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

Abstract. 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 values of the bulk Richardson number Ri_B 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 wavenumber 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 multiplicative constant being in good agreement with previous measurements in the atmosphere. We next focus on the heat flux at the top of the CBL, , which is parameterized in classical closure models for the entrainment rate w_e at the interface. We show that can be computed exactly using the method proposed by Winters et al. (1995), from which the values of a turbulent diffusivity across the IL can be inferred. These values are recovered by tracking particles within the IL using a Lagrangian stochastic model coupled with the LES. The relative difference between the Eulerian and Lagrangian values of is found to be lower than 10%. A simple expression of w_e as a function of is also proposed. Our results are finally used to assess the validity of the classical "first-order'' model for w_e. We find that, when Ri_B is varied, the values for w_e derived from the "first-order'' model with the exact computation of 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 w_e for the largest values of Ri_B only.