  The presence of magnetic fields in many astrophysical objects is due to
dynamo action, whereby a part of the kinetic energy is converted into magnetic
energy. A turbulent dynamo that produces magnetic field structures on the same
scale as the turbulent flow is known as the fluctuation dynamo. We use
numerical simulations to explore the nonlinear, statistically steady state of
the fluctuation dynamo in driven turbulence. We demonstrate that as the
magnetic field growth saturates, its amplification and diffusion are both
affected by the back-reaction of the Lorentz force upon the flow. The
amplification of the magnetic field is reduced due to stronger alignment
between the velocity field, magnetic field, and electric current density.
Furthermore, we confirm that the amplification decreases due to a weaker
stretching of the magnetic field lines. The enhancement in diffusion relative
to the field line stretching is quantified by a decrease in the computed local
value of the magnetic Reynolds number. Using the Minkowski functionals, we
quantify the shape of the magnetic structures produced by the dynamo as
magnetic filaments and ribbons in both kinematic and saturated dynamos and
derive the scalings of the typical length, width, and thickness of the magnetic
structures with the magnetic Reynolds number. We show that all three of these
magnetic length scales increase as the dynamo saturates. The magnetic
intermittency, strong in the kinematic dynamo (where the magnetic field
strength grows exponentially) persists in the statistically steady state, but
intense magnetic filaments and ribbons are more volume-filling.
