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.
 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  reported a general increase in migration rate with discharge, but Robinson and Hanson  reported a lower rate at higher discharge. Bennett et al.  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  proposed that the erosive stress on the face of a falls should be proportional to the square of the discharge.
 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.  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.
 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.
 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.
 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.
 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
 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].
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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 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.
 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.
 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.