Equatorial symmetry of Boussinesq convective solutions in a rotating spherical shell allowing rotation of the inner and outer spheres

We investigate properties of convective solutions of the Boussinesq thermal convection in a moderately rotating spherical shell allowing the respective rotation of the inner and outer spheres due to the viscous torque of the fluid. The ratio of the inner and outer radii of the spheres, the Prandtl number, and the Taylor number are fixed to 0.4, 1, and 5002, respectively. The Rayleigh number is varied from 2.6 × 104 to 3.4 × 104. In this parameter range, the behaviours of obtained asymptotic convective solutions are almost similar to those in the system whose inner and outer spheres are restricted to rotate with the same constant angular velocity, although the difference is found in the transition process to chaotic solutions. The convective solution changes from an equatorially symmetric quasi-periodic one to an equatorially symmetric chaotic one, and further to an equatorially asymmetric chaotic one, as the Rayleigh number is increased. This is in contrast to the transition in the system whose inner and outer spheres are assumed to rotate with the same constant angular velocity, where the convective solution changes from an equatorially symmetric quasi-periodic one, to an equatorially asymmetric quasi-periodic one, and to equatorially asymmetric chaotic one. The inner sphere rotates in the retrograde direction on average in the parameter range; however, it sometimes undergoes the prograde rotation when the convective solution becomes chaotic