Nonlinear Physics of Plasmas (eBook)

Nonlinear Physics of Plasmas 
3 
Preface 7
Contents 11
Part I: 
19 
Chapter 1: 
20 
1.1 Ionization Equilibrium 21
1.2 Plasma Temperature 22
1.3 Collision Frequency 23
1.4 Conditions for High Temperature Plasmas 24
1.5 Debye Screening 26
1.6 Plasma Diamagnetism 27
1.7 Single Particle Motions 28
1.7.1 Guiding Center Drifts 29
1.7.2 Gravitational Drift 31
1.7.3 Magnetic Mirrors 31
1.7.3.1 Magnetic Moment 31
1.7.3.2 Trapping in Magnetic Mirror 32
1.7.3.3 Loss Cone 32
1.7.3.4 Magnetic Pumping 33
1.7.3.5 Adiabatic Invariant J 33
1.8 Collective and Individual Motions 33
1.9 Wave-Particle Resonant Interaction 38
1.9.1 Landau Damping 38
1.9.2 Kinetic Instability 39
1.10 Dispersion and Wave Spreading 39
1.11 Nonlinearity and Wave Steepening 40
1.12 Complexity in Plasmas 41
Chapter 2: 
42 
2.1 Klimontovich Equation and Vlasov Equation 43
2.2 Linear Electrostatic Wave Without External Magnetic Field 46
2.3 Landau Damping 48
2.4 Linear Electrostatic Wave with External Magnetic Field 50
2.5 Plasma Wave Echo 52
2.6 van Kampen Mode 53
2.7 Kinetic Instability and Quasi-Linear Theory 58
2.8 Drift Kinetic Theory 61
2.9 Gyrokinetic Theory 65
Chapter 3: 
73 
3.1 The Fluid Equations for Plasmas 73
3.2 Collisional Transport in Plasmas 77
3.2.1 Collision Integral 77
3.2.2 The Langevin Equation 78
3.2.3 The Fluctuation–Dissipation Theorem 79
3.2.4 Diffusion and Mobility 80
3.2.5 The Einstein Relation 82
3.2.6 Ambipolar Diffusion 84
3.3 Fluid Drifts 84
3.3.1 Cross Magnetic Field Drifts 84
3.3.2 Collisional Cross Magnetic Field Drifts 85
3.4 Magnetohydrodynamics 87
3.5 Frozen-in and Diffusion of Magnetic Field Line 88
3.5.1 Frozen-in Field Line 88
3.5.2 Diffusion of Magnetic Field Lines 89
3.6 MHD Equilibrium 90
3.6.1 Equilibrium for Plasma Column 90
3.6.1.1 Pinch 90
3.6.1.2 z 
91 
3.6.1.3 Sausage Instability 91
3.6.1.4 Kink Instability 92
3.6.1.5 Force Free Configuration 93
3.6.2 Simple Torus 94
3.6.3 Magnetic Surface 95
3.6.4 Grad–Shafranov Equation 95
3.6.4.1 Axisymmetric Plasma 95
3.6.4.2 Helically Symmetric Plasma 97
3.7 Reduced MHD Equations 98
3.7.1 Toroidal Plasma 99
3.7.2 Flute Instability 100
3.7.3 Tearing Mode Instability 100
3.7.4 Ballooning Mode Instability 102
Chapter 4: 
104 
4.1 Electrostatic Waves 104
4.1.1 Waves Without an External Magnetic Field 104
4.1.2 Waves With an External Magnetic Field B 
106 
4.2 Electromagnetic Waves 109
4.2.1 Waves Without an External Magnetic Field 109
4.2.2 Waves With an External Magnetic Field B 
110 
4.2.2.1 Ordinary Wave (E || 
110 
4.2.2.2 Extraordinary Wave (E 
110 
4.2.2.3 L-wave and R-wave 110
4.3 MHD Wave 111
4.3.1 Alfven Wave 111
4.3.2 Magnetosonic Wave 112
4.4 Wave Energy 112
4.5 Negative Energy Wave 114
4.6 Instabilities in Plasmas 116
4.6.1 Streaming Instability 117
4.6.2 Gradient Instability 120
4.6.3 Gravitational Instability 123
Part II: 
126 
Chapter 5: 
127 
5.1 Nonlinear Wave Equation 128
5.2 Resonant Three Wave Interaction 130
5.2.1 Parametric Instability 134
5.2.2 Resonant Decay Interaction 135
5.2.3 Resonant Explosive Interaction 137
5.2.4 Chaos in Resonant Three-Wave Interaction with Dissipation and Frequency Mismatch 138
5.2.5 Coupled Solitons in Resonant Three-Wave Interaction 141
5.2.6 Spatio-Temporal Evolution in Three Wave Resonant Interaction 143
5.3 Self-Interaction and Modulation Instability 144
5.4 Nonlinear Wave-Particle Interaction 147
5.4.1 Nonlinear Landau Damping 147
5.4.2 Ponderomotive Potential Force and Magnetization 149
5.4.2.1 Ponderomotive Potential Force 149
5.4.2.2 Ponderomotive Magnetization 151
5.5 Weak Turbulence Theory 154
5.6 Kinetic Theory for Waves as Quasi-Particles 159
5.7 Modulation Instability of Plasmon Gas 160
Chapter 6: 
164 
6.1 Ion Acoustic Waves and K-dV Equation 165
6.1.1 Reductive Perturbation Theory and K-dV Equation 166
6.1.2 Kinetic Theory Derivation of K-dV Equation 167
6.1.3 Stationary Solutions of K-dV Equation 169
6.1.4 Sagdeev Potential 169
6.1.5 Fermi–Pasta–Ulam Problem 171
6.1.6 K-dV Equation in the Continuum Limit of FPU Problem and Solitons 172
6.2 Langmuir Waves and Envelope Solitons 174
6.2.1 Langmuir Waves and Nonlinear Schrödinger Equation 174
6.2.2 Modulation Instability of Finite Amplitude Langmuir Wave 175
6.3 Solitons and Inverse Scattering Method 176
6.3.1 K-dV Equation 176
6.3.1.1 One-Soliton Solution 179
6.3.1.2 Two-Soliton Solution 179
6.3.2 Nonlinear Shrödinger Equation 180
6.4 Solitons and Bilinear Transformation 184
6.4.1 K-dV Equation 184
6.4.2 Nonlinear Schrödinger Equation 185
6.5 Soliton-Like Excitations in Plasmas With Multiple Modes 186
6.5.1 Basic Equations for Nonlinear Wave Propagation in An Ion Beam-Plasma System 187
6.5.2 Characteristic Times for Onset of The Explosive Instability and Soliton Formation 190
6.5.3 Numerical Solutions 192
6.5.4 K-dV Approximation for SmallAmplitude Nonlinear Modes 194
6.5.5 Nonlinear Explosion Modes 195
6.5.6 Beam Reflection and Soliton Emission 197
6.5.6.1 Linearly Stable Cases 198
6.5.6.2 Linearly Unstable Cases 201
6.6 Soliton-Like Excitations in Linearly Unstable Plasmas 203
6.6.1 Basic Equations for Long Wave Buneman Instability 204
6.6.2 Pulsating Solitons 206
6.6.3 Ordinary Solitons 207
6.6.4 Temporally Localized and Spatially Periodic Solitons 208
Chapter 7: 
209 
7.1 Two Dimensional Vortices 211
7.2 Point Vortex 214
7.3 Vortical Motions in Plasmas 217
7.3.1 Drifts for Driving Vortical Motions 217
7.3.2 Vortices in Electron Plasmas 219
7.3.3 Convective Cells 221
7.3.4 Drift Wave Vortices 222
7.3.4.1 Dipole Vortex Solutions 224
7.3.5 Particle Transport Due to Drift Wave Vortices 226
7.3.6 Ion Heat Transport Due to Drift Wave Vortices 227
7.3.7 Zonal Flow 228
7.3.8 Self-Organization of Monopole Vorticesin Temperature Inhomogeneity 230
7.4 Collisional Drift Wave Instability and Formationof Dipole Vortices 233
7.4.1 Point Vortex Description for Drift Wave Vortices 241
7.4.1.1 Dipole Vortex Solutions 244
7.4.1.2 Collision Processes of Dipole Vortices 247
7.4.2 Kinetic Theory of Vortex Diffusion 248
7.5 Vortex Collapse Revisited 252
7.5.1 Vortex Collapse 252
7.5.2 Boomerang Interaction of Three Vortices 255
7.6 Spiral Structures in Magnetized Rotating Plasmas 258
Chapter 8: 
265 
8.1 Chaos in Conservative Systems 265
8.1.1 Pendulum 265
8.1.2 Resonance 268
8.1.3 Resonance in Multiple Periodic Systems 270
8.2 Poincarè Mapping 272
8.2.1 Integrable System 272
8.2.2 Non Integrable System 272
8.2.2.1 Perturbed Twist Map 272
8.2.2.2 Radial Twist Map 273
8.3 Standard Map 274
8.3.1 Chaos in Standard Map 275
8.3.1.1 Primary Resonance Overlap 277
8.3.1.2 Secondary Resonance Overlap 278
8.3.2 Global Chaos: Greene's Method 279
8.4 Chaos in Dissipative Systems 279
8.4.1 Attractors and Strange Attractors 279
8.4.2 Bifurcation Theory 280
8.4.2.1 Tangent Bifurcation 280
8.4.2.2 Exchange of Stability 281
8.4.2.3 Pitchfork Bifurcation 282
8.4.2.4 Reversed Pitchfork Bifurcation 282
8.4.2.5 Hopf Bifurcation 283
8.4.3 Period Doubling Route to Chaos: Logistic Map 284
8.4.3.1 Fixed Points of 
284 
8.4.3.2 Fixed Points of f2 285
8.4.3.3 Accumulation of Period Doubling and Renormalization 286
8.5 Fractal Structure 288
8.6 Lyapunov Exponents 289
8.7 Dimension of Attractor 290
8.8 Correlation Dimension 291
8.9 Construction of Attractor with Observed Signal 291
8.10 Intermittent Chaos 293
8.10.1 Type I Intermittency 294
8.10.2 Type II Intermittency 294
8.10.3 Type III Intermittency 295
8.11 Chaos in Plasmas 296
8.11.1 Stochastic Web 296
8.11.2 Chaos of Particle Motion in a Magnetic Mirror Field 299
8.11.3 Chaos in a Current-Carrying Ion Sheath 304
8.11.4 Chaos of Magnetic Field Lines 308
8.11.5 Anomalous Transport in Tokamak and Tokamap 310
8.11.5.1 Toroidal Coupling and Transport Barrier 310
8.11.5.2 Tokamap 310
8.11.6 Ponderomotive Force at the Onset of Chaos 311
Chapter 9: 
317 
9.1 Hamiltonian Formulation of Ponderomotive Interactions in a Vlasov Plasma 317
9.2 The Hydrodynamics of Ponderomotive Interactions in a Collisionless Plasma 323
9.3 Spontaneous Generation of Magnetostatic Fields 328
9.3.1 Coupled Mode Equations 329
9.3.2 Parametric Instability Analysis 329
Part III: 
332 
Chapter 10: 
333 
10.1 Introduction 333
10.2 Derivation of the Generalized Zakharov Equations 334
10.3 Adiabatic Scaling and Spherical Collapse 341
10.4 Qualitative Discussion of the Collapse 344
Chapter 11: 
346 
11.1 Langmuir Soliton Stability and Collapse 346
11.1.1 Introduction 346
11.1.2 Basic Equations 348
11.1.3 Variational Treatment of Soliton Stability 350
11.1.4 Numerical Treatment 352
11.1.5 Nonlinear Stage of Soliton Instability 356
11.1.6 Self-Similarity and Collapse Regimes 358
11.2 Virial Theory of Wave Collapse 363
11.3 Hierarchy of Collapse Regimes in a Magnetized Plasma 365
11.3.1 Introduction 366
11.3.2 Model Equation 366
11.3.3 Nonexistence of Three-Dimensional Solitons 368
11.3.4 Necessary Condition for Wave Collapse 370
11.3.5 Classification of Wave Collapse Regimes 372
11.4 Weak and Strong Langmuir Collapse 374
11.4.1 Preliminaries 375
11.4.2 Nonlinear Model Equations 375
11.4.3 Simulation Results and Discussions 377
Chapter 12: 
380 
12.1 Spatiotemporal Effects in Three-Wave Interaction 380
12.1.1 Convective and Absolute Instability 381
12.1.2 Space-Only Problem in Three Wave Interaction 383
12.1.3 Spatiotemporal Evolution in Three-Wave Interaction 385
12.2 Complexity in Laser Plasma Instabilities 388
12.2.1 Introduction to Stimulated Raman Scattering 388
12.2.2 Nonlinear Saturation of SRS 390
12.2.3 Break-up of Manley–Rowe Invariantsand Nonstationary SRS 393
12.2.4 Bifurcations and Route to Low-Dimensional Chaos 393
12.2.5 Complexity of Spatiotemporal Wave Patterns 397
12.2.6 Quantitive Signatures of Spatiotemporal Regimes 400
12.2.7 Transition from Spatiotemporal Intermittency to Spatiotemporal Chaos 403
12.2.8 Summary 406
12.3 Self-Organization in a Dissipative 3WI-Saturated SRS Paradigm 407
12.3.1 Introduction 407
12.3.2 Preliminaries on Nonlinear Kinetic SRS 408
12.3.3 Dissipative SRS Saturation Model 409
12.3.4 Kinetic-Hybrid Scheme 410
12.3.5 Open Boundary Model 413
12.3.6 Self-Organization at Micro- and Macro-Scales 414
12.3.7 Dissipative Structures and Entropy Rate 419
12.3.8 Summary 422
Chapter 13: 
424 
13.1 Electronic Parametric Instabilities 425
13.1.1 Stimulated Raman Scattering 426
13.1.2 Relativistic Dispersion Relation for Cold Plasma 426
13.1.2.1 Solutions of Relativistic Dispersion Relation 431
13.1.3 Summary 434
13.2 Computer Simulations of Relativistic Plasmas 435
13.2.1 Particle-In-Cell Simulations 436
13.3 Relativistic Electromagnetic Solitons 437
13.3.1 Dynamical Equations 438
13.3.2 Relativistic Soliton Stability 441
13.3.3 Soliton Dynamics 443
13.3.4 Strongly-Relativistic Solitons 443
13.4 Stimulated Raman Cascade into Photon Condensation 447
13.4.1 Introduction 447
13.4.2 Relativistic Fluid-Maxwell Simulation 450
13.4.3 Particle Simulations 454
13.4.3.1 Stimulated Raman Cascade into Photon Condensation 455
13.4.3.2 Effect of Laser Intensity on SRS Cascadeinto Photon Condensation 459
13.5 Relativistic EM Solitons in a Low Density Plasma 463
13.5.1 Introduction 464
13.5.2 Relativistic EM Solitons 464
13.5.2.1 Standing EM Solitons 466
13.5.2.2 Backward- and Forward Accelerated EM solitons 467
13.5.2.3 Merging of Two Relativistic EM Solitons 470
13.6 Stimulated Electron Acoustic Scattering 470
13.6.1 The Electron Acoustic Waves 471
13.6.2 Stimulated Raman and Acoustic Wave Scattering 471
13.6.3 SEAS Model 473
13.6.4 Simulations of SEAS 475
13.7 Trapped SEAS, EM Soliton and Ion-Vortices in Subcritical Plasmas 477
13.7.1 Introduction 478
13.7.2 Stimulated Trapped Electron Acoustic Wave Scattering 479
13.7.3 Electromagnetic Soliton and Ion-Vortices 482
Part IV: 
487 
Chapter 14: 
488 
14.1 Introduction 488
14.2 Edge Turbulence Datasets 489
14.3 Quantification of Long-Range Dependence 489
14.3.1 Wavelet Transform of Scaling Processes 492
14.3.2 Log-Scale Diagrams of Turbulent Datasets 493
14.3.3 Testing Time Constancy of Scaling Exponents 495
14.3.4 Randomization Method for Long Range Correlations 497
14.4 Multifractal Properties of Datasets 500
14.4.1 Basic Properties of Multifractal Processes 500
14.4.2 Multiscale Diagrams and Wavelet Coefficients 502
14.4.3 Multiscale Diagrams and MultifractalSpectra for L and H Modes 503
14.5 Coupled Effects of Long-Range Dependenceand Intermittency 510
14.6 Conclusion 513
Chapter 15: 
515 
15.1 Basics on Multi-Scale Modelling 515
15.2 Multi-Scale Plasma Models 516
15.3 Equation-Free Macro-Projective Integration Method 518
15.4 The Nonlinear Ion-Sound Wave 520
15.5 Electrostatic Particle-In-Cell Code 520
15.6 Primal Macro-Projective Simulation Method 521
References 525
Index 537