1 Introduction

Nonlinear interactions in coastal and estuarine environments result from the complex interplay between tidal waves, river discharge, bathymetry, and extreme meteorological events, leading to significantly elevated water levels and flooding. (Blanton et al. 2002; NOAA 2007). These phenomena are most pronounced where the difference between tidal amplitude and water depth is significant (Murty and El-Sabh 1981; Murty 1984; Xiao et al. 2021). The transition of tidal waves from oceans to coasts and estuaries modifies the tidal cycle due to factors like shallow water friction, basin topography, and river flow (Friedrichs and Aubrey 1994; Li and O'Donnell 2005; Lanzoni and Seminara 1998; Godin 1999; van Rijn 2011; Guo et al. 2015), resulting in energetic transference to new frequencies and the generation of shallow water harmonics and overtides (NOAA 2007; Dianápoli et al. 2020; Talke and Jay 2020).

Storms and upstream flow changes further disrupt tidal cycles, leading to asymmetries and significant water level variations (Xiao et al. 2021; Dinápoli et al. 2020). Especially during storm events, the tangential force of wind across the water surface, in combination with bottom friction, tidal waves, and river discharge, can impede the natural drainage of rivers towards the sea (García 2011), resulting in significant flooding (Dinápoli et al. 2020). These nonlinear processes influence sediment patterns, turbidity, and navigation depths, highlighting their importance for flood risk mitigation, erosion defense, and wetland conservation (Gallo and Vinzón 2005; Guo et al. 2015).

Due to nonlinear interactions play a critical role in shaping the dynamics of river systems (Murty and El – Sabh 1981; Murty 1984; Xiao et al. 2021), understanding these phenomena requires sophisticated statistical techniques capable of capturing time-varying frequency components embedded within complex time series data. Classical spectral analysis methods, such as Fast Fourier Transform (FFT) or cross-spectral analysis, are useful for analyzing stationary signals (Jenkins and Watts 1971; King 1996; Zhegulin and Zimin 2021), however, the analysis of non-stationary signals where the frequency components vary in time is not plausible based on these traditional methods. Consequently, the study of these time series is carried out based on statistical methods different from those used in the description of Gaussian processes (Swami et al. 1998). Non-stationary signals analysis can be realized through the description of the complex statistical characteristics of the time series using higher-order spectra (polyspectra) obtained by Fourier transforms of cumulative functions (Swami et al. 1998). These statistical tools reveal information about the process and make possible the analysis of phase-related spectral components embedded in a noise spectrum.

High-order spectra techniques are particularly sensitive to noise and signal non-stationarity, which can lead to unclear interpretations and erroneous outcomes due to data noise and temporal variations (Zelensky et al. 2013). Additionally, high-order spectra may have limited spectral resolution for detecting closely spaced frequency components or rapidly changing signals over time. Interpreting results can be complex due to the variety of parameters and settings involved, and these techniques often require substantial computational resources while being more sensitive to noise and data artifacts. Despite these challenges, high-order statistics offer distinct advantages in signal processing as: removing the Gaussian noise from power spectra, phase identification, and signal reconstruction, while effectively detecting and characterizing complex processes (Zhegulin and Zimin 2021). High-order spectra quantify nonlinearities and enable the characterization of different types of nonlinear interactions within time series data, which is especially valuable for understanding nonlinear influences on wave patterns (Chandran et al. 1994; Elgar et al. 1995; Aubourg et al. 2017).

Bispectrum is the third-order Fourier transform based on the decomposition of the third moment of a signal over frequency, which describes nonlinear interactions of a continuous spectrum whose application is useful for analyzing systems with asymmetric nonlinearities (Zelensky et al. 2013; Ewans et al. 2020). Trispectrum (Fourier transform of fourth-order cumulants) studies symmetric nonlinearities (Aubourg et al. 2017). Bispectral analysis identifies contributions to a signal's skewness as a function of frequency triplets; meanwhile, trispectral detects energetic contributions to a signal's kurtosis as quadruplet’s frequencies.

Internationally, several studies detected nonlinear interactions in wave-wave interactions in time series of water level using high-order spectra analysis (Ewams et al. 2020). Juantorena Alén and Rosales-Grano (2003) detected energetic cells associated with nonlinear process occurrence in bimodal sea wave series. Zhegulin and Zimin (2021) characterized the internal wave production in the southern part of the White Sea Throat using bispectral and trispectral analysis. The trispectrum detected high values at frequency quadruplets corresponding to high Fourier amplitudes and measured wave time series (Ewams et al. 2020). Xiao et al. (2021) identified an amplification of the tidal wave in the Delaware Bay Estuary (United States) that was promoted by nonlinear process during the passage of Hurricane Irene.

Different studies in Argentina delimited the importance of nonlinear processes in the estuarine dynamic of Río de la Plata (Buenos Aires) (Simionato et al. 2004, 2006; Moreira et al. 2013; Tejedor et al. 2015; Moreira and Simionato 2019). Recently, Dianápoli et al. (2020) detected and modeled the contributions of nonlinear interactions to the water level rise during a storm surge. This investigation results demonstrated that nonlinear processes occurrence produced a delay in the low tide phase and, therefore, a temporary extension of the floods in the Río de La Plata hydrographic basin. Rio Negro belongs to the second most significant hydrographic system of Argentina (Limay, Neuquén, and Negro rivers) after Rio de la Plata (AIC 2022). Particularly, its hydrographic basin has been in the news since 1899 due to the sudden variations in the river's flow that have caused numerous floods (Diario Rio Negro 2005; Brieva 2018; Zabala et al. 2021; DesInventar 2022), being its lower basin the most vulnerable (Brailovsky 2012). D’Onofrio et al. (2010) detected nonlinear processes in the hydrodynamics of the Río Negro watercourse during different periods and exposed that this phenomenon can cause severe flooding in areas with reduced slopes.

To analyze flood production in the lower basin of Rio Negro, an understanding of nonlinear processes is necessary, given the reported amplification of water level in existing literature (D’Onofrio et al. 2010). This paper aims to detect and analyze nonlinear interactions using high-order spectra. While the document does not focus on specific flood forecasts, its primary goal is to advance understanding of the underlying factors that can lead to more severe floods than expected. This research lays a solid foundation for better understanding the causes behind extreme events, essential for developing stronger predictive capabilities in the future.

2 Data and methodology

2.1 Geographical characteristics of the study area

Río Negro lower basin (Fig. 1a) belongs to the Limay, Neuquén, and Negro hydrographic system (AIC 2020) (Fig. b). The study area is located between 40°- 41°S and 63°- 64°W, in the southeast of Argentina (Northern Argentine Patagonia) (Fig. 1c). The surface extension is 3393.62 \({{\text{km}}}^{2}\) and the basin orientation is NW–SE.

Fig. 1
figure 1

Study Area, lower basin of Río Negro. a Map of the lower basin of Rio Negro (Soldano 1947; Farr et al. 2007; SSRH 2010; IGN, 2017; DPA Río Negro 2022; SNIH 2022), b Location of Limay, Neuquén and Negro hydrographic basin (Farr et al. 2007; SSRH 2010; Portal IGN 2023), c Relative localization (Soldano 1947; Portal IGN 2023)

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The presence of two regimes encourages a great hydrological variability of the Río Negro in its hydrographic lower basin (UNL-DPA Río Negro 2004b). The inflow of the tide upstream from the Atlantic Ocean promotes a fluvial-marine regime from the river mouth to San Javier (kilometer 65 from estuary mouth) (D'Onofrio et al. 2010). The tidal influence on the river discharge is so significant that during high tide, a noticeable bottleneck occurs. Upstream of San Javier the effect of the tidal wave disappears, so the regime becomes fluvial (Brieva 2018). Also, Río Negro has a double flood wave product of the overlapping of the annual flood waves of Neuquén and Limay rivers, with a maximum in July (winter) and another in November (spring) (Brieva 2018). Average discharge in Primera Angostura (90 km from the mouth of the estuary) is 846 \({{\text{m}}}^{3}{{\text{s}}}^{-1}\)(UNL-DPA Río Negro 2004b; Gianola Otamendi 2019).

While the incorporation of reservoir dams in upstream tributary rivers (Limay and Neuquén rivers) of the Río Negro has moderated the river flow (AIC 2022), storm passages, flow releases, or a combination of both during high tides can lead to flooding and waterlogging in low-lying areas near the Río Negro riverbank. According to UNL-DPA Río Negro (2004a) results, the persistence of strong winds from the S and SW quadrants between 6 and 12 h promotes a significant rise of water level for urban areas located in low altitude and low-slope land. In addition, the lower basin of the Río Negro is susceptible to flooding when the flow registered in Primera Angostura exceeds 2700 \({{\text{m}}}^{3}{{\text{s}}}^{-1}\) (UNL-DPA Río Negro 2004b).

The Río Negro flows through a diminished floodplain where the watercourse occupies the northern sector; meanwhile, the southern area is characterized by flat geoforms with beach ridges and marshes (Piccolo and Perillo 1999). Río Negro has in its lower basin numerous secondary arms, paleochannels that are activated in times of flooding, islands, and temporary lagoons (Prates et al. 2019). The relief is primarily flat, with slopes less than 3% interrupted by plateaus and salt ridges that occasionally exceed 2 m above sea level (ECYT-AR 2014). The longitudinal slope decreases from San Javier (0.14 m/km) to Carmen de Patagones (0.02 m/km) (Fig. 1), and finally, in the Rio Negro mouth is 0.012 m/km (UNL-DPA Río Negro 2004b).

According to Subsecretaría de Recursos Hídricos de la Nación (2015), at the river mouth area (where the TAC limnigraph is located) (Fig. 1a), the channel width is approximately 750 m there is minimal island development. The river channel average width of the Río Negro in the area where the PCP limnigraph is located is 250 m and numerous islands presence. Conversely, near San Javier, the channel width is 400 m. The depth varies between 5 and 10 m in PCP and TAC, respectively (Piccolo and Perillo 1999), with an average depth of 6 m (D'Onofrio et al. 2010).

The Río Negro lower basin is highly anthropized with urban centers such as Loteo de Costa de Río, Viedma, Carmen de Patagones, El Juncal, San Javier, and Guardia Mitre (Fig. 1a). Viedma is located in the Rio Negro flood valley between 3.5 and 4.5 m above mean sea level (mamsl), and its geographical location makes Viedma waterfront vulnerable to severe floods (D’Onofrio et al. 2010). Because the northern flank of the river has cliffs of the order of 20 m, Carmen de Patagones is free of floods (Brailovsky 2012).

2.2 Data

The analyzed dataset included water level data from three limnigraphs located in Toma de Agua El Condor (TAC), Prefectura de Carmen de Patagones (PCP), and San Javier (SJ) (18, 39 and 65 km from Rio Negro mouth, respectively) (Data provided by the Departamento Provincial de Aguas de Río Negro, DPA Río Negro 2022) (Fig. 1a). Days with 5 h of missing data were eliminated to reestablish the time series continuity. Punta Redonda tide gauge (PR) (SHN 2022) was employed for the study of the tidal cycle. The temporal resolution of the time series was hourly from 2003 to 2022. Finally, the river discharge analysis was realized through flow times series from Primera Angostura station (PA) (SNIH 2022). The study of flood production and its causal factors in the study area was realized through a literature review. The information was obtained from the DesInventar database (DesInventar 2022) and different press articles.

2.3 Harmonic analysis (HA)

Harmonic Analysis (HA) was employed to describe the influence of astronomical tides on the river discharge. The method of least squares (Foreman 1977) was applied to the three series of hourly heights to calculate the amplitudes and phases of the tidal component waves. The errors were calculated following the methodology proposed by Pawlowicz et al. (2002). The travel times of the diurnal and semidiurnal constituents were estimated following Zhao et al. (2017) methodology (Eq. 1)

$$\mathrm{\Delta t}=\frac{{\text{T}}}{2\uppi }[\upphi ({{\text{point}}}_{2})-\upphi ({{\text{point}}}_{1})$$
(1)

where T is the tidal period, \(\upphi\) the unwrapped phase.

2.4 Definition of Fourier high-order transforms

The power spectrum of different orders provided a complete description of the frequencies and their interaction. According to Swami et al. (1998), the Fourier spectra cumulants were formally defined on the autocorrelation function, which is a statistical moment that allows characterizing a random signal in the time domain through its variance (Eq. 2):

$${{\text{C}}}_{xn} \left({\uptau }_{1},{\uptau }_{2},\dots , {\uptau }_{{\text{n}}-1}\right)={\text{E}}\{{\text{x}}\left({\text{k}}\right){\text{x}}({\text{k}}+{\uptau }_{1}{\text{x}}\left({\text{k}}+{\uptau }_{2}\right)\dots {\text{x}}({\text{k}}+{\uptau }_{{\text{n}}-1})\}$$
(2)

where E {.} is the expected value, \(\uptau\) is the time delay, and \({{\text{C}}}_{xn}\) is the nth moment of the cumulant values. Under the hypothesis that the mean of the sequence x(k) is zero, the relationship between the cumulants and expected values was defined by (Eqs. 3, 4, 5, and 6):

$${{\text{C}}}_{1n}=\mathrm{ E}\left\{{\text{x}}\left({\text{k}}\right)\right\}=0$$
(3)
$${{\text{C}}}_{2n}={\text{E}}\{{\text{x}}\left({\text{k}}\right){\text{x}}({\text{k}}+{\uptau }_{1})\}$$
(4)
$${{\text{C}}}_{3n}={\text{E}}\left\{{\text{x}}\left({\text{k}}\right){\text{x}}\left({\text{k}}+{\uptau }_{1}\right){\text{x}}\left({\text{k}}+{\uptau }_{2}\right)\right\}$$
(5)
$${{\text{C}}}_{xn}={\text{E}}\{{\text{x}}\left({\text{k}}\right){\text{x}}({\text{k}}+{\uptau }_{1}){\text{x}}\left({\text{k}}+{\uptau }_{2}\right)\dots {\text{x}}({\text{k}}+{\uptau }_{{\text{n}}-1})\}$$
(6)

Then, the polyspectra of nth order are denoted as \({{\text{S}}}_{{\text{xn}}}({{\text{f}}}_{1}, {{\text{f}}}_{2}, \dots , {{\text{f}}}_{{\text{n}}-1})\), is defined based on the (n-1) dimensions of the Fourier transform. Polyspectral definition depends on the order that possesses the expected value. In this order, the nth-order polyspectrum is defined as the Fourier Tranforms (FTs) of the corresponding cumulant sequence (Eq. 7):

$${{\text{S}}}_{xn}\left({{\text{f}}}_{1}, {{\text{f}}}_{2}, \dots , {{\text{f}}}_{{\text{n}}-1}\right)={\sum }_{-\infty }^{\infty }\dots {\sum }_{{\uptau }_{{\text{n}}-1}}^{\infty }{{\text{C}}}_{xn}\left({\uptau }_{1},{\uptau }_{2 },\dots , {\uptau }_{{\text{n}}-1}\right)*{{\text{e}}}^{-{\text{i}}2\uppi ({{\text{f}}}_{1}{\uptau }_{1}+{{\text{f}}}_{2}{\uptau }_{2}+\dots +{{\text{f}}}_{{\text{n}}-1}{\uptau }_{{\text{n}}-1})}$$
(7)

The second-order (Eq. 8), third-order (bispectrum) (Eq. 9), and four-order (trispectrum) spectra (Eq. 10) are defined as functions of one, two, and three frequencies, respectively. In contrast with the power spectrum that is real valued and non-negative, bispectra and trispectra are complex-valued.

$${{\text{S}}}_{2n}\left({{\text{f}}}_{1}\right)={\sum }_{{\uptau }_{1=-\infty }}^{\infty }{{\text{C}}}_{2n}\left({\uptau }_{1}\right)*{{\text{e}}}^{-{\text{i}}2\uppi {{\text{f}}}_{1}{\uptau }_{1}}$$
(8)
$${{\text{S}}}_{3n}\left({{\text{f}}}_{1}, {{\text{f}}}_{2}\right)={\sum }_{{\uptau }_{1}=-\infty }^{\infty }{\sum }_{{\uptau }_{2}=-\infty }^{\infty }{{\text{C}}}_{3n}\left({\uptau }_{1},{\uptau }_{2 }\right)*{{\text{e}}}^{-{\text{i}}2\uppi \left({{\text{f}}}_{1}{\uptau }_{1}+{{\text{f}}}_{2}{\uptau }_{2}\right)}$$
(9)
$${S}_{4n}\left({{\text{f}}}_{1}, {{\text{f}}}_{2},{{\text{f}}}_{3 }\right)={\sum }_{{\uptau }_{1}, {\uptau }_{2}, {\uptau }_{3}=-\infty }^{\infty }{{\text{C}}}_{4n}\left({\uptau }_{1},{\uptau }_{2 },{\uptau }_{3 }\right)*{{\text{e}}}^{-{\text{i}}2\uppi \left({{\text{f}}}_{1}{\uptau }_{1}+{{\text{f}}}_{2}{\uptau }_{2}+{{\text{f}}}_{3}{\uptau }_{3}\right)}$$
(10)

In this work was calculated the second-order spectra based on the Fast Fourier Transform (FFT) (Blackman and Tukey 1958; Tukey 1949) to compare and identify the waves that correspond to the spectral band of the astronomical tide and river discharge in the study area and observe their energetic changes in TAC, PCP, SJ, and PA. To smooth the spectra they are convolved with a Hamming window (Oppenheim and Shafer 1989).

Although studying the second-order spectrum is suitable for investigating asymmetric time series, it is not helpful for nonlinear analysis, and the interactions between the Fourier constituents are not adequately explained (Hasselmann et al. 1963). The bispectrum is a central tool to investigate the phase coherence between the three Fourier constituents of frequencies \({{\text{f}}}_{1}\), \({{\text{f}}}_{2}\) and \({{\text{f}}}_{1+2}\) (Hasselman et al. 1963; Zengulin and Zimin 2021). In this sense, if a function is random and stationary in time φ(t), the auto-bispectrum is calculated through the Fourier Transform of third-order (Eq. 3) (Swami et al. 1998). The abbreviated representation of auto-bispectrum is (Eq. 11):

$${{\text{F}}}_{{\text{xxx}}}\left({{\text{f}}}_{1}, {{\text{f}}}_{2}\right)= {{\text{A}}}_{{{\text{f}}}_{1}}{{\text{A}}}_{{{\text{f}}}_{2}}{{\text{A}}}_{{{\text{f}}}_{1+2}}^{*}$$
(11)

where \({{\text{A}}}_{{{\text{f}}}_{1}}\) and \({{\text{A}}}_{{{\text{f}}}_{2}}\) are the Fourier complex coefficients of frequencies \({{\text{f}}}_{1}\) and \({{\text{f}}}_{2}\), and \({{\text{A}}}_{{{\text{f}}}_{1+2}}^{*}\) is the Fourier complex conjugate. The normalized form of the auto-bispectrum is the auto-bicoherence (Eq. 12)

$${{\text{b}}}_{{\text{xxx}}}^{2}\left({{\text{f}}}_{1}, {{\text{f}}}_{2}\right)=\frac{|{{\text{F}}\left({{\text{f}}}_{1}, {{\text{f}}}_{2}\right)|}^{2}}{{{\text{E}}[|{{\text{A}}}_{{{\text{f}}}_{1}}{{\text{A}}}_{{{\text{f}}}_{2}}|]}^{2}{\text{E}}{[|{{\text{A}}}_{{{\text{f}}}_{1+2}}^{*}|]}^{2}}$$
(12)

The bicoherence explains the degree of coupling of the waves relating to temporal dynamics of the phase relationship between certain components in a complex nonstationary process. A phase relationship can be found when the signal contains two harmonics with \({{\text{f}}}_{1}\) and \({{\text{f}}}_{2}\) frequencies simultaneously with their sum \({{\text{f}}}_{1+2}\), and the phase summary of these harmonics remains constant (Zelenky et al. 2013). When the signal satisfies the relation \({{\text{f}}}_{1}+ {{\text{f}}}_{2}={{\text{f}}}_{1+2}\) corresponding to bicoherence a value close to unity when the signal, contrariously if this relation is not satisfying, the bicoherence function tends to zero.

Despite the advantages represented by the study of the bispectrum in non-stationary signals, this methodology used does not provide information about the energetic contribution of interacting frequencies associated with symmetric signals. Consequently, it is necessary to implement the trispectrum as the statistic tool that detects nonlinearities due to this kind of interaction (Eq. 10). It is a function of four frequencies and provides an estimate of the degree of coupling between wave constituents at three frequencies and a fourth (Ewans et al. 2020). Four-wave interactions between Fourier constituents can involve interactions of the type where \({{\text{f}}}_{1}\)+\({{\text{f}}}_{2}\)+\({{\text{f}}}_{3}\) = \({{\text{f}}}_{4}\) and \({{\text{f}}}_{1}\)+\({{\text{f}}}_{2}\) = \({{\text{f}}}_{3}\)+\({{\text{f}}}_{4}\).

2.5 Definition of study cases

To study floods along the Río Negro lower basin driven by nonlinear interactions, days were selected where water levels exceeded riverbank and flood evacuation thresholds, as per Resolution No. 1403/2010. This selection followed criteria outlined by Departamento Provincial de Aguas de Río Negro (UNL-DPA Río Negro 2004a): resolutions No. 524 (No. 41316-IGRH-12) and No. 638 (No. 41315-IGRH-12). Analysis focused on TAC and PCP locations within the tidal influence section. Flood thresholds were set at 4.5 m in TAC and 3.5 m in PCP. Two flood events were selected based on this analysis to assess the contribution of nonlinear processes to water level signals: July 23, 2009 (Río Negro, 2009) and September 3–4, 2019. Water gauge elevations during these events were compared to Punta Redonda tide gauge records (SHN 2022). Wind speed and direction data from the Viedma Aero meteorological station were analyzed to understand the impact of intense winds on water column elevation during flooding events (SMN, 2022). The temporal series for calculate the high-order spectra during storm event covered a 6-day period, with data segmented into 2 days before, during, and after storm events.

3 Results

3.1 Flooding reports

From 1899 to 2021, there were recorded 29 flooding (Table 1). Adverse weather conditions (intense rainfall and strong winds) caused the occurrence of 50% of the flood reports. Upstream flow discharges from the dams installed on the Neuquén and Limay rivers caused 25% of the flooding.

Table 1 Flooding reports of the Rio Negro lower basin, elaborated from DesInventar (2022) and different press reports
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The interaction of high flows and the passage of extreme weather events conditioned the record of the remaining 25%. Historically, Viedma has proven to be particularly susceptible to river floods. From the flooding reports, 86% occurred in this city. Carmen de Patagones waterfront was affected by 17.2% of the flood record in the hydrological lower basin of Río Negro.

3.2 Harmonic analysis

The application of HA, based on the method of least squares (Foreman, 1977), observed the amplitudes and phases of the series of hourly water level associated with the entry of the astronomical tide wave from the Atlantic Ocean. HA detected that the maxima amplitude corresponds to the semidiurnal harmonics (Table 2). The amplitude of diurnal constituents was most significant in TAC and PCP. The effect of the astronomical tide is low in SJ, and the phase showed that the tidal direction propagation was upstream. From the Río Negro mouth to San Javier, the semi-diurnal constituents arrive in 1 h; meanwhile, the diurnal wave is delayed 2 h and 4 min.

Table 2 Amplitude and phase of harmonics constituents in the different measurement sites. Elaborate on limnigraphs data belonging to DPA Río Negro (2022)
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The components with the highest amplitudes were the semidiurnal components (\({{\text{M}}}_{2}\), \({{\text{N}}}_{2}\) and \({{\text{S}}}_{2}\)), third-diurnal \({{\text{MK}}}_{3}\) and the diurnal component \({{\text{K}}}_{1}\) (Table 2 and Fig. 2), and their maximum values were detected at TAC station (Fig. 2a and b). Shallow water components (\({{\text{M}}}_{4}\),\({{\text{MK}}}_{3}\), \({{\text{MN}}}_{4}\), \({{\text{MS}}}_{4}\), \({{\text{M}}}_{6}\), \({2{\text{MN}}}_{6}\), \({2{\text{MS}}}_{6}\) and \({{\text{M}}}_{8}\)) generated by nonlinear dynamics were visualized in TAC and PCP, being found in PCP the highest amplitude of the overtides (Fig. 2d, e, and f). In this case, the nonlinearities were represented by the combination of linear constituents \({{\text{M}}}_{2}\), \({{\text{S}}}_{2}, {{\text{N}}}_{2}\) and \({{\text{K}}}_{1}\) (Parker 2007; Pugh 2004; Dinápoli et al. 2020). Although these generally had small amplitudes in SJ, in TAC and PCP they reached significant amplitudes.

Fig. 2
figure 2

Amplitude of harmonic constituents in TAC, PCP, and SJ. Elaborate on limnigraphs data belonging to DPA Río Negro (2022)

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3.3 Spectral analysis of water level

In all cases, it is observed that most of the tidal energy is concentrated in the band of semidiurnal components, which suffers an attenuation as the tidal wave penetrates the river (Fig. 3). At the mouth of the Río Negro, the diurnal components follow in importance. Also, energy peaks were observed in the fourth, sixth, eighth, and tenth diurnal scales, the most important being the fourth-day components. The spectral energy associated with tidal constituents was several magnitudes lower in SJ than in the rest of the limnigraphs. The spectral energy of the signals in TAC and PCP was similar.

Fig. 3
figure 3

Fast Fourier Transform of water level time series for TAC, PCP, and SJ. Elaborate on limnigraphs data belonging to DPA Río Negro (2022)

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The FFT also demonstrates the particular energy contribution of high harmonics of \({{\text{M}}}_{2}\) to the spectrum. The influence of shallow water constituents \({({\text{M}}}_{4}\), \({{\text{M}}}_{6}\), \({{\text{M}}}_{8}\) and \({{\text{MK}}}_{3})\) was more significant in PCP than in TAC or SJ. These results may indicate that in PCP, the degree of coupling between wave constituents was higher, and nonlinear interaction probably contributed to producing water level anomalies.

3.4 Upstream flow

A factor that promoted the occurrence of floods was the release of water flow upstream of the study area. Mean flow spectral density function showed the existence of spectral maxima on a background of random noise (Fig. 4). The main energy maximum corresponds to the annual and semi-annual cycles. In addition, energy pulses were reported between 2 and 4 years, which belong to the interannual scale.

Fig. 4
figure 4

Fast Fourier Transform of the flow of the Río Negro at Primera Angostura Station (1930–2021). Elaborate on flow data belonging to Primera Angostura Station (SNIH, 2022)

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Given that the lower basin of the Rio Negro is susceptible to flooding when the flow registered at the Primera Angostura station exceeds 2700 \({{\text{m}}}^{3}{{\text{s}}}^{-1}\), we looked for those cases were the Q exceeded this value in the period 1930 to 2021. In total, 21 instances were found between 1930–2021 with highest flow registered in June, July, and August.

3.5 High-order spectral analysis

To identify the nonlinear process production and its properties due to interacting frequencies, the auto-bispectrum and bicoherence functions were calculated. The bispectral and bicoherence analysis identified the contributions to a signal's skewness as a function of frequency triples. In this case, TAC and PCP bispectra indicate a statistically significant nonlinear relationship between three-wave interactions (Fig. 5a and b). Local maxima of the auto-bispectrum were located in the region corresponding to semidiurnal harmonics from 0.079 (Period = 12.65 h) to 0.082 \({\mathrm{hours }({\text{H}})}^{-1}\) (Period = 12.19 h).

Fig. 5
figure 5

Auto-bispectrum and bicoherence values from TAC, PCP and SJ limnigraphs during 2003–2021. a Auto-bispectrum of TAC, b Auto-bispectrum of PCP, and c Auto-bicoherence values. Elaborate on limnigraphs data belonging to DPA Río Negro (2022)

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The auto-bispectral analysis did not detect energy cells in SJ; however high auto-bicoherence values were found. In this case, the auto-bicoherence results do not that there is a production of nonlinear interaction, rather, the high values show that a frequency coupling occurs at this point. The lower amplitude of the harmonics detected by HA (Table 2 and Fig. 2) and, auto-bispectra and bicoherence results indicate that the astronomical tidal wave and river waves were coupled in SJ (Fig. 5c).

3.6 Study cases

The series of TAC and PCP elevations between the years 2003 and 2021 were analyzed for the selection of the study cases. The study showed that on four occasions the water level exceeded the riverbank line established by the Río Negro DPA (3.7 m) in PCP, while in TAC this situation was not detected. The extreme elevation records were observed on February 20, 2003 with a maximum value of 3.8 m at 3:30 AM. The second case occurred on April 30, 2006 with 3.9 m at 2 AM. The most recent cases were detected on July 23, 2009 and on September 3, 2019 with 4.1 and 4 m, respectively. These last two cases were selected to analyze how the production of nonlinear interactions affects the occurrence of floods.

From the analysis of the tidal cycle and regime of the Río Negro between July 2009 and September 2019 (Fig. 6a and b), it was possible to observe that the water level registered by the PCP limnigraph is less than the amplitude of the tidal wave registered by the PR tide gauge. Furthermore, it was visualized that both the tidal wave and the river regime are in phase at PCP. The daily flows recorded by Primera Angostura measuring station were on average 1,318 \({{\text{m}}}^{3}{{\text{s}}}^{-1}\) between July 20 and 23, 2009, and 445 \({{\text{m}}}^{3}{{\text{s}}}^{-1}\) during September 3 and 4, 2019, which meant that the reported superelevations were exclusively due to the meteorological component.

Fig. 6
figure 6

Comparison between water level time series from the PCP limnigraph and PR tide gauge: a July, 2009, b September, 2019. Elaborate on hydrological data from the DPA Río Negro (2022) and SNH (2021)

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3.6.1 Case 1: July 20–25, 2009

Between July 22 and 23, 2009, the permanence of intense winds from the south quadrant with speeds greater than 40 km h−1 (Fig. 7a), associated with the passage of a southeasterly storm, caused the unusual growth of the Río Negro with the consequent occurrence of a severe flood that affected the Viedma waterfront (Rio Negro, 2009). The reported superelevation was recorded on the water level time series of PCP limnigraph (Fig. 7b). Asymmetries occurred in the tidal cycle due to the advancement of the high phase and a delay during the low phase. It was also evident during days 22 and 23 the overlap between the discharge of the river and the tidal wave during its ebbing phase.

Fig. 7
figure 7

Meteorological and hydrological data from July 22–24, 2009. a Wind speed recorded by Viedma Aero meteorological station, b water level registered by PCP limnigraph and astronomical tide forecast for PR. Elaborate on meteorological and hydrological data belonging to SMN (2022), DPA Río Negro (2022) and SHN (2022)

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Bispectral analysis showed the presence of energy cells centered in 0.08 (period = 12.42 h) \({{\text{H}}}^{-1}\), locating the maximum energies around the semidiurnal components (Fig. 8a). In this case, the properties of the signal indicated an intense asymmetric interaction type \({{\text{f}}}_{1}\)+\({{\text{f}}}_{2}\)=\({{\text{f}}}_{1+2}\) in the semidiurnal scale. Trispectrum detected the presence of interacting frequencies on 0.02 (period = 50 h), 0.04 \({{\text{H}}}^{-1}\) (period = 25 h), and centered in 0.08 \({{\text{H}}}^{-1}\)(period = 12.42 h), which corresponded to diurnal and semidiurnal scales, respectively (Fig. 8b). These results showed the production of symmetric nonlinear processes due to interacting frequencies type \({{\text{f}}}_{1}\)+\({{\text{f}}}_{2}\)+\({{\text{f}}}_{3}\)=\({{\text{f}}}_{4}\) or \({{\text{f}}}_{1}\)+\({{\text{f}}}_{2}={{\text{f}}}_{3}\)+\({{\text{f}}}_{4}\), was more intense in the diurnal component during the storm pass.

Fig. 8
figure 8

High-order spectra of PCP limnigraph. a Bispectrum, b Trispectrum. Elaborate on elevation water data belonging to DPA Río Negro (2022)

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3.6.2 Case 2: September 3–4, 2019

As a consequence of the production of strong south/southeast winds (Fig. 9a) in conjunction with an extraordinary tide (4.4 m) a rise in water elevation was reported in PCP (Fig. 9b). From the afternoon of September 3 until the early morning of September 4, an overlap between the river flow and the tidal waves was detected in PCP. This event was particularly significant during the ebb of September 2019. Also, a four-hour delay was observed in the appearance of the ebbing phase corresponding to the waves of the river and tide.

Fig. 9
figure 9

Meteorological and hydrological data from September 03–04, 2019. a Wind speed recorded by Viedma Aero meteorological station, b water level registerd by PCP limnigraph and astronomical tide forecast for PR tide gauge. Elaborate on meteorological and hydrological data belonging to SMN (2022), DPA Río Negro (2022) and SHN (2022)

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Again, intense asymmetric interaction was powerful on the semidiurnal scale (Fig. 10a). The auto-bispectrum also showed the presence of energy cells centered in 0.08 (period = 12.42 h) \({{\text{H}}}^{-1}.\) Compared to the previous case, the auto-bispectrum showed that the energy associated with type interactions \({{\text{f}}}_{1}\)+\({{\text{f}}}_{2}\)= \({{\text{f}}}_{1+2}\) was most intense in this study case. Trispectral analysis detected the presence of interacting frequencies centered in 0.04 \({{\text{H}}}^{-1}\)(period = 25 h), 0.08 \({{\text{H}}}^{-1}\)(period = 12.42 h) and 0.12 \({{\text{H}}}^{-1}\)(period = 8.38 h), which corresponded to the diurnal, semidiurnal and third-diurnal scales (Fig. 10b). These results showed the production of symmetric nonlinear processes due to interacting frequencies type \({{\text{f}}}_{1}\)+\({{\text{f}}}_{2}\)+\({{\text{f}}}_{3}\)= \({{\text{f}}}_{4}\) or \({{\text{f}}}_{1}\)+\({{\text{f}}}_{2}={{\text{f}}}_{3}\)+\({{\text{f}}}_{4}\), was more intense in the diurnal component during the storm pass.

Fig. 10
figure 10

High-order spectra of PCP limnigraph. a Bispectrum, b Trispectrum. Elaborate on elevation water data belonging to DPA Río Negro (2022)

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For both study cases were detected that the permanence of strong winds can produce anomalous elevation of the water column. Another result was the delay in the appearance of the low tidal phase, which may promote a greater temporal extension of the flooding time, with the largest backflow being the one registered on September 4, 2019, with a delay of 4 h. The signal structure can be asymmetric or symmetric, also when concentric cells of energy are observed in the bispectrum and trispectrum around diurnal, semidiurnal, and third-diurnal tidal components, this suggests energy transfer from the fundamental frequencies to their higher frequencies. Significant differences in the results of the bispectrum and trispectrum between the two study cases can be attributed to variations in the internal dynamics of the system, such as local bathymetry or tidal propagation patterns. Additionally, wind effects, variations in tidal phases and magnitudes, as well as temporal and spatial variability in environmental conditions, can also influence frequency interactions.

4 Discussion

Fourier's theory may be applied to the treatment of periodic signals, with higher-order Fourier analysis representing an extension of this classical approach (Hatami et al. 2019). High-order spectra describe non-stationary signal dynamics and their statistical characteristics detecting triplets or quadruplet frequencies interaction. Auto-bispectrum allows the decomposition of two frequential nonlinear interactions, while the trispectrum allows studying nonlinearities produced by three frequencies interactions, which produce a fourth. High-order spectral analysis in this study decomposes the harmonics comprising the water level series. Observing frequency interactions, the study reveals how these can elevate the water column and prolong flooding due to asymmetries in the tidal cycle.

Due to the complexity of the hydrologic regimen, climatic, and topographic characteristics that present the Rio Negro lower basin, numerous flooding has been registered in this area since the nineteenth century. The influence of the astronomical tide decreases from the mouth to SJ (UNL-DPA Río Negro 2004b) due to the interaction with the river geomorphology, a depth reduction, and river discharge effects. The tidal wave entry promotes that the watercourse adopts a semidiurnal regimen until Carmen de Patagones (Fig. 1a) (D’Onofrio et al. 2010). During high tide, the study area is susceptible to floods originating from storm pass, flow release upstream, or a combination of both events.

Following what is described in the UNL-DPA report Río Negro (2004b), upstream of San Javier, the water regime of the Río Negro is fluvial, while downstream of this locality (Fig. 1a), when the tidal wave from the Atlantic Ocean enters to the Río Negro, the watercourse adopts a fluvial—marine regime. As well, HA detected that the maxima amplitude corresponds to the semidiurnal harmonics, however, the values of tidal constituents differ from what D’Onofrio et al. (2010) calculated. Compared to 1981–1984 (PCP) and 1935 (TAC), the tidal effects were amplified until PCP during 2003–2021. Diurnal and semidiurnal constituent amplitudes increase until PCP, meanwhile, the quarter and sixth diurnal amplitudes decrease in TAC and rise in PCP (Table 2).

The water level signal upstream, from the river mouth to San Javier, displays frequencies associated with semidiurnal components (\({{\text{M}}}_{2}\), \({{\text{N}}}_{2}\), \({{\text{S}}}_{2}\)), third-diurnal \({{\text{MK}}}_{3}\), and the diurnal component \({{\text{K}}}_{1}\) (Table 2 and Fig. 2). These prominent tidal components significantly contribute to the observed frequencies, particularly at the TAC station near the river mouth, where their maximum values are recorded (Fig. 2a and b). As the wave progresses upstream and interacts with river discharge, the amplitude of these diurnal, semidiurnal, and third-diurnal components decreases. Shallow water components generated by nonlinear dynamics, including \({{\text{M}}}_{4}\), \({{\text{MN}}}_{4}\), \({{\text{MS}}}_{4}\), \({{\text{M}}}_{6}\), \({2{\text{MN}}}_{6}\), \({2{\text{MS}}}_{6}\), and \({{\text{M}}}_{8}\), were observed at both TAC and PCP (located 39 km from the mouth), with PCP exhibiting the highest amplitude of these shallow water components.

In Table 2, increasing phase values from the river mouth (TAC) to SJ suggest changes and deceleration in \({{\text{K}}}_{1}\), \({{\text{N}}}_{2}\) and \({{\text{S}}}_{2}\) wave behaviors along the river. For \({{\text{M}}}_{2}\), \({{\text{M}}}_{4}\), and \({{\text{M}}}_{6}\), if tidal phases rise and then fall upstream, this may signify multiple factors affecting tide propagation as bottom friction, given the average depth of 6 m, sinuous coastline shape, along with islets, sandbars, and irregular bottom morphology (D'Onofrio et al. 2010). The initial phase rise could stem from local river conditions temporarily slowing tide speed due to increased resistance or friction. Conversely, a downstream phase decline might indicate conditions favoring faster tide propagation, like deeper waters or reduced flow resistance. The tidal phase results (Table 2) likely correlate with channel characteristics and river topography per the Subsecretaría de Recursos Hídricos de la Nación (2015). At the river mouth (TAC location) with a 750-m channel width and minimal islands, lower resistance enables faster tide movement and lower upstream phase. In contrast, at the PCP limnigraph site, with a narrower 250-m channel and numerous islands, higher friction causes tide delays and increased upstream phase. Island presence and depth variations add to the complex interaction between tides and local river features, influencing observed phase changes along the river course.

Río Negro discharge has been regulated since 1970 by the dams installed on the Limay (Alicurá, Piedra del Águila, Mari-Menuco, El Chocón) and Neuquén rivers, and various irrigation canals and lakes (Fig. 1b) (Gianola Otamendi 2019). In addition, the watershed of the river suffered 13 years of drought (Diario Río Negro, 2021). These situations modified the hydrological and geomorphological characteristics of the watercourse and cause a significant reduction in river discharge to the Atlantic Ocean from 2013–2017 (DPA Río Negro 2022). For this reason, this investigation's results differ from the D'Onofrio et al. (2010) findings.

International investigations demonstrated that the river discharge alters tidal propagation and the resulting generation of overtides (Jay and Flinchem 1997). High river flows may contribute to reducing tidal amplitude, delaying wave propagation, and increasing tidal deformation (Godin 1991; Horrevoets et al. 2004). Guo et al. (2015) found a strong tidal deformation upriver during the dry season and an amplification in tidal wave deformation downriver in the wet season in the Yangtze River (China). Upstream of Río Negro mouth, river bottom friction, shallow water, and horizontal advection effects cause an energetic transfer from linear harmonics to their high-order constituents (D’Onofrio et al. 2010). The authors reported that the amplitude of the fourth diurnal constituent was slightly higher in PCP than in the mouth of Rio Negro, but this behavior was unobserved for the rest of the high-frequency harmonics.

The spectral density of the fourth, sixth, and tenth diurnal constituents was notably higher in PCP compared to TAC (Fig. 3). Additionally, high-order spectra analysis reveals an energetic contribution to the skewness time series in both PCP and TAC (Fig. 5a and b). Furthermore, the amplitude of the sixth and eighth diurnal harmonics did not show significance in the analysis of water level time series from 2003 to 2021 (Table 2). However, during a storm event, analyzed through the two study cases, there is a redistribution of energy with a notable increase in high harmonics at the expense of linear tidal components (Figs. 8 and 10). The difference in the energy observed in the bispectra and trispectra, as well as the interacting frequencies, between both study cases, can be attributed to the persistence of strong (wind speed > 40 km/h) south quadrant winds, which was more persistent in Case 1 than in Case 2. This energetic transfer occurs across diurnal, semidiurnal, and third-diurnal tidal constituents, resulting in an anomalous increase in water level during storm events (Figs. 7 and 9) compared to normal behavior (Fig. 5).

Nonlinear effects originating from triplets and quadruplets frequencies interactions over the river flow and tidal cycle were observed in other estuaries. Dinápoli et al. (2020) demonstrated that interacting frequencies increase the asymmetry, producing faster floods and slower ebbs in Río de la Plata (Argentina). In Delaware Bay Estuary (United States), these phenomena amplified the storm surge during the Rita hurricane (Xiao et al. 2021). This work showed an increase in the asymmetry of the tidal cycle signal during a storm in the Rio Negro. Nonlinear interactions caused by symmetric and asymmetric processes provoked delays in the ebbing phase.

5 Conclusions

This paper describes the application of high-order spectra for analyzing nonlinear processes produced by interacting frequencies in Rio Negro hydrodynamics. Harmonic Analysis and Fast Fourier Transform were utilized to decompose the signal into its principal frequency components. Bispectral and trispectral analysis were employed for studying three and four frequencies interactions, respectively.

The floods of the Río Negro in its lower basin are caused by mechanisms of a diverse nature. The river overflows are conditioned by the existence of the two hydrological regimes, being fluvio-marine influenced by the wave of the astronomical tide from its mouth in the Atlantic Ocean to San Javier, (located 65 km from the mouth of the Río Negro). In this section, the water column presents semi-diurnal fluctuations (every 6 h). This effect diminishes upstream from the mouth until it becomes negligible in San Javier, marking the transition to a purely fluvial regime.

The occurrence of nonlinear processes associated with the transferce of energy from linear tidal harmonics to those of high order can increase the height of the water column and produce asymmetries in the tidal cycle. For the study area, the production of nonlinear processes was found in areas surrounding the watermarks located in Toma de Agua El Cóndor (TAC) and Prefectura de Carmen de Patagones (PCP), being more intense in PCP. Nonlinear processes produced by storm passage or high flows release upstream can cause that encourage severe flooding production in the study area. According to the bispectrum and trispectrum, the energy transfer is carried out through the diurnal, semidiurnal and third-diurnal frequencies towards high-order harmonics. Due to this energetic transference, an increase in the tidal cycle asymmetry promotes a delay in the ebbing phase, which provokes an extension of the flood duration. Due to its topographic characteristics, the waterfront of the city of Viedma is the most affected place for these phenomena occurrence.