Advances in Water Resources 121 (2018) 9–21

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Advances in Water Resources

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A reduced complexity model of a gravel-sand bifurcation: Equilibrium states and their stability

Ralph M.J. Schielen a,b,∗, Astrid Blom c a Faculty of Engineering Technology, University of Twente, b Ministry of Infrastructure and Water Management-Rijkswaterstaat, Netherlands c Faculty of Civil Engineering and Geosciences, Delft University of Technology, Netherlands

a r t i c l e i n f o a b s t r a c t

Keywords: We derive an idealized model of a gravel-sand river bifurcation and analyze its stability properties. The model River bifurcation requires nodal point relations that describe the ratio of the supply of gravel and sand to the two downstream

Mixed-size sediment branches. The model predicts changes in bed elevation and bed surface gravel content in the two bifurcates

Idealized model under conditions of a constant water , sediment supply, , and channel width and under the Equilibrium assumption of a branch-averaged approach of the bifurcates. The stability analysis reveals more complex behavior Stability analysis than for unisize sediment: three to five equilibrium solutions exist rather than three. In addition, we find that under specific parameter settings the initial conditions in the bifurcates determine to which of the equilibrium states the system evolves. Our approach has limited predictive value for real bifurcations due to neglecting several effects (e.g., transverse bed slope, alternate bars, upstream flow asymmetry, and bend sorting), yet it provides a first step in addressing mixed-size sediment mechanisms in modeling the dynamics of river bifurcations.

1. Introduction Tarekul Islam et al., 2006 ), and the zones of flow recirculation close to the bifurcation ( Bulle, 1926; De Heer and Mosselman, 2004; Marra River bifurcations or diffluences are found in alluvial fans, braided et al., 2014; Thomas et al., 2011 ), vegetation ( Burge, 2006 ), and , anabranching rivers, deltas, cut-off channels, diversions (for flood cohesive sediment and bank ( Miori et al., 2006; Zolezzi et al., control or water intakes), and in constructed side channels that are part 2006 ); of river restoration schemes. Once a bifurcation is initiated, a down- • the conditions in the area just upstream of the bifurcation: the trans- channel (or bifurcate or ) continues to deepen as long verse distribution of water and sediment over the upstream chan- as the capacity exceeds the sediment supply to the nel, which is affected by secondary flow ( Van der Mark and Mos- channel. selman, 2013 ), a transverse bed slope induced by an inlet step Sediment transport in a channel consists of bed-material load (i.e., ( Bolla Pittaluga et al., 2003 ), alternate bars ( Bertoldi and Tubino, and suspended bed-material load) and ( Church, 2007; Bertoldi et al., 2009; Redolfi et al., 2016 ), and sediment mo- 2006; Paola, 2001 ). As wash load is typically assumed to be distributed bility ( Frings and Kleinhans, 2008 ); uniformly over the water column, it is assumed to be partitioned over • conditions extending further upstream: flow asymmetry induced by the bifurcates according to the ratio of the water discharge. Bed-material a bend, which tends to provide one bifurcate with a larger fraction load, however, partitions over the bifurcates in a less straightforward content of the flow and the other one with a larger fraction content manner. The partitioning of sediment in dominated by sus- of the sediment load ( Federici and Paola, 2003; Hardy et al., 2011; pended bed-material load depends on the initial flow depth and channel Kleinhans et al., 2008; Van Dijk et al., 2014 ) and transverse sediment slope in the bifurcates ( Slingerland and Smith, 1998 ), the grain size of sorting due to bend flow ( Frings and Kleinhans, 2008; Sloff et al., the bed sediment ( Slingerland and Smith, 1998 ), and curvature-induced 2003; Sloff and Mosselman, 2012 ). effects in the upstream channel ( Hackney et al., 2017 ). The partitioning of sediment in bed load dominated streams depends on: The partitioning of the sediment load over the bifurcates determines whether the bifurcation develops towards a stable state with two open • the conditions in the bifurcates: base level, channel width, friction, downstream branches or a state in which the water discharge in one of bifurcation angle ( Bulle, 1926; Van der Mark and Mosselman, 2013; the branches continues to increase at the expense of the other branch. The latter case may lead to the silting up of one of the downstream

∗ Corresponding author. E-mail address: [email protected] (R.M.J. Schielen). https://doi.org/10.1016/j.advwatres.2018.07.010 Received 6 January 2018; Received in revised form 13 July 2018; Accepted 17 July 2018 Available online 19 July 2018 0309-1708/© 2018 The Authors. Published by Elsevier Ltd. This is an open access article under the CC BY-NC-ND license. ( http://creativecommons.org/licenses/by-nc-nd/4.0/ ) R.M.J. Schielen, A. Blom Advances in Water Resources 121 (2018) 9–21 channels. Under such conditions a one channel configuration is a stable far the influence of noncohesive mixed-size sediment on bifurcation equilibrium solution of the bifurcation system ( Wang et al., 1995 ), yet in dynamics has not been studied explicitly. Wash load, suspended bed- literature this situation is often termed an ‘unstable bifurcation’ ( Burge, material load, and bed load ( Church, 2006; Paola, 2001 ) are expected to 2006; Federici and Paola, 2003 ), as the two channel system ceases to respond differently to the above-mentioned mechanisms ( Hackney et al., exist. 2017 ). Mixed-size sediment effects are the following: Early one-dimensional reduced complexity models describing the de- 1. As the vertical profile of sediment concentration is less uniform over velopment towards the equilibrium states of two bifurcates have been depth for coarse sediment (i.e., coarse sediment tends to concen- developed for bed load transport in sand-bed rivers ( Wang et al., 1995 ), trate more strongly near the bed), coarse sediment tends to be af- bed load transport in gravel-bed rivers ( Bolla Pittaluga et al., 2003 ), and fected more by an inlet step than fine sediment ( Slingerland and suspended bed-material load ( Slingerland and Smith, 1998 ). Such re- Smith, 1998 ). duced complexity models allow for the computation of the partitioning 2. The effect of the transverse bed slope on lateral transport upstream of the water discharge as the water surface elevation at the bifurcation of the bifurcation depends on grain size, where coarse sediment point must be equal between the three reaches. The sediment partition- is affected by the transverse bed slope more strongly than fines ing, however, depends on the geometry of the bifurcation and the three- ( Parker and Andrews, 1985 ); dimensional flow structure, which obviously cannot be reproduced by a 3. The presence of a bend upstream of the bifurcation typically leads to one-dimensional model. A one-dimensional model therefore requires a bend sorting and a coarser sediment supply to the distributary in the nodal point relation that describes the partitioning of the sediment load outer bend than to the one in the inner bend ( Frings and Kleinhans, over the bifurcates. 2008; Sloff et al., 2003; Sloff and Mosselman, 2012 ); Wang et al. (1995) were the first to introduce a nodal point relation 4. Alternate formation and geometry appear to be affected by the describing the partitioning of the sediment supplied from upstream over grain size distribution of the sediment mixture ( Bertoldi and Tubino, the bifurcates. They then apply a simpler form of their nodal point rela- 2005; Lanzoni, 2000 ). 𝑠̄ 𝑠̄ 𝑞 𝑞 𝑘 , 𝑠̄ tion ( 1 ∕ 2 = ( 1 ∕ 2 ) where 1 , 2 is the rate of sediment supply per unit Our objective is to assess the elementary consequences of the intro- width to branches 1 and 2 and q1,2 is the water supply per unit width to branches 1 and 2) to analyse the stability of the solutions to the equilib- duction of mixed-size sediment mechanisms in the modelling of the dy- rium morphodynamic state of the bifurcates. Studying devel- namics of a river bifurcation. To this end we follow Wang et al. (1995) ’s opment ( Slingerland and Smith, 1998 ) introduce a nodal point relation approach and its simple nodal point relation with associated limitations that originates from integration of the vertical concentration profile of and simplifications: we neglect the effects of vegetation, cohesive sedi- the suspended sediment. Alternative nodal point relations have been ment, bank erosion, alternate bars or a bend in the upstream channel, developed by Bolla Pittaluga et al. (2003) , who account for the effects as well as the Bulle effect and the transverse slope effect. We extend of a transverse bed slope that induces lateral sediment transport to the their model to conditions with bed-material load of a two-fraction sed- deeper bifurcate, and Kleinhans et al. (2008) , who account for the ef- iment mixture consisting of gravel and sand. This implies the need for fects of an upstream bend, both of which will be addressed in further two nodal point relations describing the ratio of, respectively, the gravel detail below. and sand supply to the two bifurcates. We study the stability of the equi- Pioneering work on bifurcation dynamics using a nodal point rela- librium states of the bifurcates in an engineered river characterized by tion was conducted by Wang et al. (1995) : they assume a constant water a fixed channel width. discharge and sediment supply rate in the upstream channel, a constant The proposed analysis and model are applicable to both cases shown and equal base level in the two bifurcating branches, and unisize sedi- in Fig. 1 : a bifurcation system with two bifurcates that are characterized ment conditions. They apply the Engelund and Hansen (1967) sediment by the same base level and a side channel system. We set up a model transport relation without a threshold for significant transport: s ∝U n , describing the equilibrium solutions of the mixed sediment bifurcation where s denotes the sediment transport capacity per unit width, U the system ( Section 2 ), we determine its equilibrium solutions ( Section 3 ), depth-averaged flow velocity, and n is the exponent in the power law we derive a system of ordinary differential equations for the flow depth load relation ( 𝑛 = 5 ). They find that for k < n /3 the equilibrium solution and bed surface texture in the bifurcates ( Section 4 ), and perform a sta- where one of the bifurcates closes is stable, whereas for k > n /3 the equi- bility analysis of the equilibrium solutions ( Section 5 ). The analysis also librium solution with two open branches is stable. Despite these early re- provides insight on the time scale of the evolution towards the stable sults a model for k is still lacking. Also Slingerland and Smith (1998) re- equilibrium solutions ( Section 6 ). veal that a bifurcation or avulsion develops towards a stable state with two open downstream branches or a state in which one channel becomes 2. Model of the equilibrium state the dominant channel at the expense of the other branch.

In such strongly idealized one-dimensional analyses, two- In this section we strongly simplify the situation of a gravel-sand dimensional and three-dimensional effects near the bifurcation river bifurcation, describe the problem from a mathematical point of point are not readily accounted for. One of these effects is the Bulle view, and list the governing equations. To this end we consider an en- effect ( Bulle, 1926; Dutta et al., 2017; Van der Mark and Mosselman, gineered river with a fixed channel width that may vary between the

2013 ), which indicates a situation where the sediment supply to a branches, a temporally constant water discharge in the upstream branch diversion channel (i.e., a channel that branches off the main channel (i.e., branch 0 in Fig. 1) and a temporally constant gravel supply rate under a certain angle) is significantly larger than the diversion chan- and constant sand supply rate to the upstream branch. 𝜕 𝜕 𝑡 nel’s fraction content of the water discharge. This effect is associated Under equilibrium conditions ( ∕ = 0) without , uplift, with secondary flow (e.g., Thomas et al., 2011 ). Another effect is the and particle , the equation describing conservation of sediment difference in bed elevation that is associated with a difference in flow mass (i.e. the ) reduces to the stationary Exner equation, 𝜕 𝑆 𝜕 𝑥 , depth between the two bifurcates (e.g., Bolla Pittaluga et al., 2003; 𝑖 ∕ = 0 where Si denotes the sediment transport capacity in branch

Kleinhans et al., 2013 ). This bed elevation difference (also denoted i, the subscript i indicates branch i, and x is the streamwise coordinate. using the term inlet step) tends to increase the sediment supply to the In other words, by definition the sediment transport capacity Si equals 𝑆̄ , deeper bifurcate (e.g., Slingerland and Smith, 1998 ), which acts as a the sediment supply to branch i, 𝑖 where the bar indicates the sediment stabilizing mechanism. supply.

Although mixed-size sediment systems may reveal behavior that is For simplicity we apply the Engelund and Hansen power law load essentially different from unisize sediment systems ( Blom et al., 2017a; relation (Engelund and Hansen, 1967): 𝑛 2017b; 2016; Mosselman and Sloff, 2008; Sinha and Parker, 1996), so 𝑆 𝑖 = 𝐵 𝑖 𝑚 𝑖 𝑈 𝑖 (1)

10 R.M.J. Schielen, A. Blom Advances in Water Resources 121 (2018) 9–21

Fig. 2. Definition of symbols.

where ii denotes the channel slope. For simplicity, the Chézy friction coefficient C is assumed independent of the bed surface texture and flow conditions and hence constant. Under mixed-size sediment conditions, the Exner equation is re- placed by the equations for the conservation of gravel and sand mass at the bed surface, i.e. the Hirano equations ( Hirano, 1971; Parker, 1991; Ribberink, 1987 ). Under equilibrium conditions the Hirano equations reduce to 𝜕 𝑆 𝑖𝑔 ∕ 𝜕 𝑥 = 𝜕 𝑆 𝑖𝑠 ∕ 𝜕 𝑥 = 0 , where the subscripts g and s indicate

gravel and sand, respectively, and Sig and Sis denote, respectively, the gravel and sand transport capacities in branch i . This implies that in an equilibrium state without particle abrasion the gravel and sand load do not vary within a branch (e.g., Blom et al., 2016 ). For simplicity we Fig. 1. Schematic of (a) a channel (branch 0) bifurcating into two channels apply the Engelund and Hansen power law load relation in a fractional

(branches 1 and 2) flowing into a lake characterized by the same base level manner ( Blom et al., 2017a; 2016 ) and replace Eq. (1) by ( Wang et al., 1995 ) and (b) a side channel system. Our analysis and model are applicable to both cases. 𝑆 𝑖 = 𝑆 𝑖𝑔 + 𝑆 𝑖𝑠 (4)

in which 𝑚 𝑖 = 𝐺 𝑖 ∕ 𝐷 with D denoting a characteristic grain size, 𝐺 𝑖 = 𝑛 𝑆 𝑖𝑔 = 𝐹 𝑖𝑔 𝐵 𝑖 𝑚 𝑔 𝑈 𝑖 (5) . 𝐶 3 𝑅 2 𝑔 1∕2 , 0 05∕( 𝑖 ) Ui the depth-averaged flow velocity, Bi the channel width, Ci the Chézy friction coefficient, g denotes the gravitational ac- celeration, and R the submerged density ( 𝑅 = ( 𝜌𝑠 − 𝜌)∕ 𝜌 where 𝜌 and 𝑛 s 𝑆 𝑖𝑠 = (1 − 𝐹 𝑖𝑔 ) 𝐵 𝑖 𝑚 𝑠 𝑈 𝑖 (6) 𝜌 are the mass density of, respectively, sediment and water). For sim- 𝑚 𝑚 where F denotes the volumetric fraction content of gravel at the bed plicity we assume that mi does not vary between the branches ( 𝑖 = ), ig which implies that also the friction coefficient and the coefficient G do surface in branch i or, briefly, the surface gravel content (Fig. 2), and 𝑚 = 𝐺∕ 𝐷 and 𝑚 = 𝐺∕ 𝐷 with D and D the grain sizes of, respec- not vary between the branches ( 𝐶 𝑖 = 𝐶, 𝐺 𝑖 = 𝐺). 𝑔 𝑔 𝑠 𝑠 g s

Combination of Eq. (1) with the stationary Exner equation illustrates tively, gravel and sand. Obviously the coefficients mg and ms have dif- that under equilibrium conditions where the channel width and friction ferent values. Similarly to the unisize case, we assume mg not to vary be- do not vary spatially, besides the sediment transport rate, also the flow tween the branches. The same holds for ms . Combination of Eqs. (5) and velocity is uniform. (6) with the stationary Hirano and Saint-Venant equations shows, anal-

The flow is described using the one-dimensional conservation equa- ogous to the unisize sediment case, that (under equilibrium conditions tions for water mass and streamwise momentum, i.e. the Saint-Venant without uplift, subsidence, and particle abrasion) the flow velocity, flow equations ( Saint-Venant, 1871 ). Under equilibrium conditions, the con- depth, and surface gravel content do not vary within a branch. 𝜕 𝑄 𝜕 𝑥 The model requires a nodal point relation that relates the ratio of the servation equation for water mass is simplified to 𝑖 ∕ = 0 (where Qi denotes the water discharge in branch i , see Fig. 1 ), which implies sediment supply to the downstream branches to the ratio of the water discharge. The nodal point relation introduced by Wang et al. (1995) is 𝑄 𝑖 = 𝐵 𝑖 𝑈 𝑖 𝐻 𝑖 = const (2) applicable to unisize sediment conditions: ( ) 𝑘 where Hi denotes the flow depth (Fig. 2). As the flow velocity is uniform 𝑘 𝑠̄ 𝑞 𝑠̄ ∗ = 𝛼𝑞 ∗ , or 1 = 𝛼 1 (7) over the branch, Eq. (2) implies that also the flow depth does not vary 𝑠̄ 𝑞 2 2 over the branch. where 𝑠̄ denotes the rate of sediment supply per unit width to branch i , Under equilibrium conditions the conservation equation for stream- 𝑖 the superscript ∗ indicates the ratio of the values of the specific variable wise momentum of the flow reduces to the backwater equation. For a for branches 1 and 2 (e.g., 𝑠̄ ∗ = 𝑠̄ ∕ 𝑠̄ ), 𝛼 is the nodal point prefactor, uniform flow depth, the backwater equation reduces to the normal flow 1 2 and q is the water discharge per unit width in branch i . Eq. (7) can also equation: i be written as ( ) 1∕3 ( ) ( ) 𝑄 2 ̄ 𝑘 1− 𝑘 𝑖 𝑆 𝑄 𝐵 𝐻 𝑆̄ ∗ 𝛼𝑄 ∗ 𝑘 𝐵 ∗1− 𝑘 , 1 𝛼 1 1 𝑖 = (3) = or = (8) 𝑖 𝐶 2 𝐵 2 𝑆̄ 𝑄 𝐵 𝑖 𝑖 2 2 2

11 R.M.J. Schielen, A. Blom Advances in Water Resources 121 (2018) 9–21

For conditions dominated by two grain size modes (gravel and sand), we introduce two nodal point relations, one describing the partitioning of the gravel load over the bifurcates and one the sand load: ( ) ( ) ̄ 𝑘 1− 𝑘 𝑔 𝑆 𝑔 𝑄 𝑔 𝐵 ̄ ∗ ∗ 𝑘 𝑔 ∗1− 𝑘 𝑔 1 1 1 𝑆 𝑔 = 𝛼𝑔 𝑄 𝐵 , or = 𝛼𝑔 (9) 𝑆̄ 𝑄 𝐵 2 𝑔 2 2

( ) 𝑘 ( ) 𝑘 𝑆̄ 𝑄 𝑠 𝐵 1− 𝑠 ̄ ∗ ∗ 𝑘 𝑠 ∗1− 𝑘 𝑠 1 𝑠 1 1 𝑆 𝑠 = 𝛼𝑠 𝑄 𝐵 , or = 𝛼𝑠 (10) 𝑆̄ 𝑄 𝐵 2 𝑠 2 2 𝛼 𝛼 where kg and ks denote the nodal point coefficients and g and s are the nodal point prefactors, both for gravel and sand, respectively. We realize that the above form of the nodal point relations is too sim- ple to cover the physics of the problem of river bifurcations adequately. In addition to the strongly simplified form of the nodal point relations for gravel and sand, the values for the nodal point coefficients kg , ks , 𝛼 𝛼 g , and s likely are not constants and models for these coefficients are needed to properly analyze the physics of the bifurcation problem. Yet despite these strong simplifications we believe that the current analy- sis provides useful insight on elementary bifurcation behavior. We will address this aspect in further detail in the discussion section. The fact that both downstream branches are governed by the same base level ( Fig. 1 ) and also the upstream water surface elevation of the two branches is equal creates the following geometrical constraint in an equilibrium state ( Wang et al., 1995 ): 𝑖 𝐿 1 1 2 𝑖 ∗ = , or = (11) Fig. 3. Sections I, IIa , IIb , and III in the stability diagram in the (kg , ks ) parameter 𝐿 ∗ 𝑖 𝐿 2 1 space for the base case. where Li is the length of branch i (Fig. 1).

We now have a set of equations that can be solved to determine the 𝛼 , 𝐵 𝐵 𝐵 𝐶 𝐶 1/2 𝑆 . 𝑠 = 1 0 = 315 m, 1 = 2 = 250 m, 1 = 2 = 50 m /s, 0 𝑔 = 0 001 equilibrium states of the two downstream branches. 3 𝑆 . 3 𝑄 3 m /s, 0 𝑠 = 0 007 m /s, and 0 = 4000 m /s. For a detailed analysis of Eq. (12) we refer to Appendix A.1 . It il- 3. The equilibrium state lustrates that we can distinguish between three sections in the (kg , ks ) parameter space (I, II, and III), each with a different number of solutions We manipulate the set of equations listed in Section 2 to find the to Eq. (12) and hence of the flow depth in the downstream branches equilibrium solutions of the bifurcation cases shown in Fig. 1 . Under ( Fig. 3 ): equilibrium conditions the sediment supply rate must be equal to the ̄ ̄ • sediment transport capacity and we therefore set 𝑆 𝑖𝑔 = 𝑆 𝑖𝑔 and 𝑆 𝑖𝑠 = 𝑆 𝑖𝑠 . I and III: There are three equilibrium solutions. Two solutions cor- In addition, we substitute Eqs. (2) –(6) and (11) into (9) and (10) . This respond with one of the downstream branches closed. The other so- yields an implicit solution to the ratio of the water discharge in the two lution corresponds with both branches open. downstream branches, Q ∗ : 3 ⎛ ⎞ 𝑛 𝑚 𝑆 𝑚 𝑆 3 ⎜ 𝑔 𝑠 0 + 𝑠 𝑔0 ⎟ 𝑄 ∗ 𝐿 ∗𝐵 ∗1− 𝑛 𝑄 ∗ = ⎜ ( ) ( ) − 1⎟ = Φ( ) (12) −1 𝑘 𝑘 −1 ⎜ 𝑚 𝑆 𝛼 𝑄 ∗ 𝑘 𝑠 𝐵 ∗1− 𝑘 𝑠 𝑚 𝑆 𝛼 𝑄 ∗ 𝑔 𝐵 ∗1− 𝑔 ⎟ ⎝ 𝑔 𝑠 0 𝑠 + 1 + 𝑠 𝑔0 𝑔 + 1 ⎠ A solution of Eq. (12) provides values for the gravel and sand load in the • II and II : There are five equilibrium solutions. Two solutions corre- downstream branches, S and S ( 𝑖 = 1 , 2 ), through the nodal point re- a b ig is spond with one of the downstream branches closed. The remaining lations in Eqs. (9) and (10) , provided that the water discharge in branch three solutions correspond with both branches open.

0, Q0 , the gravel and sand supply rates to branch 0, S0 g and S0 s , and ∗ ∗ the variables mg , ms , L , B , ks , and kg are known. We compute the flow The differences between sections I and III and between IIa and IIb depth, Hi , using Eq. (3), as well as the surface gravel content, Fig , using will be addressed in the next section.

Eq. (5) or (6). The boundaries of the sections I, IIa , IIb , and III in Fig. 3 depend Eq. (12) has at least three solutions: two solutions that are associated on the ratio of the sand load to the gravel load in the upstream branch 𝑄 ∗ , 𝑄 ∗ 𝑆̂ , with the closure of one of the branches ( = 0 = ∞) and one solu- (branch 0), which is denoted by 0 the ratio of the length of the bifur- tion in which both downstream branches remain open. Generically the cates, L ∗ , and the ratio of the channel width of the bifurcates, B ∗ ( Fig. 4 ). flow depth differs between the downstream branches, but under con- An increase of the sand load in the upstream channel at the expense of 𝐿 ∗ ditions in which = 1 the flow depth in the downstream branches is its gravel load leads to a decrease of section IIb and an increase of sec- equal, even if the width varies between the branches. tion IIa . An increase of the difference in channel length between the two We define a base case that (except for the bifurcate length) is loosely bifurcates significantly decreases section II and an increase of section I. based on the bifurcation of the Bovenrijn into the Pannerdensch Kanaal Section II tends to become negligible for values of L ∗ even mildly larger 𝑆̂ ∗ and the Waal branch. The bifurcation is located in the Netherlands than 1. The effects of 0 and L are significant, whereas the effect of a and about 10 km downstream from where the River crosses the difference in channel width between the two bifurcates, B ∗ , appears to German–Dutch border. The water discharge is set equal to the one char- be limited. acterized by a one year recurrence period (4000 m 3 /s). We simply as- The current analysis is limited to engineered rivers where the chan- 𝐿 𝐿 sume the bifurcates to have the same channel length (here 1 = 2 = 10 nel width cannot adjust to changes in the controls (i.e., statistics of the km). This yields the following parameter values for the base case: 𝛼𝑔 = water discharge, sediment supply, and base level). The analysis illus-

12 R.M.J. Schielen, A. Blom Advances in Water Resources 121 (2018) 9–21

𝑆̂ Fig. 4. Stability diagram in the (kg , ks ) parameter space, for varying (a) ratio of the sand load to the gravel load in branch 0, 0 ; (b) ratio of the length of the downstream channels, L ∗; and (c) ratio of the width of the downstream channels, B ∗. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.) trates that a mixed-size sediment two-channel system consists of three to respect to a fixed reference level. As 𝐻 = 𝜂𝑤 − 𝜂 ( Fig. 2 ) and the water 𝜂 five solutions to the morphodynamic equilibrium state. This differs from surface elevation, w , is constant due to our branch-averaged approach, a unisize sediment two-channel system, for which three equilibrium so- we find that 𝜕 𝜂𝑖 ∕ 𝜕 𝑡 = − 𝜕 𝐻 𝑖 ∕ 𝜕 𝑡 . This implies that Eq. (13) can be written lutions exist ( Wang et al., 1995 ). The existence of three to five solutions as ( Wang et al., 1995 ): also contrasts with the single solution to the morphodynamic equilib- 𝜕𝐻 𝑖 𝜕𝑆 𝑖 𝑐 𝑏 𝐵 𝑖 = (14) rium state of a one-channel system, under unisize as well as mixed-size 𝜕𝑡 𝜕𝑥 sediment conditions ( Blom et al., 2017a; 2016; Howard, 1980 ). Natural rivers, where besides the channel slope and bed surface tex- As we assume that gradients in the sediment transport rate result in a 𝜕 𝜕 ture also the channel width responds to changes in the controls, allow branch-averaged or rate, we write Si / x as ̄ for more equilibrium states than engineered rivers with a fixed chan- 𝜕𝑆 𝑖 𝑆 𝑖 − 𝑆 𝑖 nel width ( Blom et al., 2017a ). In natural rivers there exists a range of = (15) 𝜕𝑥 𝐿 𝑖 equilibrium states for which the channel is able to transport the load supplied from above ( Blom et al., 2017a ). In the current analysis we Combination of Eqs. (14) and (15) then yields have not considered the effect of erodable banks, yet one may expect 𝑑𝐻 𝑖 ( ) 1 𝑆 𝑆̄ that, just as in the single channel case, the presence of erodable banks = 𝑖 − 𝑖 (16) 𝑑𝑡 𝑐 𝑏 𝐵 𝑖 𝐿 𝑖 allows for a range of equilibrium states. We apply a simplified form of the Hirano active layer model 4. Model of the stability of the equilibrium state ( Hirano, 1971 ) to describe the temporal change of the surface gravel content in the topmost part of the bed that interacts with the flow (i.e., We set up a system of differential equations for the flow depth in the active layer). and the surface gravel content in the bifurcates to study the temporal To arrive at a simplified version of the Hirano equation, we apply changes in the bifurcation system. a similar branch-averaged approach to the migration of perturbations For simplicity, we assume that perturbations in bed elevation (i.e., in the bed surface texture as to perturbations in bed elevation. Worded aggradational and degradational waves), which arise from a difference differently, we assume surface texture perturbations, which arise from between the sediment supply to a downstream branch and its sediment a difference between the grain size distribution of the sediment supply transport capacity, move so fast along the downstream branches that to a downstream branch and the grain size distribution of the trans- we can assume a branch-averaged response of bed elevation. This im- ported sediment, to move so fast along a bifurcate that we can consider 𝜂 a branch-averaged response of the bed surface gravel content in the bi- plies that we consider branch-averaged values for bed elevation, i , flow furcate, F . The reach-averaged approach is valid provided that the per- depth, Hi , and surface texture represented by the surface gravel content, ig turbation in the bed surface texture migrates downstream fast along the Fig (Fig. 2). Another consequence of this branch-averaged approach is the fact that the channel slope in each bifurcate cannot adjust with time, channel (e.g., a small depth of reworking or a small active layer thick- as the aggradation or degradation rate does not vary within a bifurcate. ness) or if the channel is relatively short (i.e., a small value of Li ). The constant channel slope and base level imply that, although the bed In addition, we assume that the vertical sediment flux between the elevation changes with time, the water surface elevation in the bifur- active layer and the substrate that is associated with a change in ele- cates remains constant with time. vation of the interface between the active layer and the substrate has Such a reach-averaged approach is valid provided that the perturba- the same grain size distribution as the one of the active layer sediment, tion in bed elevation migrates relatively fast down in the channel or if even under conditions of degradation. Under these simplifying assumptions the Hirano active layer equa- the channel is relatively short (i.e., a small value of Li ). The Exner equation describing conservation of bed sediment is tion reduces to 𝑑𝐹 𝜕𝜂 𝜕𝑆 𝑖𝑔 ( ) 𝑖 𝑖 1 1 𝐹 𝑆 𝑆̄ 𝑆̄ 𝑆 𝑐 𝑏 𝐵 𝑖 = − (13) = 𝑖𝑔 ( 𝑖 − 𝑖 ) + ( 𝑖𝑔 − 𝑖𝑔 ) (17) 𝜕𝑡 𝜕𝑥 𝑑𝑡 𝑐 𝑏 𝐵 𝑖 𝐿 𝑖 𝐿 𝐴

where t denotes time, cb the sediment concentration within the bed where LA denotes the thickness of the active layer or the surface layer ( 𝑐 𝑏 = 1 − 𝑝 with p denoting bed porosity), and 𝜂 is bed elevation with that is reworked by the flow. 13 R.M.J. Schielen, A. Blom Advances in Water Resources 121 (2018) 9–21

The adjustment of the bed surface texture is characterized by an ex- • III: The two equilibrium solutions that correspond with one branch ponential growth of which the time scale (i.e., the e-folding time or the closed are unstable. The solution with both branches open is the time interval in which the exponentially growing quantity increases by only stable solution. This implies that for every initial condition both a factor of e) equals branches remain open.

𝑇 = 𝑐 𝑏 𝐵 𝑖 𝐿 𝑖 𝐿 𝐴 (18) For a two-channel system under unisize sediment conditions, there 𝑘 𝑛 We have derived the following system of differential equations for exists one critical value of the nodal point coefficient k ( = ∕3), be- low which the equilibrium solution with one closed bifurcation is stable the flow depth, Hi , and surface gravel content, Fig by manipulating ( Wang et al., 1995 ). For values of k larger than n /3 the equilibrium Eqs. (16) –(17) , using Eqs. (9) –(10) , for simplicity setting 𝛼𝑔 = 𝛼𝑠 = 1 , ̂ ̂ solution with two open bifurcates is stable. A case in which k < n /3 is and introducing time 𝑡 where 𝑡 = 𝑡 ∕ 𝑐 𝑏 :

𝑛 similar to the current section I and the latter case is similar to the cur- 𝑑𝐻 𝑄 ( ) 1 0 = 𝑔 ( 𝐻 , 𝐻 , 𝐹 𝑔 ) − 𝑔̄ ( 𝐻 , 𝐻 ) (19) rent section III. Under unisize sediment conditions section II does not ̂ 𝐵 𝐿 1 1 2 1 1 1 2 𝑑 𝑡 1 1 exist. For a single channel system with fixed banks the single solution to 𝑛 𝑑𝐻 𝑄 ( ) 2 0 the equilibrium state is stable ( Blom et al., 2017a; 2016; Howard, 1980 ). = 𝑔 ( 𝐻 , 𝐻 , 𝐹 𝑔 ) − 𝑔̄ ( 𝐻 , 𝐻 ) (20) ̂ 𝐵 𝐿 2 1 2 2 2 1 2 𝑑 𝑡 2 2 6. Evolution towards the stable equilibrium state

𝑑𝐹 𝑄 𝑛 ( ( ) 1 𝑔 0 1 = 𝐹 𝑔 𝑔 ( 𝐻 , 𝐻 , 𝐹 𝑔 ) − 𝑔̄ ( 𝐻 , 𝐻 ) ̂ 𝐵 𝐿 𝐿 1 1 1 2 1 1 1 2 We numerically simulate the system of Eqs. (19)–(22) to assess (1) 𝑑 𝑡 1 1 𝐴 ( )) the effects of the initial flow depth, H and H , and the initial surface 𝑔̄ 𝐻 , 𝐻 𝑔 𝐻 , 𝐻 , 𝐹 1 2 + 1 𝑔 ( 1 2 ) − 1 𝑔 ( 1 2 1 𝑔 ) (21) gravel content, F1 g and F2 g , in the bifurcating branches; (2) the mecha- nism that results in closure of one of the branches; and (3) the effects of the nodal point coefficients in combination with the sediment supply. 𝑑𝐹 𝑄 𝑛 ( ( ) 2 𝑔 0 1 We analyze these three aspects below. = 𝐹 𝑔 𝑔 ( 𝐻 , 𝐻 , 𝐹 𝑔 ) − 𝑔̄ ( 𝐻 , 𝐻 ) 𝑑 𝑡̂ 𝐵 𝐿 𝐿 2 2 1 2 2 2 1 2 2( 2 𝐴 )) 𝑔̄ 𝐻 , 𝐻 𝑔 𝐻 , 𝐻 , 𝐹 + 2 𝑔 ( 1 2 ) − 2 𝑔 ( 1 2 2 𝑔 ) (22) 6.1. Effect of the initial conditions in the bifurcating branches where the functions g , 𝑔̄ , g , and 𝑔̄ ( 𝑖 = 1 , 2 ) are defined in Appendix i 𝑖 ig 𝑖𝑔 In section II the initial conditions determine to which of the three B. stable equilibrium states the system evolves. We abbreviate Eqs. (19) –(22) by ( 𝐻̇ , 𝐹̇ ) = Ψ( 𝐻 , 𝐹 ) where the dot 𝑖 𝑖𝑔 𝑖 𝑖𝑔 Fig. 5 shows the results of two numerical runs of the system of indicates the derivative with respect to time. Naturally the equilibrium Eqs. (19) –(22) , in which we assess the effects of the initial flow depth in 𝐻 , 𝐹 solutions of Section 3 are solutions of Ψ( 𝑖 𝑖𝑔 ) = 0. the bifurcating branches, H1 and H2 . The only difference between the two runs are the initial values of the flow depth, H and H . For equal 5. Stability of the equilibrium state 1 2 length of the bifurcates ( 𝐿 ∗ = 1 ) and arbitrary value of the ratio of the bifurcate width, B ∗ , the flow depth and surface gravel content in the two

Equilibrium solutions only emerge if they are stable. The stability ∗ ∗ bifurcates evolve toward the same value ( 𝐻 = 1 and 𝐹 𝑔 = 1 ). It appears properties of the solutions of Ψ( 𝐻 , 𝐹 𝑖𝑔 ) = 0 are determined by the eigen- 𝑖 that a difference in the initial flow depth results in different behavior: values of the Jacobian J of Ψ, which is defined as: in one case both branches stay open, while in the other case one branch 𝜕 𝜕 𝜕 𝜕 ⎛ Ψ1 Ψ1 Ψ1 Ψ1 ⎞ 𝜕𝐻 𝜕𝐻 𝜕𝐹 𝜕𝐹 closes. ⎜ 1 2 1 𝑔 2 𝑔 ⎟ 𝜕 𝜕 𝜕 𝜕 In the case where both branches remain open, the surface gravel con- ⎜ Ψ2 Ψ2 Ψ2 Ψ2 ⎟ ⎜ 𝜕𝐻 𝜕𝐻 𝜕𝐹 𝑔 𝜕𝐹 𝑔 ⎟ tent in the two branches, F and F , evolves towards the same value. 𝐽 1 2 1 2 1g 2g = ⎜ 𝜕Ψ 𝜕Ψ 𝜕Ψ 𝜕Ψ ⎟ (23) ⎜ 3 3 3 3 ⎟ The surface gravel content F1 g in the closing branch evolves to 0, which 𝜕𝐻 𝜕𝐻 𝜕𝐹 𝑔 𝜕𝐹 𝑔 ⎜ 1 2 1 2 ⎟ 𝜕 𝜕 𝜕 𝜕 means that eventually the bed surface in this closing branch consists of ⎜ Ψ4 Ψ4 Ψ4 Ψ4 ⎟ 𝜕𝐻 𝜕𝐻 𝜕𝐹 𝜕𝐹 sand only. ⎝ 1 2 1 𝑔 2 𝑔 ⎠ Fig. 6 shows the results of two runs where only the initial surface If all eigenvalues at an equilibrium solution have negative (positive) gravel content in the bifurcating channels, F and F , varies between real parts, the equilibrium solutions are linearly and nonlinearly stable 1g 2g the runs. Again we observe the effect of the initial conditions: they de- (unstable), and are nodal points in the 4-dimensional phase space. If termine whether the situation evolves towards a state with either both there are positive and negative eigenvalues, the solution is unstable and branches open or one branch closed. a saddle point in the phase space (e.g., Wiggins, 1990 ). Purely imaginary eigenvalues would give rise to periodic solutions in the phase space, yet 6.2. Mechanism of closure of one of the branches this does not occur for this particular set of equations. As the system of Eqs. (19) –(22) and the associated Jacobian J in We consider the case where one of the branches closes in Fig. 5 (solid Eq. (23) are too complex to be analyzed analytically, we analyze the sys- lines) to study the mechanism of branch closure. To this end we analyze tem numerically. For the details of the analysis we refer to Appendix A.2 . the difference between the load and supply of gravel and sand for, re- In summary, the following holds for the sections I-III in Fig. 3 : spectively, branches 1 and 2 ( Fig. 7 a and b). • I: The two equilibrium solutions that correspond with one branch As the sediment supply in branch 2 exceeds the sediment trans- 𝑆 𝑆̄ < closed are stable. The other solution, where both branches are open, port capacity ( 2 − 2 0), aggradation will occur and branch 2 slowly is unstable. The initial conditions determine to which stable state the closes. The sediment supply in branch 1 approaches the sediment trans- 𝑆 𝑆̄ ↓ system evolves. port capacity ( 1 − 1 0), which implies that the flow depth in branch • IIa and IIb : The two equilibrium solutions that correspond with one 1 approaches an equilibrium (nonzero) value. branch closed are stable. The other three solutions correspond with In branch 2 the sand supply exceeds the sand transport capacity 𝑆 𝑆̄ < both branches open. The two ‘new’ solutions (compared to section ( 2 𝑠 − 2 𝑠 0) and the gravel supply is smaller than the gravel trans- 𝑆 𝑆̄ > I) are created in a blue sky bifurcation (Appendix A.2). Only one of port capacity ( 2 𝑔 − 2 𝑔 0), which implies that the bed surface of the them is stable. The initial conditions determine to which of the three branch becomes increasingly sandy. This is reflected by the fact that ↓ stable equilibrium states the system evolves. bed surface gravel content approaches zero (F2 g 0, solid line in Fig. 5b).

14 R.M.J. Schielen, A. Blom Advances in Water Resources 121 (2018) 9–21

Fig. 5. Effect of the initial flow depth in the downsteam channels on bifurcation dynamics. Flow depth H1 and H2 (left panel) and surface gravel content F1 g and F2 g 𝐻 𝐻 𝐻 𝐻 𝐹 . , 𝐹 . (right panel) in the two downstream branches. Initially, 1 = 14 m and 2 = 8 m (solid lines) and 1 = 14 m and 2 = 10 m (dashed lines), and 1 𝑔 = 0 5 2 𝑔 = 0 5. ̂ Blue and red lines indicate, respectively, branch 1 and branch 2. Results are given for 𝑘 𝑔 = 3 , 𝑘 𝑠 = 1 , 𝛼𝑔 = 𝛼𝑠 = 1 , and 𝐿 𝐴 = 1 m. Here time indicates 𝑡 . (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)

Fig. 6. Effect of the initial surface gravel content in the downsteam channels on bifurcation dynamics. Flow depth H1 and H2 (left panel) and surface gravel content 𝐹 . , 𝐹 . 𝐹 . , 𝐹 . 𝐻 𝐻 F1 g and F2 g (right panel) in the two downstream branches. Initially, 1 𝑔 = 0 8 2 𝑔 = 0 5 (dashed lines) and 1 𝑔 = 0 3 2 𝑔 = 0 5 (solid lines), and 1 = 14 m and 2 = 8 ̂ m. Blue and red lines indicate, respectively, branch 1 and branch 2. Results are given for 𝑘 𝑔 = 3 , 𝑘 𝑠 = 1 , 𝛼𝑔 = 𝛼𝑠 = 1 , and 𝐿 𝐴 = 1 m. Here time indicates 𝑡. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)

Fig. 7. Difference between the supply and transport capacity of gravel (blue line) and sand (red line) in branch 1 (left panel) and branch 2 (right panel). The conditions are equal to the case represented by the solid lines in Fig. 5 . Here time indicates 𝑡̂. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)

The bed surface of branch 1 continues to consist of a mixture of gravel 6.3. Effects of nodal point coefficients and sediment supply and sand (i.e., F1 g approaches an equilibrium nonzero value). This is understandable as in the final state the gravel and sand supply from the Finally we study the dependence of bifurcation dynamics on the upstream branch is transported by branch 1, which requires both gravel nodal point coefficients, kg and ks , and the gravel and sand supply to and sand to be represented at the bed surface (see Eqs. (5) and (6) ). the upstream branch 0. We consider two situations: 𝑘 𝑔 = 5 , 𝑘 𝑠 = 1 (sec- 𝑘 , 𝑘 tion IIa , Fig. 8) and 𝑔 = 1 𝑠 = 5 (section IIb , Fig. 9). We study two

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𝑘 , 𝑘 Fig. 8. Effect of the gravel content in the sediment supply on bifurcation evolution for 𝑔 = 5 𝑠 = 1 (section IIa ): flow depth H1 and H2 (left panels) and surface 𝛼 𝛼 𝐿 ≤ ≤ 𝐹 . 𝑆̂ 𝑆 𝑆 gravel content F1 g and F2 g (right panels). Results are given for 𝑔 = 𝑠 = 1 and 𝐴 = 1 m. Initially 0.1 F1 g 0.9 and 2 𝑔 = 0 2. In upper panels 0 = 0 𝑠 ∕ 0 𝑔 = 2 (i.e., 𝑆̂ a relatively coarse supply) and in lower panels 0 = 7 (i.e., a relatively fine supply). Blue and red lines indicate, respectively, branch 1 and branch 2. Here time indicates 𝑡̂ . (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)

𝑘 , 𝑘 Fig. 9. Effect of the gravel content in the sediment supply on bifurcation evolution for 𝑔 = 1 𝑠 = 5 (section IIb ): flow depth H1 and H2 (left panels) and surface 𝛼 𝛼 𝐿 ≤ ≤ 𝐹 . 𝑆̂ 𝑆 𝑆 gravel content F1 g and F2 g (right panels). Results are given for 𝑔 = 𝑠 = 1 and 𝐴 = 1 m. Initially 0.1 F1 g 0.9 and 2 𝑔 = 0 2. In upper panels 0 = 0 𝑠 ∕ 0 𝑔 = 2 (i.e., 𝑆̂ a relatively coarse supply) and in lower panels 0 = 7 (i.e., a relatively fine supply). Blue and red lines indicate, respectively, branch 1 and branch 2. Here time indicates 𝑡̂ . (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)

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𝑆̂ 𝑆 𝑆 𝑆̂ cases with respect to the ratio 0 = 0 𝑠 ∕ 0 𝑔 : either 0 = 2 (i.e., a rela- scribing their temporal behavior are needed to properly analyze river 𝑆̂ tively coarse supply, upper plots) or 0 = 7 (i.e., a relatively fine supply, bifurcation problems. lower plots). The simplicity of our nodal point relation, however, does enable us > If kg ks (Fig. 8) a case governed by a relatively coarse sediment sup- to assess a range of reasonable values for k in the unisize sediment case. ply to the upstream branch tends towards a system with two stable open Let us consider the general nodal point relation in Eq. (8) and assume branches, and this change happens relatively fast. A case with a much that the bed-material load is primarily transported as bed load. If we larger sand content in the sediment supply tends to develop towards a consider a bifurcation point with a flat bed just upstream of the bifur- system with one closed channel if the initial surface gravel content in cation point, the sediment supply to the downstream branches is likely branch 1 is in the range 0.1-0.4 ( Fig. 8 c–d). distributed over the downstream branches according to their values of

> 𝑆̄ ∗ 𝐵 ∗, 𝑘 Yet, if ks kg (Fig. 9), a case with a relatively coarse upstream supply the channel width and so it may be argued that = and so = 0 tends towards a system with one closed channel provided that the initial and 𝛼 = 1 ( Bolla Pittaluga et al., 2003 ). At the other side of the spectrum gravel content in branch 1 is in the range 0.2–0.9 ( Fig. 9 a–b). For a we consider a case where sediment is primarily transported as wash relatively fine sediment supply to the upstream branch, the situation load and well distributed over the water column. In such a case the with two open branches is always stable ( Fig. 9 c–d). sediment supply to the downstream branches is likely partitioned over Finally we stress the long time scale associated with the temporal the downstream branches according to the ratio of the water discharge ̄ change of the bifurcation system in our numerical runs. This seems to ( 𝑆 ∗ = 𝑄 ∗), which implies that 𝑘 = 𝛼 = 1 ( De Heer and Mosselman, 2004; be due to our assumption of branch-averaged change in the two bifur- Dutta et al., 2017 ). Thus, we expect that the finer the sediment supply cates, which slows down bifurcation adjustment. This is because in the to branch 0, the larger is the value of k (within the range of 0 to 1). branch-averaged model the result of a mismatch between the sediment In reality, however, conditions are more complex due to the presence supply to a bifurcate and its sediment transport capacity is distributed of a lateral bed slope, a bend, bars, or structures. We discuss the latter over the entire branch rather than the effect initially being limited to effects below, first for the unisize sediment case and subsequently for the upstream end of the bifurcate. In reality such a mismatch leads to an the mixed-size sediment case. aggradational or degradational wave that starts at the upstream end of Typically the difference in bed elevation between the two bifurcates the bifurcate and then migrates downstream. This feeds back to the mis- induces a lateral slope just upstream of the bifurcation. This slope effect match more strongly than in our branch-averaged approach and there- increases the sediment supply to the deeper bifurcate and therefore acts fore in reality the expected change likely occurs much faster than in our as a stabilizing mechanism, as it counteracts further deepening of the idealized model. Nevertheless, there is evidence from field data that bi- deeper bifurcate. This effect needs to be accounted for when setting up furcation change can be slow: a change to a new dominant channel may a model for the nodal point coefficient. require significant time ( Slingerland and Smith, 2004 ) and often require Bends and bars affect the flow just upstream of the bifurcation several centuries in the Rhine-Meuse delta, and durations of up to 1250 and as such may affect the sediment partitioning over the downstream years have been estimated by Stouthamer and Berendsen (2001) . branches of the bifurcation. For instance, the sediment partitioning at a bifurcation that is located just downstream of a pointbar in an inner bend is affected by the associated secondary flow and the transverse 7. Discussion gradient in bed elevation just upstream of the bifurcation. The presence of a sill in the downstream branch 1 likely reduces the 7.1. The load relation sediment supply to that branch ( Fig. 10 a). Considering the nodal point relation in Eq. (7) and assuming that the sill’s effect on the sediment In our analysis we have applied the Engelund and Hansen load rela- supply is larger than on the water discharge, k must be larger than 1 tion in a fractional manner ( Blom et al., 2017a; 2016 ). This fractional and for a relatively high sill (with barely sediment supply to the specific form of the load relation has never been properly validated and this ap- channel) k should approach ∞. proach likely is more valid in lowland rivers where partial mobility is A similar line of reasoning holds under mixed-size sediment con- less relevant. The analysis can be repeated for more complicated load ditions. Recall that the nodal point relations ( Eqs. (9) and (10) ) read relations (e.g., those including a threshold for significant transport or ∗ ∗ 𝑘 𝑔 ∗ 𝑘 𝑠 𝑠̄ = 𝛼𝑔 𝑞 and 𝑠̄ 𝑠 = 𝛼𝑠 𝑞 . The above lateral slope effect induced by hiding effects). We do not expect, however, that application of a more 𝑔 the inlet step varies with grain size: coarse sediment is affected more complicated and realistic load relation affects our analysis of the equilib- strongly than fine sediment ( Parker and Andrews, 1985 ), and the same rium states and the associated conclusions, as another load relation does holds for the lateral slope effect introduced by bends and bars. This not change the analysis in a fundamental manner. The analysis would also applies to the presence of a sill in one of the bifurcates: coarse still yield four coupled differential equations (similar to Eqs. (19) –(22) ), sediment is affected more strongly than fine sediment, as the trans- in which the coefficients are different from the ones based on the origi- port of coarse sediment concentrates more strongly near the bed. An nal load relation. This also holds for the Jacobian in Eq. (23) . The results additional effect of a river bend is bend sorting. A pointbar typically would differ somewhat from the ones based on the original load relation consists of relatively fine sediment, whereas the bed surface and trans- but we do not expect new phenomena. ported load in the outer bend are coarser ( Fig. 10 b). Hence, the sediment supply to the bifurcate located in the outer bend is coarser than the sup- 7.2. The nodal point relation ply to the other bifurcate. These effects will need to be accounted for 𝛼 𝛼 in relations for the nodal point coefficients g , s , kg , and ks . The as- A crucial point in the presented analysis is the fact that we assume sociated consequences for river bifurcation dynamics will need to be that there exists a nodal point relation of the form of Eq. (8) for the studied. unisize sediment case, or Eqs. (9) and (10) for the mixed-size sediment case. This form of the nodal point relation is too simple to describe the 7.3. The stability criterion partitioning of sediment at river bifurcations (e.g., Van der Mark and Mosselman, 2013 ). Wang et al. (1995) already suggested a nodal point Based on their mathematical stability analysis for the case of unisize relation of a more extensive form: 𝑆 ∗ = 𝑓( 𝐵 ∗ , 𝑄 ∗ , 𝐶 ∗ , 𝐻 ∗ , …) . It is, how- sediment conditions, Wang et al. (1995) found that the stability crite- ever, difficult to constrain the various parameters in the nodal point re- rion for two open branches is given by k > n /3. This can also be found lation, although an attempt may lead to a more realistic relation. Even through reasoning ( Kleinhans et al., 2008 ), which is summarized here to in the simple form of the nodal point relation used here, the values of subsequently extend this reasoning to the case of mixed-size sediment. 𝛼 𝛼 𝑠̄ , its coefficients kg , ks , g , and s are not likely constant and models de- Eq. (8) illustrates that the sediment supply to branch i, 𝑖 is proportional

17 R.M.J. Schielen, A. Blom Advances in Water Resources 121 (2018) 9–21

Fig. 10. Schematic of two bifurcating branches with (a) a sill and (b) a .

𝑘 to 𝑞 𝑖 , and Eqs. (1) –(3) show that the sediment transport capacity of a metrical solutions are stable, which may contradict the above findings. 𝑛 ∕3 Our results are similar to those of Wang et al. (1995) , who find that branch i s proportional to 𝑞 𝑖 . Now suppose that k > n /3. In that case the sediment supply per unit width to branch i increases more strongly with a symmetrical solution is stable for k > n /3. This similarity between increasing water discharge per unit width in branch i , than the sediment the results may not be surprising as our model is an extension of the transport capacity, which leads to aggradation. As a result the water highly idealized model of Wang et al. (1995) to mixed-size sediment conditions, whereas other models account for the effects of a transverse discharge per unit width, qi , decreases, which means that the situation stabilizes. On the other hand, if k < n /3, the sediment supply to branch i bed slope ( Bolla Pittaluga et al., 2015; 2003; Edmonds and Slingerland, 2008; Kleinhans et al., 2013; 2008 ), alternate bars ( Bertoldi et al., 2009; increases less strongly with increasing qi than the sediment transport ca- Redolfi et al., 2016 ), curvature-induced flow asymmetry upstream of pacity. This implies that the channel degrades, which increases qi even further. So the situation further destabilizes at the expense of the other the bifurcation ( Kleinhans et al., 2008; Van Denderen et al., 2017 ), downstream branch, which closes. suspended bed-material load ( Slingerland and Smith, 1998 ), cohesive For the case of mixed-size sediment we reason in a similar manner. sediment ( Edmonds and Slingerland, 2008; Hajek and Edmonds, 2014 ),

The supply of gravel and sand to branch i , 𝑠̄ 𝑖𝑔 and 𝑠̄ 𝑖𝑠 , are proportional bend sorting (Sloff et al., 2003; Sloff and Mosselman, 2012), and bank 𝑘 𝑔 𝑘 erosion ( Miori et al., 2006; Van Denderen et al., 2017 ). These effects to 𝑞 and 𝑞 𝑠 , respectively. The transport capacities of gravel and sand 𝑖 𝑖 are not taken into account in our analysis but may be represented by in branch i, s and s , now also depend on the surface gravel content ig is 𝑛 𝑛 appropriate future models for kg and ks . 𝐹 𝑚 𝑞 ∕3 𝐹 𝑚 𝑞 ∕3, in branch i, Fig , and are proportional to 𝑖𝑔 𝑔 𝑖 and (1 − 𝑖𝑔 ) 𝑠 𝑖 respectively. We now suppose that the water discharge per unit width in branch i, qi , increases. This implies that the gravel and sand supply 8. Conclusions to branch i increase, the manner of which depend on the values of kg and ks . As kg and ks are expected to have different values, the gravel We extend a highly idealized model of the dynamics of a river bifur- and sand supply to branch i respond differently to the increase in the cation to mixed-size sediment conditions. The model is based on nodal water discharge. Also the gravel and sand transport capacities, sig and point relations for gravel and sand that set the partitioning of gravel and sis , respond differently to the increase in the water discharge due to the sand over the downstream branches or bifurcates. The model describes mobility difference between coarse and fine sediment (i.e., grain size se- the equilibrium solutions and, based on a branch-averaged approxima- lective transport). Depending on the resulting change of the gravel and tion of aggradation and degradation, describes the temporal change of sand supply to branch i , 𝑠̄ 𝑖𝑔 and 𝑠̄ 𝑖𝑠 , and the change in the gravel and bed elevation and bed surface texture in the bifurcates of a mixed-size sand transport capacities, sig and sis , the increase in water discharge qi sediment river bifurcation. affects the surface gravel content in branch i, Fig , which is expressed by a The introduction of mixed-size sediment mechanisms to the river bi- coarsening or fining of the bed surface. Thus, based on physical reason- furcation problem introduces an additional degree of freedom: the tem- ing it is much less straightforward to draw conclusions concerning the poral adjustment of the bed surface texture in each of the bifurcates. expected temporal change and a stability criterion of a river bifurcation The dynamics of the downstream branches concerning their flow depth dominated by mixed-size sediment. and bed surface texture and the resulting stable configuration of the As a next step we recommend the formulation of submodels for the downstream branches result from differences between (a) the gravel nodal point coefficients kg and ks , which, among other parameters, likely and sand supply in each branch and (b) its gravel and sand transport depend on the transverse bed slope just upstream of the bifurcation capacity. ( Bolla Pittaluga et al., 2003 ) and hence likely are a function of the ratio We set up a mathematical model of the equilibrium states and dy- ∗ 𝐻 ∗ 𝐻 𝐻 of the flow depth in the bifurcates, H (where = 1 ∕ 2 ). In such a namics of a mixed-size sediment river bifurcation. In our analysis we case the formulation of the Jacobian in Eq. (23) becomes more compli- have neglected the effects of a transverse bed slope, alternate bars, cated because of the derivatives with respect to H1 and H2 . Analysis of curvature-induced flow asymmetry upstream of the bifurcation, sus- the Jacobian and the numerical results would provide insight on the ex- pended bed-material load, cohesive sediment, bend sorting, and bank pected temporal change and the stability criterion of a river bifurcation erosion. The proposed model therefore has limited predictive value re- dominated by mixed-size sediment. garding real river bifurcations, yet provides insight on the elementary effects of mixed-size sediment mechanisms on the river bifurcation prob- 7.4. Symmetrical bifurcations lem. Subsequent analyses may combine the analysis of mixed-size sedi- ment mechanisms with the above mentioned effects. It has been found that symmetrical bifurcations (i.e., bifurcates with Howard (1980) and Blom et al. (2017a, 2016) have shown that equal properties such as water discharge, channel width, and flow there exists one solution to the morphodynamic equilibrium state in depth) tend to be unstable more often than asymmetrical bifurcations a one-channel system with nonerodible banks. In a unisize sediment (i.e., one of the bifurcates is significantly smaller than the other one) two-channel system with fixed banks three equilibrium solutions exist, ( Bertoldi and Tubino, 2007; Bolla Pittaluga et al., 2015; Edmonds and whereas three to five solutions exist in a mixed-size sediment bifurcation Slingerland, 2008; Kleinhans et al., 2013; 2008; Miori et al., 2006 ). In system. our base case, which is characterized by equal channel width, friction, In the mixed-size sediment two-channel system we distinguish three and length of the bifurcates, we find that in sections II and III sym- sections (I, II, and III) in the parameter space related to the nodal point

18 R.M.J. Schielen, A. Blom Advances in Water Resources 121 (2018) 9–21

𝑄 ∗ 𝑄 ∗ coefficients. The specific layout of the three sections depends on the emerge at kga annilihate with = 0 and = ∞ (which correspond > ratio of the bifurcate length and the ratio of the upstream gravel supply with one branch closed). So for larger values of ks and for kg kgb there to the sand supply. are again only 3 solutions ( Fig. 11 c): two solutions correspond with the In section I the only stable solutions are the ones with one branch situation with a closed branch, and the other solutions with a situation closed, and in section III the only stable solution is the one with both with both branches open. branches open. In section II there are three stable solutions: two with one branch closed and one with two open branches. To which of these A2. Stability properties three stable equilibrium solutions the system evolves depends on the initial conditions (i.e., the initial flow depth and bed surface texture in This appendix explores the ( k , k ) parameter space and the stability s g the bifurcates). properties of the equilibrium solutions in the sections I, IIa , IIb and III.

We use the same approach as in appendix A: we fix ks at a certain value

Appendix A. Details of the equilibrium solutions and their and vary kg (i.e., we again make a horizontal transect through Fig. 3). 𝑄 ∗ 𝑄 ∗ , stability For each combination of ks and kg , we determine where − Φ( ) = 0

compute the associated flow depth in the two bifurcates, Hi , and com-

A1. Equilibrium solutions pute the other variables (e.g. Sis , Sig , Fs , Fg etc) and the eigenvalues of the Jacobian J in Eq. (23) . ∗ 𝑘 < < In this appendix we analyze Eq. (12) and explain Fig. 3 in more Fig. 12a shows the result for Q for 𝑠 = 1 and 0 kg 4, for three detail. The three solutions for Q ∗ of Eq. (12) give rise to three solutions values of L ∗ . For simplicity we do not indicate the stable solutions 𝑄 ∗ = 0 𝑄 ∗ ∗ for the combination of flow depths in the downstream branches Hi : two and = ∞. For each value of L one mathematical bifurcation occurs with one branch closed, one with both branches open. We now consider at a specific value of kg (blue sky bifurcation, corresponding with the 𝐿 ∗ . a fixed value of ks and increase the value of kg , i.e., we make a horizontal occurrence of the solid lines in Fig. 12a). For = 1 33 the bifurcation > transect in Fig. 3. into a stable solution occurs for kg 4 in Fig. 12a and is therefore not

For relatively small values of ks and kg , three equilibrium solutions visible. The values of kg for which the bifurcations occur correspond exist (Fig. 11a). For such small values of ks , there exists a threshold with the transitions from section I to IIa in Fig. 3. 𝑘 𝑘 𝑄̂ ∗ 𝑄̂ ∗ 𝑄̂ ∗ 𝑘 . value for kg ( 𝑔 = 𝑔𝑎 ) for which there is a such that − Φ( ) = 0 Fig. 12b shows a similar plot for 𝑠 = 2 5. Now we observe two bi- 𝑑 𝑄 ∗ 𝑄 ∗ 𝑑 𝑄 | > ∗ and ( − Φ( ))∕ 𝑄 ∗ = 𝑄̂ ∗ = 0. This implies that for kg kga two new furcations: one where the stable solution for Q emerges in a blue sky solutions of 𝑄 ∗ − Φ( 𝑄 ∗ ) = 0 emerge and we then find five equilibrium bifurcation (occurrence of the solid lines in Fig. 12 b), and one where solutions ( Fig. 11 b): two with one branch closed and three with both the unstable solutions of Q ∗ are annihilated in a collision with the 𝑄 ∗ 𝑄 ∗ branches open. The threshold value kga depends on ks and defines the stable solutions = 0 and = ∞ (saddle-node bifurcation, vanish- 𝑄 ∗ boundary between sections I and IIa, b in Fig. 3. ing of the dashes lines at = 0), leaving only one single stable solu-

For larger values of ks , we find another threshold value for kg , which tion in section III. We summarize the consequences of these results in 𝑄 ∗ 𝑄 ∗ we call kgb . At that value, the new solutions of − Φ( ) = 0 that Section 5.

𝑄 ∗ 𝑄 ∗ Fig. 11. Typical graphs of − Φ( ) for parameter values in sections I (left plot), IIa, b (center plot) and III (right plot). Equilibrium solutions correspond with

𝑄 ∗ − Φ( 𝑄 ∗) = 0 . 𝑄 ∗ = 0 and 𝑄 ∗ = ∞ are global solutions of 𝑄 ∗ − Φ( 𝑄 ∗) = 0 .

19 R.M.J. Schielen, A. Blom Advances in Water Resources 121 (2018) 9–21

Bertoldi, W., Tubino, M., 2007. River bifurcations: experimental ob- servations on equilibrium configurations. Water Resour. Res. 43. https://doi.org/10.1029/2007WR005907 . Bertoldi, W., Zanoni, L., Miori, S., Repetto, R., Tubino M., 2009. Interaction between mi- grating bars and bifurcations in gravel bed rivers. Water Resour. Res. 45 (6) W06418 https://doi.org/10.1029/2008WR007086 . Blom, A., Arkesteijn, L., Chavarrías, V., Viparelli, E., 2017. The equilibrium under variable flow and its channel-forming discharge. J. Geophys. Res. Earth Surf. 122. https://doi.org/10.1002/2017JF004213 . Blom, A., Chavarrías, V., Ferguson, R.I., Viparelli, E., 2017. Advance, retreat, and halt of abrupt gravel-sand transitions in alluvial rivers. Geophys. Res. Lett. 44. https://doi.org/10.1002/2017GL074231 . Blom, A., Viparelli, E., Chavarrías, V., 2016. The graded alluvial river: profile concavity and downstream fining. Geophys. Res. Lett. 43. https://doi.org/10.1002/2016GL068898 . Bolla Pittaluga, M., Coco, G., Kleinhans, M.G., 2015. A unified framework for stability of channel bifurcations in gravel and sand fluvial systems. Geophys. Res. Lett. 42 (18),

Fig. 12. Bifurcation diagram for the equilibrium values of Q ∗ as a function of 7521–7536. https://doi.org/10.1002/2015GL065175. Bolla Pittaluga, M., Repetto, R., Tubino, M., 2003. Channel bifurcation in braided k for (a) 𝑘 𝑠 = 1 and (b) 𝑘 𝑠 = 2 . 5 . Solid and dashed lines indicate, respectively, g rivers: equilibrium configurations and stability. Water Resour. Res. 39 (3), 1046. 𝑄 ∗ 𝑄 ∗ stable and unstable solutions. For simplicity the stable solutions = 0 and = https://doi.org/10.1029/2001WR001112 . ∞ are not shown. (For interpretation of the references to colour in this figure Bulle, H. , 1926. Untersuchungen über die Geschiebeableitung bei der Spaltung von legend, the reader is referred to the web version of this article.) Wasserläufen: Modellversuche aus dem Flussbaulaboratorium der Technischen Hochschule zu Karlsruhe. VDI Verlag, Berlin, (in German) . Burge, L.M., 2006. Stability, morphology and surface grain size patterns of channel bifur- Appendix B. 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