Flux-drivenglobaltransportsimulationsbasedonjointapproachwith

gyrokinetic andtransportsolvers*MotokiNakata1,MitsuruHonda2,andMasanoriNunami1

1NationalInstituteforFusionScience/TheGraduateUniversityforAdvancedStudies2NationalInstitutesforQuantumandRadiologicalScienceandTechnology

7th APTWGatNagoyaUniv.,June5-8th 2017

Contents:IntroductionAboutTRESS+GKVNumericalsimulationsSummary

turbulent-flux-tube• Categoryoftransportsimulations/analyses1) reducedtransportmodels:IFS-PPPL,GLF23,TGLF,CDBM,…2) δf gyrokinetic sim.:GYRO,GENE,GKW,GKV,…3) globalfull-f gyrokinetic sim.:GT5D,GYSELA,…

• Drawbacks1) limitedvalidityofreducedmodelonlimitedparam.space2) numericalcost,ignoringtheeffectsofturb.onprofiles3) extremenumericalcost,physicsimplementationongoing

• Directcouplingoftransportsolverandδf GKcodeenablestheflexiblestudyforplasmatransportunderrealisticheatingandsink:TRESS+GKV

• Previousstudies– TGYRO(GYRO+NEO):steady-statesolutionviatheNewton-iterationmethod– TRINITY(GS2/GENE+C.H.):evolutionofmacroscopicprofilesintime

Introduction1

ü Transporteq.-basedprofilepredictionsastheextendedflux-matchingü Emulatingglobalfull-f simulationresultswithlowernumericalcostü Advancedframeworktoconstruct/testanintegratedtransportmodel

Whydirectcouplingapproach?

• ”Flux-matching”profileevaluationsinfluxtube GKsimulations:

2

wheretheplasmaparametersaredefinedas

kineticprofiles Magneticequilibrium

Ifthecalculatedfluxesindicate,aparametervariationisdeterminedsuchthat

Whydirectcouplingapproach?

• ”Flux-matching”profileevaluationsinfluxtube GKsimulations:

3

Ifthecalculatedfluxesindicate,aparametervariationisdeterminedsuchthat

fixed

0.2 0.80.4 0.6𝜌 [Nakata&Honda,etal.NF2016/Gorler etal.,PoP2014]GKVforJT-60U

wheretheplasmaparametersaredefinedas

Whydirectcouplingapproach?

• ”Flux-matching”profileevaluationsinfluxtube GKsimulations:

4

kineticprofiles Magneticequilibrium

Ifthecalculatedfluxesindicate,aparametervariationisdeterminedsuchthat

wheretheplasmaparametersaredefinedas

• Hugeparametersurveyforover10-dimensionalspaceisimpracticable.à Difficultiesinmodelingofnonlin.functionsonwholeparameterspaceà Importanceofthecouplingbetweenkineticprofiles(e.g.,BScurrent)

andmagneticequilibriumintokamaksteadyoperations

• Hugeparametersurveyforover10-dimensionalspaceisimpracticable.à Difficultiesinmodelingofnonlin.functionsonwholeparameterspaceà Importanceofthecouplingbetweenkineticprofiles(e.g.,BScurrent)

andmagneticequilibriumintokamaksteadyplasmas

Whydirectcouplingapproach?5

multipleflux-tubes

t=t0Steadypowerbalance t=t0

Steadypowerbalance

Fast,butsometimesfailed Slow,but

consistent

GK-scan(ωs) basedinterpolation+transporteq.

every-pointGKsimxtransporteq.

Modeling/flux-matchingapproach Directcouplingapproach(TRESS+GKV)

Itisusefultomakeprofilepredictionsbasedon globaltransportequation,inadditiontotheconventionalflux-matching/modelingwithGKsimulations.Directcouplingwithtransport&equilibriumsolversincl.aux-heating

Transportcode:TRESS+GKsolver:GKV

• Simple1Ddiffusivetransportcodewith3rd orderspatialaccuracy– fandf’arebothdeterminedself-consistently.

• Solvingheattransportequations:𝜒 = 𝜒$%& + 𝜒()*

6

TRESS :TRansport EquationSolver:Stable

• Multi-specieselectro-magneticturbulencesimulationsinlocalfluxtubes– Highaccuracyandperformance,Actualequilibriumreadable

• Solvinglocalgyrokinetic equationwithPoisson-Ampèreequations

GKV:GyroKinetic Vlasov solver [e.g.,WatanabeNF2006,NakataPFR2014CPC2015]

TRESS+GKV :Findthesolution(𝑇, profiles)tosatisfy

FlowchartofTRESS+GKVcalculation

MajorfeaturesofthecurrentTRESS+GKV:

*SPMDforthetimebeing*Num.oflocationsisarbitrary

GKVρ=0.1

TRESS

GKVρ=0.2

GKVρ=0.9…

time&iteration

TRESS

7

• Realisticequilibrium(e.g.,G-EQDSK)• Multi-species(electronsandions)• Adiabaticandkineticelectronresponses• Acceleratorforfastconvergence

– Adaptivesourceandsinktermshavebeenimplementedtofillthegapbetweenthecurrentheatfluxandthetargetone.

• Arbitraryheatandsinkprofiles• Dirichlet,ZeroorfinitefluxNeumannB.C.

parallelcalc.each

Quasi-Linear thermal diffusivities χtb

Estimated at ρ=0.1, 0.2, … , and 0.9 (9 fluxtubes)GKV runs linear calculations for some ky’s and kx=0, but single-ky for now. Pick up the fastest growing mode and estimate χ using the simple mixing length rule:

𝜒, = 𝛼,./01 2

34max 8

9:;<2 , where ./

01 234is calculated by GKV and 𝛼% = 𝛼> = 2.

Akima method is used to interpolate χtb.

Adaptivesource/sinkmodel

• Adaptivesource/sinktermisevaluatedas𝑆BCDE(𝑡) = −𝑓𝜕* 𝑃BMN − 𝑃>(𝑡) .• Toverifytheeffectivenessofadaptivesource/sinkmodel,weassume

– simplifiedstifftransportmodelandzerofluxNeumanncondition– 𝑃BMN = 𝑃,>$9= 3MW.

• Adaptivemodelsucceededinacceleratingconvergencetowardsthepowerbalance(PB),twicefasterthanbefore.

8

t=0

tà∞

TRESS+GKV simulations

1. ITG:Noheating,profilerelaxationtowardsmarginallystable

2. ITG:5MWheatingwithadaptivesourcemodel

3. ITG/TEM:5MWheating

ITG-ae:RelaxingtotheITGstableprofilew/oheating

• w/oheating,neoclassicaltransport• ITGunstableinitialTi profile,Δt=1ms

10

Conditions:

• InitiallyITG-unstableTi profilereachesasteadystate,withthecriticalgrad.,whereχiGKV iszero:ITG-stable.

• SimilarsteadyprofileisobservedforeachB.C.,exceptfortheedgeregion.

Results:fixedni

flat

steep

Dirichlet

B.C.

Neum

annB.C.

ITG-ae:Heatingof5MWw/adaptivesource

• 5MWheatingsource,adaptivesourcew/𝑓 = 1.5

11

Ti ciGKV

• HeatfluxPiapproachesthepowerbalanced(PB)steadystate,aswellasTi and𝜒>.

• Adaptivesourcetermexaggeratestheoscillation,whiletheplasmarapidlyapproachestheexactpowerbalance(PB) after𝑡 ≈ 0.185𝑠.

cineoPaux,Pi,Padpt

Spatio-temporalprofilesofTi andheatflux12

Ti QiGKV

w/oada

pt.sou

rce Ti QiGKV

w/ad

apt.source

PB Steady state

Still evolving to PB Still evolving to PB

PB Steady state

ITG/TEM-ke:Heatingof5MWw/adaptivesource

• ITG/TEM:elec.&iontransport,equipartition included,5MWheatingeach

13

Te, Ti Pe, PiQe,Qi

• InitialprofilesareITG/TEMunstable.• EquipartitiondynamicallychangesthetargetPBstates.• ConsistentevolutionsofTe andTi towardthePBsteadystateare

confirmed.(incl.accelerationbyadaptiveheat/sink)

Summary14

TRESS+GKV :directcouplingframeworkofthetransportcodeandtheδfgyrokinetic code,aimedatconsistentpredictionofprofileandtransport.

Futureworks:² InvestigateparametricdependenceoftheTRESS+GKV,e.g.,accelerationfactor² CouplingwithnonlinearGKsimulations² Implementrealisticsinkmodelintheedgeregion

• Treatinge- andi-heattransportinITG(ad.elec.)andITG/TEM(ki.elec.)• Quasilinearestimateofheatdiffusivities• ConvergenceacceleratortowardsthePBm

odeling

results • FinitefluxB.C.worksandthePBisreachedforthe5MWheatingcase.

• Adaptivesourcemodelsuccessfullyacceleratestheconvergence.• FirstITG/TEMsimulationemerges,butrequiresfurtherrefurbishment.

Applicationtoexperimentalanalysiswillcomeafternonlinearimplementation.