Symmetric heat and mass transfer in a rotating spherical layer

Abstract

The general theory of heat and mass transfer maintaining rotation with slightly different velocities under conditions typical for cores of planets in the solar system is developed for the first time. The analytic solution is obtained for thermal and diffusion equations without nonlinear terms responsible for the convective transfer. This spherically symmetric basic solution is applicable when the thermal flux from a planet core is weaker than or comparable to the adiabatic (radiative) flux. In the general case, by subtracting the basic solution, we simplified the inhomogeneous system of convective equations to obtain a completely homogeneous and dimensionless system. The latter system is controlled by two asymptotically small parameters: the Rossby number ε<10⁻⁵, which characterizes the relative value of differential rotation, and the generalized Eckman number E<10⁻¹², which characterizes the relative role of viscosity-diffusion effects during rapid rotation. The principal order of the solution for ε →0 and then for $\sqrt E \to 0$, for the transfer coefficients close to molecular coefficients, results in the basic flow, which is symmetric with respect to the rotation axis and directed predominantly along the azimuth. The basic-flow liquid ascends from a solid core along spirals inside an axial cylinder in contact with the equator of the solid core and descends in a narrow layer along the cylinder walls. The moment of viscous forces in the inner boundary Eckman layer provides a faster rotation of the inner solid core of terrestrial planets compared to a massive outer mantle due to the growth of the solid core at the expense of a low-density liquid core.