Table Of Content

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 Other recent books in the series: Complex and Dusty Plasmas: From Laboratory to Space V E Fortov and G E Morﬁll Applications of Laser–Plasma Interactions Shalom Eliezer and Kunioki Mima Plasma Kinetic Theory D G Swanson High Power Microwaves, Second Edition J Benford, J A Swegle and E Schamiloglu Controlled Fusion and Plasma Physics K Miyamoto Plasma Electronics: Applications in Microelectronic Device Fabrication T Makabe and Z Petrović An Introduction to Inertial Confinement Fusion S Pfalzner Aspects of Anomalous Transport in Plasmas R Balescu Non-Equilibrium Air Plasmas at Atmospheric Pressure K H Becker, R J Barker and K H Schoenbach (Eds) Magnetohydrodynamic Waves in Geospace: The Theory of ULF Waves and their Interaction with Energetic Particles in the Solar-Terrestrial Environment A D M Walker Plasma Physics via Computer Simulation (paperback edition) C K Birdsall and A B Langdon Plasma Waves, Second Edition D G Swanson Microscopic Dynamics of Plasmas and Chaos Y Elskens and D Escande Plasma and Fluid Turbulence: Theory and Modelling A Yoshizawa, S-I Itoh and K Itoh The Interaction of High-Power Lasers with Plasmas S Eliezer Introduction to Dusty Plasma Physics P K Shukla and A A Mamun 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. CRC Press Taylor & Francis Group 6000 Broken Sound Parkway NW, Suite 300 Boca Raton, FL 33487-2742 © 2013 by Taylor & Francis Group, LLC CRC Press is an imprint of Taylor & Francis Group, an Informa business No claim to original U.S. Government works Version Date: 20120719 International Standard Book Number-13: 978-1-4398-7849-1 (eBook - PDF) This book contains information obtained from authentic and highly regarded sources. Reasonable efforts have been made to publish reliable data and information, but the author and publisher cannot assume responsibility for the valid- ity of all materials or the consequences of their use. 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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