The Kelvin-Helmholtz instability is operative in the magnetosphere
at the interface between two
adjacent flow regions when there is a velocity shear.
The classical example is the generation of water waves by wind blowing
over the surface of the water. It is also the name for a whole class of such
instabilities. In Chapter 13 of Mikhailovskii [Mik92], a number of different
cases are considered theoretically, starting from the dispersion equation of
the ordinary Kelvin-Helmholtz instability. For the simplest case of a
step-function velocity profile with 
and 
being the
velocities on the two sides of the interface, the dispersion relation is given by:
where 
is Alfven velocity given by
where B is the magnetic field and 
is the mass density of the plasma.
The instability takes place if
Factors modifying the growth rate are collisions, gradients, electric fields and the structure of the waves. For example the threshold is independent of the wavelength but the growth rate is faster if the waves are shorter. A characteristic of the Kelvin-Helmholtz instability is that it tends to produce vortices. This instability can also cause a larger patch of plasma with enhanced plasma density to break up into smaller ones [Har92, p. 42, p. 325,]. The development of such instabilities need not disrupt the overall flow. The instability is driven at the shear layer and the development of the instability broadens this layer. The system can adjust by developing a smooth velocity gradient across the shear layer, and can then approach a stable state with instabilities present [Mel86, pp. 144-146,]. Extensive numerical modeling of the Kelvin-Helmholtz instability has been performed and shows all the features noted above [Taj89, Chap. 13.2,].