A two-level implicit scheme for the numerical solution of the linearized vorticity equation

A numerical scheme is developed to study the stability of the linearized vorticity equations. The linear growth rates of Kelvin-Helmholtz instabilities resulting from a velocity shear are calculated numerically. The theory of neutrally stable waves, in two dimensions, is extended to include the cases where the stream function is zero or periodic at the boundary, and is verified numerically for the case of a velocity profile having the shape of a cosine function. The instability growth rates of the oscillations in the neighbourhood of the neutrally stable waves are calculated numerically and are shown to be in very good agreement with the theory. The results are also of interest in relation to studies on the instabilities of a two-dimensional guiding-centre plasma, and also in the study of the diocotron instability in electronics.