Foundations of Plasma Physics for Physicists and Mathematicians

Cover
Title Page
Copyright
Contents
Preface
Chapter 1 Fundamental Plasma Parameters – Collective Behaviour
	1.1 Introduction
	1.2 Cold Plasma Waves
		1.2.1 Wave Breaking
	1.3 Debye Shielding
		1.3.1 Weakly and Strongly Coupled Plasmas
		1.3.2 The Plasma Parameter
	1.4 Diffusion and Mobility
		1.4.1 Einstein–Smoluchowski Relation
		1.4.2 Ambipolar Diffusion
	1.5 Wall Sheath
		1.5.1 Positively Biased Wall
		1.5.2 Free Fall Sheath
			1.5.2.1 Pre‐sheath
		1.5.3 Mobility Limited Sheath
Chapter 2 Fundamental Plasma Parameters – Collisional Behaviour
	2.1 Electron Scattering by Ions
		2.1.1 Binary Collisions – Rutherford Cross Section
		2.1.2 Momentum Transfer Cross Section
			2.1.2.1 Dynamical Friction and Diffusion
		2.1.3 Many Body Collisions – Impulse Approximation
		2.1.4 Relaxation Times
	2.2 Collisional Transport Effects
		2.2.1 Random Walk Model for Transport Effects
		2.2.2 Maxwell\'s Mean Free Path Model of Transport Phenomena
			2.2.2.1 Flux Limitation
		2.2.3 Drude Model of Electrical Conductivity
			2.2.3.1 Alternating Electric Field, No Magnetic Field
			2.2.3.2 Steady Electric Field, Finite Magnetic Field
			2.2.3.3 Oscillatory Electric Field, Finite Magnetic Field
		2.2.4 Diffusivity and Mobility in a Uniform Magnetic Field
	2.3 Plasma Permittivity
		2.3.1 Poynting\'s Theorem – Energy Balance in an Electro‐magnetic Field
	2.4 Plasma as a Fluid – Two Fluid Model
		2.4.1 Waves in Plasma
		2.4.2 Beam Instabilities
			2.4.2.1 Plasma Bunching
			2.4.2.2 Two Stream Instability
		2.4.3 Kinematics of Growing Waves
	Appendix 2.A Momentum Transfer Collision Rate
	Appendix 2.B The Central Limit Theorem
Chapter 3 Single Particle Motion – Guiding Centre Model
	3.1 Introduction
	3.2 Motion in Stationary and Uniform Fields
		3.2.1 Static Uniform Magnetic Field – Cyclotron Motion
		3.2.2 Uniform Static Electric and Magnetic Fields
	3.3 The Guiding Centre Approximation
		3.3.1 The Method of Averaging
		3.3.2 The Guiding Centre Model for Charged Particles
	3.4 Particle Kinetic Energy
	3.5 Motion in a Static Inhomogeneous Magnetic Field
		3.5.1 Field Gradient Drift
		3.5.2 Curvature Drift
		3.5.3 Divergent Field Lines
		3.5.4 Twisted Field Lines
	3.6 Motion in a Time Varying Magnetic Field
	3.7 Motion in a Time Varying Electric Field
	3.8 Collisional Drift
	3.9 Plasma Diamagnetism
	3.10 Particle Trapping and Magnetic Mirrors
		3.10.1 Fermi Acceleration
	3.11 Adiabatic Invariance
	3.12 Adiabatic Invariants of Charged Particle Motions
	Appendix 3.A Northrop’s Expansion Procedure
		3.A.1 Drift Velocity and Longitudinal Motion along the Field Lines
Chapter 4 Kinetic Theory of Gases
	4.1 Introduction
	4.2 Phase Space
		4.2.1 Γ Phase Space
			4.2.1.1 Liouville\'s Equation
		4.2.2 μ Space
	4.3 Relationship Between Γ Space and μ Space
		4.3.1 Integrals of the Liouville Equation
	4.4 The BBGKY (Bogoliubov–Born–Green–Kirkwood–Yvon) Hierarchy
	4.5 Bogoliubov\'s Hypothesis for Dilute Gases
	4.6 Derivation of the Boltzmann Collision Integral from the BBGKY Hierarchy
	4.7 Boltzmann Collision Operator
		4.7.1 Summation Invariants
	4.8 Boltzmann\'s H Theorem
	4.9 The Equilibrium Maxwell–Boltzmann Distribution
		4.9.1 Entropy and the H function
	4.10 Hydrodynamic Limit – Method of Moments
		4.10.1 Conservation of Mass
		4.10.2 Conservation of Momentum
		4.10.3 Conservation of Energy
	4.11 The Departure from Steady Homogeneous Flow: The Chapman–Enskog Approximation
Chapter 5 Wave Propagation in Inhomogeneous, Dispersive Media
	5.1 Introduction
	5.2 Basic Concepts of Wave Propagation – The Geometrical Optics Approximation
	5.3 The WKB Approximation
		5.3.1 Oblique Incidence
	5.4 Singularities in Waves
		5.4.1 Cut‐off or Turning Point
		5.4.2 Resonance Point
		5.4.3 Resonance Layer and Collisional Damping
	5.5 The Propagation of Energy
		5.5.1 Group Velocity of Waves in Dispersive Media
		5.5.2 Waves in Dispersive Isotropic Media
	5.6 Group Velocity of Waves in Anisotropic Dispersive Media
		5.6.1 Equivalence of Energy Transport Velocity and Group Velocity
	Appendix 5.A Waves in Anisotropic Inhomogeneous Media
Chapter 6 Kinetic Theory of Plasmas – Collisionless Models
	6.1 Introduction
	6.2 Vlasov Equation
	6.3 Particle Trapping by a Potential Well
Chapter 7 Kinetic Theory of Plasmas
	7.1 Introduction
	7.2 The Fokker–Planck Equation – The Stochastic Approach
		7.2.1 The Scattering Integral for Coulomb Collisions
	7.3 The Fokker–Planck Equation – The Landau Equation
		7.3.1 Application to Collisions between Charged Particles
	7.4 The Fokker–Planck Equation – The Cluster Expansion
		7.4.1 The Balescu–Lenard Equation
	7.5 Relaxation of a Distribution to the Equilibrium Form
		7.5.1 Isotropic Distribution
		7.5.2 Anisotropic Distribution
	7.6 Ion–Electron Thermal Equilibration by Coulomb Collisions
	7.7 Dynamical Friction
	Appendix 7.A Reduction of the Boltzmann Equation to Fokker–Planck Form in theWeak Collision Limit
	Appendix 7.B Finite Difference Algorithm for Integrating the Isotropic Fokker–Planck Equation
	Appendix 7.C Monte Carlo Algorithm for Integrating the Fokker–Planck Equation
	Appendix 7.D Landau’s Calculation of the Electron–Ion Equilibration Rate
Chapter 8 The Hydrodynamic Limit for Plasma
	8.1 Introduction – Individual Particle Fluid Equations
	8.2 The Departure from Steady, Homogeneous Flow: The Transport Coefficients
	8.3 Magneto‐hydrodynamic Equations
		8.3.1 Equation of Mass Conservation
		8.3.2 Equation of Momentum Conservation
		8.3.3 Virial Theorem
		8.3.4 Equation of Current Flow
		8.3.5 Equation of Energy Conservation
	8.4 Transport Equations
		8.4.1 Collision Times
		8.4.2 Symmetry of the Transport Equations
	8.5 Two Fluid MHD Equations – Braginskii Equations
		8.5.1 Magnetic Field Equations
			8.5.1.1 Energy Balance
	8.6 Transport Coefficients
		8.6.1 Collisional Dominated Plasma
			8.6.1.1 Force Terms F
			8.6.1.2 Energy Flux Terms
			8.6.1.3 Viscosity
		8.6.2 Field‐Dominated Plasma
			8.6.2.1 Force Terms F
			8.6.2.2 Energy Flux Terms
			8.6.2.3 Viscosity
	8.7 Calculation of the Transport Coefficients
	8.8 Lorentz Approximation
		8.8.1 Electron–Electron Collisions
		8.8.2 Electron Runaway
	8.9 Deficiencies in the Spitzer/Braginskii Model of Transport Coefficients
	Appendix 8.A BGK Model for the Calculation of Transport Coefficients
		8.A.1 BGK Conductivity Model
		8.A.2 BGK Viscosity Model
	Appendix 8.B The Relationship Between the Flux Equations Given By Shkarofsky and Braginskii
	Appendix 8.C Electrical Conductivity in aWeakly Ionised Gas and the Druyvesteyn Distribution
Chapter 9 Ideal Magnetohydrodynamics
	9.1 Infinite Conductivity MHD Flow
		9.1.1 Frozen Field Condition
		9.1.2 Adiabatic Equation of State
		9.1.3 Pressure Balance
			9.1.3.1 Virial Theorem
	9.2 Incompressible Approximation
		9.2.1 Bernoulli\'s Equation – Steady Flow
		9.2.2 Kelvin\'s Theorem – Circulation
		9.2.3 Alfvén Waves
Chapter 10 Waves in MHD Fluids
	10.1 Introduction
	10.2 Magneto‐sonic Waves
	10.3 Discontinuities in Fluid Mechanics
		10.3.1 Classical Fluids
		10.3.2 Discontinuities in Magneto‐hydrodynamic Fluids
	10.4 The Rankine–Hugoniot Relations for MHD Flows
	10.5 Discontinuities in MHD Flows
	10.6 MHD Shock Waves
		10.6.1 Simplifying Frame Transformations
	10.7 Properties of MHD Shocks
		10.7.1 Shock Hugoniot
		10.7.2 Shock Adiabat – General Solution for a Polytropic Gas
	10.8 Evolutionary Shocks
		10.8.1 Evolutionary MHD Shock Waves
		10.8.2 Parallel Shock – Magnetic Field Normal to the Shock Plane
	10.9 Switch‐on and Switch‐off Shocks
	10.10 Perpendicular Shock – Magnetic Field Lying in the Shock Plane
	10.11 Shock Structure and Stability
	Appendix 10.A Group Velocity of Magneto-sonicWaves
	Appendix 10.B Solution in de Hoffman–Teller Frame
		10.B.1 Parallel Shocks
Chapter 11 Waves in Cold Magnetised Plasma
	11.1 Introduction
	11.2 Waves in Cold Plasma
		11.2.1 Cut‐off and Resonance
		11.2.2 Polarisation
	11.3 Cold Plasma Waves
		11.3.1 Zero Applied Magnetic Field
		11.3.2 Low Frequency Velocity Waves
		11.3.3 Propagation of Waves Parallel to the Magnetic Field
		11.3.4 Propagation of Waves Perpendicular to the Magnetic Field
		11.3.5 Resonance in Plasma Waves
Chapter 12 Waves in Magnetised Warm Plasma
	12.1 The Dielectric Properties of Unmagnetised Warm Dilute Plasma
		12.1.1 Plasma Dispersion Relation
			12.1.1.1 Dispersion Relation for Transverse Waves
			12.1.1.2 Dispersion Relation for Longitudinal Waves
		12.1.2 Dielectric Constant of a Plasma
			12.1.2.1 The Landau Contour Integration Around the Singularity
	12.2 Transverse Waves
	12.3 Longitudinal Waves
	12.4 Linear Landau Damping
		12.4.1 Resonant Energy Absorption
	12.5 Non‐linear Landau Damping
		12.5.1 Particle Trapping
		12.5.2 Plasma Wave Breaking
	12.6 The Plasma Dispersion Function
	12.7 Positive Ion Waves
		12.7.1 Transverse Waves
		12.7.2 Longitudinal Waves
			12.7.2.1 Plasma Waves, ζe > 1
			12.7.2.2 Ion Waves ζe < 1
	12.8 Microscopic Plasma Instability
		12.8.1 Nyquist Plot
			12.8.1.1 Penrose\'s Criterion
	12.9 The Dielectric Properties of Warm Dilute Plasma in a Magnetic Field
		12.9.1 Propagation Parallel to the Magnetic Field
		12.9.2 Propagation Perpendicular to the Magnetic Field
	Appendix 12.A Landau’s Solution of the Vlasov Equation
	Appendix 12.B ElectrostaticWaves
Chapter 13 Properties of Electro‐magnetic Waves in Plasma
	13.1 Plasma Permittivity and the Dielectric Constant
		13.1.1 The Properties of the Permittivity Matrix
	13.2 Plane Waves in Homogeneous Plasma
		13.2.1 Waves in Collisional Cold Plasma
			13.2.1.1 Isotropic Unmagnetised Plasma
			13.2.1.2 Anisotropic Magnetised Plasma
	13.3 Plane Waves Incident Obliquely on a Refractive Index Gradient
		13.3.1 Oblique Incidence at a Cut‐off Point – Resonance Absorption
			13.3.1.1 s Polarisation
			13.3.1.2 p Polarisation
	13.4 Single Particle Model of Electrons in an Electro‐magnetic Field
		13.4.1 Quiver Motion
		13.4.2 Ponderomotive Force
		13.4.3 The Impact Model for Collisional Absorption
			13.4.3.1 Electron–Electron Collisions
		13.4.4 Distribution Function of Electrons Subject to Inverse Bremsstrahlung Heating
	13.5 Parametric Instabilities
		13.5.1 Coupled Wave Interactions
			13.5.1.1 Manley–Rowe Relations
			13.5.1.2 Parametric Instability
		13.5.2 Non‐linear Laser‐Plasma Absorption
			13.5.2.1 Absorption Instabilities
			13.5.2.2 Reflection Instabilities
	Appendix 13.A Ponderomotive Force
Chapter 14 Laser–Plasma Interaction
	14.1 Introduction
	14.2 The Classical Hydrodynamic Model of Laser‐Solid Breakdown
		14.2.1 Basic Parameters of Laser Breakdown
		14.2.2 The General Theory of the Interaction of Lasers with Solid Targets
		14.2.3 Distributed Heating – Low Intensity, Self‐regulating Flow
			14.2.3.1 Early Time Self‐similar Solution
			14.2.3.2 Late Time Steady‐State Solution
		14.2.4 Local Heating – High Intensity, Deflagration Flow
			14.2.4.1 Early Time Thermal Front
			14.2.4.2 Late Time Steady‐State Flow
		14.2.5 Additional Simple Analytic Models
			14.2.5.1 Short Pulse Heating
			14.2.5.2 Heating of Small Pellets – Homogeneous Self‐similar Model
	14.3 Simulation of Laser‐Solid Target Interaction
	Appendix 14.A Non-linear Diffusion
	Appendix 14.B Self-similar Flows with Uniform Velocity Gradient
Chapter 15 Magnetically Confined Plasma
	15.1 Introduction
	15.2 Equilibrium Plasma Configurations
	15.3 Linear Devices
	15.4 Toroidal Devices
		15.4.1 Pressure Balance
			15.4.1.1 Pressure Imbalance Mitigation
		15.4.2 Guiding Centre Drift
	15.5 The General Problem: The Grad–Shafranov Equation
	15.6 Boundary Conditions
	15.7 Equilibrium Plasma Configurations
		15.7.1 Perturbation Methods
		15.7.2 Analytical Solutions of the Grad–Shafranov Equation
		15.7.3 Numerical Solutions of the Grad–Shafranov Equation
	15.8 Classical Magnetic Cross Field Diffusion
	15.9 Trapped Particles and Banana Orbits
		15.9.1 Collisionless Banana Regime (ν* << 1)
			15.9.1.1 Diffusion in the Banana Regime
			15.9.1.2 Bootstrap Current (ν* << 1)
		15.9.2 Resistive Plasma Diffusion – Collisional Pfirsch–Schlüter Regime
			15.9.2.1 Pfirsch–Schlüter Current (ν* >> 1)
			15.9.2.2 Diffusion in the Pfirsch–Sclüter Regime
		15.9.3 Plateau Regime
		15.9.4 Diffusion in Tokamak Plasmas
	Appendix 15.A Equilibrium Maintaining ‘Vertical’ Field
	Appendix 15.B Perturbation Solution of the Grad–Shafranov Equation
	Appendix 15.C Analytic Solutions of the Homogeneous Grad–Shafranov Equation
	Appendix 15.D Guiding Centre Motion in a Twisted Circular Toroidal Plasma
	Appendix 15.E The Pfirsch–Schlüter Regime
		15.E.1 Diffusion in the Pfirsch–Schlüter Regime
Chapter 16 Instability of an Equilibrium Confined Plasma
	16.1 Introduction
	16.2 Ideal MHD Instability
		16.2.1 Linearised Stability Equations
		16.2.2 Normal Mode Analysis – The Stability of a Cylindrical Plasma Column
		16.2.3 m = 0 Sausage Instability
		16.2.4 m = 1 Kink Instability
	16.3 Potential Energy
	16.4 Interchange Instabilities
Supplementary Material
	M.1 Breakdown and Discharges in d.c. Electric Fields
		M.1.1 Gas Breakdown and Paschen’s Law
		M.1.2 Similarity and Proper Variables
		M.1.3 Townsend’s First Coefficient
		M.1.4 Townsend’s Breakdown Criterion
		M.1.5 Paschen Curve and Paschen Minimum
		M.1.6 Radial Profile of Glow Discharge
		M.1.7 Collisional Ionisation Rate for Low Temperature Electrons
		M.1.8 Radio Frequency and Microwave Discharges
	M.2 Key Facts Governing Nuclear Fusion
		M.2.1 Fusion Rate
		M.2.2 Lawson’s Criterion
		M.2.3 Triple Product
	M.3 A Short Introduction to Functions of a Complex Variable
		M.3.1 Cauchy–Riemann Relations
		M.3.2 Harmonic Functions
		M.3.3 Area
		M.3.4 Cauchy Integral Theorem
		M.3.5 Morera’s Theorem
		M.3.6 Analytic Continuation
		M.3.7 Extension or Contraction of a Contour
		M.3.8 Inclusion of Isolated Singularities
		M.3.9 Cauchy Formula
			M.3.9.1 Interior Domain
			M.3.9.2 Exterior Domain
		M.3.10 Treatment of Improper Integrals
		M.3.11 Sokhotski–Plemelj Theorem
		M.3.12 Improper Integral Along a Real Line
		M.3.13 Taylor and Laurent Series
		M.3.14 The Argument Principle
		M.3.15 Estimation Lemma
		M.3.16 Jordan’s Lemma
		M.3.17 Conformal Mapping
	M.4 Laplace Transform
		M.4.1 Bromwich Contour
Problems
Bibliography
Index
EULA