Fluid dynamics predicts the onset of instability and transition to
turbulent flow within
fluids of different
densities moving at different speeds.[3] If surface tension is ignored, two fluids in parallel motion with different velocities and densities yield an interface that is unstable to short-wavelength perturbations for all speeds. However,
surface tension is able to stabilize the short wavelength instability up to a threshold velocity.
If the density and velocity vary continuously in space (with the lighter layers uppermost, so that the fluid is
RT-stable), the dynamics of the Kelvin-Helmholtz instability is described by the
Taylor–Goldstein equation:
where denotes the
Brunt–Väisälä frequency, U is the horizontal parallel velocity, k is the wave number, c is the eigenvalue parameter of the problem, is complex amplitude of the
stream function. Its onset is given by the
Richardson number. Typically the layer is unstable for . These effects are common in cloud layers. The study of this instability is applicable in plasma physics, for example in
inertial confinement fusion and the
plasma–
beryllium interface. In situations where there is a state of static stability, evident by heavier fluids found below than the lower fluid, the Rayleigh-Taylor instability can be ignored as the Kelvin–Helmholtz instability is sufficient given the conditions.[clarification needed]
Numerically, the Kelvin–Helmholtz instability is simulated in a temporal or a spatial approach. In the temporal approach, the flow is considered in a periodic (cyclic) box "moving" at mean speed (absolute instability). In the spatial approach, simulations mimic a lab experiment with natural inlet and outlet conditions (convective instability).
Discovery and history
The existence of the Kelvin-Helmholtz instability was first discovered by German physiologist and physicist Hermann von Helmholtz in 1868. Helmholtz identified that "every perfect geometrically sharp edge by which a fluid flows must tear it asunder and establish a surface of separation".[5][3] Following that work, in 1871, collaborator
William Thomson (later Lord Kelvin), developed a mathematical solution of linear instability whilst attempting to model the formation of ocean wind waves.[6]
Throughout the early 20th Century, the ideas of Kelvin-Helmholtz instabilities were applied to a range of stratified fluid applications. In the early 1920s,
Lewis Fry Richardson developed the concept that such shear instability would only form where shear overcame static stability due to stratification, encapsulated in the
Richardson Number.
Geophysical observations of the Kelvin-Helmholtz instability were made through the late 1960s/early 1970s, for clouds,[7] and later the ocean. [8]
^Helmholtz (1 November 1868). "XLIII. On discontinuous movements of fluids". The London, Edinburgh, and Dublin Philosophical Magazine and Journal of Science. 36 (244): 337–346.
doi:
10.1080/14786446808640073.
^Ludlam, F. H. (October 1967). "Characteristics of billow clouds and their relation to clear-air turbulence". Quarterly Journal of the Royal Meteorological Society. 93 (398): 419–435.
Bibcode:
1967QJRMS..93..419L.
doi:
10.1002/qj.49709339803.
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. 23: 215–228.
Hwang, K.-J.; Goldstein; Kuznetsova; Wang; Viñas; Sibeck (2012). "The first in situ observation of Kelvin-Helmholtz waves at high-latitude magnetopause during strongly dawnward interplanetary magnetic field conditions". J. Geophys. Res. 117 (A08233): n/a.
Bibcode:
2012JGRA..117.8233H.
doi:
10.1029/2011JA017256.
hdl:2060/20140009615.