Continuation and stability of convective modulated rotating waves in spherical shells

Modulated rotating waves (MRW), bifurcated from the thermal-Rossby waves that arise at the onset of convection of a fluid contained in a rotating spherical shell, and their stability, are studied. For this purpose, Newton-Krylov continuation techniques are applied. Nonslip boundary conditions, an Ekman number E=10^{-4}, and a low Prandtl number fluid Pr=0.1 in a moderately thick shell of radius ratio η=0.35, differentially heated, are considered. The MRW are obtained as periodic orbits by rewriting the equations of motion in the rotating frame of reference where the rotating waves become steady states. Newton-Krylov continuation allows us to obtain unstable MRW that cannot be found by using only time integrations, and identify regions of multistability. For instance, unstable MRW without any azimuthal symmetry have been computed. It is shown how they become stable in a small Rayleigh-number interval, in which two branches of traveling waves are also stable. The study of the stability of the MRW helps to locate and classify the large sequence of bifurcations, which takes place in the range analyzed. In particular, tertiary Hopf bifurcations giving rise to three-frequency stable solutions are accurately determined.