Series in Plasma Physics Plasma Waves, 2nd Edition D Gary Swanson Auburn University, USA Institute of Physics Publishing Bristol and Philadelphia Copyright © 2003 IOP Publishing Ltd. c IOP Publishing Ltd 2003 All rights reserved. No part of this publication may be reproduced, stored in a retrieval system or transmitted in any form or by any means, electronic, mechanical, photocopying, recording or otherwise, without the prior permission of the publisher. Multiple copying is permitted in accordance with the terms of licences issued by the Copyright Licensing Agency under the terms of its agreement with Universities UK (UUK). D G Swanson has asserted his moral rights under the Copyright, Designs and Patents Act 1998 to be identified as the author of this work. British Library Cataloguing-in-Publication Data A catalogue record for this book is available from the British Library. ISBN 0 7503 0927 X Library of Congress Cataloging-in-Publication Data are available First Edition published by Academic Press, New York, 1989 Commissioning Editor: John Navas Production Editor: Simon Laurenson Production Control: Sarah Plenty Cover Design: Victoria Le Billon Marketing: Nicola Newey and Verity Cooke Published by Institute of Physics Publishing, wholly owned by The Institute of Physics, London Institute of Physics Publishing, Dirac House, Temple Back, Bristol BS1 6BE, UK US Office: Institute of Physics Publishing, The Public Ledger Building, Suite 929, 150 South Independence Mall West, Philadelphia, PA 19106, USA Typeset in LATEX2ε by Text 2 Text, Torquay, Devon Printed in the UK by MPG Books Ltd, Bodmin, Cornwall Copyright © 2003 IOP Publishing Ltd. Contents Prefacetothefirstedition Prefacetothesecondedition Acknowledgements Acronyms 1Introduction 1.1Propertiesofplasmas 1.1.1Unmagnetizedplasmas 1.1.2Magnetizedplasmas 1.1.3Thermalplasmas 1.2Plasmawaveapplications 1.2.1Plasmawavesinionosphericphysics 1.2.2Plasmawavesinastrophysics 1.2.3Plasmawavesinmagnetizedfusionplasmas 1.2.4Plasmawavesinlaser-producedplasmas 1.3Reviewofelectromagneticwavepropagation 1.3.1TheMaxwellequations 1.3.2PropertiesoftheHelmholtzequation 1.3.3Conservationlawsforelectromagneticfields 1.3.4ConservationlawswithFourieramplitudes 1.3.5MethodsofgeometricopticsfromWKBtheory 1.4Statisticalmechanicsofplasmas 1.5Overviewofth eplasmawavezoo 2Wavesinacolduniformplasma 2.1Thecoldplasmadispersionrelation 2.1.1Equationsofmotion 2.1.2Coldplasmadielectrictensor 2.1.3Formsofthedispersionrelation 2.2TheCMAdiagram 2.2.1Principalsolutions—parallelpropagation 2.2.2Principalsolutions—perpendicularpropagation Copyright © 2003 IOP Publishing Ltd. 2.2.3CMAboundaries—cutoffsandresonances 2.2.4Wavenormalsurfacetopology—spheroidsandlemniscoids 2.2.5Labeling—leftandright,ordinaryandextraordinary 2.2.6TheCMAdiagramforaone-ionspeciesplasma 2.3Phaseandgroupvelocityinthree-dimensions 2.3.1Theone-dimensionalcase 2.3.2Thethree-dimensionalcase 2.3.3Groupvelocitysurfaces 2.4ω(k,θ)dispersionsurfaces 2.4.1Underdensecase,ωp/ωce=0.32 2.4.2Overdensecase,ωp/ωce=3.2 2.5Examplesofpropagationatarbitrary θ 2.5.1Lowfrequencywaves 2.5.2Intermediatefrequencywaves—whistlers 2.6Faradayrotation 2.6.1Highfrequencylimit—region1 2.6.2Lowfrequencylimit—region13 2.7Plasmainterferometry 2.7.1Detectingthesignal 2.7.2Interpretingthesignalwhenωωp 2.7.3Interpretingthesignalwhenω∼ωp0 2.8Electrostaticwaves 2.8.1Validityconditionsfortheelectrostaticapproximation 2.8.2Lowerhybridwaves 2.8.3Resonancecones 2.9Particlemotionsnearresonance 2.9.1Lowerhybridresonance(highdensitycase) 2.9.2Upperhybridresonance 2.9.3Cyclotronresonances 3Wavesinfluidplasmas 3.1Momentsofthedistributionfunction 3.1.1Themomentequations 3.1.2Longitudinalplasmaoscillationsfromthemoment equations 3.2Thefluidequations 3.3Lowfrequencywaves 3.3.1Thelow-frequencydispersionrelation 3.3.2StringerdiagramsoftheLFDR 3.3.3Approximatedispersionrelationsandtransitions 3.3.4Parallelandperpendicularpropagation 3.3.5Highfrequencywaves 3.3.6Summaryoffluidwaves 3.4Partiallyionizedplasmasandcollisions Copyright © 2003 IOP Publishing Ltd. 3.4.1Neutralcollisions 3.4.2Electron–ioncollisions 3.5Amplifyi ngwavesandinstabilities 3.5.1Classificationofinstabilities 3.5.2Streaminginstabilities 3.6Powerandenergyflowinfluidplasmas 4Kinetictheoryofplasmawaves 4.1Thebasicequations 4.1.1TheBoltzmannequation 4.1.2CollisionsandtheFokker–Planckequation 4.1.3BBGKYtheory 4.1.4TheVlasovequations 4.2Wavesinathermal,unmagnetizedplasma 4.2.1Vlasovmethod 4.2.2Landausolution 4.2.3APhysicalpictureofLandaudamping 4.2.4ConventionaldescriptionsofLandaudamping 4.2.5IonacousticwavesandionLandaudamping 4.2.6EffectsofcollisionsonLandaudamping 4.3Wavesinamagnetizedhotplasma 4.3.1Theevolutionofthedistributionfunction 4.3.2Integratingalongtheunperturbedorbits 4.3.3Generalf0(v⊥,vz) 4.3.4Maxwelliandistributions 4.3.5Thedielectrictensor 4.3.6Thehotplasmadispersionrelation 4.3.7Examplesofhotplasmawaveeffects 4.4Electrostaticwaves 4.4.1Perpendicularpropagation–Bernsteinmodes 4.4.2High-orderBernsteinmodes 4.5Velocityspaceinstabilities 4.5.1Anisotropictemperature 4.6Conservationofenergyandpowerflow 4.6.1Poynting’stheoremforkineticwaves 4.6.2Groupvelocityandkineticflux 4.7Relativisticplasmaeffects 4.7.1Therelativisticdielectrictensor 4.7.2Therelativisticdielectrictensorwithoutsums 4.7.3Theweaklyrelativisticdielectrictensor 4.7.4Moderatelyrelativisticexpressions 4.7.5Exactexpressionswith n=0 4.7.6TherelativisticX-wave Copyright © 2003 IOP Publishing Ltd. 5Boundedhomogeneousplasmas 5.1Introduction 5.2Boundaryconditions 5.2.1Conductingboundary 5.2.2Plasma–vacuum(ordielectric)interface 5.3Unmagnetizedplasmas 5.3.1Scatteringfromaplasmacolumn 5.3.2Surfacewavesinapartially-filledplasmawaveguide 5.4Electrostaticwavesonaplasmacolumninamagneticfield 5.5Coldplasma-filledwaveguide 5.5.1Thedispersionrelation 5.5.2Wavefieldsandboundaryconditions 5.5.3MHDapproximation—ωωci 5.5.4Intermediatefrequencycase—ωωciωp 5.5.5Modeorthogonalityandpowerflow 5.5.6Antennaproblems 5.5.7Experimentsinplasma-filledwaveguides 5.6Conductingwallwithvacuumlayer,m=0,±1 5.7Infinitemagneticfieldapproximation 5.7.1Coldplasma-filledwaveguideinaninfinitemagneticfield 5.7.2Hotplasma-filledwaveguide 6Wavesininhomogeneousplasmas 6.1Introduction 6.2WKBmethodforone-dimensionalinhomogeneities 6.2.1Behaviornearacutoff 6.2.2Tunnelingbetweenback-to-backcutoffs 6.2.3Behaviornearanisolatedresonance 6.2.4Behaviorneararesonance–cutoffpair 6.3Modeconversiontheory 6.3.1Themodeconversiontheorem 6.3.2Solutionofthetunnelingequation 6.3.3Modeconversionexamples 6.3.4Conservationofenergy 6.4Absorptionandemission 6.4.1GeneralizedKirchhoff’slaw 6.4.2Absorptionandmodeconversion 6.5WKBMethodforthree-dimensionalinhomogeneousplasmas— raytracing 6.5.1Therayequations 6.5.2Theinhomogeneousplasmadispersionrelation 6.5.3Theamplitudeequations 6.6Driftwavesandinstabilities 6.6.1Introduction—driftwaves Copyright © 2003 IOP Publishing Ltd. 6.6.2Thedriftresistiveinstability 6.6.3Kinetictheoryofdriftwaves 7Quasilineartheory 7.1Introduction 7.2Quasilineartheory 7.2.1Basicequations 7.2.2Conservationlaws 7.2.3Velocityspacediffusioninamagneticfield 7.2.4H-theoremforquasilineartheory 7.2.5Weakbump-on-the-tailinstability 7.2.6Effectsofcollisions 7.3Nonlinearwave–particle–waveapplications 7.3.1Plasmawaveechoes 8Finiteamplitudeplasmawaves 8.1Nonlinearmechanismsinplasmas 8.1.1Ponderomotiveeffects 8.2Solitarywavesandsolitons 8.2.1Ion-acousticsolitarywave 8.2.2TheKorteweg–deVries(KdV)equation 8.2.3Ionacousticsolitons 8.2.4Alfve´ nwavesolitons 8.2.5NonlinearSchr¨ odingerequation 8.3Trappedparticleeffects 8.3.1NonlinearLandaudamping 8.3.2Bernstein–Greene–Kruskal(BGK)modes 8.4Parametricinstabilities 8.4.1Themodulatedharmonicoscillatormodel 8.4.2Excitationofcoupledmodeoscillations 8.4.3Effectsoffinitepumpwavelength 8.4.4Unmagnetizedplasmaexamples AComplexvariables A.1Contourintegrals A.2Analyticcontinuation A.3Themethodofsteepestdescents—saddlepointmethod A.3.1Steepestdescentswithsaddlepointsalongtherealaxis A.3.2Saddlepointmethod A.3.3SpatialLandaudampingexample BSpecialfunctionsinplasmaphysics B.1Plasmadispersionfunction,Z(ζ) B.1.1Propertiesoftheplasmadispersionfunction B.1.2GeneralizeddispersionfunctionsandtheGordeyev integrals Copyright © 2003 IOP Publishing Ltd. B.1.3Relationtotheerrorfunctionforcomplexargument B.1.4RelationtotheYfunction B.1.5Relationtothe Wfunction B.2Weaklyrelativisticplasmadispersionfunction,Fq(z) B.2.1Relationtootherfunctions B.2.2PropertiesofFq(z) ( , ) B.3Generalizedrelativisticplasmadispersionfunction, F q z a ( , ) B.3.1Propertiesof Fq z a B.3.2RelationtoZ(ζ) B.4Gammafunction, (z) B.5Generalizedhypergeometricfunctions B.5.1Integralsleadingtohypergeometricfunctionsofthefirst type B.5.2Integralsleadingtohypergeometricfunctionsofthe secondtype CTheamplitudeequationsofgeometricoptics C.1Thecurrentdensity C.2Thewaveequation C.3Theamplitudeequation C.4Energydensityconservation DAnswerstoselectedproblems References Copyright © 2003 IOP Publishing Ltd. Prefacetothefirstedition Thisbookisanadvancedtextforfirstorsecondyeargraduatestudentswhohave hadanintroductoryplasmacourseatsomelevel.Whilelargelyself-contained, sothatitcouldprecedeageneralintroductorycourse,thefullbackgroundof kinetictheoryandcollisionsisincompleteinthistext,andthemathematicalskills requiredinthelaterchaptersgobeyondwhatisusuallyexpectedinaseniorlevel course.Itmayalsobeusedasanadjuncttoamoregeneraltextforthosewho choosetoemphasizeplasmawavesinafirstyeargraduatecourse. Despitethepresumptuoustitle,thisbookisnotthelastwordinplasma waves,astherearemanytopicsuponwhichentirebookshavebeenwritten, severalofwhichwerepartialsourcesforthisbook.Itiscomprehensive,however,