The results of a one-dimensional model of a runaway electron avalanche, initiated by a high-energy cosmic ray secondary electron, in a thunderstorm electric field are presented. The time-evolution of the electron densities, characterized by mean energies of 7.2 MeV and 1-2 eV, and the associated radiation electric fields are reported. The dominant rates that control the evolution of the electron avalanche in the troposphere and in an external electric field are: avalanche ionization, low-energy electron attachment, and high-energy electron loss. The self-consistent evolution of the electric field is included. Relativistic effects are salient features of the numerically calculated radio-frequency radiation. The peak radiation electric fields and VHF spectra are in agreement with observations of a class of lightning waveforms known as narrow bipolar pulses (NBPs). It is hypothesized that the runaway electron avalanche is responsible for the rise time, and associated VHF impulses, of the NBP signals. Further work is needed to explain other aspects of the observations.