Hostname: page-component-cd9895bd7-gbm5v Total loading time: 0 Render date: 2024-12-26T03:43:37.405Z Has data issue: false hasContentIssue false
  • English
  • Français

The equations of reduced magnetohydrodynamics

Published online by Cambridge University Press:  13 March 2009

G. P. Zank  and
W. H. Matthaeus
G. P. Zank
Affiliation:
Bartol Research Institute, The University of Delaware, Newark, Delaware 19716, U.S.A.
W. H. Matthaeus
Affiliation:
Bartol Research Institute, The University of Delaware, Newark, Delaware 19716, U.S.A.
Get access Rights & Permissions [Opens in a new window]

Abstract

The equations of high- and low-beta reduced magnetohydrodynamics (RMHD) are considered anew in order to elucidate the relationship between compressible MHD and RMHD and also to distinguish RMHD from recently developed models of nearly incompressible MHD. Our results, summarized in two theorems, provide the conditions under which RMHD represents a valid reduction of compressible MHD. The equations for low-beta RMHD and high-beta RMHD are shown to be identical. Furthermore, as a direct consequence of our analysis, the conditions under which both two-dimensional incompressible MHD (in terms of the spatial co-ordinates as well as the fluid variables) and 2½ dimensional incompressible MHD (i.e. only two-dimensional in the spatial co-ordinates) represent a valid reduction of three-dimensional compressible MHD are also formulated. It is found that the elimination of all high-frequency and long-wavelength modes from the magneto-fluid reduces the fully compressible MHD equations to either two-dimensional incompressible MHD in the plasma beta (β) limit β ≪ 1, or 2½-dimensional incompressible MHD for β ≈ 1. Our approach clarifies several inconsistencies to be found in previous investigations in that the reduction is exact. Our results and analysis are expected to be of interest for plasma fusion and space and solar physics.

Type
Research Article
Information
Journal of Plasma Physics , Volume 48 , Issue 1 , August 1992 , pp. 85 - 100
Copyright
Copyright © Cambridge University Press 1992

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

Burlaga, L. F. & Ogilvie, K. W. 1970 Solar Phys. 15, 61.CrossRefGoogle Scholar
Courant, R. & Friedrichs, K. O. 1976 Supersonic Flow and Shock Waves. Springer.CrossRefGoogle Scholar
Dahlburg, D., Montgomery, D. C. & Matthaeus, W. H. 1985 J. Plasma Phys. 34, 1.CrossRefGoogle Scholar
Dahlburg, D., Doolen, G. D. & Matthaeus, W. H. 1986 J. Plasma Phys. 35, 1.CrossRefGoogle Scholar
Hazeltine, R. D., Kotschenreuther, M. & Morrison, P. J. 1985 Phys. Fluids 28, 2466.CrossRefGoogle Scholar
Klainerman, S. & Majda, A. 1982 Comm. Pure Applied Math. 35, 629.CrossRefGoogle Scholar
Kleva, R. G. 1986 Phys. Fluids 29, 2881.CrossRefGoogle Scholar
Kraichnan, R. H. & Montgomery, D. C. 1980 Rep. Prog. Phys. 43, 547.CrossRefGoogle Scholar
Kreis, H.-O. 1980 Commun. Pure Appl. Maths 33, 399.CrossRefGoogle Scholar
Landau, L. & Lifshitz, E. M. 1979 Fluid Mechanics. Pergamon.Google Scholar
Lighthill, M. J. 1952 Proc. R. Soc. Lond. A 211, 564.Google Scholar
Lighthill, M. J. 1979 Waves in Fluids. Cambridge University Press.Google Scholar
Marsch, E. 1991 Physics of the Inner Heliosphere (ed. Schwenn, R. & Marsch, E.), pp. 159–241. Springer-Verlag, Heidelberg.CrossRefGoogle Scholar
Matthaeus, W. H., Goldstein, M. L. & Roberts, D. A. 1990 J. Geophys. Res. 95, 20673.Google Scholar
Matthaeus, W. H. & Zhou, Y. 1989 Turbulence and Nonlinear Dynamics in MHD Flows, p. 93. North-Holland.CrossRefGoogle Scholar
Montgomery, D. C. 1982 Physica Scripta T2/1, 83.CrossRefGoogle Scholar
Montgomery, D. 1989 Lecture Notes on Turbulence (ed. Herring, J. R. & McWilliams, J. C.), p. 75. World Scientific.Google Scholar
Montgomery, D. C. & Hartori, T. 1984 Plasma Phys. Contr. Fusion 26, 717.CrossRefGoogle Scholar
Rosenbluth, M. N., Monticello, D. A., Strauss, H. R. & White, R. B. 1976 Phys Fluids 19, 1987.CrossRefGoogle Scholar
Shebalin, J. V., Matthaeus, W. H. & Montgomery, D. C. 1983 J. Plasma Phys. 29, 525.CrossRefGoogle Scholar
Shebalin, J. V. & Montgomery, D. C. 1988 J. Plasma Phys. 39, 339.CrossRefGoogle Scholar
Strauss, H. R. 1976 Phys. Fluids 19, 134.CrossRefGoogle Scholar
Strauss, H. R. 1977 Phys. Fluids 20, 1354.CrossRefGoogle Scholar
Zank, G. P. & Matthaeus, W. H. 1990 Phys. Rev. Lett. 64, 1243.CrossRefGoogle Scholar
Zank, G. P. & Matthaeus, W. H. 1991 Phys. Fluids A 3, 69.CrossRefGoogle Scholar
Zank, G. P. & Matthaeus, W. H. 1992 a J. Geophys. Res. (Brief Reports) (in press).Google Scholar
Zank, G. P. & Matthaeus, W. H. 1992 b Phys. Fluids A (in press).Google Scholar
Zank, G. P., Matthaeus, W. H. & Klein, L. W. 1990 Geophys. Res. Lett. 17, 1239.CrossRefGoogle Scholar