  The existence of microgauss magnetic fields in galaxy clusters have been
established through observations of synchrotron radiation and Faraday rotation.
They are conjectured to be generated via small-scale dynamo by turbulent flow
motions in the intracluster medium (ICM). Some of giant radio relics, on the
other hand, show the structures of synchrotron polarization vectors, organized
over the scales of $\sim$ Mpc, challenging the turbulence origin of cluster
magnetic fields. Unlike turbulence in the interstellar medium, turbulence in
the ICM is subsonic. And it is driven sporadically in highly stratified
backgrounds, when major mergers occur during the hierarchical formation of
clusters. To investigate quantitatively the characteristics of turbulence
dynamo in such ICM environment, we performed a set of turbulence simulations
using a high-order-accurate, magnetohydrodynamic (MHD) code. We find that
turbulence dynamo could generate the cluster magnetic fields up to the observed
level from the primordial seed fields of $10^{-15}$ G or so within the age of
the universe, if the MHD description of the ICM could be extended down to
$\sim$ kpc scales. However, highly organized structures of polarization
vectors, such as those observed in the Sausage relic, are difficult to be
reproduced by the shock compression of turbulence-generated magnetic fields.
This implies that the modeling of giant radio relics may require the
pre-existing magnetic fields organized over $\sim$ Mpc scales.
