accessibility 121
modes accessible from free space 125
modes not accessible from free space 125
action integral 48, 49
adiabatic equation of state 154
adiabatic invariants 48–60
first 50–51
harmonic oscillator 48–50
second 51
third 53
Airy functions 130
Alfvén
index of refraction 100, 134
Mach number 262, 278
mode 135
speed 134, 196, 257
time scale 241
Alfvén waves 100
generated by Jupiter’s moon Io 203
solar wind 202
transverse 198–199, 248
Ampère’s law 36, 38, 57, 85, 86, 182
analytic continuation 293, 296, 297
analytic function 293
anisotropy factor 377
electron 381, 388, 389
ion 385
anti-shielding 9, 411
aurora 2
average
distance between neighboring particles 16
kinetic energy 6
number density 138
velocity 138
average (expectation) value 138
axially symmetric magnetic field 39, 44, 50
Bennett pinch 211, 247
Bennett relation 213
Bernstein–Green–Kruskal (BGK) electrostatic potentials 406, 409–411
Bernstein-like modes 358
Bernstein mode 350–355, 387
Bessel functions 348, 349, 352, 369
modified 351
Bohm–Gross dispersion relation 159, 284–285, 303
Boltzmann equation 140–144
bounce frequency 305
bounce resonance 58
bounded integrable orbits 68, 71
bounded volumes in CMA diagram 121
bounding surfaces in CMA diagram 121
bow shock 257, 276
bump-on-tail instability 328–330, 399–400
Buneman mode 289–290
Cauchy residue theorem 292
Cauchy velocity distribution function 298–300, 313–315
chaos 404
Hamiltonian system 68, 402
magnetic field 71
single particle orbits 68–71
velocity space diffusion 400–406
characteristic parameters of a plasma 5–18
characteristic resonance energy
ion cyclotron mode 384
whistler mode 379
charge continuity equation 147, 183
charge density 7
charge neutrality 1, 87, 182, 392
charge separation 10
Chew–Goldberger–Low (CGL) equation of state 154–155
Clemmow–Mullaly–Allis (CMA) diagram 120–127
electrons 123
electrons and ions 126
closure problem 152, 394
collective motions 2, 3
collisional drag force 162–163
collision cross-section
charged and neutral particles 13
charged particles 14, 416
total 417
collision frequency 13–15, 132
charged and neutral particles 13
charged particles 13–15, 17, 415–433
effective 163, 178
electron–ion 15, 431
collision operator 143
collisions 1, 140, 144, 415–433
fully ionized plasma 415
isotropic 163
large angle 16
partially ionized plasma 13, 163
small angle 13–16, 417–420
complex Đ-plane 320
complex frequency 395
complex p-plane 293, 296
conductivity
fully ionized gas 427–431
Hall 165
Lorentz 247
Ohm’s law 163
parallel 165
partially ionized gas 164–166
Pedersen 164–166
perpendicular 165
sth species 164
conductivity tensor 85, 86, 367
cold magnetized plasma 95
cold unmagnetized plasma 88
hot magnetized plasma 370
conjugate momenta 48, 61, 62, 129
conservation of energy 252
conservation of energy flux 254
conservation of mass 252
conservation of mass flux 253
conservation of momentum 252
conservation of momentum flux 253
constancy of the magnetic moment 41–43
constant-α surface 206
constants of motion 145
continuity equation in six-dimensional phase space 142
contour integral 292
convective derivative 150, 178, 189
co-rotational electric field 29, 30
cosmic rays 66
Coulomb
collisions 416–417
force 2, 13
potential 9
scattering 14
critical drift velocity 365
critical electric field 433
crossover frequency 104, 105, 106
current-driven electrostatic ion cyclotron waves 363–366
current-driven ion acoustic instability 331–334, 339
cusp mirror geometry 226
cyclic coordinates 62, 129
cyclotron
frequency 12, 24
harmonics 348
radius 24
resonance 55
resonance energy 380
resonance interactions 360
resonance velocity 382
cyclotron damping 341, 374–377
electron 103, 104, 376
ion 106, 376–377
cyclotron frequency 12, 24
electron 12
ion 12
cylindrical magnetic field geometries 211–215
cylindrical velocity space coordinates 342, 343
damping constant 292
de Broglie wavelength 17
Debye–Hückel potential 8, 9
Debye length 7–9, 159, 303
Debye shielding 7–10
degrees of freedom 61
de Hoffmann–Teller frame 258–260
destructive transition in CMA diagram 121
diamagnetism 26
dielectric tensor 85, 86, 367
cold magnetized plasma 95
cold unmagnetized plasma 88
hot magnetized plasma 370
differential rotation of the Sun 191
diffusion 166
ambipolar 166–169
coefficient 167
equation 168, 191
tensor 421, 424–426, 432, 434
velocity 177
dipole moment per unit volume 84
Dirac delta function 77
discontinuities 251, 255
contact 255–256
rotational 255, 256–258
shock 258
tangential 256
dispersion 79
dispersion relation 78–80, 86
electromagnetic ion cyclotron waves in a hot magnetized plasma 384
electromagnetic waves in a cold magnetized plasma 97
electromagnetic waves in a cold unmagnetized plasma 89
electromagnetic waves in a hot magnetized plasma 367–386
electron plasma oscillations 92, 159
electrostatic waves in a cold beam plasma 285–290
electrostatic waves in a hot unmagnetized plasma 283–284, 294–298, 299–303
electrostatic waves in a hot magnetized plasma 345–366
electrostatic waves in a magnetized Maxwellian plasma 350–355
electrostatic waves in an unmagnetized Maxwellian plasma 308–311
free space 79
ion acoustic waves in a hot unmagnetized plasma 160-162
Langmuir waves in a hot unmagnetized plasma 158–159, 302, 313, 316
Langmuir waves in an unmagnetized Maxwellian plasma 308–311
displacement current 85
displacement field 84
distribution function (see velocity distribution function) 5, 137–140
bi-Maxwellian 140, 377–378, 388
Cauchy 170, 298–300, 320, 321, 335
counter-streaming Cauchy 327, 328, 338
double-humped 325
kappa 336
loss-cone 359, 382
Maxwellian 5, 139, 169, 308, 337, 338, 387, 388
non-equilibrium 6
non-thermal 6
reduced 295, 312
ring-type 355
solar flare electrons 405
spatially averaged 392
spherical shell 334, 357
Doppler shift 55, 153
double layer 409
drag force 149, 432
Dreicer electric field 433
drift velocities
curvature drift 34
E × B drift 27
gradient drift 32
near an x-type neutral line 58
perpendicular force 31–32
dynamical friction vector 421, 422–424, 431
Earth’s magnetosphere 2, 29, 56, 58, 102, 225
bow shock 276
ionosphere 239
magnetopause 256–258
magnetotail 56
plasmapause 30
plasmasphere 29, 30, 225
Van Allen radiation belt 68
effective potential 43, 63, 64, 66
eigenfrequency 231
eigenmodes 231
eigenvectors 201
fast magnetosonic mode 201
slow magnetosonic mode 201
transverse Alfvén mode 198
waves in a cold magnetized plasma 98, 110, 111
waves in a cold unmagnetized plasma 89
eikonal 127
electric dipole antenna 93
electromagnetic
ion cyclotron waves 100–101, 376
waves in a cold magnetized plasma 94–127
waves in a cold unmagnetized plasma 87–94
waves in a hot magnetized plasma 367–386
electron acoustic waves 160
electron and ion pressure waves 155–162
electron cyclotron harmonics 354–355
electron phase-space holes 410
electron plasma oscillations 10–12, 158, 350
electron plasma oscillations near Jupiter’s bow shock 278
electronic charge 7
electrostatic
electron cyclotron waves 360
energy density 398
instabilities 327–334
ion cyclotron waves 361, 363
noise at Jupiter’s bow shock 278
potential 7
waves in a cold magnetized plasma 98, 111
waves in a cold unmagnetized plasma 92
waves in a hot magnetized plasma 341–386
waves in a hot unmagnetized plasma 281–334
elementary plane wave 77, 78
elliptically polarized wave 108
energy conservation equation 151
energy dissipation 273
energy equation (MHD) 192–194
energy equation for the sth species 151
energy method of stability analysis 231–235
enthalpy 194, 252, 256, 261
entropy 181, 273, 274
equations of state 152, 181–182
adiabatic 153–154, 181
Chew-Goldberger-Low (CGL) 154–155, 182
cold plasma 152–153
incompressible 181
isothermal 181
power law 181, 182
error function 431, 434
evanescent wave 91, 97, 100
external sources 92–94
extraordinary mode 111
Faraday rotation 107–109
Faraday’s law 45, 53, 86, 182
Fermi acceleration 59
field aligned currents 226
field aligned density irregularities 102
finite difference equation 401
finite Larmor radius effects 244
first law of thermodynamics 153
first-order linearized Vlasov equation 343
first-order quantity (definition) 87
fluid-like motions 161
flux
coordinates 188
function 216
ropes 208
surfaces in a toroidal magnetic field 215–219
Fokker–Planck
collision operator 421, 426, 427
equation 420–421
Fourier representation of waves 75–84
Fourier transforms 75
free energy sources 358, 362, 386
beam 386
current 363, 365–366
loss-cone 362
free space L-O mode 122–126
free space R-X mode 122–126
frozen field theorem 186–191
fusion 2–3
Gardner’s theorem 318–319, 321
gaseous state 1–2
Gaussian harmonic wave packet 76
Gauss’ law 10, 90, 182
Gauss’ theorem 147
generalized coordinates 61
generalized Ohm’s law 178–181
gradient drift current 38
Grad–Shafranov equation 218
group velocity 80–84, 91
Alfvén mode 199
whistler mode 133
growth rate
Buneman mode 290
current-driven ion cyclotron waves 365
electromagnetic ion cyclotron waves 385
electromagnetic waves in a hot magnetized plasma 375
electrostatic cold beam modes 285–290
electrostatic waves in a hot unmagnetized plasma 290–303
ion acoustic waves in a hot unmagnetized plasma 314, 332
ion acoustic waves in an unmagnetized Maxwellian plasma 315–318
Langmuir waves 303, 311
whistler waves 382
guiding center 24
gyrotropic condition 343
Hall current 244
Hall MHD 245
Hamiltonian 48, 60–62, 128
Hamiltonian for a magnetic mirror field 62–66
Hamilton’s equations 60, 61, 128
harmonic oscillator 10, 221, 292
equation 11, 48
WKB solution 49
Harris dispersion relation 345–349, 386, 387
Heisenberg’s uncertainty principle 17
helium whistler 105, 106
high density approximation 112, 119
high-phase velocity approximation 284, 339, 361
homogeneous equation 79, 86, 367
electromagnetic waves in a cold magnetized plasma 96
electromagnetic waves in a cold unmagnetized plasma 89
electromagnetic waves in a hot magnetized plasma 371
electrostatic waves in a hot unmagnetized plasma 157, 283
magnetohydrodynamic waves 197
Hooke’s law 10
Hopf’s theorem 207, 215
H-theorem 6
hybrid resonances 111, 350
ion 111, 112, 113
lower 111, 112–113
upper 111–112, 353, 355, 360, 387
ideal gas law 140, 153, 167
ideal MHD 180
ideal MHD force operator 231
index of refraction 99, 389
surface 83, 116
vector 83
inhomogeneous plasmas 120
initial value problem 291
instabilities 355–366
anisotropy-driven 355–363, 377–385
cold beam 285–290
counter-streaming beam 287
current-driven 331, 355, 363
interchange 221–227
kinetic 327
Kruskal–Schwartzschild 223, 236
Rayleigh–Taylor 223, 236–239
sausage-mode 225
whistler-mode 378, 388
integral invariant 52
integrating factor 345, 369
internal energy 194, 252
internally trapped particles 68
Io 203
Alfvén waves 203
controlled radio emissions 203
plasma torus 225
ion acoustic speed 333
ion acoustic waves 158, 160–162, 312–318, 339
ion cyclotron
damping 366, 376, 377
heating 107
resonance energy 389
resonance velocity 384
waves 101
whistler 104–107, 376
ion–ion cutoff frequencies 101
ion–neutral collisions 163
ionogram 113, 114
ionosphere 2, 239
ionospheric sounders 113–114, 354
ion-to-electron mass ratio 10
isoentropic 181
Jacobian 142
Joule heating 151
jump conditions 251
jump symbol 253, 255
Jupiter’s magnetosphere 203, 225
bow shock 277–278
Io plasma torus 225
radio emission 203
kinetic energy density 138, 151, 194
kinetic energy flux 151
kinetic theory 137–146
L = 0 cutoff frequency 101
Lagrangian 61
Landau damping 160, 162, 290–308, 315
linear 308
non-linear 305–308
physical origin 304–308
Landau resonance velocity 364
Langmuir waves 10, 158–160, 284
Langmuir waves generated by a solar flare 405
Langmuir waves in the solar wind 329, 330
Laplace’s equation 169
Laplace transform 281, 291–294, 335
first derivative 292
higher order derivatives 292
integration contour 291, 297
inverse 291, 294, 295, 298, 335
second derivative 292
large-argument expansion of the plasma dispersion function 309–310, 316, 364
lattice-like structures 17
left- and right-hand polarized free space modes 107
left-hand polarized ordinary (L-O) mode 100, 101, 123, 374
lightning 2, 101, 225
ball 2
bead 225
linear force operator 227–228, 231, 234, 248
linearization of the Vlasov equation 342–345
local thermodynamic equilibrium 153
longitudinal waves 92, 98, 157
long-range force 2, 140
long-wavelength approximation 361
loop antenna 93
Lorentz force 24, 41
Lorentz gas model 162–163, 427
loss cone 45, 172, 380
low-density approximation 113
low-phase velocity approximation 339
low-temperature, high-phase velocity limit 386
low-temperature, long wavelength limit 350, 373–374
L-shell 53
Lundquist number 241
macroscopic averages 15
magnetic
amplification 190
beach geometry 106
bottle 3, 43
confinement 3
convection 185–191
diffusion 191
energy density 194
field aligned currents 363
field strength 12
flux 63
flux annihilation 241
flux invariant 47, 53, 54
helicity 209
merging 57–58
mirror field 39
mirror force 40–41
mirror machine 362, 380
mirror ratio 45–72
moment 25, 26, 41, 66
pressure 39, 183–185
pressure tensor 184
reconnection 240–244, 245
Reynolds number 185
magnetization 26
magnetization current 26, 38, 84
magnetohydrodynamics (also see MHD) 175–245
magnetohydrodynamic waves 195–204
fast magnetosonic mode 197
slow magnetosonic mode 197
transverse (shear) Alfvén mode 197, 198, 257
magnetosonic modes (see magnetohydrodynamic waves) 200–201
magnetosphere 2
Earth 29, 56, 58, 68, 102, 225
Jupiter 203, 225
magnetostatic equilibrium 39
marginal stability condition 318, 388
marginal stability line 320
Markov process 420
mass continuity equation 176, 182
Maxwellian electron and ion velocity distribution functions 315–318
Maxwellian velocity distribution function 6, 350
Maxwell’s equation 40
mean-free path 244, 251, 419
method of successive approximations 285
MHD (also see magnetohydrodynamics) 175
diffusion time scale 241
discontinuities 251–278
equations 182–183
jump conditions 251–255
shocks 276–278
stability 219–239
MHD equilibria
force balanced 210–219
force-free 206–210
microscopic scale length 251
mirror point magnetic field 52
mobility coefficient 167
moment equations 146–151
moments of the Boltzmann equation 146–151
first 147–150
second 150–151
zeroth 146–147
momentum equation 150, 176–178, 182
multi-component plasma 311
neutron star 2, 190
Newton’s law 141
non-conservative electric field 45–46
non-destructive transition in CMA diagram 121
non-integrable system 62
non-linear effects 391–411
non-linear evolution and stability of MHD shocks 275
non-propagating wave 91
normalized one-dimensional velocity distribution function 283
normal mode method of MHD stability analysis 228–231
normal modes 79
number density 5–6, 138
number of degrees of freedom 154, 181
number of electrons per Debye cube 15–17, 144, 415
Nyquist criterion 320–326
oblique propagation 116–127
cold magnetized plasma 116–127
hot magnetized plasma 385–386
Ohmic heating 164
Ohm’s law (also see conductivity and resistivity) 163–164, 178, 180, 182
operator substitutions 78
ordinary mode 110
parallel kinetic energy 60
parallel propagation in a cold magnetized plasma 97–101
parallel propagation in a hot magnetized plasma 372–385
parallel resonance velocity 362
Penrose criterion 321–326, 327, 338
permeability of free space 36
permittivity of free space 7
perpendicular kinetic energy 60
perpendicular propagation
cold magnetized plasma 110–116
hot magnetized plasma 350–355
perturbed kinetic energy of an MHD equilibrium 232
perturbed potential energy of an MHD equilibrium 232
phase mixing 305
phase space 137
phase space trajectories 304
phase-space volume element 137, 141, 144
phase velocity 77, 91
ɸ-pinch 214
photoelectron current 10
pitch angle 43, 44, 381
pitch-angle scattering 383
Planck’s constant 18
planes of constant phase 77
plane wave 77
plasma
definition 1
fourth state of matter 1
recombination 1
representative parameters 18
sources of ionization 1
plasma β 185
plasma dispersion function 308–311, 336, 364
large-argument expansion 309
power series expansions 309
small-argument expansion 309
plasma frequency 10–12
electron 11
ion 161
sth species 11
plasma heating
via hybrid resonances 115–116
via ion cyclotron resonances 107
via Ohmic dissipation 164
plasma parameter 15, 419
plasmapause 30
plasma sheaths 9–10
plasmasphere 29, 30, 225
Plemelj relation 301, 309, 321, 357, 361, 374
Poeverlein construction 129, 130
Poincaré surface of section 70
Poisson’s equation 7, 281, 282, 392, 395, 406
polarization 105
polarization charge 31, 37, 84
polarization reversal 104, 106, 107
poles of a complex function 291, 294
polytrope index 154, 181
positive ion sheath 9
Poynting flux 92, 194, 198
pressure 39
pressure tensor 138, 148
principal value integral 301
principle of superposition 79
propagating wave 97
propagation cutoff 90, 99, 116, 129
propagation vector 77
proton whistler 105, 106, 376
Q resonances 355, 387
q-factor 215
quantum effects 17
quasi-linear
diffusion coefficient 399, 400
diffusion equation 393–399, 400, 411
theory 391–406
R = 0 cutoff frequency 99–100, 107
radial diffusion 60
radiation belt 45, 66, 362, 380
radius of curvature 34
Rankine–Hugoniot equations 261
ratio of collective to discrete interactions 17
ratio of heat capacities 153
ratio of kinetic energy to potential energy 16–17
ratio of plasma pressure to magnetic field pressure 185
Rayleigh–Taylor instability 190
ray path 82, 127–131
real charges 84
real current 84
reconnection 258
reduced mass 416
reduced velocity distribution function 283, 373, 405
refraction 127
relationship between dielectric tensor and the conductivity tensor 86
residue theorem 293, 294, 295, 298
resistive MHD 182–183
resistivity 179, 430
resonance cone 116–117, 119
resonant particles 304
reverse-field pinch 209, 210
reversibility 272–274
right-hand polarized extraordinary (R-X) mode 100, 106, 123, 374
ring current 35, 36
root-mean-square velocity 6
Rosenbluth potentials 433
first 424, 431, 434
second 426, 431, 434
rotational discontinuity 258
runaway particles 432
Rutherford scattering cross-section 14, 417, 422
scale height 236
scattering cross-section (see Rutherford) 13
screw pinch 214–215
secondary electrons 10
second law of thermodynamics 272, 273
self-adjoint operator 229, 248
separatrix
magnetic field 56
phase space 305
shear Alfvén mode (see magnetohydrodynamic waves) 197
shock
adiabatic 266, 279
coordinates 261
electrostatic 409
fast 266, 271
field geometry 259
foot 276
intermediate 266, 271
oblique 271–272
overshoot 276
parallel 268–269
perpendicular 269
propagation speed 260–266, 267
ramp 276
slow 266, 271
strong 270
switch-on 269
transition region 276–278
wave 59, 251, 255, 258–278
weak 266
short-range force 140, 144
single particle motion 23–71
dipole magnetic field 66–68
magnetic mirror field 39–68
near an x-type neutral line 69
parabolic magnetic field 69, 70
perpendicular electric and magnetic fields 26–29
static uniform magnetic field 23–26
time varying magnetic field 45–47
small amplitude waves 75
small-argument expansion of the plasma dispersion function 309, 310, 316, 364
Snell’s law 129
solar corona 58, 208
solar flare 59, 276, 329, 330, 404–405
solar wind 56, 202, 256, 257, 258
interaction with a magnetized planet 257
magnetic field 202, 246, 329
pressure 60
velocity 202, 276
Solov’ev equilibria 218
sonic Mach number 262
sound speed 251, 261
species 5
spectral density 398
Spitzer–Harm resistivity 430
stability
electromagnetic waves in a hot magnetized plasma 377–385
electrostatic waves in a hot unmagnetized plasma 318–334
magnetohydrodynamic 219–239
Standard Map 401, 402, 403, 404
static MHD equilibria 204–219
static MHD equilibrium conditions 205–206
stationary non-linear electrostatic potentials 406–411
stellarator 37
Stokes’ theorem 46
strongly coupled plasmas 17, 416
strong shock limit 270
supernova 59, 276
Sweet–Parker model of magnetic reconnection 242–244
tangent form of the cold plasma dispersion relation 97
temperature 5–6
thermal cyclotron radius 352
thermal excitation of hybrid resonances 114–115
thermal speed 6, 156
electron 9
ion 9
thermodynamic state variables 273
threshold drift velocity 366
time stationary electrostatic potentials 145, 304
tokamak 37, 216
toroidal magnetic field 36, 37, 215–219
total pressure 177
total scattering cross-section 14
transverse waves 89–92, 98, 157
trapped particles in phase space 305
trapping of resonant electrons 383
turning points 43
two-stream instability 287, 410
type III solar radio bursts 329, 330, 404
ultraviolet radiation 10
uncertainty relation of quantum mechanics 77
uncertainty theorem for Fourier transforms 77
unit tensor 86
untrapped particles in phase space 305
Van Allen radiation belts 68
variational property of the perturbed MHD potential energy 233
vector calculus identities 443
vector differential operators 441–442
vector Helmholtz equation 247
vector potential 61, 209, 216
violations of the adiabatic invariants 54–60
first 55–58
second 58–59
third 59–60
virial theorem 204
Vlasov approach to the analysis of small amplitude waves 281–290
Vlasov equation 144, 281, 392
solutions based on constants of the motion 144–146
spatially averaged 393
time stationary 406
waveguide 131
wavelength 76
wave normal angle 83
wave number 76
waves in a cold plasma 75
waves in a hot magnetized plasma 341–386
waves in a hot unmagnetized plasma 281–334
wave steepening 251, 252, 275
wave vector 77
weak-beam approximation 288–289
weak growth rate approximation 300–303, 357, 358, 361, 385
weak shock limit 266–267
Wentzel–Kramer–Brillouin (WKB) equation 49, 129, 130
whistler dispersion 104
whistler mode 99, 100, 123, 124, 125, 376
whistler-mode chorus 383
whistler-mode emissions in planetary radiation belts 383
whistler-mode hiss 383
whistlers 100, 101–107
winding number 322
winding theorem 322, 324
x-type magnetic field configuration 240
x-type neutral line 56, 58, 69, 242
x-type neutral point 56, 69
zero-order quantities 87
Z-mode 123, 124, 125
Z-pinch 211–214