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.

Linear perturbations of the Bloch type of space-periodic magnetohydrodynamic steady states. III. Asymptotics of branching

“…For instance, the kinematic dynamo problem for a parity-invariant flow, for which Table 1. Instances of branching found numerically in [Chertovskih and Zheligovsky, 2023b]. F: figures ibid.…”

“…We have developed a power series asymptotic expansion in ϑ = (η 0 − η) 1/2 of magnetic Bloch modes, kinematically generated by a parity-invariant flow and featuring locally maximum growth rates, which stem from the branch of neutral (i.e., associated with the zero eigenvalue of the magnetic induction operator D (8)) space-periodic modes for q = 0 (see Fig. 14 in [Chertovskih and Zheligovsky, 2023b]). We have shown that branching occurs at a critical molecular diffusivity η 0 for which the two eigenvalues of the operator of eddy diffusivity are imaginary and complex-conjugate.…”

“…This bifurcation is illustrated, e.g., by Fig. 14(a),(b) [Chertovskih and Zheligovsky, 2023b], where branch II bifurcates from branch III comprised of neutral (stationary) magnetic modes for q = 0. We observe that at the point of bifurcation the graph of growth rates is tangent to the zero eigenvalue for q = 0, and the locally maximum growth rates in branch II are attained for the optimal q that are order ϑ = (η 0 − η) 1/2 .…”

“…The phenomenon is very common. It is observed in 9 problems out of the 18 ones which were solved in [Chertovskih and Zheligovsky, 2023b], see figs. 6, 9, 10, 13-15, 21-23 ibid. (there are two instances of branching in figs.…”

“…We have studied the dependence of the dominant growth rates of the Bloch modes on the diffusivity parameters, the molecular magnetic diffusivity η and the molecular viscosity ν. Computations [Chertovskih and Zheligovsky, 2023b] have revealed that the Bloch modes, maximizing growth rates γ(q) over the Bloch wave vectors q, constitute branches, in which the dependence of the dominant growth rates on the diffusivity parameter is smooth. It was demonstrated in [Chertovskih and Zheligovsky, 2023a] that half-integer q (whose all components are integer or half-integer) satisfy the necessary condition for the maximum, ∂γ/∂q m = 0.…”