## Abstract

Based on the well established importance of long, non-dispersive baroclinic Kelvin and Rossby waves, a resonance of tropical planetary waves is demonstrated. Three main basin modes are highlighted through joint wavelet analyses of sea surface height (SSH) and surface current velocity (SCV), scale-averaged over relevant bands to address the co-variability of variables: (1) a 1-year period quasi-stationary wave (QSW) formed from gravest mode baroclinic planetary waves which consists of a northern, an equatorial and a southern antinode, and a major node off the South American coast that straddles the north equatorial current (NEC) and the north equatorial counter current (NECC), (2) a half-a-year period harmonic, (3) an 8-year sub-harmonic. Contrary to what is commonly accepted, the 1-year period QSW is not composed of wind-generated Kelvin and Rossby beams but results from the excitation of a tuned basin mode. Trade winds sustain a free tropical basin mode, the natural frequency of which is tuned to synchronize the excitation and the ridge of the QSWs. The functioning of the 1-year period basin mode is confirmed by solving the momentum equations, expanding in terms of Fourier series both the coefficients and the forcing terms. The terms of Fourier series have singularities, highlighting resonances and the relation between the resonance frequency and the wavenumbers. This ill-posed problem is regularized by considering Rayleigh friction. The waves are supposed to be semi-infinite, i.e. they do not reflect at the western and eastern boundaries of the basin, which would assume the waves vanish at these boundaries. At the western boundary the equatorial Rossby wave is deflected towards the northern antinode while forming the NECC that induces a positive Doppler-shifted wavenumber. At the eastern boundary, the Kelvin wave splits into coastal Kelvin waves that flow mainly southward to leave the Gulf of Guinea. In turn, off-tropical waves extend as an equatorially trapped Kelvin wave, being deflected off the western boundary. The succession of warm and cold waters transferred by baroclinic waves during a cycle leaves the tropical ocean by radiation and contributes to western boundary currents. The main manifestation of the basin modes concerns the variability of the NECC, of the branch of the South Equatorial Current (SEC) along the equator, of the western boundary currents as well as the formation of remote resonances, as will be presented in a future work. Remote resonances occur at midlatitudes, the role of which is suspected of being crucial in the functioning of subtropical gyres and in climate variability.

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## Acknowledgments

The text has benefited from the advice and recommendations of Lucy Campbell of Carleton University. She receives our gratitude.

## Author information

## Appendix

### Appendix

### Wavelet Analysis

Let us consider the time series \( X = x_{0} , \ldots ,x_{N} \) with the continuous Morlet wavelet transform \( W_{n}^{X} (s) \), where *n* and *s* represent the discrete time and the scale, respectively (Torrence and Compo, 1998). The continuous wavelet transform of the discrete sequence \( x_{n} \) is defined as the convolution of \( x_{n} \) with a scaled and translated version of the wavelet function:

where the (*) indicates the complex conjugate. Here, the Morlet wavelet function is used, being defined from a plane wave modulated by a Gaussian:

\( \omega_{0} \) is the adimensional frequency, taken to be six to satisfy the admissibility condition, i.e., the wavelet function must have zero mean and be localized in both time and frequency space (Farge, 1992). The wavelet function, the resolution of which is determined by the balance between the width in real space and the width in Fourier space, is a broad function with a poor time resolution, yet good frequency resolution. Other standard wavelet functions have a lower frequency resolution; hence, the motivation for using the Morlet wavelet function is to take advantage of its ability to highlight the resonance of long waves that is revealed by sharp peaks in both SSH and SCV Fourier spectra.

#### Highlighting the Antinodes

Highlighting quasi-periodic SSH anomalies at the antinodes of the QSWs, requires the cross-wavelet analysis of two series \( X = {\text{SSH}}(x,y) \) and \( Y = {\text{SSH}}(x_{0} ,y_{0} ) \), where \( x_{0} \) and \( y_{0} \) are the longitude and the latitude of SSH considered as the temporal reference. If \( W_{n}^{X} (s) \) and \( W_{n}^{Y} (s) \) are the wavelet transforms of series \( X \) and \( Y \), the cross-wavelet spectrum is defined as:

The cross-wavelet power is \( \left| {W_{n}^{XY} (s)} \right| \) and the coherence phase is:

Both the cross-wavelet power and the coherence phase are scale averaged over a relevant band and represented according to different longitudes, latitudes and times. The cross-wavelet power of series \( X \) and \( Y \) is normalized by the square root of the wavelet power of the reference series \( Y \):

so that this value is intrinsically the amplitude of SSH variations. Generally, \( {\text{CWP}}^{X} \) is close to the square root of the wavelet power of the series \( X \), i.e. \( \sqrt {\left| {W_{n}^{XX} (s)} \right|} \), while being more accurate when data are noisy, highlighting the characteristic frequencies in a meaningful way when the reference is relevant.

The normalized wavelet cross-spectrum is used instead of wavelet coherence as presented in Maraun and Kurths (2004), because it has the same dimension as the series \( X \) to be analyzed and, as such it represents the amplitude of the observed phenomena intrinsically, whereas the coherence is dimensionless. Indeed, dividing the cross-wavelet power by the square root of the wavelet power of the temporal reference allows the amplitude to be independent of the behavior of the reference (here, we are not interested in the coherence between \( X \) and \( Y \) since \( Y \) is just used as the temporal reference the choice of which should not appear through the results).

The phase is the propagation time of the waves between \( (x_{0} ,y_{0} ) \) and \( (x,y) \). In the case of a QSW, both \( Y = {\text{SSH}}(x_{0} ,y_{0} ) \) and \( X = {\text{SSH}}(x,y) \) are supposed to have only one maximum during the period \( T \). It is convenient to express the cross-wavelet on the time such that \( Y \) is maximal. Then the phase represents the elapsed time when \( Y \) and \( X \) reach their respective maxima. In addition, the phase can be expressed so that it is the date, within a particular cycle, when the maximum (or the minimum) \( X \) is reached. This transformation requires knowledge of the dates when the series used as the temporal reference reaches its maxima. These dates are estimated by filtering the series with a bandpass filter whose band is characteristic of the resonance, then searching for the maxima.

In this way, neither the amplitude nor the phase of QSWs depends on the reference. By construction (the QSW is symmetrical), the time elapsed between when SSH reaches a maximum and the closest minimum is \( T/2 \) where \( T \) is the period of the basin mode.

#### Highlighting the Nodes

The nodes of QSWs being characterized by modulated currents that strengthen and vanish periodically can be highlighted from the cross-wavelet analysis of SCV within a relevant band. While the direction of surface currents is not known a priori, the wavelet analysis refers to the magnitude of the velocity vector. More precisely, since the surface currents may reverse, the analysis applies to the signed magnitude of the velocity vector. Conventionally, the sign associated with the magnitude of the SCV field is the same as the sign of the longitudinal component of the velocity vector *u*, i.e., positive if the velocity vector is from west to east and negative from east to west. In such conditions, the modulated currents are represented by the ridge drawn from the highest values of the normalized cross-wavelet power of the series \( X = \pm v(x,y) \) and the temporal reference \( Y = \pm v(x_{0} ,y_{0} ) \), where ±*v* represents the signed magnitude of the SCV field, scale-averaged over a relevant band representative of the resonance. Indeed, due to the continuity of surface currents, the ridges highlight the strongest modulated currents from their magnitude, and, consequently, their directions. Like for SSH, the phase is expressed so that it is the time within a cycle during which the maximum of the magnitude of SCV is reached. The time elapsed between the maximum and the minimum SCV is \( T/2 \).

### The Forced Version of Equations of Motion

The forced versions of linearised primitive equations, that traduce momentum (14), (15) and continuity (16) equations of two superposed fluids of different density, are, with the potential vorticity Eq. (17) that follows from the previous ones:

where \( \tilde{\eta } = \eta - h \); \( \eta \) is the perturbation of the surface height and \( h \) the upward interface displacement, which is resolved into a vertical mode: \( \mu = \eta /h = - g^{\prime\prime}H_{2} /gH \).

So \( \tilde{\eta } = \eta \left[ {(\mu - 1)/\mu } \right] \), \( g^{\prime\prime} = g(1 - \rho_{1} /\rho_{2} ) \), \( g^{\prime} = g[\mu /(\mu - 1)] \)

\( f = \beta y \) (the beta plane approximation is used where \( f \) is the Coriolis parameter), and \( \xi = \partial v/\partial x - \partial u/\partial y \) is the potential vorticity. \( H_{1} \) is the depth of the upper layer, \( H_{2} \) is the depth of the lower layer and \( H \) is the total depth. \( \rho_{1} \) and \( \rho_{2} \) are the density of the upper and the lower layer, respectively.

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

As usual, supposing \( \Upomega \ll N \) where \( N \) is the buoyancy frequency, which is assumed to be constant, (\( \Upomega \) is the rotation rate of earth) and \( \beta \,c \ll N^{2} \), the previous equations remain valid on and near the equator.

To relate \( v \) to the other variables, new variables \( q \) and \( r \) are defined rather than \( u \) and \( v \) (Gill, 1982), so that:

\( q = g^{\prime}\tilde{\eta }/c + u,r = g^{\prime}\tilde{\eta }/c - u \)

By combining A14 to A17, three new independent equations are obtained:

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:

Then utilizing the property of parabolic cylinder functions, namely \( \left( {\frac{d}{d\xi } + \xi /2} \right)D_{m} = mD_{m - 1} \) and \( \left( {\frac{d}{d\xi } - \xi /2} \right)D_{m} = - D_{m + 1} \), \( \xi = (2\beta /c)^{1/2} y \), the equations for the coefficients become for the modes \( n \ge 0 \) (conventionally, coefficients whose subscript is \( n - 1 \) are zero; \( n = 0 \) corresponds to mixed planetary-gravity waves):

Without forcing, the solutions are: \( r_{n - 1} = R_{n - 1} \sin (kx - \omega t) \), \( v_{n} = V_{n} \cos (kx - \omega t) \) and \( q_{n + 1} = Q_{n + 1} \sin (kx - \omega t) \) whose amplitudes \( R_{n - 1} \) and \( Q_{n + 1} \) are (supposing \( V_{n} = 1 \)):

\( R_{n - 1} = (2\beta c)^{1/2} n/(ck + \omega ) \), \( Q_{n + 1} = (2\beta c)^{1/2} /(ck - \omega ) \) and the relation between the wave number \( k \) and the frequency \( \omega \) is:

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 \( \omega \), i.e. for small \( \omega \):

Considering Kelvin waves, for which \( v = 0 \) and \( \omega = kc \), the coefficients are obtained from the two equations:

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Pinault, J. Long Wave Resonance in Tropical Oceans and Implications on Climate: the Atlantic Ocean.
*Pure Appl. Geophys.* **170, **1913–1930 (2013). https://doi.org/10.1007/s00024-012-0635-9

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### Keywords

- Long oceanic wave resonance
- quasi-stationary waves
- wavelet analysis
- tuned basin mode