# Rossby-gravity waves

Rossby-gravity waves are equatorially trapped waves (much like Kelvin waves), meaning that they rapidly decay as their distance increases away from the equator (so long as the Brunt–Vaisala frequency does not remain constant). These waves have the same trapping scale as Kelvin waves, more commonly known as the equatorial Rossby deformation radius.[1] They always carry energy eastward, but their 'crests' and 'troughs' may propagate westward if their periods are long enough.

## Derivation

The eastward speed of propagation of these waves can be derived for an inviscid slowly moving layer of fluid of uniform depth H.[2] Because the Coriolis parameter (f = 2Ω sin(θ) where Ω is the angular velocity 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, ${\displaystyle {\frac {\partial f}{\partial y}}=\beta }$.[3] With the inclusion of this approximation, the primitive equations become (neglecting friction):

• the continuity equation (accounting for the effects of horizontal convergence and divergence and written with geopotential height):
${\displaystyle {\frac {\partial \phi }{\partial t}}+c^{2}\left({\frac {\partial v}{\partial y}}+{\frac {\partial u}{\partial x}}\right)=0}$
• the U-momentum equation (zonal wind component):
${\displaystyle {\frac {\partial u}{\partial t}}-v\beta y=-{\frac {\partial \phi }{\partial x}}}$
• the V-momentum equation (meridional wind component):
${\displaystyle {\frac {\partial v}{\partial t}}+u\beta y=-{\frac {\partial \phi }{\partial y}}}$.[2]

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:

${\displaystyle {\begin{Bmatrix}u,v,\phi \end{Bmatrix}}={\begin{Bmatrix}{\hat {u}}(y),{\hat {v}}(y),{\hat {\phi }}(y)\end{Bmatrix}}e^{i(kx-\omega t)}}$.[2]

Once the frequency relation is formulated in terms of ω, the angular frequency, the problem can be solved with three distinct solutions. These three solutions correspond to the equatorially trapped gravity wave, the equatorially trapped Rossby wave and the mixed Rossby-gravity wave (which has some of the characteristics of the former two) .[3] Equatorial gravity waves can be either westward- or eastward-propagating, and correspond to n=1 (same as for the equatorially trapped Rossby wave) on a dispersion relation diagram ("w-k" diagram). At n = 0 on a dispersion relation diagram, the mixed Rossby-gravity waves can be found where for large, positive zonal wave numbers (+k), the solution behaves like a gravity wave; but for large, negative zonal wave numbers (−k), the solution appears to be a Rossby wave (hence the term Rossby-gravity waves).[1] As mentioned earlier, the group velocity (or energy packet/dispersion) is always directed toward the east with a maximum for short waves (gravity waves).[1]

## Vertically propagating Rossby-gravity waves

As previously stated, the mixed Rossby-gravity waves are equatorially trapped waves unless the buoyancy frequency remains constant, introducing an additional vertical wave number to complement the zonal wave number and angular frequency. If this Brunt–Vaisala frequency does not change, then these waves become vertically propagating solutions.[1] On a typical "m,k" dispersion diagram, the group velocity (energy) would be directed at right angles to the n = 0 (mixed Rossby-gravity waves) and n = 1 (gravity or Rossby waves) curves and would increase in the direction of increasing angular frequency.[1] Typical group velocities for each component are the following: 1 cm/s for gravity waves and 2 mm/s for planetary (Rossby) waves.[1]

These vertically propagating mixed Rossby-gravity waves were first observed in the stratosphere as westward-propagating mixed waves by M. Yanai.[4] They had the following characteristics: 4–5 days, horizontal wavenumbers of 4 (four waves circling the earth, corresponding to wavelengths of 10,000 km), vertical wavelengths of 4–8 km, and upward group velocity.[1] Similarly, westward-propagating mixed waves were also found in the Atlantic Ocean by Weisberg et al. (1979) with periods of 31 days, horizontal wavelengths of 1200 km, vertical wavelengths of 1 km, and downward group velocity.[1] Also, the vertically propagating gravity wave component was found in the stratosphere with periods of 35 hours, horizontal wavelengths of 2400 km, and vertical wavelengths of 5 km.[1]

## References

1. Gill, Adrian E., 1982: Atmosphere-Ocean Dynamics, International Geophysics Series, Volume 30, Academic Press, 662 pp.
2. ^ a b c Zhang, Dalin, 2008: Personal Communication, “Waves in Rotating, Homogeneous Fluids,” University of Maryland, College Park.
3. ^ a b Holton, James R., 2004: An Introduction to Dynamic Meteorology. Elsevier Academic Press, Burlington, MA, pp. 394–400.
4. ^ Yanai, M. and T. Maruyama, 1966: Stratospheric wave disturbances propagating over the equatorial pacific. J. Met. Soc. Japan, 44, 291–194.
Bahama Banks

The Bahama Banks are the submerged carbonate platforms that make up much of the Bahama Archipelago. The term is usually applied in referring to either the Great Bahama Bank around Andros Island, or the Little Bahama Bank of Grand Bahama Island and Great Abaco, which are the largest of the platforms, and the Cay Sal Bank north of Cuba. The islands of these banks are politically part of the Bahamas. Other banks are the three banks of the Turks and Caicos Islands, namely the Caicos Bank of the Caicos Islands, the bank of the Turks Islands, and wholly submerged Mouchoir Bank. Further southeast are the equally wholly submerged Silver Bank and Navidad Bank north of the Dominican Republic.

Carbonate platform

A carbonate platform is a sedimentary body which possesses topographic relief, and is composed of autochthonic calcareous deposits. Platform growth is mediated by sessile organisms whose skeletons build up the reef or by organisms (usually microbes) which induce carbonate precipitation through their metabolism. Therefore, carbonate platforms can not grow up everywhere: they are not present in places where limiting factors to the life of reef-building organisms exist. Such limiting factors are, among others: light, water temperature, transparency and pH-Value. For example, carbonate sedimentation along the Atlantic South American coasts takes place everywhere but at the mouth of the Amazon River, because of the intense turbidity of the water there. Spectacular examples of present-day carbonate platforms are the Bahama Banks under which the platform is roughly 8 km thick, the Yucatan Peninsula which is up to 2 km thick, the Florida platform, the platform on which the Great Barrier Reef is growing, and the Maldive atolls. All these carbonate platforms and their associated reefs are confined to tropical latitudes. Today's reefs are built mainly by scleractinian corals, but in the distant past other organisms, like archaeocyatha (during the Cambrian) or extinct cnidaria (tabulata and rugosa) were important reef builders.

Carl-Gustaf Rossby

Carl-Gustaf Arvid Rossby (Swedish pronunciation: [kɑːɭ ²ɡɵsːtav ¹arːvɪd ¹rɔsːbʏ] 28 December 1898 – 19 August 1957) was a Swedish-born American meteorologist who first explained the large-scale motions of the atmosphere in terms of fluid mechanics. He identified and characterized both the jet stream and the long waves in the westerlies that were later named Rossby waves.

Coriolis frequency

The Coriolis frequency ƒ, also called the Coriolis parameter or Coriolis coefficient, is equal to twice the rotation rate Ω of the Earth multiplied by the sine of the latitude φ.

${\displaystyle f=2\Omega \sin \varphi .\,}$

The rotation rate of the Earth (Ω = 7.2921 × 10−5 rad/s) can be calculated as 2π / T radians per second, where T is the rotation period of the Earth which is one sidereal day (23 hr 56 m 4.1 s). In the midlatitudes, the typical value for ${\displaystyle f}$ is about 10−4 rad/s. Inertial oscillations on the surface of the earth have this frequency. These oscillations are the result of the Coriolis effect.

Equatorial wave

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. 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.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.

Index of wave articles

This is a list of Wave topics.

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 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).

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A list of active and extinct submarine volcanoes and seamounts located under the world's oceans. There are estimated to be 40,000 to 55,000 seamounts in the global oceans. Almost all are not well-mapped and many may not have been identified at all. Most are unnamed and unexplored. This list is therefore confined to seamounts that are notable enough to have been named and/or explored.

Oceanic plateau

An oceanic or submarine plateau is a large, relatively flat elevation that is higher than the surrounding relief with one or more relatively steep sides.There are 184 oceanic plateaus covering an area of 18,486,600 km2 (7,137,700 sq mi), or about 5.11% of the oceans. The South Pacific region around Australia and New Zealand contains the greatest number of oceanic plateaus (see map).

Oceanic plateaus produced by large igneous provinces are often associated with hotspots, mantle plumes, and volcanic islands — such as Iceland, Hawaii, Cape Verde, and Kerguelen. The three largest plateaus, the Caribbean, Ontong Java, and Mid-Pacific Mountains, are located on thermal swells. Other oceanic plateaus, however, are made of rifted continental crust, for example Falkland Plateau, Lord Howe Rise, and parts of Kerguelen, Seychelles, and Arctic ridges.

Plateaus formed by large igneous provinces were formed by the equivalent of continental flood basalts such as the Deccan Traps in India and the Snake River Plain in the United States.

In contrast to continental flood basalts, most igneous oceanic plateaus erupt through young and thin (6–7 km (3.7–4.3 mi)) mafic or ultra-mafic crust and are therefore uncontaminated by felsic crust and representative for their mantle sources.

These plateaus often rise 2–3 km (1.2–1.9 mi) above the surrounding ocean floor and are more buoyant than oceanic crust. They therefore tend to withstand subduction, more-so when thick and when reaching subduction zones shortly after their formations. As a consequence, they tend to "dock" to continental margins and be preserved as accreted terranes. Such terranes are often better preserved than the exposed parts of continental flood basalts and are therefore a better record of large-scale volcanic eruptions throughout Earth's history. This "docking" also means that oceanic plateaus are important contributors to the growth of continental crust. Their formations often had a dramatic impact on global climate, such as the most recent plateaus formed, the three, large, Cretaceous oceanic plateaus in the Pacific and Indian Ocean: Ontong Java, Kerguelen, and Caribbean.

Outline of oceanography

The following outline is provided as an overview of and introduction to Oceanography.

Physical oceanography

Physical oceanography is the study of physical conditions and physical processes within the ocean, especially the motions and physical properties of ocean waters.

Physical oceanography is one of several sub-domains into which oceanography is divided. Others include biological, chemical and geological oceanography.

Physical oceanography may be subdivided into descriptive and dynamical physical oceanography.Descriptive physical oceanography seeks to research the ocean through observations and complex numerical models, which describe the fluid motions as precisely as possible.

Dynamical physical oceanography focuses primarily upon the processes that govern the motion of fluids with emphasis upon theoretical research and numerical models. These are part of the large field of Geophysical Fluid Dynamics (GFD) that is shared together with meteorology. GFD is a sub field of Fluid dynamics describing flows occurring on spatial and temporal scales that are greatly influenced by the Coriolis force.

Undersea mountain range

Undersea mountain ranges are mountain ranges that are mostly or entirely underwater, and specifically under the surface of an ocean. If originated from current tectonic forces, they are often referred to as a mid-ocean ridge. In contrast, if formed by past above-water volcanism, they are known as a seamount chain. The largest and best known undersea mountain range is a mid-ocean ridge, the Mid-Atlantic Ridge. It has been observed that, "similar to those on land, the undersea mountain ranges are the loci of frequent volcanic and earthquake activity".

Wave base

The wave base, in physical oceanography, is the maximum depth at which a water wave's passage causes significant water motion. For water depths deeper than the wave base, bottom sediments and the seafloor are no longer stirred by the wave motion above.

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