Research examining the hydraulics, morphology, and stability of step-pool mountain streams has blossomed in the last decade, resulting in more than a dozen dissertations. These, along with other research projects, have transformed our understanding of step-pool channels. Contributions have been made toward understanding depositional step formation and destruction, scour downstream of steps, step-pool hydraulics, and the effect of sediment transport on step stability. We propose that depositional steps exist in a jammed state whereby the boulders are structurally arranged within the channel and thereby stabilize it. Once a step has formed, a scour pool with a characteristic length and depth develops downstream, creating a zone where additional steps are unlikely to occur. Downstream of the scour hole, steps are more likely to occur as the high energy associated with the plunge pool has dissipated. Data suggest that the presence of cobbles or boulders limits pool scour as well as the degree to which well-defined, channel-spanning step-pools form. We propose a state-space for step-pools in which conditions for a step to form include (1) the ratio between width and boulder diameter (the jamming ratio), (2) the ratio between applied shear stress and the stress needed to mobilize the bed (relative Shields number), and (3) the ratio between bed material supply and discharge (bed sediment concentration). Available data suggest this model is plausible. Emerging critical research questions are discussed.
 Since at least 1960, researchers have examined the flow of water and sediment through steep, sediment-filled channels [Peterson and Mohanty, 1960]. Such channels form important headwaters in the world's uplands, where they drain much of the landscape and are the source of water for down-valley communities and stream habitats [Beschta and Platts, 1986]. However, they pose a significant risk to people when they destabilize and flood downstream communities with water and sediment. In light of these concerns a number of important studies of steep channels have been completed within the last decade, including more than a dozen dissertations. The objective of this paper is to integrate key findings of these and other studies to appraise the current understanding of step-pool stability and channel form, and to formulate a new conceptual model that may help us to understand step-pool stability.
 The paper begins by reviewing the morphology and flow characteristics of step-pools formed by deposition of clastic sediments. In a stable state, the channel must dissipate the entire energy of the flow except for the fraction required to move the imposed sediment load. Accordingly, the partitioning of flow resistance is examined. Theories of how boulder-cobble step-pools form are reviewed, followed by a discussion of sediment transport and step-pool stability. The shape and form of scour pools downstream of steps are analyzed and step spacing is revisited. Key findings from these observations are integrated into a theory of depositional step stability incorporating the idea that granular material may be arranged in a jammed state [Cates et al., 1998, 1999; Liu and Nagel, 1998], and data from the literature are presented to support it. The jammed state is a concept that has emerged in the physics of granular media to describe the stability of certain grain arrangements that are able to resist directed forces. In the present case, the grain arrangements are the steps and the directed force is exerted by water flowing over the steps. It appears that critical factors influencing the development of a jammed state in streambed sediments include the geometrical ratio of channel size (width) to clast size, which determines the length and resilience of the force chains sustained by the jammed clasts, the relative Shields number (the ratio τ/τc), which indexes the magnitude of the applied force, and bed material concentration in the flow, which determines the potential rate of grain exchanges with the bed. Finally, emerging research questions are presented.
Chin and Wohl  have recently published a general review of empirical research on step-pool occurrence and formation. Accordingly, this review more selectively focuses on findings that lead toward the jammed state hypothesis.
2. Characterization of Step-Pools
 Step-pools are channel forms composed of alternating channel-spanning ribs (steps) and pools (Figures 1a, 1b, 2b, and 2c) with tumbling flow [Peterson and Mohanty, 1960] that oscillates between subcritical in the pool and supercritical over the step [Hayward, 1980; Grant et al., 1990; Montgomery and Buffington, 1997]. Relative roughness (D/d, where D is a diameter representative of the larger clasts and d is a measure of flow depth) is near 1 at flood flow. Montgomery and Buffington  quote a range 0.3 < D84/d < 0.8, while Comiti et al.  observed that D84/R was 0.85 for a flow event about 80% of bankfull. In both French Pete Creek, Oregon [Grant et al., 1990], and Shatford Creek, British Columbia [Zimmermann and Church, 2001], relative roughness remained >1 at calculated or measured high flows and in East Creek, British Columbia, during an event that was the largest in 34 years, D84/d was 1.55, in part due to overbank flooding that resulted in a relatively small average depth (0.33 m) despite main channel depths on the order of 0.7 m. Steps may consist of cobble or boulder chains, woody debris (often channel-spanning logs), or bedrock. In this review, we focus attention on cobble-boulder step-pools, formation of which is most consistently associated with the movement and deposition of sediment in the channel (hence “depositional” steps), that might therefore be expected to exhibit the most consistent hydraulic behavior.
 Whether step-pools constitute a distinct channel type or simply represent a channel morphological unit has been debated. Grant et al.  suggested that they are a channel unit phenomenon with a characteristic scale on the order of one bankfull width set within a larger cascade channel type defined at the reach scale. In contrast, Montgomery and Buffington  suggested that step-pools represent a channel type distinct from the cascade (Figure 1c) and “plane bed” (or rapid; Figures 1d and 2a) channel morphologies. They noted that the cascade channel type may include boulder ribs, but that they do not span the channel width, thus making them distinct from the step-pool morphology.
 In part, the difficulty in discriminating step-pool and cascade streams results from the challenge of identifying them in an objective manner. Wooldridge and Hickin  investigated four means of classifying boulder step-pool and cascade stream channels, including visual identification, bed level crossings about the mean gradient, bed elevation differencing and power spectrum analysis. They found that visual identification was most consistently able to recognize the geometry of and classify the individual bed forms. Milzow et al.  have since developed a step identification technique that classifies step-pool sections based on the occurrence of a critical slope (e.g., >15%) followed by a low-gradient section (pool). Preliminary tests that we have made with this technique show that it has promise, possibly because it does not require any spatial averaging of the long profile. Herein a distinction is made between the tumbling cascade morphology, which may be steep but exhibits poorly defined local steps that do not span the channel (e.g., Figure 1c), and the step-pool morphology that is composed of channel-spanning steps and pools, even though the distinction may reflect only the relative congestion of large clasts and may not be hydraulically fundamental.
 Individual step-pool units have been reported on gradients greater than 2° or about 4% [Whittaker and Jaeggi, 1982; Chin, 1989; Grant et al., 1990; Montgomery and Buffington, 1997] and continuous step-pool morphology on gradients greater than about 4° or 7%, where structural reinforcement becomes necessary to maintain bed stability [Church, 2002]. On lower gradients, boulder and cobble ribs (e.g., Figures 1d and 2a) grade into rapid steps [Hayward, 1980] or stone lines/stone cells [Church et al., 1998; cf. the “transverse ribs of” McDonald and Banerjee, 1971; McDonald and Day, 1978]. These lower-gradient units are distinct from step-pools as their relative roughness is generally less than 1.0 and they lack channel-spanning pools. Step-pools also present distinct hydraulic conditions in the form of a channel-spanning transition to supercritical flow at the top of the step and a turbulent plunge pool downstream (see Figure 1a). Conversely, stone lines are drowned at moderate flows and channel-spanning hydraulic jumps are not present. Some investigators [e.g., Abrahams et al., 1995; Aberle and Smart, 2003] have suggested 3% as the threshold gradient above which continuous step-pools are found. However, Comiti  and Comiti and Lenzi  noted that in flume experiments with a 3% slope, antidunes formed that did not break and tumbling flow did not occur; conversely at slopes greater than 4.5% the ribs caused a hydraulic jump to occur and flow conditions resembled those found in step-pools during flood. In the field, within a short reach many steep channels alternate between clearly defined step-pools and the less distinctive cascade or rapid-type morphologies described by Montgomery and Buffington. The discrimination of a threshold gradient for steps may depend in part on the relative size of the channel and the step-forming clasts, and it may lie in part in the eye of the beholder.
 We are unaware of any investigation that has critically examined the maximum slope at which step-pools occur. Grant et al.  show 40% as the upper limit of their observations of step-pools while Wohl and Grodek  recorded boulder steps on gradients up to 73% in a hyperarid boulder/bedrock wadi. Wohl and Grodek's Figure 6 shows, in a compilation of their own data with those of Hayward  and Grant et al. , that there is no further systematic reduction in pool length on gradients above 20%, when length averages about 1 meter. Further increase in gradient is entirely accommodated by increasing step height. At 20% gradient, pool + tread length is 5 × drop height, “tread” being defined here as the distance between the end of the scour pool and the crest of the next step (see Figure 2c). One must wonder whether this is because, at higher gradients there is no longer room for pool formation so that one is dealing rather with a series of drops and sills. Furthermore, above a certain gradient sediment movement in the channel is likely to be dominated by colluvial mass wasting processes, including debris flows, and step-pools will not persist. It appears reasonable to suppose that, for channel slopes between 7 and 20% or more, an unbroken sequence of step-pool units constitutes a distinct channel type.
 The singular geometry of step-pools has attracted significant field investigation (reviewed by Chin and Wohl ). The characteristic dimensions step height (H) and step spacing (L) (see Figure 2 for definitions) have been measured extensively and their ratio has been found generally to fall in the range 0.06 ≤ H/L ≤ 0.20, with a median value near 0.1. The range corresponds fairly well with the range of gradients on which continuous step-pools are found, suggesting that the steps control nearly all the drop in step-pool channels. Step spacing has been correlated with overall stream gradient (S) in the form H/L αSβ, in which 0.42 ≤ β ≤ 0.68 [Abrahams et al., 1995]. H is variously represented as step height or the representative dimension of a step-forming element – a distinction that will turn out to be of some importance. At present, it is sufficient to note that the correlation is a necessary consequence of the appearance that steps control the drop. It has also been asserted that steps are relatively regularly spaced along the channel, although the claim remains controversial (reviewed by Chin and Wohl ).
3. Step-Pool Hydraulics
3.1. Flow Regime
 Steps are essentially irregular drop structures, so it is worthwhile to review flow patterns and scour observations made at weirs, sills and other drop structures. Flow over discrete steps with a free fall is known as nappe flow [Chanson, 2001], while flow affected by the downstream tailwater is known as submerged flow [Wu and Rajaratnam, 1996, 1998]. It is unclear if the term tumbling flow, which has often been used to describe flow in step-pool channels, applies to one or both of these flow regimes. Flow assumes critical depth on the step and plunges onto the lower sill (or into the pool) (Figure 3d). Where the drop is not shear, a clear hydraulic jump occurs in the pool. As flow increases it proceeds toward becoming either critical or subcritical throughout the channel length. Which occurs depends on the rate at which the tailwater depth on the downstream sill (or at the pool outlet) increases compared to the critical depth and this, in turn, depends on channel slope and roughness.
 When discussing uniform flow, energy gradients are defined as mild, critical or steep according to the Froude condition. On mild slopes, flow is subcritical (F < 1) and on steep slopes flow is supercritical (F > 1). In step-pool streams uniform flow does not occur. However, it is useful to consider what bed slope would divide supercritical flow (steep slopes) from subcritical flow (mild slopes) in step-pool channels. To develop a rough estimate we start from Manning's formula, u = d2/3S1/2/n, where u is the mean velocity and n is Manning's roughness coefficient, even though it is not strictly applicable to the strongly nonuniform flow in a step-pool channel. Solving for the critical condition Fr = 1; i.e., u = (gd)1/2, the critical slope (Sc) is given by
where g is the acceleration due to gravity and dc is the critical flow depth. We introduce Strickler's equation, n ≈ 0.035 D1/6, where D is some relatively large grain diameter to yield
There is some evidence [e.g., Zimmermann and Church, 2001; Canovaro et al., 2004] that grain roughness contributes only about 20–40% of the total flow resistance in boulder-dominated streams. The result of Canovaro et al. is particularly interesting even though it was obtained in a flume experiment employing fixed stone ribs, as it is based on independent assessment of each resistance component (as opposed to assuming a residual value for form resistance). We adjust the coefficient (c = 1.23 × 10−3) in Strickler's equation accordingly. This yields c = 6.13 × 10−3 if grain resistance contributes 20% of the total (1.23 × 10−3/0.2) and c = 3.06 × 10−3 if grain resistance contributes 40% of the total. For 0.5 ≤ D/d ≤ 1.0, this leads to 6.0% > Sc > 2.4% (3.4° and 1.4°), so the critical gradients described in the last section may discriminate ranges of step-pool occurrence. A transition from submerged flow to nappe flow is expected, which is apt to result in distinctive hydraulic and morphological features. (In systems with significant woody debris, grain resistance has been estimated to be as low as 10% of the total [Curran and Wohl, 2003; Wilcox et al., 2006], which would increase the critical gradient to 12%.)
 Flow may never actually attain the supercritical state throughout; we suppose that such flow would destabilize most channels. Oscillation between supercritical and subcritical flows results in large energy losses to spill resistance and we suppose this to be a key control on channel stability. On steep slopes at higher stages, flow is launched vigorously off the step and a recirculating cell develops in the head of the pool under the drop. The supercritical jet also becomes extended (Figure 3e). This flow type, termed transition flow, is notably chaotic. If the calculations above are a useful guide, it seems quite possible that step-pool morphology adjusts to normal high flows so that the flow remains within this transitional regime. Chanson  determined that nappe flow occurs on sills when dc/H < 1, where dc is the (critical) flow depth at the brink of the step and H is the drop height, while transition flow develops when 0.85 ≤ dc/H ≤ 1.15, approximately. If dc/H > 1, skimming flow develops (Figure 3f), in which the water flows over the steps in a coherent stream with recirculation in the pools beneath. Flow is then continuously supercritical. These discriminant criteria have not been tested in step-pools but the condition dc/H ≈ 1 roughly corresponds with the observation that D/d ≈ 1 at high flow.
 On mild and critical slopes a submerged drop occurs when the water surface at the tailwater is higher than the crest height of the drop [Wu and Rajaratnam, 1996]. Submerged drops can be further subdivided into those with an impinging jet (Figure 3a) and those with a surface jet (Figure 3b). Impinging jets, in which the center of the jet points toward the bottom of the downstream pool, tend to form at lower tailwater depths. Surface jets occur when the jet is directed downstream and does not impinge directly on the bed. Wu and Rajaratnam  showed that, for broad crested weirs, the occurrence of impinging or surface jets depends on the critical depth, tailwater depth, drop height and discharge. Whether Wu and Rajaratnam's relations would hold in step-pool streams has not been tested; however, it is reasonable to surmise that both impinging jets and surface jets occur in natural step-pool channels. At Shatford Creek, British Columbia, (6.8% < S < 8.7%) the tailwater surface elevation was found to be on average 40 cm above the upstream step crest during an event with a return interval of 1.7 years, but to vary between 90 and 6 cm (standard deviation = ±23 cm; n = 22) [Zimmermann and Church, 2001]. Thus submerged jets were observed to occur on gradients similar to the critical gradients estimated above.
 Herein we have attempted to combine the jet classification schemes proposed by Chanson  and Wu and Rajaratnam , which describe nappe, transitional and skimming flow, and impinging and surface jets, respectively (Figure 3). A combined classification scheme is warranted as jet scour is increasingly thought to be a key factor controlling step-pool morphology [Comiti et al., 2005; Curran and Wilcock, 2005; Comiti and Lenzi, 2006]. In order to create a single classification scheme we have interpreted Wu and Rajaratnam's [1996, 1998] impinging jet to apply only to conditions in which the elevation of the tailwater affects flow over the upstream drop and have proposed that the term nappe flow be used for flow over a free overfall as is explicit in Chanson's  definition of nappe flow.
 Insight into the nature of the jet in step-pool streams can be gained from the work of Lee , Wohl and Thompson , and Wilcox and Wohl . Velocity profiles were measured by Lee  at a variety of discharges in flume experiments (5.5% < S < 6.7%) in which the tailwater depth was consistently well above the upstream step (i.e., the channel had a mild slope). The profiles are spatially variable, but they confirm the presence of a surface jet, as there is no obvious point where the jet impinged on the bed (Figure 4). Confirming Lee's observations, Wohl and Thompson  found in a field study (2.5% < S < 12.3%) that the velocity profile in plunge pools during floods is dominated by midprofile shear caused by negative (i.e., upstream) flow in the lower portion of the profile. They also observed that, immediately upstream of and at the lip of the step, bed-generated turbulence dominates the profile. Bed-generated turbulence is, however, a less effective energy dissipater than the wake-generated turbulence associated with form and spill drag expressed by midprofile shear in the plunge pool. Wilcox and Wohl  used an ADV to observe that flows across a range of discharges in a step-pool stream exhibited strong three-dimensional turbulent kinetic energy, including a substantial vertical component.
 These observations reveal the very nonuniform energy gradient across a step (see Figure 2c). Upstream of the step, but well downstream of the upstream pool, the energy grade line is parallel to the bed and balanced by grain friction. Beyond the step crest the energy grade line initially steepens only slightly as the decrease in bed elevation is made up for by an increase in velocity. Once the turbulent boil in the hydraulic jump is reached, the energy grade line drops significantly. Downstream of the hydraulic jump it is nearly horizontal and water velocities are at their lowest of any location along the profile. As flow exits the pool, the bed gradient increases downstream and the kinetic energy increases until it is balanced once again by grain friction.
 In summary, step-pools exhibit nappe flow at low stages and may progress to either transitional or submerged regimes at high flow depending on whether the gradient is steep or mild. Limited observations of hydraulics in relatively low gradient step-pool channels suggest that free overfalls may be rare at flood flows, while surface and impinging jets are both quite common. Transition flows may be restricted to high gradients and skimming flows may be exceptional. These inferences warrant further investigation, especially on steeper slopes.
3.2. Resistance to Flow
 At low flow, the resistance to flow in step-pools is contributed mainly by spill associated with nappe flow. Velocity profiles show that, at high flows, energy is extracted from the mean flow by underlying recirculating cells in the pools while, at all flows, boulders provide prominent form resistance as they give rise to leeside eddies. In comparison, grain resistance is of minor significance. In stylized experiments (stone “steps” placed as transverse strips across a flume), Canovaro et al.  showed that form-induced resistance amounted to as much as 80% of the total. It has been known for a long time [cf. Scheuerlein, 1973, and references therein] that the classical resistance formulations, predicated on the dominance of grain resistance, do not successfully describe resistance to flow in step-pools. A recent review has been given by Aberle and Smart .
Lee and Ferguson  analyzed observations from six field sites and from flume experiments to show that data covering essentially the full range of step-pool morphologies (0.027 < S < 0.184) fail to collapse onto the classical Keulegan equation. Aberle and Smart  developed a Keulegan-type relation from experimental data collected by Rosport  and Koll  (2% < S < 10%), but the data did not collapse satisfactorily. Given the principal sources of resistance these outcomes are not surprising. Aberle and Smart proceeded to criticize the choice of grain size to represent the roughness of steep channels, where aggregate grain structures are known to be important, and advocated instead that s, the standard deviation of the bed elevation be adopted. They then obtained 1/√f = 3.14log10(1.36 d/s), in which f is the Darcy Weisbach friction factor. The collapse is reasonable, but there is some indication of residual structure in the data. Canovaro and Solari  have recently shown that data from a number of field studies can be roughly collapsed onto 1/√f = 2.74log10(0.48 L/H), a scaling that directly emphasizes step-pool geometry.
 An alternative empirical approach is to recognize that flow resistance is related to the hydraulic geometry of the channel, hence may be summarized by equations of the form u = cQm, in which m is an index of resistance to flow and Q is discharge [Kellerhals, 1970, 1973; Bathurst, 1993] (reviewed by Aberle and Smart ). Kellerhals and Bathurst showed that we may expect higher values of m in steeper reaches (Kellerhals actually considered the exponent in A ∝ Qp, in which A is cross-section area and p varies inversely as m). Aberle and Smart generalized the approach by considering dimensionless groupings of the variates u, q (specific discharge), S, g, and l (a length scale) to obtain the general relation
which is a Froude type equation. Adopting l = s, regression of the Rosport-Koll data yielded [Aberle and Smart, 2003]
which is very similar to the rational result previously inferred by Rickenmann [1990, equation 5.20]
which is equation (3a) with α = 0.6, β = 0.2.) The coefficient in the relation is of order 1. (Rickenmann obtained equation (3b) from an equation of Takahashi  for a debris flow front and showed that it describes reasonably well experimental data of step-pool flows taken by Smart and Jäggi  on gradients 0.05 ≤ S ≤ 0.20.)
Equation (3b) can be rewritten as u/(gd)1/2 ∝ S1/2(d/D) which, in turn, can be rearranged to give
This is simply the Chezy equation adjusted by a measure of relative roughness.
Giménez-Curto and Corniero Lera  have distinguished the “jet flow” regime in which flow separates over high boundary roughness. For such flows they determined that f = 0.52(g〈H〉sinβ/u2)2/3, where 〈H〉 is the mean obstacle height, specifically equated with step height of Giménez-Curto and Corniero . fG = gdS/u2 is the friction coefficient (the familiar Darcy-Weisbach friction coefficient is 8fG, as fG is defined here) and β is the channel slope, so sinβ ≈ S. Substitution for fG again leads to equation (3c) provided D = 〈H〉. We will carry this discussion forward in section 7.
 In the foregoing equations, s can be substituted for D and S should be understood to be bed slope, which will not differ from the water surface slope if averaged over several step-pool units. u and d are averages that are best calculated from observations of tracer diffusion and measurements of channel width, the only easily accessible geometrical measure of the channel.
 Introduction of s as a measure of bed roughness opens the question of necessary survey intensity to characterize s adequately. Clearly, the survey must capture all significant bed level fluctuations, which means that survey spacing must be less than the diameter of any significant roughness element. The low-frequency limit of significant additions to aggregated roughness must also be considered. J. Aberle (personal communication, 2005) took observations every 0.24 cm along 240 cm of flume in order to calculate the standard deviation of the bed. Within this length there were about 7 step-pools. The calculated standard deviation began to fluctuate when fewer than 250 points (every fourth) was used. This suggests about 35 bed elevation measurements are need as a minimum through each step-pool unit to characterize the bed. To capture the low-frequency variance within a reach, a distance equivalent to 30 or more widths should be sampled [Trainor and Church, 2003]. Additional approaches based on spatial correlation have been essayed by Furbish , Robert [1988, 1990, 1991], Nikora et al. , and Marion et al. .
Aberle and Smart's  results derived from experimental runs that were all completed at near bed-mobilizing flows and it remains to be shown that standard deviation can be used to assess flow roughness at low-flow conditions. Others have had only limited success using standard deviation at lower flows [Lee, 1998; Lee and Ferguson, 2002] and there are theoretical reasons why standard deviation may not then work. During nappe flow the average downstream velocity may be very similar through a sequence of step-pools regardless of step height, yet a measurement of bed elevation using standard deviation would differ significantly between two channels with different step heights. Similarly, a measure of step steepness (e.g., H/L) is likely to work more effectively at higher flows. At low flows, the height of the step might not much matter once the flow is in a free drop. Conversely, when flows approach submerged or transitional states, the size of the step will have a pronounced effect on water velocities and flow resistance. Research should be conducted to examine how step height affects average velocities and the applicability of standard deviation to characterize bed roughness at low-flow conditions.
 Hydraulic theories for step formation have been promoted by the appearance of regularity in step spacing along the channel. Among authors who have argued that step-pools are a function of the flow hydraulics, at least four variants exist of a basic theory that relates steps to standing waves. Judd  (S < 5.6%) suggested that the initial step creates standing waves downstream and subsequent steps form under these waves. Conversely, McDonald and Day  suggested that step-pools form when a downstream hydraulic jump migrates upstream as sediment is deposited under standing waves. Whether the observed “steps” in McDonald and Day's study were indeed steps is, however, questionable. During their experiments sediment was deposited only where the standing waves were located: the rest of the bed was the smooth aluminum base of their flume. In addition, the slope did not exceed 2%. None of these conditions occurs in natural step-pool channels. Allen  suggested an alternative hypothesis whereby boulders are deposited under supercritical flow conditions, causing a hydraulic jump which leads to the creation of a downstream pool. The next downstream step can form only once supercritical flow is reestablished.
 The most cited theory of step formation suggests that step-pools form under antidune crests when a large stone is deposited, anchoring the antidune and initiating the deposition of other stones [Whittaker and Jaeggi, 1982]. Whittaker and Jaeggi showed this result experimentally (1% < S < 24%) and the observations have been repeated [Egashira and Ashida, 1991; Curran and Wilcock, 2005]. The deposited stones then trigger scour downstream which forms a pool. Regardless of the details, all of these theories suggest that the location of steps is a function of the hydraulics associated with standing waves so that they should be regular and predictable.
 In contrast to the “hydraulic” approach, Zimmermann and Church  suggested that the location of steps depends on the random location of keystones against which other stones come to rest. This idea was formally tested by Curran [Crowe, 2002; Curran and Wilcock, 2005]. Using a tilting 15 cm wide flume Curran found that steps could be formed when sediment in the 45–64 mm grain size class was present in the mixture and the bed slope was between 5 and 8.3%. When the sediment was exclusively finer than 45 mm and the bed slope varied between 3.5 and 5.2%, antidunes formed that were repeatedly washed out and subsequently reformed. These observations suggest that some minimum grain size is required to form step-pool morphologies, as had previously been proposed [Grant et al., 1990], as well as potentially a minimum slope. Using a wider flume (30 cm) and slopes between 5.6 and 6.7%, Lee  also came to the conclusion that the formation of step-pools depends on the ratio between the grain size and channel width. Like Curran, she observed that the location of new steps was not consistently under the crest of the standing waves but, rather, was governed by the location of other large stones against which mobile stones would stop. K. Koll (personal communication, 2005) also noted that the location of keystones governed the location of steps.
 In their experiments with 45–64 mm grains, Curran and Wilcock  recorded the formation and destruction of steps and found that no single process was responsible for the formation of the step-pools; rather four separate processes were identified. About 50% of all steps formed when a stone came to rest against an existing obstacle. An additional 20% of the steps formed when a knickpoint was created by the breaking of an existing step so that the head of the pool migrated upstream to the next pool or to a grain that could not be moved. A further 25% of the steps formed under the antidune train that formed downstream of an upstream step (cf. the hydraulic hypotheses). Finally 5% of all steps formed when the bed was locally degraded and existing steps reemerged. Curran and Wilcock also examined the position of each step relative to the upstream step and found that an exclusion zone existed where no steps occurred. Reanalysis of Curran's data suggests that this zone corresponds roughly with the scour pool downstream of the step (see further discussion in section 7).
 Downstream of the exclusion zone an exponential frequency distribution describes the probability of steps occurring at a particular location. While the length of the exclusion zone varied (23–40 cm), depending on the step formation process, few steps formed in the first 30 cm downstream of the upstream step. Figure 5 illustrates the fitted frequency distribution and the observed distribution of step spacing from Curran and Wilcock  for those steps formed through the obstacle and exhumation processes (first and fourth processes described above). Milzow et al.  examined the morphology of a 17% gradient step-pool cascade and observed that the length and height of the pool downstream of a step was related to the height of the upstream step, suggesting that a strong feedback exists between step height, pool scour and step-pool morphology.
 Surprisingly, when the largest stone in Curran's experiments was 45 mm, steps did not form, yet when w/D84 from her study is compared to values from eight other flume studies which have reportedly created steps, the eight other studies all had larger ratios of w/D84. Evidence that the entire grain size distribution affects the formation of steps comes from Tatsuzawa et al. , who performed flume experiments with three sediment mixtures. All mixtures had the same Dmax but two of them had a greater proportion of large stones than the third. They found that step-pools (which they described as anchored antidunes) formed when the two mixtures with a larger proportion of coarse material were used, but not when the third mixture was used, suggesting that it is not just the size of the largest stones in the mixture that influences the formation of step-pools, but also the number or frequency of large stones that are present.
 An extensive set of flume experiments by Rosport , Rosport and Dittrich , and Koll  showed that high flows wash out steps and create a morphology similar to riffle pools with shorter, irregular barriers. Maxwell  performed similar experiments which showed that bed stability also depends on bed slope. Using sketches of the bed structure, Weichert et al.  demonstrated that, during steep flume experiments, the type of bed form present changes as stream power increases until the bed eventually degrades to the flume floor. They also observed more frequent channel-spanning steps for narrow channels and steeper slopes. These studies suggest that stream power (the product of flow and slope) or transport stage (the ratio τ/τc, the applied shear stress compared with the critical shear stress for particle entrainment), in addition to the ratio w/D84, influences the stability of step-pool streams. A possible explanation for the failure to form steps at relatively conservative w/D ratios in the Curran experiments is that, even at the lowest flow rates (0.031–0.043 m2/s), all sizes in the mixture were readily transported.
 A number of authors have noted that the formation of steps takes on different behavior for bed slopes greater than about 7%. Whittaker and Jaeggi  noted that, below slopes of about 7.5%, the antidunes remain mobile, while at greater slopes the antidunes are quickly anchored by large stones. Likewise, Koll et al. [2000, p. 5.6] note that “with bed slopes less than 4% the self stabilizing processes under clear water conditions differ from those on slopes greater than 8%”. In light of these observations it is noteworthy that Curran's experiments at slopes ranging from 3.5 to 5.2% with sub-45 mm sediment produced only mobile, antidune-like features, while she produced steps on gradients between 5.0% and 8.3% by adding +45 mm sediment. Koll's experiments were conducted on gradients between 7.5% and 10%, Rosport's at 2, 4, 8 and 10%, Tatsuzawa et al's at 2.5, 5 and 10%, Maxwell's from 3 to 7%, Weichert's from 3.5 to 9% and Lee's from 5.6 to 6.7%. Only the experiments discussed by Whittaker and Jaeggi (1% < S < 24%) covered the entire range of gradients on which step-pools exist. It appears that quite distinct “step” forming mechanisms may occur on moderate gradients compared to truly steep ones, with the former having been chiefly investigated in the experiments reported to date, while the latter appears to depend more strictly on grain congestion and transport intensity.
5. Sediment Transport and Step-Pool Stability
5.1. Sediment Transport Through Step-Pools
 Sediment supply has been shown to affect the stability of step-pool streams. De Jong  argued that sediment starved conditions must exist in order for steps to form. Indeed, the “normal” condition in step-pool streams was well summarized by Egashira and Ashida [1991, pp. 52–53], as follows:
Sediment transport in mountain streams sometimes is very scarce or occurs much less than that expected in an equilibrium state even if the flow discharge exceeds over some critical values. Such occasions prevail in channels with stable step-pool systems or fully armored beds where the sediment transportation is controlled by the rates of sediment supply from upstream region. In such conditions sediment particles may be transported as overpassing loads or through puts, filling pools and pores in armored bed surface with themselves or eroding themselves from the bed. However, original step-pool systems with armored bed surface remain unchanged….
 Field researchers have observed a general pattern of low sediment transport rates following periods of low sediment availability and transient high sediment transport rates following sediment delivery and channel destabilizing events [e.g., Ashida et al., 1976; Sawada et al., 1983; Warburton, 1992; Adenlof and Wohl, 1994; Gintz et al., 1996; Lenzi et al., 1999, 2004]. As an extreme example, Gintz et al., using magnetically tagged tracers, observed that the mean and range of travel distance of stones increased about 10 times following a large event that obliterated the step-pool pattern. Following the flood the step-pool pattern was reestablished. Lenzi  also described a step-pool channel (Rio Cordon, Belluno, Italy) with large, well defined pools, indicating a stable, scoured channel, that was modified after a large flood in 1994 to a less stable structure with positively sloping pools. These changes promoted sediment transport, which then declined over several years as the channel reestablished a relatively stable bed. A mudflow in 2001 and a small debris flow in 2002 once again increased sediment transport rates [Lenzi et al., 2004], although not as extensively as after the 1994 event. Lenzi et al.  examined the effective flow for channel maintenance based on 17 years of flow, bed load and suspended load data in this channel and found that flows with a relatively frequent return interval (1.5–3 years) were responsible for maintaining channel form, altering minor steps and scouring pools. Large magnitude events with return intervals of approximately 30–50 years were responsible for macroscale changes in channel form.
 An increase in the supply of sediment to a step-pool reach may destabilize the bed by modifying the channel roughness. In early flume experiments, Hayward  noted that the velocity of water through pools filled with gravel was considerably greater than through pools devoid of gravel. Nevertheless, he found that conventional bed load formulas tend to overpredict sediment transport, presumably because they do not account for the energy loss associated with plunging flow in pools. In the field, Ergenzinger and Schmidt  recorded considerable variation in sediment storage over short periods of time along the Lainbach, in Bavaria. At the beginning of an event the pools tended to fill with sediment and the bed roughness was noticeably reduced. The subsequent removal of sediment may have been promoted by the reduction in roughness and/or a further increase in discharge. However, Whittaker and Davies , again in the laboratory, observed that, as gradients increase, it requires a very high feed rate to drown steps formed by fixed baffles and that, at steep slopes, a large range of feed rates produce the same water velocity.
 In flume experiments in which fed material is distinctly colored, it has been demonstrated that adding sediment to a channel with stable step-pools can cause the pools to infill, resulting in a decrease in the form roughness and mobilization of the bed [Koll et al., 2000; Koll and Dittrich, 2001; Koll, 2002, 2004]. During these experiments more sediment exited the flume than was fed in, and it was visually evident that the fed material mixed with the bed material.
 In summary, the experiments of Koll showed that an increase in bed load transport rate causes bed instabilities and this effect has been confirmed in field settings. Field studies demonstrate that sediment supply can modify the storage of material in the channel, which would be expected to modify the form roughness and may in turn modify the stability of the streambed. For a more detailed review of sediment transport studies through step-pools, see Rickenmann .
5.2. Step Destruction
 In her experiments [Crowe, 2002], Curran observed four principal methods by which steps collapsed. Seventy-seven percent of the steps were destroyed by downstream scour, with almost equal proportion of these failures occurring due to tumbling of the keystone (39%) and aggregate slumping (38%) processes. Tumbling occurs when the top keystone topples into the downstream pool due to sediment being scoured downstream of its support, while slumping occurs when the downstream pool scour leads to a general collapse of the step. Rosport and Dittrich  likewise noted that step destruction occurs when the hydraulic jump erodes material from the base of the step. Lee  observed that steps tended to fail when the largest stones protruded prominently into the flow, suggesting toppling of the keystone.
 Curran also observed that 10% of the failures occurred when a mobile large clast moving from upstream impacted and dislodged a keystone. The remaining 13% of the steps disappeared as a result of being buried by sediment, which does not really constitute step destruction. Burial has been reported in natural step-pool channels as well [De Jong, 1992; Warburton, 1992; Lenzi, 2001]. Of course, steps also fail when a debris flow runs through the channel (e.g., Lainbach/Schmiedlaine described by Ergenzinger , De Jong , and Gintz et al. ). The failure of the steps in Rio Cordon occurred during a flash flood that featured near-hyperconcentrated characteristics [Lenzi et al. 2006].
 While major floods and debris flows are obvious causes of bed instability, the observation that nearly 90% of all step failures in Curran's flume occurred due to focused downstream scour emphasizes the need to understand pool scour.
6. Scour and Self-Affinity of Pools
 Recently researchers have devoted considerable attention to the shape and size of pools downstream of steps in order to predict when sills will be undercut and steps will destabilize. Observations by Lenzi and Comiti  and by Milzow et al  that the size of the pool downstream of a step is strongly related to the drop height (Lenzi and Comiti) or to the height of the step (Milzow et al.) also emphasize the need to understand downstream scour. Comiti , following results of Gaudio et al. , found that pools are self-affine across a wide range of slopes, discharges and grain sizes. The longitudinal (x) and vertical (y) dimensions of the pool normalized by the maximum depth (ys; equivalent to step height H) and pool length (Ls), respectively are constant (Figure 6; variables defined in Figure 2). Bennett  found similar relations for scour pools downstream of knickpoints in eroding rills. In neither case, however, were pools self-similar. The maximum scour depth could not be predicted based on the maximum length of the pool.
The formula was calibrated against data of model drop structures but, inasmuch as these data extended to q = 0.42 m2 s−1, H = 2.15 m, and Dm = 28 mm (mean grain size), they appear to represent a more reasonable approximation of step-pool conditions than the data of full-scale drop structures (which yielded a different constant). Mason  later considered air entrainment but did not succeed to further improve test results. In most step-pools, with limited drop, air entrainment probably remains relatively minor. We know of no data of air entrainment in falls over steps, but Vallé and Pasternack  reported air fractions to 42% in natural hydraulic jumps. This value would adjust Mason's results by only 10%. However, Lenzi et al. [2003b] noted that the residual depth, scaled by drop height (z) approaches a fixed value as the drop height scaled by the critical flow depth increases. This is thought to occur as the increase in drop height is balanced by an increase in air entrainment [Lenzi et al., 2003b]. Other investigators have considered jet penetration angle and penetration distance into the pool [Bormann and Julien, 1991; Stein et al., 1993], but the formulations offered yield notably erratic results (C. Rennie, unpublished analysis, 1999). A significant problem in all these studies appears to be the limited investigation of a disconcertingly weak dependence on grain size which, because of competence limitations, clearly is a major factor limiting scour in natural pools.
 On the basis of flume experiments, Comiti  and Marion et al.  found that step height, ys, scaled by critical specific energy on the sill, Hs = dc + u2/2g, depends upon the relative drop energy (a1/Hs), sediment sorting coefficient, σ = [D84/D50 + D50/D16], and the scaled spacing of the steps, L/Hs, yielding
It is interesting that sediment gradation (σ) was found to be more important than grain size, presumably because of armoring effects. Following Abrahams et al. , Comiti [2003; see also Lenzi et al., 2003a] also found that when steps were sufficiently closely spaced, the downstream step limited the depth of the pool. In (5) (1 − accounts for this interference. It has the effect of reducing scour by 10% if the step length is 16 times the critical energy of the step (Hs). However, Comiti [2003; see also Comiti et al., 2005] found that a simple relation between the drop height (z) and the critical depth (dc) predicts scour depth as effectively as equation (5):
when fitted to selected data from sill measurements in flumes and streams (R2 = 0.92).
 A problematic aspect of (6) (previously noted by Lenzi et al. [2003b]) is that the drop height (z) is used to predict the scour depth (ys) which is, in effect, the sum of the drop height (z) and the residual pool depth (dr) (see Figure 2). Thus in (6) there is a degree of spurious correlation. It would be preferable to return to the use of residual depth as adopted by Jaeger  (as discussed by Hager ) if the drop height or some closely related surrogate (e.g., a1) is going to be used as an independent variable. The same problem occurs in equation (5), where a1, the “morphological jump,” is proposed to predict scour depth, and a similar problem exists in (4), where scour depth is measured as the residual depth plus the tailwater depth and is predicted based on the tailwater depth. To investigate the effect of spurious correlation, the residual depth was computed for each pool. Figure 7 illustrates the resulting correlation between normalized residual depth (dr/dc) and normalized drop height (z/dc). It is evident that much of the apparent concordance between scour depth and drop height is due to the induced correlation and that residual depth is not particularly well predicted by drop height.
b is the crest width of the sill and B is the width of the pool. As such the equation includes a three dimensional component, which V. D'Agostino (as discussed by Lenzi et al. [2003b]) noted as being important. Equation (7) has R2 = 0.85 based on 145 laboratory experiments, but it remains untested for step-pool streams.
 Grain size is notably absent from equations (5) and (6), yet many researchers have found it to have some effect, albeit modest (see equations (4) and (7)). This is likely because the largest grains found on the bed in studies examining scour downstream of artificial sills are mobile during peak flows and do not critically influence total scour. However, in some step-pool channels, particularly those that verge on being cascades, the largest stones may not be mobile, and scour predictions for such channels will need to recognize this circumstance. In East Creek, British Columbia, a relatively small stream with many boulders as large as 1.5 meters in diameter, we have observed the two largest flows during 34 years of record (see Figure 1a), yet at two of the eight steps monitored, each greater than 60 cm in height, the residual depth in the pools is less than 5 cm. It would appear that at these steps flows never reach magnitudes sufficiently large to move the boulders located downstream of the steps. R. Weichert (personal communication, 2005) confirmed in flume experiments with artificially created steps that, when large stones were present in the pool locations, the pools did not scour to the predicted dimensions. The particular morphology of a step-pool system is apt to be influenced by the abundance of large, relatively immobile stones and the supply of finer material that may infill the pools, both of which may preempt self-affinity. To predict scour dimensions in step-pool streams we require a better understanding of the threshold of grain mobility in steep streams.
 In summary, while the investigations of Comiti et al. , Marion et al. , and D'Agostino and Ferro  into scour pools have certainly pushed forward our understanding of scour, additional research will be necessary to account for jet hydraulics, clast size and the likelihood of forming a step within the interference range.
7. Step Geometry Revisited
Abrahams et al.  made a fundamental linkage between step-pool geometry and the flow resistance problem. They demonstrated by flume experiments in which step length was deliberately varied that a strong maximum in flow resistance occurs when S ≤ H/Ls ≤ 2S, indicating that the steps control the entire drop of the channel. Here Ls is the length of the scour pool (Figure 2). H/Ls/S may be greater than 1 since, where the bottom of the pool is lower than the crest of the downstream step, H/Ls exceeds the overall gradient. They found that, at very close step spacing, resistance is reduced since pools are small and scour is inhibited whereas, at very large step spacing, the scour pool represents a diminished portion of the resistance. They further showed that the morphology of natural channels is such that 1 ≤ H/Ls/S ≤ 2 when Ls = L, the distance between steps; that is, the observed morphology when the scour pool occupies the entire distance between steps coincides with the morphology observed in the flume to maximize resistance.
in which 〈H〉 is the obstacle height, that is, the step height, and 〈H〉/d is normalized mean drop height. Reversing the argument, when L = Lmin for given H (determined by available grain size), f is a maximum. This does not, however, necessarily set the actual step spacing, as emphasized by Curran and Wilcock .
Zimmermann and Church  showed that step-pools actually occupy a greater range in (H/L)/S and that the ratio appears to have a slope dependency. Comiti [2003; see also Comiti et al., 2005] subsequently demonstrated that, if one assumes that steps are closely spaced and there is no extended tread at the end of the pool (i.e., Figure 2b occurs rather than Figure 2c), (H/L)/S is indeed slope dependent:
In (9), k = Ls/dr is the ratio between pool length and the residual depth, hence k is a shape factor for pools. Comiti [2003; see also Comiti et al., 2005] found that k varied between 6 and 8 for flume and field studies with well graded sediments and was 4.7 for well sorted sediments in a flume study; that is, the ratio apparently has only a narrow range. Equation (9) can be arrived at only if one assumes that the pool length (Ls) and the step wavelength (L) are the same (see Figure 2). If k ≥ 10, (9) is essentially H/L = S.
 In Figure 8a the relation between k and the distance between successive sills (i.e., L) is plotted using data from Comiti  and from our work at Shatford and East creeks. The data from natural step-pools exhibit significantly more scatter than the flume or field data of artificial sills, due largely to the small residual depths measured in some of the natural pools because of the presence of large boulders that limit scour. Considering only the flume sill data it is evident that the pool shape factor systematically decreases as the spacing between sills increases. When data of natural step-pools are added to the comparison, it is no longer obvious that the pool shape factor varies consistently; rather, there appears to be some average pool shape with substantial variance.
 In Figure 8bH/L/S is plotted as a function of S for natural step-pools and for flume and field drop structures. The data of Zimmermann and Church  plot noticeably below Comiti's  data. More generally, it is evident that, on gradients below approximately 0.07, the augmented data stray outside the range specified by Abrahams et al. and that, at the highest gradients, they more strictly approach 1.0. An evident reason for the variation on low gradients is that steps no longer control the entire drop (a reason for the observed empirical relation between S and H/L; see section 2). The long profile of Shatford Creek, for example, shows that, from the end of the scour pool to the next step, some distance exists where there is a positively sloping bed. Inspection of Comiti's data (Figure 6b) indicates that some of his step-pools also have such a tread, possibly explaining why some of his data plot low. Therefore, in streams with an extended tread, like those depicted in Figure 2c, a portion of the overall slope and elevation loss is attributed to the tread. In such scenarios Comiti's  assumption that the elevation drop across the step is the total elevation drop is no longer true. Instead the total elevation drop is composed of both the step height (z) and the elevation drop associated with the tread (Et). Thus the slope of the channel is given by
Using the definition z = H − dr and the definition of k, (10) yields
which can be rearranged to give
Equation (11) is identical to (9) if Ls = L and Et = 0. These two conditions are met if there is no tread downstream from the pool. In that case, the data of Figure 8b would be expected to plot around the mean value of k, certainly mostly within the ±1 standard deviation range. The observation that most of the points plot below mean k indicates the common presence of treads in step-pools. It is also evident that the deviation from mean k increases on lower gradients. Treads are likely to be more common on lower gradients since there is less elevation drop and thus a reduced propensity for steps to form. On high gradients, in order to accommodate the steep slope, scour pools are arranged back to back and no tread is likely to exist. The drop structures from the field plot particularly low in Figure 8b, further emphasizing the effect of treads. On average the scour pool downstream of the sills occupies only 35% of the distance between sills.
 Thus, at lower slopes, morphologies with longer treads tend to develop with less frequent step-pools. This division may signal more significant variations in the hydraulic regime, potentially dividing mild channels (submerged jets) from steep ones (transitional and skimming flow).
8. Channel Stability: The Jammed State Hypothesis
 The studies reviewed above present some evidence for the development of step-like features from antidune-like bed forms under conditions in which there is substantial sediment movement in the channel. In particular, it seems reasonable to suppose that larger stones may stall under antidunes and, on declining flow, trap other clasts, still mobile, to form a step. This process would, however, require movement of the step-forming stones over an alluvial bed and therefore maintenance of a relatively high rate of sediment transport. These requirements set gradient limits to the channels that may exhibit such phenomena.
 The traditional approach to channel stability has been to assess the mobility of individual grains on the bed and to determine the flow required to entrain the bed material. This approach is based on Shields' criterion, which suggests that stability is governed by the ratio between the applied shear stress (τ) and grain inertia (τc). It leads to the conclusion that no mobilizable sediment should be deposited in streams steeper than about 4° (7%). Shields' criterion is written as
in which τ*c (known as the Shields number) is a dimensionless expression of the critical shear force for mobilization of a grain with diameter D. For normally graded material in streams τ*c ≈ 0.045 [Komar and Li, 1988; Wilcock and Southard, 1988]. Rearranging, replacing the constants and solving for Sm, the slope at which the bed should be mobile, yields
 In steep channels with D ≈ d at high flow, the critical angle for the stability of large clasts is 4.2° (0.074 m/m) and, if D < d, the limit gradient becomes lower, yet sediment is found in channels with considerably higher gradients. Sediment must persist because of effects in addition to the weight of the stone that act to prevent mobility.
 It is proposed that sediment is present in channels steeper than 4° because the sediment is structurally arranged in the channel and is jammed. The jammed state has recently been defined formally by physicists investigating flow in granular media [Cates et al., 1998; Bocquet et al., 2002; De-Song et al., 2003]. It can be described as a state in which granular materials form oriented “force chains,” structures that strongly resist deformation or displacement under a directed force as the result of grain-on-grain structural arrangements and/or strong frictional binding. In this case, the force chain is established across the channel (rarely directly transversely) because of compression induced by the down-channel directed shear and pressure gradient forces set up by the flow when the resisting objects are trapped in an opening too narrow for their joint passage.
 While the argument that steps are controlled by boulder jams is not new [cf. Grant et al., 1990], boulder jams in steep channels have not heretofore been analogized to jammed state physics. We propose that the propensity for jamming is controlled by three dimensionless factors which are directly analogous to the factors that control conceptual jamming. First is the ratio of channel width/grain size, w/D, herein termed the jamming ratio, which plays the role of local system density. Evidence for this factor derives from field observations of the relative dimensions of step keystones and the channels in which they are found, and evidence from experimental efforts to create steps (in particular, the experience of Crowe , also reported by Curran and Wilcock ). Second is the mobilizing force of the flow, measured as τ/τc (or, equivalently, τ*/τ*c, also called the “transport stage”), a measure analogous to the “granular temperature” of the jammed system (the granular temperature may loosely be thought of as the level of kinetic energy in the system). Evidence for the importance of relatively low transport stage for jammed structures to develop is available from both field evidence and from some flume experience. With sufficiently high shear forces, steps are broken and a channel-clearing debris flood or debris flow may develop. The third factor is the ratio between sediment supply and discharge, or sediment concentration, Qs/Q, the load in the system. Again, field and flume evidence shows that steps develop at relatively low transport rates: at high transport, steps become buried or break. Each of these factors must be of relatively low magnitude for jamming, in effect, local freezing of the bed (whence the thermal analogy), to occur. In lower gradient streams research has shown that structuring of the bed leading to the formation of stone cells and stone lines can increase the critical Shields number for entrainment. The development of these structures has been shown to be related to sediment supply [Church et al., 1998; Hassan and Church, 2000]. Similar, but more pronounced structural arrangements are believed to occur in step-pool streams in association with sediment starved conditions.
 In step-pool streams there is also a large amount of spill and form resistance that consumes energy and limits the shear force applied directly to stones. If an acceptable means of partitioning grain roughness from total roughness existed, then the jammed state should be evaluated using the ratio between applied grain stress (τ′) and the critical stress to mobilize the sediment (τc). However, no means of partitioning roughness has been shown to work effectively. As an expedient for this exploratory study we set τ′ = 0.3τ, which falls within the range reported by Canovaro et al.  and by other investigators for experimental barred and stepped flows, which we assume to be analogous to boulder steps.
8.2. Evaluating the Jammed State Hypothesis
 In order to test whether or not data from the literature support the jammed state hypothesis, data on step-pool absence and presence were collected from all known sources. In order for a data set to be included reliable measures of discharge, bed slope, channel width, sediment transport and grain size were required. In total, data from 10 studies were found to be suitable (Table 1). In order to plot these data, some important differences in measurement procedures amongst the studies had to be overcome.
Table 1. Studies Used to Evaluate the Jammed State Hypothesisa
no feed, channel stabilized after sediment transport ceased.
 One challenge was the differences amongst the 10 studies in channel width measurement. In Figure 9 a section of East Creek is shown. The 2.2 m3/s water line (highest flow in 34 years of record) is noticeably wider (7.55 m) than the edge of the potentially mobile substrate (4.47 m) as a result of overbank flooding at high flows. Furthermore, considerable variations in channel width occur. It is evident that the average width at the steps (4.0 m) is smaller than the average channel width or bankfull width that is usually measured in field studies. In flume experiments reproducing step-pools, the flume width is likely most similar to the minimum width per step-pool unit that would be measured in the field, rather than the average bankfull width or average width of the mobile channel. A second major difference between studies was how grain size was measured. In general, but not exclusively, reports of flume studies have tended to give the grain size of the bulk mixture, while reports of field studies present the grain size of either the step-forming grains or the surface material. The effect of the differences in how width and grain size were measured was estimated to cause jamming ratios to vary by as much as one half of their reported value, which varied between 3 and 20. In a crude attempt to rectify these differences, field widths were divided by a factor of 1.5 to correct for the differences between flume width and the bankfull width measured in field studies. In an attempt to rectify the different grain size measurements, where the surface of the entire stream was sampled, the reported D84 was multiplied by 1.5 to estimate D84 of the step-forming stones based on the ratio observed in East Creek.
 In order to determine the total applied shear stress flow depth must be known, yet this was rarely measured in the studies that we have incorporated, and where it was measured, the techniques used varied widely. For this reason we have rearranged equation (3c) (as Du = C1(gS)1/2d3/2) and used the flow continuity equation for specific discharge to determine depth
Here C1 is a constant that is on the order of 1. Bed slope was used along with the estimate of the D50 of the bed surface stones to determine flow depth. In the studies of Lee , Crowe , and Weichert et al. , only D50 of the bulk material was reported; these values were multiplied by 1.5 to approximate the D50 size of the surface. The comparison between measured and calculated depth in the cases for which a depth is reported (Figure 10) lends some confidence that the calculations are unbiased, although the range of variation remains large.
 On the basis of the 10 studies, 188 data points were collected, of which 31 are from field studies. Table 1 lists information about the data utilized. Table S1 (in the auxiliary material) includes all the data that we were able to collect on step stability. Of the 188 data points, 53 were not associated with steps, 26 had partial steps and 109 were associated with steps. Figure 11 illustrates the presence or absence of steps in relation to the measured or estimated values of transport stage, channel jamming ratio and the sediment concentration. To make the sediment concentration nondimensional a mineral density of 2650 kg m−3 and porosity of 0.4 were used to convert the mass sediment transport rates to volumetric sediment transport rates. For experiments in which there was no sediment feed a nominal sediment transport rate of 0.001 kg/m/s was used in order to plot the data on the logarithmic sediment concentration axis. In order to calculate the critical shear stress to mobilize the bed the surface D50 was used in equation (12). We suppose that it approximately characterizes pool sediment and, since pool scour is a commonly reported reason for step failure by undermining (see above), this seems to be a relevant quantity. Furthermore, its adoption allows us to assume a value for τ*c = 0.045 (the value for the jammed stones in the steps is undoubtedly much higher).
 The surface illustrated in Figure 11 represents an attempt to separate those conditions where steps are apt to form from those conditions where steps are absent. The equation of the surface is
implying that transport stage is the more important mediator of sediment concentration and step stability. However, the surface is merely sketched by hand and is meant to illustrate the concept, rather than being a surface that definitively delimits step-pool stability. We emphasize that in order to plot the data a number of measures have had to be rather crudely estimated from other related variables. We judge that fitting a surface and analyzing the error associated with defining step stability is premature based on these results and is better left to future research in which the variables that ostensibly influence the jammed state are directly measured.
 The surface classifies 81% of the stepped channels and 85% of the nonstepped channels successfully, while 38% of the unstable channels are classified as “stepped” and 62% as “not stepped”. Still, outliers are noticeable and the data tend to be grouped. The series of stable steps that plot at very high sediment transport rates well above the sketched surface are from Koll's  experiments. In these experiments stones of different color were used for the feed and bed material and it was observed that at high feed rates the steps that existed prior to the initiation of the feed remained stable once the feeding of sediment began. Whether the bed morphology continued to have steps, or was simply buried by the mobile material is, however, unclear. These results highlight a significant challenge that persists in this analysis, namely, researchers have used a range of definitions to define when steps are present or absent. Theory-testing experiments which cover the range of observed conditions and have a common means of defining when step-pools exist will add valuable insight into the investigation of step-pool occurrence.
9. Emerging Research Questions
 While much progress has been made in recent years in research on step-pools, some important conditions remain poorly understood.
 1. The most basic question is “Under what conditions do step-pools exist?”
 First, in particular, are “step-pool” phenomena below about 7% (4°) gradient different than those on higher gradients?
 We have quoted experimental and field evidence showing that, on lower, mild gradients and at relatively high transport rates, antidune-like features occur which are possibly not anchored. There is limited scour pool development, and step-pools are possibly not continuously distributed along the channel. High flows are apt to create submerged surface jets. Above ∼7% gradient, potentially mobilizable clasts must be jammed or otherwise anchored to remain stable, which implies a clearly different mechanism for formation on steep gradients. Is such a transition systematically associated with the transition from mild to steep gradients and/or the transition from impinging jets to persistent nappe flow?
 Second, at the other extreme, what conditions lead to cascades, that is, channels with jammed steps but without channel-spanning pools, on gradients steeper than about 20% (Grant et al.  quote 22%)?
 The answer to this question may have to do with the need for a more tightly articulated clastic structure on increasingly large gradients, but the question has not been investigated.
 2. What is the nature of step-pool flow at step-forming discharges? Is flow restricted to the transitional regime with Fr ≈ 1 [cf. Grant, 1997], or does skimming flow develop? Is the high energy dissipation associated with transitional flows an essential element in the stability of step-pools?
 3. How does step height affect average velocity – that is, how does it affect the resistance to flow? An ancillary question is how should bed roughness best be characterized [cf. Aberle and Smart, 2003]?
 4. What are the controls on scour-pool formation? In particular, what is the role of the jet type, and what is the influence of variable grain size in widely graded materials?
 5. How do step-pool-tread systems function hydraulically? In particular, does sediment transport rate influence the likelihood for a tread (or berm) to develop? Conversely, what is the probability for a step to develop within the “exclusion zone” of Curran and Wilcock ?
 These questions bear on step-pool geometry, on mean flow and hydraulic geometry, and on step formation mechanisms.
 6. Do pools exhibit a sufficiently near approach to self-similarity, or to self-affinity, to allow some predictive insight for the deliberate design of step-pool sequences?
 This is an important issue related to attempts to stabilize mountain torrents. In Crowe's  experiments, over 75% of all steps are destroyed by downstream scour. Comiti's work on scour was motivated by a need to understand scour undermining the foundations of sills, which may be applied to step-pool stability. Future research efforts should address the applicability of Comiti's [2003; see also D'Agostino and Ferro, 2004; Comiti et al., 2005, 2006] results to a range of natural step-pool channels and, in particular, should seek to identify when a scour hole becomes sufficiently deep to destabilize the upstream boulders. The apparent self-affinity of scour pools [Comiti, 2003] suggests that this limit may be some multiple of the height of the keystones making up the step. The observation that at high slopes the ratio H/L/S is always <2 might represent the maximum pool depth that is compatible with pool geometry (F. Comiti, personal communication, 2006).
 7. At what flows are steps directly broken (that is, broken by direct entrainment of the step-forming stones as opposed to undermining of the step)? Are natural steps ever directly broken, except in debris flows?
 Step destruction holds clear implications for downstream bed material transport and consequent risk to aquatic habitat, structures and people, but flows that may lead to step destruction remain undefined.
 8. Does the jammed state hypothesis represent a tenable state description for step-pools? Is it appropriately represented? If so, step-pools represent a distinct class of quasi-alluvial channels.
 An apparent contradiction emerges from the results presented here. If steps are in fact destroyed mainly by downstream scour, step destruction is most likely during low sediment transport, as high transport rates limit scour depths [Whittaker, 1987; Comiti, 2003]. Conversely, high sediment transport rates in gravel bed rivers are associated with a relatively mobile and unstable bed [Dietrich et al., 1989]. In steep streams, large transport rates might also be expected to create a less stable bed as there are apt to be shallower pools and a reduction in channel form resistance leading to an increase in bed shear stresses and an amplified ability to transport sediment [Hayward, 1980; Koll et al., 2000]. So which is it? Does an increase in sediment supply increase or decrease step stability? Is there an ideal sediment supply rate to promote step stability?
 In light of the observations that step-pool hydraulics may vary depending on whether the bed has a mild or steep slope and the observation that mobile antidunes (if such they be) do not occur at slopes greater than about 7%, there is a need for more studies covering a wider range of gradients. Jet types which may occur in step-pool channels can be classified into three distinct types, namely surface and impinging submerged jets, and jets associated with free overfall nappe flow. For step-pool streams, the gradient at which each type of jet dominates has not been evaluated and may add important insight into pool maintenance and step stability. Particular attention needs to given to hydraulics and channel conditions at high flows, which have heretofore rarely been investigated in the field.
 Research on step-pool channels has advanced in a number of key areas in recent years. A much improved understanding of scour downstream of steps and understanding of possible step-pool formation and destruction phenomena have emerged largely through flume studies. Velocity profiles, roughness characteristics and hydraulic geometry within step-pool channels have been quantified. These results have increased our understanding of the range of processes and factors interacting in step-pool channels.
 Despite these results, and supporting evidence that may be drawn from a more extensive engineering literature, it remains difficult to predict or model most aspects of step-pool channels. At present a model of step-pool stability does not exist; furthermore, the ability to predict hydraulic geometry and quantify roughness characteristics based on the morphology of the step-pool channel is lacking. Finally it is not currently possible to predict sediment transport rates as a function of flow and in-channel sediment storage, both key factors regulating sediment transport in step-pool streams. These questions drive current research on step-pool streams.
 On the basis of the shortcomings of current knowledge and to provide a new conceptual model that may drive specific research investigations, a new theory governing step-pool formation has been proposed herein and a state-space diagram presented. The theory suggests that, for a restricted range of jamming ratios, transport stage and sediment concentration, steps are likely to form. Drawing on the work of Curran and Wilcock  for more details, we suggest that, once a step is formed, a hydraulic jump and subsequent scour pool occur downstream of the step. It appears that the dimensions of the scour pool may be predicted by scour formula that depend on step height [Milzow et al., 2006]. The pool is then followed by a zone where subsequent steps may form, following a probability distribution that is again determined by the local jamming ratio, Shields ratio and sediment concentration.
 Within a given reach channel gradient, width, grain size, shear forces and sediment supply rates are likely to remain roughly similar, so macroscopic properties of step-pool reaches are likely to have meaningful averages that vary from reach to reach and stream to stream and to be well correlated with gradient. Likewise, since the occurrence of a particular step depends on the local width and availability of keystones, local step to step variability may be considerable. The jammed state concept suggests that future research should more closely examine the role of the jamming ratio (ratio between channel width and grain size), transport stage and sediment supply to improve our understanding of step stability and channel form.
 We wish to thank the step-pool investigators who have contributed data and/or discussion to facilitate our analysis. These include Jochen Aberle, Joanna Curran (née Crowe), Rob Ferguson, Katinka Koll, Colin Rennie, and Roman Weichert, who generously hosted A.Z. at his laboratory. The journal's reviewers, F. Comiti, J. Curran, and A. Wilcox, each contributed a remarkably constructive review, and we especially appreciate the willingness of each of them to share with us their most recent, unpublished results. Research for this paper was supported by the Natural Sciences and Engineering Research Council of Canada through the Strategic Project “Stability of step-pool channels” (to M.C.) and through a Graduate Research Fellowship to A.Z. We also wish to thank Joshua Caulkins and Jon Tunnicliffe for assisting us with the collection of data at East Creek and Eric Leinberger for the graphics.