  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. We investigate
rotating double-diffusive convection (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
stabilising whereas the compositional one is destabilising. 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 $Ra \sim Ek^{-1}$, where $Ek$ is the Ekman number. This differs
from rotating convection in unstably stratified spheres, for which $Ra \sim
Ek^{-4/3}$. The domain of existence of inviscid convection thus increases as
$Ek^{-1/3}$. Second, we perform nonlinear 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
anti-symmetric, 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
nonlinear 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.
