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Keywords:

  • hydraulic jump;
  • energy dissipation;
  • river step;
  • hydraulic geometry;
  • channel evolution

Abstract

  1. Top of page
  2. Abstract
  3. 1. Introduction
  4. 2. Step Systematics
  5. 3. Results
  6. 4. Discussion
  7. 5. Conclusions
  8. Acknowledgments
  9. References
  10. Supporting Information

[1] The river step is an important driver for geomorphic evolution in bedrock rivers, but the effect that variations in channel geometry upstream and downstream of a river step have on hydraulic jump regime and energy dissipation has not previously been investigated. The associated hydraulic jump is inherent to a river step and its regime is a primary control on step morphodynamics. In turn, the hydraulic jump regime is controlled by several variables as detailed in a new conceptual framework herein. Also in this study, a parsimonious semianalytical numerical model of step hydraulics is developed to quantify energy dissipation and delineate hydraulic jump regimes, accounting for discharge, jump submergence, and nonuniform channel geometry through a step. Despite remaining limitations in step theory, the model simulates how natural steps respond to a wide range of conditions. The model shows that hydraulic jump regime and energy dissipation exhibit greater sensitivity to channel nonuniformity as discharge increases and/or step height decreases. Also, channel conditions that create greater jump submergence lead to decreased energy dissipation, regardless of the discharge regime. The model also reinforces the common observation about gully erosion that downstream channel widening enhances upstream knickpoint migration. The new algorithm may be used to aid river engineering involving steps and could be useful for landscape evolution modeling.

1. Introduction

  1. Top of page
  2. Abstract
  3. 1. Introduction
  4. 2. Step Systematics
  5. 3. Results
  6. 4. Discussion
  7. 5. Conclusions
  8. Acknowledgments
  9. References
  10. Supporting Information

1.1. Background

[2] Landscape evolution models are just beginning to account for knickpoint migration in alluvial gullies [Flores-Cervantes et al., 2006], but the mechanisms by which lift and drag actually scour the bed below diverse river steps at all landscape positions from bedrock mountain tops to large alluvial rivers remain poorly understood [Pasternack et al., 2007]. A key challenge for understanding natural systems arises because experimental studies of hydromorphologic processes at river steps have previously been simplified with 2-D flume or scale model studies [e.g., McCarthy and O'Leary, 1978; Mason and Arumugam, 1985; Lenzi et al., 2002; Frankel et al., 2007]. Natural river steps, however, typically exhibit complex 3-D flow processes [e.g., Valle and Pasternack, 2002, 2006a, 2006b; Pasternack et al., 2006] and occur in nonuniformly shaped channels [e.g., Sinha et al., 1998]. A visual confirmation of the natural variability of process and form at river steps is evident in the entries to the world waterfall database (http://www.world-waterfalls.com), which documents 949 waterfalls between ∼100–1000 m high and ranging in estimated discharge from ∼150–42500 m3/s. A river step is defined herein as a vertical or near-vertical downstream drop in channel bed elevation, and may be referred to similarly as a waterfall, cascade, knickpoint, headcut, or downstep. This study addresses the fluid mechanics of steps relevant for eventual process-based simulation, regardless of their degree of complexity using a control volume approach in which upstream and downstream conditions constrain internal step processes.

[3] Studies of knickpoint migration in uniform alluvial gullies presently provide the most developed basis for proposed equations for use in landscape evolution modeling, but they only address the subset of natural steps that have significant plunge pools and they do not include the important role of the lift force exhibited by the flow on the substrate. The form of the shear-stress equation that is typically assumed to govern step migration rate, and thus long-term channel incision in such alluvial gullies, posits that the migration rate increases as discharge increases, because a higher discharge is expected to yield a higher shear stress on the bed below a step [Alonso et al., 2002; Flores-Cervantes et al., 2006]. Experimental flume studies do not consistently confirm that expectation, though there is always experimental error to consider. Slattery and Bryan [1992] reported a general increase in migration rate with discharge, but Robinson and Hanson [1996] reported a lower rate at higher discharge. Bennett et al. [2000] found no statistically significant relation among nine experimental runs, though the lowest observed migration rate did occur at the highest discharge. Beyond gullies, researchers have also applied shear stress models to bedrock rivers and determined that channel incision generally occurs locally around steepened knickpoint faces [e.g., Seidl and Dietrich, 1992; Stock and Montgomery, 1999]. On the basis of geomorphic studies of nonvertical, sloped waterfalls in Japan, Hayakawa and Matsukura [2003] proposed that the erosive stress on the face of a falls should be proportional to the square of the discharge.

[4] Two important mechanisms help explain why bed shear stress at the base of a step does not necessarily have to increase as a function of discharge when considering any arbitrary river step. First, steps with a 3-D plan view brink geometry (e.g., horseshoe falls, oblique falls, and labyrinth weirs) exhibit stage-dependent convergence and/or divergence of flow. Pasternack et al. [2006, 2007] showed that at low discharge, flow over a horseshoe falls converges strongly causing higher shear stress. As discharge increases, flow convergence decreases enough to yield a net decrease in local shear stress. Pasternack et al. [2007] experimentally observed lower downthrust stresses for correspondingly higher discharges while holding hydraulic jump regime constant. They also noted that these values could not be accurately predicted using the mathematical approaches suggested in the preceding paragraph to predict erosion in gullies.

[5] Second, shear stress on the bed below a step may decrease even as discharge increases because velocity at the bed is dependent on the hydraulic jump regime, and the latter may change as discharge increases [Pasternack et al., 2006], causing a decrease in shear stress at higher flows. Some previous geomorphic research has discussed the role that hydraulic jumps have in flow mechanics and channel evolution [e.g., Kieffer, 1987; Carling, 1995; Grant, 1997; Montgomery and Buffington, 1997]. Hydraulic jumps occur as rapid transitions from supercritical to subcritical flow [Chanson, 1999], and are recognized to be controlled by variations in channel geometry and/or stream discharge [e.g., Mossa et al., 2003], but the extent of this control is largely unknown [Balachandar et al., 2000]. In experimental flume studies, it is possible to manipulate the hydraulic jump regime independently of discharge through the use of a sluice gate downstream of the step. By lowering the gate, flow can be reduced, thereby increasing water depth downstream of the step (i.e., “tailwater” depth). In nature the analogous mechanism for tailwater control is the hydraulic geometry associated with channel size and shape as well as discharge. A detailed characterization of hydraulic jump regimes at steps is presented below in section 2.4, and an explanation of the role of hydraulic geometry at a step is presented in section 2.2. As of yet, no studies have systematically explored lift and drag forces below steps over the full range of hydraulic jump regimes. Studies of hydraulic jumps at the toe of dam spillways have shown that jumps are capable of creating hydraulic forces that can weaken and erode such structures [e.g., Smith, 1976; Fiorotto and Rinaldo, 1992; Vischer and Hager, 1998]. On the basis of experimental measurements, bed material can be plucked by turbulent pressure fluctuations and strong lift forces under hydraulic jumps [Bollaert and Schleiss, 2003; Pasternack et al., 2007]. Plucked material can then be exported by the high drag forces just downstream of jumps [Bormann and Julien, 1991; Pasternack et al., 2007]. In terms of knickpoint migration in gullies, no experimental studies have systematically manipulated hydraulic jump regime to ascertain its effect on migration rate. It is known that as the plunge pool deepens the force of the jet impinging at the bottom of the pool decreases. Similarly, it is conjectured that as a hydraulic jump or plunge pool becomes increasingly submerged with increasing discharge, the deceleration of the impinging jet would dampen pressure fluctuations on the bed and lift fluctuations above it. These effects provide another reason why the rate of knickpoint retreat would not necessarily increase with discharge. Hydraulic jump regime is therefore likely to be an important aspect in river step mechanics, but it is largely controlled by channel geometry. Lacking experimental studies to clarify these issues, the opportunity exists for new theoretical developments.

[6] In previous research on river steps, the effect of variability of channel geometry upstream and downstream of a step on step hydraulics has not been investigated. A few studies of engineered spillways have discussed the use of downstream channel widening as an energy dissipater [e.g., Ram and Prasad, 1998; Ohtsu et al., 1999]. However, studies of man-made dams and spillways address a very narrow range of channel conditions in which a single optimal design is sought. In contrast, natural channels can expand or constrict through a step to any arbitrary degree yielding diverse nappe trajectories (i.e., water profiles over the vertical drop) and hydraulic jump conditions (including the absence of a jump) whose combined effects on energy dissipation, bed scour, and step migration are presently unknown.

[7] On the basis of the above analysis of past studies, a key limiting factor in understanding and predicting scour at river steps is associated with understanding the stage dependence of river step fluid mechanics. The focus of this study was to use a numerical model to heuristically investigate channel hydraulic geometry and discharge in determining the hydraulic jump regime and energy dissipation at a river step. Specific objectives included predicting the hydraulic jump regime and energy dissipation as (1) discharge varies for a given channel geometry, (2) channel geometry varies for a given discharge, and (3) channel geometry upstream of a step varies relative to that downstream of it. The approach involved a purely theoretical framework in which available analytical and empirical equations were coupled to yield a new parsimonious model formulation. Admittedly, the resulting numerical model has several assumptions and limitations, but it does provide a strong heuristic explanation of why erosion at river steps is not a direct function of only discharge. It also elucidates the key scientific gaps that need to be addressed to promote further advancement.

[8] Although the study presents detailed fluid mechanics results, the general conclusions are relevant to a variety of applied water resources problems involving natural and man-made river steps. One value of this work is that it provides a new and different approach to predicting erosion at river steps in channels with variable geometry in landscape evolution models. This model does not yet predict scour directly, but it predicts hydraulic jump regime and energy dissipation, which are both important factors controlling bed scour below steps. Another value is that river rehabilitation and engineering project conceptual models, including those for dam removal, fish passage, and whitewater parks, often employ steps, but do not consider the importance of channel geometry in controlling the safety and functionality of these hydraulic structures. This model provides a tool that would be of immediate value in improving the safety of hydraulic structures.

1.2. Step Conceptual Framework

[9] The role of hydraulic jump regime in scour at the bottom of a river step is not widely understood. Very few experimental studies of river steps have varied the hydraulic jump regime over the full range possible to explore this factor. To promote a better understanding of the general relevance of hydraulic jumps associated with river steps for water resources management and to guide research ultimately leading to prediction of river step morphodynamics, a conceptual model encompassing independent variables and responding processes was developed in which the key dynamics were grouped into five categories (Figure 1). Evolution of a river step can be characterized by the processes of scour hole formation, upstream retreat, and change in step geometry. These processes are driven by a complex, interdependent set of hydrologic and geologic variables acting over multiple scales. Basin variables include the independent watershed inputs of water and sediment discharges as well as channel geology (Figure 1). For example, some steps may exist in various geologic conditions ranging from well-bedded sedimentary bedrock to fractured homogeneous igneous bedrock. The role of sediment supply has been an important recent addition to shear stress models [e.g., Sklar and Dietrich, 2004; Gasparini et al., 2007]. The basin variables control the channel variables, which include the cross-sectional geometries within the channel, the channel slope, and the longitudinal spacing between river steps. In this framework, a sequence of cross sections is used as an indicator of complex 3-D channel morphology, since channel width and cross-sectional area often fluctuate down a mountain river system, even as they generally increase downstream because of increasing discharge. Basin variables also help shape the step morphology components of step height, planform shape of the step, step slope, and step roughness. Step height can be dependent upon step spacing [Wohl and Grodek, 1994].

image

Figure 1. Flowchart showing interrelationships between dependent and independent processes at river steps. Rhombi are external independent processes. Bold arrows indicate processes discussed herein.

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[10] All of the step morphology variables affect step hydraulics characterized by the nappe trajectory and hydraulic jump regime, though step roughness only affects the jump regime indirectly through nappe trajectory. Nappe trajectory and hydraulic jump regime are of central importance in the conceptual model because (1) the former is a key variable controlling jet scour of bedrock near the step toe [Mason and Arumugam, 1985; Bormann and Julien, 1991; Stein et al., 1993; Alonso et al., 2002], where toe refers to the slope break at the bottom of the step and (2) the latter controls turbulent lift forces that pluck and suspend bedrock downstream of the point of jet impact [Fiorotto and Rinaldo, 1992; Pasternack et al., 2007]. Nappe trajectories for linear overfalls have been thoroughly investigated [U.S. Bureau of Reclamation, 1948b; Vischer and Hager, 1998; Chanson, 2002], while those for overfalls with 3-D brink configurations have only recently come under some scrutiny [Falvey, 2003; Pasternack et al., 2006]. Hydraulic jump regimes for a free overfall include: supercritical flow with no jump, pushed-off unsubmerged jump (defined later in section 2.4), optimal jump, submerged jump, standing waves, and subcritical flow with no jump [U.S. Bureau of Reclamation, 1948b; Leutheusser and Birk, 1991]. Hydraulic jump regime is strongly influenced by discharge and tailwater depth [U.S. Bureau of Reclamation, 1948b; Pasternack et al., 2006], with the latter in turn controlled by upstream and downstream channel configuration. Step brink planform shapes that deviate from linearity cause nappe interference and a shift in jump regime for a given discharge and channel configuration [Falvey, 2003; Pasternack et al., 2006]. Additionally, the nappe regime has been anecdotally observed to be affected by the direction and magnitude of wind impacting it, but no scientific studies have yet explored nappe response to diverse wind regimes.

[11] Step hydraulics such as jet impact, turbulent pressure fluctuations, drag, and lift drive channel morphodynamics by changing the size and shape of the scour hole [Lenzi et al., 2003; Alonso et al., 2002], step geometry [Pasternack et al., 2006], and upstream retreat of the step [Hanson et al., 1997; Bennett et al., 2000; Stein and LaTray, 2002]. Upstream retreat is determined by the relative erodibility and erosional force on the step top versus that on the face and toe of the step [e.g., Stein and Julien, 1993; Flores-Cervantes et al., 2006; Frankel et al., 2007]. The shape of the scour hole and the regime of the associated hydraulic jump affect the erosional ability of the flow below the step. Scour depth has previously been shown to be dependent on step morphology [Alexandrowicz, 1994; Lenzi et al., 2002] and sediment supply [e.g., Marion et al., 2006]. Step morphodynamics, in turn, can affect channel geometry and step morphology.

[12] The conceptual framework described above serves to organize past research, promote quantification of identified linkages, and highlight important gaps in the current understanding. A casual observer of major waterfalls and whitewater rapids in mountain rivers will quickly take note of the diversity and complexity of natural step morphologies. Addressing natural diversity presents the most important gap in the scientific understanding of river steps. This study addresses the problem of how channel expansions and constrictions through geomorphic units with steps affect step fluid mechanics.

2. Step Systematics

  1. Top of page
  2. Abstract
  3. 1. Introduction
  4. 2. Step Systematics
  5. 3. Results
  6. 4. Discussion
  7. 5. Conclusions
  8. Acknowledgments
  9. References
  10. Supporting Information

[13] Although it is impossible at this time to produce a numerical model that incorporates all of the processes discussed in section 1.2, that conceptual framework can serve as a roadmap to guide process-based research leading toward an eventual predictive capability. The first phase of developing a suitable numerical model requires starting at the step itself and adequately characterizing step hydraulics on the basis of step morphology and surrounding channel variables. The quantitative formulation used below is very different from that proposed previously for gullies [e.g., Alonso et al., 2002; Flores-Cervantes et al., 2006]. Admittedly, this approach has limitations and uncertainties, but the results in section 3 provide new insights that shed light on how river steps likely evolve over time as a result of natural spatial variation in channel geometry.

2.1. Eulerian Governing Equations

[14] As an introduction to the broader 3-D problem, consider steady energy and momentum conservation for a control volume in a level rectangular channel with clear water including a broad-crested bed overfall and the region downstream of the step (Figure 2). There holds for average conditions the following classic hydraulic equations [e.g., U.S. Bureau of Reclamation, 1948b; Ackers et al., 1978; Chaudry, 1993; Chanson, 1999; Munson et al., 2006]: the overall energy conservation equation

  • equation image

a rearrangement of equation (1) that solves for the submergence variable, hd

  • equation image

the mass conservation equation that is valid for any cross section, i

  • equation image

a special case of equation (1) that solves for the critical flow condition

  • equation image

the definition of Froude Number at location i

  • equation image

and the broad-crested weir equation derived from equations (1) to (4)

  • equation image

where Ei and hi are total energy and water depth at any location i as defined in Figure 2, H the specific energy at the upstream location (weir crest as datum), P the broad-crested step height, q the specific discharge, hL the total energy loss in the control volume, g the gravitational constant, vi the velocity at location i, and Cb = 0.848 the broad-crested weir discharge coefficient [Ackers et al., 1978; Leutheusser and Birk, 1991; Chanson, 1999]. For the equations presented above and throughout this text, the subscript “up” refers to the channel section upstream of the step, and the subscript “tail” refers to the controlling channel section downstream of the step and associated hydraulic jump regime (Figure 2). Equations (4)–(6) assume a rectangular channel, and are approximate for a wide channel of any shape. They are not, however, ultimately used in the new numerical model presented herein, thereby representing a significant advance and eliminating that concern.

image

Figure 2. Longitudinal schematic of an idealized step showing a submerged jump illustrating parameters used in the model. Not to scale to permit notation (See notation list).

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[15] In past studies momentum conservation was used to isolate the head loss solely due to the hydraulic jump for the classic hydraulic jump [Chanson, 1999] and unsubmerged jumps at overfalls [Henderson, 1966]. In those cases there exists an identifiable upstream supercritical cross section and downstream subcritical cross section bounding the jump. The Belanger equation has also been modified to describe the unsubmerged hydraulic jump for nonprismatic channels [Negm, 2000]. However, these approaches are invalid when the hydraulic jump is submerged because the equation necessitates a measurable upstream supercritical depth value, but the supercritical jet is underwater. As the model developed herein encompasses all hydraulic jump regimes, including submerged jumps, the classic Belanger equation is not used and it is thus not possible to isolate the energy dissipation of just the jump.

[16] Critical depth is often used to nondimensionalize variables; however, locating the critical point introduces error [Ackers et al., 1978], whereas defining upstream specific and total energy is more practical and certain. In addition, the variable (H + P)/H is the nondimensional energy variable accounting for both discharge and step height [U.S. Bureau of Reclamation, 1948b]. It shows that geometric scaling to yield any energy condition is achievable by holding either step height or flow constant. Higher (H + P)/H values correspond with taller steps with relatively less flow over them. As discharge increases for a given step height, so does the head on the step. Therefore decreasing (H + P)/H values represent conditions of increasing discharge and increasing energy input for a given step height. In the lower limit of no step (P = 0), the variable approaches unity.

[17] In this study, “submergence” is defined as the condition when htail is deep enough to place the leading edge of the jump upstream of the location of the free-falling nappe toe [Leutheusser and Birk, 1991], where the nappe toe is defined as the slope break in the water surface profile downstream of the step. According to classic equations (1)–(6), when (H + P)/H and the nondimensional submergence variable (hd/H) are specified, the resulting hL/(H + P) and flow kinematics at upstream and tail cross sections are independent of step geometry [U.S. Bureau of Reclamation, 1948b; Pasternack et al., 2006]. However, the degree of submergence and hydraulic jump regime are dependent on aspects of step geometry, notably planform brink shape and step slope.

[18] Whereas engineers classically solved equations (1)–(6) to find optimal spillway and overfall designs for engineered structures to minimize damage to such structures [e.g., Moore, 1943; White, 1943; Henderson, 1966], those studies did not tackle the broader water resources problem facing natural channels in which step fluid mechanics responds to a very wide range of nonoptimal upstream energy and downstream submergence conditions. For example, there exist many steps in bedrock rivers that exhibit supercritical jets impinging on exposed bedrock and continuing on as supercritical flow; peruse the world waterfall database at http://www.world-waterfalls.com to see many examples of this phenomenon. There are also many steps with submerged jumps due to the presence of a plunge pool. While transference of classic engineering foundations to the problem of understanding natural 3-D waterfalls has value, the unique aspects of natural systems warrant further investigation. Pasternack et al. [2006] addressed this broader problem by solving equations (1)–(6) for fractional energy dissipation hL/(H + P) for a larger nonclassical range of submergence hd/H and energy (H + P)/H. Unlike in laboratory flumes with sluiced inflows, the upstream Fr approaching a step in a natural channel is not an independent variable, so it is not a governing variable for natural systems. The results showed that the maximum hL/(H + P) for any (H + P)/H occurs when htail is exactly critical with no hydraulic jump present. This maximum involves a transition from supercritical to critical flow and htoe < htail. Also, as htail is decreased to less than critical depth, hL/(H + P) decreases and the flow increases its efficiency until htoe = htail. The primary conclusion from their analysis was that arbitrary htail must be included in models of flow kinematics and energy dissipation at natural river steps.

2.2. Channel Parameterization

[19] Even though Pasternack et al. [2006] used the classic equations (1)–(6) to expand the breadth of the relevance of those equations into nonclassical water resources problems, their model assumed a uniform rectangular channel, because that was their flume setup for experiments on a broad-crested horseshoe waterfall. In this study, equations (4)–(6) were replaced to further generalize the model to reduce or eliminate the assumptions of a rectangular channel, a 2-D overfall, and mutual independence of (H + P)/H and hd/H. Arbitrary upstream and downstream cross-sectional morphologies are now enabled and characterized using another classic principle from a different branch of river science, namely empirical at-a-station hydraulic geometry relations originally stemming from channel regime theory [Garde and Ranga Ragu, 1985]. Even though such equations are empirically derived geometric descriptors, they do satisfy mass conservation requirements [Leopold and Maddock, 1953] and are widely used by fluvial geomorphologists, who accept them as classic in their discipline. These equations do not assume a rectangular channel. They are used herein to replace equations (4)–(6) to obtain discharge from upstream channel conditions and add more equations to govern channel changes through the geomorphic unit containing the step. As will be shown, they also are used to directly link (H + P)/H and hd/H, so those two variables are not independent. Although brink geometry and nappe trajectory play an important role in step dynamics, they are not necessary in quantifying the net energy loss between the upstream and downstream cross sections. Admittedly, both factors affect the jump regime [Falvey, 2003; Pasternack et al., 2006] and that remains an important limitation of the numerical model discussed further below. However, the model will clearly illustrate the importance of variable cross-sectional geometry on hydraulic jump regime and energy loss at river steps.

[20] According to classic channel regime theory, at a given cross section the flow width, depth, and velocity can be related to the volumetric discharge (Q) by the power functions [Leopold and Maddock, 1953]

  • equation image
  • equation image

and

  • equation image

where w is average channel width, h is average channel depth, v is average flow velocity, {a, c, k} are empirical coefficients, and {b, f, m} are empirical exponents that largely control cross-sectional shape. Equation (3) governs that b + f + m = 1 and a × c × k = 1 [Leopold and Maddock, 1953]. In this study, b, f, m, a, c, and k are all termed “parameters” of the overall model. No general values for these parameters have been published yet for bedrock rivers with steps, though such studies appear pending. Several specific values have been published for alluvial rivers [Wilcock, 1971; Dury, 1976; Betson, 1979; Andrews, 1984; Rhoads, 1991; Singh, 2003; Stewardson, 2005], with many states in the United States now developing extensive databases stratified by physiographic province. After a review, the published values of the b exponent were found to vary from 0.04 to 0.6, and those for the f exponent from 0.20 to 0.65. Similarly, those of the a coefficient ranged from 3 to 19, with a mean value of 9.74, and those of the c coefficient from 0.474 to 0.73, with a mean value of 0.51. In this study, hydraulic geometry relations are used to describe channel size and shape at one cross section located upstream of a step and one downstream of it. When applying the numerical model developed in this study to a given real step, one can use monitoring data to parameterize the coefficients and exponents for the cross sections at that site.

[21] To interpret the results of this study, it is critical to understand the geometric patterns implied by combinations of exponents and coefficients. Dingman [2007] provides a detailed mathematical analysis of how to interpret these parameters, and the key concepts are summarized below. Channel shape changes predictably as a function of the exponents, assuming constant coefficients (Figure 3). When b = f, the channel is triangular in cross section. In the limit, as b approaches 0, width does not change as discharge increases. Geometrically, that necessitates that the channel is rectangular when b = 0. For values of b < f, the channel is concave up. Similarly, as f approaches 0, depth does not change as discharge increases. Geometrically, that necessitates that the channel is infinitely wide when f = 0. For values of b > f (Figure 3, shaded region), the channel is convex up. In summary, when bf and one of those exponents is allowed to change, then the geometric effect is one of “bending,” for increasing f bending is inward and for increasing b, bending is outward. As both b and f approach 0 together, the velocity exponent, m, increases. Geometrically, a condition of no change to depth or width as discharge increases requires an increase in energy slope and/or a decrease in channel roughness with increasing discharge. It is widely held that roughness decreases as discharge increases [Smart, 1999; Dingman, 2007]. Similarly, for a meandering channel, valley slope is higher than channel slope at the reach scale, so the energy slope is higher for out-of-bank flows than in channel flows.

image

Figure 3. Physical interpretation of hydraulic geometry exponent values for a cross section in a single_threaded stream. Shaded region is where b > f.

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[22] For any given set of exponent values and corresponding channel shape, the values of the coefficients affect the scaling of the channel. A higher value of the coefficient a stretches the banks horizontally (i.e., for a triangular channel, the angle between the banks increases), which makes the wetted channel flatter. A higher value of the coefficient c stretches the banks vertically, which makes the channel higher. Finally, a higher value of the coefficient k is associated with a steeper bed slope and/or a smoother bed at any given discharge.

[23] Although the exact geometries associated with changes in exponents versus coefficients are different (i.e., bending versus stretching, respectively), the effects of both types of changes on the simpler variables of cross-sectionally-averaged channel depth and width are similar. For example, increasing either a or b alone in equation (7a) causes the width at any given discharge greater than one to increase (Figure 4). Either change also causes the slope of the tangent line of the relation to increase. Thus in terms of width and depth, changes to both types of parameters yield similar outcomes, even though they do so in different ways geometrically.

image

Figure 4. Sensitivity analysis illustrating the similar effects on channel width of varying either (a) the coefficient a and (b) the exponent b in equation (7a), while holding the other constant.

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[24] Finally, while hydraulic geometry relations can characterize braided or multithreaded channels, which sometimes are associated with river steps, as exemplified by Great Falls on the Potomac River between Maryland and Virginia (38°59′51″N, 77°15′10″W), the parameters for multithreaded channels are nonunique and thus difficult to interpret. Although hydraulic geometry equations are empirically derived, they are well suited for parsimonious exploration of channel nonuniformity at river steps. They are also easily quantifiable at real river steps by mapping the cross sections upstream and downstream of the step when flow is low and then using automated stage recorders at those locations to record the local stage discharge responses. Accurate physics-based 3-D hydrodynamics models of the multiphase flow through hydraulic jumps at natural river steps have yet to be achieved, limiting strict physics-based alternatives.

2.3. Channel Nonuniformity

[25] For the independent variables of step height and upstream depth along with the upstream values of the geometric exponents (bup and fup) and coefficients (aup and cup) of equations (7a) and (7b), the resulting relative submergence and head loss through the step (equations (1) and (2)) are determined from the following set of equations. For the equations presented here, the subscript “up” refers to the channel section upstream of the step, and the subscript “tail” refers to the controlling channel section downstream of the step and hydraulic jump regime (Figure 2).

[26] Rearranging equation (7b), discharge is determined at the upstream cross section as

  • equation image

where hH is the effective flow depth above the step crest. Again, this equation does not assume a rectangular channel. As shown in Figure 2, hH can be determined by

  • equation image

[27] For varying effective step heights for the upstream and downstream channel sections, the definition of hH remains the same, but the calculation of depth may differ. This definition accounts for the physical reality that the water from hup = 0 to hup = P is “dead” storage that does not affect the dynamics of the step, i.e., there can be no tailwater until this volume is filled. Figure 2 represents the geometry of this dead storage with an arbitrary channel bottom, which means there could be a step up for the flow or the channel bottom could be flat and planar leading up to the step drop. In either instance, that geometry does not affect the equations used in this model. In practical application, one would develop an upstream hydraulic geometry relation referenced to the step crest elevation instead of the channel bottom as most are.

[28] Next, the tail depth is determined for the controlling downstream cross section at the exit of the tail pool as

  • equation image

where fratio is the ratio of the upstream f exponent (fup) to the downstream f exponent (ftail). An fratio of one describes a uniform channel consistent with the previous assumption of Pasternack et al. [2006]. An fratio < 1.0 refers to a system in which flow transitions from a wider, more triangular channel into a narrower, taller canyon such that the upstream depth rises slower than the downstream depth as discharge increases (Figure 5). Examples of this are ubiquitous in mountain channels (e.g., Figure 6). A similar effect on width and depth could be achieved by having a lower c value upstream and a higher c value downstream, though the exact geometric interpretation would be different, since the nonuniformity would be due to stretching, as described earlier. Also, cup is scaled by the fratio in equation (10), and this indirect effect on htail is synergistic with the direct effect of the fratio, though the relative magnitude of the effect will be evaluated in the results later. For example, if cup = 0.51 and fratio = 0.50, then cequation image = 0.26. This smaller value in the denominator makes htail bigger for any given discharge, and thus is consistent with a deeper constricted channel downstream. An fratio > 1.0 refers to the opposite effect as just described. In this case, flow transitions from a narrower, more triangular channel into a wider, flatter channel (Figure 7). Many natural example of this type of transition can be found in mountain rivers, as well (e.g., Figure 8). If cup = 0.51 and fratio = 1.5, then cequation image = 0.64, so the coefficient effect is smaller than for an equivalent incremental change of fratio when fratio < 1. In both cases, the step is located in the transitional region, and is thus impacted by the nonuniformity. To fully understand the consequences of using at a station hydraulic geometry relations to characterize channel nonuniformity, sensitivity analyses were performed over a wide range of model parameters, as discussed further in section 2.5.

image

Figure 5. Illustration of the channel nonuniformity through a reach with a river step where the fratio < 1. Widths, depths, and shapes are roughly to scale, but are not mapped quantitatively. Depth and width values for values of (H + P)/H equal to 5.81 (i.e., low discharge) and 1.98 (i.e., high discharge) are shown for fup = bup = 0.4 and fratio = 0.5.

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image

Figure 6. Example of a step unit with f > b and fratio < 1 showing channel constriction through the domain from the Upper South Fork Snoqualmie River, Washington, looking upstream.

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Figure 7. Illustration of the channel nonuniformity through a reach with a river step where the fratio > 1. Widths, depths, and shapes are roughly to scale, but are not mapped quantitatively. Depth and width values for values of (H + P)/H equal to 5.81 (i.e., low discharge) and 1.98 (i.e., high discharge) are shown for fup = bup = 0.4 and fratio = 0.5.

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Figure 8. Example of a step unit with f > b and fratio > 1 showing channel expansion through the domain from the Upper South Fork Blackwood Creek, California, looking upstream.

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[29] Upstream and downstream widths are determined from hydraulic geometry relations on the basis of the discharge obtained in equation (8)

  • equation image

and

  • equation image

Insufficient literature exists on the interactions of b, f, and m values for at a station conditions in the vicinity of river steps to precisely constrain b values at this time. However, it is conceptually evident that as a channel's f value increases a channel becomes more rectangular and thus the b value must decrease (Figure 3). Dingman [2007] quantitatively demonstrated this for three different hydraulic equations. On this basis, the ratio of upstream and downstream b values was assumed to be equal to the inverse of the fratio. Future field investigation may enable revision of this assumption, but it is expected to only affect magnitudes of effect and not cause threshold effects.

[30] The cross-sectionally-averaged upstream and downstream velocities and corresponding averaged velocity heads were determined from equation (7c) and calculated as

  • equation image

and

  • equation image

where the coefficients, k, and exponents, m, are solved from mass conservation (k = 1/ac and m = 1 − (b + f) [Leopold and Maddock, 1953]. These equations do not assume a rectangular channel.

[31] The upstream specific energy (H) is the effective head on the step (Figure 2), and is calculated as

  • equation image

Thus the key variables of submergence, hd, (equation 2) and head loss, hL, (equation 1) can now be nondimensionalized as hd/H, and hL/(H + P) in conjunction with the other key variable, nondimensional upstream energy, (H + P)/H, to define the Eulerian fluid mechanics of a river step.

2.4. Hydraulic Jump Regime Equations

[32] Previous researchers have developed threshold equations for hydraulic jump regimes. Most such equations delineate jump regimes using only the Froude number on the basis of observations of the classic hydraulic jump in a rectangular flume with no bed step, where variable Fr is imposed using an upstream sluice gate [e.g., Chanson, 1999]. For the nonclassical case of nonuniform channels with overfalls, jump regime equations must account for jump submergence. Of the published equations, only those of the U.S. Bureau of Reclamation [1948b] for an ogee-crested 2-D overfall and those of Pasternack et al. [2006] for a broad-crested, rectangular overfall do that (Figure 9). The thresholds of U.S. Bureau of Reclamation [1948b] are empirically derived and, most importantly, nondimensional. Those of Pasternack et al. [2006] are semianalytical and only address two regimes: the critical Fr and optimal jump thresholds. Conceptually similar regimes for a sharp-crest weir were described by Leutheusser and Birk [1991], but not quantified. It is possible to modify and use the model of Pasternack et al. [2006] to establish the critical Fr threshold for a sharp-crested weir. More investigations into regime equations for typical 3-D overfall brink configurations are needed.

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Figure 9. Generalized hydraulic jump regimes in reference to dimensionless upstream energy and downstream submergence, as derived from the U.S. Bureau of Reclamation [1948b].

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[33] Accepting the limited availability of relevant equations, Figure 9 shows the four empirical regime boundaries used in this study that were first described by the U.S. Bureau of Reclamation [1948b] for an ogee-crested weir. The top curve represents the threshold above which flow over the step remains supercritical and cascades past the tail cross section. The threshold curve below that delineates the optimal jump condition with the jump occurring exactly at the step toe. The condition that lies between those thresholds represents emerged, or pushed-off, hydraulic jumps with a measurable supercritical zone and rapid transition to a subcritical zone within the control volume. The next curve below delineates jumps that are sufficiently submerged to be considered “drowned,” which is undefined by U.S. Bureau of Reclamation [1948b]. Drowned jumps are interpreted to mean those submerged jumps with a strong upstream recirculation. The bottom line in Figure 9 represents the threshold below which no hydraulic jump occurs, with the flow having either standing waves or no near-critical surface expression at all. There is no regime uniquely delineated for a standing wave condition as shown by the U.S. Bureau of Reclamation [1948b] framework, which is a detriment. Overall, this quantitative framework serves to delineate well-recognized jump types, each with characteristic fluid mechanics in terms of jet impact force, turbulent pressure fluctuations, drag along the bed, and lift force.

[34] The threshold equations of the ascribed conditions for 2 ≤ (H + P)/H ≤ 6 (Figure 9), with the corresponding terminology of Leutheusser and Birk [1991] given in parentheses, are:Pushed-off jump (swept-out jump)

  • equation image

Hydraulic jump (optimum jump)

  • equation image

Drowned jump (plunging nappe)

  • equation image

No jump (surface nappe)

  • equation image

The above equations provide a reasonable characterization of the hydraulic jump regimes at or downstream of a step toe. Substituting the equations of Pasternack et al. [2006] for equations (15) and (16) produce a very small change in the lines, with minimal conceptual difference.

[35] One important limitation of these equations is that they do not account for the effects of nonuniform step brink geometry on jump regime (i.e., whether in plan view the brink is horseshoe shaped, perpendicular to the banks, oblique to the banks, etc). Pasternack et al. [2006] describe the deviations from these conditions associated with horseshoe-shaped brinks, such as that of Niagara Falls. Another limitation of the equations is that they do not account for regime deviations associated with abrupt channel expansions or constrictions. Additional factors that could affect the equations include step and plunge pool roughness, plunge pool depth, bedrock resistance, and high winds. Because the jump regime is influenced by so many variables, it is not apparent how to further generalize these equations other than by careful study of each possible archetypal configuration. Nevertheless, this study provides a first assessment as to how the most important forms of channel nonuniformity would affect hydraulic jump regimes.

2.5. Model Implementation

[36] Equations (1), (2), and (8)–(14) were programmed into Excel 2003 (Microsoft Corp, Redmond, Washington) to investigate the effect of varying upstream and downstream channel geometry, in conjunction with discharge and step height, on flow processes, hydraulic jump regime, and energy dissipation associated with a river step. The key independent variables were (H + P)/H and fratio, with fup, bup, aup, and cup serving as key parameters. The nondimensional response variables evaluated were hd/H, indicative of hydraulic jump regime, hL/(H + P), a measure of energy loss, and hv_tail/(H + P), a measure of kinetic energy.

[37] The computational procedure for the model begins with a set of starting values {aup, cup, bup, fup, fratio, hH, and P}. An arbitrary value of 5 was used in the model for the step height, P. From the upstream depth and hydraulic geometry values, a discharge is calculated (equation (8)) for that depth. The upstream width and specific energy (H) are calculated using equations (11a) and (14). Using the given fratio, the downstream hydraulic geometry parameters are calculated, and then the model calculates the downstream depth, width, and specific energy (equations (10)–(14)). Equations (1) and (2) are then used to determine the head loss, hL, and submergence, hd. These variables and the total energy (H + P) are nondimensionalized by the upstream specific energy, H.

[38] Plots of these response variables as a function of (H + P)/H were made for nine combinations of upstream f and b values from the set {0.2, 0.25, 0.4} to explore the sensitivity of the model to the absolute value of each parameter. Values of f or b higher than 0.4 did not yield substantially different results than those shown for 0.4, and hence are not presented here. For each combination of b and f, six values of fratio {0.5, 0.75, 0.9, 1.0, 1.25, 1.5} were used in the model to investigate its influence on river step response. For these ranges of f and b values that were used, the model yielded ranges of discharge from 3 to 104,000, upstream width from 12 to 400, and upstream depth from 0.8 to 5.5 among the six combinations of fratio. These variables have typical dimensions, but specific units are purposefully omitted to allow the model to be applied to any channel system within any system of units. These analyses therefore capture the essential functionalities and sensitivities of the model with respect to the exponents, and through them natural steps.

[39] To explore the effects of the coefficients of equations (7a)–(7c), a sensitivity analysis was performed in which the range of reported values for a and c were investigated. The analysis consisted of holding the upstream b and f values constant, while varying the geometric coefficients, a and c, for the range of modeled fratio. In these tests, aup = atail and cup = ctail. Two combinations of fup and bup were evaluated, namely fup = bup = 0.2 and fup = bup = 0.4. Future versions of the model could solve for the model parameters on the basis of various theories that attempt to explain their origin, but in this study the goal was to provide a characterization of the effects of their values on river step hydraulics.

[40] Currently, this model is purely theoretical. As with many models in hydrology and geomorphology, including distributed hydrological models, 3-D hydrodynamic models, and landscape evolution models, this one is out in front of the experimental capability to validate all of its components. No data sets presently exist to test the range of the model. To empirically satisfy the physicality of the model would require laboratory equipment that varies upstream and downstream channel geometries through a suite of combinations, or locating natural study sites that fit the suite of combinations. It would then be necessary to carefully measure the hydraulic terms and estimate energy dissipation, which is something that Henderson [1966] describes as extremely challenging to do accurately. The model consists of a manipulation of classic hydraulic and geomorphic equations, and thus should be technically sound. It provides first insights into energy dissipation and hydraulic jump regimes for nonuniform channels and a roadmap of key variables deserving of further experimental investigation. An attempt has been made to fully express all assumptions and limitations upfront.

3. Results

  1. Top of page
  2. Abstract
  3. 1. Introduction
  4. 2. Step Systematics
  5. 3. Results
  6. 4. Discussion
  7. 5. Conclusions
  8. Acknowledgments
  9. References
  10. Supporting Information

[41] To understand the effects of channel nonuniformity on hydraulic jump regime and energy dissipation at river steps, it is first important to understand how width and depth vary through the section in response to changes in discharge and fratio (Figure 10). As bup increases for a given fratio and constant fup, the downstream width-to-depth ratio also increases (e.g., Figures 10g, 10h, and 10i). As fup increases for a given fratio and constant bup, the downstream width-to-depth ratio decreases (e.g., Figure 10i, 10f, and 10c). In general, however, the geometric width exponent, bup, has a greater effect on the downstream width-to-depth ratio than fup (Figure 10).

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Figure 10. Width to depth ratios at the downstream cross section as a function of dimensionless upstream energy illustrating that the fratio effectively controls channel constriction and expansion as a function of discharge through a reach with a river step.

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[42] For the cases of f = b and an fratio = 1.0, the model shows the expected results of the downstream width-to-depth ratio remaining constant for the range of discharge evaluated (Figures 10c, 10e, and 10g, diamonds), while increasing the fratio increases the downstream width-to-depth ratio (e.g., Figures 10c, 10e, and 10g, circles). Decreasing the fratio, however, decreases the downstream width-to-depth ratio (e.g., Figures 10c, 10e, and 10g, triangles). The responses to either case of varying fratio become more dramatic as discharge increases.

[43] These patterns as a function of fratio comprehensively confirm the interpretations illustrated in Figures 5 and 7 over the full range of exponent values explored. When fup = bup = 0.4 and fratio = 0.5, the geometric interpretation is that the channel is bending and stretching to a more concave up cross section through the step (Figure 5). The hydraulic shape of the downstream channel shows that depth will increase much faster than width as discharge increases (Figure 10c, circles). For the same upstream model parameters, but an fratio = 1.5, the geometric interpretation is that the channel is bending and stretching to a more convex up cross section through the step (Figure 7). The downstream channel shape will create a greater response to flow width than flow depth as discharge increases (Figure 10c, triangles). As an example, the specific values of widths and depths for two flow stages shown in Figures 5 and 7 quantify the differences in the rates of change. For the low fratio example, the ratio of the upstream widths between the two discharges is about five, and this ratio decreases through the step to 2.2 at the downstream cross section (Figure 5). For the high fratio example however, those width ratios increase from five to 11 (Figure 7). In summary, the results demonstrate that manipulation of the fratio in the model effectively simulates nonuniform channel constriction and expansion through a river step. In the following subsections, the fundamental response of a river step to different flow magnitudes and different ways of manipulating channel nonuniformity is explored with respect to downstream hydraulic jump regime and energy loss at the step.

3.1. Role of Energy Input

[44] This model recaptures the basic response of the river step to increasing discharge for a uniform channel with relatively high values of f and b as previously demonstrated by Pasternack et al. [2006]. Consider the case of the uniform channel (fratio = 1.0) with f and b values of 0.4 each (Figure 11c, diamonds). For high (H + P)/H, the step is relatively high and thus there exists a drop over the step yielding a hydraulic jump at the step toe. Model results show that the highest fractional energy loss occurs at the lowest energy input regardless of geometric combination, because a high fall yields a conversion of potential energy into kinetic energy, and this large amount of converted energy cannot be recovered as potential energy through the hydraulic jump at the base of a high drop (e.g., Figure 12c). When the flow has reach the downstream cross section, the kinetic energy is greatly dissipated, representing <0.1% of the total energy (e.g., Figure 13c). As (H + P)/H decreases, the tail depth increases and the water drop height decreases, causing the jump to submerge. In the limit as (H + P)/H [RIGHTWARDS ARROW] 1, the hydraulic jump gives way to a standing wave and ultimately no surface expression at all. Correspondingly, there is less potential energy to be dissipated as the jump submerges, so the submerged hydraulic jump helps reduce the energy loss at the step (Figure 12c). Also, since the kinetic energy increases at the upstream cross section with increasing energy input, it also shows the same functionality at the downstream cross section (Figure 13c). In the limit as hd/H [RIGHTWARDS ARROW] 0, simple hydraulic geometry relations no longer apply, as the step is fully submerged and the downstream depth becomes the control for the upstream depth. The overall mechanics reported above represent the classic 2-D step behavior for a uniform channel. However, natural mountain rivers have strongly nonuniform channels and 3-D planform brink geometries, thereby creating a wide array of conditions remaining to be understood.

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Figure 11. Hydraulic jump regime model results and sensitivity analysis for the ranges of fup, bup, and fratio explored. The f and b values stated refer to upstream condition. Solid lines delineate hydraulic jump regimes.

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Figure 12. Energy dissipation model results and sensitivity analysis for the ranges of fup, bup, and fratio explored. The f and b values stated refer to upstream condition.

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Figure 13. Energy loss through the step and kinetic energy at the downstream cross section had an inverse relation for any given set of parameters explored.

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3.2. Channel Nonuniformity

[45] To fully understand the dependence of the downstream geometry on the upstream parameters and the fratio used in this model, consider a channel with the upstream hydraulic exponents (fup and bup) equal to 0.2 (Table 1 for fup = bup = 0.2). For an fratio of 1.5, the bratio is equal to the inverse, i.e., 1/1.5 = 0.67. The definition of fratio is the ratio of fup to ftail, therefore the value for the downstream f value decreases to 0.13. Since the bratio is the inverse of the fratio, the downstream b value increases from 0.2 to 0.3. Table 1 presents the range of hydraulic exponents explicitly used in this model.

Table 1. Values Used in Sensitivity Analysis of the Effect of Hydraulic Geometry Exponents on Step Hydraulics
fratiobratiofup = 0.4, bup = 0.2fup = 0.4, bup = 0.25fup = bup = 0.4
ftailbtailftailbtailftailbtail
1.50.6670.2670.30.2670.3750.2670.6
1.250.8000.3200.250.3200.3130.3200.5
11.0000.4000.20.4000.250.4000.4
0.91.1110.4440.180.4440.2250.4440.36
0.751.3330.5330.150.5330.1880.5330.3
0.52.0000.8000.10.8000.1250.8000.2
fratiobratiofup = 0.25, bup = 0.2fup = bup = 0.25fup = 0.25, bup = 0.4
ftailbtailftailbtailftailbtail
1.50.6670.1670.30.1670.3750.1670.6
1.250.8000.2000.250.2000.3130.2000.5
11.0000.2500.20.2500.250.2500.4
0.91.1110.2780.180.2780.2250.2780.36
0.751.3330.3330.150.3330.1880.3330.3
0.52.0000.5000.10.5000.1250.5000.2
fratiobratiofup = bup = 0.2fup = 0.2, bup = 0.25fup = 0.2, bup = 0.4
ftailbtailftailbtailftailbtail
1.50.6670.1330.30.1330.3750.1330.6
1.250.8000.1600.250.1600.3130.1600.5
11.0000.2000.20.2000.250.2000.4
0.91.1110.2220.180.2220.2250.2220.36
0.751.3330.2670.150.2670.1880.2670.3
0.52.0000.4000.10.4000.1250.4000.2

[46] Consider again the case of the upstream channel geometry having fup and bup values of 0.4 each (Figure 11c). Using the results associated with an fratio of 1.0 described above as the baseline, the model shows the effect of varying the downstream geometry on the hydraulic jump regime for a given discharge and step height. As the fratio for the channel increases above one, the downstream cross section bends and stretches outward, so water depth decreases, width increases, and the hydraulic jump becomes increasingly emergent. For an fratio of 1.5, the zone of the supercritical flow regime increases until (H + P)/H < 3 and there is no hydraulic jump present within the downstream control volume. In contrast, as the fratio for the channel decreases below one, the downstream cross section bends and compresses inward, so water depth increases, width decreases, and the hydraulic jump increasingly submerges for constant (H + P)/H to a point where no jump or standing wave feature is present. For example, when fratio = 0.5, there is no surface expression of the step for (H + P)/H < 3.76. As both fup and bup decrease together, the upstream velocity exponent, m, must increase. The associated acceleration of flow decreases depth and width at any given discharge, and thus the model shows that as m increases, the hydraulic jump regime becomes more emergent for all fratio (Figures 11c, 11e, and 11g). For example, when a channel with fup = bup = 0.4 is compared to one with fup = bup = 0.25 at an fratio = 0.75 (Figures 11c and 11e, crosses), the former shows a wide range of hydraulic jump regimes for 2 < (H + P)/H < 6, while the latter only exhibits the pushed-off and optimum jump regimes.

[47] A nonuniform channel with an fratio > 1 exhibits more relative head loss than one with fratio < 1 for all geometric combinations (Figure 12). The dominant cause of head loss at river steps is the difference in potential energy at the upstream and downstream cross sections, since kinetic energy is extremely difficult to recover as potential energy and tends to dissipate through multiple scales of turbulence and bed interaction. For high values of f, the amount of kinetic energy at the downstream cross section is less than 1% of total energy (Figures 13a, 13b, and 13c). These new results demonstrate a key role for channel nonuniformity in controlling the diverse conditions found in the vicinity of natural river steps, in contrast to the limited conditions reported for engineered steps.

3.3. Varying Upstream Channel Geometry

[48] A comparison among results for different sets of fup and bup combinations (Figures 11 and 12) shows that the standard concept of how steps function breaks down when at-a-station hydraulic geometry is not dominated by depth responsiveness, i.e., when the value of f is low. Because the sensitivity of the system to f and b values is codependent on the fratio, the results are explained for specified example fratio values. The most extreme condition examined in which the standard conceptual model breaks down is the case when fratio ≥ 1.5 (Figure 11, circles). Under this condition, the hydraulic jump is pushed downstream of the step toe at high (H + P)/H values for all channel geometries. As (H + P)/H decreases, the jump becomes increasingly emergent (i.e., the supercritical zone downstream of the step toe becomes longer) until no jump is present and the flow remains supercritical throughout the downstream section. The threshold for supercriticality depends on the channel geometry, occurring at lower (H + P)/H ratios as either f or b increases. Thus for fratio ≥ 1.5, the hydraulic jump regime is insensitive to the absolute values of f and b. It is sensitive, however, in terms of energy loss and kinetic energy (Figures 12 and 13). For fratio ≥ 1.5, the system in general incurs more head loss for higher values of f and b; except for cases in which b ≥ 0.4, then the system exhibits greater head loss for decreasing f values. This pattern corresponds with the general decrease in kinetic energy as f and/or b increase.

[49] The behavior of the hydraulic jump regime as a function of f and b values when fratio = 1.25 (Figure 11, squares) appears to be similar to that described above, when both f and b values are low. Under that condition, the flow becomes fully supercritical throughout the downstream section for low (H + P)/H values. At high f and b values, however, the jump exists as a pushed-off jump throughout the range of (H + P)/H values. The head loss for systems with fratio = 1.25 responds similarly to fratio ≥ 1.5 systems described above.

[50] In contrast to the insensitivity of the jump emergence regime to upstream f and b values when fratio ≥ 1.0, the fundamental behavior of the hydraulic jump regime shows strong sensitivity to those variables when fratio ≤ 1.0. When f = 0.4 and fratio = 0.75, the hydraulic jump submerges increasingly fast as a function of decreasing (H + P)/H, regardless of the value of b. A similar trend is evident when b = 0.4 and fratio = 0.75, except when f = 0.2. In that case the depth response is weak enough that the just transitions from a pushed-off to an optimum jump condition, but not to a submerged jump (Figure 11i). Holding f = 0.2 and fratio = 0.75, as b decreases from 0.4 to 0.2, the geometric effect is to bend the channel from convex up to triangular, but also to strongly raise the stage dependence of channel slope and/or roughness. These changes enable the downstream cross section to transport water away faster, which lowers the tailwater depth and emerges of the hydraulic jump for any given (H + P)/H. (Figures 11g–11i). The head loss when fratio = 0.75 tends to decrease with increasing upstream b or f values (Figure 12), since the tailwater depth is higher at those higher exponent values.

[51] When fratio = 0.5 (Figure 11, triangles), the channel choking (i.e., upstream submergence due to downstream controls) is extreme enough that the nonlinear rate of submergence increases for all values of b and f to the point that no surface feature is present when (H + P)/H = 2 unless f and b are both ≤0.2 (Figure 11g). In that case, the at-a-station hydraulic geometry requires that most of the response to increasing discharge goes to increasing flow velocity, thereby holding off complete drowning of the step. The head loss decreases rather quickly compared to the other fratio systems with decreasing (H + P)/H values. The response of the head loss is, however, insensitive to varying b and f values, except for the extreme lower b and f values (≤2) (Figure 12g). The head loss response mimics the hydraulic jump regime response.

3.4. Model Sensitivity to Geometric Coefficients

[52] The impact of the hydraulic geometry coefficients, a and c, on step mechanics is similar to that reported for manipulations of the fratio (Figures 14–17), and is consistent with intuition on the basis of the geometric explanations provided in sections 2.2, 2.3, 3.1, and 3.2. The step response to only variations in geometric coefficients is evident in the results for fratio = 1 (Figures 14–17, diamonds). For example, when fup = bup = 0.2 and fratio = 1, the hydraulic jump regimes become more supercritical as either a or c decrease with increasing discharge (i.e., decreasing (H + P)/H) (Figure 14). A similar effect was described as either b or f decreased, while coefficients were held constant (Figure 11). A decrease in either coefficient while holding the other constant and the exponents constant requires an increase in the coefficient k. Physically, that means that the channel is steeper and/or smoother, which enables water to transport out of the step region faster, lowering tailwater depth and emerging the jump regime. The same emergence response of hydraulic jump regime to decreasing a and/or c is evident regardless of fratio (Figure 14). These patterns were also observed when holding coefficients constant and varying exponents (Figure 11).

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Figure 14. Hydraulic jump regime model results and sensitivity analysis for the ranges of aup, cup, and fratio explored with fup = bup = 0.2. The a and c values stated refer to upstream condition. Solid lines delineate hydraulic jump regimes.

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image

Figure 15. Energy dissipation model results and sensitivity analysis for the ranges of aup, cup, and fratio explored with fup = bup = 0.2. The a and c values stated refer to upstream condition.

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image

Figure 16. Hydraulic jump regime model results and sensitivity analysis for the ranges of aup, cup, and fratio explored with fup = bup = 0.4. The a and c values stated refer to upstream condition. Solid lines delineate hydraulic jump regimes.

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image

Figure 17. Energy dissipation model results and sensitivity analysis for the ranges of aup, cup, and fratio explored with fup = bup = 0.4. The a and c values stated refer to upstream condition.

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[53] When fup = bup = 0.4, the sensitivity of the model to the coefficients is quite dampened (Figure 16), as compared to the case when fup = bup = 0.2. For all combinations of a and c tested at these higher exponent values, the hydraulic jump submerged with increasing discharge when fratio < 0.9. These results show that channels whose energy slope and bed roughness exhibit a sensitive discharge dependence will be more sensitive to the exact values of the coefficients than channels whose primary discharge sensitivity is exhibited in depth and/or width.

[54] The similarity of the response of the hydraulic jump regimes to changes in the coefficients versus the exponents of equation (7) holds in the energy loss relations as well (e.g., compare Figures 12, 15, and 17). When fup = bup = 0.2, energy loss is lower for higher coefficient values, since tailwater depth and velocity are lower (Figure 15). When fup = bup = 0.4, the range of variability in energy loss as a function of coefficient values is less than when fup = bup = 0.2 (Figure 17). These results confirm that either the geometric coefficients or exponents can affect the step response, even though they do so differently, by stretching versus bending of the channel cross section.

4. Discussion

  1. Top of page
  2. Abstract
  3. 1. Introduction
  4. 2. Step Systematics
  5. 3. Results
  6. 4. Discussion
  7. 5. Conclusions
  8. Acknowledgments
  9. References
  10. Supporting Information

4.1. Energy, Velocity, and Erosion

[55] For any set of model parameters, the results revealed an inverse relation between velocity at the downstream cross section and energy loss through the step for a given fratio (Figure 13). This finding has ramifications for step erosion mechanisms. At low discharge, step height is relatively high and energy dissipation is in the range of 80–95% of total energy. As water goes over the brink, it accelerates and hits the bed downstream of the step toe. For a vertical step, the angle of the nappe profile is steepest at the lowest discharge [Pasternack et al., 2006], which facilitates direct hydraulic erosion [U.S. Bureau of Reclamation, 1948a]. For the range of hydraulic geometry coefficients and exponents explored, the hydraulic jump regime for low discharge is predominantly pushed off or optimal jump, meaning that the falling jet can go through the tailwater, impact the bed, and cause erosion, promoting plunge pool formation. However, since the specific energy (H) is low at low discharge, the velocity of the water as it goes over the brink is relatively low, and even with acceleration it remains relatively low at the step toe compared to that at higher H. When the water reaches the downstream cross section, its velocity is very low, since most energy was dissipated (Figure 13). As discharge increases, upstream velocity, tailwater depth, and tailwater velocity all increase, while nappe profile angle becomes more horizontal and energy loss decreases, all driven by higher specific energy. Depending on the fratio, the hydraulic jump regime may become more emergent or more submerged. Different scour mechanisms appear to exist for the different hydraulic jump regimes [Pasternack et al., 2007], but insufficient literature exists to compare and contrast the relative magnitude of scour under pushed off versus optimum jump conditions. However, once a jump is drowned, both jet impingement and lift-stress variation on the bed likely decrease. Higher energy loss, higher velocity at the step toe, and jump emergence promote bed scour, while lower energy loss, jump submergence, and a more horizontal nappe profile diminish it. Therefore, the mechanisms and magnitude of scour vary substantially for a given channel configuration as discharge changes.

[56] Compared to the velocity, energy loss, and implied scour mechanics at an individual step, there is a highly significant effect of changing channel configuration. For all sets of model parameters explored, an increase in the fratio (or an equivalent change to geometric coefficients) for a given level of tailwater kinetic energy causes a dramatic increase in energy loss and keeps the tailwater velocity high (Figure 13). For example, when kinetic energy is held at 5% of total energy and the other parameters are as given for Figures 9e and 10e, energy loss increases from 0 for a fratio of 0.5 to 63% of total energy for a fratio of 1 to 76% for a fratio of 1.5 (Figure 18). As the fratio and energy loss increase, the hydraulic jump regime emerges, while the nappe profile angle remains the same, since H is the same. In this case, different factors and scour mechanisms are working synergistically to promote bed scour. Overall, the model results demonstrate that variation in channel configuration is a powerful variable controlling scour mechanisms at river steps.

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Figure 18. Head loss through a river step for a given magnitude of tailwater velocity is higher for reaches with a higher fratio. Relation shown is for fup = bup = 0.25, aup = atail = 9.74, cup = ctail = 0.51, and 5% of total energy at the downstream cross section exists as kinetic energy.

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4.2. Knickpoints at Canyon Entrances

[57] One physiographic domain in which knickpoints are common is the transitional canyon between high-elevation plateaus and lowlands. In such regions, there exist observations of knickpoints occurring at the upstream limit of the canyon, right at the transition point where valley and channel width suddenly changes [e.g., Snyder and Kammer, 2008]. Landscape evolution models of this domain that simulate plateau retreat by computing channel scour as a function of local slope and basin area predict higher rates of retreat during higher flows. However, the results of this study indicate that the exact opposite could hold, or scour rates could just be independent of discharge. Specifically, a knickpoint at the transition from a high plateau to a bedrock canyon exhibits valley constriction through the step unit. As discharge increases during an overbank event, the tailwater depth would quickly increase and drown the step, neutralizing the most effective scour mechanisms acting on the step toe. If the step is too high to completely submerge, it would at least drown enough to protect the bed from both hydraulic jet scour and bed load impact scour. Thus, the geomorphically significant example of plateau-to-canyon transitions illustrates the importance of the new model relative to the existing algorithms used in landscape evolution modeling. Scour at knickpoints in bedrock channels is definitely not a simple power function of slope and basin area.

4.3. Model Applications

[58] Investigation of the fluid mechanics at river steps in nonuniform channels using a parsimonious semianalytical model revealed important nonclassical behavior governing the fundamental nature of river steps. Although the model was generalized for broad scientific inquiry, model parameterization for a specific local site would enable its use for a wide range of management and engineering applications. For a specified site, such parameterization would entail observing channel width and depth upstream and downstream of the step over the range of flows under the baseline condition. Once the local flow geometry relations are known, then the model becomes easily testable for any site since the only independent input variable needed is the upstreamflow depth. Theoretical jump regimes and energy losses could then be compared to experimental or observed conditions. After satisfactory validation of the local model, alternative modifications to the channel geometry could be tested with the model to evaluate the resulting hydraulic jump and energy dissipation regimes relative to the desired goals. Common goals typically involve reducing swimmer or boater drowning hazard, decreasing the rate of knickpoint migration in gullies, and creating exciting recreational whitewater parks.

[59] Leutheusser and Birk [1991] described the drowning hazard posed by overflow hydraulic structures that have a drowned jump (a.k.a. “plunging nappe”) and they analyzed the engineering requirements to avoid this hazard. The same hazard exists at many natural steps as well. The design goal to avoid the hazard is to ensure that the step has any other hydraulic jump regime over the range of discharge and tailwater depth conditions that occur there. The solution provided by Leutheusser and Birk [1991] only allows for adjustment of step height. Using the new model proposed herein, the same hazard avoidance can be achieved using a variety of geometric adjustments. For situations where step height could not be modified, one solution would be increase the fratio of the channel to keep the step toe in the optimum jump condition over the range of discharges expected.

[60] Although the model does not directly predict scour and knickpoint migration rates at this time, it does provide the basis for inference and channel design. A key advantage of linking scour to energy dissipation instead of drag force is that existing equations used to predict scour using the latter approach have been shown to perform poorly [Pasternack et al., 2007], because they do not account for the mechanics and trade-offs discussed in section 4.1. In contrast, this new numerical model predicting energy dissipation does account for different processes, including the hydraulic jump regime. According to the model, the higher the fratio for a knickpoint, then the more the channel morphology is promoting higher energy dissipation (Figure 12) and thus higher rates of toe scour and knickpoint migration related to this process. To reduce the rate of knickpoint migration in that case, the channel manipulation would be the exact opposite of that for drowning hazard mitigation. By decreasing the fratio at a site, the magnitude of energy dissipation at the knickpoint would be decreased for a given tailwater velocity (Figures 12 and 18) and the hydraulic jump would be more submerged (Figure 11), for any given discharge. The utility of this solution depends on the actual fratio at a knickpoint and the degree to which a site can be reengineered.

[61] A third example where this model would be of value is in the construction of increasingly popular, recreational whitewater parks. The primary engineering feature used repetitively in such parks is a channel constriction at the step to produce the desired standing wave or hydraulic jump for canoe and kayak acrobatics (Figure 19). Presently such features are designed on the basis of practitioner experience, but the model could be used to accurately predict and control what the jump regime will be in advance as a cost-saving measure. Similar applications are readily apparent for dam removal, river rehabilitation, and fish passage − all situations in which human safety is jeopardized by the unknown hydraulic jump regime and geomorphic unit instability associated with step scour that is not accurately predicted by accounting for the scour associated with energy dissipation at the step.

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Figure 19. Example of engineered step with hydraulic jump in a whitewater park from the Lyons Play Park, St. Vrain River, Lyons, Colorado.

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4.4. Roadmap for Step Theory Advancement

[62] The underlying assumptions and limitations of the new model presented in this study have been described in detail throughout. Although this study explored a wide range of model parameters, there exists a dearth of data on the hydraulic geometry of channels in the vicinity of knickpoints. The exact relation between fratio and bratio in bedrock channels is unknown. The relative abundance of steps with different fratio, perhaps as a function of uplift rate and geology, is unknown. Overall, a new examination and appreciation of the real diversity in natural river steps needs to replace existing models on the basis of processes in alluvial rivers, gullies, and man-made spillways.

[63] The biggest deficiency in the model at this time is in the hydraulic jump regime equations (15)–(18), which are only strictly valid for an ogee-crested, rectangular 2-D overfall. The delineation between optimum jump and drowned jump regimes suggested by the U.S. Bureau of Reclamation [1948b] appears highly subjective. Also, the thresholds do not delineate the standing wave regime. Consequently, this theoretical study indicates that future research into the hydraulic geometry of step units in bedrock rivers and the dynamics of hydraulic jump regimes for different archetypal steps should be prioritized.

[64] Beyond the numerical model, key areas of future research are suggested by the conceptual framework (Figure 1). Most important would be to quantify the role of hydraulic jump regime and energy dissipation in step morphodynamics. In the past, studies have focused on the role of the hydraulic jet in scouring the plunge pool [Mason and Arumugam, 1985; Bormann and Julien, 1991]. Some studies have also demonstrated the importance of pressure fluctuations and lift in scour below steps [Fiorotto and Rinaldo, 1992; Fiorotto and Salandin, 2000; Pasternack et al., 2007], though not accounting for scour associated with bed load impact. As airborne laser swath mapping provides meter-scale resolution of topography in otherwise inaccessible canyons, we conjecture that it will be possible for landscape evolution models to incorporate the knickpoint mechanics associated with channel nonuniformity. Other factors in the conceptual framework that could be addressed relatively easily include step roughness and wind.

5. Conclusions

  1. Top of page
  2. Abstract
  3. 1. Introduction
  4. 2. Step Systematics
  5. 3. Results
  6. 4. Discussion
  7. 5. Conclusions
  8. Acknowledgments
  9. References
  10. Supporting Information

[65] The effect of varying channel geometry upstream and downstream of a river step on hydraulic jump regime and energy dissipation has not previously been investigated. This research has quantified the roles that hydraulic geometry and discharge have in determining the hydraulic jump regime and energy dissipation at river steps using a parsimonious model. For a given discharge and step morphology, an increase in the fratio leads to an increasingly emergent hydraulic jump. Channel nonuniformities that exhibit an fratio > 1.0 tend to be relatively insensitive to changes in the f or b parameters in terms of the hydraulic jump regime, existing as a pushed-off jump or supercritical flow for the whole range of (H + P)/H values. However, these channel nonuniformities exhibit higher energy loss through a step as f or b increases. For channel nonuniformities with an fratio < 1.0, there exists a strong sensitivity to changes in both f and b values in terms of the hydraulic jump regime ranging from a pushed-off jump to a completely drowned-out jump. The head loss of these systems decreases with either increasing f or b values. For nonuniformities with fratio > 1.0, head losses are high over the whole (H + P)/H range. For nonuniformities with fratio < 1.0, flow becomes choked and head losses decrease significantly as (H + P)/H decreases. Regardless of channel geometry, head losses due to potential energy losses dominate the flow regime at high relative step heights.

[66] Limitations in the model show that as the ratio hd/H approaches zero, the simple hydraulic geometry equations no longer apply at the upstream cross section, because the downstream depth acts as a control on the upstream depth. Because of the interplay between flow, channel geometry, and geological resistance, step toe emergence with high energy dissipation can induce scour and plunge pool formations, limiting the abundance of natural emerged jump regimes at bedrock steps. It may be that using a cross section to characterize the downstream control on the step toe is inadequate, necessitating a 3-D topographic and hydrodynamic modeling framework. However, the parsimony of the analytical approach presented herein provides a reasonable predictive foundation for further analytical investigation.

[67] A better understanding of hydrogeomorphic processes in mountain rivers, particularly at river steps, would provide advancement in the understanding of mountain river fluid mechanics, aquatic ecology, and channel evolution. Such improved understanding can aid in determining the impacts of channel engineering projects and can provide guidance to river restoration and urban stream rehabilitation efforts. Previous research has typically ignored the relevance of hydraulic jumps within channel evolution models. Since hydraulic jump regimes can be associated with steps of any size, including those induced by large wood jams and gravel bars, a better understanding of complex step processes can contribute to a better understanding of many channel features.

Notation
a

hydraulic geometry width coefficient.

bup

upstream hydraulic geometry width exponent.

btail

downstream hydraulic geometry width exponent.

c

hydraulic geometry depth coefficient.

Cb

broad-crested weir discharge coefficient.

E

total flow energy.

Eup

upstream total flow energy.

Etail

downstream total flow energy.

fup

upstream hydraulic geometry depth exponent.

ftail

downstream hydraulic geometry depth exponent.

fratio

ratio of fup to ftail.

Fr

Froude number.

g

gravitational constant.

H

specific energy upstream of step.

hb

flow depth at brink of step.

hc

critical depth.

hd

submergence variable of hydraulic jump.

hH

flow depth above step crest datum.

hup

upstreamflow depth.

htail

downstreamflow depth.

htoe

flow depth at toe of hydraulic jump.

hL

head loss.

hv

velocity head.

hv_up

upstream velocity head.

hv_tail

downstream velocity head.

kup

upstream hydraulic geometry velocity coefficient.

ktail

downstream hydraulic geometry velocity coefficient.

mup

upstream hydraulic geometry velocity exponent.

mtail

downstream hydraulic geometry velocity exponent.

P

step height.

Q

volumetric discharge.

q

specific discharge.

v

flow velocity.

w

flow width.

Acknowledgments

  1. Top of page
  2. Abstract
  3. 1. Introduction
  4. 2. Step Systematics
  5. 3. Results
  6. 4. Discussion
  7. 5. Conclusions
  8. Acknowledgments
  9. References
  10. Supporting Information

[68] This research was funded by the Hydrology Program of the National Science Foundation under agreement number EAR-0207713. Any opinions, findings, and conclusions or recommendations expressed in this material are those of the authors and do not necessarily reflect the views of the National Science Foundation. We thank Greg Tucker, Itai Haviv, Paul Carling, Kurt Frankel, Noah Snyder, Marc Parlange, Keith Richardson, and anonymous reviewers for helpful feedback.

References

  1. Top of page
  2. Abstract
  3. 1. Introduction
  4. 2. Step Systematics
  5. 3. Results
  6. 4. Discussion
  7. 5. Conclusions
  8. Acknowledgments
  9. References
  10. Supporting Information

Supporting Information

  1. Top of page
  2. Abstract
  3. 1. Introduction
  4. 2. Step Systematics
  5. 3. Results
  6. 4. Discussion
  7. 5. Conclusions
  8. Acknowledgments
  9. References
  10. Supporting Information
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jgrf468-sup-0001-t01.txtplain text document1KTab-delimited Table 1.

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