Sentis Mathematical Models and Methods for Plasma Physics, Volume 1

Fluid Models
2014
ISBN: 978-3-319-03804-9
Verlag: Springer Nature Switzerland
Format: PDF
Kopierschutz: 1 - PDF Watermark

E-Book, Englisch, 246 Seiten

Reihe: Modeling and Simulation in Science, Engineering and Technology

This monograph is dedicated to the derivation and analysis of fluid models occurring in plasma physics. It focuses on models involving quasi-neutrality approximation, problems related to laser propagation in a plasma, and coupling plasma waves and electromagnetic waves. Applied mathematicians will find a stimulating introduction to the world of plasma physics and a few open problems that are mathematically rich. Physicists who may be overwhelmed by the abundance of models and uncertain of their underlying assumptions will find basic mathematical properties of the related systems of partial differential equations. A planned second volume will be devoted to kinetic models.First and foremost, this book mathematically derives certain common fluid models from more general models. Although some of these derivations may be well known to physicists, it is important to highlight the assumptions underlying the derivations and to realize that some seemingly simple approximations turn out to be more complicated than they look. Such approximations are justified using asymptotic analysis wherever possible. Furthermore, efficient simulations of multi-dimensional models require precise statements of the related systems of partial differential equations along with appropriate boundary conditions. Some mathematical properties of these systems are presented which offer hints to those using numerical methods, although numerics is not the primary focus of the book.

Autoren/Hrsg.

Sentis, Rémi

Sentis, Remi

Inhaltsverzeichnis

1;Foreword;8
2;Contents;12
3;Chapter1 Introduction: Some Plasma Characteristic Quantities;14
3.1;1.1 Historical Account;14
3.2;1.2 Notations;16
3.3;1.3 Heuristics for Introducing Some Plasma Characteristic Quantities;18
4;Chapter2 Quasi-Neutrality and Magneto-Hydrodynamics;24
4.1;2.1 Massless-Electron Approximation;24
4.1.1;2.1.1 The Ion–Electron Electrodynamic Model;25
4.1.2;2.1.2 The Ion Euler System with Massless-Electron Approximation;29
4.2;2.2 Quasi-Neutrality Approximation;37
4.2.1;2.2.1 Asymptotic Analysis in the Nonmagnetized Case;37
4.2.2;2.2.2 Asymptotic Analysis in the Magnetized Case;40
4.2.3;2.2.3 Proofs of the Propositions of Sects.2.1 and 2.2;42
4.3;2.3 Two-Temperature Euler Models and Magneto-Hydrodynamics;50
4.3.1;2.3.1 The Two-Temperature Euler System;51
4.3.1.1;2.3.1.1 Accounting for the Thermal Conduction;53
4.3.1.2;2.3.1.2 Accounting for Radiative Coupling;55
4.3.1.3;2.3.1.3 Accounting for Electric Current;58
4.3.2;2.3.2 Electron Magneto-Hydrodynamics;60
4.3.2.1;2.3.2.1 Case with Scalar Conductivity;61
4.3.2.2;2.3.2.2 Case with a Tensor Conductivity;64
4.3.2.3;2.3.2.3 Boundary Conditions. Axi-Symmetric Geometry Case;67
4.4;2.4 Analysis of the Hyperbolic Part of Systems (E2T) and (MHD);76
4.4.1;2.4.1 On the Galilean Transformations;78
4.4.2;2.4.2 Hyperbolic Properties of Both Models;79
4.4.3;2.4.3 Proofs of the Propositions of the Section;82
5;Chapter3 Laser Propagation: Coupling with Ion Acoustic Waves;85
5.1;3.1 Laser Propagation in a Plasma;87
5.1.1;3.1.1 On the Time Envelope Models;87
5.1.1.1;3.1.1.1 Decomposition of the Electromagnetic Fields;89
5.1.1.1.1;Orientation;92
5.1.1.2;3.1.1.3 Properties of the Basic Time Envelope Model;98
5.1.2;3.1.2 Geometrical Optics;103
5.1.2.1;3.1.2.1 The WKB Expansion;105
5.1.2.2;3.1.2.2 On the Ray-Tracing Method;108
5.1.3;3.1.3 The Paraxial Approximation;111
5.1.3.1;3.1.3.1 The WKB Expansion;113
5.1.3.2;3.1.3.2 The Classical Paraxial Equation;116
5.1.3.3;3.1.3.3 Numerics for the Classical Paraxial Equation;117
5.2;3.2 The Brillouin Instability in Laser–Plasma Interaction;123
5.2.1;3.2.1 The Modified Decay Model in a Homogeneous Plasma;126
5.2.2;3.2.2 The Standard Decay System in a Homogeneous Plasma;127
5.2.3;3.2.3 Model with a Nonhomogeneous Plasma;131
5.2.4;3.2.4 A Three-Wave Coupling System and Its Analysis;132
5.2.4.1;3.2.4.1 Conservation Properties;132
5.2.4.2;3.2.4.2 Characteristic Values of the System (TWC);133
5.2.4.3;3.2.4.3 Dimensionless Form;134
6;Chapter4 Langmuir Waves and Zakharov Equations;147
6.1;4.1 Langmuir Waves Without Coupling with Ions;148
6.1.1;4.1.1 Conductivity and Dispersion Relation;150
6.1.2;4.1.2 Linear Langmuir Wave Theory;153
6.1.2.1;Energy Balance;154
6.2;4.3 The Zakharov Equations and Their Properties;161
6.2.1;Instabilities;165
7;Chapter5 Coupling ElectronWaves and LaserWaves;170
7.1;5.1 Raman Instability;171
7.1.1;5.1.1 Model with Fixed Ions;173
7.1.2;5.1.2 Reduction of the Model with Fixed Ions;180
7.1.2.1;Conclusion;183
7.1.3;5.1.3 The Raman Model with an Ion Acoustic Wave;190
7.2;5.2 The Euler–Maxwell Model for Short Ultra-High Intensity Laser Pulses;192
7.2.1;5.2.1 Well-Posedness of the Model;195
7.2.1.1;Symmetrization;196
7.2.1.2;Sketch of the Proof of Theorem 5;199
7.3;5.3 Envelope Models for Very Short High-Intensity Laser Pulses;203
8;Chapter6 Models with Several Species;210
8.1;6.1 Two-Temperature Euler System for a Mixing of Two Ion Species;210
8.1.1;6.1.1 The Three-Population Full Model;211
8.1.1.1;6.1.1.1 Conservation of Ion Momentum and Coupling with the Electron Velocity;212
8.1.1.2;6.1.1.2 Energy Balance and Statement of the Model;213
8.1.1.3;Statement of the Model;215
8.1.1.4;6.1.2.1 A Model with Mass Fraction, Average Ion Velocity and Average Ion Energy;217
8.1.1.5;6.1.2.2 Simplified Models with Mass Fraction;220
8.2;6.2 Some Models for Weakly Ionized Plasmas;223
8.2.1;6.2.1 The Multifluid Model and the Multispecies Diffusion Model;224
8.2.1.1;6.2.1.1 Derivation of the Multispecies Diffusion Model;225
8.2.1.2;6.2.1.2 Statement of the Multispecies Diffusion Model;227
8.2.2;6.2.2 The Ambipolar Diffusion Model;229
9;Appendix A;237
9.1;A.1 Tensor Analysis Formula;237
9.2;A.2 Useful Lemmas of Functional Analysis;237
10;Bibliography;240
11;Index;245