{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,6,25]],"date-time":"2025-06-25T12:40:09Z","timestamp":1750855209579,"version":"3.41.0"},"reference-count":12,"publisher":"Geophysical Center of the Russian Academy of Sciences","content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":[],"published-print":{"date-parts":[[2023,12,10]]},"abstract":"<jats:p>We consider Bloch eigenmodes of three linear stability problems: the kinematic dynamo problem, the hydrodynamic and MHD stability problem for steady space-periodic \ufb02ows and MHD states comprised of randomly generated Fourier coe\ufb03cients and having energy spectra of three types: exponentially decaying, Kolmogorov with a cut o\ufb00, or involving a small number of harmonics (\u201cbig eddies\u201d). A Bloch mode is a product of a \ufb01eld of the same periodicity as the perturbed state and a planar harmonic wave, exp(iq \u00b7 x). Such a mode is characterized by the ratio of spatial scales which, for simplicity, we identify with the length |q| &lt; 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 di\ufb00usivity and\/or viscosity from 0.03 to 0.3, and the scale separation is high (i.e., |q| is small) only for large molecular di\ufb00usivities. 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 a\ufb00ected by strong di\ufb00usion, 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; \ufb02ows or magnetic \ufb01elds 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 \u03b1-e\ufb00ect and eddy di\ufb00usivity that are based on spatial scale separation, as the primary instability or magnetic \ufb01eld generating mechanisms in astrophysical applications. The Braginskii magnetic \u03b1-e\ufb00ect in a weakly non-axisymmetric \ufb02ow, often used for explanation of the solar and geodynamo, is advantageous not being upset by a similar de\ufb01ciency.<\/jats:p>","DOI":"10.2205\/2023es000838","type":"journal-article","created":{"date-parts":[[2023,10,22]],"date-time":"2023-10-22T18:00:07Z","timestamp":1697997607000},"page":"1-20","source":"Crossref","is-referenced-by-count":2,"title":["Linear Perturbations of the Bloch Type of Space-Periodic Magnetohydrodynamic Steady States. II. Numerical Results"],"prefix":"10.2205","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-5179-4344","authenticated-orcid":false,"given":"R","family":"Chertovskih","sequence":"first","affiliation":[{"name":"Institute of Earthquake Prediction Theory and Mathematical Geophysics Russian academy of sciences","place":["ru"]}]},{"given":"V","family":"Zheligovsky","sequence":"additional","affiliation":[{"name":"Institute of Earthquake Prediction Theory and Mathematical Geophysics Russian academy of sciences","place":["ru"]}]}],"member":"1030","reference":[{"key":"ref1","unstructured":"Braginsky, S. I. (1964a), Self-excitation of a magnetic field during the motion of a highly conducting fluid, Soviet Physics JETP, 20, 726-735 (in Russian)."},{"key":"ref2","doi-asserted-by":"crossref","unstructured":"Braginsky, S. I. (1964b), Theory of the hydromagnetic dynamo, Soviet Physics JETP, 20, 1462-1471 (in Russian). Chertovskih, R., and V. Zheligovsky (2023), Linear perturbations of the Bloch type of space-periodic magnetohy-drodynamic steady states. I. Mathematical preliminaries, Russian Journal of Earth Sciences, 23, ES3001, https:\/\/doi.org\/10.2205\/2023ES000834.","DOI":"10.2205\/2023ES000834"},{"key":"ref3","doi-asserted-by":"crossref","unstructured":"Frisch, U. (1995), Turbulence: The legacy of A. N. Kolmogorov, Cambridge University Press, https:\/\/doi.org\/10.1017\/CBO9781139170666.","DOI":"10.1017\/CBO9781139170666"},{"key":"ref4","doi-asserted-by":"crossref","unstructured":"Krause, F., and K.-H. Radler (1980), Mean-Field Magnetohydrodynamics and Dynamo Theory, Elsevier, https:\/\/doi.org\/10.1016\/c2013-0-03269-0.","DOI":"10.1515\/9783112729694"},{"key":"ref5","unstructured":"Landau, L. D., and E. M. Lifshitz (1987), Fluid Mechanics. Volume 6 of Course of Theoretical Physics, 2nd ed., Pergamon Press."},{"key":"ref6","doi-asserted-by":"crossref","unstructured":"Rasskazov, A., R. Chertovskih, and V. Zheligovsky (2018), Magnetic field generation by pointwise zero-helicity three- dimensional steady flow of an incompressible electrically conducting fluid, Physical Review E, 97(4), 043,201, https:\/\/doi.org\/10.1103\/PhysRevE.97.043201.","DOI":"10.1103\/PhysRevE.97.043201"},{"key":"ref7","doi-asserted-by":"publisher","DOI":"10.1134\/S1069351317050135"},{"key":"ref8","unstructured":"Steenbeck, M., I. M. Kirko, A. Gailitis, A. P. Klyavinya, F. Krause, I. Y. Laumanis, and O. A. Lielausis (1968), Experimental observation of the electromotive force alongside of an external magnetic field induced by the flow of liquid metal (alpha effect), Doklady Akademii nauk SSSR, 180(2), 326-329 (in Russian)."},{"key":"ref9","doi-asserted-by":"publisher","DOI":"10.1086\/171494"},{"key":"ref10","doi-asserted-by":"crossref","unstructured":"Zheligovsky, V. (2011), Large-Scale Perturbations of Magnetohydrodynamic Regimes: Linear and Weakly Nonlinear Stability Theory, Springer Berlin Heidelberg, https:\/\/doi.org\/10.1007\/978-3-642-18170-2.","DOI":"10.1007\/978-3-642-18170-2"},{"key":"ref11","doi-asserted-by":"publisher","DOI":"10.1007\/BF01060831"},{"key":"ref12","doi-asserted-by":"publisher","DOI":"10.1134\/S1069351320010152"}],"container-title":["Russian Journal of Earth Sciences"],"original-title":["Linear Perturbations of the Bloch Type of Space-Periodic Magnetohydrodynamic Steady States. II. Numerical Results"],"language":"en","deposited":{"date-parts":[[2025,6,25]],"date-time":"2025-06-25T12:01:48Z","timestamp":1750852908000},"score":1,"resource":{"primary":{"URL":"http:\/\/rjes.ru\/en\/nauka\/article\/55083\/view"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2023,12,10]]},"references-count":12,"URL":"https:\/\/doi.org\/10.2205\/2023es000838","relation":{},"ISSN":["1681-1208","1681-1208"],"issn-type":[{"type":"print","value":"1681-1208"},{"type":"electronic","value":"1681-1208"}],"subject":[],"published":{"date-parts":[[2023,12,10]]}}}