Cool cores of galaxy clusters are thought to be heated by low-power active galactic nuclei (AGN), whose accretion is regulated by feedback. However, the interaction between the hot gas ejected by the AGN and the ambient intracluster medium is extremely difficult to simulate as it involves a wide range of spatial scales and gas that is Rayleigh-Taylor (RT) unstable. Here we present a series of three-dimensional hydrodynamical simulations of a self-regulating AGN in a galaxy cluster. Our adaptive-mesh simulations include prescriptions for radiative cooling, AGN heating and a subgrid model for RT-driven turbulence, which is crucial to simulate this evolution. AGN heating is taken to be proportional to the rest-mass energy that is accreted onto the central region of the cluster. For a wide range of feedback efficiencies, the cluster regulates itself for at least several $10^9$ years. Heating balances cooling through a string of outbursts with typical recurrence times of around 80 Myrs, a timescale that depends only on global cluster properties. Under certain conditions we find central dips in the metallicity of the intracluster medium. Provided the sub-grid model used here captures all its key properties, turbulence plays an essential role in the AGN self-regulation in cluster cores.
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