Analysis of the magneto-rotational instability with the effect of
  cosmic-ray diffusion
  We present the results obtained from linear stability analysis and
2.5-dimensional magnetohydrodynamic (MHD) simulations of the magnetorotational
instability (MRI), including the effects of cosmic rays (CRs). We took into
account of the CR diffusion along the magnetic field but neglect the
cross-field-line diffusion. Two models are considered in this paper: shearing
box model and differentially rotating cylinder model. We studied how MRI is
affected by the initial CR pressure (i.e., energy) distribution. In the
shearing box model, the initial state is uniform distribution. Linear analysis
shows that the growth rate of MRI does not depend on the value of CR diffusion
coefficient. In the differentially rotating cylinder model, the initial state
is a constant angular momentum polytropic disk threaded by weak uniform
vertical magnetic field. Linear analysis shows that the growth rate of MRI
becomes larger if the CR diffusion coefficient is larger. Both results are
confirmed by MHD simulations. The MHD simulation results show that the outward
movement of matter by the growth of MRI is not impeded by the CR pressure
gradient, and the centrifugal force which acts to the concentrated matter
becomes larger. Consequently, the growth rate of MRI is increased. On the other
hand, if the initial CR pressure is uniform, then the growth rate of the MRI
barely depends on the value of the CR diffusion coefficient.
