2025/02/27 by Liu, Xunchuan
#Astrophysics of Galaxies (astro-ph.GA) #FOS: Physical sciences
paper · doi:10.48550/arxiv.2502.20458
Turbulence is a complex phenomenon that plays a critical role in the interstellar medium (ISM). Previous simulations and observations show that the probability density function (PDF) of gas density in isothermal and compressible systems under turbulence exhibits a near lognormal shape, with a strong empirical relation between the variance (σ2) and Mach number (M). In this work, we aim to explain the σ2-M relation and the deviation from the lognormal shape from a thermodynamic and cascading perspective. By introducing a virtual dissipation process, during which turbulent entropy and structural dissipation are assumed to be coupled, we derive the empirical relation σ2 = ln(1 + M)2 . Additionally, by introducing a delay parameter q for the local gas temperature, we derive the deviation from the empirical relation at high M. We further argue that the exponential tails of PDFs (on the s = ln(ρ) scale) arise from the convolution of PDF kernels, which can be skewed at both the low-s and high-s ends. Skewness has limited influence on the σ2-M relation. Two density-fraction strategies--the mass-fraction and volume-fraction approaches--are introduced to explain the physical origins of the low-s and high-s skewed PDF kernels. These two types of PDF kernels are dual to each other and exhibit highly symmetric mathematical structures. We speculate that the high-s skewed PDF kernels are physical and may be analogous to the high-density tails of column-density PDFs in molecular clouds, which are influenced by gravity. Inspired by this, we propose a form of an "isothermal" turbulent system that likely favors the volume-fraction strategy.