topographic waves in barotropic flowmetnet.imd.gov.in/mausamdocs/14023_f.pdf · interaction between...

Mausam, (1989). 40, ~ . 131 ·136

551.509.313 : 551.513

Topographic waves in barotropic flowA IL KUMAR JAI 1

M eteorological Office, Agar/ala Airport, Tripura( Receil'cd 3 M orell 1988)

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r:t'~'fl'll'TFl' 'f." 3f"lJ:A lNq ~ 'If.:" ~9'f~ ~ t JfW II tn'... i llq'A'~ :i= ~PTTIl "r.I' ~of 3i..,.fuf if llT ai " q ~ llT f<rqll ~ 1 ~

vff'lo..rf'=f q~ ~Jl:f 'Ti!:9" '=f l~ 1111l9- i ~"J if fr.Q1 lTlfr ~ I

. A.I~STRI\CT-. t echnique .of, Fourier series :tn~l l.y ~is is used to study linear and non-linear effects on theIn...lablil ty of a baSIC state ~on"l" l mg of a.topographically ron.."t.;t wave in an inviscid baro tropic beta plane model.II, IS ..hown that ~ I~ an alytical study on l!l s.ta bll~ry ISpo-tslbl.c If amplit ude of wave like perturbat ion is either anC\CIl or odd function In space about ong m. 1he study is carried out in the present paper for former case.

1. Introduction

In a series of recent pap ers (C harney and Devore 1979,Ch arney and St ra uss 19HO, lI art 1979, Deinin ger 19HI,Ped losky 1981), co nce pt of to pog raphic ins tability andtheory on reso nant topog raphic waves in baroclinic andbarotropic flows arc dev eloped, Holo pa incn (I97H) hassho wn impo rtance and sign ificant relative co ntribution oftransient eddies in maintenance of horizontal flux ofrelative vorticity. Also po inted ou t by Ped losk y (19HI l,the mod els in the papers ci ted above suffer from eithersevere truncations or assumptions som etimes unrealistic.It can be argued that truncation s or assumption s areno t ado pted to simplify analysis of basic vortici ty equa­tio n by a ll the investigators. Imposition of a constraintis found unavoidable for want of an additional equationor co nditio n to conclude results from the analvsis. Thegoverning eq uation cannot impose restriction on thefield of pert urbat ion s that may' exist in the atmosphere.

used for linear a nd non-linear analyses. However theanalysis . presented here, is more rigo rous. It isbelieved in pre sent wo rk th at interact ion between sta nd­ing a nd transient eddies is func tio n of topography andperturbat ion , For all type of perturbations th at mayexis t in the a tmos phe re, top ogra ph y need not to resultinstability. However , there may exist so me perturba­lion s th at exh ibit ins ta bility d ue to int eraction wit htopogra phy. Hence, a classifica t ion of pert urbations islikely to be possible such that perturbati on s fro m aclassifie d group may result in an ins ta bi lity. Classifyingperturbat io ns in three groups. i.c. . wave like per turba­lions pn:,pagating in (k", I.) d irect ion with frequency;\ and eit her an even o r odd or mixed amplitude inspace about origin, a linear and non-linear a nalysis iscarried out for first classification in the present work.

2. TIl< model

where 'I' is th e gcos trophic st rea m fun ction whose x and)' der ivatives give " a nd - /I respectively. Th e plan etaryvorticity grad ient is fi.

If hn is height o f the topography then

The q uasi-gcos trophic vo rticity equation fo r a ho mo ­geneous, ba ro tropic fluid o n the /l-pla ne ca n be writte nin no n-di mensional for m as (Ped los ky 1979) :

(~ ,NI a a'll a )- + - -- - - - - - (V'2'1'+/ly+ h) =OCt OX ay ay OX

(2, I)

The atmosphere is baroclinic. Ho wever. the beh a­vio ur of a barot rop ic atmosphere. surely has a mappingin the baroclinic atmosphere. Hence. a result that isobta ined fo r ba rotropic model , should be trea ted as theres ult th a t pro vides basic understand ing of the physicsinvo lved.

Deininger (198 i) use d theory of pert urbation withinfinite series so lutio n of differential equation to studyinteraction between standing and transient eddies in thebarotropic flow in th e atmosph ere. He proposed trun­ca t ion to ma ke ana lysis possible. Although it is pos­sible to carry out analysis for less restr ict ive truncationyet trunca tion is a mu st for an analysis. The analy siswo uld be subjected to cr it icism for any choice of finit etruncat ion. In the present paper simila r theory is

(13t )

h = lIn/<n (2 .2)