Walker A.D.M. Magnetohydrodynamic waves in geospace
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Walker A.D.M. Magnetohydrodynamic waves in geospace: the theory of ULF waves and their interaction with energetic particles in the solar-terrestrial environment / Walker A.D.M. - Bristol: Institute of Physics, 2005. - 549 p. - (Series in plasma physics; vol.16). - ISBN 0-7503-0910-5.
 
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  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


Вверх Walker A.D.M. Magnetohydrodynamic waves in geospace: the theory of ULF waves and their interaction with energetic particles in the solar-terrestrial environment / Walker A.D.M. - Bristol: Institute of Physics, 2005. - 549 p. - (Series in plasma physics; vol.16). - ISBN 0-7503-0910-5.

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