bifurcation modelling in a meandering gravel–sand bed river

EARTH SURFACE PROCESSES AND LANDFORMSEarth Surf. Process. Landforms (2012)Copyright © 2012 John Wiley & Sons, Ltd.Published online in Wiley Online Library(wileyonlinelibrary.com) DOI: 10.1002/esp.3305

Bifurcation modelling in a meandering gravel–sandbed riverKees Sloff* and Erik MosselmanDeltares and Delft University of Technology

Received 22 March 2011; Revised 4 July 2012; Accepted 5 July 2012

*Correspondence to: Kees Sloff, Deltares and Delft University of Technology. E-mail: [email protected]

ABSTRACT: The Rhine bifurcation at Pannerden forms the major distribution point for water supply in the Netherlands, distributingnot only water and sediment but also flooding risks and navigability. Its morphological stability has been a concern for centuries. Wepresent experiences from more than two decades of numerical morphological modelling of this bifurcation with a gravel–sand bedand a meandering planform. Successive computations have shown the importance of upstream approach conditions, the necessity toinclude physical mechanisms for grain sorting and alluvial roughness, and the need to assume a thicker active layer of the river bedthan is suggested by laboratory flume experiments using a constant discharge. The active layer must be thicker in the model toaccount for river bed variations due to higher-frequency discharge variations that are filtered out inmorphological modelling.We discusslimitations in calibration and verification, but argue that, notwithstanding these limitations, 2D and 3D morphological models arevaluable tools, not only for pragmatic applications to engineering problems, but also for revealing the limitations of establishedknowledge and understanding of the relevant physical processes. The application of numerical models to the Pannerden bifurcationappeared to reveal shortcomings in established model formulations that do not pose particular problems in other cases. This applicationis therefore particularly useful for setting the agenda for further research. Copyright © 2012 John Wiley & Sons, Ltd.

KEYWORDS: river morphology; numerical modeling; graded sediment; river bifurcations; Rhine

Introduction

The river Rhine divides its course into several branches in its deltain the Netherlands. The main bifurcation is located at thevillage of Pannerden and hence known as ‘Pannerdense Kop’(Pannerden’s Head). The river, called ‘Bovenrijn’ in this reach,splits at this bifurcation into the Waal to the left and thePannerdens Kanaal to the right (Figures 1 and 2). The stability ofthe Pannerden bifurcation has been a source of concern forcenturies. This was the reason for creating Rijkswaterstaat in1798, the first centralized government body in a time when theNetherlands still operated as a union of largely autonomousprovinces with conflicting interests (Van de Ven, 1976). ThePannerden bifurcation is the major distribution point for watersupply in the Netherlands, and this point not only distributeswater and sediment, but as a consequence also flooding risksand navigability. Table I lists some characteristics of the riverbranches at the bifurcation. The characteristic average dischargesare representative values for morphological computationswhen using a constant discharge. Figure 3 shows the 2009 bedtopography from multi-beam echo-sounder measurements.Figure 4 shows the bed sediment composition measured in2000 by Gruijters et al. (2001).Works to stabilize the bifurcation at Pannerden relied for

many years on engineering judgement and trial and error. Itwas only in the twentieth century that the State Committee forthe Zuiderzee (Staatscommissie Zuiderzee, 1926), headed byNobel laureate Hendrik Lorentz, introduced physical and

numerical modelling into hydraulic engineering in theNetherlands. Modelling of the Pannerden bifurcation startedwith studying the distribution of sediment over Waal andPannerdens Kanaal in a large-scale mobile-bed physical modelat the De Voorst research station of Delft Hydraulics (Van derZwaard, 1981; Sloff et al., 2003). Figure 5 shows the modelwith a sieving facility in the background to separate differentsediment size fractions from the downstream sediment traps.The separated size fractions were subsequently mixed into thecorrect inflow dosage and composition before recirculation tothe upstream inflow section. The physical model tests inspiredthe mathematician Flokstra (1985) to carry out a nonlinearphase-plane analysis of 1D morphological equations forriver bifurcations, revealing bifurcation stability to dependsensitively on the relation between sediment distribution anddischarge distribution. Wang et al. (1995) repeated andextended this analysis.

The physical model experiments and the nonlinear phase-plane analysis were followed by numerical modelling usingthe 2D morphological model RIVCOM, a predecessor of themorphology module in Delft3D (Struiksma, 1988). The compu-tations assumed spatially constant sediment properties and aspatially constant Chézy coefficient for hydraulic resistance.Comparing the first numerical results with the results from thephysical model led to the conclusion that numerical modelsperform better than physical models. For a decade, this successwas seen as a major demonstration of the power of numericalmorphological models. However, the enthusiasm was damped

Figure 1. Aerial photograph of Rhine bifurcation at Pannerden.

Figure 2. Location map. This figure is available in colour online atwileyonlinelibrary.com/journal/espl

Table I. Characteristics of river branches near the Rhine bifurcationat Pannerden

Bovenrijn WaalPannerdens

Kanaal

Main-channel width (m) 340 260 135River gradient (m/km) 0.010 0.085 0.050Characteristic averagedischarge (m3/s)

2135 1463 672

Chézy roughness (m1/2/s) 53 50 50Flow depth (m) 5.5 5.6 5.3Width-averaged mediansediment size (mm)

2.2 1.7 2.8

Width-averaged 90thpercentile sedimentsize (mm)

9 7 10

K. SLOFF AND E. MOSSELMAN

when proper calibration of a 2D morphological modelappeared impossible after imposing a measured spatiallyvarying granulometry of bed sediment composition (Struiksma,1998). Good reproduction in a numerical model was found tobe less simple and trivial than thought previously, and thismotivated a line of research in subsequent years, using themodels RIVCOM, DELFT2DMOR and Delft3D. The modellingfocused on the development of bed topography and sedimentsorting at the scale of bars and pools, with a resolution in theorder of tens of metres. Field studies at the Pannerdense Kop

Copyright © 2012 John Wiley & Sons, Ltd.

also addressed processes at smaller scales, involving thegrowth and decay of dunes and the associated vertical sortingwith upward fining and the development of coarse layers(Kleinhans, 2001; Kleinhans et al., 2007; Frings and Kleinhans,2008). Our purpose is to present the major findings of themodelling line of research. This presentation is a review ofexperiences and corresponding conclusions on physicalprocesses rather than verification or validation of a specific model.

Numerical Model

Different numerical morphological models have been appliedto the Rhine bifurcation at Pannerden over the past 25 years.In the meantime these models changed in various ways, e.g.from only two dimensions to the option of three dimensions,from strictly Reynolds-averaged flow equations to optionallarge-eddy simulation, from only uniform sediment to theoption of graded sediment and from monodisciplinary morpho-logical models to integrated modelling systems. Changes alsooccurred in the numerical scheme and the method of combin-ing small hydrodynamic time steps with large morphologicaltime steps. Here we present mainly technical details of the finalmodel, Delft3D, as our purpose is not to present or to validatedifferent software versions, but to present modelling experi-ences with a more generic character that do not depend on aspecific implementation. All models used stem from thepioneering 2D morphological model by Van Bendegom(1947), cf. Allen (1978). This model included 3D effects fromhelical flow in a parameterized form. Subsequent researchand development resulted in RIVCOM (Struiksma, 1985;Struiksma et al., 1985), the predecessor of the morphologymodule in Delft3D, as well as in T20 (Olesen, 1987), thepredecessor of the morphology module in Mike21C. Shimizuand Itakura (1985, 1989) and Nelson and Smith (1989) areamong the early developers of similar models. The applicationof 2D morphological models with parameterizations of 3Deffects became a standard element in river engineering studies,such as those for the Waal programme (Havinga et al., 2006).More models appeared in the next two decades, albeitsometimes as simple extensions of 3D hydrodynamic modelswithout accounting for bed slope effects on sediment transportand without adequate validation (cf. Mosselman, 2010).

The equations in Delft3D hold for curvilinear co-ordinatesystems, but for clarity we present all equations here for arectangular cartesian co-ordinate system in which the x axis isaligned parallel to the banks and the y axis is perpendicularto the banks. Lesser et al. (2004) present the full hydrodynamicequations of Delft3D. These equations are quasi-3D in thesense that the vertical momentum equation has been reducedto the hydrostatic pressure equation by assuming that verticalflow accelerations are negligible compared with gravity. The3D model thus consists of several 2D layers that are coupledthrough the hydrostatic pressure equation and a continuityequation for mass conservation. This allows an approach inwhich the horizontal sizes of the computational grid are muchlarger than the vertical sizes, which suits the modelling ofnatural water bodies such as rivers where the horizontal extentof the computational domain is usually much larger than thewater depth. We use a single layer in our morphologicalcomputations, which means that we use 2D hydrodynamicequations despite the availability of quasi-3D equations in thesoftware. We do this because morphological computationsrequire a new computation of the flow field after each time stepof bed evolution, leading to long computation times. Fastcomputation of the flow field greatly improves the overallperformance of the model.

Earth Surf. Process. Landforms, (2012)

http://www.wileyonlinelibrary.com/journal/espl

Figure 4. Average sediment composition of top layer that had beenreworked during the preceding decade, measured in 2000 by Gruijterset al. (2001).

Figure 5. Physical model at De Voorst research station of DelftHydraulics (Van der Zwaard, 1981).

Figure 3. Bed topography from multi-beam echo-sounder measurements in 2009 (NAP=Amsterdam Ordnance Datum, approximately equal tomean sea level).

BIFURCATION MODELLING IN A MEANDERING GRAVEL-SAND BED RIVER

We use Reynolds-averaged flow equations and assume thattime derivatives in hydrodynamic equations can be neglected.This quasi-steady flow assumption is valid even under varying


discharges as long as morphological changes operate on amuch slower time scale than adaptations of water flow tochanging conditions. This is the case if Froude numbers arebelow 0.6 to 0.8 (De Vries, 1959, 1965; Jansen et al., 1979).Values for hydraulic roughness can be imposed directly or becalculated by an alluvial roughness predictor (Van Rijn,1984). The parameterization of helical flow is based on intro-ducing an equilibrium helical flow intensity, Ie, calculated by

Ie ¼ hffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiu2 þ v2

p

R(1)

where h denotes flow depth, u is the depth-averaged flowvelocity in x direction, v is the depth-averaged flow velocity iny direction and R is the radius of streamline curvature defined by

1R¼ � 1

u@v@x

(2)

if v≪u (flow more or less in x direction). A relaxation-diffusionequation describes how the actual helical flow intensity, I, adaptsto the equilibrium helical flow intensity, Ie. Writing onlyderivatives with respect to x for clarity, this equation reads

Lahffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiu2 þ v2

p @ huIð Þ@x

þ Laffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiu2 þ v2

p @

@xDH

@I@x

� �þ I ¼ Ie (3)

where La is a characteristic adaptation length for the helical flowintensity and DH is the horizontal eddy diffusivity. Derivativeswith respect to y lead to terms of the same form in the completeequation. The angle, at, between the near-bed flow directionand the depth-averaged flow direction is calculated from theactual helical flow intensity by

tanat ¼ �AIffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

u2 þ v2p (4)

where A is a coefficient which weighs the influence of helicalflow on the direction of bed shear stress. In case of a logarithmicvertical flow velocity profile, this coefficient is given by



A ¼ 2k2

1�ffiffiffig

p2kC

� �(5)

where k is the Von Kármán constant, g is the acceleration due togravity and C is the Chézy coefficient for hydraulic roughness(Olesen, 1987).In rivers with predominantly bedload, the near-bed flow

direction is equal to the direction of sediment transport over aflat bed. The parameterization used holds for mildly curvedbends and hence yields inaccurate results in sharp bends orclose to the banks.Sediment transport is described by semi-empirical transport

formulae and a depth-averaged sediment balance. For the totalsediment mixture, the latter reads

1� eð Þ @zb@t

þ @qsx@x

þ @qsy@y

¼ 0 (6)

in which qsx and qsy are volumetric sediment transport compo-nents per unit width (excluding pores) in the x- and y-direction,respectively, t is time, zb denotes bed level and e is the porosityof the bed. The porosity factor translates the net volume ofsediment grains into the bulk volume which corresponds tothe volumetric changes of bed topography due to erosion andsedimentation. Usually e=0.4 is assumed (Jansen et al., 1979).Graded sediments are accounted for through (i) division of

the sediment mixture into separate fractions, (ii) transportformulae and mass conservation equations for each of theseparate fractions, (iii) hiding-and-exposure corrections for thecritical shear stress of each of the fractions, (iv) an active layeror transport layer affected by erosion and sedimentation (Hirano,1972) and (v) a bookkeeping system for substratum that hasbecome inactive due to sedimentation. The thickness of theactive layer is equal to a fraction of the height of the bedforms:half the height in the case of triangular dunes (Ribberink, 1987).An exchange layer may be added under the transport layer(Ribberink, 1987, cf. Figure 6). The exchange of sedimentbetween the active layer and the substrate is thought to beassociated with the extension of sporadic deeper troughs ofthe bedforms.The relative occurrence of a sediment size fraction i in the

active layer is indicated by pi,a, and in the substratum by pi,0.By definition

Xi

pi;a ¼ 1 andXi

pi;0 ¼ 1 (7)

The mass conservation equation or sediment balance foreach fraction reads

Figure 6. Subdivision of river bed into transport layer, exchange layer and


1� eð Þ @ pi;ad� �@t

þ pi z0ð Þ @z0@t

� �þ @qsxi

@xþ @qsyi

@y¼ 0

pi z0ð Þ ¼ pi;a sedimentationpi;0 erosion

(8)

in which z0 is the upper level of the substratum, pi(z0) is therelative occurrence of a sediment size fraction i at this level(taken equal to pi,a during sedimentation and equal to pi,0during erosion), d is the thickness of the active layer, and qsxiand qsyi are volumetric sediment transport components per unitwidth for fraction i in the x and y direction, respectively. Theactual bed level is given by

zb ¼ z0 þ d (9)

The sediment transport rate per fraction is described using astandard transport formula. Here the bedload formula ofMeyer-Peter and Müller (1948) is taken as a starting point. Forgraded sediment, this formula is written as

qsi ¼ Acal

ffiffiffiffiffiffiffiffiffiffiffiffiffigΔD3

i

q CC90

� �3=2 tbrwgΔDi

� θcxi

!3=2

(10)

where qsi is the total sediment transport rate per unit width forfraction i, Acal is a calibration factor, Δ is the relative density ofthe sediment under water, defined by Δ= (rs� rw)/rw, rw andrs are mass densities of water and sediment, respectively, Di isthe grain size of fraction i, C90 is a Chézy coefficient for grainroughness, (C/C90)

3/2 is the ripple factor, tb is the magnitude ofthe bed shear stress, θc is the critical Shields parameter(nondimensional critical shear stress) for initiation of sedimentmotion and xi is the hiding-and-exposure correction. The valuesof Acal and θc are 8 and 0.047, respectively, in Meyer-Peter andMüller’s original formula. The grain roughness is based on aNikuradse roughness of 3D90 instead ofMeyer-Peter andMüller’soriginal D90:

C90 ¼ 18 log12h3D90

� �(11)

where D90 denotes the ninetieth percentile of the grain sizedistribution, i.e. the grain size exceeded by 100–90=10% ofthe sediment mixture.

The hiding-and-exposure correction is modelled accordingto the Egiazaroff (1965) formulation, adjusted by Ashida andMichiue (1972, 1973)

a substratum with sublayers.



xi ¼ 0:85Dm

Difor

Di

Dm< 0:47

xi ¼log19

log 19Di

Dm

� �2664

37752

forDi

Dm≥0:47

(12)

whereDm denotes the average grain size of the sedimentmixture.Besides helical flow, also transverse bed slopes cause a

difference between the directions of sediment transportand depth-averaged flow. This effect is modelled byapplying Koch and Flokstra’s (1981) relation to individual grainsize fractions:

tanasi ¼sinat � 1

fi@zb@y

cosat � 1fi@zb@x

(13)

where asi is the angle between the transport direction ofsediment grain size fraction i and the depth-averaged flowdirection, and fi is a dimensionless parameter weighing theinfluence of gravity pull along inclined beds on the transportdirection of sediment fraction i. It is calculated by

fi ¼ Ashtb

rwgΔDi

� �Bsh Di

h

� �Csh Dm

Di

� �Dsh

(14)

in which Ash, Bsh, Csh and Dsh are calibration parameters andtb/(rwgΔDi) represents the nondimensional Shields parameterfor the mobility of sediment grain size fraction i. Here Di /h hasbeen introduced in analogy to a similar parameter for the effectof bedforms in the case of uniform sediment where Csh=0.3(Talmon et al., 1995). The ratio Dm /Di can be considered toaccount for effects of hiding and exposure, but it does not acton the critical shear stress as in the sediment transport formulaor the relation for the effect of transverse bed slopes by Parkerand Andrews (1985). Note that Bsh=Csh�Dsh implies that allgrain size fractions move in the same direction, because theinfluence of Di vanishes. Yet this does not imply equal mobility,because these calibration parameters do not affect the transportformula and its associated hiding-and-exposure correction. Acommonly used value is Bsh=0.5, but the other parameters areessentially poorly known. Moreover, it is not clear whether theproper mobility parameter to be used in the equation is theShields parameter for each individual grain size fraction, asshown here, or an overall Shields parameter for an average grainsize. This question as well as the high number of calibrationparameters basically reflect the poor state of our presentknowledge on the effect of gravity pull along inclined beds. Wecome back to this in the discussion section.Sloff et al. (2001) tested the implementation of the graded-

sediment equations in Delft3D against Ribberink’s (1987)straight-flume experiments and Olesen’s (1987) curved-flumeexperiments.

Figure 7. Bathymetries from prototype measurements (top), physicalmodel tests (centre) and the first 2D numerical model (bottom)(Struiksma et al., 1988).

Results

Spatial heterogeneity in bed sediment compositionand hydraulic roughness

Struiksma (1988) carried out the first 2D numerical computationof the Rhine bifurcation at Pannerden. The cells of the computa-tional grid were 50 to 250m long and generally 20 to 30mwide,but narrower close to the banks. The time step for morphologicalupdating was equal to 2days. Struiksma imposed a constant


discharge of 1950m3/s at the upstream boundarywith a specifieddistribution over the cross-section. Figure 7 shows a comparisonbetween the results from this 2D numerical computation byStruiksma (1988), the mobile-bed physical model by Van derZwaard (1981) and prototype measurements. The numericalcomputation was seen as a great succes, because the resultingbed topography agreed better with prototype measurements thanthe bed topography obtained in the physical model. Struiksmahad carried out his computation by assuming, for each of thebranches separately, spatially smoothed sediment propertiesand a spatially constant Chézy coefficient for hydraulicresistance. Samples of river bed material indicated, however, thatsediment granulometry varies spatially more strongly due to grainsorting. Struiksma (1998) therefore repeated his computationusing measured spatially varying grain sizes, albeit withoutphysics-based process descriptions for changes in bed sedimentcomposition. Surprisingly, it appeared no longer possible toreproduce the bed topography correctly. Shallow crossingsbetween consecutive bends appeared too far downstreamand transverse bed slopes became too gentle. As the composi-tion of the sediment in both the bed and the material suppliedupstream had been key parameters in the calibration of thephysical model, Struiksma (personal communication)suggested that the problematic results might be explained fromgrain sorting processes responsible for the observed spatialvariations in bed sediment composition. Mosselman et al.(1999) explored this suggestion first theoretically by extendinga linear analysis of Struiksma et al. (1985) with gradedsediment. The original linear analysis shows that alluvial bedtopographies in curved laboratory flumes and meanderingrivers can be seen as a superimposition of forced point-barsand nonmigrating free alternate bars. The nonmigrating freealternate bars are characterized by a damping length, LD, anda wave length, LP, given by


igure 8. Effect of grain size variations on wave length and dampingngth of nonmigrating alternate bars (b=5).


LDlw

¼ 4

2 lwls

þ 3� b(15)

LPlw

¼ 2pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffilwls

� lwLD

�2r (16)

where b is the degree of nonlinearity in the dependence ofsediment transport rate on flow velocity, lw is the longitudinaladaptation length for transverse profiles of longitudinal depth-averaged flow velocity and ls is the longitudinal adaptationlength for transverse profiles of bed level. These parametersare defined as (Struiksma et al., 1985):

b ¼ uqs

@qs@u

(17)

lw ¼ C2h2g

(18)

ls ¼ 1p2

Bh

� �2

f h (19)

where B denotes river width, h denotes reach-averaged flowdepth and f is the dimensionless parameter weighing theinfluence of gravity pull along inclined beds on the directionof sediment transport.Struiksma et al. (1985) carried out their analysis for a constant

sediment grain size, D, and constant hydraulic roughness, C.Mosselman et al. (1999) introduced grain sorting in thisanalysis simply by assuming a dependence of grain size ondepth-averaged flow velocity: D=D(u), considering that bedsediment is coarsest in areas of the highest flow velocities. Theydefined the degree of nonlinearity, a, of this dependence as

a ¼ uD@D@u

(20)

Furthermore, the variations in grain size required accounting forthe dependence of sediment transport rate on grain size too. Thedegree of nonlinearity, ΨD, of this dependence was defined as

ΨD ¼ Dqs

@qs@D

(21)

which expresses that, under the same flow conditions, transportrates for finer grains are higher than transport rates for coarsergrains (ΨD< 0). The modified linear analysis for a river withgraded sediment changed the expression for the dampinglength into

LDlw

¼ 4

2 lwls

þ 3� b � aΨD

(22)

where aΨD essentially represents a reduction of b because a>0andΨD< 0. The expression for the wave length retained the sameform as Equation (15), but nonetheless the wave length is affectedby aΨD too through its dependence on damping length. Figure 8presents the results for b=5 and for values of ls /lw that arerepresentative for the Waal branch. Spatial variations in grain sizeappear to increasewave lengths and to decrease damping lengths,


Fle

which is consistent with the observed downstream shift ofcrossings and reduction of transverse bed slopes.

Mosselman et al. (1999) analyzed the effect of spatialvariations in hydraulic roughness in a similar way andidentified conditions under which the effect of spatial grain sizevariations is counterbalanced by the effect of spatial variationsin hydraulic roughness. The success of numerical computationsusing spatially constant sediment properties and hydraulicroughness suggested that such counterbalancing occurred inthe area of the Rhine bifurcation at Pannerden. However, thereare no reasons to assume that this counterbalancing would be ageneral phenomenon occurring under other conditions as well.The conclusion was therefore that proper modelling requiredinclusion of physical mechanisms for grain sorting andalluvial roughness.

Active layer for sediment transport

Having concluded that grain sorting is an important factor, weapplied a 2D morphological model with formulations ofphysical mechanisms for the transport of graded sediment. Weused a 0.1m thick active layer at a discharge of 2400m3/s, basedon the 0.2m high dunes measured by Wilbers and Ten Brinke(2003). Surprisingly, the computed bed topography retained itsinitial state. It hardly exhibited any erosion and sedimentationdespite substantial sediment transport and considerable gradi-ents in initial sediment transport capacity. Only a 6000m3/sflood discharge, with an active layer of 1.0m, was foundto change bed topography. Mosselman and Sloff (2007)observed, at 2400m3/s, that a pattern of bed sediment compo-sition patches, hundreds of metres long and tens of metreswide, developed so fast, that it reduced all gradients insediment transport capacity to zero before any appreciablechange in bed levels could occur. They explained this behav-iour of the model using Ribberink’s (1987) 1D theoreticalanalysis for rivers with graded sediment. Ribberink integratedmass conservation equations for individual sediment sizefractions over the full sediment mixture and combined theresulting equation for sediment composition with quasi-steadyflow equations and the sediment balance or Exner equation,assuming capacity-limited sediment transport as a function ofdepth-averaged flow velocity and bed sediment composition.He then carried out an analysis of characteristics of the set ofequations, revealing how information propagates in themathematical system. This led to identifying two referencecelerities, one for perturbations in bed level (cbed) and onefor perturbations in bed sediment composition (cmix):


Figure 9. Celerities of morphological perturbations as a function ofthe parameter x for interactions between bed topography and bedsediment composition.

igure 10. Computed bed topography after 3 years simulation timeAP=Amsterdam Ordnance Datum).


cbed ¼ bqS1� eð Þ 1� Fr2ð Þh (23)

cmix ¼ DmT

Dm

qS1� eð Þd (24)

where Fr is the Froude number, Dm is the average grain size ofbed material, DmT is the average grain size of the sediment intransport and d is the thickness of the active layer. Celeritiesrepresent propagation speeds of information. Mosselman andSloff (2007) used these expressions to derive a ratio of charac-teristic time scales:

Tmix

Tbed¼ cbed

cmix¼ b

1� Fr2Dm

DmT

dh

(25)

where Tbed denotes the characteristic time scale for bed levelevolution and Tmix is the characteristic time scale for bedsediment composition evolution. The ratio of time scales ishence proportional to the ratio of active-layer thickness to flowdepth. If d≪ h, the sediment composition of the bed respondsimmediately to the initial bed topography in a way thateliminates gradients in sediment transport capacity, afterwhich the bed topography remains unchanged. If d≫ h, theevolution of the bed topography is forced by the initialsediment composition pattern, while the bed sediment compo-sition remains unchanged. A combined evolution of bedtopography and bed sediment composition is only possible if(about five times) d and h have the same order of magnitude.This analysis led to two conclusions. First, floods appear to be

more important for the overall morphological evolution thanthought previously, because they make the active layer suffi-ciently thick for changes in bed level to occur. Second, the resultssuggest that the active-layer thickness to be adopted in the modelmight be larger than just half the bedform height as found inexperiments in laboratory flumes with a constant discharge.Physical considerations support this suggestion, because theactive layer in the model corresponds to the layer of sedimentthat is reworked by erosion and deposition within a singlemorphological time step. This reworking results not only frommigrating dunes, but also from river bed variations due tovariations in discharge, such as the variation of cross-sectionalbed tilting in river bends and the generation of erosion anddeposition waves at locations where water enters or leaves thefloodplains during floods. Some parameterization of the varia-tions would be needed for proper calculation of the active-layerthickness. Improvements could be obtained from using a morecontinuous model for vertical grain sorting instead of the rathercrude subdivision in discrete layers (Parker et al., 2000; Blom,2003; Blom et al., 2003, 2003), but even then the effects fromthe higher-frequency variations, which are filtered out in themodel, might require a parameterization. For the time being,however, the active layer thickness remains a mere calibrationparameter. We address this limitation in the discussion section.Closer scrutiny of Ribberink’s analysis suggests that interactions

between bed topography and bed sediment composition evenincrease the difference between the two celerities and, hence,increase the differences between the time scales for bed levelevolution and bed sediment composition evolution. This can beinferred from Ribberink’s general equation for the celerities, c, ofperturbations in bed level and bed sediment composition:

c � cbedð Þ c � cmixð Þ � xc ¼ 0 (26)

where the parameter for the interactions, x, is given by


x ¼ DmT �Dm

1� eð Þd@qS@Dm

(27)

Usually x≥0 because DmT≤Dm and @ qs /@Dm< 0. The plotof Equation (24) in Figure 9 shows that the difference betweenthe two celerities increases as x increases with respect to thereference situation without interactions (x=0, i.e. DmT=Dm).This underscores the importance of a proper representation ofactive-layer thickness.

State-of-the-art results

The current state of the art is presented by the results inFigures 10 and 11. We obtained these results by carrying outcomputations using Delft3D on a curvilinear grid with averagecell dimensions of 60�20m2. The hydraulic roughness wascalculated by an alluvial roughness predictor based on asimplification of Van Rijn (1984). Daily discharge measure-ments at Lobith since 1900 were schematized into a steppedannual hydrograph containing eight discrete discharge levels(Yossef et al., 2008). The upstream inflow of sediment was

F(N


Figure 11. Computed bed sediment composition after 3 yearssimulation time.


specified for each size fraction by imposing a rate of bed levelchange and a composition of the active layer, withoutaccounting for changes due to discharge variations. We keptthe upstream sediment inflow conditions close to measuredvalues. Flume experiments show these conditions to have astrong effect on downstream mixture dynamics (Parker andWilcock, 1993; Kleinhans, 2005). Downstream water levelswere calculated with discharge rating curves derived from a1D SOBEK model of the Rhine branches, calibrated at hydro-metric stations. The initial bed topography was derived frommulti-beam echo-sounder measurements in 1999 and 2002.We selected a Meyer-Peter and Müller type bedload formulawith Acal=5, a ripple factor fixed at a constant value of 0.7and a critical Shields parameter value set at 0.025. The porosityof the bed was taken equal to a constant value of 0.4. Thehelical flow intensity was computed by calculating A accordingto Equation (4), whereas the effect of gravity pull along inclinedbeds was calibrated by setting Ash=1.5, Bsh=0.5, Csh=0 andDsh=0. We schematized the vertical structure of the bed byan active layer on top of an infinitely deep substratum. Theactive layer was taken to be 1.0m thick, which is ten timesthicker than half the dune height from measurements analyzedby Wilbers and Ten Brinke (2003). We divided the sedimentmixture into the ten size fractions presented in Table II. Thesediment composition of the bed, including several metres ofthe subsoil, was derived from 102 vibro corings combined withseismic surveys by Gruijters et al. (2001). We constructed aninitial bed sediment composition for the computations byaveraging the mean grain sizes over width, by assigninguniform distributions of the resulting grain sizes to each cross-

Table II. Division of sediment mixture into ten separate size fractions

Grain sizes (mm)

Fraction number Lower limit Upper limit Geometric mean (Di)

1 0.063 0.25 0.1252 0.25 0.50 0.3543 0.50 1.00 0.7074 1.00 2.00 1.4145 2.00 2.80 2.3666 2.80 4.00 3.3477 4.00 8.00 5.6578 8.00 16.00 11.3149 16.00 32.00 22.62710 32.00 64.00 45.255


section and by smoothing the resulting longitudinal distributionof grain sizes along the branches. This schematized initial bedsediment composition showed long-stream variations with amarked segregation between sediments in the Waal and thePannerden Kanaal, but no 2D patterns within the branches inorder to eliminate the risk that the results would merely repro-duce conditions imposed initially. The percentiles of theschematized grain size distributions were converted intorelative occurrences of the ten grain size fractions by assumingthat the grain size distributions were log-normal. The computationsimulated the development during a period of 3 years.

The bend morphology in Figure 10 and the bend sorting inFigure 11 highlight the importance of upstream approachconditions for the distribution of sediment at the bifurcation.This enhances the earlier conclusion about this importanceby Kleinhans et al. (2008).

Discussion

The experiences from the Rhine bifurcation at Pannerden raisefundamental questions about the calibration and verification ofnumerical morphological models in general. Calibrationinvolves a large number of parameters. The hydrodynamicsare calibrated first by adjusting hydraulic roughness values orby adjusting the parameters of an alluvial roughness predictor.The longitudinal bed level profiles of the branches arecalibrated next by adjusting the parameters of the sedimenttransport predictor and the active-layer thickness. This stepmay also involve some adjustment of sediment grain diameterswithin the confidence interval of scattered grain size measure-ments. Finally, the patterns of bars, pools and grain sorting arecalibrated by adjusting the four parameters of the empiricalrelation for gravity pull along inclined beds on sedimenttransport direction. The coefficient for the influence of helicalflow on the direction of bed shear stress has little effect on thepatterns, but can be used for a final adjustment of the amplitudeof spatial variations in bed level. Some iterations may benecessary to optimize the different calibration steps.

The large number of calibration parameters creates aproblem of equifinality in the sense that several combinationsof parameter values may yield similar results. Calibrating onas many different morphological phenomena as possiblereduces this problem (Mosselman et al., 2008), but by nomeans solves it completely. This means that applications ofthe model to new situations may require reconsideration ofprevious calibrations. Morphological modellers should alwayscheck whether selected parameter values are still appropriate,and should be prepared to abandon earlier calibrations if newinsights require so. In principle this is true for all models,including hydrodynamic ones, but we believe this to holdstrongly for morphological models in particular. Morphologicalmodellers hence need knowledge on physical processes aswell as knowledge on model concepts and numericalimplementation. This wide range of required expertise impliesthat 2D and 3D morphological modelling is often a matter ofteamwork, with experts in different categories of knowledge.Van Zuylen et al. (1994) distinguish in this context (i) domainknowledge based on common sense and expertise from experi-ence with real rivers, (ii) knowledge about model conceptssuch as the underlying mathematical equations, (iii) knowledgeabout model constructs such as grids, time steps and initial andboundary conditions, and (iv) knowledge about model artefactssuch as user interfaces and file formats.

Verification is problematic too because the field data do notprovide sufficient information to assess the actual spatialdistributions in the river. Sediment sample sizes are too small



to derive representative local sediment properties. Bedtopography measurements with a multi-beam echo-sounderdo provide a synoptic view of the dunes on the river bed, butthese dunes are not the smaller bedforms responsible forhydraulic resistance. Improved measurement techniques areneeded for in situ measurement of grain sizes (cf. Eleftherakiset al., 2010) and high-resolution measurement of bed topogra-phy. This uncertain verification illustrates the more generalconclusion by Oreskes et al. (1994) that absolute verificationand validation of numerical models in the earth sciencesare impossible.We used 2D hydrodynamic equations despite the availability

of quasi-3D equations in the software. Using 3D equationsmight have seemed better, because they do not require anyhelical flow parameterizations that perform poorly close to banksand in complex geometries. However, the near-bed flowdirection in 3D models depends on the height of near-bedcomputational cells and converges only to the true direction ifthe vertical grid spacing near the bed is sufficiently small. Thus3D models do not calculate the essential features of 3D flowcorrectly if near-bed computational cells are too high. In manycases, parameterizations of helical flow work well in large areasaway from the banks. Whether or not a 3D model is better thana 2D model hence cannot be stated in general and needs to beevaluated on a case-to-case basis. For the Rhine bifurcation atPannerden we prefer the use of Delft3D in 2D mode, becausethis speeds up computations of long-term evolution and becausewe expect only minor inprovements from a 3Dmodel in the lightof the large uncertainties in active-layer thickness and effect ofgravity pull along inclined beds.Do our sobering observations on the problems in calibration

and verification render numerical morphological modelsuseless? We do not think so, because these models have provento be useful tools, not only in pragmatic applications toengineering problems, but also in revealing the limitations ofestablished knowledge and understanding of the relevantphysical processes. Modelling thus drives the setting of aresearch agenda, and this has been particularly true for thebifurcation modelling in the meandering gravel–sand bed partof the Rhine at Pannerden, which stretches the applicabilityof existing model concepts to the limit.

Conclusions

We have presented decades of experiences from morphologicalmodelling of the Rhine bifurcation at Pannerden, which ischaracterized by a gravel–sand bed and a meandering planform.Successive computations have shown the importance ofupstream approach conditions, the necessity to include physicalmechanisms for grain sorting and alluvial roughness, and theneed to assume a thicker active layer of the river bed than issuggested by laboratory flume experiments using a constantdischarge. The thicker active layer in themodel accounts for riverbed variations due to higher-frequency discharge variations thatare filtered out in morphological modelling. It actually calls foran active-layer thickness predictor based on some parameteriza-tion accounting for bed level variations due to discharge varia-tions, possibly improved by a continuous model for verticalgrain sorting. A major knowledge gap for horizontal grain sortingremains the formulation for the effect of gravity pull alongtransverse bed slopes on the transport directions of different grainsizes in a sediment mixture.We have used the experiences to discuss more generally the

problems of calibration and verification in 2D and 3D numericalmorphological modelling. Calibration is uncertain due to thelarge number of calibration parameters, because several


combinations of these parameters may yield similar results. Thisproblem of underdetermination may be diminished but cannotbe eliminated by calibrating a model on as many differentmorphological phenomena as possible. Verification is uncertaindue to insufficient detail in field data. We argue that, notwith-standing these limitations, 2D and 3D morphological modelsdo represent valuable tools, not only in pragmatic applicationsto engineering problems, but also in revealing the limitations ofestablished knowledge and understanding of the relevantphysical processes. We find the main gaps in our knowledge tobe related to vertical sorting and the effect of gravity pull alonginclined beds. This insight from numerical modelling sets theagenda for further research. The application to the Pannerdenbifurcation appears to be particularly useful for setting aresearch agenda, because it reveals shortcomings in establishedmodel formulations that do not pose particular problems inother cases.

Acknowledgements—Themodelling has been funded by Rijkswaterstaat,which is the executive body of the Dutch Ministry of Transport, PublicWorks and Water Management. It was part of a wider national researchprogramme on the Rhine bifurcation at Pannerden, carried out by themembers of the Morphological Triangle, a thematic subdivision of theNetherlands Centre for River studies (NCR). The field data were collectedand made available by Rijkswaterstaat and TNO-NITG (GeologicalSurvey). We express our gratitude to Arjan Sieben (Rijkswaterstaat) forhis continuous support and our fruitful discussions. We thank MicheleBernabè (University of Trento, Italy), Tilmann Baur (Delft Hydraulics),Patrick Verhaar (Delft University of Technology) and Alfons Smale (DelftUniversity of Technology) for their contributions to the numerical compu-tations. Finally, we thank Maarten Kleinhans for encouraging us tosummarize the work described in reports and conference papers, and topresent the lessons we learnt to a wider audience.

LIST OF SYMBOLS

A
coefficient for influence of helical flow on direction ofbed shear stress (�)
Acal
calibration factor in sediment transport formula (�) Ash calibration factor in function for influence of gravity
pull along inclined beds on sediment transport direction (�)
B river width (m) Bsh exponent of Shields parameter in function for influence of
gravity pull along inclined beds on sediment transportdirection (�)

b
degree of nonlinearity in qs(u): b ¼ uqs
@qs@u (�)

C
Chézy coefficient for total hydraulic roughness (m1/2/s) Csh exponent of ratio Di /h in the function for influence of
gravity pull along inclined beds on sediment transportdirection (�)

C90
Chézy coefficient for grain roughness (m1/2/s) c celerity of bed level and bed sediment composition
perturbations (m/s)
cbed reference celerity of bed level perturbations (m/s) cmix reference celerity of bed sediment composition
perturbations (m/s)
D sediment grain size (m) DH horizontal eddy diffusivity (m2/s) Di grain size of sediment fraction i (m) Dm average grain size of bed material (m) DmT average grain size of sediment in transport (m) Dsh exponent of ratio Dm /Di in the function for influence
of gravity pull along inclined beds on sedimenttransport direction (�)

D90
ninetieth percentile of sediment grain size distribution (m)ffiffiffiffiffip Fr Froude number: Fr ¼ u= gh (�) f function for influence of gravity pull along inclined
beds on sediment transport direction (�)



fi

Copyri

function for influence of gravity pull along inclinedbeds on transport direction of sediment grain sizefraction i (�)

g
acceleration due to gravity (m/s2) h flow depth (m) I actual helical flow intensity (m/s) Ie equilibrium helical flow intensity (m/s) i sediment fraction number (�) La adaptation length of secondary flow intensity (m) LD damping length of nonmigrating alternate bars (m) LP wave length of nonmigrating alternate bars (m) pi relative occurrence of sediment size fraction (�) pi,a relative occurrence of sediment size fraction in active
layer (�)
pi,0 relative occurrence of sediment size fraction in substratum (�) qs volumetric sediment transport per unit width, excluding
pores (m2/s)
qsi volumetric sediment transport per unit width for
sediment fraction i, excluding pores (m2/s)
qsx volumetric sediment transport per unit width in x
direction, excluding pores (m2/s)
qsxi volumetric sediment transport per unit width in x
direction for sediment fraction i, excluding pores (m2/s)
qsy volumetric sediment transport per unit width in
y direction, excluding pores (m2/s)
qsyi volumetric sediment transport per unit width in y
direction for sediment fraction i, excluding pores (m2/s)
R radius of curvature of depth-averaged streamline (m) Tbed characteristic time scale for bed level evolution (s) Tmix characteristic time scale for bed sediment composition
evolution (s)
t time (s) u depth-averaged flow velocity in x direction (m/s) v depth-averaged flow velocity in y direction (m/s) x longitudinal horizontal co-ordinate (m) y transverse horizontal co-ordinate (m) zb bed level (m+datum) z0 upper level of substratum (m+datum) a degree of nonlinearity in D(u): a ¼ u
D@D@u (�)

asi
angle between transport direction for sediment fractioni and depth-averaged flow direction (�)
at
angle between bed shear stress and depth-averagedflow direction (�)
Δ
relative mass density of sediment under water:Δ= (rs�rw)/rw (�)
d
thickness of active layer (m) e porosity (�) k Von Kármán constant (�) ls longitudinal adaptation length for transverse profile
of longitudinal depth-averaged flow velocity (m)
lw longitudinal adaptation length for transverse profile of bed
level (m)
x parameter representing the interactions between bed
topography and bed sediment composition (�)
xi hiding-and-exposure correction (�) rs mass density of sediment (kg/m3) rw mass density of water (kg/m3) tb bed shear stress (Pa) ΨD degree of nonlinearity in qs(D): ΨD ¼ D
qs@qs@D (�)

ReferencesAllen JRL. 1978. Van Bendegom: a neglected innovator in meanderstudies. In Fluvial Sedimentology, Miall AD (ed). Canadian Societyof Petroleum Geologists Memoir 5, University of Calgary, Calgary,Alberta, Canada, August 1978: 199–209.

ght © 2012 John Wiley & Sons, Ltd.

Ashida K, Michiue M. 1972. Study on hydraulic resistance and bedloadtransport rate in alluvial streams. Transactions JSCE 206: 59–69.

Ashida K, Michiue M. 1973. Studies on bed-load transport in openchannel flows. Proceedings of the International Symposium of RiverMechanics, IAHR, Bangkok, Thailand. A36: 407–418.

Blom A. 2003. A vertical sorting model for rivers with non-uniformsediment and dunes. PhD thesis University of Twente, UniversityPress, Veenendaal, the Netherlands, ISBN 90-9016663-7.

Blom A, Ribberink JS, de Vriend HJ. 2003a. Vertical sorting in bedforms. Flume experiments with a natural and a tri-modal sedimentmixture. Water Resources Research AGU 39(2): 1025.

Blom A, Ribberink JS, Parker G. 2003b. Sediment continuity for riverswith non-uniform sediment, dunes, and bed load transport. In Sedi-mentation and Sediment Transport: at the Crossroad of Physics andEngineering, Gyr A, KinzelbachW (eds). Kluwer Academic Publishers:Switzerland; 179–182.

De Vries M. 1959. Transients in bedload-transport (basic considerations).Report R3, Delft Hydraulics Laboratory.

De Vries M. 1965. Considerations about non-steady bed-load transportin open channels. Proceedings of the 11th Congress IAHR, Leningrad(also Delft Hydraulics Laboratory Publication No. 36).

Egiazaroff IV. 1965. Calculation of non-uniform sediment concentra-tions. Journal of Hydraulic Division ASCE 91(HY4): 225–247.

Eleftherakis D, Mosselman E, Amiri-Simkooei A, Giri S, Snellen M,Simons DG. 2010. Identifying changes in river bed morphologyand bed sediment composition using multi-beam echo-sounder mea-surements. Proceedings of the 10th European Conference on Under-water Acoustics, Istanbul, Turkey, July 5–9, 2010, Volume 2,pp.1365–1373.

Flokstra C. 1985. De invloed van knooppuntsrelaties op debodemligging bij splitsingspunten (The influence of nodal pointrelations on bed topography at bifurcation points). Rapport R2166,Delft Hydraulics.

Frings RM, Kleinhans MG. 2008. Complex variations in sediment trans-port at three large river bifurcations during discharge waves in theriver Rhine. Sedimentology 55: 1145–1171.

Gruijters SHLL, Veldkamp JG, Gunnink J, Bosch JHA. 2001. De litholo-gische en sedimentologische opbouw van de ondergrond van dePannerdensche Kop (The lithological and sedimentological composi-tion of the subsoil of the Pannerdensche Kop). Eindrapport, TNO-NITG 01-166-B.

Havinga H, Taal M, Smedes R, Klaassen GJ, Douben N, Sloff CJ. 2006.Recent training of the lower Rhine River to increase Inland WaterTransport potentials: a mix of permanent and recurrent measures. InProceedings of the River Flow 2006, Vol.1, Lisbon, 6–8 September,Ferreira RML, Alves ECTL, Leal JGAB, Cardoso AH (eds). Taylorand Francis: London; 31–50.

Hirano M. 1972. Studies on variation and equilibrium state of a riverbed composed of non-uniform material. Transactions Japanese Societyof Civil Engineers 4: 128–129.

Jansen PPh, van Bendegom L, van den Berg J, de Vries M, Zanen A.1979. Principles of River Engineering; the Non-Tidal Alluvial River.Pitman: London.

Kleinhans MG. 2001. The key role of fluvial dunes in transport anddeposition of sand-gravel mixtures, a preliminary note. SedimentaryGeology 143: 7–13.

Kleinhans MG. 2005. Upstream sediment input effects on experimentaldune through scour in sediment mixtures. Journal of GeophysicalResearch 110: F04S06. doi:10.1029/2004JF000169.

Kleinhans MG, Wilbers AWE, Ten BrinkeWBM. 2007. Opposite hyster-esis of sand and gravel transport upstream and downstream of a bifur-cation during a flood in the river Rhine, the Netherlands.NetherlandsJournal of Geosciences – Geologie en Mijnbouw 86(3): 273–285.

Kleinhans MG, Jagers HRA, Mosselman E, Sloff CJ. 2008. Bifurcationdynamics and avulsion duration in meandering rivers by one-dimen-sional and three-dimensional models. Water Resources ResearchAGU 44: W08454, doi:10.1029/2007WR005912.

Koch FG, FlokstraC. 1981. Bed level computations for curved alluvial chan-nels. Proceedings of the 19th Congress IAHR,NewDelhi, India, February1981. Also Delft Hydraulics Publication No. 240, November 1980.

Lesser GR, Roelvink JA, van Kester JATM, Stelling GS. 2004. Develop-ment and validation of a three-dimensional morphological model.Coastal Engineering 51(8–9): 883–915.



Meyer-Peter E, Müller R. 1948. Formulas for bed-load transport.Proceedings of the 2nd Congress IAHR, Stockholm, Paper No. 2,pp. 39–64.

Mosselman E. 2010. Three-dimensional (3D) modeling of non-uniformsediment transport in a channel bend with unsteady flow (discussion).Journal of Hydraulic Research 48(6): 824.

Mosselman E, Sloff CJ. 2007. The importance of floods for bed topogra-phy and bed sediment composition: numerical modelling of Rhinebifurcation at Pannerden. In Gravel Bed Rivers VI – From ProcessUnderstanding to River Restoration, Habersack H, Piégay H, Rinaldi M(eds). Developments in Earth Surface Processes 11, Elsevier:Amsterdam; 161–180, DOI: 10.1016/S0928–2025(07)11124–X.

Mosselman E, Sieben A, Sloff K, Wolters A. 1999. Effect of spatial grainsize variations on two-dimensional river bed morphology. Proceedingsof the IAHR Symposium, River, Coastal and Estuarine Morphody-namics, Genova, 6–10 Sept. 1999, Vol. I, pp. 499–507.

Mosselman E, Sloff K, van Vuren S. 2008. Different sediment mixtures atconstant flow conditions can produce the same celerity of beddisturbances. In River Flow 2008, Proceedings of the InternationalConference on Fluvial Hydraulics, Çeşme, Izmir, Turkey, Sept. 3–5,2008, Altinakar M, Kokpinar MA, Gogus M, Tayfur G, Kumcu Y,Yildirim N (eds). Kubaba Kavaklidere: Ankara; 1373–1377.

Nelson JM, Smith JD. 1989. Evolution and stability of erodible channelbeds. In River Meandering, Ikeda S, Parker G (eds). AGU, WaterResources Monograph 12: 321–377.

Olesen KW. 1987. Bed topography in shallow river bends. Communi-cations on Hydraulic and Geotechnical Engineering No.87-1, DelftUniversity of Technology ISSN 0169–6548.

Oreskes N, Shrader-Frechette K, Belitz K. 1994. Verification, validationand confirmation of numerical models in the earth sciences. Science263: 641–646.

Parker G, Andrews ED. 1985. Sorting of bed load sediments by flow inmeander bends. Water Resources Research AGU 21(9): 1361–1373.

Parker G, Wilcock PR. 1993. Sediment feed and recirculating flumes:fundamental difference. Journal of Hydraulic Engineering ASCE 119(11): 1192–1204.

Parker G, Paola C, Leclair S. 2000. Probabilistic Exner sedimentcontinuity equations for mixtures with no active layer. Journal ofHydraulic Engineering ASCE 126(11): 818–826.

Ribberink JS. 1987. Mathematical modelling of one-dimensionalmorphological changes in rivers with non-uniform sediment.Communications on Hydraulic and Geotechnical EngineeringNo.87-2, Delft University of Technology, ISSN 0169–6548.

Shimizu Y, Itakura T. 1985. Practical computation of two-dimensional flowand bed deformation in alluvial streams. Civil Engineering ResearchInstitute Report, Hokkaido Development Bureau, Sapporo, Japan.

Shimizu Y, Itakura T. 1989. Calculation of bed variation in alluvialchannels. Journal of Hydraulic Engineering ASCE 115(3): 367–384.

Sloff CJ, Jagers HRA, Kitamura Y, Kitamura P. 2001. 2D Morphody-namic modelling with graded sediment. Proceedings of the 2nd IAHRSymposium, River, Coastal and Estuarine Morphodynamics, 10–14Sept. 2001, Obihiro, Japan, pp. 535–544.

Sloff CJ, Bernabè M, Baur T. 2003. On the stability of the PannerdenseKop river bifurcation. Proceedings of IAHR Symposium River, Coastal


and Estuarine Morphodynamics, Barcelona, 1–5 Sept. 2003,Sánchez-Arcilla A, Bateman A (eds). IAHR: Madrid; 1001–1011,ISBN 90-805649-6-6.

Staatscommissie Zuiderzee. 1926. Verslag van de Staatscommissiebenoemd bij Koninklijk Besluit van 4 Juli 1918 No.30 met opdrachtte onderzoeken in hoeverre, als gevolg van de afsluiting van deZuiderzee, ingevolge de wet van 14 Juni 1918 (Staatsblad No.354),te verwachten is, dat tijdens storm hoogere waterstanden en eengrootere golfoploop, dan thans het geval is, zullen voorkomen vóórde kust van het vaste land van Noord-Holland, Friesland en Groningen,alsmede vóór de daarvoor gelegen Noordzee-eilanden. AlgemeneLandsdrukkerij, The Hague: The Netherlands.

Struiksma N. 1985. Prediction of 2-D bed topography in rivers. Journalof Hydraulic Engineering ASCE 111(8): 1169–1182.

Struiksma N. 1988. RIVCOM: a summary of results of some test compu-tations. Report Q794, Delft Hydraulics, December 1988.

Struiksma N. 1998. Morfologisch model Pannerdense Kop op basis vanDELFT2DMOR; IJking (Morphological model of Pannerdense Kop onthe basis of DELFT2DMOR; Calibration). Draft report Q2403, WL |Delft Hydraulics, June 1998.

Struiksma N, Olesen KW, Flokstra C, de Vriend HJ. 1985. Bed deforma-tion in curved alluvial channels. Journal of Hydraulic Research IAHR23(1): 57–79.

Talmon AM, Struiksma N, vanMierloMCLM. 1995. Laboratory measure-ments of the direction of sediment transport on transverse alluvial-bedslopes. Journal of Hydraulic Research IAHR 33(4): 495–517.

Van Bendegom L. 1947. Some considerations on river morphology andriver improvement. De Ingenieur, Vol.59, No.4 (in Dutch; Englishtransl.: Natl. Res. Council Canada, Tech. Translation 1054, 1963).

Van de Ven GP. 1976. Aan de wieg van Rijkswaterstaat; Wordings-geschiedenis van het Pannerdens Kanaal. Zutphen.

Van der Zwaard JJ. 1981. Splitsingspunt Pannerdense Kop; Verdelingvan het sedimenttransport bij het splitsingspunt (Pannerdense Kopbifurcation; Distribution of sediment transport at the bifurcation).Verslag modelonderzoek M932, Delft Hydraulics, July 1981.

Van Rijn LC. 1984. Sediment transport, Part III: bed forms andalluvial roughness. Journal of Hydraulic Engineering ASCE 110(12): 1733–1754.

Van Zuylen HJ, van Dee DP, Mynett AE, Rodenhuis GS, Moll JR, OginkHJM, van der Most H, Gerritsen H, Verboom GK. 1994. Hydroinfor-matics at Delft Hydraulics. Journal of Hydraulic Research IAHR32: Extra Issue Hydroinformatics 83–136.

Wang ZB, Fokkink RJ, de Vries M, Langerak A. 1995. Stability of riverbifurcations in 1D morphodynamic models. Journal of HydraulicResearch IAHR 33(6): 739–750.

Wilbers AWE, Ten Brinke WBM. 2003. The response of subaqueousdunes to floods in sand and gravel bed reaches of the Dutch Rhine.Sedimentology 50: 1013–1034.

Yossef MFM, Jagers HRA, van Vuren S, Sieben A. 2008. Innovativetechniques in modelling large-scale river morphology. In River Flow2008, Proceedings of the International Conference on FluvialHydraulics, Çeşme, Izmir, Turkey, Sept. 3–5, 2008, Altinakar M,Kokpinar AM, Gogus M, Tayfur G, Kumcu Y, Yildirim N (eds). Kubaba,Kavaklidere: Ankara, 1065–1074. ISBN 978-605-60136-2-1.


bifurcation modelling in a meandering gravel–sand bed river

Documents