Cold fronts in cluster cool cores should be erased on short timescales by thermal conduction, unless protected by magnetic fields that are "draped" parallel to the front surfaces, suppressing conduction perpendicular to the sloshing fronts. We present a series of MHD simulations of cold front formation in the core of a galaxy cluster with anisotropic thermal conduction, exploring a parameter space of conduction strengths parallel and perpendicular to the field lines. Including conduction has a strong effect on the temperature distribution of the core and the appearance of the cold fronts. Though magnetic field lines are draping parallel to the front surfaces, preventing conduction directly across them, the temperature jumps across the fronts are nevertheless reduced. The geometry of the field is such that the cold gas below the front surfaces can be connected to hotter regions outside via field lines along directions perpendicular to the plane of the sloshing motions and along sections of the front that are not perfectly draped. This results in the heating of this gas below the front on a timescale of a Gyr, but the sharpness of the density and temperature jumps may nevertheless be preserved. By modifying the gas density distribution below the front, conduction may indirectly aid in suppressing Kelvin-Helmholtz instabilities. If conduction along the field lines is unsuppressed, we find that the characteristic sharp jumps seen in Chandra observations of cold front clusters do not form. Therefore, the presence of cold fronts in hot clusters is in contradiction with our simulations with full Spitzer conduction. This suggests that the presence of cold fronts in hot clusters could be used to place upper limits on conduction in the bulk of the intracluster medium. Finally, the combination of sloshing and anisotropic thermal conduction can result in a larger flux of heat to the core than either process in isolation. While still not sufficient to prevent a cooling catastrophe in the very central (r ~ 5?kpc) regions of the cool core (where something else is required, such as active galactic nucleus feedback), it reduces significantly the mass of gas that experiences a cooling catastrophe outside those small radii.
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