Chapter 13 The Pros and Cons of Diffusion, Filters and Fixers in Atmospheric General Circulation Models ChristianeJablonowskiandDavidL.Williamson AbstractAllatmosphericGeneralCirculationModels(GCMs)needsomeformof dissipation,eitherexplicitlyspecifiedorinherentinthechosennumericalschemes forthe spatial andtemporaldiscretizations.Thisdissipationmayserve manypur- poses,includingcleaningupnumericalnoisegeneratedbydispersionerrorsorcom- putational modes, and the Gibbs ringing in spectral models. Damping processes mightalsobeusedtocrudelyrepresentsubgridReynoldsstresses,eliminateunde- sirable noise due to poor initialization or grid-scale forcing from the physics pa- rameterizations,coverup weakcomputationalstability, damptracervariance,and preventtheaccumulationofpotentialenstrophyorenergyatthesmallestgridscales. ThischaptercriticallyreviewsthewideselectionofdissipativeprocessesinGCMs. Theyaretheexplicitlyaddeddiffusionandhyper-diffusionmechanisms,divergence damping, vorticity damping, external mode damping, sponge layers, spatial and temporalfilters,inherentdiffusionpropertiesofthenumericalschemes,andaposte- riorifixersusedtorestorelostconservationproperties.Alltheoreticalconsiderations aresupportedbymanypracticalexamplesfromawideselectionofGCMs.Theex- amplesutilizeidealizedtestcasestoisolatecausesandeffects,andtherebyhighlight theprosandconsofthediffusion,filtersandfixersinGCMs. 13.1 Introduction Therearemanydesignaspectsthat needto beconsideredwhenbuildingthe fluid dynamicscomponent,theso-calleddynamicalcore,foratmosphericGeneralCircu- ChristianeJablonowski DepartmentofAtmospheric,OceanicandSpaceSciences,UniversityofMichigan,2455Hayward St.,AnnArbor,MI48109,USA,e-mail:[email protected] DavidL.Williamson NationalCenterforAtmosphericResearch, 1850TableMesaDrive,Boulder,CO80305,USA, e-mail:[email protected] 387 388 ChristianeJablonowskiandDavidL.Williamson lation Models(GCMs). Amongthem arethe choice ofthe equationset andprog- nosticvariables,thecomputationalgridandgridstaggeringoptions,thespatialand temporalnumericaldiscretizations,built-inconservationproperties,andthechoice of dissipative processes that (1) might be needed to keep a model simulation nu- mericallystableand(2)mighttruthfullymimicthecumulativeeffectsofunresolved subgrid-scale processes on the resolved fluid flow. The latter aspect is at least a “hope”in theGCM modelingcommunity.Here,thephrasesubgrid-scaledenotes thedryadiabaticunresolvedprocessesinthedynamicalcore.Thisisincontrastto allotherunresolvedprocessesthatleadtophysicalparameterizationssuchasradi- ation,convection,cloudprocessesandplanetaryboundarylayerphenomena.These arenotconsideredhere,eventhoughtheyareintimatelycoupledtotheequationsof motion.Thischaptershedslightontheprosandconsofthemostpopularprocesses to handle both “physical”or “unphysical”subgrid-scale flow and mixing,and re- viewstheuseofexplicitlyaddedandinherentdiffusion,filtersandfixersinGCMs. ThesearerarelydocumentedintherefereedGCMliteraturebutmightbedetailed intechnicalmodeldescriptions. It is commonpractice in GCMs to include a parameterizationof the effects of subgrid-scalemotionsin the horizontalmomentumand thermodynamicequations that is formulated as a local diffusive mixing. In fact, all numerical models need some form of dissipation, either explicitly specified or inherentin the chosen nu- mericalschemesforthespatialandtemporaldiscretizations.Thisdissipationmay serve manypurposes,includingcleaningup numericalnoisegeneratedby disper- sion errors,computationalmodes,or the Gibbsringing,crudelyrepresentingsub- grid Reynoldsstresses, eliminating undesirable noise dueto poor initialization or grid-scaleforcingfromthephysicsparameterizations,coveringupweakcomputa- tionalstability,dampingtracervariance,andpreventingtheaccumulationofpoten- tialenstrophyorenergyatthegridscale(Woodetal2007;Thuburn2008a).Such an accumulation of energy is due to the physical downscale cascade and can re- sult in excessive small-scale noise. It is known as spectral blocking and leads to anupturn(hook)orflatteninginthekineticenergyspectrumatthesmallestscales. Furthermore,physical“noise”in GCMs mightoriginatefromparameterizedgrid- scaleforcingsorfromsurfaceboundaryconditionssuchasorography,theland-sea orland-usemask. Anaccumulationofenergyandenstrophyatthe smallest scalesmayalsoarise duetoanumericalmisrepresentationofnonlinearinteractions,theso-calledalias- ingeffect.Nonlinearinteractionsandaliasingmostlyoriginatefromthe quadratic orhigher-ordertermsintheequationsofmotion.Inessence,productsofwavescan create new waves that are shorter than 2!x where !x is the physical grid spac- ing.Thesewavescannotberepresentedonamodelgridandarealiasedintolonger waves.Aliasing,ifleftunchecked,canleadtoablowupofthesolution.Thisphe- nomenonis characterizedas nonlinear computationalinstability as first discussed by Phillips (1959). Note that almost all GCMs suffer to some degree from alias- ing.Exceptionsarespectraltransformmodelswithquadratictransformgridswhich eliminatethealiasingofquadraticadvectionterms,themostproblematicform,but donotcompletelyeliminatealiasingfromhigher-orderterms.Nonlinearcomputa- 13 Diffusion,filtersandfixersinGCMs 389 tional instability does not occur in models that conserve quadratic quantities like enstrophyandkineticenergy(Arakawa1966;ArakawaandLamb1981).Aliasing errorsare not necessarily fatal. Whether an amplification of the waves occursde- pendsonthe phaserelationbetweenthemisrepresentedandoriginalwavesinthe model.Moreinformationonnonlinearcomputationalinstabilityandaliasingispro- videdintextbookslikeDurran(1999,2010),Kalnay(2003)orLin(2007). All mixingprocessesremoveenergyand enstrophyfrom the simulation which would otherwise build up to unrealistic proportions.Frequently, the included dis- sipation is restricted to be in the horizontal, as there is usually sufficient vertical mixing or diffusion in the physical boundarylayer turbulence and convectivepa- rameterizationsin full GCMs or sufficient inherentnumerical diffusionto control noise in the vertical direction. Sometimes, vertical diffusion is also explicitly in- cluded in the dynamical core and applied throughout the whole troposphere. An exampleisthemodelbyTomitaandSatoh(2004)whichisdiscussedlater. A common expectation might be that dissipative formulationsbased on turbu- lence theory or observations provide a physical foundation for the subgrid-scale mixing.However,suchphysicalmotivationisnotguaranteedandeachadhocmix- ingprocessinaGCMmustbecriticallyreviewed.AspointedoutbyMellor(1985) the horizontal diffusivities in use by GCMs are typically many orders of magni- tude larger than those which would be appropriate for turbulence closures. Thus, horizontaldiffusionusedbymostmodelscannotbeconsideredarepresentationof turbulencebutshouldbeviewedasasubstitutemechanismforunresolvedhorizon- tal advectiveprocesses.Awarenessof thismightoffersomeguidancein choosing anadequatesubgrid-scalemixingscheme. Mixing in GCMs generally serves as a numerical filter and neither reflects the mathematical representation of the energy or enstrophy transfer to small scales nor the representation of physical molecular diffusion (Koshyk and Boer 1995). Subgrid-scaleprocesses,althoughsmall,canhaveaprofoundimpactonthelarge- scale circulation. For example, diffusive mechanisms affect the propagation of wavesandtherebythemeanflowthroughwave-meanflowinteractions.Inaddition, bothinherentandexplicitlyaddeddissipationprocessessmearoutsharpgradients inthetracerfields,andmayleadtounphysicalandoverlystrongmixing.Suchmix- ing might provide feedbacks to the physical parameterizations. For example, the precipitationfieldmightbehighlyinfluencedbythediffusivecharacteristicsofthe moisturetransportalgorithminthedynamicalcore.Thenotionofoverlydiffusive GCMswasdiscussedbyShutts(2005).Hearguedthatnumericaladvectionerrors, horizontal diffusion and parameterization schemes like the gravity wave drag or convection,actasenergysinksandleadtoexcessiveenergydissipationinGCMs. However,suchaconclusionmightbehighlymodeldependent. In summary, some mixing processes are used for purely numerical reasons to prevent the model from becoming unstable. Others are meant to mimic subgrid- scaleturbulenceprocessesthatareunsolvedonthechosenmodelgrid.Inpractice, manyfiltersandmixingprocessesareusedatonce,whichmakesitmoredifficultto evaluatetheirindividualeffects.Theformofthediffusionprocessesinatmospheric dynamical cores varies widely and is somewhat arbitrary. There are explicit dis- 390 ChristianeJablonowskiandDavidL.Williamson sipationprocessesandfilters,inherentnumericaldissipation,andfixersin GCMs. Throughoutthechapterweassociatethephraseexplicitdiffusionwithprocessesex- plicityaddedtotheequationsofmotion.Thephraseimplicitdiffusioncharacterizes theinherentdissipationofnumericalschemes.Thesephrasesareintendedtomake adistinctionfromexplicitandimplicitnumericalapproximationstodiffusionoper- ators.Notethatthewords“diffusion”,“dissipation”and“viscosity”areoftenused interchangeablyintheliterature.Othercharacterizationsofdampingaresmoothing, filteringandmixing. 13.1.1 Modelequationsandtherepresentationofexplicitdiffusion Mixing processes in GCMs can appear in many forms. A very dominant form is based on explicit dissipation mechanisms that get appended to the equations of motion shown in chapter 15. In the continuous equations this mixing symbolizes moleculardiffusion.However,GCMsarenotcapableofrepresentingmoleculardif- fusionat thenmormmscalesincetheyaretypicallyappliedwithhorizontalgrid spacingsbetween20-300km.NonhydrostaticGCMs(Tomitaetal2005;Fudeyasu etal2008)andmesoscalelimited-areamodelsliketheWeatherResearchandFore- casting Model WRF (Skamarocket al 2008)are also run with finer grid spacings ofafewkilometers.Otheratmosphericmodelswithevenfinerscalesmightutilize theLargeEddySimulation(LES)approach.LESisamathematicalmodelfortur- bulencethatisbasedupontheNavier-Stokesequationswithbuilt-inlow-passfilter. TheunderlyingideawasinitiallyproposedbySmagorinsky(1963)andfurtherde- velopedby Deardorff(1970).LES hasbeenextensivelyused to studysmall-scale physicalprocessesandmixingintheatmosphere.Butinanycase,modelstruncate themulti-scalespectrumofatmosphericmotionswellabovethemoleculardiffusion scales.Theunresolvedpartistypicallymodeledasdissipationandonemighthope thatitadequatelycapturestheadiabaticsubgrid-scaleprocessesinsomepoorlyun- derstoodway. Explicit dissipation can be added to the momentumand thermodynamicequa- tionsinthesymbolicform "# =Dyn(#)+Phys(#)+F (13.1) # "t whereDyn(#)denotesthetimetendenciesoftheprognosticvariable#according totheresolvedadiabaticdynamics,Phys(#)symbolizesthetime tendenciesfrom thesubgrid-scalediabaticphysicalparameterizations,andF isthedissipation.The # actualformofthisdissipationismodeldependent.Forexample,modelsinmomen- tum form,that utilize the zonal and meridionalvelocitiesu, v and temperatureT, might append the diffusive terms Fu,Fv,FT. Models in vorticity-divergence($,%) formaddthediffusionF ,F ,F ,orevenreplaceF withadiffusionoftheabsolute $ % T $ vorticityF where f symbolizestheCoriolisparameter.Alternatively,ifthepo- $+f tentialtemperature&isselectedinthethermodynamicequationadiffusivetermF & 13 Diffusion,filtersandfixersinGCMs 391 mightbechosen.Dissipationmightalsobeappliedtothetracertransportequations, andincaseofnonhydrostaticmodelstotheverticalvelocity.Whetherexplicitdiffu- sionisneededforcomputationalstabilityismodeldependent.Somemodelsprefer tocontrolthesmallestscalesviainherentnumericaldissipationandselectF =0. # However,theformofF isoneofthemainfociinsections13.3-13.5,andtherefore # weintroducethegenericformoftheforcinghere. 13.1.2 Overviewofthechapter Thischapterpresentsacomprehensivereviewofdissipativeprocessesandfixersin generalcirculationmodels.Manypointerstoreferencesaregiven,andweillustrate thepracticalimplicationsofthediffusion,filtersandfixersonthefluidflowinatmo- sphericdynamicalcores.Inparticular,wereviewtheprinciplesbehindthedifferent dissipative formulations, isolate causes and effects, provide many examples from today’sGCMs and utilize idealizeddynamicalcoretest cases andso-calledaqua- planetsimulationstodemonstratetheconcepts.Thesetestcasesarebrieflyoutlined insection13.2.Overall,wequoteorshowexamplesfromover20differentdynami- calcorestohighlightthebroadspectrumofthedissipativeprocessesinGCMs.The modelsarelistedinsection13.2.Wecharacterizefourteenofthemingreaterdetail intheAppendixsincetheyareusedasexamplesthroughoutthechapter. Thechapterisorganizedasfollows.Sections13.3and13.4discussthemostpop- ularexplicitdiffusionanddampingmechanismsinthedynamicalcoresofGCMs. Section 13.3 includes the classical linear and nonlinear horizontal diffusion and hyper-diffusion,their diffusion coefficients and stability constraints, and physical consistency arguments. Section 13.4 discusses the 2D and 3D divergence damp- ing,vorticitydamping,Rayleighfrictionanddiffusivespongesnearthemodeltop, and externalmode damping.In general,it is debatablewhether filters are consid- eredexplicitdissipationorjustacomputationaltechniquetokeepamodelnumeri- callystable.Here,wechoosetopresentthemintheirowncategoryinsection13.5 wherebothtemporalandspatialfilterareassessed.Section13.6capturesthebasic ideas behind inherent numerical dissipation which is nonlinear and sometimes is interpretedasphysicallymotivateddiffusion.Section13.7shedslightonthecon- servationpropertiesofatmosphericGCMsandintroducesaposteriorifixers.They includethedryairmassfixer,fixersfortracermassesandtotalenergyfixers.Some finalthoughtsarepresentedinsection13.8. 13.2 SelectedDynamicalCoresandTest Cases We illustrate manyof the effectsof the diffusion,filters and fixersin GCMs with thehelpof2Dshallowwateror3Dhydrostaticmodelrunstodiscussthepractical implications of the theoretical concepts. Throughoutthis chapter, we point to the 392 ChristianeJablonowskiandDavidL.Williamson specificimplementationsofthedissipativeprocesses,andquotetypicalvaluesfor the empirical coefficients from a variety of models to shows their spread in the GCMs.Theintentionistogivehands-onguidanceandpresentthedesignoptions. Inparticular,thischapterfeaturesexamplesfromthedynamicalcoresCAMEu- lerian, CAM semi-Lagrangian,COSMO, ECHAM5, FV, FVcubed, GEOS, GME, HOMME,ICON,IFS,NICAM,UMandWRF.TheAppendixexplainstheacronyms andbrieflycharacterizesthenumericalschemes.Mostmodelsareglobalhydrostatic GCMsandusetheshallow-atmosphereapproximation.Theonlyexceptionsarethe nonhydrostatic models COSMO, NICAM, UM and WRF that are built upon the deepatmosphereequationset(seeWhiteetal(2005)forareviewoftheequations). WealsobrieflyrefertoothermodelssuchasNASA’sModelEbytheGoddardInsti- tuteforSpaceStudies,theAtmosphericGCMfortheEarthSimulator(AFES)de- velopedbytheCenterforClimateSystemResearchattheUniversityofTokyoand the National Institute for EnvironmentalStudies (Japan),the Global Environmen- talMultiscale(GEM)modelfromtheCanadianMeteorologicalCentre,theGlobal Forecast System (GFS) and the Eta model developedby the National Centers for EnvironmentalPrediction(NCEP).Thereferencesforthesemodelsaregivenlater. The model simulations utilize a variety of idealized test cases. They include the steady-state and baroclinic wave test cases for dynamical cores suggested by JablonowskiandWilliamson(2006a,b),selectedshallowwatertestcasesfromthe Williamsonetal(1992)testsuite,theHeld-Suarezclimateforcing(HeldandSuarez 1994), a variant of the Held-Suarez test with modified equilibrium temperatures in the stratosphere (Williamson et al 1998), and the aqua-planet configuration as proposedbyNealeandHoskins(2000).Theadiabaticdynamicalcoreandshallow watertestcasesgenerallyassessthepropertiesofthenumericalschemesinshortde- terministicmodelrunsofupto30days.TheidealizedHeld-Suarez-typesimulations utilize a prescribedNewtoniantemperaturerelaxationandboundarylayerfriction thatreplacethephysicalparameterizationpackage.Thesemodelrunsaretypically integratedformultipleyearstoassess theclimatestatistics ona dryandspherical earthwithnomountains.Theaqua-planetassessmentsarethemostcomplexsimula- tionsdiscussedinthischapter.TheyrepresentmoistGCMrunsthatincludethefull physicalparameterizationsuitebututilizeasimplifiedlowerboundarycondition.In essence, the lower boundaryis replaced by a water coveredearth with prescribed sea surface temperatures. In addition, the settings of the physical constants and a symmetricozonedatasetareprescribedinaqua-planetsimulations. 13.3 ExplicitHorizontalDiffusion Thissectiondiscussestheideasbehindexplicithorizontaldiffusionmechanismsin GCMs. Inparticular,we assess thelinearsecond-orderdiffusion,the higher-order and thereby more scale-selective hyper-diffusion, reveal the selection criteria for thediffusioncoefficientsinbothspectraltransformandgridpointmodels,discuss theconceptofspectral viscosity,andreviewthe stabilityconstraintsforthediffu- 13 Diffusion,filtersandfixersinGCMs 393 sionequation.Inaddition,weintroducetheprinciplesbehindnonlinearhorizontal diffusionandbrieflysurveythephysicalconsistencyofexplicitdiffusionschemes. 13.3.1 Genericformoftheexplicitdiffusionmechanism Thegenericformoftheexplicitlineardiffusionisgivenby F#=( 1)q+1K2q’2q# (13.2) − whereq 1isapositiveinteger,2qdenotestheorderofthediffusion,K stands 2q ≥ forthediffusioncoefficientand’isthegradientoperator.Boththehorizontal2D gradientoperatororanapproximated3Dgradientoperatorhavebeenusedforthe horizontaldiffusionasfurtherexplainedbelow.Settingq=1yieldsasecond-order diffusionthatemergesfromphysicalprinciplessuchastheheatdiffusion,molecular diffusionandBrownianmotion.However,moleculardiffusionactsonthenanome- tertomillimeterscale,andisthereforeunresolvedonaGCMmodelgrid. In practice, second-orderdiffusionis often applied as an artificial sponge near the topboundary,andhasverylittle resemblancewith itsphysicalcounterpart.In general,morescale-selectivehyper-diffusionschemeswithq=2,3,4areselected in the majorityof the modeldomain.The most popularchoiceis the fourth-order hyper-diffusionwithq=2thatisalsocalledbi-harmonicdiffusionorsuperviscos- ity.Theuseofhyper-diffusionisoftenmotivatedbytheneedtomaximizetheratio of enstrophy to energy dissipation since 2D turbulence theory predicts a vanish- ingenergydissipationrateatincreasingReynoldsnumbers(SadournyandMaynard 1997).Thehighertheorderofthehyper-diffusion,thehighertheratioofenstrophy toenergydissipationbecomes.FargeandSadourny(1989)evensuggestedusinga 16th-orderhyper-diffusion. 13.3.2 ParticularformsofexplicitdiffusioninGCMs TheexactformofthedampingvarieswidelyinGCMs.Typically,forconvenience thehorizontaloperatorsareappliedalongmodellevelswiththepossibleexception oftheformulationforthescalartemperaturediffusion.Wenowlistseveralexamples toillustratethevarietyofthediffusionmechanisms.Ourfirstexampleistakenfrom theweatherforecastmodelGMEwhichhasbeendevelopedattheGermanWeather Service.Itappliesthefourth-orderhyper-diffusion Fu = K4’4u (13.3) − Fv = K4’4v (13.4) − FT = K4’4(T Tref) (13.5) − − 394 ChristianeJablonowskiandDavidL.Williamson where’isthehorizontalgradientoperatorandTref isareferencetemperaturethat dependsonlyonheight(Majewskietal2002).Atupperlevelsnearthemodeltop, asecond-orderdiffusionisapplied.GMEutilizeslocalbasisfunctionsthatareor- thogonalandconformperfectlytothesphericalsurface.Theyarelocallyanchored ineachtriangleofGME’sicosahedralgrid.Withinthelocalneighborhoodofagrid point the coordinate system is thereforenearly Cartesian.Note that Cartesian co- ordinatessimplifytherepresentationofthe’2q operatorsincethemetrictermsare equaltounity. If the diffusion is expressed in spherical coordinates many metric terms are present.Here,wefirstshowtheoperatorsforthreespatialdimensionsbeforesim- plifyingthem.Thescalar3DLaplacian’2 operatorinsphericalcoordinatesfor (3D) aprognosticvariable#hastheform 1 1 1 ’2(3D)#= r2cos2("))#+r2cos("((cos("(#)+r2"r(r2"r#) (13.6) whererdenotestheradialdistanceinthelocalverticaldirectionfromthecenterof theearth,)and(arethelongitudeandlatitude,andthenotation"x symbolizesa partialderivativeinthexdirectionwherexisaplaceholderfor),(andr.Inaddi- tion,the3DvectorLaplacianforthethree-dimensionalwindvectorv =(u,v,w)is 3 givenby 2sin( 2 1 ’2 u " v+ " w u (3D) −r2cos2( ) r2cos( ) −r2cos2(  1 2 2sin(  ’2 v = ’2 v v+ " w+ " u . (13.7) (3D) 3  (3D) −r2cos2( r2 ( r2cos2( )   2 2 2  ’2 w w " u " (cos(v)  (3D) −r2 −r2cos( ) −r2cos( (      Theseexpressionsaree.g.showninAppendixAinSatoh(2004).Theextraterms besidesthescalar’2 operatorariseduetothespatialvariationoftheunitvectors (3D) insphericalcoordinates.Withtheexceptionoftheundifferentiatedtermineachof the componentsthe extra terms are not necessarily negligiblein comparisonwith thoseofthescalardiffusionoperator.Infact,someofthemarecrucialinensuring thatthediffusionoperatorconservesangularmomentumasoutlinedbyStaniforth etal(2006). Generally, approximated forms of (13.6) and (13.7) are chosen to express the horizontaldiffusion.Thedistancerisoftenapproximatedbytheconstantradiusof theeartha,thetermscontainingtheverticalderivative"randverticalvelocityware dropped,andtheverticalcomponentofthevectorLaplacianisneglectedtocreatea 2Ddiffusionoperator.Thereplacementofthedistancerbyaisinfactanecessityin hydrostaticmodelsbasedontheprimitiveequations(Whiteetal2005).Thescalar 2DLaplacianthensimplifiesinthefollowingway 1 1 ’2#= " #+ " (cos(" #). (13.8) a2cos2( )) a2cos( ( ( 13 Diffusion,filtersandfixersinGCMs 395 The2DvectorLaplacianforthetwo-dimensionalwindvectorv=(u,v)isgivenby 2sin( 1 ’2u " v u ’2v= −a2cos2( ) −a2cos2( . (13.9)  1 2sin(  ’2v v+ " u −a2cos2( a2cos2( )     ThisformofthevectorLaplacianleadstothe“conventionalform”ofthehorizontal momentum diffusion as characterized by Becker (2001). Unfortunately,this form doesnotconserveangularmomentumasfurtherdiscussedinsection13.3.7.Some modelsalsodroptheextratermsandonlyapplythescalar2DLaplacianoperatorto thevectorwind(u,v)insphericalgeometry.Suchasimplifiedformise.g.provided as an optionalspongelayer dampingmechanismnear the modeltop in the finite- volume(FV)dynamicalcoreintheCommunityAtmosphereModelCAM(version 5)(Nealeetal2010).ThemodelCAMFVisusedattheNationalCenterforAtmo- sphericResearch(NCAR). There is another caveat. Both formulations of the Laplacian in Eqs. (13.7) or (13.9)wouldleadto anundesireddampingofa solidbodyrotationasthoroughly analyzed by Staniforthet al (2006)for the Unified Model (UM) developedat the UKMetOffice.Thereforeinpractice,amorecomplicatedformofthemomentum diffusionischoseninthemodelUMthatisappliedtothevelocitycomponentsu,v and w (see Staniforth et al (2006) for the derivation). The NCAR CAM spectral transform Eulerian (EUL) and semi-Lagrangian (SLD) dynamical cores (Collins etal2004)alsoincludesuchacorrectionforsolidbodyrotationasexplainedlater. Concerningthe scalar diffusionin the modelUM, a form similar to Eq. (13.8) isselectedforthediffusionofpotentialtemperature.Itisappliedtwice(including a sign reversal) to resemble a fourth-orderhyper-diffusionmechanism. The main differences to Eq. (13.8) are that (1) the model UM does not utilize a shallow- atmosphereapproximationand retainsthe radialdistancer, (2)a slope correction is utilized over steep terrain to lessen the spurious mixing along UM’s deformed orography-followingverticalcoordinate,and(3)thediffusioncoefficientisdiffer- ent in the two horizontal directions. The coefficient is constant in the meridional direction, but the strength of the diffusion in longitudinal direction is allowed to varywithlatitude.Thisleadstonon-isotropicdiffusionandisfurtherexplainedin section13.3.5.NotethatthemodelUMdoesnotneedhorizontaldiffusionforcom- putationalstabilityreasonsduetotheinherentnumericaldissipationintheinterpo- lationsofitssemi-Lagrangianscheme.Inpractice,theexplicitdiffusionistherefore optionalandnotusedbydefault.Forexample,itisneverutilizedinshortweather predictionsimulations(TerryDavies,personalcommunication). Scalardiffusionoftype(13.8)isalsoappliedinothermodelssuchasthespectral transformIntegratedForecastingSystem(IFS)attheEuropeanCentreforMedium- RangeWeatherForecast(Ritchieetal1995;ECMWF2010).Themodelutilizesa second-order(q=1)diffusionschemeclosetothemodeltopandafourth-order(q= 2)hyper-diffusionoftheprognosticscalarvariablesrelativevorticity$,horizontal divergence%andtemperature.Ityieldstheexplicitdiffusion 396 ChristianeJablonowskiandDavidL.Williamson F$ =(−1)q+1K2q’2q$ (13.10) F% =(−1)q+1K2q’2q% (13.11) FT =( 1)q+1K2q’2qT. (13.12) − Thisformofthe diffusionisfurthermoreutilizedbythespectraltransformmodel ECHAM5 developed at the Max-Planck Institute for Meteorology, where even higher-order diffusion operators are chosen below the sponge layer at the model top(Roeckneretal2003). Theapplicationofthediffusionalongslopinggeneralverticalcoordinates,like thehybridpressure-based*-coordinate(SimmonsandBurridge1981),isstraight- forward to implement, but as mentioned before can cause spurious mixing over mountains, especially in the neighborhoodof steep terrain. This is largely due to thepresenceoflargeverticaltemperaturevariationsalongtheslopingsurfacesthat overlaythehorizontalgradients.Suchspuriousmixingtriggeredbytheverticalvari- ationsisundesirableandmaygrowtosignificantproportions.Thereforeinpractice, the diffusion of the temperature in the model IFS is modified to approximatethe horizontaldiffusiononsurfacesofconstantpressureratherthanonthesloping*- coordinatesurfaces.Thisisfurtherexplainedinthetechnicalmodeldocumentation of the CAM EUL andSLD dynamicalcores(Collinset al 2004).CAM EUL and SLDapplythefourth-ordertemperaturediffusion "T "p FT = K4 ’4T ps ’4lnps (13.13) − ’ − "p"ps ( where pisthepressureand p symbolizesthesurfacepressure.Thesecondtermin s FT consistsoftheleadingterminthetransformationofthe’4operatorfrom*sur- facestopressuresurfaces.CAMalsoappliesasecond-ordersponge-layerdiffusion atupperlevels.ButsincetheupperlevelsinCAMcoincidewithpurepressurelevels thecorrectionisnotneededthere.Ingeneral,itisunclearwhetherdiffusionshould beappliedalongconstantmodellevels,constantpressurelevelsorevenalongcon- stant height or isentropic levels. If the diffusion primarily counteracts numerical artifacts,argumentscanbefoundthatitshouldbeappliedalongmodellevels.How- ever,iftheprimarymotivationistocharacterizephysicalmixing,height,pressureor isentropiclevelsareadvantageousasexplainedindetailbyStaniforthetal(2006). NotethatNCAR’sEULandSLDdynamicalcoresactuallyapplyavariantofthe diffusionshowninEqs.(13.10)and(13.11)totherelativevorticityandhorizontal divergencefieldswhichgeneralizestheapproachbyBourkeetal(1977).Itisgiven by 2 q F$ =(−1)q+1K2q ’2q($+f)+(−1)q+1($+f) a2 (13.14) ’ ) * ( 2 q F% =(−1)q+1K2q ’2q%+(−1)q+1% a2 (13.15) ’ ) * (