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.
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