Department of Civil and Environmental Engineering, University of Virginia,Charlottesville, Virginia,USA
Corresponding author: J. C. Curran, Department of Civil and Environmental Engineering, University of Virginia, PO Box 400742, Thornton Hall B228, 351 McCormick Ave., Charlottesville, VA 22904, USA. (email@example.com)
 Step-pool units, common features in steep, narrow streams, are highly dynamic systems that adjust during high flows and active sediment transport conditions. Consecutive step pools in a reach form a step-pool sequence, and though these features are prevalent in nature, quantifying the stability of such systems is challenging. This study focuses on the statistical relationships between 445 stable sequences of three or more steps and nine geometric and resistance-based parameters. Step sequence stability is parameterized through a sequence stability parameter, a metric of how long a given step-pool sequence exists relative to the total run time. Despite variability in the strength of the statistical relationships, the results allow identification of dominant trends between sequence stability and both roughness and geometric parameters. The sediment concentration ratio provides a means for comparing the relative strength of these statistical relationships. As sediment concentration ratios increase, the relations between flow resistance parameters and step stability change from dominantly inverse to dominantly direct. Thus, at higher sediment concentration ratios, sequence stability is more likely to increase with flow resistance, but the reverse is more likely at low sediment concentration ratios. A connection exists between bed morphology, flow resistance, and stability. For the majority of stable step sequences, resistance parameters increase over time, indicating some sequences adjust their geometry to increase stability during flood events. The influence of sediment concentration over the stability-geometric and stability-resistance parameter relationships may enable the use of the sediment concentration ratio as a predictor of step sequence stability during a flow event.
 The step-pool bed form, a common feature in steep, narrow channels, is composed of a series of steps and pools (Figure 1). The step is formed by a number of the largest clasts in the channel aligned transverse to the flow, creating the step riser. A pool immediately downstream of the step is scoured by flow tumbling over the upstream step and is followed by the step “tread.” Flow becomes supercritical as it approaches the step crest and then tumbles into the downstream pool; a hydraulic jump returns the flow to subcritical, and flow becomes supercritical as it accelerates over the next step crest. Step sequences consist of a sequential number of steps and pools within a single river reach [Comiti and Mao, 2012]. The parameters associated with the stability of sequences composed of three or more steps and pools are the focus of this research.
 The geometry of the step-pool sequence is defined by characteristics of the steps in the sequence. One element essential to alluvial step formation is a heterogeneous grain size distribution that includes a large grain that can act as a step forming clast [Comiti et al., 2005; Curran and Wilcock, 2005; Weichert et al., 2008; Zimmermann, 2009]. The size of the step forming grain must be large enough so that a small number of grains can span the channel width to form a step. This has been verified by both flume and field experiments, and is a component of the jammed state hypothesis [Church and Zimmermann, 2007]. Step height scales with the size of the large, step-forming grain [Abrahams et al., 1995; Curran and Wilcock, 2005; Grant et al., 1990; Judd and Peterson, 1969; Lenzi, 2001; Zimmermann, 2009], but does not necessarily equal the total hydraulic loss at the step. Hydraulic losses at steps derive from nappe flow passing over the step crest and turbulence generated in the downstream pool. Some researchers separate the height parameter into a pool depth and a step height so that hydraulic losses are considered individually. For example, Chartrand et al.  define step height as the difference in elevation between successive steps and the pool depth as the elevation difference from the deepest point in the pool to the elevation of the downstream step crest. Comiti et al. [2005, 2009] define the step height as the vertical distance from the step crest to the deepest point or maximum scour depth in the downstream pool, which is equivalent to the step height definition used by Abrahams et al. . Because this distance represents the largest possible vertical drop for flow passing over a step, we define it as the maximum step height, H (Figure 1). Step spacing, L, is defined as the distance between the crests of sequential steps.
 Step sequence formation typically occurs during floods with estimated return intervals of 30 years or more and high associated sediment transport rates [Lenzi, 2001; Mao and Lenzi, 2007; Molnar et al., 2010; Turowski et al., 2009]. In a few areas debris flows and smaller floods that created high rates of sediment influx and transport have also created step sequences [Milzow et al., 2006; Sawada et al., 1983], and large flows without significant sediment transport have also been shown to mobilize steps [Rosport, 1994; Rosport and Dittrich, 1995]. Individual steps within a sequence can shift their locations after initial deposition in response to flows lower than the step setting flow. A shift, or migration, is an adjustment of the position of the maximum step height within the area defined by the exclusion zone [Curran, 2012; Recking et al., 2012]. Steps are able to shift location by the erosion or deposition of sediments smaller than the step forming grain size on the stoss and lee sides of the step. Each time an individual step shifts, the spacing of that step relative to its upstream and downstream neighbor steps is affected, as is the mean spacing of the step sequence. Not all steps experience shifts in their location, and those steps that do not migrate are eroded more quickly. Alterations in bed morphology in response to migration of the step forming grain have been observed in multiple flume experiments [Curran, 2012; Rosport and Dittrich, 1995] (see also the work of Bacchi as discussed by Recking et al. ). Field evidence for step migration comes from Turowski et al. , who report on a series of floods in the Erlenbach for which over 90% of the steps were broken and rearranged in two floods but in a third, which had a lower discharge, only 30% of the steps rearranged. Individual steps within sequences exhibited different degrees of stability such that during a single event, some steps were quickly destroyed, some were able to shift in location on the bed surface, and others were stable immediately upon deposition.
 Individual step stability and overall step sequence stability are necessarily linked topics. Step sequence stability refers to the maintenance of the spacing between steps in a sequence such that a stable step sequence has a constant spacing over time. An individual step may be eroded and replaced by a new step, but if the new step is in a location that maintains the same spacing relative to the upstream and downstream steps, the step sequence has not been significantly altered. Thus, individual step stability can affect the sequence of steps over a reach but does not necessarily do so. Hypothesized to be necessary for step sequence stability is a step spacing that maximizes overall channel flow resistance [Canovaro and Solari, 2007; Weichert et al., 2008]. This hypothesis has its roots in experiments by Abrahams et al. , who fit data from 12 flume experiments and 18 step-pool configurations in New York and England to the following relationship: 1 ≤H/L/S ≤ 2, where H is step height, L is step spacing, and Sis channel slope. Step-pool sequences best fit this relation when the distance between steps was equal to the length of the downstream pool [Abrahams et al., 1995]. More recent data from flume experiments have shown steps form with a minimum spacing set by the exclusion zone, which is larger than the length of the associated pool [Curran and Wilcock, 2005; Gimenez-Curto and Corniero, 2003]. Many steps were spaced farther apart than the minimum spacing set by the exclusion zone, and thus did not fit the Abrahams relation, despite the wide parameter range possible. As a result, research continues to explore the parameters responsible for step sequence stability, including the role and function of the pool [Comiti et al., 2005; Zimmermann and Church, 2001].
 While all steps in a step-pool sequence contribute to channel roughness, only stable steps can contribute in a predictable manner. If the number and spacing of steps frequently varies because of step instability and individual step destruction, reliable estimates of total flow resistance and sediment transport rates through the channel during high flows will be difficult to develop. When the step sequence exhibits a high level of stability, more accurate reach-averaged roughness estimates become possible. Because sediment transport rates have been shown to increase in the years immediately following a step destabilizing flow [Lenzi et al., 2004; Turowski et al., 2009], this type of predictability would prove useful. Field research has focused on the interaction of a stable step sequence with channel hydraulics [Comiti and Lenzi, 2006; Comiti et al., 2009; Wilcox et al., 2006, 2011] and sediment transport rates [Egashira, 1988; Lenzi, 2002; Mao and Lenzi, 2007; Marion, 2001; Whittaker, 1987] but has not clearly defined the factors responsible for creating and maintaining a stable sequence.
 This paper evaluates geometric and resistance parameters and varying sediment supply for their contribution to alluvial step sequence stability. A series of flume experiments testing four discharge rates against five sediment transport rates provided an opportunity to document step formation and destruction processes, changes in step spacing, and step sequence stability for 445 sequences of three or more steps each. Previously, these data were examined to test hypotheses related to step spacing, individual step formation processes, and the stability of individual steps [Curran and Wilcock, 2005; Curran, 2007, 2012]. The data are now reexamined and the step sequence stability is investigated for its relation to geometric and resistance-based sequence characteristics as well as the influence of sediment concentration ratio on stability.
2.1. Experimental Setup
 The conditions under which step-pool sequences form and break are too infrequent for a field study that would define the stability of hundreds of step sequences over a range of flow and sediment supply conditions. A useful alternative is to perform a series of step-forming experiments in a laboratory flume. Laboratory experiments not only allow for control of the flow and sediment transport conditions, but also provide the opportunity to observe the formation of many step-pool sequences directly. In these flume runs the sediment bed was fully mobile and steps continued to form and break throughout each run, creating hundreds of different step sequences.
 Data for this study were collected during experiments described in detail by Curran and Wilcock , Curran , and Crowe , and are summarized here. Experiments were conducted in a small tilting flume of 0.15 m width, 0.3 m depth, and 5.2 m length, with 3.5 m working length. The flume walls were clear acrylic, allowing direct observation of the transport and bed forms. Water was recirculated and sediment was fed into the upstream end of the flume. In total, seventeen flume runs were completed, testing combinations of four discharge rates against five sediment feed rates. Sediment feed rates (Qs) spanned an order of magnitude from 110 to 1250 g m−1 s−1 and were constant for each flume run. Discharge (Q), which was also held constant for the duration of each run, varied from 0.0046 to 0.0065 m3 s−1. This range of flow rates was selected in order to ensure a highly mobile system and maximize the number of unique step formations over time. Our interest in these experiments was to determine the factors responsible for creating step sequences which could remain stable despite the high flows and shear stresses that would otherwise mobilize the step forming grains. Thirteen of the 17 flume runs were analyzed for the purposes of this study. Those runs not included had equilibrium run times, defined by equal feed and transport rates of the largest grain size of sediment, of less than 30 min, creating data sets restricted in length. Measured and calculated experimental parameters for the 13 flume runs are shown in Table 1.
Table 1. Summary of Measured and Calculated Experimental Parameters
Average water depth, calculated from video/photo analysis.
Average hydraulic radius, calculated as flow area divided by wetted perimeter.
Average bed slope, calculated using video/photo analysis.
Average dimensionless shear stress for the D84 grain size.
 The same sediment was fed into the flume as was used to create the initial sediment bed. Sediment was coarsely graded, with a grain size distribution extending from 0.5 mm to 64 mm. Constant sediment distributions were ensured through sieving bulk samples prior to each run. Sediment characteristics included a D50 of 14 mm, where D50 is defined as the grain size for which 50% of the sediment is finer by weight; D84 of 38.6 mm, where D84is defined as the grain size for which 84% of the sediment is finer by weight; 7.4% sand, defined as sediment between 0.5 and 2.0 mm; 8.3% in the 45–64 mm size class. Sediment sizes were selected to ensure a distribution that would promote jamming of larger grains in the narrow flume and the formation of step sequences. The ratio of step-forming grain size,DSFG, to flume width was consistent with previous flume and field observations where step-pool formations were observed [seeCurran, 2007]. Zimmermann et al.  hypothesized that the ratio of channel width, b, to step-forming grain size has a critical role in creating stable steps, as steps are more likely to form and remain stable where the channel is narrow. Because of our use of a constant grain size distribution in a constant width flume, we tested only one ratio ofb/DSFG. Thus, the jammed state hypothesis was not addressed in these experiments.
2.2. Experimental Measurements
 Direct measurements of flow depth and bed elevation were made possible through the use of video analysis. Every run was recorded and a mirror on the wall behind the flume enabled measurements of both sides of the flume. A photo was taken from the video for each minute of the equilibrium portion of each run. Bed and water surface elevations were measured over both the near and far sides of the flume every 3 cm in the downstream direction using a grid overlay on each image. These measurements were taken every 2 min of flume run time. A 2 min record of flow depth along the length of the flume was calculated from the measured bed and water surface elevations. Although there was variability in the water depth between steps and pools, this was minimized by the application of high flows simulating floods. When the full flow record for all the runs was examined, the average flow depth was 0.066 m with a standard deviation of 0.006 m. Bed slope was calculated as sin α (Figure 1). One minute values were interpolated between the measured values. Reach average water surface slope, bed slope, and energy slope were calculated using values measured from the images. Discharge was measured using a mercury manometer and velocity calculated from continuity using the measured water depth. Step heights were also measured from these photos. The depth of turbulent mixing within a pool was observed to vary during the experiments with fluctuations in the local velocity and sediment dynamics. Thus, our measure of step height may over or under estimate total head loss by a small amount in any given minute. By measuring over the life of each step this error was kept to a minimum.
 The dynamic nature of the steps made the video analysis critical in our determination of step sequence stability. The videos allowed us to create a continuous record of step formation, migration, and destruction for each experiment. For example, all step migrations were verified from the videos assuring that each migration was movement of the same step and not the destruction and formation of a new step. Step locations were tracked continuously and documented across the length of the flume from the videos. Step sequence formation and the number of steps comprising each step sequence were documented from this record. Step spacing was also measured from the video analysis.
2.3. Sequence Stability Parameter
 Difficulty in analyzing step-pool stability has arisen in part because of the lack of an overall quantitative measure of stability. In order to measure step-pool sequence stability in a way in which it can be compared with flow conditions and reach characteristics, we define a sequence stability parameter,SS*. This parameter expresses step-pool sequence stability as the cumulative existence time of that sequence, divided by the total equilibrium transport run time for a given combination of flow and sediment transport rates:
where SS* is the dimensionless sequence stability parameter for sequence i, Ei is the cumulative time in minutes that sequence i exists, and Tj is the total equilibrium run time in minutes for run j. By providing a dimensionless measure of stability, this parameter can be applied to compare degrees of sequence stability across runs of varying duration. As an example of the parameter's calculation, it was observed that sequence 3 in run 23 existed for 4 min. Applying equation (1) above with the total equilibrium run time of 131 min for run 23 (Table 1), the corresponding sequence stability parameter was calculated as follows:
The stability parameter essentially provides a metric of how long a given step-pool sequence, defined as the same number of steps with established step spacing, existed relative to the total run time. To account for step migrations, a step was considered stable in its position so long as any movements did not exceed the size of the exclusion zone, defined as 30 cm for these experiments [Curran and Wilcock, 2005]. Step movement within this zone was regarded as a rearrangement or migration of the same step during calculation of the stability parameter, and thus, was considered to exist as part of the same sequence. When a step migrated a distance greater than the exclusion zone length, it was considered a new step, forming a new step sequence. A new stability parameter was calculated for each successive step-pool sequence in every run.
 During periods with step sequences present, reach parameters were calculated at 1 min intervals for the existence of each sequence of three or more steps. Calculated parameters were then averaged over the duration of the sequence to obtain the sequence average values used in our analysis. Changes in the stability parameter reflected successive changes to the system as a whole, encompassing individual step breakups and new step formations that resulted in new sequence development. Therefore, the stability parameter provided a means to evaluate dynamic shifts between stable and unstable periods within the system, as well as the overall distribution of sequences displaying varying levels of stability.
 Total flow resistance is often separated into its component parts, and in these experiments the difference between resistance generated by the step bed form (macroroughness) and the total flow resistance is of interest. Here we use macroroughness to include resistance due to the step bed form and spill resistance over the step, while the roughness of the bed surface without step forms comprises the base resistance, similar to the work of Rickenmann and Recking . Base resistance as a fraction of total flow resistance has been shown to be small in steep channels with significant bed forms [Chiari and Rickenmann, 2011; Ferguson, 2007; Reid and Hickin, 2008; Rickenmann, 2012]. In step-pool sequences flow resistance has been considered to be dominated by bed form roughness and spill resistance, together comprising macroroughness, with the base resistance a small component in comparison [Church and Zimmermann, 2007; Wilcox et al., 2006]. In contrast, in a recent flume study where the flow depth over the step grains was low, Zimmermann  found that partitioning was not applicable and that base resistance was the majority of the total flow resistance. For our experiments, it was necessary to apply flow resistance partitioning and separate macroroughness and base resistance from total flow resistance to evaluate the importance of macroroughness to step formation and step sequence stability.
 In choosing a flow resistance equation for use with our data, we considered the traditional Darcy-Weisbach and Manning approaches as well as those based on Darcy-Weisbach and Manning resistance equations, models integrating a logarithmic velocity profile, and those applying a power law associated with at-a-station hydraulic geometry. Although we were able to calculate the Darcy-Weisbach friction factor directly from our measurements, the experimental conditions precluded use of this method. The step-pool channels studied had ratios of hydraulic radius to step forming grain size that classified the flows as shallow, making a roughness layer approach for calculating the friction factor more applicable [Ferguson, 2007]. We found the most appropriate models for our data were those of the variable power equation from Ferguson , as presented by Rickenmann and Recking  and Egashira and Ashida , both of whom derived resistance equations specific for step-pool channels. Each model was evaluated on the basis of its ability to reproduce the reach average flow velocities measured during the experiments. The models gave median velocity residuals of 0.19 and 0.34 m s−1, respectively, making the model derived by Rickenmann and Recking  a better fit for our data set. Rickenmann and Recking evaluated an extensive field data set that included channels with step-pool bed forms, steep slopes, and low relative depths, and using the variable power equation fromFerguson developed flow resistance partitioning equations applicable to the step-pool channel conditions. To calculate the total flow resistance, we appliedequation (2) to our data, where equation (2) is equivalent to equation (10a) of Rickenmann and Recking , with the exception that we use hydraulic radius in place of flow depth because of the low ratio of flow width to depth ratio, which averaged 2.3 in our experiments:
where ftot is the total flow resistance, R is hydraulic radius, and D84 is the grain size for which 84% of the bed is finer (equal to 38.6 mm for these experiments). Hydraulic radius was calculated as the flow area divided by the wetted perimeter. Because the channel width was constant at 0.15m, the hydraulic radius varied with the flow depth. The portion of total flow velocity corresponding to base resistance is Uo, found using equation (20b) of Rickenmann and Recking , which is
where Se is the energy slope. The flow resistance due to base roughness was then calculated using equation (4), the values from equations (2) and (3), and the total flow velocity, Utot.
Macroroughness (fmac) was calculated as the difference between the total flow resistance and the resistance due to base roughness.
 Some authors [e.g., Yager et al., 2012] have used shear stress calculations to interpret sediment movement in step-pool channels. Both shear stress and friction factor are measures of the frictional resistance acting in the channel and each applies the assumption that frictional flow retardation is balanced by the downslope component of the gravitational forces acting on the water. We include the dimensionless shear stress acting on theD84 grain size in our analysis to enable comparison with the work of other studies. However, we focus on the friction factor rather than shear stress because it presents a nondimensional value of the shear stress acting on the boundary of an open channel [Middleton and Southard, 1984], and can be interpreted as a drag coefficient if resistance is assumed equal to the gravitational drag force acting per unit boundary area and proportional to the square of the velocity [Ferguson, 2007].
 The resistance parameters included in this study are total flow resistance, total shear stress, macroroughness, and the ratio of macroroughness to total flow resistance. The experimental conditions included constant discharge values for each run and flow rates during the experiments that ensured full bed mobility and shear stresses much larger than critical shear values. Experiments were designed to study steps and step sequences under the step mobilizing conditions of a large flood. Therefore, neither critical shear stress nor stream power was explicitly included in our analyses. The analysis instead focused on changes in the step sequence stability parameter, how it was impacted by both geometric and resistance parameters, and how step sequences adjusted to maintain stability under extreme flood conditions.
2.5. Statistical Analysis
 No a priori assumptions were made about the statistical distribution of the step-pool sequence data collected, making nonparametric tests [Siegel, 1957] the most suitable statistical approach for analyses. The primary statistical measure applied in this study was Kendall's correlation coefficient τ modified to account for the presence of ties, i.e., combinations of variables that occur more than once within a sample of paired observation data [Helsel and Hirsch, 2002]. The modified coefficient, known as Kendall's τb, determines the strength of a monotonic association between two variables arranged categorically on an ordinal scale. Because calculation of this coefficient was based on ranks of paired data, it was generally resistant to outliers within the data set. Calculation of Kendall's τb involved counting the number of observed data pairs ranked higher or lower than a predetermined combination of two variables. For a more detailed description of this statistical measure and the calculation procedure, the reader is directed to the work of Helsel and Hirsch .
 To analyze step-pool sequence stability using Kendall'sτb, the median of the calculated stability parameters for each run was used as a partitioning point to define low and high sequence stability. Median SS*was selected instead of run-averagedSS* values because median values presented a more representative division point for stability given the tendency toward lower SS* values observed in the data set. Values of SS*above and below the median value for a given run were considered high- and low-stability sequences, respectively. High and low values of sequence-averaged parameters were defined in the same way, using the median variable value of all sequences for each run as a dividing point. Consequently, data pairs could be classified as a combination of either high or low values of two parameters and categorized on the basis of the ordinal scale necessary to calculate Kendall'sτb. These variables were paired with the corresponding values of low or high stability and organized into categories for each experimental run. A Kendall's τb coefficient was then calculated on the basis of the four possible combinations of SS* for the parameter in question. For example, the association between stability and step spacing was analyzed from the observed frequencies of paired data contained within the following categories: low stability, low spacing; low stability, high spacing; high stability, low spacing; and high stability, high spacing.
 Sequence stability was statistically tested against two groups of variables, designated as geometric and roughness parameters, respectively. In addition to sequence step spacing, the geometric group of tested parameters included number of steps in the sequence (#steps), average sequence step height (H), step steepness (H/L), and ratio of step steepness to bed slope (H/L/S). The group of tested roughness parameters included total roughness (ftot), macroroughness (fmac), fraction of macroroughness to total roughness (fmac/ftot) hereafter referred to as the macroroughness fraction, and dimensionless shear stress . Altogether, SS*was analyzed against nine different parameters, including geometric and resistance-related quantities in either dimensional or dimensionless form, in order to determine correlations between sequence stability and various sequence characteristics.
3.1. Sequence Stability Parameter
 The experimental flow and sediment feed conditions ensured an active, mobile bed, designed to create a large number of steps. Distinct step sequences formed as steps developed and broke, creating between 20 and 64 step sequences per run (Table 2) that remained stable for between 1 and 19 min (Figure 2a). As mentioned in section 2.1, the experimental conditions were designed to maximize the number of unique step formations and step sequences over time by generating shear stresses within a range sufficient to mobilize the step forming grains. To evaluate the ability of the flow to mobilize the large, step-forming grains, the critical shear stress, , was compared to the average total shear stress applied during each experiment. We used two methods for calculating critical shear stresses for the D84 grain size, . From the Wilcock and Crowe  model we found = 0.022. Because steep slopes are not accounted for in the Wilcock and Crowe model, we also applied the method outlined by Lamb et al. . The proportion of the total stress attributable to the bed forms averaged 89.7% in our experiments, which is outside the range of direct application of the model presented by authors. We instead used their Figure 10 and the data match for the case where bed morphology represented 80% of the total stress to calculate = 0.2. This is an order of magnitude higher than the calculation without considering the steep slope, but Lamb et al. warn that for high bed form roughness values the model over predicts critical shear stress. These two values for critical shear stress provide a benchmark range against which we evaluated the critical shear stresses for the D84 grain size during our experiments, which were between 0.046 and 0.484, with a median of 0.13. Thus, the step forming grains in our experiments could form steps and step sequences while remaining susceptible to entrainment.
Table 2. Stability Parameter Results
Number of Sequences
Run Average SS*
75th Percentile SS*
 This mobility is reflected in the step sequences. Approximately 80% of all sequences existed for 3 min or less (Figure 2a), but because of varying run durations, these sequences ranged in SS* from under 0.010 to 0.065. In dimensionless form, approximately 80% of observed sequences showed SS* values of 0.040 or less (Figure 2b). Values of SS* varied within and between individual runs, precluding identification of a median SS* value common to all runs. Figure 3 shows examples of the SS*values for sequential step sequences within three runs. In the interest of space, only plots for runs 7B, 12B, and 22 are shown, but these are representative of all the experiments. Sequence stability within a run was variable with higher stability sequences interspersed among the more dominant lower-stability sequences throughout each of the runs. HighSS* sequences were typically followed by lower SS* sequences, and for half of the runs the maximum SS* sequence preceded a sequence matching the minimum SS* for the run (for example, sequences 24 and 25 in Figure 3b). Overall, sequences shifted between higher and lower SS* values throughout the experimental runs without following a defined pattern.
3.2. Sequence Stability and Geometric Parameters
 Correlations between step sequence stability and number of steps in a sequence, sequence average step spacing, step height, H/L, and H/L/S were identified for every run. Sequence average geometric values are those parameter values measured at each minute during sequence existence, averaged over the existence time of that sequence. Sequence averaged SS* values were tested against each of the geometric parameters and the Kendall's τb calculated for each SS* parameter pair. The average Kendall's τb coefficient provided a general measure of the statistical strengths of the dominant and subordinate correlations identified between the parameters across all runs showing a similar trend (i.e., direct or inverse) or opposite trend.
 Lower SS* values were calculated as the total number of steps within a sequence increased, particularly with respect to the maximum calculated SS* values (Figure 4). Significantly more sequences consisting of three or four steps were documented than sequences containing six or seven steps. Coefficients of variance (CVs), which provide nondimensional measures of variability calculated as the standard deviation divided by the mean of a sample [Ang and Tang, 1975], reflect the variability attributable to more observations of sequences with fewer steps and thus, a greater standard deviation of SS* values (Figure 4). High CVs were calculated for sequences of each step number, which is representative of the dynamic nature of these experiments. Nevertheless, a decreasing trend in stability for the bulk of sequences is suggested as the number of steps in a sequence increased. Within individual runs, the inverse relationship between the number of steps in a sequence and sequence stability was supported by Kendall's τb analysis for nine of the 13 experimental runs (Table 3).
Table 3. Kendall's τb Results: Relationships Between Sequence Stability and Geometric Parameters
Number of Steps
Step Spacing L
Step Height H
Bold run totals denote dominant statistical trends.
 Trends between the stability parameter and dimensional geometric variables of step-pool sequences indicated that sequence stability was generally higher for larger step spacing (Figure 5a) and step height (Figure 5b). The correlations between the stability parameter, step spacing, and step height for successive sequences presented in Figure 5 are from run 8 (Q = 0.0055 m3 s−1; Qs = 475 g m−1 s−1) and run 20 (Q = 0.0065 m3 s−1; Qs = 1000 g m−1 s−1), which were representative of the links between sequence stability, average step spacing, and average step height calculated for each run. Sequence average step height was calculated as the sum of the individual step heights from step crest to pool base for each step in the step-pool sequence at a given time (seeFigure 1), divided by the total number of steps present in the sequence at that time. Total step height did not necessarily equal the step forming grain size, as steps often grew larger through the deposition of additional grains, step forming size and smaller, around the step form. Scour depths also varied between steps, contributing to overall variability in the measured values of step height.
 Statistical analyses supported the finding of dominant direct relationships between the stability parameter and both sequence step spacing and step height. The majority of sequence average step spacing values correlated directly to SS* values, such that larger step spacing was associated with high SS* and vice versa. Results from Kendall's τb analysis for SS* and step spacing indicated this positive association was dominant for 10 out of the 13 runs with an average Kendall's τb of 0.16 (Table 3). Similarly, a direct correlation between SS* and H was supported statistically for seven of the 13 runs (Table 3). Though not as distinct as the correlation between SS* and step spacing, the direct relationship between stability and step height was observed with a considerably higher average Kendall's τb coefficient for runs directly related (τb = 0.23) than for runs showing an inverse correlation (τb = −0.14). Therefore, the relationship between SS* and step height was classified as a direct relationship.
 Analyzing SS* values relative to the dimensionless geometric parameters describing step steepness (H/L) and step steepness to bed slope ratio (H/L/S) placed the analysis in a scale-free context. Increased stability was generally observed when sequences exhibited lower values ofH/L and H/L/S. Examples of the relationships between H/L and H/L/S with sequence stability shown in Figures 5c and 5d, respectively, highlight how the respective relationships varied with different step sequence development and system changes. Though no single trend was evident for all sequences, there was an inverse relationship between SS* and each of the dimensionless parameters. Statistical analyses supported the inverse correlations, as evidenced by the Kendall's τb coefficients shown in Table 3. The parameter H/L/S showed the most pronounced inverse association of the geometric parameters, as 10 of the 13 experimental runs had negative correlations. Sequence average H/L was negatively correlated with sequence stability for eight of the 13 runs, thereby exhibiting a dominant inverse correlation, as well.
3.3. Sequence Stability and Roughness Parameters
 Three separate dimensionless measures of flow resistance were calculated and compared to sequence stability, including total flow resistance, ftot, macroroughness, fmac, and macroroughness fraction, fmac/ftot. Median values were determined from sequence averages and calculated in the same way as the geometric sequence parameters described above (Table 4). For total flow resistance and macroroughness, as well as for macroroughness fraction, the statistical analysis showed that higher sequence stability corresponded to lower-value sequence roughness, and vice versa. The Kendall'sτb coefficients measured a statistically strong inverse correlation between roughness and SS* for a majority of the runs (Table 5). Identical trends and correlation coefficients were obtained for total and macroroughness, which was not surprising given the two roughness measures are calculated from the same sequence parameters, and thus, are proportional. In each case, 9 of the 13 experimental runs had negative Kendall's τb coefficients, which showed statistical support for the inverse relationship between sequence stability and sequence roughness. An even stronger inverse relationship was measured between SS* and the macroroughness fraction, for which 10 of 13 runs had negative Kendall's τb coefficients with an average Kendall's τb coefficient 67% stronger (τb = −0.20) than that of the directly correlated runs (τb = 0.12). Dimensionless shear stress, , also had an inverse correlation to sequence stability for a majority of the runs (Table 5). Thus, a sequence was generally more likely to be stable under a lower shear stress.
Table 4. Median Sequence Roughness Results
Table 5. Kendall's τb Results: Relationships Between Sequence Stability and Resistance Parameters
Total Resistance ftot
Macroroughness Fraction fmac/ftot
Dimensionless Shear Stress
Bold run totals denote dominant statistical trends.
 Sequence flow resistance associated with sequence stability parameter values above the run-specific 75th percentileSS* (hereafter referred to as SS*-75) showed that, for the most stable sequences documented in any run, the range in total roughness values was significantly reduced (Table 6 and Figure 6). There was a 65% average reduction in roughness variability for the SS*-75 sequences and a maximum reduction in roughness variability of 85%. Thus, variability in flow resistance was much less for the most stable step sequences. For example,Figure 7 shows the range in total flow resistance and step sequence stability values for run 17B (Q = 0.0050 m3 s−1; Qs = 1000 g m−1 s−1). The most stable step sequences for this run are highlighted with cross-hatched columns. The calculated total flow resistance values for these high stability sequences fell within the narrow band onFigure 7 that identifies 27% of the total variability in flow resistance. Differences in average total resistance between the SS*-75 sequences and the remaining sequences in a run were also identifiable as average total resistance was lower forSS*-75 sequences in 10 of 13 runs (Table 6 and Figure 6). The relationship between flow depth and step sequence stability mirrored this finding. Our calculation of total flow resistance was dependent on the ratio of hydraulic radius to D84 grain size. Because we used a single grain size distribution and constant flow width in these experiments, increases in flow depth were expected to correlate with decreases in flow resistance and increases in step sequence stability. For the same ten runs discussed above, the flow averaged 1.9% deeper for sequences above the SS*-75 threshold when compared to the less stable sequences.
Table 6. Sequence Stability and Roughness: Percentile Analysis Results
Total Roughness Ranges
Percent Change ftot Range
Average Total Roughness
Percent Change ftot
SS* < SS*-75
SS* > SS*-75
SS* < SS*-75
SS* > SS*-75
 Change in the macroroughness parameter value was quantified over the existence time for those step sequences existing more than 1 min. We investigated the cumulative change in macroroughness separately for those step sequences with stability above SS*-75 values and the step sequences with maximumSS* values from each run. Of the 250 total sequences existing for at least 2 min, approximately 57% experienced a cumulative increase in macroroughness, with an overall average increase of 3.3% for all sequences (Figure 8a). Focusing on the 94 step sequences with stability greater than SS*-75 values, 61% had cumulative increases in macroroughness over sequence existence, with an average increase of 5.3% (Figure 8b). Of the step sequences with maximum SS* values from each run (22 total for the 13 runs; runs 7B, 16, 22, 23 had multiple maximum SS* sequences), 73% adjusted to increase macroroughness values during sequence existence. The cumulative change in macroroughness for these sequences was the greatest, with an overall average increase of 7.4%.
 Through statistical analysis, we identified dominant relationships between sequence stability and several geometric and resistance-based parameters. There was variability in the strength of the statistical relationships between the runs and not a single parameter sequence stability relationship was dominant across all the runs, as reflected in the calculated Kendall'sτb coefficients (Tables 3 and 5). However, statistical results allowed us to identify dominant trends in the relationships between step sequence stability and sequence averaged values of step spacing, step height, step steepness, step steepness to bed slope ratio, total and macroroughness, macroroughness fraction, and dimensionless shear stress.
4.1. Relationship Between Roughness and Stability
 Channel roughness has long been hypothesized as a governing characteristic of step-pool stability.Abrahams et al. theorized that step-pool systems evolved toward a state of maximum stability by adjusting channel form to attain maximum flow resistance. In the present study, statistical correlations between total flow resistance, macroroughness, and macroroughness fraction indicated an inverse correlation between flow resistance and step-pool sequence stability, seemingly providing a contradictory finding to that of Abrahams et al. However, the above statistical results for roughness and stability associations were based on data from all the sequences documented during the respective runs. As such, these correlations supported only the finding that sequence stability was enhanced at lower values of roughness for the bulk of sequences, but did not provide any indication concerning how an individual sequence evolved through time. In order to investigate the role of morphological adjustment of step-pool sequences to attain maximum roughness and stability, an additional roughness analysis was conducted focusing on resistance parameters calculated for individual sequences.
 Cumulative change in macroroughness over the existence of a step-pool sequence was quantified and provided the necessary information to applyAbrahams et al.’s  theory to our sequence data set. For the most stable sequences within our data set, there were considerable increases in the macroroughness. Mechanisms by which the macroroughness of a stable sequence increased included step migrations (within the exclusion zone length) and increases in step steepness, which in turn increased energy dissipation and head loss. The correlation between high step sequence stability and a considerable increase in macroroughness over time agrees with the theory put forward by Abrahams et al.  that the most stable step sequences adjusted to maximize total flow resistance, and thus, increase sequence stability. Every step sequence did not experience an increase in macroroughness, and only 45% of the step sequences fit into the stability bounds established by the relationship between step steepness and channel bed slope put forward by Abrahams et al. . We attributed this to the highly dynamic conditions present in the flume during the experiments. Our experimental conditions included mobilizing flows and variable sediment feed. In contrast, the flume experiments conducted by Abrahams et al.  did not test a mobile bed, nor did they include sediment supply, which has been shown to decrease form roughness through infilling of pools [Koll and Dittrich, 2001], to increase transport capacity [Whittaker and Davies, 1982] and to impact sequence stability (the work of Bacchi as discussed by Recking et al. ).
 The flow and sediment transport rates that composed the flood conditions tested can be parameterized by the sediment concentration ratio, Qs/Q, a dimensionless parameter proposed by Church and Zimmermann and shown to have an impact on step stability. Our results indicated a strong influence of sediment concentration on the stability of step sequences. The run-averaged change in cumulative macroroughness decreased with increasing sediment concentration ratio (Pearson correlation coefficient,r, equal to −0.64). This decrease was most noticeable in run 23, which had a high Qs/Q ratio of 12.86 (see Table 1 and Figure 8b), indicating that in flows containing higher sediment loads sequences were not able to adjust to fully maximize macroroughness before being destroyed. The same finding was surmised by field studies where the sediment supply impacted step-pool sequence adjustment by promoting downstream transport of step-forming grains and adversely affected step stability [Recking et al., 2012]. The apparent inability of some step sequences to adjust to increase their macroresistance suggested that the development state of a step-pool system [Molnar et al., 2010] may act as an indicator of sequence stability.
4.2. Role of Sediment Concentration Ratio
 Some of the observed variation in the statistical dominance of any parameter-sequence stability relationship may be due to the influence of the sediment concentration ratio under which the step sequence was formed and measured [Church and Zimmermann, 2007]. Variation in the statistical correlations with respect to sediment concentration was evident for all parameters tested (Figure 9). Therefore, the relationship between sequence stability and each of the nine tested parameters was evaluated to determine the influence of sediment concentration ratio over the identified stability trend. Average Qs/Qratios were calculated for those runs with the same statistical trends for a specified parameter-SS* relationship. The dominant trends, either direct or inverse, previously determined between sequence stability and the geometric parameters corresponded to higher Qs/Q ratios (Figure 10a). All of the resistance parameters tested had a statistically dominant inverse relationship to sequence stability. When these flow resistance–sequence stability relationships were tested against the corresponding Qs/Q ratios, the average Qs/Q ratios were between 22% and 31% lower for runs with the dominant inverse relationship when compared to runs with a statistically direct relationship (Figure 10b). The statistical trend of increased sequence stability at a reduced flow resistance remained dominant only when the sediment concentration was low. For the eight runs with a statistically dominant inverse relationship between SS* and each of the four resistance parameters, the average Qs/Q ratio was 6.44. As the sediment concentration ratio increased, statistical support for the inverse association weakened. High Qs/Q values corresponded to those runs previously determined to have a direct relationship between resistance parameters and sequence stability, and the three runs with direct relationships between SS* and all four resistance parameters had an average sediment concentration ratio of 11.82. Overall, as Qs/Q ratios increased, the dominant inverse relationships observed between sequence stability and resistance parameters were no longer statistically valid; rather, a direct relation between flow resistance parameters and step sequence stability became statistically dominant at high sediment concentration values.
 The geometric parameters examined exhibited a mix of statistically dominant inverse and direct relationships with step sequence stability. Dimensionless measures of sequence geometry had inverse relationships while the dimensioned parameters had direct relationships to sequence stability. Comparing statistical agreement to the sediment concentration ratio, the dominant trends previously determined, either direct or inverse, corresponded to higher Qs/Q ratios. Thus, the statistical trends showed that when the sequence stability was high and sediment concentration ratio low, step steepness and H/L/S were also high. From the resistance parameters, it was indicated that under this scenario the flow resistance would be low. A possible scenario of stable step sequence formation for those runs with lower sediment concentration ratios was suggested by the experimental observations and statistical findings. During runs with low sediment supply, sediment was unlikely to deposit and accumulate around the steps. At the same time, the relatively clear water passing through the initial step sequence scoured pools, which created high step steepness ratios. Increasing step steepness through pool formation adjusted the roughness geometry toward a state of maximized stability [Weichert et al., 2008]. However, the growth of the step, and by extension the step steepness ratio, was restricted by the low amount of sediment available from transport, limiting the macroroughness contribution to total flow resistance. Thus, macroroughness and total flow resistance through the step sequence remained low and step sequences were able to remain stable for a period of time for flows with low Qs/Q ratios. Extended high flows with limited sediment transport are known to destabilize steps over time [Curran, 2012; Rosport and Dittrich, 1995; Wohl and Jaeger, 2009]. As sediment concentrations in the supply increased, we measured concurrent increases in step height, steepness, and macroroughness.
 An individual sequence analysis of step height, conducted in a similar fashion to the macroroughness analysis outlined in section 4.1, showed that of the most stable sequences in each run (i.e., SS* above SS*-75) over 60% had a cumulative increase in step height, and presumably scour, over the existence of the sequence. Higher steps resulted in larger macroroughness contributions by the step form. The macroroughness contribution to total flow resistance increased over sequence existence as head loss due to step height increased and scour continued to deepen the downstream pool, a finding in agreement with research showing concurrent increases in the contribution of spill resistance and step height [Comiti et al., 2009]. Consequently, flows with a higher sediment supply allowed for accumulation of sediment at a step along with greater pool scour, which together may have enabled sequences to stabilize temporarily as macroroughness and flow resistance increased during the existence of the step sequence.
 A similar connection between sediment concentration ratio and step stability has been noted in previous flume and field studies. Flume experiments by Bacchi, as discussed by Recking et al. , investigated sediment supply effects on step stability and found that scour could significantly impact individual step destabilization, which was also shown by the step destruction mechanisms documented for these experiments [Curran, 2012]. On the basis of field results and the Bacchi flume work, Recking et al. proposed that scouring in step-pool systems “connected” to a sediment source lowered the hydrologic conditions necessary for step mobilization. For our sequence data set, the relationship between resistance and stability changed as sediment concentration ratio increased. However, lower sequence stability was observed overall for runs with higher sediment supply conditions (Tables 1 and 2), consistent with the hypothesis discussed by Recking et al. .
4.3. Form and Process in Sequence Stability
 Linking form and process in step-pool systems over temporal and spatial scales is important when investigating the stability of sequences [Chin, 1998; Wilcox et al., 2011]. The geometric parameters tested in our study acted as surrogates for the influence of form while the resistance parameters more aptly pertained to the processes potentially impacting step-pool sequence stability. Full division of form and process was not possible because the two are necessarily related. However, by independently analyzing both groups of parameters, we identified some of the factors associated with more stable sequences and, from those associations, explored possible links between form and process-based factors for a broader understanding of sequence stability.
 Form and process analysis of step-pool sequences can be appropriately conceptualized by considering the development state [Molnar et al., 2010] of a sequence. Geometric and resistance characteristics of a given sequence changed during its existence, so the manner in which observations were made became important to the stability analysis. When sequence-averaged values were considered, the statistical correlations indicated that roughness and sequence stability were inversely associated. However, the averaging process may mask within sequence adjustments. If a single time step was analyzed, the sequence measurements would not have been able to represent the potential adjustment ongoing to increase step stability [Chin, 1998, 2003; Wilcox et al., 2011]. By analyzing adjustments occurring over time within stable sequences we found step steepness and macroroughness values were able to increase as transported sediment accumulated at the step-forming grain. The progressive adjustment of sequences observed on an individual sequence basis illustrated the importance of development state on the stability of step-pool sequences. These findings apply to all the sequences without regard to the number of steps in a particular sequence. However, they may combine to explain the reduction inSS* as the number of steps in a sequence increases.
 Sequences with larger step spacing were predominantly observed as more stable, reflecting the importance of pool development during sequence existence, which has been shown to dissipate energy through turbulence [Gimenez-Curto and Corniero, 2003, 2006; Wilcox et al., 2011]. By definition, the same sequence showed little changes in step spacing, so increases in either H/L or H/L/S were a result of increases in step height and scour related to pool development. Step steepness had a statistically dominant inverse relationship with step sequence stability such that for larger H/L measurements there was lower stability. On an individual sequence basis, though, H/L values increased for the majority (approximately 57%) of sequences with SS*values larger than run-specificSS*-75 values. Similar results were obtained forH/L/Svalues of individual sequences, where 60% of these higher-stability sequences also increasedH/L/S over sequence existence time. The longer a sequence existed, the more it was exposed to flow and sediment transport processes that affected the sequence geometry [Weichert et al., 2008], including sediment deposition around an existing step and pool development downstream of a step. The measured adjustment in step steepness within a stable step sequence links the change in step form with step sequence process, as indicated by the increases in macroroughness and total flow resistance discussed above. The connection between step steepness, pool scour, and total flow resistance statistically quantified here has been suspected from field measurements in the Italian Alps and Colorado [Wilcox et al., 2011]. Parameter adjustments occurring in the more stable sequences may indicate the development state of sequences as the geometric adjustments impact channel processes and stability as a sequence persisted without destruction.
 As pointed out by Weichert et al. , the stabilization potential of a channel bed will not be fully recognized as long as sediment supply is available and transport conditions are adequate, which was the case with our study. Constant discharge and sediment feed ensured active transport conditions such that no sequence could reach its maximum stability potential, in part contributing to the highly dynamic sequence behavior encountered in each run. Our experimental conditions also precluded analysis of incremental flow change impacts on sequence stability, which has been previously investigated [e.g., Rosport and Dittrich, 1995]. The role of stream power, often linked to step sequence stability [Church and Zimmermann, 2007; Weichert et al., 2008], was not explicitly investigated because of the constant nature of flow in these experiments. All grain sizes present in these experiments were mobilized under the flows tested. Therefore, critical values of incipient motion were not used as a stability metric [Church and Zimmermann, 2007; Recking et al., 2012; Zimmermann, 2010].
 Our analysis addressed dynamic changes to step-pool sequences during a flooding event as all grain sizes employed were mobile and readily transported. The more stable sequences were those that continued to develop and adjust to increase their stability parameter until either new step formation or step destruction occurred to form a new sequence. This is reflective of the transient nature of step-pool sequences occurring during a flooding event. By observing sequence behavior under constant flows and fully mobile transport conditions, our findings on sequence stability are most appropriately applied to the adjustment of step-pool systems occurring during a flood of a set duration, analogous to one of our experiment flume runs.
 All steps in a step-pool sequence are active in channel processes, affecting the transport of flow and sediment downstream. However, only stable step sequences contribute to channel processes in a predictable manner. In this research an extensive data set collected from 13 flume runs of varying discharge and sediment transport conditions was used to analyze 445 sequences of three or more step-pools and investigate how geometric and resistance factors influenced step-pool sequence stability. To enable comparison of step sequence stability across runs, we defined a sequence stability parameter (SS*) as a metric of how long a given step-pool sequence, defined as the same number of steps with established step spacing, existed relative to the total run time. TheSS* values for each of the 445 sequences were tested against a range of geometric and resistance parameters to determine the strongest statistical relationships which would provide an indication of those parameters more influential in creating a stable step sequence. Those geometric parameters examined included dimensional values of step number (#steps), height (H) and spacing (L) and nondimensional step steepness (H/L) and the ratio of step steepness to channel bed slope (H/L/S). The resistance parameters analyzed included total flow resistance (ftot), macroroughness (fmac), dimensionless shear stress corresponding to the D84 grain size , and the ratio of macroroughness to total flow resistance (fmac/ftot). Our analysis focused on changes in the step sequence stability parameter, how stability was correlated to both geometric and resistance parameters, and how a step sequence adjusted to maintain stability under extreme flood conditions.
 The analysis showed variability in the strength of the statistical relationships, yet the results allowed us to identify dominant trends in the relationships between sequence stability and sequence average values of each of the nine parameters. The resistance parameters and dimensionless geometric parameters were predominantly inversely related to step sequence stability, meaning that sequences with higher stability parameter values were statistically more likely to have low flow resistance and low step steepness ratios. However, almost half of the step sequences included in this analysis existed for only 1 min of run time. The experimental conditions ensured formation of dynamic step-pool sequences as the runs were designed to replicate the events during a flooding event when all grain sizes were mobile and readily transported. These conditions allowed for the creation of a large number of step sequences but also ensured that none of the sequences remained stable throughout a run. To evaluate the role of morphological adjustment of step-pool sequences to attain maximum roughness and stability, the 1 min sequences were excluded from further analyses and the statistical trends were examined on an individual sequence basis.
 In contrast to the results using all step sequences, we found that the majority of stable sequences increased their roughness parameter values over time and approximately half fit within the bounds of the step steepness to bed slope relationship developed by Abrahams et al. . The correlation between high step sequence stability and an increase in macroroughness over time agreed with the theory put forward by Abrahams et al. that the most stable step sequences adjusted to maximize total flow resistance. Not all stable step sequences adjusted to increase channel macroroughness, a result that may be at least in part due to the highly dynamic conditions during the experiments. The analysis of parameter adjustments during the existence time of individual step sequences indicated one manner by which some step-pool sequences are able to increase stability during flood events.
 The sediment concentration ratio, Qs/Q, provided a means for testing the relative strength of the statistical relationships between sequence stability and both roughness and geometric parameters as well as a way to evaluate the importance of different parameter-stability relationships. Each experimental run was conducted at a different sediment concentration ratio, which had a strong influence over step sequence stability. AsQs/Qratios increased, the dominant inverse relationships observed between sequence stability and resistance parameters were no longer statistically valid; rather, a direct relation between flow resistance parameters and step sequence stability became statistically dominant at high sediment concentration values. The influence of the sediment concentration ratio over the stability-geometric and stability-resistance parameter relationships (direct at highQs/Q values and inverse at low Qs/Q values) may enable the use of the sediment concentration ratio as a predictor of step sequence stability during a flood event. For example, if a step sequence in a given area has been measured and known to have a large step steepness ratio, then when a flood event occurs that sequence is statistically more likely to remain stable when the Qs/Q ratio for that flood is low. However, if the flood event has a high Qs/Q ratio, that same step sequence is more likely to be mobilized.
 The link between step-pool sequence form and process allows us to speculate on the scenarios during which a step sequence is statistically more likely to be stable. During an event with a lowQs/Q ratio, there would be a limited amount of sediment in transport and therefore, a limited amount of sediment that could deposit around an existing step. Although flow tumbling over the steps scours downstream pools, without additional deposition at the step form, the step steepness ratio would remain restricted. The potential contribution of macroroughness to total flow resistance would be limited. The result would be a step sequence more likely to remain stable if the step steepness ratio was low and the Qs/Q ratio during the flow event was also low. During a flood characterized by a large Qs/Qratio, the step sequences would be expected to have a low overall stability. However, a significant number of these step sequences would also be expected to adjust to increase stability by increasing the macroroughness associated with the step-pool form. The large amount of sediment in transport would lead to greater deposition around the steps and allow step sequences to stabilize as macroroughness and flow resistance increased during the existence of the step sequence. These two scenarios were indicated by the data but remain speculative as we have yet to gather the field data to test them. However, parameter adjustments occurring within step sequences may illustrate a development process by which geometric and resistance parameters adjust within stable step sequences.
 This work has been aided by thoughtful discussions on the nature of step-pool systems with Peter Wilcock, Francesco Comiti, Jens Turowski, Dieter Rickenmann, Tom Lisle, Andre Zimmermann, and Mike Church. The manuscript was improved by the comments of Kristen Cannatelli, Dieter Rickenmann, Jens Turowski, Francesco Comiti, and an anonymous reviewer.