Summary
This paper is concerned with the relativistic phenomenological theory of anisotropic magnetohydrodynamics. An anisotropic fluid scheme is defined and studied. The main system of anisotropic magnetohydroldynamics is deduced. This system may describe a collisionless anisotropic plasma embedded in a strong magnetic field. The main system is shown to yield to three types of waves as in isotropic (perfect) magnetohydrodynamics: the entropic waves, the magnetosonic waves and the Alfven waves. For the rays associated respectively to the magnetosonic and Alfven waves the fundamental property concerning the propagation of infinitesimal discontinuities of variables is established. The conditions under which the velocities of propagation of magnetosonic and Alfven waves are real are derived: these conditions imply as in the classical theory the absence of fire hose and mirror instabilities in the fluid. The study of wave cones allows, on the one hand to point out some particularities of the propagation of waves in anisotropic magnetohydrodynamics, and on the other hand to clear up the hyperbolicity character of differential operators associated to various waves.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
Bibliographie
G. F. Chew —M. L. Goldberger — F. E. Low, Proc. Roy. Soc., A236 (1956), pp. 112–118.
R. V. Polovin —L. T. Tsintsadze, Ukraïnski fizitchni Zhurnal,4 (1959), pp. 30–38.
E. Lüst, in Summer Course in Plasma Physics, 1960, p. 201, ed. by C. F. Wandel Danish A.E.C.
T. F. Volkov,Problems in Plasma Theory, 1964, Vol. IV, ed. parM. A. Leontovich (Trad. par Ronson Acad. Press, London, 1966).
B. A. Shrauner, J. Plasma Phys.,1 (1967), pp. 361–381.
Y. Kato —M. Tajiri —T. Taniuti, J. Phys. Soc. Japan,21 (1967), pp. 765–777.
V. Baranow —M. D. Kartzlev,izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i gaza,6 (1970), pp. 3–10.
M. D. Kartalev,Soviet phys. Doklady,17 (1973), pp. 744–746.
A. Lichnerowicz,Théories relativistes de la gravitation et de l'électromagnétisme, Paris, Masson, 1955.
A. Lichnerowicz,Relativistic Hydrodynamics and Magnetohydrodynamics, New York, Benjamin, 1967.
Pham Mau Quan, Thèse, Paris, 1954 (Bologna, Cooperativa tipografica Azzoguidi, 1955).
A. H. Taub,Relativisitic Hydrodynamics, Arch. Rat. Mech. Anal.,3 (1959), p. 312; et in Lectures in Applied Mathematics,8 (1967).
M. Cissoko, Ann. Inst. Henri Poincaré, vol. 22 (1975), p. 1–27.
A. Lichnerowicz, Ann. Inst. Henri Poincaré,7 (1967), pp. 271–302.
A. Lichnerowicz, Comm. Math. Phys.,12 (1969), pp. 145–174.
Y. Choquet-Bruhat, Astr. Acta,6 (1960), pp. 354–365.
J. Leray,Hyperbolic differential equations, Institute for advanced Study, Princeton, 1952, multigraphié.
A. Lichnerowicz, Comm. Math. Phys.,1 (1966), pp. 328–373.
Author information
Authors and Affiliations
Additional information
Entrata in Redazione il 23 aprile 1975.
Rights and permissions
About this article
Cite this article
Cissoko, M. Sur la magnétohydrodynamique anisotrope relativiste. Annali di Matematica 111, 331–368 (1976). https://doi.org/10.1007/BF02411823
Issue Date:
DOI: https://doi.org/10.1007/BF02411823