One-dimensional models for topographic Rossby waves in elongated basins on thef-plane

Abstract

Topographic Rossby waves in elongated basins on thef-plane are studied by transforming the linear boundary-value problem for the mass transport stream function into a class of two-point boundary-value problems of which the independent spatial variable is the (curved) basin axis. The procedure for deriving the substitute problems is theMethod of Weighted Residuals. What emerges is a vector differential equation and associated boundary conditions, its dimension indicating the order of the approximate model. It is shown that each substitute problem in the class entails the qualitative features typical of topographic waves, and increasing the order of the model corresponds to higher-order approximations. Equations are explicitly presented for cross-sectional distributions of the lake topography which has a power-law representation and permits the analysis of weak and strong topographies.Straight channels in which the depth profile does not change with position along the axis are studied in detail. The dispersion relation is discussed and dispersion curves are shown for the three lowest-order models. Convergence properties are thereby uncovered and phase speed and group velocity properties are found as they depend on wavenumber and topography. Further, for the lowest two modes, cross-channel stream-function distributions are presented. Apart from further convergence properties these distributions show that forU-shaped channels wave activity is nearer to the shore than forV-shaped channels, important information in the design of mooring systems.An analysis of topographic Rossby wave reflection follows, which emphasizes the importance of the depth profile in the reflecting zone. Based on these results some lake solutions are presented.