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The warm phase of the coupled tropical Pacific El Niño–Southern Oscillation, with central and eastern equatorial Pacific surface warming and a weakened Walker circulation. Changes of trade winds, thermocline depth, and equatorial Kelvin waves and equatorial Rossby waves participate in its evolution.

The cold phase of El Niño–Southern Oscillation, with central and eastern equatorial Pacific surface cooling.

A physical oceanographer whose work includes tropical Pacific observations and El Niño–Southern Oscillation dynamics. Author of El Niño and La Niña: Causes and Global Consequences by Michael J. McPhaden.

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.

Coastal Kelvin-wave radiation of long Rossby waves requires a common phase convention to match the two waves. The supplied Rossby wave formula conventionally multiplies $e^{i(k_{R}x+{\ell }_{R}y-{\omega }_{R}t)}$, while the coastal expression uses $e^{i({\omega }_{K}t-{\ell }_{K}y)}$. Since physical fields are real, matching may use the complex-conjugate representative; in either convention the boundary forcing fixes the absolute frequency and alongshore wavenumber. Thus $|{\omega }_R|=|{\omega }_K|$ and $|{\ell }_R|=|{\ell }_K|$.

For $k_R^2+{\ell }_R^2\ll L_d^{-2}$, the shallow-water Rossby-wave dispersion relation reduces toChoosing $k_R>0$ recovers the printed magnitude relation. Both the zonal phase velocity and zonal group velocity are approximately $-\beta L_d^2$, so the long Rossby waves carry the coastal signal westwards into the basin.

The coastal condition $|{\ell }_K|L_d\ll 1$ alone does not guarantee a long zonal wavelength. Since $|{\omega }_K|=c|{\ell }_K|$, matching requires $|k_R|=c|{\ell }_K|/(\beta L_d^2)$ and hence the stronger consistency conditionFor example, at $10^{\circ }$ a coastal wavenumber with $|{\ell }_K|L_d=0.05$ gives $|k_R|L_d\simeq 0.56$, which is not asymptotically small even though the meridional wavelength is long. Thus the source's proposed excitation is possible at sufficiently low frequency; it is not an automatic consequence of its meridional long-wave assumption.

At $10^{\circ }$ N or S the beta plane parameter isTaking a representative Pacific basin width $W\simeq 1.5\times 10^7$ m givesA width of $1.4$–$1.7\times 10^7$ m gives roughly $2.0$–$2.4$ years. This is propagation near $10^{\circ }$ latitude, not an equatorial $f=0$ calculation. Slow Rossby wave adjustment supplies a basin-scale delay relevant to El Niño–Southern Oscillation: changing thermocline depth can affect later sea surface temperature and coupled wind forcing. Reflection and coupling may help produce interannual memory, but this travel-time estimate alone does not establish a climate oscillation or its period.

The temperature of a specified near-surface ocean layer. Skin and bulk measurements refer to different sampling depths. Coupled variations in sea surface temperature, winds and thermocline depth are central to El Niño–Southern Oscillation.