  Hydromagnetic dynamo theory provides the prevailing theoretical description
for the origin of magnetic fields in the universe. Here we consider the problem
of kinematic, small-scale dynamo action driven by a random, incompressible,
non-helical, homogeneous and isotropic flow. In the Kazantsev dynamo model the
statistics of the driving flow are assumed to be instantaneously correlated in
time. Here we compare the results of the model with the dynamo properties of a
simulated flow that has equivalent spatial characteristics as the Kazantsev
flow but different temporal statistics. In particular, the simulated flow is a
solution of the forced Navier-Stokes equations and hence has a finite
correlation time. We find that the Kazantsev model typically predicts a larger
magnetic growth rate and a magnetic spectrum that peaks at smaller scales.
However, we show that by filtering the diffusivity spectrum at small scales it
is possible to bring the growth rates into agreement and simultaneously align
the magnetic spectra.
