Turbulence in virtual: Origin of the variance and skewness of density function
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| Format: | Preprint |
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2025
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| _version_ | 1866913756213149696 |
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| author | Liu, Xunchuan |
| author_facet | Liu, Xunchuan |
| contents | 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 ($σ^2$) and Mach number ($M$). In this work, we aim to explain the $σ^2$-$M$ 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 $ σ^2 = \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 $σ^2$-$M$ 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. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2502_20458 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Turbulence in virtual: Origin of the variance and skewness of density function Liu, Xunchuan Astrophysics of Galaxies 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 ($σ^2$) and Mach number ($M$). In this work, we aim to explain the $σ^2$-$M$ 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 $ σ^2 = \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 $σ^2$-$M$ 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. |
| title | Turbulence in virtual: Origin of the variance and skewness of density function |
| topic | Astrophysics of Galaxies |
| url | https://arxiv.org/abs/2502.20458 |