The Vlasov Equation 1 History and General Properties

The Vlasov equation is the master equation which provides a statistical description for the collective behavior of large numbers of charged particles in mutual, long-range interaction. In other words, a low collision (or “Vlasov”) plasma. Plasma physics is itself a relatively young discipline, whose “birth” can be ascribed to the 1920s. The origin of the Vlasov model, however, is even more recent, dating back to the late 1940s. This “young age” is due to the rare occurrence of Vlasov plasma on Earth, despite the fact it characterizes most of the visible matter in the universe.

This book – addressed to students, young researchers and to whoever wants a good understanding of Vlasov plasmas – discusses this model with a pedagogical presentation, focusing on the general properties and historical development of the applications of the Vlasov equation. The milestone developments discussed in the first two chapters serve as an introduction to more recent works (characterization of wave propagation and nonlinear properties of the electrostatic limit).

Pierre Bertrand is Emeritus Professor at the University of Lorraine, France. He has been working for more than 50 years in various fields of theoretical physics, and numerical simulation involving plasma physics, quantum mechanics, signal theory and fluid mechanics.

Daniele Del Sarto is Associate Professor at the Jean Lamour Institute, University of Lorraine, France. Her main research field is fundamental physics of warm plasmas, with applications to astrophysics, magnetic confinement fusion and laser-plasma interactions.

Alain Ghizzo is Professor at the Jean Lamour Institute, University of Lorraine, France. He is active in plasma Vlasov modeling, high performance computing, laser-plasma interactions and gyrokinetic modeling.

Preface ix

Chapter 1 Introduction to a Universal Model: the Vlasov Equation 1

1.1 A historical point of view 1

1.2 Individual and collective effects in plasmas 5

1.3 Graininess parameter 7

1.4 The collective description of a Coulomb gas: an intuitive approach 8

1.5 From N-body to Vlasov 12

1.6 The graininess parameter and 1D, 2D or 3D models 16

1.7 The Vlasov equation at the microscopic fluctuations level 19

1.8 The Wigner equation (Vlasov equation for quantum systems) 21

1.9 The relativistic Vlasov–Maxwell model 26

1.10 References 28

Chapter 2 A Paradigm for a Collective Description of a Plasma: the 1D Vlasov–Poisson Equations 31

2.1 Introduction 31

2.2 The linear Landau problem 33

2.2.1 The Maxwellian case 34

2.2.2 Landau poles and others 36

2.2.3 Unstable plasma: two-stream instability 38

2.3 The 1D cold plasma model: nonlinear oscillations 39

2.3.1 Hydrodynamic description 39

2.3.2 Lagrangian formulation through the Von Mises transformation 40

2.3.3 The wave-breaking phenomenon 42

2.4 The water bag model 44

2.4.1 Basic equations 44

2.4.2 Linearized theory 47

2.4.3 Water bag hydrodynamic description 48

2.5 Connection between the hydrodynamic, water bag and Vlasov models 50

2.5.1 A Vlasov hydrodynamic description 50

2.5.2 Vlasov numerical simulations of Pn⁻³ 52

2.5.3 The fundamental contribution of poles besides Landau 56

2.6 The multiple water bag model 58

2.6.1 A multifluid description 59

2.6.2 Linearized analysis 63

2.7 Further remarks 66

2.8 References 71

Chapter 3 Electromagnetic Fields in Vlasov Plasmas: General Approach to Small Amplitude Perturbations 75

3.1 Introduction and overview of the chapter 75

3.2 Linear analysis of the Vlasov–Maxwell system: general approach 77

3.2.1 Dispersion relation and response matrix 81

3.2.2 The choice of the basis for the response tensor 83

3.2.3 About the number of “waves” in plasmas 89

3.2.4 Real or complex values of k and ω: steady state and initial value problems 92

3.3 Polynomial approximations of the dispersion relation: why and how to use them 93

3.3.1 Truncated-Vlasov and fluid–plasma descriptions for the linear analysis 96

3.3.2 Wave dispersion and resonances allowed by inclusion of high-order moments in fluid models 99

3.3.3 An example: fluid moments and Finite–Larmor–Radius effects 103

3.3.4 Key points about approximated normal mode analysis 108

3.4 Vlasov plasmas as collisionless conductors with polarization and finite conductivity: meaning of plasma’s “dielectric tensor” 109

3.4.1 Polarization charges and wave equation in dielectric materials 112

3.4.2 The “equivalent” dielectric tensor and its complex components 115

3.4.3 Temporal and spatial dispersion in plasmas 120

3.4.4 Conductivity and collisional resistivity in Vlasov plasmas 122

3.5 Symmetry properties of the complex components of the equivalent dielectric tensor and energy conservation 126

3.5.1 Onsager’s relations 126

3.5.2 Poynting’s theorem 129

3.5.3 Symmetry of the coefficients of the equivalent dielectric tensor 130

3.5.4 More about Onsager’s relations for wave dispersion 134

3.5.5 Energy dissipation versus real and imaginary parts of σ_ij and Є_ij 138

3.6 References 141

Chapter 4 Electromagnetic Fields in Vlasov Plasmas: Characterization of Linear Modes 147

4.1 Introduction 147

4.2 Characterization of electromagnetic waves and of wave-packets 148

4.2.1 Polarization of electromagnetic waves in plasmas 153

4.2.2 Phase velocity, group velocity and refractive index 156

4.2.3 Example of propagation in unmagnetized plasmas: underdense and overdense regimes 161

4.2.4 Example of propagation in magnetized plasmas: ion-cyclotron resonances and Faraday’s rotation effect 166

4.2.5 Wave–particle resonances, Landau damping and wave absorption 172

4.2.6 Resonance and cut-off conditions on the refractive index 176

4.2.7 Graphical representations of the dispersion relation 178

4.3 Instabilities in Vlasov plasmas: some terminology and general features 182

4.3.1 Linear instabilities 184

4.3.2 Absolute and convective instabilities and some other classification criteria 192

4.4 On some complementary interpretations of the collisionless damping mechanism in Vlasov plasmas 198

4.4.1 Landau damping as an inverse Vavilov–Cherenkov radiation 199

4.4.2 Landau damping in N-body “exact” models 203

4.4.3 Some final remarks about interpretative issues of collisionless damping in Vlasov mean field theory 206

4.5 References 207

Chapter 5 Nonlinear Properties of Electrostatic Vlasov Plasmas 215

5.1 The Vlasov–Poisson system 215

5.2 Invariants of the Vlasov–Poisson model 216

5.3 Stationary solutions: Bernstein–Greene–Kruskal equilibria 217

5.4 Some mathematical properties of the Vlasov equation 220

5.5 The Bernstein–Greene–Kruskal solutions 229

5.5.1 The case of (electrostatic) two-stream instability 230

5.5.2 Chain of BGK equilibria 235

5.5.3 Stability of the periodic BGK steady states 236

5.6 Traveling waves of BGK-type solutions 242

5.7 Role of minority population of trapped particles 245

5.7.1 Nonlinear Landau damping and the emergence of the nonlinear Langmuir-type wave 247

5.7.2 Electron acoustic wave in the nonlinear Landau damping regime 254

5.7.3 Kinetic electrostatic electron nonlinear waves 260

5.7.4 Emergent resonance for KEEN waves 268

5.8 Nature of KEEN waves and NMI 270

5.8.1 Adiabatic model for a single linear wave: the (electrostatic) trapped electron mode model 270

5.8.2 The Dodin and Fisch model connected to the emergence of KEEN waves 274

5.9 Electron hole and plasma wave interaction 281

5.10 References 291

Index 297

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