Waves and oscillations in plasmas

570 Pages·2013·5.656 MB·English

by  Hans L Pécseli

SERIES IN PLASMA PHYSICS
Waves and
Oscillations
in Plasmas
Hans L. Pécseli
Waves and Oscillations in Plasmas
Series in Plasma Physics
Series Editor:
Steve Cowley, Imperial College, UK and UCLA, USA
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Series in Plasma Physics
Waves and
Oscillations
in Plasmas
Hans L. Pécseli
University of Oslo
Norway
The cover photo shows the Q-machine at the Risø National Laboratory  in Denmark. The device was operating from 1967
until 1992.
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Contents
Preface
1 Introduction 1
1.1 Whatisaplasma? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.2 Wheredowe(cid:192)ndplasma? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.3 Plasmaphysics–whybother? . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2
1.4 Whystudywavesinplasmas? . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
1.4.1 Alfve´nwaves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
1.4.2 Landaudampingandinstability . . . . . . . . . . . . . . . . . . . . . . . 4
2 BasicsofContinuumModels 5
2.1 Continuityequation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
2.2 Newton’ssecondlaw . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
2.3 Equationofstate . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
2.3.1 Gaspressureonasurface. . . . . . . . . . . . . . . . . . . . . . . . . . . 10
2.3.2 Heatcapacities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
2.4 Dynamicproperties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
2.4.1 Soundwaves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
2.5 Incompressibility . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
2.6 Molecularviscosity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
2.6.1 Anexplicitcalculationoftheviscositycoef(cid:192)cient . . . . . . . . . . . . . . 18
2.7 Thermalconductivityingases . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
2.8 Diffusioningases . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22
2.9 Dimensionalanalysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22
2.10 Summationconvention . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26
3 LinearWaveDynamics 29
3.1 Dispersionrelations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30
3.1.1 Complexnotation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31
3.1.2 Characteristicvelocities . . . . . . . . . . . . . . . . . . . . . . . . . . . 32
3.1.3 Dopplershifts. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33
3.1.4 Wavepolarization. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34
3.1.5 Methodofstationaryphase . . . . . . . . . . . . . . . . . . . . . . . . . . 36
3.1.6 Absoluteandconvectiveinstabilities . . . . . . . . . . . . . . . . . . . . . 37
3.1.7 Pulseresponse . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38
3.2 Wavepropagationininhomogeneousmedia . . . . . . . . . . . . . . . . . . . . . 42
3.2.1 Snell’slaw . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42
3.2.2 WKBanalysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42
4 WeaklyNonlinearWaves 45
4.1 Nondispersivewaves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46
4.1.1 Simplewaves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47
4.1.2 Burgers’equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50
4.2 Weaklydispersivewaves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53
4.2.1 Korteweg–deVriesequation . . . . . . . . . . . . . . . . . . . . . . . . . 53
4.2.2 PerturbationsofaKdVequation . . . . . . . . . . . . . . . . . . . . . . . 58
4.2.3 Boussinesqequations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61
4.2.4 Generalizationtothreedimensions. . . . . . . . . . . . . . . . . . . . . . 61
4.3 Stronglydispersivewaves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62
4.3.1 Simpleoscillators. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62
4.3.2 Weaklynonlineardispersivewaves . . . . . . . . . . . . . . . . . . . . . 64
4.3.3 Modulationalinstability . . . . . . . . . . . . . . . . . . . . . . . . . . . 67
4.3.4 SolitonsolutionsoftheNLSequation . . . . . . . . . . . . . . . . . . . . 68
4.3.5 DerivationofthenonlinearSchro¨dingerequation . . . . . . . . . . . . . . 69
4.3.6 Generalizationtothreespatialdimensions . . . . . . . . . . . . . . . . . . 71
5 BasicsofElectromagnetism 73
5.1 Maxwell’sequationsintheirbasicform . . . . . . . . . . . . . . . . . . . . . . . 73
5.1.1 Boundaryconditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74
5.1.2 Materialrelationsforsimplemedia . . . . . . . . . . . . . . . . . . . . . 75
5.2 DiscussionsofMaxwell’sequations . . . . . . . . . . . . . . . . . . . . . . . . . 75
5.3 Potentials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79
5.4 Poynting’sidentity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79
5.4.1 Poynting’sidentityforavacuum . . . . . . . . . . . . . . . . . . . . . . . 80
5.5 Electromagneticforces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82
5.5.1 Electromagneticforcesonparticlesandcurrents . . . . . . . . . . . . . . 82
5.5.2 Electromagneticforcesonmatter . . . . . . . . . . . . . . . . . . . . . . 83
5.6 Wavesinsimpleconductingmedia (cid:2)k . . . . . . . . . . . . . . . . . . . . . . . 85
5.7 Polarizationdescription . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87
5.7.1 Methodofimages . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89
5.8 Lorentztransformations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90
5.9 Dielectricproperties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94
5.9.1 Simplemedia . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94
5.9.2 Materialrelations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95
5.9.3 De(cid:192)nitionofthedielectricfunction . . . . . . . . . . . . . . . . . . . . . 97
5.10 Energydensityindielectrics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98
5.10.1 Simplemedia . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98
5.10.2 Realdielectricfunctions—dispersivemedia . . . . . . . . . . . . . . . . . 99
5.10.2.1 Electrostaticwaves . . . . . . . . . . . . . . . . . . . . . . . . 100
5.10.3 Inclusionofspatialdispersion . . . . . . . . . . . . . . . . . . . . . . . . 101
5.10.4 Negativeenergywaves . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102
5.10.5 Complexdielectricfunctions—dispersivemedia . . . . . . . . . . . . . . 102
5.10.6 Dampingbydielectriclosses—electrostaticwaves . . . . . . . . . . . . . 103
5.11 Forceona(cid:193)uidoragas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104
6 NaturalOccurrencesofPlasmas 109
6.1 Saha’sequation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109
6.2 Coronalequilibrium (cid:2)k . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111
6.3 Chapmanionosphere (cid:2)k . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113
7 SingleParticleMotion 119
7.1 Singleparticleorbits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119
7.1.1 E(cid:3)B . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119
7.1.2 E⊥B . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121
7.1.3 F ⊥B . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122
c
7.1.4 FiniteLarmorradiuscorrectionsforinhomogeneouselectric(cid:192)elds . . . . . 125
7.1.5 Polarizationdrifts,dE/dt(cid:5)=0 . . . . . . . . . . . . . . . . . . . . . . . . 127
7.1.6 (cid:148)B⊥B . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 128
7.1.7 (cid:148)B(cid:3)B . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132
7.1.8 Magneticmoment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133
7.1.9 Magneticmirrorcon(cid:192)nement . . . . . . . . . . . . . . . . . . . . . . . . 137
7.1.10 Larmor’stheorem. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 140
7.2 Adiabaticinvariants . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141
7.3 Radiationlosses . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143
7.4 TheDessler–Parkerrelations (cid:2)k . . . . . . . . . . . . . . . . . . . . . . . . . . . 145
8 BasicPlasmaParameters 149
8.1 Plasmafrequencies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 149
8.2 TheDebyelength . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 150
8.3 Theplasmaparameter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 150
8.4 Debyeshielding . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 151
8.4.1 Immobileions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 151
8.4.1.1 Shieldinginthreespatialdimensions . . . . . . . . . . . . . . . 152
8.4.1.2 Shieldingintwospatialdimensions . . . . . . . . . . . . . . . . 153
8.4.1.3 Shieldinginonespatialdimension . . . . . . . . . . . . . . . . 153
8.4.2 Mobileions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 154
8.4.3 NonlinearDebyeshieldinginonespatialdimension (cid:2)k . . . . . . . . . . . 155
8.5 Interactionenergy (cid:2)k . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 156
8.6 EvacuationofaDebyeSphere (cid:2)k . . . . . . . . . . . . . . . . . . . . . . . . . . 158
8.7 Collisionsbetweenchargedparticles . . . . . . . . . . . . . . . . . . . . . . . . . 158
8.7.1 Simpleargumentsforcollisionalcrosssections . . . . . . . . . . . . . . . 160
8.7.2 Center-of-massdynamics. . . . . . . . . . . . . . . . . . . . . . . . . . . 161
8.8 Plasmaresistivitybyelectron–ioncollisions . . . . . . . . . . . . . . . . . . . . . 163
8.8.1 Collisionsinmagnetic(cid:192)elds . . . . . . . . . . . . . . . . . . . . . . . . . 167
8.8.2 Collisionsinelectricandmagnetic(cid:192)elds . . . . . . . . . . . . . . . . . . 168
8.8.2.1 TheapproximationE(y)≈E y/(cid:2) . . . . . . . . . . . . . . . . 169
0 1
8.8.2.2 TheapproximationE(y)≈E (y/(cid:2) +y2/(cid:2)2) . . . . . . . . . . . 170
0 1 2
8.9 Plasmaresistivitybyneutralcollisions (cid:2)k . . . . . . . . . . . . . . . . . . . . . . 172
8.9.1 Time-varyingelectric(cid:192)elds . . . . . . . . . . . . . . . . . . . . . . . . . 175
8.10 Plasmaasadielectric (cid:2)k . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 176
8.10.1 Plasmaasadielectricathighfrequencies . . . . . . . . . . . . . . . . . . 177
8.10.2 Amagnetizedplasmaasadielectricatlowfrequencies . . . . . . . . . . . 177