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 ran dom magnetic field in a random mirror symmetric flow of conducting flu id. The flow is assumed to be locally homogeneous and isotropic and th e correlation scale l is considered to be small in comparison to the s ize 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 dens ity E reaches a certain value (of the order of the kinetic energy dens ity E(c)), then the steady-state values of E are of order E(c) (the eq uipartition dynamo). Otherwise, if the generation efficiency only decl ines 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 co nstant mu of order unity] (the supra-equipartition dynamo). These gene ral properties of the steady state are illustrated by two simple model s of nonlinearity.