forced rossby wave packets in barotropic shear flows with

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Ž . Dynamics of Atmospheres and Oceans 28 1998 9–37 Forced Rossby wave packets in barotropic shear flows with critical layers L.J. Campbell, S.A. Maslowe ) Department of Mathematics and Statistics, McGill UniÕersity, Montreal, Quebec, Canada, H3A 2K6 ´ ´ Received 28 June 1997; revised 10 February 1998; accepted 10 February 1998 Abstract In this paper, we investigate the meridional propagation of a forced Rossby wave packet towards a critical layer in a zonal shear flow by solving the linearized barotropic vorticity equation. The forcing is applied north of the critical layer. Two approaches are employed for solving this problem. First, an analytic solution valid for large time is derived, using Fourier and Laplace transform techniques and asymptotic approximations. This solution exhibits the modifica- Ž . w tion due to the wave packet of the solution obtained by Warn and Warn 1976 Warn, T., Warn x H., 1976. On the development of a Rossby wave critical level. J. Atmos. Sci., 33, 2021–2024. in the monochromatic case. A numerical investigation is then carried out using a finite difference scheme and a time-dependent radiation condition. It is found that the forced wave packet is absorbed at the critical layer and the total momentum transferred to the mean flow as a result of the absorption is observed to be proportional to the length scale of the wave packet. We also consider the case of a north–south mean flow with a longitudinally propagating wave packet forced to the east or west of the critical layer. The monochromatic version of this problem has Ž been used before Geisler, J.E., Dickinson, R.E., 1975. Critical level absorption of barotropic . Rossby waves in a north–south flow. J. Geophys. Res., 80, 3805–3811. to examine the interaction of western boundary currents and oceanic Rossby waves. q 1998 Elsevier Science B.V. All rights reserved. Keywords: Rossby wave packets; Barotropic shear flows; Critical layers ) Corresponding author. Tel.: q1-514-398-3800; fax: q1-514-398-3899; e-mail: [email protected] 0377-0265r98r$19.00 q 1998 Elsevier Science B.V. All rights reserved. Ž . PII S0377-0265 98 00044-X

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Page 1: Forced Rossby wave packets in barotropic shear flows with

Ž .Dynamics of Atmospheres and Oceans 28 1998 9–37

Forced Rossby wave packets in barotropic shearflows with critical layers

L.J. Campbell, S.A. Maslowe )

Department of Mathematics and Statistics, McGill UniÕersity, Montreal, Quebec, Canada, H3A 2K6´ ´

Received 28 June 1997; revised 10 February 1998; accepted 10 February 1998

Abstract

In this paper, we investigate the meridional propagation of a forced Rossby wave packettowards a critical layer in a zonal shear flow by solving the linearized barotropic vorticityequation. The forcing is applied north of the critical layer. Two approaches are employed forsolving this problem. First, an analytic solution valid for large time is derived, using Fourier andLaplace transform techniques and asymptotic approximations. This solution exhibits the modifica-

Ž . wtion due to the wave packet of the solution obtained by Warn and Warn 1976 Warn, T., WarnxH., 1976. On the development of a Rossby wave critical level. J. Atmos. Sci., 33, 2021–2024. in

the monochromatic case. A numerical investigation is then carried out using a finite differencescheme and a time-dependent radiation condition. It is found that the forced wave packet isabsorbed at the critical layer and the total momentum transferred to the mean flow as a result ofthe absorption is observed to be proportional to the length scale of the wave packet. We alsoconsider the case of a north–south mean flow with a longitudinally propagating wave packetforced to the east or west of the critical layer. The monochromatic version of this problem has

Žbeen used before Geisler, J.E., Dickinson, R.E., 1975. Critical level absorption of barotropic.Rossby waves in a north–south flow. J. Geophys. Res., 80, 3805–3811. to examine the

interaction of western boundary currents and oceanic Rossby waves. q 1998 Elsevier ScienceB.V. All rights reserved.

Keywords: Rossby wave packets; Barotropic shear flows; Critical layers

) Corresponding author. Tel.: q1-514-398-3800; fax: q1-514-398-3899; e-mail: [email protected]

0377-0265r98r$19.00 q 1998 Elsevier Science B.V. All rights reserved.Ž .PII S0377-0265 98 00044-X

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( )L.J. Campbell, S.A. MaslowerDynamics of Atmospheres and Oceans 28 1998 9–3710

1. Introduction

In the linear, inviscid theory of wave propagation, a singularity can occur in theequation for the amplitude of the disturbance when there is a point where the phasevelocity of the disturbance is equal to the speed of the shear flow. This point is referredto by meteorologists as the critical level, and the adjacent region is called the criticallayer.

Critical layers occur in a wide variety of circumstances, including many applicationsin physics and engineering, as well as in geophysical fluid dynamics. In this paper, weshall focus on examples in the atmosphere and ocean and, specifically, on phenomena

Ž .related to horizontally propagating Rossby planetary waves. Such problems, particu-larly the case in which Rossby waves are forced at some northern latitude and propagatetoward the equator, have received a great deal of attention during the past 20 years.

Interest in the subject has been stimulated by the belief that critical layer theories helpin the interpretation of certain phenomena observed in the atmosphere and ocean. For

Ž .instance, Lindzen and Holton 1968 argued that critical layer absorption of gravitywaves might be the cause of the changes in mean flow momentum that produce the

Žquasi-biennial oscillation and it has also been suggested Matsuno, 1971; Dunkerton et.al., 1981 that critical layer effects contribute to the dynamics of stratospheric sudden

Ž .warmings in the northern hemisphere Andrews et al., 1987, Section 6.3 .From a mathematical perspective, the singularity in the steady, linear inviscid

equation for a monochromatic wave can be removed by re-introducing to the criticallayer one or more of the effects neglected in the linear, inviscid theory for a neutralmode. These include viscosity, nonlinearity and time-dependence. In geophysical appli-cations where the Reynolds number is often very large, viscosity may be the leastimportant.

The introduction of time-dependence in the context of forced waves was firstŽ .considered by Booker and Bretherton 1967 in their study of the vertical propagation of

gravity waves in a stratified shear flow. They investigated a linear, inviscid model andshowed that near the critical layer the waves tend to be absorbed and are unable to

Ž . Ž .penetrate beyond. A few years later, Dickinson 1970 and then Warn and Warn 1976carried out analyses of the time evolution of a forced Rossby wave critical layer. An

Ž .application to ocean currents was reported by Geisler and Dickinson 1975 whoexamined the problem of critical level absorption of a longitudinally propagating Rossbywave in a barotropic fluid with the basic flow directed from north to south and madesome inferences relating their problem to that of western boundary currents in oceanbasins.

The above studies all concluded that the linear critical layer acts as a perfect absorberof incident waves. However, even in the article by Booker and Bretherton, it wasrecognized that the importance of nonlinearity in the critical layer increases with time

Ž .and at some point the linearized analysis breaks down. Analyses by Stewartson 1978Ž .and Warn and Warn 1978 used the multiple scaling method to continue the linear

solution into a nonlinear critical layer regime characterized by a balance between slowtemporal evolution and nonlinearity. There have since been a number of studies, usingboth analytic and numerical techniques, on the nonlinear development of critical layers,

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( )L.J. Campbell, S.A. MaslowerDynamics of Atmospheres and Oceans 28 1998 9–37 11

Že.g., Beland, 1976, 1978; Ritchie, 1982; Haynes, 1985; and Killworth and McIntyre,´.1985 . Comprehensive surveys of the results of research on the nonlinear critical layer

Ž . Ž .are given by Stewartson 1981 and Maslowe 1986 .Most of the foregoing studies have considered disturbances that are strictly periodic

in the zonal direction. One justification for this assumption is that the disturbance takesplace on a very large scale, and the zonal wavenumber is sometimes taken to be amultiple of the inverse of the earth’s circumference. However, for a disturbance on ascale smaller than the planetary scale, there is no reason to assume periodicity, sincewave motions in the atmosphere and ocean are affected by factors such as variations intopography and thermal processes which do not change periodically. Brunet and HaynesŽ .1996 observe that though the periodic model may be appropriate for large-scaledisturbances in the stratosphere, it may be less relevant for waves in the troposphere. Amore realistic model for actual atmospheric or oceanic motions would be a wave packetconsisting of a superposition of a range of different Fourier modes rather than a singlemonochromatic wave. The main objective of this paper therefore is to study thedevelopment of a forced disturbance propagating with an amplitude that varies slowly inthe zonal direction.

The major part of our discussion focuses on the meridional propagation of large scaleRossby waves generated in the middle atmosphere; an appropriate two-dimensionalmodel for this problem consists of a forced wave packet propagating horizontally

Ž .southwards in a zonal flow u y on a beta-plane. The beta-plane approximation is validfor describing atmospheric motions in the mid-latitudes, and under certain conditions

Žcan be used as a model in the equatorial region as well Pedlosky, 1979, pp. 105–108;.Gill, 1982, Chaps. 11–12 .

The ultimate aim of this type of investigation must be to solve the nonlinear,Ž .time-dependent equations. Brunet and Haynes 1996 carried out such an analysis to

investigate the nonlinear reflection of a Rossby wave packet at low latitudes. They useda shallow-water model in their numerical solution and showed that, for finite initialamplitudes and times longer than 10 days, the packet was reflected back into themid-latitudes. In this paper, however, we shall restrict ourselves to the linear case inwhich it is possible to obtain an explicit analytic solution to the forced wave packetproblem. By making use of a long wave approximation here, we shall derive asymptoticsolutions for the forced wave packet problem analogous to those found by DickinsonŽ . Ž .1970 and Warn and Warn 1976 in the monochromatic case. Fourier and Laplacetransform techniques will be employed to determine the effect the slow amplitudemodulation has on the critical layer development in the absence of dissipation andnonlinearity. We consider the effect of a forced disturbance in the form of a wave packetA cos kx, where the amplitude A is a slowly varying function of x, on a basic shear

Ž .flow in which the velocity is given by u y sy. Numerical solutions of the initial-valueŽ .problem are then presented in which, as in Beland 1976 , the velocity profile is´

Ž .u y s tanh y. With this velocity profile, one is justified in applying a radiationŽ .condition Beland and Warn, 1975 at the outflow boundary.´

We also carry out a numerical solution of the initial value problem in which the basicŽ .flow is in the north–south direction, i.e., its velocity is of the form Õ x , and the wave

packet propagates in the longitudinal direction. This problem, which can be considered

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( )L.J. Campbell, S.A. MaslowerDynamics of Atmospheres and Oceans 28 1998 9–3712

as a model for western boundary currents in the ocean, was solved by Geisler andŽ .Dickinson 1975 for a monochromatic wave without time-dependence. We solve this

problem with a forcing function of the form A cos ly, where A is a slowly varyingfunction of y, and with the forcing applied at the western or eastern boundary of theregion under consideration.

2. Critical layer evolution

We shall consider the propagation of disturbances in a two-dimensional region whosegeometry is described by the Cartesian coordinates x and y, with the x-axis pointing inthe eastward direction and the y-axis to the north. The evolution of the flow is taken tobe described by the barotropic vorticity equation

= 2C qC = 2C yC = 2C qbC s0, 1Ž .t x y y x x

where C represents the total streamfunction and b is the gradient of planetary vorticityin the y-direction. All the variables are assumed to be dimensionless and related to the

Ž .dimensional variables with stars by

x ) y) Ut ) c ) L2 b )

yxs , ys , ts , cs , bs ,

L L L w Ux y y

where L is a typical length scale in the y-direction, U a typical velocity scale, w is they

dimensional forcing amplitude, L is a typical length scale in the x-direction, of thex

same order of magnitude as the mean zonal wavelength, and c is the streamfunction ofthe disturbance. We can also define a non-dimensional zonal wave number by

ksL k ) ,x

where k ) is the dimensional wave number. The non-dimensional Laplacian operator isthen

E 2 E 22= sa q ,2 2E x E y

where asL2 rL2 is the aspect ratio which gives a measure of the magnitude of they x

dimensional zonal wavelength relative to the length scales in the y-direction.The equation can be linearized by expressing the streamfunction as the sum of a

steady mean part and a time-dependent perturbation component that is assumed to be ofsmall amplitude. If the mean flow is directed from west to east and is a function of yonly, we can write

C x , y ,t sc y qec x , y ,t , 2Ž . Ž . Ž . Ž .where

wes

L Uy

and

u y syc y .Ž . Ž .y

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The amplitude parameter e gives a measure of the nonlinearity and if e is small, onecan be justified in ignoring the nonlinear terms, at least outside the critical layer. Thelinearized equation is

E E EcY2qu = cq byu s0. 3Ž . Ž .ž /E t E x E x

If we assume a monochromatic neutral mode solution of the form

c x , y ,t sRe f y eikŽ xyct . ,� 4Ž . Ž .then the amplitude f satisfies the Rayleigh–Kuo equation

Ybyu

Y 2f q ya k fs0 4Ž .ž /uyc

Ž .and the critical level is located at the singular point ysy , where u y sc. The twoc c

solutions of this equation obtained by the method of Frobenius are2Y

byu yyyŽ .c cf y s yyy q q . . . ,Ž . Ž . Xa c ž /u 2c

Ybyuc

< <f y s1q . . . y f log yyy q . . . , 5Ž . Ž .Xb a cž /uc

where the subscript c denotes evaluation at y . North and south of the critical level f isc

given by

aqf y qbqf y y)yŽ . Ž .a b cf y s 6Ž . Ž .y y½a f y qb f y y-yŽ . Ž .a b c

This solution is not valid at ysy since at that point the zonal component of thec

perturbation velocity uX syc has a logarithmic singularity.y

The classical theory of hydrodynamic stability overcomes this difficulty by includingŽviscosity in the critical layer region to obtain a non-singular critical layer solution see

. Ž .Lin, 1955 . The alternative approach, taken by Dickinson 1970 and Warn and WarnŽ . Ž .1976 , is to introduce time-dependence, i.e., to assume the solution of Eq. 3 takes theform

c x , y ,t sRe f y ,t e ik x .� 4Ž . Ž .They solved the initial-value problem of forced Rossby waves propagating towards acritical layer region by taking a Laplace transform of f and found that most of the waveenergy was absorbed at the critical layer and that, for large time t, the solution outsidethe critical layer reached a quasi-steady state exhibiting similar characteristics to the

Ž .viscous solutions found earlier by Lin 1955 . In the viscous theory, the viscous criticalŽ .layer solution is matched with the inviscid outer solutions 6 and the relationship

between aq,bq and ay,by is found to be

bqsbysb ,Y

byuŽ .cq ya ya s i bu ,X< <uc

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( )L.J. Campbell, S.A. MaslowerDynamics of Atmospheres and Oceans 28 1998 9–3714

Ž .where u is the logarithmic phase shift and is equal to yp . The outer solution 6 canthen be written as

" ˆf y sa f y qbf y , 7Ž . Ž . Ž . Ž .a b

whereY

byucf̂ y s1q . . . y f log yyy q . . . ,Ž . Ž .Xb a cž /uc

with the logarithmic term now defined to beX< <log yyy s log yyy q iu sgn uŽ . Ž .c c c

when y-y .c

It can be shown that the phase shift is related to the Reynolds stress or zonallyaveraged wave momentum flux. The Reynolds stress is defined as

k 2prkX X X XRsyu Õ sy u Õ d x . 8Ž .H2p 0

This can be written in terms of f as

k)Rs Im f f . 9Ž .� 4y2

In the case of a neutral mode, R is found to be constant on either side of the criticalw xlayer, and the jump R in Reynolds stress is related to the phase change by

X< <w x2 u Rcus . 10Ž .2Y < <k byu bŽ .c

Ž .In the time-dependent analysis Dickinson, 1970; Warn and Warn, 1976 as well, it wasfound that, after the steady state was attained, the Reynolds stress jump was given by the

Ž .expression 10 with usyp .

3. Formulation for a north–south flow

For the case of a north–south mean flow with a longitudinally propagating distur-bance, the streamfunction can be written as

C x , y ,t sc x qec x , y ,t . 11Ž . Ž . Ž . Ž .where

Õ x sc x .Ž . Ž .x

and the linearized equation is then

E E Ec EcY2qÕ = cqb yÕ s0. 12Ž .ž /E t E y E x E y

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( )L.J. Campbell, S.A. MaslowerDynamics of Atmospheres and Oceans 28 1998 9–37 15

By writing the disturbance streamfunction in the neutral mode form

c x , y ,t sRe f x eilŽ yyct . ,� 4Ž . Ž .we derive the amplitude equation

ibY2Õyc af y l f y f yÕ fs0, 13Ž . Ž .Ž .x x xl

Ž .which is singular at the point x where Õ x sc. Expanding about this point, it can bec

shown that the two power series solutions of this equation are

YilÕcf x s1q xyx q . . . ,Ž . Ž .a c

b

YÕ 2y ig 1q igŽ .Ž .c1qig 2qig

f x s xyx q xyx q . . . , 14Ž . Ž . Ž . Ž .Xb c c2Õ 2q igc

Xwhere gsbra lÕ and the subscript c denotes evaluation at x . The quantity f has anc c b

algebraic singularity at x and east and west of this point it can be written asc

fq x s xyx eig log < xyx c < q . . . ,Ž . Ž .b c

fy x s xyx ey< g <ueig log < xyx c < q . . . ,Ž . Ž .b c

if, when x-x , the logarithm is defined to bec

X< <log xyx s log xyx q iu sgn Õ ,Ž . Ž .c c c

with usyp . Thus the amplitude of the f solution decreases by a factor of ey< g <p asb

the wave crosses the critical level from west to east so there is also a jump in theamplitude of the y-component Õsc of the perturbation velocity and a jump in thex

Reynolds stress across the critical layer.

4. Asymptotic solution of the long wave equation

Ž .If the dimensional wavelength of the forced wave in Eq. 3 is very large relative tothe length scale in the y-direction, then the aspect ratio a will be small and thex-derivative term in the Laplacian can be neglected. For the special case of a southwardpropagating wave packet in a zonal flow with constant shear, an asymptotic solutionvalid for large time can be derived. We consider the case

u y syŽ .and, for convenience, re-define the non-dimensional variables y and t as

y) bUt)

ys , tsbL Ly y

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Then the equation for the latitudinal propagation of the wave packet in the shear flow is

E E Ecqy c q s0, 15Ž .y yž /E t E x E x

with the critical layer located at ys0.Ž . Ž .Dickinson 1970 and Warn and Warn 1976 solved this equation for a monochro-

Ž . Ž . i k xmatic disturbance c x, y,t sf y,t e on a semi-infinite region y`-x-`, y`-

yFy , where y 41. They considered the response of an initially undisturbed flow to a1 1Ž . i k xperiodic forcing c x, y se switched on at ts0 on the northern boundary. They1

< <divided the yt-domain into three regions: the outer region where y t41, the region< < Ž . < <near the critical layer where y t;O 1 and the inner region where y t<1, and

obtained solutions for large time t41 in each of these regions. We consider a forcingof the form

c x , y sA m x eik x , 16Ž . Ž . Ž .1

with A decaying to zero as x™"`. Here, we shall assume the amplitude of theforcing takes the form of a Gaussian distribution

A m x seym 2 x 2. 17Ž . Ž .

Ž .The boundary-value problem for Eq. 15 will be solved by taking a Fouriertransform in x and a Laplace transform in time. We define the Fourier transformŽ . Ž .f l, y,t of c x, y,t by

`yi l xf l, y ,t s c l, y ,t e d x ,Ž . Ž .H

y`

Ž .and the Laplace transform of f l, y,t by

`ys tf̃ l, y ,s s f l, y ,t e d t .Ž . Ž .H

0

The shape of the wave envelope at ysy suggests that c™0 as x™"`; so, on1Ž .taking first a Fourier and then a Laplace transform of Eq. 15 , we obtain

is˜ ˜yy f qfs0, 18Ž .y yž /l

with

2Ž .' lykpy˜ 2f l, y ,s s e 19Ž . Ž .4m1

ms

Ž .at ysy . The solution of this equation with the boundary condition 19 can be1

expressed in terms of modified Bessel functions of order 1 and c is found by invertingthe Laplace and Fourier transforms:

2Ž .' lyk`1 1 pi` st i l xy2c x , y ,t s e e F l,s dld s, 20Ž . Ž . Ž .H H 4m

2p i 2p msyi` y`

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where1r2 1r2is is

yy KK 2 yy1ž / ž /l lF l,s s .Ž . 1r2 1r2is is

yy KK 2 yy1 1 1ž / ž /l l

˜ Ž .The inversion of the Fourier transform is carried out first. We let C x, y,s denote˜Ž .the inverse Fourier transform of f l, y,s , then

2Ž .' lyk`1 pi l xy˜ 2c x , y ,s s e F l,s dl 21Ž . Ž . Ž .H 4m

2p ms y`

˜ ˜Ž . Ž .and c x, y,s is also equal to the Laplace transform of c x, y,t . Since the amplitude ofthe forced wave packet is Gaussian, and m is small, we are justified in assuming that thedominant contribution to the above integral comes from the values of l lying in a small

Ž .neighborhood around the point lsk. We can then expand F l,s in powers ofŽ .lyk , approximate it by the first two terms in the series and integrate with respect to

˜Ž .lyk . Then the quantity c can be written as21 E F E F2 2i k xym x 2 2 1c̃ x , y ,s ;e F k ,s q2 im x k ,s qm k ,s qO mŽ . Ž . Ž . Ž . Ž .2s El El

and it is quite straightforward to show that2F k ,s 2 im x E FŽ .2 2i k xym x y1 y1c x , y ,t se LL y LL k ,sŽ . Ž .½ s k E s

2 2m Ey1 4q LL sF k ,s qO m . 22Ž . Ž .Ž . Ž .2 2 5k E s

The details of the calculation of the inverse Laplace transforms in the expression aboveare described in Appendix A.

To examine the outer region away from the critical layer and for large time, i.e.,< <when t41 and y t41, the necessary approximations are derived in Sections A.1 and

Ž . Ž . Ž .A.2 and are given by Eqs. 39 – 41 . On substituting these equations into Eq. 22 , thecomplete expression for the perturbation streamfunction in the outer region is found tobe

eyi k y t2 2i k x ym xc x , y ,t se e h y qh yŽ . Ž . Ž .1 2 2 2½ k yt

eyi k y1 t m2 2 xyytyi k y tqh y q h y eŽ . Ž .Ž3 22 2 2 tk yt k

qh y eyi k y1 t qO m4 , 23Ž . Ž .Ž ..3 5

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( )L.J. Campbell, S.A. MaslowerDynamics of Atmospheres and Oceans 28 1998 9–3718

where1r2 1r2yy KK 2 yyŽ . Ž .1

h y sŽ . ,1 1r2 1r2yy KK 2 yyŽ . Ž .1 1 1

y1h y sŽ .2 1r2 1r22 yyy KK 2 yyyŽ . Ž .1 1 1

and1r2 1r2h y s2 y yy KK 2 y yy .Ž . Ž . Ž .3 1 1 1

< < Ž .However, near the critical layer where y t;O 1 , we make use of the expressionsŽ . Ž .42 and 43 derived in Section A.3 and find that

yi k y t ` nq1e i nq1 !Ž .2 2i k x ym xc x , y ,t se e h y qh yŽ . Ž . Ž . Ý1 2 nq1½ kt kytŽ .ns0

yi k y t ` nq1e yi nq2 !Ž .yi k y t y1 y21qh y qe O t qO tŽ . Ž . Ž .Ý4 2 2 nq1k t kytŽ .ns0

m2 2 xyytyi k y t yi k y t 41q h y e qh y e qO m ,Ž . Ž . Ž .Ž .2 32 5tk

Ž . Ž Ž . Ž ..where h y s1r4 g y qg y .We can define a family of integrals by the expres-4 1 2

sions

`yz ue

qE z s du , Re z G0 24Ž . Ž . Ž .Hk ku1

`qz ue

yE z s du , Re z G0. 25Ž . Ž . Ž .Hk ku1qŽ .with ks0, 1,2, . . . E z is the integral whose properties are given by Abramowitz andk

Ž . Ž . yŽ .Stegun 1964 Eq. 5.1.4 , and it can be shown that E z satisfies expressionsk< <analogous to some of these. For example, for large z , asymptotic expansions can be

"Ž .derived for E z by integrating by parts,k

` nqky1 !Ž ." . zE z seŽ . Ýk nqky1

"zŽ .ns0

Ž . Ž .and our solution can then be written in terms of the integrals 24 and 25 as

h y h yŽ . Ž .2 2 2 4i k x ym x Žsgn Y . sgn Y< < < <c x , y ,t se e h y q E ik y t q E ik y tŽ . Ž . Ž . Ž .1 2 32 2½ kt k t2m 2 xyyt

yi k y t y1 y2 yi k y t1qe O t qO t q h y eŽ . Ž . Ž .Ž 22 tk

qh y eyi k y1 t qO m4 . 26Ž . Ž .Ž ..3 5

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( )L.J. Campbell, S.A. MaslowerDynamics of Atmospheres and Oceans 28 1998 9–37 19

At lowest order in m, these expressions are equivalent to the single wave numberŽ .solutions of Warn and Warn 1976 except that they are multiplied by the amplitude

function eym 2 x 2. The effect of the wave packet forcing produces additional terms in our

Ž 2 .solutions, but the largest of these are O m . One would expect then that the wavepacket solutions would exhibit the same logarithmic phase shift of yp obtained in the

Ž .single wave number case. There is a logarithmic singularity in the h y term arising1

from the Bessel function KK . Expanding in powers of y, we see that1

1r2 1r2yy KK 2 yy ;1yy log yq 1y2g yq . . . ,Ž . Ž . Ž .1

< <where log ys log y q iu for y-0 and it will be shown that usyp .< <Inside the critical layer, i.e., in the region where y t<1, the solution is obtained by

a combination of the method of matched asymptotic expansions and the method ofmultiple scales. Since the amplitude of the forced wave is a function of m x, it is naturalto define a slow spatial variable Xsm x and a corresponding long-time variable Tsm t.

Ž . < < Ž . Ž .For T to be O 1 when y t;O 1 , y must be O m . Therefore, the critical layerthickness is m; a critical layer variable Ysmy1 y can be defined and the outer solutionŽ .26 can then be expanded in powers of m as

c x , X ,Y ,T sa eik xeyX 21ymY log mqm 1y2g YymY log YqO m2Ž . Ž . Ž .1 ½

sgn Y < <E ik Y TŽ .2qm 1qma a YŽ .1 2 ikT

sgn Y < < 2 3E ik Y T 2m X m YŽ .32ym 1qa a q yŽ .1 2 2 2 2 2ž /T Tk T k k

= 1qma a Y eyi kY T qO m4 , 27Ž . Ž .Ž .1 2 5with the constants a and a given by1 2

1a s1 ,1r2 1r22 yy KK 2 yyŽ . Ž .1 1 1

1r2a s2 KK 2 yyŽ .2 0 1

< <and, for Y-0, the logarithm is defined to be log Ys log Y q iu .This indicates that the inner expansion must take the form

cseik x F Ž0.qm log mF Ž1.qmF Ž2.qm2F Ž3.qO m3 ,� 4Ž .Ž i. Ž i.Ž . Ž .where F sF X,Y,T , is0,1,2, . . . . On substituting this into Eq. 15 , we obtain

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equations for F Ž0., F Ž1., F Ž2., . . . which are solved and the results matched to theŽ .expressions of corresponding order in Eq. 27 . The inner solution can then be written as

Žsgn Y . < <E ik Y TŽ .2 2i k x yXc;a e e 1yYm log mqm sgn YŽ .1 ½ ikT

< <yY log Y yY log ikTq 1yg Y qm C X ,T YqD X ,T , 28Ž . Ž . Ž . Ž .Ž .5Ž . Ž .where g is Euler’s constant and the functions C X,T and D X,T can be determined

Ž .by matching this with the outer expansion 27 on both sides of the critical layer. Wefind then that

C X ,T sa eyX 2log kTyg ,Ž . Ž .1

D X ,T s0Ž .< < "Ž .and that the phase shift is usyp . For small z , the series expansion for E z can be2

Ž .found by integrating by parts Abramowitz and Stegun, 1964, Eq. 5.1.12 :m

` yzŽ ."E z s" z log "z q gy1 z y .Ž . Ž . Ž . Ý2 my1 m!Ž .ms0

m/1

Ž .Substituting this series into the expression 28 gives the result

m°` < <m yik Y TŽ .2i k x yX ~c;a e e 1yYm log my sgn YŽ . Ý1 ikT my1 m!Ž .¢ ms0

m/1

¶2•qmY log kTygy ipr2 qO m , 29Ž . Ž .Ž .ß

the first few terms of which are32 2 2< <i ikY T k Y T2i k x yXc;a e e 1yYm log mym q q q . . .1 ½ kT 2 12

qmY log kTygy ipr2 qO m2 . 30Ž . Ž .Ž .5From the expressions for the streamfunction, we can obtain information about the

large time behavior of the wave packet and also obtain approximations for theperturbation velocity and the momentum. In the outer region, as t™`, the second and

Ž . Ž .third terms in Eq. 23 vanish and, when x;O 1 ,

m22 2i k x ym x yi k y t yi k y t1c x , y ,t ;e e h y y y h y e qh y e , 31Ž . Ž . Ž . Ž . Ž .Ž .i 2 32½ 5k

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so we can expect c to reach an almost steady state taking the form of oscillations ini k x ym2 x 2 Ž .time superimposed on the function e e h y .1

The perturbation velocity components uX syc , ÕX sc after the steady state hasy x

Ž .been reached are readily found by differentiating the real part of the expression 31 andthe total momentum flux is the integral of yuX

ÕX over the length of the packet, i.e.,

`X XMs ym Õ d x .Ž .H

y`

For m<1, the first two terms of the asymptotic expansion for M can be shown to be

1r2p k k2 2y1 ) yk r mM;m Im h h qe Re h h qO m .Ž .1 1 y 1 1 yž / ½ 52 2 2

Note that M is the total momentum flux and not an average. Therefore it is not ananalogue of the Reynolds stress in the monochromatic case. From the expression above,it can be seen that M is proportional to my1 ; this indicates that M is proportional to thelength of the packet, and in the limit as m tends to zero, the packet length becomesinfinite and the total momentum flux tends to infinity.

Ž .Inside the critical layer, the velocity components can be calculated from Eq. 29 . Atys0, the perturbation streamfunction is the real part of

i2 2i k x ym x 2c;a e e 1y qO m , 32Ž .Ž .1 ½ 5tX Ž . X Ž y1 . Ž y1 y1and it can be shown that u ;O log t , Õ ;O t and that M varies like m t

. Xlog t . Therefore, as t™`, c , Õ and M all reach steady values inside the critical layeras well as in the outer region; uX, however, increases like log t, as found by Warn and

Ž .Warn 1976 .

5. Numerical experiments

In all the computations discussed here, the initial conditions consist of a hyperbolictangent basic flow profile with zero initial disturbance amplitude. The forcing isswitched on at one boundary of the computational domain at time ts0 and theamplitude is increased linearly from zero until time ts t after which it is held steady.1

Ž .In the numerical solution of Eq. 3 , the forcing is applied at the northern boundary ofthe computational domain y GyGy :1 2

c x , y ,t seym 2 x 2e ik x f t , 33Ž . Ž . Ž .1

with

trt t- t1 1f t sŽ . ½ 1 tG t1

Ž .With this boundary condition, the Fourier transform of Eq. 3 is solved usingŽfinite-difference approximations and a time-dependent radiation condition Beland and´

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. Ž .Warn, 1975 at ysy see Appendix B . The time derivative is approximated using the2

second-order Adams–Bashforth method and the spatial derivatives using a fourth-orderapproximation at the interior points of the finite-difference grid and a second-order

Ž .one-sided approximation at the radiation boundary Campbell, 1996 . The results shownhere were obtained with the north and south boundaries located at y s5 and y sy5.1 2

A grid of 400 points was used with a mesh-size of D ys0.025 and time-steps of sizeD ts0.05.

To check the performance of the program, it was first applied to model thepropagation of a forced monochromatic wave, so that the output could be compared with

Ž .the linear results of Beland’s 1976 numerical solution and with the results of´Dickinson’s analysis. Their results indicate that, for t41, the Reynolds stress andamplitude reach a steady state in the outer region, consisting of small decayingoscillations in time superimposed on a fixed value. After the steady state has beenreached, there is a jump in the Reynolds stress across the critical layer; it is zero south ofthe critical layer, and has a non-zero constant value to the north. The phase changeacross the critical layer is yp , the thickness of the critical layer varies like ty1 and atthe critical level the zonal perturbation velocity uX is proportional to log t. We ran ourprogram several times with different values of all the relevant parameters, and found thatprovided the aspect ratio a was not chosen to be too large, all these observations weresatisfied. Moreover, we found that the time taken for the wave to reach a steady stateoutside the critical layer was dependent on the values of b and k, the steady state beingachieved most rapidly with small values of b and large values of k. This supports the

Ž .observation made by Geisler and Dickinson 1974 that the time required for the linearsteady solution to be set up is proportional to brk. In addition, there are somerestrictions on the sizes of the parameters b and a . These are described in detail by

Ž .Beland 1976 .´The Reynolds stress R and the logarithmic phase change u were calculated from

Ž . Ž . Ž . Ž .Eqs. 9 and 10 . Warn and Warn 1978 showed that Eq. 10 still holds in thetime-dependent problem, R, f, u and b then being functions of time. The quantity b

Ž . Ž .multiplying the singular Frobenius solutions was determined from Eqs. 5 and 6Ž . Ž .which are defined so that f y s0 and f y s1 at the critical level. Therefore b ata c b c

Ž .time t is given by f 0,t which is known from the numerical solution.In the wave packet simulations as well, varying k, b and a had no significant effect

except to change the length of time taken for the steady state to be attained. The graphsshown here are for ks2, bs1, as0.2 and ms0.2. For this value of m, the forcingat ys5 takes the form of a packet confined to the region y10FxF10. In general, wefound there to be extremely good agreement between these computed results and the

< <predictions of the long wave solution. As predicted in Section 3, the amplitude c of thewave packet increases initially with time and eventually reaches a steady state, both

< <inside and outside the critical layer. We found that by ts20, c had become almostindependent of time.

` Ž X X .The variation of the momentum flux yMsH u Õ d x with y is shown in Fig. 1y`

for ms0.2. Calculation of the momentum flux for the wave packet using the expression` Ž X X.yMsH u Õ d x is an expensive and time-consuming procedure since it requires they`

inversion of the Fourier transform at each time step for all values of x. To save on

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Ž .Fig. 1. Wave packet forced to the north of u y s tanh y profile: Variation of the total momentum flux with yat ts40 with ms0.2.

computing time, we calculated M using an equivalent expression described in AppendixB.

Fig. 1 shows that the momentum flux is almost constant for y)0 and drops to zerobelow the critical layer. The drop in the momentum flux across the critical layerrepresents momentum transferred from the disturbance to the mean flow as a result ofthe absorption of the packet. Repeating this calculation for ms0.05 and 0.1, we foundthat the steady state value of M above the critical layer varies almost linearly with my1 ;thus the amount of momentum transferred to the mean flow is proportional to the lengthscale of the packet.

The perturbation streamfunction at time ts40 is shown in Fig. 2, calculated withms0.2; from these plots, it is clear that the wave packet retains its shape as it movessouthward from the forced boundary, but its amplitude decays rapidly to zero. South ofthe critical layer the streamfunction vanishes, indicating the complete absorption of thepacket at the critical layer. It may also be observed that the packet, initially centered atxs0 on the forced boundary, has moved eastward and at the critical layer, it is centeredaround xs2. This eastward shift is even more apparent in the contour plots of the total

˜ 2streamfunction Cscqc and of the perturbation vorticity = c inside the critical layerat time ts40 shown in Figs. 3 and 4, respectively. Near the center of the packet, the

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Fig. 2. Same configuration as Fig. 1: Perturbation streamfunction c at ts40 with ms0.2.

streamfunction field exhibits the characteristic ‘cat’s eye’ structure observed in inviscid,shear flows with critical layers. The cat’s eyes, or closed streamline regions, becomesmaller and less well-defined farther away from the center of the packet where theamplitude of the perturbation is close to zero. The perturbation vorticity inside the

Fig. 3. Same configuration as Fig. 1: Total streamfunction C at ts40 with ms0.2.

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Fig. 4. Same configuration as Fig. 1: Vorticity =2c in the critical layer at ts40 with ms0.2.

critical layer is also represented by closed contours, but is constant in the outer regionand away from the center of the packet.

Ž .For the numerical solution of Eq. 12 , i.e., the oceanic case, we used the mean flowŽ .profile Õ x s tanh x, with the forcing

c x , y ,t seym 2 y 2e il y f t , 34Ž . Ž . Ž .1

applied at one boundary xsx . Two types of boundary conditions were employed at the1

other end of the computational domain xsx : first a solid wall, i.e., f was set to zero2

at the boundary, and then a radiation condition, the details of which are given inAppendix C.

The parameters b , a , l and c were chosen to satisfy the requirements that thedisturbance be propagating without decay away from the forced boundary and that anypart of the disturbance transmitted beyond the critical layer decays. Applying W.K.B.analyses in the regions around the boundaries, these conditions lead to the inequalities

1r2 2 < <bG2a l Õ yc 35Ž .1

and1r2 2 < <b-2a l Õ yc , 36Ž .2

Ž .where Õ is the local value of Õ x near the forced boundary xsx and Õ the local1 1 2

value at the opposite boundary xsx . For a hyperbolic tangent profile, with the forcing2

imposed at the western boundary, Õ sy1 and Õ s1 and the inequalities above can be1 2

satisfied only if c)0, i.e., the disturbance must be propagating northward, whileimposing the forcing at the eastern boundary so that Õ s1 and Õ sy1, requires1 2

c-0.Ž .Geisler and Dickinson 1975 predicted that a wave forced to the west of the critical

layer and propagating eastward would be absorbed on reaching the critical layer, but a

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westward propagating wave forced to the east of the critical layer would pass throughwithout attenuation. They used a W.K.B. analysis to show that in the region where Õ isconstant, the westward propagating wave corresponds to the solution f of the steadya

Ž .Eq. 13 and the eastward propagating wave to the solution f . Since f is non-singu-b a

lar, it can be expected that a westward propagating wave would not ‘see’ the criticallayer. Our solutions supported this observation; there was a small jump in the amplitudeof the westward wave packet, but most of the disturbance was transmitted beyond thecritical layer. Consequently, setting f to zero at the western boundary resulted in thepacket being reflected there and the solution became amplified with time in the regionwest of the critical layer. It was therefore necessary to apply the radiation condition atthis boundary to allow the disturbance to be transmitted through.

In contrast, the eastward wave packet was almost completely absorbed on reachingthe critical layer. In fact, it was not necessary for the parameters to satisfy the

Ž .requirement 36 , since the amplitude dropped to zero east of the critical layer even ifthat inequality was not satisfied. We found that the extent of the absorption was

Ž .determined by the choice of the parameters b , l and a . In the steady solution 14 , theXyp <g <singular solution f decreases by a factor of e , where gsbra l Õ , as the criticalb c

layer is crossed from west to east. Computing the numerical solution for different valuesof b while keeping the other parameters fixed, we found that the size of the jump in theamplitude does indeed increase with increasing b.

Figs. 5–8 show the behaviour of a wave packet forced at the eastern boundary x s51

and moving westward with a phase speed of cs0 towards a critical layer located at

Ž .Fig. 5. Wave packet forced to the east of Õ x s tanh x profile: Variation of the total momentum flux with xat ts40 with ms0.2.

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Fig. 6. Same configuration as Fig. 5: Perturbation streamfunction c at ts40 with ms0.2.

Fig. 7. Same configuration as Fig. 5: Total streamfunction C at ts40 with ms0.2.

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Fig. 8. Same configuration as Fig. 5: Vorticity =2c in the critical layer at ts40 with ms0.2.

xs0 and with bs4, ls1 and as4. As predicted, there is only a small jump in theamplitude of the wave at xs0, the wave continues to propagate west of the criticallayer and the radiation condition prevents it from being reflected at the boundary, but the

Ž .Fig. 9. Wave packet forced to the west of Õ x s tanh x profile: Variation of the total momentum flux with xat ts40 with ms0.2.

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Fig. 10. Same configuration as Fig. 9: Perturbation streamfunction c at ts40 with ms0.2.

Fig. 11. Same configuration as Fig. 9: Total streamfunction C at ts40 with ms0.2.

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Fig. 12. Same configuration as Fig. 9: Vorticity =2c in the critical layer at ts40 with ms0.2.

eastward wave packet, shown in Figs. 9–12 for the case x sy5, x s5 and the same1 2

values of b , l and a , is completely absorbed at the critical layer and there is a jump inthe momentum flux. In addition, one may observe a southward shift in the location ofthe center of the packet, from ys0 at the forced boundary to approximately ysy5 inthe critical layer region.

6. Conclusions

The goal of this paper was to investigate the effect of introducing slow amplitudemodulation to the linear forced Rossby wave critical layer problem. It has been shownthat on making the long wave assumption, it is possible to derive an asymptotic solutionto the linearized barotropic vorticity equation along the lines of the large time

Ž . Ž .monochromatic solution of Dickinson 1970 and Warn and Warn 1976 . The solutionhas been found to take the form

A m x eik x f Ž0. y ,t qm2 f Ž1. x , y ,t qO m4 ,Ž . Ž . Ž . Ž .Ž .Ž .where A m x is the amplitude of the forcing; in the limit as m™0, the monochromatic

i k x Ž0.Ž .solution e f y,t is recovered.

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In addition, some numerical simulations have been carried out, the results of whichhave been found to agree very closely with our asymptotic solution. Moreover, all theobservations made in the single wave number critical layer solutions have been verifiedin our wave packet simulations; for instance, the logarithmic phase shift of yp , thesolution reaching a steady state, and the jump in the momentum flux across the criticallayer, implying absorption of the disturbance at the critical layer. We also observed aneastward shift in the location of the center of the packet and that the total momentumtransferred to the mean flow as a result of the absorption of the packet is dependent onthe size of m, i.e., proportional to the length of the packet.

We have examined the case of a north–south flow as well and found that, asŽ .observed by Geisler and Dickinson 1975 in the monochromatic case, a disturbance

propagating from west to east passed through the critical layer unaffected, but awestward propagating disturbance was absorbed at the critical layer. In the former case,we applied a radiation condition at the outflow boundary to allow the packet to betransmitted through, but it might be worthwhile to investigate the effect of applying anoutflow boundary condition that would allow the packet to be reflected at this boundary.

All these considerations now suggest that we proceed to the next step, which is tosolve the nonlinear wave packet problem for both of the mean flow profiles considered

Ž .here. The nonlinear numerical solution of Brunet and Haynes 1996 used a shallow-watermodel and a basic flow taking the form of a jet centered around 328 latitude andsymmetric about the equator. It would be interesting to consider other velocity profiles,as well.

In a nonlinear solution, one would take into account not only the wave packet-meanflow interactions, but also the interactions within the packet. The structure of the flowwould be significantly altered because the phase change responsible for the absorptionwould vary with longitude and, near the center of the packet, there would be wavereflection.

Acknowledgements

The authors are grateful to Professor T. Warn of McGill University for helpfuldiscussions and for making several suggestions. They also benefited from discussionswith Dr. G. Brunet of the Atmospheric Environment Service of Canada on the subject ofnonlinear critical layers.

Appendix A. Evaluation of the inverse Laplace transforms

< <A.1. The case y t41,t41

This section gives the details of the evaluation of the first term on the right-hand sideŽ .of Eq. 22 . The other two terms are obtained in a similar manner on making use of the

expressions for inverse Laplace transforms of derivatives.

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Ž .Fig. 13. Contour of integration in the complex s-plane for Eq. 37 .

The integral in the first term is given by1r2 1r2is is

yy KK 2 yy1st ž / ž /k kF k ,s 1 eŽ . cqi`y1LL s d s, 37Ž .H 1r2 1r2s 2p i s is iscyi`yy KK 2 yy1 1 1ž / ž /k k

where c is chosen so that the contour of integration will lie to the right of all thesingularities of the integrand. The integral is evaluated along the contour shown in Fig.13, and is found to be equal to the sum of the contributions from the three singularities:the pole at ss0, and the branch points at the point P, where ssyiky, and at PX,where ssyiky .1

The residue at the pole is readily seen to be1r2 1r2yy KK 2 yyF k ,s Ž . Ž .Ž . 1

lim s s 1r2 1r2ž /ss™0 yy KK 2 yyŽ . Ž .1 1 1

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and, integrating around the branch points, we find that the integrals along the curvedcontours vanish when R™` and r™0. To calculate the contribution from the branchpoint ssyiky, we define a new variable r along each of the contours C and C by1 2

ssr e" ip y iky and integrate along C and C letting R™` and r™0. Denoting the1 2

sum of the integrals along C and C by I, we find that1 2

1r2 1r2r r3 ip r4 3 ip r4e ip II 2 eiyi k y t yr t ž / ž /` e e k k

Is d r , 38Ž .H 1r2 1r2yry ikyŽ . yir yir0qyyy KK 2 qyyy1 1 1ž / ž /k k

where II is the modified Bessel function of the first kind of order 1. The integrand1

above is too complicated to allow the integral to be evaluated exactly, but if t issufficiently large, an asymptotic approximation can be obtained. For large t it may beobserved that most of the contribution to the integral will be produced when r is small.The integrand can then be expanded in powers of r, and in the outer region far away

< <from the critical level where y t41, it can be approximated by the first term in theseries:

r3 ip r2

yi k y t yr t e` e e ž /kI; ip d r .H 1r2 1r2yiky0 yyy KK 2 yyyŽ . Ž .1 1 1

Consequently,

ip eik y t

I; .1r2 1r22 2k y yyy KK 2 yyy tŽ . Ž .1 1 1

The contribution from the branch point ssyiky is found by integrating along the1

contours CX and CX and letting R™` and r™0. On simplifying the resulting1 2

expression and denoting it by I X, we obtain

1r2 1r2yi k y t yr t 1r21` e e k yir yirXI s qy yy KK 2 qy yyH 1 1 13 ip r4 1r2 ž / ž /yry iky k ke rŽ .0 1

=

y1 y11r2 1r2r r3 ip r4 3 ip r4KK y2 e q KK 2 e d r .1 1ž / ž /½ 5 ½ 5ž /k k

For large t, the integrand can be approximated by the first term in the series expansion,and after simplifying and integrating we obtain

yi k y te 1r2 1r2XI ;y4 ip y yy KK 2 y yy .Ž . Ž .1 1 12 2k y t1

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Therefore, the inverse Laplace transform is the sum of the residue at the pole and theŽ .y1Ž X.quantity y2p i Iq I :

E Fy1LL

E s1r2 1r2 yi k y tyy KK 2 yy eŽ . Ž .1

s y1r2 1r2 1r2 1r22 2yy KK 2 yy 2k y yyy KK 2 yyy tŽ . Ž . Ž . Ž .1 1 1 1 1 1

1r2 1r2yi k y t1e y yy KK 2 y yyŽ . Ž .1 1 1q2 . 39Ž .2 2k y t1

( 2)A.2. The O m terms

Ž 2 . Ž . Ž . y1 w Ž .xTo calculate the O m terms in Eq. 22 , we define f t sLL F k,s , and againintegrate along the indented contour of Fig. 13. Then

yi k y tE F yiey1LL sytf t sŽ . 1r2 1r2E s 2k yyy KK 2 yyy tŽ . Ž .1 1 i

1r2 1r2yi k y t1e y yy KK 2 y yyŽ . Ž .1 1 1q2 i . 40Ž .

ktŽ . y1 w Ž .xSimilarly the last term in Eq. 22 can be found by first integrating LL sF k,s .

Then2 yi k y tE sF k ,s yeŽ .Ž .

y1 2 y1LL s t LL sF k ,s sŽ .2 1r2 1r2E s 2 yyy KK 2 yyyŽ . Ž .1 1 1

1r2 1r2yi k y t1y2 ye y yy KK 2 y yy . 41Ž . Ž . Ž .1 1 1

< < ( )A.3. The case y t;O 1 , t41

< < Ž . Ž .Near the critical layer where y t;O 1 , the integrand in Eq. 38 cannot beapproximated by the first term in its series expansion, since the largest of the neglectedterms would then be of order 1. Since t41, we can still conclude that most of thecontribution to the integral will come from within a small neighbourhood of the pointrs0. We therefore expand the integrand in powers of r and retain all the terms in theseries expansion that are of order ty2 . On simplifying the integrand, we obtain

nq1`` yrŽ .yi k y t yr tI;y1r2 e e g y q1r2 g yŽ . Ž .ŽÝH 1 1nq2 nq1½0 ik yŽ .ns0

nq2` yrŽ .qg y q . . . d r ,Ž . . Ý2 nq3 nq1 5ik yŽ .ns0

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where

1g y s ,Ž .1 1r2 1r2yyy KK 2 yyyŽ . Ž .1 1 1

and1r2

KK 2 yyyŽ .0 1g y s .Ž .2 21r2yyy KK 2 yyyŽ . Ž .½ 51 1 1

Ž .Application of Watson’s lemma see, for example, Nayfeh, 1981 to calculate thisintegral gives the result

` nq11 i nq1 !Ž .yi k y tI;y1r2 e g y q1r2 g yŽ . Ž .ŽÝ1 1nq1½ kt kytŽ .ns0

` nq11 yi nq2 !Ž .qg y q . . . . 42Ž . Ž .. Ý2 2 2 nq1 5k t kytŽ .ns0

It can be verified that the first neglected term in this expression is of order ty3. Thecontribution from the second branch point can be calculated in a similar manner, andgives an expression of the form

I X;eyi k y1 t O ty1 qO ty2 . 43� 4Ž . Ž . Ž .

Ž 2 . Ž . Ž .At O m , the approximations 40 and 41 are still valid.

Appendix B. Numerical computation of the momentum flux

The perturbation streamfunction c was defined to be the integral over an infinitenumber of wave numbers of functions of the form

1i l xc s f l e .Ž .l 2p

The Reynolds stress corresponding to each wave number l is given by

l 2prlX X X XR sy u Õ sy u Õ d x . 44Ž .Ž . Hl l ll 2p 0

Ž .Following Booker and Bretherton 1967 , it can be shown using Parseval’s theoremthat the total momentum flux M is equal to 2p times the integral of these Reynoldsstresses over all the wave numbers, i.e.,

` `1 l 1)R dls Im f f dls M . 45Ž .Ž .H Hl y2 2 2p4py` y`

Therefore, at each time step it was sufficient for us to calculate R for each value of ll

Ž .from the already computed values of f l, y,t , and then integrate R over l to find M;l

it was not necessary to invert the Fourier transform numerically.

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Appendix C. The radiation condition

Ž .The derivation of the radiation condition used in the solution of Eq. 3 is describedŽ . Ž .elsewhere Beland and Warn, 1975; Campbell, 1996 . The Fourier transform f l, y,t´

of the perturbation streamfunction is shown to satisfy the expression

Ef tylfs f l, y ,t h l,tyt dt , 46Ž . Ž . Ž .H

E y 0

where

bb b t b t.yilŽy1y t2h l,t s e TT q iTTŽ . 2 a l 1 01r2 ž / ž /2al 2al2a

and TT , TT are the Bessel functions of order zero and one, respectively.0 1Ž . Ž .Following Beland and Warn 1975 , the appropriate condition for Eq. 12 can be´

Ž .obtained by taking the Laplace transform of the amplitude Eq. 13 . As x™"`, themean flow profile becomes constant and its second derivative vanishes, i.e., Õ;Õ , say,2

and Õ ;0 at the boundary. Defining the Laplace transform of f byx x

`ys tf̃ x ,l,s s e f x ,l,t d t ,Ž . Ž .H

0

we obtain

˜ ˜ ˜a sq ilÕ f qbf y sq ilÕ fs0.Ž . Ž .2 x x x 2

Solutions of this equation must satisfy

˜ ˜f qH l,s fs0, 47Ž . Ž .x

where1r221 b l b2H l,s s " sq ilÕ q ,Ž . Ž .21r2 2ž /½ 5sq ilÕ 2a a 4alŽ .2

if they are to be bounded as x™"`. The inverse Laplace transform of H is given by

b l aty1 yi lÕ t2w xLL H s " TT at dt"d t e .Ž . Ž .H 11r2½ 5ž /2a ta 0

Ž 1r2 .where d is the delta function and asbr 2a l . Therefore, after taking the inverseŽ .Laplace transform of Eq. 47 and simplifying, we arrive at the time-dependent radiation

condition for the case of a north–south mean flow:

Ef t"lfs f x ,l,t h l,tyt dt . 48Ž . Ž . Ž .H

E x 0

where

tb l ayi lÕ t2h l,t s " TT at dt e .Ž . Ž .H 11r2½ 52a ta 0

Page 29: Forced Rossby wave packets in barotropic shear flows with

( )L.J. Campbell, S.A. MaslowerDynamics of Atmospheres and Oceans 28 1998 9–37 37

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