Linear perturbations of the Bloch type of space-periodic magnetohydrodynamic steady states. II. Numerical results

Abstract

We consider Bloch eigenmodes of three linear stability problems: the kinematic dynamo problem, the hydrodynamic and MHD stability problem for steady space-periodic flows and MHD states comprised of randomly generated Fourier coefficients and having energy spectra of three types: exponentially decaying, Kolmogorov with a cut off, or involving a small number of harmonics (“big eddies”). A Bloch mode is a product of a field of the same periodicity as the perturbed state and a planar harmonic wave, exp(iq · x). Such a mode is characterized by the ratio of spatial scales which, for simplicity, we identify with the length |q| < 1 of the Bloch wave vector q. Computations have revealed that the Bloch modes, whose growth rates are maximum over q, feature the scale ratio that decreases on increasing the nondimensionalized molecular diffusivity and/or viscosity from 0.03 to 0.3, and the scale separation is high (i.e., |q| is small) only for large molecular diffusivities. Largely this conclusion holds for all the three stability problems and all the three energy spectra types under consideration. Thus, in a natural MHD system not affected by strong diffusion, a given scale range gives rise to perturbations involving only moderately larger spatial scales (i.e., |q| only moderately small), and the MHD evolution consists of a cascade of processes, each generating a slightly larger spatial scale; flows or magnetic fields characterized by a high scale separation are not produced. This cascade is unlikely to be amenable to a linear description. Consequently, our results question the allegedly high role of the α-effect and eddy diffusivity that are based on spatial scale separation, as the primary instability or magnetic field generating mechanisms in astrophysical applications. The Braginskii magnetic α-effect in a weakly non-axisymmetric flow, often used for explanation of the solar and geodynamo, is advantageous not being upset by a similar deficiency.