Rossby wave

Rossby waves, referred to as planetary waves are large-scale wave motions in the ocean or the atmosphere.

The planetary waves the underlying physical principle is the conservation of potential vorticity. When a liquid particles in the air or water on the surface of a rotating ball is moved parallel to the axis of rotation, it is subjected to the Coriolis force, the parameters depends on the 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 an oscillation with a meridional westward phase velocity.

Hough formulated in 1897 as first the equations for the motion of planetary waves on a rotating sphere and discussed the solutions in spherical coordinates. Rossby et al. (1939) and Rossby (1940 ) approximated the problem in Cartesian coordinates on a beta- plane.

Oceanic Rossby waves

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

You are encouraged within the ocean basins by spatial variations of the surface wind and air pressure variations at the sea surface or radiated from the meridonal running coasts in response to temporal variations of wind and atmospheric pressure fields. Long Rossby waves are radiated from the eastern shores and short Rossby waves from the western shores. Due to their runtime by the ocean basin from east to west, they determine the characteristic response time for setting a steady state Ocean Circulation according to temporal changes of the driving wind pattern.

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

Mathematical Description

Rossby waves are subinertiale movements that are after the geostrophic adjustment on a rotating sphere in a quasigeostrophic balance. Its special feature is that the spatial extent of the associated pressure pattern and the spatial variation of the Coriolisparameters the divergence of the quasi- geostrophic flow does not exactly disappear. This has a slow time variation of the pressure field in the form of a Rossby wave result.

We consider the properties of the linear Rossby wave in an infinite, inviscid ocean with a Great Plains soil at depth on the rotating with the angular velocity of the earth, see, eg, Gill ( 1982).

Are the vertically averaged equations for the horizontal velocity components of the hydrostatic liquid

In the equations are:

• : Time
• : The coordinates of a rectangular coordinate system with the origin at the sea level at the geographical reference width, directed counter to the force of gravity, for example, positive to the east, north positive and positive.
• The horizontal components of the velocity vector in the direction of x-and y -axis.
• : The displacement of the sea surface from the rest position.
• , The Coriolis parameter.

To account for the spatial variation of the Coriolisparameters, it must be developed into a Taylor series about the reference width by using a Cartesian coordinate system, which is terminated after the linear member

Here, the radius of the sphere and the parameter beta, which is equal to the gradient of the meridional Coriolisparameters in the reference width. In the following derivations, the linear dependence of the Coriolisparameters of the y-coordinate is always used.

For the continuity equation of the respected as incompressible fluid, we obtain

To obtain an equation for the deflection of the sea surface, the divergence of the horizontal components of the pulse is formed in consideration of the meridional variation of the continuity equation and used

Where the phase speed of a long wave on the non-rotating earth is and

The vertical component of the rotational speed of the field.

In the case of a rotating liquid, the above equation suggests to consider the change of the rotational speed of the horizontal field. For this purpose we make the rotation of the momentum equations resulting in the equation for the variation of the vertical component of the rotational speed, namely

Results. This means that the temporal change in the rotating earth is equal to the negative of the horizontal divergence movement, extended by a portion proportional to the movement southward. If one uses the continuity equation to eliminate the horizontal divergence, we obtain

This equation is the linearized form of the equation for the conservation of potential vorticity a homogeneous fluid on a rotating sphere. They can be made to the following generalized form

It expresses that the potential vorticity retains its initial value at any point at all times. The linearized form of the conservation of linear vorticity is obtained if and adopted. While the first assumption is true almost everywhere in the ocean, the second assumption is only valid if and thus, that is the geostrophic velocity is small compared to the phase velocity of the long wave on the non-rotating earth. This is for barotropic Rossby waves certainly always the case, but not for baroclinic Rossby waves at the western edge of streams, such as the Gulf Stream.

Directs the equation for the deflection of the sea again after time off, resulting

We now consider subinertiale movements are adjusted geostrophic under radiation of Poincare waves. By the following omissions the Poincare waves from the equations of motion for the fluid to be filtered out of the ocean. Then apply the following approximations:

Thus we obtain an equation for the displacement of the sea surface by subinertiale movements on the rotating earth, namely, by Rossby waves,

The dispersion relation of Rossby waves

Assuming a deflection of the sea surface in the form of a horizontally propagating wave

And sets this in the form of the equation of motion for the Rossby wave, the result is the dispersion relation for the Rossby shaft

There is a barotropic and an integer multiple of baroclinic Rossby radii are given by the respective phase velocities of the respective long shaft on the non-rotating and the Coriolis earth. For the oceans is the barotropic Rossby radius of the order of 2000 km. Current maps of the global distribution of the first baroclinic Rossby radius is found in Chelton et al. (1998); is some 10 km in mid-latitudes. The barotropic mode of the Rossby wave propagates from many meters per second, so that it passes through a typical ocean basin in a few weeks. However, the slower baroclinic modes are important for the dynamics of the ocean. They propagate at speeds of the order of 1 - 10 cm / s and have to satisfy a long time ( years) to traverse an ocean basin.

The particle velocity in Rossby waves

The velocity field associated with the Rossby wave results in a good approximation of the quasigeostrophic equations

And

Due to the geostrophic adjustment of Rossby waves, the flow is directed parallel to the wave crests and valleys. The small ageostrophic portions of the particle velocities of the Rossby waves arising from the width dependence of Coriolisparameters such that the velocities are equatorward higher than poleward. This results in a convergence of a west high pressure ridge and thus to a local pressure increase, resulting in a displacement of the shaft westwärtigen pattern.

The divergence of the planetary Rossby waves

If we calculate the divergence of a geostrophic customized fluid on a rotating sphere and put the result in the continuity equation, we obtain the continuity equation

This means that the divergence of a quasi- fluid to a rotating ball geostrophic generally does not disappear, and thus has a time variation of the pressure results in causing a further movement of the shaft, that the planetary or Rossby waves. From the above equation it also follows that there are two special cases for which disappears the divergence of the quasi- geostrophic motion on a rotating sphere. One case applies to the poles, is where. The other case is for the print fields which have no zonal gradient.

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, namely

The horizontal line denotes the average over a wavelength. The kinetic energy density of the wave resulting from the integration of the local over the entire kinetic energy of the water column, so

The ratio of kinetic to potential energy density is

Here is the horizontal wave number. It follows that the potential energy density is much larger than the kinetic long Rossby waves whose wavelengths are much larger than the Rossbyradius. Both power 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 substantially shorter wavelengths than the Rossbyradius.

The frequency range of Rossby waves

From the dispersion relation of Rossby waves follows that they are dispersive in general. From it many of its properties can be derived.

Since the frequency is dependent on the dispersion relation of the square of the meridional wavenumber l a Phasenpropagation the Rossby waves both to the north and to the south is possible. The linear dependence of the frequency of the zonal wave number k of the Rossby wave, however, permitted only a Phasenpropagation to the west. Generally speaking thus a Phasenpropagation possible only in the western hemisphere.

The dispersion relation says about it in addition that Rossby waves have a maximum frequency for the wave number vector and. The maximum

Frequency of Rossby wave is for this wave vector

It decreases with increasing width towards the poles. The spectral gap between Poincare waves and Rossby waves becomes greater say in the direction of the poles. You can also say, alternatively, that a Rossby wave with a given frequency or period is associated with a reversal width, so that they can no longer exist poleward of this latitude. The figure shows that for a given period can only exist equatorward maximum width. Thus, a baroclinic Rossby wave with the period of one year only exist equatorward of approximately 45 ° latitude.

The group velocity of the Rossby waves

In contrast to the phase velocity, ie the speed of a wave crest, which only a few centimeters high at the water surface, in thermoclines, however, usually has several meters, is the group velocity, ie the direction of propagation of wave packets and thus the transport of energy, possible in any direction. Typical speeds are in the order of a few centimeters per second. The Meridionalkomponenten of group and phase velocity are always opposite. Whether a packet of Rossby waves propagate eastward or westward, depends on their wavelengths. Short wavelengths, that is spread out to the east, whereas large wavelengths, ie have a westwärtigen energy transport. The group velocity has two maxima for a given meridional wavenumber. For is one of the maxima and amounts. Since the dispersion relation is free of dispersion for this wave-number combination, spread long Rossby waves without dispersion from the maximum group velocity to the west. The second maximum of the group velocity is the wave number and amounts. From a spatially isolated quasigeostrophic pressure disturbance in the form of a wave packet, therefore propagates a front dispersion- free long Rossby waves with the maximum group velocity to the west, while a second dispersive front short Rossby waves propagating with an eighth of the group velocity of the long waves to the east. Between the two fronts of a vanishing group Rossby shaft speed remains, which has the wavelength and frequency given above.

If a pressure trough or back with the characteristic width of a Rossby radius on the eastern shore of an ocean exists, as it is typical for Kelvin waves and coastal jet streams, propagates a front long Rossby waves with the maximum group velocity westward into the open ocean beyond. If the front has advanced a Rossbyradius out into the ocean, it begins the print pattern to widen to the west which the geostrophic adjust speed decreases in the coastal boundary layer. The characteristic time for the passage of the coastal zone. The dependence of the characteristic time of the latitude is shown in the adjacent figure and is up to the proportionality factor equal to the minimum period of the Rossby wave. While in tropical latitudes, the characteristic time is only a few days, it is located in subtropical latitudes at around three weeks, and in subpolar latitudes in two months.

A pressure disturbance on a western shore of the ocean remains unaffected over an 8- times longer time of Rossby waves, as directed eastward maximum group velocity is correspondingly slower. The Rossby waves thus lead to an east-west asymmetry in the dynamic responses of an ocean.

Rossby wave and stationary ocean circulation

Rossby waves play an essential role in the establishment of a steady state of the thread -driven ocean circulation. The atmospheric circulation at the ocean surface generated by the Ekman transport an increasing with time pressure disturbance in the form of ridge of high pressure or low pressure troughs. These dissolve on the eastern shore of the ocean from a propagating with the maximum group velocity westward Rossby wave front. Behind the wave front switches the dynamics of an Ekman balance on a Sverdrup balance by, Sverdrup (1947 ), ie as long as the divergence of the Ekman transport is balanced by buoyancy or Down Welling, the pressure disturbance increases. Behind the Rossby wave front, the divergence of the Ekman transport through the planetary divergence of the meridional component of the ocean circulation is balanced and thus arises a steady state, the Sverdrup regime, a. The characteristic time for adjusting a stationary ocean circulation is thus the time required for the Rossby wave front to travel from the eastern shore of the ocean up to any point in the ocean. It grows linearly from the East to the West Bank, thereby excited by the wind pressure disturbances in the western part of the ocean can grow longer until they are fixed by the arrival of the Rossby wave front is almost. This explains the observed asymmetry in the oceans between the slow and wide and the eastern edge of the narrow and intense western boundary flow of oceanic vortices that form the various branches of the ocean circulation. The complete reaction time of hiring a stationary circulation is the propagation time of the Rossby wave front from the eastern to the western shore, which is in the equatorial latitudes of the order of months in subtropical and higher latitudes one to several years.

Observations of oceanic Rossby waves

Over many decades, oceanographers were in the difficult position of an accepted theory of Rossby waves, but have no direct confirmation of this important phenomenon by observation. The waves inherent spatial and temporal scales complicated the in-situ observation of the waves with the then- available measurement technology. The first evidence of the existence of baroclinic planetary waves in the ocean reach Emery and Magaard (1976) and White ( 1977) by measuring the variations of the depth of the isotherms in the interior of the ocean, which are on the order of 10 m. However, the remaining limitations in the spatial and temporal sampling of the propagating wave pattern could not yet provide the necessary characteristics of the wave pattern and its spread.

The use of altimeters on orbiting satellites as a support platform made ​​it possible to observe detailed properties of Rossby waves by measuring their signature at the sea surface. Altimetermessungen must have the following requirements in order to determine the properties of Rossby waves:

• The accuracy of the measurement of the displacement of the sea surface must be sufficient to detect a signal of a few cm.

• The length of the time series and the patterns of spatial and temporal sampling of the displacement of the sea surface has the characteristic spatial and temporal variations of Rossby waves correspond.

• From the obtained data other causes of displacement of the sea surface through its different to the planetary waves spatial and temporal patterns need to be identified and eliminated. This in turn requires such a scan pattern that no sampling error are projected in the scale range of planetary waves into it.

The launch 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 his to every 10 days exactly repeating earth orbit was specifically designed to avoid aliasing caused by tides in the scale range of the Rossby waves. First identified on T / P measurement-based studies Rossby waves in various regions of the world ocean. This comprehensive study of Chelton and Schlax (1996 ) showed the ubiquity of Rossby waves and proved that they tend to propagate faster in the mid-latitudes as it is predicted by the line rare theory. The TOPEX / Poseidon mission was continued by the subsequent Jason -1 and Jason -2 missions in 2001 and 2008 respectively.

Killworth et al. (1997) and Killworth and Blundell, 2003a, b extended the theory of planetary waves by including the baroclinic background flow and the variation of the bottom topography of the ocean. The predicted by the extended theory of propagation velocities of the planetary waves are broadly consistent with the observations.

Except altimeters signatures oceanic Rossby were detected from satellites, Hill et al and by the measurement of the sea surface temperature (SST). (2000). The thermal signature of Rossby waves is not such a direct representation of the wave properties such as the displacement of the sea surface. Nevertheless, it is significant because it determines temporal and spatial scales of the thermal interaction between ocean and atmosphere, which in turn is important for the variation of the climate.

Recently, the oceanic distribution of chlorophyll -a concentration were found in measurements from the satellite patterns that correspond to those of planetary waves, Cipollini et al. (2001). This suggests that planetary waves can have an impact on the dynamics of marine ecosystems.

The thermal and environmental effect of the planetary shafts on the one hand by the advection of the corresponding meridional gradient by means of particle velocity of the Rossby waves, on the other hand by the effect of the corresponding vertical flows of heat, light, and nutrients to the properties of the outer layer, the thickness of the dynamic Rossby waves is determined, take place.

Atmospheric Rossby waves

In the overall image of the planetary circulation of air masses in the earth's atmosphere are Rossby waves as meandering course of the Polarfrontjetstreams along the air mass boundary between the cold polar air of the polar cell and the warm Subtropenluft the Ferrel cell observable on the north and to a lesser extent also in the southern hemisphere of the earth.

Jet streams occur as a result of global compensation movements between different temperature regimes or high and low pressure areas. Due to irregular thermal gradient, the air mass boundary between warm subtropical and cold polar air is not straightforward, but meanders. The resulting wave-like air mass limit is called the Rossby wave and is shown in the adjacent figure. The folding of the Polarfrontjetstreams is not uniform in reality and also winds up not continuous around the entire hemisphere. A current picture of the meandering jet stream bands is visible in the web links.

The Jetstream also rips the lower atmospheric layers, where according to the turbulence of Rossby wave always dynamic low-pressure areas ( cyclones ) in the direction of Pol (counterclockwise rotated beyond the ' troughs ', so-called troughs ) and towards the equator anticyclones ( rotated clockwise among the ' crests ', veer called back). This low-pressure areas, such as the Iceland low, are significantly involved in the central European weather as they pass through their front-end systems to a characteristic change in the weather.

Since these vortices are largely brought about by continental barriers and these are much more pronounced in the northern hemisphere than in the southern hemisphere, this effect and hence the Rossby waves is on the northern hemisphere much more.