Hostname: page-component-cd9895bd7-jkksz Total loading time: 0 Render date: 2024-12-26T03:32:22.900Z Has data issue: false hasContentIssue false
  • English
  • Français

A bicharacteristic formulation of the ideal MHD equations

Published online by Cambridge University Press:  13 April 2010

HARI SHANKER GUPTA  and
PHOOLAN PRASAD
HARI SHANKER GUPTA
Affiliation:
Department of Mathematics, Indian Institute of Science, Bangalore 560 012, India (prasad@math.iisc.ernet.in)
PHOOLAN PRASAD
Affiliation:
Department of Mathematics, Indian Institute of Science, Bangalore 560 012, India (prasad@math.iisc.ernet.in)
Get access Rights & Permissions [Opens in a new window]

Abstract

On a characteristic surface Ω of a hyperbolic system of first-order equations in multi-dimensions (x, t), there exits a compatibility condition which is in the form of a transport equation along a bicharacteristic on Ω. This result can be interpreted also as a transport equation along rays of the wavefront Ωt in x-space associated with Ω. For a system of quasi-linear equations, the ray equations (which has two distinct parts) and the transport equation form a coupled system of underdetermined equations. As an example of this bicharacteristic formulation, we consider two-dimensional unsteady flow of an ideal magnetohydrodynamics gas with a plane aligned magnetic field. For any mode of propagation in this two-dimensional flow, there are three ray equations: two for the spatial coordinates x and y and one for the ray diffraction. In spite of little longer calculations, the final four equations (three ray equations and one transport equation) for the fast magneto-acoustic wave are simple and elegant and cannot be derived in these simple forms by use of a computer program like REDUCE.

Type
Papers
Information
Journal of Plasma Physics , Volume 77 , Issue 2 , April 2011 , pp. 169 - 191
Copyright
Copyright © Cambridge University Press 2010

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Arun, K. R., Kraft, M., Lukáčová-Medvid'ová, M. and Prasad, P. 2009 Finite volume evolution Galerkin method for hyperbolic conservation laws with spatially varying flux functions. J. Comput. Phys. 228, 565–590.CrossRefGoogle Scholar
Biskamp, D. 1997 Nonlinear Magnetohydrodynamics. Cambridge, U.K.: Cambridge University Press.Google Scholar
Butler, D. S. 1960 The numerical solution of hyperbolic systems of partial differential equations in three independent variables. Proc. R. Soc. 255A, 233–252.Google Scholar
Choquet-Bruhat, Y. 1969 Ondes asymptotiques et approchées pour des systèmes d'èquations aux derivées partielles nonlinéaires. J. Math. Pure Appl. 48, 117–158.Google Scholar
Courant, R. and Friedrichs, K. O. 1948 Supersonic Flow and Shock Waves. New York: Interscience Publishers.Google Scholar
Courant, R. and Hilbert, D. 1962 Methods of Mathematical Physics, Vol II: Partial Differential Equations. New York: John Wiley & Sons.Google Scholar
Courant, R. and Lax, P. D. 1949 On nonlinear partial differential equations with two independent variables. Commun. Pure Appl. Math. 2, 255–273.CrossRefGoogle Scholar
De Sterck, H. 1999 Numerical simulation and analysis of magnetically dominated MHD bow shock flows with applications in space physics, PhD thesis, Katholieke Universiteit Leuven, Belgium, and National Center for Atmospheric Research, Boulder, CO.Google Scholar
De Sterck, H., Low, B. C. and Poedts, S. 1998 Complex magnetohydrodynamic bow shock topology in field-aligned low-β flow around a perfectly conducting cylinder. Phys. Plasmas 5 (11), 4015–4027.CrossRefGoogle Scholar
De Sterck, H., Low, B. C. and Poedts, S. 1999 Characteristic analysis of a complex two-dimensional magnetohydrodynamic bow shock flow with steady compound shocks. Phys. Plasmas 6 (3), 954–969.CrossRefGoogle Scholar
De Sterck, H. and Poedts, S. 1999a Field-aligned magnetohydrodynamic bow shock flows in the switch-on regime. Parameter study of the flow around a cylinder and results for the axi-symmetrical flow over a sphere. Astron. Astrophys. 343, 641–649.Google Scholar
De Sterck, H. and Poedts, S. 1999b Stationary slow shocks in the magnetosheath for solar wind conditions with β < 2/γ: three-dimensional MHD simulations. J. Geophys. Res., 22, 22401–22406.CrossRefGoogle Scholar
Godunov, S. K. 1972 Symmetrical form of the magnetohydrodynamic equations. Computer centre of the Siberian branch of USSR Academy of Sciences, 3, 26–34.Google Scholar
Hu, Y. K. and Wu, S. T. 1984 A full-implicit-continuous-Eulerian (FICE) scheme for multidimensional transient magnetohydrodynamic (MHD) flows. J. Comput. Phys., 55, 33–64.CrossRefGoogle Scholar
Janhunen, P. 2000 A positive conservative method for magnetohydrodynamics based on HLL and Roe methods. J. Comput. Phys. 160, 649–661.CrossRefGoogle Scholar
Jeffrey, A. 1966 Magnetohydrodynamics. Edinburgh: Oliver & Boyd.Google Scholar
Kogan, M. N. 1960 Plane flows of an ideal gas with infinite electrical conductivity, in a magnetic field not parallel to the flow velocity. PMM J. Appl. Math. Mech., 24 129–143.CrossRefGoogle Scholar
Kulikovskiy, A. G. and Lyubimov, G. A. 1965 Magnetohydrodynamics. Reading, MA: Addison-Wesley.Google Scholar
Lukáčová-Medvid'ová, M., Morton, K. W. and Warnecke, G. 2000 Evolution Galerkin methods for hyperbolic systems in two space-dimensions. Math. Comput. 69, 1355–1384.CrossRefGoogle Scholar
Morrison, P. J. and Greene, J. M. 1980 Noncanonical Hamiltonian density formulation of hydrodynamics and ideal magneto hydrodynamics. Phys. Rev. Lett. 45, 790–794. Erratum (1982), Phys. Rev. Lett. 48, 569.CrossRefGoogle Scholar
Nakagawa, Y. 1981 Evolution of magnetic field and atmospheric responses. II. Formulation of proper boundary equations. Astrophys. J. 247, 719–733.CrossRefGoogle Scholar
Powell, K. G., Roe, P. L., Linde, T. J., Gombosi, T. I. and De Zeeuw, D. L. 1999. A solution-adaptive upwind scheme for ideal magnetohydrodynamics. J. Comput. Phys. 154, 284–309.CrossRefGoogle Scholar
Prasad, P. 1993, On the Lemma on bicharacteristics, Appendix in “A nonlinear ray theory”, Berichte der Arbeitsgruppe Technomathematik, Universitaet Kaiserslautern, Nr. 101.Google Scholar
Prasad, P. 2000, An asymptotic derivation of weekly nonlinear ray theory, Proc. Indian Acad. Sci. (Math. Sci.), 110, 431–447.CrossRefGoogle Scholar
Prasad, P. 2001 Nonlinear Hyperbolic Waves in Multi-dimensions. Monograph and Surveys in Pure and Applied Mathematics, 121, New York: Chapman and Hall/CRC.Google Scholar
Prasad, P. 2007 Ray theories for hyperbolic waves, kinematical conservation laws (KCL) and applications. Indian J. Pure Appl. Math. 38, 467–490.Google Scholar
Reddy, A. S., Tikekar, V. G. and Prasad, P. 1982 Numerical solution of hyperbolic equations by method of bicharacteristics. J. Math. Phys. Sci. 16, 575–603.Google Scholar
Sears, W. R. 1960 Some remarks about flow past bodies. Rev. Mod. Phys., 32, 701–705.CrossRefGoogle Scholar
Shapiro, A. H. 1954 The Dynamics and Thermodynamics of Compressible Fluid Flows. Vol. I, New York: Ronald Press Company.Google Scholar
Webb, G. M. and Zank, G. P. 2007 Fluid relabelling symmetries, Lie point symmetries and the Lagrangian map in magnetohydrodynamics and gas dynamics. J. Phys. A: Math. Theor. 40, 545–579.CrossRefGoogle Scholar
Webb, G. M., Zank, G. P., Kaghashvili, E. Kh. and Ratkiewicz, R. E. 2005a Magnetohydrodynamic waves in non-uniform flows. I. A variational approach. J. Plasma Phys. 71, 785–809.CrossRefGoogle Scholar
Webb, G. M., Zank, G. P., Kaghashvili, E. Kh. and Ratkiewicz, R. E. 2005b Magnetohydrodynamic waves in non-uniform flows. II. Stress-energy tensors, conservation laws and Lie symmetries. J. Plasma Phys. 71, 811–857.CrossRefGoogle Scholar
Whitham, G. B. 1974. Linear and Nonlinear Waves. New York: John Wiley and Sons.Google Scholar