Experimentally,theerror- - PDF document

Experimentally,theerror-Þeldpenetrationthresholdisfoundtoincreaseapproximatelylinearlywithelectronnum-berdensity.Theabovescalings,whichareallfairlysimilartooneanother,exhibitasigniÞcantlyweakerthanlinearin-creasewithelectronnumberdensityveryrapidlywithincreasing.Thetheoreticalscalingsalsoexhibitafairlystrong,approximatelylineardecreasewithincreasingtoroidalmagneticÞeldstrength.ThisscalingisconsistentwithexperimentaldataobtainedfromDIII-D,JET,andAlcatorC-Mod.Thescalingofthepenetrationthresholdwiththeplasmamajorradiusisnotdirectlymea-suredinexperiments,butis,instead,inferredfromtheob-servedscalingswithelectrondensityandtoroidalmagneticÞeldstrengthviadimensionalanalysis.Hence,ifthereisagreementbetweenpredictionandexperimentonthescalingofthepenetrationthresholdwithelectronnumberdensityandtoroidalmagneticÞeldstrength,thentherewillalsobeagreementonthescalingwithplasmamajorradius.Unfor-tunately,thepredictedandexperimentalscalingsofthepen-etrationthresholdwiththeelectronnumberdensityareinV.SUMMARYTheMHDmodeloferror-ÞeldpenetrationintokamakplasmaspresentedinRefs.7Ð9hasbeenextendedtotakedrift-MHDphysicsintoaccount.Diamagneticandsemicolli-i.e.,thelayerwidthfallingbeloweffectsarebothfoundtomodifythepenetrationthreshold.However,thesemodiÞcationsarenotparticularlydramatic.Theextendedmodelhasbeenusedtoexaminethescal-ingoftheerror-Þeldpenetrationthresholdinohmictokamakplasmas.Inordertoperformascalingstudy,itisnecessarytomakesomeassumptionsregardingthescalingofthenatu-ralfrequencyoftearingmodesintheplasma,andthatoftheplasmamomentumconÞnementtimescale.ThescalingofthepenetrationthresholdthusobtainedisfoundtobefairlyclosetothatpreviouslyobtainedfromMHD.Ingeneral,thepredictedscalingsarenotentirelyinaccordancewiththescalingofthepenetrationthresholdmeasuredinexperi-ments.Thepredictedscalingsgenerallyexhibittooslowanincreaseofthepenetrationthresholdwithincreasingelectronnumberdensity,butabouttherightdecreasewithincreasingtoroidalmagneticÞeldstrength.Thelackofagreementbetweenthepredictedandexperi-mentalpenetrationthresholdscalingsmaybeduetoincor-rectassumptionsregardingthescalingsofthenaturalmodefrequencyandthemomentumconÞnementtimescale.Alter-nately,thislackofagreementmayindicatethatmorephysicsneedstobeaddedtothemodel.Forinstance,neoclassicalßow-dampingisknowntogiverisetoalargeenhancementofioninertia.Suchanenhancementwouldshiftthebound-ariesofthevariouslinearresponseregimesseetheÞgureswhichcouldmodifythepenetrationthreshold.NeoclassicaleffectsalsoallownonresonantcomponentsofanerrorÞeldtoexertaslowing-downtorqueontheplasma,thusaffectingthepenetrationthreshold.Boththeseeffectswillbeinves-tigatedinfuturepublications.TheauthorswouldliketothankSteveWolfeandRobLaHayeforhelpfuldiscussionsduringthepreparationofthispaper.Oneoftheauthorswouldalsoliketoacknowl-edgehelpfuldiscussionswithBorisBreizman.ThisresearchwasfundedbytheU.S.DepartmentofEnergyunderContractDE-FG05-96ER-54346.J.T.Scoville,R.J.LaHaye,A.G.Kellman,T.H.Osborne,R.D.Stam-baugh,E.J.Strait,andT.S.Taylor,Nucl.Fusion,875T.C.Hender,R.Fitzpatrick,A.W.Morris,P.G.Carolan,R.D.Durst,T.Edlington,J.Ferreira,S.J.Fielding,P.S.Haynes,J.Hugill,I.J.Jenkins,R.J.LaHaye,B.J.Parham,D.C.Robinson,T.N.Todd,M.Valovi,andG.Vayakis,Nucl.Fusion,2091G.M.FishpoolandP.S.Haynes,Nucl.Fusion,109R.J.Buttery,M.DeBenedetti,D.A.Gates,Y.Gribov,T.C.Hender,R.J.LaHaye,P.Leahy,J.A.Leuer,A.W.Morris,A.Santagiustina,J.T.Scoville,andB.J.D.Tubbing,JETTeam,COMPASS-DResearchTeam,DIII-DTeam,Nucl.Fusion,1827R.J.Buttery,M.DeBenedetti,T.C.Hender,andB.J.D.Tubbing,Nucl.,807S.M.Wolfe,I.R.Hutchinson,R.S.Granetz,J.Rice,A.Hubbard,A.Lynn,P.Phillips,T.C.Hender,D.F.Howell,R.J.LaHaye,andJ.T.Scoville,Phys.Plasmas,056110R.Fitzpatrick,Nucl.Fusion,1049R.Fitzpatrick,Phys.Plasmas,3325R.Fitzpatrick,Phys.Plasmas,1782R.D.HazeltineandJ.D.Meiss,PlasmaConnementDover,Mineola,NY,2003J.F.DrakeandY.C.Lee,Phys.Fluids,1341F.L.Waelbroeck,Phys.Plasmas,4040R.FitzpatrickandF.L.Waelbroeck,Phys.Plasmas,022307H.P.Furth,J.Killeen,andM.N.Rosenbluth,Phys.Fluids,459A.Erdelyi,W.Magnus,F.Oberhettinger,andF.G.Tricomi,HigherTran-scendentalFunctionsMcGraw-Hill,NewYork,1953,Vol.II.G.Ara,B.Basu,B.Coppi,G.Laval,M.N.Rosenbluth,andB.V.Wad-dell,Ann.Phys.,443A.H.Boozer,Phys.Plasmas,4620A.B.MikhailovskiiandB.N.Kuvshinov,PlasmaPhys.Rep.,789E.Lazzaro,R.J.Buttery,T.C.Henderetal.,Phys.Plasmas,3906032503-9Drift-magnetohydrodynamicalmodeloferror-ÞeldpenetrationPhys.Plasmas,032503 br
rs crit,SCi Thisregimeisvalidprovided Finally,error-ÞeldpenetrationoccursintheHRiregimewhenthenaturalmoderotationfrequencyisreducedto1/2ofitsoriginalvalue,and crit,HRi Thisregimeisvalidprovided E.PenetrationthresholdsinohmictokamakplasmasduetodiamagneticrotationSuppose,asissomewhatmorerealistic,thatthenaturalrotationofthemodeisentirelyduetodiamagneticro-tationattherationalsurfacei.e.,thereisno.Thisimpliesthat=0.Itfollows,fromTableIandEq.,thaterror-ÞeldpenetrationoccursintheVRiregimewhenthenaturalmoderotationfrequencyisreducedto7/12ofitsoriginalvalue, crit,VRi 7/12ThisregimeisvalidprovidedError-ÞeldpenetrationoccursintheSCiregime,whenthenaturalmoderotationfrequencyisreducedto3/5ofitsoriginalvalue,and crit,SCi Thisregimeisvalidprovided Finally,error-ÞeldpenetrationoccursintheHRiregimewhenthenaturalmoderotationfrequencyisreducedto1/2ofitsoriginalvalue,and crit,HRi Thisregimeisvalidprovided Itcanbeseen,bycomparingtheresultsofthissectionwiththoseoftheprecedingone,thatitmakeslittledifferencetothepenetrationthresholdwhetherthenaturalmodefre-quencyisduetoordiamagneticrotation.Hence,inthefollowing,weshallemploytheexpressionsforthepenetra-tionthresholdoccurringinSec.IVD,withtheunderstandingthatthenaturalfrequency,,representstheunperturbedmodefrequencyattherationalsurface,andcanbemadeupofbothdiamagneticandF.ScalingofohmicpenetrationthresholdsTheparametersappearinginEqs.arenotdi-rectlymeasurableintokamakexperiments.However,wecaneasilyexpressthemintermsofthestandarddimensionless,and.Here,istheelectrontemperature.Thecolli-sionalityparameteristheratioofthetypicalelectron-ioncollisionfrequencytothetransitfrequencyi.e.,thenumberoftimespersecondatypicalelectronexecutesatoroidalcircuitoftheplasma.TheparametersappearinginEqs.canalsobeexpressedintermsoftheengineering,andConsideraclassofohmicallyheatedtokamakplasmasinwhichtheaspectratio,,andtheequilibriumproÞlesareheldÞxed.BydeÞnition, ne/
R0B 2,S B Te0/ ,and.Here,theparallelresis-tivityisassumedtoobeytheclassicalSpitzerformula,andisthemomentumconÞnementtime.Itimmediatelyfol-lows,byinspectionofthevariouspowersof,etc.,that,andLetusassume,asseemsplausible,thatthenaturalmodefrequency,,scaleslikethediamagneticfrequency,.Weshallalsoassume,withconsiderablylessjustiÞcation,thatthemomentumconÞnementtime,,scalesliketheenergyconÞnementtime,Foranohmicallyheatedtokamak,wecanbalancetheohmicheatingrateagainsttheenergylossrate.Itfollowsthat.Thisrelationcanbeusedtoeliminate,yield-Intermsofdimensionlessparameters,theerror-Þeldpenetrationthresholdisfoundtoscaleas crit,VRi crit,SCi crit,HRiintheÞrstvisco-resistive,Þrstsemicollisional,andÞrstHall-resistiveregimes,respectively.Intermsofengineeringparameters,thepenetrationthresholdisfoundtoscaleas crit,VRi crit,SCi crit,HRiAsiseasilydemonstrated,thetheoreticalscalingofthepen-etrationthresholdwithengineeringparametersisofthegen-eralform,where+0.5+1.25032503-8A.ColeandR.FitzpatrickPhys.Plasmas,032503 .Note,however,thatnoneoftheresultsobtainedinthissectionwouldchangeradicallywereConsideralargeaspectratio,lowbeta,tokamakplasmawithapproximatelycircularßuxsurfaces.Letthemajorandminorradiioftheplasma,respectively.Toagoodapproximation,theplasmacanbetreatedasaperiodiccylinderwithperiod2.Adoptingstandardcylindricalco-,andsimulatedtoroidalcoordinates,,where,theequilibriummagneticÞeldtakestheform,whereplaystheroleoftheguideÞeld.SupposethattheerrorÞeldishelical,withperiodsinthepoloidaldirection,andperiodsinthetoroidaldirection.TheerrorÞeldresonateswiththeplasmaattherationalßuxsurface,radius,where=0.Itisconvenienttospecifythescalelength,scalemag-neticÞeldstrength,andscaletimeintroducedinSec.IIAas.Here,isthemagneticshearattherationalsur-face,andisthesafety-factorproÞle.Thecharacteristichydromagnetic,resistive,andviscoustimescalesaredeÞned: ,and,respectively.Here,theplasmamassdensity.Finally,theLundquistandmagneticPrandtlnumberstaketheform:,and,re-spectively.UsingtheabovedeÞnitions,wecanidentifythefollow-ingcorrespondencesbetweenslabandtokamakquantities:.Here,istheoscillationfrequencyofancomovingwiththeframeattherationalsurfaceseeninthelabframeisthevalueofintheabsenceofanerrorÞeld,andistheelectrondiamagneticfrequencyattherationalsurface.Furthermore,theparameteristheconventionalplasmabetaevaluatedattherationalsurface=1,forthesakeofsimplicity.Finally,giventhat1,theparameterreducesto,whereistheionLarmorradiuscalculatedusingtheelectrontemperature.Wealsohave,and.Here,istheconventionaltearingstabilityindexofthemode,andvacuumradialmagneticperturbationassociatedwiththeer-rorÞeldi.e.,themagneticperturbationintheabsenceofB.TorquebalanceTheslabforcebalanceequationistransformedintothefollowingtoroidaltorquebalanceequation: +ˆ
Q=4P
Q0Q 2.
Here,rsa
rs 
r B
rs isaslab-to-tokamakcorrectionfactor.Thisfactorarises,Þrst,becausetheglobalviscousdiffusionequationhasslightlydifferentformsinslabandcylindricalgeometry,and,second,becauseatokamakplasmaissubjecttostrongpoloi-dalßowdampingwhicheffectivelyeliminatesthepoloidalcontributionstothemodefrequencyandelectromagnetictorque,leavingthemuchsmallertoroidalcontributions.isconvenienttorewritethetorquebalanceequationas +ˆ
Q=2P
Q0Q C.Error-ÞeldpenetrationthresholdsWegenerallyexpect1inahightemperaturetoka-makplasma.Moreover,,assumingthatthemodeisatearingmode.Itthereforefollowsthat1.Hence,itisagoodap-proximationtoneglectintheabovetorquebalanceequa-tion.Theerror-Þeldpenetrationthresholdcorrespondstothecriticalerror-Þeldamplitudeabovewhichtorquebalanceislost,i.e.,theapproximatedtorquebalanceequationhasnoItfollowsthat =max wherethemaximumisobtainedbyvaryingD.PenetrationthresholdsinohmictokamakplasmasduetoEBrotationSupposethatthenaturalrotationofthemodeisentirelyduetorotationattherationalsurfacethereisnodiamagneticcomponent.Thisimpliesthat=0.Now,wedonotexpecttheßowinanohmicallyheatedtokamakplasmatobesufÞcientlylargethattheerror-Þeldresponseregimeviolatestheconstant-WealsoexpectthatItfollows,fromSec.IIIG,thatthethreemostlikelyerror-ÞeldresponseregimesaretheÞrstregime,theÞrstsemicollisionalregime,andtheÞrstHall-resistiveregimeÑseeTableI.AccordingtoFig.2,theVRiregimeholdswhen,theSCiregimeholdswhen,andtheHRiregimeholdswhenItfollows,fromTableIandEq.,thaterror-ÞeldpenetrationoccursintheVRiregimewhenthenaturalmoderotationfrequencyisreducedto1/2ofitsoriginalvalue,and crit,VRi 7/12ThisregimeisvalidprovidedError-ÞeldpenetrationoccursintheSCiregimewhenthenaturalmoderotationfrequencyisreducedto3/5ofitsoriginalvalue,and032503-7Drift-magnetohydrodynamicalmodeloferror-ÞeldpenetrationPhys.Plasmas,032503 G.SummaryThevariouslineardrift-MHDresponseregimesforastaticerrorÞeldarelistedinTableI.Byreplacinginallofthevaliditycriterialistedabove,itispossibletodeterminetheextentsoftheseresponseregimesinspaceÑtheresultsareillustratedinFigs.1Ð3.Itcanbeseenthatasthedriftparameter,,increases,thefamiliarMHD,responseregimesmodiÞedbydiamagnetismi.e.,theRIandVRregimesÑaregraduallyreplacedbycompletelynewdrift-MHD,constant-regimesÑi.e.,theHRandSCregimes.Notethatthereconnectinglayerwidthfallsbelowwhichisequivalenttoinalow-inboththesenewregimes.IV.APPLICATIONTOTOKAMAKSA.IntroductionInthissection,weshalldiscusstheapplicationoftheslabanalysispresentedintheprecedingsectionstotheprob-lemofdeterminingtheerror-Þeldpenetrationthresholdinalargeaspectratio,lowbeta,ohmicallyheated,tokamakplasma.Inthefollowing,weshallassumethat=0,forthesakeofsimplicity.Thisimpliesthat=0and TABLEI.Lineardrift-MHDresponseregimesforastaticerrorÞeld.Theabbreviationsindicatethevariousdifferentresponseregimes:HRiÑÞrstHall-resistive;HRiiÑsecondHall-resistive;SCiÑÞrstsemicollisional;SCiiÑsecondsemicollisional;RIiÑÞrstresistive-inertial;RIiiÑsecondresistive-inertial;VRiÑÞrstvisco-resistive;VRiiÑsecondvisco-resistive;VIÑvisco-inertial;IÑinertial.Here,,andAbbreviationResponseLayerwidth,=2.124=2.124=3.142=3.142=2.124=2.124=2.104=2.104=4.647=3.142 FIG.1.Aschematicdiagramofthelineardrift-MHDresponseregimesinspaceforthecase.Here,,and.ThevariousregimesaretheÞrstHall-resistive,Þrst,Þrstresistive-inertial,secondresistive-inertial,Þrstvisco-resistive,secondvisco-resistive,visco-,andinertial FIG.2.Aschematicdiagramofthelineardrift-MHDresponseregimesinspaceforthecase1.Here,,and.ThevariousregimesaretheÞrstHall-resistive,second,Þrstsemicollisional,secondsemicollisional,Þrstresistive-inertial,secondresistive-inertial,Þrstvisco-,secondvisco-resistive,visco-inertial,andiner- FIG.3.Aschematicdiagramofthelineardrift-MHDresponseregimesinspaceforthecase1.Here,,and.ThevariousregimesaretheÞrstHall-resistive,second,Þrstsemicollisional,secondsemicollisional,secondresistive-inertial,secondvisco-resistive,visco-,andinertial032503-6A.ColeandR.FitzpatrickPhys.Plasmas,032503 ˆ
Q=2
Weshallrefertothisasthesecondresistive-inertialregime.Inthelimit,Eqs.,and 
i
QQe WeshallrefertothisasthesecondHall-resistiveregime,sincetheplasmaresponseisdominatedbyresistivityandtheHallterm.Supposethat.Equationducesto i
QQi Inthelimit,Eqs.,and WeshallrefertothisastheÞrstvisco-resistivesincetheplasmaresponseisdominatedbyviscosityandresistivity.Inthelimit,Eqs.,and  WeshallrefertothisastheÞrstsemicollisional11,12Supposethat.Equationducesto c2 Inthelimit,Eqs.,and Weshallrefertothisasthesecondvisco-resistiveregime.Finally,inthelimit,Eqs.,and 
i
QQe WeshallrefertothisastheÞrstHall-resistiveF.Nonconstant-Supposethat.Inthislimit,Eq. d p2 i
QQeQ
QQi+i
QQi+c2p2+c2 i
QQe+c2+i
QQiD2p2+
p4p2Y Inthevariousnonconstant-regimesconsideredbelow,reducesto p2 isrealandnon-negative,andisacomplexcon-stant.Let.Theaboveequationtransformsto ThisequationisidenticalinformtoEq.,whichwehavealreadysolved.Indeed,thesolutionwhichisboundedashasthesmall-,where ,and/2.Thelayerwidthinspaceagainscalesas.MatchingtoEq. Supposethat.Equationducesto Inthelimit,Eqs.,and 2
Weshallrefertothisasthevisco-inertialtheplasmaresponseisdominatedbyviscosityandioniner-tia.Inthelimit,Eqs.,and Weshallrefertothisastheinertialsincetheplasmaresponseisdominatedbyioninertia.NotethattheplasmaresponseintheinertialregimeisequivalenttothatoftwocloselyspacedAlfvŽnresonanceswhichstraddletheresonantsurface.032503-5Drift-magnetohydrodynamicalmodeloferror-ÞeldpenetrationPhys.Plasmas,032503 d p2 i
QQe+p2 Q
QQi+i
QQi+c2p2+c2 =0, Theboundaryconditionsarethatisboundedas 0.ThisissufÞcienttouniquelydetermine.Ingeneral,wecanÞndanalyticsolutionstoEq.,whicharewell-behavedas,andsatisfyEq.0,byoneoftwodifferentmethods.TheÞrstmethod,outlinedinSec.IIID,isatwo-stepmatchingprocessthatgeneratesaclassofsolutionswhichsatisÞestheconstant-secondmethod,outlinedinSec.IIIF,isaone-stepmatchingprocessthatgeneratesaclassofsolutions.Inthefollowing,weassumethat  ,and,forthesakeofsimplicity.D.Constant-Lettherebetwolayersinspace.Inthesmall-layer,supposethatEq.reducesto p2 i
QQe+p2 .Integratingdirectly,weÞnd 1 pp .Thisapproximation,whichisequivalenttothewell-knownconstant-isvalidpro- Inthelarge-layer,for,weobtain Q
QQi+i
QQi+c2p2+c2 boundedas.Asymptoticmatchingtothelayeryieldstheboundarycondition Inthevariousconstant-regimesconsideredbelow,Eq.reducestoanequationoftheform isrealandnon-negative,andissomecomplexconstant.Let z= ,where/2.TheaboveequationtransformsintoamodiÞedBesselequa-tionofgeneralorder, +z =0,.Thesolutionwhichisboundedashasthesmall- 
2 z 1 
z AcomparisonoftheaboveexpressionwithEq. Note,Þnally,that,wheredenotesthewidthofthelarge-layerinE.Constant-Supposethat.Equationducesto QQi Inthelimit1,Eqs.,and WeshallrefertothisastheÞrstresistive-inertialsincetheplasmaresponseisdominatedbyre-sistivityandioninertia.Inthelimit1,and WeshallrefertothisasthesecondsemicollisionalSupposethat.Equationducesto i
QQic2p2 Inthelimit,Eqs.,and032503-4A.ColeandR.FitzpatrickPhys.Plasmas,032503 EMy F.ModiÞedEBvelocityproÞleNow,wedonotexpecttheappliederrorÞeldtosigniÞ-cantlychangetheequilibriumelectron/iondiamagneticve-locityproÞles,i.e.,wedonotexpectachangeintheequilib-riumplasmapressuregradient.However,theerrorÞeldisfreetomodifytheequilibriumvelocityproÞleexertingelectromagneticforcesontheplasma.Infact,thesteady-stateperturbedvelocityproÞletakestheformisthemodiÞedvelocityat=0.TheabovevelocityproÞleisconsistentwithanerror-Þeldinducedelec-tromagneticforcewhichislocalizedaround=0,andbound-aryconditionswhicheffectivelypreventtheequilibriumelectricÞeldattheedgeoftheplasmafromchanging.G.ViscousforceThenet-directedviscousforceactingontheinnerre-gioninasteadystateisgivenbyVSy H.ForcebalanceInasteadystate,zeronetforceactsontheinnerregioninthedirection.ItfollowsthatEMyVSy=0, 2
E2Im
 I.Normalizedforcebalanceequation,and.Thenor-malizedforcebalanceequationbecomes +ˆ
Q=4P
Q0Q Tomakefurtherprogress,weneedtoevaluateThisinformationcanonlybeobtainedfromnonideallayerIII.DRIFT-MHDLAYERPHYSICSA.IntroductionSupposethattheerrorÞeldisquasistatic,sothat.LinearizingEqs.,assuminganexpikydenceofperturbedquantitiesinthedirection,normalizing,andtakingthelimits=0and1,weobtain ,
= ˜+Xd2˜ +2z+c2d2Z˜ ,
i
QQid2 ˜ =2˜ +Pd4
˜+Z˜ ,
z˜++Pd2V˜z ,andAsymptoticmatchingtotheouterregionyieldstheboundaryseeEqs. B.FouriertransformationipXetc.TheFouriertransformedlayerequationsbecome p2¯,
= ¯D2d
p2¯ c2z c2p2Z¯,
i
QQip2 ¯=d
p2¯
¯+Z¯,
z¯+ V¯z,
where ¯
p
¯0ˆ 1 C.LayerequationLetusignoretheterminEq..Thisap-proximationcanbeveriÞedaposteriori.Equationsreduceto032503-3Drift-magnetohydrodynamicalmodeloferror-ÞeldpenetrationPhys.Plasmas,032503 minologyisahighpoloidalbetaorderingschemedoesnotwhichintokamakterminologyisthetoroidaltobeeithermuchlessthanormuchgreaterthanunity.Weadoptthereduced,two-dimensional,drift-MHDequationsderivedinRef.13,
t= Z,+ J,
1
Z
t= ,Z+c2Vz,+d2J,+c2 Y,
2
U
t= ,U 22 ,Z+U,Z+Y, J,+i2
U+Y,
3
Vz ,and.Here, /
,d=cdi/ ,and.Theguiding-centervelocityiswritten.More-over,istheplasmaresistivity,ionviscosity,andthenormalizedcollisionlessionskindepth.Finally,isthemagneticßuxfunction,ispropor-tionaltotheperturbedplasmapressure,istheguiding-centerstreamfunction,andistheionveloc-ityintheNotethat,inadaptingtheaboveequationsfromRef.13,wehaveneglectedelectronviscosityandthermalconductiv-ity,forthesakeofsimplicity.Notealsothatthetransport,appearingintheseequations,isphenomeno-logicalinnature,andissupposedtorepresenttheanomaloustransportofmomentumacrossmagneticßuxsurfacesduetosmall-scaleplasmaturbulence.Equationsbothelectronandiondiamagneticeffects,includingthecon-tributionoftheanisotropiciongyroviscoustensor,butne-glectelectroninertia.OurequationsareÒreducedÓinthesensethattheydonotcontainthecompressibleAlfvŽnwave.However,theydocontaintheshear-AlfvŽnwave,themag-netoacousticwave,andthewhistler/kinetic-AlfvŽnwave.B.InitialplasmaequilibriumSupposethattheplasmaisperiodicinthewithperiod,andisboundedbyperfectlyconductingwalls=±1.Theinitialplasmaequilibriumtakestheform/2,,and=0.Thisisatearing-stableequilibrium,withtheuniform,andtheuniform-directedionandelectrondiamagneticvelocities,respectively.Here,isthetotaldiamagneticvelocity.C.ErrorÞeldTheerrorÞeldisgeneratedbyapplyingequalandoppo--directed,periodicdisplacements,,totheconductingwalls,where.Theappropriateboundaryconditiononissimply±1, Inthefollowing,isassumedtobeeitherstatic,orveryslowlyvarying,asisappropriateforanerrorÞeld.D.AsymptoticmatchingThroughoutmostoftheplasma,theresponsetotheap-pliederrorÞeldisdeterminedbytheequationsofmarginallystable,ideal-MHD.Thisregionistermedtheouterregion.Marginallystable,ideal-MHDbreaksdowninathinlayer,centeredon=0,whichistermedtheinnerregion.Intheouterregion,thevorticityequation.reducesto0.Linearizingthisequation,assumingasindependenceofperturbedquantitiesinthedirection,we intheouterregion,wheredenotesaperturbedquantity.Theboundaryconditionsare,fromsymme-try,and±1,.ThemostgeneralsolutiontoEq.takestheform tanhk +
tsinhkx istheperturbedmagneticßuxattheedgeoftheinnerregionwhichequalsthereconnectedßuxina.Theparameter measuresthestrengthofthecurrentsheetinducedintheinnerregion.Asymptoticmatchingbetweentheinnerandouterregionsyields =2/tanh,and/sinh.Notethat0istheconventionaltearingstabilityindex.E.ElectromagneticforceThemeanelectromagneticforceactingontheplasmainthedirectiontakestheform 2Im
2˜ Itisclear,fromEq.,thatzeromeanforceisexertedontheplasmaintheouterregion.Thenetmeanforceexertedontheinnerregioniswritten032503-2A.ColeandR.FitzpatrickPhys.Plasmas,032503 Wecanwrite,and,whereisthemagneticÞeld,andthetotalplasmapressure.Here,weareassumingthatareuniform, Drift-magnetohydrodynamicalmodeloferror-ÞeldpenetrationintokamakplasmasA.ColeandR.FitzpatrickInstituteforFusionStudies,DepartmentofPhysics,UniversityofTexasatAustin,Austin,Texas78712Received16September2005;accepted30January2006;publishedonline16March2006Apreviouslypublishedmagnetohydrodynamicalmodeloferror-Þeldpenetrationintokamakplasmasisextendedtotakedrift-MHDphysicsintoaccount.Inparticular,diamagneticandsemicollisionaleffectsarebothfullyincorporatedintotheanalysis.Thenewmodelisusedtoexaminethescalingofthepenetrationthresholdinohmictokamakplasmas.©2006AmericanInstituteofPhysicsI.INTRODUCTIONTokamakplasmasareinvariablysubjecttosmallampli-tude,static,resonantmagneticperturbationsÑknownaser-rorÞeldsÑwhichareprimarilygeneratedbyÞeld-coilmis-