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): <math display="block">\frac{\partial u}{\partial x} + \frac{\partial v}{\partial y} = \frac{-1}{H} \frac{\partial \eta}{\partial t}</math>
  • the u-momentum equation: <math display="block">\frac{\partial u}{\partial t} = - g \frac{\partial \eta}{\partial x} + f v</math>
  • the v-momentum equation: <math display="block">\frac{\partial v}{\partial t} = - g \frac{\partial \eta}{\partial y} - f u.</math>

in which f is the Coriolis coefficient, which depends on the latitude φ:

<math display="block">f = 2\,\Omega\,\sin \phi</math>

where Ω ≈ 2π / (86164 sec) ≈ 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: <math display="block">\frac{\partial v}{\partial y} = \frac{-1}{H} \frac{\partial \eta}{\partial t}</math>
  • the u-momentum equation: <math display="block">g \frac{\partial \eta}{\partial x} = f v</math>
  • the v-momentum equation: <math display="block">\frac{\partial v}{\partial t} = - g \frac{\partial \eta}{\partial y}</math>

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 <math>c=\sqrt{gH},</math> 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). similar to those found in a topological insulator.

See also

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

References

  • 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