Preface ...................................................... xiii
Part I Plasma physics preliminaries ............................ 1
1 Introduction ................................................. 3
1.1 Motivation .............................................. 3
1.2 Thermonuclear fusion and plasma confinement ............. 4
1.2.1 Fusion reactions ................................. 4
1.2.2 Conditions for fusion ............................ 6
1.2.3 Magnetic confinement and tokamaks ............... 10
1.3 Astrophysical plasmas .................................. 13
1.3.1 Celestial mechanics ............................. 13
1.3.2 Astrophysics .................................... 15
1.3.3 Plasmas enter the stage ......................... 18
1.3.4 The standard view of nature ..................... 21
1.4 Definitions of the plasma state ........................ 23
1.4.1 Microscopic definition of plasma ................ 23
1.4.2 Macroscopic approach to plasma .................. 27
1.5 Literature and exercises ............................... 29
2 Elements of plasma physics .................................. 34
2.1 Theoretical models ..................................... 34
2.2 Single particle motion ................................. 34
2.2.1 Cyclotron motion ................................ 34
2.2.2 Excursion: basic equations of electrodynamics
and mechanics ................................... 38
2.2.3 Drifts, adiabatic invariants .................... 41
2.3 Kinetic plasma theory .................................. 47
2.3.1 Boltzmann equation and moment reduction ......... 48
2.3.2 Collective phenomena: plasma oscillations ....... 54
2.3.3 Landau damping .................................. 58
2.4 Fluid description ...................................... 65
2.4.1 From the two-fluid to the MHD description of
plasmas ......................................... 67
2.4.2 Alfven waves .................................... 71
2.4.3 Equilibrium and stability ....................... 74
2.5 In conclusion .......................................... 79
2.6 Literature and exercises ............................... 80
3 'Derivation' of the macroscopic equations ................... 83
3.1 Two approaches ......................................... 83
3.2 Kinetic equations ...................................... 84
3.2.1 Boltzmann equation .............................. 84
3.2.2 Moments of the Boltzmann equation ............... 88
3.2.3 Thermal fluctuations and transport .............. 90
3.2.4 Collisions and closure .......................... 94
3.3 Two-fluid equations .................................... 98
3.3.1 Electron-ion plasma ............................. 98
3.3.2 The classical transport coefficients ............ 99
3.3.3 Dissipative versus ideal fluids ................ 104
3.3.4 Excursion: waves in two-fluid plasmas .......... 108
3.4 One-fluid equations ................................... 119
3.4.1 Maximal ordering for MHD ....................... 119
3.4.2 Resistive and ideal MHD equations .............. 124
3.5 Literature and exercises .............................. 126
Part II Basic magnetohydrodynamics ........................... 129
4 The MHD model ............................................. 131
4.1 The ideal MHD equations ............................... 131
4.1.1 Postulating the basic equations ................ 131
4.1.2 Scale independence ............................. 138
4.1.3 A crucial question ............................. 139
4.2 Magnetic flux ......................................... 140
4.2.1 Flux tubes ..................................... 140
4.2.2 Global magnetic flux conservation .............. 142
4.3 Conservation laws ..................................... 145
4.3.1 Conservation form of the MHD equations ......... 145
4.3.2 Global conservation laws ....................... 148
4.3.3 Local conservation laws - conservation of
magnetic flux .................................. 152
4.3.4 Magnetic helicity .............................. 155
4.4 Dissipative magnetohydrodynamics ...................... 161
4.4.1 Resistive MHD .................................. 161
4.4.2 (Non-)conservation form of the dissipative
equations ...................................... 165
4.5 Discontinuities ....................................... 167
4.5.1 Shocks and jump conditions ..................... 167
4.5.2 Boundary conditions for plasmas with an
interface ...................................... 171
4.6 Model problems ........................................ 173
4.6.1 Laboratory plasmas (models I—III) .............. 174
4.6.2 Energy conservation for interface plasmas ...... 178
4.6.3 Astrophysical plasmas (models IV-VI) ........... 180
4.7 Literature and exercises .............................. 182
5 Waves and characteristics .................................. 186
5.1 Physics and accounting ................................ 186
5.1.1 Introduction ................................... 186
5.1.2 Soundwaves ..................................... 186
5.2 MHD waves ............................................. 190
5.2.1 Symmetric representation in primitive
variables ...................................... 190
5.2.2 Entropy wave and magnetic field constraint ..... 194
5.2.3 Reduction to velocity representation: three
waves .......................................... 198
5.2.4 Dispersion diagrams ............................ 202
5.3 Phase and group diagrams .............................. 205
5.3.1 Basic concepts ................................. 205
5.3.2 Application to the MHD waves ................... 207
5.3.3 Asymptotic properties .......................... 212
5.4 Characteristics ....................................... 213
5.4.1 The method of characteristics .................. 213
5.4.2 Classification of partial differential
equations ...................................... 216
5.4.3 Characteristics in ideal MHD ................... 219
5.5 Literature and exercises .............................. 227
6 Spectral theory ............................................ 230
6.1 Stability: intuitive approach ......................... 230
6.1.1 Two viewpoints ................................. 230
6.1.2 Linearization and Lagrangian reduction ......... 233
6.2 Force operator formalism .............................. 237
6.2.1 Equation of motion ............................. 237
6.2.2 Hilbert space .................................. 242
6.2.3 Proof of self-adjointness of the force
operator ....................................... 244
6.3 Spectral alternatives ................................. 250
6.3.1 Mathematical intermezzo ........................ 250
6.3.2 Initial value problem in MHD ................... 253
6.4 Quadratic forms and variational principles ............ 256
6.4.1 Expressions for the potential energy ........... 256
6.4.2 Hamilton's principle ........................... 259
6.4.3 Rayleigh-Ritz spectral variational principle ... 259
6.4.4 Energy principle ............................... 261
6.5 Further spectral issues ............................... 263
6.5.1 Normal modes and the energy principle .......... 263
6.5.2 Proof of the energy principle .................. 266
6.5.3 or-stability ................................... 268
6.5.4 Returning to the two viewpoints ................ 271
6.6 Extension to interface plasmas ........................ 274
6.6.1 Boundary conditions at the interface ........... 276
6.6.2 Self-adjointness for interface plasmas ......... 280
6.6.3 Extended variational principles ................ 283
6.6.4 Application to the Rayleigh-Taylor
instability .................................... 287
6.7 Literature and exercises .............................. 296
7 Waves and instabilities of inhomogeneous plasmas ........... 300
7.1 Hydrodynamics of the solar interior ................... 300
7.1.1 Radiative equilibrium model .................... 301
7.1.2 Convection zone ................................ 305
7.2 Hydrodynamic waves and instabilities of
a gravitating slab .................................... 308
7.2.1 Hydrodynamic wave equation ..................... 309
7.2.2 Convective instabilities ....................... 312
7.2.3 Gravito-acoustic waves ......................... 313
7.2.4 Helioseismology and MHD spectroscopy ........... 317
7.3 MHD wave equation for a gravitating magnetized
plasma slab ........................................... 322
7.3.1 Preliminaries .................................. 322
7.3.2 Derivation of the MHD wave equation for
a gravitating slab ............................. 327
7.3.3 Gravito-MHD waves .............................. 335
7.4 Continuous spectrum and spectral structure ............ 345
7.4.1 Singular differential equations ................ 345
7.4.2 Alfven and slow continua ....................... 351
7.4.3 Oscillation theorems ........................... 357
7.4.4 Cluster spectra ................................ 363
7.5 Gravitational instabilities of plasmas with magnetic
shear ................................................. 365
7.5.1 Energy principle for a gravitating plasma
slab ........................................... 366
7.5.2 Interchange instabilities in sheared magnetic
fields ......................................... 371
7.5.3 Interchanges in the absence of magnetic
shear .......................................... 376
7.6 Literature and exercises .............................. 379
8 Magnetic structures and dynamics ........................... 384
8.1 Plasma dynamics in laboratory and nature .............. 384
8.2 Solar magnetism ....................................... 385
8.2.1 The solar cycle ................................ 387
8.2.2 Magnetic structures in the solar atmosphere .... 395
8.3 Planetary magnetic fields ............................. 407
8.3.1 The geomagnetic dynamo ......................... 409
8.3.2 Magnetic fields of the other planets ........... 413
8.4 Magnetospheric plasmas ................................ 415
8.4.1 The solar wind and the heliosphere ............. 415
8.4.2 Solar wind and planetary magnetospheres ........ 419
8.5 Perspective ........................................... 426
8.6 Literature and exercises .............................. 427
9 Cylindrical plasmas ........................................ 431
9.1 Equilibrium of cylindrical plasmas .................... 431
9.1.1 Diffuse plasmas ................................ 431
9.1.2 Interface plasmas .............................. 436
9.2 MHD wave equation for cylindrical plasmas ............. 438
9.2.1 Derivation of the MHD wave equation for
a cylinder ..................................... 438
9.2.2 Boundary conditions for cylindrical
interfaces ..................................... 445
9.3 Spectral structure .................................... 450
9.3.1 One-dimensional inhomogeneity .................. 450
9.3.2 Cylindrical model problems ..................... 453
9.3.3 Cluster spectra ................................ 462
9.4 Stability of cylindrical plasmas ...................... 462
9.4.1 Oscillation theorems for stability ............. 462
9.4.2 Stability of plasmas with shearless magnetic
fields ......................................... 469
9.4.3 Stability of force-free magnetic fields ........ 475
9.4.4 Stability of the 'straight tokamak' ............ 482
9.5 Literature and exercises .............................. 492
10 Initial value problem and wave damping ..................... 496
10.1 Implications of the continuous spectrum ............... 496
10.2 Initial value problem ................................. 497
10.2.1 Reduction to a one-dimensional
representation ................................. 498
10.2.2 Restoring the three-dimensional picture ........ 502
10.3 Damping of Alfven waves ............................... 507
10.3.1 Green's function ............................... 509
10.3.2 Spectral cuts .................................. 512
10.4 Quasi-modes ........................................... 516
10.4.1 Dispersion equation ............................ 517
10.4.2 Exponential damping ............................ 520
10.4.3 Different kinds of quasi-modes ................. 522
10.5 Leaky modes ........................................... 523
10.5.1 Model equations and boundary conditions ........ 525
10.5.2 Normal-mode analysis ........................... 528
10.5.3 Initial value problem approach ................. 529
10.6 Literature and exercises .............................. 530
11 Resonant absorption and wave heating ....................... 533
11.1 Ideal MHD theory of resonant absorption ............... 534
11.1.1 Analytical solution of a simple model
problem 5 ...................................... 534
11.1.2 Role of the singularity ........................ 541
11.1.3 Resonant'absorption'versus resonant
'dissipation' .................................. 549
11.2 Heating and wave damping in tokamaks and coronal
magnetic loops ........................................ 553
11.2.1 Tokamaks ....................................... 553
11.2.2 Coronal loops and arcades ...................... 554
11.2.3 Numerical analysis of resonant absorption ...... 555
11.3 Alternative excitation mechanisms ..................... 561
11.3.1 Foot point driving ............................. 562
11.3.2 Phase mixing ................................... 565
11.3.3 Applications to solar and magnetospheric
plasmas ........................................ 567
11.4 Literature and exercises .............................. 573
Appendices .................................................... 577
A Vectors and coordinates .................................... 577
A.l Vector identities ..................................... 577
A.2 Vector expressions in orthogonal coordinates .......... 578
A.2.1 Cartesian coordinates {x, y, z) ................ 580
A.2.2 Cylinder coordinates (x, θ, z) ................. 581
A.2.3 Spherical coordinates (x, θ, φ) ................ 582
В Tables of physical quantities .............................. 585
References .................................................... 594
Index ......................................................... 607
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