Abstract
A single-fluid version of the equations of two-fluid magnetohydrodynamics is obtained. This paper is concerned with the following topics: derivation of the energy conservation law; proof of degenerate ellipticity of the the generalized Ampère’s law; passage to the limit to the equations of the classical magnetohydrodynamics; derivation of plasmastatics equations that generalize the Grad–Shafranov equations and that belong to the class of equations of mixed type: elliptic for dense plasma and hyperbolic for rarefied plasma; analytical and numerical analysis of their solutions for the θ-pinch and z-pinch.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
L. D. Landau and E. M. Lifshitz, Theoretical Physics, Vol. 8: Electrodynamics of Continuous Media [in Russian], Nauka, Fizmatlit, Moscow (1982).
A. G. Kulikovskii and G. A. Lyubimov, Magnetohydrodynamics [in Russian], Logos, Moscow (2005).
A. I. Morozov, Introduction to Plasmadynamics [in Russian], Fizmatlit, Moscow (2006).
K. V. Brushlinskii and A. I. Morozov, in: M. A. Leontovich, ed., Problems in the Theory of Plasma, Vol. 8 [in Russian], Atomizdat, Moscow (1974), p. 88.
V. S. Imshennik and N. A. Bobrova, Dynamics of Collisional Plasma [in Russian], Energoatomizdat, Moscow (1997).
S. I. Braginskii, in: M. A. Leontovich, ed., Problems in the Theory of Plasma, Vol. 1 [in Russian], Atomizdat, Moscow (1963), p. 183.
L. S. Solov’ev, Selected Works in Two Volumes, Vol. 2 [in Russian], Nauka, Moscow (2001), pp. 7–34.
M. B. Gavrikov, Basic Equations of Two-Fluid Electromagnetic Hydrodynamics, Pt. 1 [in Russian], Preprint No. 59, Keldysh Institute of Applied Mathematics, RAS (2006).
L. A. Arzimovich, Controllable Thermonuclear Fusion [in Russian], Fizmatgiz, Moscow (1961).
V. D. Shafranov, Zh. Eksp. Teor. Fiz., 33, No. 3 (9), 710 (1957).
V. D. Shafranov, in: M. A. Leontovich, ed., Reviews of Plasma Physics, Vol. 2 [in Russian], Atomizdat, Moscow (1963), p. 92.
A. V. Bitsadze, Equations of Mathematical Physics [in Russian], Nauka, Moscow (1976), pp. 9–15.
I. A. Shishmarev, Introduction to the Theory of Elliptic Equations [in Russian], Izd. Mosk. Univ., Moscow (1979), pp. 6–12.
V. V. Stepanov, A Course in Differential Equations [in Russian], GITTL, Moscow–Leningrad (1950).
B. B. Kadomtsev, Collective Phenomena in Plasma [in Russian], Nauka, Moscow (1988).
A. I. Morozov and V. V. Savelyev, “On the magnetic trap Galatea with plasma immersed conductors,” Usp. Fiz. Nauk, 41, No. 11, 1049–1089 (1998).
K. V. Brushlinskii and V. V. Savelyev, Mat. Model., 11, No. 5, 5 (1999).
A. I. Morozov and V. V. Savelyev, Fiz. Plazm., 19, No. 8, 977 (1993).
V. V. Savelyev, Fiz. Plazm., 21, No. 3, 216 (1995).
A. I. Morozov and V. V. Savelyev, Fiz. Plazm., 21, No. 3, 209 (1995).
N. N. Kalitkin, Numerical Methods [in Russian], Nauka, Fizmatlit, Moscow (1978).
G. I. Marchuk and V. I. Agoshkov, Introduction to Projection-Grid Methods [in Russian], Nauka, Fizmatlit, Moscow (1981).
V. V. Shaidurov, Multigrid Finite Element Methods [in Russian], Nauka, Fizmatlit, Moscow (1989).
K. Fletcher, Numerical Methods Based on the Galerkin Method [Russian translation], Mir, Moscow (1988).
Author information
Authors and Affiliations
Corresponding author
Additional information
Translated from Trudy Seminara imeni I. G. Petrovskogo, No. 27, Part I, pp. 3–66, 2009.
Rights and permissions
About this article
Cite this article
Gavrikov, M.B., Savelyev, V.V. Equilibrium configurations of plasma in the approximation of two-fluid magnetohydrodynamics with electron inertia taken into account. J Math Sci 163, 1–40 (2009). https://doi.org/10.1007/s10958-009-9662-1
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10958-009-9662-1