Inverse cascades and time-dependent dynamos in MHD flows
This paper addresses the effects of two different kinds of instabilities on magnetohydrodynamical (MHD) flows. The alpha effect occurs in a three- dimensional small-scale fluid flow which is helical or otherwise lacks parity invariance) when it acts on a large-scale magnetic field and generates an electromotive force which is a function of the field itself. The alpha effect may lead to dynamo action, which is the growth of a large-scale magnetic field. An analogous transport effect occurs in ordinary hydrodynamics; when a large-scale velocity field is superimposed on a small-scale flow lacking Galilean invariance, the mean Reynolds stress can be a function of the large-scale flow. This is called the anisotropic kinetic alpha (AKA) effect, and it may lead to the growth of the large-scale flow when the original three-dimensional small-scale flow is anisotropic and lacks parity invariance. The authors examine the phenomenon known as ``inverse cascade'' whereby energy is transferred from the linearly most unstable modes to successively larger scales, and the mean fields eventually saturate in a state whose dominant mode is that of the system. The authors relate metastable states (states dominated by larger and larger scales) to the existence of a steady cellular solution. Such cellular solutions are unstable, and there is eventually transition to a cellular solution on a larger scale. This suggests the existence of heteroclinic connections going from any steady solution to a steady solution on a larger scale. This picture is supported by the analysis of a simple solvable model.
scientific article; zbMATH DE number 4190786