The second-moment equations for buoyancy flux and density-fluctuation variance in the entrainment zone of a mixed layer are combined to yield an entrainment rate (we dependence having the nature of Turner's relation. That is, weq−1 is seen as a function solely of an interfacial Richardson number (Rie) involving q where q2 is twice the interfacial turbulence kinetic energy (TKE). The mean thickness of the entrainment layer is found to be a satisfactory representation of the integral length scale near the mixed-layer interface which was used by Turner. The TKE equation in the entrainment layer is then utilized to evaluate Rie so that we can be obtained. This procedure allows the TKE equation to be solved past the point at which the interfacial-shear bulk Richardson number, Riv, becomes critical, and into the more unstable regime beyond where the experiment of Ellison and Turner can give guidance. Previous entrainment parameterizations which retain the critical-Riv concept predict infinite entrainment rates at the critical point and negative ones beyond.
The present entrainment results are shown as a function of the various Richardson numbers: Riv, RiT based on the friction velocity, and Ri* based on the convective velocity scale, for values of each ranging from near zero to 1000. Comparison is made with previous results, which the present results resemble only in narrow regions of parameter space. For typical values of RiT or Ri*, we is found to be enhanced by 10–20% if Riv drops from ∞ to 5, and by a factor of from 3 to 8 if it drops to its critical value, (Riv)crit near 0.7–1.0.