5.1. Tidal and Flow Effects on the Salt Wedge
The farthest extent reached by the parameters of stratification, buoyancy, and potential energy anomaly is interpreted as the upper limit of the influence zone of the saline wedge from three different perspectives. The first perspective simply relies on the calculation of the density gradient [
72,
73]. The second perspective explores the vertical stability between two layers of water with differing physical properties. The higher the buoyancy index, the greater the stability. Peak buoyancy values commonly occur within the pycnocline [
74,
75]. Lastly, the potential energy anomaly measures the mechanical energy needed to instantaneously homogenize a water column [
76]. A comparative assessment of the maximum reach of the wedge under syzygy and quadrature tide conditions reveals the important influence of tides on the extent of saltwater ingress (
Figure 8). The stratification index indicates that during syzygy, there is a 1.5% increase in the depth of intrusion compared to quadrature. Likewise, the buoyancy frequency exhibits a parallel trend with a 1.42% increase. The potential energy anomaly reports a slightly higher total penetration of 2.6%. Nevertheless, this estimation may be imprecise when comparing stratification strengths at the same point during different tidal phases, particularly when the ratio of tidal amplitude to the thickness of the water column is not sufficiently small [
77]. Within the Q = 2000 to 2500 m
3/s range, the uppermost reach demonstrates a distinct pattern attributed to the widening of the channel geometry spanning kilometers 12 to 18. Despite this, all parameters consistently show an inversely proportional relationship between seawater infiltration and river discharge.
Based on the stratification and buoyancy parameters, it can be reaffirmed that the magnitude of flow rate and the FSI front displays a non-linear, inversely proportional relationship (as illustrated in
Figure 9) described by the curve fitting formula FSI (Q) = 1.133 × 10
−6 Q
2 − 0.0142 Q + 46.2825. The given regression model has a coefficient of determination (R
2) of 0.9846, thereby reaffirming the river dominance in the MRE dynamics [
29,
30]. This model stands as a promising inaugural approximation for predicting the location of the FSI and, in extension, for delineating regions where the TMZ may consolidate [
34]. Formerly, ref. [
78] undertook this endeavor within the Pearl River estuary. This is a microtidal-fluvial estuary distinguished by its shallow depth and funnel-shaped geometry. They applied two regression methodologies, yielding results in alignment with our own findings. There is a non-linear, inversely proportional bond between the salt intrusion length and the river’s hydrodynamics. Furthermore, estuarine salinity exhibits variability not only due to the primary influence of the river but also in connection with the temporal alignment of peak flow events relative to the baseline flow conditions. These patterns have been observed in numerous estuaries worldwide, primarily through the application of Van der Burgh’s coefficients [
79,
80,
81].
It is essential to recognize the limitations of this statistical fitting due to the role of tides in stratification and mixing in estuaries. Examining the same flow scenario, spring tides enhance salt layering. In contrast, during quadrature tides, mixing is increased, which implies less differentiation of layers in the water column. Based on the average impact of two tidal cycles, it was discovered that the intrusion depth can differ by up to 1 km. However, if morphological changes in the channel are considered, this variation could amplify exponentially. In the Guadalquivir River estuary, an escalation in tidal inflow was observed following bed-dredging operations [
82]. In the Ems estuary, the deepening of the channel has led to heightened turbidity and sedimentation levels [
83]. The rising sea level acts as another factor contributing to the variability in tide influxes [
84,
85]. In most scenarios, it results in an augmentation in both the length and volume of salt wedge intrusion [
86,
87]. These discoveries open up new avenues of study within the dredging context of the MR [
27], encompassing short-term analyses throughout a complete tidal cycle.
The characteristics of the TMZ are determined by a composite interplay of channel morphology, sediment properties, river outflow, sea inflow, and other hydrodynamic processes. Typically, the TMZ tends to be situated in close proximity to the landward extremity of the salt wedge, and is assembled by near-bottom density gradients [
88,
89,
90]. In the MRE, the core of the TMZ is not only downstream of the FSI, but usually remains in close proximity to it, regardless of flow or tidal phase [
31]. This finding implies that the TMZ also experiences migration. Additionally, it was observed that TMZ intensifies with floc precipitation when the FSI penetrates into the countercurrent [
34]. The floc-trapping capability of the TMZ is a result of the convergent shear fluxes within its boundaries [
91]. The flocs emerge as a consequence of the aggregation of colloidal particles [
92]. Their dimensions are subject to sediment availability, organic matter concentration, as well as advection and convection velocities [
93,
94,
95]. In the MRE, a significant amount of organic and inorganic material is present in suspension [
33]. Consequently, the flocculation process is mainly conditioned by the frictional stresses in the bed, which depend mainly on the MR. The discovery of this research indicates that the bottom turbulence production (
P) and the Richardson number (
RL) can be used to infer the magnitude of the shear stress at the bottom (
τxy) given that
P and
τxy are directly proportional to each other, whereas
RL and
τxy are inversely related. The highest shear stresses occur between kilometers 0–6 and 9–13, and are greater when the river flow increases. Under quadrature tide conditions, turbulent production averages 14.2% higher than during syzygy. In discharges ranging from 2000 to 4000 m
3/s, the average water column stability increases by approximately 5.30% compared to syzygy. These enhancements are attributed to the FSI front position, and the ability of tides to counteract shear stresses at the river bottom, with a greater impact observed in low-flow scenarios. Deeper and more intense ebb and flow tidal currents lead to a reduction in bottom turbulence. Simultaneously, the increased tidal currents promote the vertical mixing of water parcels. Nevertheless, the primary factor conditioning water stability and bottom turbulent production remains the river (
Figure 10).
Estuaries can be classified as partially mixed and salt wedge types based on their buoyancy coefficients (
β). Partially mixed estuaries exhibit a
β between 0.0025 and 0.01 s
−2, while salt wedge estuaries have
β values between 0.01 and 0.1 s
−2 [
65]. The findings obtained in this research confirm that the MRE can be classified into both categories depending on its tidal phase. During quadrature and syzygy, it behaves as a wedge estuary for flows below 5500 and 6500 m
3/s, respectively. This implies that for flows above these limits, its theoretical configuration corresponds to that of a partially mixed estuary. According to [
34], this limit is set at 4000 m
3/s, regardless of the tidal cycle.
5.2. FSI Monthly Mobility
Although the Magdalena River can reach flows as high as 17,000 m3/s, analyzing its response to discharges below 6500 m3/s supplies sufficient insight into the dynamics of the salt wedge. It should be noted that it is in the lower-flow scenarios that the greatest variability in the magnitude of the FSI occurs.
Table 5 is created by integrating the stratification-mixing indicators in the estuary with a statistical analysis of flow regimes. It displays the monthly ranges where the FSI and TMZ are mobilized by proximity. For all instances, the cumulative probability (
Pa) helps to validate the feasibility of each interval in both neap tide (NT) and spring tide (ST). In this sense, it can be stated that 90.1% of the time, flows between 6000 and 14,583 m
3/s occur in December, which means that the FSI is unable to penetrate beyond 2.2 km, regardless of the tidal cycle. During this month, its lowest intrusion depth is estimated, and the highest turbulent production rate at the bottom of the estuary. As for January, the MR begins to decline, which results in a greater stratification and deepening of the saline wedge up to a maximum of 14.2 km. Approximately, this should be located around km 7.1 ± 7.1.
During February, the saline front gradually moves upstream, from oscillating around km 9.7 ± 9.7 to stopping above km 11 ± 10 in March, at which point it is expected to reach its maximum extent before receding due to increased river flow. By April, it is found above km 9.7 ± 9.7, above km 4.5 ± 4.5 in May, and at 1.8 ± 1.8 km in June. It advances against the current again for both July and August, stopping in both cases near km 5.8 ± 5.8. In September, the halocline is located at approximately km 4.5 ± 4.5 and retreats in October (km 1.8 ± 1.8), November (km 1.4 ± 1.4), and December (km 1.2 ± 1). This pattern suggests that the FSI positioning is highly responsive to changes in intra-annual scales and consistently remains focused around km 5.
Considering that accelerated floc precipitation occurs when turbulent stratification is disrupted, it can be argued that an increase in bed sedimentation is promoted during the initial phase of the dry–rainy climate transition. Two important processes take place during this phase. First, the capacity of the river to transport larger fragments increases. Second, the FSI–TMZ emerges from the stream, which favors sediment uptake. Specifically, it follows that during the most intense transition of the year (February–March–April), the precipitation volume peaks between km 1 and 9.7 ± 9.7. Furthermore, it is expected that the periods of greatest erosion are associated with the restriction of the wedge above km 2, due to the increase in Q and the intensification of bottom shear.
5.3. Probabilistic Model Validation
Although the FSI–TMZ relationship proposed here is based on the characterization of the MR regime and the application of a previously calibrated and validated numerical model, it is evident that estuarine circulation processes involve complex interactions that are difficult to synthesize using such approaches [
7]. For this reason, a case study is presented that integrates the probabilistic flow regime (refer to
Figure 11) and a multi-bathymetric analysis of the bed for the year 2016.
Ref. [
20] found that sedimentation processes were dominant during the transitions from March–February and August–July, with an average rate of 883 mm/m (March–February) and 271 mm/m (August–July). Moreover, the data collected showed that a maximum accumulation of 8628 mm/m was recorded above kilometer 4.5 in the August–July transition period. Similarly, the study discovered that the erosion processes with the highest intensity take place during the transitions of February–January, September–August, October–September, November–October, and December–November, as well as in close proximity to kilometer 4.5. The average scour rate ranged from 194 mm/m (October–September) to 952 mm/m (February–January) with a maximum of 13,222 mm/m in December–November. In April–March, May–April, June–May, and July–June, a mixed range was identified with a slight predominance of erosional processes. The range in average erosion and accommodation rates is 112–835 mm/m and 165–833 mm/m, respectively. However, there are spatial differences in the distribution of these processes. For instance, the most significant sedimentation processes occur between km 0 and 2 from June to May. In May–April, sedimentation arises on the western margin between kilometers 0 and 3. In July–June, it occurs on the eastern margin between km 0 and 3, and extending up to km 5 in April–March.
In line with the functional model presented in this research (refer to
Table 6), the salt wedge can migrate up to a maximum of 20.2 km and 5.5 km in March–February and August–July, respectively. During the first period, there was a retreat observed from ~20 km (February) to ~17.5 km (March). This movement occurred due to a rise in mean flow magnitude of 200 m
3/s (February: 2467 m
3/s, March: 2681 m
3/s) that resulted in the accumulation of particulate material without disrupting channel stratification. During the second period, there is a comparable occurrence where the average flow increases from 5268 m
3/s (July) to 5323 m
3/s (August) with a maximum amplitude of 3.6 km (July) and 5.5 km (August). Note that the average sedimentation rate in March–February is more than three times higher than in August–July, and that turbulent energy production is practically nil for flows between 2500 and 3000 m
3/s (March–February). This is different from the production related to discharges in the order of 5000 m
3/s (August–July), which has two energy maxima: at kilometer 3 (
Pmax = 1.336 × 10
−4 W/Kg) and at kilometer 11 (
Pmax = 1.05 × 10
−4 W/Kg). As mentioned, higher shear stresses on the bed promote resuspension and aggregation of material, while hindering its sedimentation. This is why the precipitated volume during March–February is much higher, and sedimentation in August–July is focused between kilometers 3 and 11. In this section, turbulent energy decreases substantially, which causes the deposition of the previously accreted flocs that remained in suspension (as seen in
Figure 4D and
Figure 5D).
During the analysis of intervals featuring the most significant erosion, it was found that a correlation exists with the periods when the wedge is constricted toward the river’s estuary. Specifically, this correlation is noticeable at km 2.7 ± 0.8 in September and 3.5 ± 1.5 in August, and during the October–September period (km 1.4 ± 0.5 and km 2.5 ± 0.9), November–October (km ~ 0 and km 1.4 ± 0.5), and December–November (km ~ 0). However, an anomaly occurs during the February–January period. Although the FSI–TMZ is capable of penetrating to a depth of approximately ~20.2 km, the river experiences a decrease in competence during this stage, with the mean flow dropping from 2706 m3/s to 2467 m3/s. As a result of this reduction in flow, suspended material is precipitated and moves toward the salt front.