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States of minimum dissipation in magnetohydrodynamics: a review

Published online by Cambridge University Press:  13 March 2009

Lee Phillips
Affiliation:
Code 6440, Naval Research Laboratory, Washington, DC 20375-5344, USA

Abstract

The hypothesis that the steady, or statistically steady, states of a magnetofluid can be predicted by minimizing the integral of the total energy dissipation is discussed in relation to theorems in fluid dynamics. A survey is made of the recent literature wherein the hypothesis is used to predict magnetofluid behaviour and some related work. A detailed discussion of boundary conditions is provided, and we close with a brief summary of the current state of research.

Type
Research Article
Copyright
Copyright © Cambridge University Press 1996

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References

REFERENCES

Agim, Y. Z. and Montgomery, D. 1991 Nonideal, helical, vortical magnetohydrodynamic steady states. Plasma Phys. Contr. Fusion 33, 881902.CrossRefGoogle Scholar
Bevir, M. K., Caloutsis, A. and Gimblett, C. G. 1993 A note on minimum dissipation plasma states. Plasma Phys. Contr. Fusion 35, 133149.CrossRefGoogle Scholar
Browning, P. K. 1988 Magnetohydrodynamics in solar coronal and laboratory plasmas: a comparative study. Phys. Rep. 169, 329384.CrossRefGoogle Scholar
Chandrasekhar, S. and Kendall, P. C. 1957 On force-free magnetic fields. Astrophys. J. 126, 457460.CrossRefGoogle Scholar
Chen, H., Shan, X. and Montgomery, D. 1990 Galerkin approximations for dissipative magnetohydrodynamics. Phys. Rev. A42, 61586165.CrossRefGoogle ScholarPubMed
Dahlburg, J. P., Montgomery, D. and Matthaeus, W. H. 1985 Turbulent disruptions from the Strauss equations. J. Plasma Phys. 34, 146.CrossRefGoogle Scholar
Dahlburg, J. P., Montgomery, D., Doolen, G. D. and Matthaeus, W. H. 1986 a Large-scale disruptions in a current-carrying magnetofluid. J. Plasma Phys. 35, 142.CrossRefGoogle Scholar
Dahlburg, J. P., Montgomery, D., Doolen, G. D. and Turner, L. 1986 b Turbulent relaxation to a force-free field-reversed state. Phys. Rev. Lett. 57, 428431.CrossRefGoogle ScholarPubMed
Dahlburg, J. P., Montgomery, D., Doolen, G. D. and Turner, L. 1987 Turbulent relaxation of a confined magnetofluid to a force-free state. J. Plasma Phys. 37, 299321.CrossRefGoogle Scholar
Dahlburg, J. P., Montgomery, D., Doolen, G. D. and Turner, L. 1988 Driven, steady-state REP computations. J. Plasma Phys. 40, 3968.CrossRefGoogle Scholar
deGroot, S. R. and Mazur, P. 1984 Non-Equilibrium Thermodynamics. Dover, New York.Google Scholar
Farengo, R. and Soběhart, J. R. 1993 Minimum dissipation states in tokamak plasmas. Plasma Phys. Contr. Fusion 35, 465471.Google Scholar
Farengo, R. and Soběhart, J. R. 1994 Minimum ohmic dissipation and dc helicity injection in tokamak-like plasmas. Plasma Phys. Contr. Fusion 36, 16911700.CrossRefGoogle Scholar
Farengo, R. and Soběhart, J. R. 1995 a Corrigendum: Minimum dissipation states in tokamak plasmas. Plasma Phys. Contr. Fusion 37, 183184.Google Scholar
Farengo, R. and Soběhart, J. R. 1995 b Determination of minimum-dissipation states with self-consistent resistivity in magnetized plasmas. Phys. Rev. E52, 21022105.Google ScholarPubMed
Ferraro, V. C. A. and Plumpton, C. 1966 An Introduction to Magneto-Fluid Mechanics, 2nd edn. Clarendon Press, Oxford.Google Scholar
Feynman, R. P., Leighton, R. B. and Sands, M. 1964 The Feynman Lectures on Physics, Vol. II, Addison-Wesley.Google Scholar
Finlayson, B. A. 1972 Existence of variational principles for the Navier-Stokes equation. Phys. Fluids 15, 963967.CrossRefGoogle Scholar
Hada, T. 1994 Evolutionary conditions in the dissipative MHD system: stability of intermediate MHD shock waves. Geophys. Res. Lett. 21, 22752278.CrossRefGoogle Scholar
Jaynes, E. T. 1980 The minimum entropy principle. Ann. Rev. Phys. Chem. 31, 579601.CrossRefGoogle Scholar
Kirby, P. 1988 Numerical simulation of the reversed field pinch. Phys. Fluids 31, 625629.CrossRefGoogle Scholar
Konopinski, E. J. 1969 Classical Descriptions of Motion. Freeman, San Francisco.Google Scholar
Krommes, J. A. and Kim, C.-B. 1990 A ‘new’ approach to the quantitative statistical dynamics of plasma turbulence: the optimum theory of rigorous bounds on steady-state transport. Phys. Fluids B2, 13311337.CrossRefGoogle Scholar
Kucinski, M. Y. and Okano, V. 1995 Relaxation of toroidal plasmas: tokamaks. Plasma Phys. Contr. Fusion 37, 116.CrossRefGoogle Scholar
Lamb, H. 1932 Hydrodynamics, 6th edn. Cambridge University Press (reprinted 1945 Dover, New York).Google Scholar
Lanczos, C. 1970 The Variational Principles of Mechanics, 4th edn. University of Toronto Press (reprinted 1986 Dover, New York).Google Scholar
Moffatt, H. K. 1964 Electrically driven steady flows in magnetohydrodynamics. Proceedings of the 11th International Congress of Applied Mechanics (ed. Görtler, H.), pp. 946953. Springer-Verlag, Berlin.Google Scholar
Montgomery, D. 1990 Relaxed states in driven, dissipative magnetohydrodynamies: helical distortions and vortex pairs. Trends in Theoretical Physics, Vol. 1 (ed. Ellis, P. J. and Tang, Y. C.), pp. 239262, Addison-Wesley, New York.Google Scholar
Montgomery, D. 1992 ‘Reduced’ magnetohydrodynamics and minimum dissipation rates. Phys. Fluids B4, 292293.CrossRefGoogle Scholar
Montgomery, D. and Phillips, L. 1988 Minimum dissipation rates in magnetohydrodynamics. Phys. Rev. A38, 29532963.CrossRefGoogle ScholarPubMed
Montgomery, D. and Phillips, L. 1989 MHD turbulence: relaxation processes and variational principles. Physica D37, 215226.Google Scholar
Montgomery, D. and Phillips, L. 1990 Minimum dissipation and maximum entropy. Maximum Entropy and Bayesian Methods (ed. Fougre, P. F.). Kluwer, Deventer.Google Scholar
Montgomery, D., Phillips, L. and Theobald, M. L. 1989 Helical, dissipative magnetohydrodynamic states with flow. Phys. Rev. A40, 15151523.CrossRefGoogle ScholarPubMed
Onsager, L. 1931 a Reciprocal relations in irreversible processes. Phys. Rev. 37, 405426.CrossRefGoogle Scholar
Onsager, L. 1931 b Reciprocal relations in irreversible processes. Phys. Rev. 38, 22652279.CrossRefGoogle Scholar
Robinson, P. C. 1987 Toroidal pinches. Technical Report CLM-P810, Culham Laboratory.Google Scholar
Serrin, J. 1959 Mathematical principles of classical fluid mechanics: IV. Variational principles. Handbuch der Physik, Vol. VIII/1 (ed. Flügge, S.), pp. 144150. Springer-Verlag, Berlin.Google Scholar
Shan, X., Montgomery, P. and Chen, H. 1991 Nonlinear magnetohydrodynamics by Galerkin-method computation. Phys. Rev. A44, 68006815.CrossRefGoogle ScholarPubMed
Storer, R. G. 1983 Spectrum of an exactly soluble resistive magnetohydrodynamic model. Plasma Phys. 25, 12791282.CrossRefGoogle Scholar
Taylor, J. B. 1974 Relaxation of toroidal plasma and generation of reverse magnetic field. Phys. Rev. Lett. 33, 11391141.CrossRefGoogle Scholar
Taylor, J. B. 1986 Relaxation and magnetic reconnection in plasmas. Rev. Mod. Phys. 58, 741763.CrossRefGoogle Scholar
Theobald, M. L. and Montgomery, D. 1990 A helically distorted MHD flux rope model. J. Ceophys. Res. 95, 1489314904.CrossRefGoogle Scholar
Theobald, M. L., Montgomery, D., Doolen, G. P. and Dahlburg, J. P. 1989 Sawtooth oscillations about helical current channels. Phys. Fluids B1, 766773.CrossRefGoogle Scholar
Truesdell, C. 1984 Rational Thermodynamics, 2nd edn. Springer-Verlag, New York.CrossRefGoogle Scholar
Wang, C. Y. and Bhattacharjee, A. 1991 Optimum theory for the energy dissipation in a turbulent pinch. Phys. Fluids 3, 34623476.CrossRefGoogle Scholar
Wang, C. Y., Bhattacharjee, A. and Hameiri, E. 1991 Upper bounds on fluctuational power absorption in a turbulent pinch. Phys. Fluids 3, 715720.CrossRefGoogle Scholar
Wei, J. 1966 Irreversible thermodynamics in engineering. Indust. Engng Chem. 58, 5560.CrossRefGoogle Scholar