The Probability Distribution of Density Fluctuations in Supersonic
  Turbulence
  We study density fluctuations in supersonic turbulence using both theoretical
methods and numerical simulations. A theoretical formulation is developed for
the probability distribution function (PDF) of the density 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 tail of the PDF and reduces the left one, the
pressure term reduces both the right and left tails, and the viscosity term,
counter-intuitively, broadens the right tail of the PDF. Despite the inaccuracy
of the velocity divergence from the Riemann runs, as found in our previous
work, we show that the density PDF from the Riemann runs is consistent with
that from the N-S runs. Taking advantage of their much higher effective
resolution, we then use the Riemann runs to study the dependence of the PDF on
the Mach number, $\mathcal{M}$, up to $\mathcal{M}\sim30$. The PDF width,
$\sigma_{s}$, follows the relation $\sigma_{s}^2 = \ln (1+b^2 {\mathcal M}^2)$,
with $b\approx0.38$. However, the PDF exhibits a negative skewness that
increases with increasing $\mathcal{M}$, so much of the growth of $\sigma_{s}$
is accounted for by the growth of the left PDF tail, while the growth of the
right tail tends to saturate. Thus, the usual prescription that combines a
lognormal shape with the standard variance-Mach number relation greatly
overestimates the right PDF tail at large $\mathcal{M}$, which may have a
significant impact on theoretical models of star formation.
