AbstractAcodethatmodelssignalsproducedbychargeexchangereactionsbetweenfastionsandinjectedneutralbeamsintokamakplasmasisdescribedWiththefastiondistributionfunctionasinputthecodepredictstheefuxtoa ID: 429445
Download Pdf The PPT/PDF document "Commun.Comput.Phys.doi:10.4208/cicp.1908..." is the property of its rightful owner. Permission is granted to download and print the materials on this web site for personal, non-commercial use only, and to display it on your personal computer provided you do not modify the materials and that you retain all copyright notices contained in the materials. By downloading content from our website, you accept the terms of this agreement.
Commun.Comput.Phys.doi:10.4208/cicp.190810.080211aVol.10,No.3,pp.716-741September2011ACodethatSimulatesFast-IonDaandNeutralParticleMeasurementsW.W.Heidbrink1,,D.Liu1,3,Y.Luo1,4,E.Ruskov1andB.Geiger21DepartmentofPhysicsandAstronomy,UniversityofCalifornia,Irvine,California,CA92697,USA.2Max-PlanckInstitutef¨urPlasmaphysik,Garching,Germany.3DepartmentofPhysics,UniversityofWisconsin-Madison,Madison,WI53706,USA.4TriAlphaEnergyCorporation,27211Burbank,FoothillRanch,CA92610,USA.Received19August2010;Accepted(inrevisedversion)8February2011Availableonline1June2011 Abstract.Acodethatmodelssignalsproducedbycharge-exchangereactionsbetweenfastionsandinjectedneutralbeamsintokamakplasmasisdescribed.Withthefast-iondistributionfunctionasinput,thecodepredictstheefuxtoaneutralparticleanalyzer(NPA)diagnosticandthephotonradianceofBalmer-alphalighttoafast-ionDa(FIDA)diagnostic.Reactionswithboththeprimaryinjectedneutralsandwiththecloudofsecondaryhaloneutralsthatsurroundthebeamaretreated.Accuratecalculationofthefractionofneutralsthatoccupyexcitedatomicstates(thecollisional-radiativetransitionequations)isanimportantelementofthecode.ComparisonwithTRANSPoutputandothertestsverifythesolutions.Judiciousselectionofgridsizeandotherparametersfacilitateefcientsolutions.TheoutputofthecodehasbeenvalidatedbyFIDAmeasurementsonDIII-Dbutfurthertestsarewarranted.PACS:52.55.Pi,52.65.Pp,52.70.KzKeywords:Fastions. 1IntroductionSupra-thermalpopulationsofenergeticionsplayanimportantroleinmagneticfusionresearch.Thesefastionsarecreatedbyneutral-beaminjection,byRFheating,andinfusionreactions.Thedistributionfunctionthatdescribesthesepopulationsgenerallyisa Correspondingauthor.Emailaddresses:Bill.Heidbrink@uci.edu(W.W.Heidbrink),dliu29@wisc.edu(D.Liu),yluo@trialphaenergy.com(Y.Luo),eruskov@uci.edu(E.Ruskov),bgeiger@ipp.mpg.de(B.Geiger)http://www.global-sci.com/716c\r2011Global-SciencePress W.W.Heidbrinketal./Commun.Comput.Phys.,10(2011),pp.716-741717complicatedfunctionofvelocityandconguration-spacevariables.Measuringthefast-iondistributionfunctionintheharshmagneticfusionenvironmentisamajordiagnosticchallenge.Oneapproachistoexploitchargeexchangereactionsbetweenenergeticdeuteriumionsandaninjectedneutralbeam.Collectionofescapingneutralsisthebasisofneutralparticleanalysis(NPA)[1],atechniquethathasbeenappliedtotokamakplasmasfornearlyvedecades[2].Amorerecenttechniqueistoanalyzethevisiblephotonsemittedbyhydrogenicfastionsthatneutralizeintheinjectedbeam[3].Areviewofthesefast-ionDa(FIDA)measurementswasrecentlypublished[4].BothNPAandFIDAdiagnosticsprovidevaluableinformationaboutthefast-iondis-tributionfunctionbutalsodependsensitivelyonotherplasmaparametersandonatomiccrosssections.Onewaytorelatethemeasuredsignalstotheoryistoconstructaphase-spaceweightfunctionforeachmeasurement[5];thesignalistheconvolutionofthefast-iondistributionfunctionwiththeweightfunction.Asillustratedbytheexamplesin[4],thisapproachisquiteusefulforrapidqualitativeinterpretationofthemeasurements.Itcanalsobethebasisforaninversionalgorithm.Althoughtheprocessesaretoocom-plicatedforauniqueinversion[6],aleast-squaresminimizationschemethatutilizesaweightfunctioncandeterminewhichmodeldistributionfunctionagreesbestwiththedata.AnexampleofinferenceofthedistributionfunctionfromcollectiveThomsonscat-teringdatawasrecentlypublished[7].Alternatively,onecanuseforwardmodeling.Inthisapproach,thedistributionfunc-tionisagivenquantitysuppliedbytheory.Thecodedescribedinthispaper,dubbedFIDASIM,takesthisapproach.FIDASIMacceptsatheoreticaldistributionfunctionasinputandpredictsFIDAandNPAspectraforcomparisonwiththedata.Thecodeisdesignedtocomputeactivesignalsproducedbyaninjectedneutralbeam.(Inreal-ity,collisionswithedgeneutralsalsoproduceFIDAandNPAsignalsbutthecodedoesnottreatthesepassivereactions.)Todate,thecodehasbeenusedtomodelmeasure-mentsontheDIII-DandASDEX-UpgradeconventionaltokamaksandontheNSTXandMASTsphericaltokamaks.AnearlyversionofthecodewasdescribedintheAppendixof[3].Thispaperdescribesversion3.0andisorganizedasfollows.Section2presentstheassumptionsandorganizationofthecode.Section3describesteststhatverifythatthecodecorrectlysolvesthedesiredequations.Section4explainstheoptimalselectionofnumericalparametersintermsofphysicalprocesses.Section5summarizesvalidationbyexperiment.Section6providesanoutlookforfurthertestsandimprovements.2ModelThecodehasfourmainsections(Fig.1).Therstsectionpreparesthedataandthesecondcalculatestheneutralpopulations.Thethirdandfourthsectionsbothrelyonthersttwosectionsbutareindependentofeachother.OnesectioncomputestheNPAuxandtheothercomputestheFIDAradiance. 718W.W.Heidbrinketal./Commun.Comput.Phys.,10(2011),pp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r 2 A D I A T I V E ) N 3 T A T E S T I M E / U T 3 T A T E S ) N T E N S I T Y 3 P E C T R U M ) N ! C C U M U L A T E 3 P E C T R A ) N D ) D , O O P O V E R C E L L S ) , ' $ 5 D G L D Q F H 0 D S 3 O D V P D 3 U R I L O H V ) D V W L R Q ' L V W U L E X W L R Q 0 D S ' L V W U L E X W L R Q 1 3 $ V S H F W U D . E U T R A L I Z A T I O N ! T T E N U A T I O N ' L U H F W & ; V S H F W U D + D O R V S H F W U D Figure1:FlowdiagramfortheFIDASIMcode.2.1InputdataandcoordinatemappingThecodebeginsbycollectingtheinputdata.Thegeometryofthesourceofinjectedneu-tralsisspeciedrst.Insomedevices(suchasNSTX)thedetectorsightlinesintersectseveralbeams,sothecodecanaccommodatemultiplebeamlines.ThecodeusestheconventionsoftheNUBEAMmodule[8]oftheTRANSPcode[9]todescribethegeome-tryoftheviewedneutralbeamsource(orsources).Eachtokamakhasitsownsubroutinecalled,e.g.,BEAM GEOMETRY D3D.AsinNUBEAM,theneutralbeamisdescribedbyrectangularsourceandaperturedimensionsandbyfocallengthsanddivergencesinboththehorizontalandverticaldirections.Thebeamenergy,power,andspeciesmixbetweenfull-energy,half-energy,andthird-energycomponentsarealsoinputparameters. W.W.Heidbrinketal./Commun.Comput.Phys.,10(2011),pp.716-741719Next,thecodecollectsinformationaboutthedetectorlocationsandsightlines.ForFIDA,thedetectorlocationisactuallythepositionoftheprimarylens(ormirror)ofthecollectionoptics,sinceitisthispositionthatdeterminestheDopplershiftoftheemittedradiation.ForanNPA,boththesightlinesandthesolidanglesarespecied.Informationontheequilibriumisinputusingtheso-calledeqdskformatproducedbytheEFITequilibriumcode[10].ForinstallationsthatdonotuseEFIT,apost-processorthatispartoftheTRANSPdistributioncanconvertTRANSPoutputlesintothedesiredformat.Thecoderequiresprolesofelectrondensityandtemperature,iontemperatureandtoroidalrotation,andimpuritydensityasafunctionofuxsurface.(Thesequantitiesareallassumedtobeuxfunctions.)AsubroutineexiststhatconvertsTRANSPoutputintothedesiredformat.Thenalmajorpieceofinputdataisthetheoreticalfast-iondistributionfunction,whichcanhaveacomplicateddependenceonenergyE,pitchp=vk/v,andspacer.(AsinTRANSP,positivepisdenedbythedirectionoftheplasmacurrentratherthanbythedirectionofthetoroidaleld.)Threedistinctcoordinatesystemsareutilizedintheinitialstagesofthecode(Fig.2).Thebeamanddetectorgeometriesarespeciedinright-handedCartesian(u,v,z)coor-dinateswithoriginthecenterofthetokamakandztheverticaldirection.Plasmapa-rametersareone-dimensionalfunctionsofuxcoordinates.Becauseneutralstravelin X X 8 , " 4 . ! 8 , " ! 0 ! 2 4 # % . ! & ) $ ! U & ) $ ! V B E A M A P E R T U R E . " I O N S O U R C E A L P H A ? N B X Y Z O R I G I N X & ) $ ! X & ) $ ! Y Figure2:PlanviewofNSTX.Geometricalneutralbeamanddetectorinputtothecodeisin(u,v,z)coordinates.Neutralbeamparameters(uppercaselabels)followtheTRANSPconventions.Thecodetransformsquantitiesinto(x,y,z)coordinatesalongtheselectedbeam. 720W.W.Heidbrinketal./Commun.Comput.Phys.,10(2011),pp.716-741straightlines,aright-handedCartesiangrid(x,y,z)isemployedinthemainsectionsofthecode.Thecenterlineoftheprimarybeamformsthexaxis,withtheoriginjustout-sidetheplasma.Aftercollectingalloftheneededinputdata,thecodetransforms(u,v)coordinatesinto(x,y)coordinatesandndstheuxcoordinatethatcorrespondstoeachpositioninthe(x,y,z)mesh.Thevaluesoftemperature,density,andplasmaowateachcellpositionarestoredinarrays;inaddition,severalotherquantitiesusedinthecal-culationsofcollisional-radiativetransitions(discussedbelow)arecomputedandstored.Fromtheequilibriumdata,themagneticandelectriceldsin(x,y,z)coordinatesateachcellpositionarecomputedandstored.Similarly,the(x,y,z)vectorsfromeachcelltotheFIDAlens(orlenses)andNPAdetectorsareplacedinarrays.Dependingonthesourceofthetheoreticalfast-iondistributionfunction,themappingintothe(x,y,z)coordinatescanbefairlycomplicated.Todate,distributionsproducedbytheTRANSPNUBEAM[8],ORBIT-RF[11],andCQL3D[12]codeshavebeenused.Thedesiredoutputisathree-dimensionalarrayoftheguiding-centerdistributionfunctioninthevariablesenergy,pitch,andcellnumber,F(E,p,cell).TheenergyandpitchvariableshaveuniformspacingdEanddp.ThenormalizationforFischosensothatthesumovervelocityspaceistheguiding-centerfast-iondensitynfineachcell,i.e.,åiåjF(Ei,pj)dEdp=nf.(2.1)ConversionoftheNUBEAMoutputintothisformisstraightforward,astheNUBEAMdistributionisalreadyafunctionofenergy,pitch,andanarrayof(R,z)positions(Risthemajorradius);also,apartfromafactoroftwo,thenormalizationisthesameasinEq.(2.1).Interpolationofthe(R,z)positionsontothe(x,y,z)gridcompletestheopera-tion.UseofresultsfromaMonteCarlodrift-orbitfollowingcodesuchasORBIT-RFisalsostraightforward.WhenrunningORBIT-RF,thephase-spacecoordinatesandweightsoftheorbitsaresampledfrequentlynearthetimeofinterest.Afterreadinganoutputlewiththisinformation,FIDASIMselectsaphase-spacecoordinate,thensearchesforpar-ticlesthatarewithinthephase-spacevolumeofthiscoordinate;thedesiredFissimplythesumoftheweightsofparticlesthatfallwithineachphase-spacebin.CQL3DisaFokker-Planckcode.Theoretically,thedistributionfunctionfinanax-isymmetrictokamakcanbedescribedbyjustthreecoordinates.CQL3DselectsthemajorradiusRmidandpitchanglecmidattheoutermidplanecrossingandthespeedvascoordi-natesforf.Althoughthreecoordinatessufcefortheory,theFIDAandNPAdiagnosticsrequirethedistributionfunctionthroughoutthe(x,y,z)volume.Aparticularlocationisconnectedtoitsmidplanecrossingthroughtheuxsurfacegeometry.Intherstcom-parisonsofCQL3DpredictionswithFIDAdata[5,13],zerobanana-widthorbitswereassumedbut,inarecentversionofCQL3D,themidplanecrossingisshiftedbyarstordercorrectionforthenitebananawidth.Withorwithoutthenitebananawidthcor-rection,becauseofconservationoftherstadiabaticinvariant,thepitchanglechangesasthefastionorbits.Therelationshipbetweenthepitchanglecatanarbitraryposition W.W.Heidbrinketal./Commun.Comput.Phys.,10(2011),pp.716-741721andthepitchangleattheoutermidplaneissin2c=B Bmidsin2cmid,(2.2)whereB/Bmidistheratioofthemagneticeldatthenewpositiontotheeldattheoutermidplane.UnlikeFIDASIM,whichincludestheJacobianofthevelocity-spacecoordi-natetransformationinthedenitionofF,finCQL3DdoesnotincludetheJacobian,soRRf(v,cmid)v2sin(cmid)dvdcmidgivesthefast-iondensityatthemidplane.Withthefurthertransformationfrom(v,c)coordinatesinto(E,p)coordinates,thedesireddistri-butionfunctionF(E,p)thathasJacobianfactorsincludedisFµvf.ThevaluesofFinthedesiredvariablesarefoundbyselectingaphasespacecoordinate,calculatingthecorrespondingvaluesofv,Rmidandcmid,andtheninterpolatingoverthemidplanedis-tributionfunction.2.2InjectedandhaloneutraldensitiesThesecondmajorsectionofFIDASIMisdevotedtocalculationoftheinjectedandhaloneutraldistributionsinrealspace,velocityspace,andenergylevels.Sincetheinjectedneutralspenetratethetorusinmicrosecondsandthehaloformsontherapidion-ioncol-lisiontime(10s),theseneutralpopulationsaretime-independentinthecode.Notethat,althoughthemajorityofneutralsoccupythegroundstate,itisessentialtocomputethefractionaloccupationofhigherenergylevels,asthisinuencesbeamattenuation,theprobabilitythatcharge-exchangeeventspopulatehigherenergylevelsand,ultimately,thefractionofneutralsthatundergothedetectedBalmeralpharadiativetransition.Intreatingtheatomicphysicsoftheneutrals,threeimportantsimplicationsarepermis-sible.First,insomeportionsofthecode,theprobabilityofareaction1 exp( x/)isapproximatedbyx/iftherelevantmean-freepathislongerthantheevaluatedsteplengthx.Second,althoughtherearemanypossibleprinciplequantumnumbersnandangularmomentumstateslavailabletotheneutrals,thestrongne-structuremixingallowstheassumptionthatthepopulationofeachquantumstatemaybegroupedasasinglepopulationbasedontheprinciplequantumnumber[14].(Evenifthisassumptionisnotstrictlyvalid,theeffectisprimarilyonthepolarizationandStarkshiftofemittedradiation,nottheoverallintensity.Theeffectonneutralizationprobabilitiesshouldalsobesmall.)Withthisassumption,theequationsthatgovernthepopulationsNjofneutralsinthedifferentenergylevelsjarewritteninmatrixformasdNj dt=åkNkMkj,(2.3)whereMkjisamatrixofratesgoverningthetransitionsfromlevelsktoj.Throughoutthecode,theNjrepresentfractionaloccupationaldensities,soåjNj1.(Whentrackinganeutral,thesummaydecreasebelowunityduetoionizationandchargeexchangelosses.) 722W.W.Heidbrinketal./Commun.Comput.Phys.,10(2011),pp.716-741RadiativeandcollisionalprocessesarebothincludedinM.Theradiativetransitionratesareindependentofplasmaparameters,sotheseEinsteincoefcients[15]areloadedintoanarrayintheinitializationstageofthecode.Incontrast,thecollisionalratesdependonplasmaparameters.Thethirdmajorsimplicationinthetreatmentofthecollisional-radiativetransitionsistoassumethatthespeedsofthevariousspeciesfollowtheorderingve vfvnvi vI,wherethesubscriptsrepresentelectrons,fastions,hydrogenicneutrals(bothfastandthermal),thermalhydrogenicions,andimpurityions,respectively.SincetheelectrondistributionfunctionisMaxwellianandtheelectronthermalspeedismuchgreaterthanthefastestneutrals,itisexpedienttoworkdirectlywiththereactivitieshsviforelectroncollisionswithneutrals.Duringinitialization,theprogramcomputesandstorestheelec-tronreactivitiesineachcell.Hydrogenicratesareevaluatedusingtherelativevelocitybetweentheionandtheneutral,jvi vnj,whereviandvnaretheionandneutralveloc-ities,respectively.Forthecaseofcollisionsofthefast-ionpopulationwithaneutral,virepresentsthefast-ionvelocity.Forcollisionsbetweenthethermal-ionpopulationandtheneutrals,itisnecessarytoaveragethereactivityovertheiondistributionfunction,whichisassumedtobeadriftedMaxwellianwithtemperatureTiandtoroidalrotationvelocityvrot.Withtheassumptionthattheimpurityspeedisnegligiblecomparedtothespeedofahydrogenicneutral,collisionswithimpuritiesonlydependontheneu-tralspeedvn.Thecurrentversionofthecodetreatsfully-strippedcarbonastheonlyimpurityspecies.Combiningthethreespecies,atypicalmatrixelementM12(inthiscase,thematrixelementforexcitationfromthegroundstatetothen=2state)isM12=nehsvicoll,e12+ndhsvicoll,d12+nCscoll,C12vn,wherene,nd,andnCaretheelectron,deuteron,andcarbondensities.Thedeuteriumdensityisnotdirectlymeasuredsoitisinferredusingquasineutrality:nd=ne 6nC.Thereisasubtletyassociatedwiththedeuteriumdensity,however.Inthecoreoflowdensityplasmaswithlargefast-ionpopulations,thefast-iondensitynfcanbecomparabletothethermaldeuteriumdensity.Inprinciple,becausethetwopopulationshavequitedifferentvelocitydistributions,theinteractionofneutralswiththesetwopopulationsshouldbecalculatedseparately.Unfortunately,acorrecttreatmentofneutralcollisionswithfastionsisquitecomplicated,requiringcalculationssimilartothoseinthenalandmosttime-consumingportionofthecode.Asanalternative,inthedepositionoftheinjectedneutrals,thecodeignoresthedistinctionbetweenfastandthermaldeuterons,effectivelyapproximatingthefast-ionstoppingcrosssectionbythethermal-ioncrosssection.Forallspecies,deexcitationratesarederivedfromtheprincipleofdetailedbalance,i.e.,hsviu!l=n2l n2uhsvil!u, W.W.Heidbrinketal./Commun.Comput.Phys.,10(2011),pp.716-741723whereuandlrepresenttheupperandlowerquantumnumbers,respectively.WiththematrixelementsMkjdened,thecodeproceedstothecalculationoftheneutralpopulationsassociatedwiththeinjectedbeamorbeams.Thiscalculationutilizesthreesubroutinesthatareemployedthroughouttheremainingportionsofthecode.Onebasicsubroutine(calledTRACK)calculatesthetrajectoryofaneutralthroughtheCarte-siangrid,returningthelengthofthetrackineachcell.Asecondsubroutine(calledCOLRAD)solvesthetime-dependentcollisional-radiativeequations(Eq.(2.3))fortheneutraldensitiesineachstateNj;thenumberofBalmer-alpharadiativetransitionsisalsocomputed.Athirdsubroutine(calledSPECTRUM)calculatestheStarkandDopplershiftsofemittedphotonsgiventhelocalelectricandmagneticelds,thevelocityoftheneutral,andthedirectionofthephoton.(Thedetectorisassumedtomeasureallemittedpolarizations.)Zeemansplittingisnegligible.TheDoppler-shiftedwavelengthis=0r 1 v 21+vk 1,(2.4)where0istherestwavelength,visthespeedoftheneutral,isthespeedoflight,andvkisthecomponentoftheneutralvelocityinthedirectionoftheemittedphoton.Theelectriceldintheneutralframe,E=Elab+vB,splitstheDopplershiftedlinethroughtheStarkeffect.Forastatisticallpopulation,therelativeintensitiesandshiftsoftheninespectrallinesaregivenin[16].Usingtheknownbeamgeometry,aMonteCarloprocedurelaunchesraysfromran-dompositionsonthesourcewithaGaussiandistributionofvelocitiesderivedfromthespeciedbeamdivergence.Raysthatclearthebeamaperturearefollowedintotheplasma.Thefull,half,andthirdenergycomponentsareassumedtofollowthesametrajectoriesbutthesethreepopulationsaretreatedseparatelyastheyprogressintotheplasma.Allneutralsareemittedinthegroundstate.Throughsolutionofthecollisional-radiativeequations,thedensitiesandvelocitiesofthefull,half,andthirdneutrals(eachasafunctionofenergyleveln)areaddedtothestructurethatdescribeseachcell.COL-RADalsocomputestheprobabilityofBalmer-alphaemissionforeachneutralrayandSPECTRUMcalculatesthespectraproducedbyeachbeamcomponentineachcell;thisbeam-emissionlightfromeachcellisaccumulatedandstored.Astheinjectedbeamattenuates,thecodekeepstrackofcharge-exchangeeventswiththedeuteriumpopulation.Theseeventsarethesourceofhaloneutrals.ThefastneutralsthatproduceFIDAlightandNPAsignalsarenotcountedashaloneutrals,sothecharge-exchangesourcerateismultipliedbynd/(nf+nd)toobtainthehalosourcerate.(Conse-quently,thecodeneglectsfastioncollisionswithsecond-generationfastneutrals.)Fromtheknownsource,thecodecomputesthecloudofhaloneutralsthatsurroundeachin-jectedbeam.Theinitialvelocityisrandomlyselectedbasedonthelocaliontemperatureandrotation.Becausetheinjectedneutralshavelargevelocitiesandthecharge-exchangecrosssectionisastrongfunctionofrelativeenergy,thetruedistributionofinitialveloc-itiesisskewedrelativetothethermal-iondistribution,butthiseffectisneglectedinthecurrentversionofthecode.(Accordingto[14],thedeviationassociatedwiththisapprox- 724W.W.Heidbrinketal./Commun.Comput.Phys.,10(2011),pp.716-741imationissmall.)Thisneutralisthenfollowedthroughthecells.Ifachargeexchangeeventhappensinacell,theparticleisrestartedwithanewrandomvelocitybasedonthelocaliontemperatureandrotation.Theneutralisfolloweduntilitionizes.Toob-tainexponentialprobabilitydistributions[17]forthechargeexchangeandionizationevents,thecodecomputestheratioofthetracklengthinthecelltothemean-freepath,r=ndsCXDl.Ifr�ln(1/h),wherehisauniformrandomlygeneratednumberontheinter-val[0,1],acharge-exchangeeventoccursinthecell.Asimilarcomparisonisperformedforionization.Forsimplicity,thisspatialdiffusioniscomputedassumingthatalloftheneutralsareinthegroundstate.(Fortypicalparameters,thisistruefor99%ofthehaloneutrals,sothisisanexcellentapproximation.)Therelativelyslowtimescaleofhalofor-mationimpliesthattheenergyoccupationlevelscanbeapproximatedbythesteady-statecollisional-radiativebalance,sothisiscomputedaftercompletionofthespatialdiffusioncalculation.Thehalodensities(asafunctionofn)foreachcellarethenstored.Thecodealsopredictsthespectrumproducedbythehaloneutrals.Thermalionsthatchargeexchangewithaninjectedneutralareinitiallyfarfromcollisional-radiativeequilibriumbutsubsequentgenerationsofhaloneutralsrelaxtowardequilibriumoccu-pancylevels,sorst-generationthermalneutralsaretreatedseparatelyfromdaughterhaloneutrals.Therst-generationordirectcharge-exchangelightiscalculatedinthesamemannerastheFIDAlightandisdescribedbelow.Forthedaughterhaloneutrals,accordingtoMandl[18],Starksplittinganddistortionsofthespectrumassociatedwiththeenergydependenceofthecrosssectionareminoreffects,sotheseeffectsareignored.ThedistributionoflightissimplyashiftedMaxwellian,g(vk)dvkµexph (vk vk,rot)2 v2tii,(2.5)wherevkandvk,rotarethecomponentsoftheemittedlightandplasmarotationinthedi-rectionofthelens,respectively,andvtiistheionthermalvelocity.TheassociatedDopplershiftrelativetotherestwavelengthisD/0'vk/.TherateofemissionfromagivencellisnHaloN3A32,wherenHaloisthehalodensityaftersubtractionofrst-generationhalos,N3isthefractionofthehalopopulationinthen=3state,andA32istheEinsteincoefcientfortheBalmer-alphatransition.2.3NPAuxTheNPAuxisfoundinthethirdstageofthecode.Owingtothelargemassdiffer-encebetweenelectronsandionsandtothesmallenergyexchangeincharge-exchangereactions,theangulardeectionassociatedwithcharge-exchangereactionsis1andisignoredthroughoutthecode.Consequently,thepitchpoftheescapingneutralisde-terminedentirelybytheNPAsightlinegeometry.TheuxofneutralswithenergyEiincidentuponacollimatedNPAofareaAandsolidangleDWisF(Ei)=ZF(Ei,p;r,)nn(r)scx(Ei,nrel)Vi,nrelDW 4pe lAdl(s 1),(2.6) W.W.Heidbrinketal./Commun.Comput.Phys.,10(2011),pp.716-741725whereFisthefast-iondistributionfunction,nnistheneutraldensity,scxisthecharge-exchangecrosssection,Vi,nrelistherelativevelocitybetweenthefast-ionandtheneutral,e lrepresentsreionizationoftheneutrals,anddlisthedifferentiallengthalongthesight-line.EvaluationofEq.(2.6)isstraightforwardbuttherearesubtleties.Thefast-ionveloc-ityvectorisdeterminedbyEiandthedetectorgeometry,sotherelativevelocityisreadilycomputed,andhencennscxVrelforeachoftheneutralspecies.Sincethebeamdivergenceissmall,therelativevelocityforthefull,half,andthirdcomponentsisapproximatedusingthenominalvelocityvectorfortheinjectedneutrals.Forthehaloneutrals,thereactivityisaveragedoveradriftedMaxwellian.Thedistributionfunctionthatisstoredintherststageofthecodeisthedistributionofguidingcenters.Thedetectorgeometrydeterminesthediagnosticviewingconebutthedetectedparticleshaveguidingcentersthatareagyroradiusfromtheviewingcone.Withthevelocity~vknownfromtheviewinggeometry,thegyroradius~ris~r=(b~v) Wci,(2.7)wherebistheunitvectorinthedirectionofthemagneticeldandWciisthecyclotronfrequency.ThedistributionfunctionusedinEq.(2.6)isevaluatedattheguidingcenterposition.TheattenuationofneutralsiscomputedinthelastpartoftheNPAcalculation.Thecodendsthecellinthedetectorsightlinethatisfarthestfromthedetector,thencom-putestheattenuationofneutralsfromthatpointallalongtheneutralpathforseveralenergies.(Thiscalculationisessentiallythesameasthedepositioncalculationforthein-jectedneutrals.)Withthetermsontheright-handsideofEq.(2.6)known,thepredictedspectrumforeachdetectorsightlineisfoundbysummingovertheviewedcells.2.4FIDAradianceThefourthstageofthecodeusesaweightedMonteCarloroutinetocalculatetheFIDAradiance.Thefast-iondensitynfandthesumofinjectedneutralandhaloneutralden-sitiesånnhavealreadybeencalculatedasafunctionofposition.Theproductnfånnprovidesaconvenientestimateoftheprobabilityofachargeexchangereaction(thatneglectsthecomputationallyintensivedependenceofthereactionrateontherelativeve-locity),sothisproductisusedtodeterminehowmanyfastneutralstolaunchfromeachcell.Theinitialpositionofthefastneutralwithinthecellisselectedrandomly.TheinitialvelocityisfoundusingaMonteCarlorejectiontestinthetwodimensionsthatdescribethevelocitydistribution(energyandpitch).Thegyroangleisrandomlygenerated,theinitialpositionofthefastionisshiftedbyagyroradius(Eq.(2.7)),andthevelocityvectoristransformedinto(x,y,z)Cartesiancoordinates.Withthevelocitynowspecied,theactualreactionrateofthefastionwitheachoftheneutralpopulationscanbecomputed;thesumoftheseratesistheweightofthisparticularfastneutral.(Infact,eachindividualfastneutralrepresentsabundleoftheentiresetofneutralnstatesalongtheselected 726W.W.Heidbrinketal./Commun.Comput.Phys.,10(2011),pp.716-741path.)Next,thetrajectoryofthefastneutralthroughthecellsiscomputedbyTRACK.Asthefastneutraltravelsthrougheachcell,thetime-dependentcollisional-radiativetran-sitionsbetweenstates(Eq.(2.3))iscomputedbyCOLRAD,includingthenumberofDaphotonsthatareemitted.Withthevelocityoftheneutralknown,thespectrumoftheemittedphotonsineachcelliscomputedbySPECTRUM.Finally,theproperlyweightedspectrumisaddedtotheaccumulatedspectraineachcell.Thecalculationofthedirectcharge-exchangelightfromrst-generationthermalneu-tralsispatternedaftertheFIDAcalculationwithtwodifferences.Sincethehalo-lightcalculationalreadytreatslightassociatedwithmultiplecharge-exchangeevents,onlytheinjectedneutraldensityisusedinthecalculationofthecharge-exchangeprobability.Thefast-iondistributionfunctionisreplacedbyadriftedMaxwellianwithdensitynd.TheoutputoftheFIDA,beam-emission,directcharge-exchange,andhalolightarestoredintwoforms.Atthemostbasiclevel,theoutputistheemissivityeiinthedirectionofthelensfromeachcellinunitsofphotons/cm3/s/nm.Thecodealsoperformsasimpleintegrationovertheseemissivitieseialongthespeciedsightlines.Forthis,TRACKcomputesthelengthofthesightlineineachcelldliandthecodesumsåieidlitoobtaintheradianceforeachviewchordinunitsofphotons/cm2/s/nm.Moresophisticatedmanipulationoftheoutputofthecodeisstraightforward.Thecomputedradiancetreatseachviewchordasaninnitely-narrowpencilbeam.Inreal-ity,actualchordshavenitetransverseextentatthefocalplaneandevenbroaderspatialextentawayfromthefocalplane.Onecaneasilyimplementmoreaccurateintegrationoverthestoredemissivitiestomodeltheactualoptics.Tomodeldatafromanimagingcameraacquiredwithanarrowbandlter[19],apost-processorusesTRACKtointe-grateoverthethousandsofcamerasightlinesandsumsthespectraoverthepassbandofthelter.Optionally,thispost-processorperformssimilarcalculationsforthehaloandinjected-neutrallight.Notethatthecomputedspectraneglectinstrumentalbroadening.Forcomparisonwithexperiment,thetheoreticalspectraareconvolvedwiththeinstrumentfunction.3Verication3.1AtomicphysicsAtomicphysicscrosssectionsareimportantintwoplacesinthecode:inthecalculationofneutralizationprobabilityandinthesolutionofEq.(2.3)inCOLRAD.TherequiredcrosssectionsandreactivitiesareavailableintheliteratureandintheAtomicDataandAnal-ysisStructure(ADAS)compilation[20,21].ForCOLRAD,Eqs.9and10of[22]givethecrosssectionforprotonexcitationandimpactionizationfromthegroundstate,while[23]containscrosssectionsforexcitationfromhigherstates.ExpressionsforelectronimpactionizationasafunctionofelectrontemperatureTeandenergylevelnappearin[24].For-mulasforelectronexcitationfromoneenergyleveltoanotherarein[25].ImpuritycrosssectionsarelistedinEqs.13-16of[22].Analternativecompilationofmanyoftherates W.W.Heidbrinketal./Commun.Comput.Phys.,10(2011),pp.716-741727 & 2 ! # 4 ) / . ! , / # # 5 0 ! 4 ) / . N N N N Figure3:SolutionofEq.(2.3)immediatelyfollowingneutralizationofthefastionforatypicalcase(densityof41013cm 3,temperatureof4keV).Thetotaltimerepresentedbytheabscissais11.1ns;thefastneutraltravels3cmduringthistime.Theevolutionofthen=3statedeterminestheintensityofFIDAlight.appearsin[26]buttheeffectofthesedifferencesissmallcomparedtootheruncertaintiessothecurrentversionofthecodeusestheoldercompilationbyJanev[24].InCOLRAD,ratesforenergylevelsupton=7arenormallyemployed.Fortheinitialneutralizationprobability,crosssectionsforthecharge-exchangereac-tionsbetweenfastionsandneutralsinstatesn=1 4aregiveninADAS[20].Charge-exchangereactionstostateswithn�4areneglectedinourcalculationsbecausetheseenergylevelsaresparselypopulatedandthecrosssectionsseemuncertain.ThecalculationsinCOLRADarethemosttimeconsumingportionofthecode.Fig.3showsatypicalsolutionfollowingneutralizationofafastion.Althoughonlyasmallfractionoftheinjectedneutralsoccupyhigherenergylevels,thecrosssectionforchargeexchangebetweenexcitedstatesisseveralorderofmagnitudeslargerthanthecrosssec-tionforchargeexchangefromthegroundstatetoanexcitedstate.Consequently,theinitialconditionsforthecollisional-radiativeequations(Eq.(2.3))arefarfromequilib-rium.Forexample,forthecaseillustratedinFig.3,thesteady-stateoccupationlevelsfortheinjectedbeamareall 1%forn�1but,whenmultipliedbythecrosssection,theinitialoccupationfractionsfortheN2,N3,andN4statesare10,3,and1%,respectively.Rapiddecayoftheexcitedstatesoccursintherstfewcells.Asaresult,itisnecessarytosolvethefullsystemofequations;areducedsetofequations,suchasthoseproposedinEqs.2-7of[15],areinaccuratewhentheinitialn=3occupationfractionN3ishigherthanthenaloccupationfraction.Afourth-orderRunge-Kuttaroutineisadoptedinthecode.AssuggestedbyHutchinson[15],thecodeselectsthetimestepbasedonthesmallerofthematrixelementtimescales,tscale=min(M11 M22) 1,(M11 M33) 1.(3.1)Usually,anaccuracyof1%inN3isobtainedwithatimestepoftscale/4but,ifthesolu-tionisunphysical(e.g.,negativedensitiesoråiNi&#x-2.4;衸1),thesolutionisrecomputedwithhalfthetimestep.Onthebasisofextensivetests,thecodeordinarilyusessevenenergy 728W.W.Heidbrinketal./Commun.Comput.Phys.,10(2011),pp.716-741levels.Surprisingly,onaverage,theuseof7energylevelsonlytakes7%longerthancal-culationswith4energylevelsbuttheaccuracyisimprovedappreciably.(FewerenergylevelsunderestimatesN3by10%.)AsHutchinsonnotes[15],accuracyisimprovedbyadjustingthecoefcientofthehighestretaineddiagonalmatrixelementsothatelectronsintheuppermostmodeledstateareimmediatelylost.Alternatively,additionalenergylevelscanberetainedsothattheuppermoststatesareverysparselypopulated;thisistheapproachadoptedinFIDASIM.ThenumberofDatransitionsisproportionaltotheintegralofN3overthetimeinthecell;thisisquicklycomputedwithadequateaccuracyusinganextendedtrapezoidalrulesummation.COLRADiscalledeachtimeaneutralentersanewcell.Thecollisional-radiativematrixislledatthebeginningofeachcall.Collisionrateswithhydrogenicandim-purityspeciesmustbereevaluatedbasedonthecurrentneutralvelocity.Ratherthanrecomputingtheseratesateachcall(particularythetime-consumingintegrationovertheMaxwellianthermaldistribution),thecodeutilizespre-computedlook-uptablesoftheratesversusneutralenergyandiontemperature.(Inregionswheresvchangesrapidly,thelook-uptablepresentlyusesstepsof0.25and1.0keVintemperatureandfast-ionenergy,respectively.)TheaccuracyofCOLRADwasveriedinseveralsteps.ThematrixelementsMagreetowithinafewpercentwiththevaluesinTable1of[15].(Asmentionedearlier,anexceptionisthediagonalelementsathighn.)Thecodecorrectlyreproducesthesteady-statefractionsinFig.1of[15].3.2BeamdepositionThecalculationoftheinjectedneutraldensitywascomparedwithTRANSPforanNSTXcase.Theneutralprolesalongtheyaxis(perpendiculartotheneutralbeamsource)andalongthezaxis(verticaltothesource)axisagreewellwiththeTRANSPsimulationresults(Fig.4)andwithneutralbeamcalibrationdata[27].TheattenuationofthebeamisalsoinreasonableagreementwithTRANSPsimulations.AsummaryofoneofthesecomparisonsisshowninFig.5.Fig.5(a)showstheneutraldensityalonganNPAsight-linefromTRANSPandFIDASIMsimulations.Whenhaloneutralsarenotconsidered,bothcodesgiveverysimilarneutraldensity.Thegurealsoshowsthatthehaloneutraldensitycanbecomparabletotheinjectedneutraldensitysothatthetotalneutralden-sityalmostdoubleswhenhaloneutralsareincludedinFIDASIM.(ThecurrentversionofTRANSPredistributeshaloneutralsovertheentireplasmavolume.ThisapproachsufcesforpowerbalancecalculationsbutisinaccurateforNPAandFIDAsimulations.)Fig.5(b)showstheattenuationfactorsfor60keVneutralsalonganNPAsightlinefortheTRANSPandFIDASIMsimulations.Thesmalldiscrepancybetweenthetwocurvesisprobablycausedbytheuseofdifferentcrosssectionsorpossiblybytheinclusionofmulti-stepionizationinFIDASIM.CalculationsoftheattenuationofinjectedneutralscomputedbyTRANSP,FIDASIM,andapencilbeamcode[28]showsimilardiscrepan-cies;ingeneral,FIDASIMpredictsslightlyfasterattenuationthantheothertwocodes, W.W.Heidbrinketal./Commun.Comput.Phys.,10(2011),pp.716-741729 \r \r 6 ( & ) $ ! S I M 4 2 ! . 3 0 ( 6 T S Figure4:Comparisonofinjectedneutraldensitypro lealongtheyandzaxesfromtheFIDASIMcode(symbols)andfromTRANSP(lines)atthebeamtangencypoint(R=70cm)forNSTXSourceA. $ I S T A N C E F R O M T H E . 0 ! P I V O T C M W O H A L O S 4 2 ! . 3 0 W O H A L O S & ) $ ! S I M W H A L O S & ) $ ! S I M 4 2 ! . 3 0 & ) $ ! S I M W O H A L O S 4 2 ! . 3 0 W O H A L O S & ) $ ! S I M W H A L O S & ) $ ! S I M Figure5:ComparisonbetweentheFIDASIMandTRANSPsimulationsinNSTXdischarge#122631att=0.1s.(a)Neutraldensity,(b)attenuationfactorfor60keVneutralsand(c)di erentialcharge-exchange\ruxalongtheNPAsightline.whichmaybeassociatedwithamoreaccuratetreatmentofmultistepionization.ComparisonsoftheNPAcalculationwithTRANSPareshowninFig.5(c)andFig.6.Fig.5(c)showsthedifferentialcontributiontothe60keVcharge-exchangeefuxalonganNPAsightlineforanNSTXcase.Withhaloneutralsneglected,theagreementissat-isfactory.Fig.6comparesspectraafterintegrationoverthesightline.Atlowdensity, 730W.W.Heidbrinketal./Commun.Comput.Phys.,10(2011),pp.716-741 W O H A L O S 4 2 ! . 3 0 W O H A L O S & ) $ ! S I M W H A L O S & ) $ ! S I M Figure6:NPAenergyspectraforasightlinewithRtan=70cmfromtheTRANSPandFIDASIMsimulationswithandwithouthalosatthreedi erenttimesinNSTXdischarge#122631.Thedensityincreasedfromapproximately21013cm 3to51013cm 3between0.1and0.5s.theagreementissatisfactorywhenhaloneutralsareneglectedinFIDASIM.Athigherdensities,thespectralshapesremainsimilarbutFIDASIMpredictssmallersignalsthanTRANSP;thisisaconsequenceofthelargerattenuationpredictedbyFIDASIM.3.3HaloneutralsThehaloneutralsimulationportionofthecodewasveriedbycomparingwithaone-dimensionaldiffusionmodel.Ifweassume:(1)auniformplasmawithacircularcross-sectionneutral-beaminjectionpatternand(2)nobeamattenuationalongtheneutral-beamcenterline,thehaloneutralswilldiffuseonlyintheradialdirectionandthedensitycanbedeterminedfromthefollowingsimple1-Ddiffusionmodel[29],D¶ r¶rr¶nh ¶r=nhnehsviei 3åk=1ninb,k(r)hsvicx,k,(3.2)whereni,ne,nb,k,andnharetheion,electron,kthcomponentofinjectedneutrals,andhalodensities,respectively,hsvieiandhsvicx,karetheelectron-impactandcharge-exchangereactivities,D=Ti/mdgcxisthediffusioncoefcientandgcxisestimatedbythemostprobablenihsvicxbasedontheiontemperature.TheleftmostterminEq.(3.2)represents W.W.Heidbrinketal./Commun.Comput.Phys.,10(2011),pp.716-741731thehaloneutraldiffusion.Thetermnhnehsvieiisthehalodecaytermduetocollisionswithelectronsandtherightmosttermisthehaloformationtermduetochargeexchangereactionsbetweentheprimarybeamneutralsandthermalions.Thehaloneutrallossesthroughimpurityimpactionizationandimpuritychargeexchangearenotincludedinthissimplediffusionmodel,buttheyareincludedinFIDASIMandtheyhaveaminoreffect(lessthan5%)onhaloneutraldensity.Eq.(3.2)isaninhomogeneousmodiedBesselequation.Itisnoteasilysolvedanalyticallyforaspatiallyvaryingbeamneutralprolebutasolutionexistsforaconstantsource.Totestthehalocalculation,theradialdirectionisdividedintomanyregionsintheFIDASIMprogramandplasmaparameterssuchasthedensitiesareconstantineachregion.Thenthehaloneutraldensitysolutionsfortheseregionscanbegenerallyexpressedasnh(r)I=C1I0(r)+gI b,(3.3a)nh(r)II=C2I0(r)+E2K0(r)+gII b,,(3.3b)nh(r)n 1=Cn 1I0(r)+En 1K0(r)+gn 1 b,(3.3c)nh(r)n=EnK0(r),(3.3d)whereI0andK0aremodiedBesselfunctions,=p nehsviei/D,gjisthesourcetermforeachregion,andb=nehsvieiisthelossterm.TheconstantcoefcientsCjandEjarealgebraicallydeterminedbymatchingthedensitiesfromdifferentregionsusingcontinu-ityofnHanditsrstderivativednH/dr.Thehaloneutraldensityinthewholeregioncanbeobtainedbycombiningalloftheseanalyticalsolutions.Fig.7showsthehaloneu- 5 D G L X V U H O D W L Y H W R 1 % F H Q W H O L Q H F P \ W L V Q H G O D U W X H 1 P F ) X O O O I 7 K L U G + D O R ' G L I X V L R Q + D O R ) , ' $ V L P Figure7:Comparisonofhaloneutraldensitiescalculatedfromthe1-Ddi usionmodel(Eq.(3.3))andtheFIDASIMhalodi usionsubroutineforplasmapro leswithni=ne=1.11014cm 3,Te=2.5keVandTi=1.25keV.Thethreedashedstepsrepresentthedensitiesofthefull,half,andthirdenergycomponentsofinjectedneutralsusedinbothcodes. 732W.W.Heidbrinketal./Commun.Comput.Phys.,10(2011),pp.716-741traldensitiescalculatedfromthediffusionmodelandtheMonte-Carlohalosimulationsubroutineforidenticalplasmaproles.Theagreementisgood.3.4FIDAspectraTheSPECTRUMsubroutineusedtocomputetheDaspectraandtheweightedMonteCarloschemewereveriedasfollows.AspartoftheinitialinvestigationofthefeasibilityofFIDA,asimpliedmodeloftheexpectedspectrawasdevelopedthatignoresatomicphysicsandassumesthatthemagneticeldispurelytoroidal;Fig.2of[3]showsaresultcalculatedbythiscode.TotestthemainFIDAsimulationloop,wereplacedthemagneticeldwithatoroidaleldandmodiedthecrosssectionstobeindependentofvelocity.Theresultingspectrawereconsistentwiththeoutputofthesimplemodel.4NumericsNumericalinputparametersaffectboththeaccuracyandcomputationalexpenseofacalculation.Uncertaintiesinplasmaparameters(particularlyelectrondensity)introduceuncertaintiesinthepredictedradianceorefuxof20%ormore(AppendixAof[30]),soextremelynegridsarewastefulandunnecessary.Moreover,uncertaintiesinatomiccrosssectionsalsointroduceconsiderableuncertaintiesinthepredictions.Thissectionshowsexamplesofthesensitivityofthepredictedradiancetonumericalinputparam-etersandprovidesrecommendationsforthesechoicesintermsoftherelevantphysi-calprocesses.Asin[30],weagainstudythecaseofanMHD-quiescentDIII-Dplasma(shot122060att=2.05s)withgoodagreementbetweentheoryandexperimentfortheFIDAspectrum.Theneutral-beaminjectionenergyis80keVinthisplasma.Thepreviousstudyexploredthedependenceofthepredictionsonuncertaintiesinplasmaparameters.Here,weexplorethedependenceonnumericalinputparameters.Notsurprisingly,thetotalnumberofgridcellsandthenumberofbeamMonteCarlo(MC)particlesaffectthecomputationaltimethemost.TheoriginalversionofthecodewaswrittenintheInteractiveDataLanguage(IDL).Recently,thefourthportionofthecode,theMonteCarlocalculationoftheFIDAspectra,wasconvertedintoFortran90.TheFortranversionofthecodeisanorderofmagnitudefasterthantheIDLversion.FortheIDLversion,onafairlymodernDual-CoreAMDOpteronProcessorrunningat2.8GHz,thewallclocktimespentperneutralis10ms.ForatypicalFIDAsimulationwith107reneutralsthistranslatestoabout28hours.About35%ofthattimeisspentsolvingcollisional-radiativeequationsinCOLRAD;calculationoftheneutralizationprobabilityisalsotime-consuming.ForastandardDIII-DFIDAgridsizechoiceoffnx,ny,nzg=f31,21,21g,thepreparatorystepsperformedinthersttwostagesofthecodetakesabout4additionalhours.Thehaloneutraldensitycalculationandthemappingofthebeamdistributionfunctiontothe(x,y,z)gridtake1houreach. W.W.Heidbrinketal./Commun.Comput.Phys.,10(2011),pp.716-7417334.1NumberofMonteCarloreneutralslaunchedThereisalineardependencebetweenthenumberoflaunchedneutralizedfastions(orreneutrals)andthecomputationaltimeinthefourthsectionofthecode.Fig.8(a)com-paresthespectraatthelocationofpeakDaemissivity(R=187.5cm)forseveralsimu-lationswithvaryingnumberoflaunchedreneutrals.Toeliminatetheinuenceoftherandomnumberseedchoice,thesameseedvaluewasusedinallvesimulations.Othersimulationswithrandomseedand107reneutralshaveshownthattheMCnoiseis5%,forspectrawithEl30keV.(Eachwavelengthisassociatedwithanequivalentenergyalongtheline-of-sightknownasEl;sincetheDopplershiftonlymeasuresonecompo-nentofthevelocity,ElistheminimumenergyofthereneutralsthatproduceaparticularDopplershift[30].)NumericalMCnoiseisproportionalto1/p N,whereNisthenumberofparticlesinthesimulation.Thecomputationallymostexpensivesimulationinourstudyused9107 650 652 654 656 658 660 662 1 2 3 4 5 650 652 654 656 658 660 662 Wavelength (nm) 0.6 0.8 1.0 1.2 1.4 (b) R=187.5 cm -30keV 30keV photons / cm / s / nm2x 1013 MC ptcls. ( 10 40 1.0 2.5 10keV -10keV Figure8:(a)ComparisonofFIDAspectrafromsim-ulationswith106to9107MonteCarloparticles.(ThereductioninFIDAlightforsmallDopplershiftsisanartifactcausedbytruncatingthefast-iondistri-butionfunctionatEmin=10keV.)(b)FIDAspectranormalizedtothespectrumcomputedwith90millionparticles.ThedashedverticallinesrelateDopplershiftstoequivalentenergiesalongtheline-of-sightEl. % O X H V K L I W H G ) , ' $ V S H F W U D S U R I L O H V ( ! N H 9 L 5 F P . $ Q U D M T C \n P H O T O N S C M P H O T O N S C M Figure9:ComparisonofFIDApro lesforsimula-tionswith106to9107MonteCarloparticles.ThespectraareintegratedoverDopplershiftslargerthan(a)El10keVand(b)El30keV. 734W.W.Heidbrinketal./Commun.Comput.Phys.,10(2011),pp.716-741reneutrals.Consideringtheresultingspectraasthemostaccurate,itisinstructivetocomparetheratiosofthespectrafromsimulationswithlessparticlestothespectrafromthisparticularsimulation(Fig.8(b)).Asexpected,thesimulationwith4107reneutralsisessentiallyindistinguishablefromthatwith9107reneutrals.TheagreementgetsprogressivelyworseasthenumberofMCparticlesinthesimulationdecreases,especiallyforwavelengthswithhighredandblueshifts,becauseonlyasmallfractionofthebeamdistributionfunctionhasenergiescloseto80keV;moreover,thelikelihoodofanenergeticn=3neutralmovingdirectlyalongtheFIDAline-of-sightisquitesmall.FIDAspectrafromseveralradialpositionsareoftenintegratedoveraspecicwave-lengthrangetoobtainspatialproles.Fig.9showsthecalculatedprolesfordifferentnumbersofMCparticles.IfthespectralintegrationisfromEl=10keVandabove,goodspatialprolesareobtainedwithjust106reneutralsbutmoreparticlesareneededifthelowerlimitofspectralintegrationisEl=30keV.Therefore,twofactorsdeterminethechoiceofthenumberofreneutralsthatthecodeneedstolaunch:theintegratingspec-tralbandofthediagnostics,andtheenergyrangewherethebeaminteractionofinteresttakesplace.ForaFIDAdiagnosticthatstartsintegratingclosetothethermalionener-gies,simulationswith106particlesaresufcient.However,iftheintegrationstartsathalfthebeaminjectionenergy,3timesmoreparticlesareneeded.Resolvingnespec-traldetailsclosetotheinjectionenergyrequiresanadditionalorderofmagnitudemoreparticles.4.2SimulationvolumeThecomputationalgridmustenclosemostoftheinteractingbeamionsandneutralpar-ticles.Fig.10illustratesthegeometryofthebeamlinethatinjects2.5MWof80keVdeu-teriumneutralsintotheDIII-Dplasmaunderconsideration.TenverticalFIDAsightlinescollectradiationalongthebeamx-axis.Thefast-iondensitycalculatedbyTRANSPandthehaloandbeamneutraldensitiescalculatedbyFIDASIMareshown.Thehaloandbeamneutraldensitiesarecomparableinmagnitude,whilethefast-iondensityisthreeordersofmagnitudehigherbecauseintokamaksthefastionsareverywellconned.Thedashedrectanglewithx-ydimensionsof120cmby60cmrepresentsthestandardhorizontalgridsizeinoursimulations.TheneutralscontourplotsshowninFig.10arefromsimulationsusingx-ygridsizeof160120cm.Thecloudofhaloneutralsisblowninthedirectionofplasmarotation.Clearly,the12060cmhorizontalgridsizetruncatessomehaloneutralsbuttheeffectontheFIDAprolesisinsignicant.Furtherreductionofthegridinthex-directionto90cmtruncatesasizablefractionofthehaloneutralsbuttheeffectonthecalculatedspectraisstillsmallbecauseDalightduetochargeexchangewiththeseneutralsrarelyreachestheverticallypositionedFIDAdetectors.Forvertically-viewingFIDAchords,thespatialextentintheverticaldirectionismostimportant.Theneutralbeamshavelowdivergenceand,apartfromsteadyattenuationalongthex-axis,theinjectedneutralproleinthey-zplanedoesnotchangeappreciably.(Forexample,theinjectedneutralproleatx=70cmisverysimilartotheproleat W.W.Heidbrinketal./Commun.Comput.Phys.,10(2011),pp.716-741735 X F P F Y Z Y # F B N M J O F 1 M B T N B n P X E J S F D U J P O F [ # F B N J P O E F O T J U Z D Z ) " - 0 O F V U S B M T E F O T J U Z Y # & " . O F V U S B M T E F O T J U Z D Z Y Y A B D C Figure10:(a)Midplane(z=0)crosssectionofaquadrantoftheDIII-Dtokamak,showingtheintersectionoftheverticalFIDAsightlines(triangles)andthe(x,y)coordinatesystemassociatedwiththeactiveneutralbeam.(b)Midplanehaloneutraldensity.Thedistributionisasymmetricalduetotheplasma\row.(c)Beam-iondensity.(d)Injectedneutraldensity.x=2cmthatisshowninFig.11(a),apartfromareductionbyafactoroffour.)However,thesituationwiththehaloneutralsisdramaticallydifferent.Whentheinjectedneutralsentertheplasma,thebeamhaloislimitedtotheregionofhighinjectedneutraldensity(Fig.11(b)).Furtherdownthex-axis,theplasmadensityincreasesand,asmoreinjectedneutralschargeexchangewiththermaldeuteriumions,thebeamhaloprolesspreadinbothtransversedirections(Fig.11(c)). -40 -20 0 20 40 -40 -20 0 20 40z (cm) 0.5 1.0 1.5 2.0 2.5 -40 -20 0 20 40 -40 -20 0 20 40 9x 10 Beam neutral density at x=2cm (a) Halo neutral density at x=2cm -60 -40 -20 0 20 40 60 -80 -60 -40 -20 0 20 40 60 80z (cm) 0.2 0.4 0.6 -60 -40 -20 0 20 40 60 -80 -60 -40 -20 0 20 40 60 80 (b) -60 -40 -20 0 20 40 60y (cm) -80 -60 -40 -20 0 20 40 60 80z (cm) 0.1 0.2 0.3 0.4 -60 -40 -20 0 20 40 60 -80 -60 -40 -20 0 20 40 60 80 Halo neutral density at x=70cm (c) Figure11:Vertical(y,z)planesof(a)injectedneutraldensityneartheplasmaedge,(b)haloneutraldensityneartheplasmaedge,and(c)haloneutraldensityintheplasmainterior. 736W.W.Heidbrinketal./Commun.Comput.Phys.,10(2011),pp.716-741Blue-shiftedFIDAprolesfromasetoffoursimulationswith107MCparticles,whereasingleverticaldimensionwasvaried,areshowninFig.12.Ourbaselinesimulationusesvaluesofthegridhalf-widthof30and40cminthey-andz-directions,respectively.ThisstandardverticalgridisillustratedinFig.11(c).Doublingthesizeofthegridinthey-directionandthusincludingessentiallyallbeamhaloneutralsinthatdirectiondoesnotchangetheFIDAprole.Whenthey-widthisreducedto20cm,theproleuniformlydecreasesby5%(notshown),whichiswithintheMCnoiselevel.Incontrast,extendingthegridinthez-directionresultsina10%increaseinthepredictedprole.Iftheverticalgriddimensionisreducedtoaz-widthof20cm,thepredictedvaluesdecreaseby25%.Tounderstandtheimpactofthesechoicesonthepredictedprole,consideranex-tremecasewithnohalo-neutralcontribution.(TheFIDASIMcodehasanoptiontoturnoffthehalo-neutralcalculation.)AsshowninFig.13,turningoffthehaloproducespro-lesandspectrathatare50%lowerthanthebaselinevalues.Neglectofthehalonotonlyaffectsthemagnitudeofthepredictedsignalbutalsoaffectstheshapeofthespa-tialproleandofthespectrum.Theprolevariationsareprincipallyduetovariationsintheratioofinjected:haloneutraldensitiesalongdifferentsightlines.Thespectralvariationsareultimatelyassociatedwiththestrongenergydependenceofthecharge-exchangecrosssections.Haloneutralshavelowervelocitiesthaninjectedneutrals,sotherelativevelocityoffastionswithdifferentvelocities(and,hence,Dopplershifts)isdifferentforcollisionswithhaloneutralsthanforinjectedneutrals.Althoughtheno-halocaseshowninFig.13isunrealistic,itisusefulasanupper-boundonthemagni-tudeofeffectsassociatedwithtruncationofthebox-size.ThevariationsinsignalshowninFig.11dependonthefractionofthehalopopulationthatisincludedinthesimula-tion.TheseresultsindicatethattheviewinggeometryoftheFIDAdiagnosticultimatelydeterminestheextenttowhichthehaloneutral'svolumemustbeincludedinthesim-ulation.ForFIDAchordsthatviewhorizontally,expandingthegridalongthey-axisismoreimportantthanexpandingitalongthez-axis.Ultimately,therequiredsimulationvolumedependsonthespatialextentofthehalo,whichscalesasthegeometricmeanofthemean-freepathsforionizationandcharge-exchangeevaluatedattheiontemperature[31].Thesemean-freepathsarebothinverselyproportionaltothedensity,withonlyweakdependenciesonTeandTi(fortypicaltem-peratures),sotherequiredsimulationvolumescalesapproximatelyas1/ne.4.3CellsizeOncethesimulationvolumeinthe(x,y,z)spaceisdened,thesizeofeachcellneedstobedetermined.Thecalculatedsignalsarelineintegralsoverthree-dimensionalemissiv-ityproles.Fig.14showsatwo-dimensionalexample.Ifthegridistoocoarse,pixelationisevidentintheresultingprole.Theseverityofthiseffectdependsprimarilyonthegradientoftheemissivityproleindirectionsperpendiculartothesightline;pixelationisalsomoreseverewhenasightlineisorientedalongoneofthe(x,y,z)axes.Physically,theemissivityprolesdependonnumerousquantitiesincludingthebeam W.W.Heidbrinketal./Commun.Comput.Phys.,10(2011),pp.716-741737 5 F P Z X J E U I [ X J E U I D N \n T U B O E B S E D I P J D F \n % O X H V K L I W H G ) , ' $ V S H F W U D S U R I L O H V ( N H 9 L P H O T O N S C M Figure12:DependenceofthecalculatedFIDAradianceonthevolumeofthesimulation.ThespectraareintegratedoverwavelengthsaboveEl=10keV. Figure13:(a)Ratiooftheblue-shiftedFIDAradianceforsimulationswithouthaloneutralsandwithhaloneutralsvs.majorradius.ThespectraareintegratedoverwavelengthsaboveEl=10keV.(b)RatiooftheFIDAspectralradianceforsimulationswithouthaloneutralsandwithhaloneutralsfortheFIDAchannelatR=187.5cm.TwovaluesofElareindicated. Figure14:(a)FIDAemissivityinan(x,z)plane,withthestandard(3321)gridboundariesoverlaid.Theredlinesillustratetherangeofsightlineanglesgraphedinthelower gure.(b)Calculatedradiancevs.sightlineanglefordi erentchoicesofgridsizeinthe(x,z)plane.The\ratportionsofthecurvesarecausedbypixelation.injectiongeometryandtheplasmaparameters;gradientsinanyofthesequantitiesim-pacttransverseemissivitygradients.TheFIDAandNPAemissivitiesareapproximatelyproportionaltotheproductoffast-ionandneutraldensities,nfånn;thesetypicallychange 738W.W.Heidbrinketal./Commun.Comput.Phys.,10(2011),pp.716-741overscalelengthsofseveralcentimeters.Theinjectedneutrallightdependsonthebeamattenuationandcanchangeappreciablyinafewcentimetersfordensities�51013cm 3.Generallyspeaking,owingtothelineintegration,thecellsizeintheapproximatedirec-tionofthesightlinescanbetwiceaslargeasintransversedirections.Finergridsincreasethecomputationalexpense.OurbaselinesimulationsforthenominallyverticalsightlinesshowninFig.10(a)employx-y-zgridcellsof434cm.Tostudytheeffectofcoarserornergrids,weusethestandard1206080cmcom-putationaldomainand107particles.Reningthegridsizeintheverticaldirectionto430.8cmor,inthehorizontaldirection,to2.51.24cmincreasesthenumberofgridcellsngbyafactorof5and3.6,respectively.Thecorrespondingcomputationaltimesare63%and22%higher.Theeffectofthesechangesonthecalculatedspatialprolesinte-gratedaboveEl=10keVisnegligible.Coarseningofthex-y-zgridsizeto645.5cmleadstoacomputationaltimesavingsof25%.Forthiscase,theFIDAspatialproleresemblesthebaselinebutdeviatesslightlymorethanexpectedfromtheMCnoiselevel.Whilefurthercoarseningofthegridto121014cmleadstoacomputationalsavingsof35%,theproleisdeformedinshapeandhasdecreasedvaluesbetween10%and40%atdifferentradialchordlocations.Toavoidpixelationinsimulationsoftwo-dimensionalimagingdatawiththousandsofsightlines,cellsizesneedtobe1-2cm.Ontheotherhand,becausethebandpassltersemployedinimagingdiagnosticsaverageoverwavelength,coarsevelocity-spacegridsarepermissible.5ValidationThecalculationoftheinjectedneutralswascomparedwithexperimentalmeasurementsofthebeam-emissionlightinaDIII-Dexperiment[32].Afterpassingthroughabandpasslter,two-dimensionalimagesofthelightweremeasuredwithaCCDcamera.Thecodepredictionsareingoodagreementwithmeasurementsoftheverticalextentofthebeamandofthebeampenetrationasafunctionofdensity.InarecentDIII-Dexperiment[33],halolightwasmeasuredforsightlinesthatareoutsidethefootprintoftheinjectedneutralbeam.Goodagreementwithcodepredictionsisobserved.TherstdetailedquantitativecomparisonofthepredictedFIDAspectrumwithex-perimentwasreportedin[34].InMHD-quiescentDIII-Dplasmas,codepredictionsbasedonthefast-iondistributionfunctionpredictedbyNUBEAMhavethesamespectralshapeasexperimentandtheintensityoftheFIDAsignalagreestowithin25%.AlaterDIII-Dexperiment[35]alsofoundagreementtowithinabout25%betweenthespectralshape,radialprole,andabsoluteintensityforquietplasmaswithiontemperaturesbe-low3keV.Atwo-dimensionalmeasurementoftheproleofFIDAlightwasmadewithaband-passlterandimagingCCDcameraonDIII-D[19].Theproleshapeanddependence W.W.Heidbrinketal./Commun.Comput.Phys.,10(2011),pp.716-741739ofthesignalonbeam-injectionangleagreedwellwithcodepredictions.FIDASIMpredictstheradiancefromtheinjectedbeam,thehalos,andthefastions.Inaddition,apost-processorthatusestheoutputofthesecondstageofthecodecalculatestheexpectedvisiblebremsstrahlungradiance.Comparisonoftherelativeintensityofthesefourfeaturesisausefulcheckthatisindependentofanyexperimentalerrorsintheintensitycalibration.Todate,themostdetailedcomparisonofthistypewasperformedonASDEX-Upgrade[36]andshowsgoodagreementforallfourspectralcontributions.6OutlookWithplasmaprolesandafast-iondistributionfunctionasinput,FIDASIMpredictstheuxmeasuredbyNPAsandtheradiancemeasuredbyaDaspectrometer.Thecodehasarathercompletephysicsmodelthattreatstheatomicphysicsoftheseprocesseswithscarcelyanyapproximations.Onepossibleareaofimprovementisapost-processorthatreplacestheapproximationofinnitesimalsightlineswithanaccuratetreatmentofthecollectionoptics.Amorechallengingupgradeisneededtotreatplasmaswherethefast-iondensityiscomparabletothethermal-iondensity.Thepresentcodeiscomputationallyintensive.ParallelizationoftheMCroutinesorfurtheroptimizationofthemosttimeconsumingsubroutinesisdesirable.Acomplemen-taryreducedmodelthatissufcientlyfasttomakepredictionsbetweendischarges(forexample)isneeded.Althoughmanyaspectsofthecodehavebeensuccessfullyvalidatedbyexperiment,additionalcomparisonsaredesirable.TheFIDApredictionshavebeenvalidatedincon-ventionaltokamaksbuthavenotyetbeenconrmedinasphericaltokamak.ValidationoftheNPAmodelisalsoafuturetask.AcknowledgmentsThisworkhasbenetedfromthecontributionsofalargenumberofscientists.KeithBur-rell,BillDavis,RainerFischer,ManuelGarc´a-Munoz,DougMcCune,andMikeVanZee-landgavevaluableadvice.ContributorstothecodeitselfincludeBrianGrierson,CliveMichael,ChrisMuscatello,NoviPablant,MarioPodest´a,andWayneSolomon.WearealsoindebtedtoourexperimentalcollaboratorsonDIII-DandNSTX.Theoriginatingde-veloperofADASistheJETJointUndertaking.ThisworkwasfundedbytheU.S.Depart-mentofEnergyunderSC-G903402andDE-FC02-04ER54698andDE-FG02-06ER54867.References[1]I.H.Hutchinson,PrinciplesofPlasmaDiagnostics,CambridgeUniversityPress,NewYork,1987.[2]L.A.Artsimovich,V.V.AfrostimovandI.P.Gladkovskijetal.,inPlasmaPhysicsandCon-trolledNuclearFusionResearch1965,Volume2,pp.595,Vienna,1966,IAEA. 740W.W.Heidbrinketal./Commun.Comput.Phys.,10(2011),pp.716-741[3]W.W.Heidbrink,K.H.Burrell,Y.Luo,N.A.PablantandE.Ruskov,Hydrogenicfast-iondiagnosticusingBalmer-alphalight,Plasma.Phys.Control.Fusion.,46(2004),18551875.[4]W.W.Heidbrink,Fast-ionDameasurementsofthefast-iondistribution(invited),Rev.Sci.Instrum.,81(2010),10D727.[5]W.W.Heidbrink,Y.LuoandK.H.Burrelletal.,Measurementsoffast-ionaccelerationatcyclotronharmonicsusingBalmer-alphaspectroscopy,Plasma.Phys.Control.Fusion.,49(2007),14571475.[6]J.EgedalandH.Bindslev,Reconstructionofgyrotropicphase-spacedistributionsfrom1Dprojections,Phys.Plasma.,11(2004),21912198.[7]M.Salewski,F.MeoandM.Stejneretal.,ComparisonoffastioncollectiveThomsonscatter-ingmeasurementsatASDEXupgradewithnumericalsimulations,Nucl.Fusion.,50(2010),035012.[8]A.Pankin,D.Mccune,R.Andre,G.BatemanandA.Kritz,ThetokamakMonteCarlofastionmoduleNUBEAMinthenationaltransportcodecollaborationlibrary,Comput.Phys.Commun.,159(2004),157184.[9]R.V.Budny,AstandardDTsupershotsimulation,Nucl.Fusion.,34(1994),1247,[10]L.L.Lao,H.St.John,R.D.Stambaugh,A.G.KellmanandW.Pfeiffer,Reconstructionofcurrentproleparametersandplasmashapesintokamaks,Nucl.Fusion.,25(1985),16111622.[11]V.S.Chan,S.C.ChiuandY.A.Omelchenko,Radio-frequency-drivenradialcurrentandplasmarotationinatokamak,Phys.Plasma.,9(2002),501510.[12]R.W.HarveyandM.G.Mccoy,TheCQL3DFokker-Planckcode,inProceedingsofIAEATechnicalCommitteeMeetingonAdvancesinSimulationandModelingofThermonuclearPlasmas,Montreal,1992,IAEA.[13]D.Liu,W.W.HeidbrinkandM.Podestaetal.,Prolesoffastionsthatareacceleratedbyhighharmonicfastwavesinthenationalsphericaltorusexperiment,Plasma.Phys.Control.Fusion.,52(2010),025006.[14]T.A.Gianakon,R.J.Fonck,J.D.Callen,R.D.DurstandJ.S.Kim,EffectsofedgeplasmaturbulenceonradialcorrelationlengthmeasurementswithBES,Rev.Sci.Instrum.,63(1992),49314933.[15]I.H.Hutchinson,Excited-statepopulationsinneutralbeamemission,Plasma.Phys.Con-trol.Fusion.,44(2002),7184.[16]G.W.Series,TheSpectrumofAtomicHydrogen,OxfordUniversityPress,NewYork,1987.[17]W.H.Press,S.A.Teukolsky,W.T.VetterlingandB.P.Flannery,NumericalRecipes:TheArtofScienticComputing,Cambridge.[18]W.Mandl,DevelopmentofactiveBalmer-alphaspectroscopyatJET,TechnicalReportJET-IR(92)05,JET,1992.[19]M.A.VanZeeland,W.W.HeidbrinkandJ.Yu,FastionDaimagingintheDIII-Dtokamak,Plasma.Phys.Control.Fusion.,51(2009),055001.[20]http://adas.phys.atrath.ac.uk/.[21]H.Anderson,M.VonHellermannandR.Hoekstraetal.,Neutralbeamstoppingandemis-sioninfusionplasmasI:deuteriumbeams,Plasma.Phys.Control.Fusion.,42(2000),781806.[22]R.K.Janev,C.D.BoleyandD.E.Post,Penetrationofenergeticneutralbeamsintofusionplasmas,Nucl.Fusion.,29(1989),2125.[23]J.G.Lodge,I.C.PercivalandD.Richards,Semi-empiricalcrosssectionsforexcitationofhydrogenatomsbyprotons,J.Phys.B.,9(1976),239254. W.W.Heidbrinketal./Commun.Comput.Phys.,10(2011),pp.716-741741[24]R.K.Janev,W.D.Langer,Jr.D.E.PostandJr.K.Evans,ElementaryProcessesinHydrogen-HeliumPlasmas,Springer-Verlag,Berlin,1987.[25]L.VriensandA.H.M.Smeets,Cross-sectionandrateformulasforelectron-impaction-ization,excitation,deexcitation,andtotaldepopulationofexcitedatoms,Phys.Rev.A.,22(1980),940951.[26]R.Janev,D.ReiterandU.Samm,TechnicalReport4105,JuelReport,2004,http://www.eirene.de.[27]R.E.Bell,CarbonionplumeemissionproducedbychargeexchangewithneutralbeamsonNSTX,TechnicalReportPPPL-4167,PrincetonPlasmaPhysicsLaboratory,2006.[28]D.F.Finkenthal,TheMeasurementofAbsoluteHeliumIonDensityProlesontheDIII-DTokamakUsingChargeExchangeRecombinationSpectroscopy,PhDthesis,UniversityofCalifornia,Berkeley,1994.[29]V.Tang,ExperimentalandNumericalCharacterizationofIon-cyclotronHeatedProtonsontheAlcatorC-ModTokamak,PhDthesis,MassachusettsInstituteofTechnology,2006.[30]Y.Luo,W.W.Heidbrink,K.H.Burrell,D.H.KaplanandP.Gohil,MeasurementoftheDaspectrumproducedbyfastionsinDIII-D,Rev.Sci.Instrum.,78(2007),033505.[31]R.J.GoldstonandP.H.Rutherford,IntroductiontoPlasmaPhysics,InstituteofPhysicsPublishing,Bristol,1995.[32]M.A.VanZeeland,J.H.YuandN.H.Brooksetal.,Activeandpassivespectroscopicimag-ingintheDIII-Dtokamak,Plasma.Phys.Control.Fusion.,52(2010),045006.[33]B.Griersonetal.,Nucl.Fusion,(2011),submitted.[34]Y.Luo,W.W.Heidbrink,E.Ruskov,K.H.BurrellandW.M.Solomon,Fast-ionDameasure-mentsandsimulationsinquietplasmas,Phys.Plasmas.,14(2007),112503.[35]W.W.Heidbrink,M.MurakamiandJ.M.Parketal.,Beam-ionconnementfordifferentinjectiongeometries,Plasma.Phys.Control.Fusion.,51(2009),125001.[36]B.Geigeretal.,Fast-ionDameasurementsatASDEXupgrade,Plasma.Phys.Control.Fu-sion.,53(2011),065010.