Second-Order Closure PBL Model with New Third-Order Moments: Comparison with LES Data

Abstract

This paper contains two parts. In the first part, a new set of diagnostic equations is derived for the third-order moments for a buoyancy-driven flow, by exact inversion of the prognostic equations for the third-order moment equations in the stationary case. The third-order moments exhibit a universal structure: they all are a linear combination of the derivatives of all the second-order moments, w2, w?, ?2, and q2. Each term of the sum contains a turbulent diffusivity Dt, which also exhibits a universal structure of the form Dt = a?t + bw?. Since the sign of the convective flux changes depending on stable or unstable stratification, Dt varies according to the type of stratification. Here ?t ? wl (l is a mixing length and w is an rms velocity) represents the ?mechanical? part, while the ?buoyancy? part is represented by the convective flux w?. The quantities a and b are functions of the variable (Nτ)2, where N2 = gα?Θ/?z and τ is the turbulence time scale. The new expressions for the third-order moments generalize those of Zeman and Lumley, which were subsequently adopted by Sun and Ogura, Chen and Cotton, and Finger and Schmidt in their treatments of the convective boundary layer. In the second part, the new expressions for the third-order moments are used to solve the ensemble average equations describing a purely convective boundary layer heated from below at a constant rate. The computed second- and third-order moments are then compared with the corresponding LES results, most of which are obtained by running a new LES code, and part of which are taken from published results. The ensemble average results compare favorably with the LES data.