Continuation of Double Hopf Points in Thermal Convection of Rotating Fluid Spheres

The thermal convection of rotating fluids in spherical geometry is a classical problem with application to many geophysical and astrophysical problems. The study of the transition to periodic solutions from the steady conduction state of a rotating and self-gravitating fluid sphere, heated uniformly from the inside, is discussed here. The continuation of double Hopf points is used to determine the region of the parameter space in which the first bifurcation is to solutions independent of the longitude (axisymmetric solutions). It is limited by three segments of curves separated by two triple Hopf points. This type of so-called torsional solutions was recently found, and it is shown here that they are the preferred solutions at the onset of convection for a wide range of fluids of Prandtl numbers, Pr, extending from Pr = 0 to Pr approximate to 0.9, which includes, for instance, liquid metals and gases. Although the corresponding interval of Ekman numbers, E, narrows when Pr -> 0, it is shown that there is always a small gap of parameters, relevant to geophysics and astrophysics, where the torsional solutions are preferred. The limits of the double Hopf curves when Pr -> 0 follow linear laws of the form E = c(m(1), m(2))Pr, c(m(1), m(2)) being constant depending on the two azimuthal wavenumbers, m(1) and m(2), of the eigenfunctions that define the double Hopf problem.