Rossby wave
Overview
 
Atmospheric Rossby waves are giant meanders in high-altitude
Altitude
Altitude or height is defined based on the context in which it is used . As a general definition, altitude is a distance measurement, usually in the vertical or "up" direction, between a reference datum and a point or object. The reference datum also often varies according to the context...

 wind
Wind
Wind is the flow of gases on a large scale. On Earth, wind consists of the bulk movement of air. In outer space, solar wind is the movement of gases or charged particles from the sun through space, while planetary wind is the outgassing of light chemical elements from a planet's atmosphere into space...

s that are a major influence on weather
Weather
Weather is the state of the atmosphere, to the degree that it is hot or cold, wet or dry, calm or stormy, clear or cloudy. Most weather phenomena occur in the troposphere, just below the stratosphere. Weather refers, generally, to day-to-day temperature and precipitation activity, whereas climate...

.
They are not to be confused with oceanic Rossby waves, which move along the thermocline
Thermocline
A thermocline is a thin but distinct layer in a large body of fluid , in which temperature changes more rapidly with depth than it does in the layers above or below...

: 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.
Atmospheric Rossby waves emerge due to shear in rotating fluids
Vortex
A vortex is a spinning, often turbulent,flow of fluid. Any spiral motion with closed streamlines is vortex flow. The motion of the fluid swirling rapidly around a center is called a vortex...

, so that the Coriolis force changes along the sheared coordinate.
Encyclopedia
Atmospheric Rossby waves are giant meanders in high-altitude
Altitude
Altitude or height is defined based on the context in which it is used . As a general definition, altitude is a distance measurement, usually in the vertical or "up" direction, between a reference datum and a point or object. The reference datum also often varies according to the context...

 wind
Wind
Wind is the flow of gases on a large scale. On Earth, wind consists of the bulk movement of air. In outer space, solar wind is the movement of gases or charged particles from the sun through space, while planetary wind is the outgassing of light chemical elements from a planet's atmosphere into space...

s that are a major influence on weather
Weather
Weather is the state of the atmosphere, to the degree that it is hot or cold, wet or dry, calm or stormy, clear or cloudy. Most weather phenomena occur in the troposphere, just below the stratosphere. Weather refers, generally, to day-to-day temperature and precipitation activity, whereas climate...

.
They are not to be confused with oceanic Rossby waves, which move along the thermocline
Thermocline
A thermocline is a thin but distinct layer in a large body of fluid , in which temperature changes more rapidly with depth than it does in the layers above or below...

: 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.

Atmospheric waves

Atmospheric Rossby waves emerge due to shear in rotating fluids
Vortex
A vortex is a spinning, often turbulent,flow of fluid. Any spiral motion with closed streamlines is vortex flow. The motion of the fluid swirling rapidly around a center is called a vortex...

, so that the Coriolis force changes along the sheared coordinate. In planetary
Planet
A planet is a celestial body orbiting a star or stellar remnant that is massive enough to be rounded by its own gravity, is not massive enough to cause thermonuclear fusion, and has cleared its neighbouring region of planetesimals.The term planet is ancient, with ties to history, science,...

 atmospheres
Atmosphères
Atmosphères is a piece for full orchestra, composed by György Ligeti in 1961. It is noted for eschewing conventional melody and metre in favor of dense sound textures...

, they are due to the variation in the Coriolis effect
Coriolis effect
In physics, the Coriolis effect is a deflection of moving objects when they are viewed in a rotating reference frame. In a reference frame with clockwise rotation, the deflection is to the left of the motion of the object; in one with counter-clockwise rotation, the deflection is to the right...

 with latitude
Latitude
In geography, the latitude of a location on the Earth is the angular distance of that location south or north of the Equator. The latitude is an angle, and is usually measured in degrees . The equator has a latitude of 0°, the North pole has a latitude of 90° north , and the South pole has a...

. The waves were first identified in the Earth's atmosphere
Earth's atmosphere
The atmosphere of Earth is a layer of gases surrounding the planet Earth that is retained by Earth's gravity. The atmosphere protects life on Earth by absorbing ultraviolet solar radiation, warming the surface through heat retention , and reducing temperature extremes between day and night...

 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
Phase velocity
The phase velocity of a wave is the rate at which the phase of the wave propagates in space. This is the speed at which the phase of any one frequency component of the wave travels. For such a component, any given phase of the wave will appear to travel at the phase velocity...

 (that of the wave crests) always has a westward component. However, the wave's group velocity
Group velocity
The group velocity of a wave is the velocity with which the overall shape of the wave's amplitudes — known as the modulation or envelope of the wave — propagates through space....

 (associated with the energy flux
Energy flux
Energy flux is the rate of transfer of energy through a surface. The quantity is defined in two different ways, depending on the context:# Rate of energy transfer per unit area...

) can be in any direction. In general, shorter waves have an eastward group velocity and long waves a westward group velocity.

The terms "barotropic
Barotropic
In meteorology, a barotropic atmosphere is one in which the pressure depends only on the density and vice versa, so that isobaric surfaces are also isopycnic surfaces . The isobaric surfaces will also be isothermal surfaces, hence the geostrophic wind is independent of height...

" 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 speed
Speed
In kinematics, the speed of an object is the magnitude of its velocity ; it is thus a scalar quantity. The average speed of an object in an interval of time is the distance traveled by the object divided by the duration of the interval; the instantaneous speed is the limit of the average speed as...

s. 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
Jet stream
Jet streams are fast flowing, narrow air currents found in the atmospheres of some planets, including Earth. The main jet streams are located near the tropopause, the transition between the troposphere and the stratosphere . The major jet streams on Earth are westerly winds...

. When these loops become very pronounced, they detach the masses of cold, or warm, air that become cyclone
Cyclone
In meteorology, a cyclone is an area of closed, circular fluid motion rotating in the same direction as the Earth. This is usually characterized by inward spiraling winds that rotate anticlockwise in the Northern Hemisphere and clockwise in the Southern Hemisphere of the Earth. Most large-scale...

s and anticyclone
Anticyclone
An anticyclone is a weather phenomenon defined by the United States' National Weather Service's glossary as "[a] large-scale circulation of winds around a central region of high atmospheric pressure, clockwise in the Northern Hemisphere, counterclockwise in the Southern Hemisphere"...

s and are responsible for day-to-day weather patterns at mid-latitudes. Rossby waves may be partly responsible for the fact that the Northeast United States is colder than Europe at the same latitudes.

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

Let us start with perturbing a flow that with only a time and spatially invariant zonal flow U with no meridional component.



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


Relative Vorticity , U and V can be written in the form stream function
Stream function
The stream function is defined for two-dimensional flows of various kinds. The stream function can be used to plot streamlines, which represent the trajectories of particles in a steady flow. Streamlines are perpendicular to equipotential lines...

 () (assuming non-divergent flow which 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 we make the approximation that , so the perturbation flow does not advect relative vorticity.


which , and plug in the definition of stream function to obtain:


Guess
Method of undetermined coefficients
In mathematics, the method of undetermined coefficients, also known as the lucky guess method, is an approach to finding a particular solution to certain inhomogeneous ordinary differential equations and recurrence relations...

 a traveling wave solution with wave numbers k and l, and frequency omega:


We obtain the dispersion relation
Dispersion relation
In physics and electrical engineering, dispersion most often refers to frequency-dependent effects in wave propagation. Note, however, that there are several other uses of the word "dispersion" in the physical sciences....

 of:


The zonal phase speed and group speed are given by


where c is the phase speed, is the group speed, u is the mean westerly flow, is the Rossby parameter
Rossby parameter
The Rossby parameter is a number used in geophysics and meteorology which arises due to the meridional variation of the Coriolis force caused by the spherical shape of the Earth...

, and k is the zonal wave number. The above proves that phase speed is always westward relative to mean flow, but group speed can travel both ways depending on the wave number; large zonal wave number waves (short waves) leads the mean flow, and small zonal wave number waves (long wave) retrogrades. The meaning of large and small only depends on the value of , if , then the group speed is the same as the mean zonal flow.

Meaning of Beta

The Rossby parameter
Rossby parameter
The Rossby parameter is a number used in geophysics and meteorology which arises due to the meridional variation of the Coriolis force caused by the spherical shape of the Earth...

 is defined:
is the latitude, ω is the angular speed of the Earth's rotation, and a is the mean radius of the Earth.

If , 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 but except at the poles, one can still have Rossby Waves (Equatorial Rossby wave).

Poleward-propagating atmospheric Rossby 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). Pole-ward propagating Rossby waves are an important and unambiguous part of the variabiity in the Northern Hemisphere, as expressed in the Pacific North America pattern. Similar mechanisms apply in the Southern Hemisphere and partly explain the strong variabiity in the Amundsen Sea region of Antarctica. In 2011, a Nature Geoscience
Nature Geoscience
Nature Geoscience is a monthly, peer reviewed, scientific journal published by Nature Publishing Group. The Chief Editor is Heike Langenberg. The first issue was published in January 2008.-Scope:...

study using general circulation models linked Pacific Rossby waves generated by increasing central tropical Pacific temperatures to warming of the Amundsen Sea
Amundsen Sea
The Amundsen Sea is an arm of the Southern Ocean off Marie Byrd Land in western Antarctica. It is bounded by Cape Flying Fish, the northwestern tip of Thurston Island to the east and Cape Dart on Siple Island to the west. East of Cape Flying Fish starts the Bellingshausen Sea. West of Cape Dart is...

 region, leading to winter and spring continental warming of Ellsworth Land
Ellsworth Land
Ellsworth Land is that portion of the Antarctic continent bounded on the west by Marie Byrd Land, on the north by Bellingshausen Sea, on the northeast by the base of Antarctic Peninsula, and on the east by the western margin of Ronne Ice Shelf. It extends between 103°24'W and 79°45'W...

 and Marie Byrd Land
Marie Byrd Land
Marie Byrd Land is the portion of West Antarctica lying east of the Ross Ice Shelf and the Ross Sea and south of the Pacific Ocean, extending eastward approximately to a line between the head of the Ross Ice Shelf and Eights Coast. It stretches between 158°W and 103°24'W...

 in West Antarctica
West Antarctica
West Antarctica, or Lesser Antarctica, one of the two major regions of Antarctica, is the part of the continent that lies within the Western Hemisphere including the Antarctic Peninsula.-Location and description:...

 via an increase in advection
Advection
Advection, in chemistry, engineering and earth sciences, is a transport mechanism of a substance, or a conserved property, by a fluid, due to the fluid's bulk motion in a particular direction. An example of advection is the transport of pollutants or silt in a river. The motion of the water carries...

.

Oceanic waves

Oceanic Rossby waves are thought to communicate climatic changes due to variability in forcing
Harmonic oscillator
In classical mechanics, a harmonic oscillator is a system that, when displaced from its equilibrium position, experiences a restoring force, F, proportional to the displacement, x: \vec F = -k \vec x \, where k is a positive constant....

, due to both the wind
Wind
Wind is the flow of gases on a large scale. On Earth, wind consists of the bulk movement of air. In outer space, solar wind is the movement of gases or charged particles from the sun through space, while planetary wind is the outgassing of light chemical elements from a planet's atmosphere into space...

 and buoyancy
Buoyancy
In physics, buoyancy is a force exerted by a fluid that opposes an object's weight. In a column of fluid, pressure increases with depth as a result of the weight of the overlying fluid. Thus a column of fluid, or an object submerged in the fluid, experiences greater pressure at the bottom of the...

. 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
Satellite
In the context of spaceflight, a satellite is an object which has been placed into orbit by human endeavour. Such objects are sometimes called artificial satellites to distinguish them from natural satellites such as the Moon....

 altimetry. Observations by the NASA
NASA
The National Aeronautics and Space Administration is the agency of the United States government that is responsible for the nation's civilian space program and for aeronautics and aerospace research...

/CNES
CNES
The is the French government space agency . Established under President Charles de Gaulle in 1961, its headquarters are located in central Paris and it is under the supervision of the French Ministries of Defence and Research...

 TOPEX/Poseidon
TOPEX/Poseidon
Launched in 1992, TOPEX/Poseidon was a joint satellite mission between NASA, the U.S. space agency, and CNES, the French space agency, to map ocean surface topography. The first major oceanographic research vessel to sail into space, TOPEX/Poseidon helped revolutionize oceanography by proving the...

 satellite
Satellite
In the context of spaceflight, a satellite is an object which has been placed into orbit by human endeavour. Such objects are sometimes called artificial satellites to distinguish them from natural satellites such as the Moon....

 confirmed the existence of oceanic Rossby waves.

Baroclinic waves also generate significant displacements of the oceanic thermocline
Thermocline
A thermocline is a thin but distinct layer in a large body of fluid , in which temperature changes more rapidly with depth than it does in the layers above or below...

, 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
Europa (moon)
Europa Slightly smaller than Earth's Moon, Europa is primarily made of silicate rock and probably has an iron core. It has a tenuous atmosphere composed primarily of oxygen. Its surface is composed of ice and is one of the smoothest in the Solar System. This surface is striated by cracks and...

's ocean.

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