# Kelvin wave

A Kelvin wave is a wave in the ocean, a large lake or the atmosphere that balances the Earth's Coriolis force against a topographic boundary such as a coastline, or a waveguide such as the equator. A feature of a Kelvin wave is that it is non-dispersive, i.e., the phase speed of the wave crests is equal to the group speed of the wave energy for all frequencies. This means that it retains its shape as it moves in the alongshore direction over time.
A Kelvin wave (fluid dynamics) is also a long scale perturbation mode of a vortex in superfluid 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 and continuity equations much simpler). This wave is named after the discoverer, Lord Kelvin (1879).

## Coastal Kelvin wave

In a stratified ocean of mean depth H, whose height is perturbed by some amount η (a function of position and time), free waves propagate along coastal boundaries (and hence become trapped in the vicinity of the coast itself) in the form of Kelvin waves. These waves are called coastal Kelvin waves. 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 of coastal Kelvin waves, which are among the class of waves called boundary waves, edge waves, trapped waves, or surface waves (similar to the Lamb waves). Assuming that the depth H is constant, the (linearised) primitive equations then become the following:
* the continuity equation (accounting for the effects of horizontal convergence and divergence):
${\frac {\partial u}{\partial x}}+{\frac {\partial v}{\partial y}}={\frac {-1}{H}}{\frac {\partial \eta }{\partial t}}$
* the u-momentum equation:
${\frac {\partial u}{\partial t}}=-g{\frac {\partial \eta }{\partial x}}+fv$
* the v-momentum equation:
${\frac {\partial v}{\partial t}}=-g{\frac {\partial \eta }{\partial y}}-fu.$
in which f is the Coriolis coefficient, which depends on the latitude φ:
$f=2\,\Omega \,\sin \phi$
where Ω ≈ 2π / (86164 sec) ≈ 7.292×10$^{−5}$ rad/s is the angular speed of rotation of the earth.
If one assumes that u, the flow perpendicular to the coast, is zero, then the primitive equations become the following:
* the continuity equation:
${\frac {\partial v}{\partial y}}={\frac {-1}{H}}{\frac {\partial \eta }{\partial t}}$
* the u-momentum equation:
$g{\frac {\partial \eta }{\partial x}}=fv$
* the v-momentum equation:
${\frac {\partial v}{\partial t}}=-g{\frac {\partial \eta }{\partial y}}$
The first and third of these equations are solved at constant x by waves moving in either the positive or negative y direction at a speed $c={\sqrt {gH}},$ the speed of so-called shallow-water gravity waves without the effect of Earth's rotation. However, only one of the two solutions is valid, having an amplitude that decreases with distance from the coast, whereas in the other solution the amplitude increases with distance from the coast. 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 – at the western side of an ocean and poleward – positive phase speed – at the eastern boundary; the waves move cyclonically around an ocean basin). If we assume constant f, the general solution is an arbitrary wave form $W(y-ct)$ propagating at speed c multiplied by $\exp(\pm fy/{\sqrt {gH}}),$ with the sign chosen so that the amplitude decreases with distance from the coast.

## Equatorial Kelvin wave

Kelvin waves can also exist going eastward parallel to the equator. Although waves can cross the equator, the Kelvin wave solution does not. The primitive equations are identical to those used to develop the coastal Kelvin wave solution (U-momentum, V-momentum, and continuity equations). Because these waves are equatorial, the Coriolis parameter vanishes at 0 degrees; therefore, it is necessary to use the equatorial beta plane approximation:
$f=\beta y,$
where β is the variation of the Coriolis parameter with latitude. The wave speed is identical to that of coastal Kelvin waves (for the same depth H), indicating that the equatorial Kelvin waves propagate toward the east without dispersion (as if the earth were a non-rotating planet). The dependence of the amplitude on x (here the north-south direction) though is now $\exp(-x^{2}\beta /(2{\sqrt {gH}})).$
For a depth of four kilometres, the wave speed, ${\sqrt {gH}},$ is about 200 metres per second, but for the first baroclinic 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 wave at the Equator is moving to the east, a height gradient going downwards toward the north is countered by a force toward the Equator because the water will be moving eastward and the Coriolis force acts to the right of the direction of motion in the Northern Hemisphere, and vice versa in the Southern Hemisphere. Note that for a wave moving 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, 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.
In 2017, using data from ERA5, equatorial Kelvin waves were shown to be a case of classical topologically protected excitations, similar to those found in a topological insulator.