By Vladislav Zheligovsky
New advancements for hydrodynamical dynamo conception were spurred via contemporary facts of self-sustained dynamo task in laboratory experiments with liquid metals.
The emphasis within the current quantity is at the advent of robust mathematical concepts required to take on smooth multiscale research of continous structures and there software to a few real looking version geometries of accelerating complexity.
This introductory and self-contained study monograph summarizes the theoretical state of the art to which the writer has made pioneering contributions.
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Additional resources for Large-Scale Perturbations of Magnetohydrodynamic Regimes: Linear and Weakly Nonlinear Stability Theory
S. is a sum of the identity and a compact operator , and hence the Fredholm alternative theorem [154, 171] is applicable. s. 15). 15) has a unique solution in the subspace of L-periodic zero-mean vector fields, as required. 13) at successive orders n in two steps: 1°. Consider the spatial mean of the equation and satisfy the solvability condition. 2°. Solve the resultant partial differential equation in the fast spatial variables. 1 Solution of the Order e0 Equations Step 1° for n = 0. Averaging of this equation yields 0 ¼ k0 hh0 i; and two possibilities arise: k0 ¼ 0; and/or hh0 i ¼ 0: As discussed in Sect.
15) is an eigenvalue equation for a shortscale stability mode q0 ðxÞ. Stability with respect to both modes, the large-scale one and the small-scale q0 ðxÞ, is then determined by the sign of the real part of k0 . Thus, if Re k0 6¼ 0, construction of the large-scale mode hardly provides any new information on linear stability of passive scalar transport. Since the operator of passive scalar transport, L, does not possess imaginary eigenvalues (see Sect. 1), the only case of interest is k0 ¼ 0. 3 Perturbation of the Neutral Mode In this section we will consider the problem for k0 ¼ 0, which guarantees solvability of Eq.
14) at order e1 reduces to Lfq1 g ¼ ðV Á rX Þhq0 i þ k1 hq0 i: ð2:16Þ Step 1° for n = 1. s. 16) is zeromean. s. 16) vanishes as long as k1 ¼ 0. 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)].