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

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Ralph

M.J.

Schielen

a,b,∗

_,

_Astrid

_Blom

c

a Faculty of Engineering Technology, University of Twente, Netherlands b Ministry of Infrastructure and Water Management-Rijkswaterstaat, Netherlands c Faculty of Civil Engineering and Geosciences, Delft University of Technology, Netherlands

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Keywords: River bifurcation Mixed-size sediment Idealized model Equilibrium Stability analysis

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We derive an idealized model of a gravel-sand river bifurcation and analyze its stability properties. The model requires nodal point relations that describe the ratio of the supply of gravel and sand to the two downstream branches. The model predicts changes in bed elevation and bed surface gravel content in the two bifurcates under conditions of a constant water discharge, sediment supply, base level, and channel width and under the assumption of a branch-averaged approach of the bifurcates. The stability analysis reveals more complex behavior 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

Riverbifurcationsordiffluencesarefoundinalluvialfans,braided rivers,anabranchingrivers,deltas,cut-off channels,diversions(forflood controlorwaterintakes),andinconstructedsidechannelsthatarepart ofriverrestorationschemes.Onceabifurcationis initiated,a down-streamchannel(orbifurcateordistributary)continuestodeepenaslong asthesedimenttransportcapacityexceedsthesedimentsupplytothe channel.

Sedimenttransportinachannelconsistsofbed-materialload(i.e., bed loadandsuspended bed-material load)and washload (Church, 2006;Paola,2001).Aswashloadistypicallyassumedtobedistributed uniformlyoverthewatercolumn,itisassumedtobepartitionedover thebifurcatesaccordingtotheratioofthewaterdischarge.Bed-material load,however,partitionsoverthebifurcatesinalessstraightforward manner.The partitioning of sedimentin streams dominatedby sus-pendedbed-materialloaddependsontheinitialﬂowdepthandchannel slopeinthebifurcates(SlingerlandandSmith,1998),thegrainsizeof thebedsediment(SlingerlandandSmith,1998),andcurvature-induced eﬀectsintheupstreamchannel(Hackneyetal.,2017).Thepartitioning ofsedimentinbedloaddominatedstreamsdependson:

• theconditionsinthebifurcates:baselevel,channelwidth,friction, bifurcationangle(Bulle,1926;VanderMarkandMosselman,2013;

∗_{Corresponding}_author.

E-mailaddress:r.m.j.schielen@utwente.nl(R.M.J. Schielen).

TarekulIslametal.,2006),andthezonesofﬂowrecirculationclose tothebifurcation(Bulle,1926;DeHeerandMosselman,2004;Marra etal.,2014; Thomasetal., 2011),vegetation(Burge,2006),and cohesivesedimentandbankerosion(Miorietal.,2006;Zolezzietal., 2006);

• theconditionsintheareajustupstreamofthebifurcation:the trans-versedistributionofwaterandsedimentovertheupstream chan-nel,whichisaffectedbysecondaryflow(VanderMarkand Mos-selman, 2013), a transverse bed slope induced by an inlet step (BollaPittalugaetal.,2003),alternatebars(BertoldiandTubino, 2007;Bertoldietal.,2009;Redolfi etal.,2016),andsediment mo-bility(FringsandKleinhans,2008);

• conditionsextendingfurtherupstream:flowasymmetryinducedby abend,whichtendstoprovideonebifurcatewithalargerfraction contentoftheflowandtheotheronewithalargerfractioncontent ofthesedimentload(FedericiandPaola,2003;Hardyetal.,2011; Kleinhansetal.,2008;VanDijketal.,2014)andtransversesediment sortingduetobendflow(FringsandKleinhans,2008;Sloff etal., 2003;Sloff andMosselman,2012).

Thepartitioningofthesedimentloadoverthebifurcatesdetermines whetherthebifurcationdevelopstowardsastablestatewithtwoopen downstreambranchesorastateinwhichthewaterdischargeinoneof thebranchescontinuestoincreaseattheexpenseoftheotherbranch. Thelatter casemayleadtothesiltingupof oneof thedownstream

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

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channels.Undersuchconditionsaonechannelconﬁgurationisastable equilibriumsolutionofthebifurcationsystem(Wangetal.,1995),yetin literaturethissituationisoftentermedan‘unstablebifurcation’(Burge, 2006;FedericiandPaola,2003),asthetwochannelsystemceasesto exist.

Earlyone-dimensionalreducedcomplexitymodelsdescribingthe de-velopmenttowardstheequilibriumstatesoftwobifurcateshavebeen developedforbedloadtransportinsand-bedrivers(Wangetal.,1995), bedloadtransportingravel-bedrivers(BollaPittalugaetal.,2003),and suspendedbed-materialload(SlingerlandandSmith, 1998).Such re-ducedcomplexitymodelsallowforthecomputationofthepartitioning ofthewaterdischargeasthewatersurfaceelevationatthebifurcation pointmustbeequalbetweenthethreereaches.Thesediment partition-ing,however,dependsonthegeometryofthebifurcationandthe three-dimensionalﬂowstructure,whichobviouslycannotbereproducedbya one-dimensionalmodel.Aone-dimensionalmodelthereforerequiresa nodalpointrelationthatdescribesthepartitioningofthesedimentload overthebifurcates.

Wangetal.(1995)weretheﬁrsttointroduceanodalpointrelation describingthepartitioningofthesedimentsuppliedfromupstreamover thebifurcates.Theythenapplyasimplerformoftheirnodalpoint rela-tion(̄𝑠1∕̄𝑠2=(𝑞1∕𝑞2)𝑘,wherē𝑠1,2istherateofsedimentsupplyperunit

widthtobranches1and2andq1,2isthewatersupplyperunitwidthto

branches1and2)toanalysethestabilityofthesolutionstothe equilib-riummorphodynamicstateofthebifurcates.Studyingavulsion devel-opment(SlingerlandandSmith,1998)introduceanodalpointrelation thatoriginatesfromintegrationoftheverticalconcentrationproﬁleof thesuspendedsediment.Alternative nodalpointrelationshave been developedbyBollaPittalugaetal.(2003),whoaccountfortheeﬀects ofatransversebedslopethatinduceslateralsedimenttransporttothe deeperbifurcate,andKleinhansetal.(2008),whoaccountforthe ef-fectsofanupstreambend,bothofwhichwillbeaddressedinfurther detailbelow.

Pioneeringworkonbifurcationdynamicsusinganodalpoint rela-tionwasconductedbyWangetal.(1995):theyassumeaconstantwater dischargeandsedimentsupplyrateintheupstreamchannel,aconstant andequalbaselevelinthetwobifurcatingbranches,andunisize sedi-mentconditions.TheyapplytheEngelundandHansen(1967)sediment transportrelationwithoutathresholdforsigniﬁcanttransport:s∝Un_,

wheresdenotesthesedimenttransportcapacityperunitwidth,Uthe depth-averagedﬂowvelocity,andnistheexponentinthepowerlaw loadrelation(𝑛=5).Theyﬁndthatfork<n/3theequilibriumsolution whereoneofthebifurcatesclosesisstable,whereasfork>n/3the equi-libriumsolutionwithtwoopenbranchesisstable.Despitetheseearly re-sultsamodelforkisstilllacking.AlsoSlingerlandandSmith(1998) re-vealthatabifurcationoravulsiondevelopstowardsastablestatewith twoopendownstreambranchesorastateinwhichonechannelbecomes thedominantchannelattheexpenseoftheotherbranch.

In such strongly idealized one-dimensional analyses, two-dimensional and three-dimensional effects near the bifurcation pointarenot readilyaccountedfor.One ofthese effectsistheBulle effect(Bulle,1926;Duttaetal.,2017;VanderMarkandMosselman, 2013), which indicates a situation where the sediment supply to a diversionchannel(i.e.,achannel thatbranchesoff themainchannel underacertainangle)is significantlylargerthanthediversion chan-nel’sfractioncontentofthewaterdischarge.Thiseffectisassociated withsecondaryflow(e.g.,Thomasetal.,2011).Anothereffectisthe differenceinbedelevationthatisassociatedwithadifferenceinflow depth between thetwo bifurcates(e.g., BollaPittalugaet al., 2003; Kleinhans et al., 2013).This bed elevationdifference (also denoted usingtheterminletstep)tendstoincreasethesedimentsupplytothe deeperbifurcate (e.g.,SlingerlandandSmith,1998),whichactsasa stabilizingmechanism.

Althoughmixed-sizesedimentsystemsmayrevealbehaviorthatis essentiallydiﬀerentfromunisizesedimentsystems(Blometal.,2017a; 2017b;2016;MosselmanandSloﬀ,2008;SinhaandParker,1996),so

far theinfluence of noncohesive mixed-size sedimenton bifurcation dynamicshasnotbeenstudiedexplicitly.Washload,suspended bed-materialload,andbedload(Church,2006;Paola,2001)areexpectedto responddifferentlytotheabove-mentionedmechanisms(Hackneyetal., 2017).Mixed-sizesedimenteffectsarethefollowing:

1. Astheverticalproﬁleofsedimentconcentrationislessuniformover depthfor coarsesediment(i.e., coarsesedimenttends to concen-tratemorestronglynearthebed),coarsesedimenttendstobe af-fectedmore byaninletstep thanﬁnesediment(Slingerland and Smith,1998).

2. Theeffectofthetransversebedslopeonlateraltransportupstream of the bifurcation depends on grain size, wherecoarse sediment is affected by the transversebed slope more strongly thanfines (ParkerandAndrews,1985);

3. Thepresenceofabendupstreamofthebifurcationtypicallyleadsto bendsortingandacoarsersedimentsupplytothedistributaryinthe outerbendthantotheoneintheinnerbend(FringsandKleinhans, 2008;Sloﬀ etal.,2003;Sloﬀ andMosselman,2012);

4. Alternatebarformationandgeometryappeartobeaﬀectedbythe grainsizedistributionofthesedimentmixture(BertoldiandTubino, 2005;Lanzoni,2000).

Ourobjectiveistoassesstheelementaryconsequencesofthe intro-ductionofmixed-sizesedimentmechanismsinthemodellingofthe dy-namicsofariverbifurcation.TothisendwefollowWangetal.(1995)’s approachanditssimplenodalpointrelationwithassociatedlimitations andsimplifications:weneglecttheeffectsofvegetation,cohesive sedi-ment,bankerosion,alternatebarsorabendintheupstreamchannel, aswellastheBulleeffectandthetransverseslopeeffect.Weextend theirmodeltoconditionswithbed-materialloadofatwo-fraction sed-imentmixtureconsistingofgravelandsand.Thisimpliestheneedfor twonodalpointrelationsdescribingtheratioof,respectively,thegravel andsandsupplytothetwobifurcates.Westudythestabilityofthe equi-libriumstatesofthebifurcatesinanengineeredrivercharacterizedby afixedchannelwidth.

Theproposedanalysisandmodelareapplicabletobothcasesshown inFig.1:abifurcationsystemwithtwobifurcatesthatarecharacterized bythesamebaselevelandasidechannelsystem.Wesetupamodel describingtheequilibriumsolutionsofthemixedsedimentbifurcation system(Section2),wedetermineitsequilibriumsolutions(Section3), wederiveasystemofordinarydiﬀerentialequationsfortheﬂowdepth andbedsurfacetextureinthebifurcates(Section4),andperforma sta-bilityanalysisoftheequilibriumsolutions(Section5).Theanalysisalso providesinsightonthetimescaleoftheevolutiontowardsthestable equilibriumsolutions(Section6).

2. Modeloftheequilibriumstate

Inthissection westronglysimplifythesituation ofagravel-sand riverbifurcation,describetheproblemfromamathematicalpointof view,andlistthegoverningequations.Tothisendweconsideran en-gineeredriverwithaﬁxedchannelwidththatmayvarybetweenthe branches,atemporallyconstantwaterdischargeintheupstreambranch (i.e.,branch0inFig.1)andatemporallyconstantgravelsupplyrate andconstantsandsupplyratetotheupstreambranch.

Underequilibriumconditions(𝜕∕𝜕𝑡=0)withoutsubsidence,uplift, andparticleabrasion,theequationdescribingconservationofsediment mass(i.e.theExnerequation)reducestothestationaryExnerequation, 𝜕𝑆𝑖∕𝜕𝑥=0,whereSidenotesthesedimenttransportcapacityinbranch

i,thesubscriptiindicatesbranchi,andxisthestreamwisecoordinate. Inotherwords,bydeﬁnitionthesedimenttransportcapacitySiequals

thesedimentsupplytobranchi, ̄𝑆_𝑖,wherethebarindicatesthesediment supply.

ForsimplicityweapplytheEngelundandHansenpowerlawload relation(EngelundandHansen,1967):

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Fig.1. Schematic of (a) a channel (branch 0) bifurcating into two channels (branches 1 and 2) ﬂowing into a lake characterized by the same base level ( Wangetal.,1995) and (b) a side channel system. Our analysis and model are applicable to both cases.

inwhich𝑚𝑖=𝐺𝑖∕𝐷 withDdenotingacharacteristicgrainsize,𝐺𝑖=

0.05∕(𝐶3

𝑖𝑅2𝑔1∕2), Ui the depth-averagedﬂowvelocity, Bi the channel

width,Ci theChézyfrictioncoeﬃcient,gdenotesthegravitational

ac-celeration,andRthesubmergeddensity(𝑅=(𝜌_𝑠−𝜌)∕𝜌 where𝜌_sand 𝜌 arethemassdensityof,respectively,sedimentandwater).For sim-plicityweassumethatmidoesnotvarybetweenthebranches(𝑚𝑖=𝑚), whichimpliesthatalsothefrictioncoeﬃcientandthecoeﬃcientGdo notvarybetweenthebranches(𝐶𝑖=𝐶,𝐺𝑖=𝐺).

CombinationofEq.(1)withthestationaryExnerequationillustrates thatunderequilibriumconditionswherethechannelwidthandfriction donotvaryspatially,besidesthesedimenttransportrate,alsotheﬂow velocityisuniform.

Theﬂowisdescribedusingtheone-dimensionalconservation equa-tionsforwatermassandstreamwisemomentum,i.e.theSaint-Venant equations(Saint-Venant,1871).Underequilibriumconditions,the con-servationequationforwatermassissimpliﬁedto𝜕𝑄𝑖∕𝜕𝑥=0(whereQi

denotesthewaterdischargeinbranchi,seeFig.1),whichimplies

𝑄𝑖=𝐵𝑖𝑈𝑖𝐻𝑖=const (2)

whereHidenotestheﬂowdepth(Fig.2).Astheﬂowvelocityisuniform

overthebranch,Eq.(2)impliesthatalsotheﬂowdepthdoesnotvary overthebranch.

Underequilibriumconditionstheconservationequationfor stream-wisemomentumoftheflowreducestothebackwaterequation.Fora uniformflowdepth,thebackwaterequationreducestothenormalflow equation: 𝐻𝑖= ( 𝑄2 𝑖 𝑖𝑖𝐶2𝐵2𝑖 )1∕3 (3)

Fig.2. Deﬁnition of symbols.

whereii denotesthechannelslope.Forsimplicity,theChézyfriction

coeﬃcientCisassumedindependentofthebedsurfacetextureandﬂow conditionsandhenceconstant.

Under mixed-size sediment conditions, the Exner equation is re-placedbytheequationsfortheconservationofgravelandsandmassat thebedsurface,i.e.theHiranoequations(Hirano,1971;Parker,1991; Ribberink,1987).UnderequilibriumconditionstheHiranoequations reduceto𝜕𝑆_𝑖𝑔∕𝜕𝑥=𝜕𝑆_𝑖𝑠∕𝜕𝑥=0,wherethesubscriptsgandsindicate gravelandsand,respectively,andSigandSisdenote,respectively,the

gravelandsandtransportcapacitiesinbranchi.This impliesthatin anequilibriumstatewithoutparticleabrasionthegravelandsandload donotvarywithinabranch(e.g.,Blometal.,2016).Forsimplicitywe applytheEngelundandHansenpowerlawloadrelationinafractional manner(Blometal.,2017a;2016)andreplaceEq.(1)by

𝑆𝑖=𝑆𝑖𝑔+𝑆𝑖𝑠 (4)

𝑆𝑖𝑔=𝐹𝑖𝑔𝐵𝑖𝑚𝑔𝑈𝑖𝑛 (5)

𝑆𝑖𝑠=(1−𝐹𝑖𝑔)𝐵𝑖𝑚𝑠𝑈𝑖𝑛 (6)

whereF_ig denotesthevolumetricfractioncontentofgravelatthebed surfaceinbranchior,brieﬂy,thesurfacegravelcontent(Fig.2),and 𝑚𝑔=𝐺∕𝐷_𝑔 and𝑚𝑠=𝐺∕𝐷_𝑠withDgandDs thegrainsizesof,

respec-tively,gravelandsand.Obviouslythecoeﬃcientsm_gandm_shave dif-ferentvalues.Similarlytotheunisizecase,weassumemgnottovary

be-tweenthebranches.Thesameholdsforms.CombinationofEqs.(5)and

(6)withthestationaryHiranoandSaint-Venantequationsshows, anal-ogoustotheunisizesedimentcase,that(underequilibriumconditions withoutuplift,subsidence,andparticleabrasion)theﬂowvelocity,ﬂow depth,andsurfacegravelcontentdonotvarywithinabranch.

Themodelrequiresanodalpointrelationthatrelatestheratioofthe sedimentsupplytothedownstreambranchestotheratioofthewater discharge.ThenodalpointrelationintroducedbyWangetal.(1995)is applicabletounisizesedimentconditions:

̄𝑠∗₌_𝛼𝑞∗𝑘_, _or ̄𝑠1 ̄𝑠2 =𝛼 ( 𝑞1 𝑞2 )𝑘 (7) wherē𝑠𝑖denotestherateofsedimentsupplyperunitwidthtobranchi,

thesuperscript∗_indicates_the_ratio_of_the_values_of_the_speciﬁc_variable

forbranches1and2(e.g., ̄𝑠∗₌_̄𝑠

1∕̄𝑠2),𝛼 isthenodalpointprefactor,

andq_iisthewaterdischargeperunitwidthinbranchi.Eq.(7)canalso bewrittenas ̄𝑆∗₌_𝛼𝑄∗𝑘_𝐵∗1−𝑘_, _or ̄𝑆1 ̄𝑆2 =𝛼 (_𝑄 1 𝑄2 )𝑘(_𝐵 1 𝐵2 )1−𝑘 (8) 11

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Forconditionsdominatedbytwograinsizemodes(gravelandsand), weintroducetwonodalpointrelations,onedescribingthepartitioning ofthegravelloadoverthebifurcatesandonethesandload:

̄𝑆∗ 𝑔=𝛼𝑔𝑄∗𝑘𝑔𝐵∗1−𝑘𝑔, or ̄𝑆1𝑔 ̄𝑆2𝑔 =𝛼_𝑔 (_𝑄 1 𝑄2 )𝑘𝑔(_𝐵 1 𝐵2 )1−𝑘𝑔 (9) ̄𝑆∗ 𝑠=𝛼𝑠𝑄∗𝑘𝑠𝐵∗1−𝑘𝑠, or ̄𝑆1𝑠 ̄𝑆2𝑠 =𝛼_𝑠 (_𝑄 1 𝑄2 )𝑘𝑠(_𝐵 1 𝐵2 )1−𝑘𝑠 (10) wherekgandksdenotethenodalpointcoeﬃcientsand𝛼gand𝛼sare

thenodalpointprefactors,bothforgravelandsand,respectively. Werealizethattheaboveformofthenodalpointrelationsistoo sim-pletocoverthephysicsoftheproblemofriverbifurcationsadequately. Inadditiontothestronglysimpliﬁedformofthenodalpointrelations forgravelandsand,thevaluesforthenodalpointcoeﬃcientskg,ks,

𝛼g,and𝛼slikelyarenotconstantsandmodelsforthesecoeﬃcientsare

neededtoproperlyanalyzethephysicsofthebifurcationproblem.Yet despitethesestrongsimpliﬁcationswebelievethatthecurrent analy-sisprovidesusefulinsightonelementarybifurcationbehavior.Wewill addressthisaspectinfurtherdetailinthediscussionsection.

Thefactthatbothdownstreambranchesaregovernedbythesame baselevel(Fig.1)andalsotheupstreamwatersurfaceelevationofthe twobranchesisequalcreatesthefollowinggeometricalconstraintinan equilibriumstate(Wangetal.,1995):

𝑖∗₌ 1 𝐿∗, or 𝑖1 𝑖2 =𝐿2 𝐿1 (11) whereL_iisthelengthofbranchi(Fig.1).

Wenowhaveasetofequationsthatcanbesolvedtodeterminethe equilibriumstatesofthetwodownstreambranches.

3. Theequilibriumstate

WemanipulatethesetofequationslistedinSection2toﬁndthe equilibriumsolutionsofthebifurcationcasesshowninFig.1.Under equilibriumconditionsthesedimentsupplyratemustbeequaltothe sedimenttransportcapacityandwethereforeset𝑆𝑖𝑔= ̄𝑆𝑖𝑔and𝑆𝑖𝑠= ̄𝑆𝑖𝑠.

Inaddition,wesubstituteEqs.(2)–(6)and(11)into(9)and(10).This yieldsanimplicitsolutiontotheratioofthewaterdischargeinthetwo downstreambranches,Q∗_: 𝑄∗₌_𝐿∗_𝐵∗1−3_𝑛 ⎛ ⎜ ⎜ ⎜ ⎝ 𝑚𝑔𝑆𝑠0+𝑚𝑠𝑆𝑔0 𝑚𝑔𝑆𝑠0 ( 𝛼𝑠𝑄∗𝑘𝑠𝐵∗1−𝑘𝑠+1)−1+𝑚𝑠𝑆𝑔0 ( 𝛼𝑔𝑄∗𝑘𝑔𝐵∗1−𝑘𝑔+1 )−1−1 ⎞ ⎟ ⎟ ⎟ ⎠ 3 𝑛 =Φ(𝑄∗₎ ₍₁₂₎

AsolutionofEq.(12)providesvaluesforthegravelandsandloadinthe downstreambranches,SigandSis(𝑖=1,2),throughthenodalpoint

re-lationsinEqs.(9)and(10),providedthatthewaterdischargeinbranch 0,Q0,thegravelandsandsupplyratestobranch0,S0gandS0s,and

thevariablesm_g,m_s,L∗_,_B∗_,_k

s,andkgareknown.Wecomputetheﬂow

depth,Hi,usingEq.(3),aswellasthesurfacegravelcontent,Fig,using

Eq.(5)or(6).

Eq.(12)hasatleastthreesolutions:twosolutionsthatareassociated withtheclosureofoneofthebranches(𝑄∗₌₀_,_𝑄∗₌_∞)_and_one

solu-tioninwhichbothdownstreambranchesremainopen.Genericallythe flowdepthdiffersbetweenthedownstreambranches,butunder con-ditionsinwhich𝐿∗₌₁_the_flow_depth_in_the_downstream_branches_is

equal,evenifthewidthvariesbetweenthebranches.

Wedeﬁneabasecasethat(exceptforthebifurcatelength)isloosely basedonthebifurcationoftheBovenrijnintothePannerdenschKanaal andthe Waal branch.The bifurcation is located in the Netherlands andabout10kmdownstreamfromwheretheRhineRivercrossesthe German–Dutchborder.Thewaterdischargeissetequaltotheone char-acterizedbyaoneyearrecurrenceperiod(4000m3_/s)._We_simply

as-sumethebifurcatestohavethesamechannellength(here𝐿1=𝐿2=10

km).Thisyieldsthefollowingparametervaluesforthebasecase:𝛼𝑔=

Fig.3. Sections I, II a, II b, and III in the stability diagram in the ( kg,ks) parameter

space for the base case.

𝛼𝑠=1,𝐵0=315m,𝐵1=𝐵2=250m,𝐶1=𝐶2=50m1/2/s,𝑆0𝑔=0.001

m3_/s,_𝑆

0𝑠=0.007m3/s,and𝑄0=4000m3/s.

ForadetailedanalysisofEq.(12)werefertoAppendixA.1.It il-lustratesthatwecandistinguishbetweenthreesectionsinthe(kg,ks)

parameterspace(I,II,andIII),eachwithadiﬀerentnumberofsolutions toEq.(12)andhenceoftheﬂowdepthinthedownstreambranches (Fig.3):

• IandIII:Therearethreeequilibriumsolutions.Twosolutions cor-respondwithoneofthedownstreambranchesclosed.Theother so-lutioncorrespondswithbothbranchesopen.

• II_aandII_b:Thereareﬁveequilibriumsolutions.Twosolutions corre-spondwithoneofthedownstreambranchesclosed.Theremaining threesolutionscorrespondwithbothbranchesopen.

ThediﬀerencesbetweensectionsIandIIIandbetweenII_aandII_b willbeaddressedinthenextsection.

Theboundaries ofthesectionsI,IIa,IIb,andIIIin Fig.3depend

ontheratioofthesandloadtothegravelloadintheupstreambranch (branch0),whichisdenotedby ̂𝑆0,theratioofthelengthofthe

bifur-cates,L∗_,_and_the_ratio_of_the_channel_width_of_the_bifurcates,_B∗₍_Fig.₄_).

Anincreaseofthesandloadintheupstreamchannelattheexpenseof itsgravelloadleadstoadecreaseofsectionIIbandanincreaseof

sec-tionIIa.Anincreaseofthediﬀerenceinchannellengthbetweenthetwo

bifurcatessigniﬁcantlydecreasessectionIIandanincreaseofsectionI. SectionIItendstobecomenegligibleforvaluesofL∗_even_mildly_larger

than1.Theeffectsof ̂𝑆0andL∗aresignificant,whereastheeffectofa

diﬀerenceinchannelwidthbetweenthetwobifurcates,B∗_,_appears_to

belimited.

Thecurrentanalysisislimitedtoengineeredriverswherethe chan-nelwidthcannotadjusttochangesinthecontrols(i.e.,statisticsofthe waterdischarge,sedimentsupply,andbaselevel).Theanalysis

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illus-Fig.4. Stability diagram in the ( k_g,k_s) parameter space, for varying (a) ratio of the sand load to the gravel load in branch 0, ̂𝑆0; (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.)

tratesthatamixed-sizesedimenttwo-channelsystemconsistsofthreeto fivesolutionstothemorphodynamicequilibriumstate.Thisdiffersfrom aunisizesedimenttwo-channelsystem,forwhichthreeequilibrium so-lutionsexist(Wangetal.,1995).Theexistenceofthreetofivesolutions alsocontrastswiththesinglesolutiontothemorphodynamic equilib-riumstateofaone-channelsystem,underunisizeaswellasmixed-size sedimentconditions(Blometal.,2017a;2016;Howard,1980).

Naturalrivers,wherebesidesthechannelslopeandbedsurface tex-turealsothechannelwidthrespondstochangesinthecontrols,allow formoreequilibriumstatesthanengineeredriverswithaﬁxed chan-nelwidth(Blometal.,2017a).Innaturalriversthereexistsarangeof equilibriumstatesforwhichthechannelisabletotransporttheload suppliedfrom above(Blom etal., 2017a).Inthecurrentanalysiswe havenotconsideredtheeﬀectoferodablebanks,yetonemayexpect that,justasinthesinglechannelcase,thepresenceoferodablebanks allowsforarangeofequilibriumstates.

4. Modelofthestabilityoftheequilibriumstate

We set up a system of diﬀerentialequations for the ﬂow depth andthesurfacegravelcontentinthebifurcatestostudythetemporal changesinthebifurcationsystem.

Forsimplicity,weassumethatperturbationsinbedelevation(i.e., aggradationalanddegradationalwaves),whicharisefromadiﬀerence betweenthesedimentsupplytoadownstreambranchanditssediment transportcapacity,movesofastalongthedownstreambranchesthat wecanassumeabranch-averagedresponseofbedelevation.This im-pliesthatweconsiderbranch-averagedvaluesforbedelevation,𝜂i,ﬂow

depth,H_i,andsurfacetexturerepresentedbythesurfacegravelcontent, Fig(Fig.2).Anotherconsequenceofthisbranch-averagedapproachis

thefactthatthechannelslopeineachbifurcatecannotadjustwithtime, astheaggradationordegradationratedoesnotvarywithinabifurcate. Theconstantchannelslopeandbaselevelimplythat,althoughthebed elevationchangeswithtime,thewatersurfaceelevationinthe bifur-catesremainsconstantwithtime.

Suchareach-averagedapproachisvalidprovidedthatthe perturba-tioninbedelevationmigratesrelativelyfastdowninthechannelorif thechannelisrelativelyshort(i.e.,asmallvalueofLi).

TheExnerequationdescribingconservationofbedsedimentis

𝑐𝑏𝐵𝑖𝜕𝜂_𝜕𝑡𝑖 =−𝜕𝑆_𝜕𝑥𝑖 (13)

wheretdenotestime, cb thesedimentconcentrationwithin the bed

(𝑐𝑏=1−𝑝withpdenotingbedporosity),and𝜂 isbedelevationwith

respecttoaﬁxedreferencelevel.As𝐻=𝜂_𝑤−𝜂 (Fig.2)andthewater surfaceelevation,𝜂w,isconstantduetoourbranch-averagedapproach,

weﬁndthat𝜕𝜂𝑖∕𝜕𝑡=−𝜕𝐻𝑖∕𝜕𝑡.ThisimpliesthatEq.(13)canbewritten

as(Wangetal.,1995):

𝑐𝑏𝐵𝑖𝜕𝐻_𝜕𝑡𝑖= 𝜕𝑆_𝜕𝑥𝑖 (14)

Asweassumethatgradientsinthesedimenttransportrateresultina branch-averageddegradationoraggradationrate,wewrite𝜕Si/𝜕xas

𝜕𝑆𝑖

𝜕𝑥 = 𝑆𝑖− ̄𝑆𝑖

𝐿𝑖 (15)

CombinationofEqs.(14)and(15)thenyields 𝑑𝐻𝑖 𝑑𝑡 = 1 𝑐𝑏𝐵𝑖𝐿𝑖 ( 𝑆𝑖− ̄𝑆𝑖) (16)

We apply a simpliﬁed form of the Hirano active layer model (Hirano,1971) todescribethetemporalchange ofthesurfacegravel contentinthetopmostpartofthebedthatinteractswiththeﬂow(i.e., intheactivelayer).

ToarriveatasimplifiedversionoftheHiranoequation,weapply asimilarbranch-averagedapproachtothemigrationofperturbations inthebedsurfacetextureastoperturbationsinbedelevation.Worded differently,weassumesurfacetextureperturbations,whicharisefrom adifferencebetweenthegrainsizedistributionofthesedimentsupply toadownstreambranchandthegrain sizedistributionofthe trans-portedsediment,tomovesofastalongabifurcatethatwecanconsider abranch-averagedresponseofthebedsurfacegravelcontentinthe bi-furcate,Fig.Thereach-averagedapproachisvalidprovidedthatthe

per-turbationinthebedsurfacetexturemigratesdownstreamfastalongthe channel(e.g.,asmalldepthofreworkingorasmallactivelayer thick-ness)orifthechannelisrelativelyshort(i.e.,asmallvalueofLi).

Inaddition,weassumethattheverticalsedimentﬂuxbetweenthe activelayerandthesubstratethatisassociatedwithachangein ele-vationoftheinterfacebetweentheactivelayerandthesubstratehas thesamegrainsizedistributionastheoneoftheactivelayersediment, evenunderconditionsofdegradation.

UnderthesesimplifyingassumptionstheHiranoactivelayer equa-tionreducesto 𝑑𝐹𝑖𝑔 𝑑𝑡 = 1 𝑐𝑏𝐵𝑖𝐿𝑖 1 𝐿𝐴 ( 𝐹𝑖𝑔(𝑆𝑖− ̄𝑆𝑖)+(̄𝑆𝑖𝑔−𝑆𝑖𝑔)) (17)

whereLAdenotesthethicknessoftheactivelayerorthesurfacelayer

thatisreworkedbytheﬂow. 13

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Theadjustmentofthebedsurfacetextureischaracterizedbyan ex-ponentialgrowthofwhichthetimescale(i.e.,thee-foldingtimeorthe timeintervalinwhichtheexponentiallygrowingquantityincreasesby afactorofe)equals

𝑇=𝑐_𝑏𝐵_𝑖𝐿_𝑖𝐿_𝐴 (18)

We have derived the following system of diﬀerential equations for the ﬂow depth,Hi, and surfacegravel content, Fig bymanipulating

Eqs.(16)–(17),usingEqs.(9)–(10),forsimplicitysetting 𝛼𝑔=𝛼𝑠=1,

andintroducingtimê𝑡wherê𝑡=𝑡∕𝑐_𝑏: 𝑑𝐻1 𝑑̂𝑡 = 𝑄𝑛 0 𝐵1𝐿1 ( 𝑔1(𝐻1,𝐻2,𝐹1𝑔)−̄𝑔1(𝐻1,𝐻2) ) (19) 𝑑𝐻2 𝑑̂𝑡 = 𝑄𝑛 0 𝐵2𝐿2 ( 𝑔2(𝐻1,𝐻2,𝐹2𝑔)−̄𝑔2(𝐻1,𝐻2) ) (20) 𝑑𝐹1𝑔 𝑑̂𝑡 = 𝑄𝑛 0 𝐵1𝐿1 1 𝐿𝐴 ( 𝐹1𝑔 ( 𝑔1(𝐻1,𝐻2,𝐹1𝑔)−̄𝑔1(𝐻1,𝐻2) ) +(̄𝑔1𝑔(𝐻1,𝐻2)−𝑔1𝑔(𝐻1,𝐻2,𝐹1𝑔))) (21) 𝑑𝐹2𝑔 𝑑̂𝑡 = 𝑄𝑛 0 𝐵2𝐿2 1 𝐿𝐴 ( 𝐹2𝑔(𝑔2(𝐻1,𝐻2,𝐹2𝑔)−̄𝑔2(𝐻1,𝐻2) ) +(̄𝑔2𝑔(𝐻1,𝐻2)−𝑔2𝑔(𝐻1,𝐻2,𝐹2𝑔))) (22)

wherethefunctionsg_i, ̄𝑔𝑖,gig,and̄𝑔𝑖𝑔(𝑖=1,2)aredeﬁnedinAppendix

B.

WeabbreviateEqs.(19)–(22)by( ̇𝐻_𝑖,𝐹𝑖𝑔̇ )=Ψ(𝐻_𝑖,𝐹_𝑖𝑔)wherethedot indicatesthederivativewithrespecttotime.Naturallytheequilibrium solutionsofSection3aresolutionsofΨ(𝐻_𝑖,𝐹_𝑖𝑔)=0.

5. Stabilityoftheequilibriumstate

Equilibriumsolutionsonlyemergeiftheyarestable. Thestability propertiesofthesolutionsofΨ(𝐻_𝑖,𝐹_𝑖𝑔)=0aredeterminedbythe eigen-valuesoftheJacobianJofΨ,whichisdeﬁnedas:

𝐽= ⎛ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎝ 𝜕Ψ1 𝜕𝐻1 𝜕Ψ1 𝜕𝐻2 𝜕Ψ1 𝜕𝐹1𝑔 𝜕Ψ1 𝜕𝐹2𝑔 𝜕Ψ2 𝜕𝐻1 𝜕Ψ2 𝜕𝐻2 𝜕Ψ2 𝜕𝐹1𝑔 𝜕Ψ2 𝜕𝐹2𝑔 𝜕Ψ3 𝜕𝐻1 𝜕Ψ3 𝜕𝐻2 𝜕Ψ3 𝜕𝐹1𝑔 𝜕Ψ3 𝜕𝐹2𝑔 𝜕Ψ4 𝜕𝐻1 𝜕Ψ4 𝜕𝐻2 𝜕Ψ4 𝜕𝐹1𝑔 𝜕Ψ4 𝜕𝐹2𝑔 ⎞ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎠ (23)

Ifalleigenvaluesatanequilibriumsolutionhavenegative(positive) realparts,theequilibriumsolutionsarelinearlyandnonlinearlystable (unstable),andarenodalpoints inthe4-dimensionalphasespace.If therearepositiveandnegativeeigenvalues,thesolutionisunstableand asaddlepointinthephasespace(e.g.,Wiggins,1990).Purelyimaginary eigenvalueswouldgiverisetoperiodicsolutionsinthephasespace,yet thisdoesnotoccurforthisparticularsetofequations.

Asthesystemof Eqs.(19)–(22) andtheassociated JacobianJin Eq.(23)aretoocomplextobeanalyzedanalytically,weanalyzethe sys-temnumerically.ForthedetailsoftheanalysiswerefertoAppendixA.2. Insummary,thefollowingholdsforthesectionsI-IIIinFig.3:

• I:Thetwoequilibriumsolutions thatcorrespondwithonebranch closedarestable.Theothersolution,wherebothbranchesareopen, isunstable.Theinitialconditionsdeterminetowhichstablestatethe systemevolves.

• IIaandIIb:Thetwoequilibriumsolutionsthatcorrespondwithone

branchclosedarestable.Theotherthreesolutionscorrespondwith bothbranchesopen.Thetwo‘new’solutions(comparedtosection I)arecreatedinablueskybifurcation(AppendixA.2).Onlyoneof themisstable.Theinitialconditionsdeterminetowhichofthethree stableequilibriumstatesthesystemevolves.

• III:Thetwoequilibriumsolutionsthatcorrespondwithonebranch closedareunstable.Thesolutionwithboth branchesopenis the onlystablesolution.Thisimpliesthatforeveryinitialconditionboth branchesremainopen.

Foratwo-channelsystemunderunisizesedimentconditions,there existsonecriticalvalueofthenodalpointcoeﬃcientk(𝑘=𝑛∕3), be-lowwhichtheequilibriumsolutionwithoneclosedbifurcationisstable (Wangetal., 1995).Forvalues ofk largerthann/3theequilibrium solutionwithtwoopenbifurcatesisstable.Acaseinwhichk<n/3is similartothecurrentsectionIandthelattercaseissimilartothe cur-rentsectionIII.UnderunisizesedimentconditionssectionIIdoesnot exist.

Forasinglechannelsystemwithﬁxedbanksthesinglesolutionto theequilibriumstateisstable(Blometal.,2017a;2016;Howard,1980). 6. Evolutiontowardsthestableequilibriumstate

WenumericallysimulatethesystemofEqs.(19)–(22)toassess(1) theeﬀectsoftheinitialﬂowdepth,H1andH2,andtheinitialsurface

gravelcontent,F1gandF2g,inthebifurcatingbranches;(2)the

mecha-nismthatresultsinclosureofoneofthebranches;and(3)theeﬀectsof thenodalpointcoeﬃcientsincombinationwiththesedimentsupply. Weanalyzethesethreeaspectsbelow.

6.1. Eﬀectoftheinitialconditionsinthebifurcatingbranches

InsectionIItheinitialconditionsdeterminetowhichofthethree stableequilibriumstatesthesystemevolves.

Fig. 5shows the results of two numerical runs of thesystem of Eqs.(19)–(22),inwhichweassesstheeffectsoftheinitialflowdepthin thebifurcatingbranches,H1andH2.Theonlydifferencebetweenthe

tworunsaretheinitialvaluesoftheﬂowdepth,H₁andH₂.Forequal lengthofthebifurcates(𝐿∗₌₁₎_and_arbitrary_value_of_the_ratio_of_the

bifurcatewidth,B∗_,_the_ﬂow_depth_and_surface_gravel_content_in_the_two

bifurcatesevolvetowardthesamevalue(𝐻∗₌₁_and_𝐹∗

𝑔 =1).Itappears

thatadifferenceintheinitialflowdepthresultsindifferentbehavior: inonecasebothbranchesstayopen,whileintheothercaseonebranch closes.

Inthecasewherebothbranchesremainopen,thesurfacegravel con-tentinthetwobranches,F1gandF2g,evolvestowardsthesamevalue.

ThesurfacegravelcontentF1gintheclosingbranchevolvesto0,which

meansthateventuallythebedsurfaceinthisclosingbranchconsistsof sandonly.

Fig.6showstheresultsoftworunswhereonlytheinitialsurface gravelcontentinthebifurcatingchannels,F₁_gandF₂_g,variesbetween theruns.Againweobservetheeﬀectoftheinitialconditions:they de-terminewhetherthesituationevolvestowardsastatewitheitherboth branchesopenoronebranchclosed.

6.2. Mechanismofclosureofoneofthebranches

WeconsiderthecasewhereoneofthebranchesclosesinFig.5(solid lines)tostudythemechanismofbranchclosure.Tothisendweanalyze thediﬀerencebetweentheloadandsupplyofgravelandsandfor, re-spectively,branches1and2(Fig.7aandb).

As thesediment supply in branch2 exceedsthe sediment trans-portcapacity(𝑆2− ̄𝑆2<0),aggradationwilloccurandbranch2slowly

closes.Thesedimentsupplyinbranch1approachesthesediment trans-portcapacity(𝑆1− ̄𝑆1↓ 0),whichimpliesthattheﬂowdepthinbranch

1approachesanequilibrium(nonzero)value.

In branch2thesand supplyexceeds thesand transportcapacity (𝑆2𝑠− ̄𝑆2𝑠<0)andthegravelsupplyissmallerthanthegravel

trans-portcapacity(𝑆2𝑔− ̄𝑆2𝑔>0),whichimpliesthatthebedsurfaceofthe

branchbecomesincreasinglysandy.This isreﬂectedbythefactthat bedsurfacegravelcontentapproacheszero(F₂_g↓0,solidlineinFig.5b).

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Fig.5. Eﬀect of the initial ﬂow depth in the downsteam channels on bifurcation dynamics. Flow depth H₁and H₂(left panel) and surface gravel content F₁_gand F₂_g (right panel) in the two downstream branches. Initially, 𝐻1= 14 m and 𝐻2= 8 m (solid lines) and 𝐻1= 14 m and 𝐻2= 10 m (dashed lines), and 𝐹1𝑔= 0 .5 ,𝐹2𝑔= 0 .5 .

Blue and red lines indicate, respectively, branch 1 and branch 2. Results are given for 𝑘𝑔= 3 ,𝑘𝑠= 1 ,𝛼𝑔= 𝛼𝑠= 1 ,and 𝐿𝐴= 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 H1and H2(left panel) and surface gravel content F1gand F2g(right panel) in the two downstream branches. Initially, 𝐹1𝑔= 0 .8 ,𝐹2𝑔= 0 .5 (dashed lines) and 𝐹1𝑔= 0 .3 ,𝐹2𝑔= 0 .5 (solid lines), and 𝐻1= 14 m and 𝐻2= 8 m. Blue and red lines indicate, respectively, branch 1 and branch 2. Results are given for 𝑘𝑔= 3 ,𝑘𝑠= 1 ,𝛼𝑔= 𝛼𝑠= 1 , and 𝐿𝐴= 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 ̂𝑡. (For interpretation of the references to colour in this ﬁgure legend, the reader is referred to the web version of this article.)

Thebedsurfaceofbranch1continuestoconsistofamixtureofgravel andsand (i.e.,F1gapproachesanequilibriumnonzerovalue).Thisis

understandableasintheﬁnalstatethegravelandsandsupplyfromthe upstreambranchistransportedbybranch1,whichrequiresbothgravel andsandtoberepresentedatthebedsurface(seeEqs.(5)and(6)).

6.3. Eﬀectsofnodalpointcoeﬃcientsandsedimentsupply

Finally we studythe dependenceof bifurcation dynamics onthe nodalpointcoeﬃcients,kgandks,andthegravelandsandsupplyto

theupstreambranch0.Weconsidertwosituations:𝑘𝑔=5,𝑘_𝑠=1 (sec-tionII_a,Fig.8)and𝑘𝑔=1,𝑘𝑠=5(sectionIIb,Fig.9).Westudytwo

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Fig.8. Eﬀect of the gravel content in the sediment supply on bifurcation evolution for 𝑘𝑔= 5 ,𝑘𝑠= 1 (section II a): ﬂow depth H1and H2(left panels) and surface gravel content F1gand F2g(right panels). Results are given for 𝛼𝑔= 𝛼𝑠= 1 and 𝐿𝐴= 1 m. Initially 0.1 ≤ F1g≤ 0.9 and 𝐹2𝑔= 0 .2 . In upper panels ̂𝑆0= 𝑆0𝑠∕ 𝑆0𝑔= 2 (i.e.,

a relatively coarse supply) and in lower panels ̂𝑆0= 7 (i.e., a relatively ﬁne supply). Blue and red lines indicate, respectively, branch 1 and branch 2. 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.9. Eﬀect of the gravel content in the sediment supply on bifurcation evolution for 𝑘𝑔= 1 ,𝑘𝑠= 5 (section II b): ﬂow depth H1and H2(left panels) and surface gravel content F1gand F2g(right panels). Results are given for 𝛼𝑔= 𝛼𝑠= 1 and 𝐿𝐴= 1 m. Initially 0.1 ≤ F1g≤ 0.9 and 𝐹2𝑔= 0 .2 . In upper panels ̂𝑆0= 𝑆0𝑠∕ 𝑆0𝑔= 2 (i.e.,

a relatively coarse supply) and in lower panels ̂𝑆0= 7 (i.e., a relatively ﬁne supply). Blue and red lines indicate, respectively, branch 1 and branch 2. 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.)

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Finallywestressthelongtimescaleassociatedwiththetemporal changeofthebifurcationsysteminournumericalruns.Thisseemsto beduetoourassumptionofbranch-averagedchangeinthetwo bifur-cates,whichslowsdownbifurcationadjustment.Thisisbecauseinthe branch-averagedmodeltheresultofamismatchbetweenthesediment supplytoabifurcateanditssedimenttransportcapacityisdistributed overtheentirebranchratherthantheeffectinitiallybeinglimitedto theupstreamendofthebifurcate.Inrealitysuchamismatchleadstoan aggradationalordegradationalwavethatstartsattheupstreamendof thebifurcateandthenmigratesdownstream.Thisfeedsbacktothe mis-matchmorestronglythaninourbranch-averagedapproachand there-foreinrealitytheexpectedchangelikelyoccursmuchfasterthaninour idealizedmodel.Nevertheless,thereisevidencefromfielddatathat bi-furcationchangecanbeslow:achangetoanewdominantchannelmay requiresignificanttime(SlingerlandandSmith,2004)andoftenrequire severalcenturiesintheRhine-Meusedelta,anddurationsofupto1250 yearshavebeenestimatedbyStouthamerandBerendsen(2001). 7. Discussion

7.1. Theloadrelation

InouranalysiswehaveappliedtheEngelundandHansenload rela-tioninafractionalmanner(Blometal.,2017a;2016).Thisfractional formoftheloadrelationhasneverbeenproperlyvalidatedandthis ap-proachlikelyismorevalidinlowlandriverswherepartialmobilityis lessrelevant.Theanalysiscanberepeatedformorecomplicatedload relations(e.g.,thoseincludingathresholdforsignificanttransportor hidingeffects).Wedonotexpect,however,thatapplicationofamore complicatedandrealisticloadrelationaffectsouranalysisofthe equilib-riumstatesandtheassociatedconclusions,asanotherloadrelationdoes notchangetheanalysisinafundamentalmanner.Theanalysiswould stillyieldfourcoupleddifferentialequations(similartoEqs.(19)–(22)), inwhichthecoefficientsaredifferentfromtheonesbasedonthe origi-nalloadrelation.ThisalsoholdsfortheJacobianinEq.(23).Theresults woulddiffersomewhatfromtheonesbasedontheoriginalloadrelation butwedonotexpectnewphenomena.

7.2. Thenodalpointrelation

Acrucialpointinthepresentedanalysisisthefactthatweassume thatthereexistsanodalpointrelationoftheformofEq.(8)forthe unisizesedimentcase,orEqs.(9)and(10)forthemixed-sizesediment case.Thisformofthenodalpointrelationistoosimpletodescribethe partitioningofsedimentatriverbifurcations(e.g.,VanderMarkand Mosselman,2013).Wangetal.(1995)alreadysuggestedanodalpoint relationofamoreextensiveform:𝑆∗₌_𝑓(_𝐵∗_,_𝑄∗_,_𝐶∗_,_𝐻∗_,_…)_._It_is,

how-ever,diﬃculttoconstrainthevariousparametersinthenodalpoint re-lation,althoughanattemptmayleadtoamorerealisticrelation.Even inthesimpleformofthenodalpointrelationusedhere,thevaluesof itscoeﬃcientsk_g,k_s,𝛼g,and𝛼sarenotlikelyconstantandmodels

de-thedownstreambranchesaccordingtotheratioofthewaterdischarge (̄𝑆∗₌_𝑄∗_),_which_implies_that_𝑘₌_{𝛼 =}₁₍_De_Heer_and_Mosselman,_2004;

Duttaetal.,2017).Thus,weexpectthatthefinerthesedimentsupply tobranch0,thelargeristhevalueofk(withintherangeof0to1). Inreality,however,conditionsaremorecomplexduetothepresence ofalateralbedslope,abend,bars,orstructures.Wediscussthelatter effectsbelow,firstfortheunisizesedimentcaseandsubsequentlyfor themixed-sizesedimentcase.

Typicallythedifferenceinbedelevationbetweenthetwobifurcates inducesalateralslopejustupstreamofthebifurcation.Thisslopeeffect increasesthesedimentsupplytothedeeperbifurcateandthereforeacts asastabilizingmechanism,asitcounteractsfurtherdeepeningofthe deeperbifurcate.Thiseffectneedstobeaccountedforwhensettingup amodelforthenodalpointcoefficient.

Bends and bars affect the flow just upstream of the bifurcation andassuchmayaffectthesedimentpartitioningoverthedownstream branchesofthebifurcation.Forinstance,thesedimentpartitioningat abifurcationthatislocatedjustdownstreamofapointbarinaninner bendis affectedbytheassociatedsecondary flowandthetransverse gradientinbedelevationjustupstreamofthebifurcation.

Thepresenceofasillinthedownstreambranch1likelyreducesthe sedimentsupplytothatbranch(Fig.10a).Consideringthenodalpoint relationinEq.(7)andassumingthatthesill’seﬀectonthesediment supplyislargerthanonthewaterdischarge,kmustbelargerthan1 andforarelativelyhighsill(withbarelysedimentsupplytothespeciﬁc channel)kshouldapproach∞.

A similarlineof reasoningholds undermixed-size sediment con-ditions. Recallthatthenodalpointrelations(Eqs.(9)and(10))read

̄𝑠∗

𝑔=𝛼𝑔𝑞∗𝑘𝑔 and ̄𝑠𝑠=𝛼_𝑠𝑞∗𝑘𝑠.Theabovelateralslopeeffectinducedby theinletstepvarieswithgrain size:coarsesedimentisaffectedmore stronglythanfinesediment(ParkerandAndrews,1985),andthesame holds forthelateralslope effectintroduced bybendsandbars. This alsoapplies tothepresenceof asill inone ofthebifurcates: coarse sediment is affectedmore strongly thanfine sediment,as the trans-portofcoarse sedimentconcentratesmorestronglynearthebed.An additionaleffectofariverbendisbendsorting.Apointbartypically consistsofrelativelyfinesediment,whereasthebedsurfaceand trans-portedloadintheouterbendarecoarser(Fig.10b).Hence,thesediment supplytothebifurcatelocatedintheouterbendiscoarserthanthe sup-plytotheotherbifurcate.Theseeffectswillneedtobeaccountedfor inrelationsforthenodalpointcoefficients𝛼g,𝛼s,kg,andks.The

as-sociated consequencesforriverbifurcation dynamicswillneedtobe studied.

7.3. Thestabilitycriterion

Basedontheirmathematicalstabilityanalysisforthecaseofunisize sedimentconditions,Wangetal.(1995)foundthatthestability crite-rionfortwoopenbranchesisgivenbyk>n/3.Thiscanalsobefound throughreasoning(Kleinhansetal.,2008),whichissummarizedhereto subsequentlyextendthisreasoningtothecaseofmixed-sizesediment. Eq.(8)illustratesthatthesedimentsupplytobranchi,̄𝑠𝑖,isproportional

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

to𝑞𝑘

𝑖,andEqs.(1)–(3)showthatthesedimenttransportcapacityofa

branchisproportionalto𝑞𝑛∕3

𝑖 .Nowsupposethatk>n/3.Inthatcasethe

sedimentsupplyperunitwidthtobranchiincreasesmorestronglywith increasingwaterdischargeperunitwidthinbranchi,thanthesediment transportcapacity,which leadstoaggradation.Asaresultthewater dischargeperunitwidth,qi,decreases,whichmeansthatthesituation

stabilizes.Ontheotherhand,ifk<n/3,thesedimentsupplytobranchi increaseslessstronglywithincreasingq_ithanthesedimenttransport ca-pacity.Thisimpliesthatthechanneldegrades,whichincreasesqieven

further.Sothesituationfurtherdestabilizesattheexpenseoftheother downstreambranch,whichcloses.

Forthecaseofmixed-sizesedimentwereasoninasimilarmanner. Thesupplyofgravelandsandtobranchi,̄𝑠𝑖𝑔and̄𝑠𝑖𝑠,areproportional

to𝑞𝑘𝑔

𝑖 and𝑞𝑖𝑘𝑠,respectively.Thetransportcapacitiesofgravelandsand

inbranchi,s_igands_is,nowalsodependonthesurfacegravelcontent inbranchi,F_ig,andareproportionalto𝐹𝑖𝑔𝑚𝑔𝑞𝑖𝑛∕3and(1−𝐹𝑖𝑔)𝑚𝑠𝑞𝑖𝑛∕3,

respectively.Wenowsupposethatthewaterdischargeperunitwidth inbranchi,qi,increases.Thisimpliesthatthegravelandsandsupply

tobranchiincrease,themannerofwhichdependonthevaluesofkg

andk_s.Ask_gandk_s areexpectedtohavediﬀerentvalues,thegravel andsandsupplytobranchiresponddiﬀerentlytotheincreaseinthe waterdischarge.Alsothegravelandsandtransportcapacities,sigand

s_is,responddifferentlytotheincreaseinthewaterdischargeduetothe mobilitydifferencebetweencoarseandfinesediment(i.e.,grainsize se-lectivetransport).Dependingontheresultingchangeofthegraveland sandsupplytobranchi, ̄𝑠𝑖𝑔 and̄𝑠𝑖𝑠,andthechangeinthegraveland

sandtransportcapacities,sigandsis,theincreaseinwaterdischargeqi

aﬀectsthesurfacegravelcontentinbranchi,Fig,whichisexpressedbya

coarseningorﬁningofthebedsurface.Thus,basedonphysical reason-ingitismuchlessstraightforwardtodrawconclusionsconcerningthe expectedtemporalchangeandastabilitycriterionofariverbifurcation dominatedbymixed-sizesediment.

Asanextstepwerecommendtheformulationofsubmodelsforthe nodalpointcoeﬃcientskgandks,which,amongotherparameters,likely

dependon thetransversebed slope justupstream ofthe bifurcation (BollaPittalugaetal.,2003)andhencelikelyareafunctionoftheratio oftheﬂowdepthinthebifurcates,H∗_(where_𝐻∗₌_𝐻

1∕𝐻2).Insucha

casetheformulationoftheJacobianinEq.(23)becomesmore compli-catedbecauseofthederivativeswithrespecttoH₁andH₂.Analysisof theJacobianandthenumericalresultswouldprovideinsightonthe ex-pectedtemporalchangeandthestabilitycriterionofariverbifurcation dominatedbymixed-sizesediment.

7.4. Symmetricalbifurcations

Ithasbeenfoundthatsymmetricalbifurcations(i.e.,bifurcateswith equal properties such as water discharge, channel width, and flow depth)tendtobeunstablemoreoftenthanasymmetricalbifurcations (i.e.,oneofthebifurcatesissignificantlysmallerthantheotherone) (BertoldiandTubino,2007;BollaPittalugaetal.,2015;Edmondsand Slingerland,2008;Kleinhansetal.,2013;2008;Miorietal.,2006).In ourbasecase,whichischaracterizedbyequalchannelwidth,friction, andlengthof thebifurcates,we findthatin sectionsIIandIII

sym-metricalsolutionsarestable,whichmaycontradicttheabovefindings. Ourresults aresimilar tothoseofWangetal.(1995),whofindthat a symmetrical solutionis stable fork>n/3. This similaritybetween theresultsmaynotbesurprisingasourmodelisanextensionofthe highlyidealizedmodelofWang etal.(1995)tomixed-sizesediment conditions,whereasothermodelsaccountfortheeffectsofatransverse bedslope(BollaPittalugaetal.,2015;2003;EdmondsandSlingerland, 2008;Kleinhansetal.,2013;2008),alternatebars(Bertoldietal.,2009; Redolfi etal., 2016),curvature-inducedflowasymmetry upstreamof the bifurcation (Kleinhans etal., 2008; Van Denderenet al., 2017), suspendedbed-materialload(SlingerlandandSmith,1998),cohesive sediment(EdmondsandSlingerland,2008;HajekandEdmonds,2014), bendsorting(Sloff etal.,2003;Sloff andMosselman,2012),andbank erosion(Miorietal.,2006;VanDenderenetal.,2017).These effects arenottakenintoaccountinouranalysisbutmayberepresentedby appropriatefuturemodelsforkgandks.

8. Conclusions

Weextendahighlyidealizedmodelofthedynamicsofariver bifur-cationtomixed-sizesedimentconditions.Themodelisbasedonnodal pointrelationsforgravelandsandthatsetthepartitioningofgraveland sandoverthedownstreambranchesorbifurcates.Themodeldescribes theequilibriumsolutionsand,basedonabranch-averaged approxima-tionofaggradationanddegradation,describesthetemporalchangeof bedelevationandbedsurfacetextureinthebifurcatesofamixed-size sedimentriverbifurcation.

Theintroductionofmixed-sizesedimentmechanismstotheriver bi-furcationproblemintroducesanadditionaldegreeoffreedom:the tem-poraladjustmentofthebedsurfacetextureineachof thebifurcates. Thedynamicsofthedownstreambranchesconcerningtheirflowdepth andbedsurfacetexture andtheresulting stableconfigurationof the downstream branchesresult fromdifferencesbetween (a)the gravel andsandsupplyineachbranchand(b)itsgravelandsandtransport capacity.

Wesetupamathematicalmodeloftheequilibriumstatesand dy-namicsofamixed-sizesedimentriverbifurcation.Inouranalysiswe have neglected theeffects of a transverse bed slope,alternate bars, curvature-induced flow asymmetry upstream of the bifurcation, sus-pendedbed-materialload,cohesivesediment,bendsorting,andbank erosion.Theproposedmodelthereforehaslimitedpredictivevalue re-gardingrealriverbifurcations,yetprovidesinsightontheelementary effectsofmixed-sizesedimentmechanismsontheriverbifurcation prob-lem.Subsequentanalysesmaycombinetheanalysisofmixed-size sedi-mentmechanismswiththeabovementionedeffects.

Howard (1980) and Blom et al.(2017a, 2016) have shown that there existsone solutiontothemorphodynamicequilibriumstatein a one-channelsystem withnonerodiblebanks.Inaunisize sediment two-channelsystemwithﬁxedbanksthreeequilibriumsolutionsexist, whereasthreetoﬁvesolutionsexistinamixed-sizesedimentbifurcation system.

Inthemixed-sizesedimenttwo-channelsystemwedistinguishthree sections(I,II,andIII)intheparameterspacerelatedtothenodalpoint

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A1. Equilibriumsolutions

Inthisappendixwe analyzeEq.(12) andexplainFig.3in more detail.ThethreesolutionsforQ∗_of_Eq.₍₁₂₎_give_rise_to_three_solutions

forthecombinationofﬂowdepthsinthedownstreambranchesHi:two

withonebranchclosed,onewithbothbranchesopen.Wenowconsider aﬁxedvalueofk_sandincreasethevalueofk_g,i.e.,wemakeahorizontal transectinFig.3.

Forrelativelysmallvaluesofksandkg,threeequilibriumsolutions

exist(Fig.11a).Forsuchsmallvalues of k_s,there existsathreshold valueforkg(𝑘𝑔=𝑘𝑔𝑎)forwhichthereisa ̂𝑄∗suchthat ̂𝑄∗−Φ(̂𝑄∗)=0

and𝑑(𝑄∗₋_Φ(_𝑄∗_))∕_𝑑_𝑄_|

𝑄∗₌̂𝑄∗=0.Thisimpliesthatforkg>kgatwonew

solutionsof𝑄∗₋_Φ(_𝑄∗₎₌₀_emerge_and_we_then_ﬁnd_ﬁve_equilibrium

solutions(Fig.11b):twowithonebranchclosedandthreewithboth branchesopen.Thethresholdvaluekgadependsonks anddeﬁnesthe

boundarybetweensectionsIandIIa,binFig.3.

Forlargervaluesofks,weﬁndanotherthresholdvalueforkg,which

we callkgb. At thatvalue, thenew solutionsof 𝑄∗−Φ(𝑄∗)=0that

computetheassociatedﬂowdepthinthetwobifurcates,H_i,and com-putetheothervariables(e.g.Sis,Sig,Fs,Fgetc)andtheeigenvaluesof

theJacobianJinEq.(23).

Fig.12ashowstheresultforQ∗_for_𝑘

𝑠=1and0<kg<4,forthree

valuesofL∗_._For_simplicity_we_do_not_indicate_the_stable_solutions_𝑄∗₌₀

and𝑄∗₌_∞._For_each_value_of_L∗_one_mathematical_bifurcation_occurs

ataspeciﬁcvalueofkg (blueskybifurcation,correspondingwiththe

occurrenceofthesolidlinesinFig.12a).For𝐿∗₌₁_.₃₃_the_bifurcation

intoastablesolutionoccursforkg>4inFig.12aandisthereforenot

visible.Thevaluesof kgforwhich thebifurcationsoccur correspond

withthetransitionsfromsectionItoII_ainFig.3.

Fig.12bshowsasimilarplotfor𝑘𝑠=2.5.Nowweobservetwo

bi-furcations:onewherethestablesolutionforQ∗_emerges_in_a_blue_sky

bifurcation(occurrenceofthesolidlinesinFig.12b),andonewhere the unstable solutions of Q∗ _are _annihilated _in _a _collision _with _the

stablesolutions 𝑄∗₌₀_and_𝑄∗₌_∞_(saddle-node_bifurcation,

vanish-ingofthedasheslinesat𝑄∗₌₀_),_leaving_only_one_single_stable

solu-tioninsectionIII.Wesummarizetheconsequencesoftheseresultsin Section5.

Fig.11. Typical graphs of 𝑄∗₋_Φ(_𝑄∗_{) for parameter values in sections I (left plot), II}

a,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_.

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Fig.12. Bifurcation diagram for the equilibrium values of Q∗_as_a_{function of} kgfor (a) 𝑘𝑠= 1 and (b) 𝑘𝑠= 2 .5 . Solid and dashed lines indicate, respectively,

stable and unstable solutions. For simplicity the stable solutions 𝑄∗₌₀_and_𝑄∗₌ ∞are not shown. (For interpretation of the references to colour in this ﬁgure legend, the reader is referred to the web version of this article.)

AppendixB. Functionsinthesystemofequations

Thisappendixprovidesthefunctionsforg1,g1g,g2,g2g, ̄𝑔1, ̄𝑔1𝑔, ̄𝑔2,

and̄𝑔2𝑔,requiredinthesystemdescribedbyEqs.(19)–(22):

𝑔1(𝐻1,𝐻2,𝐹1𝑔)=(𝑚𝑔𝐹1𝑔+𝑚𝑠(1−𝐹1𝑔)) 𝛾1𝐻 (𝑛∕2) 1 ( 𝛽1𝐻 (3∕2) 1 +𝛽2𝐻 (3∕2) 2 )𝑛 𝑔1𝑔(𝐻1,𝐻2,𝐹1𝑔)=𝑚𝑔𝐹1𝑔 𝛾1𝐻 (𝑛∕2) 1 ( 𝛽1𝐻 (3∕2) 1 +𝛽2𝐻 (3∕2) 2 )𝑛 𝑔2(𝐻1,𝐻2,𝐹2𝑔)=(𝑚𝑔𝐹2𝑔+𝑚𝑠(1−𝐹2𝑔)) 𝛾2𝐻 (𝑛∕2) 2 ( 𝛽1𝐻 (3∕2) 1 +𝛽2𝐻 (3∕2) 2 )𝑛 𝑔2𝑔(𝐻1,𝐻2,𝐹2𝑔)=𝑚𝑔𝐹2𝑔 𝛾2𝐻 (𝑛∕2) 2 ( 𝛽1𝐻 (3∕2) 1 +𝛽2𝐻 (3∕2) 2 )𝑛 ̄𝑔1(𝐻1,𝐻2)= 𝐵(1−𝑛) 0 𝐻𝑛 0 (_𝑓 𝑔(𝐻1,𝐻2)𝐹0𝑔𝑚𝑔 1+𝑓_𝑔(𝐻1,𝐻2) +𝑓𝑠(𝐻1,𝐻2)(1−𝐹0𝑔)𝑚𝑠 1+𝑓_𝑠(𝐻1,𝐻2) ) ̄𝑔1𝑔(𝐻1,𝐻2)= 𝐵(1−𝑛) 0 𝐻𝑛 0 (_𝑓 𝑔(𝐻1,𝐻2)𝐹0𝑔𝑚𝑔 1+𝑓_𝑔(𝐻1,𝐻2) ) ̄𝑔2(𝐻1,𝐻2)= 𝐵(1−𝑛) 0 𝐻𝑛 0 ( _𝐹 0𝑔𝑚𝑔 1+𝑓_𝑔(𝐻1,𝐻2) + (1−𝐹0𝑔)𝑚𝑠 1+𝑓_𝑠(𝐻1,𝐻2) ) ̄𝑔2𝑔(𝐻1,𝐻2)= 𝐵(1−𝑛) 0 𝐻𝑛 0 ( _𝐹 0𝑔𝑚𝑔 1+𝑓_𝑔(𝐻₁,𝐻₂) ) where 𝛽𝑖=𝐵𝑖𝐶𝐿(−1∕2)𝑖 , 𝑖=1,2 𝛾𝑖=𝐵_𝑖𝐶𝑛𝐿_𝑖(−𝑛∕2), 𝑖=1,2 and 𝑓𝑔(𝐻1,𝐻2)= ̄𝑆_𝑔∗= ̄𝑆1𝑔 ̄𝑆2𝑔 =𝐻∗(3𝑘𝑔∕2)_𝐵∗(1−𝑘𝑔) ( 𝐵∗ √ 𝐿∗ )𝑘𝑔 𝑓𝑠(𝐻1,𝐻2)= ̄𝑆𝑠∗= ̄𝑆_̄𝑆1𝑠 2𝑠 =𝐻∗(3𝑘𝑠∕2)_𝐵∗(1−𝑘𝑠) ( 𝐵∗ √ 𝐿∗ )𝑘𝑠

whichfollowsfromsubstitutionofEqs.(2)–(3)and(11)intoEqs.(9)– (10).

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