The Amazon shelf break is a key region for internal tide (IT) generation. It also shows a large seasonal variation in circulation and associated stratification. This study, based on a high-resolution model (

The passage of barotropic tidal currents over a sloping bottom or topographic feature in a stratified fluid generates
internal waves that propagate at a tidal frequency and are called internal tides or baroclinic tides. Internal tides induce (vertical) isopycnal displacements of up to tens of metres and are distributed into a set of vertical modes. The low modes can propagate horizontally over hundreds to thousands of kilometres, carrying most of the generated baroclinic energy away from the internal tide generation sites

Contrary to barotropic tides, which are extremely stable with time (except in some very particular locations), the
baroclinic tides are permanently modulated by the background ocean variability. Consequently, internal tide amplitudes and phases can be seen as the resulting sum of a “stable” or phase-locked component, called coherent tides, and a “variable-with-time” non-phase-locked component, called incoherent tides. In practice, the coherent tide is obtained by harmonic analysis of variables such as sea surface height (SSH) from altimetric observations and numerical models

The Amazon shelf is a shallow wide shelf extending off the north Brazilian coast in the western tropical Atlantic. The shelf break occurs along the 100 m isobath (Fig.

Temperature and salinity (the stratification) along the north Brazilian continental shelf vary under the influence of the freshwater discharge of the Amazon and Para rivers, the trade winds, the North Brazil Current (NBC), and the tidal forcing, primarily the semi-diurnal M2

During its annual cycle, the NBC develops a double retroflection, first into the Equatorial Undercurrent (EUC) in winter–spring and second into the North Equatorial Countercurrent (NECC) at about 5–8

Our work is done as part of a project related to the future SAR-interferometry wide-swath altimeter mission SWOT (Surface Water and Ocean Topography). SWOT is designed to provide global 2D SSH observations for the spatial scale down to the sub-mesoscale of 15–30 km

Our study is based on a high-resolution ocean numerical model presented in Sect. 2. Section 2 is also dedicated to Argo and altimetric data used for the model validation and to the method of separating barotropic and baroclinic tides. The model is validated over the MAMJJ and ASOND seasons in Sect. 3 where the contrasting EKE characteristics are explored. The generation, propagation, and dissipation of the coherent internal tide M2 are presented in Sect. 4, along with some snapshots of the baroclinic flux and currents that illustrate the interaction of the internal tide with the circulation for each season. The SSH characteristics are analysed in Sect. 5. A summary of the paper is given in Sect. 6. The paper ends with Sect. 7 on discussion and perspectives.

The numerical model used in this study is NEMOv3.6 (Nucleus for European Modeling of the Ocean,

Model validation was performed by comparing model outputs with observations. The model potential density and stratification were compared to the CORA

Barotropic and baroclinic tides must be clearly separated to derive a correct internal tide energy budget. Baroclinic pressure and horizontal velocity are commonly defined as the difference between the total field and the depth-averaged field in a stratified ocean. This definition proposed by

In practice, to carry out the vertical mode decomposition, we solve the eigenfunctions for 10 modes at each point of the model using the local mean stratification over the analysed periods (the entire period, March to December, or the seasons MAMJJ and ASOND). We then fit the U eigenmodes to each harmonic constant of the 3D velocity and pressure fields and used the modal amplitudes and phase in the energy analysis (see Eqs. 1 to 5 in Sect. 4). This provides the description of the barotropic tide (mode 0) and the coherent baroclinic tide that can be analysed for each mode or as the sum of the nine baroclinic modes. The M2 wavelength varies spatially and temporally between 90–125 km for mode 1 and 12–15 km for mode 9 (not shown). The horizontal resolution of the model allows us to solve for the first eight modes

We first evaluated the ability of the model to correctly simulate the barotropic and baroclinic tide. For this purpose only, the barotropic and baroclinic tides are evaluated over the entire simulation period from March to December (Figs.

The barotropic tide evolves freely in the model after it has been forced at its lateral boundaries by FES2012. The resulting modelled M2 barotropic is maximum near the northwest and southeast of the Amazon mouth because of the landward propagation and convergence of the barotropic tide coming from the open ocean (Fig.

In this subsection, we illustrate the contrasts in ocean conditions (circulation and stratification) between MAMJJ and ASOND in the model. The surface current, the EKE, and density profiles are validated by comparison with AVISO and Argo observations. The 5-month “seasons” of MAMJJ and ASOND correspond to 1752 h covering the periods shown in Table 1. The MAMJJ shift of 1 week in August is necessary to have the same number of spring and neap tide cycles, which is necessary for the comparison of tidal harmonics.

First, 25 h running means were performed to separate tide and high frequency from the low-frequency mesoscale variability in the model. Then EKE was evaluated using the anomaly of the 25 h running mean current relative to the mean current from March to December. During MAMJJ, the current is weak, the NBC is trapped along the coast, and the EKE is between 900–1200

About 50 Argo vertical profiles of potential density were selected between March and December 2015. The selection criterion was the stability of the Brunt–Väisälä frequency (hereafter

Mean vertical profiles from Argo (red) and model (blue) during

The model and observations are collocated in time and space. The mean potential density profiles from March to December (annual, in Fig.

Overall, the model reproduces the vertical and temporal variations in the potential density and

The first 50 m of depth was not taken into account when determining the depth of

Table 1 summarizes the circulation and stratification contrasts between MAMJJ and ASOND. MAMJJ is the season of low current, low EKE, and a shallower and stronger pycnocline with weak spatial gradient. In ASOND, the currents are stronger, the retroflection is well developed, the EKE is strong, and the pycnocline is deeper and weaker with stronger horizontal gradient.

Circulation and stratification characteristics during the MAMJJ and ASOND seasons.

Assuming that the energy tendency and nonlinear advection are small, the barotropic and baroclinic tide energy budget equations reduce to a balance between the conversion rate (CVR), the divergence of the energy flux, and the dissipation

Top: M2 conversion rate (CVR, colour, units: W m

For MAMJJ (Fig.

M2 total

Internal tide generation occurs along the shelf break (Fig.

After generation, M2 internal tide mainly propagates to the open ocean in the northeast direction (Fig.

A proxy of the dissipation is given in Fig.

The M2 conversion rate is integrated in the same way as the dissipation, having a maximum at 10 km distance from the shelf break and a zero crossing at 50 km from the shelf break (Fig.

The internal tide generated on the Amazon shelf propagates through a complex environment of strong boundary currents
(NBC, NECC, EUC), eddies, and salinity plumes associated with strong frontal structures and density gradients. It is not excluded that changes in oceanic conditions from MAMJJ to ASOND have an impact on the trajectory of the internal tides through the interaction between the internal tide and the background circulation (eddies, current, or stratification). To more precisely investigate the internal tide interactions with the circulation, we make the choice to leave aside the harmonic analysis approach, which does not allow us to depict short-term changes in the internal tide propagation characteristics. Instead, we make use of time filtering over a 25 h period, which provides a fair separation of tidal and non-tidal processes, at the sacrifice of individual tidal constituent diagnostics, leaving the neap and spring tide modulation in the filtered tidal signal. In Fig.

Examples of 25 h mean snapshots of depth-integrated baroclinic flux (colours and arrows, left, units: W m

During ASOND, the very intense currents delimit a frontal line with a steep pycnocline slope. Along the 1025 kg m

During MAMJJ, the currents are weaker and the eddies less intense and of smaller diameter (Fig.

According to Fig.

Since the differences between the M2 baroclinic fluxes of MAMJJ and ASOND are strongly linked to the interactions with the circulation, a fraction of the internal tide has become incoherent (non-phase-locked). The term incoherent is not limited to the internal tide; it also encompasses internal gravity waves (IGWs), which constitute a continuum of energy over a wide range of spatial and temporal scales. This study is conducted as part of a SWOT project, so we evaluate the incoherent components based on their SSH signatures.

As mentioned in the introduction, SSH from altimetric observations or models includes high-frequency unbalanced (non-geostrophic) components from the barotropic tides, from the coherent and incoherent internal tides, and from IGWs. Global model estimates of the barotropic tide are applied as a correction to altimetric SSH before the data are used for ocean circulation studies (e.g. FES2014,

SSHBT and SSHBC are respectively the coherent barotropic and baroclinic SSH, and they constitute mode 0 and the sum of the nine baroclinic modes remaining after projection on the vertical mode (see Sect. 2.3). They contain both the diurnal and semidiurnal tide components by which the model was forced. SSH1 corresponds to the usual processing of altimeter observations from which the barotropic tide correction is removed from the total SSH (Eq. 6). The coherent part of internal tides (SSHBC) is removed from SSH1 to obtain SSH2 (Eq. 7). SSH1 and SSH2 have similar low-frequency (here

To study the spatio-temporal scales of the coherent and incoherent SSH, spectral analyses are performed on SSHBC, SSH1, and SSH2. Before the fast Fourier transform (FFT) calculation, SSH is detrended and windowed with a Tukey 0.5 window, as previously done in

The frequency spectra of the total baroclinic tides, SSH1, are integrated at each point of the model to deduce the geographical distribution of the total (full, Fig.

Root means square (rms) of SSH1 for

Root mean square of

For both seasons the maximum variations in SSH1 occur north of 6

SSH1 includes the coherent baroclinic SSH (SSHBC) and the incoherent SSH (SSH2, Eq. 7). The coherent part (Fig. 11a and b) and incoherent part (Fig. 11c and d) of SSH1 at tidal frequencies (Fig. 10c and d) are separately evaluated in Fig. 11. M2 being the dominant component of the internal tide, the geographical distributions of the rms in Fig.

During ASOND, the tidal incoherence dominates north of 4

Rms of SSH1 at subtidal frequencies; coherent (SSHBC) and incoherent (SSH2) at tidal frequencies; and SSH1, SSHBC, and SSH2 at super tidal frequencies. Mean refers to the mean of rms in Figs.

In preparation for SWOT, it is important to know how the spatio-temporal SSH structures of the model depicted in Figs.

Meridional frequency–wavenumber of

Examples of frequency–wavenumber spectra of hourly SSH1 (Fig.

The altimetry data (Saral_full, black) and SSH1_full (blue) have both been corrected for the barotropic tide only. They show flatter SSH power spectrum density (PSD) spectral slopes over the 20–300 km wavelength range and are characterized by spectral peaks around 120 and 70 km. Despite the discrepancies at large scales and at scales smaller than 60 km, the agreement between altimetry and model reinforces our confidence in the model. At subtidal frequencies, the baroclinic SSH1_subtidal (red) is closer to SSH1_full (blue) from 1000 to 300 km in Fig.

SSH meridional wavenumber spectra separated into different frequency bands during

The baroclinic contributions to the spectral PSD are shown in the lower panels of Fig.

Finally, it is relevant to know up to what wavelengths the geostrophic balance relation is still valid and to determine the wavelength of transition from which the mesoscale and sub-mesoscale dominate over non-geostrophic movements including the internal tide and the IGWs. The SSH1_subtidal spectrum associated with the mesoscale and sub-mesoscale first intersects the SSH1_tidal spectrum (dominated by the internal tide) around 250 km during MAMJJ and ASOND; it then intersects the SSH1_supertidal spectrum (dominated by the IGWs) at 166 km in MAMJJ and 142 km in ASOND (see Table 3). For both seasons, the spectra of SSH1_subtidal and SSH1_supertidal are such that the variance of SSH1 at tidal frequencies dominates the supertidal ones for scales above 60 km. It is therefore reasonable to set the transition scale at 250 km given the behaviour of the spectra of SSH1_subtidal and SSH1_tidal during the MAMJJ and ASOND seasons. This is similar to the transition scale in the Amazon region found by

Transition length scale between balanced and unbalanced motion.

One of the challenges for the future SWOT mission is to propose appropriate processing to filter out most of the internal tide signals in the SSH products. Such an objective requires a clearer knowledge of internal tide dynamics including their temporal variability in various regions of the ocean. This study focuses on the Amazon shelf, one of the hotspots of M2 internal tide generation in the tropical Atlantic. The Amazon shelf is influenced by freshwater from river flow and precipitation below the ITCZ, as well as strong currents and eddies. The seasonal cycles of these oceanic, continental, and atmospheric forcings lead to two contrasting seasons (March to July – MAMJJ and August to December – ASOND) for which the properties of the M2 internal tide, the interaction of the internal tide with the circulation, and the SSH imprint of the internal tide have been explored. Barotropic and baroclinic tides were separated using vertical mode decomposition

The analyses are based on 9.5 months (March to December 2015) of hourly outputs of a high-resolution (

For both seasons, we have shown that the M2 barotropic tide originating from the southeastern open ocean is converted to M2 internal tide between the 100 m (the shelf break reference) and the 1000 m isobaths, with the maximum conversion occurring 10 km from the shelf break. The generated M2 internal tide then propagates mainly offshore in a northeasterly direction from sites A and B as in

The SSH has been separated into its coherent (phase-locked to barotropic forcing) and incoherent (with variable amplitude and phase) components. For each of the MAMJJ and ASOND seasons, the frequency and frequency–wavenumber spectra have been integrated for different frequency bands: the subtidal band for periods greater than 28 h counting for intraseasonal and mesoscale and sub-mesoscale variations, the tidal band between 28 and 11 h dominated by internal tide motions, and the supertidal band for periods less than 11 h where the inertial gravity waves are prominent. On the wavenumber spectra, it appears that the SSH variability for scales larger than 300 km is due to the intraseasonal and mesoscale and sub-mesoscale variability. Between 250 and 60 km, the SSH wavenumber spectra are flattened with peaks at mode 1 (150–100 km) and mode 2 (100–60 km) wavelength bands, and the SSH variance is related to the internal tide of tidal frequency. The supertidal and thus inertial gravity waves dominate scales under 60 km. At tidal and supertidal frequencies, the incoherent SSH induces SSH variations are of an order equal to or even greater than the coherent SSH. In the mode 1 wavelength band, the incoherent fraction (measuring how incoherent SSH is) is 0.4 during MAMJJ and 0.6 during ASOND. For mode 2 and wavelengths under 60 km, the incoherence fraction is higher than 0.5, marking a predominance of the incoherent tide. The transition scale corresponding to the wavelength at which the balanced (geostrophic) motion becomes more important than the unbalanced (non-geostrophic) motion was defined as the crossing wavelength of the SSH wavenumber spectra for subtidal and tidal frequencies. The transition scale is 250 km during MAMJJ for both coherent and incoherent SSHs at tidal frequencies. During ASOND, the transition scale is shifted from 200 km with the coherent to 250 km with the incoherent SSH. Even if coherent internal tide corrections are made available for conventional altimetry and SWOT data in this region, incoherent tides will still be present out to the transition scale wavelength of 250 km and will pollute the calculation of geostrophic currents at smaller scales.

Although this study provides some answers on the dynamics of the internal tide in this region of the tropical Atlantic, it raises other questions. The impression of non-propagation of the baroclinic tidal fluxes from sites E and D on the shelf break is, in our opinion, linked to the merging of these baroclinic fluxes with others. The branching of the baroclinic flux is probably an effect of refraction. However, the refraction here can be related to the density gradient at the front of the NBC retroflection or to the internal tidal interaction with the circulation (current and eddies). Much remains to be done to clearly describe the interaction of the internal tide with the background circulation in this area. An eastern extension of the model is being developed to distinguish whether the eastward deviation of the baroclinic flux from A is related to advection by the current or to strong refraction. With this new simulation, we hope to look at what happens to the baroclinic fluxes coming from C and B. It also remains to quantitatively determine the conditions under which the current advects the internal tide. According to

Intense semidiurnal internal solitary waves (ISWs, up to hundreds of kilometres from the shelf break) are consistently observed with SAR images propagating toward the open ocean in the Amazon area

At the sites of internal tide generation, changes in stratification from MAMJJ to ASOND had an impact on the generation of higher modes, which is not surprising given that higher modes are best projected on density profiles with a stratification maximum near the ocean surface. Stratification has certainly played a role in the dissipation and propagation of the internal tide. In fact, the hotspots of M2 dissipation have been observed along propagating beams distant from about 90 to 120 km, in good agreement with previous simulations

The energy level of the SSH wavenumber spectra at subtidal frequencies is not exactly the same in the models with and without tide, especially at large scales and slightly at small scales. This is not surprising since the interactions between internal waves and eddies can enhance the forward energy cascade

In the past decade, many investigations have been motivated by the internal tide surface signature corrections for all altimetry missions but especially for the future wide-swath altimetry SWOT mission. Various empirical atlases for surface internal tides have been derived from nearly 30 years of multi-mission altimetry, which reveal the coherent part of this signal over the altimetry era. The altimetry community's more pressing issue is the non-coherent part that is left aside in these atlases, whose magnitude and variability are the main concerns today as they will significantly contribute to the conventional altimetry and SWOT error budgets. Our investigations are a contribution to their quantification in a specific area and demonstrate the large variability of the internal tide dynamics at seasonal timescales. They also suggest even higher variability when considering shorter timescales because of the interaction with the upper ocean circulation, indicating clearly that the internal tide correction will be one of the most challenging problems for future altimetry data processing. In tropical regions with high seasonal variability, it is possible that internal tidal predictions at seasonal frequencies are more effective for altimetry data correction than annual prediction maps as currently proposed.

Data are available upon request by contacting the corresponding authors.

The supplement related to this article is available online at:

This work is part of the MT postdoc supervised by FL and AKL. JJ performed the numerical simulations, and MT made the analysis. The Argo data were pre-processed by SB. MT wrote the paper with contributions from all co-authors.

The contact author has declared that none of the authors has any competing interests.

Publisher's note: Copernicus Publications remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

We thank Damien Allain for helpful discussions. This work is a contribution to the project “Amazomix”.

Michel Tchilibou's postdoc is funded by CNRS. Ariane Koch Larrouy, Yves Morel, and Julien Jouanno are funded by IRD. Rosemary Morrow is funded by CNAP. Florent Lyard is funded by CNRS. Simon Barbot is funded by CNES (grant no. 2884), CLS, CNRS (grant no. 167349), and UPS.

This paper was edited by Neil Wells and reviewed by two anonymous referees.