Patterns of thermal convection in slowly rotating,weakly magnetic fluid shells

The effects of rotation and a toroidal magnetic field on the preferred pattern of small amplitude convection in spherical fluid shells are considered. The convective motions are described in terms of associated Legendre functions P_l^|m| (cos θ). For a given pair of Prandtl number P and magnetic Prandtl number $\text{P}$_m the physically realized solution is represented either by m = 0 or |m| = l depending on the ratio of the rotation rate Λ to the magnetic field amplitude H. The case of m = 0 is preferred if this ratio ranges below a critical value, which is a function of the shell thickness, and |m| = l otherwise.