STEADY TETRAHEDRAL AND CUBIC PATTERNS OF SPHERICAL-SHELL CONVECTION WITH TEMPERATURE-DEPENDENT VISCOSITY

Steady thermal convection of an infinite Prandtl number, Boussinesq fl uid with temperature-dependent viscosity is systematically examined in a three-dimensional, basally heated spherical shell with isothermal a nd stress-free boundaries. Convective flows exhibiting cubic (l = 2, m = {0,4}) and tetrahedral (l = 3, m = 2) symmetry are generated with a finite-volume numerical model for various combinations of Rayleigh nu mber Ra (defined with viscosity based on the average of the boundary t emperatures) and viscosity contrast tau(mu) (ratio of maximum to minim um viscosities). The range of Ra for which these symmetric flows in sp herical geometry can be maintained in steady state is sharply reduced by even mild viscosity variations (tau(mu) less than or equal to 30), in contrast with analogous calculations in Cartesian geometry in which relatively simple, three-dimensional convective planforms remain stea dy for tau(mu) approximate to 10(4). The mild viscosity contrasts empl oyed place some solutions marginally in the sluggish-lid transition re gime in Ra-tau(mu) parameter space. Global heat transfer, given by the Nusselt number Nu, is found to obey a single power law relation with Ra when Ra is scaled by its critical value. A power law of the form Nu similar to (Ra/Ra-crit)(1/4) (Ra-crit is the minimum critical value o f Ra for the onset of convection) is obtained, in agreement with previ ous results for isoviscous spherical shell convection with cubic and t etrahedral symmetry. The calculations of this paper demonstrate that t emperature-dependent viscosity exerts a strong control on the nature o f three-dimensional convection in spherical geometry, an effect that i s likely to be even more important at Rayleigh numbers and viscosity c ontrasts more representative of the mantles of terrestrial planets. Th e robustness of the Nu-Ra relation, when scaled by Ra-crit, is importa nt for studies of planetary thermal history that rely on parameterizat ions of convective heat transport and account for temperature dependen ce of mantle viscosity.