  We consider the problem of self-regulated heating and cooling in galaxy
clusters and the implications for cluster magnetic fields and turbulence.
Viscous heating of a weakly collisional magnetised plasma is regulated by the
pressure anisotropy with respect to the local direction of the magnetic field.
The intracluster medium is a high-beta plasma, where pressure anisotropies
caused by the turbulent stresses and the consequent local changes in the
magnetic field will trigger very fast microscale instabilities. We argue that
the net effect of these instabilities will be to pin the pressure anisotropies
at a marginal level, controlled by the plasma beta parameter. This gives rise
to local heating rates that turn out to be comparable to the radiative cooling
rates. Furthermore, we show that a balance between this heating and
Bremsstrahlung cooling is thermally stable, unlike the often conjectured
balance between cooling and thermal conduction. Given a sufficient (and
probably self-regulating) supply of turbulent power, this provides a physical
mechanism for mitigating cooling flows and preventing cluster core collapse.
For observed density and temperature profiles, the assumed balance of viscous
heating and radiative cooling allows us to predict magnetic-field strengths,
turbulent velocities and turbulence scales as functions of distance from the
centre. Specific predictions and comparisons with observations are given for
several different clusters. Our predictions can be further tested by future
observations of cluster magnetic fields and turbulent velocities.
