Physics:Kelvin wave

Short description: Type of wave in geophysics

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 (a coastal Kelvin wave), or a waveguide like the equator (an equatorial Kelvin wave). A feature of a Kelvin wave is that it is non-dispersive (in the shallow water model), i.e., the phase speed of the wave crests is equal to the group speed of the wave energy for all frequencies, thus the wave retains its profile as it moves; unlike beach waves, Kelvin waves do not "curl over". The direction of the wave is fixed: the coastal wave moves towards equator along the western coasts, towards the poles along the eastern coasts (like California), and cyclone-like (counterclockwise in the Northern hemisphere, clockwise in the Southern one) in the closed bodies of water; the equatorial waves always travel in the eastern direction.

Observations of the equatorial Kelvin wave are important for predictions about El Niño–Southern Oscillation (ENSO), thus the phenomenon attracts attention of the mainstream media. Coastal Kelvin waves are an important component of tide waves.

The Kelvin waves have planetary scale, sometimes thousands of miles long. The equatorial waves move large amounts of heat and are known to temporarily reverse some currents in the upper ocean. At the same time, the height of the equatorial wave (observed changes in the ocean level) is very small: in the Pacific, downwelling Kelvin waves raise sea level by just a few centimetres.

During the meteorological or oceanographical analysis, it can be assumed that one velocity component of a Kelvin wave vanishes (for example, for an equatorial wave there is no flow in the north–south direction), thus making the momentum and continuity equations much simpler). This wave is named after its discoverer, Lord Kelvin (1879).

Surface and internal waves

Kelvin wave. Both the surface and the internal wave are shown, not to scale

The Kelvin waves can occur as surface waves (also referred to as barotropic) manifesting themselves as a change in ocean level, and internal (baroclinic) waves, where the interface of a wave is a subsurface change of the water density caused, for example, by a temperature gradient (thermocline).

The surface and internal waves travel in the same direction, but at different speeds:

• the surface waves are fast: in the deep ocean, about 200 m/sec (similar to a tsunami or passenger jet), much slower in the shallow water;
• the internal ocean waves carrying the heat have speed of 1-3 m/sec (similar to a walking or running human). For example, the equatorial internal waves cross the Pacific ocean in about two months.

The Rossby radius for the coastal barotropic waves is about 3000 kilometers in the deep ocean, 200 kilometers on the continental shelf. The wave height increases as the wave moves into shallower water.

Equatorial Kelvin wave

File:Equatorial Kelvin wave.ogv

An equatorial Kelvin wave, captured through sea surface height anomalies in 2010

Kelvin waves can exist going eastward parallel to the equator (equatorially trapped).

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 (x direction U-momentum, y direction 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 Niño–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.

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: $$f=2\mathrm{\Omega }\sin \varphi$$ 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 $\exp (\pm fx/\sqrt{gH}),$ with the sign chosen so that the amplitude decreases with distance from the coast.

Atmospheric Kelvin wave

The atmospheric coastal Kelvin waves are primarily internal (at the inversion layer interface). The coastal waves form at the steep topography features, like meridional edges of a plateau (for example, the Tibetan Plateau) or mountain range (for example, at the coasts of South Africa or California). Equatorial atmospheric Kelvin waves occur in continuously stratified air and propagate vertically.

Media reactions

Equatorial Kelvin wave attracted attention of journalists in 2014, when a particularly large wave was moving eastward across the tropical Pacific Ocean. At the time, it was not as popular as the "polar vortex".

See also

• Rossby wave
• Rossby-gravity waves
• Equatorial Rossby wave
• Kelvin–Helmholtz instability
• Edge wave
• Tropical wave

Sources

• L'Heureux, Michelle (January 22, 2015). "Oceanic Kelvin waves: The next polar vortex". National Oceanic and Atmospheric Administration. https://content-drupal.climate.gov/news-features/blogs/enso/oceanic-kelvin-waves-next-polar-vortex. 
• Rydbeck, A. V.; Jensen, T. G.; Flatau, M. (2019). "Characterization of Intraseasonal Kelvin Waves in the Equatorial Pacific Ocean". Journal of Geophysical Research: Oceans 124 (3): 2028–2053. doi:10.1029/2018JC014838. 
• Wang, Bin (2003). "Kelvin Waves". in Holton, James R.; Curry, Judith A.; Pyle, John A. (PDF). Encyclopedia of Atmospheric Sciences. San Diego: Academic Press. doi:10.1006/rwas.2002.0191. https://www.soest.hawaii.edu/MET/Faculty/bwang/bw/paper/wang_103.pdf. Retrieved 30 September 2026. 

This article incorporates text from a publication now in the public domain: L'Heureux, Michelle (January 22, 2015). "Oceanic Kelvin waves: The next polar vortex". National Oceanic and Atmospheric Administration. https://content-drupal.climate.gov/news-features/blogs/enso/oceanic-kelvin-waves-next-polar-vortex. 

External links

• Overview of Kelvin waves from the American Meteorological Society
• US Navy page on Kelvin waves.
• Slideshow at utexus.edu about Kelvin waves.
• Kelvin Wave Renews El Niño – NASA Earth Observatory, image of the day, 2010 March 21