  A number of studies have shown that the convective stability criterion for
the intracluster medium (ICM) is very different from the Schwarzchild criterion
due to the effects of anisotropic thermal conduction and cosmic rays. Building
on these studies, we develop a model of the ICM in which a central active
galactic nucleus (AGN) accretes hot intracluster plasma at the Bondi rate and
produces cosmic rays that cause the ICM to become convectively unstable. The
resulting convection heats the intracluster plasma and regulates its
temperature and density profiles. By adjusting a single parameter in the model
(the size of the cosmic-ray acceleration region), we are able to achieve a good
match to the observed density and temperature profiles in a sample of eight
clusters. Our results suggest that convection is an important process in
cluster cores. An interesting feature of our solutions is that the cooling rate
is more sharply peaked about the cluster center than is the convective heating
rate. As a result, in several of the clusters in our sample, a compact cooling
flow arises in the central region with a size R that is typically a few kpc.
The cooling flow matches onto a Bondi flow at smaller radii. The mass accretion
rate in the Bondi flow is equal to, and controlled by, the rate at which mass
flows in through the cooling flow. Our solutions suggest that the AGN regulates
the mass accretion rate in these clusters by controlling R: if the AGN power
rises above the equilibrium level, R decreases, the mass accretion rate drops,
and the AGN power drops back down to the equilibrium level.
