Thermal convection in rotating spherical geometry: A numerical overview of the transitions from periodic axisymmetric to temporally complex three-dimensional flows.

The aim of this work is to elucidate the type of transitions that take place when the periodic axisymmetric flows, which can set up at the onset of thermal convection in rotating fluid spheres, lose stability and to study the behavior of the new stable velocity fields until the flows become temporally chaotic. The computations for Prandtl numbers Pr=0.715, 0.1, and 0.01 show that when it decreases, the range of stability of these flows becomes narrower because the kinetic energy of the axisymmetric periodic solutions increases very fast, favoring their instability. From the stability analysis and direct three-dimensional simulations it is found that the transition to stable quasiperiodic flows through Neimark-Sacker bifurcations is supercritical when Pr≥0.01. For Pr=0.1 two branches of stable periodic flows emerging from the conduction state have been found due to the proximity to a double Hopf bifurcation. However, only the branches bifurcating from the azimuthal rotating waves are stable at large Rayleigh numbers. Far from this bifurcation the stable flows keep the influence of the axisymmetric dynamics up to large Rayleigh numbers. For small Pr they behave as repeated transients of mixed dynamics, controlled by the azimuthal wave numbers m=0, m=1, and m=2.