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Mean-field equations for weakly nonlinear two-scale perturbations of forced hydromagnetic convection in a rotating layer
Authors:Vladislav Zheligovsky
Institution:1. International Institute of Earthquake Prediction Theory and Mathematical Geophysics , Moscow, Russian Federation Observatoire de la C?te d'Azur, CNRS, U.M.R. 6529, Nice Cedex 4, France vlad@mitp.ru
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 α-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 nonsteady and insignificance of the α-effect in the system does not rely on the presence of a spatial symmetry, the combined eddy diffusivity operator also involves a nonlocal pseudodifferential operator. If the perturbed CHM state is almost symmetric, α-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.
Keywords:Hydromagnetic convection  Weakly nonlinear stability  Mean-field equations  Amplitude equation  Homogenisation  α-Effect  Eddy diffusivity
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