Resonance of baroclinic waves in the tropical oceans: The Indian Ocean and the far western Pacific

Journal Pre-proof Resonance of Baroclinic Waves in the Tropical Oceans: the Indian Ocean and the far western Pacific Jean-Louis Pinault

PII:

S0377-0265(19)30076-4

DOI:

https://doi.org/10.1016/j.dynatmoce.2019.101119

Reference:

DYNAT 101119

To appear in:

Dynamics of Atmospheres and Oceans

Received Date:

15 June 2019

Revised Date:

19 September 2019

Accepted Date:

27 October 2019

Please cite this article as: Pinault J-Louis, Resonance of Baroclinic Waves in the Tropical Oceans: the Indian Ocean and the far western Pacific, Dynamics of Atmospheres and Oceans (2019), doi: https://doi.org/10.1016/j.dynatmoce.2019.101119

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Resonance of Baroclinic Waves in the Tropical Oceans: the Indian Ocean and the far western Pacific Jean-Louis Pinault 1,* 1

Independent Scholar; 96, rue du Port David, 45370, Dry, France.

* Correspondence: [email protected]; Tel.: +33 7 89 94 65 42

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Received: date; Accepted: date; Published: date

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Abstract: The Indian Ocean has a particularity, its width is close to half the wavelength of a Rossby wave of biannual frequency, this coincidence having been capitalized on by several authors to give the observations a physical basis. The purpose of this article is to show that this is not the case since the

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resonance of tropical baroclinic waves occurs in all three oceans. This is because the westward-propagating Rossby wave is retroflexed at the western boundary to form off-equatorial Rossby waves dragged by

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countercurrents before receding and turning back as a Kelvin wave. Thus a quasi-stationary baroclinic wave is formed, whose mean period is tuned to the forcing period. Two independent basin modes resonantly

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forced are highlighted – 1) a nearly symmetric zonal 1/2-yr period Quasi-Stationary Wave (QSW) that is resonantly forced by the biannual monsoon. It is formed from first baroclinic mode equatorial-trapped Rossby and Kelvin waves and off-equatorial Rossby waves at the western antinode. This QSW controls the

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Equatorial Counter Current at the node. The Indian Ocean Dipole (IOD) results from a subharmonic mode locking resulting from the coupling of this QSW and the 2d, 3rd and 4th baroclinic modes - 2) a 1-yr period QSW formed from an off-equatorial baroclinic Rossby wave, which is induced from the southernmost current of the Indonesian Throughflow through the Timor passage, propagating in the southern and northern hemispheres: the drivers are south-easterlies in the southern hemisphere and monsoon wind in

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the northern hemisphere.

Keywords: resonantly forced waves; coupling of basin modes; tropical Indian Ocean; Indian Oscillation Dipole

Highlights: In the Indian Ocean, two independent basin modes resonantly forced are highlighted

1.

Introduction

Like the tropical Atlantic and Pacific oceans, the Indian Ocean is subject to the resonance of nondispersive Kelvin and Rossby waves equatorially trapped as well as off-equatorial Rossby waves, resulting in a Quasi-Stationary Wave (QSW). The QSW exhibits a particularly strong semiannual (180-day period) zonal surface current. The resonant nature of the Atlantic and Pacific oceans is poorly documented while it is well known for the Indian Ocean that has a particularity, its width is close to half the wavelength of a Rossby wave of biannual frequency, that is, 6,300 km from the east coast of Africa to the west coast of Sumatra, and 12,100 km, respectively. Biannual Rossby waves have been studied in the equatorial Indian Ocean by several authors (e.g. Gent et al., 1983, Han et al., 1999, Hase et al., 2008, Jensen, 1993, Luyten and

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Roemmich, 1982, Nagura and McPhaden, 2010, Qiu and Yu, 2009), emphasizing that the first-baroclinicmode is subject to a half-wave resonance (Cane and Moore, 1981, Jin, 2001, Han et al., 2011). The annual

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variability has been investigated by McCreary et al., 1993, Masumoto and Meyers, 1998, Hermes and Reason, 2008.

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The semiannual modulated zonal current is much stronger than the annual one, whereas the semiannual zonal wind amplitude is comparable to the annual component, which has led many authors to

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invoke a resonance phenomenon by matching the width of the basin, the period of forcing and the phase velocities of Kelvin and Rossby waves. The notion of no-normal flow boundary conditions at both eastern and western boundaries had been introduced (Cane and Moore, 1981, Gent, 1981). In such a way the

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condition for resonance is 𝑇 = 4𝐿⁄𝑚𝑐𝑛 where 𝑇 is the forcing period, 𝐿 is the equatorial basin width, 𝑐𝑛 is

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the Kelvin wave speed of the 𝑛𝑡ℎ baroclinic mode (𝑛=1, 2, . . .), and 𝑚=1, 2, . . . is a positive integer.

Figure 1 - Amplitude (left) and phase (right) in 2004, averaged over the 5-7 month band of - a, b) Sea Surface Height (SSH) anomalies (the antinodes of the QSW) - c, d) velocity u (facing east) at the modulated surface currents (the nodes of the QSW).

The concept of no-normal-flow boundary conditions 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 conditions of resonance results from the boundary condition where the zonal flow is supposed to vanish. As has been shown in the tropical Atlantic (Pinault, 2013) and Pacific (Pinault, 2015a, b) oceans, the resonance of non-dispersive baroclinic waves is ubiquitous. This is because the westward-propagating Rossby wave equatorially trapped is retroflexed at the western boundary to form off-equatorial Rossby waves dragged by countercurrents. Owing to the geostrophic forces acting at the tropical basin scale, these

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off-equatorial Rossby waves recede to the western boundary then turn back as an eastward-propagating Kelvin wave. This phenomenon also occurs in the Indian Ocean as show the two off-equatorial Sea Surface

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Height (SSH) anomalies on both sides of the equator at the western boundary (Figure 1a). In this way the period of the QSW resulting from the superposition of Kelvin and Rossby waves fits the forcing period and

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resonance occurs without involving the width of the basin; only the dispersion relation of free waves establishes a link between the wavelength and the resonance period. It is the time propagation of the off-

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equatorial Rossby waves the phase speed of which slows down significantly when they move away from the equator that regulates the period of what forms the QSW. Note also the possible role of the eastern

forces in the eastern tropical basin.

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boundary in the tuning of the period of resonance due to the coastal Kelvin waves that alter geostrophic

In this context the resolution of the equations of motion must be deeply rethought because QSWs

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represent a single dynamical system within a characteristic bandwidth. The key time scale for the establishment of resonances is the time it takes a Kelvin wave to cross the basin and a first-meridional-mode Rossby wave to turn back to the western boundary, irrespective of the baroclinic modes. This enables the tuning of natural and forcing periods due to the delayed response of off-equatorial waves at the western antinode. The resulting basin mode is tuned to the monsoon winds so that the period of forcing coincides

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with the period of free baroclinic waves that fulfil the dispersion relation. In this way, resonantly forced QSWs are observed. As ready stated (Pinault, 2015b) only one basin mode can subsist to a particular frequency, competition between resonant and non-resonant modes acting in favor of resonant mode for which the maximum energy is captured. QSWs of different frequencies corresponding to the different baroclinic modes of Kelvin and Rossby waves are coupled. Consequently they are subject to a subharmonic mode locking (Pinault, 2018a). It will be shown that the Indian Ocean Dipole (IOD) results from the coupling of the first, second, third and fourth baroclinic mode Rossby and Kelvin waves, which involves a train of consequences as to the frequency of

different modes. Furthermore, this way of approaching the physical oceanography of the tropical Indian Ocean allows a better understanding of the functioning of the reversal geostrophic currents that are the Somali and the monsoon drift as a result of an annual off-equatorial Rossby wave propagating in both hemispheres. The paper is organized as follows: 1) the methods are recalled 2) the biannual QSW is investigated and the observations are interpreted by solving the equations of motion 3) the IOD is interpreted as resulting from the coupling of baroclinic modes 4) the annual QSW is investigated, its leading role in the circulation

Method

2.1

Cross-wavelet analysis

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2.

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of the Somali and the monsoon drift currents, as well.

Applied to Sea Surface Height (SSH), Surface Current Velocity (SCV) and Sea Surface Temperature

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(SST) 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 anomalies, the cross-wavelet

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analysis highlights quasi-stationary-waves (QSWs) that represent a single dynamical phenomenon within a characteristic bandwidth.

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The amplitude of the anomalies is given by the cross-wavelet power (more exactly, the normalized cross-wavelet power) and their phase by the coherence phase, both of these being performed according to a reference, hence the term ‘‘cross-wavelet analysis’’. Thus, figures are paired, where the first exhibits

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anomalies regardless of the time when the maximum occurs, and the second is the phase, i.e. the time when the maximum anomaly occurs over a period. In order to take into account the natural broadening of the frequency band associated with the oscillation, both the normalized cross-wavelet power and the coherence phase are scale-averaged over relevant bands. Every series is analyzed according to latitude and longitude, with a fixed temporal

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reference: SSH, SCV or SST at a particular location. 2.2

Data

Sea Surface Height (SSH) is provided by the French CNES (Centre National d’Etudes Spatiales):

www.aviso.oceanobs.com. Geostrophic surface current velocity field (SCV) is obtained through the OSCAR (Ocean Surface Current Analyses: Real time) program and provided by the NOAA (National Oceanic and Atmospheric

Administration):

www.oscar.noaa.gov/datadisplay/datadownload.htm.

Sea

Surface

Temperature (SST) is issuing from the Extended Reconstructed Sea Surface Temperatures, version

3(ERSST.v3), provided by the NOAA: www.emc.ncep.noaa.gov/research/cmb/sst_analysis. The profiles of seawater temperature are provided by the NOAA from the global network of tropical moored buoys: Research Moored Array for African-Asian-Australian Monsoon Analysis and Prediction (RAMA) https://www.pmel.noaa.gov/tao/drupal/disdel/. Monthly zonal and meridional wind stresses (Surface Gauss U and V momentum fluxes) at the surface of the oceans are provided by the NOAA: www.esrl.noaa.gov/psd/data/timeseries/. 2.3

The forced version of linear equations of motion for the biannual QSW

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The equations of the wave propagation in two stratified fluids are recalled in the appendix. Here three coupled systems of equations are solved simultaneously, representing 1) the equatorially trapped first

equatorial Rossby wave 3) the southern off-equatorial Rossby wave.

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baroclinic mode Kelvin waves and first baroclinic, first meridional mode Rossby waves 2) the northern off-

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The following reasoning aims at taking into account the boundary conditions, in the western part of the basin, of coupled systems. In this case the wave is simply deflected (not reflected) to the limits of the

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basin, unless it leaves the basin. The coexistence of off-equatorial and equatorial waves requires the continuity of the zonal currents u off and ueq and the thermocline depth (i.e. conservation of warm water

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transferred) ˆoff and ˆeq at the interface. Additional forcing term is to be introduced in the equations of motion: the forcing term X in (A1) is replaced by

X  1H1u 0 and E in (A3) by  1ˆ 0 where u 0

and ˆ represent the zonal modulated current and the oscillation of the thermocline along the trajectory of 0

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the wave at the western boundary (the evaporation is not taken into account in the current model): both are zero everywhere else.

r j ( x, t ) , q j ( x, t ) and v j ( x, t ) are solved by expanding in terms of Fourier series both the coefficients and the forcing terms X , Y and E . So:

(v j , q j , r j )  l 0 m0 (vm,l , q m,l , rm,l ) expi (mkx  lt )

(1)

( X j , Y j , E j )  l 0 m0 ( X m,l , Ym,l , E m,l ) expi(mkx  lt )

(2)

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







The coefficients v n , m ,l , rn 1, m ,l and q n 1, m ,l can be expressed according to forcing terms X n 1, m ,l ,

X n1,m,l , Yn ,m ,l , E n 1,m ,l , E n ,m ,l and E n 1,m ,l . According to (A10), (A11) and (A12):

v n , m ,l 

(l  cmk )Φ  l (2c)1 / 2  X n 1  cE n 1  cl (2n  1)  cmk   l (cmk ) 2  (l ) 2





(3)

Φ  (cmk  l)Yn  (c / 2)1 / 2 (n  1) X n1  X n1   fEn



rn 1,m ,l 

i X n 1  cE n 1  (2 c)1 / 2 nvn ,m ,l cmk  l

q n 1,m ,l 

i X n 1  cE n 1  (2 c)1 / 2 v n ,m ,l cmk  l



(4)



(5)



(6)

Off-equatorial and equatorial forced planetary waves are ruled by equations (3) to (6). Forcing

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maintains the component v n , m ,l in quadrature with the components rn 1, m ,l and q n 1, m ,l for Rossby waves.

propagation). On the other hand, q 0 , m ,l is undefined for

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According to (A14) rn 1, m ,l is undefined if l  m( 2n  1) for Rossby waves (westward

l  m for Kelvin waves (eastward propagation).

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Singularities highlight resonances. So, this ill-posed problem needs to be regularized to be solved, which is done when Rayleigh friction is considered. To introduce friction terms into (A1), (A2) and (A3):

case

r   / t everywhere, r being the decay rate of the friction r   where 0    1 (the

  1 is singular). In this way, l

 r  l in (3), (4), (5) and (6).

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replaced by

 / t is

has to be replaced by

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On the equator, forced Kelvin waves ruled by (7) and (8) propagate eastward. The coefficients r0 , m ,l and q 0 , m ,l are, according to (A15) and (A16):

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r0,m ,l 

q 0 , m ,l 

Results and discussion

3.1.

The bi-annual quasi-stationary wave

(7)

i X 0  cE 0  cmk  l

(8)

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3.

i X 0  cE 0  cmk  l

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. 3.1.1

The observations The biannual QSW seesaws from the western part of the basin to its eastern part, the equator acting as

a waveguide, as shown by the amplitude and phase of SSH (Figure 1a, b). The geostrophic component of

the modulated zonal current, which is the Equatorial Counter-Current, preferably flows east between longitudes 50°E and 90°E (Figure 1c, d and Figure 2a), but may reverse (see also Scharffenberg and Stammer,

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2010): during a cycle exchanges occur between western and eastern antinodes.

Figure 2 - a) Geostrophic current speed at 71.5°E, 0.5°S measured and filtered in the 5-7 month band. The

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speed is facing east when it is positive - b) Fourier spectrum (frequency representation) of the speed.

The QSW is the superposition of a Rossby and Kelvin wave in the opposite direction, both being

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retroflexed to the limits of the basin that are the coast of Eastern Equatorial Africa on the one hand, Malacca and Sumatra secondly. The western antinode forms a ridge in March and September, and the Eastern

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antinode in May and November, whence a slight asymmetry in the duration of transfers between the eastern and western tropical basins due to the difference in phase velocity of the Rossby and Kelvin waves. The

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speed of the modulated current at the node is maximal in May and November when directed eastward, i.e. it is in phase with the eastern antinode.

The resulting basin mode is tuned to the monsoon winds. The equatorial-trapped Rossby wave is

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retroflexed as two off-equatorial Rossby waves both sides of the equator whose westward phase velocity is 0.3 m/s at 8°N or S. They are embedded into the Equatorial Counter-Current whose eastward velocity at that time is higher as shown in Figure 1, Figure 2. This suggests geostrophic forces oppose to the reflection of the Rossby wave to form an eastward propagating Kelvin wave as often hypothesized but a Doppler shift occurs along the Equatorial Counter-Current, resulting in the eastward propagation of troughs and

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ridges off the coasts. In turn, growing of the western antinode generates a westward modulated current nearly in phase (Figure 1a, b), which intensifies the Equatorial Counter-Current that can therefore be considered as a component of the node of the biannual QSW in its western part. As such off-equatorial waves in the western antinode play the role of “tuning slides” as already observed in the Atlantic and Pacific oceans (Pinault, 2013, Pinault, 2015a, b). The existence of QSWs requires that the antinodes have balanced amplitudes and are in opposite phase. Indeed, the volumes of SSH anomalies are approximatively proportional to the mass of warm water displaced, supposing a constant ratio μ throughout the tropical Indian Ocean when the equations are

resolved into a single vertical mode μ=η/h: η is the perturbation of the surface height and h the downward pycnocline displacement. Estimated from different realizations the volumes of antinodes are V ± σ = 101 ± 30 km3 to the west (both off-equatorial QSWs combined) and 89 ± 25 km3 to the east, which indicates considerable variability (V is the mean volume and σ is the estimated standard deviation). However, these volumes are highly correlated since the ratio of the western volume on the eastern volume is 1.2 ± 0.2. The difference between these volumes is not significant, especially because the contour of the antinodes is imprecise. Another characteristic of QSWs is the stability of the phase when the modulated currents at the nodes

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reach their maximum velocity, whatever the magnitude. The mean SCV in the area [4.5°S, 1.5°N] × [63.5°E, 73.5°E] is 0.087 ± 0.016 m/s. The standard deviation of the coherence phase is 11 days, which confirms high

First baroclinic mode Kelvin waves, first baroclinic mode, first meridional mode Rossby waves

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3.1.2

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reproducibility in the time evolution of the zonal current.

The solution of the equations of motion allows to check and specify prior assumptions quantitatively.

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Like in the Atlantic and the Pacific oceans (Pinault, 2013, Pinault, 2015b), separate equations of motion represent the equatorial and off-equatorial waves. The waves are supposed to be retroflexed at the limits of the basin (without the zonal current vanishing at the boundaries), taking advantage of geostrophic forces

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resulting from the oscillation of the tilt of the sea surface along the equator.

Figure 3 – Amplitude (left) and phase (right) of the solution of the equations of motion for the biannual QSW (only the equatorial waves are represented). Monsoon wind stress is considered for the period 2000-2001 - a,

b) antinodes - c, d) velocity (facing east) of the zonal current u - e, f) velocity (facing north) of the meridional current v, about 10 times lower than the zonal flow. The average depth of the thermocline is assumed to be 150 m, the phase velocity for the first baroclinic mode 2.3 m/s. The length of the partial sums of the Fourier series is 12 with respect to the longitude and 20 with respect to time. Periods are 𝑇= 6 and 2 months for Rossby and Kelvin waves, respectively. The decay rate of Rayleigh friction are 4 10−8 s-1 and 2 10−8 s-1 for Rossby and Kelvin waves, respectively. This method differs from usual assumptions that consist in vanishing the zonal current to the boundaries, which is based on no physical basis, since the wave is not reflected strictly to the limits of the basin but is retroflexed to produce off-equatorial waves ruled by geostrophic forces

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before receding and turning back.

This is what shows the solution of the forced version of the equations of motion in Figure 3a, b, from

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both eastern and western antinodes in phase opposition. The amplitude, of the order of 0.05 m, and the phase of antinodes are consistent with observations, within the limits of variability. Regarding the

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modulated zonal current, the solution shows that it is divided into two parts, one eastern, the other western, substantially in phase opposition (Figure 3c, d), whereas only one node is observed in Figure 1c, d. This disparity results from some elements of simplification as concerns the forcing terms in the equations of

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motion. The actual western node is more extended than the modeled one to the detriment of the eastern node, forming the Equatorial Counter-Current whose phase is close to that of the modeled western node.

3.1.3

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This demonstrates that the involved geostrophic forces extend beyond the tropical ocean. Evolution of the bi-annual wave

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During the eastward phase propagation, warm water is transferred from the western to the eastern antinode where it partially leaves the tropical basin to join the eastern boundary current while cold water replaces warm water to the west of the basin by stimulating upwelling off the eastern coast of Africa, leading to the rise of the thermocline. This phase, during which upwelling off the coast of Sumatra is reduced, ends in spring or autumn (Figure 4a). The speed of the Equatorial Counter-Current, which flows preferentially

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to the east, increases in spring and autumn: the Kelvin wave is diverted along the west coast of the Indonesian archipelago, forming coastal waves that propagate poleward. During the summer and winter, the modulated current at the node vanishes or reverses. Warm water replaces cold water at the western antinode while upwelling is reinforced along the coast of Sumatra, causing the rise of the thermocline. During a period the mixed layer, warm, is translated from the western antinode where it is formed to the eastern antinode. According to the geostrophy of the tropical ocean, advection may also be performed back, when the modulated current reverses as shown in Figure 2a. Thus, the biannual basin mode induces

reversible heat transfer between the western and eastern parts of the tropical Indian Ocean while stimulating or reducing upwelling at the boundaries of the basin. Due to the seasonal reversal of monsoon winds, forcing mainly occurs at the eastern antinode and southern India. Indeed, northwest winds reach their maximum in April-May and October-November, and are reversed in March and September, in phase

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with the eastern antinode.

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Figure 4 - Schematic representation of the ridge of the Rossby waves (red) and Kelvin waves (blue) - a) The biannual QSW - b) The annual QSW. The months of the year when the ridge is formed are indicated (the

3.1.4

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time evolution of the troughs is in phase opposition with respect to the ridges).

The Indian Ocean Dipole (IOD)

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Meteorological consequences of the IOD are of primary importance in the regions of the Indian Ocean (Saji et al., 1999, Saji and Yamagata, 2003). The link between baroclinic Indian Ocean response to wind stress and thermal forcing, and the Indian Ocean dipole (IOD) has been subject of particular attention (Nagura and McPhaden, 2010, Rao et al., 2001, Vinayachandran et al., 2002, Yuan and Liu, 2009, Gnanaseelan, 2008, Gnanaseelan and Vaid, 2010), but ignoring the coupling of QSWs of different frequencies.

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The IOD involving Sea Surface Temperature (SST) anomalies at the western and eastern antinodes can be studied from the available time series, covering nearly a century and a half. SST anomalies disclose the antinodes since convective processes occur in the mixed layer when antinodes are growing. The amplitude of the IOD shows a significant variability over time (Figure 5), which may be a factor of 3 or 4 of a cycle to another. This is what makes meteorologists say it is an aperiodic variable. But this behavior is only apparent because the IOD is actually the superposition of four coupled oscillators whose periods are 1/2, 1, 2 and 4 years, i.e. the biannual fundamental wave that is resonantly forced by wind stress and its subharmonics

whose average periods are multiples of the shorter periods (Pinault, 2018a). Indeed, the subharmonic mode

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locking of QSWs of different frequencies results from their coupling where their nodes are merging.

Figure 5 - Thermal gradient between the western equatorial anomaly (50°E-70°E and 10°S-10°N) and the south-eastern equatorial anomaly (90°E-110°E and 10°S-0°N), to which is applied a moving average (the

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mean is performed over 7 month intervals), and its components obtained by filtering within the different bands expressed in months. These correspond to the 6, 12, 24 and 48 months periods. The sum of these four

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components is also represented, showing that the original signal is substantially reconstructed. The dashed lines are located at temperature cut-offs.

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As shown in Figure 6, the representation of SST anomalies for different periods brings out a western and a south-eastern anomaly. The amplitude of the thermal anomalies significantly varies from one mode

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to another. As shown in Figure 6e, f the biennial oscillation is predominant during the years 1997-1998, displaying large SST anomalies in opposite phase both in the western basin where the anomaly extends to the Australian coast over more than 40° latitude, and throughout the Timor Passage where the anomaly

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extends over more than 40° longitude. During this period the western and south-eastern temperature anomalies associated with the other modes not only have a low amplitude but they are almost in phase (Figure 6). In 1994 the quadrennial component induces the positive anomaly (Figure 5). Sometimes it is the annual component that causes the anomaly; it follows a positive anomaly flanked by two negative anomalies shifted of 6 months. Note that the SST anomaly observed for the biannual mode (Figure 6a, b) in

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the Arabian Sea is not directly involved in the east-west seesaw because it is out of phase with the western anomaly. Moreover, the southern annual anomaly highlighted in Figure 6c does not result from the annual subharmonic of the bi-annual QSW but from the annual off-equatorial QSW as will be shown farther.

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Figure 6 - Amplitude (left) and phase (right) of sea surface temperature of the Indian Ocean, scale-averaged over the bands 3-9 (a, b), 9-18 (c, d), 18-36 (e, f) and 36-72 (g, h) months. The period of the oscillations is

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biannual in a, b), annual in c, d) 2 years in e, f) and 4 years in g, h). The phase is expressed relative to the

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maximum of the positive IOD anomaly in 01/1998.

To involve phase velocities that decrease with respect to the periods, we have to refer to the stimulation of the first, second, third and fourth baroclinic mode Rossby and Kelvin waves. The first mode corresponds to the oscillation of the thermocline, more exactly of the interface at the base of the pycnocline, 180 m deep, for which the phase velocity is c = 2.3 m/s. The fourth mode corresponds to the oscillation of the interface at the top of the pycnocline, 50 m deep, with a phase velocity of 0.29 m/s, i.e. almost 8 times lower than that of the first baroclinic mode. These two modes being set by the pycnocline are found in all three oceans. As against the intermediate oscillation modes involve interfaces within the pycnocline. Due to the given periods, their phase velocities are in the order of one half and one quarter of that of the first baroclinic mode.

layers', and the boundary conditions

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Taking account of 5 layers within a 4& 1⁄2 layer model with 'motion-less lower layer and active upper

w  0 at the surface and the bottom of the ocean, phase speeds are

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calculated by recurrence from the bottom to the top, the upper layer being split into two sub-layers at each step (Pinault, 2018a). The phase velocities corresponding to the 2d and 3rd vertical modes are 0.98 and 0.55

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m/s considering superposed layers 0-50-75-150-180-1750 m whose respective densities averaged over the equatorial basin are 1022.48, 1023.08, 1024.98, 1026.49 and 1031.45 kg/m 3. It is supposed eigenvectors vanish

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around 1750 m (Valsala, 2008). Therefore, these modes probably owe more their existence to the coupling of the QSWs considered as oscillators rather than a supposed stratification into the mixed layer. In this case, the depth of the interfaces is determined based on the phase velocity and not the inverse. This hypothesis

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is suggested by the fact that this problem is ' ill-posed ', i.e. the depth of the interface may vary significantly for a small change in the profile of the pycnocline, which does not apply to the first and fourth modes. This

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issue has already been addressed by Iskandar et al., 2008, who have suggested that the increase in the contribution from higher baroclinic modes during the IOD event is associated with the change in the background stratification for which the projection of the wind forcing is in favor to the higher baroclinic modes.

This hypothesis, somewhat artificial, would amount to ignoring the subharmonic modes of the tropical

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Indian basin, for which the stability of the dynamical system formed of the four coupled oscillators is optimum. The four oscillators are coupled because they share the same node, the Equatorial CounterCurrent, whose Fourier spectrum shows four distinct peaks at the above periods (Figure 2b).

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Figure 7 - Amplitude (top) and phase (bottom), scale-averaged over the 5-7 month band, of profiles of seawater temperature anomalies at 2°N 90°E (a, e), 2°S 80°E (b, f), 12°N 90°E (c, g) and 15°N 90°E (d, h). The phase expresses the date during which the thermal anomaly is maximal. The minimum is reached 3 months

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later.

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As shown in Figure 7a, b, e, f the low SST variability in the eastern part of the equatorial Indian Ocean results from a barrier layer formed nearly 60 m deep that isolates shallow water and the advected layer at the antinode. Barrier layers form preferentially at the antinodes, the warm water transferred being saltier

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and denser than the upper layer so that the stratification persists even when the thermal anomaly reaches its maximum along the profile obtained from the Global Tropical Moored Buoy Array (Figure 7a, b, e, f). At 2°N 90°E the warm advected layer is transferred uniformly in July (or in January) whereas at 2°S 80°E the

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bottom of the warm water layer is transferred one month and half earlier than the top between 80 and 125 m deep. The temperature of the upper layer above 60 m remains nearly constant during a cycle, the temperature variation being generally lower than 0.2°C. In the Bay of Bengal the thermal profile exhibits the superposition of two advected layers nearly in opposite phase. The isotherm between the two layers is located between 80 and 100 m deep (Figure 7c, d, g, h).

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As done for the ENSO (Pinault, 2015a, b, Pinault, 2018b), the histogram of positive and negative events

versus their time lag within consecutive 4-year length time intervals defined from 01/1852, that is, 01/1852 to 01/1856, 01/1856 to 01/1860..., 01/1996 to 01/2000..., gives a representation of the time evolution of events, in a statistical way. This analysis is performed by considering only the most significant events between 1854 and 2010. Their number is 62 or 60 depending on whether positive or negative events occur, temperature cut-offs considered being 0.4°C and -0.35°C, respectively.

Figure 8 - Histogram of positive and negative events versus their time lag. In red is shown the frequency of events; in blue the frequency is weighted by the amplitude of events. The biannual mode does not appear

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explicitly in the histograms (its amplitude remains relatively constant from one cycle to another and never

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reaches the temperature cut-offs).

Both histograms show the modes of 1, 2 and 4 years average period with a time resolution of 3 months. The same 4-year time intervals being used for ENSO and IOD, the histograms show that the quadrennial

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positive IOD events are in phase with ENSO, triggering in the middle of the time intervals. Negative events are shifted of 6 months compared to positive events. The half-time of 4-year length intervals being in

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January, Figure 8 shows that the positive and negative dipoles grow during boreal winter and summer, respectively.

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Moreover, the amplitudes of the two histograms show a great similarity provided the central bars of the histogram of positive events are merged. Then, the histogram of the negative events is deduced from that of the positive events by operating a translation of a half year. This confirms that the negative SST

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anomaly is the counter-part of the positive anomaly with an annual, biennial and quadrennial periodicity. Since each subharmonic QSW behaves as a single dynamical system with equatorial Rossby and Kelvin waves and off-equatorial Rossby waves, in IOD years SST anomalies result from second, third and fourth baroclinic mode Rossby and Kelvin waves, depending on the period of the dominant QSW that is at the origin of the anomalies. It can be assumed that the climate impact varies greatly depending on the dominant

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baroclinic mode, that is to say the time lag of the IOD. This would deserve more investigations. 3.1.5

The ENSO and the IOD

We can draw a parallel between ENSO and IOD whose meteorological consequences are of primary

importance, and that result of the coupling of baroclinic waves. In the tropical Pacific the subharmonic of 4 year-average-period, which induces the ENSO, is coupled to the fourth meridional mode Rossby wave forced by seasonal winds (Pinault, 2018b). In the Indian Ocean the second, third and fourth baroclinic mode

Rossby and Kelvin waves, which induce the IOD, are coupled to the biannual first baroclinic mode forced by the monsoon winds. In either case forcing of the lower baroclinic mode QSW occurs resonantly and the QSWs of different frequencies are subject to a subharmonic mode locking that ensure their optimal stability. Moreover, the western and eastern thermal anomalies in the Indian Ocean induce a coupling between the ocean and atmosphere reminiscent of the Walker circulation in the equatorial Pacific (Schott et al., 2009). The oceanatmosphere coupling self-sustains subharmonics of 4-year period in the Pacific (ENSO events), of 1, 2 and 4-years period in the Indian Ocean, phenomena of evaporation, thus of cooling, leading to the uprising of

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the thermocline at a critical stage in the evolution of the resonantly forced wave. However, the variability concerns both the frequency and amplitude of the quadrennial subharmonic of the Pacific Ocean but only

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the amplitude of subharmonics of 1, 2 and 4-years average period in the Indian Ocean (Figure 8).

Another likeness is about the climate impact of ENSO and IOD events that should greatly depends on

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their time lag. This link is proven for ENSO since central (CP) or eastern (EP) Pacific events occur according

The annual quasi-stationary wave

3.2.1

The observations

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3.2.

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to their time lag (Pinault, 2018b). Although not founded, this link is highly likely for IOD.

Figure 9 - Amplitude (left) and phase (right), averaged over the 8-16 month band of - a, b) antinodes whose phase is expressed relative to 01/2007 - c, d) velocity u (facing west) at the nodes whose phase is relative to 01/2008: only the phase of modulated currents in the southern hemisphere, the Somali along the African coast, and the monsoon drift south of Sri Lanka are reproducible from one cycle to another.

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Figure 10 – a, c, e, g, i) Representation of SSH and SCV data, filtered and unfiltered – b, d, f, h, j)

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Corresponding Fourier power spectra and confidence spectrum assuming a red-noise with a lag-1 (autocorrelation) α.

Two main antinodes are visible in both hemispheres (Figure 4b, Figure 9a, b). The southernmost antinode extends westward from the Timor passage, longitude 80°E, following first the Indonesian Throughflow then the South Equatorial Current. The northernmost antinode follows the monsoon drift off

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the east coast of Africa, south of the Arabian Sea, to the southern tip of the Indian subcontinent. Less extended, antinodes develop along the coast of the Bay of Bengal. To the east they are formed from the coastal Kelvin waves, as evidenced by the phase change north of the bay. Three main nodes are recognizable in Figure 9c, d. To the south is the South Equatorial Current between the Timor passage and longitude 60°E, to the west the Somali, a current that follows the eastern African coast, to the north the monsoon drift that is mostly visible south of the Indian subcontinent.

The Fourier power spectra of both SSH at 10.5°S 86.5°E (Figure 10a, b) and SCV at 4.5°N 79.5°E, 7.5°N 54.5°E, and 11.5°S 113.5°E (Figure 10c, d, e, f, g, h) exhibit a sharp peak centered at one year, at the antinodes and nodes of the 1-yr period QSW. The current at 11.5°S 113.5°E south of the southern coast of Java is a modulated current flowing westward, vanishing periodically (Figure 10g, h), whereas the Monsoon Drift at 4.5°N 79.5°E south of Sri Lanka and the Somali Current 7.5°N 54.5°E are reversing currents (Figure 10c, d, e, f). First baroclinic mode, first meridional mode Rossby wave

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3.2.2

Figure 11 - Amplitude (left) and phase (right) of the solution of the equations of motion for the annual QSW, that is, for the off-equatorial Rossby wave. Wind stress is considered for the period 2003-2004. The phase velocity is assumed to be independent of longitude; the solution is truncated at 90°E - a, b) antinodes - c, d) velocity (facing west) of the zonal current u - e, f) velocity (facing south) of the meridional current v. The

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length of the partial sums of the Fourier series is 12 with respect to the longitude and 16 with respect to time. Period 𝑇= 1 year. The momentums associated with surface stress forcing are rotated so as to represent them parallel and perpendicular to the direction of propagation of the wave. These rotated moments are used to calculate the forcing terms in the equations of motion.

As shown in Figure 11, the QSW is represented by an annual first baroclinic mode, first meridional mode Rossby wave, propagating in the southern hemisphere, turning back off the western boundary, then propagating in the northern hemisphere. It is resonantly forced by the trade winds in the southern hemisphere and the monsoon winds in the northern hemisphere. Since this off-equatorial Rossby wave does

not propagate parallel to the equator, the phase velocity varies with the latitude. However, to understand how the modulated currents are formed involving the geostrophy of the tropical band, it is convenient to assume the phase velocity is constant, being averaged along the path of the traveling waves. Although it is simplistic, such a model has the merit of highlighting the geostrophic forces that are causing the transient adjustment of tropical ocean circulation to changes in wind forcing at the sea surface. In addition, mechanisms involved in resonant forcing at the basin scale are evidenced. The solution of the equations of motion shows the Rossby wave is very sensitive to the latitude on both sides of the equator in the western part of the basin. Indeed, a small variation in latitude causes significant

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change in the phase velocity. In this way tuning of the Rossby wave to seasonal winds results from the path of the wave in the vicinity of the equator.

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The solution reproduces the modulated component of geostrophic currents satisfactorily as evidenced in Figure 9c, d and Figure 11c, d: the maximum speed of modulated currents at the nodes, of the order of

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0.25 m/s, and their phase are in agreement with observations. The speed of the geostrophic component of the South Equatorial Current is maximum in June-July when the trade winds in the southern hemisphere

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peak. The Somali and the monsoon drift south of Sri Lanka are maximum in May-June when facing east, i.e. during the monsoon winds.

However, the rapid variation of the phase of the southern antinode does not appear in the solution

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(Figure 9b, Figure 11b): only the northern edge of the observed antinode is in phase with the trade winds, reaching a maximum in July. This phase rotation within the southern antinode reflects the decrease in the

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phase velocity of the Rossby wave when the latitude increases: 2.3 m/s at the equator, it is 0.08 m/s at 15°S. In contrast, the node indicates a constant phase in the southern hemisphere (Figure 9d), as a result of geostrophic forces acting throughout the basin. In this case, everything happens as if the phase velocity did not depend on the latitude.

This approach differs from that more widespread about the Ekman transport, horizontal movement of

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the layers of the ocean surface waters by the action of wind stress. Geostrophic currents as seen in Figure 9c, d result from antinodes that affect geostrophic forces at the tropical basin scale, these SSH anomalies being resonantly generated by wind stress. In particular solving the equations of motion allows to explain the geostrophic component of the Somali and the monsoon drift south of Sri Lanka. 3.2.3

Evolution of the annual wave

The annual Rossby wave is formed at the outlet of the Timor passage to cross the Indian Ocean. It propagates across the Indian Ocean from the outlet of the Timor passage along the South Equatorial

Current. It is then deflected northward approaching the western boundary of the Indian Ocean, then follows the Somali and the Southwest Monsoon Current, avoids the Indian subcontinent south of Sri Lanka acting as a waveguide to go along the coast of the Bay of Bengal. The propagation direction is reversed with the Northeast Monsoon Current (Figure 4b). The wave propagation in the northern hemisphere requires the phase velocity of the westward propagating Rossby wave is lower than the speed of the current flowing eastward that drags it so that the apparent velocity of the wave is eastward. When the monsoon drift is reversed, the apparent velocity of the Rossby wave is westward, and the Somali along the coast of Somalia is reversed, too, a part of this current

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leaving the tropical ocean to feed the western boundary current along the eastern coast of Madagascar and the coast of southeast Africa to form the Mozambique current.

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Antinodes show a north-south seesaw of warm waters between the northern and the southern hemispheres, being nearly in opposite phase (Figure 9b). From the Pacific they accumulate during the boreal

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summer to form the southern antinode whereas, due to upwelling that is stimulated in the Bay of Bengal and the Arabian Sea, cold water overruns the northern part of the basin. In spring the phenomenon is

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reversed, warm water accumulating in the north of the basin. Upwelling weakens as well as the South Equatorial Current; reversing of monsoon drift promotes seesaw of warm waters. So, the southernmost current of the Indonesian Throughflow through the Timor passage contributes

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significantly to the propagation of long waves between the western Pacific and the Indian Ocean (Potemra, 2001; Potemra and Schneider, 2007; Vaid et al., 2007). The modulated current vanishes in January while

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reversing some years at the western end of the node (Figure 10i). Thus, although the modulated current at the node flows mainly to the west, it behaves as a reversing current at its western end: more, the magnitude of the velocity oscillates at low frequencies as shown in Figure 10j. The amplitude of SCV averaged over the area [12.5°S, 10.5°S]×[108.5°E, 118.5°E] is 0.090 ± 0.007 m/s and the standard deviation of the coherence phase is 15 days, which points out the stability of the easternmost end of the southern node.

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SCV being considered at the reversing Monsoon Drift south of Sri Lanka, its amplitude averaged over the area [3.5°N, 4.5°N]×[76.5°E, 83.5°E] is 0.236 ± 0.036 m/s. The standard deviation of the coherence phase is 25 days, which indicates the poor stability of the node. In contrast, SCV being considered at the Somali current, the same reasoning that refers to the area [4.5°N, 12.5°N]× [48.5°E, 56.5°E] gives 0.106 ± 0.006 m/s and 6 days, which indicates the extreme stability of this reversing current. To summarize, the thermal energy is transferred from the western basin in the Pacific, which acts as a heat sink, to the Indian Ocean via the Timor passage. Then, heat exchange occurs between the two hemispheres via the Somali and the monsoon drift, each reversing periodically in phase. The annual wave

feeds the western boundary current, i.e. the Agulhas, through a succession of warm and cold waters. The solution of the equations of motion emphasizes the evolution of the annual off-equatorial quasi-stationary Rossby wave which propagates in both hemispheres. In the southern hemisphere, the northeastward propagation of the Rossby wave, then its spreading in the northern hemisphere, occur when the speed of the reversing Monsoon Drift in which it is dragged is in opposite direction and higher than the phase velocity of the westward-propagating wave. Resonant forcing of the wave results from the dephasing of each main antinodes in both hemispheres and the timing of the winds: the natural period of the wave tunes to the forcing period essentially by the route of its path in the vicinity of the equator. The driver of reversing

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currents is not Ekman transport but rather geostrophic forces resulting from the resonant forcing of the

3.2.4

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Rossby wave. Vertical temperature gradient and barrier layer

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Advection resulting from the resonance of baroclinic waves induces stratification of layers, as a result of strong salinity gradients in the tropical Indian Ocean. This has a significant impact on the SST behavior



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during cycles.

In the southern hemisphere, deep mechanisms involved into the storage of warm water in

November are demonstrated from the amplitude and phase of the profiles of seawater temperature

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anomalies obtained from the Global Tropical Moored Buoy Array (Figure 12a, b, e, f). A barrier layer is located between 20 and 40 m, so that the underlying advected layer at the antinode is bordered by this barrier at the top and the thermocline more than 300 m deep at the bottom. Due to the buoyancy of the

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upper layer and the associated mixing processes when the halocline is shallow, seawater temperature in shallow water remains nearly constant during the year. 

In the northern hemisphere, warm water at the antinode alongside the eastern part of the Bay of

Bengal is advected in August, in conformity with the propagation of the ridge (Figure 6c, d, Figure 12d, h, Figure 7c, d, g, h). The underlying advected layer is under a barrier layer, 60 m deep, resulting from the

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buoyancy of the upper advected layer, both in opposite phase. Sea-surface salinity is particularly low in the Bay of Bengal where Himalayan Rivers flow, mainly the Ganga and the Brahmaputra. So the density difference between the upper and the underlying layers induce the duplication of the advected layer at the antinode, hence the seasonal variations in SST about 2°C where the upper advected layer rises to the surface in the extreme northern part of the bay.

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Figure 12 - Amplitude (top) and phase (bottom), scale-averaged over the 8-16 month band, of profiles of seawater temperature anomalies at 8°S 80°E (a, e), 8°S 67°E (b, f), 0°N 90°E (c, g) and 15°N 90°E (d, h). The phase is calibrated so that it expresses the date during which the thermal anomaly is maximal. The minimum

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is reached 6 months later.

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The decoupling of the upper and the deeper layers also occurs on the equator at the same longitude (Figure 12c, g) but, here, the underlying advected layer that belongs also to the northern antinode exhibits large temperature variations, the amplitude of seawater temperature anomaly reaching 2.7°C, while there

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are limited exchanges with the upper layer due to the barrier layer, 100 m deep. Several studies have highlighted the importance of the Seychelles-Chagos thermocline ridge (SCTR), that is, what turns out to be the southern antinode of the annual QSW, for the regional climate [Hermes and

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Reason, 2008, 2009] and the intensification of tropical cyclones [Xie et al., 2002; Reason and Keibel, 2004; Malan et al., 2013]. The impact of the ocean during the passage of a tropical cyclone is also influenced by the barrier layer [Sengupta et al., 2008; McPhaden et al., 2009. Wang et al., 2011; Mawren, and Reason, 2017] that can inhibit cyclone intensification resulting from heat transfer from the ocean to the atmosphere. In the absence of a barrier layer, which is mistaken for a thick barrier layer in previous works, the cyclone

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intensification results from the vertical temperature gradient into the antinode and the induced convective processes, mainly when the thermocline is rising between October and March. Monsoons, which results from the larger amplitude of the seasonal cycle of land temperature compared

to that of nearby oceans, mainly occur where SST remains nearly constant during annual or semi-annual cycles, increasing therefore the thermal gradient between the land and the oceans. Both the 1-yr and the 1/2yr period basin modes play an important role in reducing SST variations where the salinity of the advected layer at the antinodes makes it sink under a differentiated surface layer. This is particularly true for the 1/2-

yr period basin mode, the area where SST is seasonal-independent coinciding with the zone of low salinity in the south-eastern tropical Ocean. This phenomenon is pointed out off the western coast of Sumatra. 4

Conclusion Through resonantly forced QSWs, this study attempts to synthesize and unify our knowledge of the

physical oceanography of the tropical Indian Ocean whose dynamics exhibits two independent basin modes. Indeed, the Equatorial Counter-Current is not part of the annual off-equatorial QSW, being out of phase with the two modulated currents that are the Somali and the monsoon drift as shown in Figure 8d.

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The biannual QSW and subharmonics thus operate independently of the annual QSW, which itself propagates out of the equator. These two systems have no modulated current in common, contrarily to

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what occurs in the Atlantic and Pacific.

The biannual mode and its subharmonics produces the IOD, which is actually the superposition of

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four coupled oscillators whose periods are 1/2, 1, 2 and 4 years, i.e. the biannual fundamental wave that is resonantly forced by monsoon wind stress and its subharmonics whose average periods are multiples of

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the shorter periods.

As a result of the annual off-equatorial basin mode, thermal energy is transferred from the western basin of the Pacific to the Indian Ocean via the Timor passage. Then, heat exchange occurs between the two

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hemispheres via the Somali and the monsoon drift. The annual wave feeds the western boundary current, i.e. the Agulhas, through a succession of warm and cold waters. Resonantly forced QSWs are ubiquitous in the three tropical oceans, forming single dynamical systems.

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Apart from the annual off-equatorial QSW in the Indian Ocean, equatorial waves play a central role in the evolution of tropical QSWs. In any case, their natural period tunes to the forcing period so that the resonance conditions remain fulfilled during the successive cycles. Such resonantly forced QSW make the most of the energy dissipated by forcing. Otherwise, the forcing would lose its efficiency in the absence of synchronization, and the non-resonant QSW would disappear in favor of a resonant QSW.

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Zonal modulated currents produced by tropical QSWs have a key role in the general ocean circulation.

Feeding the western boundary currents, they induce the resonance of gyral Rossby waves, producing longperiod subharmonics (Pinault, 2018c). Their direct or indirect incidence on climate is considerable, being at the origin of the ENSO in the tropical Pacific, the IOD in the Indian Ocean and, more generally, to the longterm climate evolution. 5

Appendix: The forced version of equations of motion

The forced versions of linearized primitive equations, that traduce momentum (A1), (A2) and continuity (A3) equations of two superposed fluids of different density are, with the potential vorticity equation (A4) that follows from the previous ones:

u / t  yv   g ' ~ / x  X / 1 H1

A9

 / t  yu   g ' ~ / y  Y / 1 H1

A2

~ / t  H1 u / x  v / y   E / 1

A3

   f~ / H 1   v  1 Y / x  X / y  fE  t 1 H 1

is the perturbation of the surface height and h the upward interface displacement,

which is resolved into a vertical mode:

~ So 

of



   / h   g" H 2 / gH

 (  1) /  , g"  g (1  1 /  2 ) , g '  g /(  1)

  v / x  u / y

is the Coriolis parameter), and

H 1 is the depth of the upper layer, H 2 is the depth of the

H is the total depth.  1 and  2 are the density of the upper and the lower layer,

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lower layer and

is the potential vorticity.

f

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f  y (the beta plane approximation is used where

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where ~    h ;

A4

respectively.

As usual, supposing

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The forcing terms X and Y represent a surface stress and E an evaporation rate.

  N where N is the buoyancy frequency, which is assumed to be constant, ( 

is the rotation rate of earth) and

c  N 2 , the previous equations remain valid on and near the equator.

that:

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To relate v to the other variables, new variables

q and r are defined rather than u and v (Gill, 1982), so

q  g '~ / c  u , r  g '~ / c  u

By combining A1 to A4, three new independent equations are obtained: A5

r r v 1  c  c   yv   ( X  cE ) t x y 1 H 1

A6

 Y X    r v v  1  Y  c  yr   c   cv   c   fE   t  y t x  1 H 1  t  x y 

A7

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q q v 1 c  c   yu  ( X  cE ) t x y 1 H 1

The shallow water equations are solved by expanding in terms corresponding to normal modes, i.e. the parabolic cylinder functions that appear in the wave solution. So:



(v, q, r )  n 0 (v n , q n , rn ) Dn (2 / c)1 / 2 y 





( X , Y , E )  1 H 1 n 0 ( X n , Yn , E n ) Dn (2 / c)1 / 2 y 

A8



A9

 d    / 2  Dm  mDm 1 and  d 

Then utilizing the property of parabolic cylinder functions, namely 

 d     / 2  Dm   Dm 1 ,   (2 / c)1 / 2 y , the equations for the coefficients become for the modes  d 

n  0 (conventionally, coefficients whose subscript is n  1 are zero; n  0 corresponds to mixed

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planetary-gravity waves):

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( / t  c / x)rn1  (2c)1 / 2 nvn  ( X n1  cEn1 ) ( / t  c / x)qn1  (2c)1/ 2 vn  X n1  cEn1

A10 A11

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        1/ 2 1/ 2  (2c) rn 1    c v n   cv n    c Yn  ( c / 2) (n  1) X n 1  X n 1   fEn t   t  x  t  x     

forcing,

the

solutions

are:

rn 1  Rn 1 sinkx  t  ,

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Without

vn  Vn coskx  t 

A12 and

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q n1  Qn1 sinkx  t  whose amplitudes Rn 1 and Qn 1 are (supposing Vn  1 ):

Rn1  (2c)1 / 2 n /(ck  ) , Qn 1  ( 2  c)1 / 2 /( ck   ) and the relation between the wave number k and the frequency



is:

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    4 2 k  1  1  2 2  c(2n  1)   2   2  c  

1/ 2



   

A13

The negative sign before the bracket corresponds to gravity waves and the positive sign to planetary waves: in this case, the wave number k tends to 0 with the frequency

 c

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k

 , i.e. for small  :

(2n  1)

Considering Kelvin waves, for which

A14

v  0 and   kc , the coefficients are obtained from the two

equations:

r0 / t  cr0 / x  ( X 0  cE 0 )

A15

q0 / t  cq0 / x  X 0  cE 0

A16

Funding: This research received no external funding

Acknowledgments: We thank the editor and the reviewers for their helpful comments. Conflicts of Interest: The author declares no conflict of interest.

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