Quantifying the effect of turbulent magnetic diffusion on the growth rate of the magneto−rotational instability^⋆

¹ Department of Physics, Gustaf Hällströmin katu 2a, PO Box 64, 00014 University of Helsinki, Finland
e-mail: miikka.vaisala@helsinki.fi
² Nordita, KTH Royal Institute of Technology and Stockholm University, Roslagstullsbacken 23, 10691 Stockholm, Sweden
³ Department of Astronomy, AlbaNova University Center, Stockholm University, 10691 Stockholm, Sweden
⁴ Aalto University, ReSoLVE Centre of Excellence, Department of Information and Computer Science, PO Box 15400, 00076 Aalto, Finland

Received: 11 October 2013
Accepted: 9 May 2014

Abstract

Context. In astrophysics, turbulent diffusion is often used in place of microphysical diffusion to avoid resolving the small scales. However, we expect this approach to break down when time and length scales of the turbulence become comparable with other relevant time and length scales in the system. Turbulent diffusion has previously been applied to the magneto-rotational instability (MRI), but no quantitative comparison of growth rates at different turbulent intensities has been performed.

Aims. We investigate to what extent turbulent diffusion can be used to model the effects of small-scale turbulence on the kinematic growth rates of the MRI, and how this depends on angular velocity and magnetic field strength.

Methods. We use direct numerical simulations in three-dimensional shearing boxes with periodic boundary conditions in the spanwise direction and additional random plane-wave volume forcing to drive a turbulent flow at a given length scale. We estimate the turbulent diffusivity using a mixing length formula and compare with results obtained with the test-field method.

Results. It turns out that the concept of turbulent diffusion is remarkably accurate in describing the effect of turbulence on the growth rate of the MRI. No noticeable breakdown of turbulent diffusion has been found, even when time and length scales of the turbulence become comparable with those imposed by the MRI itself. On the other hand, quenching of turbulent magnetic diffusivity by the magnetic field is found to be absent.

Conclusions. Turbulence reduces the growth rate of the MRI in the same way as microphysical magnetic diffusion does.