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Kelvin-Helmholtz Instability

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 tex2html_wrap_inline4724 and tex2html_wrap_inline4726 being the velocities on the two sides of the interface, the dispersion relation is given by:


where tex2html_wrap_inline4728 is Alfven velocity given by


where B is the magnetic field and tex2html_wrap_inline4732 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,].

Andreas Schiffler
Wed Oct 9 10:05:17 CST 1996