Rotating double-diffusive convection in stably stratified planetary cores

In planetary fluid cores, the density depends on temperature and chemical composition, which diffuse at very different rates. This leads to various instabilities, bearing the name of double-diffusive convection (DDC). We investigate rotating DDC (RDDC) in fluid spheres. We use the Boussinesq approximation with homogeneous internal thermal and compositional source terms. We focus on the finger regime, in which the thermal gradient is stabilizing whereas the compositional one is destabilizing. First, we perform a global linear stability analysis in spheres. The critical Rayleigh numbers drastically drop for stably stratified fluids, yielding large-scale convective motions where local analyses predict stability. We evidence the inviscid nature of this large-scale double-diffusive instability, enabling the determination of the marginal stability curve at realistic planetary regimes. In particular, we show that in stably stratified spheres, the Rayleigh numbers Ra at the onset evolve like Ra similar to Ek(-1), where Ek is the Ekman number. This differs from rotating convection in unstably stratified spheres, for which Ra similar to Ek(-4/3). The domain of existence of inviscid convection thus increases as Ek(-1/3). Secondly, we perform non-linear simulations. We find a transition between two regimes of RDDC, controlled by the strength of the stratification. Furthermore, far from the RDDC onset, we find a dominating equatorially antisymmetric, large-scale zonal flow slightly above the associated linear onset. Unexpectedly, a purely linear mechanism can explain this phenomenon, even far from the instability onset, yielding a symmetry breaking of the non-linear flow at saturation. For even stronger stable stratification, the flow becomes mainly equatorially symmetric and intense zonal jets develop. Finally, we apply our results to the early Earth core. Double diffusion can reduce the critical Rayleigh number by four decades for realistic core conditions. We suggest that the early Earth core was prone to turbulent RDDC, with large-scale zonal flows.