  The conventional theory of small-scale magnetic field generation in a
turbulent flow considers time-reversible random flows. However, real turbulent
flows are known to be time irreversible: the presence of energy cascade is an
intrinsic property of turbulence. We generalize the 'standard' model to account
for the irreversibility. We show that even small time asymmetry leads to
significant suppression of the dynamo effect at low magnetic Prandtl numbers,
increases the generation threshold and may even make generation impossible for
any magnetic Reynolds number. We calculate the magnetic energy growth rate as a
function of the parameters of the flow.
