Structure of iso-density sets in supersonic isothermal turbulence

TL;DR

This paper develops a theoretical framework to analyze the structure of iso-density regions in supersonic isothermal turbulence, revealing their fractal nature and the scale-dependent behavior of the turbulent cascade.

Contribution

It introduces a novel correlation and structure function approach to characterize iso-density set geometries and their evolution in turbulence, supported by high-resolution simulation data.

Findings

Iso-density sets are surface fractals with varying dimensions.

Turbulent cascade transfers surface area from large to small scales.

No constant transfer rate observed across scales, challenging existing turbulence models.

Abstract

Context: The gas density structure of the cold molecular phase of the interstellar medium is the main controller of star formation. Aims: A theoretical framework is proposed to describe the structural content of the density field in isothermal supersonic turbulence. Methods: It makes use of correlation and structure functions of the phase indicator field defined for different iso-density values. The relations between these two-point statistics and the geometrical features of iso-density sets such as the volume fraction, the surface density, the curvature, and fractal characteristics are provided. An exact scale-by-scale budget equation is further derived revealing the role of the turbulent cascade and dilation on the structural evolution of the density field. Although applicable to many flow situations, this tool is here first invoked for characterising supersonic isothermal turbulence, using data from the currently best-resolved numerical simulation. Results: We show that iso-density sets are surface fractals rather than mass fractals, with dimensions that markedly differ between dilute, neutral, and dense regions. The surface-size relation is established for different iso-density values. We further find that the turbulent cascade of iso-density sets is directed from large towards smaller scales, in agreement with the classical picture that turbulence acts to concentrate more surface into smaller volumes. Intriguingly, there is no range of scales that complies with a constant transfer rate in the cascade, challenging our fundamental understanding of interstellar turbulence. Finally, we recast the virial theorem in a new formulation drawing an explicit relation between the aforementioned geometrical measures and the dynamics of iso-density sets.