Amplitude equations for weakly nonlinear two-scale perturbations of free hydromagnetic convective regimes in a rotating layer

TL;DR

This paper derives amplitude equations for weakly nonlinear, large-scale perturbations in free hydromagnetic convection within a rotating layer, highlighting differences from forced regime models and emphasizing anisotropic eddy effects.

Contribution

It introduces a novel system of amplitude equations for free convective regimes, accounting for anisotropic eddy diffusivity and advection, with a distinct structure from forced convection models.

Findings

Amplitude equations involve anisotropic eddy diffusivity and advection operators.

The system differs qualitatively from mean-field equations for forced regimes.

Equations for magnetic perturbations are evolutionary, others are not.

Abstract

Weakly non-linear stability of regimes of free hydromagnetic thermal convection in a rotating horizontal layer with free electrically conducting boundaries is considered in the Boussinesq approximation. Perturbations are supposed to involve large spatial and temporal scales. Applying methods for homogenisation of parabolic equations, we derive the system of amplitude equations governing the evolution of perturbations under the assumption that the alpha-effect is insignificant in the leading order. The amplitude equations involve the operators of anisotropic combined eddy diffusivity correction and advection. The system is qualitatively different from the system of mean-field equations for large-scale perturbations of forced convective hydromagnetic regimes. It is mixed: equations for the mean magnetic perturbation are evolutionary, all the rest involve neither time derivatives, nor the molecular diffusivity operator.