Statistical properties of compressible isothermal turbulence from sub- to supersonic conditions

¹ CNRS, CORIA, UMR 6614, Normandy Univ., UNIROUEN, INSA Rouen, 675 Avenue de l’université, BP 12, 76801 Saint Etienne du Rouvray Cedex, France
² Research School of Astronomy and Astrophysics, Australian National University, Cotter Road, Canberra, ACT 2611, Australia
³ Australian Research Council Centre of Excellence in All Sky Astrophysics (ASTRO3D), Cotter Road, Canberra ACT 2611, Australia

^⋆ Corresponding authors: fabien.thiesset@cnrs.fr; christoph.federrath@anu.edu.au

Received: 24 May 2025
Accepted: 14 October 2025

Abstract

Context. Turbulence is one of the key processes that control the spatial and temporal evolution of matter and energy of many astrophysical systems.

Aims. This paper investigates the statistical properties of isothermal turbulence in both the subsonic and supersonic regimes. The focus is on the influence of the Mach number (Ma) and the Reynolds number (Re) on both the space-local and scale-dependent fluctuations of relevant gas variables, the density, velocity, their derivatives, and the kinetic energy.

Methods. We carried out hydrodynamical simulations of driven turbulence with explicit viscosity and therefore controlled Re, at converged numerical resolutions up to 1920³ grid cells.

Results. We confirm previous work that the probability density functions (PDFs) of the gas density are approximately log-normal and depend on Ma. We provide a new detailed quantification of the dependence of the PDFs of density and velocity on Re, finding a relatively weak dependence, provided Re > 200. In contrast, derivatives of the density and velocity field are sensitive to Re, with the probability of extreme events (the tails of the PDFs) growing with Re. The PDFs of the density gradient and velocity divergence (dilatation) exhibit increasingly heavy tails with growing Ma, signalling enhanced internal intermittency. At sufficiently high Ma, the statistics of dilatation are observed to saturate at a level determined solely by Re, suggesting that turbulent dilatation becomes limited by viscous effects. We also examine the scale-by-scale distribution of kinetic energy through a compressible form of the Kármán-Howarth-Monin (KHM) equation that incorporates density variations. In the intermediate range of scales, a marked difference is found between subsonic and supersonic turbulence: while Kolmogorov-like scaling applies in the sub- and transonic regimes, supersonic turbulence aligns more closely with Burgers turbulence predictions. The analysis of individual terms in the KHM equation highlights the role of the pressure-velocity coupling as an additional mechanism for converting and transferring kinetic energy from large to small scales. Moreover, the contributions of the KHM terms exhibit non-monotonic behaviour with increasing Ma, with dilatational effects becoming more pronounced and acting to oppose the cascade of kinetic energy.