Linear Growth of the Kelvin-Helmholtz Instability with an Adiabatic Cosmic-ray Gas

We investigate effects of cosmic-rays on the linear growth of the Kelvin-Helmholtz instability. Cosmic-rays are treated as an adiabatic gas and allowed to diffuse along magnetic field lines. We calculated the dispersion relation of the instability for various sets of two free parameters, the ratio of the cosmic-ray pressure to the thermal gas pressure and the diffusion coefficient. Including cosmic-ray effects, a shear layer is more destabilized and the growth rates can be enhanced in comparison with the ideal magnetohydrodynamical case. Whether the growth rate is effectively enhanced or not depends on the diffusion coefficient of cosmic-rays. We obtain the criterion for effective enhancement by comparing the growing time scale of the instability with the diffusion time scale of cosmic-rays. These results can be applied to various astrophysical phenomena where a velocity shear is present, such as outflows from star-forming galaxies, AGN jet, channel flows resulting from the nonlinear development of the magnetorotational instability, and galactic disks.