Preface ...................................................... xiii
Acknowledgements .............................................. xix
1 Introduction ............................................... 1
1.1 Fermat's principle of stationary time ...................... 2
1.1.1 General comments .................................... 2
1.1.2 Uniform media ....................................... 3
1.1.3 Snell's Law ......................................... 4
1.1.4 Distributed sources ................................. 5
1.1.5 Stationarity vs. minimization: the law of
reflection .......................................... 6
1.1.6 Smoothly varying media .............................. 9
1.2 Hamilton's principle of stationary phase .................. 11
1.2.1 Phase speed ........................................ 12
1.2.2 Phase integrals and rays ........................... 14
1.3 Modern developments ....................................... 18
1.3.1 Quantum mechanics and symbol calculus .............. 18
1.3.2 Ray phase space and plasma wave theory ............. 22
1.4 One-dimensional uniform plasma: Fourier methods ........... 27
1.4.1 General linear wave equation: D(—i∂x, i∂t)ψ = 0 .... 28
1.4.2 Dispersion function: D(k, ω) ....................... 29
1.4.3 Modulated wave trains: group velocity and
dispersion ......................................... 31
1.4.4 Weak dissipation ................................... 34
1.4.5 Far field of dispersive wave equations ............. 35
1.5 Multidimensional uniform plasma ........................... 38
1.6 One-dimensional nonuniform plasma: ray tracing ............ 40
1.6.1 Eikonal equation for an EM wave .................... 40
1.6.2 Wave-action conservation ........................... 42
1.6.3 Eikonal phase θ(х) ................................. 43
1.6.4 Amplitude A(x) ..................................... 44
1.6.5 Hamilton's equations for rays ...................... 45
1.6.6 Example: reflection of an EM wave near the plasma
edge ............................................... 46
1.7 Two-dimensional nonuniform plasma: multidimensional
ray tracing ............................................... 47
1.7.1 Eikonal equation for an EM wave .................... 48
1.7.2 Wave-action conservation ........................... 48
1.7.3 Eikonal phase θ(x,y) and Lagrange manifolds ........ 49
1.7.4 Hamilton's equations for rays ...................... 49
Problems .................................................. 51
References ................................................ 55
2 Some preliminaries ........................................ 62
2.1 Variational formulations of wave equations ................ 62
2.2 Reduced variational principle for a scalar wave equation .. 63
2.2.1 Eikonal equation for the phase ..................... 64
2.2.2 Noether symmetry and wave-action conservation ...... 64
2.3 Weyl symbol calculus ...................................... 66
2.3.1 Symbols in one spatial dimension ................... 66
2.3.2 Symbols in multiple dimensions ..................... 72
2.3.3 Symbols for multicomponent linear wave equations ... 74
2.3.4 Symbols for operator products: the Moyal series .... 74
Problems .................................................. 76
References ................................................ 78
3 Eikonal approximation ..................................... 80
3.1 Eikonal approximation: x-space viewpoint .................. 81
3.2 Eikonal approximation: phase space viewpoint .............. 84
3.2.1 Lifts and projections .............................. 89
3.2.2 Matching to boundary conditions .................... 92
3.2.3 Higher-order phase corrections ..................... 94
3.2.4 Action transport using the focusing tensor ......... 95
3.2.5 Pulling it all together ............................ 97
3.2.6 Frequency-modulated waves ......................... 102
3.2.7 Eikonal waves in a time-dependent background
plasma ............................................ 104
3.2.8 Symmetries ........................................ 105
3.2.9 Curvilinear coordinates ........................... 108
3.3 Covariant formulations ................................... 111
3.3.1 Lorentz-covariant eikonal theory .................. 111
3.3.2 Energy-momentum conservation laws ................. 119
3.4 Fully covariant ray theory in phase space ................ 121
3.5 Special topics ........................................... 128
3.5.1 Weak dissipation .................................. 129
3.5.2 Waveguides ........................................ 132
3.5.3 Boundaries ........................................ 134
3.5.4 Wave emission from a coherent source .............. 139
3.5.5 Incoherent waves and the wave kinetic equation .... 142
Problems ................................................. 146
References ............................................... 151
4 Visualization and wave-field construction ................ 154
4.1 Visualization in higher dimensions ....................... 155
4.1.1 Poincare surface of section ....................... 155
4.1.2 Global visualization methods ...................... 157
4.2 Construction of wave fields using ray-tracing results .... 170
4.2.1 Example: electron dynamics in parallel electric
and magnetic fields ............................... 173
4.2.2 Example: lower hybrid cutoff model ................ 173
References ............................................... 182
5 Phase space theory of caustics ........................... 183
5.1 Conceptual discussion .................................... 187
5.1.1 Caustics in one dimension: the fold ............... 187
5.1.2 Caustics in multiple dimensions ................... 191
5.2 Mathematical details ..................................... 193
5.2.1 Fourier transform of an eikonal wave field ........ 194
5.2.2 Eikonal theory in k-space ......................... 196
5.3 One-dimensional case ..................................... 198
5.3.1 Summary of eikonal results in x and k ............. 198
5.3.2 The caustic region in x: Airy's equation .......... 200
5.3.3 The normal form for a generic fold caustic ........ 205
5.3.4 Caustics in vector wave equations ................. 210
5.4 Caustics in n dimensions ................................. 212
Problems ................................................. 218
References ............................................... 226
6 Mode conversion and tunneling ............................ 228
6.1 Introduction ............................................. 228
6.2 Tunneling ................................................ 242
6.3 Mode conversion in one spatial dimension ................. 247
6.3.1 Derivation of the 2 × 2 local wave equation ....... 247
6.3.2 Solution of the 2 × 2 local wave equation ......... 252
6.4 Examples ................................................. 258
6.4.1 Budden model as a double conversion ............... 259
6.4.2 Modular conversion in magnetohelioseismology ...... 261
6.4.3 Mode conversion in the Gulf of Guinea ............. 263
6.4.4 Modular approach to iterated mode conversion ...... 269
6.4.5 Higher-order effects in one-dimensional
conversion models ................................. 273
6.5 Mode conversion in multiple dimensions ................... 276
6.5.1 Derivation of the 2 × 2 local wave equation ....... 276
6.5.2 The 2 × 2 normal form ............................. 279
6.6 Mode conversion in a numerical ray-tracing algorithm:
RAYCON ................................................... 283
6.7 Example: Ray splitting in rf heating of tokamak plasma ... 295
6.8 Iterated conversion in a cavity .......................... 301
6.9 Wave emission as a resonance crossing .................... 303
6.9.1 Coherent sources .................................. 304
6.9.2 Incoherent sources ................................ 308
Problems ................................................. 310
Suggested further reading ................................ 322
References ............................................... 323
7 Gyroresonant wave conversion ............................. 327
7.1 Introduction ............................................. 327
7.1.1 General comments .................................. 329
7.1.2 Example: Gyroballistic waves in one spatial
dimension ......................................... 331
7.1.3 Minority gyroresonance and mode conversion ........ 333
7.2 Resonance crossing in one spatial dimension: cold-
plasma model ............................................. 335
7.3 Finite-temperature effects in minority gyroresonance ..... 348
7.3.1 Local solutions near resonance crossing for
finite temperature ................................ 359
7.3.2 Solving for the Bernstein wave .................... 373
7.3.3 Bateman-Kruskal methods ........................... 379
Problems ................................................. 385
References ............................................... 392
Appendix A Cold-plasma models for the plasma dielectric
tensor ........................................................ 394
A.l Multifluid cold-plasma models ............................ 395
A.2 Unmagnetized plasma ...................................... 397
A.3 Magnetized plasma ........................................ 399
A.3.1 k  B0 ............................................. 400
A.3.2 k || B0 ............................................ 401
A.4 Dissipation and the Kramers-Kronig relations ............. 403
Problems ................................................. 404
References ............................................... 405
Appendix В Review of variational principles .................. 406
B.1 Functional derivatives ................................... 406
B.2 Conservation laws of energy, momentum, and action for
wave equations ........................................... 408
B.2.1 Energy-momentum conservation laws ................. 408
B.2.2 Wave-action conservation .......................... 409
References ............................................... 411
Appendix С Potpourri of other useful mathematical ideas ...... 412
C.l Stationary phase methods ................................. 412
C.l.l The one-dimensional case .......................... 412
C.1.2 Stationary phase methods in multidimensions ........ 416
C.2 Some useful facts about operators and bilinear
forms .................................................... 421
Problem .................................................. 424
References ............................................... 424
Appendix D Heisenberg-Weyl group and the theory of
operator symbols .............................................. 426
D.1 Introductory comments .................................... 426
D.2 Groups, group algebras, and convolutions on groups ....... 427
D.3 Linear representations of groups ......................... 430
D.3.1 Lie groups and Lie algebras ....................... 434
D.4 Finite representations of Heisenberg-Weyl ................ 436
D.4.1 The translation group on n points .................. 436
D.4.2 The finite Heisenberg-Weyl group ................... 439
D.5 Continuous representations ............................... 442
D.6 The regular representation ............................... 445
D.7 The primary representation ............................... 445
D.8 Reduction to the Schrodinger representation .............. 446
D.8.1 Reduction via a projection operator ................ 446
D.8.2 Reduction via restriction to an invariant
subspace ........................................... 446
D.9 The Weyl symbol calculus ................................. 447
References ............................................... 452
Appendix E Canonical transformations and metaplectic
transforms .................................................... 453
E.1 Examples ................................................. 453
E.2 Two-dimensional phase spaces ............................. 457
E.2.1 General canonical transformations ................. 457
E.2.2 Metaplectic transforms ............................ 459
E.3 Multiple dimensions ...................................... 466
E.3.1 Canonical transformations ......................... 466
E.3.2 Lagrange manifolds ................................ 467
E.3.3 Metaplectic transforms ............................ 469
E.4 Canonical coordinates for the 2 × 2 normal form .......... 471
References ............................................... 476
Appendix F Normal forms ...................................... 479
F.1 The normal form concept .................................. 479
F.2 The normal form for quadratic ray Hamiltonians ........... 482
F.3 The normal form for 2 × 2 vector wave equations .......... 488
F.3.1 The Braam-Duistermaat normal forms ................ 497
F.3.2 The general case .................................. 497
References ............................................... 499
Appendix G General solutions for multidimensional
conversion .................................................... 500
G.1 Introductory comments .................................... 500
G.2 Summary of the basis functions used ...................... 500
G.3 General solutions ........................................ 504
G.4 Matching to incoming and outgoing fields ................. 506
Reference ................................................ 510
Glossary of mathematical symbols .............................. 511
Author index .................................................. 514
Subject index ................................................. 517
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