Dynamics in planetary Atmospheric physics: comparative studies … · 2009-11-04 · 164 Johns hopkins ApL TechnicAL DigesT, VoLume 26, number 2 (2005) X. Zhu T Dynamics in planetary

164 JohnshopkinsApLTechnicALDigesT,VoLume26,number2(2005)

X. Zhu

T

DynamicsinplanetaryAtmosphericphysics:comparativestudiesofequatorialsuperrotationforVenus,Titan,andearth

Xun Zhu

helong-standingproblemofequatorialsuperrotationinaslowlyrotatingplanetaryatmospherehasbeensolvedrecently.Thisarticlepresentsasystematic,comprehensive,and analytical examination of the momentum budget that maintains stable superrota-tionalwindsinagivenplanetaryatmosphere.Afterreviewinggeneralapproachesusedinmodelingcomparativeplanetaryatmospheres, idescribe themajordynamicalprocessesthat maintain stable equatorial superrotation for the atmospheres of Venus, saturn’smoonTitan,andearth.underappropriateconditions,allthreebodiescouldhaveequa-torial superrotationalwindsgreater than100ms21.Venus’ equatorial superrotationof118ms21ismaintainedatitscloud-toplevelat≈65km.mymodelpredictsthat,tomain-tainanequatorialsuperrotationaljetstreaminTitan’sstratosphere,itsmainhazelayer,whichabsorbssolarradiation,wouldneedtobecenteredatanaltitudeof288km(corre-spondingto10papressure).itishopedthatongoingobservationsbythecassini-huygensmissiontosaturnwillbeabletovalidatethismodelprediction.strongequatorialsuperro-tationalwindscouldalsobeinducedonearthbyopticallythickdustclouds,suchasthosearisingfromanasteroidimpactorextremevolcanicevent.superrotationwouldgreatlyenhancemeridional transport in theearth’satmosphere,whichcouldhaveacceleratedtheglobalizationofanenvironmentunfavorabletothesurvivalofthedinosaurs.

THE SCIENCE OF COMPARATIVE PLANETARY ATMOSPHERES

Thisarticlegivesabriefoverviewofmodelingstud-ies of planetary atmospheric physics, focusing mainlyon atmospheric motions associated with dynamics,and reports significant progress in solving the long-standingdynamicalproblemofequatorialsuperrotation

inaslowlyrotatingplanetaryatmosphere.1Thediscus-sionsarepresentedfromtheperspectiveofcomparativeplanetaryatmospheresratherthanforaspecificplanet.Thescientificmeritsofconductingstudiesofcompara-tiveplanetaryatmospheresareillustratedbasedonbothgeneralrationalesandspecificexamples.

JohnshopkinsApLTechnicALDigesT,VoLume26,number2(2005) 165

DYnAmicsinpLAneTArYATmosphericphYsics

overthepastfewdecades,ourunderstandingoftheplanets (including planetary bodies such as saturn’smoonTitan)andtheiratmosphereshasadvancedfromahighlysubjectivearttoapurelyscientificprocess,sup-ported by a large number of measurements made byvarious space missions. our current knowledge indi-cates that while planetary atmospheres share physicalprocesses,theyexhibitdiversecharacteristics.Thegoalof modeling comparative planetary atmospheres is tounderstand a variety of atmospheric phenomena gov-ernedbycommonphysicallaws.

onewaytoillustratetheimportanceofcompara-tiveplanetaryatmospheres istoconsidertheearth’satmospherefromfirstprinciples.Therearetwoclear-cutmethodologiesinthephysicalsciences:(1)theo-reticalmodelingbasedonobservationsand(2)labo-ratoryexperiments.Thefirstmethodstartsfromasetof fundamental physical laws that govern the statesandmotionsoftheatmosphere.Thesephysical lawsareexpressedmathematicallyandareoftenanalyzedandsolvednumerically.solutionsarecomparedwithmeasurements,fromwhichourunderstandingoftheearth’s atmosphere can be steadily improved. Theessenceofthesecondmethodologyisthesimulationof processes in the laboratory, where one can con-trolandchangetheexternalconditionsthatgovernthebehaviorof theparticularphysical systemunderstudy. one must therefore be able to reproduce anapproximatephysicalmodeloftherealsysteminthelaboratory where it can be controlled—a require-mentthatclearlycannotbemetfortheearth’sentireatmosphere. in fact, atmospheric research is mostlyanobservationalscienceratherthananexperimentalscience.

With an observational science, the critical processof hypothesis testing is severely curtailed because ofourinabilitytocontroltheexternalvariablesinlabora-toryexperimentsdesignedtosimulateatmosphericpro-cesses. however, planetary atmospheres with remark-ably different astronomical conditions, which resultin significantlydifferentatmospheres in termsof theirphysical andchemicalproperties, serveas ideal “labo-ratories.” They provide a natural set of experiments,allowingustotestourbasicunderstandingofphysicalprocessesthataregovernedbythesamesetofphysicallaws,yetunderdifferentexternalconditions.Therefore,comparingandcontrastingdifferentatmospheresinthesolarsystemenablesustotestourbasicunderstandingofatmosphericprocessesingeneralandcanalsohelpusgaininsight intotheearth’satmosphere.Forexample,the well-known phenomenon of the Antarctic “ozonehole”was foundtobecausedbythecatalyticdestruc-tion of ozone by chlorine and other halogen-contain-ingcomponents.Thisdiscoverybenefitedgreatly fromtheoreticalstudiesofthephotochemistryofhclintheatmosphereofVenus.2

Theearth’satmospherehasbeenstudiedextensively,mainly by meteorologists. Yet meteorologists have dif-ferent objectives and approaches from those studyingcomparativeplanetaryatmospheres.Themainobjectiveofstudyingearth’satmosphereistomakebetterpredic-tionsofitsfuturestate.inweatherforecastingaswellasclimateprediction,onelikestoknowthephysicalstateof the atmosphere at any given time. The predictionproblem is often treated as an “initial value problem”inmathematical terms.so-calleddata-modelassimila-tion,whichappropriatelyincorporatescurrentmeasure-ments into a continuously running numerical model,oftenplaysanimportantroleinprediction.conversely,incomparativeplanetaryatmospheres,oneoftencaresmoreaboutthephysicalmechanismsthatexplainorpre-dictatmosphericphenomena.inthesecasesof“mecha-nisticstudies,”theproblemisoftentreatedasaso-calledforcing-response problem, and data-model assimilationusuallydoesnotenterdirectlyintotheformulationandsolutionproceduresinnumericalmodeling.Thephysicalmechanismsincomparativeplanetaryatmospheresusu-allyincludethosethatexplainthecausesofatmosphericcomposition,origin,evolution,anddynamics.

This article focuses mainly on modeling studies ofatmospheric motions associated with dynamics andreportsonrecentprogressinsolvingthelong-standingdynamicalproblemoftheequatorialatmosphererotat-ingmuchfasterthanitssolidplanetarybody(superrota-tion)inaslowlyrotatingplanetaryatmosphere.

OVERVIEW OF ATMOSPHERIC MODELING STUDIES

in modeling studies of atmospheric states andmotions,significantadvancescanbemadebyinvestigat-ingcommonatmosphericprocesses,suchasenergybal-ance, the momentum budget, the interaction betweenwavesandthemeanflow,etc.,astheyapplytodifferentplanetary settings. Atmospheric features observed ondifferentplanetscanfrequentlybeexplainedbyunder-standingcommonprocessesgovernedbycommonphysi-callaws.

Thebasicphysicallawsdescribingtheconservationofmomentum,energy,andmassarethesameforallplane-taryatmospheres.Themathematicalexpressionsoftheselawsarecalled“primitiveequations”(seebox1,“invari-antprimitiveequations”).often,onedoesnotsolvethecompletesetofprimitiveequationsdirectlywhenstudy-ing atmospheric motions. box 1 gives one example ofderivinganimportantfeatureofplanetaryatmosphericmotionsbyasetofsimplesigntransformations.numeri-cal models for other planetary atmospheres are usuallyderivedfromthosefortheearth’satmosphere.Aninvari-antcharacteristicdemonstratedinbox1indicatesthatitisthemagnituderatherthanthedirectionofplanetaryrotationthatmatters.


X. Zhu

intermsofthenumberofspacevariablesexplicitlyincluded,thestate,structure,andcirculationofaplan-etary atmosphere can be studied by 0-, 1-, 2-, or 3-Dmodels.Asthenumberofmodeldimensionsincreases,morerealisticphysicalprocessesthatareoftenparame-terized(i.e.,thedetailedprocessesareexpressedintermsofsimplifiedempiricalformulaswithafewbulkparam-eters) in lower-dimensionalmodels are included.eventhoughmorephysical processes are explicitlymodeledin a higher-dimensional model, the number of adjust-ableandempiricalmodelparametersstillincreasesforafixednumberofobservationsbecauseofthemuchfasterincrease in the number of model gridpoints. There-fore,thecomputationalresourcerequirementsincrease

dramatically as one proceeds to higher-dimensionalmodels.

There is always a fundamental trade-off betweenmodel efficiency and model robustness in terms ofchoosingtheappropriatemodeldimensionalityandtheappropriatephysicalprocesses tobeexplicitly includedin themodel.Three-dimensionalmodelsare themostcomplicated,addressingallinteractionsamongdynam-ics,photochemistry,radiation,andtransport.however,themeritandsuccessofusing3-Dmodelsforstudyingcomparative planetary atmospheres depend largely ontheavailabilityofdatathatcanbeusedtoinitializeandconstrainthemodel;thatis,theavailabilityofdatapro-videsconstraintsonthecomplexityofmodels.here,i

BOx 1: INVARIANT PRIMITIVE EQUATIONS UNDER A COORDINATE TRANSFORMATIONThe so-called atmospheric primitive equations for the

earth’satmosphereareexpressedinsphericalcoordinatesthatdescribe the fundamentalconservation laws foratmosphericmomentum,energy,andmass,3thatis,

momentum equations

DuDt

ua

v w

uwa

p

− +

+

+ = − ∂

2 2

1

Ω Ωsintan

cosff

f

r ∂∂+

xFrx ,

(1)

DvDt

ua

uvwa

py

Fry+ +

+ = − ∂∂

+21Ω sin

tan,f

f

r (2)

DwDt

uu v

apz

g Frz− − + = − ∂∂

− +212 2

Ω cos ;fr

(3)

the thermodynamic energy equation

D c T

Dtpv Q

( );+ ∇ =

r. V (4)

andthecontinuity equation

DDt

rr+ ∇ =. V 0 . (5)

intheseequations,

a = earthradius, cv = specificheatatconstantvolume, cvT = internalenergy, g = gravity, p = airpressure, T = airtemperature, t = time, u, v, w = eastward,northward,andupwardvelocity,

respectively,

V = velocityvector(u, v, w), x, y = eastwardandnorthwarddistance,respectively, z = altitude, r = airdensity, f = latitude,and V = earth’srotationrate.

ThetermsFrx, Fry, and Frzineqs.1–3andQineq.4rep-resentthenetforcingtothemomentumandinternalenergy,respectively. The total derivativeD/Dt with respect to timet and wind divergence =.V in spherical coordinates thatadopted appropriate approximations for the earth’s atmo-spherearedefinedasfollows:

DDt t

ux

vy

wz

;∂∂

+ ∂∂

+ ∂∂

+ ∂∂

, (6a)

∇ ∂∂

+ ∂∂

+ ∂∂

.cos

(cos ).V =

ux

vy

wz

1f

f (6b)

equations1–6canalsobe applied to studying theatmo-sphericmotionsofotherplanetarybodies.Theinclinationoftheearth’sequatortoitseclipticis23.45°anditsrotationaldirectionisfromwesttoeast.someplanetsorplanetarybodieshavenegative inclinations. For example,Venus’ inclinationis−2.7°andpluto’sis−57.54°.Venusandplutorotateretro-gradewithrespecttoearth:fromeasttowest.

it can be easily verified that eqs. 1–5 are invariant aftermakingthefollowingsigntransforms: Ω Ω→ − → −, andx x, u u;→ − i.e.,ifourearthwererotatingfromeasttowest,themagnitudeofitsatmosphericmotionwouldremainthesameexcept that its wind direction would be reversed. in otherwords,theatmosphericstatesassolutionsofarotatingdynam-icalsystemcanbemoreeffectivelydescribedbyacoordinatesystem based on a rotating axis rather than fixed east–westcoordinates. Therefore, when studying atmospheric motionsfrom the standpoint of comparative planetary atmospheres,onecandefineprograde (retrograde)wind tobemotions inthe same (opposite) direction as the rotational direction ofa planet. Likewise, prograde (retrograde) acceleration corre-spondstoamomentumsource(sink)thatgenerates(destroys)the east-to-west winds in the atmosphere of Venus and thewest-to-eastwindsintheatmospheresofearthandTitan.



giveabriefdescriptionofatmosphericmodelsinlowerdimensions.

Zero-dimensionalmodelsdescribeatmosphericstatesthatareindependentofspatialvariables(i.e.,location).Asimpleexampleofa0-DmodelisthecalculationoftheeffectivetemperatureTeofaplanet,whichisdefinedbyanenergybalancebetweentheincomingsolarfluxandoutgoingthermalradiation4:

a a S

da T

2

22 4

14

( ),p

spe

−= s (1)

where

a=planetradius(6371kmforearth), ap=planetaryalbedo(0.30forearth), dsp=sun–planetdistance(1.00Auforearth), s=solarconstantat1Au(1366Wm22), Te=effectivetemperature,and s=stefan-boltzmannconstant(5.67310−8Wm22).

The left-hand side of eq. 1 denotes the solar energyintercepted and absorbed by the planet disk, whereastheright-handsidedenotesthethermalenergyradiatedbackouttospacebytheplanetasablackbodyradiator.equation1canbeconsideredadegenerateformofeq.4inbox1:Q=0.

Table1listsseveralvaluesoftheeffectivetempera-tureTeandthemeansurfacetemperatureTs fora fewselected planets. because of the greenhouse effect, Tsis often greater thanTe. “greenhouse effect” refers tothe planet’s surface receiving both direct shortwavesolarradiationthattransparentlypassesthroughtheair,whichwouldyieldTe,andlongwaveirradiationemit-tedfromtheairthatabsorbsandthuspreventstheirradiation frombelow from returning to space, leadingtoTs>Te.ThehigherTs thanTe for Jupiter ismainlydue to its internal heat source. in addition to Jupiter,saturn and neptune also possess substantial internalheatsourcescomparedtothesolarenergytheyreceive.Foraplanetwithaverythinatmospheresuchaspluto,Te is approximately the same as the planetary Ts. Forpluto’s icy surface, i assume a relatively large value ofalbedo:ap=0.67.onthebasisoftheaboveenergybal-anceequation,itcanbefoundthatTe=38.7k,whichis consistent with the value derived from coincidentmeasurements.5

one-dimensionalmodelsoftenrefertoradiative-con-vectiveor radiative-conductivemodels thatdeterminethe vertical (z) thermal structure or photochemical(box) models that determine the vertical distributionofchemicalsintheatmosphere.Atmosphericradiativetransferiskeytounderstandinganddevelopingvarious1-D models of the vertical thermal structure of plan-etary atmospheres.6 measured vertical profiles of tem-perature,airdensity,andchemicalcompositionprovide

constraints on atmospheric energy exchange, photo-chemistry, and transport within the atmosphere andpossiblyalsoonthermodynamicandchemicalstatesatthe lower boundary or in the deep atmosphere. sincethe distribution of minor photochemical species oftenexhibitsthelargestgradientinaltitude,1-Dmodelsareusefultoolsforinterpretingthemeasurementsofminorchemicalconstituents.notethatthemajorcomponentsof planetary atmospheres are generally uniformly dis-tributedglobally,suchasn2forearthandTitan,co2forVenusandmars,andh2forgiantplanets.Therefore,their distribution can often be derived by 0-D modelsalone on the basis of thermodynamics and from theperspectiveofsolarsystemevolution.2sincetheminorconstituentsofplanetaryatmospheres(e.g.,o3forearthand mars, hcl for Venus, and ch4 for giant planets)varythegreatestintheverticaldirectionandaresensi-tivetotheverticalthermalanddynamicalstructureoftheatmosphere,a1-Dmodel is theminimumrequire-mentfordeterminingtheverticalprofilesofminorspe-cies in photochemical-diffusive equilibrium in a plan-etaryatmosphere.Various1-Dmodelsfordifferentplan-ets arepresented in a recentmonographbyYung andDemore.7

Two-dimensionalmodels,whichcontaintwospatialvariables(altitudeandlatitude),havebeenusefultoolsformechanisticstudiesoftheatmospheresofearthandotherplanets.Forarotatingplanetaryatmosphere,zonalmean(latitudebandsaveragedacrossalllongitudes)2-Dmodels can be considered a good trade-off, mainlybecauselarge-scaleatmosphericstructureandcirculationvarytheleastalonglatitudebands.inazonalmean2-Dmodel,atmosphericstatevariablessuchastemperature,winds,andchemicalspeciesdistributionsareexpressedas functionsof time t, latitudef, andaltitude z.cur-rentcomputationalcapacityallowsonetocouplecom-plicatedphotochemistrywithdynamicsthroughtrans-port ina2-Dmodel.however,thiskindofmodelcan

Table1. Effective temperatures Te and mean surface temperatures Ts for select planets.

sun–planet distance, Albedo, dsp(Au) ap Te (k) Ts (k)

Venus 0.72 0.77 227.0 750.0earth 1.00 0.30 255.0 280.0mars 1.52 0.15 217.0 225.0Jupiter 5.20 0.58 98.0 138.0a

Titan 9.57 0.30 82.0 94.0pluto 29.70b 0.67 38.7 38.7aAtcloudtop(pressure=1baror105pa).binmay1993.


X. Zhu

onlybewelltestedfortheearth’satmospherebecauseoftheavailabilityofdatabasesofdynamicalstatevariablesand chemical species.Themost observable and stabledynamicalfeaturesforplanetaryatmospheresarezonaljets atvarious altitudes and latitudes. importantques-tions inplanetaryatmosphericdynamics remain:howisanatmosphericjetstreammaintainedonaparticularplanet?Whyisthejetstreamforthatplanetlocatedataparticularlatitudeandaltitude?Whatdeterminesthemagnitudeof themaximumwind at the jet center? istheobservedjetstreamastable,steadyphysicalstateoratransientrealizationofanoscillatorystate?

MAINTENANCE OF EQUATORIAL SUPERROTATION IN A PLANETARY ATMOSPHERE

A long-standing problem in planetary atmosphericdynamics is the maintenance of equatorial super-rotation in the atmospheres of slowly rotating plan-ets. strong equatorial superrotational zonal winds of≈100ms−1(Fig.1)werefirstobservedatVenus’cloud-toplevelinthe1960s,8consistentwithanatmosphericangularvelocitynearly60timesgreaterthantherota-tionofthesolidplanet!severalmechanismshavebeenproposed toexplain thegenerationormaintenanceofthesuperrotationofcloudsonVenus.

i have already shown in box 1 the primitive equa-tions for atmosphericmotionapplicable to all planets.To examine the dynamics of a zonal jet stream, oneneeds to take a zonal average over eqs. 1–5 in box 1andevaluatethezonalmomentumbudgetofallterms.since the zonal momentum equation describing thezonalmeanwindscontains10terms(seeeq.2below),arigorousandquantitativestudyofthesuperrotationhasalways been difficult. most theories have qualitativelyorquantitativelydescribedpossiblemechanismsarisingfromoneor twoparticulardynamicalprocesses corre-spondingtooneortwotermsinthezonalmomentumequation. Also, most theories have studied equatorialsuperrotationbasedonasetofatmosphericparametersspecifictoVenus.Twowell-knowntheoriesthatexplainthemaintenanceofVenus’equatorialsuperrotationare(1)meridionalmomentumtransportfrommid-latitudesbyeddymixing9,10and(2)momentumpumpingbyther-maltides.11,12carefulexaminationoftheexistingtheo-ries shows that they either have difficulties leading toa contradiction with fundamental physical lawsor areincomplete,leadingtoincorrectpredictionsofthemag-nitudeofthesuperrotatingwinds.

To rigorously solve problems of large-scale circula-tion, such as equatorial superrotation in a planetaryatmosphere, a 2-D model is necessary to describe thezonallyaveragedwinds.bytakingazonalaverageofthefluxformofeq.1inbox1,iarriveatthefollowingmeanzonalmomentumequation1:

(i) (ii) (iii) (iV)

− − +2 2Ω Ωcos ( ) sin ( )

(V) (Vi) (Vii)rf r r f rw u va

(2)

− − ′ ′ −( cos )

cos,

(Viii) (iX) (X)

u

av w u

awua

f

fr r rf

where theoverbars andprimesdenote the zonal aver-agesanddeviationsfromthem,respectively.ihavealsoreplaced the momentum drag term Frx by a rayleighfrictionaldragterm −a rr u withtherayleighfrictionalcoefficient ar. equation 2 shows that the mean zonalmomentumequationcontains10terms(i–X),eachrep-resentingadifferentdynamicalprocesscontributingtothe momentum sources and sinks. These terms haveneverbeensystematicallyandself-consistentlyevaluatedinthesamedynamicalframe.Therefore,theexactphys-icalmechanismthatmaintainsequatorialsuperrotationon Venus has remained a mystery, and the previouslyproposed hypotheses are mutually inconsistent. This

Figure 1. A UV image of thick clouds on Venus taken by the Pio-neer Venus Orbiter (26 February 1979). Venus rotates very slowly, taking 243 days to complete one revolution. By tracking the cloud features, it is found that the clouds travel quickly around the planet, taking only 4 days to circulate once, indicating superrotational winds at the cloud level. Both the winds and the solid planet rotate from east to west, opposite to the direction on Earth. The Y-shaped cloud patterns move from right to left in the figure.

rf

r f r rf

ua

v u w u wut z z= − ′ ′ − ′ ′ −12

2

cos( cos ) ( )



and internalparameters.Thisgeneral theoryhasbeenappliedtotheatmospheresofVenus,Titan,andearth.

For the atmospheres of these three bodies, whichhaveorcouldhaveequatorialsuperrotation,thetermsthatmakethemajorcontributionstothemomentumbudget in eq. 2 are iii, Vi, and Viii. Dynamically,iii ( )= − ′ ′rw u z

representstheconvergenceoftheverti-caleddymomentumfluxcausedbytidalwavepump-ing, i.e., thedynamical effectofwavesonmeanflowthrough the nonlinear interaction between the two.The physical mechanism is shown schematically inFig.2a.Whenanoscillatoryheatingsourcecausedbytheabsorptionofsolarradiationmovesinadirectionopposite to the zonal flow, the phase surfaces (tiltedlines)andairparceltrajectories(arrows)oftheexcitedtidalwavesslantdifferentlyaboveandbelowtheheat-inglayer.Asaresult,averticalmomentumfluxconver-gencearises across theheating layer,whichproducesan acceleration of the mean zonal flow. Leovy12 alsonoted that the momentum flux convergence can berewrittenas

− ′ ′ = ′ ′( ) ,r r jw u wz (3)

where ′ = − ′ + ′j u wz x isthehorizontalvorticity(alongthe y direction) of the perturbation field that corre-spondstotidesinthisstudy.equation3showsthatthemomentum flux convergence within the heating layercanalsoberepresentedasavorticityflux.Figure2billus-trates how a positive correlation between ′w and ′j leads toanaccelerationof themeanzonalflowat thecenterasthevortexparcelsmoveinanorganizedway.ThoughtheschematicdepictionsofiiishowninFigs.2aand2bareapparentlydifferent,theyareactuallyidenti-calintermsofmathematicsandphysics.

since the momentum source iii produces progradeacceleration of the equatorial superrotational winds,terms Vi and Viii produce retrograde acceleration tobalanceiii.Themomentumsink Vi r= − a ru duetoparameterized rayleigh friction represents the effectofsmall-scaleeddies.Theseeddiescascadethekineticenergyof theflow from large-scalemotions to smalleronesuntilcompletelydissipatedbythemolecularviscos-ity.ThemomentumsinkViii ( cos ) ( cos )= − −r f f

fv a u1

represents the advection of horizontal wind shear bythehorizontalbranchesof thehadleycirculationcell.The dynamical mechanism of momentum sink Viii

(a) z

z z

Longitude

Longitude Latitude

0

Heated layer

Zonal wind

ρ (wu) < 0

ρ (wu) > 0

ρ ξ (w ) > 0

+

+

– + –

(b) (c)

–

lackof resolutionof thesuperrota-tionissueonVenushasalsocausedfurther confusion and debate con-cerningequatorialsuperrotationinTitan’s stratosphere (e.g., refs. 13–15). Titan is also a slowly rotatingplanetary body, which, similar toVenus’cloudlayer,hasahazelayerthatabsorbsasignificantportionofitsincidentsolarradiation.

recently,isolvedtheproblemofequatorial superrotation in a plan-etary atmosphereby specifying the2-D zonal winds in an analyticalform that resembles a jet stream.1Asaresult,eachofthe10termsineq.2canbeanalyzedsystematically,firstbyscaleanalysis,thenthroughanalyticalformulationsandnumer-ical evaluations, and finally bycomparison with well-known plan-etary atmospheres. key parametersof the analytical formulations areconsistentwith fundamentalphysi-cal concepts andagreewithobser-vationalconstraints.Furthermore,thistheoryhasbeendevelopedfromthegeneralperspectiveofequatorialsuperrotation for any given plane-tary atmosphere. The analyticalforms of the approach explicitlyshow how an equatorial superrota-tion depends on various external

Figure 2. (a) Oscillating heating (1) and cooling (2) centers move opposite to the direc-tion of the zonal wind. The phase surfaces (tilted lines) and air parcel trajectories (arrows) of the excited tidal waves cause a negative mean momentum flux above the heating layer and a positive one below the heating layer. Vertical momentum flux convergence across the heating layer produces an acceleration of the mean zonal flow. (b) Momentum flux convergence within the heating layer can also be represented as the vorticity flux. A posi-tive correlation between w 9 and j9 also leads to the acceleration of the mean zonal flow. (c) The incoming branches of the Hadley circulation cell cause the convergence of the momentum-depleted air mass from the outside into the region of the superrotational jet, whereas the outgoing branches cause the divergence of the momentum surplus air mass from the jet center to the region outside. The Hadley circulation is a zonally averaged thermal circulation that consists of warm air rising at the equator, moving poleward, then descending at the high latitudes and moving equatorward at the lower altitudes.


X. Zhu

decelerates the wind at the jetcenter as shown schematically inFig. 2c. The incoming branches ofthe hadley circulation cause theconvergence of the momentum-depletedairmassintotheregionofthesuperrotationaljet,whereastheoutgoingbranchescausethediver-genceofthemomentumsurplusairmassfromthejetcentertotheout-sideregion.

As a result of wave/mean-flowinteraction, the functional depen-dence of the wave pumping iii onthezonalwindatthejetcenter u0 ishighlynonlinear,asshownsche-maticallyinFig.3bytheredcurve(see box 2, “stability Analysis”),which vanishes when u0 gets toosmall or too large. however, bothVi and Viii are linearly propor-tionalto u0 , sothesumofthetwois a straight line passing throughtheorigin.Asteadyandstablesolu-tion exists where the momentumsinklineintersectsthemomentumsource curve (Fig. 3). The peakvalueoftheredcurveandslopeofthe straight lines in Fig. 3 dependonthestrengthofthesolarheatingrate Q, since both the amplitudeof the tidal waves ( , , )′ ′ ′u v w andthestrengthofthehadleycircula-

ThemodeledVenus u0 (118ms21)isveryclosetothemeasuredzonalwindof≈115ms21atthe65-kmcloud-toplevel.16

ForTitan’sstratosphere,limitedmeasurements17andnumerical simulationsbya3-Dmodel13 suggest super-rotationalwindsof≈100ms21centeredatthe1.0-mbar(100-pa) pressure level. my analytic model shows thatno solution is possible for equatorial superrotation atthislevel,correspondingtothecaseofnonintersectionbetweenthebluedashedlineandtheredcurveinFig.3.Titanisfarfromthesunandonlyinterceptsasmallpor-tionofthesun’senergy.however,ifthemainsolarradia-tionabsorptionlayerinTitan’sstratosphereisliftedfrom1.0 mbar (≈185 km) to 0.1 mbar (≈288 km), whichincreases the solar heating rate per unit volume by afactor10becauseofdecreasingairdensity,anequatorialsuperrotationof106ms21couldbemaintainedatthe0.1-mbar(10-pa)level(Table2).itishopedthatongoingobservationbythecassini-huygensmissionwillprovidesuitabledatatovalidatethismodelprediction.

ThelastthreerowsinTable2showtherelativecon-tributionstothemomentumbudgetfromthreetermsineq.2.negativevalues indicatemomentumsinks.The

2.0

Momentum sink

Momentum source

Critical solution

Unstablesolution

Stablesolution

1.5

1.0

0.5

00 0.5 1.0

Zonal wind at jet center

Mom

entu

m s

ourc

e an

d si

nk it

ems

1.5 2.0

Figure 3. Dimensionless relationship between the wave pumping and momentum sinks as a function of the magnitude of the zonal wind at the superrotational jet center. The red curve represents the momentum source caused by wave pumping. Three straight lines represent relative momentum sinks under different parameter settings. A steady and stable superrotational solution is possible when two intersections exist between the source curve and a sink line.

tion ( , )v w arelinearlyproportionaltoQ.however,iiidependsquadraticallyonthewaveamplitude,whereasViiidependslinearlyon v. hence,therelativestrengthofthemomentumsinkViiiisinverselyproportionaltoQ.notethatViisindependentofQ.Thus,theslopeofthestraightlinesinFig.3,whichmeasurestherelativestrength of the total momentum sinks, increases withincreasingdsp,sincelesssolarenergyisreceivedwhenaplanetgetsfartherawayfromthesun.

i show in Table 2 an example of equatorial super-rotation corresponding to the steady and stable solu-tions illustrated in Fig. 3, typically at the intersectionoftheredcurveandthesolidblueline(box2).here,usisthesurfaceangularvelocityattheequatorforthesolidplanet, u0 isthezonalwindattheatmosphericjetcenterlocatedataltitudez0withapressurep0,and u0c isthecriticalzonalwindbelowwhichthereisnosuper-rotation(box2). in theanalyticmodel, the jetcenteris characterizedbya layerwhere the solar radiation isstronglyabsorbed:acloudlayerforVenus,ahazelayerforTitan,andadustcloud layer forearth.ForVenus,themodeledatmosphericangularvelocityatthecloud-toplevelis65timesgreaterthanitssolidbodyrotation.



momentum source driving equatorial superrotation iswavepumping(iii)duetothestrongabsorptionofsolarradiationinanabsorbinglayerwithintheatmosphere.Windshearadvectionbythehorizontalbranchesofthehadleycirculation(Viii)isamajormomentumsinkforVenus.FortheatmospheresofTitanandearth,frictionaldrag(Vi)mainlybalancesthemomentumsource.

BOx 2: STABILITY ANALYSIS OF A BALANCED SUPERROTATIONAL JET AT THE EQUATOR

Thesourceofzonalmomentumiswavepumping,whichisthedynamicaleffectofthenonlinearwave/meanflowinterac-tion.itisanonlinearfunctionofthezonalwindasshownbytheredcurveinFig.2.however,themomentumsinkcanbeexpressedasalinearfunctionofthestrengthofthejet.Thestrength of the momentum sink relative to the momentumsourceismeasuredbytheslopeofthestraightlinesinFig.2.Thejetcenteraccelerationcanbedescribedbythedifferencebetweenthewavepumpingandtherelativemomentumsink,

ε ∂∂

= −− −h

teh ahh3 1 2 2/ , (1)

wherehisthedimensionlesswindatthejetcenterandthetwopositivecoefficientsandadenotetherelativestrengthsoftheaccelerationandmomentumsink,respectively.Whenthewavepumpingisexactlybalancedbythemomentumsink,zonalwindaccelerationvanishes:

ε∂∂

= − =− −h00

3 1 20

02

0t

eh ahh/ . (2)

Toseewhethersuchabalanceleadstoastableorunstablesolutionh0,iintroduceasmallperturbationh9thatissuper-imposedonh0toexaminehowh9willchangewithtime.by

substitutingh=h01h9 intoeq.1,usingeq.2,andkeepingonlythelineartermsonh9sinceitisasmallperturbation,iobtainthefollowinglinearizedequationgoverningtheevolu-tionofh9:

ε ∂ ′∂

= − − ′−h

ta h h( ) .4 0

2 (3)

Whenh0=h0c;1/2, the right-hand sideofeq.3vanishes.correspondingtothetangentpointoftheblueandredlinesinFig.3,h0cisthecriticalzonalwindbelowwhichthereisno solution fora steadyand stableequatorial superrotation.Thedimensionalvaluesofcriticalzonalvelocity u0c forthreeplanetsarelistedinTable2.

Thesolutionofeq.3canbewrittenas

′ = ′ − −=− −h h a ht t0

10

24exp[ ( ) ] ,ε (4)

where ′ =ht 0 denotes the initial perturbation. equation 4indicates that when two solutions exist, say, h01<h0c andh02>h0c,fromeq.2,asshowninFig.3,thesmallerh01corre-spondstoanunstablesolutionbecauseaninitialperturbation

′ =ht 0 willgrowexponentially,whereasthelargerh02leadstoastablesolutionbecauseofitsexponentialsuppressionofaninitialperturbation.

Table 2. Equatorial superrotation and momentum budgets for the atmospheres of three bodies.

bodies Venus Titan earth

rotationrate,V(s21) 3.00131027 4.57331026 7.29231025

radius,a(m) 6.0523106 2.5753106 6.3713106

surfaceangularvelocity,us(ms21) 1.82 11.8 465.0Zonalwindatjetcenter, u0 (ms21) 118 106 120criticalzonalwind, u0c (ms21) 14.5 86.3 55.7Altitude,z0(km) 65 288 17pressure,p0(mbar) 100 0.1 100momentumsource,iii 100% 100% 100%momentumsink,Vi 231.5% 281.9% 269.5%momentumsink,Viii 267.2% 217.7% 229.8%

EQUATORIAL SUPERROTATION IN EARTH’S LOWER STRATOSPHERE AND IMPLICATIONS FOR THE ExTINCTION OF THE DINOSAURS

There is no steady equatorial superrotation in theearth’s atmospherebecause it is almost transparent to


X. Zhu

visiblesolarradiation.Themajorityofthesolarenergyiseitherabsorbedatearth’ssurfaceorreflectedbacktospace by clouds. superrotational winds in the earth’satmosphere(Table2)could,however,existifalayerofsuspendingdustwere injected intothestratospherebyhuge meteorite impacts or by massive volcanic erup-tions,whichwouldleadtothestrongabsorptionofsolarradiationbytheresultingdustcloud.

hugemeteoriteimpactsandmassivevolcanicerup-tionshaveemergedasthedominanttheoriestoexplainpaleo-extinction events, specifically the extinctionofdinosaurs (e.g., ref. 18) at the cretaceous/Tertiary(k/T)boundary,about65millionyearsago.Alvarezetal.19proposedthata10-km-dia.asteroidstruckearthintheYucatanpeninsulainmexicoandinjectedalayerofdebriscloudsintothelowerstratosphere,blockingsun-lightandcausingarapidcoolingthatwipedoutabout50%ofthebiota,includingnearlyallofthedinosaurs.iftheextinctionsnearthek/Tboundaryweregradualorstepwise,avolcanicexplanationfortheextinctionswould be more consistent (e.g., ref. 18).however, ineithertheory,anenvironmentfavorabletothesurvivalof dinosaurswasdisrupted locally andquickly spreadacrosstheentireplanet,resultinginglobalmassextinc-tion.numericalsimulationsbyanaerosolmodelshowthat the environmental consequences of the impact-generated dust cloud are sensitive to the horizontaltransporttime,sincetheeffectondinosaurextinctionrequiresthedustcloudtoremaininthestratosphereforalongenoughperiodtohaveaglobalimpact.20

basedonmyanalyticmodel,1itisfoundthatanopti-callythickdustcloudintheearth’s lowerstratospherecenteredatthe100-mbarlevel,suchasonecreatedbyanasteroidimpact,couldproduceanequatorialsuper-rotation of≈120m s21 (Table 2).such a strong equa-torial superrotational jet would greatly enhance themeridional transport of the dust cloud. Two processesassociatedwithequatorialsuperrotationcancausethisenhancedtransport.First,enhancedheatingofthedustcloud increases the strength of the radiatively drivenhadleycirculationandreducesthemeridionaltransporttime.second,global-scalewavesareexpectedtobegen-eratedfromthejetstream,possiblyasaresultofinstabil-ity,whichcanalsoeffectivelytransportthedustcloudpoleward.The largemeridional spreadingoforganizedcloud features shown by the dark horizontal Y-shaped4-daywavesapparentnearVenus’cloudtop(Fig.1)canbeconsideredanexampleofthepresenceofglobal-scalewavesinastronglysuperrotationalatmosphere.8

in the earth’s current atmosphere (with no strongequatorial superrotational winds), large-scale wavesexcited in theequatorial loweratmosphereareusuallyconfined within a narrow band about 15° north andsouthoftheequator.Theseequatoriallytrappedwavesplayamajorroleindrivingequatorialcirculationinthestratosphere through wave/mean-flow interaction, but

havelittledirecteffectonmid-latitudeatmosphericcir-culation.21 since the zonal mean winds are driven byequatoriallytrappedwaves,theireffectsarealsomainlyconfinedtolowerlatitudes.Fromtheviewpointoftracertransport in theearth’s lower stratosphere, the tropicscanbeconsideredisolatedfromthemid-latitudes.

measurements and model simulations suggest thatthetransporttimefromtheequatortohighlatitudesis1to2yearsinthelowerstratosphere.22,23ontheotherhand,becauseof thepresenceof superrotationalwindandtheassociatedglobal-scalewavemodesnearVenus’cloud-toplevel,itonlytakesafewdaystotransportcloudfeaturesfromlow-tohigh-latituderegionsontheplanet.onewouldexpectsimilarmodesofunstableglobal-scalewaves to appear on earth if the meridional variationsofstrongsuperrotationalwindinthelowerstratospherewere large enough to cancel out the curvature effectthattrapslarge-scaleequatorialwaves.Therefore,moreefficientmeridionaltransportofadustcloudwouldbeexpectedtoacceleratetheglobalizationofasubstantialmeteoriticdustcloud,rapidlyspreadinganenvironmentunfavorabletothesurvivalofthedinosaursonascaleofafewdaysratherthanoverafewyearsasintheearth’spresentatmosphere.

CONCLUDING REMARKSThecomparativeapproachtostudyingplanetarysci-

encehasreceivedincreasingattentionandhasbecomefruitfulasmoredatafromdifferentplanetshavebecomeavailable.24–26 because of the extensive diversity ofplanetarybodies,in-depthstudiesofcomparativeplan-etaryatmospheresareoftenconductedonaone-to-onecomparison basis. in this article, atmospheric equato-rial superrotation is investigated forVenus,Titan,andearth. by a brief and hierarchical overview of variousmodelingmethods, ihavedemonstrated the feasibilityandusefulnessofthecomparativeapproachofstudyingplanetaryatmospheresusingacommonsetofphysicallaws.ThemainresultsaresummarizedinTable2,whichshowsthatunderappropriateconditionsallthreebodiescouldmaintainstableequatorial superrotationalwindsgreater than100m s21.The singlemomentum sourcethatmaintainsthesuperrotationisthermalpumpingbytidalwaves,balancedbythetwomomentumsinksoffric-tionaldragandtheadvectionofthehorizontalshearbythehadleycirculation.Furthermore,sinceihavesolvedtheproblemanalytically,thegeneralparameterdepen-dence of equatorial superrotation on various externalandinternalparametersbecomesapparent.Thedetailedderivationsandexplicitexpressionsforthesolutionaregiven in a comprehensive paper.1 The analytic modelincludesthefollowingimportantparameters:

• sun–planetdistance• radiusoftheplanet• rotationrate



• inclinationoftheequatorialplane• gravity• Atmosphericscaleheight• Atmosphericbuoyancyfrequency• rayleighfrictioncoefficient• Albedo• pressure level at which the absorption of the solar

radiationoccurs

studies of the atmospheres of other planets alsobenefitourunderstandingoftheearth’satmosphere.ihavealreadypointedoutoneexample:theunderstand-ingoftheAntarctic“ozonehole”associatedwiththecatalytic destruction of ozone by chlorine and otherhalogen-containing components benefited in partfromtheoreticalstudiesofthephotochemistryofhclintheatmosphereofVenus.2oneimportantby-prod-uct of investigating the equatorial superrotation of aslowlyrotatingplanetaryatmosphere,suchasVenusorTitan,istherealizationthatearth’satmospherecouldalso have had equatorial superrotation 65 millionyearsago.sinceatmosphericequatorial superrotationwasfirstobservedonVenus, ithasalwaysbeenasso-ciatedwitha slowplanetary rotation rate.Therefore,the possibility of superrotation and its consequenceson the faster-rotatingearthhaveneverbeenconsid-ered before in a similar perspective as that of Venusor Titan. my analytic solution to the problem, i.e.,evaluatingallof themomentumterms ineq.2, sug-geststhatthemagnitudeoftheplanetaryrotationrateisnotcriticalfortheexistenceofequatorialsuperrota-tiononaslowlyrotatingplanet.Thedominantcauseofequatorialsuperrotationislayeredsolarheating,therateofwhichisinverselyproportionaltotheairdensityandincreasesexponentiallywithaltitude.Asaresult,iamalsoabletoexaminethepossibilityofsuperrota-tionanditsconsequencesonearth.Thisleadstomyconclusionthatequatorialsuperrotationintheearth’slowerstratosphere followinganasteroid impactcouldhaveacceleratedtheglobalizationofanenvironmentunfavorabletothesurvivalofthedinosaurs.

planetaryatmosphericresearchisamultidisciplinedfield. Among the three major disciplines—dynamics,radiation,andphotochemistry—ihavefocusedonatmo-sphericdynamicsinthisarticleanddiscussedradiativeenergybalanceverybriefly.TheApLspaceDepartmenthasauniquescienceteamwhosemembersareactivelyengaged inall threedisciplines.understandingplane-taryatmospheresisimportantbothforitsownscientificmeritsandforgaininginsightintotheearth’sbiosphere.Looking at atmospheric processes on earth from theperspectiveoftheentiresolarsystemwilltransformourviewofatmosphericscience.Thepossibilityofequato-rialsuperrotationintheearth’slowerstratosphereanditsimplicationsfortheextinctionofdinosaursdiscussedinthisarticleisonesuchexample.

irecentlyencounteredthefollowingparagraphthatdescribesthepossiblegcm(generalcirculationmodel)simulation of equatorial superrotation in the martianatmosphere.

The grid point dynamic model is based on the LmD ter-restrialmodeldescribedbySadourny and Laval [1984].Thediscretization scheme conserves both potential enstrophyforbarotropicnondivergentflows[Sadourny,1975]andtotalangularmomentumforconstantsurfacepressureaxisymmet-ricflow.Thelatterpropertywasnotincludedintheoriginalterrestrial version but was found to be very important forsimulating the martian atmosphere in order to avoid spu-rious prograde zonal winds in equatorial regions [Hourdin,1992b].27

because themartian atmosphere is known tohavesuspendeddustthatabsorbssolarradiationandproducesstrongtides,theoriginallysimulatedprogradeequatorialzonalwinds intheLmD(LaboratoiredemétéorologieDynamique)martianmodelmaynotbeentirelyduetothemodel’sspuriousmomentumsource.

AcknoWLeDgmenTs: This work was supportedbynAsAgrantnAg5-11962toApL.Valuablecom-mentsbyDrs.stevenA.Lloyd,Williamh.swartz,andtwo anonymous reviewers on the original manuscriptledtosignificantimprovementsinthearticle.

reFerences 1Zhu,X.,Maintenance of Equatorial Superrotation in a Planetary Atmo-

sphere: Analytic Evaluation of the Zonal Momentum Budgets for the Strato-spheres of Venus, Titan and Earth, srA-2005-01, Jhu/ApL, Laurel,mD(2005).(pDFfileavailableuponrequestfromtheauthor.)

2Lewis,J.s.,andprinn,r.g.,Planets and Their Atmospheres: Origin and Evolution,Academicpress(1984).

3holton,J.r.,An Introduction to Dynamic Meteorology, 4thed.,elsevierAcademicpress(2004).

4houghton,J.T.,The Physics of Atmospheres,3rded.,cambridgeuniv.press,newYork(2002).

5Tryka,k.A.,brown,r.h.,cruikshank,D.p.,owen,T.c.,geballe,T. r., and Debergh, c., “Temperature of nitrogen ice on plutoand its implications to Flux measurements,” Icarus 112, 513–527(1994).

6Zhu,X.,“radiativeTransferinthemiddleAtmosphereandplanetaryAtmospheres,inObservation, Theory and Modeling of Atmospheric Vari-ability,X.Zhuetal.(eds.),Worldscientific,singapore,pp.359–396(2004).

7Yung, Y. L., and Demore, W. b., Photochemistry of Planetary Atmo-spheres,oxforduniv.press(1999).

8schubert,g.,“generalcirculationandDynamicalstateoftheVenusAtmosphere,”inVenus,D.m.huntenetal.(eds.),univ.ofArizonapress,Tucson,pp.681–765(1983).

9gierasch, p., “meridional circulation and the maintenance of theVenusAtmosphericrotation,”J. Atmos. Sci.32,1038–1044(1975).

10rossow,W.b.,andWilliams,g.p.,“Large-scalemotionintheVenusstratosphere,” J. Atmos. Sci.36,377–389(1979).

11Fels,s.b.,andLindzen,r.s.,“TheinteractionofThermallyexcitedgravity Waves with mean Flows,” Geophys. Fluid Dyn. 6, 149–191(1974).

12Leovy, c. b., “Zonal Winds near Venus’ cloud Top Level: AnAnalyticmodelof theequatorialWindspeed,”Icarus69,193–201(1987).

13hourdin, F.,Talagrand,o.,sadourny,r.,courtin,r.,gautier,D.,andmckay,c.p.,“numericalsimulationofthegeneralcirculationoftheAtmosphereofTitan,”Icarus117,358–374(1995).


X. Zhu

THE AUTHOR

XunZhu

xun Zhu graduated fromnanjing institute ofmeteorology,china, inFebruary1982with ab.s. inmeteorology andreceivedaph.D.degreeinatmosphericsciencesfromtheuniversityofWashington,seattle,inJuly1987.hejoinedApLin1997asanatmosphericphysicist.Dr.ZhuiscurrentlyamemberoftheseniorprofessionalstaffintheAtmosphericand

ionosphericremotesensinggroupofthespaceDepartment.hisareasofexpertiseareanalyticalandnumericalmodelingofearth’smiddleatmosphereand theatmospheresofotherplanetarybodies.hise-mailaddressisxun.zhu@jhuapl.edu.

14Tokano,T.,neubauer,F.m.,Laube,m.,andmckay,c.p.,“seasonalVariationofTitan’sAtmosphericstructuresimulatedbyageneralcirculationmodel,”Planet. Space Sci.47,493–520(1999).

15Zhu,X.,andstrobel,D.F.,“onthemaintenanceofThermalWindbalanceandequatorialsuperrotationinTitan’sstratosphere,”Icarus176,331–350(2005).

16gierasch,p.J.,goody,r.m.,Young,r.e.,crisp,D.,edwards,c.,etal.,“ThegeneralcirculationoftheVenusAtmosphere:AnAssess-ment,”inVenus II: Geology, Geophysics, Atmosphere, and Solar Wind Environment, J.W.bucher,D.m.hunten, andr. J.philips (eds.),Arizonauniv.press,Tucson,pp.459–500(1997).

17kostiuk,T.,Fast,k.e.,Livengood,T.A.,hewagama,T.,goldstein,J. J., espenak, F., and buhl, D., “Direct measurement of Winds onTitan,”Geophys. Res. Lett.28,2361–2364(2001).

18oard, m. J., “The extinction of the Dinosaurs,” CEN Tech. J. 11,137–154(1997).

19Alvarez,L.W.,Alvarez,W.,Asaro,F.,andmichel,h.V.,“extrater-restrialcause for thecretaceous–Tertiaryextinction,”Science208,1095–1108(1980).

20pollack,J.b.,Toon,o.b.,Ackerman,T.p.,mckay,c.p.,andTurco,r. p., “environmental effects of an impact-generated Dust cloud:

implications for the cretaceous–Tertiary extinctions,” Science219,287–289(1983).

21Andrews,D.g.,holton,J.r.,andLeovy,c.b.,Middle Atmosphere Dynamics,Academicpress,newYork(1987).

22hall, T. m., Waugh, D. W., boering, k. A., and plumb, r. A.,“evaluationofTransport instratosphericmodels,” J. Geophys. Res.104(D15),18,815–18,839(1999).

23Zhu,X.,Yee,J.-h.,andstrobel,D.F.,“middleAtmosphereAgeofAirinagloballybalancedTwo-dimensionalmodel,”J. Geophys. Res.105(D12),15,201–15,212(2000).

24Luhmann, J.g.,Tatrallyay,m.,andpepin,r.o.(eds.),Venus and Mars: Atmospheres, Ionospheres, and Solar Wind Interactions, Agu,Washington,Dc(1992).

25mendillo,m.,nagy,A.,andWaite,J.h.(eds.),Atmospheres in the Solar System: Comparative Aeronomy,Agu,Washington,Dc(2002).

26irwin,p.g.J.,Giant Planets of Our Solar System: Atmospheres, Compo-sition and Structure,springer-Verlag,newYork(2003).

27Forget,F.,hourdin,F.,Fournier,r.,hourdin,o.,Talagrand,o.,etal., “improved general circulation models of the martian Atmo-spherefromthesurfacetoAbove80km,”J. Geophys. Res.104(e10),24,155–24,175(1999).

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Dynamics in planetary Atmospheric physics: comparative studies … · 2009-11-04 · 164 Johns hopkins ApL TechnicAL DigesT, VoLume 26, number 2 (2005) X. Zhu T Dynamics in planetary