Foundations of Plasma Physics for Physicists and Mathematicians

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 𝜇Space 70 4.3 Relationship Between Γ Space and 𝜇Space 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 Alfvén 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, 𝜁e > 1 257 12.7.2.2 Ion Waves 𝜁e < 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 (𝜈∗ ≪1) 354 15.9.1.1 Diffusion in the Banana Regime 355 15.9.1.2 Bootstrap Current (𝜈∗ ≪1) 355 15.9.2 Resistive Plasma Diffusion – Collisional Pfirsch–Schlüter Regime 356 15.9.2.1 Pfirsch–Schlüter Current (𝜈∗ ≫1) 357 15.9.2.2 Diffusion in the Pfirsch–Sclüter 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–Schlüter Regime 368 15.E.1 Diffusion in the Pfirsch–Schlüter 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