A model for (non-lognormal) density distributions in isothermal turbulence
We propose a new, physically motivated fitting function for density PDFs in turbulent gas. Although it is known that when gas is isothermal, the PDF is approximately lognormal in the core, high-resolution simulations show large deviations from exact log-normality. The proposed function provides an extraordinarily accurate description of the density PDFs in simulations with Mach numbers ~0.1-15 and dispersion in log(rho) from ~0.01-4 dex. Compared to a lognormal or lognormal-skew-kurtosis model, the fits are improved by orders of magnitude in the wings of the distribution (with fewer free parameters). This is true in simulations using distinct numerical methods, including or excluding magnetic fields. Deviations from lognormality are represented by a parameter T that increases with the compressive Mach number. The proposed distribution can be derived from intermittent cascade models of the longitudinal (compressive) velocity differences, which should be directly related to density fluctuations, and we provide a simple interpretation of the density PDF as the product of a continuous-time relaxation process. As such the parameter T is consistent with the same parameter needed to explain the (intermittent) velocity structure functions; its behavior is consistent with turbulence becoming more intermittent as it becomes more dominated by strong shocks. It provides a new and unique probe of the role of intermittency in the density (not just velocity) structure of turbulence. We show that this naturally explains some apparent contradictory results in the literature based on use of different moments of the density PDF, as well as differences based on whether volume-weighted or mass-weighted quantities are measured. We show how these are fundamentally related to the fact that mass conservation requires violations of log-normal statistics.