Simple Models of Coastal-Trapped Waves Based on the Shape of the Bottom Topography

Solutions of barotropic coastal-trapped waves in the shallow-water context are discussed for different shapes of the bottom topography. In particular, an infinite family of topographic waves over continental shelves characterized by a shape parameter is considered. The fluid depth is proportional to xs, where x is the offshore coordinate and s is a real, positive number. The model assumes the rigid-lid approximation and a semi-infinite domain 0 ≤ x ≤ ∞. The wave structure and the dispersion relation depend explicitly on the shape parameter s. Essentially, waves over steeper shelves possess higher frequencies and phase speeds. In addition, the wave frequency is independent of the alongshore wavenumber k, implying a zero group velocity component along the coast. The advantages and limitations of this formulation, as well as some comparisons with other models, are discussed in light of numerical simulations for waves over arbitrary topography within a finite domain. The numerical calculations show that the frequency of the waves present a nondispersive regime at small wavenumbers (observed by several authors), followed by a constant value predicted by the analytical solutions for larger k. It is concluded that these frequencies can be considered as an upper limit reached by barotropic coastal-trapped waves over the infinite family of xs-bottom profiles, regardless of the horizontal and vertical scales of the system. The modification of the dispersion curves in a stratified ocean is briefly discussed.