# The Probability Distribution of Density Fluctuations in Supersonic Turbulence

## TL;DR

This paper develops a theoretical model for the density fluctuation PDF in supersonic turbulence, validates it with simulations, and explores how the PDF depends on Mach number, revealing limitations of existing models for star formation.

## Contribution

It introduces a new theoretical formulation for the density PDF in supersonic turbulence and compares it with high-resolution simulations, highlighting the effects of dynamical processes and Mach number dependence.

## Key findings

- Theoretical PDF matches N-S simulation results.
- PDF width follows a specific relation with Mach number, σ_s^2 = ln(1 + b^2 M^2).
- The PDF exhibits increasing negative skewness with Mach number.

## Abstract

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.