Energy- and flux-budget (EFB) turbulence closure m

Energy-andflux-budget(EFB)turbulenceclosuremodelforthestablystratifiedflows.PartI:Steady-state,homogeneousregimesS.S.Zilitinkevich1,2,3,T.Elperin4,N.Kleeorin4andI.Rogachevskii41DivisionofAtmosphericSciences,UniversityofHelsinki,Finland2FinnishMeteorologicalInstitute,Helsinki,Finland3NansenEnvironmentalandRemoteSensingCentre/BjerknesCentreforClimateResearch,Bergen,Norway4PearlstoneCenterforAeronauticalEngineeringStudies,DepartmentofMechanicalEngineering,Ben-GurionUniversityoftheNegev,Beer-Sheva,Israel“Boundary-layerMeteorology”Submittedon28.09.06;revisedon11.02.07Boun-S-06-00194Energy-andflux-budget(EFB)turbulenceclosuremodelforthestablystratifiedflows.PartI:Steady-state,homogeneousregimesS.S.Zilitinkevich1,2,3,T.Elperin4,N.Kleeorin4andI.Rogachevskii41DivisionofAtmosphericSciences,UniversityofHelsinki,Finland2FinnishMeteorologicalInstitute,Helsinki,Finland3NansenEnvironmentalandRemoteSensingCentre/BjerknesCentreforClimateResearch,Bergen,Norway4PearlstoneCentreforAeronauticalEngineeringStudies,DepartmentofMechanicalEngineering,Ben-GurionUniversityoftheNegev,Beer-Sheva,IsraelAbstractWeproposeanewturbulenceclosuremodelbasedonthebudgetequationsforthekeysecondmoments:turbulentkineticandpotentialenergies:TKEandTPE(comprisingtheturbulenttotalenergy:TTE=TKE+TPE)andverticalturbulentfluxesofmomentumandbuoyancy(proportionaltopotentialtemperature).BesidestheconceptofTTE,wetakeintoaccountthenon-gradientcorrectiontothetraditionalbuoyancyfluxformulation.TheproposedmodelgrantstheexistenceofturbulenceatanygradientRichardsonnumber,Ri.Insteadofitscriticalvalueseparating–asusuallyassumed–theturbulentandthelaminarregimes,itrevealsatransitioninterval,0.1Ri1,whichseparatestworegimesofessentiallydifferentnaturebutbothturbulent:strongturbulenceatRi1;andweakturbulence,capableoftransportingmomentumbutmuchlessefficientintransportingheat,atRi1.Predictionsfromthismodelareconsistentwithavailabledatafromatmosphericandlabexperiments,directnumericalsimulation(DNS)andlarge-eddysimulation(LES).Keywords:TurbulenceclosureStablestratificationCriticalRichardsonnumberKinetic,potentialandtotalturbulentenergiesTurbulentfluxesEddyviscosityandheat-conductivityAnisotropyofturbulenceTurbulentlengthscaleTurbulentboundarylayers21.IntroductionMostofthepracticallyusedturbulenceclosuremodelsarebasedontheconceptofthedown-gradienttransport.Accordinglytheyexpressturbulentfluxesofmomentumandscalarsasproductsofthemeangradientofthetransportedpropertyandthecorrespondingturbulenttransportcoefficient(eddyviscosity,,heatconductivity,,ordiffusivity,).FollowingKolmogorov(1941),thelatteraretakenproportionaltotheturbulentvelocityscale,,andlengthscale,:MKHKDKTuTlMK~~~.HKDKTTlu(1)Usuallyisidentifiedwiththeturbulentkineticenergy(TKE)perunitmass,.ThelatteriscalculatedfromtheTKEbudgetequationusingtheKolmogorov’sclosurefortheTKEdissipationrate:2TuKEKε~,TKtE/(2)where~/istheturbulentdissipationtimescale.TtTlTuThisapproachisjustifiedwhenappliedtotheneutral-stabilityflows(wherecanbetakenproportionaltothedistanceformthenearestwall).TlHowever,itfacesdifficultiesinstratifiedflows(bothstableandunstable).TheturbulentPrandtlnumber=/exhibitsessentialdependenceonthestratificationandcannotbeconsideredasconstant.Next,asfollowsfromthebudgetequationsfortheverticalturbulentfluxes,thevelocityscalecharacterisingverticalturbulenttransportsisdeterminedastherootmeansquare(r.m.s.)verticalvelocity~TPrMKHKTuTuzE(whereistheenergyoftheverticalvelocityfluctuations).Inneutralstratification~,whichiswhythetraditionalformula~zEzEKETuKEholdstrue.Butinstronglystablestratificationitbecomesinsufficientlyaccuratebecauseofstabilitydependenceoftheanisotropyofturbulence/:generallydecreaseswithincreasingstability.zAzE≡KEzAToreflecttheeffectofstratification,theturbulentlengthscalesforthemomentum,,andheat,,aretakendifferent.Asaresulttheaboveclosurescheme(formulatedbyKolmogorovfortheneutralstratificationandwell-groundedinthisandonlyinthiscase)losesitsconstructiveness:theunsolvedpartoftheproblemissimplydisplacedfrom{,}to{,}.TheTKEbudgetequationbecomesinsufficienttodetermineadditionalunknownparameters.TMlTHlMKHKTMlTHlNumerousalternativeturbulenceclosureswereformulatedusingequationsforotherturbulentparameters(besidesTKE)togetherwithheuristichypothesesandempirical3relationships,butnoconsensusisstillachieved;seeoverviewsbyWengandTaylor(2003),andUmlaufandBurchard(2005).Inthispaperweanalysetheeffectsofthedensitystratificationontheturbulentenergiesandverticalturbulentfluxesinthestablystratifiedatmospheric(oroceanic)boundary-layerflows,inwhichthehorizontalvariationsofthemeanvelocityandtemperaturearemuchweakerthantheverticalvariations.TheproposedtheoryprovidesrealisticstabilitydependenciesoftheturbulentPrandtlnumber,theverticalanisotropy,andtheverticalturbulentlengthscale.Belowwepresentmaterialinmeteorologicalterms,butallresultscanbeeasilyreformulatedintermsofwatercurrentsintheoceanorlakes,expressingthebuoyancythroughthetemperatureandsalinityinsteadofthetemperatureandhumidity.Weconsideraminimalsetofthebudgetequationsfortheseco