Orthogonality of coastal trapped waves

Abstract

Coastal trapped wave modes are shown to be orthogonal in the sense that they make independent contributions to the energy of the wave field. The hydrostatic Boussinesq dynamics on an $f$-plane, linearized around a state of rest, are formulated as a (generalized) Schrödinger equation, which exposes the Hermitian structure of the wave operator that implies the orthogonality of eigenmodes. This formulation, which becomes particularly simple in weak form, is parlayed into a finite-element discretization that preserves the symmetries of the original problem and therefore the energy conservation and orthogonality of modes. The geostrophic momentum approximation, under which the orthogonality of modes has been recognized previously, is reprised and discussed in the present context to emphasize that low-frequency coastal trapped waves are edge waves. Their dynamics are governed by boundary potential-vorticity anomalies, both Bretherton-type contributions familiar from the quasi-geostrophic equations and lateral contributions that are important for steep slopes and coastal walls.