A consistent theory for linear waves of the Shallow Water Equations on a rotating plane in mid-latitudes

The present study provides a unified and consistent theory for the three types of linear waves of the Shallow Water Equations (SWE) on the β-plane – Kelvin, inertia-gravity (Poincare) and planetary (Rossby). The unified theory obtains by formulating 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 a1-sided Gaussian on the β-plane. Similarly, inertia-gravity waves are only slightly modified in the new theory compared to their characteristics on the f-plane. However, for planetary (Rossby) waves (that exist only on the β-plane) the new theory yields similar dispersion relation to the classical theory only for large values of gravity waves' speed. On the other hand, for low gravity-wave phase speed, i.e. in equivalent-barotropic cases with small density jump at the interface, the dispersion relation of the new theory has phase speeds that are twice larger than in the classical theory for realistic widths and up to 3.3 times larger for wide channels. 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 Topex/Poseidon satellite. The unified theory also admits inertial waves, i.e. waves that oscillate at the local inertial frequency, as a consistent solution of the eigenvalue problem. 3