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We analyse the results of direct numerical simulations of rotating convectionin spherical shell geometries with stress-free boundary conditions, whichdevelop strong zonal flows. Both the Ekman number and the Rayleigh number arevaried. We find that the asymptotic theory for rapidly rotating convection canbe used to predict the Ekman number dependence of each term in the governingequations, along with the convective flow speeds and the dominant lengthscales. Using a balance between the Reynolds stress and the viscous stress,together with the asymptotic scaling for the convective velocity, we derive anasymptotic prediction for the scaling behaviour of the zonal flow with respectto the Ekman number, which is supported by the numerical simulations. We do notfind evidence of distinct asymptotic scalings for the buoyancy and viscousforces and, in agreement with previous results from asymptotic plane layermodels, we find that the ratio of the viscous force to the buoyancy forceincreases with Rayleigh number. Thus, viscosity remains non-negligible and wedo not observe a trend towards a diffusion-free scaling behaviour within therapidly rotating regime.

Asymptotic scaling relations for rotating spherical convection with strong zonal flows