Mathematical Models and Methods for Plasma Physics, Volume 1 (eBook)

Foreword 8
Contents 12
Chapter1 Introduction: Some Plasma Characteristic Quantities 14
1.1 Historical Account 14
1.2 Notations 16
1.3 Heuristics for Introducing Some Plasma Characteristic Quantities 18
Chapter2 Quasi-Neutrality and Magneto-Hydrodynamics 24
2.1 Massless-Electron Approximation 24
2.1.1 The Ion–Electron Electrodynamic Model 25
2.1.2 The Ion Euler System with Massless-Electron Approximation 29
2.2 Quasi-Neutrality Approximation 37
2.2.1 Asymptotic Analysis in the Nonmagnetized Case 37
2.2.2 Asymptotic Analysis in the Magnetized Case 40
2.2.3 Proofs of the Propositions of Sects.2.1 and 2.2 42
2.3 Two-Temperature Euler Models and Magneto-Hydrodynamics 50
2.3.1 The Two-Temperature Euler System 51
2.3.1.1 Accounting for the Thermal Conduction 53
2.3.1.2 Accounting for Radiative Coupling 55
2.3.1.3 Accounting for Electric Current 58
2.3.2 Electron Magneto-Hydrodynamics 60
2.3.2.1 Case with Scalar Conductivity 61
2.3.2.2 Case with a Tensor Conductivity 64
2.3.2.3 Boundary Conditions. Axi-Symmetric Geometry Case 67
2.4 Analysis of the Hyperbolic Part of Systems (E2T) and (MHD) 76
2.4.1 On the Galilean Transformations 78
2.4.2 Hyperbolic Properties of Both Models 79
2.4.3 Proofs of the Propositions of the Section 82
Chapter3 Laser Propagation: Coupling with Ion Acoustic Waves 85
3.1 Laser Propagation in a Plasma 87
3.1.1 On the Time Envelope Models 87
3.1.1.1 Decomposition of the Electromagnetic Fields 89
Orientation 92
3.1.1.3 Properties of the Basic Time Envelope Model 98
3.1.2 Geometrical Optics 103
3.1.2.1 The WKB Expansion 105
3.1.2.2 On the Ray-Tracing Method 108
3.1.3 The Paraxial Approximation 111
3.1.3.1 The WKB Expansion 113
3.1.3.2 The Classical Paraxial Equation 116
3.1.3.3 Numerics for the Classical Paraxial Equation 117
3.2 The Brillouin Instability in Laser–Plasma Interaction 123
3.2.1 The Modified Decay Model in a Homogeneous Plasma 126
3.2.2 The Standard Decay System in a Homogeneous Plasma 127
3.2.3 Model with a Nonhomogeneous Plasma 131
3.2.4 A Three-Wave Coupling System and Its Analysis 132
3.2.4.1 Conservation Properties 132
3.2.4.2 Characteristic Values of the System (TWC) 133
3.2.4.3 Dimensionless Form 134
Chapter4 Langmuir Waves and Zakharov Equations 147
4.1 Langmuir Waves Without Coupling with Ions 148
4.1.1 Conductivity and Dispersion Relation 150
4.1.2 Linear Langmuir Wave Theory 153
Energy Balance 154
4.3 The Zakharov Equations and Their Properties 161
Instabilities 165
Chapter5 Coupling ElectronWaves and LaserWaves 170
5.1 Raman Instability 171
5.1.1 Model with Fixed Ions 173
5.1.2 Reduction of the Model with Fixed Ions 180
Conclusion 183
5.1.3 The Raman Model with an Ion Acoustic Wave 190
5.2 The Euler–Maxwell Model for Short Ultra-High Intensity Laser Pulses 192
5.2.1 Well-Posedness of the Model 195
Symmetrization 196
Sketch of the Proof of Theorem 5 199
5.3 Envelope Models for Very Short High-Intensity Laser Pulses 203
Chapter6 Models with Several Species 210
6.1 Two-Temperature Euler System for a Mixing of Two Ion Species 210
6.1.1 The Three-Population Full Model 211
6.1.1.1 Conservation of Ion Momentum and Coupling with the Electron Velocity 212
6.1.1.2 Energy Balance and Statement of the Model 213
Statement of the Model 215
6.1.2.1 A Model with Mass Fraction, Average Ion Velocity and Average Ion Energy 217
6.1.2.2 Simplified Models with Mass Fraction 220
6.2 Some Models for Weakly Ionized Plasmas 223
6.2.1 The Multifluid Model and the Multispecies Diffusion Model 224
6.2.1.1 Derivation of the Multispecies Diffusion Model 225
6.2.1.2 Statement of the Multispecies Diffusion Model 227
6.2.2 The Ambipolar Diffusion Model 229
Appendix A 237
A.1 Tensor Analysis Formula 237
A.2 Useful Lemmas of Functional Analysis 237
Bibliography 240
Index 245