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Simplified variational principles for barotropic magnetohydrodynamics

Published online by Cambridge University Press:  30 June 2008

ASHER YAHALOM  and
DONALD LYNDEN-BELL
ASHER YAHALOM
Affiliation:
Institute of Astronomy, University of Cambridge, Madingley Road, Cambridge CB3 0HA, UK Ariel University Center of Samaria, Ariel 40700, Israel
DONALD LYNDEN-BELL
Affiliation:
Institute of Astronomy, University of Cambridge, Madingley Road, Cambridge CB3 0HA, UK Clare College, University of Cambridge, Cambridge CB, UK
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Abstract

Variational principles for magnetohydrodynamics have been introduced by previous authors both in Lagrangian and Eulerian form. In this paper we introduce simpler Eulerian variational principles from which all the relevant equations of barotropic magnetohydrodynamics can be derived. The variational principle is given in terms of six independent functions for non-stationary barotropic flows with trivial topologies and three independent functions for stationary barotropic flows. This is less than the seven variables which appear in the standard equations of barotropic magnetohydrodynamics, which are the magnetic field B the velocity field v and the density ρ.

For non-trivial topologies it is necessary to assume that some of the variables introduced in the non-stationary formalism are non-single-valued. That is, it is necessary to introduce a number of branch cuts in order to define single-valued branches of the field variables. In turn, these cuts along with the six field variables constitute an extended number of dynamic variables. The number of cuts necessary depends on the flow. The relations between barotropic magnetohydrodynamics topological constants and the functions of the present formalism will be elucidated.

The equations obtained for non-stationary barotropic magnetohydrodynamics resemble the equations of Frenkel et al. (Phys. Lett. A, vol. 88, 1982, p. 461). The connection between the Hamiltonian formalism introduced in Frenkel et al. (1982) and the present Lagrangian formalism (with Eulerian variables) will be discussed.

Type
Papers
Information
Journal of Fluid Mechanics , Volume 607 , 25 July 2008 , pp. 235 - 265
Copyright
Copyright © Cambridge University Press 2008

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References

REFERENCES

Almaguer, J. A., Hameiri, E., Herrera, J. & Holm, D. D. 1988 Phys. Fluids 31, 1930.CrossRefGoogle Scholar
Arnold, V. I. 1965 a Appl. Math. Mech. 29, 846.Google Scholar
Arnold, V. I. 1965 b Dokl. Acad. Nauk USSR 162, 975.Google Scholar
Arnold, V. I. 1966 J. Méc. 5, 19.Google Scholar
Bekenstein, J. D. & Oron, A. 2000 Phys. Rev. E 62, 5594.Google Scholar
Dungey, J. W. 1958 Cosmic Electrodynamics. Cambridge University Press.Google Scholar
Frenkel, A., Levich, E. & Stilman, L. 1982 Phys. Lett. A 88, 461.CrossRefGoogle Scholar
Kats, A. V. 2001 Physica D 459, 152.Google Scholar
Kats, A. V. 2003 JETP Lett. 77, 657.CrossRefGoogle Scholar
Kats, A. V. 2004 Phys. Rev. E 69, 046303.Google Scholar
Katz, J., Inagaki, S. & Yahalom, A. 1993 Pub. Astron. Soc. Japan 45, 421.Google Scholar
Kats, A. V. & Kontorovich, V. M. 1997 Low Temp. Phys. 23, 89.CrossRefGoogle Scholar
Katz, J. & Lynden-Bell, D. 1985 Geophys. Astrophys. Fluid Dyn. 33, 1.CrossRefGoogle Scholar
Lynden-Bell, D. 1996 Current Sci. 70, 789.Google Scholar
Lynden-Bell, D. & Katz, J. 1981 Proc. R. Soc. Lond. A 378, 179.Google Scholar
Moffatt, H. K. 1969 J. Fluid Mech. 35, 117.CrossRefGoogle Scholar
Moreau, J. J. 1977 Seminaire D'analyse Convexe, Montpellier Expose no: 7.Google Scholar
Ophir, D., Yahalom, A., Pinhasi, G. A. & Kopylenko, M. 2005 A combined variational and multi-grid approach for fluid simulation. Proc. Intl Conf. on Adaptive Modelling and Simulation (ADMOS), Barcelona, Spain, p. 295.Google Scholar
Prix, R. 2004 Phys. Rev. D 69, 043001.Google Scholar
Prix, R. 2005 Phys. Rev. D 71, 083006.Google Scholar
Sakurai, T. 1979 Pub. Astron. Soc. Japan 31, 209.Google Scholar
Seliger, R. L. & Whitham, G. B. 1968 Proc. R. Soc. Lond. A 305, 1.Google Scholar
Sturrock, P. A. 1994 Plasma Physics. Cambridge University Press.CrossRefGoogle Scholar
Vladimirov, V. A. & Moffatt, H. K. 1995 J. Fluid Mech. 283, 125.CrossRefGoogle Scholar
Vladimirov, V. A., Moffatt, H. K. & Ilin, K. I. 1996 J. Fluid Mech. 329, 187.CrossRefGoogle Scholar
Vladimirov, V. A., Moffatt, H. K. & Ilin, K. I. 1997 J. Plasma Phys. 57, 89.CrossRefGoogle Scholar
Vladimirov, V. A., Moffatt, H. K. & Ilin, K. I. 1999 J. Fluid Mech. 390, 127.CrossRefGoogle Scholar
Yahalom, A. 1995 J. Math. Phys. 36, 1324.CrossRefGoogle Scholar
Yahalom, A. 1996 Energy principles for barotropic flows with applications to gaseous disks. Thesis submitted as part of the requirements for the degree of PhD to the Senate of the Hebrew University of Jerusalem.Google Scholar
Yahalom, A. 2003 Method and system for numerical simulation of fluid flow. US patent 6,516,292.Google Scholar
Yahalom, A., Katz, J. & Inagaki, K. 1994 Mon. Not. R. Astron. Soc. 268, 506.CrossRefGoogle Scholar
Yahalom, A., Pinhasi, G. A. & Kopylenko, M. 2005 A numerical model based on variational principle for airfoil and wing aerodynamics. Proc. AIAA Conf., Reno, USA.CrossRefGoogle Scholar
Yahalom, A. & Pinhasi, G. A. 2003 Simulating fluid dynamics using a variational principle. Proc. AIAA Conf., Reno, USA.CrossRefGoogle Scholar
Yang, W. H., Sturrock, P. A. & Antiochos, S. 1986 Astrophys. J. 309, 383.CrossRefGoogle Scholar
Zakharov, V. E. & Kuznetsov, E. A. 1997 Usp. Fiz. Nauk 40, 1087.CrossRefGoogle Scholar