Equatorial wave
Equatorial waves are oceanic and atmospheric waves trapped close to the
Equatorial waves may be separated into a series of subclasses depending on their fundamental dynamics (which also influences their typical periods and speeds and directions of propagation). At shortest periods are the equatorial gravity waves while the longest periods are associated with the equatorial Rossby waves. In addition to these two extreme subclasses, there are two special subclasses of equatorial waves known as the mixed Rossby-gravity wave (also known as the Yanai wave) and the equatorial Kelvin wave. The latter two share the characteristics that they can have any period and also that they may carry energy only in an eastward (never westward) direction.
The remainder of this article discusses the relationship between the period of these waves, their wavelength in the
Equatorial Rossby and Rossby-gravity waves
Rossby-gravity waves, first observed in the stratosphere by M. Yanai,
- the continuity equation (accounting for the effects of horizontal convergence and divergence and written with geopotential height):
- the U-momentum equation (zonal wind component):
- the V-momentum equation (meridional wind component):
- .[3]
We may seek travelling-wave solutions of the form[4]
- .
Substituting this exponential form into the three equations above, and eliminating and leaves us with an eigenvalue equation
for . Recognizing this as the Schrödinger equation for a quantum harmonic oscillator of frequency , we know that we must have
for the solutions to tend to zero away from the equator. For each integer therefore, this last equation provides a dispersion relation linking the wavenumber to the angular frequency .
In the special case the dispersion equation reduces to
but the root has to be discarded because we had to divide by this factor in eliminating , . The remaining pair of roots correspond to the Yanai or mixed Rossby-gravity mode whose group velocity is always to the east [1] and interpolates between two types of modes: the higher frequency Poincaré gravity waves whose group velocity can be to the east or to the west, and the low-frequency equatorial Rossby waves whose dispersion relation can be approximated as
.
The Yanai modes, together with the Kelvin waves described in the next section, are rather special in that they are topologically protected. Their existence is guaranteed by the fact that the band of positive frequency Poincaré modes in the f-plane form a non-trivial bundle over the two-sphere . This bundle is characterized by Chern number . The Rossby waves have , and the negative frequency Poincaré modes have Through the bulk-boundary connection[5] this necessitates the existence of two modes (Kelvin and Yanai) that cross the frequency gaps between the Poincaré and Rossby bands and are localized near the equator where changes sign.[6][7]
Equatorial Kelvin waves
Discovered by Lord Kelvin, coastal Kelvin waves are trapped close to coasts and propagate along coasts in the Northern Hemisphere such that the coast is to the right of the alongshore direction of propagation (and to the left in the Southern Hemisphere). Equatorial Kelvin waves behave somewhat as if there were a wall at the equator – so that the equator is to the right of the direction of along-equator propagation in the Northern Hemisphere and to the left of the direction of propagation in the Southern Hemisphere, both of which are consistent with eastward propagation along the equator.[1] The governing equations for these equatorial waves are similar to those presented above, except that there is no meridional velocity component (that is, no flow in the north–south direction).
- the continuity equation (accounting for the effects of horizontal convergence and divergence):
- the u-momentum equation (zonal wind component):
- the v-momentum equation (meridional wind component):
The solution to these equations yields the following
Like other waves, equatorial Kelvin waves can transport energy and momentum but not particles and particle properties like temperature, salinity or nutrients.
Connection to El Niño Southern Oscillation
Kelvin waves have been connected to
Changes associated with the waves and currents can be tracked using an array of 70 moorings which cover the equatorial Pacific Ocean from Papua New Guinea to the Ecuador coast.[8] Sensors on the moorings measure the sea temperature at different depths and this is then sent by satellite to ground stations where the data can be analysed and used to predict the possible development of the next El Niño.
During the strongest
The overall ENSO cycle is usually explained as follows (in terms of the wave propagation and assuming that waves can transport heat): ENSO begins with a warm pool travelling from the western Pacific to the eastern Pacific in the form of Kelvin waves (the waves carry the warm SSTs) that resulted from the MJO.[9] After approximately 3 to 4 months of propagation across the Pacific (along the equatorial region), the Kelvin waves reach the western coast of South America and interact (merge/mix) with the cooler Peru current system.[9] This causes a rise in sea levels and sea level temperatures in the general region. Upon reaching the coast, the water turns to the north and south and results in El Niño conditions to the south.[9] Because of the changes in sea-level and sea-temperature due to the Kelvin waves, an infinite number of Rossby waves are generated and move back over the Pacific.[9] Rossby waves then enter the equation and, as previously stated, move at lower velocities than the Kelvin waves and can take anywhere from nine months to four years to fully cross the Pacific Ocean basin (from boundary to boundary).[9] And because these waves are equatorial in nature, they decay rapidly as distance from the equator increases; thus, as they move away from the equator, their speed decreases as well, resulting in a wave delay.[9] When the Rossby waves reach the western Pacific they ricochet off the coast and become Kelvin waves and then propagate back across the Pacific in the direction of the South America coast.[9] Upon return, however, the waves decrease the sea-level (reducing the depression in the thermocline) and sea surface temperature, thereby returning the area to normal or sometimes La Niña conditions.[9]
In terms of climate modeling and upon coupling the atmosphere and the ocean, an ENSO model typically contains the following dynamical equations:
- 3 primitive equations for the atmosphere (as mentioned above) with the inclusion of frictional parameterizations: 1) u-momentum equation, 2) v-momentum equation, and 3) continuity equation
- 4 primitive equations for the ocean (as stated below) with the inclusion of frictional parameterizations:
- u-momentum,
- v-momentum,
- continuity,
- thermodynamic energy,
- .[10]
Note that h is the depth of the fluid (similar to the equivalent depth and analogous to H in the primitive equations listed above for Rossby-gravity and Kelvin waves), KT is temperature diffusion, KE is eddy diffusivity, and τ is the wind stress in either the x or y directions.
See also
References
- ^ a b c d e f g Holton, James R., 2004: An Introduction to Dynamic Meteorology. Elsevier Academic Press, Burlington, MA, pp. 394–400.
- ^ Yanai, M. and T. Maruyama, 1966: Stratospheric wave disturbances propagating over the equatorial pacific. J. Met. Soc. Japan, 44, 291–294. https://www.jstage.jst.go.jp/article/jmsj1965/44/5/44_5_291/_article
- ^ WP:RS)
- ^ T. Matsuno, Quasi-Geostrophic Motions in the Equatorial Area, Journal of the Meteorological Society of Japan. Ser. II, vol. 44, no. 1, pp. 25–43, 1966.
- ^ Y. Hatsugai, Chern number and edge states in the integer quantum Hall effect, Physical Review Letters, vol. 71, no. 22, p. 3697, 1993.
- ^ Pierre Delplace, J.B. Marston, Antoine Venaille, Topological Origin of Equatorial Waves, arXiv:1702.07583.
- S2CID 206661727.
- ^ a b c d “El Niño and La Nina,” 2008: Stormsurf, http://www.stormsurf.com/page2/tutorials/enso.shtml.
- ^ a b c d e f g h The El Niño/Earth Science Virtual Classroom, 2008: “Introduction to El Niño,” http://library.thinkquest.org/3356/main/course/moreintro.html Archived 2009-08-27 at the Wayback Machine.
- ^ Battisti, David S., 2000: "Developing a Theory for ENSO," NCAR Advanced Study Program, "David Battisti: Developing a Theory for ENSO". Archived from the original on 2010-06-10. Retrieved 2010-08-21.