The resolution of global spectral models

Abstract

See, stats, and : https : / / www . researchgate . net / publication / 234882313 The Article CITATIONS 39 READS 72 1 : Some : Empirical - simulated - surface René Université 190 , 306 SEE All - text , letting . Available : René Retrieved : 10TheletterofProfessorPielke(1991)suggestingadefinitionofhorizontalresolutionofgridpointmodelspromptsmetoreciprocatewithashortessayonacorrespondinganalysisforglobalspectralmodels.ThehorizontalstructureofdependentvariablesinspectralmodelsisrepresentedbyseriesexpansionofsphericalharmonicsY;;',where0::sJml::sn(e.g.,Kubota1959);thetransformmethod(MachenhauerandRasmussen1972)isusedtocalculatenonlineartermsonaGaussianlatitude-longitudegridinallmodernspectralmodels.Triangulartruncation(O.$:n.$:N)oftheseriesisoftenused,asitoffersuniformresolutiononasphericaldomain.Asthisnotewilltrytoshow,theredoesnotseemtobeanystraightfor-wardwaytodefineanequivalentmeshsizeforspec-tralmodels;thismakestheestimationofeffectiveresolutionevenmoredifficultthaningridpointmodels.Asometimesquotedestimateofspectral-modelresolutionconsistsintheaveragespacingbetweenGaussianlatitudesofthetransformgrid;fortriangulartruncation,thisspacingisequaltothatbetweenlongi-tudesattheequator:L1=21Ta/(3N+1),withatheradiusoftheearth.Hence,L1=13.3/Ninunitsofthousandsofkilometers;foraT31model,L1=426km.Thisestimateofresolutionisoverlyoptimisticbecausethedimensionsofthetransformgridarechosentoallowcalculationofquadratictermswithoutaliasingoftheresolvedspectralfields,andhencethetransformgridisfinerthanrequiredbytheinformationcontentofthecorrespondingspectralseries.Amorerealisticestimateofresolutionisgivenbythesizeofhalfawavelengthoftheshortestresolvedzonalwaveattheequator:L2=1Ta/N=20/Ninunitsofthousandsofkilometers.Hence,aT31modelwouldhavearesolutionofL2=646km,accordingtothismeasure.Triangulartruncationprovidesanisotropicanduniformresolutiononasphere.Theshortestresolvedzonalwave(Iml=N)usedtodetermineL2correspondstoamodewiththegravestmeridionalstructure,becausemodesthatareveryshortinonedirectionmustbeelongatedintheotherdirection:sectorialsphericalharmonics(Iml=n)havemodalstructuresshapedasorangesegmentsandhavetheirlargestamplitudeinthetropics,hardlyanadequatemeasureofresolutionforgeneralcirculationmodels.Analternativewaytoestimatetheresolutionofspectralmodelsisasfollows.Considerthattheareaoftheearth'ssurfaceisgivenby47TB2;thereare(N+1)2realcoefficientstoasphericalharmonicseriesattriangulartruncationwithmaximumindexN,thatis,N(N+1)/2complexcoefficientsforthemodes1:SIml:sN,plus(N+1)realcoefficientsforthemodeswithm=O.Ifanequalareaonthesurfaceoftheearthisassignedtoeverypieceofinformationcontainedintheseries,thisgivesafootprintofsurfaceareaequalto41Ta2/(N+1)2foreveryrealcoefficient.ResolutioninaspectralmodelcouldbedefinedasthewidthL3ofaflat,rectangulartileofthesamesurfacearea:L3=(41T)1/2a/(N+1)=22.6/Ninunitsofthousandsof