Critical Shields values in coarse-bedded steep streams

Authors


Abstract

[1] Critical Shields values ( inline image) suitable for specific applications are back-calculated from representative bed load samples in mountain streams and a flow competence/critical flow approach. The general increase of inline image (for the bed D50 size) as well as inline image and inline image (for the bed D16 and D84 sizes) with stream gradient Sx and also the stratification of inline image by relative flow depth and relative roughness are confirmed. Critical Shields values inline image are shown to exceed inline image by about threefold, while those for inline image are nearly half of inline image. However, it remains unclear to what extent physical processes or numerical artifacts contribute to determining critical Shields values. Critical bankfull Shields values ( inline image) back-computed from the average largest particles mobile at bankfull flow DBmax,bf approach inline image at steep gradients and inline image at low gradients and therefore increase very steeply with Sx. The relation inline image is stratified by bed stability (D50/DBmax,bf) and predictable if bed stability can be field categorized. Noncritical Shields values ( inline image) computed from bankfull flow depth and the D50 size differ from inline image and inline image. Only in bankfull mobile streams where D50/DBmax = 1 can τ*cbf, inline image, and inline image be used interchangeably. In highly mobile streams, substituting inline image by inline image overpredicts the DBmax,bf size by up to fivefold and underpredicts DBmax,bf by the same amount in highly stable streams. A value of 0.03 is appropriate for inline image only on low stability beds with Sx ≅ 0.01, but overpredicts DBmax,bf by 30-fold on highly stable beds with Sx ≅ 0.1. Differences in field and computational methods also affect critical Shields values.

1. Introduction

[2] Estimating the largest bed material particle size mobile at a specified flow in coarse-bedded steep streams is a common task in many geomorphological environments and for various engineering problems, and the Shields equation is commonly employed for this purpose. A wide variety of Shields-type parameters that differ in computational details and numerical values have been proposed for steep and coarse-grained channels. However, there is not much guidance as to which values should be selected for a specific purpose, and large errors may result when critical particle size or critical flow is computed from an inappropriate Shields value. It is the aim of this empirically oriented study to improve user choices for selecting appropriate critical Shields values for use in steep channels. The study further aims to alert the user to differences among various Shields-type parameters and to provide choices for critical Shields values for bankfull flow.

[3] The Shields equation is based on flume experiments in which Shields [1936] quantified the critical dimensionless shear stress inline image as the numerical value of the term:

display math(1)

at incipient motion of rounded to angular particles from relatively well-sorted beds with a specified Dm size in the absence of bed forms; ρf and ρs denote the water and sediment densities, g is acceleration due to gravity, Rc is the hydraulic radius at incipient motion of Dm, Dm is the mean bed material particle size that in well-sorted distributions is similar to the median (D50) size, and Sf is the friction gradient which under conditions of uniform flow is similar to the channel bed gradient Sx that is typically determined along the channel waterlines. Shields [1936] presented inline image values in relation to the particle Reynolds number inline image that is controlled mainly by Dm and secondly by Sx which covers a much wider range among the study streams than Rc; ν is kinematic viscosity that decreases with water temperature Tw from 1.5 · 10−6 to 1.0 · 10−6 m2/s for Tw from 5 to 20°C. Shields' [1936] flume experiments employed beds with rather uniformly sized particles of less than 4 mm and ended near Rep = 500, but he (and especially later researchers) assumed that the nonmonotonic, spoon-shaped curve inline image approaches a constant value of inline image for Rep much larger than 500, extending the Shields relationship considerably beyond its original range. With application of the Shields equation to natural streams where the central tendency of the bed material particle-size distribution is better characterized by the median (D50) than the mean (Dm) due to the distribution skewness, the critical Shields value at incipient motion of the bed D50 size (a.k.a. Shields stress) became denoted as inline image. In this study, the terms Shields value or Shields number refers to the critical value at incipient motion unless otherwise noted.

[4] Subsequent redrawings and analytical expressions of the Shields curve as well as additional experiments [Rouse, 1939; Meyer-Peter and Müller, 1948; Vanoni et al., 1966; Neill, 1968; Paintal, 1971; Miller et al., 1977; Yalin and Karahan, 1979; Brownlie 1981; Parker et al., 2003] suggested that inline image values for Rep > 500 range within 0.03–0.06, rather than being the single value of 0.056 proposed by Shields. This variability invalidates the assumption of a unique relation inline image, a requirement, if the Shields equation (equation (1)) is to be solved for either Rc, Sx, or D50. Shields values compiled by Buffington and Montgomery [1997] extended over an even wider range from 0.01 to 0.09 in coarse-bedded steep streams where Rep takes values within 4000–100000. Buffington and Montgomery [1997] attribute the variability to differences in bed stability (or bed mobility) as well as computations and field methods. Many previous and later studies have shown that Shields values are affected by whether bed particles are loose and easily mobilized or restrained by bed structures and by particle interlock. Particle entrainment has been reported to be affected by particle imbrication [Komar and Li, 1986; Gordon et al., 1992], by different particle shapes [Li and Komar, 1986], the presence of particle clusters [Brayshaw et al., 1983], by particle hiding and protrusion, bed sorting, pocket, and pivot angles [Isbash, 1936; White, 1940; Fenton and Abbott, 1977; Parker et al., 1982; Andrews, 1983; Fisher et al., 1983; Wiberg and Smith, 1987; Kirchner et al., 1990; Andrews and Smith, 1992; Carling et al., 1992; Andrews, 1994; Ferguson, 2012]; by packing density [Gordon et al., 1992], stone structures [Church et al., 1998; Hassan and Church, 2000], as well as by antecedent flow and bed conditions [e.g., Gomez, 1983; Reid et al., 1985; Beschta, 1987; Turowski et al., 2011]. An increasingly wider range of Shields values from <0.03 to >0.5 has been reported for steep channels [Bathurst et al., 1983, 1987; Lepp et al., 1993; Rosgen, 1994, 1996; Shvidchenko and Pender, 2000; Buffington and Montgomery, 2001; Buffington et al., 2004, 2006; Zimmermann et al., 2010; Bunte et al., 2010a; Comiti and Mao, 2012]. Finally, various flume, modeling, and field studies have shown that Shields values for the bed D50 size increase with stream gradient Sx in coarse-bedded steep streams as shown in Figure 1 [Shvidchenko and Pender, 2000; Mueller et al., 2005, Buffington et al., 2006; Parker et al., 2011; Pitlick et al., 2008; Lamb et al., 2008; Recking, 2009; Camenen, 2012; Comiti and Mao, 2012; Buffington, 2012; Ferguson, 2012; Bunte, 2012a; Recking and Pitlick, 2013; J. M. Schneider et al., Field data based bed load transport prediction for mixed size sediments, submitted to WRR, 2013]. The reported correlation of Shields values with Sx offers an opportunity for prediction of Shields values. However, individual relations of Shields values versus Sx likewise involve scatter, and the proposed relations differ among studies as presented in Figure 1. The inter-study variability in relations of inline image is attributable not only to differences in stream conditions (structural bed stability, bed material size composition, sediment supply, flow hydraulics, and stream morphology), but also to methodological and computational diversity. Field measurements of Sx, D50, Q, d, and R may differ among studies, but differences are typically less than a factor of 2. Possibly larger variations result from methodological differences for quantifying particle entrainment in the field [Wilcock, 1988]: there is a direct, field-based flow competence approach in which either the average or the absolute largest bed load particle size DBmax mobile at a specific flow is quantified from repeated bed load samples or from tracer particles (largest grain method), and a relation of critical flows versus entrained particle sizes is established. For this approach, the emplacement depth and location of tracer particles as well as the ability of a sampling device to representatively include the largest mobile particle sizes, the sampling time, and sampling frequency become important. The “small transport method” quantifies critical flow as the flow at which a preset (low) transport rate (typically computed rather than measured) is exceeded for a specified size class using, for example, the reference transport rate approach introduced by Parker et al. [1982] (not discussed here). Apart from scatter and between-study variability in the reported relation of inline image versus Sx, another challenge concerning use of the Shields approach is its (mis-) application to predict the largest particle size mobile at bankfull flow in coarse-bedded streams. In this context, bankfull flow is broadly considered as the flow that inundates point bars and lateral bars sufficiently to shape them and extends laterally to the onset of perennial near-bank vegetation. Such flows tend to correspond with the 1.5–2 year recurrence interval flow. Bankfull flow is considered a reference state for high flows [Leopold, 1994; Buffington and Montgomery, 1999; Trush et al., 2000; Parker et al., 2007], and prediction of the largest particle size mobile at bankfull flow (DBmax,bf) is of interest for restoration projects and for evaluation of streambed stability or mobility. In assessments of the relative bed stability index in gravel/cobble/bed streams, the D50 particle size is set in relation to the DBmax,bf size [e.g., Olsen et al., 1997; Kappesser, 2002, Lorang and Hauer, 2003; Kaufmann et al., 2008, 2009]. For Rep > 500, these applications use a constant Shields value, e.g., 0.03 or 0.056 to compute DBmax,bf. However, the original Shields equation was not developed to predict the particle size mobile at bankfull (or any other) flow. When used to predict Dc, the Shields equation iteratively determines whether a (near-) uniform bed with a specific mean particle size (Dm) can be mobilized given R and Sx (ignoring ρf and ρs here). Hence, solving the Shields equation for Dc given R and Sx cannot accurately predict which of the many particle sizes in a mixed-size bed will become entrained because entrainment depends on the bed material composition and structure, sediment supply, flow hydraulics, and stream morphology. Not knowing which of the bed material sizes can be entrained by a specific flow, it is inappropriate to use the bed D50 size to determine the Rep- and hence the inline image-value to be used when solving the Shields equation for the largest mobile particles at bankfull flow (DBmax,bf) given Sx and Rbf. This approach presupposes mobility of the bed D50 size at bankfull flow. Users often ignore these limitations [e.g., Kaufmann et al., 2008, 2009]. Our study will show that solving equation (1) for Dc based on Rbf and Sx and a Shields value from the curve provides an unbiased estimate of DBmax,bf only in streams that move their bed D50 size at bankfull flow. For the many (frequently steep) streams that do not behave in this manner, application of the Shields equation assuming mobility of the D50 size is not correct.

Figure 1.

Empirical and modeled [Recking, 2009; Ferguson, 2012 (for D84/D50 = 2)] relations of inline image, inline image, inline image, and inline image with gradient in steep streams compiled from multiple sources. inline image is the critical Shields value for the bed D50 size; inline image refers to the reference transport method not discussed here; inline image and inline image refer to non-critical Shields values computed from bankfull flow hydraulics and the bed surface D50 and subsurface size D50sub, respectively. For clarity of the figure, flume-derived relations inline image [Bathurst et al., 1983, 1987; Shvidchenko and Pender, 2000; Lamb et al., 2008; Parker et al., 2011] are not plotted individually but contained within the shaded area.

[5] This study employed a direct approach for quantifying particle entrainment. Extensive measurements of gravel bed load particle sizes in various mountain streams were used to quantify relations of critical flow for incipient motion (Qc, dc, and Rc) for specified particle sizes as well as critical (i.e., entrained) particle sizes (Dc) for a specified flow for each study stream. Critical Shields values were then back-calculated from those relations. Characterizing relations between Rc and Dc exclusively from a flow competence approach—as well as its inverse, the critical flow curve—rather than compiling incipient motion information from various sources (e.g., tracer particles, critical flows, and reference transport rates) reduces the inherent methodological variability that may underlie published Shields values. The back-computed critical Shields values inline image are then regressed against stream gradient Sx. Scatter in the relations of inline image is reduced via stratification by relative flow depth (dbf /D50) and relative bankfull roughness (D84/dbf), and fitted regression functions provide guidance for prediction of critical Shields values using channel and flow parameters that are relatively easy to obtain. The study presents critical Shields values for the D16, D50, D50sub, and D84 bedmaterial percentile particle sizes as well as for the largest particle sizes DBmax,bf entrained at bankfull flow ( inline image) and compares measured critical Shields values inline image and inline image with noncritical Shields values derived from bankfull flow and the D50 size ( inline image). Based on the observed differences in sampling results among bed load samplers [e.g., Bunte et al. 2008, 2010b], Shields values back-computed from bed load traps and similarly suited samplers are compared to those back-computed from the frequently used Helley-Smith [1971] bed load samples. The effects of input parameter error and bias on critical Shields values are also assessed.

2. Methods

2.1. Bed load Measurements of Particle Mobility

[6] Computations of inline image that are based on a flow competence or critical flow approach require bed load transport measurements that accurately represent the largest bed load particle size DBmax mobile at a specified flow [Wilcock, 2001]. In steep streams with unsteady flow and a wide range of bedmaterial particle sizes, those measurements require bed load samplers with large openings and large sample volumes in order to sample over a long time to catch the infrequently moving large particles. In addition, the sampler's interaction with bed particles and ambient flow hydraulics must be minimal. The sampling scheme, i.e., the number of samples taken over a highflow day and across the stream, must be appropriately intense to cover the temporal and lateral variability of bed load transport over the highflow season. With these criteria in mind, bed load traps were designed and deployed in this study to representatively collect DBmax particles [Bunte et al., 2004, 2007, 2008, 2010c; Bunte and Abt, 2005]. Bed load traps have a sampler capacity of about 20 liters. For an individual sample, four to six lateral traps were deployed for 1 h or more when transport rates were low; during this time fluctuations in discharge were typically low as well. At high transport rates and when snowmelt discharge changed quickly, sampling times were reduced to 30 min, and to as little as 5 min to avoid overfilling the net when bed load particles were streaming into the sampler.

2.2. Bed load Data Sets

[7] Field data used for the computations of Shields values comprised bed load samples collected with bed load traps in 10 Rocky Mountain streams [Bunte et al., 2004, 2008, 2010c, 2011, 2012; Potyondy et al., 2010] and those collected in the Riedbach, a proglacial stream in the Swiss Alps, by Schmid [2011] as well as by J. Schneider (personal communication 2012). Also used were gravel bed load data from studies with other samplers that likely provide representative samples of the rarely moving largest clasts. These samplers include the large 1 m3 perforated moving steel baskets placed under a weir overfall at the Erlenbach torrent in the Swiss Pre-Alps [Rickenmann et al., 2012; J. Turowski and D. Rickenmann, personal communication 2012] and the large net-frame samplers at Squaw Creek, SW Montana [Bunte, 1996] and at Dupuyer Creek, NW Montana [Whitaker, 1997; Whitaker and Potts, 1996, 2007a, 2007b], the vortex sampler at Oak Creek, NW Oregon [Milhous, 1973], and nonweighing pit traps (locations 3B and 1A) at Harris Creek in British Columbia [Hassan and Church, 2000, 2001; Church and Hassan, 2002].

[8] The characteristics of all study sites are presented in Table 1. Most of the study streams are located in the US Rocky Mountains, some are located in other mountain ranges of the Western US and Canada and in the Swiss Alps. Except for the pristine Riedbach, study watersheds have experienced some logging, roading, mining, or direct stream management, but are not considered notably impaired. All sites have snowmelt regimes except Oak Cr. and Erlenbach which have pluvio-nival regimes and Riedbach which has a glacial-nival regime. As is typical of steep Rocky Mountain streams, channels are incised into a vegetated and morphologically largely inactive floodplain, and flows of 150% of bankfull (i.e., 150% the 1.5 or 2 year recurrence interval) cause only minor overbank spill.

Table 1. Study Site Characteristics
Stream and Year(s) SampledA (km2)Qbkf(m3/s)wbf (m)Sx(m/m)dbf (m)D16 (mm)D50 (mm)D84 (mm)D50s (mm)Dominant LithologyDominant Morphology
  1. a

    Estimated. Descriptions (low) and (high) refers to relatively low and high sediment supply.

Dupuyer 1995836.5100.0100.5216429037Mixed sedimentsMixed plane-bed and pool riffle
Little Granite 2002132.86.30.0120.32236713834Mixed sediments
Harris 1991, (3B and 1A)22019200.0130.729.5a48a9532Gneiss, Basalt
Oak 1969–197171.34.50.0140.3215a49a76a20Volcanic
Halfmoon riffle 2004616.28.60.0140.71144911926Granite, GneissPlane-bed with isolated,forced pool-riffle sequences
Halfmoon bar 2004616.28.60.0140.55144911926Granite, Gneiss
East Dallas 2007 (high)343.77.50.0170.31125712321Andesite, other volcanic
East Dallas 2007 (high)343.77.50.0170.31165912521
Little Granite 1999555.714.30.0170.39205913342Mixed sediments
St. Louis 1998344.06.50.0170.38227616341Gneiss, Granite
Squaw 1988 and 19911055.7110.0210.3313439927Andesite, Gneiss
Cherry ‘99413.19.50.0250.426.45214027Andesite, other volcanicPlane-bed with low steps
Riedbach 2011163.0110.0280.417.15616328Gneiss, Schist
NF Swan 2011 (high)161.35.60.0300.237.63912422Gneiss, Schist
NF Swan 2011 (low)161.35.60.0300.235.85213222Gneiss, Schist
Hayden 2005 (low & high)401.97.00.0380.26146316436Mixed sedimentsStep-pool
Fool 2009 and 201030.31.30.0440.20125212225Gneiss, schist
E. St. Louis '01 (high), E. St. Louis 2003 (low)80.83.70.0930.441510825854Gneiss, schist
Erlenbach 2009–20110.81.53.70.1050.318.76421527Mixed sediments
            

[9] At most of the authors' bed load trap study sites, bed load was also collected using a “3 inch” (7.6 by 7.6 cm) opening size sheet-metal Helley-Smith (HS) sampler with a standard 0.25 mm mesh bag, placing the HS sampler directly onto the bed at 15−20 locations across the stream for 2 min per location or less if the bag filled. Typically, fewer samples were collected with a Helley-Smith sampler (HS) than with bed load traps during a field season. At sites where the authors' HS samples fit well within the larger data sets of HS samples collected by Ryan et al. [2005] at the same site or nearby, the two sets of HS samples were combined to improve the flow competence/critical flow relations fitted to HS data sets. Because bed load traps are designed to collect particles >4 mm, all sediment <4 mm was mathematically removed from the Helley-Smith samples such that only gravel bed load of particle sizes >4 mm is compared for both samplers.

[10] A change in steepness of bed load transport (Qb) rating curves, often between rising and falling limbs of the high flow season (hysteresis), is a known phenomenon [e.g. Beschta, 1987; Bunte and MacDonald, 1999; Mao, 2012]. Changes in the steepness of flow competence curves within and between events can likewise occur [Mao, 2012; Swingle et al., 2012]. In two of the authors' bed load trap study streams, the flow competence relation changed during the high flow season. At North Fork (NF) Swan Cr., DBmax sizes dropped abruptly before peakflow when a beaver dam intercepted upstream sediment supply [Bunte et al., 2011] as presented in Figure 2. At Hayden Creek, DBmax sizes decreased considerably after peakflow, likely due to upstream particle entrapment. At Squaw Cr. and East St. Louis Cr., flow competence and critical flow relations differed between years of repeated sampling [see also Turowski et al., 2011], and at Halfmoon Cr., flow competence curves diverged between two neighboring sampling locations (riffle and bar site). Because changes in the flow competence/critical flow relation are brought about by changes in sediment supply and particle mobility, which are likewise reflected in critical Shields values inline image back-calculated from field data, separate regressions were fitted to sites where changes in the flow competence relation were observed. Altogether, 22 data sets were compiled from these coarse-bedded steep streams. The typically wider data scatter of the HS data did not permit segregation within a highflow period, among neighboring sites, or between years. As a result, there is only one HS data set per site.

Figure 2.

Examples of flow competence relations at North Fork Swan Creek, 2011 not available in Bunte et al. [2004, 2008, 2010c, 2011] or Potyondy et al. [2010]. Field data are segregated into two parts, one for the beginning of the highflows season and one during the highflow season when the site experienced a depletion of sediment supply due to sediment retained in an upstream beaver dam.

2.3. Quantification of Bed Material Particle Sizes, Stream Gradient, and Discharge

[11] At almost all of the bed load trap study sites as well as at the Erlenbach, bed material D50 sizes were quantified from reach-spanning pebble counts using the SFT procedure [Bunte et al., 2009] that minimizes observer bias and samples 400+ particles within the bankfull channel. Subsurface D50sub sizes were obtained from 2 to 3 large volumetric samples of the subsurface sediment (sand included) with a joint weight of 200–300 kg. The subsurface samples were collected on riffles and runs under water within a 0.6 by 0.6 m plywood shield enclosure after removal of surface particles [Bunte and Abt, 2001a]. All particle sizes, i.e., surface and subsurface sediment as well as the bed load DBmax sizes were quantified using a square-hole sieve stack or a template with opening sizes progressing in 0.5 ϕ. At East Dallas and NF Swan Creeks, bed material was sampled before and after changes in sediment supply. The remaining studies employed other sampling techniques. Drawing upon the text or photos, as well as information obtained from other studies conducted at or near the same site, reported bed material data were converted or adjusted to align with an SFT pebble count or a large subsurface sample. For example, Milhous [1973] obtained the bed D50 size originally from volumetric armor samples that were converted to grid-by-number samples [Kellerhals and Bray, 1971]. Hassan and Church [2000, 2001] and Church and Hassan [2002] determined the surface D50 size from a pebble count truncated at 2 mm, while reporting the presence of sand on the bed. To compare only untruncated distributions, a sand portion slightly smaller than that reported for the subsurface was mathematically added to the surface sediment, and the size distribution was recalculated.

[12] Stream gradient Sx in the bed load trap study streams was determined as the arithmetic mean of the gradients computed from longitudinal profiles surveyed along both waterlines at low or moderate flows over reach lengths of 7–12 stream widths (w), hence Sx denotes a reach-averaged value. At most of the study streams, 20–30 discharge measurements were taken over the range of observed flows; the mean flow velocity v per vertical was measured at 0.6 of the local depth along 15–20 verticals in a transect. Mean flow depth dm was computed from the cross-sectional flow area AQ/w for each measurement, and a relation between dm and discharge Q was established for each site. Given that cross-sectional shapes varied over the reach and that different cross sections were used to facilitate discharge measurements at low and high flows, the hydraulic radius R was determined from dm by assuming a trapezoidal channel cross section with a 45° bank angle rather than from the ratio AQ/wetted perimeter at each measurement location. Relations of dm and R versus discharge in other studies were either used as reported, estimated based on the text or topographic site maps, or back-calculated if shear stresses τ = ρf · g · d (or R) · S were presented. Particle and water densities of 2650 and 1000 kg/m3 were used for all study streams when computing Shields values because bed particles were either dominated by silica-rich granite/gneiss/schist lithologies or comprised particle mixes of various densities for which silica density was considered a workable estimate. Viscosity ν was set to 1.3 · 10−6 m2/s in the computation of Rep. Only a few of the bed load trap study streams had long flow records; hence the bankfull estimate considered both the flow that inundates unvegetated gravel point and/or lateral bars by several cm and the Q1.5 flow from reasonably close gauged basins.

2.4. Flow Competence, Critical Flow, and Shields Values

[13] A flow competence curve is the relation between the largest measured bed load particle size (DBmax′) encountered in a bed load sample collected over a sufficiently long time period and discharge (Q′) at the time of sampling, where ′ denotes individually measured data. In order to predict the critical, i.e., largest, particle sizes mobile at a specified flow, power function regressions were fitted to log-transformed values of DBmax′ and Q′ for all study streams:

display math(2)

where c is the regression coefficient, and d is the exponent. Most flow competence curves established from bed load trap samples were well defined. Measured flows within a highflow season spanned a 1.2 to 6.3 fold range (mean of 3; see also Table 2); the largest particles collected in the bed load traps were in the 90–128 mm class; sampled DBmax sizes spanned a factor of 4–32 depending on encountered high flows, and r2 values were within 0.43–0.90 (see example in Figure 2).

Table 2. Sampled Range of Flows (in % of Bankfull Flow) and Parameters of Power-Function Flow-Competence and Critical-Flow Curves Fitted to Bed load Samples Collected With Bed load Traps and Similar Samplers as Well as With a 7.62 cm (3-inch) Helley-Smith Samplera
Study Stream, Year(s) Sampled, and Reference(s)Sampler UsedSampled Range of %QbfBed load Trap and Other SamplersHelley-Smith Sampler
Flow Competence Curve DBmax = c Q dCritical Flow Curve inline imageFlow Competence Curve DBmax = g Q hCritical Flow Curve inline image
MinMaxr2nd-exp.c-coeff.CFf-exp.e-coeff.CFr2nh-exp.g-coeff.CFj-exp.i-coeff.CF
  1. a

    The c-, e-, g-, and i-coefficients are reported antilogged.

  2. b

    Authors' unpublished data. e = estimated; NF = net-frame sampler; BT = bed load traps; PT = pit traps; Vor = vortex sampler; MB = moving baskets.

  3. c

    This study.

Dupuyer 1995 [Whitaker, 1997; Whitaker and Potts, 1996, 2007a, 2007b]NF71208NA1201.623.251.1e0.5680.4311.1e        
Little Granite 2002[Bunte et al., 2008, 2010a]BT281020.66531.705.151.070.3890.6231.02650.320.5077.061.090.6330.3341.11
Harris 1991 (3b and 1A) [Hassan and Church, 2000, 2001; Church and Hassan, 2002]PT3971NA102.730.061.0e0.3382.2931.0e        
Oak 1969–1971 [Milhous, 1973; Komar, 1987]Vor111930.67920.7746.51.110.8710.0311.12        
Halfmoon riffle 2004 [Bunte et al., 2008, 2010a]BT18880.571491.403.981.090.4060.7791.032040.730.9717.941.060.7530.2761.05
Halfmoon bar 2004 [Bunte et al., 2008, 2010a]BT23760.75932.281.801.090.3291.0311.01        
East Dallas 2007 [Bunte et al., 2010a; Potyondy et al., 2010]BT181130.871,601.3210.51.040.6590.2251.02600.540.67016.71.070.8780.0941.09
Little Granite 1999 [Bunte et al., 2008, 2010a]BT611310.67553.340.141.100.2012.5531.012870.480.9743.701.230.4901.0631.11
St. Louis 1998 [Bunte et al., 2008, 2010a]BT27650.43402.381.261.070.1811.5211.002300.531.134.391.120.4700.8101.05
Squaw 1988 (high) [Bunte et al., 2010a]NF761010.62234.270.05751.070.2332.0001.00        
Squaw 1991 (low)bNF96118NA255.500.00401.2e0.1843.0001.2e        
Cherry 1999 [Bunte et al., 2008, 2010a]BT491450.90182.710.741.040.3331.2211.01460.570.6705.701.120.8500.2691.23
Riedbach 2011 [Schmid, 2011]BT7440.84531.0214.51.040.8210.1111.03        
NF Swan 2011 (high)cBT321320.75662.3020.81.100.3260.3821.01        
NF Swan 2011 (low)cBT911490.561384.802.591.120.1171.0781.00        
Hayden 2005 (high) [Bunte et al., 2008, 2010a; Potyondy et al., 2010]BT281560.67572.165.091.080.3080.6751.011080.520.77714.51.080.6700.1691.07
Hayden 2005 (low) [Bunte et al., 2008, 2010a; Potyondy et al., 2010]BT521200.871211.5113.61.050.5740.2371.02
Fool 2009 and 2010 [Bunte et al., 2010 a]bBT22790.401651.4798.31.180.2720.0991.031900.300.78539.21.160.3820.0771.07
E.St. Louis 2001 (high) [Bunte et al., 2008, 2010a]BT27710.60911.5232.01.030.3930.1771.01910.451.29323.31.080.3510.1941.02
E.St. Louis 2003 (low) [Bunte et al., 2008, 2010a]BT441250.681331.9741.41.060.3430.2411.01400.501.06825.01.040.4720.1701.02
Erlenbach 2009–2011 [Rickenmann et al., 2012]MB2372NA420.821051.1e0.8900.0131.0e        

[14] For Helley-Smith samples, the largest DBmax particle size was in the 64–90 mm size class, and the sampled range of DBmax was typically narrower. The HS sampler occasionally collects large particles in relatively low flows, overrepresenting the mobile particle size. At high flows, the short 2 min sampling time reduces the chance of collecting the infrequently moving largest particles [Bunte and Abt, 2005], while the HS opening is too small to accommodate large gravels, both of which cause undersampling of the true DBmax. Consequently, flow competence curves derived from HS samples are notably flatter than those from derived from bed load traps (Bunte et al., 2004, 2008, 2010b] (Table 2), and the less steep increase together with a larger data scatter dropped HS r2 values to 0.27–0.73.

[15] At study sites where numerous samples cover a wide range of flows and where gravel supply does not change much over the highflow season, flow competence and critical flow relations in log-log space exhibit straight trends parallel to the upper and lower envelope curves. A few of the data sets from sites not sampled with bed load traps covered a rather narrow range of sampled flows. Here, an overly flat trendline fitted from a least square regression that would mispredict flow competence was avoided by fitting a trendline line through the central data cloud parallel to the upper and lower data envelopes.

2.4.1 Computation of Critical Particle Sizes and Critical Flows

[16] The average largest bed load particle sizes DBmax,bf mobile at bankfull flow Qbf [Bunte et al., 2010a; Bunte, 2012a] were predicted from the flow competence relations (equation (2)) established for each study stream. The general underprediction of the y estimates (here DBmax,bf) from x values (here Q) in fitted power functions was addressed by multiplying the prediction of DBmax,bf by a bias correction factor CF = e (2.651 sy2) [Miller, 1984; Ferguson, 1986, 1987] where sy is the standard error of the y estimate (DBmax,bf):

display math(3)

[17] Other correction factors are available [e.g., Duan, 1983; Koch and Smillie, 1986; Hirsch et al., 1993], but since sample size was >30 and sy < 0.5 in most cases, CF was selected for computational simplicity.

[18] Critical flows Qc50 required to move the bed surface D50 sizes were predicted from power function regressions fitted to log transformed values of flows Qc′ at which a particle size DBmax was found mobile against log transformed DBmax′ sizes inline image; e and f are empirically determined. The critical flow Qc50 for the bedmaterial D50 size was computed by solving the critical flow curve for:

display math(4)

[19] The Qc16, Qc84, and Qc50sub for the bed surface D16 and D84 and the subsurface D50sub sizes were computed accordingly.

2.4.2 Computation of Shields Values

[20] Once the bankfull mobile particle size DBmax,bf as well as the Rc for mobility of preset particle sizes were known, critical bankfull Shields values inline image and critical inline image values for specified percentile particle sizes were computed by solving equation (1). The notation inline image refers to mobility of the bed surface D50 size, while inline image and inline image refer to mobility of the D16 and D84 sizes, respectively, and inline image to the subsurface bed D50 size as presented in Table 3. The notation inline image refers to unspecified particle sizes. Noncritical bankfull Shields values inline image were computed from the bed D50 size, Rbf and Sx (besides ρs, ρf, and g). This study presents an empirical approach and—similar to Mueller et al. [2005], Pitlick et al. [2008], and Recking [2009]—has computed Shields values without roughness corrections, accepting that Shields values in rough mountain streams back-calculated from field-measured, reach-averaged critical flow parameters and from reliable field observations of particle mobility may be higher than if computed with roughness correction.

Table 3. Input Parameters for Computation of Various Shields Values
Shields ValueInput Particle SizeInput FlowRequired Bed Load Transport ObservationsUnderlying Assumptions
For Specified Particle Size
inline imageBed surface D50Rc at average mobility of D50 sizeFull field-measured critical flow curve; Repeated sampling to cover wide range of Qc and DBmax for good regression fit and to recognize changes in trendBed load samples reliably represent the relation Qc = f(DBmax)
inline image, inline imageBed sub-surface D50subRc at average mobility of D50sub size
inline image, inline imageBed surface D16Rc at average mobility of D16 size
inline image, inline imageBed surface D84Rc at average mobility of D84 size
inline imageBed surface D50Rc at absolute largest mobile D50 size
For Specified Flow
inline image, inline imageDBmax,bf mobile at QbfRbf at average mobility of DBmax,bfField-measured flow competence curve, esp. near QbfBed load samples reliably represent DBmax,bf
inline imageDBmax,bf mobile at QbfRbf at absolute largest DBmax,bf size
For Specified Flow and Specified Particle Size (=Noncritical Shields Value)
inline image, inline imageBed surface D50RbfNoneBed D50 is largest size transported at Qbf (=bankfull mobility)

[21] The definition of particle mobility affects Shields values, and the range of incipient motion criteria extends from an occasional roll-over of an isolated particle to full participation of a specified particle size in gravel bed load transport [Stelczer, 1981]. In this study, a particle size class was considered mobile at a given flow when it was found to be transported based on the flow competence/critical flow curve (equations (3) and (4)). Consequently, DBmax represents the average largest mobile size of a wide range of DBmax′ sizes collected at the same flow in repeated bed load samples. It might be argued that flow competence and critical flow should rather refer to the absolutely largest particle size found mobile at a specified flow which would be indicated by the upper envelope of the flow competence and the lower envelope of the critical flow data. To investigate the effects of these differences, Shields values inline image and inline image were computed from particle mobility indicated by the lower envelope of the critical flow curves Qc,lo.env and the upper envelope curves DBmax,bf,up.env, respectively, using equation (1) (Table 3). Use of different sampler types affects the particle size measured as mobile. To determine the effect of sampler type on calculated Shields values, values of inline image, inline image, inline image, inline image, inline image, and inline image from bed load trap samples were compared to the corresponding Shields values inline image determined from Helley-Smith samples (Table 3).

[22] Critical flow conditions and critical dimensionless Shields values computed from bed load trap and other representative samplers for the average mobile DBmax particle size are compiled in Table 4. Linear and power functions are used to predict relations of various critical Shields values to stream gradient Sx (Table 5) because both regression types are presented in the literature, and both yield similar predictions for the central data range. Power functions are preferable on physical grounds. Neither of the functions should be extrapolated beyond the range of Sx{0.01–0.105}, especially not the linear functions which would provide unreasonably low predictions of inline image for small gradients.

Table 4. Critical Flow Depth dc, Critical Hydraulic Radius Rc, and Critical Dimensionless Shields Values for the Bed Surface D50, D50sub, D16, and D84 Particle Sizes as Well as for Bankfull Flow at the Study Streams
Stream and Year(s) Sampleddc50 (m)Rc50 (m) inline imagedc50sub (m)Rc50sub (m) inline imagedc16 (m)Rc16 (m) inline imagedc84 (m)Rc84 (m) inline imagedcbf (m)Rcbf (m) inline imageRbf (m) inline imagea
  1. a

    See Table 1 for dbf and D50.

Dupuyer 19950.430.390.0570.410.380.0630.340.320.1210.510.460.0310.520.480.0390.480.069
Little Granite 20020.340.310.0330.300.280.0590.280.260.0810.390.350.0180.320.290.0660.290.032
Harris 1991 (3B and 1A)0.280.270.0450.240.230.0570.150.140.1190.370.350.0290.720.670.0280.670.111
Halfmoon (riffle) 20040.530.470.0820.450.410.1340.390.360.2120.660.580.0410.710.610.0920.610.106
Halfmoon (bar) 20040.430.390.0680.390.360.1180.350.330.1960.500.450.0320.550.490.0330.490.085
Oak 1969–19710.230.260.0450.160.190.0800.140.170.0960.280.300.0340.320.290.0380.290.050
East Dallas 2007 (high)0.290.270.0480.210.200.0940.170.160.1420.370.340.0280.310.280.0470.280.050
East Dallas 2007 (low)0.290.270.0470.210.200.0940.190.180.1130.380.340.0280.310.280.0470.280.049
Little Granite 19990.400.380.0660.380.360.0890.350.330.1740.440.410.0320.390.370.0760.370.065
St. Louis 19980.360.330.0440.350.320.0790.340.310.1410.380.340.0210.380.340.0990.340.046
Squaw Cr. 19880.300.290.0850.290.280.1290.270.260.2500.330.310.0400.330.310.0390.310.091
Squaw Cr. 19910.360.340.1000.350.330.1540.330.310.3050.390.360.0470.330.310.0590.310.091
Cherry Cr. 19980.480.440.1280.450.410.2310.380.360.8410.540.480.0520.420.390.3630.390.114
Riedbach 20110.410.380.1190.340.320.1960.220.220.5230.570.520.0550.410.390.1460.390.121
NF Swan 2011 (high)0.230.210.0990.210.190.1570.170.160.3870.280.250.0370.230.220.0880.220.101
NF Swan 2011 (low)0.270.240.0850.250.230.1890.230.220.6690.280.260.0350.230.220.3450.220.075
Hayden 2005 (high)0.290.270.0990.270.250.1620.230.220.3630.330.310.0430.260.240.2500.240.089
Hayden 2005 (low)0.300.280.1020.260.240.1560.200.190.3120.390.350.0500.260.240.1470.240.089
E. St. Louis 2001 (high)0.530.410.2120.460.370.3830.360.301.1360.630.460.1000.440.350.9280.350.183
E. St. Louis 2003 (low)0.550.420.2180.490.380.3990.390.321.2190.640.460.1010.440.350.7760.350.183
Fool 2009-20100.200.150.0760.180.140.1480.160.130.2950.220.160.0350.200.150.2020.150.076
Erlenbach 2009–20110.220.190.2120.170.150.3990.120.110.9020.310.270.0860.310.290.1220.290.319
Table 5. Regressions Fitted to Relations Between Various Shields Values (Based on Bed load Trap Field Data) and Stream Gradient
Shields ValueStratification RangeLinear Function τ* = a + b SxPower Function τ* = g Sx h
a-const.b-coeff.r2g-coeff.ah-exp.r2
  1. a

    The power-function g-coefficients are reported anti-logged. The a-constant in linear fits indicates τ* values for zero-gradient channels, while the b-coefficient (actually b − a) indicates τ* for a 100% slope (45° angle). In the power fit, the g-coefficient indicates τ* at a 100% slope, while the h-exponent indicates how steeply the function τ* = f(Sx) curves upward.

  2. b

    Erlenbach excluded from regression.

  3. c

    Handfitted.

inline image 0.0371.740.850.980.670.76
 dbf /D50  Stratification by dbf /D50  
 3.8–4.10.00162.280.952.110.970.88
 4.5–6.0b0.0241.680.964.721.130.98
 6.5–8.1−0.0295.670.8826.01.470.92
 11–15−0.10212.2-c3151.95-c
 D84/dbf  Stratification by D84/dbf  
 0.13−0.24−0.1113.0-c5722.12-c
 0.30−0.43−0.0526.520.8772.91.760.88
 0.53−0.700.0311.770.911.170.760.86
inline image 0.0210.710.720.300.540.72
inline image 0.0523.370.882.290.760.83
inline image 0.06410.00.7212.51.030.74
inline image −0.00125.620.486.801.130.60
inline imageb −0.0969.740.8825.21.460.77
 D50/DBmax,bf  Stratification by bed mobility D50/DBmax,bf  
 0.2–0.40.0190.901.000.500.660.99
 0.6–1.20.0271.940.280.750.630.32
 1.2–2.80.0184.480.753.140.860.87
 3.2–5.10.1477.560.954.480.700.96
inline image 0.0381.880.780.710.570.58
inline image 0.0421.880.760.950.640.64
inline image 0.0260.810.710.330.530.59
inline image 0.0523.650.842.060.720.77
inline image −0.01512.40.8114.51.090.76
inline image −0.06311.60.8910.61.060.75
inline image 0.0381.510.780.740.620.65
inline image 0.0241.410.850.740.900.76
inline image −0.00572.800.483.401.130.60

[23] We suggest that our results are applicable to mountain streams in temperate climates in the indicated range of gradients, under conditions of frequently occurring highflows (20–150% Qbf) and sediment supply conditions associated with those events. Established torrents with high sediment supply may have a different relation of inline image versus Sx; Catastrophic debris torrents as well as streams with virtually no sediment supply are not included in the study. Note also that bankfull flow is morphologically not well represented in such channels.

3. Results

3.1. Critical Shields Values for the Bed D50 and Other Particle Sizes

3.1.1. inline image Values From Steep Streams in the Shields Diagram

[24] Particle Reynolds numbers Rep reach 6000–50000 for the mountain study streams. Hence critical Shields values inline image from this study plot on the right side of the Shields diagram as shown in Figure 3, far beyond Rep = 500 where Shields [1936] ended his experiments and assumed constancy for inline image. The inline image values back-computed for the study streams cover a 6.5-fold range from 0.033 to 0.22, a range spanned by extrapolating to Rep 50000 both the steep increase of the Shields curve observed within Rep 30–70 and the minimum Shields value just above 0.03. The wide range shows that a single value of inline image cannot describe incipient motion of the bed D50 size in mountain streams and that Rep is not a good predictor of inline image.

Figure 3.

Critical Shields values inline image computed for the mountain study streams plot to the right side of the Shields diagram, covering a range between the steepest upward trend of the Shields curve within Rep of 30–70 and the lowest Shields values (thick dashed lines).

3.1.2. inline image Values Increase With Stream Gradient and are Stratified by Relative Depth and Relative Roughness

[25]  inline image values from the study presented here have a strong positive relation with Sx (Figure 4) that is well described by a linear and a by power function (Table 5). Compared to other empirical relations for mountain streams [Rosgen, 1994; Mueller et al., 2005; Pitlick et al., 2008; Schneider et al., submitted manuscript, 2013] and to those modeled [Recking, 2009; Ferguson, 2012], the relation inline image from this study ranges within the center of those depicted in Figure 1 when Sx is smaller than 0.05, but is higher than the reported relations for Sx > 0.05.

Figure 4.

Relations of inline image with stream gradient and the Montgomery-Buffington [1997] stream types. The solid line shows the fitted power function, and thin dashed lines show 95% confidence intervals. The ribbon line shows the fitted linear regression.

[26] The alignment of inline image values with the Montgomery and Buffington [1997] stream type is loose (Figure 4), suggesting that each stream type may require a range of inline image values for particle motion, but the alignment provides first estimates of inline image in the field. For mixed pool-riffle/plane-bed streams and plane-bed streams with forced pool-riffle sequences, respectively, average critical Shields values of 0.055 and 0.065 are a good starting point. For plane-beds with low steps and low gradient step-pool streams (Sx < 0.05) respectively, average values of 0.108 and 0.092 are good estimates, and an average value of 0.21 may be suitable for steep step-pool streams (Sx close to 0.1). However, the association of inline image values with stream type offers no improvement in prediction because the stream type classification is mainly based on Sx.

[27] The increase of inline image values with stream gradient might be ascribed to several factors. The increase is numerically explainable because Sx (numerator in equation (1)) spans a wider (10-fold) range in the steep study streams than Rc and particle sizes D50, D84, etc., that each span a 2 to 4-fold range, hence Sx exerts the largest control over computed inline image values. By contrast, Wiberg and Smith [1987], Mueller et al. [2005], Lamb et al. [2008], Recking [2009], and Ferguson [2012] attributed the increase of inline image with stream gradient to physical causes, including near-bed hydraulic conditions, grain force balance, and total flow resistance. Other factors that are commonly associated with steep streams are increasing relative protrusion or relative roughness (D84/d), bed material sorting, a decrease in relative flow depth (d/D50), and an increase in the structural stability of the bed (e.g., by imbrication, steps, particle wedging, and stone structures [Church et al., 1998]), all of which could also act on their own to increase inline image without the additional effects of Sx. Teasing out the effects of poor bed mobility and Sx on inline image would be desirable and a subject for further research.

[28] Similar to results shown by Mueller et al. [2005], scatter in the relation of inline image versus Sx for the study streams is narrowed when the data are stratified by relative bankfull flow depth (dbf /D50) as well as by relative roughness (D84/dbf) (Figure 5). Hence, stratification improves the prediction of inline image from Sx, as shown in Table 5. The ratio dbf /D50 best stratifies the relation of inline image versus Sx, whereas D84/dbf explains the Erlenbach outlier that is not explained by dbf /D50. Stratifications by dbf /D50 and D84/dbf are well developed for moderately steep channels with Sx in the range of 0.02–0.04, whereas the stratification lines converge as Sx approaches 0.01 m/m. In general, inline image increases as streams become shallower and rougher (Figure 5). This is expected because shallow relative flow depths and rough beds in steep channels are associated with poor bed mobility. However, for stream beds of equal steepness, deeper (dbf /D50) and smoother (D84/dbf) channels yield higher values for inline image than shallower and rougher streams, the opposite of the general trend, which might suggest that other physical effects may overwhelm the effects of relative depth and roughness. However, being part of the Shields equation (equation (1)), the ratio dbf /D50 (and the related term D84/dbf) can also numerically control the Shields value. Besides the two ratios above, other channel parameters that correlate with Sx and that might be influencing the relation inline image versus Sx were found to show little effect. Bed material sorting which typically increases with Sx and may be expressed as the ratio D84/D50 as used by Ferguson [2012] or the Inman [1952] sorting coefficient (∣ϕ84 − ϕ16∣/2) does not stratify the inline image versus Sx relation in this study. Similarly, the parameters Qbf, % subsurface fines <8 mm, basin area A, bankfull stream width wbf, and the bed D16 size that decrease with Sx had weak negative relation with inline image but also do not improve prediction of the inline image versus Sx relation. Nevertheless, those weak relations suggest that inline image is somewhat affected by characteristics of stream steeps in ways other than Sx exerting a numerical artifact.

Figure 5.

Stratification of the relationship inline image versus Sx by (a) relative bankfull depth (dbf /D50) and (b) relative bankfull roughness (D84/dbf). The outlier in the group 4.5–6.0 dbf /D50 is the Erlenbach, a steep mountain torrent with high sediment supply [Turowski et al., 2009, 2011] that was not included in the regression. Thin dashed lines show 95% confidence intervals. Inset tables provide parameters for stratifying power functions. (1)Excl. Erlenbach; (2)handfitted in accordance with other relations. See Table 5 for linear regressions.

[29] Bed mobility or stability is a parameter that should affect Shields values, associating higher critical Shields values with less mobile streams. Bed mobility can be quantified as the ratio inline image [e.g., Buffington, 2012] which is essentially the depth ratio dbf/dc50. Alternatively, bed stability may be quantified by the grain size ratio D50/DBmax,bf where DBmax,bf is the largest particle size mobile at bankfull flow. The mobility and the stability parameters are inversely related to each other (r2 = 0.59 for the study streams). The grain-size ratio is preferred over the depth ratio in this study because the parameter DBmax,bf is used to assess stream stability, (e.g., Kaufmann et al. [2008, 2009] who predict DBmax,bf from the bed D50 size and an analytical expression of the Shields curve) as well as for stream restoration. D50/DBmax,bf ≅ 1 defines bankfull stability for particles ≥ D50 or bankfull mobility for particles ≤D50, whereas ratios >1 indicate stable beds and ratios <1 indicate unstable beds. In contrast to relative depth dbf /D50 and the relative roughness D84/dbf (Figure 5), neither the depth ratio for bed mobility nor the grain-size ratio expression for bed stability stratifies the relation of inline image versus S. High bed stability or low mobility is only weakly associated with higher critical Shields values inline image as shown in Figure 6. The poorly developed relations of inline image with D50/DBmax,bf and dbf /D50 suggest that bed stability/mobility has a minor influence on inline image compared to the major (and perhaps largely numerical) effect exerted by stream gradient. Figure 6 shows that each stream type experiences a wide range of bed stability (perhaps due to differences in sediment supply), hence the relation of bed stability to inline image and to stream type remains unclear [Buffington, 2012]. Similarly, the diagonal trends of the ratios dbf /D50 and D84/dbf in the plots of inline image versus Sx (Figure 5) suggest that relative depth and relative roughness are related to stream type in very general terms only and that any stream type (or stream gradient class) may experience a range of different dbf /D50 and D84/dbf conditions at incipient motion of the D50 particle size. More field data on measured incipient motion and further development in the morphological description of stream types regarding bed mobility/stability and sediment supply might help to bring clarification of this issue.

Figure 6.

Relations of inline image with bed stability/mobility expressed by (a) D50/DBmax,bf and (b) dbf/dc50. Dotted lines in Figure 6a depict fitted trendlines for individual stream types; plane-bed streams with low steps and low-gradient step-pool streams were grouped. Thick dashed lines indicate the trend fitted by eye. The Sx ranges indicated for individual stream types refer to those encountered in this study.

3.1.3. Critical Shields Values for Different Particle Sizes

[30] Depending on the application, entrainment might need to be predicted for particle sizes other than the bed D50 size, necessitating size-specific Shields values. inline image values are required to predict mobility of the bed surface D84 or when analyzing streambed erosion thresholds; inline image values back-computed from the bed D16 size may be required for determining flows that flush fine gravel from a coarse bed. Similar to inline image, critical Shields values inline image, inline image, and inline image in the study streams also increase with stream gradient (Figure 7 and Table 5), but the relations inline image differ among particle sizes. inline image values are on average half as large as inline image, while inline image values for the subsurface bed material D50 size are nearly twice as large as inline image; inline image values exceed inline image on average by a factor of 3.5. The relations inline image double in steepness from inline image to inline image (Figure 7). Progressively higher values for inline image, inline image and inline image are also shown by Recking [2009]. The linear regressions for inline image and inline image versus Sx fitted in the current study are generally similar to Recking's, albeit somewhat steeper and higher. The inline image and inline image values based on bed load trap samples exceed Recking's by about 20%, whereas the relation inline image versus Sx is up to 3 times higher and considerably steeper (Figure 8) (see Discussion for further exploration of the differences between the two studies).

Figure 7.

Relations of inline image values for the bed material D84, D50, D50sub, and D16 sizes with stream gradient (Sx). Fitted power functions are shown in solid lines. The relation of inline image with Sx (diagonal ribbon line, from Figure 9) is shown for comparison. The inset table provides the regression parameters, the Erlenbach is excluded for inline image. See Table 5 for linear regression functions.

Figure 8.

Linear regressions of inline image for the bed material D84, D50, D50sub, and D16 sizes fitted to bed load trap and Helley-Smith samples (Table 5) compared with inline image relations modeled by Recking [2009] for the D84, D50, D16 particle sizes. Empirical HS-based relations from Figure 1 are contained within the encircled area but exclude the flume data, the noncritical inline image and inline image [Mueller et al., 2005; Pitlick et al., 2008], and the mainly bed load trap based inline image data from J. M. Schneider et al. (submitted manuscript, 2013).

3.2. Critical Bankfull Shields Values

3.2.1. Scattered Relation of τ inline image With Stream Gradient

[31] Critical Shields values inline image backcalculated from the average largest bed load particle sizes collected at bankfull flows (DBmax,bf) (equation (3)) span a 32-fold range for the study streams, five times wider than the range of inline image values. inline image values increase steeply with Sx as portrayed in Figure 9. The increase of inline image with Sx, that is steeper than for inline image of any individual particle size, is caused by the extreme differences in bankfull flow competence among mountain streams.

Figure 9.

Critical Shields values inline image computed from the largest particle sizes mobile at bankfull flow versus stream gradient Sx. The low inline image value for the Erlenbach torrent reduces the correlation coefficient. Excluding the Erlenbach, the best-fit regression (thin line) is more suited for steep streams with low sediment supply. Thin dashed lines show 95% confidence intervals. See Table 5 for linear regression functions.

[32] A low and very low bankfull flow competence with values of inline image > 0.1 is typical where Sx is larger than 0.02 or 0.03 m/m and sediment supply is low—hence effects of structural bed stability set in. Given that bankfull flow in those plane-bed and step-pool streams moves only particles that were 20–50% of the bed surface D50 size, computed inline image values are 2 to 5 times higher than inline image, and inline image approaches the inline image values (see Figure 7). About half of the study streams are incapable of moving their bed D50 sizes at bankfull flow, and those streams may be classified as semi-alluvial [Meshkova et al., 2012; Bunte, 2012b]. The bed is rough and has a high degree of structural stability from particle interlock and wedging, from low (about 1 particle high) steps and high immobile boulder steps. The channels are typically incised and gravel bars are lacking, but most of those streams have occasional hydraulically forced patches of mobile gravel [Yager et al., 2007, 2012; Nelson et al., 2009, 2010]. By contrast, in some of the lower-gradient mountain streams where Sx is 0.01–0.02, as well as in the steep Erlenbach torrent with its high sediment supply, its highly erodible bed sediment and propensity to summer flash floods [Molnar et al., 2010; Turowski et al., 2009], bankfull flow transports particles up to twice the bed D50 size. Consequently, inline image values are about half those for inline image, approaching the inline image values computed for low Sx. Presence of active gravel bars, often within a channel incised into a largely inactive floodplain, is a common feature for this group of streams.

3.2.2. Stratification of τ inline image by Bed Stability

[33] With data scatter over an order of magnitude, the relation inline image is not well suited to provide guidance for selection of suitable values for inline image (Table 5), and narrowing the scatter via stratification is desirable. Because the measured bankfull mobile particle size DBmax,bf from which inline image is computed is part of the bed stability term (D50/DBmax,bf), it is not surprising that scatter in the relation of inline image (Figure 9) can be explained by D50/DBmax,bf as shown in Figure 10. For a specified gradient, the less stable streams (lower values of D50/DBmax,bf) have lower inline image values. Low bed stability also explains the rather low inline image value in Figure 9 for the highly erodible sediment step-pool Erlenbach torrent [Turowski et al., 2009, 2011; Molnar et al., 2010]. Compared to the wide scatter in the relation of inline image, stratification by bed stability improved the predictions (Figure 10 and Table 5). We are aware of the circularity here: DBmax,bf is used to compute inline image, hence the ratio D50/DBmax,bf) stratifies the relation inline image. However, the aim is to provide an estimate for the unknown inline image with which to compute DBmax,bf in unmeasured streams. The fitted stratifying relations can be practically employed to improve the prediction of inline image from stream gradient if estimates of relative bed mobility or stability can be gleaned from a visual assessment of morphological and granulometric channel features. Especially after performing a 400-particle pebble count, an operator has some idea of which particle sizes are mobile or are stuck in the bed and whether particles much larger or smaller than the bed D50 size might move at bankfull flow. Hence, bed stability expressed by the grain size ratio D50/DBmax,bf is easier to assess in the field than the depth ratio expression for bed mobility dbf /dc50 which also did not stratify inline image. Indicators of low bankfull bed stability in steep coarse-bedded streams include abundance of active gravel bars with particle sizes finer than the thalweg bed material, obvious near-stream sediment sources, as well as a high percent of surface and subsurface sand and pea gravel, and relatively many large particles that lie fully exposed on top of the bed. Signs of low sediment supply are low presence of fines on and in the bed, sediment retention by nearby dams, possibly presence of algae and moss cover, particles that are stuck deeply in the gravel/cobble bed, and indicators of structural bed stability (e.g., particle wedging, imbrication, stone structures). If bed stability can be roughly categorized as very high, high, bankfull stability, and low, critical bankfull Shield values inline image can be estimated accordingly. Adopting a finer morphological stream classification that considers bars and sediment supply as well as bed material texture and structure would be helpful to illuminate bed mobility/stability (e.g., torrent and low-step plane-bed as subtypes of step-pool; alternate bars and plane-bed without bars but meandering thalweg as subtypes of pool-riffle).

Figure 10.

Stratifications of the relations inline image by relative bed mobility (solid lines). Streams for which D50/Dbmax,bf is within 0.6–1.2 may be considered bankfull mobile streams. Inset tables provide parameters for stratifying power regressions, and thin dashed lines show 95% confidence intervals. See Table 5 for linear regressions. Ribbon lines show relations inline image established for various particle size percentiles in Figure 7.

[34] Interestingly, the relations inline image stratified by bed stability align with the relations established for mobility of specific percentile particle sizes presented in Figure 10: inline image values for very high, high, and low bed stability approximate those for inline image, inline image, and inline image, respectively. In bankfull mobile/stable streams, inline image values are similar to inline image values. These results suggest that streams with high bed stability transport their subsurface D50sub size at bankfull flow; highly mobile streams transport their bed surface D84, and very stable streams transport their D16 sizes at bankfull flow. Hence, inline image, inline image, and inline image provide estimates of inline image for very high, high, and low bed stability, respectively, while inline image is an estimate of inline image for bankfull mobile/stable streams. The illustrated correspondence between bankfull mobility/stability and mobility of specific bed material percentile sizes would benefit from more field data and may serve as a rule of thumb only. Nevertheless, estimating bankfull inline image values from a inline image value in reference to estimated bed stability is an improvement over the sweeping use of inline image or the noncritical value inline image as an estimate of bankfull inline image without regard to bed stability.

4. Discussion

4.1. Uncertainty in Computed Shields Values

[35] Random errors in the input parameters to the Shields equation (equation (1)) within and among studies contribute to uncertainty of inline image values as well as to scatter in relations of inline image with Sx and other variables. Yet with a sufficiently large number of inline image values, random errors should on average cancel out and not alter those relations. Bias, by contrast, systematically increases or decreases computed Shields values and changes the relations of inline image with Sx and other variables. Both error types are addressed below.

4.1.1. Effects of Unbiased Errors in ρs, R, Sx D50, and DBmax on τ inline image

[36] Errors in ρs typically result from using the particle density for quartz (2650 kg/m3) instead of the actual density for a specific rock type. Not accounting for a channel bed made up entirely of sandstone particles with ρs = 2300 kg/m3 would overpredict inline image by 27%, while not accounting for a basalt ρs of 3000 kg/m3 would underpredict inline image by 21%. Hence a user may adjust results presented here for particular rock densities. However, streambeds commonly comprise a mix of different rock densities. In aggregate, they may approach quartz density, or necessitate a more accurate but rather involved analysis that apportions ρs to the percentage frequency of different lithologies and to the different composition of lithologies in each size class, an unlikely routine.

[37] Stream gradients can be determined accurately when survey equipment is used, but hand levels can introduce errors as large as 30% [Isaak et al., 1999; Halwas and Church, 2002]. Errors in field-determined bankfull flow can also be large [Roper et al., 2008; Buffington et al., 2009], but the resulting errors in dbf (and Rbf) are only about half as large because d = k Q l, where l is typically within 0.3–0.6. Letting Qbf in the study streams vary by ±20 and ±40% changed Rbf on average by ±9 and ±20%, respectively. Without doing a formal propagation of error calculation, the combined random errors in ρs, R, and Sx are estimated to be approximately 15% for the bed load trap study streams.

[38] In coarse-bedded streams, the bed D50 as well as D84 sizes may differ by a factor of two among studies that use different field methods, stemming mainly from differences in sampling location within the reach, particle selection and measurement, and sample size [Bunte et al., 2009]. Errors in the bed D16 may be twice as high because the D16 size typically constitutes the fine tail of a skewed frequency distribution that is associated with higher variability [Rice and Church, 1996; Bunte and Abt, 2001a, 2001b]. Subsurface D50sub sizes reported among studies likely differ more than surface D50 sizes because differences in sample volume and sampled depth further contribute to variability [Bunte and Abt, 2001a]. Detailed bed material sampling efforts in the bed load trap study streams kept errors in the D50 and D84 bed material sizes at an estimated 5%, and at 10–20% for the D16 and D50sub sizes.

[39] The error potential for the DBmax size from misfitted flow competence/critical flow curves is high unless a large number of bed load samples have been collected. Individual days of a snowmelt highflow season typically exhibit well-defined relations of DBmax versus Q and of Qc versus DBmax, but changes in sediment supply cause the relations to differ over consecutive days and within the highflow season (e.g., Figure 2), resulting in daily and seasonal hysteresis [e.g., Mao, 2012; Swingle et al., 2012]. Combining all daily relations of DBmax versus Q over a highflow season produces a scatter of 1–2 ϕ units (a factor of 2–4) between the smallest and the largest instantaneous DBmax collected for a given flow [Bunte et al., 2008, 2012a] (e.g., Figure 2). Given the high natural variability of the largest transported bed load particle sizes, taking only a few isolated samples over a highflow season yields a scattered flow competence relation, while concentrating field sampling on a single day produces a well-defined relation of DBmax versus Q, but records only one of the many daily relations that exist over the highflow season. Intensive sampling that covers the highflow season is the only way to avoid this problem. In those study streams where 6–10 daily bed load measurements covered most days of the highflow season and where the flow-competence and critical-flow relations were segregated to accommodate periods of different sediment supply (e.g., rising and falling limbs of flow), errors in Qc or DBmax from (mis-) fitted flow-competence and critical-flow relations are guessed to be 5–10%. With lower sampling intensity and less highflow coverage, errors in fitted flow-competence and critical-flow curves may increase to 20–50%, and errors are typically largest early in the highflow season when DBmax sizes tend to be most variable.

[40] The combined errors from several input parameters in the current study are estimated to introduce 5–50% errors to computed inline image values. In studies where sampling methods produce inappropriate estimates of bed material size or where too few bed load samples are available to accurately define the flow-competence and critical-flow relations, errors in inline image values may amount to a factor of 2.

4.1.2. Effects of Biased Errors in DBmax Sizes on τ inline image

[41] Biased errors in the DBmax size or in critical flow Qc can stem from bed load samplers that produce biased samples. Flow competence and critical flow curves determined from bed load traps and a 7.6 by 7.6 cm Helley-Smith (HS) sampler at the same site differ considerably: the HS placed directly onto the stream bed collects larger gravel particles than bed load traps at low flows but smaller gravel particles at high flows [Bunte et al., 2004, 2008, 2010b, 2010c, 2012]. Consequently, flow-competence curves from the two samplers intersect (mostly within 60 and 80% of bankfull flow and mostly within DBmax sizes of 10 to 30 mm). HS flow-competence curves in this study have exponents of 0.5–1.3 and are less than half as steep as those from bed load traps deployed at the same sites (exponents of 1.3–3.5) (Table 2). Similarly, HS critical-flow curves have exponents of 0.35–0.88 (mean of 0.59) that are on average 1.6 times steeper than those from bed load traps (0.18–0.66, mean of 0.37). Those sampler-induced differences are reflected in back-computed Shields values. At study sites where both samplers were employed, the relations of inline image computed for the D50sub, D50, and D84 sizes are similarly steep for both samplers, but inline image values are on average higher by 6, 9, and 15%, respectively (Figure 8). The relations inline image intersect for the two samplers, and inline image values are smaller than the bed load trap inline image values for Sx < 0.03. The HS sampling bias against large gravel and cobbles overpredicts critical flow mobilizing coarse particles (and hence Rc) which, in turn, overpredicts Shields values for coarse particles. By analogy, the HS propensity for sampling overly coarse particles at low flows and on lower gradient streams explains the average lower inline image values for the HS. Similar to bed load trap results for inline image (see Figure 7), inline image values resemble inline image in steep streams (Sx ≅ 0.1). In gentler gradient streams with Sx = 0.01, inline image values approach inline image whereas the bed load trap inline image approaches inline image. Hence, HS samples suggest bankfull mobility of the bed D50sub size rather than D84 particles in lower gradient streams and underpredict bed mobility. Many of the field data used by Recking [2009] for his relations inline image are derived from HS samplers and tracer studies. However, Recking's relations (Figure 8)—which are lower than those from bed load traps—differ more from the HS-derived inline image relations in this study than from the bed load trap relations of inline image. Computational differences other than use of a particular bed load sampler obviously contribute to the differences in the inline image relations between the studies.

[42] A relatively large bias is introduced to inline image values if computations are based on either the absolutely largest particle size observed to be mobile at a specific flow or on the average largest size. The decision to use either one may depend on the study aim: averaging over a number of DBmax observations may be useful for regime characterizations, whereas the absolutely largest mobile particle size may be important for design questions that require a safety margin. The absolutely largest mobile particle size is indicated by the upper envelope of the flow competence data. For large flow competence data sets, that envelope is typically well defined rather than constituting an individual outlier data point. By extension, where particle mobility is expressed in terms of critical flow, mobility may refer to the average critical flow (regression function) or the lowest critical flow at which a specific particle size was observed to be mobile (lower envelop curve). When sufficient field data cover a wide flow range, the envelopes have the same slopes as fitted regression functions but are shifted, on average, upward by a factor of 2 (1.6–2.3 in the study streams) for flow competence curves, and downward by a factor of nearly two for critical flow curves.

[43] The effects of using the absolutely largest versus the average largest mobile particle size on inline image values can be estimated by halving the e-coefficients of the fitted power function critical flow curves (equation (4)) (Table 2). Halving the e-coefficients halves Qc for a given DBmax. The resulting reduction of dc (and hence Rc) is about half as much because d increases with Q with the power of 0.3–0.6 (see above). Not unexpectedly then, inline image values computed from the absolutely largest mobile particle sizes in the study streams were on average 25% lower than inline image computed from the average largest mobile particle size as presented in Figure 11a. The 25% reduction was observed for all particle sizes ( inline image, inline image, and inline image) and could explain the 25 and 38% lower inline image and inline image relations shown by Recking [2009] (Figure 8), suggesting that his inline image values might be derived from absolutely largest rather than average largest mobile particle sizes. Reasons for the up to 3-fold difference in the inline image relations between the two studies for steep gradients (Sx > 5%), however, remain unclear. Using the absolutely largest versus the average largest particle sizes has a larger effect on inline image values determined from the flow competence equation (equation (3)). Doubling the c-coefficient doubles the DBmax,bf size which causes inline image values to be halved (Figure 11b).

Figure 11.

(a) Values of inline image obtained from the regression function (gray circles) and the lower envelope (open squares) of the critical flow curve and fitted power regressions inline image (Table 5). Inset figure shows regression (solid line) and the lower envelope Qc50,lo.env (dashed line) of the critical flow curve. (b) Values inline image obtained from the regression function (gray circles) and the upper envelope curve (open squares) of the flow competence curve and fitted power regressions inline image. Inset figure shows flow competence regression function (solid line) and the upper envelope (dashed line). Inset tables in (a) and (b) provide power function regression parameters.

[44] Finally, biases may occur if the flow competence/critical flow approach is applied to natural streams as opposed to flumes where bed morphology and flow hydraulics are less spatially varied. In many steep streams it may be more likely that a collected DBmax particle was not entrained from the bed in front of the sampler but originated from a more mobile gravel patch upstream, and once entrained, continued its downstream path, perhaps aided by secondary currents. Hence inline image values derived from the flow competence/critical flow approach in natural streams may be lower for a specified particle size, R and Sx than in flume experiments conducted on a bed material with similar grain size distribution. By contrast, surface particles in beds of natural streams with low sediment supply may arrange to maximize bed stability (e.g., imbrication, wedging, stone structures) which may make those beds less entrainable than less structured flume beds with similar bed particle sizes.

4.2. Hiding and Exposure

4.2.1. Difference in Critical Shields Values Between Particle Sizes

[45] Our study shows a systematic increase of inline image values for the D84 to the D50 and the D16 sizes (Figure 7). Wiberg and Smith [1987] attributed higher inline image values for smaller particles to the effects of small particles hiding between larger ones and being therefore less well entrainable than from uniformly sized fine beds. Similarly, lower inline image values for larger particles were attributed to exposure above neighboring smaller particles which enhances entrainability of large particles in mixed sized beds compared to entrainment from uniformly sized coarse beds. However, higher inline image values for smaller particles do not necessarily indicate that smaller particles require deeper flows for mobilization than coarser ones. One might also consider that the difference between small and large particle sizes spans a wider range than critical flow depths and influences inline image (equation (1)) more than Rc. Values of inline image in the study streams are on average 3.5 times larger than those of inline image, whereas critical flow depths for the bed D16 size (Rc16) are on average 73% of those of Rc50, and the D50 sizes are on average 4.7 times larger than the D16 sizes. Hence, inline image exceeding inline image is more attributable to dissimilarity between the two grain sizes rather than to changes in Rc. Similarly, inline image values are on average 0.48 times those of inline image, while the Rc84 values are 1.12 times those of Rc50, and D50 values are 0.41 times those of D84; again differences in grain-size (where D84 is typically >2 D50) contribute more to inline image exceeding inline image than differences in critical flow. That higher inline image values for smaller particles are attributable to hiding, and lower inline image values for larger particles to exposure is also not supported by our field observations of bedmaterial surfaces during pebble counts at low flows and by visual observations of bed load dynamics at moderate high flows. Typically, we do not encounter many small surface gravels (e.g., 4–16 mm) that are lodged deeply in interstices between large particles (coarse gravel and small cobble) inaccessible to flow. Small gravels mobile during moderate high flows most likely originated their downstream journey on a localized, recently flooded, relatively mobile deposit of small gravels somewhere upstream. Once within the main channel bed, small gravel particles scurry around immobile larger gravels and cobbles, traveling over low lying (large) particles that are flanked by protruding (large) particles. Secondary currents aid transport of small particles by sweeping them to smoother transport paths along the banks, while eddies sweep fines gravels in and out of cobble wake deposits. In pebble counts we typically find relatively few large particles that are fully exposed on top of the bed ready for transport. Instead, many coarse gravels and small cobbles are either stuck deeply in the bed with only a small portion of the particle exposed to flow or are locked in bed structures (imbrication, stone structures, wedged between neighboring particles). For coarse material, entrainability appears to be controlled less by the degree to which the particle is exposed above its neighbors than by the extent to which the particle reaches below the bed surface.

4.2.2. Hiding Function Exponents

[46] Hiding functions in the form inline image [Parker et al., 1982; Andrews, 1983] are sometimes used to assess mobility of particle sizes other than the D50. The value estimated for inline image depends on the β-exponent for which a range from about −0.5 to −1.1 is commonly reported, but no relation exists from which to predict β (e.g., Schneider et al., submitted manuscript, 2013). The back-calculated values of inline image, inline image, and inline image in this study were used to estimate hiding function exponents β for all study streams. The range of Di/D50 extends from 0.1–0.3 at the low end to 2–3 at the high end, and none of the hiding functions show a change in trend. Values of β were found to range from −0.51 to −0.92. This study determined empirically that β is related to the exponents of the critical flow and flow competence curves. Hiding function exponents β closer to −1 are associated with flatter critical flow relations (lower exponents f) and steeper flow competence curves (higher exponents d). For β from −0.92 to −0.66, the best fit logarithmic functions between β and the exponents of the critical flow (d) and flow competence (f) curves are β = −0.72 − 0.28 log(d) and β = −0.68 + 0.27 log(f); r2 values are 0.76 and 0.75, respectively, and n = 17. The relations fall apart for β{−0.66 to −0.51}. Power functions can be fitted over the entire range of β when using the term β +1 (adding 1 seems to better preserve the relation between individual β than multiplication by −1). β is related to the exponents d of the flow competence curves by β + 1 = 0.36 d−0.78 and by β + 1 = 0.44 f0.75 to the exponents of the critical flow curves; r2 = 0.64, n = 22, and sy = 0.137 for both regressions. The relations of β with the steepness of the critical flow/flow competence curves are not unexpected. For a given stream, the hiding function may be simplified to (Rci/Di)/(Rc50/D50) = (Di/D50)β, assuming constancy of ρf, ρs, and Sx. If Di and D50 are eliminated from the left hand side term, the resulting equation represents a dimensionless curve of critical flow depth Rci/Rc50 = (Di/D50)φ which has a positive exponent φ and is related to the critical flow and flow competence curves. Although determinable, the relations of β or β +1 with the f and d exponents are not practical to predict the hiding function exponent β given the large field effort required to obtain a well-defined critical flow or flow competence curve. β + 1 was also found to increase with dbf /D50, but not well correlated, and appears to be nonmonotonically related to D84 and D84/D50. Those moderate correlations make prediction of β a problem for unsampled streams.

4.3. Specific Shields Values Are Required for Specific Purposes

4.3.1. τ inline image No Substitute for τ inline image

[47] In the absence of field-measured particle mobility, a study might estimate critical Shields values for the bed surface D50 size ( inline image) from noncritical bankfull flow and the bed D50 size ( inline image) [e.g., Buffington et al., 2004, 2006; Buffington, 2012]. In the mountain study streams, inline image values increase with stream gradient as shown by Buffington [2012], although with larger data scatter than inline image (Table 5). Noncritical dimensionless bankfull shear stress values ( inline image) provide accurate estimates of inline image in streams that can just transport their bed D50 particles at bankfull flow where inline image bankfull stability. The study presented here found the ratios inline image to decrease systematically with bed mobility or stability (Figure 12), reaching 1.6 for the least stable streams and decreasing to around 0.9 for the most stable ones, whereas Parker et al. [2007] suggested a general ratio of 1.6 for inline image. With a ratio of 1.6, inline image overpredicts inline image by approximately 60% in highly mobile streams; in very stable ones, inline image underpredicts inline image by 10%.

Figure 12.

Ratios of various Shields values plotted versus relative bed stability D50/DBmax,bf for the study streams: noncritical bankfull inline image to critical inline image, critical inline image to critical bankfull inline image; and noncritical bankfull inline image to critical bankfull inline image.

4.3.2. τ inline image Required to Predict the Bankfull Mobile Particle Size

[48] Only critical bankfull Shields values inline image (which are based on the largest particle size measured to be mobile at bankfull flow) can predict the largest bankfull mobile particle size (DBmax,bf). As shown for the study streams, inline image differed from inline image for other particle sizes (Figure 7). Therefore, no single value ( inline image, inline image, inline image, or inline image) can be used as a general predictor for DBmax,bf, except for bankfull mobile/stable streams where inline image. Otherwise, values of inline image differ widely from inline image depending on bed stability (D50/DBmax,bf). The negative relation of the ratio inline image with bed stability presented in Figure 12 shows that inline image is less than half the value of inline image on poorly stable beds (low values of D50/DBmax,bf). On highly stable beds, inline image is more than three times the value of inline image. If inline image was used instead of inline image to estimate DBmax,bf, inline image would underestimate DBmax,bf by a factor of more than 2 in the least stable study streams and overestimate DBmax,bf by more than 3 times in the most stable study streams.

[49] Because bankfull flow depth is used for computing both the noncritical inline image and the critical inline image, the impression may arise that inline image and inline image are closely related. However, computation of inline image and inline image differ greatly with respect to the particle sizes used, D50 in the first case and DBmax,bf in the latter. As a result, and except for their point of equality in bankfull mobile/stable streams, inline image and inline image are inversely related to bed stability; the ratio of inline image is near 4 for poorly stable beds (Figure 12) and drops to 0.2 for highly stable beds. Use of inline image instead of inline image to predict the largest bankfull mobile particle size DBmax,bf is therefore a poor option. In the least stable study streams, inline image underpredicts DBmax,bf by a factor of four while overpredicting DBmax,bf by fivefold in the most stable study streams.

[50] Estimating inline image in a variety of steep, coarse streams from a fixed Shields value is not advised. Taking for example the original value near 0.06 [Shields, 1936], the ratio inline image decreases with both bed stability and Sx and thus with the associated stream type. For the study streams, a value of 0.06 may be an appropriate estimate for inline image in plane-bed streams with forced pool-riffle sequences (Sx = 0.014–0.021 m/m) under bankfull mobility/stability (Figure 13a). However, 0.06 is less than one-tenth of inline image in the most stable step-pool study streams and would overpredict DBmax,bf there by more than 10 times. Another example is a Shields value of 0.03 that is proposed by Parker [1979] and Parker et al. [2007] for bankfull conditions in low-gradient gravel-bed streams based on hydraulic geometry considerations, and by Parker et al. [2003] for high Rep (actually the explicit Reynolds number inline image, as well as by CIRIA, CUR, CETMEF [2007] for steep earthen dams. A value of 0.03 is a suitable estimate of inline image in mixed plane-bed/pool-riffle streams (Sx = 0.01–0.014 m/m) of low bed stability (Figure 13b) but not generally suitable for bankfull mobile/stable streams where inline image exceeds 0.03 by 1.4 to 7 times depending on stream gradient. Using 0.03 in place of the actual inline image value for predicting the bankfull mobile particle size DBmax,bf [Kaufmann et al., 2008, 2009] biases the prediction of DBmax,bf size in all but plane-bed/pool-riffle streams with low bed stability and overpredicts DBmax,bf by as much as 30 times in highly stable step-pool streams. Such mispredictions then introduce considerable errors if bed stability is evaluated from a predicted D50/DBmax,bf.

Figure 13.

(a) Ratios of inline image and (b) ratios inline image, plotted versus bed stability for the study streams and stratified by stream gradient classes. Trendlines are fitted by eye.

5. Summary and Conclusions

[51] Predicting particle mobility in steep gravel-bed streams becomes increasingly important, yet wide scatter of reported inline image values around the Shields diagram and inter-study variability in relations of inline image versus stream gradient Sx challenge the selection of appropriate Shields values. Prediction of the largest bankfull mobile particle size DBmax,bf from the Shields approach is especially problematic because Shields values were not designed with that task in mind. This study raised awareness to the differences among Shields values computed for different purposes and from different field and computational methods and to the fact that specific Shields values are required for specific purposes. The study selected field methods which minimize errors in the characterization of bed material and mobile particle sizes, both of which commonly have a high error potential. Based on detailed field data sets, the study used a flow-competence/critical-flow approach to back-calculate critical Shields values inline image for various particle sizes and at bankfull flow. The computed values of inline image were related to stream gradient and then stratified by various parameters. The resulting relations provide guidance for selecting Shields values suitable for specific applications. Specific study results are listed below.

[52] 1. Average inline image values for streams with Sx = 0.01 were near 0.05 and increased to 0.2 for Sx of 0.1. The over 6-fold range spanned by individual inline image values indicates that a single value cannot represent inline image in steep streams. A positive relation of inline image and thus with stream type was confirmed. Stratification by relative depth (dbf /D50) and relative roughness (D84/dbf) improved the prediction of inline image but was unrelated to stream type. However, it remains unclear whether (or to what degree) the relations of inline image with Sx, dbf /D50 and D84/dbf are attributable to numerical artifact or physical processes.

[53] 2. Unexpectedly, inline image values were not statistically related to bed mobility or stability quantified either by a depth ratio dbf /D50 or a grain-size ratio D50/DBmax,bf, suggesting that each stream-type (or each stream gradient class) can experience a wide range of bed mobility, relative depths, and relative roughness at incipient motion. Expanding stream type classifications to include sediment supply and bed stability could advance the relation between critical Shields values and stream morphology.

[54] 3. inline image values back-computed from critical flows entraining the bed D84 size were about half as large as inline image values, while inline image values exceeded inline image by about 3.5 times. The steepness of the relations τ*c for various particle sizes with Sx progressively increased from inline image to inline image. Again, how much these differences are attributable to numerical artifact or natural processes remains unclear.

[55] 4. Hiding function exponents are positively related to the steepness of the critical flow and negatively related to the steepness of the flow competence curve, hence poorly predictable without a large field study.

[56] 5. Prediction of DBmax,bf requires critical Shields values inline image. inline image values approximate inline image when Sx ≅ 0.1 and inline image when Sx ≅ 0.01, thus spanning an over 30-fold range. inline image increases steeply with Sx; stratification by bed stability D50/DBmax,bf narrows and explains the scatter. inline image may be estimated from Sx if field evidence permits classification of bed stability as very high, high, bankfull stable, and low. inline image, as well as the noncritical inline image are equal to inline image at bankfull mobility/stability. inline image, inline image, and inline image may predict inline image in streams with very high, high, and low bed stability, respectively. Employing inline image or inline image instead of inline image to estimate DBmax,bf underpredicts DBmax,bf by 2 or 4 times, respectively, in poorly stable beds and overpredicts by 5-fold in highly stable streams. Setting inline image may predict DBmax,bf in poorly stable streams with Sx near 0.01, but overpredicts DBmax,bf by up to 30 times in highly stable streams with Sx near 0.1.

[57] 6. Bed load samplers that are unsuitable for collecting the largest mobile particles bias collection results and consequently the Shields values computed from them. Helley-Smith derived relations of inline image for the D50sub, D50, and D84 sizes are of similar steepness as those derived from bed load traps, but are 6, 9, and 15% larger due to undersampling of large bed load particles by the HS and a subsequent overprediction of critical flow. For Sx < 0.03, HS derived inline image values are smaller than those from bed load traps which reflects the oversampling of midsized particles on finer and shallower beds during lower flows.

[58] 7. Shields values computed from the absolutely largest mobile particle size as opposed to the average largest mobile particle size are systematically lower: inline image values are halved, and inline image values for various particle sizes are reduced by about 25%.

[59] 8. More reliable field data sets of flow competence and other stream parameters from diverse stream environments are desirable to ascertain predictive relations of critical Shields values with stream parameters.

Acknowledgments

[60] We dedicate this paper to John Potyondy (ret., USFS) who initiated the study. We appreciate John's foresight and longterm support of this work. We are grateful to Dieter Rickenmann and Jens Turowski (Swiss Federal Research Institute for Forest, Snow and Landscape, WSL) for giving us access to their gravel bed load data obtained from moving baskets at the Erlenbach. Field data from the studies by Milhous [1973], Whitaker [1997], Hassan and Church [2001], and Schmid [2011] were likewise very useful. Discussions with Tom Lisle, John Buffington, and Peter Wilcock provided food for thought that went into this study. Comments from Mike Church, Alain Recking, Dieter Rickenmann, Rob Ferguson, and an anonymous reviewer helped to improve the manuscript. This study as well as the collection of detailed bed load data over many years was funded by the United States Forest Service Stream Systems Technology Center.

Notations
A

basin area.

AQ

cross-sectional flow area.

a, c, e, g, i, k

regression coefficients.

b, d, f, h, j, l

regression exponents.

dbf

bankfull flow depth.

dc

critical flow depth for entrainment of an unspecified particle size.

dc50, dc50sub

critical flow depth for entrainment of the bed D50 and D50sub particle sizes.

dc16, dc84

critical flow depth for entrainment of the bed D16 and D84 particle sizes.

DBmax

average largest bed load particle size, determined from a flow competence curve.

DBmax,bf

average largest bed load particle size at bankfull flow, determined from a flow competence curve.

Dc

critical, i.e., mobile particle size at specified flow.

Dm

mean bed material particle size.

D50

median bed surface particle size.

D16, D84

bed surface particle size of which 16% and 84% of the distribution is finer.

D50sub

median bed subsurface particle size.

g

acceleration due to gravity.

n

sample size.

Qbf

bankfull flow.

Qc

critical flow for entrainment of unspecified D.

Qc50, Qc50sub

critical flow for D50 and D50sub.

Qc16, Qc84

critical flow for D16 and D84.

Rbf

hydraulic radius at Qbf.

Rc

critical hydraulic radius for unspecified D.

Rc50, Rc50sub

critical hydraulic radius for D50 and D50sub.

Rc16, Rc84

critical hydraulic radius for D16 and D84.

Rep

Particle Reynolds number (g · Rc · S)0.5 · Dm/ν.

r2

coefficient of determination.

sy

standard error of the y estimate.

Sf

friction gradient.

Sx

stream gradient.

Tw

water temperature.

v

mean flow velocity per vertical.

w

stream width.

wbf

bankfull stream width.

β

exponent of the hiding function.

δs

bed material sorting coefficient (∣ϕ16 − ϕ84∣/2).

ϕ

grain-size unit; ϕ = −3.3219·log(D[mm]).

ν

kinematic viscosity.

ρf, ρs

densities of water and sediment.

inline image

noncritical dimensionless shear stress from Qbf and D50.

inline image

noncritical dimensionless shear stress from Qbf and D50sub.

inline image

critical dimensionless shear stress for unspecified D.

inline image

critical dimensionless shear stress for D50 and D50sub.

inline image

critical dimensionless shear stress for D16 and D84.

inline image

critical dimensionless shear stress for Dcbf.

inline image

critical dimensionless shear stress for Dm.

inline image

critical dimensionless shear stress for Dc indicated by the lower envelope of the critical flow curves Qc,low env.

inline image

critical dimensionless shear stress for Dcbf indicated by the upper envelope curves DBmax,bf,up env.

inline image

critical dimensionless shear stress for unspecified D based on Helley-Smith bed load samples.

Ancillary