Bifurcations to quasiperiodicity of the torsional solutions of convection in rotating fluid spheres: Techniques and results

Abstract

The linear stability of the periodic and axisymmetric solutions of the convection in rotating, internally heated, and self-gravitating fluid spheres is presented. The transition to quasiperiodic flows via Neimark–Sacker bifurcations of different azimuthal wave numbers, m, is studied using matrix-free continuation and Floquet theory. Several pairs of Ekman and Prandtl numbers are considered in the region where the first bifurcation from the conduction state gives rise to the axisymmetric solutions. It is shown that the azimuthal wave numbers m = 1 and m = 2 are preferred and that, for small Ekman and Prandtl numbers, the secondary bifurcations to different m accumulate close to the onset of convection. This study confirms some results previously found just by direct simulations. The methods presented can be applied to systems of parabolic partial differential equations with O(2) or SO(2) symmetry group, when a periodic orbit, invariant under the group, loses stability through a Neimark–Sacker bifurcation.