WATER RESOURCES RESEARCH, VOL. 45, W06418, doi:10.1029/2008WR007086, 2009
Interaction between migrating bars and bifurcations in gravel bed rivers W. Bertoldi,1 L. Zanoni,1 S. Miori,1 R. Repetto,2 and M. Tubino1 Received 2 April 2008; revised 19 March 2009; accepted 31 March 2009; published 13 June 2009.
[1] In the present work we investigate the interaction between migrating alternate bars
and the dynamics of river bifurcations. Laboratory experiments are carried out to study a Y-shaped bifurcation with fixed banks and erodible bed composed of well-sorted sand. The problem is also analyzed by developing a theoretical, one-dimensional model. Results show the occurrence of regular fluctuations in the discharge distribution at the bifurcation node, which are strictly related to bar migration. The effectiveness of bars in conditioning the bifurcation behavior increases with bar amplitude and decreases with bar migration speed. Four qualitatively different behaviors of the system are observed as the controlling parameters of the flow are varied within a range significant for gravel bed rivers. The theoretical predictions are in good qualitative agreement with the experimental observations. Citation: Bertoldi, W., L. Zanoni, S. Miori, R. Repetto, and M. Tubino (2009), Interaction between migrating bars and bifurcations in gravel bed rivers, Water Resour. Res., 45, W06418, doi:10.1029/2008WR007086.
1. Introduction [2] Gravel bed braided rivers are extremely dynamic morphological systems, characterized by a continuous rearrangement of the channel network. Channel bifurcations play a crucial role on their behavior as they control the division of water and sediment discharge throughout the network. Even slight variations in the shape of a bifurcation may induce significant changes in the downstream part of the network [see, e.g., Luchi et al., 2007]. [3] The dynamics of bifurcations is extremely complicated and, in spite of many efforts, is far from being satisfactorily understood. Such a complexity arises from a full coupling between hydrodynamic and morphodynamic processes, whereby flow characteristics at the bifurcation (and consequently water discharge splitting into the downstream branches) is strongly dependent on the local morphology of the bifurcation, which in turn is shaped by the flow [Ferguson et al., 1992]. Another effect, with which the present contribution is concerned, is the fact that, in gravel bed braided rivers, the characteristic time scale of bifurcation development is typically of the same order of morphological changes in single channels, namely large-scale bed forms formation and migration. The present work represents an attempt to systematically investigate the effect of migrating bed forms, in particular alternate bars, on the evolution of a bifurcation. [4] Previous research on river bifurcations mainly focused on the following basic problem: under which conditions is a symmetrical configuration of a bifurcation stable (where
1 Department of Civil and Environmental Engineering, University of Trento, Trento, Italy. 2 Department of Engineering of Structures, Water and Soil, University of L’Aquila, L’Aquila, Italy.
Copyright 2009 by the American Geophysical Union. 0043-1397/09/2008WR007086
symmetrical means that both branches are approximately fed with equal water and sediment discharges)? [5] Research was carried out both theoretically/numerically [Wang et al., 1995; Bolla Pittaluga et al., 2003; Hirose et al., 2003; Slingerland and Smith, 2004; Miori et al., 2006; Kleinhans et al., 2008; Edmonds and Slingerland, 2008] and experimentally [Federici and Paola, 2003; Bertoldi and Tubino, 2007]. A few field observations are also available, from which some insight on the stability of bifurcations can be inferred [Ferguson et al., 1992; Burge, 2006]. [6] Federici and Paola [2003] performed a series of experiments with bifurcations on gravel cohesionless bed and concluded that symmetrical bifurcations tend to be stable for relatively large values of the Shields number. Conversely, for small values of the Shields parameter (typical of gravel bed braided rivers), the symmetrical configuration is found to be unstable and bifurcations evolve either toward strongly unbalanced states, with one of the two branches being fed by a significantly larger amount of water and sediment than the other, or to the abandonment of one branch. [7] Such a finding was confirmed by another experimental work, performed by Bertoldi and Tubino [2007]. The authors also provided an alternative, yet not contradictory, interpretation of the bifurcation stability problem in terms of morphodynamic influence of the bifurcation. [8] All authors point out the importance of the upstream channel morphology and characteristics in determining sediment partition and bifurcation morphology. For instance, the presence of a bend just upstream of a bifurcation [Burge, 2006; Kleinhans et al., 2008], of vegetation [Burge, 2006], of suspended sediment load and cohesiveness [Edmonds and Slingerland, 2008] and, especially in gravel bed rivers, transverse bed channel deformation [Zolezzi et al., 2006; Bertoldi and Tubino, 2007] are recognized to crucially affect the bifurcation. [9] The above contributions focus on the stability of bifurcations and their possible equilibrium conditions. How-
W06418
1 of 12
BERTOLDI ET AL.: INTERACTION BETWEEN BARS AND BIFURCATION
W06418
ever, a common feature of all the above observations is that bifurcation morphology is actually strongly variable in time. Such an unsteadiness is typically induced by flood pulses, but it is also largely related to the generation and migration of large-scale sediment waves (in particular alternating bars) in the upstream channel. [10] Ferguson et al. [1992] described in detail the morphological behavior of a reach that constitutes a chute and an expansion in the Sunwapta River in Alberta, Canada. In particular, their measurements show the effect of a sediment wave traveling along the upstream channel as it approaches the bifurcation. The sediment wave was characterized by an uneven transverse structure and therefore it may be thought of as an alternate bar front. As the wavefront approached the bifurcation point, the authors observed that its effect on sediment and water discharge partition into the downstream branches progressively increased. [11] Also Burge [2006] found that the destabilization of the symmetrical configuration, and possibly the tendency to closure of one of the two branches, are invariably related to the formation of a bar (associated with a transverse structure of the bed) upstream of the bifurcation. Bolla Pittaluga et al. [2003] and Bertoldi and Tubino [2007] also recognized the importance of bar migration on bifurcation morphology. [12] In this work we propose a systematical investigation (numerical and experimental) of the role of migrating bars on water and sediment distribution in a bifurcation and on its morphological evolution. We focus on gravel bed bifurcations with cohesionless bed and consider bed load transport only. In section 2 we define the parameters used in the paper and recall some results on bar formation. In section 3 we formulate the theoretical model and discuss the results. The experiments are then described in section 4. The main outputs of the work are discussed and compared with the existing literature in section 5, where theoretical and experimental results are also compared. Finally, some conclusions follow in section 6.
2. Theoretical Framework [13] In this section we first define the controlling parameters that will be employed in the rest of the paper. We then summarize some results on bar formation and, in particular, briefly describe the analytical model for finite amplitude bars proposed by Colombini et al. [1987]. Results from this model will be employed to account for the effect of bar migration on the morphological evolution of a bifurcation, extending the model of Bolla Pittaluga et al. [2003] for gravel bed river bifurcations. [14] By purely dimensional arguments it can be shown that the state of a channel transporting sediment as bed load can be completely characterized using three dimensionless parameters. In this paper the following parameters will be employed:
J¼
ru2* ; ðrs rÞgDs
b¼
b ; 2D
ds ¼
Ds ; D
ð1Þ
where J is the Shields number, b the channel aspect ratio and ds the dimensionless sediment size. In (1) r denotes water
W06418
density, u* friction velocity, rs sediment density, g gravitational acceleration, b free surface width, D water depth and Ds median grain size. [15] The morphological adaptation of a channel reach scale takes place on a time scale Tm which can be inferred by the Exner equation in the following form: Tm ¼
Db ; qs
ð2Þ
where qs is sediment discharge per unit width. Following Miori et al. [2006] we assume the value of Tm of the upstream channel as a reference time scale for the morphological adaptation of the bifurcation. [16] Moreover, in order to characterize the state of the bifurcation, and in particular its degree of asymmetry, we introduce the following two dimensionless parameters DQ ¼
Qb Qc ; Qa
Dh ¼
hb hc ; Da
ð3Þ
where the subscripts indicate the branch of the bifurcation (with a denoting the upstream channel and b and c the downstream branches, as shown in Figure 1), Q is water discharge and h is average bed elevation at the bifurcation point. These indicators will be referred to as ‘‘discharge asymmetry’’ and ‘‘inlet step’’, respectively. The parameter DQ measures the degree of asymmetry of discharge distribution between the two branches and provides an easy-tomeasure indication of how much a bifurcation is unbalanced toward one of the two downstream channels. DQ is equal to zero when the discharge is evenly distributed and is equal to 1 (or 1) when all water is diverted into one branch. Dh quantifies the transverse difference in bed elevation between the inlets of the downstream branches, scaled with the flow depth of the upstream branch, and is therefore a measure of bed elevation asymmetry between the distributaries. A difference of bed level is almost invariably observed to characterize the region close to the bifurcation. [17] We finally recall some results relative to bar formation. Migrating alternate bars are known to form in straight and sufficiently long channels, provided channel width is large enough. Various stability analyzes showed that the relevant parameter to predict bar formation is the channel aspect ratio b (see Tubino et al. [1999] for a review on the topic). In particular, bars form if b exceeds a threshold critical value b c, which depends on the Shields stress J and the dimensionless sediment size ds. [18] Colombini et al. [1987] (hereinafter referred to as CST87) performed a weakly nonlinear analysis for finite amplitude bars, by employing a perturbation technique based on the expansion of all variables in the neighborhood of critical conditions (i.e., when b = bc). They introduced a small parameter ec, defined as ec = (b b c)/b c, which quantifies the distance from critical conditions, and sought a solution of the problem by employing an asymptotic expansion of all variables in terms of ascending powers of e1/2. At the third order of approximation the authors obtained a reconstruction of the three-dimensional structure of finite amplitude alternating bars and the corresponding velocity and sediment transport distribution in the cross section.
2 of 12
W06418
BERTOLDI ET AL.: INTERACTION BETWEEN BARS AND BIFURCATION
W06418
Figure 1. Scheme of a bifurcation. CST87’s solution is rigorously valid only for ec 1, however the authors found that comparisons with field and laboratory data support the convergence of the approach within a fairly wide range of values of ec, even for ec O(1).
upstream end of each cell was set proportional to cell width by imposing
3. A 1-D Model for Bifurcations With Alternate Migrating Bars 3.1. Model Description [19] In this section, we briefly recall the main ideas behind the work of Bolla Pittaluga et al. [2003], as this is the basis for our model. We then illustrate how the model of Colombini et al. [1987] is employed in order to account for the effect of bar migration on the morphological evolution of a bifurcation. [20] Bolla Pittaluga et al. [2003] (hereinafter referred to as BRT03) proposed a set of nodal point conditions to be adopted at the bifurcation point in one-dimensional models. Their formulation is based on the observation that the influence of a bifurcation extends upstream of the nodal point for a length of order a ba, with ba width of the upstream channel and a an order 1 parameter, to be determined empirically. This conception is justified by experimental and numerical observations [e.g., Bertoldi and Tubino, 2005; Kleinhans et al., 2006]. Following this observation the authors assumed that, at an upstream distance aba from the bifurcation, water and sediment discharge are uniformly distributed in the cross section. Getting closer to the bifurcation point a transverse bed gradient might form, along with transverse water and sediment fluxes. These are important as they control sediment and water division in the downstream channels. To account for these effects in the context of a onedimensional model, BRT03 introduced two cells, of length aba, splitting longitudinally the most downstream reach of channel a (see Figure 1). In the authors’ approach water and sediment transverse exchange is allowed between the two cells and the transverse bed gradient is related to the difference between bed levels at the inlets of the two branches and to the width of the upstream channel. Moreover, following the above discussion, water and sediment feeding at the
QIN i ¼
bi Qa bb þ bc
ð4aÞ
QIN si ¼
bi Qsa ; bb þ bc
ð4bÞ
where QIN and QIN s denote water and sediment discharges entering the cells, respectively, and i = b, c indicates the branch. Within this framework the authors proposed the following nodal point conditions: conservation of water discharge (condition 1), constancy of water level (conditions 2 and 3), and sediment continuity equation applied to the two cells (conditions 4 and 5), which reads ba qsy 1 ba dhi qsi qsa bb þbc ð1 p Þ 1 þ ¼0 þ 2 aba bb þ bc dt bi ði ¼ b; cÞ;
ð5Þ
where t is time, p is sediment porosity, h is bed elevation at the entrance of the downstream channels and qsy represents the transverse sediment flux per unit width between the two cells. Again, the subscript a denotes the upstream channel and i = b, c the two downstream branches. [21] In the conditions (5) the transverse sediment flux between the two adjacent cells was evaluated by BRT03 on the basis of a well established approach whereby such flux consists of a contribution due to transverse exchange of water discharge and a contribution due to the transverse gradient of bed elevation. In the latter term a constant, say r, appears, which has to be determined empirically, and according to estimates provided in the literature ranges between 0.3 and 1 [Ikeda et al., 1981; Talmon et al., 1995]. We refer the reader to BRT03 Miori et al. [2006] and Kleinhans et al. [2008] for further details on this approach.
3 of 12
BERTOLDI ET AL.: INTERACTION BETWEEN BARS AND BIFURCATION
W06418
[22] BRT03 employed the above conditions to study the equilibrium configurations and stability of a bifurcation. They found that for low values of the channel aspect ratio and large values of the Shields parameter of the flow, the symmetrical configuration of the bifurcation is stable. As the channel aspect ratio increases (or the Shields number decreases) this configuration loses stability. In this case, according to BRT03’s model the system evolves toward a strongly unbalanced configuration, in which one branch is fed with a much larger water and sediment discharge than the other one. [23] Migrating alternate bars along the upstream channel influence the bifurcation evolution because they produce an uneven and time-dependent transverse distribution of water and sediment flux. This effect can be included in the model proposed by BRT03 modifying the upstream feeding conditions of the cells. This implies replacing equations (4a) and (4b) with suitable relationships. [24] The model by Colombini et al. [1987] allows to obtain an expression for the spatial structure and time evolution of bars of finite amplitude. At a given section CST87’s model predicts that any variable z has the following time dependence and spatial distribution in the transverse direction: 2 z ð y; tÞ ¼ z 0 þ e1=2 c A½z 1 cosðwt þ f1 Þ sinð yp=2Þ þ ec A
½z 22 cosð2wt þ f22 Þ cosð ypÞ þ z 02 cosð2wt þ f02 Þ þ z 20 cosð ypÞ þ z 00 þ O e3=2 ; c
ð6Þ
where y is the normalized transverse coordinate with origin at the center of the channel (so that the two banks are at y = ±1), z 0 is the value of the variable z in uniform flow conditions (i.e., in the unperturbed flow without bars) and all other terms account for the effect of bars. In the above equation A is the equilibrium amplitude of the perturbation, w its frequency and, along with the constants z 1, f1, z 22, f22, z 02, f02, z 20 and z 00, they are the output of CST87’s model. The values of all such quantities depend only on the controlling dimensionless parameters J, and ds and on the length of the bar which, in CST87’s model, is taken to be that corresponding to the most linearly unstable bar. [25] In order to couple our 1-D bifurcation model to CST87’s bar model we need to compute water and sediment fluxes entering each of the two cells at the bifurcation, at each time. Assuming, for the sake of simplicity, bb = bc, with reference to Figure 1, we integrate in the transverse direction y the liquid and sediment discharge per unit width q(y, t) and qs(y, t) (computed with CST87’s model) from 1 to 0 (to evaluate the flux entering the cell upstream of channel c), and from 0 to 1 (to evaluate the flux entering the cell upstream of channel b). Therefore we have
QIN c ðt Þ
Z0 ¼ ba =2
qð y; t Þdy; 1
QIN sc ðt Þ ¼ ba =2
QIN b ðt Þ
Z1 ¼ ba =2
qð y; tÞdy ð7aÞ 0
Z0 qs ð y; tÞdy; 1
QIN sb ðt Þ ¼ ba =2
Z1 qs ð y; tÞdy: 0
ð7bÞ
W06418
Note that all the above functions of time are harmonic and therefore can be characterized by amplitude, period and phase. [26] Given the structure of the solution (6), each of the two branches of the bifurcation is fed alternatively by a larger amount of water and sediment and the time period of this feeding is set by the ratio between bar length and migration speed w. [27] It is worth observing that the values of q(y, t) and qs(y, t) presented in (6) are proportional to powers of ec. IN When ec is large enough the values of QIN si and Qi can become negative in time, which is physically meaningless. In our model these situations are dealt with by considering the last valid solution, before values become negative, i.e., diverting all water and sediment discharges alternatively into each one of the two cells. [28] Notice that, coherently with the nodal point conditions (5), the solid and liquid discharges (7) are assumed independent from the state of evolution of the bifurcation as the cell length (aba) is defined as the upstream distance from the bifurcation at which flow conditions and bed topography are not affected by the bifurcation itself. [29] In the following the time evolution of the system will be studied under the assumption of uniform flow conditions in all branches. Moreover, following Miori et al. [2006], we will employ a ‘‘local approach’’ whereby the time scale of the system evolution is fixed by the nodal point conditions. Note that two independent time scales appear in the above formulation of the nodal point conditions: the time scale of bifurcation evolution, which is set by the nodal point conditions [see Miori et al., 2006], and the time scale of bar migration. 3.2. Model Results [30] We study the evolution of a bifurcation with symmetrical planimetry where bb = bc = 0.65 ba, according to the regime formula of Ashmore [2001]. The length of the two cells upstream of the bifurcation is assumed equal to the upstream channel width (a = 1). Sediment transport is computed through the formula of E. Meyer-Peter and R. Mu¨ller (Formulas for bed-load transport, paper presented at 2nd Congress, International Association for Hydraulic Research, Stockholm, 1948), and the coefficient r characterizing the transverse sediment transport is set equal to 0.85. [31] Starting from a slightly unbalanced state of the bifurcation we study the time behavior of the system employing the nodal conditions described above with the two upstream cells fed according to equation (7). In Figure 2 results are presented in terms of the time evolution of DQ(t). [32] The fluctuation of the sediment and water input fluxes induce periodic variations of bed elevation at the inlet of the downstream branches and fluctuations of the downstream discharges. The interaction between bar migration and the inherent dynamics of the bifurcation produces scenarios of evolution of the system significantly different from those described by BRT03 in the absence of bars. Changing the values of b a and Ja within a range typical for gravel bed rives we identify four qualitatively different behaviors of the system, which are described below and are shown, in terms of DQ(t), in Figure 2. In Figure 2 solid curves refer to the behavior of the system in the presence of bars and dashed lines to the corresponding behavior in the absence of bars.
4 of 12
W06418
BERTOLDI ET AL.: INTERACTION BETWEEN BARS AND BIFURCATION
W06418
Figure 2. The four behavioral classes of bifurcation evolution in terms of DQ as reproduced by the theoretical model (dsa = 0.05). (a) At b a = 12.5 and Ja = 0.1 the bifurcation is ‘‘balanced’’ (class 1); (b) at ba = 7.5 and Ja = 0.055 the bifurcation is ‘‘bar perturbed’’ (class 2); (c) at ba = 20 and is Ja = 0.1 the bifurcation is ‘‘bar dominated’’ (class 3); and (d) at ba = 35 and Ja = 0.06 the bifurcation closes (class 4). Solid curves refer to the behavior of the system in the presence of bars, and dashed lines refer to the corresponding behavior in the absence of bars. [33] Category 1 consists of bifurcations, with balanced equilibrium configuration, whose state is perturbed by bar migration (Figure 2a). Bar migration induces oscillations of DQ about zero. We refer to bifurcations falling into this category as to ‘‘balanced’’ bifurcations. [34] Category 2 consists of bifurcations, with unbalanced equilibrium configuration, whose state is perturbed by bar migration. In this case the bifurcation evolves toward a strongly unbalanced state as predicted by BRT03, and the trend of DQ is affected by superimposed fluctuations related to the migration of bed forms (Figure 2b). However, barinduced oscillations are not strong enough to make the main flow switching from one branch to the other. These bifurcations are referred to as ‘‘bar perturbed’’. Note that in this case bars have a net effect on the state of the bifurcation as the average of DQ over a period, when the system has reached a periodic state, differs from the value that D Q reaches in the absence on bars. [35] Category 3 consists of bifurcations, with unbalanced equilibrium configuration, strongly affected by the presence of bars. Bifurcations belonging to this category would assume an unbalanced configuration in the absence of bars (dashed curves). However, bar-induced oscillations of water and sediment distribution at the bifurcation point are large enough to make the main flow switching between the two branches at any period of oscillation. In most cases this leads to symmetrical and large-amplitude fluctuations of DQ about
0 (Figure 2c). Bifurcations behaving in this way are considered ‘‘bar-dominated’’ bifurcations. [36] Category 4 consists of bifurcations which evolve toward the closure of one branch. When sediment and flow division into the bifurcates is highly asymmetrical and such asymmetry is persistent in time, as it happens when large and slow alternating bars migrate along the upstream channel, one of the branches closes (Figure 2d). [37] Obviously, for ec < 0 no bars form in the upstream channel and the bifurcation behavior is the same as that predicted by BRT03. [38] Both the reference equilibrium configuration (according to BRT03, i.e., in the absence of bars) and the amplitude and frequency of the bar-induced perturbation (according to CST87) depend on the controlling parameters of flow in the upstream channel b a, Ja and dsa. In Figure 3 we show the above classification of bifurcation behavior as a function of the controlling parameters ba and Ja. In order to obtain the plot we spanned the b J plane (keeping dsa fixed) and studied the time evolution of the bifurcation in each point. Figure 3 shows that bars become more effective in controlling the morphodynamic evolution of a bifurcation (i.e., there is a change from category 2 to category 3 and, eventually, category 4) for increasing values of the aspect ratio of the upstream channel. This happens because large values of b a imply large amplitude bars. [39] Note that bar amplitude predicted by CST87 represents an equilibrium value which is reached asymptotically in
5 of 12
W06418
BERTOLDI ET AL.: INTERACTION BETWEEN BARS AND BIFURCATION
Figure 3. The four possible behaviors of the bifurcation in the plane b J. The solution is proposed for two different bar amplitudes: dashed lines separate the areas of different behavior when bar amplitude is equal to A, and dotted lines separate areas when amplitude is set to 0.75 A. Each area is marked by a circled number identifying the corresponding behavioral class: solid markers refer to amplitude 0.75 A (i.e., they correspond to the dotted lines), and open markers refer to amplitude A, corresponding to the dashed lines. time (parameter A in equation (6)). Thus it represents an upper bound as, in reality, bars may not be able to reach equilibrium. Having this in mind we show in Figure 3 the bifurcation behavior for two different values of bar amplitude equal to A (dashed line and open circled markers) and 0.75 A (dotted line and solid circled markers). From the qualitative point of view results are similar. The main effect induced by a change in bar amplitude is to modify the size of zones 2 and 4. The areas with ‘‘no bars’’ (where ec 0) and of ‘‘balanced’’ conditions (where the equilibrium in the absence of bars corresponds to DQ = 0), are unchanged in all cases as they do not depend on bar amplitude.
W06418
computer controlled laser profiler in order to automatically measure bed elevation on a regular grid (in the present case the adopted spacing was 1 cm in the transverse direction and 10 cm in the longitudinal direction). [43] We built a Y-shaped bifurcation with fixed banks channels, characterized by an initially rectangular cross section and mobile sand bed. The upstream channel was 0.36 m wide and 10 m long. The two downstream branches were 0.24 m wide and 6 m long. The bifurcation angle was set to 30° (see Figure 4). The presence of a sharp angle slightly affected the flow field and the morphology at the bifurcation, but did not change significantly the experimental results, as we observed performing similar runs without the fixed banks. [44] Using this configuration, different experimental runs were performed (denoted by F, see Table 1), changing water discharge Qa and the longitudinal bed slope Sa, in order to reproduce values of the relevant dimensionless parameters (the aspect ratio ba, the Shields parameter Ja and dimensionless sediment size dsa) typical of gravel bed braided rivers. Table 1 reports also information about the computed averaged depth Da in the upstream branch and the values of the Froude number Fr. Averaged Reynolds number ranged from 2000 to 15000. [45] In order to measure the discharge flowing in each downstream channel, the flow was conveyed into two different tanks equipped with a triangular mill weir, where flow depth was measured by a pressure sensor device. In all experiments water discharge in the upstream channel was kept constant throughout the run; furthermore, sediment supply was imposed according to the transport capacity of the incoming flow. This was first estimated using the Parker [1990] and Bagnold [1980] formulas and then adjusted in order to avoid local scour or deposition at the inlet. We found that the Bagnold [1980] formula, with a critical Shields stress for sediment motion equal to 0.04, best approximated the observed sediment transport rates. [46] A symmetrical configuration was used as initial condition for all runs, with equal slopes and bed elevation in the
4. Experimental Investigation 4.1. Experimental Setup and Procedure [40] Two series of experiments were performed in order to investigate the effect of bar migration on bifurcation dynamics. The experiments were carried out in a large laboratory flume, 25 m long and 2.90 m wide, which was constructed in the Hydraulics Laboratory of the University of Trento (Italy). The bed was composed of well-sorted quartz sand. Two different grain sizes were used for two different sets of experiments. Grain size distributions were quite narrow and respectively characterized by a mean diameter Ds equal to 0.63 mm (D10 = 0.5 mm, D90 = 0.75 mm) and 1.05 mm (D10 = 0.92 mm, D90 = 1.19 mm). [41] Water discharge was supplied by a pump, regulated by an inverter, that could set discharge values ranging from 0.5 to 20 L/s. Furthermore, the sediment input was provided by a volumetric sand feeder with three screws conveying sand into the flume through a diffuser. [42] The measuring instruments were moved along the longitudinal, transverse and vertical directions through a carriage driven by electric motors and supported by a highaccuracy rail. Bed topography was surveyed employing a 6 of 12
Figure 4. Panoramic view of the experimental flume.
BERTOLDI ET AL.: INTERACTION BETWEEN BARS AND BIFURCATION
W06418
W06418
Table 1. Relevant Parameters of the Experiments Run
Ds (mm)
Sa
Qa (L/s)
Da (m)
Fr
ba
Ja
dsa
F5-25 F5-30 F5-40 F5-50 F7-06 F7-07 F7-08 F7-09 F7-10 F7-12 F7-13 F7-15 F7-17 F7-20 F7-24 F10-14 F10-30 B5-20 B5-25 B5-30 B5-40 B7-10 B7-12 B7-15 B7-18 B7-20
1.05 1.05 1.05 1.05 0.63 0.63 0.63 0.63 0.63 0.63 0.63 0.63 0.63 0.63 0.63 1.05 1.05 1.05 1.05 1.05 1.05 0.63 0.63 0.63 0.63 0.63
0.0052 0.0047 0.0042 0.0058 0.0065 0.0066 0.0077 0.0078 0.0067 0.007 0.0076 0.0076 0.0068 0.0078 0.0072 0.0097 0.0096 0.0054 0.0053 0.0052 0.0057 0.0066 0.0069 0.0068 0.0070 0.0067
2.5 3.0 4.0 5.0 0.6 0.7 0.8 0.9 1.0 1.2 1.3 1.5 1.7 2.0 2.4 1.4 3.0 2.0 2.5 3.0 4.0 1.0 1.2 1.5 1.8 2.0
0.0203 0.0233 0.0289 0.0301 0.0068 0.0075 0.0078 0.0083 0.0093 0.0102 0.0105 0.0113 0.0127 0.0135 0.0154 0.0119 0.0188 0.0175 0.0201 0.0226 0.0264 0.0095 0.0105 0.0120 0.0133 0.0144
0.79 0.77 0.74 0.87 0.96 0.96 1.03 1.06 0.99 1.03 1.07 1.11 1.05 1.13 1.12 0.99 1.06 0.79 0.80 0.80 0.85 0.98 1.01 1.04 1.07 1.05
8.61 7.50 6.05 5.82 26.30 23.91 23.01 21.66 19.38 17.71 17.21 15.88 14.17 13.35 11.66 14.68 9.31 10.01 8.70 7.75 6.64 18.46 16.73 14.56 13.17 12.18
0.0545 0.0557 0.0600 0.0857 0.0415 0.0462 0.0559 0.0600 0.0573 0.0652 0.0727 0.0785 0.0781 0.0947 0.0991 0.0623 0.0938 0.0500 0.0552 0.0608 0.0752 0.0575 0.0657 0.0741 0.0836 0.0863
0.052 0.045 0.037 0.035 0.092 0.084 0.081 0.076 0.068 0.062 0.060 0.056 0.050 0.047 0.041 0.089 0.056 0.060 0.052 0.047 0.040 0.066 0.060 0.052 0.047 0.044
two distributaries. At the end of each run the final bed topography was surveyed in detail using the laser profiler. [47] A second series of experiments (denoted by B, see Table 1) was carried out in a single thread straight channel with fixed banks, erodible bed and the same width as channel a in the F series with the aim to measure alternate bar height, length and migration speed in a configuration not affected by the bifurcation. The dimensionless parameters of the F series were reproduced in order to compare data surveyed in the two series. [48] Bar morphology was measured referring to the parameters defined in section 2 and computed from bed topography survey (length and amplitude) and from visual observation during the runs (migration speed). The parameter DQ, characterizing the bifurcation evolution was calculated directly from the measured discharge data. The time series were then plotted and the maximum and minimum points were individuated manually. In particular, for each run the average values of the fluctuation period t DQ and of the fluctuation amplitude ADQ were computed. [49] An alternative, more objective, procedure to evaluate t DQ was also used based on wavelet transform analysis which is known to be a valid technique for the characterization of irregularly oscillating time signals [see, e.g., Lau and Weng, 1995; Torrence and Compo, 1998]. This second procedure always led essentially to the same results as the first one, but made it possible to quantify the time irregularity of the signal. The wavelet analysis pointed out the existence of a dominant harmonic even in cases in which time signal D Q(t) appeared very irregular. 4.2. Experimental Results [50] The experimental runs showed different behaviors of the bifurcation, with complex interactions between alternate bar migration and discharge distribution in the distributaries. During each run, bar morphology was not perfectly constant,
as both amplitude and migration speed varied. As a result, the effect on the node configuration was obviously not as regular as that predicted by the theoretical model (see Figure 2). In all runs, the fluctuation period of discharge distribution into the downstream branches corresponded closely to the period of alternate bar migration. In Figure 5 the average period of oscillation t DQ of discharge asymmetry is plotted versus the Shields number of upstream flow (with diamonds) together with the average period of bar migration t bars (with asterisks). Points refer to the F series for bifurcation data and to B series for bar period (see Table 1). The two periods show a similar trend with Ja and close values. [51] Greater variability of the feeding conditions in the downstream branches was detected in runs with larger values of the aspect ratio. This is essentially due to the fact that bars grow in amplitude as the channel aspect ratio increases.
Figure 5. Average period of oscillation of the discharge asymmetry t DQ and average period of bar migration t bars versus the Shields stress in the upstream channel.
7 of 12
W06418
BERTOLDI ET AL.: INTERACTION BETWEEN BARS AND BIFURCATION
W06418
Figure 6. Temporal evolution of the discharge asymmetry DQ as a function of time as observed in four different experimental runs with the four different behaviors. (a) In run F10–30 the bifurcation is ‘‘balanced’’ (class 1); (b) in run F5– 25 the bifurcation is ‘‘bar perturbed’’ (class 2); (c) in run F7 – 09 the bifurcation is ‘‘bar dominated’’ (class 3); and (d) in run F7 –07 the bifurcation closes (class 4). Another effect is probably related to the morphodynamic influence of the bifurcation on bars. Theoretical [e.g., Zolezzi et al., 2006] and experimental [e.g., Bertoldi and Tubino, 2007] works have shown that transverse disturbance on bed level has an upstream influence if the width to depth ratio of the channel is large enough. In this case bar migration is possibly influenced by the presence of the bifurcation and this may result in a stronger interaction between bar migration and morphodynamic evolution of the bifurcation. [52] Experimental runs can be classified in the four classes described in section 3.1 (see Figure 2) through the theoretical model. Four examples of the identified behavioral categories are shown in Figure 6, where discharge asymmetry DQ(t) is plotted as a function of time. The unbalanced runs were classified as bar perturbed or bar dominated depending on the occurrence of frequent shifts between positive and negative values of DQ (see Figures 6b and 6c). [53] In Figure 7 we report all experimental runs in the b J plane using different symbols for the four classes above individuated. It appears that at large values of Shields stress and low values of the aspect ratio, bifurcations keep a balanced configuration. This is in agreement with previous experimental observations [Federici and Paola, 2003; Bertoldi and Tubino, 2007] and theoretical predictions [Bolla Pittaluga et al., 2003; Miori et al., 2006]. However, this symmetrical configuration is perturbed in time by the presence of migrating bars in the upstream channel, which produce oscillations of DQ from positive to negative values (category 1, open circles in Figures 7 – 11). The amplitude of
such oscillations depends on the amplitude and migration speed of bars. [54] At lower sediment mobility (small J) and higher values of the channel aspect ratio b, discharge distribution tends to become unbalanced. Moreover, as b increases the effect of bars grows and the transition from bar-perturbed bifurcations (category 2, solid circles in Figures 7– 11) to bardominated bifurcations (crosses in Figures 7 – 11, category 3) is observed. This is what one would expect by intuition since
Figure 7. Classification of the experimental runs in the plane b J.
8 of 12
W06418
BERTOLDI ET AL.: INTERACTION BETWEEN BARS AND BIFURCATION
W06418
Figure 8. Comparison of bar amplitude Hm and inlet step Dh for the whole set of experimental runs (a) as a function of the relative distance from critical conditions ec and (b) in a direct comparison. increasing b implies increasing bar amplitude [e.g., Colombini et al., 1987] and, obviously, large bars are likely to be more efficient in controlling the bifurcation dynamics. Finally, for very large values of aspect ratio b bifurcations tend to close one of the two branches because of the presence of very large bars in the upstream channel (open squares in Figures 7 –11, category 4). [55] The above findings confirm, at least qualitatively, the theoretical predictions described in section 3. A more quantitative comparison between theoretical and experimental results is postponed to the next section. [56] In order to understand the physical processes acting in the various interactions between bar migration and bifurcation evolution, we analyze in detail bed topography of the upstream channel and temporal discharge fluctuations in the two branches for each experimental run. The relationship between bar migration and morphological behavior of the bifurcation is investigated by comparing bar amplitude Hm with the magnitude of bed level fluctuations given by the inlet step Dh, defined by equation (3). Bar amplitude has been computed as the maximum difference between mean bed elevation of left and right half cross section, normalized by Da. In Figure 8a values of Hm and Dh are reported as a function of ec and the different classes of bifurcations are indicated with different symbols. Points referring to bar amplitude have been measured in the B series (see Table 1), whereas values of D h have been monitored in the F runs,
with similar initial conditions. As expected, bar amplitude increases with ec (i.e., as the channel aspect ratio grows with respect to its critical value for bar formation) and so does the inlet step. Moreover, the values attained by Hm and D h are typically quite close to each other. This provides a further indication that morphological changes of the bifurcation are strictly related with the presence of bars. Also notice that the bifurcation tends to be dominated for values of ec larger than about 2. Eventually, for values of ec 5 4 bars are large enough to induce the closure of one downstream branch. [57] The different behavior between perturbed and dominated bifurcations can be highlighted plotting the comparison between the bar height Hm and the inlet step amplitude D h. Figure 8b clearly shows that bifurcations dominated by bar migration are characterized by values of bar height larger than those reached in the perturbed runs. [58] Discharge asymmetry fluctuations were analyzed referring to the mean value and standard deviation of DQ (t) (respectively plotted in Figure 9a and 9b). These values were computed considering the DQ (t) signal after three times the morphological time scale Tm. [59] In Figures 9a and 9b the time average and standard deviation of DQ are plotted against ec. Figures 9a and 9b point out a clear difference between the four behavioral classes. Balanced bifurcations display values of the average discharge asymmetry very close to zero and fairly low values of the corresponding standard deviation. Perturbed runs are
Figure 9. (a) Mean value and (b) standard deviation of discharge asymmetry DQ for the whole set of experimental runs as a function of the relative distance from critical conditions ec. 9 of 12
W06418
BERTOLDI ET AL.: INTERACTION BETWEEN BARS AND BIFURCATION
Figure 10. Dimensionless average period of discharge oscillations t DQ/Tm as a function of the relative distance from critical conditions ec in the laboratory experiments. characterized by a more unbalanced average configuration, with values of the average DQ up to 0.4. The standard deviation in these cases keeps low, with values not greater than 0.06. This means that bars are not able to switch often the main flow from one distributary to the other. On the contrary, bar-dominated runs are characterized by low values of the mean discharge distribution, but show a much larger amplitude of discharge fluctuations, with standard deviation ranging from 0.18 up to 0.38. Bifurcation in the last category, where one of the downstream branches closes, are characterized by values of DQ up to 1. These runs do not show clear fluctuations and consequently are not reported in Figure 9b. [60] Finally, Figure 10 reports the values of t DQ/Tm versus ec. Again, different symbols indicate different behavior classes. The analysis shows that bar-dominated runs are invariably characterized by longer time scales (approximately greater than 2 Tm). This suggests that, given the correlation between period of bifurcation oscillations and bar migration, slower bars affect more the system evolution.
5. Discussion of the Results [61] Both the theoretical model and the experimental investigation point out that alternate migrating bars affect
W06418
significantly the node configuration of a bifurcation, inducing fluctuations of the discharge distribution. At low values of the Shields parameter, which are typical in gravel bed braided rivers, the main controlling parameter is the aspect ratio of the incoming flow ba. In particular, there is a strong enhancement of bar effect on the bifurcation evolution at fairly large values of the aspect ratio of the upstream channel b a. According to our experimental results bars dominate the dynamics of the bifurcation for ba 5 15. This value is easily encountered in gravel bed braided rivers, as also reported by field measurements performed by Burge [2006] and by Zolezzi et al. [2006]. In their data set incoming branches are generally characterized by an aspect ratio ranging from 10 to 25. Therefore, flow conditions in real gravel bed braided rivers are such that migrating bars could affect strongly the node behavior or just perturb its equilibrium configuration. This has been highlighted also by Burge [2006], who observed that river bars generate bifurcation instability. In some cases he also noted the complete closure of one distributary due to the migration into the bifurcation of a migrating bar (see the description of the fourth bifurcation considered by the author). Burge [2006] found that bifurcations are more balanced at low values of the Shields stress, that is in contrast with the present findings. This can be due, as also the author suggests, to the presence of vegetation, or to armoring processes that can affect sediment mobility particularly in the case of low bed shear stress. [62] We now compare experimental and theoretical results in more detail. In particular we verify if the model is able to predict the correct behavior of system (i.e., if it predicts the correct classification into the four categories described above) for the values of the controlling parameters employed in the experiments. In Figure 11 the theoretical results are plotted together with all experimental points in the plane b J. Theoretical curves have been computed keeping constant the ratio between the upstream channel width ba and sediment size Ds, as this is the only parameter that does not change between all the experiments. Bar amplitude has been calibrated with the values of Hm measured for the free bars reproduced in the series B experiments, finding that the best fit in terms of bar amplitude between CST87’s model and our experiments is obtained with 0.78 A. [63] Figure 11 shows that there is an overall good agreement, as the large majority of experimental points, reported in
Figure 11. Comparison between numerical simulations and experimental results in the plane b J: (a) ba/Ds = 343 and (b) ba/Ds = 572. The comparison is obtained with a calibrated bar amplitude equal to 0.78 A. 10 of 12
W06418
BERTOLDI ET AL.: INTERACTION BETWEEN BARS AND BIFURCATION
Figure 11 with different symbols for each behavioral category, fall in the correct region. The theory is also able to correctly predict that the effect of bars grows with the aspect ratio of the upstream channel. This suggests that the model captures the essential physics of the phenomenon, since it predicts a sensible dependence of the bifurcation behavior from the controlling parameters. [64] We have to note, however, that some possibly important effects are not reproduced by the theoretical model. In particular, as discussed above, when the channel aspect ratio b a is larger than its resonant value b r (as defined by Blondeaux and Seminara [1985] and Zolezzi and Seminara [2001]), i.e., when the effect of morphodynamic disturbances is felt upstream, we expect the bifurcation to significantly influence bar migration. The upstream overdeepening effect on bifurcations has been highlighted by Bertoldi and Tubino [2007] and by Kleinhans et al. [2008], and in both cases it was found to be stronger for larger values of er = (b a b r)/b r . As a consequence, wide channels are more likely to produce unbalanced bifurcations. In this case, bed topography of the upstream channel shows a steady transverse pattern, similar to that of a long alternate bar. This bed configuration interacts with free migrating bars. Kleinhans et al. [2008] pointed out this effect analyzing the long-term evolution of the Rhine River bifurcation near the border between Netherlands and Germany. The effect of a bifurcation on migrating bars is generally that of inducing a lower migration speed [Bertoldi et al., 2005]. This effect is not accounted for in our theoretical model.
6. Conclusions [65] In the present work the attention was focused on the interaction between bar migration and bifurcation evolution in gravel bed rivers. The understanding of this interaction is crucial for the correct prediction of the evolution of a bifurcation and, consequently, of flow and sediment distribution in a braided network. [66] We tackled the problem through theoretical modeling and experimental investigation, analyzing water and sediment distribution in a Y-shaped bifurcation, where a straight channel divides in two symmetrical distributaries, with fixed banks and erodible bed. [67] We observed a strong link between bar height and amplitude of discharge oscillations into the two distributaries. Results suggest that four qualitatively different behaviors of the system can be identified. In particular the bifurcation can show oscillations of the parameter DQ(t) about the symmetrical configuration 1, or about an asymmetrical configuration 2, where, on average, one of the two branches is fed with a significantly larger sediment and water discharge. Moreover, bifurcations can be dominated by bar migration (category 3) and, in this case, the main flow frequently switches from one branch to the other. Finally, the effect of bar presence in the upstream channel can be so large as to induce the closure of one of the two branches (category 4). [68] The main controlling parameters of the problem are the Shields parameter and the aspect ratio of the upstream channel, with the latter playing the major role. Larger values of the aspect ratio imply a stronger effect of bed form migration on the system evolution. In general the effect of bars on the bifurcation dynamics grows if either the bar amplitude or the bar period increases.
W06418
[69] The present investigation refers to a simple case, with fixed banks and straight channels. Further studies, particularly field measurements, are needed to improve the description of the interaction mechanisms between bar migration and bifurcation evolution. Monitoring bifurcations in braided rivers will allow the investigation of the role of effects not considered here, such as the angle between the distributaries, bed armoring and planform evolution of the branches. Moreover, the numerical model employed in this work is based on a one dimensional scheme. In order to model in more detail the evolution of a bifurcation subject to bed form migration a fully 2-D (3-D) approach would be necessary, able to grasp also the morphodynamic influence of the node on bars characteristics. Finally, the proposed model is limited by the validity of CST87’s model, which strictly applies only close to threshold conditions for bar formation, i.e., when the channel aspect ratio b is close to bc.
Notation A bar equilibrium amplitude [dimensionless]. Aq fluctuation amplitude of sediment input normalized with qsa [dimensionless]. ADQ averaged fluctuation amplitude of discharge asymmetry [dimensionless]. b free surface width [m]. ds dimensionless sediment size [dimensionless]. D maximum water depth [m]. Ds median grain size [m]. Fr Froude number [dimensionless]. g gravitational acceleration [m/s2]. Hm dimensionless bar amplitude. p sediment porosity [dimensionless]. q water discharge per unit width [m2/s]. qs sediment discharge per unit width [m2/s]. qsy transverse sediment flux per unit width between the two cells [m2/s]. Q water discharge [m3/s]. QIN water discharge entering the computational cells [m3/s]. IN Qs sediment discharge entering the computational cells [m3/s]. r coefficient for sediment transport due to transverse bed slope [dimensionless]. S longitudinal bed slope [dimensionless]. t time [s]. t bars average period of bar migration [s]. t DQ average fluctuation period of discharge asymmetry [s]. Tm Exner time scale [s]. Tq fluctuation period of sediment input normalized with Tm [dimensionless]. u* friction velocity [m/s]. y transverse coordinate normalized with the half width of the upstream channel [dimensionless]. DQ discharge asymmetry between the two branches [dimensionless]. Dh transverse difference in bed elevation of the downstream branches [dimensionless]. h bed elevation at the bifurcation point [m]. a coefficient quantifying the upstream influence of the bifurcation [dimensionless].
11 of 12
W06418
BERTOLDI ET AL.: INTERACTION BETWEEN BARS AND BIFURCATION
b channel aspect ratio [dimensionless]. b c threshold value of the aspect ratio for bar formation [dimensionless]. b r resonant value of the aspect ratio [dimensionless]. ec relative distance from critical conditions [dimensionless]. er relative distance from resonant conditions [dimensionless]. J Shields stress number [dimensionless]. w bar frequency [1/s]. r liquid density [kg/m3]. rs sediment density [kg/m3]. [70] Acknowledgments. This work has been developed within the framework of the Centro di Eccellenza Universitario per la Difesa Idrogeologica dell’Ambiente Montano and of the project Morphodynamical processes in river and riparian ecosystems—COFIN 2006, cofunded by the Italian Ministry of University and Scientific Research and the University of Trento and of the projects RIMOF and MODITE financed by the Fondazione Cassa di Risparmio di Verona, Vicenza, Belluno e Ancona. The paper has benefited from the comments by Guido Zolezzi, by two anonymous referees and by the appreciated work of Marteen Kleinhans, who carefully reviewed the manuscript, providing many useful suggestions. The authors gratefully acknowledge Stefania Baldo, David Marchiori, Alessio Pasetto, Andrea Casarin, and the staff of the Hydraulic Laboratory, who helped in the execution of the experimental runs.
References Ashmore, P. E. (2001), Braiding phenomena: Statics and kinetics, in GravelBed Rivers V, edited by M. P. Mosley, pp. 95 – 121, N. Z. Hydrol. Soc., Wellington, New Zealand. Bagnold, R. (1980), An empirical correlation of bedload transport rates in flumes and natural rivers, Proc. R. Soc. London, Ser. A, 372, 453 – 473. Bertoldi, W., and M. Tubino (2005), Bed and bank evolution of bifurcating channels, Water Resour. Res., 41, W07001, doi:10.1029/2004WR003333. Bertoldi, W., and M. Tubino (2007), River bifurcations: Experimental observations on equilibrium configurations, Water Resour. Res., 43, W10437, doi:10.1029/2007WR005907. Bertoldi, W., A. Pasetto, L. Zanoni, and M. Tubino (2005), Experimental observations on channel bifurcations evolving to an equilibrium state, in Proceedings of the 4th IAHR Symposium on River, Coastal and Estuarine Morphodynamics, Urbana, Illinois, USA, edited by G. Parker and M. Garcia, pp. 409 – 419, Taylor and Francis, London. Blondeaux, P., and G. Seminara (1985), A unified bar-bend theory of river meanders, J. Fluid Mech., 112, 363 – 377. Bolla Pittaluga, M., R. Repetto, and M. Tubino (2003), Channel bifurcation in braided rivers: Equilibrium configurations and stability, Water Resour. Res., 39(3), 1046, doi:10.1029/2001WR001112. Burge, L. M. (2006), Stability, morphology and surface grain size patterns of channel bifurcation in gravel-cobble bedded anabranching rivers, Earth Surf. Processes Landforms, 31, 1211 – 1226. Colombini, M., G. Seminara, and M. Tubino (1987), Finite-amplitude alternate bars, J. Fluid Mech., 181, 213 – 232. Edmonds, D. A., and R. L. Slingerland (2008), Stability of delta distributary networks and their bifurcations, Water Resour. Res., 44, W09426, doi:10.1029/2008WR006992. Federici, B., and C. Paola (2003), Dynamics of channel bifurcations in noncohesive sediments, Water Resour. Res., 39(6), 1162, doi:10.1029/ 2002WR001434.
W06418
Ferguson, R. I., P. E. Ashmore, C. Paola, and K. L. Prestegaard (1992), Measurements in a braided river chute and lobe: 1. Flow pattern, sediment transport and channel change, Water Resour. Res., 28(7), 1885 – 1977. Hirose, K., K. Hasegawa, and H. Meguro (2003), Experiment and analysis of mainstream alternation in a bifurcated channel in mountain rivers, in Proceedings of 3rd IAHR Symposium on River, Coastal and Estuarine Morphodynamics, vol. 1, edited by A. Sa´nchez-Arcilla and A. Bateman, pp. 571 – 583, International Association of Hydraulic Engineering and Research, Madrid. Ikeda, S., G. Parker, and K. Sawai (1981), Bend theory of river meanders. Part 1—Linear development, J. Fluid Mech., 112, 363 – 377. Kleinhans, M., B. Jagers, E. Mosselman, and K. Sloff (2006), Effect of upstream meanders on bifurcation stability and sediment division in 1D, 2D and 3D models, in River Flow 2006, edited by R. Ferreira et al., p. 1355, Taylor and Francis, London. Kleinhans, M. G., H. R. A. Jagers, E. Mosselman, and C. J. Sloff (2008), Bifurcation dynamics and avulsion duration in meandering rivers by onedimensional and three-dimensional models, Water Resour. Res., 44, W08454, doi:10.1029/2007WR005912. Lau, K., and H. Weng (1995), Climate signal detection using wavelet transform: How to make a time series sing, Bull. Am.n Meteorol. Soc., 76(12), 2391 – 2402. Luchi, R., W. Bertoldi, G. Zolezzi, and M. Tubino (2007), Monitoring and predicting channel change in a free-evolving, small alpine river: Ridanna Creek (north east Italy), Earth Surf. Processes Landforms, 32, 2104 – 2119. Miori, S., R. Repetto, and M. Tubino (2006), A one-dimensional model of bifurcations in gravel bed channels with erodible banks, Water Resour. Res., 42, W11413, doi:10.1029/2006WR004863. Parker, G. (1990), Surface-based bedload transport relation for gravel rivers, J. Hydraul. Res., 28, 417 – 436. Slingerland, R., and N. D. Smith (2004), River avulsions and their deposits, Annu. Rev. Earth Planet. Sci., 32, 257 – 285. Talmon, A., M. C. L. M. van Mierlo, and N. Struiskma (1995), Laboratory measurements of direction of sediment transport on transverse alluvialbed slopes, J. Hydraul. Res., 33, 519 – 534. Torrence, C., and G. Compo (1998), A practical guide to wavelet analysis, Bull. Am. Meteorol. Soc., 79(1), 61 – 79. Tubino, M., R. Repetto, and G. Zolezzi (1999), Free bars in rivers, J. Hydraul. Res., 37, 759 – 775. Wang, Z. B., R. J. Fokkink, M. D. Vries, and A. Langerak (1995), Stability of river bifurcations in 1D morphodynamics models, J. Hydraul. Res., 33, 739 – 750. Zolezzi, G., and G. Seminara (2001), Downstream and upstream influence in river meandering. Part one: General theory and application of overdeepening, J. Fluid Mech., 438, 183 – 211. Zolezzi, G., W. Bertoldi, and M. Tubino (2006), Morphological analysis and prediction of channel bifurcations, in Braided Rivers: Process, Deposits, Ecology and Management, edited by G. Sambrook-Smith et al., Spec. Publ. Int. Assoc. Sedimentol., 36, pp. 227 – 250.
W. Bertoldi, S. Miori, M. Tubino, and L. Zanoni, Department of Civil and Environmental Engineering, University of Trento, Via Mesiano 77, I-38100 Trento, Italy. (
[email protected]; stefano.miori@ing. unitn.it;
[email protected];
[email protected]) R. Repetto, Department of Engineering of Structures, Water and Soil, University of L’Aquila, Monteluco di Roio, I-67040 L’Aquila, Italy. (
[email protected])
12 of 12
