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Simplified variational principles for non-barotropic magnetohydrodynamics

Published online by Cambridge University Press:  08 March 2016

Asher Yahalom Open the ORCID record for Asher Yahalom [Opens in a new window]
Asher Yahalom*
Affiliation:
Department of Electrical and Electronic Engineering, Ariel University, Ariel 40700, Israel
*
†Email address for correspondence: asya@ariel.ac.il
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Abstract

Variational principles for magnetohydrodynamics were 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 non-barotropic magnetohydrodynamics can be derived for certain field topologies. The variational principle is given in terms of five independent functions for non-stationary barotropic flows. This is less than the eight variables which appear in the standard equations of barotropic magnetohydrodynamics which are the magnetic field $\boldsymbol{B}$ the velocity field $\boldsymbol{v}$, the entropy $s$ and the density ${\it\rho}$.

Type
Research Article
Information
Journal of Plasma Physics , Volume 82 , Issue 2 , April 2016 , 905820204
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Copyright
© Cambridge University Press 2016 

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References

Almaguer, J. A., Hameiri, E., Herrera, J. & Holm, D. D. 1988 Lyapunov stability analysis of magnetohydrodynamic plasma equilibria with axisymmetric toroidal flow. Phys. Fluids 31, 1930–1939.Google Scholar
Arnold, V. I. 1965a A Variational principle for three-dimensional stationary flows of the ideal fluid. Z. Angew. Math. Mech. 29 (5), 846–851.Google Scholar
Arnold, V. I. 1965b On the conditions of the nonlinear stability of flat stationary curvilinear flows of the ideal fluid. Dokl. Akad. Nauk USSR 162 (5), 975–978.Google Scholar
Bateman, H. 1931 On dissipative systems and related variational principles. Phys. Rev. 38, 815–820.CrossRefGoogle Scholar
Bekenstein, J. D. & Oron, A. 2000 Conservation of circulation in magnetohydrodynamics. Phys. Rev. E 62 (4), 5594–5602.Google ScholarPubMed
Bernstein, I. B., Frieman, E. A., Kruskal, M. D. & Kulsrud, R. M. 1958 An energy principle for hydromagnetic stability problems. Proc. R. Soc. Lond. A 244 (1236), 17–40.Google Scholar
Clebsch, A. 1857 Uber eine allgemeine transformation der hydro-dynamischen Gleichungen. J. Reine Angew. Math. 54, 293–312.Google Scholar
Clebsch, A. 1859 Uber die Integration der hydrodynamischen Gleichungen. J. Reine Angew. Math. 56, 1–10.Google Scholar
Faber, J. A., Baumgarte, T. W., Shapiro, S. L. & Taniguchi, K. 2006 General relativistic binary merger simulations and short gamma-ray bursts. Astrophys. J. Lett. 641 (2), L93.CrossRefGoogle Scholar
Frenkel, A., Levich, E. & Stilman, L. 1982 Hamiltonian description of ideal MHD revealing new invariants of motion. Phys. Lett. A 88, 461.CrossRefGoogle Scholar
Hoyos, J., Reisenegger, A. & Valdivia, J. A. 2007 Simulation of the magnetic field evolution in neutron stars VI. In Reunion Anual Sociedad Chilena de Astronomia (SOCHIAS), vol. 1, p. 20. Sociedad Chilena de Astronomia (SOCHIAS).Google Scholar
Igumenshchev, I. V., Narayan, R. & Abramowicz, M. A. 2003 Three-dimensional magnetohydrodynamic simulations of radiatively inefficient accretion flows. Astrophys. J. 592 (2), 1042.Google Scholar
Kats, A. V. 2001 Variational principle and canonical variables in hydrodynamics with discontinuities. Physica D 459, 152–153.Google Scholar
Kats, A. V. 2002 Variational principle in canonical variables, weber transformation, and complete set of the local integrals of motion for dissipation-free magnetohydrodynamics. Los Alamos Archives Physics 0212023.Google Scholar
Kats, A. V. 2003 Variational principle in canonical variables, Weber transformation, and complete set of the local integrals of motion for dissipation-free magnetohydrodynamics. JETP Lett. 77 (12), 657–661.Google Scholar
Kats, A. V. 2004 Canonical description of ideal magnetohydrodynamic flows and integrals of motion. Phys. Rev. E 69, 046303.Google Scholar
Kats, A. V. & Kontorovich, V. M. 1997 Hamiltonian description of the motion of discontinuity surfaces. J. Low Temp. Phys. 23, 89.Google Scholar
Katz, J., Inagaki, S. & Yahalom, A. 1993 Energy principles for self-gravitating barotropic flows: I. General theory. Pub. Astro. Soc. Japan 45, 421–430.Google Scholar
Mignone, A., Rossi, P., Bodo, G., Ferrari, A. & Massaglia, S. 2010 High-resolution 3D relativistic MHD simulations of jets. Mon. Not. R. Astron. Soc. 402 (1), 7–12.Google Scholar
Miyoshi, T. & Kusano, K. 2005 A multi-state HLL approximate Riemann solver for ideal magnetohydrodynamics. J. Comput. Phys. 208 (1), 315–344.CrossRefGoogle Scholar
Morrison, P. J. 1982 Poisson brackets for fluids and plasmas. AIP Conf. Proc. 88, 13–46, Table 2.Google Scholar
Ophir, D., Yahalom, A., Pinhasi, G. A. & Kopylenko, M.2005 A combined variational and multi-grid approach for fluid simulation. In Proceedings of the International Conference on Adaptive Modelling and Simulation (ADMOS), Barcelona, Spain, pp. 295–304.Google Scholar
Sakurai, T. 1979 A new approach to the force-free field and its application to the magnetic field of solar active regions. Pub. Astro. Soc. Japan 31, 209.Google Scholar
Sturrock, P. A. 1994 Plasma Physics. Cambridge University Press.CrossRefGoogle Scholar
Vladimirov, V. A. & Moffatt, H. K. 1995 On general transformations and variational-principles for the magnetohydrodynamics of ideal fluids. 1. Fundamental principles. J. Fluid Mech. 283, 125–139.Google Scholar
Vladimirov, V. A., Moffatt, H. K. & Ilin, K. I. 1996 On general transformations and variational principles for the magnetohydrodynamics of ideal fluids. (Part 2). J. Fluid Mech. 329, 187.Google Scholar
Vladimirov, V. A., Moffatt, H. K. & Ilin, K. I. 1997 On general transformations and variational principles for the magnetohydrodynamics of ideal fluids. (Part 3). J. Plasma Phys. 57, 89.Google Scholar
Vladimirov, V. A., Moffatt, H. K. & Ilin, K. I. 1999 On general transformations and variational principles for the magnetohydrodynamics of ideal fluids. (Part 4). J. Fluid Mech. 390, 127.Google Scholar
Yahalom, A.1994 Los-Alamos Archives (preprint) solv-int9407001.Google Scholar
Yahalom, A. 1995 Helicity conservation via the Noether theorem. J. Math. Phys. 36, 1324–1327.Google Scholar
Yahalom, A.2003 Method and system for numerical simulation of fluid flow. US patent 6,516,292.Google Scholar
Yahalom, A. 2010 A four function variational principle for barotropic magnetohydrodynamics. Europhys. Lett. 89, 34005.Google Scholar
Yahalom, A. 2011 Stability in the weak variational principle of barotropic flows and implications for self-gravitating discs. Mon. Not. R. Astron. Soc. 418, 401–426.CrossRefGoogle Scholar
Yahalom, A. 2013a Aharonov–Bohm effects in magnetohydrodynamics. Phys. Lett. A 377 (31–33), 1898–1904.CrossRefGoogle Scholar
Yahalom, A. 2013b Using fluid variational variables to obtain new analytic solutions of self-gravitating flows with nonzero helicity. Procedia IUTAM 7, 223–232.CrossRefGoogle Scholar
Yahalom, A.2013c A new diffeomorphism symmetry group of magnetohydrodynamics. In Lie Theory and Its Applications in Physics: IX International Workshop, Springer Proceedings in Mathematics & Statistics, vol. 36, pp. 461–468.Google Scholar
Yahalom, A., Katz, J. & Inagaki, K. 1994 Energy principles for self-gravitating barotropic flows II. The stability of Maclaurin discs. Mon. Not. R. Astron. Soc. 268, 506–516.Google Scholar
Yahalom, A. & Lynden-Bell, D. 2008 Simplified variational principles for barotropic magnetohydrodynamics. J. Fluid Mech. 607, 235–265; (Los-Alamos Archives physics/0603128).Google Scholar
Yahalom, A. & Pinhasi, G. A.2003 Simulating fluid dynamics using a variational principle. In Proceedings of the AIAA Conference, Reno, USA.Google Scholar
Yahalom, A., Pinhasi, G. A. & Kopylenko, M.2005 A Numerical model based on variational principle for airfoil and wing aerodynamics. In Proceedings of the AIAA Conference, Reno, USA.CrossRefGoogle Scholar
Yang, W. H., Sturrock, P. A. & Antiochos, S. 1986 Force-free magnetic fields – the magneto-frictional method. Astrophys. J. 309, 383–391.Google Scholar
Zakharov, V. E. & Kuznetsov, E. A. 1997 Hamiltonian formalism for nonlinear waves. Usp. Fiz. Nauk 40, 1087.Google Scholar