The Kelvin–Helmholtz instability (after
Lord Kelvin
William Thomson, 1st Baron Kelvin (26 June 182417 December 1907), was a British mathematician, Mathematical physics, mathematical physicist and engineer. Born in Belfast, he was the Professor of Natural Philosophy (Glasgow), professor of Natur ...
and
Hermann von Helmholtz
Hermann Ludwig Ferdinand von Helmholtz (; ; 31 August 1821 – 8 September 1894; "von" since 1883) was a German physicist and physician who made significant contributions in several scientific fields, particularly hydrodynamic stability. The ...
) is a
fluid instability
In physics, a fluid is a liquid, gas, or other material that may continuously move and deform (''flow'') under an applied shear stress, or external force. They have zero shear modulus, or, in simpler terms, are substances which cannot res ...
that occurs when there is
velocity shear in a single
continuous fluid or a velocity difference across the interface between two
fluid
In physics, a fluid is a liquid, gas, or other material that may continuously motion, move and Deformation (physics), deform (''flow'') under an applied shear stress, or external force. They have zero shear modulus, or, in simpler terms, are M ...
s. Kelvin-Helmholtz instabilities are visible in the
atmosphere
An atmosphere () is a layer of gases that envelop an astronomical object, held in place by the gravity of the object. A planet retains an atmosphere when the gravity is great and the temperature of the atmosphere is low. A stellar atmosph ...
s of planets and moons, such as in
cloud formations on
Earth
Earth is the third planet from the Sun and the only astronomical object known to Planetary habitability, harbor life. This is enabled by Earth being an ocean world, the only one in the Solar System sustaining liquid surface water. Almost all ...
or the
Red Spot on
Jupiter
Jupiter is the fifth planet from the Sun and the List of Solar System objects by size, largest in the Solar System. It is a gas giant with a Jupiter mass, mass more than 2.5 times that of all the other planets in the Solar System combined a ...
, and the
atmospheres of the Sun and other stars.
Theory overview and mathematical concepts
Fluid dynamics
In physics, physical chemistry and engineering, fluid dynamics is a subdiscipline of fluid mechanics that describes the flow of fluids – liquids and gases. It has several subdisciplines, including (the study of air and other gases in motion ...
predicts the onset of instability and transition to
turbulent flow
In fluid dynamics, turbulence or turbulent flow is fluid motion characterized by Chaos theory, chaotic changes in pressure and flow velocity. It is in contrast to laminar flow, which occurs when a fluid flows in parallel layers with no disrupt ...
within
fluid
In physics, a fluid is a liquid, gas, or other material that may continuously motion, move and Deformation (physics), deform (''flow'') under an applied shear stress, or external force. They have zero shear modulus, or, in simpler terms, are M ...
s of different
densities moving at different speeds.
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
Surface tension is the tendency of liquid surfaces at rest to shrink into the minimum surface area possible. Surface tension (physics), tension is what allows objects with a higher density than water such as razor blades and insects (e.g. Ge ...
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
In fluid dynamics, two types of stream function (or streamfunction) are defined:
* The two-dimensional (or Lagrange) stream function, introduced by Joseph Louis Lagrange in 1781, is defined for incompressible flow, incompressible (divergence-free ...
. Its onset is given by the
Richardson number
The Richardson number (Ri) is named after Lewis Fry Richardson (1881–1953). It is the dimensionless number that expresses the ratio of the buoyancy term to the flow velocity, flow shear (fluid), shear term:
:
\mathrm = \frac = \frac \frac
wh ...
. 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
Inertial confinement fusion (ICF) is a fusion energy process that initiates nuclear fusion reactions by compressing and heating targets filled with fuel. The targets are small pellets, typically containing deuterium (2H) and tritium (3H).
Typical ...
and the
plasma–
beryllium
Beryllium is a chemical element; it has Symbol (chemistry), symbol Be and atomic number 4. It is a steel-gray, hard, strong, lightweight and brittle alkaline earth metal. It is a divalent element that occurs naturally only in combination with ...
interface. In situations where there is a state of static stability (where there is a continuous density gradient), the
Rayleigh–Taylor instability is often insignificant compared to the magnitude of the Kelvin–Helmholtz instability.
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".
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.
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
Lewis Fry Richardson, Fellow of the Royal Society, FRS (11 October 1881 – 30 September 1953) was an English mathematician, physicist, meteorologist, psychologist, and Pacifism, pacifist who pioneered modern mathematical techniques of weather ...
developed the concept that such shear instability would only form where shear overcame static stability due to stratification, encapsulated in the
Richardson Number
The Richardson number (Ri) is named after Lewis Fry Richardson (1881–1953). It is the dimensionless number that expresses the ratio of the buoyancy term to the flow velocity, flow shear (fluid), shear term:
:
\mathrm = \frac = \frac \frac
wh ...
.
Geophysical observations of the Kelvin-Helmholtz instability were made through the late 1960s/early 1970s, for clouds, and later the ocean.
See also
*
Rayleigh–Taylor instability
*
Richtmyer–Meshkov instability
*
Mushroom cloud
A mushroom cloud is a distinctive mushroom-shaped flammagenitus cloud of debris, smoke, and usually condensed water vapour resulting from a large explosion. The effect is most commonly associated with a nuclear explosion, but any sufficiently e ...
*
Plateau–Rayleigh instability
In fluid dynamics, the Plateau–Rayleigh instability, often just called the Rayleigh instability, explains why and how a falling stream of fluid breaks up into smaller packets with the same total volume but less surface area per droplet. It is ...
*
Kármán vortex street
*
Taylor–Couette flow
In fluid dynamics, the Taylor–Couette flow consists of a viscous fluid confined in the gap between two rotating cylinders. For low angular velocities, measured by the Reynolds number ''Re'', the flow is steady and purely azimuthal. This laminar ...
*
Fluid mechanics
Fluid mechanics is the branch of physics concerned with the mechanics of fluids (liquids, gases, and plasma (physics), plasmas) and the forces on them.
Originally applied to water (hydromechanics), it found applications in a wide range of discipl ...
*
Fluid dynamics
In physics, physical chemistry and engineering, fluid dynamics is a subdiscipline of fluid mechanics that describes the flow of fluids – liquids and gases. It has several subdisciplines, including (the study of air and other gases in motion ...
*
Reynolds number
In fluid dynamics, the Reynolds number () is a dimensionless quantity that helps predict fluid flow patterns in different situations by measuring the ratio between Inertia, inertial and viscous forces. At low Reynolds numbers, flows tend to ...
*
Turbulence
In fluid dynamics, turbulence or turbulent flow is fluid motion characterized by chaotic changes in pressure and flow velocity. It is in contrast to laminar flow, which occurs when a fluid flows in parallel layers with no disruption between ...
Notes
References
*
*
* Article describing discovery of K-H waves in deep ocean:
External links
*
Giant Tsunami-Shaped Clouds Roll Across Alabama Sky- Natalie Wolchover,
Livescience
Live Science is a science news website. The publication features stories on a wide range of topics, including space, animals, health, archaeology, human behavior, and planet Earth. It also includes a reference section with links to other websites. ...
via Yahoo.com
Tsunami Cloud Hits Florida CoastlineVortex formation in free jet- YouTube video showing Kelvin Helmholtz waves on the edge of a free jet visualised in a scientific experiment.
Wave clouds over Christchurch CityKelvin-Helmholtz clouds, in Barmouth, Gwynedd, on 18 February 2017
{{DEFAULTSORT:Kelvin-Helmholtz instability
1868 introductions
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Fluid dynamics
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Fluid dynamic instabilities
Articles containing video clips
Hermann von Helmholtz
William Thomson, 1st Baron Kelvin
Plasma instabilities