EFFECTSOFNESTINGFREQUENCYANDLATERALBOUNDARYPERTURBATIONS
ONTHEDISPERSIONOFLIMITED-AREAENSEMBLEFORECASTS
PaulNutter∗1,2,DavidStensrud3,andMingXue1,2
1
SchoolofMeteorologyand2CenterforAnalysisandPredictionofStorms,
UniversityofOklahoma,Norman,Oklahoma3
NationalSevereStormsLaboratory,Norman,Oklahoma
1.Introduction
Itisknownthat”one-way”lateralboundarycon-ditionsconstrainthegrowthofinitialperturba-tionsinlimited-areaensembleforecasts(Pae-gleetal.,1997,referencestherein).Externalboundaryconditions(LBCs)typicallylackfine-scalefeatures,andinthecaseofensemblefore-casts,alsolackconsistentperturbations.Pertur-bationsgrowingonthenesteddomainbecomedisplacedbythecoarselyresolvedLBCs(ErricoandBaumhefner,1987;VukicevicandPaegle,1989)whilethedomainsizeitselfdeterminesthemaximumwavelengthattainablebytheperturba-tions(VukicevicandErrico,1990).
Anotheraspectoftheboundaryconditionprob-lemthathaspreviouslyreceivedlittleattentionistheimpactofLBCupdateinterval(Warneretal.,1997).Commonlyusedlinearinterpolationbe-tweenrelativelyinfrequentLBCupdatesactsasafilterthatexacerbatesthescaledeficiencyprob-lem.
Short-rangeensembleforecastexperimentshaveshownthattheensemblesoftenareunderdispersive.Thatis,theverifyinganalysisdoesnotfallwithintherangeofpossibilitiesforecastbytheensemble.DuandTracton(1999)foundthataregionalensemblewithalargerdomainproducesgreaterspreadthandoesanensem-blewithasmallerdomain.Furthermore,theyfoundthatthecontributionofdifferentLBCstoensemblespreadincreaseswithtimewhilethatofinitialconditionperturbationsdecreaseswithtime.Theseandothersimilarresults(HamillandColucci,1997;Houetal.,2001;Stensrudetal.,2000)demonstratethat,withtime,thespreadoftheLAMensembleforecastbecomesincreas-inglydeterminedbythespreadintheglobalen-sembleashighfrequencyandsmallscalecom-ponentsare“swept”fromtheLAMdomain.
Tohelprestoretheerrorvariancelostatsmallscales,weproposetoapplycoherentperturba-∗Corresponding
authoraddress:PaulNutter,CAPS,Uni-versityofOklahoma,100E.BoydSt.,Rm.1110,Norman,OK73019;e-mail:pnutter@ou.edu.
tionstotheLBCsatscalesunresolvedbytheex-ternalmodel.Theamplitudeofsuchperturba-tionswillbechosentomimictheexponentialer-rorgrowthcurvesgeneratedfromsimulationsinunboundeddomains.Theuseofmorefrequentlyupdatedboundaryconditionsandtheinclusionofsmall-scaleboundaryperturbationswillbeshowntoenhancethedispersionforlimited-areaensem-bleforecasts.
Inthispaper,necessaryanalysistoolsarein-troducedalongwiththeparameterizedpotentialvorticitymodelusedinthisstudy.WewillthenpresentresultsontheeffectofLBCupdatein-tervalonthenested-gridensembledispersion.Otherresultswillbepresentedattheconference.2.
EnsembleMSEandDispersion
Beforeintroducingtheunderlyinghypothesisforthisworkinsection3.,itisusefultoreviewstan-dardensemblestatisticalmethods.ConsiderNmembersofanensembleforecastf(t)andcor-respondinganalysesa(t)givenasvectorsonap-elementgrid.Althoughidenticalinaregularensembleconfiguration,lettheanalysescorre-spondingtoeachforecastbeuniqueforgener-ality.Theensemblemean-squareerror(MSE)is
V2
=1Nfi−ai2
N(1)
i=1where·2istheaveragesumofsquares(dotproduct)overthegrid(StephensonandDoblas-Reyes,2000).Togainfurtherinsight,addandsubtracttheensemblemeanforecast¯fandanal-ysis¯ainside(1),thenexpandandmanipulatesothat
V2
=1NNfi−¯f2
+1Nai−¯a
2(2)
i=1Ni=1−2NN1
(fi−¯f)·(ai−¯a)+¯f−¯a
2.i=1p
Notethatananalogousexpressionmaybefoundbyaddingandsubtractingthegridmean(scalar)
forecastandanalysisforeachensemblemember,multipliedbytheidentityvector.Thevarianceandcovariancetermsin(2)maybecombinedtowritethetotalbiasederrorvariance
Nσ2≡V2−¯f−¯a2=1N(fi−ai)−(f−a)i=1
2.(3)
Ensembleforecastsusuallyareverifiedagainst
oneanalysis.Inthatcase,ai=¯a
=a,andboth(2)and(3)reduceto
V
2
=
1
Nfi−¯f2+¯f−a2
Ni=1
=D2+¯f−a2,
(4)
whereD2istheensembledispersion,orspread.
ThisresultshowsthatthesquarederroroftheensemblemeanislessthantheensembleMSEV2becauseensembledispersionallowsunpre-dictablecomponentsofflowtobeaveragedoutintheensemblemean(Leith,1974;StephensonandDoblas-Reyes,2000).Notethatwhencom-paringallensemblememberstooneanalysisasin(4),D2=σ2.
Asinitialforecasterrorsgrowwithtime,ensem-bleforecastmembersbecomeuncorrelatedwithanalysesandtheircovarianceiszero.Iffore-castsareunbiasedandhavethesamevarianceastheanalyses,thentheexpectedvalueof(2)convergesasV2=2D2.ThisistheclassicresultobtainedbyLeith(1974),andmotivatesthecom-monpracticethattotalerrorvariancesbenormal-izedbytheclimatevarianceofanalyses.3.ImpactofLBCsonTotalErrorVarianceEquation(4)revealsalinear,additiverelationbe-tweenV2andD2,therebyprovidingadirectlinkbetweenensembleMSEandensembledisper-sion.ThefollowingargumenthighlightschangesinensembleMSE(hence,ensembledispersion)resultingfromcoarselyresolvedlateralboundaryforcing.
Supposewedecomposetheforecastandanal-ysisfieldsonthenesteddomainintolongwaveandshortwavecomponentssothatf=fl+fsanda=al+as.Theshortwavecomponentsarethosescalesnotresolvedbytheexternalmodel,whilelongwavecomponentsarethosescalescommonlyresolvedonbothexternalandinternalgrids.
Lettheshortwavecomponentsoftheforecastbedecomposedfurthertoincludelossesbyad-
vective“sweeping”andregenerationbynonlin-eardynamicdownscalingsothatfs=fα+fη.Withtime,fαcomponentsaredisplacedbythecoarselyresolvedLBCs,suggestingtheassump-tionthattheiramplitudestendtowardzero.NotethattemporalinterpolationofLBCsactsasafilterthatmaylengthenthescalefαcomponents.Fol-lowingfrom(1)whiledroppingisubscripts,theresultingimpactofLBCsontheensembleMSEisV
2
=1NN0fl+f α+fη−al−as2
i=1
=
1N(f1N
l−al)2
+fη−as2
Ni=1Ni=1+2NN
1(fl−al)·(fη−as).i=1p
(5)
Thisequationprovidesinsighttothehypothe-sisthatlimited-areamodelsareabletoproducefine-scaleinformationnotpresentinthecoarse-gridexternalmodel.Theextenttowhichsmall-scalewavecomponentsareskillfullyregeneratedwilldependonthesizeandlocationofthenesteddomainandontheLBCupdateinterval.Thus,animportantaspectofthisworkistodeterminetherelativecontributionoftheregeneratedvariancecomparedtothetotalvariance.
Toquantifythecontributionsofeachtermin(5),variancesarecomputedspectrallyusingthemethodoutlinedbyErrico(1985).Specifically,ifF(k)isthediscreteFouriertransformoftheerrorfield,then(3)maybeobtainedas
σ2=1NNK−12|Fi(k)|2
,
(6)
i=1
k=1
wherek=1,...,K−1arethesetofNyquistresolvedwavenumbers.
Skillismeasuredbynormalizingvarianceswithindifferentwavelengthbandsbytheclimato-logicalvariancesobtainedspectrally.Whennor-malizedinthismanner,resultsmaybecomparedtootherstudiessuchasLapriseetal.(2000).4.
ParameterizedPotentialVorticityModel
AsimplifiedmodelisusedforthisworktohelpisolateonlythoseerrorsassociatedwithLBCsinacontrolledandefficientmanner.Emphasisisdi-rectedtowardslargescalemid-troposphericflowsincethesearethepatternsthatareimportantforaccurateplacementofdevelopingmesoscale
Figure1:PPVmodelsimulationona25kmgrid,
15daysafterinitializingwithaperturbedshearflow.Streamfunctioncontoursareshowninbothpanels(ψ×106m2s−1).Shadingin(a)indicatesrelativevor-ticitygreaterthan±2×10−5s−1.Shadingin(b)showswindspeedsgreaterthan20ms−1.Boxesshowout-linesoffourdifferentsub-gridsusedforsingly-nestedmodelconfigurations.
andsmallerfeatures(Paegleetal.,1997).Fur-thermore,wewishtoexcludefromconsiderationinthisworkthehypothesisthatpredictabilityisenhancedundertheinfluenceofsurfaceforcing(VanTuylandErrico,1989;Warneretal.,1989).Thesingle-levelgridpointmodelrunsonamid-latitudebetachannelandisbasedonanapprox-imationofthequasi-geostrophicpotentialvortic-ityequation.Letξ≡ζ−λ2ψdefineapa-rameterizedrelativepotentialvorticity,whereζistherelativevorticity,ψisthestreamfunction,andλ=7.071×10−7m−1isaninverselengthscalebasedontheRossbyradiusofdeformation(Holton,1979).Theparameterizationrepresentsthefirst-ordereffectsofverticalmotionsinabaro-clinicatmosphere(vortexstretching).Theparam-eterizedpotentialvorticity(PPV)modelis
∂ξ
=−∂ψ∂ξ−∂ψ∂ξ∂x−β∂ψ∂x
−ν∇4∂t∂x∂y∂yξ.(7)
Ifλ=0thePPVmodelreducestothestandardbarotropicvorticitymodel,exceptforthe4thordernumericaldiffusionterm.Anexampleofthevor-ticityandstreamfunctionfieldsproducedbythePPVmodelisshowninFig.(1).5.ImpactofNestingInterval
Theimpactofnestingintervalisexploredinaper-fectmodelconfigurationfollowingthemethodde-scribedbyLapriseetal.(2000).PerfectinitialconditionsandLBCsareprovidedbyanexternal
controlsimulationthatmaybelow-passfilteredtoremovepoweratsmallscales.Initialcondi-tionsarenotperturbed,sotheonlysourceofer-roristhatduetointerpolationbetweenavailableLBCupdatesandthewaveabsorbingzoneusedfor“one-way”nesting.Normalizedvariancesfortheresultingfieldswerecomputedspectrallyasdiscussedaboveandaveragedover100cases.Dispersionstatisticsfromplannedensembleex-perimentsshouldappearsimilartoresultsshownhere.
Afrequencyanalysisofatimeseriesofξatasinglegridpointrevealsthat99.6%ofthevari-anceisexplainedbywaveshavingperiodslongerthan3hours.Theperfectmodelsimulationswereconsistent,showingthatboundary-inducederrorswereminimalwhenLBCsareupdatedatintervalsof3hoursorless.ErrorsbecomemuchlargerwhenLBCsareupdatedevery6hoursasshowninFig.2.
Inanotherwiseperfectmodelsimulation,LBCinterpolationcausesinconsistenciesbetweentheexternalandinternalfields.Attemptstosmooththediscontinuityacrosstheboundaryzonegen-eratesasmallscalewavethatentersthenesteddomain.Fromthere,thedisturbancemayei-therdissipateoramplify,dependingonitsam-plitudeandstabilityoftheambientflow.Com-paredtothelargedomain,boundary-induceder-rorsonthemediumandsmall(center)domainsgrowfasterwithtimeandstabilizeatdifferentsat-urationpoints(Fig.2).Onthesmall(south)do-main,locatedoutsidethemainjetprofile,theflowisweakanderrorsgrowmoreslowlywithtime.Inotherexperiments,LBCswerelow-passfil-teredtoexaminetheabilityofthenestedgridtoregenerateshortwavelengths,eveninthepresenceofcontinuedcoarse-resolutionbound-aryforcing.Figure(3)showssomerecoveryofskillatsmallscalesduetodynamicdownscaling.However,afterfourdaystheerrorshaveaboutthesameleveloferrorasthepreviouscase,in-dicatingthatLBCerrorshavesweptthroughthedomain.Inallcases,longwavecomponentsareminimallyaffectedsincetheLBCiswell-sampledatthesescales.6.
Summary
ThegoalofthisworkistoquantifytheextenttowhichcoarselyresolvedLBCsmodifyerrorvari-ances(hence,ensembledispersion)forlimited-domainsolutions.Ifitcanbeshownthatensem-bledispersionisdeficientdue,inpart,tocoarsely
Large domain
11Medium domain
0.80-250 km250-500 km0.80.6500-750 km> 750 km
0.60.40.40.20.2000
1224364860728496
0
1224364860728496
Small domain, center
Small domain, south
110.80.80.60.60.40.40.20.2000
1224364860728496
0
1224364860728496
Time (hours)Time (hours)
Figure2:Normalizedvorticityvariancesat250km
wavelengthintervalsforperfectmodelsimulationsrunona50-kmgridwithLBCslow-passfilteredtoex-cludewavelengths<400-km.LBCsareupdatedevery6hours.
Large domain
11Medium domain
0.80-250 km250-500 km0.80.6500-750 km> 750 km
0.60.40.40.20.2000
1224364860728496
0
1224364860728496
Small domain, center
Small domain, south
110.80.80.60.60.40.40.20.2000
1224364860728496
0
1224364860728496
Time (hours)Time (hours)
Figure3:AsinFig.2,exceptinitialconditionsarealso
low-passfiltered.
resolvedLBCs,thenaprocedurewillbedevel-opedtoaddsmallscaleLBCperturbationsusingparametricexponentialerrorgrowthcurves.
Errorgrowthcurvesobtainedfromensemblesimulationsontheexternalperiodicchanneldo-mainwerenotyetcompleteatthetimeofwrit-ing.Theseresults,alongwithamoredevelopedmethodforperturbingLBCsatunresolvedscalesusingerrorgrowthcurveswillbepresentedattheconference.
AcknowledgementThefirstauthorissupportedby
aWilliamsGraduateResearchFellowshipthroughtheCenterforAnalysisandPredictionofStormsattheUni-versityofOklahoma.Computationalsupportwaspro-vided,inpart,byNSSL.
References
Du,J.andM.S.Tracton,1999:Impactoflateralbound-aryconditionsonregional-modelensemblepredic-tion.Researchactivitiesinatmosphericandoceanicmodeling,Report28,H.Ritchie,ed.,WMO,TD-942,6.7–6.8.
Errico,R.andD.Baumhefner,1987:Predictabil-ityexperimentsusingahigh-resolutionlimited-areamodel.Mon.Wea.Rev.,115,488–504.
Errico,R.M.,1985:Spectracomputedfromalimitedareagrid.Mon.Wea.Rev.,113,1554–1562.
Hamill,T.M.andS.J.Colucci,1997:Verificationofeta-rsmshort-rangeensembleforecasts.Mon.Wea.Rev.,125,1312–1327.
Holton,J.R.,1979:AnIntroductiontoDynamicMete-orology,volume23ofInternationalGeophysicsSe-ries.AcademicPress,2ndedition.
Hou,D.,E.Kalnay,andK.K.Drogemeier,2001:Objec-tiveverificationofthesamex’98ensembleforecasts.Mon.Wea.Rev.,129,73–91.
Laprise,R.,M.R.Varma,B.Denis,D.Caya,andI.Za-wadzki,2000:Predictabilityofanestedlimited-areamodel.Mon.Wea.Rev.,128,4149–4154.
Leith,C.E.,1974:Theoreticalskillofmontecarlofore-casts.Mon.Wea.Rev.,102,409–418.
Paegle,J.,Q.Yang,andM.Wang,1997:Predictabilityinlimitedareaandglobalmodels.Meteorol.Atmos.Phys.,63,53–69.
Stensrud,D.J.,J.-W.Bao,andT.T.Warner,2000:Us-inginitialconditionandmodelphysicsperturbationsinshort-rangeensemblesimulationsofmesoscaleconvectivesystems.Mon.Wea.Rev.,128,2077–2107.
Stephenson,D.B.andF.J.Doblas-Reyes,2000:Sta-tisticalmethodsforinterpretingmontecarloensem-bleforecasts.Tellus,52A,300–322.
VanTuyl,A.H.andR.M.Errico,1989:Scaleinterac-tionandpredictabilityinamesoscalemodel.Mon.Wea.Rev.,117,495–517.
Vukicevic,T.andR.M.Errico,1990:Theinfluenceofartificalandphysicalfactorsuponpredictabilityes-timatesusingacomplexlimited-areamodel.Mon.Wea.Rev.,118,1460–11482.
Vukicevic,T.andJ.Paegle,1989:Theinfluenceofone-wayinteractinglateralboundaryconditionsonpredictabilityofflowinboundednumericalmodels.Mon.Wea.Rev.,117,340–350.
Warner,T.T.,L.E.Key,andA.M.Lario,1989:Sen-sitivityofmesoscale-modelforecastskilltosomeinitial-datacharacteristics,datadensity,dataposi-tion,analysisprocedureandmeasurementerror.Mon.Wea.Rev.,117,1281–1310.
Warner,T.T.,R.A.Peterson,andR.E.Treadon,1997:Atutorialonlateralboundaryconditionsasabasicandpotentiallyseriouslimitationtoregionalnumer-icalweatherprediction.Bull.Amer.Met.Soc.,78,2599–2617.
因篇幅问题不能全部显示，请点此查看更多更全内容