Symmetry-breaking bifurcations of central forced and heated convection in a spherical fluid shell

We study convection in a spherical shell under a gravitational force designed to mimic the GeoFlow microgravity experiment, using a combination of time-dependent simulation and path-following methods. With an outer radius which is twice that of the inner radius, the critical modes are spherical harmonics with $\ell=4$, leading generically to transcritical bifurcations involving axisymmetric and octahedral branches, in agreement with predictions by Michel, Ihrig \& Golubitsky, Chossat, Matthews, and Busse \& Riahi. A secondary bifurcation involving the $\ell=5$ mode leads to an additional seven-cell branch. All three steady patterns are simultaneously stable for $7 150 < Ra < 17 450$. For $Ra > 18 710$, simulations lead to time-dependent states, some periodic and some chaotic. The period varies greatly: some of the orbits belong to different branches and a global bifurcation is suspected of delimiting the lower limit of periodic states.