Saturation of the magnetothermal instability in three dimensions

In dilute astrophysical plasmas, thermal conduction is primarily along magnetic field lines, and therefore highly anisotropic. As a result, the usual convective stability criterion is modified from a condition on entropy to a condition on temperature. For small magnetic fields or small wavenumbers, instability occurs in any atmosphere where the temperature and pressure gradients point in the same direction. We refer to the resulting convective instability as the magnetothermal instability (MTI). We present fully three-dimensional simulations of the MTI and show that saturation results in an atmosphere with different vertical structure, dependent on the boundary conditions. When the temperature at the boundary of the unstable layer is allowed to vary, the temperature gradient relaxes until the unstable region is almost isothermal. When the temperature at the boundary of the unstable region is fixed, the magnetic field is reoriented to an almost vertical geometry as a result of buoyant motions. This case exhibits more vigorous turbulence. In both cases the resulting saturated heat flux is almost one-half of the value expected if the conduction were purely isotropic, Q̃ ∼ ΧΔT/L, where Χ is the thermal conductivity, ΔT is the fixed temperature drop across the simulation domain, and L is the temperature gradient scale length. The action of the MTI results in dynamical processes that lead to significant transport perpendicular to the initial direction of the magnetic field. The resulting magnetoconvection in both cases amplifies the magnetic field until it is almost in equipartition with sustained subsonic turbulence. These results are relevant to understanding measurements of the temperature profiles of the intracluster medium of clusters of galaxies as well as the structure of radiatively inefficient accretion flows.