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 timescales .
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 timescales of at least tens of microseconds , with an upper limit of \sim 100 \mu s .