  AGN feedback is responsible for maintaining plasma in global thermal balance
in extended halos of elliptical galaxies and galaxy clusters. Local thermal
instability in the hot gas leads to the formation of precipitating cold gas
clouds that feed the central supermassive black holes, thus heating the hot gas
and maintaining global thermal equilibrium. We perform three dimensional MHD
simulations of self-regulated AGN feedback in a Perseus-like galaxy cluster
with the aim of understanding the impact of the feedback physics on the
turbulence properties of the hot and cold phases of the ICM. We find that, in
general, the cold phase velocity structure function (VSF) is steeper than the
prediction from Kolmogorov's theory. We attribute the physical origin of the
steeper slope of the cold phase VSF to the driving of turbulent motions
primarily by the gravitational acceleration acting on the ballistic clouds. We
demonstrate that, in the pure hydrodynamical case, the precipitating cold
filaments may be the dominant agent driving turbulence in the hot ICM. The
arguments in favor of this hypothesis are that: (i) the cold phase mass
dominates over hot gas mass in the inner cool core; (ii) hot and cold gas
velocities are spatially correlated; (iii) both the cold and hot phase velocity
distributions are radially biased. We show that, in the MHD case, the
turbulence in the ambient hot medium (excluding the jet cone regions) can also
be driven by the AGN jets. The driving is then facilitated by enhanced coupling
due to magnetic fields of the ambient gas and the AGN jets. In the MHD case,
turbulence may thus be driven by a combination of AGN jet stirring and filament
motions. We conclude that future observations, including those from high
spatial and spectral resolution X-ray missions, may help to constrain
self-regulated AGN feedback by quantifying the multi-temperature VSF in the
ICM.
