An Improved Fit to the Density Distribution in Supersonic Isothermal Turbulence

TL;DR

The paper demonstrates that the conventional lognormal description of the density PDF in isothermal, supersonic turbulence is inadequate when compressive driving and driving correlation time ${τ_a}$ are significant. By performing a large suite of 512^3 simulations with varied driving types and ${τ_a}$, the authors derive empirical scaling relations showing that the volume-weighted density variance ${σ_{s,V}^2}$ scales roughly linearly with the Mach number ${M_V}$ and includes a ${τ_a}$-dependent term: ${σ_{s,V}^2 ≈ M_V [1 + \frac{2}{3}(1 + λ_a) Θ(1 + λ_a)]}$, with ${λ_a = \ln(τ_a/τ_e)}$; in contrast, solenoidal driving yields ${σ_{s,V}^2 ≈ \frac{1}{3} M_V}$ for ${M_V \lesssim 8}$, and mixed driving sits between these limits. The results reveal that ${P_V}(s)$ and ${P_M}(s)$ exhibit strong skewness and broadening, particularly under compressive driving with large ${τ_a}$, challenging Gaussian-based inferences and informing more accurate models of astrophysical turbulence in molecular clouds and the ISM. The work thus provides a refined framework for predicting density statistics in turbulent astrophysical environments, improving connections to observations and theoretical models of star formation and ISM structure.

Abstract

The density distribution of supersonic isothermal turbulence plays a critical role in many astrophysical systems. It is commonly approximated by a lognormal distribution with a variance of $σ_{s,{\rm V}}^2 \approx \ln(1 + b^2 M_{\rm V}^2),$ where $s \equiv \ln ρ/ρ_0,$ $M_{\rm V}$ is the rms volume-weighted Mach number, and $b$ is a parameter that depends on the driving mechanism, which can be solenoidal (divergence-free), compressive (curl-free), or a mix of both. However, this fit neglects the driving correlation time, $τ_{\rm a}$, which plays a key role when compressive driving is significant. Here we conduct turbulence simulations spanning a wide range of Mach numbers, driving mechanisms, and $τ_{\rm a}$ values. In the compressive case, $σ_{s,{\rm V}}^2$ is not well fit by the standard expression. Instead, it scales approximately linearly with $M_{\rm V},$ and its dependence on $τ_{\rm a}$ is $σ_{s,{\rm V}}^2 \approx M_{\rm V} [1 + \frac{2}{3}(1 + λ_{\rm a})Θ(1 + λ_{\rm a})]$, where $λ_{\rm a} \equiv \ln(τ_{\rm a}/τ_{\rm e})$, $τ_{\rm e}$ is the eddy turnover time, and $Θ$ is the Heaviside step function. Mixed-driven turbulence shows a weak dependence on $τ_{\rm a},$ and for solenoidally-driven turbulence, $σ_{s,{\rm V}}^2 \approx \frac{1}{3}M_{\rm V}$, which is consistent with the standard expression when $M_{\rm V} \lesssim 8.$ The volume-weighted mean and skewness also show systematic trends with $M_{\rm V}$ and $τ_{\rm a}$, deviating from lognormal expectations. The mass-weighted density distribution displays significant broadening and skewness in compressively-driven cases, especially at large $τ_{\rm a}/τ_{\rm e}$. These results provide a refined framework for modeling astrophysical turbulence.

Figures (7)

• Figure 1: Representative results from our simulations. From left to right, columns show results slices of $s$ from runs with compressive driving, and Mach numbers $M_{\rm V} \approx$ 2, 4, and 8, mixed-driving runs with $M_{\rm V} \approx 4$ runs, and purely solenoidal runs with $M_{\rm V} \approx 4.$ From top to bottom, the rows show cases with $\tau_{\rm a}$ = 0.01, 0.1, and 1. The corresponding run names are (top left to bottom right): $\mathtt{Ms1.9\_C\_\lambda-3.1}$, $\mathtt{Ms4.7\_C\_\lambda-2.1}$, $\mathtt{Ms10.4\_C\_\lambda-1.3}$, $\mathtt{Ms5.0\_M\_\lambda-2.1}$, $\mathtt{Ms5.3\_S\_\lambda-2.0}$, $\mathtt{Ms2.0\_C\_\lambda-0.7}$, $\mathtt{Ms4.3\_C\_\lambda-0.1}$, $\mathtt{Ms9.2\_C\_\lambda-0.9}$, $\mathtt{Ms5.0\_M\_\lambda-0.2}$, $\mathtt{Ms6.2\_S\_\lambda-0.4}$, $\mathtt{Ms1.4\_C\_\lambda-1.4}$, $\mathtt{Ms4.0\_C\_\lambda-2.4}$, $\mathtt{Ms10.6\_C\_\lambda-3.3}$, $\mathtt{Ms5.2\_M\_\lambda-2.6}$, $\mathtt{Ms6.9\_S\_\lambda-2.8}$. Snapshots are shown at $t \approx 8 \tau_e$. At that time, the results are stable and do not vary.
• Figure 2: Volume-weighted PDFs from a representative subset of our simulations. Here, the top row shows the results of simulations with fully compressive driving and average volume-weighted Mach numbers of $M_{\rm V} \approx 2,4,6,$ and 9. The lower row shows simulation results with mixed driving, and $M_{\rm V} \approx 2,5,$ and 10, as well as results from solenoidally-driven simulations with $M_{\rm V} \approx 5.$ In each panel, the colored lines show $P_{\rm V}(s)$ for runs with $\tau_{\rm a}$ = 0.01 (cyan), 0.03 (blue), 0.1 (purple), 0.3 (magenta), 1.0 (red), and 3.0 (orange). Increasing $\tau_{\rm a}$ has a strong effect on the compressive runs, broadening $P_{\rm V}(s)$ and moving the peak to the left, consistent with the formation of large voids. These effects are seen to a limited degree in the mixed driving runs, while the differences in the solenoidal results are consistent with those expected due to the small differences in $M_{\rm V}$ between the various runs.
• Figure 3: Volume-weighted variance of $s,$$\sigma^2_{\rm s,V}$ (top row), and standard deviation of $\rho/\rho_0,$$\sigma_{\rho,V}$ (center row), as a function of Mach number, and $\sigma^2_{\rm s,V}$ as a function of $\sigma_{\rho,V}$ (bottom row). In all rows, the filled circles are the results of our simulations, with the colors corresponding to the $\lambda_{\rm a}$ values. The other points are taken from Lemaster08, Federrath10, Price11, Konstandin12, and Pan18. In the upper row, the dashed lines show fits of the form $\sigma^2_{s,{\rm V}} = \ln(1+M_{\rm V}^2 b_s^2 )$, and the solid lines show fits of the form $\sigma^2_{s,{\rm V}} = B \, M_{\rm V}$ with $b_s$ and $B$ values labeled in each panel. In the center row, the dashed lines show fits of the form $\sigma_{\rho,{\rm V}} = b_{\rho} M_{\rm V}$. In the bottom row, the dashed lines show $\sigma^2_{s,{\rm V}}= \ln(1+\sigma_{\rho,{\rm V}}^2)$ as expected for a Gaussian distribution of $s$ and the solid lines show $\sigma^2_{s,{\rm V}}= \sigma_{\rho,{\rm V}}.$ These panels show that the primary problem with eq. (\ref{['eq:standardfit']}) is that $P_{\rm V}(s)$ cannot be adequately approximated by a Gaussian.
• Figure 4: Top: The volume-weighted variance of $s$, normalized by the volume-weighted Mach number squared, $\sigma^2_{s,{\rm V}} M_{\rm V}^{-2},$ (stars) and by the volume-weighted Mach number, $B \equiv \sigma^2_{s,{\rm V}} M_{\rm V}^{-1},$ (circles), which provides a much better description of the data. In all cases, the colors correspond to the Mach number. Columns show results from compressively-driven turbulence (left), mixed-driving turbulence (center), and solenoidally-driven turbulence (right). The solid lines are fits $B = 1 + \frac{2}{3}(1+ \lambda_{\rm a}) \Theta(1+\lambda_{\rm a})$, $B= \frac{3}{4},$ and $B=\frac{1}{3},$ where $\lambda_{\rm a} \equiv \ln(\tau_{\rm e}/\tau_{\rm a})$ and $\Theta$ is the Heaviside step function. Middle: The volume-weighted mean values of $s$, normalized by the volume-weighted Mach number squared, $\left< s \right>_{\rm V} M_{\rm V}^{-2},$ (stars) and by the volume-weighted Mach number, $\left< s \right>_{\rm V} M_{\rm V}^{-1},$ (circles). Again, normalizing by $M_{\rm V}^{-1}$ provides a better description of the data, and from left to right the lines give fits of $\left<s\right>_{\rm V} M_{\rm V}^{-1}=-\frac{1}{3}- \frac{1}{3}( 1+\lambda_{\rm a}) \Theta(1+\lambda_{\rm a}),$$\left<s\right>_{\rm V} M_{\rm V}^{-1}=-\frac{1}{3}$, and $\left<s\right>_{\rm V} M_{\rm V}^{-1}=-\frac{1}{6}.$Bottom: Skewness of $P_{\rm V}(s).$ For most cases, the distributions are negatively skewed, although $\mu_{s,\rm V}$ is the largest in the compressively-driven and mixed simulations with small driving correlation times. Unlike $\sigma_{s,{\rm V}}$ and $\left<s\right>_{\rm V}$, skewness shows no strong trends with Mach number.
• Figure 5: Mass-weighted PDFs from a representative subset of our simulations. Columns, rows, and line styles are as in Fig. \ref{['fig:PSV']}. For the compressive runs, increasing $\tau_{\rm a}$ shifts the peak to the right, broadens the distribution and leads to a large negative skewness. As in the volume-weighted case, similar effects are seen to a limited degree in the mixed driving runs, while the differences between the solenoidal runs are consistent with small changes in the Mach number.