Force balances in spherical shell rotating convection

TL;DR

This study systematically analyzes force balances in spherical shell rotating convection models across various regimes, revealing distinct force behaviors and transitions, with implications for understanding planetary core dynamics.

Contribution

It provides a comprehensive comparison of integrated and scale-dependent force representations, highlighting the different force balances in radial, azimuthal, and co-latitudinal components across regimes.

Findings

Mean forces exhibit a primary thermal wind balance.

Fluctuating forces are in a quasi-geostrophic balance.

Curl forces transition between different force balances with increasing Rayleigh number.

Abstract

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 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 $Ra/Ra_c=1.2-1967$ and Ekman number $E=10^{-3}-10^{-6}$, with fixed Prandtl number $Pr=1$, no-slip and fixed flux boundaries. We exclude $10$ velocity boundary layers from the spherical shell boundaries, to get a consistent representation of the dynamics between the force and curled force balance. Radial, azimuthal and co-latitudinal components exhibit distinct balances. The total magnitudes of the mean forces and mean curled forces exhibit a primary thermal wind (TW) balance; the corresponding fluctuating forces are in a quasi-geostrophic (QG) primary balance, while the fluctuating curled forces transition from a Viscous-Archimedean-Coriolis balance to an Inertia-Viscous-Archimedean-Coriolis balance with increasing $Ra/Ra_c$. The curled force balances are weakly scale-dependent compared to the forces and do not show clear cross-over scales. The fluctuating force and curled force balances are consistent with three regimes of RC (weakly nonlinear, rapidly rotating, and weakly rotating), but do not exhibit sharp changes with $Ra/Ra_c$, which inhibits the identification of precise regime boundaries from these balances.