Quantifying the effect of turbulent magnetic diffusion on the growth
  rate of the magneto-rotational instability
  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. 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.
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. 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. Turbulence reduces the growth rate of the MRI in a way that is the
same as microphysical magnetic diffusion.
