Collisionless Plasmas in Astrophysics

Collisionless Plasmas in Astrophysics examines the unique properties of media without collisions in plasma physics. Experts in this field, the authors present the first book to concentrate on collisionless conditions in plasmas, whether close or not to thermal equilibrium. Filling a void in scientific literature, Collisionless Plasmas in Astrophysics explains the possibilities of modeling such plasmas, using a fluid or a kinetic framework. It also addresses common misconceptions that even professionals may possess, on phenomena such as "collisionless (Landau) damping". Abundant illustrations are given in both space physics and astrophysics.

Table of Contents:
About the Authors XI 1 Introduction 1 1.1 Goals of the Book 1 1.2 Plasmas in Astrophysics 5 1.2.1 Plasmas Are Ubiquitous 5 1.2.2 The Magnetosphere of Stars 6 1.2.3 Shock Waves 6 1.2.4 Planetary Magnetospheres 7 1.3 Upstream of Plasma Physics: Electromagnetic Fields and Waves 9 1.3.1 Electromagnetic Fields 9 1.3.2 Transverse and Longitudinal Electromagnetic Field 16 1.3.3 Electromagnetic Fields in Vacuum 17 1.3.4 Plane Waves in a Plasma 24 1.3.5 Electromagnetic Components of Plane Plasma Waves 25 1.3.6 Some General Properties of PlaneWave Polarization and Dispersion 26 1.3.7 Electrostatic Waves 27 1.3.8 Wave Packets and Group Velocity 28 1.3.9 Propagation of Plane Waves in a Weakly Inhomogeneous Medium 28 1.3.10 Useful Approximations of the Maxwell Equations in Plasma Physics 30 1.4 Upstream of Plasma Physics: The Motion of Charged Particles 35 1.4.1 The Motion of the Guiding Center 35 1.4.2 Adiabatic Invariants 39 1.4.3 The Motion of a Particle in a Wave 42 2 Plasma Descriptions and Plasma Models 47 2.1 Distribution Function and Moments 47 2.1.1 From Individual Particles to Kinetic Description 47 2.1.2 Kinetic Description and First Order Moments 50 2.1.3 Higher-Order Moments 53 2.1.4 Moments for a Mixture of Populations 54 2.1.5 Nontrivial Generalization of the Fluid Concepts 55 2.1.6 Fluid vs. Kinetic Description: An Example 57 2.2 From Kinetic to Fluid Equations 58 2.2.1 Moment Equations 58 2.2.2 Lagrangian Form of the Moment Equations 62 2.2.3 Fluid Equations: Necessity of a Closure Equation 63 2.2.4 Collisional Limit: Fluid Dynamics and Thermodynamics 65 2.3 Numerical Methods 67 2.3.1 Vlasov Codes 67 2.3.2 Particle in Cell Codes (PIC) 69 2.3.3 Perturbative PIC Codes 73 2.4 Fluid Codes 74 2.5 Hybrid Codes 75 3 The Magnetized Plasmas 77 3.1 Ideal MHD 77 3.1.1 The Ideal MHD System 77 3.1.2 The Ideal Ohm's Law 79 3.2 Establishing the MHD Model 82 3.2.1 Large-Scale Conditions of Validity 85 3.2.2 Departures from MHD: Multi-Fluid and Kinetic Effects 86 3.3 Dimensional Analysis and Plasma Characteristic Scales 88 3.3.1 Dimensional Analysis: The General Methods 88 3.3.2 Temporal and Spatial Scales, Adimensional Numbers 92 3.3.3 Dispersive and Dissipative Effects 102 3.3.4 Physical Importance of the Dimensionless Parameters 103 4 Collisional-Collisionless 105 4.1 Notion of Collisions in Plasma Physics 105 4.1.1 Coulomb Interaction: A Long Range Interaction 105 4.1.2 Mean Free Path 109 4.1.3 The Debye Length and the Notion of Debye "Screening" 113 4.1.4 Knudsen Number 115 4.1.5 Plasma Regimes 117 4.2 Notion of Dissipation 119 4.2.1 Transfers of Energy and Dissipation 119 4.2.2 The Concept of Dissipation in Collisional Fluids 120 4.2.3 Reversibility 123 4.2.4 Irreversibility and Damping 125 4.2.5 The Notion of Reversibility Depends on the Description 127 4.2.6 Entropy 132 5 Waves in Plasmas 135 5.1 MHD Waves 136 5.1.1 Polarization of the MHD Waves 137 5.1.2 Application: Alfven and MHD Waves in the Earth's Magnetosphere 139 5.2 Transport Induced by Waves 140 5.2.1 Alfven Wave Pressure 141 5.3 High-FrequencyWaves 146 5.3.1 Cold Plasma Model 146 5.3.2 Parallel Propagation 148 5.3.3 Perpendicular Propagation: Ordinary and ExtraordinaryWaves 150 5.3.4 Application: Plasma Cut-offs and Limits to the Radio Astronomy 151 5.3.5 Application: The Dispersion of Radio Waves from Pulsars 152 5.3.6 Application: Faraday Rotation in the Interstellar Medium 153 5.4 Whistler Mode 154 5.5 Collisional Damping in Fluid Theories 158 5.5.1 Dissipative Effects and Entropy 158 5.5.2 Dissipation and Collisions 161 5.5.3 Strongly Collisional Systems 163 5.5.4 Heat Conduction: From Collisional to Collisionless 164 5.5.5 The Thermoelectric Field: Another Consequence of Collisions between Ions and Electrons 167 5.6 Collisionless Damping 168 5.6.1 Number of Eigenmodes: Fluid vs. Kinetic 168 5.6.2 A Simple Example: The LangmuirWave, from Fluid to Kinetic 169 5.6.3 Fluid Treatment of the LangmuirWave: Choice of a Closure 170 5.6.4 Kinetic Treatment of the Langmuir Wave: Landau Damping 176 5.6.5 Other Types of Kinetic Damping 192 5.7 Instabilities 193 5.7.1 Real Space Instabilities: Fluid Treatment 193 5.7.2 Velocity Space Instabilities: Kinetic Treatment 193 5.7.3 Weak Kinetic Effects 194 5.7.4 An Example: The Two-Stream Instability 196 6 Nonlinear Effects, Shocks, and Turbulence 199 6.1 Collisionless Shocks and Discontinuities 199 6.1.1 Nonlinear Propagation, Discontinuities, Jumps 199 6.1.2 Shocks and Other Discontinuities in a Magnetized Plasma 205 6.1.3 The Unmagnetized Shock Wave 208 6.1.4 A Particular Case: The Tangential Discontinuity 208 6.1.5 Example: The Terrestrial Bow Shock, the Foreshocks 211 6.2 Turbulence (Mainly MHD) 213 6.2.1 Hydrodynamics: Equations, Shocks 216 6.2.2 Hydrodynamics: 3D Incompressible Turbulence 220 6.2.3 MHD Turbulence - Introduction 225 6.2.4 Weak Isotropic (IK) Regime 230 6.2.5 Anisotropic Regimes 235 6.2.6 Discussion 250 6.3 Nonlinear Kinetic Physics 253 6.3.1 Nonlinear Electrostatic Waves 254 6.3.2 Particle Trapping 255 6.3.3 The Nonlinear Interaction of Many Electrostatic Waves of Low Amplitude 263 6.3.4 Quasi-Linear Theory 265 6.3.5 Trapping versus Quasi-Linear Diffusion 269 7 Flow and Particle Acceleration Processes 275 7.1 Flow Acceleration and Heating in a Collisional Fluid 275 7.1.1 Basic Equations 275 7.1.2 Expressions for the Polytropic Fluids 279 7.1.3 Bernoulli's Principle 281 7.1.4 Venturi Effect 281 7.1.5 De Laval Nozzle 283 7.1.6 Stellar Winds 284 7.1.7 Possible Routes to Turbulence in Stellar Winds 291 7.1.8 Accretion 293 7.2 Magnetic Reconnection 294 7.2.1 Conservation of Connections vs. Reconnection 294 7.2.2 Departure from the Ideal Ohm's Law: Microscopic Mechanisms and Macroscopic Consequences 296 7.2.3 Flow Acceleration by Reconnection 297 7.2.4 Tearing Instability 302 7.2.5 3D Reconnection 304 7.3 Kinetic Acceleration Processes in Magnetospheres 305 7.3.1 Substorms and Auroras in the Earth's Magnetosphere 305 7.3.2 Fermi Acceleration in the Magnetosphere 307 7.3.3 Acceleration by a Forced Current Forced along Convergent Magnetic Field Lines 307 7.3.4 Acceleration by an Electric Current Forced by a Wave 311 7.3.5 Acceleration by an Alfven Wave (NonMHD) Parallel Electric Field 312 7.3.6 Resonant Acceleration by a Wave 315 7.3.7 Acceleration by a Wave of Short Length 316 7.3.8 Application: Acceleration in the Earth's Magnetosphere 317 8 Transport and Acceleration of Cosmic Rays 319 8.1 The Problem of Transport 320 8.1.1 The Magnetic Field: Obstruction to Transport 320 8.1.2 Magnetic Irregularities: Transport Agent 322 8.1.3 Other Diffusion Coefficients 327 8.1.4 Transport Equation of Cosmic Rays 330 8.1.5 Distribution of Suprathermal Particles Crossing a Shock 334 8.1.6 From Transport to Acceleration 336 8.2 Fermi Acceleration of Cosmic Rays 336 8.2.1 The Basic Fermi Process 338 8.2.2 Fermi Process at a Nonrelativistic Shock 346 8.2.3 Astrophysical Application: Cosmic Rays and Supernovae 350 8.2.4 Astrophysical Application: Synchrotron Sources 351 8.2.5 Generation of Magnetic Turbulence 353 8.2.6 Why Are Fermi Processes Favored at Shocks? 354 8.2.7 What about the Relativistic Regime of Fermi Acceleration? 355 9 The Kinetic-Fluid Duality 357 9.1 Toy Models 357 9.1.1 Small Amplitude Ballistic Fluctuations 358 9.1.2 Large-Amplitude Ballistic Fluctuations 361 9.1.3 Quasi-Fluid Behavior of a Collisionless Plasma: Launching a 2D Plasma Bullet 366 9.2 Solar and Stellar Wind Expansion 368 9.2.1 A Simple NoncollisionalWind 368 9.2.2 More Sophisticated NoncollisionalWind Models 370 9.2.3 Charge Neutralizing Field for a Plasma in a Gravitational Field 371 9.2.4 Qualitative Radial Profile of the Total Proton Potential 374 9.2.5 Charge Neutralizing Electric Field and Dreicer Field 376 9.2.6 Electric Field Intensity at the Sonic Radius rs 377 9.2.7 Effective Closure for the Solar Wind 377 Appendix 383 A.1 Notation 383 A.1.1 Vectors and Tensors 383 A.1.2 Derivatives 383 A.1.3 List of Notation 384 A.2 Asymptotic Expansions and Adiabatic Invariants 385 A.2.1 Multiscale Expansion 385 A.2.2 The Adiabatic Invariants 388 A.2.3 Derivation of the Guiding Center Equations 390 A.3 Fokker-Planck Equation, First Order Term 393 References 395 Index 405

About the Author :
Gerard Belmont works as a "Directeur de Recherches" at the French CNRS for twenty years. He is a specialist of collisionless media, and their description through kinetic and fluid theories. Roland Grappin is Astronomer at the Paris Observatory since 1979. His scientific activity covers turbulence in fluids and plasmas, dynamics of the solar wind, corona and transition region. Fabrice Mottez is a scientist at the Paris Observatory. He has devoted his career to collisionless space plasmas, the terrestrial and Jovian magnetospheres, fundamental plasma physics, and numerical simulation. Filippo Pantellini is a scientist at the Paris Observatory. His main research fields cover the theoretical and numerical investigation of collisionless and weakly collisional space plasmas, with a particular interest for the solar wind and the solar corona. Guy Pelletier is a professor at the University Joseph Fourier in Grenoble. He founded the theoretical group of the Laboratory for Astrophysics. He accessed to all the levels of professorship and got the status of Emeritus Professor in 2009