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Linear and nonlinear stability criteria for compressible MHD flows in a gravitational field

Published online by Cambridge University Press:  14 June 2013

S. M. MOAWAD
S. M. MOAWAD*
Affiliation:
Department of Mathematics, Faculty of Science, Beni-Suef University, Beni-Suef, Egypt (smmoawad@hotmail.com)
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Abstract

The equilibrium and stability properties of ideal magnetohydrodynamics (MHD) of compressible flow in a gravitational field with a translational symmetry are investigated. Variational principles for the steady-state equations are formulated. The MHD equilibrium equations are obtained as critical points of a conserved Lyapunov functional. This functional consists of the sum of the total energy, the mass, the circulation along field lines (cross helicity), the momentum, and the magnetic helicity. In the unperturbed case, the equilibrium states satisfy a nonlinear second-order partial differential equation (PDE) associated with hydrodynamic Bernoulli law. The PDE can be an elliptic or a parabolic equation depending on increasing the poloidal flow speed. Linear and nonlinear Lyapunov stability conditions under translational symmetric perturbations are established for the equilibrium states.

Type
Papers
Information
Journal of Plasma Physics , Volume 79 , Issue 5 , October 2013 , pp. 873 - 883
Copyright
Copyright © Cambridge University Press 2013 

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References

Agim, Y. Z. and Tataronis, J. A. 1985 J. Plasma Phys. 34, 337.CrossRefGoogle Scholar
Alfvén, H. 1950 Cosmical Electrodynamics. Oxford: Clarendon Press.Google Scholar
Alfvén, H. and Falthammar, C. G. 1963 Cosmical Electrodynamics: Fundamental Principles, 2nd edn.Oxford: Clarendon Press.Google Scholar
Almaguer, J. A., Hameiri, E., Herrera, J. and Holm, D. D. 1988 Phys. Fluids 31, 1930.CrossRefGoogle Scholar
Andreussi, T., Morrison, P. J. and Pegoraro, F. 2010 Plasma Phys. Control. Fusion 52, 055001.CrossRefGoogle Scholar
Andreussi, T. and Pegoraro, F. 2008 Phys. Plasmas 15, 092108.CrossRefGoogle Scholar
Arnold, V. I. 1966 IZV. Vyssh. Uchebn. Zaved. Mathematika 54, 3; English Translation: 1969 Am. Math. Soc. Transl. 79, 267.Google Scholar
Asakura, N., Fonck, R. J., Jaehnig, K. P., Kaye, S. M., LeBlanc, B. and Ok-abayashi, M. 1993 Nucl. Fusion 33, 1165.CrossRefGoogle Scholar
Bernstein, I. B., Frieman, E. A., Kruskal, M. D. and Kulsrud, R. M. 1958 Proc. R. Soc. London A 244, 17.Google Scholar
Betti, R. and Freidberg, J. P. 2000 Phys. Plasmas 7, 2439.CrossRefGoogle Scholar
Blandford, R. D. and Payne, D. G. 1982 Mon. Not. R. Astron. Soc. 199, 883.CrossRefGoogle Scholar
Bogovalov, S. V. 1994 Mon. Not. R. Astron. Soc. 270, 721.CrossRefGoogle Scholar
Branover, H. and Unger, Y. (eds.) 1993 Metallurgical Technologies, Energy Conversion, and Magnetohydrodynamic Flows, Progress in Astronautics and Aeronautics Series, Vol. 148. Reston, VA: American Institute of Aeronautics and Astronautics.Google Scholar
Chandrasekhar, S. 1956 Proc. Natl. Acad. Sci. 42, 273CrossRefGoogle Scholar
Chandrasekhar, S. 1957 Proc. Natl. Acad. Sci. 43, 24.CrossRefGoogle Scholar
Chandrasekhar, S. and Predergast, K. H. 1956 Proc. Natl. Acad. Sci. 42, 5.CrossRefGoogle Scholar
Chandrasekhar, S. and Woltjer, L. 1958 Proc. Natl. Acad. Sci. 44, 285.CrossRefGoogle Scholar
Cheviakov, A. F. and Bogoyavlenskij, O. I. 2004 J. Phys. A: Math. Gen. 37, 7593.CrossRefGoogle Scholar
Contopoulos, J. and Lovelace, R. V. E. 1994 Astrophys. J. 425, 139.CrossRefGoogle Scholar
Cowling, T. G. 1957 Magnetohydrodynamics. New York: Interscience.Google Scholar
Del Zanna, L. and Chiuderi, C. 1996 Astron. Astrophys. 310, 341.Google Scholar
Edenstrasser, J. W. 1980a J. Plasma Phys. 24, 299.CrossRefGoogle Scholar
Edenstrasser, J. W. 1980b J. Plasma Phys. 24, 515.CrossRefGoogle Scholar
Eriksson, L.-G., Righi, E. and Zastrow, K.-D. 1997 Plasma Phys. Control. Fusion 39, 27.CrossRefGoogle Scholar
Freidberg, J. 1982 Rev. Mod. Phys. 54, 801.CrossRefGoogle Scholar
Frieman, E. and Rotenberg, M. 1960 Rev. Mod. Phys. 32, 898.CrossRefGoogle Scholar
Fung, Y. T. 1984 Phys. Fluids 27, 838.CrossRefGoogle Scholar
Garofalo, A. M., Turnbull, A. D., Strait, E. J., Austin, M. E., Bialek, J., Chu, M. S., Fredrickson, E., La Haye, R. J., Navratil, G. A., Lao, L. L., Lazarus, E. A., Okabayashi, M., Rice, B. W., Sabbagh, S. A., Scoville, J. T., Taylor, T. S. and Walker, M. L. 1999 Phys. Plasmas 6, 1893.CrossRefGoogle Scholar
Grad, H. 1960 Rev. Mod. Phys. 32, 830.CrossRefGoogle Scholar
Grad, H. and Rubin, H. 1958 Proc. Second United Nations Int. Conf. on the Peaceful Uses of Atomic Energy, Vol. 31. Geneva: United Nations, p. 190.Google Scholar
Guazzotto, L., Betti, R., Manickam, J. and Kaye, S. 2004 Phys. Plasmas 11, 604.CrossRefGoogle Scholar
Hameiri, E. 1983 Phys. Rev. A 27, 1259.CrossRefGoogle Scholar
Heinemann, M. and Olbert, S. 1978 J. Geophys. Res. 83, 2457.CrossRefGoogle Scholar
Hofmann, F. 2003 Fundamental Principles of Electromagnetic Flow Measurement, 3rd edn.Duisburg: KROHNE Messtechnik GmbH.Google Scholar
Holm, D. D., Marsden, J. E., Ratiu, T. and Weinstein, A. 1985 Phys. Rep. 123, 1.CrossRefGoogle Scholar
Hughes, W. F. and Young, F. J. 1966 The Electromagnetodynamics of Fluids. New York: John Wiley.Google Scholar
Khater, A. H. and Moawad, S. M. 2003 Plasma Phys. Control. Fusion 45, 265.CrossRefGoogle Scholar
Khater, A. H. and Moawad, S. M. 2009a Phys. Plasmas 16, 052504.CrossRefGoogle Scholar
Khater, A. H. and Moawad, S. M. 2009b Phys. Plasmas 16, 122506.CrossRefGoogle Scholar
Kruskal, M. D. and Kulsrud, R. M. 1958 Phys. Fluids 1, 265.CrossRefGoogle Scholar
Lovelace, R. V. E., Mehanian, C., Mobarry, C. M. and Sulkanen, M. E. 1986 Astrophys. J. 62 (Suppl.), 1.CrossRefGoogle Scholar
Lüst, R. and Schlüter, A. 1957 Z. Naturforschung 12, 850.CrossRefGoogle Scholar
McClements, K. G. and Hole, M. J. 2010 Phys. Plasmas 17, 082509.CrossRefGoogle Scholar
McClements, K. G. and Thyagaraja, A. 2001 Mon. Not. R. Astron. Soc. 323, 733.CrossRefGoogle Scholar
Mitchner, M. and Kruger, C. H. 1973 Partially Ionized Gases. New York: John Wiley.Google Scholar
Miyamoto, K. 1976 Plasma Physics for Nuclear Fusion. Cambridge: MIT Press.Google Scholar
Moawad, S. M. 2012 Can. J. Phys. 90, 305.CrossRefGoogle Scholar
Morrison, P. J. 1982 Mathematical Methods in Hydrodynamics and Integrability in Dynamical Systems (AIP Conf. Proc. Vol. 88) eds. Tabor, M. and Treve, Y.). New York: AIP, p. 13.Google Scholar
Morrison, P. J. 1998 Rev. Mod. Phys. 70, 467.CrossRefGoogle Scholar
Morrison, P. J. and Greene, J. M. 1980 Phys. Rev. Lett. 45, 790.CrossRefGoogle Scholar
Morrison, P. J. and Greene, J. M. 1982 Phys. Rev. Lett. 48, 569.CrossRefGoogle Scholar
Morrison, P. J. and Hazeltine, R. D. 1984 Phys. Fluids 27, 886.CrossRefGoogle Scholar
Nakariakov, V. M. and Verwichte, E. 2005 Living Rev. Solar Phys. 3, 1.Google Scholar
Nakariakov, V. M., Verwichte, E., Berghmans, D. and Robbrecht, E. 2000 Astron. Astrophys. 362, 1151.Google Scholar
Rice, J. E., Greenwald, M., Hutchinson, I. H., Marmar, E. S., Takase, Y., Wolfe, S. M. and Bombarda, F. 1998 Nucl. Fusion 38, 75.CrossRefGoogle Scholar
Rosa, R. J. 1968 Magnetohydrodynamic Energy Conversion. New York: McGraw-Hill.Google Scholar
Shafranov, V. D. 1957 JETP 33, 710.Google Scholar
Shercliff, J. A. 1962 The Theory of Electromagnetic Flow-Measurement. Cambridge: Cambridge University Press.Google Scholar
Shercliff, J. A. 1965 A Textbook of Magnetohydrodynamics. Oxford, New York: Pergamon Press.Google Scholar
Simintzis, Ch., Throumoulopoulos, G. N., Pantis, G. and Tasso, H. 2001 Phys. Plasmas 8, 2641.CrossRefGoogle Scholar
Spies, G. 1980 Phys. Fluids 23, 2017.CrossRefGoogle Scholar
Suckewer, S., Eubank, H. P., Goldston, R. J., Hinnov, E. and Sauthoff, N. R. 1979 Phys. Rev. Lett. 43, 207.CrossRefGoogle Scholar
Takechi, M., Matsunaga, G., Aiba, N., Fujita, T., Ozeki, T., Koide, Y., Sakamoto, Y., Kurita, G., Isayama, A. and Kamada, Y. 2007 Phys. Rev. Lett. 98, 055002.CrossRefGoogle Scholar
Tasso, H. and Throumoulopoulos, G. N. 1998 Phys. Plasmas 5, 2378.CrossRefGoogle Scholar
Taylor, T. S.et al. 1995 Phys. Plasmas 2, 2390.CrossRefGoogle Scholar
Throumoulopoulos, G. N. and Pantis, G. 1989 Phys. Fluids B 1, 1827.CrossRefGoogle Scholar
Throumoulopoulos, G. N. and Pantis, G. 1996 Plasma Phys. Control. Fusion 38, 1817.CrossRefGoogle Scholar
Throumoulopoulos, G. N., Poulipoulis, G., Pantis, G. and Tasso, H. 2003 Phys. Lett. A 317, 463.CrossRefGoogle Scholar
Throumoulopoulos, G. N. and Tasso, H. 1997 Phys. Plasmas 4, 1492.CrossRefGoogle Scholar
Throumoulopoulos, G. N. and Tasso, H. 1999 J. Plasma Phys. 62, 449.CrossRefGoogle Scholar
Throumoulopoulos, G. N. and Tasso, H. 2001 Geophys. Astrophys. Fluid Dyn. 94, 249.CrossRefGoogle Scholar
Throumoulopoulos, G. N., Tasso, H. and Poulipoulis, G. 2009 J. Phys. A: Math. Theor. 42, 335501.CrossRefGoogle Scholar
Tsinganos, K. C. 1981 Astrophys. J. 245, 764.CrossRefGoogle Scholar
Tsinganos, K. C. 1982a Astrophys. J. 252, 775.CrossRefGoogle Scholar
Tsinganos, K. C. 1982b Astrophys. J. 259, 820.CrossRefGoogle Scholar
Tsinganos, K. C. 1982c Astrophys. J. 259, 832.CrossRefGoogle Scholar
Verwichte, E., Foullon, C. and Nakariakov, V. M. 2006a Astron. Astrophys. 446, 1139.CrossRefGoogle Scholar
Verwichte, E., Foullon, C. and Nakariakov, V. M. 2006b Astron. Astrophys. 449, 769.CrossRefGoogle Scholar
Vlahakis, N. and Tsinganos, K. 1998 Mon. Not. R. Astron. Soc. 298, 777.CrossRefGoogle Scholar
Vlahakis, N. and Tsinganos, K. 1999 Mon. Not. R. Astron. Soc. 307, 279.CrossRefGoogle Scholar
Whitham, G. B. 1974 Linear and Nonlinear Waves. New York: Wiley.Google Scholar
Woltjer, L. 1959a Astrophys. J. 130, 400.CrossRefGoogle Scholar
Woltjer, L. 1959b Astrophys. J. 130, 405.CrossRefGoogle Scholar
Woltjer, L. 1959c Proc. Natl. Acad. Sci. 45, 769.CrossRefGoogle Scholar
Zehrfeld, H. P. and Green, B. J. 1972 Nucl. Fusion 12, 569.CrossRefGoogle Scholar