Equatorial wave

In oceanography and meteorology, equatorial waves are waves that propagate along the equator. The limitation to the vicinity of the equator is caused by the zero crossing of the Coriolis force at the equator. Limits to 5–15 ° north or south latitude are typical.

The waves have wavelengths on the order of 1000 km and are related to the planetary waves that were investigated by Carl-Gustaf Rossby . Rossby was originally interested in the north-south deflection of the jet stream , which he described as a wave .

The same wave equations can also be used in the ocean. Horizontal deflections of the thermocline are considered, i.e. the water layer between the warm surface water and the cold deep water. The amplitude of the change in depth of the thermocline is in the order of 10–50 m, while the surface amplitude is only a few centimeters. The wavelengths are 100–1000 km, with a propagation speed of a few cm / s, i. H. it takes several months to spread across the Pacific. Such a shift of a 50 m thick layer of warm water from the South China Sea off the west coast of Central America is one of the attempts to explain the El Niño phenomenon.

To observe the waves, in -situ measurements are used to represent the thermocline by measuring the water depth of the 18 ° isotherm, i.e. the water depth at which the water temperature is 18 ° C. However, satellite measurements are now also accurate enough to resolve the few centimeters of the water surface height (SSH for sea-surface height ). At the same time, other parameters such as temperature (SST for sea-surface temperature ) are also measured.

Among the equatorial waves, a distinction is made between the equatorial Rossby waves and the inertial gravity waves with modes m = 1, 2,… . In addition, Yanai waves occur, which can be understood as m = 0 mode of the same dispersion relations. The equatorial Kelvin waves are described by simplified wave equations.

Wave equations

The equations of motion for the waves are obtained by considering a layer of an incompressible liquid of constant density with a free surface. The continuity equations also contain a contribution from the Coriolis force , which is considered in the so-called -plane model: The meridional dependence of the Coriolis parameter is developed as a Taylor series in the distance to the equator : ${\displaystyle \beta }$ ${\displaystyle y}$

${\displaystyle f_{\text{C}}=2\Omega \sin \varphi +{\frac {2\Omega \cos \varphi }{a}}y+{\mathcal {O}}(y^{2})}$

${\displaystyle \Omega }$is the angular velocity of the earth's rotation, the latitude and the earth's radius. ${\displaystyle \varphi }$ ${\displaystyle a}$

Because of the proximity to the equator, and . We therefore get the wave equations: ${\displaystyle \sin \varphi \approx 0}$ ${\displaystyle \cos \varphi \approx 1}$

${\displaystyle {\frac {\partial u}{\partial t}}-\beta yv=-g{\frac {\partial \zeta }{\partial x}}}$

${\displaystyle {\frac {\partial v}{\partial t}}+\beta yu=-g{\frac {\partial \zeta }{\partial y}}}$

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

Where:

${\displaystyle x}$ zonal coordinate

${\displaystyle y}$ meridional coordinate

${\displaystyle u}$ zonal speed component

${\displaystyle v}$ meridional velocity component

${\displaystyle g}$ Gravity acceleration

${\displaystyle \zeta }$ Surface deflection

${\displaystyle \beta ={\frac {2\Omega }{a}}}$ Rossby parameter, a constant in this approximation

${\displaystyle H}$ "Water depth" or layer thickness.

In the literature, the equations are mostly converted into a dimensionless form. The units of time and length are included

${\displaystyle \left[T\right]={\sqrt {\frac {1}{c\beta }}}\mathrm {\ und\ } \left[L\right]={\sqrt {\frac {c}{\beta }}}}$,

where is the speed of gravity waves in this "water depth". The model is also referred to as a channel model , or if the constant density is to be emphasized as a barotropic model . ${\displaystyle c={\sqrt {gH}}}$

To solve the above wave equations, one uses the - and - dependency of , and as , e.g. B. . After eliminating and results ${\displaystyle x}$ ${\displaystyle t}$ ${\displaystyle u}$ ${\displaystyle v}$ ${\displaystyle \zeta }$ ${\displaystyle e^{i(kx-\omega t)}}$ ${\displaystyle v=e^{i(kx-\omega t)}{\bar {v}}}$ ${\displaystyle u}$ ${\displaystyle \zeta }$

${\displaystyle {\frac {d^{2}{\bar {v}}}{dy^{2}}}+\left({\frac {\omega ^{2}}{c^{2}}}-k^{2}-{\frac {\beta k}{\omega }}-{\frac {\beta ^{2}}{c^{2}}}y^{2}\right){\bar {v}}=0\ ,}$

thus the Schrödinger equation of the harmonic oscillator . It is therefore

${\displaystyle {\bar {v}}(y)=\mathrm {const} \cdot e^{-{\frac {1}{2}}{\frac {\beta }{c}}y^{2}}H_{m}({\sqrt {\frac {\beta }{c}}}y)}$

with the Hermite polynomials and the modes m = 0, 1, 2,… . ${\displaystyle H_{m}}$

Dispersion relations

Dispersion relation of equatorial waves according to Matsuno, Wheeler-Kiladis and Judt

The dispersion relations are obtained

${\displaystyle {\frac {\omega ^{2}}{c_{s}^{2}}}-k^{2}-{\frac {\beta k}{\omega }}={\frac {\beta }{c_{s}}}(2m+1)\ .}$

The speed of free gravity waves introduced above is referred to as a to distinguish between group and phase speeds of the waves under consideration . Matsuno classified the waves by representing the solutions of the dispersion relation approximated for large wave numbers : ${\displaystyle c_{s}={\sqrt {gH}}}$

${\displaystyle \omega _{1/2}=\pm c_{s}{\sqrt {k^{2}+{\frac {\beta }{c_{s}}}(2m+1)}}\ ,}$

${\displaystyle \omega _{3}=-{\frac {\beta k}{k^{2}+{\frac {\beta }{c_{s}}}(2m+1)}}\ .}$

Note that this also applies to solutions of the dispersion relations. ${\displaystyle k\to 0}$

In the selected representation, the phase velocity is directed eastward for positive wavenumber and westward for negative wavenumber.

Inertial gravity waves

The first two solutions are called inertio-gravity waves and have westward and eastward phase velocity . The solutions have minima at around the y- axis . Depending on the direction of the phase velocity, these waves are abbreviated as EIG ( eastward inertio-gravity wave ) or TIG . ${\displaystyle m\geq 1}$ ${\displaystyle c_{1/2}=\pm {\frac {c_{s}}{k}}{\sqrt {k^{2}+{\frac {\beta }{c_{s}}}(2m+1)}}}$ ${\displaystyle {\sqrt {{\frac {\beta }{c_{s}}}(2m+1)}}}$

Equatorial Rossby waves

Matsuno identified the third solution for Rossby waves with slower, westward running , whereby, as indicated above, in contrast to Rossby's description, the Coriolis force near the equator is only included by the . The phase velocity is . ${\displaystyle m\geq 1}$ ${\displaystyle \beta }$ ${\displaystyle c_{3}=-{\frac {\beta }{k^{2}+{\frac {\beta }{c_{s}}}(2m+1)}}}$

Note that there is a clear gap between the maximum speed of the equatorial Rossby wave and the minimum speed of the inertial gravity waves. ${\displaystyle m\geq 1}$

Yanai waves, also mixed Rossby gravity waves

The dispersion relation can be factored for, whereby the lines for the roots for the inertial gravity waves on the one hand and for the Rossby wave on the other hand intersect. There is a solid line east of the solution for an inertial gravity line and west of a Rossby wave. The third solution must be ruled out because it does not approach zero towards the poles. ${\displaystyle m=0}$

The line for thus composed closes the gap in the spectrum which, as mentioned above, also occurs for the modes . These waves were named Yanai waves in honor of Michio Yanai . ${\displaystyle m=0}$ ${\displaystyle m\geq 1}$

The Yanai wave behaves like a Rossby wave with westward phase velocity for low frequencies and like a gravity wave with eastward phase velocity for high frequencies . Because of this mixed behavior, the Yanai wave is also called mixed Rossby gravity wave ( MRG ). The group speed for Yanai waves is always directed to the east and is of the order of 2 to 3 m / s. This means that these waves can move eastward across the equatorial ocean relatively quickly and that, for example, it takes them about a month to cross the Pacific at the equator.

Equatorial Kelvin waves

Another solution to the wave equations is obtained by not allowing any meridional velocity components and by excluding the v in the wave equations from the start. These solutions are called equatorial Kelvin waves . Even if they have nothing to do with the above approach, Matsuno already gave them the mode number " ". ${\displaystyle m=-1}$

Kelvin waves usually occur near the coast, where they are given a clear boundary condition by the coast. The change in sign of the Coriolis force forms a "wall" at the equator and therefore represents a corresponding boundary condition for the Kelvin waves like a coast. The solutions have different directions of rotation in the northern hemisphere and the southern hemisphere.

literature

• John R. Apel: Principles of Ocean Physics. Academic Press, London et al. 1987, ISBN 0-12-058865-X .
• Adrian E. Gill: Atmosphere-Ocean Dynamics (= International Geophysics Series 30). Academic Press, New York NY et al. 1982, ISBN 0-12-283520-4 .
• Joseph Pedlosky: Geophysical Fluid Dynamics. 2nd edition. Springer, New York NY et al. 1987, ISBN 0-387-96388-X .
• M. Wheeler, GN Kiladis: Convectively Coupled Equatorial Waves . Analysis of Clouds and Temperature in the Wavenumber Frequency Domain. In: J. Atmospheric Sci. tape 56 , no. 3 , 1999, p. 374-399 ( ametsoc.org [PDF]).
• A. Solodoch, WR Boos, Z. Kuang, E. Tziperman: Excitation of Intraseasonal Variability in the Equatorial Atmosphere by Yanai Wave Groups via WISHE-Induced Convection . In: J. Atmospheric Sci. tape 68 , 2011, p. 210-225 , doi : 10.1175 / 2010JAS3564.1 .

Individual evidence

1. ↑ ^{a b} M. Kappas: Climatology: Climate Research in the 21st Century . Challenge for natural and social sciences. Springer-Verlag, 2009, ISBN 978-3-8274-2242-2 ( limited preview in the Google book search).
2. ↑ T.Shinoda: Observation of first and second baroclinic mode Yanai waves in the ocean . In: Qu.J.Roy. Meteorological Soc. tape 138 , no. 665 , 2012, p. 1018-1024 , doi : 10.1002 / qj.968 ( wiley.com [PDF]).
3. ↑ C. Jacobi, Institute for Meteorology, University of Leipzig: Equatorial waves - summary. Retrieved December 29, 2016 . 
4. ↑ G.Siedler, W.Zenk: Oceanography . In: W. Raith (ed.): Textbook of Experimental Physics - Ludwig Bergmann . 7 (earth and planets). Walter de Gruyter, 2001, ISBN 978-3-11-016837-2 ( limited preview in the Google book search).
5. ↑ ^{a b c d e f} T. Matsuno: Quasi-Geostrophic Motions in the Equatorial Area . In: J. Meteorological Soc. Japan. Ser. II . tape 44 , no. 1 , 1966, p. 25-43 ( jst.go.jp [PDF]).
6. ^ ^{A b} F. Judt: Equatorial Wave Theory. April 23, 2007, archived from the original on December 30, 2016 ; accessed on December 29, 2016 (presentation in the Geophysical Fluid Dynamics 2 seminar , spring 2007). 
7. ↑ M. Hantel: Introduction to Theoretical Meteorology . Springer, 2013, ISBN 978-3-8274-3056-4 ( limited preview in Google book search).
8. ↑ ^{a b} M. Wheeler, GN Kiladis: Convectively Coupled Equatorial Waves: Analysis of Clouds and Temperature in the Wavenumber Frequency Domain . In: J. Atmosperic Sci. tape 56 , no. 3 , 1999, p. 374-399 , doi : 10.1175 / 1520-0469 (1999) 056 <0374: CCEWAO> 2.0.CO; 2 .