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Finite-amplitude thermal convection and geostrophic flow in a rotating magnetic system

Published online by Cambridge University Press:  19 April 2006

A. M. Soward
A. M. Soward
Affiliation:
Institute of Geophysics and Planetary Physics and Department of Earth and Space Sciences, University of California, Los Angeles, California 90024
Now at the School of Mathematics, University of Newcastle upon Tyne, England.
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Abstract

An electrically conducting Boussinesq fluid is confined between two rigid horizontal planes. The fluid is permeated by a strong uniform horizontal magnetic field and the entire system rotates rapidly about a vertical axis. By heating the fluid from below and cooling it from above the system becomes unstable to small perturbations when the adverse temperature gradient becomes sufficiently large. Attention is restricted to small values of the Ekman number E and the ratio q of the thermal and magnetic diffusivities (see (1.2) and (1.3) below). In this parameter range marginal convection is steady and its character depends on the relative sizes of the Coriolis and Lorentz forces as measured by the parameter λ (see (1.1) below). When λ [ges ] 2/3½, motion consists of a single roll, whose axis is perpendicular to the applied magnetic field. On the other hand, when λ < 2/3½, two distinct rolls are possible: the axis of each roll lies oblique but with equal angle to the applied magnetic field. Only the latter case is discussed here.

Once the Rayleigh number R exceeds its critical value Rc only one of the two sets of single rolls remains stable, while its squared amplitude increases linearly with R – Rc. For certain values of the parameters λ and τ (see (1.6) below) a second critical value may be isolated at which the system becomes unstable to unidirectional geostrophic flow perturbations aligned with the applied magnetic field. The instability sets in as either a steady or oscillatory shear flow dependent on the values taken by λ and τ. In both cases, after the secondary instability sets in, the roll amplitude remains largely insensitive to further increase in the Rayleigh number with the consequence that the geostrophic flow is stabilized. The amplitude of the shear, on the other hand, increases with R, adjusting its magnitude to ensure stability of the convection rolls.

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 98 , Issue 3 , 12 June 1980 , pp. 449 - 471
Copyright
© 1980 Cambridge University Press

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References

Braginsky, S. I. 1964 Self-excitation of a magnetic field during the motion of a highly conducting fluid. Zh. Eksp. teor. Fiz. 47, 1084. (Trans. Sov. Phys., J. Exp. Theor. Phys. 20, 726–735 (1965)Google Scholar
Braginsky, S. I. 1970 Torsional magnetohydrodynamic vibrations in the Earth's core and variations in day length. Geomag. i Aeronomiya (U.S.S.R.) 10, 3. (Trans. Geomag. Aero. 10, 1–8Google Scholar
Busse, F. H. 1975 A model of the geodynamo. Geophys. J. Roy. Astron. Soc. 42, 437–459Google Scholar
Busse, F. H. 1978a Magnetohydrodynamics of the Earth's dynamo. Ann. Rev. Fluid Mech. 10, 435–462Google Scholar
Busse, F. H. 1978b Introduction to the theory of geomagnetism. Rotating Fluids in Geophysics (ed. P. H. Roberts & A. M. Soward), pp. 361–388. Academic.
Chandrasekhar, S. 1961 Hydrodynamic and Hydromagnetic Stability. Oxford University Press.
Eltayeb, I. A. 1972 Hydromagnetic convection in a rapidly rotating fluid layer. Proc. Roy. Soc. A 326, 229–254Google Scholar
Fearn, D. R. 1979 Thermal and magnetic instabilities in a rapidly rotating fluid sphere. Geophys. Astrophys. Fluid Dyn. 14, 103–126Google Scholar
Moore, D. W. 1978 Homogeneous fluids in rotation: A. Viscous effects. Rotating Fluids in Geophysics (ed. P. H. Roberts & A. M. Soward), pp. 29–66. Academic
Roberts, P. H. 1978 Magneto-convection in a rapidly rotating fluid. Rotating Fluids in Geophysics (ed. P. H. Roberts & A. M. Soward), pp. 420–436. Academic
Roberts, P. H. & Loper, D. E. 1979 On the diffusive instability of some simple steady magnetohydrodynamic flows. J. Fluid Mech. 90, 641–668Google Scholar
Roberts, P. H. & Soward, A. M. 1972 Magnetohydrodynamics of the Earth's core. Ann. Rev. Fluid Mech. 4, 117–153Google Scholar
Roberts, P. H. & Stewartson, K. 1974 On finite amplitude convection in a rotating magnetic system. Phil. Trans. Roy. Soc. A 277, 287–315Google Scholar
Roberts, P. H. & Stewartson, K. 1975 Double roll convection in a rotating magnetic system. J. Fluid Mech. 68, 447–466Google Scholar
Soward, A. M. 1979 Thermal and magnetically driven convection in a rapidly rotating fluid layer. J. Fluid Mech. 90, 669–684Google Scholar
Taylor, J. B. 1963 The magnetohydrodynamics of a rotating fluid and the Earth's dynamo problem. Proc. Roy. Soc. A 274, 274–283Google Scholar