Editing Rossby wave (section)

==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>