SIMPLE-MODELS OF NONLINEAR FLUCTUATION DYNAMO

We discuss asymptotic solutions of nonlinear steady-state equations of the fluctuation dynamo, i.e. equations describing generation of a random magnetic field in a random mirror symmetric flow of conducting fluid. The flow is assumed to be locally homogeneous and isotropic and the correlation scale l is considered to be small in comparison to the size of the region occupied by the flow, L. These presumptions admit a closed nonlinear equation for the mean energy density of the magnetic field whose solutions are considered here for l/L much less than 1. If the generation efficiency drops to zero when the magnetic energy density E reaches a certain value (of the order of the kinetic energy density E(c)), then the steady-state values of E are of order E(c) (the equipartition dynamo). Otherwise, if the generation efficiency only declines monotonically with E remaining positive, the steady-state values of E can strongly exceed E(c) [by the factor (L/l)2/mu with certain constant mu of order unity] (the supra-equipartition dynamo). These general properties of the steady state are illustrated by two simple models of nonlinearity.