Kelvin wave

A coastal Kelvin wave is trapped within a Rossby deformation radius of a boundary. It propagates with the boundary on its right in the Northern Hemisphere and on its left in the Southern Hemisphere, and its linear shallow-water potential-vorticity anomaly vanishes.

For a harmonic coastal Kelvin wave, $v=(c/H)\eta$, $c=\sqrt{gH}$ and the offshore decay length is $c/|f|$. Integrating the period-mean pressure-work flux over the offshore direction gives $\mathcal{F}=\rho g^{2}H{A}_0^2/(4|f|)$ for peak coastal amplitude $A_0$. Slow changes with negligible dissipation or scattering conserve this flux, giving $A_0\propto \sqrt{|f|/H}$. The changing mode width explains why depth scaling differs from fixed-width wave shoaling. The approximation is not uniform at the equator.

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.

Exponentially weighted cross-channel integrals of velocity and pressure isolate the two oppositely propagating boundary Kelvin waves. Their conserved characteristic amplitudes supply the two integral constraints missing from a channel geostrophic-adjustment problem.