Entrainment parameterization based on entrainment zone Richardson number

Entrainment at the top of a convective or transitional boundary layer is a first-order term in the budgets of heat, moisture, momentum, and pollutants. Model treatments of entrainment, however, tend to be somewhat haphazard. Either entrainment is expected to emerge correctly from a 1.5 or higher order boundary layer scheme, or it is crudely parameterized as a fraction of the surface flux. The latter approach has some justification for buoyancy flux only, and only under pure free convection conditions. Because entrainment is difficult to measure, only a few observational studies are available to test model formulations, but those measurements that are available strongly suggest that entrainment as a fraction of the surface flux is quite variable. Large-eddy simulations tend to show less entrainment than the measurements, even under conditions that depart substantially from free convection (strong shear). Furthermore, LES show entrainment heat flux increasing with inversion strength, while the only available measurements show the opposite. This discrepancy needs further exploration.

Based on the few available measurements, I propose a parameterization of entrainment heat flux based on a Richardson number computed across the entrainment zone. The parameterization is a non-linear function of the Richardson number. It may be usefully incorporated directly into some types of models, but at least should serve to stimulate further research into how modeled boundary layers entrain.