Destruction of advective heat flux by toroidal flow in a non-rotating internally heated full sphere

We study thermal convection in a non-rotating internally heated full sphere at a Prandtl number of unity with fixed temperature boundary conditions using 3D Direct Numerical Simulations. The onset mode of this system is purely poloidal, steady-state and has spherical harmonic degree l=1 both for stress-free and no-slip boundary conditions. As the Rayleigh number is increased to slightly supercritical values, the poloidal flow intensifies, but no toroidal flow is produced. We analytically demonstrate the existence of a large class of poloidal modes whose non-linear interactions do not generate toroidal flow. The onset of toroidal flow occurs as a secondary instability at roughly 10Ra_c. In the case of stress-free boundary conditions, this leads to a decrease in the Nusselt number measuring the strength of advective heat flux. The resulting flow is still steady-state. As the Rayleigh number is increased further, there is a range of the Rayleigh number over which the share of kinetic energy of the toroidal flow continues to grow and the Nusselt number continues to decrease. We believe that this effect, which occurs only for stress-free boundary conditions, has not been demonstrated before in 3D simulations.