A one-dimensional model of bifurcations in gravel bed channels with erodible banks



[1] In a recent paper, Bolla Pittaluga et al. proposed a physically based formulation for the nodal point conditions to be adopted at a channel bifurcation in the context of a one-dimensional approach. They employed such conditions to study the equilibrium configurations of a simple bifurcation and their stability, assuming erodible bed and fixed banks. With the present work we extend the model proposed by Bolla Pittaluga et al. to the case of channels with erodible banks, i.e., channels which may adapt their width to the actual flow conditions. Such an extension is of a particular interest for river bifurcations in gravel bed braided streams, in which bed evolution and bank erosion processes occur over comparable timescales. Moreover, to study the morphological evolution of a bifurcation, we employ a different approach with respect to Bolla Pittaluga et al. and introduce a “local analysis” which does not account for the influence exerted on channel morphology by downstream conditions on longer timescales. For values of the controlling parameters typical of gravel bed braided streams, the model shows that the stable equilibrium solutions of a bifurcation are invariably characterized by a strongly unbalanced partition of water and sediment discharges in the two branches. Numerical simulations, based on a simplified model of the bifurcation evolution, allow us to investigate the role of the mechanism of generation of a bifurcation on its final equilibrium configurations. It is shown that bifurcations which form through the incision of a new, initially narrow, channel may lead to significantly different equilibrium configurations with respect to bifurcations which generate from a central deposition in a wide channel. In spite of its simplicity the model seems to retain the most relevant effects which govern the behavior of gravel bed channel bifurcations.

1. Introduction

[2] Channel bifurcations in gravel bed braided rivers are almost invariably asymmetrical: Water and sediment discharges in downstream branches are typically unbalanced and the channel carrying the larger discharge is, in general, wider and deeper. A first attempt to reproduce such an asymmetry within the context of a one-dimensional approach is due to Bolla Pittaluga et al. [2003] (hereinafter referred to as BRT). Their analysis follows the original approach of Wang et al. [1995], who studied the equilibrium configurations of channel divisions along with their stability, with particular reference to fine sediment rivers. Within the context of a one-dimensional approach five internal conditions are required at the bifurcation point in order for the problem to be well posed: Among these conditions the so-called nodal point condition, which sets the partition of sediment discharge within the downstream branches, plays a crucial role. In the model of Wang et al. [1995] such a condition has an empirical foundation, while in BRT an alternative formulation is proposed, based on physical considerations.

[3] Both Wang et al.'s [1995] and BRT's models are based on the fundamental assumption that channel banks are fixed. Such an assumption is equivalent to consider that the timescale of planimetrical changes is much larger than the timescale of bed evolution. This is a quite common way of proceeding for single thread meandering channels where both cohesion and vegetation may reinforce the strength of channel banks. In gravel bed braided rivers, however, the timescales of bank and bed erosion may be comparable: Field observations suggest that during a formative event the overall geometry of the braided network may experience significant changes [Ashmore, 2001]. It is then reasonable to expect the width of each individual distributary to be, if not in equilibrium with, at least strongly affected by the actual flow conditions. In this respect, setting a priori the value of branches width, without concern to the actual hydraulic conditions as done by BRT, may turn out to be a too crude approximation.

[4] A further fundamental assumption is adopted in previous works. In order to investigate the stability of the equilibrium configurations of a channel bifurcation both Wang et al. [1995] and BRT assume that the time evolution of the system can be described as a sequence of uniform flow conditions. In other words, variations of water depth along each channel are neglected and scour and deposition in the downstream branches are treated as uniformly distributed along the whole channel lengths. Therefore sediment continuity is applied to the downstream branches in global form, accounting for the sediment discharge entering each channel from its upstream end and leaving it from the outlet section. An alternative approach has been recently proposed in simplified form by Hirose et al. [2003]: In their analysis the problem is cast in terms of local variables and the effect of downstream boundary conditions is discarded.

[5] In the present contribution we build upon the above ideas and extend the model proposed by BRT to the case of channels with erodible banks. Hence we allow the width of each distributary splitting at the bifurcation to change according to actual flow conditions. We also introduce a local analysis to investigate the stability of the bifurcation; hence we do not account for the control which is exerted on channel morphology by the downstream boundary conditions on a longer timescale. One could argue that the same arguments which justify the adoption of a local analysis should, in principle, discourage one from the use of a regime relationship to link channel width to flow discharge, which would imply to assume the instantaneous adaptation of the channel geometry to the actual flow conditions. We can also expect that a gravel bed channel widens when the discharge is increased, while channel narrowing as a consequence of a discharge fall is not physically justified.

[6] In order to account for the above features a simple differential approach is introduced to evaluate width adjustment to flow conditions. The model also accounts, in a simplified way, for the different response of channel width to discharge raise and fall.

[7] The nonsymmetrical character of bifurcations in braided networks, which is widely documented both in the field and in laboratory investigations (see section 2), is reproduced by our model. In fact, when the channel width is allowed to change according to flow discharge the stable equilibrium configurations of the bifurcation are found to be invariably characterized by a strong unbalance of water and sediment discharges in the two branches, for values of the controlling parameters typical of gravel bed channels.

[8] A consequence of the different response of channel width to increasing or decreasing water discharge is that the equilibrium configurations are found to depend on the mechanism through which the bifurcation is formed. More specifically, a bifurcation originating from the incision of an initially narrow branch leaving the main channel tends to an equilibrium configuration which may significantly differ from the asymptotic limit of a bifurcation which forms as a consequence of a central deposition in a wide channel. The model thus shows that even under steady forcing conditions, a range of possible equilibrium solutions is possible, depending on the initial state of the system.

[9] The present paper is organized as follows. In section 2 we give a summary of some recent field and laboratory observations on gravel bed bifurcations, with the aim of identifying their main features. The nodal point condition proposed by BRT is briefly recalled in section 3 along with the available criteria to relate channel width to local flow conditions. In section 4 we investigate the equilibrium configurations of a simple Y-shaped bifurcation along with their stability, accounting for the width adaptation process to flow conditions. We then analyze the role of the inception mechanism in section 5. Finally, some concluding remarks follow in section 6.

2. Field and Laboratory Observations on Gravel Bed Bifurcations

[10] The recognition of the role of bifurcations as relevant unit processes characterizing the dynamic behavior of multichannel river systems has recently motivated various laboratory investigations. Few detailed observations are also presently available concerning natural bifurcations in gravel bed braided rivers, i.e., bifurcations characterized by a nonfixed planimetrical geometry and subject to natural hydraulic forcing.

[11] Before revisiting these results it is worth recalling the classic experimental work by Bulle [1926], which has been recently reconsidered by de Heer and Mosselman [2004]. Bulle [1926] carried out experiments measuring flow and bed load distribution in alluvial diversions with fixed sidewalls. He considered a straight main channel from which a minor branch diverted at a specified angle and observed that close to the bifurcation, the flow was strongly three-dimensional, with liquid particles moving along helical trajectories. Such a complex flow field significantly affected the bed load transport and the so-called “Bulle effect” occurred, according to which a disproportionate amount of bed load feeds the diverting channel. Another distinctive feature of the flow field was the formation of an eddy downstream of the diversion point. These effects were probably enhanced by the quite large angle that the diverting branch formed with the main channel and by the fact that the deviation of the sidewall was sharp. The above observations, which have been qualitatively, even not quantitatively, reproduced by de Heer and Mosselman [2004], who employed a fully three-dimensional turbulence-averaged model for the flow field, are of a great importance in order to understand the complex flow field at a bifurcation. However, in a gravel bed bifurcation the deviations of sidewalls are never sharp and the flow depth is relatively small. Therefore it is reasonable to expect both the helical flow and, in particular, the eddy at the bifurcation to be less intense in natural contexts.

[12] More recently, Federici and Paola [2003] have performed a series of experiments on gravel bed braided streams. In a first set of experiments they built up a laboratory model of a braided river and monitored its development starting from a single straight channel of trapezoidal cross section. After some time such a configuration invariably broke up into a multichannel pattern, following a sequence of events resulting from the interaction between bed and channel evolution processes. A detailed quantitative analysis of the above process is also given by Bertoldi and Tubino [2005]. Federici and Paola [2003] mainly focused their attention on bifurcations originating from the formation of central bar deposits, which seemed to have reached a “stable” configuration. Typically, the average flow velocities and water depths at the mouth of the two downstream channels were found to be comparable and the axes of the two branches approximatively formed the same angle with the centerline of the upstream reach. However, the widths of the two branches were in general markedly unequal, as well as water and sediment discharges. Federici and Paola [2003] also followed each monitored bifurcation in time and observed that some of them were stable; that is, they remained open in time, while others were “unstable” and evolved into a single-thread channel with one of the two branches being abandoned.

[13] Such an instability process led Federici and Paola [2003] to perform a second series of experiments within a flume consisting of a straight constant width channel, which was followed by a linearly diverging reach, ending in a much wider, constant width channel. Banks were everywhere erodible except for the upstream reach of the straight incoming channel where they were kept fixed and vertical. For any hydraulic condition and slope the diverting streamlines produced the formation of a bifurcation. They observed that for relatively large values of the Shields parameter of the incoming flow, the bifurcation was stable and, typically, fairly symmetrical. At low values of the Shields parameter, say smaller than 0.15, bifurcations still formed; however, they were unstable since one of the two branches was often closed, to be eventually reopened later. This process occurred repeatedly and randomly. Federici and Paola [2003] termed such configuration “switch bifurcation.” They also observed that at the bifurcation the water level was approximatively constant while the local bed elevation was characterized by a sudden step between the mouths of the two downstream branches. We note that in the experiments it was possible to reproduce fairly large values of Shields parameter since the upstream reach had fixed banks; in case of erodible banks the upstream channel would have probably widened, with a consequent decrease of the Shields stress. Furthermore, the upstream channel was too short for large-scale bed forms to form and migrate through it.

[14] We finally report on some very recent field observations on channel bifurcations in braided networks conducted by Zolezzi et al. [2006]. The observations were carried out during a series of field campaigns in two streams: the Sunwapta river (Alberta, Canada) and the Ridanna creek (South Tirol, Italy). The two field sites are located in mountain areas, a few kilometers downstream of the present position of the glacier snout. In the study reach in the Sunwapta River the sole contribution due to glacier melting may produce formative conditions, while in the Ridanna creek meteoric events are also required. In both reaches the bed is mainly constituted of gravel, the grain size distribution is strongly heterogeneous and sediment transport mainly consists of bed load. Both reaches are braided, though the braiding index in the Sunwapta river is significantly higher.

[15] Zolezzi et al. [2006] selected seven bifurcations in the two reaches (three in the Sunwapta river and four in the Ridanna creek) and surveyed their hydraulic and morphometric characteristics in detail. The main recurring feature of the observed bifurcations is a strong asymmetry of their morphological configuration: This result is common to all collected data and is in agreement with the observations of Mosley [1983] and Federici and Paola [2003]. Zolezzi et al. [2006] invariably found that one of the two distributaries was carrying a far larger discharge, the ratio rQ between the discharges in downstream branches ranging between 0.05 (in the Ridanna creek at a very low stage) and 0.65. Measurements were carried out in each bifurcation at different stages. The branch carrying the largest discharge was found to be invariably wider and deeper. As a result of the strong asymmetry of the configuration often only one of the two branches was found to be morphologically active, i.e., to transport sediment.

[16] A further recurring element identified by Zolezzi et al. [2006] was a local aggradation just upstream of the bifurcation point, which reflected the tendency toward deposition due to local flow divergence. Downstream of such a deposit a sudden degradation was invariably observed in the larger branch. The formation of an “inlet step” was detected in all the monitored bifurcations. Its amplitude, given in terms of the difference between the local values of bed elevation measured at the mouth of downstream branches, was found to be related to the discharge ratio rQ and to scale with the mean upstream water depth. The inlet step provides a measure of the transverse bottom inclination which characterizes the bifurcation region, whose effect may extend for a length of few channel widths upstream of the bifurcation point. According to field observations such a transverse slope is mainly responsible for the partition of flow and bed load within the downstream distributaries, at least at intermediate and low stages.

[17] Results of laboratory and field investigations reported in this section provide a consistent picture of the morphology of bifurcations and give information on the relevant processes which must be retained in order to build up sound nodal point conditions to be used in theoretical analyzes. Summarizing the main points, we can say that (1) bifurcations in gravel bed rivers are almost invariably asymmetrical and convey different flow and sediment discharges in downstream distributaries; (2) bifurcations are highly unstable, particularly at low values of Shields parameters, and often tend to close one of the two branches; (3) just upstream of the bifurcation the bed undergoes aggradation and an inlet step establishes at the mouth of the two branches, with lower bed elevation in the larger branch; and (4) the inlet step produces a transverse bed slope extending upstream for a length of a few channel widths.

3. Nodal Point Conditions for Sediment Transport and the Regime Relationship

[18] We now attempt to extend the one-dimensional model of BRT to the case of channels with erodible banks, i.e., to channels which may adjust their width to the actual flow conditions. Let us consider a simple Y-shaped bifurcation and denote with b channel width and with the subscript letters a, b and c the upstream channel and the downstream distributaries, respectively (see Figure 1).

Figure 1.

Scheme of the nodal point relationship.

[19] As pointed out in section 2, the flow field at the bifurcation is complex and three dimensional, and bed topography is rapidly varying. Such effects cannot be reproduced through a simple one-dimensional model. On the other hand, such a simplified model provides a useful tool to analyze complex patterns typical of braided networks. BRT attempted to account for the complexity of such a flow employing a “quasi two-dimensional approach” close to the bifurcation. They considered the last reach of the upstream channel, with a length αba where α is an order one coefficient, as virtually divided into two adjacent cells, as shown in Figure 1. They applied to both cells the Exner equation, accounting for the transverse exchange of sediment discharge between the two cells due to flow exchange and to the gravitational effect on transverse slope. The resulting nodal point conditions for sediment transport take the following form:

equation image

where p is sediment porosity, η is bed elevation, qa is the incoming sediment discharge per unit width, qb and qc are sediment discharges per unit width feeding the downstream branches. Furthermore, qy represents the transverse exchange of sediments per unit width between the two cells and t is time; the following expression is adopted in BRT:

equation image

where the first term in brackets accounts for the deviation of the velocity vector with respect to the longitudinal direction, while the second term accounts for the effect of gravity on transverse slope, which is responsible for the deviation of particle trajectories from flow direction. Moreover, Qy is the flow exchange between the two cells, Dabc is an average depth along the cells, ϑ is the Shields parameter and r is an empirical constant ranging between 0.3 and 1 [e.g., Ikeda et al., 1981; Talmon et al., 1995].

[20] The above procedure provides suitable nodal point conditions through which it is possible to account for the topographically driven effects on flow and bed load induced by the bed pattern which establishes immediately upstream of the bifurcation.

[21] We note that the longitudinal length αba of the “two-dimensional reach” must be of the order of the upstream distance from the bifurcation at which its effect is no longer felt by bed morphology. Experimental investigations by BRT, along with field observations reviewed in section 2, suggest that α ranges between 1 and 3; therefore the length of the reach is small if compared with the longitudinal scale of flow and bed variations. This means that the flow in the upstream and downstream channels can be reasonably described through a one-dimensional model, regardless of the actual flow conditions at the bifurcation, which are accounted for, in the simplified manner described above, only to formulate the nodal point conditions. In the following, we have assumed α = 1.

[22] In the analysis of BRT, channel widths are fixed and are prescribed as input data. In order to account for channel width adaptation a further ingredient is needed, namely, a suitable relationship between channel width and hydraulic and morphological characteristics. For the case of single-thread channels, many so-called “regime relationships” have been proposed in the literature. They can be distinguished in two different classes: rational and empirical. The former are based on somewhat simplified solutions of the governing equations for water and sediment motion, while the latter are based on regressions on field data. As an example, the rational regime relationships proposed by Henderson [1966], Parker [1979], and Griffiths [1981] are

equation image
equation image
equation image

where br denotes channel width in regime conditions, S is the average slope, and ds the grain size. Notice that the above formulas have an identical structure as the exponents of dimensional variables coincide. Griffiths [1981] points out that the differences in the coefficients are due to the use of different physical constants, resistance relations, and sediment entrainment criteria.

[23] On the other hand, few estimates exist for channels in gravel bed braided rivers. Chew and Ashmore [2001] have tested various regime formulas on field data collected on a braided reach of the Sunwapta river, showing that empirical formulas based on data from single-thread channels fail in describing longitudinal river width changes. According to Chew and Ashmore [2001] this is mainly due to the fact that discharge dominates empirical regime relations for single-thread channels, while in gravel-braided rivers the effect of grain size seems to play an important role. Rational regime formulas perform better to reproduce the dependence of width on the controlling parameters, though the predicted local values of channel width are often incorrect.

[24] An empirical relationship has been recently proposed by Ashmore [2001] on the basis of a statistical regression on field data mainly from gravel bed rivers. Such a regression yields

equation image

where Ω = γSQ is the stream power, with γ water specific weight.

[25] In principle, the adoption of any of the relationships reported above to evaluate the width of a channel evolving in time would imply to assume an instantaneous adaptation of channel width to flow conditions. Even if such a process is quite intense and relatively fast in gravel bed braided rivers under formative conditions, it can hardly be thought of as instantaneous. On the other hand, the empirical formula (6) is more likely to be representative of typical states of gravel bed rivers, rather then constituting a “true” regime relationship. In order to preserve the evolutionary character of the width adaptation process, a further differential equation for channel width would be required. The following simplified form can be adopted:

equation image

where b denotes the actual channel width, Tw is the time scale of the width adaptation process and br is computed through one of the above regime formulas. The above equation is consistent with the two limiting cases of fixed width (Tw → ∞) treated by BRT and of instantaneous width adaptation to flow conditions (Tw → 0), which implies b = br.

4. Equilibrium Configurations and Stability of a Simple Channel Bifurcation

4.1. Formulation of the Problem

[26] We now employ the nodal point conditions described above to study the equilibrium configurations and stability of a channel bifurcation with erodible banks.

[27] As pointed out in section 1, we adopt a different approach with respect to that originally proposed by Wang et al. [1995] and later extended by BRT. More specifically, we still assume uniform flow conditions in the upstream channel; however, we assume its width to be in equilibrium with hydraulic conditions according to a regime relationship. Furthermore, we employ the nodal point conditions (1) along with equation (7) to relate the width of downstream branches to flow discharge and set sediment discharges qb and qc equal to the transport capacity at the inlet of each branch. Moreover, instead of setting the flow and bed topography in downstream branches in equilibrium with the downstream boundary conditions, as done in BRT, we perform a local analysis and assume the flow in downstream branches to be in equilibrium with the local bed slope. To this end, a stage-discharge relationship for each branch must be supplemented, for which the uniform flow condition represents a suitable choice. It is worth noting that this condition could be replaced by other relationships without any conceptual change in the analysis.

[28] One can readily argue that the above approach corresponds to a different choice of the timescale adopted in the analysis. BRT's model follows the classical viewpoint of long-term analysis, in that the equilibrium and the stability of the channel bifurcation is investigated on a relatively long timescale, namely, that required for the morphological bed response of the downstream branches to adapt to the flowing discharge and to the effect of downstream boundary conditions. The above approach doesn't recognize the fact that the local channel morphology in the neighborhood of a node may develop on a much faster timescale than that required for the morphodynamic influence of downstream conditions to be felt. Given the relatively small values of the Shields parameters typical of gravel bed channels, the latter scale may turn out to be quite large compared to the time span of existence of a bifurcation in braided networks. Hence a (short term) local analysis seems more suitable to describe the equilibrium configuration of a bifurcation and its stability in gravel bed braided rivers, unless backwater effects are significant, which may be the case when a strong upstream influence is determined under formative conditions by physical constraints imposed by valley geometry.

[29] In conclusion, in order to study the time evolution of the Y-shaped bifurcation sketched in Figure 1 we solve the five nodal point conditions which are required at the bifurcation along with two stage-discharge relationships for the downstream branches and two further relationships of the form (7). We thus impose (1) the water discharge balance at the node

equation image

(2) the constancy of water level H at the node

equation image

(3) the nodal point conditions (1) which govern the partition of sediment discharge in downstream branches; (4) uniform flow conditions in downstream branches equation

equation image

where C is Chezy coefficient and R the hydraulic radius; and (5) the adaptation of channel width of downstream branches through equation (7).

[30] We finally note that according to the “local” character of our analysis, the slopes Sb and Sc of downstream branches must be prescribed as input data. Physically, these values are likely to be controlled by the mechanism of generation of the bifurcation and by the local topography of the network rather than by the evolution of the bifurcation itself.

[31] The above equations must be solved for the nine unknowns Qb, Qc, Db, Dc, bb, bc, ηa, ηb, and ηc, for given values of the water and sediment discharge in the upstream channels. However, the use of stage-discharge relationships of the form (10), which is consistent with the short-term viewpoint adopted in our analysis, does not allow one to properly account for the effect of downstream conditions, which would set the reference value for bed and water level. This implies, that according to our formulation, a bed level value, say ηa, is left undetermined and the mathematical problem is over-specified by the above conditions. Hence a given value of the ratio Sb/Sc, or Sb/Sa, is sufficient to close the mathematical problem.

4.2. Equilibrium Configurations

[32] To compute the equilibrium solutions, the time derivatives in (1) and (7) are set to zero and the resulting nonlinear algebraic system is solved through the Newton-Rapson method. Notice that setting db/dt = 0 in equation (7) implies b = br, hence the value of the timescale Tw does not affect the equilibrium configurations.

[33] If channel widths are fixed we recover BRT's analysis. In this case the equilibrium configurations can be determined, under the assumptions of uniform flow conditions throughout the network, in terms of the dimensionless parameters ϑa, Sa and βa, namely, the Shields stress, the bed slope and the aspect ratio of the upstream channel. The above parameters provide a complete representation, in dimensionless form, of the incoming uniform flow. Results of BRT for a symmetrical configuration (bb = bc) suggest that for given values of βa and Sa, a symmetrical solution always exists, in which the two downstream branches are fed with the same water and sediment discharges. (In BRT the Chezy coefficient Ca was used instead of the slope Sa, but their results also hold for a given Sa.) However, such a solution is stable only at relatively large values of ϑa; for smaller values of ϑa, say ϑa ≲ 0.1, two further equilibrium solutions appear, one reciprocal to the other, which are found to be invariably stable and are characterized by a strong imbalance of discharge partition in the downstream branches.

[34] The above scenario changes when the channels are allowed to adjust their width to flow discharge. A first novelty is brought in the dimensionless representation of the incoming flow, in that the system loses one degree of freedom when a regime relationship of the form (6) is included. Hence the values of ϑa, Sa and βa can no longer be selected arbitrarily since they are related through the above relationships. In Figures 2a and 2b we plot the relationship between the Shields parameter ϑ and the aspect ratio β, for different values of the slope S, as obtained according to the regime formulas proposed by Ashmore [2001] and Griffiths [1981], respectively. The former implies a slight increase of the Shields stress as the width ratio increases. This is also true for any rational formula, though the variability is quite limited in this case, as shown in Figure 2b.

Figure 2.

Dependence of the Shields parameter ϑ on the channel aspect ratio β, for different values of channel slope S, as obtained according to different regime relationship: (a) Ashmore [2001] (ds = 0.05 m), (b) Griffiths [1981].

[35] We note that both regime formulas predict very small values of the Shields stress, which are close to the condition of incipient sediment motion, whatever the value of the aspect ratio. Hence one could argue that according to the results of BRT, a gravel bed bifurcation with erodible banks should almost invariably admit of the unbalanced equilibrium solutions. Such an intuitive statement is confirmed by the results of our model, which are shown in Figures 3a and 3b. Results of the present model are reported with solid lines and are obtained using the sediment transport formula proposed by Parker [1990]. In Figure 3a the ratio rQ = Qb/Qc between water discharges flowing into the downstream branches at equilibrium is plotted versus the aspect ratio of channel a. In Figure 3b the same equilibrium configurations are shown in terms of the ratio rD = Db/Dc between water depths. We can see that along with the symmetrical solution, the model predicts the existence of unbalanced solutions for any value of βa. Furthermore, the degree of asymmetry increases, that is, the above ratios diverge from the unity, as the aspect ratio of channel a increases. In the same plots we have also reported, with dotted lines, results obtained keeping the widths of the two branches fixed and equal to 1/2 ba and setting the Shields stress of the upstream channel ϑa equal to the case of variable width. It appears that at equilibrium, water discharges are significantly more unbalanced if channel widths are allowed to adapt to flow conditions than in the corresponding case of fixed banks. The opposite is true for the ratio between water depths rD in the downstream branches which, in equilibrium conditions, attains larger values if the banks are kept fixed.

Figure 3.

Equilibrium values of (a) the discharge ratio rQ = Qb/Qc and (b) the depth ratio rD = Db/Dc as functions of the aspect ratio of the upstream channel βa (Sa = 0.01, Sb/Sc = 1). Solid lines indicate erodible banks (b = br, br computed through equation (6)); dotted lines indicate fixed banks (bb = bc = ba/2, ϑa computed through equation (6)).

[36] The present model also computes the width of downstream channels: The equilibrium solutions are plotted in Figure 4 in terms of the ratio rb = bb/bc. It appears that for typical values of βa, the main downstream channel at equilibrium may be 2–4 times wider than the other channel.

Figure 4.

Ratio between the channel widths of the two downstream branches rb = bb/bc at equilibrium as a function of the aspect ratio of the upstream channel βa (Sa = 0.01, Sb/Sc = 1, br computed through equation (6)).

[37] The “inlet step” observed by Zolezzi et al. [2006] in natural bifurcations is also reproduced by the model. Its height is related to the degree of asymmetry of the bifurcation and it increases as βa increases, in qualitative agreement with field observations. Theoretical predictions are reported, with a solid line, in Figure 5, where the dimensionless parameter Δ, defined as the ratio ∣ηbηc∣/Da, is plotted versus the aspect ratio βa. The correspondent case of fixed banks is also reported in the plot with a dotted line. Results show that channel width adaptation to flow conditions in the downstream branches acts so as to decrease the height of the inlet step.

Figure 5.

Dimensionless inlet step height Δ = ∣ηbηc∣/Da at equilibrium as a function of the aspect ratio of the upstream channel βa (Sa = 0.01, Sb/Sc = 1). Solid lines indicate erodible banks (b = br, br computed through equation (6)); dotted lines indicate fixed banks (bb = bc = ba/2, ϑa computed through equation (6)).

[38] Finally, in Figures 6a and 6b we show the effect of changing the parameter Sb/Sc on the equilibrium configurations; again results of the present model are reported with solid lines, while dotted lines refer to the case of fixed banks. It appears that a range of values of Sb/Sc exists, close to the value Sb/Sc = 1, where three solutions are possible. This range is slightly larger in the case of fixed banks. Notice that one of these solutions implies that the channel with a milder slope is the larger one and carries most of the water discharge. If the slopes of the downstream branches are different enough one from the other, only one equilibrium solution survives which is such that the steeper branch is much larger than the other.

Figure 6.

Equilibrium values of (a) the discharge ratio rQ and (b) the width ratio rb as functions of Sb/Sc (βa = 20, Sa = 0.01). Solid lines indicate erodible banks (b = br, br computed through equation (6)); dotted lines indicate fixed banks (bb = bc = ba/2, ϑa computed through equation (6)).

[39] We note that the unbalanced equilibrium configurations obtained when the banks are erodible, almost invariably imply that sediment load vanishes in the smaller branch; hence, in this case even fairly small perturbations of the bed topography at the bifurcation point may lead to the complete abandonment of one branch.

4.3. Stability of the Equilibrium Configurations

[40] Let us now investigate the stability of the above equilibrium configurations. To determine the time evolution of the system, the governing equations (1), (8), (9), and (10), coupled with equation (7) applied to channels b and c, are solved starting from an initial condition and advancing in time using a fourth-order Runge-Kutta scheme.

[41] We first consider the case of instantaneous width adaptation to flow conditions. As discussed in section 3, this case is described by equation (7) in the limit Tw → 0, which implies that equation (7) reduces to the algebraic form b = br.

[42] Results of numerical simulations suggest that as in the case of a fixed banks bifurcation treated by BRT, for Sb/Sc = 1 the symmetrical solution is unstable and the unbalanced solutions are stable. For Sb/Sc ≠ 1, when three solutions exist the middle one (that is the less unbalanced configuration) is unstable and the other two are stable, while when only one solution exists such a solution is found to be invariably stable. Examples of evolution of the system in time are reported in Figures 7a and 7b for the case Sb/Sc = 1, in terms of the ratios rQ = Qb/Qc and rb = bb/bc. Here and in the following br is evaluated through (6) and T is a dimensionless time, scaled with the morphological timescale Tm which can be readily obtained from Exner equation. Using the present notation it reads

equation image

Notice that in the plot we consider the unbalanced solution with rQ < 1 and rb < 1; obviously a reciprocal stable solution also exists. In Figures 7a and 7b, the occurrence of an inflection point in the curves starting close to the points rQ = 1 and rb = 1 is due to the fact that the symmetrical configuration of the bifurcation corresponds to an unstable solution of the system.

Figure 7.

Behavior of the bifurcation starting form different initial conditions. (a) The discharge ratio rQ and (b) the width ratio rb are plotted versus the dimensionless time T (Sa = 0.01, βa = 20, Sb/Sc = 1, Tw → 0).

[43] A first clear outcome of the model is that whatever the initial condition, it predicts that a natural bifurcation in gravel bed rivers is always expected to assume an unbalanced configuration, with a larger water and sediment discharge feeding one of the two distributaries. This result agrees, at least qualitatively, with the experimental and field observations described in section 2.

[44] As previously stated, one of the novelties introduced in present analysis with respect to BRT is the local character of the analysis, which implies that the time evolution of the system is no longer dependent on the length L of downstream branches. This dependence, which is brought in BRT's analysis by the inclusion of downstream boundary conditions, crucially affects the timescale of the process, as can be inferred from equations (19a) and (19b) of BRT. In the present approach such a timescale is set by the conditions (1) and, through the remaining set of nodal equations, it is inherently related to the local characteristics of the bifurcation.

[45] The difference between the two approaches can be appreciated from Figure 8a, where the temporal evolution of the ratio rQ is given in terms of the dimensionless time T. Results corresponding to the present “local approach” are reported with a solid line. The other curves in the plot are obtained following the approach proposed by BRT and imposing different values of length L of the downstream branches. Figure 8a clearly shows that according to the latter approach, the time evolution of the system is crucially affected by the parameter L/ba (notice that all curves refer to the limiting case of instantaneous width adaptation Tw → 0). In Figure 8b the evolutionary timescale τ of the system is plotted versus L/ba. Such a timescale has been computed imposing a small perturbation to the symmetrical equilibrium configuration and fitting the time evolution of rQ with an exponential function. Figure 8b suggests that the dimensionless timescale τ increases linearly with L/ba, such that for values of L/ba which are typical of braided networks (say 10 < L/ba < 30) it may attain much larger values than those obtained with the present local approach (L/ba = 0).

Figure 8.

(a) Transition from the unstable symmetrical solution to the stable asymmetrical solution. The discharge ratio rQ is plotted versus the dimensionless time T, for different values of the length of the downstream branches L, scaled with the upstream channel width ba. The present model (local analysis, L/ba → 0) is reported with solid line (Sa = 0.01, βa = 20, Sb/Sc = 1, Tw → 0). (b) Dimensionless timescale of the bifurcation τ plotted versus the dimensionless length of downstream channels L/ba for different choices of Tw (Sa = 0.01, βa = 20, Sb/Sc = 1).

[46] Let us now finally investigate the role of the timescale of width adaptation Tw, as defined through equation (7), on the behavior of the system. A proper choice of Tw would require ad hoc field or experimental investigations, since it may depend on various factors. However, we may expect that the actual configuration attained by the bifurcation is bounded within those obtained in the two limiting cases of instantaneous width adaptation (Tw → 0) and of fixed banks (Tw → ∞). Figure 8b suggests that the qualitative behavior of the solution doesn't depend on the choice of Tw: In fact, whatever the value of Tw, the adaptation process of the bifurcation, as predicted in terms of the local analysis in the limit L/ba → 0, is invariably quite fast as compared with the time required to feel the effect of downstream boundary conditions (note that the case Tw = 1 corresponds to a width adaptation process occurring on a timescale comparable to the morphological timescale). Furthermore, as Tw varies in the range (0, ∞) the order of magnitude of the timescale τ doesn't change. This is also confirmed by the results reported in Figure 9 where the time evolution of the bifurcation starting from the symmetrical equilibrium configuration is shown, for L/ba = 0, in terms of the ratio rQ. Each line corresponds to a different value of the timescale of width adaptation Tw. The limiting cases of fixed banks (Tw → ∞) and of instantaneous width adaptation to flow conditions (Tw → 0) are reported with thicker lines, as well as the case Tw = 1. It appears that the qualitative behavior of the solution does not depend on the choice of Tw, since the system invariably evolves toward an unbalanced configuration. The value of Tw only affects the time required to reach such solution.

Figure 9.

Transition from the unstable symmetrical solution to the stable asymmetrical solution. The discharge ratio rQ is plotted versus the dimensionless time T, for different values of Tw (Sa = 0.01, βa = 20, Sb/Sc = 1). Thicker lines denote the limiting cases Tw → 0, Tw → ∞ and the case Tw = 1.

5. Role of the Genesis of the Bifurcation

[47] In section 4 we have discussed the equilibrium configurations of a simple Y-shaped bifurcation and their stability, using a relationship of the form (7) to account for width adjustment to the local flow conditions. We have seen that results do not differ qualitatively, whatever the value of Tw, which implies that a simple algebraic relationship b = br, with br given by (6), can provide a suitable, albeit simplified, model of the widening process of a distributary in braided networks in reply to a discharge increase. On the other hand, field evidence and the results of some recent laboratory experiments [Bertoldi et al., 2005] suggest that the counterpart process, i.e., a discharge reduction in the channel, can hardly be handled in the same way. In this case the channel has no way to reduce its width accordingly, at least on a relatively short timescale, apart from shrinking effects induced, locally, by lateral deposition at the inside bank of the narrower channel, which are typically also responsible for the increase of the bifurcation angle. Further modification of the overall channel geometry typically results, on longer timescales, from depositional processes and the reworking of the network.

[48] Moreover, according to the approach employed in section 4 no information is incorporated in the model concerning the mechanism of generation of the bifurcation. In other words, the equilibrium configurations which have been determined are independent of the initial condition of the system. However, one could wonder whether differences may occur in the final configuration toward which the system is driven in its evolution, which might depend on whether the bifurcation is generated from the incision of a new channel, through a chute cutoff or an avulsion process, or it is formed as a consequence of the deposition of a central bar. In the former case the new channel is expected to be initially much narrower than the main one, while in the latter case the two branches, which bifurcate due to a central bar deposit, may initially have comparable widths.

[49] The different response of channel width to discharge rise or fall can be readily incorporated in our model through an approximate procedure: At each time step, channel width is computed through equation (7) in the limit Tw → 0; if, however, according to the solution of equation (7) channel width decreases, such a narrowing process is not permitted and the width is kept fixed. Such a procedure implies that the dependence of the equilibrium state on the initial condition is felt through the adaptation process of channel width to the flow. In fact, our procedure implies that the asymptotic behavior of the bifurcation predicted by the model changes because the system cannot reach a final configuration in which one branch is narrower than its initial value. Obviously, the given initial conditions are arbitrarily imposed and reflect the occurrence of different inception mechanisms.

[50] In Figure 10a we plot the time evolution of a Y-shaped bifurcation, starting from different initial values of channel widths, in terms of the ratio rQ = Qb/Qc between water discharges in the downstream channels. The corresponding values of the ratio between channel widths rb = bb/bc are plotted in Figure 10b. The dimensionless parameters governing the problem have been chosen identical to those used to produce Figures 7a and 7b so as to make the comparison between the two cases straightforward. Figure 10b shows that when the initial value of the ratio rb is larger than the corresponding equilibrium value reached in Figure 7b, which will be referred to as equation imageb, the equilibrium configuration strongly depends on the initial conditions; indeed the range of equilibrium values asymptotically reached by the ratio rb is quite large. On the other hand, when the initial value of rb is smaller than equation imageb the system invariably evolves toward an equilibrium solution which is close to equation imageb. In this case the bifurcation is initially characterized by a strong imbalance, which implies that in the subsequent evolution, the smaller channel experiences the largest width variation and widens according to equation (6), while the other branch keeps its width constant.

Figure 10.

Time behavior of the bifurcation. (a) The discharge ratio rQ and (b) the width ratio rb plotted versus the dimensionless time T for different initial conditions (Sa = 0.01, βa = 20, Sb/Sc = 1, Tw → 0). At each time step, channel width evolves according to equation (7) if the model predicts channel widening and is kept constant if the model predicts channel narrowing.

[51] It is interesting to notice that as shown in Figure 10a, similar results also hold for the discharge ratio rQ. However, even in the case of (rb)T=0 > equation imageb, the values of rQ attained at equilibrium fall within a relatively narrow range, which implies that the discharge ratio is quite less sensitive to the initial conditions than the width ratio.

[52] Finally, in Figures 11a and 11b we have plotted (in gray) the range of possible values of rQ and rb which are asymptotically reached starting from different initial conditions, if the different response of the system to discharge rise or fall is accounted for. It clearly appears that the range of possible equilibrium solutions is quite narrow for the ratio between water discharges rQ while it is very large for the ratio between the widths of the downstream branches rb.

Figure 11.

Range of possible final states of the system in terms of (a) the discharge ratio rQ and (b) the width ratio rb (Sa = 0.01, Sb/Sc = 1).

6. Concluding Remarks

[53] In the present work we have employed a one-dimensional model to analyze the equilibrium configurations of a simple channel bifurcation in a gravel bed river and to investigate its time evolution. We have extended the model proposed by BRT to account for width adjustments to actual flow conditions. Moreover the time evolution of the system has been computed employing a local analysis whereby system behavior is not affected by downstream conditions. This seems a suitable approach to describe bifurcations in gravel bed mountainous rivers, whose dynamics is essentially controlled by local parameters while downstream effects are likely to play a fairly minor role.

[54] The difference between the present local approach and that proposed by BRT mainly resides in a different choice of timescale for the system evolution. In BRT's model such a timescale is that required for the downstream reaches to morphologically adapt to downstream boundary conditions while, in the present approach, it is set by the nodal point conditions.

[55] The response of channel width to flow discharge has been computed through an empirical regime relationship like that proposed by Ashmore [2001]. This may turn out to be a quite strong assumption even in the case of formative conditions, though such relationship can be more properly interpreted as a suitable measure of the instantaneous state of a channel in a braided network. Furthermore, an attempt has been pursued to determine the influence of the timescale of width adaptation process on the behavior of the bifurcation. A more detailed investigation of such process would require further experimental observations.

[56] The model shows that when the hypothesis of fixed banks is removed, the system only admits of stable configurations which are strongly asymmetrical: One of the two branches is larger than the other and is fed with greater water and sediment discharges. The model also predicts the generation, at the bifurcation point, of an inlet step, i.e., a difference between bed elevations at the mouths of the downstream branches. The presence of such an inlet step generates, just upstream of the bifurcation, a transverse bed slope which crucially affects the partition of sediment discharges into the downstream branches. Both the above findings are in good qualitative agreement with field and laboratory observations.

[57] It is worth noting that the present model is unable to reproduce the disturbance which may be induced on the bifurcation by the migration of bars in the upstream channel, which may strongly affect the bifurcation behavior and the inlet step height. The above effect, which has been documented by some experiments reported in BRT and recently investigated in a physical model of a braided network [Bertoldi et al., 2005], would also deserve further attention.

[58] Finally, through the inclusion of the mechanism whereby channel width evolves differently as a response to water discharge rise or fall, the model predicts that the initial morphology of the bifurcation influences its final equilibrium configuration. Thus, even in the fairly idealized case of constant discharge considered herein, a range of possible equilibrium solutions exists: A bifurcation generated from the incision of a new, initially narrow, channel evolves toward a different configuration with respect to a bifurcation which, at its initial stage, is fairly symmetrical.

[59] Some, possibly important, effects have been disregarded in the present model, namely, the role of planimetrical geometry, the three dimensionality as well as the unsteady character of the flow field.


[60] This work has been developed within the framework of the “Centro di Eccellenza Universitario per la Difesa Idrogeologica dell'Ambiente Montano-CUDAM” of the project “La risposta morfodinamica di sistemi fluviali a variazioni di parametri ambientali COFIN 2003” cofunded by the Italian Ministry of University and Scientific Research and the University of Trento and of the project “Rischio idrogeologico e morfodinamica fluviale RIMOF” cofunded by Fondazione Cassa di Risparmio di Verona Vicenza Belluno e Ancona and the University of Trento. The authors thank the anonymous referees for their constructive comments which contributed to improve the manuscript.