The Turbulent Magnetic Prandtl Number of MHD Turbulence in Disks

The magnetic Prandtl number Pr M is the ratio of viscosity to resistivity. In astrophysical disks the diffusion of angular momentum (viscosity) and magnetic fields (resistivity) are controlled by turbulence. Phenomenological models of the evolution of large-scale poloidal magnetic fields in disks suggest that the turbulent magnetic Prandtl number Pr M,T controls the rate of escape of vertical field from the disk; for Pr M,T ⩽ R / H vertical field diffuses outward before it can be advected inward by accretion. Here we measure field diffusion and angular momentum transport due to MHD turbulence in a shearing box, and thus Pr M,T , by studying the evolution of a sinusoidal perturbation in the magnetic field that is injected into a turbulent background. We show that the perturbation is always stable, decays approximately exponentially, has decay rate ∝ k 2 , and that the implied Pr M,T ∼ 1.