Ray tracing and beyond: phase space methods in plasma wave theory (Cambridge, 2014). - ОГЛАВЛЕНИЕ / CONTENTS

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ОбложкаRay tracing and beyond: phase space methods in plasma wave theory / E.R.Tracy, A.J.Brizard, A.S.Richardson, A.N.Kaufman. - Cambridge: Cambridge university press, 2014. - xxi, 521 p.: ill. - Incl. bibl. ref. - Auth. ind.: p.514-516. - Sub. ind.: p.517-521. - ISBN 978-0-521-76806-1
 

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Оглавление / Contents
 
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 fig.3 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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