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.
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).
[1][2] 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.
[3] Assuming that the depth H is constant, the (linearised) primitive equations then become the following:
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 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
the speed of so-called shallow-water gravity waves without the effect of Earth's rotation.
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).
[3] If we assume constant f, the general solution is an arbitrary wave form
with the sign chosen so that the amplitude decreases with distance from the coast.
Kelvin waves can also exist going eastward parallel to the equator.
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.
[3] 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.
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.
[3] 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,[5] similar to those found in a topological insulator.