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Erschienen in:
Introduction: Some Plasma Characteristic Quantities Kapitel lesen Erstes Kapitel lesen

2014 | OriginalPaper | Buchkapitel

2. Quasi-Neutrality and Magneto-Hydrodynamics

verfasst von : Rémi Sentis

Erschienen in: Mathematical Models and Methods for Plasma Physics, Volume 1

Verlag: Springer International Publishing

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Abstract

In this chapter, we justify firstly the massless-electron approximation from the general ion–electron electrodynamic model. Secondly, we present the quasi-neutrality approximation, which is the heart of most of the fluid models presented in this book; this approximation is rigorously proved by an asymptotic analysis where a small parameter related to the Debye length goes to zero. We then present the two-temperature Euler system which is the basic model for quasi-neutral plasmas; in this framework we deal also with thermal conduction and radiative coupling. Lastly, we introduce the well-known model called electron magneto-hydrodynamics (MHD) which is the fundamental model for all magnetized plasmas. We give some details about the related boundary conditions.Some crucial mathematical properties related to the “ideal part” of the previous models are displayed at the end of this chapter.

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Vorheriges Kapitel Introduction: Some Plasma Characteristic Quantities
Nächstes Kapitel Laser Propagation: Coupling with Ion Acoustic Waves
Fußnoten
1
A sequence un converges weakly to u, if \(\int u_{n}v \rightarrow \int uv\) for any v in L 2.
 
2
Let S be a mapping from a convex subset of a Banach space into itself, if S is continuous and compact with respect to the Banach topology, then S has a fixed point.
 
3
See result 1 in the Appendix.
 
4
As a matter of fact, the electric resistivity is defined as μ 0 χ; it is also denoted by η in some physics textbooks.
 
5
For a one-dimensional space variable, a system of the form \(\partial _{t}\mathbf{Y} + \frac{\partial } {\partial x}(\mathbf{F}(\mathbf{Y})) = 0\) is called hyperbolic if all the eigenvalues of the Jacobian matrix \(\partial \mathbf{F}/\partial \mathbf{Y}\) are real and there exists a complete set of eigenvectors. For a three-dimensional space variable, a system \(\partial _{t}\mathbf{Y} + \Sigma _{j} \frac{\partial } {\partial x_{j}}(\mathbb{F}_{j}(\mathbf{Y})) = 0\) is called hyperbolic if one has the analogous property for the Jacobian matrix \(\frac{\partial } {\partial \mathbf{Y}}(\omega _{1}\mathbb{F}_{1} +\omega _{2}\mathbb{F}_{2} +\omega _{3}\mathbb{F}_{3})\) for all coefficients \(\omega _{1},\omega _{2},\omega _{3}\).
 
6
The Aubin–Lions lemma says that if \(\theta \in {L}^{2}(0,t,{H}^{1})\) and \(\partial _{t}\theta \in {L}^{2}(0,t,{H}^{-1}),\) then θ ∈ C(0, t, L 2). (H −1 is the dual space of H 1).
 
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Metadaten
Titel
Quasi-Neutrality and Magneto-Hydrodynamics
verfasst von
Rémi Sentis
Copyright-Jahr
2014
Buch
Mathematical Models and Methods for Plasma Physics, Volume 1

Print ISBN: 978-3-319-03803-2

Electronic ISBN: 978-3-319-03804-9

Copyright-Jahr: 2014

DOI
https://doi.org/10.1007/978-3-319-03804-9_2