INFLUENCE OF LINEAR DEPTH VARIATION ON POINCARE, KELVIN, AND ROSSBY WAVES

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 generaliza tion to finite s (depth variation parameter) of the approximate soluti ons for infinitesimal s. The equations also respect an energy conserva tion principle (and the normal modes are thus neutrally stable) in con tradistinction to those of previous studies. They are evaluated numeri cally for a range in s from s = 0.1 to s = 1.95, and the range of vali dity of previously derived approximate solutions is established. For s mall s the Kelvin and Poincare solutions agree well with those of Hyde , which were obtained by expanding in s. For finite s the solutions di ffer 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 dif ference 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 unu sual behavior as s increases.