Fundamentals of Plasma Physics

Cover......Page 1
Half-title......Page 3
Title......Page 5
Copyright......Page 6
Dedication......Page 7
Contents......Page 9
Preface......Page 15
1.2 Brief history of plasma physics......Page 21
1.3 Plasma parameters......Page 23
1.5 Logical framework of plasma physics......Page 24
1.6 Debye shielding......Page 27
1.7 Quasi-neutrality......Page 31
1.8 Small- vs. large-angle collisions in plasmas......Page 32
1.9 Electron and ion collision frequencies......Page 36
1.10 Collisions with neutrals......Page 39
1.11 Simple transport phenomena......Page 40
1.12 A quantitative perspective......Page 43
1.13 Assignments......Page 45
2.2 Phase-space......Page 54
2.3 Distribution function and Vlasov equation......Page 55
2.4 Moments of the distribution function......Page 58
2.4.1 Treatment of collisions in the Vlasov equation......Page 59
2.5 Two-fluid equations......Page 61
2.5.1 Entropy of a distribution function......Page 66
2.5.2 Effect of collisions on entropy......Page 69
2.5.3 Relation between pressure and Maxwellian......Page 71
2.6 Magnetohydrodynamic equations......Page 72
2.6.2 MHD equation of motion......Page 73
2.6.3 MHD Ohm’s law......Page 74
2.6.4 Ideal MHD and frozen-in flux......Page 76
Double adiabatic laws......Page 77
2.6.6 MHD approximations for Maxwell’s equations......Page 79
2.7 Summary of MHD equations......Page 81
2.8 Classical transport......Page 82
2.9 Sheath physics and Langmuir probe theory......Page 84
2.10 Assignments......Page 89
3.1 Motivation......Page 95
3.2 Hamilton–Lagrange formalism vs. Lorentz equation......Page 96
3.3 Adiabatic invariant of a pendulum......Page 100
3.4 Extension of WKB method to general adiabatic invariant......Page 103
3.4.1 Proof for the general adiabatic invariant......Page 105
3.5.1 Simple E×B and force drifts......Page 108
3.5.2 Drifts in slowly changing arbitrary fields......Page 111
3.5.3 Mu conservation......Page 116
3.5.4 Relation of Mu conservation to other conservation relations......Page 118
3.5.5 Drift equations, frozen-in flux, and frozen-in field lines......Page 120
3.5.6 Magnetic mirrors - a consequence of Mu conservation......Page 121
3.5.8 Consequences of …-invariance......Page 125
3.5.9 The third adiabatic invariant......Page 126
3.6 Relation of drift equations to the double adiabatic MHD equations......Page 128
3.7 Non-adiabatic motion in symmetric geometry......Page 135
3.7.1 Temporal reversal of magnetic field-energy gain......Page 142
3.7.2 Spatial reversal of field-cusps......Page 143
3.7.3 Stochastic motion in large-amplitude, low-frequency waves......Page 146
3.8 Particle motion in small-amplitude oscillatory fields......Page 149
3.9.1 “Average velocity”......Page 151
3.9.2 Motion of particles in a sawtooth potential......Page 152
3.9.4 Wave–particle energy transfer in a sinusoidal wave......Page 155
3.10 Assignments......Page 162
4.1 General method for analyzing small-amplitude waves......Page 166
4.2 Two-fluid theory of unmagnetized plasma waves......Page 167
4.2.1 Electrostatic (compressional) waves......Page 168
4.2.2 Electromagnetic (incompressible) waves......Page 173
4.3.1 Overview of Alfvén waves......Page 175
4.3.2 Zero-pressure MHD model......Page 176
4.3.4 Comparison of the two modes......Page 179
4.3.5 Finite-pressure analysis of MHD waves......Page 180
4.3.6 MHD compressional (fast) mode......Page 181
4.3.8 Limitations of the MHD model......Page 183
4.4 Two-fluid model of Alfvén modes......Page 184
4.4.1 Two-fluid slow (shear) modes......Page 188
4.5 Assignments......Page 192
5.2 Streaming instabilities......Page 194
5.3.1 Attempt to solve the linearized Vlasov–Poisson system of equations using Fourier analysis......Page 200
5.3.2 Landau method: Laplace transforms......Page 202
5.3.3 The relationship between poles, exponential functions, and analytic continuation......Page 207
5.3.4 Asymptotic long-time behavior of the potential oscillation......Page 209
5.3.5 Evaluation of the plasma dispersion function......Page 212
5.3.6 Landau damping of electron plasma waves......Page 215
5.3.7 Power relationships......Page 217
5.3.8 Landau damping for ion acoustic waves......Page 218
5.3.9 The Plemelj formula......Page 219
5.4 The Penrose criterion......Page 220
5.5 Assignments......Page 223
6.2 Redundancy of Poisson’s equation in electromagnetic mode analysis......Page 226
6.3 Dielectric tensor......Page 228
6.3.1 Mode behavior at Theta = 0......Page 231
6.3.3 Mode behavior at Theta…......Page 234
6.3.4 Very-low-frequency modes where Theta is arbitrary......Page 235
6.3.5 Modes where Omega and Theta are arbitrary......Page 236
6.3.6 Wave normal surfaces......Page 237
6.3.7 Taxonomy of modes – the CMA diagram......Page 238
6.3.8 Use of the CMA diagram......Page 242
6.4 Dispersion relation expressed as a relation between…......Page 243
6.5 A journey through parameter space......Page 245
6.6 High-frequency waves: Altar–Appleton–Hartree dispersion relation......Page 248
6.6.1 Quasi-transverse modes…......Page 250
6.6.2 Quasi-longitudinal dispersion…......Page 251
6.7 Group velocity......Page 253
6.8 Quasi-electrostatic cold plasma waves......Page 254
6.9 Resonance cones......Page 256
6.10 Assignments......Page 260
7.1 Wave propagation in inhomogeneous plasmas......Page 262
7.2 Geometric optics......Page 265
7.3 Surface waves – the plasma-filled waveguide......Page 267
7.4 Plasma wave-energy equation......Page 273
7.5 Cold plasma wave-energy equation......Page 275
7.6 Finite-temperature plasma wave-energy equation......Page 279
7.7 Negative energy waves......Page 280
7.8 Assignments......Page 283
8.1 Solving the Vlasov equation by tracking each particle’s history......Page 285
8.2 Analysis of the warm plasma electrostatic dispersion relation......Page 292
8.3 Bernstein waves......Page 293
8.4 Finite K dispersion: linear mode conversion......Page 296
8.5.1 Airy equation......Page 299
8.5.2 Steepest descent contour......Page 300
8.5.3 Relationship between saddle-point solutions and WKB modes......Page 303
8.5.4 Using boundary conditions to choose contours......Page 304
8.6 Drift waves......Page 309
8.6.1 Simple two-fluid model of drift waves......Page 311
8.6.2 Two-fluid drift wave model with collisions......Page 312
8.6.3 Vlasov theory of drift waves: collisionless drift waves......Page 319
8.7 Assignments......Page 324
9.1 Why use MHD?......Page 325
9.2 Vacuum magnetic fields......Page 326
9.3 Force-free fields......Page 329
9.4 Magnetic pressure and tension......Page 330
9.5 Magnetic stress tensor......Page 332
9.6 Flux preservation, energy minimization, and inductance......Page 334
9.8.1 Static equilibria in two dimensions: the Bennett pinch......Page 336
9.8.2 Impossibility of self-confinement of current-carrying plasma in three dimensions: the virial theorem......Page 338
9.8.3 Three-dimensional static equilibria: the Grad–Shafranov equation......Page 340
9.9.1 Incompressible plasma with self-field only......Page 348
9.9.2 Compressible plasma and applied poloidal field......Page 354
9.10 Assignments......Page 358
10.1 Introduction......Page 362
10.2 The Rayleigh–Taylor instability of hydrodynamics......Page 363
10.3 MHD Rayleigh–Taylor instability......Page 366
10.4 The MHD energy principle......Page 371
10.4.1 Energy equation for a magnetofluid......Page 372
10.4.2 Self-adjointness of potential energy as a consequence of energy integral......Page 376
10.4.3 Formal solution for perturbed potential energy......Page 377
10.4.4 Evaluation of Delta W......Page 379
10.5 Discussion of the energy principle......Page 385
10.6 Current-driven instabilities and helicity......Page 386
10.7 Magnetic helicity......Page 387
10.8 Characterization of free-boundary instabilities......Page 390
10.8.1 Qualitative examination of the sausage instability......Page 391
10.8.2 Qualitative examination of the kink instability......Page 392
10.8.3 Qualitative examination of wall stabilization......Page 393
10.9 Analysis of free-boundary instabilities......Page 394
Sausage modes......Page 400
Kink modes......Page 401
10.10 Assignments......Page 403
11.2.1 Linkage helicity......Page 405
11.2.2 Twist helicity......Page 407
11.2.3 Conservation of magnetic helicity during magnetic reconnection......Page 410
11.3 Woltjer–Taylor relaxation......Page 412
11.4 Kinking and magnetic helicity......Page 414
11.4.1 K helicity content of the flux tube......Page 417
11.4.2 Evaluation of K......Page 421
11.4.3 Evaluation of K......Page 423
11.5 Assignments......Page 427
12.1 Introduction......Page 430
12.2 Water-beading: an analogy to magnetic reconnection......Page 432
12.3 Qualitative description of sheet current instability......Page 433
12.4 Semi-quantitative estimate of the tearing process......Page 436
12.5 Generalization of tearing to sheared magnetic fields......Page 444
12.6 Magnetic islands......Page 450
12.7 Assignments......Page 452
13.1 Introduction......Page 456
13.2 Statistical argument for the development of the Fokker–Planck equation......Page 458
13.2.1 Slowing down......Page 465
Intermediate case: beam faster than ions, slower than electrons......Page 469
13.3 Electrical resistivity......Page 470
13.4 Runaway electric field......Page 471
13.5 Assignments......Page 473
14.1 Introduction......Page 476
14.2.1 Derivation of the quasi-linear diffusion equation......Page 478
Conservation of momentum......Page 485
Conservation of energy......Page 486
14.2.3 Energy exchange with resonant particles......Page 487
Behavior of the resonant particles......Page 489
Behavior of the non-resonant particles......Page 491
14.3 Echoes......Page 493
14.3.1 Ballistic terms and Laplace transforms......Page 497
14.3.3 Beam echoes......Page 498
14.3.4 The self-consistent nonlinear Vlasov–Poisson problem......Page 499
The linear problem......Page 501
Fourier–Laplace transform of the non-linear equation......Page 502
Double-impulse source function and its transform......Page 505
14.3.6 Higher order echoes......Page 508
14.4 Assignments......Page 509
Mode–mode coupling instabilities......Page 511
15.2 Manley–Rowe relations......Page 513
15.3 Application to waves......Page 519
15.3.1 Examples of nonlinearities......Page 520
15.3.2 Possible types of wave interaction......Page 522
Low-frequency wave is an ion acoustic wave......Page 525
Low-frequency wave is an electron plasma wave......Page 526
High-frequency wave is an electron plasma wave......Page 527
Coupled oscillator formulation......Page 528
15.4 Instability onset via nonlinear dispersion method......Page 531
15.5 Digging a density hole via ponderomotive force......Page 537
Caviton instability......Page 539
Stationary Envelope Soliton......Page 541
Propagating envelope soliton......Page 542
15.6 Ion acoustic wave soliton......Page 543
15.7 Assignments......Page 547
16.2 Brillouin flow......Page 550
16.3 Isomorphism to incompressible 2-D hydrodynamics......Page 553
16.4 Near-perfect confinement......Page 555
16.5 Diocotron modes......Page 557
16.5.2 The l = 1 diocotron mode: a special case......Page 559
16.5.3 Energy analysis of diocotron modes......Page 561
16.5.4 Resistive wall......Page 564
16.5.5 Diocotron modes with…......Page 566
16.5.6 Phase mixing and relation to Landau damping......Page 569
16.6 Assignments......Page 570
17.1 Introduction......Page 576
17.2 Electron and ion current flow to a dust grain......Page 577
17.3 Dust charge......Page 579
17.4 Dusty plasma parameter space......Page 583
17.5 Large P limit: dust acoustic waves......Page 584
17.6 Dust ion acoustic waves......Page 588
17.7 The strongly coupled regime: crystallization of a dusty plasma......Page 589
17.8 Assignments......Page 599
Vector algebra triple product......Page 602
Derivation of the vector calculus identities......Page 603
Summary of vector identities......Page 605
Generalized gradient operator......Page 606
Generalized divergence and curl......Page 607
Generalized Laplacian of a scalar......Page 608
Application to cylindrical coordinates......Page 609
Laplacian of a vector in cylindrical coordinates......Page 610
Application to spherical coordinates......Page 611
Lengths......Page 613
Frequencies......Page 614
Dimensionless......Page 615
Warm plasma waves......Page 616
Bibliography and suggested reading......Page 617
References......Page 619
Index......Page 624