We conduct a comprehensive theoretical and numerical investigation of the pollution of pristine gas in turbulent flows, designed to provide new tools for modeling the evolution of the first generation of stars. The properties of such Population III (Pop III) stars are thought to be very different than later generations, because cooling is dramatically different in gas with a metallicity below a critical value Z_c, which lies between ~10^-6 and 10^-3 solar value. Z_c is much smaller than the typical average metallicity,View original: http://arxiv.org/abs/1306.4663
, and thus the mixing efficiency of the pristine gas in the interstellar medium plays a crucial role in the transition from Pop III to normal star formation. The small critical value, Z_c, corresponds to the far left tail of the probability distribution function (PDF) of the metallicity. Based on closure models for the PDF formulation of turbulent mixing, we derive equations for the fraction of gas, P, lying below Z_c, in compressible turbulence. Our simulation data shows that the evolution of the fraction P can be well approximated by a generalized self-convolution model, which predicts dP/dt = -n/tau_con P (1-P^(1/n)), where n is a measure of the locality of the PDF convolution and the timescale tau_con is determined by the rate at which turbulence stretches the pollutants. Using a suite of simulations with Mach numbers ranging from M = 0.9 to 6.2, we provide accurate fits to n and tau_con as a function of M, Z_c/ , and the scale, L_p, at which pollutants are added to the flow. For P>0.9, mixing occurs only in the regions surrounding the pollutants, such that n=1. For smaller P, n is larger as mixing becomes more global. We show how the results can be used to construct one-zone models for the evolution of Pop III stars in a single high-redshift galaxy, as well as subgrid models for tracking the evolution of the first stars in large cosmological simulations.