Weakly nonlinear stability of convective magnetohydrodynamic systems without alpha-effect to perturbations involving large scales

Abstract

I consider the problem of weakly nonlinear stability of three-dimensional convective magnetohydrodynamic systems, where there is no alpha-effect or it is insignificant, to perturbations involving large scales. I assume that the convective MHD state (steady or evolutionary), the stability of which I investigate, does not involve large spatio-temporal scales, and it is stable to perturbations involving the same small spatial scales, as the perturbed state. Mean-field equations, which I derive for the perturbation using asymptotic techniques for multiscale systems, are a generalization of the equations of magnetohydrodynamics (the Navier-Stokes and magnetic induction equations). The operator of combined eddy diffusivity emerges, which is in general anisotropic and not necessarily negatively defined, as well as new quadratic terms analogous to the ones describing advection.