Preface xvii
1 Fundamental Plasma Parameters – Collective Behaviour 1
1.1 Introduction 1
1.2 Cold Plasma Waves 2
1.2.1 Wave Breaking 3
1.3 Debye Shielding 4
1.3.1 Weakly and Strongly Coupled Plasmas 6
1.3.2 The Plasma Parameter 7
1.4 Diffusion and Mobility 8
1.4.1 Einstein–Smoluchowski Relation 8
1.4.2 Ambipolar Diffusion 9
1.5 Wall Sheath 9
1.5.1 Positively Biased Wall 10
1.5.2 Free Fall Sheath 10
1.5.2.1 Pre-sheath 11
1.5.3 Mobility Limited Sheath 11
2 Fundamental Plasma Parameters – Collisional Behaviour 13
2.1 Electron Scattering by Ions 13
2.1.1 Binary Collisions – Rutherford Cross Section 13
2.1.2 Momentum Transfer Cross Section 15
2.1.2.1 Dynamical Friction and Diffusion 16
2.1.3 Many Body Collisions – Impulse Approximation 16
2.1.4 Relaxation Times 20
2.2 Collisional Transport Effects 21
2.2.1 Random Walk Model for Transport Effects 22
2.2.2 Maxwell’s Mean Free Path Model of Transport Phenomena 23
2.2.2.1 Flux Limitation 25
2.2.3 Drude Model of Electrical Conductivity 26
2.2.3.1 Alternating Electric Field, No Magnetic Field 27
2.2.3.2 Steady Electric Field, Finite Magnetic Field 27
2.2.3.3 Oscillatory Electric Field, Finite Magnetic Field 28
2.2.4 Diffusivity and Mobility in a Uniform Magnetic Field 29
2.3 Plasma Permittivity 30
2.3.1 Poynting’s Theorem – Energy Balance in an Electro-magnetic Field 31
2.4 Plasma as a Fluid – Two Fluid Model 32
2.4.1 Waves in Plasma 33
2.4.2 Beam Instabilities 36
2.4.2.1 Plasma Bunching 36
2.4.2.2 Two Stream Instability 36
2.4.3 Kinematics of Growing Waves 37
Appendix 2.A Momentum Transfer Collision Rate 39
Appendix 2.B The Central Limit Theorem 41
3 Single Particle Motion – Guiding Centre Model 43
3.1 Introduction 43
3.2 Motion in Stationary and Uniform Fields 44
3.2.1 Static Uniform Magnetic Field – Cyclotron Motion 44
3.2.2 Uniform Static Electric and Magnetic Fields 45
3.3 The Guiding Centre Approximation 45
3.3.1 The Method of Averaging 46
3.3.2 The Guiding Centre Model for Charged Particles 48
3.4 Particle Kinetic Energy 51
3.5 Motion in a Static Inhomogeneous Magnetic Field 52
3.5.1 Field Gradient Drift 53
3.5.2 Curvature Drift 53
3.5.3 Divergent Field Lines 55
3.5.4 Twisted Field Lines 55
3.6 Motion in a Time Varying Magnetic Field 56
3.7 Motion in a Time Varying Electric Field 56
3.8 Collisional Drift 58
3.9 Plasma Diamagnetism 58
3.10 Particle Trapping and Magnetic Mirrors 59
3.10.1 Fermi Acceleration 61
3.11 Adiabatic Invariance 61
3.12 Adiabatic Invariants of Charged Particle Motions 63
Appendix 3.A Northrop’s Expansion Procedure 64
3.A.1 Drift Velocity and Longitudinal Motion along the Field Lines 65
4 Kinetic Theory of Gases 67
4.1 Introduction 67
4.2 Phase Space 68
4.2.1 Γ Phase Space 68
4.2.1.1 Liouville’s Equation 69
4.2.2 ?Space 70
4.3 Relationship Between Γ Space and ?Space 71
4.3.1 Integrals of the Liouville Equation 72
4.4 The BBGKY (Bogoliubov–Born–Green–Kirkwood–Yvon) Hierarchy 73
4.5 Bogoliubov’s Hypothesis for Dilute Gases 74
4.6 Derivation of the Boltzmann Collision Integral from the BBGKY Hierarchy 76
4.7 Boltzmann Collision Operator 78
4.7.1 Summation Invariants 79
4.8 Boltzmann’s H Theorem 79
4.9 The Equilibrium Maxwell–Boltzmann Distribution 80
4.9.1 Entropy and the H function 81
4.10 Hydrodynamic Limit – Method of Moments 81
4.10.1 Conservation of Mass 83
4.10.2 Conservation of Momentum 83
4.10.3 Conservation of Energy 84
4.11 The Departure from Steady Homogeneous Flow: The Chapman–Enskog Approximation 84
5 Wave Propagation in Inhomogeneous, Dispersive Media 89
5.1 Introduction 89
5.2 Basic Concepts of Wave Propagation – The Geometrical Optics Approximation 90
5.3 The WKB Approximation 92
5.3.1 Oblique Incidence 93
5.4 Singularities in Waves 93
5.4.1 Cut-off or Turning Point 94
5.4.2 Resonance Point 96
5.4.3 Resonance Layer and Collisional Damping 99
5.5 The Propagation of Energy 100
5.5.1 Group Velocity of Waves in Dispersive Media 100
5.5.2 Waves in Dispersive Isotropic Media 101
5.6 Group Velocity of Waves in Anisotropic Dispersive Media 102
5.6.1 Equivalence of Energy Transport Velocity and Group Velocity 106
Appendix 5.A Waves in Anisotropic Inhomogeneous Media 107
6 Kinetic Theory of Plasmas – Collisionless Models 111
6.1 Introduction 111
6.2 Vlasov Equation 111
6.3 Particle Trapping by a Potential Well 114
7 Kinetic Theory of Plasmas 121
7.1 Introduction 121
7.2 The Fokker–Planck Equation – The Stochastic Approach 122
7.2.1 The Scattering Integral for Coulomb Collisions 124
7.3 The Fokker–Planck Equation – The Landau Equation 128
7.3.1 Application to Collisions between Charged Particles 130
7.4 The Fokker–Planck Equation – The Cluster Expansion 131
7.4.1 The Balescu–Lenard Equation 132
7.5 Relaxation of a Distribution to the Equilibrium Form 135
7.5.1 Isotropic Distribution 135
7.5.2 Anisotropic Distribution 137
7.6 Ion–Electron Thermal Equilibration by Coulomb Collisions 139
7.7 Dynamical Friction 140
Appendix 7.A Reduction of the Boltzmann Equation to Fokker–Planck Form in the Weak Collision Limit 142
Appendix 7.B Finite Difference Algorithm for Integrating the Isotropic Fokker–Planck Equation 144
Appendix 7.C Monte Carlo Algorithm for Integrating the Fokker–Planck Equation 145
Appendix 7.D Landau’s Calculation of the Electron–Ion Equilibration Rate 147
8 The Hydrodynamic Limit for Plasma 149
8.1 Introduction – Individual Particle Fluid Equations 149
8.2 The Departure from Steady, Homogeneous Flow: The Transport Coefficients 150
8.3 Magneto-hydrodynamic Equations 151
8.3.1 Equation of Mass Conservation 151
8.3.2 Equation of Momentum Conservation 152
8.3.3 Virial Theorem 154
8.3.4 Equation of Current Flow 154
8.3.5 Equation of Energy Conservation 155
8.4 Transport Equations 156
8.4.1 Collision Times 157
8.4.2 Symmetry of the Transport Equations 158
8.5 Two Fluid MHD Equations – Braginskii Equations 161
8.5.1 Magnetic Field Equations 162
8.5.1.1 Energy Balance 164
8.6 Transport Coefficients 165
8.6.1 Collisional Dominated Plasma 165
8.6.1.1 Force Terms F 165
8.6.1.2 Energy Flux Terms 165
8.6.1.3 Viscosity 166
8.6.2 Field-Dominated Plasma 166
8.6.2.1 Force Terms F 166
8.6.2.2 Energy Flux Terms 167
8.6.2.3 Viscosity 168
8.7 Calculation of the Transport Coefficients 168
8.8 Lorentz Approximation 170
8.8.1 Electron–Electron Collisions 173
8.8.2 Electron Runaway 174
8.9 Deficiencies in the Spitzer/Braginskii Model of Transport Coefficients 177
Appendix 8.A BGK Model for the Calculation of Transport Coefficients 178
8.A.1 BGK Conductivity Model 178
8.A.2 BGK Viscosity Model 180
Appendix 8.B The Relationship Between the Flux Equations Given By Shkarofsky and Braginskii 181
Appendix 8.C Electrical Conductivity in a Weakly Ionised Gas and the Druyvesteyn Distribution 182
9 Ideal Magnetohydrodynamics 187
9.1 Infinite Conductivity MHD Flow 188
9.1.1 Frozen Field Condition 189
9.1.2 Adiabatic Equation of State 190
9.1.3 Pressure Balance 191
9.1.3.1 Virial Theorem 191
9.2 Incompressible Approximation 192
9.2.1 Bernoulli’s Equation – Steady Flow 192
9.2.2 Kelvin’s Theorem – Circulation 193
9.2.3 Alfven Waves 193
10 Waves in MHD Fluids 197
10.1 Introduction 197
10.2 Magneto-sonic Waves 198
10.3 Discontinuities in Fluid Mechanics 203
10.3.1 Classical Fluids 203
10.3.2 Discontinuities in Magneto-hydrodynamic Fluids 204
10.4 The Rankine–Hugoniot Relations for MHD Flows 205
10.5 Discontinuities in MHD Flows 206
10.6 MHD Shock Waves 207
10.6.1 Simplifying Frame Transformations 207
10.7 Properties of MHD Shocks 208
10.7.1 Shock Hugoniot 208
10.7.2 Shock Adiabat – General Solution for a Polytropic Gas 209
10.8 Evolutionary Shocks 212
10.8.1 Evolutionary MHD Shock Waves 213
10.8.2 Parallel Shock – Magnetic Field Normal to the Shock Plane 214
10.9 Switch-on and Switch-off Shocks 216
10.10 Perpendicular Shock – Magnetic Field Lying in the Shock Plane 217
10.11 Shock Structure and Stability 218
Appendix 10.A Group Velocity of Magneto-sonic Waves 218
Appendix 10.B Solution in de Hoffman–Teller Frame 220
10.B.1 Parallel Shocks 222
11 Waves in Cold Magnetised Plasma 223
11.1 Introduction 223
11.2 Waves in Cold Plasma 223
11.2.1 Cut-off and Resonance 226
11.2.2 Polarisation 227
11.3 Cold Plasma Waves 227
11.3.1 Zero Applied Magnetic Field 227
11.3.2 Low Frequency Velocity Waves 228
11.3.3 Propagation of Waves Parallel to the Magnetic Field 229
11.3.4 Propagation of Waves Perpendicular to the Magnetic Field 232
11.3.5 Resonance in Plasma Waves 234
12 Waves in Magnetised Warm Plasma 237
12.1 The Dielectric Properties of Unmagnetised Warm Dilute Plasma 237
12.1.1 Plasma Dispersion Relation 238
12.1.1.1 Dispersion Relation for Transverse Waves 239
12.1.1.2 Dispersion Relation for Longitudinal Waves 239
12.1.2 Dielectric Constant of a Plasma 239
12.1.2.1 The Landau Contour Integration Around the Singularity 241
12.2 Transverse Waves 243
12.3 Longitudinal Waves 244
12.4 Linear Landau Damping 245
12.4.1 Resonant Energy Absorption 245
12.5 Non-linear Landau Damping 248
12.5.1 Particle Trapping 248
12.5.2 Plasma Wave Breaking 250
12.6 The Plasma Dispersion Function 252
12.7 Positive Ion Waves 256
12.7.1 Transverse Waves 256
12.7.2 Longitudinal Waves 256
12.7.2.1 Plasma Waves, ?e > 1 257
12.7.2.2 Ion Waves ?e < 1 257
12.8 Microscopic Plasma Instability 258
12.8.1 Nyquist Plot 259
12.8.1.1 Penrose’s Criterion 260
12.9 The Dielectric Properties of Warm Dilute Plasma in a Magnetic Field 262
12.9.1 Propagation Parallel to the Magnetic Field 269
12.9.2 Propagation Perpendicular to the Magnetic Field 270
Appendix 12.A Landau’s Solution of the Vlasov Equation 274
Appendix 12.B Electrostatic Waves 276
13 Properties of Electro-magnetic Waves in Plasma 281
13.1 Plasma Permittivity and the Dielectric Constant 281
13.1.1 The Properties of the Permittivity Matrix 284
13.2 Plane Waves in Homogeneous Plasma 286
13.2.1 Waves in Collisional Cold Plasma 287
13.2.1.1 Isotropic Unmagnetised Plasma 287
13.2.1.2 Anisotropic Magnetised Plasma 289
13.3 Plane Waves Incident Obliquely on a Refractive Index Gradient 290
13.3.1 Oblique Incidence at a Cut-off Point – Resonance Absorption 293
13.3.1.1 s Polarisation 293
13.3.1.2 p Polarisation 293
13.4 Single Particle Model of Electrons in an Electro-magnetic Field 295
13.4.1 Quiver Motion 295
13.4.2 Ponderomotive Force 297
13.4.3 The Impact Model for Collisional Absorption 298
13.4.3.1 Electron–Electron Collisions 301
13.4.4 Distribution Function of Electrons Subject to Inverse Bremsstrahlung Heating 301
13.5 Parametric Instabilities 305
13.5.1 Coupled Wave Interactions 305
13.5.1.1 Manley–Rowe Relations 306
13.5.1.2 Parametric Instability 307
13.5.2 Non-linear Laser-Plasma Absorption 308
13.5.2.1 Absorption Instabilities 309
13.5.2.2 Reflection Instabilities 310
Appendix 13.A Ponderomotive Force 310
14 Laser–Plasma Interaction 313
14.1 Introduction 313
14.2 The Classical Hydrodynamic Model of Laser-Solid Breakdown 314
14.2.1 Basic Parameters of Laser Breakdown 315
14.2.2 The General Theory of the Interaction of Lasers with Solid Targets 316
14.2.3 Distributed Heating – Low Intensity, Self-regulating Flow 318
14.2.3.1 Early Time Self-similar Solution 319
14.2.3.2 Late Time Steady-State Solution 319
14.2.4 Local Heating – High Intensity, Deflagration Flow 321
14.2.4.1 Early Time Thermal Front 321
14.2.4.2 Late Time Steady-State Flow 323
14.2.5 Additional Simple Analytic Models 324
14.2.5.1 Short Pulse Heating 324
14.2.5.2 Heating of Small Pellets – Homogeneous Self-similar Model 325
14.3 Simulation of Laser-Solid Target Interaction 325
Appendix 14.A Non-linear Diffusion 327
Appendix 14.B Self-similar Flows with Uniform Velocity Gradient 329
15 Magnetically Confined Plasma 337
15.1 Introduction 337
15.2 Equilibrium Plasma Configurations 337
15.3 Linear Devices 338
15.4 Toroidal Devices 340
15.4.1 Pressure Balance 341
15.4.1.1 Pressure Imbalance Mitigation 342
15.4.2 Guiding Centre Drift 343
15.5 The General Problem: The Grad–Shafranov Equation 344
15.6 Boundary Conditions 345
15.7 Equilibrium Plasma Configurations 347
15.7.1 Perturbation Methods 348
15.7.2 Analytical Solutions of the Grad–Shafranov Equation 349
15.7.3 Numerical Solutions of the Grad–Shafranov Equation 350
15.8 Classical Magnetic Cross Field Diffusion 351
15.9 Trapped Particles and Banana Orbits 352
15.9.1 Collisionless Banana Regime (?∗ ?1) 354
15.9.1.1 Diffusion in the Banana Regime 355
15.9.1.2 Bootstrap Current (?∗ ?1) 355
15.9.2 Resistive Plasma Diffusion – Collisional Pfirsch–Schluter Regime 356
15.9.2.1 Pfirsch–Schluter Current (?∗ ?1) 357
15.9.2.2 Diffusion in the Pfirsch–Scluter Regime 357
15.9.3 Plateau Regime 357
15.9.4 Diffusion in Tokamak Plasmas 358
Appendix 15.A Equilibrium Maintaining ‘Vertical’ Field 359
Appendix 15.B Perturbation Solution of the Grad–Shafranov Equation 360
Appendix 15.C Analytic Solutions of the Homogeneous Grad–Shafranov Equation 363
Appendix 15.D Guiding Centre Motion in a Twisted Circular Toroidal Plasma 364
Appendix 15.E The Pfirsch–Schluter Regime 368
15.E.1 Diffusion in the Pfirsch–Schluter Regime 369
16 Instability of an Equilibrium Confined Plasma 371
16.1 Introduction 371
16.2 Ideal MHD Instability 371
16.2.1 Linearised Stability Equations 371
16.2.2 Normal Mode Analysis – The Stability of a Cylindrical Plasma Column 375
16.2.3 m = 0 Sausage Instability 379
16.2.4 m = 1 Kink Instability 380
16.3 Potential Energy 381
16.4 Interchange Instabilities 382
Supplementary Material 387
M.1 Breakdown and Discharges in d.c. Electric Fields 387
M.1.1 Gas Breakdown and Paschen’s Law 387
M.1.2 Similarity and Proper Variables 388
M.1.3 Townsend’s First Coefficient 388
M.1.4 Townsend’s Breakdown Criterion 389
M.1.5 Paschen Curve and Paschen Minimum 389
M.1.6 Radial Profile of Glow Discharge 390
M.1.7 Collisional Ionisation Rate for Low Temperature Electrons 391
M.1.8 Radio Frequency and Microwave Discharges 392
M.2 Key Facts Governing Nuclear Fusion 393
M.2.1 Fusion Rate 393
M.2.2 Lawson’s Criterion 396
M.2.3 Triple Product 398
M.3 A Short Introduction to Functions of a Complex Variable 400
M.3.1 Cauchy–Riemann Relations 401
M.3.2 Harmonic Functions 402
M.3.3 Area Rule 402
M.3.4 Cauchy Integral Theorem 402
M.3.5 Morera’s Theorem 403
M.3.6 Analytic Continuation 403
M.3.7 Extension or Contraction of a Contour 404
M.3.8 Inclusion of Isolated Singularities 404
M.3.9 Cauchy Formula 404
M.3.9.1 Interior Domain 404
M.3.9.2 Exterior Domain 405
M.3.10 Treatment of Improper Integrals 405
M.3.11 Sokhotski–Plemelj Theorem 406
M.3.12 Improper Integral Along a Real Line 407
M.3.13 Taylor and Laurent Series 407
M.3.14 The Argument Principle 408
M.3.15 Estimation Lemma 408
M.3.16 Jordan’s Lemma 409
M.3.17 Conformal Mapping 409
M.4 Laplace Transform 410
M.4.1 Bromwich Contour 410
Problems 413
Bibliography 427
Index 437
