  Stably-stratified layers may be present at the top of the
electrically-conducting fluid layers of many planets either because the
temperature gradient is locally subadiabatic or because a stable composition
gradient is maintained by the segregation of chemical elements. Here we study
the double-diffusive processes taking place in such a stable layer, considering
the case of Mercury's core where the temperature gradient is stable but the
composition gradient is unstable over a 800km-thick layer. The large difference
in the molecular diffusivities leads to the development of buoyancy-driven
instabilities that drive radial flows known as fingering convection. We model
fingering convection using hydrodynamical simulations in a rotating spherical
shell and varying the rotation rate and the stratification strength. For small
Rayleigh numbers (i.e. weak background temperature and composition gradients),
fingering convection takes the form of columnar flows aligned with the rotation
axis and with an azimuthal size comparable with the layer thickness. For larger
Rayleigh numbers, the flows retain a columnar structure but the azimuthal size
is drastically reduced leading to thin sheet-like structures that are elongated
in the meridional direction. The azimuthal length decreases when the thermal
stratification increases, following closely the scaling law expected from the
linear non-rotating planar theory (Stern, 1960). We find that the radial flows
always remain laminar with local Reynolds number of order 1-10.
Equatorially-symmetric zonal flows form due to latitudinal variations of the
axisymmetric composition. The zonal velocity exceeds the non-axisymmetric
velocities at the largest Rayleigh numbers. We discuss plausible implications
for planetary magnetic fields.
