Tropical Pacific Ocean

As the Atlantic Ocean, the Pacific is subject to the resonance of equatorial waves formed from the first baroclinic mode (vertical) Kelvin waves, and the first baroclinic mode, first meridional mode Rossby waves. However, while under the effect of forcing resulting from wind stress the resonance occurs at a frequency of one cycle per year in the Atlantic Ocean, due to the width of the basin, 17,760 km instead of 6,500 km, the period of tropical waves in the Pacific is necessarily multiyear. The resonant basin mode produces the El Niño phenomenon well known for its meteorological effects on a global scale (Pinault, 2015).


The annual wave



On the other hand, an annual quasi-stationary wave is observable north of the equator between latitudes 0°N and 12°N. The dynamic representation of the quasi-stationary wave shows two long-crested waves, almost in opposite phase, which extend from the eastern coasts of Southeast Asia to Central America, suggesting that they result from resonant forcing of the first baroclinic mode, fourth meridional mode Rossby wave. Meridional modes indeed show an increasing number of zonal energetic strips as the mode increases (one for the first mode, two strips for the second and so on).

The nodes are the two zonal surface currents flowing north and south of the equator, i.e. between 4.5°N and 7.5°N, forming the North Equatorial Counter-Current, and 0°N, 4.5°S, forming the South Equatorial Current. The period of the modulated currents is one year, too. The modulated South Equatorial Current feeds the western boundary currents, the outlet being located near the equator, i.e. off North Maluku and north Sulawesi islands in the Indonesian archipelago.

Evolution of the annual quasi-stationary wave

Only the stimulation of a high meridional mode can indeed explain the structure in strips of the wave and the two nodes whose analysis reveals that they are only one, being highly correlated. The antinodes are mainly visible in the northern hemisphere when they should appear antisymmetric in the southern hemisphere: only the southernmost antinode is visible west of 160°W, in phase with the southernmost antinode in the northern hemisphere, as the trade winds are weaker south of the equator.

The wavelength, which is 9400 km, is less than the width of the basin. The phase of the quasi-stationary wave thus reverse at both ends, which may explain the large variability from one cycle to the other of the observed waveform. In the illustrated realization, the speed of the modulated component of the Counter-Current peaks in September-October in the northern hemisphere, as it flows eastward, in phase with the South Equatorial Current in the southern hemisphere. This occurs while warm water has been transferred to the southernmost antinode in each hemisphere, then a few months later at the northernmost antinode.

The quadrennial wave



The pattern of antinodes and nodes of the quadrennial wave recalls what is observed in the tropical Atlantic (the frequency representation of the sea surface height and the speed of the geostrophic current near the equator west of the basin show that the period is about 4 years with a high variability). As in the Atlantic basin this mode shows a main node where the modulated equatorial current extends into the western half of the basin, two antinodes on both sides of the equator west of the basin and an equatorial antinode: the central-eastern antinode results from the superposition of the first baroclinic mode, first meridional mode Rossby wave and a Kelvin wave, both trapped by the equator, but propagating in opposite directions. Western antinodes on the one hand and the central-eastern antinode secondly separate the Pacific into two parts where the thermocline oscillates almost in phase opposition.

The north-western antinode forms a curl joining the eastern coast of central and southern Philippines, overlapping the North Equatorial Current in its northern part, and riding the North Equatorial Counter-Current and the South Equatorial Current in the Southern part. A Rossby wave, which is reflected against the eastern coast of the Indonesian archipelago, spreads along the curl, driven by the North Equatorial Counter-Current.

The south-western antinode forms a tongue extending to the coast of north-eastern New Guinea through the Solomon Islands. It also proceeds from a Rossby wave that is reflected against New Guinea, driven by the South Equatorial Current.

The main node is the modulated component of the South Equatorial Current flowing westward along the equator between latitudes 0°N and 8°S.

Evolution of the quadrennial quasi-stationary wave

The evolution of quasi-stationary waves during a cycle can be expressed relative to the ENSO event that occurred during the cycle (Pinault, 2018b). The westward phase propagation of the quasi-stationary wave along the equator begins when the ridge is reflected against the South American coast, during the maturation stage of the ENSO event. It lasts almost two years during which the thermocline rises along the central-eastern antinode. At the end of the westward phase propagation, the main modulated current, which then flows to the west, reaches its maximum speed and partially leaves the equatorial belt to feed the western boundary currents. In the absence of a powerful counter-current south of the equator, which prevents any reflection of the equatorial ridge to the south-western antinode, the resonant wave is partly reflected to the curl forming the north-western antinode. The westward phase propagation of the resonant wave along the equator stimulates upwelling at the eastern limit of the basin, i.e. the South American coast, while the trough of the wave is formed along the central-eastern antinode. Thus, in the central and eastern part of the basin cold water gradually replaces the warm water leaving the equatorial belt.

When the phase is ± 2 years relative to the ENSO event, under the effect of winds western antinodes form a ridge with the deepening of the thermocline in the « warm water pool » down to 250 m. The ridge of the north-western antinode as well as the speed of the North Equatorial Counter-Current flowing eastward reach their maximum at the same time as the south-western antinode. Meanwhile, the trough deepens at the central-eastern antinode, stimulating the migration of warm water from the western antinodes, replacing cold water while upwelling weakens off the South American coast.

As in the Atlantic Ocean, the north-western antinode plays the role of « tuning slide », but the propagation time along the curl is short compared to the period of the quasi-stationary wave, so that only a fine tuning of the period occurs. Again, the south-western antinode acts as a heat sink.

Coupling of basin modes

The functioning of the quadrennial quasi-stationary wave cannot be dissociated from the ENSO. Indeed, El Niño events are triggered at a critical time in the cycle of the wave when, at the end of the eastward phase propagation, the ridge reaches the west coast of South America. These El Niño events stimulate evaporation from the surface of the central-eastern equatorial anomaly, which cools the mixed layer, and thus raises the thermocline. ENSO is also a way of forcing of the resonant wave because it stimulates the propagation of the ridge to the west. More La Niña, which announces the resumption of the Walker circulation with increased surface stress from easterlies, is also a way of forcing because it becomes effective after El Niño, so during the westward phase propagation of the ridge.

However, the equations of motion show that forcing associated with ENSO is not sufficient to explain the amplitude of antinodes and speed of modulated currents. Thus, a coupling between annual and quadrennial basin modes has to be invoked. The modulated components of the North Equatorial Counter-Current and South Equatorial Current, which form an integral part of the annual quasi-stationary wave, indeed merge with the main node of the quadrennial quasi-stationary wave along a narrow equatorial strip west of 150°W. The special location of the island of Papua New Guinea, near the equator, modifies the current lines of zonal flows and instabilities are exacerbated along a narrow line between 136°E and 141°E longitude and between 0 N° and 2°N latitude.

These instabilities result from the North Equatorial Counter-Current that flows closer to the equator and in the amplification of the modulated current acceleration common to both basin modes. Off the Cape d’Urville 137.5°E 0.5°N the North Equatorial Counter-Current may accelerate from 0 to 1.5 m/s within one month, which is considerable. Some of these accelerations are harbingers of an ENSO event. In this case the current accelerates rapidly eastward and its speed decreases before increasing again, reaching a maximum two months later. The geostrophic current off the Cape d’Urville is perturbed at an early stage of the development of an ENSO event, and then returns to its original speed. This occurs 7 times late 1992 to mid-2015: the corresponding ENSO events occur in 08/1994, 11/1997, 12/2002, 12/2004, 12/2006, 11/2009 and 09/2012 (an event is maturing in June 2015).

a) The geostrophic current observed off the Cape d'Urville 137.5°E, 0.5°N. The dotted lines indicate the different ENSO events to their maturity. The arrows show the acceleration of the current associated with each of the ENSO events – b) Gross signal –SOI and filtered in the band 1.5-15 years. ENSO events occur to the maxima of the filtered signal.
a) The geostrophic current observed off the Cape d’Urville 137.5°E, 0.5°N. The dotted lines indicate the different ENSO events to their maturity. The arrows show the acceleration of the current associated with each of the ENSO events – b) Gross signal –SOI and filtered in the band 1.5-15 years. ENSO events occur to the maxima of the filtered signal.

On the other hand, these instabilities anticipate the maturation stage of ENSO events of 4 to 6 months, this time depending on how the surface temperature anomalies develop during the evolution of ENSO. The acceleration of the North Equatorial Counter-Current that flows closer to the equator at the western part of the basin stimulates a baroclinic Kelvin wave, crossing the basin from west to east in two months and causing a deepening of the thermocline in the central-eastern part of the basin. Geostrophic forces in the tropical basin make that two Kelvin waves cannot succeed in less than one year and a half. This is the minimum time required for a complete cycle of the quasi-stationary wave to occur, the tilt along the equator of the sea surface having to promote the propagation of Kelvin waves to the east. Therefore all the current accelerations, whose average period is one year, do not produce a Kelvin wave. This is why some accelerations, even of large amplitude as that of 2003, does not produce ENSO event. In this case geostrophic forces remain confined to the west of the basin.

The periods of coupled oscillators

Applied to the case of coupled ocean waves, the theory of Sub-harmonic mode locking in coupled oscillators with inertia indicates that the average periods of the coupled waves are multiples of the average period of the fundamental wave, i.e. one year here when we consider that the trade wind cycle is the temporal reference of the tropical basin. The quadrennial wave is indeed subject to a sub-harmonic mode locking of the annual mode (Pinault, 2018c). This holds true irrespective of the variability of the period from a cycle to another. As a result, the average periods of the two basin modes are precisely 1 and 4 years. This latter period is unambiguously determined from the distribution of ENSO events.

The 8-year period sub-harmonic



A sub-harmonic whose average period is 8 years exhibits three antinodes and a main node like the quadrennial quasi-stationary wave. The central-eastern antinode extends between 160°E and 130°W. The north-western antinode is located off the Philippines against which Rossby waves are reflected. The south-western antinode, parallel to the equator, stretches from the eastern Australian coast to 130°W between latitudes 25°S and 20°S. The functioning of this basin mode is reminiscent of the quadrennial mode. The central-eastern antinode is formed in the waveguide formed by the equator, the north-western antinode plays the role of « tuning slide » and the south-western antinode is a heat sink.

This basin mode involves Rossby and Kelvin waves whose phase velocities are necessarily lower than these of the quadrennial basin mode: the second baroclinic mode Rossby and Kelvin waves have to be invoked. The phase Velocity is nearly 1 m/s, i.e. less than half that of the first baroclinic mode. This second baroclinic mode (or vertical) not correspond to the oscillation of the thermocline, more precisely the interface at the basis of the pycnocline, depth of 200 to 250 m, but the oscillation of the interface atop the pycnocline at the depth of 125 m.

The main node coincides with that of the quadrennial basin mode west of 170°E, so that this mode is coupled to the previous basin modes of 1 and 4-year period: the average period is 8 years since a sub-harmonic mode locking occurs. This basin mode contributes to the ENSO less than the quadrennial basin mode because the central-eastern antinode interferes little with the cold currents in the eastern basin.

In this way, the outlet of the tropical Pacific, where the modulated currents leave the basin to supply the western boundary currents, is common to the three basin modes. Thus the Pacific Ocean differs from the Atlantic Ocean, due to the superposition of strong modulated currents at the output of the tropical basin whose average periods are 1, 4 and 8 years.


A standing wave is the phenomenon resulting from the simultaneous propagation in different directions of several waves of the same frequency. In a standing wave nodes remain fixed, alternating with antinodes. A quasi-stationary wave acts as a standing wave but the antinodes and nodes may overlap.

Upwelling. Here, this term indicates an upward flow of deep water, therefore cold. The upwelling is associated with the functioning of tropical resonant waves.

In a homogeneous medium, propagation in a given direction of a monochromatic wave (or sine) results in a simple translation of the sinusoid at a speed called phase velocity or celerity. In a non-dispersive medium, the speed does not depend on the frequency (or wavelength). In this case every complex wave is the sum of several monochromatic waves that also undergo an overall translation of its profile, this without deformation. In contrast, in a dispersive medium the phase velocity depends on the frequency and the energy transported by the wave moves at a speed lower than the phase velocity, said group velocity.

SOI (Southern Oscillation Index). The SOI is the amplitude of the Southern Oscillation; it is a measure of the monthly change in the normalized atmospheric pressure difference at sea level between Tahiti and Darwin (Australia).

Geostrophic currents are derived from measurements of wind, temperature and satellite altimetry. The calculation uses a quasi-stationary geostrophic model while incorporating a wind-driven component resulting from wind stress. Geostrophic current thus obtained is averaged over the first 30 meters of the ocean.

The Walker Circulation, El Niño, La Niña. In the tropics, the direct airflow on the surface towards the equator (called Hadley cell) forms the inter-tropical convergence zone. The Coriolis force is small at these latitudes but enough to divert to the west the circulation, giving the trade winds (north-east in the northern hemisphere and south -east in the south). The Humboldt Current from Antarctica cools the coast of South America. So there is a large temperature difference between the western and eastern Pacific that leads to direct circulation similar to Hadley circulation (air masses rise close to Asia and Australia and down along the coast of South America). If convective activity decreases in the western Pacific, the eastward aloft flow decreases or stops, cutting cold air intake in the eastern Pacific and the return surface flow weakens. The opposite of El Niño is La Niña. Convection in the western Pacific increases in this case which amplifies cell Walker bringing colder air along the coast of America.

The fundamental quasi-stationary wave is in phase with forcing. In sound pipes, strings and vibrating membranes form harmonics whose period is a divisor of that of the fundamental wave. As regards the long ocean waves, sub-harmonics are formed whose period is a multiple of that of the fundamental wave as occurs for high rank baroclinic modes.

Like any system of resonantly forced coupled oscillators, quasi-stationary baroclinic waves oscillate in subharmonic modes, whether tropical or at mid-latitude. Their coupling occurs when they share the same modulated current (the node) at the origin of the exchanges between the antinodes (where the thermocline oscillates) in opposite phase.
The average period τ0 of the fundamental wave being annual according to the declination of the sun, the average periods of the subharmonics are deduced by recurrence. The period τm + 1 is deduced from the period τm so that τm+1 = nm τm where nm is an integer. The average periods of the main modes observed are 1, 4 and 8 years in the tropics (the average period of 4 years paces the El Nino phenomenon in the tropical Pacific). At mid-latitudes these are (in years) 1, 4, 8 = 4 × 2, 64 = 8 × 8, 128 = 64 × 2, 256 = 128 × 2 (solar forcing, Gleissberg cycle), 768 = 256 × 3 (solar forcing), 24576 = 768 × 32 (orbital forcing, precession), 49152 = 24576 × 2 (orbital forcing, obliquity), 98304 = 49152 × 2 (orbital forcing, eccentricity). The forcing efficiency is all the stronger as its period is closer to one of the periods of resonance of the climatic system.
To the long periods corresponds an integer number of turns made by the gyral Rossby wave around the gyre (anticyclonically) during half a period. This number of turns is the subharmonic mode. For the 128 year period the gyral Rossby wave travels 2 turns except in the South Pacific where it is 1 and the south of the Indian Ocean where it is 3/2.