Influence of Linear Depth Variation on Poincaré, Kelvin, and Rossby Waves

Abstract

Exact solutions to the linearized shallow-water equations in a channel with linear depth variation and a mean flow are obtained in terms of confluent hypergeometric functions. These solutions are the generalization to finite s (depth variation parameter) of the approximate solutions for infinitesimal s. The equations also respect an energy conservation principle (and the normal modes are thus neutrally stable) in contradistinction to those of previous studies. They are evaluated numerically for a range in s from s = 0.1 to s = 1.95, and the range of validity of previously derived approximate solutions is established. For small s the Kelvin and Poincaré solutions agree well with those of Hyde, which were obtained by expanding in s. For finite s the solutions differ significantly from the Hyde expansions, and the magnitude of the phase speed decreases as s increases. The Rossby wave phase speeds are close to those obtained when the depth is linearized although the difference increases with s. The eigenfunctions become more distorted as s increases so that the largest amplitude and the smallest scale occur near the shallowest boundary. The negative Kelvin wave has a very unusual behavior as s increases.