A model for (non-lognormal) density distributions in isothermal turbulence

We propose a new, physically motivated fitting function for density probability distribution functions (PDFs) in turbulent, ideal gas. Although it is generally known that when gas is isothermal, the PDF is approximately lognormal in the core, high-resolution simulations show large deviations from exact lognormality. The proposed function provides an extraordinarily accurate description of the density PDFs in simulations with Mach numbers similar to 0.1-15 and dispersion in log (rho) from similar to 0.01 to 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 a variety of distinct numerical methods, including or excluding magnetic fields. Deviations from lognormality are represented by a parameter T that appears to increase systematically with the compressive Mach number of the simulations. 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 also provide a simple interpretation of the density PDF as the product of a continuous-time relaxation process. As such this parameter T is consistent with the same single parameter needed to explain the (intermittent) velocity structure functions; its behaviour 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 (for example, in the dispersion-Mach number relation) 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 lognormal statistics.