  Turbulent heating may play an important role in galaxy-cluster plasmas, but
if turbulent heating is to balance radiative cooling in a quasi-steady state,
some mechanism must set the turbulent velocity to the required value. This
paper explores one possible regulating mechanism associated with an active
galactic nucleus at cluster center. A steady-state model for the intracluster
medium is presented in which radiative cooling is balanced by a combination of
turbulent heating and thermal conduction. The turbulence is generated by
convection driven by the buoyancy of cosmic rays produced by a central radio
source. The cosmic-ray luminosity is powered by the accretion of intracluster
plasma onto a central black hole. The model makes the rather extreme assumption
that the cosmic rays and thermal plasma are completely mixed. Although the
intracluster medium is convectively unstable near cluster center in the model
solutions, the specific entropy of the thermal plasma still increases outwards
because of the cosmic-ray modification to the stability criterion. The model
provides a self-consistent calculation of the turbulent velocity as a function
of position, but fails to reproduce the steep central density profiles observed
in clusters. The principal difficulty is that in order for the fully mixed
intracluster medium to become convectively unstable, the cosmic-ray pressure
must become comparable to or greater than the thermal pressure within the
convective region. The large cosmic-ray pressure gradient then provides much of
the support against gravity, reducing the thermal pressure gradient near
cluster center and decreasing the central plasma density gradient. A more
realistic AGN-feedback model of intracluster turbulence in which relativistic
and thermal plasmas are only partially mixed may have greater success.
