Scaling of Turbulent Viscosity and Resistivity: Extracting a Scale-dependent Turbulent Magnetic Prandtl Number

TL;DR

This paper develops a method to determine scale-dependent turbulent viscosity and resistivity in magnetized turbulence, revealing how their ratio varies with scale and suggesting it approaches unity at high Reynolds numbers.

Contribution

It introduces a coarse-graining approach to extract scale-dependent turbulent transport coefficients from high-resolution simulations, advancing understanding of magnetic Prandtl number behavior.

Findings

Pr_t varies from ~1-2 at small scales to ~5 at large scales

Pr_t tends to become scale-independent and near unity at high Reynolds numbers

Velocity and magnetic spectra exponents become equal in the decoupled range

Abstract

Turbulent viscosity $\nu_t$ and resistivity $\eta_t$ are perhaps the simplest models for turbulent transport of angular momentum and magnetic fields, respectively. The associated turbulent magnetic Prandtl number $Pr_t\equiv \nu_t/\eta_t$ has been well recognized to determine the final magnetic configuration of accretion disks. Here, we present an approach to determining these ''effective transport'' coefficients acting at different length-scales using coarse-graining and recent results on decoupled kinetic and magnetic energy cascades [Bian & Aluie 2019]. By analyzing the kinetic and magnetic energy cascades from a suite of high-resolution simulations, we show that our definitions of $\nu_t$, $\eta_t$, and $Pr_t$ have power-law scalings in the ''decoupled range.'' We observe that $Pr_t\approx1 \text{~to~}2$ at the smallest inertial-inductive scales, increasing to $\approx 5$ at the largest scales. However, based on physical considerations, our analysis suggests that $Pr_t$ has to become scale-independent and of order unity in the decoupled range at sufficiently high Reynolds numbers (or grid-resolution), and that the power-law scaling exponents of velocity and magnetic spectra become equal. In addition to implications to astrophysical systems, the scale-dependent turbulent transport coefficients offer a guide for large eddy simulation modeling.