The Wind-Current-Water Levels Effect over Surface Wave Parameters Nearby the Magdalena River Delta: A Numerical Approach

The growing worldwide awareness to reduce CO₂ emissions and the recent proposition of China in considering the dual carbon goals for reaching a carbon neutrality [1], motivated the development of new projects for renewables, considering the impacts of these technologies on the human health and environment [2] and a fair access to it [3]. Herein, the marine energy technologies [4] participate in the world energy transition, where the maturity and development of these depends on applied research.

There exist limited but very important studies performed in Colombia about marine power potential [5], salinity gradient power [6, 7], wave power [8], offshore wind [9-11], tidal energy [12] and ocean thermal energy [13], what evidence the interest of the engineering community for the non-conventional energies. In this sense, engineering applications such as wave power extraction at rivers deltas require the understanding of the effects of atmospheric and ocean forcers over the surface wave parameters to reduce uncertainties of electricity generation because of the high nonlinearity of waves (e.g., whitecapping, quadruplets, triads) [14], where the proportion of the forcers over the wave parameters depends of the river delta configuration and climate variability [15, 16]. The effects of wind-waves-water levels nearby a river delta generate non-linear interactions over the wave hydrodynamics, that difficult the understanding of transport processes due to the complex met-ocean interactions [17].

The Magdalena River delta is located in the Colombian Caribbean shelf, and drives fresh water to the Caribbean Colombian coast, generating a triangular delta of 1,690 km2 [18]. The coastline associated with the deltaic system showed intense wave activity with 206 ergs s-1m-1 of power [19]. Coleman and Wright [19] categorized the river delta as a wave-dominated system. The Magdalena River mouth during January to April (windiest season) discharges 4360 m3s-1 of fresh water into the ocean [20], and its nearby wave field shows high energy because the surface wind stress generates wave heights up to 2.5 m [21]. During September to December (rainiest season), the high stream river flow of the Magdalena River delta discharges 8063 m3s-1 of fresh water [20], and its ocean hydrodynamic field exhibits low wave energy because moderate surface winds produce wave heights lower than 1.5 m [21]. A transition season occurs during May to August where river flows and wave heights are moderate with records within the limit range [22]. Local government studies established that the Magdalena River delta is influenced by three climate seasons governed by the local rainfall and surface winds [23, 24]: December-March (dry season), April-July (transition season), and August-November (wet season).

Bastidas-Salamanca and Rueda-Bayona [25] argued that the mesoscale climate of the Magdalena River delta is modulated by the Caribbean Low level jet which produces two representative climate seasons: dry (February) and wet (October). That research concluded that the climate variability in the study area is dominated by short, mid and long term scenarios, where the surface winds weaken during positive Atlantic Multidecadal Oscillation (AMO) and La Niña events. The same researchers pointed out that the surface winds in the Magdalena River delta shows a bimodal annual cycle with a maximum in February and the lowest magnitudes in October [26].

Studies related to the Magdalena River delta are limited because the zone exhibits a high-energy hydrodynamic field which triggers several security risks for the field campaigns. For example, the equipment deployment and measurements to perform hydrographic and oceanographic surveys in the Magdalena River mouth are highly restricted due to navigation security considerations, that limit the data collection periods. Restrepo et al. [27] measured just one-half of the tidal period because of the security restrictions in the channel. These restrictions have been overcome with the utilization of remote images such as the study of Moreno-Madriñán et al. [28] who used MODIS reflectance images as an indicator of the superficial sediment concentration of the Magdalena River discharge. Restrepo and López [29] characterized the marine climate in the Magdalena River mouth through remote sensing and time series analysis of river flows, sea levels, and wave fields. They utilized water level data (1954-2000) and wave data (1963-2000) for the climate analysis. These authors reported a significant wave height (Hs) at the delta of 2.1 m, an M2 tidal amplitude of 0.3 m, a mean sea water level of 0.62 m, a spring tide of 0.64 m, and a neap tide of 0.48 m. Restrepo et al. [30] applied MODIS satellite images to capture the space-time variability of the distribution of suspended sediment of the Magdalena River delta.

The application of hydrodynamic modelling is another alternative to face technical restrictions at river deltas and marine areas [31-39]. At the Magdalena River delta, Urbano-Latorre et al. [40] modelled the currents and wave field using the H2D [41] and SWAN models [42] but the model was not calibrated, then, the reported Hs and velocity currents were not compared with in-situ data to verify the numerical model accuracy. Ospino et al. [43] performed hydrodynamic modelling to analyse the saltwater intrusion into the Magdalena River delta and pointed that water levels and surface winds modulated the saltwater intrusion.

Advanced statistical techniques such as DOE-ANOVA are commonly used for analysing complex non-linear interactions among independent variables and their effect on dependent variables, hence for the proper application of this technique it must be verify the Central Limit Theorem (samples < 30) and the homoscedasticity, normality and independency tests [44]. The application of factorial DOE-ANOVA analysis is evidenced in several research topics of Oceanography and Ocean Engineering such as wave run-up, debris loads, coastal protection, naval design, and ocean modelling [45-51] However, the use of time series as factor and responses for factorial design and the use of DOE-ANOVA for analysing wind-currents-water levels effect over wave field have not been reported.

The studies of the Magdalena River abovementioned showed the complexity of the local hydrodynamic processes, the limitation of installing and deploying instruments for mid or long-term periods in the river channel and delta, the access restrictions to local databases, and the need for understanding the dynamic of surface wave parameters at river deltas. In this sense, the novelty of this research is the application of the DOE-ANOVA factorial design using continuous time-series as inputs for understanding complex met-ocean interactions at river deltas. Then, this study used time series with 1-hour time interval of surface winds, currents and water levels as factors and significant wave heights and peak periods as responses to identify the effect of wind-water level–current effects over the wave hydrodynamics in the nearby field of the Magdalena River delta.

After the definition of the study area and the implementation of the numerical model, a hydrodynamic numerical validation is performed through the comparison of numerical results against hydrodynamic in situ records and satellite imagery. During the validation, the numerical results of water levels, and the modelled sea surface salinity (SSS) and temperature (SST) considering local in-situ databases and the results of other studies are revised. After the numerical validation, the wave fields of the three seasons for the further DOE-ANOVA analysis are analysed.

Hydrodynamic numerical validation

This study validated the water levels for June and October 2010 (Figure 5); the water level simulation results (Figure 5a) for June 2010 compared to the in-situ measurements of the Casa Pilotos station (Figure 1) indicated a determination coefficient of 0.78. Additionally, the validation of October 2010 (Figure 5b) showed a lower determination coefficient of 0.70, which could be generated by human errors because the water level was measured through direct observations using a fixed rule. Furthermore, the Form Factor F [64] calculated from the simulated water level indicated a mixed semi and micro diurnal tide.

The validation of SSS and SST evidenced that during February 2010, the surface modelled thermohaline values (Figure 6) from K0 to K6 of the Magdalena River channel (Figure 2) were similar to the findings of Alvarez-Silva and Osorio [65] and Restrepo et al. [55]. On February 4^th, the surface salinity distribution along the river channel from the river mouth (K0) up K10 (Figure 2) varied between 0.1 and 5 (Figure 6a). The modelled salinity of the surface Magdalena River plume also showed values between 10 ppt and 32 ppt, which were like the numerical results [43].

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Figure 5. Water level validation from in-situ records of Casa Pilotos station; (a) June 1^st to 17^th of 2010; (b) October 9^th to 25^th of 2010

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Figure 6. Surface hydrodynamic modelling for February 4^th of 2010 at 2300 hr (a) Salinity (ppt); (b) Temperature (℃), and February 24^th of 2010 at 0000 hr; (c) Salinity (ppt); (d) Temperature (℃)

Note: The white area represents the dry zone delimited by the hydrodynamic model.

Salamanca et al. [66] measured surface salinity and temperature at two stations in the Magdalena River mouth; one located inside the channel (river area) and the other was located beside the channel (sea area). The salinity in the river area during February 4^th and 24^th reported in that study was like the findings of this research (Figure 6 a, c) with values varying between 0.1 ppt and 0.5 ppt. The modelled salinity on February 4^th and 24^th of this study was also similar to the measurements performed by Salamanca et al. [66] in the sea locations with 34 ppt and 30 ppt of salinity respectively. Besides, the surface temperature for February 4^th and 24^th measured in that study varied between 26℃ and 28℃, that range was like the numerical results of this study (Figure 6 b, d).

This research analysed two MODIS reflectance images (modis.gsfc.nasa.gov/data/dataprod/mod09.php) of February 2010 to visualize the effect of winds over the surface Magdalena River plume. (Figure 7). Low reflectance values are an indicator of a more turbid surface water because of the surface albedo reduction, so the purple values indicated a darker surface colour that could be associated with a high turbidity due to the effect of the surface sediment plume. In this sense, the image of February 4, 2010 (Figure 7 a) showed low reflectance values (purple) pointing that the river plume turned to the west.

From January to March, northeast trade winds predominate over the study area with maximum velocities between 4.5 m/s and 6.1 m/s [23]. As a result, the local hydrodynamics under the effect of winds (Figure 6) depicts a curvature of the surface Magdalena River plume (Figure 7 a).

Figure 8 shows wind vectors from north-northeast during February 2010, with maximum velocities up to 13 m/s during the first and third week. From the 22^nd to the 28^th, the wind vectors decreased in magnitude and changed the direction for the 23^rd and 24^th. The change in wind direction (Figure 8) from northeast to the south allowed to the Magdalena River discharge arriving at the ocean without the curvature (Figure 6 c,d). Considering the influence of wind on the river discharge, it is important noticing that the seasonal variability of local winds strongly modulates the hydrodynamics and surface transport in the study area.

The modelled surface salinity distribution of February 2010 (Figure 6 a, c) showed the wind effect on the study area during February 4^th; the wind predominated from the northeast (Figure 8), which curved the surface river plume to the west. In contrast, on February 24^th, the wind predominated from the south, allowing a uniform discharge (radial) and reducing the salinity values in the eastern coastal zones (Figure 6b). The comparison between in situ measurements in Bocas de Ceniza station and modelled salinity and temperature (Figure 6) showed that the implemented model was able to simulate the surface thermohaline levels associated with the local characteristics. Additionally, the modelled surface thermohaline of the Punta Roca station, located further from the river mouth, performed similarly to the in-situ REDCAM information for February 24^th (Figure 6 c, d). Consequently, this study evidenced the effect of wind, water level and currents over the nearby wave field and the surface Magdalena River plume.

After the validation of water levels and surface salinity-temperature numerical results in the study area, the nearby wave field from the river delta was analysed. As a result, it was observed that wave vectors in the access channel predominated from the north-northeast during all the sea states (Figure 9), entering the river and diffracting to the left side of the channel (left jetty). The wave field during the dry season of 2010 (February) showed wave heights with maximum values of 2 m in front of K0 (Figure 9 a). Then, the numerical model indicated wave heights with a mean value of 0.37 m and a minimum of 0.05 m inside the river channel; the modelled area nearby the Casa Pilotos station showed a wave height of 0.77 m. The wave heights in the access channel (K-1.5) were maximum during the dry season (Figure 9 d) due to the wind effect from January to March.

For the transition season (June) of 2010 at the Casa Pilotos station, maximum, mean, and minimum wave heights were 0.62 m, 0.34 m, and 0.16, respectively (Figure 9 b), with vector directions like February (Figure 9 a). Finally, the wet season at the Casa Pilotos station (Figure 9 c and Figure 9 f) featured wave heights of 0.6 m, 0.35 m, and 0.01 m, respectively, with vectors’ directions similar to the other two seasons (Figure 9 a, b).

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Figure 7. Data pixels (a) MODIS band 2 satellite image for February 4^th 2010; (b) MODIS band 2 satellite image for February 24^th 2010

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Figure 8. Surface wind vectors during February 2010 over Magdalena River delta

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Figure 9. Modelling of wave field during 2010; (a) vectors on February 21th 1300 hr; (b) vectors in June 16^th 1300 hr; (c) vectors in October 21th 1300 hr; (d) Hs (m) in February 21th 1300 hr; (e) Hs (m) in June ¹6th 1300 hr; (f) Hs (m) in October 21th 1300 hr.

Note: Projected coordinates in Magna-Sirgas Bogota Zone (units in meters).

DOE-ANOVA analysis

To analyze the effect of wind-currents-water level on the surface wave parameters at the study area, a multivariate statistical analysis was performed in this study. The first (e.g., A) and second order effects (e.g., AA) of the independent variables produced a higher effect over the response when compared to the combine effects (e.g., AB). In this sense. the combined effects scatter the influence of the independent variables over the response of the dependent variables. Herein, the DOE-ANOVA showed that during February (dry season), the Hs of the Point 1 and 2 is positive and directly affected by v-wind speed (+D, +DD), and by u-wind speed (+C) only at Point 1 (Table 4). The Tp at Point 1 decreases with the increment of current velocity (-AA) and u-wind speed (-CC), in contrast, the Tp at Point 2 decreases when v-wind speed increases (-C, -D).

Table 4. Standardized effects of wind, water level and currents (factors) over Hs and Tp (responses)

Note: The + and - signs indicates a positive and negative standardized effect. A second order effect of factors over the responses is identified by a double letter (i.e., AA, BB, CC, DD) and combined effects by two letters (i.e., +AD, -AC).

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Figure 10. Pareto charts and main effect plots of Point 1 (a, b, c, d) and Point 2 (e, f, g, h) for June (transition season) in 2010

In June (transition season), the Hs at Point 1 decreases due to the increment of current velocity and water levels (-B, -A), with a reduced wind speed effect (+AC, +CC) (Table 4, Figure 10). At the Point 2, the Hs changes non-linearly (convex) with the increment of current velocity (+AA) and behaves linearly positive with the direct effect of wind speed (+CC, +DD). The Tp at Point 1 rises with the increment of current velocity (+A) and falls when water levels increase (-B, -BB). For the Point 2, Tp changes non-linearly (concave) because of the rise of current velocity (-AA) and water levels (-BB). In the wet season (October), the Hs at the Point 1 (Table 4,) decreases when the current velocity increases, and behaves non-linear (convex) predominantly positive with the increment of wind-speed (+C, +CC). The Hs at the Point 2 rises and performed convex because of wind speed (+CC). The response of Tp at Points 1 and 2 was not statistically significant, hence, the effect of factors on the Tp was not proved.

The higher standardized effects were found in June when the flow river discharge and winds were not stronger than Februarys’ nor lower than Octobers’. In this sense, the Pareto charts and main effect plots of June are depicted in Figure 10 to visualize the statistical significance and the linear and non-linear behaviour (convex, concave) of the Hs and Tp due to the effect of winds, water levels and currents.

According to the DOE-ANOVA for Points 1 and 2 (Table 4), it was evidenced that the factors (wind, water level, currents) induced differently the responses of Hs and Tp. The Point 1 is located aside of the Magdalena River’s mouth (Figure 1) under the direct effect of currents of surface plume which increase the sea water level in the nearby field, then, waves at this point are dissipated due to the flow river discharge and because of changes in bathymetry that produces energy dissipation (shoaling and refraction). The Point 2 is located further from the river mouth at a zone without important changes of bathymetry toward the river mouth. In this sense, this location is less affected by the surface river plume that reduces the wave energy due to the opposite current and the increment of the water level. When direction of currents is opposite to the wave direction, occurs a choke because of the encountering (convergence) of kinetic energy vector of currents and waves. This choke provokes a redistribution of kinetic wave energy to potential, what is observed in the increment of wave heights and the decrement of wave celerity. When currents and waves propagate with same direction, the kinetic energy of wave increase, extracting its potential energy and reducing its wave heights.

The Magdalena River mouth increases the freshwater discharge from February (minimum) to December (Figure 4), being October the starting month of the wet season (high precipitations). Because the winds of the study area are associated with the Interconvergence Tropical Zone (ICTZ) [67, 68], the winds and flow river are directly opposite. In this sense, the low flow river discharge during February (windiest season) allows to the strong winds forcing to curve the surface river plume to left, easing the wave propagation into the coast. After the dry season, the winds decrease up to December and the river flow increases, causing the reduction of the wave energy due to a stronger opposite current. Regarding to the response of Tp, it was evidenced that the current velocity and water level increase it with some non-linearities due to the complex hydrodynamic interactions. In contrast, the wind speed reduces the Tp because the local wind forcing generates momentum that redistributes the wave energy frequencies.

The main and second-order effects found in the DOE-ANOVA of this study (Table 4) evidenced that river discharge transferred energy into the ocean, through the velocity currents (kinetic energy) and water level changes (potential energy). That energy transfer reduces notably the energy frequency distribution of waves during most of the year, which is evidenced in the increment of Tp and the reduction of Hs when flow river discharge increases.

The local wind field significantly affects the wave field during the dry season (February) increasing the Hs and reducing the Tp by the momentum transfer. Similarly, to the findings reported by Rueda-Bayona et al. [51], DOE-ANOVA analysis showed second-order standardized effects related to extreme wave events that occurred in February, probably pointing to the presence of quadruplets wave-wave interactions and white-capping dissipation that redistribute the frequencies along the wave spectra.

The reduction of water levels and the Magdalena River flow allowed to the wind intensifying the local wave field, consequently, the wave height increased, and the breaking zone moved closer to the access river channel (K0). During the transition season, winds decreased, and river flow increased, generating a decrement of the wave potential energy at the river delta. In the wet season, the wind speed decreased but the river flow showed maximum records, which reduced the wave heights at the river mouth. Accordingly, the wave field of the Magdalena River delta was modulated by the wind and river flows during all the three seasons. In this sense, the winds modulated (rise) the wave energy potential during the dry season (Figure 9d), and during the wet season, the river flows reduced the wave heights that could move the breaking zone further from K0 (Figure 9f).

Considering that the study area is dominated by the Caribbean Low level jet, the intrusion and curvature of the Magdalena River delta will depend of the river’s flow controlled by the precipitations of the upper basin during the wet season (October), and the highest wind stress over the river delta during the dry season (February). That explain the predominance of the river’s flow and surface winds over the wave parameters because of the first and second order effects seen in February and October (Figure 10, Table 4). The water level factor during these two seasons (dry, wet) had not an important contribution over the response of wave parameters. However, during the transition season (June), the water levels had a more important contribution being the second factor, after the velocity currents, which influenced the behaviour of the Hs and Tp.

The results of DOE-ANOVA (Table 3) allowed to identify the effects of the three main factors (A, B, C and D) over the wave parameters in the study area (Hs, Tp), where the first and second order effects of one factor (e.g., C, CC) revealed the direct effect of the factor (e.g., u-wind) on the response (e.g., Hs). Then, the three performed experiments showed that dry season exhibited the most statistical significance of the effect of winds and currents compared to the transition and wet season. In this sense, during the dry season at the two points (Table 4), the winds intensify the Hs (+C) and reduce the Tp (-C, -D), which suggest that kinetic wave energy is reduced and transferred to the potential wave energy. In the transition season, currents, and water level controls the wave energy, where the kinetic energy (in terms of Tp) at Point 1 is reduced and potential energy is incremented (in terms of Hs), and the kinetic energy at Point 2 decreases and the potential energy rises. In the wet season, the winds (+C) increase the potential energy at Points 1 and 2, due to the wind stress push the waves against to the intensified flow of the river delta, which increase the wave heights and provoke a rapid redistributed wave frequencies because of the river-waves encounter. This rapid redistribution of wave frequencies associated to a turbulent and complex hydrodynamic field was evidenced using the results of the DOE-ANOVA for October (Table 4), where there was not significant statistical evidence that wind, currents and water levels influenced the Tp in the study area.