Numerical study of the onset of thermosolutal convection in rotating spherical shells

The influence of an externally enforced compositional gradient on the onset of convection of a mixture of two components in a rotating fluid spherical shell is studied for Ekman numbers E = 10(-3) and E = 10(-6), Prandtl numbers sigma = 0.1, 0.001, Lewis numbers tau = 0.01, 0.1, 0.8, and radius ratio eta = 0.35. The Boussinesq approximation of the governing equations is derived by taking the denser component of the mixture for the equation of the concentration. Differential and internal heating, an external compositional gradient, and the Soret and Dufour effects are included in the model. By neglecting these two last effects, and by considering only differential heating, it is found that the critical thermal Rayleigh number R-e(c) depends strongly on the direction of the compositional gradient. The results are compared with those obtained previously for pure fluids of the same sigma. The influence of the mixture becomes significant when the compositional Rayleigh number R-c is at least of the same order of magnitude as the known R-e(c) computed without mixture. For positive and sufficiently large compositional gradients, R-e(c) decreases and changes sign, indicating that the compositional convection becomes the main source of instability. Then the critical wave number m(c) decreases, and the drifting waves slow down drastically giving rise to an almost stationary pattern of convection. Negative gradients delay the onset of convection and determine a substantial increase of m(c) and omega(c) for R-c sufficiently high. Potential laws are obtained numerically from the dependence of R-e(c) and of the critical frequency omega(c) on R-c, for the moderate and small Ekman numbers explored. (C) 2012 American Institute of Physics. [http://dx.doi.org/10.1063/1.4723865]