Whistler-regulated MHD: Transport equations for electron thermal
  conduction in the high $\beta$ intracluster medium of galaxy clusters
  Transport equations for electron thermal energy in the high $\beta_e$
intracluster medium (ICM) are developed that include scattering from both
classical collisions and self-generated whistler waves. The calculation employs
an expansion of the kinetic electron equation along the ambient magnetic field
in the limit of strong scattering and assumes whistler waves with low phase
speeds $V_w\sim{v}_{te}/\beta_e\ll{v}_{te}$ dominate the turbulent spectrum,
with $v_{te}$ the electron thermal speed and $\beta_e\gg1$ the ratio of
electron thermal to magnetic pressure. We find: (1) temperature-gradient-driven
whistlers dominate classical scattering when $L_c>L/\beta_e$, with $L_c$ the
classical electron mean-free-path and $L$ the electron temperature scale
length, and (2) in the whistler dominated regime the electron thermal flux is
controlled by both advection at $V_w$ and a comparable diffusive term. The
findings suggest whistlers limit electron heat flux over large regions of the
ICM, including locations unstable to isobaric condensation. Consequences
include: (1) the Field length decreases, extending the domain of thermal
instability to smaller length-scales, (2) the heat flux temperature dependence
changes from $T_e^{7/2}/L$ to $V_wnT_e\sim{T}_e^{1/2}$, (3) the magneto-thermal
and heat-flux driven buoyancy instabilities are impaired or completely
inhibited, and (4) sound waves in the ICM propagate greater distances, as
inferred from observations. This description of thermal transport can be used
in macroscale ICM models.
