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Trapped continental shelf waves with a free surface

Abstract A number of recent results have shown that within the shallow water, rigid-lid approximation, alongshore variations in the bathymetry, i.e. a submerged ridge, can lead to continental shelf waves (CSWs) that are localised geographically at the ridge and then decay both along and away from the coast. Removing the rigid-lid assumption, introduces the superinertial Poincaré waves and a Kelvin wave that is present at all frequencies. This implies that the spectrum of the associated wave operator, which is bounded for the rigid lid case, is continuous and unbounded for the free-surface case. In the rigid-lid case the localisation of modes are isolated eigenvalues lying above the continuous spectrum whereas any localised modes for the free-surface problem must necessarily be embedded in the continuous spectrum. The purpose of this work is to construct trapped CSWs analytically and numerically for a non-rectilinear shelf. A regular asymptotic method is employed by considering a slowly varying, non-rectilinear shelf with an approximate boundary condition at the shelf–ocean boundary. It is shown that even with the free-surface present, trapped CSWs do indeed exist for the submerged ridge topography. Comparison with highly accurate numerical results demonstrates the accuracy of the asymptotic method and also allows the consideration of shelves that abut an open ocean so avoiding the approximate boundary condition at the shelf–ocean boundary.