The fastest growing modes in rotating magnetoconvection with anisotropic diffusivities and their links to the Earth's core conditions

Abstract

ABSTRACT The fastest growing modes in anisotropic rotating magnetoconvection (RMC) processes are presented. After reminding the state of the art, we present our new approach that applies to isotropic as well as anisotropic diffusivities' conditions. We describe three cases: T (only temperature perturbation is time dependent), Q (temperature and velocity field perturbations are time dependent, but magnetic field perturbation is time independent) and G (temperature, magnetic and velocity field perturbations are time dependent). Isotropies as well as anisotropies are further distinguished by values of molecular and turbulent diffusivities. We show that T case does not describe properly convection in the Earth's outer core conditions, because it implies too huge Ekman numbers for the transition between the RMC modes of weak and strong magnetic field types. In G case, the convection is usually much more facilitated than in the T case: instabilities may arise with much smaller values of Ekman number and, in general, all types of convections occur with values of the physical parameters of the Earth consistent with the most reliable estimations. We demonstrate (and indicate) that Q and G cases can be well suited for the magnetic field of the Earth (and for other planetary magnetic fields), but only G case may correspond to turbulent stay of the Earth's core. We prove that, analogously like in the marginal modes, the value of anisotropic parameter α (the ratio between horizontal and vertical diffusivities) crucially influences the convection. The cases of α < 1 ( 1 $ ]]> α > 1 ) strongly decrease (increase) the Ekman numbers at which the RMC modes of weak and strong magnetic field types change between each other. Finally, we show and stress that not all types of anisotropies in the fastest growing modes can be equally strong. More specifically, we show that a fixed Rayleigh number puts a constraint on the maximum value of α, but do not put any lower positive limit on the minimum value of α. This special constraint is given by the necessary positiveness of the growth rate of the fastest growing modes. Our RMC 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.