  Terrestrial gamma ray flashes (TGFs) are very short bursts of gamma radiation
associated to thunderstorm activity and are the manifestation of the
highest-energy natural particle acceleration phenomena occurring on Earth.
Photon energies up to several tens of megaelectronvolts are expected, but the
actual upper limit and high-energy spectral shape are still open questions.
Results published in 2011 by the AGILE team proposed a high-energy component in
TGF spectra extended up to $\approx$100 MeV, which is difficult to reconcile
with the predictions from the Relativistic Runaway Electron Avalanche (RREA)
mechanism at the basis of many TGF production models. Here we present a new set
of TGFs detected by the AGILE satellite and associated to lightning
measurements capable to solve this controversy. Detailed end-to-end Monte Carlo
simulations and an improved understanding of the instrument performance under
high-flux conditions show that it is possible to explain the observed
high-energy counts by a standard RREA spectrum at the source, provided that the
TGF is sufficiently bright and short. We investigate the possibility that
single high-energy counts may be the signature of a fine-pulsed time structure
of TGFs on time scales $\approx$4 {\mu}s, but we find no clear evidence for
this. The presented data set and modeling results allow also for explaining the
observed TGF distribution in the (Fluence x duration) parameter space and
suggest that the AGILE TGF detection rate can almost be doubled. Terrestrial
gamma ray flashes (TGFs) are very short bursts of gamma radiation associated to
thunderstorm activity and are the manifestation of the highest-energy natural
particle acceleration phenomena occurring on Earth. (...continues)
