Vlasov Equation 1

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 ixChapter 1 Introduction to a Universal Model: the Vlasov Equation 11.1 A historical point of view 11.2 Individual and collective effects in plasmas 51.3 Graininess parameter 71.4 The collective description of a Coulomb gas: an intuitive approach 81.5 From N-body to Vlasov 121.6 The graininess parameter and 1D, 2D or 3D models 161.7 The Vlasov equation at the microscopic fluctuations level 191.8 The Wigner equation (Vlasov equation for quantum systems) 211.9 The relativistic Vlasov–Maxwell model 261.10 References 28Chapter 2 A Paradigm for a Collective Description of a Plasma: the 1D Vlasov–Poisson Equations 312.1 Introduction 312.2 The linear Landau problem 332.2.1 The Maxwellian case 342.2.2 Landau poles and others 362.2.3 Unstable plasma: two-stream instability 382.3 The 1D cold plasma model: nonlinear oscillations 392.3.1 Hydrodynamic description 392.3.2 Lagrangian formulation through the Von Mises transformation 402.3.3 The wave-breaking phenomenon 422.4 The water bag model 442.4.1 Basic equations 442.4.2 Linearized theory 472.4.3 Water bag hydrodynamic description 482.5 Connection between the hydrodynamic, water bag and Vlasov models 502.5.1 A Vlasov hydrodynamic description 502.5.2 Vlasov numerical simulations of Pn−3 522.5.3 The fundamental contribution of poles besides Landau 562.6 The multiple water bag model 582.6.1 A multifluid description 592.6.2 Linearized analysis 632.7 Further remarks 662.8 References 71Chapter 3 Electromagnetic Fields in Vlasov Plasmas: General Approach to Small Amplitude Perturbations 753.1 Introduction and overview of the chapter 753.2 Linear analysis of the Vlasov–Maxwell system: general approach 773.2.1 Dispersion relation and response matrix 813.2.2 The choice of the basis for the response tensor 833.2.3 About the number of “waves” in plasmas 893.2.4 Real or complex values of k and ω: steady state and initial value problems 923.3 Polynomial approximations of the dispersion relation: why and how to use them 933.3.1 Truncated-Vlasov and fluid–plasma descriptions for the linear analysis 963.3.2 Wave dispersion and resonances allowed by inclusion of high-order moments in fluid models 993.3.3 An example: fluid moments and Finite–Larmor–Radius effects 1033.3.4 Key points about approximated normal mode analysis 1083.4 Vlasov plasmas as collisionless conductors with polarization and finite conductivity: meaning of plasma’s “dielectric tensor” 1093.4.1 Polarization charges and wave equation in dielectric materials 1123.4.2 The “equivalent” dielectric tensor and its complex components 1153.4.3 Temporal and spatial dispersion in plasmas 1203.4.4 Conductivity and collisional resistivity in Vlasov plasmas 1223.5 Symmetry properties of the complex components of the equivalent dielectric tensor and energy conservation 1263.5.1 Onsager’s relations 1263.5.2 Poynting’s theorem 1293.5.3 Symmetry of the coefficients of the equivalent dielectric tensor 1303.5.4 More about Onsager’s relations for wave dispersion 1343.5.5 Energy dissipation versus real and imaginary parts of σij and Єij 1383.6 References 141Chapter 4 Electromagnetic Fields in Vlasov Plasmas: Characterization of Linear Modes 1474.1 Introduction 1474.2 Characterization of electromagnetic waves and of wave-packets 1484.2.1 Polarization of electromagnetic waves in plasmas 1534.2.2 Phase velocity, group velocity and refractive index 1564.2.3 Example of propagation in unmagnetized plasmas: underdense and overdense regimes 1614.2.4 Example of propagation in magnetized plasmas: ion-cyclotron resonances and Faraday’s rotation effect 1664.2.5 Wave–particle resonances, Landau damping and wave absorption 1724.2.6 Resonance and cut-off conditions on the refractive index 1764.2.7 Graphical representations of the dispersion relation 1784.3 Instabilities in Vlasov plasmas: some terminology and general features 1824.3.1 Linear instabilities 1844.3.2 Absolute and convective instabilities and some other classification criteria 1924.4 On some complementary interpretations of the collisionless damping mechanism in Vlasov plasmas 1984.4.1 Landau damping as an inverse Vavilov–Cherenkov radiation 1994.4.2 Landau damping in N-body “exact” models 2034.4.3 Some final remarks about interpretative issues of collisionless damping in Vlasov mean field theory 2064.5 References 207Chapter 5 Nonlinear Properties of Electrostatic Vlasov Plasmas 2155.1 The Vlasov–Poisson system 2155.2 Invariants of the Vlasov–Poisson model 2165.3 Stationary solutions: Bernstein–Greene–Kruskal equilibria 2175.4 Some mathematical properties of the Vlasov equation 2205.5 The Bernstein–Greene–Kruskal solutions 2295.5.1 The case of (electrostatic) two-stream instability 2305.5.2 Chain of BGK equilibria 2355.5.3 Stability of the periodic BGK steady states 2365.6 Traveling waves of BGK-type solutions 2425.7 Role of minority population of trapped particles 2455.7.1 Nonlinear Landau damping and the emergence of the nonlinear Langmuir-type wave 2475.7.2 Electron acoustic wave in the nonlinear Landau damping regime 2545.7.3 Kinetic electrostatic electron nonlinear waves 2605.7.4 Emergent resonance for KEEN waves 2685.8 Nature of KEEN waves and NMI 2705.8.1 Adiabatic model for a single linear wave: the (electrostatic) trapped electron mode model 2705.8.2 The Dodin and Fisch model connected to the emergence of KEEN waves 2745.9 Electron hole and plasma wave interaction 2815.10 References 291Index 297