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

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, ${\mathrm{Pm}}_{\mathrm{t}}$, 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 ($\mathrm{Re}$) 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 $\mathrm{Re}$ and the scale separation ratio. However, a weak decreasing trend is found even at the largest values of $\mathrm{Re}$. The turbulent magnetic Prandtl number converges to a value that is slightly below unity for large $\mathrm{Re}$ whereas for small $\mathrm{Re}$, 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 ${\mathrm{Pm}}_{\mathrm{t}}$ found from the simulations ($\approx 0.9\dots 0.95$) at large $\mathrm{Re}$ and scale separation ratio is higher than any of the analytic predictions.

13 pages, 12 figures, accepted in Astron. Astrophys

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• Interplay between tidal flows and magnetic fields in nonlinear simulations of stellar and planetary convective envelopes
• Turbulent Prandtl number from isotropically forced turbulence
• Multiscale Turbulence Synthesis: Validation in 2D Hydrodynamics
• Generation of mean flows in rotating anisotropic turbulence: The case of solar near-surface shear layer