  We present an extended version of an invited talk given on the International
Conference "Turbulent Mixing and Beyond". The dynamical and statistical
description of stably stratified turbulent boundary layers with the important
example of the stable atmospheric boundary layer in mind is addressed.
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 an
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-Helmoholtz waves and their interaction with the vortical
turbulence.
