Abstract Direct numerical simulations of the time evolution of homogeneous stably stratified turbulent sheer flows have been performed for several Richardson numbers Ri and Reynolds numbers Rλ. The results show excellent agreement with length scale models developed from laboratory experiments to characterize oceanic turbulence. When the Richardson number Ri is less than the stationary value Ris, the turbulence intensity grows at all scales; the growth rate is a function of Ri. The size of the vertical density inversions also increases. When Ri ≥ Ri, the largest turbulent eddies become vertically constrained by buoyancy when the Ellison (turbulence) scale LEand the Ozmidov (buoyancy) scale LO are equal. At this point the mixing is most efficient and the flux Richardson number or mixing efficiency is Rf ≈ 0.20 for the stationary Richardson number Ris = 0.21. The vertical mass flux becomes countergradient when ϵ ≈ 19vN2, and vertical density overturns are suppressed in few than half of a Brunt-Vaisala period...