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

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 $P{r}_t\equiv {\nu }_t/{\eta }_t$ 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 $P{r}_t$ have power-law scalings in the ''decoupled range.'' We observe that $P{r}_t\approx 1\text{ to }2$ at the smallest inertial-inductive scales, increasing to $\approx 5$ at the largest scales. However, based on physical considerations, our analysis suggests that $P{r}_t$ 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.

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