On Small Roberts Number Magnetoconvection in Rapidly Rotating Systems

Details On Small Roberts Number Magnetoconvection in Rapidly Rotating Systems On Small Roberts Number Magnetoconvection in Rapidly Rotating Systems

Apr 1996

adsabs.harvard.edu/cgi-bin/nph-data_query?bibcode=1996rspsa.452..981z&link_type=abstract

Proceedings: Mathematical, Physical and Engineering Sciences, Volume 452, Issue 1947, pp. 981-995

Physics

12

Scientific paper

Convection of an electrically conducting fluid of magnetic diffusivity eta and thermal diffusivity kappa in rapidly rotating systems in the presence of an imposed toroidal magnetic field is investigated. The motivation for this study comes from the study of convection in planetary cores. Two important parameters of the system are the Elsasser number Λ , which measures the strength of the imposed field, and the modified Rayleigh number R, which measures the amplitude of buoyancy forces. In this system both magnetically driven instability due to the field curvature and thermally driven instability due to buoyancy can occur. Attention is focused on the behaviour of linear magnetoconvection at small Roberts number, q = kappa /eta -> 0, appropriate for the Earth's core. Two different approaches are adopted for investigation. First, an asymptotic analysis with q -> 0 is carried out to show that Rc = O(1/q), q -> 0, Λ -> Λ c, where Rc is the critical value of the modified Rayleigh number and Λ c denotes the critical value for the purely magnetic instabilities. Moreover, the nature of the transition between magnetically driven modes and thermally driven modes is investigated. Second, numerical solutions at different values of q in a rapidly rotating spherical shell are obtained for two different cases: (i) stress-free boundary conditions with an insulating inner sphere and (ii) no-slip boundary conditions with a conducting inner sphere. Both numerical solutions confirm the singular behaviour of magnetoconvection in the limit q -> 0, as predicted by the asymptotic analysis. In consequence, the transition from the thermally dominant mode to the magnetically dominant mode has a rather complex structure in the limit q -> 0. It is shown that there is no uniform scaling that is appropriate for all O(1) values of Λ . The results shed new light on well-known numerical difficulties in the problem of magnetoconvection at the small Roberts number limit.

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