Axisymmetric convection in a rotating sphere. Part 1. Stress-free surface

This paper examines large-scale nonlinear thermal convection in a rotating selfgravitating sphere of Boussinesq fluid containing a uniform distribution of heat sources. Conservative finite-difference forms of the equations of axisymmetric laminar motion are marched forward in time. The surface is assumed to be stress free and at constant temperature. Numerical solutions are obtained for Taylor numbers in the range 0 ≤ Λ ≤ 10⁴ and Rayleigh numbers with \[ R_c \leqslant R\lesssim 10R_c. \] For high Prandtl number (P > 5) the solutions are steady and most of them resemble the solutions of the linear stability equations, though other steady solutions are also found. For P [lsim ] 1, the steady solutions have horizontal wavenumber l = 1 and nearly uniform angular momentum per unit mass, rather than nearly uniform angular velocity. This rotation law seems to be independent of many details of the model and may hold in the convective core of a rotating star.