Vladislav Zheligovsky's Large-Scale Perturbations of Magnetohydrodynamic Regimes: PDF

By Vladislav Zheligovsky

New advancements for hydrodynamical dynamo concept were spurred by way of fresh facts of self-sustained dynamo job in laboratory experiments with liquid metals.

The emphasis within the current quantity is at the creation of robust mathematical recommendations required to take on glossy multiscale research of continous structures and there software to a few sensible version geometries of accelerating complexity.

This introductory and self-contained examine monograph summarizes the theoretical cutting-edge to which the writer has made pioneering contributions.

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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.

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