The Probability Distribution of Density Fluctuations in Supersonic Turbulence

A theoretical formulation is developed for the probability distribution function (pdf) of gas density in supersonic turbulence at steady state, connecting it to the conditional statistics of the velocity divergence. Two sets of numerical simulations are carried out, using either a Riemann solver to evolve the Euler equations or a finite-difference method to evolve the Navier-Stokes (N-S) equations. After confirming the validity of our theoretical formulation with the N-S simulations, we examine the effects of dynamical processes on the pdf, showing that the nonlinear term in the divergence equation amplifies the right pdf tail and reduces the left one, the pressure term reduces both the right and left tails, and the viscosity term, counterintuitively, broadens the right tail of the pdf. Despite the inaccuracy of the velocity divergence from the Riemann runs, we show that the density pdf from the Riemann runs is consistent with that from the N-S runs. Taking advantage of their higher effective resolution, we use Riemann runs with resolution up to 2048(3) to study the dependence of the pdf on the Mach number, M, up to M similar to 30. The pdf width, sigma(s), follows the relation (sigma(2)(s) = ln(1 b(2)M(2)), with b approximate to 0.38. However, the pdf exhibits a negative skewness that increases with increasing M, as the growth of the right tail with increasing M tends to saturate. Thus, the usual prescription that combines a lognormal shape with a variance-Mach number relation greatly overestimates the right pdf tail at large M, with important consequences for star formation models.