Geomorphology 163–164 (2012) 10–25

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Geomorphology

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A simplified 2D model for meander migration with physically-based bank evolution

Davide Motta a,⁎, Jorge D. Abad b, Eddy J. Langendoen c, Marcelo H. Garcia a

a Dept. of Civil and Environmental Engineering, University of Illinois at Urbana-Champaign, Urbana, IL, 61801, United Statesb Dept. of Civil and Environmental Engineering, University of Pittsburgh, Pittsburgh, PA, 15261, United Statesc US Department of Agriculture, Agricultural Research Service, National Sedimentation Laboratory, Oxford, MS, 38655, United States

⁎ Corresponding author.E-mail addresses: dmotta2@illinois.edu (D. Motta), j

eddy.langendoen@ars.usda.gov (E.J. Langendoen), mhga

0169-555X/$ – see front matter © 2011 Elsevier B.V. Adoi:10.1016/j.geomorph.2011.06.036

a b s t r a c t

a r t i c l e i n f o

Article history:Received 1 July 2010Received in revised form 18 January 2011Accepted 1 June 2011Available online 16 August 2011

Keywords:Meander migrationMigration coefficientBank erosionPlanform shapeComputer model

The rate of migration, calculated by numerical models of river meandering, is commonly based on a methodthat relates the rate of migration to near-bank excess velocity multiplied by a dimensionless coefficient.Notwithstanding its simplicity, since the early 1980s this method has provided important insight into thelong-term evolution of meander planforms through theoretical exercises. Its use in practice has not been assuccessful, because the complexity of the physical processes responsible for bank retreat, the heterogeneity infloodplain soils, and the presence of vegetation, make the calibration of the dimensionless coefficient ratherchallenging. This paper presents a new approach that calculates rates of meander migration using physically-based streambank erosion formulations. The University of Illinois RVR Meander model, which simulatesmeandering-river flow and bedmorphodynamics, is integrated with algorithms for streambank erosion of theUS Department of Agriculture channel evolution computer model CONCEPTS. The performance of theproposed approach is compared to that of the more simple classic method through the application to severaltest cases for idealized and natural planform geometry. The advantages and limitations of the approach arediscussed, focusing on simulated planform pattern, the impact of soil spatial heterogeneity, the relativeimportance of the different processes controlling bank erosion (hydraulic erosion, cantilever, and planarfailure), the requirements for obtaining stable migration patterns (centerline filtering and interpolation ofbank physical properties), and the capability of predicting the planform evolution of natural rivers overengineering time scales (i.e., 50 to 100 years). The applications show that the improved physically-basedmethod of bank retreat is required to capture the complex long-term migration patterns of natural channels,which cannot be merely predicted from hydrodynamics only.

abad@pitt.edu (J.D. Abad),rcia@illinois.edu (M.H. Garcia).

ll rights reserved.

© 2011 Elsevier B.V. All rights reserved.

1. Introduction

The modeling of meandering-river migration requires the simu-lation of the following processes: hydrodynamics, sediment transport,bedmorphodynamics, and bank erosion. The hydrodynamicmodelingresolves the mean and turbulent flow fields: e.g., primary andsecondary flows, Reynolds stresses, and turbulent kinetic energyamong other hydrodynamic parameters. In bends, curvature-driven(Prandtl's first kind) and turbulence-driven (Prandtl's second kind)secondary flow can be present and alter the morphology of the bedand banks, which then affects the anisotropy of the flow (Akahori andSchmeeckle, 2002; Blanckaert and de Vriend, 2005). Because the shearstresses exerted by the flow on the bed and banks control the erosionand transport of the boundary materials, their modeling is critical.

Modeling of the sediment transport in meandering streamssimulates the transport of sediments as a combination of bed andsuspended load, because of the complex flow and the possible large

difference between bed and bank material particle sizes. Bed load isquantified using empirical formulations (Garcia, 2008), and itsdirection is determined by the near-bed flow direction corrected bythe effect of bed slopes (Seminara and Tubino, 1989; Kovacs andParker, 1994; Talmon et al., 1995; Mosselman, 2005; Abad et al.,2008). The suspended load is calculated using an advection–diffusionequation, where the diffusion coefficient is related to the turbulencecharacteristics of the flow (Lyn, 2008). Abad et al. (2008) show theapplication of this methodology for the case of laboratory meanderingchannels.

Modeling of the bed morphodynamics provides the bed morphol-ogy at different spatial scales, which allows for reproducing thefeedback between bed structures and flow field (Best, 2005), like thedisruption of secondary flows because of migrating bedforms (Abadet al., 2010) and the interactions between suspended sedimentparticles and bed morphology (Schmeeckle et al., 1999).

Modeling bank erosion allows for simulating the migration of themeandering channel, which in turn affects hydrodynamics, sedimenttransport and bed morphodynamics. In a bend, faster and deeper flowdevelops near the outer bank, which causes bank erosion (Thomson,1879). At the inner bank, a point bar commonly forms and promotes

11D. Motta et al. / Geomorphology 163–164 (2012) 10–25

bank accretion. Widening in meandering channels may happen whenouter bank retreat exceeds the rate of accretion of the opposite bank(Nanson and Hickin, 1983).

Models of different degrees of sophistication have been used tosimulate freely meandering channels. For example, Ruether and Olsen(2007) performed three-dimensional (3D) numerical modeling usingReynolds Averaged Navier–Stokes (RANS) equations, a k−εmodel forturbulence, an advection–diffusion equation for suspended sediment,and van Rijn's (1984) formulation for bed load. Bolla Pittaluga et al.(2009) developed a 3D analytical model for low-sinuosity meandersand steady bed morphology. These models need coupling with bankerosion and meander evolution submodels, however, to simulateplanform changes for engineering and geological time scales.

Two-dimensional (2D) analytical models for long-term rivermigration, again valid for low-sinuosity meanders and steady bedmorphology, were developed, among others, by Ikeda et al. (1981),Blondeaux and Seminara (1985), Johannesson and Parker (1989b),Howard (1992, 1996), Sun et al. (1996, 2001), Zolezzi and Seminara(2001), and Lancaster and Bras (2002). These models calculate therate of migration based on a method independently introduced byHasegawa (1977) and Ikeda et al. (1981). This method relates the rateof migration to the near-bank excess velocity multiplied by adimensionless coefficient, and is referred to as the classic or MC(Migration Coefficient) approach hereafter. The dimensionless coef-ficient is obtained by means of calibration against field data and istypically a very small number (10−7–10−8). From a theoreticalperspective, this method has provided fundamental insight into theplanform evolution of meandering channels, but it has not been assuccessful in practical applications, where the calibration of thedimensionless coefficient can be challenging and unable to capturethe observed migration patterns. Constantine et al. (2009) sought toestablish a relation between the migration coefficient and measurablephysical characteristics of thematerials of the channel boundary usingdata from the Sacramento River, California, USA. This enables theestimation of the migration coefficient directly from field data forstreams where historical data are unavailable or controlling condi-tions have changed.

Natural meander patterns show that meander migration is notcontinuous in time or space, giving rise to spiky or, in general,irregular and complex planform shapes. Part of this complexityderives from the hydrodynamic conditions. Frascati and Lanzoni(2009) were able to reproduce features observed in nature such asupstream- or downstream-skewed simple bends, compound bends,and multiple loops using a suitable hydrodynamic model thataccounts for the full range of morphodynamic regimes in combinationwith the classic migration-coefficient approach for bank erosion. Onthe other hand, the complexity of the bank erosion processes becauseof heterogeneity of floodplain soils and vegetation can also producecomplex patterns of meander planforms and shapes of bends. Thesimple approach using a calibrated migration coefficient cannotadequately capture the complexity of bank erosion at the sub-bendscale, because it predicts a smooth centerline. Many additionallimitations occur in the MC approach. The linearity of the expressionimplies that the only bank retreat mechanism considered is particle-by-particle erosion (also termed hydraulic or fluvial erosion). It doesnot explicitly account for local, episodic mass failure mechanisms likecantilever, planar, rotational, and seepage-induced failures, which cantemporarily change rates of local bank retreat, thereby, alteringmigration patterns. The formulation does not account for an erosionthreshold. Further, it does not consider the effect of the bankgeometry either, because it assumes vertical sidewalls. Finally, theclassic approach does not consider the impact of the verticalheterogeneity of the bank materials and the associated differencesin erodibility and shear-strength of the soils.

With the ongoing effort in the United States and Europe to re-naturalize highly modified streams, it cannot be expected that

assessment studies, using the classic migration method, will accu-rately simulate the response of meandering streams to in-stream andriparianmanagement practices over engineering time scales (that is, afew years to decades or, in general, periods before cutoff occurrence).A new physically-based approach is, therefore, needed, whichexplicitly relates meander migration to the processes controllingstreambank erosion.

This paper presents a new modeling approach that merges thefunctionalities of the RVR Meander toolbox (Abad and Garcia, 2006),which is a 2D long-term meander migration model based on Ikeda etal.'s (1981) model, with the physically-based streambank erosionalgorithms of the CONCEPTS (CONservational Channel Evolution andPollutant Transport System) channel evolution model (Langendoenand Alonso, 2008; Langendoen and Simon, 2008; Langendoen et al.,2009). Darby et al. (2002) and Rinaldi et al. (2008) carried out similarefforts, however, only for short reaches and simulation periods. Thepaper describes the new physically-based methodology for comput-ing river migration and presents model tests for idealized (sine-generated and Kinoshita curve) and observed planforms. Thecomputed planform evolution is compared to that obtained with theclassic method based on a migration coefficient.

2. Model description

The modeling platform is composed of two main components. Thefirst component simulates the hydrodynamics and bed morphody-namics. The second component simulates the channel migration.Because the main goal of this paper is to evaluate the performance ofthe new approach for bank retreat as compared to the classic method,it is coupled with a simple physically-based analytical model forhydrodynamics and bed morphodynamics of meandering streams,which is based on the model of Ikeda et al. (1981).

2.1. Hydrodynamics and bed morphodynamics model

Ikeda et al.'s (1981) model for hydrodynamics and bed morpho-dynamics provides an analytical solution of the 2D depth-averagedshallow water equations through linearization and adimensionaliza-tion techniques. The model does not, however, explicitly solve for themorphodynamics of the bed, but prescribes it. The model hereinadopted is a slightly modified version of that developed by Ikeda et al.(1981) and Johannesson and Parker (1985), and details on thederivation of the solution are presented by Garcia et al. (1994). Itsmain theoretical limitations are that coupling between hydrodynam-ics and sediment dynamics is absent and that the lateral redistributionof streamwise momentum because of secondary currents is not takeninto consideration (Camporeale et al., 2007). The first issue can onlybe overcome by using a more refined model which fully couples flow,sediment transport, and bed morphodynamics (Johannesson andParker, 1989b; Zolezzi and Seminara, 2001). Without such coupling,only the sub-resonant response can be described and resonancecannot emerge, because curvature is the only forcing for the flow(Lanzoni et al., 2006; Frascati and Lanzoni, 2009). The second issuecan be indirectly addressed in Ikeda et al.'s (1981) model byincreasing the factor controlling the bed transverse slope, which itselfdepends on the secondary flow (Johannesson and Parker, 1989a). Thishas justified the use of these models in the study of long-termmeandering river dynamics (Howard and Knutson, 1984; Stolum,1996; Sun et al., 1996; Edwards and Smith, 2002) and in practicalapplications (Johannesson and Parker, 1985; Garcia et al., 1994;Camporeale et al., 2007). Below we only present the 2D solutions offlow velocity, flow depth, bed elevation, and bed shear stress used byour model.

Fig. 1 defines the coordinate system and planform and cross-section configurations. The governing equations and the solution areexpressed in intrinsic coordinates: s* is streamwise coordinate and n*

left bank at t*

right bank at t*

right bank at t*+dt*

θvalley centerline

A

A

n*

2B*

left bank at t*+dt*dt*

s*

Ro*ξ∗

a

bleft bank at t* right bank at t*

)renni()retuo( n* −n*LC

D*

H*

2B*

transversal

η∗

slope (St)

reference level

A−A

Fig. 1. Definition sketch of: (a) planform configuration and migration from time t* tot⁎+Δt⁎, and (b) cross-section configuration.

12 D. Motta et al. / Geomorphology 163–164 (2012) 10–25

is transverse coordinate. From hereon the superscript star indicatesvalues with dimensions, whereas the omission of the superscript starindicates a dimensionless quantity.

Assuming that the ratio of channel width to radius of curvature ismuch smaller than one, it is possible to derive a solution of the depth-averaged flow velocity components in s- and n-directions (U and V,respectively) and theflowdepth (D) that is composed of the solution fora straight channel (identified by the subscript ch) and a perturbationbecause of channel curvature (identified by the subscript 1)

U s;nð Þ;V s;nð Þ;D s;nð Þð Þ = 1;0;1ð Þ + U1 s;nð Þ;V1 s;nð Þ;D1 s;nð Þð Þ: ð1Þ

The solution is normalized as U=U⁎/Uch⁎ , V=V⁎/Uch

⁎ , andD=D⁎/Dch

⁎ , where Uch⁎ and Dch

⁎ are the reach-averaged velocity andflow depth at a particular time. The perturbation variables U1, V1, andD1 are

U1 s;nð Þ = a′1 nð Þe−a′2s + n a′3C sð Þ + a′4e−a′2s∫ s

0 C sð Þea′2sdsÞ�

ð2Þ

V1 s;nð Þ = a′22e−a′2s 2∫ n

1 U1 0;nð Þdn−nU1 0;nð Þ + U1 0;1ð Þ� �

+ ð3Þ

+a′22

nU1 s;nð Þ−U1 s;1ð Þð Þ + a′52

n2−1� �

D1 s;nð Þ = C sð Þn F2ch + α� �

ð4Þ

where C=B⁎C⁎=−B⁎dθ/ds⁎=B⁎/R0⁎ is curvature, B⁎ is channel half-width, θ is the angle between the channel centerline axis and thehorizontal axis, R0⁎ is local radius of curvature, Fch is reach-averagedFroude number, α is a coefficient relating transverse bed slope tocurvature, and

a′1 nð Þ = U1 0;nð Þ + nC s = 0ð Þ ð5Þ

a′2 = Cf ;chs1β ð6Þ

a′3 = −1 ð7Þ

a′4 = βCf ;ch F2ch + α−1 + s1ð Þ−D⁎ch

U⁎ch

s2 F2ch + α� � !

ð8Þ

a′5 = 1−α−F2ch� � ∂C

∂s −Cf ;cha′6B⁎

U⁎ch

C ð9Þ

a′6 =U⁎ch

D⁎ch

F2ch + α−1ð ÞU⁎ch

D⁎ch

−s2 F2ch + α� �

ð10Þ

s1 = 2 ð11Þ

s2 =U⁎ch

Cf ;ch

∂Cf

∂D⁎

!ch

= −5U⁎ch

D⁎ch

ffiffiffiffiffiffiffiffiffiCf ;ch

qð12Þ

where β=B⁎/Dch⁎ and the friction coefficient Cf reads (following

Engelund and Hansen (1967))

Cf = 6:0 + 2:5lnD⁎

2:5d⁎s

! !−2

ð13Þ

where ds⁎ is sediment particle size.The bed shear stress components in s and n directions are

calculated as

τ⁎s ; τ⁎n� �

= ρCf U⁎ ;V⁎� � ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

U⁎2 + V⁎2p

ð14Þ

where ρ is the density of water.The parameter s2, defined by Eq. (12), derives from the assumption

that the friction coefficient varies in space according to the variations inthe water depth D⁎. Assuming a spatially constant friction coefficient(equal to the reach-averaged value calculated with Eq. (13) using thereach-averaged depth), then s2=0 and the hydrodynamic solution byJohannesson and Parker (1985) is recovered.

The above expressions allow for calculating the depth-averagedflow, bed shear stress, and bed elevation distributions at any giventime step and for any given channel planform shape represented bythe curvature of the channel centerline. Following Johannesson andParker (1985)

C = − dxds

d2yds2

− d2xds2

dyds

!ð15Þ

where x and y are the dimensionless Cartesian coordinates.To avoid the propagation of numerical errors related to the

computation of the channel curvature, we use the three-point curvaturesmoothing method suggested by Crosato (1990) after every time step

Ci =Ci−1 + 2Ci + Ci + 1

4ð16Þ

where i is cross-section index. This filter removes spurious node-to-node oscillations in calculated curvature caused by the inaccuracies ofthe curvature calculation method (Eq. (15)).

b short bankswith steepprofile

planarfailure surface

a

failure surface

cohesive layer

noncohesive layer

overhanggenerated onupper bank

preferentialretreat oferodiblebasal layer

Fig. 2. Bank failure mechanisms: (a) cantilever failure and (b) planar failure.

13D. Motta et al. / Geomorphology 163–164 (2012) 10–25

2.2. Channel migration

The normal rate of bank retreat ξ⁎ is defined as

ξ⁎ =dn⁎

b

dt⁎ð17Þ

where nb⁎ is the transverse coordinate of the outer bank. Assuming thatchannel width is locally constant, the normal displacement of ameandering channel centerline equals that of the outer bank(Eq. (17)). The coordinates of a point P located on the channelcenterline then migrate as follows

dxPdt

= −ξsinθ ð18Þ

dyPdt

= + ξcosθ ð19Þ

where xP=xP⁎/B⁎ and yP=yP⁎/B⁎ are the dimensionless coordinates ofthe point P, and time and displacement are normalized as t= t⁎Uch

⁎ /B⁎

and ξ=ξ⁎/Uch⁎ , respectively.

2.2.1. Classic approach for migrationHasegawa (1977) and Ikeda et al. (1981) introduced the classic

meander migration approach, which linearly relates ξ(s) to thedimensionless perturbation velocity at the outer bank U1(s,n=1)

ξ sð Þ = E0U1 s;n = 1ð Þ ð20Þ

where E0 is a calibrated erosion coefficient which depends on banksoil properties and riparian vegetation as well as the hydraulicproperties of the flow. Eq. (20) assumes that the force of the flowexerted on the bank erodes the soil particles. Odgaard (1987),Hasegawa (1989), and Pizzuto and Meckelnburg (1989) have shownthat this assumption agrees with observations of natural meanders.

2.2.2. Proposed approach for migrationOur new methodology relates the rate of migration directly to the

physical processes controlling bank retreat, i.e. hydraulic erosion andmass failure, using the bank erosion methods of Langendoen andSimon (2008) and Langendoen et al. (2009). The new approach is alsoreferred to as PB (Physically-Based) method hereafter. The bankerosionmethod accounts for natural bank profiles and the presence ofhorizontal soil layers. This is more realistic than the classic approach,which assumes vertical banks with vertically homogeneous soilproperties represented by the erosion coefficient E0. In the PBapproach, the simulated retreat of the banks determines themigration of the centerline. Thus, the use of a migration coefficientcalibrated against historic centerlines is avoided. We further assumethat eroded bank material is carried in suspension by the flow andtransported out of the modeled channel.

The lateral rate of hydraulic erosion E⁎ for each bank-material layeris modeled using an excess shear stress relation, typically used forfine-grained materials

E⁎ = M⁎ τ⁎ = τ⁎c−1� � ð21Þ

where M⁎ is the erosion-rate coefficient (with dimensions of lengthover time) and τc⁎ is the critical shear stress. We assume that the bankshear stress τ⁎ equals the near-bank bed shear stress predicted by thehydrodynamic model at n=±1. At the banks the magnitude of theshear stress is equal to that of the shear stress in the streamwisedirection (τ=τs) since V=0.

Cantilever failures occur when overhanging blocks of bankmaterial, generated by preferential retreat of more erodible layers atdepth or simply by the erosion of the bank below the water surface,

fail (Fig. 2a). For given unit weight and shear-strength properties, theextent of the overhang (or undercut) determines its stability. Becausefailed material is immediately transported out of the modeling reach,we can, therefore, assume that stability can be assessed using anarbitrary undercut threshold.

Planar failure (Fig. 2b) is analyzed using a limit equilibriummethod in combination with a search algorithm to determine thesmallest factor of safety (stability factor), which is the ratio ofavailable shear strength to mobilized shear strength. Shear strength isa combination of cohesive and frictional forces. The bank is unstable ifthe factor of safety is smaller than one, and a failure is then simulated.Potential failure blocks are subdivided in vertical slices, and a stabilityanalysis is performed for each slice and for the entire failure block.Three different methods can be used for the computation of the factorof safety (Langendoen and Simon, 2008): (1) ordinary method, whichdoes not consider interslice forces; (2) Janbu simple method, whichconsiders only interslice normal forces; and (3) Morgenstern–Pricemethod, which considers interslice normal and shear forces. Theanalysis also considers the possible formation of a tension crack on thefloodplain behind the eroding bank face.

Whereas hydraulic erosion is a continuous process in time (as longas the critical shear stress is exceeded), cantilever and planar failureprocesses are episodic. Details of the bank stability analysis are foundin Langendoen and Simon (2008).

Two alternatives exist to compute the centerline migration. Thefirst option (Option 1, Fig. 3a) consists of calculating the centerlinedisplacement ξ⁎Δt⁎ at each cross-section from the lateral displace-ment of the toe of the right bank Si, right⁎ and that of the toe of the leftbank Si, left⁎

ξ⁎Δt⁎ =S⁎i;lef t t⁎−Δt⁎

� �−S⁎i;lef t t⁎

� �h i2

−S⁎i;right t⁎

� �−S⁎i;right t⁎−Δt⁎

� �h i2

:

ð22Þ

Alternatively, the intersect of the bank and the water surface can beconsidered instead of the bank toe. After each time step Δt⁎, the newwidth of each i-th cross-section is then:

B⁎i

� �new = S⁎i;right

� �new

− S⁎i;lef t� �

new: ð23Þ

To migrate the dimensionless centerline its displacement isnormalized by the half width of the channel B⁎. The new dimensionedgeometry can be recovered by multiplying by the half width. Because

CL

EL

0.5(ER−EL)

CL

CL

ER

(A) Initial cross−section

CL

(B) Bank erosion

(C) New centerline

CL

ER

CL

ERb

(A) Initial cross−section

(B) Bank erosion

(C) New centerline

a

Fig. 3. Centerline migration options for the proposed physically-based approach: (a) Option 1 and (b) Option 2. Bank toe displacements (EL and ER at left and right banksrespectively) determine centerline (CL) migration. Alternatively, the intersect points between banks and water surface can be used.

14 D. Motta et al. / Geomorphology 163–164 (2012) 10–25

the channel width changes after each iteration the dimensionlesscenterline coordinates x and y are rescaled as:

xnew; ynewð Þ = xold; yoldð Þ B⁎old

B⁎new

: ð24Þ

Use of this option will result in a new channel width that variesalong the stream. The hydrodynamic model assumes, however, aconstant width, which we define as the minimum width among allcross-sections:

B⁎new = min B⁎

i;new

� �: ð25Þ

The choice of the minimum width is governed by the modelimposing a slip boundary condition at the sidewalls, whichmeans thatonly the central region of the channel, where the effect of thesidewalls is not present, is actually modeled. Hence, for a series ofcross-sections characterized by different widths, the only constant-width channel which can represent the central region for all cross-sections is that having a width equaling the minimum width amongall the cross-sections. Moreover, a sensitivity analysis of thehydrodynamic solution in Eqs. (2–4) showed that changes in bedshear stress (τs⁎) along the outer bank are relatively larger forincreasing channel width than decreasing channel width. Therefore,the selection of minimum width minimizes possible errors incalculated long-term rates of migration introduced by this method.

The second centerline-migration option (Option 2, Fig. 3b) equatesthe dimensioned centerline displacement ξ⁎Δt⁎ to the displacement ofthe outer bank. We define the outer bank of a cross-section as thebank which experiences more erosion. If the outer bank is the leftbank, the dimensioned centerline displacement is

ξ⁎Δt⁎ = S⁎i;lef t t⁎−Δt⁎� �

−S⁎i;lef t t⁎� � ð26Þ

otherwise it is

ξ⁎Δt⁎ = S⁎i;right t⁎−Δt⁎� �

−S⁎i;right t⁎� �

: ð27Þ

Again, S⁎ can indicate either the bank toe or the intersect betweenbank and water surface. As in the classic approach we assume that thewidth of the channel is constant. Therefore, the inner bankdisplacement equals that of the outer bank.

The above migration options satisfy the constant-width require-ment of Ikeda et al.'s (1981) model. A more robust description ofchannel migration, however, requires a hydrodynamic model, whichconsiders, besides the interactions between channel curvature andflow-bed topography, also the interactions associated with widthvariations and the mutual width–curvature interactions (Luchi et al.,2010).

2.2.3. Regridding and smoothingCrosato (2007) demonstrated that simulating long-term channel

migration eventually introduces spurious oscillations into the centerlinegeometry because of non-equidistant, increasing cross-section spacingcaused by centerline elongation. Following Sun et al. (1996) we useParametric Cubic Splines (PCS) to extract a set of equally-spaced nodesafter each migration step. When the amplitude of meander bendsincreases, regridding of the nodes using PCS introduces additionalcenterline nodes and reduces the spacing between consecutive nodes.

Redistribution and addition of centerline nodes require theinterpolation of bank geometry and materials. Right and left bankgeometry at a newly introduced node are obtained by interpolation ofthe bank geometry of the two existing centerline nodes locatedimmediately upstream and downstream of the new node. Fig. 4illustrates the interpolation procedure. First, main chords are definedthat connect the toe and floodplain points of the existing banks. Minorchords are then generated by connecting existing points on one left or

Minor chord

Bank 2

Interpolated bank

Bank 1

Point of interpolated bank

Main chordBank profile

Existing pointInterpolated point

Fig. 4. Sketch of the procedure for bank geometry interpolation.

15D. Motta et al. / Geomorphology 163–164 (2012) 10–25

right bank profile to interpolated points on the other left or right bankprofile. The two points defining a minor chord have the sameproportional distance to the main chords bounding the bank profiles.A tolerance on minimum distance avoids closely-spaced minorchords. Linear interpolation along minor and main chords providesthe bank geometry at the new centerline node. Soil properties for eachbank-material layer are similarly obtained.

The approach of physically-based migration may locally producelarge changes in centerline curvature, which requires the use of a filterto smooth the migrated centerline to avoid numerical instability. Ourmodel adopts the Savitzky–Golay smoothing filter (Savitzky andGolay, 1964), which is a low-pass filter for smoothing noisy data and isapplied to a series of equally-spaced data values, in this case thecoordinates of the centerline nodes. The main advantage of thisapproach is that it tends to preserve features of the distribution suchas relative maxima, minima and width, which are usually flattened bymoving averaging techniques. The Savitzky–Golay filter is a general-ization of the moving window averaging method. It uses a polynomialof degree k to perform a local regression on a series of at least k+1nodes. Following Fagherazzi et al. (2004) and Legleiter and Kyriakidis(2006), the model employs 2nd and 4th-degree polynomial filterswith averaging windows of 5 to 13 nodes. In contrast to the curvaturefilter in Eq. (16), the use of the Savitzky–Golay filter is optionalbecause it is not meant to filter spurious node-to-node oscillations inchannel centerline geometry but to smooth very large curvaturegradients that may arise, for example, for highly skewed bends.

3. Model applications

We evaluate the performance of the new physically-based channelmigration method for three different cases. Table 1 reports the run

Table 1Test case run parameters. (Q is discharge; 2B* is channel width; S0 is valley slope; F02 is sqcoefficient for straight channel characterized by valley slope; β0 is half-width to depth raparameter).

Case Shape Q(m3/s)

2B*(m)

1 Sine-generated 20 302 Kinoshita 20 303 Mackinaw River 46.2 38

parameters. Because reach-averaged Froude number, friction coeffi-cient, and half-width to depth ratio change over time as the reach-averaged depth varies with changing sinuosity, Table 1 reports thevalues corresponding to a straight channel inclined at the valley slope.The channel centerline alignment of the first two cases follows a sine-generated and a Kinoshita curve, respectively. The goal of these twocases is to illustrate the features and limitations of the new approach,the variety in calculated planform shapes, and the differencesbetween the planform shapes simulated by the classic approach andour new approach. The third case tests themodel against the observedmigration of a reach on the Mackinaw River in Illinois, USA, andcompares the results to those obtained using the classic approach.Because high curvatures may develop during the simulations, thefriction coefficient computed using Eq. (13) can become undefinedbecause of a negative local depth at inner banks. Therefore, weassumed s2=0 and a spatially constant, reach-averaged frictioncoefficient was used in the simulations.

3.1. Case 1: sine-generated channel

The centerline of a sine-generated meandering channel isexpressed as

θ = θ0sin2πsΛ

� �ð28Þ

where θ0 is the angle θ at the crossover point and Λ is the length of thechannel centerline over one meander wavelength.

This case considers a 2040-meter long (2000-meter long sinuoussection with 20-meter long straight entrance and exit sections) and30-meter wide channel with Λ=250 m and θ0=55°, which corre-sponds to a relatively low sinuosity Ω=1.27 (Fig. 5). 511 equally-spaced nodes describe the initial centerline, yielding a node spacing of4 m. To accurately represent the channel planform evolution we limitthemaximum grid spacing by imposingΔs⁎b0.9B⁎. The initial channelcross-section geometry is trapezoidal with a bottom width of 30 m, atop width of 34 m, and a bank height of 2 m (therefore, the bank slopeis 45°).

We simulated the centerline evolution for a 300-year period andemployed a time step of 0.2 years and centerline-migration option 2using the displacement of the bank toes to compute the migrationdistance. We did not apply the Savitzky–Golay smoothing filter.Table 1 lists the various simulation parameters. We conducted tworuns in which only fluvial erosion and cantilever failures ofhomogeneous banks were simulated for two different sets of criticalshear stress (τc⁎) and erosion-rate coefficient (M⁎) values. The erosion-rate coefficient was 5⋅10−7 m/s for each run, while critical shearstress equalled 11.5 and 12 Pa, respectively. The maximum size of thecantilever overhang was arbitrarily set to 0.1 m.

Fig. 5 shows the simulated shear stress distribution in streamwisedirection (τs⁎) at the start of the simulation. The combination of widthto depth ratio, sinuosity, Froude number, and friction coefficientproduces a peak bank shear stress (N12 Pa) just upstream of thecrossovers, whereas it is below τc⁎ at the channel apices. This case

uared Froude number for straight channel characterized by valley slope; Cf0 is frictiontio for straight channel characterized by valley slope; and α is transverse bed slope

S0 F02 Cf0 β0 α

0.0011 0.162 0.0070 22.96 50.0010 0.146 0.0069 22.16 50.0009 0.109 0.0082 17.06 5

x [m]y

[m]

0 100 200 300 400 500 600 700 800 900 1000 1100 1200 1300 1400 1500 1600

0

100

200

1 2 3 4 5 6 7 8 9 10 11 12 13 14

Tau*_s [Pa]

Fig. 5. Simulated shear stress distribution in streamwise direction (τs⁎) for the initial planform of the sine-generated channel. Flow is from left to right.

16 D. Motta et al. / Geomorphology 163–164 (2012) 10–25

therefore resembles that of a confined meandering river, e.g. Nicolland Hickin (2010).

Fig. 6 shows the simulated migration pattern. The use of a criticalshear stress threshold for hydraulic erosion prevents some channel

x

y [m

]

800 850 900 950 100

0

40

80

120Initial

After 300 years, T

After 300 years, T

Fig. 6. Simulated migration pattern of a sine-generated channel using the physically-based bFlow is from left to right.

0 1 2 3 4

Fig. 7. Meander bends along the confined meandering Beaver River in Alberta, Canada (54°Distribution Spot Image/Astrium Services).

x

y [m

]

800 850 900 950 100

0

40

80

120Initial

After 300 years, P

Fig. 8. Comparison between the simulated migration patterns of a sine-generated channel uand MC 3, E0=4 ⋅10−9. Flow is from left to right.

portions from migrating. As a consequence, the simulated migrationpattern after 300 years is strongly skewed and locations characterizedby high curvature gradients arise. The lower critical shear stressresults in a higher rate of downstream migration. The simulated

[m]0 1050 1100 1150 1200 1250

au*_c = 12 Pa

au*_c = 11.5 Pa

ank retreat method with two different critical shear stress values for hydraulic erosion.

miles

15′ 35″ N and 109° 59′ 08″ W). The image is provided by Google Earth (© CNES 2011.

[m]0 1050 1100 1150 1200 1250

After 300 years, MC 1

After 300 years, MC 3

After 300 years, MC 2B

sing the PB (τc⁎=11.5 Pa) and MC approaches. MC 1, E0=2 ⋅10−9; MC 2, E0=3 ⋅10−9;

17D. Motta et al. / Geomorphology 163–164 (2012) 10–25

planform is very similar to that of confined meandering channels as isshown by the observed series of river bends along the Beaver River,Canada (Fig. 7).

Fig. 8 shows the comparison between the simulated migrationpattern obtainedusing the classicmethod and thenewphysically-based

x

y [m

]

250 300 350 400 450 500 550 600 650

50

100

150

200

250

InitialAfter 5 years, PB

Fig. 9. Simulated migration pattern and shear stresses using the physically-based migra

x

y [m

]

250 300 350 400 450 500 550 600 650

0

50

100

150

200

250 InitialAfter 5 years, PB

Fig. 10. Simulated migration pattern and shear stresses using the physically-based migra

x

y [m

]

400 500 600 700 8000

50

100

150

200

250

300

After 5 years,Initial

Fig. 11. Comparison of the simulated migration patterns obtained with the MC and PB

x

y [m

]

400 500 600 700 8

0

50

100

150

200Initial After 5 years,

Fig. 12. Comparison of the simulated migration patterns obtained with the MC and PB a

method. The migration coefficient values used in the MC approach areE0=2⋅10−9, E0=3⋅10−9, and E0=4⋅10−9. The PB and MCmethodspredicted different planform shapes. The MC approach cannot repro-duce the pattern predicted by the PB approach. As the PB approach, theMC method also predicted a downstream migration of the meander

[m]700 750 800 850 900 950 1000 1050 1100

x [m]

y [m

]

350 400 450 500 550 600 650

50

100

150

200

1 2 3 4 5 6 7

Tau*_s [Pa]

tion method for an upstream-skewed Kinoshita channel. Flow is from left to right.

[m]700 750 800 850 900 950 1000 1050 1100

x [m]

y [m

]

350 400 450 500 550 600 650

0

50

100

150

200 1 2 3 4 5 6 7

Tau*_s [Pa]

tion method for a downstream-skewed Kinoshita channel. Flow is from left to right.

[m]900 1000 1100 1200 1300

MC After 5 years, PB

approaches for an upstream-skewed Kinoshita channel. Flow is from left to right.

[m]00 900 1000 1100 1200

After 5 years, PBMC

pproaches for a downstream-skewed Kinoshita channel. Flow is from left to right.

x [m]

y [m

]

500 550 600 650 700 750 800 850 900 950

-50

0

50

100

150

200

250

300

Initial After 5 years, PB

BA

a

x [m]

y [m

]

500 550 600 650 700 750 800 850 900 950

-50

0

50

100

150

200

250

300

Initial After 5 years, PB

AB

b

Fig. 14. Simulated migration pattern of an upstream-skewed Kinoshita channelusing the physically-based migration method. The floodplain comprises two differentsoils: in zone A, τc⁎=10 Pa and M⁎=5 ⋅10−7 m/s; and in zone B, τc⁎=4 Pa andM⁎=5 ⋅10−7 m/s. Flow is from left to right.

18 D. Motta et al. / Geomorphology 163–164 (2012) 10–25

bends with little growth of the meander amplitude, however, it cannotproduce the strong asymmetry simulated by the PB approach. From Fig.8 it is evident that the bank retreatmodel strongly affects themigrationpattern. The MC approach is only capable of simulating such strongplanform asymmetry by assuming non-erodible valley boundaries.Howard (1992) and Howard (1996) did follow this approach for theconfined Beaver River.

3.2. Case 2: Kinoshita channel

Kinoshita-generated meandering channels are described byKinoshita (1961)

θ = θ0 sin κsð Þ + θ30 Jscos 3κsð Þ− Jf sin 3κsð Þ� �

ð29Þ

where Js and Jf are the skewness and flatness coefficients respectively,and κ=2π/Λ is the wave number.

We consider a 4200-meter long (4000-meter long meanderingsection with 100-meter long straight entrance and exit sections) and30-meter wide channel with Λ=500 m, Js=±1/32 (positive forupstream-skewed configuration and negative for downstream-skewed configuration), Jf=1/192 and θ0=110°, which correspondsto a high sinuosity Ω=3.28. These parameters result in locally high-curvature reaches which violate the assumption of mild curvature ofIkeda et al.'s (1981) model. The below model tests were designed,however, tominimize the impact of the high-curvature reaches on themodel results. The initial cross-section geometry is trapezoidal with abottom width of 30 m, a top width of 34 m, and a bank height of 2 m.421 equally-spaced nodes describe the initial channel centerline,which yields a node spacing of 10 m.

We assessed the performance of the new approach for fourdifferent scenarios: (1) a comparison with the MC method regardingthe evolution of upstream- and downstream-skewedmeander bends;(2) spatially heterogenous floodplain soils; (3) sensitivity analysis ofthe centerline migration method (cf. Section 2.2.2) and centerlinesmoothing method (cf. Section 2.2.3); and (4) influence of planarfailures on centerline migration. Table 1 lists the main parametersused in the simulations.

3.2.1. Evolution of upstream- and downstream-skewed meander bendsWe performed a 5-year simulation using: a time step of 0.2 years;

centerline migration option 2 using the displacement of the bank toeto compute the migration distance; and the Savitzky–Golay smooth-ing filter with second order polynomial regression, an averagingwindow of 5 nodes, and applied every 10 iterations. We onlyconsidered hydraulic erosion and cantilever failure processes ofhomogeneous banks with τc*=5 Pa and M⁎=5⋅10−7 m/s.

Fig. 13. Preferential migration of bend portions at lobes A and B in the Pembina River in Albestrong local curvature and curvature gradient is present in the downstream portion of lobe

Figs. 9 and 10 show the simulated centerlinemigration patterns forupstream- and downstream-skewed meander bends, respectively.The downstream-skewed meander bends migrate faster than thoseskewed upstream, as was already speculated by Abad and Garcia(2009). Also, whereas the upstream-oriented bends tend to preservethe orientation, the downstream-oriented bends are changing theorientation towards upstream. As pointed out by Lanzoni et al. (2006),

rta, Canada (Parker et al., 1982). Flow is from left to right. A reach characterized by veryB.

x [m]

y [m

]

500 550 600 650 700 750 800 850 900 950

-50

0

50

100

150

200

250

300

Initial After 5 years, PB

A BA B AB BA

Fig. 15. Simulated migration pattern of a downstream-skewed Kinoshita channelusing the physically-based migration method. The floodplain comprises two differentsoils: in zone A, τc⁎=8 Pa and M⁎=5 ⋅10−7 m/s; and in zone B, τc⁎=5 Pa andM⁎=5 ⋅10−7 m/s. Flow is from left to right.

x [m]

y [m

]

550 600 650 700 750 800

50

100

150

200

InitialAfter 10 years, PB, Case 9

After 10 years, PB, Case 12After 10 years, PB, Case 11After 10 years, PB, Case 10

a

x [m]

y [m

]

550 600 650 700 750 800

50

100

150

200

InitialAfter 10 years, PB, Case 3

After 10 years, PB, Case 15After 10 years, PB, Case 11After 10 years, PB, Case 7

b

Fig. 16. Comparison of simulated migration patterns for the different test cases listed inTable 2. Flow is from left to right. (a) Cases 9, 10, 11, and 12. (b) Cases 3, 7, 11, and 15.

19D. Motta et al. / Geomorphology 163–164 (2012) 10–25

the hydrodynamic model used here can only describe sub-resonantmorphologic regimes and is, therefore, bound to produce, inhomogeneous soil, upstream-skewed planform features. As laterillustrated in the paper, however, floodplain soil heterogeneity canpreserve downstream-oriented bends (Fig. 15), which motivated theuse of an initial downstream-skewed configuration for the simula-tions in Figs. 10 and 15. Compared to the PB approach, the MCapproach, with E0=2.5 ⋅10−7, simulates similar trends, see Figs. 11and 12. The results of the PB approach, however, show a moredramatic tendency towards the formation of necks for the down-stream-skewed configuration, which is commonly observed fornatural rivers. In Fig. 9 and especially in Fig. 10 some portions ofbend are below the critical shear stress for hydraulic erosion,enhancing the spatial discontinuity of the migration process andgenerating zones characterized by very strong local curvature andcurvature gradient. Although the above scenarios are idealized, theycapture the preferential migration of only portions of a bend, which isoften observed in nature such as in the case of the Pembina River inAlberta, Canada (Fig. 13).

3.2.2. Spatially heterogenous floodplain soilsWe conducted two simulation scenarios to study the effect of

floodplain soil heterogeneity onmeander migration. The first scenarioconsiders an upstream-skewed Kinoshita channel with different soilcritical shear stresses for the left-most 67% and right-most 33% of the

Table 2Sensitivity analysis test cases (IF=Iteration Frequency, i.e. iteration interval for the applica

Case Centerline migration method Bank point used

1 Option 1 Toe2 Option 1 Toe3 Option 1 Toe4 Option 1 Toe5 Option 1 Water surface in6 Option 1 Water surface in7 Option 1 Water surface in8 Option 1 Water surface in9 Option 2 Toe10 Option 2 Toe11 Option 2 Toe12 Option 2 Toe13 Option 2 Water surface in14 Option 2 Water surface in15 Option 2 Water surface in16 Option 2 Water surface in

river valley (Fig. 14). The erosion-rate coefficient is the same for eachsoil (M⁎=5⋅10−7 m/s), whereas the critical shear stresses are 4 and10 Pa, respectively. In the second scenario the Kinoshita channel isdownstream-skewed and the upper and lower halves of eachmeander bend are located in two different floodplain soils. The twosoils alternate in downvalley direction (Fig. 15). The rate of erosioncoefficient is the same for each soil (M⁎=5⋅10−7 m/s), whereas the

tion of centerline filtering or bank interpolation).

for migration Filtering Bank interpolation

No NoYes (IF=10) NoNo Yes (IF=10)Yes (IF=10) Yes (IF=10)

tersect No Notersect Yes (IF=10) Notersect No Yes (IF=10)tersect Yes (IF=10) Yes (IF=10)

No NoYes (IF=10) NoNo Yes (IF=10)Yes (IF=10) Yes (IF=10)

tersect No Notersect Yes (IF=10) Notersect No Yes (IF=10)tersect Yes (IF=10) Yes (IF=10)

20 D. Motta et al. / Geomorphology 163–164 (2012) 10–25

critical shear stresses are 5 and 8 Pa, respectively. These idealizedscenarios are able to show that differences in bank soil properties caninvert the migration pattern of upstream- and downstream-skewedKinoshita channels in homogeneous soils (cf. Figs. 9 and 10).

Fig. 14 shows the results of the first scenario. The more erosion-resistant soil (i.e., soil A) locally inhibits channel migration. Fig. 14ashows that upstream skewness increases compared to the homoge-neous scenario (cf. Fig. 9) if the majority of the channel is locatedwithin floodplain soil B (the more erodible soil). If the majority of thechannel is located within soil A, the migration of the meander bendslocated within floodplain soil B causes these bends to becomedownstream-oriented (see Fig. 14b).

Fig. 15 shows the results of the second scenario. The cyclic variation insoil erosion resistance increases the rate of migration of every othermeander. As a consequence, the tendency of the downstream-orientedbends to change their orientation towards upstream is moderated (cf.Fig. 10).

3.2.3. Sensitivity to migration and smoothing methodsWe performed a sensitivity analysis to study the effects of the

centerlinemigration(seeSection2.2.2)andsmoothing(seeSection2.2.3)methods. Table 2 lists the 16 cases evaluated. We also analyzed theimpact of using either the displacement of the bank toe or that of theintersectbetweenwater surfaceandbankprofile in calculating centerlinemigration. The Kinoshita channel was upstream-skewed and itsconfiguration is the same as for the above test cases. The simulationperiod was 10 years.

x [m]

y [m

]

400 450 500 550 600 650 700 750

50

100

150

200

250

InitialAfter 10 years, PB, Soil 1After 10 years, PB, Soil 2

a

Station [m]

Ele

vati

on

[m

]

-35 -30 -25 -20 -15 -10

-5

0

5

10

15

t = 10 years

t = 1 years

t = 0 years

Soil 1c

Fig. 17. Impact of the planar failure mechanism on the simulated migration pattern for an upattern after 10 years with soils 1 and 2; (b) detail of the simulated migration pattern (centmonitoring node; (c) evolution of the left bank geometry at the monitoring node with soil

Fig. 16 presents the most relevant results of the analysis. Onlyminor differences occur between the various cases in the zonescharacterized by high curvature (indicated with circles in Fig. 16a),because shear stress only exceeds the critical shear stress downstreamof the bend apex (cf. Fig. 9). Fig. 16b shows that near the crossoverpoints, however, the centerline migration method has the largesteffect on the planform evolution. The simulated centerline migrationin cases 3 and 7 is significantly different from that in cases 11 and 15.The rates of migration are lower when using Option 1 because thedisplacement of the eroding outer bank is partially counteracted bythat of the eroding inner bank (Fig. 3a). This then raises the question:which method is more appropriate? Observations show that manymeandering alluvial streams, in the long term, maintain a roughlyconstant width even while actively migrating (Ikeda et al., 1981).Alluvial streams accomplish this by balancing erosion at one bankwith deposition at the opposite bank. We, therefore, suggest Option 2for long-term simulations in which the assumption of constant widthis reasonable from empirical observations. The model, however, doesnot provide a physically-based description of the deposition processeswhich lead to the reconstruction of the inner bank. Option 1 requiresthe selection of a representative channel width to calculate thehydrodynamics and the bed morphodynamics because Ikeda et al.'s(1981) analytical model assumes a constant channel width (see alsoSection 2.2.2). Obviously this introduces an error, which however canbe considered negligible compared to other approximations intro-duced in the modeling if the simulation period is not too long (a fewyears). Then, the differences between the hydrodynamics and bedmorphodynamics computed for a constant-width channel are not too

x [m]

y [m

]

420 440 460 480 500

140

160

180

b

Node at t = 0 years

Station [m]

Ele

vati

on

[m

]

-35 -30 -25 -20 -15 -10

-5

0

5

10

15

t = 10 years

t = 1 years

t = 0 years

Soil 2

d

pstream-skewed Kinoshita channel. Flow is from left to right. (a) Simulated migrationerlines every one year for soil 2 and final centerline only for soil 1) with location of the1; and (d) evolution of the left bank geometry at the monitoring node with soil 2.

21D. Motta et al. / Geomorphology 163–164 (2012) 10–25

different from those calculated for a varying-width channel. We,therefore, suggest to use Option 1 for relatively short-term simula-tions, which are associatedwith small longitudinal changes in channelwidth resulting from net erosional processes. A more accuratedescription of the hydrodynamics and morphodynamics of varying-width channels requires either a fully non-linear 2D depth-averagednumerical model or, as previously mentioned, an analytical solutionwhich also considers width variations, such as that developed byLuchi et al. (2010). The cross-section geometry obtained with Option1 is not that corresponding to morphologic equilibrium, because onlythe flow in the central region of the channel is actually computed.Bank shape, however, is in equilibrium after the bank erosion processesare solved.

3.2.4. Effects of planar failure mechanics on channel centerline migrationWe studied the impact of planar bank failures on migration

patterns by varying the shear strength of the bank soil. The Kinoshitachannel is upstream-skewed with a similar geometry as used earlier.The initial streambanks are vertical, however, and comprise a singlesoil with τc⁎=5 Pa, M⁎=5⋅10−7 m/s, and a saturated unit weight of18 kN/m3. The cohesion of each soil was 5 kPa, whereas the frictionangle was varied as 26° (soil 1) and 20° (soil 2). We used theMorgenstern–Price bank stability analysis method for the computa-

0 2 41 Kilom

Study reach

Mackinaw

River

Illinois

Mac

Peoria

Fig. 18. Mackinaw Ri

tion of the factor of safety. The lateral displacement of the intersect ofbank and water surface determined centerline migration.

At a monitoring cross-section located in the downstream portionof one of the Kinoshita bends, a planar failure (factor of safety is 0.91)occurs at the beginning of the simulation in the case of soil 2(Fig. 17d). The bank then retreats because of the combined action ofhydraulic erosion and cantilever failure and preserves the shape of thebank profile. No planar failures occur in the case of soil 1 during thesimulation period at the cross-section considered (Fig. 17c); the soilshear-strength and limited bank height inhibit bank failure. In bothcases the rate of bank retreat decreases in time because the channeltends to straighten at the location considered with consequentdecrease of the streamwise shear stress at the bank. At the end ofthe first year of simulation, bank retreat is greater in the case of soil 2,because of the occurrence of planar failure. After 10 years, however,bank retreat is greater for the case of soil 1, because of the evolution ofthe local hydrodynamic conditions, which are determined by localand upstream curvatures.

The impact of planar failures is mainly limited to the shape of thecross-section, whereas the rates and distances of migration are onlyweakly affected (Fig. 17a and b). Mass failures will generally,however, affect the local flow field and failed material maytemporarily protect the bank toe from eroding. We do not consider

eters

kinaw River

South Pekin

Green Valley

ver study reach.

22 D. Motta et al. / Geomorphology 163–164 (2012) 10–25

these effects here because the focus is on long-term migrationpatterns at the reach scale. The results shown in Fig. 17 seem toconfirm the observations by Constantine et al. (2009), who found acorrelation between migration coefficient (E0) and erosion-ratecoefficient (M⁎), implying that the effects of mass failure mechanismson the rate of channel migration can be accounted for by adjustingM⁎.

3.3. Case 3: Mackinaw River

We tested the newmeandermigrationmodel against the observedmigration of a reach on the Mackinaw River in Illinois, USA, andcompared the results to those obtained using the MC approach. Thereach is located in Tazewell County about 15 km upstream of thejunction of the Mackinaw River with the Illinois River between thetowns of South Pekin and Green Valley (Fig. 18). Fig. 19 shows aerialimagery of the study reach in the years 1951 and 1988.

An equidistant grid of 350 nodes with a spacing of 11.4 mdescribes the initial channel centerline (year 1951). Channel widthis 38 m and valley slope is 0.0009 m/m (Garcia et al., 1994). The meansinuosity of the study reach is 1.34. From an analysis of the dischargerecord between 1922 and 1956 at the USGS station 05568000 nearGreen Valley, we derived a model discharge of 46.2 m3/s, which isbetween the average value (20.9 m3/s) and the maximum value(61.6 m3/s) of the mean annual streamflow over the period. Table 1lists the values of the other parameters which characterize thesimulation. The upstream boundary for modeling was set in a straightreach. Because the velocity distribution is not known there, a uniformprofile in the transverse direction was assumed for simplicity.

Simulated bank retreat in the PB method is a combination ofhydraulic erosion and cantilever failures. Because no bank-materialdata are available, we assumed homogeneous bank material andcalibratedM⁎=1.2 ⋅10−6 m/s and τc⁎=9 Pa. This rather large value ofcritical shear stress accounts for the effects of temporary bank toe

Fig. 19. Historic aerial photographs of the Mackinaw River study

protection by failed bank materials and the presence of riparianvegetation visible in Fig. 19, and implicitly accounts for the absence ofa transfer function from near-bed to near-bank shear stress and theomission of shear-stress partitioning between skin friction, respon-sible for hydraulic erosion, and bedform friction. Further, we used:(a) channel centerline migration Option 2 using bank toe displace-ment; (b) a 2nd-order Savitzky–Golay filter with an averagingwindow of 5 nodes applied every 10 iterations; and (c) interpolationof bank physical properties every 10 iterations. For the MC approach,we calibrated two alternative values of migration coefficient E0:5.0⋅10−7 and 6.5 ⋅10−7.

Fig. 20 compares the simulated centerlines using the MC and PBmethods to that observed in 1988. The simulated channel centerlineusing the PB method agrees well with that observed away from theboundaries of the model reach. In terms of planform shapes, the PBapproach can capture the growth of the four upstream lobes (L1, L2,L3, and L4), which preserve the symmetry while migrating. The MCmethod produces shapes for lobes L1, L3, and L4, which arecharacterized by strong upstream skewness. Lobes L1 and L3 developa compound-loop shape which cannot be reproduced by the MCapproach. The PB method also performs remarkably well also in thedownstream portion of the reach (lobes L5, L6, and L7). Tuning thevalue of the migration coefficient (MC 1 or MC2 in Fig. 20) can matchthe observed pattern in one bend or the other, but in general thepredicted migration is biased in terms of lateral migration (especiallyin the downstream portion of the reach) and downstream migration.

As a measurement to quantify model performance, we calculatedthe ratio of the area between simulated and observed centerlines tothe length of the observed centerline, which is equivalent to anaverage distance between simulated and observed centerlines. Thisdistance is 94.6 m (about 2.5 times the channel width) for the MC 1simulation, 126.5 m (3.3 times the channel width) for the MC 2simulation, and 67.1 m (1.8 times the channel width) for the PB

reach in the years 1951 and 1988. Flow is from right to left.

x[m]

y[m

]

-3000 -2800 -2600 -2400 -2200 -2000 -1800 -1600 -1400 -1200 -1000 -800 -600 -400 -200 0

x[m]-3000 -2800 -2600 -2400 -2200 -2000 -1800 -1600 -1400 -1200 -1000 -800 -600 -400 -200 0

x[m]-3000 -2800 -2600 -2400 -2200 -2000 -1800 -1600 -1400 -1200 -1000 -800 -600 -400 -200 0

0

200

400

600

800

y[m

]

0

200

400

600

800

y[m

]

0

200

400

600

800

1951 historic

1988 predicted, MC 11988 historic

a

1951 historic

1988 predicted, MC 21988 historic

b

1951 historic

1988 predicted, PB1988 historic

cL3

L2

L1L7

L4

L6

L5

Fig. 20. Comparison between historic and simulated 1988 channel centerlines of the Mackinaw River study reach. MC: migration coefficient method (MC 1, E0=5.0 ⋅10−7; MC 2,E0=6.5 ⋅10−7). PB: physically-based method (M*=1.2 ⋅10−6 m/s and τc*=9 Pa). Flow is from right to left.

23D. Motta et al. / Geomorphology 163–164 (2012) 10–25

simulation. Therefore, the prediction error using our new method is,respectively, 29 and 47% smaller than that of the classic method.

4. Conclusions

To quantify the migration of meandering streams researchers havebeen using an empirical formulation that relates the rate of channelmigration to excess near-bank velocity and a migration coefficient.This approach requires the calibration of the migration coefficientagainst historic channel centerlines, and, therefore, does not explicitlyrelate channel migration to the processes controlling streambankretreat. A new physically- and process-based method was developedthat relates channel migration to the streambank erosion processes ofhydraulic erosion andmass failure. Hence, channel migration dependson measurable soil properties, natural bank geometry, distribution ofriparian vegetation, and vertical and horizontal heterogeneity offloodplain soils. This approach is suitable for long-term simulation ofmigration patterns of natural rivers.

The presented test cases show that the planform shapes obtainedwith the physically-based migration method differ from thoseproduced using the classic approach. The new approach is able tosimulate features such as high skewness and sharp necks, which arecommonly observed in nature. In particular, it is capable of modelingdownstream skewness of meander bends (Fig. 14b), compound loops(Fig. 20), “rectangular” shapes (Fig. 8), and preferential migration ofsome portions of a bend (Figs. 9 and 10). The test cases also show that

spatial heterogeneity of floodplain soils is as important as thesimulated hydrodynamics in determining the planform evolution.Mechanisms of mass failure, like planar failures, are important but canbe represented by modifying the resistance to erosion parameters τc⁎

and M⁎ to quantify long-term rates of migration. Application of theproposed approach to the Mackinaw River in Illinois, USA, showedsignificant improvements over the classic approach in predicting themigration of natural streams.

Notwithstanding the above improvements, the model still haslimitations regarding the simulation of hydrodynamics and bedmorphodynamics. Most importantly, the model assumes: (a) uniformbed material; (b) constant water discharge (therefore, the impact ofunsteady flows on river migration is not taken into account); (c)constant channel width to compute the hydrodynamics and bedmorphodynamics; (d) eroded bank soils are transported as suspendedload out of the modeling reach; and (e) no net aggradation ordegradation occurs along the channel. In particular, the last twolimitations highlight the need of coupling sediment transport andbank erosion: because gravitational bank failure depends on bankheight, bed aggradation/degradation because of sediment transportimbalance or deposition of failed bank material at the toe could affectbank erosion.

Future studies will evaluate: (1) the effects of vertical bank-soilheterogeneity and bank shear stress distribution on migrationpatterns; (2) the impact of horizontal soil heterogeneity on migrationpatterns; (3) the need of more advanced linear models of meandering

24 D. Motta et al. / Geomorphology 163–164 (2012) 10–25

river hydrodynamics (e.g., Zolezzi and Seminara (2001)), possiblyaccounting for width variations (Luchi et al., 2010); (4) the impact ofcenterline filtering on the migration patterns, especially for very long-term scenarios; and (5) model applications to natural rivers withmeasured bank-soil physical properties.

The current version is a stand-alone platform for Windows andLinux operating systems. It is currently being migrated to a GIS(Geographical Information Systems) environment. It can handle theincorporation of other linear and fully nonlinear 2D numerical models.The model presented can be applied for river restoration andremeandering projects, in particular to assess the degree of instabilityof meandering channel designs.

Acknowledgments

This research was supported by an agreement from the U.S.Department of Agriculture, Forest Service, Pacific Southwest ResearchStation and using funds provided by the Bureau of Land Managementthrough the sale of public lands as authorized by the Southern NevadaPublic Land Management Act. This work was performed underSpecific Cooperative Agreement No. 58-6408-8-265 between theDepartment of Civil and Environmental Engineering at the Universityof Illinois at Urbana-Champaign and the U.S. Department ofAgriculture, Agricultural Research Service, National SedimentationLaboratory. Inci Guneralp is gratefully acknowledged for providing theaerial pictures and the channel centerlines of the Mackinaw River.

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