On the entrainment law at the top of the convectively-driven atmospheric boundary layer

We discuss the entrainment law at the top of the convectively-driven boundary layer (see Stevens et al., 2002, for an introduction and Fedorovich et al., 2004, for a thorough review), in which the dimensionless entrainment rate scales like the inverse of a Richardson number at the interface (Deardorff et al., 1980). We show that, for strong enough stratification of the interfacial layer, this law can be rederived from basic physical assumptions using energy considerations and the concept of mixing efficiency. This law then simply expresses that a fraction γ, equal to the mixing efficiency, of the potential energy made available by ground surface heating is used to lift the interface. The value of γ is well-known to be in the order of 0.2 in stably-stratified flow. When this law is expressed as a function of the Froude number squared at the interface, which involves the finite thickness of the interfacial layer, experimental and numerical data yield a scaling factor close to 1.2.

This discussion and reasoning are supported by both data from measurements and large-eddy simulation, including our recent high-resolution (256³ grid points) numerical experiment, using both Eulerian and Lagrangian approaches to investigate the entrainment law. We use our high-resolution simulation to analyze the mixed-layer dynamics from a deterministic (organized structure identification) and statistical point of view (1D and 2D spatial as well as temporal spectra) and characterize the destabilization mechanism of the interfacial layer. We found that the mixed layer may be regarded as locally homogeneous and isotropic and that the Taylor's (1938) hypothesis holds, based on the horizontal mean flow and the vertical velocity of the convective cells. A particle dispersion approach using LES coupled with a Lagrangian stochastic model is also used to investigate the entrainment law. The Froude number squared dependence of the dimensionless entrainment rate, with a multiplicative constant of order 1.2, is retrieved.

These results may help to suggest entrainment parameterization that work well at the interface and may also have applications for remote sensing of the mixed-layer top (for instance to compute the entrainment velocity proceeding from a particle dispersion approach).