  We study the interplay between turbulent heating, mixing, and radiative
cooling in an idealized model of cool cluster cores. Active galactic nuclei
(AGN) jets are expected to drive turbulence and heat cluster cores. Cooling of
the intracluster medium (ICM) and stirring by AGN jets are tightly coupled in a
feedback loop. We impose the feedback loop by balancing radiative cooling with
turbulent heating. In addition to heating the plasma, turbulence also mixes it,
suppressing the formation of cold gas at small scales. In this regard, the
effect of turbulence is analogous to thermal conduction. For uniform plasma in
thermal balance (turbulent heating balancing radiative cooling), cold gas
condenses only if the cooling time is shorter than the mixing time. This
condition requires the turbulent kinetic energy to be $\gtrsim$ the plasma
internal energy; such high velocities in cool cores are ruled out by
observations. The results with realistic magnetic fields and thermal conduction
are qualitatively similar to the hydrodynamic simulations. Simulations where
the runaway cooling of the cool core is prevented due to {\em mixing} with the
hot ICM show cold gas even with subsonic turbulence, consistent with
observations. Thus, turbulent mixing is the likely mechanism via which AGN jets
heat cluster cores. The thermal instability growth rates observed in
simulations with turbulence are consistent with the local thermal instability
interpretation of cold gas in cluster cores.
