Rossby wave

 ( Physical Oceanography ) 

One can identify a terrestrial Rossby wave as its phase velocity, marked by its wave crest, always has a westward component. 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, 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.

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 and , 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 , and why the Mediterranean is dry during summer (Rodwell–Hoskins mechanism).

Poleward-propagating Rossby waves explain many of the observed statistical connections between low- and high-latitude climates. 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. In 2011, a Nature Geoscience study using general circulation models 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.

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.

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 , particularly at low- and mid-latitudes. Due to the Beta plane, transit times of Rossby waves increase with latitude. In a basin like the Pacific Ocean, waves travelling at the equator may take months, while closer to the poles transit may take decades.

Rossby waves have been suggested as an important mechanism to account for the heating of the ocean on Europa, a moon of Jupiter.

Normally freely travelling 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 (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".

A 2017 study by Mann, Rahmstorf, et al. connected the phenomenon of anthropogenic Arctic amplification to planetary wave resonance and extreme weather events.

The perturbation is assumed to be much smaller than the mean zonal flow. $$U\; \backslash gg\; u\text{'},v\text{'}\backslash !$$

The relative vorticity $\backslash eta$ and the perturbations $u\text{'}$ and $v\text{'}$ can be written in terms of the stream function $\backslash psi$ (assuming non-divergent flow, for which the stream function completely describes the flow):

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.

$$\backslash frac\{d\; (\backslash eta\; +\; f)\; \}\{dt\}\; =\; 0\; =\; \backslash frac\{\backslash partial\; \backslash eta\}\{\backslash partial\; t\}\; +\; U\; \backslash frac\{\backslash partial\; \backslash eta\}\{\backslash partial\; x\}\; +\; \backslash beta\; v\text{'}$$ with $\backslash beta\; =\; \backslash frac\{\backslash partial\; f\}\{\backslash partial\; y\}$. Plug in the definition of stream function to obtain: $$0\; =\; \backslash frac\{\backslash partial\; \backslash nabla^2\; \backslash psi\}\{\backslash partial\; t\}\; +\; U\; \backslash frac\{\backslash partial\; \backslash nabla^2\; \backslash psi\}\{\backslash partial\; x\}\; +\; \backslash beta\; \backslash frac\{\backslash partial\; \backslash psi\}\{\backslash partial\; x\}$$

Using the method of undetermined coefficients one can consider a traveling wave solution with zonal and meridional wavenumbers k and ℓ, respectively, and frequency $\backslash omega$: $$\backslash psi\; =\; \backslash psi\_0\; e^\{i(kx+\backslash ell\; y-\backslash omega\; t)\}\backslash !$$

This yields the dispersion relation: $$\backslash omega\; =\; Uk\; -\; \backslash beta\; \backslash frac\; k\; \{k^2+\backslash ell^2\}$$

The zonal ( x-direction) phase speed and group velocity of the Rossby wave are then given by where c is the phase speed, c _g is the group speed, U is the mean westerly flow, $\backslash beta$ is the Rossby parameter, k is the zonal wavenumber, and ℓ is the 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.

If $\backslash beta\; =\; 0$, 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 $f\; =\; 0$, because $\backslash beta\; >\; 0$. These are known as Equatorial Rossby waves.

• Atmospheric wave
• Equatorial wave
• Equatorial Rossby wave – mathematical treatment
• Harmonic
• Kelvin wave
• Polar vortex
• Rossby whistle
• Sudden stratospheric warming

• Description of Rossby Waves from the American Meteorological Society
• An introduction to oceanic Rossby waves and their study with satellite data
• Rossby waves and extreme weather (Video)