  Magnetic fluctuations with a zero mean field in a random flow with a finite
correlation time and a small yet finite magnetic diffusion are studied.
Equation for the second-order correlation function of a magnetic field is
derived. This equation comprises spatial derivatives of high orders due to a
non-local nature of magnetic field transport in a random velocity field with a
finite correlation time. For a random Gaussian velocity field with a small
correlation time the equation for the second-order correlation function of the
magnetic field is a third-order partial differential equation. For this
velocity field and a small magnetic diffusion with large magnetic Prandtl
numbers the growth rate of the second moment of magnetic field is estimated.
The finite correlation time of a turbulent velocity field causes an increase of
the growth rate of magnetic fluctuations. It is demonstrated that the results
obtained for the cases of a small yet finite magnetic diffusion and a zero
magnetic diffusion are different. Astrophysical applications of the obtained
results are discussed.
