Fingering convection in a spherical shell

We use 120 three dimensional direct numerical simulations to study fingering convection in non-rotating spherical shells. We investigate the scaling behaviour of the flow lengthscale, mean velocity and transport of chemical composition over the fingering convection instability domain defined by $1 \leq R_\rho \leq Le$, $R_\rho$ being the ratio of density perturbations of thermal and compositional origins. We show that the horizontal size of the fingers is accurately described by a scaling law of the form $\mathcal{L}_h/d \sim |Ra_T|^{-1/4} (1-\gamma)^{-1/4}/\gamma^{-1/4}$, where $d$ is the shell depth, $Ra_T$ the thermal Rayleigh number and $\gamma$ the flux ratio. Scaling laws for mean velocity and chemical transport are derived in 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 show that the transport follows power laws of a small parameter $\epsilon^\star$ measuring the distance to onset. For the latter, we find that the Sherwood number $Sh$, which quantities the chemical transport, 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, a secondary instability may occur taking the form of large-scale toroidal jets. 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.