Turbulent Thermal Convection in the Presence of Rotation and a Magnetic Field: A Heuristic Theory

Abstract

The growth of linear, Boussinesq convective modes for a superadiabatic fluid in a planar geometry with negligible thermal diffusion or molecular viscosity is analysed. Small and large magnetic diffusivity limits are both considered. A finite amplitude theory is constructed by assuming that non-linearities (e.g. shear instabilities) limit the flow amplitude, and the dominant modes are assumed to be those which transport most heat. This leads to a unique prescription for the average velocity, superadiabaticity and horizontal lengthscale as a function of the imposed heat flux, rotation rate, large scale magnetic field and vertical lengthscale. Simple analytical results are provided (in terms of the relevant dimensionless numbers) which are as easy to apply as the conventional mixing length theory (MLT). The conclusions are: (i) For no rotation or magnetic field, the model reproduces MLT except that it predicts about an order of magnitude lower heat flux. (ii) Rotation alone or magnetic field alone inhibits the convection. For a given heat flux, the convective velocity and horizontal wavelength are reduced, and the temperature gradient increases. (iii) If R_0 ≾ 0.1, where R_0 is the nominal Rossby number, then the combined effect of rotation and magnetic field enhances the convection, and the maximum heat flux occurs at non-zero field. (iv) The rotational inhibition of convection in rapidly rotating stars does not substantially modify their static structure. (v) Convective velocities of around 0.02 cm/sec are predicted for the Earth's core, consistent with geomagnetic secular variation timescales, provided the toroidal field is less than about three times the poloidal field. (vi) In both the Earth and Jupiter, the actual magnetic field is within an order of magnitude of the field which minimizes the temperature gradient for a given heat flux. Application of these results to stellar and planetary dynamos is briefly discussed and critically assessed.

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