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This introductory and self-contained examine monograph summarizes the theoretical cutting-edge to which the writer has made pioneering contributions.
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Additional info for Large-Scale Perturbations of Magnetohydrodynamic Regimes: Linear and Weakly Nonlinear Stability Theory
Rx þ erX ð3:7Þ (the indices x and X denote differentiation in the respective variables). The scale ratio e of the slow and fast variables is a small parameter. s. of the resultant equation as a power series in e: Considering independently the mean and the fluctuating part of its coefficients, we obtain solenoidality conditions for a large-scale magnetic mode, rX Á hhn i ¼ 0; ð3:11Þ rx Á fhn g þ rX Á fhnÀ1 g ¼ 0 ð3:12Þ holding for all n ! 0: (By definition, hn 0 for n\0:) Let L denote the restriction of the magnetic induction operator M to the subspace of vector fields, which are L-periodic in space and have a zero spatial mean.
Dkj denotes the Kronecker symbol. r2 denotes the Laplace operator and rÀ2 its inverse. C stands for the operator of the inverse curl, which maps a solenoidal zero-mean (in space) vector field to its solenoidal zero-mean vector potential, satisfying boundary conditions independently detailed in each chapter. I is the identity operator. In each chapter, A and E denote the a-effect and eddy diffusion operators, respectively, emerging in the large-scale stability problem under consideration in this chapter (as a consequence, the operators E introduced in different chapters are different, as are the operators A).
Step 2° for n = 1. L is a linear partial differential operator in the fast spatial variables; hq0 i does not depend on them. s. s. 7). 2 Eddy Diffusion The operator of eddy diffusion emerges as the solvability condition of the next, e2 , order equation À Á ð2:19Þ Lfq2 g ¼ Àl 2ðrx Á rX Þfq1 g þ r2X hq0 i þ ðV Á rX Þq1 þ k2 hq0 i: Step 1° for n = 2. 17)]. 20) we call mean-field equations. s. 20). Since it involves only second-order derivatives in spatial variables, it can be interpreted as the operator of eddy diffusion, generically anisotropic.