Equatorial Symmetry of Chaotic Solutions in Boussinesq Convection in a Rotating Spherical Shell

Abstract

We investigate properties of convective solutions of the Boussinesq thermal convection in a moderately rotating spherical shell allowing the inner and outer sphere rotation 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 500, respectively. The inertial moments of the inner and outer spheres are fixed to about 0.22 and 100, respectively. The Rayleigh number is varied from 2.6 × 10 to 3.4 × 10. In this parameter range, convective solutions transit from equatorially symmetric quasiperiodic ones to equatorially asymmetric chaotic ones as the Rayleigh number is increased. The transition route in the system allowing rotation of both the spheres is different from that in the co-rotating system, which means the inner and outer spheres rotate with the same constant angular velocity: the convective solutions transit as equatorially symmetric quasi-periodic solution → equatorially symmetric chaotic solution → equatorially asymmetric chaotic solution in the system allowing both the spheres rotation, while equatorially symmetric quasi-periodic solution → equatorially asymmetric quasiperiodic solution → equatorially asymmetric chaotic solution in the co-rotating system. Keywords—thermal convection, numerical simulation, equatorial symmetry, quasi-periodic solution, chaotic solution