Aspects of Anomalous Transport in Plasmas (Series in Plasma Physics)

Aspects of Agnomalous Transport in Plasmas......Page 1
Table of Contents......Page 3
Preface......Page 7
1.1 The Problem of Anomalous Transport......Page 10
1.2 Microscopic and Macroscopic Levels of Description......Page 14
1.3 Closure of the Plasmadynamical Equations......Page 19
1.4 Experimental Aspects......Page 21
1.5 The Turbulent State of the Plasma......Page 24
1.6 Statistical Description of a Turbulent Plasma......Page 26
1.7 The Averaging Operation......Page 29
2.1 The Inhomogeneous Plasma......Page 31
2.2 Characteristic Quantities of a Fusion Plasma......Page 37
2.3 Reduced Macroscopic Equations......Page 41
2.4 The Hamaguchi-Horton Equations......Page 42
2.5 The Hasegawa-Wakatani Equations......Page 47
2.6 The Hasegawa-Mima Equation......Page 52
3.1 Introduction......Page 54
3.2 Homogeneous plasmas......Page 56
3.3 Density Gradient (DG) Drift Waves......Page 57
3.4 Collisional DG Drift Waves......Page 58
3.5 DG Drift-Ion Acoustic Waves......Page 60
3.6 Ion Temperature Gradient (ITG) Modes......Page 61
3.7 The ITG Mode in Different Geometries......Page 66
4.1 Guiding Centre Variables......Page 68
4.2 Electrostatic Fluctuations in Shearless Slab Geometry......Page 74
4.3 The Gyrokinetic Equation......Page 76
4.4 The Fourier Representation......Page 82
5.1 Introduction......Page 88
5.2 The Particle Number Balance......Page 90
5.3 The Thermal Energy Balance......Page 92
5.4 The Electrostatic Energy Balance......Page 95
6.1 Kinetic Equation and Langevin Equations......Page 99
6.2 The Non-Markovian Kinetic Equation......Page 102
6.3 Eulerian Correlations......Page 106
6.4 Lagrangian Correlations......Page 112
6.5 Fluxes and Lagrangian Correlations......Page 114
6.6 The Local Equilibrium Approximation......Page 116
7.1 The Quasilinear Approximation......Page 119
7.2 The Fourier Representation......Page 122
7.3 The Fourier-Laplace Representation......Page 124
7.4 Quasilinear Anomalous Fluxes (I)......Page 128
7.5 The Poisson Equation......Page 133
7.6 Self-consistent Quasilinear Theory......Page 135
7.7 Solution of the Dispersion Equation......Page 138
7.8 DG Drift-Ion Acoustic Modes......Page 141
7.9 Quasilinear Anomalous Fluxes (II)......Page 144
7.10 Discussion of the Quasilinear Fluxes......Page 146
8.1 The Basic Equations of Drift Wave Turbulence......Page 148
8.2 Potential and Response Correlations......Page 153
8.3 The Renormalization Programme......Page 158
8.4 Short History of Microscopic Plasma Turbulence Theory......Page 163
8.5 General Relations between Renormalized Quantities......Page 167
8.6 Coherent and Incoherent Responses......Page 172
8.7 The Direct Interaction Approximation (DIA)......Page 176
8.8 The Renormalized Gyrokinetic Equation......Page 180
8.9 Realizability and Langevin Representation......Page 183
8.10 Energy Conservation......Page 185
8.11 Markovian Closures......Page 188
9.1 The Renormalized Quasilinear Approximation......Page 190
9.2 Turbulent Resonance Broadening......Page 192
9.3 The Threshold of Diffusivity......Page 199
9.4 The Renormalized Dispersion Equation......Page 201
9.5 Anomalous Fluxes in the RQL Approximation......Page 203
9.6 The Marginal Stability Constraint......Page 205
10.1 Nonlinear Mode Coupling. The Spectrum Balance Equation......Page 208
10.2 Trajectory Correlation......Page 213
10.3 Relative Diffusion......Page 217
10.4 Partial Conclusions......Page 222
11.1 A-Langevin Equations......Page 224
11.2 Mean Square Displacement and Diffusion Coefficient......Page 230
11.3 V-Langevin Equations......Page 236
11.4 Statistical Definition of Magnetic Field Fluctuations......Page 238
11.5 Langevin Equation for Test Particle in a Fluctuating Magnetic Field......Page 242
11.6 Magnetic Subdiffusion......Page 245
11.7 Magnetic Field Line Diffusion......Page 251
11.8 The Concept of a Hybrid Kinetic Equation......Page 253
11.9 The Hybrid Kinetic Equation for Magnetic Fluctuations......Page 255
11.10 Hybrid Kinetic Equation for Collisional Diffusion......Page 257
12.1 Classical Random Walks......Page 260
12.2 Stable Distributions......Page 262
12.3 Continuous Time Random Walks (CTRW)......Page 268
12.4 The Standard Long-Tail CTRW (SLT-CTRW)......Page 275
12.5 SLT-CTRW: The Density Profile......Page 277
12.6 SLT-CTRW: The Non-Markovian Diffusion Equation......Page 281
Construction of the SLT-CTRW......Page 287
Rechester - Rosenbluth Theory and SLT-CTRW......Page 289
Markovianization of the SLT-CTRW Diffusion Equation......Page 291
12.8 Hybrid Kinetic Equation and SLT-CTRW Subd-iffusion Equation......Page 295
12.9 Conclusions......Page 305
13.1 Langevin Equations for Relative Motion......Page 307
13.2 The Clump Effectxs......Page 315
13.4 Real Time Dependence......Page 319
13.5 Numerical Simulation......Page 322
14.1 Statement of the problem......Page 324
14.2 Collisionless Drift Motion in Two Dimensions......Page 327
14.3 Collisionless Drift Motion: The Corrsin Approximation......Page 330
14.4 Qualitative Discussion of Trapping......Page 334
14.5 Eulerian Statistics in Subensembles......Page 338
14.6 Lagrangian Statistics in Sub ensembles. The DCT Method......Page 342
14.7 The Decorrelation Trajectory......Page 343
14.8 Hybrid Kinetic Equation for Collisionless 2D Drift Motion......Page 354
14.9 Hybrid Kinetic Equation and Continuous Time Random Walks......Page 357
15.1 Statement of the problem......Page 364
15.2 V-Langevin Equations for Magnetic Fluctuations......Page 365
15.3 Average over Perpendicular Collisions......Page 367
15.4 The DCT Method for Magnetic Diffusion......Page 368
a. The limit of infinite perpendicular correlation length......Page 372
b. The Collisionless limit......Page 375
c. The Kadomtsev-Pogutse limit......Page 376
15.6 The Subdiffusive Regime......Page 380
15.7 The Diffusive Regime......Page 386
16.1 Qualitative Overlook (I)......Page 392
16.2 The Modified Hasegawa-Mima Equation......Page 394
16.3 The Wave Liouville Equation......Page 397
16.4 Zonal Flows and Reynolds Stress......Page 400
16.5 Qualitative Overlook (II)......Page 401
16.6 The Wave Kinetic Equation......Page 403
16.7 The DOT Method for Zonal Flows......Page 408
16.8 The Decorrelation Trajectories......Page 416
16.9 Diffusion Coefficients......Page 420
17.1 Confinement Degradation......Page 426
17.2 Perturbative Transport Experiments......Page 427
17.3 Profile Consistency......Page 429
17.4 Attempts at a Theoretical Explanation......Page 431
17.5 Lévy Flights and Nonlocal Transport......Page 432
17.6 The van Milligen-Sanchez-Carreras Model......Page 439
a. Steady State Density Profile......Page 443
b. Confinement Degradation......Page 445
c. Perturbative Transport......Page 447
d. Profile Consistency......Page 448
Appendix A: The Electrostatic Approximation......Page 453
Appendix B: The Cauchy Integral......Page 455
Appendix C: The Plasma Dispersion Function......Page 459
Appendix D: Lévy Probability Distributions......Page 462
Bibliography......Page 467