  Turbulence is believed to play important roles in the origin of cosmic
magnetism. While it is well known that turbulence can efficiently amplify a
uniform or spatially homogeneous seed magnetic field, it is not clear whether
or not we can draw a similar conclusion for a localized seed magnetic field.
The main uncertainty is the rate of magnetic field diffusion on scales larger
than the outer scale of turbulence. To measure the diffusion rate of magnetic
field on those large scales, we perform a numerical simulation in which the
outer scale of turbulence is much smaller than the size of the system. We
numerically compare diffusion of a localized seed magnetic field and a
localized passive scalar. We find that diffusion of the magnetic field can be
much faster than that of the passive scalar and that turbulence can efficiently
amplify the localized seed magnetic field. Based on the simulation result, we
construct a model for fast diffusion of magnetic field. Our model suggests that
a localized seed magnetic field can fill the whole system in (L_sys/L) times
the large-eddy turnover time and that growth of the magnetic field stops in
max(15, L_sys/L) times the large-eddy turnover time, where L_sys is the size of
the system and $L$ is the driving scale. Our finding implies that, regardless
of the shape of the seed field, fast magnetization is possible in turbulent
systems, such as large-scale structure of the universe or galaxies.
