Subcritical convection in a rapidly rotating sphere at low Prandtl numbers

We study non-linear convection in a low Prandtl number fluid ($Pr = 0.01-0.1$) in a rapidly rotating sphere with internal heating. We use a numerical model based on the quasi-geostrophic approximation, in which variations of the axial vorticity along the rotation axis are neglected, whereas the temperature field is fully three-dimensional. We identify two separate branches of convection close to onset: (i) a well-known weak branch for Ekman numbers greater than $10^{-6}$, which is continuous at the onset (supercritical bifurcation) and consists of a superposition of thermal Rossby waves, and (ii) a novel strong branch at lower Ekman numbers, which is discontinuous at the onset. The strong branch becomes subcritical for Ekman numbers of the order of $10^{-8}$. On the strong branch, the Reynolds number of the flow is greater than $10^3$, and a strong zonal flow with multiple jets develops, even close to the non-linear onset of convection. We find that the subcriticality is amplified by decreasing the Prandtl number. The two branches can co-exist for intermediate Ekman numbers, leading to hysteresis ($Ek = 10^{-6}$, $Pr = 0.01$). Non-linear oscillations are observed near the onset of convection for $Ek = 10^{-7}$ and $Pr = 0.1$.