Channel bifurcation in braided rivers: Equilibrium configurations and stability

Authors

Errata

This article is corrected by:

  1. Errata: Correction to “Channel bifurcation in braided rivers: Equilibrium configurations and stability” Volume 39, Issue 11, Article first published online: 25 November 2003

Abstract

[1] We investigate the equilibrium configurations and the stability of river bifurcations in gravel braided networks. Within the context of a one-dimensional approach, the nodal point conditions play a crucial rule, as pointed out by Wang et al. [1995] who propose an empirical relationship relating water and sediment flow rates into the downstream branches. In the present paper, an alternative formulation of nodal point conditions is proposed based on a quasi two-dimensional approach. The results show that, if the Shields parameter of the upstream channel is large enough, the system only admits of one solution with both branches open, which is invariably stable. As the Shields parameter of the upstream channel decreases, two further stable solutions appear characterized by a different partition of water discharge into the downstream branches: in this case, the previous solution becomes unstable. Theoretical findings are confirmed by the numerical solution of the nonlinear one-dimensional equations.

1. Introduction

[2] Bifurcations are unit processes of fundamental importance for the behavior of braided river networks; in particular, weak variations of the geometry of a bifurcation may strongly affect water and sediment partition into the downstream branches, thus crucially controlling the development of braiding networks. The sensitive dependence of braided rivers on conditions at bifurcations has been highlighted in many contributions [e.g., Hoey, 1992]. In spite of their importance bifurcations have not been as widely studied as confluences, their counterpart processes [Mosley, 1976; Ashmore and Parker, 1983] and few experimental and field data are available in literature [Dancy, 1947; Federici, 1999].

[3] Bifurcations are common features of different fluvial systems, like braiding and anabranching river networks and river deltas. The morphology and development of a bifurcation is mainly governed by the dominant sediment transport mechanism, which determines the partitioning of sediment discharge into the two downstream branches. If suspended load is dominant, whatever the bed topography at the bifurcation, the sediment discharge rates into the downstream channels are closely related to the partition of water discharge. On the other hand, if bed load is dominant, the effect of local bed slope may play a crucial role in governing sediment partition, in particular at low values of Shields stress which typically characterize gravel braided rivers.

[4] In the present contribution we study the equilibrium configurations and the stability of gravel bed river bifurcations through a one-dimensional approach. The formulation of the problem is given in section 2. Though one-dimensional models do not allow a detailed description of flow field and bed topography, they are widely used in river engineering for long-term predictions of river morphological development. They also represent a useful tool for the investigation of complex systems, like braided networks, whose planimetric structure is extremely complex [Howard et al., 1970; Sapozhnikov and Foufoula-Georgiou, 1996] and whose dynamics are highly unstable and characterized by many recurring processes [Ashmore, 1982, 1991]. In fact, several conceptual models have been proposed to describe the dynamics of braided rivers, which are able to reproduce the gross features of the network. They are based on simplified approaches whereby the network is schematized in terms of cells or channel units, which are subject to exchanges and dynamical rules based on experimental observations and theoretical models [Murray and Paola, 1994; Satofuka, 2001]. Numerical approaches have also been attempted in the context of two-dimensional shallow water models [Enggrob and Tjerry, 1999; McArdell and Faeh, 2001; Kurabayashi and Shimizu, 2001], however a fully mechanical description of braiding networks is still not available.

[5] In one-dimensional network confluences and bifurcations constitute nodal points where suitable internal conditions must be imposed. However, important differences exist in modeling the two processes as pointed out by Wang et al. [1995]: in the former case water and sediment discharges are known in both of the upstream branches, on the contrary a further relationship is required in the latter case, which governs water and sediment distribution in the two downstream channels.

[6] A first attempt to describe river bifurcations with movable bed through a one-dimensional model is due to Wang et al. [1995] who devoted particular attention to the case of river deltas; a brief overview of their work is reported in section 3. A further contribution, in the case of dominant suspended load, is due to Slingerland and Smith [1998] who investigated the conditions leading to avulsion of a meandering river.

[7] Wang et al. [1995] introduce an empirical nodal point condition which relates water and sediment discharges into the downstream branches. Their analysis suggests that such a condition plays a crucial role on the long-term evolution of the system. However, the nodal point condition depends on an empirical parameter which may vary significantly according to the estimates provided by the Authors; hence, the above approach can hardly be applied to predict the evolution of a real bifurcation.

[8] In order to overcome this difficulty an alternative formulation for the nodal point conditions, based on a quasi two-dimensional approach, is proposed in section 4 of the present paper. The equilibrium configurations of a simple channel loop are theoretically investigated in section 5. Section 6 is devoted to the linear stability analysis of the above solutions. Moreover theoretical results are compared with the full numerical solution of one-dimensional equations in section 7. Finally, section 8 is devoted to some concluding remarks.

2. Formulation of the Problem

[9] The one-dimensional model is formulated in terms of the dependent variables D, H, and Q, namely water depth, free surface level and flow discharge, respectively. In the following we assume a wide rectangular cross section with fixed banks. The above hypothesis may turn out to be not completely adequate when applied to self-formed channels where channel width may adjust to water discharge. The flow equations are written in the form

equation image
equation image

where b is channel width, Ω is the cross-sectional area, g the gravitational acceleration and j = τ/(ρgD), with τ bottom shear stress and ρ water density. Finally t is time and x the longitudinal coordinate. Equations (1a) and (1b) are coupled with the sediment continuity equation which reads

equation image

with q sediment discharge per unit width, p sediment porosity and η = HD bottom elevation. In order to complete the mathematical formulation of the problem, closure relationships for j and q are introduced as follows

equation image

where Φ is the dimensionless sediment transport formula, expressed in terms of the local value of Shields parameter

equation image

and of its threshold value for sediment mobilization ϑcr. Furthermore, ds and ρs represent the mean diameter and density of the sediment and C is the Chezy coefficient. In the following we evaluate the bed load function Φ through the formula proposed by Meyer-Peter and Müller [1948] which reads

equation image

while the friction coefficient C is evaluated in terms of the local depth through a standard logarithmic law.

[10] The system (1a), (1b), and (2) can be readily written in the form

equation image
equation image
equation image

where

equation image

The system (6a)(6c) is hyperbolic and admits of three characteristic curves: two of them describe downstream propagation of information (positive celerity) while the third one describes upstream propagation (negative celerity). Hence, in order to determine the solution in each branch it is necessary to impose, in addition to the initial condition, two boundary conditions at the channel inlet and one at the channel outlet. As a consequence five relationships are required at the bifurcation both in the case of subcritical and supercritical flow, namely one for the outlet of the upstream channel and two for the inlet of each of the downstream branches.

3.

[11] In this section we give a brief overview of the work of Wang et al. [1995]. The Authors consider the simple geometry sketched in Figure 1: channel a bifurcates into two symmetrical branches (b and c) which flow into a lake, which imposes a constant water level. All branches have constant width and bed slope is constant throughout the network. At the inlet section of channel a water and sediment discharges are prescribed. Wang et al. [1995] propose the following nodal point conditions at the bifurcation:

  1. water discharge balance
    equation image
  2. sediment discharge balance
    equation image
  3. , 4. constancy of water level
    equation image
    equation image
    where subscripts denote the branch as shown in Figure 1.
  4. In addition, the Authors introduce a nodal point relationship governing water and sediment partitioning in channels b and c in the form
    equation image
    The reader should note that relationship (11) is empirical and requires the knowledge of the exponent k. The empirical estimates provided by the authors, which are based on measurements in a real bifurcation and in a laboratory flume, are largely different, being equal to 2.2 and 6, respectively.
Figure 1.

Sketch of the geometry considered by Wang et al. [1995].

[12] The first problem tackled by Wang et al. [1995] is that of finding the equilibrium configurations of the system. Their analysis shows that the following equilibria are possible:

  1. both branches are open;
  2. , 3. one of the two branches is closed and all water and sediment flow into the other branch.

Notice that in the case of a symmetrical bifurcation (constant slope throughout the network and bb = bc) the solution 1 implies the same flow conditions in the two branches. Wang et al. [1995] then perform a linear stability analysis of the configuration 1 under the hypothesis of uniform flow in the whole network. Water and sediment discharges are evaluated through the following relationships:

equation image
equation image

where Si represents the slope of channel i (b or c) and M is a constant.

[13] The configuration with both branches open is found to be stable provided k > n/3, with k the exponent of the nodal point relationship (11) and n the exponent appearing in the sediment transport formula (12b). On the other hand, for values of k < n/3, the bifurcation is unstable; this implies that the system should develop toward one of the other equilibrium configurations. It is noted that Wang et al. [1995] use the sediment transport formula proposed by Engelund and Hansen [1967]; hence in their work n is equal to 5. With a different bed load formula the threshold value of k may be strongly dependent on Shields parameter ϑ, in particular at relatively small values of ϑ as typically occurs in braided rivers.

4. A Physically Based Nodal Point Relationship

[14] The approach by Wang et al. [1995] can hardly be applied to predict the evolution of a real bifurcation: the parameter k, which is found to govern the system development, is unknown and it is neither related to the local hydraulic conditions at the nodal point nor to the geometry of the bifurcation.

[15] To overcome the above difficulties an alternative formulation for the nodal point conditions is proposed herein. At first we note that, in the context of a two-dimensional model, specific nodal point conditions for water and sediment discharges division would no longer be required: referring to the sketch reported in Figure 2a, whatever the computational mesh, the solution is obtained solving the flow equations and sediment continuity equation in each cell with suitable boundary conditions at the channel banks. Also notice that two-dimensional effects are mainly felt close to the bifurcation where strong deviations from the one-dimensional flow configuration and significant transverse bottom gradients are likely to occur. This suggests that a simplified approach can be adopted whereby a quasi two-dimensional scheme is introduced close to the bifurcation (Figure 2b). More precisely we assume that the effect of the bifurcation is felt within the final reach of the upstream channel, with length equal to αba, which is considered as virtually divided into two adjacent cells. Sediment continuity is applied separately to each cell. The incoming sediment discharge is assumed to be uniformly distributed along the upstream section of the reach, i. e. channel a feeds each cell (say cell i) with a fraction of the total sediment discharge equal to bi/(bb + bc). Furthermore, the transverse exchange of sediment between the two adjacent cells is taken into account. The solid discharges leaving both cells feed the two downstream branches of the network.

Figure 2.

Scheme of the nodal point relationship.

[16] The longitudinal length αba of the final reach of the upstream channel, where two-dimensional effects are included, should have the same order of magnitude of the upward distance from the nodal point required for the effects of the bifurcation on bed topography to decay. To determine the order of magnitude of the parameter α, experiments have been carried out in a flume, with a length of 18 m and a width of 60 cm, in the Laboratory of the Department of Environmental Engineering of Genova University. The downstream part of the flume was divided into two parts of equal width, through a longitudinal Plexiglas panel, thus simulating a bifurcation. In Figure 3 the measured bed topography close to the bifurcation is reported: the local amplitude A1 of the leading transverse mode of the Fourier representation of bed elevation, scaled with its amplitude at the bifurcation, is plotted versus the upward distance from the nodal point s. It appears that the upstream influence of the bifurcation almost vanishes at a distance of few channel widths, say 2–3 times ba, which suggests that α is an order 1 parameter. Notice that a further contribution to transverse bed deformation in the upstream channel was also due to the presence of migrating alternate bars. Also notice that only the results of experiments in which a significant difference between bed elevations at the inlets of the two branches was observed are reported in Figure 3.

Figure 3.

The amplitude of the first transverse mode of the Fourier representation of measured bed profile, scaled with its value at the nodal point, is plotted versus the dimensionless upward distance from the nodal point s/ba (ba = 0.6 m, ds = 1.2 mm; run 1: Qa = 15.0 l/s, Sa = 0.002; run 2: Qa = 24.9 l/s, Sa = 0.002; run 3: Qa = 15.0 l/s, Sa = 0.002). The transverse structure of the first Fourier mode is sketched in the lower left-hand part of the figure.

[17] In the present model the transverse exchange of sediments just upstream of the bifurcation is evaluated on the basis of a well established procedure which has been widely adopted to describe two-dimensional bed load transport over an inclined bed [Ikeda et al., 1981]. The bed load vector can be expressed through the relationship

equation image

where x and y represent the longitudinal and transverse coordinates, respectively. In the case of mild bed slope and quasi unidirectional flow cos δ ∼ 1 and the following expression is obtained

equation image

where U and V are longitudinal and transverse velocity components, respectively, and the last term accounts for the effect of gravity related to transverse bed slope ∂η/∂y, which makes particle trajectories deviate with respect to bottom stress direction. The constant r in (14) has been experimentally determined and it ranges between 0.3 and 1 [e.g., Ikeda et al., 1981; Talmon et al., 1995].

[18] According to (13) and (14) the transverse sediment exchange between the two adjacent cells of the final reach of channel a (see Figure 2) is estimated in our model through the bulk relationship

equation image

The transverse velocity V has been evaluated as the ratio Qy/(αbaDabc), where the transverse flow discharge Qy is computed through a mass balance applied to each of the two cells

equation image

and Dabc is the average depth

equation image

Furthermore (U2 + V2)−1/2 is estimated as the bulk velocity of the incoming flow and the transverse bed slope ∂η/∂y is calculated in terms of the difference between bed elevations at the inlet of channels b and c.

[19] In summary, the proposed nodal point conditions at the bifurcation are the following:

  1. water discharge balance (8);
  2. , 3. water level constancy (10a) and (10b);
  3. , 5. Exner equation applied to both cells of the final reach of channel a
equation image
equation image

Notice that the resulting scour or deposition dηi/dt is assigned to the inlet section of the downstream branch i.

[20] The adoption of (18a) and (18b) implies that sediment transport balance (9) at the nodal point is no longer satisfied: therefore a discontinuity may be originated in bed elevation at the inlets of channels b and c. This result seems sensible when applied to braided rivers in which strong variations of bed profile are often encountered close to channel bifurcations.

[21] It is worth noticing that, in spite of its approximate character, the above formulation seems able to reproduce the main effects on flow and sediment transport induced by a bifurcation. However, several important local effects are neglected, which may be associated with the angle of bifurcation, the planimetrical development of the branches, the strong three-dimensionality of the flow and the occurrence of large-scale bar forms in the channel. An example of an experimental run in which alternate bars formed in channel a is given in Figure 4. The partition of flow discharge into the downstream branches displays an oscillating behavior as the result of the local periodic change of transverse bed slope close to the nodal point induced by the migration of alternate bar fronts along the upstream channel.

Figure 4.

The water discharges measured in the two branches of the experimental flume are plotted versus time. Alternate bars were migrating along channel a.

5. A Simple Channel Loop: Equilibrium Configurations

[22] In this section we apply the nodal point relationships proposed above to study the equilibrium configurations of the simple channel loop sketched in Figure 5. At the confluence point the surface level is assumed to be constant; hence, the geometry considered herein is equivalent to that analyzed by Wang et al. [1995] and depicted in Figure 1.

Figure 5.

Sketch of the channel loop.

[23] We assume given values of water and sediment discharges (Qa, baqa) in the upstream channel; furthermore the geometrical characteristics of the network, i. e. channel lengths Lb and Lc and channel widths ba, bb, and bc are fixed. As mentioned in section 2, in the present model the adjustment of channel width to discharge is not considered; as a consequence the width of each branch may be arbitrarily given. For the sake of simplicity, most of the results presented in the following are obtained with bb = bc = ba/2; a different choice will be explicitly specified.

[24] In a single channel, uniform flow over a flat bed is the steady equilibrium solution of the one-dimensional equations (6a)(6c), provided the channel has constant width and it is transporting sediment. Hence, to find the possible equilibrium configurations of the loop we assume uniform flow conditions in channels b and c. Further relationships are given by the nodal point conditions (8), (10a), (10b), (18a), and (18b) where, at the equilibrium, we impose dηi/dt = 0 (i = b, c). Notice that in the steady case considered herein (18a) and (18b) impose the constancy of sediment discharge throughout the nodal point, as expressed by (9). In conclusion, we need to solve a nonlinear system of seven algebraic equations for the unknowns Qb, Qc, Db, Dc, Ha, Hb, and Hc. The solution is found numerically using the Newton–Raphson method.

[25] Results are discussed in terms of the relevant dimensionless parameters of the uniform flow in the upstream channel, for given flow discharge, sediment discharge, channels width and grain size, namely the Shields stress ϑa, the aspect ratio βa, defined as the half-width to depth ratio, and the Chezy coefficient Ca. We note that the above parameters completely determine in dimensionless form the flow in the upstream channel.

[26] At first we note that for all possible geometrical configurations of the loop and flow conditions in channel a, the system admits of at least two trivial solutions in which either channel b or channel c is closed and all water and sediment flow into the other channel.

[27] The equilibrium configurations of the single channel loop sketched in Figure 5 are found to be crucially dependent on the value of Shields stress of the upstream channel ϑa.

[28] For relatively high values of Shields parameter ϑa of channel a the system only admits of one solution with both channels b and c open. In Figure 6 the ratio Qb/Qc between the water discharges in channels b and c, at equilibrium, is plotted versus the ratio between channel lengths Lb/Lc. In the same figure the ratio Db/Dc between the water depths is also reported. Notice that, since the water level at the confluence point is constant, increasing Lb/Lc is equivalent to decreasing the ratio between the channel slopes Sb/Sc. Figure 6 shows that at equilibrium the ratio Qb/Qc (as well as Db/Dc) is equal to 1 when the lengths of channels b and c are equal (Lb = Lc), which implies that the solution is symmetrical: flow conditions are exactly the same in the two branches of the loop. Increasing Lb/Lc, the ratio Qb/Qc (and consequently Db/Dc) progressively decreases, such that the water discharge is lower in channel b where the slope is milder with respect to channel c.

Figure 6.

The ratios Qb/Qc and Db/Dc between equilibrium values of water discharges and water depths in channels b and c are plotted versus the ratio between channel lengths Lb/Lca = 0.3, βa = 8, Ca = 12.5).

[29] At low values of the Shields parameter ϑa, for a given aspect ratio βa two further equilibrium solutions (s1 and s3) appear as shown in Figures 7a and 7b. It is worth noticing that, while the previous solution (s2) is symmetrical when Lb = Lc, these further solutions are characterized by values of water and sediment discharge and flow depth significantly different in the two downstream branches, even in the case of a symmetrical loop (Lb = Lc). In particular, when Lb = Lc the two new solutions s1 and s3 are reciprocal one to the other. Figures 7a and 7b show that the solution s1 is very weakly dependent on the ratio Lb/Lc. On the contrary the solution s2, which is symmetrical if Lb = Lc, is strongly modified by an increase of Lb/Lc. In particular it displays an opposite behavior with respect to the case shown in Figure 6, which refers to large values of Shields stress ϑa: the more Lb/Lc is increased the more water flows into channel b, which is characterized by a milder slope. Finally, in the equilibrium solution s3 the difference between water levels and water discharges in channels b and c reduces for increasing values of the ratio Lb/Lc. The occurrence of higher values of the discharge flowing into the milder channel, which are observed under suitable conditions, is not surprising since the discharge is invariably larger into the branch where scour occurs, which implies a larger cross-sectional area (see Figure 7b).

Figure 7.

The ratios between equilibrium values of water discharges Qb/Qc (a) and water depths Db/Dc (b) in downstream channels are plotted versus the ratio between channel lengths Lb/Lca = 0.1, βa = 8, Ca = 12.5).

[30] If the ratio Lb/Lc between the lengths of channels b and c exceeds a threshold value, defined by the intersection between the solutions s2 and s3, the problem only admits of the solution s1 with both branches open. However, the solutions with either channel b or channel c closed still exist.

[31] Notice that in the case reported in Figures 7a and 7b, for Lb/Lc = 1 both downstream branches are transporting sediment, even in the case of the unbalanced solutions s1 and s3. However, as Lb/Lc increases, the value of Shields parameter ϑb corresponding to the equilibrium solution s1 decreases and channel b gradually approaches the critical conditions for sediment motion (see Figure 7a). Similar equilibrium solutions are found for higher values of the aspect ratio βa of channel a: however, as βa increases the maximum value of the ratio Lb/Lc compatible with sediment motion in channel b decreases, such that for a given value of Shields parameter ϑa a threshold value of βa exists above which branch b is not transporting sediment, according to solution s1. Under such a condition the theoretical procedure is no longer able to find all the possible equilibrium conditions of the system.

[32] The threshold value of Shields stress ϑa, below which the channel loop admits of three solutions with both branches open, increases with the aspect ratio βa of the upstream channel, as shown in Figure 8a for different values of the Chezy coefficient Ca of channel a and for Lb = Lc. This implies that, for a given value of ϑa, a threshold value of βa can also be determined, above which three solutions exist. Similar results are obtained in the case of different widths of the two downstream branches (bbbc) as shown in Figure 8b.

Figure 8.

The threshold value of ϑa above which the channel loop only admits of one solution with both branches open and below which three solutions are found is plotted in the plane (βa, ϑa) for different values of the Chezy coefficient Ca (a) and for different widths in channels b and c (b) (Ca = 12.5).

[33] Note that through (15) the above results depend on the choice of the parameter α, which represents the dimensionless length of the final reach of the upstream channel, where a quasi two-dimensional model for the sediment transport has been adopted. However, for α > 4 the results are no longer dependent on this parameter.

6. Stability of the Equilibrium Solutions

[34] We now study the stability of the equilibrium solutions found in the previous section. Following the work of Wang et al. [1995], we assume uniform flow conditions throughout the network even if the system is not at equilibrium: each channel (b and c) is thus characterized by a single value of water depth, which is eventually a function of time. Under this hypothesis the time evolution of the network is described by a system of two ordinary differential equations, which impose the sediment continuity equation (2) to channels b and c

equation image
equation image

where qj(i) and qj(o) represent sediment discharge per unit width entering (i) and leaving (o) channel j, respectively. Notice that in the present analysis water level variations in time at the nodal point are accounted for; hence, the derivative dηj/dt appears in (19a) and (19b), instead of dDj/dt as it was proposed by Wang et al. [1995]. The sediment discharge leaving each channel is set equal to the transport capacity of the flow (5) while the incoming solid discharge is determined through the nodal point conditions (18a) and (18b). Using (12a) and (5) and the nodal point conditions (8), (10a), (10b), (18a), and (18b), the system (19a) and (19b) can be written in the following form

equation image
equation image

where the time derivative of the water depth in each branch is a function of the water depths in both channels; at equilibrium we have ℱ1= ℱ2 = 0. The eigenvalues of the Jacobian associated to system (20a) and (20b) govern the stability of the solution: if both the eigenvalues are negative the equilibrium configuration is stable; if one of the two eigenvalues is positive the equilibrium is unstable and the system evolves toward another equilibrium solution.

[35] We now consider the stability of the equilibrium solutions of the channel loop sketched in Figure 5, in the case in which channels b and c have the same width and length; hence, we assume bb = bc = ba/2 and Lb = Lc.

[36] In the previous section we have shown that, provided the Shields stress ϑa in the upstream channel is greater than a threshold value, which depends on the aspect ratio βa and on the Chezy coefficient Ca, the loop only admits of one solution with both branches open. This solution is symmetrical, i. e. both branches have the same hydraulic conditions: Db = Dc and Qb = Qc. The linear stability analysis suggests that this solution is invariably stable since both the eigenvalues of the Jacobian are negative. Hence, the loop keeps both branches open in time.

[37] At relatively small values of Shields parameter ϑa two further solutions appear (s1 and s3), which are characterized by greatly different values of water discharge in the two branches of the loop. Under such conditions the symmetrical solution (s2) is linearly unstable (one of the eigenvalues of the Jacobian is positive); hence, the threshold curve reported in Figure 8 can be thought of as the neutral stability curve for the symmetrical equilibrium configuration s2. If the Shields stress ϑa falls above the curve the solution s2 is stable; on the other hand, if ϑa falls below the marginal curve the solution s2 is unstable and the system either develops toward another equilibrium solution (s1 or s3) or it tends to close one of the two branches of the loop.

[38] Furthermore, the theory shows that the unbalanced equilibrium solutions s1 and s3 are invariably stable. However, it is worth noticing that though mathematically stable the solutions s1 and s3 may be physically unstable. In fact, as we discussed in the previous section, at relatively large values of the aspect ratio βa, sediment transport vanishes either in channel b or in channel c at equilibrium (solutions s1 and s3). Under this condition a small perturbation of channel geometry at the bifurcation may easily cause the system to abandon one of the two branches leading to its complete closure. Notice, furthermore, that the solutions with one of the two branches closed are always stable.

[39] The above results do not change qualitatively when a nonsymmetrical loop is considered (LbLc).

[40] The results presented above are summarized in Figures 9a and 9b. In Figure 9a the ratio Qb/Qc at equilibrium is plotted versus the Shields stress ϑa of the upstream channel, for given values of the aspect ratio βa and of Chezy coefficient Ca. Stable and unstable solutions are indicated with continuous and dotted lines, respectively. At relatively high values of ϑa a single solution is found; decreasing the Shields stress of the upstream channel a bifurcation occurs which leads to the generation of two new solutions (s1 and s3). The figure also shows that the unbalance of water discharges into the downstream branches associated to the latter solutions occurs gradually for decreasing values of ϑa, and it grows until the Shields stress in one branch reaches the critical value and sediment transport vanishes. The figure also shows that when only one equilibrium solution exists, such a solution is invariably stable; on the other hand once two further solutions occur they are invariably stable while the previous solution becomes unstable. The bifurcation diagram associated with the mathematical system (20a) and (20b) is also reported in Figure 9 in terms of the aspect ratio βa of the upstream channel.

Figure 9.

Stability diagrams of system (20a) and (20b): the ratio Qb/Qc between water discharges in channels b and c at equilibrium is plotted versus the Shields stress ϑa (a) and the aspect ratio βa (b) of the upstream channel. Continuous lines indicate stable solutions; dotted lines indicate unstable solutions. (a) βa = 8, Ca = 12.5; (b) ϑa = 0.1, Ca = 12.5.

[41] The existence of stable equilibrium solutions characterized by unbalanced values of flow discharge in the downstream branches is mainly related to the effect of transverse sediment transport qy, which is accounted for through the nodal point conditions (18a) and (18b). In fact the solutions s1 and s3 are characterized by different transport capacities in the downstream branches. Equilibrium is achieved when the transverse sediment flux induced by transverse bed slope and by flow exchange at the bifurcation is such to lead each branch to be fed with the required sediment supply. As a consequence qy plays a stabilizing role. In fact if we neglect this contribution and set qy = 0 in (18a) and (18b), the one-dimensional model invariably predicts instability of the symmetrical solution s2, for any value of the Shields stress ϑa of the upstream channel. Furthermore, the unbalanced equilibrium solutions no longer exist.

[42] The linear stability analysis suggests that at low values of the Shields parameter ϑa the symmetrical solution of the simple loop sketched in Figure 5 is unstable. This finding is, at least qualitatively, confirmed by the experimental observations of Federici [1999] who studied the behavior of simple symmetrical bifurcation shaped into a cohesionless bed made of a well sorted sediment with mean diameter equal to 0.5 mm. The geometry of the experimental bifurcations was not fixed, contrary to the assumption made in our theory, except for the position of the inlet and the outlet sections; in particular channel widths were arbitrarily chosen by the system. The bifurcation was defined stable if both branches remained open in time and it was defined unstable when one of the two channels tended to be closed. The experimental results, summarized in Table 1 (ϑ, β, and C represent the Shields stress, the width ratio and the Chezy parameter at the inlet section, respectively) display the same tendency as the theoretical predictions: at high values of the Shields stress in the upstream channel the bifurcation always keeps both branches open while, at low values of the Shields parameter, the bifurcation is unstable.

Table 1. Summary of the Experimental Results of Federici [1999]
ϑβC 
0.069.410.40unstable
0.0730.19.23unstable
0.099.410.40unstable
0.0910.710.0unstable
0.1210.710.07stable
0.1812.59.68stable
0.2730.07.95stable

7. The Numerical Model

[43] In this section the theoretical findings discussed in sections 5 and 6 are compared with the results of a fully numerical model. The network is solved following the technique introduced by Schaffranek et al. [1981]: at each time step the boundary values of the unknown variables, including the internal nodal points, are preliminary computed for all the branches of the network; then (6a)(6c) are solved along each branch, with the given boundary conditions, using the “box scheme,” originally proposed by Preissmann [1961], whereby time and space derivatives are computed as weighted average of finite differences, evaluated in adjacent points along the perimeter of each computational cell.

[44] The stability of the equilibrium solutions of the simple loop sketched in Figure 5 is investigated numerically by perturbing the initial equilibrium condition with the introduction of an arbitrarily small sediment bump in one of the two downstream branches and following the behavior of the system in time.

[45] At first we consider the case of a symmetrical network, so that bb = bc = ba/2 and Lb = Lc. The initial condition is the symmetrical configuration with Qb = Qc and Db = Dc. In Figures 10 and 11 the bottom profile throughout the network is plotted at different times, for two different values of the Shields parameter ϑa in the upstream channel. The geometrical and hydraulic conditions of channel a are the same as those adopted to compute the equilibrium configurations reported in Figures 6 and 7 (for Lb/Lc = 1).

Figure 10.

Longitudinal profile of bed elevation under stable conditions (the initial bed slope has been filtered out from the results): Da = 1.87 m, Qa = 365 m3/s, ba = 30 m, ds = 0.056 m. The corresponding values of the dimensionless parameters of the upstream channel are ϑa = 0.3, βa = 8, Ca = 12.5.

Figure 11.

Longitudinal profile of bed elevation under unstable conditions (the initial bed slope has been filtered out from the results): Da = 1.87 m, Qa = 211 m3/s, ba = 30 m, ds = 0.056 m. The corresponding values of the dimensionless parameters of the upstream channel are ϑa = 0.1, βa = 8, Ca = 12.5.

[46] In agreement with the theoretical findings, the numerical solution predicts the stability of the loop for ϑa = 0.3, as reported in Figure 10: the sediment bump, introduced in channel c, is progressively damped, hence, after some time, deposition occurs along the whole channel c, whose depth slowly decreases in time until the unperturbed initial configuration is reestablished in the system. Channel b is almost unaffected by the presence of the perturbation induced in channel c except for a weak erosion process which characterizes the initial stage of the numerical run.

[47] In the case of ϑa = 0.1 the numerical model predicts instability of the symmetrical equilibrium configuration in agreement with the theoretical results. The network development is shown in Figure 11: the initial transient does not differ significantly from the previous stable case; however, deposition in branch c grows in time and consequently branch b undergoes an erosion process. In Figure 12 the ratio Qb/Qc between the discharges at the middle point of channels b and c is plotted versus time. It appears that the loop evolves from the initial symmetrical configuration (Qb/Qc = 1) to the unbalanced equilibrium condition referred to as solution s3 in Figure 7, which is thus stable as predicted by the linear theory. Notice that the loop evolves alternatively toward the solution s1 or s3 depending on the channel (b or c) in which the perturbation is introduced.

Figure 12.

The ratio Qb/Qc between water discharges in channels b and c is plotted versus time (ϑa = 0.1, βa = 8, Ca = 12.5).

[48] We now finally consider the case of a nonsymmetrical loop, with branches characterized by different lengths and we still assume for simplicity bb = bc = ba/2. A nonsymmetrical loop is equivalent to a system with two branches with the same length (Lb = Lc) and different water levels imposed at the outlet of each channel. The latter procedure is more efficient numerically since it allows us to follow the behavior of the system under different geometrical configurations simply changing the water level at the end of one branch.

[49] In Figure 13 examples of numerical runs are reported: the ratio between the water discharges in channels b and c is plotted versus time. At the beginning of the simulations, water level is kept constant at the outlet section of each downstream channel; furthermore, the initial condition is given by the equilibrium configuration s3 reported in Figure 7 for Lb/Lc = 1. At a certain time water level is raised in channel b to a prescribed value such that the system is equivalent to the case in which the ratio Lb/Lc is equal to 1.05 (continuous line). It appears that the ratio between the water discharges Qb/Qc decreases until the new equilibrium solution is reached (see solution s3 in Figure 7).

Figure 13.

The ratio Qb/Qc between water discharges in channels b and c is plotted versus time. Water level at the outlet of channel b has been raised to simulate branches with different lengths (ϑa = 0.1, βa = 8, Ca = 12.5).

[50] The dotted line in Figure 13 represents the case in which water level in channel b is raised to a value which corresponds to Lb/Lc = 1.08. According to Figure 7, under such a condition, the equilibrium solution s3 does not exist anymore: the only possible equilibrium configuration of the loop is given by the solution s1. In agreement with theoretical results the loop rapidly evolves toward the s1 solution, moving from an initial configuration in which channel b is fed with more water than channel c to an opposite configuration whereby larger water discharge occurs in channel c.

8. Conclusions

[51] The present model, without the claim to describe in detail the flow field and bed topography in channel bifurcations, is proposed as a useful tool to predict the long-term evolution of a channel network through a simple one-dimensional approach.

[52] The model is based on a nodal point condition, which allows for the inclusion of two-dimensional effects close to the bifurcation. In the case of a simple channel loop the model predicts the existence of a threshold value of the Shields parameter in the upstream channel above which the loop only admits of one equilibrium solution with both branches open; this solution is found to be invariably stable. For values of the Shields parameter below the above threshold two further equilibrium configurations appear, which are characterized by different values of water discharge flowing into the two downstream branches. Under these conditions the two new solutions are stable while the previous solution becomes unstable.

[53] Notice that braided gravel rivers are typically characterized by low values of Shields stress, even at high stages. Furthermore, stable, symmetrical bifurcations are seldom observed. Hence, the present model seems to provide a sound interpretation for one of the leading mechanisms, which are responsible for the highly unstable character of braided networks.

[54] The model also provides useful information on the physical mechanism that governs the development of a bifurcation in gravel braided rivers, in which sediment transport mainly occurs as bed load. In particular it is shown that the transverse exchange of sediment, induced by topographical effects close to the channel division, plays a crucial role on the stability of the bifurcation and allows the system to admit of equilibrium configurations characterized by different values of flow and sediment discharges into the downstream branches. Also notice that the relatively small length of each branch in a braided network, due to the continuous interplay of channels, might suggest that nonuniform downstream boundary conditions significantly affect the development of the network. However, provided the length of the upstream channel is sufficient to prevent changes of boundary conditions at the inlet, the present work suggests that the behavior of a bifurcation is mainly related to the actual hydraulic conditions and geometry at the nodal point. Under such conditions backwater effects may strongly affect the equilibrium configurations but do not influence their stability.

[55] Finally, it is worth recalling that several important local effects have been neglected in the present approach, such as the planimetrical geometry of the bifurcation, the strong three-dimensionality of the flow and the occurrence of large-scale bar forms. In this respect the present model has to be considered as a suitable starting point for more refined analyses, adequately supported by experimental and field observations.

Notation
b

channel width

C

Chezy coefficient

D

water depth

ds

sediment diameter

g

gravitational acceleration

H

free surface elevation

j

τ/(ρgD)

k

coefficient of the nodal point relationship by Wang et al. [1995]

L

channel length

n

power of the sediment transport law by Wang et al. [1995]

p

sediment porosity

Q

water discharge

Qs

sediment discharge

Qy

transverse water discharge

q

sediment discharge per unit width

qy

transverse sediment discharge per unit width

q(i)

sediment discharge per unit width entering the channel

q(o)

sediment discharge per unit width leaving the channel

r

empirical constant for transverse slope effect

S

channel slope

U = (U,V)

depth averaged velocity vector

t

time

x

longitudinal coordinate

y

transverse coordinate

α

dimensionless length of the final reach of channel a

β

aspect ratio

δ

deviation between sediment transport and bottom stress

Φ

bed load function

η

bed elevation

Ω

cross-section area

ϑ

Shields parameter

ϑcr

critical value of Shields parameter for sediment movement

ρ

water density

ρs

sediment density

τ

bottom shear stress

Acknowledgments

[56] This work has been developed within the framework of the “Centro di Eccellenza Universitario per la Difesa Idrogeologica dell'Ambiente Montano-CUDAM” and of the project “Morfodinamica delle reti fluviali-COFIN2001” cofunded by the Italian Ministry of University and Scientific Research and the University of Trento. A preliminary version of the present work has been presented at RCEM 2001 [Bolla Pittaluga et al., 2001]. The authors thank the anonymous referees for their constructive criticism, which contributed to the improvement of the manuscript.

Ancillary