Classification of eastward propagating waves on the spherical Earth

Abstract

Observational evidence for an equatorial non-dispersive mode propagating at the speed of gravity waves is strong, and while the structure and dispersion relation of such a mode can be accurately described by a wave theory on the equatorial β-plane, prior theories on the sphere were unable to find such a mode except for particular asymptotic limits of gravity wave phase speeds and/or certain zonal wave numbers. Here, an ad hoc solution of the linearized rotating shallow-water equations (LRSWE) on a sphere is developed, which propagates eastward with phase speed that nearly equals the speed of gravity waves at all zonal wave numbers. The physical interpretation of this mode in the context of other modes that solve the LRSWE is clarified through numerical calculations and through eigenvalue analysis of a Schrödinger eigenvalue equation that approximates the LRSWE. By comparing the meridional amplitude structure and phase speed of the ad hoc mode with those of the lowest gravity mode on a non-rotating sphere we show that at large zonal wave number the former is a rotation-modified counterpart of the latter. We also find that the dispersion relation of the ad hoc mode is identical to the n = 0 eastward propagating inertia-gravity (EIG0) wave on a rotating sphere which is also nearly non-dispersive, so this solution could be classified as both a Kelvin wave and as the EIG0 wave. This is in contrast to Cartesian coordinates where Kelvin waves are a distinct wave solution that supplements the EIG0 mode. Furthermore, the eigenvalue equation for the meridional velocity on the β-plane can be formally derived as an asymptotic limit (for small (Lamb Number) -1/4 ) of the corresponding second order equation on a sphere, but this expansion is invalid when the phase speed equals that of gravity waves i.e. for Kelvin waves. Various expressions found in the literature for both Kelvin waves and inertia-gravity waves and which are valid only in certain asymptotic limits (e.g. slow and fast rotation) are compared with the expressions found here for the two wave types.

The mixed Rossby–gravity wave on the spherical Earth

“…In addition, it was also shown in Garfinkel et al. s*() that the LSWEs on the sphere reduce to those on the equatorial β ‐plane only for ϵ 1/4 ≫ 1, where ϵ is Lamb's parameter (see definition in Section 2 below). However a comparison with numerical solutions on a sphere suggests that the dispersion relation obtained by Matsuno is valid even for small values of ϵ .…”

“…In order to restrict the integration to the regular solution, we set (which is a normalization of the linear differential system) and to ensure regularity at the singular poles (see (Garfinkel et al. , )). Note the following.…”

“…(a) A different time (and velocity) scale is used in Garfinkel et al. s*(). (b) The above relation is only valid for k ≠ 0, while for k = 0 the same regularity requirement yields , where again .…”

“…Recently, the authors of the present article used this approach in the study of eastward‐propagating waves, including the nondispersive Kelvin wave, on the sphere (Garfinkel et al. , ). In that study it was found that, in contrast to the equatorial β ‐plane, where the Kelvin wave is a particular solution of the LSWEs that is not associated with an energy level of the Schrödinger eigenvalue equation, on a sphere the “Kelvin” wave should be classified as the lowest EIG wave mode derived from the ground‐state energy level of the Schrödinger eigenvalue problem.…”

The Teleconnection of El Niño Southern Oscillation to the Stratosphere

“… Aleutian low Quasi‐stationary low pressure system in the North Pacific near the Aleutian Islands. Amundsen Sea low Quasi‐stationary low pressure system in the South Pacific in the Amundsen Sea. Arctic Oscillation The dominant mode of climate variability in the Northern Hemisphere extratropics, representing a seesaw in mass between the extratropics and the poles, here used synonymously with the Northern Annular Mode. Atmospheric Kelvin wave Classically considered to be a nondispersive, equatorially confined wave with zero meridional velocity that propagates eastward at the same speed a gravity wave would propagate in the absence of rotation. See Garfinkel, Fouxon, et al () for a geophysical fluid dynamics definition of Kelvin waves. Brewer‐Dobson Circulation (BDC) The mean mass transport within the stratosphere, which consists of rising motion in the tropics and subsiding motion over the poles. Canonical El Niño A “regular”/Eastern Pacific El Niño event. Cold point The coldest region of the tropical tropopause where temperatures can drop below 190 K. El Niño Modoki An El Niño event that peaks further west in the equatorial Pacific in comparison to the canonical El Niño. ENSO flavor The type of ENSO event in terms of its peak location in the equatorial Pacific. Final warming The final reversal of zonal mean zonal winds in the stratospheric polar vortex of both hemispheres at the end of each winter. The winds then remain easterly until the next fall. Hadley cell A thermally driven and zonally symmetric circulation consisting of the equatorward movement of the trade winds between about latitude 30 ∘ and the equator in each hemisphere, ascent near the equator, poleward flow aloft, and, finally, descent peaking near latitude 30 ∘ . Hypsometric equation By taking the vertical integral of the hydrostatic equation, it can be shown that vertical distance between two pressure levels is proportional to the mean temperature. Madden‐Julian Oscillation (MJO) The dominant organized fluctuation in tropical weather on weekly to monthly time scales.…”