Form and stability of step-pool channels: Research progress

Transcription

1 Click Here for Full Article WATER RESOURCES RESEARCH, VOL. 43,, doi: /2006wr005037, 2007 Form and stability of step-pool channels: Research progress Michael Church 1 and André Zimmermann 1 Received 13 March 2006; revised 14 August 2006; accepted 19 September 2006; published 13 March [1] 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, channelspanning 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. Citation: Church, M., and A. Zimmermann (2007), Form and stability of step-pool channels: Research progress, Water Resour. Res., 43,, doi: /2006wr Introduction [2] 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. [3] 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, 1 Department of Geography, University of British Columbia, Vancouver, British Columbia, Canada. Copyright 2007 by the American Geophysical Union /07/2006WR005037$09.00 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 t/t c ), 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. [4] Chin and Wohl [2005] have recently published a general review of empirical research on step-pool occurrence and formation. Accordingly, this review more selec- 1of21

2 CHURCH AND ZIMMERMANN: RESEARCH PROGRESS ON STEP-POOL CHANNELS 2of21 tively focuses on findings that lead toward the jammed state hypothesis. 2. Characterization of Step-Pools [5] 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 [1997] quote a range 0.3 < D 84 /d <0.8,while Comiti et al. [2006] observed that D 84 /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, D 84 /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. [6] Whether step-pools constitute a distinct channel type or simply represent a channel morphological unit has been debated. Grant et al. [1990] 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 [1997] 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. [7] 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 [2002] 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. [2006] have since developed a step identifi- Figure 1. Steep stream channel morphologies. (a) A steppool unit at East Creek, in the Coast Range of British Columbia, during flood (Q = 2.2 m 3 /s, the largest recorded flow in 34-years of record). (b) Same location at low flow (Q 0.01 m 3 /s). While wood is present in the channel, the steps are composed entirely of boulders. (c) Cascade channel unit in Giveout Creek, in the interior of British Columbia. (d) Stone lines in a rapid unit of the Lainbach, Germany. (All photos by A. Zimmermann.)

3 CHURCH AND ZIMMERMANN: RESEARCH PROGRESS ON STEP-POOL CHANNELS Figure 2. Illustrations with definitions of (a) rapid channel morphology, (b) a step-pool unit with no tread between successive pools (definitions given in the downstream pool are used particularly in studies of pool scour), and (c) a step-pool unit with a tread, extended forms of which may be considered equivalent to a run. cation 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. [8] 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 [2003] and Comiti and Lenzi [2006] 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. [9] We are unaware of any investigation that has critically examined the maximum slope at which step-pools occur. Grant et al. [1990] show 40% as the upper limit of their observations of step-pools while Wohl and Grodek [1994] 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 [1980] and Grant et al. [1990], 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 3of21

4 CHURCH AND ZIMMERMANN: RESEARCH PROGRESS ON STEP-POOL CHANNELS 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. [10] The singular geometry of step-pools has attracted significant field investigation (reviewed by Chin and Wohl [2005]). 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 as b, in which 0.42 b 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 [2005]). 3. Step-Pool Hydraulics 3.1. Flow Regime [11] 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. [12] 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 steppool channels. To develop a rough estimate we start from Manning s formula, u = d 2/3 S 1/2 /n, where u is the mean Figure 3. Illustrations of hydraulics over stepped beds. In this diagram, we analogize step-pool flow to flow through a series of drop structures and sills; hence no pool is illustrated. At low stages, all channels exhibit nappe flow. 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 (S c )is given by S c gn 2 =d 1=3 c where g is the acceleration due to gravity and d c is the critical flow depth. We introduce Strickler s equation, n D 1/6, where D is some relatively large grain diameter to yield S c ¼ g 1: ð D=dc Þ 1=3 ð2þ 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. ð1þ 4of21

5 CHURCH AND ZIMMERMANN: RESEARCH PROGRESS ON STEP-POOL CHANNELS 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 = ) in Strickler s equation accordingly. This yields c = if grain resistance contributes 20% of the total ( /0.2) and c = if grain resistance contributes 40% of the total. For 0.5 D/d 1.0, this leads to 6.0% > S c > 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%.) [13] 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 [2001] determined that nappe flow occurs on sills when d c /H <1, where d c is the (critical) flow depth at the brink of the step and H is the drop height, while transition flow develops when 0.85 d c /H 1.15, approximately. If d c /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 steppools but the condition d c /H 1 roughly corresponds with the observation that D/d 1 at high flow. [14] 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 [1998] 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. [15] Herein we have attempted to combine the jet classification schemes proposed by Chanson [2001] and Wu and Rajaratnam [1996], 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 [2001] definition of nappe flow. [16] Insight into the nature of the jet in step-pool streams can be gained from the work of Lee [1998], Wohl and Thompson [2000], and Wilcox and Wohl [2007]. Velocity profiles were measured by Lee [1998] 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 [2000] 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 [2007] used an ADV to observe that flows across a range of discharges in a step-pool stream exhibited strong threedimensional turbulent kinetic energy, including a substantial vertical component. [17] 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. [18] 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 5of21

6 CHURCH AND ZIMMERMANN: RESEARCH PROGRESS ON STEP-POOL CHANNELS Figure 4. Velocity profiles measured during Lee s [1998] flume experiments over a step-pool unit. Dashed lines indicate the water surface. Where the velocity profile is not affected by the step, the profile is logarithmic (see highlighted profile A). At the top of the step, where water is accelerating into the pool and passing from subcritical to supercritical, the bottom of the profile is very steep (profile B). When the water reaches the pool, only the upper portion of the water is traveling at high velocity, and strong midprofile shear is generated (profile C). Finally, at the end of the pool, the upper portion of the profile decelerates and near bed shear resumes (profile D). Redrawn from Lee [1998, Figure 10.1]. may be restricted to high gradients and skimming flows may be exceptional. These inferences warrant further investigation, especially on steeper slopes Resistance to Flow [19] 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. [2004] 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 [2003]. [20] Lee and Ferguson [2002] 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 [2003] developed a Keulegan-type relation from experimental data collected by Rosport [1998] and Koll [2002] (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/ p f = 3.14log 10 (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 [2005] have recently shown that data from a number of field studies can be roughly collapsed onto 1/ p f = 2.74log 10 (0.48 L/H), a scaling that directly emphasizes step-pool geometry. [21] 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 = cq m, in which m is an index of resistance to flow and Q is discharge [Kellerhals, 1970, 1973; Bathurst, 1993] (reviewed by Aberle and Smart [2003]). Kellerhals and Bathurst showed that we may expect higher values of m in steeper reaches (Kellerhals actually considered the exponent in A / Q p, in which A is cross-section area and p varies 6of21

7 CHURCH AND ZIMMERMANN: RESEARCH PROGRESS ON STEP-POOL CHANNELS 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 u= ðgl h. Þ 0:5 / q g 0:5 l 1:5 i a; i S b ð3aþ which is a Froude type equation. Adopting l = s, regression of the Rosport-Koll data yielded [Aberle and Smart, 2003] u ¼ 1:06g 0:18 q 0:64 S 0:26 s 0:46 which is very similar to the rational result previously inferred by Rickenmann [1990, equation 5.20] u / g 0:2 q 0:6 S 0:2 D 0:4 90 ð3bþ which is equation (3a) with a = 0.6, b = 0.2.) The coefficient in the relation is of order 1. (Rickenmann obtained equation (3b) from an equation of Takahashi [1978] for a debris flow front and showed that it describes reasonably well experimental data of step-pool flows taken by Smart and Jäggi [1983] on gradients 0.05 S 0.20.) [22] Equation (3b) can be rewritten as u/(gd) 1/2 / S 1/2 (d/d) which, in turn, can be rearranged to give u / ðgdsþ 1=2 ðd=dþ 1 ð3cþ This is simply the Chezy equation adjusted by a measure of relative roughness. [23] Giménez-Curto and Corniero Lera [1996] have distinguished the jet flow regime in which flow separates over high boundary roughness. For such flows they determined that f = 0.52(ghHisinb/u 2 ) 2/3, where hhi is the mean obstacle height, specifically equated with step height of Giménez-Curto and Corniero [2006]. f G = gds/u 2 is the friction coefficient (the familiar Darcy-Weisbach friction coefficient is 8f G,asf G is defined here) and b is the channel slope, so sinb S. Substitution for f G again leads to equation (3c) provided D = hhi. We will carry this discussion forward in section 7. [24] 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. [25] 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 [1987], Robert [1988, 1990, 1991], Nikora et al. [1998], and Marion et al. [2003]. [26] Aberle and Smart s [2003] results derived from experimental runs that were all completed at near bedmobilizing 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. 4. Formation of Step-Pools [27] Some of the first research on step-pools was concerned with how step-pools form [Judd, 1963] and this continues to be a topic of much research [Lee, 1998; Comiti, 2003; Curran and Wilcock, 2005]. While it is generally accepted that step height is governed by the grain size of the step-forming clasts [Judd, 1963; McDonald and Day, 1978; Whittaker and Jaeggi, 1982; Allen, 1983; Wohl et al., 1997; Lee, 1998; Chin, 1999; Chartrand and Whiting, 2000], the controls on step length remain widely debated. Central to many theories is that large, immobile clasts form anchor points against which other stones become imbricated and stop moving. This yields an initial step. The controls on the formation of additional steps are, however, greatly contested; two general schools of thought have emerged. Following hydraulic tradition, step-pools have been supposed to be controlled by the flow field [Whittaker and Jaeggi, 1982; Grant, 1994; Comiti et al., 2005] but, more recently, it has been claimed that the chance location of steps depends on both the flow and the location of keystones within the channel [Lee, 1998; Zimmermann and Church, 2001; Crowe, 2002; Curran and Wilcock, 2005]. [28] 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 [1963] (S < 5.6%) suggested that the initial step creates standing waves downstream and subsequent steps form under these waves. Conversely, McDonald and Day [1978] suggested that step-pools form when a down- 7of21

8 CHURCH AND ZIMMERMANN: RESEARCH PROGRESS ON STEP-POOL CHANNELS stream 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 [1983] 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. [29] 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. [30] In contrast to the hydraulic approach, Zimmermann and Church [2001] 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 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 [1998] 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. [31] In their experiments with mm grains, Curran and Wilcock [2005] 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 Figure 5. Histogram of measured step spacing for steps formed through the obstacle and exhumation processes. A Poisson distribution is fitted to the data. Numbers on the abscissa indicate the lower value of each 10 cm spacing bin. From Curran and Wilcock [2005]. The second abscissa scales the data by L. Similar distributions have been shown from field studies by Zimmermann and Church [2001] and by Milzow [2004]. 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). [32] 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 [2005] for those steps formed through the obstacle and exhumation processes (first and fourth processes described above). Milzow et al. [2006] examined the morphology of a 17% gradient steppool 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. [33] Surprisingly, when the largest stone in Curran s experiments was 45 mm, steps did not form, yet when w/d 84 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/d 84. Evidence that the 8of21

9 CHURCH AND ZIMMERMANN: RESEARCH PROGRESS ON STEP-POOL CHANNELS entire grain size distribution affects the formation of steps comes from Tatsuzawa et al. [1999], who performed flume experiments with three sediment mixtures. All mixtures had the same D max 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. [34] An extensive set of flume experiments by Rosport [1994], Rosport and Dittrich [1995], and Koll [2002] showed that high flows wash out steps and create a morphology similar to riffle pools with shorter, irregular barriers. Maxwell [2000] performed similar experiments which showed that bed stability also depends on bed slope. Using sketches of the bed structure, Weichert et al. [2004] 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 t/t c, the applied shear stress compared with the critical shear stress for particle entrainment), in addition to the ratio w/d 84, influences the stability of steppool 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 ( m 2 /s), all sizes in the mixture were readily transported. [35] 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 [1982] 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, antidunelike 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 [36] Sediment supply has been shown to affect the stability of step-pool streams. De Jong [1995] 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 ], 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... [37] 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 [2001] 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. [2006] 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 years were responsible for macroscale changes in channel form. [38] 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 [1980] 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 [1990] 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 [1982], 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. 9of21

10 CHURCH AND ZIMMERMANN: RESEARCH PROGRESS ON STEP-POOL CHANNELS [39] 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. [40] 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 [2001] Step Destruction [41] In her experiments [Crowe, 2002], Curran observed four principal methods by which steps collapsed. Seventyseven 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 [1995] likewise noted that step destruction occurs when the hydraulic jump erodes material from the base of the step. Lee [1998] observed that steps tended to fail when the largest stones protruded prominently into the flow, suggesting toppling of the keystone. [42] 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 [1992], De Jong [1995], and Gintz et al. [1996]). The failure of the steps in Rio Cordon occurred during a flash flood that featured near-hyperconcentrated characteristics [Lenzi et al. 2006]. [43] 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 [44] 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 [2003] and by Milzow et al [2006] 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.) Figure 6. Affinity in scour holes from laboratory studies at Padova, Italy [Comiti, 2003]. Solid circles refer to experiments S4-M1-M2 (spacing of sills = L = 0.5 m), open circles refer to experiments S2 S3 (L = 0.75 m), and crosses refer to experiments S1-M3-M4 (L = 1.5 m). Distances beyond 1.0 indicate the presence of a tread (see Figure 2c). also emphasize the need to understand downstream scour. Comiti [2003], following results of Gaudio et al. [2000], 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 (y s ; equivalent to step height H) and pool length (L s ), respectively are constant (Figure 6; variables defined in Figure 2). Bennett [1999] 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. [45] Scour depth in plunge pools has been investigated for over 75 years, the motivation being to predict scour on natural streambeds or constructed aprons downstream from dam overfalls and other engineered free drops (reviewed by Mason and Arumugam [1985] and D Agostino and Ferro [2004]; see also Chanson [2001]). Interest in scour below culvert outfalls has also given rise to a number of experimental investigations (reviewed by Doehring and Abt [1994]). Mason and Arumugam [1985] arrived at a rational relation for individual drop structures (see Figure 2 for definitions):. d s ¼ 3:27q 0:60 h 0:05 dt 0:15 g 0:30 D 0:10 The formula was calibrated against data of model drop structures but, inasmuch as these data extended to q = 0.42 m 2 s 1, H = 2.15 m, and D m = 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 [1989] 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 ð4þ 10 of 21