Rossby wave

Rossby waves , also known as planetary waves , are large-scale wave movements in the ocean or the earth's atmosphere .

The physical principle underlying the planetary waves is the conservation of potential vorticity . If a liquid particle in the air or water on the surface of a rotating sphere is not displaced parallel to the axis of rotation , it will be deflected by the Coriolis acceleration, the parameters of which depend on the geographical latitude . The changing potential vorticity causes a restoring force that is proportional to the displacement from the initial position of the particle. This leads to a meridional oscillation with a westward phase velocity .

Sydney Samuel Hough was the first to formulate the equations for the motion of planetary waves on a rotating sphere in 1897 and discussed the solutions in spherical coordinates . The waves are named after Carl-Gustaf Rossby , who developed an approximate solution in Cartesian coordinates for the problem .

Oceanic Rossby waves

Rossby waves play an important role in the subinertial dynamics of the ocean. They enable a stationary wind-driven ocean circulation and shape their shape in a characteristic way, they influence the properties of mesoscale eddies in the ocean and play an important role in the propagation of ocean-climate signals, for example in the ENSO events (El Niño-Southern Oscillation ).

They are stimulated in the interior of the ocean basin by spatial variations in the surface wind and by air pressure fluctuations on the sea ​​surface or emitted from the meridonally running coasts as a reaction to temporal fluctuations in the wind and air pressure fields. Long Rossby waves are emitted from the eastern banks and short Rossby waves are emitted from the western banks. Due to their transit time through the ocean basin from east to west, they determine the characteristic reaction time for the setting of a stationary ocean circulation after changes in the driving wind pattern over time.

Although the existence of Rossby waves was theoretically proven more than 100 years ago, it was not until the end of the 20th century that their existence was confirmed by means of conventional oceanographic observation methods inside the water column and by means of satellite altimetry on the sea surface in every ocean and at all latitudes .

Mathematical description

Rossby waves are sub-inertial movements that are in quasi- geostrophic equilibrium after geostrophic adjustment on a rotating sphere. Its special feature is that the divergence of the quasi-geostrophic flow does not exactly disappear due to the spatial expansion of the associated pressure pattern and the spatial change in the Coriolis parameter . This results in a slow change in the pressure field over time in the form of a Rossby wave.

We consider the properties of the linear Rossby wave in an infinitely extensive, frictionless ocean with a flat bottom at depth on the earth rotating at angular velocity ${\displaystyle z=-H}$ ${\displaystyle {\vec {\Omega }}}$

The vertically averaged equations for the horizontal velocity components of the hydrostatic fluid are

${\displaystyle {\frac {\partial u}{\partial t}}-fv=-g{\frac {\partial \eta }{\partial x}}}$,

${\displaystyle {\frac {\partial v}{\partial t}}+fu=-g{\frac {\partial \eta }{\partial y}}}$.

In the equations are:

• ${\displaystyle t\,}$: the time
• ${\displaystyle x,y,z\,}$: the coordinates of a right-angled coordinate system with the zero point at sea level on the geographical reference latitude , e.g. B. positive to the east, positive to the north and positive directed against gravity. ${\displaystyle \varphi _{0}}$
• ${\displaystyle u,v\,}$: the horizontal components of the velocity vector in the direction of the x and y axes.
• ${\displaystyle \eta }$: the deflection of the sea surface from its resting position.
• ${\displaystyle f=2\Omega \sin \varphi }$, the Coriolis parameter .

In order to take the spatial change of the Coriolis parameter into account, it must be developed into a Taylor series around the reference width when using a Cartesian coordinate system , which is broken off after the linear term ${\displaystyle \varphi _{0}}$

${\displaystyle f(y)=f(\varphi _{0})+{\frac {2\Omega \cos \varphi _{0}}{R}}y=f(\varphi _{0})+\beta y}$.

Here is the radius of the sphere and the beta parameter, which is equal to the meridional gradient of the Coriolis parameter in the reference width. The following derivations are always based on the linear dependence of the Coriolis parameter on the y-coordinate. ${\displaystyle R\,}$ ${\displaystyle \beta ={\frac {2\Omega \cos \varphi _{0}}{R}}}$

For the continuity equation of the fluid considered incompressible, we get

${\displaystyle {\frac {\partial \eta }{\partial t}}+H\left({\frac {\partial u}{\partial x}}+{\frac {\partial v}{\partial y}}\right)=0}$,

To obtain an equation for the deflection of the sea surface, the divergence of the horizontal components of the momentum is formed taking into account the meridional variation of and the continuity equation is used ${\displaystyle f(y)}$

${\displaystyle {\frac {\partial ^{2}\eta }{\partial t^{2}}}-c^{2}\left({\frac {\partial ^{2}\eta }{\partial x^{2}}}+{\frac {\partial ^{2}\eta }{\partial y^{2}}}\right)+fH\zeta -\beta Hu=0}$,

where is the phase velocity of a long wave on the non-rotating earth and ${\displaystyle c^{2}=gH}$

${\displaystyle \zeta ={\frac {\partial v}{\partial x}}-{\frac {\partial u}{\partial y}}}$,

the vertical component of the rotation of the velocity field.

In the case of a rotating liquid, the above equation suggests taking into account the change in rotation of the horizontal velocity field. For this purpose, we form the rotation of the momentum equations from which the equation for the time change of the vertical component of the rotation of the velocity, viz

${\displaystyle {\frac {\partial \zeta }{\partial t}}+f\left({\frac {\partial u}{\partial x}}+{\frac {\partial v}{\partial y}}\right)+\beta v=0}$,

results. This means that the change in time on the rotating earth is equal to the negative divergence of the horizontal movement, expanded by a proportion proportional to the southward movement. If the continuity equation is used to eliminate the horizontal divergence, the result is ${\displaystyle {\tfrac {\zeta }{f}}}$

${\displaystyle {\frac {\partial }{\partial t}}\left({\frac {\zeta }{f}}-{\frac {\eta }{H}}\right)+{\frac {\beta }{f}}v=0}$.

This equation is the linearized form of the equation for the conservation of the potential vorticity of a homogeneous liquid on a spinning sphere. It can be summarized in the following generalized form

${\displaystyle {\frac {\mathrm {d} }{\mathrm {d} t}}{\frac {f+\zeta }{H+\eta }}=0}$

It expresses that the potential vorticity retains its initial value at every point at all times. The linearized form of the conservation of the linear vorticity is obtained if and is assumed. While the first assumption is valid almost everywhere in the ocean, the second assumption is only valid if and thus is, i.e. H. the geostrophic velocity is small compared to the phase velocity of the long wave on the non-rotating earth. This is certainly always the case for barotropic Rossby waves, but not for baroclinic Rossby waves in the area of ​​western edge currents, e.g. B. the Gulf Stream . ${\displaystyle \eta \ll H}$ ${\displaystyle \zeta \ll f}$ ${\displaystyle {\frac {u}{R_{R}}}\ll f}$ ${\displaystyle {\frac {u}{c}}\ll 1}$

If one derives the equation for the deflection of the sea surface again according to the time, the result is

${\displaystyle \left({\frac {\partial ^{2}}{\partial t^{2}}}+f^{2}\right){\frac {\partial \eta }{\partial t}}-c^{2}{\frac {\partial }{\partial t}}\left({\frac {\partial ^{2}\eta }{\partial x^{2}}}+{\frac {\partial ^{2}\eta }{\partial y^{2}}}\right)-H\beta \left(fv+{\frac {\partial u}{\partial t}}\right)=0}$.

We now consider sub-inertial motions that are geostrophically adapted under the radiation of Poincaré waves . Due to the following neglect, the Poincaré waves are filtered out of the equations of motion for the liquid of the ocean. The following approximations then apply:

${\displaystyle {\frac {\partial ^{2}}{\partial t^{2}}}\ll f^{2}}$,

${\displaystyle fv=g{\frac {\partial \eta }{\partial x}}}$

O {u} = O {v}.

This gives us an equation for the displacement of the sea surface by sub-inertial movements on the rotating earth, namely by Rossby waves,

${\displaystyle f^{2}{\frac {\partial \eta }{\partial t}}-c^{2}{\frac {\partial }{\partial t}}\left({\frac {\partial ^{2}\eta }{\partial x^{2}}}+{\frac {\partial ^{2}\eta }{\partial y^{2}}}\right)-\beta c^{2}{\frac {\partial \eta }{\partial x}}=0}$.

The dispersion relation of the Rossby waves

If one takes a deflection of the sea surface in the form of a horizontally propagating wave

${\displaystyle \eta =\eta _{0}e^{i(\omega t-kx-ly)}}$

and inserts this form into the equation of motion for the Rossby wave, the dispersion relation for the Rossby wave results in

${\displaystyle \omega =-\beta R_{R}{\frac {kR_{R}}{1+(k^{2}+l^{2})R_{R}^{2}}}}$

There is a barotropic and an integral multiple of baroclinic Rossby radii , which are given by the respective phase velocities of the corresponding long wave on the non-rotating earth and the Coriolis parameter. For the oceans, the Rossby barotropic radius is on the order of 2000 km. Current maps of the global distribution of the first baroclinic Rossby radius can be found in Chelton et al. (1998); is some 10 km in mid-latitudes. The barotropic mode of the Rossby wave spreads many meters per second, so that it crosses a typical ocean basin in a few weeks. However, the slower baroclinic modes are important to the dynamics of the ocean. They spread at speeds in the order of magnitude of 1 - 10 cm / s and take a longer time (years) to cross an ocean basin. ${\displaystyle R_{\mathrm {R} }={\frac {c}{f}}}$ ${\displaystyle R_{R0}}$ ${\displaystyle R_{R1}}$

The particle velocity in Rossby waves

The velocity field associated with the Rossby wave results to a good approximation from the quasi-geostrophic equations

${\displaystyle u={\frac {gl}{f}}\eta \left(\omega t-kx-ly+{\frac {\pi }{2}}\right)}$

and

${\displaystyle v=-{\frac {gk}{f}}\eta \left(\omega t-kx-ly+{\frac {\pi }{2}}\right)}$.

Due to the geostrophic adaptation of the Rossby waves, the current is directed parallel to the wave crests and valleys. The small ageostrophic proportions of the particle velocities of the Rossby waves result from the latitude dependence of the Coriolis parameter in such a way that the velocities towards the equator are higher than towards the poles. This leads to a convergence west of a high pressure ridge and thus to a pressure increase there with the consequence of a westward shift of the wave pattern.

The planetary divergence of the Rossby waves

If we calculate the divergence for a geostrophically adapted liquid on a rotating sphere and insert the result into the continuity equation, we get the continuity equation

${\displaystyle {\frac {\partial \eta }{\partial t}}-{\frac {\beta }{f^{2}}}c^{2}{\frac {\partial \eta }{\partial x}}=0}$.

This means that the divergence of a quasi-geostrophic fluid on a rotating sphere does not generally disappear and thus results in a change in pressure over time, which causes another wave movement, namely the planetary or Rossby waves. From the above equation it also follows that there are two special cases for which the divergence of the quasi-geostrophic motion on a rotating sphere vanishes. One case applies to the poles where is. The other case applies to pressure fields that do not have a zonal gradient. ${\displaystyle \beta =0}$

The potential and kinetic energy of the Rossby waves

The potential energy density of the Rossby wave is given by the corresponding expression for the shallow water wave, viz ${\displaystyle E_{\mathrm {p} }\,}$

${\displaystyle E_{\mathrm {p} }={\frac {\rho }{2}}g{\bar {\eta ^{2}}}}$.

The dash denotes the mean value over a wavelength. The kinetic energy density of the wave results from the integration of the local kinetic energy over the entire water column, i.e. ${\displaystyle E_{\mathrm {k} }}$

${\displaystyle E_{\mathrm {k} }={\frac {\rho }{2}}H\left({\bar {u^{2}}}+{\bar {v^{2}}}\right)={\frac {\rho }{2f^{2}}}g^{2}H\left(k^{2}+l^{2}\right){\bar {\eta ^{2}}}}$.

The ratio of kinetic to potential energy density is

${\displaystyle {\frac {E_{k}}{E_{p}}}={\frac {gH}{f^{2}}}\left(k^{2}+l^{2}\right)=\kappa _{H}^{2}R_{R}^{2}}$.

Here is the horizontal wavenumber. It follows from this that the potential energy density is much greater than the kinetic one for long Rossby waves, the wavelengths of which are much greater than the Rossby radius. Both energy densities are the same for Rossby waves with the maximum frequency and the kinetic energy density is higher than the potential for Rossby waves with wavelengths much smaller than the Rossby radius. ${\displaystyle \kappa _{H}}$

The frequency range of the Rossby waves

It follows from the dispersion relation of the Rossby waves that they are generally dispersive. A large number of its properties can be derived from it.

Dispersion relationship of the first modes of a baroclinic Rossby wave at 30 ° latitude.

Since the frequency in the dispersion relationship depends on the square of the meridional wave number l, phase propagation of the Rossby waves is possible both to the north and to the south. The linear dependence of the frequency on the zonal wave number k of the Rossby wave, on the other hand, only allows phase propagation in a westerly direction. Generally speaking, phase propagation is only possible in the western half-space.

Minimal period of the first baroclinic mode of a Rossby wave of the middle latitudes.

The dispersion relation also states that Rossby waves have a maximum frequency for the wavenumber vector and . The maximum frequency of the Rossby wave is for this wave number vector ${\displaystyle l_{\mathrm {max} }=0}$ ${\displaystyle k_{\mathrm {max} }=-{\frac {1}{R_{R}}}}$

${\displaystyle \omega _{\mathrm {max} }={\frac {1}{2}}\beta R_{\mathrm {R} }={\frac {1}{2}}\beta {\frac {c}{f}}}$.

It decreases with increasing width towards the poles. The spectral gap between Poincaré waves and Rossby waves increases in the direction of the poles. Alternatively, one can also say that a Rossby wave with a given frequency or period is assigned an inverse width, so that it can no longer exist towards the pole of this width. The adjacent figure shows that a given period can only exist equatorially towards a maximum width. For example, a baroclinic Rossby wave with a period of one year can only exist towards the equator of approximately 45 ° latitude.

The group speed of the Rossby waves

In contrast to the phase velocity , i.e. the velocity of a wave crest, which is only a few centimeters high on the water surface, but in which thermoclines are usually several meters, the group velocity , i.e. the direction of propagation of wave packets and thus of the energy transport, is possible in any direction . Typical speeds are on the order of a few centimeters per second. The meridional components of group and phase velocities are always opposite. Whether a packet of Rossby waves propagates east or west depends on their wavelengths . Short wavelengths, i.e. H. spread to the east, whereas long wavelengths, i.e. H. have a westward energy transport. The group velocity has two maxima for a given meridional wave number . For is one of the maxima and is . Since the dispersion relation for this wave number combination is dispersion-free, long Rossby waves propagate dispersion-free with the maximum group velocity to the west. The second maximum of the group speed is at the wave number and and is . From a spatially isolated quasi-geostrophic pressure disturbance in the form of a wave packet, a front of dispersion-free long Rossby waves spreads to the west with the maximum group speed , while a second, dispersive front of short Rossby waves spreads to the east with one eighth the group speed of the long waves . Between these two fronts there remains a Rossby wave with vanishing group velocity, which has the wavelength and frequency given above . ${\displaystyle k^{2}+l^{2}>R_{\mathrm {R} }^{-2}}$ ${\displaystyle k^{2}+l^{2}<R_{R}^{-2}}$ ${\displaystyle l\,}$ ${\displaystyle l=0\,}$ ${\displaystyle k=0\,}$ ${\displaystyle c_{\mathrm {gr,max,1} }=-\beta R_{\mathrm {R} }^{2}}$ ${\displaystyle l=0\,}$ ${\displaystyle k^{2}R_{\mathrm {R} }^{2}=3}$ ${\displaystyle c_{\mathrm {gr,max,2} }={\frac {\beta }{8}}R_{\mathrm {R} }^{2}}$ ${\displaystyle c_{\mathrm {gr,max,1} }=-\beta R_{\mathrm {R} }^{2}}$ ${\displaystyle k_{\mathrm {max} }}$ ${\displaystyle \omega _{\mathrm {max} }}$

Propagation time of the front of the first fashions of baroclinic long Rossby waves over the distance of a Rossby_Radius as a function of the geographical latitude.

If a pressure trough or ridge with the characteristic width of a Rossby radius exists on the east bank of an ocean, as is typical for Kelvin waves or coastal jet currents , a front of long Rossby waves propagates westward out into the open ocean at the maximum group velocity. When the front has penetrated beyond a Rossby radius into the ocean, it begins to broaden the pressure pattern to the west, reducing the geostrophically adjusted speed in the coastal boundary layer. The characteristic time for traversing the coastal zone is . The dependence of this characteristic time on the geographical latitude is shown in the adjacent figure and, apart from the proportionality factor, is equal to that of the minimum period of the Rossby wave. While the characteristic time in tropical latitudes is only a few days, it is around three weeks in subtropical latitudes and two months in sub-polar latitudes. ${\displaystyle T_{R}={\frac {R_{\mathrm {R} }}{c_{\mathrm {gr,max,1} }}}={\frac {1}{\beta R_{\mathrm {R} }}}}$ ${\displaystyle 4\pi }$

A pressure disturbance on a western shore of the ocean remains unaffected by Rossby waves for an 8 times longer time, since the maximum group speed directed to the east is correspondingly slower. The Rossby waves thus lead to an east-west asymmetry in the dynamic reactions of an ocean.

Rossby wave and stationary ocean circulation

Rossby waves play an essential role in establishing a steady state of wind-driven ocean circulation . The atmospheric circulation on the sea surface generates a pressure disturbance in the form of high pressure ridges or low pressure troughs that increases over time via the Ekman transport . On the east bank of the ocean, these trigger a Rossby wave front propagating to the west at maximum group speed. Behind the wave front, the dynamics switch from an Ekman balance to a Sverdrup balance, i.e. H. as long as the divergence of the Ekman transport is compensated by buoyancy or downwelling, the pressure disturbance increases. Behind the Rossby wavefront, the divergence of the Ekman transport is balanced out by the planetary divergence of the meridional component of the ocean circulation and a steady state, the Sverdrup regime, is established. The characteristic time for the establishment of a stationary ocean circulation is thus the time it takes for the Rossby wave front to travel from the east bank of the ocean to any point in the ocean. It grows linearly from the east to the west bank, which means that the wind-induced pressure disturbances in the western part of the ocean can grow longer until they are more or less fixed by the arrival of the Rossby wave front.

This explains the asymmetry observed in the oceans between the slow and broad east rim and the narrow and intense west rim currents of the oceanic eddies that make up the various branches of ocean circulation. The complete response time for the cessation of stationary circulation is the propagation time of the Rossby wavefront from the east to the west bank, which is on the order of months in the equatorial latitudes and one to several years in subtropical and higher latitudes.

Observations of oceanic Rossby waves

For many decades, oceanographers have found themselves in the difficult position of having an accepted theory of Rossby waves but no direct observation of this important phenomenon. The spatial and temporal scales inherent in the waves made in-situ observation of the waves difficult with the measurement technology available at the time. Emery and Magaard (1976) and White (1977) obtained the first evidence of the existence of baroclinic planetary waves in the ocean by measuring the variations in the depth of the isotherms in the interior of the ocean, which are in the order of 10 m. However, the remaining restrictions in the spatial and temporal sampling of the propagating wave patterns could not yet provide the required characteristics of the wave patterns and their propagation.

The use of altimeters on orbiting satellites as a carrier platform made it possible to observe detailed properties of the Rossby waves by measuring their signature on the surface of the sea. Altimeter measurements must meet the following requirements to determine the properties of Rossby waves:

• The accuracy of the measurement of the deflection of the sea surface must be sufficient to record a signal of a few cm.
• The length of the time series and the pattern of spatial and temporal sampling of the deflection of the sea surface must correspond to the characteristic spatial and temporal variations of the Rossby waves.
• From the data set obtained, it must be possible to identify and eliminate other causes of the deflection of the sea surface due to its spatial and temporal patterns that differ from the planetary waves. This in turn requires such scanning patterns that no scanning errors are projected into the scale range of the planetary waves.

The start of TOPEX / Poseidon (T / P) in 1992 marked the beginning of a new era in the observation of planetary waves from space. The pattern of its Earth orbits, which repeat itself exactly every 10 days, was specially designed to avoid tidal aliasing in the scale range of the Rossby waves. First investigations based on T / P measurements identified Rossby waves in various regions of the world ocean. The extensive study by Chelton and Schlax (1996) showed the ubiquity of Rossby waves and proved that they tend to propagate faster in mid-latitudes than is predicted by linear theory. The TOPEX / Poseidon mission was continued by the subsequent Jason 1 and Jason 2 missions in 2001 and 2008, respectively.

The theory of planetary waves has been expanded to include the background baroclinic current and the variation in the bottom topography of the ocean. The propagation speeds of the planetary waves predicted by the extended theory are largely in agreement with the observations.

In addition to altimeters, the signatures of oceanic Rossby waves have also been detected by measuring the sea surface temperature (SST) from satellites. The thermal signature of Rossby waves is not as direct a representation of the wave properties as that of the displacement of the sea surface. Nevertheless, it is important because it determines the temporal and spatial scales of the thermal interaction between ocean and atmosphere, which in turn is important for the variation of the climate.

In measurements of the oceanic distribution of chlorophyll-a concentration, the satellite found patterns that correspond to those of planetary waves. This suggests that planetary waves can influence the dynamics of marine ecosystems.

The thermal and ecological effect of the planetary waves can on the one hand by the advection of the corresponding meridional gradients by means of the particle speed of the Rossby waves, on the other hand by the effect of the corresponding vertical flows of heat, light and nutrients on the properties of the surface layer, whose thickness is determined by the dynamics of Rossby Waves is destined to take place.

Atmospheric Rossby waves

In the overall picture of the planetary circulation of air masses of the Earth's atmosphere Rossby waves are the meandering course of the Polarfrontjetstreams along the air mass boundary between the cold polar air of the polar cell and the warm subtropical air the Ferrel cell on the North - and to a lesser extent also on the southern hemisphere of the earth observable .

Rossby waves in the jet stream:
a, b: Onset of wave formation
c: Beginning separation of a cold air drop
blue / orange: cold / warm air masses

Jet streams arise as a result of global compensatory movements between different temperature regimes or high and low pressure areas . Due to the irregular thermal gradient, the air mass boundary between warm subtropical and cold polar air is not straight, but meandering. The resulting undulating air mass limit is called the Rossby wave and is shown in the adjacent figure. The folding of the polar front jet stream is inconsistent in reality and does not wind continuously around the entire hemisphere. A current picture of the meandering jet stream bands can be viewed in the web links.

The jet stream also pulls the lower layers of air with it, with dynamic low pressure areas (cyclones) in the direction of the pole (twisted counterclockwise above the 'wave troughs', so-called troughs ) and in the direction of the equator high pressure areas ( twisted clockwise below ) , depending on the swirl of the Rossby wave the 'wave crests', so-called ridges ). These low pressure areas, such as the Icelandic Depression, play a major role in the Central European weather, as their front systems lead to a characteristic change in weather .

Since these turbulences are mainly caused by continental obstacles and these are much more pronounced in the northern hemisphere than in the southern hemisphere, this effect and thus also the Rossby waves are much stronger in the northern hemisphere.

"A subtle resonance mechanism that holds waves in the middle latitudes and significantly amplifies them," became the cause of the increase in the number of extreme weather events in summer since the turn of the millennium - such as the record heatwave of 2010 in Eastern Europe, which resulted in crop losses and devastating forest fires around Moscow - connected.

literature

• Chelton, DB & Schlax, MG (1996): Global observations of oceanic Rossby waves . Science, 272: 234-238.
• Chelton, DB, Szoeke, RA de, Schlax, MG, Naggar, KE & Siwertz, N. (1998): Geographical variability of the first baroclinic Rossby radius of deformation . Journal of Physical Oceanography, 28: 433-460.
• Cipollini, P., Cromwell, D., Challenor, PG & Raffaglio, S. (2001): Rossby waves detected in global ocean color data . Geophysical Research Letters, 28: 323-326.
• Emery, W. and Magaard, L. (1976): Baroclinic Rossby waves as inferred from temperature fluctuation in the eastern Pacific . Journal of Marine Research, 34: 365-385.
• Gill, AE (1982): Atmosphere-Ocean Dynamics . Academic Press, San Diego.
• Hill, KL, Robinson, IS & Cipollini, P. (2000): Propagation characteristics of extratropical planetary waves observed in the ATSR global sea surface temperature record . Journal of Geophysical Research - Oceans, 105 (C9): 21927-21945.
• Hough, S. (1897): On the application of harmonic analysis to the dynamical theory of the tides, Part I. On Laplace's oscillations of the first species, and on the dynamics of ocean currents. Philos. Trans. Roy. Soc. , A, 189: 201-257.
• Killworth, PD, Chelton DB & Szoeke, R. de (1997): The speed of observed and theoretical long extra-tropical planetary waves . J. Phys. Oceanogr., 27: 1946-1966.
• Killworth, PD and JR Blundell (2003a): Long extra-tropical planetary wave propagation in the presence of slowly varying mean flow and bottom topography . I: the local problem. J. Phys. Oceanogr., 33: 784-801.
• Killworth, PD and JR Blundell (2003b): Long extra-tropical planetary wave propagation in the presence of slowly varying mean flow and bottom topography . II: ray propagation and comparison with observations, J. Phys. Oceanogr., 33: 802-821.
• Rossby, CG et al. (1939): Relations between variations in the intensity of the zonal circulation of the atmosphere and the displacements of the semi-permanent centers of action . In: Journal of Marine Research 2: 38-55.
• Rossby, CG (1940): Planetary flow patterns in the atmosphere . In: Quarterly Journal of the Royal Meteorological Society 66: 68-87.
• Harald Ulrik Sverdrup (1947): Wind-driven currents in a baroclinic ocean: with application to the equatorial currents of the eastern pacific . Proceedings of the National Academy of Sciences 33 (11): 318-326.
• White, WB (1977): Annual forcing of baroclinic long waves in the tropical North Pacific Ocean . Journal of Physical Oceanography, 7: 50-61.

Web links

Commons : Rossby Waves  - Collection of Images, Videos and Audio Files

• Graphic "current JetStreams", Lyndon State College (English)
• astronews.com: Huge eddies discovered on the Sun May 8, 2018

Individual evidence

1. ↑ B. Gill (1982)
2. ↑ | Sverdrup (1947)
3. ↑ Killworth et al., 1997; Killworth and Blundell, 2003.
4. ↑ Hill et al., 2000.
5. ↑ Cipollini et al., 2001.
6. ↑ More extreme weather due to the rocking of huge waves in the atmosphere. Potsdam Institute for Climate Impact Research, press release from August 11, 2014
7. ↑ Dim Coumou et al .: Quasi-resonant circulation regimes and hemispheric synchronization of extreme weather in boreal summer. In: Proceedings of the National Academy of Sciences . Volume 111, No. 34, 2014, pp. 12331-12336, doi: 10.1073 / pnas.1412797111