Equatorial waves

Equatorial waves are oceanic and atmospheric waves trapped close to the equator, meaning that they decay rapidly away from the equator, but can propagate in the longitudinal and vertical directions. Wave trapping is the result of the Earth's rotation and its spherical shape which combine to cause the magnitude of the Coriolis force to increase rapidly away from the equator. Equatorial waves are present in both the tropical atmosphere and ocean and play an important role in the evolution of many climate phenomena such as El Niño. Many physical processes may excite equatorial waves including, in the case of the atmosphere, diabatic heat release associated with cloud formation, and in the case of the ocean, anomalous changes in the strength or direction of the trade winds. Equatorial waves are oceanic and atmospheric waves trapped close to the equator, meaning that they decay rapidly away from the equator, but can propagate in the longitudinal and vertical directions. Wave trapping is the result of the Earth's rotation and its spherical shape which combine to cause the magnitude of the Coriolis force to increase rapidly away from the equator. Equatorial waves are present in both the tropical atmosphere and ocean and play an important role in the evolution of many climate phenomena such as El Niño. Many physical processes may excite equatorial waves including, in the case of the atmosphere, diabatic heat release associated with cloud formation, and in the case of the ocean, anomalous changes in the strength or direction of the trade winds. Equatorial waves may be separated into a series of subclasses depending on their fundamental dynamics (which also influences their typical periods and speeds and directions of propagation). At shortest periods are the equatorial gravity waves while the longest periods are associated with the equatorial Rossby waves. In addition to these two extreme subclasses, there are two special subclasses of equatorial waves known as the mixed Rossby-gravity wave (also known as the Yanai wave) and the equatorial Kelvin wave. The latter two share the characteristics that they can have any period and also that they may carry energy only in an eastward (never westward) direction. The remainder of this article discusses the relationship between the period of these waves, their wavelength in the zonal (east-west) direction and their speeds for a simplified ocean. Rossby-gravity waves, first observed in the stratosphere by M. Yanai, always carry energy eastward. But, oddly, their 'crests' and 'troughs' may propagate westward if their periods are long enough. The eastward speed of propagation of these waves can be derived for an inviscid slowly moving layer of fluid of uniform depth H. Because the Coriolis parameter (ƒ = 2Ω sin(θ) where Ω is the angular velocity of the earth, 7.2921 × {displaystyle imes } 10−5 rad/s, and θ is latitude) vanishes at 0 degrees latitude (equator), the “equatorial beta plane” approximation must be made. This approximation states that “f” is approximately equal to βy, where “y” is the distance from the equator and 'β' is the variation of the coriolis parameter with latitude, ∂ f ∂ y = β {displaystyle {frac {partial f}{partial y}}=eta } . With the inclusion of this approximation, the governing equations become (neglecting friction): We may seek travelling-wave solutions of the form Substituting this exponential form into the three equations above, and eliminating u , {displaystyle u,} and ϕ {displaystyle phi } leaves us with an eigenvalue equation for v ^ ( y ) {displaystyle {hat {v}}(y)} .Recognizing this as the Schrödinger equation for a quantum harmonic oscillator of frequency Ω = β / c {displaystyle Omega =eta /c} , we know that we must have for the solutions to tend to zero away from the equator. For each integer n {displaystyle n} therefore, this last equation provides a dispersion relation linking the wavenumber k {displaystyle k} to the angular frequency ω {displaystyle omega } . In the special case n = 0 {displaystyle n=0} the dispersion equation reduces to