Volume 41, Issue 7
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Bed and bank evolution of bifurcating channels

W. Bertoldi

W. Bertoldi

Department of Civil and Environmental Engineering, University of Trento, Trento, Italy

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M. Tubino

M. Tubino

Department of Civil and Environmental Engineering, University of Trento, Trento, Italy

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First published: 06 July 2005
Citations: 48

Abstract

[1] The evolution of natural river channels is strongly affected by the interplay between the altimetric pattern and the planimetric configuration acting contemporary in branches of braided rivers. Very few quantitative observations are presently available to characterize such interactions. In the present contribution we discuss the results of experimental runs performed with both uniform and graded sediments. The experiments were aimed at describing quantitatively the evolution of a single laterally unconstrained channel until the occurrence of the first bifurcation. An objective criterion for the occurrence of the bifurcation has been established using the data provided by the Fourier analysis of the evolving bank profiles; the procedure enabled us to characterize the morphodynamic sequence leading to flow and channel bifurcation. The sought outcome of the investigation is to derive a suitable description of the bifurcation process to be implemented in predictive models for braiding evolution, for which physically based nodal point conditions would be highly desirable.

1. Introduction

[2] Modeling planform changes of natural rivers is gaining a renewed relevance in the context of fluvial research due to the progressive shift of the engineering practice toward the issues of renaturalization and restoration that represent the most recent trends of river management. The possibility of predicting the planimetric and altimetric evolution of natural rivers is still an ongoing debate and strongly depends at present on the type of river we are dealing with. While the essential processes characterizing the dynamics of single thread meandering channels have received much attention in the last two decades (the subject has been recently reviewed by Seminara et al. [2001]), an effective modeling of channel adjustment in braided rivers can only be achieved over short prediction spans [e.g., Jagers, 2003].

[3] A major reason for this difficulty must be sought in a crucial difference between single and multiple channel rivers, which essentially involves the timescales of bed and bank evolution. In river meanders bank erosion is mainly controlled by sediment cohesion and vegetation, which forces the planform to evolve on a much longer timescale than those of processes of bed deformation. This implies that as a first approximation, the corresponding mathematical problem can be decoupled, which provides a much simpler description. On the contrary, in braided rivers each channel can be often considered, up to a certain extent, as laterally unconstrained [Murray and Paola, 1994]. Hence their dynamics depends strongly on the interaction between the altimetric patterns and the planimetric configuration. Both laboratory models and field studies suggest that under these conditions the planform of single channels is often unstable as bifurcations are promoted. This occurs preferentially through the mechanism of chute cutoff due to a local flow acceleration [see Ashmore, 1991].

[4] The above process is also common to other river morphologies: striking evidence is offered by meandering channels, where the occurrence of chute cutoff gives rise to a cyclic reduction of channel sinuosity [Howard, 1996]. Besides being common in meandering [Gay et al., 1998], pseudomeandering [Bartholdy and Billi, 2002], and anastomosing rivers [Makaske, 1998], chute cutoff is quite frequent and ubiquitous in braided streams as witnessed by the weakly meandering character of each single branch.

[5] Very few attempts have been made until now to explain the mechanics of chute cutoff [Klaassen and van Zanten, 1989; Slingerland and Smith, 1998; Jagers, 2003]. Such knowledge appears even more relevant when we consider that most braided rivers often seem to reflect the history of few active branches, though the number of wet channels can be quite large [Mosley, 1983; Ashmore, 2001]. This suggests that the evolution of a braided network could be reconstructed in terms of the dynamics of single thread weakly meandering channels interacting at joining points, like confluences and bifurcations, provided bed and bank evolution processes are not decoupled. This “synthetic” schematization of braiding, according to which network development results from the interaction among single objects like channels and nodes [see Jagers, 2001, 2003], might provide a reliable and manageable way to predict the time evolution of such complex systems over a relatively long time span.

[6] The dynamics of channels and nodes is mutually dependent. Channel adjustment is largely controlled by the processes of node shifting, creation or annihilation; in turn, channel migration may affect the movement of the nodes. Furthermore, bifurcations often occur after a well defined sequence of in-channel events [Ashmore, 1991] which reflect the strong interaction between the planimetric and the altimetric response.

[7] Most of these processes have been independently investigated and understood with enough detail, particularly those related with the effect on the bed deformation of planimetric nonuniformities, such as channel curvature and width variations. Kinoshita and Miwa [1974] and Tubino and Seminara [1990] investigated the interaction between free migrating bars and steady point bars in meandering channels, showing that a threshold value for channel curvature exists, above which free bars cease migrating and bed topography is characterized by steady patterns. A similar suppressive effect on migrating bars is also exerted by periodic width variations, as shown by Repetto and Tubino [1999] and Repetto et al. [2002]. In this case, provided the amplitude of bank oscillation is large enough, the resulting bed topography displays a steady central bar pattern. Under suitable conditions, this in turn enhances the amplitude of the width variations, which may lead the channel to bifurcate. The above findings generally agree qualitatively with the results of the laboratory investigations of Ashworth [1996] on the evolution of a confluence-diffluence unit. The formation of a central bar downstream of the confluence induces the flow to diverge and to concentrate toward the banks, determining the instability of the planimetric configuration.

[8] Notice however that most of the above results on bed dynamics mainly refer to fixed-bank channels, which implies that a detailed knowledge of the simultaneous development of the bed and the banks of laterally unconstrained channels is presently not available. In particular, the following questions need to be addressed: how are free-forced bed interactions affected by the erodible character of the boundaries? How can the altimetric pattern modify the evolution of channel planform? What conditions define the occurrence of channel bifurcations?

[9] In the present work we attempt to provide quantitative answers to the above questions. We have performed four sets of experimental runs, with both uniform and graded sediments, tracing the evolution of laterally unconstrained channels until the occurrence of the first bifurcation, with the aim of ascertaining the combined role of free and forced altimetric bed responses. In each experimental run the following sequence of processes has been invariably detected: the initially straight channel first widens, then forms an alternate pattern of bars, which determines the occurrence of a regular sequence of erosional bumps along both banks. Therefore a slightly meandering configuration establishes, displaying fairly large width oscillations. Under these conditions the pattern of previously formed bars appears highly reworked and soon leads to the occurrence of flow bifurcation through a chute cutoff mechanism.

[10] Experimental findings on planform development suggest that a suitable criterion for channel bifurcation can be given, through the Fourier analysis of bank profiles, in terms of the relevant dimensionless parameters, namely the Shields stress and the width to depth ratio of the channel.

[11] The rest of the paper is organized as follows: in 2, 3, and 4 we give a description of the experimental setup and of the data analysis procedure. In 5 we present the results of the altimetric and planimetric evolution of the channel and we finally analyze the bifurcation process. In 9 a summary of experimental results is included along with some concluding remarks.

2. Experimental Setup

[12] A laboratory model reproduces the main features of water and sediment motion of a gravel bed braided river provided it satisfies the Froude similarity and the flow is fully turbulent, hydraulically rough and the dominance of bed load transport is ensured [Yalin, 1971]. A laboratory flume that meets the above criteria reproduces the behavior and the processes of a gravel bed stream in general [Ashmore, 1982]. The main advantages in physical model investigations are the direct control of specific variables and the possibility to observe and to measure the development of planimetric and altimetric patterns, even if all the characteristics of a prototype stream cannot be modeled exactly [Young and Warburton, 1996].

[13] The experiments were carried out in a laboratory flume, 12 m long and 0.6 m wide, in the Hydraulic Laboratory of the University of Trento (Figure 1). A constant water discharge, measured through an electromagnetic meter on the delivery pipe, and a constant sediment rate were supplied into the channel. The sand was fed through a volumetric sand feeder and dropped into the channel via a diffuser, in order to avoid local disturbances. The appropriate sediment discharge in equilibrium with the values of water discharge and channel slope for each experimental run was fed into the channel, in order to achieve an overall equilibrium, i.e., to avoid, on the average, bed degradation or aggradation. At the channel inlet a system of metallic meshes regularized the incoming flow and at the downstream end of the flume a tailgate was constructed to maintain the outlet elevation. Along both sides of the channel a 0.3 m high rail supported a carriage used for levelling the bed and measuring bottom topography and the channel planimetric evolution. The rail slope could be adjusted to the prescribed value.

Details are in the caption following the image
The initial configuration of the channel.
[14] The experiments involved different values of water discharge (Q), sediment rate (Qs) and initial slope (S), which were chosen to obtain values of dimensionless parameters typical of gravel bed rivers (Table 1). The above initial values were calculated under the assumption of uniform flow with reference to the initial trapezoidal section. In Table 1 the following notation is employed: Ds is the mean grain diameter, s is the geometric standard deviation of the grain size distribution, and the dimensionless parameters ?, β, and ds are the Shields stress, the width ratio of the channel, and the relative roughness, respectively, which are defined as follows:
urn:x-wiley:00431397:media:wrcr10063:wrcr10063-math-0001
where b is the half free surface width, H is the reach averaged value of water depth, ?s and ? are the sediment and water density, g is gravity and t is the average bed shear stress. The subscript o denotes the initial (reference) values of parameters and variables.
Table 1. Experimental Conditions of the Performed Runs
Run Ds, mm s S, % Q × 10−3, m3/s Qs, g/s βo equation imageo dso
A1-10 0.5 1 1 0.167 0.567 4.74 0.086 0.059
A1-15 0.5 1 1 0.250 0.833 4.03 0.104 0.047
A1-20 0.5 1 1 0.333 1.517 3.61 0.118 0.040
A1.5-7 0.5 1 1.5 0.117 0.267 6.03 0.099 0.081
A1.5-10 0.5 1 1.5 0.167 0.767 5.16 0.117 0.066
A1.5-15 0.5 1 1.5 0.250 1.600 4.36 0.142 0.053
A1.5-20 0.5 1 1.5 0.333 2.333 3.90 0.162 0.045
B1.5-20 1.3 1 1.5 0.333 0.583 3.58 0.069 0.103
B1.5-25 1.3 1 1.5 0.417 1.350 3.31 0.076 0.091
B1.5-30 1.3 1 1.5 0.500 1.517 3.12 0.082 0.083
B2-15 1.3 1 2 0.250 0.467 4.20 0.076 0.130
B2-20 1.3 1 2 0.333 1.050 3.77 0.086 0.111
B2-25 1.3 1 2 0.417 1.650 3.48 0.095 0.099
B2-30 1.3 1 2 0.500 1.900 3.27 0.103 0.090
MB1.5-7 0.8 1.7 1.5 0.117 0.350 5.71 0.066 0.120
MB1.5-10 0.8 1.7 1.5 0.167 0.450 4.91 0.077 0.099
MB1.5-15 0.8 1.7 1.5 0.250 0.783 4.17 0.093 0.079
MB1.2-10 0.8 1.7 1.2 0.167 0.400 4.69 0.065 0.093
MB1.2-15 0.8 1.7 1.2 0.250 0.700 4.00 0.078 0.075
MC1.5-15 1.04 2.1 1.5 0.250 0.783 4.07 0.074 0.099
MC1.5-20 1.04 2.1 1.5 0.333 1.717 3.65 0.084 0.085
MC1.5-25 1.04 2.1 1.5 0.417 2.133 3.38 0.092 0.075
MC2-12 1.04 2.1 2 0.200 1.283 4.70 0.083 0.121
MC2-15 1.04 2.1 2 0.250 2.000 4.30 0.092 0.107
MC2-17 1.04 2.1 2 0.283 2.283 4.10 0.098 0.100
MC2-22 1.04 2.1 2 0.367 3.000 3.72 0.109 0.087

[15] The experimental investigation consisted of four sets of experiments. In the first two sets (denoted by A and B) we used two different well sorted quartz sand distributions with values of the mean diameter Ds of 0.5 mm and 1.3 mm, respectively. The restriction to almost uniform sediments allows for a closer comparison between experimental data and existing theories, since most theoretical results have been derived with reference to uniform grain size.

[16] The other two sets of runs were performed with two different bimodal mixtures, denoted MB and MC. MB was a weakly bimodal mixture, obtained with equal percentages of sands A and B, whereas MC was a strongly bimodal mixture, resulting from equal percentages of sand A and another uniform sand (C) whose diameter was 1.9 mm. The degree of bimodality of MB (MC) was chosen to be lower (higher) than the threshold value beyond which the critical shear stress for the incipient motion of each fraction becomes dependent on the size according to Wilcock [1993].

3. Experimental Procedure

[17] In each experiment we adopted the following procedure. The bed was flattened to the prescribed slope using a wide scraper attached to a carriage that ran along the rails; at the same time a straight narrow channel of trapezoidal shape was traced into the cohesionless sloping surface, with base width of 6 cm and sloping banks such that the initial width of the free surface was 8–12 cm (see Figure 1). Then a very low discharge was passed over the bed to prepare a smoothly saturated surface.

[18] During the runs, the planimetric development of the channel was continuously monitored and documented through series of pictures taken from a digital camera mounted on a carriage that ran along the longitudinal rails, every picture covering 1 m of the channel length. Dry bed topography was surveyed periodically with a laser scanning device, on a regular grid spacing 10 cm in the longitudinal direction and 1 cm in the transverse direction.

[19] Each experimental run was managed through a well defined sequence of actions, until the occurrence of the first channel bifurcation that set the end of the run. At first each run was performed without interruptions until the final stage, continuously monitoring its planimetric evolution. Then the experimental run was repeated, starting from the same initial condition but with a regular sequence of stops to perform laser surveys of the dry bed topography. The experiment was restarted after any intermediate stop: the effect of bar dissection induced by the withdrawal of the water was always found to be negligible.

[20] Additional measurements of surface flow velocity were made at fixed locations along the channel using a high-speed video camera and light particles as flow tracers. During the experiments the presence of bed forms was observed and their wavelength and migration speed estimated. The sediment discharge was collected periodically using a trap placed at the downstream end of the channel and compared with the sediment supply Qs at the inlet.

[21] In a few runs the planimetric evolution of the system led the channel banks to reach the fixed walls of the flume before the occurrence of a bifurcation. These runs are not considered in the subsequent data analysis.

4. Data Analysis

[22] In order to characterize quantitatively the conditions that determine channel bifurcation, we have analyzed both planimetric and altimetric data, namely the evolution in time of channel width, bank profiles and bed elevation.

[23] As shown in Figure 2, prior to the occurrence of flow bifurcation the channel exhibited a fairly regular, periodic pattern, displaying a negligible longitudinal variation of the overall geometry within the measuring reach; this enabled us to employ a Fourier transform procedure to process the above data. The analysis of bank and bottom configuration was performed both on the whole channel and per single wavelength: no significant differences were detected in the computed results. The Fourier analysis allowed us to determine the wave number ?w and the amplitude d of bank oscillations, defined in terms of the length L and amplitude A of the leading component of the Fourier representation of the bank profile; in dimensionless form they read:
urn:x-wiley:00431397:media:wrcr10063:wrcr10063-math-0003
Details are in the caption following the image
A step of the evolution of the channel: a slow meandering channel displaying regular width variations.

[24] A suitable procedure was adopted to define the longitudinal length of the record looking for the condition which maximized the amplitude of the leading component of the spectrum. In all cases this component was clearly distinguishable as shown in Figure 3. This fact is closely related to the control exerted on bank profile by the development of free bars in the channel.

Details are in the caption following the image
A typical Fourier spectrum of the longitudinal bank profile (run B2-20).

[25] The bed topography elevation was analyzed through a 2D Fourier procedure in order to recognize the contribution of different bar patterns to the overall bed morphology; in the following we refer to the first transverse mode to denote the alternate bar pattern and to second transverse mode to denote a central bar structure. The geometrical characteristics of such modes are given in terms of the dimensionless values of bar wave number ?b (scaled with the reach averaged value of half channel width b) and of the corresponding amplitude Ab (scaled with the initial depth of the flow Ho).

[26] The major problem encountered in the analysis of topographical data is the definition of suitable reach averaged values of the relevant dimensionless parameters, namely, the Shields stress ? and the width ratio β. Their evaluation for a given channel configuration would also require direct measure of flow depth or velocity. In the absence of local measures of such variables a possible way to determine the average depth is to refer to the uniform flow that would occur in a rectangular channel with the same average width and for the same values of water discharge and longitudinal slope. The latter was not found to vary appreciably during the experimental run with respect to the initial prescribed value. Notice that the above procedure does not take into account the actual geometry of channel cross sections and the consequent nonuniform lateral distribution of bottom stress; at low values of Shields stress this may imply a strong underestimate of the average sediment flux due to its nonlinear dependence on shear stress, as first pointed out by Paola and Seal [1995] [see also Ferguson, 2003]. This is shown in Figure 4 where predicted values of the sediment discharge obtained with the above simplified procedure are compared with observed values (open symbols).

Details are in the caption following the image
Comparison between the measured solid discharge and that calculated according to two different estimates of the Shields stress.

[27] To overcome the above difficulty we adopted an alternative procedure whereby the measured bottom topography was used to define a transverse partition of the cross section into narrow strips in order to compute the sediment rate as the sum of the contributions associated with the local values of flow depth. A suitable reach averaged value of ? was then defined as the value corresponding to the computed solid discharge in a rectangular channel with the same width. In this way more realistic values of ?, and hence of Qs can be obtained, as shown in Figure 4 (solid symbols). It is worth noticing that the agreement with the observed data also depends on the sediment transport formula adopted in the computation. Results reported in Figure 4 are obtained using Parker's [1990] relationship, which performs better at relatively low values of Shields stress, which are more relevant for present analysis. The use of a bed load transport formula of Meyer-Peter and Müller [1948] type, which includes a threshold value, would cause an overall overestimate of measured sediment discharge. The same procedure was adopted to compute the reach averaged value of the width ratio β.

5. Results

[28] In this section a quantitative description of the observed channel dynamics is presented, focusing on the interaction between bar structures and planimetric patterns. Experimental data are also compared with theoretical results on both bar and channel planform evolution in straight and weakly meandering channels [Colombini et al., 1987; Tubino and Seminara, 1990]. The discussion of experimental results also includes further information on the configuration of channel bifurcations, namely the angle between the two downstream channels and the relationship among the relevant flow parameters at the onset of bifurcation.

5.1. Altimetric Evolution

[29] Since the initial width of the channel was not in equilibrium with the imposed flow discharge, an initial, almost uniform widening of the straight channel occurred. The width of the channel at the beginning of each experiment was set so that the value of the width ratio βo was lower than the threshold value βc for the formation of free alternate bars [Colombini et al., 1987], which implies a stable plane bed configuration.

[30] The formation of regular trains of migrating alternate bars was then observed as channel widening and the consequent reduction of the average water depth caused the width ratio β to exceed the threshold value (Figure 5).

Details are in the caption following the image
Formation of alternate bars in the early stage of channel development.

[31] Bar amplitude and wave number were determined processing the output of the laser bed survey through a Fourier transform procedure. In the first two series of the runs (A and B) the bars were fairly regular, with dimensionless wave number ?b ranging between 0.35 and 0.45, which roughly corresponds to the typical range of values of free alternate bars in channels with fixed banks and uniform sand [Tubino et al., 1999].

[32] A slightly different behavior was observed in the two series of runs with bimodal mixtures. In this case bar morphology was less regular, especially in the MC runs (Table 1) characterized by a higher degree of bimodality where the lengths of single bar units were different and the migration speed was considerably reduced. The dimensionless wave number was on average lower compared with the case of uniform sediments, ranging between 0.25 and 0.3. Moreover, the presence of graded sediment caused the formation of regular sorting patterns, characterized by the selective deposition of coarse particles on the bar fronts (Figure 6). These findings agree with the theoretical and experimental results of Lanzoni and Tubino [1999] and Lanzoni [2000], who highlighted the role played by sediment nonuniformity on the formation and equilibrium configuration of alternate bars in straight channels.

Details are in the caption following the image
Example of sorting pattern for three subsequent configurations of the channel. Dark regions denote the accumulation of coarse particles.

[33] The observed morphological features of bars are also compared with those predicted by the weakly nonlinear theory of Colombini et al. [1987] in Figure 7 in terms of theoretical and observed values of bar height. It is worth noticing that the theoretical results refer to a sequence of alternate bars that have reached an equilibrium amplitude in straight channel with fixed banks. This configuration differs from that of present experiments, where the channel banks are subject to lateral erosion; in the latter case the role of the forcing effects of planform nonuniformities, such as channel curvature and width variations, may prevent the achievement of an equilibrium amplitude and cause modification of bar structures through nonlinear interaction [Tubino and Seminara, 1990; Repetto and Tubino, 1999].

Details are in the caption following the image
Comparison between the measured values of bar height and theoretical predictions of Colombini et al. [1987]. Open symbols correspond to values of β > 3βc, for which the weakly nonlinear theory is no longer valid.

[34] Despite these differences, in the early stage of channel evolution, when the amplitude of such planimetric forcing remains relatively small, bars may undergo a finite amplitude development such that their amplitude is quite well predicted by the theory, at least within the weakly nonlinear regime, (β < 2βc), in which the theory is applicable.

[35] As the experimental run proceeds, the adjustment of the flow field to the presence of alternate bars determines the lateral shift of the main flow close to the banks and the formation of a regular sequence of bumps along the bank profiles (Figure 2) whose length scale coincides with that of the alternate bars. The resulting planimetric configuration is a weakly meandering channel that also exhibits regular width variations. As pointed out in the Introduction these planimetric nonuniformities may strongly affect bed topography, as they promote the transition from the migrating free response (alternate bars) to a steady forced bed deformation. Bar structures then progressively become fixed with respect to the planimetric configuration; this enhances local bank erosion which, in turn, implies a further increase of the planimetric forcing effects.

[36] Because of the above processes, not only the bar migration speed vanishes, but also the bar pattern is strongly modified. The analysis of the Fourier spectra of bed topography (Figure 8) reveals the tendency of bed topography to evolve from alternate to central bar patterns as the run proceeds. Figures 9 and 10 summarize the results of the whole set of experiments for the amplitude of the leading components of bed topography measured at the initial stage and at the final stage of the experimental runs, respectively (notice that in the plots the cumulative effect of higher-order transverse modes, second and third, is reported). 8, 910 show that a similar tendency was displayed in almost all the experiments, with a decrease of the amplitude of the alternate bars component (first mode) and the simultaneous increase of the amplitude of higher-order transverse modes (second and third). This embodies the fundamental mechanism leading to channel bifurcation: the resulting topography forced by planimetric nonuniformities progressively leads to the process of chute cutoff of the alternate bars formed in the initial stage of the experiment.

Details are in the caption following the image
The amplitude of leading components of the Fourier spectrum of bed topography measured at three subsequent stages (t1, t2, and t3) during the experimental run A1.5-10.
Details are in the caption following the image
Comparison between the amplitude of alternate bars and that of transverse modes 2 + 3 in the initial stage of experimental runs.
Details are in the caption following the image
Comparison between the amplitude of alternate bars and that of transverse modes 2 + 3 at the onset of the bifurcation.

[37] One might be tempted to explain the above process in terms of the standard approach based on linear theories, the so-called “bar theory of river meandering” [Fredsæ, 1978; Kuroki and Kishi, 1985] according to which the onset of braiding is related to the amplification of higher-order modes. As shown in Figure 11, this typically occurs for relatively large values of the width ratio β, provided it exceeds a threshold value. Indeed the observed values of flow parameters at the onset of bifurcation (in particular the width ratio) fall within the range of amplification of higher-order modes. Furthermore, the value of the dimensionless longitudinal wave number of bars ?b was found to invariably increase during the experiments since the length of bars was almost fixed and equal to the “initial length”, while the channel width increased (equation (2)). According to linear theories this also promotes the instability of higher-order transverse modes [Tubino et al., 1999]. However, the present experimental findings suggest that the modification of previously formed bars and the transition to a central bar pattern occur when alternate bars have already undergone a finite amplitude development, as shown in Figure 7, which implies that linear theories no longer apply. Moreover, the experimental results show that the modification of bar structures is mainly associated with the forcing effects of planform, which are not included in linear theories.

Details are in the caption following the image
Threshold values of the width ratio for the occurrence of different river regimes according to the linear theory of Colombini et al. [1987]. Parker's [1990] bed load transport relation has been employed.

[38] It is possible to observe that the Fourier spectrum reported in Figure 8 shows that the third transverse mode can attain a relatively high amplitude. Its role can be related to the asymmetry of the planimetric configuration and to the history of the bed evolution. In fact, due to channel curvature lateral erosion is alternatively shifted toward the left and right bank; hence width variations are not symmetrically displaced with respect to the channel axis. The presence of third harmonics has then to be considered in the light of the overall channel evolution, the final configuration being the result of the gradual drifting of alternate bars toward the channel axis.

[39] Finally, it is worth mentioning that in the runs with graded sediments the evolution of bed topography was slightly different. In particular, the development of higher-order modes was less prominent, the alternate bar mode being invariably dominant until the onset of the bifurcation (Figure 10). This was mainly related to the deposition of coarse particles on bar fronts (Figure 6) which caused a local decrease of Shields stress, thus slowing down the evolution of the bed configuration.

5.2. Planimetric Evolution

[40] In all the experiments the overall channel alignment continuously adapted to macroscale perturbations of bed topography. As a result, the evolution of channel planform was crucially controlled by the migration speed of alternate bars. Indeed, two different evolutionary scenarios (termed as “slow” and “fast” in the following) were detected, that are closely related to the ratio between the migration speed of bars and the bank erosion rate. In the case of cohesionless banks the latter process is mainly controlled by the intensity of local erosion induced by migrating bars, which can be related to the excess of flow velocity at the bank. Hence both effects contributing to the above ratio are related to the topographic expression of bars. The above ratio is a key parameter that affects the subsequent development of the channel. Bars that migrate fast are unable to produce high localized bank erosion, which implies that the amplitude of planform nonuniformities (width variations and curvature) cannot reach a value high enough to suppress bar migration and to enhance the development of a central pattern. In this case the slowing down of bars is mainly related to continuous channel widening, with a consequent reduction of the averaged bed shear stress.

[41] The experimental runs can then be divided into two groups, namely slow runs and fast runs (Figure 12), which involve different mechanisms of channel bifurcation. The slow runs are characterized by slowly migrating bars, so that bend amplification and channel migration are almost of the same order of magnitude; consequently width variations and channel curvature may strongly affect bar patterns and bifurcation occurs through the mechanism of chute cutoff. On the contrary, fast runs are characterized by a much faster bar migration. This is shown in Figure 13 where measured values of downstream and lateral migration of the channel for the B runs are reported as a function of the reach averaged value of the Shields stress. In the case of fast runs bifurcation mainly occurs due to the different mobility of single bar units which may lead a bar front to merge into a scour hole left by the preceding bar. Most of the performed experiments were slow, in particular those with bimodal sediments. These were invariably characterized by lower bar migration speed, due to the sorting effects [Lanzoni and Tubino, 1999].

Details are in the caption following the image
Examples of the planimetric development in a slow run and in a fast run.
Details are in the caption following the image
Downstream and lateral migration of the channel observed in the experimental runs B. Measured values are scaled by the reach averaged flow velocity.

[42] A qualitative explanation of the observed shift from slow to fast runs, which occurs as Shields stress increases, can be given through the results of linear stability analysis of free bars [e.g., Colombini et al., 1987; Seminara and Tubino, 1989]. In Figures 14a and 14b we plot the theoretical values of the migration speed of bars and of the local maximum of the excess of longitudinal velocity at the bank for the same set of data of Figure 13: the estimates are based on the values of the dimensionless parameters reported in Table 1. It appears that while the former exhibits an overall increasing trend as the Shields parameter increases, the latter keeps almost constant.

Details are in the caption following the image
(a) The migration speed of bars, c, and (b) the local excess of longitudinal velocity at the bank, umax, for the experimental runs B are computed according to linear stability analysis of free bars [Colombini et al., 1987]. Theoretical values are scaled by the reach averaged flow velocity.

[43] The evolution of channel planform can be described through the Fourier analysis of bank profiles, which clearly shows the presence of a dominant harmonic: its amplitude increases in time and its length coincides with that of alternate bars (Figure 3). The interrelation between altimetric and planimetric patterns is quantitatively revealed by the high correlation between the length of width variations and that of bars. This is shown in dimensionless form in Figure 15, which compares the values of the wave number of width variations ?w and of bars ?b, measured at the initial stage and at the onset of bifurcation. As pointed out before, the increase of the dimensionless value of the wave numbers is mainly related to channel widening, since ? is scaled through the actual width of the channel (see (2)). On the contrary, the physical length of bank oscillation remains almost constant and coincides with the length of bars that formed at the initial stage of the process. Hence, according to our observations the longitudinal spacing of subsequent bifurcations depends on the length of initially formed bars.

Details are in the caption following the image
Comparison between the wave numbers of bars (?b) and of the bank profiles (?w).

[44] The amplitude of the leading component of bank oscillations is plotted in Figure 16 versus the width ratio, i.e., for increasing times. It is worth noticing that all the slow runs show a similar behavior, characterized by an initial growth, until a peak value is reached, followed by a stage of slow decay. The occurrence of this maximum is of crucial importance as it provides an objective criterion to set the onset of the bifurcation. In fact, on the rising limb, bank oscillations increase their amplitude as the main flow is shifted toward the bank by the presence of the alternate bar. When the channel bifurcates, the flow erodes also the opposite banks, the location of maximum bank erosion shifts along the channel, leading to a more irregular bank line. The process is depicted in Figure 17. As a consequence the amplitude of width variations starts decreasing immediately after the occurrence of flow bifurcation.

Details are in the caption following the image
Evolution of the dimensionless amplitude of bank oscillations as a function of the width ratio.
Details are in the caption following the image
The onset of flow bifurcation.

[45] The maximum values of the amplitude of bank oscillations are plotted in Figure 18 for all of the experiments. These values are greater than the threshold values above which migrating alternate bars are suppressed in variable width channels as predicted by Repetto and Tubino [1999]. Furthermore, the present experimental results agree qualitatively with the above theoretical analysis, which predicts that the threshold amplitude of width variations that marks the transition from migrating alternate bars to steady central bars is a decreasing function of the Shields stress ?. Also notice that the two fast runs show a different behavior, reaching a maximum value of approximately 0.1. In this case width variations are unable to stop the migration of the bars. The maximum amplitude of bank oscillations attains, on the average, higher values in the runs with graded sediments, which may be seen as a further indirect effect of the reduced mobility of bars in this case; as a result, the bifurcation occurs at higher values of the width ratio and consequently at lower values of the Shields stress.

Details are in the caption following the image
Peak values of the dimensionless amplitude of bank oscillations as a function of Shields stress.

5.3. Flow Parameters at Incipient Bifurcation

[46] Once an objective criterion for the occurrence of the bifurcation has been established, as discussed in the preceding Subsection, it is then possible to describe channel geometry and characterize channel and flow at the onset of flow bifurcation.

[47] We measured the angles between the streamlines of the two main branches, analyzing the planimetric configuration. The observed values range between 35° and 55°, displaying a weakly positive dependence on the width ratio of the channel. In the runs with graded sediments, we invariably observed the formation of a central wedge shaped deposit of coarse particles (as shown in Figure 19). The characteristic angles of these sorting patterns were slightly greater than the streamline angles, ranging between 40° and 60° (Figure 20). These findings are in fairly good agreement with other experimental and field observations. In particular, Federici and Paola [2003], investigating the occurrence of bifurcations in diverging channels, found that the angle between the two branches was typically about 50°, with a larger value of the angle formed by the central bar deposit.

Details are in the caption following the image
The central, wedge shaped deposit of coarse particles.
Details are in the caption following the image
(left) Angles of bifurcations measured by the planimetric configuration and (right) angles of the central deposit.

[48] Finally, in Figure 21 we trace the path of each experimental run in the ? — β plane. The plot describes the instantaneous reach-averaged hydraulic conditions. Each run is represented in this plane by a decreasing curve, the last point of which roughly corresponds to the flow conditions at the onset of flow bifurcation. The experimental runs in which bifurcation did not occur before the channel reached the fixed flume banks are plotted in Figure 21 with dashed lines.

Details are in the caption following the image
Bifurcation points on the plane Shields stress—width ratio.

[49] It appears that the values of the Shields stress and of the width ratio at the onset of bifurcation can be somehow related, at least for the runs with uniform sediments. The points sit along a critical curve, whereby larger values of the Shields stress are associated with larger values of the width ratio. The diameter of the sediments does not seem to affect this relationship, in that series A and B display a similar behavior.

[50] It must be pointed out that results obtained with graded sediments do not conform to the above behavior. The paths on the ? — β plane show a similar trend, but the bifurcation points are more scattered and correspond to lower values of Shields stress and higher values of width ratio. Finally, we note that the two fast runs display a different behavior and bifurcate at lower values of the width ratio, which confirms a different bifurcation mechanism.

6. Discussion

[51] In this work the attention has been focused on the interaction between bed and bank processes which characterize the evolution of laterally unconstrained channels. The main results can be summarized as follows.

[52] 1. In a laterally unconstrained planform the development of forcing effects, which are mainly related to width variations, lead to a strong modification of the bar structures, driving the transition from spontaneously developing migrating bars to fixed steady bars.

[53] 2. The main effect of bars on the planimetric pattern is the generation of channel curvature and width variations through local bank erosion; bar migration speed is a crucial parameter which controls the subsequent development of the channel, in that channel dynamics may be strongly or weakly conditioned by planimetric forcing depending on the ratio of bar migration speed to lateral bank erosion rate (this sensitive dependence on the migration properties of bars may result in a severe restriction on the applicability of numerical models to predict channel changes with cohesionless boundaries).

[54] 3. The analysis of the development of the planimetric configuration allowed us to define an objective criterion for the occurrence of channel bifurcation; in particular we have observed that the amplitude of width variations and channel sinuosity increase until the onset of bifurcation and then decrease as the concentration of the main flow is shifted toward bank lines that were previously undisturbed.

[55] 4. The planimetrically driven modification of bar structures is responsible for flow and channel bifurcation, occurring mainly through the mechanisms of chute cutoff and of bar dissection.

[56] 5. The longitudinal spacing of bifurcation points is essentially related to the length of bars that formed in the channel at the initial stage.

[57] The above findings may shed some light on the debated question of identifying typical length scales in braided networks [Ashmore, 2001]. In particular a strong indication is obtained on the role of bars on the definition of the scale of link length in braided rivers. Present results, however, refer to individual channels evolving to a bifurcated state subject to a constant flow and sediment supply. Hence this picture neglects the reworking effects which arise from the interaction between different branches. They imply a hierarchy of scales of bar forms migrating through the system, which is also due to historical legacies related to the variations of flow regime and sediment supply. Further evidence is needed to ascertain up to what extent the initial control of bars developing in individual distributaries is felt in actively braided systems.

[58] The experimental investigation also allowed us to characterize the flow and channel geometry at the onset of bifurcation: the configuration was described in terms of both geometrical properties, as the angle between the two main flow directions downstream, and by hydrodynamical parameters, like the average Shields stress and channel width ratio. This characterization provides useful data and possible rules to be implemented in predictive models of channel changes in braided systems [e.g., Jagers, 2003], with the aim of ensuring a more physical base to the prescribed rules. Indeed, understanding and predicting the occurrence of channel bifurcations is a crucial step to improve morphological predictions in braided networks, as chute cutoffs control the position of the main flow and therefore channel adjustment and the location of bank erosion. This requires integration of the present findings with a theoretical framework able to consider the instantaneous interaction of channel curvature and width variations with bed deformation.

Acknowledgments

[59] 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,” cofunded by the Italian Ministry of University and Scientific Research (MIUR) and the University of Trento and of the project “Rischio Iidraulico e Morfodinamica Fluviale” financed by the Fondazione Cassa di Risparmio di Verona, Vicenza, Belluno e Ancona. The authors gratefully acknowledge the fundamental support of Guido Zolezzi and the tireless help of Guido Pettinacci, Kilian Albergati, Anita Ravagnani, Edi Meneguz, and the whole staff of the Hydraulic Laboratory in carrying out the experimental work. The paper has also benefited from the comments of Peter Ashmore and the thorough review of Stuart Lane and Chris Paola.