  We consider the evolution of two incompressible fluids with homogeneous
densities $\rho_-<\rho_+$ subject to gravity described by the inviscid
Boussinesq equations and provide the explicit relaxation of the associated
differential inclusion. The existence of a subsolution to the relaxation allows
one to conclude the existence of turbulently mixing solutions to the original
Boussinesq system. As a specific application we investigate subsolutions
emanating from the classical Rayleigh-Taylor initial configuration where the
two fluids are separated by a horizontal interface with the heavier fluid being
on top of the lighter. It turns out that among all self-similar subsolutions
the criterion of maximal initial energy dissipation selects a linear density
profile and a quadratic growth of the mixing zone. The subsolution selected
this way can be extended in an admissible way to exist for all times. We
provide two possible extensions with different long-time limits. The first one
corresponds to a total mixture of the two fluids, the second corresponds to a
full separation with the lighter fluid on top of the heavier. There is no
motion in either of the limit states.
