  Terrestrial gamma-ray flashes (TGFs) are short intense flashes of gamma rays
associated with lightning activity in thunderstorms. Using Monte Carlo
simulations of the relativistic runaway electron avalanche (RREA) process,
theoretical predictions for the temporal and spectral evolution of TGFs are
compared to observations made with the Gamma-ray Burst Monitor (GBM) on board
the Fermi Gamma-ray Space Telescope. Assuming a single source altitude of 15
km, a comparison of simulations to data is performed for a range of empirically
chosen source electron variation time scales. The data exhibit a clear
softening with increased source distance, in qualitative agreement with
theoretical predictions. The simulated spectra follow this trend in the data,
but tend to underestimate the observed hardness. Such a discrepancy may imply
that the basic RREA model is not sufficient. Alternatively, a TGF beam that is
tilted with respect to the zenith could produce an evolution with source
distance that is compatible with the data. Based on these results, we propose
that the source electron distributions of TGFs observed by GBM vary on time
scales of at least tens of microseconds, with an upper limit of approx. 100
microseconds.
