# Turbulent viscosity and effective magnetic Prandtl number from simulations of isotropically forced turbulence

## TL;DR

This study uses simulations of isotropically forced turbulence to compute turbulent viscosity, magnetic diffusivity, and their ratio, revealing that the turbulent magnetic Prandtl number is slightly below unity at high Reynolds numbers, exceeding many analytic predictions.

## Contribution

First direct numerical simulations of turbulent viscosity and magnetic diffusivity in isotropic turbulence, providing detailed scale dependence and convergence behavior of the turbulent magnetic Prandtl number.

## Key findings

- Turbulent viscosity and magnetic diffusivity are of similar magnitude.
- Turbulent magnetic Prandtl number converges to ~0.9-0.95 at high Re.
- Scale dependence of turbulent viscosity is Lorentzian.

## Abstract

(abridged) Context: Turbulent diffusion of large-scale flows and magnetic fields play major roles in many astrophysical systems. Aims: Our goal is to compute turbulent viscosity and magnetic diffusivity, relevant for diffusing large-scale flows and magnetic fields, respectively, and their ratio, the turbulent magnetic Prandtl number, ${\rm Pm}_{\rm t}$, for isotropically forced homogeneous turbulence. Methods: We use simulations of forced turbulence in fully periodic cubes composed of isothermal gas with an imposed large-scale sinusoidal shear flow. Turbulent viscosity is computed either from the resulting Reynolds stress or from the decay rate of the large-scale flow. Turbulent magnetic diffusivity is computed using the test-field method. The scale dependence of the coefficients is studied by varying the wavenumber of the imposed sinusoidal shear and test fields. Results: We find that turbulent viscosity and magnetic diffusivity are in general of the same order of magnitude. Furthermore, the turbulent viscosity depends on the fluid Reynolds number (${\rm Re}$) and scale separation ratio of turbulence. The scale dependence of the turbulent viscosity is found to be well approximated by a Lorentzian. The results for the turbulent transport coefficients appear to converge at sufficiently high values of ${\rm Re}$ and the scale separation ratio. However, a weak decreasing trend is found even at the largest values of ${\rm Re}$. The turbulent magnetic Prandtl number converges to a value that is slightly below unity for large ${\rm Re}$ whereas for small ${\rm Re}$, we find values between 0.5 and 0.6. Conclusions: The turbulent magnetic diffusivity is in general consistently higher than the turbulent viscosity. The actual value of ${\rm Pm}_{\rm t}$ found from the simulations ($\approx0.9\ldots0.95$) at large ${\rm Re}$ and scale separation ratio is higher than any of the analytic predictions.