Large-scale balances and asymptotic scaling behaviour in spherical dynamos

The large-scale dynamics of convection-driven dynamos in a spherical shell, as relevant to the geodynamo, is analysed with numerical simulation data and asymptotic theory. An attempt is made to determine the asymptotic size (with the small parameter being the Ekman number, Ek) of the forces, and the associated velocity and magnetic fields. In agreement with previous work, the leading order mean force balance is shown to be thermal wind (Coriolis, pressure gradient and buoyancy) in the meridional plane and Coriolis-Lorentz in the zonal direction. The Lorentz force is observed to be weaker than the mean buoyancy force across a range of Ek and thermal forcing; the relative difference in these forces appears to be O(Ek^1/6) within the parameter space investigated. We find that the thermal wind balance requires that the mean zonal velocity scales as O(Ek^-1/3), whereas the meridional circulation is asymptotically smaller by a factor of O(Ek^1/6). The mean temperature equation shows a balance between thermal diffusion and the divergence of the convective heat flux, indicating the presence of a mean temperature length scale of size O(Ek^1/6). Neither the mean nor the fluctuating magnetic field show a strong dependence on the Ekman number, though the simulation data shows evidence of a mean magnetic field length scale of size O(Ek^1/6). A consequence of the asymptotic ordering of the forces is that Taylor's constraint is satisfied to accuracy O(Ek^1/6), despite the absence of a leading-order magnetostrophic balance. Further consequences of the force balance are discussed with respect to the large-scale flows thought to be important for the geodynamo.