Finite amplitude convection and magnetic field generation in a rotating spherical shell

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Abstract
Finite amplitude solutions for convection in a rotating spherical fluid shell with a radius ratio of η=0.4 are obtained numerically by the Galerkin method. The case of the azimuthal wavenumber m=2 is emphasized, but solutions with m=4 are also considered. The pronounced distinction between different modes at low Prandtl numbers found in a preceding linear analysis (Zhang and Busse, 1987) is also found with respect to nonlinear properties. Only the positive-ω-mode exhibits subcritical finite amplitude convection. The stability of the stationary drifting solutions with respect to hydrodynamic disturbances is analyzed and regions of stability are presented. A major part of the paper is concerned with the growth of magnetic disturbances. The critical magnetic Prandtl number for the onset of dynamo action has been determined as function of the Rayleigh and Taylor numbers for the Prandtl numbers P=0.1 and P=1.0. Stationary and oscillatory dynamos with both, dipolar and quadrupolar, symmetries are close competitors in the parameter space of the problem.

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