Equatorial waves
Encyclopedia
Equatorial waves are waves trapped close to the equator
, meaning that they decay rapidly away from the equator, but can propagate in the longitudinal and vertical directions. Wave trapping is the result of the Earth's rotation and its spherical shape which combine to cause the magnitude of the Coriolis Force to increase rapidly away from the Equator. Equatorial Waves are present in both the tropical atmosphere and ocean and play an important role in the evolution of many climate phenomena such as El Nino. Many physical processes may excite equatorial waves including, in the case of the atmosphere, diabatic heat release associated with cloud formation, and in the case of the ocean, anomalous changes in the strength or direction of the trade winds. Equatorial waves may be separated into a series of sub-classes 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 wave
s. In addition to these two extreme sub-classes, there are two special sub-classes 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 zonal direction and their speeds for a simplified ocean.
of the earth, 7.2921
10−5 rad/s, and θ is latitude) vanishes at 0 degrees latitude (equator), the “equatorial beta plane
” approximation must be made. This approximation states that “f” is approximately equal to βy, where “y” is the distance from the equator and "β" is the variation of the coriolis parameter with latitude,
. With the inclusion of this approximation, the governing equations become (neglecting friction):
These three equations can be separated and solved using solutions in the form of zonally-propagating waves, which are analogous to exponential solutions with a dependence on x and t and the inclusion of structure functions that vary in the y-direction:
.
Once the frequency relation is formulated in terms of ω, the angular frequency, the problem can be solved with 3 distinct solutions. These three solutions correspond to the equatorial gravity waves, the equatorially trapped Rossby waves and the mixed Rossby-gravity wave (which has some of the characteristics of the former two). Equatorial gravity waves can be either westward- or eastward-propagating.
s 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. 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 solution to these equations yields the following phase speed: c2 = gH; this result is the same speed as for shallow-water gravity waves without the effect of Earth’s rotation. Therefore, these waves are non-dispersive (because the phase speed is not a function of the zonal wavenumber
). Also, these Kelvin waves only propagate towards the east (because as Φ approaches zero, y approaches infinity).
(due to the MJO) typically propagates eastward into the North Pacific Ocean and can produce easterly winds. These easterly winds can transfer West Pacific warm water toward the east, thereby exciting a Kelvin wave, which in this sense can be thought of as a warm-water anomaly that travels under the ocean’s surface at a depth of about 150 meters. This wave can be observed at the surface by a slight rise in sea surface height of about 8 cm (associated with a depression of the thermocline) and an SST increase that covers hundreds of square miles across the surface of the ocean.
If the Kelvin wave hits the South American coast (specifically Ecuador
), its warm water gets transferred upward, which creates a large warm pool at the surface. That warm water also starts to flow southward along the coast of Peru
and north towards Central America
and Mexico
, and may reach parts of Northern California
; the wave can then be tracked primarily using an array of 70 buoys anchored along the entire width of equatorial Pacific Ocean, from Papua New Guinea
to the Ecuador coast. Temperature sensors are placed at different depths along the buoys' anchor-lines and are then able to record sub-surface water temperature. The sensors send their data in real-time using a satellite to a central processing facility. These temperature measurements are then compared and contrasted to historically- and seasonally-adjusted average water temperatures for each buoy location. Some results indicate deviations from the 'normal' expected temperatures. Such deviations are referred to as anomalies and can be thought of as either warmer-than-normal (El Nino) or cooler-than-normal (La Nina
) conditions.
The overall ENSO cycle can be explained as follows (in terms of the wave propagation throughout the Pacific Ocean): ENSO begins with a warm pool traveling 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. 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. 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 Nino conditions to the south. 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. 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). 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. 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. 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 Nina conditions.
In terms of climate modeling and upon coupling the atmosphere and the ocean, an ENSO model typically contains the following dynamical equations:
Note that h is the depth of the fluid (similar to the equivalent depth and analoguous 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.
Equator
An equator is the intersection of a sphere's surface with the plane perpendicular to the sphere's axis of rotation and containing the sphere's center of mass....
, meaning that they decay rapidly away from the equator, but can propagate in the longitudinal and vertical directions. Wave trapping is the result of the Earth's rotation and its spherical shape which combine to cause the magnitude of the Coriolis Force to increase rapidly away from the Equator. Equatorial Waves are present in both the tropical atmosphere and ocean and play an important role in the evolution of many climate phenomena such as El Nino. Many physical processes may excite equatorial waves including, in the case of the atmosphere, diabatic heat release associated with cloud formation, and in the case of the ocean, anomalous changes in the strength or direction of the trade winds. Equatorial waves may be separated into a series of sub-classes 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 wave
Rossby wave
Atmospheric Rossby waves are giant meanders in high-altitude winds that are a major influence on weather.They are not to be confused with oceanic Rossby waves, which move along the thermocline: that is, the boundary between the warm upper layer of the ocean and the cold deeper part of the...
s. In addition to these two extreme sub-classes, there are two special sub-classes of equatorial waves known as the mixed Rossby-gravity wave (also known as the Yanai wave) and the equatorial Kelvin Wave
Kelvin wave
A Kelvin wave is a wave in the ocean or 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...
. 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 zonal direction and their speeds for a simplified ocean.
Equatorial Rossby and Rossby-gravity waves
Rossby-gravity waves, first observed in the stratosphere by M. Yanai, always carry energy eastward. But, oddly, their 'crests' and 'troughs' may propagate westward if their periods are long enough. The eastward speed of propagation of these waves can be derived for an inviscid slowly moving layer of fluid of uniform depth H. Because the Coriolis parameter (ƒ = 2Ω sin(θ) where Ω is the angular velocityAngular velocity
In physics, the angular velocity is a vector quantity which specifies the angular speed of an object and the axis about which the object is rotating. The SI unit of angular velocity is radians per second, although it may be measured in other units such as degrees per second, revolutions per...
of the earth, 7.2921
10−5 rad/s, and θ is latitude) vanishes at 0 degrees latitude (equator), the “equatorial beta planeBeta plane
In geophysical fluid dynamics, an approximation whereby the Coriolis parameter, f, is set to vary linearly in space is called a beta plane approximation...
” approximation must be made. This approximation states that “f” is approximately equal to βy, where “y” is the distance from the equator and "β" is the variation of the coriolis parameter with latitude,
. With the inclusion of this approximation, the governing equations become (neglecting friction):- 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):
-
.
These three equations can be separated and solved using solutions in the form of zonally-propagating waves, which are analogous to exponential solutions with a dependence on x and t and the inclusion of structure functions that vary in the y-direction:
.Once the frequency relation is formulated in terms of ω, the angular frequency, the problem can be solved with 3 distinct solutions. These three solutions correspond to the equatorial gravity waves, the equatorially trapped Rossby waves and the mixed Rossby-gravity wave (which has some of the characteristics of the former two). Equatorial gravity waves can be either westward- or eastward-propagating.
Equatorial Kelvin waves
Discovered by Lord Kelvin, coastal Kelvin waveKelvin wave
A Kelvin wave is a wave in the ocean or 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...
s 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
Equator
An equator is the intersection of a sphere's surface with the plane perpendicular to the sphere's axis of rotation and containing the sphere's center of mass....
– 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. 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 equationContinuity equationA continuity equation in physics is a differential equation that describes the transport of a conserved quantity. Since mass, energy, momentum, electric charge and other natural quantities are conserved under their respective appropriate conditions, a variety of physical phenomena may be described...
(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 phase speed: c2 = gH; this result is the same speed as for shallow-water gravity waves without the effect of Earth’s rotation. Therefore, these waves are non-dispersive (because the phase speed is not a function of the zonal wavenumber
Wavenumber
In the physical sciences, the wavenumber is a property of a wave, its spatial frequency, that is proportional to the reciprocal of the wavelength. It is also the magnitude of the wave vector...
). Also, these Kelvin waves only propagate towards the east (because as Φ approaches zero, y approaches infinity).
Connection to El Nino Southern Oscillation
Kelvin waves have been connected to El Nino (beginning in the Northern Hemisphere winter months) in recent years in terms of precursors to this atmospheric and oceanic phenomenon. Many scientists have utilized coupled atmosphere–ocean models to simulate an El Nino Southern Oscillation (ENSO) event and have stated that the Madden–Julian oscillation (MJO) can trigger oceanic Kelvin waves throughout its 30–60 day cycle or the latent heat of condensation can be released (from intense convection) resulting in Kelvin waves as well; this process can then signal the onset of an El Nino event. The weak low pressure in the Indian OceanIndian Ocean
The Indian Ocean is the third largest of the world's oceanic divisions, covering approximately 20% of the water on the Earth's surface. It is bounded on the north by the Indian Subcontinent and Arabian Peninsula ; on the west by eastern Africa; on the east by Indochina, the Sunda Islands, and...
(due to the MJO) typically propagates eastward into the North Pacific Ocean and can produce easterly winds. These easterly winds can transfer West Pacific warm water toward the east, thereby exciting a Kelvin wave, which in this sense can be thought of as a warm-water anomaly that travels under the ocean’s surface at a depth of about 150 meters. This wave can be observed at the surface by a slight rise in sea surface height of about 8 cm (associated with a depression of the thermocline) and an SST increase that covers hundreds of square miles across the surface of the ocean.
If the Kelvin wave hits the South American coast (specifically Ecuador
Ecuador
Ecuador , officially the Republic of Ecuador is a representative democratic republic in South America, bordered by Colombia on the north, Peru on the east and south, and by the Pacific Ocean to the west. It is one of only two countries in South America, along with Chile, that do not have a border...
), its warm water gets transferred upward, which creates a large warm pool at the surface. That warm water also starts to flow southward along the coast of Peru
Peru
Peru , officially the Republic of Peru , is a country in western South America. It is bordered on the north by Ecuador and Colombia, on the east by Brazil, on the southeast by Bolivia, on the south by Chile, and on the west by the Pacific Ocean....
and north towards Central America
Central America
Central America is the central geographic region of the Americas. It is the southernmost, isthmian portion of the North American continent, which connects with South America on the southeast. When considered part of the unified continental model, it is considered a subcontinent...
and Mexico
Mexico
The United Mexican States , commonly known as Mexico , is a federal constitutional republic in North America. It is bordered on the north by the United States; on the south and west by the Pacific Ocean; on the southeast by Guatemala, Belize, and the Caribbean Sea; and on the east by the Gulf of...
, and may reach parts of Northern California
California
California is a state located on the West Coast of the United States. It is by far the most populous U.S. state, and the third-largest by land area...
; the wave can then be tracked primarily using an array of 70 buoys anchored along the entire width of equatorial Pacific Ocean, from Papua New Guinea
Papua New Guinea
Papua New Guinea , officially the Independent State of Papua New Guinea, is a country in Oceania, occupying the eastern half of the island of New Guinea and numerous offshore islands...
to the Ecuador coast. Temperature sensors are placed at different depths along the buoys' anchor-lines and are then able to record sub-surface water temperature. The sensors send their data in real-time using a satellite to a central processing facility. These temperature measurements are then compared and contrasted to historically- and seasonally-adjusted average water temperatures for each buoy location. Some results indicate deviations from the 'normal' expected temperatures. Such deviations are referred to as anomalies and can be thought of as either warmer-than-normal (El Nino) or cooler-than-normal (La Nina
La Niña
La Niña is a coupled ocean-atmosphere phenomenon that is the counterpart of El Niño as part of the broader El Niño-Southern Oscillation climate pattern. During a period of La Niña, the sea surface temperature across the equatorial Eastern Central Pacific Ocean will be lower than normal by 3–5 °C...
) conditions.
The overall ENSO cycle can be explained as follows (in terms of the wave propagation throughout the Pacific Ocean): ENSO begins with a warm pool traveling 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. 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. 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 Nino conditions to the south. 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. 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). 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. 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. 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 Nina conditions.
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,
-
-
.
-
Note that h is the depth of the fluid (similar to the equivalent depth and analoguous 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
- Primitive equationsPrimitive equationsThe primitive equations are a set of nonlinear differential equations that are used to approximate global atmospheric flow and are used in most atmospheric models...
- Beta planeBeta planeIn geophysical fluid dynamics, an approximation whereby the Coriolis parameter, f, is set to vary linearly in space is called a beta plane approximation...
- Rossby Waves
- El Nino
- Equatorial Rossby wave