A Consistent Theory for Linear Waves of the Shallow-Water Equations on a Rotating Plane in Midlatitudes

Abstract

Abstract 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 β 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 β 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 β 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 β 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 10−3, 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.