  We investigate transient radiation processes in the non-linear superradiance
(SR) regime of the Doppler broadened Maxwell-Bloch equations when the velocity
distribution is of total bandwidth greatly exceeding that of the transient
process itself. We demonstrate the formation of global polarisation phase
correlation and the quenching of temporal structure if a smooth distribution is
inverted above the critical threshold required to enter the non-linear SR
regime. We propose candidate stochastic velocity distributions capable of
sustaining finite temporal structure in the non-linear emission process. We
develop a novel algorithm for simulating the Doppler broadened Maxwell-Bloch
equations which is $O(n)$ complex in the number of velocity channels $n$
whenever the emerging polarisation correlation is of moderate bandwidth, and we
apply it to a stochastic velocity distribution in order to demonstrate
sustained delay and duration of peak intensity in the widely Doppler broadened
limit. We discuss the transverse inversion process and recognise an
autoregulation mechanism on the number of molecules cooperatively participating
in SR emission. This mechanism has the effect of limiting the temporal duration
of the intensity pulse to a lower bound proportional to the length of the
sample, which we confirm through simulation.
