Scaling Relations for Bedload Transport and Flow Competence Relations In

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SCALING RELATIONS FOR EXPONENTS AND COEFFICIENTS OF BEDLOAD TRANSPORT AND FLOW COMPETENCE CURVES IN COARSE-BEDDED STREAMS WITH CHANNEL GRADIENT, RUNOFF YIELD, BASIN AREA AND SUBSURFACE FINES

Kristin Bunte, Research Scientist, Colorado State University, Engineering Research Center, 1320 Campus Delivery, Fort Collins, Colorado 80523, [email protected]; Steven R. Abt, Professor emeritus, Colorado State University, Engineering Research Center, 1320 Campus Delivery, Fort Collins, Colorado 80523, [email protected]; Kurt W. Swingle, Environmental Scientist; 630 Iris Ave., Boulder, Colorado 80304, [email protected]; Daniel A. Cenderelli, Fluvial Geomorphologist/Hydrologist, USDA Forest Service, National Stream and Aquatic Ecology Center, 2150 Centre Ave., Fort Collins, Colorado 80526, [email protected]; Dieter Rickenmann, Research Scientist, Swiss Federal Research Institute WSL, Mountain Hydrology and Mass Movements, Züricherstrasse 111, 8903 Birmensdorf, Switzerland; [email protected]; Dave Gaeuman, Research Scientist, Trinity River Restoration Program, Weaverville, California, [email protected].

Abstract Relations of bedload transport (QB) and flow competence (Dmax,QB) with water discharge Q are difficult to measure or to accurately estimate in steep mountain streams. To further the understanding of how gravel transport and flow competence curves differ among streams, this empirical study relates exponents and coefficients of power b g functions fitted to gravel transport (QB = aQ ) and flow competence curves (Dmax,QB = fQ ) to watershed, channel, and bed material parameters. Exponents b and g are found to systematically increase with basin area, the bankfull width/depth ratio and stream width, and to decrease with the bankfull unit runoff yield per basin area as well as the percentage of subsurface fines. The exponents are non-monotonically (convexly) related to stream gradient, unit stream power, relative bankfull depth, and roughness. A tight linear relation between the b- and g-exponents reveals that steep QB curves result when increasing flows find increasingly larger particles to transport, a feature best expressed in steep plane-bed streams. Coefficients a and f are inversely related to the b and g exponents, respectively, hence scaling relations positive for exponents are negative for coefficients and vice versa. Overall, r2- values of scaling relations are within 0.3 to 0.6 and significant, sufficient to determine the range of expected exponents and coefficients, but not tight enough to predict them with certainty. Scaling relations are tightened, and their practical applicability for predicting QB and Dmax,QB curves enhanced, when stream types are segregated based stream gradient and basin area.

INTRODUCTION

Particle entrainment and subsequent bedload transport is difficult to predict in steep mountain streams because entrainment and transport are caused by complex interactions between flow hydraulics and sediment transport dynamics. Flow hydraulics, including turbulence, scour jets, wake eddies, and secondary circulation, exert fluctuating and variable forces on coarse channel beds. Coarse beds are typically comprised of not only a wide mix of particle sizes ranging from sand to boulders, but also contain a variety of structural characteristics that develop in response to the interactions between flow hydraulics and sediment supply (Church et al., 1998). Particles may be exposed on top of the bed, embedded firmly in the bed, integrated in steps, particle clusters or stone structures, or anchored to other particles, tree roots, or bedrock. Hence, bed particle entrainability ranges from highly mobile to very stable (e.g., Bunte et al., 2013). The variability of flow hydraulics, sediment supply, and channel bed characteristics between streams controls the rate at which the quantity and size of gravel in motion increases as discharge increases. As a result, each stream has a specific bedload transport and flow competence relation. The variability of transport and flow competence curves is evident in studies that bring together observations from numerous streams (Barry et al., 2004; Bunte et al., 2008, 2013, 2014; Bathurst, 2013; Schneider et al., 2014). Case studies also show that bedload transport and flow competence curves can vary for a given stream in response to changes in sediment supply which alter bed material conditions and particle transportability within or between events (e.g., Beschta, 1987; Lenzi, et al., Mao et al., 2012; Yu et al., 2009; Hassan et al., 2014). The complexities of flow hydraulics, sediment supply, and bed structure—all of which are difficult to quantify—cannot be accounted for in bedload transport equations typically used for estimating transport rates or incipient motion. To be simple enough for practical application, transport equations must be limited to a few parameters (typically flow depth, stream gradient, and the median size of bed surface particles). Those bedload equations therefore cannot capture the processes that determine how gravel transport rates and flow competence relate to flow in a specific stream, and computed transport rates can deviate from measured gravel transport rates by several orders of magnitude (e.g., Bathurst, 2007; Barry et al., 2008; Schneider et al., 2015). Numerous applications in fluvial geomorphology and channel management would benefit from a method to quickly and accurately estimate the steepness (exponent) and y-axis intercept (coefficient) of gravel transport and flow competence relations from simple field-measured parameters, or more conveniently from parameters quantified from maps and areal photographs. This study uses an empirical approach similar to that used for quantifying suspended sediment transport relations (e.g., Asselmann, 1999, 2000; Syvitski et al., 2000; Dodov and Foufoula- Georgiou, 2005; Yang et al., 2006) and provides a comparison and prediction of exponents and coefficients of bedload transport and flow competence curves. Earlier studies that empirically predicted exponents and coefficients of bedload transport curves from watershed, flow, channel, and bed material parameters were based on a narrow range of b-exponents (Barry and Buffington, 2002; Barry et al., 2004), an artifact of the Helley-Smith samplers used (Bunte et al., 2004, 2008), or the empirical studies lacked robustness due to data paucity (Bunte and Swingle, 2003; Bunte and Abt, 2003; Bunte et al., 2006). For this study, we gathered a large dataset from a wide range of coarse- bedded mountain streams where gravel transport was measured with samplers suitable for gravel bedload to 1) show how watershed, flow, channel, and bed material parameters affect exponents and coefficients of bedload transport and flow competence curves; 2) to assess which parameters best capture the characteristics of the flow- and bed material interactions; and 3) to explore the potential for improving the prediction of bedload transport. While simple watershed, flow, channel morphometry, and bed material parameters are obviously not the direct factors entraining or restraining bed particles, this study proposes that those parameters have either provided conditions for actual channel processes or been influenced by them and hence serve as easily quantifiable proxies for actual gravel transport processes, especially when streams are segregated according to gradient and size.

METHODS

Compilation of datasets Of the 45 datasets of gravel bedload transport relations (QB curves) from wadeable, steep gravel- and cobble-bed streams worldwide compiled in this study, 29 datasets also include flow competence relations (Dmax,QB curves). Forty percent of the datasets are derived from gravel transport and flow competence measured mainly in the Rocky Mountain streams with snowmelt regimes. (Bunte et al., 2004, 2008, 2010a; Potyondy et al., 2010). The other datasets include streams in alpine and arctic pro-glacial environments, densely forested watersheds, and high desert environments. Sediment supply ranges from very high to very low with bankfull unit transport rates that extend over ten orders of magnitude from 1E-4 to 1E6 g/m·s. Study sites span a wide range of basin areas (1-260 km2), bankfull stream widths (1-20 m), stream gradients (0.007-0.136 m/m), and bed surface sizes (D50 of 20 - 200 mm, D84 of 33-360 mm). Gravel transported in the study streams was collected either with bedload traps, vortex, net-frame, basket, and pit samplers (see references in Ryan et al. (2005a) and Bunte et al. (2013) for those samplers), as well as with hanging baskets, including those that deploy automatically (Rickenmann et al., 2012), or was quantified from continuous, automated surveys of an accumulating debris pile volume (Lenzi et al., (1999, 2004, 2006). Bedload data from pressure difference samplers were not included in this compilation because they yield QB and Dmax,QB curves that are significantly flatter that those from bedload traps (Bunte et al., 2004, 2008, 2010b) and are not thought to accurately represent the transport-discharge relation. Also not included are steep streams in which wedges of fine-grained sediment build up low-gradient sections between log steps (Green et al., 2014). Most of the gravel bedload datasets used in this study are described in Bunte et al. (2014), and many of the flow competence data are summarized in Bunte et al. (2013).

Fitted bedload transport and flow competence curves The increase of transport rates QB with water discharge Q (QB curves) is typically (and admittedly somewhat simplistically) described by power functions in the form QB = a Q b, where a and b are empirically determined (e.g., Barry et al., 2004; Bunte et al., 2008, 2014). Exponents b, which refer to flow quantified by Q in this study, are lower than exponents of bedload transport curves obtained when water flow is quantified by flow depth d (or shear stress). Exponents b are independent of the units used to quantify QB and Q, while a-coefficients are unit-dependent. All the a-coefficients in this study are based on QB units 3 of g/s and discharge in units of m /s. Flow competence relations (Dmax,QB curves) are also described by power g functions as Dmax,QB = f Q (e.g., Mao, 2012; Bunte et al., 2013), where Dmax,QB is the largest bedload particle size 3 collected in a sample, and f and g are empirically determined; Q is quantified in units of m /s, while Dmax,QB (often measured in 0.5 phi increments) is expressed in units of mm. For some of the study streams, sediment supply changed between the rising and falling limbs of flow during a highflow season, or between individually sampled years. When this caused notable differences, two separate QB and/or Dmax,QB curves were fitted. Mathematically complex function types may describe gravel transport or flow competence relations more accurately (e.g., Gaeuman et al., 2014), but simple power functions described all of the authors’ QB and Dmax,QB curves. Power functions are

commonly reported for QB and Dmax,QB curves in other studies, and the empirically quantified exponents and coefficients are fully adequate to describe the scaling relations developed in this study.

Stream parameters Four stream parameters were quantified for the study datasets:  basin area (A),  bankfull unit runoff yield (qbf/A) where bankfull flow Qbf was field-determined or based on the 1.5-year recurrence interval flood (Castro and Jackson 2001), and bankfull unit flow is qbf = Qbf/wbf,  stream gradient (S), and  the percentage of subsurface sediment < 8 mm (%Dsub<8) (not available for all the bedload dataset in this study).

Study stream grouping The progression from step-pool to plane-bed morphology in mountain streams is generally concomitant with an increase in basin area. However, the lower gradient plane-bed streams with occasional riffle- pool sequences occur in small drainages <10 km2 as well as in large drainages of 60 - 260 km2. Similarly, some step-pool streams in this study have small basin areas of only a few km2, while others have areas up to 27 km2. To reflect the different sizes of streams with similar gradients, study reaches were first categorized into three stream gradient classes to denote the basic channel morphology (Montgomery and Buffington, 1997):  step-pool,  steep plane-bed with low, narrow steps but no pools, and  low gradient plane-bed with occasional pool-riffle sequences, especially forced around sharp channel bends.

The three channel types were then segregated into those with small and large basin areas along a cutoff value of 15 km2 (Table 1) which is slightly less than the dataset’s geometric mean drainage size. Those group thresholds are somewhat arbitrary and are reflective of conditions encountered among the study streams and could shift if more data sets become available for this line of analyses.

Table 1 Stream type grouping based on basin area size and stream gradient.

Stream type Drainage Gradient range Secondary morphological Abbreviation area (km2) (mm) features Small step-pool < 15 rare, short plane-bed small s-p 0.040 – 0.136 Large step-pool > 15 reaches large s-p Small steep plane-bed 1) < 15 some low, narrow steps, no small steep p-b 0.016 – 0.038 Large steep plane-bed > 15 plunge pools large steep p-b Small plane-bed < 15 forced or occasional pool- small p-b w/p-r 0.007 – 0.014 Large plane-bed > 15 riffle sequences large p-b w/p-r 1) not encountered among the study streams

Regression analyses Exponents b and g as well as coefficients a and f from all datasets and for the stream groups delineated in Table 1 were regressed against the various watershed, channel, and bed material parameters. Power 1 2 3 4 functions in the form of b = α1·X , a = α2·X , g = α3·X , and f = α4·X generally obtained the best fit, where X denotes a watershed or streambed parameter;  and  are empirically determined.

RESULTS AND DISCUSSION

Scaling relations of exponents from QB and Dmax,QB curves with watershed, channel, flow, and bed material parameters The steepness of the QB and Dmax,QB curves compiled in this study differed widely between streams. Exponents b covered a seven-fold range from 2.6 to 18.2, while exponents g spanned a 20-fold range from 0.27 to 5.5. These exponents vary systematically with watershed, flow, channel, and bed material parameters. Exponents b and g are both positively, moderately, but nevertheless significantly related to basin area size A with r2-values of 0.32 and 0.40 (Figure 1a). Exponents b and g decrease with unit bankfull runoff yield (qbf/A) (Figure 1b), a parameter that indicates the flow per drainage area available to supply and transport bedload. Scaling relations are likewise negative with %Dsub<8 (Figure 1c). An abundance of subsurface fine gravel and a well-developed armor in coarse bedded streams typically suggest that a large supply of fine gravel is transported. The negative scaling relations 2 (Figure 1b and c) are moderately well-defined (r of 0.20 to 0.43) and likewise significant, and they confirm that QB (and Dmax,QB) curves are flatter in streams with higher sediment supply and more bed mobility (Dietrich et al., 1989; Laronne and Reid, 1993; Lisle, 1995; Reid and Laronne, 1995; Laronne et al., 2001; Hayes et al., 2002; Lisle and

Church, 2002; Barry et al., 2004; Ryan et al., 2005b; Bunte et al., 2006; Gran et al., 2006; Hassan et al., 2008; Diplas and Shaheen, 2008).

100 100 a) 0.21 b) -0.25 y = 3.92x 0.21 b-exp. y = 2.82x y =2 3.92x R = 0.40 g-exp. y =2 2.82x-0.25 2 R = 0.48 p R< 0.0001 = 0.40 2 pR < 0.0001= 0.48 10 10

Exponents Exponents 1 1 0.27 y = 0.386x-0.35 g y = 0.69x 0.27 y = 20.69x 2 R = 0.32 R = 0.39 R2 = 0.32 p = 0.0003 p < 0.0001 0.1 0.1 0 1 10 100 1000 0.001 0.01 0.1 1 Basin area (km²) Unit bankfull runoff yield (m³/m·s·km²) 100 100 c) -0.58 d) y = 50.3x mean = 10.8 R2 = 0.203 stdev = 4.7 p = 0.0059 10 10 mean = 8.4 stdev = 2.8 mean = 5.0 stdev = 2.2

Exponents 1 Exponents 1 mean = 1.9 stdev = 0.7 y = 336x-1.62 mean = 2.7 mean = 1.2 2 stdev = 1.5 stdev = 0.6 R = 0.43 plane-bed steep p < 0.0001 w/ pool-riffleplane-bed step-pool 0.1 0.1 10 100 0.001 0.01 0.1 Percent subsurface fines <8 mm StreamStream gradient

Figure 1 Scaling relation of b- and g-exponents with the four stream parameters from the 44 datasets show a positive trend with A (a), negative trends with qbf/A and %Dsub<8 (b, c), and a non-monotonic, convex trend with S (d). Yellow symbols denote b-exponents of QB curves and purple symbols denote g-exponents of Dmax,QB curves. All relations are significant.

Exponents non-monotonically related to stream type Among the study streams, the highest exponents coincide with streams that are moderately steep (S of 0.016 to 0.039 m/m), moderately rough (D84/dbf of 0.03 to 0.04), moderately deep (dbf/D50 6 to 8) and moderately powerful (bf of 7 to 11 kg/m·s) (Figure 1d). Those reaches have steep plane-bed channels with occasional low, narrow steps, but no plunge pools. Exponents are lower in the lower gradient study reaches where S 0.014 m/m, D84/dbf <0.03, dbf/D50 >8, and stream power <7 kg/m·s. Those conditions are typical of the lower gradient plane-beds with occasional or forced riffle-pool sequences. The trend of rating curve flattening for lower gradient streams is attributed to higher sediment supply with less transport capacity and competence (see above). This study identifies a second, opposing trend: Compared to the steep plane-bed reaches, exponents are about half as large in the steepest streams where S >0.04 m/m. These channels are coarse, rough, and shallow; they have high stream power, and exhibit step-pool morphology. As a result of two opposing trends, exponents b and g have non-monotonic (convex) relations with S (Figure 1d).

Occurrence of the steepest QB curves in steep plane-bed streams is attributed to the combination of two bed material conditions: The inundated bed surface harbors minimal fine gravels, and their scarcity keeps transport very low early in the highflow season. Transport is still relatively low at moderate flows, but at flows above bankfull, coarse gravel and cobbles on the upper surface of bar deposits—inaccessible at moderate flows—become available for transport. The large difference between very low transport rates of small gravels early in a highflow season and cobble transport at above bankfull flow may span 7-8 orders of magnitude and causes steeply increasing QB and Dmax,QB curves

in steep plane-bed streams. By contrast, flat QB curves result in step-pool streams where small gravel pockets provide ample bedload supply early in the highflow season, but flows above bankfull do not produce cobble transport because bar deposits are scarce and bed particles are restrained by structural bed stability. Low gradient plane-bed streams take an intermediate position. The low flow sediment supply is moderate, and an increase in QB and Dmax,QB at the highest flows may be slowed if moderate flows have already transported all but the very largest bed particles.

Scatter around the scaling relations Scaling relations in Figure 1 provide suitable estimates of the range of exponents to be expected for QB and Dmax,QB curves for a given parameter value. However, even for the best- developed scaling relations, measured b- and g-exponents vary by a factor of about 2 around the predicted value. Much of this variability is attributed to natural inter-annual variability in QB and Dmax,QB curves that is not related to or not yet reflected in watershed, channel, runoff, or bed material parameters. Variability in exponents also arises from less than optimally measured QB and Dmax,QB relations due to sampling errors which may include a low number of field samples, a narrow range of measured flows, poor quantification of the lowest or highest QB, unfortunate timing of samples within the high-flow season, or a poor fit of measured data to a power function. The remainder of the variability reflects the suitability of a watershed, channel, flow or bed material parameter to serve as a proxy for processes that cause gravel transport, and the parameter’s ability to provide or respond to the specific conditions that supply, entrain, or restrain particles and determine gravel transport.

Scaling relations for exponents segregated by stream type and stream size Prediction of b- and g-exponents is improved when scaling relations in Figure 1 are segregated by stream type. The discussion here is limited to b- exponents because the effects of stream-type segregation are very similar for b- and g-exponents, but b-exponents have richer datasets.

For scaling relations of b with A (Figure 2a), step-pool and steep plane-bed streams follow the same positive trend, irrespective of stream gradient. However, a secondary stratification by S occurs in the low-gradient plane-bed

100 100 large p-b w/ p-r a) b) large p-b with p-r lrg.steep p-b y = 3.33x0.28 lrg.steep p-b large s-p R2 = 0.42 large s-p small s-p p = 0.012 small s-p small p-b w/p-r small p-b w/p-r

10 10

y = 4.10 x-0.17 b-Exponents b-Exponents 0.14 y = 4.90 x R2 = 0.52 R2 = 0.46 p = 0.0036 p <0.0001 y = 2.28x-0.31 y = 2.28x-0.31 R2 = 0.50 R2 = 0.50 p <0.0001 1 1 0.1 1 10 100 1000 0.001 0.01 0.1 1 3 2 Basin area (km2) Unit bankfull runoff yield (m /m·s·km )

100 100 large p-b w/ p-r large p-b w/ p-r d) lrg.steep p-b c) large steep p-b large s-p large s-p -1.15 small s-p y = 406 x small s-p 2 small p-b w/p-r R = 0.42 small p-b w/ p-r p = 0.012 y = 62.5 x-0.61 2 10 y =R 62.5 = 0.51 x-0.61 10 p =2 0.0041 R = 0.51 -0.41

b-Exponents y = 2.02x b-Exponents R2 = 0.31 p = 0.0009 y = 58.5x-0.77 R2 = 0.47 plane-bed steep step-ppol p = 0.041 w/ pool-riffle plane-bed 1 1 10 100 0.001 0.01 0.1 1

Subsurface percent fines < 8 mm Stream gradient (m/m)

Figure 2 Scaling relations for b-exponents shown in Figure 1 segregated by stream type.

streams and describes a scaling relation with a flatter trend. The same segregation patterns hold true for scaling relations of b with qbf/A (Figure 2b), which are better defined than for A and make qbf/A a better predictor of b. The non-monotonic trend of b-exponents vs. S (Figure 2d) is not further segregated by basin area. However, gradient and basin area act together and define three distinct scaling relations for b with the %Dsub<8 (Figure 2c). The b- exponents for step-pool and steep plane-bed streams are clearly differentiated by stream size, such that large, steep plane-bed and step-pool stream define a scaling relation of b with subsurface fines that is notably higher than that for small step-pool streams. The low-gradient plane-bed streams follow a flatter scaling relation, but again with higher exponents for larger, and slightly lower exponents for smaller streams. Segregation by stream type is especially effective for the parameter %Dsub<8 as it narrows the range of b-exponents to a factor of about 2 for a given %Dsub< 8 and stream type, rendering %Dsub<8 a parameter suitable for the prediction of b.

Interrelatedness of steepness and coefficients of QB and Dmax,QB curves Exponents with exponents and coefficients with coefficients The a-coefficients and b-exponents of QB curves as well as the f-coefficients and g-exponents of Dmax,QB curves are strongly related. Exponents b and g increase in direct proportion with a well defined linear relation (r2 = 0.91) as shown in Figure 3a. This proportionality explains the almost parallel trend of the scaling relations for b- and g-exponents shown in Figure 1 and lets g be computed from b and vice versa. The tight, positive relation between b- and g-exponents is not segregated by stream types and clearly shows that the steepness of QB curves depends on whether increasing flows find increasingly larger particles to entrain and transport. If that is the case, Dmax,QB and QB curves steepen.

6 1000 large p-b with p-r a) large p-b with p-r b) p-b, low steps p-b, low steps y = 0.29 x – 0.60 small s-p y = 0.2923x - 0.5992 small s-p 5 R2 = 0.91 100 small p-b w/p-r R2 = 0.9129 small p-b w/p-r p < 0.0001 large s-p curves large s-p curves B 4 10

max,Qb max,Q D D

3 f 1

2 0.1 0.21 y = 18.4x0.21 icients o icients y = 218.4x ff R = 0.90 R2 = 0.90 1 0.01 p < 0.0001

-coe -exponents of f g 0 0.001 1E-14 1E-12 1E-10 1E-08 1E-06 0.0001 0.01 1 100 10000 1E+06 0 5 10 15 20 b -exponents of Q curves B a -coefficients of Q B curves

1.E+06 -2.22x 1000 y = 312554e large p-b with p-r 1.E+05 c) -1.65x d) 2 y = 109e 1.E+04 R = 0.61 lrg. steep p-b R2 = 0.62 1.E+03 p < 0.0001 small s-p 100 1.E+02 p < 0.0001 small p-b w/p-r curves 1.E+01 curves B 1.E+00 large s-p

Q 10

f 1.E-01 1.E-02 max,QB s o

t 1.E-03 D 1.E-04 1 y = 93.3e-1.51x en i 1.E-05 2 c -1.97x 1.E-06 y = 35089e R = 0.69 ffi large p-b with p-r 2 p = 0.0008 1.E-07 R = 0.80 0.1 1.E-08 lrg. steep p-b p < 0.0001 -coe 1.E-09 small s-p a 1.E-10 -4.71x y = 307e-2.96x 1.E-11 y = 1E+12e 0.01 small p-b w/p-r 2 -coefficients of 2 1.E-12 R = 0.85 f large s-p R = 0.80 1.E-13 p = 0.0011 p = 0.016 1.E-14 0.001 1101000.1 1 10

b -exponents of Q B curves g-exponents of D max,QB curves

Figure 3 Interrelatedness of QB and Dmax,QB curves: among the two exponents (a), among the two coefficients (b), and among coefficients and exponents (c and d).

Inherent inverse relations between exponents and coefficients Power function coefficients represent not only the y-axis intercept for x = 1 but are also inversely related to the function’s steepness. Negative relations between exponents and coefficients are a general feature of power functions. Consequently, all positive scaling relations for exponents (Figure 1) become negative for coefficients and vice versa. For the study stream QB curves, an increase of b-exponents by 1 is met by a 1 to 2 order of magnitude decrease in a-coefficients (Figure 3c); f-coefficients of Dmax,QB curves decrease less, by about half an order of magnitude as g-exponents increase by 1 (Figure 3d). Coefficients are also directly controlled by the magnitude of bedload transport rates. For example, a doubling in transport rates for all Q doubles the coefficient. The two-fold control of coefficients (by exponent steepness and by transport rates) complicates a comparison of coefficients among streams: A QB curve with a coefficient 10 times larger than that of another QB curve at a specified parameter x-value does not mean that transport rates of the two curves differ by a factor of 10.

Values of a- and f-coefficients from fitted QB and Dmax,QB curves extend over about 10 and over 4 orders of magnitude, respectively. Nevertheless, the negative relation between a-coefficients and b-exponents is fairly well and significantly described by an exponential function (r2 of 0.61) (Figure 3c), as observed for suspended sediment (Syvitski et al. 2000; Yang et al. 2007). The relation between a-coefficients and b-exponents is clearly affected by stream type. Segregation by stream gradient raises the correlation for large, steep, plane-bed streams to an r2 of 0.80 and to an r2 of 0.85 for large, low-gradient plane-bed streams with occasional riffles and pools (Figure 3c). Hence for the large study streams, segregation by S narrows the range of a-coefficients for a given b-exponent from 10 to about 6 orders of magnitude. Stream size segregates the a-b relation for step-pool streams; large step-pool streams have a steep relation, while small step-pool streams have a flatter one. Stream gradient segregates the a-b relation between small streams with higher a-coefficient for small step-pool streams than for small, more gently sloped plane-bed streams.

For Dmax,QB curves, the trend between f-coefficients with g-exponents is likewise fairly well and significantly described by a negative exponential function (r2 of 0.62) (Figure 3d). Segregation by stream gradient improves r2 from 0.62 for all data to 0.69 for large steep plane-beds, and to 0.80 for large low-gradient plane-bed streams with occasional riffles and pools. However, a further segregation of step-pool streams by size or small streams by gradient is not evident.

Scaling relations of QB and Dmax,QB curve coefficients with watershed, channel, flow, and bed material parameters Coefficients of the QB and Dmax,QB curves are negatively related to basin area A (Figure 4a). A negative relation of a with A was also reported by Barry et al. (2004) for streams in Idaho. Following the inherent inverse relation between exponents and coefficients, coefficients are positively related to qbf/A (Figure 4b). A positive scaling relation had also been expected for the bed material parameters %Dsub<8 (Figure 4c) but is not revealed by the data. Similarly, the convex non-monotonic scaling relations observed for exponents with stream gradient does not turn into concave a non-monotonic relation for coefficients. Instead, coefficients display poorly to moderately well-developed positive trends with S (Figure 4d).

Predictability of coefficients Scaling relations for coefficients with the parameters A and qbf/A (Figure 4a, b) are significant and tend to yield higher r2-values than scaling relations for exponents (Figure 1a-b), but that does not render coefficients more predictable. Scatter around the fitted regression function is large, about  1 order of magnitude for coefficients of the Dmax,QB curves and  4 to 5 orders of magnitude for QB curves, which might preclude a meaningful prediction of QB coefficients from watershed and channel parameters.

Fitting QB and Dmax,QB curves with unit discharge q = Q/w instead of total Q does not narrow the scatter in the scaling relations for coefficients. Compared to a, the range of aq-coefficients obtained from QB curves with q instead of Q extend over a range about one order of magnitude wider than a, and neither the general trends of scaling relations (Figure 4), nor the patterns observed for individual stream groups change in a significant way between a and aq.

2 Coefficients a of QB and f of Dmax,QB curves, like exponents, show a strong positive correlation (r = 0.90) (Figure 3b), corroborating that gravel transport and flow competence increase in unison. The relation of f- vs. a-coefficients is not differentiated by stream type.

1.E+06 1.E+06 y = 1679x 1.47 1.E+05 a) y = 328x-1.377 1.E+05 1.47 b) y =R 1679x2 = 0.47 1.E+04 2 1.E+04 2 1.E+03 R = 0.57 1.E+03 Rp < = 0.0001 0.47 1.E+02 p < 0.0001 1.E+02 1.E+01 1.E+01 1.E+00 1.E+00 1.E-01 f 1.E-01 1.E-02 1.E-02 1.E-03 1.E-03 1.E-04 1.E-04 1.E-05 1.E-05 1.E-06 1.E-06

Coefficients 1.E-07 1.E-07 1.E-08 Coefficients -5.64 1.E-08 5.76 1.E-09 y = 58186x a 1.E-09 y = 1E+07x 1.E-10 R2 = 0.54 1.E-10 R2 = 0.47 1.E-11 1.E-11 1.E-12 p < 0.0001 1.E-12 p < 0.0001 1.E-13 1.E-13 1.E-14 1.E-14 0.1 1 10 100 1000 0.001 0.01 0.1 1 Basin area (km²) Unit bankf.runoff yield (m³/m·s/km²)

1.E+06 1.E+06 1.79 1.E+05 c) 1.E+05 y = 2282x d) 2 1.79 1.E+04 1.E+04 y =R 2 =282x 0.3 4 1.E+03 1.E+03 pR =2 =0.0002 0.34 1.E+02 1.E+02 1.E+01 1.E+01 f) 1.E+00 1.E+00 1.E-01 1.E-01 1.E-02 1.E-02 1.E-03 1.E-03 1.E-04 1.E-04 1.E-05 1.E-05 1.E-06 1.E-06 1.E-07 Coefficients

1.E-07 Coefficients 1.E-08 1.E-08 1.E-09 1.E-09 1.E-10 1.E-10 y = 4E+08x7.23 1.E-11 1.E-11 2 1.E-12 1.E-12 R = 0.33 1.E-13 1.E-13 p < 0.0001 1.E-14 1.E-14 10 100 0.001 0.01 0.1 1

Percent subsurface fines <8 mm Stream gradient (m/m)

Figure 4 Scaling relations for a- and f-coefficients with the four stream parameters showing a negative trend with A (a), a positive trend with qbf/A (b), a poorly defined positive trend with %Dsub<8 (c) and a non-monotonic trend with S (d). Yellow symbols denote a-coefficients and purple symbols f-coefficients of Dmax,QB curves.

Differentiation of coefficients by stream type and stream size The prediction of a- and f-coefficients from basin area, runoff yield, stream gradient, and the percentage of subsurface fines < 8 mm is improved when scaling relations are segregated by stream type; the discussion here is limited to a-coefficients for which datasets are richer than for f-coefficients. Segregation of the data scatter is primarily by stream size and indicates steep scaling relations for a-coefficients with all parameters for large streams, and flatter scaling for small streams. The resulting intersection of the two scaling relations for a-coefficients with A (Figure 5a) reflects two opposing segregation effects by stream gradient; among small streams, a-coefficients are clearly lower for plane-bed streams than for step-pool streams. The effect of gradient is reversed among large plane- bed streams; those with low-gradients have higher coefficients than those that are steeper. These segregation trends are not evident for all parameters. Trends for scaling relations of a-coefficients with A are reversed for the parameter qbf/A (Figure 5b) except for the differentiation among gentle and steep-gradient plan-bed streams. Secondary stratification by stream gradient is consistent only among small streams and for all four parameters (Figure 5 a-d), with higher coefficients for the steeper streams. Being stratified by stream type in multiple ways makes A a better suited parameter for predicting a-coefficients.

1.E+06 1.E+06 1.E+05 a) 1.E+05 b) 1.E+04 1.E+04 1.E+03 1.E+03 1.E+02 1.E+02 1.E+01 1.E+01 1.E+00 1.E+00 1.E-01 1.E-01 1.E-02 1.E-02 1.E-03 1.E-03 1.E-04 1.E-04 1.E-05 1.E-05 1.E-06 1.E-06 1.E-07 1.E-07 a-Coefficients a-Coefficients 1.E-08 large p-b 1.E-08 large p-b 1.E-09 lrg. steep p-b 1.E-09 lrg.steep p-b 1.E-10 1.E-10 large s-p 1.E-11 large s-p 1.E-11 1.E-12 small p-b 1.E-12 small p-b 1.E-13 small s-p 1.E-13 small s-p 1.E-14 1.E-14 0.1 1 10 100 1000 0.001 0.01 0.1 1 2 3 2 Basin area (km ) Unit bankfull runoff yield (m /m·s·km )

1.E+06 1.E+06 1.E+05 c) 1.E+05 d) 1.E+04 1.E+04 1.E+03 1.E+03 1.E+02 1.E+02 1.E+01 1.E+01 1.E+00 1.E+00 1.E-01 1.E-01 4.61 1.E-02 1.E-02 y = 2E-09x 1.E-03 1.E-03 R2 = 0.54 1.E-04 1.E-04 1.E-05 1.E-05 1.E-06 1.E-06 1.E-07 1.E-07 large p-b a-Coefficients a-Coefficients 1.E-08 1.E-08 large p-b lrg.steep p-b 1.E-09 lrg.steep p-b 1.E-09 1.E-10 1.E-10 large s-p large s-p 1.E-11 1.E-11 small p-b 1.E-12 small p-b 1.E-12 1.E-13 small s-p 1.E-13 small s-p 1.E-14 1.E-14 0.001 0.01 0.1 1 10 100 Subsurface percent fines < 8 mm Stream gradient (m/m)

Figure 5 Scaling relations for a-coefficients shown in Figure 4 segregated by stream type. Dotted lines indicate possible trends.

SUMMARY AND CONCLUSION

 Study results demonstrate that exponents and coefficients of QB and Dmax,QB curves are systematically and significantly related to four parameters: basin area, runoff yield, stream gradient, and the percentage of subsurface fines < 8 mm. This finding is not reflected in current bedload transport equations and may serve to improve their application to gravel-bed rivers in mountain areas.  Even though the data scatter is wide and the r2-values are only moderate, the fairly large sample size of the study ensures significance and robustness of the scaling relations.  The scaling relations of exponents with basin area, unit runoff yield, stream gradient, and percent subsurface fines scatter widely; hence a user can confidently determine the range of expected exponents but not accurately predict its value. Scaling relations of coefficients with those four parameters are slightly better correlated and likewise significant, but the wide scatter that extends over several orders of magnitude (especially for QB curves) appears to preclude any practical prediction of coefficients.  Scatter can be narrowed when scaling relations for exponents and coefficients are differentiated by stream types.  The specific effects of segregation by stream type on scaling relations differ among stream parameters as well as between exponents and coefficients, but segregation is primarily by basin area, and secondarily by stream gradient.  At this stage of research, an iterative approach is suggested to estimate exponents and coefficients from scaling relations; the estimate is initially based on a parameter’s x-value, and then narrowed based on stream type group. After repeating the process for all parameters, estimates are either averaged or weighted based on the parameter considered most relevant or reliable.

 Segregation by stream type can improve the r2 of a scaling relation but decreases its significance. However, the lower sample size in data sets segregated by stream type lets an individual data point have influence on a scaling relation, hence a larger data base to increase certainty would be desirable.

Acknowledgment We thank all researchers who shared their hard-won bedload data with us: Gustavo Merten (Univ. of Minnesota, Duluth, USA) for data from Arvorezinha Creek, Francesco Comiti (Free University of Bolzano/Bozen, Italy) for Saldurbach data, and Luca Mao (Pontificia Universidad Catolica de Chile, Santiago, Chile) for Estero Morales data; Johannes Schneider and Jens Turowski (Swiss Federal Research Institute, WSL, Birmensdorf, Switzerland; Jens also at Helmholtz Centre Potsdam, GFZ, Potsdam, Germany) for data from the Riedbach and the Erlenbach. The National Stream and Aquatic Ecology Center (formerly Stream Systems Technology Center) of the USDA Forest Service, Fort Collins, Colorado, USA funded this study and the many years of field data collection that provided the basis for the analyses. We are indebted to John Potyondy (retired, Forest Service) for his vision and long-term support of our data collection efforts without which this study could not have been done.

REFERENCES

Barry, J.J. and Buffington, J.M. (2002). “A general power function for predicting bedload transport rates,” Eos, Transactions, AGU, 83 (47), Fall Meeting Supplement, Abstract H21C-0817. Barry, J.J., Buffington, J.M., and King, J.G. (2004). “A general power equation for predicting bed load transport rates in gravel bed rivers,“ Water Resources Research, 40, W10401, doi:10.1029/2004WR003190. Barry, J.J., Buffington, J.M., Goodwin, P., King, J.G., and Emmett, W.W. (2008). “Performance of bed-load transport equations relative to geomorphic significance: Predicting effective discharge and its transport rate,” Journal of Hydraulic Engineering, 134 (5), pp 601-615. Bathurst, J.C. (2007). “Effect of coarse surface layer on bed-load transport,” Journal of Hydraulic Engineering, 133 (11), pp 1192-1205. DOI: 10.1061/ASCE0733-9429(2007) 133:11(1192). Bathurst, J.C. (2013). “Critical conditions for particle motion in coarse bed materials of non-uniform size distribution,” Geomorphology, 197, pp 170–184; http://dx.doi.org/10.1016/j.geomorph.2013.05.008. Bunte, K. and Abt, S.R. (2003). “Bedload rating and flow competence curves vary with watershed and bed material parameters,” EOS, Trans. AGU, 84 (46) Fall Meeting Supplement, Abstract H32I-01. http://www.agu.org/meetings/fm03/waisfm03.html Bunte, K. and Swingle, K. (2003). Results from testing the bedload traps at Little Granite Creek, 2002: effect of sampling duration and sampler type on bedload transport rates and systematic variability of rating curves with basin area and stream bed parameters. Report submitted to the Stream Systems Technology Center, USDA Forest Service, Rocky Mountain Research Station, Fort Collins, Colorado, 110 pp. Bunte, K., Abt, S.R., Swingle, K.W., Cenderelli, D.A., Schneider, J.M. (2013). “Critical Shields values in coarse- bedded steep streams,” Water Resources Research, 49(11), pp 7427-7447. http://dx.doi:10.1002/2012WR012672, 2013 Bunte, K., Abt, S.R. and Swingle, K.W. (2006). “Predictability of bedload rating and flow competence curves from bed armoring, stream width and basin area,” In: Proceedings of the 8th Federal Interagency Sedimentation Conference, April 2-6, 2006 in Reno, Nevada., Session 4A-2, 9 pp. [CD ROM]. http://pubs.usgs.gov/misc_reports/FISC_1947-2006/html/pdf.html Bunte, K., Swingle, K. and Abt, S.R. (2007). Guidelines for Using Bedload Traps in Coarse-bedded Mountain Streams: Construction, Installation, Operation and Sample Processing. General Technical Report RMRS- GTR-191, USDA, Forest Service, Rocky Mountain. Research Station, 91 pp, Fort Collins, Colordo. http://www.fs.fed.us/rm/pubs/rmrs_gtr191.pdf Bunte, K., Swingle, K.W. and Abt, S.R. (2010a). “Necessity and difficulties of field calibration of signals from surrogate techniques in gravel-bed streams: possibilities for bedload trap samples,” In: Bedload-surrogate monitoring technologies, Gray, J.R., Laronne, J.B., and Marr, J.D.G (eds.), USGS Scientific Investigation Report 2010-5091, 17 pp. http://pubs/usgs.gov/sir/2010/5091 Bunte, K., Abt, S.R., Potyondy, J.P., and Swingle, K.W. (2008). “A comparison of coarse bedload transport measured with bedload traps and Helley-Smith samplers,” Geodinamica Acta, 21(1/2), pp 53-66. Bunte, K., Abt, S.R., Potyondy, J.P., and Ryan, S.E. (2004). “Measurement of coarse gravel and cobble transport using a portable bedload trap,” Journal of Hydraulic Engineering, 130(9), pp 879-893. Bunte, K., Abt, S.R., Swingle, K.W., and Cenderelli, D.A. (2014). “Effective discharge in Rocky Mountain headwater streams”, Journal of Hydrology, 519, pp 2136-2147, http://dx.doi.org/10.1016/j.jhydrol.2014.09.080

Bunte, K., Abt, S.R., Swingle, K.W., and Potyondy, J.P. (2010b). “Functions to adjust transport rates from a Helley-Smith sampler to bedload traps in coarse gravel-bed streams (rating curve approach),” In: Proceedings 9th Federal Interagency Sedimentation and 4th Federal Interagency Hydrologic Modeling Conference, Las Vegas, NV, June 27 – July 1, 2010, Session: Fluvial Geomorphology, CD-ROM ISBN 978-0-9779007-3-2. http://acwi.gov/sos/pubs/2ndJFIC/Contents/10D_Bunte_03_01_10.pdf Bunte, K., Abt, S.R., Swingle, K.W., and Potyondy, J.P. (2009). “Comparison of three pebble count protocols (EMAP, PIBO, and SFT) in two mountain gravel-bed streams,” Journal of the American Water Resources Association, 45(5), pp 1209-1227. DOI: 10.1111 ⁄ j.1752-1688.2009.00355.x http://www.stream.fs.fed.us/publications/PDFs/Bunte_others_2009.pdf Castro J.M., and Jackson, P.L. (2001). “Bankfull discharge recurrence intervals and regional hydraulic geometry relationships: patterns in the Pacific Northwest, USA”, Journal of the American Water Resources Association, 37 (5), pp 1249-1262. Church, M., Hassan, M.A., and Wolcott, J.F. (1998). “Stabilizing self-organized structures in gravel-bed stream channels: field and experimental observations,” Water Resources Research, 34(11), pp 3169-3179. Dietrich, W.E., Kirchner, J.W., Ikeda, H., Iseya, F. (1989). “Sediment supply and the development of the coarse surface layer in gravel-bedload rivers,” Nature, 340, pp 215-217. Diplas, P., and Shaheen, H. (2008). “Bed load transport and streambed structure in gravel streams,” in: Gravel-Bed Rivers VI: From Process Understanding to River Restoration. H. Habersack, H. Piegay, and M. Rinaldi (eds.), Elsevier, Amsterdam, pp 291-312. Dodov, B., and Foufoula-Georgiou, E. (2005). “Fluvial processes and streamflow variability: Interplay in the scale- frequency continuum and implications for scaling,” Water Resources Research, 41, W05005, doi:10.1029/2004WR003408. Gaeuman, D., Holt, C.R., and Bunte, K. (2014). “Maximum Likelihood Parameter Estimation for Fitting Bedload Rating Curves,” Water Resources Research , 51, doi:10.1002/2014WR015872. Gran, K.B., Montgomery D.R., and Sutherland, D.G. (2006). “Channel bed evolution and sediment transport under declining sand inputs,” Water Resources, Research, 42, W10407, doi:10.1029/2005WR004306. Hassan, M.A., Smith, B.J., Hogan, D.L., Luzi, D.S., Zimmermann, A.E., and Eaton, B.C. (2008). “Sediment storage and transport in coarse bed streams: scale considerations,” in: Gravel-Bed Rivers VI: From Process Understanding to River Restoration. H. Habersack, H. Piegay, and M. Rinaldi (eds.). Elsevier, Amsterdam, pp 473-497. Hayes, S.K., Montgomery, D.R., Newhall, C.G. (2002). “Fluvial sediment transport and deposition following the 1991 eruption of Mount Pinatubo,” Geomorphology, 45, pp 211– 224. Laronne, J.B., Garcia C., and Reid, I. (2001). “Mobility of patch sediment in gravel bed rivers: Patch character and its implications for bedload,” in: Gravel-Bed Rivers V, M.P. Mosley (ed.), New Zealand Hydrological Society, Wellington, New Zealand, pp 249-290. Laronne, J.B., and Reid, I. (1993). “Very high rates of bedload sediment transport by ephemeral desert rivers,” Nature 366, pp 148-150. Lenzi, M. A., Mao, L., and Comiti, F. (2004). “Magnitude-frequency analysis of bed load data in an Alpine boulder bed stream,” Water Resources Research, 40, W07201, doi:10.1029/2003WR002961. Lenzi, M.A., Agostino, V.D., Billi, P. (1999). “Bedload transport in the instrumented catchment of the Rip Cordon. Part I: Analysis of bedload records, conditions and threshold of bedload entrainment,” Catena, 36, pp 171-190. Lenzi, M.A., Mao, L., and Comiti, F. (2006). “Effective discharge for sediment transport in a mountain river: Computational approaches and geomorphic effectiveness,” Journal of Hydrology, 326, pp 257–276, doi:10.1016/j.jhydrol.2005.10.031 Lisle, T.E., 1995. “Particle size variations between bed load and bed material in natural gravel bed channels,” Water Resources Research, 31(4), pp 1107-1118. Lisle, T.E., and Church, M. (2002). “Sediment transport-storage relations for degrading, gravel bed channels,” Water Resources Research, 38 (11), 1219, doi:10.1029/2001WR0001086, 2002. Montgomery, D.R., and Buffington, J.M. (1997). “Channel-reach morphology in mountain drainage basins.,” Geol. Soc. Am. Bull., 109 (5), pp 596-611. Potyondy, J.P., Bunte, K., Abt, S.R., and Swingle, K.W. (2010). Bedload movement in mountain channels: insights gained from use of portable bedload traps. Proc. 4th Fed. Interagency Hydrologic Modeling Conference and 9th Fed. Interagency Sedimentation Conference, Las Vegas, NV, 2010. CD-ROM ISBN 978-0-9779007-3-2. http://acwi.gov/sos/pubs/2ndJFIC/Contents/10D_Potyondy_03_01_10.pdf Reid, I., and Laronne, J.B. (1995). “Bed load sediment transport in an ephemeral stream and a comparison with seasonal and perennial counterparts,” Water Resources Research, 31(3), pp 773-781.

Rickenmann, D., Turowski, J.M., Fritschi, B., Klaiber A., and Andreas, L. (2012). “Bedload transport measurements at the Erlenbach stream with geophones and automated basket samplers,” Earth Surface Processes and Landforms, 37(9), pp 1000-1011, DOI: 10.1002/esp.3225 Ryan, S.E., Bunte, K., and Potyondy, J.P. (2005a). Breakout Session II: Bedload Transport Measurements, Data Uncertainty, and New Technologies. Proceedings of the Federal Interagency Sedimentation Monitoring Instrument and Analysis Workshop, September 9-11, 2003, Flagstaff, Arizona, J.R. Gray (ed.). USGS Survey Circular 1276, pp 16-28. http://water.usgs.gov/pubs/circ/2005/circ1276/ Ryan S.E., Porth, L.S., and Troendle, C.A. (2005b). “Coarse sediment transport in mountain streams in Colorado and Wyoming, USA,” Earth Surface Processes and Landforms, 30(3), pp 269-288. DOI: 10.1002/esp.1128 Schneider, J.M., Rickenmann, D., Turowski, J.M., Bunte, K., Kirchner, J.W. (2015). “Applicability of bedload transport models for mixed size sediments in steep streams considering macro-roughness,” Water Resources Research, in revision. Schoklitsch, A. (1962). Handbuch des Wasserbaus. [Handbook of Hydraulic Engineering.]. Springer Verlag, Wien. Yang, G., Chen, Z., Yu, F., Wang, Z., Zhao, Y., and Wang Z. (2007). “Sediment rating parameters and their implication: Yangtze River, China,” Geomorphology, 85, pp 166-175. Yu, G., Wang, Z., Zhang, K., Chang T., and Liu, H. (2009). “Effect of incoming sediment on the transport rate of bed load in mountain streams,” International Journal of Sediment Research, 24(3), pp 260–273.