a systematic analysis of eight decades of incipient motion ... · a systematic analysis of eight...

A systematic analysis of eight decades of incipient motionstudies, with special reference to gravel-bedded rivers

John M. Buffington and David R. MontgomeryDepartment of Geological Sciences, University of Washington, Seattle

Abstract. Data compiled from eight decades of incipient motion studies were used tocalculate dimensionless critical shear stress values of the median grain size, t*c50

.Calculated t*c50

values were stratified by initial motion definition, median grain size type(surface, subsurface, or laboratory mixture), relative roughness, and flow regime. Atraditional Shields plot constructed from data that represent initial motion of the bedsurface material reveals systematic methodological biases of incipient motion definition;t*c50

values determined from reference bed load transport rates and from visualobservation of grain motion define subparallel Shields curves, with the latter generallyunderlying the former; values derived from competence functions define a separate butpoorly developed field, while theoretical values predict a wide range of generally higherstresses that likely represent instantaneous, rather than time-averaged, critical shearstresses. The available data indicate that for high critical boundary Reynolds numbers andlow relative roughnesses typical of gravel-bedded rivers, reference-based and visually basedstudies have t*c50

ranges of 0.052–0.086 and 0.030–0.073, respectively. The apparent lack of auniversal t*c50

for gravel-bedded rivers warrants great care in choosing defendable t*c50values for

particular applications.

Introduction

Incipient motion of streambeds is a fundamental processwith applications to a wide variety of research problems, suchas paleohydraulic reconstructions [Church, 1978], placer for-mation [Komar and Wang, 1984; Li and Komar, 1992], canaldesign [Lane, 1955], flushing flows [Milhous, 1990; Kondolf andWilcock, 1992], and assessment of aquatic habitat [Buffington,1995; Montgomery et al., 1996]. Regardless of whether oneadvocates equal mobility [Parker et al., 1982], selective trans-port [e.g., Komar, 1987a, b], or some other style of sedimentmovement, most investigators use a standard or modified formof the critical Shields parameter to define incipient motion ofa grain size of interest. The Shields parameter, or dimension-less critical shear stress, is defined as t*ci

5 tci/(rs 2 r) gDi,

where tciis the critical shear stress at incipient motion for a

grain size of interest, Di; g is the gravitational acceleration;and rs and r are the sediment and fluid densities, respectively.Of particular interest for fluvial geomorphologists is determi-nation of dimensionless critical shear stress values of the me-dian grain size, t*c50

, for high boundary Reynolds numberscharacteristic of gravel-bedded streams.

Shields [1936] demonstrated that t*c50of near-uniform grains

varies with critical boundary Reynolds number, Re*c, and hy-pothesized on the basis of an analogy with Nikuradse’s [1933]findings that t*c50

attains a constant value of about 0.06 aboveRe*c 5 489 (Figure 1). The critical boundary Reynolds num-ber is defined as Re*c 5 u*cks/n , where u*c is the critical shearvelocity for incipient motion (u*c [ (tc/r)1/ 2), ks is theboundary roughness length scale, and n is the kinematic vis-cosity; Shields [1936] set ks 5 D50, the median grain size of thesediment. Although Shields’ [1936] boundary Reynolds num-

bers differ from Nikuradse’s [1933], the general form of Shields’[1936] curve (Figure 1) is quite similar to Nikuradse’s [1933]curve, indicating regions of hydraulically smooth, transitional,and rough turbulent flow. The commonly quoted value of t*c50

' 0.06 for rough turbulent flow reflects a single data pointwithin the overall swath of Shields’ [1936] data (Figure 1).

There have been numerous additions, revisions, and modi-fications of the Shields curve since its original publication.Shields [1936], Grass [1970], Gessler [1971], and Paintal [1971]recognized that incipient motion of a particular grain size isinherently a statistical problem, depending on probability func-tions of both turbulent shear stress at the bed and intergranu-lar geometry (i.e., friction angles) of the bed material, thelatter being controlled by grain shape, sorting, and packing[Miller and Byrne, 1966; Li and Komar, 1986; Kirchner et al.,1990; Buffington et al., 1992]. Consequently, there is a fre-quency distribution of dimensionless critical shear stresses forany grain size of interest. Reanalyzing Shields’ [1936] data andcorrecting for sidewall effects and form drag, Gessler [1971]reported t*c50

' 0.046 for a 50% probability of movement inrough turbulent flow. Without consideration of the probabilityof movement, Miller et al. [1977] arrived at a similar value oft*c50

' 0.045 for rough turbulent flow using compiled flumedata from various sources. Miller et al. [1977, p. 507] employeddata from “flumes with parallel sidewalls where flows wereuniform and steady over flattened beds of unigranular,rounded sediments”; sidewall corrections were applied andeach source used a consistent definition of incipient motion.Although Miller et al. [1977] used carefully selected data toensure compatibility within their compilation, scrutiny of theirdata shows use of both uniform and nonuniform sedimentmixtures, differing incipient motion definitions between stud-ies, and in some cases bed load transport rates influenced bybed forms.

Using a larger data set and ignoring differences in sediment

Copyright 1997 by the American Geophysical Union.

Paper number 96WR03190.0043-1397/97/96WR-03190$09.00

WATER RESOURCES RESEARCH, VOL. 33, NO. 8, PAGES 1993–2029, AUGUST 1997

1993

characteristics, channel roughness, or definition of incipientmotion, Yalin and Karahan [1979, Figure 5] also report t*c50

'0.045 for rough turbulent flow. They further demonstrate theexistence of a second Shields curve for fully laminar flow,which for the same Re*c values behaves differently than thetraditional Shields curve derived from turbulent flow with vari-able hydrodynamic boundary conditions (i.e., smooth, transi-tional, or rough). The t*c50

values reported by Miller et al. [1977]and Yalin and Karahan [1979] are average values of data setswith considerable scatter; in both studies individual t*c50

valuesfor rough turbulent flow range from about 0.02 to 0.065.

Previous compilations of t*c50values combine data derived

from quite different experimental conditions and methodolo-gies with little assessment of compatibility. Continued prolif-eration of incipient motion studies using new definitions ofinitial motion further complicates comparison and understand-ing of published studies. Although differences in experimentalcondition and methodology have been recognized and dis-cussed [e.g., Tison, 1953; Miller et al., 1977; Carson and Grif-fiths, 1985; Lavelle and Mofjeld, 1987; Wilcock, 1988, 1992b],their influence on reported t*c50

values has not been well ex-amined. Here we compile eight decades of incipient motiondata and stratify calculated t*c50

values by (1) initial motiondefinition, (2) choice of surface, subsurface or laboratory mix-ture median grain size, (3) relative roughness, and (4) flowregime, providing a systematic reanalysis of the incipient mo-tion literature. We also evaluate the compatibility of differentinvestigative methodologies and interpret the range of re-ported t*c50

values.

Data Compilation and StratificationAll available incipient motion data are summarized in Ta-

bles 1a–1e. Values of t*c50, critical boundary Reynolds number

(Re*c), and median grain size (D50) are reported for eachsource, as well as experimental conditions and dimensionlesscritical shear stress equations where these are different thanShields’ [1936]. Where available, the graphic sorting coefficient(sg [Folk, 1974]), sediment density (rs), and relative rough-ness (D50/hc, where hc is the critical flow height at incipientmotion) are also reported. In many cases values of Re*c andt*c50

(or particular types of t*c50, as discussed later) were not

reported but could be calculated from the data and equationspresented by the author(s); to be consistent with Shields [1936],we used ks 5 D50 when calculating Re*c. The graphic sorting

coefficient is defined as (f84 2 f16)/2, where f84 and f16 arethe 84th and 16th percentiles of the grain size distributionexpressed in units of the phi (log2) scale. Values of hc used todetermine relative roughness were back-calculated fromdepth-slope products where sufficient data were reported. De-tailed notes regarding both our calculations and the investiga-tive procedures used by each source are presented byBuffington [1995] and are abbreviated in the appendix.

The data compiled in Tables 1a–1e are stratified by incipientmotion definition. The four most common methods of definingincipient motion are: (1) extrapolation of bed load transportrates to either a zero or low reference value (Table 1a) [e.g.,Shields, 1936; Day, 1980; Parker and Klingeman, 1982]; (2)visual observation (Table 1b) [e.g., Gilbert, 1914; Kramer, 1935;Yalin and Karahan, 1979]; (3) development of competencefunctions that relate shear stress to the largest mobile grainsize, from which one can establish the critical shear stress for agiven size of interest (Table 1c) [e.g., Andrews, 1983; Carling,1983; Komar, 1987a]; and (4) theoretical calculation (Table 1d)[e.g., White, 1940; Wiberg and Smith, 1987; Jiang and Haff, 1993].

Dimensionless critical shear stresses determined from thefirst method are based on critical shear stresses associated witheither a zero or low reference transport rate extrapolated frompaired shear stress and bed load transport measurements. Val-ues determined from this approach are sensitive to the extrap-olation method [cf. Parker and Klingeman, 1982; Diplas, 1987;Ashworth and Ferguson, 1989; Ashworth et al., 1992] and theparticular reference transport value that is chosen [Wilcock,1988].

Visual observation, used in the second method, is direct butcan be subjective depending on one’s definition of how muchmovement constitutes initial motion [e.g., Gilbert, 1914;Kramer, 1935; Neill and Yalin, 1969; Wilcock, 1988]. Paintal[1971] argues that there will always be some probability ofgrain movement as long as there is any fluid motion; hence thethreshold of movement becomes a definitional construct [seealso Lavelle and Mofjeld, 1987]. Standardized definitions ofincipient motion have been proposed on the basis of the num-ber of grains in motion, the area of bed observed, and theduration of observation [Neill and Yalin, 1969; Wilcock, 1988];however, these definitions have not been widely adopted.

Competence functions, used in the third method, are sensi-tive to the size and efficiency of the sediment trap, sample size,sampling strategy, availability of coarse grain sizes, and curve-

Figure 1. Shields’ [1936] curve redrafted from Rouse [1939].

BUFFINGTON AND MONTGOMERY: REVIEW1994

fitting technique [Wilcock, 1992b; Wathen et al., 1995]. Further-more, the competence method is inappropriate for sedimentthat exhibits equal mobility, as the competence approach relieson selective transport [Wilcock, 1988, 1992b].

The fourth method utilizes simple force balance argumentsto predict initial motion thresholds and is sensitive to modelparameters such as grain protrusion, packing, and friction an-gle. In our analysis, t*c50

values corresponding to these fourmethods of measuring incipient motion are symbolized as t*cr50

(reference), t*cv50(visual), t*cq50

(competence), and t*ct50(theo-

retical).Data compiled for each definition of incipient motion are

further subdivided by median grain size type (e.g., Table 1a).Values of t*c50

have been variously reported in the literature forthe median grain size of the surface (D50s), subsurface(D50ss), and laboratory sediment mixture (D50m), the threeof which are equal only for uniform-sized sediment; corre-sponding dimensionless critical shear stresses for these threemedian grain size types are denoted here as t*c50s

, t*c50ss, and

t*c50m. Expression of dimensionless critical shear stress in terms

of the subsurface grain size distribution was popularized byAndrews [1983], who expressed the Shields stress of a givengrain size of interest (t*ci

) as a power law function of the ratioDi/D50ss; Andrews [1983] found that for his data, Di/D50ss wasbetter correlated with t*ci

than was Di/D50s (r2 5 0.98 versus0.89 [Andrews, 1983]). Although expression of bed load trans-port formulations in terms of D50ss [e.g., Parker et al., 1982]seems reasonable because of the general correspondence ofbed load and subsurface grain size distributions [Milhous, 1973;Kuhnle, 1993a], it is counterintuitive to use the dimensionlesscritical shear stress of the subsurface to define thresholds ofmotion and the onset of bed load transport in gravel-beddedchannels [e.g., Andrews, 1983; Parker et al., 1982]. It is wellknown that most gravel-bedded rivers are armored and thatthe surface and subsurface grain size distributions can differsignificantly [e.g., Leopold et al., 1964; Milhous, 1973]. Analysisof incipient motion of gravel-bedded rivers therefore shouldemploy surface values of critical shear stress. No matter howwell the subsurface grain size distribution correlates with thebed load transport size distribution, the initiation of bed loadtransport is controlled by bed surface grains. Nevertheless, thecorrespondence of subsurface and transport size distributionsindicates that subsurface-based mobility values are appropriatefor describing bed load transport beyond incipient motion.Because the difference between subsurface and surface grainsize distributions is unpredictable, there is no a priori conver-sion of subsurface-based incipient motion values to surfaceones.

There is currently little recognition in the incipient motionliterature of the difference between t*c50

values for the variousD50 types (i.e., t*c50s

, t*c50ss, and t*c50m

). As such, careful eval-uation of reported values is necessary in order to compare andchoose appropriate values of t*c50

. For example, using an An-drews-type power function expressed in terms of a genericmedian grain size, Komar [1987a] reports a generic t*c50

valueof 0.045 for three gravel channels studied by Milhous [1973],Carling [1983], and Hammond et al. [1984]. It is only upon closeinspection of Komar’s [1987a] analysis that it becomes appar-ent that this value represents incipient motion of grain sizessimilar or equal to the median subsurface size (note 32, ap-pendix); the corresponding unreported surface values (t*c50s

)range from 0.021 to 0.027 (Table 1c), roughly half that re-ported by Komar [1987a]. Scaling critical shear stresses by

subsurface median grain sizes generally produces t*c50values

larger than surface-based ones because of bed surface armor-ing (compare t*c50s

and t*c50ssvalues of Parker and Klingeman

[1982], Wilcock and Southard [1988], Kuhnle [1992], Andrewsand Erman [1986], Komar [1987a], and Komar and Carling[1991], given in Tables 1a and 1c).

We emphasize that the dimensionless critical shear stressvalues reported here are for the median grain size only. Shieldsparameters of other grain sizes of interest will vary as a func-tion of size-specific friction angle, grain protrusion, and mo-bility of neighboring grains.

AnalysisOf the 613 dimensionless critical shear stress values com-

piled in Tables 1a–1e, we examined only those that representincipient motion of the bed surface, because of their relevancefor determining sediment transport thresholds in armoredgravel-bedded channels. Subsurface dimensionless criticalshear stress values (t*c50ss

) were removed from the database, asthey were all derived from armored channels and thus do notrepresent initial motion of the streambed surface.

Sorting coefficients (sg) were used to establish conditions inwhich initial motion of laboratory mixtures could be used as ameasure of surface mobility (i.e., establishing when t*c50m

ap-proximates t*c50s

). Poorly sorted laboratory sediment mixtureshave the potential to exhibit textural response and reworkingprior to measurement of incipient motion in both reference-based and visually based studies. Reference-based laboratorystudies commonly employ shear stress and bed load transportdata collected after attainment of equilibrium conditions ofslope, bed form character, and transport rate [e.g., Gilbert,1914; Shields, 1936; Guy et al., 1966; Williams, 1970; Wilcock,1987], prior to which considerable reworking of the bed surfacemay occur [e.g., Wilcock and Southlard, 1989; Wilcock andMcArdell, 1993]. Visually based studies also allow varying de-grees of water working and sediment transport depending onthe specific definition of initial motion employed [e.g., Kramer,1935, U.S. Waterways Experimental Station (USWES), 1935].Consequently, the actual surface grain size distribution of ini-tially poorly sorted mixtures may not resemble the originalmixture distribution at the time of incipient motion measure-ment. This causes potentially erroneous results when measuredshear stresses for water-worked sediments are combined withunworked mixture distributions to determine dimensionlesscritical shear stress, as is commonly done in laboratory studies.

Textural response of laboratory mixtures is controlled byrelative conditions of transport capacity and sediment supply[e.g., Dietrich et al., 1989; Kinerson, 1990; Buffington. 1995].Depending on the direction of textural response, t*c50m

valuescould overestimate or underestimate actual dimensionless crit-ical shear stress values of the surface (t*c50s

). Mixture mediangrain sizes will approximate surface median grain sizes onlywhen laboratory sediment mixtures are well sorted, as there islittle potential for textural response of a well-sorted bed ma-terial. Hence only under these conditions will dimensionlesscritical shear stresses of the mixture approximate those of thesurface. We confined our use of mixture-based studies to thoseusing well-sorted material, where “well sorted” is defined assg # 0.5. Under this definition, some of the laboratory sed-iment mixtures used by Shields [1936] are mixed-grain (Table1a). Mixture-based studies with unknown sg values were notused.

1995BUFFINGTON AND MONTGOMERY: REVIEW

We further screened the data for relative roughness effects(D50/hc). Bathurst et al. [1983] demonstrated that for a givengrain size, t*c50

systematically increases with greater relativeroughness and that the rate of increase depends on channelslope [see also Shields, 1936; Cheng, 1970; Aksoy, 1973; Mi-zuyama, 1977; Torri and Poesen, 1988]. The increase in t*c50

with greater relative roughness can be explained through theconcept of shear stress partitioning

t0 5 t9 1 t0 1 z z z t n (1)

which is predicated on the hypothesis that the total channelroughness and shear stress (t0) can be decomposed into lin-early additive components (t9, t0, etc.), each characterizing aparticular roughness element [Einstein and Barbarossa, 1952;Engelund, 1966; Hey, 1979, 1988; Parker and Peterson, 1980;Brownlie, 1983; Prestegaard, 1983; Dietrich et al., 1984; Griffiths,1989; Nelson and Smith, 1989; Petit, 1989, 1990; Robert, 1990;Clifford et al., 1992; Millar and Quick, 1994; Shields and Gippel,1995]. The effective shear stress (t9) is defined here as thatwhich is available for sediment transport after correction forother roughness elements (i.e., t9 5 t0 2 t0 2 z z z tn).

Based on this concept of shear stress partitioning, greaterform drag caused by increased relative roughness (D50/hc)will decrease the shear stress available at the bed for sedimenttransport (t9), resulting in a higher total shear stress (t0) re-quired for incipient motion and thus an apparently greater t*c50

value. The compiled data demonstrate a general positive cor-relation between t*c50

and D50/hc over the range 0.01 #D50/hc # 2 (Figure 2). The apparent inverse correlation oft*c50

and D50/hc for D50/hc , 0.01 is a coincident effect offlow regime and is not a relative roughness effect. The com-piled data with D50/hc , 0.01 generally have Re*c # 20,which corresponds with the hydrodynamically transitional andsmooth portions of the Shields [1936] curve for which t*c50

isnegatively correlated with Re*c (Figure 1). It is the associationwith these low Re*c flow regimes, not the relative roughnessitself, that causes the apparent inverse correlation of t*c50

andD50/hc for D50/hc , 0.01. Because of the influence of rela-tive roughness on t*c50

, we restricted our analysis to data withD50/hc # 0.2, a value that we chose to be generally repre-sentative of gravel-bedded streams. We excluded data fromstudies with unknown D50/hc values.

We also excluded data from convergent-wall flume studiesbecause of their apparent incompatibility with those from par-allel-wall flumes [Vanoni et al., 1966; Miller et al., 1977]. The

above screening results in a database of 325 t*c50values,

roughly half of the total compilation.A traditional Shields plot constructed from data represent-

ing initial motion of the bed surface exhibits the expectedgeneral form of the original Shields curve but reveals system-atic methodological biases of incipient motion definition (Plate1). Values of t*c50

determined from reference bed load trans-port rates (t*cr50

) and from visual observation of grain motion(t*cv50

) define subparallel Shields curves, with the visual datagenerally underlying the reference data. Reference-based di-mensionless critical shear stress values determined from well-sorted laboratory mixtures (t*cr50m

) and from surface grains ofnatural channels (t*cr50s

) dovetail quite well. Dimensionlesscritical shear stress values derived from competence functions(t*cq50s

) define a separate but poorly developed field. Althoughnot shown, theoretical values (t*ct50s

) exhibit no trend in rela-tion to Re*c and are widely variable depending on choice ofintergranular friction angle and grain protrusion (Table 1d).Furthermore, the theoretical values generally predict highstresses that likely represent instantaneous, rather than time-averaged, critical shear stresses [Buffington et al., 1992]. Scatterwithin Shields curves has long been attributed to methodolog-ical differences between experiments [e.g., Tison, 1953; Milleret al., 1977; Carson and Griffiths, 1985; Lavelle and Mofjeld,1987], but our reanalysis presents the first comprehensive sup-port for this hypothesis.

The data in Plate 1 are also segregated by flow condition(i.e., fully laminar versus hydraulically smooth, transitional, orrough turbulent flow). We did not limit the laminar data toD50/hc # 0.2, as relative roughness effects are unlikely forlaminar flow conditions. As demonstrated by Yalin and Kara-han [1979], two Shields curves are defined for laminar versusturbulent flow conditions over similar Re*c values. The lower-angle trend of our compiled laminar curve is similar to thatidentified by Yalin and Karahan [1979].

Despite eight decades of incipient motion studies there re-mains a lack of t*c50

values representative of fully turbulentflow and low relative roughness typical of gravel-bedded rivers(Plate 1). The available data indicate that for such conditionsreference-based and visually based studies have t*c50

ranges of0.052–0.086 and 0.030–0.073, respectively (Figure 3). The vi-sual range, however, is rather speculative because of the lack ofdata for high critical boundary Reynolds numbers.

Scatter within the stratified data sets likely reflects a varietyof factors, such as differences in bed material properties (i.e.,

Figure 2. Variation of dimensionless critical shear stress with relative roughness. Only data with knownD50/hc are shown here; all mixture data have sgm # 0.5.


1989]. Each of these methods can result in different estimatesof shear stress, and thus distinct t*c50

values, particularly ifshear stress is nonuniform through a study reach.

Use of different sampling techniques to characterize grainsize distributions, and hence D50, may also cause some of theobserved scatter. The compiled studies use a variety of areal,grid, and volumetric sampling techniques each of which canyield different results [e.g., Kellerhals and Bray, 1971; Diplasand Sutherland, 1988; Diplas and Fripp, 1992; Fripp and Diplas,1993]. Reference-based studies that use Parker and Klinge-man’s [1982] method are particularly sensitive to grain sizesampling technique, as their method employs the proportion ofeach size class of the grain size distribution.

Although dimensionless critical shear stress is trigonometri-cally related to bed slope [e.g., Wiberg and Smith, 1987] (afactor not accounted for in the traditional Shields equation), itseffect on the compiled t*c50

values is minimal, as most of thedata are derived from experiments with bed slopes less than0.01. The data of Bathurst et al. [1987] and Mizuyama [1977] arenotable exceptions; however, t*c50

values reported for thesestudies are based on modified Shields stresses that account forboth bed slope and bulk friction angle of the sediment (Table1a).

Use of appropriate ks values when calculating Re*c mayreduce some of the observed scatter [e.g., Ippen and Verma,1953]. There have been numerous ks empiricisms proposed [cf.Einstein and Barbarossa, 1952; Leopold et al., 1964; Kamphuis,1974; Hey, 1979; Bray, 1980], most of which are greater than

D50s for heterogeneous bed surfaces; Whiting and Dietrich[1990], for example, suggest ks 5 3D84s. Although we use ks

5 D50s for comparison with historical Shields curves, ks 5D50s is only appropriate for uniformly sized sediment. How-ever, Re*c correction using appropriate ks values will not im-prove the t*c50

uncertainty, which accounts for most of theobserved scatter.

Differences in the scale and duration of observation withinand between methodologies may also contribute to the scatterof compiled data [e.g., Neill and Yalin, 1969; Fenton and Ab-bott, 1977; Wilcock, 1988]. For example, the spatial scale ofobservation in visually based studies varies from the entire bedsurface [e.g., Gilbert, 1914] to that viewed from a microscope[White, 1970]. Similarly, reference- and competence-basedstudies may employ channel-spanning bed load traps that sam-ple all material passing a cross section [e.g., Milhous, 1973] orthey may combine several point measures of bed load transport[e.g., Ashworth and Ferguson, 1989], representing a smallerscale of observation. Temporal scales of observation also varyamong and within methodologies. For example, visually basedstudies are typically of short duration and made while thechannel adjusts to perturbations of slope or hydraulic dis-charge. In contrast, reference-based studies conducted influmes employ data collected over long time periods and afterattainment of equilibrium conditions of channel morphologyand hydraulics. However, reference- and competence-basedstudies conducted in the field are influenced by nonequilibriumconditions and may require shorter periods of data collection

Figure 3. Comparison of (a) reference-based and (b) visually based t*c50data shown in Plate 1. Circled

triangles in Figure 3a are discussed in caption to Plate 1. See appendix note 12 regarding open circles ofFigure 3a.


because of logistics and safety during high flows [e.g., Ashworthand Ferguson, 1989; Wilcock et al., 1996]. Differences in spatialand temporal scales of observation can yield different esti-mates of critical conditions for incipient motion, particularly inchannels that exhibit nonuniformities of shear stress, grainsize, and bed material properties that cause spatial differencesin mobility [e.g., Powell and Ashworth, 1995; Wilcock et al.,1996]. Rules for standardizing incipient motion definition andthe spatial and temporal scales of observation between inves-tigations have been proposed [e.g., Neill and Yalin, 1969; Yalin,1977; Wilcock, 1988] but are not widely used. Wilcock [1988]proposed a standard definition of incipient motion for mixed-grain sediments that accounts for the number of grains moved,their size and proportion of the grain size distribution, and thearea and duration of observation. Even when such rules areapplied, channels with identical reach-average shear stressesand grain size distributions may demonstrate different t*c50

values because of subreach differences in the spatial variabilityof shear stress, sediment supply, and bed surface textures (i.e.,grain size, sorting, and packing).

Differences of incipient motion definition within each meth-odology may also contribute to the observed scatter. For ex-ample, Kramer’s [1935] three definitions of visual grain motion(weak, medium, and general) represent a two-fold differencein t*c50

values. Similarly, differences in the choice of dimen-sionless reference transport rate used to define incipient mo-tion can result in a three-fold variation of reference-based t*c50

values [Paintal, 1971, Figure 6]. Despite this potential for vari-ation, Wilcock [1988] found that reference-based t*c50

valuesdetermined from the Parker and Klingeman [1982] and Ackersand White [1973] methods differed by only 5% for the samedata set.

Scatter within the reference- and competence-based datamay also reflect choice of curve fitting technique [Diplas, 1987;Ashworth and Ferguson, 1989; Ashworth et al., 1992; Wathen etal., 1995]. In an extreme example, Paintal [1971, Figure 8]demonstrates that nonlinear relationships between bed loadtransport rate and dimensionless shear stress that are mistak-enly fit with a linear function can cause up to a five-fold over-estimation of reference-based t*c50 values. In many cases it isdifficult to assess or correct differences in curve-fitting tech-nique between investigations due to incomplete documenta-tion of measurements and analysis procedure. The results ofShields [1936], in particular, are often used as the standard forcomparison, yet Shields’ [1936] basic measurements and curvefitting technique are unreported, making it difficult to fullyassess the causes of discrepancy between Shields’ [1936] dataand those reported by others.

The above influences on t*c50values can be of comparable

magnitude and can easily account for the observed scatter ofvalues within methodologies. For example, neglect of rough-ness effects and natural variation of bed surface characteristicshave the potential to cause similar magnitudes of scatter. Vari-ation in particle protrusion and packing can result in an orderof magnitude range in t*c50

[Fenton and Abbott, 1977; Powelland Ashworth, 1995], while bed form drag in natural rivers cancomprise 10%–75% of the total channel roughness [Parker andPeterson, 1980; Prestegaard, 1983; Dietrich et al., 1984; Hey1988], indicating a similar range of t*c50

variation if bed formresistance is not accounted for. Nonetheless, despite poten-tially similar sources and/or magnitudes of scatter, the com-piled data demonstrate distinct methodological biases (Plate1).

DiscussionOur reanalysis and stratification of incipient motion values

reveal systematic methodological biases and highlight funda-mental differences of median grain size type and their associ-ated values of dimensionless critical shear stress. The Shieldscurve constructed from our data compilation (Plate 1) (1)specifically represents incipient motion of the bed surface, (2)excludes data associated with large relative roughness valuesthat are uncharacteristic of gravel-bedded rivers, and (3) in-cludes for the first time reference- and competence-based t*c50

values for surface material (t*cr50sand t*cq50s

). We find that therough, turbulent flow value of t*c50

' 0.045 reported in pre-vious compilation studies [Miller et al., 1977; Yalin and Kara-han, 1979] is typical of visually determined mobility thresholdsof laboratory mixtures (t*cv50m

), but underestimates dimension-less critical shear stresses determined from reference transportrates (t*cr50m

and t*cr50s) (Figure 3).

Although methodological bias explains much of the scatterin our constructed Shields curve, one is still faced with decidingwhich investigative method to rely on when choosing a t*c50

value. None of the four investigative methods is demonstrablysuperior; each has its strengths and weaknesses. However,some methods may be more appropriate for particular appli-cations [Carson and Griffiths, 1985]. For example, because ref-erence- and competence-based values are derived from bedload transport measures, they may be more well suited toapplication in bed load transport investigations. Depending onthe bed load sampling strategy, reference- and competence-based methods may also integrate differential bed mobilityresulting from bed surface textural patches and reach-scaledivergence of shear stress and sediment supply and thus maybe more appropriate for representing reach-average bed mo-bility. In contrast, visually based methods typically record localincipient motion and are best applied to mobility studies ofdiscrete bed surface textural patches. Because of methodolog-ical biases, care should be taken to choose t*c50

values from aninvestigative method that represents the scale and type of in-cipient motion needed for one’s particular study goals. Con-versely, study results should be interpreted in light of the in-cipient motion method used and the sediment transportprocesses that it measures. For example, t*c50

values fromeither competence- or reference-based methods could be usedto predict reach-average incipient motion, but competence-based values describe motion of the coarsest bed load sizes,whereas reference-based values describe motion of the full bedload distribution; the two methods describe the mobility ofdifferent subpopulations of the bed material and may yielddifferent results if equivalent scaling factors are not used[Wilcock, 1988].

Of the four methods from which to choose t*c50values,

competence-based and theoretically based methods can be ex-cluded because of a paucity of data that precludes confidentinterpretation of the functional relationship between t*c50

andRe*c. Nevertheless, the competence-based data define a hori-zontal band of roughly constant dimensionless critical shearstresses at high Re*c values as expected for a Shields-typerelationship (Plate 1). Furthermore, this band of data generallylies within the Shields curve defined by visually based methodsand systematically underlies the reference-based data (cf. Plate1 and Figure 3), contrary to expectations that competencevalues should be greater than reference-based ones due tounderrepresentation of coarse grain sizes [Wilcock, 1992b].


The fact that some of the data in both methods are derivedfrom the same study sites [Milhous, 1973; Ashworth et al., 1992;Wathen et al., 1995] makes this difference between compe-tence- and reference-based approaches credible. For the samestudy site, competence-based t*c50

values are roughly 15%–30% smaller. The systematically lower incipient motion valuesdetermined from the competence approach may reflect aninherent bias associated with use of the largest mobile grainsize. Larger bed surface grains may have lower mobility thresh-olds because of greater protrusion and smaller intergranularfriction angles [e.g., Buffington et al., 1992]. Komar andCarling’s [1991] variant of the competence approach using themedian grain size of the load, rather than the maximum grainsize, produces t*c50

values similar to reference-based ones (Ta-ble 1e).

In contrast to theoretically based and competence-basedmethods, functional relationships between t*c50

and Re*c arewell defined for reference-based and visually based ap-proaches. Both the reference-based and visually based studiesexhibit a roughly twofold range in t*c50

values for conditionstypical of gravel-bedded channels (Figure 3), which representssignificant uncertainty in dimensionless critical shear stress.Many bed load transport equations are based on the differencebetween the applied and critical grain shear stresses raised tosome power greater than 1 (see the review by Gomez andChurch [1989]). Differences between applied and critical shearstresses are typically small in gravel-bedded channels [Parker etal., 1982] because of the approximately bankfull-threshold na-ture of bed mobility (see the review by Buffington [1995]).Consequently, small errors in tc50

due to uncertainty in t*c50can

cause significant errors in calculated bed load transport rates.Consideration of the sources of scatter and their systematic

influence on the reference-based and visually based data pro-vides further guidance in choosing specific t*c50

values. In par-ticular, neglect of form drag effects may cause systematic over-estimation of t*c50

values. It is commonly implied that becauseflume-based studies of incipient motion employ initially planarbed surfaces they are free of form drag influences caused bybed forms [e.g., Miller et al., 1977]. This is true for the visuallybased studies, but it is not so for most of the reference-basedinvestigations, such as Shields’ [1936]. In the visual studies, flowis typically increased gradually until grains are observed tomove from a plane-bed surface [e.g., Kramer, 1935; White,1970; Yalin and Karahan, 1979]. In contrast, most of the ref-erence studies are based on bed load transport data collectedafter attainment of equilibrium conditions, which in many in-stances are characterized by the presence of bed forms [e.g.,Gilbert, 1914; Shields, 1936; Guy et al., 1966; Wilcock andSouthard, 1988]. Because bed form resistance can comprise upto 75% of the total channel roughness [Hey, 1988], there is apotentially significant difference between t9 and t0, and hencethe calculated t*c50

value, if bed form roughness is not ac-counted for. Moreover, it is uncertain whether bed load trans-port data from surfaces characterized by bed forms can providea meaningful extrapolation to conditions of initial motion froma lower-regime plane bed, as is commonly intended in labora-tory reference-based studies. Although bed form resistance isnot an issue in visually based studies, relative roughness effectscommon to these studies (i.e., lineaments of Figure 3b) mayprovide an equally important source of form drag and overes-timation of t*c50

values if not accounted for.In analyzing incipient motion data it is common practice to

fit a single average curve through the scatter of data. However,

a Shields curve defined by minimum t*c50values will minimize

overestimation of t*c50caused by neglect of form drag resis-

tance in both reference-based and visually based studies (Ta-bles 1a and 1b) and may be more representative of poorlysorted sediments typical of gravel channels. Poorly sorted sed-iments tend to have lower intergranular friction angles andthus lower incipient motion thresholds [Buffington et al., 1992].The necessarily narrow range of sorting (sg # 0.5) of themixture data, however, may preclude any meaningful analysisof sorting effects. The t*c50

values for rough turbulent flowderived from minimum Shields curves are 0.052 and 0.030 forreference-based and visually based studies, respectively (Fig-ure 3). However, thorough accounting of roughness effectsmay produce even lower values.

Regardless of whether an average or minimum curve is cho-sen for the reference-based data, Oak Creek is an outlier(Figure 3a). It does not fit with the expected general form ofthe traditional Shields curve as defined by the other data. Thisis somewhat disconcerting, as Oak Creek is believed to be oneof the best bed load transport data sets available for naturalchannels and has been used by many authors as the standardfor comparison. Although the issue warrants further investiga-tion, the discrepancy between Oak Creek and the other refer-ence-based data may be due to unaccounted for differences inchannel roughness (Table 1a).

ConclusionsOur reanalysis of incipient motion data for bed surface ma-

terial indicates that (1) much of the scatter in Shields curves isdue to systematic biases that investigators should be aware ofwhen choosing and comparing dimensionless critical shearstress values from the literature; and (2) there is no definitivet*c50

value for the rough, turbulent flow characteristic of gravel-bedded rivers, but rather there is a range of values that differsbetween investigative methodologies. Our analysis indicatesthat less emphasis should be placed on choosing a universalt*c50

value, while more emphasis should be placed on choosingdefendable values for particular applications, given the ob-served methodological biases, uses of each approach, and sys-tematic influences of sources of uncertainty associated withdifferent methods and investigative conditions.

Note added in proof. During the time this article was inpress we discovered several other referenced-based values sim-ilar to those of Oak Creek [see Andrews, 1994; Andrews andNankervis, 1995].

NotationD50, D84, Di grain size for which 50%,

84%, and i% of the grainsare finer.

D50s, D50ss, D50m, D50l median grain sizes of thesurface, subsurface,laboratory mixture, and bedload.

Dgs geometric mean surfacegrain size.

g gravitational constant.hc critical flow depth for

incipient motion.ks boundary roughness length

scale (equivalent to


Tab

le1a

.Pr

evio

usly

Rep

orte

dt

* c50

Val

ues:

Ref

eren

ceT

rans

port

Rat

e

Sour

ceN

ote*

Surf

ace

Gra

inSi

zeD

istr

ibut

ion

t* c

r50s

Re*

c†

Prop

osed

t* c

Fun

ctio

nO

ther

Tha

nSh

ield

s’D

50

s,m

ms

gs

(f)

rs,

kg/m

3D

50

s/h

c‡

Exp

erim

enta

lCon

ditio

ns

Par

ker

and

Klin

gem

an[1

982]

Ref

eren

ceT

rans

port

Rat

eP

arke

ran

dK

linge

man

[198

2]1

0.03

5§6,

744

t* c

ri5

0.03

5(D

i/D5

0s)

20

.94

541.

0928

500.

15na

tura

lpoo

l-riffl

ech

anne

l(O

akC

reek

);no

form

drag

orsi

dew

allc

orre

ctio

nD

ipla

s[1

987]

0.03

4§6,

647

t* c

ri5

0.08

7(D

i/D5

0ss

)20

.94

541.

0928

500.

16sa

me

asP

arke

ran

dK

linge

man

[198

2]F

rom

Wilc

ock

and

Sout

hard

[198

8]2

Milh

ous

[197

3]0.

027§

5,92

3t

* cri

50.

073(

Di/D

50

ss)2

0.9

854

1.09

2850

0.20

sam

eas

Par

ker

and

Klin

gem

an[1

982]

Ash

wor

than

dF

ergu

son

[198

9]3

0.07

2§;

7,77

3t

* cri

50.

072(

Di/D

50

s)2

0.6

5;

50m

2540

0.11

natu

ralp

ool-r

iffle

chan

nel(

Alt

Dub

haig

),va

riab

lesi

nuos

ity;s

idew

all

corr

ectio

nim

plic

it0.

054§

;8,

463

t* c

ri5

0.05

4(D

i/D5

0s)

20

.67

;57

.5m

2600

0.10

natu

ralp

ool-r

iffle

chan

nel(

Riv

erF

eshi

e),m

ildly

brai

ded;

side

wal

lco

rrec

tion

impl

icit

0.08

7§;

16,1

38t

* cri

50.

087(

Di/D

50

s)2

0.9

2;

69m

3090

0.13

natu

ralb

raid

edch

anne

l(L

yngs

dals

elva

);si

dew

allc

orre

ctio

nim

plic

itP

arke

r[1

990]

0.03

4§6,

731

t* c

ri5

0.03

9(D

i/Dg

s)2

0.9

054

1.02

2850

0.16

sam

eas

Par

ker

and

Klin

gem

an[1

982]

Ash

wor

thet

al.[

1992

]0.

061§

2,46

3t

* cri

50.

061(

Di/D

50

s)2

0.7

924

m26

500.

06na

tura

l,br

aide

d,gr

avel

chan

nel

(Sun

wap

taR

iver

);fo

rmdr

agan

dsi

dew

allc

orre

ctio

nas

inno

te3

Kuh

nle

[199

2]0.

065§

869

t* c

ri5

0.08

6(D

i/D5

0ss

)20

.81

11.7

30.

90zz

z0.

03na

tura

lcha

nnel

with

mix

edgr

avel

and

sand

bed

exhi

bitin

gm

acro

scal

edu

nes

[Kuh

nle

and

Bow

ie,1

992]

(Goo

dwin

Cre

ek);

nosi

dew

allo

rfo

rmdr

agco

rrec

tion

Fro

mW

ilcoc

k[1

993]

4M

ilhou

s[1

973]

0.01

2§3,

949

t* c

ri5

0.03

3(D

i/D5

0ss

)20

.98

541.

0928

500.

20sa

me

asP

arke

ran

dK

linge

man

[198

2];

Ein

stei

n[1

950]

side

wal

land

form

drag

corr

ectio

n¶W

ilcoc

kan

dM

cArd

ell[

1993

]5

0.02

8§88

t* c

ri5

0.02

8(D

i/D5

0s)

20

.45,

0.77

#D

i/D5

0s#

17.3

2.6

2.51

2610

0.12

stra

ight

,rec

tang

ular

flum

e;va

riab

lebe

dfor

ms;

Ein

stei

n[1

950]

side

wal

land

form

drag

corr

ectio

n¶W

athe

net

al.[

1995

]0.

086§

2,44

5t

* cri

50.

086(

Di/D

50

s)2

0.9

021

.3;

1.6

2650

0.01

–0.1

4na

tura

lpoo

l-riffl

ech

anne

l(A

ltD

ubha

ig),

vari

able

sinu

osity

;no

side

wal

lor

form

drag

corr

ectio

n

Ack

ers

and

Whi

te[1

973]

Ref

eren

ceT

rans

port

Rat

eD

ay[1

981]

60.

035

2,12

9zz

z20

0.92

zzz

$0.

24–0

.41

stra

ight

,rec

tang

ular

flum

e;pl

ane

bed;

nosi

dew

allc

orre

ctio

n0.

036

2,49

1zz

z22

0.86

zzz

$0.

28–0

.45

asab

ove

Zer

oR

efer

ence

Tra

nspo

rtIp

pen

and

Ver

ma

[195

3]0.

045

30zz

z2.

0#

0.13

1280

0.23

stra

ight

,rec

tang

ular

flum

e;pl

astic

test

grai

npl

aced

onfix

edpl

ane

bed;

sign

ifica

ntgr

ain

proj

ectio

nan

dex

posu

re;*

*si

dew

alle

ffec

tsin

sign

ifica

nt


Tab

le1a

.(C

ontin

ued)

Sour

ceN

ote*

Surf

ace

Gra

inSi

zeD

istr

ibut

ion

t* c

r50s

Re*

c†

Prop

osed

t* c

Fun

ctio

nO

ther

Tha

nSh

ield

s’D

50

s,m

ms

gs

(f)

rs,

kg/m

3D

50

s/h

c‡

Exp

erim

enta

lCon

ditio

ns

0.04

4§33

zzz

2.0

#0.

1312

800.

19as

abov

e0.

045§

35zz

z2.

0#

0.13

1280

0.16

asab

ove

Mel

and

and

Nor

rman

[196

6]7

0.00

8§33

zzz

2.09

025

60$

0.05

stra

ight

,rec

tang

ular

flum

e;fix

edpl

ane

bed;

glas

ste

stgr

ain

plac

edon

rhom

bohe

dral

lypa

cked

bed;

sign

ifica

ntgr

ain

proj

ectio

nan

dex

posu

re;*

*si

dew

allc

orre

ctio

nim

plic

it0.

022§

390

zzz

7.76

025

10$

0.16

asab

ove

0.01

6§33

0zz

z7.

760

2510

$0.

16as

abov

e,bu

tw

itha

loos

ebe

d

Sour

ceN

ote*

Lab

orat

ory

Mix

ture

Gra

inSi

zeD

istr

ibut

ion

t* c

r50m

Re*

c†

Prop

osed

t* c

Fun

ctio

nO

ther

Tha

nSh

ield

s’D

50

m,

mm

sg

m(f

)r

s,kg

/m3

D5

0m

/hc‡

Exp

erim

enta

lCon

ditio

ns

Zer

oR

efer

ence

Tra

nspo

rtR

ate

Fro

mSh

ield

s[1

936]

8G

ilber

t[1

914]

††0.

059§

489

zzz

7.01

,0.

2226

90?

0.16

stra

ight

,rec

tang

ular

flum

e;be

dfo

rmty

pes

not

repo

rted

;par

tials

idew

all

corr

ectio

n(?

);‡‡

nofo

rmdr

agco

rrec

tion;

subr

ound

edgr

ains

0.05

1§22

7zz

z4.

94?

,0.

2626

90?

0.18

asab

ove

Cas

ey[1

935]

0.06

7§1.

9zz

z0.

170.

4126

50#

0.03

stra

ight

,rec

tang

ular

flum

e;va

riou

sbe

dfo

rms;

part

ials

idew

allc

orre

ctio

n(?

);‡‡

nofo

rmdr

agco

rrec

tion;

suba

ngul

arto

roun

ded

grai

ns0.

052§

3.3

zzz

0.27

0.42

2650

#0.

04as

abov

e0.

035§

11zz

z0.

680.

1626

50#

0.10

asab

ove

0.03

5§16

zzz

0.87

0.17

2650

#0.

12as

abov

e0.

038§

18zz

z0.

940.

1926

50#

0.05

asab

ove

0.04

2§31

zzz

1.26

0.20

2650

,0.

20?

asab

ove

0.04

3§49

zzz

1.75

0.19

2650

#0.

10as

abov

e0.

039§

80zz

z2.

460.

2226

50#

0.10

asab

ove

Kra

mer

[193

5]0.

033

6.6

zzz

0.53

0.81

2700

?#

0.04

stra

ight

,rec

tang

ular

flum

e;va

riou

sbe

dfo

rms;

part

ials

idew

allc

orre

ctio

n(?

);‡‡

nofo

rmdr

agco

rrec

tion;

wel

l-ro

unde

dgr

ains

0.03

97.

6zz

z0.

510.

7427

00?

#0.

04as

abov

e0.

033

7.8

zzz

0.55

0.62

2700

?#

0.04

asab

ove

USW

ES

[193

5]§§

?0.

051§

2.2

zzz

0.21

?0.

3226

500.

02st

raig

ht,r

ecta

ngul

arflu

me;

vari

ous

bed

form

s;pa

rtia

lsid

ewal

lcor

rect

ion

(?);

‡‡no

form

drag

corr

ectio

n0.

046

4.1

zzz

0.31

?0.

53?

2650

0.02

asab

ove


0.03

9§4.

4zz

z0.

35?

0.37

?26

500.

02as

abov

e0.

035

7.4

zzz

0.52

?0.

53?

2650

0.03

asab

ove

0.03

67.

4zz

z0.

51?

0.82

?26

500.

03as

abov

e0.

036

7.8

zzz

0.52

?0.

53?

2650

0.03

asab

ove

Shie

lds

[193

6]0.

037§

6.3

zzz

0.36

0.30

4300

#0.

04?

stra

ight

,rec

tang

ular

flum

e;va

riou

sbe

dfo

rmty

pes;

form

drag

and

side

wal

lco

rrec

tion

asin

note

3;an

gula

rba

rite

grai

ns0.

036§

56zz

z1.

520.

3542

00#

0.04

?as

abov

e0.

042§

124

zzz

2.46

?0.

2241

90#

0.04

?as

abov

e0.

046§

219

zzz

3.44

0.16

4200

#0.

04?

asab

ove

0.04

4§14

2zz

z2.

76?

0.41

4200

#0.

04?

asab

ove

0.03

88.

8zz

z1.

560.

5910

60#

0.04

?as

abov

e,bu

tw

ithan

gula

ram

ber

grai

ns;

part

ials

idew

allc

orre

ctio

n;‡‡

nofo

rmdr

agco

rrec

tion

0.04

18.

9zz

z1.

560.

5910

60#

0.04

?as

abov

e0.

045

9.3

zzz

1.56

0.59

1060

#0.

04?

asab

ove

0.03

0§16

zzz

0.85

0.23

2700

#0.

04?

asab

ove,

but

with

angu

lar

gran

itic

grai

ns0.

035§

29zz

z1.

230.

2327

10#

0.04

?as

abov

e0.

049§

100

zzz

2.44

0.23

2690

#0.

04?

asab

ove

0.03

015

zzz

1.77

?0.

78?

1270

#0.

04?

asab

ove,

but

with

angu

lar

coal

grai

ns0.

038

20zz

z1.

770.

7812

70#

0.04

?as

abov

e0.

040

23zz

z1.

880.

7212

70#

0.04

?as

abov

e0.

049

38zz

z2.

530.

5612

70#

0.04

?as

abov

eF

rom

John

son

[194

3]9

Gilb

ert

[191

4]††

100.

297

12zz

z0.

305

,0.

0926

900.

02st

raig

ht,r

ecta

ngul

arflu

me;

vari

ous

bed

form

s;E

inst

ein

[194

2]si

dew

all

corr

ectio

n;¶

nofo

rmdr

agco

rrec

tion;

suba

ngul

argr

ains

0.26

215

zzz

0.37

5,

0.11

2690

0.02

asab

ove

Mac

Dou

gall

[193

3];R

iver

Hyd

raul

ics

Lab

orat

ory,

(unp

ublis

hed

repo

rt)

0.05

5§16

zzz

0.67

0.38

zzz

0.03

stra

ight

,rec

tang

ular

flum

e;be

dfo

rmty

pes

not

repo

rted

;sid

ewal

lcor

rect

ion

impl

icit

(?);

roun

ded

grai

nsC

hyn

[193

5]0.

062

14zz

z0.

601.

23zz

z0.

02as

abov

e,bu

tw

ithpl

ane-

bed

ordu

nem

orph

olog

y;no

form

drag

corr

ectio

n0.

044§

23zz

z0.

830.

30zz

z0.

02as

abov

e,bu

tw

ithdu

nes

only

Joris

sen

[193

8]0.

064

16zz

z0.

620.

8526

700.

02as

abov

e,bu

tw

ithpl

anar

orri

pple

dbe

d;no

form

drag

corr

ectio

n0.

044

24zz

z0.

920.

6126

700.

03as

abov

eM

elan

dan

dN

orrm

an[1

969]

110.

047

207

zzz

3.9

zzz

2560

,0.

08?

stra

ight

,rec

tang

ular

flum

e;va

riou

sbe

dfo

rms;

form

drag

and

side

wal

lcor

rect

ion

sam

eas

inno

te3;

glas

sgr

ains

Pai

ntal

[197

1]12

0.05

0§63

8zz

z7.

950.

1826

500.

12st

raig

ht,r

ecta

ngul

arflu

me;

plan

ebe

d(?

);Jo

hnso

n[1

942]

side

wal

lcor

rect

ion¶

0.05

0§11

2zz

z2.

50.

1426

500.

02as

abov

eSt

ernb

erg

[197

1]§§

,\\0.

048

4.8

zzz

0.42

1.09

zzz

,2

310

25

sand

ym

arin

ech

anne

l(Pi

cker

ing

Pass

age)

with

ripp

led

bed;

roug

hnes

sco

rrec

tion

sim

ilar

toth

atfo

rno

te3

Miz

uyam

a[1

977]

††,¶

¶13

0.03

8§(0

.049

)43

7t* c r

50m

5t c

r50m

/[(r s

2r)

gD50

m(t

anF

cosu

2si

nu)]

6.4

,0.

1225

070.

12st

raig

ht,r

ecta

ngul

arflu

me;

bed

form

type

not

repo

rted

;Ein

stei

n[1

942]

side

wal

lco

rrec

tion;

¶no

form

drag

corr

ectio

n0.

047

(0.0

60)

481

asab

ove

6.4

,0.

1225

070.

27as

abov

e0.

053

(0.0

66)

507

asab

ove

6.4

,0.

1225

070.

49as

abov

e0.

042§

(0.0

41)

1,07

6as

abov

e12

.0,

0.21

2656

0.12

asab

ove

0.04

4(0

.043

)1,

096

asab

ove

12.0

,0.

2126

560.

33as

abov

e


Tab

le1a

.(C

ontin

ued)

Sour

ceN

ote*

Lab

orat

ory

Mix

ture

Gra

inSi

zeD

istr

ibut

ion

t* c

r50m

Re*

c†

Prop

osed

t* c

Fun

ctio

nO

ther

Tha

nSh

ield

s’D

50

m,

mm

sg

m(f

)r

s,kg

/m3

D5

0m

/hc‡

Exp

erim

enta

lCon

ditio

ns

0.05

2(0.

049)

1,17

4as

abov

e12

.0,

0.21

2656

0.59

asab

ove

0.05

8(0

.054

)1,

227

asab

ove

12.0

,0.

2126

560.

83as

abov

e0.

067

(0.0

60)

1,29

7as

abov

e12

.0,

0.21

2656

1.00

asab

ove

0.08

3(0

.072

)1,

419

asab

ove

12.0

,0.

2126

561.

04as

abov

e0.

043

(0.0

42)

2,63

7as

abov

e22

.5,

0.08

2490

0.25

asab

ove

0.05

1(0

.048

)2,

841

asab

ove

22.5

,0.

0824

900.

63as

abov

e0.

057

(0.0

53)

2,97

1as

abov

e22

.5,

0.08

2490

0.90

asab

ove

0.06

9(0

.061

)3,

199

asab

ove

22.5

,0.

0824

901.

05as

abov

e0.

079

(0.0

66)

3,32

0as

abov

e22

.5,

0.08

2490

1.47

asab

ove

0.09

8(0

.077

)3,

580

asab

ove

22.5

,0.

0824

901.

73as

abov

eP

azis

and

Gra

f[1

977]

14;

0.02

06.

2–92

zzz

0.49

–3.0

2;

u?26

50–

1410

,0.

02st

raig

ht,r

ecta

ngul

arflu

me;

sand

and

plas

ticgr

ains

;bed

form

type

sun

repo

rted

;Ein

stei

n[1

950]

side

wal

lan

dfo

rmdr

agco

rrec

tion

(?)¶

Fro

mB

athu

rst

etal

.[19

87]

15E

cole

Poly

tech

niqu

eF

eder

ale

deL

ausa

nne

0.05

2§(0

.036

)90

5t

* cr5

0m5

tc

r50m

/[(r

s2

r)g

D5

0m

(tan

Fco

su2

sinu

)]11

.5;

u?26

500.

08st

raig

ht,r

ecta

ngul

arflu

me;

vari

ous

bed

form

s;E

inst

ein

[195

0](?

)si

dew

alla

ndfo

rmdr

ag(?

)co

rrec

tion¶

0.06

3§(0

.043

)94

4as

abov

e11

.5;

u?26

500.

10as

abov

e0.

070§

(0.0

48)

1,03

6as

abov

e11

.5;

u?26

500.

13as

abov

e0.

062§

(0.0

53)

3,23

1as

abov

e22

.20.

3425

700.

12as

abov

e0.

087

(0.0

71)

3,43

6as

abov

e22

.20.

3425

700.

27as

abov

e0.

102

(0.0

82)

3,62

1as

abov

e22

.20.

3425

700.

39as

abov

e0.

113

(0.0

88)

3,75

3as

abov

e22

.20.

3425

700.

50as

abov

e0.

115

(0.0

88)

3,88

6as

abov

e22

.20.

3425

700.

65as

abov

e0.

061

(0.0

49)

7,97

6as

abov

e44

.3;

u?27

500.

35as

abov

e0.

068

(0.0

54)

8,36

6as

abov

e44

.3;

u?27

500.

53as

abov

e0.

088

(0.0

68)

8,71

4as

abov

e44

.3;

u?27

500.

59as

abov

e0.

087

(0.0

65)

9,13

1as

abov

e44

.3;

u?27

500.

79as

abov

eB

athu

rst

etal

.[19

79]

160.

094§

(0.0

92)

881

asab

ove

8.8

0.42

2629

0.13

stra

ight

,rec

tang

ular

flum

e;pl

ane-

bed

orlo

wam

plitu

debe

dfo

rms;

nofo

rmdr

agor

side

wal

lcor

rect

ion

(?)

0.12

6(0

.113

)97

4as

abov

e8.

80.

4226

290.

27as

abov

e0.

185

(0.1

70)

1,19

4as

abov

e8.

80.

4226

290.

28as

abov

e0.

095(

0.07

9)6,

198

asab

ove

340.

4426

290.

61as

abov

eL

iand

Kom

ar[1

992]

170.

048§

3.3

zzz

0.24

#0.

13zz

z0.

03st

raig

ht,r

ecta

ngul

arflu

me;

nonu

nifo

rmflo

w;b

edfo

rmty

pes

unre

port

ed;n

ofo

rmdr

agor

side

wal

lcor

rect

ion

Ack

ers

and

Whi

te[1

973]

Ref

eren

ceT

rans

port

Rat

eF

rom

Day

[198

0]U

SWE

S[1

935]

180.

050

9.1

zzz

0.42

0.86

2650

0.02

stra

ight

,rec

tang

ular

flum

e;va

riou

sbe

dfo

rms;

nosi

dew

allo

rfo

rmdr

agco

rrec

tion;

suba

ngul

arto

subr

ound

edgr

ains

0.04

77.

3zz

z0.

440.

5926

500.

02as

abov

e,bu

tw

ithan

gula

rto

subr

ound

edgr

ains


0.03

418

9zz

z4.

100.

5426

500.

07as

abov

e,bu

tw

ithsu

brou

nded

tosu

bang

ular

grai

nsD

ay[1

980]

0.02

437

zzz

1.75

2.10

zzz

0.04

stra

ight

,rec

tang

ular

flum

e;be

dfo

rmty

peno

tre

port

ed;n

osi

dew

allo

rfo

rmdr

agco

rrec

tion

0.02

934

zzz

1.55

1.70

zzz

0.03

asab

ove

Day

[198

1]19

0.04

51,

025

zzz

11.3

1.41

zzz

$0.

24–

0.47

stra

ight

,rec

tang

ular

flum

e;pl

ane

bed;

nosi

dew

allc

orre

ctio

n

Par

ker

and

Klin

gem

an[1

982]

Ref

eren

ceT

rans

port

Rat

eL

eopo

ldan

dE

mm

ett

[197

6,19

77]\

\20

0.07

232

t* c

ri5

0.07

2(D

i/D5

0m

)20

.86

1.25

2.8

2650

0.00

6na

tura

l,du

ne-r

ippl

ech

anne

l(E

ast

For

k);

nosi

dew

allo

rfo

rmdr

agco

rrec

tion

Fro

mW

ilcoc

kan

dSo

utha

rd[1

988]

Day

[198

0]0.

037

48t

* cri5

0.03

7(D

i/D5

0m

)20

.81

1.82

2.09

zzz

0.03

stra

ight

,rec

tang

ular

flum

e;be

dfo

rmty

peno

tre

port

ed;n

ofo

rmdr

agor

side

wal

lcor

rect

ion

(?)

0.03

739

t* c

ri5

0.03

7(D

i/D5

0m

)20

.95

1.57

1.73

zzz

0.03

asab

ove

Dha

mot

hara

net

al.[

1980

]0.

071

108

t* c

ri5

0.07

1(D

i/D5

0m

)21

.12.

161.

4326

300.

07st

raig

ht,r

ecta

ngul

arflu

me;

bed

form

type

not

repo

rted

byW

ilcoc

kan

dSo

utha

rd[1

988]

;no

form

drag

orsi

dew

allc

orre

ctio

n(?

)M

isri

etal

.[19

84]

0.04

810

1t

* cri

50.

048(

Di/D

50

m)2

1.0

2.36

1.05

2650

,0.

05st

raig

ht,r

ecta

ngul

arflu

me;

bed

form

type

not

repo

rted

;Ein

stei

n[1

950]

side

wal

land

form

drag

corr

ectio

n¶0.

042

194

t* c

ri5

0.04

2(D

i/D5

0m

)20

.95

3.81

1.65

2650

,0.

07as

abov

e0.

037

196

t* c

ri5

0.03

7(D

i/D5

0m

)20

.92

4.00

1.29

2650

,0.

07as

abov

eW

ilcoc

k[1

987]

0.03

061

t* c

ri5

0.03

0(D

i/D5

0m

)21

.01.

830.

5326

500.

04st

raig

ht,r

ecta

ngul

arflu

me;

vari

ous

bed

form

s;E

inst

ein

[195

0]si

dew

alla

ndfo

rmdr

agco

rrec

tion¶

0.03

667

t* c

ri5

0.03

6(D

i/D5

0m

)20

.97

1.83

1.06

2650

0.03

asab

ove

0.02

3§12

t* c

ri5

0.02

3(D

i/D5

0m

)20

.98

0.67

0.29

2650

0.02

asab

ove

0.03

7§33

2t

* cri

50.

037(

Di/D

50

m)2

1.1

5.28

0.20

2650

0.06

stra

ight

,rec

tang

ular

flum

e;pl

ane

bed;

Ein

stei

n[1

950]

side

wal

land

form

drag

corr

ectio

n¶W

ilcoc

k[1

992a

]21

0.04

911

5t

* cri

50.

049(

Di/D

50

m)2

1.0

42.

550.

89zz

z0.

07st

raig

ht,r

ecta

ngul

arflu

me;

vari

ous

bed

form

s;E

inst

ein

[195

0]si

dew

alla

ndfo

rmdr

agco

rrec

tion;

¶bi

mod

alse

dim

ent

??

t* c

ri5

0.01

7(D

i/D5

0m

)21

.25,

0.27

#D

i/D5

0m

#0.

39;

t* c

ri5

0.06

3(D

i/D5

0m

)21

.14,

2.1

#D

i/D5

0m

#3.

1

2.00

1.65

zzz

zzz

asab

ove,

but

with

stro

ngly

bim

odal

sedi

men

t;D

50

mis

fictit

ious

,not

asi

zeoc

curr

ing

inth

em

ixtu

re

0.05

219

t* c

ri5

0.52

(Di/D

50

m)2

0.7

3,

0.7

#D

i/D5

0m

#1.

0;t

* cri

50.

042(

Di/D

50

m)2

1.1

7,

5.8

#D

i/D5

0m

#8.

3

0.75

1.67

zzz

0.01

asab

ove,

but

D5

0m

isre

al

Kuh

nle

[199

3b]

0.03

58.

4t

* cri

50.

035(

Di/D

50

m)2

1.0

0.47

2.18

zzz

0.02

stra

ight

,rec

tang

ular

flum

e;pl

ane

bed

orlo

wam

plitu

debe

dfo

rms;

Van

onia

ndB

rook

s[1

957]

side

wal

lcor

rect

ion¶

,***

0.03

9§40

4t

* cri

50.

039(

Di/D

50

m)2

1.1

5.58

0.36

zzz

0.12

asab

ove

0.04

38.

7t

* cri

50.

043(

Di/D

50

m)2

1.1

,0.

45#

Di/D

50

m#

2.5;

t* c

ri5

0.01

9(D

i/D5

0m

)20

.32,

3.6

#D

i/D5

0m

#20

.4

0.47

0.87

zzz

0.02

asab

ove


Tab

le1a

.(C

ontin

ued)

Sour

ceN

ote*

Lab

orat

ory

Mix

ture

Gra

inSi

zeD

istr

ibut

ion

t* c

r50m

Re*

c†

Prop

osed

t* c

Fun

ctio

nO

ther

Tha

nSh

ield

s’D

50

m,

mm

sg

m(f

)r

s,kg

/m3

D5

0m

/hc‡

Exp

erim

enta

lCon

ditio

ns

0.04

111

t* c

ri5

0.04

1(D

i/D5

0m

)21

.0,

0.37

#D

i/D5

0m

#3.

0;t

* cri

50.

035(

Di/D

50

m)2

0.5

5,

4.2

#D

i/D5

0m

#11

.9

0.57

1.89

zzz

0.02

asab

ove

0.04

529

t* c

ri5

0.04

5(D

i/D5

0m

)21

.0,

0.22

#D

i/D5

0m

#1.

2;t

* cri

50.

037(

Di/D

50

m)2

0.4

2,

1.8

#D

i/D5

0m

#7.

1

0.95

2.03

zzz

0.03

asab

ove

Wilc

ock

and

McA

rdel

l[19

93]

220.

020

219

t* c

ri5

0.02

8(D

i/D5

0s)

20

.45,

0.77

#D

i/D5

0s

#17

.35.

32.

8826

100.

16st

raig

ht,r

ecta

ngul

arflu

me;

vari

able

bed

form

s;E

inst

ein

[195

0]si

dew

alla

ndfo

rmdr

agco

rrec

tion¶

Oth

erR

efer

ence

Tra

nspo

rtD

efini

tion

Fro

mB

ridge

and

Dom

inic

[198

4]G

ilber

t[1

914]

††0.

040§

4.2

zzz

0.30

,0.

0926

900.

08st

raig

ht,r

ecta

ngul

arflu

me;

uppe

rst

age

plan

ebe

d;W

illia

ms

[197

0]si

dew

allc

orre

ctio

n;su

bang

ular

rive

rsa

nd0.

052§

6.9

zzz

0.38

,0.

1126

900.

13as

abov

e0.

042§

9.6

zzz

0.51

,0.

2626

900.

17as

abov

e0.

030

16zz

z0.

79,

0.32

2690

0.24

asab

ove

0.04

014

5zz

z3.

17,

0.24

2690

0.24

asab

ove,

but

low

erst

age

plan

ebe

dan

dsu

brou

nded

rive

rgr

avel

0.04

1§28

6zz

z4.

94,

0.26

2690

0.19

asab

ove

Will

iam

s[1

970]

0.04

036

zzz

1.35

0.20

zzz

0.40

stra

ight

,rec

tang

ular

flum

e;up

per

stag

epl

ane

bed;

Will

iam

s[1

970]

side

wal

lcor

rect

ion

0.04

0§41

zzz

1.35

0.20

zzz

0.07

asab

ove,

but

low

erst

age

plan

ebe

dG

uyet

al.[

1966

]0.

040§

2.1

zzz

0.19

0.45

zzz

0.02

stra

ight

,rec

tang

ular

flum

e;up

per

stag

epl

ane

bed;

nosi

dew

allc

orre

ctio

n0.

040

3.3

zzz

0.28

0.81

zzz

0.03

asab

ove

0.04

04.

4zz

z0.

320.

65zz

z0.

07as

abov

e0.

040

22zz

z0.

930.

59zz

z0.

008

asab

ove,

but

low

erst

age

plan

ebe

d

Sour

ceN

ote*

Subs

urfa

ceG

rain

Size

Dis

trib

utio

n

t* c

r50s

sR

e*c†

Prop

osed

t* c

Fun

ctio

nO

ther

Tha

nSh

ield

s’D

50

ss,

mm

sg

ss(f

)r

s,kg

/m3

D5

0ss

/hc‡

Exp

erim

enta

lCon

ditio

ns

Par

ker

and

Klin

gem

an[1

982]

Ref

eren

ceT

rans

port

Rat

eP

arke

ran

dK

linge

man

[198

2]0.

088

2,41

0t

* cri

50.

088(

Di/D

50

ss)2

0.9

820

2.20

2850

0.06

sam

eas

Par

ker

and

Klin

gem

an[1

982]

,firs

ten

try

ofT

able

1aF

rom

Wilc

ock

and

Sout

hard

[198

8]M

ilhou

s[1

973]

0.07

32,

113

t* c

ri5

0.07

3(D

i/D5

0ss

)20

.98

19.5

2.20

2850

0.07

asab

ove

Kuh

nle

[199

2]0.

086

596

t* c

ri5

0.08

6(D

i/D5

0ss

)20

.81

8.31

2.77

zzz

0.02

sam

eas

Kuh

nle

[199

2],s

even

then

try

ofT

able

1aK

uhnl

e[1

993b

]0.

089

606

t* c

ri5

0.08

0(D

i/D5

0ss

)20

.94,

0.04

#D

i/D5

0m

#0.

34;

t* c

ri5

0.08

9(D

i/D5

0ss

)20

.81,

0.67

#D

i/D5

0m

#5.

4

8.31

2.77

zzz

0.02

asab

ove

seve

nth

entr

yof

Tab

le1a


Not

eth

atsy

mbo

lsfo

rsi

mila

rfo

otno

tes

may

bedi

ffer

ent

inT

able

s1a

–1e.

Whi

lem

ost

t* c

r50

valu

esar

ede

term

ined

from

extr

apol

atio

nof

bed

load

tran

spor

tra

tes,

som

ear

eba

sed

onex

trap

olat

ion

ofpa

rtic

leor

bed

form

velo

city

[e.g

.,Ip

pen

and

Ver

ma,

1953

;Mel

and

and

Nor

rman

,196

6;St

ernb

erg,

1971

].Se

eno

tatio

nse

ctio

nfo

rsy

mbo

lsno

tpr

evio

usly

defin

edin

text

.See

resp

ectiv

eap

pend

ixno

tes

for

valu

esin

pare

nthe

ses.

Her

e“u

”de

note

sun

iform

grai

nsi

zes

(sg

#0.

5),

and

“m”

deno

tes

mix

edgr

ain

size

s(s

g.

0.5)

,w

here

sg

isth

egr

aphi

cst

anda

rdde

viat

ion

defin

edas

(f84

2f

16)/

2[F

olk,

1974

].*S

eeap

pend

ix.

†Re*

c5

u* c

D5

0/n

.Mos

tval

ues

are

back

-cal

cula

ted

from

repo

rted

data

.For

exam

ple,

usin

gt

* cr5

0san

dD

50

sre

port

edby

Par

ker

and

Klin

gem

an[1

982]

,we

calc

ulat

edt

cr5

0sfr

omSh

ield

s’[1

936]

equa

tion,

allo

win

gde

term

inat

ion

ofu

* c5

(tc

r50s

/r)0

.5,

and

henc

eR

e*c,

with

nes

timat

edfr

omw

ater

tem

pera

ture

sre

port

edby

Milh

ous

[197

3].W

here

unre

port

edby

the

orig

inal

sour

ce,i

tw

asas

sum

edth

atr

san

dr

wer

e26

50an

d10

00kg

/m3 ,r

espe

ctiv

ely,

and

that

n5

102

6m

2 /sfo

rla

bora

tory

and

theo

retic

alst

udie

san

d1.

53

102

6m

2 /sfo

rfie

ldst

udie

s.‡W

here

unre

port

edby

aso

urce

,hc

valu

esar

eba

ck-c

alcu

late

dfr

omth

eSh

ield

s[1

936]

equa

tion

with

the

tc

expr

esse

das

ade

pth-

slop

epr

oduc

t(h

c5

t* c

(rs

2r

)D5

0/r

S,w

ithS

dete

rmin

edfr

omth

eor

igin

also

urce

,whe

reav

aila

ble)

.For

exam

ple,

we

used

the

aver

age

slop

eof

Gilb

ert’s

[191

4]7.

01-m

mex

peri

men

tsfo

rth

efir

stG

ilber

ten

try

from

Shie

lds

[193

6];f

orG

ilber

ten

trie

sfr

omB

ridge

and

Dom

inic

[198

4]w

eus

edth

esu

bset

ofsl

opes

corr

espo

ndin

gw

ithpl

ane

bed

mor

phol

ogie

sfo

rea

chse

dim

ent.

Thi

spr

oced

ure

may

caus

eov

eres

timat

ion

ofD

50/h

cw

here

dept

h-sl

ope

prod

ucts

have

been

redu

ced

for

roug

hnes

sef

fect

s.§U

sed

inPl

ate

1.\R

epor

ted

data

are

with

resp

ect

toth

ege

omet

ric

mea

ngr

ain

size

.¶U

seof

the

aver

age

velo

city

inth

eE

inst

ein

[194

2;19

50],

John

son

[194

2]an

dV

anon

iand

Bro

oks

[195

7]eq

uatio

nslik

ely

over

estim

ates

t9(s

eeno

te3)

.**

Her

ew

ede

scri

begr

ain

prot

rusi

onin

term

sof

proj

ectio

nan

dex

posu

re[s

ensu

Kirc

hner

etal

.,19

90].

††R

epor

ted

data

are

with

resp

ect

toth

em

ean

nom

inal

grai

ndi

amet

er.N

omin

aldi

amet

ers

are

assu

med

equi

vale

ntto

inte

rmed

iate

grai

ndi

amet

ers

[Cui

and

Kom

ar,1

984]

.Mea

nan

dm

edia

nsi

zes

are

sim

ilar

for

near

-uni

form

sedi

men

t.‡‡

Side

wal

lcor

rect

ion

for

the

prox

imity

ofw

alls

(i.e

.,W

/h),

but

not

for

the

diff

eren

cein

wal

land

bed

grai

nro

ughn

ess.

§§R

epor

ted

data

are

with

resp

ect

tom

ean

grai

nsi

zes.

Mea

nan

dm

edia

ngr

ain

size

sar

eas

sum

edsi

mila

rfo

rne

ar-u

nifo

rmse

dim

ent.

\\R

epor

ted

data

are

dete

rmin

edfr

ombu

lk(i

.e.,

surf

ace

and

subs

urfa

cem

ixtu

re)

grai

nsi

zesa

mpl

ing,

trea

ted

here

asm

ixtu

re-b

ased

valu

es.

¶¶A

lso

give

nby

Ash

ida

and

Bay

azit

[197

3].

***S

idew

allc

orre

ctio

nfo

rth

edi

ffer

ence

inw

alla

ndbe

dgr

ain

roug

hnes

sbu

tno

tfo

rth

epr

oxim

ityof

wal

ls(i

.e.,

W/h

).

Tab

le1b

.Pr

evio

usly

Rep

orte

dt

* c50

Val

ues:

Vis

ualO

bser

vatio

n

Sour

ceN

ote*

Surf

ace

Gra

inSi

zeD

istr

ibut

ion

t* c

v50s

Re*

c†

Prop

osed

t* c

Fun

ctio

nO

ther

Tha

nSh

ield

s’D

50

s,m

ms

gs

(f)

rs,

kg/m

3D

50

s/h

c‡

Exp

erim

enta

lCon

ditio

ns

Var

ious

Mov

emen

tD

efini

tions

Col

eman

[196

7]23

0.28

4(0

.005

)6.

2zz

z12

.70

1278

zzz

stra

ight

rect

angu

lar

flum

e;si

dew

all

corr

ectio

nim

plic

it;pl

astic

test

grai

non

fixed

plan

ebe

dof

like

grai

ns;

sign

ifica

ntpr

ojec

tion

and

expo

sure

;§sa

ddle

rota

tion;

shea

rm

easu

red

with

stra

inga

uge;

wat

eror

sodi

umca

rbox

ymet

hyl-c

ellu

lose

fluid

med

ium

0.28

4(0

.005

)7.

3zz

z12

.70

1278

zzz

asab

ove

0.25

5(0

.004

)7.

3zz

z12

.70

1278

zzz

asab

ove

0.23

0(0

.004

)10

zzz

12.7

012

78zz

zas

abov

e0.

195

(0.0

04)

13zz

z12

.70

1278

zzz

asab

ove

0.16

6(0

.004

)12

zzz

12.7

012

78zz

zas

abov

e0.

160

(0.0

04)

15zz

z12

.70

1278

zzz

asab

ove

0.14

2(0

.004

)19

zzz

12.7

012

78zz

zas

abov

e0.

121

(0.0

05)

31zz

z12

.70

1278

zzz

asab

ove

0.11

3(0

.009

)13

0zz

z12

.70

1278

zzz

asab

ove

0.11

9(0

.011

)16

0zz

z12

.70

1278

zzz

asab

ove

0.12

9(0

.011

)19

0zz

z12

.70

1278

zzz

asab

ove

0.13

8(0

.014

)38

0zz

z12

.70

1278

zzz

asab

ove


Tab

le1b

.(c

ontin

ued)

Sour

ceN

ote*

Surf

ace

Gra

inSi

zeD

istr

ibut

ion

t* c

v50s

Re*

c†

Prop

osed

t* c

Fun

ctio

nO

ther

Tha

nSh

ield

s’D

50

s,m

ms

gs

(f)

rs,

kg/m

3D

50

s/h

c‡

Exp

erim

enta

lCon

ditio

ns

0.12

3(0.

012)

360

zzz

12.7

012

78zz

zas

abov

e0.

116(

0.00

4)19

zzz

12.7

078

22zz

zas

abov

e,bu

tw

ithst

eelt

est

grai

n0.

117(

0.00

4)23

zzz

12.7

078

22zz

zas

abov

e0.

084(

0.00

4)43

zzz

12.7

078

22zz

zas

abov

e0.

098(

0.00

6)58

zzz

12.7

078

22zz

zas

abov

e0.

099(

0.00

7)82

zzz

12.7

078

22zz

zas

abov

e0.

108(

0.01

0)14

0zz

z12

.70

7822

zzz

asab

ove

0.10

7(0.

011)

500

zzz

12.7

078

22zz

zas

abov

e0.

116(

0.01

2)61

0zz

z12

.70

7822

zzz

asab

ove

0.10

7(0.

011)

1340

zzz

12.7

078

22zz

zas

abov

eF

ento

nan

dA

bbot

t[1

977]

0.24

866

zzz

2.5

0?zz

zzz

zst

raig

htre

ctan

gula

rflu

me;

angu

lar

poly

styr

ene

grai

non

fixed

plan

ebe

dof

like

grai

ns;*

*si

dew

allc

orre

ctio

nim

plic

it;pr

otru

sion

of2

0.20

0.16

454

zzz

2.5

0?zz

zzz

zas

abov

e,bu

tw

ithpr

otru

sion

of2

0.06

0.08

539

zzz

2.5

0?zz

zzz

zas

abov

e,bu

tw

ithpr

otru

sion

of0.

300.

050

30zz

z2.

50?

zzz

zzz

asab

ove

0.04

027

zzz

2.5

0?zz

zzz

zas

abov

e,bu

tw

ithpr

otru

sion

of0.

340.

149

51zz

z2.

50?

zzz

zzz

asab

ove,

but

with

prot

rusi

onof

0.04

0.11

946

zzz

2.5

0?zz

zzz

zas

abov

e,bu

tw

ithpr

otru

sion

of0.

160.

070

35zz

z2.

50?

zzz

zzz

asab

ove,

but

with

prot

rusi

onof

0.25

0.07

272

zzz

2.5

0?zz

zzz

zas

abov

e,bu

tw

ithle

ad-fi

lled

test

grai

nsan

dpr

otru

sion

of0.

340.

032

48zz

z2.

50?

zzz

zzz

asab

ove,

but

with

prot

rusi

onof

0.41

0.08

378

zzz

2.5

0?zz

zzz

zas

abov

e,bu

tw

ithpr

otru

sion

of0.

080.

107

88zz

z2.

50?

zzz

zzz

asab

ove,

but

with

prot

rusi

onof

0.05

0.05

361

zzz

2.5

0?zz

zzz

zas

abov

e,bu

tw

ithpr

otru

sion

of0.

440.

071

70zz

z2.

50?

zzz

zzz

asab

ove,

but

with

prot

rusi

onof

0.35

0.07

673

zzz

2.5

0?zz

zzz

zas

abov

e,bu

tw

ithpr

otru

sion

of0.

160.

009

1690

zzz

380

zzz

0.23

stra

ight

rect

angu

lar

flum

e;fix

edpl

ane

bed

ofw

oode

ngr

ains

;tab

lete

nnis

test

grai

nsfil

led

with

lead

shot

orpo

lyst

yren

egr

ains

and

sand

;sid

ewal

lco

rrec

tion

impl

icit;

prot

rusi

onof

0.82

0.01

017

00zz

z38

0zz

z0.

23as

abov

e0.

010

1760

zzz

380

zzz

0.36

asab

ove

0.01

132

00zz

z38

0zz

z0.

36as

abov

e0.

012

3280

zzz

380

zzz

0.24

asab

ove

Pet

it[1

994]

0.05

814

03t

* cvi

50.

058(

Di/D

50

s)2

0.6

612

.8,

0.20

?26

50zz

zst

raig

htre

ctan

gula

rflu

me;

natu

ralg

rain

son

plan

ebe

dof

like

grai

ns;V

anon

ian

dB

rook

s[1

957]

side

wal

lco

rrec

tion\

,¶0.

047

3284

t* c

vi5

0.04

7(D

i/D5

0s)

20

.73

24.2

,0.

34?

2650

zzz

asab

ove

0.04

566

24t

* cvi

50.

045(

Di/D

50

s)2

0.8

139

.2,

0.15

?26

50zz

zas

abov

e0.

049

2444

t* c

vi5

0.04

9(D

i/D5

0s)

20

.68

19.6

,0.

54?

2650

zzz

asab

ove,

but

with

fixed

bed


Sour

ceN

ote*

Lab

orat

ory

Mix

ture

Gra

inSi

zeD

istr

ibut

ion

t* c

v50m

Re*

c†

Prop

osed

t* c

Fun

ctio

nO

ther

Tha

nSh

ield

s’D

50

m,

mm

sg

m(f

)r

s,kg

/m3

D5

0m

/hc‡

Exp

erim

enta

lCon

ditio

ns

“Med

ium

Mov

emen

t”[K

ram

er,1

935]

orIt

sE

quiv

alen

tG

ilber

t[1

914]

**24

0.05

2††

322

zzz

4.94

,0.

2626

900.

11st

raig

ht,r

ecta

ngul

arflu

me;

plan

ebe

d;Sh

imiz

u[1

989]

side

wal

lcor

rect

ion;

‡‡Su

brou

nded

grai

ns0.

052†

†32

3zz

z4.

94,

0.26

2690

0.16

asab

ove

0.06

9††

372

zzz

4.94

,0.

2626

900.

15as

abov

e0.

048†

†31

0zz

z4.

94,

0.26

2690

0.17

asab

ove

0.05

8††

341

zzz

4.94

,0.

2626

900.

18as

abov

e0.

046†

†30

4zz

z4.

94,

0.26

2690

0.12

asab

ove

0.04

7††

308

zzz

4.94

,0.

2626

900.

16as

abov

e0.

062†

†35

2zz

z4.

94,

0.26

2690

0.18

asab

ove

0.05

332

6zz

z4.

94,

0.26

2690

0.27

asab

ove

0.03

2††

52zz

z1.

71,

0.48

2690

0.09

asab

ove

0.05

7††

69zz

z1.

71,

0.48

2690

0.20

asab

ove

0.04

9††

160

zzz

3.17

,0.

2426

900.

11as

abov

e0.

049†

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1zz

z3.

17,

0.24

2690

0.11

asab

ove

0.06

0††

587

zzz

7.01

,0.

2226

900.

08as

abov

eK

ram

er[1

935]

0.04

38.

2zz

z0.

530.

8127

000.

03st

raig

ht,r

ecta

ngul

arflu

me;

plan

ebe

d;Sh

imiz

u[1

989]

side

wal

lcor

rect

ion;

‡‡w

ell-r

ound

edgr

ains

0.04

88.

7zz

z0.

530.

8127

000.

02as

abov

e0.

039

7.7

zzz

0.53

0.81

2700

0.02

asab

ove

0.04

08.

0zz

z0.

530.

8127

000.

01as

abov

e0.

037

7.3

zzz

0.51

0.74

2700

0.04

asab

ove

0.03

77.

1zz

z0.

510.

7427

000.

02as

abov

e0.

034

7.0

zzz

0.51

0.74

2700

0.02

asab

ove

0.03

06.

5zz

z0.

510.

7427

000.

02as

abov

e0.

035

7.8

zzz

0.55

0.62

2700

0.04

asab

ove

0.03

98.

2zz

z0.

550.

6227

000.

02as

abov

e0.

038

7.9

zzz

0.55

0.62

2700

0.02

asab

ove

0.03

78.

0zz

z0.

550.

6227

000.

01as

abov

eU

SWE

S[1

935]

0.05

29.

5zz

z0.

430.

9526

500.

01st

raig

ht,r

ecta

ngul

arflu

me;

plan

ebe

d;Sh

imiz

u[1

989]

side

wal

lcor

rect

ion;

‡‡su

bang

ular

tosu

brou

nded

rive

rsa

nd0.

044

8.8

zzz

0.43

0.95

2650

0.02

asab

ove

0.05

29.

6zz

z0.

430.

9526

500.

02as

abov

e0.

051

10zz

z0.

450.

6626

500.

01as

abov

e0.

067

12zz

z0.

450.

6626

500.

01as

abov

e0.

043

8.0

zzz

0.48

0.53

2650

0.01

asab

ove,

but

subr

ound

edto

roun

ded

grai

ns0.

048

9.0

zzz

0.48

0.53

2650

0.02

asab

ove

0.04

98.

6zz

z0.

480.

5326

500.

02as

abov

e0.

052

7.8

zzz

0.44

0.82

2650

0.01

sam

eas

abov

e,bu

tan

gula

rto

subr

ound

edgr

ains

0.04

97.

6zz

z0.

440.

8226

500.

02as

abov

e0.

063

8.2

zzz

0.44

0.82

2650

0.02

asab

ove

0.05

36.

6zz

z0.

400.

6926

500.

01as

abov

e,bu

tan

gula

rto

suba

ngul

argr

ains

0.05

36.

7zz

z0.

400.

6926

500.

02as

abov

e0.

059

7.0

zzz

0.40

0.69

2650

0.02

asab

ove

0.04

8††

5.5

zzz

0.34

0.37

2650

0.02

asab

ove,

but

subr

ound

edto

suba

ngul

argr

ains


Tab

le1b

.(C

ontin

ued)

Sour

ceN

ote*

Lab

orat

ory

Mix

ture

Gra

inSi

zeD

istr

ibut

ion

t* c

v50m

Re*

c†

Prop

osed

t* c

Fun

ctio

nO

ther

Tha

nSh

ield

s’D

50

m,

mm

sg

m(f

)r

s,kg

/m3

D5

0m

/hc‡

Exp

erim

enta

lCon

ditio

nsgr

ains

0.07

65.

1zz

z0.

250.

5326

500.

008

asab

ove

0.03

819

6zz

z4.

00.

5626

500.

04as

abov

e0.

042

200

zzz

4.0

0.56

2650

0.05

asab

ove

0.04

521

2zz

z4.

00.

5626

500.

06as

abov

e0.

074†

†2.

4zz

z0.

180.

3226

500.

008

asab

ove,

but

suba

ngul

arto

angu

lar

grai

nsC

hang

[193

9]**

0.06

11.

5zz

z0.

134

2520

0.00

08st

raig

ht,r

ecta

ngul

arflu

me,

and

conv

erge

nt-w

alle

dflu

me;

plan

ebe

d;Sh

imiz

u[1

989]

side

wal

lcor

rect

ion‡

‡

“Gen

eral

Mov

emen

t”[K

ram

er,1

935]

Fro

mK

ram

er[1

935]

Scha

ffern

ak[1

916]

0.03

041

zzz

1.5

0.5

2650

zzz

unre

port

edby

Kra

mer

[193

5],b

utex

peri

men

talp

roce

dure

dire

ctly

com

para

ble

tohi

s[K

ram

er,1

935]

H.K

rey

(unp

ublis

hed

Elb

eex

peri

men

tsre

port

)0.

057

14zz

z0.

600.

9426

50zz

zas

abov

e

0.06

013

zzz

0.57

0.69

2650

zzz

asab

ove

0.05

615

zzz

0.63

0.89

2650

zzz

asab

ove

0.05

313

zzz

0.57

0.71

2650

zzz

asab

ove

0.03

421

zzz

0.92

1.20

2650

zzz

asab

ove,

but

less

com

para

ble

[Kra

mer

,19

35]

0.03

814

zzz

0.67

1.19

2650

zzz

asab

ove

0.06

93.

2zz

z0.

210.

6726

50zz

zas

abov

eH

.Kre

y(u

npub

lishe

dre

port

)0.

038

5.8

zzz

0.38

0.21

2680

zzz

asab

ove

0.03

38.

8zz

z0.

530.

1826

10zz

zas

abov

e0.

025

14zz

z0.

80.

2425

70zz

zas

abov

eSc

hokl

itsch

[191

4]0.

052

476

zzz

6.52

026

00zz

zst

raig

ht,r

ecta

ngul

arflu

me

[Man

tz,1

977]

;sl

ate

grai

ns(?

);ex

peri

men

tal

proc

edur

ele

ssco

mpa

rabl

e[K

ram

er,

1935

]0.

041

206

zzz

4.05

026

00zz

zas

abov

e0.

031

75zz

z2.

260

2600

zzz

asab

ove

0.02

226

zzz

1.24

026

00zz

zas

abov

e0.

019

15zz

z0.

920

2600

zzz

asab

ove

Eng

els

[193

2]0.

061

31zz

z1.

001.

3526

50,

0.02

mea

nder

ing,

larg

e-sc

ale

flum

e[E

ngel

san

dK

ram

er,1

932]

;pla

nebe

d;no

corr

ectio

nfo

rsi

dew

alls

orch

anne

lcu

rvat

ure

(?);

expe

rim

enta

lpro

cedu

rele

ssco

mpa

rabl

e[K

ram

er,1

935]


Oth

erV

isua

lMov

emen

tD

efini

tion

Fro

mO

’Brie

nan

dR

indl

aub

[193

4]an

dM

avis

etal

.[1

935]

Ho

[193

3]0.

015

5zz

z0.

5#

0.25

2620

zzz

stra

ight

,rec

tang

ular

flum

e;pl

ane

bed;

nosi

dew

allc

orre

ctio

n(?

);ri

ver

sand

0.01

49

zzz

0.7

#0.

2526

80zz

zas

abov

e0.

024

20zz

z1.

0#

0.25

2630

zzz

asab

ove

0.02

835

zzz

1.4

#0.

2526

30zz

zas

abov

e0.

030

62zz

z2.

0#

0.25

2620

zzz

asab

ove

0.02

910

2zz

z2.

8#

0.25

2660

zzz

asab

ove

0.02

415

9zz

z4.

0#

0.25

2650

/266

0zz

zas

abov

e,bu

tun

cert

ain

whe

ther

grai

nsar

eri

ver

sand

orcr

ushe

dlim

esto

ne0.

036

194

zzz

4.0

#0.

2526

50/2

660

zzz

asab

ove

0.02

828

7zz

z5.

7?#

0.25

2650

/266

0zz

zas

abov

e0.

028

288

zzz

5.7

#0.

2526

50/2

660

zzz

asab

ove

Fro

mM

avis

etal

.[19

37]

Liu

[193

5]0.

022†

†19

8zz

z4.

30.

3126

600.

03st

raig

ht,r

ecta

ngul

arflu

me;

plan

ebe

d;Sh

imiz

u[1

989]

side

wal

lcor

rect

ion‡

‡0.

026†

†21

3zz

z4.

30.

3126

600.

05as

abov

e0.

024†

†20

7zz

z4.

30.

3126

600.

12as

abov

e0.

032†

†23

5zz

z4.

30.

3126

600.

19as

abov

e0.

019†

†11

0zz

z3.

10.

3326

600.

04as

abov

e0.

028†

†13

5zz

z3.

10.

3326

600.

05as

abov

e0.

029†

†13

7zz

z3.

10.

3326

600.

10as

abov

e0.

037†

†15

6zz

z3.

10.

3326

600.

16as

abov

e0.

024†

†75

zzz

2.2

0.35

2660

0.03

asab

ove

0.02

8††

82zz

z2.

20.

3526

600.

05as

abov

e0.

029†

†82

zzz

2.2

0.35

2660

0.10

asab

ove

0.03

5††

90zz

z2.

20.

3526

600.

17as

abov

e0.

022†

†37

zzz

1.4

0.31

2660

0.03

asab

ove

0.02

3††

37zz

z1.

40.

3126

600.

06as

abov

e0.

026†

†39

zzz

1.4

0.31

2660

0.11

asab

ove

0.03

0††

43zz

z1.

40.

3126

600.

20as

abov

e0.

021†

†15

8zz

z3.

80.

3626

600.

03as

abov

e0.

023†

†16

5zz

z3.

80.

3626

600.

06as

abov

e0.

033†

†19

8zz

z3.

80.

3626

600.

09as

abov

e0.

034†

†20

4zz

z3.

80.

3626

600.

17as

abov

e0.

026†

†53

zzz

1.7

0.45

2660

0.03

asab

ove

0.02

8††

55zz

z1.

70.

4526

600.

05as

abov

e0.

026†

†53

zzz

1.7

0.45

2660

0.11

asab

ove

0.03

0‡‡

57zz

z1.

70.

4526

600.

20as

abov

eW

hite

[194

0]25

0.33

8††

0.04

3zz

z0.

21;

u?zz

z0.

05st

raig

ht,r

ecta

ngul

arflu

me;

plan

ebe

d;no

side

wal

lcor

rect

ion;

oilfl

uid

med

ium

;en

tirel

yla

min

arflo

w0.

168†

†0.

30zz

z0.

90;

u?zz

z0.

11as

abov

e0.

180

2.4

zzz

0.12

2;

uzz

zzz

zco

nver

gent

-wal

led

nozz

le;p

lane

bed;

nosi

dew

allc

orre

ctio

n;w

ater

fluid

med

ium

;lam

inar

boun

dary

laye

rw

ithin

stea

dy“i

nvis

cid”

flow

0.11

933

zzz

0.90

;u

2600

zzz

asab

ove,

but

with

turb

ulen

tbo

unda

ryla

yer

0.10

148

0zz

z5.

6;

u26

00zz

zas

abov

e0.

064

35zz

z0.

71m

7900

zzz

asab

ove,

but

with

stee

lsho

tgr

ains


Tab

le1b

.(C

ontin

ued)

Sour

ceN

ote*

Lab

orat

ory

Mix

ture

Gra

inSi

zeD

istr

ibut

ion

t* c

v50m

Re*

c†

Prop

osed

t* c

Fun

ctio

nO

ther

Tha

nSh

ield

s’D

50

m,

mm

sg

m(f

)r

s,kg

/m3

D5

0m

/hc‡

Exp

erim

enta

lCon

ditio

ns

0.09

880

zzz

0.90

;u

2100

zzz

asab

ove,

but

with

natu

ralg

rain

san

dflu

idm

ediu

mof

air

0.10

212

80zz

z5.

6;

u21

00zz

zas

abov

eM

eyer

-Pet

eran

dM

ulle

r[1

948]

ii0.

033†

†13

3t* c v

50m

5[t

c v50

m(Q

b/Q

)(n g

/nb)

3/2 ]/

[(r s

2r)

gD50

m]

3.2

,0.

12?

2680

0.01

stra

ight

,rec

tang

ular

flum

e;pl

ane

bed

orve

rylo

wam

plitu

debe

dfo

rms;

form

drag

and

side

wal

lcor

rect

ion;

roun

ded

grai

ns0.

032†

†10

1as

abov

e2.

7,

0.15

?26

800.

02as

abov

e0.

037†

†18

3as

abov

e3.

8,

0.16

?26

800.

02as

abov

e0.

030†

†15

1as

abov

e3.

6,

0.15

?26

800.

01as

abov

e0.

039†

†17

8as

abov

e3.

66,

0.13

?26

800.

03as

abov

e0.

040†

†17

9as

abov

e3.

66,

0.13

?26

800.

06as

abov

e0.

037†

†60

9as

abov

e8.

5,

0.19

?26

800.

06as

abov

e0.

050†

†57

5as

abov

e7.

4,

0.28

?26

800.

10as

abov

e0.

047†

†68

6as

abov

e8.

5,

0.23

?26

800.

10as

abov

e0.

040†

†14

2as

abov

e3.

14,

0.15

?26

800.

13as

abov

e0.

033†

†59

asab

ove

1.86

,0.

33?

2680

0.01

asab

ove,

but

with

angu

lar

grai

ns0.

025†

†11

3as

abov

e3.

14,

0.15

?26

800.

01as

abov

e0.

033†

†12

7as

abov

e3.

1,

0.17

?26

800.

02as

abov

e0.

038†

†17

6as

abov

e3.

66,

0.13

?26

800.

03as

abov

e0.

040†

†13

7as

abov

e3.

06,

0.19

?26

800.

02as

abov

e0.

040†

†10

6as

abov

e2.

58,

0.18

?26

800.

02as

abov

e0.

048†

†11

9as

abov

e2.

61,

0.20

?26

800.

05as

abov

e0.

050†

†23

3as

abov

e4.

04,

0.22

?26

800.

05as

abov

e0.

043†

†24

1as

abov

e4.

34,

0.22

?26

800.

10as

abov

e0.

043†

†81

asab

ove

2.10

,0.

16?

2680

0.11

asab

ove

0.04

3††

147

asab

ove

3.14

,0.

15?

2680

0.10

asab

ove

Wol

man

and

Bru

sh[1

961]

0.02

0††

9zz

z0.

670.

26zz

z0.

05st

raig

ht,r

ecta

ngul

arflu

me;

plan

ebe

d;er

odib

leba

nks;

Shim

izu

[198

9]si

dew

all

corr

ectio

n‡‡

0.02

4††

10zz

z0.

670.

26zz

z0.

07as

abov

e0.

030†

†12

zzz

0.67

0.26

zzz

0.07

asab

ove

0.02

4††

11zz

z0.

670.

26zz

z0.

12as

abov

e0.

048†

†86

zzz

20.

5zz

z0.

16as

abov

e0.

049†

†91

zzz

20.

5zz

z0.

06as

abov

e0.

052†

†94

zzz

20.

5zz

z0.

12as

abov

e0.

036†

†80

zzz

20.

5zz

z0.

05as

abov

e0.

044†

†86

zzz

20.

5zz

z0.

09as

abov

e0.

030†

†71

zzz

20.

5zz

z0.

04as

abov

eR

audk

ivi[

1963

]§§

0.03

6††

6.0

zzz

0.40

0.42

2600

0.00

3st

raig

ht,r

ecta

ngul

arflu

me;

plan

ebe

d;V

anon

iand

Bro

oks

[195

7]si

dew

all

corr

ectio

n\,¶

Van

oni[

1964

]§§

0.09

7††

1.3

zzz

0.10

20.

1926

500.

0009

stra

ight

rect

angu

lar

flum

e;pl

ane

bed;

side

wal

lcor

rect

ion

impl

icit;

quar

tzgr

ains

0.12

6††

1.5

zzz

0.10

20.

1926

500.

001

asab

ove

0.12

0††

1.5

zzz

0.10

20.

1926

500.

001

asab

ove


0.22

6††

0.43

zzz

0.03

70.

2524

900.

0005

asab

ove,

but

with

glas

sbe

ads

0.22

8††

0.43

zzz

0.03

70.

2524

900.

0005

asab

ove

Nei

ll[1

967]

**0.

030†

†32

8zz

z6.

2;

0.48

?25

400.

04st

raig

ht,r

ecta

ngul

arflu

me;

plan

ebe

d;Sh

imiz

u[1

989]

side

wal

lcor

rect

ion‡

‡0.

032†

†34

0zz

z6.

2;

0.48

?25

400.

04as

abov

e0.

030†

†32

7zz

z6.

2;

0.48

?25

400.

05as

abov

e0.

041†

†38

2zz

z6.

2;

0.48

?25

400.

05as

abov

e0.

038†

†37

2zz

z6.

2;

0.48

?25

400.

06as

abov

e0.

032†

†33

9zz

z6.

2;

0.48

?25

400.

07as

abov

e0.

036†

†35

9zz

z6.

2;

0.48

?25

400.

07as

abov

e0.

032†

†33

7zz

z6.

2;

0.48

?25

400.

11as

abov

e0.

023†

†28

9zz

z6.

2;

0.48

?25

400.

12as

abov

e0.

050†

†67

5zz

z8.

5;

0.48

?25

200.

05as

abov

e0.

056†

†71

3zz

z8.

5;

0.48

?25

200.

07as

abov

e0.

037†

†58

4zz

z8.

5;

0.48

?25

200.

14as

abov

e0.

039

595

zzz

8.5

;0.

48?

2520

0.28

asab

ove

0.03

7††

2099

zzz

20.0

;0.

48?

2520

0.11

asab

ove

0.03

1††

1938

zzz

20.0

;0.

48?

2520

0.14

asab

ove

0.03

6††

2066

zzz

20.0

;0.

48?

2520

0.18

asab

ove

0.03

119

35zz

z20

.0;

0.48

?25

200.

23as

abov

e0.

027

1802

zzz

20.0

;0.

48?

2520

0.33

asab

ove

0.03

219

50zz

z20

.0;

0.48

?25

200.

39as

abov

e0.

031

1928

zzz

20.0

;0.

48?

2520

0.55

asab

ove

0.03

7††

259

zzz

5.0

024

900.

03as

abov

e,bu

tw

ithgl

ass

grai

ns0.

039†

†26

6zz

z5.

00

2490

0.04

asab

ove

0.04

2††

278

zzz

5.0

024

900.

04as

abov

e0.

038†

†26

4zz

z5.

00

2490

0.04

asab

ove

0.03

7††

261

zzz

5.0

024

900.

05as

abov

e0.

027†

†22

1zz

z5.

00

2490

0.08

asab

ove

Rat

hbur

nan

dG

uy[1

967]

0.02

63.

4zz

z0.

31.

2626

500.

005

stra

ight

,rec

tang

ular

flum

e;pl

ane

bed

Van

onia

ndB

rook

s[1

957]

side

wal

lco

rrec

tion

\,¶W

ard

[196

8]\\

0.10

221

zzz

0.70

,0.

526

40zz

zco

nver

gent

-wal

led

flum

e;pl

ane

bed;

sidew

all

corr

ectio

nim

plic

it;w

ater

fluid

med

ium

0.08

456

zzz

1.43

,0.

526

40zz

zas

abov

e0.

092

119

zzz

2.29

,0.

526

40zz

zas

abov

e0.

110

25zz

z0.

70,

0.5

3100

zzz

asab

ove,

butw

ithta

coni

tegr

ains

0.07

585

zzz

1.80

,0.

531

00zz

zas

abov

e0.

079

125

zzz

2.29

,0.

531

00zz

zas

abov

e0.

098

21zz

z0.

70,

0.5

2710

zzz

asab

ove

butw

ithD

ougl

assa

nd#

20.

086

56zz

z1.

43,

0.5

2560

zzz

asab

ove

butw

ithD

ougl

assa

nd#

10.

075

73zz

z1.

80,

0.5

2560

zzz

asab

ove

0.03

615

9zz

z2.

04,

0.5

11,3

40zz

zas

abov

e,bu

twith

lead

shot

grai

ns0.

489

577

zzz

5.03

,0.

57,

830

zzz

asab

ove,

butw

ithst

eels

hotg

rain

s0.

170

0.63

zzz

0.24

,0.

526

40zz

zco

nver

gent

-wal

led

flum

e;pl

ane

bed;

sidew

all

corr

ectio

nim

plic

it;oi

lflui

dm

ediu

m;f

ully

lam

inar

flow

0.10

32.

4zz

z0.

70,

0.5

2640

zzz

asab

ove

0.06

311

zzz

2.29

,0.

526

40zz

zas

abov

e0.

066

7.8

zzz

1.80

,0.

526

40zz

zas

abov

e0.

148

1.2

zzz

0.37

,0.

529

50zz

zas

abov

e,bu

twith

glas

sbe

ads

0.36

80.

91zz

z0.

52,

0.5

1050

zzz

asab

ove,

butw

ithst

yren

egr

ains

0.38

72.

2zz

z0.

91,

0.5

1050

zzz

asab

ove

0.10

42.

7zz

z0.

70,

0.5

3100

zzz

asab

ove,

butw

ithta

coni

tegr

ains


Tab

le1b

.(C

ontin

ued)

Sour

ceN

ote*

Lab

orat

ory

Mix

ture

Gra

inSi

zeD

istr

ibut

ion

t* c

v50m

Re*

c†

Prop

osed

t* c

Fun

ctio

nO

ther

Tha

nSh

ield

s’D

50

m,

mm

sg

m(f

)r

s,kg

/m3

D5

0m

/hc‡

Exp

erim

enta

lCon

ditio

ns

0.06

18.

4zz

z1.

80,

0.5

3100

zzz

asab

ove

0.05

111

zzz

2.29

,0.

531

00zz

zas

abov

e0.

114

2.6

zzz

0.70

,0.

527

10zz

zas

abov

e,bu

tw

ithD

ougl

assa

nd#

20.

065

7.9

zzz

1.80

,0.

527

10zz

zas

abov

e0.

081

6.1

zzz

1.43

,0.

525

60zz

zas

abov

e,bu

tw

ithD

ougl

assa

nd#

10.

620

9.7

zzz

3.05

,0.

593

0zz

zas

abov

e,bu

tw

ithpo

lyet

hyle

negr

ains

Whi

te[1

970]

0.05

3††

36zz

z2.

2;

u10

50,

0.2

stra

ight

,rec

tang

ular

flum

e;pl

ane

bed;

side

wal

lcor

rect

ion;

poly

styr

ene

grai

ns0.

037†

†53

zzz

2;

u15

40,

0.2

asab

ove,

but

with

PVC

grai

ns0.

055†

†1.

7zz

z0.

137

;u

2600

,0.

2as

abov

e,bu

tw

ithgl

ass

ballo

tinig

rain

s0.

073†

†0.

94zz

z0.

088

;u

2600

,0.

2as

abov

e0.

055†

†2.

2zz

z0.

17;

u25

20,

0.2

asab

ove,

but

with

natu

ralg

rain

s0.

058†

†1.

9zz

z0.

153

;u

2520

,0.

2as

abov

e0.

071†

†1.

6zz

z0.

133

;u

2520

,0.

2as

abov

e0.

066†

†0.

92zz

z0.

093

;u

2520

,0.

2as

abov

e0.

073†

†0.

73zz

z0.

077

;u

2520

,0.

2as

abov

e0.

125†

†0.

42zz

z0.

044

;u

2520

,0.

2as

abov

e0.

102†

†0.

21zz

z0.

030

;u

2520

,0.

2as

abov

e0.

103†

†0.

20zz

z0.

029

;u

2520

,0.

2as

abov

e0.

146†

†0.

23zz

z0.

028

;u

2520

,0.

2as

abov

e0.

110†

†0.

16zz

z0.

024

;u

2520

,0.

2as

abov

e0.

112†

†0.

26zz

z0.

033

;u

2550

,0.

2as

abov

e,bu

tw

ithcr

ushe

dsi

lica

grai

ns0.

151†

†0.

10zz

z0.

016

;u

2550

,0.

2as

abov

e0.

037†

†16

zzz

2.2

;u

1050

,0.

2as

abov

e,bu

tw

ithpo

lyst

yren

egr

ains

inoi

l;fu

llyla

min

arflo

w(?

)0.

034†

†40

zzz

2;

u15

40,

0.2

asab

ove,

but

with

PVC

grai

ns0.

143†

†0.

20zz

z0.

088

;u

2600

,0.

2as

abov

e,bu

tw

ithgl

ass

ballo

tinig

rain

s0.

132†

†0.

32zz

z0.

133

;u

2520

,0.

2as

abov

e,bu

tw

ithna

tura

lgra

ins

0.12

2††

0.19

zzz

0.09

3;

u25

20,

0.2

asab

ove

0.16

6††

0.16

zzz

0.07

7;

u25

20,

0.2

asab

ove

0.21

8††

0.08

6zz

z0.

046

;u

2520

,0.

2as

abov

e0.

254†

†0.

055

zzz

0.03

3;

u25

20,

0.2

asab

ove

0.28

8††

0.04

8zz

z0.

030

;u

2520

,0.

2as

abov

e0.

219†

†0.

033

zzz

0.02

5;

u25

20,

0.2

asab

ove

Gra

ss[1

970]

\\26

0.14

1††

(0.1

74)

0.87

(0.9

7)zz

z0.

090

,0.

2426

50,

0.02

?st

raig

ht,r

ecta

ngul

arflu

me;

plan

ebe

d;si

dew

allc

orre

ctio

nim

plic

it0.

131†

†(0

.154

)0.

84(0

.91)

zzz

0.09

0,

0.24

2650

,0.

02?

asab

ove

0.11

0††

(0.1

31)

1.6

(1.7

)zz

z0.

115

,0.

1326

50,

0.02

?as

abov

e0.

086†

†(0

.095

)1.

8(1

.9)

zzz

0.13

8,

0.13

2650

,0.

03?

asab

ove

0.06

9††

(0.0

93)

2.1

(2.5

)zz

z0.

165

,0.

1326

50,

0.03

?as

abov

e0.

072†

†(0

.079

)2.

8(2

.9)

zzz

0.19

5,

0.11

2650

,0.

04?

asab

ove

0.05

8(0

.091

)1.

6(2

.0)

zzz

0.14

3,

0.74

2650

,0.

03?

asab

ove

Ster

nber

g[1

971]

\\,¶

¶0.

094

5.9

zzz

0.42

1.09

zzz

,2

310

25

sand

y,ri

pple

d,m

arin

ech

anne

l(P

icke

ring

Pass

age)

;for

mdr

agco

rrec

tion

sim

ilar

toth

atin

note

30.

048

6.2

zzz

0.50

1.0

zzz

,2

310

25

asab

ove

0.03

85.

5zz

z0.

501.

0zz

z,

23

102

5as

abov

e


0.04

13.

1zz

z0.

331.

12zz

z!

0.20

sand

ym

arin

ech

anne

l(C

lovi

sPa

ssag

e)w

ithra

ndom

roug

hnes

sel

emen

ts;f

orm

drag

corr

ectio

nsi

mila

rto

that

inno

te3

Eve

rts[1

973]

\\0.

023†

†14

5zz

z3.

57,

0.25

?26

500.

06st

raig

ht,r

ecta

ngul

arflu

me;

plan

ebe

d;W

illia

ms

[197

0]si

dew

allc

orre

ctio

n0.

027†

†15

7zz

z3.

57,

0.25

?26

500.

08as

abov

e0.

029†

†16

2zz

z3.

57,

0.25

?26

500.

19as

abov

e0.

023†

†51

zzz

1.79

,0.

25?

2650

0.03

asab

ove

0.02

5††

54zz

z1.

79,

0.25

?26

500.

04as

abov

e0.

019†

†47

zzz

1.79

,0.

25?

2650

0.06

asab

ove

0.01

7††

44zz

z1.

79,

0.25

?26

500.

08as

abov

e0.

017†

†16

zzz

0.89

5,

0.25

?26

500.

01as

abov

e0.

021†

†17

zzz

0.89

5,

0.25

?26

500.

02as

abov

e0.

018†

†16

zzz

0.89

5,

0.25

?26

500.

02as

abov

e0.

020†

†17

zzz

0.89

5,

0.25

?26

500.

04as

abov

e0.

020†

†7.

2zz

z0.

508

,0.

25?

2650

0.00

6as

abov

e0.

024†

†8.

0zz

z0.

508

,0.

25?

2650

0.00

9as

abov

e0.

022†

†7.

7zz

z0.

508

,0.

25?

2650

0.01

asab

ove

0.02

5††

8.1

zzz

0.50

8,

0.25

?26

500.

02as

abov

e0.

029†

†5.

2zz

z0.

359

,0.

25?

2650

0.00

5as

abov

e0.

026†

†4.

9zz

z0.

359

,0.

25?

3650

0.00

6as

abov

e0.

027†

†5.

1zz

z0.

359

,0.

25?

2650

0.00

9as

abov

e0.

023†

†4.

6zz

z0.

359

,0.

25?

2650

0.04

asab

ove

0.03

2††

3.3

zzz

0.25

4,

0.25

?26

500.

003

asab

ove

0.03

5††

3.4

zzz

0.25

4,

0.25

?26

500.

004

asab

ove

0.04

1††

3.7

zzz

0.25

4,

0.25

?26

500.

008

asab

ove

0.03

9††

2.1

zzz

0.18

,0.

25?

2650

0.00

2as

abov

e0.

046†

†2.

3zz

z0.

18,

0.25

?26

500.

005

asab

ove

0.05

2††

1.5

zzz

0.12

7,

0.25

?26

500.

002

asab

ove

0.05

2††

1.5

zzz

0.12

7,

0.25

?26

500.

002

asab

ove

0.04

0††

1.3

zzz

0.12

7,

0.25

?26

500.

01as

abov

e0.

056†

†3.

9zz

z0.

18,

0.25

?47

000.

003

asab

ove,

but

with

ilmen

itegr

ains

0.05

6††

3.9

zzz

0.18

,0.

25?

4700

0.00

5as

abov

e0.

062†

†4.

1zz

z0.

18,

0.25

?47

000.

006

asab

ove

0.05

8††

2.3

zzz

0.12

7,

0.25

?47

000.

002

asab

ove

0.06

0††

2.4

zzz

0.12

7,

0.25

?47

000.

002

asab

ove

0.05

9††

2.4

zzz

0.12

7,

0.25

?47

000.

003

asab

ove

0.06

3††

2.4

zzz

0.12

7,

0.25

?47

000.

004

asab

ove

0.05

7††

1.4

zzz

0.09

,0.

25?

4700

0.00

1as

abov

e0.

070†

†1.

5zz

z0.

09,

0.25

?47

000.

002

asab

ove

0.05

8††

1.4

zzz

0.09

,0.

25?

4700

0.00

2as

abov

e0.

081†

†1.

6zz

z0.

09,

0.25

?47

000.

004

asab

ove

Fern

ande

zL

uque

and

van

Bee

k[1

976]

\\28

0.03

9††

18zz

z0.

90.

2526

400.

01st

raig

ht,r

ecta

ngul

arflu

me;

plan

ebe

d;si

dew

allc

orre

ctio

nim

plic

it;be

dsl

ope

of08

0.03

8††

48zz

z1.

80.

2526

400.

02as

abov

e0.

047†

†12

7zz

z3.

30.

2526

400.

04as

abov

e0.

043†

†74

zzz

1.8

0.25

4580

0.02

asab

ove,

but

with

mag

netit

egr

ains

0.03

9††

16zz

z1.

50.

2513

400.

01as

abov

e,bu

tw

ithw

alnu

t(s

hell?

)gr

ains

0.03

116

zzz

0.9

0.25

2640

0.01

stra

ight

,rec

tang

ular

flum

e;pl

ane

bed;

side

wal

lcor

rect

ion

impl

icit;

bed

slop

eis

128

0.03

043

zzz

1.8

0.25

2640

0.02

asab

ove

0.03

711

4zz

z3.

30.

2526

400.

04as

abov

e


Tab

le1b

.(C

ontin

ued)

Sour

ceN

ote*

Lab

orat

ory

Mix

ture

Gra

inSi

zeD

istr

ibut

ion

t* c

v50m

Re*

c†

Prop

osed

t* c

Fun

ctio

nO

ther

Tha

nSh

ield

s’D

50

m,

mm

sg

m(f

)r

s,kg

/m3

D5

0m

/hc‡

Exp

erim

enta

lCon

ditio

ns

0.03

567

zzz

1.8

0.25

4580

0.02

asab

ove,

but

with

mag

netit

egr

ains

0.03

114

zzz

1.5

0.25

1340

0.01

asab

ove,

but

with

wal

nut

(she

ll?)

grai

ns0.

027

14zz

z0.

90.

2526

400.

01st

raig

ht,r

ecta

ngul

arflu

me;

plan

ebe

d;si

dew

allc

orre

ctio

nim

plic

it;be

dsl

ope

is18

80.

026

39zz

z1.

80.

2526

400.

02as

abov

e0.

031

112

zzz

3.3

0.25

2640

0.04

asab

ove

0.02

859

zzz

1.8

0.25

4580

0.02

asab

ove,

but

with

mag

netit

egr

ains

0.02

613

zzz

1.5

0.25

1340

0.01

asab

ove,

but

with

wal

nut

(she

ll?)

grai

ns0.

025

14zz

z0.

90.

2526

400.

01st

raig

ht,r

ecta

ngul

arflu

me;

plan

ebe

d;si

dew

allc

orre

ctio

nim

plic

it;be

dsl

ope

is22

80.

021

36zz

z1.

80.

2526

400.

02as

abov

e0.

027

99zz

z3.

30.

2526

400.

04as

abov

e0.

023

55zz

z1.

80.

2545

800.

02as

abov

e,bu

tw

ithm

agne

tite

grai

ns0.

024

12zz

z1.

50.

2513

400.

01as

abov

e,bu

tw

ithw

alnu

t(s

hell?

)gr

ains

Fro

mM

antz

[197

7]Sc

hokl

itsch

[191

4]**

0.06

617

6zz

z3.

060

2700

zzz

stra

ight

,rec

tang

ular

flum

e;pl

ane

bed;

slat

egr

ains

;no

side

wal

lcor

rect

ion

(?)

0.04

137

zzz

1.24

027

00zz

zas

abov

eH

o[1

939]

**0.

018

265

zzz

6.71

0.56

2660

#0.

17st

raig

ht,r

ecta

ngul

arflu

me;

plan

ebe

d;sl

ate

grai

ns;n

osi

dew

allc

orre

ctio

n(?

)0.

025†

†22

5zz

z5.

710.

4626

60#

0.12

asab

ove

0.02

110

2zz

z3.

261.

1327

00#

0.09

asab

ove

0.03

465

zzz

2.21

0.90

2490

#0.

20as

abov

e0.

024

33zz

z1.

630.

9824

50#

0.05

asab

ove

Man

tz[1

975]

0.09

4††

0.70

zzz

0.06

6;

0.20

2650

0.00

07st

raig

ht,r

ecta

ngul

arflu

me;

plan

ebe

d;Sh

imiz

u[1

989]

side

wal

lcor

rect

ion;

‡‡qu

artz

grai

ns0.

121†

†0.

79zz

z0.

066

;0.

2026

500.

0008

asab

ove

0.12

7††

0.81

zzz

0.06

6;

0.20

2650

0.00

1as

abov

e0.

122†

†0.

79zz

z0.

066

;0.

2026

500.

002

asab

ove

0.12

6††

0.81

zzz

0.06

6;

0.20

2650

0.00

3as

abov

e0.

155†

†0.

48zz

z0.

045

;0.

2026

500.

0004

asab

ove

0.14

7††

0.47

zzz

0.04

5;

0.20

2650

0.00

06as

abov

e0.

134†

†0.

45zz

z0.

045

;0.

2026

500.

0008

asab

ove

0.12

7††

0.44

zzz

0.04

5;

0.20

2650

0.00

1as

abov

e0.

140†

†0.

46zz

z0.

045

;0.

2026

500.

002

asab

ove

0.14

0††

0.27

zzz

0.03

0;

0.20

2650

0.00

03as

abov

e0.

133†

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27zz

z0.

030

;0.

2026

500.

0004

asab

ove

0.16

4††

0.29

zzz

0.03

0;

0.20

2650

0.00

05as

abov

e0.

161†

†0.

29zz

z0.

030

;0.

2026

500.

0008

asab

ove

0.16

5††

0.28

zzz

0.03

0;

0.20

2650

0.00

1as

abov

e0.

198†

†0.

12zz

z0.

015

;0.

2026

500.

0002

asab

ove

0.07

0††

0.77

zzz

0.07

6;

0.2?

2740

#0.

004

asab

ove,

but

with

mic

aceo

usfla

kes*

*an

dno

side

wal

lcor

rect

ion

0.06

2††

0.71

zzz

0.07

6;

0.2?

2740

#0.

004

asab

ove


0.05

5††

0.39

zzz

0.05

3;

0.2?

2740

#0.

003

asab

ove

0.07

0††

0.44

zzz

0.05

3;

0.2?

2740

#0.

003

asab

ove

0.06

1††

0.29

zzz

0.04

2;

0.2?

2740

#0.

002

asab

ove

0.06

8††

0.30

zzz

0.04

2;

0.2?

2740

#0.

002

asab

ove

0.06

3††

0.20

zzz

0.03

3;

0.2?

2740

#0.

002

asab

ove

0.07

6††

0.22

zzz

0.03

3;

0.2?

2740

#0.

002

asab

ove

0.09

2††

0.15

zzz

0.02

4$

0.2?

2740

#0.

001

asab

ove

0.08

7††

0.15

zzz

0.02

4;

0.2?

2740

#0.

001

asab

ove

0.14

0††

0.10

zzz

0.01

6;

0.2?

2740

#0.

001

asab

ove

Wim

bush

and

Les

ht[1

979]

\\,¶

¶27

0.03

02.

4zz

z0.

35#

0.84

2930

(?)–

1100

53

102

7sa

ndy,

ripp

led,

mar

ine

chan

nel(

Stra

ight

sof

Flo

rida

);fo

rmdr

agco

rrec

tion

sim

ilar

toth

atin

note

3Y

alin

and

Kar

ahan

[197

9]0.

038†

†25

zzz

1.00

,0.

34–,

0.54

?26

500.

16st

raig

ht,r

ecta

ngul

arflu

me;

plan

ebe

d;si

dew

allc

orre

ctio

nim

plic

it?0.

030†

†9.

4zz

z0.

56,

0.34

–,0.

54?

2650

0.12

asab

ove

0.11

3††

1.5

zzz

0.10

,0.

34–,

0.54

?26

500.

02as

abov

e0.

036†

†6.

1zz

z0.

40,

0.34

–,0.

54?

2650

0.09

asab

ove

0.05

3††

2.5

zzz

0.19

,0.

34–,

0.54

?26

500.

03as

abov

e0.

079†

†1.

8zz

z0.

140

2500

0.03

asab

ove,

but

with

glas

sbe

adgr

ains

0.17

8††

0.13

zzz

1.00

,0.

34–,

0.54

2650

0.09

asab

ove,

but

with

natu

ralg

rain

sin

aw

ater

-gl

ycer

inm

ixtu

rean

dfu

llyla

min

arflo

w0.

156†

†0.

30zz

z1.

88,

0.34

–,0.

5426

500.

24as

abov

e0.

135†

†0.

55zz

z2.

86,

0.34

–,0.

5426

500.

38as

abov

e0.

172†

†0.

054

zzz

0.56

,0.

34–,

0.54

2650

0.04

asab

ove

0.11

0††

0.78

zzz

1.00

,0.

34–,

0.54

2650

0.10

asab

ove

0.09

2††

1.9

zzz

1.88

,0.

34–,

0.54

2650

0.30

asab

ove

0.14

1††

0.38

zzz

0.56

,0.

34–,

0.54

2650

0.04

asab

ove

0.08

6††

3.4

zzz

2.86

,0.

34–,

0.54

2650

0.41

asab

ove

0.09

1††

3.4

zzz

2.86

,0.

34–,

0.54

2650

0.25

asab

ove

0.13

4††

0.86

zzz

1.00

,0.

34–,

0.54

2650

0.07

asab

ove

0.14

3††

0.38

zzz

0.56

,0.

34–,

0.54

2650

0.03

asab

ove

0.11

0††

2.0

zzz

1.88

,0.

34–,

0.54

2650

0.16

asab

ove

0.14

0††

0.80

zzz

0.56

,0.

34–,

0.54

2650

0.04

asab

ove

0.06

9††

6.4

zzz

2.86

,0.

34–,

0.54

2650

0.31

asab

ove

0.08

6††

3.8

zzz

1.88

,0.

34–,

0.54

2650

0.18

asab

ove

0.10

6††

1.6

zzz

1.00

,0.

34–,

0.54

2650

0.09

asab

ove

Iked

a[1

982]

0.02

0††

8.7

zzz

0.42

0.36

zzz

0.00

6la

tera

llytil

ting

win

dtu

nnel

;str

aigh

t,re

ctan

gula

rw

alls

;pla

nebe

d;A

dach

i[19

62]

side

wal

lcor

rect

ion

0.04

7††

72zz

z1.

30.

31zz

z0.

02as

abov

eYo

ung

and

Man

n[1

985]

¶¶28

0.03

02.

5zz

z0.

361.

815

001.

53

102

5st

raig

ht,r

ecta

ngul

arse

aflu

me;

plan

ebe

dor

low

-am

plitu

debe

dfo

rms;

side

wal

lco

rrec

tion

impl

icit;

skel

etal

carb

onat

egr

ains

0.02

84.

3zz

z0.

53.

2.1

1500

2.3

310

25

asab

ove

0.10

91.

6zz

z0.

18,

2.0

1500

1.4

310

25

asab

ove

0.07

51.

5zz

z0.

19,

2.1

1500

7.3

310

26

asab

ove

0.02

60.

58zz

z0.

14,

2.0

1500

5.3

310

26

asab

ove

0.10

51.

4zz

z0.

16,

2.1

1500

5.3

310

26

asab

ove

0.05

92.

0zz

z0.

25,

2.2

1500

1.0

310

25

asab

ove

0.07

03.

6zz

z0.

352.

015

001.

53

102

5as

abov

e0.

047

3.0

zzz

0.35

2.2

1500

2.3

310

25

asab

ove


Tab

le1b

.(C

ontin

ued)

Sour

ceN

ote*

Lab

orat

ory

Mix

ture

Gra

inSi

zeD

istr

ibut

ion

t* c

v50m

Re*

c†

Prop

osed

t* c

Fun

ctio

nO

ther

Tha

nSh

ield

s’D

50

m,

mm

sg

m(f

)r

s,kg

/m3

D5

0m

/hc‡

Exp

erim

enta

lCon

ditio

ns

Pra

ger

etal

.[19

96]¶

¶0.

025†

†7.

1zz

z0.

5#

0.5?

2660

0.04

stra

ight

,rec

tang

ular

flum

e;pl

ane

bed;

Shim

izu

[198

9]si

dew

allc

orre

ctio

n;‡‡

wel

l-ro

unde

dqu

artz

grai

ns0.

021†

†6.

6zz

z0.

5#

0.5?

2770

0.04

asab

ove,

but

with

ooid

grai

ns0.

025†

†7.

3zz

z0.

5#

0.5?

2730

0.04

asab

ove,

but

with

roun

ded,

mix

edca

rbon

ate

and

terr

igen

ous

grai

ns0.

022†

†6.

2zz

z0.

5#

0.5?

2500

0.04

asab

ove,

butw

ithpl

aty,

skel

etal

carb

onat

egr

ains

0.02

011

zzz

0.7

0.80

2700

?0.

05as

abov

e0.

017

19zz

z1.

10.

9326

70?

0.09

asab

ove

Not

eth

atsy

mbo

lsfo

rsi

mila

rfo

otno

tes

may

bedi

ffer

enti

nT

able

s1a

–1e.

See

nota

tion

sect

ion

for

sym

bols

notp

revi

ousl

yde

fined

inte

xt.S

eere

spec

tive

appe

ndix

note

sfo

rva

lues

inpa

rent

hese

s.H

ere

‘‘u’’

deno

tes

unifo

rmgr

ain

size

s(s

g#

0.5)

and

‘‘m’’

deno

tes

mix

edgr

ain

size

s(s

g.

0.5)

,whe

res

gis

the

grap

hic

stan

dard

devi

atio

nde

fined

as(f

842

f16

)/2

[Fol

k,19

74].

*See

appe

ndix

.†R

e*c

5u

* cD

50/n

.W

here

unre

port

edby

aso

urce

,Re*

cva

lues

are

back

-cal

cula

ted

from

avai

labl

eda

ta(s

eefo

otno

teke

yed

to†,

Tab

le1a

).‡W

here

unre

port

edby

aso

urce

,hc

valu

esar

eba

ck-c

alcu

late

dfr

omcr

itica

ldep

th-s

lope

prod

ucts

usin

gre

port

edda

ta(s

eefo

otno

teke

yed

to‡,

Tab

le1a

).§H

ere

we

desc

ribe

grai

npr

otru

sion

inte

rms

ofpr

ojec

tion

and

expo

sure

[sen

suK

irchn

eret

al.,

1990

].\S

idew

allc

orre

ctio

nfo

rth

edi

ffer

ence

inw

alla

ndbe

dgr

ain

roug

hnes

sbu

tno

tfo

rth

epr

oxim

ityof

wal

ls(i

.e.,

W/h

).¶U

seof

the

aver

age

velo

city

inth

eE

inst

ein

[194

2;19

50],

John

son

[194

2]an

dV

anon

iand

Bro

oks

[195

7]eq

uatio

nslik

ely

over

estim

ates

t9(s

eeno

te3)

.**

Rep

orte

dda

taar

ew

ithre

spec

tto

the

mea

nno

min

algr

ain

diam

eter

.Nom

inal

diam

eter

sar

eas

sum

edeq

uiva

lent

toin

term

edia

tegr

ain

diam

eter

s[C

uian

dK

omar

,198

4].M

ean

and

med

ian

size

sar

esi

mila

rfo

rne

ar-u

nifo

rmse

dim

ent.

††U

sed

inPl

ate

1.‡‡

Side

wal

lcor

rect

ion

appl

ied

bycu

rren

tau

thor

s.Sh

imiz

u’s

[198

9]co

rrec

tion

acco

unts

for

both

the

diff

eren

cein

wal

land

bed

grai

nro

ughn

ess

and

the

prox

imity

ofw

alls

(i.e

.,W

/h).

We

assu

med

abe

dgr

ain

roug

hnes

s10

0tim

esgr

eate

rth

anw

allr

ough

ness

for

smoo

thflu

me

wal

ls.

§§R

epor

ted

data

are

with

resp

ect

toth

ege

omet

ric

mea

ngr

ain

size

.\\

Rep

orte

dda

taar

ew

ithre

spec

tto

mea

ngr

ain

size

s.M

ean

and

med

ian

grai

nsi

zes

are

assu

med

sim

ilar

for

near

-uni

form

sedi

men

t.¶¶

Rep

orte

dda

taar

ede

term

ined

from

bulk

(i.e

.,su

rfac

ean

dsu

bsur

face

mix

ture

)gr

ain

size

sam

plin

g,tr

eate

dhe

reas

mix

ture

-bas

edva

lues

.


Tab

le1c

.Pr

evio

usly

Rep

orte

dt

* c50

Val

ues:

Com

pete

nce

(Lar

gest

Mob

ileG

rain

)

Sour

ceN

ote*

Surf

ace

Gra

inSi

zeD

istr

ibut

ion

t* c

q50s

Re*

c†

Prop

osed

t* c

Fun

ctio

nO

ther

Tha

nSh

ield

s’D

50

s,m

ms

gs

(f)

rs,

kg/m

3D

50

s/h

c‡

Exp

erim

enta

lCon

ditio

ns

Car

ling

[198

3]29

0.02

06,

629

t* c

qi5

1.17

Re*

i20

.46

62;

227

100.

29na

tura

l,st

eep,

grav

el-b

edde

dch

anne

l(G

reat

Egg

lesh

ope

Bec

k);c

oars

e-gr

aine

dpl

ane

bed/

alte

rnat

eba

rm

orph

olog

y(?

);no

form

drag

orsi

dew

allc

orre

ctio

n0.

080

18,6

34t

* cqi

54.

99R

e*i2

0.4

277

;1.

724

600.

34na

tura

l,st

eep,

narr

ow,g

rave

l-bed

ded

chan

nel(

Car

lBec

k);c

oars

e-gr

aine

dpl

ane

bed

mor

phol

ogy

(?);

nofo

rmdr

agor

side

wal

lcor

rect

ion

Ham

mon

det

al.[

1984

]30

0.02

5§87

7t

* cqi

50.

025(

Di/D

50

s)2

0.6

015

.50.

9026

50,,

0.20

natu

ral,

plan

ar,t

idal

chan

nel(

Wes

tSo

lent

)A

ndre

ws

and

Erm

an[1

986]

310.

050§

8,37

7t

* cqi

50.

101(

Di/D

50

ss)2

1.0

758

0.97

zzz

0.14

natu

ral,

mea

nder

ing,

pool

-riffl

ech

anne

l(S

ageh

enC

reek

);no

form

drag

orsi

dew

allc

orre

ctio

nF

rom

Kom

ar[1

987a

]32

Milh

ous

[197

3]33

0.02

7§7,

277

t* c

qi5

0.04

4(D

i/D5

0ss

)20

.43

63,

0.71

2850

0.20

sam

eas

first

entr

yof

Tab

le1a

Milh

ous

[197

3]33

0.02

46,

861

t* c

qi5

0.04

4(D

i/D5

0ss

)20

.53

63,

0.71

2850

0.23

asab

ove

Car

ling

[198

3]34

0.02

16,

738

t* c

qi5

0.04

5(D

i/D5

0)2

0.6

862

;2

2710

0.28

sam

eas

Car

ling

[198

3],fi

rst

entr

yof

Tab

le1c

(Gre

atE

ggle

shop

eB

eck)

Car

ling

[198

3]34

0.02

26,

896

t* c

qi5

0.04

5(D

i/D5

0)2

0.6

462

;2

2710

0.27

asab

ove

Ham

mon

det

al.[

1984

]35

0.02

7§91

1t

* cqi

50.

045(

Di/D

50)2

0.7

115

.50.

9026

50,,

0.20

sam

eas

Ham

mon

det

al.[

1984

],se

cond

entr

yof

Tab

le1c

Fro

mK

omar

[198

7b]

Fah

nest

ock

[196

3]36

0.03

121

,033

zzz

134

1.79

zzz

0.61

natu

ral,

prog

laci

al,b

raid

edch

anne

l(W

hite

Riv

er);

nofo

rmdr

agor

side

wal

lco

rrec

tion

Fer

guso

net

al.[

1989

]0.

047§

37,8

80t

* cqi

50.

047(

Di/D

50

s)2

0.8

873

zzz

2800

0.20

natu

ral,

prog

laci

al,b

raid

ed,g

rave

lcha

nnel

(Whi

teR

iver

);fo

rmdr

agan

dsi

dew

all

corr

ectio

nas

inno

te3

Fro

mK

omar

and

Car

ling

[199

1]37

Milh

ous

[197

3]0.

028

7,60

1t

* cqi

50.

059(

Di/D

50

ss)2

0.6

463

,0.

7128

500.

27sa

me

asfir

sten

try

ofT

able

1aK

omar

and

Car

ling

[199

1]0.

039§

9,18

2t

* cqi

50.

039(

Di/D

50

s)2

0.8

262

;2

2710

0.15

sam

eas

Car

ling

[198

3]fr

omK

omar

[198

7a]

Ash

wor

thet

al.[

1992

]0.

049§

1,80

7t

* cqi

50.

049(

Di/D

50

s)2

0.6

921

zzz

2650

0.07

sam

eas

Ash

wor

thet

al.[

1992

],si

xth

entr

yof

Tab

le1a

Lep

pet

al.[

1993

]38

0.14

928

,421

t* c

qi5

0.14

9(D

i/D5

0s)

20

.40

910.

40zz

z0.

35N

atur

al,s

teep

,gra

vel-b

edde

dri

ver;

bed

form

type

not

repo

rted

;Shi

miz

u[1

989]

side

wal

lcor

rect

ion,

\bu

tno

form

drag

corr

ectio

n0.

143

24,6

93t

* cqi

50.

143(

Di/D

50

s)2

0.5

084

0.33

zzz

0.45

asab

ove

0.15

540

,645

t* c

qi5

0.15

5(D

i/D5

0s)

20

.47

114

0.38

zzz

0.48

asab

ove


Tab

le1c

.(C

ontin

ued)

Sour

ceN

ote*

Surf

ace

Gra

inSi

zeD

istr

ibut

ion

t* c

q50s

Re*

c†

Prop

osed

t* c

Fun

ctio

nO

ther

Tha

nSh

ield

s’D

50

s,m

ms

gs

(f)

rs,

kg/m

3D

50

s/h

c‡

Exp

erim

enta

lCon

ditio

ns

Fer

guso

n[1

994]

0.07

414

,096

t* c

qi5

0.07

4(D

i/D5

0s)

20

.87

721.

5626

500.

38na

tura

lbou

lder

-bed

stre

am(R

oari

ngR

iver

);be

dfo

rmty

peno

tre

port

ed;n

osi

dew

allc

orre

ctio

n;T

hom

pson

and

Cam

pbel

l[19

79]

form

drag

corr

ectio

nfo

rre

lativ

ero

ughn

ess

0.07

816

,005

t* c

qi5

0.07

8(D

i/D5

0s)

20

.88

771.

1426

500.

40as

abov

e0.

061

34,7

01t

* cqi

50.

061(

Di/D

50

s)2

0.8

914

0?1.

06?

2650

0.47

?as

abov

e0.

070

24.6

06t

* cqi

50.

070(

Di/D

50

s)2

0.7

810

61.

0626

500.

31as

abov

e0.

047§

11,9

43t

* cqi

50.

047(

Di/D

50

s)2

0.6

975

zzz

2650

0.03

asab

ove,

but

for

the

Gau

laR

iver

Wat

hen

etal

.[19

95]

0.05

9§2,

025

t* c

qi5

0.05

9(D

i/D5

0s)

20

.70

21.3

;1.

626

500.

02–0

.21

sam

eas

Wat

hen

etal

.[19

95],

tent

hen

try

ofT

able

1a

Sour

ceN

ote*

Subs

urfa

ceG

rain

Size

Dis

trib

utio

n

t* c

q50s

sR

e*c†

Prop

osed

t* c

Fun

ctio

nO

ther

Tha

nSh

ield

s’D

50

ss,

mm

sg

ss(f

)r

s,kg

/m3

D5

0ss

/hc‡

Exp

erim

enta

lCon

ditio

ns

And

rew

san

dE

rman

[198

6]0.

101

4,42

9t

* cqi

50.

101(

Di/D

50

ss)2

1.0

730

2.25

zzz

0.07

sam

eas

And

rew

san

dE

rman

[198

6],t

hird

entr

yof

Tab

le1c

Fro

mK

omar

[198

7a]

Milh

ous

[197

3]0.

044

1,66

2t

* cqi

50.

044(

Di/D

50

ss)2

0.4

320

,2.

6728

500.

12sa

me

asP

arke

ran

dK

linge

man

[198

2],fi

rst

entr

yof

Tab

le1a

Car

ling

[198

3]39

0.03

72,

570

t* c

qi5

0.04

5(D

i/D5

0)2

0.6

827

zzz

2710

0.16

sam

eas

Car

ling

[198

3],fi

rste

ntry

ofT

able

1cFr

omK

omar

and

Car

ling

[199

1]M

ilhou

s[1

973]

0.05

91,

974

t* c

qi

50.

059(

Di/D

50

ss)2

0.6

420

,2.

6728

500.

13sa

me

asPa

rker

and

Klin

gem

an[1

982]

,firs

ten

try

ofT

able

1aK

omar

and

Car

ling

[199

1]0.

077

3,70

7t

* cq

i5

0.03

9(D

i/D5

0s)

20

.82

27zz

z27

100.

08sa

me

asC

arlin

g[1

983]

,firs

ten

try

ofT

able

1c

Not

eth

atsy

mbo

lsfo

rsi

mila

rfo

otno

tes

may

bedi

ffer

ent

inT

able

s1a

–1e.

See

nota

tion

sect

ion

for

sym

bols

not

prev

ious

lyde

fined

inte

xt.

*See

appe

ndix

.†R

e*c

5u

* cD

50/n

.W

here

unre

port

edby

aso

urce

,Re*

cva

lues

are

back

-cal

cula

ted

from

avai

labl

eda

ta(s

eefo

otno

teke

yed

to†,

Tab

le1a

).‡W

here

unre

port

edby

aso

urce

,hc

valu

esar

eba

ck-c

alcu

late

dfr

omcr

itica

ldep

th-s

lope

prod

ucts

usin

gre

port

edda

ta(s

eefo

otno

teke

yed

to‡,

Tab

le1a

).§U

sed

inPl

ate

1.\S

idew

allc

orre

ctio

nap

plie

dby

curr

enta

utho

rs.S

him

izu’

s[1

989]

corr

ectio

nac

coun

tsfo

rbo

thth

edi

ffer

ence

inw

alla

ndbe

dgr

ain

roug

hnes

san

dth

epr

oxim

ityof

wal

ls(i

.e.,

W/h

).W

eas

sum

eda

bed

grai

nro

ughn

ess

100

times

grea

ter

than

wal

lrou

ghne

ssfo

rsm

ooth

flum

ew

alls

.


Tab

le1d

.Pr

evio

usly

Rep

orte

dt

* c50

Val

ues:

The

oret

ical

Sour

ceN

ote*

Surf

ace

Gra

inSi

zeD

istr

ibut

ion

t* c

t50s

Re*

c†

Prop

osed

t* c

Fun

ctio

nO

ther

Tha

nSh

ield

s’D

50

s,m

ms

gs

(f)

rs,

kg/m

3D

50

s/h

c‡

Exp

erim

enta

lCon

ditio

ns

Whi

te[1

940]

400.

209

44se

ere

fere

nce

0.9

;u

2600

nath

eore

tical

plan

ebe

d;si

ngle

grai

nw

ithin

like

bed;

vari

able

pack

ing,

proj

ectio

nan

dex

posu

re;§

F5

458

0.13

651

see

refe

renc

e0.

71;

u79

00na

asab

ove

0.21

069

3se

ere

fere

nce

5.6

;u

2600

naas

abov

e0.

209

117

see

refe

renc

e0.

9;

u21

00na

asab

ove,

but

with

air

fluid

med

ium

0.20

81,

829

see

refe

renc

e5.

6;

u21

00na

asab

ove

Che

pil[

1959

]0.

018

zzz

see

refe

renc

ezz

zu,

mzz

zna

theo

retic

alpl

ane

bed

unde

rgoi

ngen

mas

sem

otio

n;va

riab

lepa

ckin

g,pr

ojec

tion

and

expo

sure

;§F

524

8E

giaz

arof

f[1

965]

\0.

060

1,00

0se

ere

fere

nce

zzz

u,m

zzz

nath

eore

tical

plan

ebe

d;si

ngle

grai

non

like,

unifo

rm,o

rm

ixed

-gra

inbe

d;si

gnifi

cant

proj

ectio

nan

dex

posu

re§

Iked

a[1

982]

410.

061

(0.0

34)

1,00

0se

ere

fere

nce

zzz

0zz

zna

theo

retic

alpl

ane

bed;

sing

legr

ain

onlik

e,un

iform

bed;

sign

ifica

ntpr

ojec

tion

and

expo

sure

;§F

545

8N

aden

[198

7]42

0.04

6(0

.030

,0.0

21)

.77

(62,

52)

see

refe

renc

e.

20

2650

nath

eore

tical

plan

ebe

d;si

ngle

grai

non

like,

unifo

rmbe

d;fu

llpr

ojec

tion

and

expo

sure

;§F

535

80.

216

(0.1

24,0

.080

).

167

(127

,102

)se

ere

fere

nce

.2

026

50na

asab

ove,

but

with

two

grai

nson

like,

unifo

rmbe

d;fu

llpr

ojec

tion,

but

noex

posu

re§

for

grai

nof

inte

rest

0.06

3(0

.036

,0.0

23)

.90

(68,

55)

see

refe

renc

e.

20

2650

naas

abov

e,bu

tw

ithsi

ngle

grai

nw

ithin

like,

unifo

rmbe

d;no

proj

ectio

nor

expo

sure

§W

iber

gan

dSm

ith[1

987]

0.06

01,

000

see

refe

renc

ezz

z0

2650

nath

eore

tical

plan

ebe

d;si

ngle

grai

non

like,

unifo

rmbe

d;si

gnifi

cant

proj

ectio

nan

dex

posu

re;§

F5

608

0.04

01,

000

see

refe

renc

ezz

z0

2650

naas

abov

e,bu

tw

ithF

550

8Ja

mes

[199

0]0.

046

100

see

refe

renc

ezz

z0

zzz

nath

eore

tical

plan

ebe

d;si

ngle

grai

nw

ithin

bed;

mod

erat

epr

ojec

tion

and

expo

sure

;§F

564

80.

011

130

see

refe

renc

ezz

z0?

zzz

nath

eore

tical

plan

ebe

d;si

ngle

grai

non

like,

unifo

rmbe

d;si

gnifi

cant

proj

ectio

nan

dex

posu

re;§

F5

198

Kirc

hner

etal

.[19

90]

430.

125

325–

480

see

refe

renc

e3.

74–4

.85

;0.

94–1

.18

2650

nath

eore

tical

plan

ebe

d;si

ngle

grai

non

like,

mix

ed-g

rain

bed;

dist

ribu

tion

offr

ictio

nan

gle,

prot

rusi

onan

dex

posu

re§

Buf

fingt

onet

al.[

1992

]44

0.10

033

4–12

,145

see

refe

renc

e4.

1–45

0.67

–1.5

026

50na

theo

retic

alpl

ane

bed;

sing

legr

ain

onlik

e,m

ixed

-gra

inbe

d;di

stri

butio

nof

fric

tion

angl

e,pr

otru

sion

and

expo

sure

§B

ridge

and

Ben

nett

[199

2]0.

060

1,00

0se

ere

fere

nce

zzz

0zz

zna

theo

retic

alpl

ane

bed

with

F'

398,

prot

rusi

onof

0.5

[sen

suF

ento

nan

dA

bbot

t,19

77],

Cor

eysh

ape

fact

orof

1,an

dPo

wer

sro

undn

ess

of6

sing

legr

ain

onlik

e,un

iform

bed


Tab

le1d

.(C

ontin

ued)

Sour

ceN

ote*

Surf

ace

Gra

inSi

zeD

istr

ibut

ion

t* c

t50s

Re*

c†

Prop

osed

t* c

Fun

ctio

nO

ther

Tha

nSh

ield

s’D

50

s,m

ms

gs

(f)

rs,

kg/m

3D

50

s/h

c‡

Exp

erim

enta

lCon

ditio

ns

0.04

210

,000

see

refe

renc

ezz

z0

zzz

naas

abov

eJi

ang

and

Haf

f[1

993]

450.

086–

0.16

170

0–1,

000

see

refe

renc

e7.

2(?

)m

2650

nasi

mul

ated

,het

erog

eneo

us,p

lane

bed

surf

ace

unde

rgoi

nga

“sla

b”sh

ear;

vari

able

proj

ectio

nan

dex

posu

re§

Lin

g[1

995]

460.

050

500

see

refe

renc

ezz

z0

zzz

nath

eore

tical

plan

ebe

d;si

ngle

grai

non

like,

unifo

rmbe

d;si

gnifi

cant

proj

ectio

nan

dex

posu

re§

Not

eth

atsy

mbo

lsfo

rsi

mila

rfo

otno

tes

may

bedi

ffer

enti

nT

able

s1a

–1e.

See

nota

tion

sect

ion

for

sym

bols

notp

revi

ousl

yde

fined

inte

xt.S

eere

spec

tive

appe

ndix

note

sfo

rva

lues

inpa

rent

hese

s.H

ere

“u”

deno

tes

unifo

rmgr

ain

size

s(s

g#

0.5)

,an

d“m

”de

note

sm

ixed

grai

nsi

zes

(sg

.0.

5),

whe

res

gis

the

grap

hic

stan

dard

devi

atio

nde

fined

as(f

842

f16

)/2

[Fol

k,19

74];

na,n

otap

plic

able

.*S

eeap

pend

ix.

†Re*

c5

u* c

D5

0/n

.W

here

unre

port

edby

aso

urce

,Re*

cva

lues

are

back

-cal

cula

ted

from

avai

labl

eda

ta(s

eefo

otno

teke

yed

to†,

Tab

le1a

).‡W

here

unre

port

edby

aso

urce

,hc

valu

esar

eba

ck-c

alcu

late

dfr

omcr

itica

ldep

th-s

lope

prod

ucts

usin

gre

port

edda

ta(s

eefo

otno

teke

yed

to‡,

Tab

le1a

).§H

ere

we

desc

ribe

grai

npr

otru

sion

inte

rms

ofpr

ojec

tion

and

expo

sure

[sen

suK

irchn

eret

al.,

1990

].\R

epor

ted

data

are

with

resp

ect

tom

ean

grai

nsi

zes.

Mea

nan

dm

edia

ngr

ain

size

sar

eas

sum

edsi

mila

rfo

rne

ar-u

nifo

rmse

dim

ent.

Tab

le1e

.Pr

evio

usly

Rep

orte

dt

* c50

Val

ues:

Oth

er

Sour

ce

Surf

ace

Gra

inSi

zeD

istr

ibut

ion

t* c

50s

Re*

c†

Prop

osed

t* c

Fun

ctio

nO

ther

Tha

nSh

ield

s’D

50

s,m

ms

gs

(f)

rs,

kg/m

3D

50

s/h

c‡

Exp

erim

enta

lCon

ditio

ns

Equ

ival

ence

ofB

edL

oad

and

Surf

ace

Gra

inSi

zeC

ecen

and

Bay

azit

[197

3]0.

044

1,48

1zz

z14

.60.

72zz

z0.

26st

raig

ht,r

ecta

ngul

arflu

me;

plan

ebe

d;no

side

wal

lcor

rect

ion

Fro

mK

omar

and

Car

ling

[199

1]M

ilhou

s[1

973]

0.03

78,

738

t* c

q50l

50.

104(

D5

0l/D

50

ss)2

0.8

963

,0.

7128

500.

20sa

me

asP

arke

ran

dK

linge

man

[198

2],fi

rst

entr

yof

Tab

le1a

Kom

aran

dC

arlin

g[1

991]

0.05

510

,904

t* c

q50l

50.

055(

D5

0l/D

50

s)2

0.8

962

;2

2710

0.11

sam

eas

Car

ling

[198

3],fi

rst

entr

yof

Tab

le1c

Smal

lest

Tra

nspo

rtC

aptu

red

byB

edL

oad

Tra

pP

owel

land

Ash

wor

th[1

995]

0.01

02,

790

zzz

49zz

z25

860.

20na

tura

l,st

raig

ht,g

rave

l-bed

ded

chan

nel

with

low

-am

plitu

dem

edia

lbar

(Riv

erW

harf

e);l

oose

,ope

nfr

amew

ork,

suba

ngul

argr

ains

;no

side

wal

lor

form

drag

corr

ectio

n0.

011

2,86

0zz

z49

zzz

2586

0.18

asab

ove

0.05

56,

700

zzz

49zz

z25

860.

03as

abov

e,bu

tw

ithim

bric

ated

alga

l-cov

ered

grai

ns0.

067

7,59

0zz

z49

zzz

2586

0.03

asab

ove

Not

eth

atsy

mbo

lsfo

rsi

mila

rfo

otno

tes

may

bedi

ffer

ent

inT

able

s1a

–1e.

See

nota

tion

sect

ion

for

sym

bols

not

prev

ious

lyde

fined

inte

xt.

*See

appe

ndix

.†R

e*c

5u

* cD

50/n

.W

here

unre

port

edby

aso

urce

,Re*

cva

lues

are

back

-cal

cula

ted

from

avai

labl

eda

ta(s

eefo

otno

teke

yed

to†,

Tab

le1a

).‡W

here

unre

port

edby

aso

urce

,hc

valu

esar

eba

ck-c

alcu

late

dfr

omcr

itica

ldep

th-s

lope

prod

ucts

usin

gre

port

edda

ta(s

eefo

otno

teke

yed

to‡,

Tab

le1a

).


Nikuradse [1933] sand grainroughness).

m mixed grain size.ng, nb Manning roughnesses due

to grains and the combinedeffects of grains and bedforms [Meyer-Peter andMuller, 1948].

Q, Qb total volumetric fluiddischarge and that actingon the grains and bedforms [Meyer-Peter andMuller, 1948].

Re*c, Re*ci critical grain/boundaryReynolds number for D50

and Di.u uniform grain size.

u*, u*c general and critical shearvelocities.

W/h width-to-depth ratio.a coefficient.b exponent.u angular bed surface slope.n kinematic viscosity.

r , rs fluid and sedimentdensities.

sg, sgs, sgss, sgm sorting coefficient (Folk’s[1974] graphic standarddeviation) and those for thesurface, subsurface, andlaboratory mixtures.

t0 total boundary shear stress.t9, t0, z z z components of total

boundary shear stress dueto roughness elements suchas grains (t9), bed forms,walls, large woody debris,etc.

tc critical shear stress forincipient motion.

t*c dimensionless critical shearstress for incipient motion.

t*ci, t*c50

, t*c50s, t*c50ss

, t*c50mt*c of Di, D50, D50s,D50ss, and D50m.

t*cqi, t*cq50

, t*cq50s, t*cq50ss

, t*cq50lt*c of Di, D50, D50s,D50ss, and D50l based onempirical competenceequations determined fromcoupled bed load samplingand shear stressmeasurement.

t*cri, t*cr50

, t*cr50s, t*cr50m

t*c of Di, D50, D50s, andD50m for a specifiedreference bed loadtransport rate.

t*cvi, t*cv50

, t*cv50mt*c of Di, D50, and D50m

based on visual observationof incipient motion.

f16, f84 log2 grain sizes for which16% and 84% of the grainsare finer.

F intergranular friction angle.

Appendix: Notes for Tables 1a–1e1. We estimated the exponent of the Parker and Klingeman

[1982] t*crifunction from their Figure 3. To calculate D50s/hc,

we used a slope of 0.01 based on armor-breaching discharges(over 40 feet3/s (1.13 m3/s)) during the winter of 1971 [Milhous,1973, Table I-3]; except where reported differently, we as-sumed this same slope for all sources using Milhous’ [1973]data.

2. We calculated t*cr50swith Di 5 D50s 5 54 mm [Parker

and Klingeman, 1982] and D50ss 5 19.5 mm [Wilcock andSouthard, 1988, Table 1].

3. D50s values are averages of those reported in Ashworthand Ferguson’s [1989] Table 1. Although Ashworth and Fergu-son [1989] used local velocity profiles rather than depth-slopeproducts to determine shear stress, they used the full velocityprofile rather than just near-bed values. The full profile in-cludes all local roughness effects (bed form drag, etc.) andlikely overestimates the effective shear stress (t9) (see discus-sion of segmented velocity profiles by Middleton and Southard[1984] and Smith and McLean [1977]). Their local velocitymeasures do, however, implicitly account for sidewall effects.

4. We estimated t*cr50sfrom Wilcock and Southard’s [1988]

Oak Creek equation using the same Di and D50ss values asthose in note 2 but with the coefficient of the equation reducedby 55% for bed form drag [Wilcock, 1993].

5. We calculated t*cr50sfrom the Shields equation using

tcr50sdetermined from Wilcock and McArdell’s [1993] Figure 8

expression, with D50s estimated by averaging their Figure 5data for runs 7b, 7c, 2, 4, 5, 6, and 14c (assumed to be equal to‘‘start-up’’).

6. We estimated t*cr50sfor runs 4 and 5 from Day’s [1981]

Figure 2 using D50s values [Day, 1981, Figure 1] from theimmediately preceding runs (i.e., 3 and 4, respectively) andrecognizing that t*c is the square of the Ackers-White mobilitynumber.

7. We calculated t*cr50sfrom the Shields equation using

tcr50sregressed from particle velocity and u* data for D50s

values in Meland and Norrman’s [1966] Figure 4.8. We estimated t*cr50m

and Re*c values from Shields’ [1936]Figure 6. Because the corresponding grain sizes are unre-ported by Shields, we assigned D50m values reported by eachsource based on a sensible match of grain size with estimatedt*cr50m

and Re*c pairs. Kramer’s data are included here, how-ever, it is uncertain if they are reference- or visual-based val-ues; Kramer measured bed load transport rates but did notreport them. D50m and hc values for Casey [1935] are fromTison [1953].

9. Using data in Johnson’s [1943] Tables 27–30, we linearlyextrapolated tc values from plots of bed load transport rateversus shear stress for each sediment mixture and used thesedata to calculate t*cr50m

values from the Shields equation. Lackof bedload size distributions precluded analysis of nonlinearrelationships between bedload transport and shear stress usinga Parker and Klingeman [1982] type method.

10. Curiously, Gilbert’s [1914] data analyzed in this fashionare very different than the other reference-based data and areexcluded from our analysis.

11. We applied the method of note 7 to Meland and Norr-man’s [1969] Figure 5, with D50m ' 3.9 mm [Meland andNorrman, 1969].

12. Although Paintal [1971] questions the existence of adefinitive threshold for mobility [see also Lavelle and Mofjeld,


1987], two potential t*cr50mvalues can be estimated from his

analysis. Extrapolating high bed load transport rates to a zerovalue yields t*cr50m

' 0.05 for D50m values of 2.5 and 7.95 mm[Paintal, 1971, Figure 8]. The resultant t*c50

and Re*c pairsagree with other referenced-based values determined by unre-ported curve-fitting techniques [e.g., Shields, 1936]. However, amore appropriate nonlinear fit of Paintal’s [1971, Figure 8]data can yield t*c50

values as low as 0.01, depending on thechosen reference bedload transport rate.

13. We corrected Mizuyama’s [1977] modified Shieldsequation for a neglected buoyancy term (left-hand side of hisequation (3.27); see work by Wiberg and Smith [1987] for asimilar correct derivation). Using this corrected equation, wecalculated t*cr50m

values with data from Mizuyama’s [1977] Ta-bles 3.1 and 3.2; t*cr50m

values using a traditional Shields equa-tion are shown in parentheses for comparison. F values usedby Mizuyama [1977] are mass angles of repose for the bulksediment mixture [sensu Miller and Byrne, 1966], rather thanintergranular values.

14. The t*cr50mis for the lowest dimensionless bed load

transport rate (1026) of the composite data set [Pazis and Graf,1977, Figure 3].

15. We calculated t*cr50mvalues using Bathurst et al.’s [1987]

equation (15.1) and values in their Table 15.3; equivalent t*cr50m

values determined from a traditional Shields expression areshown in parentheses for comparison. We estimated Re*c val-ues from Bathurst et al.’s [1987] Figure 15.3. As with Mizuyama[1977], F is the mass angle of repose of the sediment mixture.

16. We estimated t*cr50mvalues from Bathurst et al.’s [1987]

Figure 15.3. Equivalent t*cr50mvalues using a traditional Shields

expression are shown in parentheses [Bathurst et al., 1979,Table 6].

17. We calculated t*cr50mfrom the Shields equation using

tcr50mof Li and Komar’s [1992] Figure 1b.

18. Using Day’s [1980] Figure 9 we determined t*cr50mval-

ues for dimensionless particle sizes corresponding to reportedD50m values [Day, 1980, Table 1], recognizing that t*c is thesquare of the Ackers-White mobility number.

19. We estimated t*cr50mfor run 3 from Day’s [1981] Figure

2 using D50m of Day’s [1981] Figure 1 and recognizing that t*cis the square of the Ackers-White mobility number.

20. We calculated t*cr50mfrom the Parker and Klingeman

[1982] method as modified by Ashworth and Ferguson [1989]using data reported in Leopold and Emmett’s [1976, 1977]Tables 1 and 2.

21. Using the Shields equation we developed power lawfunctions for t*cri

from data in Wilcock’s [1992a] Figure 6.5 andTable 6.2.

22. We used the same procedure as in note 5, but with Di

5 D50m estimated from the bulk bed distribution of Wilcockand McArdell’s [1993] Figure 5.

23. Coleman [1967] reports critical boundary Reynoldsnumbers in terms of u (the flow velocity measured at a heightof 0.5D50s) rather than u*. Consequently, his Rec values mustbe converted to Re*c values by replacing u with u*. We usedRe*c values estimated from Coleman’s [1967] data by Fentonand Abbott [1977], but we did not use their corrected t*c50

values, as Coleman’s [1967] shear stresses are not calculatedfrom measures of u but instead are based on direct measuresof strain and can be read from his Figure 3 without need forconversion. Fenton and Abbott’s [1977] t*c50

values are shownfor comparison in parentheses.

24. Reported data were derived from Gilbert’s [1914] Table

10 using his definition of incipient motion (i.e., “several grainsmoving” from a plane-bed surface [Gilbert, 1914, pp. 68, 71])which we consider similar to Kramer’s [1935] “medium move-ment.”

25. We calculated the first two t*cv50mand Re*c pairs from

White’s [1940] Table 1 (experiments 1a, 1bii, and 2a) assumingrs 5 2650 kg/m3 and r ' 900 kg/m3; r was estimated bycomparing the reported depth-slope products [White, 1940,Table 1, “from d”] with those calculated for a fluid medium ofwater. We calculated the third t*cv50m

and Re*c pair from White’s[1940, p. 328] data assuming n 5 1026 m2/s. All other reportedvalues were taken from White’s [1940] Table 2.

26. Reported data are derived from Grass’ [1970] averagesof instantaneous shear stresses and are assumed equivalent totime-averaged values; instantaneous equivalents are shown inparentheses for comparison. Grass’ [1970] direct shear stressmeasures implicitly account for sidewall effects.

27. We calculated t*cv50mfrom the Shields equation using

the reported u*c value for initiation of grain motion [Wimbushand Lesht, 1979, Table 1], r 5 1027.6 kg/m3 (sea water of 3.5%salinity and 7.38C [Todd, 1964]) and rs 5 2015 kg/m3. Weestimated rs by averaging the densest possible carbonate (ara-gonite) with the least dense skeletal test measured by Wimbushand Lesht [1979].

28. We estimated t*cv50mby replacing the mean sand diam-

eter in reported Shields stresses with D50m values determinedfrom the full grain size data of Young and Mann’s [1985] Table1. We calculated corresponding Re*c values in a similar fashion.

29. The t*cq50sof Great Eggleshope Beck was calculated

from the Shields equation, with D50s taken as the median grainsize of the framework distribution [Komar and Carling, 1991,Table 1] and tcq50s

determined from Carling’s [1983] equation(7) using the above D50s value. The framework distribution isobservationally similar to the censored (i.e., armored) surfacelayer distribution [Carling and Reader, 1982]; t*cq50s

for CarlBeck was calculated in a similar fashion using a median frame-work gravel size estimated from Carling’s [1989] Figure 2, withthe distribution truncated at 4 mm [Carling, 1989].

30. We developed a power law function for t*cqiusing Di

and t*cqidata from Hammond et al.’s [1984] Table 1 and D50s

5 15.5 mm estimated from the grab sample data of theirFigure 3 truncated at 2 mm; the surface material was observa-tionally devoid of sand [Hammond et al., 1984].

31. Rather than using the Andrew’s [1983] equation, weregressed a power law function through Andrews and Erman’s[1986] Figure 7 data and evaluated t*cq50s

with Di 5 D50s 5 58mm and D50ss 5 30 mm [Andrews and Erman, 1986].

32. Using bed load transport data from Milhous [1973],Carling [1983], and Hammond et al. [1984], Komar [1987a]developed empirical competence equations for each study site,expressing competence as a power law function between shearstress and the largest mobile grain size [Komar, 1987a, Table1]. To facilitate convergence of different data sets, Komar[1987a] proposed that Shields stress be expressed as a powerlaw function of the form t*cqi

5 a(Di/D50)b (similar to thatused by Parker et al. [1982] and Andrews [1983]), where D50 isa generic term that Komar [1987a] inconsistently evaluated aseither D50ss or the ‘‘crossover’’ grain size [Komar, 1987a, p.205]. For Milhous’ [1973] data, Komar [1987a] set D50 5 D50ss

and algebraically manipulated the competence equation intothe desired form of t*cqi

[Komar, 1987a, p. 207]. For the Carling[1983] and Hammond et al. [1984] data, Komar [1987a] choseD50 values based on the empiricism that competence curves


cross the Miller et al. [1977] incipient motion curve at values ofDi ' D50, as demonstrated by Day’s [1980] data [Komar,1987a, Figure 2], where D50 5 D50m for Day’s [1980] data.However, the crossover value for Carling’s [1983] data (20 mm[Komar, 1987a, Figure 3]) is similar to the D50ss value for thatsite (27 mm [Komar and Carling, 1991, Table 1]); while theHammond et al. [1984] D50ss value is not known, the crossovervalue (7.5 mm [Komar, 1987a, Table 1]) is less than D50s (15.5mm, note 30) and more like the D50 of their grab sample (10.3mm [Hammond et al., 1984, Figure 3]) which is likely to becomposed predominantly of subsurface material. These obser-vations indicate that Komar’s [1987a] t*c qi

equations forMilhous [1973], Carling [1983], and Hammond et al. [1984][Komar, 1987a, Table 1] are functions of median grain sizessimilar or equal to those of the subsurface (i.e., Komar’s[1987a] generic D50 is defined as D50ss in the Milhous equa-tion and is roughly equivalent to D50ss in the Carling andHammond et al. equations).

Komar [1987a, Figure 4] fixed a values for the Carling [1983]and Hammond et al. [1984] t*cqi

equations from crossoverpoints with the Miller et al. [1977] curve, forcing a 5 0.045 andleaving b to be back-calculated. Komar [1987a] found that theresultant t*cqi

equation for Carling’s [1983] data appeared todescribe Milhous’ [1973] data quite well, so he used it to rep-resent Milhous’ [1973] data, abandoning the algebraically de-fined Milhous equation (cf. p. 207, Figure 5, and Table 1 ofKomar [1987a]). Rejecting this unsound rationale, we havemaintained the original algebraically defined Milhous equationin Table 1c (as also preferred by Komar and Shih [1992]).

We calculated t*cq50svalues from these “derived” t*cqi

equa-tions using values of Di 5 D50s culled from other sources (seefollowing notes) and D50 as defined by Komar [1987a, Table 1].Komar’s [1987a] t*cqi

equations for Day’s [1980] data were notused, as they represent Di/D50m . 1 only [Komar, 1987a, p.209].

33. We calculated t*cq50sfrom the algebraically manipu-

lated Milhous equation [Komar, 1987a, p. 207] and values ofD50 5 D50ss 5 20 mm [Komar, 1987a, Table 1] and Di 5D50s 5 63 mm [Milhous, 1973, Table 2; Winter, 1971]. Weestimated a second set of values from a power law fit of datapresented in Komar’s [1987a] Figure 5.

34. We calculated t*cq50sfrom the Carling equation [Komar,

1987a, Table 1] and values of D50 ' 20 mm [Komar, 1987a,Table 1] and Di 5 D50s 5 62 mm, the median grain size ofthe framework distribution [Komar and Carling, 1991, Table 1](see also note 29). We estimated a second set of values from apower law fit of data presented in Komar’s [1987a] Figure 6.

35. We calculated t*cq50sfrom the Hammond et al. [1984]

t*cqiequation [Komar, 1987a, Table 1] and values of D50 5 7.5

mm [Komar, 1987a, Table 1] and Di 5 D50s 5 15.5 mm (note30).

36. We calculated tcq50sfrom the Fahnestock competence

equation reported by Komar [1987b, Table 1], and used thisvalue to determine t*cq50s

from the Shields equation. For thesecalculations D50s was determined from the composite WhiteRiver grain size data [Fahnestock, 1963, Table 2].

37. We calculated t*cq50svalues from relevant equations

[Komar and Carling, 1991, Figure 9] and grain sizes [Milhous,1973, p. 16; Komar and Carling, 1991, Table 1]. Although thelegend for Komar and Carling’s [1991] Figure 9 indicates thatthe Carling equation is expressed as a function of D50ss, thenormalizing grain size used by the authors (62 mm) is that ofthe framework gravel [Komar and Carling, 1991, p. 498], which

is equivalent to the censored (i.e., armored) surface size[Carling and Reader, 1982].

38. Values of t*cq50swere determined from power functions

for t*cqideveloped from Lepp et al.’s [1993, Tables 2–4] data.

39. We used Di 5 D50ss 5 27 mm [Komar and Carling,1991, Table 3] and D50 5 20 mm (see note 32).

40. We determined t*ct50svalues using the Shields equation

and tct50svalues presented in White’s [1940] Table 2. However,

he erroneously adds tanu to tanF, rather than subtracting it [cf.Wiberg and Smith, 1987]; our reported values reflect this cor-rection.

41. The t*ct50svalues were estimated from Ikeda’s [1982]

Figure 5 for dimensionless lift-to-drag ratios of 0 and 0.8(shown in parentheses).

42. Reported t*ct50svalues are means for velocity fluctua-

tions of zero, one, and two standard deviations, respectively[Naden, 1987, Table 2].

43. F and grain protrusion measurements were made fromflume-worked heterogeneous bed surfaces with plane-bed orlow-amplitude topography. The specific t*ct50s

value reportedhere was read from Kirchner et al.’s [1990] Figure 18 for D/K50

5 1 and n 5 10.44. F measurements were made from bed surfaces of a

natural pool-riffle stream (Wildcat Creek), and protrusion val-ues were derived from Kirchner et al.’s [1990] experiments. Thespecific t*ct50s

value reported here was read from Buffington etal.’s [1992] Figure 13 for D/K50 5 1 and n 5 0.1.

45. Reported t*ct50svalues bracket the calculated threshold

of “continuous” motion, defined as one or more particles mov-ing at all times during a simulation [Jiang and Haff, 1993].

46. Ling [1995] proposes two Shields curves representingsediment motion characterized by lifting and rolling, respec-tively. These curves coalesce at high Re*c values. The t*ct50s

value reported here is half that of Ling’s [1995] Figure 3,corresponding to ks/D50s 5 1 [Ling, 1995, p. 477].

Acknowledgments. This work was funded by the Pacific NorthwestResearch Station of the U.S. Department of Agriculture Forest Ser-vice (cooperative agreement PNW 94-0617) and by the WashingtonState Timber, Fish, and Wildlife agreement (TFW-SH10-FY93-004,FY95-156). Bill Dietrich, Peter Wilcock, and two anonymous reviewersprovided insightful criticism that significantly improved this work.

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(Received April 17, 1996; revised August 9, 1996;accepted October 18, 1996.)


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