Small-amplitude coastally trapped disturbances and the reduced-gravity shallow-water approximation

Abstract

Solutions are obtained for linear hydrostatic disturbances propagating parallel to the face of an uninterrupted topographic step in an infinitely deep, stably stratified fluid on an β-plane. These waves are vertically trapped because their frequencies are smaller than the Coriolis parameter and the height of the topographic step is finite. These waves are referred to as step-trapped Kelvin waves, because they are dynamically similar to internal Kelvin waves throughout the layer of fluid below the top of the topographic step. These waves appear to provide an idealized, semi-analytic model for the coastally trapped disturbances observed to propagate parallel to mountainous coastlines in several parts of the world. Computations are performed for a basic state with uniform static stability and for a three-layer basic state in which the two lowest layers represent the marine boundary layer and a strong capping inversion. One might suppose that the linear dynamics of hydrostatic disturbances in the three-layer basic state could be well approximated by a reduced-gravity shallow-water model, but this is not the case. In particular, the reduced-gravity shallow-water model does not provide reliable estimates for the phase speed of linear step-trapped Kelvin waves. This defect suggests that detailed quantitative comparisons between marine boundary-layer flows and the reduced-gravity shallow-water system may not have any intrinsic physical significance. Nevertheless, these results do not preclude the possibility of constructing useful qualitative analogies between marine boundary-layer flows and the reduced-gravity shallow-water model.