# Rossby wave

Atmospheric Rossby waves are giant meanders in high-altitude winds with major influence on weather. Rossby waves are associated with pressure systems and the jet stream.[1] Oceanic Rossby waves move along the thermocline: that is, the boundary between the warm upper layer of the ocean and the cold deeper part of the ocean. Rossby waves are a subset of inertial waves.

## Rossby wave types

### Atmospheric waves

Meanders of the northern hemisphere's jet stream developing (a, b) and finally detaching a "drop" of cold air (c). Orange: warmer masses of air; pink: jet stream.

Atmospheric Rossby waves emerge due to shear in rotating fluids, so that the Coriolis force changes along the sheared coordinate. In planetary atmospheres, they are due to the variation in the Coriolis effect with latitude. The waves were first identified in the Earth's atmosphere in 1939 by Carl-Gustaf Arvid Rossby who went on to explain their motion.

One can identify a Rossby wave in that its phase velocity (that of the wave crests) always has a westward component. However, the wave's group velocity (associated with the energy flux) can be in any direction. In general, shorter waves have an eastward group velocity and long waves a westward group velocity.

The terms "barotropic" and "baroclinic" Rossby waves are used to distinguish their vertical structure. Barotropic Rossby waves do not vary in the vertical, and have the fastest propagation speeds. The baroclinic wave modes are slower, with speeds of only a few centimetres per second or less.

Most work on Rossby waves has 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, they detach the masses of cold, or warm, air that become cyclones and anticyclones and are responsible for day-to-day weather patterns at mid-latitudes. Rossby waves may be partly responsible for the fact that eastern continental edges, such as the Northeast United States and Eastern Canada, are colder than Western Europe at the same latitudes.[2]

#### Poleward-propagating atmospheric waves

Deep convection and heat transfer to the troposphere is enhanced over anomalously warm sea surface temperatures in the tropics, such as during, but by no means limited to, El Niño events. This tropical forcing generates atmospheric Rossby waves that propagates poleward and eastward and are subsequently refracted back from the pole to the tropics.

Poleward propagating Rossby waves explain many of the observed statistical teleconnections between low latitude and high latitude climate, as shown in the now classic study by Hoskins and Karoly (1981).[3] 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.[4] 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.[5]

### Oceanic waves

Oceanic Rossby waves gain momentum from wind stress at the Ocean surface layer and are thought to communicate climatic changes due to variability in forcing, due to both the wind and buoyancy. 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. Observations by the NASA/CNES TOPEX/Poseidon satellite confirmed the existence of oceanic Rossby waves.[6]

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 basins, particularly at low- and mid-latitudes. These waves can take months or even years to cross a basin like the Pacific.

Rossby waves have been suggested as an important mechanism to account for the heating of Europa's ocean.[7]

## Definitions

### Free barotropic Rossby waves under a zonal flow with linearized vorticity equation

Let us start by perturbing a zonal mean flow, "U", where "U" is constant in time and space. Let $\vec{u} = $ 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'".

$u = U + u'(t,x,y)\!$
$v = v'(t,x,y)\!$

We assume the perturbation to be much smaller than the mean zonal flow.

$U \gg u',v'\!$

Relative Vorticity $\eta$, u and v can be written in term of the stream function $\psi$ (assuming non-divergent flow, for which the stream function completely describes the flow):

$u' = \frac{\partial \psi}{\partial y}$
$v' = -\frac{\partial \psi}{\partial x}$
$\eta = \nabla \times (u' \mathbf{\hat{\boldsymbol{\imath}}} + v' \mathbf{\hat{\boldsymbol{\jmath}}}) = -\nabla^2 \psi$

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.

$\frac{d (\eta + f) }{dt} = 0 = \frac{\partial \eta}{\partial t} + U \frac{\partial \eta}{\partial x} + \beta v'$

with $\beta = \frac{\partial f}{\partial y}$. Plug in the definition of stream function to obtain:

$0 = \frac{\partial \nabla^2 \psi}{\partial t} + U \frac{\partial \nabla^2 \psi}{\partial x} + \beta \frac{\partial \psi}{\partial x}$

Consider a traveling wave solution with zonal and meridional wavenumbers k and l, respectively, and frequency $\omega$:

$\psi = \psi_0 e^{i(kx+ly-\omega t)}\!$

We obtain the dispersion relation:

$\omega = Uk - \beta \frac {k}{k^2+l^2}$

The zonal (x-direction) phase speed and group velocity of the Rossby wave are then given by

$c \ \equiv\ \frac {\omega}{k} = U - \frac{\beta}{(k^2+l^2)},$
$c_g \ \equiv\ \frac{\partial \omega}{\partial k}\ = U - \frac{\beta (l^2-k^2)}{(k^2+l^2)^2},$

where c is the phase speed, $c_g$ is the group speed, U is the mean westerly flow, $\beta$ 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.

### Meaning of Beta

The Rossby parameter is defined:

$\beta = \frac{\partial f}{\partial y} = \frac{1}{a} \frac{d}{d\phi} (2 \omega \sin\phi) = \frac{2\omega \cos\phi}{a}$

$\phi$ is the latitude, ω is the angular speed of the Earth's rotation, and a is the mean radius of the Earth.

If $\beta = 0$, 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 near the equator on Earth where $f = 0$ but $\beta > 0$ except at the poles, one can still have Rossby Waves (Equatorial Rossby wave).

## References

1. ^ Holton, James R. (2004). Dynamic Meteorology. Elsevier. p. 347. ISBN 0-12-354015-1.
2. ^ Kaspi, Yohai; Schneider, Tapio (2011). "Winter cold of eastern continental boundaries induced by warm ocean waters". Nature 471 (7340): 621–4. doi:10.1038/nature09924. PMID 21455177.
3. ^ Hoskins, Brian J.; Karoly, David J. (1981). "The Steady Linear Response of a Spherical Atmosphere to Thermal and Orographic Forcing". Journal of the Atmospheric Sciences 38 (6): 1179. doi:10.1175/1520-0469(1981)038<1179:TSLROA>2.0.CO;2.
4. ^ Lachlan-Cope, Tom; Connolley, William (2006). "Teleconnections between the tropical Pacific and the Amundsen-Bellinghausens Sea: Role of the El Niño/Southern Oscillation". Journal of Geophysical Research 111. doi:10.1029/2005JD006386.
5. ^ Ding, Qinghua; Steig, Eric J.; Battisti, David S.; Küttel, Marcel (2011). "Winter warming in West Antarctica caused by central tropical Pacific warming". Nature Geoscience 4 (6): 398. doi:10.1038/ngeo1129.
6. ^ Chelton, D. B.; Schlax, M. G. (1996). "Global Observations of Oceanic Rossby Waves". Science 272 (5259): 234. doi:10.1126/science.272.5259.234.
7. ^ Tyler, Robert H. (2008). "Strong ocean tidal flow and heating on moons of the outer planets". Nature 456 (7223): 770–2. doi:10.1038/nature07571. PMID 19079055.

## Bibliography

• Rossby, C.-G. (1939). "Relation between variations in the intensity of the zonal circulation of the atmosphere and the displacements of the semi-permanent centers of action". Journal of Marine Research 2: 38. doi:10.1357/002224039806649023.
• Platzman, G. W. (1968). "The Rossby wave". Quarterly Journal of the Royal Meteorological Society 94 (401): 225. doi:10.1002/qj.49709440102.
• Dickinson, R E (1978). "Rossby Waves--Long-Period Oscillations of Oceans and Atmospheres". Annual Review of Fluid Mechanics 10: 159. doi:10.1146/annurev.fl.10.010178.001111.