The family of anisotropically scaled equatorial waves (original) (raw)
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The present study provides a consistent and unified theory for the three types of linear waves of the shallow-water equations (SWE) in a zonal channel on the  plane: Kelvin, inertia-gravity (Poincaré), and planetary (Rossby). The new theory is formulated from the linearized SWE as an eigenvalue problem that is a variant of the classical Schrödinger equation. The results of the new theory show that Kelvin waves exist on the  plane with vanishing meridional velocity, as is the case on the f plane, without any change in the dispersion relation, while the meridional structure of their height amplitude is trivially modified from exponential on the f plane to a one-sided Gaussian on the  plane. Similarly, inertia-gravity waves are only slightly modified in the new theory in comparison with their characteristics on the f plane. For planetary waves (which exist only on the  plane) the new theory yields a similar dispersion relation to the classical theory only for large gravity wave phase speed, such as those encountered in a barotropic ocean or an equivalent barotropic atmosphere. In contrast, for low gravity wave phase speed, for example, those in an equivalent barotropic ocean where the relative density jump at the interface is 10 Ϫ3 , the phase speed of planetary waves in the new theory is 2 times those of the classical theory. The ratio between the phase speeds in the two theories increases with channel width. This faster phase propagation is consistent with recent observation of the westward propagation of crests and troughs of sea surface height made by the altimeter aboard the Ocean Topography Experiment (TOPEX)/Poseidon satellite. The new theory also admits inertial waves, that is, waves that oscillate at the local inertial frequency, as a genuine solution of the eigenvalue problem.
Tropical Stationary Waves in a Nonlinear Shallow-Water Model with Realistic Basic States
Journal of the Atmospheric Sciences, 2007
The nonlinear shallow-water equations are used to study the tropical stationary wave response to steady thermal forcing near the equator in earthlike zonally symmetric basic states. A thin (200 m) fluid layer is superimposed over a large (1500 m) zonally symmetric topography distribution that decreases smoothly from the Tropics to the Poles, thus providing the large meridional height gradients required to maintain realistic zonal-mean zonal winds without introducing unrealistically large tropical wave speeds. A mean meridional circulation is maintained by relaxing the fluid toward its initial, global-mean depth. Both hemispherically symmetric (equinoctial) and hemispherically asymmetric (solstitial) basic states are considered. Stationary waves are generated by adding a fixed mass source/sink distribution near the equator. The presence of westerly zonal-mean winds in the subtropics amplifies the steady eddy response to the tropical mass forcing and shifts the Rossby gyres poleward a...
Modulation of shallow water equatorial waves due to a varying equivalent height background
2013
The dynamics of convectively coupled equatorial waves (CCEWs) is analyzed in an idealized model of the large-scale atmospheric circulation. The model is composed of a linear rotating shallow-water system with a variable equivalent height, or equivalent gravity wave speed, which varies in space. This model is based on the hypothesis that moist convection acts to remove convective instability, therefore modulating the equivalent height of a shallow-water system. Asymptotic solutions are derived in the case of a small perturbation around a constant coefficient, which is assumed to be a mean moist equivalent height derived from satellite observations. The first-order solutions correspond to the free normal modes of the linear shallow-water system and the second-order flow is derived solving a perturbation eigenvalue problem. The asymptotic solutions are documented in the case of a zonally varying equivalent height and for wavenumbers and frequencies that are consistent with observations of CCEWs. This analysis shows that the dynamics of the secondary divergence and its impact on the full divergence varies mode by mode. For instance, for a negative equivalent height anomaly, which is interpreted as a moister background, the secondary divergence is nearly in phase with the primary divergence in the case of Kelvin waves-in contrast to mixed Rossby-gravity waves where the secondary divergence acts to attenuate the primary divergence. While highly idealized, the modeled waves share some features with observations, providing a mechanism for the relationship between CCEWs phase speed, amplitude, and horizontal structure.
Slow oscillations in an ocean of varying depth Part 1. Abrupt topography
Journal of Fluid Mechanics, 1969
This paper is part of a study of quasigeostrophic waves, which depend on the topography of the ocean floor and the curvature of the earth.In a homogeneous, β-plane ocean, motion of the fluid across contours of constant f/h releases relative vorticity (f is the Coriolis parameter and h the depth). This well-known effect provides a restoring tendency for either Rossby waves (with h constant) or topographic waves over a slope. The long waves in general obey an elliptic partial differential equation in two space variables. Because the equation has been integrated in the vertical direction, the exact inviscid bottom boundary condition appears in variable coefficients.When the depth varies in only one direction the equation is separable at the lowest order in ω, the frequency upon f. With a simple slope, |[xdtri ]h/h| = constant, the transition from Rossby to topographic waves occurs at |[xdtri ]h| ∼ h/Re, where Re is the radius of the earth. Isolated topographic features are considered i...
Asymptotic approach for the nonlinear equatorial long wave interactions
Journal of Physics: Conference Series, 2011
In the present work we use an asymptotic approach to obtain the long wave equations. The shallow water equation is put as a function of an external parameter that is a measure of both the spatial scales anisotropy and the fast to slow time ratio. The values given to the external parameters are consistent with those computed using typical values of the perturbations in tropical dynamics. Asymptotically, the model converge toward the long wave model. Thus, it is possible to go toward the long wave approximation through intermediate realizable states. With this approach, the resonant nonlinear wave interactions are studied. To simplify, the reduced dynamics of a single resonant triad is used for some selected equatorial trios. It was verified by both theoretical and numerical results that the nonlinear energy exchange period increases smoothly as we move toward the long wave approach. The magnitude of the energy exchanges is also modified, but in this case depends on the particular triad used and also on the initial energy partition among the triad components. Some implications of the results for the tropical dynamics are disccussed. In particular, we discuss the implications of the results for El Niño and the Madden-Julian in connection with other scales of time and spatial variability.
A Filtered Model of Tropical Wave Motions
Journal of Advances in Modeling Earth Systems, 2009
Large-scale tropical phenomena such as the Madden-Julian Oscillation (MJO) and El Niño-Southern Oscillation (ENSO) are often studied using the longwave approximation to equatorial β-plane theory. This approximation involves the neglect of the (∂v/∂t) term in the meridional momentum equation. The approximation does not distort Kelvin waves, completely filters inertia-gravity waves, is reasonably accurate for long Rossby waves, but greatly distorts short Rossby waves. Here we present an improvement of the longwave model, based on an approximation of the (∂v/∂t) term rather than its complete neglect. The new model is similar to the longwave model in the sense that it does not distort Kelvin waves and completely filters inertia-gravity waves. However, it differs from the longwave model in the sense that it accurately describes Rossby waves of all wavelengths, thus making it a useful tool for the study of a wider range of tropical phenomena than just the MJO and ENSO. Although most of the mathematical analysis performed here is in the context of equatorial β-plane theory, we briefly discuss how the ideas can be generalized to spherical geometry.
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Dispersive shallow water wave modelling. Part III: Model derivation on a globally spherical geometry
2017
The present article is the third part of a series of papers devoted to the shallow water wave modelling. In this part, we investigate the derivation of some long wave models on a deformed sphere. We propose first a suitable for our purposes formulation of the full Euler equations on a sphere. Then, by applying the depth-averaging procedure we derive first a new fully nonlinear weakly dispersive base model. After this step, we show how to obtain some weakly nonlinear models on the sphere in the so-called Boussinesq regime. We have to say that the proposed base model contains an additional velocity variable which has to be specified by a closure relation. Physically, it represents a dispersive correction to the velocity vector. So, the main outcome of our article should be rather considered as a whole family of long wave models.
Balance model for equatorial long waves
Journal of Fluid Mechanics, 2013
Geophysical fluid models often support both fast and slow motions. As the dynamics are often dominated by the slow motions, it is desirable to filter out the fast motions by constructing balance models. An example is the quasi-geostrophic (QG) model, which is used widely in meteorology and oceanography for theoretical studies, in addition to practical applications such as model initialization and data assimilation. Although the QG model works quite well in the mid-latitudes, its usefulness diminishes as one approaches the equator. Thus far, attempts to derive similar balance models for the tropics have not been entirely successful as the models generally filter out Kelvin waves, which contribute significantly to tropical low-frequency variability. There is much theoretical interest in the dynamics of planetary-scale Kelvin waves, especially for atmospheric and oceanic data assimilation where observations are generally only of the mass field and thus do not constrain the wind field w...