Coastal imbalance: generation of oceanic Kelvin waves by atmospheric perturbations

The response of a semi-infinite ocean to a slowly travelling atmospheric perturbation crossing the coast provides a simple example of the breakdown of nearly geostrophic balance induced by a boundary. We examine this response in the linear shallow-water model at small Rossby number $\varepsilon \ll 1$ . Using matched asymptotics, we show that a long Kelvin wave, with $O(\varepsilon ^{-1})$ length scale and $O(\varepsilon )$ amplitude relative to quasi-geostrophic response, is generated as the perturbation crosses the coast. Accounting for this Kelvin wave restores the conservation of mass that is violated in the quasi-geostrophic approximation.