Pure Appl. Geophys.  2015 Springer International Publishing DOI 10.1007/s00024-015-1212-9

Pure and Applied Geophysics

Long Wave Resonance in Tropical Oceans and Implications on Climate: The Pacific Ocean JEAN-LOUIS PINAULT1 Abstract—The dynamics of the tropical Pacific can be understood satisfactorily by invoking the coupling between the basin modes of 1-, 4- and 8-year average periods. The annual quasistationary wave (QSW) is a first baroclinic-mode, fourth meridional-mode Rossby wave resonantly forced by easterlies. The quadrennial QSW is built up from a first baroclinic-mode Kelvin wave and a first baroclinic-mode, first meridional-mode Rossby wave equatorially trapped and two off-equatorial Rossby waves, their dovetailing forming a resonantly forced wave (RFW). The 8-year period QSW is a replica of the quadrennial QSW for the second-baroclinic mode. The coupling between basin modes results from the merging of modulated currents both in the western part of the North Equatorial Counter Current and along the South Equatorial Current. Consequently, a sub-harmonic mode locking occurs, which means that the average period of QSWs is 1-, 4- and 8-year exactly. The quadrennial sub-harmonic is subject to two modes of forcing. One results from coupling with the annual QSW that produces a Kelvin wave at the origin of transfer of the warm waters from the western part of the basin to the central-eastern Pacific. The other is induced by El Nin˜o and La Nin˜a that self-sustain the subharmonic by stimulating the Rossby wave accompanying the westward recession of the QSW at a critical stage of its evolution. The interpretation of ENSO from the coupling of different basin modes allows predicting and estimating the amplitude of El Nin˜o events a few months before they become mature from the accelerations of the geostrophic component of the North Equatorial Counter Current. Key words: Resonantly forced waves, coupling between basin modes, tropical Pacific, El Nin˜o.

1. Introduction Like the Atlantic and the Indian oceans (PINAULT 2013, 2015, 2016), the Pacific is subject to the resonance of non-dispersive equatorial long waves. Nevertheless, the resonant nature of the Atlantic and Pacific oceans is poorly documented while it is well

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known for the Indian Ocean. It is because a feature of the tropical Indian Ocean is that its width, which is 6300 km from the east coast of Africa to the west coast of Sumatra, is close to half the wavelength of a Rossby wave of biannual frequency, i.e. 12,100 km: biannual Rossby waves have been investigated in the equatorial Indian Ocean by several authors, suggesting a half-wave resonance for the first baroclinic mode (e.g. CANE and MOORE 1981; GNANASEELAN et al. 2008; GNANASEELAN and VAID 2010; HAN et al. 2011). Because the authors assume the zonal current u vanishes at the boundaries of the tropical basin, they introduce the concept of no-normal flow to skirt the fact that the basin width is not exactly equal to half the wavelength. This concept is artificial and is not based on a realistic physical basis since it supposes the Kelvin wave speed adapts to the width of the basin. What appears as a trick to satisfy the boundary conditions is not essential because the resonance of non-dispersive baroclinic waves is ubiquitous and, therefore, does not rely on the particularities of the Indian Ocean. This is because the westward-propagating Rossby wave is not really reflected at the western boundary but diverted to form off-equatorial Rossby waves carried by countercurrents. Owing to the geostrophic forces acting at the tropical basin scale, these Rossby waves recede to the western boundary to form an eastward-propagating Kelvin wave. Therefore, in average the natural period of baroclinic waves that fulfil the dispersion relation is tuned to the period of forcing. The ubiquity of resonantly forced waves (RFW) in the three tropical oceans is based on the fact that the wavelengths of the waves are of the same order or higher than the width of the basins. So, only one basin mode can survive to a particular frequency, competition between resonant and non-resonant modes acting in favor of resonant

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mode for which the maximum energy is captured. RFWs generate geostrophic currents that require an outlet to leave the tropical basin, which mainly occurs eastward in the Indian Ocean and westward in the Atlantic and Pacific oceans. In no case the resonance results from the spatial structure of the forcing. Simply, the forcing occurs at a critical stage of the RFWs. In the Pacific Ocean a resonant basin mode occurs due to the deflection close to the western boundary of the basin of equatorial and off-equatorial Rossby waves and the reflection against the eastern boundary of equatorial Kelvin waves. Same phenomena are observed in the Atlantic, however, While the Atlantic Ocean is resonantly forced under the effect of wind stress at a frequency of one cycle per year, due to the width of the basin, 17,760 km instead of 6500 km, the period of tropical waves in the Pacific is necessarily multiyear. The resonant basin mode produces the El Nin˜o phenomenon well known for its meteorological effects on a global scale. In fact, this statement concerns the first-baroclinic mode equatorial waves formed from the Kelvin waves, and the first-meridional mode Rossby waves. These waves have a simple latitudinal structure because they are symmetrical about the equator and form a single narrow zonal strip. But the equatorial Pacific Ocean holds two well-known particularities, namely an equatorial asymmetry and phase speed of Rossby waves slower than what is expected for the first-meridional mode, which has been attributed to a variety of different effects. However, the previous studies ignore the energetic band north of 4.5N that evidences resonant forcing of the fourth-meridional mode equatorial Rossby wave. Being anti-symmetric with respect to the equator and sea surface height (SSH) anomalies forming several zonal strips, longcrested baroclinic waves propagate, exchanging warm water from an energetic band to another. Such a resonant wave is forced by the winds and tuned to an annual period. The previous works, which try to show the influence of background currents on meridional modes, are very conceptual since either they apply to the meridional mode 0, which is symmetrical about the equator (e.g. CHELTON et al. 2003) or to

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meridional modes 1, 2 that are not observed (e.g. DURLAND et al. 2011). In this way, no concrete information can be inferred. In addition, the concept of background currents, which are part of the solution of the equations of motion, is somewhat ambiguous. This is why it is substituted by large-scale geostrophic forces here, to express exogenous influences acting in the tropical basin. Thus two quasi-stationary waves (QSW) can be clearly differentiated in the tropical Pacific, sharing the same modulated currents at the nodes. Coupling of these two basin modes has significant repercussions on the frequency of occurrence of ENSO. These new insights in physical oceanography of the tropical Pacific justify the reinterpretation of concepts that are currently admitted. The paper is organized as follows: (1) the methods that have already been used in PINAULT (2013) are recalled; (2) the annual RFW is investigated and the observations are interpreted by solving the equations of motion; (3) the same reasoning is applied to the quadrennial RFW; (4) the implications resulting from the coupling of the two basin modes are exposed; (5) an 8-year period RFW is investigated, and (6) it is shown how the ENSO is connected to the quadrennial cycle.

2. Method A wavelet analysis (PINAULT 2012) is preferred to a complex EOF analysis to investigate the resonance of long waves in the tropical ocean. If the two methods are similar for typical frequency domain analyses, i.e. the power spectral and coherency analyses, the time domain EOF analysis, which is basically the computation of eigenvector and eigenvalue of a covariance or a correlation matrix computed from a group of original time series data, is far different from the wavelet analysis that is well suited to highlight the time lag between time series when it varies continuously. Applied to SSH and surface current velocity (SCV) data, the wavelet analysis brings out the propagation of the waves as well as their variability from a cycle to another. In particular, by representing the amplitude and phase of SSH and SCV anomalies, the cross-wavelet analysis highlights quasi-stationary-waves (QSWs) that

Long Wave Resonance in Tropical Oceans and Implications on Climate: The Pacific Ocean

represent a single dynamical phenomenon within a characteristic bandwidth. Geostrophic forces closely constrain the behavior of the baroclinic waves at the limits of the basin, forming antinodes at the place of SSH anomalies and nodes where modulated geostrophic currents ensure the transfer of warm water from an antinode to another. Although these terms node and antinode are abusive because the phase of QSWs is not uniform, which supposes some overlapping, the paired figures devoted to the amplitude and phase of these QSWs merely reflect their evolution during a cycle. As concerns the equations of motion, they are solved by considering long-wavelength waves are infinite, i.e. the waves are simply deflected (not reflected) to the limits of the basin, unless they feed the boundary currents (PINAULT 2013). Thus the wave resonance conditions result from the match between the wavenumber k and the pulsation x that must satisfy the dispersion relation k = -(2 m ? 1)(x/ c) where c is the phase speed of the first-baroclinic mode Kelvin wave and m the meridional mode of Rossby waves. In particular, the zonal and the meridional geostrophic currents u and v do not vanish at the western limit of the basin. Furthermore, the waves are resonantly forced when the period of free waves coincides with the forcing period and the solution would be infinite if it were not regularized because of the Rayleigh friction.

3. Results and Discussion 3.1. The Annual Quasi-Stationary-Wave 3.1.1 The Observations The 1-year period QSW is represented in Fig. 1, amplitude and phase being averaged in the 8–16 months band (see also SCHARFFENBERG and STAMMER 2010). In Fig. 1a, b antinodes reveal two long-crested waves between latitudes 0N and 12N, almost in opposite phase, which extend from the eastern coasts of Southeast Asia to Central America. The period of the wave is proved to be 1 year from the Fourier power spectrum of SSH and SCV as shown in Fig. 2: the Fourier power spectrum of SSH at 6.5N 140.5W in the central Pacific exhibits a sharp peak centered at 1 year (Fig. 2a, b), as well as the Fourier power spectrum of the North Equatorial Counter Current (NECC) at 8.5N 129.5W (Fig. 2c, d) whose filtered signal reaches a maximum in October close to 0.5 m/s and vanishes in April. Those observations suggest that both nodes and antinodes result from resonant forcing of the firstbaroclinic mode, fourth-meridional mode Rossby wave. This hypothesis is supported by the two modulated zonal currents flowing north and south of the equator, i.e. between 4.5N and 7.5N, merging with the NECC, and between 0N, 4.5S, merging

Figure 1 Amplitude (left) and phase (right) relative to 08/1999, averaged in the 8–16 months band of a, b antinodes, c, d velocity u (facing east) at the nodes. Sea surface height (SSH) is provided by the French Centre National d’Etudes Spatiales (CNES): http://www.aviso.oceanobs.com. Geostrophic surface current velocity field (SCV) is obtained through the Ocean Surface Current Analyses: Real time (OSCAR) program and provided by the National Oceanic and Atmospheric Administration (NOAA): http://www.oscar.noaa.gov/datadisplay/datadownload.htm

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Figure 2 a, c, e Representation of 1-year period SSH and SCV series, filtered and unfiltered. b, d, f Corresponding Fourier power spectra (period of observation: 1992–2009) and confidence spectrum assuming a red-noise with a lag-1 (autocorrelation) a

Figure 3 a The geostrophic currents observed off the Cape d’Urville 137.5E, 0.5N (in red) and in the eastern Pacific (in blue). The dotted lines indicate the different El Nin˜o events during their maturity. b The correlogram of the currents: at 107.5W, 0.5N the current is 4–5 months ahead of the current at 137.5E, 0.5N

with the South Equatorial Current (SEC), as shown in Fig. 1c, d. Only the stimulation of a high meridional mode Rossby wave can explain the strip structure of the wave and the two nodes whose analysis reveals that they are only one, being highly correlated. The

magnitude-squared coherence of the modulated components of the NECC and the SEC is 0.5, which suggests both nodes are coupled (Fig. 3). The southern node is broad compared to the northern one and becomes narrower towards the west of 180 (Fig. 2c,

Long Wave Resonance in Tropical Oceans and Implications on Climate: The Pacific Ocean

d). As shown in Fig. 2e, f, the oscillations of the southern current velocity are erratic, which is highlighted by the broadness of the peak centered at 1 year in the Fourier spectrum (a modulated current produced by a 1/2-year period harmonic can also be hypothesized from Fig. 2e, f). Yet it remains to be explained why the antinodes are mainly visible in the northern hemisphere when they should appear as anti-symmetric in the southern hemisphere: only the western part of the more poleward strip in the southern hemisphere is observable between 140E and 165W, in phase with the more equatorward strip in the northern hemisphere. In fact, these anti-symmetric antinodes exist, as shown by the amplitude of SSH anomalies averaged along parallels. Mean SSH anomalies between longitudes 180E and 100W reach 0.028 m 8.5S, 0.057 m 0.5S, 0.056 m 0.5N and 0.059 m 8.5N. The noise is 0.017 m, and the standard deviation is 0.009 m in each case, which evinces two energetic bands in the southern hemisphere. Consequently, antinodes are to be considered as nearly antisymmetrical in each side of the equator, though of lower amplitude south of 5S (Fig. 4b).

The northern node, i.e. the modulated component of the NECC, flowing eastward (Fig. 2c), requires an outlet east of the basin. In the absence of strong modulated currents flowing poleward along the American coasts the modulated return current merging with the SEC is essential. So, the modulated components of the NECC and the SEC appear as forming a single stream. The modulated component of the SEC feeds the western boundary currents, the outlet of the tropical basin being located near the equator, i.e. off North Maluku and north Sulawesi islands in the Indonesian archipelago (Fig. 1c). The assumption that both SSH and SCV anomalies observed in the 8–16 month band represent a single dynamical phenomenon requires that the mass of warm water exchanged between the two antinodes evidenced north of the equator is conserved. Effectively, the volumes involved into these two antinodes are almost balanced, i.e. v ± r = 553 ± 41 km3 north and 470 ± 48 km3 south. These volumes are approximatively proportional to the mass of warm water displaced, supposing a constant ratio l throughout the tropical Pacific when the equations are resolved into a single vertical mode l = g/h, where g is the perturbation of the surface height and h the upward thermocline displacement. The mean SCV in the area [4.5N, 7.5N] 9 [120.5W, 140.5W] is v ± r = 0.115 ± 0.035 m/s and the standard deviation of the coherence phase is 14 days, which denotes the high variability of the amplitude of the node. In contrast the phase is reproducible, a characteristic property of QSWs. 3.1.2 The First-Baroclinic Mode, Fourth-Meridional Mode ROSSBY Wave

Figure 4 Schematic representation of the ridge of the Rossby wave (red) and Kelvin wave (blue)—a the waves associated with El Nin˜o events whose average period is 4 years. The time when the ridge is formed is expressed in years relative to the El Nin˜o event that occurs during the cycle. The time evolution of the trough is in phase opposition with respect to the ridge—b the fourth-meridional mode Rossby wave whose period is 1 year; the phase reverses at both ends. The numbers represent the months of the year when the crest is formed

Let us get to solve the equations of motion (‘‘Appendix 1’’) to check and specify prior assumptions quantitatively. The mathematical treatment of RFWs consists in solving the forced version of the linearized equations of motion, i.e. the two momentum equations, the equation of continuity and the potential vorticity equation (GILL 1982). However, the resolution of the equations of motion relies on a particularity, i.e. geostrophic forces acting at the basin scale are taken into account through additional forcing terms, added up to surface stress forcing

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Figure 5 Amplitude and phase relative to 08/1997 of the solution of the equations of motion for the first-baroclinic mode, fourth-meridional mode Rossby wave, resulting from resonant forcing of long ocean waves by wind stress—a, b antinodes, c, d velocity u (facing east) at the nodes, e, f velocity v (facing north) at the nodes. Monthly zonal and meridional wind stresses (surface Gauss U and V momentum fluxes) at the surface of the oceans are provided by the NOAA: http://www.esrl.noaa.gov/psd/data/timeseries/

terms (PINAULT 2013). The zonal velocity field u of the fourth-meridional mode Rossby wave is the modulated component of the NECC and the SEC that superimposes on the background currents. To obtain a realistic solution, geostrophic forces are taken into account, as shown in Fig. 21 (‘‘Appendix 2’’). The geostrophic constrain is introduced from forcing terms (1) and (2). The first one adds to the longitudinal component of wind forcing, and the second one to evaporation. They do not depend on the longitude. The forcing term (1) informs about the annual periodicity, phase and amplitude of longitudinal geostrophic forces. The term u3, that is fixed empirically (see Fig. 21), has the dimension of a velocity. The forcing term (2) informs about the annual periodicity, phase and amplitude of vertical geostrophic forces. The term g3, which is expressed according to u3, has the dimension of a length. The solution shows two ridges almost in phase opposition (Fig. 5a, b) and a main modulated zonal

current in each of the two hemispheres (Fig. 5c, d). The speed of the NECC, which flows eastward, reaches a maximum in May in the northern hemisphere; the SEC flows westward (in November in Fig. 5) in the southern hemisphere. This occurs while warm water has been transferred to the northernmost antinodes located in both hemispheres. These directions are reversed in November when warm water has been transferred to the southernmost antinodes in both hemispheres. Like the sea height and the zonal current anomalies, meridional currents, which are nearly five times lower than the zonal currents, are subject to a phase inversion at each end of the basin because the wavelength of the QSW is shorter than the width of the basin (Fig. 5e, f). These meridional currents converge towards zonal currents and diverge periodically in quadrature (ahead of a quarter period) compared to zonal currents and antinodes. The solution reproduces the observations quite accurately as concerns the amplitude and phase of the

Long Wave Resonance in Tropical Oceans and Implications on Climate: The Pacific Ocean

Figure 6 Amplitude (top) and phase (bottom) averaged in the 8–16 months band of the profiles of temperature anomalies at 5N latitude and 137E to 95W longitude. The phases are expressed in months relative to 03/2009. The data are provided by the NOAA from the global network of tropical moored buoys: Tropical Atmosphere Ocean Project TAO project http://www.pmel.noaa.gov/tao/

antinodes in the northern hemisphere and nodes. However, the mitigation of the antinodes, observed south of 5S because of the weakness of wind stress forcing, is not reflected by the solution. Furthermore, the solution exhibits four nodes whereas only two are observed. The main discrepancies between the solution and the observations relate to the meridional structure of the fourth-meridional mode Rossby wave that does not depend on the forcing terms in the solution of the linearized equations of motion as outlined in GILL (1982) and PINAULT (2013), a limitation of the classical theory. Concerning the longitudinal structure of the solution, in the central Pacific in the northern hemisphere, the observed phase of the more equatorward antinode is not uniform (Fig. 1b), unlike what shows the solution (Fig. 5d), which may be due to effects resulting from the phase reversal at the two ends of the basin. Measurements of the temperature of seawater carried out by a network of moored buoys at 5N (Global Tropical Moored Buoy Array) confirm the phase inversion of the thermocline depth at both ends of the basin while the thickness of the thermocline increases to the west, but decreases to the east: the warm water layer is advected in October in the central Pacific, in January to the west and in June to

the east. The wavelength is close to what is expected, i.e. 9800 km (Fig. 6), a confirmation of the resonance of the fourth-meridional mode Rossby wave and, consequently, of the adequacy of the observed and computed phase velocities. 3.2. Quadrennial Quasi-Stationary-Waves The pattern of antinodes and nodes in the broad band 1.5–7 years as shown in Fig. 7 recalls what is observed in the tropical Atlantic, which confirms the implication of first meridian mode Rossby waves in the resonance of quadrennial QSW. Indeed, the frequency representation of the SSH 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 (Fig. 8). Like in the Atlantic basin the QSW shows a main node, extending into the western half of the basin (Fig. 7c, d), two antinodes on both sides of the equator west of the basin and an equatorial antinode (Fig. 7a, b). However, unlike what is observed in the Atlantic Ocean, antinodes and nodes exhibit a broad band in the Fourier power spectrum. In particular, the Fourier power spectrum of SSH at 2.5N 135.5E in the western basin shows a main peak centered at 4 years

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Figure 7 Amplitude (left) and phase relative to 11/1997 (right) averaged in the broadband 1.5–7 years of a, b antinodes, c, d velocity u (facing east) at the nodes; -SOI (the opposite of SOI) is used as the reference signal so that the phase is expressed relative to the maximum of -SOI, i.e. the maturation phase of El Nin˜o in 11/1997. The monthly Southern Oscillation Index (SOI) is provided by the National Centre for Atmospheric Research: http://www.cgd.ucar.edu/cas/catalog/climind/soi.html

(Fig. 8a, b). It is the same for the South Equatorial Current (SEC) at 0.5S 155.5E, a modulated current flowing mostly westward (Fig. 8c, d). Although the geostrophic current reverses erratically, the velocity filtered in the 8–16 month band shows a quasiperiodicity, vanishing regularly. The 4-year period sub-harmonic of the NECC at 5.5N 135.5E is better resolved as shown in Fig. 8e, f. Due to the variability of the period of the node and antinodes, restitution of SSH and SCV signals after filtering requires a 1.5- to 7-year bandwidth (Fig. 8d). Like in the Atlantic Ocean, the central-eastern antinode results from the superposition of the firstbaroclinic mode, first-meridional mode Rossby wave and a Kelvin wave, both trapped by the equator, but propagating in opposite directions (DELCROIX et al. 1994). Western antinodes on the one hand and the central-eastern antinode on the other separate the tropical Pacific into two parts where the thermocline seesaws almost in phase opposition. Here again the QSW represents a single dynamical phenomenon as evidenced from the estimation of the volumes of antinodes (two cycles are available). They are 147 and 98 km3 for the north-western antinode, 165 and 114 km3 for the south-western one, i.e. 312 and 212 km3 for the western Pacific while they are 303 and 196 km3 for the central-eastern antinode. This

shows a large variability in the amplitude of the antinodes but a good balance between warm water masses during a cycle. The velocity at the node averaged in the area [2.5S, 0.5N] 9 [150.5E, 175.5E] is v ± r = 0.050 ± 0.016 m/s and the standard deviation of the phase is 12 days. This shows the large variability of the amplitude of the node, whereas the phase evidences a high reproducibility. The north-western antinode forms a curl joining the eastern coasts of the central and southern Philippines included between 8N and 15N, i.e. Bicol Region, Eastern Visayas and Caraga, and eastern Islands of the Indonesian Archipelago included between 2S and 5N, i.e. West Irian Papoazia, North Maluku and North Sulawesi. The eastern turning point of the curl is located between 145E and 170E in longitude. The southern branch straddles the NECC and the SEC. The northern branch merges with the North Equatorial Current (NEC). The coherence phase around the curl is nearly constant, i.e. in opposite phase with the eastern end of the central-eastern antinode. A Rossby wave, which is reflected against the eastern coast of the Indonesian archipelago, spreads along the curl, driven by the NECC (PAZAN et al. 1986). The south-western antinode forms a tongue extending southeast northwest, from 20S 160W to

Long Wave Resonance in Tropical Oceans and Implications on Climate: The Pacific Ocean

Figure 8 a, c, e, g Representation of 4-year period SSH and SCV series, filtered and unfiltered. The response of SSH to the ENSO event occurring in 11/1997 is visible in a—b. d, f, h Corresponding Fourier power (period of observation: 1992–2009) and confidence spectra assuming a rednoise with a lag-1 a

the north-eastern coasts of Papua New Guinea, through the Solomon Islands. It is in phase with the north-western antinode although the southern edge of the tongue is lagging behind the main antinode by 0.33 years. Contrarily to the eastern boundary, the western boundary is not an impenetrable barrier to the planetary waves; the western antinodes expand beyond this barrier in the Indonesian Archipelago and they envelop the north western coast of Australia (Fig. 7a, b). The south-western antinode results from

the formation of baroclinic Rossby waves that reflect against Papua New Guinea (PAZAN et al. 1986) while inducing the reversal of the SEC, like what occurs in the southern branch of the north-western antinode. This similarity is reflected in the Fourier power spectrum of the SCV at the node, exhibiting two peaks well resolved at 1 and 4 years (Fig. 8g, h). A major node is evidenced in Fig. 7c, d, i.e. a reversing current that spans the Pacific, whose velocity increases towards the west. It flows to the easternmost

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Island of the Indonesian Archipelago straddling the equator, i.e. the West Irian Papoazia and the North Maluku. It follows the equator between latitudes 0 and 8S merging with the SEC. The node associated with the southern branch of the curl at the north-western antinode, which merges with the NECC, is an eastward modulated current that grows off the coasts of Philippines and follows the parallel 8N, in phase with the antinode (Figs. 8a, b, e, f). The phase is indeed nearly opposite to that of the SEC flowing westward south of the curl. In the same way, the reversing current associated with the south-western antinode (Fig. 8g, h) is nearly in phase with the propagation of the crest along the antinode, i.e. in phase opposition with the SEC. 3.2.1 Evolution of the Quadrennial QSW In the absence of a significant counter-current south of the equator, which prevents any possibility of reflection of the equatorial ridge to the south-western antinode, the resonant wave is partly deflected to the north-western antinode. Approaching the western boundary, the westward propagating Rossby wave, equatorially trapped, is poorly reflected as a Kelvin wave: ZANG et al. (2002) estimate the reflection efficiency at the western boundary of the Pacific (for semi-annual Rossby waves reflecting into Kelvin waves) to be no more than 31 %. It is probably close to zero for the quadrennial Rossby wave for which the equatorial Rossby wave is mainly diverted to the north as an offequatorial Rossby wave. This is enabled because the Rossby wave whose westward phase velocity is 0.3 m/s at 8N is embedded into the NECC whose eastward velocity at that time is higher as shown in Fig. 8e. This suggests geostrophic forces oppose to the refection of the Rossby wave and to the eastward propagation of a Kelvin wave while a Doppler shift occurs along the NECC, which allows the eastward propagation of troughs and ridges off the coasts. In turn, growing of the north-western antinode generates an eastward current, which intensifies the NECC that can, therefore, be considered as a component of the node of the quadrennial QSW in its western part (Fig. 8e, f). One now turns to the evolution of QSWs during a cycle by expressing the phase of the nodes and antinodes relative to the signal -SOI, i.e. the El Nin˜o event that occurs during the cycle as shown in

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Fig. 4a. The westward phase propagation of the QSW along the equator begins when the ridge is reflected against the South American coast, during the maturation stage of the El Nin˜o event. It lasts almost 2 years during when the thermocline rises along the central-eastern antinode. At the end of the westward phase propagation, the main node, which then flows to the west, reaches its maximum speed and partially leaves the equatorial belt to feed the western boundary currents. 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 El Nin˜o 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 (Fig. 6). The ridge of the north-western antinode and the speed of the NECC flowing eastward reach their maximum at the same time as the south-western antinode. Meanwhile, the trough deepens at the central-eastern antinode, prompting 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 is short compared to the period of the QSW so that only a fine tuning of the period occurs. Again, the south-western antinode acts as a heat sink. 3.2.2 Coupling of Basin Modes The functioning of the quadrennial QSW cannot be dissociated from the ENSO. Indeed, El Nin˜o events are triggered at a critical time in the cycle of the QSW when, at the end of the eastward phase propagation, the ridge reaches the west coast of South America. These El Nin˜o events stimulate evaporation from the surface of the central-eastern antinode, which cools the mixed layer, and thus raises the thermocline while stimulating the propagation of the ridge to the west. Moreover, La Nin˜a, which announces the resumption of the Walker

Long Wave Resonance in Tropical Oceans and Implications on Climate: The Pacific Ocean

circulation with increased surface stress from easterlies, is also a way of forcing because it becomes effective after El Nin˜o, so during the westward phase propagation of the ridge. Strengthening winds during La Nin˜a reduce cloud cover in the tropical Pacific, which allows shortwave (visible sunlight) to warm the ocean. Thus, La Nin˜a helps recharge the warm water pool in the Western Pacific. It is this mass of warm water, which is under a thermal barrier formed by a water layer of low density (less salty), which will arouse the following El Nin˜o episode, transport to the central-eastern antinode being performed by a Kelvin wave. The quadrennial sub-harmonic appears as selfsustained because forcing associated with El Nin˜o and La Nin˜a occurs during a critical phase of its evolution, which explains the high variability of its period in the absence of an exogenous periodic forcing. However, the equations of motion show that forcing associated with the ENSO is not sufficient to explain the amplitude of observed antinodes and nodes. Thus, a coupling between annual and quadrennial basin modes has to be invoked, because of the merging of the nodes of the annual and quadrennial QSWs. In particular, Figs. 1c and 7c evidence the nodes merge along a narrow equatorial strip west of 150W. The special location of the island of Papua New Guinea, near the equator, modifies the current lines of the NECC that shift over time along a narrow line between 136E and 141E longitude and between 0N and 2N latitude (Fig. 3a). Off the Cape d’Urville 137.5E 0.5N the NECC may accelerate from 0 to 1.5 m/s within 1 month while it approaches the equator (Fig. 3). Some of these accelerations are harbingers of an El Nin˜o event. In this case the current accelerates rapidly eastward and its speed decreases before increasing again, reaching a maximum 2 months later. The momentary fall of the geostrophic current reflects the change in geostrophic forces, thus altering the tilt along the equator of the sea surface when the Walker circulation and, consequently, easterlies weaken. This occurs six times between late 1992 to late 2012: in 08/1994, 11/1997, 12/2002, 12/2004, 12/2006 and 11/2009. Geostrophic forces responsible for the acceleration of the NECC and its approach to the equator promote the rebound of Rossby waves from

the western antinodes to form an equatorial Kelvin wave crossing the basin from west to east in 2 months and causing a deepening of the thermocline at the central-eastern antinode. The acceleration of the NECC off the Cape d’Urville anticipates the maturation stage of El Nin˜o events of 4–6 months, this time depending on how the surface temperature anomalies develop during the evolution of ENSO. Geostrophic forces in the tropical basin involve two Kelvin waves cannot succeed in less than the minimum time required for a complete cycle of the quadrennial QSW, which is reflected by the tilt along the equator of the sea surface. This time can be estimated around 1.5 years, which is indicative since it is included between 1 year and 2 years from observation of the time elapsed between two successive El Nin˜o events. Therefore, all the current accelerations, whose average period is 1 year, do not produce a Kelvin wave. This is why some accelerations, even of large amplitude as that of 2003, do not produce El Nin˜o event. In this case geostrophic forces remain confined to the west of the basin. 3.2.3 First-Baroclinic Mode, First-Meridional Mode Rossby Waves Like for the Atlantic Ocean, three sets of equations of motion are solved in relation with the south-western, north-western and equatorial antinodes (‘‘Appendix 2’’). In the equatorial belt, computed and observed phase of antinodes and nodes are very similar (Figs. 7b, d, 9b, d). However, the observed velocities of nodes are lower than those calculated because the observed central-eastern antinode is less extended to the west than that calculated, probably because the forcing terms used in the equations of motion do not accurately reflect large-scale geostrophic forces in the equatorial belt. However, the solution allows supporting the assumptions on the functioning of the coupled modes. Dynamics of equatorial waves can be summarized as follows: 1. The modulated component of the NECC observed off the Cap d’Urville may indicate the propagation

J.-L. Pinault

Pure Appl. Geophys.

Figure 9 Amplitude and phase relative to 11/1997 of the solution of the equations of motion for the quadrennial RFW—a, b antinodes, c, d velocity u (facing east) at the nodes, e, f velocity v (facing north) at the nodes

of a Kelvin wave, which supposes geostrophic forces extend beyond the western basin. Otherwise, geostrophic forces are confined in the western basin. 2. An El Nin˜o event is triggered a few months later. 3. The Kelvin wave ‘‘rebounds’’ against the eastern boundary as a Rossby wave. Actually this Kelvin wave partly propagates to the north along the American coast as a coastal trapped Kelvin wave and then recedes at the discretion of coastal geostrophic currents as shown in Fig. 7a, b. 4. The Rossby wave is diverted to the North when approaching the western boundary, following the NECC, then recedes to the west nearly in phase with the southern antinode. 3.2.4 The Periods of Coupled Oscillators Using the equations of motion, JIN et al. (1996) showed that, under certain conditions, the average

frequency of El Nin˜o events is a sub-harmonic frequency of seasonal winds. In other words, the average period of ENSO is a rational number of years. Although the assumptions formulated by the authors are different from those presented here, in particular the model they use is not based on the coexistence of the fourth and first-meridional mode Rossby waves, some properties are universal when they relate to coupled oscillators. Viscosity that justifies the coupling between basin modes does not appear in the equations of motion that suppose sea water inviscid. CHOI and THOULESS (2001) showed that in very general terms, a sub-harmonic mode locking occurs in coupled oscillators (‘‘Appendix 4’’). Applied to the case of coupled QSWs of different frequencies, the coupling results from processes involving viscosity of seawater. Consequently, the average period of the quadrennial QSW, a sub-harmonic of the annual QSW that is resonantly forced by trade wind stress, is an integer number of

Long Wave Resonance in Tropical Oceans and Implications on Climate: The Pacific Ocean

Figure 10 Amplitude (left) and phase relative to the El Nin˜o that is mature in 12/2004 (right) averaged in the 7.5- to 8.5-year band of—a, b antinodes, c, d velocity u (facing east) at the nodes

years. As such, the annual QSW can be considered as the fundamental wave. 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 El Nin˜o events as one will see. 3.3. The 8-Year Period Sub-Harmonic A sub-harmonic whose average period is 8 years is shown in Figs. 10 and 11, with three antinodes and a main node like the quadrennial QSW. The centraleastern antinode extends between 160E and 130W. The north-western antinode is located off the Philippines against which Rossby waves are deflected. The south-western antinode, parallel to the equator, stretches from the eastern Australian coast to 130W between latitudes 25S and 20S. The functioning of this QSW is reminiscent of the quadrennial QSW. 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. The volumes involved in antinodes are (only one realization is available) 255 km3 in the central Pacific, 105 km3 north-west and 75 km3 south-west. They are roughly balanced between the two western antinodes (the total volume is 180 km3) and the

central-eastern antinode, which confirms that the QSW represents a single dynamical phenomenon. Moreover, the modulated current at the main node coincides with that of the 4-ye¨ar period QSW to the west of 170E (Fig. 10c, d). This basin mode involves Rossby and Kelvin waves whose phase velocities are necessarily lower than these of the quadrennial basin mode: the secondbaroclinic mode Rossby and Kelvin waves have to be invoked in this basin mode. The phase velocity is nearly 1 m/s, i.e. less than half that of the firstbaroclinic mode: it is obtained by considering three superposed layers 0-125-255-1750 m whose respective densities are 1023.03, 1025.52 and 1031.58 kg/ m3. It is supposed eigenvectors vanish around 1750 m (VALSALA 2008). The main node coincides with that of the quadrennial QSW west of 170E, so this mode is coupled to the previous basin modes of 1- and 4-year average period: the mean period is 8 years since a sub-harmonic mode locking occurs. This QSW contributes to the ENSO less than the quadrennial QSW because the central-eastern antinode that vanishes east of longitude 110W 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

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Pure Appl. Geophys.

Figure 11 a, c, e Representation of 8-year period SSH and SCV series, filtered and unfiltered. b, d, f Corresponding Fourier power (period of observation: 1992–2009) and confidence spectra assuming a red-noise with a lag-1 a

from the Atlantic Ocean, due to the superposition of strong nodes at the output of the tropical basin whose average periods are 1, 4 and 8 years. 3.4. The Major El Nin˜o Events from 1867 to 2013 One of the main characteristics of the tropical Pacific is the genesis of El Nin˜o events whose study is facilitated by the availability of long time series of the Southern Oscillation Index (SOI) and the sea surface temperature (SST). These series are much longer than those of SSH, thus improving the representativeness of what is deduced from the observations of ENSO. First, the frequency representation of the Southern Oscillation (Fig. 12) shows that El Nin˜o events have a large bandwidth, which denotes the variability of their period. The Fourier spectrum of the SOI indeed shows a peak centered about 4 years and a broad peak extending beyond 6.5 years, frequencies ranging from 1 cycle/1.5 years to about 1 cycle/30 years. The ENSO reflects the seesaw of the thermocline whose deepening alternates between the western and eastern tropical Pacific, which induces changes in the atmospheric pressure difference between these two parts of the basin at the same time as the SST

Figure 12 Fourier spectrum (frequency representation) of the SOI over the period 1867–2013. The confidence spectrum assumes a red-noise with a lag-1 a = 0.61

anomaly in the central-eastern part of the basin. Filtering the signal -SOI within a bandwidth included between 1.5 and 15 years and taking for selection criterion that the filtered value -SOI is higher than 0.85, the 36 most significant events from 1867 to 2013 have an average period of about 4 years. Filtering allows to select events not only according to their amplitude but also their duration (Fig. 13; Table 1). Since they measure the fall in the pressure difference between the eastern and western parts of the Pacific during ENSO, the maxima of the filtered

Long Wave Resonance in Tropical Oceans and Implications on Climate: The Pacific Ocean

Figure 13 Raw signal -SOI and filtered in the band 1.5–15 years

signal are proportional to the amplitude of these episodes. The classification of events according to their amplitude A allows determining their return period. Three events whose return period is the highest (higher than or equal to 50 years) occur in 05/1905, 08/1877 and 12/1982 (Figs. 13, 14). 3.4.1 The Outbreak of ENSO During the Quadrennial Cycle Deep phenomena associated with ENSO events are represented from the amplitude and the phase of monthly seawater temperature profiles (Fig. 15). Although the four major ENSO events are triggered at different timing within the 4-year length intervals (Table 1), the phases associated with these events at 0N 110W present some similarities. The lag between when the core of the warm water layer is translated to the central-eastern antinode and the ENSO event becomes mature is a few months, varying according to the event. Such profiles are observable in the eastern Pacific, the depth of the

thermocline varying from 150 to 200 m. In the central Pacific, the thermocline is deeper, as shown if Fig. 16e, f. As shown in Figs. 15 and 16, deepening of the thermocline follows the advection of the core of the warm water layer as well as the expansion of the warm water layer up to the surface. Figure 15 evidences the temporal variability of the advection of warm water to the central-eastern antinode. Furthermore, the thermal energy transferred may vary substantially from a cycle to another, which results from both the thickness of the warm advected layer and its average temperature. Peaks in the amplitude of SOI associated with events occurring in 11/1997 and 12/2006 are among the highest whilst the maximum temperature anomaly overreaches 3 and 2 C, respectively, just before the events are mature (Fig. 16). Finally, the profiles of thermal anomalies in the western Pacific show warm water pool is contained between a thermal barrier through which thermal transfer to the surface is slow, which gives the western

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Pure Appl. Geophys.

Table 1 ENSO events N

Date

Lag (years)

Amplitude

RI (years)

15 24 25 2 32 6 34 7 3 11 14 9 26 8 27 10 29 17 21 28 35 12 23 5 16 1 18 19 39 31 4 30 20 22 13 36

07/1868 12/1870 11/1873 08/1877 07/1885 10/1888 06/1891 11/1896 05/1905 12/1911 09/1914 03/1919 10/1923 01/1926 08/1932 03/1941 08/1946 10/1951 07/1953 11/1957 10/1963 10/1965 08/1969 09/1972 09/1977 12/1982 05/1987 12/1991 04/1993 08/1994 11/1997 12/2002 12/2004 12/2006 11/2009 08/2012

-1.50 0.92 -0.17 -0.42 -0.50 -1.25 1.42 -1.17 -0.67 1.92 0.67 1.17 1.75 0.00 -1.42 -0.83 0.58 1.75 -0.50 -0.17 1.75 -0.25 -0.42 -1.33 -0.33 0.92 1.33 1.92 -0.75 0.58 -0.17 0.92 -1.08 0.92 -0.17 -1.42

2.07 1.65 1.63 3.66 1.05 2.75 0.93 2.47 3.63 2.17 2.07 2.35 1.57 2.40 1.57 2.22 1.42 2.01 1.78 1.57 0.90 2.13 1.71 2.81 2.04 3.81 2.00 1.93 0.87 1.36 3.09 1.42 1.83 1.72 2.10 0.89

9.9 6.2 5.9 74.0 4.6 24.7 4.4 21.1 49.3 13.5 10.6 16.4 5.7 18.5 5.5 14.8 5.1 8.7 7.0 5.3 4.2 12.3 6.4 29.6 9.3 148.0 8.2 7.8 3.8 4.8 37.0 4.9 7 6.7 11.4 4.1

N is the rank of the event with respect to its amplitude listed in descending order. The last column represents the recurrence interval (RI) of events

antinodes the role of heat reservoirs. The thermocline deepens nearly 1 year and a half before events occurring in 11/1997 and 12/2006 as shown in the Fig. 16g, h. Deep mechanisms involved in SST variability into the 1.5–7 year band are highlighted in Fig. 16c, d, g, h; the low SST variability results from a thermal barrier more or less close to the surface, depending on the buoyancy of the upper layer that isolates the warm water transferred. Such thermal barriers form preferentially at the antinodes when the warm advected layer is more salted and denser than the

Figure 14 The return period of the 36 most significant El Nin˜o events versus their amplitude (maximum of signal -SOI filtered in the band 1.5–15 years)

upper layers. In this way, the stratification persists even when the thermal anomaly along the profile is maximal around 150 m deep. As occurs in the Atlantic Ocean, attenuation of seasonal SST variations results from slow heat exchange between the upper and the advected layers through the thermal barrier, which causes a phase shift (Fig. 17). In contrast, the magnitude of the 1-year period oscillation of SST is prominent in the eastern Pacific, south of the equator, where the buoyancy of the warm advected layer makes it rise to the surface. The warm advected layer floats in the area of upwelling, due to the high density of cold water, which is a determinant phenomenon in the maturation of ENSO events. The warm advected layer sinks under a 20-m thick layer further north (Fig. 6). The highest SST anomalies occur off the Peruvian and the northern Chilean coasts in April (in May close to the coast and along the equator) as shown in Fig. 17. If one looks at the seven most significant El Nin˜o events observed since the commissioning of moored buoys (TAO), a period extending from 1980 to 2013, the estimated correlation coefficient between the amplitude of El Nin˜o events and the temperature at 60 m depth in the eastern part of the basin (e.g., 0N 110W) is 0.96. This correlation close to 1 indicates that the amplitude of ENSO is determined from the thermal anomaly of the mixed layer translated to the east of the basin. In other words, not only does the quadrennial QSW trigger an El Nin˜o event as soon as a ridge is formed at the central-eastern antinode, but also the mixed layer is fully involved in the genesis and support of the event (the coefficient of correlation

Long Wave Resonance in Tropical Oceans and Implications on Climate: The Pacific Ocean

Figure 15 Profiles of temperature anomalies at 0N 110W associated with the events mature in 12/1982 (a, e), 05/1987 (b, f), 11/1997 (c, g) and 12/2006 (d, h)—amplitude of (T, -SOI) in a–d and phase in e–h, scale-averaged over the 1.5- to 7-year band. The phase expresses the time elapsed from when the thermal anomaly is maximal and the ENSO event is mature, versus the depth. The anomaly is minimal 6 months later. Time elapses from the bottom to the top of the ordinates. Dotted arrows emphasize deepening of the thermocline and dashed arrows the expansion of the warm water layer from the core up to the surface

with the temperature at 200 m depth is close to 1, too). This deduction implies that the convection within the mixed layer is effective regardless of its thickness and its average temperature, and the energy released into the atmosphere is proportional to the thermal anomaly in the mixed layer of the centraleastern antinode (the correlation is confirmed independently of the coordinates of buoys located along this antinode). The almost concomitance of the advection of warm water in the central-eastern antinode and the onset of El Nin˜o events highlights a positive feedback between the ocean and atmosphere during these events. Warming of the upper part of the ocean promotes deep convection in the atmosphere, thus weakening the Walker circulation. Conversely, attenuation of winds promotes convection processes in the atmosphere and, as a corollary, the process of convection in the mixed layer of the ocean due to the release of latent heat that cools the surface.

The end of the El Nin˜o event leads to La Nin˜a as soon as the westward phase propagation of the quadrennial QSW occurs, which strengthens upwelling off the South American coast. The cold ascending column reduces evaporation process, which causes an increase in atmospheric pressure and the resumption of trade winds, due to the strengthening of the Walker circulation and expansion of the Hadley cell. In the central Pacific the Rossby wave reinforces mixing of shallow water, which reduces the SST and also reduces the evaporation process. 3.4.2 Distribution of El Nin˜o Events To confirm that the average period of El Nin˜o events is 4 years, their time lag is represented within successive intervals of 4 years defined from 01/1868, i.e. 01/1868 to 01/1872, 01/1872 to 01/1876…, 01/1996 to 01/2000… If one selects among the most significant El Nin˜o as many events as the number of 4-year length intervals, namely 36, two

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Pure Appl. Geophys.

Figure 16 Profiles of temperature anomalies at 0N 140W associated with the events mature in 11/1997 (a, e), 12/2006 (b, f), and at 0N 156E associated with the same events mature in 11/1997 (c, g) and 12/2006 (d, h)—amplitude of (T, -SOI) in a–d and phase in e–h, scale-averaged over the 1.5- to 7-year band. Same convention as in the Fig. 15

Figure 17 Amplitude (a) and phase (b) of SST anomalies relative to 05/2001, scale-averaged over the 8–16 months band. SST is issuing from the Extended Reconstructed Sea Surface Temperatures, version 3(ERSST.v3), provided by the NOAA: http://www.emc.ncep.noaa.gov/research/ cmb/sst_analysis

events occur during the same interval three times in 1868–1872, 1888–1892 and 2004–2008, while intervals 1892–1896, 1928–1932 and 1936–1940 contain none, which reflects the variability of the period (Table 1). The distribution of the time lags associated with these events is illustrated by the histogram in Fig. 18. It clearly shows a periodicity of 4 years because time intervals of length different from 4 years would lead to an almost uniform distribution,

which confirms and specifies what was deduced from the sub-harmonic mode locking. No element occurs when the nodes common to the basin modes are in opposite phase (0 \ time lag \ 0.5 years). The most favorable period to the extension of the centraleastern antinode to the eastern boundary of the basin is such that -0.5 \ time lag \ 0 years, i.e. when both the events are shifted slightly and the common nodes are in phase.

Long Wave Resonance in Tropical Oceans and Implications on Climate: The Pacific Ocean

Moreover, events that are strongly shifted are little energetic (Fig. 18). 3.4.3 Evolution of El Nin˜o Events

Figure 18 Histogram of the number of major El Nin˜o events (or available energy) versus the time lag. The available energy is assumed to be proportional to the energy released during ENSO, which in turn is proportional to the amplitude of the event

Despite the high variability of the period it appears that the quadrennial sub-harmonic is strongly influenced by the annual basin mode since about 90 % of events occur when -1.5 \ time lag \ 1 year, -0.5 \ time lag \ 0 years, 0.5 \ time lag \ 1 and 1.5 \ time lag \ 2 years, i.e. when the annual component of the NECC flows eastward. Atypical events that do not obey this rule result from the duration of maturation of El Nin˜o events, which is variable. Finally, the sub-harmonic is rarely strongly shifted, as shown by the histogram. A large positive time lag also produces atypical El Nin˜o events.

The amplitude and phase, averaged in the band 1.5–7 years, of SST anomalies during the 36 major El Nin˜o events observable from 1867 to 2013 shows a very marked typology depending on whether the events occur in the first or in the second half of the consecutive 4-year length time intervals (only two representative events are represented in Fig. 19). 3.4.3.1 El Nin˜o Events Whose Time Lag is Negative As shown in Fig. 19a, b they are characterized by SST anomalies in the eastern tropical Pacific, which gradually shift to the west, independently of the available energy. The surface temperature anomalies are triggered off the coast of Peru and Chile, where upwelling is linked to the Humboldt Current, which shows that at the end of the eastward phase propagation, a large warm water pool is formed in the eastern Pacific (the thickness of the mixed layer is about 80 m, Fig. 6). Thus the atmospheric pressure decreases, which weakens the Walker circulation: the strength of trade winds is sensitive to upwelling off

Figure 19 Amplitude (left) and phase (right), averaged in the band 1.5–7 years of SST anomalies. The phase is relative to El Nin˜o events that are mature in 09/1972 (a, b) and in 12/1982 (c, d). The time lag of the first event is negative, and that of the second is positive. Clear strips in the western part of the basin that extend eastward on both sides of the SST anomaly show that the western equatorial Pacific is very little involved in the ENSO

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Pure Appl. Geophys.

Figure 20 Amplitude (on the left) and phase relative to the El Nin˜o mature in 11/1997 (on the right) of SST anomalies (a, b), RRH anomalies gridded with a 0.5 9 0.5 resolution (c, d). Both are scale-averaged over the 3.5–4.5 years band. Monthly rainfall data are issued from the University of East Anglia Climate Research Unit (CRU): http://badc.nerc.ac.uk/data/cru. Reduced rainfall height (RRH), i.e. rainfall height divided by its standard deviation, is used in calculations to overcome the variations in the rainfall amount linked to the local context, especially the relief, the latitude and the distance from the ocean

the coast of South America that maintains an area of high pressure. Since upwelling is masked during ENSO, SST anomalies extend from the South American coast to the equator. Deepening of the thermocline allows supporting sustainable evaporation due to convective effects. When the SST anomaly extends westward along the equator beyond 160E, the ocean–atmosphere coupling becomes fully effective, and El Nin˜o reaches its mature phase. The SST anomalies are then in phase with SOI, i.e. with the drop in atmospheric pressure. Therefore, much of the warm waters of the western equatorial Pacific have been transferred to the central-eastern antinode, marking the beginning of the westward reflux of the QSW after it has ‘rebounded’ against the eastern boundary of the basin. These events usually occur when the velocity of the annual modulated component of the NECC is close to its maximum. Three events violate this rule in 05/1905, 03/1941 and 04/1993 (time lags are -0.67, -0.83 and -0.75 year); in these three cases the maturity of the El Nin˜o events occurs late after the surface temperature anomaly has spread from the coast of Chile to the equator, thus revealing the complexity of the convective process at the end of the eastward phase propagation of the QSW. In particular, the ENSO of 03/1941 remained mature for almost 2 years instead of the generally observed 4–6 months (Fig. 13).

3.4.3.2 El Nin˜o Events Whose Time Lag is Positive These events, which are fewer than the previous ones, are generally triggered from SST anomalies between latitudes 5N and 20N in the central Pacific (Fig. 19c, d). These anomalies may join others in the eastern basin, which in this case reports the maturity of El Nin˜o. As shown in Fig. 20, SST advection and associated evaporative processes developing in the central Pacific induce decreasing in atmospheric pressure in a narrow strip included between 10S and 10N in Southeast Asia and Oceania to the west and in South America to the east. The impact of SST anomalies in the central Pacific on the atmospheric pressure at both the western and the eastern equatorial Pacific are represented from the amplitude of the 4-year period inland rainfall oscillation. Reduced rainfall height (RRH) anomalies are in opposite phase relative to the 4-year average period SST anomalies in the central Pacific, which suggests that a quasi-geostrophic motion of the atmosphere is produced from heating in the Central equatorial Pacific. To the east, there are easterly trade winds set up by atmospheric Kelvin waves emanating from the heating zone. Westerlies are produced on the west side of the heating zone as an atmospheric planetary wave response (GILL 1982). Both Kelvin and Rossby waves are equatorialtrapped; hence, only a narrow strip is involved in the see-saw of air surface pressure. The increase in

Long Wave Resonance in Tropical Oceans and Implications on Climate: The Pacific Ocean

atmospheric pressure in Southeast Asia and Oceania while SST anomalies are growing in the Central Pacific weakens the Walker Circulation because of the decrease in atmospheric pressure difference between the western Pacific and the area where upwelling occurs off the coasts of Chile and Peru. Inversely, as a result of the weakening of trade wind, SST anomalies expand along the equator both eastward and westward. Owing to this positive feedback, El Nin˜o reaches its mature phase when the SST anomaly expands to 100W and then the expansion slows down up to the South American coast. These observations confirm and extend the work of KAO and YU (2009) who suggest that the difference observed between the events that trigger from SST anomalies in the central and eastern Pacific may result from the timing.

4. Summary and Conclusion The dynamics of the tropical Pacific is dominated by geostrophic forces at the basin scale and beyond the basin. It can be understood by invoking the coupling between the basin modes. This coupling results from the merging of nodes produced by the different QSWs both in the western part of the NECC and along the SEC. Consequently a sub-harmonic mode locking occurs: because the SSH perturbation g and the modulated currents u and v have a zero mean, it follows that the average periods of sub-harmonics are multiple of the period of the fundamental QSW, i.e. 1, 4 and 8 years. The annual QSW results from a first baroclinic, fourth meridional mode Rossby wave resonantly forced by easterlies. The nodes of this QSW are the modulated components of the SEC and the NECC. The quadrennial sub-harmonic results from the superposition of equatorial first baroclinic mode Kelvin waves and equatorial as well as off-equatorial first baroclinic mode, first meridional mode Rossby waves. This QSW is subject to two modes of forcing. One results from coupling with the annual basin mode that produces a Kelvin wave at the origin of transfer of the warm waters from the western part of the basin to the central-eastern antinode. The other is induced by El Nin˜o and La Nin˜a that self-sustain the

sub-harmonic by stimulating the Rossby wave accompanying the westward recession of the QSW at a critical stage of its evolution. The sub-harmonic of 8-year average period recalls the quadrennial QSW, involving second baroclinic mode Kelvin and Rossby waves. The central-eastern antinode is less extended eastward than the quadrennial QSW and hence its weak contribution to ENSO events. ENSO events occur at the end of the eastward phase propagation of the quadrennial QSW: the resonance of long equatorial waves induces atmospheric phenomena and not the inverse as often hypothesized. The interpretation of ENSO from the coupling of different basin modes allows predicting and estimating the amplitude of El Nin˜o events a few months before they become mature from the accelerations of the geostrophic component of the NECC while it flows closer to the equator off the Cape d’Urville, i.e. from the onset of discharge of the western basin. This precursory signal may indicate the propagation of a Kelvin wave, which is subject to geostrophic forces throughout the tropical basin: two successive Kelvin waves cannot be stimulated within less than 1 year and a half during which geostrophic forces remain confined to the western basin. Finally, the difference observed between the ENSO events that are triggered from SST anomalies in the central and eastern Pacific clearly result from the timing. So the genesis and the evolution of ENSO appear as a response to geostrophic forces acting in the tropical Pacific.

Appendixes Appendix 1: The Equations of Motion of Resonantly Forced Waves Under the Approximation of b Plane In the following equations x and y represent the longitude and latitude relative to the central point of the b plane tangent to the terrestrial sphere. The representation of movement in this plane rather than on the sphere simplifies the equations of motion. Their linearized version with forcing terms includes the momentum Eqs. (1), (2), the equation of continuity (3) and the equation of potential vorticity (4). Assuming to simplify writings two superimposed

J.-L. Pinault

fluids, the components u and v of the velocity vector (u is the longitudinal or zonal component, v is the latitudinal or meridional component) and the perturbation g of the height of the ocean surface are solutions of the system of equations (GILL 1982) ou=ot  fv ¼ gon=ox þ X=q1 H1

ð1Þ

ov=ot þ fu ¼ gon=oy þ Y=q1 H1

ð2Þ

og=ot þ H1 ðou=ox þ ov=oyÞ ¼ E=q1

ð3Þ

o 1 ðn  f g=H1 Þ þ bv ¼ ðoY=ox  oX=oy þ fEÞ ot q1 H1 ð4Þ l ¼ g=h ¼ g0 H2 =gH g0 ¼ ð1  q1 =q2 Þ; where h is the displacement of the interface up, which is resolved in vertical mode: f is the Coriolis parameter, b the gradient of the Coriolis parameter and n ¼ ov=ox  ou=oy the potential vorticity. H1 is the depth of the upper layer, H2 that of the lower layer and H the total depth. q1 and q2 are the density of the upper and lower layer. The forcing terms X and Y represent the surface stress and E the evaporation rate. As usual we assume that X  N where N is the buoyancy frequency, which is supposed to be constant (X is the speed of rotation of the earth) and bc  N 2 . The Coriolis parameter f is expanded with respect to y such that f ¼ f0 þ by; where f0 is the Coriolis parameter at the central point of the b plane whose latitude is /0 , with b ¼ ð2X=RÞ cos /0 (in the case of equatorial waves f0 ¼ 0). Under the approximation of the b plane, u and v are the components of the velocity vector u expressed relative to the longitude and latitude. Similarly, X and Y are the forcing terms relative to the surface stress, expressed in relation to the longitude and the latitude. In order to link v to the other variables, new variables q and r are defined and replace u and v (GILL 1982), such that q ¼ gg=c þ u, r ¼ gg=c  u where c is the phase velocity for the baroclinic mode considered. Combining (1) and (4), three new independent equations are obtained: oq oq ov 1 þ c þ c  fv ¼ ðX  cEÞ ot ox oy q1 H1

ð5Þ

or oq ov 1  c þ c þ fv ¼  ðX þ cEÞ ot ox oy q1 H1

ð6Þ

Pure Appl. Geophys.

  o or ov ov c  fr þ þ c þ bcv ot oy  ot  ox  1 oY oY oX þc  þ fE ¼ q1 H1 ot ox oy

ð7Þ

In a shallow water model, inviscid, equations are solved by expanding r, q, v, X, Y and E in terms corresponding to the meridional modes, i.e. parabolic cylinder functions: ðv; q; rÞ ¼

1 X

  ðvn ; qn ; rn ÞDn ð2b=cÞ1=2 y

ð8Þ

n¼0

ðX; Y; EÞ ¼ q1 H1

1 X

  ðXn ; Yn ; En ÞDn ð2b=cÞ1=2 y ð9Þ

n¼0

Then, using the property of the parabolic cylinder functions 

 d þ n=2 Dm ¼ mDm1 and dn   d  n=2 Dm ¼ Dmþ1 ; dn

n ¼ ð2b=cÞ1=2 y, the equations of the coefficients become for the modes n  0 (coefficients with a negative index are zero; n ¼ 0 corresponds to mixed Rossby-gravity waves). ðo=ot  co=oxÞrn1 þ f0 vn1 þ ð2bcÞ1=2 nvn ¼ ðXn1 þ cEn1 Þ

ð10Þ

ðo=ot þ co=oxÞqnþ1  f0 vnþ1  ð2bcÞ1=2 vn ¼ Xnþ1  cEnþ1

ð11Þ

  o o o f0 rn  ð2bcÞ1=2 rn1 þ þc vn þ bcvn ot ot ox   o o Yn  ðbc=2Þ1=2 ½ðn þ 1ÞXnþ1  Xn1  þ fEn ¼ þc ot ox 



ð12Þ Without forcing the solutions are rn1 ¼ Rn1 sinðkx  xtÞ, vn ¼ Vn cosðkx  xtÞ and qnþ1 ¼ Qnþ1 sinðkx  xtÞ whose amplitudes Rn1 and Qnþ1 are (supposing Vn ¼ 1)Rn1 ¼ ð2bcÞ1=2 n= ðck þ xÞ, Qnþ1 ¼ ð2bcÞ1=2 =ðck  xÞ and the relation between the wave number k and the pulsation x is (for long-period long-wavelength waves) k  ð2n þ 1Þðx=cÞ

ð13Þ

Long Wave Resonance in Tropical Oceans and Implications on Climate: The Pacific Ocean

Considering Kelvin waves for which v ¼ 0 and x ¼ kc, the coefficients are or0 =ot  cor0 =ox ¼ ðX0 þ cE0 Þ

ð14Þ

oq0 =ot þ coq0 =ox ¼ X0  cE0

ð15Þ

While Fourier or Laplace transforms are typically used to calculate the coefficients rj ðx; tÞ, qj ðx; tÞ and vj ðx; tÞ (which assume that the waves vanish at infinity), here the equations are solved by expanding in Fourier series versus x and t both the coefficients and the forcing terms ri , qi , vi , Xi , Yi and Ei 

1 X 1  X vj ; qj ; rj ¼ vj;m;l ; qj;m;l ; rj;m;l l¼0 m¼0

 exp½iðmkx  lxtÞ 

ð16Þ

1 X 1  X Xj;m;l ; Yj;m;l ; Zj;m;l Xj ; Yj ; Zj ¼ l¼0 m¼0

 exp½iðmkx  lxtÞ

ð17Þ

The coefficients vn;m;l , rn1;m;l and qnþ1;m;l can be expressed in terms of forcing terms Xnþ1;m;l , Xn1;m;l , Yn;m;l , En1;m;l En;m;l and Enþ1;m;l : the terms of the Fourier series can be considered as normal modes of forced waves, the solution for the free waves corresponding to the mode m ¼ 1 and l ¼ 1. vn;m;l ¼

ðlx þ cmkÞU  lxð2bcÞ1=2 ðXn1 þ cEn1 Þ   bcðlxð2n þ 1Þ þ cmkÞ þ lx ðcmkÞ2 ðlxÞ2 ð18Þ

U ¼ ðcmk  lxÞYn  ðbc=2Þ1=2 ððn þ 1ÞXnþ1  Xn1 Þ þ fEn h i Xnþ1 þ cEn1 þ ð2bcÞ1=2 nvn;m;l cmk þ lx ð20Þ h i i Xnþ1  cEnþ1 þ ð2bcÞ1=2 vn;m;l ¼ cmk  lx ð21Þ

rn1;m;l ¼

qnþ1;m;l

ð19Þ i

Forced equatorial and off-equatorial planetary waves are governed by Eqs. (18)–(21). The coefficient vn;m;l is in quadrature relative to the coefficients rn1;m;l and qnþ1;m;l .

On the equator, forced Kelvin waves are governed by (14) and (15), propagating eastward. The coefficients r0;m;l and q0;m;l are r0;m;l ¼

i ½X0 þ cE0  cmk þ lx

ð22Þ

q0;m;l ¼

i ½X0 þ cE0  cmk  lx

ð23Þ

According to the dispersion relation (13) rn1;m;l is infinite if l ¼ mð2n þ 1Þ for the westward propagating planetary waves. Similarly, q0;m;l is infinite if l ¼ m for eastward propagating Kelvin waves. These singularities highlight the resonances. This so-called ‘ill-posed’ problem as having singularities has to be regularized to be resolved, which is done by considering the Rayleigh friction. From a physical point of view the Rayleigh friction moderates variations in wave amplitude in the vicinity of resonances, assigning a finite value in all circumstances. To introduce Rayleigh friction terms in (10)–(12), (14), (15) o=ot is replaced by r þ o=ot everywhere, r being the decay rate of the friction r ¼ ax where 0\a\1 (the case a ¼ 1 is singular). In this way, lx has to be replaced by r þ lx in (18)–(23). Appendix 2: Forcing Terms of the Annual RFW The solution of the equations of motion is vn;m;l , rn1;m;l and qnþ1;m;l , where the meridional mode n ¼ 4 indicates the rank of the parabolic cylinder functions with respect to the latitude y; m and l are the ranks of the terms of the Fourier series with respect to the longitude x and the time t (e.g. PINAULT 2013). The forcing terms corresponding to surface stress are X3;m;l , X5;m;l and Y4;m;l , and those relating to evaporation are E3;m;l and E5;m;l (f ¼ 0 on the equator). As in the tropical Atlantic, additional forcing terms are necessary to model the modulated components of the NECC and the SEC. Thus the forcing term X3;m;l is replaced by X3;m;l þ q1 H1 xu3 sinðxt  uu3 Þ

ð24Þ

and E3;m;l by:  q1 xg3 sin xt  ug3 =l;

ð25Þ

where x ¼ 2p=T (the period T is 1 year); u3 and g3 =l represent the amplitude of the third term of the zonal modulated current and thermocline depth

J.-L. Pinault

m

(a)

Pure Appl. Geophys.

m/s

0.4

2

0.2

1

0.0

0

-0.2

-1

-0.4 01/1997

01/1998

01/1999

01/2000

m

(c)

0.10 0.00 -0.10 -0.20 120

170

220

270

longitude (°)

-2 01/1997

(b)

01/1998

01/1999

01/2000

m/s 1.0 0.5 0.0 -0.5 -1.0 120

(d)

170

220

270

longitude (°)

Figure 21 Representation of the solution of the equations of motion. The blue curves are obtained by introducing the surface stress alone, the red curves by introducing forcing terms (24) and (25): a antinode at 160W, 3N, b node at 160W, 3N, c antinode in 10/1998, d node in 10/1998

(which does not depend on longitude) expanded in series of parabolic cylinder functions (evaporation is not explicitly taken into account in the model). The positive forcing term (25) indicates that the current carries warm water. Only additional forcing terms of rank 3 are important to inform about the two modulated forcing currents both sides of the equator. As shown in Fig. 21 the solution varies significantly, even when the forcing terms are small; the phases uu3 and ug3 are set such that the forcing terms (24) and (25) reach a maximum in August, as observed (uu3 ¼ ug3 ). Under these conditions, the period of the solution is 1 year and the phase of the antinode and the node is uniform in the central Pacific, but is reversed at both ends of the basin (Fig. 21c, d), the wavelength of the annual RFW being close to 9800 km (Fig. 5). In Fig. 5 the average depth of the thermocline is assumed to be 255 m, the phase velocity for the firstbaroclinic mode 2.8 m/s. The length of the partial sums of the Fourier series is 20 with respect to the longitude and 20 with respect to time. T = 12 months. xu3 = 3.5E-07 m/s2 xg3 =l ¼ ðH1 =cÞxu3 = 3.1E-05 m/s.

Appendix 3: The Forcing Terms of the Quadrennial RFW Like for the Atlantic Ocean (e.g. PINAULT 2013) three sets of equations of motion are solved, in relation with the north-western, the south-western and the equatorial antinodes, i.e. the two off-equatorial Rossby waves and the equatorially trapped Kelvin wave that reflect off the eastern boundary to produce an equatorially trapped Rossby wave. Additional forcing terms are to be introduced in the equations of motion to account for geostrophic forces at the basin scale. Geostrophic forces result from the merging of the nodes of the annual and quadrennial QSWs, which form the North Equatorial Counter-Current. By stimulating a Kelvin wave from the extreme western part of the basin, geostrophic forces initiate the eastward phase propagation of the quadrennial RFW. The forcing term X is replaced by X ± q1H1xu0 and E by ±q1xg0/l where u0 and g0/l refer to the forcing of the zonal modulated current and the thermocline depth (the direct impact of evaporation due to El Nin˜o events is ignored). These forcing terms are not dependent on the longitude: u0 is proportional to the dimensionless current

Long Wave Resonance in Tropical Oceans and Implications on Climate: The Pacific Ocean

ur = ub/max(|ub|) where ub is the zonal current observed at the western boundary 0.5N, 137.5E (Fig. 8c) and xg0/l = (H1/c)xu0. The signs depend on the direction of forcing of the current and the up or down movement of the thermocline. When the forcing current accelerates at the western boundary, either it produces a Kelvin wave propagating eastward when qg/ qx \ 0 along the equator, or it leaves the basin to join the western boundary currents when qg/qx [ 0 (implying qu/qt \ 0) that occurs when two accelerations of the forcing current are too close. In Fig. 9 the length of partial sums of Fourier series is 10 versus the longitude and 20 versus the time. The Rayleigh friction parameter a is set equal to 0.4 for Rossby waves and 0.06 for Kelvin waves so that the decay rate of friction r = ax are 2E-08 s-1 and 2.3E-08 s-1, respectively: x = 2p/T, T = 4 years for the Rossby waves and T = 0.5 year for the Kelvin wave. The forcing terms are u0 = -2.4E-04 ur m/s for the northern antinode, u0 = -7.8E-05 ur m/s for the southern antinode and u0 = 1.2E-06 ur m/s for the equatorial antinode. Appendix 4: Sub-Harmonic Mode Locking in Coupled Oscillators with Inertia The following is deducted from the works of Choi and Thouless in the particular case of quasi-stationary waves that share the same modulated current at the node, the coupling resulting from processes involving viscosity l of seawater. Indeed, viscosity is a means of removing mechanical energy from the system by giving rise to stresses on the surface of material volume elements related to the rate of strain. In this way, the terms (l/q1) 9 (q2u/qx2 ? q2u/qy2) and (l/ q1) 9 (q2v/qx2 ? q2v/qy2) have to be added to the right-hand side of the two momentum Eqs. (1) and (2) in Appendix 1 to introduce the coupling terms in the equations of motion: u and v are the zonal and meridional currents; l/q1 is the kinematic viscosity (GILL 1982). The following results are deduced from the general equations of coupled oscillators with inertia without having to solve the particular equations of motion of barocline waves. Let us suppose the set of equations of motion for a system of N coupled oscillators:

X

 €j þ cMij u_ j þ Jij sin ui  uj ¼ Ii þ e; Mij u

j¼1;N

ð26Þ where ui represents the phase of the ith oscillator, i.e. the coordinate of the crest of the quasi-stationary wave (ui increases by 2p every time the crest covers a complete cycle along the equatorial and the offequatorial waves in the case of tropical waves), u_ i the €i first derivative versus time, i.e. the phase velocity, u the second derivative, Mij ¼ Mdij the inertia matrix, c the damping parameter, and Jij the coupling strength between the oscillators i and j. The right-hand side describes the periodic forcing with frequency X: Ii  Ii;a cos Xt where Ii;a is the amplitude of forcing on the ith oscillator. Each of the coupled oscillator possesses inertia M and suffers from dissipation of strength cM. e is a white noise of zero mean and variance he2 i that reflects the variability of resonant forcing from cycle to cycle. An element of simplification of the theory relies on the fact that forcing terms in (26) have a zero mean. Indeed, at any positive antinode of a quasistationary wave corresponds a negative antinode half a period later, and vice versa, offsetting the effects over time. This also applies to the modulated zonal and meridional currents at the nodes of the quasistationary waves, the sign of the velocity of which is reversed during each half period. Using the canonical transformation u_ i ¼ oH=opi and p_ i ¼ oH=o/i (H is the Hamiltonian), then Eq. (1) can be written in the following form where (/i ,pi ) are conjugate variables: h_ i ¼ M 1 pi ect p_ i ¼ 

X

Jij ect sinðhi  hj  aij Þ þ eect ;

ð27Þ ð28Þ

j

where hi  /i þ ai , M a_ i  ect Qi ,Qi is defined so that Q_ i e  ct ¼ Ii and

 1 Ia c sin Xt  cos XtÞ Ia aij ¼ ai  aj ¼ ð M c2 þ X 2 X ¼ Ii;a  Ij;a ð29Þ apart from an arbitrary constant.

J.-L. Pinault

The time evolution of the probability distribution Pðfhi g; fpi g; tÞ of phases and momenta at time t is described by the Fokker–Planck equation (RISKEN 1989) for the inertial behavior of the system: X pi op oP X oP ¼ ect þ Jij ect sinðhi  hj  aij Þ ot opi M ohi i ij þ

he2 i X 2ct o2 P e 2 i op2i ð30Þ

Equation (30) yields the stationary solution valid in the limit t ! 1 2

Pðfhi g; fpi g; tÞ ¼ Ae2cMH=he iÞ ;

ð31Þ

where A is the normalization constant; the Hamiltonian is given by H¼

e2ct X 2 X p  Jij cosðhi  hj  aij Þ 2M i i i\j

ð32Þ

so that (30) gives op he2 i X 2ct o2 P 2 ¼ e ¼ Ace2cMH=he i ot 2 i op2i X 2cp2i e2ct =he2 i  1 i

The first term of (32) becomes vanishingly small in the stationary state (t ? ?). One thus obtains the Hamiltonian X X H¼ Ji;j cosðhi  hj  aij Þ ¼  Ji;j cosðui  uj Þ i\j

i

ð33Þ Equation (29) shows that aij , defined modulo 2p, is periodic in time with period s ¼ 2p=X. Since the Hamiltonian as well as aij have the periodicity s the Floquet theorem is applicable to the corresponding Schro¨dinger equation for the wave function Wi ðt þ sÞ  eiEs Wi ðtÞ with the quasi-energy E (e.g. WHITTAKER and WATSON 1952). This imposes that, apart from the dynamical contribution Es, the corresponding change in the phase of the wave function Wi should be a whole number of 2p hi ðt þ sÞ  hi ðtÞ ¼ 2ni p  Es; where ni is an integer depending on the form of aij , i.e. of the driving Ii , and E has been

Pure Appl. Geophys.

assumed to be real. One thus has the average change rate of the phase D E 1Z s h_ i dt ¼ ni X  E; hu_ i i ¼ h_ i ¼ ð34Þ s 0 where ni is the winding number. So, the difference in phase velocities hu_ i i  u_ j ¼ ðni  nj ÞX that is a multiple of the frequency X displays a particular subharmonic mode locking. REFERENCES CANE M.A. and MOORE D.W. (1981) A note on low-frequency equatorial basin modes, J. Phys. Oceanogr., 11, 1578–1584). CHELTON D.B., SCHLAX M.G., LYMAN J.M., JOHNSON G.C. (2003). Equatorially trapped Rossby waves in the presence of meridionally sheared baroclinic flow in the Pacific Ocean. Progress in Oceanography, 56, 323–380. CHOI M.Y. and THOULESS D.J. (2001) Topological interpretation of sub-harmonic mode locking in coupled oscillators with inertia, Physical Review B, doi:10.1103/PhysRevB.64.014305. DELCROIX, T., BOULANGER, J. -P., MASIA, F., and MENKES, C. (1994). Geosat-derived sea level and surface current anomalies in the equatorial Pacific during the 1986-1989 El Nin˜o and La Nin˜a. Journal of Geophysical Research, 99, 25093–25107. DURLAND, T.S., SAMELSON R.M., CHELTON D.B., DE SZOEKE R.A. (2011). Modification of Long Equatorial Rossby Wave Phase Speeds by Zonal Currents. J. Phys. Oceanogr., 41, 1077–1101. doi:10.1175/2011JPO4503.1. GILL A.E. (1982) Atmosphere–Ocean Dynamics, International Geophysics Series, 30, Academic Press, 662 pp. GNANASEELAN C., VAID B. H., and POLITO P. S. (2008) Impact of Biannual Rossby Waves on the Indian Ocean Dipole, IEEE Geoscience Remote Sensing Letters, 5(3), doi:10.1109/LGRS. 2008.919505. GNANASEELAN C. and VAID B.H. (2010) Interannual variability in the Biannual Rossby waves in the tropical Indian Ocean and its relation to Indian Ocean Dipole and El Nin˜o forcing. Ocean Dynamics, 60 (1). 27–40. doi:10.1007/s10236-009-0236-z. HAN W., MCCREARY J. P., MASUMOTO Y., VIALARD J., DUNCAN B. (2011) Basin Resonances in the Equatorial Indian Ocean. J. Phys. Oceanogr., 41, 1252–1270. JIN F.F., NEELIN D. and GHIL M. (1996) El Nin˜o/Southern Oscillation and the annual cycle: sub-harmonic frequency-locking and aperiodicity, Physica, D 98, 442–465. KAO H.Y. and YU J.Y. (2009) Contrasting Eastern-Pacific and Central-Pacific types of ENSO, J. Climate, 22 (3), 615–632, doi:10.1175/2008JCLI2309.1. PAZAN, S.E., WHITE, W.B., INOUE, M., O’BRIEN, J.J. (1986). Offequatorial influence upon Pacific equatorial dynamic height variability during the 1982–1983 El Nin˜o/Southern Oscillation event, Journal of Geophysical Research: Oceans, 91, C7, 8437–8449. PINAULT J.L. (2012) Global warming and rainfall oscillation in the 5–10 year band in Western Europe and Eastern North America, Climatic Change, doi:10.1007/s10584-012-0432-6.

Long Wave Resonance in Tropical Oceans and Implications on Climate: The Pacific Ocean PINAULT J.L. (2013) Long wave resonance in tropical oceans and implications on climate: the Atlantic Ocean, Pure and Applied Geophysics, doi:10.1007/s00024-012-0635-9. PINAULT J.L. (2015) Explain with realism climate variability (http:// climatorealist.neowordpress.fr/2015/06/11/climate-change/). PINAULT J.L. (2016) Long Wave Resonance in Tropical Oceans and Implications on Climate: the Indian Ocean and the far western Pacific, submitted. RISKEN H. (1989) The Fokker-Planck Equation: Methods of Solution and Applications, Springer-Verlag, Berlin. SCHARFFENBERG and STAMMER (2010) ‘‘Seasonal variations of the large-scale geostrophic flow field and eddy kinetic energy

inferred from the TOPEX/Poseidon and Jason-1 tandem mission data’’, J. Geophys. Res., doi:10.1029/2008JC005242. VALSALA V. (2008) First and second-baroclinic mode responses of the tropical Indian Ocean to interannual equatorial wind anomalies, Journal of oceanography, 64, 4, 479–494. WHITTAKER E.T. and WATSON G.N. (1952) A Course of Modern Analysis, Cambridge Univ. Press, Cambridge. ZANG X, FU LL, WUNSCH C (2002) Observed reflectivity of the western boundary of the equatorial Pacific Ocean, Journal of Geophysical Research - Wiley Online Library, doi:10.1029/ 2000JC000719.

(Received February 5, 2015, revised November 16, 2015, accepted November 19, 2015)