We consider the inhomgeneous incompressible Euler equations describing two fluids with different constant densities under the influence of gravity. Initially the fluids are supposed to be at rest and separated by a flat horizontal interface with the heavier fluid being on top of the lighter one. Due to gravity this configuration is unstable, the two fluids begin to mix in a more and more turbulent way. This is one of the most classical instances of the Rayleigh-Taylor instability. In the talk we will see how weak locally dissipative solutions to the Euler equations reflecting a turbulent mixing of the two fluids in a quadratically growing zone can be constructed. If time allows, we will discuss an arising selection problem for the averaged motion of solutions. The core of the talk is based on a joint work with József Kolumbán.