The (3,4) Mode Interaction in the GeoFlow Framework

The problem of convection in a self‐gravitating spherical shell of fluid is commonly encountered in sciences like astrophysics and geophysics (earth's liquid core). The GEOFLOW‐experiment is a project of the European Space Agency in order to perform the spherical Rayleigh‐Bénard convection problem on the International Space Station in a micro‐gravity environment: the central force field is simulated by a dielectrophoretic one. Beyond a critical Rayleigh number Ra_c, generically an unique spherical ℓ mode becomes unstable and only stationary or travelling waves solutions are expected near the onset. But, for a critical aspect ratio η_c two consecutive modes (ℓ, ℓ + 1) are unstable. The (1,2) and (2,3) interactions have showed a rich bifurcation diagram, in particular, we have found heteroclinic cycles predicted by the theoretical study. Because of the experiment requirements, only the (3,4) one is possible. So, this paper purposes to analyse this bifurcation in non‐rotating case in the GEOFLOWframework using the theory of bifurcation with the spherical symmetry.