The exchange of fluid mass between quasi-geostrophic and ageostrophic motions during the reflection of Rossby waves from a coast. I. The case of an infinite rectilinear coast

Abstract

When the problem of the reflection of spatially localized Rossby waves from a coast is treated using the quasigeostrophic (QG) approximation, the total fluid mass and the along-shore circulation calculated from the geostrophic height field are not conserved. To understand the correct mass balance and the degree to which the QG equations and boundary conditions may be in error, we analyze an initial-value problem for the Laplace tidal equations on a β-plane in the asymptotic limit ϵ ⪡ 1, where ϵ is the ratio of the spatial scale of the motion to the Earth's radius. It is shown that there is a coupling between QG and O(ϵ) fields. Physically, the coupling occurs by a peculiar adjustment process in the O(ϵ) approximation in which fast gravity waves are permanently generated to build up a quasi-stationary edge Kelvin wave. Different temporal scales (large for O(1) Rossby waves and small for the O(ϵ) gravity waves make comparable the contributions of the waves to the mass and circulation balance equations. However, QG analysis itself describes the reflection of Rossby waves correctly, but is incomplete, and for satisfactory balances one has to take into account the fields of both orders of the approximation. Applications of the results to closed basins, baroclinicity, and variable bottom topography are discussed. It is conjectured that an interaction of strong oceanic eddies with a coast (continental slope) may give rise to noticeable along-shore jet currents.