An Improved Fit to the Density Distribution in Supersonic Isothermal Turbulence

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 $\sigma_{s,{\rm V}}^2 \approx \ln(1 + b^2 M_{\rm V}^2),$ where $s \equiv \ln \rho/\rho_0,$ $M_{\rm V}$ 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, $\tau_{\rm a}$, 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 $\tau_{\rm a}$ values. In the compressive case, $\sigma_{s,{\rm V}}^2$ is not well fit by the standard expression. Instead, it scales approximately linearly with $M_{\rm V},$ and its dependence on $\tau_{\rm a}$ is $\sigma_{s,{\rm V}}^2 \approx M_{\rm V} [1 + \frac{2}{3}(1 + \lambda_{\rm a})\Theta(1 + \lambda_{\rm a})]$, where $\lambda_{\rm a} \equiv \ln(\tau_{\rm a}/\tau_{\rm e})$, $\tau_{\rm e}$ is the eddy turnover time, and $\Theta$ is the Heaviside step function. Mixed-driven turbulence shows a weak dependence on $\tau_{\rm a},$ and for solenoidally-driven turbulence, $\sigma_{s,{\rm V}}^2 \approx \frac{1}{3}M_{\rm V}$, 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 $\tau_{\rm a}$, deviating from lognormal expectations. The mass-weighted density distribution displays significant broadening and skewness in compressively-driven cases, especially at large $\tau_{\rm a}/\tau_{\rm e}$. These results provide a refined framework for modeling astrophysical turbulence.