The growth rate of Rayleigh-Taylor turbulence depends on the large scale structures of the mixing

The growth rate $\alpha_n$ of a turbulent Rayleigh-Taylor (RT) mixing layer is defined such that the mixing layer width $L(t)=\alpha_n\,A\,g(t)\,t^2$, where $A$ is the Atwood number and $g(t)\sim t^n$ is the time history of the acceleration. We will show that the ensemble averaged growth rate of Rayleigh-Taylor can be inferred theoretically from first principle assuming a low Atwood mixing, analyticity of large scale turbulent spectra (for small $k$ the spectra behave like $E(k)\sim k^p$) and self-similarity at late time. The expression of $\alpha_n$ depends on the value of $n$ and $p$. Although it can be counter intuitive, the evolution of the mixing zone width is proved to depend most importantly upon what happens at the center of the mixing zone.