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Quasi-Neutrality and Magneto-Hydrodynamics

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Mathematical Models and Methods for Plasma Physics, Volume 1

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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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Notes

  1. 1.

    A sequence un converges weakly to u, if \(\int u_{n}v \rightarrow \int uv\) for any v in L 2.

  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. 3.

    See result 1 in the Appendix.

  4. 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. 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. 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).

Bibliography

  1. G. Ali, L. Chen, A. Juengel, Y.J. Peng, The zero-electron-mass limit in the hydrodynamic model for plasmas. Nonlin. Anal. T.M.A. 72, 4415–4427 (2010)

    Google Scholar 

  2. A. Ambroso, F. Mehats, P.-A. Raviart, On singular perturbation problems for the nonlinear Poisson equation. Asymptotic Anal. 25, 39–91 (2001)

    MATH  MathSciNet  Google Scholar 

  3. C. Bardos, F.Golse, B. Perthame, R. Sentis, The nonaccretive radiative transfer equation; global existence and Rosseland approximation. J. Funct. Anal. 77, 434–460 (1988)

    Article  MATH  MathSciNet  Google Scholar 

  4. J.P. Berenger, A Perfectly Matched Layer for the absorption of electromagnetic waves. J. Comp. Phys. 114, 185 (1994)

    Article  MATH  MathSciNet  Google Scholar 

  5. C. Berthon et al., Mathematical Models and Numerical Methods for Radiative Transfer (Panoramas et synthèses SMF, Paris, 2009)

    MATH  Google Scholar 

  6. F. Bethuel, H. Brezis, F. Helein, Asymptotics for the minimization of Ginzburg–Landau. Calc. Var. 1, 123–148 (1993)

    Article  MATH  MathSciNet  Google Scholar 

  7. R.L. Bowers, J.R. Wilson, Numerical Modeling in Applied Physics (Jones-Bartlett, Boston, 1991)

    MATH  Google Scholar 

  8. H. Brezis, F. Golse, R. Sentis, Analyse asymptotique de l’équation de Poisson couplée à la relation de Boltzmann. Note C. R. Acad. Sci. Ser. I 321, 953–959 (1995)

    MATH  MathSciNet  Google Scholar 

  9. J. Castor, Radiation Hydrodynamics (Cambridge University Press, Cambridge, 2007)

    Google Scholar 

  10. F.F. Chen, Introduction to Plasma Physics (Academic, New York, 1974)

    Google Scholar 

  11. F.F. Chen, J.P. Chang, Lectures Notes on Principle of Plasma Processing (Kluver-Plenum, New York 2003)

    Book  Google Scholar 

  12. S. Cordier, E. Grenier, Quasi-neutral limit of a Euler–Poisson system arising from plasma physics. Commun. Part. Differ. Equ. 25, 1099–1113 (2000)

    Article  MATH  MathSciNet  Google Scholar 

  13. S. Cordier, P. Degond, P. Markowich, C. Schmeiser, Travelling waves analysis and jump relations for Euler–Poisson model in the quasineutral limit. Asymptotic Anal. 11, 209–240 (1995)

    MATH  MathSciNet  Google Scholar 

  14. P. Crispel, P. Degond, M.-H. Vignal, Quasi-neutral fluid models for current-carrying plasmas. J. Comput. Phys. 205, 408–438 (2005)

    Article  MATH  MathSciNet  Google Scholar 

  15. P. Crispel, P. Degond, M.-H. Vignal, An asymptotic preserving scheme for the two-fluid Euler-Poisson model in the quasineutral limit. J. Comput. Phys. 223, 208–234 (2007)

    Article  MATH  MathSciNet  Google Scholar 

  16. G. DalMaso, P. LeFloch, F. Murat, Definition and weak stability of nonconservative product. J. Math. Pures et Appl. 74, 483–548 (1995)

    MathSciNet  Google Scholar 

  17. A. Decoster, Fluid equations and transport coefficient of plasmas, in Modelling of Collisions, ed. by P.A. Raviart (Elsevier/North-Holland, Paris, 1997)

    Google Scholar 

  18. P. Degond, J.-G. Liu, M.-H. Vignal, Analysis of an asymptotic preserving scheme for the two-fluid Euler–Poisson model in the quasineutral limit. SIAM J. Numer. Anal. 46, 1298 (2008)

    Article  MATH  MathSciNet  Google Scholar 

  19. B. Desprès, Lagrangian systems of conservation law. Numer. Math. 89, 99 (2001)

    Article  MATH  MathSciNet  Google Scholar 

  20. B. Desprès, Lois de conservation, Méthodes Numériques (Springer, Berlin, 2008)

    Google Scholar 

  21. B. Despres, R. Sart, Reduced resistive MHD in Tokamak ESAIM: Math. Model. Numer. Anal. 46, 1021–1105 (2012)

    Article  MathSciNet  Google Scholar 

  22. E.M. Epperlein, R.W. Short, A pratical nonlocal model for electron heat transport in laser plasmas. Phys. Fluids B 3, 3092–3098 (1991)

    Article  Google Scholar 

  23. E. Godlevsky, P.A. Raviart, Numerical Approximation of Hyperbolic Systems (Springer, Berlin, 1996)

    Google Scholar 

  24. T. Goudon, A. Jungel et al., Zero-mass electron limits. Appl. Math. Lett. 12, 75 (1999)

    Article  MATH  MathSciNet  Google Scholar 

  25. S.Y. Ha, M. Slemrod, Global existence of plasma ion-sheaths and their dynamics. Commun. Math. Phys. 238, 149–186 (2003)

    MATH  MathSciNet  Google Scholar 

  26. G.J.M. Hagelaar, How to normalize Maxwell–Boltzmann relation. J. Comput. Phys. 227, 871–876 (2007)

    Article  MATH  MathSciNet  Google Scholar 

  27. A. Jungel, Y.J. Peng, A hierarchy of hydrodynamics models for plasmas, zero-electron-mass limits. Ann. Inst. Henri Poincare (C) Non Lin. Anal. 17, 83–118 (2000)

    Article  MathSciNet  Google Scholar 

  28. E.W. Larsen, J. Morel, Asymptotic solutions of numerical transport problems in optically thick diffusive regimes II. J. Comput. Phys. 83, 212 (1989)

    Article  MATH  MathSciNet  Google Scholar 

  29. E.W. Larsen, A. Kumar, J. Morel, Properties of the implicitly time-differenced equations of thermal radiation transport. J. Comput. Phys. 238, 82–96 (2013)

    Article  MathSciNet  Google Scholar 

  30. Y.T. Lee, R.M. More, An electron conductivity model for dense plasmas. Phys. Fluids 27, 1273–1286 (1984)

    Article  MATH  Google Scholar 

  31. D. Mihalas, B.W. Mihalas, Foundations of Radiation Hydrodynamics (Oxford University Press, Oxford, 1984)

    MATH  Google Scholar 

  32. F. Nataf, A new approach to Perfectly Matched Layers for Euler equations. J. Comp. Phys. 214, 757–772 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  33. M. Parisot, T. Goudon, On the Spitzer–Harm regime and nonlocal approximation: modeling, analysis, and numerical simulations…SIAM Multiscale Model. Sim. 9, 568–600 (2011)

    MATH  MathSciNet  Google Scholar 

  34. H. Poincaré, Les fonctions fuchsiennes et une équation avec Laplacien. J. Math. Pures Appl. (5e serie) 4, 137–230 (1898)

    Google Scholar 

  35. G.C. Pomraning, Equations of Radiation Hydrodynamics (Pergamon, Oxford 1973)

    Google Scholar 

  36. G. Schurtz, P. Nicolaï, M. Busquet, A nonlocal electron conduction model. Phys. Plasmas 7, 4238–4249 (2000)

    Article  Google Scholar 

  37. A. Sitenko, V. Malnev, Plasma Physics Theory (Chapman-Hall, London, 1995)

    MATH  Google Scholar 

  38. M. Slemrod, Shadowing and the plasma-sheath transition layer. J. Nonlin. Sci. 11, 397–414 (2001)

    Article  MATH  MathSciNet  Google Scholar 

  39. M. Slemrod, N. Sternberg, Quasi-neutral limit for Euler-Poisson System. J. Nonlin. Sci. 11, 193–209 (2001)

    Article  MATH  MathSciNet  Google Scholar 

  40. L. Spitzer, Physics of Fully Ionized Gases (Academic, New York, 1956)

    MATH  Google Scholar 

  41. J.A. Stratton, Electromagnetic Theory (McGraw-Hill, New-York, 1941)

    MATH  Google Scholar 

  42. S. Wang, Quasineutral limit of Euler–Poisson System with and without Viscosity. Commun. Part. Differ. Eq. 29, 419–456 (2004)

    Article  MATH  Google Scholar 

  43. T. Zel’dovich, Y. Raizer, Physics of Shock Waves and High-Temperature Hydrodynamic Phenomena, 1st (1963) Russian edn. (Academic, New York, 1967). 2nd edn. (Dover, New York, 2002)

    Google Scholar 

  44. M. Bostan, B. N’Konga, R. Sentis. Mathematical Models and Methods for Plasma Physis. Volume 2, Kinetic Models. (To appear)

    Google Scholar 

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Authors and Affiliations

  1. Commissariat Energie Atomique DAM-Ile-de-France, Paris, France

    Rémi Sentis

  2. Laborartoire Jacques-Louis-Lions, Université Pierre-et-Marie-Curie, Paris, France

    Rémi Sentis

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Sentis, R. (2014). Quasi-Neutrality and Magneto-Hydrodynamics. In: Mathematical Models and Methods for Plasma Physics, Volume 1. Modeling and Simulation in Science, Engineering and Technology. Birkhäuser, Cham. https://doi.org/10.1007/978-3-319-03804-9_2

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