Spherical convective dynamos in the rapidly rotating asymptotic regime

Abstract

Self-sustained convective dynamos in planetary systems operate in an asymptotic regime of rapid rotation, where a balance is thought to hold between the Coriolis, pressure, buoyancy and Lorentz forces (the MAC balance). Classical numerical solutions have previously been obtained in a regime of moderate rotation where viscous and inertial forces are still significant. We define a unidimensional path in parameter space between classical models and asymptotic conditions from the requirements to enforce a MAC balance and to preserve the ratio between the magnetic diffusion and convective overturn times (the magnetic Reynolds number). Direct numerical simulations performed along this path show that the spatial structure of the solution at scales larger than the magnetic dissipation length is largely invariant. This enables the definition of large-eddy simulations resting on the assumption that small-scale details of the hydrodynamic turbulence are irrelevant to the determination of the large-scale asymptotic state. These simulations are shown to be in good agreement with direct simulations in the range where both are feasible, and can be computed for control parameter values far beyond the current state of the art, such as an Ekman number E = 10 −8 . We obtain strong-field convective dynamos approaching the MAC balance and a Taylor state to an unprecedented degree of accuracy. The physical connection between classical models and asymptotic conditions is shown to be devoid of abrupt transitions, demonstrating the asymptotic relevance of classical numerical dynamo mechanisms. The fields of the system are confirmed to follow diffusivity-free, power-based scaling laws along the path.

The cross-over to magnetostrophic convection in planetary dynamo systems

“…Thus, it is hypothesized that a magnetostrophically balanced convective state can exist only on scales below X ( figure 9). Because the system-scale value of Λ is typically less than unity in planetary dynamo models (table 1), our analysis implies that their large-scale convective flows must be in quasigeostrophic balance [28,34,56]. Estimates suggest that X < 1 for planets as well (table 2).…”

Stochastic generation of MAC waves and implications for convection in Earth’s core

“…Finally, the induction source depends onV andB , as well as a length scale d I and a time scale τ I = d I /V , which is taken to be an advective timescale. Combining these results gives Aubert et al (2017) argue that the length scales in (55)-(57) (denoted collectively as d ⊥ ) are nearly invariant once the dynamo solution reaches a MAC force balance with realistic values for Rm and . Fixing these length scales allows us to predict the change in the amplitude of the sources from Solution 2 to Solution 1 using only the values from Table 1 (with Ro = RmEPm −1 ).…”

“…Several things could alter the preceding estimates. First, convective temperature anomalies are predicted to become smaller as conditions become more realistic (Aubert et al 2017). Such small anomalies would not penetrate as deeply into the stratified layer and the source would be confined to the lowermost region of the stratified layer.…”

“…A second consideration concerns the uncertainty in estimates for . Numerical models that achieve realistic values for Rm = 10 3 often produce values for between 20 and 30 (Aubert et al 2017), depending on the nature of the convective forcing (see Solution 5). Larger values for (relative to our choice of = 10) would imply a larger wave amplitude for both magnetic and induction sources, although the greatest increase occurs in the magnetic source.…”

“…However, there is no assurance that the amplitudes of the individual forces achieve Earth-like values. A recent study of Aubert et al (2017) confronts this question by formulating a strategy for extrapolating numerical solutions to Earth-like conditions while maintaining the so-called MAC force balance. A surprising outcome of applying this general approach to the problem of MAC waves is that the dominant source terms from the dynamo model have a greatly reduced role once the results are extrapolated to more realistic conditions for Earth's core.…”