Thermal convection in the internally heated sphere

We present the first nonlinear results on the problem of non-rotating thermal convection in an internally heated full sphere. A nonlinear stability analysis by the energy method yields that, at least for no-slip boundary conditions, the critical Rayleigh numbers for linear stability and nonlinear stability coincide. We then explore different ranges of the parameter regime using direct numerical simulations. We first report on the system behaviour for a fixed Prandtl number of unity and both stress-free and no-slip boundary conditions up to very high forcing, reaching Rayleigh number $Ra=2\times 10^{12}$ , approximately 250 million times the critical value ( $Ra_c$ ) for the onset of convection under no-slip conditions. For both boundary conditions, we observe a scaling for the advective heat transfer measured by the Nusselt number $Nu$ close to $Nu \sim Ra^{1/4}$ . This is consistent with a scaling prediction that we formulate analogously to the classical scaling in Rayleigh–Bénard convection. We then investigate the Prandtl number dependence at low to intermediate forcing for stress-free boundary conditions in the ranges $0.1 \leq Pr \leq 30$ and $Ra_c=3091\leq Ra \leq 3\times 10^5 \approx 100Ra_c$ . We find five distinct dynamical regimes depending on the Prandtl number, describe each regime individually and issue heuristic interpretations of the system behaviour where possible.