River bifurcations: Experimental observations on equilibrium configurations

River bifurcations: Experimental observations

on equilibrium configurations

W. Bertoldi1 and M. Tubino1

Received 23 January 2007; revised 27 June 2007; accepted 2 August 2007; published 30 October 2007.

[1] In this work we have investigated the equilibrium configurations of a Y-shaped fluvialbifurcation through a laboratory analysis. Three series of experimental runs have beenperformed in a wide flume, where a symmetrical bifurcation has been constructedjoining three branches with fixed banks and movable bed made of a well sorted quartzsand; the angle between the two downstream distributaries was equal to 30 degrees. Theexperiments have been carried out with different values of longitudinal bed slope andwater discharge, in order to investigate a range of the relevant morphodynamicparameters typical of gravel bed braided rivers. The equilibrium configuration of thebifurcation has been characterized through the measure of the discharge partition indownstream branches and of the local bed structure at the node. The existence ofunbalanced equilibrium configurations has been observed and the role of migratingalternate bars has been pointed out. The experimental results confirm the theoreticalpredictions which have been recently obtained through the simple model of Bolla Pittalugaet al. (2003). Moreover, interpreting the measured data in the light of the concept ofmorphodynamic influence provides a new perspective in the analysis of theequilibrium configurations of a bifurcation.

Citation: Bertoldi, W., and M. Tubino (2007), River bifurcations: Experimental observations on equilibrium configurations, Water

Resour. Res., 43, W10437, doi:10.1029/2007WR005907.

1. Introduction

[2] Field inspection of channel bifurcations within braid-ed rivers often reveals the occurrence of markedly unbal-anced configurations (Figure 1). Such an asymmetry istypically displayed in terms of the geometrical propertiesof downstream branches, namely channels alignment, bedtopography and width [Federici and Paola, 2003; Miori etal., 2006]. Furthermore, it is also reflected by the structureof the flow field and the uneven partition of flow andsediment discharge at the node [Zolezzi et al., 2006].Though the above unbalance is more evident at low stages,it also characterizes active bifurcations undergoing forma-tive events.[3] One could argue that the asymmetry of single nodes

within a braided reach is somehow related to the highlyunsteady character of the network and mainly results fromthe interference of various processes, such as bed formsmigration and channel shift, which continuously force thesystem to depart from the condition of local equilibrium.Conversely, asymmetry could be inherently related to thefree response of the system, the unbalanced configurationbeing a possible final equilibrium state of freely evolvingbifurcations.[4] The analysis of the above aspects and the identifica-

tion of the recurring processes characterizing the timeevolution of a bifurcation are key ingredients in order to

understand and predict the dynamics of a braided network[Ashmore, 2001; Bolla Pittaluga et al., 2001]. In fact, thebifurcation mechanism represents the main cause of thebraided nature of a river [Leopold and Wolman, 1957;Ashmore, 1991; Richardson and Thorne, 2001; Bertoldiand Tubino, 2005] and strongly influences the way throughwhich water and sediments are distributed and deliveredfurther downstream [Ferguson et al., 1992; Bristow andBest, 1993; Burge, 2006]. However, modeling the morpho-logical evolution of a bifurcation and predicting the waterand sediment partition at the node are still challengingissues for existing analytical and numerical models, evenin simple configurations [Paola, 2001; Klaassen et al.,2002; Jagers, 2003; Hardy et al., 2005].[5] Some attempts have been recently pursued to inves-

tigate the equilibrium and the stability of single nodesthrough a one-dimensional model, coupled with a suitabledescription of the flow and sediment exchange at thebifurcation [Bolla Pittaluga et al., 2003; Miori et al.,2006]. In spite of the strongly simplified character of theapproach, the model seems able to capture the mainingredients embodied in bifurcations dynamics. The abovetheoretical results, which will be briefly summarized in thenext section, claim for a laboratory validation, which is alsoneeded in order to calibrate the parameters adopted in themodel to describe flow and sediment partition at the node.[6] In the present work we report on the results of a

systematic set of experimental runs that we have performedwith the aim of providing a quantitative description of thepossible equilibrium states of a simple ‘Y-shaped’ bifurca-tion with movable bed and fixed banks. The attention hasbeen mainly focused on the measure of the equilibrium

1Department of Civil and Environmental Engineering, University ofTrento, Trento, Italy.

Copyright 2007 by the American Geophysical Union.0043-1397/07/2007WR005907

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WATER RESOURCES RESEARCH, VOL. 43, W10437, doi:10.1029/2007WR005907, 2007

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properties of the bifurcation, namely the discharge partitionat the node and the altimetric pattern, for different hydraulicconditions of the upstream channel. Furthermore, we havealso experimentally investigated the stability of such equi-librium states and analyzed the effect induced by theoccurrence of bars propagating in the upstream channel.[7] Experimental data allow for a direct validation of the

theoretical predictions of Bolla Pittaluga et al. [2003].Moreover, a novel feature which arises from the experi-mental results is the observation of the upstream influenceexerted by the bifurcation. Analyzing such process in thelight of the concept of morphodynamic influence, as it hasbeen defined by Zolezzi and Seminara [2001], opens a newperspective in the interpretation of the dynamic of bifurcat-ing streams and greatly facilitates the analysis of experi-mental results.[8] The theoretical framework to which we refer is

reported in the next section; in the subsequent sections theexperimental set up, procedures and results are described. Acomparison between the theoretical predictions of BollaPittaluga et al. [2003] and experimental data is includedin the last section along with some concluding remarks.

2. Theoretical Framework

[9] The analysis of the equilibrium configurations andstability of a single bifurcation has been recently tackledwithin the context of a one-dimensional approach by Wanget al. [1995] and by Bolla Pittaluga et al. [2003] (hereinafterreferred to as BRT). In the above works a simple ‘Y-shaped’configuration is considered, in which the upstream channel,named a, divides into two downstream branches (b and c).All branches have constant width (fixed walls) and longitu-dinal bed slope, measured along the axis of each branch. Thetheoretical analysis seeks the equilibrium states of the systemsubject to given values of water discharge Q and sedimentdischarge Qs.[10] Within the framework of a one-dimensional model

with movable bed, five nodal conditions are needed at thebifurcation. In this respect, results of theoretical modelssuggest that the system response is quite sensitive to therelationship adopted to distribute the sediment load in thedownstream branches. BRT have introduced a suitablenodal condition, based on a quasi two-dimensional scheme,

to determine the discharge partition at the node. In theirmodel the last reach of the upstream channel is divided intotwo adjacent cells, whose longitudinal length is equal toaba, where ba is the upstream channel width and a is anorder-one parameter. The sediment inputs in downstreambranches are then computed solving the mass balanceequation applied to each cell and accounting for the lateralexchange of sediments which is driven both by flowpartition and by the difference in bed elevation betweenthe two cells. It is worth noticing that through this term themodel retains the required coupling between the exchangeprocesses at the node and the local bed structure. Thesystem of nodal point conditions adopted in BRT is thencompleted imposing the water balance at the node andsetting the constancy of water levels in the three channelsjoining at the node. The model of BRT has been recentlyextended by Miori et al. [2006] to the case of self-formingchannels with erodible banks.[11] We refer the interested reader to the original paper of

BRT for further details. Here we recall that the theoreticalresults obtained by BRT can be expressed in terms of therelevant dimensionless parameters characterizing the up-stream flow, namely the longitudinal bed slope S, theShields stress # and the aspect ratio b:

Ja ¼ta

�s � rð ÞgDs

; ba ¼ba

2Da

; ð1Þ

where D is the reach averaged value of water depth, Ds isthe mean grain size, rs and r are the sediment and the waterdensity, respectively, g is gravity and t is the average bedshear stress. The subscript a refers to the upstream channel.[12] For a symmetrical bifurcation (that is, when down-

stream branches have identical width and slope) BRT havefound that a unique equilibrium solution exists provided theShields stress Ja attains relatively high values (or the aspectratio ba is sufficiently low): in this case the dischargedistribution is balanced. On the other hand, for relativelylow values of the Shields parameter (or for larger values ofthe aspect ratio) three possible equilibrium solutions occur.One is the balanced solution, as in the previous case; theother two are reciprocal and correspond to unbalancedconfigurations characterized by an uneven distribution offlow and sediment discharge in the downstream branches.Moreover, the stability analysis of these equilibrium con-figurations has revealed that, when three solutions exist,only the unbalanced states are stable.[13] An example of the equilibrium configurations

obtained through the model of BRT is reported in Figure 2,in terms of the ratio Qb/Qc which measures the dischargepartition in the downstream channels.[14] The theoretical model of BRT also predicts that the

unbalanced solutions are invariably characterized by a nonsymmetrical altimetric configuration, such that the bedlevels at the inlet of the two downstream branches aredifferent. Furthermore, the larger is the imbalance of flowpartition at the node (i.e., the smaller is Qb/Qc), the larger isthe amplitude of this gap between the two branches. Therole played by the inlet step in the nodal point condition isquite relevant since it determines a transverse slope whichredistributes the excess of sediment load toward the inlet of

Figure 1. Bifurcation in a braided river (Sunwapta River,Canada).

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the downstream branch that carries much water and conse-quently requires a larger sediment supply.[15] The above results document the ability of the theo-

retical model of BRT to reproduce the asymmetry which isoften displayed by the bifurcations within a braided net-work. They also suggest that, under suitable conditions,such an asymmetry is inherently related to the dynamics ofthe node and can be produced even in the absence ofexternal forcings, like that related to the migrationof alternate bars in the upstream flow. The latter effect,which will be analyzed in the next sections, has beenneglected in the theoretical model of BRT; however, intheir paper some experimental observations are reported forthe case of a bifurcation angle equal to zero (a channeldivided by a longitudinal wall in the downstream reach),which suggest that the migration of bars can induce fluctu-ations of water and sediment discharge in the two down-stream branches (see Figure 4 of BRT). Similar results havebeen also found by Hirose et al. [2003].[16] The theoretical results of BRT highlight the role of

topographical effects acting within the final reach of up-stream channel as the main drivers of the dynamicalbehavior of the node. The above finding sets the opportu-nity to investigate such behavior in the light of the conceptof morphodynamic influence. Within a two-dimensionalframework such influence manifests itself through theoccurrence of bed responses displaying a lateral bed struc-ture, namely large-scale sediment waves termed bars. Sincethe original work of Olesen [1983], it has been recognizedthat a non uniform initial condition as well as a variation inchannel geometry or curvature is able to induce the forma-tion of stationary two-dimensional perturbations of bedtopography (see also the recent review paper of Lanzoniet al. [2006]).[17] Stationary undulations are typically triggered by

planimetric discontinuities which cause an abrupt changein the alignment of the main flow. A notable example is the‘‘overdeepening’’ phenomenon produced by a sharp varia-tion of curvature: Struiksma et al. [1985] first documented,in a straight channel followed by a curved reach, theoccurrence of a sequence of non migrating alternate barswhose amplitude was decaying slowly in the downstreamdirection. Blondeaux and Seminara [1985] have shown that

the analysis of stationary two-dimensional perturbations isalso relevant for river meandering since such modes can beresonantly excited in meandering channels with a periodicdistribution of channel curvature, as the aspect ratio bapproaches a resonant value br. Under bed load dominatedconditions such resonant value depends on the Shieldsstress and the relative roughness [Seminara and Tubino,1992, Figure 4a]: in particular, at relatively low values ofthe Shields stress #, say less than 0.1, br falls in the range5–20; its value increases sharply for larger values of #.[18] In subsequent works Zolezzi and Seminara [2001]

and Zolezzi et al. [2005] have shown both theoretically andexperimentally that the resonant value of the aspect ratioalso controls the direction toward which the morphody-namic influence is dominantly propagated. In particular theyhave shown that such influence is mainly felt upstream ordownstream depending upon b being larger (super-resonantchannels) or smaller (sub-resonant channels) than br. It isworth pointing out that the above two-dimensional view-point offers a complementary view on the general issue ofmorphodynamic influence with respect to the classical one-dimensional theory [e.g., de Vries, 1965]. In particular, insuch more general context the existence of a wider spectrumof both migrating and stationary sediment waves carryinginformation related to bed topography has been recognized.Furthermore, both theoretical results and experimental ev-idence suggest that the aspect ratio of the channel is likely toplay for two-dimensional bed forms (namely, those control-ling the planform shape of channels) a role similar to that ofthe Froude number for long purely longitudinal one-dimen-sional bed waves. Examples in this respect are the criticalvalue of the aspect ratio bc which sets the occurrence ofmigrating alternate bars in straight channels [see Colombiniet al., 1987] or the resonant aspect ratio.[19] The above distinction between sub-resonant and

super-resonant cases may turn out to be relevant also inthe present analysis of the equilibrium configuration ofbifurcations. In fact, like an abrupt change of curvature,also a bifurcation acts as a planimetric discontinuity whichcauses a sudden deviation of the flow direction with respectto the incoming flow. Furthermore, the uneven distributionof flow discharge at the node, with the consequent asym-metry of bed level at the inlet of downstream branches, ismore likely to produce an upstream influence on bedtopography in super-resonant conditions. Hence we mayexpect that a stronger topographic control can be exerted onflow and sediment distribution at the node when b > br.

3. Experimental Set Up and Procedure

[20] The experimental runs were performed in the ‘pflume’, a large facility (25 m long and 3.14 m wide) formovable bed experiments located in the Hydraulic Labora-tory of the University of Trento.[21] The ‘p flume’ is equipped with an instrumentation to

monitor the bed topography. The survey is carried outthrough a laser device, supported by a carriage. Highaccuracy is ensured by a system of rails and motors, whichallows a precise positioning of the measuring instrumenta-tion along the longitudinal, transversal and vertical direc-tions. On the same carriage a water gauge is positioned, inorder to measure the free surface level. The management ofthe whole system (water and sediment feeding, measuring

Figure 2. The discharge distribution in downstreamchannels of a Y-shaped bifurcation, as predicted by themodel of Bolla Pittaluga et al. [2003], as a function of theaspect ratio, for different values of the Shields parameter.

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system) is completely automated and controlled by asoftware.[22] The flume was filled with a well-sorted, sieved

quartz sand. The water discharge was supplied by a pump,regulated with an inverter, that allowed to set dischargevalues from 0.5 to 20 liters per second. At the upstream endof the flume, the first meter was devoted to regulate thekinetic energy of the incoming flow, while at the down-stream end a tailgate was placed in order to fix the bedelevation, after which a chute conveyed the flow to asubmerged tank. The sediment input was provided by anopen circuit, consisting of a volumetric sand feeder, threeelectric motors, and a diffuser that conveyed the sand intothe flume. Dry sand was supplied in order to ensure aconstant and well defined input, a condition particularlyrelevant for the case of low sediment rates.[23] A ‘Y-shaped’ bifurcation was built inside the flume,

constructing three channels with fixed walls, movable bedand rectangular cross section joining at the node. Channelwidth was set to 0.36 m and to 0.24 m for the upstreamchannel and for the two downstream channels, respectively.The above values were chosen on the basis of the outcomesof rational regime theories [see for example Griffiths, 1981],that suggest a non-linear relationship between the flowdischarge and the channel width, implying a ratio of 1.3between the total width of the downstream channels andthe upstream width, in the case of symmetrical configura-tions [Ashmore, 2001]. The bifurcation angle was set to30 degrees, with the two distributaries diverging symmetri-cally from the direction of the upstream flow. The wholesystem was approximately 12 m long and ended with aseries of tanks and mill weirs, which allowed the measure ofthe water discharges flowing into the two downstreambranches.[24] Two sets of runs were performed using a sieved

sand with a 0.63 mm mean diameter and grain density of2630 kg/m3. In the third set a coarser sediment wasemployed, with a 1.05 mm mean diameter. Such values ofgrain size were chosen in order to reproduce in the modeltransport conditions similar to those occurring in gravel bedbraided rivers. In this respect the laboratory model can beviewed as a generic model with a scaling factor falling inthe range of 50–100.[25] In the first set of runs the slope was regulated to the

value of 0.3% in order to ensure relatively low values of theaspect ratio ba of upstream flow. Consequently the occur-rence of free bars in the upstream channel was inhibited,since ba did not exceed the threshold value for bar forma-tion as predicted by the linear theory [see for exampleColombini et al., 1987]. In the other two series the slopewas set to 0.5% and to 0.7%, which implied lower values ofthe flow depth. Consequently in almost all runs the forma-tion of alternate migrating bars was observed, whoseamplitude was larger in the third series, characterized byhigher values of the aspect ratio.[26] The experimental runs were performed according to

the following procedure. At first, the bed was flattened tothe prescribed longitudinal slope using a scraper attachedto a carriage running along adjustable rails. The bed wasthen saturated with a very low water discharge in order tohave a smooth surface and then surveyed with the laserprofiler to check the initial conditions. Water discharge was

then set to the prescribed value and the sediment dischargewas regulated estimating the equilibrium bed load transportcapacity under uniform flow conditions through Parker[1990] formula for gravel bed rivers.[27] During the run water discharge was gauged through

a pressure sensor device positioned in the right tank,monitoring the water depth at 1-min intervals. Additionalmeasures were performed with graduated rods in both tanks.When flow partition and bed topography reached an equi-librium configuration the run was stopped. In the runswhere migrating bars did not occur, such equilibrium wasdefined as the final steady state reached by the dischargedistribution in downstream branches. In the other cases, dueto the oscillatory behavior of discharge partition induced bythe migration of bars in the upstream channel, an averageequilibrium state was considered.[28] Finally bed topography was surveyed with the laser

profiler, on a grid spacing 10 cm in the longitudinaldirection and 1 cm in the transverse direction.

4. Experiments Description

[29] Three sets of experiments were performed with theaim of determining the equilibrium configuration of thebifurcation, described in terms of discharge partition andbed topography, and assessing the effect on such equilibriuminduced by the migration of alternate bars in the upstreamchannel. Different hydraulic and transport conditions in theupstream flow were reproduced by varying the water dis-charge Qa, keeping the longitudinal bed slope S fixed foreach set of runs. The resulting values of the relevantdimensionless parameters describing the flow and sedimenttransport, namely the aspect ratio ba and the Shields stress#a, fall in the following ranges

4 < ba < 25; 0:045 < #a < 0:11; ð2Þ

which reproduce typical conditions encountered in natural,gravel bed, braided networks.[30] The experimental conditions and the values of

the dimensionless parameters characterizing the differentruns are reported in Table 1, where Da is mean flow depthin channel a, computed through a uniform flow stage-discharge relationship, and ds is the relative roughness,defined as the ratio between the mean grain size and theflow depth. The roughness coefficient was calibratedthrough a series of previous measurements on a straightchannel with bed composed of the same material used in theexperiments. Furthermore, br is the resonant aspect ratio forthe upstream branch defined in Section 2, which has beencomputed in terms of the Shields stress and relative rough-ness. We note that, unlike the study of Seminara and Tubino[1992], where Meyer-Peter Muller formula was used toobtain the plot reported in Figure 4a, here Parker [1990]bed load formula has been adopted, which performs betterfor gravel bed rivers, particularly at low transport rate. In afew cases, the runs were repeated in order to check thereproducibility of the results and to assess the absence ofany influence due to possible asymmetries in the initialconfiguration.[31] Table 1 also reports the measured equilibrium values

of the discharge ratio rQ = Qc/Qb, where channel ‘b’ is themain downstream channel.

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[32] Figure 3 shows the time evolution of the dischargeratio rQ observed in three different experiments. These plotshighlight the different behavior detected in the runs withoutfree bars (a), as compared to that observed in the runscharacterized by the presence of migrating alternate bars inthe upstream channel (b, c).[33] In the former case (see Figure 3a) the system reached

an equilibrium configuration in a relatively short time (sayfew hours, depending on the sediment mobility), followingan asymptotic trend. Moreover in few runs, namely thosecharacterized by higher values of the Shields parameter ofthe incoming flow, the bifurcation kept balanced (rQ beingapproximately equal to 1). The stability of the equilibriumconfiguration reached at the end of these runs was alsoverified perturbing the final equilibrium state with anamount of sand fed in one of the downstream branches: inevery tested configuration the system returned to the sameequilibrium condition reached before the perturbation.[34] The evolution of the second and third sets of runs

was strongly affected by the development and migration of

alternate bars, mainly in the upstream channel. As aconsequence, the recorded evolutionary pattern of dischargepartition at the node was more complex (two examples arereported in Figures 3b and 3c), resulting in a less regulartrend of the measured data. The first example (Figure 3b)shows the results of a run with a relatively high value of theShields stress Ja (run F7-24), in which rQ displayed smalloscillations around the final equilibrium value; in this casethe discharge distribution kept almost balanced and thepresence of bars affected the bifurcation only slightly. Onthe contrary, the second example (run F7-08, Figure 3c)shows the results of an unbalanced run. Here, the occur-rence of migrating alternate bars marked the subsequentbehavior at the node since the early stage, causing a suddeninstability of the bifurcation. Because of bed forms migra-tion the flow then switched from one channel to the other,leading to the closure and re-opening of the downstreambranches. We note that in this case the oscillation betweentwo unbalanced states, induced by bar migration, oftencorresponded to mirror solutions, the discharge ratio oscil-

Table 1. Experimental Conditions and Relevant Dimensionless Parameters

Run Ds, mm S Q, liters/s Da, m ba Ja dsa br rQ Dh

F3-18 0.63 0.0031 1.8 0.0167 10.77 0.0459 0.038 5.75 0.29 0.694F3-20 0.63 0.0026 2.0 0.0189 9.55 0.0425 0.033 5.81 0.46 0.594F3-21 0.63 0.0027 2.1 0.0191 9.45 0.0453 0.033 5.89 0.56 0.514F3-23 0.63 0.0031 2.3 0.0194 9.26 0.0524 0.032 6.46 0.73 0.365F3-25 0.63 0.0026 2.5 0.0215 8.38 0.0487 0.029 6.22 0.65 0.312F3-29 0.63 0.0031 2.9 0.0223 8.08 0.0599 0.028 8.47 0.80 0.153F3-37 0.63 0.0033 3.7 0.0254 7.09 0.0721 0.025 10.50 0.91 0.043F3-45 0.63 0.0037 4.5 0.0277 6.50 0.0873 0.023 12.11 0.99 0.022F3-61 0.63 0.0029 6.1 0.0363 4.95 0.0855 0.017 12.61 0.97 0.019F5-20 1.05 0.0046 2.0 0.0184 9.53 0.0439 0.057 5.21 0.17 0.871F5-25 1.05 0.0041 2.5 0.0217 8.07 0.0461 0.048 5.47 0.61 0.350F5-30 1.05 0.0047 3.0 0.0233 7.50 0.0557 0.045 6.60 0.74 0.291F5-40 1.05 0.0042 4.0 0.0289 6.06 0.0603 0.036 8.32 0.95 0.139F5-50 1.05 0.0058 5.0 0.0300 5.84 0.0860 0.035 11.11 0.96 0.107F7-06 0.63 0.0065 0.6 0.0068 26.30 0.0420 0.092 4.61 0.00 1.307F7-07 0.63 0.0066 0.7 0.0075 23.91 0.0462 0.084 4.81 0.00 1.370F7-08 0.63 0.0077 0.8 0.0077 23.30 0.0559 0.082 5.72 0.00 1.726F7-09 0.63 0.0078 0.9 0.0083 21.66 0.0606 0.076 6.86 0.25 1.318F7-10 0.63 0.0067 1.0 0.0093 19.38 0.0574 0.068 6.24 0.05 1.228F7-12 0.63 0.0070 1.2 0.0102 17.71 0.0655 0.062 8.07 0.45 0.821F7-13 0.63 0.0076 1.3 0.0105 17.21 0.0734 0.060 8.84 0.50 0.773F7-15 0.63 0.0076 1.5 0.0113 15.88 0.0787 0.056 9.49 0.50 0.397F7-17 0.63 0.0068 1.7 0.0126 14.28 0.0784 0.050 9.69 0.45 0.722F7-20 0.63 0.0078 2.0 0.0134 13.45 0.0943 0.047 11.11 1.00 0.485F7-24 0.63 0.0072 2.4 0.0153 11.73 0.0985 0.041 11.73 1.00 0.198

Figure 3. The time evolution of the discharge ratio rQ of downstream branches measured in threedifferent runs:(a) F3-21; (b) F7-24; (c) F7-13.


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lating between two reciprocal values. As said before, whenbar migration dominated the evolutionary process at thenode, the identification of the equilibrium state was morecomplex and the quantification of the equilibrium values ofthe bifurcation parameters was obtained in terms of theaveraged values reached by the system.[35] We invariably observed that the unbalanced runs

were characterized by a distinctive asymmetry in the bedtopography of downstream distributaries. In particular thebranch carrying a lower value of the discharge was gener-ally subject to an aggrading process, such that, at equilib-rium, the main downstream branch exhibited in all cases alower mean bed elevation. The maximum value of theabove difference can be used as a characteristic measureof the equilibrium morphological response of the bifurca-tion. Measured values of such ‘inlet step’Dh at equilibrium,normalized with the flow depth Da, are also reported inTable 1. Figure 4 shows an example of the longitudinal bedprofiles measured along the three channels joining at thenode.

[36] In order to estimate the inlet step Dh we tested threedifferent procedures. First, we determined Dh as the reachaveraged value of the difference in bed elevation of down-stream branches; the length of the test reach was chosenapproximately equal to two times the downstream channelwidth. In a second procedure, the longitudinal profiles of thedownstream channels were linearly interpolated, over alength of approximately 10 widths, and Dh was computedas the relative distance of these two regression lines at thebifurcation section (see Figure 4). A third estimate of theamplitude of the inlet step was obtained through a FourierTransform analysis of bed elevation data collected at singlecross sections upstream of the bifurcation. In this case Dhwas computed as the amplitude of the principal harmonic,with transverse wavelength equal to two channel widths. Itis worth pointing out that the three selected procedures leadto very similar values for Dh. Results reported in thefollowing section have been obtained through linear inter-polation of the bed profiles of downstream branches.[37] We also observed that, in the unbalanced runs, the

asymmetrical bed configuration at the node induced theformation of a steady altimetric pattern in the regionupstream of the bifurcation, with a transverse structuresimilar to that of alternate bars (i.e., non symmetrical withrespect to the channel axis), but with a wavelength rangingabout 15 times the channel width (almost twice the length ofmigrating bars). Figure 5 shows an example of the bedtopography of the upstream channel with migrating alter-nate bars (a) and with the steady pattern induced by theimbalanced bifurcation (b). The latter configuration, whoseamplitude decays slowly in the upstream direction, wit-nesses the existence of an upstream morphodynamic influ-ence induced by the bifurcation.

5. Experimental Results

[38] The measured values of rQ and Dh for the threeseries of runs are reported in Figures 6 and 7 as functions of

Figure 4. The longitudinal bed profiles of the downstreambranches measured at equilibrium in run F3-21.

Figure 5. Pictures and bed topography maps of the final reach of the upstream channel showing thepresence of migrating bars (a) and of steady longer bars caused by the morphodynamic influence of thebifurcation (b).

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both the Shields stress and the aspect ratio of the upstreamchannel. We note that, due to the fixed width of thebranches, higher values of Ja correspond to lower valuesof ba.[39] The main outcome of present experimental observa-

tions is the recognition that for given slope, grain size andchannel width the ability of the bifurcation to keep a moreor less balanced discharge partition depends on flow dis-charge. In particular, the smaller is the discharge of theincoming flow, the larger is the observed imbalance at thenode. In dimensionless form this implies that the node cankeep balanced (i.e., rQ ’ 1 and Dh ’ 0) only for relativelylarge values of the Shields stress of the upstream flow (or,equivalently, for relatively small values of the aspect ratio).Furthermore, the discharge ratio becomes smaller and theamplitude of inlet step larger for decreasing (increasing)values of Ja (ba). The above finding doesn’t seem to beappreciably affected by the occurrence of migrating bars inthe upstream flow, in spite of their strong influence on nodedynamics, as described in the preceding section. It is worthnoting that the theoretical findings of BRT appear quiteconsistent with the above experimental scenario.[40] Figure 6 shows that, in a few runs characterized by

extremely low values of the Shields parameter, one of thedownstream branches was almost completely dry (rQ ’ 0).In such conditions, as suggested by Figure 7, the transversaldifference of bed elevation at the bifurcation attained, atequilibrium, a value comparable with the flow depth in the

upstream channel (Dh ’ 1). We note, however, that in theruns characterized by the formation of alternate bars eventhis strongly asymmetrical configuration was not observedto keep invariably stable. In fact, the sudden aggradationprocess induced in the only (temporarily) active branch bythe migration through it of two consecutive bars was oftenable to activate again the closed branch, shifting the node tothe opposite equilibrium configuration (see Figure 3 for anexample).[41] Experimental results highlight the close relationship

between the discharge distribution in downstream channelsand the amplitude of the inlet step. A linear regression of thedata set of the two first series (S = 0.3% and S = 0.5%) leadsto a simple relationship (Figure 8):

Dh ¼ 1� rQ: ð3Þ

stating that the sum of rQ and Dh is approximately equalto 1. The equilibrium configurations attained in the thirdseries of runs do not show such a close relationship, due tothe strong influence of migrating alternate bars on the bedtopography near the node.[42] As pointed out in Section 2, it is of interest to

analyze present experimental results in terms of the theoryof morphodynamic influence [Zolezzi and Seminara, 2001].On the basis of the computed values of the resonant aspectratio br reported in Table 1 the experimental runs can bedivided into two groups: sub-resonant and super-resonant

Figure 6. Equilibrium values of the discharge ratio rQ of downstream branches as a function of theShields stress Ja and of the aspect ratio ba of the upstream channel.

Figure 7. Equilibrium values of the dimensionless inlet step, Dh, as a function of the Shields stress baand of the aspect ratio Ja of the upstream channel.


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runs. The measured values of rQ and Dh are reported inFigures 9 and 10 as a function of the relative distance fromthe resonant conditions (b � br)/br. It appears that themeasured values display a quite distinct behavior dependingupon b being smaller or larger than br: in sub resonant runsthe bifurcation keeps almost balanced and symmetrical,whereas in super-resonant conditions the bifurcationevolves toward an unbalanced configuration; furthermore,the degree of asymmetry of the bifurcation becomes largeras the distance from the resonant value increases.[43] It is important to point out that taking the above

viewpoint provides a unified interpretation of experimentalresults, as the relative distance from the resonant rangeemerges as the controlling parameter of the bifurcationasymmetry: in fact, Figures 9 and 10 suggest that the pointsof the whole set of runs, which correspond to differentvalues of longitudinal slope and grain size, lay approxi-mately on the same curve.[44] The above results can be given the following simple

interpretation. In super resonant conditions any departurefrom the balanced and symmetrical configuration at thenode can produce an upstream influence which leads to asteady perturbation in the upstream channel causing atransverse bed deformation. This in turn induces a topo-graphical forcing on the approach flow and diverts a greaterpercentage of the water discharge in one of the downstreambranches, further promoting the development of an unbal-anced configuration. On the other hand, in sub-resonantconditions the bed topography just upstream the bifurcationcan keep nearly flat because the morphodynamic influenceof the node is not felt upstream; hence, the dischargedistribution is more likely to keep symmetrical. Figure 10suggests that for values of the aspect ratio lower than br theamplitude of the inlet step is generally negligible.[45] In the light of the above considerations we can

conclude that the present results provide a strong experi-mental support to the theory of morphodynamic influence,as it has been recently proposed by Zolezzi and Seminara[2001].

6. Comparison With the BRT Theoretical Model

[46] The experimental findings have been used to test theaccuracy of the theoretical predictions of the BRT model.

Unlike in BRT, where Meyer-Peter and Muller relationshipwas used, here Parker [1990] bed load formula has beenadopted in order to avoid discontinuity at low transportrates.[47] The theory of BRT allows one to determine a

threshold curve on the plain ba � Ja that separates theregion where a symmetrical bifurcation can keep balancedand stable from the region where the system gets toward anunbalanced configuration, characterized by a dominantdownstream branch. In Figure 11 the whole set of experi-mental points are reported, where the runs with values of thedischarge ratio rQ larger than 0.95 have been considered as‘‘balanced’’. The comparison with the theoretical curvedetermined through BRT model (for the same set of con-ditions of the upstream channel) shows a good agreement,also in the case of runs strongly affected by bar migration.[48] It is worth recalling that BRT model assumes that the

transverse exchange of flow and sediment transport justupstream the node is concentrated over a length of fewchannel widths; such length, which provides an estimate ofthe upstream distance over which the effect of the bifurca-tion is mainly felt, is set by BRT equal to (a ba), where a isan order one parameter. We may note that the predictedequilibrium values of rQ and Dh depend appreciably on thechoice of the a value. Hence such parameter has beencalibrated in order to fit the experimental data and toquantify correctly the degree of asymmetry of the bifurca-

Figure 8. Relationship between the measured equilibriumvalues of the discharge ratio and of the dimensionlessamplitude of the inlet step.

Figure 9. The discharge ratio in the downstream branchesrQ as a function of the relative distance from resonantconditions.

Figure 10. The amplitude of inlet step Dh as a function ofthe relative distance from resonant conditions.

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tion. The optimal value of a for the overall comparisonreported in Figure 11 has been found to be equal to 6.Furthermore, different theoretical predictions have beencomputed, changing the value of a, in order to fit theexperimental values of the discharge asymmetry rQ atequilibrium. We have found that the values of a thatoptimize the comparison with laboratory results dependon the relative distance from the resonant conditions (seeFigure 12). Data of the third series of experiments have notbeen considered in the above analysis, since these runs werestrongly affected by the migration of bars in the upstreamchannel and the measured equilibrium values are, therefore,less reliable. Furthermore, the model of BRT didn’t accountfor two-dimensional effects associated with bar occurrence.The optimal value of a increases sharply as ba approachesthe resonant range, which implies that under these condi-tions the length over which the morphodynamic influence ofthe node is felt may be relatively large. The above finding isnot surprising: in fact, theoretical results [Seminara andTubino, 1992; Zolezzi and Seminara, 2001] suggest that atresonance (b = br) the spatial damping of bed perturbationsforced by the discontinuity should vanish. Furthermore,Figure 12 shows that in sub-resonant and strongly super-resonant conditions the optimal value of a tends to 1. Theabove results seem to indicate that the use of an appropriaterelationship to estimate the parameter a, where its depen-dence on (b � br) is accounted for, could incorporate inBRT model further two-dimensional ingredients otherwiseneglected.[49] Finally, it is worth mentioning that the predicted

values of Dh, computed with the optimized values of aobtained through the above procedure, are in a goodagreement with the experimental results. This confirms theability of BRT model to correctly reproduce the relationshipbetween the local bed topography and the discharge parti-tion at the node.

7. Conclusions

[50] In the present work the attention has been focused onthe morphodynamics of a ‘Y-shaped’ bifurcation, constitut-ed by an upstream channel dividing into two symmetricaldownstream branches. The study has been carried out

through three sets of experimental runs, in a flume withmovable bed and fixed walls. A systematic investigation ofthe equilibrium configurations of the bifurcation has beenperformed, with a quantitative description of both thedischarge distribution in the downstream branches and thebed topography in the region affected by the channeldivision.[51] From the analysis of experimental data and the

comparison with theoretical predictions of BRT model thefollowing outcomes can be pointed out.[52] 1. The present experimental results clearly demon-

strate the existence of unbalanced equilibrium configura-tions for high values of the aspect ratio and low values ofthe Shields stress. These observations can physically ex-plain why natural braided rivers are likely to concentrate thedischarge and consequently the morphodynamic activity ina few channels [Mosley, 1983; Stojic et al., 1998].[53] 2.The predictions of the one-dimensional theory

recently proposed by BRT are confirmed, at least in aqualitative way. The experimental observations highlightthe crucial role played by the local bed structure justupstream the bifurcation in governing the dynamics of thebifurcation. They also suggest that the theoretical findingsare not significantly altered by the presence of alternate barsin the upstream channel, at least on the average, thoughtheir migration can affect strongly the evolution of thebifurcation.[54] 3.The analysis of the morphodynamic influence of

the bifurcation, with reference to the theoretical frameworkproposed by Zolezzi and Seminara [2001], provides aunified interpretation of the experimental data measured atequilibrium. Experimental findings indicate that the finalconfiguration toward which the system is driven cruciallydepends on the distance (ba � br) of the aspect ratio ofupstream flow from a threshold resonant value (the lattercoincides with the resonant aspect ratio first discovered byBlondeaux and Seminara [1985] as that corresponding to aresonant behavior of the linear solution for flow and bedtopography in meanders with periodic curvature distribu-tion). In sub-resonant runs the node keeps a balanced waterdistribution in the downstream branches, whereas super-resonant conditions lead to unbalanced configurations,which are driven by the steady bed deformation induced

Figure 11. The occurrence criterion of balanced andunbalanced equilibrium configurations, as predicted by thetheoretical model of Bolla Pittaluga et al. [2003], is testedagainst the experimental results on the ba � Ja plane.

Figure 12. The values of the parameter of the model ofBolla Pittaluga et al. [2003] that optimize the theoreticalestimate of rQ.


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in the upstream channel by the morphodynamic influence ofthe bifurcation.[55] 4.The parameter a of BRT model, which provides a

measure of the longitudinal extent of the morphodynamicinfluence of the bifurcation, has been calibrated throughexperimental data. Present results confirm that the regionwhere two-dimensional effects induced by the bifurcationare mainly felt spreads over a length of few (say, from 1 to 7)channel widths; moreover they suggest that a is a functionof the distance (ba � br) and reaches a maximum as theaspect ratio of upstream flow approaches its resonant value.

[56] Acknowledgments. This work has been developed within theframework of the ‘Centro di Eccellenza Universitario per la DifesaIdrogeologica dell’Ambiente Montano - CUDAM’ and of the project ‘Larisposta morfodinamica di sistemi fluviali a variazioni di parametri ambi-entali - COFIN 2003’, co-funded by the Italian Ministry of University andScientificResearch (MIUR) and the University of Trento and of the project‘Rischio Idraulico e Morfodinamica Fluviale’ financed by the FondazioneCassa di Risparmio di Verona, Vicenza, Belluno e Ancona. The paper hasbenefited from the comments of Rodolfo Repetto, Guido Zolezzi andtwo anonymous reviewers. The authors gratefully acknowledge StefanoMiori, Luca Zanoni, Stefania Baldo, David Marchiori, Andrea Casarin andthe staff of the Hydraulic Laboratory, who helped in the execution of theexperimental runs.

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River bifurcations: Experimental observations on equilibrium configurations