  A statistical mechanics theory for a fluid stratified in density is
presented. The predicted statistical equilibrium state is the most probable
outcome of turbulent stirring. The slow temporal evolution of the vertical
density profile is related to the presence of irreversible mixing, which alters
the global distribution of density levels. We propose a model in which the
vertical density profile evolves through a sequence of statistical equilibrium
states. The theory is then tested with laboratory experiments in a two-layer
stably stratified fluid forced from below by an oscillating grid. Quantitative
measurements of density fluctuations across the interface are made by planar
laser induced fluorescence. These fluctuations are splitted in a "wave" part
and a "turbulent" part. The wave part of the density field is well described by
a previous theory due to Phillips. We argue that statistical mechanics
predictions apply for the turbulent part of the density field sufficiently
close to the interface. However inside the mixed layer density fluctuations are
instead controlled by a balance between eddy flux downward and dissipation by
cascade to small scales. We report exponential tails for the density pdf in
this region.
