Planetary (Rossby), Inertia-Gravity (Poincaré) and Kelvin waves on the f-plane in the presence of a uniform zonal flow

Abstract:

A linear wave theory of the Rotating Shallow Water Equations (RSWE) is developed in a channel on the mid-latitude f-plane in the presence of a uniform mean zonal flow that is balanced geostrophically by a meridional gradient of the fluid surface height. We show that this surface height gradient is a potential vorticity (PV) source that generates Rossby waves even on the f-plane similar to the generation of these waves by PV sources such as the –effect, shear of the mean flow and bottom topography. Numerical solutions of the RSWE show that the resulting planetary (Rossby), Inertia-Gravity (Poincaré) and Kelvin-like waves differ from their counterparts without mean flow in both their phase speeds and meridional structures. Doppler shifting of the “no mean-flow” phase speeds does not account for the difference in phase speeds, and the meridional structure is often trapped near one the channel's boundaries and does not oscillate across the chancel. The phase speed of Kelvin-like waves is modified by the presence of a mean flow compared to the classical gravity wave speed but their meridional velocity does not vanish. The gaps between the dispersion curves of adjacent Poincaré modes are not uniform but change with the zonal wavenumber, and the convexity of the dispersion curves also changes with the zonal wavenumber. In some cases, the Kelvin-like dispersion curve crosses those of Poincaré modes. The attached figure shows the dispersion relations of the first meridional modes of Rossby waves on the f-plane, in both wide channel (where the channel width is larger than the radius of deformation, left panel) and narrow channel (right panel). The structure of these curves is similar to that of the classical theory on the -plane without mean flow but the values of the frequency are not.