Each SWOT track was subdivided into several windows located before and after the interaction zone between ISWs and the targeted eddy (Table 1). At first, for each window, the mean along-track wavelength spectrum was computed from ADT_swot signal to identify dominant wavelengths (Fig. 2a, black spectrum). In this region, ISWs come from the IT disintegration so we consider that distance between ISW packets correspond to typical wavelengths of IT-modes (Magalhaes et al., 2022; de Macedo et al., 2023; Tchilibou et al., 2022). Based on this, in second step, the dominant wavelengths of the spectrum (Fig. 2a, black spectrum) were isolated using a band-pass filter (Fig. 2a, blue spectrum), based on ranges corresponding to typical IT-modes: mode 1 (180–100 km), mode 2 (100–60 km), and mode 3 (60–30 km) (de Macedo et al., 2023; Tchilibou et al., 2022). The filter was constructed in the frequency domain by applying a Fast Fourier Transform (FFT), retaining only spectral components corresponding to the targeted wavelengths (Fig. 2a, blue spectrum). The filtered ADT_swot signal (Fig. 2b, blue line, bottom panel) was then reconstructed via inverse FFT. In the third step, local positive maxima in the filtered signal (Fig. 2b, blue line, bottom panel) were extracted (Fig. 2b, black crosses on blue line, bottom panel). Finally, each extracted pixels was mapped back on raw ADT_swot (Fig. 2b, black crosses on grey line bottom panel). This is done for all along track pixels of ADT_swot (Fig. 2b, black crosses on ADT swot swath, top panel). We coupled sigma 0 measurements for verify that the detected crests correspond to ISWs. Note that we tested different window sizes, which are presented in the Appendix. We specifically verified that windows smaller than 500 km provide the best correlations between the filtered signal (i.e., the detected ISWs) and the raw signal. In this study, we chose the largest possible window size to minimizing edge effects while avoiding the inclusion of additional submesoscale processes.

After identifying the position of ISWs occurrence (Fig. 2c, black crosses), ISWs crests were reconstructed by interpolating the local maxima using a second-degree polynomial (Fig. 2c, orange line) of the form Eq. (1) 1f(x)=ax2+bx+c The geometry of the reconstructed wavecrest was described using several parameters: the propagation axis, the concave/plane/convex geometry, the curvature intensity, the azimuth and crest-to-crest distance (Fig. 3).

The concave, plane, or convex nature of the crests was determined by the sign of the quadratic coefficient “a” in the interpolation function. If a was negative, the interpolation curve had a concave geometry, whereas a positive a indicated Fig. 3, (1), (2), (3) have a concave geometry because a1, a2, a3 are less than 0. On the contrary, (4), (5), (6) have a convex geometry because a4, a5, a6 are greater than 0.

Moreover, the curvature intensity was defined by the absolute value of a. When a was close to 0, the curve tended to be plane, and the curvature increased as a deviated from 0. For example, in Fig. 3 a4>a6; hence, (6) has a lower curvature than (4).

Finally, the distance between successive ISWs crests along the SWOT track (DSWOT) is obtained from spectral analysis. Since ISWs are highly anisotropic and the propagation direction of the ISW packets is not always aligned with the SWOT track. In order detect the ISW wavelength, one needs to apply a rotation of  θISW/SWOT which represents the angle between the ISW propagation direction and the SWOT track direction. The crest-to-crest distance of the  ISWs (DISW) can be determined as Eq. (2): 2DISW=DSWOT⋅cosθISW/SWOT

The extraction of ISW crests on SWOT ADT_swot field reveals the geometry of ISW and provides an initial insight into their response upon interaction with eddies. In this section, we present the results of the detection method described in Sect. 2.2. Each track was divided into several regions associated with different dynamics before and after interaction with the eddies (Fig. 7, Table 1).

Before any interaction with the seamount or the eddies, three packets of ISWs were detected, exhibiting relatively large crest ADT signatures. In the NE case, the main crests reach ADT amplitudes between 10 and 14 cm (Fig. 7g). In the CE case, ADT amplitudes range from 6.3 to 8.5 cm (Fig. 7h), while in the AE case they vary between 4.7 and 8.6 cm (Fig. 7i).

After crossing the seamount, the NE case shows 13 crests (Fig. 7g), whose surface expression is reduced, with ADT amplitudes between 3 and 8 cm (Fig. 7g). In the CE case, the interaction with the cyclone leads to the detection of 7 crests (Fig. 7h) in the refracted wave field, with an average ADT amplitude of about 5.8 cm (Fig. 7h). In the AE case, the diffracted wave field to the north contains 9 crests (Fig. 7i) sampled by track 074 and 4 crests along track 046, with a low ADT signature of 0.2 cm. The refracted wave field to the east includes 3 crests sampled along track 046 and another 3 crests along track 018. These ISWs nevertheless remain energetic, with ADT signatures of 6.1 and 9.3 cm, respectively (Fig. 7i).

The spectral analysis of the ISWs fluxes sampled before seamount/eddy interaction area (Fig. 7a, b, c; black spectrum) shows similar patterns: each spectrum highlights two peaks at wavelengths of 170–140 and 75 km, corresponding to modes-1 and 2 IT wavelengths, respectively.

A major finding is that after crossing the seamount and interacting with eddies, the spectral analyses of the three cases differ markedly. None of the spectra show peaks associated with mode-1 IT wavelengths between 180 and 100 km (Fig. 7a, b, c; red spectrum). All spectra exhibit high energy levels at smaller scales. Specifically, in the NE case, after the ISWs cross the seamount, the spectrum shows elevated energy at scales below 50 km, with peaks between 30–40, suggesting the presence of mode-3 IT (Fig. 7a; red spectrum #2). For CE case, spectrum shows highers peaks between 50 and 25 km (Fig. 7b; red spectrum #4). For AE case, the spectra of the branches refracted to the north display generally higher energy levels at wavelengths below 30 km (Fig. 7c; red spectrum #6 and #8).

Then, the analysis was extended to characterize wave trains detected in SWOT data.In NE case, the wavelength spectra show no important peaks in area 1^′, indicating the absence of secondary structures near the principal wave crest in ISWs packet (Fig. 7d, black spectrum #1'). But the spectra around the next individual ISWs packets from A-D flux are dominated by components at 20 km (Fig. 7d, black spectrum #1”) . In CE case, the spectra around individual ISWs packets from A-D flux are dominated by components at 12 km (Fig. 7e, black spectrum #3') and by peacks around 20 km (Fig. 7e, black spectrum #3”). In the AE case, the spectra associated with the eastward-deflected branch (Fig. 7f, black spectrum #8' and #9') and with the main flux from A to D (Fig. 7f, black spectrum #5') show energy peaks between 20 and 40 km in wavelength

The wave crest detection shows that, in all three cases, the ISWs generated close to the IT source sites exhibit inter-packet distances comparable to the wavelength of IT mode-1 aroud 130 km. (Fig. 7g, h, i). In NE case, wave packet signature is observed before seamount with crest-to-crest distance around 19 km (Fig. 7g).

In addition, in case CE, where the fluxes from sites A and D converge, wave packet signatures emerge with decreasing crest-to-crest distances, ranging from 14 km down to 18 km away from the carrier wave (Fig. 7h, areas 3^′′ and 3^′). Similarly, in case AE (Fig. 7i), the merging of A and D fluxes is associated with wave packets signature characterized by distances between crest ranging from of 13–18 km (Fig. 7i, area 5^′).

After the interaction with eddy or seamount, the distance between wave packets differs significantly across the three cases. In NE (Fig. 7g), the wave crests are spaced approximately 30 km (Fig. 7g, area 2), suggesting a transformation from IT-mode-1 to IT mode-3. In CE, the distance between crests in the refracted flux are shorter than in the incident flux, around 30–35 km (Fig. 7h, area 4). In AE, the portion of the flux that is refracted northward shows waves packets with 10–12 km crest spacing inside each wave packet. (Fig. 7i, area 6). Due to data gaps between SWOT ground tracks, it is not possible to resolve the wavelengths of the flux from A that is deflected eastward by the anticyclone. However, we identify ISWs wave packets sea surface signature. The main wave gradually degenerates into smaller secondary waves characterized by distance between crests between 20 and 12 km (Fig. 7i, areas 8^′ and 9^′). Finally, the third branch, refracted northward along the edge of the anticyclone, is characterized by a short distance between crests of 10–12 km (Fig. 7i, area 8)

After reconstructing the wave crests using a second-degree polynomial fit, it is observed that in the NE case, the ISWs corresponding to IT mode-1 generated at sites A and D propagate northeastward θISW1 =24° (Fig. 8, area 1, black circles). The wavecrest has a relatively plane geometry, with an average curvature coefficient of a1=0.10. During the crossing of the seamount, the propagation direction remained unchanged (ΔθISW1-2<6°) (Fig. 8, area 2, red circles). However, a decrease in the coefficient a is observed, reaching an average of a2=-0.14 (Fig. 8, area 2, red circles). This indicates an increase in curvature and a more pronounced concavity of the wavefront.

In the CE case, the waves sampled before interaction also propagate northeastward, with θISW3=29° and an average curvature coefficient of a3=-0.07 (Fig. 8, area 3–3^′′–3^′, black triangles). These characteristics are similar to those observed in the NE case. After interaction with the cyclone and the seamount, a significant change in propagation direction is observed: the wavecrest is refracted by ΔθISW3-4=50° toward the west. The crests of the refracted ISWs then propagate northeastward, with θISW4=-20° (Fig. 8, area 4, red triangles). Compared to the reference case, this deflection cannot be attributed to bathymetric effects, supporting the hypothesis that the refraction is induced by the cyclone. Furthermore, a strong increase in curvature is measured, reaching a2=-0.64 (Fig. 8, area 4, red triangles), nearly ten times higher than that of the incident wavefront.

In the AE case, the ISWs originating from point A initially propagated northwestward with θISW5-east=25°. These waves encountered those from site D, which propagated at θISW5′-west=40° (Fig. 8, area 5^′ – west, black cross). The wavecrest exhibited a relatively plane geometry (a5=-0.07) (Fig. 8, area 5, black cross. Near the eastern edge of the anticyclone, ISWs packet was refracted northward. The first three reconstructed crests were characterized by increased curvature (a6=-0.23), then, during their northward propagation, the ISWs gradually flattened, eventually exhibiting a curvature similar to that of the incident ISWs (a6=-0.08) (Fig. 8, area 6, brown cross) and an azimuth of θISW6=-2° (Fig. 8, area 6, brown cross). According to two scenarios, if the ISW branch originates from site A, it refracts northward with ΔθISW5-east-6=27°. In contrast, if it originates from site D, the refraction is stronger, with ΔθISW5-west-6=42°. An other part of the incident ISWs packet was refracted eastward and propagated along the edge of the anticyclone with θISW8′&9′=51° (Fig. 8, areas 8^′–9^′, red cross). According two scenarios, if it come from site A, ISWs was refract with ΔθISW5-east-8′=27°. In contrast if ISWs provide from site D it refract with ΔθISW5-west-8′=12°. The wavefronts in this region exhibited the highest curvature values, with a coefficient of a8&9=-0.8 (Fig. 8, areas 8^′–9^′, red cross). Part of ISWs was refracted northward with ΔθISW8′-8=43,5 (Fig. 8, area 8, red cross), and the wavefronts were characterized by a plane front a8=-0.013.

These three cases demonstrate that the ISWs originating from sites A and D were relatively plane and that the combined effects of the seamount and the refraction induced by the cyclonic and anticyclonic eddies deflected the wave trajectories and modulated the crest curvature.

In our study, we observe that the ISWs emitted from sites A and D predominantly exhibit an inter-packet spacing characteristic of mode-1 IT and converge in both case studies (AE and CE) between 3–5° N and 44–46° W. These results are in good agreement with modeling outcomes (Tchilibou et al., 2022) that demonstrated this convergence, as well as the study by de Macedo et al. (2023), which identified the convergence region of these fluxes as a hotspot of intense mode-1 ISW activity. Ultimately, this convergence region may play a role in mixing intensification, as suggested by the work of Kouogang et al. (2025a), based on recent direct in-situ measurements from the AMAZOMIX program.

In the interaction region between fluxes A and D, we observe a V-geometry wave crest suggestive of an oblique interaction in both AE and CE cases. According to the literature (Yuan et al., 2018, 2023; Wang and Pawlowicz, 2012; Helfrich, 2007; Shimizu and Nakayama, 2017), oblique interactions described here can mainly be indentified in two forms:

A merging of wave packets leading to the formation of a longer packet (Fig. 9a),

Both solitary wave packets passing through each other without significant alteration in geometry or amplitude (Fig. 9b).

In our study, for the AE case, following the oblique interaction between ISWs originating from sites A and D, the lack of data just after the convergence point prevents us from concluding the type of interaction and the subsequent ISW trajectories. Several scenarios are possible. It could involve a merging of wave fronts (Fig. 9a) followed by divergence under the influence of an anticyclone: ISWs from A could be deflected eastward, while those from D would be deflected westward. A second possible scenario in the AE case (Fig. 9b) is that ISWs from A and D meet, cross paths, and are deflected in opposite directions – northward for those from A, and eastward for those from D.

The different case studies in our research illustrate the complexity of possible outcomes and suggest that the offshore Amazon region is particularly favorable for studying these complex nonlinear wave interaction phenomena.

The NECC is closely linked to mesoscale eddy dynamics, which makes it difficult to isolate its specific impact on ISW propagation. Nevertheless, its role deserves particular attention. Indeed, in the AE case, ISWs appear to be deflected eastward toward the edge of the eddy prior to the interaction, following the NECC streamlines. This intense zonal current, with a variable path, could indeed influence ISW trajectories. This observation supports the hypothesis proposed by Tchilibou et al. (2022), de Macedo et al. (2023) and Magalhães et al. (2016) suggesting that the strengthening of the NECC plays an important role in the acceleration, refraction, diffraction and shift ITs associated to ISWs in the northeastern region.. For comparison, it has been showed that the meandering Gulf Stream significantly refracts and traps ITs (Duda et al., 2018), highlighting the ability of such a current to act as a barrier or waveguide for ITs. By analogy, the contribution of the NECC in the deflection or distortion of ISW crests cannot be overlooked using satellite observation, and a dedicated analysis would be necessary to assess its relative contribution using idealized numerical experiments.

This study shows that ISW deflections substantially increase wave crest curvature, in agreement with idealized simulations (Guo et al., 2023). This is particularly evident in the AE case, when the center of the ISW front is aligned with the current along the northern edge of the anticyclone (Fig. 10c), and in the CE case, when the ISW tip is opposed to the current along the western edge of the cyclone (Fig. 10b). Eddies feature strong gradients in both velocity and stratification, with maximum speed at the edges and minimum at the center. When ISWs interact with this spatially varying current field, their local phase speed can be altered (Lamb, 2014; Dunphy and Lamb, 2014). Some sections of the wave front may be “accelerated” where the current aligns with the direction of ISWs propagation (Fig. 10c), while others may be “slowed down” in the presence of opposing or weaker currents (Fig. 10b). This spatial variation in ISW phase speed leads to significant distortion of the wave front, which becomes markedly more curved than the initially planar incident front.

In contrast, in the AE case, after interaction, part of the ISW flux is refracted northward (Fig. 7i). This portion of the flux maintains a relatively flat curvature throughout its propagation. One possible explanation for the absence of wavefront distortion after refraction is that the ISWs do not interact directly with the intense edge currents of the eddy, but are instead reflected upon encountering a kind of “physical wall” (Fig. 10d). This interpretation is supported by recent studies (Guo et al., 2023; Wang and Legg, 2023), which show that stratification changes induced by eddies can refract ISWs.

This observation contributes to an ongoing debate within the scientific community regarding the relative roles of stratification and currents in the processes of internal tide refraction and ISW distortion. Several studies (Bendinger et al., 2025; Guo et al., 2023; Wang and Legg, 2023; Xu et al., 2024) suggest that eddy-related currents play the dominant role in ISW refraction, relegating the effect of local stratification variations to a secondary role. However, in other oceanic contexts, it is precisely these stratification changes that appear to be the main driver of internal tide refraction and coherence loss, surpassing the influence of relative vorticity gradients and subtidal currents (Zaron and Egbert, 2014). These findings highlight the importance of regional conditions and the complex interaction between stratification and currents in shaping ISW trajectories and behavior.

The scenarios explored underscore the need for accurate estimates of eddy locations to assess their influence on ISW dynamics. However, uncertainty exists regarding the precise positioning of these eddies. They are identified from daily altimetric maps that combine measurements from several satellites passing at different times. The interpolation of these data inevitably leads to smoothing of small scales and a loss of spatial and temporal resolution. Consequently, when compared with instantaneous measurements from SWOT and MODIS/NOAA20, it becomes difficult to determine whether the ISW–eddy interaction occurs at the eddy core or periphery. This uncertainty can limit the interpretation of observations.

In the AE case, the lack of data between SWOT tracks prevents observation of the wavefront evolution after the wave–wave interaction. This makes it impossible to precisely characterize the type of wave–wave interaction and also limits observations of wave–eddy interactions.

More generally, several factors hinder the observation of wave–eddy interactions: the temporal resolution of satellite sensors, environmental conditions (cloud cover, wind, ocean surface state, sun angle), and the viewing angle. These limitations also affect the interpretation of physical processes, including ISW refraction and the detection of ISW packets. In unfavorable conditions, only the leading wave with strong contrast is visible in sunglint or σ0 images.

Additionally, what is interpreted in the CE case as westward refraction of the wave flux by the cyclone may in fact only represent part of a diffracted wave flux, of which only the western branch is captured. Other branches to the east may exist but are unfortunately not captured in our case studies. Therefore, we cannot conclude on the full extent of the flux, as satellite images do not capture it in its entirety.

Finally, satellite altimetry and surface geostrophic velocities primarily capture the barotropic signal and the baroclinic mode-1 structure, which remains a substantial limitation for the detection and characterization of baroclinic eddies. Yet, the vertical structure and intensity of baroclinic eddies likely condition their interactions with the surrounding flow and with propagating internal solitary waves, potentially leading to distinct dynamical responses. Our results provide qualitative and first-order evidence of eddy–ISW interactions as observed by SWOT, but the absence of in situ measurements prevents us from assessing the vertical structure of the eddies involved. It is therefore reasonable to expect that these interactions may differ when eddies exhibit a more complex vertical configuration, such as a dipolar structure, and future work should aim to resolve and analyze this vertical structure to better quantify these processes.

These observational limitations may lead to an underestimation of the extent and complexity of eddy-induced ISW modifications and emphasize the need to complement the analysis with in situ measurements or 3D high-resolution modeling to better capture the full scale and dynamics of ISWs and their interactions with eddies

In the NE case, an energy transfer from mode-1 IT to higher IT modes, particularly mode-3, was observed. This phenomenon aligns with numerous studies showing that steep bathymetry can disperse internal tide energy into higher modes and enhance mixing (Johnston and Merryfield, 2003; Johnston et al., 2003; Mathur et al., 2014). One hypothesis is that the presence of a seamount can locally alter stratification and the effective water column depth, thus affecting internal wave phase speed and reducing wavelength. A broader analysis of SWOT data would be valuable to systematically confirm the impact of topography on ISW behavior.

In addition to the effect of seamounts, a similar mode transfer from mode-1 to mode-3 was observed during wave deflection by a cyclone. These findings are consistent with earlier studies (Guo et al., 2023; Dunphy and Lamb, 2014), which demonstrated that a mode-1 internal tide interacting with an eddy can transfer energy to higher modes, thus reducing the main mode's energy flux. Moreover, in the AE case, we observe the emergence of wave trains (three crests following the main one, with inter-packet distances on the order of 10 km) in the deviated branch after interaction with the anticyclone. In contrast, in the NE case, the main crest appears alone, i.e., without a trailing wave train. Our results raise the hypothesis that the eddy may destabilize the ISW's main crest and enhance wave train formation. More quantitative studies are needed to refine this hypothesis.