DynamicalcoresforNWP: Anuncertainlandscape *
NigelWood
MetOffice,DynamicsResearch,Exeter,UnitedKingdom
1Introduction
Thedynamicalcoreofanumericalweatherpredictionmodelisthatpartofthemodelthat isresponsibleforthefluid-dynamicalprocessesthatcanberepresented(withwhateverdegreeofaccuracy)bythespatialandtemporaldiscretizationofthegoverningequations.Thisis distinctfromthe“physics”packages,whichrepresenttheeffectsofnonfluid-dynamicalprocesses,suchasradiation,andoffluid-dynamicalprocessesthatoccuratscalesthatcannotbe resolvedbythesmallestscalesrepresentedbythediscretizedequations,typicallysuchas boundary-layerturbulence.
Thefirstoperationalnumericalweatherforecastwascreatedinthesummerof1954atthe SwedishHydrometeorologicalInstitute,followedlessthanayearlaterbytheUSJointNumericalWeatherPredictionUnit(e.g., Persson,2005).Creatingthoseforecastsrequiredenormousamountsofinsightintotheunderpinningphysicsoftheproblemand,usingthat insight,ingenuityastohowtoderiveequationsthatcouldbesuccessfullysolvedusing thelimitedandemergingcomputerresourcesavailable.Thosefirstmodelswerebasedon manipulationsto,andapproximationsof,theinviscid,Eulerequationsinarotatingframe ofreference.Thosefirstmodelsneglectedtheeffectsofunresolvedprocessesandsorepresent whatwillbereferredtoasanintegrationofadynamicalcore,thatis,theinviscid,energyconservingaspectsofthecompletesystem.Overthelast60years,thedevelopmentofthe dynamicalcorepartsofnumericalweather(andclimate)predictionmodelscanberegarded astheprogressiverelaxationoftheapproximationsmadeinthoseearlydays.Thishasbeen
* #BritishCrownCopyright.
madepossibleprincipallybytheexponentialincreaseinthespeedofcomputersaswellas improvedmethodsofderivingtheinitialconditionsforthosemodels;inasensethecunning ofnumericalmodelershasbeenreplacedbythecunningofcomputationalscientists!However,weareperhapsattheendofthatjourneyandneedagaintoexplorehowtosqueezethe mostoutofalimitedresource.
Initially,ofcoursetherewereonlyaverysmallnumberofdifferentmodels.Astime progressedthatnumberincreasedasdifferencesinbothapproach(numericalschemeand levelofapproximation)andapplication(weather,climate,global,andregional)increased. Butasthenumberofapproximationsappliedhasreducedandtheunderstandingofthenumericalmethodsavailablehasincreased,onemighthaveanticipatedthatthenumberofdifferentmodelswouldnowbereducing.Thatisnotthecase.Anindicationofthisisgivenby Tables1and2of Marrasetal.(2016) inwhicharelisted,respectively,26existingNWPsystemsand28thatareunderdevelopment.Infact,intheearlydecadesofthe21stcentury,we seemtobeinanerawherethenumberofapproachesandhencedifferentmodelsappeartobe increasing.Thefollowingsectionsareaimedatgivingapotentialarchitectofanewdynamicalcoreanideaoftherangeofchoicesthatareavailableandthatneedtobemade.Fromthis, itwillperhapsbecomeclearwhyweappeartonotyetbeconvergingononeapproachto modelingofthedynamicalcore.
Thediscussioncoversarangeoftopicseachofwhichiscoveredinamostlyqualitative,nonanalyticalmanner,withtheinten tionofgivingabroadoverviewoftheissues involvedratherthanspecificdetailandderivation.Theinterestedreaderisreferredto anyoftheexcellenttextbooksavailablethatcovereachofthetopicsinmuchmoreanalyticdetail.Twoofparticularrelevancetothistopicare Durran(2010) and Lauritzen etal.(2011) .
2Governingequations
TheunapproximatedEulerequationscanbewrittensymbolicallyintermsofthephysical terms.Startingwiththewindfield:
•Horizontalwinds:
Tendency+Transport ¼ Coriolis+PressureGradient : (1)
•Verticalwinds:
Tendency+Transport ¼ Coriolis+PressureGradient+Gravity: (2)
Theearliestdynamicalcoresappliedfourkeyapproximationstotheseequations.Thefirst isthegeostrophicapproximation.ThisassumesthattheCoriolistermsareinexactbalance withthepressuregradienttermsand,therefore,neglectthetendencyandtransportofthe horizontalwinds(theleft-handsideof Eq.1).Subsequentmodelsrelaxedthisbyallowing forsomeaspectsofthetransportterm:thequasigeostrophicmodelsallowfortransportof thegeostrophicwindbythegeostrophicwindandsemigeostrophicmodelsallowfor
transportofthegeostrophicwindbythefullwind.ButnocontemporaryNWPmodelsmake anyoftheseapproximations;theysolvethefullEq.(1).
Thesecondistheshallow-atmosphereapproximation.Thisassumesthatthedepthofthe atmosphereissmallcomparedtotheradiusoftheEarth.Thisallowscertainsimplificationsin theequationstobemade.Manymodelsstillmakethisapproximation(togetherwiththetraditionalapproximationdiscussedlater)thoughthereareexceptionssuchastheseriesofdynamicalcoresusedintheMetOffice’sUnifiedModel(WhiteandBromley,1995; Daviesetal., 2005; Woodetal.,2014).
Thethirdisthehydrostaticapproximation.Thisneglectsthetendencyandtransportofthe verticalcomponentofthewind.Modelsemployingthisapproximationoften,butnotalways, alsoapplytheshallow-atmosphereapproximation(andtheresultingequationsaretermedas theprimitiveequations).Inthiscase,itisassumedthattheverticalpressuregradientbalances theaccelerationduetogravity.Whentheshallow-approximationisnotmade,thehydrostatic approximationisreferredtoasthequasihydrostaticapproximation.Thehydrostaticapproximationisonlyvalidforhorizontalscalesthataresignificantlylongerthanverticalscales(see, e.g., Holton,1992; Daviesetal.,2003).Therefore,ashorizontalresolutioncontinuestoincrease,mostnewmodelformulationsarenothydrostatic(butitisnotablethattheEuropean Center’smodelremainshydrostaticdespiterunningverysuccessfullyatthehighest,global, operationalresolution).
Thefourthisthetraditionalapproximation. Whiteetal.(2005) discusstheapplication(or not)oftheshallowandhydrostaticapproximationsandassociatedissuesofconsistencyofthe equationsetswithvariousphysicalconservationprinciples(namelyconservationofmass, energy,axialangularmomentum,andpotentialvorticity).Inparticular,theauthorsfound thatwhentheshallowapproximationismadethentheCoriolistermsalsoneedtobeapproximatedinorderfortheequationstoretaintheprincipleofconservationofaxialangularmomentum.Specifically,axialangularmomentumisconservedifthehorizontalcomponentof theEarth’srotationvectorisneglected,meaningthattheCoriolisterminEq.(2)vanishes completelyandtheterminEq.(1)ismodified.Thisapproximationisreferredtoasthetraditionalapproximationandhasgenerallybeenmadeinconjunctionwiththeshallowatmosphereapproximation.Itfollowedthattherewerefourchoicesofconsistentequation sets,determinedbywhethereachofthehydrostaticandshallowapproximationsareor arenotmadeindependently(withthetraditionalapproximationbeingmadewhentheshallowapproximationismade).
However,therehasbeensomethingofarevivalofinterestinthisareaintherecentyears withanumberofrecentpapersrevisit ingandgeneralizingthisworksuchas Charronand Zadra(2014) and Staniforth(2014a).Inparticular, TortandDubos(2014) showedthatconsistencycaninfactbeachievedwhileretainingahorizontalcomponentoftheEarth’srotationvector,albeitinanapproximateform.Thismeansthattherearethensix consistentequationsets,determinedbywhetherthehydrostaticapproximationisoris notmade,togetherwithoneof:deepatmosphereandfullCoriolisterms;shallowatmosphereandtraditionalapproximation;andshallow-atmosphereandapproximated fullCoriolis.
Forthermodynamicvariables,theequationsaresimpler,allhavingtheform:
Thisequation(anditssimplifiedforminwhichthedivergencetermisnotpresent)appliesto anyofthethermodynamicvariablessuchastemperature, T,pressure, p,anddensity, ρ and anyfunctionofanyofthese,forexample, Π , θ ,ln θ ,ln p,etc.,where θ denotesthepotential temperatureand Π istheExnerfunction.Thesearerelatedtoeachotherbytheequationof state
where R isthespecificgasconstant,andthedefinitions
where p0 isareferencepressure,typicallytakentobe1000hPaand κ ≡ R/cp with cp thespecificheatcapacityofairatconstantpressure,and
Theequationshavebeengivensymbolicallyatthisstagetoavoidhavingtomakeanyspecificchoicesaboutexactlywhichvariablesareusedinthenumericalimplementationofthe equations.Fortheanalystworkingwiththecontinuousequations,thechoiceofvariablesis oneofconvenience,thesolutionisindependentofhowtheequationsarewritten.However, numericalmodelersanddiscreteanalystsdonothavethisluxury;whentheequationsare discretizedonlya(small)finitenumberofthepropertiesofthecontinuousequationsarepreservedbyanyparticulardiscretization.And,thepropertiesthatarepreservedarestrongly relatedtothechoiceofhowthediscreteequationsarewritten.So,totiedownexactlythedesignofthedynamicalcore,achoicehastobemadeforhowthewindswillberepresentedand whichthermodynamicvariableswillbeusedasprognosticvariables.
Forthewinds(andforthemomentassumingthehorizontalsurfaceisflat),thereare broadlytwochoices:
• Thecomponentsofthevelocity.Letthesebedenotedby u, v, w ðÞ,where u, v ðÞ representthe horizontalcomponents(perpendiculartogravity)and w representstheverticalcomponent paralleltogravity.
• Thecomponentsofthemomentum.Thesearesimplythecomponentsofthevelocity multipliedbythedensity, ρ
Theadvantageofusingthemomentumisthatitsimplifiesthepressuregradientterm(itis alinearterminvolvingonlythepressure)anditallowsthetransporttermstobewrittenina formthatallowsexactnumericalconservationoflinearmomentum.Thelatterpropertyis usefulforlocalregionalmodelsforwhichlinearmomentumisanaturalquantity.Thisform isused,forexample,intheWeatherResearchandForecastingmodel(see Skamarocketal., 2019 andalsotheWRFweb-basedresourcesat https://doi.org/10.5065/D6MK6B4K).However,onthesphere,itisaxialangularmomentumthatistheconservedquantity.Although someconsiderationtousingangularmomentumhasbeengiven,theauthorisnotawareof anymodelthatusesthisforthewindprognosticvariables.Thedisadvantageofusingmomentumisthatthevelocityneedstobediagnosedfromittoevaluatethetransportterm. Howmuchofanissuethisisdependsonthespecificdiscretizationused.
Theadvantageofusingvelocityisthatthisisdirectlythequantityneededtoevaluatethe transport.Thisenablestheuseofthesemi-Lagrangianscheme,ofwhichmorelater,anditalso enablestheequationstobewrittendirectlyinvectorform(oftenreferredtoasthevectorinvariantform),whichavoidstheneedfortheexplicitinclusionofpotentiallycomplicated termsthatarespecifictothespecificchoiceofcoordinatesused.Thedisadvantagesofthe velocityformareanonlinearpressuregradientterm,thatis,onethatinvolvestheproduct ofthegradientofthepressurevariable(e.g., p or Π )withanotherthermodynamicquantity (e.g.,1/ρ or θ ),andthelackofastraightforwardwayofobtainingdiscreteconservationof linear,andangular,momentum.
Inthecasethatthecoordinatelinesareperpendiculartoeachother(anorthogonalcoordinatesystem),thenthereisnoambiguityinwhatthecomponents u, v, w ðÞ represent(other thanchoicesofscalingfactors).Whenthecoordinatesarenotorthogonal,though,thereare furtherchoicestobemadeaboutwhatwindcomponentstouse.Thetwonaturalonesare:the contravariantcomponentsinwhichthewindvectorisexpressedintermsofbasevectorsthat areparalleltothecoordinatelinesandthecovariantcomponentsinwhichthebasevectorof onecoordinatedirectionisperpendiculartotheplanedefinedbytheothercoordinatedirections.However,inmeteorology,athirdalternativeisoftenused.WhentheEarth’ssurfaceis notflatbecauseoforography(hillsandmountains),theverticalcoordinateisoftenchosen suchthat,nearthesurfaceatleast,surfacesofconstantverticalcoordinateareparallelto theorography.Suchacoordinateistermedaterrain-followingcoordinate,forexample,that of Gal-ChenandSomerville(1975).Althougheitherthecontra-orcovariantcomponents couldbeused, Clark(1977) foundthatgoodnumericalconservationoflinearmomentum andenergywasmucheasiertoachieveifasetoforthogonalbasevectorsareusedtodescribe thewindfield,withtheverticalbasevectorparalleltothegravityvector,andthisisacommon practiceinmanycontemporaryNWPmodels(e.g., Woodetal.,2014).
Itisworthnotingthat,forhydrostaticmodels(forwhichtheverticalvelocityisnotaprognosticvariable),thereisathirdchoiceforthewindvariables.Inplaceofthetwohorizontal componentsofvelocity,thisapproachusestheverticalcomponentofvorticityandthedivergenceofthehorizontalwind(i.e.,thehorizontalwindissplitintoitsrotationaland nonrotationalcomponents).Thisisaparticularlyattractivechoiceformodelsthatarefocused onthelarge-scaledynamics.Duetothethermalstratificationoftheatmosphere,thelargescaledynamicsaredominatedbytheeffectsoftheplanetaryrotationwhicharecaptured bytheevolutionoftheverticalvorticity,andinparticularthepotentialvorticitywhichjudiciouslycombinesthevorticitywiththepotentialtemperature(theinterestedreaderisreferredtoanybooksondynamicalmeteorologysuchas Holton,1992 and Vallis,2006).The divergenceofthehorizontalwindplaysanessentialroleinthedynamicsofthegravitywaves (andneglectingthetendencyofthedivergenceisawayoffilteringoutthegravitywaves). However,insuchmodels,thewindcomponentsarestillrequiredinorder,forexample,to evaluatehowthevorticityistransported.Inavorticity-divergence-basedmodel,theseare obtainedbyinvertingtherelationsbetweenthewindsandthevorticityanddivergence.These operationsareexpensiveforallbutspectralmodels(forwhichthecostiseffectivelyalready paidforintheFourierandLegendretransformations).Fornonhydrostaticmodels,forwhich theinterestincludesmuchsmallerscalesthanjustplanetaryscales,theverticalvelocityisa prognosticvariableandallthreecomponentsofthevorticitybecomerelevant.Thesplitinto theverticalvorticityanddivergencebecomesamuchlessnaturalchoice.
Forthethermodynamicvariables,thereisalargerangeofchoices.Essentially,anytwo independentchoicesoutof T, p, ρ,andanyfunctionofanyofthese,forexample, Π , θ , ln θ ,ln p,etc.
Indiscussingtheprosandconsofvariouschoicesofthermodynamicvariables,itisuseful tointroducetheisothermalprofile.IntheUSStandardAtmosphere,thevalueoftemperature departsfromavalueof238.5Kbyonly 20%betweenthesurfaceand100km.Between 10and50km,thevariationfrom243.5Kisonly 11%.Areasonablefirstestimateofhow thermodynamicquantitiesvaryintheatmosphereis,therefore,toassumethat TzðÞ¼ T0 ¼ constant,where z denotesheightabovethesurface.Togetherwiththeequation ofstateandtheassumptionthattheatmosphereisinhydrostaticbalancetheverticalvariation ofalltheotherthermodynamicvariablescanthenbeobtained.Itisfoundthat
Theseall,therefore,decayexponentially(exceptfor θ whichincreasesexponentially)withthe exponentof Π and θ beingrathersmaller(slowerdecay/increase)thanthatfor p and ρ bythe factor κ ,whichhasthevalue2/7foridealdryair.Thescaleheightoverwhich p and ρ decayis givenby RT0/g whichhasatypicalvalueof7km.
Whatthenofthechoicesforthermodynamicvariable?
Animportantquestioniswhetherthedynamicalcoreshouldpreservethemassoftheatmosphere.Ifitisrequiredtodoso,thenoneoftheprognosticthermodynamicvariablesneeds tobedensity and thedensityneedstobeevolvedusinganumericallyconservativescheme.If aquantityotherthanalinearfunctionofdensityisused,orifanonconservativenumerical schemeisused,thenmassconservationisonlyachievedbysomeformofglobalfixerthat addsorremovestherequisiteamountofmasstorestorethetotalmass.Suchschemesinevitablyleadtovaryingdegreesofnonphysicaltransportofmass.Anearlyexampleofascheme thattriedtolocalizethattransportasmuchaspossibleistheschemeof Priestley(1993) with recentvariantsonthatby,forexample, ZerroukatandAllen(2015)
Ifdensityisnotchosenasaprognosticvariable,thenanaturalchoiceforoneofthethermodynamicvariablesispressure,eitherdirectlyas p oras Π or,toavoidanynumericalinaccuracieswithitsexponentialdecay,ln p.
Thereseemtobetwoalternativeroutestofollowforwhichothervariabletochoose.Numericalschemesaremostaccuratewhenappliedtoquantitiesthatvarysmoothly.Ithasalreadybeennotedthattemperaturedoesnotvarydramaticallywithheightsothismightbea naturalchoicethatavoidstheexponentialchanges.Theotherispotentialtemperature, θ .The appealofthisisthatbecauseln θ isproportional(inadryatmosphere)totheentropyofthe system,itispreservedintheabsenceofanydiabaticprocesses,thatis,thedivergencetermin Eq.(3)vanishesfor θ anditispurelytransported.Thedisadvantageistheexponentialvariationof θ butthiscanbeavoidedbyusingln θ ,whichvariesinamannercloserto temperature.
Itisworthnotingthatatemptingchoiceistouse ρθ asthetransportedvariablesincethe equationforitsevolutioncanbewritteninconservativeform(i.e.,aformthatanumerical schemecanstraightforwardlyensuretheglobalintegralof ρθ isexactlyconserved).However, itisimportanttonotethatfromEqs. (4)–(6) ρθ isafunctiononlyof p (orequivalently Π ). Therefore,theonlyremainingchoiceofindependentthermodynamicvariablewouldbea functiononlyof T.
Thesechoiceshavetobeconsideredalsoinabroadersense,intermsofwhatimpactthey haveonthedesignasawhole.Aparticularaspectistheformofthepressuregradienttermin Eqs.(1),(2).Thisdependsonthechoicemadeforthewindcomponents.Ifmomentumisused, thenPressureGradient ¼ Grad p ðÞ directly(whereGraddenotesevaluationofthespatialgradientofitsargument).Therefore,useof p mightbemorenaturalasoneofthethermodynamic variablesratherthanusingaderivedquantitysuchas Π orln p,bothofwhichwould reintroduceanunnecessarynonlinearityintothescheme.Ifvelocityisused,thenPressure Gradient ¼ ρ 1Grad p ðÞ.Then,perhaps ρ and p arenaturalchoices.However,usingtheequationofstatethiscanalsobewrittenaseither RTGradln p ðÞ or cpθ Grad Π ðÞ.Thechoicesseemto beincreasingnotreducing!However,helpisathand.
Itwillbeseenbelowthatanimportantroleofthedynamicalcoreisthepropagationofa varietyofdifferentoscillatorymodes,orwaves.Theseminalworkof ThuburnandWoollings (2005) showedthatthechoiceofthermodynamicvariables(andhowtorepresentthemspatially)playsacriticalroleinhowwellthedynamicalcorerepresentsthosewaves,andhence howaccuratetheschemeis.Inparticular,theauthorsidentified(foragivenchoiceofvertical coordinate)asinglechoiceofthermodynamicvariablethatgavewhattheydescribedasthe optimalconfiguration.Inafollow-uppieceofwork, Thuburn(2006) showedhow,byusing specificnumericalformsofthepressuregradient,someofthesuboptimalconfigurationscan infactbemadetobeoptimal.Dependingonone’sperspective,thisiseitherunfortunate(it doesnothelpnarrowthepossibilities!)orfortunate(itleavesopenmorepossibilities!).
Itisclearthatbeforeoneevengetsclosetodefininganumericalschemeforadynamical core,thereisaverylargerangeofoptionsforthechoiceofcontinuousequationsetandits constituentvariables.Toclosethissection,itisworthgivingsomeexamplesoftherange ofchoicesthathavebeenmadeinwidelyusedandwidelyknownmodels:
•TheoperationaldynamicalcoreofECMWF’sIFSmodelisaglobalNWPmodel(see ECMWF’sweb-baseddocumentationat https://www.ecmwf.int/en/publications/ifsdocumentation).Itmakesthetraditional,shallow-atmosphere,andhydrostaticapproximations.Itusesapressure-basedverticalcoordinate.Thewindisrepresentedusing velocitycomponents.Itstwothermodynamicvariablesaretemperatureandgeopotential height(ameasureoftheheightassociatedwithagivenvalueofpressure).
•TheWRFdynamicalcore(Skamarocketal.,2019)isprincipallyusedasaregionalNWP model(butitdoeshaveaglobalcapability).Itmakesaformoftheshallowapproximation butnotthetraditionalnorhydrostaticapproximations.Itusesapressure-basedvertical coordinate.Thewindisrepresentedusingmomentumcomponents.Itpredictsthree thermodynamicvariables:densityanddensity-weightedpotentialtemperature(bothina conservativeform),togetherwiththegeopotentialheightwhichisusedinthepressure gradientterm.(Theuseofthreeprognosticthermodynamicvariablesmeansthatinsteadof theequationofstatebeingusedtoeliminateoneofthethermodynamicvariablesfromthe
equationset,orbeingusedasaconstraintontheevolutionofthethermodynamic variables,itsderivativeintimeisusedtocreateanadditionalprognosticequation.)
•ThedynamicalcoreoftheMetOffice’sUnifiedModel(e.g., Waltersetal.,2017)isbothan operationalglobalandoperationalregionalNWPmodel.Itmakesneithertheshallowatmosphere,thetraditional,northehydrostaticapproximations.Itusesaheight-based verticalcoordinate.Thewindisrepresentedusingvelocitycomponents.Itstwo thermodynamicvariablesaredensity(butnotinaconservativeform)andpotential temperature.However,thepressuregradienttermiswrittenintermsoftheExner function,whichisderivedfromdensityandpotentialtemperatureusingtheequationof state.
3Somephysicalproperties
Beforeconsideringwhatnumericalschemestoapplytothegoverningequations,itisimportanttohaveaclearideaofthenatureofthephysicalphenomenonrepresentedbythose equations.ThereareperhapsthreekeypropertiesoftheequationsrepresentedbyEqs. (1)–(3). Twoofthemarerelated:theequationsareenergypreserving(theycanbederivedfrom Hamilton’sprinciple,e.g., Salmon,1988; Shepherd,1990; Staniforth,2014a);andtheyconstituteahyperbolicsystemofequationswhichmeansthattheyadmitwave-likesolutions.The thirdproperty,andonethatsetsthemapartfrommanyothersimilarequationsets,isthatthe presenceofrotationmeansthattheequationsadmitnontrivialsteadystates,inparticular steadystateswithnonzerowindfields.Thelarge-scalestatesoftheatmospherecanoften beregardedastheevolutionfromonenear-steadystatetoanother.Further,thetransition fromonenear-steadystatetoanotherisachievedbythewave-likesolutionstotheequations. Therefore,itisessentialfortheaccuracyofadynamicalcore(a)thatthosesteadystatesare wellrepresentedandmaintainedbythenumericalschemesemployedand(b)thattheadjustmentbythewavesiswellcapturedbythenumericalschemes.
Theleadingordersteadystateshavealreadybeendiscussed.Inthehorizontal,itisgeostrophicbalancewhichcansymbolicallyberepresentedas
SincetheCoriolistermsactperpendiculartotheflowdirection,geostrophicbalancerequires thewindfieldtobeperpendiculartothepressuregradienttermsothatflowisparalleltolines ofconstantpressure.
Inthevertical,itishydrostaticbalancewhichcanberepresentedas
wherethetraditionalapproximationhasbeenassumedsotheCorioliseffectonverticalmotionhasbeenneglected.Iftheflowisinbothgeostrophicandhydrostaticbalance,thenthere isalsothermalwindbalancewhichrequiresthatanyverticalshearinthehorizontalwind mustbebalancedbyahorizontalgradientoftemperatureperpendiculartothewind direction.
Wavemotionsformwhentheinertiaofanairparcelisopposedbyarestoringforce.By consideringthegoverningequationsinthepresenceofperturbationstoageostrophically
andhydrostaticallybalancedstate,itisfound(see,e.g.,thecomprehensivetextbooksof Gill, 1982,Holton,1992,and Vallis,2006)thatthefiveequations(threeforthewindcomponents andtwoforeachofthechosenthermodynamicvariables)supportfivemodesofoscillations, asthreedifferenttypesofwave:
• OneRossbywave.ThisisduetothevariationoftheCorioliseffectwithlatitude(thereneeds tobeahorizontalgradientinthebackgroundpotentialvorticityfield).Rossbywaveshave apropagationspeed,relativetothemeanwind,thatisclosetozero.Indeed,onan f-plane, forwhichtheCoriolistermisconstant,theRossbywaveispurelytransportedbythemean wind,thatis,itisstationaryrelativetothemeanwind.Rossbywavesarethemost importantwavesfordeterminingthelarge-scaleevolutionoftheweatherandare predominantlyresponsibleforthelarge-scalepatternsseenatmid-latitudesinsatellite images.
• Twogravitywaves.Theseareduetothebuoyancycreatedbytheeffectsofgravity(there needstobeaverticalgradientinthebackgroundpotentialtemperaturefield).They propagateanisotropicallyandcomeinpairsthatpropagateinoppositedirections.They areimportantfortheadjustmentofflowduetothepresenceoforographyandstrong convectivedisturbances.Theyhaveapropagationspeeddeterminedbythestratification oftheatmosphere.Atypicalspeedmightbe 50ms 1 (relativetothewindspeed). However,forthedeepestmodesthatextendthroughasignificantproportionofthe troposphere,themaximumpropagationspeedisaround320ms 1,similartothespeedof sound.
• Twoacousticwaves.Theseareduetothecompressibilityoftheatmosphere.Theypropagate isotropicallyandcomeinpairsthatpropagateinoppositedirections.Thedirecteffectof acousticwavesonweatherforecastmodelsisnegligible.Indeed,thehydrostatic approximationfiltersoutallacousticmodesexceptforonepairofmodesthatpropagate purelyhorizontally.TheyappearinNWPmodelseffectivelyasaby-productofnot wantingtomakethehydrostaticapproximationbecauseofhowthatapproximation distortsthepropagationofthegravitywaves.(Acousticwavescanalsobefilteredby removing,oratleastapproximating,thecompressibilityoftheatmospherebymakingthe anelasticapproximation.However,thisisofquestionableaccuracyforthelargestscales.
See Daviesetal.(2003) forfurtherdiscussion.)Thespeedofsoundvarieswithheightand temperaturebutaccordingtotheUSStandardAtmosphere,itvariesbetweenamaximum of340ms 1 (relativetothewindspeed)nearthesurfaceandaminimumofaround 275ms 1 atheightsaround80–90km.
Hence,adynamicalcorehastohandletransportbythemeanwindasaccuratelyaspossible,whilerepresentingbothaccuratelyandstablythepropagationofgravitywaves.Unless theequationsethasbeenappropriatelyapproximated,thedynamicalcorealsohastohandle stablythepropagationoftheacousticwavesbuttheaccuracywithwhichitdoesthisisof secondaryimportance.
Afurtherimportantaspecttoconsiderforanysystembeforetryingtomodelitistheenergyspectrumofitsmotions,thatis,howmuchenergythereisinagivenrangeofspatial (usuallyhorizontal)scales.Variouseffortshavebeenmadetoobservetheatmosphere’sspectrum,themostfamousbeingtheworkof NastromandGage(1985).Thespectrumisimportantindecidingwhatrangeofscalesitisimportantforamodeltoresolve.Itisclearly importanttocaptureaccuratelythosescaleswherethereisthemostenergyasthosearelikely
tobethemostimportant.Intheatmosphere,thesescalesareverylarge,oftheorderofthousandsofkilometers.Therewillalsogenerallybealengthand/oratimescalebelowwhichthe flowhasvirtuallynoenergy.Thesescalesthenplacealowerboundonthespaceand/ortime scalesthatitissensibleforamodeltoresolve.Intheatmosphere,itisthemolecularviscosity thatultimatelyputsalowerboundonthescalesofmotion;kinematicmolecularviscosityof airhasavaluethatisoforder10 5 m 2 s 1.Asaresult,acompleterepresentationoftheatmospherewouldhavetorepresentallscalesfromthemolecular(atmostofordermillimeters) totheplanetary(oforder10,000kminthehorizontal),thatis,scalescoveringatleasteight ordersofmagnitude.Thisis,andwillforalongtime,remainbeyondthereachofsupercomputers.Therefore,simulationshavetoparameterizetheeffectsofmotionsbelowsomenominallysmallscale.Itis,therefore,importanttounderstandhowenergyflowsfromonerange ofscalestoanother.Forexample,ifenergyonlyflowsfromlargescalestosmallscales(called adownwardcascade),thenthesmallscalesaresaidtobeslavedtothelargeronesandparameterizingtheeffectsofthesmallscalesintermsofthelargeronesissensible.Itturnsout thattoalargeextentthisiswhathappensintheatmosphereanditisthisthathasallowed accurateweatherpredictionswithrelativelylimitedcomputationalresource.Butasever moreaccurateforecastsaresoughtthefactthatthereissomeupwardenergycascadefrom smallscalestolargerscalesstartstobecomeimportant.Italsoturnsout(Holdawayetal., 2008)thattheslopeofthespectrumcaninfluence,andevendetermine,therateatwhichnumericalmethodsconvergetothecorrectanswer;iftheslopeofthespectrumisinsomesense shallowthenincreasingtheaccuracyofthenumericalschemewillnotnecessarilyimprove theaccuracyoftheresults—itis,inthatcase,moreeffectivetoimprovetheresolutionof themodel.
4Discretizingintime
Thissectiongivesaverysimpleintroductiontosomebasictime-steppingschemes.Much moredetailcanbefoundelsewheresuchasthetextbook Durran(2010).Oncewehavethe basicschemestheirsuitabilityasbuildingblocksforadynamicalcorewillbeexamined.
First,wetakeasteptowardamoremathematicaldescriptionoftheequations.Eqs. (1)–(3) canberepresentedas
(9) where dF/dt representstherateofchangeof F withtime, F representsa“statevector,”thatis, avectorwhosecomponentsarethesetofallprognosticvariables(whichthemselvesmightbe avectorofanumberofdiscretevariables,e.g.,thetemperatureateachdiscretepointin space),and G isthevectorofallthetermsthatcontributetothetendencyof F.Thetermscomprising G aresometimesreferredtoasthesourcetermsorforcingterms.However,thiscanbe misleadingsincemostoftheelementsof G arefunctionsof F andsoarenotexternaltothe problem.
Theroleofthedynamicalcoreistostepforwardintimefromsomegivenstateattime t toa newstateatadiscretetimelater,say t + Δt,where Δt isthetimeinterval,ortimestep.Let t ¼ nΔt forsomeinteger n,thenumberoftimestepsalreadytakenfromsomeinitialconditionat
time t ¼ 0,andlet Fn denotethestateofthemodelattime t.Then,thestateattime t + Δt,after n +1timesteps,canbeobtainedbyintegratingEq.(9)fromtime t totime t + Δt,thatis
Inthiscontext,thechallengeoftemporaldiscretizationsisthentoestimatetheintegralof G overatimeintervalandsomedifferentapproachesarenowexplored.
4.1Somedifferentapproaches
4.1.1Single-stage,single-stepschemes
Thesimplesttime-steppingschemeisobtainedbyapproximatingtheintegralontherighthandsideofEq.(10)assimply ΔtGtðÞ,whichcanbedenotedas ΔtGn.Eq.(10)thenbecomes
Inthisscheme,thetendencyof F isestimatedentirelyfromquantitiesattimelevel n.Itis, therefore,calledanexplicitscheme(sinceallcontributionstothetendencyof F areknown explicitly)anditisalsocalledaforward-in-timeschemeorforwardEuler.
Analternativeistoinsteadestimatetheintegralas ΔtGt + Δt ðÞ,thatis, ΔtGn+1.Theexpressionfor Fn+1,thatis,
isanimplicitexpressionfor Fn+1 sinceingeneral Gn+1 dependson Fn+1,whichisunknown. Thisschemeis,therefore,calledanimplicitschemeandisalsoknownasabackward-in-time schemeorbackwardEuler.
Bothoftheseschemesareonlyfirst-orderaccurate,whichmeansthattheaccuracyoftheir solutionsimproveslinearlyasthetime-step Δt isreduced.
Asecond-orderschemeisonewhoseaccuracyimprovesquadraticallyas Δt isreduced, thatis,halvingthetimestepreducestheerroroftheschemebyafactorof4.Anexample ofsuchaschemeisobtainedbyapproximatingtheintegralontheright-handsideofEq. (10)usingthetrapezoidalrule,thatis,approximatingtheintegralas Δt=2 ðÞ Gt + Δt ðÞ + GtðÞ ½ .Theresultingscheme,
+1 ¼ Fn + Δt 2 Gn +1 + Gn , (13) istheCrank-Nicolsonscheme.
Theseschemesareallsingle-stepschemessinceonlyonetimelevel, Fn,isneededtoadvancetothenexttimelevel.
4.1.2Single-stage,multistepschemes
TheCrank-Nicolsonschemecanalternativelybethoughtofasfirstapproximatingtheintegralusingthemid-pointrule,thatis,approximatingtheintegralas ΔtGt + Δt=2 ðÞ,andsecondasapproximating Gt + Δt=2 ðÞ Gt + Δt ðÞ + GtðÞ ½ =2.Withthisviewinmind,another
Fn
schemeisobtainedif,insteadofintegratingEq.(9)overonetimestep,itisintegratedovertwo timesteps,thatis,overatimeintervalof2Δt.Then,
Applyingthemidpointruletothisgives
Thisisnowanexplicitschemefor Fn+2 since,once Fn+1 hasbeenevaluated,theright-handside termisexplicitlyknown.Itiscalledtheleap-frogschemesincetheschemeadvancesfrom Fn to Fn+2 leap-froggingover Fn+1 (though Fn+1 isneededtoevaluatethetendency).Thescheme istermedamultistepschemesincetwotimelevels, Fn and Fn+1,areneededtoadvancetothe nexttimelevel.
Analternativefamilyofschemescanbeachievedif,insteadofintegratingthetimederivativeovermultipletimesteps(twointhecaseof Eq.14),multipletime-stepvaluesareusedto estimatetheintegralof G.Thetwo-stepAdams-Bashforthschemeisoneexample.Inthis scheme,thevaluesof G attime tn Δt (denotedby Gn 1)andtime tn (Gn)areusedtocreate alinearextrapolationof G validovertheinterval t [t, t + Δt],namely:
Usingthisexpression,theintegralontheright-handsideofEq.(10)canbeevaluatedas
sothatthetwo-stepAdams-Bashforthschemeis
4.1.3Multistagesingle-stepschemes
Theforward-Eulerschemeisonlyfirst-orderaccuratebutitisaverysimpleand,therefore, cheapscheme.Theideaofthenextschemeisthatthisestimatefor Fn+1 canbeusedasafirst predictorof Fn+1 intheCrank-Nicolsonschemetocreateanexplicitschemebutonethatis moreaccuratethantheforward-Eulerscheme.Theresultisatwo-stagescheme:
where F 1 ðÞ isthe(first)predictorfor Fn+1 and G 1 ðÞ ≡ GF 1 ðÞ .Thisschemecanberegardedalternativelyasthefirsttwostagesofamultistageiterativeschemeinwhicheachiterationis givenby
where F k ðÞ isthe kthpredictorfor Fn+1 and G k 1 ðÞ ≡ GF k 1 ðÞ and F 0 ðÞ ≡ Fn .Itcanalsobe regardedasapredictor-correctorschemeinwhich F 1 ðÞ isthepredictortowhichacorrector ΔtG 1 ðÞ Gn =2isapplied.Indeed,theiterativeschemecanalsoberegardedasamultiple predictor-correctorschemewhere,for k > 1:
Thisschemeislimitedtosecond-orderaccuracynomatterhowmanyiterations,orcorrectors,areapplied.Atconvergence,itconvergestothesecond-orderCrank-Nicolsonscheme. Animportantclassoftime-steppingschemescanbeintroducedbytakinganalternative routetoEq.(19).Havingevaluatedanestimate, F 1 ðÞ ,for F attime t + Δt,thisestimatecan beusedwiththeforward-Eulerschemetogenerateanestimate, F n +2 ,for F attime t +2Δt (thetildeisusedtodistinguishthisfromtheactualsolutionattime-step n +2):
Then, Fn+1 canbeobtainedasthelinearinterpolationbetween Fn and F n +2 ,thatis,
Onsubstitutingfor F 1 ðÞ fromEq.(18),thiscanbeseentobeexactlythesame,second-order schemeasEq.(19).Butfollowingasimilarprinciple,ahigher-orderschemecanbeobtained. Again,thesameestimatefor F n +2 ≡ Ft +2Δt ðÞ isobtainedbutthistimeitisusedtolinearly interpolatetoanestimate, F 2 ðÞ ,for
Thethirdandfinalpredictorisobtainedinasimilarmannerby:firstusing F 2 ðÞ andthe forward-Eulermethodtoobtainanestimate, Fn+3/2,for F attime t +3Δt/2,namely
andthenlinearlyinterpolatingbetweenthisand Fn toobtaintheestimatefor Fn+1
Thisscheme,knownastheShu-Oshermethod,isaparticularformofaclassofschemes knownasexplicitRunge-Kuttaschemes.Thereisaverylargeliteratureonsuchschemesbut fordiscussioninthecontextofNWPsee,forexample, Baldauf(2008), Welleretal.(2013),and Locketal.(2014).TheShu-Oshermethodhasthepropertythateachstageisbuiltexclusively fromapplicationofthesame,forward-Euler,scheme.Thispermitsthedevelopmentofwhat aretermedstrongstabilitypreservingschemes(Durran,2010).Further,theseeminglyarbitrarychoiceofstepshasbeenmadedeliberatelysothattheschemehasthedesirableproperty ofbeingthird-orderaccurateafterthreestagesevenwhen G isanonlinearfunctionof F (Baldauf,2008).Amorenaturalchoicemightbetheiterativescheme
with F 0 ðÞ ≡ Fn .Thefirstthreestagesofthisscheme(whichcanbecompared,respectively,with Eqs.18, 24, 26)are
Thiswouldappeartobeamorenaturalschemebecauseif G isalinearfunctionof F theneach iterationofthisschemeaddsthenexttermintheseriesexpansionfortheoperatorexp ΔtG ðÞ appliedto Fn,forexample,inthatcase,Eqs.(29),(30)become
However,incontrasttotheShu-Oshermethod,the kthstageofthisschemeis kth-orderaccurateonlywhen G isalinearfunctionof F.
4.2Somenumericalproperties
Anaturalquestiontoaskishowgoodorbadareeachoftheseschemes?
Ithasbeenseenthattheimportantaspectsofanyschemeforadynamicalcoreishowit handlestransportandalsowave-likemotion.Theessentialtemporalaspectsofbothofthese processesarecapturedbyEq.(9)withthechoice GFðÞ¼ iΩF,where F cannowbecomplex with i denotingtheimaginaryunitand Ω isthefrequencyofthemotion.Fortransportof awavewithwavenumber k (i.e.,wavelength2π /k)byawindofspeed U, Ω ¼ Uk,while forthepropagationofasoundwaveofthesamewavenumber Ω ¼ csk,where cs isthespeed ofsound.
Eq.(9)isthen
anditsanalyticalevolutionoveratimestepoflength Δt isgivenby
Fn +1 ana ≡ Fana t + Δt ðÞ¼ Fana t ðÞ exp iΩΔt ðÞ ≡ Fnana exp iΩΔt ðÞ (34)
Thisexpressesthefactthatoverthetimestepthewaveistransported,orpropagated,without achangeinamplitudeanditsphaseisshiftedbytheamount ΩΔt.If ΩΔt isasmallquantity (i.e.,ifthetimestepismuchlessthanthetimescaleimpliedbythefrequencyofthewave), thenaTaylorseriesexpansionin ΩΔt showsthat
where O ΩΔt ðÞ4 denotesalltermsinvolvingatleastfourthpowersof ΩΔt
4.2.1Single-stage,single-stepschemes
Applyingtheforward-EulerschemetoEq.(33),thediscretesolutionis
ComparingEq.(36)withEq.(35)itisseen,aspreviouslystated,thattheforward-Euler schemeisfirst-orderaccurate;onlythetermsinthefirsttwopowersof ΩΔt,thatis, ΩΔt ðÞ0 and ΩΔt ðÞ1 ,matchthosetermsintheanalyticexpression. Further,Eq.(36)canbewrittenas
where,whatistermedtheamplificationfactorofthescheme, AFE ¼ 1+ iΩΔt jj (definedtobe positive),isgivenby
¼ + 1+ ΩΔt ðÞ2 q ,
and,whatistermedthephaseshiftofthescheme, ωFE Δt ¼ arg1+ iΩΔt ðÞ,isgivenby
FromEq.(34),theidealvalueof AFE is AFE ¼ exp iΩΔt ðÞ jj¼ 1andtheidealvalueof ωFEΔt is ωFE Δt ¼ argexp iΩΔt ðÞ ½ ¼ ΩΔt.Inparticular,itisseenthat AFE > 1independentlyofhow smallthetimestepis.Thismeansthatateachtimesteptheamplitudeof FFE increases.This isincontrasttotheanalyticsolutionforwhichtheamplitudeisunchanged.TheforwardEulerschemeis,therefore,saidtobeunconditionallyunstableforthisproblem(thereis noconditionthatthetimestepcansatisfy,otherthanbeingzero,thatallowsthescheme tobestable);itgeneratesunrealisticsolutionsthatbecomearbitrarilylargeaftersufficient time.Forsufficientlysmalltimesteps,suchthatthemagnitudeof ΩΔt issmall,thephaseshift ofthewaveisreasonablywellcapturedbytheforward-Eulerscheme(sincetan α ! α as α ! 0)butformagnitudesof ΩΔt largerthan1thenthephaseshiftofthenumericalschemeis reduced.Itisconstrainedtoalwaysbelessthan π /2nomatterhowlargetheactualshift ΩΔt is.
Forthebackward-Eulerscheme,thesolutiontoEq.(33)is
Gatheringtogetherthetwotermsinvolving Fn +1 BE gives
andthesolutionisfinallyobtainedbyapplyingtheinverseof1 iΩΔ
When ΩΔt issmall,theright-handsidecanbewrittenas
andbycomparingEq.(43)withEq.(35)itisseenthatthebackward-Eulerschemeisalsofirstorderaccurate.
Eq.(42)canbewrittenas
wheretheamplificationfactor, ABE,isgivenby
andthephaseshift, ωBE,isagaingivenby
Thenumericalphaseofthebackward-Eulerschemeis,therefore,thesameasthatforthe forward-Eulerscheme.But,importantly,theamplitudeisalwayslessthan1,nomatter howlargeatimestepisused.Thismeansthatwitheachsuccessivetimesteptheamplitude ofthewavewillbereduced,tendingaftersufficientlymanytimesteps,tozero.Theschemeis, therefore,saidtobeunconditionallystable.Butitisalsosaidtobedamping;incontrasttothe physicalsystemwhichpreservestheamplitudeofthewave,thenumericalschemeadds dampingtothesystemandeventuallythewavewillbeeliminated.
Theunconditionalstabilityoftheschemecomeswithacost.Ingeneral, G isanoperator, usuallyoneinvolvingaspatialderivative,whosediscreterepresentationwillbeamatrix. Then,multiplicationof Fn +1 BE by iΩΔt representsapplicationofthatoperator,ormultiplication bythematrix.Solutionoftheimplicitschemetheninvolvesevaluatingtheinverseofthat operator.Thisisoftenaglobalproblem,solvingforonepointinspaceinvolvesthevalues atall,oratleastalargenumberof,otherpoints.Thisisamoreexpensiveoperationthansolvingforeachvaluelocally,particularlyonmassivelyparallelcomputersinwhichtheproblem issplitacrossalargenumberofprocessors.
ApplyingtheCrank-NicolsonschemetoEq.(33),andusingthesamemethodologyas earlier,resultsin
where ACN isgivenby
and ωCN isgivenby
Usingthehalf-angletrigonometricrelationships,thesingularityat
avoidedbyrewritingthisas
Forsmall ΩΔt (Eq.47)canbeexpandedas
SinceEq.( 52 )matchesthefirstthreetermsintheexpansionoftheanalyticsolution(35 ),the schemeisseentobesecond-orderaccurate.Als o,theamplificationfactoris1independent ofthesizeofthetimestep.Thescheme,therefore,retainstheanalyticamplitudewithout growthordecay;theschemeissaidtobeneutrallystable.Thesetwopropertiesmakeit averyattractiveschemefornumericalsolutionofwave-likeprocesses.Ifitisimportant togetthephasepropertiesofthewavecorrect (i.e.,itspropagationspeed),thenthetime stepmustbechosensmallenoughtoresolvethewave’sfrequency,thatis,chosentoensure thatthemagnitudeof ΩΔt islessthan1sothat,fromEq.(51 ), ωCN Ω.However,by comparingEq.( 51)withEqs.(46 ),(39 ),itisseenthattheCrank-Nicolsonschemematches thephaseerrorofthebackward-orforward-Eulerschemesbutwithatimestepthatistwice aslarge.
4.2.2Single-stage,multistepschemes
Applyingtheleap-frogschemetoEq.(33)gives
Thepreviousmethodologyisnowappliedtotwosuccessivetimesteps.Asolutionissought thathastheform
and
SubstitutingtheseintoEq.(53)leadstotherequirement
Fromthisitfollowsthat,provided
then
(i.e.,theschemeisneutrallystable)and
However,if
If ΩΔt > +1,thenthesolutionwiththepositiveroothasamplitudegreaterthan1;if ΩΔt < 1, thenthesolutionwiththenegativeroothasamplitudegreaterthan1.Therefore,whateverthe signof ΩΔt,thereisarootforwhich ALF > ΩΔt jj > 1andtheschemeisunstable.Theschemeis saidtobeconditionallystable:itisstableif ΩΔt ðÞ2 1butunstableotherwise.
Whatisadditionallydifferentaboutthisschemeisthattherearetwosolutions.By expandingtheright-handsideofEq.(57)forsmall ΩΔt,itisfoundthatthesolutionwith thepositiverootcorrespondstothephysicalsolution(andissecond-orderaccurateintime), whilethesolutionwiththenegativerootisapurelycomputationalsolutionthatarisesentirelyduetothechosenmethodofsolution.Further,becauseofEq.(59),thissolutionis notdampedatall,itpropagateswithoutchangeofamplitude.Itischaracteristicofall multistepmethodsthat,inadditiontothephysicalsolution,theysupportadditional,spurious,orcomputationalsolutions.Suchsolutionsoftenmanifestthemselvesashigh-frequency oscillations,referredtoasnoise.However,manysuchschemesaredesignedsothatthecomputationalmodesarestronglydampedintime, A < 1.Then,theirpresencemightbeacceptableprovidedthattherearenosourcetermsorforcingoftheequationsthatwouldactto continuallyexcitesuchsolutions.
Applyingthetwo-stepAdams-BashforthschemetoEq.(33)gives
SubstitutingtheseintoEq.(62)leadstotherequirement
Solvingthisquadraticfor AAB exp iωAB Δt ðÞ gives
Asforthemultistepleap-frogscheme,thisschemehastwosolutions.Byexpandingthe right-handsideofEq.(66)forsmall ΩΔt itisfoundthatthetwosolutionsevolveas
and
Therefore,thesolutionwiththepositiverootcorrespondstothephysicalsolutionanditis seentobesecond-orderaccurateintime.Thisistobeexpectedsince3FnAB Fn 1 AB =2isa second-orderestimateof Fn 1=2 AB .Theamplificationfactorforthissolutionisfoundby:gatheringalltherealtermsinEq.(67)together;squaringtheresult;gatheringalltheimaginary termstogether;squaringtheresult;summingthesetwosquares,keepingonlytermsthat arelessthan O ΩΔt ðÞ5 hi;andthenapproximatingthesquarerootoftheresultfromaTaylor seriesexpansion,againonlykeepingtermsthatarelessthan O ΩΔt ðÞ5 hi.Theresultis
and,forsmall ΩΔt,itisseentobeunstable.Notsurprisingly,itremainsunstableforlarger valuesof ΩΔt asisshownin Durran(2010).
Thesolutionwiththenegativerootisapurelycomputationalsolutionthatarisesentirely duetothechosenmethodofsolution.Forsmall ΩΔt,thisevolvesas
Thisis,therefore,anexampleofamultistepschemeforwhichthecomputationalmodeis stronglydampedintime,atleastforsmall ΩΔt.
4.2.3Multistagesingle-stepschemes
Thelow-orderaccuracyoftheforward-Eulerscheme(andwecannowsayitslackof stability)motivatedthemultistagescheme(Eq.20).ApplyingthisschemetoEq.(33)gives
Thefirstfouriterationsoftheschemeare
F 1 ðÞ iter isfirst-orderaccuratebutalltheothersareallseentobesecond-orderaccurate.Their amplificationfactorsarefoundtobe
Therefore,thefirsttwoiterationsareunconditionallyunstable,whilethelasttwoarestable providedthat ΩΔt=2 ðÞ2 1.Byminimizing A 4 ðÞ iter asafunctionof ΩΔt,itisfoundthatthemaximumdampingoccurswhen ΩΔt ðÞ2 ¼ 3forwhich A 4 ðÞ iter takesthevalue 37p =8 0 76,thatis,a reductioninamplitudeofnearlyaquarterpertimestep.For ΩΔt 1,themaximumreduction islessthan2.5%pertimestep.
Itislefttothereadertoexplorehowtheschemebehaveswithfurtheriterationsforwhich aninterestingpatternquicklyemerges.
