Wave Intro

7/30/2019 Wave Intro

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ATMOSPHERIC AND OCEANIC

FLUID DYNAMICS

Supplementary Material for 2nd Edition

Geoffrey K. Vallis


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ii


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Contents

Preface vii

Part I GEOPHYSICAL FLUID DYNAMICS 1

1 Wave Fundamentals and Rossby Waves 3

1.1 Wave Fundamentals 41.1.1 Denitions and kinematics 41.1.2 Wave propagation and phase speed 51.1.3 The dispersion relation 6

1.2 Group Velocity 71.2.1 Superposition of two waves 91.2.2 Superposition of many waves 101.2.3 The method of stationary phase 12

1.3 Ray Theory 131.3.1 Ray theory in practice 15

1.4 Rossby Waves 16

1.4.1 Waves in a single layer 161.4.2 The mechanism of Rossby waves 181.4.3 Rossby waves in two layers 19

1.5 * Rossby Waves in Stratied Quasi-Geostrophic Flow 211.5.1 Setting up the problem 211.5.2 Wave motion 22

1.6 Energy Flux of Rossby Waves 231.6.1 Rossby wave reection 26

1.7 Rossby-gravity Waves: an Introduction 31

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iv Contents

1.7.1 Wave properties 341.7.2 Planetary geostrophic Rossby waves 36

1.8 The Group Velocity Property 381.8.1 Group velocity in homogeneous media 38

1.8.2 Group velocity property: a general derivation 391.8.3 Group velocity property for Rossby waves 41

4 Gravity Waves 554.1 Surface gravity waves 56

4.1.1 Boundary conditions 564.1.2 Wave solutions 574.1.3 Properties of the solution 58

4.2 Internal Gravity waves in a Non-Rotating Boussinesq uid 624.3 Energetics of Poincar Waves 64

4.3.1 One-dimensional problem 64

4.3.2 Two-dimensional Poincar waves 664.4 Waves on Fluid Interfaces 67

4.4.1 Equations of motion 684.4.2 Dispersion relation 69

4.5 Internal waves in a Continuously Stratied Boussinesq uid 704.6 Properties of Internal Waves 72

4.6.1 A few interesting properties 724.6.2 Group velocity and phase speed 744.6.3 Energetics of internal waves 76

4.7 Internal Wave Reection 784.7.1 Properties of internal wave reection 79

4.8 Internal Waves in a Fluid with Varying Stratication 814.8.1 An alternative derivation 834.8.2 An atmospheric case 844.8.3 An atmospheric waveguide 84

4.9 Internal Waves in a Rotating Frame 854.9.1 Equations of motion 854.9.2 Dispersion Relation 864.9.3 Polarization relations 884.9.4 Geostrophic motion and vortical modes 88

4.10 Generation of Internal Waves 914.10.1 The problem and its solution 91

4.10.2 Energy Propagation 924.10.3 Lee waves and ow over topography 944.10.4 Gravity waves in the atmosphere 94

4.11 Acoustic-Gravity Waves in an Ideal Gas 944.11.1 Interpretation 95

4.12 The Moving Flame Effect 984.13 Breaking of Internal Waves 98

4.13.1 Relation to diapycnal diffusivity 984.13.2 The Garrett-Munk spectrum 98


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Contents v

References 101


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vi CONTENTS


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Preface

September 27, 2012

This is some additional material related to the book Atmospheric and Oceanic Fluid Dynamics (AOFD). Eventually the material will be incorporated into a second edition of that book, but that is a couple of years away.

The major items to be added may include:(i) The material on waves will be consolidated, and most of it will be moved out of Part

I into Part II. Part II will begin with a chapter on wave basics and Rossby waves.(ii) A chapter on gravity waves including some material on their importance to the

general circulation.(iii) A chapter on linear dynamics at low latitudes (equatorial waves and the Matsuno

Gill problem).(iv) A chapter on the tropical atmosphere, if it can be made coherent.(v) A chapter on the equatorial ocean and El Nio (probably two chapters in total, one

being the chapter already posted above).(vi) Up to a chapter on stratospheric dynamics.

(vii) Tentatively, a chapter on dynamical regimes of planetary atmospheres. This might

have to wait until a third edition.Some of these items are present in this document. Others remain to be started. A numberof corrections will be made throughout the existing book, and some other sections willbe shortened, claried or omitted.

In general I will post new items to the web when there is something reasonably substantial to be read, typically half a chapter or so of new material. The material willrst be posted when it is readable, but before it is complete or nalized. (There is no pointin asking for comments on material that is nished.) I would appreciate any comments you, the reader, may have whether major or minor. Suggestions are also welcome on

vii


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viii Preface

material to include or omit. There is no need, however, to comment on typos in the text these will be cleaned up in the nal version. However, please do point out typos inequations and, perhaps even more importantly, thinkos, which are sort of typos in thebrain.

Student Edition

As the second edition of the book will perforce be rather long (perhaps close to 1000pages), it may not be appropriate for graduate students who do not plan a career indynamics. Thus, the publisher (CUP) and I are considering a shorter student edition, which would have the advanced or more arcane material omitted and some of the expla-nations simplied. The resulting would likely be about 500 pages. Please let me know if you have any comments on this.

Problem SetsOne omission in the rst edition is numerically-oriented problems that graphically illus-trate some phenomena using Matlab or Python or similar. If you have any such problemsor would like to develop some that could be linked to this book, please let me know. Ad-ditional problems of a conventional nature would also be welcome. Again, please contactme.

Thank you!Geoff [email protected]


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Part I

GEOPHYSICAL FLUID DYNAMICS

1


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Catch a wave and youre sitting on top of the world.The Beach Boys

CHAPTER

ONE

Wave Fundamentals and Rossby Waves

In this chapter we provide an introduction to wave motion and a description of perhaps

the most important kind of wave motion occurring at large scales in the ocean and at-mosphere, namely Rossby waves. 1 The chapter has three main parts to it. In the rst, we provide a brief discussion of wave kinematics and dynamics, introducing such basicconcepts as phase speed and group velocity. The second part, beginning with section3.4 , is a discussion of the dynamics of Rossby waves; this may be considered to be thenatural follow-on from the previous chapter. Finally, in section 3.8 , we return to group velocity in a somewhat more general way. Wave kinematics is a somewhat formal topic, yet closely tied to wave dynamics: kinematics without a dynamical example is jejune anddry, yet understanding wave dynamics of any sort is hardly possible without appreciatingat least some of the formal structure of waves. Readers should ip pages back and forthas necessary.

Those readers who already have a knowledge of wave motion, or those who wish tocut to the chase quickly, may wish to skip the rst few sections and begin at section 3.4 .Other readers may wish to skip the sections on Rossby waves altogether and, after ab-sorbing the sections on the wave theory move on to chapter 4 on gravity waves, returningto Rossby waves (or not) later on. The Rossby wave and gravity wave discussions arelargely independent of each other, although they both require that the reader is familiar with the basic ideas of wave analysis such as group velocity and phase speed. Close tothe equator Rossby waves and gravity waves become more intertwined and we deal withthe ensuing waves in chapter ?? .

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4 Chapter 1. Wave Fundamentals and Rossby Waves

1.1 FUNDAMENTALS AND FORMALITIES

1.1.1 Denitions and kinematics

What is a wave? Rather like turbulence, a wave is more easily recognized than dened.Perhaps a little loosely, a wave may be considered to be a propagating disturbance thathas a characteristic relationship between its frequency and size; more formally, a waveis a disturbance that satises a dispersion relation. In order to see what this means,and what a dispersion relation is, suppose that disturbance, ( x , t) (where might be velocity, streamfunction, pressure, etc), satises some equation

L() =0, (1.1) where L is a linear operator, typically a polynomial in time and space derivatives; anexample is L() =2 /t +/x . We will mainly deal with linear waves for whichthe operator L is linear. Nonlinear waves certainly exist, but the curious reader must look elsewhere to learn about them. 2 If (3.1 ) has constant coefcients (if is constant in this

example) then solutions may often be found as a superposition of plane waves, each of which satisfy

=Re e i(x,t) =Re e i( k xt) . (1.2) where is a constant, is the phase, k is the vector wavenumber (k,l,m) , and is the wave frequency. We also often write the wave vector as k =(k x , k y , k z ) .

Earlier, we said that waves are characterized by having a particular relationship be-tween the frequency and wavevector known as the dispersion relation . This is an equationof the form

=( k ) (1.3) where ( k ) [meaning (k,l,m) ] is some function determined by the form of L and so

depends on the particular type of wave the function is different for sound waves, light waves and the Rossby waves and gravity waves we will encounter in this book (peak ahead to ( 3.56 ) are ( 4.42 ), and there is more discussion in section 3.1.3 ). Unless it isnecessary to explicitly distinguish the function from the frequency , we will often write =( k ) .

If the medium in which the waves are propagating is inhomogeneous, then ( 3.1 ) will probably not have constant coefcients (for example, may vary meridionally).Nevertheless, if the medium is slowly varying, wave solutions may often still be found although we do not prove it here with the general form

=Re a( x , t) e i( x ,t) , (1.4) where a( x , t) varies slowly compared to the variation of the phase, . The frequency and wavenumber are then dened by

k , t

, (1.5)

which in turn implies the formal relation between k and :

kt + =0 . (1.6)


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1.1 Fundamentals and Formalities 5

1.1.2 Wave propagation and phase speed

An almost universal property of waves is that they propagate through space with some velocity (which might be zero). Waves in uids may carry energy and momentum butnot normally, at least to a rst approximation, uid parcels themselves. Further, it turnsout that the speed at which properties like energy are transported (the group speed) may be different from the speed at which the wave crests themselves move (the phase speed).Lets try to understand this beginning with the phase speed.

Phase speed

Let us consider the propagation of monochromatic plane waves, for that is all that isneeded to introduce the phase speed. Given ( 3.2 ) a wave will propagate in the directionof k (Fig. 3.1 ). At a given instant and location we can align our coordinate axis alongthis direction, and we write k x = Kx, where xincreases in the direction of k andK 2 = |k |2 is the magnitude of the wavenumber. With this, we can write ( 3.2 ) as

=Re e i(Kx

t)

=Re e iK(x

ct ), (1.7)

where c =/K . From this equation it is evident that the phase of the wave propagatesat the speed c in the direction of k , and we dene the phase speed by

c p K

. (1.8)

The wavelength of the wave, , is the distance between two wavecrests that is, thedistance between two locations along the line of travel whose phase differs by 2 andevidently this is given by

=2 K

. (1.9)

In (for simplicity) a two-dimensional wave, and referring to Fig. 3.1 (a), the wavelengthand wave vectors in the x- and y -directions are given by,

x =

cos , y =

sin

, kx =K cos , k y =K sin . (1.10)

In general, lines of constant phase intersect both the coordinate axes and propagate alongthem. The speed of propagation along these axes is given by

c xp =c plx

l =c p

cos =c pK kx =

kx

, c y p =c ply

l =c p

sin =c pK ky =

ky

, (1.11)

using ( 3.8 ) and ( 3.10 ). The speed of phase propagation along any one of the axis isin general larger than the phase speed in the primary direction of the wave. The phasespeeds are clearly not components of a vector: for example, c xp c p cos . Analogously,the wavevector k is a true vector, whereas the wavelength is not.

To summarize, the phase speed and its components are given by

c p =K

, c xp =kx

, c y p =ky

. (1.12)


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Fig. 1.1 The propagation of a two-dimensional wave. (a) Two lines of constant phase(e.g., two wavecrests) at a time t 1 . The wave is propagating in the direction k with

wavelength . (b) The same line of constant phase at two successive times. The phasespeed is the speed of advancement of the wavecrest in the direction of travel, and soc p = l/(t 2 t 1 ) . The phase speed in the x-direction is the speed of propagation of thewavecrest along the x-axis, and c xp = lx /(t 2 t 1 ) =c p / cos .

Phase velocity

Although it is not particularly useful, there is a way of dening a phase speed so that is atrue vector, and which might then be called phase velocity. We dene the phase velocity to be the phase speed in the direction in which the wave crests are propagating; that is

c p K

k

|K | , (1.13) where k / |K | is the unit vector in the direction of wave-crest propagation. The compo-nents of the phase velocity in the the x- and y -directions are then given by

c xp =c p cos , c y p =c p sin . (1.14)

Dened this way, the quantity given by ( 3.14 ) is indeed a true vector velocity. However,the components in the x- and y -directions are manifestly not the speed at which wavecrests propagate in those directions. It is therefore a misnomer to call these quantitiesphase speeds, although it is helpful to ascribe a direction to the phase speed and so the

quantity given by ( 3.14 ) can be useful.

1.1.3 The dispersion relation

Much of above description is mostly kinematic and a little abstract, applying to almostany disturbance that has a wavevector and a frequency. The particular dynamics of a wave are determined by the relationship between the wavevector and the frequency;that is, by the dispersion relation. Once the dispersion relation is known a great many of the properties of the wave follow in a more-or-less straightforward manner, as we will


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1.2 Group Velocity 7

see. Picking up from ( 3.3 ), the dispersion relation is a functional relationship betweenthe frequency and the wavevector of the general form

=( k ). (1.15)Perhaps the simplest example of a linear operator that gives rise to waves is the one-

dimensional equationt +c

x =0. (1.16)

Substituting a trial solution of the form = Re A e i(kx t) , where Re denotes the realpart, we obtain (i +c ik)A =0, giving the dispersion relation

=ck. (1.17)The phase speed of this wave is c p =/k =c .

A few other examples of governing equations, dispersion relations and phase speedsare:

t +c =0, =c k , c p = |c | cos , c

xp =

c kk

, c y p =c k

l(1.18a)

2t 2 c

2

2 =0, 2 =c 2 K 2 , c p = c, c xp = cK k

, c y p = cK

l, (1.18b)

t

2 +x =0, =

kK 2

, c p =K

, c xp =

K 2, c y p =

k/lK 2

. (1.18c)

where K 2 =k2 +l2 and is the angle between c and k . A wave is said to be nondispersive or dispersionless if the phase speed is independent

of the wavelength. This condition is clearly satised for the simple example ( 3.16 ) butis manifestly not satised for ( 3.18 c), and these waves (Rossby waves, in fact) are dispersive. Waves of different wavelengths then travel at different speeds so that a groupof waves will spread out or disperse (hence the name), even if the medium is homo-geneous. When a wave is dispersive there is another characteristic speed at which the waves propagate, known as the group velocity, and we come to this in the next section.

Most media are, of course, inhomogeneous, but if the medium varies sufciently slowly and in particular if the variations are slow compared to the wavelength wemay still have a local dispersion relation between frequency and wavevector,

=( k ; x , t). (1.19)

Although is a function of k , x and t the semi-colon in ( 3.19 ) is used to suggest that xand t are slowly varying parameters of a somewhat different nature than k . Well pick up our discussion of this in section 3.3 , but before that we must introduced the group velocity.

1.2 GROUP VELOCITY

Information and energy travel clearly cannot travel at the phase speed, for as the direc-tion of propagation of the phase line tends to a direction parallel to the y -axis, the phase


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Wave Fundamentals

A wave is a propagating disturbance that has a characteristic relationship betweenits frequency and size, known as the dispersion relation. Waves typically arise assolutions to a linear problem of the form

L() =0 , (W.1) where L is (commonly) a linear operator in space and time. Two examples are

2 t 2 c

2

2 =0 andt

2 +x =0. (W.2)

The rst example is so common in all areas of physics it is sometimes called the waveequation. The second example gives rise to Rossby waves.

Solutions to the governing equation are often sought in the form of plane waves thathave the form =Re A e i( k xt) , (W.3)

where A is the wave amplitude, k the wavevector, k =(k,l,m) , and is the frequency.The dispersion relation is a functional relationship between the frequency and wavevector of the form = ( k ) where is a function. It arises from substitut-ing a trial solution like ( W.3 ) into the governing equation ( W.1 ). For the examplesof (W.2 ) we obtain = c 2 K 2 and = k/K 2 and K 2 = k2 + l2 +m 2 or, in twodimensions, K 2 =k2 +l2 .

The phase speed is the speed at which the wave crests move. In the direction of propagation and in the x, y and z directions group speed is given by, respectively,

c p =K

, c xp =k

, c y p =l

, c zp =m

. (W.4)

where K =2 / where is the wavelength. Phase speed is not a vector.The group velocity is the velocity at which a wave packet or wave group moves. It isa vector and is given by

c g =k

, c xg =k

, c y g =l

, c zg =m

. (W.5)

Energy and various other physical quantities are also transported at the group velocity.

If the medium is inhomogeneous but only slowly varying in space and time, thenapproximate solutions may sometimes be found in the form

=Re A( x , t) e i( x ,t) , (W.6) where the amplitude A is also slowly varying and the local wavenumber and frequency are related to the phase, , by k = and = /t . The dispersion relation isthen a local one of the form =( k ; x,t) .


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Fig. 1.2 Superposition of two sinusoidal waves with wavenumbers k and k +k , pro-ducing a wave (solid line) that is modulated by a slowly varying wave envelope or wavepacket (dashed line). The envelope moves at the group velocity, c g =/k and thephase of the wave moves at the group speed c p =/k .

speed in the x-direction tends to innity! Rather, it turns out that most quantities of interest, including energy, propagate at the group velocity, a quantity of enormous impor-tance in wave theory. Rather roughly, this is the velocity at which a packet or a group of waves will travel, whereas the individual wave crests travel at the phase speed. To intro-duce the idea we will consider consider the superposition of plane waves, noting that amonochromatic plane wave already lls space uniformly so that there can be no propa-gation of energy from place to place. We will restrict attention to waves propagating inone direction, but the argument may be extended to two or three dimensions.

1.2.1 Superposition of two waves

Consider the linear superposition of two waves. Limiting attention to the one-dimensionalcase for simplicity, consider a disturbance represented by

=Re ( e i(k 1 x 1 t) +e i(k 2 x 2 t) ). (1.20)Let us further suppose that the two waves have similar wavenumbers and frequency, and,in particular, that k1 = k +k and k2 = k k , and 1 = + and 2 = .With this, ( 3.20 ) becomes

=Re e i(kx t) [e i(kx t)

+ei(kx t) ]

=2 Re e i(kx t) cos (kx t). (1.21)The resulting disturbance, illustrated in Fig. 3.2 has two aspects: a rapidly varying com-ponent, with wavenumber k and frequency , and a more slowly varying envelope, with wavenumber k and frequency . The envelope modulates the fast oscillation, andmoves with velocity /k ; in the limit k 0 and 0 this is the group velocity,c g =/k . Group velocity is equal to the phase speed, /k , only when the frequency isa linear function of wavenumber. The energy in the disturbance must move at the group velocity note that the node of the envelope moves at the speed of the envelope and no


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energy can cross the node. These concepts generalize to more than one dimension, and if the wavenumber is the three-dimensional vector k = (k,l,m) then the three-dimensionalenvelope propagates at the group velocity given by

c g =k

k ,

l ,

m . (1.22)

1.2.2 Superposition of many waves

Now consider a generalization of the above arguments to the case in which many wavesare excited. In a homogeneous medium, nearly all wave patterns can be represented as asuperposition of an innite number of plane waves; mathematically the problem is solvedby evaluating a Fourier integral. For mathematical simplicity well continue to treat only the one-dimensional case but the three dimensional generalization is straightforward.

A superposition of plane waves, each satisfying some dispersion relation, can be rep-

resented by the Fourier integral

(x,t) = A(k) e i(kx t) dk. (1.23a)The function A(k) is given by the initial conditions:

A(k) =1

2 (x, 0) eikx dx. (1.23b) As an aside, note that if the waves are dispersionless and = ck where c is a constant,then

(x,t)

= +

A(k) e ik(x ct ) dk=

(x

ct , 0), (1.24)

by comparison with ( 3.23a ) at t =0. That is, the initial condition simply translates at aspeed c , with no change in structure.

Although the above procedure is quite general it doesnt get us very far: it doesntprovide us with any physical intuition, and the integrals themselves may be hard to eval-uate. A physically more revealing case is to consider the case for which the disturbanceis a wave packet essentially a nearly plane wave or superposition of waves conned toa nite region of space. We will consider a case with the initial condition

(x, 0) =a(x) e ik0 x (1.25)

where a(x) , rather like the envelope in Fig. 3.3 , modulates the amplitude of the wave ona scale much longer than that of the wavelength 2 /k 0 , and more slowly than the waveperiod. That is,

1a

ax

k0 ,1a

at

k0c, (1.26a,b)

and the disturbance is essentially a slowly modulated plane wave. We suppose that a(x)is peaked around some value x0 and is very small if |x x0 | k10 ; that is, a(x) is smallif we are several wavelengths of the plane wave away from the peak. We would like toknow how such a packet evolves.


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a(x)

0 = 2 /k 0

L x

L x 0

c p

c g

Fig. 1.3 A generic wave packet. The envelope, a(x) , has a scale Lx that is much largerthan the wavelength, 0 , of the wave embedded within in. The envelope moves at the

group velocity, c g , and the phase of the waves at the phase speed, c p .

We can express the envelope as a Fourier integral by rst noting that that we can write the initial conditions as a Fourier integral,

(x, 0) = A(k) e ikx dk where A(k) = 12 +(x, 0) eikx dx, (1.27a,b)so that, using ( 3.25 ),

A(k) =1

2

+

a(x) e i(k 0k)x dx and a(x) =

A(k) e i(k k0 )x dk. (1.28a,b)

We still havent made much progress beyond ( 3.23 ). To do so, we note rst that a(x) isconned in space, so that to a good approximation the limits of the integral in ( 3.28 a)can be made nite, L say, provided L k10 . We then note that when (k 0 k)x is largethe integrand in ( 3.28 a) oscillates rapidly; successive intervals in x therefore cancel eachother and make a small net contribution to the integral. Thus, the integral is dominatedby values of k near k0 , and A(k) is peaked near k0 . (Note that the nite spatial extent of a(x) is crucial for this argument.)

We can now evaluate how the wave packet evolves. Beginning with ( 3.23a ) we have

(x,t) =

A(k) exp{i(kx (k)t) }dk (1.29a)

A(k) exp i[k 0 x (k 0 )t] +i(k k0 )x i(k k0 ) k k=k0 t dk (1.29b)having expanded (k) in a Taylor series about k and kept only the rst two terms, notingthat the wavenumber band is limited. We therefore have

(x,t) =exp {i[k 0 x (k 0 )t] } A(k) exp i(k k0 ) x k k=k0 t dk (1.30a)=exp {i[k 0 x (k 0 )t] }a x c g t (1.30b)


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where c g = /k evaluated at k = k0 . That is to say, the envelope a(x) moves at thegroup velocity.

The group velocity has a meaning beyond that implied by the derivation above: thereis no need to restrict attention to narrow band processes, and it turns out to be a quite

general property of waves that energy (and certain other quadratic properties) propagateat the group velocity. This is to be expected, at least in the presence of coherent wavepackets, because there is no energy outside of the wave envelope so the energy mustpropagate with the envelope.

1.2.3 The method of stationary phase

We will now relax the assumption that wavenumbers are conned to a narrow band andlook for solutions at large t ; that is, we will be seeking a description of waves far fromtheir source. Consider a disturbance of the general form

(x,t) = A(k) e

i[kx

(k)t]

dk = A(k) e

i(k ;x,t)t

dk (1.31) where (k : x,t) kx/t (k) . (Here we regard as a function of k with parametersx and t ; we will sometimes just write (k) with (k) =/k .) Now, a standard resultin mathematics (known as the RiemannLebesgue lemma) states that

I = limt f(k) e ikt dk =0 (1.32)provided that f(k) is integrable and f(k) dk is nite. Intuitively, as t increases theoscillations in the integral increase and become much faster than any variation in A(k) ;successive oscillations thus cancel and the integral becomes very small (Fig. 3.4 ).

Looking at ( 3.31 ), the integral will be small if is everywhere varying with k . How-ever, if there is a region where does not vary with k that is, if there is a region wherethe phase is stationary and /k =0 then there will be a contribution to the integralfrom that region. Thus, for large t , an observer will predominantly see waves for which (k) =0 and so, using the denition of , for which

xt =

k

. (1.33)

In other words, at some space-time location (x,t) the waves that dominate are those whose group velocity /k is x/t . In the example plotted in Fig. 3.4 , = /k so thatthe wavenumber that dominates, k0 say, is given by solving /k 20 =x/t , which for x/t =1and =400 gives k0 =20.

We may actually approximately calculate the contribution to (x,t) from waves mov-ing with the group velocity. Let us expand (k) around the point, k0 , where (k) =0.We obtain

(x,t) = A(k) exp it (k 0 ) +(k k0 ) (k 0 ) + 12 (k k0 ) 2 (k 0 ) . . . dk (1.34)The higher order terms are small because k k0 is presumed small (for if it is large theintegral vanishes), and the term involving (k 0 ) is zero. The integral becomes

(x,t) =A(k 0 )e i(k 0 ) exp it 12 (k k0 ) 2 (k 0 ) dk. (1.35)


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1.3 Ray Theory 13

0 5 10 15 20 25 30 35 40 45 501

0.5

0

0.5

1

0 5 10 15 20 25 30 35 40 45 501

0.5

0

0.5

1

Wavenumber, k

Wavenumber, k

k 0

t = 1

t = 12

k 0

A ( k )

e i t ( k

; x , t

)

A ( k )

e i t ( k

; x , t

)

A(k)

A(k)

Fig. 1.4 The integrand of (3.31 ), namely the function that when integrated over wave-number gives the wave amplitude at a particular x and t . The example shown is for a

Rossby wave with = /k , with =400 and x/t =1 , and hence k0 =20 , for twotimes t =1 and t =12 . (The envelope, A(k) , is somewhat arbitrary.) At the later timethe oscillations are much more rapid in k, so that the contribution is more peakedfrom wavenumbers near to k0 .

We therefore have to evaluate a Gaussian, and because ecx 2 dx = /c we obtain(x,t) A(k 0 )e i(k 0 ) 2 /( it (k 0 ))

1 / 2=A(k 0 )e i(k 0 x(k 0 )t) 2i /(t (k 0 ))

1 / 2 .(1.36)

The solution is therefore a plane wave, with wavenumber k0 and frequency (k 0 ) , slowly modulated by an envelope determined by the form of (k 0 ; x,t) , where k0 is the wave-

number such that x/t =c g =/k |k=k0

1.3 RAY THEORY

Most waves propagate in a media that is inhomogeneous. In the Earths atmosphereand ocean the stratication varies with altitude and the Coriolis parameter varies withlatitude. In these cases it can be hard to obtain the solution of a wave problem by Fourier methods, even approximately. Nonetheless, the ideas of signals propagating tthe group velocity is a very robust one, and it turns out that we can often obtain much


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of the information we want and in particular the trajectory of a wave using anapproximate recipe known as ray theory, using the word theory a little generously. 3

In an inhomogeneous medium let us suppose that the solution to a particular waveproblem is of the form

( x , t) =a(x,t) ei(x,t)

, (1.37) where a is the wave amplitude and the phase, and a varies slowly in a sense we willmake more precise shortly. The local wavenumber and frequency are dened by

k , t

. (1.38)

We suppose that the amplitude a varies slowly over a wavelength and a period; that is

|a |/ |a | is small over the length 1 /k and the period 1 / or

|a/x |a |k|, |

a/t |a

(1.39)

We will assume that the wavenumber and frequency as dened by ( 3.38 ) are the same asthose that would arise if the medium were homogeneous and a were a constant. Thus, we may obtain a dispersion relation from the governing equation by keeping the spatially (and possibly temporally) varying parameters xed and obtain

=( k ; x,t), (1.40)and then allow x and t to vary, albeit slowly. (This procedure may be formalized using atwo-scale approximation, or equivalently using WKB methods.)

Let us now consider how the wavevector and frequency might change with positionand time. We recall from ( 3.6 ) that the wavenumber and frequency are related by

k it +

x i =0 , (1.41)

where we use a subscript notation for vectors, and in what follows repeated indices willbe summed. Using ( 3.41 ) and ( 3.40 ) gives

k it +

x i +

k j

k j x i =0 . (1.42)

This may be written k

t +c

g k

= (1.43)

wherec g =

k =

k

,l

,m

, (1.44)

is, again, the group velocity, sometimes written as c g = k or, in subscript notation,as c gi = /k i . The left-hand side of ( 3.23a ) is similar to an advective derivative, but with the velocity being a group velocity not a uid velocity. Evidently, if the dispersionrelation for frequency is not an explicit function of space the wavevector is propagatedat the group velocity.


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1.3 Ray Theory 15

The frequency is, in general, a function of space, wavenumber and time, and fromthe dispersion relation, ( 3.40 ), its variation is governed by

t =

t +

k

i

k it =

t

k

i

x

i

(1.45)

using ( 3.41 ). Thus, using the denition of group velocity, we may write

t +c g =

t

. (1.46)

As with ( 3.43 ) the left-hand side is like an advective derivative, but with the velocity being a group velocity. If the dispersion relation is not a function of time, the frequency propagates at the group velocity.

Motivated by ( 3.43 ) and ( 3.46 ) we dene a ray as the trajectory traced by the group velocity. Noting and if the frequency is not an explicit function of space or time, thenboth the wavevector and the frequency are constant along a ray.

1.3.1 Ray theory in practice

What use is ray theory? The idea is that we use ( 3.43 ) and ( 3.46 ) to track a group of waves from one location to another without solving the full wave equations of motion.Indeed, it turns out that we can sometimes solve problems using ordinary differentialequations (ODEs) rather than partial differential equations (PDEs).

Suppose that the initial conditions consist of a group of waves at a position x0 , for which the amplitude and wavenumber vary only slowly with position. We also supposethat we know the dispersion relation for the waves at hand; that is, we know the func-

tional form of (k ; x,t) . Let us dene the total derivate following the group velocity asdd t

t +c g , (1.47)

so that ( 3.43 ) and ( 3.46 ) may be written as

d kd t = , (1.48a)

d d t =

t

. (1.48b)

These are ordinary differential equations for wavevector and frequency, solvable pro-

vided we know the right-hand sides; that is, provided we know the space and time loca-tion at which the dispersion relation [i.e., (k ; x,t) ] is to be evaluated. But the locationis known because it is moving with the group velocity and so

d xd t =c g . (1.48c)

where c g =/ k |x ,t (i.e., c gi =/k i |x ,t ).The set ( 3.48 ) is a triplet of ordinary differential equations for the wavevector, fre-

quency and position of a wave group. They may be solved, albeit sometimes numerically,


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Fig. 1.5 Schema of the trajec-tory of two wavepackets, eachmoving with a different groupvelocity, as might be calculatedusing ray theory. If the wavepackets collide ray theory mustfail.

T i m e

x

Wave packet collision.Ray theory fails.

Trajectory 1Trajectory 2c g 1c g 2

to give the trajectory of a wave packet or collection of wave packets, as schematically il-lustrated in Fig. 3.5 . Of course, if the medium or the wavepacket amplitude is not slowly varying ray theory will fail, and this will perforce happen if two wave packets collide.

The amplitude of the wave packet is not given by ray theory. However, given ourearlier discussions, it should come as no surprise to nd that a quantity related to theamplitude of a wave packet specically, the wave activity, which is quadratic in the wave amplitude is also propagated at the group velocity, but we leave our discussionof that for later sections. We now turn our attention to a specic form of wave Rossby waves but the reader whose interest is more in the general properties of waves may skip forward to section 3.8

1.4 ROSSBY WAVES

We now shift gears and consider in some detail a particular type of waves, namely Rossby waves in a quasi-geostrophic system. These waves are perhaps the most important large-scale wave in the atmosphere and ocean (although gravity waves, discussed in the nextchapter, are arguably as important in some contexts). 4

1.4.1 Waves in a single layer

Consider ow of a single homogeneous layer on a at-bottomed -plane. The unforced,inviscid equation of motion is

DDt

( +f k2d ) =0 , (1.49) where =2 is the vorticity and is the streamfunction and kd =1 /L d is the inverseradius of deformation. (Note that denitions of kd and Ld vary, even )


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1.4 Rossby Waves 17

Innite deformation radius

If the scale of motion is much less than the deformation scale then on the -plane theequation of motion is governed by

DDt ( +y) =0 . (1.50)Expanding the material derivative gives

t +u +v =0 or

t +J(,) +

x =0 . (1.51)

We now linearize this equation; that is, we suppose that the ow consists of a time-independent component (the basic state) plus a perturbation, with the perturbationbeing small compared with the mean ow. Such a mean ow must satisfy the time-independent equation of motion, and purely zonal ow will do this. For simplicity wechoose a ow with no meridional dependence and let

= + (x,y,t), (1.52) where = Uy and | | | |. (The symbol U represents the zonal ow of the basicstate, not a magnitude for scaling purposes.) Substitute ( 3.52 ) into ( 3.51 ) and neglectthe nonlinear terms involving products of to give

t +J(, ) +

x =0 or

t

2 +U

2 x +

x =0. (1.53a,b)

Solutions to this equation may be found in the form of a plane wave,

=Re e i(kx +ly t) , (1.54)

where Re indicates the real part of the function (and this will sometimes be omittedif no ambiguity is so-caused). Solutions of the form ( 3.54 ) are valid in a domain withdoubly-periodic boundary conditions; solutions in a channel can be obtained using ameridional variation of sin ly , with no essential changes to the dynamics. The amplitudeof the oscillation is given by and the phase by kx +ly t , where k and l are the x-and y -wavenumbers and is the frequency of the oscillation.

Substituting ( 3.54 ) into ( 3.53 ) yields

[( +Uk)( K 2 ) +k] =0 , (1.55)

where K 2

=k2

+l2. For non-trivial solutions this implies

=Uk kK 2

. (1.56)

This is the dispersion relation for Rossby waves. The phase speed, c p , and group velocity,c g , in the x-direction are

c xp k =U

K 2

, c xg k =U +

(k 2 l2 )(k 2 +l2 ) 2

. (1.57a,b)


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The velocity U provides a uniform translation, and Doppler shifts the frequency. Thephase speed in the absence of a mean ow is westwards, with waves of longer wave-lengths travelling more quickly, and the eastward current speed required to hold the waves of a particular wavenumber stationary (i.e., c xp =0) is U =/K 2 . We discuss themeaning of the group velocity in the appendix.Finite deformation radius

For a nite deformation radius the basic state = Uy is still a solution of the originalequations of motion, but the potential vorticity corresponding to this state is q =Uy/L 2d +y and its gradient is q =( +U/L 2d ) j. The linearized equation of motion is thus

t +U

x

(

2 /L 2d ) +( +U/L 2d )x =0 . (1.58)

Substituting = e i(kx +ly t) we obtain the dispersion relation,

=k(UK 2 )K 2 +1/L 2d =

Uk k +U/L 2dK 2 +1/L 2d

. (1.59)

The corresponding x-components of phase speed and group velocity are

c xp =U +Uk 2dK 2 +k2d =

UK 2 K 2 +k2d

, c xg =U +( +Uk 2d )(k 2 l2 k2d )

(k 2 +l2 +k2d ) 2, (1.60a,b)

where kd =1/L d . The uniform velocity eld now no longer provides just a simple Dopplershift of the frequency, nor a uniform addition to the phase speed. From ( 3.60 a) the wavesare stationary when K 2 = K 2s /U ; that is, the current speed required to hold wavesof a particular wavenumber stationary is U = /K

2. However, this is not simply themagnitude of the phase speed of waves of that wavenumber in the absence of a current

this is given by

c xp =

K 2s +k2d = U

1 +k2d /K 2s. (1.61)

Why is there a difference? It is because the current does not just provide a uniform trans-lation, but, if Ld is non-zero, it also modies the basic potential vorticity gradient. Thebasic state height eld 0 is sloping, that is 0 = (f 0 /g)Uy , and the ambient potential vorticity eld increases with y , that is q = ( +U/L 2d )y . Thus, the basic state denes apreferred frame of reference, and the problem is not Galilean invariant. 5 We also notethat, from ( 3.60 b), the group velocity is negative (westward) if the x-wavenumber is suf-

ciently small, compared to the y -wavenumber or the deformation wavenumber. Thatis, said a little loosely, long waves move information westward and short waves move information eastward, and this is a common property of Rossby waves The x-component of the phase speed, on the other hand, is always westward relative to the mean ow.

1.4.2 The mechanism of Rossby waves

The fundamental mechanism underlying Rossby waves is easily understood. Consider amaterial line of stationary uid parcels along a line of constant latitude, and suppose that


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1.4 Rossby Waves 19

Fig. 1.6 The mechanism of a two-dimensional ( xy ) Rossby wave. An initial distur-bance displaces a material line at constant latitude (the straight horizontal line) to thesolid line marked (t = 0) . Conservation of potential vorticity, y + , leads to theproduction of relative vorticity, as shown for two parcels. The associated velocity eld

(arrows on the circles) then advects the uid parcels, and the material line evolves intothe dashed line. The phase of the wave has propagated westwards.

some disturbance causes their displacement to the line marked (t =0) in Fig. 3.6 . In thedisplacement, the potential vorticity of the uid parcels is conserved, and in the simplestcase of barotropic ow on the -plane the potential vorticity is the absolute vorticity,y + . Thus, in either hemisphere, a northward displacement leads to the productionof negative relative vorticity and a southward displacement leads to the production of positive relative vorticity. The relative vorticity gives rise to a velocity eld which, inturn, advects the parcels in material line in the manner shown, and the wave propagates westwards.

In more complicated situations, such as ow in two layers, considered below, or ina continuously stratied uid, the mechanism is essentially the same. A displaced uidparcel carries with it its potential vorticity and, in the presence of a potential vorticity gradient in the basic state, a potential vorticity anomaly is produced. The potential vorticity anomaly produces a velocity eld (an example of potential vorticity inversion) which further displaces the uid parcels, leading to the formation of a Rossby wave. The vital ingredient is a basic state potential vorticity gradient, such as that provided by thechange of the Coriolis parameter with latitude.

1.4.3 Rossby waves in two layers

Now consider the dynamics of the two-layer model, linearized about a state of rest. Thetwo, coupled, linear equations describing the motion in each layer are

t

2 1 +F 1 ( 2 1 ) + 1x =0 , (1.62a)

t

2 2 +F 2 ( 1 2 ) + 2x =0 , (1.62b)

where F 1 =f 20 /g H 1 and F 2 =f 20 /g H 2 . By inspection ( 3.62 ) may be transformed into twouncoupled equations: the rst is obtained by multiplying ( 3.62 a) by F 2 and ( 3.62 b) by


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F 1 and adding, and the second is the difference of ( 3.62 a) and ( 3.62 b). Then, dening

=F 1 2 +F 2 1

F 1 +F 2, =

12

( 1 2 ), (1.63a,b)

(think for temperature), ( 3.62 ) become

t

2 +x =0, (1.64a)

t

(

2 k2d ) +x =0, (1.64b)

where now kd = (F 1 +F 2 ) 1/ 2 . The internal radius of deformation for this problem is theinverse of this, namely

Ld =k1d =1f 0

g H 1 H 2H 1 +H 2

1/ 2. (1.65)

The variables and are the normal modes for the two-layer model, as they oscillateindependently of each other. [For the continuous equations the analogous modes are theeigenfunctions of z [(f 20 /N 2 ) z ] = 2 .] The equation for , the barotropic mode, isidentical to that of the single-layer, rigid-lid model, namely ( 3.53 ) with U = 0, and itsdispersion relation is just

= kK 2

. (1.66)

The barotropic mode corresponds to synchronous, depth-independent, motion in the twolayers, with no undulations in the dividing interface.

The displacement of the interface is given by 2 f 0 /g and so proportional to theamplitude of , the baroclinic mode . The dispersion relation for the baroclinic mode is

= kK 2 +k2d. (1.67)

The mass transport associated with this mode is identically zero, since from ( 3.63 ) wehave

1 = +2F 1

F 1 +F 2, 2 =

2F 2 F 1 +F 2

, (1.68a,b)

and this impliesH 1 1 +H 2 2 =(H 1 +H 2 ). (1.69)

The left-hand side is proportional to the total mass transport, which is evidently associ-ated with the barotropic mode.

The dispersion relation and associated group and phase velocities are plotted inFig. 3.7 . The x-component of the phase speed, /k , is negative (westwards) for bothbaroclinic and barotropic Rossby waves. The group velocity of the barotropic wavesis always positive (eastwards), but the group velocity of long baroclinic waves may benegative (westwards). For very short waves, k2 k2d , the baroclinic and barotropic ve-locities coincide and their phase and group velocities are equal and opposite. With adeformation radius of 50 km, typical for the mid-latitude ocean, then a non-dimensionalfrequency of unity in the gure corresponds to a dimensional frequency of 5 10 7 s1or a period of about 100 days. In an atmosphere with a deformation radius of 1000 km


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1.5 * Rossby Waves in Stratied Quasi-Geostrophic Flow 21

Fig. 1.7 Left: the dispersion relation for barotropic ( t , solid line) and baroclinic( c , dashed line) Rossby waves in the two-layer model, calculated using (3.66 ) and(3.67 ) with k y = 0 , plotted for both positive and negative zonal wavenumbers andfrequencies. The wavenumber is non-dimensionalized by kd , and the frequency isnon-dimensionalized by /k d . Right: the corresponding zonal group and phase ve-locities, c g = /k x and c p =/k x , with superscript t or c for the barotropic orbaroclinic mode, respectively. The velocities are non-dimensionalized by /k 2d .

a non-dimensional frequency of unity corresponds to 1 10 5 s1 or a period of about 7days. Non-dimensional velocities of unity correspond to respective dimensional velocitiesof about 0 .25ms 1 (ocean) and 10ms 1 (atmosphere).

The deformation radius only affects the baroclinic mode. For scales much smallerthan the deformation radius, K 2 k2d , we see from ( 3.64b ) that the baroclinic modeobeys the same equation as the barotropic mode so that

t

2 +x =0 . (1.70)

Using this and ( 3.64a ) implies that

t

2 i + ix =0, i =1 , 2 . (1.71)

That is to say, the two layers themselves are uncoupled from each other. At the otherextreme, for very long baroclinic waves the relative vorticity is unimportant.

1.5 * ROSSBY WAVES IN STRATIFIED QUASI-GEOSTROPHIC FLOW

1.5.1 Setting up the problem

Let us now consider the dynamics of linear waves in stratied quasi-geostrophic owon a -plane, with a resting basic state. (In chapter ?? we explore the role of Rossby waves in a more realistic setting.) The interior ow is governed by the potential vorticity equation, ( ?? ), and linearizing this about a state of rest gives

t

2 +1

(z)

z(z)F(z)

z +

x =0 , (1.72)


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where is the density prole of the basic state and F(z) =f 20 /N 2 . (F is the square of theinverse Prandtl ratio, N/f 0 .) In the Boussinesq approximation = 0 , i.e., a constant.The vertical boundary conditions are determined by the thermodynamic equation, ( ?? ).If the boundaries are at, rigid, slippery surfaces then w = 0 at the boundaries and if there is no surface buoyancy gradient the linearized thermodynamic equation is

t

z =0 . (1.73)

We apply this at the ground and, with somewhat less justication, at the tropopause the higher static stability of the stratosphere inhibits vertical motion. If the ground isnot at or if friction provides a vertical velocity by way of an Ekman layer, the boundary condition must be correspondingly modied, but we will stay with the simplest case hereand apply ( 3.73 ) at z =0 and z =H .

1.5.2 Wave motion

As in the single-layer case, we seek solutions of the form

=Re (z) e i(kx +ly t) , (1.74)

where (z) will determine the vertical structure of the waves. The case of a sphere ismore complicated but introduces no truly new physical phenomena.

Substituting ( 3.74 ) into ( 3.72 ) gives

K 2 (z) + 1 z F(z) z k (z) =0 . (1.75)

Now, if satises1

z

F(z) z = , (1.76)

where is a constant, then the equation of motion becomes

K 2 + k =0 , (1.77)

and the dispersion relation follows, namely

= k

K 2 + . (1.78)

Equation ( 3.76 ) constitutes an eigenvalue problem for the vertical structure; the bound-ary conditions, derived from ( 3.73 ), are /z = 0 at z = 0 and z = H . The resultingeigenvalues, are proportional to the inverse of the squares of the deformation radii forthe problem and the eigenfunctions are the vertical structure functions.


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1.5 * Rossby Waves in Stratied Quasi-Geostrophic Flow 23

A simple example

Consider the case in which F(z) and are constant, and in which the domain is connedbetween two rigid surfaces at z = 0 and z = H . Then the eigenvalue problem for the vertical structure is

F 2 z 2 = (1.79a)

with boundary conditions of

z =0 , at z =0 , H. (1.79b)

There is a sequence of solutions to this, namely

n (z) =cos (n z/H), n =1, 2 . . . (1.80)

with corresponding eigenvalues

n =n 2F 2

H 2 =(n )2 f 0

NH

2, n =1, 2 . . . . (1.81)

Equation ( 3.81 ) may be used to dene the deformation radii for this problem, namely

Ln 1

n =NH

n f 0. (1.82)

The rst deformation radius is the same as the expression obtained by dimensional analysis, namely NH/f , except for a factor of . (Denitions of the deformation radii both with and without the factor of are common in the literature, and neither is obviously more correct. In the latter case, the rst deformation radius in a problem with uniformstratication is given by NH/f , equal to / 1 .) In addition to these baroclinic modes,the case with n =0, that is with =1, is also a solution of ( 3.79 ) for any F(z) .

Using ( 3.78 ) and ( 3.81 ) the dispersion relation becomes

= k

K 2 +(n ) 2 (f 0 /NH) 2, n =0 , 1 , 2 . . . (1.83)

and, of course, the horizontal wavenumbers k and l are also quantized in a nite domain.The dynamics of the barotropic mode are independent of height and independent of thestratication of the basic state, and so these Rossby waves are identical with the Rossby waves in a homogeneous uid contained between two at rigid surfaces. The structureof the baroclinic modes, which in general depends on the structure of the stratication,becomes increasingly complex as the vertical wavenumber n increases. This increasingcomplexity naturally leads to a certain delicacy, making it rare that they can be unam-biguously identied in nature. The eigenproblem for a realistic atmospheric prole isfurther complicated because of the lack of a rigid lid at the top of the atmosphere. 6


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Rossby Waves

Rossby waves are waves that owe their existence to a gradient of potential vorticity in the uid. If a uid parcel is displaced, it conserves its potential vorticity and so itsrelative vorticity will in general change. The relative vorticity creates a velocity eldthat displaces neighbouring parcels, whose relative vorticity changes and so on.

A common source of a potential vorticity gradient is differential rotation, or the -effect. In the presence of non-zero the ambient potential vorticity increases northward and the phase of the Rossby waves propagates westward. In general, Rossby waves propagate pseudo-westwards, meaning to the left of the direction of the poten-tial vorticity gradient.

In the simple case of a single layer of uid with no mean ow the equation of motionis

t (

2

+k2d ) +

x =0 (RW.1)

with dispersion relation

= k

k2 +l2 +k2d. (RW.2)

The phase speed is always negative, or westward. If there is no mean ow, the com-ponents of the group velocity are given by

c xg =(k 2 l2 k2d )(k 2 +l2 +k2d ) 2

, c y g =2kl

(k 2 +l2 +k2d ) 2, (1.3a,b)

The group velocity is westward if the zonal wavenumber is sufciently small, andeastward if the zonal wavenumber is sufciently large.

The reection of such Rossby waves at a wall is (like light) specular, meaning thatthe group velocity of the reected wave makes the same angle with the wall as thegroup velocity of the incident wave. The energy ux of the reected wave is equaland opposite to that of the incoming wave in the direction normal to the wall.

Rossby waves exist in stratied uids, and have a similar dispersion relation to ( RW.3 ) with an appropriate redenition of the inverse deformation radius, kd .

1.6 ENERGY FLUX OF ROSSBY WAVES

We now consider how energy is uxed in Rossby waves. To keep matters reasonably simple from an algebraic point of view we will consider waves in a single layer and without a mean ow, but we will allow for a nite radius of deformation. To remindourselves, the dynamics are governed by the evolution of potential vorticity equation


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1.6 Energy Flux of Rossby Waves 25

and the linearized evolution equation is

t

2 k2d +x =0. (1.84)

The dispersion relation follows in the usual way and is =

kK 2 +k2d

. (1.85)

which is a simplication of ( 3.59 ), and the group velocities are

c xg =(k 2 l2 k2d )

(K 2 +k2d ) 2, c y g =

2kl(K 2 +k2d ) 2

, (1.86a,b)

which are simplications of ( 3.60 ), and as usual K 2 =k2 +l2 .To obtain an energy equation multiply ( 3.84 ) by to obtain, after a couple of lines

of algebra,

12 t ()2 +k2d 2 t +i 2 =0 . (1.87)

where i is the unit vector in the x direction. The rst group of terms are the energy itself,or more strictly the energy density. (An energy density is an energy per unit mass or perunit volume, depending on the context.) The term (

) 2 / 2 = (u 2 +v 2 )/ 2 is the kinetic

energy and k2d 2 / 2 is the potential energy, proportional to the displacement of the freesurface, squared. The second term is the energy ux, so that we may write

E t + F =0. (1.88)

where E = () 2 / 2 +k2d 2 and F = /t +i 2 . We havent yet used the factthat the disturbance has a dispersion relation, and if we do so we may expect, followingthe derivations of section 3.2 , that the energy moves at the group velocity. Let us nowdemonstrate this explicitly.

We assume solution of the form

=A(x) cos ( k x t) =A(x) cos (kx +ly t) (1.89) where A(x) is assumed to vary slowly compared to the nearly plane wave. (Note that kis the wave vector, to be distinguished from k , the unit vector in the z -direction.) Thekinetic energy in a wave is given by

KE

=

A2

2 2x

+ 2y (1.90)

so that, averaged over a wave period,

KE =A2

2(k 2 +l2 )

2 2 /0 sin 2 (k x t) d t . (1.91)

The time-averaging produces a factor of one half, and applying a similar procedure1 tothe potential energy we obtain

KE =A2

4(k 2 +l2 ), PE =

A2

4k2d , (1.92)


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so that the average total energy is

E =A2

4(K 2 +k2d ), (1.93)

where K 2

=k2

+l2 .

The ux, F , is given by

F = t +i

2 = A2 cos2 ( k x t) k i2

, (1.94)

so that evidently the energy ux has a component in the direction of the wavevector,k , and a component in the x-direction. Averaging over a wave period straightforwardly gives us additional factors of one half:

F = A2

2k +i

2

. (1.95)

We now use the dispersion relation = k/(K 2 +k2d ) to eliminate the frequency, givingF =

A2 2

k kK 2 +k2d

i 12, (1.96)

and writing this in component form we obtain

F = iA2

4k2 l2 k2d

K 2 +k2d + j

2klK 2 +k2d

. (1.97)

Comparison of ( 3.97 ) with ( 3.86 ) and ( 3.93 ) reveals that

F =c g E (1.98)so that the energy propagation equation, ( 3.88 ), when averaged over a wave, becomes

E t + c g E =0 . (1.99)

It is interesting that the variation of A plays no role in the above manipulations, sothat the derivation appears to go through if the amplitude A( x , t) is in fact a constant andthe wave is a single plane wave. This seems to y in the face of our previous discussion,in which we noted that the group velocity was the velocity of a wave packet or at least of a superposition of plane waves. Indeed, the derivative of the frequency with respect to wavenumber means little if there is only one wavenumber. In fact there is nothing wrong with the above derivation if A is a constant, and the resolution of the paradox arises by noting that a plane wave lls all of space and time. In this case there is no convergenceof the energy ux and the energy propagation equation is trivially true.

1.6.1 Rossby wave reection

We now consider how Rossby waves might be reected from a solid boundary. The topichas particular oceanographic relevance, for the reection of Rossby waves turns out toone way of interpreting why intense oceanic boundary currents form on the western sidesof ocean basins, not the east. As a preliminary, let us give a useful graphic interpretationof Rossby wave propagation. 7


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l

k

k c = 2

+

2

2

k 2d

1 / 2

k c

c g

C

W k

O

C = 2

, 0

| W C | =

2

2

k 2d

1 / 2

Fig. 1.8 The energy propagation diagram for Rossby waves. The wavevectors of agiven frequency all lie in a circle of radius [(/ 2) 2 k2d ] 1/ 2 , centered at the point C .The closest distance of the circle to the origin is k c , and if the deformation radius isinnite k c the circle touches the origin. For a given wavenumber k , the group velocityis along the line directed from W to C .

The energy propagation diagram

The dispersion relation for Rossby waves, = k/(k 2 +l2 +k2d ) , may be rewritten as(k +/ 2)

2+l2 =(/ 2)

2k2d . (1.100)

This equation is the parametric representation of a circle, meaning that the wavevector(k,l) must lie on a circle centered at the point (/ 2, 0) and with radius [(/ 2) 2 k2d ] 1 / 2 , as illustrated in Fig. 3.8 . If the deformation radius is zero the circle touches theorigin, and if it is nonzero the distance of the closest point to the circle, k c say, is givenby k c = / 2 +[(/ 2) 2 k2d ] 1 / 2 . For low frequencies, specically if / 2k , thenk c k 2d / . The radius of the circle is a positive real number only when < / 2kd .This is the maximum frequency possible, and it occurs when l =0 and k =kd and whenc xg =c

y g =0.

It turns out that the group velocity, and hence the energy ux, can be visualizedgraphically from Fig. 3.8 . By direct manipulation of the expressions for group velocity and frequency we nd that

c xg =2

K 2 +k2d2 k +

2

, (1.101a)

c y g =2

K 2 +k2d2 l. (1.101b)

(To check this, it is easiest to begin with the right-hand sides and use the dispersionrelation for .) Now, since the center of the circle of wavevectors is at the position


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Fig. 1.9 The reection of a Rossby wave at awestern wall, in physical space. A Rossby wavewith a westward group velocity impinges at anangle i to a wall, inducing a reected wavemoving eastward at an angle r . The reectionis specular, with r = i , and energy conserv-ing, with |c g r | = |c gi | see text and Fig. 3.10 .

i r

c g i

c gr

x

y

(/ 2, 0) , and referring to Fig. 3.8 , we have that

c g =2

(K 2 +k2d ) 2R (1.102)

where R = W C is the vector directed from W to C , that is from the end of the wavevector

itself to the center of the circle around which all the wavevectors lie.Eq. (3.102 ) and Fig. 3.8 allow for a useful visualization of the energy and phase.

The phase propagates in the direction of the wave vector, and for Rossby waves this isalways westward. The group velocity is in the direction of the wave vector to the centerof the circle, and this can be either eastward (if k2 > l 2 +k2d ) or westward ( k2 < l 2 +k2d ).Interestingly, the velocity vector is normal to the wave vector. To see this, consider apurely westward propagating wave for which l = 0. Then v = /x = ik and u =/y = il =0. We now see how some of these properties can help us understandthe reection of Rossby waves.

[Do we need a gray box summarizing some of the properties of reection? xxx]

Reection at a wall

Consider Rossby waves incident on wall making an angle with the x-axis, and supposethat somehow these waves are reected back into the uid interior. This is a reasonableexpectation, for the wall cannot normally simply absorb all the wave energy. We rst notea couple of general properties about reection, namely that the incident and reected wave will have the same wavenumber component along the wall and their frequenciesmust be the same To see these properties, consider the case in which the wall is orientedmeridionally, along the y -axis with =90. There is no loss of generality in this choice,because we may simply choose coordinates so that y is parallel to the wall and the -effect, which differentiates x from y , not enter the argument. The incident and reected waves are

i (x,y,t) =A i e i(k i x+li y i t) , r (x,y,t) =Ar e i(k r x+lr y r t) , (1.103)


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with subscripts i and r denoting incident and reected. At the wall, which we take to beat x =0, the normal velocity u = /y must be zero so that

A i li e i(l i y i t) +A r lr e i(l r y r t) =0 . (1.104)For this equation to hold for all y and all time then we must have

lr = li , r = i . (1.105)This result is independent of the detailed dynamics of the waves, requiring only that the velocity is determined from a streamfunction. When we consider Rossby-wave dynamicsspecically, the x- and y -coordinates are not arbitrary and so the wall cannot be takento be aligned with the y -axis; rather, the result means that the projection of the incident wavevector, k i on the wall must equal the projection of the reected wavevector, k r . Themagnitude of the wavevector (the wavenumber) is not in general conserved by reection.Finally, given these results and using ( 3.104 ) we see that the incident and reectedamplitudes are related by

Ar = A i . (1.106)Now lets delve a little deeper into the wave-reection properties.

Generally, when we consider a wave to be incident on a wall, we are supposing thatthe group velocity is directed toward the wall. Suppose that a wave of given frequency, , and wavevector, k i , and with westward group velocity is incident on a predominantly western wall, as in Fig. 3.9 . (Similar reasoning, mutatis mutandis, can be applied to a wave incident on an eastern wall.) Let us suppose that incident wave, k i lies at the point I on the wavenumber circle, and the group velocity is found by drawing a line from I tothe center of the circle, C (so c gi

IC ), and in this case the vector is directed westward.The projection of the k i must be equal to the projection of the reected wave vector,k r , and both wavevectors must lie in the same wavenumber circle, centered at / 2 ,

because the frequencies of the two waves are the same. We may then graphically deter-mines the wavevector of the reected wave using the construction of Fig. 3.10 . Giventhe wavevector, the group velocity of the reected wave follows by drawing a line fromthe wavevector to the center of the circle (the line

RC ). We see from the gure thatthe reected group velocity is directed eastward and that it forms the same angle to the wall as does the incident wave; that is, the reection is specular. Since the amplitudeof the incoming and reected wave are the same, the components of the energy uxperpendicular to the wall are equal and opposite. Furthermore, we can see from thegure that the wavenumber of the reected wave has a larger magnitude than that of

the incident wave. For waves reecting of an easter boundary, the reverse is true. Putsimply, at a western boundary incident long waves are reected as short waves, whereasat an eastern boundary incident short waves are reected as long waves.

Quantitatively solving for the wavenumbers of the reected wave is a little tedious inthe case when the wall is at angle, but easy enough if the wall is a meridional, along they -axis. We know the frequency, , and the y -wavenumber, l, so that the x-wavenumberis may be deduced from the dispersion relation

= k i

k2i +l2 +k2d =k r

k2r +l2 +k2d. (1.107)


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l

kC O

k i

k r

I

Rc gi

c gr

x

y

Fig. 1.10 Graphical representation of the reection of a Rossby wave at a westernwall, in spectral space. The incident wave has wavevector k i , ending at point I . Con-struct the wavevector circle through point I with radius (/ 2) 2 k2d and centerC = (/ 2, 0) ; the group velocity vector then lies along

IC and is directed west-ward. The reected wave has a wavevector k r such that its projection on the wall isequal to that of k i , and this xes the point R. The group velocity of the reected wavethen lies along

RC , and it can be seen that c g r makes the same angle to the wall asdoes c g i , except that it is directed eastward. The reection is therefore both specularand is such that the energy ux directed away from the wall is equal to the energyux directed toward the wall.

We obtain

k i =

2 + 2 2 l2 +k2d , k r = 2 2 2 l2 +k2d . (1.108a,b)The signs of the square-root terms are chosen for reection at a western boundary, for which, as we noted, the reected wave has a larger (absolute) wavenumber than theincident wave. For reection at an eastern boundary, we simply reverse the signs.

Oceanographic relevance

The behaviour of Rossby waves at lateral boundaries is not surprisingly of some oceano-graphic importance, there being two particularly important examples. One of them con-cerns the equatorial ocean, and the other the formation of western boundary currents,common in midlatitudes. We only touch on these topics here, deferring a more extensivetreatments to later chapters.

Suppose that Rossby waves are generated in the middle of the ocean, for exampleby the wind or possibly by some uid dynamical instability in the ocean. Shorter waves


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will tend to propagate eastward, and be reected back at the eastern boundary as long waves, and long waves will tend to propagate westward, being reected back as short waves. The reection at the western boundary is believed to be particularly importantin the dynamics of El Nio. although the situation is further complicated because the

reection may also generate eastward moving equatorial Kelvin waves, which we discussmore in the next chapter.In mid-latitudes the reection at a western boundary generates Rossby waves that

have a short zonal length scale (the meridional scale is the same as the incident waveif the wall is meridional), which means that their meridional velocity is large. Now, if the zonal wavenumber is much larger than both the meridional wavenumber l and theinverse deformation radius kd then, using either ( 3.57 ) or ( ?? ) the group velocity in thex-direction is given by cg x =U +/k 2 , where U is the zonal mean ow. If the mean owis westward, so that U is negative, then very short waves will be unable to escape fromthe boundary; specically, if k > / U then the waves will be trapped in a westernboundary layer. [More here ?? xxx]1.7 ROSSBY-GRAVITY WAVES: AN INTRODUCTION

We now consider Rossby waves and shallow water gravity waves together, keeping f constant except where differentiated, following Hendershott (5.11). The equations of motion are the shallow water equations in Cartesian coordinates in a rotating frame of reference, namely

ut f v = g

x

,v t +f u = g

y

, (1.109a,b)

t +c

2 ux +

v y =0 (1.109c)

where, in terms of possibly more familiar shallow water variables, =g and c 2 =g H , where is the kinematic pressure, c will be a wave speed, is the free surface height, H is the reference depth of the uid and g is the reduced gravity.

After some manipulation (described more fully in section ?? ) we may obtain, withoutadditional approximation, a single equation for v , namely

1c 2

3v t 3 +

f 2

c 2v t

t

2v v x =0 . (1.110)

In this equation the Coriolis parameter is given by the -plane expression f = f 0 +y ;thus, the equation has a non-constant coefcient, entailing considerable algebraic dif-culties. We will address some of these difculties in chapter ?? , but for now we take asimpler approach: we assume that f is constant except where differentiated, an approx-imation that is reasonable in mid-latitudes provided we are concerned with sufciently small variations in latitude. Equation ( 3.110 ) then has constant coefcients and we may look for plane wave solutions of the form v =v exp [ i(k x t)] , whence

2 f 20c 2 (k

2 +l2 ) k =0. (1.111a)


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Rossby waves

l

k

k

O

C =

2

, 0

C

k c

R

R =

2

2

+ 2 f 20

c 2

1 / 2

2

2

f 20c 2

1 / 2

k c = 2

+ R f 20c 2

k

k

OC

R

R = 2

2

+ 2 f 20

c 2

1/

2

2 f 20

c 21 / 2

C = 2

, 0

k c = 2

+ R R

Gravity waves

Fig. 1.11 Wave propagation diagrams for Rossby-gravity waves, obtained using(3.111 ). The top gure shows the diagram in the low frequency, Rossby wave limit,and the bottom gure shows the high frequency, gravity wave limit. In each case thethe locus of wavenumbers for a given frequency is a circle centered at C = (/ 2, 0)with a radius R given by (3.117 ), but the approximate expressions differ signicantlyat high and low frequency.


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Gravity waves

l = 0

l = 1

l = 2

Wave numb er , kc/f 0

F r e q u e n c y

,

/ f 0

Pla n etary waves , l = 0 , 1 , 2

Fig. 1.12 Dispersion relation for Rossby-gravity waves, obtained from (3.123 ) with =0 .2 for three values of l. There a frequency gap between the Rossby or planetarywaves and the gravity waves. For the stratied mid-latitude atmosphere or ocean thefrequency gap is in fact much larger.

or, written differently,

k +

2

2

+l2 =

2

2

+ 2 f 20

c 2. (1.111b)

This equation may be compared to ( 3.100 ): noting that k2d = f 20 /g H = f 20 /c 2 , the twoequations are identical except for the appearance of a term involving the frequency onthe right-hand side in ( 3.117 ). The wave propagation diagram is illustrated in Fig. ?? .The wave vectors a a given frequency all lie on a circle centered at (/ 2, 0) and withradius r given by

R

=

2

2

+ 2 f 20

c 2

1 / 2

, (1.112)

and the radius must be positive in order for the waves to exist. The wave propagationdiagram is illustrated in Fig. 3.11 . In the low frequency case the diagram is essentially the same as that shown in Fig. 3.8 , but is quantitatively signicantly different in the highfrequency case. These limiting cases are discussed further in section 3.7.1 below.

To plot the full dispersion relation it is useful to nondimensionalize using the follow-ing scales for time ( T ), distance ( L) and velocity ( U )

T = f 10 , L =Ld =k1d =c/f 0 , U =L/T =c, (1.113a,b)


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so that, denoting nondimensional quantities with a hat,

=f 0 , (k, l) = (k, l)k d , =f 20c =

f 0Ld =f 0 kd . (1.114)

The dispersion relation ( 3.111 ) may then be written as

2 1 ( k2 +l2 ) k =0 (1.115)

This is a cubic equation in , as might be expected given the governing equations(3.109 ). We may expect that two of the roots correspond to gravity waves and the thirdto Rossby waves. The only parameter in the dispersion relation is = c/f 20 = L d /f 0 .In the atmosphere a representative value for Ld is 1000km, whence = 0 .1. In theocean Ld 100km, whence =0.01. If we allow ourselves to consider external Rossby waves (which are of some oceanographic relevance) then c = gH = 200ms 1 andLd =2000km, whence =0.2.

To actually obtain a solution we regard the equation as a quadratic in k and solve interms of the frequency, giving

k =

2 12

2

2 +4( 2 l2 1)

1 / 2

. (1.116)

The solutions are plotted in Fig. 3.12 , with =0 .2, and we see that the waves fall intotwo groups, labelled gravity waves and planetary waves in the gure. The gap betweenthe two groups of waves is in fact far larger if a smaller (and generally more relevant) value of is used. To interpret all this let us consider some limiting cases.

1.7.1 Wave propertiesWe now consider a few special cases of the dispersion relation.

(i) Constant Coriolis parameter

If =0 then the dispersion relation becomes 2 f 20 (k 2 +l2 )c 2 =0 , (1.117)

with the roots = 0 and 2 = f 20 +c 2 (k 2 + l2 ) . The root = 0 corresponds togeostrophic motion (and, since = 0, Rossby waves are absent), with the otherroot corresponding to Poincar waves, considered at length in chapter ?? , for which 2 > f 20 .

(ii) High frequency waves

If we take the limit of f 0 then ( 3.111a ) gives

2

c 2 (k2 +l2 )

k =0. (1.118)

To be physically realistic we should also now eliminate the term, because if f 0then, from geometric considerations on a sphere, k2 k/ . Thus, the dispersionrelation is simply 2 =c 2 (k 2 +l2 ) . These waves are just gravity waves uninuencedby rotation, and are a special case of Poincar waves.


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Rossby-Gravity Waves

Generically speaking, Rossby-gravity waves are waves that arise under the com-bined effects of a potential vorticity gradient and stratication. The simplest set-ting in which they occur is in the linearized shallow water equations which may be written as a single equation for v , namely

1c 2

3 v t 3 +

f 2

c 2v t

t

2 v v x =0 . (RG.1)

If we take both f and to be constants then the equation above admits of plane- wave solutions with dispersion relation

2 kc 2

=f 20 +c 2 (k 2 +l2 ). (RG.2)

In Earths atmosphere and ocean is common for there to be a frequency separationbetween the two classes of solution. To a good approximation, high frequency,gravity waves then satisfy

2 =f 20 +c 2 (k 2 +l2 ), (RG.3)and low frequency, Rossby waves satisfy

= kc 2

f 20 +c 2 (k 2 +l2 ) =k

k2d +k2 +l2(RG.4)

where k2d = f 20 /c 2 . Rossby-gravity waves also exist in the stratied equations. Solutions may be found

be decomposing the vertical structure into a series of orthogonal modes, and asequence of shallow water equations for each mode results, with a different c foreach mode. Solutions may also be found if f is allowed to vary in ( RG.1), at theprice algebraic complexity, as discussed in chapter ?? .

(iii) Low frequency waves

Consider the limit of f 0 . The dispersion relation reduces to

= kk2 +l2 +k2d. (1.119)

This is just the dispersion relation for quasi-geostrophic waves as previously ob-tained see ( 3.59 ) or ( 3.85 ) In this limit, the requirement that the radius of thecircle be positive becomes

2 < 2

4k2d. (1.120)

That is to say, the Rossby waves have a maximum frequency, and directly from


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Fig. 1.13 The locus of points onplanetary-geostrophic Rossby waves.Waves of a given frequency all have thesame x-wavenumber, given by xxx

l

kOk 1

k 2

k 3

Locus of wavenumbers of given frequency

k = f 20

c 2

(3.120 ) this occurs when k =kd and l =0.The frequency gap

The maximum frequency of Rossby waves is usually much less than the frequency of the Poincar waves: the lowest frequency of the Poincare waves is f 0 and the highestfrequency of the Rossby waves is / 2kd . Thus,

Low gravity wave frequencyHigh Rossby wave frequency =

f 0/ 2kd =

f 202c

. (1.121)

If f 0 = 10 4 s1 , = 10 11 m1 s1 and kd = 1/ 100km 1 (a representative oceanic

baroclinic deformation radius) then f 0 /(/ 2kd ) =200. If Ld =1000km (an atmosphericbaroclinic radius) then the ratio is 20. If use a barotropic deformation radius of Ld =2000 km then the ratio is 10. Evidently, for most midlatitude applications there is a largegap between the Rossby wave frequency and the gravity wave frequency.

Because of this frequency gap, to a good approximation Fig. 3.12 may be obtained by separately plotting ( 3.117 ) for the gravity waves, and ( 3.120 ) for the Rossby or planetary waves. The differences between these and the exact results become smaller as gets

smaller, and are less than the thickness of a line on the plot shown.

1.7.2 Planetary geostrophic Rossby waves

A good approximation for the large-scale ocean circulation involves ignoring the time-derivatives and nonlinear terms in the momentum equation, allowing evolution only tooccur in the thermodynamic equation. This is the planetary-geostrophic approximation,introduced in section ?? ?? , and it is interesting to see to what extent that system sup-ports Rossby waves. 8 It is easiest just to begin with the linear shallow water equations


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themselves, and omitting time derivatives in the momentum equation gives

f v = x

, f u = y

, (1.122a,b)

t +c

2 ux +

v y =0 . (1.122c)

From these equations we straightforwardly obtain

t

c 2 f 2

x =0. (1.123)

Again we will treat both f and as constants so that we may look for solutions in theform = exp [ i( k x t)] . The ensuing dispersion relation is

= c 2

f 20

k

= k

k2d

(1.124)

which is a limiting case of ( 3.120 ) with k2 , l 2 k2d . The waves are a form of Rossby waves with phase and group speeds given by

c p = c 2 f 20

, c xg = c 2 f 20

. (1.125)

That is, the waves are non-dispersive and propagate westward. Eq. ( 3.123 ) has thegeneral solution = G(x +c 2 /f 2 t) , where G is any function, so an initial disturbance will just propagate westward at a speed given by ( 3.125 ), without any change in form.

Note nally that the locus of wavenumbers in kl space is no longer a circle, as itis for the usual Rossby waves. Rather, since the frequency does not depend on the y - wavenumber, the locus is a straight line, parallel to the y -axis, as in Fig. 3.13 . Waves of agiven frequency all have the same x-wavenumber, given by k = f 20 /(c 2 ) = k 2d / ,as shown in Fig. 3.13 .

Physical mechanism

Because the waves are a form of Rossby wave their physical mechanism is related tothat discussed in section 3.4.2 , but with an important difference: relative vorticity is nolonger important, but the ow divergence is. Thus, consider ow round a region of highpressure, as illustrated in Fig. 3.14 . If the pressure is circularly symmetric as shown, theow to the south of H in the left-hand sketch, and to the south of L in the right-handsketch, is larger than that to the north. Hence, in the left sketch the ow converges atW and diverges at E , and the ow pattern moves westward. In the ow depicted in theright sketch the low pressure propagates westward in a similar fashion.


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H LW E W E

convergenceat W

divergenceat E

divergenceat W

convergenceat E

N

S

W E

Fig. 1.14 The westward propagation of planetary-geostrophic Rossby waves. The cir-cular lines are isobars centered around high and low pressure centres. Because of thevariation of the Coriolis force, the mass ux between two isobars is greater to thesouth of a pressure center than it is to the north. Hence, in the left-hand sketch there

is convergence to the west of the high pressure and the pattern propagates westward.Similarly, if the pressure centre is a low, as in the right-hand sketch, there is diver-gence to the west of the pressure centre and the pattern still propagates westward.

1.8 THE GROUP VELOCITY PROPERTY

In the last section of this chapter we return to a more general discussion of group velocity.Our goal is to show that the group velocity arises in fairly general ways, not just frommethods stemming from Fourier analysis or from ray theory. In a purely logical sense thisdiscussion does follow most naturally from the end of the section on ray theory (section3.3 ), but for most humans it is helpful to have had a concrete introduction to at least one

nontrivial form of waves before considering more abstract material. We rst give a simpleand direct derivation of group velocity that valid in the simple but important special caseof a homogeneous medium., 9 Then, in section 3.8 , we give a rather general derivationof the group velocity property, namely that conserved quantities that are quadratic in the wave amplitude quantities known as wave activities are transported at the group velocity.

1.8.1 Group velocity in homogeneous media

Consider, 10 waves propagating in a homogeneous medium in which the wave equationis a polynomial of the general form

L() =t

,

x(x,t) =0. (1.126)

where is a polynomial operator in the space and time derivatives. For algebraic sim-plicity we again restrict attention to waves in one dimension, and a simple example isL() = ( xx )/t +/x or = ( xx )/t +/x . We will seek a solution of theform [c.f., ( 3.4 )]

(x,t) =A(x,t) e i(x,t) , (1.127)


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1.8 The Group Velocity Property 39

where is the phase of the disturbance and A(x,t) is the slowly varying amplitude, sothat the solution has the form of a wave packet. The phase is such that k = /xand = /t , and the slowly varying nature of the envelope A(x,t) is formalized by demanding that

1A

Ax k,

1A

At , (1.128)

The space and time derivatives of are then given by

x =

Ax +iA

x

e i =Ax +iAk e

i , (1.129a)

t =

At +iA

t

e i =At iA e

i , (1.129b)

so that the wave equation becomes

=

t i,

x +ik A

=0 . (1.130)

Noting that the space and time derivative of A are small compared to k and weexpand the polynomial in a Taylor series about (, k) to obtain

( i, ik)A +

( i)At +

( ik)

Ax =0 . (1.131)

The rst terms is nothing but the linear dispersion relation; that is ( i, ik)A =0 is thedispersion relation for plane waves. Taking this to be satised, ( 3.131 ) gives

At

/k/

Ax =

At +

k

Ax =0 . (1.132)

That is, the envelope moves at the group velocity /k .

1.8.2 Group velocity property: a general derivation

In our discussion of Rossby waves in section 3.6 in ( 3.99 ) we showed that the energy of the waves is conserved in the sense that

E t + F =0, (1.133)

where E is the energy density of the waves and F is its ux. In ( 3.99 ) we further showedthat, when averaged over a wavelength and a period, the average ux was related to theenergy by F = c g E . This property is called the group velocity property and it is a very general property, not restricted to Rossby waves or even to energy. Rather, it is a property of almost any conserved quantity that is quadratic in the wave amplitude, which is hedening property of a wave activity, and we now demonstrate this in a rather general way. 11 It is a useful property, because if we can use observations to deduce c g then wecan determine how wave activity density propagates.

Our derivation holds generally for waves and wave activities that satisfy the followingthree assumptions.


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(i) The wave activity, A, and ux, F , obey the general conservation relation

At + F =0 . (1.134)

(ii) Both the wave activity and the ux are quadratic functions of the wave amplitude.

(iii) The waves themselves are of the general form

= e i( x ,t) +c.c. , =k x t, =( k ), (1.135a,b,c) where ( 3.135 c) is the dispersion relation, and is any wave eld. We will carry outthe derivation in case in which is a constant, but the derivation may be extendedto the case in which it varies slowly over a wavelength.

Given assumption (ii) , the wave activity must have the general form

A =b +a e2i( k

x

t)

+ae2i(k

x

t)

, (1.136a) where the asterisk, , denotes complex conjugacy, and b is a real constant and a is acomplex constant. For example, suppose that A = 2 and =c e i( k xt) +c ei( k xt) ,then we nd that ( 3.136a ) is satised with a = c 2 and b =2cc . Similarly, the ux hasthe general form

F =g +f e2i( k xt) +f e2i( k xt) . (1.136b) where g is a real constant vector and f is

Wave Intro

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