+353-1-416-8900REST OF WORLD
+44-20-3973-8888REST OF WORLD
1-917-300-0470EAST COAST U.S
1-800-526-8630U.S. (TOLL FREE)

Foundations of Plasma Physics for Physicists and Mathematicians. Edition No. 1

  • Book

  • 464 Pages
  • April 2021
  • John Wiley and Sons Ltd
  • ID: 5837797

A comprehensive textbook on the foundational principles of plasmas, including material on advanced topics and related disciplines such as optics, fluid dynamics, and astrophysics

Foundations of Plasma Physics for Physicists and Mathematicians covers the basic physics underlying plasmas and describes the methodology and techniques used in both plasma research and other disciplines such as optics and fluid mechanics. Designed to help readers develop physical understanding and mathematical competence in the subject, this rigorous textbook discusses the underlying theoretical foundations of plasma physics as well as a range of specific problems, focused on those principally associated with fusion.

Reflective of the development of plasma physics, the text first introduces readers to the collective and collisional behaviors of plasma, the single particle model, wave propagation, the kinetic effects of gases and plasma, and other foundational concepts and principles. Subsequent chapters cover topics including the hydrodynamic limit of plasma, ideal magneto-hydrodynamics, waves in MHD plasmas, magnetically confined plasma, and waves in magnetized hot and cold plasma. Written by an acknowledged expert with more than five decades’ active research experience in the field, this authoritative text:

  • Identifies and emphasizes the similarities and differences between plasmas and fluids
  • Describes the different types of interparticle forces that influence the collective behavior of plasma
  • Demonstrates and stresses the importance of coherent and collective effects in plasma
  • Contains an introduction to interactions between laser beams and plasma
  • Includes supplementary sections on the basic models of low temperature plasma and the theory of complex variables and Laplace transforms

Foundations of Plasma Physics for Physicists and Mathematicians is the ideal textbook for advanced undergraduate and graduate students in plasma physics, and a valuable compendium for physicists working in plasma physics and fluid mechanics.

Table of Contents

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 (𝜈1) 354

15.9.1.1 Diffusion in the Banana Regime 355

15.9.1.2 Bootstrap Current (𝜈1) 355

15.9.2 Resistive Plasma Diffusion - Collisional Pfirsch-Schlüter Regime 356

15.9.2.1 Pfirsch-Schlüter Current (𝜈1) 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

Authors

Geoffrey J. Pert University of York, UK.