Rossby wave

Atmospheric Rossby waves result from the conservation of potential vorticity and are influenced by the Coriolis force and pressure gradient.[3] 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. 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.[4]

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 cyclones and anticyclones, 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 latitudes,[5] and why the Mediterranean is dry during summer (Rodwell–Hoskins mechanism).[6]

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.[7] 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.[8] 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.[9]

Atmospheric Rossby waves, like Kelvin waves, can occur on any rotating planet with an atmosphere. The Y-shaped cloud feature on Venus is attributed to Kelvin and Rossby waves.[10]

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 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. Satellite observations have confirmed the existence of oceanic Rossby waves.[11]

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 the ocean on Europa, a moon of Jupiter.[12]

Rossby wave instabilities are also thought to be found in astrophysical discs, for example, around newly forming stars.[13][14]

To start with, a zonal mean flow, U, can be considered to be perturbed where U is constant in time and space. Let ${\displaystyle {\vec {u}}=\langle u,v\rangle }$ 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′.

${\displaystyle u=U+u'(t,x,y)\!}$

${\displaystyle v=v'(t,x,y)\!}$

The perturbation is assumed to be much smaller than the mean zonal flow.

${\displaystyle U\gg u',v'\!}$

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

${\displaystyle {\begin{aligned}u'&={\frac {\partial \psi }{\partial y}}\\[5pt]v'&=-{\frac {\partial \psi }{\partial x}}\\[5pt]\eta &=\nabla \times (u'\mathbf {\hat {\boldsymbol {\imath }}} +v'\mathbf {\hat {\boldsymbol {\jmath }}} )=-\nabla ^{2}\psi \end{aligned}}}$

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.

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

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

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

Using the method of undetermined coefficients one can consider a traveling wave solution with zonal and meridional wavenumbers k and ℓ, respectively, and frequency ${\displaystyle \omega }$:

${\displaystyle \psi =\psi _{0}e^{i(kx+\ell y-\omega t)}\!}$

This yields the dispersion relation:

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

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

${\displaystyle {\begin{aligned}c&\equiv {\frac {\omega }{k}}=U-{\frac {\beta }{k^{2}+\ell ^{2}}},\\[5pt]c_{g}&\equiv {\frac {\partial \omega }{\partial k}}\ =U-{\frac {\beta (\ell ^{2}-k^{2})}{(k^{2}+\ell ^{2})^{2}}},\end{aligned}}}$

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

The Rossby parameter is defined as the rate of change of the Coriolis frequency along the meridional direction:

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

where ${\displaystyle \varphi }$ is the latitude, ω is the angular speed of the Earth's rotation, and a is the mean radius of the Earth.

If ${\displaystyle \beta =0}$, 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 ${\displaystyle f=0}$, because ${\displaystyle \beta >0}$. These are known as Equatorial Rossby waves.