A multiscale dynamo model driven by quasi-geostrophic convection

Abstract

A convection-driven multiscale dynamo model is developed in the limit of low Rossby number for the plane layer geometry in which the gravity and rotation vectors are aligned. The small-scale fluctuating dynamics are described by a magnetically-modified quasigeostrophic equation set, and the large-scale mean dynamics are governed by a diagnostic thermal wind balance. The model utilizes three timescales that respectively characterize the convective timescale, the large-scale magnetic evolution timescale, and the large-scale thermal evolution timescale. Distinct equations are derived for the cases of order one and low magnetic Prandtl number. It is shown that the low magnetic Prandtl number model is characterized by a magnetic to kinetic energy ratio that is asymptotically large, with ohmic dissipation dominating viscous dissipation on the large-scales. For the order one magnetic Prandtl number model the magnetic and kinetic energies are equipartitioned and both ohmic and viscous dissipation are weak on the large-scales; large-scale ohmic dissipation occurs in thin magnetic boundary layers adjacent to the solid boundaries. For both magnetic Prandtl number cases the Elsasser number is small since the Lorentz force does not enter the leading order force balance. The new models can be considered fully nonlinear, generalized versions of the dynamo model originally developed by Childress and Soward [Phys. Rev. Lett., 29, p.837, 1972]. These models may be useful for understanding the dynamics of convection-driven dynamos in regimes that are only just becoming accessible to direct numerical simulations.

Convection-driven kinematic dynamos at low Rossby and magnetic Prandtl numbers

“…Dormy [19] argues that a distinguished limit should be sought where both P m and Ek are small with 1 ≫ P m ≫ Ek for the dynamics to begin to replicate that of planetary interiors. As shown by Calkins et al [10], the dynamo utilized here, in contrast with virtually all numerical models of planetary dynamos, does satisfy this inequality.…”

“…A brief overview of the derivation and main features of the QGDM are given in the present section; for a detailed discussion the reader is referred to Calkins et al [10]. …”

“…Like the asymptotic theory utilized in the development of the QGDM, these previous investigations required scale separation to justify the splitting of the dependent variables into mean and fluctuating components. In Calkins et al [10] it was shown that scale separation can also be assured when Rm = O(1) (i.e. P m = O(1)) for the case of rapidly rotating, anisotropic motions, though for this case α can no longer be written in a form as simple as that given by equation (32).…”

“…As a result of these simplifications, reduced models can provide enhanced physical interpretation and insight into observed hydrodynamic and magnetohydrodynamic processes. In the present work we utilize an asymptotic model recently developed by Calkins et al [10] to examine the behavior of convection-driven kinematic dynamo action in the limit of rapid rotation.…”

“…The quasi-geostrophic dynamo model (QGDM) recently developed by Calkins et al [10] is a fully nonlinear generalization of the Childress-Soward dynamo model, and is valid in the limit (Ro, Ek) → 0 and Re = O(1), where we stress again that this is the Reynolds number based on the small, convective scale. The QGDM can thus be used to explore both linear (i.e.…”