A formulation of convective entrainment in terms of mixing efficiency

The entrainment across a stably stratified interface forced by convective motions is discussed in the light of the mixing efficiency of the entrainment process. The context is the convectively driven atmospheric boundary layer and we focus on the regime of equilibrium entrainment, i.e. when the boundary-layer evolution is in a quasi-steady state. The entrainment law is classically based on the ratio R of the negative of the heat flux at the interface to the heat flux at the ground surface. We propose a parameterization for R that involves the mixing efficiency and the thickness of the interface, which matches well the direct computation of R from a high-resolution large-eddy simulation. This result enables us to derive modified expressions for the classical entrainment laws (the so-called zeroand first-order models) as a function of the mixing efficiency. We show that, when the thickness of the interface is ignored (zero-order model), the scaling factor A in the entrainment law is the flux Richardson number. This parameterization of A is further improved when the thickness of the interface is considered (first-order model), as shown by the direct computation of A from the large-eddy simulation.