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 lengthscale, the non-dimensional heat and compositional fluxes $Nu$ and $Sh$ and the mean convective velocity over the fingering convection instability domain defined by $1 \leq R_\rho < Le$, $R_\rho$ 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 $R_\rho \lesssim Le$ and $R_\rho \gtrsim 1$. For the former, we develop novel power laws of a small parameter $\epsilon$ 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 $Sh\sim Ra_\xi^{1/3}$ when $Ra_\xi \gg 1$; and that the P\'eclet number accordingly follows $Pe \sim Ra_\xi^{2/3} |Ra_T|^{-1/4}$, $Ra_\xi$ being the chemical Rayleigh number. 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.