Fingering convection in a spherical shell

We use 123 three-dimensional direct numerical simulations to study fingering convection in non-rotating spherical shells. We investigate the scaling behaviour of the flow length scale, the non-dimensional heat and compositional fluxes Nu and Sh and the mean convective velocity over the fingering convection instability domain defined by 1 R_ < Le, R_ being the ratio of density perturbations of thermal and compositional origins and Le the Lewis number. We show that the chemical boundary layers are marginally unstable and adhere to the laminar Prandtl–Blasius model, hence explaining the asymmetry between the inner and outer spherical shell boundary layers. We develop scaling laws for two asymptotic regimes close to the two edges of the instability domain, namely R_ Le and R_ 1. For the former, we develop novel power laws of a small parameter measuring the distance to onset, which differ from theoretical laws published to date in Cartesian geometry. For the latter, we find that the Sherwood number Sh gradually approaches a scaling Sh Ra_ ^1/3 when Ra_ 1 ; and that the Péclet number accordingly follows Pe Ra_ ^2/3 |RaT|^-1/4, RaT and Ra_ being the thermal and chemical Rayleigh numbers. When the Reynolds number exceeds a few tens, we report on a secondary instability which takes the form of large-scale toroidal jets which span the entire spherical domain. Jets distort the fingers, resulting in Reynolds stress correlations, which in turn feed the jet growth until saturation. This nonlinear phenomenon can yield relaxation oscillation cycles.