Past exam of the physics of atmospheres and oceans course of the University of Oxford / 2018 / 5 / e

The Kelvin-wave coastal amplitude scaling follows from the flux. For a wave with peak coastal amplitude $A_0$, $\overline{v\eta }=(c/H)A_0^2{e}^{-2r/L_R}/2$. Integrating the supplied flux over the offshore coordinate gives its magnitudeThe direction is the alongshore propagation direction; the expression above is positive magnitude. For slowly varying depth, negligible dissipation/reflection and no other source or sink, the transported energy per time is conserved. At fixed latitude,The coastal amplitude increases on entering shallower water. This is stronger than the familiar fixed-width $H^{-1/4}$ shoaling law because the Kelvin mode's offshore width also contracts, $L_R\propto \sqrt{H}$. The comparison is a slowly varying trapped-mode result; abrupt topography can scatter or reflect energy and invalidate simple flux conservation for one outgoing mode. The amplitude panel in solution illustrates the scaling.

Retaining the integrated energy flux from part (i), slow changes of depth and latitude obeyAt fixed depth and higher absolute latitude in the same hemisphere, $|f|=2\Omega |\sin \phi |$ increases, so $A_0\propto \sqrt{|f|}$ increases. The offshore trapping width $L_R=c/|f|$ narrows, concentrating the same transported energy closer to the coast.

This uses a local Kelvin-mode approximation on a slowly varying background, despite the original constant-$f$ derivation. The scale of variation must be long enough for the wave to adjust without substantial reflection, scattering or dissipation. It is not a rule to extrapolate through the equator, where the coastal trapping scale and hemisphere-dependent propagation rule change.