Onset of Rayleigh-Bénard convection between spherical shells for different gravity profiles

Rayleigh-Bénard convection is a fundamental phenomenon that is widely present in engineering, geophysics, and astrophysics. A spherical geometry is particularly of interest in geo- and astrophysics, such as the Earth's mantle and the convection within the Sun. In the present study, we investigate how the radial gravity profile affect the onset by considering g(r) ∼ r^a for a = 0, 1, -2. We formulate the problem in the Boussinesq approximation for non-rotating spherical shells. The fluid is heated at the inner shell and cooled at the outer shell. The radius ratio of inner and outer shells, η, varies from 0.2 to 0.7, which implicitly indicates the change of shell curvature. The choice of Rayleigh number spans from O(10³) to O(10⁶) near the onset of laminar convection, with Prandtl number fixed at unity. To predict the laminar onset, we first carry out a linear stability analysis to obtain the critical Rayleigh number and critical modes. Then we examine how the radius ratio and gravity profile affect the critical Rayleigh number. Furthermore, nonlinear evolutions are investigated through numerical simulations using spectral method. The numerical result further reveals that the ratio of inner/outer thermal boundary layer thickness exhibits different dependencies on η for different gravity profiles. We also investigate how the gravity profile influences the buoyancy-driven power, which in turn affects the thermal convection.