The problem of turbulence closure for convective boundary layer (CBLs) is considered by the example of the vertical turbulent flux of potential temperature. The chief aim is to advance in understanding the nature of non-local transport due to large-scale semi-organized structures. An overview is given of earlier closure schemes ranging from comparatively simple counter-gradient-correction closures to sophisticated closures based on budget equations for the second order fluxes and variances. It is emphasized that the key role in the non-local transport is played by third-order moments (fluxes of fluxes), in particular, by the skewness of vertical velocity.
A turbulent advection + diffusion parameterization for the flux of flux of potential temperature is developed and validated using data from large-eddy simulation (LES) of the CBL. As a result the potential-temperature-flux budget equation provides an integral non-local turbulence closure for the flux in question. In particular cases it reduces to a number of closure schemes proposed earlier, namely, the counter-gradient correction closure (Deardorff, 1972), the transport-asymmetry closure employing the second derivative of transported scalar (Wyngaard and Weil, 1991), and integral closure principally similar to that for passive scalars (Berkowicz and Prahm, 1979). Moreover, the Green-function solution to the flux budget equation provides simple procedure for decomposition of the potential temperature flux into the bottom-up and top-down components, resembling the Wyngaard (1983) bottom-up/top-down decomposition. The proposed Green-function decomposition exhibits unexpected, essentially non-linear profiles of the bottom-up and top-down components of the potential temperature flux in the well-mixed layer in sharp contrast to universally adopted linear profiles.
Physical ideas underlying our approach diverge fundamentally from classical turbulence-closure philosophy, according to which a flux in question is expressed as a function of turbulence moments of the same or the lower order. Our parameterization for the potential temperature flux(the second-order moment) involves not only the first-order moment(potential temperature gradient) and the second-order moments (r.m.s.vertical velocity and temperature), but also the third-order moment, namely, the vertical velocity triple correlation, or alternatively the vertical velocity skewness.