Kelvin-Helmholtz instability of a plasma with anisotropic and polytropic pressure laws

The Kelvin-Helmholtz (K-H) instability of two fluids of plasma streaming in opposite directions with the same velocity and in the presence of an external magnetic field is investigated. The usual magnetohydrodynamic equations with anisotropic pressure are considered. In the present problem, the two pressures parallel and perpendicular to the direction of the magnetic field are defined by polytropic pressure laws. The generalized pressure relations are used, and two equations of state for two pressures are assumed. The equations are linearized, and initially two different flow velocities are taken for the system. The flow is assumed to be in the direction perpendicular to the magnetic field. The problem is solved and a dispersion relation is obtained. From the dispersion relation, the II-H instability condition is obtained. It is found that the instability condition depends upon the polytropic indices of the pressure relations. The condition of instability is further obtained for MHD and Chew-Goldberger-Low systems. It is also found that the growth rate of the instability depends upon various polytropic indices.