Morphologically based model of bed load transport capacity ... · PDF fileElsewhere, fluvial sediment transport dominates and channel morphology reflects the orderly transfer of water

Morphologically based model of bed load transport capacity in a

headwater stream

Erich R. Mueller and John PitlickDepartment of Geography, University of Colorado, Boulder, Colorado, USA

Received 17 December 2003; revised 14 March 2005; accepted 31 March 2005; published 25 June 2005.

[1] Field measurements of channel and bed material properties are used to develop anetwork-based model of bed load transport capacity in Halfmoon Creek, a high-gradientstream in the Rocky Mountains of Colorado. The formerly glaciated watershedcontains self-formed reaches that are generally disconnected from hillslopes and majorsediment perturbations, strung together by a sequence of bedrock and cascade channels.Measurements of channel geometry, slope, and bed sediment texture (surface andsubstrate) were taken in 27 reaches with gradients ranging from 0.6 to 6% to examineinteractions among key variables affecting bed load transport. The analysis shows thatthe grain size distribution of the substrate is relatively uniform throughout the basin;however, the distinction between surface and substrate texture varies systematically withchannel gradient and shear stress. Reaches with high shear stress are characterized bycoarse surface layers, with a high threshold for transport, limiting the mobility of thesediment supplied. In reaches with lower gradient and shear stress the difference betweensurface and substrate texture is less, increasing both the frequency of transport and themobility of the bed load. While it might appear that the steeper reaches of Halfmoon Creekhave a high capacity for carrying fine-medium gravel, the relative mobility of these sizesis affected by the presence of very coarse sediment on the bed surface. As a result,modeled estimates of bed load transport rate indicate that average annual bed loadsediment yields increase downstream nearly linearly with discharge, suggesting a simplescaling of bed load transport capacity with discharge and drainage area.

Citation: Mueller, E. R., and J. Pitlick (2005), Morphologically based model of bed load transport capacity in a headwater stream,

J. Geophys. Res., 110, F02016, doi:10.1029/2003JF000117.

1. Introduction

[2] Headwater streams are frequently described as com-plex systems that evolve over time in response to a mix ofgeomorphic processes [Knighton, 1998; Wohl, 2000]. Hill-slope transport of sediment from landslides and debris flowsmay dominate sediment supply in some watersheds, locallyforcing changes in channel gradient and grain size [Grant etal., 1990; Montgomery and Buffington, 1997; Brummer andMontgomery, 2003]. Elsewhere, fluvial sediment transportdominates and channel morphology reflects the orderlytransfer of water and sediment according to establishedprinciples [Emmett, 1975; Pizzuto, 1992; Whiting et al.,1999]. These and many other observations have led togeneralized conceptual models and stream classificationsystems in which the headwater reaches of rivers arecharacterized as being primarily erosional, whereas inter-mediate or distal reaches are considered transitional todepositional [Schumm, 1977; Church, 1992; Montgomeryand Buffington, 1997].[3] Modeling erosion and sediment transport across these

different domains is problematic, and perhaps only loosely

tied to position within the drainage basin. In headwaterstreams, the most important conceptual issue concerns thedistinction between alluvial channels in which the sedimenttransport capacity balances the sediment supply and non-alluvial channels in which the capacity exceeds the supply.There is a tendency to lump mountain streams into the lattercategory for two reasons: (1) Channel gradients are gener-ally high (>2%), and therefore the shear stress or streampower available to move sediment appears to be quite high[Montgomery and Buffington, 1997]; and (2) the sedimentseen on the bed (cobbles and gravels) is typically muchcoarser than the sediment collected during short samplingintervals (bed load), and is potentially coarser than thematerial sampled from the substrate (bulk bed material).Observations that the bed load is finer than the substrate[Church et al., 1991; Lisle, 1995; Wathen et al., 1995;Whiting et al., 1999; Habersack and Laronne, 2001; Ryanand Emmett, 2002; Church and Hassan, 2002] imply thatthe load is derived from sources other than the bed, and thattransport rates are driven more by sediment supply than byflow conditions [Dietrich et al., 1989; Buffington andMontgomery, 1999b]. However, a number of other fieldstudies and data sets indicate that transport in headwaterstreams is closely coupled to flow properties and bedmaterial characteristics [Milhous, 1973; Parker et al.,

JOURNAL OF GEOPHYSICAL RESEARCH, VOL. 110, F02016, doi:10.1029/2003JF000117, 2005

Copyright 2005 by the American Geophysical Union.0148-0227/05/2003JF000117$09.00

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1982; Andrews, 1984, 1994; Lisle, 1995; Powell et al.,2001; Torizzo and Pitlick, 2004]. It appears, therefore, thatthe range of conditions in headwater basins is quite large,and it is not evident that bed load transport data, bythemselves, convey the information needed to analyzewatershed-scale sediment supply.[4] In this paper we propose an alternative approach for

assessing sediment transport capacity in headwater streams.We start with the premise that the volume or mass of bedload carried in alluvial rivers is governed by a handful ofinterrelated factors, including the water discharge, thechannel geometry (width and depth), the slope, and thegrain sizes available for transport [Gilbert, 1914; Mackin,1948; Rubey, 1952; Henderson, 1966; Ferguson, 1986;Carson and Griffiths, 1987; Howard et al., 1994]. Noneof the factors above can be excluded in assessing thesediment transport capacity of a stream system. It isparticularly important to include grain size as a variablebecause of its importance in scaling the transport rate and indefining thresholds for motion. The distinction betweensurface and substrate (or subsurface) grain sizes is of furtherimportance in gravel bed rivers because local coarsening ofthe bed surface limits the mobility of finer sizes that wouldotherwise move quite easily [Parker, 1990]. Modificationsof bed surface textures provide a mechanism for channels toadjust to changes in transport capacity and sediment supply[Dietrich et al., 1989; Lisle et al., 1993; Buffington andMontgomery, 1999a, 1999b; Wilcock, 2001; Lisle andChurch, 2002], and, in the absence of major sedimentinputs, may result in downstream fining within sedimentarylinks [Rice, 1999] or reaches of rapidly declining shearstress [Ferguson et al., 1996]. As the variables needed toestimate sediment transport rates (width, depth, slope, grainsize) are readily measured through a channel network, wepropose that their joint variation should reflect changes insediment transport capacity. We focus here on bed loadtransport because of its importance in determining channelform in headwater streams. Our approach and analysis isconsistent with transport-based theories for hydraulic ge-ometry [Parker, 1979; Pizzuto, 1992], and takes advantageof the fact that channel properties are not nearly as variablein space or time as is the process of bed load transport.[5] Linkages between channel morphology, grain size and

sediment transport capacity are investigated here usingmeasurements from Halfmoon Creek, a headwater streamlocated in the Sawatch Range of central Colorado. Half-moon Creek contains a mix of self-formed and forced(nonalluvial) reaches. This setting is typical of mountainstreams where channel and sediment properties changerapidly over short distances. Because of the glacial historyof the basin, alluvial stream reaches are typically discon-nected from hillslopes, with infrequent debris fansdistributed relatively evenly throughout the basin. Ourmeasurements emphasize geomorphic conditions in self-formed reaches with average gradients of less than about10%. These reaches are identified in the field by thepresence of alluvial bed and bank materials where fluvialprocesses are the primary controls on morphology. The self-formed reaches are discontinuous, but they occur through-out the watershed, including the lowest-order tributarysegments. The forced reaches are associated with glacialsteps or moraines. These reaches are very steep (>10%), and

act effectively as conduits that pass sand- and gravel-sizedsediment to adjoining alluvial reaches. Estimates of themass of bed load transported through the alluvial reachesare computed from a bed load transport relation and adimensionless flow frequency curve. These estimates arethen used to assess changes in the mass balance of sedimentthrough the stream network. We hypothesize that if thestream network is approximately in steady state, and ifthe sediment supply is roughly uniform throughout thebasin, then the sediment input to the channel system shouldscale linearly with drainage area, and each alluvial channelsegment will have adjusted its hydraulic geometry and slopeto carry proportionally that much more sediment than thesegment immediately above.

2. Study Area

[6] Halfmoon Creek is located in the Sawatch Mountainsof central Colorado (Figure 1). The drainage basin ranges inelevation from 4400 to 2990 m, and includes the twohighest peaks in Colorado (Mt. Elbert and Mt. Massive).The entire basin has been glaciated. Talus slopes andbedrock exposures of Precambrian gneiss and schist occurthroughout the headwaters (>3500 m elevation); the lowerbasin includes open, U-shaped valleys bordered by reces-sional and lateral moraines, which are now forested. Thesetting is typical of alpine basins in Colorado, whereQuaternary glaciation has had a primary influence on basinmorphology. Annual peak flows in Halfmoon Creek arederived from snowmelt runoff, resulting in a regular inter-annual hydrograph with a mean annual flow of 0.85 m3/sand a mean annual flood of 8.0 m3/s, as recorded at theUSGS gauge near the downstream end of the study area(period of record 1947–2002). Summer thunderstorms arecommon but they are of short duration and typically do notproduce much runoff. At the furthest downstream site(Figure 1) the creek drains an area of 64 km2 and the flowis unregulated.[7] Stream sediment is derived from localized bank

erosion and mass wasting of hillslopes, glacial morainesand debris fans. Overall, the basin is relatively stable againsterosion, and the prior effects of glaciation limit the couplingbetween hillslope processes and channel morphology in anumber of places [Church, 2002]. Nevertheless, there istrace evidence of small hillslope failures resulting fromlocalized storms in the basin, and evidence of activesediment transport along most of the main channel. Bedload sediment yields have not been measured, but depositsleft in a retention basin at the outlet of the basin suggest theannual sediment yield is on the order of 10 m3/yr.[8] Measurements were taken in 27 alluvial reaches along

the main channel and in three tributary valleys, hereindifferentiated as South, Middle, and North Halfmoon creeks(Figure 1) (see also Figure 5 in section 4.1). The study sitesare located at closely spaced intervals (every 0.5–1.5 km)along the main stem and tributary segments. Site character-istics vary considerably (Table 1). In general, the studyreaches can be discriminated between those occurring inhanging valleys, which are separated from the main valleyby cascade-type channels at glacial steps, and those occur-ring in the main valley itself (Figure 2). Although many ofthe reaches are bordered by forest, woody debris is not

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prominent in the channel. Twelve of the study sites arelocated in tributary valleys. These sites are located in planebed reaches that are steep, but otherwise, self-formed(Figure 2a). Fifteen study sites lie along the main channelof Halfmoon Creek. These sites are located in alluvial planebed reaches or pool riffle reaches where the gradients aresomewhat less (Figure 2b). Downstream of the confluencewith the south fork, the slope of the main channel decreasesconsistently as the creek flows from steep, forested terraininto a wide, sparsely forested alluvial valley through whichthe stream meanders. The valley then narrows where a smalltributary, Elbert Creek, has produced an alluvial fan and asteepening of the valley profile. Downstream of the fan, thevalley again opens up, allowing the stream to meanderacross a floodplain (Figure 2c). The east end of the valleyis bordered more closely by lateral moraines which restrictthe channel and steepen the gradient through the twodownstream sites.

3. Methods

[9] Basin-wide trends in sediment transport capacity weredetermined by calculating the annual bed load sedimentyield at each of the 27 study sites. Consistent measurementsof channel and bed material properties were made at eachsite in order to estimate parameters for calculating transportrates (details of the calculations are discussed below). Inaddition to the field data, a key element of the analysisincludes the development of an empirical relation foradjusting the threshold for bed load transport in steepstreams.

[10] The study sites were located in relatively straightreaches to limit complications arising from changes inchannel curvature and topography. A minimum of threecross sections was surveyed with a level at each site in orderto determine the bank-full channel geometry. Measurementsof both the bed and water surface profile were made over anapproximate length of ten times the channel width in orderto determine the reach average slope. As statistical tests ofthe regression coefficients indicated there was no significantdifference in bed and water surface slopes (p < 0.05), thelatter were used to approximate the energy slope. Bank-fullchannel dimensions were determined by a distinct break inbank slope. Measurements of bank-full depth at the leftbank and right bank were generally similar (within 5 cm),with the lower value being used in calculations to define abank-full hydraulic radius and width (Table 1).[11] Surface sediment samples were taken at each site

using a variation of the Wolman [1954] method, with aminimum of 100 particles sampled at random. Particle sizeswere measured at 1/2-phi intervals using a metal template(gravelometer). Bulk samples of the subsurface sedimentwere also obtained at all but one site in order to determinethe ratio of surface to subsurface grain size fractions, degreeof bed armoring, sorting, and trends in downstream fining.Bulk samples were large enough to ensure that the largestgrain was no more than 5% of the total weight and usuallymuch less [Church et al., 1987]. The coarse fraction(>32 mm) of the subsurface samples was sieved in the fieldand the fine fraction (<32 mm) was sieved in the lab, againat 1/2-phi intervals. Graphical plots of grain size versuscumulative frequency were used to determine individual

Figure 1. Location map of study sites within the Halfmoon Creek basin generated from 10-m DEMs.Numbers represent the 27 study reaches. Note distinct lateral moraines downstream from site 7.


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percentiles (D84, D50, D16) of the grain size distribution ofeach sample (Table 1).[12] Average annual bed load sediment yields were esti-

mated for each site by calculating the instantaneous bedload transport rate for each of several increments of dis-charge (discharge classes), then summing those values toget the annual load. As the procedure for estimating annualloads is somewhat involved, it is perhaps useful at this pointto lay out the individual steps: (1) Bed load transport rateswere calculated using an empirical transport function basedon excess shear stress; (2) the transport function wasmodified to include an adjustment for the threshold shearstress, which accounts for differences in flow properties inhigh-gradient streams; (3) the frequencies of sediment-transporting events were determined from a dimensionlessflow duration curve, developed from gauging stationrecords and an empirical relation for discharge as a functionof drainage area; and last, (4) bed load transport rates werecalculated for each increment of discharge, then weightedby the frequency of that discharge, p(Q), and summed to getthe average annual bed load sediment yield. Comments onthe assumptions and uncertainties of the analysis are givenin the discussion which follows.[13] The estimates of bed load transport are based on the

equation of Parker [1979], which can be written in a generalform as:

q* ¼ 11:2t*� tc*ð Þ4:5

t*3ð1Þ

where q* is a dimensionless transport parameter, t* is thedimensionless shear stress, and tc* is the critical dimen-sionless shear stress. We chose to leave tc* as free

parameter rather set it to a specific value (0.03) as wasdone originally [Parker, 1979]. The dimensionless shearstress is

t* ¼ trs � rð ÞgD50

; ð2Þ

where t is the boundary shear stress, rs and r are thedensities of sediment and water, respectively, g is thegravitational acceleration, and D50 is the median grain sizeof the bed sediment. The transport parameter can beexpanded as follows:

q* ¼ qb

s� 1ð ÞgD350

� �0:5 ; ð3Þ

where qb is the volumetric transport rate per unit width, ands is the specific gravity of sediment, assumed to be 2.65.Combining (1) and (3) yields an equation for the totalvolumetric bed load transport capacity,

Qs ¼ 11:2t*� tc*ð Þ4:5

t*3s� 1ð Þg D3

50

� �0:5B; ð4Þ

where B is the channel width. Written in this form (4)illustrates that downstream variations in bed load transportcapacity are driven by the interaction of four main variables:t*, tc*, D50, and B. The effect of these interactions isevident in most streams and rivers, where, for example,downstream reductions in grain size are generally accom-

Table 1. Study Site Characteristicsa

Site Dist., km A, km2 Qb, m3/s S D50, m D50s, m R, m B, m t, N/m2 t* Qs, t/yr

1 14.6 64.0 9.5 0.0180 0.072 0.036 0.49 10.4 87 0.075 1722 14.2 63.7 7.3 0.0140 0.061 0.023 0.46 10.8 63 0.064 1713 13.3 62.5 6.4 0.0084 0.050 0.018 0.47 9.7 39 0.048 1714 12.3 61.7 6.8 0.0085 0.052 0.019 0.58 7.8 48 0.057 1445 11.7 61.2 5.5 0.0086 0.055 0.020 0.49 8.0 41 0.046 1606 10.6 59.7 6.5 0.0120 0.062 0.019 0.48 9.3 56 0.056 1707 9.3 54.2 6.5 0.0200 0.078 0.033 0.52 7.6 102 0.081 1318 8.5 53.0 5.0 0.0064 0.030 0.019 0.37 10.7 23 0.048 1079 7.5 51.0 7.1 0.0097 0.038 0.019 0.42 10.4 40 0.065 10810 6.7 49.1 8.4 0.0130 0.053 0.018 0.45 11.8 57 0.067 15911 6.4 48.6 7.8 0.0160 0.066 0.023 0.44 10.9 69 0.065 17412 6.0 47.3 8.3 0.0220 0.068 0.018 0.50 8.3 108 0.097 10913 5.6 46.1 8.0 0.0230 0.072 0.027 0.48 8.4 108 0.093 11714 5.3 30.3 7.0 0.0370 0.086 N.A. 0.47 6.9 170 0.122 96.515 4.6 28.4 5.7 0.0440 0.078 0.030 0.41 7.0 177 0.140 79.8N1 2.5 13.4 2.6 0.0450 0.081 0.036 0.35 4.5 154 0.118 54.3N2 2.1 12.1 3.5 0.0220 0.066 0.019 0.44 3.8 95 0.089 44.2N3 1.2 9.8 1.7 0.0230 0.066 0.026 0.31 3.8 70 0.066 47.1N4 0.8 4.6 1.5 0.0460 0.055 0.034 0.26 3.6 117 0.133 24.2M1 3.3 11.1 2.1 0.0400 0.060 0.020 0.32 3.9 125 0.129 30.9M2 1.8 9.3 2.5 0.0360 0.066 0.020 0.28 5.5 99 0.092 52.7M3 0.9 6.7 1.6 0.0320 0.065 0.017 0.29 4.3 91 0.087 42.6M4 0.0 3.2 1.0 0.0470 0.060 0.020 0.29 2.5 134 0.138 18.9S1 4.2 13.8 2.3 0.0410 0.069 0.019 0.35 4.2 141 0.127 41.1S2 2.9 10.5 1.4 0.0400 0.071 0.020 0.35 2.5 137 0.119 25.2S3 1.9 7.0 1.0 0.0200 0.051 0.035 0.28 3.0 55 0.067 27.9S4 1.4 3.1 0.7 0.0670 0.074 0.026 0.21 2.7 138 0.115 26.9

aDist. is the distance downstream, A indicates the drainage area, Qb is the bank-full discharge estimated from the continuity equation, S is the slope, D50

is the median grain size of the surface bed material, D50s is the median grain size of the subsurface bed material, R is the bank-full hydraulic radius, B is thebank-full width, t is the bank-full shear stress, t* is the bank-full dimensionless shear stress, and Qs is the modeled annual sediment yield.


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panied by increases in B and/or increases in the excess shearstress, t* � tc*. If the stream system is approximately insteady state, the variables on the right-hand side of (4)should vary in ways that minimize differences in transportcapacity, @Qs/@x, otherwise there would be clear evidenceof aggradation or degradation. It is also important to notethat (4) includes two nonlinearities, thus bed load transportcomputations are very sensitive to uncertainties in theindividual variables. Among the variables in (4) we weremost concerned with uncertainty in the estimates of t* andtc*. Methods for estimating these parameters are discussedbelow; however, it is worth noting that, in the case of gravelor cobble bed rivers, there is a strong theoretical andempirical basis for suggesting that the difference betweent* and tc* should be relatively small over the range ofchannel-forming flows [Parker, 1979; Andrews, 1984;Pizzuto, 1992; Parker and Toro-Escobar, 2002; Pitlickand Cress, 2002].[14] Site-specific estimates of t* and tc* were found

using two empirical relations, one developed from measure-ments of channel properties in Halfmoon Creek, the otherdeveloped from an analysis of bed load transport measure-ments taken elsewhere. The estimates of t* are derivedfrom a basin-wide relation for the bank-full dimensionlessshear stress, tb*, obtained from field measurements of the

bank-full hydraulic radius, reach average slope and bedsurface D50, with one site excluded (S4, Table 1); thatrelation is

tb* ¼ 2:14S þ 0:034; ð5Þ

with r2 = 0.91 and p � 0.001.[15] Estimates of the threshold Shields stress, tc*, were

made for each site using a relation developed specificallyfor high-gradient gravel and cobble bed stream channels[Mueller et al., 2005]. This relation is based on ananalysis of flow and bed load transport measurementstaken in 45 streams and rivers in the western USA andCanada. The analysis focused on variations in the thresh-old for bed load transport, which arise from changes inflow properties and bed structure as channel gradient andrelative roughness increase [Bathurst et al., 1987; Jarrett,1984; Wiberg and Smith, 1991; Church et al., 1998;Buffington and Montgomery, 2001]. For each of thesedata sets, Mueller et al. [2005] plotted the relationbetween dimensionless bed load transport rate and dimen-sionless shear stress, and, following the procedure ofParker et al. [1982], estimated the reference dimension-less shear stress, tr*, associated with a small, nonzero

Figure 2. Characteristic examples of study reaches. (a) Downstream view of site N3, north fork ofHalfmoon Creek. Site is located in a broad glacial valley at an elevation of 3473 m; S = 0.023 and B = 3.8m. (b) Upstream view of site 12, main channel of Halfmoon Creek. Site is located in a forested area at anelevation of 3105 m; S = 0.022 and B = 8.3 m. (c) Upstream view of site 3, main channel of HalfmoonCreek. Mt. Elbert in the background. Site is located in an open meadow at an elevation of 3031 m; S =0.0084 and B = 9.7 m.


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transport rate. The resulting estimates of tr* were thencorrelated to reach average slope, giving

tr* ¼ 2:18S þ 0:021; ð6Þ

with r2 = 0.70 and p � 0.001. For the purposes of thepresent study, we assume tr* � tc*. It is worth noting thateven though equations (5) and (6) were developedindependently of each other, they form essentially parallellines, suggesting the difference between tb* and tc* is nearlyconstant over a wide range of channel slopes.[16] The next step involved formulation of two interme-

diate relations linking the estimates of t* derived from fieldmeasurements to the probability density function for flowfrequency, which was derived from the gauge record. Thefirst intermediate relation is used to predict the bank-fulldischarge at individual sites on the basis of drainage area.Initial estimates of Qb were made for each site usingmeasured values of bank-full width and hydraulic radius,R, and calculated values of bank-full velocity, U (i.e., Qb =UBR). The estimates of velocity were based on a flowresistance relation developed specifically for HalfmoonCreek from current-meter measurements of discharge, takenover a range of flows (0.2–5.0 m3/s) at 11 sites [Mueller,2002]. Four flows were measured at six of the main channelsites and two flows were measured at one main channel siteand four of the tributary sites, yielding data for a range ofdepths and slopes. Values of the Darcy-Weisbach frictionfactor, f = 8gRS/U2, were computed for each dischargemeasurement, and then correlated to relative roughness, R/D84, where D84 is the particle size for which 84% of thesediment is finer. A least squares fit of these data yields

1ffiffiffif

p ¼ 2:26 logR

D84

� �þ 1:00 ð7Þ

with r2 = 0.70 and p � 0.001, excluding outliers from onetributary site where very large boulders produced unusuallyhigh values of f. Over the range of observed flows the flowresistance relation for Halfmoon Creek is very similar to aprevious relation formulated by Limerinos [1970]. The

calculated values of bank-full discharge were then pooledwith data from other gauged gravel bed streams in Colorado[Andrews, 1984; Torizzo and Pitlick, 2004] to form aregional relation between Qb and drainage area, A. Figure 3shows a plot of the combined data, fit with the relation,

Qb ¼ 0:35A0:725; ð8Þ

with r2 = 0.88 and p � 0.001; A is the drainage area, insquare kilometers, and Qb is the bank-full dischargeexpected for a given drainage area. We use the fittedrelation for Qb rather than the cross section–based estimatesbecause this reduces inconsistencies arising from anom-alously high or low values of the calculated bank-full flowvelocity.[17] The second intermediate relation is used to specify

the frequency of individual discharges, Qi, relative to thebank-full discharge, Qb. To do this we developed a dimen-sionless flow duration curve based on the record of dailyflows at the Halfmoon Creek gauge (record length = 55 yr).Daily discharge values were grouped into 40 intervals, andthe interval values, Qi, were normalized by the bank-full Qat the gauge (7 m3/s). Figure 4 shows the upper end of thenormalized flow frequency distribution, fit with the follow-ing exponential relation,

p Qi=Qbð Þ ¼ 44:7e�4:85 Qi=Qbð Þ: ð9Þ

This equation was then used to determine the frequency ofindividual discharges at each site, assuming that the shapeof the flow duration curve does not vary with drainage area.This is essentially the same as assuming that the same unithydrograph applies to each site. This is a relatively goodassumption for small basins in snowmelt-dominated regionsbecause the timing and intensity of runoff tend to followvery consistent patterns, which are reflected by regionalsimilarities in the statistical distributions of flood peaks anddaily streamflows [Pitlick, 1994; Gupta and Dawdy, 1995;Torizzo and Pitlick, 2004].

Figure 3. Regional relation between measured bank-fulldischarge, Qb, and drainage area used to predict down-stream changes in discharge for Halfmoon Creek.

Figure 4. Relation between flow frequency and scaleddischarge, Qi/Qb, for flows greater than 1/2 bank-full Q, asdetermined from the USGS gauge record.


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[18] Next, we formulated an equation for estimatingindividual values of t* at each site for each increment ofdischarge, Qi. This equation has the form of an at-a-stationhydraulic geometry relation, with the local Shields stressincreasing as a power function of discharge,

t* ¼ tb*

Qi

Qb

� �0:5

; ð10Þ

where tb* is found with (5) and Qb is found with (8). Theexponent of 0.5 in (10) is the average of 12 values obtainedby Torizzo and Pitlick [2004] in their analysis of the relationbetween flow and bed load transport in gravel bed streamsin Colorado; the value of 0.5 is also consistent with theexponent for depth in a typical at-a-station hydraulicgeometry relation [Leopold and Maddock, 1953]. Equation(10) has the desirable property that t* ! tb* as Qi ! Qb.[19] Instantaneous bed load transport rates, Qs, were

calculated for each site and each increment of dischargeusing (4), (6), and (10), and the measured values of D50 andB. The calculated values of Qs, were then weighted by theannual frequency of that discharge, p(Qi/Qb), and summedto estimate the annual bed load sediment yield.[20] Given the number of assumptions and steps used in

the analysis, it is worth commenting on some of thepotential sources of error and uncertainty. Probably the

biggest source of error is associated with the estimatesof tb* and tc*; small differences in these variables canproduce large differences in transport rate, and the choiceof individual values is, therefore, very important. Thechoice of sediment transport equation also has some influ-ence on the results; we used the Parker [1979] relationbecause it gave reasonable estimates for the total sedimentyield at the outlet of the basin (170 t/yr, which comparesfavorably with our visual estimate of the annual accumula-tion of sediment in a settling pond located above a diversionstructure near the downstream study site). In contrast, theMeyer-Peter and Muller [1948] equation gave sedimentyields exceeding 1000 t/yr, which would be unreasonablefor a stream of this size in this area. The potential errorintroduced by using a standardized flow-frequency relationfor the entire basin is unknown, and, of course, there issome error associated with the field data; however, theuncertainty in these measurements and assumptions isprobably small in comparison to the errors noted above thatcarry over into the transport calculations.

4. Results

4.1. Channel Slope

[21] The longitudinal profile of Halfmoon Creek is con-cave overall (Figure 5a), but marked by several distinctinflections associated primarily with glacial steps at tribu-tary junctions [MacGregor et al., 2000], but also debris fansand moraines. Headwater segments of the creek includeshort alluvial reaches separated by cascades, which appearas steps in the profile. The most abrupt change in gradientoccurs in the area from 4 to 6 km downstream where thethree tributary forks come together. The creek then flowsthrough an open alluvial valley and the profile is quitesmooth. At a distance of 9.5 km there is another smallinflection in the profile where the creek is incising throughglacial deposits and a debris fan produced by Elbert Creek.Below that point the valley again opens up, and for another4 km, the slope decreases systematically downstream. In thelower 1 km of the study area, the creek is more confined bylateral moraines near the downstream extent of glaciation,resulting in another increase in slope.[22] In general, average gradients of tributary channel

reaches are higher than main channel reaches (Table 1). Aplot of average gradients of the study sites (Figure 5b)shows that slopes of the tributary reaches generally exceed0.025, whereas slopes of the main channel reaches aretypically less than 0.025. As noted above, systematicdecreases in slope are interrupted in several locations byglacial steps associated with tributary junctions and con-finement between moraines and debris fans.

4.2. Bed Material

[23] Median grain size of the surface bed material, D50, inHalfmoon Creek varies from 0.086 to 0.030 m (Table 1 andFigure 6). Bed surface samples from the tributary channelreaches show little systematic variation or downstreamreduction in grain size (Figure 6), whereas samples fromthe main channel show systematic fining of the bed surfacein two separate reaches, from 5.3–8.5 km and 9.3–13.3 km(Figure 6). Downstream fining in these reaches mimics therapid changes in slope noted in Figure 5, which reflect local

Figure 5. (a) Longitudinal profile of Halfmoon Creekbasin. (b) Downstream changes in reach-averaged slope.


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flattening of the valley profile. Rice and Church [1998]describe such reaches as sedimentary links where changesin grain size are punctuated by sediment inputs fromtributaries or hillslopes. For purposes of comparison wefit the surface grain size data within these reaches withseparate exponential functions:

D ¼ Do exp� x�xoð Þ=Lð Þ ð11Þ

where Do is the grain size at the start of a fining sequence,x � xo is the distance downstream from the start of a finingsequence, xo, and L is the decay length scale in kilometers.This is equivalent to the original Sternberg [1875] equationwhere L equals 1/a, and a is the fining parameter (km�1).Table 2 compares regression estimates of L and a derivedfor surface and subsurface fining relations within links, aswell as the basin as a whole. The results show that thesurface D50 within the two main stem sedimentary linksfines very rapidly over short distances (L = 3 and 9, anda = 0.33 and 0.11, respectively, p < 0.01, Table 2). Theestimates of a for these two links are comparable tosedimentary links in active gravel bed rivers wherelocalized sediment supply and effects of past glaciationsometimes force abrupt transitions in valley morphologyand slope [Ferguson and Ashworth, 1991; Knighton, 1998;Morris and Williams, 1999; Rice, 1999]. In contrast to theindividual links, the basin-wide trend in surface D50 inHalfmoon Creek is weak (L = 83; a = 0.012), and therelation is not statistically significant (p = 0.23).[24] Downstream fining of the subsurface sediment is

subtle compared to that of the surface sediment discussedabove. The median grain size of the subsurface sediment,D50s, ranges from 0.017 to 0.036 m (Table 1). A plot of thedownstream trends in D50s (Figure 6) displays minorinflections at some of the same points noted above.However, fining below these points is not strong; valuesof L and a for the subsurface D50 within individual linksare not statistically significant, nor is the coefficient for thebasin-wide relation (p > 0.2, Table 2).

[25] Figure 7 shows the full grain size distribution of allsediment samples, differentiated by reach average gradient.We differentiated the samples to examine connectionsbetween channel gradient and size separation of the indi-vidual fractions. The split was made at S = 2.5% because itis approximately at this point where we start to see majordifferences in hydrodynamic processes related to slope andrelative submergence (R/D84); this split also results inroughly equal sample sizes. Close inspection of the datain Figure 7 suggests that the differences in sedimentsamples from high- and low-gradient reaches mostly resultfrom variations in the coarser size fractions. To evaluate thesignificance of these variations we performed a series of ttests comparing differences among the individual grain sizeparameters (D84, D50, and D16). The results are presented inFigure 8, which shows a series of box plots comparing sizeparameters of surface and subsurface samples, differentiatedby gradient. Figure 8a indicates that the average surface D84

of high- and low-gradient reaches differs by 46 mm (38%),which is statistically significant (t = 3.56, p = 0.0015).Figure 8b shows that there is also a difference in the averagesurface D50 of high- and low-gradient reaches, but theseparation is much less (11 mm or 18%) and the statisticalsignificance of the difference is lower (t = 2.39, p = 0.025).Figure 8c shows that there is no significant difference in theaverage D16 of the surface sediment (t = 0.79, p = 0.44).Figures 8d–8f compare parameters of the subsurface sizedistributions. Figure 8d indicates that the average subsur-face D84 of high- and low-gradient reaches differs by 17 mm(23%), which is marginally significant (t = 2.06, p = 0.05).Figures 8e and 8f comparing values of the subsurface D50

and D16 show that there is no significant difference (t =0.52, p = 0.61; t = 1.52, p = 0.14) between samples taken inhigh- and low-gradient reaches.

4.3. Channel Morphology and Hydraulics

[26] The average bank-full width, B, of Halfmoon Creekincreases from about 3 m in headwater reaches to about 11 min lower reaches (Table 1 and Figure 9a). Over the samedistance, the average bank-full hydraulic radius, R,increases from about 0.3 to 0.5 m (Table 1, Figure 9b).The data follow more irregular trends in the middle of thestudy area where the slope and grain size are changingrapidly. From 5.3 to 8.5 km downstream, for example, thereis a decrease in bank-full R and a slight increase in bank-full

Figure 6. Downstream changes in the median grain size ofthe surface and subsurface bed material. Solid points aresurface D50, and open points are subsurface D50.

Table 2. Comparison of Surface and Substrate Fining Relations

for Halfmoon Creeka

Link n L a r2 F p

Surface5.3–8.5 km 7 3.0 0.33 0.97 187 �0.0019.3–13.3 km 5 9.0 0.11 0.94 44 0.007Overall 27 83 0.012 0.06 1.4 0.23

Substrate5.3–8.5 km 6 13 0.076 0.25 1.3 0.329.3–13.3 km 5 5.0 0.201 0.43 2.3 0.23Overall 26 71 0.014 0.04 1.1 0.31

aThe length scale, L (km), and fining coefficient, a (km�1), weredetermined from linear regression of ln-transformed values of grain sizeversus distance; n is the number of values, r2 is the coefficient ofdetermination, F is the value of the F distribution, and p is the significanceprobability.


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B coinciding with fining through the upper sedimentary linkand a transition in planform from straight plane bed reachesto more sinuous pool riffle reaches. This trend reverses atsite 7 (9.3 km downstream) where the Elbert Creek fanconfines the valley and supplies coarser bed material. Thetrends of decreasing bank-full R and increasing bank-full Bare crudely repeated through the lower sedimentary link(from 9.3 to 13.3 km) again without much change in waterdischarge.[27] Downstream trends in the bank-full shear stress, tb,

and the bank-full Shield’s stress, tb*, are plotted inFigure 10. Coding the individual values by reach averageslope, S, shows that gradient has a disproportionate influ-ence on the local estimates of both tb and tb*. Shearstress and channel gradient are clearly correlated (as t isproportional to RS), however, as is apparent in the fielddata, variations in shear stress (or Shields stress) are drivenmuch more by changes in slope than changes in depth (orgrain size). This also suggests that these slope-driven

changes in shear stress interact with mobile fractions ofthe bed sediment to give the textural trends discussedearlier. These interactions are examined in Figure 11,which plots four separate relations between the bank-fullshear stress and sediment size, measured in terms of twoparameters (D84 and D50) of the surface and subsurfacesediment. Among the four parameters, the surface D84 ismost highly correlated to tb (Table 3); the subsurface D84

and the surface D50 are likewise correlated with tb,although the trends are not quite as strong as for thesurface D84 (Table 3). Note that the relations for thesubsurface D84 (Figure 11a) and surface D50 (Figure 11b)nearly coincide, suggesting that, for a given tb, the coarserfractions of the bed substrate actively exchange with aver-age-sized particles on the bed surface. The last relationshown at the bottom of Figure 11 indicates that there isessentially no correlation between the subsurface D50 andtb. The significance probability of this relation ( p = 0.11;Table 3) suggests that the slope of the line is not

Figure 7. Full grain size distribution of all sediment samples. (a) Surface and (b) subsurface samplestaken in high-gradient reaches (S > 2.5%). (c) Surface and (d) subsurface samples taken in low-gradientreaches (S < 2.5%). The dashed shaded lines represent the average grain size distributions of the high-gradient reaches.


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significantly different from zero, meaning, the variation insubsurface D50 is independent of the bank-full shear stress.[28] The relations shown in Figure 11 suggest that the

separation between intermediate and coarse size fractionsbecomes increasing large as the shear stress increases.These differences are particularly evident in comparingthe substrate D50 with the size parameters of the bedsurface: in reaches with relatively low shear stress (t <50 N/m2), the substrate D50 differs from the surface D50 bya factor of about 2, whereas in reaches with high shearstress (t > 150 N/m2), the difference exceeds a factor of 3.Thus, while it might appear that the steeper reaches ofHalfmoon Creek should have a high capacity for carryingfine to medium gravel, the relative mobility of these sizesis affected by the presence of very coarse gravels andcobbles on the bed surface. Although, the proportion ofsurface layer grains moving during high flows is probablynever very large, it was evident during field sampling thatmany of the coarsest grains on the bed were loose andlikely to move at high flow. The substrate exists presum-ably because there is exchange between the sediment inmotion and the sediment stored in the bed. Exchangebetween the bed load and the substrate cannot occur unlessthere is some movement of the surface layer grains, and assuggested below, this appears to occur at all of the studysites at least a few days per year.

4.4. Sediment Transport Model

[29] Modeled annual sediment yields are plotted in rela-tion to bank-full discharge in Figure 12. Although themodeled values exhibit appreciable scatter, the overall trendis consistent with the expectation that sediment transportcapacity should increase monotonically downstream. Thesedata were fit with a power law relation:

Qs ¼ 23:2Q0:96b ; ð12Þ

Figure 8. Box plots illustrating the median, quartiles, minimum, maximum, and outliers in (a) surfaceD84, (b) surface D50, (c) surface D16, (d) subsurface D84, (e) subsurface D50, and (f) subsurface D16

between high- and low-gradient reaches.

Figure 9. Downstream trends in (a) width and (b) hydraulicradius.


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with r2 = 0.92 and p � 0.001, where Qs is in metric tonsper year. The observed trend in Qs reflects potentiallycomplex interactions among the variables B, D, and t* �tc* in (4). In high-gradient reaches with coarse bedmaterial, instantaneous unit transport rates (load per unitwidth per unit time) tend to be very high; however, since arelatively high discharge is required to exceed theestimated threshold Shield’s stress, tc*, bed load transportin these reaches occurs perhaps only a few days per year,and as a result, the average annual load is not particularlyhigh. In low-gradient reaches with finer bed material, tc*is exceeded more often and sediment is moved perhapsmany days per year. The frequency of sediment transport-ing flows ranges from just under 3 days/yr at the steepestsite, to nearly 18 days/yr at the lowest-gradient site. Thevariations in flow frequency are primarily related to thenonlinear relation between Qi/Qb and t* because at steepersites where tc* is higher, a slightly higher unit discharge isnecessary to initiate motion. The effects of low shear stressand small grain size in low-gradient reaches are thus morethan made up for by an increase in the frequency of bedload transport, resulting in proportionally higher annualsediment yields.[30] Interestingly, the estimated annual sediment yield

increases nearly linearly with discharge (Figure 12),suggesting a simple scaling of sediment yield as dischargeincreases downstream. Despite variations in grain size,width, flow frequency, and channel gradient, the modeled

sediment yield suggests that the alluvial reaches conveysediment in some near-equilibrium steady state where thetotal bed load increases in proportion to the waterdischarge.

5. Discussion

[31] The principal goals of this research were to investi-gate interactions among channel and bed material propertieswithin a headwater stream, and then use those properties tomodel basin-wide trends in annual bed load sediment yield.The results indicate that in spite of significant variations inchannel morphology and sediment texture, the modeledsediment yields define a clear trend, with bed load transportcapacity increasing nearly linearly with discharge. A num-ber of empirical relations and assumptions went into thedevelopment of the model, thus the question arises: Is itpossible to validate the results in any way? Since we did notmeasure bed load at any of the sites, we cannot assess theaccuracy of the estimates directly. However, a reasonablecomparison can be made using downstream relations forsuspended sediment, for which there are several examples.Emmett [1975] measured suspended sediment at 20 stationsin the upper Salmon River drainage basin in Idaho. He usedthe at-a-station relations for suspended sediment to formu-

Figure 10. Downstream trends in (a) bank-full shearstress, t, and (b) bank-full dimensionless shear stress, t*,with data stratified by slope.

Figure 11. Relation between bank-full shear stress, t, and(a) surface and subsurface D84 and (b) surface andsubsurface D50, where solid and open symbols representhigh- and low-gradient reaches, respectively.


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late a downstream relation for load as a function of bank-full discharge,

Qss ¼ 0:89Q0:75b ð13Þ

where Qss is the suspended sediment load, in tons per day,and Qb is the bank-full discharge, in cubic feet per second.Leopold and Maddock [1953] developed flow andsuspended sediment relations for 20 rivers in the westernUSA, and suggested that the exponent in (13) should have avalue of �0.80. Since sediment load equals water dischargetimes concentration, an exponent of less than 1.0 indicatesthat sediment concentration is decreasing downstream,perhaps because of floodplain deposition. The exponent of0.96 in our relation implies that concentration is roughlyconstant downstream, as expected in a system where waterand sediment are supplied in equal proportions. Similarly,Rickenmann [2001] found that bed load volumes werenearly linearly proportional to effective runoff volumesfrom bed load sampling in 19 mountain streams, 13 ofwhich had gradients above 5%.[32] The downstream trend in bed load sediment yield

of Halfmoon Creek reflects a trade-off between the fre-quency and intensity bed load transport, controlled in largepart by adjustments in bed surface texture. We found thatwhile there was not much variation in the texture of thesubstrate sediment, there was a clear correlation betweenthe bed surface texture and the local shear stress. Itappears, therefore, that the effects of grain hiding andexposure, caused by changes in surface texture, may eitherreduce or enhance the mobility of the bed load as it movesthrough high- and low-gradient reaches. In the same waythat Ferguson and Ashworth [1991] describe longitudinalchanges on the Allt Dubhaig in Scotland, we interpret thedownstream trends in surface texture (fining or coarsen-ing) as a mobile bed response that maintains equilibriumtransport through reaches of rapidly changing shear stress.If the bed load grain size distribution in this stream is atall similar to the substrate (which is consistent throughoutthe basin), then changes in surface layer texture andtransport frequency may serve to equalize transport ratesthrough individual reaches, as proposed by Parker et al.[1982].[33] Variations in bed sediment texture have also been

described in the literature as a response to changes in thesediment supply. Results from a number of laboratoryexperiments using sediment feed flumes generally indicatethat the bed becomes coarser as the sediment supplydecreases [see Lisle and Church, 2002, and references

therein] or finer as the supply increases [Dietrich et al.,1989; Lisle et al., 1993; Buffington and Montgomery,1999b]. The surface-based transport model of Parker[1990] likewise suggests that the surface texture will be-come finer or coarser as necessary to balance the sedimentsupply. However, experiments conducted by Wilcock et al.[2001] showed that, if anything, the bed surface becomescoarser as discharge and transport rate are increased. Thecontrast in results stems from differences in the operation ofsediment feed versus sediment-recirculating flumes inwhich the size distribution of the sediment supplied eitherstays the same (feed) or evolves over time (recirculating).Natural streams are hybrids of these two systems: streamsare likely to respond to changes in sediment supply throughevolution of the bed surface texture (sediment feed), but asthe bed evolves, the sediment output to downstream reacheseventually becomes disconnected from the initial input(recirculating). Clearly, dramatic increases in sediment sup-ply would likely alter sediment textures and sediment yieldsin the study basin, with the extent of this effect dictated bythe size and caliber of the input. Yet given the relativeinfrequency of channel-impacting debris flows and land-slides, conditions in Halfmoon Creek are probably moreanalogous to a sediment-recirculating system than a sedi-ment feed system. Glaciation has left a strong imprint on thelandscape, producing a mix of reaches with locally highslope and shear stress, coupled to reaches in broad alluvialvalleys. The ‘‘feed’’ to any particular reach is thereforedetermined largely by the output from the reaches upstream.The data clearly show that there is not much difference inthe D50 of the substrate throughout the basin, suggestingthat the sediment supplied from hillslopes and channel bedsources has a similar size distribution. However, the coarserfractions of the substrate and of the bed surface clearly varywith the local shear stress. It appears that the shear stressand grain size vary jointly in order to balance the frequencyand intensity of bed load transport; adjacent reaches with

Table 3. Linear Regression Statistics of Grain Size Versus Bank-

Full Shear Stress for the Surface and Subsurface D50 and D84 for

Data in Figure 11a

Grain Size a b r2 p

D50 0.00023 0.041 0.62 �0.001D50s 0.00005 0.019 0.11 0.11D84 0.00074 0.070 0.64 0.001D84s 0.00033 0.051 0.42 0.0004aThe slope is a, and the intercept of the of the regression equations is b;

the coefficient of determination is r2; and the significance probability is p.

Figure 12. Modeled annual sediment yield, Qs, plottedversus estimated bank-full discharge, Qb, for each studyreach.


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very different slopes and grain sizes appear to carry roughlythe same load.[34] Our results indicate that Halfmoon Creek and its

tributaries appear to function more or less as an alluvialstream system, even at slopes as high as 6%. However, theanalysis does not provide unambiguous evidence, one wayor the other, that Halfmoon Creek transports bed load atcapacity. Gilbert [1914, p. 35] defined sediment transportcapacity as ‘‘the maximum load a stream can carry,’’presumably for a given condition; load was defined as thequantity (mass or volume) of sediment carried through across section per unit time. In addition to discharge andslope, transport capacity varies with grain size [Gilbert,1914, 1917; Mackin,1948; Rubey, 1952]. The load can bemeasured, whereas the transport capacity can only beinferred. Typically, those inferences are made on the basisof comparisons between measured and predicted transportrates. Since we did not measure transport rates, we cannot,in the strictest sense, make that comparison. For the samereason, however, there is a danger in basing estimates oftransport capacity on measures of flow strength (e.g., streampower, or a surrogate variable, such as drainage area timesslope), because without reference to a threshold flow or acharacteristic grain size, there is no clear basis for compar-ison there either. The processes that govern erosion andsediment transport in mountain streams can, therefore, bemore fully understood if we begin to couple high-qualitytransport measurements, obtained over a full range of flows,with basin-wide measurements of channel morphology andgrain size.

6. Conclusion

[35] The channel and bed material properties of Half-moon Creek, a headwater stream in the Sawatch Range ofthe southern Rocky Mountains, reflect a combination of freeand forced responses to changes in valley slope, waterdischarge and sediment supply. Short steep reaches thatbegin in the vicinity of glacial steps, moraines or debris fansare characterized by very coarse bed material and high shearstress. Despite this, sediment within these reaches appears tobe mobile for at least a few days per year. Selectivetransport of finer sediment (medium-fine gravel) throughthe steep reaches produces rapid downstream fining, anddecreases the contrast between the bed surface layer (armor)and the substrate sediment. We postulate that, with areduction in bed armoring, the overall mobility of the bedload is enhanced, which increases the frequency of trans-port, and thereby allows increases in sediment transportthrough reaches despite declining shear stress. Sedimenttransport modeling suggests that forced changes in channelgradient are offset by changes in channel width, depth, andbed surface texture to maintain a rough balance between theintensity and frequency of bed load transport. The modeledsediment transport relation indicates that the current channelmorphology of Halfmoon Creek is adjusted such that bedload transport capacity increases systematically down-stream, resulting in a simple scaling between dischargeand bed load sediment yield throughout the basin.

[36] Acknowledgments. Financial support for this study wasprovided by the Stream Systems Technology Center of the U.S. Forest

Service; we would like to thank Larry Schmidt and John Potyondy for theiradvice and support. We would also like to thank Aaron Cloud, Casey Ward,Jordan Clayton, Peter Wilcock, and Mike McGowan for their help incollecting field data; Mike Smith for graphics help; Jonathan Nelson andNel Caine for comments during the early phase of the analysis; and DarrylGranger, John Buffington, Martin Doyle, Robert Anderson, and Jim Pizzutofor their extremely thoughtful reviews of earlier versions of this manuscript.

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Morphologically based model of bed load transport capacity ... · PDF fileElsewhere, fluvial sediment transport dominates and channel morphology reflects the orderly transfer of water