Abstract
This paper addresses the question of how the zero and small diffusivity solutions to the kinematic magnetic induction equation are related. It is shown that, in the case of perturbed linear toral automorphisms, hyperbolicity properties allow a connection between the zero diffusivity Cauchy solution and the non-zero diffusivity Wiener ensemble solution using shadowing theory. A formula is derived that calculates over finite times the small diffusivity magnetic field in terms of the local zero diffusivity magnetic field by averaging against a Gaussian density with variance proportional to diffusivity. For linear toral automorphisms, it is proven that the infinite time fast dynamo growth rate can be calculated using a local Cauchy flux average in agreement with a conjecture by Finn and Ott (1988).
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