Abstract
The system of the magnetohydrodynamic equations for a heavy fluid has been analyzed in the shallow water approximation. All discontinuous self-similar solutions and all continuous centered self-similar solutions have been found. It has been shown that magnetogravity compression waves are broken with the formation of a magnetogravity shock wave. The initial decay discontinuity problem for the magnetohydrodynamic equations has been solved in the explicit form in the shallow water approximation. The existence of five different configurations implementing the solution of the decay of an arbitrary discontinuity has been demonstrated. The conditions necessary and sufficient for the implementation of each configuration have been found.
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References
P. A. Gilman, Astrophys. J. 544, L79 (2000).
M. Miesch and P. Gilman, Sol. Phys. 220, 287 (2004).
T. V. Zaqarashvili, R. Oliver, and J. L. Ballester, Astrophys. J. 691, L41 (2009).
N. A. Inogamov and R. A. Sunyaev, Astron. Lett. 25(5), 269 (1999).
N. A. Inogamov and R. A. Sunyaev, Astron. Lett. 36(12), 848 (2010).
A. Spitkovsky, Y. Levin, and G. Ushomirsky, Astrophys. J. 566, 1018 (2002).
K. Heng and A. Spitkovsky, Astrophys. J. 703, 1819 (2009).
J. Cho, Philos. Trans. R. Soc. London, Ser. A 366, 4477 (2008).
V. Bojarevics and K. Pericleous, in Proceedings of the Joint 15th Riga and 6th Pamir Conference on Fundamental and Applied MHD Aluminum Reduction Cells, Riga, Latvia, June 27–July 1, 2005 (Riga, 2005), Vol. II, p. 87.
K. V. Karelsky, V. V. Papkov, A. S. Petrosyan, and D. V. Tsygankov, Phys. Lett. A 271, 341 (2000).
K. V. Karelsky and A. S. Petrosyan, Fluid Dyn. Res. 38, 339 (2006).
V. I. Petviashvili and O. A. Pokhotelov, Solitary Waves in Plasmas and in the Atmosphere (Energoatomizdat, Moscow, 1989; Gordon and Breach, Philadelphia, Pennsylvania, United States, 1992).
P. J. Dellar, Phys. Plasmas 9, 1130 (2002).
Ya. B. Zel’dovich and Yu. P. Raizer, Physics of Shock Waves and High-Temperature Hydrodynamic Phenomena (Nauka, Moscow, 1966; Academic, New York, 1966–1967).
L. D. Landau and E. M. Lifshitz, Course of Theoretical Physics, Vol. 8: Electrodynamics of Continuous Media (Nauka, Moscow, 1982; Butterworth-Heinemann, Oxford, 1984).
Plasma Electrodynamics, Ed. by A. I. Akhiezer (Nauka, Moscow, 1974; Pergamon, London, 1975).
H. De Streck, Phys. Plasmas 8, 3293 (2001).
K. V. Karelsky, V. V. Papkov, and A. S. Petrosyan, Phys. Lett. A 271, 349 (2000).
G. Whitham, Linear and Nonlinear Waves (Wiley, New York, 1974; Mir, Moscow, 1977).
K. V. Karelsky, A. S. Petrosyan, and A. G. Slavin, Russ. J. Numer. Anal. Math. Modell. 21(6), 539 (2006).
K. V. Karelsky, A. S. Petrosyan, and A. G. Slavin, Russ. J. Numer. Anal. Math. Modell. 22(6), 543 (2007).
K. V. Karelsky, A. S. Petrosyan, and A. G. Slavin, Russ. J. Numer. Anal. Math. Modell. 24, 229 (2009).
K. V. Karelsky, A. S. Petrosyan, and A. G. Slavin, Mat. Model. 21(6), 41 (2009).
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Original Russian Text © K.V. Karelsky, A.S. Petrosyan, S.V. Tarasevich, 2011, published in Zhurnal Eksperimental’noi i Teoreticheskoi Fiziki, 2011, Vol. 140, No. 3, pp. 606–620.
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Karelsky, K.V., Petrosyan, A.S. & Tarasevich, S.V. Nonlinear dynamics of magnetohydrodynamic flows of a heavy fluid in the shallow water approximation. J. Exp. Theor. Phys. 113, 530–542 (2011). https://doi.org/10.1134/S106377611107003X
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DOI: https://doi.org/10.1134/S106377611107003X