  In weakly collisional extragalactic plasmas such as the intracluster medium,
viscous stress and the rate of change of the magnetic-field strength are
proportional to the local pressure anisotropy, so subject to constraints
imposed by the pressure-anisotropy-driven mirror and firehose instabilities and
controlled by the local instantaneous plasma beta. The dynamics of such plasmas
is dramatically different from a conventional MHD fluid. The plasma is expected
to stay locally in a marginal state with respect to the instabilities, but how
it does this is an open question. Two models of magnetic-field evolution are
investigated. In the first, marginality is achieved via suppression of the rate
of change of the field. In the second, the instabilities give rise to anomalous
collisionality, reducing pressure anisotropy to marginal - at the same time
decreasing viscosity and so increasing the turbulent rate of strain.
Implications of these models are studied in a simplified 0D setting. In the
first model, the field grows explosively but on a time scale that scales with
initial beta, while in the second, dynamical field strength can be reached in
one large-scale turbulence turn-over time regardless of the initial seed. Both
models produce very intermittent fields. Both also suffer from strong
constraints on their applicability: for typical cluster-core conditions, scale
separation between the fluid motions and the microscale fluctuations breaks
down at beta~10^5-10^4. At larger beta (weaker fields), a fully collisionless
plasma dynamo theory is needed in order to justify the growth of the field from
a tiny primordial seed. However, the models discussed here are appropriate for
studying the structure of the currently observed field as well as large-scale
dynamics and thermodynamics of the magnetized ICM or similarly dilute
astrophysical plasmas.
