Large-scale balances and asymptotic scaling behaviour in spherical dynamos

Abstract

Summary The large-scale dynamics of convection-driven dynamos in a spherical shell, as relevant to the geodynamo, is analyzed 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, 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(Ek1/6) 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(Ek1/6). 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(Ek1/6). 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(Ek1/6). A consequence of the asymptotic ordering of the forces is that Taylor’s constraint is satisfied to accuracy O(Ek1/6), 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.

Asymptotic behaviour of rotating convection-driven dynamos in the plane layer geometry

“…This behaviour is similar to that observed in spherical dynamo simulations, which show a saturation of the axisymmetric component of the magnetic field (Calkins et al. 2021; Orvedahl et al. 2021).…”

“…In the large-scale dynamo regime, the mean magnetic energy appears to saturate and then decreases slightly as Ra increases. This behaviour is similar to that observed in spherical dynamo simulations, which show a saturation of the axisymmetric component of the magnetic field (Calkins et al 2021;Orvedahl et al 2021).…”

“…grow with increasing Ra. Similar behaviour of the mean magnetic field is observed in spherical dynamos, where the azimuthally averaged component of the magnetic field shows a saturation with Rayleigh number (Calkins et al 2021;Orvedahl et al 2021). The exact cause of this saturation is not currently known, but it may be due to a breakdown of the α 2 -dynamo mechanism that operates only when Rm O(1) (Calkins et al 2015).…”

“…In contrast, simulations show that the large-scale dynamics in spherical dynamos is semi-magnetostrophic with a thermal wind balance in the meridional plane and a Coriolis–Lorentz force balance in the zonal direction (Aubert 2005; Calkins et al. 2021). This latter balance allows for saturation of the large-scale magnetic field via the Malkus–Proctor mechanism (Malkus & Proctor 1975).…”

“…Recent work in the spherical geometry points to a leading-order geostrophic balance on the small-scale motions (Yadav, Gastine & Christensen 2016; Aubert, Gastine & Fournier 2017; Schaeffer et al. 2017; Schwaiger, Gastine & Aubert 2021) and a semi-magnetostrophic force balance on the large scales (Aubert 2005; Calkins, Orvedahl & Featherstone 2021; Orvedahl, Featherstone & Calkins 2021). Here, the term ‘semi-magnetostrophic’ is used because the Lorentz force only enters the leading-order force balance in the zonal component of the large-scale momentum equation, and it is of smaller magnitude than the mean buoyancy force.…”