Scaling Laws of Passive-Scalar Diffusion in the Interstellar Medium

TL;DR

This paper investigates the fundamental scaling laws of passive scalar diffusion in the turbulent, supersonic, magnetized interstellar medium, proposing a fractional diffusion model validated by simulations.

Contribution

It introduces a simple fractional diffusion equation for scalar transport in the ISM, accounting for scale-dependent turbulent diffusivity and Mach number effects, validated by simulations.

Findings

Scaling laws depend on Mach number and shear flow.

Scalar distribution develops non-Gaussian tails over time.

Ensemble averages differ from individual scalar patch behavior.

Abstract

Passive scalar mixing (metals, molecules, etc.) in the turbulent interstellar medium (ISM) is critical for abundance patterns of stars and clusters, galaxy and star formation, and cooling from the circumgalactic medium. However, the fundamental scaling laws remain poorly understood in the highly supersonic, magnetized, shearing regime relevant for the ISM. We therefore study the full scaling laws governing passive-scalar transport in idealized simulations of supersonic turbulence. Using simple phenomenological arguments for the variation of diffusivity with scale based on Richardson diffusion, we propose a simple fractional diffusion equation to describe the turbulent advection of an initial passive scalar distribution. These predictions agree well with the measurements from simulations, and vary with turbulent Mach number in the expected manner, remaining valid even in the presence of a large-scale shear flow (e.g. rotation in a galactic disk). The evolution of the scalar distribution is not the same as obtained using simple, constant "effective diffusivity" as in Smagorinsky models, because the scale-dependence of turbulent transport means an initially Gaussian distribution quickly develops highly non-Gaussian tails. We also emphasize that these are mean scalings that only apply to ensemble behaviors (assuming many different, random scalar injection sites): individual Lagrangian "patches" remain coherent (poorly-mixed) and simply advect for a large number of turbulent flow-crossing times.