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

Turbulent viscosity μ_t and resistivity η_t are perhaps the simplest models for turbulent transport of angular momentum and magnetic fields, respectively. The associated turbulent magnetic Prandtl number Pr_t = μ_t/η_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. By analyzing the kinetic and magnetic energy cascades from a suite of high-resolution simulations, we show that our definitions of μ_t, η_t, and Pr_t  have power-law scalings in the "decoupled range." We observe that Pr_t ≈1 to 2 at the smallest inertial-inductive scales, increasing to ≈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.