A Kelvin wave is a
wave
In physics, mathematics, and related fields, a wave is a propagating dynamic disturbance (change from equilibrium) of one or more quantities. Waves can be periodic, in which case those quantities oscillate repeatedly about an equilibrium (re ...
in the ocean or atmosphere that balances the Earth's
Coriolis force
In physics, the Coriolis force is an inertial or fictitious force that acts on objects in motion within a frame of reference that rotates with respect to an inertial frame. In a reference frame with clockwise rotation, the force acts to the ...
against a
topographic
Topography is the study of the forms and features of land surfaces. The topography of an area may refer to the land forms and features themselves, or a description or depiction in maps.
Topography is a field of geoscience and planetary scie ...
boundary such as a coastline, or a
waveguide
A waveguide is a structure that guides waves, such as electromagnetic waves or sound, with minimal loss of energy by restricting the transmission of energy to one direction. Without the physical constraint of a waveguide, wave intensities de ...
such as the equator. A feature of a Kelvin wave is that it is
non-dispersive, i.e., the
phase speed
The phase velocity of a wave is the rate at which the wave propagates in any medium. This is the velocity at which the phase of any one frequency component of the wave travels. For such a component, any given phase of the wave (for exampl ...
of the wave crests is equal to the
group speed of the
wave energy
Wave power is the capture of energy of wind waves to do useful work – for example, electricity generation, water desalination, or pumping water. A machine that exploits wave power is a wave energy converter (WEC).
Waves are generated by win ...
for all frequencies. This means that it retains its shape as it moves in the alongshore direction over time.
A Kelvin wave (
fluid dynamics
In physics and engineering, fluid dynamics is a subdiscipline of fluid mechanics that describes the flow of fluids— liquids and gases. It has several subdisciplines, including ''aerodynamics'' (the study of air and other gases in motion) a ...
) is also a long scale perturbation mode of a
vortex
In fluid dynamics, a vortex ( : vortices or vortexes) is a region in a fluid in which the flow revolves around an axis line, which may be straight or curved. Vortices form in stirred fluids, and may be observed in smoke rings, whirlpools in ...
in
superfluid
Superfluidity is the characteristic property of a fluid with zero viscosity which therefore flows without any loss of kinetic energy. When stirred, a superfluid forms vortices that continue to rotate indefinitely. Superfluidity occurs in two ...
dynamics; in terms of the meteorological or oceanographical derivation, one may assume that the meridional velocity component vanishes (i.e. there is no flow in the north–south direction, thus making the
momentum
In Newtonian mechanics, momentum (more specifically linear momentum or translational momentum) is the product of the mass and velocity of an object. It is a vector quantity, possessing a magnitude and a direction. If is an object's mass ...
and
continuity equation
A continuity equation or transport equation is an equation that describes the transport of some quantity. It is particularly simple and powerful when applied to a conserved quantity, but it can be generalized to apply to any extensive quantity. ...
s much simpler). This wave is named after the discoverer,
Lord Kelvin
William Thomson, 1st Baron Kelvin, (26 June 182417 December 1907) was a British mathematician, mathematical physicist and engineer born in Belfast. Professor of Natural Philosophy at the University of Glasgow for 53 years, he did important ...
(1879).
Coastal Kelvin wave
In a stratified ocean of mean depth ''H'', perturbed by some amount ''η'', free waves propagate along coastal boundaries (and hence become trapped in the vicinity of the coast itself) in the form of internal Kelvin waves on a scale of about 30 km. These waves are called coastal Kelvin waves, and have propagation speeds of approximately 2 m/s in the ocean. Using the assumption that the cross-shore velocity ''v'' is zero at the coast, ''v'' = 0, one may solve a frequency relation for the
phase speed
The phase velocity of a wave is the rate at which the wave propagates in any medium. This is the velocity at which the phase of any one frequency component of the wave travels. For such a component, any given phase of the wave (for exampl ...
of coastal Kelvin waves, which are among the class of waves called boundary waves,
edge wave
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s, trapped waves, or surface waves (similar to the
Lamb waves Lamb waves propagate in solid plates or spheres. They are elastic waves whose particle motion lies in the plane that contains the direction of wave propagation and the direction perpendicular to the plate. In 1917, the English mathematician Horace ...
).
[Gill, Adrian E., 1982: ''Atmosphere–Ocean Dynamics,'' International Geophysics Series, Volume 30, Academic Press, 662 pp.] The (
linearised)
primitive equations
The primitive equations are a set of nonlinear partial differential equations that are used to approximate global atmospheric flow and are used in most atmospheric models. They consist of three main sets of balance equations:
# A '' continuity e ...
then become the following:
* the
continuity equation
A continuity equation or transport equation is an equation that describes the transport of some quantity. It is particularly simple and powerful when applied to a conserved quantity, but it can be generalized to apply to any extensive quantity. ...
(accounting for the effects of horizontal convergence and divergence):
::
* the ''u''-momentum equation (zonal wind component):
::
* the ''v''-momentum equation (meridional wind component):
::
If one assumes that the
Coriolis coefficient
The Coriolis frequency ''ƒ'', also called the Coriolis parameter or Coriolis coefficient, is equal to twice the rotation rate ''Ω'' of the Earth multiplied by the sine of the latitude \varphi.
:f = 2 \Omega \sin \varphi.\,
The rotation rate ...
''f'' is constant along the right boundary conditions and the zonal wind speed is set equal to zero, then the primitive equations become the following:
* the continuity equation:
::
* the ''u''-momentum equation:
::
* the ''v''-momentum equation:
::
.
The solution to these equations yields the following phase speed: ''c''
2 = ''gH'', which is the same speed as for shallow-water gravity waves without the effect of Earth's rotation.
[Holton, James R., 2004: ''An Introduction to Dynamic Meteorology''. Elsevier Academic Press, Burlington, MA, pp. 394–400.] It is important to note that for an observer traveling with the wave, the coastal boundary (maximum amplitude) is always to the right in the northern hemisphere and to the left in the southern hemisphere (i.e. these waves move equatorward – negative phase speed – on a western boundary and poleward – positive phase speed – on an eastern boundary; the waves move cyclonically around an ocean basin).
Equatorial Kelvin wave
The equatorial zone essentially acts as a waveguide, causing disturbances to be trapped in the vicinity of the Equator, and the equatorial Kelvin wave illustrates this fact because the Equator acts analogously to a topographic boundary for both the Northern and Southern Hemispheres, making this wave very similar to the coastally-trapped Kelvin wave.
The primitive equations are identical to those used to develop the coastal Kelvin wave phase speed solution (U-momentum, V-momentum, and continuity equations) and the motion is unidirectional and parallel to the Equator.
Because these waves are equatorial, the
Coriolis parameter
The Coriolis frequency ''ƒ'', also called the Coriolis parameter or Coriolis coefficient, is equal to twice the rotation rate ''Ω'' of the Earth multiplied by the sine of the latitude \varphi.
:f = 2 \Omega \sin \varphi.\,
The rotation rate ...
vanishes at 0 degrees; therefore, it is necessary to use the equatorial
beta plane
In geophysical fluid dynamics, an approximation whereby the Coriolis parameter, ''f'', is set to vary linearly in space is called a beta plane approximation.
On a rotating sphere such as the Earth, ''f'' varies with the sine of latitude; in the so ...
approximation that states:
::
where ''β'' is the variation of the Coriolis parameter with latitude. This equatorial beta plane assumption requires a geostrophic balance between the eastward velocity and the north–south pressure gradient. The phase speed is identical to that of coastal Kelvin waves, indicating that the equatorial Kelvin waves propagate toward the east without dispersion (as if the earth were a non-rotating planet).
For the first
baroclinic
In fluid dynamics, the baroclinity (often called baroclinicity) of a stratified fluid is a measure of how misaligned the gradient of pressure is from the gradient of density in a fluid. In meteorology a baroclinic flow is one in which the densi ...
mode in the ocean, a typical phase speed would be about 2.8 m/s, causing an equatorial Kelvin wave to take 2 months to cross the Pacific Ocean between New Guinea and South America; for higher ocean and atmospheric modes, the phase speeds are comparable to fluid flow speeds.
When the motion at the Equator is to the east, any deviation toward the north is brought back toward the Equator because the
Coriolis force
In physics, the Coriolis force is an inertial or fictitious force that acts on objects in motion within a frame of reference that rotates with respect to an inertial frame. In a reference frame with clockwise rotation, the force acts to the ...
acts to the right of the direction of motion in the Northern Hemisphere, and any deviation to the south is brought back toward the Equator because the Coriolis force acts to the left of the direction of motion in the Southern Hemisphere. Note that for motion toward the west, the Coriolis force would not restore a northward or southward deviation back toward the Equator; thus, equatorial Kelvin waves are only possible for eastward motion (as noted above). Both atmospheric and oceanic equatorial Kelvin waves play an important role in the dynamics of
El Nino-Southern Oscillation
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, by transmitting changes in conditions in the Western Pacific to the Eastern Pacific.
There have been studies that connect equatorial Kelvin waves to coastal Kelvin waves. Moore (1968) found that as an equatorial Kelvin wave strikes an "eastern boundary", part of the energy is reflected in the form of planetary and gravity waves; and the remainder of the energy is carried poleward along the eastern boundary as coastal Kelvin waves. This process indicates that some energy may be lost from the equatorial region and transported to the poleward region.
Equatorial Kelvin waves are often associated with anomalies in surface wind stress. For example, positive (eastward) anomalies in wind stress in the central Pacific excite positive anomalies in 20 °C isotherm depth which propagate to the east as equatorial Kelvin waves.
Recently, equatorial Kelvin waves were shown to be a case of classical topologically protected excitations,
[Delplace, P., Marston, J. B., & Venaille, A. (2017). Topological origin of equatorial waves. Science, 358(6366), 1075–1077. https://doi.org/10.1126/science.aan8819] [Tong,D. (2022). A Gauge Theory for Shallow Water. Arxiv https://arxiv.org/abs/2209.10574] similar to those found in a
topological insulator
A topological insulator is a material whose interior behaves as an electrical insulator while its surface behaves as an electrical conductor, meaning that electrons can only move along the surface of the material.
A topological insulator is an ...
.
See also
*
Rossby wave
Rossby waves, also known as planetary waves, are a type of inertial wave naturally occurring in rotating fluids. They were first identified by Sweden-born American meteorologist Carl-Gustaf Arvid Rossby. They are observed in the atmospheres an ...
*
Rossby-gravity waves
Rossby-gravity waves are equatorially trapped waves (much like Kelvin waves), meaning that they rapidly decay as their distance increases away from the equator (so long as the Brunt–Vaisala frequency does not remain constant). These waves have ...
*
Equatorial Rossby wave
Equatorial Rossby waves, often called planetary waves, are very long, low frequency water waves found near the equator and are derived using the equatorial beta plane approximation.
Mathematics
Using the equatorial beta plane approximation, f ...
*
Kelvin–Helmholtz instability
The Kelvin–Helmholtz instability (after Lord Kelvin and Hermann von Helmholtz) is a fluid instability that occurs when there is velocity shear in a single continuous fluid or a velocity difference across the interface between two fluids. K ...
*
Edge wave
Edge or EDGE may refer to:
Technology Computing
* Edge computing, a network load-balancing system
* Edge device, an entry point to a computer network
* Adobe Edge, a graphical development application
* Microsoft Edge, a web browser developed by ...
*
Tropical wave
References
External links
Overview of Kelvin waves from the American Meteorological Society.Kelvin Wave Renews El Niño- NASA, Earth Observatory, image of the day, 2010 March 21
{{DEFAULTSORT:Kelvin Wave
Gravity waves
Tropical meteorology
Atmospheric dynamics
Fluid mechanics
Wave
In physics, mathematics, and related fields, a wave is a propagating dynamic disturbance (change from equilibrium) of one or more quantities. Waves can be periodic, in which case those quantities oscillate repeatedly about an equilibrium (re ...