  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.
