Stability of a plasma stream in a magnetic field

The system studied is a plasma streaming parallel to a uniform magnetic field with a velocity which varies in a transverse direction. The flow is bounded at y =± a by perfectly conducting planes.

The Poisson-Vlasov equations are used to derive an integro-differential equation for ø the Fourier transform of the electrostatic potential. The kernel of this equation is expanded using a small Larmor radius expansion for ø and for the equilibrium distribution function f₀.