Comparison of quasi-geostrophic, hybrid and 3D models of planetary core convection

We present investigations of rapidly-rotating convection in a thick spherical shell geometry relevant to planetary cores, comparing results from Quasi-Geostrophic (QG), 3D and hybrid QG-3D models. The 170 reported calculations span Ekman numbers, Ek, between 10⁻⁴ and 10⁻¹⁰, Rayleigh numbers, Ra, between 2 and 150 times supercritical, and Prandtl numbers, Pr, between 10 and 10⁻². The default boundary conditions are no-slip at both the ICB and the CMB for the velocity field, with fixed temperatures at the ICB and the CMB. Cases driven by both homogeneous and inhomogeneous CMB heat flux patterns are also explored, the latter including lateral variations, as measured by Q*, the peak-to-peak amplitude of the pattern divided by its mean, taking values up to 5. The Quasi-Geostrophic (QG) model is based on the open-source pizza code. We extend this in a hybrid approach to include the temperature field on a 3D grid. In general, we find convection is dominated by zonal jets at mid-depths in the shell, with thermal Rossby waves prominent close to the outer boundary when the driving is weaker. For the thick spherical shell geometry studied here the hybrid method is best suited for studying convection at modest forcing, Ra ≤ 10 Ra_c when Pr = 1, and departs from the 3D model results at higher Ra, displaying systematically lower heat transport characterized by lower Nusselt and Reynolds numbers. We find that the lack of equatorially anti-symmetric and z-correlations between temperature and velocity in the buoyancy force contributes to the weaker flows in the hybrid formulation. On the other hand, the QG models yield broadly similar results to the 3D models, for the specific aspect ratio and range of Rayleigh numbers explored here. We cannot point to major disagreements between these two datasets at Pr ≥ 0.1, although the QG model is effectively more strongly driven than the hybrid case due to its cylindrically-averaged thermal boundary conditions. When Pr is decreased, the range of agreement between the Hybrid and 3D models expands, e.g. up to Ra ≤ 15 Ra_c at Pr = 0.1, indicating the hybrid method may be better suited to study convection in the regime Pr ≪ 1. We effectively observe two regimes: (i) at Pr ≥ 0.1 the QG and 3D models agree in the studied range of Ra/Ra_c while the hybrid model fails when Ra > 10 Ra_c; (ii) at Pr = 0.01 the QG and 3D disagree above Ra/Ra_c = 10 while the hybrid and 3D models agree fairly well up to Ra ∼ 20 Ra_c. Models that include laterally-varying heat flux at the outer boundary reproduce regional convection patterns that compare well with those found in similarly forced 3D models. Previously proposed scaling laws for rapidly-rotating convection are tested; our simulations are overall well described by a triple balance between Coriolis, inertia and Archimedean (CIA) forces with the length-scale of the convection following the diffusion-free Rhines-scaling. The Prandtl number, Pr, affects the number and the size of the jets with larger structures obtained at lower Pr; higher velocities and lower heat transport are also seen on decreasing Pr. The scaling behaviour of the convective velocity shows a strong dependence on Pr. This study is an intermediate step towards a hybrid model of core convection also including 3D magnetic effects.