Mean-field equations for weakly nonlinear two-scale perturbations of forced hydromagnetic convection in a rotating layer

TL;DR

This paper derives mean-field equations for weakly nonlinear, large-scale perturbations in rotating hydromagnetic convection, introducing new anisotropic terms like eddy diffusivity and advection, with methods to evaluate their coefficients.

Contribution

It extends standard hydromagnetic convection equations by including new anisotropic eddy effects and non-local operators, especially near symmetry-breaking bifurcations.

Findings

Derived generalized mean-field equations with new eddy terms.

Identified conditions for the emergence of alpha-effect and non-local operators.

Presented an efficient method to compute coefficients of new terms.

Abstract

We consider stability of regimes of hydromagnetic thermal convection in a rotating horizontal layer with free electrically conducting boundaries, to perturbations involving large spatial and temporal scales. Equations governing the evolution of weakly nonlinear mean perturbations are derived under the assumption that the alpha-effect is insignificant in the leading order (e.g., due to a symmetry of the system). The mean-field equations generalise the standard equations of hydromagnetic convection: New terms emerge -- a second-order linear operator representing the combined eddy diffusivity, and quadratic terms associated with the eddy advection. If the perturbed CHM regime is non-steady and insignificance of the alpha-effect in the system does not rely on the presence of a spatial symmetry, the combined eddy diffusivity operator also involves a non-local pseudodifferential operator. If the perturbed CHM state is almost symmetric, alpha-effect terms appear in the mean-field equations as well. Near a point of a symmetry-breaking bifurcation, cubic nonlinearity emerges in the equations. All the new terms are in general anisotropic. A method for evaluation of their coefficients is presented; it requires solution of a significantly smaller number of auxiliary problems than in a straightforward approach.