Water Resources Research

River bifurcations: Experimental observations on equilibrium configurations

Authors


Abstract

[1] In this work we have investigated the equilibrium configurations of a Y-shaped fluvial bifurcation through a laboratory analysis. Three series of experimental runs have been performed in a wide flume, where a symmetrical bifurcation has been constructed joining three branches with fixed banks and movable bed made of a well sorted quartz sand; the angle between the two downstream distributaries was equal to 30 degrees. The experiments have been carried out with different values of longitudinal bed slope and water discharge, in order to investigate a range of the relevant morphodynamic parameters typical of gravel bed braided rivers. The equilibrium configuration of the bifurcation has been characterized through the measure of the discharge partition in downstream branches and of the local bed structure at the node. The existence of unbalanced equilibrium configurations has been observed and the role of migrating alternate bars has been pointed out. The experimental results confirm the theoretical predictions which have been recently obtained through the simple model of Bolla Pittaluga et al. (2003). Moreover, interpreting the measured data in the light of the concept of morphodynamic influence provides a new perspective in the analysis of the equilibrium configurations of a bifurcation.

1. Introduction

[2] Field inspection of channel bifurcations within braided rivers often reveals the occurrence of markedly unbalanced configurations (Figure 1). Such an asymmetry is typically displayed in terms of the geometrical properties of downstream branches, namely channels alignment, bed topography and width [Federici and Paola, 2003; Miori et al., 2006]. Furthermore, it is also reflected by the structure of the flow field and the uneven partition of flow and sediment discharge at the node [Zolezzi et al., 2006]. Though the above unbalance is more evident at low stages, it also characterizes active bifurcations undergoing formative events.

Figure 1.

Bifurcation in a braided river (Sunwapta River, Canada).

[3] One could argue that the asymmetry of single nodes within a braided reach is somehow related to the highly unsteady character of the network and mainly results from the interference of various processes, such as bed forms migration and channel shift, which continuously force the system to depart from the condition of local equilibrium. Conversely, asymmetry could be inherently related to the free response of the system, the unbalanced configuration being a possible final equilibrium state of freely evolving bifurcations.

[4] The analysis of the above aspects and the identification of the recurring processes characterizing the time evolution of a bifurcation are key ingredients in order to understand and predict the dynamics of a braided network [Ashmore, 2001; Bolla Pittaluga et al., 2001]. In fact, the bifurcation mechanism represents the main cause of the braided nature of a river [Leopold and Wolman, 1957; Ashmore, 1991; Richardson and Thorne, 2001; Bertoldi and Tubino, 2005] and strongly influences the way through which water and sediments are distributed and delivered further downstream [Ferguson et al., 1992; Bristow and Best, 1993; Burge, 2006]. However, modeling the morphological evolution of a bifurcation and predicting the water and sediment partition at the node are still challenging issues for existing analytical and numerical models, even in simple configurations [Paola, 2001; Klaassen et al., 2002; Jagers, 2003; Hardy et al., 2005].

[5] Some attempts have been recently pursued to investigate the equilibrium and the stability of single nodes through a one-dimensional model, coupled with a suitable description of the flow and sediment exchange at the bifurcation [Bolla Pittaluga et al., 2003;Miori et al., 2006]. In spite of the strongly simplified character of the approach, the model seems able to capture the main ingredients embodied in bifurcations dynamics. The above theoretical results, which will be briefly summarized in the next section, claim for a laboratory validation, which is also needed in order to calibrate the parameters adopted in the model to describe flow and sediment partition at the node.

[6] In the present work we report on the results of a systematic set of experimental runs that we have performed with the aim of providing a quantitative description of the possible equilibrium states of a simple ‘Y-shaped’ bifurcation with movable bed and fixed banks. The attention has been mainly focused on the measure of the equilibrium properties of the bifurcation, namely the discharge partition at the node and the altimetric pattern, for different hydraulic conditions of the upstream channel. Furthermore, we have also experimentally investigated the stability of such equilibrium states and analyzed the effect induced by the occurrence of bars propagating in the upstream channel.

[7] Experimental data allow for a direct validation of the theoretical predictions of Bolla Pittaluga et al. [2003]. Moreover, a novel feature which arises from the experimental results is the observation of the upstream influence exerted by the bifurcation. Analyzing such process in the light of the concept of morphodynamic influence, as it has been defined by Zolezzi and Seminara [2001], opens a new perspective in the interpretation of the dynamic of bifurcating streams and greatly facilitates the analysis of experimental results.

[8] The theoretical framework to which we refer is reported in the next section; in the subsequent sections the experimental set up, procedures and results are described. A comparison between the theoretical predictions of Bolla Pittaluga et al. [2003] and experimental data is included in the last section along with some concluding remarks.

2. Theoretical Framework

[9] The analysis of the equilibrium configurations and stability of a single bifurcation has been recently tackled within the context of a one-dimensional approach by Wang et al. [1995] and by Bolla Pittaluga et al. [2003] (hereinafter referred to as BRT). In the above works a simple ‘Y-shaped’ configuration is considered, in which the upstream channel, named a, divides into two downstream branches (b and c). All branches have constant width (fixed walls) and longitudinal bed slope, measured along the axis of each branch. The theoretical analysis seeks the equilibrium states of the system subject to given values of water discharge Q and sediment discharge Qs.

[10] Within the framework of a one-dimensional model with movable bed, five nodal conditions are needed at the bifurcation. In this respect, results of theoretical models suggest that the system response is quite sensitive to the relationship adopted to distribute the sediment load in the downstream branches. BRT have introduced a suitable nodal condition, based on a quasi two-dimensional scheme, to determine the discharge partition at the node. In their model the last reach of the upstream channel is divided into two adjacent cells, whose longitudinal length is equal to αba, where ba is the upstream channel width and α is an order-one parameter. The sediment inputs in downstream branches are then computed solving the mass balance equation applied to each cell and accounting for the lateral exchange of sediments which is driven both by flow partition and by the difference in bed elevation between the two cells. It is worth noticing that through this term the model retains the required coupling between the exchange processes at the node and the local bed structure. The system of nodal point conditions adopted in BRT is then completed imposing the water balance at the node and setting the constancy of water levels in the three channels joining at the node. The model of BRT has been recently extended by Miori et al. [2006] to the case of self-forming channels with erodible banks.

[11] We refer the interested reader to the original paper of BRT for further details. Here we recall that the theoretical results obtained by BRT can be expressed in terms of the relevant dimensionless parameters characterizing the upstream flow, namely the longitudinal bed slope S, the Shields stress ϑ and the aspect ratio β:

equation image

where D is the reach averaged value of water depth, Ds is the mean grain size, ρs and ρ are the sediment and the water density, respectively, g is gravity and τ is the average bed shear stress. The subscript a refers to the upstream channel.

[12] For a symmetrical bifurcation (that is, when downstream branches have identical width and slope) BRT have found that a unique equilibrium solution exists provided the Shields stress ϑa attains relatively high values (or the aspect ratio βa is sufficiently low): in this case the discharge distribution is balanced. On the other hand, for relatively low values of the Shields parameter (or for larger values of the aspect ratio) three possible equilibrium solutions occur. One is the balanced solution, as in the previous case; the other two are reciprocal and correspond to unbalanced configurations characterized by an uneven distribution of flow and sediment discharge in the downstream branches. Moreover, the stability analysis of these equilibrium configurations has revealed that, when three solutions exist, only the unbalanced states are stable.

[13] An example of the equilibrium configurations obtained through the model of BRT is reported in Figure 2, in terms of the ratio Qb/Qc which measures the discharge partition in the downstream channels.

Figure 2.

The discharge distribution in downstream channels of a Y-shaped bifurcation, as predicted by the model of Bolla Pittaluga et al. [2003], as a function of the aspect ratio, for different values of the Shields parameter.

[14] The theoretical model of BRT also predicts that the unbalanced solutions are invariably characterized by a non symmetrical altimetric configuration, such that the bed levels at the inlet of the two downstream branches are different. Furthermore, the larger is the imbalance of flow partition at the node (i.e., the smaller is Qb/Qc), the larger is the amplitude of this gap between the two branches. The role played by the inlet step in the nodal point condition is quite relevant since it determines a transverse slope which redistributes the excess of sediment load toward the inlet of the downstream branch that carries much water and consequently requires a larger sediment supply.

[15] The above results document the ability of the theoretical model of BRT to reproduce the asymmetry which is often displayed by the bifurcations within a braided network. They also suggest that, under suitable conditions, such an asymmetry is inherently related to the dynamics of the node and can be produced even in the absence of external forcings, like that related to the migration of alternate bars in the upstream flow. The latter effect, which will be analyzed in the next sections, has been neglected in the theoretical model of BRT; however, in their paper some experimental observations are reported for the case of a bifurcation angle equal to zero (a channel divided by a longitudinal wall in the downstream reach), which suggest that the migration of bars can induce fluctuations of water and sediment discharge in the two downstream branches (see Figure 4 of BRT). Similar results have been also found by Hirose et al. [2003].

[16] The theoretical results of BRT highlight the role of topographical effects acting within the final reach of upstream channel as the main drivers of the dynamical behavior of the node. The above finding sets the opportunity to investigate such behavior in the light of the concept of morphodynamic influence. Within a two-dimensional framework such influence manifests itself through the occurrence of bed responses displaying a lateral bed structure, namely large-scale sediment waves termed bars. Since the original work of Olesen [1983], it has been recognized that a non uniform initial condition as well as a variation in channel geometry or curvature is able to induce the formation of stationary two-dimensional perturbations of bed topography (see also the recent review paper of Lanzoni et al. [2006]).

[17] Stationary undulations are typically triggered by planimetric discontinuities which cause an abrupt change in the alignment of the main flow. A notable example is the “overdeepening” phenomenon produced by a sharp variation of curvature: Struiksma et al. [1985] first documented, in a straight channel followed by a curved reach, the occurrence of a sequence of non migrating alternate bars whose amplitude was decaying slowly in the downstream direction. Blondeaux and Seminara [1985] have shown that the analysis of stationary two-dimensional perturbations is also relevant for river meandering since such modes can be resonantly excited in meandering channels with a periodic distribution of channel curvature, as the aspect ratio β approaches a resonant value βr. Under bed load dominated conditions such resonant value depends on the Shields stress and the relative roughness [Seminara and Tubino, 1992, Figure 4a]: in particular, at relatively low values of the Shields stress ϑ, say less than 0.1, βr falls in the range 5–20; its value increases sharply for larger values of ϑ.

[18] In subsequent works Zolezzi and Seminara [2001] and Zolezzi et al. [2005] have shown both theoretically and experimentally that the resonant value of the aspect ratio also controls the direction toward which the morphodynamic influence is dominantly propagated. In particular they have shown that such influence is mainly felt upstream or downstream depending upon β being larger (super-resonant channels) or smaller (sub-resonant channels) than βr. It is worth pointing out that the above two-dimensional viewpoint offers a complementary view on the general issue of morphodynamic influence with respect to the classical one-dimensional theory [e.g., de Vries, 1965]. In particular, in such more general context the existence of a wider spectrum of both migrating and stationary sediment waves carrying information related to bed topography has been recognized. Furthermore, both theoretical results and experimental evidence suggest that the aspect ratio of the channel is likely to play for two-dimensional bed forms (namely, those controlling the planform shape of channels) a role similar to that of the Froude number for long purely longitudinal one-dimensional bed waves. Examples in this respect are the critical value of the aspect ratio βc which sets the occurrence of migrating alternate bars in straight channels [see Colombini et al., 1987] or the resonant aspect ratio.

[19] The above distinction between sub-resonant and super-resonant cases may turn out to be relevant also in the present analysis of the equilibrium configuration of bifurcations. In fact, like an abrupt change of curvature, also a bifurcation acts as a planimetric discontinuity which causes a sudden deviation of the flow direction with respect to the incoming flow. Furthermore, the uneven distribution of flow discharge at the node, with the consequent asymmetry of bed level at the inlet of downstream branches, is more likely to produce an upstream influence on bed topography in super-resonant conditions. Hence we may expect that a stronger topographic control can be exerted on flow and sediment distribution at the node when β > βr.

3. Experimental Set Up and Procedure

[20] The experimental runs were performed in the ‘π flume’, a large facility (25 m long and 3.14 m wide) for movable bed experiments located in the Hydraulic Laboratory of the University of Trento.

[21] The ‘π flume’ is equipped with an instrumentation to monitor the bed topography. The survey is carried out through a laser device, supported by a carriage. High accuracy is ensured by a system of rails and motors, which allows a precise positioning of the measuring instrumentation along the longitudinal, transversal and vertical directions. On the same carriage a water gauge is positioned, in order to measure the free surface level. The management of the whole system (water and sediment feeding, measuring system) is completely automated and controlled by a software.

[22] The flume was filled with a well-sorted, sieved quartz sand. The water discharge was supplied by a pump, regulated with an inverter, that allowed to set discharge values from 0.5 to 20 liters per second. At the upstream end of the flume, the first meter was devoted to regulate the kinetic energy of the incoming flow, while at the downstream end a tailgate was placed in order to fix the bed elevation, after which a chute conveyed the flow to a submerged tank. The sediment input was provided by an open circuit, consisting of a volumetric sand feeder, three electric motors, and a diffuser that conveyed the sand into the flume. Dry sand was supplied in order to ensure a constant and well defined input, a condition particularly relevant for the case of low sediment rates.

[23] A ‘Y-shaped’ bifurcation was built inside the flume, constructing three channels with fixed walls, movable bed and rectangular cross section joining at the node. Channel width was set to 0.36 m and to 0.24 m for the upstream channel and for the two downstream channels, respectively. The above values were chosen on the basis of the outcomes of rational regime theories [see for example Griffiths, 1981], that suggest a non-linear relationship between the flow discharge and the channel width, implying a ratio of 1.3 between the total width of the downstream channels and the upstream width, in the case of symmetrical configurations [Ashmore, 2001]. The bifurcation angle was set to 30 degrees, with the two distributaries diverging symmetrically from the direction of the upstream flow. The whole system was approximately 12 m long and ended with a series of tanks and mill weirs, which allowed the measure of the water discharges flowing into the two downstream branches.

[24] Two sets of runs were performed using a sieved sand with a 0.63 mm mean diameter and grain density of 2630 kg/m3. In the third set a coarser sediment was employed, with a 1.05 mm mean diameter. Such values of grain size were chosen in order to reproduce in the model transport conditions similar to those occurring in gravel bed braided rivers. In this respect the laboratory model can be viewed as a generic model with a scaling factor falling in the range of 50–100.

[25] In the first set of runs the slope was regulated to the value of 0.3% in order to ensure relatively low values of the aspect ratio βa of upstream flow. Consequently the occurrence of free bars in the upstream channel was inhibited, since βa did not exceed the threshold value for bar formation as predicted by the linear theory [see for example Colombini et al., 1987]. In the other two series the slope was set to 0.5% and to 0.7%, which implied lower values of the flow depth. Consequently in almost all runs the formation of alternate migrating bars was observed, whose amplitude was larger in the third series, characterized by higher values of the aspect ratio.

[26] The experimental runs were performed according to the following procedure. At first, the bed was flattened to the prescribed longitudinal slope using a scraper attached to a carriage running along adjustable rails. The bed was then saturated with a very low water discharge in order to have a smooth surface and then surveyed with the laser profiler to check the initial conditions. Water discharge was then set to the prescribed value and the sediment discharge was regulated estimating the equilibrium bed load transport capacity under uniform flow conditions through Parker [1990] formula for gravel bed rivers.

[27] During the run water discharge was gauged through a pressure sensor device positioned in the right tank, monitoring the water depth at 1-min intervals. Additional measures were performed with graduated rods in both tanks. When flow partition and bed topography reached an equilibrium configuration the run was stopped. In the runs where migrating bars did not occur, such equilibrium was defined as the final steady state reached by the discharge distribution in downstream branches. In the other cases, due to the oscillatory behavior of discharge partition induced by the migration of bars in the upstream channel, an average equilibrium state was considered.

[28] Finally bed topography was surveyed with the laser profiler, on a grid spacing 10 cm in the longitudinal direction and 1 cm in the transverse direction.

4. Experiments Description

[29] Three sets of experiments were performed with the aim of determining the equilibrium configuration of the bifurcation, described in terms of discharge partition and bed topography, and assessing the effect on such equilibrium induced by the migration of alternate bars in the upstream channel. Different hydraulic and transport conditions in the upstream flow were reproduced by varying the water discharge Qa, keeping the longitudinal bed slope S fixed for each set of runs. The resulting values of the relevant dimensionless parameters describing the flow and sediment transport, namely the aspect ratio βa and the Shields stress ϑa, fall in the following ranges

equation image

which reproduce typical conditions encountered in natural, gravel bed, braided networks.

[30] The experimental conditions and the values of the dimensionless parameters characterizing the different runs are reported in Table 1, where Da is mean flow depth in channel a, computed through a uniform flow stage-discharge relationship, and ds is the relative roughness, defined as the ratio between the mean grain size and the flow depth. The roughness coefficient was calibrated through a series of previous measurements on a straight channel with bed composed of the same material used in the experiments. Furthermore, βr is the resonant aspect ratio for the upstream branch defined in Section 2, which has been computed in terms of the Shields stress and relative roughness. We note that, unlike the study of Seminara and Tubino [1992], where Meyer-Peter Muller formula was used to obtain the plot reported in Figure 4a, here Parker [1990] bed load formula has been adopted, which performs better for gravel bed rivers, particularly at low transport rate. In a few cases, the runs were repeated in order to check the reproducibility of the results and to assess the absence of any influence due to possible asymmetries in the initial configuration.

Table 1. Experimental Conditions and Relevant Dimensionless Parameters
RunDs, mmSQ, liters/sDa, mβaϑadsaβrrQΔη
F3-180.630.00311.80.016710.770.04590.0385.750.290.694
F3-200.630.00262.00.01899.550.04250.0335.810.460.594
F3-210.630.00272.10.01919.450.04530.0335.890.560.514
F3-230.630.00312.30.01949.260.05240.0326.460.730.365
F3-250.630.00262.50.02158.380.04870.0296.220.650.312
F3-290.630.00312.90.02238.080.05990.0288.470.800.153
F3-370.630.00333.70.02547.090.07210.02510.500.910.043
F3-450.630.00374.50.02776.500.08730.02312.110.990.022
F3-610.630.00296.10.03634.950.08550.01712.610.970.019
F5-201.050.00462.00.01849.530.04390.0575.210.170.871
F5-251.050.00412.50.02178.070.04610.0485.470.610.350
F5-301.050.00473.00.02337.500.05570.0456.600.740.291
F5-401.050.00424.00.02896.060.06030.0368.320.950.139
F5-501.050.00585.00.03005.840.08600.03511.110.960.107
F7-060.630.00650.60.006826.300.04200.0924.610.001.307
F7-070.630.00660.70.007523.910.04620.0844.810.001.370
F7-080.630.00770.80.007723.300.05590.0825.720.001.726
F7-090.630.00780.90.008321.660.06060.0766.860.251.318
F7-100.630.00671.00.009319.380.05740.0686.240.051.228
F7-120.630.00701.20.010217.710.06550.0628.070.450.821
F7-130.630.00761.30.010517.210.07340.0608.840.500.773
F7-150.630.00761.50.011315.880.07870.0569.490.500.397
F7-170.630.00681.70.012614.280.07840.0509.690.450.722
F7-200.630.00782.00.013413.450.09430.04711.111.000.485
F7-240.630.00722.40.015311.730.09850.04111.731.000.198

[31] Table 1 also reports the measured equilibrium values of the discharge ratio rQ = Qc/Qb, where channel ‘b’ is the main downstream channel.

[32] Figure 3 shows the time evolution of the discharge ratio rQ observed in three different experiments. These plots highlight the different behavior detected in the runs without free bars (a), as compared to that observed in the runs characterized by the presence of migrating alternate bars in the upstream channel (b, c).

Figure 3.

The time evolution of the discharge ratio rQ of downstream branches measured in three different runs:(a) F3-21; (b) F7-24; (c) F7-13.

[33] In the former case (see Figure 3a) the system reached an equilibrium configuration in a relatively short time (say few hours, depending on the sediment mobility), following an asymptotic trend. Moreover in few runs, namely those characterized by higher values of the Shields parameter of the incoming flow, the bifurcation kept balanced (rQ being approximately equal to 1). The stability of the equilibrium configuration reached at the end of these runs was also verified perturbing the final equilibrium state with an amount of sand fed in one of the downstream branches: in every tested configuration the system returned to the same equilibrium condition reached before the perturbation.

[34] The evolution of the second and third sets of runs was strongly affected by the development and migration of alternate bars, mainly in the upstream channel. As a consequence, the recorded evolutionary pattern of discharge partition at the node was more complex (two examples are reported in Figures 3b and 3c), resulting in a less regular trend of the measured data. The first example (Figure 3b) shows the results of a run with a relatively high value of the Shields stress ϑa (run F7-24), in which rQ displayed small oscillations around the final equilibrium value; in this case the discharge distribution kept almost balanced and the presence of bars affected the bifurcation only slightly. On the contrary, the second example (run F7-08, Figure 3c) shows the results of an unbalanced run. Here, the occurrence of migrating alternate bars marked the subsequent behavior at the node since the early stage, causing a sudden instability of the bifurcation. Because of bed forms migration the flow then switched from one channel to the other, leading to the closure and re-opening of the downstream branches. We note that in this case the oscillation between two unbalanced states, induced by bar migration, often corresponded to mirror solutions, the discharge ratio oscillating between two reciprocal values. As said before, when bar migration dominated the evolutionary process at the node, the identification of the equilibrium state was more complex and the quantification of the equilibrium values of the bifurcation parameters was obtained in terms of the averaged values reached by the system.

[35] We invariably observed that the unbalanced runs were characterized by a distinctive asymmetry in the bed topography of downstream distributaries. In particular the branch carrying a lower value of the discharge was generally subject to an aggrading process, such that, at equilibrium, the main downstream branch exhibited in all cases a lower mean bed elevation. The maximum value of the above difference can be used as a characteristic measure of the equilibrium morphological response of the bifurcation. Measured values of such ‘inlet step’ Δη at equilibrium, normalized with the flow depth Da, are also reported in Table 1. Figure 4 shows an example of the longitudinal bed profiles measured along the three channels joining at the node.

Figure 4.

The longitudinal bed profiles of the downstream branches measured at equilibrium in run F3-21.

[36] In order to estimate the inlet step Δη we tested three different procedures. First, we determined Δη as the reach averaged value of the difference in bed elevation of downstream branches; the length of the test reach was chosen approximately equal to two times the downstream channel width. In a second procedure, the longitudinal profiles of the downstream channels were linearly interpolated, over a length of approximately 10 widths, and Δη was computed as the relative distance of these two regression lines at the bifurcation section (see Figure 4). A third estimate of the amplitude of the inlet step was obtained through a Fourier Transform analysis of bed elevation data collected at single cross sections upstream of the bifurcation. In this case Δη was computed as the amplitude of the principal harmonic, with transverse wavelength equal to two channel widths. It is worth pointing out that the three selected procedures lead to very similar values for Δη. Results reported in the following section have been obtained through linear interpolation of the bed profiles of downstream branches.

[37] We also observed that, in the unbalanced runs, the asymmetrical bed configuration at the node induced the formation of a steady altimetric pattern in the region upstream of the bifurcation, with a transverse structure similar to that of alternate bars (i.e., non symmetrical with respect to the channel axis), but with a wavelength ranging about 15 times the channel width (almost twice the length of migrating bars). Figure 5 shows an example of the bed topography of the upstream channel with migrating alternate bars (a) and with the steady pattern induced by the imbalanced bifurcation (b). The latter configuration, whose amplitude decays slowly in the upstream direction, witnesses the existence of an upstream morphodynamic influence induced by the bifurcation.

Figure 5.

Pictures and bed topography maps of the final reach of the upstream channel showing the presence of migrating bars (a) and of steady longer bars caused by the morphodynamic influence of the bifurcation (b).

5. Experimental Results

[38] The measured values of rQ and Δη for the three series of runs are reported in Figures 6 and 7 as functions of both the Shields stress and the aspect ratio of the upstream channel. We note that, due to the fixed width of the branches, higher values of ϑa correspond to lower values of βa.

Figure 6.

Equilibrium values of the discharge ratio rQ of downstream branches as a function of the Shields stress ϑa and of the aspect ratio βa of the upstream channel.

Figure 7.

Equilibrium values of the dimensionless inlet step, Δη, as a function of the Shields stress βa and of the aspect ratio ϑa of the upstream channel.

[39] The main outcome of present experimental observations is the recognition that for given slope, grain size and channel width the ability of the bifurcation to keep a more or less balanced discharge partition depends on flow discharge. In particular, the smaller is the discharge of the incoming flow, the larger is the observed imbalance at the node. In dimensionless form this implies that the node can keep balanced (i.e., rQ ≃ 1 and Δη ≃ 0) only for relatively large values of the Shields stress of the upstream flow (or, equivalently, for relatively small values of the aspect ratio). Furthermore, the discharge ratio becomes smaller and the amplitude of inlet step larger for decreasing (increasing) values of ϑa (βa). The above finding doesn't seem to be appreciably affected by the occurrence of migrating bars in the upstream flow, in spite of their strong influence on node dynamics, as described in the preceding section. It is worth noting that the theoretical findings of BRT appear quite consistent with the above experimental scenario.

[40] Figure 6 shows that, in a few runs characterized by extremely low values of the Shields parameter, one of the downstream branches was almost completely dry (rQ ≃ 0). In such conditions, as suggested by Figure 7, the transversal difference of bed elevation at the bifurcation attained, at equilibrium, a value comparable with the flow depth in the upstream channel (Δη ≃ 1). We note, however, that in the runs characterized by the formation of alternate bars even this strongly asymmetrical configuration was not observed to keep invariably stable. In fact, the sudden aggradation process induced in the only (temporarily) active branch by the migration through it of two consecutive bars was often able to activate again the closed branch, shifting the node to the opposite equilibrium configuration (see Figure 3 for an example).

[41] Experimental results highlight the close relationship between the discharge distribution in downstream channels and the amplitude of the inlet step. A linear regression of the data set of the two first series (S = 0.3% and S = 0.5%) leads to a simple relationship (Figure 8):

equation image

stating that the sum of rQ and Δη is approximately equal to 1. The equilibrium configurations attained in the third series of runs do not show such a close relationship, due to the strong influence of migrating alternate bars on the bed topography near the node.

Figure 8.

Relationship between the measured equilibrium values of the discharge ratio and of the dimensionless amplitude of the inlet step.

[42] As pointed out in Section 2, it is of interest to analyze present experimental results in terms of the theory of morphodynamic influence [Zolezzi and Seminara, 2001]. On the basis of the computed values of the resonant aspect ratio βr reported in Table 1 the experimental runs can be divided into two groups: sub-resonant and super-resonant runs. The measured values of rQ and Δη are reported in Figures 9 and 10 as a function of the relative distance from the resonant conditions (ββr)/βr. It appears that the measured values display a quite distinct behavior depending upon β being smaller or larger than βr: in sub resonant runs the bifurcation keeps almost balanced and symmetrical, whereas in super-resonant conditions the bifurcation evolves toward an unbalanced configuration; furthermore, the degree of asymmetry of the bifurcation becomes larger as the distance from the resonant value increases.

Figure 9.

The discharge ratio in the downstream branches rQ as a function of the relative distance from resonant conditions.

Figure 10.

The amplitude of inlet step Δη as a function of the relative distance from resonant conditions.

[43] It is important to point out that taking the above viewpoint provides a unified interpretation of experimental results, as the relative distance from the resonant range emerges as the controlling parameter of the bifurcation asymmetry: in fact, Figures 9 and 10 suggest that the points of the whole set of runs, which correspond to different values of longitudinal slope and grain size, lay approximately on the same curve.

[44] The above results can be given the following simple interpretation. In super resonant conditions any departure from the balanced and symmetrical configuration at the node can produce an upstream influence which leads to a steady perturbation in the upstream channel causing a transverse bed deformation. This in turn induces a topographical forcing on the approach flow and diverts a greater percentage of the water discharge in one of the downstream branches, further promoting the development of an unbalanced configuration. On the other hand, in sub-resonant conditions the bed topography just upstream the bifurcation can keep nearly flat because the morphodynamic influence of the node is not felt upstream; hence, the discharge distribution is more likely to keep symmetrical. Figure 10 suggests that for values of the aspect ratio lower than βr the amplitude of the inlet step is generally negligible.

[45] In the light of the above considerations we can conclude that the present results provide a strong experimental support to the theory of morphodynamic influence, as it has been recently proposed by Zolezzi and Seminara [2001].

6. Comparison With the BRT Theoretical Model

[46] The experimental findings have been used to test the accuracy of the theoretical predictions of the BRT model. Unlike in BRT, where Meyer-Peter and Muller relationship was used, here Parker [1990] bed load formula has been adopted in order to avoid discontinuity at low transport rates.

[47] The theory of BRT allows one to determine a threshold curve on the plain βa − ϑa that separates the region where a symmetrical bifurcation can keep balanced and stable from the region where the system gets toward an unbalanced configuration, characterized by a dominant downstream branch. In Figure 11 the whole set of experimental points are reported, where the runs with values of the discharge ratio rQ larger than 0.95 have been considered as “balanced”. The comparison with the theoretical curve determined through BRT model (for the same set of conditions of the upstream channel) shows a good agreement, also in the case of runs strongly affected by bar migration.

Figure 11.

The occurrence criterion of balanced and unbalanced equilibrium configurations, as predicted by the theoretical model of Bolla Pittaluga et al. [2003], is tested against the experimental results on the βa − ϑa plane.

[48] It is worth recalling that BRT model assumes that the transverse exchange of flow and sediment transport just upstream the node is concentrated over a length of few channel widths; such length, which provides an estimate of the upstream distance over which the effect of the bifurcation is mainly felt, is set by BRT equal to (αba), where α is an order one parameter. We may note that the predicted equilibrium values of rQ and Δη depend appreciably on the choice of the α value. Hence such parameter has been calibrated in order to fit the experimental data and to quantify correctly the degree of asymmetry of the bifurcation. The optimal value of α for the overall comparison reported in Figure 11 has been found to be equal to 6. Furthermore, different theoretical predictions have been computed, changing the value of α, in order to fit the experimental values of the discharge asymmetry rQ at equilibrium. We have found that the values of α that optimize the comparison with laboratory results depend on the relative distance from the resonant conditions (see Figure 12). Data of the third series of experiments have not been considered in the above analysis, since these runs were strongly affected by the migration of bars in the upstream channel and the measured equilibrium values are, therefore, less reliable. Furthermore, the model of BRT didn't account for two-dimensional effects associated with bar occurrence. The optimal value of α increases sharply as βa approaches the resonant range, which implies that under these conditions the length over which the morphodynamic influence of the node is felt may be relatively large. The above finding is not surprising: in fact, theoretical results [Seminara and Tubino, 1992; Zolezzi and Seminara, 2001] suggest that at resonance (β = βr) the spatial damping of bed perturbations forced by the discontinuity should vanish. Furthermore, Figure 12 shows that in sub-resonant and strongly super-resonant conditions the optimal value of α tends to 1. The above results seem to indicate that the use of an appropriate relationship to estimate the parameter α, where its dependence on (ββr) is accounted for, could incorporate in BRT model further two-dimensional ingredients otherwise neglected.

Figure 12.

The values of the parameter of the model of Bolla Pittaluga et al. [2003] that optimize the theoretical estimate of rQ.

[49] Finally, it is worth mentioning that the predicted values of Δη, computed with the optimized values of α obtained through the above procedure, are in a good agreement with the experimental results. This confirms the ability of BRT model to correctly reproduce the relationship between the local bed topography and the discharge partition at the node.

7. Conclusions

[50] In the present work the attention has been focused on the morphodynamics of a ‘Y-shaped’ bifurcation, constituted by an upstream channel dividing into two symmetrical downstream branches. The study has been carried out through three sets of experimental runs, in a flume with movable bed and fixed walls. A systematic investigation of the equilibrium configurations of the bifurcation has been performed, with a quantitative description of both the discharge distribution in the downstream branches and the bed topography in the region affected by the channel division.

[51] From the analysis of experimental data and the comparison with theoretical predictions of BRT model the following outcomes can be pointed out.

[52] 1. The present experimental results clearly demonstrate the existence of unbalanced equilibrium configurations for high values of the aspect ratio and low values of the Shields stress. These observations can physically explain why natural braided rivers are likely to concentrate the discharge and consequently the morphodynamic activity in a few channels [Mosley, 1983; Stojic et al., 1998].

[53] 2.The predictions of the one-dimensional theory recently proposed by BRT are confirmed, at least in a qualitative way. The experimental observations highlight the crucial role played by the local bed structure just upstream the bifurcation in governing the dynamics of the bifurcation. They also suggest that the theoretical findings are not significantly altered by the presence of alternate bars in the upstream channel, at least on the average, though their migration can affect strongly the evolution of the bifurcation.

[54] 3.The analysis of the morphodynamic influence of the bifurcation, with reference to the theoretical framework proposed by Zolezzi and Seminara [2001], provides a unified interpretation of the experimental data measured at equilibrium. Experimental findings indicate that the final configuration toward which the system is driven crucially depends on the distance (βaβr) of the aspect ratio of upstream flow from a threshold resonant value (the latter coincides with the resonant aspect ratio first discovered by Blondeaux and Seminara [1985] as that corresponding to a resonant behavior of the linear solution for flow and bed topography in meanders with periodic curvature distribution). In sub-resonant runs the node keeps a balanced water distribution in the downstream branches, whereas super-resonant conditions lead to unbalanced configurations, which are driven by the steady bed deformation induced in the upstream channel by the morphodynamic influence of the bifurcation.

[55] 4.The parameter α of BRT model, which provides a measure of the longitudinal extent of the morphodynamic influence of the bifurcation, has been calibrated through experimental data. Present results confirm that the region where two-dimensional effects induced by the bifurcation are mainly felt spreads over a length of few (say, from 1 to 7) channel widths; moreover they suggest that α is a function of the distance (βaβr) and reaches a maximum as the aspect ratio of upstream flow approaches its resonant value.

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 ‘La risposta morfodinamica di sistemi fluviali a variazioni di parametri ambientali - COFIN 2003’, co-funded by the Italian Ministry of University and ScientificResearch (MIUR) and the University of Trento and of the project ‘Rischio Idraulico e Morfodinamica Fluviale’ financed by the Fondazione Cassa di Risparmio di Verona, Vicenza, Belluno e Ancona. The paper has benefited from the comments of Rodolfo Repetto, Guido Zolezzi and two anonymous reviewers. The authors gratefully acknowledge Stefano Miori, Luca Zanoni, Stefania Baldo, David Marchiori, Andrea Casarin and the staff of the Hydraulic Laboratory, who helped in the execution of the experimental runs.

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