 Solution

Use the standard Earth radius $a_E=6.371\times 10^6\mathrm{m}$ and rotation rate $\Omega =7.292\times 10^{-5}{\mathrm{s}}^{-1}$. At $10^{\circ }$S,Take a predominantly zonal wave train, $l\simeq 0$, and the gravest baroclinic mode from part (c). ThenThe background current is westward, so $U=-0.025\mathrm{m}{\mathrm{s}}^{-1}$. The pulse uses the baroclinic Rossby-wave group velocity, not the crest speed:ThereforeFor comparison $c_{px}\simeq -0.373\mathrm{m}{\mathrm{s}}^{-1}$ would give about $0.85$ years, but that is a crest transit time and not the requested pulse delay.

The approximately three-year group delay is comparable to the stated three-to-five-year El Niño–Southern Oscillation period. Wind-driven thermocline disturbances can store and convey oceanic adjustment westward; interaction with the western boundary and subsequent faster signals can contribute a delayed feedback on the tropical coupled system. This calculation supports a relevant time scale, not a complete ENSO oscillator: realistic meridional structure, variable stratification and currents, damping, boundary conversion and equatorial coupling are omitted. A nonzero unspecified meridional wavenumber would also change the numerical group speed, so the $l\simeq 0$ assumption is material.