Estimating the diffusive heat flux across a stable interface forced by convective motions

Entrainment at the top of the convectively-driven boundary layer (CBL) is revisited using data from a high- resolution large-eddy simulation (LES). In the range of val- ues of the bulk Richardson number RiB studied here (about 15-25), the entrainment process is mainly driven by the scouring of the interfacial layer (IL) by convective cells. We estimate the length and time scales associated with these convective cells by computing one-dimensional wavenum- ber and frequency kinetic energy spectra. Using a Taylor assumption, based upon transport by the convective cells, we show that the frequency and wavenumber spectra follow the Kolmogorov law in the inertial range, with the multiplica- tive constant being in good agreement with previous mea- surements in the atmosphere. We next focus on the heat flux at the top of the CBL, Fi , which is parameterized in classical closure models for the entrainment rate we at the interface. We show that Fi can be computed exactly using the method proposed by Winters et al. (1995), from which the values of a turbulent diffusivity K across the IL can be inferred. These values are recovered by tracking particles within the IL us- ing a Lagrangian stochastic model coupled with the LES. The relative difference between the Eulerian and Lagrangian values of K is found to be lower than 10%. A simple expres- sion of we as a function of K is also proposed. Our results are finally used to assess the validity of the classical "first- order" model for we. We find that, when RiB is varied, the values for we derived from the "first-order" model with the exact computation of Fi agree to better than 10% with those computed directly from the LES (using its definition). The simple expression we propose appears to provide a reliable estimate of we for the largest values of RiB only.