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We consider the evolution of two incompressible fluids with homogeneousdensities $\rho_-<\rho_+$ subject to gravity described by the inviscidBoussinesq equations and provide the explicit relaxation of the associateddifferential inclusion. The existence of a subsolution to the relaxation allowsone to conclude the existence of turbulently mixing solutions to the originalBoussinesq system. As a specific application we investigate subsolutionsemanating from the classical Rayleigh-Taylor initial configuration where thetwo fluids are separated by a horizontal interface with the heavier fluid beingon top of the lighter. It turns out that among all self-similar subsolutionsthe criterion of maximal initial energy dissipation selects a linear densityprofile and a quadratic growth of the mixing zone. The subsolution selectedthis way can be extended in an admissible way to exist for all times. Weprovide two possible extensions with different long-time limits. The first onecorresponds to a total mixture of the two fluids, the second corresponds to afull separation with the lighter fluid on top of the heavier. There is nomotion in either of the limit states.

Relaxation of the Boussinesq system and applications to the Rayleigh-Taylor instability