Propagation of an Equatorial Kelvin Wave in a Varying Thermocline

We look at the effect of a slow zonal variation in thermocline depth on the propagation of a finite-amplitude Kelvin wave pulse in a single layer model. Dispersive effects are included by also allowing a weak meridional variation in background state. Analytical results are obtained using the method of multiple scales—in essence a WKB approximation. The evolution of wave amplitude riding with the Kelvin wave is found to be governed by a KdV equation with variable coefficients. As expected from energy conservation, the amplitude must increase as the thermocline depth decreases; however, the power appearing in the analog of “Green's Law” is different than that found for shallow water waves impinging on a beach. This modified “Green's Law” is verified using a numerical model. The most interesting conclusion, which is also checked numerically, is that a significant portion of the mass flux carried by a Kelvin wave pulse propagating eastward into a shoaling thermocline (the oceanographically relevant solution) is reflected by westward-propagating Rossby and gravity modes. This is not true of the energy flux, and we explain this seeming paradox using scaling arguments.