Mathematical models and methods for plasma physics. Volume 1. Fluid models
The first volume of this monograph is devoted to fluid models in plasma physics. The author says to have about two motivations for undertaking this work: to understand and justify from the mathematical point of view the derivations of some classical-specific models from more general ones, and to give the precise statements of the PDFs systems with the appropriated boundary conditions which are necessary to perform efficient numerical simulations mainly in multi-dimensional cases. The short Chapter 1, after a historical information, gives some classic characteristic quantities appearing in plasma physics. In Chapter 1, the massless-electron approximation is justified from the general ion-electron electrodynamic model. The quasi-neutrality approximation is presented as the essence of most fluid models considered further. This approximation is rigorously proved by an asymptotic analysis where a small parameter related to the Debye length tends to zero. Then the two-temperature Euler system is considered as the basic model for quasi-neutral plasmas, presented in this framework with thermal conduction and radiative coupling. The known model of electron magnetohdrodynamics (MHD) is introduced as the fundamental model for all magnetized plasmas. In a conclusion section the analyses of the hyperbolic part for (E2T) and (MHD) systems is given. Chapter 3 contains most of the fundamental concepts for the laser-plasma interaction. The paraxial approximation for the laser propagation is derived from the full Maxwell equations by using a time envelope model and performing the Wenzel-Kramer-Brillouin (WKB) expansion. Here, the geometrical optics approximation and the paraxial approximations are compared. Then the Brillouin instability corresponding to a coupling of the laser waves and an ion acoustic wave are considered that lead to the three-wave coupling system introduced by Kadomtsev. Some crucial mathematical properties of this system are deduced for the better understanding of the structure of three-wave coupling systems. In Chapter 4, the modelling of the electron plasma waves is presented. It is shown how the coupling of these waves with the ion population leads to the system of Zakharov equations and the different approximations that are made for this derivation. Chapter 5 goes back to the laser-plasma interaction by the coupling of the laser waves with the electron plasma waves with the derivation of the Raman instability model. In the case of the fixed-ion assumption it leads to a three-wave coupling system with the same structure as the system of the Brillouin instability. The second part of Chapter 5 deals with the modelling of the interaction of an ultra-intense laser pulse and a plasma, that leads to the Euler-Maxwell system, for which some mathematical properties of this system are given. The first part of Chapter 6 contains the modelling of hot plasmas with different species of ions in quasi-neutrally approximation with the aim to derive the equation system describing the averaged ion fluid and its coupling with the electron temperature equation. The second part of Chapter 6 is devoted to models of weakly ionized plasmas.
scientific article; zbMATH DE number 6243678
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