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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. The authors and publishers have attempted to trace the copyright holders of all material reproduced in this publication and apologize to copyright holders if permission to publish in this form has not been obtained. If any copyright material has not been acknowledged please write and let us know so we may rectify in any future reprint. Except as permitted under U.S. Copyright Law, no part of this book may be reprinted, reproduced, transmitted, or uti- lized in any form by any electronic, mechanical, or other means, now known or hereafter invented, including photocopy- ing, microfilming, and recording, or in any information storage or retrieval system, without written permission from the publishers. For permission to photocopy or use material electronically from this work, please access www.copyright.com (http:// www.copyright.com/) or contact the Copyright Clearance Center, Inc. (CCC), 222 Rosewood Drive, Danvers, MA 01923, 978-750-8400. CCC is a not-for-profit organization that provides licenses and registration for a variety of users. For organizations that have been granted a photocopy license by the CCC, a separate system of payment has been arranged. Trademark Notice: Product or corporate names may be trademarks or registered trademarks, and are used only for identification and explanation without intent to infringe. Visit the Taylor & Francis Web site at http://www.taylorandfrancis.com and the CRC Press Web site at http://www.crcpress.com To myfamily Aslongasthecenturiescontinuetounfold,thenumberofbookswillgrowcontinually,andone canpredictthatatime willcome, whenitwill be almostasdif(cid:192)cultto learnanythingfrom books asfromthedirectstudyofthewholeuniverse.Itwillbealmostasconvenienttosearchforsomebit oftruthconcealedinnatureasitwillbeto(cid:192)ndithiddenawayinanimmensemultitudeofbound volumes.DenisDiderot,In:Encyclope´die,1755. 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

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