  In the intracluster medium (ICM) of galaxy clusters, heat and momentum are
transported almost entirely along (but not across) magnetic field lines. We
perform the first fully self-consistent Braginskii-MHD simulations of galaxy
clusters including both of these effects. Specifically, we perform local and
global simulations of the magnetothermal instability (MTI) and the
heat-flux-driven buoyancy instability (HBI) and assess the effects of viscosity
on their saturation and astrophysical implications. We find that viscosity has
only a modest effect on the saturation of the MTI. As in previous calculations,
we find that the MTI can generate nearly sonic turbulent velocities in the
outer parts of galaxy clusters, although viscosity somewhat suppresses the
magnetic field amplification. At smaller radii in cool-core clusters, viscosity
can decrease the linear growth rates of the HBI. However, it has less of an
effect on the HBI's nonlinear saturation, in part because three-dimensional
interchange motions (magnetic flux tubes slipping past each other) are not
damped by anisotropic viscosity. In global simulations of cool core clusters,
we show that the HBI robustly inhibits radial thermal conduction and thus
precipitates a cooling catastrophe. The effects of viscosity are, however, more
important for higher entropy clusters. We argue that viscosity can contribute
to the global transition of cluster cores from cool-core to non cool-core
states: additional sources of intracluster turbulence, such as can be produced
by AGN feedback or galactic wakes, suppress the HBI, heating the cluster core
by thermal conduction; this makes the ICM more viscous, which slows the growth
of the HBI, allowing further conductive heating of the cluster core and a
transition to a non cool-core state.
