Equatorial wave

Last updated

Equatorial waves are oceanic and atmospheric waves trapped close to the equator, meaning that they decay rapidly away from the equator, but can propagate in the longitudinal and vertical directions. [1] 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 Niño. 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. [1]

Contents

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 zonal (east-west) 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, [2] 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. [3] [ unreliable source? ] Because the Coriolis parameter (ƒ = 2Ω sin(θ) where Ω is the angular velocity of the earth, 7.2921 105 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, . [1] With the inclusion of this approximation, the governing equations become (neglecting friction):

. [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

.

Dispersion relations for equatorial waves with different values of
n
{\displaystyle n}
: The dense narrow band of low-frequency Rossby waves and the higher frequency Poincare gravity waves are in blue. The topologically protected Kelvin and Yanai modes are highlighted in magenta Equatorial-wave-dispersion.jpg
Dispersion relations for equatorial waves with different values of : The dense narrow band of low-frequency Rossby waves and the higher frequency Poincaré gravity waves are in blue. The topologically protected Kelvin and Yanai modes are highlighted in magenta

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).

[1]

The solution to these equations yields the following phase speed: ; this result is the same speed as for shallow-water gravity waves without the effect of Earth's rotation. [1] 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). [3]

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 El Niño (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 Niño Southern Oscillation (ENSO) event and have stated that the Madden–Julian oscillation (MJO) can trigger oceanic Kelvin waves throughout its 30- to 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 Niño event. [8] The weak low pressure in the Indian Ocean (due to the MJO) typically propagates eastward into the North Pacific Ocean and can produce easterly winds. [8] These easterly winds can force West Pacific warm surface water eastwards, and also excite Kelvin waves, which in this sense can be thought of as warm-water anomalies that affect the top few hundred metres of the ocean. [8] As the surface warm water is less dense than the underlying watermasses, this increased thickness of the near surface thermocline results in a slight rise in sea surface height of about 8 cm.

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 El Niños the strength of the cold Equatorial Undercurrent drops as does the trade wind in the eastern Pacific. As a result cold water is no longer upwelled along the Equator in the eastern Pacific, resulting in a large increase of sea surface temperatures and a corresponding sharp rise in sea surface height near the Galapagos Islands. The resulting increase in sea surface temperatures also affects the waters off the South American coast (specifically Ecuador), and can also influence temperatures southward along the coast of Peru and north towards Central America and Mexico, and may reach parts of Northern California.

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:

. [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

Related Research Articles

Linear elasticity is a mathematical model of how solid objects deform and become internally stressed by prescribed loading conditions. It is a simplification of the more general nonlinear theory of elasticity and a branch of continuum mechanics.

Rossby waves, also known as planetary waves, are a type of inertial wave naturally occurring in rotating fluids. They were first identified by Sweden-born American meteorologist Carl-Gustaf Arvid Rossby in the Earth's atmosphere in 1939. They are observed in the atmospheres and oceans of Earth and other planets, owing to the rotation of Earth or of the planet involved. Atmospheric Rossby waves on Earth are giant meanders in high-altitude winds that have a major influence on weather. These waves are associated with pressure systems and the jet stream. Oceanic Rossby waves move along the thermocline: the boundary between the warm upper layer and the cold deeper part of the ocean.

In physics, the Hamilton–Jacobi equation, named after William Rowan Hamilton and Carl Gustav Jacob Jacobi, is an alternative formulation of classical mechanics, equivalent to other formulations such as Newton's laws of motion, Lagrangian mechanics and Hamiltonian mechanics.

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.

The Rossby parameter is a number used in geophysics and meteorology which arises due to the meridional variation of the Coriolis force caused by the spherical shape of the Earth. It is important in the generation of Rossby waves. The Rossby parameter is given by

In differential geometry, the four-gradient is the four-vector analogue of the gradient from vector calculus.

The electromagnetic wave equation is a second-order partial differential equation that describes the propagation of electromagnetic waves through a medium or in a vacuum. It is a three-dimensional form of the wave equation. The homogeneous form of the equation, written in terms of either the electric field E or the magnetic field B, takes the form:

In superconductivity, a long Josephson junction (LJJ) is a Josephson junction which has one or more dimensions longer than the Josephson penetration depth . This definition is not strict.

In fluid mechanics, potential vorticity (PV) is a quantity which is proportional to the dot product of vorticity and stratification. This quantity, following a parcel of air or water, can only be changed by diabatic or frictional processes. It is a useful concept for understanding the generation of vorticity in cyclogenesis, especially along the polar front, and in analyzing flow in the ocean.

<span class="mw-page-title-main">Mild-slope equation</span> Physics phenomenon and formula

In fluid dynamics, the mild-slope equation describes the combined effects of diffraction and refraction for water waves propagating over bathymetry and due to lateral boundaries—like breakwaters and coastlines. It is an approximate model, deriving its name from being originally developed for wave propagation over mild slopes of the sea floor. The mild-slope equation is often used in coastal engineering to compute the wave-field changes near harbours and coasts.

Rossby-gravity waves are equatorially trapped waves, meaning that they rapidly decay as their distance increases away from the equator. These waves have the same trapping scale as Kelvin waves, more commonly known as the equatorial Rossby deformation radius. They always carry energy eastward, but their 'crests' and 'troughs' may propagate westward if their periods are long enough.

Equatorial Rossby waves, often called planetary waves, are very long, low frequency water waves found near the equator and are derived using the equatorial beta plane approximation.

A Sverdrup wave is a wave in the ocean, or large lakes, which is affected by gravity and Earth's rotation.

In mathematics, and more precisely, in functional Analysis and PDEs, the Schauder estimates are a collection of results due to Juliusz Schauder concerning the regularity of solutions to linear, uniformly elliptic partial differential equations. The estimates say that when the equation has appropriately smooth terms and appropriately smooth solutions, then the Hölder norm of the solution can be controlled in terms of the Hölder norms for the coefficient and source terms. Since these estimates assume by hypothesis the existence of a solution, they are called a priori estimates.

While geostrophic motion refers to the wind that would result from an exact balance between the Coriolis force and horizontal pressure-gradient forces, quasi-geostrophic (QG) motion refers to flows where the Coriolis force and pressure gradient forces are almost in balance, but with inertia also having an effect.

Menter's Shear Stress Transport turbulence model, or SST, is a widely used and robust two-equation eddy-viscosity turbulence model used in Computational Fluid Dynamics. The model combines the k-omega turbulence model and K-epsilon turbulence model such that the k-omega is used in the inner region of the boundary layer and switches to the k-epsilon in the free shear flow.

<span class="mw-page-title-main">Trochoidal wave</span> Exact solution of the Euler equations for periodic surface gravity waves

In fluid dynamics, a trochoidal wave or Gerstner wave is an exact solution of the Euler equations for periodic surface gravity waves. It describes a progressive wave of permanent form on the surface of an incompressible fluid of infinite depth. The free surface of this wave solution is an inverted (upside-down) trochoid – with sharper crests and flat troughs. This wave solution was discovered by Gerstner in 1802, and rediscovered independently by Rankine in 1863.

In physical oceanography and fluid mechanics, the Miles-Phillips mechanism describes the generation of wind waves from a flat sea surface by two distinct mechanisms. Wind blowing over the surface generates tiny wavelets. These wavelets develop over time and become ocean surface waves by absorbing the energy transferred from the wind. The Miles-Phillips mechanism is a physical interpretation of these wind-generated surface waves.
Both mechanisms are applied to gravity-capillary waves and have in common that waves are generated by a resonance phenomenon. The Miles mechanism is based on the hypothesis that waves arise as an instability of the sea-atmosphere system. The Phillips mechanism assumes that turbulent eddies in the atmospheric boundary layer induce pressure fluctuations at the sea surface. The Phillips mechanism is generally assumed to be important in the first stages of wave growth, whereas the Miles mechanism is important in later stages where the wave growth becomes exponential in time.

<span class="mw-page-title-main">Topographic Rossby waves</span> Waves in the ocean and atmosphere created by bottom irregularities

Topographic Rossby waves are geophysical waves that form due to bottom irregularities. For ocean dynamics, the bottom irregularities are on the ocean floor such as the mid-ocean ridge. For atmospheric dynamics, the other primary branch of geophysical fluid dynamics, the bottom irregularities are found on land, for example in the form of mountains. Topographic Rossby waves are one of two types of geophysical waves named after the meteorologist Carl-Gustaf Rossby. The other type of Rossby waves are called planetary Rossby waves and have a different physical origin. Planetary Rossby waves form due to the changing Coriolis parameter over the earth. Rossby waves are quasi-geostrophic, dispersive waves. This means that not only the Coriolis force and the pressure-gradient force influence the flow, as in geostrophic flow, but also inertia.

The recharge oscillator model for El Niño–Southern Oscillation (ENSO) is a theory described for the first time in 1997 by Jin., which explains the periodical variation of the sea surface temperature (SST) and thermocline depth that occurs in the central equatorial Pacific Ocean. The physical mechanisms at the basis of this oscillation are periodical recharges and discharges of the zonal mean equatorial heat content, due to ocean-atmosphere interaction. Other theories have been proposed to model ENSO, such as the delayed oscillator, the western Pacific oscillator and the advective reflective oscillator. A unified and consistent model has been proposed by Wang in 2001, in which the recharge oscillator model is included as a particular case.

References

  1. 1 2 3 4 5 6 7 Holton, James R., 2004: An Introduction to Dynamic Meteorology. Elsevier Academic Press, Burlington, MA, pp. 394400.
  2. Yanai, M. and T. Maruyama, 1966: Stratospheric wave disturbances propagating over the equatorial pacific. J. Met. Soc. Japan, 44, 291294. https://www.jstage.jst.go.jp/article/jmsj1965/44/5/44_5_291/_article
  3. 1 2 3 Zhang, Dalin, 2008: Personal Communication, “Waves in Rotating, Homogeneous Fluids,” University of Maryland, College Park (not a WP:RS)
  4. 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.
  5. Y. Hatsugai, Chern number and edge states in the integer quantum Hall effect, Physical Review Letters, vol. 71, no. 22, p. 3697, 1993.
  6. Pierre Delplace, J.B. Marston, Antoine Venaille, Topological Origin of Equatorial Waves, arXiv:1702.07583.
  7. Delplace, Pierre; Marston, J. B.; Venaille, Antoine (2017). "Topological origin of equatorial waves". Science. 358 (6366): 1075–1077. arXiv: 1702.07583 . Bibcode:2017Sci...358.1075D. doi:10.1126/science.aan8819. PMID   28982798. S2CID   206661727.
  8. 1 2 3 4 “El Niño and La Nina,” 2008: Stormsurf, http://www.stormsurf.com/page2/tutorials/enso.shtml.
  9. 1 2 3 4 5 6 7 8 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 .
  10. 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.