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DARTTutorialSec'on7:SomeAddi'onalLow-OrderModels

LowOrderModelsinDART

Model Size Features

lorenz_63 3 Chao'c,nearlyintegralaSractor,bifurca'ons

lorenz_84 3 MorecomplexaSractor,notasperiodic

9var 9 Transientoff-aSractordynamics

lorenz_96 40(variable) Higherdimensionalsystem.ASractordimension13with40variablesandstandardforcing.

forced_lorenz_96 80(variable) Allowsassimila'onofmodelparameter(seeSec'on20)

lorenz_96_2scale 396(variable) Twoprimaryinterac'ngspa'al/temporalscales

lorenz_04 variable Mul'scaledynamics

DARTTutorialSec'on7:Slide2

Lorenz84Model

ASractornotsheet-like.Raresignificantdevia'ons.Trajectoriesalongdevia'onsdon’t‘mesh’backupwiththerestoftheaSractor.Thisbehaviorcanbechallengingforcertainfiltervariants.

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x DARTTutorialSec'on7:Slide3

models/lorenz_84/work/

Lorenz84Model

3-variables:Parameters

a=0.25,b=4,f=8,g=1.25

cansetfrommodel_nml

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x

dx1dt

= −x22 − x3

2 − ax1 + af

dx2dt

= x1x2 − bx1x3 − x2 + g

dx3dt

= bx1x2 + x1x3 − x3

DARTTutorialSec'on7:Slide4

models/lorenz_84/work/

Lorenz84Model

Exercise:cd models/lorenz_84/workcsh workshop_setup.cshEachstatevariableisobservedeveryonceeveryhour.Observa'onalerrorvarianceis1.UseMatlabtoexaminetheoutput.There’sanewtypeoffilterchallengerepresentedhere.Canyouiden'fyit?Canyouproposewaystoaddressitwithtechniqueslearnedtodate?

DARTTutorialSec'on7:Slide5

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9VariableModel

ThreegroupsofvariablesVariables1-3:DivergenceVariables4-6:Vor'city.Variables7-9:Height.Ingeneral,divergenceissmall.Heightandpressuresimilar.HeightandpressurehaveaSractorsimilartoLorenz63.

DARTTutorialSec'on7:Slide6

models/9var/work/

9VariableModel

!Xi =UjUk +VjVk−v0aiXi +Yi +aizi

!Yi =UjYk +YjVk−Xi−v0aiYi

!zi =Uj zk−hk( )+ z j−hj( )Vk−g0Xi−K0aizi +Fi

Ui =−bjxi + cyi

Vi =−bkxi−cyi

Xi =−aixi

Yi =−aiyi

Xisdivergence,Yisvor'city,ZisheightAllparameterscanbeadjustedfrommodel_mod.nml

i=1,2,3 j =mod(i, 3)+1 k =mod(i+1,3)+1

DARTTutorialSec'on7:Slide7

9VariableModel

WhenperturbedofftheaSractor,mimics‘gravitywaves’.Transient,highfrequencyoscilla'onsdominatedivergencevariables.Canalsoappearinheightandpressurevariables.cd models/9var/workcsh workshop_setup.cshY1,Y2,Y3(the‘vor'city’variables)areobservedonceevery6hoursObserva'onalerrorvarianceis0.4.UseMatlabtoexaminetheoutput.Howdodifferentfilterkindsinteractwith‘gravity’waves?

DARTTutorialSec'on7:Slide8

Lorenz96(40-variable)Model

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Lorenz_96 state varnum 1 Ensemble Members of ./Prior_Diag.nc

True StateEnsemble MeanEnsemble Members (20)

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True StateEnsemble MeanEnsemble Members (20)

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True StateEnsemble MeanEnsemble Members (20)

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The Attractor

state variable # 2

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riabl

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1 2 3 True_State.nc Lorenz_96 true state

DARTTutorialSec'on7:Slide9

preassim.nc

preassim.nc

preassim.nc

true_state.nc

models/lorenz_96/work/

Lorenz96(40-variable)ModelASractordimension13bysomemeasureswith40variablesandstandardforcing(forcing=8.00in&model_nml).Starttoexploremodelsizesclosertoensemblesize.Canexaminepossibledegeneracyissueswithsamplecovariance.Naiveapplica'onofsmallensemblesdivergesinmanycases.

DARTTutorialSec'on7:Slide10

Lorenz96(40-variable)Model

cd models/lorenz_96/workcsh workshop_setup.csh40observa'ons,randomlylocatedinspace,equallyspacedin'me.Observedonceanhour;observa'onalerrorvarianceis1.0.UseMatlabtoexaminetheoutput.Neednewtechniquestofixproblemseenhere.Forplot_ens_.me_series,plot_ens_mean_.me_series:

Canselectsubsetofvariablestoplot,Defaultselec'onofvariables1,13,and27areapproximately

equallyspacedaroundthecyclicdomain.

DARTTutorialSec'on7:Slide11

1.   FilteringForaOneVariableSystem2.   TheDARTDirectoryTree3.   DARTRunAmeControlandDocumentaAon4.   HowshouldobservaAonsofastatevariableimpactanunobservedstatevariable?

MulAvariateassimilaAon.5.   ComprehensiveFilteringTheory:Non-IdenAtyObservaAonsandtheJointPhaseSpace6.   OtherUpdatesforAnObservedVariable7.   SomeAddiAonalLow-OrderModels8.   DealingwithSamplingError9.   MoreonDealingwithError;InflaAon10.   RegressionandNonlinearEffects11.   CreaAngDARTExecutables12.   AdapAveInflaAon13.   HierarchicalGroupFiltersandLocalizaAon14.   QualityControl15.   DARTExperiments:ControlandDesign16.   DiagnosAcOutput17.   CreaAngObservaAonSequences18.   LostinPhaseSpace:TheChallengeofNotKnowingtheTruth19.   DART-CompliantModelsandMakingModelsCompliant20.   ModelParameterEsAmaAon21.   ObservaAonTypesandObservingSystemDesign22.   ParallelAlgorithmImplementaAon23.  Loca'onmoduledesign(notavailable)24.  Fixedlagsmoother(notavailable)25.   Asimple1DadvecAonmodel:TracerDataAssimilaAon

DARTTutorialIndextoSec'ons

DARTTutorialSec'on7:Slide12