Characterization of the hydraulics at natural step crests in step-pool streams via weir flow concepts

Transcription

1 WATER RESOURCES RESEARCH, VOL. 48, W09542, doi: /2011wr011724, 2012 Characterization of the hydraulics at natural step crests in step-pool streams via weir flow concepts David Dust 1 and Ellen Wohl 1 Received 9 December 2011; revised 8 August 2012; accepted 11 August 2012; published 25 September [1] The hydraulics of step-pool streams are characterized by rapidly varied flow at the step crest, a hydraulic jump, and gradually varied flow in the pool unit of the step-pool sequence. The flow characteristics at the step crests act as the hydraulic control for the water surface profile within the upstream pool unit. Using both field and flume investigations, we demonstrate the use of weir flow concepts for assessing and categorizing the hydraulic characteristics of natural step-crests in step-pool streams. We categorize the results of our investigations in terms of the crest-clast, planform, longitudinal, and instream wood geometries of the step crests. The broad-crested weir equation can be expressed as Q ¼ C g 0.5 Wh 3/2, where Q is the flowrate, C is a dimensionless discharge coefficient, W is the crest width, g is the acceleration of gravity, and h is the upstream flow depth above the step crest. Although the flow over a natural step is generally more complex than for an engineered weir, the results of our investigations indicate that the C -value for simulated and natural steps increases linearly as a function of the upstream head (h), with C values ranging from 0.15 to As a result, the application of weir flow concepts to natural steps provides means for (1) indirectly estimating flow rates; (2) characterizing the hydraulics for individual steps; (3) defining external and/or internal boundary conditions at step crests for hydraulic model simulations of natural or restored step-pool streams; and (4) estimating the upstream pressure force acting on step-crest clasts. Citation: Dust, D., and E. Wohl (2012), Characterization of the hydraulics at natural step crests in step-pool streams via weir flow concepts, Water Resour. Res., 48, W09542, doi: /2011wr Introduction 1.1. Objectives and Hypotheses [2] It is generally agreed in the literature that the defining characteristic of step-pool streams is downstream alternating channel-spanning steps and plunge pools (Figure 1a) [e.g., Montgomery and Buffington, 1997; Church and Zimmermann, 2007]. However, Church and Zimmermann [2007] have suggested that step-pool streams can be further classified in terms of characteristic flow regimes and the corresponding valley slope within which the step-pool sequences have formed, based on assessing approximately 20 years of research. Furthermore, there is continuing debate regarding appropriate approaches for computing flow characteristics and/or simulating the wide range of complex water surface profiles characteristic of step-pool streams [e.g., Aberle and Smart, 2003; Church and Zimmermann, 2007; Ferguson, 2010]. Hence, it is important to clearly describe the hydraulic setting when proposing any new approach for modeling 1 Department of Geosciences, Colorado State University, Fort Collins, Colorado, USA. Corresponding author: E. Wohl, Department of Geosciences, Colorado State University, Fort Collins, CO , USA. (ellen.wohl@ colostate.edu) American Geophysical Union. All Rights Reserved /12/2011WR the hydraulics at natural step crests and defining the corresponding flow regimes for step-pool streams. [3] We have been investigating the hydraulic and geomorphic characteristics of step-pool streams in the Colorado Rocky Mountains (United States) for over 20 years. The hydraulics for at least a significant subset, if not the majority, of these step-pool sequences can be characterized by rapidly varied flow at the step crest, followed by a hydraulic jump, and gradually varied flow within the pool unit of the step-pool sequence (Figures 1a and 2), as similarly described in a number of investigations [e.g., Grant et al., 1990; Grant, 1994; Montgomery and Buffington, 1997; Zimmermann and Church, 2001; Church and Zimmermann, 2007]. Under these circumstances, the flow characteristics of the step crests control the subcritical water surface profile within the upstream pool unit, which typically have M1 drawdown or M2 backwater type water surface profiles [e.g., Chow, 1959] for at least up to bankfull flow conditions (Figure 2). When viewed from this perspective, step-pool streams with these characteristics share the same fundamental hydraulic characteristics as a series of broad-crested weirs (i.e., Figure 1a versus 1b) [e.g., Chow, 1959; Bos, 1988; Sturm, 2001] for at least some range of flow rates, rather than the fundamental hydraulic characteristics of modeling approaches based on uniform flow [e.g., Einstein and Barbarossa, 1952; Thorne and Zevenbergen; 1985; Marcus, 1992; Wilcox et al., 2006; Ferguson, 2007; David et al., 2010] or stepped W of14

2 Figure 1. Schematic illustrating similarities in flow characteristics between a series of broad-crested weir structures and step-pool channels (V is velocity, h is upstream head, y critical is critical flow depth). spillway hydraulics [e.g., Chanson, 1994, 2001; Church and Zimmermann, 2007; Comiti et al., 2009]. [4] Few complicated hydraulic problems can be solved deterministically using the fundamental laws of physics without making significant simplifying assumptions. Hence, the ultimate applicability of any hydraulic modeling approach lies in the validity of the underlying assumptions for a specific application. Although application of weir-flow concepts to an array of flow measurement structures has an extensive history [e.g., Bélanger, 1828; Horton, 1907; Woodburn, 1932; U.S. Bureau of Reclamation, 1948; Tracy, 1957; U.S. Bureau of Reclamation, 1987], the application of weir-flow concepts to describe the hydraulic characteristics of natural step-crest geometries does not appear to have been previously investigated. Contrary to investigations by Pasternack et al. [2006, 2007] and Wyrick and Pasternack [2008] that implicitly assumed that the discharge coefficient in the broad-crested weir equation is insensitive to variations in natural step-crest geometries, we use field and flume investigations to evaluate the applicability of using weir flow concepts for assessing and categorizing the hydraulic characteristics of natural step-crest geometries in step-pool streams in terms of the crest-clast, planform, longitudinal and instream-wood geometry of the step-crests. [5] The hypotheses that guided our investigations are: (H1) the assumptions associated with the derivation of the broad-crested weir flow equation are sufficiently valid for at least a subset of the possible natural step-crest geometries such that the step-crest coefficient (C ) varies in a predictable manner over a useful range of flow rates; (H2) the lower and/or upper thresholds for this range of flow rates can be identified in terms of hydraulic and/or geometric measures of the downstream step crest in a step-pool sequence; (H3) the geometry of the step crest can have a significant influence on the hydraulic characteristics of a step crest; and (H4) simulated steps in the flume can meaningfully represent the hydraulic characteristics of natural steps. In our hypotheses, we introduce the designation of step-crest coefficient to clearly differentiate between discharge coefficients for natural step-crest geometries and those for engineered weir steps Motivation for These Investigations [6] As a part of our broader investigations into the fundamental processes governing the geomorphic characteristics Figure 2. Corral Creek. An example of a step-pool stream in the Colorado Rocky Mountains. 2of14

3 of the steep mountain streams in the Colorado Rocky Mountains, we have encountered many challenges in our endeavors to assess and quantitatively describe various hydraulic characteristics of step-pool streams with and without significant instream wood [e.g., Wilcox et al., 2006; Wilcox and Wohl, 2006, 2007; David et al., 2010; Wohl, 2010; Wilcox et al., 2011;Wohl and Dust, 2012]. The two key challenges that have been motivating our research into the hydraulic characteristics of natural step crests include: (1) an indirect means for estimating flow rates in step-pool streams [e.g., Bathurst, 1985; Thorne and Zevenbergen; 1985; Wohl, 2010]; and (2) a means for computing the flow characteristics (i.e., flow depth, velocity and specific energy) just upstream of step crests with and without significant instream wood. [7] If the hydraulic characteristics of natural step-crest geometries can be quantitatively represented with weir flow concepts in terms of C values, flow rates at step crests can be indirectly estimated as a function of C and basic field measurements of the step crest. Consequently, the flow characteristics just upstream of natural step crests can also be estimated as a function of C, flowrate (Q) and basic field measurements of the step crest. [8] Just upstream of step crests, the flow transitions from gradually varied to rapidly varied flow (Figure 1). As a result, the flow characteristics just upstream of step crests correspond to unique hydraulic control points within the complex water surface profiles characteristic of step-pool sequences. Therefore, being able to compute the flow characteristics just upstream of step crests can provide: (1) the downstream boundary conditions required for computing water surface profiles and the resulting tailwater depths that influence hydraulic jumps in pool-units [e.g., Chow, 1959; Mossa et al., 2003; Pasternack et al., 2006]; (2) the hydraulic parameters necessary for computing the energy gradeline and the corresponding energy dissipation for individual or groups of step-pool sequences [e.g., Church and Zimmerman, 2007] and (3) the hydraulic parameters needed for assessing potential scour depths in plunge-pools [e.g., Mason and Arumugam, 1985; Hager, 1998; D Agostino and Ferro, 2004; Comiti et al., 2005; Church and Zimmerman, 2007; Tregnaghi et al., 2011]. [9] Some researchers characterize the hydraulics at engineered and natural step crests in terms of critical flow conditions [e.g., Comiti et al., 2005; Church and Zimmerman, 2007; Comiti et al., 2009; Tregnaghi et al., 2011]. Compared to measuring the upstream gauge head (h) for weir flow analyses, direct field measurements of critical flow conditions for natural step crests can be significantly more problematic and prone to uncertainties, due to the complex flow patterns along irregular step crests. The practical problems associated with measuring critical flow conditions for even engineered step crests prompted the development of an assortment of weir flow equations and the corresponding use of the upstream flow characteristics as the reference point for computing flow rates over weirs, spillways and dam crests from the 1800s through to the present [e.g., Bélanger, 1828; Horton, 1907; Woodburn, 1932; U.S. Bureau of Reclamation, 1948; Tracy, 1957; Neogy, 1972; Ackers et al., 1978; U.S. Bureau of Reclamation, 1987; Bos, 1988; Sturm, 2001]. Therefore, the application of weir flow concepts to estimate flow rates at natural step crests can complement the use of critical flow conditions to describe step-crest hydraulics, because representative critical flow conditions can be computed using given flow rates [e.g., Comiti et al., 2005] and, thereby, avoid the problems associated with direct measurements. 2. Theoretical Foundation: Physics of Broad-Crested Weir Flow [10] The general form of the broad-crested weir flow equation can be derived from both conservation of momentum and conservation of energy principles [e.g., Doeringsfeld and Barker, 1941; Chow, 1959; Ackers et al., 1978; Bos, 1988]. Even though application of the two principles can be used to derive nearly identical broad-crested weir equations, the inherent distinction between applications of the conservation of momentum and energy principles is that momentum is a vector quantity (i.e., forces) and energy is a scalar quantity. This distinction is important to our broader investigations, because we are also interested in the forces acting on step-crest clasts. Hence, the control volume and forces appropriate for deriving the broad-crested weir flow equation via conservation of momentum principles are depicted in Figure 3. [11] Equations (1) and (2) are the basic broad-crested weir equations, as can be derived from conservation of momentum and energy principles, respectively. Broad-crested weir equation via conservation of momentum [e.g., Doeringsfeld and Barker, 1941]: Q ¼ C f C w ð3=8þ 1=2 g 1=2 Wh 3=2 ¼ C f C w ð0:6124þg 1=2 Wh 3=2 (1) Broad-crested weir equation via conservation of energy [e.g., Ackers et al., 1978; Bos, 1988]: Q ¼ C d C v ð2=3þ 3=2 g 1=2 Wh 3=2 ¼ C d C v ð0:5443þg 1=2 Wh 3=2 (2) where Q is flowrate, g is the acceleration of gravity, W is the channel and step crest width, h is the gauge head above the step crest as denoted in Figure 3, C f ¼ 1=ðC 0 F Þ 1=2 (3) is a friction-force compensation coefficient. C 0 F is a drag coefficient that compensates for friction forces F 0 f and F 00 f acting on the approach channel bottom and along the length of the step crest (L), C W ¼ y 1=2 1 ; (4) ðy 1 þ h w Þ is the vertical flow-contraction coefficient; C d ¼ f h=l; h=ðh þ h w Þ is an energy-loss compensation coefficient [Singer, 1964]; (5) C v ¼ h þ V 1 2 3=2 =h (6) 2g 3of14

4 Figure 3. Schematic of broad-crested weir showing conservation of momentum terms [after Chow, 1959] where P 1, P 2, and P w are the hydrostatic forces; F 1 and F 2 are hydrodynamic forces; F 0 f and F 00 f are friction forces; A i, y i, V i, and i are the flow area, depth, cross-section averaged velocity, and momentum coefficient at section i ; g is the acceleration of gravity; is the density of water; h w is the height of step; W is the channel and step-crest width; and y c is the critical flow depth. is the velocity-head compensation coefficient, is the kinetic energy coefficient which compensates for a nonuniform velocity distribution, L is the length of the step in the direction of flow, h w is the height of the step crest, and y 1 and V 1 are the approach flow depth and velocity, respectively. [12] For C ¼ C f C w ð3=8þ 1=2 ¼ C d C v ð2=3þ 3=2, equations (1) and (2) simplify to the following basic form of the broad-crested weir equation: Q ¼ C g 1=2 Wh 3=2 (7) where C is a dimensionless discharge or step-crest coefficient. [13] As reflected in equations (3) (6), the discharge coefficient (C ) for a rectangular weir reflects the energy and momentum losses due to both friction forces and, more notably, the vertical contraction of the flow as it approaches and accelerates over the horizontal step crest, regardless of the approach used to derive the broad-crested weir equation. As a consequence of the irregularities in natural stepcrest geometries, the step-crest coefficient (C ) has the added role of also reflecting the energy and momentum losses due to the horizontal contraction of the flow as it approaches and accelerates over an irregular step crest. 3. Methods 3.1. Flume and Field Investigations [14] We used a combination of flume and field investigations to characterize the hydraulics for simulated and natural steps, in terms of the step-crest coefficient (C ). The flume experiments were conducted in a m flume with a bed slope of 14% (Figure S1 S5 in the auxiliary material), where each flume setup allowed the testing of at least two simulated step geometries in series for flow rates up to 0.11 m 3 s 1. 1 Each of the simulated step crests was constructed with cobbles and boulders of similar shape to a height sufficient to prevent submergence of the step crest by the downstream water surface for the full range of flow rates investigated. The floor of the flume upstream of the simulated steps was lined with a mixture of gravels and cobbles (Figure S1 in the auxiliary material). [15] To verify that the simulated steps constructed in the flume can meaningfully represent the hydraulic characteristics of natural steps, we compare the flume results with field investigations. Geometric and flow data were collected for three sequential natural steps along the East Fork of Roaring Creek (Figures S6 S8 in the auxiliary material), Colorado, for flows ranging from to m 3 s 1, which correspond to flow depths ranging from approximately 40% to 75% of bankfull depth as measured from the low point of the step crest. The East Fork of Roaring Creek has a gradient of approximately 20% and a drainage area of 15 km 2. The drainage basin is located entirely above an altitude of 2400 m and, as a result, the creek has a snowmelt flow regime [Jarrett, 1993]. [16] For the flume experiments, the simulated step crests had characteristic crest-clast diameters ranging from to m. The natural steps had characteristic crest-clast diameters ranging from to m. Hydraulic data were collected for unit discharges ranging from to 0.14 m 2 s 1 for the simulated steps and from to 0.12 m 2 s 1 for the natural steps. Although it was not our specific intention to model the East Fork Roaring Creek steps in the flume, the simulated steps in these experiments were at or nearly at a 1:1 scale (i.e., length ratio for all 3 dimensions ¼ L model /L prototype 1) with the natural steps, in 1 Auxiliary materials are available in the HTML. doi: / 2011WR of14

5 terms of the crest-clast diameter, plus the hydraulic data were collected for a range of unit discharge rates that spans those for the field investigations. Furthermore, the simulated pool unit lengths (3 m) and step heights (0.7 m) in our flume investigations are within the range observed within the Rocky Mountains, even when considered at 1:1 scale [e.g., David et al., 2010; Wohl and Dust, 2012]. [17] Unlike the experiments used to assess discharge coefficients for engineered weir crests, the simulated step crests in our experiments were always evaluated in series (Figure S1 in the auxiliary material), similar to natural step-pool sequences. This aspect of our flume investigations is important and unique, because it allowed us to observe and assess the influence of flow characteristics in the pool unit (e.g., standing waves associated with hydraulic jumps) on the gauge head (h) and the corresponding C values, as can occur in natural step-pool sequences. [18] Even though the geometric characteristics of our simulated step-pool sequences are comparable with observed natural systems at a 1:1 scale, our rigid-bed model for step crests constructed with cobbles and boulders is conducive to maintaining exact geometric and Froude similitude criteria for a relatively wide range of length ratios (i.e., L model /L prototype ) [Julien, 2002]. Therefore, the results of our flume experiments can be scalable to natural step crests with smaller or larger crest-clast diameters to a significant degree Categorization of Step Geometries [19] We categorized the simulated and natural step crests in terms of the crest-clast, planform, longitudinal and/or instream wood geometries of the step crest, as shown schematically in Figure 4. The primary classification level of the simulated steps was in terms of the crest-clast geometry (Figure 4), with the subrounded and rounded designations based on criteria developed by Powers [1953]. [20] A key concept associated with applying weir flow to natural steps is that a natural step crest is defined and formed by a channel-spanning sequence or line of adjacent clasts over which the flow transitions through critical depth, as reflected schematically in Figures 4d 4f. However, natural step crests can often be more complex by having one or more individual clasts positioned immediately upstream of the channel-spanning sequence of clasts that transitions the flow through critical depth, as reflected schematically in Figure 4i (Figures S2a versus S2b in auxiliary material). Hence, we introduce the terminology of a staggered clast, a staggered crest and crest staggering to describe aspects of this longitudinal geometry. [21] Another aspect of longitudinal geometry defined in Figure 4h is that of approach slope (S a ). We define the approach slope for a step crest as that portion of the channel bed immediately upstream of the step-crest. [22] The crest-clast, planform, longitudinal and instreamwood geometries of the simulated steps constructed in the flume were systematically varied to allow the incremental influence of the various step geometries on the step-crest coefficient (C ) to be quantified. For example, we evaluate the influence of crest staggering on the step-crest coefficient by collecting data for a given step crest with and without a staggered clast (Figures 4i and Figure S2a versus S2b in Figure 4. Hierarchical classification system for step-crest and instream wood geometries. 5of14

6 auxiliary material). A total of 29 simulated and 3 natural step crest configurations were evaluated and categorized as various combinations of the crest-clast, planform, longitudinal and/or instream wood geometries shown schematically in Figure 4 (Figures S2 S4 in auxiliary material) Data Collection Procedures Unique to Natural Step Geometries [23] Computing C values per equation (7) for either a simulated or natural step crest requires collecting upstream gauge head (h), step-crest width (W), and discharge (Q) data. However, the specific definitions for the upstream gauge head (h) and the flow or step-crest width (W) as defined for an engineered weir are not adequate for either simulated or natural steps, because the geometry of natural steps is typically far more irregular and complicated than that of an engineered weir step. Hence, criteria and/or procedures are needed for establishing a vertical datum and measuring flow depths and widths for natural step crests. [24] The upstream gauge head (h) for an engineered weir would simply be the elevation difference between the top of the horizontal weir step (Figure 3) and the water surface at a distance of at least 2.5 times the head (h) upstream of the weir step [Brater and King, 1976; Ackers et al., 1978; Bos, 1988]. The upper surface of a natural step crest is typically neither flat nor horizontal (Figures S2 S8 in auxiliary material); therefore, we define the low point of the step crest as the vertical datum for measuring the upstream gauge head (h) for both natural and simulated natural steps. [25] In the derivation of the weir equation, the weir step is assumed to be straight and oriented perpendicular to the flow direction. As a result, the width of the crest (W crest )is assumed to be the same as both the flow width and the width of the upstream approach channel (W) (Figure 3). The effective crest width (W crest ) of a natural step as measured along the top of the crest can be significantly greater than that of the approach channel because of the oblique angle and/or curvature of the step crest (Figures 4e and 4f). We computed the step-crest coefficients (C ) using only the wetted crest width as measured along the top of the crest (W crest ). [26] In the flume, we measured water surface levels upstream of the simulated step crests with a point gauge to a thousandth of foot (0.3 mm) via a Vernier scale at three equally spaced positions across the 0.61 m wide flume. We then used an average water surface level to compute the upstream gauge head (h) and corresponding step-crest coefficient (C ) for the associated flowrate Flow Measurements [27] For the flume investigations, we used an electromagnetic flowmeter on the supply line to measure discharge in the flume. For the field investigations, measurements from a Marsh-McBirney velocity meter mounted to a top-setting wading rod were used to estimate the flow rates upstream of each of the three consecutive natural steps. The cross sections along which the velocity measurements were collected were all located approximately a meter upstream of each of the step crests. When possible, the precise alignment of the flow measurement cross sections were shifted slightly upstream or downstream to avoid the influence of obstacles as the flowrate changed over the approximately 2 month period during which field data were collected. Velocity measurements were collected at a point below the water surface corresponding to 0.6 of the total flow depth and at 0.1 m or less increments across the stream, depending upon the specific geometry of the cross section Data Analysis [28] Inspection of equations (3) (6) demonstrates that C for rectangular weir steps is a function of the approach velocity (V 1 ), the upstream flow depth in terms of y 1 or h, and the geometry of the weir-step in terms of both h w and L (Figure 3). Singer [1964] demonstrated conclusively that coefficient C d (equation (5)) can effectively be described as a function of length ratios h/l and h/(h þ h w ), where L is the length of the weir crest in the direction of flow and h w is the height of the weir step at the upstream face (Figure 3). As a result, the investigations into the discharge coefficients for engineered weir steps are commonly presented as plots of coefficient C d versus ratio h/l or H/L [e.g., Ackers et al., 1978; Bos, 1988], where H is the total head upstream of the weir step (i.e., H ¼ h þ (V 2 /2g)). [29] To generate similar plots of C for natural step geometries, we must define a length ratio for natural step crests that is analogous to the h/l ratio for rectangular weir-steps. The hydraulic significance of length L is linked to a key assumption in the derivation of the broad-crested weir equation; that is, L is sufficiently long in the direction of flow to allow the flow to be parallel and have a hydrostatic pressure distribution (Figure 3). In recognition that natural step crests have a wide range of irregular geometries, we introduce the relative submergence ratio h/d step for natural and simulated natural step crests, where the definition for D step is varied depending upon the geometry of the clasts, instream wood, and/or bedrock forming the step crest to permit the parameter to be as hydraulically analogous to L for engineered weir steps as possible. [30] For the planar steps, we define D step as the weighted crest length in the direction of flow (D l ) (Figure S2c in auxiliary material). For simulated and natural step crests composed of subrounded and/or rounded clasts, we define D step in terms of a characteristic diameter for the step-crest clasts. Particle-size cumulative frequency curves and individual values on these curves (i.e., percentiles) are commonly used to quantitatively characterize the size of bed material in streams [e.g., Bunte and Abt, 2001]. For the natural step crests in our investigations, we define D step as the clast diameter corresponding to the 50th percentile or median, which is typically denoted as D 50. [31] Application of this approach to the simulated step crests constructed in the flume is problematic, because the simulated step crests were commonly composed of only two or three clasts and the corresponding median diameter is not representative for hydraulic purposes as a result of the very small sample size. Hence, we define D step as the mean crest-clast diameter (i.e., D mean ) for the simulated step crests. [32] For the log step crest, as shown schematically in Figure 4k, we define D step as the mean diameter of the log (i.e., D wood ). However, we define D step as the mean crestclast diameter (i.e., D mean ) for simulated step crests with channel spanning instream wood or logjam, as shown schematically in Figures 4l and 4m. 6of14

7 4. Results 4.1. Observed Flow Regimes and Thresholds [33] The results for each of our 29 flume and 3 field investigations were plotted in terms of relative submergence (h/d step ) versus the step-crest coefficient (C ). The data for each of the simulated and natural steps are plotted in the same basic pattern as the example set of data (Experiment R3.2) shown in Figure 5a. The pattern of the example flume experiment results is significant in clearly illustrating that the flow transitions through three distinct flow regimes in terms of the weir flow model and that there are upper and lower limits in the applicability of the weir flow model to natural steps. We refer to the three flow regimes as the interstitial, weir, and oscillating flow regimes (Figures 5b and Figure S5 in auxiliary material). [34] In the interstitial flow regime, flow rates are so low that the flow predominantly passes through gaps between and under the clasts that form the step crest and not over the step-crest clasts, as assumed in the weir flow equation. As a result, the weir flow equation expressed as a function of upstream gauge head (i.e., equation (7)) is not suitable for evaluating flow characteristics in the interstitial flow regime without accounting for the flow leaking through and/ or under the step crest in some manner. [35] In the weir flow regime, a clearly defined hydraulic jump occurs at the base of the steps, the jump length is generally much shorter than the pool-unit length, and the flow approaching the step crests is subcritical (Froude Number ranging from 0.05 to 0.75), with a relatively smooth water surface (Figure S3 in auxiliary material). In this flow regime, the hydraulic jump generally displays the characteristics of what is referred to as a weak jump in engineering texts [e.g., Chow, 1959]. In terms of the step-crest coefficient, the weir flow regime corresponds to the midrange flow conditions, where the C value increases with increasing flowrate and relative submergence (Figure 5a). This pattern of the step-crest or discharge coefficient increasing with increasing relative submergence is identical to that observed during experiments with broad-crested weirs of various geometries [e.g., Tracy, 1957; Singer, 1964; Crabbe, 1974; Ackers et al., 1978; Bos, 1988]. [36] In our flume experiments, the hydraulic jump at the base of a step transitioned from a weak to an oscillating hydraulic jump [e.g., Bradley and Peterka, 1957; Chow, 1959] as flow rates were increased. Although the flow approaching the step-crest remained subcritical on a crosssection average basis, a unique characteristic of oscillating jumps is the presence of standing waves and oscillating jets of flow that typically extend relatively long distances downstream of the jump face [e.g., Bradley and Peterka, 1957; Chow, 1959]. As the standing waves and oscillating jets of flow associated with the oscillating jump extended over the step crest in our investigations (Figure S3c in auxiliary material), the flow at the step crest transitioned from the weir to the oscillating flow regime. In terms of the stepcrest coefficient, the weir to oscillating flow regime threshold corresponds to flow conditions where the C value increases sharply with decreasing or only slightly increasing relative submergence, as the discharge is incrementally increased (Figure 5a). As a result, the weir-flow equation expressed as a function of upstream gauge head (i.e., equation (7)) is not suitable for evaluating flow characteristics in the oscillating flow regime. [37] Although the flow characteristics associated with the oscillating jumps observed in our flume investigations limited the flow range of the weir flow regime, it is not our intent to suggest that this will generally be the case for natural step-pool sequences, because the hydraulic jump type and the longitudinal position of a hydraulic jump are controlled by both hydraulic characteristics of the flow and the geometric characteristics of the pool unit [e.g., Chow, 1959; U.S. Bureau of Reclamation, 1987; Mossa et al., 2003]. However, the results of our investigations validate hypothesis H2 by demonstrating that: (1) there can be both a low-flow and high-flow threshold for the applicability of the weir flow model (in the form of equation (7)) to natural Figure 5. Example flume investigation results (experiment R3.2) and schematics of observed step-pool flow regimes. 7of14

8 step-pool sequences; and (2) these thresholds can be identified by evaluating the hydraulic characteristics of a natural step-pool sequence in terms of C versus h/d step. [38] The flow regimes we identified are based on quantitative thresholds (Figure 5a) associated with a specific mathematical model (equation (7)). As a result, the interstitial, weir, and oscillating flow regimes that we define do not correlate well on a one-to-one basis with flow regimes previously defined in the literature, but instead appear to span some portion of the nappe flow regime described by Chanson [1994, 2001], Church and Zimmerman [2007] and Comiti et al.[2009] Step-Crest Coefficients for the Weir Flow Regime [39] Using the thresholds observed by plotting the data points for each of the simulated and natural step crests in terms of C versus relative submergence (h/d step ) (Figure 5a), the data points corresponding to the weir flow regime were separated from those corresponding to the other flow regimes and further analyzed using linear regression techniques. Corresponding to only the weir flow regime, the C versus h/d step data points and the results of the linear regression analyses are provided in: Figure 6 and Table 1 for the planar-clast steps; Figure 7 and Table 2 for the subrounded-clast steps; Figure 8 and Table 3 for the roundedclast steps; Figure 9 and Table 4 for the log step and crests with instream wood; and Figure 10 and Table 5 for the natural steps. In Tables 1 5, the step geometry for each of the flume and field investigations is categorized in terms of the crest-clast, planform, longitudinal, and/or instream wood geometries shown schematically in Figure 4. [40] As also provided in Tables 1 5, the R 2 values for the C versus h/d step linear regression analyses corresponding to the 32 step geometries investigated ranged from 0.80 to 0.99, with a mean R 2 value of The results of the linear regression analyses indicate a high level of linear correlation between the step-crest coefficient (C ) and relative submergence (h/d step ) data in the weir flow regime for all of the step geometries investigated. Therefore, the results presented in Figures 6 10 and Tables 1 5 validate hypothesis H1 by demonstrating that the step-crest coefficient (C ) varies in a predictable manner over a useful range of flow rates and thereby further demonstrates that the assumptions associated with the derivation of the broad-crested weir flow equation are sufficiently valid for at least a subset of possible natural step-crest geometries Comparison of Incremental Changes in Crest Geometries [41] The step-crest coefficient (C ) can be viewed in terms of efficiency, because the flowrate over a step increases with increasing C values (equation (7)). This means that the higher the C values are for a step crest, the more efficient the step crest is at passing flow for a given upstream head. Comparison of the results in Figures 6 10 indicates that some step geometries are consistently more efficient than others within the crest-clast geometry groupings. The data for the planar-clast steps can be segregated into three basic groups, as shown in Figure 6, with the straight-notched step crests being the most efficient and the oblique angle steps being the least efficient per unit length of crest width (W crest ). The data for the subrounded-clast steps can be segregated into two basic groups (Figure 7), with the straight steps being more efficient than the arched steps. However, close inspection of Figure 8 and Table 3 indicates that the data for the rounded-clast steps cannot be easily segregated based on step geometry. In this case, altering the step geometry only resulted in relatively subtle variations in the corresponding C values for rounded-clast steps. [42] To evaluate the influence of crest staggering, we collected data for the same step crest with and without a staggered clast on the upstream face of the step (Figure S3 in auxiliary material). The intermediate axis of the staggered clasts ranged from 0.18 to 0.26 m in the 0.61 m wide flume and therefore represented a high level of crest staggering. Within Tables 2 and 3, there are 10 pairs of data for nonstaggered and staggered step-crests. Close comparison of these 10 pairs of data indicates that the nonstaggered step crests are consistently more efficient than staggered steps by approximately 2 to 12%, on average, with crest Figure 6. Planar-clast crests (Figure 4a): C values and regression lines for weir flow conditions as listed in Table 1. 8of14

9 Table 1. Planar-Clast Crests: C Value Regression Line and Correlation Coefficient Data Experiment ID Approach Slope (S a ) a Geometric Classification per Figure 4 Regression Line Parameters: C ¼ m (h/d l ) þ b m b D l (m) Min h/d l Max h/d l R 2 P1.1, Straight Step 5H:1V 4a and 4d P1.2, Straight-notched Step 5H:1V 4a, 4d, and 4j P1.3, Oblique Angle Step 6H:1V 4a and 4f P2.1, Straight Step 10H:1V 4a and 4d P2.2, Straight-notched Step 10H:1V 4a, 4d, and 4j P2.3, Oblique Angle Step 11H:1V 4a and 4f a S a is the slope of the bed upstream of the step crest. staggering having the lowest influence when the crests were composed of rounded clasts. [43] Figure 9 shows three sets of data points corresponding to a straight step constructed of rounded clasts, the same step crest with a single piece of channel-spanning wood (as shown schematically in Figure 4l), and the same step crest with a channel-spanning logjam (as shown schematically in Figure 4m). Figure 9 illustrates quantitatively how a step crest can become incrementally less efficient at passing flow for a given upstream head (h) as a logjam forms. [44] The results provided in Tables 1 5 and Figures 6 10 demonstrate that both the magnitude of the step-crest coefficients (C ) and the slope of the C versus h/d step data points can vary significantly, depending upon the geometry of the step crest in terms of the crest-clast, planform, longitudinal, and/or instream wood geometries shown schematically in Figure 5. Therefore, the results validate hypothesis H3 by demonstrating that the geometry of a step crest can have a significant influence on the hydraulic characteristics of the step Comparison of Flume and Field Investigations [45] For the weir flow regime, the results for all of the flume and field investigations are shown in Figures 11 and listed in Tables 1 5. As reflected in Table 5, the C values for the three natural steps range from to and have corresponding regression line slopes that range from to In comparison, the C values for the simulated natural steps range from to and have corresponding regression line slopes that range from to Hence, the C values for the natural steps are primarily within and only extend slightly below the range for the simulated steps investigated, and the regression line slopes for the natural steps all fall within the range of the simulated steps. Therefore, the results of our investigations indicate that the C values for the natural and simulated step-crests are comparable, and support hypothesis H4 in that simulated steps in the flume investigations can meaningfully represent the hydraulic characteristics of natural steps. 5. Discussion 5.1. Comparison of Step-Crest Coefficients With Discharge Coefficients [46] Typical discharge coefficients (C ) for a rectangular broad-crested weir range from approximately 0.47 to 0.59 [Brater and King, 1976, p. 5 40, L ¼ 0.31 m]. As shown in Figure 11, the C values for only a few of the simulated natural steps constructed of clasts extend above 0.47 and into the engineered weir range; however, the C values for the log step (experiment W1.1, Figure 4k) start and extend above those for both the engineered weir steps and all of the simulated/natural step crests composed of clasts. Therefore, the results of our investigations demonstrate quantitatively Figure 7. Subrounded-clast crests (Figure 4b): C values and regression lines for weir flow conditions as listed in Table 2. 9of14

10 Table 2. Subrounded-Clast Crests: C Value Regression Line and Correlation Coefficient Data Experiment ID Approach Slope (S a ) a Geometric Classification per Figure 4 Regression Line Parameters: C ¼ m (h/d mean ) þ b m b D mean (m) Min h/d mean Max h/d mean R 2 SR1.1, Straight-Stag. Step 12H:1V 4b, 4d, and 4i SR1.2, Straight Step 12H:1V 4b and 4d SR1.3, Arched-Stag. Step 6H:1V 4b, 4e, and 4i SR1.4, Arched Step 6H:1V 4b and 4e SR2.1, Straight-Stag. Step 6H:1V 4b, 4d, and 4i SR2.2, Straight Step 6H:1V 4b and 4d SR2.3, Arched-Stag. Step 3H:1V 4b, 4e, and 4i SR2.4, Arched Step 3H:1V 4b and 4e a S a is the slope of the bed upstream of the step crest. Figure 8. Rounded-clast crests (Figure 4c): C values and regression lines for weir flow conditions as listed in Table 3. Table 3. Rounded-Clast Crests: C Value Regression Line and Correlation Coefficient Data Experiment ID Approach Slope (S a ) a Geometric Classification per Figure 4 Regression Line Parameters: C ¼ m (h/d mean ) þ b m b D mean (m) Min h/d mean Max h/d mean R 2 R1.1, Straight-Stag. Step 6H:1V 4c, 4d, and 4i R1.2, Straight Step 6H:1V 4c and 4d R1.3, Arched-Stag. Step 7H:1V 4c, 4e, and 4i R1.4, Arched Step 7H:1V 4c and 4e R2.1, Straight-Stag. Step 10H:1V 4c, 4d, and 4i R2.2, Straight Step 10H:1V 4c and 4d R2.3, Arched-Stag. Step 12H:1V 4c, 4e, and 4i R2.4, Arched Step 12H:1V 4c and 4e R3.1, Straight-Stag. Step 70H:1V 4c, 4d, and 4i R3.2, Straight Step 70H:1V 4c and 4d R3.3, Arched-Stag. Step 40H:1V 4c, 4e, and 4i R3.4, Arched Step 40H:1V 4c and 4e a S a is the slope of the bed upstream of the step crest. 10 of 14

11 Figure 9. Step crests with instream wood (Figures 4k, 4l, and 4m): C values and regression lines for weir flow conditions as listed in Table 4. C values and regression line for weir flow conditions for experiment R3.1 are also shown for comparison purposes. Table 4. Log Step and Crests With Instream Wood: C Value Regression Line and Correlation Coefficient Data Experiment ID Geometric Classification per Figure 4 Regression Line Parameters: C ¼ m (h/d step ) þ b m b D step a,b (m) Min h/d step Max h/d step R 2 W1.1, Straight-Log Step (S a ¼ 8H:1V) c 4d and 4k WR3.1.1, Step R3.1 þspanning log 4c, 4d, 4i, and 4l WR3.1.2, Step R3.1 þspanning logjam 4c, 4d, 4i, and 4m a D step ¼ D wood ¼ m ¼ mean diameter of the log step in experiment W1.1. b D step ¼ D mean for experiment R3.1 in experiments WR3.1.1 and WR c S a is the slope of the bed upstream of the step crest. Figure 10. Natural step crests composed of subangular to subrounded clasts: C values and regression lines for weir flow conditions as listed in Table of 14

12 Table 5. Natural Step Crests: C Value Regression Line and Correlation Coefficient Data Experiment ID Geometric Classification per Figure 4 Regression Line Parameters: C ¼ m (h/d 50 ) þ b m b D 50 (m) Min h/d 50 Max h/d 50 R 2 E. Fork Roaring Creek, upstream site 4b, 4e, and 4i E. Fork Roaring Creek, middle site 4b, 4e, and 4i E. Fork Roaring Creek, downstream site 4b, 4e, and 4i two intuitive findings: (1) natural step-crest geometries composed of clasts (with and without instream wood) can be as much as 70% less efficient than rectangular weir steps; and (2) the log step (free of branches and bark) is more efficient than rectangular weir steps (and all of the simulated/natural step crests composed of clasts) by a factor of approximately 1.3 at low flows to 1.9 at higher flows Effects of Weir Submergence [47] The definition for weir or step-crest submergence corresponds to the condition where the tailwater depth downstream of the weir or step is at an elevation higher than the crest elevation [U.S. Bureau of Reclamation, 1948]. Our simulated natural steps were intentionally constructed such that the downstream tailwater would not be above the upstream step crest for the full range of flow rates investigated. Similarly, the step-pool reach along the East Fork of Roaring Creek has very high steps (i.e., > 2 m) in comparison to the bankfull depths at the step crests of approximately 0.4 m, so that step submergence cannot possibly occur even during an above-bankfull event. Hence, our flume and field investigations have specifically not considered the effects of step submergence on the step-crest coefficient (C ). [48] The U.S. Bureau of Reclamation [1948] investigated the effects of step submergence on the discharge coefficients for two dam crest geometries in a series of flume experiments. The results of these investigations are presented in terms of the degree of submergence ratio or h D /H, where H is the specific energy above the dam crest and h D is the vertical distance between the energy gradeline upstream of the dam crest to the tailwater depth. The degree of submergence (h D /H) value ranges from 1 to 0 as the crest becomes more submerged, where h D /H ¼ 1 (i.e., h D ¼ H) when the tailwater is at the same elevation as the crest (i.e., the point of crest submergence) and h D /H 0 (i.e., h D 0) when the dam or step crest is so submerged that it has essentially no effect on the water surface profile. [49] The results of the investigations indicate that for dam crest configurations with a substantial downstream drop similar to the steps in our investigations, the percent decrease in the discharge coefficient was less than approximately 1% and 5% for degrees of submergence (h D /H) as low as 0.63 and 0.30, respectively. This means that the errors in the discharge coefficients for nonsubmerged conditions are less than approximately 1% and 5%, even though the drops from the upstream energy gradeline to the tailwater are only approximately 60% and 30% of the head upstream of the dam crest (H), respectively. Therefore, the results of the USBR [1948] investigations demonstrate that even when a step crest is fully submerged by the downstream tailwater: (1) a step crest can still be a hydraulic control point in the water surface profile for step-pool sequences; (2) the weir flow model can be applicable with Figure 11. Comparison of C -value regression lines for all simulated and natural step crests under weir flow conditions as listed in Tables 1 5, including the straight-log step crest (experiment W1.1) and published discharge coefficient data for rectangular broad-crested weir [Brater and King, 1976, p. 5 40]. 12 of 14

13 only small to no adjustment to the free-flow discharge coefficients; and (3) the weir flow regime described in our investigations can extend from the nappe flow regime [Chanson, 1994, 2001; Church and Zimmermann, 2007] into the submerged flow regime [Wu and Rajaratnam, 1996, 1998; Church and Zimmermann, 2007] New Indirect Method for Estimating Flow Rates [50] We envision at least two basic approaches for applying weir-flow concepts for indirectly estimating flow rates in step-pool streams. The simplest approach involves applying the weir flow equation (equation (7)) using an estimated value of C based on observed characteristics of the step-crest geometry and field measurements of the effective width of the step crest (W), the D step appropriate for the step-crest and the upstream gauge head (h). To obtain more accurate flow estimates for relatively high flow conditions at a specific step, field measurements of flow rates and step-crest geometry can be collected under relatively low flow conditions (when collecting instream data is least hazardous) and used to estimate the linear C versus h/d step relation for the step. Bearing in mind the limitations described in section 4.1, this linear relation can then be extrapolated to allow estimation of a wider range of flow rates, including those with potentially more geomorphological significance, such as bankfull conditions Application in Hydraulic Model Simulations [51] The results of our investigations demonstrate that weir-flow concepts can provide an effective means for computing the upstream gauge head (h) for individual step crests within a step-pool sequence, by correlating the energy/ momentum losses due to flow contracting and accelerating over a natural step crest with step-crest coefficients (C ). This approach for simulating the hydraulics at individual step crests in HEC-RAS or other hydraulic modeling software could be useful in research, floodplain delineation and restoration projects. The current version of HEC-RAS includes an algorithm for simulating inline broad-crested weirs [Brunner, 2010], where the discharge coefficient (C) in HEC-RAS corresponds to C ¼ðC g 1=2 Þ=C v. The results of our investigations can be used to estimate the discharge coefficients needed to simulate the hydraulics at individual natural and/or restored steps using the inline broad-crested weirs routine in HEC-RAS. The broadcrested weir equation (equation (7)) and the C values from our investigations can also be used to manually compute external (i.e., upstream and/or downstream) and/or internal boundary conditions for use in various 1-, 2- and 3-dimensional model analyses Estimation of Upstream Pressure Force Acting on Step-Crest Clasts [52] In our broader investigations, the flow characteristics associated with step-crest formation and mobilization are of interest; hence, the forces acting on step-crest clasts are also of interest. Even though natural step-crest clasts only occasionally have vertical upstream faces as assumed in the weir flow model (Figure 3), characterizing the hydraulics at step crests in terms of C and weir flow concepts provides a means for at least estimating the upstream pressure force acting on a natural step-crest clast (i.e., force P w in Figure 3). That is, pressure force P w acting on a crest clast can also be expressed as a function of C and unit discharge (q) using the broad-crested weir flow equation (equation (7)) as: ðp w Þ clast ffið1=2þgw clast h clast ð2y 1 h clast Þ ffið1=2þgw clast h clast 2 q=ðc pffiffiffi 2=3 (8) g Þ þ hclast where: W clast and h clast are the width and height of the upstream face of the step-crest clast, respectively. 6. Conclusions [53] Recognizing that the basic underlying physics for shallow flow over step crests in step-pool streams can be very similar to that for rapidly varied flow over weirs, we have used flume and field investigations to evaluate the applicability of broad-crested weir flow concepts to natural step-crest geometries. The results of our investigations demonstrate that the assumptions associated with the derivation of the broad-crested weir flow equation are sufficiently valid for at least a subset of the possible natural step-crest geometries such that the step-crest coefficient (C ) varies linearly over a useful range of flow rates. This research can be of interest and immediate use to floodplain, stream restoration, and aquatic/riparian habitat managers, because it can provide a means to describe the hydraulics at key control points along the complex water surface profile of a step-pool sequence. This ability opens up many avenues for application including: (1) an indirect method for flow estimation specifically suited for step-pool streams; (2) hydraulic characterization of both natural and restored step-crest geometries; (3) selection of step-crest coefficients that can be used directly in HEC-RAS for modeling steps via the inline broad-crested-weir feature; (4) a means for computing both internal and external boundary conditions for 1-, 2- and 3-dimensional hydraulic modeling simulations; and (5) a means for estimating the upstream pressure force acting on step-crest clasts. [54] Acknowledgments. The comments of Graham Sander, C. Ancey, Greg Pasternack, A. Maxwell, and two anonymous reviewers improved the manuscript. References Aberle, J., and G. M. Smart (2003), The influence of roughness structure on flow resistance on steep slopes, J. Hydraul. Res., 41, Ackers, P., W. White, J. Perkins, and A. Harrison (1978), Weirs and Flumes for Flow Measurement, John Wiley and Sons, Chichester, N.Y. Bathurst, J. C. (1985), Flow resistance estimation in mountain rivers, J. Hydraul. Eng., 111, Bélanger, J. B. (1828), Essai sur la solution numérique de quelques problèmes relatifs au mouvement permanent des eaux courantes (Essay on the numerical solution of some problems relative to steady flow of water), Carilian-Goeury, Paris, France. Bos, M. G. (1988), Discharge measurement structures, ILRI Publ. 20, Int. Inst. for Land Reclam. and Improv., Wageningen, Netherlands. Bradley, J. N., and A. J. Peterka (1957), The hydraulic design of stilling basins: Hydraulic jumps on a horizontal apron (Basin I), J. Hydraul. Div., Am. Soc. Civ. Eng., 83(HY5), pp Brater, E. F., and H. W. King (1976), Handbook of Hydraulics for the Solution of Hydraulic Engineering Problems, McGraw-Hill, New York. 13 of 14