A graded stream response relation for bed load–dominated streams

A graded stream response relation for bed load––dominated streams

Brett C. Eaton and Michael ChurchDepartment of Geography, University of British Columbia, Vancouver, British Columbia, Canada

Received 27 June 2003; revised 26 May 2004; accepted 8 July 2004; published 1 September 2004.

[1] The relation between equilibrium stream channel morphology and three independentgoverning variables, discharge, sediment supply, and valley slope, is investigatedusing a stream table. The experiments are based on a generic physical model of a laterallyactive gravel bed stream and directed by a recently published rational regime model.While a variety of channel adjustments are possible, the primary adjustment observedwas in the channel slope. Other adjustments, such as surface armor composition andcross-sectional shape, were minor in comparison. The equilibrium slope is well predictedby a linear function of sediment concentration which is bounded by two thresholds:one associated with the minimum stream power necessary to deform the bed and the otherassociated with the maximum sediment feed that can be transported by the availableflow. The results suggest that the system tends to move toward the minimum slope capableof transporting the sediment supply, in the process increasing the flow resistance forthe system. Whether or not a minimum slope or a maximum system-scale flow resistanceis reached cannot be determined given the available data, but there appears to be a uniqueslope (or, at least, a limited range of slopes) for which a stable alluvial channelmorphology can be established for a given sediment concentration. The implications ofthis behavior are evaluated for the theoretical basis of the extremal hypothesis used toclose the formulation of a rational regime model. INDEX TERMS: 1815 Hydrology: Erosion and

sedimentation; 1824 Hydrology: Geomorphology (1625); KEYWORDS: channel response, Froude scaling,

flume experiments, graded river, flow competence

Citation: Eaton, B. C., and M. Church (2004), A graded stream response relation for bed load–dominated streams, J. Geophys. Res.,

109, F03011, doi:10.1029/2003JF000062.

1. Introduction

[2] Alluvial channel morphology has long been under-stood to reflect the action of several governing conditions[Lane, 1957; Mackin, 1948], the most important of whichare (1) the time distribution of fluid flow, Q(t); (2) the timeand size distribution of sediment flux, Qb(t, D); (3) thelongitudinal valley slope, Sv; (4) the boundary materials andstrength, including vegetation; and (5) the geological his-tory. However, current understanding of channel dynamicsremains primarily qualitative. For example, while it ispossible to distinguish single-thread from multiple-threadchannels based on the characteristics of a stream and itsfloodplain [Begin, 1981; Henderson, 1961; Lane, 1957;Leopold and Wolman, 1957; Parker, 1976], it is much moredifficult to predict the sinuosity of single-thread channels[e.g., Dade, 2000; van den Berg, 1995]. This situationarises from the near-impossibility to collect appropriatedata in the field. The primary problem is adequate charac-terization of the boundary conditions, particularly bankstrength and sediment supply rate. In any case, there isno assurance that channels in the field are the product of anequilibrium response to constant governing conditions or,

indeed, that the governing conditions remain constant[Howard, 1988].[3] Similarly, our understanding of channel response to

changes in the governing conditions is incomplete. Currentunderstanding of channel response has generally been sum-marized in tables showing the possible direction of anadjustment in some aspect of the channel geometry (e.g.,width, depth, slope) to changes in, primarily, discharge andsediment supply [Schumm, 1971]. These summaries areindeterminate since one or more channel response remainsundefined. For example, Schumm’s [1971] analysis suggeststhat in response to a simultaneous increase or decreasein both Q and Qb, mean depth and sinuosity may eitherincrease or decrease. These summaries are based on(1) empirical hydraulic geometry relations which describethe outcome of some set of channel processes but which arenot themselves expressions of physical laws; (2) fieldevidence from individual case studies; and (3) the typicalform of relations predicting sediment transport and flowresistance [Kellerhals and Church, 1989; Montgomery andBuffington, 1998; Schumm, 1971]. They do not includeany constraints on the stability of the resultant form, whichis one factor contributing to their relative ambiguity withrespect to the nature of channel response. The matter iscompounded by the effect of geomorphic history, whichmay restrict the range of adjustments available to a stream

JOURNAL OF GEOPHYSICAL RESEARCH, VOL. 109, F03011, doi:10.1029/2003JF000062, 2004

Copyright 2004 by the American Geophysical Union.0148-0227/04/2003JF000062$09.00

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channel by forcing the stream into a configuration fromwhich other theoretically stable configurations are notaccessible.[4] Eaton et al. [2004] have recently demonstrated that a

rational regime model incorporating a bank stability analy-sis of the form described by Millar and Quick [1993] canconstrain the possible channel response to changes in thegoverning conditions and thus can provide a more completeand informative framework for evaluating river channelresponse. The key to their approach is the definition ofthe ‘‘fluvial system,’’ representing a unit length of flood-plain rather than a unit length of channel. A stream mayfollow paths of varying sinuosity and length to achieve thesame net down-valley transport, and thus channel lengthbecomes a variable that can be adjusted by the system toreach equilibrium. Eaton et al. [2004] postulate that stabilityis approached by maximizing flow resistance for the fluvialsystem, thereby minimizing the energy available to deformthe system, subject to the ability to pass the sedimentsupplied with the available discharge. The system-scaleflow resistance is resolved into three separate components:

fsys ¼ f 0 þ f 00 þ f 000; ð1Þ

where f 0 is the grain resistance; f 00 is the within-channelform resistance due to bars, dunes, and other in-channelfeatures; and f 000 is the reach-scale form resistance due tochannel sinuosity. Variable fsys is calculated per unit headdrop along the floodplain, not along the channel; thus

fsys ¼8gRSv

u2; ð2Þ

where R is the hydraulic radius corresponding to theformative discharge, Sv is the valley slope, and u is theaverage velocity, also for formative discharge. Eaton et al.[2004] show that if f 0 and f 00 are held constant, fsys ismaximized by increasing the path length taken by thechannel (or equivalently, its sinuosity) to achieve the samenet down-valley sediment and fluid transfer, therebyminimizing the channel slope. This is equivalent to applyinga minimum slope hypothesis [Chang, 1979] or any of theequivalent extremal hypotheses [Davies and Sutherland,1983]. In the context of the fluvial system presented byEaton et al. [2004], Q, Qb, and Sv are defined to beindependent variables, and the channel morphology isassumed to adjust to equilibrium by changing the flowresistance components in equation (1).[5] The objectives of this work are (1) to explore the

relation between equilibrium single-thread channel geome-try and the governing conditions (Q, Qb, Sv) and (2) toelucidate some of the principles underlying channelresponse to changes in these variables in light of the theorypresented by Eaton et al. [2004]. Two sets of stream tableexperiments are presented. In one set, Sv was held constant,while Q and Qb were varied. In the other set, channelsinuosity (L*) was held approximately constant by covary-ing Qb and Sv in accordance with the understandinggenerated from the first set of experiments. The secondset was run in order to investigate the generality of therelation established from the first set (further justificationsfor these experiments are discussed in section 3). The

imposed range of Q (2.6–4.3 L s�1) and Qb (128–214 g min�1) are meant to represent a reasonable rangethat a prototype stream might experience as a result ofclimate change and/or changes in land use.[6] To the authors’ knowledge, previous stream table

experiments [e.g., Ackers and Charlton, 1970; Ashmore,1982; Schumm and Khan, 1972] have not treated Q, Qb, andSv as independent variables; either Qb and Sv have beencovaried, according to no particular theoretical basis, or thesediment transported by the stream has been recirculated.Nor have all of them been able to demonstrate whether ornot equilibrium and/or steady state conditions were reached[e.g., Schumm and Khan, 1972]. These types of experimentsdo not permit the relation between the governing conditions(namely Q and Qb in the experimental context) to bediscerned.

2. Experimental Methods

[7] The experiments were conducted on a tilting streamtable that is 30 cm deep, 3 m wide, and 20 m long using asand-granule mixture. This apparatus provides a reasonablescale model for a moderately steep (�1%) gravel bedstream. For example, at a scale of 1:32 the model can beinterpreted as a generic representation of a stream with aD50 of about 32 mm. The goal of the generic modelingapproach adopted here is to recreate some reasonableprocess similarity to a prototype river or class of rivers,without modeling the detailed morphology of any onechannel. Such an approach was discussed by Hooke[1968] and reported by Schumm et al. [1987]. It remainsnecessary to maintain the values of key dimensionlessparameters in the model, such as grain Reynolds number(Re*) and Froude number (Fr), within a reasonable range ofplausible values and to replicate a suitable range of proto-type grain sizes in the model. In the suite of gravel bedprototypes considered here it is reasonable to require thatFr < 1, while Yalin [1971] claims that reasonable dynamicsimilarity is approximated so long as Re* > 70–150 (Re*was on the order 110 for these experiments). The grain sizedistribution is based on bed material samples of a mean-dering gravel bed stream reported by Eaton and Lapointe[2001]. The design grain size distribution excluded sedi-ment <177 mm in order to maintain approximate similitudein the bed material transport process since below that size,the dimensionless shear stress for entrainment changesrapidly (see reformulated Shield’s curve in the work ofYalin [1971, p. 154]), and the particles may no longer beexpected to behave in a manner similar to their positedequivalents in the generic prototype. To accommodate thislower limit, the original prototype distribution is, in effect,truncated at 5.6 mm. A commercially available sand mix-ture matches the design distribution tolerably well, as shownin Figure 1.[8] The stream table was filled with sand to approximately

12 cm uniform depth and an initially straight rectangularchannel was cut with a width of between 40 and 65 cm, adepth of between 4 and 5 cm, and an initial slope equal to Sv.For the first experiments the initial channel dimensionswere held constant (40 � 5 cm), but, later they were variedso as to produce an initial specific discharge (q) of about0.65 L s�1 dm�1, which helped to prevent vertical incision

F03011 EATON AND CHURCH: A GRADED STREAM RESPONSE RELATION

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during the runs (Table 1). Each experiment began with aninitial Q that was about 10% of the design Q to wet thechannel boundary. The channel and lower parts of the‘‘floodplain’’ were wetted up under these conditions, then

the flow was increased rapidly to the design Q, and thesediment feeder was turned on. Water flowed onto thecenter of the stream table at the upstream end through a1-m-long metal tray with a rectangular section (40 � 5 cm)oriented at 25� relative to the stream table centerline soas to generate an initial bend at the inlet [after Schummand Khan, 1972]. A sediment feed unit introduced thedesired sediment supply to the system at the inlet trayusing material with the same grain size distribution as thefloodplain sediment. Sediment supply was held constantfor the duration of each experiment.[9] Clearly, the grain size distribution of the sediment

supply to a river channel may be different from that of thebed material, but the simplest assumption that can be madein the laboratory is that all of the sediment supplied to areach is the product of nonselective erosion of alluvialbanks upstream, and thus the assumption that the distri-bution of the supply and the bed material are identical isconsistent with the idea that the stream is truly alluvial,provided the suspended sediment load is negligible.[10] All sediment leaving the stream table at the down-

stream end was captured in a sediment trap. A load cell anddata logger recorded the weight of sediment in the trapevery 2 min during the experiment. These readings wereused to generate average transport rates for 30 min intervals,which are sufficiently long to average out most of theshort-term variation in Qb.[11] Samples of the bed and bank surface material were

collected following each experiment using a flexible rubberplate covered with a layer of wet clay. The objective was tosample the surface layer, one grain thick, but there wasinvariably some plastic deformation of the clay paste aroundthe surface grains. To avoid the errors associated withtransforming these surface-based samples to bulk equiva-

Figure 1. Design and experimental grain size distribu-tions. The design grain size distribution represents the bedmaterial from a meandering gravel bed stream in Quebec,Canada. The experimental grain size distribution is that of acommercially available sand mixture.

Table 1. Governing Conditions and Equilibrium Response for All Experimentsa

Experiment Initial q, L s�1 dm�1

Governing Conditions Equilibrium Response

Qb, g min�1 Q, L s�1 Sv, %Time,b

hours S, % Lc W, cm A, cm2

Constant Sv1-1 0.85 128 3.4 1.09 8.5 0.97 1.122 99.4 1291-2 0.75 128 3.0 1.09 17.5 1.00 1.012 48.3 80.91-3 1.08 128 4.3 1.09 24 0.96 1.124 93.9 1571-4 0.66 214 4.3 1.09 1 1.00 1.108 91.8 1521-5a 0.68 214 3.4 1.09 4 1.04 1.029 61.7 1071-5bc n/a 186 3.4 1.09 4 1.01 1.050 64.8 1011-6 0.66 186 4.3 1.09 1 0.99 1.072 78.4 1251-7d 0.65 128 2.6 1.09 . . . 1.08 1.067 59.4 87.2

Constant L*2-1 0.61 434 3.4 1.28 1 1.18 1.051 67.1 95.52-2 0.65 370 3.4 1.22 1 1.15 1.074 70.2 1112-3 0.65 270 3.4 1.12 1 1.09 1.049 65.8 1072-4 0.65 320 3.4 1.15 1 1.10 1.052 66.8 1082-5e 0.62 540 3.4 1.30 . . . 1.17 1.115 72.2 1122-6e 0.62 537 3.4 1.28 . . . 1.12 1.083 79.6 1202-7e 0.62 520 3.4 1.28 . . . 1.09 1.136 82.0 1312-8e 0.65 560 3.4 1.20 . . . 1.11 1.083 73.8 1092-9e 0.65 560 3.4 1.20 . . . 1.15 1.082 72.4 110aAll nonequilibrium results are shown in italics.bTime to reach graded channel configuration.cThis experiment is the template for the constant L* experiments and is plotted with them on the relevant figures and included in

the relevant analysis.dThe flow strength for this experiment was below the competence limit (section 3.5).eThese experiments did not reach equilibrium between the sediment feed rate and the sediment output.


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lents, reference surface samples of the undisturbed bedmaterial were taken prior to the beginning of the experi-ment. These samples represent the distribution of thesubsurface material in the channel and of the sedimentsupply, against which the surface samples were compared.[12] Measurements of the water surface elevation and of

the bed topography were made using a point gauge mountedon a cart which, in turn, was mounted on rails parallel to thefloodplain surface. The x position (along the stream table)was recorded within a resolution of ±2 mm, the y position(across the stream table) within ±1 mm, and z within±0.2 mm. The water surface elevation at the channel thalwegwas measured at the apices of the meander bends, thecrossover points, and at locations approximately halfwayfrom one apex/crossover to the next crossover/apex. Thechannel slope (S) was estimated based on linear regressionlines fit to the recorded water surface elevation plottedagainst distance along the thalweg. A measurement uncer-tainty for S is estimated to be ±0.015 (e.g., S = 0.97 ± 0.015,where S is expressed in units of %), based on the precisionwith which the stream table can be positioned at a given Svand from the uncertainty associated with the regression ofwater surface height against distance down the thalweg.[13] Channel sinuosity was estimated by forming the ratio

of the thalweg length to the valley length (a straight lineparallel to the x axis) using the water surface survey data.Surveyed cross sections were located at each apex andcrossover to characterize the channel bed topography.Descriptions of the alluvial system presented herein arebased on a study reach comprising the middle 10 m of thestream table, away from any potential inlet and outleteffects. The shape of the channel cross section, the degreeof armoring, and the degree of channel sinuosity as mea-sured within this study reach are interpreted to represent thefluvial system adjustments to produce equilibrium.[14] Use of the term ‘‘equilibrium’’ in geomorphology

tends to be imprecise and occasionally confused, as pointedout by Thorn and Welford [1994], so it is necessary to defineit carefully. We adoptHoward’s [1988] terminology, whereinequilibrium refers to the behavior of a single system propertydescribed by some variable, not to the system as a whole.Before an equilibrium value for any of the system propertiescan be reached, steady state, with respect to the governingconditions, must be achieved so that we can be certain thegoverning conditions are acting over the length of the stream,not just near the inlet. In this study, steady state is said tooccur when the mass fluxes at the outlet are equal to thoseat the inlet, under appropriate averaging. Steady state dis-charge is established very quickly, after about 1 or 2 min.Steady state sediment transport takes longer to establish andcan feasibly be determined only after the experiment iscompleted and the sediment output data records are ana-lyzed. In practice, experiments were run until equilibriumchannel sinuosity (L*) was observed. We also define anequilibrium water surface slope, but this condition was notnecessarily established at the same time as equilibrium L*.[15] In the first set of experiments the valley slope was

held constant and the sediment supply and discharge werevaried over a range similar to that which might be expectedof the long-term mean sediment supply and discharge for ariver in the field as a result of changes in the climate and/orland use: accordingly, the discharge was varied by ±25%

about a mean value of 3.4 L s�1, and the sediment supplywas varied by ±50% about a mean of 170 g min�1. Table 1summarizes the governing conditions for these experimentsand presents the initial specific discharge (q), which is auseful index of the shear force at the beginning of theexperiments.[16] From the initial experiments a strong relation between

the thalweg slope and the sediment concentration emerged.A second set of experiments was conducted wherein thevalley slope and the sediment supply were covariedaccording to this relation so as to produce a selecteddegree of channel sinuosity. That is, the system constantswere manipulated according to the apparent relation ob-served in the first series of experiments to produce adesired single-thread channel pattern. Each experiment inthis second phase, then, represents a test of the initialrelation. The justification for this additional set of experi-ments is further discussed in the results section. The initialconditions are shown in Table 1.[17] Limitations of the model must be acknowledged.

Ultimately, all experiments with significant channel sinuos-ity ended up producing a multiple-thread channel. Theinitial response of the system was (usually) to maintain asingle channel with a coherent thalweg, which evolvedrapidly toward an equilibrium sinuosity, then maintainedthat state as the channel bars migrated downstream. Oncethe migrating thalweg encountered the downstream bar top,the flow stripped the bar surface, and a multiple-threadchannel formed as the back of the bar eroded. The long-terminstability of sinuous, single-thread channels in laboratoryexperiments is discussed by Paola [2002], who claimed thateither vegetation or cohesive sediment is necessary tomaintain a single-thread channel, primarily because theyintroduce a significant difference between the depositionaland subsequent erosional thresholds. This condition wouldprevent the sort of process described above. Simply put, ourlaboratory model was an incomplete physical model of asingle-thread prototype since it did not construct banks.This limitation is not specific to our study [cf. Friedkin,1945], and it is reasonable to assume that previous streamtable experiments suffered from the same limitation. Forexample, Ackers and Charlton [1970] reported that theirmeandering channels were inevitably at least twice as wideas their straight ones but were unable to provide a mean-ingful explanation of this observation.[18] Furthermore, owing to the dramatic change in parti-

cle behavior in fluid transport for sizes much smaller thanmedium sand, it is not possible to reproduce the entire rangeof prototype particle sizes and still maintain similarity of thegrain behavior and interaction. Particles in the model thatare smaller than medium sand become more difficult toentrain, in a relative sense, than their equivalents in theprototype since the thickness of the laminar sublayer is onthe order of the particle size in the model, while in theprototype the particle size is much greater than sublayerthickness [Yalin, 1971]. The grain size distribution for thisphysical model, as for most other models, is truncated. Mostof the prototype particle sizes that contribute to develop-ment of the upper bar and ultimately result in bankconstruction are absent from the model since only 3% ofthe bed material is <5.6 mm prototype equivalent (Figure 1),whereas, in the field, this fraction accounts for about 30% of


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the distribution. The bar tops encountered by the migratingthalweg in the model are lower than they would be in thefield prototype, the processes and size fractions responsiblefor floodplain formation being absent.[19] Provided Froude scaling is maintained, models with

the necessarily truncated grain size distributions appearto accurately reproduce the behavior of the equivalentprototype grain sizes [Ashmore, 1991], which correspondapproximately to the bed load component of the load. Thisundoubtedly represents an important aspect of channeldynamics in gravel bed streams. As long as the limb ofthe migrating thalweg along the eroding bank does notencounter the incompletely formed bar tops, the modelrepresents an approximation of the bed load component ofsingle-thread prototypes; that is, the model represents aprototype which is not significantly influenced by vegeta-tion or cohesive sediment deposition but rather is dominatedby the bed load transport dynamics.[20] While we acknowledge that equilibrium single-

thread configurations were eventually abandoned as thebars migrated downstream, we believe that since thalwegsinuosities did reach constant values after initial periods ofrapid adjustment, they are sufficient to study the systemdynamics in single-thread prototypes in a general way. Theeffect of not reproducing bank advance is analyzed andappears to make no discernable difference to the equilibriumchannel slope that is reached. The thalweg, then, is taken torepresent the path for the bulk of both water and sedimentflux and to represent the prototype channel sinuosity,though, of course, the presence of floodplain load, cohesivesediment, and/or vegetation in the field may introduce asystematic bias between model and prototype.

3. Results

3.1. Trajectories Toward Equilibrium

[21] The equilibrium channel planform was approachedby a number of paths. Experiment 1-1 provides a particu-

larly good example of the typical response of the system.The sediment supply concentration for this experiment was0.63 g L�1, which is near the lower end of the range forthese experiments. Figure 2 presents the thalweg location atvarious times during the experiment. During the first4 hours, channel sinuosity increased rapidly. Figure 3 showsthe channel morphology at low flow after 4 hours, lookingupstream. Between 4 and 8 hours, the average sinuositychanged only slightly as the system stabilized, but by12 hours into the experiment, the bar tops were exposedand eroded, leading to a multiple-thread pattern. For the sake

Figure 2. Thalweg locations during experiment 1-1. Flow is from right to left. The equilibrium single-thread channel occurred at 8 hours. The thalweg at 12 hours reflected the occurrence of chute cutoffsacross the top of the point bars as the channel develops a multiple-thread pattern.

Figure 3. Photograph of channel planform after 4 hoursduring experiment 1-1. This photograph shows the channelat low flow: at the design discharge the bars were entirelycovered by water. The approximate thalweg location andflow direction are shown.


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of simplicity, only the dominant thread at 12 hours is shownin Figure 2; its location clearly indicates that the bar topshave been exploited during the change in channel pattern.[22] For this experiment the system configuration after

8 hours is considered to represent the equilibrium state, aninterpretation that is supported by the sediment transportdata. The average sediment export from the system between7.5 and 8 hours was approximately 134 g min�1; quite closeto the sediment feed of 128 g min�1 and consistent with asteady state mass flux. The 30 min average transport rate forthe earlier stages of the experiment were much higher,though they declined steadily after 2.5 hours (Figure 4a).Since the recorded transport represents only the net exportfrom the system, it may not describe the transport rateupstream of the outlet, and the stable thalweg configurationafter about 4 hours may reflect a local transport rate in themiddle part of the stream tray that was similar to thesediment supply. Since steady state mass flux is generallyfirst established near the inlet and then progresses down-stream from there, the relatively high transport ratesbetween 4 and 7.5 hours likely indicate that the steady statecondition had not yet propagated all the way to the outlet.[23] The material transported out of the stream table

between 4 and 8 hours had a grain size distribution thatwas very similar to the material used to construct the bedand provide the sediment supply (Figure 4b). Since thesediment trap used a 160 mm mesh screen, both distributionsshown in Figure 4b were truncated at 180 mm. The maindifference between them is a slight underrepresentation ofthe particles at either end (less than about 500 mm andgreater than about 4 mm) in the transported material. Thefiner particles may be missing due to some inefficiency inthe trap since water occasionally overflowed the trap whenthe mesh became clogged with fine sediment. The largestparticles in the bed material were absent from the trans-

ported material and remained on the bed surface as an armorlayer. However, the D25, D50, and D90 of both distributionsremained nearly identical.[24] The rate at which a given experiment approaches

equilibrium was observed to depend strongly on the initialconditions. For initial channels in which the shear force waslow (as indexed by q in Table 1), channel sinuosityincreased very rapidly and stabilized after 1–4 hours(experiments 1-4, 1-5a, 1-6, and 2-1 to 2-9). Unfortunately,these experiments moved just as quickly on to multiple-thread configurations, making data collection problematic.When the initial shear force was higher, the system some-times initially degraded while remaining relatively straight,without widening its channel appreciably. Only after manyhours, when the channel slope was substantially reduced,did the channel become sinuous.[25] Experiment 1-3 demonstrates this two-phase system

response, wherein an initial degradational response gaveway to a lateral channel pattern response. Figure 5 presentsthe thalweg locations and water surface elevations for theexperiment. For the first 7 hours, the primary response wasvertical degradation, especially near the inlet, while thalwegsinuosity changed only slightly. Between 7 and 16 hours,the trend was reversed, with most of the adjustment occur-ring after 11 hours (thalweg profile not shown since it isnearly identical to the profile at 7 hours). After 11 hours, thechannel shifted laterally, increasing the sinuosity. Only afterabout 24 hours did the channel sinuosity reach equilibrium.[26] The grain size distribution of the transported material

is similar to that for experiment 1-1, wherein the primarydifferences between the bed material and transport grainsize distributions occur in the tails. The sediment transportrate for experiment 1-3 (see Figure 4a) exhibits a range ofvalues similar to that for experiment 1-1. The pattern ofchange is also similar, exhibiting a continuous decrease in

Figure 4. (a) Sediment transport (30 min averages) recorded at the stream table outlet for experiments1-1 and 1-3. For both experiments the sediment feed rate is 128 g min�1, shown with a horizontal dashedline. (b) Comparison of the bed material and the transported material for experiment 1-1. The sample ofthe transport between 4 and 8 hours is presented. Other samples of the transported material for differenttime periods and for the other experiments presented here have similar distributions.


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the sediment transport rate over time, but transport duringexperiment 1-3 reached steady state after about 20 hours,before the equilibrium channel pattern was attained. Simi-larly, an equilibrium thalweg slope was reached by about20 hours, well before the channel planform became stable.The configuration surveyed after 24 hours was used torepresent the equilibrium configuration.[27] The more rapid transitions toward equilibrium

configurations were more difficult to document sinceadjustment occurred more rapidly than the data could becollected. The channel response for these experimentsinvolved only lateral channel adjustment, wherein a sinuousthalweg developed very rapidly so the system quicklyreached steady state mass flux. Experiment 1-5 providesthe best example of the rapid transition. Since there wasvery little difference in any of the thalweg profiles, thechanges over time are best observed by examining thethalweg slope. Figure 6 presents the thalweg slope at 1 hourintervals. For the first 4 hours of the experiment (1-5a), Qwas held constant at 3.4 L s�1 and Qb at 214 g min�1. Theslope converged to a constant value after about 3 hours:equilibrium was assumed to occur at 4 hours. Almost all the

sinuosity adjustment occurred during the first hour, reachinga value of about 1.025; during the next 3 hours the sinuosityincreased by less than 0.5% to 1.029. The changes in channelslope between the surveys at 1 hour and 4 hours were justslightly larger, at 1.5%.[28] In the second half of the experiment (1-5b), Qb was

reduced to 186 g min�1. The response in channel sinuosityand slope was immediate. By 6 hours into the experiment(2 hours after the change in Qb), the channel sinuositystabilized at 1.05, and the channel slope reached what isbelieved to be the equilibrium value. The channel slope fellbelow that slope between 6 and 7 hours but returned to it by8 hours; channel sinuosity was constant throughout, so thesmall variation in slope can be attributed only to verticalchanges in the local bed height and/or to changes in localflow resistance. The second equilibrium configuration ischaracterized by the channel condition at 8 hours.

3.2. Constant Valley Slope Experiments

[29] In the first set of experiments, in which discharge andsediment supply were varied at constant valley gradient,seven of eight runs reached a steady state configuration,

Figure 5. Thalweg locations and elevations during experiment 1-3. The x and y coordinates aremeasured in meters, relative to the datum parallel to the floodplain surface, and the z coordinate ismeasured in millimeters.


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wherein the transport rate and grain size distribution at theoutlet were consistent with the sediment feed rate at theinlet. For experiment 1-7 the measured transport rate atthe outlet was consistent with the sediment feed rate,but the grain size distribution of the transported materialwas different than that of the bed material and from thetransported material for the other seven experiments. Wewill defer an extended discussion of this experiment tosection 3.4.[30] The equilibrium results are best understood within a

theoretical context that makes reference to a possiblephysical basis underlying the channel adjustments. Theleading hypothesis underlying the channel stability theorypresented by Eaton et al. [2004] is that flow resistanceshould be highest for the equilibrium channel configurationand that transitions toward equilibrium should be charac-terized by a consistently increasing fsys. It is known that anequivalent conjecture is that channel gradient is minimizedsubject to the capacity to pass the imposed sediment load,provided that f 0 and f 00 remain essentially constant. While itis not possible to demonstrate directly that the flow resist-ance for the equilibrium condition is at a maximum or thatthe equilibrium slopes are at a minimum, we can investigatethe trajectories toward equilibrium using the results ofexperiments 1-3 and 1-5a, for which the appropriate dataexist.[31] Since net degradation occurs during some of the

experiments, the valley slope, Sv, for the experiments isnot always consistent with the interpretation of Sv uponwhich the derivation of equation (2) is based [Eaton et al.,2004], which really refers to the water surface slope for achannel running straight down the valley. This slope can be

estimated for the experiments using a regression of thewater surface elevation data against distance down thestream table. If we define this quantity to be the systemslope (Ssys), then

Ssys ¼ Sv � DSdeg; ð3Þ

where DSdeg represents the effect of degradation on thewater surface relative to the valley surface (i.e., thedistribution of vertical incision). This is effectively apartitioning of Sv into a component related to the reach-scale flow resistance, Ssys, and a component associated withdegradation. The changes in the system associated withDSdeg may be incorporated into the adjustments of f 0 and/orf 00, while the adjustment of the channel slope relative toSsys is incorporated into f 000. This partitioning of Sv istheoretically important, but practically, the difference issmall, with an average difference of 2.8% between theequilibrium values of Sv and Ssys and a maximum differenceof 7.5% during the degradational phases of experiments 1-2and 1-3. Since fsys is linearly dependent on the value ofslope used, the differences associated with this correctionare of the same magnitude. So, when calculating fsys, theslope Ssys is used in equation (2).[32] The most easily interpreted data come from experi-

ment 1-5a. A plot of fsys against time shows a consistentincrease over time, with the maximum value occurring atequilibrium (Figure 7). Data from experiment 1-3 exhibitthe same general behaviour, but the pattern is complicatedby the positive correlation between W/d and sinuosity,discussed later in this section. It is argued that this positivecorrelation is the result of the absence of bank advance inthe model and should not be expected of the prototype.[33] To mitigate this channel width distortion, a modified

fsys was also calculated. It is assumed that while the channelwidth is unrealistically large, the mean velocity, calculatedby Q/A, is more-or-less representative since the additionalwidth over the bar top results in a very minor increase in thetotal cross-sectional area. Furthermore, it was estimatedfrom observations during the course of the experiments thatabout 90% of the discharge is transmitted through an areawith an effective width (Weff) of about 50 cm. By substi-tuting the approximation

R ¼ d ¼ 0:9Q

Weffuð4Þ

into equation (2) and including the appropriate slope term,we get

f sys* ¼ 8g 0:9Qð ÞSsysWeffu3

� �: ð5Þ

[34] Thus one can recalculate fsys by imposing a constanteffective width. A key assumption in the above argument isthat the mean velocity based on the average dimensions ismore-or-less correct, particularly given the dependence off *sys on u3 in equation (5). To verify the validity of thisapproach, the cross-sectional area associated with the spec-ified effective width was calculated for the experiment 1-3surveys at 16 hours and 24 hours, for which Weff was mostdifferent from the total W. On the basis of the assumption

Figure 6. Water surface slope along the thalweg duringexperiments 1-5a and 1-5b. For the governing conditions for1-5a and 1-5b, refer to Table 1.


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that 90% of the discharge was passed through this part of thechannel, the average velocity was calculated. In both casesthe difference in the calculated velocities was less than 3%.[35] This alternate definition ( f *sys) is also shown in

Figure 7. For experiment 1-5a, there is little apparentchange in the data, but for experiment 1-3 a divergenceoccurs between the two. The onset of this divergencecoincides with the onset of significant lateral activity, andf *sys based on the effective width definition increases con-sistently to a local maximum at equilibrium. The divergenceis the result of a strong bias in the estimate of the meandepth produced by the inclusion of extensive shallow areason the bar tops that do not contribute much to the overallcross-sectional area but which greatly inflate the wettedwidth. The consistency of the trends in Figure 7 providesome support for the principle, presented by Eaton et al.[2004], of flow resistance maximization as a means ofachieving channel stability.[36] There are insufficient bed survey data to analyze the

trajectories for the other experiments, particularly those thatreached equilibrium after about 1 hour. However, thecalculated values of f *sys using equation (5) are similar tothe equilibrium values in Figure 7 for all the other experi-ments, ranging between 0.13 and 0.19, except for experi-ment 1-2, for which f *sys is anomalously low (0.08). This isprobably due to the fact that this experiment was terminatedbefore significant lateral deformation occurred, and theanomalous f *sys value may indicate that there was furtherpotential for the system to deform and thereby increase thesystem scale flow resistance.[37] The channel slope, S, represents the reach-scale

component of the adjustment of fsys in equation (1). Inour experiments, channel slope is positively correlated with

the sediment supply and negatively correlated with thedischarge and is well described by a multiple regressionof the form

S ¼ 0:0368 Qbð Þ � 0:0244 Qð Þ þ 0:983;

SE ¼ 0:0084; R2 ¼ 0:93; ð6Þ

where S is expressed as a percent, Qb is reported in g s�1,and Q is reported in L s�1. SE is the standard error for theestimate S. Both coefficients and the intercept arestatistically significant at a = 0.02. The slope data areplotted first against Qb and then against Q in Figure 8, as arelines representing the predicted slopes from equation (6).Since the x axis is scaled to represent doubling of the lowestQ and Qb, it is clear that the range of slopes achieved isdriven more by changes in Qb than by changes in Q.However, it is equally clear that a proportional change in Qor Qb produces a similar change in S: for example, a 50%increase in Qb, relative to the lowest specified Qb of about2 g s�1 (Q held constant), results in an increase in thepredicted slope of about 0.0368%, while a 50% increase inQ (Qb constant) results in a 0.0374% decrease in S.[38] However, a slightly better statistical relation exists

between slope and sediment concentration, expressed as adimensionless quantity (g g�1). The equilibrium slope forthe first set of experiments is plotted against the sedimentconcentration in Figure 9a: the equilibrium slope is wellpredicted by a linear function of the form

S ¼ 136 Qb

�Q

� �þ 0:892; SE ¼ 0:0068; R2 ¼ 0:94: ð7Þ

Figure 7. System-scale flow resistance, indexed by fsys, plotted against time. Equilibrium was reachedafter 4 hours for experiment 1-5 and after 24 hours for experiment 1-3. Two definitions were used: fsys isbased on the raw data and f *sys on the assumption that 90% of the flow is carried within a cross sectionwith an effective width of 50 cm. The values of fsys closest to the y axis represent the flow resistance inthe initial rectangular channel cut straight down the stream table.


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[39] This expression has the advantage of being moreeasily interpreted since as Qb/Q drops to zero, the value of Sapproaches the threshold for transport. The constant in theregression therefore represents a critical slope for entrain-ment for our physical model (i.e., the slope of a self-formedthreshold channel). Standard errors for the coefficient andthe intercept of this regression are provided in Table 2.[40] The equilibrium channel sinuosity, which is another

way of expressing the reach-scale flow resistance, is also

plotted against the sediment supply concentration (Qb/Q), asshown in Figure 9b. The observed channel sinuosity iscorrelated with Qb/Q, but the trajectories by which equilib-rium was reached, involving changes in Ssys relative to Svdue to vertical incision (DSdeg), introduce scatter that is notevident in the plot of channel slope. The clearest examplesof this are experiments 1-2 and 1-3, which result from aprolonged period of vertical degradation at the beginning ofthe experiments, issuing in Ssys values of 1.01% and 1.04%,

Figure 8. Equilibrium slope for constant valley slope experiments plotted against (a) sediment feed rate(downward triangle = 3.0 L s�1, upward triangle = 3.4 L s�1, and circle = 4.3 L s�1) and (b) discharge(downward triangle = 214 g min�1, upward triangle = 186 g min�1, and circle = 128 g min�1). Themultiple regression model is shown for comparison, and the x axis is scaled to represent a range of 2 timesthe lowest Qb and Q for Figures 8a and 8b, respectively.

Figure 9. Constant valley slope equilibrium channel configuration: (a) thalweg water surface slope (%)and (b) sinuosity (m m�1) versus sediment concentration (g L�1): the line in Figure 9b has been fit by eyeto the data having nearly the same Ssys.


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respectively. Experiment 1-6, which had nearly the sameimposed sediment concentration as experiment 1-2 butreached equilibrium quickly by primarily lateral adjust-ments (owing to the lower q at the beginning of theexperiment), reached a statistically identical channel slopebut a much higher channel sinuosity. Experiment 1-4, whichexhibits a higher sinuosity than seems consistent with theremaining data, reached a value of Ssys of about 1.11% byaggrading the upper part of the stream table. The remainingexperiments, for which a line has been fit by eye onFigure 9b, have Ssys values between 1.05% and 1.07%.[41] Another often-cited possible adjustment that could

stabilize the stream channel is a change in the surfacearmoring [Talbot and Lapointe, 2002]. Surface sampleswere used to estimate the average armor ratio, based onboth the D50 and the D90 of the bed material and of thearmor that developed in the channel as the system adjusted.Neither definition of the armor ratio (Table 3) shows adiscernable correlation with the sediment concentration forthese experiments, probably because of the narrow range insediment concentration employed. However, both measuresof the armor ratio show a consistent (but statisticallyinsignificant at a < 0.125 for the D50-based ratio and a <0.154 for the D90 ratio) correlation with the time required toreach equilibrium (Figure 10a).[42] However, armoring clearly plays an important role in

the trajectory toward equilibrium. The surface grain sizedistributions for experiment 1-3 at various times during theexperiment are shown in Figure 10b, along with the averagegrain size distribution for the experiments with a duration of1 hour. The surface in the straight reach immediately beforethe onset of lateral activity was clearly coarser than thesurfaces typical for shorter experiments. It was this degreeof armoring that stopped the initial degradation and permit-ted the lateral activity to commence. The subsequent armorsurfaces developed during this experiment are finer butremain coarser than for the shorter experiments. Despitethis, the development of differential degrees of surfacearmor does not seem to have introduced a detectable biasinto the relation between the slope and the sedimentconcentration, though it may contribute to the varianceabout the relation. That there is very little scatter about thatrelation suggests that the range of armoring that developedduring these experiments was not sufficiently large tointroduce a detectable effect.[43] There is also the possibility that systematic changes

in the average channel dimensions contributed to theachievement of equilibrium [Eaton et al., 2004; Schumm,1993]. In that case the parameters describing the channelcross section, representing secondary adjustments, shouldbe correlated with the channel slope, which appears to bethe primary adjustment. The width-to-depth ratio (W/d) isthe common measure of channel shape. Figure 11a summa-rizes the W/d ratio of all the surveyed cross sections and for

just the bend apex sections. Both the average W/d and theapex W/d are negatively correlated with S, with low R2

values (0.13 and 0.38, respectively). Since S is a surrogatefor Qb/Q, according to Figure 9 (equation (7) is essentially arating relation between total stream power and sedimentload), this type of relation runs counter to the observedbehavior of alluvial systems in the field, where experiencesuggests that as sediment concentration increases, so toodoes the W/d ratio [Schumm, 1969]. There is a bettercorrelation between the observed W/d ratios and channelsinuosity (Figure 11b), which is the result of the limitationof the physical model, since it cannot recreate bank advanceso that the more sinuous the channel becomes, the wider itbecomes. However, the effective width (based on visualobservations), through which most of the sediment and fluidflux pass, does not seem to be similarly distorted and wasestimated to be about 50 cm for all of these experiments. Toresolve the relative contribution of channel slope andchannel shape adjustments to the achievement of an equi-librium condition, it is therefore necessary to attempt tocontrol channel sinuosity and the attendant change in theaverage channel shape. This is an objective of the constantsinuosity experiments discussed in section 3.3.

3.3. Constant Sinuosity Experiments

[44] As a test of the observation that slope increases withsediment concentration, additional experiments were con-ducted wherein the sediment supply and the valley slopewere covaried according to the relation in equation (7), whileQ was held constant, so as to produce the same degree ofchannel sinuosity (1.05). The channel pattern was allowed todevelop as in the experiments previously described. Thedesign sinuosity was kept deliberately low so as to minimizethe effect of not reproducing bank advance in the physicalmodel. These experiments also represent a deliberate testof the notion presented by Schumm and Khan [1972] thatthere is some inherent relation between valley slope andchannel sinuosity, independent of sediment concentration.They constitute a test of the sensitivity of the slope adjust-ment to the degree of sinuosity achieved (in comparison with

Table 2. Regressions of S Against Qb/Q

Data Set Coefficienta Intercepta n R2

Constant Sv 136 ± 30 0.892 ± 0.024 7 0.94Constant L* 133 ± 31 0.901 ± 0.024 5 0.96All data 139 ± 11 0.891 ± 0.014 11 0.98

aThe ranges correspond to a 95% confidence interval.

Table 3. Armor Ratios for All Experiments

Experiment Time,a hoursD50surf :D50b:m:

b D90surf :D90b:m:

b

Constant Sv1-1 8.5 1.57 1.291-2 17.5 1.76 1.441-3 24 1.73 1.411-4 1 1.56 1.361-5a 4 1.68 1.341-5b 8 1.73 1.361-6 1 1.53 1.36

Constant L*2-1 1 1.49 1.302-2 1 1.49 1.312-3 1 1.54 1.342-4 1 1.55 1.402-5 1 1.35 1.292-6 1 1.61 1.342-7 1 1.56 1.342-8 1 1.46 1.342-9 1 1.46 1.31

aFor armor development.bRatios are based on D50 and D90 for the surface and the bed material.


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the first set of experiments), provide an opportunity toinvestigate further the relation between sediment concentra-tion and channel cross section adjustment, and representan attempt to reveal the relation between the effective widthand sediment concentration.

[45] In four of the experiments, steady state sedimenttransport was reached, and in each case the achievedequilibrium channel sinuosity was not statistically differentfrom the design value, based on equation (7) (the 95%confidence interval for sinuosity is estimated to be ±0.030).

Figure 10. (a) Armor ratios for the constant Sv experiments, based on two separate definitions, plottedagainst the time available for armor development. Regression lines relating the armor ratio and theindependent variable are shown. None are significant at a = 0.05. (b) Grain size distributions forexperiment 1-3 prior to the development of alternate bar morphology (lower part of the stream table at11 hours) and during the development of appreciable sinuosity (surface at 16 and 24 hours). The averagesurface grain size distribution for the graded channel experiments with a duration of 1 hour is shown forcomparison.

Figure 11. W/d ratio is plotted against (a) channel slope and then against (b) sinuosity for the constantSv experiments. Data calculated from all cross sections and from only the apex cross sections are plottedseparately, and linear trends are fit to each data series.


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The results are shown in Figure 12. A regression similar tothat in equation (7) was fit to the data: the relations for thetwo different sets of experiments are not statistically differ-ent from each other, nor are they different from a regressionline fit to a data set combining both sets of experiments, asshown by the regression statistics in Table 2. In contrast tothe constant valley slope experiments, all of the variation inS for this set of experiments is produced by changing Qb,not Q: we have chosen to plot the data against sedimentconcentration simply to facilitate comparison with the mostpertinent previous results.[46] By producing channels with similar (and relatively

low) sinuosity, we have presumably removed at least partof the effect related to the inability of our physical modelto produce bank advance. The resultant channels remainedrelatively straight so that the equilibrium form did notinvolve much cutbank retreat and hence would haverequired little inner bank advance (in the prototype, wherebank advance is possible). Thus the results should moreclearly reveal the predicted correlation between the govern-ing conditions and the average channel dimensions. Indeed,the relation between W/d and Qb/Q now exhibits theexpected positive correlation (Figure 13b). It is likely thatthe wetted channel widths for these experiments moreclosely represent meaningful (effective) channel widths,and the scaling shown in Figure 13b is probably similarto that for streams in the field.

3.4. Nongraded Responses

[47] For a subset of the experiments in Table 1 the watersurface slope along the thalweg for both the constant Sv andthe constant L* experiments is well predicted by a linear

Figure 12. S is plotted against sediment concentration forboth sets of experiments. Symbols for equilibrium experi-ments (solid circles) are scaled to represent a measurementuncertainty of ±1 standard error for estimates of S. The solidtriangles represent Qb/Q based on the imposed governingconditions, while the open ones represent Qb/Q based on themeasured sediment output in the same experiments. Notethat transport records exist for only four of the fivenonequilibrium experiments. The anomalous experiment1-7 is indicated by an open circle.

Figure 13. (a) Armor ratio for the graded channels that reached equilibrium after 1 hour (circles) and forthe experiments above the maximum sediment feed threshold (open triangles, defined in section 3.4)plotted against sediment concentration using the measured transport at the outlet. (b) Average W/d ratiosfor the constant sinuosity graded channels and the above-threshold channels plotted against sedimentconcentration. Sinuosity for the above-threshold channels is presented in parentheses.


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function (Figure 12). These stream channels are at-grade inthe sense that their slopes are adjusted to the sediment supplyand fluid flow, which is approximately the definition of agraded river proposed by Mackin [1948]. The form of therelation (equation (7)) is identical to the governing relationproposed by Lane [1955] for constant grain size. For anothersubset, there is no evidence of a graded channel adjustment.When the sediment feed rate exceeds about 450 g min�1, thegraded relation overpredicts the slope at which the morphol-ogy in the study reach appears to become stable, and thesystem fails to reach a steady state sediment transportcondition during the run. At the other end of the spectrum,when the specified discharge is below 3.0 L s�1, the gradedrelation predicts a much lower slope than is reached by thestream channel. We refer to the constraints implied by theseresults as the ‘‘sediment feed threshold’’ and the ‘‘streampower threshold,’’ respectively. The upper value of the feedrate and the lower value of the discharge quoted aboverepresent limits to the range of conditions within whichgraded channel behavior occurs for our physical model. Theexistence of the thresholds is inferred on the basis of theobserved deviation from the graded relation.[48] For the experiments above the maximum sediment

feed threshold (experiments 2-5 through 2-9), Sv, rangedfrom 1.20 to 1.30%. In all cases the initial channel slope wasgreater than the equilibrium slope predicted by equation (7).Rather than remaining relatively straight, the channels over-adjusted, developing a larger degree of sinuosity and reach-ing a lower channel slope than expected. The slopes reachedwere lower than that for experiment 2-1 (Qb = 430 g min�1),which had an initial channel slope of 1.28%, implying thatthe breakdown of the graded relation is not related to theshear force acting on the boundary but somehow to thesediment supply rate. For these experiments the channelslopes measured in the study reach are not representative ofconfigurations at equilibrium with the sediment supply at theinlet since steady state transport was not achieved. Channelsinuosity in the study section did stabilize (probably tempo-rarily) at a constant value, as did channel slope, albeit atlower-than-expected values (Figure 12). Since sedimentsupply exceeded sediment output, net aggradation must haveoccurred by definition, but this was not evident in the studysection 5 m downstream of the inlet.[49] However, when the slopes for these channels are

plotted against Qbo/Q, where Qbo is based on the sedimentoutput rather than the sediment supply, the points collapseonto the graded relation. Although the scatter in the datausing the measured sediment output is greater, owing to thetemporal variations in sediment transport, the measuredsediment output likely represents a more accurate valuefor the sediment supply to the study reach. Fan-styleaggradation was limited to the upper 5 m of the streamtable during these relatively short duration experiments.[50] The degree of surface armoring that occurred during

these experiments is similar to that achieved during theconstant Sv experiments of the same duration (Table 3).There is no statistically significant correlation between thetwo measures of the degree of armoring and the sedimentconcentration in these experiments, and differential degreesof armoring is not an avenue by which these experimentsreached a stable channel morphology. Armor ratio is plottedagainst sediment concentration in Figure 13a, from which it

is clear that the surface texture of the experiments above thesediment feed threshold is similar to the surface texture forthe graded channels with comparable run time. Similarly,the average W/d ratios are plotted against sediment concen-tration (Figure 13b). While the W/d ratios are all higher thanthe trend established for the constant L* graded channels,this appears to be the likely effect of an increased degree ofchannel sinuosity, as implied by Figure 11b, since thedepartures from the trend line are approximately propor-tional to the measured channel sinuosity. The channel shapeadjustments for these experiments are thus consistent withthe graded channel experiments, though their W/d ratiosrepresent the combined effect of varying both sinuosity andsediment concentration.[51] Below the stream power threshold (experiment 1-7)

the equilibrium channel slope was substantially higher thanthat predicted by equation (7), and the lower part of thechannel did not become appreciably sinuous, nor did thetypical sequence of lateral bars develop.[52] While the nature of the change in alluvial behavior at

both thresholds is incompletely understood, since theexperiments were not designed to study them, the thresholdsseem to be related to the mobility of the bed material. InFigure 14 the ratio of the transported material and the bedmaterial is plotted for each 1/2 phi size class greater than theD50. Most of the size fractions near D50 were transported inproportion to their occurrence in the bed, while the coarsestsize fractions tended to be underrepresented, implying somedegree of size-selective transport. However, there was asmall but consistent difference in the degree of underrepre-sentation between the at-grade experiments (those channelsconsistent with the graded relation shown in Figure 12) andthe experiments above the maximum sediment feed thresh-old (Qb > 450 g min�1). The largest three size classes (2.8–4 mm, 4–5.6 mm, 5.6–8 mm) for the at-grade experimentswere underrepresented in the samples of the transportedmaterial (Figure 14). The degree of underrepresentationranges from about 20% for the 2.8–4 mm size class toessentially 100% for the 5.6–8 mm class, reflecting the factthat this sediment size was almost never present in thetransported load. For the experiments above the sedimentfeed threshold, only the largest size class was underrepre-sented in the load, and then by only about 50%, on average.The inferred stream power threshold is also distinctive onthe plot of bed material mobility (Figure 14) in that all thebed material fractions coarser than 2 mm are underrepre-sented in the transport sample.

4. Discussion

[53] On the basis of our experiments, the form of thegraded relation between S and Qb/Q does not depend on theaverage W/d ratio or the degree of surface armoring, nor isthe relation detectably influenced by the time-to-equilibriumor the sequence of vertical and lateral adjustments by whichequilibrium is achieved. Accordingly, S is interpreted to bea functional relation of Qb/Q, as illustrated in Figures 9and 12. That is, there is a cause-and-effect link between thechannel slope and the imposed sediment concentration. Wemust, however, acknowledge that the range of governingconditions and possible channel responses examinedremains rather narrow.


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[54] The analysis presented by Eaton et al. [2004] dem-onstrates that equilibrium alluvial states can be described indimensionless form in a four-dimensional space defined byrelative roughness (D/d), width/depth ratio (W/d), slope (S),and bank strength relative to the bed (f0). The gradedrelation between Qb/Q and S emerges once relative rough-ness is held constant. Since, in a physical model (and in itsprototype), D is held constant but d varies in proportion to afractional power of Q, then the graded relation holds only ifQ is nearly constant. The appropriate interpretation of Qb/Qin this context is as a dimensionless measure of Qb. Theadvantage of expressions using dimensionless quantities isthat they preserve the one-to-one correspondence betweenmodel and prototype.[55] Strictly speaking, our data most clearly demonstrate

a relation between S and Qb since, in the first set ofexperiments, the range of imposed discharge was limitedsuch that the effect of relative roughness was apparentlyundetectable and, in the second set, discharge was heldconstant. We propose that the link between S and Qb/Qworks through the maximization of the system-scale flowresistance, mainly by changing the reach-scale flow resist-ance. We consider that S reflects the reach-scale flowresistance component from equation (1) ( f 000), whichappears to be the most significant component of the systemscale flow resistance, fsys.[56] The other possible adjustments ( f 0 and f 00) were less

important in our experiments. Despite the reversal in thesense of the correlation between W/d (an approximate indexof f 00) and S between the constant Sv and constant L*experiments, there is no statistical difference between thegraded relations for the two different sets of experiments(Table 2): channels that were wider and shallower, onaverage, did not stabilize at higher slopes than did narrower,deeper channels with the same imposed sediment concen-tration. Perhaps the best example of this comes from acomparison of experiments 1-2 and 1-6. Both experimentshave similar sediment concentrations (�0.71 g L�1) and

reach statistically identical channel slopes (0.01 m m�1).However, the stable W/d ratio for experiment 1-6 was justover 70% larger than the W/d ratio for experiment 1-2. Ifnearly doubling the average W/d ratio does not produce anydetectable effect on the equilibrium slope reached, thenobviously W/d has a very small second-order effect. Thisoutcome suggests that the limitations of the physical model,which does not reproduce bank advance, are not nearly sosevere as they may at first appear. Indeed, visual observa-tions (and the supporting analysis in the discussion of f *sys)suggest that the effective width for these two experimentswere likely of comparable magnitude.[57] Similarly, the magnitude of the surface armoring

(an index of f 0) does not seem to be an importantadjustable variable in these experiments. No systematictrends occurred in armor ratios. Thus f 0 does not adjustsufficiently during our experiments to have an appreciableeffect on the equilibrium channel configuration.[58] While neither f 0 nor f 00 seem to be important com-

ponents of the adjustments documented in these experi-ments, this is almost certainly not the case in the field.There, resistant channel banks may limit the degree towhich channel sinuosity (and hence f 000) can adjust, whilethe geomorphic history can create highly armored andstructured surfaces that make the sort of adjustments docu-mented in these experiments impossible because the armorcannot be breached. Our experiments do demonstrate,however, that the order of system adjustments is significantin determining the final channel geometry. In two cases inwhich early degradation and bed armoring occurred, theequilibrium configuration remained less sinuous than inother cases because part of the slope adjustment was takenup by the early degradation.[59] Rational regime models predict that for a specified Q

and Qb, there is a range of possible slopes and stablechannel configurations that can accommodate the specifiedfluxes, ranging from channels with steep slopes wheresporadic transport occurs over a relatively large width to

Figure 14. Bed mobility for at-grade channels, for channels below the lower threshold (Q < 3.0 L s�1),and for channels above the upper threshold (Qb > 450 g L�1). The proportion of the transported grain sizein each size fraction (Li) is normalized by the proportion of the bed material grain size in the same sizefraction (Pi), after Wilcock and Southard [1989]. A ratio near 1 indicates that the size fraction istransported in proportion to its occurrence in the bed material.


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channels with lower slopes having intense sediment trans-port over a much more restricted part of the channel. Ifalluvial states are, in reality, randomly distributed in thespace of stable solutions, there is no reason why the behaviorshown in Figure 12 should necessarily arise: the channelsshould be able to stabilize over a range of slopes by adjustingtheir width and surface texture rather than just their slope.Certainly, given the experimental design, the channels didhave access to these higher slope solutions. This implies thatthere should have been substantial scatter about the S versusQb/Q relation and that the residuals should be correlated withdifferences in W/d and/or the armor ratio. There were nosignificant residuals (see Figure 12), despite significantvariations, particularly in the W/d ratio. This implies thatnot all parts of the theoretical stability curve based onrational regime theory are equally attractive, absent otherconstraints. It also implies that there is no obvious correla-tion between the average W/d ratio and the width of thechannel active in transporting sediment.[60] Our data demonstrate that graded channels consist-

ently bypass higher slopes that, according to rational regimetheory, are capable of transmitting the sediment supplygiven the available discharge [see Eaton et al., 2004]. Werefer to this tendency for the systems to move towardminimum slope as slope minimizing behavior (SMB). Sincethere is very little scatter in the S versus Qb/Q relation andsince these slopes can be reached by various combinationsof vertical and lateral adjustments, we suppose that thechannels inhabit the same part of their respective stabilitycurves. This behavior is relevant to the stability of solutionspredicted by a rational regime formulation and is at the rootof a physical explanation for extremal hypotheses. Extremalhypotheses simply choose points having the same relativeposition on their stability curves (i.e., homologous points);thus the action of extremal hypotheses such as the minimumslope hypothesis [Chang, 1979], or any of the equivalentextremal hypotheses [Davies and Sutherland, 1983], areimplicit in the observed SMB. Whether a true minimumslope, or a true maximum flow resistance for the system, isactually reached cannot be decided. Indeed, it is difficult toimagine an experimental test that would permit such a claimto accepted. However, the monotonic approach of ourmodel systems toward equilibrium from slopes higher thanthe equilibrium slope represents suggestive evidence forsuch behavior.[61] However, some of the experiments did not behave in

accordance with the graded relation described by theregressions in Table 2, and these deserve some attention.There appear to be thresholds related to a minimum streampower and a maximum sediment feed rate beyond which thesystem response changes fundamentally.[62] Below the stream power threshold the channel failed

to erode its banks in the lower reaches, did not develop anyappreciable bar forms there, and could not transport materialmuch coarser than the D90 of the bed material and feed(experiment 1-7). This threshold is independent of thesediment feed rate since other experiments with the samesupply did adjust their sinuosity and develop typical alluvialbed features throughout their length. The only differencebetween experiment 1-7 and the at-grade experiments wasits lower initial stream power (QSv) and lower ultimatestream power (QS ).

[63] While the transport rates at the outlet during exper-iment 1-7 were similar to the supply rates at the inlet, thesystem was not at equilibrium with the sediment feed. It islikely that the coarsest part of the feed grain size distributionwas deposited in the channel upstream of the study reach.The difference between the imposed feed rate and themeasured transport at the outlet is, by definition, small(we are considering the upper 10% of the distribution)and could not be detected given the fluctuations in themeasurements at the outlet. The system was most likelyaggrading in response to a competence limit that becameevident once the imposed total stream power fell below acritical level. This aggradation seems to have been limitedto that part of the stream table above the study reach. Eventhough this experiment was run for 16 hours, an assumed10% disparity between feed rate and the measured sedimentoutput results in a net deposition of only 12 kg which, whendistributed evenly throughout the upper 5 m of the streamtray, results in an average change in bed elevation of only3 mm. Such a small change in bed elevation could not bedetected without high-resolution topographic survey data.However, if the experiment had continued for a sufficientlength of time, we presumably would have observed anaggradational adjustment to an equilibrium state.[64] In run 1-7 the study reach had a bed that was

significantly coarser than the sediment that could be trans-ported by the available flow and, if the competence-induceddeposition was limited to the upper 5 m of the stream tableas we suspect, the study reach was likely supplied withsediment having a grain size distribution that was funda-mentally different from the material forming the channelboundary. Since there was a significant difference betweenthe grain size distribution of the bed material in the studyreach and the grain size distribution that the flow wascompetent to transport, experiment 1-7 does not representa fully alluvial system. It represents an experimental limitbeyond which the combination of bed material, slope, anddischarge is no longer reasonable.[65] The behavior above the sediment feed threshold is

less easily explained. Upstream of the study reach, it seemsthat aggradation was again occurring but this time inresponse to an excess sediment supply (a transport capacitylimitation). As a result of this aggradation, the sedimentsupply rate at the upstream boundary of the study reach waslikely similar to the sediment transport rate recorded at theoutlet, though we have no proof of this (beyond thesatisfactory collapse that its assumption produces). Giventhe short duration of the experiment, even the largestdifference between the sediment feed rate and the sedimenttransport rate amounts to an average elevation increase ofonly about 2.5 mm over the upper 5 m of the stream table.Let us assume that the stream tray upstream of the studyreach was behaving like a fan, storing between 30 and 40%of the sediment supplied to it. The study reach, then, can becompared to an alluvial channel downstream of a fan, whichwas supplied with sediment at a rate similar to that recordedat the outlet. This scenario would explain why the channelconfiguration stabilized in the study reach at a slope thatwas consistent with the graded relation in Figure 12.[66] The other system adjustments (bed armor, W/d ratio),

when analyzed on the assumption that the measured sedi-ment output actually represents the supply to the study


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reach, are entirely consistent with the other graded channels.If we accept that the sediment supply at the upstream endof the study reach was probably closer to the sedimenttransport measured at the outlet, the response of thesechannels ceases to be a mystery since they simply representadditional examples of the graded-style adjustment.[67] We think that it is not reasonable to interpret the

difference in behavior above the threshold as simply theresult of insufficient run time, after which the graded slopecould have been reached via aggradation: the initial channelslope in all cases was well above the predicted equilibriumslope, but rather than stabilizing at that slope, it wasbypassed. If, in fact, the graded relation holds and thechannel would stabilize at that slope after aggrading, whywould it not simply stabilize there in the first place? Wespeculate that violation of the sediment feed threshold maybe associated with a change in the ultimately stable channelmorphology to a braided stream pattern. Since such adramatic change in channel pattern would be expected tofundamentally change the graded relation (primarily byincreasing the within-channel form resistance, f 00), it is alsolikely that the equilibrium slope would be higher than thatpredicted by equation (7). The sediment feed threshold may,in fact, be associated with the transition between meander-ing and braided stream patterns.[68] However, two issues remain unclear: (1) why did the

sediment output from the upper part of the stream table dropbelow the maximum transport rate for the upper limit of thegraded experiments (2-4); and (2) what is the significance ofthe differences in bed mobility between the experimentsabove the sediment feed threshold and the experimentsbelow the threshold (Figure 14). These are likely relatedissues and may imply that full mobility of the entire grainsize distribution, in this case, destabilizes the channelmorphology (because of the noncohesive nature of thebanks): if all the grains on the bed can be moved, thennoncohesive banks certainly can be eroded. The end resultis that a change in channel pattern must result in order toachieve stability. We have insufficient data from the sectionof the stream table upstream of the study reach to elaboratethese problems, so we can do no more than point out theseperplexing aspects of channel adjustment as a topic forfurther study.[69] Physical models are powerful tools for the study of

complex natural phenomena. Their power derives from thefact that they are natural systems, not numerically generatedapproximations of them: the water and sediment are real,and the laws of physics remain unchanged. There are somescaling issues that do affect the applicability of the models.These issues have no bearing on the truth of such physicalmodels since they are natural systems in their own right, butthey do impact the relation between a model and its fieldprototype. Essentially, the physical model discussed in thispaper represents the bed load component of a prototype(with grain sizes larger than 5.6 mm), having all suspendedload deposits stripped away. Herein lies an important scale-mediated distortion since those particles that typically formthe upper parts of the fluvial stratigraphy are absent fromthe model. The absence of this ‘‘floodplain load,’’ as wellas the absence of vegetation and cohesive sediment, has theconsequence that the physical model cannot reproducethe processes of bank advance that act in the field prototype.

As a result, the physical model can persist as a single-threadchannel only until meander progression encounters theincompletely formed bar tops, where a fully formed flood-plain deposit would exist in the field analogue. At this pointthe sedimentological differences between the model and theintended prototype overwhelm the process similarities, andthe single-thread pattern gives way to a multiple-threadform where its prototype would not. Interpretation of theresults from physical models must be tempered with thisunderstanding of the relation between model and prototypesince it guides our expectations of when similarities anddifferences between the two may exist.

5. Conclusions

[70] Stream table experiments using a generic physicalmodel of a steep gravel bed stream show that stable,equilibrium channel morphology has a water surface slopealong the thalweg that is determined by the imposedsediment supply and discharge. This behavior is mostsuccinctly captured in a relation between the dimensionlessquantities of slope and sediment concentration, as proposedby Mackin [1948] and Lane [1957]. It does not appear to beinfluenced by the imposed valley slope, the degree ofthalweg sinuosity, or the degree of surface coarsening.[71] The analysis of these experiments is based on the

assumption that the thalweg path in the model representssomething close to the prototype channel path and that thethalweg slope represents a quantity that is adjusted toproduce an equilibrium channel configuration. The equilib-rium water surface slope (along the thalweg) is consideredto represent a regime (or graded) slope (Sr). It also repre-sents an index of the reach-scale flow resistance proposedby Eaton et al. [2004] and appears to be the dominantcomponent of the system-scale adjustment of flow resist-ance, which they argue is maximized as alluvial systemsapproach equilibrium.[72] Our experiments demonstrate that Sr may be

approached either by vertical degradation or by lateralerosion leading to increase of sinuosity. The system mayinitially respond by degrading vertically with little changein the channel sinuosity or channel pattern. However, once Sapproaches Sr by this degradational mechanism, sedimentbegins to deposit locally in the channel, and lateral defor-mation is initiated. However, under the appropriate initialconditions, the adjustment may be entirely lateral. Regard-less of the response trajectory, channel slope is the primaryadjustable variable, consistent with the concept of thegraded stream advanced by Mackin [1948]. Indeed, it isthe only systematic adjustment observed in these experi-ments. Other responses, such as increased flow resistanceper unit length of channel, may occur when lateral channelmigration is inhibited by resistant banks and/or the devel-opment of highly armored beds, but they are not evident inour experiments.[73] The graded response is bound by thresholds beyond

which the graded relation breaks down. Between thesethresholds, S can be predicted by the sediment concentra-tion, Qb/Q. The implication is that this analysis is restrictedto alluvial channels with similar characteristics and witherodible channel banks. Sand bed rivers, where the bedmaterial is fully mobile for much of the year and where


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bank stability completely controls the system dynamics,may exhibit quite different behavior. Similarly, in (steep)channels where locked arrangements of individual particlesdominate, the dynamics are very different, though theprinciple of system-scale flow resistance maximization,proposed by Eaton et al. [2004], may still hold.[74] For channels that are similar to the generic prototype

studied here, slope is adjusted, relative to Sv, mainly bychanging the channel sinuosity, and a range of gradedchannels can exist for a given Sv in accordance with afourfold range of sediment concentration. That is, there isno unique pairing of sediment concentration and valleyslope, contra the suggestion of Schumm and Khan [1972],and the degree of channel sinuosity depends on the sedimentconcentration for a given Sv. Within this fourfold range ofsediment concentration our experimental results suggest thatchanges in Qb/Q will produce channel adjustments consist-ent with the regime relation summarized in Figure 12,primarily by changing the channel sinuosity, unless thegeomorphic history of the system and/or particularlyinerodible boundaries render this response inaccessible.

Notation

d flow depth, m.D grain diameter (subscripts indicate corresponding

percentile), mm.f flow resistance.

L* channel sinuosity, m/m.Li proportion of the ith size fraction in the transported

load.Pi proportion of the ith size fraction in the bed material.Q discharge, L s�1.q specific discharge, L s�1 dm�1.

Qb bed material sediment supply, g min�1.R hydraulic radius.S slope, %.t time, hours.u mean velocity, m s�1.W wetted width, m.t shear stress, N m�2.

[75] Acknowledgments. We thank Tim Davies for giving us access tothe stream table at Lincoln University, without which this work would nothave been done, as well as for the many stimulating discussions of theresults and their implications. Warwick Hill provided invaluable assistancein making things go in the lab; we were very fortunate to have his talents atour disposal. Rob Millar and Marwan Hassan both played key roles inidentifying the need for such experiments, for which we thank them,as well. We also thankfully acknowledge funding provided by theNatural Sciences and Engineering Research Council of Canada throughM. Church’s Discovery Grant and B. Eaton’s postgraduate scholarship.Comments by the journal’s reviewers, T. Hoey and J. Pizzuto, havecontributed greatly to the clarification of our thinking and of the paper,for which we are grateful.

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��M. Church and B. C. Eaton, Department of Geography, University of

British Columbia, 1984 West Mall, Vancouver, BC, Canada V6T 1Z2.([email protected]; [email protected])


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A graded stream response relation for bed load–dominated streams

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