The most unstable modes in rotating magnetoconvection with anisotropic diffusion in the Earth’s outer core

It was often shown how the anisotropy (due to turbulence) in the Earth’s outer core strongly influences some convection processes very important in the Core Dynamics. For instance, it was described how some instabilities in rotating magnetoconvection, described as usually by the analysis in term of normal modes, depend strictly on the anisotropic diffusion. Thus, we developed many models concerning the marginal modes (stationary and oscillating modes) of rotating magnetoconvection with different cases of anisotropy in the viscosity, thermal and magnetic diffusivities. In all cases, an anisotropy greater in the vertical direction parallel to gravity (“atmospheric anisotropy”) facilitates the convection, while an anisotropy greater in horizontal directions (“oceanic anisotropy”) inhibits some types of convection. This is linked with the balance among Magnetic, Archimedean and Coriolis forces in the Earth’s outer core.  

After recalling these former results concerning marginal modes, we present new results concerning the most unstable modes, namely the ones with maximum growth rate, with isotropic and anisotropic diffusivities.

Firstly, the state of the art about this topic in isotropic conditions is reminded, then our new approach on it is presented. We show that assuming a time-dependence only in the temperature perturbation (we call it T-case), like it was done in some former works, does not describe properly these modes in the Earth’s outer core. Indeed, this implies that some types of convection would occur only with some values of the dimensionless numbers unrealistic for the Earth (e.g., with too huge values of the Ekman numbers). We study the most general isotropic case (and we christen it G-case), namely the most unstable modes of convection with temperature, velocity and magnetic perturbations time-dependent. In this case the convection is much more facilitated than in the T-case: it occurs with much smaller values of Ekman and Elsasser numbers. Another model (named by us Q-case) with very small magnetic Prandtl number, namely with magnetic diffusivity much greater than viscosity, is considered. The Q-case results are very similar to the G-case ones. We demonstrate (and indicate) that Q and G cases can hold for the Earth (and for other planets).

We show that the anisotropy strongly influences the most unstable modes. Indeed, like in the marginal ones, the atmospheric anisotropy facilitates the occurrence of the most unstable modes convection, while the oceanic one inhibits it. Furthermore, we prove that, in contrast with isotropic case, in case of strong oceanic anisotropy the differences between Q and G cases can be significant for the Geodynamo.

Our approach allows to easily deal with very huge wave numbers and Rayleigh numbers as well as with very small Ekman numbers, what is usually not possible in the standard geodynamo simulations. This aspect and the growth rates search are useful to look for possible connections with small length and time scale analysis of the Geomagnetic field.