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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.

Coupled variability of the tropical Pacific ocean and terrestrial atmosphere, including the warm El Niño phase, the cold La Niña phase, and associated changes in surface pressure, trade winds and the thermocline.

Kelvin waves: weakened trade winds, or a westerly wind burst, excite a downwelling equatorial Kelvin wave. Its positive upper-layer thickness anomaly travels eastward and deepens the eastern Pacific thermocline. Upwelling then entrains warmer water, favouring eastern Pacific surface warming and the development of El Niño. This is a propagating adjustment of the thermocline, rather than transport of an unchanged parcel of warm water across the entire basin.^[ref]

Gravity waves: changed tropical convection and its latent heat release force atmospheric internal gravity waves. Their rapid propagation spreads the associated pressure and buoyancy disturbances, allowing the tropical terrestrial atmosphere to adjust to the shifted heating and contributing to the divergent circulation and surface wind response coupled to the ocean. This is one atmospheric adjustment mechanism; it does not assert that all wind variability during a particular El Niño is caused by this wave family.

Rossby waves: wind forcing during El Niño also launches westward equatorial Rossby waves with a shoaling thermocline signal. Reflection at the western Pacific boundary can produce an eastward upwelling equatorial Kelvin wave, which shoals the eastern thermocline and promotes cooling. This supplies a delayed negative feedback that can help terminate the event.^[ref] These examples describe mechanisms relevant to events such as 2023/24, rather than claiming an event-specific attribution from the exam data.

On a beta plane, $f=f_0+\beta y$, the preceding vorticity equation isIn the broad ocean interior, assume a statistically steady flow, weak advection of relative vorticity, and negligible friction compared with planetary-vorticity advection. The constant-depth, constant-density assumptions of 2(a) exclude topographic and baroclinic vorticity generation contributions. More precisely, require $|{\xi }_t|,|\mathbf{u}\cdot \nabla \xi |,|r\xi |\ll |\beta v|$. Then Sverdrup balance isSmall Rossby number and slow forcing help justify these conditions in the interior; they do not justify them inside an intense western boundary current. For fluctuations, the mean eddy contribution to relative-vorticity advection must also be negligible.

The depth-integrated northward volume flux per unit zonal distance is $V=Hv$. For a basin $0<x<L$ and wind stress independent of $x$, the northward interior volume flux isA negative value denotes southward transport. The cancellation of $H$ is why this interior volume flux is insensitive to the chosen constant depth. In a closed, steady basin the net volume flux across a full latitude section is zero: a boundary return must compensate $T$. Thus the displayed formula cannot represent the total closed-basin transport including that return.

For a concrete circulation sketch, use a northern-hemisphere subtropical basin $0<y<B$ andThis represents westward trade winds at the southern edge and eastward westerlies at the northern edge. The negative wind stress curl drives southward interior flow. Define an ocean transport streamfunction by $Hu=-{\Psi }_y$, $Hv={\Psi }_x$ and impose no normal flow at the eastern wall through $\Psi (L,y)=0$. Integrating Sverdrup balance westward givesSince $V=0$ at $y=0,B$, this also gives no normal flow at the north and south walls. Interior water moves westward in the southern part, southward across the basin, and eastward in the northern part. A northward western boundary current completes the clockwise ocean gyre.

Pure Sverdrup balance cannot satisfy the western impermeability condition as well: ${\Psi }_I(0,y)=-LV(y)$ varies along that wall. Indeed imposing no normal flow on both east and west walls would require their constant streamfunction values to agree in this simply connected closed basin, whereas their difference here is the nonzero $T(y)$. The shaded return in the sketch represents a boundary region where the interior approximation breaks down. It is supplied by friction in 2(c), rather than inferred from Sverdrup balance alone. A southern-hemisphere subtropical ocean gyre has the reflected, anticlockwise circulation.

For an order-of-magnitude estimate, use ${\tau }_0=0.1\mathrm{N}{\mathrm{m}}^{-2}$, ${\rho }_0=10^3\mathrm{kg}{\mathrm{m}}^{-3}$, $\beta =2\times 10^{-11}{\mathrm{m}}^{-1}{\mathrm{s}}^{-1}$, $L=5\times 10^6\mathrm{m}$ and a wind stress variation scale ${\ell }_{\tau }=10^6\mathrm{m}$. ThenHere $1\mathrm{Sv}=10^6{\mathrm{m}}^3{\mathrm{s}}^{-1}$ is a Sverdrup. The sinusoidal example has maximum $|{\tau }_{s,y}|=\pi {\tau }_0/B$; taking $B=\pi \times 10^6\mathrm{m}$ makes it agree with the chosen variation scale. Other reasonable basin widths and wind profiles change the coefficient, but retain a transport of order tens of Sverdrups.