Preface ..................................................................... xix
PART 1
Fundamentals of MHD wave theory ............................................... 1
1 Basic ideas of thermodynamics and electrodynamics ............................. 3
1.1 Introduction .............................................................. 3
1.2 Elementary ideas of thermodynamics and kinetic theory ..................... 4
1.2.1 Equation of state of an ideal gas ................................... 4
1.2.2 Comparison with kinetic theory ...................................... 4
1.2.3 First law of thermodynamics ......................................... 4
1.2.4 Second law of thermodynamics ........................................ 5
1.2.5 Ratio of specific heats of a gas .................................... 5
1.2.6 State variables and Maxwell's relations ............................. 5
1.2.7 Rate of change of entropy in reversible processes ................... 6
1.2.8 Specific energy, entropy, and enthalpy .............................. 6
1.3 Maxwell's equations in the presence of currents and charges ............... 6
1.4 The Lorentz force law ..................................................... 8
1.5 Low-velocity approximation to Maxwell's equations - Ampere's law .......... 8
1.6 Motion of charged particles in uniform electric and magnetic fields ....... 9
1.6.1 Equation of motion .................................................. 9
1.6.2 Cyclotron motion .................................................... 9
1.6.3 Electric field drift ............................................... 11
1.6.4 Drifts due to an external force .................................... 12
1.7 Electromagnetic energy ................................................... 13
1.7.1 Joule energy transfer .............................................. 13
1.7.2 Physical interpretation of the flux vector ......................... 14
1.8 Electromagnetic momentum ................................................. 15
1.9 Summary .................................................................. 17
2 The magnetohydrodynamic approximation ........................................ 19
2.1 Introduction ............................................................ 19
2.2 Fluid equations for the particle species ................................ 20
2.2.1 The continuity equation ........................................... 20
2.2.2 The momentum equation ............................................. 21
2.2.3 Adiabatic law ..................................................... 23
2.3 Characteristic lengths and frequencies .................................. 23
2.3.1 The Debye length .................................................. 23
2.3.2 The plasma frequency .............................................. 24
2.3.3 The electron and ion gyrofrequencies .............................. 25
2.3.4 Characteristic speeds ............................................. 25
2.3.5 Parameters for approximation ...................................... 26
2.4 The MHD equations for a fully ionized plasma ............................ 27
2.4.1 MHD variables ..................................................... 28
2.4.2 Continuity equation ............................................... 29
2.4.3 Momentum equation ................................................. 29
2.4.4 Adiabatic law ..................................................... 29
2.4.5 Generalized Ohm's law ............................................. 30
2.4.6 Reduced MHD equations ............................................. 31
2.5 Gravitation ............................................................. 32
2.6 Frozen-in magnetic fields ............................................... 32
2.7 Losses within plasmas ................................................... 33
2.7.1 Resistive effects ................................................. 33
2.7.2 Viscous effects ................................................... 34
2.8 Partially ionized plasma ................................................ 34
2.8.1 Current density in a partially ionized plasma ..................... 34
2.8.2 The conductivity tensor ........................................... 36
2.9 Conservation laws ....................................................... 39
2.9.1 MHD energy conservation ........................................... 39
2.9.2 Momentum conservation ............................................. 40
2.10 Summary ................................................................. 41
3 *Single-particle motion in electromagnetic fields ............................ 43
3.1 *Introduction ............................................................ 43
3.2 *Guiding-centre motion - heuristic approach .............................. 43
3.2.1 *Qualitative description of guiding-centre motion .................. 43
3.2.2 *Drift due to a magnetic field gradient ............................ 44
3.2.3 *Drifts due to the variation of the zero-order drift velocity ...... 46
3.2.4 *Parallel drift due to magnetic field shear ........................ 47
3.2.5 *The drift velocity of the guiding centre .......................... 47
3.3 *General motion in a varying field ....................................... 47
3.3.1 *Equations of motion ............................................... 47
3.4 *Theory of motion in a slowly varying field - the guiding-centre
approximation ............................................................ 50
3.4.1 *Slowly varying fields ............................................. 50
3.4.2 *The particle phase ................................................ 51
3.4.3 *The averaging process ............................................. 52
3.4.4 *Equations of motion for v^ and v|| ................................. 53
3.4.5 *The magnetic moment, an adiabatic invariant ....................... 54
3.4.6 *Drift velocity - the motion of the guiding centre ................. 55
3.4.7 *The energy equation ............................................... 57
3.5 *Motion in a dipole field—second and third adiabatic invariants and
constants of the motion .................................................. 58
3.5.1 *Natural periodicities ............................................. 58
3.5.2 *Second and third adiabatic invariants ............................. 59
3.5.3 *Energy and L-shell as constants of the motion ..................... 59
3.5.4 *Bounce motion ..................................................... 60
3.5.5 *Azimuthal drifts .................................................. 60
3.5.6 *Cross-L drifts .................................................... 60
3.6 *Summary ................................................................. 61
4 *Kinetic theory of plasmas ................................................... 62
4.1 *Introduction ............................................................ 62
4.2 *The distribution function ............................................... 63
4.3 *Mean values of the particle properties .................................. 63
4.3.1 * Averages over the velocity ....................................... 63
4.3.2 * Averages over the gyrophase ...................................... 64
4.3.3 *Directional average ............................................... 65
4.4 *Fluid and MHD variables ................................................. 65
4.4.1 *Mass density ...................................................... 66
4.4.2 *Drift velocity and current density ................................ 66
4.4.3 *Pressure tensor ................................................... 67
4.4.4 *Energy density and temperature .................................... 69
4.4.5 *Energy flux ....................................................... 70
4.5 * Kinetic equations ...................................................... 71
4.5.1 *Conservation of particles in phase space .......................... 71
4.5.2 *Boltzmann and Vlasov equations .................................... 73
4.6 *Approximations to the kinetic equation .................................. 74
4.6.1 *Low-frequency average of Vlasov equation .......................... 74
4.6.2 *Drift kinetic equations ........................................... 75
4.7 *Collisions and equivalent processes ..................................... 76
4.7.1 *The nature of the collision term in the Boltzmann equation ........ 77
4.8 *Equilibrium states ...................................................... 81
4.8.1 *Time scales to reach equilibrium and quasi-equilibrium ............ 81
4.8.2 *The Maxwell-Boltzmann and Maxwellian distributions ................ 83
4.8.3 *Jeans's theorem and quasi-equilibrium states ...................... 84
4.9 *Summary ................................................................. 86
5 *Fluid behaviour ............................................................. 88
5.1 *Introduction ............................................................ 88
5.2 *Distribution functions and their moments ................................ 89
5.3 *Evolution of particle properties ........................................ 89
5.3.1 *Moments of the particle distribution for a single fluid ........... 90
5.3.2 *Rate of change of a particle property ............................. 91
5.4 *Moment equations ........................................................ 93
5.4.1 *Moment equations for a single species ............................. 93
5.4.2 *Moment equations for a multi-ion plasma ........................... 95
5.5 *Closure of the moment equations ........................................ 100
5.5.1 *Successive approximations to the Boltzmann equation .............. 101
5.5.2 *Orders of magnitude .............................................. 102
5.5.3 *Truncation of the electromagnetic hierarchy ...................... 102
5.5.4 *Cold plasma ...................................................... 103
5.5.5 Thermal equilibrium ............................................... 103
5.5.6 *The fluid equations in the absence of collisions between
species ........................................................... 106
5.6 *Summary ................................................................ 110
6 Equilibrium and steady-state conditions ..................................... 112
6.1 Introduction ............................................................ 112
6.2 MHD equilibrium ......................................................... 112
6.3 MHD in the steady state ................................................. 113
6.4 Boundary conditions ..................................................... 114
6.5 Discontinuities and shocks .............................................. 117
6.5.1 Classification of discontinuities ................................. 117
6.5.2 Tangential discontinuity .......................................... 118
6.5.3 Rotational or Alfven discontinuity ................................ 118
6.5.4 Contact discontinuities ........................................... 120
6.5.5 MHD shocks ........................................................ 120
6.6 Summary ................................................................. 121
7 Harmonic plane waves in a uniform loss-free plasma .......................... 122
7.1 Introduction ............................................................ 122
7.2 Wave equations .......................................................... 123
7.2.1 Wave equation for a non-dispersive medium ......................... 123
7.2.2 Dispersive media .................................................. 124
7.3 Simple examples of waves ................................................ 125
7.3.1 Waves on strings and in gases ..................................... 125
7.3.2 Simple transverse Alfven waves .................................... 126
7.3.3 Simple compressional Alfven waves ................................. 127
7.4 General wave equation for MHD waves ..................................... 127
7.4.1 Linearization of the MHD equations ................................ 127
7.4.2 Wave equation ..................................................... 128
7.5 Harmonic waves .......................................................... 129
7.5.1 Equations for harmonic waves ...................................... 130
7.5.2 Dispersion relations .............................................. 131
7.5.3 Phase velocity .................................................... 132
7.5.4 Refractive index .................................................. 134
7.5.5 Relations between field components ................................ 137
7.6 Waves for non-scalar pressure ........................................... 138
7.7 Summary ................................................................. 144
8 *Collisionless damping of MHD waves ......................................... 145
8.1 *Introduction .......................................................... 145
8.2 *Specification of the problem .......................................... 145
8.3 *Single-particle motion in a wave ...................................... 147
8.4 * Kinetic effects ...................................................... 148
8.4.1 *First-order perturbation of the Vlasov equation ................. 148
8.4.2 *Integration over the unperturbed orbits ......................... 149
8.4.3 *Evaluation of moments of the perturbed distribution function .... 154
8.5 *The pressure tensor ................................................... 156
8.6 *Wave equations and dispersion relation ................................ 157
8.7 *Special cases of the dispersion relation .............................. 159
8.7.1 *Small β ......................................................... 159
8.7.2 *Parallel propagation ............................................ 160
8.7.3 *Perpendicular propagation ....................................... 161
8.8 *Physical picture of collisionless damping ............................. 161
8.9 * Wave properties ...................................................... 162
8.10 *Discussion ............................................................ 166
8.11 *Summary ............................................................... 166
9 Wavepackets and energy propagation in uniform media ......................... 168
9.1 Introduction ............................................................ 168
9.2 Wavepackets and rays .................................................... 169
9.2.1 Superpositions of one-dimensional harmonic waves .................. 169
9.2.2 Angular spectrum of plane waves ................................... 169
9.2.3 One-dimensional, spatially limited plane waves .................... 170
9.2.4 Rays and the group velocity ....................................... 171
9.2.5 Ray velocity and ray surface ...................................... 173
9.3 Propagation of energy by waves .......................................... 175
9.3.1 Energy conservation for waves ..................................... 175
9.3.2 Alternative form of the energy flux vector ........................ 176
9.3.3 Computing energy flux and energy density for harmonic waves ....... 177
9.3.4 Energy flux and energy density for quasimonochromatic waves in
a uniform medium .................................................. 177
9.3.5 Relation between total MHD energy and wave energy ................. 180
9.4 Summary ................................................................. 180
10 Reflection and transmission at sharp boundaries in stationary media ........ 182
10.1 Introduction .......................................................... 182
10.2 Equations for the field components .................................... 182
10.3 Boundary conditions ................................................... 185
10.3.1 Coordinate system .............................................. 185
10.3.2 Snell's law and the law of reflection .......................... 185
10.3.3 Boundary conditions at the sharp boundary ...................... 186
10.3.4 Boundary conditions at infinity ................................ 186
10.4 Reflection and transmission ........................................... 188
10.4.1 Reflection and transmission coefficients ....................... 188
10.5 Computation of reflection and transmission coefficients ............... 190
10.5.1 Partial reflection and Brewster angle .......................... 190
10.5.2 Critical angle; Total internal reflection ...................... 192
10.5.3 A more general case ............................................ 193
10.5.4 Energy flux .................................................... 196
10.6 Summary ............................................................... 196
11 Slowly varying media ....................................................... 198
11.1 Introduction .......................................................... 198
11.2 Phase integral solutions in a stratified medium ....................... 199
11.2.1 Phase integral solutions for the transverse Alfven wave ........ 200
11.2.2 Plane stratified media ......................................... 200
11.3 Huygens' construction ................................................. 202
11.4 Ray-tracing ........................................................... 203
11.4.1 The ray-tracing equations ...................................... 203
11.4.2 Ray-tracing and Huygens' construction .......................... 205
11.5 Summary ............................................................... 205
PART 2
The solar-terrestrial environment .......................................... 207
12 The Sun, the solar wind, and the magnetosphere ............................. 209
12.1 Introduction .......................................................... 209
12.2 The Sun ............................................................... 209
12.2.1 The visible outer regions ...................................... 210
12.2.2 Sunspots and the solar cycle ................................... 210
12.2.3 Helioseismic oscillations ...................................... 212
12.2.4 The solar magnetic field ....................................... 212
12.3 The solar wind ........................................................ 213
12.3.1 The quiet solar wind ........................................... 214
12.3.2 The fast solar wind ............................................ 216
12.4 Structure of the Earth's magnetosphere ................................ 217
12.4.1 Formation of the Earth's magnetosphere ......................... 217
12.4.2 Magnetic structure of the magnetosphere ........................ 221
12.5 Cold plasma populations in the magnetosphere .......................... 223
12.5.1 The Ionosphere ................................................. 223
12.5.2 The plasmasphere ............................................... 224
12.5.3 Cold plasma outside the plasmapause ............................ 225
12.5.4 The polar wind ................................................. 225
12.6 Hot plasma populations ................................................ 225
12.6.1 Boundary layers ................................................ 225
12.6.2 Tail lobes and plasma sheet .................................... 228
12.6.3 Radiation belts and ring current ............................... 228
12.7 Summary ............................................................... 229
13 Observations of ultra-low-frequency oscillations and waves ................. 230
13.1 Introduction .......................................................... 230
13.2 Waves and turbulence in the solar wind ................................ 230
13.3 Historical observations of ULF pulsations: 1861 - 1970 ................ 231
13.4 Physical understanding of ULF pulsations: 1971 to the present ......... 233
13.4.1 Global pulsations arising from sources at or beyond the
magnetopause ................................................... 234
13.4.2 Global pulsations arising from wave-particle interaction ....... 241
13.4.3 Pi2 pulsations ................................................. 244
13.5 Instrumentation ....................................................... 244
13.5.1 Modern instrumentation and the internet ........................ 244
13.5.2 Satellites and spacecraft ...................................... 245
13.5.3 Magnetometer arrays and other ground-based instrumentation ..... 245
13.5.4 Auroral radar arrays ........................................... 245
13.6 Summary ............................................................... 247
PART 3
Waves in solar-terrestrial physics ......................................... 249
14 MHD wave equations in non-uniform media .................................... 251
14.1 Introduction .......................................................... 251
14.2 Models ................................................................ 251
14.2.1 The magnetosphere .............................................. 251
14.2.2 Cylindrical models—sunspots and coronal loops .................. 253
14.3 Coupled wave equations in a plane-stratified medium ................... 254
14.3.1 First-order wave equations ..................................... 254
14.3.2 Polarization relations ......................................... 257
14.3.3 Second-order wave equation ..................................... 257
14.4 Wave equations for a cold plasma in a dipole field .................... 260
14.5 Multicomponent plasmas ................................................ 262
14.5.1 Background model ............................................... 263
14.5.2 Linearized equations ........................................... 264
14.6 Wave equations in a cylindrically stratified medium ................... 265
14.7 Summary ............................................................... 267
15 Propagation in a plane-stratified medium ................................... 269
15.1 Introduction .......................................................... 269
15.2 Wave propagation through numerical computation ........................ 270
15.3 WKBJ solutions of the wave equation ................................... 270
15.3.1 Energy flux .................................................... 272
15.3.2 Error terms in the differential equation ....................... 273
15.4 Cumulative error in the WKBJ solutions ................................ 273
15.4.1 Method of variation of parameters .............................. 274
15.4.2 Cumulative error ............................................... 274
15.5 Reflection ............................................................ 276
15.5.1 Geometrical optics of reflection ............................... 276
15.6 Full wave theory of reflection ........................................ 277
15.6.1 Stokes' equation and Airy functions ............................ 278
15.6.2 WKBJ solutions of the Stokes equation .......................... 279
15.6.3 The Stokes phenomenon .......................................... 280
15.6.4 WKBJ approximations to the Airy functions ...................... 280
15.6.5 Approximate WKBJ representation of a general wave .............. 281
15.6.6 Error near the zeros of q ...................................... 283
15.7 Connection relations .................................................. 284
15.7.1 Stokes and anti-Stokes lines ................................... 284
15.7.2 Connection relations ........................................... 285
15.7.3 Range of validity of asymptotic approximations in the complex
plane .......................................................... 286
15.8 Summary ............................................................... 287
16 Standing waves and oscillations in a cold plasma ........................... 288
16.1 Introduction .......................................................... 288
16.2 Transverse Alfven oscillations ........................................ 289
16.2.1 Uniform medium ................................................. 289
16.2.2 Medium with a transverse gradient of Alfven speed .............. 291
16.3 The Earth-ionosphere system as a boundary ............................. 293
16.3.1 Height-integrated conductivity of the ionosphere ............... 294
16.3.2 Fields below the ionosphere .................................... 296
16.3.3 Transmission through the ionosphere ............................ 297
16.4 Lossy oscillations in a uniform medium ................................ 299
16.5 Alfven oscillations in a dipole-like field ............................ 300
16.5.1 Cylindrically symmetric oscillations ........................... 300
16.5.2 Oscillations with large azimuthal wavenumber ................... 301
16.6 Properties of localized field-line oscillations ....................... 302
16.6.1 Basic equations ................................................ 302
16.6.2 Numerical solutions when the ionosphere has very large
conductivity ................................................... 304
16.6.3 Numerical solutions for finite ionospheric conductivity ........ 305
16.6.4 Perturbation solution of the azimuthal equation ................ 308
16.6.5 WKBJ solutions ................................................. 311
16.7 Summary ............................................................... 315
17 Standing waves and oscillations in a compressional plasma .................. 317
17.1 Introduction .......................................................... 317
17.2 Localized oscillations ................................................ 318
17.3 Models ................................................................ 320
17.3.1 Ring current ................................................... 320
17.3.2 Boundary conditions ............................................ 323
17.4 Solutions of the coupled equations .................................... 324
17.4.1 Phase integral solutions ....................................... 324
17.4.2 Numerical solutions ............................................ 328
17.5 Summary ............................................................... 333
18 Field-line resonance in low-pressure plasmas ............................... 334
18.1 Introduction .......................................................... 334
18.2 Basic ideas of field-line resonance ................................... 335
18.2.1 Standing waves and field-line resonance ........................ 338
18.2.2 Loss mechanisms ................................................ 338
18.3 Waves and conservation laws ........................................... 340
18.3.1 Definition of wave invariant ................................... 340
18.3.2 Conservation and non-conservation at singular points ........... 341
18.4 Modelling resonance in a dipole geometry .............................. 342
18.5 Summary ............................................................... 346
19 Mathematics of field-line resonance in compressible media .................. 348
19.1 Introduction .......................................................... 348
19.2 Field-line resonance in a compressible plane-stratified plasma ........ 348
19.2.1 The resonance equation ......................................... 349
19.2.2 Series solution to the resonance equation ...................... 351
19.2.3 WKBJ approximations ............................................ 353
19.2.4 The Stokes phenomenon .......................................... 354
19.3 Solutions of the resonance equation ................................... 357
19.3.1 Numerical computation of the solutions ......................... 357
19.3.2 Accuracy of the WKBJ solutions ................................. 360
19.3.3 Approximate solutions of the wave equation ..................... 361
19.4 Reflection coefficients ............................................... 361
19.5 Resonance in cylindrical geometries ................................... 362
19.5.1 Equations describing dissipation in cylindrical flux tubes ..... 364
19.5.2 Solutions of the dissipative equations in cylindrical
geometry ....................................................... 366
19.5.3 Resonance heating .............................................. 371
19.6 Summary ............................................................... 373
20 Cavity oscillations and waveguide modes .................................... 374
20.1 Introduction .......................................................... 374
20.2 The magnetospheric cavity or waveguide ................................ 375
20.3 Lossy modes ........................................................... 379
20.4 Time-dependent behaviour .............................................. 380
20.4.1 Time-varying behaviour in a closed cavity ...................... 381
20.4.2 The Green's function ........................................... 382
20.4.3 Some applications of the Green's function method ............... 384
20.5 Leaky cavities and waveguides ......................................... 395
20.5.1 Reflection and transmission coefficients at a leaky boundary ... 395
20.6 Excitation of the magnetospheric cavity ............................... 398
20.6.1 Time development of the reflected and transmitted waves ........ 399
20.6.2 Discussion ..................................................... 402
20.7 The waveguide picture ................................................. 403
20.8 Summary ............................................................... 403
21 Waves in moving media ...................................................... 405
21.1 Introduction .......................................................... 405
21.2 Wave propagation in a uniform moving medium ........................... 406
21.2.1 Qualitative picture ............................................ 406
21.2.2 Modification of the wave equations for a moving medium ......... 407
21.2.3 Harmonic waves ................................................. 408
21.2.4 The entropy wave ............................................... 411
21.3 Energy relations in a uniform medium .................................. 411
21.3.1 Energy conservation equation ................................... 411
21.4 Reflection and transmission of a plane wave at a tangential
discontinuity ......................................................... 413
21.4.1 Reflection and transmission coefficients ....................... 413
21.4.2 Numerical results .............................................. 416
21.4.3 Energy conservation at the boundary ............................ 417
21.4.4 Reflection and transmission coefficients for the energy ........ 418
21.5 Energy balance in a non-uniform plasma ................................ 420
21.5.1 Relation between total MHD energy and wave energy .............. 424
21.6 Ray-tracing in a moving medium ........................................ 429
21.7 The negative energy wave picture ...................................... 430
21.8 Over-reflection in solar-terrestrial physics .......................... 434
21.8.1 Excitation of long period pulsations ........................... 434
21.8.2 Resonant absorption in coronal plumes .......................... 434
21.9 Summary ............................................................... 435
22 Shock waves ................................................................ 436
22.1 Introduction .......................................................... 436
22.2 Properties of shock waves ............................................. 437
22.2.1 Change of properties across a shock in a gas ................... 437
22.2.2 Changes through MHD shocks ..................................... 440
22.3 Waves in the neighbourhood of shocks .................................. 440
22.3.1 Coordinate system and boundary conditions ...................... 440
22.3.2 Dispersion relations ........................................... 441
22.3.3 Relation between field components .............................. 442
22.3.4 Direction of propagation relative to the shock ................. 442
22.4 Classification of shocks .............................................. 444
22.4.1 Fast and slow shocks ........................................... 444
22.4.2 Perpendicular and parallel shocks .............................. 445
22.5 Propagation of MHD waves through shocks ............................... 447
22.5.1 Boundary conditions ............................................ 447
22.5.2 Behaviour of the waves on either side of the boundary .......... 451
22.6 Summary ............................................................... 452
23 Magnetohydrodynamic instability ............................................ 454
23.1 Introduction .......................................................... 454
23.2 Nature of instability ................................................. 455
23.2.1 Growth of a spatial perturbation ............................... 455
23.2.2 Convected and non-convected instability ........................ 456
23.2.3 Macroscopic and microscopic instability ........................ 457
23.3 Fluid instabilities ................................................... 457
23.4 The Kelvin-Helmholtz instability ...................................... 457
23.4.1 Physical basis ................................................. 457
23.4.2 Sharp boundary between two counterstreaming MHD media .......... 459
23.4.3 Effect of finite boundary thickness ............................ 462
23.4.4 Applications in magnetospheric conditions ...................... 466
23.5 *Pressure anisotropy .................................................. 466
23.5.1 *Firehose instability .......................................... 467
23.5.2 *Mirror instability ............................................ 469
23.6 *Summary .............................................................. 471
24 *Wave-particle interactions and kinetic effects ............................ 472
24.1 *Particle resonance ................................................... 472
24.1.1 *Uniform medium ................................................ 472
24.1.2 *Dipole field .................................................. 473
24.2 *Wave-particle interaction in a uniform medium ........................ 476
24.3 *Wave-particle interaction in a dipole field .......................... 480
24.3.1 *Linearization of the drift kinetic equation ................... 480
24.3.2 *Evaluation of moments of the perturbed distribution
function ....................................................... 484
24.4 *Quasilinear theory: a tutorial example ............................... 487
24.4.1 *Model ......................................................... 487
24.4.2 *Basic equations of the disturbance ............................ 488
24.4.3 *Boundary conditions ........................................... 489
24.4.4 *Zero-order solution ........................................... 489
24.4.5 *Boundary perturbation ......................................... 490
24.4.6 *Energy at marginal stability .................................. 490
24.4.7 *Relationship Between ω and ky ................................. 491
24.5 *Quasilinear theory in a dipole field ................................. 491
24.6 *Limits of magnetohydrodynamics ....................................... 493
24.7 *Summary .............................................................. 493
25 Last words ................................................................. 495
PART 4
Appendices .................................................................. 497
A Some mathematical techniques ................................................ 499
A.1 The essence of Cartesian tensors ........................................ 499
A.2 Vector operators in curvilinear coordinates ............................. 500
A.3 Properties of the bi-Maxwellian distribution ............................ 500
A.4 The Stokes equation and Airy functions .................................. 502
A.5 The plasma dispersion function .......................................... 503
A.6 Method of stationary phase .............................................. 506
B Properties of the geomagnetic field ......................................... 508
B.1 Properties of a dipole field ............................................ 508
B.2 Geometry of field lines ................................................. 509
B.3 Magnetic field coordinates .............................................. 512
B.3.1 Cylindrical symmetry in the absence of currents ................... 512
B.3.2 Local coordinates in the presence of currents ..................... 515
C Fourier analysis techniques ................................................. 516
C.1 Some generalized functions .............................................. 516
C.1.1 The Dirac delta-function .......................................... 516
C.1.2 The Heaviside unit step function .................................. 516
C.2 The Fourier transform ................................................... 516
C.2.1 Derivatives of the Fourier transform .............................. 517
C.2.2 Convolution and the convolution theorem ........................... 517
C.2.3 Theorems relating to Fourier transforms ........................... 517
C.2.4 Symmetry properties of the Fourier transform ...................... 518
C.2.5 Some Fourier transforms ........................................... 518
C.3 The modified Laplace transform .......................................... 518
C.3.1 Laplace transforms of derivatives of f(t) ......................... 519
D Wave analysis techniques .................................................... 520
D.1 Introduction ............................................................ 520
D.2 Integral representations ................................................ 520
D.2.1 Transform methods for the solution of the one-dimensional wave
equation .......................................................... 520
D.3 Wavepacket analysis ..................................................... 523
D.3.1 The analytic signal ............................................... 523
D.3.2 Quasiperiodic functions ........................................... 524
Bibliography .................................................................. 528
Index ......................................................................... 541
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