  The small-scale dynamo is typically studied by assuming that the correlation
time of the velocity field is zero. Some authors have used a smooth renovating
flow model to study how the properties of the dynamo are affected by the
correlation time being nonzero. Here, we assume the velocity is an
incompressible Gaussian random field (which need not be smooth), and derive the
lowest-order corrections to the evolution equation for the two-point
correlation of the magnetic field in Fourier space. Using this, we obtain the
evolution equation for the longitudinal correlation function of the magnetic
field ($M_L$) in nonhelical turbulence, valid for arbitrary Prandtl number. The
non-resistive terms of this equation do not contain spatial derivatives of
$M_L$ of order greater than two. We further simplify this equation in the limit
of high Prandtl number, and find that the growth rate of the magnetic energy is
much smaller than previously reported. Nevertheless, the magnetic power
spectrum still retains the Kazantsev form at high Prandtl number.
