Introduction to Modern Magnetohydrodynamics

281 Pages·2016·26.264 MB·English

by  Sébastien Galtier

Introduction to Modern Magnetohydrodynamics
Ninety-nine percent of ordinary matter in the Universe is in the form of ion-
izedfluids,orplasmas.Thestudyofthemagneticpropertiesofsuchelectrically
conducting fluids, magnetohydrodynamics (MHD), has become a central theory
in astrophysics, as well as in areas such as engineering and geophysics. This
textbookoffersacomprehensiveintroductiontoMHDanditsrecentapplications,
innatureandinlaboratoryplasmas;fromthemachineryoftheSunandgalaxies,
tothecoolingofnuclearreactorsandthegeodynamo.Itexposesadvancedunder-
graduate and graduate students to both classical and modern concepts, making
them aware of current research and the ever-widening scope of MHD. Rigorous
derivations within the text, supplemented by over 100 illustrations and followed
byexercisesandworkedsolutionsattheendofeachchapter,provideanengaging
and practical introduction to the subject and an accessible route into this wide-
rangingfield.
Se´bastien Galtier is a Professor of astrophysics at the Universite´ Paris–Saclay,
France. His research focuses on magnetohydrodynamic turbulence, and he has
published widely in the field. He was President of the French National Program
inSolarPhysics(CNRS)duringtheyears2010–2014,andisanhonorarymember
oftheprestigiousInstitutUniversitairedeFrance.
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Introduction to
Modern
Magnetohydrodynamics
Se´bastien Galtier
Universite´ Paris–Saclay
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©EditionsVuibert–Paris2013
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Firstpublished2013asMagne´tohydrodynamique–Desplasmasdelaboratoirea`l’astrophysique
UpdatedEnglisheditionpublished2016
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Names:Galtier,Se´bastien,author.
Title:Introductiontomodernmagnetohydrodynamics/Se´bastienGaltier,
Universite´Paris–Saclay
Description:Cambridge,UnitedKingdom;NewYork,NY:CambridgeUniversity
Press,2016.|Includesbibliographicalreferencesandindex.
Identifiers:LCCN2016014636|ISBN9781107158658(Hardback)|ISBN1107158656(Hardback)
Subjects:LCSH:Magnetohydrodynamics.
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Tomyfamily
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Contents
Preface pagexiii
TableofPhysicalQuantities xv
PartI Foundations 1
1 Introduction 3
1.1 SpaceandLaboratoryPlasmaPhysics 3
1.2 WhatIsaPlasma? 7
1.3 KineticDescription 9
1.3.1 CollisionlessPlasma 10
1.3.2 PlasmawithCollisions 11
1.3.3 Non-linearities 12
1.4 TimeScalesandLengthScales 13
1.4.1 PlasmaOscillations 13
1.4.2 ElectricScreening 14
1.4.3 MagneticScreening 15
1.4.4 Cyclotronfrequency 16
1.5 FromKinetictoFluids 17
1.5.1 Multi-fluidEquations 17
1.5.2 Mono-fluidEquations 18
2 Magnetohydrodynamics 19
2.1 Introduction 19
2.1.1 ElectromagneticInduction 19
2.1.2 ExtensiontoConductingFluids 21
vii
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viii Contents
2.2 TowardsaFormulationofMHD 21
2.2.1 Maxwell’sEquations 21
2.2.2 GeneralizedOhm’sLaw 23
2.3 Quasi-neutrality 24
2.4 Generalized(Hall)MHDEquations 25
2.4.1 TheIncompressibleLimit 27
2.4.2 ElectronMHD 28
2.4.3 IdealMHD 28
2.5 ExamplesofElectricallyConductingFluids 29
3 Conservation Laws 32
3.1 Mass 32
3.2 Momentum 33
3.3 Energy 34
3.4 Cross-helicity 36
3.5 MagneticHelicity 38
3.6 Alfve´n’sTheorem 39
3.6.1 MagneticFluxConservation 39
3.6.2 Kelvin’sTheorem 40
3.6.3 Alfve´n’sTheorem 41
3.7 MagneticTopology 42
3.8 TopologyatSub-ionScales 45
ExercisesforPartI 47
PartII FundamentalProcesses 49
4 Magnetohydrodynamic Waves 51
4.1 MagneticTension 51
4.2 Alfve´nWaves 53
4.3 MagnetosonicWaves 55
4.4 Whistler,Ion-Cyclotron,andKineticAlfve´nWaves 58
4.4.1 IncompressibleHelicalWaves 58
4.4.2 CompressibleHallMHDWaves 61
5 Dynamos 65
5.1 Geophysics,Astrophysics,andExperiments 65
5.1.1 ExperimentalDynamos 65
5.1.2 NaturalDynamos 70
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Contents ix
5.2 TheCriticalMagneticReynoldsNumber 73
5.3 TheKinematicRegime 74
5.4 Anti-dynamoTheorems 76
5.5 ThePonomarenkoDynamo 79
5.6 TheTurbulentDynamo 83
5.6.1 KinematicMeanFieldTheory 83
5.6.2 Theα-effect 84
5.7 Conclusion 85
6 Discontinuities and Shocks 86
6.1 Rankine–HugoniotConditions 86
6.2 Discontinuities 91
6.2.1 TangentialandContactDiscontinuities 92
6.2.2 RotationalDiscontinuity 92
6.3 Shocks 94
6.3.1 IntermediateShocks 94
6.3.2 TrueShock 94
6.4 CollisionlessShocks 94
7 Magnetic Reconnection 96
7.1 ACurrentSheetinIdealMHD 96
7.2 TheSweet–ParkerModel 99
7.3 CollisionlessHallMHDReconnection 103
7.4 Perspectives 104
ExercisesforPartII 107
PartIII InstabilitiesandMagneticConfinement 111
8 Static Equilibrium 113
8.1 EquilibriumEquations 113
8.2 MagneticConfinementbyθ-Pinch 114
8.3 MagneticConfinementbyz-Pinch 116
8.4 ToroidalTokamakConfiguration 117
8.4.1 TheGrad–ShafranovEquation 118
8.4.2 TheSolovievExactSolution 121
8.5 Force-FreeFields 122
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x Contents
9 Linear Perturbation Theory 125
9.1 Instabilities 125
9.1.1 Classification 125
9.1.2 ConditionofExistence 126
9.2 KineticVersusFluid 127
9.2.1 TheKineticApproach 127
9.2.2 TheFluidApproach 128
9.3 TheEnergyStabilityCriterion 129
9.3.1 AOne-DimensionalExample 129
9.3.2 Two-DimensionalExamples 131
9.3.3 TheMHDCase 132
9.4 PerturbationTheory 133
9.4.1 TheSmall-DisplacementOperator 133
9.4.2 SolutiontoInitialValues 135
9.4.3 TheEquationoftheNormalModes 136
9.4.4 PropertiesoftheOperatorF 136
9.4.5 TheReturnontheEnergyIntegral 138
10 Study of MHD Instabilities 139
10.1 StabilityofMHDWaves 139
10.1.1 Alfve´nWaves 140
10.1.2 MagnetosonicWaves 141
10.2 Rayleigh–TaylorInstability 143
10.2.1 TheFirstMethod:EnergyIntegrals 143
10.2.2 TheSecondMethod:NormalModes 144
10.3 Kruskal–SchwarzschildInstability 146
10.4 z-PinchInstability 152
10.4.1 StaticEquilibrium 152
10.4.2 InstabilityModes 154
10.4.3 ResolutionbyNormalModes(Casem = 0) 155
10.4.4 Configurationm = 1 160
10.5 z–θ PinchInstability 160
10.6 Magneto-rotationalInstabilityinAccretionDisks 161
ExerciseforPartIII 168
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