An Improved Fit to the Density Distribution in Supersonic Isothermal Turbulence

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 ${\sigma }_{s,\mathrm{V}}^2\approx \ln (1+b^2{M}_{\mathrm{V}}^2),$ where $s\equiv \ln \rho /{\rho }_0,$ $M_{\mathrm{V}}$ 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, ${\tau }_{\mathrm{a}}$, 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 ${\tau }_{\mathrm{a}}$ values. In the compressive case, ${\sigma }_{s,\mathrm{V}}^2$ is not well fit by the standard expression. Instead, it scales approximately linearly with $M_{\mathrm{V}},$ and its dependence on ${\tau }_{\mathrm{a}}$ is ${\sigma }_{s,\mathrm{V}}^2\approx M_{\mathrm{V}}[1+\frac{2}{3}(1+{\lambda }_{\mathrm{a}})\Theta (1+{\lambda }_{\mathrm{a}})]$, where ${\lambda }_{\mathrm{a}}\equiv \ln ({\tau }_{\mathrm{a}}/{\tau }_{\mathrm{e}})$, ${\tau }_{\mathrm{e}}$ is the eddy turnover time, and $\Theta$ is the Heaviside step function. Mixed-driven turbulence shows a weak dependence on ${\tau }_{\mathrm{a}},$ and for solenoidally-driven turbulence, ${\sigma }_{s,\mathrm{V}}^2\approx \frac{1}{3}{M}_{\mathrm{V}}$, which is consistent with the standard expression when $M_{\mathrm{V}}\lesssim 8.$ The volume-weighted mean and skewness also show systematic trends with $M_{\mathrm{V}}$ and ${\tau }_{\mathrm{a}}$, deviating from lognormal expectations. The mass-weighted density distribution displays significant broadening and skewness in compressively-driven cases, especially at large ${\tau }_{\mathrm{a}}/{\tau }_{\mathrm{e}}$. These results provide a refined framework for modeling astrophysical turbulence.

21 pages, 7 figures, ApJ, in press

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