Asymptotics of spherical dynamos exhibiting a small-scale MAC balance

TL;DR

The paper analyzes how convection-driven spherical dynamos reach MAC balance on convective scales and contrasts this with non-magnetic convection, focusing on Ekman-number dependence. Using a spherical-shell model with $Pm=2$ (and variable $Pm$ in parts), it demonstrates that velocity follows quasi-geostrophic scaling while buoyancy scales with Coriolis in dynamos, leading to CIA-like flow speeds; a consistent $O(Ek^{1/3})$ velocity-length scale arises alongside $O(Ek^{1/6})$ for energy- and dissipation-related length scales. The results show that advection remains comparable to inertia and viscous terms, indicating persistent nonlinear convective Rossby waves even in MAC regimes, and reveal distinct temperature and dissipation scalings between dynamo and non-magnetic cases due to Ohmic dissipation. Collectively, these findings advance understanding of ambipolar-like MAC dynamics in spherical geometries and provide scaling insights for extrapolating geodynamo behavior to Earth-like conditions.

Abstract

Understanding the asymptotic behaviour of numerical dynamo models is critical for extrapolating results to the physical conditions that characterise terrestrial planetary cores. Here we investigate the behaviour of convection-driven dynamos reaching a MAC (magnetic-Archimedes-Coriolis) balance on the convective length scale and compare the results with non-magnetic convection cases. In particular, the dependence of physical quantities on the Ekman number, $Ek$, is studied in detail. The scaling of velocity dependent quantities is observed to be independent of the force balance and in agreement with quasi-geostrophic theory. The primary difference between dynamo and non-magnetic cases is that the fluctuating temperature is order unity in the former such that the buoyancy force scales with the Coriolis force. The MAC state yields a scaling for the flow speeds that is identical to the so-called CIA (Coriolis-inertia-Archimedes) scaling. There is an $O(Ek^{1/3})$ length scale present within the velocity field irrespective of the leading order force balance. This length scale is consistent with the asymptotic scaling of the terms of the governing equations and is not an indication that viscosity plays a dominant role. The peak of the kinetic energy spectrum and the ohmic dissipation length scale both exhibit an Ekman number dependence of approximately $Ek^{1/6}$, which is consistent with a scaling of $Rm^{-1/2}$, where $Rm$ is the magnetic Reynolds number. For the dynamos, advection remains comparable to, and scales similarly with, both inertia and viscosity, implying that nonlinear convective Rossby waves play an important role in the dynamics even in a MAC regime.

Figures (14)

• Figure 1: Influence of the magnetic Prandtl number ($Pm$) for fixed Ekman number ($Ek=10^{-5}$) and Rayleigh number ($Ra=1.5\times 10^{8}$) ($\widetilde{Ra} = Ra Ek^{4/3} = 32.3$): (a) radial components of the fluctuating forces; (b) viscous ($\epsilon_u$) and magnetic ($\epsilon_b$) dissipation. Points to the left of the vertical dashed lines are non-magnetic.
• Figure 2: Reynolds number for both dynamo (filled symbols) and non-magnetic (open symbols) cases: (a) convective Reynolds number; (b) rescaled convective Reynolds number; (c) rescaled convective Reynolds number versus supercriticality, $Ra/Ra_c$; (d) compensated convective Reynolds number; (e) rescaled zonal Reynolds number; (f) compensated zonal Reynolds number.
• Figure 3: Forces versus $\widetilde{Ra}$ for (a) non-magnetic cases and (b) dynamo cases. The Ekman number is fixed at $Ek=10^{-5}$.
• Figure 4: (a) Advection and (b) viscous force from the fluctuating radial momentum equation for all cases. The insets show the raw data with no rescaling. The symbols are the same as defined in figure \ref{['F:Rec']}.
• Figure 5: (a) Buoyancy force; (b) buoyancy force rescaled by $Ek$; (c) buoyancy force rescaled by $Ek^{4/3}$ for the dynamo cases; (d) ratio of the buoyancy force to the Coriolis force. The filled symbols denote dynamo cases and the unfilled symbols denote non-magnetic cases.