Kelvin–Helmholtz instability

Numerical simulation of a temporal Kelvin–Helmholtz instability

The Kelvin–Helmholtz instability, after Lord Kelvin and Hermann von Helmholtz, can occur when velocity shear is present within a continuous fluid, or when there is sufficient velocity difference across the interface between two fluids. One example is wind blowing over a water surface, where the wind causes the relative motion between the stratified layers (i.e., water and air). The instability will manifest itself in the form of waves being generated on the water surface. The waves can appear in numerous fluids and have been spotted in clouds, Saturn's bands, waves in the ocean, and in the sun's corona.^[1]

The theory can be used to predict the onset of instability and transition to turbulent flow in fluids of different densities moving at various speeds. Helmholtz studied the dynamics of two fluids of different densities when a small disturbance such as a wave is introduced at the boundary connecting the fluids.

Stability

A KH instability rendered visible by clouds over Mount Duval in Australia

For some short enough wavelengths, if surface tension can be ignored, two fluids in parallel motion with different velocities and densities will yield an interface that is unstable for all speeds. The existence of surface tension stabilises the short wavelength instability however, and theory then predicts stability until a velocity threshold is reached. The theory with surface tension included broadly predicts the onset of wave formation in the important case of wind over water.

A KH instability on the planet Saturn, formed at the interaction of two bands of the planet's atmosphere

In presence of gravity, for a continuously varying distribution of density and velocity, (with the lighter layers uppermost, so the fluid is RT-stable), the dynamics of the KH instability is described by the Taylor–Goldstein equation and its onset is given by a suitably defined Richardson number, Ri. Typically the layer is unstable for Ri<0.25. These effects are quite common in cloud layers. Also the study of this instability becomes applicable in plasma physics, e.g. inertial confinement fusion and the plasma–beryllium interface.

The classic textbooks by Chandrasekhar and Drazin & Reid consider the KH and RT instabilities in much detail.

From a numerical point of view, the KH instability is simulated either in a temporal or a spatial way. In the temporal approach, one considers the flow in a periodic (cyclic) box "moving" at the mean speed (absolute instability). In the spatial approach, one tries to simulate a lab experiment with natural inlet and outlet conditions (convective instability).

See also

• Rayleigh–Taylor instability
• Richtmyer–Meshkov instability
• Mushroom cloud
• Plateau–Rayleigh instability
• Kármán vortex street
• Taylor–Couette flow
• Fluid mechanics
• Fluid dynamics

Notes

1. ^ Fox, Karen C.. "NASA's Solar Dynamics Observatory Catches "Surfer" Waves on the Sun". NASA-The Sun-Earth Connection: Heliophysics. NASA. http://www.nasa.gov/mission_pages/sunearth/news/sun-surfing.html. 

References

• Lord Kelvin (William Thomson) (1871). "Hydrokinetic solutions and observations". Philosophical Magazine 42: 362–377. 
• Hermann von Helmholtz (1868). "Über discontinuierliche Flüssigkeits-Bewegungen [On the discontinuous movements of fluids]". Monatsberichte der Königlichen Preussische Akademie der Wissenschaften zu Berlin [Monthly Reports of the Royal Prussian Academy of Philosophy in Berlin] 23: 215–228. 
• Article describing discovery of K-H waves in deep ocean: Broad, William J. (April 19, 2010). "In Deep Sea, Waves With a Familiar Curl". New York Times. http://www.nytimes.com/2010/04/20/science/20waves.html?src=sch&pagewanted=all. Retrieved April, 2010.