Structure of iso-density sets in supersonic isothermal turbulence

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