ContentslistsavailableatScienceDirect
Advances
in
Water
Resources
journalhomepage:www.elsevier.com/locate/advwatres
A
regularization
strategy
for
modeling
mixed-sediment
river
morphodynamics
Víctor
Chavarrías
a,∗,
Guglielmo
Stecca
b,
Annunziato
Siviglia
c,
Astrid
Blom
a a Faculty of Civil Engineering and Geosciences, Delft University of Technology, Delft, The Netherlandsb National Institute of Water and Atmospheric Research (NIWA), Christchurch, New Zealand
c Laboratory of Hydraulics, Hydrology and Glaciology, Swiss Federal Institute of Technology (ETH), Zurich, Switzerland
a
b
s
t
r
a
c
t
Anotabledrawbackinmixed-sizesedimentmorphodynamicmodelingisthefactthatthemostcommonlyusedmathematicalmodelinthisfield(i.e.,theactivelayer modelHirano,1971)canbeill-posedundercertaincircumstances.Undertheseconditionsthemodellosesitspredictivecapabilities,asnegligibleperturbationsin theinitialorboundaryconditionsproducesignificantdifferencesinthesolution.Inthispaperweproposeapreconditioningmethodthatregularizesthemodelto recoverwell-posednessbyalteringthetimescaleofthesedimentmixingprocesses.Wecompareresultsoftheregularizedmodeltodatafromfournewlaboratory experimentsconductedunderconditionsinwhichtheactivelayermodelisill-posed.Theregularizedactivelayermodelcapturesthechangeofbedelevationand surfacetextureaveragedoverthepassageofseveralbedforms.Neithertheactivelayermodelnortheregularizedoneaccountforsmallscalechangesduetoindividual bedforms.
1. Introduction
Thepresenceofmixed-sizesedimentisakeyfeatureofrivers. Sed-iment sortingpatternsdevelopin thestreamwisedirection (e.g.,the characteristicdownstreamfiningprofileSternberg,1875),inthe trans-versedirection(e.g.,bendsortingAllen,1970),andinthevertical di-rection(e.g.,bedarmoringParkerandKlingeman,1982anddune sort-ingBlometal.,2003).Modelingapplicationsinwhichthemixed-size characterofrivermorphodynamicsisnotnegligiblemandatetheuseof asuitablecontinuitymodelaccountingformassconservationofeach oftheconsideredsedimentsizefractions.Hirano(1971) wasthefirst todevelopa mixed-sedimentcontinuitymodel. Heassumed thatthe river bed can be vertically dividedinto an active top part(the ac-tivelayer),which interacts withtheflow,andaninactivesubstrate. Inthemodel,sedimenttransport andfrictiondependonthetexture oftheactivelayer,whereas thesedimentin thesubstrateonlyplays aroleifnetaggradationcreatesnewsubstratesedimentornet degra-dationleadstotheentrainmentofsubstratesedimentintotheactive layer.
Althoughithasbeenfruitfullyusedtorepresentphysicalphenomena relatedtomixed-sedimentfornearlyhalfacentury(seeChavarríasetal., 2018), the activelayer model suffers from a drawback. Under cer-tainconditionsitbecomesill-posed(Chavarríasetal.,2018;Ribberink, 1987; Steccaet al., 2014).Amodel isill-posed ifaunique solution doesnotexist,orifthesolutiondoesnotdependcontinuouslyonthe initialandboundaryconditions(Hadamard,1923).Ifa modelis ill-posed, infinitesimal variations in theinitial or boundary conditions
∗Correspondingauthor.
E-mailaddress:v.chavarriasborras@tudelft.nl(V.Chavarrías).
yieldasignificantdeviationofthesolutionwithinaninfinitesimaltime (Hadamard,1923).Whensolvingthemathematicalmodelby numeri-calapproximations,perturbationsintheinitialandboundaryconditions simplyarisebytruncationerrors.Thismakesanill-posedmodel unsuit-ableinpractice.
Theproblemofill-posednessarisesfromaninaccurate representa-tionof thephysical processes(JosephandSaut,1990).Forinstance, atwo-fluid modelforincompressibleandinviscidflow intwolayers withavelocitydiscontinuityisill-posed(vonHelmholtz,1868;Kelvin, 1871).Itisregularized(i.e.,becomeswell-posed)ifviscouseffectsare takenintoaccount(JosephandSaut,1990).Fromthisperspective,the preferredapproachtoregularizetheactivelayermodelwouldbe the developmentofanewmodelthatincludesthosephysicalmechanisms thatarenotaccountedforbytheactivelayermodel.
There exist alternatives to the active layer that typically aim to improve the physical description of sediment mixing process. Ribberink(1987) introducedasecondlayertoaccountforthemixing duetodunesexceptionallylargerthantheaverageduneheight.Besides producingaverticalsortingprofilethatbetterreproducestheresults ofalaboratoryexperiment(Blom,2008),Ribberink’stwo-layermodel makestheoccurrenceofill-posednesslesslikely,althoughitdoesnot completelyavoidit(Sieben,1994).Luuetal.(2006,2004)proposeda modelinwhichtheactivelayerisreplacedbythesedimenttransport layerrepresentingthesedimentintransportratherthanthesediment atthebedsurface.Thethicknessofthesedimenttransportlayeris esti-matedwithaclosurerelationsuchastheonedevelopedbyEgashira and Ashida (1992). Although conceptually different, the model by
https://doi.org/10.1016/j.advwatres.2019.04.001
Received12June2018;Receivedinrevisedform20February2019;Accepted2April2019 Availableonline5April2019
0309-1708/© 2019TheAuthors.PublishedbyElsevierLtd.ThisisanopenaccessarticleundertheCCBY-NC-NDlicense. (http://creativecommons.org/licenses/by-nc-nd/4.0/)
Luuetal.(2006,2004) ismathematicallyequivalenttotheactivelayer model,whichimpliesthatitcanalsobeill-posed.
BlomandParker(2004) andBlometal.(2006,2008) developeda modelinwhichbothbedelevationandbedgrainsizedistributionare treatedusingaverticallycontinuousformulation(Parkeretal.,2000). Thisimpliesthatthereisnodistinctionbetweentheactiveand inac-tivepartofthebed.ThemodelbyBlomandcoauthorssatisfactorily describestheverticalstratigraphyduetodunesatlaboratoryscale,but itrequiresatimesteptoosmalltobeapplicableatlargescale. More-over,itswell-posednesshasnotbeenstudied.Simplifyingthe contin-uousframeworkproposedbyParkeretal.(2000),thevertically con-tinuousmodelbyViparellietal.(2017) overcomestheneedforasmall timestep.Althoughapplicableatlargespatialandtemporalscales,their modeldoesnotsolvetheproblemofill-posedness(Chavarríasetal., 2018).
Giventhefacts that:(a)There isnotyeta practicallyfeasible al-ternativetotheactivelayermodel,(b)theactivelayermodelremains well-posedoveralargerangeofapplications(Chavarríasetal.,2018), and(c)itisacomputationallycheapmodelimplementedinseveral soft-warepackages,hereourobjectiveistodevelopastrategytoavoid ill-posednesswhilemaintainingtheconceptualframeworkoftheactive layermodel.Tothisend,wedeveloparegularizationstrategythat re-coverswell-posednessoftheactivelayermodelandweconduct4 labo-ratoryexperimentsunderconditionsinwhichtheactivelayermodelis ill-posedtoobtaindatatowhichwecomparetheresultsofour regular-izedmodel.
Thepaperisorganized asfollows.InSection2 we review strate-giesforregularizingill-posedmodels.InSection3 wepresentthe reg-ularizationstrategy.Section4 presentsthelaboratoryexperimentsand Section5 focusesonthenumericalrunstoreproducethe experimen-talresults.InSection6 wediscussthelimitationsoftheregularization strategy,aswellasotherpossiblemodelingstrategies.
2. Overviewofregularizationtechniques
Inthis section we review techniquesused toregularize ill-posed problems.Propagationproblemsaremost completelymathematically representedby a setof partialdifferential equationsconstituting an initialvalueproblem. Intheseproblemsaninitialstatechangeswith timesubjecttoconditionsattheboundariesofthedomain.The matrix-vectorformulationprovidesacompactexpressionofthesetof equa-tions(e.g.,CourantandHilbert,1989;LynandGoodwin,1987;Toro, 2001):
𝜕𝐐
𝜕𝑡 +𝐀𝜕𝐐𝜕𝑥 =𝐒, (1)
whereQisthevectorofdependentvariables,Aisthesystemmatrix,and Sisthevectorofsourceterms.Thevelocityatwhichsmallwaves prop-agatethroughoutthedomain(i.e.,theeigenvaluesofmatrixA)must berealfortheproblemtobewell-posed(e.g.,Hadamard,1923;Ivrii andPetkov,1974;Kabanikhin,2008;Lax,1957;1958;1980;Mizohata, 1961).Whentheeigenvaluesarereal,theproblemishyperbolic.Ifthe eigenvalueshaveanimaginarycomponent(theproblembeingelliptic orofmixed-type),aninitialvalueproblemisill-posed.
Thetwo-fluidshallowflowmodel(i.e.,amodeloftheflowoftwo layers of superimposedfluids at different velocities) is knowntobe ill-posedwhenthedifferenceinflowvelocitybetween theupperand lowerlayers exceedsa certainthreshold(Ardron,1980; Armi,1986; Lawrence,1990;Long,1956;Pelantietal.,2008).Ingeneralterms ill-posednessarisesinmultiphasemodels(e.g.,bubblesinafluid)(Harlow andAmsden,1975;KumbaroandNdjinga,2011;Murray,1965; Stew-art,1979;StewartandWendroff,1984).Multiphasemodelsare regu-larizedbyaccountingfortheforcesattheinterfacebetweenthetwo fluids(AbgrallandKarni,2009;Drewetal.,1979;Liskaetal.,1995; Lyczkowskiet al., 1978;Ramshaw andTrapp, 1978;Stewart, 1979;
Stuhmiller,1977;TiseljandPetelin,1997;Travisetal.,1976).Although thephysicsofmultiphaseproblemsisbetterrepresentedwhen includ-ingtheeffectsoftheinterfaceforces,thisapproachdoesnotcompletely eliminatethepossibilityoftheproblembeingill-posed.
Fernández Nieto (2003), Castro Díaz et al. (2011), and Sarno et al. (2017) introduce an additional term in the momen-tumequations toaccount forfrictionbetweenthefluidlayers.Their regularization strategyyieldsa well-posedmodelandhasaphysical origin.However,theadditionalphysicaltermdependsonthetimestep ofthenumericalsolution,whichimpliesthatitcannotbeconsidereda fullyphysically-basedsolution.
Thenumericalsolutionofamathematicallyill-posedmodelcanbe well-posed(ChenandPeng,2006;Chenetal.,2007;SavaryandZech, 2007;Spinewineetal.,2011)ifthenumericalsolutionneglects informa-tioninthephysicalequations(Grecoetal.,2008).Wordeddifferently, insuchacasethephysicalequationsareill-posed,butthenumerical equationsthatweactuallysolvearewell-posed.Inparticular,when us-ingtheHLLsolver(acommonapproximateRiemannsolverproposed byHartenetal.,1983,seeToro,2009),oneonlyusesthefastestand slowesteigenvaluesofthesystem,whichimpliesthatthedynamicsdue totheintermediateceleritiesarenotresolved.Thishidestheproblem ofill-posednessratherthansolvesit.
Indetermining thesteady(equilibrium)stateofa fluiddynamics problem,acommonlyadoptedstrategytoachievefastconvergenceis tomodifytheceleritiesatwhichinformationpropagates(i.e.,the sys-temeigenvalues)(Chorin,1967;GrabowskiandBerger,1976;Plows, 1968;SohandBerger,1984).Forinstance,inaerodynamics,thespeed of soundmaydiffersignificantlyfrom theair velocity,which causes aslowconvergencetosteadystate(ChoiandMerkle,1993;Fengand Merkle,1990;Godfreyetal.,1993;vanLeeretal.,1991). Precondition-ingmethods(Turkel,1987;1993;1999)aimatbringingthe eigenval-ues ofthesystemclosertoeachothersuchthatalargertimestepis allowed.
Analogously,the“bedcelerity” (i.e.,thespeedofthewaverelatedto changesinbedelevation(LynandAltinakar,2002;MorrisandWilliams, 1996;Steccaetal.,2014;DeVries,1965))isgenerallyslowcompared totheceleritiesassociatedwithperturbationsoftheflow.Thisfacthas encouragedtheuseofa“morphodynamicaccelerationfactor” in mor-phodynamicmodelingtoreducethecomputationaltime(Latteux,1995; Lesseretal.,2004;Ranasingheetal.,2011;Roelvink,2006). Mathemat-ically,aswewillshowlater,theuseofamorphodynamicacceleration factorisequivalenttotheapplication ofaparticularpreconditioning method.
Byalteringthecelerityatwhichinformationpropagates,the tran-sientstateofthepreconditionedproblemisalteredwithrespecttothe originalproblem,butbothproblemsconvergetothesamesteadystate solutioniftheboundaryconditionsaresteady.Adrawbackof precondi-tioningisthefactthat,whentheproblemissubjecttounsteady bound-aryconditions,preconditioningmethodsmodifythesteadystate,asthey indirectlymodifythetimingoftheboundaryconditions(Turkel,1999). Forthisreason,theboundaryconditionsofapreconditionedmodelneed tobeadjustedifthesevarywithtime.
Thefactthatapreconditioningmethodaltersthetransientstatewas usedbyZanottietal.(2007) toregularizethetwo-fluidmodel.They modifiedthesystemofequationsbyintroducingtwoparameters.One parametermodifiesthecontinuityequation,whichaffectstheimaginary partoftheeigenvalues.Dependingontherelationsbetweenvelocities anddensitiesofthetwofluids,aspecificvalueofthisparametermakes theimaginarypartequaltozero.Apartfrommodifyingtheimaginary component,theparameteralsomodifiestherealpartoftheeigenvalues. Theyintroduceasecondparameterthataffectsallequationstorecover theoriginalrealpartoftheeigenvalues.Theycomparethesolutionof theregularizedmodeltoanalyticalsolutionsandtheyshow thatthe regularizedtwo-fluidmodelisstable.Inthenextsectionwewillfollow asimilarapproachtoderivearegularizationstrategyfortheactivelayer model.
3. Regularizationstrategyfortheactivelayermodel
Inthissectionweproposeastrategyforrecoveringthewell-posed character of the system of equations for modeling mixed-sediment rivermorphodynamics.Themodifiedsetofequationsispresentedin Section3.1.InSection3.2 -wederivetheparametersusedtorecover thewell-posedcharacterofthemodelconsideringasimplifiedcasewith twosedimentsizefractionsandsteadyflow,whichallowsustoobtain analyticalexpressions.Wethenextendthevaliditytounsteadyflow con-ditions(Section3.3)andtoconditionswithmorethan2sedimentsize fractions(Section3.4).InSection3.5wediscusstheimplementationof thestrategy.
3.1. Modifiedsystemofequations
Weconsiderone-dimensionalhydrostaticflowoverabedcomposed ofNnon-cohesive sizefractions.Theflow isdescribed bythe Saint-Venant(1871) equations.We assumea Chézy-typefrictionin which thenondimensionalfrictioncoefficientisindependentoftheflowand bed parameters. The sediment transport rateis considered toadapt instantaneouslytochanges in thebed shear stress(Belland Suther-land,1983).Themassconservationofthebedsedimentisdescribed bythe Exner (1920) equation,andthe 𝑁−1 activelayerequations (Hirano,1971)accountfortheconservationofthemassofeachgrain sizefractionwithinadiscretetoplayerofthebedsurface(i.e.,the ac-tivelayer).Giventheflow,friction,andsedimenttransportassumptions, themodelcannotrepresentsmall-scaleprocesses(i.e.,processesatthe scaleof bedelevationfluctuationsdue thestochasticnatureof sedi-menttransport,ripples,dunes,orbedloadsheets).Inotherwords,the variablesrepresentparametersaveragedoveraperiodlargerthanthe characteristictimeofsmall-scalebedelevationfluctuations(Armanini anddiSilvio,1988;Blometal.,2008;Parkeretal.,2000;Ribberink, 1987;WongandParker,2006).WerefertoAppendixA forthemodel equationsandthematrix-vectorformulationofthesystem.
AnalogoustoZanottietal.(2007) (Section2),thesystemof equa-tionsinEquation(1) ismodifiedmultiplyingthetimederivativeterm byadiagonalmatrixMtoregularizetheproblem:
𝐌𝜕𝐐𝜕𝑡 +𝐀𝜕𝐐
𝜕𝑥 =𝐒. (2)
MatrixMmodifiesthetransientstateonly.Thepreconditioning tech-niquedoesnotaffectthesolutionofthesteadystate(i.e.,𝜕∕𝜕𝑡=0).
Themorphodynamicmodelunderunisizeconditionswasanalyzed byCordieretal.(2011).TheyfoundthattheSaint-Venant-Exnermodel isalwayswell-posedassumingaChézy-typefriction.Thisconfirmsthat theill-posedcharacterofthemixed-sizesedimentmodelresultsfromthe inappropriaterepresentationofthemixingprocessesbytheactivelayer model(Chavarríasetal.,2018).Forthisreason,weproposea regulariza-tionstrategythatrecoversthewell-posedcharactermodifyingthe celeri-tiesatwhichmixedsedimentprocessesoccur.Thisisdonebymeansofa setofparameters𝛼k[– ]for1≤𝑘≤𝑁−1thatmultiplythetime deriva-tiveofeachactivelayerequation.SimilarlytoZanottietal.(2007),we consideraparameter𝛽 [– ]thatcanbeusedtorescalethecelerities af-terbeingmodifiedby𝛼k.Westipulatethatthisparameter𝛽 affectsonly thesedimentprocesses(includingtheExner(1920) equation)butnot theflow.
Themodifiedsystemofequationsmustbemassconservativewith respecttothesediment.Thisimpliesthat𝛼kcannotbegrainsize de-pendent(i.e.,𝛼𝑘=𝛼 ∀k)andthatthepreconditioningtechniqueisonly applicablewhentheactivelayerthicknessisconstant(AppendixB).
3.2. Derivationoftheregularizationcoefficients
Inthissectionwederivethevaluesofthecoefficients𝛼 and𝛽 that
enableregularizationoftheactivelayermodel.Weconsidera simpli-fiedcasewithtwosedimentsizefractionsundersteadyflowconditions,
asthisallowsustoobtainanalyticalexpressionsoftheregularization parameters.
Inthiscase,thedependentvariablesofthesystemarethebed eleva-tion𝜂 [m]andthevolumeoffinesedimentintheactivelayerperunit ofbedarea,Ma1[– ](Chavarríasetal.,2018andAppendixA): 𝐐s2=
[
𝜂,𝑀a1
]⊺
. (3)
Thesystemmatrixis: 𝐀s2=𝑢 ⎡ ⎢ ⎢ ⎣ 𝜆b 𝜆s1 𝜇1,1 𝜆b𝛾1 𝜆s1 ⎤ ⎥ ⎥ ⎦, (4)
wheretheparameters𝜆b [−]and𝜆s1[−]arethenondimensional
ap-proximated bed andsorting celerities,which (approximately) repre-senttheceleritiesatwhichinfinitesimalperturbationsinbedleveland grain size distributionof thebed surfacepropagatethrough the do-main(Chavarríasetal.,2018;Steccaetal.,2014;DeVries,1965 and AppendixA.5),andu[m/s]isthemeanflowvelocity.Theparameters
𝛾1 [−]and𝜇1,1[−]relate thechangesin thesedimenttransport rate
tothepropertiesofthebed(AppendixA.5).Subscriptsindicatesthat themodelissteadyandsubscript2highlightsthatitaccountsfortwo sedimentsizefractionsonly.
Thepreconditioningmatrixis: 𝐌s2=𝛽 [ 1 0 0 𝛼 ] . (5)
Notethat𝛽 doesnotaffectthemathematicalcharacterofthesystem, asitmodifiesallequationsequally.Wordeddifferently,theparameter
𝛽 changesthemagnitudeoftheeigenvaluesbutnotthetype(realor complex). Wecomputetheeigenvalues(𝜆k for𝑘=1,2)of the modi-fiedsystemofequationsastherootsofthecharacteristicpolynomial det(𝐌s2𝜆 −𝐀s2)=0: 𝜆𝑘=2𝑢𝛽 ( 𝜆b+ 𝜆s1 𝛼 ± √ Δ 𝛼 ) for 𝑘=1,2, (6)
wherethediscriminantΔisaseconddegreepolynomialon𝛼 equalto: Δ =𝜆2b𝛼2+2𝜆b𝜆s1 (2𝛾 1 𝜇1,1 −1 ) 𝛼 +𝜆2 s1. (7)
Weconsiderasituationwhichisill-posediftheregularization strat-egyisnotapplied.Thisimpliesthatwhen𝛼 =1(theregularization strat-egyisnotapplied),Δ<0(theeigenvaluesarecomplex).Weaimto mod-ifythesystemofequationsaslittleaspossibleinregularizingit.Worded differently,weaimatchanging𝛼 aslittleaspossiblefrom1.The mini-mummodificationisobtainedwhenthediscriminantisequalto0(i.e., theeigenvaluesareinthelimitforhavinganimaginarypartdifferent than0).Thethresholdvalues𝛼cthatmodifythesystemofequationsas
littleaspossiblearefoundbyequating(7) tozero:
𝛼c= 𝜆s1 𝜆b ( 1−2 𝛾1 𝜇1,1 ±2 √ 𝛾1 𝜇1,1 ( 𝛾 1 𝜇1,1 −1 )) . (8)
Therearetwopossiblevaluesof𝛼cthatyieldrealeigenvalues.The
discriminant(Eq.(7))asafunctionof𝛼 isaconcaveparabolaas𝜆2 b>0.
Moreover,when𝛼 =0,Δ =𝜆2s1>0andwhen𝛼 =1,Δ<0.Thisshows thatonecriticalvalueofparameter𝛼 isbetween0and1,andthesecond valueislargerthan1(i.e.,0<𝛼c1<1<𝛼c2).
Wecomputethevalueofparameter𝛽 that,assuming𝛼 =𝛼c,
recov-erstherealpartthattheeigenvalueswouldhaveiftheyhadnotbeen modifiedusingtheparameter𝛼:
𝛽 = 𝜆b+𝜆s1∕𝛼c
𝜆b+𝜆s1
. (9)
Inthiscase,irrespectiveof thevalueof 𝛼,theeigenvaluesof the regularizedsystemareequalto:
𝜆𝑘=𝑢2(𝜆b+𝜆s1
)
Ifwedonotuse𝛽 torecovertheoriginalrealpartoftheeigenvalues (i.e.,if𝛽 =1),theeigenvaluesoftheregularizedsystemareequalto
𝜆𝑘= 𝑢2 ( 𝜆b+ 𝜆s1 𝛼 ) for 𝑘=1,2. (11)
Parameter𝛼 canbeselectedtobelargerorsmallerthan1andif wechoosetouse𝛽 (i.e.,if𝛽 ≠ 1)theeigenvaluesareindependentof𝛼.
Summarizing,wefindthreepossibleregularizationstrategies: 1. 𝛼 ≠ 1and𝛽 ≠ 1
2. 𝛼 <1and𝛽 =1 3. 𝛼 >1and𝛽 =1
Ingeneralterms,theapproximatedsortingceleritiesarepositive, andundersubcriticalflowconditions(i.e,Fr<1)theapproximatedbed celerityisalsopositive.However,duetohidinginthesediment trans-portrelation,underconditionsinwhichill-posednesslikelyoccurs,𝜆s1
maybenegativeregardless oftheFroudenumber(Chavarríasetal., 2018).Inthiscase,Strategies1and2donotguaranteethatthe eigen-values𝜆k>0.Weconsiderthatitisphysicallyunrealisticthat morpho-dynamicinformationtravelsintheupstreamdirectionundersubcritical flowconditions.Anegativeeigenvaluewouldimplythattheboundary conditionformorphologyneedstobeimposedatthedownstreamend toyieldawell-posedmodel,andthisiscontradictorytothefactthat themorphodynamicstateundersubcriticalflowconditionsdependson theloadcomingfromupstream(Blometal.,2017a;2016).Ontheother hand,Strategy3guaranteesthat𝜆k>0(AppendixAofthe supplemen-tarymaterial).Thus,weconsiderthattheonlypossibleregularization strategyistheoneinwhich𝛼 >1and𝛽 =1.
Weneedtoguaranteethattheceleritiesofthesystemofequations modifiedbytheregularizationstrategyarenotphysicallyunrealistic.In particular,underasufficientlysmallFroudenumber,themodifiedbed andsortingceleritiesmustbesignificantlysmallerthanthecelerities oftheflow.Theregularizationtechniquedoesnotmodifythe approx-imatedcelerityassociatedwithbedelevationchanges(i.e.,𝛽 =1)and decreasesthecelerityassociatedwithmixingprocesses(i.e.,𝛼 >1,we willdiscussthispointinSection6.1).Forthisreason,theregularization techniquedoesnotcausetheceleritiestobephysicallyunrealistic.
Theregularizationstrategyisnotlimitedtoaparticularrangeof pa-rametersettings.Yet,whenusingthevalueof𝛼 derivedinthissection, theFroudenumbercannotbeinthetranscriticalregion,asinthiscase thequasi-steadyapproximationisnotvalid(CaoandCarling,2002b; Caoetal.,2002;ColombiniandStocchino,2005;Lyn,1987;Lynand Altinakar,2002;Sieben, 1999).Inthefollowing sectionweconsider unsteadyflow,whichextendstheregularizationtechniquetothe trans-criticalregion.
3.3. Validityunderunsteadyflowconditions
Inthissectionweextendthevalidityoftheregularizationparameter
𝛼 foundforsteadyflowcases(Section3.2)tounsteadyflowconditions.
Fig. 1. Maximumimaginarypartofalltheeigenvaluesofthereferencecase (Table1)asafunctionof𝛼.Inthiscase𝛼c=16.1isthesmallestvalueof𝛼 >1
thatyieldsawell-posedmodel(i.e.,alleigenvaluesarereal). Table 1
Referencevaluesinthecomparisonofthevalueof𝛼ccomputedanalyticallyand
numerically.
u [m/s] h [m] Cf [ − ] La [m] Fa1 [ − ] 𝑓 I
1 [ − ] d1 [m] d2 [m]
1 1 0.01 0.20 0 1 0.001 0.005
Whenconsideringunsteadyflowconditions,wecannotobtainan an-alyticalexpressionof𝛼cforregularizingthesystemofequations.
Nev-erthelesswecannumericallyfindthesmallestvalueof𝛼 >1forwhich therootsofthecharacteristicpolynomialofdet(𝐌u𝜆 −𝐀u)=0arereal values(i.e.,theeigenvaluesarereal),wheresubscriptuindicatesthat themodelisunsteady.MatricesMuandAuarelistedinAppendixA.5.
Thisprocedureisnonethelessexpensivecomputationallyincomparison withanalgebraiccalculation.Fig.1showsthemaximumimaginarypart ofalleigenvaluesofareferenceill-posedcase(Table1)considering un-steadyflowforvarying𝛼.Thesedimenttransportrateiscomputedusing afractionalversionoftheEngelundandHansen(1967)sediment trans-portrelation(Blometal.,2017a).Avalue𝛼 >16.1yieldsawell-posed model(i.e.,alleigenvaluesarereal).
Totestthevalidityofthealgebraicvalueof𝛼c obtainedassuming
steadyflowweconsiderthesamereferencecase(Table1)andwevary theflowvelocitytoobtainarangeofconditions.InFig.2(a)wepresent thevalueof𝛼cnecessarytoobtainawell-posedmodelcomputed
assum-ingsteadyflow(Eq.(8))andnumericallyconsideringunsteadyflow.We concludethatforaFroudenumberbelowapproximately0.6,thereisno significantdifferencebetweenthevaluesforsteadyandunsteadyflow. Thisimpliesthat,forFr<0.6,thevalueof𝛼cobtainedanalytically
as-sumingsteadyflowisagoodapproximationoftheactualvalue. Fig. 2. Comparisonbetween(a )thesteadyandunsteady val-uesoftheregularizationparameter𝛼c,and(b )theexactand
3.4. Validityundermultiplesizefractionsconditions
Inamodelwithmorethan2sizefractions,wecannotanalytically obtainthevalueof𝛼cthatregularizestheactivelayermodel.Similar
totheunsteadycase,itispossibletonumericallyobtainthesmallest valueof𝛼 >1thatyieldsrealeigenvaluescomputedastherootsofthe characteristicpolynomialdet(𝐌s𝜆 −𝐀s)=0(matricesMsandAsare
pre-sentedinAppendixA.5).Again,thisprocessisrelativelyexpensivein computationalterms.Inthissectionweproposeamethodtoobtainan approximatevalueof𝛼cforsuchcasesandcompareittotheexactvalue
obtainednumerically.
Assumingsteadyflow,asystemthatmodelsNsedimentsize frac-tionshasNequations(AppendixA).WereducethesystemofN equa-tionstoanapproximatesystemof2equationsfollowingtheapproach ofRibberink(1987).WesumtheNactivelayerequationstoobtainone equationthatmodelsthechangesofthemeangrainsizeofthebed sur-facesediment(AppendixA.3).Subsequently,weapplythesame tech-niqueastheonewehaveusedinthecaseof2sizefractionstoobtaina criticalvalueof𝛼 thatguaranteesthattheapproximatemodelis well-posed: 𝛼cm= 𝜆m 𝜆b ⎛ ⎜ ⎜ ⎝ 1−2𝛾m 𝜇m ±2 √ 𝛾m 𝜇m (𝛾 m 𝜇m −1 )⎞ ⎟ ⎟ ⎠, (12)
wherethesymbolsaretheequivalentofthecasefortwosizefractions intheapproximatemodel(AppendixA.5)
Weconsideracasewith3sedimentsizefractions,wherethefine andcoarsefractionshavecharacteristicsizesequalto𝑑1=0.001mand
𝑑3=0.005m,respectively.Thevolumefractioncontentsofthe3size
fractionsintheactivelayerare𝐹a1=0,𝐹a2=0.9,and𝐹a3=0.1.The
substrateisfullycomposedoffinesediment.Wevarythemediumgrain size(d2)toobtainarangeofconditions.Theremainingparametersare
thesameastheonespresentedinTable1.InFig.2(b)wecomparethe exactvalueof𝛼c(computednumerically)totheapproximatedone
(com-putedusingEq.(12)).Theapproximatedvalueof𝛼cfollowsthesame
trendastheexactone.However,theapproximatedvalueisbothlarger andsmallerthantheexactonedependingonthesedimentconditions. Thisimpliesthatthecurrentapproximateapproachmaybeinsufficient toregularizetheactivelayermodelinthecaseofmorethan2sediment sizefractions.
Theapproximatesystemofequationscanbeill-posedunder degra-dationalconditionsintoafinesubstrateonly(Chavarríasetal.,2018; Ribberink,1987).However,a3sizefractionscasecanbeill-posed un-derdegradationalconditionsintoacoarsesubstrate(Chavarríasetal., 2018),whichfurtherlimitstheapplicabilityoftheapproximatesolution forthethresholdvalueof𝛼.
3.5. Implementation
Inthissectionwedescribeourapproachfornumericallysolvingthe systemofequationsandapplytheregularizationstrategy.
WehavedevelopedthenumericalresearchcodeElvtomodel mixed-sizesedimentrivermorphodynamics(Blometal.,2017a;2017b)which solvestheequationsforflow,bedelevation,andthebedsurfacegrain sizedistributioninadecoupledmanner(i.e.,inseriesandnotasa cou-pledsystemofequations).Thus,ourcodeisnotappropriatefor solv-ingtranscriticalsituations(Lyn,1987;LynandAltinakar,2002;Sieben, 1999)orcaseswithahighsedimentconcentration(CaoandCarling, 2002a;MorrisandWilliams,1996).
Theone-dimensionalspatialdomainis discretizedusing an equis-pacedgrid.Allvariablesarecomputedatthecellcentersandare con-sideredconstantineachtimestep.Hereweassumesteadyflow,whichis representedbythebackwaterequation(Eq.(16)).Thisordinary differ-entialequationisintegratedusingthestandardfourth-orderfinite differ-enceRunge–Kuttamethod(RK4).TheExner(1920) equation(Eq.(17)) andactivelayerequation(Eq.(19))aresolvedinconservativeform
us-ingafirstorderupwindschemeincombinationwithforwardEulerto integrateintime.Wediscretizetheverticaldomaininafinitenumber ofcellshavingacertainthicknesstoaccountforstratigraphicchanges inthesubstrate.Ourschemeisbalancedfortheverticalfluxesbetween theactivelayerandthesubstrate(Steccaetal.,2016).Thismeansthat massconservationisguaranteedindependentofthesubstrate discretiza-tion.ThetimestepvarieswithtimeandiscomputedsuchthattheCFL number(Courantetal.,1928)isconstantandequalto0.9(Toro,2009; Wu,2007).Thedetailsofthenumericalimplementationaredescribed inAppendixBofthesupplementarymaterial.
Whentheregularizationstrategyisapplied,wefirstdeterminethe mathematicalcharacter ofthemodel(i.e.,well-posedorill-posed)at eachnodeusingtheapproachproposedbyChavarríasetal.(2018).For thecaseof2sizefractions,thisisdoneevaluatinganalgebraicequation, andformorethan2sizefractionswenumericallycomputethe eigen-valuesofthesystemmatrix.Atcontinuation,foreachnodewecompute thethreshold value𝛼c that guaranteesthatthe modeliswell-posed.
Again,this isdoneevaluatinganalgebraicexpression(Eq.(8)) for2 sizefractionsanditisdonenumericallyformorethan2sizefractions (Section3.4).
Theregularizationstrategyyieldsequaleigenvalues(i.e.,inatwo sizefractionscase𝜆1=𝜆2,Eq.(11)).Thisimpliesthattheproblemis
hyperbolicbutnotstrictlyhyperbolic(Cordieretal.,2011;Lax,1980; Toro,2009).Inanon-strictlyhyperbolicproblem,thesolutionmaynot be uniqueandresonancemayoccur, whichgivesrisetostrong non-linearinteractions(IsaacsonandTemple,1992;Liu,1987).Inavoidinga non-strictlyhyperbolic-problem,wemodifythevalueof𝛼cusingasmall
parameter𝜖 >0[– ]suchthat𝛼∗=𝛼
c(1+𝜖),where𝛼∗[– ]isthevalue
usedforupdatingthebedsurfacegrainsizedistribution.Forthecases wehavestudiedavalueof𝜖 =0.005issufficienttoavoidtheproblems associatedwithnon-stricthyperbolicity.
Ill-posednesscausesshort-waveinstability(Chavarríasetal.,2018; JosephandSaut,1990 andSection5.1.2)meaningthatperturbations willgrowunstableatratesdependingontheinverseoftheirlength. Dif-fusioncounteractstheseeffectsbydampeningperturbations(Grayand Ancey, 2011). Regularization of the problem can be provided by numericaldiffusionifafirst-order(diffusive)methodisused.However, iftheunderlying problemis ill-posed,cellrefinementwillbeableto revealitsill-posedcharacterevenifafirst-ordermethodisusedinits solution,aswedointhispaper.Thisisbecause,withdecreasingcellsize, thenumericaldiffusioncoefficientofafirst-ordermethodwillgenerally decrease,whileatthesametimeshorter(moreunstable)perturbations willbe solved.Therefore,anill-posedproblemwillshowno conver-genceduetoitsinherentinstabilitywhenthemeshisprogressively re-fined,regardlessofthelow-ordermethodinuse.
Weobservesuchabehaviorin Section5.1.2 whereweshowthat ourlow-ordernumericalschemesufficestocapturetheconsequences ofill-posednessbyrevealinginstabilityandnon-convergingcharacter in simulationsconductedwithintheill-posedrange. Itislikely that, withahigher-order(non-diffusive)method,thesefeatureswouldhave becomeapparentevenatlowermeshresolutionduetoabsenceof spuri-ousdiffusiondampeningperturbations.However,itmustbeconsidered thatourupwindschemeischaracterizedbysmallnumericaldiffusion coefficient,andthatSteccaetal.(2016) andSivigliaetal.(2017)have shownthatafirst-orderupwindschemewithafinegridresolutionis sufficienttocapturethemainfeaturesofmixed-sizesediment morpho-dynamicsimulationssuchastheonesweconduct.
4. Laboratoryexperiments
Inthis sectionwe describethelaboratoryexperiments conducted underconditionsinwhichtheactivelayermodel(Hirano,1971)is ill-posedinordertoobtainadatasettowhichwecancomparetheresults oftheproposedregularizationstrategy.Wedescribetheexperimental plan, materials,andmeasurementsin Section4.1.InSection4.2 we presenttheexperimentalresults.
4.1. Experimentalplanandmeasurements
Weconducted4laboratoryexperiments(I1,I2,I3,andI4).The ex-periments reproduceddegradational conditionsinto afine substrate, whichareconditionspronetobeill-posed(Chavarríasetal.,2018; Rib-berink,1987;Steccaetal.,2014).Theexperimentswereconductedin a14mlong,0.40mwide,and0.45mhightiltingflumeintheWater LaboratoryoftheFacultyofCivilEngineeringandGeosciencesofDelft UniversityofTechnology.Attheupstreamend,aturbulencedissipation devicewasinstalled(item(a)inFig.3).Aninclinedplanewasplaced downstreamfromtheturbulencedissipationdevice(item(b)inFig.3) toallowforanalluvialbed(item(c)inFig.3).Thestructurewas cov-eredwithgluedsedimentsuchthatfrictionwassimilartotheoneofthe alluvialbed.Itselevationcouldbeadjusted.
Weconsiderareferencesystemwithcoordinateoriginatthebottom oftheflumeatthedownstreamendofthemetalstructure.Thez-axis isparalleltogravityandpointingup.Thex-axisfollowsthestreamwise directionoftheflume,beingpositiveinthedirectionoftheflow.The
y-axisisperpendiculartotheothertwoaxesformingarighthanded orthonormalbasis.
Weusedtwosedimentsizefractions(fineandcoarse)with charac-teristicgrainsizes(computedasthearithmeticmeanin𝜙 scale)equalto 2.1mmand5.5mm.Thestandarddeviationofthetwosizefractionsis 1.1mmand1.2mm,respectively.Thebedsurfacewasinitiallyflat,with aconstantslope,andcomposedofcoarsesedimentonly.Belowa0.03m thicklayerofcoarsesediment,weinstalledapatchoffinesedimentof varyinglengthLp[m](Fig.3 andTable2).Weimposedaconstant
wa-terdischargeandaconstantsedimentfeedrateofthecoarsefraction only,whichwasinequilibriumwiththeinitialcondition(Table3 and AppendixC ofthesupplementarymaterial).Thesedimentwas intro-ducedusingafeederplacedontopoftheflume(item(d)inFig.3).The downstreamwaterlevelwasloweredatarateof0.01m/hduring8h byadjustingasharp-crestedweirat𝑥=12.60m(item(g)inFig.3).The loweringofthewaterlevelledtobeddegradationandentrainmentof thefinesedimentinthepatch.Wehavetestedthatintheseconditions theactivelayermodelisill-posedregardlessoftheactivelayerthickness andsedimenttransportrelation.
Sedimentwascollectedinasandtrap(item(e)inFig.3)atthe down-streamendoftheflume(𝑥=12.10m).Thesedimentwaspumpedfrom thesandtrap(item(f)inFig.3)intoatankpositionedonaweight bal-ancenexttotheflume.Thissystemallowedustocontinuouslymeasure thesedimenttransportrate.Thewaterinflowwasmeasuredusingan
Table 2
Length(Lp)andposition(initialxp0andfinalxpfcoordinates)
ofthepatchoffinesedimentbelowthecoarsebedsurface. Experiment Lp [m] xp0 [m] xpf [m] I1 0.50 4.70 5.20 I2 1.00 4.49 5.49 I3 2.00 4.47 6.47 I4 4.00 4.47 8.47 Table 3
Experimentalconditions,whereqdenoteswaterdischargeperunitwidth,s0 initialbedslope,qb0sedimentfeedrateperunitwidth,hflowdepth,umean
flowvelocity,andFristheFroudenumber. q [m 2 /s] s
0 [ − ] qb0 [m 2 /s] h [m] u [m/s] Fr [ − ] 0.150 3.50 × 10 −3 7.86 × 10 −6 0.187 0.799 0.59
ultrasonicflowmeterandthedownstreamwaterlevelusingaposition sensor.Weobtainedprofilesofthewaterandbedelevationusinglaser sensorsthatwerefixedtoacarriage(item(h)inFig.3).Acamerawas mountedonthecarriagetomeasurethegrainsizedistributionofthebed surfaceusingthetechniquedevelopedbyOrrú etal.(2016a,b)(item(i) inFig.3).Tothisend,thecoarsesedimentwaspaintedredandthefine sedimentblue.Ourexperimentalset-upallowedustomeasureeithera profileofbedandwatersurfaceelevationorthebedsurfacegrainsize distributionatacertainlocationwithtime.
Forthemodelingofthelaboratoryexperiments(Section5),itis im-portanttoobtainturbulentflow conditionsofa relativelydeepflow (i.e.,flowcannotbeaffectedbyindividualgrains),wheresedimentis predominantlytransportedasbedload.Theconcentrationofsediment needstobesosmallthatwecanassumeclearwater.Theseconditions weresatisfied(AppendixCofthesupplementarymaterial).
4.2. Results
Allexperimentsweregovernedbythesameconditionsbefore the finesedimentinthepatchwasentrained.Weobservedthesuperposition ofbedformsoftwodifferentlengthscales(Fig.4).Secondarybedforms approximately0.5mlongand0.01mhighweresuperimposedon pri-marylongerbedformsoftheorderof3mandtwiceashigh.Theprimary
Fig. 3. Sketchoftheflumeset-up:(a )Turbulencedissipator,(b )metalplatewithgluedsediment,(c )alluvialbed,(d )feeder,(e )sandtrap,(f )sedimentpump,(g ) weir,(h )lasersensorsforwaterandbedsurfaceelevation,and(i )cameraformeasuringthebedsurfacegrainsizedistribution.
Fig. 4. Measuredbedelevationbeforefinesedimentofthepatchesisentrainedshowingthesuperpositionofbedformsoftwodifferentlengthscales(ExperimentI4 at1:51h).
bedformsareinterpretedasincipientgraveldunes(Carling,1999; Car-lingetal.,2005).Thecharacteristicsofthesefeaturesremainedsteady asthebeddegraded.Thesteadinessofthefeatures’characteristicsis confirmedinapreparatoryexperimental runwithoutapatchof fine sediment(AppendixCofthesupplementarymaterial).
Afterapproximately2hthebedhaddegradeduptoapointatwhich thetroughofalongbedformwaslowerthanthetoppartofthepatch (Fig.5(a)).Atthatmomentfinesedimentwasexposed,entrained,and transported.Thelargermobilityofthefinesedimentcreateda down-streammovingdegradationalwave(Fig.5(b)).Aserosionproceeded, theshearstresswasreduced(duetotheincreasedflowdepth),which reducedthedegradationrate.Meanwhile,thesubsequentbedform ad-vancedandstartedtofillthedegradationalpitwithcoarsesediment (Fig.5(c)). Overall,thepassageof bedformsinducedentrainmentof finesedimentandsubsequentcoarseningofthetoppartofthesubstrate. Sincethedegradationalwaveincreasedindepthindownstream direc-tion,alsothethicknessofthecoarsetoplayerincreasedindownstream direction(Fig.5(d)).
Thesubstratecoarseningmechanismcreatedanirregularinterface between coarse andfine sedimentcompared tothe initial situation wheretheinterfacewasparalleltothebedsurface.Asaconsequence, theentrainmentoffinesedimentbecameapseudo-randomprocessin spaceandtime.Degradationalwavesformedatthoselocationswhere finesedimentwasclosesttothebedsurface.Yet,mostofthewavesgrew foronlyalimitedlength,as,duetotheirregularinterface,atsomepoint thesedimentpresentatthetroughwascoarseratherthanfine. Some-timestheinterface wassufficientlyparalleltothebedsurfaceanda largedegradationalwaveformed.Thisisseeninthecontentofcoarse sedimentatthebed surfaceof thepatch (Fig.6(a),(c),(e)and(g)) andinthebedelevation(Fig.7).Oneortwosmalldegradationalwaves formedafterthepassageofalargedegradationalwave,characterized bythefactthatthebedsurfaceiscomposedofmainlyfinesediment andthetroughofabedformreacheselevationssignificantlylowerthan average.
Alongerpatchallowedforthedevelopmentoflonger(inspaceand time)anddeepererosionalwaves(Figs.6 and7).Yet,thedecreasein degradationrateasthewaveadvancedactedasasaturationmechanism limitingtheheightofthewave.Thus,theprobabilityoflowerbed el-evationatthepatchzonewasnotsignificantlylargerforanincreasing patchlength(Fig.8).Afterthepatch,wherethesubstratewascomposed ofcoarsesedimentonly,waveheightdecreasedandthebedelevation profiletendedtotheoneupstreamofthepatch(Fig.7).Yet,the pres-enceoffinesedimentdownstreamofthepatchslightlyincreasedthe heightofthebedformswithrespecttothebedformsupstreamofthe patch(Fig.8(a)and(c)).Bedformsdownstreamofthepatchwere char-acterizedbyacoarsefrontandfinetail,andwereapproximately2grain sizesofthecoarsesedimenthigh.Thesecharacteristicsmayindicatethe presenceofbedloadsheets(Dietrichetal.,1989;Reckingetal.,2009; Whitingetal.,1988)orbedformsinatransitionalphasetosmalldunes. Thedomaindownstreamfromthepatchwasnotlongenoughto
pre-Fig. 5. Sketchofthecyclicentrainmentofsubstratesediment:(a )Bedforms formedoutofcoarsesedimentsonly,(b )finesedimentfromthepatchis en-trainedinthetroughofabedform,(c )adegradationalwaveformsandtravels downstream,(d )coarsesedimentfromupstreamfillsthepitleftbythe degra-dationalwave.
ciselyconcludeonthetypeofbedforms.Thechangesinvolume frac-tioncontentofcoarsesedimentatthebedsurfacewerelesspronounced downstreamofthepatchcomparedtoatthepatch(Fig.6(b),(d),(f) and(h)).Thisisbecausefinesedimententrainedatthepatchdispersed inthedownstreamdirection.
Fig. 6. MeasuredsurfacefractioncontentofcoarsesedimentasafunctionoftimeforvariouslengthsofthepatchLp:Atthecenterofthepatch(a,c,e,g ),andatthe
downstreamend(b,d,f,h ).Notethatthestreamwiselocationofthecenterofthepatchvariesforeachexperimentwhilethedownstreampositionisthesameforall cases(𝑥=9.15m).
5. Numericalmodeling
Inthissectionweapplytheregularizationstrategyinmodelingthe laboratoryexperimentsconductedunderconditionsinwhichtheactive layermodelisill-posed(Section5.1).InSection5.2 wecomparethe re-sultsoftheregularizedactivelayermodeltotheresultsofthetwo-layer modeldevelopedbyRibberink(1987)byapplyingthemtoathought ex-perimentunderconditionsinwhichtheactivelayermodelisill-posed.
5.1. Modelingofourlaboratoryexperiments
In Section 5.1.1 we calibrate the numerical model. In Section 5.1.2 weconductaconvergencetesttoshowtheconsequencesof ill-posednessandthe benefits of theregularization strategy. InSection 5.1.3 weapplythenumericalmodeltothelaboratoryexperiments de-scribedintheprevioussection.InSection5.1.4 wetestthe regulariza-tionstrategyassumingthreesedimentsizefractions.
5.1.1. Calibration
Modelingthelaboratoryexperimentsrequiresvaluesfortheactive layerthicknessandthefrictioncoefficient,andthechoiceofasediment transportrelation.Tothisendweusetheresultsofasetof prepara-toryexperiments(AppendixCofthesupplementarymaterial).Tochose asedimenttransportrelation,weruntwoexperimentsconducted un-derequilibriumconditions,whilefeedingthefineandthecoarse sed-iment size fractions. Thesediment transport relation byAshida and Michiue(1971) reproduces ourresults reasonably well (Appendix D ofthesupplementarymaterial).Toobtaintheskinfrictioncoefficient (Cfb)forcomputingthesedimenttransportrate(AppendixA.4)we
cor-rectthetotalmeasuredfrictioncoefficientCfforsidewallfrictionwith
themethoddevelopedbyJohnson(1942) (seeGuo,2015).Weobtain thevalues𝐶f=0.0104and𝐶f b=0.0084. Bedformdragwasnegligible
duringtheinitialphaseasbedformswerelow.Whenfinesedimentwas entrained,bedformsgrewandbedformdragmayhaveplayedarole.Itis notreasonabletomodelthisadditionalfrictionusingstandardrelations (e.g.EngelundandHansen,1967;HaqueandMahmood,1983;Wright andParker,2004),astheserelationsprovideabedform-averaged fric-tioncoefficient,whileinourcaselargebedformswereisolatedinspace andtime.Wedecidetouseaconstantfrictioncoefficientandwethink thatthemostsensibleapproachistoneglectbedformdrag.
Areasonablevaluefortheactivelayerthicknessis0.01m,which correspondstothedistancebelowthemeanbedelevationwitha proba-bilityofentrainmentbelowapproximately5%(Blom,2008;Ribberink, 1987). This valueis alsoin accordance with1–3 times D90 as
pro-posedby,forinstance,Hirano(1971);HoeyandFerguson(1994),and Seminaraetal.(1996).
Inonepreparatoryexperimentunderequilibriumconditions,wefed coarsesedimentonlyand,fromsomepoint,westartedfeedingtracer sediment(i.e.,sedimentofadifferentcolor).Modelingthepropagation ofthefrontoftracersediment,weconfirmthat0.01misareasonable valuefortheactivelayerthickness(AppendixDofthesupplementary material).
5.1.2. Convergencetest
First we aim to show the consequencesof ill-posedness.To this end, we simulate conditions similar tothe ones of the experiments using theactivelayermodel. Intheexperiments,degradationintoa coarse substrate(i.e., under well-posedconditions) occurred for ap-proximately2h, asthepatchof finesedimentwas placed3cm be-low theinitialbedsurface.Inorder toobtainill-posedconditionsat
Fig. 7. DetrendedbedelevationasafunctionoftimeinExperiment(a )I1,(b )I2,(c )I3,and(d )I4.Thedashedblacklinesindicatetheboundariesofthepatch. Thebedelevationisdetrendedsubtractingthebedslopeofeachprofileindividually,obtainedfittingafirstdegreepolynomial.
Fig. 8. Probabilitydensityofdetrendedbedelevation:(a ) Up-streamofthepatch,(b )atthepatch,(c )anddownstreamof thepatch.
thestartofthesimulations,thepatchoffinesedimentisplacedright belowtheactivelayer.Inthis manner,300ssimulationssuffice for ourpurpose.Moreover,thepatchextendsoveradistanceof8m(from
𝑥=1mto𝑥=9m)tomaximizethedomainoverwhichthemodelis ill-posed.
Weconduct13simulationsusingcellsizesrangingfrom0.1mdown to2.44× 10−5m.Theresultsdonotconvergeandcontinuetochangeas thegridisrefined(Fig.9(a)).Wecomputetheerrorasafunctionofthe cellsizetoquantifythe(lackof)convergence.Asthereisnoanalytical solutiontowhich wecancomparetheresults ofthenumericalruns,
Fig. 9. Bedelevationat𝑡=300spredictedusingthe(a )activelayermodel(Hirano,1971)and(b )regularizedactivelayermodel.Eachofthe13linespresentsthe resultscomputedusingadifferentcellsize(rangingfrom0.1mdownto2.44× 10−5m,wheredarkercolorsrepresentsmallercellsizes).Panels(c )and(d )present
theerroratacertaintimeusingaparticularcellsize(seeEquation(13))whenusingtheactivelayermodelandtheregularizedactivelayermodel,respectively.In panels(b )and(d )onlyonelineisvisible,asitoverlapsallotherlines.
wecomputetheerrorbetweentheresultsoftwosuccessivesimulations
sand𝑠+1(LoveandRider,2013; Roy,2005).Tothisend,firstwe interpolatethebedelevationresultsofallsimulationsusingthesmallest cellsize.Theinterpolation,ratherthanlinear,takesintoconsideration thateachvalueisconstantinsideacell.Second,wecomputetheerror asthenorm1ofthedifferencebetweenbedelevationsoftwosuccessive simulationsatacertaintimet:
error𝑡𝑠= 1 𝐿𝑁x 𝑁x ∑ 𝑟=1 || |𝜂𝑟𝑡𝑠−𝜂𝑡𝑟𝑠+1|||, (13)
whereNxdenotesthenumberofcellsofthesimulationwiththesmallest
cellsize,andL[m]thedomainlength.Fig.9(c)showstheerrorasa func-tionofthecellsizeforseveraltimes.Ifthecellsizeislarge(forinstance, largerthan0.01m),forshortsimulationtimes(forinstance,shorten than10s),theresultsseemtoconverge.Yet,usingthesamecellsize, theresultsdonotconvergeifoneconsidersalongersimulationtime. Similarly,consideringasimulationtimeequalto10s,theresultsdonot convergewhenthecellsizeissmallerthan0.002m.Thisbehavioris characteristicofill-posedsimulations.Thegrowthrateofperturbations increaseswithdecreasing cellsize.Forthis reason,theconsequences ofill-posednessariseearlierforsmallercellsizes.Givenacertaincell size,ifthesimulationisshortenough,perturbationsdonothavetimeto growandthesolutionseemstoconverge.Forafixedtime,simulations seemtoconvergeaftertheerrorgrows(forinstance,for𝑡=120s, sim-ulationsseemtoconvergeforcellsizesbetween0.001mand0.01m). Thisisdue tothefactthat,atthegiventime,perturbations have al-readygrownsignificantlyandhavecoarsenedthebedmaterialcausing thesimulationtobewell-posed.Afurtherdecreaseofthecellsizeoran analysisatadifferenttimeshowsthattheactivelayermodeldoesnot converge.
Werepeatthesamesimulationsapplyingtheregularization strat-egy.The initialvalue of theparameterthat recovers thewell-posed characterofthesystemis𝛼c=11.6.Inthiscasethesolutiondoesnot
showoscillations(Fig.9(b)).Moreover,thesolutionconvergesfora de-creasingcellsizeindependentlyfromthetimeatwhichconvergenceis tested(Fig.9(d)).Thissupportsthefactthattheregularizedmodelis well-posed,contrarytotheactivelayermodel.Therateatwhichthe solutionconvergesconfirmsthatthenumericalschemeisfirst-order ac-curate(Section3.5).
5.1.3. Twosedimentsizefractions
Wereproducealllaboratoryexperimentsusingacellsizeequalto 0.05m.Theregularizedmodelshowsspatialortemporaloscillations innoneofthecases(Fig.10).Forallcasesthebedelevationdecreases smoothlyinthestreamwisedirection(Fig.10(b),(f),(j)and(n)).This contrastswiththemeasuredtemporalchangeofbedelevation,which presentsbedformsandtheformationofdegradationalwavesatthe up-streamendof thepatch (Fig.10(a),(e),(i)and(m)).Themeasured increaseinwaveheightatthepatch(Fig.10(a),(e),(i)and(m))and Section4.2)isnotcaptured.Theeffectofthepatchisobservedinthe modelresultsinthefactthatdegradationoccursfasterforalongpatch (Fig.10)thanforashortone(Fig.10(b)).
Thecontinuousandsmoothpredictedentrainmentofsubstrate sed-imentyieldsanalmoststeadyvolumefractioncontentofsedimentin theactivelayerbothatthepatch(Fig.10(c),(g),(k)and(o))andatthe downstreamend(Fig.10(d),(h),(l)and(p)).Themeasureddatashows, ontheotherhand,avariablevolumefractioncontentatthebedsurface. Themodelcorrectlycapturesthemeanvalueandnicelyreproducesthat alongerpatchcausesanincrease intheamount offine sedimentat thebedsurface.Thefactthatthemodeldoesnotcapturebedformsis notsurprising,asthemechanismsnecessaryforbedformformationare notpresentinthemodel.Forinstance,thefactthattheflowmodelis basedonthehydrostaticpressureassumptionpreventsmodelling pro-cessessuchasflowseparation.Thepossibilityofcapturingthe forma-tionofthedegradationalwavesatthepatchisalsodiscarded,asfrom theanalysisofwell-posednessweseethattheregularizedmodeldoes
not showany instabilitymechanismthatcouldinducewavegrowth. Forthisreason,themodelresultsrepresentvaluesaveragedover the passageofseveralbedformsanddegradationalwaves.Wechoosenot tofilterthemeasuredbedelevationdata,asgiventhecharacteristicsof thebedforms,itwouldintroducealargeamountofspuriousinformation (e.g.,thedegradationalwavewouldstartatthewronglocation)andwe wouldloseasignificantamountofdataatthebeginningandendofthe domain.
Overalltheregularizedmodelyieldsareasonableapproximationof the meantemporalchange of themeasureddata. Thedegradational trendiscapturedandthesurfacegrainsizedistributionapproximates theaveragemeasuredvalues.Thesubstrateisnotunrealisticallyaltered astherearenooscillationsinthesolution.
5.1.4. Threesedimentsizefractions
Totesttheregularizationstrategyformultiplegrainsizes,wemodel ExperimentI4(Table2)using3different grainsizesbyapplyingthe exactsolutiontoobtaintheregularizationparameter.Thefinesize frac-tionremainsthesameandthepreviouscoarse sizefractionis repre-sentedinthiscasebytwocharacteristicgrainsizesequalto4.895mm and5.895mm.Foraninitialvolumefractioncontentatthebed sur-faceofthemediumsizesedimentequalto0.375,theinitialbedslope isthesameaswhenusingtwocharacteristicsizesandthesumofthe sedimenttransportrateofthemediumandcoarsefractionswhenusing three sizesisequaltothesedimenttransportrateofthecoarse frac-tionwhenusingtwosizes.Inthismannerthesimulationaccountingfor threesedimentfractionsiscomparabletotheoneaccountingfortwosize fractions.
InFig.11 wecomparethebedelevationandmeangrainsizeofthe bedsurfacesedimentpredictedbytheregularizedmodelusing2and3 sedimentsizefractions.Theevolutionofthebedelevationshowsonlya weakdependenceonthenumberofsizefractionsusedtodiscretizethe sedimentmixture.Themodelwith3sizefractionspresentsamild coars-ening(0.2%increaseinmeangrainsize)withtimebeforesedimentfrom thepatchisentrained(after2h).Thiscoarseningisnotvisiblewhen us-ing2sizefractions,becauseinthiscase,duringtheinitialstate,thebed surfacesedimentconsistsofonesinglegrainsizeonly.Weconcludethat theregularizationtechniqueisapplicableforageneralcasewithmore than2sizefractions.
5.2. ComparisonbetweenRibberink’s(1987)two-layermodelandthe regularizedmodel
Toourknowledgethereisnootherlaboratorydatasetapartfromthe onepresentedinSection4towhichwecanapplytheregularizedactive layermodeltotestitsperformance.Thisis becauseeitherthe condi-tionsthatotherresearchershavestudiedyieldawell-posedactivelayer model(e.g.AshidaandMichiue,1971)ortheactivelayermodelis ill-posedbuttheactivelayerthicknessvarieswithtimeduetodunegrowth (Blometal.,2003).Thelattercaseisasituationthattheregularization strategycannotdealwith(Section3.1).However,Ribberink(1987) ap-plieshistwo-layermodeltoathoughtexperimentunderconditionsin whichtheactivelayermodelisill-posed.Inthissectionweapplythe regularizedactivelayermodeltohisthoughtexperimentandcompare ittothetwo-layermodel.
Ribberink(1987) conductedalaboratoryexperimentwith mixed-sizesediment,whichwasdominatedbyaggradationafteraperiodof degradation(ExperimentE8-E9).Theinitialbedwascharacterizedbya uniformslope,composedofabimodalmixture(acoarseandfine frac-tion),andwellmixedbothinthestreamwiseandverticaldirection.The sedimentsupply wasinitially inequilibrium. Atemporalincrease of theproportionofthecoarsefractioninthesedimentsupplyperturbed theequilibriumconditionandinducedthedownstreampropagationof acoarseningwave.Thedownstreammigrationofthecoarseningfront causedaprecedingandtemporarybeddegradationasaresultof the
Fig. 10. Comparisonbetweenmeasureddataandregularizedmodelresults:ExperimentI1(a -d ),ExperimentI2(e -h ),ExperimentI3(i -l ),andExperimentI4(m -p ). Thefirstandsecondcolumnsshowthemeasuredandpredictedbedelevationwithtime,respectively.Theverticaldashedlinesindicatethepositionofthepatchof finesediment.Thethirdandfourthcolumnspresentthesurfacefractioncontentofcoarsesedimentatthecenterofthepatchoffinesedimentandatthedownstream endoftheflume,respectively.
differenceinsedimentmobilitybetweenthecoarsesedimentforming thewedgeandthefinesedimentdownstreamofthefrontofthewedge. Eventually,thebedaggradedandwascharacterizedbyalargerslope thantheinitialone,soastoallowforthetransportofthecoarserfed sedimentunderequilibriumconditions.
Duringtheshortdegradationalpartoftheexperiment,thebed sur-facewascoarserthanthesubstrate(i.e.,conditionsinwhichtheactive layermodelispronetobeill-posedChavarríasetal.,2018;Ribberink, 1987;Steccaetal.,2014).However,whilereproducingtheexperiment
numerically, Ribberink(1987) foundthattheactivelayermodelwas well-posed.Subsequently,Ribberink(1987)appliedhistwo-layermodel toathoughtexperimentthatwasequaltoE8-E9exceptforthefactthat thesubstratesedimentwasfinerthanintheflumeexperimentsuchthat theactivelayermodelisill-posed.Anumericalsimulationofthethought experimentusingtheactivelayermodelshowedoscillationsthat even-tuallymadethecodecrash(Ribberink,1987).Thethoughtexperiment was reproduced well bya numericalcode implementing Ribberink’s (1987) two-layermodel.
Fig. 11. Bedelevation(a )andmeangrainsizeatthebedsurface(b )asa func-tionoftimepredictedinExperimentI4usingtheregularizedactivelayermodel using2and3sedimentsizes.
Herewerunanumericalsimulationofthethoughtexperiment us-ingourregularizedactivelayermodelandcompareittotheresults of Ribberink’s (1987) two-layer model reported in Fig. 7.9 of Ribberink(1987).SimulationdetailscanbefoundinAppendixC.
Fig.12presentsthetimeseriesofbedelevationandmeangrainsize ofthebedsurfacesedimentatalocation20mdownstreamfromthe inlet.Duringthefirst20htheeffectsofthecoarseningofthefed sedi-mentarenotfelt20mdownstreamfromtheinlet.Whiletheregularized activelayermodelpredictsaconstantbedelevationandgrainsize dis-tributionofthebedsurfacesedimentduringthisperiodof time,the two-layermodelpredictsafiningofthebedsurface(Fig.12(b)).This isduetothefactthattheinitialgrainsizedistributionoftheexchange layerisnotinequilibriumwiththeoneattheactivelayerandcausesa verticalfluxofsediment.However,thebedelevationremainsconstant aspredictedbybothmodels(Fig.12(a)).
Theaggradationalphaseisprecededbyadegradationalwave,which ismuchmorepronouncedintheregularizedactivelayermodelthan inthetwo-layermodel.Thisisbecauseintheregularizedactivelayer modeldegradationcausesentrainmentofthefinesubstratesediment, whereas in thetwo-layer model theexchange layeracts asa buffer thatslows down theprocess. The coarseningof thebed surface be-tweenapproximately 25 hand40 haspredicted byboth modelsis verysimilar.Whileafter40htheregularizedactivelayermodel pre-dictsa constantgrain sizedistribution of thebed surface sediment, thetwo-layer modelpredictsan asymptoticadaptationtoward equi-libriumconditions.Thiseffect isagaincausedbytheexchangelayer thatcoarsensslowly comparedtotheactivelayeron topof it,asit accounts forthe effectsof occasionally large bedforms.The equilib-riumstatediffersbetweenthetwomodels.Webelievethatthisisdue tothe factthat we do not knowexactly what values wereused by Ribberink(1987) fortheconstantsinthesedimenttransportrelation (AppendixC).
Theregularizedactivelayermodelcapturesthedynamicspredicted bythetwo-layermodelofRibberink(1987).Theadvantageofthe two-layermodelisthatitaccountsforasourceofverticalmixingthatthe regularizedactivelayermodeldoesnottakeintoconsideration(i.e.,the mixingduetooccasionallylargebedforms).Ontheotherhand,the two-layermodelmaybecomeill-posed(Sieben,1994)whiletheregularized activelayermodelisalwayswell-posed.
Fig. 12. Bedelevation(a )andmeangrainsizeofthebedsurfacesediment (b )withtimepredictedforthethoughtexperimentbasedonExperiment E8-E9conductedby Ribberink(1987) usingRibberink’stwo-layermodelandthe regularizedactivelayermodel.Theresultsofthetwo-layermodelareextracted fromFig.7.9ofRibberink(1987).
6. Discussion
Inthissectionwediscussthephysicalinterpretationofthe regular-izationstrategy(Section6.1),aswellaspossibleextensionsandfurther development(Section6.2).
6.1. Physicalinterpretationoftheregularizationstrategy
Theill-posedsolutionpredictedbytheactivelayermodelis charac-terizedbyoscillationsthattemporarilyfinethebedsurfaceandcoarsen thesubstrate.Thisbehaviorisalsoobservedinourlaboratory experi-ments(Figs.6 and7).Onemaybetemptedtoconcludethattheactive layermodel,althoughbeingmathematicallyill-posed,provides reason-ableresults.Thisargumentiswrongfortworeasons.Thefirstreason isthatthenumericalsolutiondoesnotconvergeforadecreasingmesh size.Thesolutionkeepschangingandoscillationsbecomelargerwhen thecellsizeisreduced(Chavarríasetal.,2018;JosephandSaut,1990). Suchasolutioncannotberepresentativeofphysicalphenomena. Sec-ond,thephysicalprocessesresponsibleforthesmallscalevariabilityin bedelevation(i.e.,ripples,bedloadsheets)arenotaccountedforbythe activelayermodel(Section3.1).Anyresemblanceofthemodelresults withbedelevationfluctuationsduetosmallscalebedformsistherefore coincidence.
The frequently used morphodynamic factor(Φ𝜂) (Latteux, 1995; Ranasingheetal.,2011;Roelvink,2006)isaparticularcaseofa pre-conditioningmatrixwithparameters𝛽 =1∕Φ𝜂and𝛼𝑘=1∀k.The pro-posedregularizationstrategycanbeconsideredastheuseofa morpho-dynamicfactornotonlyforthechangesinbedelevation(𝜂)butalso forthechangesingrainsizedistributionofthebedsurface(Mak).The “sortingmorphodynamicfactor” (Φsk)isthendefinedasΦs𝑘=1∕(𝛼𝑘𝛽). Wehaveseenthattheonlyapplicableregularizationstrategyisthatin which𝛼𝑘=𝛼 >1∀kand𝛽 =1,whichisequivalenttosayingthatthe regularizationstrategyisbasedona“sortingmorphodynamicfactor” 0<Φs<1.Thisimpliesthatthemixingorsortingprocessesassociated withchangesingrainsizedistributionofthebedsurfacesedimentare sloweddownwithrespecttotheceleritypredictedbytheactivelayer model.
Theeffectofapplyingtheregularizationstrategyisaslowdownof thesedimentmixingprocessesinthemodelcomputations.Thiseffectis similartotheeffectofa(temporary)increaseoftheactivelayer thick-ness.Fromaphysical perspectivethisslowdownofmixingprocesses maybeassociatedwitha(temporary)increaseoftherangeofelevations coveredbythebedlevelfluctuations(Blometal.,2008).Theslowdown ofmixingprocessesresultingfromapplyingtheregularizationstrategy impliesthattheregularizedactivelayermodelcanbeappliedtoawider rangeof physicalproblems (i.e.,alsothose characterizedbya fairly smalltimescaleofmixing)thantheactivelayermodel.
6.2. Alternativestotheregularizationstrategy
Our regularization strategy is applied locally and temporally. Wordeddifferently,onlywhenandwherethemodelisill-posed,we up-datethegrainsizedistributionofthebedsurfacesedimentusingthe pa-rameter𝛼c.Moreover,𝛼cdependsonlyontheconditionsatthelocation
underconsideration(notethatthepreconditioningmatrixisdiagonal). Thisisthesimpleststrategybutonecoulddecidetoavoid discontinu-itiesinthevalueof𝛼cthroughoutthedomainbycouplingneighboring
nodes.
Carraroetal.(2018) proposeatechniquetodecreasethe computa-tionalcostofmorphodynamicsimulations.Asinourcase,theirstrategy canbeseenasapreconditioningtechnique.Theyconsiderunisize sedi-mentconditionsandmodifynotonlytheExner(1920)equationbutalso thecontinuityequation.Herewemodifytheactivelayerequationbut nottheflowequationsortheExner(1920) equation.Acombinationof bothstrategiescouldyieldatechniquethatbothdecreasesthecostof numericalsimulationsandguaranteesthatthemodeliswell-posed.
Wehavefocusedonrestoringthehyperboliccharacterofthesystem ofequationsandtothisendwebasedourstudyonthelinearsolution (i.e.,shortwaves).Thisfocussufficeshere,asshortwavesaremost sen-sitivetoill-posedness(JosephandSaut,1990).However,the regular-izationstrategymodifiesthecelerityandgrowthratenotonlyofshort wavesbutalsooflongones.Forthisreason,wesuggesttofurtherstudy howlongwavesareaffectedandwhethertheresultsofthe regulariza-tionstrategyarephysicallyrealisticbasedonasimilaranalysistothat ofLanzonietal.(2006).
We have assumeda constant activelayerthickness toavoid the addedcomplexityduetoacumbersomeclosurerelationlinkingthe pre-conditioningparameterstothechangeintimeoftheactivelayer thick-ness.Itmaybepossibletoextendourregularizationstrategyto situa-tionsinwhichtheactivelayerthicknesschangeswithtime(e.g.,dueto dunegrowth)byprovidingsuchaclosurerelation.Ontheotherhand,it isreasonablethattheregularizationstrategyrequiresaconstantactive layerthicknessgiventhefactthatmathematicallythestrategyhasthe sameeffectasanincreaseintheactivelayerthickness(i.e.,adecrease inthecelerityofthemixingprocesses).
Wehaveconcludedthattheregularizationstrategyneedsto slow-downthemixingprocesses(i.e.,𝛼c>1)toguaranteethatthe
eigenval-uesarealwayspositiveregardlessofthevalueofthesortingcelerity
𝜆s1.However,ifthesortingcelerityisguaranteedtobepositive(e.g.,
becausehidingisnegligible),theaccelerationofthemixingprocesses alsoyieldspositiveeigenvaluesandawell-posedmodel.Theremaybe casesinwhichthelatterstrategyyieldsmorerealisticresults.Moreover, wehavechosentoguaranteethattheregularizedeigenvaluesare pos-itivereasoningthatmorphodynamicinformationtravelsinthe down-streamdirectionundersubcriticalconditions(Lanzonietal.,2006;Lyn andAltinakar,2002;Steccaetal.,2014;Suzuki,1976).Thisstatement ispartiallycontradictorytorecentstudiesthatconsidersediment trans-portasastochasticprocess(AnceyandHeyman,2014;Furbishetal., 2012).Thestochasticnatureofsedimenttransportyieldsan advection-diffusionequationthatmodelstheamountofmovingparticlesperunit ofbedarea.Thediffusivecharacterimpliesthatinformationalsotravels intheupstreamdirection.Forthisreason,aregularizationstrategyin
whichinformationtravelsintheupstreamdirectionmaybephysically realisticundercertaincircumstances.
Foracasewithmorethantwosedimentsizefractions(Section3.4), theapproximatevalueof theparameter𝛼c isnot (completely)
satis-factoryaswell-posednessisnotguaranteed.Wehaveobservedinour teststhatill-posednessoccurswhen(atleast)twoeigenvaluesofthebed andsortingeigenvaluesaresimilarwithrespecttotheotherbedand sortingeigenvalues.Foracaseconsideringtwosedimentsizefractions this isreferred inliteratureasthe“crossing ofeigenvalues” (Sieben, 1997;Steccaetal.,2014).Wordeddifferently,thedifferencebetween twoeigenvaluesmustbelargeenoughforthemodeltobewell-posed.A regularizationstrategybasedonguaranteeingaminimumdistance be-tweeneigenvaluescouldyieldaninexpensivesolutionforthecasewith morethantwosedimentsizefractions.
7. Conclusions
Wehavedevelopedapreconditioningmethodforregularizingthe activelayermodel(Hirano,1971)used inmodelingmixed-sediment rivermorphodynamics.Ourmethodrecoversthewell-posedcharacter ofthesystemofequationsbymeansofoneparameterthatmodifiesthe celerityofthemixingprocesses.Physicallythismeansthatthemixing processesaresloweddownorthetimescaleofthemixingprocessesis increased.
Weconduct4laboratoryexperimentsunderconditionsinwhichthe activelayermodelisill-posedandwecomparetheobservationstothe predictionsoftheregularizedactivelayermodel.Theregularizedactive layermodelcapturesthemeanbehaviorobservedintheexperiments associatedwithchangesaveragedoverthepassageofseveralbedforms. Acknowledgments
This research is partof the research programme RiverCare, sup-portedbytheDutchAppliedandEngineeringSciences(AES)domainof theNetherlandsOrganizationforScientificResearch(NWO),andwhich ispartlyfundedbytheMinistryofEconomicAffairs undergrant num-ber P12-14 (PerspectiveProgramme).The involvementof Guglielmo SteccawassupportedbyNIWAundertheSustainableWaterAllocation Programme.Thefruitfuldiscussionsandcommentsonthemanuscript ofLiselotArkesteijnaregratefullyacknowledged.WethanktheEditor Prof.Dr.G.C.Sander,AssociateEditorDr.C.Manes,andthree anony-mousreviewersfortheircomments,whichhavesignificantlyimproved themanuscript.
AppendixA. Modelequations
In this section we present the system of equations for model-ingmixed-sedimentrivermorphodynamics.InSectionA.1 wepresent the flow equations. In Section A.2 we present the active layer model (Hirano, 1971).A simplified activelayer model is presented in Section A.3. In Section A.4 we show the closure relations. In SectionA.5 we presentthesystemofequationsin matrix-vector for-mulation.
A.1. Flowequations
Weconsideraone-dimensionalmixtureofwaterandsediment flow-ingoveramobilebed.Asetofpartialdifferentialequationsthat ac-countsfortheinteractionsbetweensedimentandwaterisfoundby ap-plyingmassandmomentumconservationprinciplesforthemixtureof sedimentandwater(e.g.,Garegnanietal.,2011;Grecoetal.,2012). ThecompletesystemofequationsreducestotheSaint–Venant–Exner model (i.e.,clearwater approximation)underlow sediment concen-trations(𝑐=𝑞b∕𝑞<0.006,whereqb[m2/s]andq[m2/s]arethe
Garegnanietal.,2011;Garegnanietal.,2013).Intheremainingwewill assumethattheclearwaterapproximationisvalid.
TheflowismodeledusingtheSaint-Venant(1871)equations:
𝜕ℎ 𝜕𝑡 +𝜕𝑞𝜕𝑥=0, (14) 𝜕𝑞 𝜕𝑡+𝜕 (𝑞2∕ℎ+𝑔ℎ2∕2) 𝜕𝑥 +𝑔ℎ𝜕𝜕𝜂𝑥=−𝑔ℎ𝑆f, (15)
wheret[s]denotesthetimecoordinate,x[m]thestreamwise coordi-nate,h[m]theflowdepth,g[m/s2]theaccelerationduetogravity,𝜂
[m]thebedelevation,andSf[– ]thefrictionslope.
Theflowequationscanbefurthersimplifiedassumingsteadyflow. Underthis conditiontheconservationofwatermassandmomentum reducetoaspatiallyconstantdischarge,andthebackwaterequation:
𝜕ℎ 𝜕𝑥= −1 1−Fr2 𝜕𝜂 𝜕𝑥− 𝑆f 1−Fr2, (16)
whereFr=𝑞∕√𝑔ℎ3istheFroudenumber.
A.2. Activelayermodel
Tomodelchangesin bedelevationwe assume thatthesediment transportrateadaptsinstantaneouslytochangesinbedshearstress. Spa-tialand/ortemporaladaptationtocapacityload(BellandSutherland, 1983;PhillipsandSutherland,1989;1990)isnotconsidered.Neglecting mechanismssuchassubsidenceanduplift(PaolaandVoller,2005),and assumingaconstantbedporosity,weobtaintheExner(1920)equation:
𝜕𝜂 𝜕𝑡+
𝜕𝑞b
𝜕𝑥 =0, (17)
whereforsimplicitythesedimenttransportrateincludesthepores. ThesedimentphaseiscomposedofamixtureofNnon-cohesive sed-imentsizefractions.Eachfractionis characterizedbyagrain sizedk [m]wherekisanindexidentifyingasizefraction.Thetotalsediment transportrateperunitwidthisthesumofthesedimenttransportrate ofsizefractionk,qbk[m2/s]: 𝑞b= 𝑁 ∑ 𝑘=1 𝑞b𝑘. (18)
Theconservationofthevolumeofsedimentofsizefractionkinthe activelayerperunitofbedarea(𝑀a𝑘=𝐹a𝑘𝐿a[m])isexpressed
math-ematicallyas(Hirano,1971): 𝜕𝑀a𝑘 𝜕𝑡 +𝑓𝑘I𝜕(𝜂 −𝐿a ) 𝜕𝑡 + 𝜕𝑞b𝑘 𝜕𝑥 =0 for1≤𝑘≤𝑁−1, (19)
whereFak ∈[0,1][– ]isthevolumefractioncontentofsizefractionk
intheactivelayer,𝑓I
𝑘 ∈[0,1][– ]isthevolumefractioncontentofsize fractionkattheinterfacebetweentheactivelayerandthesubstrate, andLa[m]istheactivelayerthickness.Bydefinition,
𝑁 ∑ 𝑘=1 𝐹a𝑘=1, 𝑁 ∑ 𝑘=1 𝑓I 𝑘=1. (20)
FromthefirstconstraininEq.(20) oneobtains thechangeofthe volumeofsedimentintheactivelayeroftheNthgrainsizewithtime.
Thesystemis complete withanequation forthe conservationof massin thesubstrate.Yet,thisequation islinearlydependentonthe Exner(1920) andHirano(1971) equationswhichimpliesthatitdoes notplayaroleinthemathematicalcharacterofthesystem(Chavarrías etal.,2018;Steccaetal.,2014).
A.3. SimplifiedActiveLayerModel
Tosimplifythesystemofequationswereplacethe𝑁−1equations thataccountforthechangeinbedsurfacevolumefractioncontentof theNfractionsbyoneequationthatmodelstheaveragegrainsize fol-lowingtheapproachofRibberink(1987).Wemultiplyeachregularized
activelayerequationbyitscharacteristicgrainsizeandweaddallthe equations: 𝛼𝜕𝐷ma 𝜕𝑡 − 𝐷I m 𝐿a 𝜕𝑞b 𝜕𝑥 + 1 𝐿a 𝑁 ∑ 𝑘=1 𝑑𝑘𝜕𝑞𝜕𝑥b𝑘 =0, (21) where𝐷ma= ∑𝑁
𝑘=1𝑑𝑘𝐹a𝑘[m]isthemeangrainsizeofthesedimentin
theactivelayerand𝐷I
m=
∑𝑁
𝑘=1𝑑𝑘𝑓𝑘I[m]isthemeangrainsizeofthe
sedimentattheinterfacebetweentheactivelayerandthesubstrate.The meangrainsizeiscomputedarithmeticallyasitisanecessarystepto obtainanapproximateequation.Yet,weconsiderthatthemeangrain sizeisbetterapproximatedassumingthegrainsizedistributiontobe logarithmicallydistributed.
Wewritethesedimenttransportrateqbkasafunctionoftheflow depthhandthemeangrainsizeofthesedimentintheactivelayerDma
suchthat: 𝜕𝑞b𝑘 𝜕𝑥 = 𝜕𝑞b𝑘 𝜕ℎ 𝜕ℎ𝜕𝑥+ 𝜕𝑞b𝑘 𝜕𝐷ma 𝜕𝐷ma 𝜕𝑥 . (22) Weusethat: 𝐷ma= 𝑁 ∑ 𝑘=1 𝑑𝑘𝐹a𝑘=𝑑𝑁+𝐿1 a 𝑁−1∑ 𝑘=1 𝑀a𝑘(𝑑𝑘−𝑑𝑁), (23) wherewehaveusedtheconstrainthat∑𝑘𝑁=1𝐹a𝑘=1.Thus,
𝜕𝑞b𝑘 𝜕𝐷ma = 𝑁−1∑ 𝑙=1 𝜕𝑞b𝑘 𝜕𝑀a𝑙 𝜕𝑀a𝑙 𝜕𝐷ma =𝐿a 𝑁−1∑ 𝑙=1 1 𝑑𝑙−𝑑𝑁 𝜕𝑞b𝑘 𝜕𝑀a𝑙, (24) wherewehaveusedthat:
𝜕𝑀a𝑙
𝜕𝐷ma
= 𝐿a
𝑑𝑙−𝑑𝑁. (25)
We substitute the backwater equation (Eq. 16) in the Exner(1920)equation(Eq.17)andtheequationofthemeangrainsize (Eq.21)toobtainthefinalsetofequations:
𝜕𝜂 𝜕𝑡 − 1 1−Fr2 𝜕𝑞b 𝜕ℎ𝜕𝑥𝜕𝜂+ 𝜕𝑞b 𝜕𝐷ma 𝜕𝐷ma 𝜕𝑥 = 1 1−Fr2 𝜕𝑞b 𝜕ℎ𝑆f, (26) 𝛼𝜕𝐷ma 𝜕𝑡 + 1 𝐿a 1 1−Fr2 [ 𝐷I m 𝜕𝑞b 𝜕ℎ − 𝑁 ∑ 𝑘=1 𝑑𝑘𝜕𝑞𝜕ℎb𝑘 ] 𝜕𝜂 𝜕𝑥 − 1 𝐿a [ 𝐷I m 𝜕𝑞b 𝜕𝐷ma − 𝑁 ∑ 𝑘=1 𝑑𝑘𝜕𝐷𝜕𝑞b𝑘 ma ] 𝜕𝐷ma 𝜕𝑥 = −1 𝐿a ( 1−Fr2) [ 𝐷I m 𝜕𝑞b 𝜕ℎ − 𝑁 ∑ 𝑘=1 𝑑𝑘𝜕𝑞𝜕ℎb𝑘 ] 𝑆f. (27)
A.4. Closurerelations
Toclosethesystemof equationsweprovide closurerelationsfor thefrictionterm,thesedimenttransportrate,andthefluxbetweenthe activelayerandthesubstrate.
WeadoptthefollowingChézyclosurerelationforthefrictionterm:
𝑆f=
𝐶f𝑢2
𝑔ℎ , (28)
whereCf[– ]isanondimensionalfrictioncoefficientthatweassumeto
beconstant(i.e.,independentofthefloworbedproperties),and𝑢=𝑞∕ℎ [m/s]isthemeanflowvelocity.
The sediment transport rate of size fraction k per unit width is assumed tobe the product of a nondimensional sediment trans-port rate (𝑞∗
b𝑘 [−]) and the bed surface fraction content. The latter
we assume equal to the active layer volume fraction content. The Einstein (1950) parameter(√𝑔𝑅𝑑3
𝑘)scalesthenondimensional quan-titysuchthat:
𝑞b𝑘=𝐹a𝑘 √
𝑔𝑅𝑑3
