Water Resources Research

Dynamics of channel bifurcations in noncohesive sediments

Authors


Abstract

[1] Braided rivers produce flow confluences and divergences (diffluences) in roughly equal numbers, but braided river research has focused mainly on confluences. Divergences, however, are of equal importance dynamically: they are sites of formation of central bars and the development of flow divisions that can steer the flow toward one part of the braided network or another. In a series of experiments on flow in isolated, well-defined bifurcations, we found that although a central bar always develops, the divided flow may continue to flow on both sides of the bar (“stable” bifurcation) or may eventually be forced entirely to one side of the bar or the other (“unstable” bifurcation). We found that an unstable bifurcation forms when the flow field is characterized by both a low Shields stress and a nonuniform incoming flow. We also found that divergences with erodible banks tend to an equilibrium configuration that depends mainly on the widening ratio of the channel.

1. Introduction

[2] The practical need to investigate the fundamental processes that control the apparently chaotic morphodynamics of braided rivers arises in the context of river management. Braided rivers pose difficult problems of river regulation because of rapid, extreme, and unpredictable changes in stream patterns due to bed and bank scour during floods of even moderate intensity.

[3] The complexity of braided river patterns is such that it may be useful as a first step to identify single unit-processes or elementary “building blocks” governing the generation and development of the network, and to study them in isolation. In this sense, the two most important “unit processes” of braiding are bed scour and channel narrowing at confluences, and expansion and flow bifurcation around bars. Mosley [1976] has termed the confluence-bar couplet the “atom” of braiding. Of the two parts of this couplet, however, attention has so far been disproportionately focused on the confluence side [e.g., Ashmore and Parker, 1983; Best, 1988; Roy and Bergeron, 1990; Biron et al., 1993, Best and Ashworth, 1997]. The dynamics of expansions and flow bifurcations are equally interesting, however, and may ultimately be more important in determining the evolution of braided reaches than confluences. As we will show, bifurcations can act as switches that steer flow to one side of the channel or the other. In an attempt to redress this imbalance we report here on an experimental study of the behavior of channel bifurcations, with particular attention to the development and effects of emergent central bars.

[4] A channel bifurcation can be initiated either by deposition in midstream topographic lows or by erosional dissection of topographic highs. Ashmore [1991] identified two mechanisms of depositional bifurcation: central bars and transverse bar conversion (termed also “chutes and lobes”), and two erosional bifurcation processes: chute cutoff of point bars, and multiple bar dissection. Moreover, Ashmore [1982] described another erosional mechanism for producing a bifurcation, namely avulsion, in which a new channel is produced but the old one remains active.

[5] Briefly, central bar deposition is the process whereby an elongated, more or less symmetric, medial bar without avalanche face develops in the middle of a channel. This mechanism, apparently first described by Leopold and Wolman [1957], is the initiation mechanism of braiding most frequently cited in fluvial geomorphology texts, even though Ashmore [1991] found it uncommon in his flume experiments. Qualitative geomorphic models have described central bar initiation and growth as caused by stalling of bed load around the channel centerline [e.g., Leopold and Wolman, 1957; Ashmore, 1991; Ferguson, 1993] due to the fact that the mean bed shear stress is very close to the threshold for motion.

[6] Transverse bar conversion can be observed when flow expands out of a pool (or chute) because pools at sites of flow convergence can scour appreciably, generating enough sediment for substantial downstream deposition where the flow subsequently diverges. The lobe so created is characterized by avalanche faces and by thin bed load sheets stalling on the top or front of it. If the lobe emerges, the flow is deflected off its edges.

[7] The chute cutoff mechanism, instead, consists of a headward incision by flow taking a shortcut across an alternate or point bar. The new path has a steeper water surface slope than the flow in the thalweg, and so a potential competitive advantage.

[8] The multiple-bar dissection process occurs in multiple-bar braiding, in which local decline of competence in a flow expansion or over a bar top initiates dissection of the submerged bars.

[9] Finally, avulsion consists of the overtopping of part of the stream over a channel bank, resulting typically from bank erosion or from a rise in the local water level due to local deposition or increasing discharge. One hallmark of braided rivers is that it is relatively easy to form new channels. Thus the new channel often does not have a significant advantage over the old one, in which case both channels may remain active.

[10] Laboratory research on depositional processes [Ashmore, 1991; Ashworth, 1996] has involved analysis of conditions under which a midchannel bar is likely to occur, though this work focused mainly on the formation of a bifurcation immediately downstream of an anabranch confluence. The rationale behind this approach was that a confluence-diffluence unit is needed in order to develop a bifurcation. On the contrary the latter work did not explore the conditions under which a single channel tends to divide into two branches or the mechanisms that govern the evolution of channel bifurcations.

[11] Only recently, Richardson and Thorne [2001] have tackled the bifurcation problem and suggested that division of the velocity field into multiple threads within a single channel precedes a division in the cross-sectional morphology of the channel and appears to be a necessary prerequisite for development of a bifurcation.

[12] The present work is motivated by the above observations and by the lack of a satisfactory understanding of the mechanisms of formation and evolution of bifurcations within a single channel. The investigation proceeded in the following steps. We first built a laboratory model of a braided river (section 2) [see also Federici, 1999]: observations suggested that streamline divergence is the most common mechanism leading to the creation of a central bar, as already pointed out by Mosley [1976]; moreover, the bifurcations created in this way are not always stable, as one of the two branches may close. We were then led to perform experiments where the effect of streamline divergence on bar formation was isolated. This goal was pursued by using an erodible channel whose width increased linearly downstream (section 3). We observed that flow divergence invariably leads to the formation of central bars for any Froude number and aspect ratio (width/depth) of the incoming stream. Moreover, bifurcations created by this mechanism may be subject to two distinct modes of instability that seem to depend on the value of Shields stress, defined as: τ* = τo/[(ρs − ρ)gd], where τo is bottom stress, d is sediment diameter, ρ and ρs are water and sediment density respectively, and g is the gravitational acceleration. In the case of relatively high values of Shields stress (τ* > 0.15) we have observed stable bifurcations, that is bottom topography evolved toward a bifurcation pattern which laterally oscillated in time but kept both branches open throughout the experiment. On the contrary, in the case of low values of Shields stress (τ* < 0.15) we have observed what we term a “switch” bifurcation, i.e., a bifurcation displaying an oscillatory instability, such that one of the two branches closed, leaving the stream confined within a single branch. Note that the latter process occurred repeatedly and randomly: in other words, closure of one channel was typically followed by the flattening of the bed in the divergent reach, then by the formation of a new bifurcation followed by the closure of one of the two branches (not necessarily coincident with the branch previously closed). Finally, we have focused our attention on the process of erosion of the banks of the divergent channel due to the deviation of the flow around the central bar (section 4). We have also observed the existence of an equilibrium configuration of the banks depending on the channel divergence and on Shields stress, but independent of the initial value of the planform divergence angle and of the degree of asymmetry of the divergent reach. In section 5 some conclusions complete the paper.

2. Laboratory Observations on Bifurcations in a Model Braided River

2.1. Experimental Equipment and Procedure

[13] It is well known that the principles of hydraulic modeling can be applied to gravel bed braided streams to produce laboratory channels that are Froude models of prototype streams [e.g., Parker, 1978; Ashmore, 1982]. For a generic process study, the main ingredients that must be reproduced in the model are the hydraulically rough character of boundary, the fully turbulent character of the flow, dominance of bed load transport, and a Froude number in the same range as typical natural cases. Using these general criteria, one can develop a laboratory stream aimed to reproduce the features and the processes of gravel bed braided streams in general, rather than the exact details of a specific river section.

[14] The first set of experiments were carried out in a laboratory flume 9.5 m long and 2 m wide at St. Anthony Falls Laboratory, University of Minnesota (Figure 1) [see also Federici, 1999]. The flume was filled with nearly uniform quartz sand, characterized by a d50 of 5 × 10−4 m, and supplied with a constant water discharge of 3.5 × 10−4 m3/s, in an open loop through a constant head tank located at the head of the flume. The flume slope was set to 1.4%, sufficiently steep to generate a braided pattern at the imposed discharge. Dry sediment was fed at a constant rate from a sediment feeder, which consists of a conical hopper with a vibrating feed slot. The appropriate sediment discharge was estimated, using the formula of Meyer-Peter and Muller [1948] for given water discharge and flume slope such that the system was in equilibrium, i.e., the bed of the channel was neither degrading nor aggrading in time. This sediment discharge was equal to 100 g/min for all the experiments in this experimental set. We also performed some preliminary experiments to check the establishment of this equilibrium state.

Figure 1.

Laboratory flume 9.5 m long and 2 m wide, filled with quasi-uniform quartz sand, at St. Anthony Falls Laboratory, University of Minnesota.

[15] For this set the flow Reynolds number had a mean value ranging about 8000, the relative smoothness (Y/d50) varied between 8 and 16, Froude numbers were slightly above 1, and Shields stress had mean value of 0.15 so the sediment particles were moved as bed load. Except for the Reynolds number, these conditions are typical of natural gravel braided rivers. To characterize the bifurcations observed in the model braided river, we measured their geometry, topography, flow depth, flow velocity and sediment discharge in the upstream and in the two downstream reaches shown in Figure 2.

Figure 2.

A typical bifurcation observed in the model braided river. Solid lines indicate the upstream and the two downstream sections in which we measured bed topography, flow depth, flow velocity, and sediment discharge.

[16] Planform geometry was analyzed using vertical-incidence photographs, which included reference marks for alignment.

[17] Bed topography was surveyed using 10–15 thin metric rules distributed over the bed at fixed locations. The rules were oriented parallel to flow streamlines so that the flow was not significantly disturbed. Their absolute elevation was measured using a point gage. Bed elevations η were read on the upstream side of the rules to avoid wake effects.

[18] Flow depth, Y, was measured using dye and a digital camera, associating color density of the dye with depth through calibration and image analysis, using the technique described by Gran and Paola [2001]. We checked the dye measurement by reading water elevations directly on the rules described above. The measurement error was about 0.5 mm.

[19] Data on surface flow velocity Us were collected by a particle tracking technique using polystyrene beads as surface tracers and a digital camera with a long exposure time (Figure 3a). Assuming the vertical distribution of velocity as logarithmic, the depth averaged flow velocity, U, was calculated using the following relationship:

equation image

where k is Von Kàrmàn constant (=0.4) and u* is the friction velocity. The latter was estimated as follows: u* = Us/C, with C local Chezy coefficient given by: C = 6 + 2.5 ln(R/ε), R hydraulic radius, and ε roughness parameter, set equal to (2.5d50) after Engelund and Hansen [1967]. Note that because of the assumption of logarithmic vertical distribution of velocity, in relationship (1) the depth averaged flow velocity is assumed to be reached at a distance of 0.4 flow depth. The flow velocity averaged in a cross-section Um was evaluated as the arithmetic mean of all the measurements collected at different times and only for uniform cross-sections, i.e., those with only minor variation in depth. This measurement technique was tested in a straight channel characterized by constant flow depth and known water discharge and was found to be accurate within 6.5%.

Figure 3.

(a) Example of pictures taken by a digital camera with a long time frame, in which traces of polystyrene beads used to evaluated the superficial flow velocity are visible. The black lines indicate the thin metric rules used to survey the topography. Notice the wake phenomenon downstream the rules. The color assumed by the water is due to the dyer used to measure the flow depth. (b) Sediment trap that lies on the sand bed to estimate the local sediment discharge. Notice the light interference with the flow field.

[20] To estimate the sediment discharge we built small sediment traps to be laid on the sand bed (Figure 3b). Measurements were performed at several locations for each cross section, moving the trap from one point to another every minute. Because a slight interference with the flow field was observed downstream of the trap, the measurements were performed a little downstream of the cross-section in question. The trapped sediments were then dried and weighed.

2.2. Formation of a Braided Network

[21] Following the lead of most previous investigations, we started experiments in this set from a straight, narrow channel with trapezoidal cross section and aspect ratio (width/depth) equal to 12. The channel was cut into a flat sloping surface.

[22] Provided a sufficient discharge flows through the initial channel, the channel is not in equilibrium in Parker's [1978] sense. Thus it erodes its banks, increases its width, and decreases its depth, progressively increasing its aspect ratio. Once the stream approaches the critical conditions for alternate bar formation, alternate bars form and initially migrate downstream (Figure 4a).

Figure 4.

(a) Alternate bars form in the straight channel initially cut through the cohesionless flat sloping surface (t ≅ 0). (b) A rhythmic sequence of bumps forms on both banks while the stream displays a tendency to meander (t ≅ 20 min). (c) The stream continues to widen and central bars form through by-pass mechanisms similar to chute cutoffs typical of meandering streams. Also note the attempt of the stream to avulse (t ≅ 50 min). (d) The stream continues to widen, while irregular small scale bars form in the central part of the stream. Avulsion is regrading (t ≅ 7 h). (Flow from up to down.)

[23] However, after about 15 minutes, bar pools provide preferential sites for bank erosion. A rhythmic sequence of indentations then forms on both banks, while the stream displays a tendency to meander (Figure 4b). The periodic variation of channel width and the onset of channel curvature associated with bank erosion have an important consequence: alternate bars cease their migration, i.e., they become fixed bars. This observation was made by Ashmore [1991]. Ashmore also suggested that suppression of alternate bar migration requires that the transport capacity of the stream vanish over the bar. However, recently Repetto and Tubino [1999] have shown that bar migration may be suppressed in transporting streams provided the channel width undergoes periodic variations of sufficiently large amplitude.

[24] The development of curvature of the main stream enhances its eroding capacity; hence the bank indentations grow and migrate downstream. As a result, the stream continues to widen while central bars form through the mechanism of chute cutoff (Figure 4c).

[25] Further channel widening eventually leads to the appearance of irregular smaller scale bars in the central part of the stream (Figure 4d). Up to this stage the braided character of the stream is not yet fully developed as bars have not emerged: water essentially flows within a single, albeit irregular, channel, unless avulsion occurs (note that Figures 4c and 4d clearly show an aborted avulsion).

[26] The last stage of the process is the transformation of the multibar pattern into a multichannel pattern induced by the emergence of bars (Figure 5). The nature of the channel morphodynamics then changes: further development of the channel and bar structure arises from a continuous rearrangement of the stream network, controlled by erosion of channel banks, abandonment or merging of old channels, and creation of new channels.

Figure 5.

The multibar pattern has evolved into a multichannel pattern (t ≅ 20 h). (Flow from up to down.)

2.3. Formation of Bifurcations and Their Evolution

[27] When the stream began to develop a multichannel pattern, we focused our attention on the mechanism of central bar formation, which followed the mechanism described by Leopold and Wolman [1957]. This involves the widening and shallowing of high gradient channels containing noncohesive material. This shallowing results in a decrease of bed shear stress and the deposition of the coarser portion of bed load near the center of the channel. Gradual enlargement of this deposit, by the entrapment of other particles, ultimately results in flow diversion, incision of the divided flow and exposure of a midchannel bar (Figure 6).

Figure 6.

Observed mechanism of central bar formation in the model of braided river. Channel widening and shallowing led to the deposition of the bed load near the center of the channel. Gradual enlargement of this deposit, by the entrapment of other particles, ultimately results in diversion of the flow and exposure of a midchannel bar. (Flow from up to down.)

[28] The central bars we observed developed in the first 2 m downstream of the flume entrance, within a channel devoid of any other braiding activity except for avulsion. This allowed us to study well defined bifurcations, with only two downstream channels and no lateral inflow or outflow (Figure 2).

[29] We tried to characterize the geometry and hydrodynamics of the bifurcations observed once they seemed to have reached a stable configuration [see also Federici, 1999]. Typically, the two downstream branches of the bifurcation were characterized by the same flow velocity U and flow depth Y (both averaged in the cross section), by the same bottom slope if and by the same angle γ between the centerline of each branch and the centerline of the upstream reach. However, the widths b of the two branches were generally unequal (typically in a ratio 1:2) as were water and sediment discharge, Qw and Qs respectively. The observed values of parameters characteristic of the various bifurcations are reported in Table 1.

Table 1. Observed Values of Parameters Characteristic of the Two Downstream Branches of the Bifurcations Observeda
RunUa, cm/sUb, cm/sYa, cmYb, cmifaifbγaγbba, cmbb, cmQwa, cm3/sQwb, cm3/sQsa, g/minQsb, g/min
  • a

    The two downstream branches are indicated with the subscripts a and b. All the values are average in the cross section.

1Bif29340.50.80.0180.0162330723101.5625.511.553.3
2Bif33280.50.40.0760.07429351117181.5190.516.544.2
3Bif33270.450.50.0380.0403135725104.0337.523.251.0
4Bif32300.750.60.0100.0243424818192.0324.023.152.3
5Bif30320.60.50.0220.0301847187324.0112.572.216.8
6Bif38410.70.450.0230.032357914239.5258.551.329.2
7Bif38330.60.80.0200.0263021169364.5237.551.728.9
8Bif42370.450.550.0260.017321151694.5325.521.053.0

[30] The emerging portion of the central bar always had a well defined triangular shape characterized by an angle δ varying between 60° and 90°, about 1.5 times the angle (γa + γb) formed by the centerlines of the two branches (Figures 7 and 8).

Figure 7.

In the observed bifurcations the emerging portion of the central bar had always a well-defined triangular shape characterized by an angle δ, about twice the angle γ = (γa + γb) formed by the centerline of the two downstream branches.

Figure 8.

Comparison between the values of the angle δ, characterizing the emerging portion of the central bar in a bifurcation, and the values of the angle γ = (γa + γb), formed by the centerlines of the two branches of a bifurcation, observed in the bifurcations studied.

[31] Finally, we followed the morphodynamic evolution of the bifurcations in order to ascertain qualitatively whether they would asymptotically reach a stable configuration. It turned out that in some cases bifurcations are stable (both branches remain open); in other cases they were found to be unstable (a single channel on one side captured all the flow), as shown in Figure 9. This instability mechanism will be explained further in the following sections.

Figure 9.

Observed morphodynamic evolution of an unstable bifurcation: a single channel on one side captured all the flow. (Flow from up to down.)

3. Laboratory Observations on Bed Topography in a Divergent Erodible Channel

[32] We then focused our attention on the effect of streamline divergence on bar formation by performing experiments in a channel whose width increased linearly downstream. The initial experiments were performed at St. Anthony Falls Laboratory, University of Minnesota, and the later ones at the Hydraulic Laboratory of the Department of Environmental Engineering, University of Genoa.

[33] The first set of experiments was designed to confirm the observation that streamline divergence induces the formation of a central bar and observe the evolution of bed topography in a divergent channel.

[34] In the second set of experiments we explored how the evolution of the bed topography depends on the geometry of the divergent reach and on the hydraulic-sedimentologic conditions of the incoming stream.

3.1. St. Anthony Falls Experiments

[35] We went on using the flume described in section 2.1, which had the advantage of being 2 m wide so that the stream was nearly unconfined laterally. The bed material consisted again of quartz sand with d50 of 5 × 10−4 m, and the initial configuration of the bed was plane with a constant slope. A divergent stream was obtained by constructing a straight channel 0.15 m wide and 0.7 m long that was linearly widened from its initial width to the full 2 m width of the flume. The angle between the centerline of the flume and either divergent bank was 20°. The first 0.45 m long reach of the narrow straight channel was characterized by fixed banks, while the following 0.25 m long straight reach and the whole divergent reach had erodible banks. Obviously, from the beginning of the experiments the initial sharp angle formed by the upstream straight reach and the divergent reach was slightly smoothed. The channel was supplied with a constant water discharge (3.5 × 10−4 m3/s) and a constant sediment flux that was calculated using the formula of Meyer-Peter and Muller [1948] for given initial slope of the bed such that the system was in equilibrium. The achievement of equilibrium was verified by checking that the average slope of the bed neither increased nor decreased. During the experiments the development of bottom topography and free surface elevation was monitored for several hours on a fixed grid, as described above.

[36] We confirmed the observation that the streamline divergence always induces the formation of a central bar. Moreover, we followed the evolution of the bifurcations thus created and made the following observations. In the first experiment (run 1Mn), with constant initial slope and sediment discharge set equal to 0.04 and 200 g/min respectively, bottom topography evolved toward a bifurcating pattern that oscillated laterally in time but in which no branch ever closed completely throughout the experiment. We term these “stable” bifurcations. On the contrary, in the second experiment (run 2Mn), in which the initial slope and the sediment discharge were reduced to 0.02 and 100 g/min respectively, the bifurcation displayed an oscillatory instability in that one of the two branches was closed, leaving the stream confined within a single branch. Note that the latter process occurred repeatedly and randomly: in other words, closure of one channel was typically followed by the flattening of the bed in the divergent reach, then by the formation of a new bifurcation followed by the closure of one of the two branches (not necessarily coincident with the branch previously closed; Figure 10). Hence the experiments suggest that diverging streams on erodible beds are subject to at least two distinct modes of instability: in the former mode a stable bifurcation arises with two channels invariantly separated by a central bar; in the latter mode, what we call a “switch” bifurcation develops with all the flow alternately confined to one of the two single channels. Because the conditions of the two experiments differ essentially in having different values of the Shields stress τ* of the upstream straight reach (Table 2), these data suggest that a high value of τ* (>0.15) leads to a “stable” bifurcation while a lower value of τ* (<0.15) leads to a “switch” bifurcation. As we will discuss below, this initial inference must be checked further and qualified by a dependence on conditions upstream of the bifurcation revealed by the next series of experiments.

Figure 10.

The evolution of bed topography in a divergent channel. At the beginning the whole divergent reach is covered by water; after 30 minutes, a bifurcation formed, and after 30 more minutes, only one branch remained open. The sequence repeated in time. (The intense gray denotes scour and the lighter gray denotes deposit; the arrows indicate the position of the main flow.)

Table 2. Parameters Characteristic of the Upstream Incoming Flow in the St. Anthony Falls Experiments and the Relative Mode of Instabilitya
RunQw, cm3/sQs, g/minifb, cmY, cmD50/Yτ*βStable or Switch
  • a

    The first two run were performed in the 2 m wide flume; the others were performed in the 1 m wide flume.

1Mn3502000.04150.550.090.2714stable
2Mn3501000.02150.650.080.1412switch
3Mn300700.015150.650.080.1212switch
4Mn300600.0075150.80.060.079switch
5Mn3501000.035300.40.130.1738stable
6Mn3502000.035150.60.080.2513stable
7Mn3501000.015300.50.100.0930switch

[37] We also performed five more experiments in a laboratory flume 1 m wide, mainly changing the initial bed slope, in order to vary the Shields stress of the upstream straight reach, and we observed the same dependence of the bifurcation stability on Shields stress (Table 2).

[38] In all the experiments we observed that a branch closed as soon as the bed elevation in its upstream part increased by about half the local flow depth. This occurred as the local free-surface elevation remained roughly constant. This situation forces all the flow into the other branch; hence the shallower branch progressively loses all its discharge without further significant changes of bed topography.

3.2. Genoa Experiments

[39] The aim of the last set of experiments was to concentrate our observations on bar formation in the divergent reach under varying hydraulic and sedimentologic parameters (namely the Froude number Fr and the Shields stress τ*) as well as geometric parameters, like the aspect ratio β of the stream, the widening ratio r of the channel (the ratio between the widths of the straight reaches of the channel downstream and upstream of the divergent segment respectively) and the angle α that the banks of the divergent reach form with the channel centerline.

[40] The experiments were carried out in a laboratory flume 0.6 m wide and 12 m long that was divided into three parts (Figure 11). Upstream, the flume was narrowed by nonerodible banks to 0.2 m in the first set of experiments and to 0.05 m in the second set, so as to produce two different widening ratios r (respectively equal to 3 and 12). Moreover, the upstream reach was long enough (3 m) that the flume entrance would not disturb the flow in the divergent reach immediately downstream. In the middle part of the flume, the channel widened linearly with an angle α attaining values ranging from 8° to 25° in the different experiments. Hence the divergent reach was never longer than 2 m, and was confined by nonerodible symmetrical banks in the first set of experiments, erodible symmetrical banks in the second set, and erodible nonsymmetrical banks in the third set. Downstream, the straight reach was at least 6 m long, always confined by nonerodible banks, and ended with a weir. In the second and third sets of experiments, the connection between nonerodible and erodible banks generates local scour; in particular, the scour was significant close to the junctions between the nonerodible walls of the upstream reach and the erodible walls of the divergent reach (Figure 12a). We then smoothed out the above junctions by introducing short curved walls to connect the straight reach to the divergent reach. These adjustments helped considerably to reduce the localized scour (Figure 12b).

Figure 11.

Laboratory flume 0.6 m wide and 12 m long, filled with quasi-uniform sand, at the Hydraulic Laboratory of the Department of Environmental Engineering, University of Genoa. (In this picture the upstream reach of the flume is narrowed to 0.2 m, so the widening ratio is equal to 3.)

Figure 12.

(a) Localized scour at the junctions between the nonerodible walls of the upstream reach and the erodible walls of the divergent reach before the introduction of short curved walls to connect smoothly the two reach. (b) Short curved walls used to reduce the local scour.

[41] The flume was filled with nearly uniform sand, characterized by d50 equal to 10−3 m, and was supplied with a constant water discharge in an open loop. Dry sediment was fed from a sediment feeder at a constant rate chosen, depending on the given water discharge and on the initial slope of the bed, to maintain equilibrium as described in section 3.1. The parameters characterizing the experiments are given in Table 3. Note that the upstream straight reach was always characterized by values of the aspect ratio β (hereafter the ratio between the channel half-width and the flow depth) lower than 10, that is, lower than the threshold value required for bar formation according to theoretical and experimental findings [Colombini et al., 1987].

Table 3. Parameters Characterizing Genoa Experimentsa
Runbup,b cmαO,c degif, %Qw, L/sQs, m3/sFrupbβupbτ*upbFrdownbβdownbτ*downbtrund
  • a

    In the runs from 1 to 8 included, the divergent reach was confined by nonerodible symmetrical banks, in the runs from 9 to 20 included, the divergent reach was confined by erodible symmetrical banks; in the runs from 21 to 25 included, the divergent reach was confined by erodible nonsymmetrical banks.

  • b

    The subscripts “up” and “down” indicate that the parameters are evaluated in the upstream or downstream straight reach respectively.

  • c

    Here αo indicates the initial value of the angle that the banks of the divergent reach form with the channel centerline.

  • d

    Here trun indicates the time at which each run was stopped to survey the bed topography. In some cases, we stopped the run at different times to survey the intermediate bed topography.

1Ge201022.51.2 × 10-51.3950.21.31300.1245 min
2Ge201522.91.4 × 10-51.44.50.221.32270.135 min–1.25 h
3Ge201525.52.6 × 10-51.4430.31.45190.1945 min
4Ge2015318 × 10-61.5100.161.44600.0825 min
5Ge2015332.6 × 10-51.750.31.6300.1725 min
6Ge2015117 × 10-70.9370.170.88430.076 h
7Ge20150.502.35 × 10-70.7230.070.71210.045 h
8Ge20150.5052.1 × 10-60.7220.10.77130.0615 min
9Ge208117 × 10-70.9270.070.88430.041 h
10Ge20820.93.5 × 10-51.2790.121.05500.073 h
11Ge51520.42.4 × 10-61.481.70.180.9750.0515–30 min–1.5 h
12Ge515217 × 10-61.6610.31.18500.0715 min–1.5 h
13Ge510217 × 10-61.6610.31.18500.071.5 h
14Ge520217 × 10-61.6610.31.18500.071.5 h
15Ge52520.42.4 × 10-61.481.70.180.9750.053 h
16Ge52020.42.4 × 10-61.481.70.180.9750.054 h
17Ge51021.51 × 10-51.740.80.371.22400.091.5 h
18Ge510221.4 × 10-51.80.70.441.25330.145 min–1.5 h
19Ge510112.4 × 10-61.220.80.180.9430.042.5 h
20Ge511128 × 10-71.320.50.270.91270.061.5–3 h
21Ge525–1020.42.4 × 10-61.481.70.180.9750.051 h
22Ge525–10217 × 10-61.6610.31.18500.0715 min–1 h
23Ge525–1021.51 × 10-51.740.80.371.22400.0915 min–1–1.5 h
24Ge525–10128 × 10-61.320.50.270.91270.0615 min–1 h
25Ge525–11112.4 × 10-61.220.80.180.9430.041.5–6.5 h

[42] At the end of each experiment, the bed topography was surveyed by means of a laser distance sensor (M5LASER/200) after the channel had been slowly drained. In some cases, when the evolution of the pattern seemed particularly interesting, we repeated the run, stopping it at different times to survey the intermediate bed topography. Moreover, runs were replicated under identical conditions to demonstrate that the evolution process and the final pattern were reproducible.

[43] Firstly, the observation that streamline divergence induces the formation of a central bar was confirmed in the somewhat different geometry of the Genoa experiments. This bar directs flow toward the banks of the channel, creating channels on either side of the deposit and hence a bifurcation. This process was invariably observed for all Froude numbers of the upstream flow (in the experiments it varied between 0.7 and 1.8). More significantly, this was true for all aspect ratios β of the flow in the downstream reach, which varied between 10 and 75 in the experiments. It is well known that the bar mode in straight channels arises in response to intrinsic instabilities of the flow-sediment system (“free” bars [Seminara and Tubino, 1989]), and depends principally on the aspect ratio of the flow. Moreover, as the value of β increases, bars evolve from alternate (β ≅ 10–20), to central and eventually to multiple row bars (β > 20). In our divergent reach this sequence was not observed; rather, for the range of aspect ratios we used, the channel divergence invariably forced the formation of a single central bar (Figure 13).

Figure 13.

A central bar forms in the divergent reach whatever aspect ratio β the flow in the downstream reach had (flow from up to down). (a) β = 13, (b) β = 27, and (c) β = 60.

[44] In the downstream straight reach, forced bars originate from the nonuniform topography at the exit cross-section of the divergent reach. As a consequence, a steady bar pattern is observed with a transverse mode selected by the system depending on the aspect ratio of the downstream reach (Figure 14). In other words, the central bar formed in the divergent reach is felt by the downstream straight reach as an imposed nonuniform initial condition that induces the phenomenon of downstream overdeepening [Zolezzi and Seminara, 2001].

Figure 14.

Downstream the divergent reach, the transversal mode selected by the system is stationary and variable with β. (a) β= 13, (b) β = 27, and (c) β = 60.

[45] The bifurcations formed in the divergent reach were always stable in the sense defined above. Then the question arises: why did we not observe unstable bifurcations like those in the St. Anthony Falls experiments even when the Shields stress fell in the same range of values? A reasonable explanation seems related to the fact that, in the Genoa experiments, the incoming stream was always fairly uniform because the banks of the upstream reach were not erodible and moreover the upstream reach itself was long enough not to be influenced by the upstream inflow conditions. Instead, in the St. Anthony Falls experiments, all the channel banks were erodible and so the incoming stream sometimes concentrated on one side of the channel; hence the branch of the bifurcation on the opposite side was perturbed, i.e., it lost discharge and, when the Shields stress of the upstream flow assumed relatively low values, it closed. The main implication of this is that the evolution of bifurcations can be influenced strongly by conditions upstream of the flow expansion.

4. Laboratory Observations on Equilibrium Configurations of the Banks in a Divergent Erodible Channel

[46] During the Genoa experiments (Table 3), we also focused our attention on the process of erosion of the divergent channel banks.

[47] It is well known that bank erosion is due either to the removal of superficial particles by the flow, or to bank collapse. However the temporal scale of the process is controlled by the bed load transport capacity of the stream, i.e., the rate of sediment removal from the bank toes. Hence the downstream migration, the altimetric evolution, and the emergence of bars can strongly influence the erosion process.

[48] In our particular case, the deviation of flow due to the central bar induces bank erosion and, more importantly, it causes the angle α, between the banks of the divergent reach and the channel centerline, to evolve toward an equilibrium configuration. Equilibrium is attained as the process of erosion stops.

[49] The equilibrium configuration depends on the rate of widening described by the widening ratio r. In fact, if the widening ratio is small (r = 3) the equilibrium value of α is lower (≅10°) because the effect of the divergence is not felt within the divergent reach but in the downstream straight reach so that the lateral scour induced by the central bar acts weakly on the divergent walls. On the contrary, if the widening ratio is larger (r = 12) the central bar emerges in the divergent reach and so it induces a strong erosion of the divergent banks; hence the equilibrium value of α is higher (≅20°).

[50] The equilibrium configuration also depends on the value attained by the Shields stress τ* in the downstream straight reach. In fact, the value of τ* affects the position of the bar front at its equilibrium configuration, i.e., when the bar is stationary and emergent. In particular, for higher values of τ* (≅0.1), the front is located farther downstream than for values of τ* close to τ*c. So, for high values of τ*, the banks are eroded less, because the deviation of the flow in the divergent reach driven by the central bar is weaker. As a result, the equilibrium value of α is lower. In conclusion, the dependence of α on Shields stress is weaker than the dependence on r, as shown in Figure 15.

Figure 15.

The equilibrium value of the angle α depending on the value of the Shields stress of the downstream straight stream, for r = 12.

[51] On the contrary, the equilibrium configuration of the banks does not depend on the initial value of α. In fact, if the initial value is lower than the equilibrium value, the stream erodes the banks as explained above, while if the initial value is greater than the equilibrium value, the stream tends to deposit sediments close to the banks and so it effectively reduces the angle to the equilibrium value (Figure 16). This mechanism is also clear in the experiments performed with nonerodible divergent banks because the angles (≅15°) were always greater than the equilibrium value (≅10° for r = 3).

Figure 16.

Picture of the equilibrium configuration of the banks of the divergent erodible channel in the run 23Ge, in which the initial configuration was nonsymmetric, i.e., the left bank was longer than the right one. The stream eroded the left bank because the initial value of the angle α (≅10°) was lower than the equilibrium one (≅20° for r = 12) and tended to deposit sediments close to the right bank to reduce the initial value of the angle (≅25°) to the equilibrium one. In the picture, the main flow is confined inside the white lines, while the water has velocity close to zero outside them, i.e., close to the banks.

[52] Finally, we performed six experiments with an asymmetric divergent reach. We observed a combined mechanism of erosion along the longer bank and deposition along the shorter bank, so that at equilibrium the banks had again recovered a symmetric configuration (Figure 17). Moreover, the initial asymmetry always led to the formation of an asymmetric central bar in the divergent reach and to alternate spatial bars downstream. With the restoration of symmetry we observed a symmetric central bar in the divergent reach and spatial central bars downstream (Figure 18).

Figure 17.

The combined mechanism of erosion along the longer bank and deposition along the shorter bank led to a symmetric configuration at equilibrium. (a) t = 15 min; (b) t = 1.5h.

Figure 18.

Asymmetric divergent reach. The initial asymmetry always led to the formation of an asymmetric central bar in the divergent reach and to alternate spatial bars downstream. With the restoration of symmetry we observed a symmetric central bar in the divergent reach and spatial central bars downstream. (a) t = 15 min; (b) t = 1.5 h.

5. Conclusions

[53] 1. In all the experiments performed, streamline divergence produces flow bifurcations by inducing the formation of a central bar, as suggested by Mosley [1976], for any Froude number and aspect ratio (width/depth) of the incoming stream. Downstream of this forced central bar in the divergent reach, stationary bars, whose mode depends on the aspect ratio β, form.

[54] 2. The stability of bifurcations created by the formation of a central bar appears to depend on the value of the Shields stress τ* of the upstream reach: a relatively high value of τ* (>0.15 in our experiments) leads to a “stable” bifurcation while a lower value of τ* (<0.15) can produce a “switch” bifurcation. Moreover, if the value of Shields stress is low, the character of a bifurcation seems to be influenced also by the upstream conditions: if the incoming stream concentrates on one side of the channel, the branch of the bifurcation on the opposite side is perturbed, i.e., it loses discharge and may close; instead, if the incoming stream is fairly uniform, as it happened in our Genoa experiments where the upstream reach banks were not erodible and the upstream reach was long enough, the bifurcations remained stable also at low values of Shields stress. Hence we found that an unstable bifurcation can form only provided that the flow field is characterized contemporarily by a low Shields stress and by a nonuniform incoming flow.

[55] This observation suggests that a bifurcation in a braided network is typically unstable: the direction of the incoming stream usually deviates from perfectly straight due to the curvature of the upstream network branches, and the value of Shields stress is typically low and close to the threshold for motion.

[56] 3. The banks in a divergent reach tend toward an equilibrium configuration in which there is no bank erosion. The equilibrium configuration depends strongly on the widening ratio and weakly on the value of the Shields stress of the downstream straight stream. It does not depend on the initial value of the angle α and on any initial asymmetry of the divergent reach.

Acknowledgments

[57] This work has been partially developed within the framework of the National Project cofunded by the Italian Ministry of University and Research and the University of Genoa (COFIN 2001) “Morphodynamics of fluvial networks,” partially funded by NSF BRAID GRANT, NCED, and by the University of Genoa with a grant in the project “Training of undergraduate student to the scientific research.” Partial support has also come from Fondazione Cassa di Risparmio di Verona, Vicenza, Belluno e Ancona (Progetto RIMOF). This work is also part of the PhD thesis of Bianca Federici, to be submitted to the University of Padova. The authors are grateful to Giovanni Seminara for support and insight throughout this work, and to Alessandro Cantelli, Silvia Degli Esposti and Karen Gran for help with the experiments.

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