  Turbulent mixing layers in nature are often characterized by the presence of
a mean shear and an unstable buoyancy gradient between two streams of different
velocity. Depending on the relative strength of shear versus buoyancy, either
the former or the latter may dominate the turbulence and mixing between the two
streams. In this paper, we present a phenomenological theory that leads to the
identification of two distinct turbulent regimes: an early regime, dominated by
the mean shear, and a later regime dominated by the buoyancy. The main
theoretical result consists of the identification of a cross-over time-scale
that discerns between the shear- and the buoyancy-dominated turbulence. This
cross-over time depends on three large-scale constants of the flow, namely the
buoyancy difference, the velocity difference between the two streams, and the
gravitational acceleration. We validate our theory against direct numerical
simulations (DNSs) of a temporal turbulent mixing layer compounded with an
unstable stratification. We observe that the cross-over time correctly predicts
the transition from shear to buoyancy driven turbulence, in terms of turbulent
kinetic energy production, energy spectra scaling and mixing layer thickness.
