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On hydromagnetic instabilities driven by the Hartmann boundary layer in a rapidly rotating sphere

Published online by Cambridge University Press:  26 April 2006

Keke Zhang  and
F. H. Busse
Keke Zhang
Affiliation:
Department of Mathematics, University of Exeter, UK
F. H. Busse
Affiliation:
Institute of Physics, University of Bayreuth, Germany
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Abstract

The instability of an electrically conducting fluid of magnetic diffusivity λ and viscosity v in a rapidly rotating spherical container of magnetic diffusivity $\hat{\lambda}$ in the presence of a toroidal magnetic field is investigated. Attention is focused on the case of a toroidal magnetic field induced by a uniform current density parallel to the axis of rotation, which was first studied by Malkus (1967). We show that the internal ohmic dissipation does not affect the stability of the hydromagnetic solutions obtained by Malkus (1967) in the limit of small λ. It is solely the effect of the magnetic Hartmann boundary layer that causes instabilities of the otherwise stable solutions. When the container is a perfect conductor, $\hat{\lambda}$ = 0, the hydromagnetic instabilities grow at a rate proportional to the magnetic Ekman number of the fluid Eλ; when the container is a nearly perfect insulator, $\lambda/\hat{\lambda}\ll 1$, the hydromagnetic instabilities grow at a rate proportional to E1/2λ; when the container is a nearly perfect conductor, λ 1, the growth rates are proportional to λ, where λ is the magnetic Ekman number based on the diffusivity λ of the container. The main characteristics of the instabilities are not affected by varying magnetic properties of the container. In light of the destabilizing role played by the Hartmann boundary layer, we also examine the corresponding magnetoconvection in a rapidly rotating fluid sphere with the perfectly conducting container and stress-free velocity boundary conditions. Analytical magnetoconvection solutions in closed form are obtained and implications are discussed.

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 304 , 10 December 1995 , pp. 263 - 283
Copyright
© 1995 Cambridge University Press

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