  The hot intracluster medium (ICM) is thought to be quiescent with low
observed velocity dispersions. Surface brightness fluctuations of the ICM also
suggest that its turbulence is subsonic with a Kolmogorov scaling relation,
indicating that the viscosity is suppressed and the kinetic energy cascades to
small scales unscathed. However, recent observations of the cold gas filaments
in galaxy clusters find that the scaling relations are steeper than that of the
hot plasma, signaling kinetic energy losses and the presence of supersonic
flows. In this work we use high-resolution simulations to explore the turbulent
velocity structure of the cold filaments at the cores of galaxy clusters. Our
results indicate that supersonic turbulent structures can be "frozen" in the
cold gas that cools and fragments out of a fast, $10^7$ K outflow driven by the
central active galactic nucleus (AGN), when the radiative cooling time is
shorter than the dynamical sound-crossing time. After the cold gas formation,
however, the slope of the velocity structure function (VSF) flattens
significantly over short, 10 Myr timescales. The lack of flattened VSF in
observations of H$\alpha$ filaments indicates that the H$\alpha$-emitting phase
is short-lived for the cold gas in galaxy clusters. On the other hand, the
ubiquity of supersonic turbulence revealed by observed filaments strongly
suggests that supersonic outflows are an integral part of AGN-ICM interaction,
and that AGN activity plays a crucial role at driving turbulence in galaxy
clusters.
