Coastal Kelvin-wave radiation of long Rossby waves ($|\omega |\simeq \beta L_d^2|k|$)

A slowly varying coastal Kelvin wave can force long Rossby waves at the same absolute frequency and alongshore wavenumber. In the regime $k^2+{\ell }^2\ll L_d^{-2}$, the offshore group velocity is westward with magnitude $\beta L_d^2$. With coastal phase velocity magnitude $c$, matching gives $|k|=c|\ell |/(\beta L_d^2)$. The zonal long-wave condition therefore additionally requires $|\ell |\ll \beta L_d/c$; $|\ell |L_d\ll 1$ alone is insufficient. This mechanism transmits coastal thermocline anomalies into an ocean basin over a much slower timescale than coastal propagation.