Feedback from the active galactic nuclei (AGNs) is one of the most promising heating mechanisms to circumvent the cooling-flow problem in galaxy clusters. However, the role of thermal conduction remains unclear. Previous studies have shown that anisotropic thermal conduction in cluster cool cores (CCs) could drive the heat-flux-driven buoyancy instabilities (HBIs) that reorient the field lines in the azimuthal directions and isolate the cores from conductive heating from the outskirts. However, how the AGN interacts with the HBI is still unknown. To understand these interwined processes, we perform the first 3D magnetohydrodynamic simulations of isolated CC clusters that include anisotropic conduction, radiative cooling, and AGN feedback. We find the following: (1) For realistic magnetic field strengths in clusters, magnetic tension can suppress a significant portion of HBI-unstable modes, and thus the HBI is either completely inhibited or significantly impaired, depending on the unknown magnetic field coherence length. (2) Turbulence driven by AGN jets can effectively randomize magnetic field lines and sustain conductivity at ~1/3 of the Spitzer value; however, the AGN-driven turbulence is not volume?filling. (3) Conductive heating within the cores could contribute to ~10% of the radiative losses in Perseus-like clusters and up to ~50% for clusters twice the mass of Perseus. (4) Thermal conduction has various impacts on the AGN activity and intracluster medium properties for the hottest clusters, which may be searched by future observations to constrain the level of conductivity in clusters. The distribution of cold gas and the implications are also discussed.
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