Dynamo Action of Fluid Motions with Two-Dimensional Periodicity

G. O. Roberts

Abstract

In a previous paper it has been established that almost all spatially periodic motions of an infinite homogenous conducting fluid give magnetohydrodynamic dynamo action for almost all values of the magnetic diffusivity or resistivity $\lambda $. It was shown that there is a dynamo action if and only if for some real vector j there is a growing magnetic field solution of the form B(x, t) = H(x) exp (pt + ij$\cdot $x), where the complex vector function H(x) has the same periodicity as the motion. The complex growth rate p was studied in a first-order limit of small j to obtain the above result. In this sequel the special case of spatially periodic motions with their three components functions of the two Cartesian coordinates y and z only is considered. The first-order method establishes dynamo action for only half, in a certain sense, of the motion-resistivity combinations. It is shown that the two-dimensional spatially periodic motion u = (cos y - cos z, sin z, sin y) is a first-order dynamo, at least for almost all resistivities. Three similar motions, which are not first-order dynamos for any resistivity, are also studied. It is proved that multiple-scale versions of all three can give growing magnetic fields for certain resistivities when terms of higher order in j are included. Heuristic descriptions of all four dynamo mechanisms are given. A numerical method is described for determining the most rapidly growing magnetic field solution of the above form, and results for all four motions are presented, giving the growth rate Re p as a function of j, for a range of resistivities $\lambda $ down to about 10$^{-2}$. The first motion gives growing solutions for all resistivities in this range, the others give dynamo action only for resistivities below critical values near unity. The numerical and analytic results agree.

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