Fast dynamos with finite resistivity in steady flows with stagnation points

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Abstract
Results of the kinematic fast dynamo problem for two classes of steady incompressible flows are presented. These flows are the ABC flow and the spatially aperiodic flow of Lau and Finn [Physica D 57, 238 (1992)]. In a range of parameters, these three‐dimensional flows have stagnation points (A and B type) and there are chaotic streamlines. The chaos is associated with the intermingled stable and unstable manifolds of the stagnation points. In the aperiodic flow the chaos takes the form of chaotic scattering. The growth rate of the dynamos for the aperiodic flow is found to obey a certain scaling law with resistivity η (as η→0), from which the results are extrapolated to the limit η→0 (infinite magnetic Reynolds number). Numerical results are presented indicating that fast kinematic dynamos can exist in these flows and that chaotic flow is a necessary condition. The structure of the dynamo magnetic fields is also shown, in particular, the relationship between the regions of maximal field strength and the invariant dynamical structures of the aperiodic flow. For the aperiodic flow, the unstable mode has a real frequency and these regions consist of two fingers of oppositely directed field. These regions rotate about a streamline (the one‐dimensional unstable manifold) coming out of the type A stagnation point. For the ABC flow with A=B=C, it is found that there are two dynamo modes: an oscillating mode and a purely growing mode. The mode crossing occurs at magnetic Reynolds number between 300 and 350, with the purely growing mode dominating for larger magnetic Reynolds numbers. For the oscillating mode, the region of large ‖B‖ is similar to that for the aperiodic flow. For the purely growing mode, the region of large ‖B‖ is localized in single fingers about the one‐dimensional unstable manifolds. The distribution function of ln‖B‖ is observed to be approximately Gaussian for both modes of the ABC flow. The distribution function for the mode found for the aperiodic flow has a much more complex structure, apparently associated with the escape of streamlines in chaotic scattering.

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