The slope of the power spectrum of the density field in isothermal supersonic compressible turbulence

TL;DR

This paper explains how the slope of the density power spectrum in supersonic turbulence varies with Mach number, providing a model that matches simulations and observations, but warns against using the slope to estimate Mach number.

Contribution

The authors present a quantitative model linking the density power spectrum slope to Mach number in isothermal turbulence, validated by simulations and interpreted for interstellar medium observations.

Findings

Model accurately reproduces measured slopes across different inertial ranges.

The slope of the density power spectrum is not a reliable indicator of Mach number.

A resolution criterion for simulating turbulence with specific spectral slopes is discussed.

Abstract

The power spectrum (PS) of the density field in supersonic turbulence is a fundamental quantity that characterizes the statistical properties of the structures formed in compressible flows. It is also widely used to estimate the Mach number in the interstellar medium from simulation-derived relations. In this paper, we provide a first quantitative explanation for the evolution of the slope of the PS of the density field with the Mach number in homogeneous isotropic isothermal turbulence using a time-invariant quantity derived by Chandrasekhar (1951). For simulated turbulent flows, the model reproduces the measured slopes for different widths of the inertial range and density variances very well. Our model also provides a comprehensive interpretation of the characteristic slopes of the PS of the density field measured in the interstellar medium. Based on these results, we stress that the Mach number cannot be reliably deduced from the slope of the PS of the density field. In closing, we discuss a resolution criterion that must be fulfilled to correctly simulate a turbulent flow with a given density PS slope.