Springer Series on ATOMIC, OPTICAL, AND PLASMA PHYSICS 71 Springer Series on ATOMIC, OPTICAL, AND PLASMA PHYSICS TheSpringerSeriesonAtomic,Optical,andPlasmaPhysicscoversinacomprehensivemanner theory and experiment in the entire field of atoms and molecules and their interactionwith electromagneticradiation.Booksintheseriesprovidearichsourceofnewideasandtechniques with wide applications in fields such as chemistry, materials science, astrophysics, surface science, plasma technology, advanced optics, aeronomy, and engineering. Laser physics is a particularconnectingthemethathasprovidedmuchofthecontinuingimpetusfornewdevel- opmentsinthefield.Thepurposeoftheseriesistocoverthegapbetweenstandardundergrad- uatetextbooksandtheresearchliteraturewithemphasisonthefundamentalideas,methods, techniques,andresultsinthefield. PleaseviewavailabletitlesinSpringerSeriesonAtomic,Optical,andPlasmaPhysicsonseries homepagehttp://www.springer.com/series/411 Jan Weiland Stability and Transport in Magnetic Confinement Systems With 51 Figures JanWeiland ChalmersUniversityofTechnology andEURATOMVRAssociation Gothenburg,Sweden ISSN1615-5653 ISBN978-1-4614-3742-0 ISBN978-1-4614-3743-7(eBook) DOI10.1007/978-1-4614-3743-7 SpringerNewYorkHeidelbergDordrechtLondon LibraryofCongressControlNumber:2012935694 #SpringerScience+BusinessMediaNewYork2012 Thisworkissubjecttocopyright.AllrightsarereservedbythePublisher,whetherthewholeorpartof the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation,broadcasting,reproductiononmicrofilmsorinanyotherphysicalway,andtransmissionor informationstorageandretrieval,electronicadaptation,computersoftware,orbysimilarordissimilar methodologynowknownorhereafterdeveloped.Exemptedfromthislegalreservationarebriefexcerpts inconnectionwithreviewsorscholarlyanalysisormaterialsuppliedspecificallyforthepurposeofbeing enteredandexecutedonacomputersystem,forexclusiveusebythepurchaserofthework.Duplication ofthispublicationorpartsthereofispermittedonlyundertheprovisionsoftheCopyrightLawofthe Publisher’s location, in its current version, and permission for use must always be obtained from Springer.PermissionsforusemaybeobtainedthroughRightsLinkattheCopyrightClearanceCenter. ViolationsareliabletoprosecutionundertherespectiveCopyrightLaw. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publicationdoesnotimply,evenintheabsenceofaspecificstatement,thatsuchnamesareexempt fromtherelevantprotectivelawsandregulationsandthereforefreeforgeneraluse. While the advice and information in this book are believed to be true and accurate at the date of publication,neithertheauthorsnortheeditorsnorthepublishercanacceptanylegalresponsibilityfor anyerrorsoromissionsthatmaybemade.Thepublishermakesnowarranty,expressorimplied,with respecttothematerialcontainedherein. Printedonacid-freepaper SpringerispartofSpringerScience+BusinessMedia(www.springer.com) Preface This book presents the collective drift and MHD type modes in inhomogeneous plasmas from the point of view of two fluid and kinetic theory. It is based on a lectureseriesgivenatChalmersUniversityofTechnology.Thetitleofthelecture notes is Low frequency modes associated with drift motions in inhomogeneous plasmas. The level is undergraduate to graduate. Basic knowledge of electrody- namicsandcontinuummechanics isnecessaryandanelementarycourseinPlasma Physics is a desirable background for the student. The author is grateful to A.Zagorodny,I.Holod,VZasenko,H.Nordman,A.Jarme´n,R.Singh,P.Anders- son,J.P.Mondt,H.Wilhelmsson,V.P.Pavlenko,H.SanukiandC.S.Liuformany enlighteningdiscussions,toG.Bateman,A.KritzandP.Strandforcollaborationon transport simulation, to my collaborators at JET, J.Christiansen, P. Mantica, V. Naulin,T.Tala,K.Crombe,E.AspandL.GarzottiinmodellingJETdischargesand toH.G.Gustavssonforhelpwithproofreading.ThanksarealsoduetotheAmeri- can Institute of Physics, the American Physical Society and Nuclear Fusion for allowing the use of several figures. Finally I extend my gratitude to my family, Wivan,HenrikandHelenafortheircontinousencouragementandsupport. Gothenburg,Sweden JanWeiland v Contents 1 Introduction.................................................................. 1 1.1 PrinciplesforConfinementofPlasma byaMagneticField.................................................... 1 1.2 EnergyBalanceinaFusionReactor................................... 4 1.3 MagnetohydrodynamicStability....................................... 7 1.4 Transport............................................................... 8 1.5 ScalingLawsforConfinementofPlasma inToroidalSystems.................................................... 9 1.6 TheStandpointofFusionResearchToday............................ 9 References................................................................... 10 2 DifferentWaysofDescribingPlasmaDynamics........................ 11 2.1 GeneralParticleDescription,Liouville andKlimontovichEquations.......................................... 11 2.2 KineticTheoryasGenerallyUsed byPlasmaPhysicists.................................................. 13 2.3 GyrokineticTheory.................................................... 14 2.4 FluidTheoryasObtainedbyTakingMoments oftheVlasovEquation................................................ 15 2.4.1 TheMaxwellEquations....................................... 16 2.4.2 TheLowFrequencyExpansion............................... 16 2.4.3 TheEnergyEquation ......................................... 18 2.5 GyrofluidTheoryasObtainedbyTakingMoments oftheGyrokineticEquation........................................... 20 2.6 OneFluidEquations.................................................. 21 2.7 FiniteLarmorRadiusEffectsinaFluidDescription................. 22 2.7.1 EffectsofTemperatureGradients ............................ 25 References................................................................... 26 vii viii Contents 3 FluidDescriptionforLowFrequencyPerturbations inanInhomogeneousPlasma.............................................. 27 3.1 Introduction.......................................................... 27 3.2 ElementaryPictureofDriftWaves.................................. 29 3.2.1 EffectsofFiniteIonInertia ................................. 32 3.2.2 DriftInstability.............................................. 34 3.2.3 ExcitationbyElectron-IonCollisions....................... 35 3.3 MHDTypeModes................................................... 36 3.3.1 Alfve´nWaves................................................ 37 3.3.2 InterchangeModes .......................................... 37 3.3.3 TheConvectiveCellMode.................................. 40 3.3.4 ElectromagneticInterchangeModes........................ 40 3.3.5 KinkModes.................................................. 43 3.3.6 StabilizationofElectrostaticInterchangeModes byParallelElectronMotion................................. 45 3.3.7 FLRStabilizationofInterchangeModes.................... 45 3.3.8 KineticAlfve’nWaves...................................... 47 3.4 QuasilinearDiffusion................................................ 49 3.5 ConfinementTime................................................... 52 3.6 Discussion............................................................ 53 References................................................................... 55 4 KineticDescriptionofLowFrequencyModes inInhomogeneousPlasma................................................. 57 4.1 IntegrationAlongUnperturbedOrbits............................... 57 4.2 UniversalInstability................................................. 63 4.3 InterchangeInstability............................................... 65 4.4 DriftAlfve’nWavesandbLimitation.............................. 67 4.5 LandauDamping..................................................... 70 4.6 TheMagneticDriftMode............................................ 71 4.7 TheDriftKineticEquation.......................................... 72 4.8 DielectricPropertiesofLowFrequencyVortexModes............ 73 4.9 FiniteLarmorRadiusEffectsObtained byOrbitAveraging.................................................. 76 4.10 Discussion............................................................ 80 4.11 Exercises............................................................. 80 References................................................................... 81 5 KineticDescriptionsofLowFrequencyModesObtained byGyroaveraging.......................................................... 83 5.1 TheDriftKineticEquation.......................................... 83 5.1.1 MomentEquations........................................... 87 5.1.2 TheMagneticDriftMode................................... 88 5.1.3 TheTearingMode........................................... 89 Contents ix 5.2 TheLinearGyrokineticEquation.................................... 90 5.2.1 Applications.................................................. 94 5.3 TheNonlinearGyrokineticEquation................................ 96 5.4 Gyro-FluidEquations................................................ 99 References................................................................. 100 6 LowFrequencyModesinInhomogeneousMagneticFields.......... 101 6.1 AnomalousTransportinSystemswithInhomogeneous MagneticFields.................................................... 101 6.2 ToroidalModeStructure........................................... 103 6.3 CurvatureRelations................................................ 107 6.4 TheInfluenceofMagneticShearonDriftWaves................. 110 6.5 InterchangePerturbationsAnalysed bytheEnergyPrincipleMethod................................... 113 6.6 Eigenvalue Equations for MHD TypeModes..................... 116 6.6.1 StabilizationofInterchangeModes byMagneticShear ....................................... 116 6.6.2 BallooningModes........................................ 119 6.7 TrappedParticleInstabilities...................................... 128 6.8 ReactiveDriftModes.............................................. 131 6.8.1 IonTemperatureGradientModes........................ 132 6.8.2 ElectronTemperatureGradientMode ................... 135 6.8.3 TrappedElectronModes................................. 136 6.9 CompetitionBetweenInhomogeneitiesinDensity andTemperature................................................... 139 6.10 AdvancedFluidModels........................................... 140 6.10.1 The Development of Research........................... 141 6.10.2 Closure.................................................... 144 6.10.3 Gyro-LandauFluidModels............................... 146 6.10.4 NonlinearKineticFluidEquations....................... 147 6.10.5 ComparisonswithNonlinearGyrokinetics............... 148 6.11 ReactiveFluidModelforStrongCurvature....................... 150 6.11.1 TheToroidalZ Mode.................................... 151 i 6.11.2 ElectronTrapping......................................... 154 6.11.3 Transport.................................................. 156 6.11.4 NormalizationofTransportCoefficients................. 158 6.11.5 FiniteLarmorRadiusStabilization....................... 159 6.11.6 TheEigenvalueProblemforToroidal DriftWaves............................................... 160 6.11.7 EarlyTestsoftheReactiveFluidModel................. 163 6.12 ElectromagneticModesinAdvancedFluidDescription.......... 164 6.12.1 EquationsforFreeElectrons IncludingKinkTerm..................................... 165 6.12.2 KineticBallooningModes................................ 167 x Contents 6.13 ResistiveEdgeModes............................................. 168 6.13.1 ResistiveBallooningModes.............................. 170 6.13.2 TransportintheEnhancedConfinementState........... 173 6.14 Discussion.......................................................... 175 References................................................................. 176 7 Transport,OverviewandRecentDevelopments....................... 181 7.1 StabilityandTransport............................................. 181 7.2 MomentumTransport............................................. 181 7.2.1 SimulationofanInternalBarrier......................... 183 7.2.2 SimulationofanEdgeBarrier............................. 184 7.3 Discussion.......................................................... 187 References................................................................. 187 8 InstabilitiesAssociatedwithFastParticles inToroidalConfinementSystems....................................... 191 8.1 GeneralConsiderations............................................ 191 8.2 TheDevelopmentofResearch..................................... 192 8.3 DilutionDuetoFastParticles..................................... 193 8.4 FishboneTypeModes............................................. 194 8.5 ToroidalAlfve´nEigenmodes...................................... 195 8.6 Discussion.......................................................... 197 References................................................................. 198 9 NonlinearTheory......................................................... 199 9.1 TheIonVortexEquation.......................................... 199 9.2 TheNonlinearDielectric.......................................... 207 9.3 Diffusion........................................................... 208 9.4 Fokker-PlanckTransitionProbability............................. 212 9.5 Discussion.......................................................... 215 References................................................................. 215 GeneralReferences........................................................... 219 AnswerstoExercises ........................................................ 221 Index........................................................................... 225