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On the theory of magneto-sound double simple waves

Published online by Cambridge University Press:  01 August 2008

DAVY D. TSKHAKAYA  and
HOMAYOON ESHRAGHI
DAVY D. TSKHAKAYA
Affiliation:
Institute for Theoretical Physics, University of Innsbruck, A-6020 Innsbruck, Austria ([email protected])
HOMAYOON ESHRAGHI
Affiliation:
Physics Department, Iran University of Science and Technology (IUST), Narmak, Tehran, Iran Institute for Studies in Theoretical Physics and Mathematics (IPM), Tehran PO Box 19395-5531, Iran ([email protected])
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Abstract

A two-dimensional double simple wave solution is given for both weakly and highly magnetized non-relativistic plasmas moving across the magnetic field. The dependence of the density and the magnetic field on the two independent phases, namely, components of the fluid velocity, is derived. It is shown that initial spatial distributions must satisfy a definite equation whose solution determines a special category for initial conditions. The time of blow up for any fixed value of the pair phase is found. A large general class of solutions for initial distributions is obtained. For any chosen initial distribution, the physical plane of flow at any instant of time splits into two regions, one forbidden and the other permitted. These regions are obtained numerically at a typical time for a special initial distribution. For this double wave solution, differential equations for streamlines and fluid trajectories are derived. Only for the simplest cases can the corresponding curves be completely integrated and these are given in this paper. The results are qualitatively similar to the one-dimensional case derived by Stenflo and Shukla.

Type
Papers
Information
Journal of Plasma Physics , Volume 74 , Issue 4 , August 2008 , pp. 455 - 471
Copyright
Copyright © Cambridge University Press 2007

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References

[1]Landau, L. D. and Lifshitz, E. M. 1984 Electrodynamics of Continuous Media, 2nd edn.Oxford: Pergamon.Google Scholar
[2]Akhiezer, A. I., Akhiezer, I. A., Polovin, R. V., Sitenko, A. G. and Stepanov, 1975 Plasma Electrodynamics. Oxford: Pergamon.Google Scholar
[3]Stenflo, L., Shvartsburg, A. B. and Weiland, J. 1997 On shock wave formation in a magnetized plasma. Phys. Lett. A 225, 113–116.Google Scholar
[4]Shukla, P. K., Eliasson, B., Marklund, M. and Bingham, R. 2004 Nonlinear model for magnetosonic shocklets in plasmas. Phys. Plasmas 11, 2311–2313.Google Scholar
[5]Sagdeev, R. Z. 1960 Collective processes and shock waves in a rarefied plasma. Reviews of Plasma Physics, Vol. 4 (ed. Leontovich, M. A.). New York: Consultants Bureau.Google Scholar
[6]Jeffrey, A. and Taniuti, T. 1964 Nonlinear Wave Propagation with Application to Physics and Magnetohydrodynamics. New York: Academic Press.Google Scholar
[7]Cabannes, H. 1970 Theoretical Magnetohydrodynamics. New York: Academic Press.Google Scholar
[8]Zajaczkowski, W. 1979 Riemann invariants interaction in MHD double waves. Demonstratio Math. 12, 543–563.Google Scholar
[9]Zajaczkowski, W. 1980 Demonstratio Math. 13, 317–333.Google Scholar
[10]Webb, G. M., Ratkiewicz, R., Brio, M. and Zank, G. P. 1995 Solar Wind 8 (ed. Winterhalter, D., Gosling, J. T., Habbal, S. R., Kurth, W. S. and Neugebauer, M.), (AIP Conference Proceedings, 382). New York: American Institute of Physics, pp. 335–338.Google Scholar
[11]Grundland, A. M. and Lalague, L. 1995 Lie subgroups of symmetry groups of fluid dynamics and magnetohydrodynamics equations. Can. J. Phys. 73, 463–477.CrossRefGoogle Scholar
[12]Courant, R. and Friedrichs, K. O. 1976 Supersonic Flow and Shock Waves. New York: Springer.Google Scholar
[13]Landau, L. D. and Lifshitz, E. M. 1987 Fluid Mechanics, 2nd edn.Oxford: Pergamon.Google Scholar
[14]Chorin, A. J. and Marsden, J. E. 1979 A Mathematical Introduction to Fluid Mechanics. New York: Springer.Google Scholar
[15]Von Mises, R. 1958 Mathematical Theory of Compressible Flow. New York: Academic Press.Google Scholar
[16]Boillat, G. 1970 Simple waves in N-dimensional propagation. J. Math. Phys. 11, 1482–1483.Google Scholar
[17]Webb, G. M., Ratkiewicz, R., Brio, M. and Zank, G. P. 1998 Multidimensional simple waves in gas dynamics. J. Plasma Phys. 59, 417–460.Google Scholar
[18]Burnat, M. 1965 Cauchy's problem for the compressible flow of simple wave type (Compressible flow of simple wave type in gas dynamics, solving Cauchy problem). Proc. 5th Symp. on Fluid Dynamics Transactions, Jablonna, Poland, 26 August–2 September 1961 edited by Fiszdon, W., Pergamon Press, pp. 31–57.Google Scholar
[19]Burnat, M. 1970 The method of characteristics and Riemann invariants for multi-dimensional hyperbolic systems. Siberian Math. J. 11, 210–232.Google Scholar
[20]Burnat, M. 1971 Geometrical methods in fluid mechanics. Fluid Dyn. Trans. 6, 115–186.Google Scholar
[21]Peradzynski, Z. 1971 Riemann invariants for the nonplanar k-waves. Bull. Acad. Polon. Sci., Ser. Sci. Techn. 19, 717–724.Google Scholar
[22]Yanenko, N. N. 1956 Dokl. Akad. Nauk SSSR 109, 44–47 (in Russian).Google Scholar
[23]Sidorov, A. F. and Yanenko, N. N. 1958 On the problem of nonstationary plane fluxes of polytropic gases with straight line characteristics. Dokl. Akad. Nauk SSSR 123, 832–834 (in Russian).Google Scholar
[24]Komarovskii, L. V. 1960 An accurate solution of the three-dimensional equations for a nonsteady gas-flow of the double wave type. Sov. Phys. Dokl. 135, 1163–1165.Google Scholar
[25]Grundland, A. M. 1974 Riemann invariants for nonhomogeneous systems of quasilinear partial differential equations. Conditions of involution. Bull. Acad. Polon. Sci., Ser. Sci. Techn. 4, 177–185.Google Scholar
[26]Grundland, A. M. 1980 Nonlinear superpositions of simple waves in nonhomogeneous systems. Polish J. Nucl. Res. 7, 1–46.Google Scholar
[27]Grundland, A. M. and Lalague, L. 1996 Invariant and partially invariant solutions of the equations describing nonstationary and isentropic flow for an ideal and compressible fluid in (3+1)dimensions. J. Phys. A: Math. Gen. 29, 1723–1739.CrossRefGoogle Scholar
[28]Tskhakaya, D. D. and Eshraghi, H. 2002 Two-dimensional double simple waves in a pair plasma at relativistic temperatures. Phys. Plasmas 9, 2518–2525.Google Scholar