Table Of ContentSeries 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.
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D G Swanson has asserted his moral rights under the Copyright, Designs and
PatentsAct1998tobeidentifiedastheauthorofthiswork.
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Contents
Prefacetothefirstedition
Prefacetothesecondedition
Acknowledgements
Acronyms
1Introduction
1.1Propertiesofplasmas
1.1.1Unmagnetizedplasmas
1.1.2Magnetizedplasmas
1.1.3Thermalplasmas
1.2Plasmawaveapplications
1.2.1Plasmawavesinionosphericphysics
1.2.2Plasm awavesinastrophysics
1.2.3Plasmawavesinmagnetizedfusionplasmas
1.2.4Plasmawavesinlaser-producedplasmas
1.3Reviewofelectromagneticwavepropagation
1.3.1TheMaxwellequations
1.3.2PropertiesoftheHelmholtzequation
1.3.3Conservationlawsforelectromagneticfields
1.3.4ConservationlawswithFourieramplitudes
1.3.5 MethodsofgeometricopticsfromWKBtheory
1.4Statisticalmechanicsofplasmas
1.5 Overviewofth eplasmawavezoo
2Wavesinacolduniformplasma
2.1Thecoldplasmadispersionrelation
2.1.1Equationsofmotio n
2.1.2Coldplasmadielectrictensor
2.1.3Formsofthedispersionrelation
2.2TheCMAdiagram
2.2.1Principalsolutions—parallelpropagation
2.2.2Principalsolutions—perpendicularpropagation
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2.2.3CMAboundaries—cutoffsandresonances
2.2.4Wavenormalsurfacetopology—spheroidsandlemniscoids
2.2.5 Labeling—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,ω /ω =0.32
p ce
2.4.2Overdensecase,ω /ω =3.2
p ce
2.5 Examplesofpropagationatarbitraryθ
2.5.1Lowfrequencywaves
2.5.2Intermediatefrequencywaves—whistlers
2.6Faradayrotation
2.6.1Highfrequencylimit—region1
2.6.2Lowfrequencylimit—region13
2.7Plasm ainterferometry
2.7.1Detectingthesignal
2.7.2Interpretin gthesignalwhenω(cid:2)ω
p
2.7.3 Interpretingthesignalwhenω∼ω
p0
2.8 Electrostaticwaves
2.8.1Validityconditionsfortheelectrostaticapproximation
2.8.2L owerhybridwaves
2.8.3Resonancecones
2.9Particlemotionsnearresonance
2.9.1L owerhybridresonance(highdensitycase)
2.9.2Upperhybridresonance
2.9.3Cyclotronresonances
3Wavesinfluidplasmas
3.1Momentsofthedistributionfunction
3.1.1Themomentequations
3.1.2Longitudinalplasmaoscillationsfromthemoment
equations
3.2Theflu idequations
3.3L owfrequencywaves
3.3.1Thelow-frequencydispersionrelation
3.3.2StringerdiagramsoftheLFDR
3.3.3Approximatedispersionrelationsandtransitions
3.3.4Parallelandperpendicularpropagation
3.3.5 Highfrequencywaves
3.3.6Summaryoffluidwaves
3.4Partiallyionizedplasmasandcollisions
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3.4.1Neutralcollisions
3.4.2Electron–ioncollisions
3.5 Amplifyingwavesandinstabilities
3.5.1Classificationofinstabilities
3.5.2Streaminginstabilities
3.6Powerandenergyflowinfluidplasmas
4Kinetictheoryofplasmawaves
4.1Thebasicequations
4.1.1TheBoltzmannequation
4.1.2Collisionsandt heFokker–Planckequation
4.1.3BBGKYtheory
4.1.4TheVlasovequations
4.2Wavesinathermal,unmagnetizedplasma
4.2.1Vlasovmethod
4.2.2Landausolution
4.2.3APhysicalpictureofLandaudamping
4.2.4ConventionaldescriptionsofLandaudamping
4.2.5 IonacousticwavesandionLandaudamping
4.2.6EffectsofcollisionsonLandaudamping
4.3Wavesinamagnetizedhotplasma
4.3.1Theevolutio nofthedistribu tionfunction
4.3.2Integratingalongtheunperturbedorbits
4.3.3Generalf 0(v⊥,vz)
4.3.4Maxwelliandistributions
4.3.5 Thedielectrictensor
4.3.6Thehotplasmadispersionrelation
4.3.7Examplesofhotplasmawaveeffects
4.4Electrostaticwaves
4.4.1Perpendicularpropagation–Bernsteinmodes
4.4.2High-orderBernsteinmodes
4.5 Velocityspaceinstabilities
4.5.1Anisotropictemperature
4.6Conservatio nofenergyandpowerflow
4.6.1Poynting’stheoremforkineticwaves
4.6.2Groupvelocityandkineticflux
4.7Relativisticplasmaeffects
4.7.1Therelativisticdielectrictensor
4.7.2Therelativisticdielectrictensorwithoutsums
4.7.3Theweaklyrelativisticdielectrictensor
4.7.4Moderatelyrelativisticexpressions
4.7.5 Exactexpressionswith n(cid:5) =0
4.7.6TherelativisticX-wave
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5Boundedh omogeneousplasmas
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
5.5.1Thedispe rsionrelation
5.5.2Wavefieldsandboundaryconditions
5.5.3MHDapproximation—ω(cid:6)ω
ci
5.5.4 Intermediatefrequencycase—ω(cid:7)ω (cid:6)ω
ci p
5.5.5 Modeorthogonalityandpowerflow
5.5.6Antennaproblems
5.5.7Experimentsinplasma-filledwaveguides
5.6Conductingwallwithvacuumlayer,m=0,±1
5.7Infinitemagneticfieldapproximation
5.7.1Coldplasma-filledwaveguideinaninfinitemagneticfield
5.7.2Hotplasma-filledwaveguide
6Wavesininhomogeneousplasmas
6.1Introductio n
6.2WKBmethodforone-dimensionalinhomogeneities
6.2.1Behaviornearacutoff
6.2.2Tunnelingbetweenback-to-backcutoffs
6.2.3Behaviornearanisolatedresonance
6.2.4Behaviorneararesonance–cutoffpair
6.3Modeconversiontheory
6.3.1Themodeconversiontheorem
6.3.2Solutio nofthetunnelin gequation
6.3.3Modeconversionexamples
6.3.4Conservationofenergy
6.4Absorptionandemission
6.4.1GeneralizedKirchhoff’slaw
6.4.2Absorptio nandmodeconversion
6.5 WKBMethodforthree-dimensionalinhomogeneousplasmas—
raytracing
6.5.1Therayequations
6.5.2Theinhomogeneousplasmadispersionrelation
6.5.3Theamplitudeequations
6.6Driftwavesandinstabilities
6.6.1Introduction—driftwaves
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6.6.2Thedriftresistiveinstability
6.6.3Kinetictheoryofdriftwaves
7Quasilineartheory
7.1Introductio n
7.2Quasilineartheory
7.2.1Basicequations
7.2.2Conservationlaws
7.2.3Velocityspacediffusioninamagneticfield
7.2.4H-theoremforquasilineartheory
7.2.5 Weakbump-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–deVr ies(KdV)equation
8.2.3Ionacousticsolitons
8.2.4Alfv´enwavesolitons
8.2.5 NonlinearSchr¨odingerequation
8.3Trappedparticleeffects
8.3.1NonlinearLandaudamping
8.3.2Bernstein–Greene–Kruskal(BGK)modes
8.4Parametricinstabilities
8.4.1Themodulatedharmonicoscillatormodel
8.4.2Excitationofcoupledmodeoscillations
8.4.3E ffectsoffinitepumpwavelength
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
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B.1.3Relationtotheerrorfunctionforcomplexargument
B.1.4RelationtotheYfunction
B.1.5 Relationtothe Wfunction
B.2Weaklyrelativisticplasmadispersionfunction,F (z)
q
B.2.1Relationtootherfunctions
B.2.2Propertieso fF (z)
q
B.3Generalizedrelativisticplasmadispersionfunction, (z,a)
q
B.3.1Propertiesof (z,a) F
q
B.3.2RelationtoZ(ζ) F
B.4Gammafunction,(cid:10)(z)
B.5 Generalizedhypergeometricfunctions
B.5.1Integralsleadingtohypergeometricfunctionsofthefirst
type
B.5.2Integralsleadingtohypergeometricfunctionsofthe
secondtype
CTheamplitudeequationsofgeometricoptics
C.1Thecurrentdensity
C.2Thewaveequatio n
C.3Theamplitudeequatio n
C.4Energydensityconservation
DAnswerstoselectedproblems
References
Copyright © 2003 IOP Publishing Ltd.
Prefacetothefirstedition
Thisbookisanadvancedtextforfirsto rsecondyeargraduatestudentswhohave
hadanintroductoryplasmacourseatsomelevel.Whilelargelyself-contained,
sothatitcouldprecedeageneralintroductorycourse,thefullbackgroundof
kinetictheoryandcollisionsisincompleteinthistext ,andthemathematicalskills
requiredinthelaterchaptersgobeyondwhatisusuallyexpectedinaseniorlevel
course.Itmayalsobeusedasanadjuncttoamoregeneraltextforthosewho
choosetoemphasizeplasmawavesinafirstyeargraduatecourse.
Despitethepresumptuoustitle,thisbookisnotthelastwordinplasma
waves,astherearemanytopicsuponwhichentirebookshavebeenwritten,
severalofwhichwerepartialsourcesforthisbook.Itiscomprehensive,however,
inthesensethatfewwavetopicsarenotatleastintroduced,anditdoesnottreat
everysubjectinanintroductorymanner.Thegeneralphilosophyhasbeento
includethebasicdevelopmentandformulasfora smanya spectsofplasmawaves
aspossible,anddevelopmorefullythosetopicswhichhavenotbeentreated
inothertexts.Thisleadstononuniformdeptho fcoverage,butothertex tsare
presumablyavailablefortheinstructortoexpandonfavoritetopics.Becauseof
theunevencoverage,somemayfindt heirfavoritetopi ctreatedeitherinacavalier
fashiono rnotatall.Inordertoproducethebookatareasonablelengthand
price,however,Ihaveattemptedtoincludeonlyafewexamplesafterthegeneral
developmentofeachtopic,leavingtheinstructortodevelopotherexamples.
Someadditionaltopicsareincludedintheproblems,bu tprobablynotenoughto
indicatethebreadthofpossibleapplications.Answerstosomeofth eproblemsare
includedinappendixD,andaninstructormayobtainasolutionsmanualdirectly
fromtheauthor.
Sincemyownbackgroundhasbeenalmostexclusivelyinlaboratoryplasma
physics,boththeoreticalandexperimental,alargepartofwhichhasbeenfusion
related, the book is primarily addressed to this audience. As such, the original
ambitionofmakingabookthationosphericandastrophysicalscientistscouldalso
useisunsatisfied,butIbelievethereismuchherefromwhichthelatteraudience
maybenefit,sincenotallofthesetopicsarewellcoveredintheirliterature.
This book has many beginnings1, starting from notes on a course by
1 Sincemyowncareerhaslargelybeendevotedtowaveheatingforfusionapplications,andhence
totheheatingofdeuterium,itcouldbesaidthattheGenesisofthisbookisDeuteronomy.
Copyright © 2003 IOP Publishing Ltd.
R W Gould, following through notes on a course I taught at the University of
Texas from Stix’s book (which is undoubtedly the book to which I am most
indebted),andculminatingwithareviewarticleIwroteonmodeconversionand
ion wave heating. But the determination to actually write the book came from
conversationswithDQHwang,withwhomIoriginallybegantowritethebook.
Wehaddeterminedthetimewasrightforageneralbookonplasmawaves,andI
tookasixmonthssabbaticalin1987atUCDavis,Livermore,soIcouldwriteit
withhim. Unfortunately,theLivermoreLaboratorydidnotpermithimthetime
to pursue this adventure, so he had to withdraw from the project, extendingthe
time an extra six months while I learned and wrote what was to have been his
contribution.
In addition to D Q Hwang, I would like to acknowledge the patience and
helpfulnessofmycolleagues, J-MWersinger, JD HansonandRFGandy,who
have had to carry the major responsibility for our mutual projects during the
preparation of this book, and to my former student, Suwon Cho, who assisted
me andhelpedkeepsomeof myresearchalive duringthisperiod. I wouldalso
like to thankWalter Sadowskiforstimulatingand supportingthe earlier review,
and for his patience and supportduring the preparation of this book. Finally, I
wouldlike to thankLeslie Lamportand DonaldKnuthfortheir developmentof
LATEXandTEX,respectively,withoutwhichIwouldneverhaveattemptedtowrite
thisbook,muchlessprepareitcameraready.
DGarySwanson
DepartmentofPhysics
AuburnUniversity,AL36849-5311
Copyright © 2003 IOP Publishing Ltd.
