Energy conservation and second-order statistics in stably stratified turbulent boundary layers

We address the dynamical and statistical description of stably stratified turbulent boundary layers with the important example of the atmospheric boundary layer with a stable temperature stratification in mind. Traditional approaches to this problem, based on the profiles of mean quantities, velocity second-order correlations, and dimensional estimates of the turbulent thermal flux run into a well-known difficulty, predicting the suppression of turbulence at a small critical value of the Richardson number, in contradiction with observations. Phenomenological attempts to overcome this problem suffer from various theoretical inconsistencies. Here we present a closure approach taking into full account all the second-order statistics, which allows us to respect the conservation of total mechanical energy. The analysis culminates in an analytic solution of the profiles of all mean quantities and all second-order correlations removing the unphysical predictions of previous theories. We propose that the approach taken here is sufficient to describe the lower parts of the atmospheric boundary layer, as long as the Richardson number does not exceed an order of unity. For much higher Richardson numbers the physics may change qualitatively, requiring careful consideration of the potential Kelvin-Helmholtz waves and their interaction with the vortical turbulence