Weakly nonlinear stability to large-scale perturbations in convective magnetohydrodynamic systems without the α-effect

The problem of weakly nonlinear stability with respect to large-scale perturbations in 3-D convective magnetohydrodynamic (MHD) states in which the α-effect is absent or insignificant (e.g., because the system has symmetry relative to a center or a vertical axis) is examined. It is assumed that the MHD state whose stability is studied is free from large spatiotemporal scales and is insensitive to perturbations with the same small spatial scale as in the state under study. The equations for mean perturbation fields derived by asymptotic methods generalize the standard equations of magnetohydrodynamics (the Navier-Stokes and magnetic induction equations). A combined eddy diffusion operator, generally anisotropic and not necessarily negative definite, and additional quadratic terms similar to advective terms arise in the inferred generalized equations.