高等大气动力学Rossbywave.ppt

PhysicsofRossbyWave,YihuaLinAOPP,ClarendonLaboratoryUniversityofOxford,UK4th,June,2008,WaveRossbywaveinatmosphere03March200512Z.,9,10,The2smallblackcirclesoffthecoastoftheAmericascorrespondtosealevelhighsonthetrailingedgeofKelvinwaves.Theleadingedgeofthiswavegrouphasbouncedoffthecoastline,creatingRossbywaveswhoserisingandfallingsealevelsaremarkedbysolidanddashedlines,respectively.,IntheJulyimage,thecirclesaremovingwestwiththeRossbywaves.TheXmarksarelativesealevellowcausedbyaKelvinwavemovingeast.,InDecember,theRossbywavescontinuetomovewestward.TheshapesofthesolidanddashedlinesindicatethattheRossbywavesaremovingawayfromtheAmericasfasterattheequatorthanathigherlatitudes.,Whiteandredindicatedhigherthanaveragelevels;purpleandmagentaindicatelowerthanaveragelevels.Thesescenesshoweastward-movingKelvinwavesandwestward-movingRossbywaves.SuperimposedblackcirclesshowhowtheelevatedseasurfacemoveseastinApril,thenwestasRossbywavesduringJulyandDecember.,Ocean,RossbyWavesThemostimportantclassoflarge-scaleatmosphericwavesarecalledplanetarywaves,orRossbywaves.Thesewavesarecharacterizedbyoscillationsintherotationalpartofthehorizontalwindthatareparalleltothehorizontalgradientinthepotentialvorticity.,11,MechanismofRossbyWave,Restoringforceisthenorth-southgradientofbackgroundpotentialvorticity(f/H).Thatgradientcanbeduetoeitherthevariationinfwithlatitude,ortoavariationinH(“topographicRossbywaves”andwhentrappednearthecoast“continentalshelfwaves”).NotethatRossbywavesaretranversewaves,thatistheparticlesmoveperpendiculartothedirectionofpropagation.ConservesPV=(f+)/H.,12,13,Sketch:northernhemisphere,Imaginethatsomethingpushesawatercolumnperpendiculartotheplanetaryvorticity.Whatwillhappen?Theparcelwillconservetotalpotentialvorticity(f+/H).Asitmovesintoaregionofhigherf,itwilltendtodecreaserelativevorticityatendencytoclockwisemotion.Likewiseifitweredisplacedtowardslowerrelativevorticity,itwillincreaserelativevorticityatendencytowardscounterclockwisemotion.,14,15,NowletslookattheCoriolisforcesontheinducedcirculationpattern.,Sizeofrestoringforce(foragivendisplacement)isproportionalto=df/dy.PVequation:D/Dt+v=0,ThesamethingwillholdtrueonaslopepushedtowardsshallowerwaterwillrequiredecreasedrelativevorticityandviceversasoshallowerwateristheanalogofnorthwardoveraflatbottomandthereareanalogouswavesgeneratedonstrongtopographicslopescalledtopographicRossbywaves.Hf.,16,Inlastsection,wederivedthepropertiesofRossbywaves.OurkeyfindingswasthatRossbywavesrelyonthe-effectfortheirexistence.WealsodiscussedthemechanismforRossbywavepropagationthroughpotentialvorticityconservation.,17,Rossbywavepropagation,PotentialvorticityequationConsidersmall-amplitudedisturbancestoauniformzonalbackgroundflow(U,0,0)forwhichthestreamfunctionisandthequasigeostrophicpotentialvorticityisThestreamfunctionforthetotalflow(backgroundplussmalldisturbance)isthusandthepotentialvorticity,18,WhereWenowsubstitutetheaboveinthequasigeostrophicpotentialvorticityequationandneglecttermsthatarequadraticinthedisturbancequantities:Thefirsttermrepresentstherateofchangeoftheperturbedpotentialvorticityfollowingthemeanflow,andthesecondtermrepresentstheadvectionofmeanpotentialvorticitybytheperturbedflow.,19,DispersionrelationWenowlookforplane-wavesolutions,Weshallalsoassumeconstantstratification,i.e.,N2=constant.ThisgivesthedispersionrelationforquasigeostrophicRossbywaves:Asinlastsection,the-effectiscrucialfortheseRossbywaves.Setting=0gives=Uk,whichimpliesc(x)p=c(x)g=U,i.e.,thewavesaremerelycarriedalongbythebackgroundflow,U.,20,Forthebarotropicmotion,wegetthesimplerdispersionrelationforRossbywaves(onamotionlessstate,U=0)Thisfunctionisplottedasbelow(Notethatl/=x/(x2+1),wherex=k/l.),21,Thefunctionx/(x2+1).,1./k=/(k2+l2)0fork/l1,andd/dk0)thatarenottoostrong(U/(Uc(x)p)thenmisimaginary.Ifwesetm=i,whereisrealandpositive,thenthewavesolutionsareoftheformi.e.,thewavespropagatehorizontallybutnotvertically,anddecaywithheight.Equivalently,thishappensifForstationarywaves(c(x)p=0),verticaltrappingoccursifthebackgroundwindsarewestward(UUc).,29,VerticalmodesIntheocean,Rossbywavesarestronglyconstrainedbythesurfaceandbottomboundaries.WecanseekRossbywavesolutionswithmodalstructuresinthevertical.ThusweseeksolutionsoftheformSubstitutingtheaboveintothepotentialvorticityequationgivesThisseparablesolutiononlyworksifboththelargerscaleofthesewillpropagaterapidlywestward.Beforelong,theywillreachthewesternboundaryoftheoceanwheretheywillbereflected.Unlikegravity(orlight)waves,thereflectedwaveswillnotsimplybeamirrorimageoftheincidentwaves:thereflectedwavesmusthaveaneastwardcomponentofgroupvelocityandsomustbeofshortzonalwavelength.,36,Moreover,theywillpropagaterelativelyslowly,moresothantheincomingwaves.Thus,therewillbeakindoftrafficjamatthewesternboundaryinformationcangetinmorereadilythatitgetsout.Theinformationthataccumulatestherewillinvolvemotionsofsmallzonalscale.Thisistheunderlyingdynamicalreasonfortheexistenceofstrongboundarycurrentsonthewestern,ratherthaneaste