Growth rates for fast kinematic dynamo instabilities of chaotic fluid flows

Abstract

It is shown that the exponential growth rate of the fast kinematic dynamo instability can be related to the Lagrangian stretching properties of the underlying chaotic flow. In particular, a formula is obtained relating the growth rate to the finite time Lyapunov numbers of the flow and the cancellation exponent κ. (The latter quantity characterizes the extremely singular nature of the magnetic field with respect to fine-scale spatial oscillation in orientation.) The growth rate formula is illustrated and tested on two examples: an analytically soluble model, and a numerically solved spatially smooth temporally periodic flow.