Force balances in spherical shell rotating convection

Significant progress has been made in understanding planetary core dynamics using numerical models of rotating convection (RC) in spherical shell geometry. However, the behaviour of forces in these models within various dynamic regimes of RC remains largely unknown. Directional anisotropy, scale dependence and the role of dynamically irrelevant gradient contributions in incompressible flows complicate the representation of dynamical balances in spherical shell RC. In this study, we systematically compare integrated and scale-dependent representations of mean and fluctuation forces and curled forces (which contain no gradient contributions) separately for the three components ( $\hat {r},\hat {\theta },\hat {\phi }$ ). The analysis is performed with simulations in a range of convective supercriticality $Ra_T/Ra_T^{c}=1.2{-}297$ where $Ra_T$ and $Ra^{c}_T$ are the Rayleigh and critical Rayleigh numbers, respectively and Ekman number $E=10^{-3}{-}10^{-6}$ , with fixed Prandtl number $Pr=1$ , along with no-slip and fixed flux boundaries. We have excluded regions from each boundary of the spherical shell, with a thickness equivalent to ten velocity boundary layers, which provides a consistent representation of the bulk dynamics between the volume-averaged force and curled force balance in the parameter space studied. Radial, azimuthal and co-latitudinal components exhibit distinct force and curled force balances. The total magnitudes of the mean forces and mean curled forces exhibit a primary thermal wind balance; the corresponding fluctuating forces are in a quasi-geostrophic primary balance, while the fluctuating curled forces transition from a Viscous–Archimedean–Coriolis balance to an Inertia–Viscous–Archimedean–Coriolis balance with increasing $Ra_T/Ra_T^{c}$ . The curled force balances are more weakly scale-dependent compared to the forces, and do not show clear cross-over length scales. The fluctuating force and curled force balances are broadly consistent with three regimes of RC (weakly nonlinear, rapidly rotating and weakly rotating), but do not exhibit sharp changes with $Ra_{T}/Ra_{T}^{c}$ , which inhibits the identification of precise regime boundaries from these balances.