Onset of convection in internally heated, temperature-dependent, power-law viscosity fluids at large viscosity contrasts

We use two-dimensional numerical simulations to investigate the finite-amplitude onset of convection in internally heated, infinite Prandtl number fluids with strongly temperature-dependent, power-law viscosity. We focus on the stagnant-lid regime which is relevant to planetary interiors. We find that convection can occur in both the usual, widespread convection planform, where the convection cells form in the entire layer, as well as the localized convection planform characterized by a single upwelling surrounded by a nearly stationary fluid. The critical Rayleigh number for the onset of convection by finite-amplitude perturbations is nearly independent of the convection planform in a broad range of the spacings between upwellings, from of the order of the depth of the layer to up to infinity. A simple heuristic analysis suggests scaling relationships which fit the numerical results in the stagnant-lid regime for an arbitrary power-law exponent, $n>1$. The results of this study provide new fluid dynamical constrains on the threshold for convection in planetary interiors.