Coupled Morphodynamics of River Bifurcations and Confluences
Abstract
Multithread fluvial environments like anastomosing and braided rivers are fundamentally directed by the continuous concatenation of channel bifurcations and confluences, which distribute flow and sediment among different branches that are reconnecting further downstream. A large number of theoretical, experimental, and numerical studies conducted in the last two decades have provided a clear picture of stability conditions for river bifurcations. However, most analyses are focused on the dynamics of bifurcations alone, ignoring the possible mutual interaction with downstream confluences. In this work, we study the morphodynamic equilibrium and stability conditions of a bifurcationconfluence loop, where flow splits in two secondary anabranches that rejoin after a prescribed distance. Through the formulation of a novel theoretical model for mobile bed confluences, we show that the dominating anabranch (i.e., that carrying most discharge) is subject to an increase of the water surface elevation that is proportional to the square of the Froude number. This effect causes a decrease of the slope of the dominating anabranch, which acts as a negative feedback that increases the stability of the bifurcationconfluence system. A linear analysis of the coupled model reveals that the stabilizing effect exerted by the confluence depends on the ratio between the length of the anabranches and the average water depth, independently of channel slope and Froude number. Ultimately, this effect is potentially able to stabilize the loop even when the sediment is mainly transported in suspension, a condition which makes the classic stabilizing mechanism (i.e., the topographic effect at the bifurcation node) practically ineffective.
Key Points

The dominating branch of a channel confluence shows a higher water level, depending on the square of the Froude number

The confluence exerts a negative feedback on the upstream bifurcation, which promotes the stability of the system

The bifurcationconfluence system can be stable even when the bedload steering at the bifurcation is very weak
1 Introduction
The study of multithread systems like braided and anastomosing rivers, deltas, alluvial fans, represents a fascinating topic in the vast world of river patterns. The flow splits around exposed bars, islands, ridges, and often reconnects a little further downstream. Water and sediment partition in the bifurcates has a fundamental control of the river morphological evolution and ecological functionality (Ashmore, 2013; Nanson & Knighton, 1996). At the channel scale, bifurcations and confluences play the role of basic unit processes of multithread systems. Understanding their own distinct morphology, flow structure and dynamics, as well as their mutual interplay, is therefore of crucial importance for managing water resources, mitigating the impacts of anthropic pressure [e.g., dams construction (Graf, 2006; TongHuan et al., 2020)], ensuring flood protection and adopting river restoration measures suitable for recovering deteriorated ecosystems (e.g., Habersack & Piégay, 2007; Wohl et al., 2015).
Almost all studies performed to date consider the two processes separately, although they frequently appear as closely interconnected. Figure 1 shows some illustrative examples of the morphological bond between bifurcations and confluences in different fluvial environments. Braided rivers like the Rakaia River in New Zealand (Figure 1a) are characterized by a complex, highly dynamic planform, where the water and sediment fluxes divide among multiple channels, although just few of them are morphologically active at a given time (Ashmore, 2001; Bertoldi et al., 2009a). In these fluvial systems, sequences of confluencebifurcation units are ubiquitous features that control channel morphology and the spatial/temporal patterns of sediment transport (Ashmore, 2001; Ashworth, 1996). Midchannels bars and vegetated islands frequently recur in natural meandering rivers (Figure 1b), often associated with width fluctuations or chute and neck cutoff (Grenfell et al., 2012; Zolezzi et al., 2012). They also serve as key elements in the restoration of pristine multithread patterns, as in the case of the Drau River in Austria (Figure 1c), where a former sidechannel was reopened with the aim of improving the habitat conditions, stabilize the river bed, and ensure flood protection through channel widening (Formann et al., 2007; Habersack & Piégay, 2007). A further example comes from a novel interest of river restoration practices, namely the construction of the socalled longitudinal training dams to mitigate the effects of shipping on riverine habitat biodiversity and functionality and, at the same time, guarantee the fluvial navigation (Collas et al., 2018; de Ruijsscher et al., 2020) (Figure 1d).
In this work, we tackle the problem of analyzing the coupled morphodynamical response of such bifurcationconfluence systems. We therefore consider a simple “channel loop,” where a singlethread channel divides into two secondary anabranches that reconnect further downstream, and we try to identify, within the framework of a simplified quasi twodimensional approach, the key parameters that rule the mutual interplay between the upstream and downstream nodes.
River bifurcations often display an unbalanced distribution of water and sediment fluxes, with the two anabranches showing significant differences in their width and bed elevation (Zolezzi et al., 2006). This behavior has been highlighted by a large number of theoretical studies (Bolla Pittaluga et al., 2003, 2015; Redolfi et al., 2019; Salter et al., 2018; Wang et al., 1995), field observations (e.g., Zolezzi et al., 2006), numerical (Miori et al., 2012; Siviglia et al., 2013), and laboratoryscale physical models (Bertoldi & Tubino, 2007; Bertoldi et al., 2009b; Federici & Paola, 2003). These studies provide clear evidence on the existence of an inherent (free) instability mechanism that drives such unbalanced configuration. Most of the above analysis place emphasis on the bifurcation node and treat the downstream boundary condition as independently fixed. Therefore, they often simply assume a constant water level at the downstream end of the bifurcating channels, which implies that in the long term there are not slope differences that may advantage one of the two channels. In other words, they assume that the bifurcating system is “upstream controlled.” This approach is also typically adopted for modeling channel networks (e.g., Kleinhans et al., 2012), sidechannel systems (van Denderen et al., 2018), and channel loops (Bolla Pittaluga et al., 2003), where bifurcations are followed by confluences. However, bifurcation stability can be highly influenced by the downstream conditions, as clearly shown by recent analysis (Ragno et al., 2020; Salter et al., 2019, 2018). Specifically, the stability of a bifurcation is found to be very sensitive to a small water slope advantage of one of the two anabranches. Therefore, the assumption of water level constancy at a channel confluence (e.g., van Denderen et al., 2018) may be inadequate to analyze the morphodynamics of bifurcationconfluence loops, as it ignores the feedback mechanism on the bifurcation driven by backwater effects induced by variations of the downstream condition.
Comparatively, fluvial confluences have been widely investigated in the last 40 years through physically scaled laboratory experiments, field investigations, numerical simulations, and conceptual models (e.g., Ashmore & Parker, 1983; Ashmore et al., 1992; Best, 1986, 1988; Best & Rhoads, 2008; Mosley, 1976; Paola, 1997). These studies have been mainly focused on studying their “local” hydrodynamics and morphodynamics, in order to analyze flow structures, bed morphology (e.g., scour depth) and sediment transport. However, much less research has been devoted to investigating the effect of the confluence on the upstream water and bed levels. Specifically, there are currently no theoretical models able to predict the upstream effect exerted by a confluence node when the bed is erodible.
In order to fill this gap and explore possible feedback mechanisms in bifurcationconfluence loops, in this work, we pursue the following objectives:

formulate a physically based model that is able to predict variations of the water surface elevation in mobile bed confluences

analyze whether the presence of a downstream confluence has an effect on the stability of the upstream bifurcation

identify the key parameters that control the stability of the coupled bifurcationconfluence system
To achieve the above aims, we first develop a novel theoretical model for an erodiblebed confluence, and subsequently we couple it with the bifurcation model proposed by Bolla Pittaluga et al. (2003). After a brief review on the state of the art of fluvial bifurcations (Section 2), the formulation of the confluence model is introduced in Section 3. The outcomes of the model are reported in Section 4. In Section 5, the governing equations of the coupled problem are analyzed by means of a linearization procedure. We then discuss the main implications of our results in Section 6, while Section 7 is devoted to some concluding remarks.
2 Background
Channel bifurcations show an intrinsic tendency to evolve toward unbalanced states. Specifically, even a geometrically symmetric bifurcation can spontaneously produce an uneven distribution of flow and sediment discharge in downstream anabranches because of an instability mechanism, which manifests itself with the formation of an upstream steady bar that deviates water and sediment fluxes toward the dominating branch (e.g., Edmonds & Slingerland, 2008; Le et al., 2018; Salter et al., 2019). As shown by Redolfi et al. (2016) such morphodynamical requirement sets an intriguing bond between the stability of bifurcations and the phenomenon of 2D morphodynamic influence (Zolezzi & Seminara, 2001).
Bifurcation instability arises from an imbalance between the sediment distributed by the bifurcation and the transport capacity of the downstream branches (e.g., Bolla Pittaluga et al., 2003; Wang et al., 1995). In bedloaddominated rivers, this mechanism can be compensated by the gravitational effect of lateral bed slope on the sediment transport, which is more effective when the widthtodepth ratio is small. For this reason, the unbalanced configuration is observed to occur at relatively high values of the widthtodepth ratio (e.g., Bertoldi & Tubino, 2007; Bolla Pittaluga et al., 2015; Redolfi et al., 2019). These effects can be parsimoniously modeled following the approach of Bolla Pittaluga et al. (2003), who proposed to take into account the flow and sediment redistribution immediately upstream the bifurcation node by introducing a twocell model, as sketched in Figure 2a. The length of the two cells, measured by the empirical parameter α, physically represents the upstream extension of the effect of the bifurcation.
2.1 The Role of the Downstream Boundary Condition
3 Formulation of the Problem
In this analysis, we refer to the bifurcationconfluence loop illustrated in Figure 3. We assume that the banks are fixed, so that the planform geometry does not change in time. We formulate the model by considering a generic configuration, where the channel widths (W_{a}, W_{b}, W_{c}, W_{d}), the confluence angles (δ_{b}, δ_{c}), and the length of the connecting branches (L_{b}, L_{c}) can be freely chosen. For modeling the response of the bifurcation, we rely on the abovedescribed model of Bolla Pittaluga et al. (2003). Therefore, we do not need to add new ingredients with respect to the existing literature. Conversely, for the confluence node, we need to formulate a new model, which is described in the following subsection.
3.1 A Model for Mobile Bed Confluences
In this section, we formulate a onedimensional model for mobile bed channel confluences, which allows for determining how the water surface elevation adjusts depending on the incoming water and sediment fluxes. The model is based on the momentum balance in the streamwise direction on two distinct control volumes, as originally proposed by Shabayek et al. (2002) for a fixed bed channel confluence. We extend this model to mobile bed confluences by including the effect of the pressure forces acting on the bed, as needed to close the momentum balance when the bed is not flat. To this aim, we also provide a rigorous demonstration of how the momentum equation can be applied to a curved control volume (see Appendix A).
Figure 4 illustrates the confluence geometry adopted in the present formulation, with two channels b and c with rectangular cross section, having width W_{b} and W_{c} and slope S_{b} and S_{c}, respectively, merging in a downstream channel d of width W_{d} and slope S_{d}, and forming angles δ_{b} and δ_{c} with the downstream direction. Two control volumes CV_{b} and CV_{c} are defined in proximity of the node, whose downstream widths depend on the discharge asymmetry ΔQ.
The sharp deviation of the flow at the confluence often induces the formation of a separation zone. This area, characterized by lower pressure, reduced velocity, and flow recirculation, represents a sort of “sediment trap,” possibly leading to the formation of a bar within the postconfluence channel (Best, 1988; Leite Ribeiro et al., 2012). The width and length of the separation zone are found to increase with both the junction angle and the discharge asymmetry of the two incoming flows (Best & Reid, 1984; Hager, 1987).
The solid red line between the two control volumes in Figure 4 schematically represents the shear layer that develops along the interface between the two colliding streams, which is characterized by vortex generation and strong threedimensionality of the flow structure (Best, 1986; Bradbrook et al., 2000; Rhoads & Sukhodolov, 2004). Another fundamental component of the flow structure at confluences is the presence of a doublehelical circulation, characterized by two rotating cells that are converging at the stream surface and diverging near the bed (Biron & Lane, 2008). These cells drive the sediment toward the banks, forming a central scour whose depth and shape mainly depend on the discharge asymmetry and the junction angle (Ashmore & Parker, 1983; Best, 1986; Mosley, 1976). The mechanism that leads to the formation of the helicoidal cells has been a widely discussed topic (e.g., Lane et al., 2000). Ashmore et al. (1992) attributed this phenomenon to the curvature of flow trajectories at the junction, which causes a secondary flow circulation due the imbalance between the centrifugal force and pressure gradient, while Paola (1997) suggested that the formation of helicoidal cells originates from the opposite components of momentum between the two colliding streams. Recently Sukhodolov and Sukhodolova (2019) verified the validity of the analogy to curved flows in open channels in a fieldbased study in the absence of a central scour.
4 Results
In equilibrium conditions, both water and sediment fluxes in channel d coincide with those in channel a. For the simple, equalwidth configuration examined, this implies that all the hydraulic parameters, including water depth, aspect ratio, Shields and Froude numbers are also equal. Therefore, in the following, we will employ the suffix _{0} to denote generically the flow and sediment parameters in both upstream and downstream channels, when used as reference values in the analysis of model results.
This section is organized in two parts. We first apply our model to analyze the upstream effect of the confluence for varying flow conditions (Section 4.1). Then, we fully exploit the mutual interplay between bifurcation and confluence nodes and we analyze the equilibrium configurations of a channel loop (Section 4.2).
4.1 The Upstream Effect of a Confluence with Erodible Bed
The flow conditions at equilibrium, both in the main channel and in the tributary branches, are uniquely determined by means of the sediment and water continuity equations as a function of the dimensionless parameters β_{0}, θ_{0}, Fr_{0}, and of the water and sediment discharge asymmetry (ΔQ and ΔQs, respectively). However, the parameters ΔQ and ΔQs are usually covarying (i.e., the branch carrying more water also transports more sediment), and therefore we find more informative to adopt ΔQ and the slope asymmetry ΔS as independent parameters. This choice is fully equivalent, as ΔS and ΔQs are clearly related.
In Figure 6a, the equilibrium solution for Δη is shown as a function of the discharge asymmetry ΔQ, for increasing values of downstream Froude number Fr_{0}. When ΔQ = 0 (i.e., equal discharge in channel b and c), the hydrodynamic conditions in the two branches are the same, and the confluence does not introduce any asymmetry in the water surface and bed elevation. The shear forces coincide for both control volumes, whereas F^{I} is equal to zero (because U_{b} = U_{c}). In this condition, the confluence simply produces a difference between the upstream and the downstream bed elevation, which is induced by the energy dissipation at the separation zones (see Equation 28). When the incoming discharge partition is unequal, aggradation characterizes the branch that carries less discharge, leading to a higher value of the bed pressure force , which is essential to maintain the momentum balance within the two control volumes. The resulting differences in the bed elevation between the two upstream branches are weakly affected by the Froude number, and gradually increase with the discharge asymmetry up to values comparable with the downstream water depth D_{0}.
In general, the resulting differences in water surface elevation are rather modest, being on the order of onetenth of the water depth. For this reason, since the pioneering work of Taylor (1944) it is typically assumed that the water surface elevations of the two converging channels are equal (e.g., Gurram et al., 1997; Hsu et al., 1998). For most applications, this assumption can be fully accepted, as highlighted by several experimental and numerical studies (e.g., Coelho, 2015). However, as pointed out in Section 2.1, the sensitivity of bifurcations to variations of downstream conditions suggests that even narrow differences of the water surface elevation at the confluence node are likely to affect significantly the dynamic behavior of the whole system.
Results reported in Figure 6 are obtained by considering the Parker (1990) transport formula, and given values of the Shields parameter θ_{0} = 0.08, aspect ratio β_{0} = 20 and junction angle 2δ = 40°. However, it is worth observing that model results are fully independent of the channel aspect ratio β_{0}, and negligibly affected by the Shields number θ_{0} and the choice of the sediment transport predictor. In particular, when the Engelund and Hansen (1967) transport formula is employed, results are exactly independent of the Shields parameter θ_{0}. These observations highlight the primary importance of the Froude number Fr_{0} in determining the dynamic response of the confluence.
Equation 26 suggests that the results can be conveniently analyzed in terms of the scaled water level asymmetry, . In Figure 7, we perform a sensitivity analysis due to understand the importance of the two empirical parameters K_{I} and K_{b,c}. Due to the symmetry properties highlighted earlier, we report only half of the plot. Figure 7a reveals that the scaled water level asymmetry increases with K_{b,c}. This can be explained by considering that the dissipative shear force at the separation zone, as given by Equation 18a, increases more at the side where discharge (and velocity) are higher, thus requiring a higher pressure force toward the dominant channel. Noteworthy, the water level asymmetry ΔH positively correlates with the discharge asymmetry ΔQ also when the coefficients K_{b,c} vanish. According to Equation 20, this occurs when the junction angle is fairly small (δ_{b,c} ≤ 14°). However, this situation is probably more common in natural confluences, in which the separation zone may be often absent (Ashmore et al., 1992; Biron et al., 1993).
Similarly, in Figure 7b, we explore the role of different values of the interfacial shear coefficient, from the extreme case K_{I} = 0 (hence, neglecting the effect of the interfacial shear term in the momentum balance) to the more realistic values of 0.2–0.3 reported by Shabayek et al. (2002) and Luo et al. (2018) (see supporting information). This analysis reveals the fundamental role of the parameter K_{I}, as a value of at least 0.1 is needed to obtain the same qualitative behavior depicted in Figure 6, where the confluence tilts toward the dominant channel (i.e., that carrying more water and sediment).
The above findings are supported by the results of the recent field experiments of Sukhodolov and Sukhodolova (2019), who measured discharge, water depth, velocity, and free surface elevation, in a 10 m wide symmetric confluence with fixed, artificial banks but erodible bed, for three different confluence angles at different values of discharge asymmetry. We calculated ΔH on the basis of the contour maps reported in their Figure 6, by estimating the average water surface elevation in the two branches at the entrance of the confluence (i.e., at a distance of about 0.7–0.9 W_{0} with respect to the confluence node). Results reported in Figure 8 highlight four remarkable characteristics of the confluence response to varying conditions: (i) the water surface level is always higher in the branch that carries most of the flow; (ii) the scaled water surface asymmetry increases with the discharge asymmetry, by following a roughly linear trend; (iii) the ratio tends to increase with the confluence angle, although no significant differences can be appreciated when comparing cases with δ_{b,c} = 0° and δ_{b,c} = 20°; (iv) values of the scaled water level asymmetry corresponding to ΔQ ≃ 0.8 are in the order of 0.6 for the two lower angles, and about 1.3 for δ_{b,c} = 35°. The above results are not directly comparable with those reported in Figures 6 and 7, because the asymmetry of the energy slope in the experiments was not vanishing. Nevertheless, the experimental outcomes suggest that our approach based on a global momentum balance is able to capture the essential response of the confluence asymmetry to varying flow and geometrical conditions.
4.2 Equilibrium Configurations of the BifurcationConfluence Loop
On the basis of river bifurcation theory described in Section 2, and the confluence model formulated in Section 3, we now connect the two systems, exploring the morphodynamics of the bifurcationconfluence loop sketched in Figure 3. This is achieved by coupling the nodal point relation at the bifurcation (1) with the momentum balance at the confluence (7) by means of Equation 5, which relates the (downstream) water level differences with the (upstream) slope asymmetry at equilibrium. Differently from Section 4.1, we determine the reference flow conditions by setting the dimensionless parameters β_{0}, θ_{0}, and c_{0}. The rational motivation behind this choice will be more clear after Section 5, keeping in mind the link between c_{0} and Froude number Fr_{0} expressed by Equation 24.
An example of the equilibrium diagram of a bifurcationconfluence loop is shown in Figure 9, where the equilibrium values of discharge asymmetry ΔQ are reported as a function of the aspect ratio β_{0} and for different values of the scaled length L*, defined as the ratio between the length of the bifurcates (L) and the reference water depth (D_{0}). When the connecting branches are infinitely long (L* → ∞), the presence of the confluence is not felt by the bifurcation, and the model gives exactly the same solution of the sole bifurcation problem as obtained by Bolla Pittaluga et al. (2003), which is reported in Figure 2. Conversely, gradually decreasing the anabranches length, the critical value of the aspect ratio at which the balanced solution (ΔQ = 0) becomes unstable gets larger. This behavior is associated with the negative feedback exerted by the confluence on the upstream bifurcation, which increases the stability of the system. When the aspect ratio β_{0} exceeds the critical value β_{C}, the geometric discontinuity that originates at the inlet of channels b and c steers a greater fraction of water and sediment fluxes to one of the two branches, promoting the development of an unbalanced configuration. In this case, the downstream confluence responds by increasing the water level of the dominant branch (Figure 6), therefore reducing its slope. This effect tends to reequilibrate the bifurcation, so that the system has the possibility to maintain the balanced state for higher values of the aspect ratio β_{0} with respect to the case of the sole bifurcation.
5 Linear Analysis of the Coupled System
To determine the marginal stability conditions and the associated critical parameters, we perform a linear analysis of the mathematical problem. When the system is in marginal stability conditions, small initial perturbations with respect to the equilibrium configuration neither grow nor decay in time. Therefore, in these conditions, a nontrivial steady solution of the linearized system exists. With reference to the diagrams of Figures 2 and 9, this occurs when the equilibrium solution shows a pitchfork bifurcation (i.e., when the channel aspect ratio equals the critical value β_{C}).
The first two equations of system (31) represent the flow and sediment rating curves. The third equation is the nodal relation proposed by Bolla Pittaluga et al. (2003). The fourth equation is the linearized momentum balance applied to the control volume CV_{b} at the confluence, as formulated in Equation 7a. The last equation defines the relationship between the channel slope and the downstream water levels, which is directly obtained from Equation 5.
6 Discussion
This work represents the first attempt to include within a unified theoretical framework the morphodynamic response of river bifurcations and confluences. This has been done by considering the state of the art of theoretical models for both bifurcations and confluences, which are coupled together to analyze the global dynamics of the system. Specifically, we have followed a quasi2D approach, where the complex, highly threedimensional flow and sediment transport pattern at the nodes are modeled by considering couples of cells (or control volumes) that can exchange mass and momentum fluxes. A key novelty of this work is represented by our confluence model. Starting from the approach proposed by Shabayek et al. (2002), we have provided a sound formulation of the streamwise momentum balance on a curved control volume (see Appendix A), and introduced an additional term representing the pressure force acting on the sloping bottom, which results from the morphological adjustments of the mobile bed.
6.1 The Stabilizing Effect of the Confluence
The present work indicates that water level variations at a fluvial confluence affect the stability and the evolution of an upstream bifurcation. In particular, when the distance between the two nodes is sufficiently short, the effect of the confluence contributes to maintain a stable balanced configuration, where the flow is equally partitioned in the downstream branches. This effect is clearly related to the fact that the confluence tends to increase the water elevation of the channel that is carrying the higher discharge fraction. This causes a decrease of the slope of the dominating branch, which acts as a negative feedback that tends to reestablish a balanced distribution of water and sediment fluxes.
6.2 Key Controls on the Dynamic of the System
Equation 42 suggests that the correct scale for the channel length is given by the product . This represents an important result of our analysis, because it marks a clear difference with respect to the classic scaling based on the backwater length, defined as the ratio between the water depth D_{0} and channel slope S_{0} (e.g., Paola & Mohrig, 1996), which represents the typical length scale of backwater effects in fluvial and deltaic systems. For example, the backwater length determines the extent of the propagation of tidal currents into rivers (e.g., Ragno et al., 2020; Seminara et al., 2012), the length of upstream flow profiles generated by instream structures (e.g., Samuels, 1989), and it has been employed for studying deltaic avulsions (e.g., Chadwick et al., 2019; Moodie et al., 2019). In these cases, the effect of the channel slope is clear, as a given elevation difference (say ΔH = 0.1 m) can be felt nearly 10 times further upstream when S reduces, e.g., from 1% to 0.1%.
A similar effect also occurs in bifurcationconfluences loops, in which, when slope decreases, the bifurcation is potentially more sensitive to variations of the downstream boundary conditions. However, the variations of the water surface elevation produced by the confluence, being proportional to , also reduce for lower slopes, and these two effects tend to compensate. For the same reason, but from an opposite point of view, when Froude number is small, the water surface asymmetry at the confluence node is nearly vanishing. In these conditions, it seems reasonable to assume that the water surface level is the same in the two branches (e.g., Kleinhans et al., 2012; Schaffranek et al., 1981). Nevertheless, the low channel slope associated with the small value of the Froude number makes the bifurcation more sensitive to downstream effects, so that even small variations of the confluence level are sufficient to cause a significant effect on the water and sediment distribution. The independence of the loop solution from the Froude number also implies that, once the Chézy coefficient c_{0} and the Shields number θ_{0} are fixed, calculation of the grain size from Equation 23 is not needed. This suggests that the above results are also valid in the case of dunecovered channels, provided suitable sediment transport relations are chosen.
The stability diagram of the bifurcationconfluence loop becomes more clear and significant when represented in terms of the two key parameters (39), as reported in Figure 11, for several reasons. First, it does not show singularities: the asymptotic behavior appearing in Figures 10b and 10c does indeed depend on the fact that the parameters β_{0} and L* appear at the denominator of Equation 39, so that conditions where the marginal stability curve approaches Λ = 0 (i.e., dominating “bifurcation effect”) or (i.e., dominating “confluence effect”) are attained for values of the critical parameters and β_{C} that tend to infinity. Second, it is independent of the empirical parameters r and α, as their effect is fully incorporated in the definition of the key controlling parameters (39). Third, it includes part of the effect of the Shields number and the Chézy coefficient. Specifically, the residual effect of c_{0} is very weak, to the point that it exactly vanishes when adopting a friction formula having the form of a powerlaw (e.g., the Manning formula). Similarly, there is no residual effect of the Shields number when considering sand bed cases, where the transport rate is given by the powerlaw transport formula of Engelund and Hansen (1967), while a rather important effect remains in gravel bed cases. This is clear from Figure 11a, which shows that the region of instability narrows when increasing the Shields number, consistently with existing theoretical models for gravel bed bifurcations (Bolla Pittaluga et al., 2003; Redolfi et al., 2016).
We note that the present model is based on the commonly adopted assumption that the water level at the bifurcation node is the same in the two branches. This was originally introduced by Bolla Pittaluga et al. (2003) by assuming the conservation of energy across the bifurcation and neglecting the kinetic terms, which is strictly valid only at low Froude numbers. However, the reason behind this assumption is more profound, and therefore, the uniformity of water levels is more generally legitimated in this case, as the presence of a largescale, upstream steady bar generated by the bifurcation (Bertoldi & Tubino, 2007; Redolfi et al., 2016) allows the flow to gradually adapt to the bottom topography without significant deformations of the free surface, even at moderate Froude numbers. Specifically, calculations based on the steady bar model of Zolezzi and Seminara (2001) reveal that differences in the water surface elevation are at least 1 order of magnitude smaller than those reported in Figure 6b, independently of the Froude number.
6.3 Implication for SuspensionDominated Systems
In sand bed channels, the sediment transport is often dominated by the presence of suspended load. As suggested by several authors (e.g., Kleinhans et al., 2008; Redolfi et al., 2019), suspended load is substantially not deflected by the gravitational pull, which makes bifurcations more unstable. Specifically, in the limiting case where the gravitational effect is negligible (i.e., r = 0, ), the sediment flux is distributed proportionally to the water discharge, and the sole bifurcation turns out to be invariably unstable, also consistently with the classic analysis of Wang et al. (1995). Nonetheless, bifurcations in sand bed rivers are often observed to be stable over an extended period of time. This contradiction may be at least partially explained by considering that the presence of the confluence induces a distinct stabilizing effect, fully independent on the mechanism of gravitational pull. If the connecting branches are sufficiently short (i.e., Λ ≃ 1.6), this effect is sufficient to stabilize the bifurcationconfluence loop, even in the limiting case r = 0.
6.4 Relevance in Fluvial Systems
From the results reported in Figure 11, we can derive an estimate of the magnitude of the channel length L for which the presence of a downstream confluence plays an in important role in determining the stability of the bifurcation. Specifically, we can see that values of Λ of order one are needed to produce a significant interaction between the effect of the bifurcation and that of the confluence. Since c_{0} ∼ 10, the corresponding values of L* are of order 100, which, assuming W_{0}/D_{0} ∼ 10, gives lengthtowidth ratios L/W_{0} of order 10. Similar values are often observed in natural settings, as reported, e.g., by Nanson and Knighton (1996), who measured the ratio between island length and main channel width in different types of multithread rivers (see their Figure 2). Furthermore, it is worth noting that comparable values of L/W_{0} result from the empirical expressions proposed by Hundey and Ashmore (2009) for the link length (i.e., the mean length of subsequent confluencebifurcation units, see Ashmore, 2001) of braided systems.
To test more directly the possible effect of the presence of a downstream confluence in natural bifurcations, we take a few examples from the literature, where data for different channel loops in gravel bed streams are reported. Specifically, we consider four natural cases in the wandering Renous River (Canada), studied in detail by Burge (2006), and the case of an artificial, sidechannel reconstruction along the Drau River (Austria, see Figure 1c), which has been investigated, among the others, by Formann et al. (2007). For a specific estimation of the parameter α, we adopt the method of Redolfi et al. (2019), which is based on the assumption that should be equal to the resonant threshold value β_{R}. Results of the linear analysis, reported in Table 1, reveal that the critical aspect ratio β_{C} is nearly doubled with respect to the value given by the formulation of Bolla Pittaluga et al. (2003). This highlights the important role played by the downstream confluence in determining the stability of the system and controlling its longterm configuration.
River  Case  β_{0}  θ_{0}  c_{0}  Fr_{0}  L/D_{0}  α  Λ  β_{C}  

Renous  Bif. 2  17.0  0.035  11.3  0.58  167  7.3  1.14  0.41  5.5  12.1 
Bif. 3  21.5  0.051  10.7  0.55  297  6.6  0.68  0.37  5.5  10.9  
Bif. 4  9.5  0.040  12.8  0.65  86  8.4  2.21  1.30  6.1  14.1  
Bif. 6  16.5  0.034  11.4  0.58  134  7.2  1.18  0.57  5.6  12.9  
Drau  27.4  0.210  16.4  0.63  263  3.6  0.15  1.03  22.3  61.4 
 The critical aspect ratio β_{C} turns out to be significantly higher than the corresponding value calculated considering for the sole bifurcation (Bolla Pittaluga et al., 2003), which highlights the significant effect of the confluence on the stability of the system.
6.5 Limitations and Potential for Further Studies
For the sake of simplicity, we limited our analysis to a symmetrical case, where the connecting channels have the same width and length, and reconnect at the confluence with the same angle. Moreover, we assumed that the width of the main upstream and downstream branch is the same, and equal to twice the width of the two secondary channels. However, these hypotheses can be easily relaxed, as the model is formulated in general terms, with values of angles, lengths and widths that can be freely assigned.
The steady assumption could also be relaxed by considering an evolving system, where the sediment flux is not constant along the channels. A first approximation could be obtained by assuming that the channels evolve as a sequence of quasiequilibrium states (e.g., Bolla Pittaluga et al., 2003), while a more general approach would require a description of the channels through a onedimensional differential problem (see the cell discretization of Salter et al. (2018)). Studying the unsteady dynamics of the bifurcationconfluence loop would allow for exploring the initial transitory phase, the response to discharge variations, and the interaction with timedependent forcing effects due to the migration of bars (e.g., Bertoldi et al., 2009b) or downstream tidal fluctuations. Moreover, the unsteady analysis would allow for detecting the presence of possible autogenic oscillations, similar to those revealed by Salter et al. (2018, 2019) for bifurcations in depositional river deltas.
The present model is based on the assumption of fixed banks. If this can be considered valid for the case of artificial structures (see Figure 1d), this is never exactly the case for natural bifurcations. However, we expect that the approach is more generally applicable to cases where bank erosion processes (and the associated time scale of planform evolution) are comparatively slow with respect to the bed adaptation (see Monegaglia & Tubino, 2019). This condition is typically fulfilled in anastomosing rivers, where the individual channels, often separated by vegetated islands, are usually persisting for decades or centuries (Nanson & Knighton, 1996). Conversely, more caution needs to be taken when applying our approach to braided channels, where the lateral channel and confluence migration can be relatively fast (see Ashmore, 2013; Dixon et al., 2018; Sambrook Smith et al., 2019).
Lastly, the empirical expressions for the interfacial shear and separation coefficients proposed by Shabayek et al. (2002) and Luo et al. (2018) (see Equation 20) are based on experimental data taken in conditions far from those observed in natural confluences. Our comparison with the field experiment of Sukhodolov and Sukhodolova (2019) suggests that this laboratorybased coefficients are also adequate to represent (at least qualitatively) the confluence dynamics in much more realistic conditions. However, there is a demand for a more specific calibration of the coefficients K_{b,c} and K_{I}, depending on confluence geometry, bed morphology, and flow characteristics. For example, in cases where the junction corners are not sharp but characterized by a more gentle curvature, as typical of natural settings (Ashmore et al., 1992), the importance of the separation zone is likely to reduce, and therefore the appropriate value of the coefficients K_{b,c} is probably lower. Similarly, when the bed is erodible, the interfacial shear coefficient K_{I}, physically representing the lateral exchange of momentum along the shear layer, may significantly vary with respect to the value 0.21 adopted in the present formulation. However, from a qualitatively point of view, a value of K_{I} such that ΔH > 0 (see Figure 7b) is highly expectable, in as much as the interaction between the bed morphology and the flow field that defines the shear layer structure is likely to even reinforce the lateral momentum exchange and increase the shear stress at the interface.
7 Conclusions
In this work, we propose a novel theoretical modeling framework for investigating the effect of a downstream confluence on the stability of a river bifurcation. On the basis of model results and their interpretation, we can draw the following conclusions:

The proposed confluence model allows for predicting differences in the water surface elevation between the two merging channels, depending on geometrical and hydrodynamical conditions. The model reveals that the dominating branch (i.e., that carrying more water and sediment) is subject to a slightly higher water surface elevation, showing a water level asymmetry that is proportional to the square of the Froude number

The increase in the water surface elevation at the confluence node tends to reduce the slope of the dominating branch, which therefore becomes “less dominant.” This negative feedback is responsible for a clear stabilizing effect on the bifurcationconfluence loop

The linear analysis of the coupled system allows for identifying the key controlling parameters of the stability of a bifurcationconfluence loop. The stabilizing effect of the confluence turns out to depend on the ratio between the length of the connecting channels and the average water depth, while it is almost independent of channel slope and Froude number, which clearly shows that the backwater length is not the correct scaling length for this kind of systems

The effect of the confluence is potentially able to stabilize the bifurcationconfluence loop even in conditions where the classic mechanism described by Bolla Pittaluga et al. (2003) (i.e., the topographical effect related to the gravitational pull on the sediment transport) is very weak, as in the case when most of the sediment is transported in suspension

Typical lengths of connecting branches observed in natural rivers suggest that the effect of the confluence is usually important in determining the stability of natural bifurcationconfluence loops. Noteworthy, this effect is equally relevant in lowgradient channels, despite the fact that variations of the water surface elevation, being proportional to the square of the Froude number, may appear as negligible
In our analysis, we restricted the attention to the simplest geometrical configuration. However, our model is also suitable for examining the effect of different channel lengths, widths, and confluence angles. Moreover, the proposed model could allow for investigating the stability of a freeforced system, where the presence of “external” disturbances (e.g., downstreammigrating bars in the upstream channel) may alter the stability of the loop, possibly leading to multiple, counterintuitive solutions (Redolfi et al., 2019). Ultimately, this work provides a relatively simple, theoretically based, modeling framework that can be used to design and interpret results from physical experiments and numerical simulations.
Acknowledgments
This work has been supported by the Italian Ministry of Education, University and Research (MIUR) in the frame of the “Departments of Excellence” grant L. 232/2016 and by the “Agenzia Provinciale per le Risorse Idriche e l'Energia” (APRIE) of the Province of Trento (Italy). The work has benefited from the thoughtful comments by Maarten Kleinhans and Gerard Salter.
Appendix A: Streamwise Momentum Equation on a Curved Control Volume
The nodal point relation we propose, as well as the original formulation of Shabayek et al. (2002), are based on the momentum balance on the curved control volumes illustrated in Figure 5. The application of such a balance is not straightforward, due to the fact that forces and momentum fluxes are acting in different directions. Therefore, we need to formalize the procedure adopted to derive an integral momentum equation for an arbitrary, curved control volume. For the sake of notational compactness, we here neglect the effects of the bottom friction, as justified when the domain length is relatively short.
As illustrated in Figure A1, we consider a generic open channel, and we define a volume whose lateral boundaries follow the flow streamlines (or the channel banks). For each point in the domain, it is possible to define cross sections so that the associated crosssectionally averaged momentum vector () is always orthogonal to the sections themselves. This allows us to draw a curvilinear axis that is everywhere aligned with the momentum vector, and to define the coordinate s ∈ [0, l] as the distance along the axis itself.
Open Research
Data Availability Statement
A reprinted version of the Figure 2 of Shabayek et al. (2002), where experimental data used for the calibration of the empirical coefficients K_{b,c} and K_{I} are illustrated, is reported in supporting information. A Matlab code for computing the marginal stability conditions for the bifurcationconfluence loop is made available in a public repository at https://doi.org/10.5281/zenodo.3939277.