6.3 Zonally asymmetric circulations in the tropics: the Gill model

Transcription

1 6.3 Zonally asymmetric circulations in the tropics: the Gill model [Gill, Quart. J. R. Meteor. Soc., 106, (1980).] We consider the zonally varying circulation forced by steady thermal forcing on the equatorial β-plane. We linearize about a stably stratified state of no motion. The perturbation thermodynamic equation is wheres= we replace T t +w S= J, ρc p ( dt 0 + g dz c p ). TorelatethistoashallowwatermodelofdepthD, T h S ( ) u w D x + v J Dρc p S Q to give a set of perturbation shallow water equations u t βyv = g h x v t +βyu = g h ( ) h u t +D x + v = DQ whicharethesameastheunforcedsetbutwiththeadditionofthe thermal forcing. In order to incorporate dissipation(we ll see that this is important to the solutions) we include both linear drag and Newtonian cooling, each withthesamerate ε,toget u t βyv = g h x εu v t +βyu = g h εv ( ) h u t +D x + v = DQ εh 15

2 Now we non-dimensionalize 1, using the deformation radius L E = (c/β) 1/2, andthetimescaleτ E =L E /c=(cβ) 1/2,wherec=(gD) 1/2 isthegravity wavespeed. Notethat,withc=55.7ms 1,andβ= m 1 s 1, wehavel E =1560km,andτ E =28000s=7.78hr. Then,withu =cu, v =cv,y L E y,x L E x,h =Dh,weget u t v t h t + u x + v yv = h x εu +yu = h εv = Q εb whereε=τ E ε. Finally, we makethe longwave approximationthatthe zonallengthscale L E ;thisimpliesthat u v andso,providedε 1, wecanneglectεvinthesecondequation. Then,insteadystate,wehave εu yv = h x εh+ u x + v yu = h = Q. (29) Now, if we associate ε with iω, we know that the homogeneous (unforced) version of(29) have eigenfunctions that are Hermite functions, so we anticipate that expanding in these functions will be a good approach to the problem. First, however, we follow Gill by writing u = 1 2 (q r) h = 1 2 (q+r) 1 OurdefintionforL E followsamore conventionalapproachthan thatofgill, whose definition differs from ours by afactor of 2. Accordingly, differences of factors of 2 appear between our equations and those of Gill. 16

3 when(29) can be rewritten and reorganized to Now we expand εq+ q x + v yv = Q εr r x + v +yv = Q (30) q +yq+ r yq = 0. q r v Q = n=0 q n (x) r n (x) v n (x) Q n (x) V n(y) (31) wherev n arethehermitefunctionsweencounteredpreviously. Theyhave the recurrence relations dv n dy +yv n = 2nV n 1 dv n dy yv n = V n+1 (32) V n+1 2yV n +2nV n 1 = 0 Substituting into(30) and using(32), we get [( εq n + q ] n x +Q n )V n v n V n+1 n=0 [( εr n r ] x +Q n )V n +2nv n V n 1 n=0 = 0 [2nq n V n 1 r n V n+1 ] = 0 n=0 = 0 (33) Given the forcingq(x,y) = Q n (x)v n (y), equations (33) can be used to build the solution. Note that the Hermite functions are normalized such that so that Q n (x)= V m (y)v n (y)dy=n!2 n πδ nm, (34) 1 n!2 n Q(x,y)V n (y)dy. (35) π 17

4 6.3.1 Localized forcing on the equator SupposethatQislocalizedinx,beingzeroexceptin L<x<L,say,and thatinlatitudeitissimplydescribedbyv 0 (y)=exp ( 1 2 y2), sothatonly Q 0 (x)isnonzero. Then(33)givesus εq 0 + q 0 x = Q 0 εq 2 + q 2 x v 1 = 0 εr 0 r 0 x +2v 1 = Q 0 4q 2 r 0 = 0 andallothercoefficientsarezero. Wecanusethe2nd,3rdand4thofthese toget 3εq 2 q 2 x = 1 2 Q 0. Taking the first equation first, then outside the forcing region, εq 0 + q 0 x =0 i.e.,q 0 e εx. Demandingaboundedsolutionleadsustoconcludethat q 0 (x)= 0, x< L x Q L 0(x )e ε(x x) dx, L<x<0 e εx L Q L 0(x )e εx dx, x>l (36) Thispartofthesolution, whichhasu=h exp ( 1 2 y2) andv =0isthe Kelvinwavepart. ThereisnoKelvinwaveresponsetothewestoftheforcing simply because the Kelvin wave group velocity is eastward. To the east the solutiondecaysase x/λ where λ=ε 1 = groupvelocity disspation rate (recall that in dimensional terms ω = k for the Kelvin wave, whence the group velocity is unity). 18

5 Thesecondpartofthesolutionhas 1 2 e3εx L L Q 0(x )e 3εx dx, x< L q 2 (x)= 1 L Q 2 x 0(x )e 3(x x ) dx, L<x<L. (37) 0, x>l Hereboundednessdemandsthatthereisnocomponentlikee 3εx inx>l, totheeastoftheforcing. Thispartofthesolutionalsoincludesv 1 (x)and r 0 (x),where v 1 = εq 2 + q 2 x r 0 = 4q 2 Notethatthiscomponentiszerototheeastoftheforcingand,tothewest, varies as e 3εx. This is the n = 1 Rossby wave component. Recall that, inthelongwavelimitunderconsiderationhere,then=1rossbywavehas ω= k/3,sothatitsgroupvelocityis 1/3. Hencethereisnocontribution totheeastoftheforcingandtothewestthesolutiondecaysase x/λ,where now λ=(3ε) 1 = groupvelocity disspationrate. Sincethegroupvelocityofthen=1Rossbywaveis1/3ofthatoftheKelvin wave,itattenuatesafactorof3morerapidlythanthekelvinwave. All other components of the expansion(31) are zero, a consequence of the factthatwechoseaforcingstructure(iny)thatdoesnotprojectontothe highermodes. ThefullsolutionisshowninFig. 1;thecentralframe,showing low-level pressure and wind, clearly shows the predicted characteristics: zonal flowtothe east, with a Gaussian latitudinalprofileof u andp( h), and twincyclonesstraddlingtheequatortothewest,asexpectedforthen=1 Rossby wave,which has h structure h = 1 2 (q r) = 1 2 [q 2(x)V 2 (y)+r 0 (x)v 0 (y)] = 1 2 [q 2(x)[V 2 (y)+4v 0 (y)]] = q 2 (x) ( 2y 2 +1 ) e y2 /2. 19

6 Figure 1: Gill s solution for forcing symmetric about the equator. 20

7 Thisfunctionhasextremainyaty=0andy=± 3/2=±1.22,consistent with the locations of the twin low level cyclones on the figure. [Note that Gill svaluesforxandymustbedividedby 2toconverttoournotation.] Withournumbers,L E 1600km,sothesevaluesofycorrespondtodimensionaldistanceof1950km,orabout±17.5 o latitude. Sinceweareassuming first baroclinic mode structure, we also expect to see twin anticyclones at these locations in the upper troposphere. The latitudinally averaged streamfunction in the x z plane, shown in the bottom frame of Fig. 33, shows the twin circulation cells in this plane. The more extensive and(slightly) stronger cell is located to the east, and appears to correspond to the Walker circulation in the atmosphere. It extends a distanceoforder7l E 11100km,aboutenoughtostraddletheequatorial Pacific Ocean. Note, however, that this depends entirely on the value chosen for ε, the dissipation rate. Gill used ε = (in our notation), so the predictede-foldingdistanceisjustl E /ε=11200 km. Givenourtimescale of7.78hr,gill svalueforεcorrespondstoadissipationtime,forbothwind andtemperature,of55hr,ratherashorttimeforboth Antisymmetric forcing Iftheforcingisantisymmetricabouttheequator,suchthatQ(x,y)hasno projection onto the even Hermite functions, the the response is qualitatively different. Gill s solution for the response to such forcing is shown in Fig. 2. (The imposed distribution of Q is approximately indicated by the distribution inthetopframe.) SincethereisnoprojectionofQontoV 0 (y),thereisno Kelvin wave component to the solution and, since this component is the only onewitheastwardgroupvelocity,therecanbenoresponsetotheeastofthe forcing. Nearandtothewestoftheforcing,theresponse(dominatedbythe n = 2 Rossby wave) takes the form of a cyclone/anticyclone pair straddling theequatorneary=±2 ±3200km ±29 o latitude,withcross-equatorial flowaroundandoutoftheanticycloneandaroundandintothecyclone Forcing centered off the equator Fig. 3 shows the response to forcing centered in the northern subtropics infact,itissimplythesumofthefirsttwocases. (Again,Qisapproximately shownbythedistributionofw.) Theresponseshowsmanyofthefeatures characteristic of the Indian Ocean region in northern summer: low level 21

8 Figure 2: Gill s solution to forcing localized in x and antisymmetric about y=0. 22

9 Figure 3: Gill s solution for response to forcing centered in the northern subtropics. 23

10 cyclone/upper level anticyclone slightly NW of the upwelling, and crossequatorial flow to the west of the forcing, becoming a southwesterly inflow into the region of low-level convergence WhatdoestheGillmodelmean? The Gill model is remarkably successful at reproducing major characteristics of the tropical flow, yet it is built on a demonstrably false premise, that one can treat the dynamics a priori as a first baroclinic mode structure. Further, it appears to rely in unreasonably strong dissipation (in order to get reasonable zonal length scales). So is the model s success a fluke, or is it somehow doing the right things for the wrong reason? One possible explanation (Lindzen & Nigam, 1987; Neelin 1989) is that the single-node vertical structure of the 1st baroclinic mode is really representing flow in the boundary layer, and flow of the opposite sign in the free atmosphere; moreover, the importance of the boundary layer then makes Gill s dissipation seem less unreasonable. More recently(e.g., Neelin& Zeng 2000) it seems better to think of it as a consequence of the quasi-equilibrium nature of the tropical atmosphere, where the vertical stratification is constrained by deep moist convection, in which case 1st baroclinic mode-like dynamical structures are imposed by the vertical structure of the convection, rather than by real modal constraints. 24