旋转层中受迫磁流体对流弱非线性双尺度扰动的平均场方程
Mean-field equations for weakly nonlinear two-scale perturbations of forced hydromagnetic convection in a rotating layer
摘要：
我们考虑在旋转水平层与自由导电边界的磁流体热对流制度的稳定性，涉及大的空间和时间尺度的扰动。弱非线性平均扰动的演化方程是在α效应在首阶中不显著的假设下导出的（例如，由于系统的对称性）。平均场方程概括了磁流体对流的标准方程：新的术语出现-一个二阶线性算子代表的组合涡扩散，二次项与涡平流。如果扰动CHM制度是非稳定的，在系统中的α效应不依赖于空间对称性的存在下，不重要的，组合涡扩散算子也涉及到一个非局部的伪微分算子。如果微扰CHM态是几乎对称的，α效应项也出现在平均场方程中。在临界分岔点附近，方程出现了立方非线性。所有新项一般都是各向异性的。提出了一种评价其系数的方法，它需要解决的辅助问题比在一个简单的方法显着较少。
英文摘要：
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.
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