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Magneto-Thermal Instability In Galaxy Clusters II: Three-Dimensional Simulations

In the intracluster medium (ICM) of galaxies, exchanges of heat across magnetic field lines are strongly suppressed. This anisotropic heat conduction, in the presence of a large-scale temperature gradient, destabilizes the outskirts of galaxy clusters via the magneto-thermal instability (MTI), and thus supplies a source of observed ICM turbulence. In this paper we continue our investigation of the MTI with 3D simulations using the Boussinesq code SNOOPY. We redress two issues intrinsic to our previous 2D study: an inverse energy cascade and the impossibility of dynamo action. Contrary to 2D simulations, we find inconsequential transport of energy across scales (most energy is dissipated at the same scale as its injection), and that turbulent eddies are vertically elongated at or below the thermal conduction length, but relatively isotropic on larger scales. Similar to 2D, however, the saturated turbulent energy levels and the integral scale follow clear power-laws that depend on the thermal diffusivity, temperature gradient, and buoyancy frequency. We also show that the MTI amplifies magnetic fields, through a fluctuation dynamo, to equipartition strengths provided that the integral scale of MTI turbulence is larger than the viscous dissipation scale. Finally, we show that our scaling laws are consistent with extant observations of ICM turbulence if the thermal conductivity is reduced by a factor of $\sim 10$ from its Spitzer value, and that on global cluster scales the stable stratification significantly reduces the vertical elongation of MTI motions.