Whistler-Regulated Magnetohydrodynamics: Transport Equations for Electron Thermal Conduction in the High-Beta Intracluster Medium of Galaxy Clusters

Transport equations for electron thermal energy in the high-β_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 ~ v_te/β_e « v_te dominate the turbulent spectrum, with v_te the electron thermal speed and β_e » 1 the ratio of electron thermal to magnetic pressure. We find: (1) temperature-gradient-driven whistlers dominate classical scattering when L_c > L/β_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 ~ 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.