## 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 as the numerical value of the term:

at incipient motion of rounded to angular particles from relatively well-sorted beds with a specified *D _{m}* size in the absence of bed forms;

*ρ*and

_{f}*ρ*denote the water and sediment densities,

_{s}*g*is acceleration due to gravity,

*R*

_{c}is the hydraulic radius at incipient motion of

*D*,

_{m}*D*is the mean bed material particle size that in well-sorted distributions is similar to the median (

_{m}*D*

_{50}) size, and

*S*is the friction gradient which under conditions of uniform flow is similar to the channel bed gradient

_{f}*S*that is typically determined along the channel waterlines.

_{x}*Shields*[1936] presented values in relation to the particle Reynolds number that is controlled mainly by

*D*and secondly by

_{m}*S*

_{x}which covers a much wider range among the study streams than

*R*

_{c};

*ν*is kinematic viscosity that decreases with water temperature

*T*from 1.5 · 10

_{w}^{−6}to 1.0 · 10

^{−6}m

^{2}/s for

*T*from 5 to 20°C.

_{w}*Shields'*[1936] flume experiments employed beds with rather uniformly sized particles of less than 4 mm and ended near

*Re*= 500, but he (and especially later researchers) assumed that the nonmonotonic, spoon-shaped curve approaches a constant value of for

_{p}*Re*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 (

_{p}*D*

_{50}) than the mean (

*D*) due to the distribution skewness, the critical Shields value at incipient motion of the bed

_{m}*D*

_{50}size (a.k.a. Shields stress) became denoted as . 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 values for *Re _{p}* > 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 , a requirement, if the Shields equation (equation (1)) is to be solved for either

*R*

_{c},

*S*, or

_{x}*D*

_{50}. 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

*Re*takes values within 4000–100000.

_{p}*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

*D*

_{50}size increase with stream gradient

*S*in coarse-bedded steep streams as shown in Figure 1 [

_{x}*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

*S*offers an opportunity for prediction of Shields values. However, individual relations of Shields values versus

_{x}*S*likewise involve scatter, and the proposed relations differ among studies as presented in Figure 1. The inter-study variability in relations of 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

_{x}*S*,

_{x}*D*

_{50},

*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

*D*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

_{Bmax}*Parker et al*. [1982] (not discussed here). Apart from scatter and between-study variability in the reported relation of versus

*S*, 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 [

_{x}*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 (

*D*) 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

_{Bmax}_{,}_{bf}*D*

_{50}particle size is set in relation to the

*D*size [e.g.,

_{Bmax}_{,}_{bf}*Olsen et al*., 1997;

*Kappesser*, 2002,

*Lorang and Hauer*, 2003;

*Kaufmann et al*., 2008, 2009]. For

*Re*> 500, these applications use a constant Shields value, e.g., 0.03 or 0.056 to compute

_{p}*D*. However, the original Shields equation was not developed to predict the particle size mobile at bankfull (or any other) flow. When used to predict

_{Bmax}_{,}_{bf}*D*

_{c}, the Shields equation iteratively determines whether a (near-) uniform bed with a specific mean particle size (

*D*) can be mobilized given

_{m}*R*and

*S*(ignoring

_{x}*ρ*and

_{f}*ρ*here). Hence, solving the Shields equation for

_{s}*D*

_{c}given

*R*and

*S*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

_{x}*D*

_{50}size to determine the

*Re*- and hence the -value to be used when solving the Shields equation for the largest mobile particles at bankfull flow (

_{p}*D*) given

_{Bmax}_{,}_{bf}*S*and

_{x}*R*. This approach presupposes mobility of the bed

_{bf}*D*

_{50}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

*D*

_{c}based on

*R*

_{bf}and

*S*and a Shields value from the curve provides an unbiased estimate of

_{x}*D*only in streams that move their bed

_{Bmax}_{,}_{bf}*D*

_{50}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

*D*

_{50}size is not correct.

[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 (*Q*_{c}, *d*_{c}, and *R*_{c}) for specified particle sizes as well as critical (i.e., entrained) particle sizes (*D*_{c}) for a specified flow for each study stream. Critical Shields values were then back-calculated from those relations. Characterizing relations between *R*_{c} and *D*_{c} 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 are then regressed against stream gradient *S _{x}*. Scatter in the relations of is reduced via stratification by relative flow depth (

*d*

_{bf}/

*D*

_{50}) and relative bankfull roughness (

*D*

_{84}/

*d*

_{bf}), 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

*D*

_{16},

*D*

_{50},

*D*

_{50sub}, and

*D*

_{84}bedmaterial percentile particle sizes as well as for the largest particle sizes

*D*entrained at bankfull flow ( ) and compares measured critical Shields values and with noncritical Shields values derived from bankfull flow and the

_{Bmax}_{,}_{bf}*D*

_{50}size ( ). 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.