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.