SHADOWING AND THE DIFFUSIONLESS LIMIT IN FAST DYNAMO THEORY

This paper addresses the question of how the zero and small diffusivit y 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 diffusiv ity Cauchy solution and the non-zero diffusivity Wiener ensemble solut ion using shadowing theory. A formula is derived that calculates over finite times the small diffusivity magnetic field in terms of the loca l zero diffusivity magnetic field by averaging against a Gaussian dens ity with variance proportional to diffusivity. For linear toral automo rphisms, it is proven that the infinite time fast dynamo growth rate c an be calculated using a local Cauchy flux average in agreement with a conjecture by Finn and Ott.