i n which every contour lying

i n the upper and lower boundary surface i s a geostrophic contour:

an example of t h i s i s a c y l i n d e r .

Geostrophically guided i n which it i s possible t o cover the boundary with a unique s e t of geostrophic contours:

a n example of t h i s i s a sphere.

Geostrophically blocked where geostrophic contours do not e x i s t :

an example of t h i s i s a s l i c e d

cylinder i n which the top and bottom a r e not p a r a l l e l . The problem of steady flow i n a cylinder, which i s geostrop h l c a l l y f r e e , i s degenerate because the height constant.

h

is a

I n t h i s case the geostrophic equations give l e s s

information, and w i l l be s a t i s f i e d by any pressure

cg(x,y).

However, as we i l l u s t r a t e d i n Chapter 5, i t i s possible t o solve the problem by introducing the EKman boundary l a y e r .

The geos t rophic mode

63

I n the case where the boundary i s geostrophically blocked i t i s impossible t o o b t a i n exact geostrophic balance and the

steady geostrophic mode b r e a m down i n t o an i n f i n i t e s e t of' low frequency waves.

These a r e c a l l e d Rossby waves which we

w i l l study i n Chapter 8.

For a container whose boundary i s geostrophically guided, a n a l y s i s s i m i l a r t o t h a t given f o r a sphere shows t h a t the

# o can be w r i t t e n as a f u n c t i o n purely of

pressure

(note f o r a sphere radius where

R

2

=

a'

- %). 2

a , we can w r i t e

#o

=

h

P0(h)

The v e l o c i t y i s then given by the

express i o n

(6.10) Where

&,

and

n,g

bottom such t h a t

a r e the vectors normal t o the top and

&-k

= 1 and

= -1.

C irc u l a t ion Let us consider the c i r c u l a t i o n around a geostrophic contour

r = 4; 9 . i ~ . C

I n the case of a sphere we use expression (6.9) t o o b t a i n

(6.11)

Hence the geostrophic mode possesses non-zero c i r c u l a t i o n

64

Circulation of the geostrophic mode

around each geostrophic contour.

I n the case of a general geostrophically guided geometry, the vector p a r a l l e l t o the geostrophic contour w i l l be perpendicular t o both

*&,

and

&.

Thus we can write

and expression (6.10) implies (6.12)

Hence, i n general the geostrophic mode possesses c i r c u l a t i o n .

Problems

65

ChaDter 6 Problem 6.1)

Consider flow i n a geostrophically guided region. Show t h a t no geostrophic flow can have any n e t f l u x through any s u r f a c e h(x,y) curve

6.2)

X

of constant height

associated w i t h a c l o s e d geostrophic C.

I d e n t i f y the labeled regions i n the following diagram as e i t h e r geostrophically guided, free o r blocked.

3-dimensional figure

l e v e l curves

h(x,y)

66

Problems

6.3)

Let

I: be t h e s u r f a c e of a g e o s t r o p h i c a l l y guided

c o n t a i n e r where t h e top and bottom a r e defined by z = z,(x,y)

and

z = z,(x,y)

respectively.

Write t h e normal v e c t o r s ~

Let

C

f

~

=

v kz T - and

=

-(k

- ozg).

be a geostr-phic contour and l e t

h

be

t h e height of t h e column bounded by t h e a s s o c i a t e d p a i r of geostrophic contours. a)

Show t h a t t h e geostrophic p r e s s u r e f u n c t i o n purely of

b)

.

Show t h a t t h e geostrophic v e l o c i t y &O=

6.4)

h

- 1 3 2 ah

go

is a i s given by

ax&*

Consider a r o t a t i n g sphere of f l u i d .

I n the

r o t a t i n g co-ordinate system t h e i n i t i a l v e l o c i t y i s given by & C

=

Ak,

where

A

i s a constant.

Determine t h e geostrophic mode.

CHAPTER 7 INERTIAL MODES

I n S e c t i o n 6 we assumed t h a t t h e time dependent p a r t of t h e flow can be represented by a s u p e r p o s i t i o n of normal modes. These a r e c a l l e d i n e r t i a l modes and s a t i s f y equations ( 6 . 4 ) and ( 6 . 5 ) , t o g e t h e r with t h e boundary c o n d i t i o n

9.A

t h e prescribed i n i t i a l conditions f o r t h e problem.

and

= 0

Again t h e

t a n g e n t i a l boundary conditions can not be s a t i s f i e d by t h e

O( I) problem which i s e s s e n t i a l l y i n v i s c i d .

The following

manipulation, which should by now be f a m i l i a r t o t h e r e a d e r , gives t h e equation f o r t h e p r e s s u r e f i e l d We compute

and

k.(6.4)

ix,

:“,am

and

k

x (6.4)

Bm. t o give

+ “r~k-$,-~~l (7.2) =

-kxm,.

S u b s t i t u t i o n of (6.4) and (7.1) i n t o ( 7 . 2 ) gives

(7.3) Because

v .Q = 0, t h e divergence of equation ( 7 . 3 ) g i v e s -m “m z (7.4) - ‘ 2 V B r n

67

x

68 Hence

Bm

1x1

r e a l and

<

2

satisfies

which I s Known a s Poincark's equation. We will l a t e r show t h a t

A

equation (7.5) is hyperbolic.

9m-g=

0

is r e a l and

IAI

<

2,

thus

However t h e boundary condition

on the closed boundary

C

of t h e container is of

the type u s u a l l y associated with a n e l l i p t i c equation.

In

general, Poincar'e's equation would be i n c o n s i s t a n t with the boundary condition values

Am

9.;

= 0

on

C, but a t p a r t i c u l a r eigen-

there e x i s t s a s o l u t i o n t o t h e problem:

this

property, of course, is used t o determine .,A A rigorous a n a l y s i s of the i n i t i a l value problem with prescribed boundary conditions, including t h e study of the inv i s c i d spectrum as the l i m i t of the VISCOUS s o l u t i o n and a determination of t h e completeness of t h e normal modes, is very d i f f i c u l t and w i l l not be attempted here.

However, i t

is possible t o e x h i b i t c e r t a i n f a i r l y simple p r o p e r t i e s of the normal modes.

u Let [ (6.4)

3:

be the complex conjugate of

Qm.

Computing

gives

I n t e g r a t i n g (7.6) over the volume of the container gives

69

I n e r t i a l modes

From Stokes

However,

theorem

V*C&

*

= 0,

*

and

Sm-n = 0

on the boundary, hence

(7.7) becomes

We write

-

= A

Q -Dm

+

*

iB, Q r

rm

-- -

= A

LB.

Thus

Hence we conclude t h a t

am is r e a l and

It is possible t o eliminate t h e case

manner.

Assume

x

= 2

Computing

s*

+ ax$

x (7.9)

iS*XS + Hence

< 2.

A = 2

in the following

is an eigenvalue; t h e eigenfunction then

s a t is f i e s is

lXmI

=

-

+

va.

(7.9)

gives iQCJ1z

-

(cJ* k)g =

4i.s*xg + 191’ - Iq.2

L? =

-

+

9*xoa.

- 5 (irX9*).VP.

and from t h e complex conjugate of (7.10) we have

(7.10)

(7.11)

O r thogona l i t y

70

icxLJ*

V.S* =

since

(7.12)

.V@

0.

7.12) and i n t e g r a t i n g over

Combining (7.11) and

Evaluating (7.8) a t

shows t h a t the

km = 2

gives

V

L.H.S.

of (7.13)

i s zero, hence

v#

and

= 0

Q*E = 0.

Thus (7.10) becomes

which coupled with t h e divergence equation implies t h a t

v-s

-

Since

Q-G = 0

v.kxg

and

= 0

we can write

Q =

above conditions we observe t h a t functions of

x

and

y

= 0.

ui

+

u

and

and applying the

v?

a r e analytic

v

f o r each value of

z.

9-i=

E.

Thus

$. must

satisfy

v 29

= 0

with

T h i s problem has the unique s o l u t i o n

0

on

Q s 0.

-?

no n o n - t r i v i a l eigenfunction corresponding t o

Thus t h e r e i s A = 2.

Orthoaonality Let

(SmjAmm)and

eigenvalues

(6.4)

.

, hence

Both

$

Sn,Xn) and

be d i s t i n c t eigenfunctions and

%

s a t i s f y a form of equation

I n e r t i a l modes

71 (7.14)

and

A d d i t i o n of ( 7 . 1 4 ) and (7.15) and i n t e g r a t i o n over t h e volume

v.C&

Now

gives

V

*

=

v.9 n

= 0

and

r

*

n-C.Jrn=

= 0

on

2 , thus

a p p l y i n g StOKeS' theorem t o (7.16) g i v e s

i -n .Q*dv Q

= 0,

rn

n,

V

i.e.,

d i s t i n c t eigenfunction a r e orthogonal.

We note t h a t t h e proof f o l l o w s through if

A n = 0, hence

t h e g e o s t r o p h i c mode i s o r t h o g o n a l t o each i n e r t i a l mode i n t h e s e n s e of t h e energy i n n e r product g i v e n by ( 7 . 1 6 ) . &an C i r c u l a t i o n Theorem A s we showed i n S e c t i o n 6 , t h e c i r c u l a t i o n

r

of t h e

g e o s t r o p h i c mode i n t e g r a t e d around a g e o s t r o p h i c c o n t o u r i s non-zero.

We w i l l now show t h a t t h e c i r c u l a t i o n of a d e p t h

averaged i n e r t i a l mode i s z e r o ; thus t h e g e o s t r o p h i c mode must c a r r y a l l t h e c i r c u l a t i o n p r e s e n t i n t h e i n i t i a l conditions. Let mode,

<sm>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<2J

+

2;

X

<9,>

= -0

sz,

ZT

idz

+ PT&

+ o*%

z

(7.17)

Hence, iXmV

since

v.<,&J

x <€J&=

= 0

and

v x (IT* + P*%),

&

= 0.

(7.18)

Computing t h e s u r f a c e

i n t e g r a l of (7.18) on a cap bounded by a g e o s t r o p h i c contour C

on t h e s u r f a c e of t h e c o n t a i n e r , and applying Gauss'

theorem g i v e s

However a s we remarked i n

S e c t i o n 6, t h e v e c t o r p a r a l l e l t o

a g e o s t r o p h i c contour w i l l be p e r p e n d i c u l a r t o hence

-+ads

provided t h a t

=

-%.cis

X,

= 0.

0.

% -

and

-%,

Thus (7.19) g i v e s t h e r e s u l t t h a t

Hence only t h e depth averaged g e o s t r o -

phic mode can c a r r y c i r c u l a t i o n . J n i t i a l Value Pr oblem I f we assume t h e completeness of t h e s e t of modes we a r e now a b l e in p r i n c i p l e , i f not i n p r a c t i c e , t o w r i t e down a

I n e r t i a l modes

73

s o l u t i o n t o t h e i n i t i a l value problem f o r flow i n a r o t a t i n g c o n t a i n e r whose boundary i s composed of geostrophic curves. We can attempt t o give t h e s o l u t i o n a s a Fourier s e r i e s expansion i n terms of t h e orthogonal i n e r t i a l modes and t h e geos t rophic mode :

Let t h e i n i t i a l c o n d i t i o n be of t h e modes we then have

-

q = u(r).

- *

From o r t h o g o n a l i t y

And from t h e mean c i r c u l a t i o n theorem

C

where

C

C

is a geostrophic contour.

Thus

so

i s t h e unique

geostrophic flow t h a t c a r r i e s t h e mean c i r c u l a t i o n of t h e i n i t i a l depth averaged flow.

I n o t h e r words, t h i s i s t h e

property t h a t f o r a p a r t i c u l a r i n l t i a l value problem, d e t e r mines t h e s p e c i f i c value of t h e f u n c t i o n

I O ( h ) , from a l l t h e

p o s s i b l e geostrophic p r e s s u r e f i e l d s allowed by t h e c o n t a i n e r , which i s t h e unique geostrophic s o l u t i o n . Let us r e c a l l t h e problem i n t h e case of a g e o s t r o p h i c a l l y f r e e boundary (e.g.,

the c y l i n d e r ) .

I n t h i s case any p a i r of

closed contours i n t h e upper and lower bounding s u r f a c e s a r e geostrophic contours.

Hence we s e e t h a t f o r such geometry

74

I n e r t i a l modes i n a c y l i n d e r

i t i s necessary t o Know t h e i n i t i a l mean c i r c u l a t i o n about

every c l o s e d c o n t o u r , which ( f o r a simply connected r e g i o n ) , i s e q u i v a l e n t t o r e q u i r i n g t h a t t h e v e r t i c a l component of t h e i n i t i a l v o r t i c i t y be s p e c i f i e d .

I n o t h e r words, the unique

s o l u t i o n c a n only be determined when s t r o n g e r requirements a r e placed on t h e i n i t i a l d a t a .

From t h e s e o b s e r v a t i o n s and t h o s e

i n Chapter 6 , we conclude t h a t although t h e concept of geos t r o p h y is v a l u a b l e i n computing t h e s t e a d y flow i n a geos t r o p h i c a l l y guided geometry, i t i s l e s s powerful when t h e problem is such t h a t t h e f l u i d i s s u b j e c t t o two c o n s t r a i n t s , namely t h e geometry and t h e e f f e c t s of r o t a t i o n , t h e p o s s i b l e flow p a t t e r n s a r e s t r i c t l y l i m i t e d .

However i f t h e two con-

s t r a i n t s c o i n c i d e , a s they do i n a c y l i n d e r , l e s s i n f o r m a t i o n i s immediately a v a i l a b l e from t h e g e o s t r o p h i c a n a l y s i s . D e r t i a l Modes i n a C . v l i n d u We have shown t h a t t h e i n i t i a l v a l u e problem c a n be solved

i n terms of a F o u r i e r s e r i e s expansion i n t h e i n e r t i a l modes and t h e g e o s t r o p h i c mode.

However t h e d e t e r m i n a t i o n of t h e

e i g e n f l m c t i o n s a s s o l u t i o n of P o i n c a r e ' s e q u a t i o n (7.5) with z e r o normal v e l o c i t y i s i n g e n e r a l , v e r y d i f f i c u l t , and has only been c a r r i e d out f o r a few r e l a t i v e l y simple geometries. Aldridge and Toomre [ 11 , have g i v e n g e n e r a l polynomial formulas f o r t h e s p h e r i c a l and s p h e r o i d a l modes.

In the case

of a s p h e r e , i t i s a l s o p o s s i b l e t o compute t h e e f f e c t of v i s c o s i t y on model decay ( s e e Greenspan [ 2 6 ] ) ,

and r e s u l t s

a r e i n good agreement with experimental d a t a .

The s p h e r i c a l

75

I n e r t i a l modes problem has a t t r a c t e d considerable a t t e n t i o n because of i t Is obvious geophysical relevance.

We w i l l now give a s an example of t h e mathematics involved

in a simple geometry, t h e computation of t h e normal modes in a cylinder.

This problem was f i r s t considered by Kelvin [ 3 3 ]

and t h e r e have been many recent additions t o t h e theory.

We

wish t o solve

(7.20)

with boundary conditions

~ a =k o

-

e

6

Q - r= 0

at

z = 0,1;

at

r = a.

The co-ordinates a r e c y l i n d r i c a l co-ordinates w r i t e t h e boundary conditions i n t e r m of

@

(r,B , z )

.

To

we observe

equation (7.1) gives

hence,

ar az =

0

at

z = 0,l.

(7.21)

Also the h o r i z o n t a l components of the momentum equation (6 . 4 )

s a t i s f i e d by the i n e r t i a l mode give

76

Cylindrical solution

Hence t h e boundary c o n d i t i o n a t

r = a

becomes

We w i l l now s e e k s o l u t i o n s t o e q u a t i o n (7.20) with boundary

c o n d i t i o n s ( 7 . 2 1 ) and ( 7 . 2 2 ) u s i n g t h e method of s e p a r a t i o n of v a r i a b l e s .

We w r i t e 0 = F(r)eiKe cos mz,

where

K = $1, 2 2 , .

..

[Note t h e boundary c o n d i t i o n s a t

n = +1, +2,.

z = 0

and

that a Fourier cosine s e r i e s i s appropriate].

.. 1 indicate

Substitution

of t h i s form of s o l u t i o n i n t o e q u a t i o n (7.20) g i v e s

-5 F + (+ r 2

-1 b r4 r

( y ) ,where

Hence

and

F = JIKl

J

IKI

is t h e

k)-th

a

2 2

1) n n F = 0 .

(7.23)

satisfies

Bessel function.

From t h e boundary

c o n d i t i o n ( 7 . 2 2 ) we o b t a i n t h e t r a n s c e n d e n t a l e q u a t i o n f o r namely iA

;

u,

I n e r t i a l modes Thus

a = 5 mntc

which i s the

and the eigenvalue

,,x

m-th

77

p o s i t i v e zero of

i s given by

=

e(l+

k? -1/2 k 2 z) .

nn a

We note t h a t as one would expect i n a 3-dimensional problem, the eigenvalue spectrum is a t r i p l y i n f i n i t e s e t . Hence i t i s c l e a r t h a t even i n the r e l a t i v e l y simple geometry of a cylinder the eigenfunctions giving the i n e r t i a l modes a r e f a r from simple. J?&gg W a ve S o l u t i o a

We w i l l now i n v e s t i g a t e i n e r t i a l waves by SeeKing a plane wave s o l u t i o n i n a n unbounded f l u i d .

[We note t h a t i n a n

unbounded f l u i d our arguments f o r excluding hold.]

X = 2

no longer

We seeK a s o l u t i o n t o Poincarkls equation of the form

where the wave vector

-5

frequency of the wave.

=

+ u2j + K3L,

and

i s the

I n an unbounded f l u i d i t i s no longer

necessary t o s a t i s f y boundary conditions, thus be t r e a t e d a s constants.

A

and

i

can

S u b s t i t u t i o n of t h e plane wave form

i n t o Poincark's equation gives

Plane wave s o l u t i o n

78

L 2 ( K 12

+ u22 +

ns> 2

-

4~~ 2 =

0,

and

(7.24)

Thus t h e frequency depends on t h e d i r e c t i o n b u t not t h e magnitude of t h e wave v e c t o r .

From t h e d i s p e n s i o n r e l a t i o n (7.24),

t h e phase v e l o c i t y and t h e group v e l o c i t y c a n be computed. The phase v e l o c i t y constant

-

is t h e v e l o c i t y of s u r f a c e s of

C i s the ‘g v e l o c i t y with which energy is t r a n s p o r t e d by t h e wave. ($a$

Lt)

EP

and t h e group v e l o c i t y

The phase v e l o c i t y

c -“2=+2. -P - In1

2U

I* I

(7.25)

Thus t h e system i s d i s p e r s i v e with t h e phase v e l o c i t y b e i n g i n v e r s e l y p r o p o r t i o n a l t o t h e magnitude of t h e wave v e c t o r . Hence t h e long waves t r a v e l f a s t e s t and i n t h i s r e s p e c t i n e r t i a l waves behave liKe s u r f a c e waves. The group v e l o c i t y

C

-g

=

vK-A.

Hence

(7.26) Thus We n o t e t h a t

c

Q ‘

= -

151

2 - c

-P.

I n e r t i a l modes

79

= 0.

Hence energy i s transported a t right angles t o the d i r e c t i o n i n which the phase of the wave i s seen t o move. of the curious p r o p e r t i e s of an i n e r t i a l wave.

T h i s i s one A further

unusual property ( s e e P h i l l i p s [ 5 4 ] ) i s t h a t r e f l e c t i o n does not s a t i s f y S n e l l ' s Law, but r a t h e r t h e r e f l e c t e d wave and incident wave maKe the same angle w i t h the a x i s of r o t a t i o n , which i s independent of the alignment of the r e f l e c t i n g surface !

Problems

80

ChaDter 7 Problem

7.1)

For an incompressible, i n v i s c i d , r o t a t i n g f l u i d show t h a t the plane wave s o l u t i o n

satisfies

9-3 =

0.

Is t h i s r e s u l t t r u e more generally? 7.2)

Show t h a t i f K

m

p

i s any vector perpendicular t o

then

$.=%xg+ie. 7.3)

-

I n problem 7 . 2 ) choose 2

K

and the outward normal

plane.

t o be co-planar with t o an i n f i n i t e

Hence show t h a t a wave incident upon t h i s

plane is r e f l e c t e d i n such a d i r e c t i o n t h a t the Incident wave and the r e f l e c t e d wave make the same angle with t h e d i r e c t i o n of a x i s of r o t a t i o n .

k

of' t h e

Problems

7.4)*

81

An i n e r t i a l wave with group v e l o c i t y the

x

-

If

in

z plane bounces back and f o r t h between

t h e walls of a wedge

13).

2g

z

AB = d , and

=

x t a n Y.

BC = 4 ' .

( s e e Figure

Show t h a t

L L lSgl

\

=

l$l

k

FIGURE 13 *[The problems t h a t appear i n the t e x t marKed w i t h a n a s t e r i s k o r i g i n a t e from a course on r o t a t i n g f l u i d dynamics taught by the M.I.T.

Mathematics department .]

Problems

82 #

Consider Poincare's problem f o r i n e r t i a l modes i n a sphere. Write equation (7.5) and the boundary condition *

% an

on

= 0

r

2

+

z'

=

i n c y l i n d r i c a l co-

a'

ordinates. Introduce oblate spheroidal co-ordinates (b ,q)

defined by

=

[*

.=[-)-

1

y qp

.

Obtain the separable s o l u t i o n f o r

I

as

where

Obtain the transcendental equation t h a t defines the eigenvalue

Am.

a3

Problems

7.6)

Derive t h e frequency of t h e i n e r t i a l waves f o r a homogeneous f l u i d bounded by a r i g i d w a l l a t z = 0

and a f r e e s u r f a c e a t

z = Ho

[see

f i g u r e 141.

z = H 0 + q

z = H

0

F l u i d of density

FIGURE 14

Note:

The f r e e s u r f a c e e l e v a t i o n i s

The boundary conditions a t are:

w = a

at

z = Ho

+

q(x,y,t). q(x,y,t)

This Page Intentionally Left Blank

CHAPTER 8 ROSSBY WAVES

I n the previous s e c t i o n s we have discussed the geostrophic mode which e x i s t s when the boundary of t h e container can be covered with geostrophic contours, e.g.,

t h e sphere, the e l l i p -

soid, and i n a degenerate sense, the c y l i n d e r .

We w i l l now

consider t h e problem of flow i n a container i n which t h e r e a r e no geostrophic contours, namely t h e s l i c e d cylinder ( s e e Pedlosky and Greenspan [ 491 )

.

1

z

El

E

A s l i c e d cylinder FIGURE 15

85

S l i c e d cylinder

86

I n t h i s geometry t h e r e a r e no contours around which a column of f l u i d can move and r e t a i n a constant height, hence the [See Figure 151

geostrophic mode can no longer e x i s t .

.

How-

ever we have shown t h a t t h e i n e r t i a l modes c a r r y no mean c i r c u l a t i o n : thus if the i n i t i a l flow possesses mean c i r c u l a t i o n , i t must e x c i t e a mode t h a t is d i s t i n c t from the i n e r t i a l modes.

We w i l l show t h a t i n t h e absence of geostrophic con-

tours the c i r c u l a t i o n i s c a r r i e d by low frequency waves which a r e c a l l e d Rossby waves.

Let us assume t h a t the s l i c e angle consider a perturbation expansion i n tions are

as + at

I

a a.

i s small and we w i l l The i n v i s c i d equa-

-

2k x j = -VP

v.q = 0 w

2.;

with boundary condition

= 0.

We seek an o s c i l l a t o r y

solution

where

k,

9

and

I

can be expanded i n powers of

x -

9 I We note t h a t we s e t

Xla

a

as

+. ..

so + Sla +... = R0 + +... .

=

Lo = 0

ilU

since we a r e i n v e s t i g a t i n g t h e

e f f e c t on the geostrophic mode of the existence of the s l i c e

Rossby waves a t angle

87

a.

S u b s t i t u t i o n of t h i s form of s o l u t i o n i n t o t h e equations and equating successive powers of

Thus

&o

m

The

a

gives t o

O(1)

has t h e p r o p e r t i e s of a geostrophic mode. O(a)

problem i s

with

(zo+

and

*

(-.k +

a;) = o

at

z = t a n ay

-

ay.

Now Hence t o

O(a)

&o.;

.

The lower boundary c o n d i t i o n i s

-

&-(Ql

+

y

o 9) 39 =

at

z = 0.

(8.5)

We manipulate equations ( 8 . 2 ) i n the f a m i l i a r way t o o b t a i n t h e equation f o r t h e pressure: Taking

Since obtain

v x (8.2) gives

$

= 0 , we can i n t e g r a t e (8.6) with r e s p e c t t o

$1 =

ix +

(z-1)v x

so + $X,Y)l

z

to

88

Pressure equation f o r Rossby waves

hence

ihl and

4.; =

=

ix +

( z - 1 ) v x $o

from boundary condition ( 8 . 4 ) .

0

(8.7) S u b s t i t u t i o n of

(8.7) i n t o (8.5) gives

Recalling t h a t

Qo s a t i s f i e s the geostrophic equations, we

m

know t h a t

.%b = az

0

and

$o

=

x vio(x,y).

Hence (8.8) becomes

This is the equation s a t i s f i e d by t h e Rossby wave.

We note

t h a t we derived t h i s equation from the r e l a t i o n between and

k-Sl a t

t h e boundary

only terms t h a t a r e independent of holds f o r a l l

0

i

z

I

So

z = 0, however (8.9) involves z,

hence t h e equation

1.

The above procedure i s r a t h e r t y p i c a l of the manner i n which r e s u l t s a r e obtained i n r o t a t i n g f l u i d problems.

The

reader may draw a n analogy with t h e Ekman l a y e r a n a l y s i s of Section 5 where t h e equation defining t h e i n t e r i o r flow was obtained by studying the boundary conditions.

The f a c t t h a t a

f i r s t order problem can be obtained from the boundary condit i o n s on the higher order problem i s of course dependent on the Taylor-Proudman theorem.

This c l e a r l y gives a valuable

technique t h a t should be exploited when the Taylor-Proudman

89

Rossby waves theorem i s v a l i d . 8-Plane Probleg We have derived t h e equation s a t i s f i e d by Rossby waves

which r e p l a c e t h e geostrophic mode when a c y l i n d e r i s s l i c e d . We w i l l show t h a t we can o b t a i n e x a c t l y t h e same equation t o d e s c r i b e t h e low frequency modes t h a t e x i s t on a

8-plane.

I n o t h e r words, t h e r e i s dynamic equivalence between t h e waves t h a t e x i s t i n a geometry where t h e h e i g h t v a r i e s with a h o r i z o n t a l co-ordinate, and t h e waves t h a t e x i s t on a where C o r i o l i s f o r c e v a r i e s with l a t i t u d e .

p-plane

I n S e c t i o n 4 we

derived t h e geostrophic equations t h a t approximate flow i n a t h i n s h e l l on a s e c t i o n of a r o t a t i n g sphere.

Equations

(4.12) and (4.13) i n v e c t o r form g i v e

ft x where

f = fo

+

By, [ y

-

(8.10)

q = -vp,

i n c r e a s e s towards t h e n o r t h ( s e e

Figure 3 ) ] . We a l s o r e c a l l t h a t t h e p o t e n t l a l v o r t i c i t y was shown t o be

a conserved q u a n t i t y i n S e c t i o n 2.

Clearly we could follow

through t h e same s t e p s i n t h e proof of E r t e l ' s theorem i n t h e p-plane context and o b t a i n (8.11)

where

X

i s any s c a l a r such t h a t

p a r t i c u l a r expression f o r

X.

2

= 0.

We w i l l choose a

Consider a t h i n l a y e r of f l u i d

above a bottom s u r f a c e of v a r i a b l e h e i g h t

H(x,y).

The

E f f e c t s of topography

90

following a n a l y s i s shows t h a t t h e q u a n t i t y

(y)i s

con-

served and hence t h i s q u a n t i t y i s a s u i t a b l e candidate f o r t h e r o l e of t h e s c a l a r

X.

Variable depth ocean FIGURE 16

Let

z = h(x,y)

denote t h e bottom s u r f a c e and l e t

t h e l e v e l s u r f a c e of t h e top. L-H(x,y).

The height

The divergence equation gives

H(x,y)

z = L

is

be

ROssby waves

(ux-vy),

(8.12)

w=Uhx+vh. Y

(8.13)

az

and a t

z

-

91

=

= h,

Since the flow i s geostrophic,

and hy = 0.

= 0

az

Thus (8.12) can be i n t e g r a t e d using t h e boundary condition (8.13) to give

w

=

(h-z)(ux+v ) Y

A t the upper surface

z

= L

uhx

+

we have

u + v = X

+

uh

+

w

Vhy

-

(8.14)

0, hence

I

vh

H

Y

Therefore (8.14) can be w r i t t e n i n the form

from which we conclude t h a t The

9 L = z] = dt [

0.

@-plane approximation was derived assuming flow i n a

t h i n s h e l l , hence the h o r i z o n t a l components dominate t h e v e r t i c a l component

w.

Thus

u

i s conserved.

v(?)

i(f+

v

,B i s approximately

and the p o t e n t i a l v o r t i c i t y , Ti = 2 P

and

- E)

92

Analogy between v a r i a b l e

and v a r i a b l e

f

M.

I n the case of a homogeneous f l u i d t h i s r e s u l t gives

From t h i s equation we can derive Rossby waves supported by or variable

e i t h e r variable

f

&& a)

be constant and l e t

Let

H

H.

+ py.

f = fo

Lineariza-

t i o n of equation (8.15) gives

u

Substituting f o r

and

v

i n terms of

5 V2P + p 2 Writing

P = @(x,y)eut

from (8.10) gives

P

(8.17)

= 0.

gives the wave equation

which is equivalent t o the Rossby wave equation given by (8.9).

U b) Treat

f

as a constant and l e t

H

E

L

-

ay.

Again

l i n e a r i z a t i o n of equation (8.15) gives

From t h e geostrophic r e l a t i o n s the equation f o r

4 at .v 2 P

+ ia %

= 0,

P

is

(8.20)

Rossby waves

93

which is once again the Rossby wave equation. Thus we see t h a t t h e s l i c e d c y l i n d e r i s a dynamic model f o r the behavior of flow i n an ocean b a s i n on a r o t a t i n g sphere, and both s i t u a t i o n s s u s t a i n Rossby waves.

I n view of

t h e analogy between the two problems we have labeled t h e shallow s i d e of the s l i c e d c y l i n d e r i n Figure 15 the north. Clearly i n the geophysical context t h e C o r i o l i s parameter increases towards the north.

f

Our a n a l y s i s o f equation (8.15)

shows t h a t t h e laboratory model of the s l i c e d c y l i n d e r and t h e p-plane a r e i n dynamic correspondence when t h e height decreases towards the north.

H

Experimentally, i t i s of course

very u s e f u l t o be a b l e t o model an ocean basin by considering a s l i c e d cylinder:

we remind the reader t h a t the b a s i s f o r

t h i s modeling i s dynamic r a t h e r than geometric. I n the general problem, where C o r i o l i s parameter

f

is a

f u n c t i o n of l a t i t u d e , and the height describes an a r b i t r a r y topography

H(x,y), l i n e a r i z a t i o n of equation (8.15) again

gives r i s e t o an equation t h a t supports wave-1iKe s o l u t i o n s . These w i l l be somewhat more complicated, but of t h e same b a s i c s t r u c t u r e a s t h e Rossby waves of cases ( a ) and ( b ) .

The

pressure equation i n the more general case i s

(8.21) We remarK t h a t the Rossby wave node, l i k e the geostrophic mode, has non-zero mean c i r c u l a t i o n , and i s orthogonal t o each i n e r t i a l mode. cause the

O(1)

These properties of the Rossby wave hold beflow f i e l d s a t i s f i e s the equations of geostro-

94

Helmholtz e q u a t i o n

phic balance g i v e n by (8.1).

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

which t h e C o r i o l i s parameter, o r t h e h e i g h t , i s non-constant, i t i s p o s s i b l e t o s e e k a s o l u t i o n i n terms of normal modes.

However, i f t h e r e a r e no g e o s t r o p h i c c o n t o u r s , any i n i t i a l c i r c u l a t i o n w i l l e x c i t e Rossby waves r a t h e r t h a n t h e g e o s t r o We n o t e t h a t i t i s p o s s i b l e t h a t t h e boundary

phic mode.

c o n t a i n s g e o s t r o p h i c c o n t o u r s and y e t Rossby waves could be excited.

For example, l e t

H(x,y) = L - a ( ~ ~ + y ~ ) ~ /I n ' . this

c a s e t h e geometry s u p p o r t s both a g e o s t r o p h i c mode and Rossby waves, however t h e g e o s t r o p h i c mode would c a r r y a l l t h e i n i t i a l circulation. Let u s c o n s i d e r t h e s o l u t i o n t o t h e Rossby wave e q u a t i o n

(8.9) i n t h e geometry of t h e s l i c e d c y l i n d e r .

The boundary

condition i s

-2-

We seeK a s o l u t i o n

#o

= A(x,Y)e

ibl.

Substitution into

(8.9) g i v e s t h e Helmholtz e q u a t i o n :

(8.22)

with

A = 0

on

r = a.

The s o l u t i o n s f o r t h e e i g e n f u n c t i o n

a r e g i v e n by t h e doubly i n f i n i t e family

A,

where

Kmn

= cos m0 Jm(K,r/a),

i s the

n-th

s i n me J,(K,r/a),

p o s i t i v e z e r o of

J,.

The eigen-

Rossby waves

95

values a r e given by

Hence,

Plane Wave S o l u t i o n The plane wave s o l u t i o n t o t h e Rossby wave equation d i s plays a c h a r a c t e r i s t i c f e a t u r e of Rossby waves, namely t h e r e e x i s t s a mode t h a t propagates t o t h e west ( s e e Figure 15) w i t h speed

ax1.2

Consider a s o l u t i o n of t h e form

-is

02 and 3.s = K1x + K2y. S u b s t i t u t i n g t h i s where I 0 = c e form f o r @ o i n t o equation (8.9) gives t h e d i s p e r s i o n r e l a tion

(8.24) Hence t h e phase v e l o c i t y

The phase of t h e Rossby wave t h e r e f o r e t r a v e l s i n t h e d i r e c -

96

Plane wave s o l u t i o n

tiOn of negative

x , o r i n t h e context of t h e

p-plane model

A d e t a i l e d d e s c r i p t i o n , with i l l u s t r a t i o n s ,

towards the west.

of t h e propagation of a Rossby wave towards t h e west i n a d i f f e r e n t i a l l y r o t a t e d s l i c e d c y l i n d e r is given i n Pedlosky and Greenspan [49]. The group v e l o c i t y

C

-g

=

ouaXl(~)

Thus energy w i l l propagate towards t h e e a s t i f towards t h e west i f

2

K2

2

> ul.

2

U1

>

2 K2,

and

It has been shown by Longuet-

Higgins [43] t h a t waves i n c i d e n t on a western boundary with 2

K~

>

2 ulJ

2 g1

r e f l e c t as waves with

2

> u2

(and vice-versa on

an e a s t e r n boundary).

It can a l s o be shown ( s e e Pedlosky

[ 4 8 ] ) t h a t waves with

'cl

2

>>

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 ) and from (9.18), = O(E m w = O ( E 1-23). m

<

Since

p

1/3, the dominant term on t h e R.H.S.

E-8?n,

hence balance w i t h

2;

requires

of (9.15) i s

El/‘- l a y e r equations

104

h

P = O(EB),

II

and (9.17) reduces t o

I f we again impose t h e condi-

Pz = 0.

O ( E ’I2)we o b t a i n

t i o n t h a t t h e v e r t i c a l mass f l u x i s

6

hence

= 1/4.

Thus t h e second boundary l a y e r has dimensionless thickness 0(EV4) : t h e components a r e given by an asymptotic expansion i n powers of

as:

EV4

h

v = vo

h

-

U2 E

u = h

.....

+

h

w =

h

wlE114

+

I*

F1E114

+

P =

112+

.....

.....

.....

And the highest order equations a r e h

u2 51 h

+ EilZ

= 0

(9.19)

h

(9.20)

-2v0 = -pl rl 2C02 = vo h

nn

h

0=P1

z

.

We note t h a t (9.22) implies a weak form of t h e Taylor-Proudman theorem holds i n t h e the

E’/’-layer

El’‘-layer.

However, we observe t h a t

equations have a d i s t i n c t l y d i f f e r e n t char-

a c t e r from t h e i n t e r i o r equations, and i n f a c t , t h e possesses a high degree of v e r t i c a l s t r u c t u r e .

E1”-layer

We thus have a

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

105

progression from the i n t e r i o r where t h e r e i s no v e r t i c a l s t r u c ture, t o the

z and

m

w

E1/4-layer

where

depends l i n e a r l y on

CI

and

z,

( s e e equations (9.19)

. . )

v

-

a r e independent of

u

(9.22)), t o t h e s t r o n g v e r t i c a l s t r u c t u r e of the

E1I3-layer.

These v e r t i c a l shear l a y e r s a r e c a l l e d Stewartson l a y e r s : they can e x i s t a t any localized d i s c o n t i n u i t y , f o r example as i n t e r n a l shear l a y e r s bounding a Taylor column o r a s boundary l a y e r s a t t h e walls of a c y l i n d e r . We w i l l consider i n a l i t t l e more d e t a i l the equations

-

describing the

We wish t o solve equations (9.19)

E1l4-layer.

(9.22) with t h e appropriate boundary conditions.

the Ekman l a y e r i s much thinner than the

E1/4-layer,

Since the

Ekman l a y e r s u c t i o n condition holds here a s i t does i n the interior.

Hence we o b t a i n t h e boundary condition =

Where

vB

1/2

&

E 2

. E-l/4

A a,, (vo-vB)

at

z = 0,1.

i s t h e imposed wall v e l o c i t y a t t h e top and bottom

of the c y l i n d e r .

Now

vB

=

v B ( r ) , hence

Thus balancing terms of the same order i n the EKman l a y e r s u c t i o n condition gives m

w1 = i 2 aq v~ L,

at

z

= 0,1.

(9.23)

The s i d e wall boundary condition i s m

v0 = V,

- v,(a)

at

q =

o

(9.24)

106

V e r t i c a l mass flux

where

V,

i s t h e imposed s i d e w a l l v e l o c i t y and

vI

is the

i n t e r i o r v e l o c i t y t h a t can be determined u s i n g t h e a n a l y s i s given i n S e c t i o n 5 . From equations (9.19) and (9.21) we o b t a i n v,

+

N

rlrlrl

ZGl

= 0. Z

Since (9.20) and ( 9 . 2 2 ) imply t h a t

-

vo

(9.25) i s independent of

z,

we have

Hence from t h e boundary c o n d i t i o n s (9.23) we o b t a i n t h e equation for

#u

vo, namely

from boundary c o n d i t i o n ( 9 . 2 4 ) and e q u a t i o n ( 9 . 2 7 ) .

vo

t u t i n g t h i s expression f o r

Substi-

i n t o e q u a t i o n s (9.19) and

(9.21) g i v e s

and

u2

= [VW’VI(

w1

=

- JZ

a ) ] e- J2TJ

(z

- $1

[vw-v1(a)le - J 2 q

Thus t h e s o l u t i o n f o r t h e v e l o c i t y components i n t h e layer is e a s i l y obtained.

(9.29) (9.30) E 1/4-

We n o t e t h a t a s c o n s t r u c t e d , t h i s

l a y e r s a t i s f i e s t h e boundary c o n d i t i o n on t h e azimuthal v e l o c i t y component.

However, t h e t o t a l v e r t i c a l mass f l u x

Vertical shear layers

107

= 0.

It i s n e c e s s a r y t h a t t h e narrower s i d e - w a l l l a y e r o f

O(E

1/31

a l s o e x i s t s t o r e t u r n t h e i n t e r i o r v e r t i c a l mass f l u x . Let u s now 1ooK a t t h e

E1’3-layer

The e q u a t i o n s d e s c r i b i n g t h e flow a r e g i v e n by ( 9 . 9 ) (9.12).

-

i n a l i t t l e more d e t a i l .

From (9.9) and ( 9 . 1 1 ) we o b t a i n

-

-2w1

z

-

=

v

=

w

lxxx ’

and from (9.10) and ( 9 . 1 2 ) ,

-

ZVl

Z

Hence

3%

--$+4w1 ax

zz

-

(9.32)

lxxx

=o.

(9.33)

We observe t h a t t h e coupled e q u a t i o n s (9.31) and (9.32) have a somewhat s i m i l a r s t r u c t u r e t o t h o s e t h a t d e s c r i b e t h e EKman l a y e r , w i t h t h e r o l e s of t h e h o r i z o n t a l and the v e r t i c a l i n t e r changed.

These e q u a t i o n s c a n be i n t e r p r e t e d i n terms of gene-

r a t i o n of v o r t i c i t y :

w r i t e t h e v o r t i c i t y i n component form

as : 0

xg=_B=xS+Yi+zk.

Then t o h i g h e s t o r d e r

Y =

-; and 1,

and (9.32) can be r e w r i t t e n as:

Z =

7

lX

.

Thus (9.31)

Problem f o r the

108

-

-2w1

-

and

-2V1

Z

z

E1/’-layer

= zxx

(9.34)

= Yxx.

(9.35)

Hence shear i n the v e r t i c a l component of v e l o c i t y (Or s t r e t c h i n g of the flow l i n e s ) produces a v e r t i c a l component of v o r t i c i t y which determines t h e c i r c u l a t i o n of h o r i z o n t a l velocity.

Also shear i n the h o r i z o n t a l v e l o c i t y ( o r t i l t i n g

of the flow l i n e s ) produces a h o r i z o n t a l v o r t i c i t y which determines the c i r c u l a t i o n of v e r t i c a l v e l o c i t y . To determine the boundary conditions we again observe t h a t

t h e Ekman layer is much thinner than t h e

E1’3-layer,

the Ekman layer suction condition holds.

Thus the upper and

lower boundary conditions a r e

-

w1

hence

z = 0,l (9.36)

= i

I n the p a r t i c u l a r example I l l u s t r a t e d i n Figure 6, where V w = r

at

z = O

and

at

Vw = 0

z

= 1,

the boundary con-

d i t i o n becomes t o t h e highest order

-

at

w1 = 0

z

x

(9.37)

0,l.

We therefore seeK the s o l u t i o n t o the equation

a%

-++4G1 ax

ZZ

= o

w i t h boundary conditions (9.37) and the condition

a

together w i t h

J;

s,il

w dxdz -1

= 0

at

=

x

Ja,

r

= 0,

winterier and

-

dr,

0

as

x

-

a,.

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

109

3 w1 = e -mx s i n m ( z - 1 )

where Since t h e r e a r e two eigenvalues with p o s i t i v e r e a l p a r t f o r each i n t e g e r

n, i t is possible t o s a t i s f y both t h e boundary

condition on the f l u x on

vl,

Jy

wldx,

and the boundary condition

and hence determine the s o l u t i o n t o t h e problem.

We have described the s t r u c t u r e of two side-wall l a y e r s t h a t e x i s t t o r e t u r n the i n t e r i o r mass f l u x and t o match the i n t e r i o r and side-wall boundary conditions.

However, t h e r e

a r e a l s o o t h e r p o s s i b i l i t i e s f o r side-wall boundary l a y e r s . More complicated geometries and boundary conditions involving mass i n j e c t i o n can lead t o asymptotic expansions i n powers of

or

E '15 o r

Ell7.

I n general, the boundary layer

problems t h a t occur on surfaces where t h e normal v e c t o r and the r o t a t i o n vector a r e perpendicular ( i . e . , much l e s s t r a c t a b l e than those where

G-2

A-9 =) 0.

= 0)

are

I n t h e sphere

the troublesome region i s a t t h e equator where the Ekman l a y e r a n a l y s i s breaks down.

The e q u a t o r i a l boundary l a y e r has been

examined by Stewartson [6 2 ], who showed t h a t , i n a band of l a t i t u d i n a l width

O ( E 'I5)about t h e equator, the boundary

l a y e r thickens t o become

O(E215).

When considering flow

between concentric spheres, Stewartson's a n a l y s i s shows the need f o r a f u r t h e r boundary layer

O(E117),

t o remove a

s i n g u l a r i t y i n the azimuthal v e l o c i t y , a t t h e equator of t h e inner sphere.

Clearly the s u b j e c t of v e r t i c a l boundary l a y e r s

i n a r o t a t i n g f l u i d can lead t o very complex asymptotic

-

Boundary l a y e r s i n a s l i c e d c y l i n d e r

110

analysis.

We w i l l now i n v e s t i g a t e t h e

c a s e of a s l i c e d c y l i n d e r .

E l”

boundary l a y e r i n t h e

We have a l r e a d y s e e n t h a t i n t h i s

geometry, where t h e r e a r e no g e o s t r o p h i c c o n t o u r s , t h e beh a v i o r of t h e flow i s markedly d i f f e r e n t from t h a t of a r i g h t circular cylinder.

We w i l l show t h a t t h e r e i s a l s o a d i f f e r -

ence i n t h e v e r t i c a l s h e a r l a y e r s t h a t can e x i s t i n t h e two geometries. The s l i c e d c y l i n d e r i s no longer axisymmetric, hence i t i s necessary t o c o n s i d e r t h e equations of motion where not s e t e q u a l t o z e r o .

The

-2G1

=

2ii3 =

E1”-layer

equations t h e n become

-F

-

(9.381

3,

+ Fj e + G

o=-P

3Z

1 -

‘3x

+;

+ a ‘le +

(9.39)

IXX

( 9 -40)

lxx

’

1,

(9 -41)

= 0.

The boundary c o n d i t i o n , i n g e n e r a l , w i l l a g a i n be t h e t o p and bottom.

is

ae

*

3.n

= 0

However, i n t h e s l i c e d c y l i n d e r problem

t h e normal v e c t o r i s no longer p a r a l l e l t o t h e a x i s of r o t a tion

k. From t h e geometry i l l u s t r a t e d i n Figure

11, we

observe t h a t a t t h e bottom s u r f a c e i n t h e boundary l a y e r a t r = a, z

-

ay = a a s i n

e , ( f o r small a)

at

V e r t i c a l shear layers hence

n -*q . I

A t t h e s ie-wall

vl,

on

x

=

-

+

-wl

= 0, we

111

-

a cos 0 v l .

a g a i n have a boundary c o n d i t i o n

together with the condition t h a t the flux

returns the i n t e r i o r v e r t i c a l flux.

The boundary l a y e r

q u a n t i t i e s decay away from t h e boundary, hence

3,Eliminating

us

0

and

il -.

as

0

x

-

OD.

from e q u a t i o n s ( 9 . 3 9 ) and (9.41) g i v e s

?(- al -PJg

+ c

IXX

) + 1a -

Vlg

x

+

ijl

= 0, Z

and s u b s t i t u t i o n of (9.38) g i v e s , - a1 Vlg + 2V

Thus

lxxx +

-

V

lxxx

a1 - + Wlz Vlg

- .

= 0.

(9.42)

= -2w1

Z

From (9.38) and (9.40) we o b t a i n t h e e q u a t i o n (9.43) Hence t h e same coupled e q u a t i o n s f o r

-

w1

and

3, hold i n

both t h e symmetric and non-axisymmetric c a s e s . We must t h e r e f o r e s e e k a s o l u t i o n t o t h e problem g i v e n by

(9.44) w i t h boundary c o n d i t i o n s

Western boundary l a y e r

112

-il +

a cos e

G1

-

=

o

wl-0 : 1 = 0

at

x

= 0,

o

at

z =

as

x-CO

and

Jr

( f o r small

ildx

a)

prescribed.

we again have a s o l u t i o n of the form

(9.45) and The lower boundary condition now requires tan?

3

=

a cos e .

The condition of exponential decay r e q u i r e s

small

>

0, thus, f o r

m, we have

mand

Rm

120 cos

e I 1/3

when

cos 0

>

0

m-

However, i n order t o s a t i s f y both the boundary condition on

-

v1

at

x = 0, and the condition on

J:

-

Gldx, i t is of course

necessary t o have two eigenvalue s o l u t i o n s f o r

rn.

problem can only be solved i n the region where

cos 0

negative.

Thus the

is

I n t e r p r e t i n g t h i s r e s u l t when the s l i c e d cylinder

i s viewed as a dynamic model f o r an ocean on a

@-plane [ s e e

Figure 141, we expect t o f i n d a boundary layer only on the

western s i d e of a n ocean basin ( i n t h e northern hemisphere). This phenomenon i s i n f a c t , r e a d i l y observed:

the i n t e r i o r

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

113

ocean c i r c u l a t i o n i s returned i n narrow boundary l a y e r s on the western s i d e of an ocean.

The narrowness of the boundary

layer induces high speed c u r r e n t s :

i n the A t l a n t i c t h i s

current i s c a l l e d the Gulf Stream and i n the P a c i f i c i t i s c a l l e d the Kuroshio c u r r e n t .

A s we remarked a t t h e end of

Section 8, the energy for the boundary l a y e r c u r r e n t s can be deduced from t h e r a t h e r curious r e f l e c t i o n p r o p e r t i e s of the Rossby waves. Further l i g h t can be shed on the nature of the s i d e wall l a y e r i n a s l i c e d cylinder by t r e a t i n g t h e s l i d e angle a power of

E.

We w r i t e

a = EY

boundary l a y e r equations a s when

Y

>

1/2

a

as

and we i n v e s t i g a t e t h e

Y varies.

We f i r s t note t h a t

the p e r t u r b a t i o n of the bottom of the cylinder

i s of smaller order than the thickness of the Eeman l a y e r . Hence i n t h i s case the s l i c e angle i s s o small t h a t t h e bottom does not penetrate t h e Ekman l a y e r and t o t h e f i r s t order our a n a l y s i s of the s t r u c t u r e of the s i d e wall l a y e r s remains unchanged. We now consider the range

0

<

Y

<

1/2.

From equations

(9.45) and ( 9 . 4 6 ) we observe t h a t t h e dependence of the boundary l a y e r q u a n t i t i e s on the s t r e t c h e d co-ordinate, x = E-1’3(a-r),

is of the form ,-I2

cos 8 ) 1/3 EY/3 E-1/3(r-a)

Hence the thickness of the side-wall l a y e r i s

O(E(1-Y)’3).

The Ekman l a y e r s u c t i o n condition on the s l i c e d bottom should t h e r e f o r e be w r i t t e n i n the form

Sv erdrup s r e l a t i o n

114

The balance of terms i n t h e boundary c o n d i t i o n (9.47) w i l l be determined by t h e s i z e of

Y. When Y

<

1/4, t h e boundary

c o n d i t i o n becomes

which i s t h e boundary c o n d i t i o n t h a t was v a l i d i n t h e preceeding a n a l y s i s of t h e s i d e w a l l l a y e r i n a s l i c e d c y l i n d e r . When

Y

>

1/4, t h e boundary c o n d i t i o n becomes

-

-wl

= E

1/6

+

~ / 3L Za ax

’

which i s t h e boundary c o n d i t i o n t h a t i s c h a r a c t e r i s t i c of t h e side wall layer.

Eli4

(See 9 . 2 3 ) r a t h e r t h a n a n

We can summarize our o b s e r v a t i o n s a s f o l l o w s . continuously r a i s i n g t h e s l o p e of t h e bottom.

of s l o p e

i s l e s s than

a

u n a f f e c t e d by t h e s l o p e . O(E1’2)

of a n O(E1l4)

<

a

<

O(E1j4),

E1/4-layer.

<

a

<

0(E1I2) As

a

E1”-layer. Imagine

When t h e a n g l e

the side wall layers are

increases, so t h a t

t h e s i d e w a l l l a y e r has t h e s t r u c t u r e

When t h e s l o p e becomes g r e a t e r with

1, t h e s i d e w a l l l a y e r h a s t h i c k n e s s

and e x i s t s only on t h e western s i d e of t h e c y l i n d e r . t r a n s i t i o n p o i n t when EW4

a =

O(E 1/4

O(E/a)’/’ A t the

t h i s l a y e r has t h i c k n e s s

and hence t h e two s i d e - w a l l l a y e r s c o i n c i d e .

An Ocean Model:

Sver-D’s

Relac

We w i l l c o n s i d e r a s t r e s s - d r i v e n ocean model and d e r i v e a well-known e x p r e s s i o n i n oceanography f o r t h e i n t e r i o r northward t r a n s p o r t .

The r e t u r n of t h i s t r a n s p o r t , v i a a boundary

115

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

l a y e r , which our previous a n a l y s i s has shown must l i e on the western s i d e , gives r i s e t o the western i n t e n s i f i c a t i o n of the ocean c u r r e n t . I n t h e i n t e r i o r the

p-plane equations a r e 1

-

fk x q =

-

-vP

(9.48)

v.q = 0

where

f = fo

-

A

By, q = u i

+

A

vj

+

A

wk.

E

Components of wind s t r e s s on the ocean FIGURE 18

116

S t r e s s d r i v e n flow

The boundary c o n d i t i o n s a r e

[E-1/2(~1,~2)

a r e t h e s c a l e d components of t h e a p p l i e d s u r f a c e

wind s t r e s s ] with

u

=

v = w = 0

at

z = 1.

From e q u a t i o n (9.48) we have t h e f a m i l i a r i n t e r i o r r e s u l t

I n t h e upper Ekman l a y e r we w r i t e

and t h e equations become

We can t h e r e f o r e d e r i v e a n e x p r e s s i o n f o r 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 f o r s t r e s s d r i v e n flow, namely

V e r t i c a l shear layers

117

Also the northward mass t r a n s p o r t i n t h e EKman l a y e r i s given by

T1

(9.52)

=-T

We r e t u r n t o the i n t e r i o r s o l u t i o n . hence

&Y dz

are

w

and

o r smaller ( s e e ( 9 . 5 1 ) ) J thus equa-

O(Eli2)

t i o n (9.50) shows t h a t

We note t h a t

v

i s also

O(E1/‘).

The lower EKman

layer i s t h e r e f o r e r e l a t i v e l y unimportant because i t w i l l only induce a s u c t i o n of

smaller than t h e order of

O(E1”)

v.

Thus the I n t e r i o r problem i s given by equation (9.50) w i t h boundary conditions (9.511)~and t i o n of (9.48),

w = 0

at

together w i t h the f a c t t h a t

S u b s t i t u t i n g t h i s value f o r

w

z = 0. = 0,

Integragives

i n t o the boundary condition

(9.51) gives (9.53) We can combine expressions (9.55) and (9.53) t o o b t a i n the t o t a l northward mass transport

namely

Streamlines i n an ocean model

118

-1/2

-

0

;I;.oxz

.

(9.54)

This I s Known a s the Sverdrop r e l a t i o n f o r the northward mass

transport.

The general wind s t r e s s

5

has non-zero c u r l :

hence f o r a f i n i t e ocean with side walls, i t i s necessary t o consider a side-wall boundary layer capable of returning t h e northward mass t r a n s p o r t .

A s we have previously shown, t h i s

boundary layer may e x i s t only on the western s i d e of the ocean.

Figure 19 shows a sketch of the streamlines of an

ocean model constructed by Stommel [63], where a p l a u s i b l e f’unctlon i s chosen f o r the wind s t r e s s The dynamic analogy between the

&.

p-plane model f o r ocean

c i r c u l a t i o n and motion I n a s l i c e d cylinder has stimulated considerable experimental work.

Some of the most important

laboratory models studied i n the past t e n years a r e reviewed by Maxworthy and Browand

[44], who comment on the success and

the l i m i t a t i o n s of such models i n simulating oceanographic phenomena.

Their a r t i c l e a l s o gives an idea of t h e complexi-

t i e s of the apparatus involved i n such experiments.

Vertical shear layers

Stream lines showing the western boundary current in an ocean model due to Stommel. FIGURE 19

Problems

120

t e r 9 Problew

9.1)

Discuss the time-scale on which the v e r t i c a l boundary layers form.

9.2)

Consider the s t r u c t u r e of the s i d e wall l a y e r when t h e r e i s both v a r i a t i o n i n the bottom topography and v a r i a t i o n i n C o r i o l i s force with l a t i t u d e .

fo +

Write

f =

and

z = aly

surface where

BY

+

alJ a Z J and

a2x

B

on the bottom

a r e small parameters.

Discuss 1) the cases where one small parameter dominates. 2)

The case where a l l the small parameters a r e of the same order.

Would you be surprised t o find the boundary l a y e r on the e a s t e r n s i d e of a lake i n the northern hemisphere?

9.3)

Obtain t h e highest order terms i n the v e r t i c a l boundary layers f o r steady flow i n a cylinder w i t h the following boundary conditions.

a)

2

b)

--

o

-r8

at

z =

q = +re

n

at

z = 1

q = o

at

r - a

ri

at

2 - 0

at

z = 1

at

r = a

=

-I

q =

0

n

= re

q = o

c)

3

= re

at

z =

q = o

at

z = l

q-re

at

r = a

w

A

0

P r ob lems

9.4)

121

Consider flow i n a rectangular ocean b a s i n 0

x

< a,

0


g b

with constant depth.

Assume the s u r f a c e wind s t r e s s has t h e form r = -A

Compute

cos

Jf

i.

a)

The t o t a l northward mass t r a n s p o r t .

b)

The northward mass t r a n s p o r t i n the EBman l a y e r .

c)

The t o t a l upward t r a n s p o r t i n t o t h e Ekman l a y e r .

Assume

*

9.1

= 0

at

x = a.

Is i t possible t o

compute the eastward mass t r a n s p o r t ?

SKetch

the t o t a l mass t r a n s p o r t p a t t e r n .

9.5)*

A h o r i z o n t a l c i r c u l a r d i s K immersed i n a r o t a t i n g

cylinder of f l u i d i s rising very slowly.

Describe

the quasi-steady flow when the d i s c is h a l f way between the top and bottom boundaries.

This Page Intentionally Left Blank

CHAPTER 10

ANALOGIES BETWEEN ROTATION AND STRATIFICATION

I n Sections ( 3 )

-.

( g ) , we have considered problems of

flow i n a r o t a t i n g homogeneous f l u i d .

We w i l l now i n v e s t i g a t e

the e f f e c t of s t r a t i f i c a t i o n on a r o t a t i n g f l u i d .

Clearly,

t h i s is important i n many geophysical problems, s i n c e both the ocean and the atmosphere a r e composed of f l u i d of v a r i a b l e density.

For example, t h e r e i s a v e r t i c a l d e n s i t y gradient i n

the ocean caused by s o l a r heating of the surface:

t h e non-

uniform concentration of s a l t in s e a water a l s o gives r i s e t o a nonhomogeneous d e n s i t y f i e l d . The Navier-Stokes equation t h a t describes the motion of a r o t a t i n g f l u i d with nonconstant d e n s i t y a r e given by ( 1 . 4 ) and

(1.5).

To c l o s e t h e system i t i s necessary t o add t o these

equations a f u r t h e r equation s a t i s f i e d by t h e temperature f i e l d T

( i n the case of thermally induced s t r a t i f i c a t i o n ) and a l s o

the equation of s t a t e , s a t i s f i e d by t h e f l u i d , which gives the density as a function of temperature and pressure.

We s h a l l

assume t h a t the thermodynamics of the f l u i d can be described by the simple p l a u s i b l e equation t h a t leads t o a l i n e a r temperature-density r e l a t i o n p =

-

;U(T-T),

(10.1)

124

Equilibrium s o l u t i o n

where

i s t h e c o e f f i c i e n t of thermal expansion.

a

temperature

The

s a t i s f i e s t h e equation

T

(10.2) where

p

and

v

c o s i t i e s and

K

a r e t h e kinematic and kinematic bulk v i s -

i s t h e c o e f f i c i e n t of thermal d i f f u s i b i l i t y .

We f i r s t make t h e following observation.

-

Let us seek a

s t e a d y s o l u t i o n i n which t h e f l u i d i s i n equilibrium, i . e . , q = 0.

Equations ( 1 . 4 ) ,

(1.5) and (10.2) t h e n reduce t o

-vP + pv (G

(10.3) (10.4)

v x (10.3) gives 2 2 - 9) = 0,

v p x v (0

thus

=

V2 T = 0 .

and Computing

2 2 - y) 0

p = p ( G - q ) .

Let us consider t h e case where

G = gz; equation (10.5) then

implies 2 2 p = p ( z - * ) .

Hence from (10.1),T = T ( z

-

q),

however t h e r e i s no non-

t r i v i a l s o l u t i o n t o Laplaces equation (10.4) which has t h i s

Analogies between r o t a t i o n and s t r a t i f i c a t i o n f u n c t i o n a l form.

125

We t h e r e f o r e conclude t h a t a steady s o l u t i o n

i n which t h e r e i s no motion is not possible and hence some convection must occur.

i.e.,

However, i n most geophysical problems

the g r a v i t a t i o n a l force i s much l a r g e r than t h e c e n t r i -

f i c a l f o r c e , hence i t i s reasonable t o approximate the potent i a l by

gz

and neglect t h e centrofugal c o n t r i b u t i o n

2 2

.

When t h i s i s done, t h e convection induced by the c e n t r i f u g a l force i s neglected and a s t a t e of s t a t i c equilibrium can be This approximate i n i t i a l s t a t e i s given by

assumed t o e x i s t .

q = 0, p = po(z), -.

T = T0(z).

I m p l i c i t i n t h i s equilibrium s t a t e is the assumption t h a t t h e s t r a t i f i c a t i o n i s s t a t i c a l l y s t a b l e with a heavier f l u i d p a r t i c l e lying below a l i g h t e r p a r t i c l e .

I n order t o o b t a i n a system of non-dimensional l i n e a r i z e d equations we consider a small p e r t u r b a t i o n about t h e approximate equilibrium s t a t e .

The p e r t u r b a t i o n i n the v e l o c i t y from

r i g i d r o t a t i o n i s characterized by scales are

L

and

E ~ L : the length and time

respectively.

The d e n s i t y and tempera-

t u r e f i e l d s a r e given a s the composite of t h e average value, the f i e l d due t o v e r t i c a l s t r a t i f i c a t i o n , and t h e d e v i a t i o n of

O( E )

produced by a v e l o c i t y

O( E O L ) .

We t h e r e f o r e w r i t e

126

Equations f o r a Boussinesq f l u i d

T = ?

+

AT T 0 ( z )

+

E

. I2lQ k2

AP

T'

The l i n e a r i z e d p e r t u r b a t i o n equations a r e obtained by s u b s t i t u t i n g t h e above expansion i n t o equations (1.4), (1.5), ( 1 0 . 2 ) and using (10.11, and neglecting terms

O(E').

We also make

the Boussinesq approximation which follows from our choice of the simple equation of s t a t e (10.1) and leads t o t h e r e s u l t t h a t t h e perturbation d e n s i t y is only retained where i t appears i n conjunction w i t h t h e g r a v i t y .

A systematic

d e s c r i p t i o n of the Boussinesq approximation is given by Spiegel and Veronis [ 6 1 ] .

Thus the l i n e a r i z e d non-dimensional

equations f o r a Boussinesq f l u i d a r e

0.3

2

+

2i; x C J =

= 0

-oP

+

(10.6) Ti;

+

Ev 2j

(10.7)

(10.8)

(The primes have been dropped f o r convenience). dimensional parameters a r e 0

= v/K,

t h e Prandtl Number

The new non-

1 27

Analogies between r o t a t i o n and s t r a t i f i c a t i o n

f

N2 = A&

and

P

[We note t h a t

, the

dimensionless B r u n d t Vaisala frequency, which i s a l s o referred t o a s the buoyancy frequency

n

.

can a l s o be w r i t t e n i n terms of t h e tempera-

N2

t u r e gradient a s

&T

thermal expansion

:3 , *

a =

2

a l t e r n a t i v e l y denoted by

:]. 1 fR

where the c o e f f i c i e n t of The parameter

, where

N2

(which i s

f R i s c a l l e d the in-

t e r n a l Froude number) i s a measure of the s t r e n g t h of the stratification.

I n many of the problems t h a t we w i l l consider

i s t r e a t e d a s a constant, however i n p r a c t i c a l geophysical

N2

problems t h e r e i s frequently v a r i a t i o n i n v e r t i c a l co-ordinate

N2

with the

z.

For convenience we w i l l a t present choose t h e equilibrium temperature f i e l d potential vTo =

To

t o be proportional t o t h e g r a v i t a t i o n a l

gz, hence i n the non-dimensional co-ordinates

g.

Before we continue t o study i n some d e t a i l t h i s system of equations describing t h e flow 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 , we w i l l i n v e s t i g a t e t h e analogy between t h e r o t a t i o n a l and the stratified constraints.

This i n t e r e s t i n g analogy has been

described by Veronis [ 6 8 ] and we w i l l now summarize h i s observations. Let us consider a f l u i d t h a t i s s t r a t i f i e d (with t h e approximations assumed i n the previous pages) but Bo_fL r o t a t ing.

Since t h e r e i s no longer a c h a r a c t e r i s t i c v e l o c i t y

nL,

i t i s necessary t o s c a l e t h e parameters by dimensions governed by the d e n s i t y gradient

Ap.

-

We then o b t a i n a system of equa-

t i o n s which i s equivalent t o (10.6)

(10.8),

with Coriolis

128

P o t e n t i a l v o r t i c i t y equation f o r a s t r a t i f i e d f l u i d

f o r c e removed.

(10.10)

V ' j = 0

:f = +

q-k =

-VP

Q

+

Tk

+

-

Ro 2 q

(10.11)

(10.12)

v2T

where We now manipulate t h i s system of equations i n t h e f a m i l i a r manner t o o b t a i n t h e p o t e n t i a l v o r t i c i t y equation f o r t h e pressure f i e l d .

Computing

k

(5- RV2)w Substitution for

Now

v

(10.11)

Hence by taking

w

(10.11) g i v e s =

-

+

T.

(10.13)

from (10.12) gives

implies

&

of (10.14) and s u b s t i t u t i n g (10.15) we

o b t a i n t h e equation

(& - RV2)(& -

v2)v2P

+

2

vHP = 0 .

(10.16)

We r e c a l l t h e p o t e n t i a l v o r t i c i t y equation f o r a r o t a t i n g

but not s t r a t i f i e d f l u i d given by ( 5 . 7 ) , namely ( S - E V 2) 2 v 2P + 4 $ = 0 . az

(10.17)

Analogies between r o t a t i o n and s t r a t i f i c a t i o n

129

It i s t h e r e f o r e obvious t h a t t h e r e i s a n exact analogy between t h e two physical system i n t h e case of two-dimensional motion

(& =

0)

when e i t h e r or

a ) motion i s steady (and l i n e a r ) b ) the P r a n d t l number

Q

= 1.

I n such cases equations (10.16) and (10.17) a r e equivalent, with t h e r o l e s of the changed

.

x

and

z

co-ordinates being i n t e r -

We remind the reader t h a t many of the p r o p e r t i e s of a rot a t i n g f l u i d discussed i n the previous s e c t i o n s were derived from the p o t e n t i a l v o r t i c i t y equation (10.17).

Thus we would

expect t h a t an a n a l y s i s of the s t r a t i f i e d equation (10.16) would lead t o analogous r e s u l t s .

A general review of pheno-

mena associated with a s t r a t i f i e d f l u i d i s given by Yih “761. We w i l l describe t h r e e examples of the equivalence between two-dimensional s t r a t i f i e d flow and homogeneous r o t a t i n g flow. (1)

Corresponding t o a Taylor column, t h e r e e x i s t s the phenomenon of blocking i n which a column of f l u i d ( p a r a l l e l t o the the

A

j

a x i s and i n f i n i t e i n

y-direction) moves a s a r i g i d h o r i z o n t a l

slab. (2)

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 f o r c e t o support waves.

I n two dimensions they a r e

analogous t o i n e r t i a l w a v e s and they a r e c a l l e d i n t e r n a l g r a v i t y waves.

The plane

wave s o l u t i o n i s

( 10.18)

Boundary layers

130

which is equivalent (with interchange of axes) t o the dispersion r e l a t i o n of i n e r t i a l waves given by ( 7 . 2 4 ) .

(3)

Clearly, the steady form of equations (10.16) and (10.17) w i l l admit t o t h e same boundary layer analysis.

For a two-dimensional s t r a t i -

f i e d f l u i d the equivalent of a n Ekman l a y e r e x i s t s on v e r t i c a l boundaries a t

x = 0,1,

and the equivalent of Stewartson layers e x i s t a t horizontal boundaries

z = 0,l.

There i s

a l s o a two-dimensional s t r a t i f i e d analogy t o the spin-up problem which is c a l l e d the "heatup" problem where t h e imposed v e r t i c a l temperat u r e gradient is suddenly increased a l i t t l e . Circulation driven by the v e r t i c a l boundary l a y e r s u c t i o n then e f f i c i e n t l y heats up t h e interior fluid. The analogy between r o t a t i n g and s t r a t i f i e d flows breaks down i n three dimensions.

T h i s is e s s e n t i a l l y because rota-

t i o n i t s e l f constrains t h e f l u i d t o motions t h a t p r e f e r t o be two-dimensional, however a s t r a t i f i e d f l u i d t h a t is f r e e t o move i n t h r e e dimensions w i l l do so without a prefered horizontal direction. I n the remaining sections we w i l l consider problems i n which both r o t a t i o n and s t r a t i f i c a t i o n a r e s i g n i f i c a n t f a c t o r s . We w i l l be p a r t i c u l a r l y i n t e r e s t e d i n the r o l e t h a t s t r a t i f i c a t i o n plays i n modifying the r e s u l t s f o r homogeneous r o t a t i n g flow

Problems

191

ChaPter 10 Problema From equations (10.10)

(10.12) obtain t h e

equation s a t i s f i e d by t h e pressure f i e l d f o r i n t e r n a l waves i n a s t r a t i f i e d f l u i d . Assume a s o l u t i o n t o the equation obtained i n (10.1) of t h e form

P = eiAt#(,r).

Show t h a t , i n a closed domain with zero flow normal t o the boundary, the eigenvalue

X

must be r e a l . Eigenfunctions corresponding t o d i s t i n c t eigenvalues a r e orthogonal. Compare these r e s u l t s w i t h those obtained i n Chapter 7 f o r i n e r t i a l waves i n a r o t a t i n g fluid

.

Would you expect low frequency waves analogous t o Rossby waves t o e x i s t i n a s t r a t i f i e d f l u i d ? Discuss your reasoning. a)

Obtain the equations s a t i s f i e d by two-

dimensional flow i n a s t r a t i f i e d f l u i d i n a v e r t i c a l boundary l a y e r a t

x = 0.

Obtain a n analogous condition t o the Ekman l a y e r s u c t i o n condition s a t i s f i e d by the I n t e r i o r flow at

x = 0.

What i s t h e "heat-up" time s c a l e ? Solve f o r the f i r s t order steady i n t e r i o r and boundary l a y e r flow i n a s t r a t i f i e d f l u i d with boundary conditions a s shown.

Problems

132

z = 1

T = To q = o

-+

z = o

x = o

T = O q = o

+

FIGURE 20

Assume the flow is two-dimensional. is constant.

Case a )

T~

Case b )

To = TO(z).

x =

CHAPTER 11

THE NORMAL MODE PROBLEM FOR ROTATING STRATIFIED FLOW

I n chapter 7 we considered the i n i t i a l value problem f o r r o t a t i n g homogeneous flow.

We proved orthogonality f o r t h e

s e t of i n e r t i a l modes together with t h e geostrophic mode.

We

a l s o showed t h a t t h e geostrophic mode must c a r r y a l l the i n i t i a l depth averaged c i r c u l a t i o n .

I f we assumed completeness

of t h i s s e t of normal modes, i t is possible t o represent t h e s o l u t i o n t o an i n i t i a l value problem i n terms of the superp o s i t i o n of the i n t r i n s i c modes of the container.

This notion

has been generalized by Howard and Siegmann [35] t o the s i t u a t i o n of a r o t a t i n g compressible f l u i d under t h e influence of a gravitational field.

They study t h e l i n e a r i z e d equations

of motion f o r a non-dissipative r o t a t i n g s t r a t i f i e d flow and they prove t h a t t h e motions of such f l u i d s s a t i s f y c e r t a i n general p r o p e r t i e s even i n the case of a compressible f l u i d . We w i l l now consider some of the i n t e r e s t i n g f e a t u r e s of their results. We r e c a l l the equations of motion given by ( 1 . 4 ) and (1.5). For a non-dissipative f l u i d , the l i n e a r i z e d non-dimensional equations f o r a small perturbation from t h e basic s t a t e become

(11.1)

Energy equation

134

(11.2) Where G(z)

pO(G)

is t h e d e n s i t y f i e l d of t h e basic s t a t e , and

is t h e s c a l e d g r a v i t a t i o n a l p o t e n t i a l .

The h e a t -

equation (10.2) becomes (11.3) Where

TO(G) i s t h e basic temperature f i e l d .

For a n a r b i -

t r a r y f l u i d t h e equation of s t a t e i s given by determining t h e d e n s i t y as a f’unction of t h e p r e s s u r e and t h e temperature, i.e.,

p =

f(T,P).

The p e r t u r b a t i o n f i e l d then s a t i s f i e s t h e

l i n e a r i z e d equation

+

p = f T

T

(11.4)

fpP.

[The simplest thermodynamics leads t o t h e equation of s t a t e given by (10.1) where

fp

B

0

and

f T = -a].

The boundary conditions a r e those a p p r o p r i a t e t o nond i s s i p a t i v e flow i n a r i g i d c o n t a i n e r , namely

6

t h e boundary

I: of the c o n t a i n e r , where

u n i t normal.

I n t h e i n i t i a l value problem

prescribed a t

t = 0.

Manipulation of equations (11.1) following equation f o r t h e “energy“.

-

-

q*h = 0

on

i s t h e outward q, p

and

T

are

I)

(11.4) y i e l d s t h e

dG

I n t e g r a t i n g (11.5) over t h e volume of the c o n t a i n e r and using

135

Normal mode problem Gauss' theorem coupled with t h e boundary c o n d i t i o n on

r:

CJ-?I

= 0

gives ( 11.6)

The f i r s t term i n t h e i n t e g r a l i s t h e k i n e t i c energy and t h e second two terms r e p r e s e n t t h e p o t e n t i a l energy of t h e flow.

fT(2)-' <

Hence equation (11.6) s t a t e s t h a t energy i s conserved with time.

It i s u s u a l l y t h e case t h a t

(i.e.,

statically stable).

r a t i c form

E

fp

2 0 and

0

Under t h e s e conditions t h e quad-

i s positive definite.

The s t r u c t u r e of

suggests t h e d e f i n i t i o n of an energy i n n e r product

E

of

t h e following form

=

s

C P O g l ' 4 e*

+

f fn P o PIP; - -2- TIT;]dv. dTn

-

(11.7)

dG

Again a l i t t l e manipulation of (11.1)

% E1,2

(11.4) shows t h a t

= O.

(11.8)

we w i l l now use t h i s r e s u l t t o prove t h e following g e n e r a l p r o p e r t i e s f o r model s o l u t i o n s whose time dependence has t h e form e i x t

.

(1)

The frequency

A

is real.

Consider t h e r e s u l t (11.8) i n t h e p a r t i c u l a r case

Orthogonality and uniqueness

136

From (11.7) and (11.8) we obtain

Since t h e integrand i s p o s i t i v e , we conclude t h a t and hence

b

*

b =

is real.

Two normal modes of d i f f e r e n t frequencies a r e

(2)

orthogonal: Let and where

b1

+

A29

Equations (11.7) and (11.8) then give t h e r e s u l t n

Since

b1

4

n

b 2 , we conclude t h a t the i n t e g r a l i s zero and

hence the two modes a r e orthogonal i n the sense of the inner product ( 11.7).

(3)

The s o l u t i o n t o t h e i n i t i a l value problem i s unique :

Let

(s,F,?)

be the d i f f e r e n c e of two s o l u t i o n s t o the

i n i t i a l value problem.

Since the d i f f e r e n t i a l equations a r e

l i n e a r , t h i s quantity is c l e a r l y a s o l u t i o n t o the equations and boundary conditions with zero i n i t i a l condition.

The

Normal mode problem energy

E

137

s a t i s f i e s (11.6) w i t h i n i t i a l condition

E(0) = 0.

We t h e r e f o r e conclude t h a t

dG

f o r a l l time

t 1 0.

Since the integrand i s p o s i t i v e , i t

follows t h a t

Thus the two s o l u t i o n s t o the i n i t i a l value problem must be identical. A s i s s o o f t e n the case, i t i s much more d i f f i c u l t t o prove

-

the existence of the s o l u t i o n t o the i n i t i a l boundary value problem f o r equations (11.1)

(11.4).

I n the case of a

s t r a t i f i e d , but incompressible, r o t a t i n g f l u i d , Friedlander and Siegmann [ 2 2 ] demonstrate t h e existence of s o l u t i o n s i n p a r t i c u l a r geometries by obtaining exact expressions f o r the normal modes. Steadv Flow I n our study of a homogeneous r o t a t i n g f l u i d we have seen t h a t the steady o r geostrophic mode corresponding t o the f r e quency

= 0,

i s of p a r t i c u l a r importance and i n t e r e s t .

i s also true f o r a stratified rotating fluid.

This

We w i l l show

t h a t t h e geostrophic node i s c r u c i a l l y dependent on the nature of the boundary.

Again i t i s s e n s i b l e t o c h a r a c t e r i z e the

boundary i n t o t h r e e regions i n the same manner a s we described

138

Steady flow

f o r homogeneous flow i n Chapter 6 .

However, f o r s t r a t i f i e d

flow t h i s c h a r a c t e r i z a t i o n i s dependent on t h e n a t u r e of t h e gravitational potential. The time-dependent forms of equations (11.1)

-

(11.4) are (11.11)

= 0

V*(P0$

(11.12)

p = fTT

The c r o s s product o f ( 1 1 . 1 2 ) with 2[k3*0G

-

J

g] +

+

fpP.

oG

(11.14)

gives

1 VG x VP = 0.

Since t h e b a s i c s t a t e temperature

i s a f u n c t i o n dependent

To

only on the g r a v i t a t i o n a l p o t e n t i a l

(11.3) r e q u i r e s

G,

(11.16)

J - V G = 0,

i.e., 0.

t h e v e l o c i t y f i e l d i s p a r a l l e l t o s u r f a c e s of c o n s t a n t From (11.15) and (11.16) we o b t a i n t h e r e s u l t q =

(11.17)

V G x vP,

and s u b s t i t u t i o n of (11.7) i n t o ( 1 1 . 1 2 ) g i v e s

PO%

[VG*PZ

-

G,VP]

+

PO

vP

+

PO

vG = 0.

139

Normal mode problem P

Hence

(11.18)

p - 2 .

Gz

Equations (11.17) and (11.18) a r e t h e equivalent of t h e geostrophic r e l a t i o n s f o r a compressible r o t a t i n g f l u i d . Clearly these equations, together w i t h the equation of s t a t e

(11.4) determine the v e l o c i t y , temperature and d e n s i t y f i e l d s f o r the steady flow once the pressure

P

i s Known.

We there-

f o r e ask what a r e t h e conditions t h a t t h e geostrophic pressure must s a t i s f y ?

S u b s t i t u t i o n of (11.17) i n t o (11.11) gives

v

.I-[

Gz

i

= 0,

which can be r e w r i t t e n i n the form

Hence t h e r e a r e no r e s t r i c t i o n s on appears t h a t any function

(11.14).

P

P

if

VG

x vGz

5

It

0.

w i l l s a t i s f y equations (11.11)

This s i t u a t i o n i s analogous t o homogeneous

geostrophic flow i n

a

cylinder where the geometry alone does

not uniquely determine the s o l u t i o n . the regions i n which

V G x VG,

Siegmann, geos t rophlcally f r e e

= 0

.

Because of t h i s analogy a r e c a l l e d by Howard and

I n t h e regions t h a t a r e not geostrophically f r e e , i n t e g r a l curves of

VG x

vG,

geostrophic curves.

can be constructed.

They a r e c a l l e d

The regions where the geostrophic curves

cross the boundary a r e c a l l e d geostrophically blocKed.

The

regions which we a r e covered by geostrophic curves t h a t do not i n t e r s e c t t h e boundary a r e c a l l e d geostrophically guided.

In

140

Guided and blocKed regions

a blocIced o r guided r e g i o n e q u a t i o n (11.19) r e q u i r e s t h a t i s c o n s t a n t along g e o s t r o p h i c c u r v e s .

P

I n t h i s c a s e (11.17)

becomes

-

V G x vG,.

2pOGz a G Z

(11.20)

Hence t h e v e l o c i t y v e c t o r i s p a r a l l e l t o t h e g e o s t r o p h i c curves.

This i s a g a i n analogous t o homogeneous flow i n a

a sphere).

g e o s t r o p h i c a l l y guided c o n t a i n e r ( e . g . ,

For a

given c o n t a i n e r t h e s p e c i f i c g e o s t r o p h i c mode t h a t s a t i s f i e s t h e boundary v a l u e problem must have t h e form of (11.20) t o ge t h e r with t he boundary c o n d i t i o n

-

L

qen = 0

on

I n a guided r e g i o n t h e v e c t o r

(11.21)

Z.

VG x VGZ

is parallel to

t h e s u r f a c e , hence t h e v e l o c i t y given by (11.20) a u t o m a t i c a l l y s a t i s f i e s t h e boundary c o n d i t i o n ( 1 1 . 2 1 ) . of

3

Also, t h e magnitude

on a given g e o s t r o p h i c curve i s p r o p o r t i o n a l t o

(VG x vGz

I. In

a blocKed r e g i o n the boundary c o n d i t i o n

(11.21) applied t o (11.20) requires A * ( V G x VG,)

= 0

a t some p o i n t on t h e g e o s t r o p h i c curve, hence t h e c o n s t a n t of p r o p o r t i o n a l i t y must be z e r o . i s zero.

Thus t h e s t e a d y v e l o c i t y v e c t o r

This s i t u a t i o n i s analogous t o homogeneous flow i n

a sliced cylinder.

We note t h a t when

2

= 0, e q u a t i o n (11.12)

Normal mode problem

141

becomes VP

Thus

v G x VP = 0

+

(11.22)

~ v G= 0 .

and we conclude t h a t

P

i s constant not

only on geostrophic curves, but a l s o on t h e l e v e l surfaces G = constant.

potent i a 1 Vort i c i t v I n Chapter 2 we showed t h a t t h e r e e x i s t s a n a t u r a l extens i o n of the concept of v o r t i c i t y t o r o t a t i n g s t r a t i f i e d flows. We derived an expression c a l l e d t h e p o t e n t i a l v o r t i c i t y

ll

and we proved t h a t f o r a non-dissipative f l u i d the p o t e n t i a l v o r t i c i t y i s conserved.

We have made use of the concept, a s

i t a p p l i e s t o a r o t a t i n g homogeneous f l u i d , on a number of

occasions i n the preceeding chapters.

We w i l l now derive a

p a r t i c u l a r form f o r the p o t e n t i a l v o r t i c i t y of a r o t a t i n g compressible flow i n an a r b i t r a r y g r a v i t a t i o n a l f i e l d , i n

-

which the equations of motion a r e approximated by t h e l i n e a r i z e d equations (11.1)

(11.4).

Because t h e b a s i c s t a t e q u a n t i t i e s a r e functions t h a t depend only on

G , equation (11.3) can be r e w r i t t e n a s dT

+ 9*vG $=

0.

Consider the c r o s s product of ( 1 1 . 2 ) with

(11.23) vG:

(11.24) S u b s t i t u t i o n of

j-vG

from ( 1 1 . 2 3 )

gives

Potential v o r t i c i t y i n a rotating s t r a t i f i e d f l u i d

142

-

&[qXVG

+

21 +

0

ZqG,

dG

+

PO

x V G = 0.

(11.25)

Hence

(11.26) dG

TaKing t h e divergence of (11.26) and using (11.1) gives

I n g e o s t r o p h l c a l l y free regions ( i . e . ,

where

VG x

vG,

= 0),

equation (11.27) shows t h a t t h e q u a n t i t y

( 11.28)

i s pointwise conserved.

This q u a n t i t y t h e r e f o r e p l a y s t h e

r o l e of a p o t e n t i a l v o r t i c i t y .

I n a g e o s t r o p h i c a l l y guided or a g e o s t r o p h i c a l l y blocked region the conserved q u a n t i t y i s a l i t t l e more complicated. Howard and Siegmann show t h a t i n a guided region

where

r

i s a closed geostrophic curve, s a t i s f i e s

Normal mode problem

145

I n a blocked region a f u r t h e r modification i s necessary t o o b t a i n a quantity t h a t i s conserved f o r each h o r i z o n t a l surface. It is an easy matter t o construct the p o t e n t i a l v o r t i c i t y i n c e r t a i n p a r t i c u l a r examples.

Let us consider f i r s t t h e

example of a laboratory experiment with v e r t i c a l g r a v i t a t i o n a l force whose p o t e n t i a l

G = +gz.

We assume i ) t h a t t h e

Boussinesq approximation i s v a l i d , ii) t h a t f T = a, a constant.

fp = 0

and

of motion a r e those derived i n Chapter 10 given by (10.6)

(10.8).

-

The l i n e a r i z e d , non-dimensional equations

For an i n v i s c i d f l u i d the equations become

v.3

(11.31)

= 0

( 11.32)

$$ + N2g.k [Recall

N2

=

= 0

.

(11.33)

y]. n~

I n t h i s problem

GZ

i s constant, hence

the s i t u a t i o n is geostrophically f r e e .

vG x v G z

= 0

and

We use (11.28) t o

o b t a i n the expression f o r the p o t e n t i a l v o r t i c i t y .

I n terms

of the non-dimensional s c a l i n g c o n s i s t a n t w i t h t h e q u a n t i t i e s i n equations (11.31) potential vorticity i s

-.

(11.33), t h e expression f o r the

Geostrophic and h y d r o s t a t i c balance

144

[Note:

i n t h i s case

V-2

= 0, hence t h e term

Zp

disappears

from e q u a t i o n (11.2'01. The g e o s t r o p h i c , o r s t e a d y mode s a t i s f i e s (11.31) (11.33) with

& = 0.

-D

Thus t h e g e o s t r o p h i c flows a r e g i v e n

by

3

- x VP = 2 2 k

T = 9p az

and

*

Equation (11.55) is e q u i v a l e n t t o g e o s t r o p h i c balance and e q u a t i o n (11.56) is c a l l e d h y d r o s t a t i c b a l a n c e .

Substitution

of t h e s e e x p r e s s i o n s i n t o (11.54) g i v e s t h e e x p r e s s i o n f o r t h e p o t e n t i a l v o r t i c i t y of a g e o s t r o p h i c flow,

In Chapters 1 2 and 13 we w i l l s t u d y t h e waves supported by 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 a g r a v i t a t i o n a l f i e l d of p o t e n t i a l G = +gz.

We w i l l n o t e t h a t t h e e x p r e s s i o n f o r t h e p o t e n t i a l

v o r t i c i t y g i v e n by (11.34) i s e q u i v a l e n t t o t h e p o t e n t i a l v o r t i c i t y t h a t c a n be d e r i v e d from E r t e l ' s theorem g i v e n i n Chapter 2. A second i n t e r e s t i n g , but simple example i s flow of a

Boussinesq f l u i d i n a s p h e r e i n a r a d i a l g r a v i t a t i o n a l f i e l d with p o t e n t i a l

G = g r2

.

This could be considered a simple

Normal mode problem

145

model f o r an astronomical body where the g r a v i t a t i o n a l force points r a d i a l l y inwards. f

T

We again assume

fp = 0

and

I n non-dimensional form the l i n e a r i z e d equations

= -a.

-

v.q = 0

+ We compute

vG

and

Hence , (where

+

y2

+

P N 2g.2 = 0.

(11.40)

t o be

G,

VG

r2 = x2

(11.38)

x vGz = ( 2 g l P 3 x

k.

(11.41)

z2).

Thus t h i s problem i s not geostrophically f r e e .

The geostro-

phic curves a r e c i r c l e s of l a t i t u d e on spheres of constant radius.

Therefore the boundary of the sphere

geostrophically guided.

r

= 1

is

The p o t e n t i a l v o r t i c i t y i n terms of

the non-dimensional q u a n t i t i e s is (11.42) I n t h e case of axisymmetric flow v o r t i c i t y can be w r i t t e n

($

= 0)

the potential

(11.43) ( (r,O ,+)

a r e s p h e r i c a l p o l a r co-ordinates)

.

The geostrophic

146

Conserved p o t e n t i a l v o r t i c i t y

mode s a t i s f i e s (11.44) Thus v e l o c i t y and temperature f o r the geostrophic flow can be w r i t t e n i n terms of the pressure:

(geostrophic balance)

1s

and

T = 22 a z

(11.46)

(hydrostatic balance). We s u b s t i t u t e (11.45) and (11.46) I n t o (11.43) t o o b t a i n the following expression f o r the geostrophic p o t e n t i a l v o r t i c i t y : "g

" + 4 J J J U a2 2 2 sin ae cos2 ae

+

b 4 1a]' N2 az z 2 az

(11.47)

I n t h e i r study of the i n i t i a l value problem Howard and Siegmann [ 3 5 ] show t h a t the concept of a conserved p o t e n t i a l v o r t i c i t y allows the complete c h a r a c t e r i z a t i o n of the geos t r o p h i c p a r t of the flow. (%a

P g a Tg )

T h i s steady c o n t r i b u t i o n

t o t h e s o l u t i o n of an i n i t i a l boundary value

problem i s the unique geostrophic flow t h a t has t h e same p o t e n t i a l v o r t i c i t y a s the i n i t i a l flow.

Problems

147

Chapter 11 Problems

-, (11.4) show t h a t

11.1) Using equations (11.1)

Lat J2{

Igl2po

f + -2

- fr

P2

3 2

PO

11.2)

TZ}

+

v.(pq)

= 0.

aG

Consider the problem of flow i n a cylinder with gravitational potential

G = gz.

The i n i t i a l

conditions a r e

q(z,O) a)

=

So(,r)

and

T ( 2 , O ) = To(:)

.

What is the equation s a t i s f i e d by t h e geostro-

phic pressure f i e l d ? b)

The boundary conditions can be shown t o be

pazp

= To

So

ar

and

on do

=

2n

J,

z = 0,1,

Q,.;

P = constant on

dB

r = a

on

r = a,

f o r fixed

z.

Determine the geostrophic flow i n the case where the i n i t i a l conditions a r e axisymmetric with

Qo = v ( r , z ) B , To = T o ( Z ) .

[Assume the

s o l u t i o n is axisymmetric].

11.3)

Show t h a t the s o l u t i o n t o the i n i t i a l value problem given i n ( 1 1 . 2 ) is unique.

148

Problems

11.4)

Consider the problem of flow in a s p h e r i c a l s h e l l with g r a v i t a t i o n a l p o t e n t i a l

a)

Describe the geostrophic contours.

b)

Assume t h a t the Boussinesq approximation i s v a l i d and t h a t

f p = 0, f

T = -a. Assume

the non-dimensional s c a l i n g of Chapter 10 i s valid. Compute the p o t e n t i a l v o r t i c i t y a s a funct i o n of v e l o c i t y and temperature.

c)

Express the p o t e n t i a l v o r t i c i t y f o r a geostrophic flow a s a f u n c t i o n a l of the geostrophic pressure.

11.5)

Consider the problem of flow i n a sphere with G = gr2

discussed i n t h i s chapter.

The

viscous c o r r e c t i o n t o the equations of motion gives the following equations

(U

is the dimensionless parameter c a l l e d

the Prandtl number).

P r ob l e as

a)

Show t h a t these equations can be manipulated t o give

at b)

n(q,T,N2)

-

= E F(q,T,N

Determine the time s c a l e f o r a n

2

u). O(1)

change i n the p o t e n t i a l v o r t i c i t y .

Is

t h i s time-scale dependent on the s i z e of

o?

This Page Intentionally Left Blank

CHAPTER 1 2

ROSSBY WAVES I N A ROTATING STRATIFIED FLUID

m e P o t e n t i a l V o r t i c i t v Fguation We have already seen a number of instances of the importance of p o t e n t i a l v o r t i c i t y i n c h a r a c t e r i z i n g the behavior of a rotating fluid.

I n Chapter 11 we derived a g e n e r a l i z a t i o n

of p o t e n t i a l v o r t i c i t y f o r a r o t a t i n g compressible f l u i d i n an a r b i t r a r y g r a v i t a t i o n a l f i e l d .

We w i l l now r e t u r n t o the

expression f o r the p o t e n t i a l v o r t i c i t y 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 derived i n Chapter 2 , namely

n = where

h

P

(,B + 29)

is any s c a l a r such t h a t

= 0

(12.l)

and

1 = x (P,p)

.

I n Chapter 8 we used the f a c t t h a t ( f o r a n i n v i s c i d f l u i d ) p o t e n t i a l v o r t i c i t y i s conserved, t o study Rossby waves i n a homogeneous r o t a t i n g f l u i d .

We w i l l now apply t h i s p r i n c i p l e

t o i n v e s t i g a t e Rossby 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 .

%

Let us consider a s t r a t i f i e d , incompressible f l u i d , i . e . , = 0.

Obviously the density i t s e l f is then a s u i t a b l e

candidate f o r the s c a l a r

A.

We follow the a n a l y s i s given i n

Chapter 10 and we assume t h a t the perturbations from the equilibrium s t a t e a r e small.

The l i n e a r i z e d equation of conserva-

t i o n of the p o t e n t i a l v o r t i c i t y becomes

P o t e n t i a l v o r t i c i t y equation

152

(12.2)

I n order t o study Rossby waves i n a s t r a t i f i e d f l u i d we cons i d e r motion on a

@-plane by w r i t i n g

29 = 2n cos

ek

-

n(fo+py)i.

Repeating t h e non-dimensional l i n e a r i z a t i o n given i n Chapter 10, ( w i t h the

@-plane modification), t h e d e n s i t y f i e l d be-

comes

( f o r convenience we have chosen the equilibrium s t r a t i f i c a t i o n t o be proportional t o

z).

Thus the l i n e a r i z e d p o t e n t i a l

vorticity is

n

(€3' +

=

where

fk)

f = fo

(k +

+

2

vp')

(12.4)

By.

The l i n e a r i z a t i o n of equation (12.2) gives n

[For convenience we have again dropped the primes on the perturbation quantities].

We note t h a t t h i s expression f o r

is equivalent t o (11.31) when Coriolis parameter

f

T[

is a

constant. We consider t h e case where the g r a v i t a t i o n a l p o t e n t i a l G(r) = gz.

To f i r s t order t h e non-dimensional perturbation

v e l o c i t y and density s a t i s f y the steady and i n v i s c i d form of equation (10.7),

namely

153

Baroclinic Rossby waves

3

A

fk X

L.g

Thus and

= -VP

=

1

+

A

(12.6)

pK.

2 VHP

(12.7)

P = , , *

(12.8)

S u b s t i t u t i o n of (12.7) and (12.8) i n t o equation (12.5) gives the l i n e a r i z e d p o t e n t i a l v o r t i c i t y equation

+

a-[V2P

at

9 1+ g

fZ 2

-4j N

(3

az

= 0.

(12.9)

This i s the equation t h a t describes the behavior of Rossby waves i n a s t r a t i f i e d f l u i d .

bv Waves f o r a StratifkLE&&l We o b t a i n a wave-like s o l u t i o n t o equation (12.9) by writing

P where The function

5.3

=

-

i ( z ) ei ( f . J A t )

= lrlx

+

(12.lo)

K2Y.

i (z) c a r r i e s the v e r t i c a l s t r u c t u r e a r i s i n g

from the s t r a t i f i c a t i o n . w - 0

at

As boundary conditions we take z = O

and

z = h .

(12.11)

Equation (10.8), together with t h e assumption t h a t t h e equilibrium f i e l d i s such t h a t

VTO =

c,

implies f o r a n i n v i s c i d

fluid (12.12)

From (10.9) and the equation of h y d r o s t a t i c balance (12.8) we

154

Eigenvalue problem

have T =

hE

(12 .lj)

az

Combining (12.11), ( 1 2 . 1 2 ) and (12.13) gives t h e boundary cond i t i o n on

namely

# (z),

at

az = 0

z = 0,h.

(12.14)

S u b s t i t u t i o n of expression (12.10) i n t o the Rossby wave equat i o n ( 1 2 . 9 ) gives

+

-A[-(K1+K2)I 2 2

Hence

+ f2

N

2

3+] az

+

BK,#

satisfies

i

(12.15) with boundary conditions ( 1 2 . 1 4 ) .

This eigenvalue problem has

the s o l u t i o n 0,

= cos

y

,

2 BK (yy = J$ (++ #f +

where

fo

(12.16)

I(;>,

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 and

jo i n these two cases.

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

the volume of the container.

=

-s

2

- $J

s*.v*dv

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

V

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

<

1Q1

<

169

P.

2:

Clearly (13.19) can not have a s o l u t i o n of the form 2

x 2 4 >

N2, s i n c e i n t h i s case both terms w i l l be p o s i t i v e

and hence the sum can not be zero. 2 ?, < N2 i s a p o s s i b i l i t y . m e b)

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

2 N2 > 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 <

Max(N2,4).

It i s therefore possible t h a t t h e d e f i n i n g equation (13.9) f o r

the pressure f i e l d can be e i t h e r hyperbolic o r e l l i p t i c .

We

w i l l now show t h a t t h e r e e x i s t two c l a s s e s of wave-like solu-

t i o n s t o equation (13.9) w i t h boundary condition ( 1 3 . 1 4 ) . The f i r s t c l a s s of waves a r e purely o s c i l l a t o r y i n t h e i r s p a t i a l dependence: Max(N,2)

and

hyperbolic.

t h e i r frequencies

Min(N,2)

A

a r e bounded by

and hence the equation f o r

I

Waves of the f i r s t c l a s s a r e analogous t o

is

Waves i n a cylinder

170

i n t e r n a l waves t h a t e x i s t i n the case of pure r o t a t i o n o r pure stratification.

However the second c l a s s of waves, a s we w i l l

show, a r e d i s t i n c t l y d i f f e r e n t i n nature; t h e i r frequencies a r e below

Min(N,2)

and the equation i s e l l i p t i c .

Because

of t h e i r s i m i l a r i t y t o Kelvin waves t h a t a r i s e i n t h e s t u d y of edge waves i n shallow water, waves of the second c l a s s have been c a l l e d by Krauss, I n t e r n a l Kelvin waves. I n h i s book, “Methoden und Ergebnisse d e r Theoretischen Ozeanographie 11, I n t e r n e Wellen” [ 391 , Krauss catalogoues the eigenfunctions f o r both c l a s a e s of i n t e r n a l waves.

He

e x h i b i t s t h e s o l u t i o n s i n rectangular co-ordinates and by a r a t h e r complicated superposition of these solutions he obtains the eigenmnctions

for i n t e r n a l waves i n a rectangular

I

box. We w i l l now o b t a i n the eigenfunctions t i e s of the eigenvalues

X

0

and the proper-

for i n t e r n a l waves i n a cylinder.

We consider a r o t a t i n g s t r a t i f i e d f l u i d bounded by r i g i d walls at

z

= 0,l

and

r

=

a.

I n c y l i n d r i c a l co-ordinates equa-

t i o n (13.9) and boundary condition (13.14) become

(13.20) with

and I A % + : ~ = O

at

r = a .

(13.22)

I n t e r n a l waves When

l i e s between

A2

(31.20) us hyperbolic.

Max(4,N2)

171 and

Min(4,N2)

equation

I n f a c t , the problem c l o s e l y resembles

t h a t given by equation ( 7 . 2 0 ) w i t h boundary conditions (7.21) and ( 7 . 2 2 ) which a r e s a t i s f i e d by an i n t e r n a l wave I n homogeneous r o t a t i n g f l u i d .

Hence we can Immediately w r i t e down

the s o l u t i o n f o r waves of t h e f i r s t c l a s s , namely

B where and

=

i s the

Jlkl

eiK'

cos mz J K ( y m r ) ,

(13.25)

k - t h Bessel function of the f i r s t Kind

(13.24) The boundary condition ( 1 3 . 2 2 ) requires t h a t m-th

I s the

y-

p o s i t i v e s o l u t i o n of the transcendental equation

Clearly these waves of t h e f i r s t c l a s s can e x i s t i f e i t h e r the s t r a t i f i c a t i o n o r r o t a t i o n i s zero

( N ~ o or

N~

m)

and t h e s o l u t i o n s reduce t o those predicted by t h e r e s u l t s obtained i n Chapter 7.

When both r o t a t i o n and s t r a t i f i c a t i o n

a r e p r e s e n t , t h e i n t e r n a l wave of the f i r s t c l a s s can be viewed a s a r o t a t i o n a l wave modified by s t r a t i f i c a t i o n ( o r vice-versa)

.

We now consider the case where Min(2,N).

The parameter

2

1x1

' 2 2 ( A -4)/(X -N )

I s l e s s than

i s then p o s i t i v e

I n t e r n a l Kelvin waves

17 2

We seeK a s o l u t i o n of t h e

and e q u a t i o n (13.20) i s e l l i p t i c . form cp = e l k e cos

where

G( r )

Hence

G ( r ) = Ik(ar) where

mz

G(r)

satisfies

Ik i s t h e

f u n c t i o n of t h e f i r s t k i n d .

k-th

modified Besael

The boundary c o n d i t i o n (13.22)

requires AaaIL(aa)

+

2 k I k ( a a ) = 0.

( k X a -0,

( 1 3- 2 8 )

40)

Since t h e modified Bessel f u n c t i o n has e x p o n e n t i a l r a t h e r t h a n o s c i l l a t o r y form, equations (13.27) and (13.28) w i l l not have a n i n f i n i t y of s o l u t i o n s .

Rather i t can be shown t h a t t h e r e

e x i s t , a t most, one s o l u t i o n t o t h i s coupled s e t of e q u a t i o n s . D e t a i l s of t h e following r e s u l t s a r e g i v e n i n F r i e d l a n d e r and Siegmann [ 2 2 ] .

They show t h a t f o r

a t most, one eigenvalue

Ank

k

positive there exist,

which w i l l be n e g a t i v e .

Hence

t h e wave t r a v e l s around t h e c y l i n d e r i n t h e p o s i t i v e d i r e c t i o n i n t h e sense of t h e p r e s c r i b e d r o t a t i o n .

The v a l u e of

i s obtained by c o n s i d e r i n g t h e i n t e r s e c t i o n of two curves

I n t e r n a l waves

173

It is convenient t o consider s e p a r a t e l y the two cases N2

>4

and

the curves

N2

<

4.

y,(X)

Figure 22 i l l u s t r a t e s t h e behavior of

and

y,(A)

i n these two cases.

An

analysis of the curves leads t o t h e following conclusions, I n t h e f i r s t case we seek s o l u t i o n s

0

can be shown t h a t t h e r e e x i s t s a mode

< 1x1 A

g

2

<

N.

It

t h a t is a

= -2

t r a n s i t i o n mode between the two c l a s s e s of waves, provided there e x i s t integers

For those i n t e g e r s

n

n

and

and

such t h a t

k

such t h a t

k

2 2 2

t h e r e e x i s t s a unique eigenvalue

Ank

(13.31) where

0

<

<

2.

There i s no s o l u t i o n f o r those wave numbers where the ine q u a l i t y (13.31) i s reversed. solutions

0

<

IX

1

N

<

I n t h e second case we seek Figure 2 2 shows t h a t i n t h i s

2.

case t h e r e i s no t r a n s i t i o n mode a unique

with

IAnkl

<

N

X = -N,

however t h e r e e x i s t s

f o r a l l integers

An i n t e r e s t i n g degenerate case i s

N = 2.

n

and

k.

I n t h i s case

a l l the waves of' the f i r s t c l a s s reduce t o a s i n g l e mode with X2 = N

2

= 4

which i s characterized by zero pressure g r a d i e n t .

174

I n t e r n a l Kelvin waves

fi

//

I

I

I I

Case

‘

2

I

I N > 4

I

I

I

I I

The behavior of the curves

yl(X)

FIGURE 2 2

I

I I

and

y2(A).

175

I n t e r n a l waves The frequency

i s a wave of t h e second c l a s s ( a n i n t e r -

Xm

n a l Kelvin wave) is given e x p l i c i t l y a s

xm

= -2

Similar r e s u l t s can be obtained f o r i n t e r n a l waves i n a sphere w i t h s o l u t i o n s t o equation (13.9) i n s p h e r i c a l coordinates being given i n terms of Legendre polynomials. Friedlander and Siegmann [ 221 3 .

[See

Again t h e r e e x i s t a t h r e e

fold i n f i n i t e s e t of eigenvalues

AmK

of the f i r s t c l a s s ,

with 2

Min(N ,4)

< x2-

<

Max N2,4),

t h a t degenerate t o a s i n g l e eigenvalue when

N

2

=

4. These

i n t e r n a l waves a r e analogous t o those t h a t e x i s t i n the case of pure r o t a t i o n o r pure s t r a t i f i c a t i o n . numbers

n

and

k

For c e r t a i n wave

there a l s o e x i s t a doubly i n f i n i t e s e t

of i n t e r n a l Kelvin waves w i t h the azimuthal wave number

K,

am and

negative with respect t o A,

2

2

Min(N ,4).

Internal

Kelvin waves e x i s t only i n the presence of both r o t a t i o n and stratification. I n many oceanographic problems the non-dimensional buoyancy frequency

is s u f f i c i e n t l y much g r e a t e r than

4 that

the dominant waves w i l l be those of the f i r s t c l a s s .

However

N

i n the deep ocean t h e value of

4.

N

decreases and i s c l o s e t o

Hence t h e i n t e r n a l Kelvin waves may be important when

deep ocean phenomena a r e under i n v e s t i g a t i o n .

Variable

176

N( z )

W l e A s we have p r e v i o u s l y mentioned, t h e buoyancy frequency

N

i n t h e ocean ( o r atmosphere) i s not a c o n s t a n t , but r a t h e r , a z , with t h e s h a r p changes i n t h e value

f u n c t i o n of t h e d e p t h of

N

d e f i n i n g t h e p o s i t i o n of t h e thermoclines.

g i v e s a sKetch of t h e curve observational data.

When

N(z) N

Figure 2 5

that is consistent with

i s no longer c o n s t a n t t h e equa-

t i o n f o r t h e p r e s s u r e f i e l d becomes a p a r t i a l d i f f e r e n t i a l e q u a t i o n with non-constant c o e f f i c i e n t s .

Equation (13.9) can

be rederived t o g i v e (13.32)

We w i l l now i n v e s t i g a t e t h i s non-constant c o e f f i c i e n t e q u a t i o n i n a r e c t a n g u l a r box t h a t i s bounded by s i d e w a l l s a t but unbounded i n t h e

y-direction.

x = 0,L

The c o n d i t i o n o f z e r o

no r ma 1 v e loc i t y g i v e s boundary c ond 1t i o n s il),

+

2)

Y

= 0

at

x = 0,L

( 15.33)

(13.34) We seek a s o l u t i o n t o e q u a t i o n (15.32) o f t h e form

2

= (Az

-

NP)A(x,z)eiky.

The e q u a t i o n and boundary c o n d i t i o n s f o r

A(x,z)

become

I n t e r n a l waves

N(z)

177

sec

-1

thermoc ine)

100

200

1000

2000

3000

400C

T

-

1 day

The v a r i a t i o n of N(z) with depth in the ocean. The period T = 2n/N. FIGURE 2 3

Separation of v a r i a b l e s

178 with

AAx

+

2kA = 0

at

x

A = 0

at

z = 0,l.

=

(13.36)

0,L

( 1 3 -37)

The above problem can be solved by separation of v a r i a b l e s by writing A with

=

Fxx

F(x)G(z)

-

K

2

(13.38)

F = -YF

and

where

Y

i s the separation constant.

There e x i s t two c l a s s e s of solutions t o equation (13.38) with boundary condition (13.36), the f i r s t corresponding t o

the i n t e r n a l wave s o l u t i o n and the second t o i n t e r n a l Kelvin waves.

The f i r s t c l a s s of solutions is given by

F(x)

sin

= 2k

with

Y

Hence t h e problem f o r

X

=

- 17

COS

X

( 13.40)

(yy + K2.

G(z)

becomes (13.41)

with

G

= 0

at

z = 0,l.

The second c l a s s of s o l u t i o n s , which a r e non-oscillatory in

x, is given by

I n t e r n a l waves

FK(X)

-

= cash 2k x A 2

with

y =

Hence the problem f o r

with [The s u b s c r i p t

K

(x' -

K -2 A

sinh

yx

(13.42)

4).

becomes

G(z)

GK = 0

-

179

at

z = 0,l.

denotes t h e s o l u t i o n s corresponding t o in-

t e r n a l Kelvin waves .] Both the problem (13.41) f o r

G

and (13.43)

for

GK

can

be w r i t t e n i n the form of a standard Sturm-Liouville problem. For the f i r s t problem we w r i t e

w2

c ome s Gzz

+

- - and A2 - 4

2 [(y) + i21[(N2 - 4)w' - 13G -

(15.41) be-

= 0

( 13 -44)

with G = 0

at

For the second problem we w r i t e

GKzz

with When

+

~'[N'M~

G = 0

N2 ( z )

>

at

z = 0,l.

v2 =

-

-$

and (13.43) becomes

k '

l]GK = 0

( 13.45

z = 0,l.

4, both (13.44) and (13.45) a r e Sturm-Liouville

equations of the form

1

Analogy w i t h t h e Schrodinger e q u a t i o n

180 with

p

>

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

<

f i x e d , when

K

2

Nulax*

The eigenvalues accumulate from above and from below a t

FIGURE 2 4

4.

I n t e r n a l waves

183

It i s possible t o obtain an asymptotic estimate f o r the behavior of t h e eigenmnction f o r problems (13.44) and (13.45) by using

This is a method whereby i t i s

techniques.

W.K.B.

assumed t h a t the o s c i l l a t o r y s o l u t i o n s t o a non-constant coe f f i c i e n t d i f f e r e n t i a l equation vary much more rapidly than the c o e f f i c i e n t s .

The s o l u t i o n

changing with r e s p e c t t o

i s large.

The

f o r large

Y

W.K.B.

G( z)

t o (13.44) i s r a p i d l y

when the wave number

z

techniques seek a s o l u t i o n of t h e form

where

S u b s t i t u t i n g t h i s asymptotic form of s o l u t i o n i n t o equation (13.44) and equating powers of and

$,(z)

$l(z).

G(z)

-

y-l

give the equations f o r

These can be solved t o give

e&iyJ[u2(N2 [m2(N2

-

-

4)

- 4) -

11

13

1/2

dz

(13.49)

1J2

There i s a d i f f i c u l t y t h a t a r i s e s i n using

W.K.B.

methods

t o o b t a i n t h i s asymptotic s o l u t i o n , namely t h e s o l u t i o n t o (13.44) is o s c i l l a t o r y only when the c o e f f i c i e n t

[(N2

-

4)m2

-

13

i s positive. (N2

-

The value of

4)m2

-

z

f o r which

1= 0

i s c a l l e d a turning point of the equation.

I n terms of t h e

184

Behavior a t a turning point

, a turning point e x i s t s when frequency i2

i.e.,

N

2

(2)

= X

2

Hence f o r a wave of given frequency

. A

i n the spectrum

(13.46) the v e r t i c a l s t r u c t u r e ceases t o be o s c i l l a t o r y a t those depths where the l o c a l buoyancy frequency t o the wave frequency N 2 (2)

i s l e s s than

exponential.

The

A.

i s equal

N

For those values of

where

z

1, the depth dependence of the wave i s

W.K.B.

approximation (13.49) i s v a l i d

away from the turning point where

N2(z)

> 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\

<

*ik

e

j’(r2N2-

1)1/2dz

-

1)U2

(rZN”

Min(4,Nmin), the function

f o r large (p2N2

k.

-

1) i s

never zero, hence the turning point a n a l y s i s i s unnecessary. The q u a l i t a t i v e behavior of the eigenfunction be examined even when the h o r i z o n t a l wave numbers

G(z) m

can and

K

185

I n t e r n a l waves a r e small and hence the asymptotic estimates given by the methods a r e not v a l i d .

W.K.B.

Equation (13.44) can be

w r i t t e n i n i n t e g r a l form a s

=

dz

s[

-y&

(N2

-

4)w2

-

l]Gdz

where f o r n o t a t i o n a l convenience we have replaced

We w i l l consider t h e lowest modes corresponding t o

k = 0,

m = 1. Equation (13.50) is s a t i s f i e d by a n i n f i n i t e family of eigenvalues eigenvalue being satisfy

GIOn(0)

‘

mlOn u

and e i g e n h n c t i o n s 2

~ The ~ eigenfunction ~ ~ .

= GIOn(l)

i f the coefficient

[(N2

-

= 0.

4)w2

-

G

~ the~ smallest ~ ,

Glen( z)

must

This c l e a r l y can not happen

13

i s always negative s i n c e

equation (13.50) then represents a f u n c t i o n whose d e r i v a t i v e increases as the function increases and hence would never r e t u r n t o i t s i n i t i a l value.

Thus

hence

2 2 In fact, Xloo ] <,,N mst be s u f f i c i e n t l y g r e a t e r t h a n [ 1 / ( N L x 4)] t o

[This is equivalent t o requiring

u’loo

ensure t h a t

dGIOO dz

intersects the

.

-

becomes zero and t h a t t h e curve f o r

z-axis a t

z

= 1.

The next eigenvalue

GIOO

+=

0

rt

z = 1

z =

I

-

The upper curve gives [N2(z) 41 as a function of z . The lower curces show the s t r u c t u r e of the first t h r e e modes. FIGURE 25

187

I n t e r n a l Waves t o allow the curve U’101 i s s u f f i c i e n t l y l a r g e r than wlo0 Or G l O l t o t u r n around twice i n t h e i n t e r v a l ( 0 , l ) and again i n t e r s e c t the

z-axis a t

z = 1.

We can successively

c o n s t r i c t t h e q u a l i t a t i v e behavior of the modes

n = 2,3,.

.. .

This i s i l l u s t r a t e d i n Figure 25.

There have been many developments concerning i n t e r n a l waves

i n the Ocean s i n c e t h e f i r s t reported observation of t h e Norwegian explorer F r i d t j o f Nansen i n 1893.

A considerable

body of more recent r e s u l t s is t o be found i n Vol. 80 of t h e Journal of Geophysical Research [ 19751, which comprises a c o l l e c t i o n of papers on oceanic i n t e r n a l waves.

These include

a h i s t o r i c a l introduction which i s extended by Briscoe [4] and a broad review by Thorpe [66] of t h e t h e o r e t i c a l s t u d i e s of the physical processes a f f e c t i n g t h e 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. This l a t e r paper contains a d e l i g h t f u l a r t i s t i c rendering of the physical processes, including bottom topography, r a i n , f i s h , etc., waves.

t h a t could a f f e c t t h e generation of i n t e r n a l

A paper by Wunsch [741 reviews t h e s t a t e of observa-

t i o n of i n t e r n a l waves and poses a number of important quest i o n s which have stimulated a c t i v i t y i n the f i e l d .

The

current “ s t a t e of t h e a r t ” i s surveyed i n 1979 by t h e review a r t i c l e of Garret and Munk [24].

They discuss experimental

evidence concerning i n t e r n a l oceanic waves and attempt t o construct a compatible model f o r the d i s t r i b u t i o n of energy i n wave-number frequency space.

A broad survey of the theory

188

Review A r t i c l e s

of ocean waves w i t h emphasis on r e c e n t developments i s given

by LeBlond and rvZysaK [ 411.

They c o n s i d e r f i n i t e amplitude

e f f e c t s and n o n - l i n e a r i n t e r a c t i o n s .

Of p a r t i c u l a r i n t e r e s t

i s t h e s e c t i o n concerning unresolved problems which i n c l u d e

a s p e c t s of non-linear s u r f a c e waves; t i d a l d i s s i p a t i o n ; t h e energy sources g e n e r a t i n g long period oceanic waves; and trapped waves. LeBlond and MysaK g i v e a b r i e f review of t h e c l a s s i c a l wave modes i n the ocean.

The i n t e r e s t e d s t u d e n t c a n f i n d

f u r t h e r d e t a i l s i n P h i l l i p s [55] and Krauss [ 591.

I n a recent

t e x t , Kamenkovich [ 571, g i v e s a u n i f i e d treatment of t h e v a r i o u s types of wave8 t h a t e x i s t i n a n unbounded r o t a t i n g , s t r a t i f i e d ocean.

The f r e q u e n c i e s of t h e f r e e o s c i l l a t i o n s

of t h e ocean a r e obtained a s t h e i n t e r s e c t i o n of t h e eigenv a l u e curves of two d i s t i c n t problems which a r e l a b e l e d problem H and problem V .

The e f f e c t s of t h e e a r t h s r o t a t i o n

and s p h e r i c a l geometry appear e x p l i c i t l y only i n problem H and t h e e f f e c t s of s t r a t i f i c a t i o n appear e x p l i c i t l y only i n problem V .

Thus i n t h i s c a s e t h e f o r c e s of r o t a t i o n and

s t r a t i f i c a t i o n a r e e f f e c t i v e l y decoupled i n t h e mathematical d e sc r i p t i o n .

Problems

189

I;haDter 1 3 P r o b l e m 15.1)

Does t h e i n t r o d u c t i o n of s t r a t i f i c a t i o n change t h e r e f l e c t i o n p r o p e r t i e s of i n t e r n a l waves? Note t h a t i n a homogeneous f l u i d t h e i n c i d e n t wave and t h e r e f l e c t e d wave maKe t h e same a n g l e w i t h 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 ( s e e problem 7.5)

13.2)

.

From e q u a t i o n s (13.1) and (13.5) show t h a t t h e n

boundary c o n d i t i o n 9 - n = 0 terms of

15.3)

can be w r i t t e n i n

as

0

Determine t h e i n t e r n a l modes 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 bounded by r i g i d walls z = O,H

and

x = O,L, and unbounded i n t h e

y-direction.

with

2

and

iA

(treat a)

The problem f o r

N~

=

o at z

+

f

O,H,

=

a =

ay

i s g i v e n by

i

o

at

x

=

O,L.

as a constant).

Obtain t h e eigenvalues and e i g e n f u n c t i o n s f o r t h e two c l a s s e s of waves.

190

Problems b)

Sketch a n eigenfunction of the f i r s t c l a s s and an eigenfunction of the second c l a s s as

a function of c)

x

f o r a fixed value of

z.

Compare your s o l u t i o n s with those obtained i n the t e x t f o r the c y l i n d r i c a l problem.

13.4)

Derive the equation s a t i s f i e d by when the buoyancy frequency a function of the depth

13.5)

I($)

N(2)

is

z.

Consider the equation Gz z

Using

+

-

y2[(N‘(z)

W.K.B.

4)m2

-

1 ] G = 0.

techniques f o r l a r g e

Y

seek a s o l u t i o n of the form

+

q,(z)

By equating powers of

y-l

tions f o r

$,(z).

G =

exp[Yrlo(z)

q0(z)

and

+

t

e,(z)

+...I.

o b t a i n the equaHence show t h a t

(13.49) is an asymptotic s o l u t i o n t o t h e d i f f e r e n t i a l equation.

CHAPTER 1 4 BOUNDARY LAYERS I N A ROTATING STRATIFIED FLUID

Let us consider a r o t a t i n g s t r a t i f i e d viscous f l u i d and

-

assume t h a t the motions a r e steady ( o r almost s t e a d y ) . equations of motion a r e the steady form of (10.6) with vTo = 6 , namely

v-3 2; x

9

The (10.8)

(14.1)

= 0

+

= -VP

w =

Ti;

+

EV 2J

(14.2)

V2T.

(14.3)

N U

Before we analyze the boundary l a y e r s t r u c t u r e supported by these equations, we w i l l observe the importance of t h e

r e l a t i v e s i z e s of the dimensionless parameters Consider the case (1) where if

K

>>

v, i . e . ,

O(N2cr)

<

O(E):

E

and

N

2

0 .

t h i s could occur

thermal d i f f u s i o n strongly dominates viscous

d i f f u s i o n ; i t could a l s o occur i f t h e s t r a t i f i c a t i o n parameter N2

were very small.

T o the f i r s t order equation ( 1 2 . 3 ) then

reduces t o V

2

T

= 0.

Thus the temperature f i e l d i s determined by solving Laplace's equation w i t h appropriate boundary conditions.

9

The v e l o c i t y

can then be determined from ( 1 4 . 1 ) and ( 1 4 . 2 ) which a r e

N2 o = E a

192

equivalent t o the equations f o r a n o n - s t r a t i f i e d r o t a t i n g f l u i d , but with t h e a d d i t i o n of an inhomogeneous term Thus, when

2

O(N a )

<

Ti.

O ( E ) , the problems f o r the temperature

and v e l o c i t y f i e l d s a r e e s s e n t i a l l y uncoupled. We w i l l discuss the behavior of equations (14.1)

0

(14.3) and the r e l a t e d boundary layers when the parameter N 2u

i s l a r g e r than

I n t h i s case, presence of s t r a t l -

O(E).

f i c a t i o n i s s i g n i f i c a n t , however i t ' s exact r o l e depends on

i t ' s s i z e r e l a t i v e t o t h e Ekman number

E.

To understand how

the boundary l a y e r s vary with the r e l a t i v e s t r e n g t h s of E and N 2u we w r i t e N 20 = Ea We w i l l examine the possible

.

boundary layers t h a t e x i s t as

a

decreases from u n i t y .

I n order t o reduce the system of vector equations (14.1), ( 1 4 . 2 ) and (14.3) t o a s i n g l e equation f o r t h e pressure

P

we

carry out the now f a m i l i a r manipulation of the equations. From (14.3) and

ie(14.2) w =

we obtain

+ v2[gN u

Ev2w]

.

(14.4)

vn(14.2) gives

x

-2k.v II

and

E*v x (14.2)

3

+

= -V 2 P h az T

gives (14.6)

Thus combining (14.5) and (14.6) we obtain

4

g

= E 2 ( -v2P

+

5).

( 1 4 -7)

S t r a t i f i e d boundary l a y e r s v 2T

Substituting f o r

193

from (12.3) gives

(4-N2a )

= -Ev

4

(14.8)

P.

Hence ( 1 4 . 4 ) and ( 1 4 . 8 ) combine t o give V

(where

vH

=

v

2

2 6

2

2

[E V P + N oVHP + 4Pzz]

(14.9)

= 0

-

T h i s e i g h t h order p a r t i a l d i f f e r e n t i a l equation describes the

The f i r s t term

steady flow of a viscous, s t r a t i f i e d f l u i d .

represents t h e viscous d i f f i s i v e e f f e c t , the second term t h e buoyancy e f f e c t , and the t h i r d term C o r i o l i s force.

The solu-

t i o n of t h i s equation which s a t i s f i e s prescribed boundary conditions f o r t h e normal and t a n g e n t i a l v e l o c i t y and t h e temperature i s the exact s o l u t i o n t o w e l l posed problem f o r t h e steady l i n e a r i z e d flow.

To solve a n eighth order

P.D.E.

with general boundary conditions i s , of course, very d i f f i cult.

We w i l l seek a n asymptotic s o l u t i o n using boundary

l a y e r techniques. S t r a t i f i e d Ekman I&ygz As we have s t a t e d , we write

N

2

0

= E

a

.

We i n v e s t i g a t e the

boundary l a y e r a t a h o r i z o n t a l s u r f a c e by w r i t i n g z = Ea5

hence

2

= E-a

a5

.

From equation (14.9) we observe t h a t t h e highest order terms i n the boundary l a y e r equation a r e E2'6a

a5

+

E$;P

+

4E'2a

2

a5

= 0.

(14.10)

194

S t r a t i f i e d EKman Layer

Thus we have a balance of terms

6

2

5x6

a5

u + + 0

with

a = 1/2, provided

>

a

(14.11)

Hence t h e boundary l a y e r

-1.

has t h e same dynamics a s the homogeneous Ekman Layer provided

N 2a

the parameter

i s l e s s than 2 N a

t i o n i s so strong t h a t

>

O(E-l).

If the s t r a t i f i c a -

then t h e Ekman l a y e r

O(E’l)

s t r u c t u r e is destroyed. There i s a f u r t h e r f e a t u r e t o note.

We have shown i n

Chapter 5 t h a t t h e Ekman l a y e r , which i s characterized by equation (14.11), induces an i n t e r i o r v e r t i c a l v e l o c i t y w I n the s t r a t i f i e d problem, equation (14.3) w = O(E 1-a ) . Thus t h e r e i s a second c r i t i c a l value

of

0(E1I2).

requires of N2a i n terms of the r o l e of the Ekman layer, namely When

N2a = O ( E U 2 ) .

a

<

i s g r e a t e r than t h i s value

N2a

(i.e.,

1 / 2 ) s t r a t i f i c a t i o n i n h i b i t s t h e important mechanism of

Ekman l a y e r suction. non-divergent

.

I n t h i s case the Ekman l a y e r i s c a l l e d

When N2a

<

O(E112)

t h e Ekman l a y e r plays

the same r o l e i n d r i v i n g the i n t e r i o r flow a s i t does i n the case of a homogeneous f l u i d . To examine the Ekman layer i n a l i t t l e more d e t a i l , l e t us

consider an asymptotic expansion i n powers of

Ell2.

To t h e

highest order the Ekman l a y e r components s a t i s f y the now f a m i l i a r equations n

a

-2v = u

2u = v

55

55

(14.12) (14-13)

S t r a t i f i e d boundary layers

195

(14.14)

Let us take t h e boundary conditions t o be a n imposed horizon-

(&, zero

t a l velocity field.

From (14.12)

-

normal v e l o c i t y , and zero temperature

(14.14) we can derive the Ekman

l a y e r s u c t i o n condition a t

5

= 0,

However, returning t o the i n t e r i o r problem we repeat t h e observation t h a t equation (14.3) implies t h a t w = O ( E 1-a) Thus i f

a

<

1/2,

.

t h e condition of zero normal v e l o c i t y must

be s a t i s f i e d by the Ekman l a y e r alone.

Therefore, t o t h e

f i r s t order, t h e s u c t i o n condition (14.15) degenerates t o

Hence when

0(E1")

<

N20

<

O(E-')

t h e h o r i z o n t a l boundary

l a y e r 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 has the form of a n Ekman layer, but i t s r o l e i s l e s s important because s t r a t i f i c a t i o n prevents t h e Ekman l a y e r from inducing an i n t e r i o r v e r t i c a l v e l o c i t y of

o(Ey2).

To consider the case (14.10).

with

N

2

0

>

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

<

O(E213),

the

E1’3-layer

197

i n a homogeneous f l u i d i s

unchanged t o t h e h i g h e s t o r d e r .

&ae 2:

a

<

2/3

There a r e now two p o s s i b l e boundary l a y e r s .

The

buoyancy l a y e r is c h a r a c t e r i z e d by t h e balance between t h e viscous term and t h e buoyancy term,

This balance r e q u i r e s t h a t

thus t h e boundary l a y e r has thickness

O(+).

(N20

l1

The second l a y e r i s c a l l e d t h e h y d r o s t a t i c l a y e r and i t i s c h a r a c t e r i z e d by t h e balance between t h e buoyancy term and Coriolis force,

a2s + 4Pzz

= 0.

(14.20)

ax

This balance r e q u i r e s t h a t

thus t h e boundary l a y e r has thickness

O( (N2g)l12).

t h e t h i c k n e s s does not depend e x p l i c i t l y on

E.

Note t h a t

198

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

We a l s o observe t h a t e q u a t i o n (14.17) reduces t o t h e equat i o n t h a t c h a r a c t e r i z e s t h e homogeneous

E1/4-layer,

namely

Pzz = 0 , with

p

Thus we expect t h e

=

(14.21)

, provided

1/4

a

<

1/2.

t o play i t s r o l e of matching

E1’4-layer

t h e i n t e r i o r a z i m u t h a l v e l o c i t y w i t h t h e p r e s c r i b e d boundary v e l o c i t y f o r N20 i n t h e range N20 < O ( E 1/2 )

.

We t h e r e f o r e observe t h e f o l l o w i n g p r o g r e s s i o n a s t h e parameter N 2Q i n c r e a s e s from z e r o . The E1/4-layer i s e s s e n t i a l l y unchanged u n t i l

2

However t h e

2

A t t h i s value the layer

N 0 = 0(E1j2).

changes e a r l i e r when

N 0 = O(E2/’).

E1I3-layer

is r e p l a c e d by two new l a y e r s , t h e buoyancy l a y e r of t h i c k -

, and

ness

O(E1’2/(N2~)1’4)

ness

O((N‘U)~/~).

t h e h y d r o s t a t i c l a y e r of t h i c k -

The f i r s t o f t h e s e l a y e r s becomes pro-

g r e s s i v e l y t h i n n e r while t h e second becomes p r o g r e s s i v e l y thicKer a s

N

2

Q

increases.

l a y e r i s of thicKness E1I4-layer.

For l a r g e r

When

O ( E ’I4)

N 2 Q = O(El/’),

t h e second

and c o i n c i d e s w i t h t h e

2

N 0 , t h i s layer continues t o thicken

u n t i l i t becomes p a r t of t h e v i s c o u s - d i f f u s i v e p r o c e s s e s which effect the i n t e r i o r regions as

N2u

i n c r e a s e s t o be

O(1).

Figure 26 g i v e s a schematic diagram i l l u s t r a t i n g t h e behavior of t h e boundary l a y e r s i n d i f f e r e n t regimes of t h e parameter N

2

0.

A d e t a i l e d d i s c u s s i o n of t h e c h a r a c t e r i s t i c s o f t h e be-

h a v i o r of t h e boundary l a y e r s i n d i f f e r e n t parameter ranges

is g i v e n by B a r c i l o n and Pedlosky [ 3 1 . We w i l l now examine i n a l i t t l e more d e t a i l , t h e e q u a t i o n s of motion i n t h e s i d e - w a l l l a y e r s when t h e parameter

N 2u

2

N o side-wall layers

Interior

1 I

horizontal

Ekman

Ilayer

I

I 1

1

I

I

non-divergent

I

I / / I / / / / / /

I

I

S t e w a r t sontype layers

The buoyancy l a y e r

200

l i e s In t h e range

<

O(E2’3)

N2u

<

O(E 1/2 )

.

0(E1l2/(N 2u ) 1/41

( i ) The Buoyancy Layer of

.

To study t h i s l a y e r we w r i t e

where

r

-

a

= [E1/2/(N2~)1’4]x.

The standard s c a l i n g a n a l y s i s shows t h a t I n t h e boundary layer

2

T = (Nu)

We note t h a t we have chosen w

v 4 ij, t o be

O( (N2u)1/4)

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

so t h a t 0(Eu2).

This must be t h e case because t h e buoyancy l a y e r replaces the

E1I3-layer

return the

(when NZo

0(E1I2)

l a y e r (when N 2Q

<

>

O(E2I3)) whose r o l e I s t o

I n t e r i o r mass f l u x Induced by t h e Ekman 0(E1l2)).

To t h e f i r s t order t h e boundary l a y e r equations a r e =

-$

(14.22)

2ii =

vxx

-

(14-23)

T

(14.24)

-2;

0 =

- + wxx -

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

201

ii+iz=o

(14.25) (14.26)

We note t h a t equations ( 1 4 . 2 4 ) and (14.26) have a s i m i l a r coupled s t r u c t u r e t o t h a t of the EKman l a y e r equations ( 1 4 . 1 2 ) and (14.13) w i t h t h e roles of

fi

and

0

talcen by

?.

and

It can be shown t h a t the r o l e of t h i s side-wall l a y e r is

analogous t o t h a t of the EKman l a y e r i n t h a t i t induces a small r a d i a l v e l o c i t y

6 which i s normal t o the side-wall

boundary. The balance of terms i n equations ( 1 4 . 2 2 )

-

(14.26)

shows t h a t both r o t a t i o n and s t r a t i f i c a t i o n a r e important i n the buoyancy l a y e r .

We note t h a t t h e dimensional thiclvless of

t h i s l a y e r i s given by

[we have s u b s t i t u t e d f o r the values of Chapter 10, and

E

i n Chapter 31.

N2

and

0

given i n

Hence t h e thicKness of the

boundary l a y e r i s independent of the magnitude of r o t a t i o n The dimensionless parameter =

l?&d is KVP

c a l l e d the Rayleigh number.

We w r i t e the thickness of t h e boundary l a y e r

6

as

n.

The h y d r o s t a t i c l a y e r

2 02

( i i ) The Hydrostatic Layer o f TO

.

O( ( N 20 ) 1/2)

study t h i s l a y e r we w r i t e

The s c a l e a n a l y s i s shows t h a t t h e r e l a t i v e orders of the boundary l a y e r q u a n t i t i e s a r e

2 -1/2 v=NoE

where t h e order of

O(E1’2).

w

3

is chosen s o t h a t t h e v e r t i c a l flux i s

The boundary l a y e r equations become

23 =

3YY

(14.28)

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

20 3

The d i f f e r e n c e between t h i s s e t of equations and those f o r t h e buoyancy l a y e r occurs i n the v e r t i c a l component of t h e momentum equation.

Comparing (14.29) with (14.24) we see t h a t t h e

h y d r o s t a t i c l a y e r has a higher degree of v e r t i c a l s t r u c t u r e than t h e buoyancy l a y e r and t h a t t h e equations a r e no longer analogous t o those of t h e EKman l a y e r . c r i t i c a l value c i d e i n an

N

2

= 0(EZi3),

Q

We note t h a t a t t h e

t h e two boundary l a y e r s coin-

l a y e r with a l l t h r e e terms i n t h e

0(Eli3)

v e r t i c a l component of the momentum equation being of t h e same o r d e r , namely (14.32)

(iii) The

Eli4-Layer.

A s we previously remarked, Ekman l a y e r s u c t i o n

plays an important r o l e i n d r i v i n g t h e i n t e r i o r c i r c u l a t i o n when t h e parameter

N20

t h e r e w i l l e x i s t an c i t y component is

i s l e s s than

E1I4-layer

O(1).

O(Eli2).

I n t h i s range

i n which the azimuthal velo-

The dynamics of t h e

E1I4-layer

are

very s i m i l a r t o those t h a t e x i s t i n t h e homogeneous problem. We w r i t e

d-= ar

1

L a

an

The orders of magnitude of t h e boundary l a y e r q u a n t i t i e s a r e

The

204

E1l4-layer

u = E1/2 g N

v = v w

Ell4 I ;

2 -1/2 T = NuE

and

; ,

Thus t h e v e l o c i t y s c a l i n g agrees with t h a t of the homogeneous EV4-layer.

The introduction of s t r a t i f i c a t i o n a f f e c t s t h e O(N2~E-1/2), and determines the

second order terms which a r e

magnitude of the temperature f i e l d ,

The boundary l a y e r equa-

tions a r e m

m

-2v = -p m

T

m

2u = v

vl

N

0 = -P, m

u

rl

m +w z=o

m

and

m

w = T

tlrl

These equations show t h a t t h e

. E1/4-layer

has a weak v e r t i c a l

s t r u c t u r e and t h a t the problem f o r the temperature i s decoupled from the problem f o r t h e v e l o c i t y . . I )

Equations (14.33)

(14.37) w i t h boundary conditions given by the Ekman l a y e r

condition a t the upper and lower boundaries, and the matching condition on the

v-component of v e l o c i t y a t t h e s i d e walls,

can be solved t o determine Once

m

m

u, v

and

G.

i s calculated, the temperature

from (14.37).

(See Chapter 9.)

m

T

i s determined

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

205

We note one f i n a l f e a t u r e concerning boundary l a y e r s i n a rotating stratified fluid.

I n a homogeneous f l u i d the s i d e

wall l a y e r s a r e much t h i c k e r than t h e EKman l a y e r s

and

O(Eu3)

i n comparison with

O(E'")).

(O(E114)

Hence the Ekman

l a y e r boundary condition could be applied t o the s i d e wall flow i n the same manner as t o the i n t e r i o r flow. s t r a t i f i e d problem the h y d r o s t a t i c l a y e r of and t h e layer.

E1'4-layer

I n the

O( (N20)1/2)

a r e a l s o much t h i c k e r than t h e Ekman

However, the buoyancy l a y e r i s of thickness

o ( E ~ ~ / (0N) v4) which i s comparable t o t h e Ekman l a y e r 2 thickness when N 0 = O ( 1 ) . I n t h i s case, t h e corner regions

formed by the i n t e r s e c t i o n of the two boundary l a y e r s must be t r e a t e d more c a r e f u l l y .

I n these corner regions the d i f f u s i v e

term, t h e buoyancy term and the C o r i o l i s term i n equation

(14.9) a r e a l l important. I n t h e next s e c t i o n we w i l l give a n example of a p a r t i c u l a r problem where we uae the preceeding a n a l y s i s of the boundary layers t o o b t a i n the quasi-steady flow i n a r o t a t i n g s t r a t i fied fl ui d.

P r ob lems

z 06

14.1)

Describe t h e boundary l a y e r s 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 when t h e parameter

14.2)

N2u = O(1).

Show t h a t i n t h e buoyancy l a y e r

w

-(

+

i T = A(z)e

y 9 X

(Assume a x i a l symmetry). 14.3)

Consider t h e problem given by equations

(14.26) w i t h boundary condi-

(14.22) tions

where 7 ( Z)

i s t h e i n t e r i o r temperature and

TI

is t h e imposed boundary temperature.

I n t h i s problem

ii

Determine and 14.4)

N 20

and

= O(1).

a6 f u n c t i o n s of

T

TI.

With t h e c o n d i t i o n s s t a t e d i n problem 3, t o g e t h e r with t h e boundary c o n d i t i o n

ii+

uI = 0

at

r=a

show t h a t t h e i n t e r i o r v e l o c i t y

uI:

s a t i s f i e s the lhoyancy layer suction c ond it i o n "

Problems

14.5)

207

I n what parameter ranges of of

Ea

N

2

0

i n terms

is the behavior 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 e s s e n t i a l l y equivalent t o

a)

a homogeneous r o t a t i n g f l u i d ?

b)

a s t r a t i f i e d non-rotating f l u i d ? Discuss your answers.

14.6)

Analyze the dynamics of the h o r i z o n t a l boundary l a y e r when N20

>

O(E-l),

What i s the important d r i v i n g mechanism f o r t h e i n t e r i o r flow i n t h i s case?

14.7)

Determine t h e steady flow t o the f i r s t order f o r a rotating s t r a t i f i e d f l u i d i n a cylinder. The walls a r e r i g i d and s t a t i o n a r y , r e l a t i v e t o t h e r o t a t i n g co-ordinate system; t h e s i d e walls a r e thermally i n s u l a t e d ; the p e r t u r b a t i o n temperature a t t h e bottom i s

T = 0

and a t t h e

top, T = 1. Discuss your s o l u t i o n a s the parameter

N20

increases from zero.

See Figure 27.

208

Problems

a

g,= 0 ,

T = 1

z = 1

I

I

I I

I I

I I

I

q = o

-t

-aT =

ar

i I

I

I I

I

I I

I

I

I

I

FIGURE 27

2=0,

T = O

z = o

0

CHAPTER 15 SPIN-DOWN I N A ROTATING STRATIFIED FLUID

I n Chapter 5 we discussed the mechanism of spin-down ( o r up) i n a homogeneous r o t a t i n g f l u i d .

We showed t h a t Ekman l a y e r

suction provided a very e f f i c i e n t way f o r a boundary condition on the v e l o c i t y t o be communicated t o the i n t e r i o r .

This pro-

cess leads t o a homogeneous spin-down (up) time-scale of O(Em112) which i s much f a s t e r than the viscous time-scale of 0(Em1).

However, the problem i s r a t h e r more complicated f o r

a rotating stratified fluid.

I n t h i s case, t o complete t h e

c i r c u l a t i o n driven by the Ekman l a y e r , a f l u i d p a r t i c l e must move across a d e n s i t y gradient and thus the motion is i n h i b i t e d by d e n s i t y f o r c e s .

I n our a n a l y s i s of the s t r a t i f i e d Ekman

l a y e r we have already observed t h a t t h e r e i s no i n t e r i o r v e r t i c a l v e l o c i t y of

O(E

and hence i t i s t o be expected t h a t

t h e r o l e of the Ekman l a y e r i n d r i v i n g the i n t e r i o r c i r c u l a t i o n

is reduced by the e f f e c t s of s t r a t i f i c a t i o n .

I n f a c t , the

d e f i n i t i o n of a s t r a t i f i e d spin-down time is not u n i v e r s a l l y accepted.

For a d e t a i l e d discussion of the meaning of t h i s

term we r e f e r the reader t o Buzyna and Veronis [ 71.

They

observe t h a t f o r a s t r a t i f i e d f l u i d s i g n i f i c a n t cnages i n the angular v e l o c i t y of a p a r t i c l e can taKe place between the homogeneous spin-down time-scale of

2 09

O ( E -'I2) and the viscous

Solar Spin-down

2 10

d i f f u s i v e s c a l e of

O(E-l).

There has been some controversy connected with t h e idea of s t r a t i f i e d spin-down.

It was claimed by Pedlosky [ 5 O ] , t h a t

s t r a t i f i c a t i o n prevented t h e closure, i n the side-walls layer, of the EKman l a y e r c i r c u l a t i o n .

However, walin [ 6 9 ] and

Sakurai [58], showed independently t h a t t h i s was not i n f a c t the case and t h a t although s t r a t i f i c a t i o n i n h i b i t e d the Ekman layer c i r c u l a t i o n , i t was not completely prevented.

A recent

review of the topic i s given by Benton and Clark [6]. One p a r t i c u l a r reason w h y people have been i n t e r e s t e d i n the problem of s t r a t i f i e d spin-down i s t h e controversy concerning s o l a r oblateness.

It was conjectured by Dicke [12]

t h a t t h e s o l a r i n t e r i o r might be r o t a t i n g twenty times more rapidly than the observed r o t a t i o n r a t e of t h e o u t e r s h e l l . I f t h i s were i n f a c t t h e case, then t h e r e s u l t i n g s o l a r oblateness would d i s t o r t the suns) g r a v i t a t i o n a l f i e l d s u f f i c i e n t l y t o account f o r t h e precession of the p e r h i l i o n of Mercury. However, such a theory has been disputed by a number of people, including Howard, Moore and Spiegel [ 3 4 ]

, who

suggested t h a t

some form of spin-down process would provide e f f i c i e n t coupling between the more slowly r o t a t i n g outer convection zone and the radiative i n t e r i o r .

Hence, even i f t h e postulated angular

v e l o c i t y d i s c o n t i n u i t y e x i s t e d i n i t i a l l y , i t would not p e r s i s t over t h e l i f e time of the sun.

This argument was countered

by Dicke [13], who reasoned t h a t the s t r o n g d e n s i t y gradient

i n the core and t h e l a r g e s c a l e s t e l l a r dimensions would negate t h e e f f e c t of the Ekman l a y e r c i r c u l a t i o n and hence permit a d i f f e r e n t i a l r o t a t i o n t o e x i s t on a much longer time-

S t r a t i f i e d spin-down scale.

2 11

This notion was summed-up by Dicke i n t h e words, "the

sun i s no cup of t e a " , i . e . ,

the spin-down e f f e c t described i n

the tea-cup experiment i n Chapter 5 was not relevant t o t h e

sun.

Several mathematical models f o r the s o l a r spin-down prob-

lem have now been presented, including those of Friedlander [20] and [21] and Sakurai, C l a m and Clark [ 5 9 ] .

These i n d i -

c a t e t h a t a d i s c o n t i n u i t y i n angular v e l o c i t y a s postulated by Dicke would be smoothed out on a time s c a l e

O(N20E'l)

i s given the name, Eddington-Sweet time-scale. s o l a r parameters, N2

which

I n t h e case of

i s r a t h e r l a r g e , but t h e s t r o n g radia-

t i v e thermal d i f f u s i v i t y K implies t h a t a ( = V / K ) small. This means t h a t N'0E-l l i e s between E

i s very and E-l

-v2

and hence spin-down occurs on a time-scale t h a t l i e s between t h e homogeneous spin-down s c a l e and t h e d i f f u s i v e s c a l e .

In

f a c t , i n dimensional t e r m , t h i s time-scale is of t h e order of lo9 years which i s j u s t within t h e l i f e - t i m e of t h e sun.

On a time-scale s h o r t e r t h a t

0(1$)

years, the e f f e c t s of

the s u r f a c e s t r e s s would not f u l l y p e n e t r a t e t h e i n t e r i o r and a c e n t r a l p o r t i o n of the core could be r o t a t i n g more r a p i d l y than the surrounding s o l a r m a t e r i a l .

However, on a time-scale

close t o t h e l i f e - t i m e of the sun, the r a p i d l y r o t a t i n g region would be much too small t o induce s u f f i c i e n t oblateness t o account f o r t h e precession of the p e r i h e l i o n of Mercury.

It

i e f u r t h e r remarked by Howard, Moore and Spiegel [ J 4 ] , t h a t i t i s probably more r e a l i s t i c t o consider the boundary l a y e r s

a s turbulent, which considerably shortens t h e spin-down time. Thus t h e a n a l y s i s of spin-down 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 i n d i c a t e s t h a t Dicke's hypothesis, a s applied t o a simple

Spin-down i n a cylinder

2 12

s o l a r model, i s highly implausible.

We w i l l now o u t l i n e the techniques t h a t lead t o e x p l i c i t r e s u l t s describing quasi-steady flow i n a r o t a t i n g s t r a t i f i e d cylinder and apply the r e s u l t s t o the s o l a r spin-down a n a l y s i s . Further d e t a i l s a r e given i n Friedlander [ 2 0 ] .

I n t h e context

of t h e s o l a r problem, the viscous coupling of t h e more slowly r o t a t i n g s h e l l on the core i s represented by a steady s t r e s s T

( r ) a c t i n g on t h e h o r i z o n t a l boundaries.

For mathematical

s i m p l i c i t y we take the walls t o be thermally insulated and assume t h a t t h e side-walls a r e r i g i d , massless and f r e e t o

-

move with the f l u i d .

(10.6)

conditions.

We then seek the s o l u t i o n t o equations

(10.8) [with vTo =

k]

with appropriate boundary

For the convenience of the reader, we w i l l re-

write these basic equations :

i

+ N'9-k

=

v2T.

The boundary conditions a r e

z = 0,1,

az

(15.4)

S t r a t i f i e d spin-down and

where

j

=

u^r + v8

+

WE.

Since t h e boundary conditions a r e axisyrnmetric, we assume t h a t t h e long time s o l u t i o n t h a t we a r e seeking also s a t i s f i e s

$=

0.

We note t h a t t h i s assumption w i l l r u l e out t h e possi-

-

b i l i t y of wave-like s o l u t i o n s . Examination of equations (15.1)

(15.3) shows t h a t the

order of the i n t e r i o r flow i n powers of u =

v

= v0

+

+

E112v1

w = T = T~

+

E

Eu2

+

...

Ev2

+

...

Ew2

+

...

+

E ~ / ~ ET T~ ~+

... .

The highest order i n t e r i o r equations a r e

1

'0

av 2

= E(v2

= 2

5 ar

- +)vo r

To

5

= az

E-2u2

i s given a s

Boundary layers

2 14

aT

0=

at

o2T0

a

-

2 E - N w2.

Equations (15.7) and ( 1 5 . 9 ) show t h a t t o the f i r s t order the i n t e r i o r flow i s i n geostrophic and hydrostatic balance. note t h a t equations time-scale

vo

and

We

(15.8) and (15.10) imply t h a t on a s h o r t a r e independent of time.

To

we w i l l observe t h a t t h e r e i s no

O(1)

In fact,

steady flow on a s h o r t

time-scale and hence we w i l l be considering t h e behavior of the f l u i d on a long time-scale. The boundary conditions a t f i e d v i a an Ekman l a y e r .

z

= 0

z

and

=

1

are satis-

The analysis is s i m i l a r t o t h a t

given i n Chapter 1 4 , modified f o r the s t r e s s boundary condition.

The d e r i v a t i o n of the Ekman l a y e r suction condition

i n t h i s problem i s l e f t a s a n exercise f o r the reader. can be shown t h a t

=

O(E)

It

i n the Ekman layer, and the suc-

t i o n condition gives a n o n - t r i v i a l r e l a t i o n between

wz

and

Vo:

T(r))

at

z

(15.11)

= 0,1.

The thermal boundary condition i s unaffected by the Ekman layer t o the f i r s t order, hence

"To = az

o at z

= 0,l.

The analysis of the side-wall layers i s again s i m i l a r t o t h a t of Chapter 1 4 w i t h the necessary modification f o r the insulated boundary condition.

It can be shown t h a t

i

= O(E)

2 15

S t r a t i f i e d spin-down

i n t h e boundary layer and the r e l a t i o n s h i p between

u2

and

becomes

To

aZTo

= Ni30 -araz Lat

r = a.

(15.13)

A n d t h e v e l o c i t y condition remains vo = V

at

r

a.

=

(15.14)

The value of t h e constant wall v e l o c i t y

V

w i l l emerge from

the c a l c u l a t i o n s from t h e requirement t h a t t h e torque due t o t h e imposed surface s t r e s s i s i n balance with t h e r a t e of change of angular momentum of the f l u i d . We wish t o w r i t e t h e problem i n terms of a s i n g l e equation f o r one v a r i a b l e . equations (15.1)

-

We do t h i s by a l i t t l e manipulation of (15.3).

TaKing v x (15.2) and s u b s t i -

t u t i n g (15.3) gives

This is, of course, the p o t e n t i a l v o r t i c i t y equation f o r a viscous f l u i d .

To rewrite t h e i n t e r i o r equation i n terms of

we s u b s t i t u t e

Po

(15.9).

vo

and

To

from equations (15.7) and

Thus the problem f o r t h e

0(1)

i n t e r i o r pressure i s

a t H o + LL N2

-yv2P

10; r

9

denotes t h e horizontal Laplacian]

.

We observe t h a t on a time-scale s h o r t e r than equation (15.16) reduces t o

2

O(N 0E-l)

P o t e n t i a l v o r t i c i t y equation

2 16

= 0.

have shown t h a t t h i s equation with Howard and Siegmann [El, prescribed i n i t i a l condition, and boundary conditions consistant w i t h those t h a t we have considered, has a unique s o l u t i o n . [See problems ( 1 1 . 2 ) and (11.3)].

Hence, i f our i n i t i a l con-

d i t i o n is chosen t o be zero perturbation v e l o c i t y and temperat u r e , the unique s o l u t i o n of the problem on a time-scale s h o r t e r than

is

O(N2uE-l)

P0 = 0, implying vo

= 0, To = 0.

time-scale s h o r t e r than

Thus there is no steady flow on a O(N2aE-l).

We will therefore examine

the problem on t h e long time-scale by w r i t i n g t

2

= NuE

-1

t’.

The p o t e n t i a l v o r t i c i t y equation then becomes

We w r i t e the boundary conditions i n t e r m of

Po.

We

f i r s t observe t h a t on t h i s long time-scale (15.10) gives

Hence the Ekman l a y e r suction condition (15.ll) becomes

S t r a t i f ied spin-down

2 17

The thermal condition (15.12) gives

r

To o b t a i n the expression f o r t h e boundary conditions a t

=

a

we use (15.8) on t h e long time-scale t o give u2 =

r

Thus ( 15.13) becomes

And (15.14) gives (15. a )

-

We have w r i t t e n t h e problem i n a w e l l posed form, namely equat i o n (15.17) with boundary conditions (15.19)

(15.22).

It

is c l e a r from t h e complexity of t h e expression involved t h a t i t w i l l not be simple t o o b t a i n a s o l u t i o n .

-

However, boundary

l a y e r a n a l y s i s has enabled us t o reduce the full viscous d i f f u s i v e system of equations (15.1)

(15.3) with conditions (15.4)

and (15.5) t o a well posed boundary value problem f o r the

O(1)

2 18

Modal representation

i n t e r i o r pressure

Po.

The equation i s time-dependent and

6-th order i n space and t h e boundary conditions involve d e r i v a t i v e s of

Po

i n time and space.

The complexity of the

system i s r a t h e r t y p i c a l of the mathematical formulation of problems i n r o t a t i n g , s t r a t i f i e d , viscous flows.

One method

of t a c k l i n g such an equation and boundary conditions I s t o seek a numerical s o l u t i o n .

A d e s c r i p t i o n of the techniques

involved i n t h i s approach i s given by Williams [ 7 0 , 711. I n t h e p a r t i c u l a r problem t h a t we a r e i n v e s t i g a t i n g , it i s possible t o o b t a i n an e x p l i c i t s o l u t i o n f o r

Po

as a function

? ( r ) provided a n assumption i s made a s t o the

of the s t r e s s

form of the time-dependence of the s o l u t i o n .

We assume t h a t

t h e flow can be resolved i n t o a p a r t t h a t grows l i n e a r l y w i t h t'

,a

steady flow, and a decaying flow represented by a sum

of exponential modes, we therefore w r i t e

This form of s o l u t i o n i s p l a u s i b l e since the steady s t r e s s is

feeding angular momentum i n t o the f l u i d and hence w i l l r e s u l t i n a portion of the v e l o c i t y t h a t increases l i n e a r l y with time.

The assumption i s a l s o supported by good agreement be-

tween the r e m l t i n g s o l u t i o n s and those obtained by numerical methods. We s u b s t i t u t e expression (15.23) f o r (15.17) and the boundary conditions. in

t'

gives the problems f o r

Po

i n t o equation

Equating c o e f f i c i e n t s

Po, P1

and

P.

Each of these

problems can be solved e x p l i c i t l y [ s e e Friedlander [ZO]] problem f o r

Po

involves a homogeneous

P.D.E.

.

The

which leads

S t r a t i f i e d spin-down

2 19

t o the simple s o l u t i o n of r i g i d r o t a t i o n . The problem f o r 1 P , however, i s inhomogeneous and hence r a t h e r more d i f f i c u l t , although standard techniques lead t o a s o l u t i o n a s a s e r i e s of Bessel f’unctions with c o e f f i c i e n t s determined a s the Bessel transform of the s t r e s s boundary condition. F i n a l l y , a n a l y s i s of the eigenvalue problem f o r

P

shows

t h a t the modes a r e purely decaying and a l s o t h a t ( f o r

N 2u

<<

1 ) the smallest eigenvalue i s g r e a t e r than

Thus on a time-scale where 2

O(N 0E-I)

, the

t’

is

0(1), i . e . ,

t

O(1).

is

time dependent modes have decayed, leaving a

r e s i d u a l flow t h a t we c a l l quasi-steady.

I

S e c u l a r G r owth The problem f o r

Po

0 2[PHP 2

becomes 0

+

+

P;,]

= 0

N u

with boundary conditions

We have assumed a s i m i l a r expansion for the time dependent behavior of the wall v e l o c i t y

220

Secular and steady s o l u t i o n s

The s o l u t i o n t o (15.24), (15.25) and (15.26) i s =

vo,2 a

(15.27)

*

From (15.27) combined w i t h (15.7) and (15.9) i t follows t h a t and

TO = 0 .

(15.28)

Thus the portion of the s o l u t i o n t h a t grows l i n e a r l y w i t h time i s represented purely as r i g i d r o t a t i o n .

m e Stegdv s o l u t i o n The problem f o r

P1 i s the forced problem given by

w i t h boundary conditions

and

S t r a t i f i e d spin-down

221

The s o l u t i o n t o t h i s problem i s somewhat messy.

Particular

care must be taken i n applying the boundary conditions i n the corner regions near

r = a

and

or

z = 0

1.

This is done

by matching the flux i n t o and out of the corner Elcman l a y e r s . The s o l u t i o n may be obtained i n terms of a Fourier-Bessel s e r i e s with

z-dependence of the form

problem 15.41.

When the parameter

cosh Y n(z -1/2).

N 2o

[See

i s small, which i s

the case f o r values of the s o l a r parameters, t h i s s o l u t i o n can be expanded i n powers of

(D

+ z

2

N o

t o give

8J,( anr ) Ja Jb( anr ) r T ( r ) d r 2 5 2

n=1

a anJo(ana)

where

Jb(ana) = 0 .

(15.32)

(Vl depends on the i n i t i a l c o n d i t i o n s ) . The

0(1)

steady v e l o c i t y and temperature can be computed

from (15.7) and (15.9). of

Clearly the v e l o c i t y i s independent

z , and t h e temperature is zero.

The v e l o c i t y

Vo

of the r i g i d r o t a t i o n t h a t grows l i n e a r l y

with time i s determined by imposing t h e condition of zero integrated s t r e s s on the side-walls,

r=a

Decaying modes

222

This condition i s automatically s a t i s f i e d by t u t i n g t h e expression f o r

Po.

On s u b s t i -

P1 given by (15.32) we obtain

T h i s i s exactly the value of

Vo

required t o balance the

torque due t o the applied surface s t r e s s with the r a t e of change of angular momentum of the f l u i d . Decay inn Modeg

2h.e

The problem f o r the decaying modes

P

i s given by

with boundary c ond i t ions pzz = O

+(% pz) N o N

+

+ No

z v2pz

+

=

091, (15.35)

-1-hrAp r ar ar z

and

r = a.

(15.36)

S t r a t i f i e d spin-down

22 3

An energy i n t e g r a l can be constructed i n t h e u s u a l fashion by multiplying (15.34) by the complex conjugate of g r a t i n g over the c y l i n d e r .

r

and i n t e -

I n t e g r a t i n g by p a r t s and using the

boundary conditions (15.35) and (15.56) gives

From t h i s we can conclude t h a t l e a s t i n the case where

X

b = 0, X

must be r e a l and t h a t , a t must be p o s i t i v e .

Examina-

t i o n s of the e x p l i c i t s o l u t i o n s t o t h e forced problem where

b

0

shows t h a t t h e eigenvalues a r e i n f a c t p o s i t i v e i n t h i s

case too.

Thus t h e model s o l u t i o n s c o n s i s t of purely decaying

It i s i n t e r e s t i n g t o note t h a t the o s c i l l a t o r y solu-

modes.

t i o n s a r e excluded from our formulation of the problem because we a r e seeking t h e s o l u t i o n on the long-time s c a l e with the dominant balance of terms given by geostrophic balance (15.7) and h y d r o s t a t i c balance (15.9). It can be shown ( s e e Friedlander [20]) t h a t t h e r e e x i s t t h r e e s e t s of s o l u t i o n s t o equation (15.34) with boundary conditions (15.35) and (15.36).

The f i r s t a r e barotropic

modes excited by a n i n i t i a l non-zero wall v e l o c i t y . The 2 elgenvalues a r e O ( N 0 ) and hence t h e modes decay on a time scale

O(E’I).

dent of erature.

r

The second modes have eigenfunctions indepen-

and a r e excited by an i n i t i a l non-zero wall tempThe eigenvalues a r e

O(N2)

and hence t h e modes

Eigenvalue problem

224

decay on a time-scale scale.

which i s the thermal diff’usive

O(0E-l)

The major i n t e r e s t however l i e s i n the modes t h a t e x i s t

when t h e r e i s no i n i t i a l side-wall temperature or v e l o c i t y . The f i r s t two c l a s s e s of modes a r e p e c u l i a r t o the p a r t i c u l a r geometry of t h e cylinder i n which t h e r e e x i s t s i d e walls and a r e not relevant t o the s o l a r spin-down problem although they could have importance i n o t h e r problems. The s o l u t i o n f o r the t h i r d s e t of modes, which a r e the f r e e modes, excited by the stress boundary condition a r e obtained by considering the homogeneous problem w i t h

b = 0.

The s o l u t i o n i s given a s a Fourier-Bessel s e r i e s

P

Q)

=

r An(z)Jo(anr) where n=1

and A,(z)

= 0

a(xn)(z

-

1/2)

F2 cash b(&,)(z

-

1/2

G1 sinh a ( x n ) ( z

-

lh)+ G2 cosh b ( h n ) ( z

-

1/2

= F1 cash

+

Jb(ana)

The c o e f f i c i e n t s

a(An)

and

+

b(Xn)

a r e the non-conjugate

roots of the q u a r t i c equation

The eigenvalues

An

a r e determined by applying t h e boundary

conditions (15.35).

T h i s procedure gives two transcendental

equations f o r

a ( x n ) and

b($)

each of which has an i n f i -

n i t e damily of SOlUtiOnS corresponding t o the odd and even modes excited by t h e surface s t r e s s .

These eigenvalues a r e

S t r a t i f i e d spin-down N 2CJ

i n terms of the parameter

O(1)

225

( s e e equation (15.38)

and hence t h e modes t h a t they represent decay on a time-scale of

O(N2uEe1).

[Recall

t = N2uE-’t’

3.

I n t h e case where t h e r e i s no i n i t i a l side-wall temperat u r e o r v e l o c i t y , the time-dependent modes decay on a time2 s c a l e of O ( N ~JE-’) leaving the r e s i d u a l quasi-steady flow given by a v = E t ’ *

a

where

OD

X n=l

16

0

Jb(anr)m(r)dr 3

a anJo(ana)

1

+IhE z ar

( 15.39 1

i s given by (15.32).

P1

We note t h a t t h e term t h a t grows l i n e a r l y w i t h time does not become of elapsed.

0 ( 1 ) u n t i l t h e viscous time-scale of

has

I n the case of s o l a r parameters

Our r e s u l t s show t h a t on the time-scale between

and

O(E-’)

O(E’l)

c i t y given by

O(N‘UE‘~)

t h e flow i s approximately steady with the velo-

ar

.

This steady flow i s not r i g i d rota-

t i o n , but t h e r e is no sharp d i s c o n t i n u i t y i n v e l o c i t y between t h e core and the s h e l l a s would be required i n Dicke’s hypothesis.

We show t h a t i f an i n i t i a l d i f f e r e n t i a l r o t a t i o n be-

tween the core and the outer s h e l l e x i s t e d , i t would be which i n the case smoothed out on a time-scale of O ( N 2 0 E - l ) of s o l a r parameters i s o(109 ) years. AS we previously remarked, t h i s time-scale is c l o s e t o t h e l i f e - t i m e of t h e sunl however i t i s very l i k e l y t h a t

lo9

years i s i n f a c t an

upper bound f o r t h e s o l a r spin-down time-scale.

F u r t h e r comments

226 &&her

Comments

We have i l l u s t r a t e d t h e method by which a n e x p l i c i t solut i o n can be obtained t o a problem of geophysical ( o r perhaps i n t h i s c a s e we should say a s t r o p h y s i c a l ) o r i g i n s u s i n g mathem a t i c a l techniques f o r s t u d y i n g r o t a t i n g s t r a t i f i e d f l u i d s . We mentioned b r i e f l y t h a t numerical methods were used t o produce a computer s i m u l a t i o n 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 :

a

very good account of t h i s work i s given i n t h e papers of

Williams, [TO, 71, 72, 731.

The whole f i e l d of numerical

modeling i s extremely important i n t h e s t u d y of geophysical f l u i d dynamics where t h e mathematical d e s c r i p t i o n of a problem i s f r e q u e n t l y f a r t o o complicated t o y i e l d t o nonnumerical t o o l s .

I n f a c t , von Neumann brought Charney t o t h e

I n s t i t u t e f o r Advanced Study a t P r i n c e t o n i n t h e n i n e t e e n f o r t i e s ' t o develop t h e a p p r o p r i a t e mathematical models f o r t h e g e n e r a l c i r c u l a t i o n of t h e atmosphere.

The development

of numerical methods f o r t h e s o l u t i o n of t h e s e models provided one impetus f o r t h e c o n s t r u c t i o n of t h e f o r e r u n n e r t o t h e modern powerful computer.

A seminal paper concerning t h e s e

techniques i s t h a t of Charney and E l i a s s e n [ l l ] .

It i s c l e a r

t h a t even a f t e r many s i m p l i f i c a t i o n s of t h e u n d e r l y i n g physics, t h e mathematical systems d e s c r i b i n g t h e behavior of t h e ocean o r atmosphere a r e of g r e a t complexity.

I n many

cases it i s e s s e n t i a l t o u s e a computer t o o b t a i n even p a r t i a l results.

However, we should observe t h a t a n a n a l y t i c a l t r e a t -

ment of t h e fundamental p r o p e r t i e s of t h e mathematics i s o f t e n of g r e a t value i n s e t t i n g up t h e computer model and i n t h e i n t e l l i g e n t i n t e r p r e t a t i o n of t h e numerical r e s u l t s .

A fuller

S t r a t ifi e d spin-down

227

d i s c u s s i o n of numerical techniques i s , however, beyond t h e scope of t h i s booK. We should a l s o a t l e a s t b r i e f l y mention a t h i r d approach t o t h e problems of geophysical f l u i d dynamics, namely t h a t of t h e experimentalist.

Much of t h e e a r l y worR i n t h e s u b j e c t i s

e n t i r e l y experimental and p h y s i c a l o b s e r v a t i o n s provide a b a s i s on which a l l mathematical work i s b u i l t .

EKman was

himself a n oceanographer whose o b s e r v a t i o n s of t h e b e h a v i o r of t h e s e a El”], provided t h e impetus f o r much of t h e l a t e r mathematical s t u d i e s .

Experimental worK can, of c o u r s e , be

d i r e c t l y concerned with o b s e r v a t i o n and measurements of t h e ocean o r atmosphere, but i t can a l s o i n v o l v e l a b o r a t o r y models of s p e c i f i c p h y s i c a l phenomena.

T h i s work c a n l e a d t o a n

u n d e r s t a n d i n g of problems t h a t would o t h e r w i s e i n t r a c t a b l e , suggest i n t e r e s t i n g d i r e c t i o n s f o r f u t u r e r e s e a r c h , and also provide u s e f u l v e r i f i c a t i o n of t h e o r e t i c a l worK.

A good exam-

p l e of t h e l a t t e r a s p e c t is g i v e n i n t h e r e c e n t work of Linden

[42], concerning flow of a s t r a t i f i e d f l u i d i n a r o t a t i n g annulus.

Problems

228

t e r 15 ProblemE Derive t h e EKman l a y e r s u c t i o n condition appropriate on the h o r i z o n t a l surface

z = 0

for a rotating stratified fluid

described by equations

(15.1)

I.

(15.3)

w i t h boundary condition (15.4).

Discuss the behavior of the side-wall

r = a

boundary l a y e r a t t o equations (15.1)

-t

appropriate

(15.3)

with

boundary condition (15.5). that

N ‘20 =

O(1)

boundary l a y e r .

I n the case analyze the O ( E 1/2 )

Hence derive the r e l a t i o n -

ship (15.13) f o r the i n t e r i o r flow a t

r

Is t h i s r e l a t i o n s h i p v a l i d f o r small

N2u?

=

15.3)a) Compute the value of the constant angular velocity

Vo

t h a t occurs i n the portion

of t h e s o l u t i o n t h a t grows l i n e a r l y with time, by equating t h e torque due t o the applied s t r e s s with the r a t e of change of angular momentum of the f l u i d

.

b ) Show t h a t t h e r e can be no c o n t r i b u t i o n t o

the flow driven by a steady s t r e s s t h a t grows a s

tn, n 2 2 .

a,

Problems 15.4)

a)

229

Prove t h a t the s o l u t i o n

t o equation

P1

(15.29) w i t h boundary conditions (15.30) and (15.31) i s given by p1 =

1

(y- - -) a

0

0 4

+

8N2g

+

A.

+

BO(z-1/2)

co 4Jo(anr) cosh an(z-1/2) +

2 n = l a 2Jo(ana)

Ja

X

ta&

where

>-

cosh

9

- N2 u a2

2

1/2

cosh

an(z-1/2)

( N2u ) 12 '%

Jb( a n r ) r t ( r ) d r

8

( ~~u 'l2an

tanh

Jb(ana) b)

a:

cosh

4

= 0.

Show t h a t t h i s reduces t o (15.32) when

N 20

i s small.

Explain why the

0(1)

temperature i s a constant i n t h i s case. c)

15.5 1 a)

what is the behavior of t h e v e l o c i t y and temperature when

N

Show t h a t the eigenvalue

A

2

Q

i s large?

f o r the

t ime-depend ent modes s a t i s f y the transcendent a 1 e qua t ions

Problems

or

where

a ( k ) and

b(k)

are the nOn-ConJUgate

roots of ( 1 5 . 3 8 ) . b)

In the case when

N

2

Q

<<

1

approximate expression f o r b(X)

from ( 1 5 . 3 8 ) .

obtain an a(k)

and

Hence obtain an

upper bound and a lower bound for the smallest eigenvalue.

Obtain an asymptotic

expression f o r the behavior of large eigenvalues.

CHAPTER 16 BAROC L I N I C INSTABILITY

The i n v e s t i g a t i o n of t h e s t a b i l i t y of a flow i s an importa n t aspect of a l l f l u i d problems:

i n h i s work on hydrodynamic

s t a b i l i t y Chandrasekhar [ 91, provides a n e x c e l l e n t r e f e r e n c e f o r t h i s subject.

We w i l l now d i s c u s s one p a r t i c u l a r , but very

i n t e r e s t i n g , form of i n s t a b i l i t y t h a t can occur i n a r o t a t i n g s t r a t i f i e d f h i d , namely baroc l i n i c i n s t a b i l i t y

.

Inves t i g a -

t i o n of t h i s f i e l d has led t o important c o n t r i b u t i o n s t o our knowledge of t h e behavior of t h e atmosphere.

The mechanism of

b a r o c l i n i c i n s t a b i l i t y merits d i s c u s s i o n both because of i t s s i g n i f i c a n t geophysical a p p l i c a t i o n s and because of i t s r a t h e r curious dynamic and mathematical p r o p e r t i e s .

The o r i g i n a l

treatment of t h e s u b j e c t was given by s e v e r a l people, including Eady [ 151, Charney [ lo], S u t c l i f f e [ 6 4 ] and F j o r t o f t [ 181. We w i l l follow t h e mathematical a n a l y s i s of Eady. I n Chapters 12 and 1 5 we described waves supported by 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 c l u d i n g Rossby waves which a r e observable i n t h e atmosphere and propagate towards t h e west. We w i l l show t h a t t h e atmosphere, which is heated by t h e sun more s t r o n g l y a t t h e equator than t h e poles, i s a system t h a t i s inherently unstable.

The mechanism of b a r o c l i n i c i n s t a -

b i l i t y gives a method whereby a small p e r t u r b a t i o n of t h e b a s i c s t e a d y flow can generate l a r g e s c a l e waves mown i n t h e atmos-

The Eady model

2 32

Three ingredients a r e necessary and s u f f i -

phere as cyclones.

c i e n t f o r b a r o c l i n i c i n s t a b i l i t y , namely, r o t a t i o n , s t r a t i f i c a t i o n and a horizontal temperature gradient

5 .

ay A simple form of i n s t a b i l i t y i n a s t r a t i f i e d f l u i d i s the

obvious one t h a t occurs when heavier p a r t i c l e s l i e above

For example, consider heating a d i s h of

lighter particles. water from below:

the hot l i g h t e r f l u i d a t t h e bottom r i s e s

and t h e cold heavier r l u i d f a l l s , giving the c e l l u l a r flow /

p a t t e r n known a s Benard convection.

However, t h i s I s not the

underlying p r i n c i p l e of b a r o c l i n i c i n s t a b i l i t y s i n c e i n the problem t h a t we w i l l now consider, the i n i t i a l s t a t e i s one

i n which heavier f l u i d always l i e s below l i g h t e r f l u i d .

For

b a r o c l i n i c i n s t a b i l i t y , the a v a i l a b l e p o t e n t i a l energy of the basic flow comes from the mriz o n t a temperature gradient aTWe w i l l now show t h a t f o r the unstable modes t h i s energy 3Y can be converted I n t o waves.

.

We w i l l describe the simplest model t h a t e x h i b i t s the fundamental features of b a r o c l i n i c i n s t a b i l i t y . box with r i g i d walls a t

z = 0, z = 1 and

allow the f l u i d t o be i n f i n i t e i n the

We consider a y = 0, y = 1, and

x-direction.

We model

the s o l a r heating t h a t i s weaker a t the poles than a t t h e equator by considering a basic s t a t e i n which the North-South temperature gradient

i3T0 is 3Y

a constant

C.

The b a s i c steady

flow i s i n geostrophic and hydrostatic balance, hence the steady, inviscid reduction of (10.7) gives

Baroclinic i n s t a b i l i t y

233

(16 .l) and T

o

3 . az

=

(16.2)

Because t h e v a r i a t i o n of C o r i o l i s force w i t h l a t i t u d e is not fundamental t o t h e mechanism of b a r o c l i n i c i n s t a b i l i t y , i n t h i s simple model we w i l l t r e a t

f

a s constant.

From (16.1)

and (16.2) we o b t a i n

thus

uo

=

5 , by the n a t u r e of t h e basic steady s t a t e ,

=

’f

z.

(16.3)

We use t h e standard techniques t o i n v e s t i g a t e t h e s t a b i l i t y of the flow by considering a small p e r t u r b a t i o n about t h e

steady s t a t e .

The l i n e a r i z a t i o n follows the l i n e s of t h a t

given i n Chapter 10, but w i t h t h e modification t h a t t h e basic s t a t e is no longer a t r e s t .

We w r i t e

Potential v o r t i c i t y

234

9

=

s z i + €3'

+ ET'

T = Cy

P = czy

+

(16.4)

EP'.

The p o t e n t i a l v o r t i c i t y f o r an inviscid r o t a t i n g s t r a t i f i e d f l u i d i s derived i n Chapter 11. The appropriate expression f o r the p o t e n t i a l v o r t i c i t y i n t h i s present case i s given by

(11.37).

Hence t h e equation representing conservation of

potential vorticity i s

+

9-[v2P dt

4 9 1 N az 2

2

= 0.

We l i n e a r i z e t h i s equation by s u b s t i t u t i n g (16.4) and r e t a i n ing terms of

O(E).

This procedure gives the equation

[A + 'f

+4 N 2

2

&][v;PI

2

-a1z

The boundary conditions a r e obtained i n terms of the following arguments.

(16.6)

= 0.

P'

by

We have assumed t h a t the upper and

lower boundaries a r e r i g i d , thus

w'=o

at

z = O ,

2 3 1 .

(16.7)

I n f a c t , i t i s possible t o extend the a n a l y s i s without too much d i f f i c u l t y t o include t h e cases of f r e e surfaces and nonh o r i z o n t a l boundaries.

This i s summarized by Hide [31], i n

h i s chapter concerning planetary atmospheres.

Linearization

of the inviscid temperature equation (10.8) about t h e steady s t a t e gives

Baroclinic i n s t a b i l i t y c + ~ ~ + v ' c + N2 w =' O . at f ax

2 35

(16.8)

I n keeping with t h e d e r i v a t i o n i n Chapter 1 2 of the p o t e n t i a l v o r t i c i t y equation, we assume t h a t the

O(E)

perturbed flow

i s quasi-geostrophic, hence

and T' =

=. az

(16.9)

Thus the boundary condition (16.7) combined with (16.8) and (16.9) gives (16.10)

and ( 16.11)

Again f o r s i m p l i c i t y we assume t h a t t h e r e a r e r i g i d sidewalls a t

y = O,L,

thus v' = 0 a t

y = O,L,

which implies

AIL ax

=

o at

y = 0,1.

(16.12)

O f course t h i s boundary condition i s u n r e a l i s t i c i n t h e atmos-

phere where r i g i d l a t i t u d i n a l walls do not e x i s t , but we wish

'L 36

The s t a b i l i t y c r i t e r i o n

t o i l l u s t r a t e t h e fundamental mechanism of b a r o c l i n i c i n s t a b i l i t y without obscuring t h e mathematics by incorporating

o t h e r physical phenomena. T h e S t a b i 1 i t . v C r i t e r i 0g

Following t h e standard methods of s t a b i l i t y analysis we

seek a s o l u t i o n t o equation (16.6) with boundary conditions (16.10), (16.11) and (16.12) of the form

When t h e eigenvalue A like:

i s r e a l , the motion i s purely wave-

however, complex values of

can lead t o exponential

h

growth w i t h time, and we c a l l such modes unstable.

Substitu-

t i o n of (16.13) i n t o equation (16.6) gives i[-k

In general, A

+

k][-(k2

.+ y , thus

2 2 ) F L

F

s a t i s f i e s an equation of the

9-

I orm

a : F

az

+

f

Be

-a z

n

=

0.

(16.14)

= 0,

implying F = Man'

+

2

y2 ]

+9

,

where an

and

A

and

B

=

$ f

(k2

+ =) L2

a r e constants t o be determined from the bound-

Baroclinic i n s t a b i l i t y a r y conditions (16 .lo) and (16.11).

'2 37

Applying t h e s e boundary

conditions t o the expression (16.15) f o r

gives t h e p a i r

F(z)

of equations

A[F + Xun] + B[y - &an]

= 0

T h i s system has a n o n t r i v i a l s o l u t i o n f o r

only i f

A

[(I

A

and

B

i f and

s a t i s f i e s the eigenvalue equation

+

+

ad(?)

A

ck 2 y an -

2 2 'an A an].

(16.16)

-[(I Hence

&

- aJ(y) + x 2

2

an

2 a - A 2 an]e

= 0.

is given by t h e quadratic equation

A2-A$?+(F)

2

which has two roots

A = @[l

1/2 2 2 a, (an - 4an coth an + 4) 1.

(16.17)

Clearly the system is s t a b l e and purely wave-like when 2

an

-

4an coth an

+

4 2 0,

and represents an unstable wave whose amplitude grows exponen-

C r i t i c a l value of

2 38

a

t i a l l y when a:

- 4an coth an + 4 < 0.

T h i s expression defining the s t a b i l i t y c r i t e r i o n can be re-

w r i t t e n using the r e l a t i o n 2 an

-

4an coth an

+

4 = (an

a 2 tanh $)(an

-

- 2 coth 2) .

Because the f i r s t braclcet i s always p o s i t i v e the system i s s t a b l e providing

and could be unstable when 2

Thus the c r i t i c a l value of a 2

<

coth

an

=

3. ac

is

coth

where

2.

From t h i s i t follows t h a t ac er 2.399.

W e have therefore shown t h a t unstable disturbances a r i s e when

Baroclinic i n s t a b i l i t y

2 39

Hence t h e r e e x i s t s a t l e a s t one unstable mode

( k = 0,

n

= 1)

provided t h a t

If

1. nL < t h e parameters a r e such t h a t 1 .

( 16.18)

i s g r e a t e r than t h i s

c r i t i c a l value, a l l i n f i n i t e s i o n a l perturbations of the basic state are stable. Let us examine i n a l i t t l e more d e t a i l the p r o p e r t i e s of

<

an unstable mode t h a t e x i s t s when

ac

.

From expression

(16.17) the phase speed of an unstable mode has the form

where

g(an)

i s a r e a l valued function of

s t a b l e disturbances d r i f t w i t h v e l o c i t y v e l o c i t y of the b a s i c flow.

an.

Thus the un-

which i s t h e mean

The p e r t u r b a t i o n pressure f i e l d

is given a s

which represents an unstable wave propagating i n t h e d i r e c t i o n of the mean flow.

The constant

c

gives the s t r e n g t h

of the mean shear, thus both t h e magnitude of the d r i f t v e l o c i t y and the growth r a t e of the amplitude of t h e unstable mode i n c r e a s e with the mean shear

%. ay

I n f a c t , t h i s hori-

z o n t a l temperature gradient provides t h e source of energy f o r the growth of the i n s t a b i l i t y .

A d e s c r i p t i o n of exactly how,

and i n what regions, the a v a i l a b l e p o t e n t i a l energy is re-

Rossby radius of defomration

2 40

leased from the mean shear and converted i n t o perturbation k i n e t i c energy i s given i n the a r t i c l e of Pedlosky [sl]. We now remarK t h a t the s t a b i l i t y c r i t e r i o n (16.18) can be r e w r i t t e n i n terms of the Rossby radius of deformation.

We

r e c a l l from Chapter 11 t h a t for a given s t r e n g t h of a t r a t i f i cation N

and depth

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

L*(= h L )

exceeds t h e value

=

h.

The c r i t e r i o n (16.18)

f o r b a r o c l i n i c i n s t a b i l i t y requires

which can be r e w r i t t e n i n the form

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

%,

L*

exceeds

the conditions a r e

favorable f o r the growth of a n unstable wave, and such Rossby wave8 w i l l be influenced by t h e presence of v e r t i c a l s t r a t i fication.

Our a n a l y s i s of the generation of waves from a

wave-free basic flow by meana of b a r o c l i n i c I n s t a b i l i t y i s therefore consistent w i t h t h e f a c t t h a t modes whose horizont a l length s c a l e s a r e of the order of the Rossby radius of deformation

(1)

1000km)

a r e exactly those which occur most

frequently i n the atmosphere. We w i l l make a f u r t h e r observation about the s t a b i l i t y c r i t e r i o n (16.18).

We r e c a l l t h a t

N/f

i s the r a t i o of the

buoyancy frequency, which measures the s t r e n g t h of the s t a b l e

Baroclinic i n s t a b i l i t y

24 1

g r a v i t a t i o n a l s t r a t i f i c a t i o n , t o t h e angular v e l o c i t y of rotation.

Thus the e f f e c t of a n increase i n the v e r t i c a l

s t r a t i f i c a t i o n is t o s t a b i l i z e the flow.

We r e c a l l , however,

t h a t t h e presence of the h o r i z o n t a l s t r a t i f i c a t i o n is e s s e n t i a l f o r t h e existence of b a r o c l i n i c i n s t a b i l i t y .

The

slope of the surfaces of equal d e n s i t y is given by t h e expression

The i n s t a b i l i t y can e x i s t only i f a t y p i c a l p a r t i c l e t r a j e c t o r y s a t i s f i e s two conditions: (i)

it has slope

e g r e a t e r than zero,

so t h a t

t h e r e e x i s t s a v a i l a b l e p o t e n t i a l energy t o feed t h e i n s t a b i l i t y : (ii) t h e slope

0

is l e s s than

(- %,% ay az )

t h a t t h e s t a b i l i z i n g e f f e c t of t h e v e r t i c a l d e n s i t y gradient can be overcome. i.e.,

f o r s t a b i l i t y we must require

so

4 1

-cc

/

/

l i n e s of constant density temperature trajectory 1 (unstable) trajectory 2 ( s t a b l e )

FIGURE 28

Baroclinic i n s t a b i l i t y Trajectory 1 is unstable s a t i s f y i n g both conditions. associated h o r i z o n t a l length s c a l e Jl f

L1

243

The

i s such t h a t

= L, < 2.399.

Trajectory 2 is s t a b l e and t h e h o r i z o n t a l length s c a l e

L2

is such t h a t Jl f

= > L2

2.399.

We have discussed a simple model t h a t e x h i b i t s t h e main f e a t u r e s of b a r o c l i n i c i n s t a b i l i t y .

m c h work on t h i s s u b j e c t

has been stimulated by a d e s i r e t o understand t h e l a r g e s c a l e motions i n the atmosphere and we have seen how even a simple model can give valuable information.

However, the atmosphere

is a system t h a t i s physically much more complicated than t h e one we described.

It is t h e r e f o r e necessary t o consider ex-

tensions of the simple Eady treatment of b a r o c l i n i c i n s t a b i l i t y and laboratory experiments a r e a n e s s e n t i a l f e a t u r e of t h i s work.

The basic experimental apparatus c o n s i s t s of a

uniformly r o t a t i n g f l u i d contained i n a c y l i n d r i c a l annulus. A h o r i z o n t a l temperature gradient is produced by impOSing a

temperature d i f f e r e n c e between the inner and the o u t e r s i d e walls, with t h e inner wall representing p o l a r l a t i t u d e s and being colder than the o u t e r wall which corresponds t o equat o r i a l l a t i t u d e s [ s e e Figure 291.

The f i r s t experiments of

t h i s type were c a r r i e d out by Hide [ 2 8 ] and F u l t z [ 2 3 ] .

2 44

Laboratory models

I n the experiments It was u s u a l t o vary e i t h e r the angular velocity

n

o r t h e temperature gradient

found t h a t increasing

n

To

(Or

s i o n of d i f f e r e n t s t a t e s .

- T1)

To

-

T1.

It was

lead t o a progres-

For low angular v e l o c i t y , the flow

was a steady s p i r a l ; as i t increased t o a c r i t i c a l value ( s e e expression (16.18)) , t h e r e was onset of i n s t a b i l i t y and the b a r o c l l n i c mode of wave number 2 appeared.

in n

Further increases

gave r i s e t o progressively higher wave numbers with

s h o r t e r horizontal length s c a l e s , u n t i l a t l a r g e enough values the steady p a t t e r n began t o waver o r v a c i l l a t e .

A further

increase i n

then Produced i r r e g u l a r p a t t e r n s with turbu-

l e n t eddies.

This progression is c l e a r l y consistent with the

s t a b i l i t y c r i t e r i o n t h a t b a r o c l i n i c i n s t a b i l i t y can only occur when

The experiments not only give r e s u l t s t h a t agree with t h e theory of baroclinic I n s t a b i l i t y derived from a simple mathematical mode, but they can a l s o be used t o i n v e s t i g a t e nonl i n e a r e f f e c t s which a r e p a r t i c u l a r l y important i n the v a c l l l a t i n g and i r r e g u l a r flows.

A f u r t h e r relevant f e a t u r e

neglected i n the simple model i s v i s c o s i t y .

Barcilon [ 2 ] and

Hide [ 3 0 ] consider the r o l e of boundary l a y e r s in the s t a b i l i t y problem.

Some progress has been made using experimental

techniques i n the understanding of the way the s t a b i l i t y curve is modified by v i s c o s i t y .

A systematic study of the

e f f e c t of v i s c o s i t y on the t r a n s i t i o n from axisymmetric t o

Baroclinic i n s t a b i l i t y

Laboratory model for l a t i t u d i n a l d i f f e r e n t i a l s o l a r heating of t h e atmosphere. FIGURE 29

2 46

Laboratory experiments

non-axisymmetrlc flow can be found i n the paper by Fowlis and Hide [ l g ] .

T h i s work has been continued by Hide and Mason

[ 3 2 ] , who give p a r t i c u l a r a t t e n t i o n t o t h e h y s t e r e s i s pheno-

menon which i s the consequence of t h e e f f e c t s of v i s c o s i t y and p o t e n t i a l v o r t i c i t y gradients on b a r o c l i n i c waves. A summary of laboratory experiments concerning b a r o c l i n i c

I n s t a b i l i t y and i t s a p p l i c a t i o n t o the theory of the general c i r c u l a t i o n of the atmosphere i s given i n the a r t i c l e by Hide [30]. T h i s i s a l s o a most u s e f u l source of references f o r

the seminal work on the s u b j e c t a s a whole.

Problems

247

t e r 16 Problem 16.1)

Consider the e f f e c t of introducing a small v i s c o s i t y i n t o the problem of baroclinic instability.

a)

Assume t h e r e e x i s t s an EKman l a y e r a t t h e upper and lower surfaces are rigid.)

z

= 0,l.

(Which

Obtain the appropriate m o d i f i -

c a t i o n of the boundary conditions (16.10) and (16.11). b)

Obtain a s o l u t i o n of t h e form (16.13) t o equation (16.6) w i t h t h e modified boundary conditions i n ( a ) .

c)

How does the presence of a non-zero Ekman number a f f e c t the s t a b i l i t y c r i t e r i o n .

[This problem i s considered i n d e t a i l by Barcilon [2]. 16.2)

We have given two c r i t e r i a f o r t h e existence of b a r o c l i n i c i n s t a b i l i t y , namely (16 .l8)

and (16.19).

Following t h e o u t l i n e given

below, show t h a t these two c r i t e r i a a r e consistent. a)

Write t h e v e l o c i t y i n terms of t h e d e n s i t y gradient a s

b)

uo = 1.g%. f i aY

z,

Recall t h e dimensional buoyancy frequency N2 =

-

2 aP

(chapter 1 0 ) .

Problems c)

Define the slope

where

d)

of a t r a j e c t o r y as

a r e the root mean square

( W',V')

averages of

8

(w' ,v' )

.

Write t h e p o t e n t i a l v o r t i c i t y equation (16.6) i n terms of the v e l o c i t y components (u' ,v' ,w' )

e)

.

Take the average of the dominant terms i n t h i s equation and hence show t h a t the s t a b i l i t y c r i t e r i a (16.18) and (16.19) a r e equivalent.

APPENDIX BOUNDARY LAYER METHODS

Throughout t h e discussion of geophysical f l u i d dynamics we have made use of boundary l a y e r techniques t o approximate t h e behavior of a viscous f l u i d .

Such methods a r e c h a r a c t e r i s t i c

of the t o o l s used t o study c e r t a i n s i n g u l a r p e r t u r b a t i o n problems i n f l u i d mechanics.

For t h e reader who i s not

f a m i l i a r with these methods we w i l l give a s h o r t i n t r o d u c t i o n t o the t o p i c of boundary l a y e r s .

A much more extensive t r e a t -

ment can be found i n the books by M. S c h l i c h t i n g [60] and M. VanDyKe [671.

The o r i g i n a l idea, t h a t a f l u i d of small v i s c o s i t y could be approximated by a n i n v i s c i d f l u i d i n almost every s p a t i a l region except f o r narrow boundary l a y e r s , was f i r s t presented by Prandtl.

T h i s treatment has been given the name, "the

method of inner and outer expansions'', o r i n more recent l i t e r a t u r e , "the method of matched asymptotic expansions.

I'

Although these methods have not yet been shown t o be completely rigorous i n a l l s i t u a t i o n s , boundary l a y e r techniques have proved t o be extremely important i n the a n a l y s i s of t h e Navier-Stokes equations.

I n general, i t may be s t a t e d ,

boundary l a y e r theory i s asymptotically c o r r e c t .

2 49

Singular perturbations

250

l a r Perturbation Problem& We w i l l f i r s t discuss what we mean by a s i n g u l a r perturbat i o n problem.

I n many physical problems the d e s c r i p t i v e

d i f f e r e n t i a l equation, a f t e r i t has been made non-dimensional, includes small ( o r l a r g e ) dimensionless parameters.

Our pre-

s e n t i n t e r e s t l i e s i n the Navier-Stokes equations f o r the motion of an incompressible f l u i d .

I n a n i n e r t i a l co-ordinate

system we r e c a l l t h a t t h e equations can be w r i t t e n i n nondimensional form a s

The two dimensionless parameters a r e

R

and

E.

i s called

R

the Reynolds number and the r e c i p r o c a l of the Reynolds number measures the r e l a t i v e s t r e n g t h of the viscous f o r c e s . parameter

E

measure the r e l a t i v e s t r e n g t h of the non-linear

advective term.

[ I n a r o t a t i n g co-ordinate system

replaced by the EKman number number]. l/R

and

The

E

and

E

l/R

is

i s c a l l e d t h e Rossby

I n many important physical problems both parameters E

a r e small.

I n a general d i f f e r e n t i a l equation a small parameter is referred t o a s the perturbation quantity o r perturbation parameter.

An asymptotic expansion f o r t h e s o l u t i o n t o the

d i f f e r e n t i a l equation i s a formal power s e r i e s i n t h e perturb a t i o n parameter.

This approximate s o l u t i o n becomes more and

more accurate as t h e perturbation quantity tends t o zero. The power s e r i e s is not necessarily convergent, however i n

Boundary l a y e r methods

25 1

many problems t h e f i r s t couple of terms give a good approximation t o t h e exact s o l u t i o n .

I f t h e truncated s e r i e s gives

a good approximation ( f o r a reasonably small p e r t u r b a t i o n parameter) t h a t is v a l i d wverywhere i n t h e domain of d e f i n i -

-

t i o n of t h e equation, then the problem i s c a l l e d a r e a l = Perturbation uoblem.

I f , however, the truncated series is

not v a l i d everywhere i n t h e domain, and f a i l s t o give a

reasonable approximation t o the s o l u t i o n is some s p a t i a l region, then t h e problem is c a l l e d a g i m l a x problem.

This is exactly what occurs when we seek a s o l u t i o n

t o the Navier-Stokes equations (A.l) and (A.2) i n terms of a n asymptotic expansion i n powers of

(or E).

l/R

I n t h i s case

the p e r t u r b a t i o n of the b a s i c i n v i s c i d flow ( i . e . , t i o n with

t h e solu-

l / R = 0) is v a l i d away from the boundaries but

f a i l s near a bounding surface.

It is t h e r e f o r e necessary t o

supplement t h e f i r s t expansion with a second asymptotic expansion which is v a l i d i n the region c l o s e t o t h e boundary. This region is c a l l e d a

boundar.v ;Laver.

The two asymptotic

expansions, sometimes r e f e r r e d t o as t h e o u t e r and t h e i n n e r s o l u t i o n s , must match together i n t h e region where t h e boundary l a y e r meets t h e i n t e r i o r :

hence the name, t h e method of

matched asymptotic expansions. It is possible t o appreciate t h e n e c e s s i t y f o r a viscous

boundary l a y e r by i n t e r p r e t i n g t h e terms i n equation (A.l) i n l i g h t of the physical forces t h a t they represent. (l/R)

v

2

3

which describes the viscous force is t h e most highly

d i f f e r e n t i a t e d term i n t h e equation.

so

The term

The b a s i c i n v i s c i d flow

s a t i s f i e s t h e reduced equation obtained by

1/R = 0.

A simple example

252

Because the reduced equation is of lower order than equation (A.l),

only a normal boundary condition can be imposed on the

-%.

6 olu t i o n

f i r s t term i s

Thus, i n general, an asymptotic expansion whoee

90

w i l l not s a t i s f y the boundary condition on

the t a n g e n t i a l v e l o c i t y component ( e .g.

,a

no-slip condition).

In

Hence t h i s asymptotic expansion is not uniformly v a l i d .

order t o s a t i s f y both t a n g e n t i a l and normal boundary condit i o n s i t i s necessary t o consider an equation t h a t includes t h e most highly d i f f e r e n t i a t e d term, namely the viscous term

(VR) v

2

-

q.

I n our asymptotic a n a l y s i s of the motion of a

r o t a t i n g f l u i d we have shown t h a t exactly such an equation describes the motion i n a viscous boundary layer; e.g., Ekman l a y e r constructed i n Chapter 5.

the

The asymptotic solu-

t i o n t o the problem t h a t s a t i s f i e s a l l the boundary conditions i s composed of a c o n t r i b u t i o n from t h e i n t e r i o r and a c o n t r i bution from the boundary l a y e r . The Navier-Stokes equations ( A . l ) second small parameter term.

E

and ( A . 2 ) involve a

which multiples the non-linear

I n c e r t a i n problems a s o l u t i o n is sought i n terms of

an asymptotic expansion i n powers of

E.

T h i s problem 1s i n

general a singular perturbation problem and i t i s necessary t o consider a second s e r i e s i n l i n e a r boundary l a y e r .

E

t h a t i s v a l i d i n a non-

However our major i n t e r e s t i n t h i s

present book l i e s i n the a n a l y s i s of viscous boundary l a y e r s . l e Example Rather than consider the complicated system of p a r t i a l d i f f e r e n t i a l equations t h a t describes f l u i d motion, we w i l l look a t a much simpler ordinary d i f f e r e n t i a l equation which

25 3

Boundary l a y e r methods

has some of the e s s e n t i a l f e a t u r e s of a s i n g u l a r p e r t u r b a t i o n problem.

We w i l l examine the following i l l u m i n a t i n g example

of Friedrichs ; euXX+ ux = a with boundary conditions u ( 0 ) = 0, u ( 1 ) = 1. Equation ( A . 3 ) can be integrated t o give ux = a

+

ce -X/E

hence u

=

ax

The constants of i n t e g r a t i o n

+

c 1e -X/E

c1

and

+ c2

c2. a r e determined

from the boundary conditions ( A . 4 ) t o give the exact s o l u t i o n t o the problem

u = ax

+

-X/E

( l - a ) ( W ) . - 1 e

L e t us now consider the problem when

E

(A.5)

<<

1 and 6eeK a

s o l u t i o n t o the problem i n terms of an asymptotic expansion. We w r i t e u = u0 f o r some power

a

>

0

+

EQUl

+

E%2

+

....

which w i l l be determined.

(A.6) Substituting

254

Stretched co-ordinate

t h i s s e r i e s i n t o equation ( A . 3 )

and equating powers of

gives successively t h e equations s a t i s f i e d by The terms of order

E

... .

u0,u1,u2,

give

EO

uOX

= a

hence

uo

=

ax = bo.

Since the reduced d i f f e r e n t i a l equation ( A . 7 )

i s of order 1,

which i s of lower order than t h e P u l l d i f f e r e n t i a l equation

(A.3),

the solution

gration.

uo

contains only one constant of i n t e -

It i s t h e r e f o r e not possible t o s a t i s f y both bound-

a r y conditions

uo(0) = 0

and

u o ( l ) = 1. The question

a r i s e s as t o which boundary condition should be used t o determine the constant

Without f u r t h e r knowledge i t would

bo.

However, i t i s i n f a c t

appear t h a t t h e problem i s symmetric.

necessary t o s a t i s f y the condition u o ( l ) = 1. We w i l l pres e n t l y show why t h i s i s the case. When we s a t i s f y

u o ( l ) = 1 equation ( A . 7 ) becomes

uo = ax

+

1

- a.

( A -8)

We note t h a t t h i s l i n e a r expression i s a good approximation t o the exact s o l u t i o n ( A . 5 ) where the co-ordinate

x

f o r small

is

O(E).

E

except i n the region

T h i s region near

i n which t h e s o l u t i o n t o t h e problem ( A . 3 ) approximately given by ( A . 8 ) ,

and ( A . 4 )

x = 0,

i s not

is c a l l e d a boundary l a y e r .

order t o i n v e s t i g a t e more c l o s e l y what i s happening i n t h i s boundary layer we introduce a s t r e t c h e d co-ordinate writing

q

by

In

255

Boundary l a y e r methods

T h i s co-ordinate transformation s t r e t c h e s t h e i n t e r v a l i n

which

x

is

O(ea)

i n t o a n i n t e r v a l i n which

51

is

O(1).

I n t h i s p a r t i c u l a r problem we can e a s i l y determine t h e exact We have already

s o l u t i o n , namely t h e expression ( A . 5 ) .

observed t h a t the i n t e r i o r s o l u t i o n ( A . 8 )

approximates ( A . 5 )

everywhere except i n a boundary l a y e r of

O(e).

problem we expect t h e number

t o equal

a

1.

Thus i n t h i s I n many prob-

lems, however, i t is not possible t o o b t a i n t h e exact s o l u t i o n and hence deduce t h e power of thickness of t h e boundary l a y e r .

E

which gives t h e appropriate We w i l l t h e r e f o r e determine

a by a l t e r n a t i v e means ( a s an i l l u s t r a t i o n the g e n e r a l techniques )

.

The transformation of co-ordinates ( A . 9 )

changes t h e o r i -

g i n a l problem t o (A.lO)

There e x i s t t h r e e terms labeled ( I ) , (11), and (111) i n equat i o n ( A . l O ) which a r e r e s p e c t i v e l y , O( We wish t o choose

a

SO

O( E ' ~ ) ,

O(.)'E

t h a t term ( I ) , which is t h e most

highly d i f f e r e n t i a t e d term, is a dominant term i n balance with a t l e a s t one of t h e o t h e r terms. and (11) requires,

A balance between ( I )

Inner and outer expansions

256

thus Terms ( I ) and (11) a r e of

O(E*).

O( E-

1)

which dominates term

A balance between (I) and (111) requires 1

-

2a = 0 ,

In t h i s case (I) and (111) a r e

-

but term (11) i s

O(E')

and hence i s the dominant term.

O(E-'")

(111)

Thus we can only

achieve a balance of terms i n which term (I) i s retained as E

0

by choosing

a = 1. The boundary layer equation is

then (A.12)

where

x =

Q€.

We have now determined the s u i t a b l e power of asymptotic expansion.

E

f o r the

We seek an i n t e r i o r ( o r o u t e r ) expan-

s ion

uI

=

ax

+

1-a

+

E U ~

1

+

E

2 uI

2

+

...

(A.13)

and a boundary layer ( o r i n n e r ) expansion

ii

=

ii0 +

Eiil

+

E

2-

u2

We s u b s t i t u t e t h e expansion ( A . 1 4 )

... .

+

( A .14)

i n t o t h e boundary layer

equation ( A . 1 2 ) and equate powers of

E.

The

O(1)

equation

Boundary l a y e r methods

25 7

is

+

iio

rlrl

Go

n

= 0.

To o b t a i n t h e boundary c o n d i t i o n on

(A.16)

-

uo we w i l l seeK a comp o s i t e asymptotic expansion f o r t h e s o l u t i o n t o t h e o r i g i n a l

problem.

We w r i t e

a s t h e a d d i t i v e composition of

U(X,E)

the i n n e r and o u t e r expansion.

We r e q u i r e t h a t t h e sum of

the two expansions s a t i s f y t h e boundary c o n d i t i o n a t q = 0.

Hence

Go

must s a t i s f y

C, +

(1-a) =

The equation (A.16) f o r layer.

x = 0,

Go

o

at

q = 0.

(A.17)

i s only v a l i d i n t h e boundary

We w i l l t h e r e f o r e impose t h e c o n d i t i o n t h a t as

becomes l a r g e ( i . e . ,

11

t h e boundary l a y e r merges with t h e in-

t e r i o r ) , t h e boundary l a y e r c o n t r i b u t i o n t o t h e composite s o l u t i o n decays t o zero. on

ii,

Hence t h e second boundary c o n d i t i o n

is lim

rl-m

-

uo = 0 .

(A.18)

The s o l u t i o n t o equation (A .16) with boundary c o n d i t i o n s

(A.17) and (~.18)is

Go

=

-(l-a)e-q.

Thus t h e f i r s t approximation f o r small

E

of t h e composite

s o l u t i o n is U ( X , E )rr ax

+

( l - a ) ( l - e -X/E ) .

(A - 2 0 )

The composite and exact solutions

258

1

1 - a

\

0

f

O(E)

\

\

\

\ , , -

+

outer solution

X

-

-

inner solution

[UI + (-Go)

-

U(XIE)I

FIGURE 30

1

exact s o l u t i o n

Boundary l a y e r methods We note t h a t f o r small exact s o l u t i o n ( A . 5 ) .

E

259

t h i s i s a good approximation of the

Figure [ 3 0 ] i l l u s t r a t e s t h e inner and

o u t e r s o l u t i o n s and compares the composite approximation with the exact s o l u t i o n . There is an a l t e r n a t i v e convention i n which the asymptotic expansion is not w r i t t e n a s a composite expression, but r a t h e r a s an outer s o l u t i o n which is v a l i d outside the boundary l a y e r and an inner s o l u t i o n which is v a l i d i n s i d e the boundary l a y e r . The matching p r i n c i p l e is used t o ensure t h a t t h e two expansions match t o each order i n merges with t h e i n t e r i o r .

E

where t h e boundary l a y e r

The boundary l a y e r s o l u t i o n s a t i s -

f i e s the appropriate boundary condition a t

q = 0.

Written

i n t h i s convention, the uniform f i r s t approximation f o r our present problem is u(x,e)

LI

ax

+

(1-a)

as

(1-a) (1-e-v)

E-0, x

>

E-0, q =

0

’ E

fixed. fixed.

I n general we w i l l seek a n asymptotic expansion t h a t is w r i t t e n i n terms of the a d d i t i v e composition of t h e inner and outer expansions. Cole gives the following d e f i n i t i o n of a composite expansion:

Any s e r i e s t h a t reduces t o t h e o u t e r expansion when ex-

panded asymptotically f o r

E-0

i n o u t e r v a r i a b l e s , and t o t h e

inner expansion i n inner v a r i a b l e s . I n our use of boundary l a y e r techniques t o o b t a i n a uniform

f i r s t approximation f o r the s o l u t i o n

U(X,E)

(A.3) and (A.4) we made a c e r t a i n choice.

t o the problem

Namely, we imposed

the boundary condition a t x = 1 on t h e outer s o l u t i o n and constructed a boundary l a y e r near x = 0 t o s a t i s f y the

Behavior a t

260

second boundary condition.

x

= 1

Could we reverse t h i s procedure?

Observing t h e form of the exact s o l u t i o n given i n Figure [30], i t becomes apparent t h a t t h e answer t o t h i s question is no.

The exact s o l u t i o n i s almost a s t r a i g h t l i n e from x a region close t o x = 0: zero a t

x = 0.

= 1

to

i t then exponentially decays t o

This is the type of behavior that i n d i c a t e s

a boundary layer a t only one end of the i n t e r v a l , namely near

x = 0.

I f we were unaware of the exact s o l u t i o n and

attempted t o construct a boundary layer a t

x = 1 we would

obtain the following equation: n

u

n

66

- u

where

s

=€a

1-x = € 5 .

Thus the f i r s t term i n a n asymptotic expansion f o r t h e bounda r y l a y e r quantity

3 must have the form

Such a term grows exponentially a s the boundary l a y e r merges w i t h the i n t e r i o r and hence can not be p a r t of a uniform com-

p o s i t e expansion f o r u ( x , E ) . We t h e r e f o r e conclude t h a t t h e r e is no boundary l a y e r a t

x

= 1.

A s a n exercise we

suggest t h a t the reader use boundary l a y e r methods t o cons t r u c t the f i r s t approximation for small of the problem.

-

euXX uX = a u(1)

= 1, u ( 0 ) = 0.

E

t o the s o l u t i o n

Boundary l a y e r methods

26 1

I n the preceeding chapters we have made much use of the techniques of matched asymptotic expansions.

These methods

have enabled us t o o b t a i n approximate s o l u t i o n s f o r the Navier-Stokes equations i n various d i f f e r e n t contexts.

The

boundary l a y e r s t h a t we have considered have been viscous boundary layers t h a t a r e characterized by t h e importance of t h e viscous f o r c e i n the boundary l a y e r region.

One of the

simplest, although perhaps the most important boundary l a y e r t h a t we studied was the EKman l a y e r i n which the viscous force and C o r i o l i s force a r e i n balance.

We sought a solu-

t i o n t o t h e p a r t i a l d i f f e r e n t i a l equation 2 E2 v 6P + 4 ? = 0

(A.21)

az

i n terms of an o u t e r ( o r i n t e r i o r ) expansion and a n inner ( o r boundary l a y e r expansion). pI = p0 and

B

=

+

Po +

E

~

Eagl

We wrote

+P

..~..

( i n t e r i o r expansion),

+

....

(boundary l a y e r expansion),

where t h e boundary layer co-ordinate

5 = E-Oz.

We performed

the matched asymptotic expansion i n t h e manner described i n the preceeding simple example t o o b t a i n the r e s u l t s given i n Chapter 5 .

This Page Intentionally Left Blank

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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