Decker September 2003 Plasma Science and Fusion Center Massachusetts Institute of Technology Cambridge MA 02139 USA This work was supported by the US Department of Energy Grant No DEFG0291ER54109 by the US Department of Energy jointly with the Natio ID: 49691
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detrapped.Bananaorbitsarethenwell-deÞnedand,tothelowestorderinthecollisionalityparameter,isconstantalongtheÞeldlinesandsymmetricinthetrappedregion.Applyingthebounce-averagingoperator{A} b1 2T ccd 2r v B theÞrsttermin(2)isannihilated.In(3),istheturninganglefortrappedparticles,andthesumoverappliesonlytotrappedelectrons,forwhichtheaveragemustbeperformedoverboththeforwardandbackwardmotions.Asaresult,weobtainthebounce-averaged,Fokker-Planckequationwhichmustbesolvednumericallyinthe2Dmomentumspace.NumericalCodeEquation(4)issolvedusingthecodeDKE[11].ThiscodeusesthefullyrelativisticcollisionaloperatordevelopedbyBraamsandKarney[12]andtherelativisticRFquasilinearoperatorproposedbyLerche[13].Becausethedistributionfunctionissymmetricinthetrappedregion,onlyonehalfofthetrappedregionistobeconsideredintheFPcalculations[14].However,thisrequiresaparticulartreatmentoftheparticleßuxesinmomentumspaceatthetrapped/passingboundary,sinceelectronsthatarebarelytrappedcanbedetrappedeitherontheco-orcounter-passingside,giventhatthebounceperiodismuchshorterthanthecollisionaldetrappingtime.Withthisscheme,thebounce-averageddynamicsincludetrappingeffectsimplicitly,leadingtoveryfastcomputercalculations.Inaddition,non-uniformgridsareusedbothinmomentumandpitch-anglecoordinates,allowingforÞnercalculationsinthemostimportantregionsofmomentumspaceforECCDandOKCD,particularlynearthetrapped/passingboundary.ThecalculationspresentedinthispaperwerecarriedoutforatypicalDIII-Dplasma,withmajorradiusm,minorradiusm,andmagneticÞeldonaxisT.ThetemperatureanddensityproÞlesweretakentobeparabolic,withrespectivecoreandedgetemperatureskeVandkeV,anddensities.TheeffectiveionchargewastakenuniformlyeffTheECwavewasconsideredtobeaGaussianbeamofwidthcm,polarizedinthequasi-Xmode.Thewave-particleinteractionoccurednearthesecondharmonic,Momentsofthedistributionfunctioncalculatedfrom(4)givetheßux-surfaceaveragedcurrentdensityandtheßux-surfaceaveragedabsorbedpowerdensity.AnormalizedintrinsicCDÞgureofmeritisdeÞnedby istheelectroncollisionalfrequencyln) - 10 - 5 0 5 5 ///p/p - 10- 50510 - 10- 50510- 10- 50510 (a)(b)(c)(d) Figure1:2Ddistribution(a)-(b)andcorrespondingparalleldistribution(c)-(d)forthecasesofECCDandOKCD,respectively.Thedashedlinesongraphs(a)-(b)areacontourplotofthediffusioncoefÞcient.NotethatforaMaxwellian,thecontourswouldbeequidistantconcentriccircles.OntheLFS,thereisalargefractionoftrappedparticles,sothat,inthepresentcase,theresonantregioninvelocityspace-indashedcontoursongraph(b)-islocatedrightunderthetrappedpassingboundary.Asaconsequence,barelycounter-passingelectronsaretrappedundertheactionoftheECwave,whichincreasestheirperpendicularmomentum.Becauseofthefastbouncemotion,thedistributionfunctionisrapidlysymmetrizedinthetrappedregion,andtheseelectronscanbedetrappedeitherontheco-orcounter-passingside.Theconsequenceofanasymmetrictrappingduetothewave-whichcreatesasinkofelectronsonthecounter-passingside-,andthesymmetricdetrapping,isanaccumulationofelectronsontheco-passingside.Thisaccumulationisvisibleontheparalleldistribution,shownongraph(d),andgeneratestheOhkawacurrent.TheOKCDdrivencurrentdensityisandthedensityofpowerabsorbedis.TheÞgureofmerit(5)isthen.ThecurrentdensitydrivenbyOKCDissensiblylargerthanECCD;however,thedensityofpowerabsorbedisalsolarger,sothattheÞguresofmeritarecomparable.ThiscomparisondoesnotpredicthowmuchtotalcurrentwouldbedrivenbyaECbeamusingtheECortheOKmethod;however,thelargerpowerabsorbedandcurrentdensitiessuggestthatOKCDmayleadtonarrowercurrentproÞles,whichisimportantfortheaccuracyofcurrentproÞlecontrolandNTMstabilization. 0 0.25 0.5 0.75 1 0 20 40 60 80 100 = r/aI (kA)ECCD ( = 180o) OKCD ( = 0o) ECCD ( = 90o) Figure3:TotalcurrentgeneratedbyaECbeamMWlaunchedhorizontallyfromtheLFS.TwoECCDcasesareconsidered,withdepo-sitionat=180=90,andaOKCDcaseWecanseethattheECCDcurrentdecreasessteadilywiththenormalizedradius.Thede-creaseisfasterforECCDabovethemagneticaxis(=90),whichbecomesimpossiblebe-yondacertainradius(),inaccordancewithexperimentalobservations[7].OKCDcur-rentcanbedrivenwhenthenumberoftrappedparticlesbecomessufÞcient()andbe-comesrapidlylargerthanECCDat=90Beyondsomepoint(),itbecomesevenlargerthanECCDat=180.Itisinterestingtonotethat,inthepresentcase,OKCDwoulddriveasmuchcurrentatasECCD=90woulddriveat;therefore,OKCDcandriveappreciablecurrentsfaroff-axis,whereECCDcannot.WhencomparedtoECCDat=180,OKCDgivesslightlyhighercurrentsforVaryingtheRadialLocationofDepositioninOKCDInECCDandOKCD,thecurrentisdepositedneartheintersectionoftheraypathwiththecyclotronresonancelayer.Experimentally,theradiallocationofdepositionmayhavetobecontrolledandmodiÞedduringtheoperation.Thishasbeendonebychangingthelocationoftheresonancelayer,bymovingtheplasma,orchangingthemagneticÞeld[4][3];ithasalsobeendonebysteeringlaunchingmirrors,inordertomodifytheraypath[5]. = 0o = 60o - 0.5 0.5 ce Figure4:SimulationofOKCDwithvaryingpoloidallaunchingangle.TheresonancelayerismaintainedÞxed. 0.65 0.7 0.75 0.8 0 10 20 30 I (kA)OKCD Currentpeak Figure5:OKCDtotalcurrentasafunc-tionofthecurrentproÞlepeakTheeffectofsteeringthepoloidallaunchingangleisinvestigatedbycalculatingOKCDwithlaunchingfromtheLFSatananglewithrespecttothehorizontalplane.TheECWfrequency,andthereforetheresonancelayer,isÞxed,asshownonFig.4.TheECbeam ECCDforAdvancedTokamakOperationsFisch-BoozerversusOhkawaMethodsJoanDeckerPlasmaScienceandFusionCenter,MIT,CambridgeMA02139Currentcanbedrivenusingelectroncyclotronwaves(ECW)byoptimizingeithertheFisch-Boozermechanism(ECCD)ortheOhkawamechanism(OKCD).InECCD,perpendicularheatingduetoECWcreatesanasymmetricresistivity.InOKCD,currentisgeneratedbyECW-inducedasymmetricelectrontrapping.OKCDisagoodcandidateforoff-axisCDwheretheECCDeffectivenessisreducedduetotrappedelectrons.Thetwomechanismsaredescribedusingthekinetic,bounce-averaged,Fokker-PlanckcodeDKEwithaquasilinearECWoperator.CurrentsandCDefÞcienciesforthetwomethodsarecalculatedandcomparedindifferentregionsofanadvancedtokamakplasma.NumericalresultsconÞrmtheexperimentalobservationsthatECCDisbestforcentralCDbutbecomesineffectivebeyondacertainradialdistancefromtheplasmacenter.Onthelow-Þeldside(LFS)ofthisoutboardregion,OKCDcanveryeffectivelygeneratelocalizedcurrents.AsitisoptimizedontheLFS,OKCDrequireslowerwavefrequenciesthanECCD-anadvantagewhenconsideringECWsources.INTRODUCTIONElectroncyclotroncurrentdrive(ECCD)hasbeensuccessfullyusedforfullcurrentdrive[1],currentproÞlecontrol[2],andthestabilizationofMHDinstabilities,particularlyneoclassicaltearingmodes(NTM)[3][4][5].InaccordancewiththeFisch-Boozermethod[6],ECWareusedtotransferperpendicularenergytotheresonantelectrons.Thiscreatesanasymmetricresistivity,becausemoreenergeticelectronsarelesscollisional.Thisasymme-tryintheresistivitygeneratesaelectronßowinthesameparalleldirectionastheresonantelectronvelocity.However,ithasbeenfoundexperimentally[7][1]thattheECCDefÞ-ciencydecreasesascurrentisdrivenfurtheroff-axis.ThisdecreaseintheECCDefÞciencyhasbeenunderstoodtobeduetotheeffectoftrappedparticles.BecauseECCDincreasestheperpendicularenergyofelectrons,italsodiffusesthemtoaregionofvelocityspacethatisclosertothetrappedregion.Afractionoftheseelectronsarepitch-anglescatteredintothetrappedregion.Sincethebounceperiodoftrappedelectronsismuchshorterthanthecollisionaldetrappingtime,halfoftheelectrondetrappingwilloccurthroughtheoppositesideofthetrappedregion,thuscreatingacountercurrent.TheresultingCDefÞciencycanthusbestronglyreduced.TheOhkawamethod[8]forcurrentdrive(OKCD)makesuseofelectrontrappingtogeneratecurrent.TheECWarelaunchedintheoppositedirectionfromECCDandthewaveparametersarechosensothatthewave-particleinteractioninducestrapping.Thistrapping WorkcarriedoutincollaborationwithAbrahamBersandAbhayK.Ram,PSFC,MIT,andYvesPeyssonCEA-Cadarache,France PSFC/JA-03-17 ECCD for Advanced Tokamak Operations Fisch-Boozer versus Ohkawa Methods J. Decker September 2003 Plasma Science and Fusion Center Massachusetts Institute of Technology Cambridge, MA 02139 U.S.A. This work was supported by the U.S. Department of Energy, Grant No. DE-FG02-91ER-54109, by the U.S. Department of Energy jointly with the National Spherical Torus Experiment, Grant No. DE-FG02-99ER-54521, and by the U.S. Department of Energy for the United States government is permitted. Proceedings of the 15th Topical Conference on Radio Moran, Wyoming, May 19–21,