  It has recently been suggested that conduction-driven magnetohydrodynamic
(MHD) instabilities may operate at all radii within an intracluster medium
(ICM), and profoundly affect the structure of a cluster's magnetic field. Where
MHD instabilities dominate the dynamics of an ICM, they will re-orient magnetic
field lines perpendicular to the temperature gradient inside a cooling core, or
parallel to the temperature gradient outside it. This characteristic structure
of magnetic field could be probed by measurements of polarized radio emission
from background sources. Motivated by this possibility we have constructed 3-d
models of a magnetized cooling core cluster and calculated Faraday rotation
measure (RM) maps in the plane of the sky under realistic observing conditions.
We compare a scenario in which magnetic field geometry is characterized by
conduction driven MHD instabilities to that where it is determined by isotropic
turbulent motions. We find that future high-sensitivity spectro-polarimetric
measurements of RM, such as will be enabled by the Expanded Very Large Array
and Square Kilometer Array can distinguish between these two cases with
plausible exposure times. Such observations will test the existence of
conduction-driven MHD instabilities in dynamically relaxed cooling core
clusters. More generally, our findings imply that observations of Faraday RM
should be able to discern physical mechanisms that result in qualitatively
different magnetic field topologies, without a priori knowledge about the
nature of the processes.
