We conducted a series of flume experiments to investigate the response of self-formed gravel-bed channels to floods of varying magnitude and duration. Floods were generated by increasing the discharge into a channel created in sand- and gravel-sized sediment with a median grain size of 2 mm. Flooding increased the Shields stress along the channel perimeter, causing bank erosion and rapid channel widening. The sediment introduced to the channel by bank erosion was not necessarily deposited on the channel bed, but was rather transported downstream, a process likely facilitated by transient fining of the bed surface. At the end of each experiment, bank sediments were no longer in motion, “partial bed load transport” characterized the flat-bed portion of the channel, and the Shields stress approached a constant value of 0.056, about 1.2 times the critical Shields stress for incipient motion. Furthermore, the discharge was entirely accommodated by flow within the channel: the creation of a stable channel entirely eliminated overbank flows. We speculate that similar processes may occur in nature, but only where bank sediments are non-cohesive and where channel-narrowing processes cannot counteract bank erosion during overbank flows. We also demonstrate that a simple model of lateral bed load transport can reproduce observed channel widening rates, suggesting that simple methods may be appropriate for predicting width increases in channels with non-cohesive, unvegetated banks, even during overbank flows. Last, we present a model for predicting the equilibrium width and depth of a stable gravel-bed channel with a known channel-forming Shields stress.
 The hydraulic geometry of a river channel in equilibrium with the water and sediment supply is determined by interactions between the flow and the sediment forming the bed and banks. Details of these interactions have been investigated in a number of studies in which analytical solutions for the stable width and depth of straight channels are obtained by coupling relations for flow and sediment transport with a criterion for bank stability [Parker, 1978a, 1978b; Ikeda et al., 1988; Diplas, 1990; Pizzuto, 1990; Millar and Quick, 1993; Darby and Thorne, 1996; Cao and Knight, 1998; Eaton et al., 2004; Millar, 2005]. These models differ primarily in the types of assumptions and constraints imposed on the system [see Ferguson, 1986], and tests of the different models against field and laboratory data generally show good agreement between predicted and observed values of bankfull width and depth.
 For gravel-bed channels, current theory posits that the equilibrium bankfull hydraulic geometry is adjusted to maintain sediment transport through the center of the channel while limiting transport along the banks [Parker, 1978b; Parker, 1979; Ikeda et al., 1988; Diplas, 1990; Cao and Knight, 1998]. The boundary shear stress must exceed the threshold for transport in the center of the channel, but fall below that threshold in the vicinity of the banks, otherwise they erode [Parker, 1978b, 1979]. In the absence of bank cohesion or vegetation, there is, therefore, a narrow range of flows over which the conditions of a mobile bed with stable banks can be satisfied. Results from previous experiments indicate that channels formed in noncohesive sediment will equilibrate to stresses that are ~20% higher than the threshold for transport [Ikeda et al., 1988; Diplas, 1990; Macky, 1999]. In natural gravel channels, bankfull shear stresses are typically 30–60% higher than the threshold shear stress [Andrews, 1984; Dade and Friend, 1998; Pitlick and Cress, 2002; Parker et al., 2007], and flows that reach this stage typically occur at least a few days per year.
 The development of process-based models for hydraulic geometry represents a clear improvement over previous empirical approaches [Leopold and Maddock, 1953]. There is, however, an inherent problem in the work described above: specifically, a “channel-forming” discharge must be specified at the outset. In laboratory experiments, the discharge can be adjusted at will, thus the flow used in any particular experiment is, by default, the channel-forming discharge, assuming it can move the bed sediment. In nature, the channel-forming discharge cannot be prescribed in advance, because it depends on the modeled values of bankfull width, depth, and velocity, which are the channel properties we would like to predict. In practice, therefore, the channel-forming discharge is assumed to be equal to a flow with a specific return period, such as the 2 year flood [Bray, 1982; Ikeda et al., 1988]. The distinction here is not simply a matter of semantics or a detail of model formulation. Bankfull discharge is the product of three dependent variables (width, depth and velocity); therefore, it cannot also be treated as an independent variable and used as input to a model for hydraulic geometry. The channel-forming discharge must be specified independently, and the solution for equilibrium channel geometry must proceed from there. Unfortunately, there is no consensus on what value of discharge should be used in developing and testing models of hydraulic geometry [Church, 1992], nor is it clear that a single discharge with a set frequency and duration can adequately represent the range of flows that form and maintain the channel [Pickup and Warner, 1976; Ashmore and Day, 1988; Emmett and Wolman, 2001; Radecki-Pawlik, 2002; Segura and Pitlick, 2010]. Finally, there are contrasting approaches for modeling the hydraulic geometry of mobile-bed channels: One group of models seeks solutions for the stable width and depth based on some optimality criterion, where, for example, channels adjust to maximize flow resistance [Eaton et al., 2004] or sediment-transport capacity [White et al., 1982; Millar and Quick, 1993]. Another group of models focuses on the distribution of shear stress acting along the channel perimeter, and the conditions under which the bed sediment is mobile, but the bank sediment is not [Parker, 1978b; Parker, 1979; Ikeda et al., 1988; Pizzuto, 1990; Cao and Knight, 1998]. These distinctions are particularly important in practical applications, such as channel restoration, where a channel-forming discharge and design methodology must be specified beforehand.
 In this paper, we present results from a series of flume experiments designed to investigate the response of straight, single-thread, gravel-bed channels in equilibrium with the water and sediment discharge to over-bank flows (Figure 1). The test channels were formed in noncohesive sediment, thus the results are applicable to sand or gravel channels lacking cohesive banks or vegetation. Our goal is to describe how equilibrium channels respond during overbank flows, and to assess the extent to which available theories can explain the phenomena associated with varying flows. Our results suggest that gravel-bed rivers will widen during steady overbank flows until the discharge is entirely conveyed within the main channel. The final equilibrium form is well-predicted by current theories of hydraulic geometry, presenting an interesting paradox: If equilibrium channels evolve to eliminate overbank flows, why do floods continue to occur in natural gravel-bed rivers?
2 Experimental Design and Methods
 The experiments were designed to measure the response of stable, self-formed channels to floods of arbitrary size and duration. The channels formed in the experiments were not intended to be scale models of any specific stream or river. However, as described in Appendix A, we did use the principles of Froude-scale modeling in selecting discharges, sediment feed rates, grain sizes and slopes representative of self-formed gravel channels [Graf, 1971; Peakall et al., 1996; Parker et al., 2003]. Table 1 lists data for five gravel-bed streams that could be considered prototypes of the model channels. The first three variables listed (median grain size, depth and width) have the dimensions of length, thus if the scaling between model and prototype channels is exact, these variables would be related by a common length-scale ratio, λ. A quick look at the values listed in Table 1 suggests that while the scaling is not exact, a length-scale ratio of λ = 1/10 is appropriate for our experiments. Scaling other dimensional variables (velocity and discharge) is slightly more involved, thus we leave these details for Appendix A. The remaining variables (slope, Shields stress and transport intensity) are dimensionless, and presumably not influenced by scale. The prototypes listed are representative of single-thread gravel-bed channels that transport bed load at low to moderate intensities at peak flows. These are the conditions we have tried to replicate in the experiments.
Table 1. Characteristics of Gravel-Bed Channels Considered as Prototypes for the Channels Described in This Paper
Column headings correspond to sites referenced in the following papers: HM8, Halfmoon Creek, site 8 [Mueller and Pitlick, 2005]; WF5, Williams Fork, site 5 [Segura, 2008]; RC, Rock Creek [Torizzo and Pitlick, 2004]; SRO, Salmon River near Obsidian, ID [Mueller et al., 2005]; JC, Jacoby Creek (T. Lisle, personal communication); and Run 4-1, this paper. Note that the values of width and discharge listed for this experiment are for a half-width channel in which the flume wall represents the center of an imaginary full-width channel.
1.6 × 10−2
1.5 × 10−1
4.7 × 10−1
8.1 × 10−1
2.0 × 10−3
0.9 × 10−1
 All experiments were conducted in an 18-m long, 1.5-m wide non-tilting flume at St. Anthony Falls Laboratory, University of Minnesota (Figure 1). Water was circulated with a centrifugal pump capable of delivering discharges of up to 30 l/s. Flows were adjusted with an in-line valve, and discharges were determined with an orifice meter in the return line between the pump and the head of the flume. Sediment was fed continuously into the head of the flume with a mechanical feeder, and feed rates were held constant during individual experiments. In most experiments, we used the same feed rate of 3 g/s. We settled on this particular value because it did not appear to overload, or starve, the channel of sediment.
 The bulk sediment used in the experiments consisted of a mix of sand and fine gravel with a median grain size, D50, of 2.0 mm (Figure 2) and a graphic standard deviation, σg [Folk, 1974] of 1.55 (ψ units). The same sediment was used throughout the study. During the experiments, we observed that the bed quickly coarsened to form a mobile surface layer. Samples of the bed surface, taken after selected runs when the flume was drained, showed that the surface layer was coarser, D50 = 3.3 mm, and better sorted, σg = 0.63 (ψ units), than the bulk sediment (Figure 2). We ran several separate tests on the threshold for motion of water-worked beds, and determined from visual observations that the critical Shields stress, , for the D50 of the bed surface sediment was approximately 0.047.
 The same initial setup and procedures were used in each experiment: Sediment was placed in the flume to a thickness of 15–20 cm, and the surface was graded to a constant slope; for most experiments, we started with an initial slope between 0.0055 and 0.0060 (Table 2). We then excavated a 35-cm wide trapezoidal half-channel along the right wall of the flume (Figure 3). The reason for placing the channel along the wall of the flume, rather than in the middle, was to suppress the development of sinuous flow paths, which would have lead to braiding [Bertoldi et al., 2009; Egozi and Ashmore, 2009]. The left (free) bank of the trapezoidal half-channel was formed at an initial angle of ~30° to prevent slumping in early phases of the experiments; this side of the channel was free to adjust as necessary. We assumed that, with the exception of the area close to the flume wall, hydraulic conditions within the half-channel were representative of a full-width channel with two free banks and twice the width. Velocity and bed shear stress were likely affected by the wall, but results presented by Houjou et al.  suggest that, in rectangular channels with width-depth ratios similar to ours (~10), the influence of a vertical lateral boundary on the distributions of velocity and shear stress in the center of the channel is small. In addition, we did not observe significant scour or fill along the wall when the flume was drained to suggest that the flume wall influenced the flow in a significant way.
Table 2. Summary of Data From Mobile-Bed Experimentsa
aValues listed indicate conditions at the end of an interval involving a flood, or the end of the run
 At the beginning of each experiment the channel was slowly back-filled with water to keep the free bank from slumping. The sediment feeder and pump were then turned on and the discharge was increased slowly to a prescribed flow; adjustments in flow were often made over the first 30 min to raise the water level slightly. The discharge and feed rate were held steady for 4–6 h while the channel came into equilibrium (Figure 1a). At that point, the discharge was increased to produce an overbank flood, as shown in Figure 1b. Flow depths in the main channel varied from 6–7 cm, while depths over the floodplain were typically less than 1 cm and not capable of entraining anything other than minor amounts of silt. Floods were maintained for several more hours, and channel adjustments were documented, as explained below. In some experiments, a second or third flood was generated to assess the response of the channel to repeated high flows.
 Changes in channel geometry and bed- and water-surface elevations were measured throughout the experiments using a point gauge (micrometer) mounted on a mobile carriage. Channel cross sections were measured at a series of stations spaced 2 m apart. Distinguishing the break in slope between the channel bank and the floodplain was generally straightforward, and we kept the spacing between points at 5 cm or less in order to resolve details of the bed and bank topography. All of the data reported in Table 2 come from measurements at cross sections located in the 6-m central section of the flume (Figure 3). Bed- and water-surface profiles were measured down the full length of the channel along a line located 10–20 cm away from the right edge of the flume. Measurements of slope and channel geometry were taken every 10–20 min in the early phases of an experiment, and roughly every 60 min thereafter. To determine whether the channel was reaching equilibrium, we compared successive measurements of channel geometry and average bed slope. Channels that exhibited little change in width, depth or slope after 4–6 h were considered to be in equilibrium (Figure 3), whereas channels that experienced uneven scour or fill, or uneven widening, were considered to be out of equilibrium, and thus not included in the analysis. It was not always clear what caused uneven widening, bed erosion, or deposition, but often it was a matter of one section eroding faster than another, or an entrance-exit effect that resulted in scour or fill, which then propagated downstream or upstream into the central section of the flume. Our main criterion for deciding which measurements or runs to include or exclude was the extent to which bed and water-surface slopes diverged from each other as the experiment progressed. It was not uncommon for both bed and water-surface slopes to change slightly over the course of an experiment (Table 2), however, if the difference between the two slopes started to exceed about 10%, we generally terminated the experiment because we were no longer confident that the flow was uniform. Of the 34 experiments initiated, 19 ended in disequilibrium; consequently, we focus on results from the 15 experiments that generated the most consistent high-quality data (Table 2).
 Relevant properties of the flow were determined by averaging measurements from three cross sections positioned at distances of 8, 10 and 12 m from the head of the flume (Figure 3). The derived flow properties include the mean velocity,
 In the above equations, Q is the discharge; A is the cross-sectional area; g is the gravitational acceleration; R is the hydraulic radius; τ is the bed shear stress, H is the water depth; S is the water surface slope; s is the specific gravity of sediment (ratio of sediment density to water density, ρs/ρ); and D50 is the 50th percentile of the sediment size distribution. Unless otherwise noted, the variables τ*, τ and H, are each formulated or measured with respect to the flat-bed portion of the channel cross sections, and D50 refers to the median grain size of the bed surface.
3.1 Discharge Sequences
 Figure 4 shows examples of flow sequences (hydrographs) for four different experiments that ran for at least 12 h. The step changes in discharge shown here are unlike hydrographs that occur in nature (except perhaps for flash floods), but we found that if we increased the discharge slowly, the channel would simply widen at the same pace, and no flood would be generated. In most experiments, we limited the increases in discharge to less than 50% of the background flow rate (Table 2), and only in Run 5-3 did we come close to doubling the discharge in one pulse. In one experiment, Run 4-2, we raised the discharge to generate a flood, then dropped the flow to the initial equilibrium “bankfull” value to examine how the channel would respond in this new configuration. We observed that, because the channel had widened in response to the flood, the discharge that originally filled the channel to capacity was no longer able to transport the supply (due to the reduced depth and shear stress), consequently, the channel began aggrading, and would have continued to do so, had we not terminated the experiment. In all other experiments, we kept the discharge at the same level for as long as necessary, exceeding 24 h in the case of Run 6-3 (Table 2). Channel responses to floods are discussed in detail in the next three sections.
3.2 Changes in Morphology With Time
 Each of the 15 successful experiments was characterized by an initial transient phase in which the channel width and depth adjusted rapidly to the flow, and a longer quasi-steady phase in which width and depth continued to adjust, but at slower rates. When discharge was raised to produce a flood, the two phases were repeated, with the channel widening rapidly at first, and much more slowly thereafter. The most important outcome of experiments involving sustained floods is that, during a flood, the channel would continue to widen slowly until all of the flow was contained within the now-larger channel, in effect ending the flood. A typical sequence is illustrated in Figure 5, which shows results from an 18 h experiment involving three changes in discharge. In this experiment, the initial transient phase (t = 0–120 min) was characterized by a 20% increase in average channel width, B (Figure 5b), and a 10% decrease in the Shields stress, τ* (Figure 5c) (the curved lines in Figures 5b and 5c are added only to emphasize overall trends; general relations for width and shear stress involving the complete data set are presented in the next section). Over the next 5 h of this experiment, the change in width and shear stress slowed substantially. At t = 455 min, the discharge was increased to 15.9 l/s to produce an overbank flood, which immediately increased the depth and shear stress in the main channel, and initiated a second phase of widening. As the channel equilibrated with this higher flow, the shear stress and rate of bank erosion declined steadily, and the flow was gradually contained within the channel boundaries. At t = 695 min, the discharge was increased to 21.1 l/s to produce a third overbank flow. The response was similar to the previous increase in discharge, with initially rapid increases in width, depth and shear stress, followed by a longer period of relaxation. At the end of the experiment (t = 1066 min), the width had increased to the point where all of the flow was contained within the main channel, and the floodplain was no longer inundated (Figure 5d).
3.3 Time-Dependent Changes in Shields Stress, Transport Intensity and Channel Width
 In order to compare results from different phases of the experiments, we standardized the measurement time, setting t = 0 as either the beginning of an experiment, or the beginning of a step-change in discharge. This expanded data set (Figure 6) includes more than 100 observations of changes in bed shear stress and channel top width over time. Each point shown in these figures represents the average of measurements taken at three cross sections in the 6 m central section of the flume. The first of these plots (Figure 6a) shows the time-evolution of the Shields stress, τ*, and bed load transport stage, , where τ* is computed from ((4)) assuming no change in bed surface texture, and is taken as a constant (0.047) based on separate tests of the threshold for motion. The data in Figure 6a follow a well-defined trend, with τ* decreasing from peak values of about 0.075 (ϕ = 1.6) at the start of a run, or a flood, to a value of about 0.056 (ϕ = 1.2) at t ≈ 600 min. Fitting these data with a power law yields the following relations:
 We chose to fit these data with a power law primarily because this function provides a better fit than other candidate functions. We should note, however, that the power-law fits imply that the experimental channels never really reached a point where the Shields stress and width were no longer changing. We explored the alternative of fitting the data with a function that approaches an asymptote as t → ∞, but we found that to get an acceptable fit, the function required three parameters, one being the asymptote, which must be known a priori. We expect that longer-duration experiments would indeed reveal small changes over time; however, the data presented here suggest these changes would be very small for run times > 600 min (10 h). To get a sense for what the time-to-equilibrium might be in natural channels, we use the scaling relation for time given in Appendix A, tp = tm λ1/2 (A12b), with the assumed length-scale ratio of λ =10, and infer that a time-to-equilibrium of 10 h in the model channels equates to 10√10 ≈ 32 h in natural channels.
 Figure 6b shows the time-evolution of channel top width, B(t) − Bo, where Bo is the width at the start of a run, or the start of a flood. The scatter in these data is greater than the preceding relation primarily because the width adjusts much more than the average depth to changes in discharge and shear stress. In addition, when generating floods, we increased the discharge by varying amounts, which resulted in variable changes in width.
 The results presented above can be used to develop and test a simple model for predicting width adjustment in non-cohesive channels where the Shields stress is known. First, we observe that the lateral (cross-stream) bed load transport rate per unit channel length, qn, generated by bank erosion can be approximated as
where Hb is the height of the bank (~H at equilibrium), p is the bed sediment porosity (taken as a constant of 0.4), and dB / dt is the change in width per unit time. KA is a shape factor, defined as the ratio of the volume of bank erosion per unit downstream distance divided the product of dB and Hb. For a trapezoidal channel with bank erosion dB and a constant bed elevation (hence a constant Hb), KA is equal to 1, and we adopt this value in our computations following Duan and Julien . Rearranging equation ((6)) provides a simple first-order differential equation for the rate of change of channel width:
 Equation ((7)) can be solved for dB / dt using a suitable equation for the lateral transport rate, qn, and the time-dependent relation for τ* presented above (Figure 6a). In straight channels with negligible influence from secondary flows, the equation for lateral transport on a gently sloping surface can be written as [García, 2008]:
where is the downstream (streamwise) bed load transport rate per unit width on the sloping bank, r is a ratio of lift to drag forces, μ is a dynamic coefficient of friction, ϕb is the bed load transport stage evaluated on the sloping bank region, no is an empirically derived constant, and tan ω is the lateral bed slope. For typical values of r, μ, no, tan ω, and ϕb > 1, the product of the terms in brackets is < 1, thus qn < . We estimate the streamwise bed load transport rate on the bank, , using Parker's  approximation of the Einstein bed load transport relation:
where W* = [(s−1)g]/(τb /ρ)1.5, and τb is the streamwise boundary shear stress acting on the bank region.
 To evaluate the lateral bed load transport rate using equations ((8)) and ((9)), the shear stress on the bank region and the critical shear stress required to initiate motion on the bank must both be adjusted for side-slope effects. To determine the shear stress on the bank region, we follow the approach of Cantelli et al. , who suggest that
where τ is the bed shear stress and φ is a dimensionless parameter relating the streamwise shear stress on the bank to the streamwise shear stress in the channel. To evaluate incipient motion of the bank sediment, we define a second dimensionless parameter, ε, which relates the critical Shields stress on the bank region to the critical Shields stress on the flat-bed region:
 The parameters in equations ((7))–((11)) are specified as follows: The time-dependent change in Shields stress is modeled using the results presented in Figure 6a. The critical Shields stress is treated as a constant, = 0.047, however, we include results showing how the selection of this parameter influences the modeled widening rates. The parameter φ is adjusted within a narrow range from 0.90 to 0.80 to obtain the best fit to measured widening rates (we tried proportioning the shear stress on the banks using the equation of Flintham and Carling , but found this gave relatively low bank shear stresses, which resulted in much lower rates of bank erosion than we observed). The parameter ε is set to a value of 0.85 using the relation for critical shear stress on an inclined bank, developed by the U.S. Bureau of Reclamation [Lane, 1955], and presented graphically in a recent paper by Garcia [2008, Figure 2.26]. The other parameters are set to constant values: r = 0.85, μ = 0.43, no = 0.5, and tan ω = 0.31, where the value of ω ≈ 17° is based on measurements of slope angles at representative cross sections.
 Figure 7 compares predicted rates of width adjustment with measurements from the experiments. The results shown in the top panel (Figure 7a) were obtained from solutions to ((7))–((11)) with constant values of φ = 0.85 and ε = 0.85, but slightly different values of . The results shown in the bottom panel (Figure 7b) were obtained using the same approach, but with a constant value of = 0.047, and slightly different values of φ and ε. The optimum solution, indicated in both panels by the solid line, is obtained by setting = 0.047, ε = 0.85, and φ = 0.85. Of these three parameters, the first two are relatively well-constrained by other data and measurements, whereas the basis for setting the third parameter at φ = 0.85 is that it appears to give the best fit. It turns out that, in this case, where ε and φ have the same value, their influence on transport rates cancels out; this occurs because the bank-region transport stage, ϕb, is formed as a ratio between ((10)) and ((11)), hence the ratio of φ to ε is 1.0. Whether this condition is generally true of natural channels is unknown. In any case, the results presented in Figure 7 show that, when properly formulated, a simple model based on the equation for lateral bed load transport can be used to predict the rate of channel widening in response to changes in discharge and bed shear stress. Predicted rates of widening appear to be very sensitive to assumed values of the critical Shields stress, as well as the relation between the bed shear stress and the bank-region shear stress, but if parameter values can be constrained either in the field or in the laboratory, this approach offers a straightforward method for predicting the time-evolution of channel width in response to variations in discharge.
3.4 Sediment Mass Balance
 The mass of sediment supplied to the channel from bank erosion during floods represents anywhere from two to three times the background feed rate. Some of the sediment eroded from the banks is deposited, leading to slight increases in bed elevation (Figure 5d), but the majority of the sediment supplied continues in transport along the channel. This observation suggests that during periods of bank erosion, the total flux of bed load must increase downstream. A question thus arises: What adjustments must take place for the channel to remain in equilibrium with the increase in supply? To answer this question we use measured changes in cross-sectional area to estimate the streamwise increase in sediment supply during intervals of bank erosion then apply a bed load transport equation to model the conditions that would produce an equivalent flux. In modeling transport, we assume that the sediment supplied from the banks has the same grain size as the feed, D50s = 2.0 mm, which is finer than the equilibrium bed surface texture, D50 = 3.3 mm. The addition of this finer sediment must alter the composition of the bed for a brief period of time, and, as explained below, we attempt to capture this effect by adjusting the bed texture to enhance the transport rate.
 The equation for sediment conservation in two dimensions can be written as
where z is the bed elevation, and s and n refer to down-stream and cross-stream directions, respectively. During intervals of bank erosion, qs is unknown, except at the head of the flume where it equals the feed rate. To estimate the change in qs produced by bank erosion, we rearrange ((12)), and separate this unknown term from the two terms which we can estimate from sequential measurements of channel geometry,
 From (13) it is evident that the change in streamwise sediment flux should vary linearly with the rate sediment is supplied by bank erosion, ∂qn/∂n, and the change in bed elevation, ∂z/∂t. In most runs we observe that bank erosion contributes much more sediment to the channel than is eroded from or stored in the bed, nonetheless we took these changes into account in estimating the net sediment flux. We use results from Run 6–2 as an example. In this experiment, the discharge was increased at t = 455 min to bring the flow up to bankfull, then increased again at t = 695 min to generate an overbank flood (Figure 5). Based on cross section measurements taken during these events, we estimate that bank erosion increased the sediment supply by 0.16 and 0.57 g/s per meter of channel length, respectively. Summing these inputs over the 12.5 m working length of the flume, and adding the feed rate of 3 g/s, gives width-integrated bed load transport rates at the outlet of the flume of 5 g/s and 10 g/s, respectively.
 To estimate the change in the bed texture required to accommodate twofold to threefold increases in bed load transport rates, we use equation ((9)) and back-calculate the change in surface grain size required to maintain the bed load flux for a given shear stress. In applying ((9)) in this case, we formulate the transport stage, ϕ, with respect to the bed shear stress, and the grain size of the sediment supply,
where the subscript s indicates the D50 of the bulk sediment (2 mm). The numerator in ((14)) is calculated from measurements of depth and water-surface slope taken during the initial phase of a flood. To estimate the denominator in ((14)), we assume that the sediment supplied from the banks has the same size distribution as the bulk sediment, and that it moves as bed load over a surface that is coarser overall, but getting finer downstream. To account for the fining effect, we form a ratio between the grain size of the bed surface and the supply, D50 / D50s, and allow that ratio to vary linearly with distance. We then adjust the critical Shields stress for the supply using a hiding function,
where is the critical Shields stress for the surface D50 and the exponent γ determines the extent to which transport is size-selective. For the purposes of this exercise, we use the same value of as before (0.047) and set γ = 0.8 so that transport is somewhat size selective. Working backwards, the net effect of the assumed changes in bed texture is for the ratio of D50 to D50s to decrease downstream (consistent with a surface that is becoming finer), which decreases the critical Shields stress for the supply, , which, in turn, increases the transport stage, ϕ, and the total width-integrated transport rate, Qs. The change in D50 / D50s is adjusted downstream through trial and error until the sediment flux obtained from ((9)) matches the inferred flux from cross-section measurements.
 The results of these calculations are summarized in Table 3 and illustrated in Figure 8, where we plot the proportional change in surface grain size and bed load transport rate for the two separate phases of Run 6-2. The top panel in this figure shows that with a 10% decrease in the ratio of D50 to D50s, the transport stage increases by roughly 12%, giving a width-integrated load of 5 g/s, compared to the background feed of 3 g/s. The bottom panel shows similar results, only here, the ratio of D50 to D50s is decreased more substantially to produce larger changes in ϕ and Qs. In this case, a 28% decrease in D50 / D50s is required to get a transport rate that matches the inferred load at the outlet of the flume of 10 g/s. These results suggest that, for conditions where localized bank erosion supplies sediment similar in size to the bulk sediment, transient fining of the bed surface can lead to modest reductions in the critical shear stress, enhancing the mobility of the bed load without any significant change in bed shear stress.
Table 3. Sediment Budget Calculations for Two Phases of Bank Erosion During Run 6-2
 Linkages between sediment supply, bed texture and bed load transport rate have been investigated in a number of previous studies [Dietrich et al., 1989; Lisle et al., 1993; Wilcock, 1998; Buffington and Montgomery, 1999; Eaton and Church, 2009], and results from this work generally show that if the supply is decreased the bed coarsens, and if the supply is increased, the bed fines. This depends, of course, on the grain size of the supply, and the length scale over which the channel can adjust to the change in supply. In our experiments, the sediment supplied by bank erosion had the same size distribution as the feed, but the flume was limited in length. To explore the influence of channel-length scale and variations in the supply rate, we carried out a second set of calculations for an eroding channel of infinite length to determine the point at which the cumulative increase in sediment supply would produce a bed surface texture equal the bulk sediment mix. The results of these calculations, also listed in Table 3, suggest that in the first phase of Run 6-2, where the increase in discharge resulted in a moderate amount of bank erosion, the grain size of the bed surface, would converge to the supply at a distance of about 62.5 m, which is four times the length of the flume and about 200 times the width of the channel. In the second phase of Run 6-2, where higher flows generated higher bank erosion rates, the texture of the surface would converge to the supply at a distance of about 23.5 m, which is less than two times the length of the flume, and about 60 times the width of the channel. It seems likely that, at that point, any additional sediment supplied to the channel would lead to aggradation and lateral instability, and perhaps force a change in channel pattern.
3.5 Characteristics of Stable Channels
 One of the key results to come out of these experiments is our observation that most of the change in channel geometry produced by floods is taken up by adjustments in width rather than depth, and that with time the channel reverts to the same bankfull depth and Shields stress (0.056) as before. The inferred equilibrium transport stage of ϕ = 1.2 matches exactly the value obtained by Parker [1978b; 1979] in developing a turbulent diffusion model for distributing the bed and bank shear stresses in bed-load dominated channels. In the section below, we compare our results with results from earlier experiments [Ikeda et al., 1988; Diplas, 1990; and Macky, 1999], and show that, in situations where we can specify appropriate values for the channel-forming Shields stress, we can predict the stable width and depth of bed-load dominated channels. The basic set-up and methods used in these other experiments are very similar to ours, and the data sets encompass a broader range of conditions than those described here (Table 4).
Table 4. Comparison of Conditions Modeled in Previous Experiments With Conditions Modeled in This Study
 In the experiments, four variables are known: discharge, Q, sediment supply, Qs, grain size, D, and slope, S. The problem is thus over-specified in comparison to the field case, where the sediment supply is generally not known. The strategy adopted here, therefore, is to mimic the field case by limiting the number of independent variables to three, Q, D and S, and solve for the remaining variables, H, B, U and Qs. The solution is obtained by coupling the relations for dimensionless shear stress, flow resistance, continuity, and bed load transport, ((4)), ((3)), ((1)) and ((9)), respectively. In addition, we assume that, for a given slope and grain size, the bankfull depth will equilibrate to a constant Shields stress. In laboratory studies, that value has been determined experimentally. In other applications, however, such as morphodynamic modeling, the channel-forming Shields stress must be specified in advance; consequently, any uncertainty associated with specific values will carry over into estimates of bankfull depth, velocity and so on. Keeping this in mind, the depth of the channel is estimated by rearranging ((4)),
where τ* is the channel-forming Shields stress. The mean velocity is then estimated using a flow resistance equation [Keulegan, 1938],
where ks is the equivalent roughness, assumed here to equal 3D50. The equilibrium bankfull width is the sum of the width across the central section of the channel, plus the width of the section forming the banks; we estimate these components using equation 26 of Ikeda et al. ,
where Bc and Bb are the widths over the central and bank portions of the channel, respectively.
 To test the approach outlined above, we pooled data from the previous experiments with ours, and calculated bankfull channel depths and widths using two different assumptions about the channel-forming Shields stress. In the first set of calculations, we applied a common value of τ* = 0.047 across all data sets (note: the value of 0.047 used here should not be confused with the value of the critical Shields stress used in the analyses discussed in sections 3.3 and 3.4). A channel-forming Shields stress of 0.047 is chosen because comparable values have been used previously in developing morphodynamic models [e.g., Paola and Seal, 1995; Cui et al., 2003; Millar, 2005; Ferguson and Church, 2009], and because this value is representative of bed-load dominated channels [Andrews, 1984; Dade and Friend, 1998; Pitlick and Cress, 2002; Mueller et al., 2005; Church, 2006; Parker et al. 2007]. In the second set of calculations, we use specific values of the channel-forming Shields stress determined from data presented in the original papers; these values range from 0.054–0.065, depending on the experiment (Table 4). The results are presented in Figure 9, where the two graphs in the top panel show the effect of applying a common value of τ* = 0.047 across all data sets, and the two graphs in the bottom panel show the effect of using values of τ* specific to the individual study. In the first plot (Figure 9a), the predicted bankfull depths parallel the 1:1 line, but fall consistently below it, suggesting that the assumed channel-forming value of τ* = 0.047 is, on average, too low. The lower depths in turn lead to lower velocities, which lead to higher bankfull widths for a given discharge (Figure 9b). In the second set of calculations, where we use experiment-specific values of τ*, the agreement between predicted and observed depths is quite good (Figure 9c), as is the agreement between predicted and observed widths (Figure 9d). We obtain similar fits (not shown) by applying a constant value of τ* = 0.060 across all data sets, but this result is perhaps not unexpected, given that the flume data straddle this particular number.
 In our experiments, as well as these other experiments, the equilibrium bed load transport rate was presumably equal to the sediment supply, and in most cases, this was known. However, in natural channels, or channels that are undergoing restoration, the bed load transport capacity will rarely be known. The only practical solution to this problem is to estimate the transport capacity with a transport function, given values of the relevant variables. We conclude this section, therefore, with a comparison between the transport capacity estimated from a bed load transport function, and the equilibrium transport rate observed in our experiments. We restrict the comparison to our experiments because the other researchers either did not consistently report all the information needed to model transport, or they did not report the sediment feed.
 For consistency, we use ((9)) to calculate the streamwise bed load transport rate, only, we modify the function to compute transport rates for each size fraction,
where ϕ is the bankfull transport stage, and the subscript i refers to the ith size fraction of the bulk sediment mix. The main reason for computing fractional transport rates with respect to the bulk sediment mix is that this sediment includes the finer sizes (medium to very fine sand) that are commonly transported as bed load [Lisle, 1995; Wathen et al., 1995; Mueller et al., 2005; Clayton and Pitlick, 2007; Recking, 2010], but not very abundant on the bed surface.
 Transport stages for each size fraction are computed with a hiding function,
where is the critical Shields stress for the median grain size of the bed surface, D50. The parameter γ is set to the same value (0.8) as in section 3.3, and the critical Shields stress is set to the same value as before, 0.047. Transport rates are calculated for each size fraction then weighted by the proportion of that size fraction in the bulk sediment mix, fi:
 The fractional transport rates, , are summed over all size classes and multiplied by the bed width to get the total bed load transport rate, Qs, in grams per second.
 Figure 10 shows modeled fractional transport rates for Run 6-2 of the present study. The separate symbols indicate: the fraction of the bulk sediment in each size class, fi; the scaled transport rates for each size fraction, ; and the cumulative sum of the scaled transport rates, averaged over a bed width of 40 cm. The gradation of the bed material (open crosses) is not unlike natural channels, where the fraction in each size class increases with grain size up to some point, then drops off sharply. The transport stage at bankfull flow (ϕ = 1.2) is sufficient to mobilize much of the fine sediment but relatively little coarse sediment, hence the relation formed by the scaled transport rates (open circles) slopes sharply downward, indicating that the coarser sizes are not being transported in proportion to their abundance in the bed; this condition is referred to as partial transport [Wilcock and McArdell, 1993]. Consequently, the cumulative sum of the scaled transport rates (closed circles) rises very steeply with increasing grain size, and the first four size fractions (0.15 < Di < 0.42 mm) comprise almost 70% of the total bed load. The calculated width-integrated bed load transport rate is 1.84 g/s, which compares favorably to the measured feed rate of 3 g/s. The discrepancy here is not especially surprising, nor important, because we could tune any one of the parameters in ((19)) or ((20)) to match the sediment feed precisely. What is more important is our observation that conditions of partial transport are likely to dominate in channels that adjust morphologically to transport bed load at shear stresses not far above critical, i.e., ϕ = 1.2.
 The results of the experiments described here show that channels formed in noncohesive sediment (sand or gravel) respond consistently and rapidly to increases in discharge and shear stress caused by floods. In the absence of vegetation or cohesive sediment that might otherwise limit bank erosion, the response to experimental floods is almost instantaneous: bank sediment is rapidly entrained and transported to the channel by high bank shear stresses, forcing an increase in width. With time, however, the shear stress and width equilibrate to the higher discharge until eventually all of the flow is contained within the banks. This sequence, if repeated over and over with increasingly higher discharges, would result in a channel that is adjusted to the highest possible discharge. In nature, this end-member case might describe conditions in braided rivers, but not in single-thread rivers, where bankfull is associated with floods that occur roughly one out of every 2 or 3 years. It could be argued that the stability and morphologic consistency we associate with single-thread rivers is due largely to the effects of riparian vegetation [Gran and Paola, 2001; Tal and Paola, 2007; Braudrick, et al., 2009; Davies and Gibling, 2011], but even in these channels, bank erosion is not completely absent. There is some evidence to suggest that the effects of riparian vegetation are scale-dependent. Beechie et al. , for example, have identified a threshold in channel width (~20 m) above which riparian vegetation appears to have limited influence on the lateral migration of gravel-bed rivers in Washington, USA. These authors suggest that, because wider channels are also deeper, there is a scale beyond which flows become deep enough to erode below the rooting zone of the bank vegetation.
 In sinuous channels, the increase in cross-sectional area created by bank erosion is often counteracted by deposition on the opposite bank, resulting in the lateral growth and migration of point bars, and the construction of inset benches [Pizzuto, 1994; Moody et al., 1999]. In many rivers, the pace of narrowing is accelerated by the establishment of woody plants, which greatly increase flow resistance and promote further deposition [Scott et al., 1996; Friedman et al., 1996; Allred and Schmidt, 1999; Griffin, et al., 2005; Kean and Smith, 2005]. The processes of lateral and vertical accretion do not begin to resolve questions regarding the frequency of bankfull discharge, however, these processes provide an efficient mechanism for narrowing channels in the aftermath of floods.
 By pooling data from many experiments, we were able to develop empirical functions showing that gravel channels subject to floods will equilibrate to the higher discharges and shear stresses within 8–10 h of the start of a flood. We estimate that the time-to-equilibrium in the experimental channels equates to about 24–32 h in the field. The duration of flows capable of mobilizing the bed and bank sediment in gravel channels varies widely depending on the hydroclimatic setting. In rivers where runoff is generated primarily by snowmelt, flows may be at or near bankfull anywhere from 6 to 15 days per year [Segura and Pitlick, 2010], thus there is generally ample time in this type of setting for channels to adjust to higher-than-average shear stresses. The limiting factor in snowmelt-dominated systems is not duration but intensity: Floods generated purely by snowmelt (not involving rain) rarely exceed the mean annual flood by more than a factor of two [Church, 1988; Pitlick, 1994]; consequently, the stresses acting on the bed and banks vary within relatively narrow limits [Segura and Pitlick, 2010]. In rivers where the largest floods are generated by rain-on-snow, or high intensity rainfall, peak flows may exceed the mean annual flood by a factor of 10 or more [Pitlick, 1994]; in the most extreme cases, these floods lead to catastrophic widening of the channel [see Schumm and Lichty, 1963, for example]. The tradeoff between flow duration and intensity may shed some light on the paradox noted in the introduction: Depending on the hydroclimatic setting, there are some rivers that appear to be adjusted to discharges with a common frequency [Wolman and Miller, 1960], and there are other rivers that might be considered to be in a perpetual or long-term state of adjustment [Stevens et al., 1975; Pizzuto, 1994] [see also Buffington, 2012]. The latter state is not necessarily one of disequilibrium if we consider the potentially greater length of time required for rivers in some hydroclimatic settings to readjust to the changes caused by floods.
 Predicting the erosion of non-cohesive sediment on sloping river banks is a complex problem. The fluid shear stresses that entrain particles on sloping banks can be influenced by undulations in bank topography [Kean and Smith, 2006] and variations in the boundary shear stress due to secondary flows [Cao and Knight, 1998]. Non-cohesive bank material can be removed by slumping [Duan, 2005, Duan and Julien, 2010; Pizzuto, 1990] in addition to being entrained by fluid shear stresses. As a result, models of non-cohesive bank erosion are typically fairly complex, and often require significant computational resources to implement [Kovacs and Parker, 1994; Nagata et al., 2000].
 The results presented in Figure 7 demonstrate that a simple approach can successfully reproduce observed rates of bank erosion in non-cohesive sediment, even during overbank flows, if the Shields stress acting over the bed is known. A similar approach to bank erosion has also been used by Cantelli et al.  to explain the evolution of non-cohesive channels following dam removal.
 In many of our experiments, the volume of sediment introduced to the channel by bank erosion was substantial, yet we did not observe significant aggradation. It was evident during these experiments that the concentration of bed load was increasing downstream, yet the flow was capable of carrying the additional load with little change in bed shear stress. We did not measure changes in bed surface texture during floods, but we were able to show with a transport relation that transient fining of the bed surface can potentially generate much higher transport rates for the same shear stress. Experiments conducted by Eaton and Church  reveal similar responses to increases in sediment supply in a sinuous channel with fixed banks. Results from these experiments show that adjustments in the “bed state”, reflected by changes in bed surface texture, are sufficient to accommodate two to fourfold increases in sediment supply with little change in discharge. Results from other experiments [Dietrich et al., 1989; Buffington and Montgomery, 1999; Wilcock et al., 2001] likewise suggest that a wide range in transport rates is possible for relatively small changes in discharge and bed surface texture depending on the sediment supply. In natural channels, the sediment introduced by bank erosion is not likely to double or quadruple the sediment supply because the forces driving bank erosion are more localized than the conditions modeled in our experiments. Nonetheless, we showed that transient fining of the bed surface provides an efficient mechanism for natural channels to transport the higher volumes of sediment introduced by bank erosion without causing widespread changes in channel morphology. Exceptions to this generalization occur in active braided rivers where bank erosion is more pervasive, and where higher width-depth ratios favor deposition of mid-channel bars which then force lateral instability.
 The results presented here serve as a useful test of the hypothesis that gravel-bed channels will adjust their hydraulic geometry to a constant Shields stress, as suggested in previous studies [Andrews, 1984; Dade and Friend, 1998; Pitlick and Cress, 2002; Parker et al., 2007]. In our experiments, channels that were pushed out of equilibrium by overbank flows gradually adjusted their width and depth until the Shields stress approached a constant value of 0.056, the same as before the change in discharge. While this result is consistent with the constant-Shields-stress-hypothesis, results from other experiments with different discharges, slopes and sediment mixtures suggest that the range in bankfull τ* for laboratory channels is somewhat broader, between 0.054 and 0.065 (Table 3). Perhaps a better way to evaluate the constant-stress-hypothesis is with respect to the bankfull transport stage, , as this accounts for variations in the critical Shields stress due to differences in sediment sorting [Ikeda et al., 1988], sand content [Wilcock and Crowe, 2003], and/or relative roughness [Mueller et al., 2005]. Theories developed previously [Parker, 1978b, 1979], as well as the results presented here, indicate that, in straight channels with uniform flow, the equilibrium transport stage indeed tends toward a constant, ϕ = 1.2. Results from studies emphasizing conditions in natural channels suggest that, in gravel-bed rivers, the bankfull Shields stress clusters around a common value of about 0.048 [Andrews, 1984; Dade and Friend, 1998; Parker et al., 2003; Pitlick and Cress, 2002; Parker et al., 2007]. If this value is then normalized by a critical τ* of 0.03, we might infer that the equilibrium transport stage in gravel channels likewise tends toward a constant, ϕ = 1.6. But herein lies the problem: It now appears that there is a much wider range in field-derived values of the bankfull τ* and the critical τ* than previously realized. Data compiled by Buffington , for example, show that, in plane-bed pool-riffle channels, field-based estimates of the bankfull τ* which are uncorrected for effects of large roughness elements, variations in bed texture, and differences in sediment supply, span a surprisingly large range from 0.006 to about 0.6. Other research on bed load entrainment thresholds indicates that, at high slopes and low relative submergence, changes in near-bed flow structure strongly influence the forces acting on particles [Armanini and Gregoretti, 2005; Lamb et al., 2008; Recking, 2009], and as a result, the stress associated with incipient motion can be much higher than is commonly assumed [Mueller et al., 2005; Ferguson, 2012]. There is some evidence suggesting that the bankfull τ* and the critical τ* co-vary at the same rate [Mueller et al., 2005], and this would support the hypothesis that gravel- and cobble-bed channels adjust their hydraulic geometry to a common bed load transport stage, but for now, we leave this as an open question requiring further research.
 The measurements of channel evolution presented here show that any flow that is capable of generating shear stresses exceeding the limit of bank stability will cause bank erosion. The net effect of bank erosion is to increase the cross-sectional area, until eventually all of the flow is contained within the widened channel, and the banks are once again at the limit of stability. In our experiments, the condition of bank stability and morphologic equilibrium occurs when the ratio of bankfull Shields stress to critical Shields stress (transport stage) reaches a value of 1.2. At these flows, bed load transport is limited to the flat-bed portion of the channel, and the total bed load is dominated by partial transport.
 One of the main inferences we can draw from these experiments is that the time required for channels formed in noncohesive sediment to adjust to increases in discharge is not great. We estimate that the time-to-equilibrium of 8–10 h in the experimental channels scales to about 24–32 h in the field, and perhaps much less if the flood discharge exceeds the channel-forming discharge by more than a factor of about two. We speculate that in rivers where runoff is generated primarily by snowmelt or low intensity rainfall, there is ample time for channels to adjust to higher-than-average shear stresses; however, it is not often that stresses reach the level required to mobilize the bank sediment and widen the channel; this would be especially true of small channels with thick bank vegetation, hence high resistance to flow. In other rivers where the largest floods are generated by rain-on-snow or high-intensity rainfall, channels are subjected to higher stresses, but perhaps for shorter periods of time, so some floods do not last long enough for the width to fully adjust. In rivers that are free to adjust laterally, a variety of narrowing processes may counteract widening, and thus reduce the cross-sectional area created by bank erosion. All of these factors contribute to the observed variability in bankfull Shields stress in gravel-bed channels. A clearer understanding of the sources of variability in this parameter, and the Shields entrainment parameter, would be particularly useful in formulating quantitative models of channel evolution in rivers that actively transport bed load.
Appendix A: Scaling and Relevant Variables
 The section below briefly outlines the principles of Froude-scale modeling used in selecting discharges, slopes and grain sizes for the experiments [Graf, 1971; Peakall et al., 1996; Parker et al., 2003]. Channel geometry is described by two main variables, bankfull width, B, and depth, H, which have the dimensions of length; the grain size, represented by the diameter, D, also has the dimensions of length. In a geometrically undistorted model, these variables are related by a common linear scale
where the subscripts p and m refer to prototype and model channels, respectively, and λ is the length-scale ratio. Flow conditions are described by the mean velocity, U, the discharge, Q, and the reach-average boundary shear stress, τ = ρgRS. The scaling relations for velocity and discharge are
 Because slope is dimensionless, and ρ and g are independent of scale, the shear stress in model and prototype channels should scale linearly with R (or H, if the channel is wide in relation to the depth):
 Bed load transport thresholds and transport rates are described by two dimensionless parameters,
Where s is the specific gravity of sediment, and qs is the streamwise bed load transport rate per unit width of channel. Assuming ρs and s are independent of scale, the relations given by ((A5)) and ((A6)) should be the same for both model and prototype channels,
 A time scale for channel responses, t, was derived by coupling relations for channel widening and lateral bed load transport,
where ∆A is the change in cross sectional area of the channel, qn is the lateral bed load transport rate per unit length of channel, and KA is a shape factor for the bank profile. The lateral bed load transport rate, qn , is expressed in terms of the streamwise bed load transport rate, qs, as follows:
where r is the ratio of lift to drag forces, μ is the dynamic friction coefficient, ϕ is the transport stage, , and tan ω is the lateral bed slope. We assume that the coefficients and ratios in ((A8)) and ((A9)) are the same in model and prototype channels and infer from ((A6)) that both qs and qn scale with τ3/2. The processes governing lateral and streamwise transport should be the same in model and prototype channels,
 Rewriting ((A8)) as a ratio between prototype and model channels, and substituting ((A10)) for qn, the time-scale for width adjustment is
 Substituting the previous relations for B and H into ((A11)), we get
cross-sectional area of the flow.
average channel width.
average channel width at the start of a run or the start of a flood.
grain size of fraction i.
median grain size of the bed surface.
median grain size of the bulk sediment (subsurface).
proportion of fraction i in the bulk sediment size distribution.
water depth, averaged over the central portion of the bed.
height of the bank.
bank erosion volume per unit channel length divided by dB times Hb.
exponent in the equation for lateral bed load transport.
lateral (cross stream) bed load transport rate per unit length of channel.
streamwise bed load transport rate per unit width of channel.
streamwise bed load transport rate per unit width on the sloping bank.
unit-width transport rate of the ith size fraction of the bed load.
width-integrated bed load transport rate.
ratio of lift to drag forces.
specific gravity of sediment.
water surface slope.
mean velocity of the flow.
dimensionless transport parameter.
dimensionless parameter relating the critical Shields stress on the channel bank to the critical Shields stress on the flat-bed portion of the channel.
hiding function exponent.
length scale ratio.
dynamic coefficient of friction.
inclination of the channel bank.
bed load transport stage.
bed load transport stage evaluated on the sloping bank region.
density of water.
density of sediment.
graphic standard deviation, (ψ84 − ψ16)/2 [Folk, 1974].
streamwise boundary shear stress acting on the flat-bed portion of the channel.
streamwise boundary shear stress acting on the channel bank.
critical shear stress for entrainment of sediment on the flat-bed portion of the channel.
critical shear stress for entrainment of sediment on the channel bank.
Shields stress, averaged over the flat-bed portion of the channel.
Shields stress for individual size fraction, i.
critical Shields stress for the median grain size of the bed surface sediment.
Shields stress of the median grain size of the bulk sediment (subsurface).
critical Shields stress for the median grain size of the bulk sediment.
dimensionless parameter relating the streamwise shear stress on the bank to the streamwise shear stress on the flat-bed portion of the channel.
 This work was supported by grants from the National Science Foundation to J. Pitlick (BCS-9986338) and J. Pizzuto (BCS-9986238). We are especially grateful to Chris Paola and Gary Parker for numerous thoughtful discussions that helped clarify our thinking and analysis. We thank the many people who helped with the experiments, including Nick Allmendinger, Sara Johnson, Mike Lamb, Michal Tal and Rebecca Thomas. We would also like to thank Brett Eaton, Jason Kean, John Buffington and Alex Densmore for providing very thoughtful and perceptive reviews of earlier versions of this manuscript. Revisions to the manuscript were completed while the first author (J. Pitlick) was on sabbatical leave at IRSTEA, Grenoble, FR.