  Runaway electrons with strongly anisotropic distributions present in
post-disruption tokamak plasmas can destabilize the extraordinary electron
(EXEL) wave. The present work investigates the dynamics of the quasi-linear
evolution of the EXEL instability for a range of different plasma parameters
using a model runaway distribution function valid for highly relativistic
runaway electron beams produced primarily by the avalanche process. Simulations
show a rapid pitch-angle scattering of the runaway electrons in the high energy
tail on the $100-1000\;\rm \mu s$ time scale. Due to the wave-particle
interaction, a modification to the synchrotron radiation spectrum emitted by
the runaway electron population is foreseen, exposing a possible experimental
detection method for such an interaction.
