Boundary layer asymmetry in turbulent spherical Rayleigh–Bénard convection: combined dependence on Prandtl number and radius ratio

Direct numerical simulations (DNS) are performed to investigate the dependence of the Prandtl number ( $\textit{Pr}$ ) and radius ratio ( $\eta =r_{i}/r_{o}$ ) on the asymmetry of the mean temperature radial profiles in turbulent Rayleigh–Bénard convection (RBC) within spherical shells. Unlike planar RBC, the temperature drop, and the thermal and viscous boundary layer thicknesses, at the inner and outer boundaries are not identical in spherical shells. These differences in the boundary layer properties in spherical RBC contribute to the observed asymmetry in the radial profiles of temperature and velocity. The asymmetry originates from the differences in curvature and gravity at the two boundaries, and in addition, is influenced by $\textit{Pr}$ . To investigate the $\eta$ and $\textit{Pr}$ dependence of these asymmetries, we perform simulations of Oberbeck–Boussinesq convection for $\eta = 0.2,0.6$ and $0.1 \leqslant Pr \leqslant 50$ , and for a range of Rayleigh numbers ( $Ra$ ) varying between $5 \times 10^{6}$ and $5 \times 10^{7}$ . The Prandtl numbers that we choose cover a broad range of geophysical relevance, from low- $\textit{Pr}$ regimes ( $\textit{Pr}=0.1$ ) representative of gas giants such as Jupiter and Saturn, to high- $\textit{Pr}$ regimes characteristic of organic flows used in the convection experiments ( $\textit{Pr}=50$ ). A centrally condensed mass, with the gravity profile $g \sim 1/r^{2}$ , is employed in this study. Our results show that the asymmetry at smaller $\eta$ exhibits a stronger $\textit{Pr}$ dependence than at larger $\eta$ . Various assumptions for quantifying this asymmetry are evaluated, revealing that different assumptions are valid in different $\textit{Pr}$ regimes. It is shown that the assumption of the equal characteristic plume separation at the inner and outer boundaries, as well as the assumption of the identical thermal fluctuation scales between the two boundary layers, is valid only for $0.2 \lesssim Pr \lesssim 1$ . In contrast, assumptions based on the equivalency of the local thermal boundary layer Rayleigh numbers and laminar natural-convective boundary layers are validated at $\textit{Pr}=50$ for the explored parameter space. Furthermore, new assumptions based on the statistical analysis of the inter-plume islands are proposed for $\textit{Pr}=0.1$ and $50$ , and these are validated against the DNS data. These findings provide insights into the $(Pr,\eta)$ dependence of asymmetry in spherical RBC, and offer a framework for studying similar systems in geophysical and astrophysical contexts.