  We perform direct numerical simulations of three dimensional Rayleigh-Taylor
turbulence with a nonuniform singular initial temperature background. In such
conditions, the mixing layer evolves under the driving of a varying effective
Atwood number; the long-time growth is still self-similar, but not anymore
proportional to $t^2$ and depends on the singularity exponent $c$ of the
initial profile $\Delta T \propto z^c$. We show that the universality is
recovered when looking at the efficiency, defined as the ratio of the variation
rates of the kinetic energy over the heat flux. A closure model is proposed
that is able to reproduce analytically the time evolution of the mean
temperature profiles, in excellent agreement with the numerical results.
Finally, we reinterpret our findings in the light of spontaneous stochasticity
where the growth of the mixing layer is mapped into the propagation of a wave
of turbulent fluctuations on a rough background.
