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

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.