WATER RESOURCES RESEARCH, VOL. 48, W07505, doi:10.1029/2011WR011436, 2012
Investigating step-pool sequence stability
Kevin A. Waters1 and Joanna Crowe Curran1
Received 22 September 2011; revised 17 May 2012; accepted 21 May 2012; published 7 July 2012.
[1] Step-pool units, common features in steep, narrow streams, are highly dynamic systems
that adjust during high flows and active sediment transport conditions. Consecutive step
pools in a reach form a step-pool sequence, and though these features are prevalent in
nature, quantifying the stability of such systems is challenging. This study focuses on the
statistical relationships between 445 stable sequences of three or more steps and nine
geometric and resistance-based parameters. Step sequence stability is parameterized through
a sequence stability parameter, a metric of how long a given step-pool sequence exists
relative to the total run time. Despite variability in the strength of the statistical
relationships, the results allow identification of dominant trends between sequence stability
and both roughness and geometric parameters. The sediment concentration ratio provides a
means for comparing the relative strength of these statistical relationships. As sediment
concentration ratios increase, the relations between flow resistance parameters and step
stability change from dominantly inverse to dominantly direct. Thus, at higher sediment
concentration ratios, sequence stability is more likely to increase with flow resistance, but
the reverse is more likely at low sediment concentration ratios. A connection exists between
bed morphology, flow resistance, and stability. For the majority of stable step sequences,
resistance parameters increase over time, indicating some sequences adjust their geometry to
increase stability during flood events. The influence of sediment concentration over the
stability-geometric and stability-resistance parameter relationships may enable the use of the
sediment concentration ratio as a predictor of step sequence stability during a flow event.
Citation: Waters, K. A., and J. C. Curran (2012), Investigating step-pool sequence stability, Water Resour. Res., 48, W07505,
doi:10.1029/2011WR011436.
1.
Introduction
[2] The step-pool bed form, a common feature in steep,
narrow channels, is composed of a series of steps and pools
(Figure 1). The step is formed by a number of the largest
clasts in the channel aligned transverse to the flow, creating
the step riser. A pool immediately downstream of the step
is scoured by flow tumbling over the upstream step and is
followed by the step ‘‘tread.’’ Flow becomes supercritical
as it approaches the step crest and then tumbles into the
downstream pool; a hydraulic jump returns the flow to subcritical, and flow becomes supercritical as it accelerates
over the next step crest. Step sequences consist of a sequential number of steps and pools within a single river reach
[Comiti and Mao, 2012]. The parameters associated with
the stability of sequences composed of three or more steps
and pools are the focus of this research.
[3] Step-pool systems are common worldwide and have
been reported in a number of different settings. Bedrock
1
Department of Civil and Environmental Engineering, University of
Virginia, Charlottesville, Virginia, USA.
Corresponding author: J. C. Curran, Department of Civil and Environmental Engineering, University of Virginia, PO Box 400742, Thornton
Hall B228, 351 McCormick Ave., Charlottesville, VA 22904, USA.
([email protected])
©2012. American Geophysical Union. All Rights Reserved.
0043-1397/12/2011WR011436
step systems have developed in such diverse settings as
Israel and the Oregon High Cascades [Bowman, 1977;
Duckson and Duckson, 2001; Pasternack et al., 2006;
Wohl and Grodek, 1994]. Step-pool systems are also
reported in heavily forested watersheds in the western
United States, where large woody debris contributes to step
formation [e.g., Wohl et al., 1997]. The more common step
system is that formed from alluvium [Chartrand et al.,
2011; Comiti et al., 2005; Gomi et al., 2003; Milzow et al.,
2006; Molnar et al., 2010; Zimmermann and Church,
2001]. These are depositional step systems, where the step
forming material derives from the local channel and can be
mobilized in large flows. Only depositional, alluvial step
systems are considered here.
[4] The geometry of the step-pool sequence is defined by
characteristics of the steps in the sequence. One element
essential to alluvial step formation is a heterogeneous grain
size distribution that includes a large grain that can act as a
step forming clast [Comiti et al., 2005; Curran and Wilcock,
2005; Weichert et al., 2008; Zimmermann, 2009]. The size
of the step forming grain must be large enough so that a
small number of grains can span the channel width to form a
step. This has been verified by both flume and field experiments, and is a component of the jammed state hypothesis
[Church and Zimmermann, 2007]. Step height scales with
the size of the large, step-forming grain [Abrahams et al.,
1995; Curran and Wilcock, 2005; Grant et al., 1990; Judd
and Peterson, 1969; Lenzi, 2001; Zimmermann, 2009], but
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Figure 1. Definition diagram of a step-pool sequence under high-flow conditions. The bed slope is
indicated by and was calculated over the entire step-pool reach.
does not necessarily equal the total hydraulic loss at the step.
Hydraulic losses at steps derive from nappe flow passing
over the step crest and turbulence generated in the downstream pool. Some researchers separate the height parameter
into a pool depth and a step height so that hydraulic losses
are considered individually. For example, Chartrand et al.
[2011] define step height as the difference in elevation
between successive steps and the pool depth as the elevation
difference from the deepest point in the pool to the elevation
of the downstream step crest. Comiti et al. [2005, 2009]
define the step height as the vertical distance from the step
crest to the deepest point or maximum scour depth in the
downstream pool, which is equivalent to the step height definition used by Abrahams et al. [1995]. Because this distance
represents the largest possible vertical drop for flow passing
over a step, we define it as the maximum step height, H
(Figure 1). Step spacing, L, is defined as the distance
between the crests of sequential steps.
[5] Step sequence formation typically occurs during
floods with estimated return intervals of 30 years or more
and high associated sediment transport rates [Lenzi, 2001;
Mao and Lenzi, 2007; Molnar et al., 2010; Turowski et al.,
2009]. In a few areas debris flows and smaller floods that
created high rates of sediment influx and transport have
also created step sequences [Milzow et al., 2006; Sawada
et al., 1983], and large flows without significant sediment
transport have also been shown to mobilize steps [Rosport,
1994; Rosport and Dittrich, 1995]. Individual steps within
a sequence can shift their locations after initial deposition
in response to flows lower than the step setting flow. A
shift, or migration, is an adjustment of the position of the maximum step height within the area defined by the exclusion
zone [Curran, 2012; Recking et al., 2012]. Steps are able to
shift location by the erosion or deposition of sediments
smaller than the step forming grain size on the stoss and lee
sides of the step. Each time an individual step shifts, the spacing of that step relative to its upstream and downstream neighbor steps is affected, as is the mean spacing of the step
sequence. Not all steps experience shifts in their location, and
those steps that do not migrate are eroded more quickly. Alterations in bed morphology in response to migration of the step
forming grain have been observed in multiple flume experiments [Curran, 2012; Rosport and Dittrich, 1995] (see also
the work of Bacchi as discussed by Recking et al. [2012]).
Field evidence for step migration comes from Turowski et al.
[2009], who report on a series of floods in the Erlenbach for
which over 90% of the steps were broken and rearranged in
two floods but in a third, which had a lower discharge, only
30% of the steps rearranged. Individual steps within sequences
exhibited different degrees of stability such that during a single event, some steps were quickly destroyed, some were able
to shift in location on the bed surface, and others were stable
immediately upon deposition.
[6] Individual step stability and overall step sequence stability are necessarily linked topics. Step sequence stability
refers to the maintenance of the spacing between steps in a
sequence such that a stable step sequence has a constant
spacing over time. An individual step may be eroded and
replaced by a new step, but if the new step is in a location
that maintains the same spacing relative to the upstream and
downstream steps, the step sequence has not been significantly altered. Thus, individual step stability can affect the
sequence of steps over a reach but does not necessarily do
so. Hypothesized to be necessary for step sequence stability
is a step spacing that maximizes overall channel flow resistance [Canovaro and Solari, 2007; Weichert et al., 2008].
This hypothesis has its roots in experiments by Abrahams
et al. [1995], who fit data from 12 flume experiments and 18
step-pool configurations in New York and England to the following relationship: 1 H/L/S 2, where H is step height,
L is step spacing, and S is channel slope. Step-pool sequences
best fit this relation when the distance between steps was
equal to the length of the downstream pool [Abrahams et al.,
1995]. More recent data from flume experiments have shown
steps form with a minimum spacing set by the exclusion
zone, which is larger than the length of the associated pool
[Curran and Wilcock, 2005; Gimenez-Curto and Corniero,
2003]. Many steps were spaced farther apart than the minimum spacing set by the exclusion zone, and thus did not fit
the Abrahams relation, despite the wide parameter range possible. As a result, research continues to explore the parameters responsible for step sequence stability, including the role
and function of the pool [Comiti et al., 2005; Zimmermann
and Church, 2001].
[7] While all steps in a step-pool sequence contribute to
channel roughness, only stable steps can contribute in a
predictable manner. If the number and spacing of steps frequently varies because of step instability and individual
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step destruction, reliable estimates of total flow resistance
and sediment transport rates through the channel during
high flows will be difficult to develop. When the step
sequence exhibits a high level of stability, more accurate
reach-averaged roughness estimates become possible.
Because sediment transport rates have been shown to
increase in the years immediately following a step destabilizing flow [Lenzi et al., 2004; Turowski et al., 2009], this
type of predictability would prove useful. Field research
has focused on the interaction of a stable step sequence
with channel hydraulics [Comiti and Lenzi, 2006; Comiti
et al., 2009; Wilcox et al., 2006, 2011] and sediment transport rates [Egashira, 1988; Lenzi, 2002; Mao and Lenzi,
2007; Marion, 2001; Whittaker, 1987] but has not clearly
defined the factors responsible for creating and maintaining
a stable sequence.
[8] This paper evaluates geometric and resistance parameters and varying sediment supply for their contribution to
alluvial step sequence stability. A series of flume experiments testing four discharge rates against five sediment
transport rates provided an opportunity to document step
formation and destruction processes, changes in step spacing, and step sequence stability for 445 sequences of three
or more steps each. Previously, these data were examined
to test hypotheses related to step spacing, individual step
formation processes, and the stability of individual steps
[Curran and Wilcock, 2005; Curran, 2007, 2012]. The data
are now reexamined and the step sequence stability is
investigated for its relation to geometric and resistancebased sequence characteristics as well as the influence of
sediment concentration ratio on stability.
2.
Methods
2.1. Experimental Setup
[9] The conditions under which step-pool sequences
form and break are too infrequent for a field study that
would define the stability of hundreds of step sequences
over a range of flow and sediment supply conditions. A
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useful alternative is to perform a series of step-forming
experiments in a laboratory flume. Laboratory experiments
not only allow for control of the flow and sediment transport conditions, but also provide the opportunity to observe
the formation of many step-pool sequences directly. In
these flume runs the sediment bed was fully mobile and
steps continued to form and break throughout each run, creating hundreds of different step sequences.
[10] Data for this study were collected during experiments described in detail by Curran and Wilcock [2005],
Curran [2007], and Crowe [2002], and are summarized
here. Experiments were conducted in a small tilting flume
of 0.15 m width, 0.3 m depth, and 5.2 m length, with 3.5 m
working length. The flume walls were clear acrylic, allowing direct observation of the transport and bed forms. Water
was recirculated and sediment was fed into the upstream
end of the flume. In total, seventeen flume runs were completed, testing combinations of four discharge rates against
five sediment feed rates. Sediment feed rates (Qs) spanned
an order of magnitude from 110 to 1250 g m1 s1 and
were constant for each flume run. Discharge (Q), which
was also held constant for the duration of each run, varied
from 0.0046 to 0.0065 m3 s1. This range of flow rates was
selected in order to ensure a highly mobile system and
maximize the number of unique step formations over time.
Our interest in these experiments was to determine the factors responsible for creating step sequences which could
remain stable despite the high flows and shear stresses that
would otherwise mobilize the step forming grains. Thirteen
of the 17 flume runs were analyzed for the purposes of this
study. Those runs not included had equilibrium run times,
defined by equal feed and transport rates of the largest grain
size of sediment, of less than 30 min, creating data sets restricted in length. Measured and calculated experimental
parameters for the 13 flume runs are shown in Table 1.
[11] The same sediment was fed into the flume as was
used to create the initial sediment bed. Sediment was
coarsely graded, with a grain size distribution extending
from 0.5 mm to 64 mm. Constant sediment distributions
Table 1. Summary of Measured and Calculated Experimental Parameters
Run
Run Timea (min)
Qb (m3 s1)
Qsc (gm1 s1)
Qs/Qd
Average he (cm)
Average Rf (cm)
Average Sg
Average 84 h
1B
2B
7B
8
11
12B
15
16
17B
18
20
22
23
130
124
97
92
57
99
46
46
98
84
68
98
131
0.0046
0.0050
0.0050
0.0055
0.0046
0.0050
0.0065
0.0046
0.0050
0.0055
0.0065
0.0050
0.0055
110
110
475
475
750
750
750
1000
1000
1000
1000
1250
1250
1.35
1.25
5.38
4.89
9.23
8.49
6.53
12.31
11.32
10.29
8.71
14.15
12.86
6.1
6.2
5.7
5.9
5.5
5.9
6.5
5.7
5.6
6.3
5.8
6.2
6.1
3.35
3.38
3.22
3.31
3.16
3.28
3.48
3.22
3.19
3.41
3.27
3.40
3.35
0.095
0.081
0.068
0.070
0.080
0.080
0.059
0.076
0.079
0.067
0.083
0.078
0.063
0.101
0.116
0.153
0.159
0.143
0.135
0.162
0.131
0.159
0.134
0.245
0.107
0.144
a
Equilibrium run time, defined by equal feed and transport rates of largest grain size of sediment.
Discharge, as measured by a mercury manometer.
Sediment feed rate.
d
Dimensionless sediment concentration ratio, as defined by Church and Zimmerman [2007].
e
Average water depth, calculated from video/photo analysis.
f
Average hydraulic radius, calculated as flow area divided by wetted perimeter.
g
Average bed slope, calculated using video/photo analysis.
h
Average dimensionless shear stress for the D84 grain size.
b
c
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were ensured through sieving bulk samples prior to each
run. Sediment characteristics included a D50 of 14 mm,
where D50 is defined as the grain size for which 50% of the
sediment is finer by weight; D84 of 38.6 mm, where D84 is
defined as the grain size for which 84% of the sediment is
finer by weight; 7.4% sand, defined as sediment between
0.5 and 2.0 mm; 8.3% in the 45–64 mm size class. Sediment sizes were selected to ensure a distribution that would
promote jamming of larger grains in the narrow flume and
the formation of step sequences. The ratio of step-forming
grain size, DSFG, to flume width was consistent with previous flume and field observations where step-pool formations were observed [see Curran, 2007]. Zimmermann et al.
[2010] hypothesized that the ratio of channel width, b, to
step-forming grain size has a critical role in creating stable
steps, as steps are more likely to form and remain stable
where the channel is narrow. Because of our use of a constant grain size distribution in a constant width flume, we
tested only one ratio of b/DSFG. Thus, the jammed state hypothesis was not addressed in these experiments.
2.2. Experimental Measurements
[12] Direct measurements of flow depth and bed elevation were made possible through the use of video analysis.
Every run was recorded and a mirror on the wall behind the
flume enabled measurements of both sides of the flume. A
photo was taken from the video for each minute of the
equilibrium portion of each run. Bed and water surface elevations were measured over both the near and far sides of
the flume every 3 cm in the downstream direction using a
grid overlay on each image. These measurements were
taken every 2 min of flume run time. A 2 min record of
flow depth along the length of the flume was calculated
from the measured bed and water surface elevations.
Although there was variability in the water depth between
steps and pools, this was minimized by the application of
high flows simulating floods. When the full flow record for
all the runs was examined, the average flow depth was
0.066 m with a standard deviation of 0.006 m. Bed slope
was calculated as sin (Figure 1). One minute values were
interpolated between the measured values. Reach average
water surface slope, bed slope, and energy slope were calculated using values measured from the images. Discharge
was measured using a mercury manometer and velocity
calculated from continuity using the measured water depth.
Step heights were also measured from these photos. The
depth of turbulent mixing within a pool was observed to
vary during the experiments with fluctuations in the local
velocity and sediment dynamics. Thus, our measure of step
height may over or under estimate total head loss by a
small amount in any given minute. By measuring over the
life of each step this error was kept to a minimum.
[13] The dynamic nature of the steps made the video
analysis critical in our determination of step sequence stability. The videos allowed us to create a continuous record
of step formation, migration, and destruction for each
experiment. For example, all step migrations were verified
from the videos assuring that each migration was movement of the same step and not the destruction and formation
of a new step. Step locations were tracked continuously
and documented across the length of the flume from the
videos. Step sequence formation and the number of steps
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comprising each step sequence were documented from this
record. Step spacing was also measured from the video
analysis.
2.3. Sequence Stability Parameter
[14] Difficulty in analyzing step-pool stability has arisen
in part because of the lack of an overall quantitative measure of stability. In order to measure step-pool sequence
stability in a way in which it can be compared with flow
conditions and reach characteristics, we define a sequence
stability parameter, SS . This parameter expresses steppool sequence stability as the cumulative existence time of
that sequence, divided by the total equilibrium transport
run time for a given combination of flow and sediment
transport rates :
SSi ¼
Ei
Tj
(1)
where SS is the dimensionless sequence stability parameter for sequence i, Ei is the cumulative time in minutes that
sequence i exists, and Tj is the total equilibrium run time in
minutes for run j. By providing a dimensionless measure of
stability, this parameter can be applied to compare degrees
of sequence stability across runs of varying duration. As an
example of the parameter’s calculation, it was observed
that sequence 3 in run 23 existed for 4 min. Applying
equation (1) above with the total equilibrium run time of
131 min for run 23 (Table 1), the corresponding sequence
stability parameter was calculated as follows:
SS ¼
4 min
¼ 0:0305
131 min
The stability parameter essentially provides a metric of
how long a given step-pool sequence, defined as the same
number of steps with established step spacing, existed relative to the total run time. To account for step migrations, a
step was considered stable in its position so long as any
movements did not exceed the size of the exclusion zone,
defined as 30 cm for these experiments [Curran and
Wilcock, 2005]. Step movement within this zone was
regarded as a rearrangement or migration of the same step
during calculation of the stability parameter, and thus, was
considered to exist as part of the same sequence. When a
step migrated a distance greater than the exclusion zone
length, it was considered a new step, forming a new step
sequence. A new stability parameter was calculated for each
successive step-pool sequence in every run.
[15] During periods with step sequences present, reach
parameters were calculated at 1 min intervals for the existence of each sequence of three or more steps. Calculated
parameters were then averaged over the duration of the
sequence to obtain the sequence average values used in our
analysis. Changes in the stability parameter reflected successive changes to the system as a whole, encompassing
individual step breakups and new step formations that
resulted in new sequence development. Therefore, the stability parameter provided a means to evaluate dynamic
shifts between stable and unstable periods within the system, as well as the overall distribution of sequences displaying varying levels of stability.
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2.4. Roughness
[16] Characterizing the flow resistance in a gravel bed
channel remains challenging. The traditional flow resistance equations, the Chezy, Manning, and Darcy-Weisbach
equations, calculate flow resistance through a combination
of flow depth, energy slope, and the acceleration of gravity.
In narrow channels, the hydraulic radius is typically applied
in place of flow depth. Application of these equations over
many years to a range of alluvial channels has shown that
they are most reliable when the relative flow depth of the
channel, as measured by the ratio of hydraulic radius to
characteristic bed roughness size, is large [Bathurst, 1978;
Bathurst et al., 1982; Ferguson, 2007; Rickenmann and
Recking, 2011; Robert, 2011; Thorne and Zevenbergen,
1985] and the channel bed slope is moderate or low [Aberle
et al., 1999; Jarrett, 1984; Marcus et al., 1992; Rosport,
1997; Smart, 1984; Suszka, 1991].
[17] Modifications to the traditional equations become
necessary where large bed forms are present and the channel slope is steep, as hydraulic and sediment transport processes vary greatly between low-gradient and steep-gradient
channels [Chiari and Rickenmann, 2011; Comiti et al.,
2009; Rickenmann, 2012]. Quantifying flow resistance in
steeper streams can be particularly difficult when large
roughness elements such as steps are present [Comiti et al.,
2009; Rickenmann and Recking, 2011; Zimmermann,
2010], complicating efforts to model roughness over a steppool sequence [Comiti et al., 2009]. A number of studies
have evaluated existing flow resistance equations and either
adjusted these equations or derived new models specific to
steep channels with large bed roughness [Canovaro and
Solari, 2007; Comiti et al., 2009; Egashira and Ashida,
1991; Ferguson, 2007; Lee and Ferguson, 2002; Nitsche
et al., 2011; Yager et al., 2012; Zimmermann, 2010]. This
work has been carried out in both flume [e.g., Wilcox et al.,
2006; Zimmermann, 2010] and field [e.g., Comiti et al.,
2007; David et al., 2011; Yager et al., 2012] studies where
methodologies include modifying traditional roughness
approaches [e.g., Comiti et al., 2009; Lee and Ferguson,
2002] or developing hydraulic geometry relationships to
model roughness [e.g., Reid and Hickin, 2008; Zimmermann,
2010; Rickenmann and Recking, 2011].
[18] Total flow resistance is often separated into its component parts, and in these experiments the difference
between resistance generated by the step bed form (macroroughness) and the total flow resistance is of interest. Here
we use macroroughness to include resistance due to the
step bed form and spill resistance over the step, while the
roughness of the bed surface without step forms comprises
the base resistance, similar to the work of Rickenmann and
Recking [2011]. Base resistance as a fraction of total flow
resistance has been shown to be small in steep channels
with significant bed forms [Chiari and Rickenmann, 2011;
Ferguson, 2007; Reid and Hickin, 2008; Rickenmann,
2012]. In step-pool sequences flow resistance has been considered to be dominated by bed form roughness and spill
resistance, together comprising macroroughness, with the
base resistance a small component in comparison [Church
and Zimmermann, 2007; Wilcox et al., 2006]. In contrast,
in a recent flume study where the flow depth over the step
grains was low, Zimmermann [2010] found that partitioning
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was not applicable and that base resistance was the majority of the total flow resistance. For our experiments, it was
necessary to apply flow resistance partitioning and separate
macroroughness and base resistance from total flow resistance to evaluate the importance of macroroughness to step
formation and step sequence stability.
[19] In choosing a flow resistance equation for use with
our data, we considered the traditional Darcy-Weisbach
and Manning approaches as well as those based on DarcyWeisbach and Manning resistance equations, models integrating a logarithmic velocity profile, and those applying a
power law associated with at-a-station hydraulic geometry.
Although we were able to calculate the Darcy-Weisbach
friction factor directly from our measurements, the experimental conditions precluded use of this method. The steppool channels studied had ratios of hydraulic radius to step
forming grain size that classified the flows as shallow, making a roughness layer approach for calculating the friction
factor more applicable [Ferguson, 2007]. We found the
most appropriate models for our data were those of the
variable power equation from Ferguson [2007], as presented by Rickenmann and Recking [2011] and Egashira
and Ashida [1991], both of whom derived resistance equations specific for step-pool channels. Each model was evaluated on the basis of its ability to reproduce the reach
average flow velocities measured during the experiments.
The models gave median velocity residuals of 0.19 and
0.34 m s1, respectively, making the model derived by
Rickenmann and Recking [2011] a better fit for our data set.
Rickenmann and Recking [2011] evaluated an extensive
field data set that included channels with step-pool bed
forms, steep slopes, and low relative depths, and using the
variable power equation from Ferguson [2007] developed
flow resistance partitioning equations applicable to the
step-pool channel conditions. To calculate the total flow
resistance, we applied equation (2) to our data, where
equation (2) is equivalent to equation (10a) of Rickenmann
and Recking [2011], with the exception that we use hydraulic radius in place of flow depth because of the low ratio
of flow width to depth ratio, which averaged 2.3 in our
experiments:
R
sffiffiffiffiffiffi
6:5 2:5 8
D84
¼ sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
5=3ffi
ftot
R
6:52 2:52 D84
(2)
where ftot is the total flow resistance, R is hydraulic radius,
and D84 is the grain size for which 84% of the bed is finer
(equal to 38.6 mm for these experiments). Hydraulic radius
was calculated as the flow area divided by the wetted perimeter. Because the channel width was constant at 0.15m,
the hydraulic radius varied with the flow depth. The portion
of total flow velocity corresponding to base resistance is
Uo, found using equation (20b) of Rickenmann and Recking
[2011], which is
pffiffiffiffiffiffiffiffiffiffi
U0 ¼ 6:5 gRSe
R
D84
0:167
(3)
where Se is the energy slope. The flow resistance due to
base roughness was then calculated using equation (4), the
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values from equations (2) and (3), and the total flow velocity, Utot.
f0 ¼ ftot
2
Utot
Uo2
(4)
Macroroughness ( fmac) was calculated as the difference
between the total flow resistance and the resistance due to
base roughness.
[20] Some authors [e.g., Yager et al., 2012] have used
shear stress calculations to interpret sediment movement in
step-pool channels. Both shear stress and friction factor are
measures of the frictional resistance acting in the channel and
each applies the assumption that frictional flow retardation is
balanced by the downslope component of the gravitational
forces acting on the water. We include
the dimensionless
shear stress acting on the D84 grain size 84 in our analysis
to enable comparison with the work of other studies. However, we focus on the friction factor rather than shear stress
because it presents a nondimensional value of the shear stress
acting on the boundary of an open channel [Middleton and
Southard, 1984], and can be interpreted as a drag coefficient
if resistance is assumed equal to the gravitational drag force
acting per unit boundary area and proportional to the square
of the velocity [Ferguson, 2007].
[21] The resistance parameters included in this study are
total flow resistance, total shear stress, macroroughness, and
the ratio of macroroughness to total flow resistance. The experimental conditions included constant discharge values for
each run and flow rates during the experiments that ensured
full bed mobility and shear stresses much larger than critical
shear values. Experiments were designed to study steps and
step sequences under the step mobilizing conditions of a
large flood. Therefore, neither critical shear stress nor stream
power was explicitly included in our analyses. The analysis
instead focused on changes in the step sequence stability parameter, how it was impacted by both geometric and resistance parameters, and how step sequences adjusted to
maintain stability under extreme flood conditions.
2.5. Statistical Analysis
[22] No a priori assumptions were made about the statistical distribution of the step-pool sequence data collected,
making nonparametric tests [Siegel, 1957] the most suitable
statistical approach for analyses. The primary statistical
measure applied in this study was Kendall’s correlation
coefficient modified to account for the presence of ties,
i.e., combinations of variables that occur more than once
within a sample of paired observation data [Helsel and
Hirsch, 2002]. The modified coefficient, known as Kendall’s b, determines the strength of a monotonic association between two variables arranged categorically on an
ordinal scale. Because calculation of this coefficient was
based on ranks of paired data, it was generally resistant to
outliers within the data set. Calculation of Kendall’s b
involved counting the number of observed data pairs
ranked higher or lower than a predetermined combination
of two variables. For a more detailed description of this statistical measure and the calculation procedure, the reader is
directed to the work of Helsel and Hirsch [2002].
[23] To analyze step-pool sequence stability using Kendall’s b, the median of the calculated stability parameters
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for each run was used as a partitioning point to define low
and high sequence stability. Median SS was selected
instead of run-averaged SS values because median values
presented a more representative division point for stability
given the tendency toward lower SS values observed in the
data set. Values of SS above and below the median value
for a given run were considered high- and low-stability
sequences, respectively. High and low values of sequenceaveraged parameters were defined in the same way, using
the median variable value of all sequences for each run as a
dividing point. Consequently, data pairs could be classified
as a combination of either high or low values of two parameters and categorized on the basis of the ordinal scale necessary to calculate Kendall’s b. These variables were paired
with the corresponding values of low or high stability and
organized into categories for each experimental run. A
Kendall’s b coefficient was then calculated on the basis of
the four possible combinations of SS for the parameter in
question. For example, the association between stability and
step spacing was analyzed from the observed frequencies of
paired data contained within the following categories: low
stability, low spacing; low stability, high spacing; high stability, low spacing; and high stability, high spacing.
[24] Sequence stability was statistically tested against
two groups of variables, designated as geometric and
roughness parameters, respectively. In addition to sequence
step spacing, the geometric group of tested parameters
included number of steps in the sequence (#steps), average
sequence step height (H), step steepness (H/L), and ratio of
step steepness to bed slope (H/L/S). The group of tested
roughness parameters included total roughness ( ftot), macroroughness ( fmac), fraction of macroroughness to total
roughness ( fmac/ftot) hereafter referred to as the macro roughness fraction, and dimensionless shear stress 84 .
Altogether, SS was analyzed against nine different parameters, including geometric and resistance-related quantities
in either dimensional or dimensionless form, in order to
determine correlations between sequence stability and various sequence characteristics.
3.
Results
3.1. Sequence Stability Parameter
[25] The experimental flow and sediment feed conditions
ensured an active, mobile bed, designed to create a large
number of steps. Distinct step sequences formed as steps
developed and broke, creating between 20 and 64 step
sequences per run (Table 2) that remained stable for
between 1 and 19 min (Figure 2a). As mentioned in section
2.1, the experimental conditions were designed to maximize the number of unique step formations and step
sequences over time by generating shear stresses within a
range sufficient to mobilize the step forming grains. To
evaluate the ability of the flow to mobilize the large, stepforming grains, the critical shear stress, c , was compared
to the average total shear stress applied during each experiment. We used two methods for calculating critical shear
stresses for the D84 grain size, c 84 . From the Wilcock and
Crowe [2003] model we found c 84 ¼ 0.022. Because steep
slopes are not accounted for in the Wilcock and Crowe
model, we also applied the method outlined by Lamb et al.
[2008]. The proportion of the total stress attributable to the
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Table 2. Stability Parameter Results
Run
Number of
Sequences
Run Average SS
Median SS
Maximum SS
75th Percentile SS
SD SS
1B
2B
7B
8
11
12B
15
16
17B
18
20
22
23
30
23
45
43
29
35
23
20
41
34
21
37
64
0.0308
0.0266
0.0215
0.0235
0.0351
0.0234
0.0435
0.0554
0.0239
0.0249
0.0413
0.0212
0.0144
0.0308
0.0242
0.0206
0.0217
0.0175
0.0202
0.0217
0.0435
0.0204
0.0238
0.0294
0.0204
0.0153
0.1462
0.0726
0.0619
0.0870
0.1228
0.1111
0.1087
0.1304
0.0918
0.0833
0.1324
0.0714
0.0305
0.0385
0.0323
0.0309
0.0326
0.0351
0.0303
0.0652
0.0652
0.0306
0.0238
0.0441
0.0204
0.0229
0.0274
0.0189
0.0139
0.0177
0.0304
0.0196
0.0278
0.0342
0.0209
0.0156
0.0341
0.0167
0.0079
bed forms averaged 89.7% in our experiments, which is
outside the range of direct application of the model presented by authors. We instead used their Figure 10 and the
data match for the case where bed morphology represented
80% of the total stress to calculate c 84 ¼ 0.2. This is an
order of magnitude higher than the calculation without considering the steep slope, but Lamb et al. warn that for high
bed form roughness values the model over predicts critical
shear stress. These two values for critical shear stress provide a benchmark range against which we evaluated the
critical shear stresses for the D84 grain size during our
experiments, which were between 0.046 and 0.484, with a
median of 0.13. Thus, the step forming grains in our experiments could form steps and step sequences while remaining
susceptible to entrainment.
[26] This mobility is reflected in the step sequences.
Approximately 80% of all sequences existed for 3 min or less
(Figure 2a), but because of varying run durations, these
sequences ranged in SS from under 0.010 to 0.065. In dimensionless form, approximately 80% of observed sequences
Figure 2. Sequence distribution histograms for all documented sequences : (a) total sequence existence
time (in min) and (b) dimensionless sequence stability parameter, SS .
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WATERS AND CURRAN: STEP SEQUENCE STABILITY
showed SS values of 0.040 or less (Figure 2b). Values of SS
varied within and between individual runs, precluding identification of a median SS value common to all runs. Figure 3
shows examples of the SS values for sequential step sequences within three runs. In the interest of space, only plots for
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runs 7B, 12B, and 22 are shown, but these are representative
of all the experiments. Sequence stability within a run was
variable with higher stability sequences interspersed among
the more dominant lower-stability sequences throughout each
of the runs. High SS sequences were typically followed by
Figure 3. Sequence plots for (a) run 7B (Q ¼ 0.0050 m3 s1, Qs ¼ 475 g m1 s1, Qs/Q ¼ 5.38), (b) run
12B (Q ¼ 0.0050 m3 s1, Qs ¼ 750 g m1 s1, Qs/Q ¼ 8.49), and (c) run 22 (Q ¼ 0.0050 m3 s1,
Qs ¼ 1250 g m1 s1, Qs/Q ¼ 14.15). Sequences shown are successive sequences documented during
the specified runs. These sequences may not have necessarily existed in an uninterrupted fashion relative
to run time as only sequences consisting of at least three steps were considered in this study. The lowest
bars in each plot signify run-specific minimum values of SS , which in all cases correspond to 1 min
duration sequences.
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lower SS sequences, and for half of the runs the maximum
SS sequence preceded a sequence matching the minimum
SS for the run (for example, sequences 24 and 25 in Figure
3b). Overall, sequences shifted between higher and lower SS
values throughout the experimental runs without following a
defined pattern.
3.2. Sequence Stability and Geometric Parameters
[27] Correlations between step sequence stability and
number of steps in a sequence, sequence average step spacing, step height, H/L, and H/L/S were identified for every
run. Sequence average geometric values are those parameter values measured at each minute during sequence existence, averaged over the existence time of that sequence.
Sequence averaged SS values were tested against each of
the geometric parameters and the Kendall’s b calculated
for each SS parameter pair. The average Kendall’s b coefficient provided a general measure of the statistical strengths
of the dominant and subordinate correlations identified
between the parameters across all runs showing a similar
trend (i.e., direct or inverse) or opposite trend.
[28] Lower SS values were calculated as the total number of steps within a sequence increased, particularly with
respect to the maximum calculated SS values (Figure 4).
Significantly more sequences consisting of three or four
steps were documented than sequences containing six or
seven steps. Coefficients of variance (CVs), which provide
nondimensional measures of variability calculated as the
standard deviation divided by the mean of a sample [Ang
and Tang, 1975], reflect the variability attributable to more
observations of sequences with fewer steps and thus, a
greater standard deviation of SS values (Figure 4). High
CVs were calculated for sequences of each step number,
which is representative of the dynamic nature of these
experiments. Nevertheless, a decreasing trend in stability for
the bulk of sequences is suggested as the number of steps in
a sequence increased. Within individual runs, the inverse
relationship between the number of steps in a sequence and
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sequence stability was supported by Kendall’s b analysis
for nine of the 13 experimental runs (Table 3).
[29] Trends between the stability parameter and dimensional geometric variables of step-pool sequences indicated
that sequence stability was generally higher for larger step
spacing (Figure 5a) and step height (Figure 5b). The correlations between the stability parameter, step spacing, and
step height for successive sequences presented in Figure 5
are from run 8 (Q ¼ 0.0055 m3 s1 ; Qs ¼ 475 g m1 s1)
and run 20 (Q ¼ 0.0065 m3 s1 ; Qs ¼ 1000 g m1 s1),
which were representative of the links between sequence
stability, average step spacing, and average step height calculated for each run. Sequence average step height was calculated as the sum of the individual step heights from step
crest to pool base for each step in the step-pool sequence at
a given time (see Figure 1), divided by the total number of
steps present in the sequence at that time. Total step height
did not necessarily equal the step forming grain size, as
steps often grew larger through the deposition of additional
grains, step forming size and smaller, around the step form.
Scour depths also varied between steps, contributing to
overall variability in the measured values of step height.
[30] Statistical analyses supported the finding of dominant direct relationships between the stability parameter
and both sequence step spacing and step height. The majority of sequence average step spacing values correlated
directly to SS values, such that larger step spacing was
associated with high SS and vice versa. Results from Kendall’s b analysis for SS and step spacing indicated this
positive association was dominant for 10 out of the 13 runs
with an average Kendall’s b of 0.16 (Table 3). Similarly, a
direct correlation between SS and H was supported statistically for seven of the 13 runs (Table 3). Though not as distinct as the correlation between SS and step spacing, the
direct relationship between stability and step height was
observed with a considerably higher average Kendall’s b
coefficient for runs directly related ( b ¼ 0.23) than for runs
showing an inverse correlation ( b ¼ 0.14). Therefore, the
Figure 4. Box plot of sequence stability for sequences composed of varying numbers of steps. Plots
include mean, median, lower and upper quartiles, and minimum and maximum SS values. Numbers
located at the top left of each quartile box are the coefficients of variance for each subset of sequences,
shown as percentages. Each of the 445 documented step sequences in the data set is accounted for in
these plots.
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Table 3. Kendall’s b Results: Relationships Between Sequence Stability and Geometric Parameters
Run
Number of Steps
Step Spacing L
Step Height H
Steepness H/L
H/L/S
1B
2B
7B
8
11
12B
15
16
17B
18
20
22
23
Totals
Directa
Direct average b
Inversea
Inverse average b
0.03
0.09
0.21
0.13
0.12
0.23
0.15
0.32
0.15
0.02
0.21
0.04
0.25
0.21
0.05
0.12
0.33
0.10
0.17
0.05
0.10
0.05
0.28
0.04
0.01
0.31
0.07
0.23
0.32
0.12
0.41
0.04
0.15
0.10
0.12
0.14
0.63
0.03
0.11
0.35
0.12
0.03
0.09
0.24
0.20
0.05
0.31
0.05
0.28
0.43
0.25
0.11
0.35
0.05
0.03
0.09
0.24
0.20
0.22
0.31
0.09
0.14
0.43
0.12
0.11
4
0.12
9
0.16
10
0.16
3
0.09
7
0.23
6
0.14
5
0.21
8
0.18
3
0.29
10
0.15
a
Bold run totals denote dominant statistical trends.
relationship between SS and step height was classified as a
direct relationship.
[31] Analyzing SS values relative to the dimensionless
geometric parameters describing step steepness (H/L) and
step steepness to bed slope ratio (H/L/S) placed the analysis
in a scale-free context. Increased stability was generally
observed when sequences exhibited lower values of H/L
and H/L/S. Examples of the relationships between H/L and
H/L/S with sequence stability shown in Figures 5c and 5d,
respectively, highlight how the respective relationships
varied with different step sequence development and system changes. Though no single trend was evident for all
sequences, there was an inverse relationship between SS
and each of the dimensionless parameters. Statistical analyses supported the inverse correlations, as evidenced by the
Kendall’s b coefficients shown in Table 3. The parameter
H/L/S showed the most pronounced inverse association of
the geometric parameters, as 10 of the 13 experimental
runs had negative correlations. Sequence average H/L was
negatively correlated with sequence stability for eight of
the 13 runs, thereby exhibiting a dominant inverse correlation, as well.
3.3. Sequence Stability and Roughness Parameters
[32] Three separate dimensionless measures of flow resistance were calculated and compared to sequence stability, including total flow resistance, ftot, macroroughness,
fmac, and macroroughness fraction, fmac/ftot. Median values
were determined from sequence averages and calculated in
the same way as the geometric sequence parameters
described above (Table 4). For total flow resistance and
macroroughness, as well as for macroroughness fraction,
the statistical analysis showed that higher sequence stability corresponded to lower-value sequence roughness, and
vice versa. The Kendall’s b coefficients measured a statistically strong inverse correlation between roughness and
SS for a majority of the runs (Table 5). Identical trends
and correlation coefficients were obtained for total and
macroroughness, which was not surprising given the two
roughness measures are calculated from the same sequence
parameters, and thus, are proportional. In each case, 9 of
the 13 experimental runs had negative Kendall’s b coefficients, which showed statistical support for the inverse relationship between sequence stability and sequence
roughness. An even stronger inverse relationship was measured between SS and the macroroughness fraction, for
which 10 of 13 runs had negative Kendall’s b coefficients
with an average Kendall’s b coefficient 67% stronger
( b ¼ 0.20) than that of the directly correlated runs ( b ¼
0.12). Dimensionless shear stress, 84 , also had an inverse
correlation to sequence stability for a majority of the runs
(Table 5). Thus, a sequence was generally more likely to be
stable under a lower shear stress.
[33] Sequence flow resistance associated with sequence
stability parameter values above the run-specific 75th percentile SS (hereafter referred to as SS -75) showed that,
for the most stable sequences documented in any run, the
range in total roughness values was significantly reduced
(Table 6 and Figure 6). There was a 65% average reduction
in roughness variability for the SS -75 sequences and a
maximum reduction in roughness variability of 85%. Thus,
variability in flow resistance was much less for the most stable step sequences. For example, Figure 7 shows the range
in total flow resistance and step sequence stability values
Figure 5. Sequence stability and geometric parameter plots for (a) run 8 average step spacing, L, (b) run 20 average step
height, H, (c) run 1B step steepness, H/L, and (d) run 16 H/L/S. Lighter blue bars represent high-stability sequences,
defined as sequences with SS values above the observed median SS for the respective runs. Darker blue bars are lowstability sequences, defined as sequences with SS values below the median SS for each run. The sequence-averaged geometric parameter calculated for each sequence of the run is the red line in each plot, which varies across all sequences for
each parameter tested. The dashed black line in each plot represents the median value of the respective geometric parameter, which was used to classify high and low values of that parameter in the same way that high and low SS values were
determined.
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Figure 5.
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Table 4. Median Sequence Roughness Results
Run
Median ftot
Median fmac
Median fmac/ftot
1B
2B
7B
8
11
12B
15
16
17B
18
20
22
23
1.912
1.909
2.036
1.936
2.165
1.972
1.795
2.067
2.105
1.873
1.960
1.827
1.864
1.713
1.711
1.835
1.737
1.962
1.772
1.598
1.866
1.903
1.675
1.761
1.630
1.666
0.896
0.895
0.901
0.897
0.906
0.898
0.890
0.902
0.903
0.894
0.898
0.892
0.894
for run 17B (Q ¼ 0.0050 m3 s1 ; Qs ¼ 1000 g m1 s1).
The most stable step sequences for this run are highlighted
with cross-hatched columns. The calculated total flow resistance values for these high stability sequences fell within
the narrow band on Figure 7 that identifies 27% of the total
variability in flow resistance. Differences in average total
resistance between the SS -75 sequences and the remaining
sequences in a run were also identifiable as average total resistance was lower for SS -75 sequences in 10 of 13 runs
(Table 6 and Figure 6). The relationship between flow depth
and step sequence stability mirrored this finding. Our calculation of total flow resistance was dependent on the ratio of
hydraulic radius to D84 grain size. Because we used a single
grain size distribution and constant flow width in these
experiments, increases in flow depth were expected to correlate with decreases in flow resistance and increases in
step sequence stability. For the same ten runs discussed
above, the flow averaged 1.9% deeper for sequences above
the SS -75 threshold when compared to the less stable
sequences.
[34] Change in the macroroughness parameter value was
quantified over the existence time for those step sequences
existing more than 1 min. We investigated the cumulative
change in macroroughness separately for those step sequences with stability above SS -75 values and the step sequences
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with maximum SS values from each run. Of the 250 total
sequences existing for at least 2 min, approximately 57%
experienced a cumulative increase in macroroughness, with
an overall average increase of 3.3% for all sequences (Figure
8a). Focusing on the 94 step sequences with stability greater
than SS -75 values, 61% had cumulative increases in macroroughness over sequence existence, with an average increase
of 5.3% (Figure 8b). Of the step sequences with maximum
SS values from each run (22 total for the 13 runs; runs 7B,
16, 22, 23 had multiple maximum SS sequences), 73%
adjusted to increase macroroughness values during sequence
existence. The cumulative change in macroroughness for
these sequences was the greatest, with an overall average
increase of 7.4%.
4.
Discussion
[35] Through statistical analysis, we identified dominant
relationships between sequence stability and several geometric and resistance-based parameters. There was variability in
the strength of the statistical relationships between the runs
and not a single parameter sequence stability relationship
was dominant across all the runs, as reflected in the calculated Kendall’s b coefficients (Tables 3 and 5). However,
statistical results allowed us to identify dominant trends in the
relationships between step sequence stability and sequence
averaged values of step spacing, step height, step steepness,
step steepness to bed slope ratio, total and macroroughness,
macroroughness fraction, and dimensionless shear stress.
4.1. Relationship Between Roughness and Stability
[36] Channel roughness has long been hypothesized as a
governing characteristic of step-pool stability. Abrahams
et al. [1995] theorized that step-pool systems evolved toward a state of maximum stability by adjusting channel
form to attain maximum flow resistance. In the present
study, statistical correlations between total flow resistance,
macroroughness, and macroroughness fraction indicated an
inverse correlation between flow resistance and step-pool
sequence stability, seemingly providing a contradictory
finding to that of Abrahams et al. However, the above
Table 5. Kendall’s b Results: Relationships Between Sequence Stability and Resistance Parameters
Run
Total Resistance ftot
Macroroughness fmac
Macroroughness
Fraction fmac/ftot
Dimensionless
Shear Stress 84
1B
2B
7B
8
11
12B
15
16
17B
18
20
22
23
Totals
Direct
Direct average b
Inversea
Inverse average b
0.05
0.30
0.10
0.26
0.28
0.00
0.05
0.10
0.27
0.21
0.21
0.11
0.10
0.05
0.30
0.10
0.26
0.28
0.00
0.05
0.10
0.27
0.21
0.21
0.11
0.10
0.13
0.41
0.10
0.29
0.33
0.08
0.05
0.04
0.25
0.15
0.28
0.14
0.16
0.19
0.16
0.10
0.24
0.32
0.12
0.05
0.18
0.15
0.25
0.01
0.02
0.10
4
0.12
9
0.18
4
0.12
9
0.18
3
0.12
10
0.21
5
0.14
8
0.15
a
Bold run totals denote dominant statistical trends.
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Table 6. Sequence Stability and Roughness: Percentile Analysis Results
Total Roughness Ranges
SS < SS -75
Average Total Roughness
SS > SS -75
Run
Minimum ftot
Maximum ftot
Minimum ftot
Maximum ftot
Percent Change
ftot Range
1B
2B
7B
8
11
12B
15
16
17B
18
20
22
23
1.739
1.740
1.761
1.607
1.741
1.643
1.567
1.755
1.792
1.460
1.839
1.572
1.596
2.374
2.047
2.472
2.532
3.112
2.291
2.049
2.633
2.771
2.253
2.353
2.665
2.270
1.773
1.798
1.830
1.798
1.924
1.869
1.690
2.021
1.985
1.645
1.929
1.734
1.840
1.946
1.997
2.250
2.049
2.287
2.112
1.845
2.253
2.131
1.986
2.113
1.908
2.151
72.7%
35.2%
40.9%
72.8%
73.5%
62.4%
67.9%
73.6%
85.1%
57.0%
64.2%
84.1%
53.8%
statistical results for roughness and stability associations
were based on data from all the sequences documented during the respective runs. As such, these correlations supported only the finding that sequence stability was enhanced
at lower values of roughness for the bulk of sequences, but
did not provide any indication concerning how an individual
sequence evolved through time. In order to investigate the
role of morphological adjustment of step-pool sequences to
attain maximum roughness and stability, an additional
roughness analysis was conducted focusing on resistance
parameters calculated for individual sequences.
[37] Cumulative change in macroroughness over the existence of a step-pool sequence was quantified and provided
the necessary information to apply Abrahams et al.’s [1995]
theory to our sequence data set. For the most stable sequences
within our data set, there were considerable increases in the
macroroughness. Mechanisms by which the macroroughness
SS < SS -75
SS > SS -75
Average ftot
Average ftot
Percent
Change ftot
1.963
1.921
2.071
2.007
2.23
1.985
1.772
2.05
2.149
1.834
2.054
1.872
1.897
1.861
1.863
2.048
1.924
2.099
1.962
1.763
2.123
2.056
1.887
1.998
1.809
1.926
5.2%
3.0%
1.1%
4.1%
5.9%
1.2%
0.5%
3.6%
4.3%
2.9%
2.7%
3.4%
1.5%
of a stable sequence increased included step migrations
(within the exclusion zone length) and increases in step steepness, which in turn increased energy dissipation and head
loss. The correlation between high step sequence stability and
a considerable increase in macroroughness over time agrees
with the theory put forward by Abrahams et al. [1995] that
the most stable step sequences adjusted to maximize total
flow resistance, and thus, increase sequence stability. Every
step sequence did not experience an increase in macroroughness, and only 45% of the step sequences fit into the stability
bounds established by the relationship between step steepness
and channel bed slope put forward by Abrahams et al.
[1995]. We attributed this to the highly dynamic conditions
present in the flume during the experiments. Our experimental conditions included mobilizing flows and variable sediment feed. In contrast, the flume experiments conducted by
Abrahams et al. [1995] did not test a mobile bed, nor did they
Figure 6. Total flow resistance averages and ranges for sequences above and below 75th percentile
SS values (SS -75). Red bars represent flow resistance ranges observed for more stable sequences and
show the large reductions in roughness variability corresponding to these sequences relative to the less
stable sequence ranges (gray bars) for each run. Similarly, red triangles and black diamonds represent
the total flow resistance values averaged for the more stable and less stable sequences, respectively.
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Figure 7. Sequence stability plot with total sequence roughness and high-stability roughness range for
run 17B. Cross-hatched bars represent high-stability sequences corresponding to SS values greater than
the 75th percentile SS (SS -75) for the run. Bars without fill represent all other sequences, which show
SS at or below the SS -75 threshold. The red line with points is the total flow resistance calculated for
each sequence. This line fluctuates across all sequences, but values stay within the narrow resistance
range shown as dashed black lines for sequences with higher SS . These two lines correspond to the
bounds of total resistance variability observed for the sequences above SS -75 and thus delineate the
range referred to as the ‘‘high-stability roughness range.’’
include sediment supply, which has been shown to decrease
form roughness through infilling of pools [Koll and Dittrich,
2001], to increase transport capacity [Whittaker and Davies,
1982] and to impact sequence stability (the work of Bacchi as
discussed by Recking et al. [2012]).
[38] The flow and sediment transport rates that composed the flood conditions tested can be parameterized by
the sediment concentration ratio, Qs/Q, a dimensionless parameter proposed by Church and Zimmermann [2007] and
shown to have an impact on step stability. Our results indicated a strong influence of sediment concentration on the
stability of step sequences. The run-averaged change in cumulative macroroughness decreased with increasing sediment concentration ratio (Pearson correlation coefficient, r,
equal to 0.64). This decrease was most noticeable in run
23, which had a high Qs/Q ratio of 12.86 (see Table 1 and
Figure 8b), indicating that in flows containing higher sediment loads sequences were not able to adjust to fully maximize macroroughness before being destroyed. The same
finding was surmised by field studies where the sediment
supply impacted step-pool sequence adjustment by promoting downstream transport of step-forming grains and
adversely affected step stability [Recking et al., 2012]. The
apparent inability of some step sequences to adjust to
increase their macroresistance suggested that the development state of a step-pool system [Molnar et al., 2010] may
act as an indicator of sequence stability.
4.2. Role of Sediment Concentration Ratio
[39] Some of the observed variation in the statistical
dominance of any parameter-sequence stability relationship
may be due to the influence of the sediment concentration
ratio under which the step sequence was formed and measured [Church and Zimmermann, 2007]. Variation in the statistical correlations with respect to sediment concentration
was evident for all parameters tested (Figure 9). Therefore,
the relationship between sequence stability and each of the
nine tested parameters was evaluated to determine the
influence of sediment concentration ratio over the identified
stability trend. Average Qs/Q ratios were calculated for
those runs with the same statistical trends for a specified
parameter-SS relationship. The dominant trends, either
direct or inverse, previously determined between sequence
stability and the geometric parameters corresponded to
higher Qs/Q ratios (Figure 10a). All of the resistance
parameters tested had a statistically dominant inverse relationship to sequence stability. When these flow resistance–
sequence stability relationships were tested against the
corresponding Qs/Q ratios, the average Qs/Q ratios were
between 22% and 31% lower for runs with the dominant
inverse relationship when compared to runs with a statistically direct relationship (Figure 10b). The statistical trend
of increased sequence stability at a reduced flow resistance
remained dominant only when the sediment concentration
was low. For the eight runs with a statistically dominant
inverse relationship between SS and each of the four resistance parameters, the average Qs/Q ratio was 6.44. As
the sediment concentration ratio increased, statistical support for the inverse association weakened. High Qs/Q values corresponded to those runs previously determined to
have a direct relationship between resistance parameters
and sequence stability, and the three runs with direct relationships between SS and all four resistance parameters
had an average sediment concentration ratio of 11.82.
Overall, as Qs/Q ratios increased, the dominant inverse
relationships observed between sequence stability and resistance parameters were no longer statistically valid;
rather, a direct relation between flow resistance parameters
and step sequence stability became statistically dominant at
high sediment concentration values.
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Figure 8. Cumulative percent change in macroroughness during the existence of (a) all sequences lasting longer than 1 min and (b) sequences showing SS above the run-specific 75th percentile SS (blue
bars) or a run maximum SS value (red bars). Negative changes in macroroughness are shown as light
gray bars in Figure 8a. Figure 8b is a subset of Figure 8a but only shows the change in macroroughness
corresponding to higher-stability sequences (SS greater than SS -75 or maximum run SS ). The sequences in each plot are arranged in order of increasing Qs/Q from left to right along the horizontal axis.
[40] The geometric parameters examined exhibited a mix
of statistically dominant inverse and direct relationships with
step sequence stability. Dimensionless measures of sequence
geometry had inverse relationships while the dimensioned
parameters had direct relationships to sequence stability.
Comparing statistical agreement to the sediment concentration ratio, the dominant trends previously determined, either
direct or inverse, corresponded to higher Qs/Q ratios. Thus,
the statistical trends showed that when the sequence stability
was high and sediment concentration ratio low, step steepness and H/L/S were also high. From the resistance parameters, it was indicated that under this scenario the flow
resistance would be low. A possible scenario of stable step
sequence formation for those runs with lower sediment concentration ratios was suggested by the experimental observations and statistical findings. During runs with low sediment
supply, sediment was unlikely to deposit and accumulate
around the steps. At the same time, the relatively clear water
passing through the initial step sequence scoured pools,
which created high step steepness ratios. Increasing step
steepness through pool formation adjusted the roughness geometry toward a state of maximized stability [Weichert et al.,
2008]. However, the growth of the step, and by extension the
step steepness ratio, was restricted by the low amount of sediment available from transport, limiting the macroroughness
contribution to total flow resistance. Thus, macroroughness
and total flow resistance through the step sequence remained
low and step sequences were able to remain stable for a period of time for flows with low Qs/Q ratios. Extended high
flows with limited sediment transport are known to destabilize steps over time [Curran, 2012; Rosport and Dittrich,
1995; Wohl and Jaeger, 2009]. As sediment concentrations
in the supply increased, we measured concurrent increases in
step height, steepness, and macroroughness.
[41] An individual sequence analysis of step height, conducted in a similar fashion to the macroroughness analysis
outlined in section 4.1, showed that of the most stable
sequences in each run (i.e., SS above SS -75) over 60%
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Figure 9. Scatterplots of Kendall’s b correlation coefficients relative to sediment concentration ratios
for (a) dimensional geometric parameters, (b) dimensionless geometric parameters, and (c) dimensionless resistance-based parameters. Separate plots are included to divide data according to type of sequence
parameter and dimensional nature, as well as for purposes of clarity.
had a cumulative increase in step height, and presumably
scour, over the existence of the sequence. Higher steps
resulted in larger macroroughness contributions by the step
form. The macroroughness contribution to total flow resistance increased over sequence existence as head loss due to
step height increased and scour continued to deepen the
downstream pool, a finding in agreement with research
showing concurrent increases in the contribution of spill resistance and step height [Comiti et al., 2009]. Consequently,
flows with a higher sediment supply allowed for accumulation of sediment at a step along with greater pool scour,
which together may have enabled sequences to stabilize temporarily as macroroughness and flow resistance increased
during the existence of the step sequence.
[42] A similar connection between sediment concentration ratio and step stability has been noted in previous
flume and field studies. Flume experiments by Bacchi, as
discussed by Recking et al. [2012], investigated sediment
supply effects on step stability and found that scour could
significantly impact individual step destabilization, which
was also shown by the step destruction mechanisms documented for these experiments [Curran, 2012]. On the basis
of field results and the Bacchi flume work, Recking et al.
[2012] proposed that scouring in step-pool systems ‘‘connected’’ to a sediment source lowered the hydrologic conditions necessary for step mobilization. For our sequence
data set, the relationship between resistance and stability
changed as sediment concentration ratio increased. However, lower sequence stability was observed overall for runs
with higher sediment supply conditions (Tables 1 and 2),
consistent with the hypothesis discussed by Recking et al.
[2012].
4.3. Form and Process in Sequence Stability
[43] Linking form and process in step-pool systems over
temporal and spatial scales is important when investigating
the stability of sequences [Chin, 1998; Wilcox et al., 2011].
The geometric parameters tested in our study acted as surrogates for the influence of form while the resistance parameters more aptly pertained to the processes potentially
impacting step-pool sequence stability. Full division of
form and process was not possible because the two are necessarily related. However, by independently analyzing both
groups of parameters, we identified some of the factors
associated with more stable sequences and, from those
associations, explored possible links between form and
process-based factors for a broader understanding of sequence
stability.
[44] Form and process analysis of step-pool sequences
can be appropriately conceptualized by considering the development state [Molnar et al., 2010] of a sequence. Geometric and resistance characteristics of a given sequence
changed during its existence, so the manner in which observations were made became important to the stability
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Figure 10. Effect of sediment concentration ratio (Qs/Q) on associative trends between sequence stability and (a) geometric parameters and (b) roughness parameters. The numbers to the right of the bars
indicate the total number of runs, showing the trends that were used to calculate the average Qs/Q ratio,
with bold, italicized numbers denoting dominant statistical trends (Tables 3 and 5).
analysis. When sequence-averaged values were considered,
the statistical correlations indicated that roughness and
sequence stability were inversely associated. However, the
averaging process may mask within sequence adjustments.
If a single time step was analyzed, the sequence measurements would not have been able to represent the potential
adjustment ongoing to increase step stability [Chin, 1998,
2003; Wilcox et al., 2011]. By analyzing adjustments
occurring over time within stable sequences we found step
steepness and macroroughness values were able to increase
as transported sediment accumulated at the step-forming
grain. The progressive adjustment of sequences observed on
an individual sequence basis illustrated the importance of development state on the stability of step-pool sequences.
These findings apply to all the sequences without regard to
the number of steps in a particular sequence. However, they
may combine to explain the reduction in SS as the number
of steps in a sequence increases.
[45] Sequences with larger step spacing were predominantly observed as more stable, reflecting the importance of
pool development during sequence existence, which has been
shown to dissipate energy through turbulence [GimenezCurto and Corniero, 2003, 2006; Wilcox et al., 2011]. By
definition, the same sequence showed little changes in step
spacing, so increases in either H/L or H/L/S were a result of
increases in step height and scour related to pool development. Step steepness had a statistically dominant inverse
relationship with step sequence stability such that for larger
H/L measurements there was lower stability. On an individual sequence basis, though, H/L values increased for the majority (approximately 57%) of sequences with SS values
larger than run-specific SS -75 values. Similar results were
obtained for H/L/S values of individual sequences, where
60% of these higher-stability sequences also increased H/L/S
over sequence existence time. The longer a sequence existed,
the more it was exposed to flow and sediment transport processes that affected the sequence geometry [Weichert et al.,
2008], including sediment deposition around an existing
step and pool development downstream of a step. The
measured adjustment in step steepness within a stable step
sequence links the change in step form with step sequence
process, as indicated by the increases in macroroughness
and total flow resistance discussed above. The connection
between step steepness, pool scour, and total flow resistance statistically quantified here has been suspected from
field measurements in the Italian Alps and Colorado
[Wilcox et al., 2011]. Parameter adjustments occurring in
the more stable sequences may indicate the development
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state of sequences as the geometric adjustments impact
channel processes and stability as a sequence persisted
without destruction.
[46] As pointed out by Weichert et al. [2008], the stabilization potential of a channel bed will not be fully recognized as long as sediment supply is available and transport
conditions are adequate, which was the case with our study.
Constant discharge and sediment feed ensured active transport conditions such that no sequence could reach its maximum stability potential, in part contributing to the highly
dynamic sequence behavior encountered in each run. Our
experimental conditions also precluded analysis of incremental flow change impacts on sequence stability, which
has been previously investigated [e.g., Rosport and
Dittrich, 1995]. The role of stream power, often linked to
step sequence stability [Church and Zimmermann, 2007;
Weichert et al., 2008], was not explicitly investigated
because of the constant nature of flow in these experiments.
All grain sizes present in these experiments were mobilized
under the flows tested. Therefore, critical values of incipient motion were not used as a stability metric [Church and
Zimmermann, 2007; Recking et al., 2012; Zimmermann,
2010].
[47] Our analysis addressed dynamic changes to steppool sequences during a flooding event as all grain sizes
employed were mobile and readily transported. The more
stable sequences were those that continued to develop and
adjust to increase their stability parameter until either new
step formation or step destruction occurred to form a new
sequence. This is reflective of the transient nature of steppool sequences occurring during a flooding event. By
observing sequence behavior under constant flows and fully
mobile transport conditions, our findings on sequence stability are most appropriately applied to the adjustment of
step-pool systems occurring during a flood of a set duration, analogous to one of our experiment flume runs.
5.
Conclusions
[48] All steps in a step-pool sequence are active in channel processes, affecting the transport of flow and sediment
downstream. However, only stable step sequences contribute to channel processes in a predictable manner. In this
research an extensive data set collected from 13 flume runs
of varying discharge and sediment transport conditions was
used to analyze 445 sequences of three or more step-pools
and investigate how geometric and resistance factors influenced step-pool sequence stability. To enable comparison
of step sequence stability across runs, we defined a
sequence stability parameter (SS ) as a metric of how long
a given step-pool sequence, defined as the same number of
steps with established step spacing, existed relative to the
total run time. The SS values for each of the 445 sequences
were tested against a range of geometric and resistance parameters to determine the strongest statistical relationships
which would provide an indication of those parameters
more influential in creating a stable step sequence. Those
geometric parameters examined included dimensional values of step number (#steps), height (H) and spacing (L) and
nondimensional step steepness (H/L) and the ratio of step
steepness to channel bed slope (H/L/S). The resistance
parameters analyzed included total flow resistance (ftot),
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macroroughness (fmac), dimensionless
shear stress corresponding to the D84 grain size 84 , and the ratio of macroroughness to total flow resistance (fmac/ftot). Our analysis
focused on changes in the step sequence stability parameter, how stability was correlated to both geometric and resistance parameters, and how a step sequence adjusted to
maintain stability under extreme flood conditions.
[49] The analysis showed variability in the strength of the
statistical relationships, yet the results allowed us to identify
dominant trends in the relationships between sequence stability and sequence average values of each of the nine parameters. The resistance parameters and dimensionless
geometric parameters were predominantly inversely related
to step sequence stability, meaning that sequences with
higher stability parameter values were statistically more
likely to have low flow resistance and low step steepness
ratios. However, almost half of the step sequences included
in this analysis existed for only 1 min of run time. The experimental conditions ensured formation of dynamic steppool sequences as the runs were designed to replicate the
events during a flooding event when all grain sizes were
mobile and readily transported. These conditions allowed
for the creation of a large number of step sequences but also
ensured that none of the sequences remained stable throughout a run. To evaluate the role of morphological adjustment
of step-pool sequences to attain maximum roughness and
stability, the 1 min sequences were excluded from further
analyses and the statistical trends were examined on an individual sequence basis.
[50] In contrast to the results using all step sequences,
we found that the majority of stable sequences increased
their roughness parameter values over time and approximately half fit within the bounds of the step steepness to
bed slope relationship developed by Abrahams et al.
[1995]. The correlation between high step sequence stability and an increase in macroroughness over time agreed
with the theory put forward by Abrahams et al. [1995] that
the most stable step sequences adjusted to maximize total
flow resistance. Not all stable step sequences adjusted to
increase channel macroroughness, a result that may be at
least in part due to the highly dynamic conditions during
the experiments. The analysis of parameter adjustments
during the existence time of individual step sequences indicated one manner by which some step-pool sequences are
able to increase stability during flood events.
[51] The sediment concentration ratio, Qs/Q, provided a
means for testing the relative strength of the statistical relationships between sequence stability and both roughness
and geometric parameters as well as a way to evaluate the
importance of different parameter-stability relationships.
Each experimental run was conducted at a different sediment concentration ratio, which had a strong influence over
step sequence stability. As Qs/Q ratios increased, the dominant inverse relationships observed between sequence stability and resistance parameters were no longer statistically
valid; rather, a direct relation between flow resistance parameters and step sequence stability became statistically
dominant at high sediment concentration values. The influence of the sediment concentration ratio over the stabilitygeometric and stability-resistance parameter relationships
(direct at high Qs/Q values and inverse at low Qs/Q values)
may enable the use of the sediment concentration ratio as a
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predictor of step sequence stability during a flood event.
For example, if a step sequence in a given area has been
measured and known to have a large step steepness ratio,
then when a flood event occurs that sequence is statistically
more likely to remain stable when the Qs/Q ratio for that
flood is low. However, if the flood event has a high Qs/Q ratio, that same step sequence is more likely to be mobilized.
[52] The link between step-pool sequence form and process allows us to speculate on the scenarios during which a
step sequence is statistically more likely to be stable. During an event with a low Qs/Q ratio, there would be a limited
amount of sediment in transport and therefore, a limited
amount of sediment that could deposit around an existing
step. Although flow tumbling over the steps scours downstream pools, without additional deposition at the step
form, the step steepness ratio would remain restricted. The
potential contribution of macroroughness to total flow resistance would be limited. The result would be a step
sequence more likely to remain stable if the step steepness
ratio was low and the Qs/Q ratio during the flow event was
also low. During a flood characterized by a large Qs/Q ratio, the step sequences would be expected to have a low
overall stability. However, a significant number of these
step sequences would also be expected to adjust to increase
stability by increasing the macroroughness associated with
the step-pool form. The large amount of sediment in transport would lead to greater deposition around the steps and
allow step sequences to stabilize as macroroughness and
flow resistance increased during the existence of the step
sequence. These two scenarios were indicated by the data
but remain speculative as we have yet to gather the field
data to test them. However, parameter adjustments occurring within step sequences may illustrate a development
process by which geometric and resistance parameters
adjust within stable step sequences.
[53] Acknowledgments. This work has been aided by thoughtful discussions on the nature of step-pool systems with Peter Wilcock, Francesco
Comiti, Jens Turowski, Dieter Rickenmann, Tom Lisle, Andre Zimmermann, and Mike Church. The manuscript was improved by the comments
of Kristen Cannatelli, Dieter Rickenmann, Jens Turowski, Francesco Comiti, and an anonymous reviewer.
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