Coastal imbalance: generation of oceanic Kelvin waves by atmospheric perturbations

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