= Coastal Kelvin-wave radiation of long Rossby waves
{title2=$|\omega|\simeq\beta L_d^2|k|$}
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 $|\ell|\ll\beta L_d/c$; $|\ell|L_d\ll1$ alone is insufficient. This mechanism transmits coastal <thermocline> anomalies into an ocean basin over a much slower timescale than coastal propagation.
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