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Rossby wave
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{{Short description|Inertial wave occurring in rotating fluids}} {{Use dmy dates|date=March 2021}} [[File:Jetstream - Rossby Waves - N hemisphere.svg|thumb|451x451px|Meanders of the Northern Hemisphere's [[jet stream]] developing around the northern [[polar vortex]] (a, b) and finally detaching a "drop" of cold air (c). Orange: warmer masses of air; pink: jet stream; blue: colder masses of air.]] '''Rossby waves''', also known as '''planetary waves''', are a type of [[inertial wave]] naturally occurring in rotating fluids.<ref name="WhatIs">{{cite web |title=What is a Rossby wave? |url=https://oceanservice.noaa.gov/facts/rossby-wave.html |website=National Oceanic and Atmospheric Administration }}</ref> 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 [[atmosphere]]s and [[ocean]]s 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]] [[wind]]s that have a major influence on [[weather]]. These waves are associated with [[pressure systems]] and the [[jet stream]] (especially around the [[Polar vortex|polar vortices]]).<ref>{{cite book |title=Dynamic Meteorology |page=347 |year=2004 |first=James R. |last=Holton |publisher=Elsevier |isbn=978-0-12-354015-7}}</ref> Oceanic Rossby waves move along the [[thermocline]]: the boundary between the warm upper layer and the cold deeper part of the ocean. ==Rossby wave types== ===Atmospheric waves=== [[File:Sketches of Rossby wave's fundamental principles..png|thumb|458x458px|Sketches of Rossby waves’ fundamental principles. '''a''' and '''b''' The restoring force. '''c'''–'''e''' The waveform’s velocity. In '''a''', an air parcel follows along latitude <math>\varphi_0</math> at an eastward velocity <math>v_E</math> with a meridional acceleration <math>a_N=0</math> when the pressure gradient force balances the Coriolis force. In '''b''', when the parcel encounters a small displacement <math>\delta\varphi</math> in latitude, the Coriolis force’s gradient imposes a meridional acceleration <math>a_N</math> that always points against <math>\delta\varphi</math> when <math>v_E>0</math>. Here, <math>\Omega</math> denotes the Earth’s angular frequency and <math>a_N</math> is the northward Coriolis acceleration. While the parcel meanders along the blue arrowed line <math>l</math> in '''b''' , its waveform travels westward as sketched in '''c'''. The absolute vorticity composes the planetary vorticity <math>f</math> and the relative vorticity <math>\zeta</math>, reflecting the Earth’s rotation and the parcel’s rotation with respect to the Earth, respectively. The conservation of absolute vorticity <math>\eta</math> determines a southward gradient of <math>\zeta</math>, as denoted by the red shadow in '''c'''. The gradient’s projection along the flow path <math>l</math> is typically not zero and would cause a tangential velocity <math>v_t</math>. As an example, the path <math>l</math> in '''c''' is zoomed in at two green crosses, displayed in '''d''' and '''e'''. These two crosses are associated with positive and negative gradients of <math>\zeta</math> along <math>l</math>, respectively, as denoted by the red and pink arrows in '''d''' and '''e'''. The black arrows <math>v_t</math> denote the vector sums of the red and pink arrows bordering the crosses, both of which project zonally westward. The parcels at these crosses drift toward the green points in '''c''' and, visually, the path <math>l</math> drifts westward toward the dotted line.<ref name=":0">{{Cite journal |last1=He |first1=Maosheng |last2=Forbes |first2=Jeffrey M. |date=2022-12-07 |title=Rossby wave second harmonic generation observed in the middle atmosphere |journal=Nature Communications |language=en |volume=13 |issue=1 |pages=7544 |doi=10.1038/s41467-022-35142-3 |issn=2041-1723 |pmc=9729661 |pmid=36476614|bibcode=2022NatCo..13.7544H }}{{Creative Commons text attribution notice|cc=by4|from this source=yes}} </ref>|center]]Atmospheric Rossby waves result from the conservation of [[potential vorticity]] and are influenced by the [[Coriolis force]] and pressure gradient.<ref name=":0" /> The image on the left sketches fundamental principles of the wave, e.g., its restoring force and westward phase velocity. The rotation causes fluids to turn to the right as they move in the northern hemisphere and to the left in the southern hemisphere. For example, a fluid that moves from the equator toward the north pole will deviate toward the east; a fluid moving toward the equator from the north will deviate toward the west. These deviations are caused by the Coriolis force and conservation of potential vorticity which leads to changes of relative vorticity. This is analogous to conservation of [[angular momentum]] in mechanics. In planetary atmospheres, including Earth, Rossby waves are due to the variation in the Coriolis effect with [[latitude]]. One can identify a terrestrial Rossby wave as its [[phase velocity]], marked by its wave crest, always has a westward component.<ref name="WhatIs"/><ref name="Rossby1939">{{cite journal | last=Rossby | first=C. G. | last2=Willett | first2=H. C. | last3=Holmboe | first3=Messrs. J. | last4=Namias | first4=J. | last5=Page | first5=L. | last6=Allen | first6=R. | title=Relation between variations in the intensity of the zonal circulation of the atmosphere and the displacements of the permanent centers of action atmosphere and the displacements of the permanent centers of action | journal=Journal of Marine Research| volume=2|issue=1|pages=38-55| date=1939 | url=https://elischolar.library.yale.edu/journal_of_marine_research/544/ | access-date=4 July 2024}}</ref> However, the collected set of Rossby waves may appear to move in either direction with what is known as its [[group velocity]]. In general, shorter waves have an eastward group velocity and long waves a westward group velocity. The terms "[[barotropic]]" and "[[baroclinic]]" are used to distinguish the vertical structure of Rossby waves. Barotropic Rossby waves do not vary in the vertical{{clarify|what does "vary in the vertical" mean?|date=March 2024}}, and have the fastest propagation speeds. The baroclinic wave modes, on the other hand, do vary in the vertical. They are also slower, with speeds of only a few centimeters per second or less.<ref name=Shepherd2006>{{cite journal |last1=Shepherd |first1=Theodore G. |title=Rossby waves and two-dimensional turbulence in a large-scale zonal jet |journal=Journal of Fluid Mechanics |date=October 1987 |volume=183 |pages=467–509 |doi=10.1017/S0022112087002738 |url=http://centaur.reading.ac.uk/32992/ |bibcode=1987JFM...183..467S |s2cid=9289503 }}</ref> Most investigations of Rossby waves have been done on those in Earth's atmosphere. Rossby waves in the Earth's atmosphere are easy to observe as (usually 4–6) large-scale meanders of the [[jet stream]]. When these deviations become very pronounced, masses of cold or warm air detach, and become low-strength [[cyclone]]s and [[anticyclone]]s, respectively, and are responsible for day-to-day weather patterns at mid-latitudes. The action of Rossby waves partially explains why eastern continental edges in the Northern Hemisphere, such as the Northeast United States and Eastern Canada, are colder than Western Europe at the same [[latitude]]s,<ref>{{cite journal |last1=Kaspi |first1=Yohai |last2=Schneider |first2=Tapio |title=Winter cold of eastern continental boundaries induced by warm ocean waters |journal=Nature |date=March 2011 |volume=471 |issue=7340 |pages=621–624 |doi=10.1038/nature09924 |pmid=21455177 |bibcode=2011Natur.471..621K |s2cid=4388818 |url=https://authors.library.caltech.edu/23384/2/nature09924-s1.pdf }}</ref> and why the Mediterranean is dry during summer ([[Rodwell–Hoskins mechanism]]).<ref>{{cite journal |last1=Rodwell |first1=Mark J. |last2=Hoskins |first2=Brian J. |title=Monsoons and the dynamics of deserts |journal=Quarterly Journal of the Royal Meteorological Society |date=1996 |volume=122 |issue=534 |pages=1385–1404 |doi=10.1002/qj.49712253408 |bibcode=1996QJRMS.122.1385R |url=https://doi.org/10.1002%2Fqj.49712253408 |issn=1477-870X|url-access=subscription }}</ref> ====Poleward-propagating atmospheric waves==== Deep [[convection]] ([[heat transfer]]) to the [[troposphere]] is enhanced over very warm sea surfaces in the tropics, such as during [[El Niño]] events. This tropical forcing generates atmospheric Rossby waves that have a poleward and eastward migration. Poleward-propagating Rossby waves explain many of the observed statistical connections between low- and high-latitude climates.<ref>{{cite journal |last1=Hoskins |first1=Brian J. |last2=Karoly |first2=David J. |title=The Steady Linear Response of a Spherical Atmosphere to Thermal and Orographic Forcing |journal=Journal of the Atmospheric Sciences |date=June 1981 |volume=38 |issue=6 |pages=1179–1196 |doi=10.1175/1520-0469(1981)038<1179:TSLROA>2.0.CO;2 |bibcode=1981JAtS...38.1179H |doi-access=free }}</ref> One such phenomenon is [[sudden stratospheric warming]]. Poleward-propagating Rossby waves are an important and unambiguous part of the variability in the Northern Hemisphere, as expressed in the Pacific North America pattern. Similar mechanisms apply in the Southern Hemisphere and partly explain the strong variability in the [[Amundsen Sea]] region of Antarctica.<ref>{{cite journal |last1=Lachlan-Cope |first1=Tom |last2=Connolley |first2=William |title=Teleconnections between the tropical Pacific and the Amundsen-Bellinghausens Sea: Role of the El Niño/Southern Oscillation |journal=Journal of Geophysical Research: Atmospheres |date=16 December 2006 |volume=111 |issue=D23 |doi=10.1029/2005JD006386 |bibcode=2006JGRD..11123101L |doi-access=free }}</ref> In 2011, a ''[[Nature Geoscience]]'' study using [[general circulation model]]s linked Pacific Rossby waves generated by increasing central tropical Pacific temperatures to warming of the Amundsen Sea region, leading to winter and spring continental warming of [[Ellsworth Land]] and [[Marie Byrd Land]] in [[West Antarctica]] via an increase in [[advection]].<ref>{{cite journal |last1=Ding |first1=Qinghua |last2=Steig |first2=Eric J. |last3=Battisti |first3=David S. |last4=Küttel |first4=Marcel |title=Winter warming in West Antarctica caused by central tropical Pacific warming |journal=Nature Geoscience |date=June 2011 |volume=4 |issue=6 |pages=398–403 |doi=10.1038/ngeo1129 |bibcode=2011NatGe...4..398D |citeseerx=10.1.1.459.8689 }}</ref> ====Rossby waves on other planets==== Atmospheric Rossby waves, like [[Kelvin wave]]s, can occur on any rotating planet with an atmosphere. The Y-shaped cloud feature on [[Venus]] is attributed to Kelvin and Rossby waves.<ref>{{cite journal |last1=Covey |first1=Curt |last2=Schubert |first2=Gerald |title=Planetary-Scale Waves in the Venus Atmosphere |journal=Journal of the Atmospheric Sciences |date=November 1982 |volume=39 |issue=11 |pages=2397–2413 |doi=10.1175/1520-0469(1982)039<2397:PSWITV>2.0.CO;2 |bibcode=1982JAtS...39.2397C |doi-access=free }}</ref> ===Oceanic waves=== Oceanic Rossby waves are large-scale waves within an ocean basin. They have a low amplitude, in the order of centimetres (at the surface) to metres (at the thermocline), compared with atmospheric Rossby waves which are in the order of hundreds of kilometres. They may take months to cross an ocean basin. They gain [[momentum]] from [[wind stress]] at the ocean surface layer and are thought to communicate climatic changes due to variability in [[harmonic oscillator|forcing]], due to both the [[wind]] and [[buoyancy]]. Off-equatorial Rossby waves are believed to propagate through eastward-propagating [[Kelvin wave|Kelvin waves]] that upwell against [[Ocean current|Eastern Boundary Currents]], while equatorial Kelvin waves are believed to derive some of their energy from the reflection of Rossby waves against Western Boundary Currents.<ref>{{Cite journal |last=Battisti |first=David S. |date=April 1989 |title=On the Role of Off-Equatorial Oceanic Rossby Waves during ENSO |url=https://journals.ametsoc.org/view/journals/phoc/19/4/1520-0485_1989_019_0551_otrooe_2_0_co_2.xml?tab_body=pdf |journal=Journal of Physical Oceanography |volume=19.4 |pages=551-560}}</ref> Both barotropic and baroclinic waves cause variations of the sea surface height, although the length of the waves made them difficult to detect until the advent of [[satellite]] [[altimetry]]. [[Satellite]] observations have confirmed the existence of oceanic Rossby waves.<ref name="Chelton1996">{{cite journal |doi=10.1126/science.272.5259.234 |title=Global Observations of Oceanic Rossby Waves |year=1996 |last1=Chelton |first1=D. B. |last2=Schlax |first2=M. G. |journal=Science |volume=272 |issue=5259 |pages=234|bibcode = 1996Sci...272..234C |s2cid=126953559 }}</ref> Baroclinic waves also generate significant displacements of the oceanic [[thermocline]], often of tens of meters. Satellite observations have revealed the stately progression of Rossby waves across all the [[ocean basin]]s, particularly at low- and mid-latitudes. Due to the [[Beta plane|beta effect]], transit times of Rossby waves increase with latitude. In a basin like the [[Pacific Ocean|Pacific]], waves travelling at the equator may take months, while closer to the poles transit may take decades.<ref>{{Cite journal |last=Chelton |first=Dudley B. |last2=Schlax |first2=Michael B. |date=1996 |title=Global Observations of Oceanic Rossby Waves |url=https://www.ocean.washington.edu/courses/oc513/Chelton.Science.1996.pdf |journal=Science |volume=272 |issue=5259 |pages=234-238}}</ref> Rossby waves have been suggested as an important mechanism to account for the heating of [[Europa (moon)#Subsurface ocean|the ocean on Europa]], a moon of [[Jupiter]].<ref name="Tyler2008">{{cite journal |doi=10.1038/nature07571 |title=Strong ocean tidal flow and heating on moons of the outer planets |year=2008 |last1=Tyler |first1=Robert H. |journal=Nature |volume=456 |issue=7223 |pages=770–2 |pmid=19079055|bibcode = 2008Natur.456..770T |s2cid=205215528 }}</ref> ===Waves in astrophysical discs=== [[Rossby wave instability|Rossby wave instabilities]] are also thought to be found in astrophysical [[Accretion disk|discs]], for example, around newly forming stars.<ref>{{cite journal |last1=Lovelace |first1=R. V. E. |last2=Li |first2=H. |last3=Colgate |first3=S. A. |last4=Nelson |first4=A. F. |title=Rossby Wave Instability of Keplerian Accretion Disks |journal=The Astrophysical Journal |date=10 March 1999 |volume=513 |issue=2 |pages=805–810 |doi=10.1086/306900 |arxiv=astro-ph/9809321 |bibcode=1999ApJ...513..805L |s2cid=8914218 }}</ref><ref>{{cite journal |last1=Li |first1=H. |last2=Finn |first2=J. M. |last3=Lovelace |first3=R. V. E. |last4=Colgate |first4=S. A. |title=Rossby Wave Instability of Thin Accretion Disks. II. Detailed Linear Theory |journal=The Astrophysical Journal |date=20 April 2000 |volume=533 |issue=2 |pages=1023–1034 |doi=10.1086/308693 |arxiv=astro-ph/9907279 |bibcode=2000ApJ...533.1023L |s2cid=119382697 }}</ref> == Amplification of Rossby waves == {{Anchor|Planetary wave resonance|Planetary wave quasiresonance|Quasiresonant amplification}} It has been proposed that a number of regional weather extremes in the Northern Hemisphere associated with blocked atmospheric circulation patterns may have been caused by '''quasiresonant amplification of Rossby waves'''. Examples include the [[2013 European floods]], the [[2012 China floods]], the [[2010 Russian heat wave]], the [[2010 Pakistan floods]] and the [[2003 European heat wave]]. Even taking [[global warming]] into account, the 2003 heat wave would have been highly unlikely without such a mechanism. Normally freely travelling [[synoptic scale meteorology|synoptic]]-scale Rossby waves and [[quasistationary]] planetary-scale Rossby waves exist in the [[mid-latitudes]] with only weak interactions. The hypothesis, proposed by [[Vladimir Petoukhov]], [[Stefan Rahmstorf]], [[Stefan Petri]], and [[Hans Joachim Schellnhuber]], is that under some circumstances these waves interact to produce the static pattern. For this to happen, they suggest, the [[zonal (Earth sciences)|zonal]] (east-west) [[wave number]] of both types of wave should be in the range 6–8, the synoptic waves should be arrested within the [[troposphere]] (so that energy does not escape to the [[stratosphere]]) and mid-latitude [[waveguides]] should trap the quasistationary components of the synoptic waves. In this case the planetary-scale waves may respond unusually strongly to [[orography]] and thermal sources and sinks because of "quasiresonance".<ref>{{cite journal |last1=Petoukhov |first1=Vladimir |last2=Rahmstorf |first2=Stefan |last3=Petri |first3=Stefan |last4=Schellnhuber |first4=Hans Joachim |title=Quasiresonant amplification of planetary waves and recent Northern Hemisphere weather extremes |journal=Proceedings of the National Academy of Sciences of the United States of America |date=2 April 2013 |volume=110 |issue=14 |pages=5336–5341 |doi=10.1073/pnas.1222000110 |pmid=23457264 |pmc=3619331 |bibcode=2013PNAS..110.5336P |doi-access=free }}</ref> A 2017 study by [[Michael E. Mann|Mann]], Rahmstorf, et al. connected the phenomenon of anthropogenic [[Arctic amplification]] to planetary wave resonance and [[extreme weather]] events.<ref>{{cite journal |last1=Mann |first1=Michael E. |last2=Rahmstorf |first2=Stefan |last3=Kornhuber |first3=Kai |last4=Steinman |first4=Byron A. |last5=Miller |first5=Sonya K. |last6=Coumou |first6=Dim |title=Influence of Anthropogenic Climate Change on Planetary Wave Resonance and Extreme Weather Events |journal=Scientific Reports |date=30 May 2017 |volume=7 |issue=1 |pages=45242 |doi=10.1038/srep45242 |pmid=28345645 |pmc=5366916 |bibcode=2017NatSR...745242M }}</ref> ==Mathematical definitions== ===Free barotropic Rossby waves under a zonal flow with linearized vorticity equation=== To start with, a zonal mean flow, ''U'', can be considered to be perturbed where ''U'' is constant in time and space. Let <math>\vec{u} = \langle u, v\rangle</math> be the total horizontal wind field, where ''u'' and ''v'' are the components of the wind in the ''x''- and ''y''- directions, respectively. The total wind field can be written as a mean flow, ''U'', with a small superimposed perturbation, ''u′'' and ''v′''. <math display="block"> u = U + u'(t,x,y)\!</math> <math display="block"> v = v'(t,x,y)\!</math> The perturbation is assumed to be much smaller than the mean zonal flow. <math display="block"> U \gg u',v'\!</math> The relative vorticity <math>\eta</math> and the perturbations <math>u'</math> and <math>v'</math> can be written in terms of the [[stream function]] <math>\psi</math> (assuming non-divergent flow, for which the stream function completely describes the flow): <math display="block"> \begin{align} u' & = \frac{\partial \psi}{\partial y} \\[5pt] v' & = -\frac{\partial \psi}{\partial x} \\[5pt] \eta & = \nabla \times (u' \mathbf{\hat{\boldsymbol{\imath}}} + v' \mathbf{\hat{\boldsymbol{\jmath}}}) = -\nabla^2 \psi \end{align} </math> Considering a parcel of air that has no relative vorticity before perturbation (uniform ''U'' has no vorticity) but with planetary vorticity ''f'' as a function of the latitude, perturbation will lead to a slight change of latitude, so the perturbed relative vorticity must change in order to conserve [[potential vorticity]]. Also the above approximation ''U'' >> ''u''' ensures that the perturbation flow does not advect relative vorticity. <math display="block">\frac{d (\eta + f) }{dt} = 0 = \frac{\partial \eta}{\partial t} + U \frac{\partial \eta}{\partial x} + \beta v'</math> with <math>\beta = \frac{\partial f}{\partial y} </math>. Plug in the definition of stream function to obtain: <math display="block"> 0 = \frac{\partial \nabla^2 \psi}{\partial t} + U \frac{\partial \nabla^2 \psi}{\partial x} + \beta \frac{\partial \psi}{\partial x}</math> Using the [[method of undetermined coefficients]] one can consider a traveling wave solution with [[zonal and meridional]] [[wavenumbers]] ''k'' and ''ℓ'', respectively, and frequency <math>\omega</math>: <math display="block">\psi = \psi_0 e^{i(kx+\ell y-\omega t)}\!</math> This yields the [[dispersion relation]]: <math display="block"> \omega = Uk - \beta \frac k {k^2+\ell^2}</math> The zonal (''x''-direction) [[phase speed]] and [[group velocity]] of the Rossby wave are then given by <math display="block"> \begin{align} c & \equiv \frac \omega k = U - \frac \beta {k^2+\ell^2}, \\[5pt] c_g & \equiv \frac{\partial \omega}{\partial k}\ = U - \frac{\beta (\ell^2-k^2)}{(k^2+\ell^2)^2}, \end{align} </math> where ''c'' is the phase speed, ''c''<sub>''g''</sub> is the group speed, ''U'' is the mean westerly flow, <math>\beta</math> is the [[Rossby parameter]], ''k'' is the [[Zonal and meridional|zonal]] wavenumber, and ''ℓ'' is the [[Zonal and meridional|meridional]] wavenumber. It is noted that the zonal phase speed of Rossby waves is always westward (traveling east to west) relative to mean flow ''U'', but the zonal group speed of Rossby waves can be eastward or westward depending on wavenumber. ===Rossby parameter=== The [[Rossby parameter]] is defined as the rate of change of the [[Coriolis frequency]] along the meridional direction: <math display="block">\beta = \frac{\partial f}{\partial y} = \frac 1 a \frac d {d\varphi} (2 \omega \sin\varphi) = \frac{2\omega \cos\varphi} a,</math> where <math>\varphi</math> is the latitude, ''ω'' is the [[angular speed]] of the [[Earth's rotation]], and ''a'' is the mean [[radius of the Earth]]. If <math>\beta = 0</math>, there will be no Rossby waves; Rossby waves owe their origin to the gradient of the tangential speed of the planetary rotation (planetary vorticity). A "cylinder" planet has no Rossby waves. It also means that at the equator of any rotating, sphere-like planet, including Earth, one will still have Rossby waves, despite the fact that <math>f = 0</math>, because <math>\beta > 0</math>. These are known as [[Equatorial Rossby wave]]s. ==See also== *[[Atmospheric wave]] *[[Equatorial wave]] *[[Equatorial Rossby wave]] – mathematical treatment *[[Harmonic]] *[[Kelvin wave]] *[[Polar vortex]] *[[Rossby whistle]] *[[Sudden stratospheric warming]] == References == {{Reflist}} ==Bibliography== * {{cite journal |last1=Rossby |first1=C.-G. |title=Relation between variations in the intensity of the zonal circulation of the atmosphere and the displacements of the semi-permanent centers of action |journal=Journal of Marine Research |date=21 June 1939 |volume=2 |issue=1 |pages=38–55 |doi=10.1357/002224039806649023 |s2cid=27148455 }} * {{cite journal |last1=Platzman |first1=G. W. |title=The Rossby wave |journal=Quarterly Journal of the Royal Meteorological Society |date=July 1968 |volume=94 |issue=401 |pages=225–248 |doi=10.1002/qj.49709440102 |bibcode=1968QJRMS..94..225P }} * {{cite journal |last1=Dickinson |first1=R. E. |title=Rossby Waves—Long-Period Oscillations of Oceans and Atmospheres |journal=Annual Review of Fluid Mechanics |date=January 1978 |volume=10 |issue=1 |pages=159–195 |doi=10.1146/annurev.fl.10.010178.001111 |bibcode=1978AnRFM..10..159D }} ==External links== *[http://glossary.ametsoc.org/wiki/Rossby_wave Description of Rossby Waves from the American Meteorological Society] *[https://web.archive.org/web/20151201005036/http://www.noc.soton.ac.uk/JRD/SAT/Rossby/Rossbyintro.html An introduction to oceanic Rossby waves and their study with satellite data] *[https://www.youtube.com/watch?v=MzW5Isbv2A0 Rossby waves and extreme weather] (Video) {{physical oceanography}} {{Authority control}} {{DEFAULTSORT:Rossby Wave}} [[Category:Physical oceanography]] [[Category:Atmospheric dynamics]] [[Category:Fluid mechanics]] [[Category:Waves]]
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