  The present work proposes a new formalism for the inner regions of a
neutrino-dominated accretion flows (NDAFs) by considering the self-gravity,
where the neutrino opacity is high enough to make neutrinos trapped becoming a
dominant factor in the transportation of energy and angular momentum over the
magneto rotational instability. We investigate the possibility of gravitational
instability and fragmentation to model the highly variable structure of the
prompt emission in gamma-ray bursts (GRBs). The results lead us to introduce
the gravitational instability, in these inner regions, as a source of a new
viscosity which is of the same functional form as that of the
$\beta$-prescription of viscosity. Such a consideration brings about
fragmentation in the unstable inner disk. In addition, we find the consequent
clumpy structure of this area capable to account for the temporal variability
of GRB's light curve, especially for the lower choices of the parameter
$\beta$, $\sim 10^{-5}$. Finally, we predict the formation of gravitational
waves through the migration of fragments before being tidally disrupted. These
waves appear to be detectable via a range of current and future detectors from
LIGO to Cosmic Explorer.
