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On the diffusive instability of some simple steady magnetohydrodynamic flows

Published online by Cambridge University Press:  19 April 2006

P. H. Roberts  and
David E. Loper
P. H. Roberts
Affiliation:
School of Mathematics, University of Newcastle upon Tyne, England
David E. Loper
Affiliation:
Department of Mathematics, Forida State University, Tallahassee
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Abstract

The stability characteristics of some simple steady magnetohydrodynamic flows within an axisymmetric container of arbitrary electrical conductivity are investigated. Attention is focused upon rapidly rotating fluids in which the unperturbed velocity and magnetic field are axially symmetric and purely zonal. Detailed solutions are obtained for the particularly simple basic state representing a rigidly rotating homogeneous fluid with a uniform axial current. The theory of dynamic (dissipationless) instabilities is reviewed and its shortcomings are elucidated. A stability criterion is derived for an inviscid fluid of small electrical conductivity within a perfectly conducting axisymmetric container and it is shown that a certain class of inertial modes is unstable for any non-zero magnetic field strength. When the effects of container conductivity are included it is found that a class of slow modes with westward phase speed may be unstable. These modes are shown to be unstable within a cylinder but appear to be stable within a sphere. The influence of density gradients within a spherical container is investigated and it is found that for a certain class of exceptional slow modes with westward phase speed, a bottom-heavy density gradient is destabilizing. This surprising behaviour is explained in terms of a new branch of the stability curve developed by Eltayeb & Kumar (1977).

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 90 , Issue 4 , 27 February 1979 , pp. 641 - 668
Copyright
© 1979 Cambridge University Press

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References

Acheson, D. J. 1972 On the hydromagnetic stability of a rotating fluid annulus. J. Fluid Mech. 52, 529–541.Google Scholar
Baldwin, P. & Roberts, P. H. 1972 On resistive instabilities. Phil. Trans. Roy. Soc. A 272, 303–330.Google Scholar
Braginskii, S. I. 1964 Magnetohydrodynamics of the earth's core. Geomag. Aeron. 4, 898–912.Google Scholar
Braginskii, S. I. 1967 Magnetic waves in the earth's core. Geomag. Aeron. 7, 851–859.Google Scholar
Braginskii, S. I. & Roberts, P. H. 1975 Magnetic field generation. Proc. Roy. Soc. A 347, 125–140.Google Scholar
Eltayeb, I. A. 1972 Hydromagnetic convection in a rapidly rotating fluid layer. Proc. Roy. Soc. A 326, 229–254.Google Scholar
Eltayeb, I. A. 1975 Overstable hydromagnetic convection in a rotating fluid layer. J. Fluid Mech. 71, 161–179.Google Scholar
Eltayeb, I. A. & Kumar, S. 1977 Hydromagnetic convective instability of a rotating, selfgravitating fluid sphere containing a uniform distribution of heat sources. Proc. Roy. Soc. A 353, 145–162.Google Scholar
Eltayeb, I. A. & Roberts, P. H. 1970 On the hydromagnetics of rotating fluids. Astrophys. J. 162, 699–701.Google Scholar
Frieman, E. & Rotenberg, M. 1960 On hydromagnetic stability of stationary equilibria. Rev. Mod. Phys. 32, 898–902.Google Scholar
Furth, H. P., Killeen, J. & Rosenbluth, M. N. 1963 Finite-resistivity instabiliies of a sheet pinch. Phys. Fluids 6, 459–484.Google Scholar
Gibson, R. D. & Kent, A. 1971 On the tearing-mode instability. Quart. J. Mech. Appl. Math. 24, 63–79.Google Scholar
Gilman, P. A. 1970 Instability of magnetohydrodynamic stellar interiors from the magnetic buoyancy. I. Astrophys. J. 162, 1019–1029.Google Scholar
Hide, R. 1966 Free hydromagnetic oscillations of the earth's core and the theory of the geomagnetic secular variation. Phil. Trans. Roy. Soc. A 259, 615–647.Google Scholar
Hide, R. & Stewartson, K. 1972 Hydromagnetic oscillations of the earth's core. Rev. Geophys. Space Phys. 10, 579–598.Google Scholar
Kumar, S. & Roberts, P. H. 1975 A three dimensional kinematic dynamo. Proc. Roy. Soc. A 344, 235–258.Google Scholar
Malkus, W. V. R. 1967 Hydromagnetic planetary waves. J. Fluid Mech. 28, 793–802.Google Scholar
Pekeris, C. L., Accad, Y. & Shkoller, B. 1973 Kinematic dynamos and the earth's magnetic field. Phil. Trans. Roy. Soc. A 275, 425–461.Google Scholar
Roberts, P. H. 1967 An Introduction to Magnetohydrodynamics. Elsevier.
Roberts, P. H. 1971 Dynamo theory. Lect. in Appl. Math. 14, 129–206.Google Scholar
Roberts, P. H. 1977 MAC waves. Conf. Geofluiddyn. Wave Math. (ed. W. O. Criminale), pp. 216–221.
Roberts, P. H. 1978a Diffusive instabilities in magnetohydrodynamic convection. In Les instabilités Hydrodynamiques en Convection Libre, Forcée et Mixte (ed. J. C. Legros & J. K. Platten), pp. 102–111. Springer.
Roberts, P. H. 1978b Magneto-convection in a rapidly rotating fluid. In Rotating Fluids in Geophysics (ed. P. H. Roberts & A. M. Soward), pp. 421–435. Academic Press.
Roberts, P. H. & Soward, A. M. 1972 Magnetohydrodynamics of the earth's core. Ann. Rev. Fluid Mech. 4, 117–154.Google Scholar
Roberts, P. H. & Stewartson, K. 1974 On finite amplitude convection in a rotating magnetic system. Phil. Trans. Roy. Soc. A 277, 287–315.Google Scholar
Roberts, P. H. & Stewartson, K. 1975 On finite amplitude convection in a rotating magnetic system. J. Fluid Mech. 68, 447–466.Google Scholar
Soward, A. M. 1979 Thermal and magnetically driven convection in a rapidly rotating fluid layer. J. Fluid Mech. 90, 669–684.Google Scholar
Starr, V. P. 1968 Physics of Negative Viscosity Phenomena. McGraw-Hill.
Stewartson, K. 1967 Slow oscillations of fluid in a rotating cavity in the presenc e of a toroidal magnetic field. Proc. Roy. Soc. A 299, 173–187.Google Scholar
Wood, W. W. 1977 A note on the westward drift of the earth's magnetic field. J. Fluid Mech. 82, 389–400.Google Scholar