Preface to the first edition ................................................ xi Preface to the second edition ............................................. xiii Acknowledgements ........................................................... xiv Acronyms .................................................................... xv 1 Introduction ............................................................... 1 1.1 Properties of plasmas .................................................. 1 1.1.1 Unmagnetized plasmas ............................................. 2 1.1.2 Magnetized plasmas ............................................... 2 1.1.3 Thermal plasmas .................................................. 3 1.2 Plasma wave applications ............................................... 3 1.2.1 Plasma waves in ionospheric physics .............................. 3 1.2.2 Plasma waves in astrophysics ..................................... 4 1.2.3 Plasma waves in magnetized fusion plasmas ........................ 4 1.2.4 Plasma waves in laser-produced plasmas ........................... 5 1.3 Review of electromagnetic wave propagation ............................. 5 1.3.1 The Maxwell equations ............................................ 5 1.3.2 Properties of the Helmholtz equation ............................. 7 1.3.3 Conservation laws for electromagnetic fields .................... 10 1.3.4 Conservation laws with Fourier amplitudes ....................... 12 1.3.5 Methods of geometric optics from WKB theory ..................... 13 1.4 Statistical mechanics of plasmas ...................................... 14 1.5 Overview of the plasma wave zoo ....................................... 17 2 Waves in a cold uniform plasma ............................................ 21 2.1 The cold plasma dispersion relation ................................... 22 2.1.1 Equations of motion ............................................. 22 2.1.2 Cold plasma dielectric tensor ................................... 24 2.1.3 Forms of the dispersion relation ................................ 25 2.2 The CMA diagram ....................................................... 27 2.2.1 Principal solutions—parallel propagation ........................ 27 2.2.2 Principal solutions—perpendicular propagation ................... 29 2.2.3 CMA boundaries—cutoffs and resonances ........................... 33 2.2.4 Wave normal surface topology—spheroids and lemniscoids .......... 34 2.2.5 Labeling—left and right, ordinary and extraordinary ............. 36 2.2.6 The CMA diagram for a one-ion species plasma .................... 38 2.3 Phase and group velocity in three-dimensions .......................... 42 2.3.1 The one-dimensional case ........................................ 44 2.3.2 The three-dimensional case ...................................... 45 2.3.3 Group velocity surfaces ......................................... 46 2.4 ω(k, θ) dispersion surfaces ........................................... 48 2.4.1 Underdense case, ωp / ωce = 0.32 ................................. 49 2.4.2 Overdense case, ωp / ωce = 3.2 ................................... 53 2.5 Examples of propagation at arbitrary θ ................................ 54 2.5.1 Low frequency waves ............................................. 54 2.5.2 Intermediate frequency waves—whistlers .......................... 60 2.6 Faraday rotation ...................................................... 62 2.6.1 High frequency limit—region 1 ................................... 64 2.6.2 Low frequency limit—region 13 ................................... 65 2.7 Plasma interferometry ................................................. 66 2.7.1 Detecting the signal ............................................ 67 2.7.2 Interpreting the signal when ω >>ωp ............................. 67 2.7.3 Interpreting the signal when ω ~ωp0 ............................. 68 2.8 Electrostatic waves ................................................... 69 2.8.1 Validity conditions for the electrostatic approximation ......... 70 2.8.2 Lower hybrid waves .............................................. 71 2.8.3 Resonance cones ................................................. 72 2.9 Particle motions near resonance ....................................... 74 2.9.1 Lower hybrid resonance (high density case) ...................... 75 2.9.2 Upper hybrid resonance .......................................... 77 2.9.3 Cyclotron resonances ............................................ 78 3 Waves in fluid plasmas .................................................... 80 3.1 Moments of the distribution function .................................. 80 3.1.1 The moment equations ............................................ 80 3.1.2 Longitudinal plasma oscillations from the moment equations ...... 82 3.2 The fluid equations ................................................... 86 3.3 Low frequency waves ................................................... 87 3.3.1 The low-frequency dispersion relation ........................... 87 3.3.2 Stringer diagrams of the LFDR ................................... 90 3.3.3 Approximate dispersion relations and transitions ................ 95 3.3.4 Parallel and perpendicular propagation .......................... 99 3.3.5 High frequency waves ............................................ 99 3.3.6 Summary of fluid waves ......................................... 103 3.4 Partially ionized plasmas and collisions ............................. 103 3.4.1 Neutral collisions ............................................. 104 3.4.2 Electron-ion collisions ........................................ 106 3.5 Amplifying waves and instabilities ................................... 108 3.5.1 Classification of instabilities ................................ 108 3.5.2 Streaming instabilities ........................................ 116 3.6 Power and energy flow in fluid plasmas ............................... 122 4 Kinetic theory of plasma waves ........................................... 124 4.1 The basic equations .................................................. 124 4.1.1 The Boltzmann equation ......................................... 124 4.1.2 Collisions and the Fokker-Planck equation ...................... 125 4.1.3 BBGKY theory ................................................... 127 4.1.4 The Vlasov equations ........................................... 130 4.2 Waves in a thermal, unmagnetized plasma .............................. 131 4.2.1 Vlasov method .................................................. 132 4.2.2 Landau solution ................................................ 136 4.2.3 A Physical picture of Landau damping ........................... 143 4.2.4 Conventional descriptions of Landau damping .................... 150 4.2.5 Ion acoustic waves and ion Landau damping ...................... 157 4.2.6 Effects of collisions on Landau damping ........................ 160 4.3 Waves in a magnetized hot plasma ..................................... 162 4.3.1 The evolution of the distribution function ..................... 163 4.3.2 Integrating along the unperturbed orbits ....................... 164 4.3.3 General f0 (v^, vz) ............................................. 166 4.3.4 Maxwellian distributions ....................................... 170 4.3.5 The dielectric tensor .......................................... 175 4.3.6 The hot plasma dispersion relation ............................. 177 4.3.7 Examples of hot plasma wave effects ............................ 178 4.4 Electrostatic waves .................................................. 182 4.4.1 Perpendicular propagation-Bernstein modes ...................... 183 4.4.2 High-order Bernstein modes ..................................... 185 4.5 Velocity space instabilities ......................................... 189 4.5.1 Anisotropic temperature ........................................ 189 4.6 Conservation of energy and power flow ................................ 191 4.6.1 Poynting's theorem for kinetic waves ........................... 191 4.6.2 Group velocity and kinetic flux ................................ 193 4.7 Relativistic plasma effects .......................................... 196 4.7.1 The relativistic dielectric tensor ............................. 196 4.7.2 The relativistic dielectric tensor without sums ................ 200 4.7.3 The weakly relativistic dielectric tensor ...................... 201 4.7.4 Moderately relativistic expressions ............................ 205 4.7.5 Exact expressions with n|| = 0 .................................. 207 4.7.6 The relativistic X-wave ........................................ 211 5 Bounded homogeneous plasmas .............................................. 213 5.1 Introduction ......................................................... 213 5.2 Boundary conditions .................................................. 213 5.2.1 Conducting boundary ............................................ 214 5.2.2 Plasma-vacuum (or dielectric) interface ........................ 214 5.3 Unmagnetized plasmas ................................................. 216 5.3.1 Scattering from a plasma column ................................ 216 5.3.2 Surface waves in a partially-filled plasma waveguide ........... 225 5.4 Electrostatic waves on a plasma column in a magnetic field ........... 228 5.5 Cold plasma-filled waveguide ......................................... 230 5.5.1 The dispersion relation ........................................ 230 5.5.2 Wave fields and boundary conditions ............................ 234 5.5.3 MHD approximation - ω << ωci .................................... 236 5.5.4 Intermediate frequency case - ω ≈ ωci << ωp ..................... 237 5.5.5 Mode orthogonality and power flow .............................. 239 5.5.6 Antenna problems ............................................... 241 5.5.7 Experiments in plasma-filled waveguides ........................ 246 5.6 Conducting wall with vacuum layer, m = 0, ±1 ......................... 249 5.7 Infinite magnetic field approximation ................................ 251 5.7.1 Cold plasma-filled waveguide in an infinite magnetic field ..... 251 5.7.2 Hot plasma-filled waveguide .................................... 253 6 Waves in inhomogeneous plasmas ........................................... 257 6.1 Introduction ......................................................... 257 6.2 WKB method for one-dimensional inhomogeneities ....................... 257 6.2.1 Behavior near a cutoff ......................................... 260 6.2.2 Tunneling between back-to-back cutoffs ......................... 262 6.2.3 Behavior near an isolated resonance ............................ 264 6.2.4 Behavior near a resonance-cutoff pair .......................... 267 6.3 Mode conversion theory ............................................... 268 6.3.1 The mode conversion theorem .................................... 268 6.3.2 Solution of the tunneling equation ............................. 269 6.3.3 Mode conversion examples ....................................... 279 6.3.4 Conservation of energy ......................................... 290 6.4 Absorption and emission .............................................. 297 6.4.1 Generalized Kirchhoff's law .................................... 297 6.4.2 Absorption and mode conversion ................................. 298 6.5 WKB Method for three-dimensional inhomogeneous plasmas—ray tracing ... 301 6.5.1 The ray equations .............................................. 301 6.5.2 The inhomogeneous plasma dispersion relation ................... 305 6.5.3 The amplitude equations ........................................ 306 6.6 Drift waves and instabilities ........................................ 307 6.6.1 Introduction—drift waves ....................................... 307 6.6.2 The drift resistive instability ................................ 308 6.6.3 Kinetic theory of drift waves .................................. 311 7 Quasilinear theory ....................................................... 320 7.1 Introduction ......................................................... 320 7.2 Quasilinear theory ................................................... 321 7.2.1 Basic equations ................................................ 321 7.2.2 Conservation laws .............................................. 324 7.2.3 Velocity space diffusion in a magnetic field ................... 326 7.2.4 H-theorem for quasilinear theory ............................... 333 7.2.5 Weak bump-on-the-tail instability .............................. 335 7.2.6 Effects of collisions .......................................... 339 7.3 Nonlinear wave-particle-wave applications ............................ 340 7.3.1 Plasma wave echoes ............................................. 340 8 Finite amplitude plasma waves ............................................ 351 8.1 Nonlinear mechanisms in plasmas ...................................... 351 8.1.1 Ponderomotive effects .......................................... 352 8.2 Solitary waves and solitons .......................................... 353 8.2.1 Ion-acoustic solitary wave ..................................... 353 8.2.2 The Korteweg-de Vries (KdV) equation ........................... 355 8.2.3 Ion acoustic solitons .......................................... 356 8.2.4 Alfven wave solitons ........................................... 358 8.2.5 Nonlinear Schrodinger equation ................................. 359 8.3 Trapped particle effects ............................................. 363 8.3.1 Nonlinear Landau damping ....................................... 363 8.3.2 Bernstein-Greene-Kruskal(BGK) modes ............................ 373 8.4 Parametric instabilities ............................................. 376 8.4.1 The modulated harmonic oscillator model ........................ 378 8.4.2 Excitation of coupled mode oscillations ........................ 381 8.4.3 Effects of finite pump wavelength .............................. 384 8.4.4 Unmagnetized plasma examples ................................... 385 A Complex variables ........................................................ 391 A.1 Contour integrals .................................................... 391 A.2 Analytic continuation ................................................ 393 A.3 The method of steepest descents—saddle point method .................. 394 A.3.1 Steepest descents with saddle points along the real axis ....... 394 A.3.2 Saddle point method ............................................ 397 A.3.3 Spatial Landau damping example ................................. 399 B Special functions in plasma physics ...................................... 401 B.1 Plasma dispersion function, Ζ(ζ) ..................................... 401 B.1.1 Properties of the plasma dispersion function ................... 401 B.1.2 Generalized dispersion functions and the Gordeyev integrals .... 404 B.1.3 Relation to the error function for complex argument ............ 405 B.1.4 Relation to the Y function ..................................... 406 B.1.5 Relation to the W function ..................................... 406 B.2 Weakly relativistic plasma dispersion function, Fq(z) ................ 407 B.2.1 Relation to other functions .................................... 407 B.2.2 Properties of Fq(z) ............................................ 408 B.3 Generalized relativistic plasma dispersion function, Fq(z,a) ......... 408 B.3.1 Properties of Fq(z,a) .......................................... 409 B.3.2 Relation to Ζ(ζ) ............................................... 409 B.4 Gamma function, Γ(z) ................................................. 410 B.5 Generalized hypergeometric functions ................................. 411 B.5.1 Integrals leading to hypergeometric functions of the first type ........................................................... 411 B.5.2 Integrals leading to hypergeometric functions of the second type ........................................................... 412 C The amplitude equations of geometric optics .............................. 414 C.1 The current density .................................................. 414 C.2 The wave equation .................................................... 416 C.3 The amplitude equation ............................................... 416 C.4 Energy density conservation .......................................... 420 D Answers to selected problems ............................................. 422 References ............................................................... 442 Index .................................................................... 446