Earth Surface Processes and Landforms
Flow
in Landforms
steep mountain
streams (2008)
Earth resistance
Surf. Process.
33, 2211–2240
Published online 13 June 2008 in Wiley InterScience
(www.interscience.wiley.com) DOI: 10.1002/esp.1682
2211
Flow resistance in steep mountain streams
Donald E. Reid1* and Edward J. Hickin2
1
2
BC Hydro, Burnaby, British Columbia, Canada
Department of Geography, Simon Fraser University, Burnaby, British Columbia, Canada
*Correspondence to: Donald E.
Reid, BC Hydro, 6911
Southpoint Drive, Burnaby,
BC, V3N 4X8, Canada.
E-mail: donald.reid@bchydro.com
Received 18 January 2007;
Revised 14 December 2007;
Accepted 7 January 2008
Abstract
Resistance to flow at low to moderate stream discharge was examined in five small (12–
77 km2 drainage area) tributaries of Chilliwack River, British Columbia, more than half of
which exhibit planar bed morphology. The resulting data set is composed of eight to 12
individual estimates of the total resistance to flow at 61 cross sections located in 13 separate
reaches of five tributaries to the main river. This new data set includes 625 individual
estimates of resistance to flow at low to moderate river stage. Resistance to flow in these
conditions is high, highly variable and strongly dependent on stage. The Darcy–Weisbach
resistance factor (ff) varies over six orders of magnitude (0·29–12 700) and Manning’s n
varies over three orders of magnitude (0·047–7·95). Despite this extreme range, both power
equations at the individual cross sections and Keulegan equations for reach-averaged values
describe the hydraulic relations well. Roughness is divided into grain and form (considered
as all non-grain sources) components. Form roughness is the dominant component, accounting for about 90% of the total roughness of the system (i.e., form roughness is on average
8.6 times as great as grain roughness). Of the various quantitative and qualitative formroughness indicators observed, only the sorting coefficient (σ = D84/D50) correlates well with
form roughness. Copyright © 2008 John Wiley & Sons, Ltd.
Keywords: flow resistance; mountain streams; grain roughness; form roughness
Introduction
The resistance offered to water as it flows through open channels is a fundamentally controlling characteristic of the
hydraulics of river channels. For more than a century the mean-flow characteristics of rivers have been measured and
predicted in terms of the Manning roughness coefficient, n:
n=
Rh2/3S1/2
v
(1)
or in terms of the dimensionless Darcy–Weisbach resistance coefficient, ff:
ff =
8gdS
v2
(2)
where Rh, S, v, g and d are respectively hydraulic radius, water-surface slope, mean flow velocity, acceleration due to
gravity and mean depth of flow. Manning’s n remains the instrument of choice of river engineers in solving practical
design problems relating to flow modification.
Manning’s n was conceived as a measure of boundary roughness controlling steady uniform flow through a channel.
In application, however, it is used as a coefficient of proportionality and therefore becomes a measure of all sources of
energy loss in the hydraulic system. Energy drained from the mean flow to overcome flow resistance is energy
not used to drive the mean-flow velocity. These losses include friction related to boundary roughness that is composed
of individual grains (grain roughness) and grain aggregates (form roughness), but they also include energy loss related
to internal distortion resistance (the energy needed to drive secondary circulation in river bends, for example) and to
generate turbulence in its various forms (Leopold et al., 1960). Major energy loss in steep streams occurs where flow
Copyright © 2008 John Wiley & Sons, Ltd.
Earth Surf. Process. Landforms 33, 2211–2240 (2008)
DOI: 10.1002/esp
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D. E. Reid and E. J. Hickin
impacts on standing water (such as spill resistance in a chute or waterfall) and in the breaking waves of supercritical
flow.
Although it has been recognized for many years that flow resistance in rivers has these several components, in large
low-slope rivers resistance to flow tends to be dominated by boundary (or skin) resistance. Thus, the almost exclusive
concern of engineers and river scientists with relatively large channels carrying near-bankfull flows has meant that
thinking about flow resistance has centred on understanding the friction related to grain roughness and, to a much
lesser extent, form roughness and other sources of flow resistance.
This traditional focus on the resistance behaviour of large rivers at high stage has begun to shift in recent years,
however, as interest in the morphodynamics of mountain streams has increased (Wohl, 2000). The change of focus
reflects the normal scientific exploration of uncharted fluvial domains but it has also been accelerated by practical
engineering concerns. For example, small steep headwater channels are now being exploited for electrical power
generation in ‘run-of-the-river’ hydro-electric plants. Hundreds of these projects are under development in the small
streams draining the Coast Mountains of British Columbia in western Canada. Their development has led to many
design questions relating to the morphodynamics of these channels and to the environmental impacts of engineered
changes, particularly at relatively low flows when removing a portion of the flow leads to a loss of instream fish
habitat.
It has been noted recently (for example, see Curran and Wohl, 2003; MacFarlane and Wohl, 2003) that, in these
small steep mountain streams with their high relative roughness, skin resistance can be small relative to other sources
of resistance as a result of the irregular form of the boundary (for example, steps and pools and the highly turbulent
flow in cascades). Estimates of the effective Manning n or Darcy–Weisbach ff in these circumstances are very difficult
to make accurately because engineers and river scientists as a group do not have the experience to readily make the
transition from the large-river environment to those of small mountain streams. Standard flow-resistance guidelines
(such as those in the work of Cowan (1956), Chow (1959), Barnes (1967), Hicks and Mason (1991) and Coon (1998))
are misleading because there are very few accurate measured data available on which to base assessments for these
mountain streams; extreme underestimates of flow resistance are common. These guides generally focus on high
instream flows and ‘conservative’ values for flood design. Barnes (1967) states ‘At the present state of knowledge, the
selection of the roughness coefficients for natural channels remains chiefly an art’; this statement remains true today.
It is the primary purpose of this paper to introduce new high-quality measured data on the hydraulics of a set of
representative steep mountain channels located in the Coast Mountains of British Columbia in western Canada as a
guide to assessing flow resistance in this type of environment. These data have been obtained for a range of discharges
sufficient to define, for the low-flow domain, the at-a-station hydraulic geometry at each measurement site. A secondary purpose of this paper is to discuss the nature of total flow resistance in relation to that which might be attributed
to grain roughness alone within the low-flow discharge domain.
Attempts to separate total resistance into form and grain components build on the classic pipe-flow studies of
Nikuradse (1933). These studies determined that, in the absence of form roughness (a circular pipe with immobile
boundary), the effective roughness length (ks) is correlated to the resistance to flow in the system and the median size
of the boundary material (D50). Extrapolating these findings to natural streams, where the boundary is composed of
heterogeneous sediment and both grain and form roughness components are present, the roughness length (ks) can be
expressed as a function of the bed-material size, and is often set as the 50th or 84th percentile of the size distribution
(Dx):
ks = CxDx
(4)
where Cx represents the form roughness component of the system expressed as a multiplier of the bed-material size
(Bray, 1982; Millar and Quick, 1994). In mountainous settings, form resistance can significantly outweigh the grain
resistance, contributing up to 90% of the total resistance (Millar, 1999).
Field Sites
The study was conducted within the Chilliwack River watershed, a steep, mountainous region in southern, coastal
British Columbia (Figure 1). Detailed descriptions of these field sites are available in the work of Reid (2005).
The geology of the basin varies from erosion-resistant granodiorite batholiths in the upper watershed to older,
and more easily weathered, metamorphic, volcanic and sedimentary rocks in the study portion of the watershed. The
valley was heavily glaciated during the Pleistocene, leaving thick glacial deposits as tills, kame terraces, moraines,
glaciolacustrine and outwash material on the lower valley and hillslopes (Saunders, 1985; Saunders et al., 1987;
Copyright © 2008 John Wiley & Sons, Ltd.
Earth Surf. Process. Landforms 33, 2211–2240 (2008)
DOI: 10.1002/esp
Flow resistance in steep mountain streams
2213
Figure 1. Chilliwack watershed and study basins.
Clague, 1981). In addition to acting as sediment sources, these various glacial deposits act as groundwater reservoirs
that supply water to lowland stream channels during periods of drought, offsetting low flows. Other sources of
baseflow in the watershed are slopes more than 800 m above sea level that are covered by a persistent snowpack
through the winter (B.C. Ministry of Forests, 1995) and numerous areas of permanent ice that occur on the higher
elevation mountaintops across the basin. The watershed lies in the Coastal Western Hemlock (CWH) biogeoclimatic
zone, which is characterized by a dense coniferous forest and wet, mild winters.
The hydrologic regime of the Chilliwack watershed is typical of coastal British Columbia. Peak flows are generated
in two seasons: in June and July by snowmelt and in the months of October through January by large winter storms.
The largest flows in the river are produced by infrequent rain-on-snow events in the fall and winter, when large, warm
Pacific storms affect the coast. These storms produce extreme rainfall intensities and sudden warming that result in
rain falling on the high-elevation snow pack, causing widespread melting. The flood of record in the valley occurred
on 10 November 1990 during these rain-on-snow conditions.
Like peak flows, low flows occur during two separate periods through the year: the summer and winter low-flow
seasons. Typically, the summer low-flow period from August through October is the more severe season in any given
year, with smaller minimum flows that occur over longer periods. These conditions are produced during droughts,
when summer high-pressure ridges dominate the local weather. Usually, base flow from the watershed sustains stream
flow through these relatively short droughts before early fall storms end the summer low-flow period. However, rare
events, such as the summer and fall 2003 drought, when very little measurable rain fell during three months (July,
August and September), occur periodically. These infrequent conditions produce a long, slow recession in base flow.
In contrast, winter low flows are produced during cold, dry periods when the upper portions of the watersheds are
frozen and precipitation falls as snow.
In addition to extreme low-flow conditions, the study period includes a large flood event on 17 October 2003
(Figure 2(A)). The storm that produced this event delivered 53 mm of rain in a 24-hour period on 16 October and
a further 95 mm of rain on 20 October at the Chilliwack River Hatchery climate station (DFO, 2003; Figure 2(B)).
Copyright © 2008 John Wiley & Sons, Ltd.
Earth Surf. Process. Landforms 33, 2211–2240 (2008)
DOI: 10.1002/esp
2214
D. E. Reid and E. J. Hickin
Figure 2. Streamflow in Slesse Creek (A) and total precipitation at the Chilliwack River Hatchery (B) during the study period.
In all, the event supplied 275 mm of rain to the watershed during the nine-day storm. This extreme rainfall produced
a 15-year flood event in nearby Slesse Creek and likely a similar event in the study streams.
Five tributaries to Chilliwack River were selected for study (Figure 1). This data set includes a combination of
purposely selected and randomly selected basins. The purposely selected basins are Frosst Creek and Liumchen Creek,
which have been previously gauged by the Water Survey of Canada, and therefore a continuous record of streamflow
is available for these two basins. To complete the data set, a set of three additional basins were selected at random
from the remaining watershed. The randomly selected basins were subject to access, size and channel morphology
constraints. Channels exhibiting multi-channel form, alluvial fans, bedrock canyons and channels that are artificially
constrained for long distances at the access point were excluded from the study. Stream size was limited to those
streams small enough to wade during most flow stages, yet large enough to contain perennial flow. The random
selection yielded Borden, Foley and Chipmunk Creeks in basins ranging in size from 11·9 to 78·6 km2.
Thirteen reaches in the five study basins were selected for detailed study. Nominally, two reaches were surveyed in
each of the five tributaries except Borden Creek and Chipmunk Creek, where varied conditions near the access point
allowed three and four reaches respectively to be included in the study. For the purposes of this study, a reach is
defined as a morphologically homogeneous length of channel within which the controlling factors of stream morphology do not change appreciably (after Church, 1992). Stream gradients range from 0·017 to 0·075 and stream morphologies
are varied: pool-riffle, plane-bed, step-pool and cascade (Table I).
The study reaches are chosen for their gradient, accessibility etc. to meet practical and hydraulic geometry requirements (Reid, 2005). However, they have several features that make them well suited to the study of flow resistance.
Seven of the 13 study reaches have steep, plane beds, low channel-sinuosity and insignificant presence of bars. They
also have little encroaching vegetation or large woody debris, thus removing these confounding sources of form
roughness from the analysis. This leaves the main sources of resistance to flow as the bed material and the bedmaterial size distribution.
The field sites do, however, exhibit a natural variation in channel form and therefore include some additional
sources of resistance to flow. Six of the 13 study reaches include some obvious form of channel roughness in addition
to the bed material itself (Figure 3). Lower Frosst, Lower Borden and Upper Borden reaches have different morphologies
from the plane-bed or cascade morphologies that dominate the other reaches. Step-pool and pool-riffle morphologies
are thought to have higher longitudinal profile irregularity than plane-bed channels. Chipmunk Tributary reach
has high vegetation effects at both low and moderate flow. In this reach, willow branches project into the channel
Copyright © 2008 John Wiley & Sons, Ltd.
Earth Surf. Process. Landforms 33, 2211–2240 (2008)
DOI: 10.1002/esp
Copyright © 2008 John Wiley & Sons, Ltd.
Lower
Upper
Lower
Upper
Lower
Middle
Upper
Lowest
Lower
Upper
Tributary
Lower
Upper
Reach
Name
P-R
PB
PB
C
F(P-R)
PB
S-P
C
PB/P-R
PB/C
PB
C
PB/F(P-R)
Dominant
Morphologya
Straight/Buffered
Sinuous/Buffered
Sinuous/Buffered
Straight/Buffered
Sinuous/Buffered
Sinuous/Buffered
Sinuous/Buffered
Sinuous/Coupled
Sinuous/Intermittently Coupled
Straight/Buffered
Sinuous/Buffered
Sinuous/Intermittently Coupled
Sinuous/Intermittently Coupled
Channel Planform/
Coupling to Hillslopes
a – P-R, Pool-Riffle; PB, Plane Bed; S-P, Step Pool; C, Cascade; F(P-R), Forced Pool-Riffle
(Montgomery and Buffington, 1997).
Foley Creek
Chipmunk Creek
Borden Creek
Liumchen Creek
Frosst Creek
Stream
Table I. Study reach geomorphology
1·2
2·1
0·9
1·0
0·0
0·2
0·5
4·5
4·8
5·5
0·1
0·6
1·0
Location –
Distance
Upstream
from Mouth
(km)
5
5
3
3
5
5
5
5
5
5
5
5
5
Number
of Cross
Sections
25
35
25
25
20
15
20
50
45
25
25
75
75
Average
Cross
Section
Spacing
(m)
215
188
74
79
100
112
105
250
290
134
119
367
392
Reach
Length
(m)
30·2
27·1
54·5
54·5
17·8
17·8
17·7
33·5
33·0
12·8
11·9
76·9
75·9
Drainage
Area to
Head of
Reach
(km2)
760
790
1090
1090
1260
1260
1270
1320
1320
1280
1400
1300
1310
Mean
Basin
Elevation
(m)
1·9
3·1
3·5
4·9
3·5
2·9
7·5
3·4
1·7
2·3
2·5
2·5
1·9
Bed
Slope
(%)
10·0
8·6
24·4
17·7
9·9
9·4
10·2
15·5
15·9
9·6
8·4
22·8
23·6
Mean
Bankfull
Width
(m)
Flow resistance in steep mountain streams
2215
Earth Surf. Process. Landforms 33, 2211–2240 (2008)
DOI: 10.1002/esp
2216
D. E. Reid and E. J. Hickin
Figure 3. Examples of other roughness effects in the study reaches. This figure is available in colour online at www.interscience.
wiley.com/journal/espl
from both banks. Lower Borden reach has moderate large woody-debris effects because several logs project out into
the flow and span the bed of the creek. Lower Liumchen, Lower Borden and Upper Foley reaches have significant bars along the edges of the channel (lateral bars) or single large bars in the centre of the channel (mid-channel
bars).
On 23 October 2003, a large storm produced a flood event in the study sites. These flows altered the channel
boundary of Lower Frosst, Lower Borden and Middle Borden reaches. As a result, channel roughness and resistance to
flow were altered in these reaches to varying degrees. While there has been no attempt to quantify or separate the
resistance values before and after this event, it is likely that some of the scatter seen in the power and logarithmic
curves can be attributed to this cause. A discussion of the effect of this flood on the at-a-station hydraulic geometry of
the cross sections is presented in the work of Reid (2005).
Methods
A detailed account of the field procedures adopted in this study is available in the work of Reid (2005); a brief summary
serves the present purpose. To calculate resistance to flow, hydraulic measurements were made over a six-month period
from 19 June 2003 to 16 December 2003 at multiple stream discharges at set cross-section locations within the study
basins. Section layout was carried out in a consistent manner for each study reach. First, each reach was walked to
ensure that it contained a consistent stream morphology for more than ten bankfull widths. The relative input of each
tributary was assessed and changes in stream morphology were noted. Only Liumchen Creek, a stream gauged by the
Copyright © 2008 John Wiley & Sons, Ltd.
Earth Surf. Process. Landforms 33, 2211–2240 (2008)
DOI: 10.1002/esp
Flow resistance in steep mountain streams
2217
Water Survey of Canada, did not meet the general length criteria. Two shorter reaches are included here to ensure that
this important gauged basin was included in the study. After walking the entire reach, an arbitrary starting point near
the downstream end of the reach was selected. Cross sections were located from the starting point at equally spaced
intervals of more than two wetted bankfull-widths. Again, Liumchen Creek was an exception to this criterion due to
its shortened reaches; here only three cross sections spaced one bankfull width apart are included in the study.
After locating the appropriate number of cross sections within each reach, the bed, banks and longitudinal profile
were surveyed by total station. To ensure that every subsequent survey reproduced the same cross section, survey pins
were installed on both banks as part of the initial survey. These control pins were used as position, elevation and
distance control for the remainder of the study. The initial survey of the cross sections included all topographic breaks
across the section and included any individual large boulders located on the cross-section line. A longitudinal profile
of each reach was surveyed along the stream thalweg with the intention of defining the dominant bedform; thus
spacing between adjacent data points is variable. The mean reach gradient is defined as the slope of a best-fit line fitted
to these data while the bed-profile regularity is defined as the standard deviation of the residuals of the data to that
best-fit line (see, e.g., Lee and Ferguson, 2002). As a result, larger values of bed-profile regularity correspond to
rougher beds.
At each cross section, several measurements of the hydraulic geometry of the flow were made. Water-surface widths
and depths were determined using one of two methods: a level survey of the water surface or multiple direct
measurements of depth along the survey line. Method one, usually employed at higher stages when wading the entire
cross section was difficult, used a combination of the surveyed cross-section boundary, a surveyed water-surface
elevation and a measurement of the water surface width (w) at the observation date. The water-surface elevation was
plotted on the surveyed cross-section boundary and the total flow area (A) calculated in a computer aided drafting
program. Mean depth (∂ ) was calculated from
∂ =
A
w
(5)
where w is the water-surface width. The mean flow velocity (v) was calculated from
v=
Q
A
(6)
where Q is the stream discharge. Method two, used more often than method one when the section could be waded,
employed a technique similar to stream-discharge gauging where multiple direct measurements of the flow depth were
made across the section using a stadia rod. Distance across the section was measured on an overhead tape for each
depth measurement and for measurements of the edges of rocks protruding through the surface of the flow. From the
depth and distance measurements the area of flow for individual cells was calculated and summed across the entire
cross section to compute the total flow area. As in method one, relations among continuity, stream discharge, watersurface width and total flow area were used to compute the mean depth and mean velocity.
Stream discharge was measured using standard wading techniques and summing the individual discharge cells
(Rantz, 1982). Where possible all discharge estimates are composed of 20 or more individual cells with a minimum
separation of 0·15 m. When wetted width at the gauging section was less than 3·0 m, and 20 individual measurements
were not possible, a set interval of 0·15 m was adopted. To avoid the problems of flow metering in sections not well
suited to measurement, discharge in each reach was measured at an ideal gauging section and the discharge through
the reach assumed to be constant. These ‘ideal’ flow-metering sections are located within or close to the study reach
so that significant gains or losses of streamflow between the gauging location and the study cross sections are unlikely.
Ideal sections were selected for their smooth flow characteristics and the sections were physically altered to remove
nearby large boulders to improve the gauging conditions. The ideal gauging sections were located in large channelspanning pools with the preferred hydraulic conditions.
The average velocity in the vertical for each of the 20 discharge cells was measured at the standard 0·6 depth using
a Swoffer 2100 current meter. Each velocity measurement was calculated as the average of at least three, six second
measurements that were observed to vary by less than 10% from the mean during their measurement. If the fourth
measurement was observed to vary by less than 10% from the mean of the previous three then the first three
measurements were adopted. If the fourth measurement varied by more than 10% from the mean of the previous three
then the gauging duration was extended until a stable mean was achieved. Although several researchers have found
that the mean velocity in the vertical is more likely to be found at 0·5 depth (0·5d) in steep mountain streams because
of the s-shaped vertical velocity distribution (Jarrett, 1990), this was not the case here. There is no statistical difference
Copyright © 2008 John Wiley & Sons, Ltd.
Earth Surf. Process. Landforms 33, 2211–2240 (2008)
DOI: 10.1002/esp
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D. E. Reid and E. J. Hickin
between the velocity at 0·6d, 0·5d or the calculated mean for the vertical in the pools measured for this study (Reid,
2005). Without clear evidence that it was inappropriate, the standard gauging depth of 0·6d was adopted.
To test the accuracy of the discharge measurements, and to test the assumption that standard gauging techniques
could produce a precise discharge estimate using a Swoffer current meter, a single cross section was repeatedly
gauged on 10 and 11 June 2003. To ensure a steady-flow state, the test was conducted during a dry period and the
stage at the measuring section was noted at the start and end of the test; no stage change was noted. The three
discharge estimates made on each day were observed to vary by less than 5% from the mean of that day and less than
5% from independent repeated discharge measurements made with a separate Price pygmy-style current meter. While
the test does not confirm the accuracy of the measurements, the close agreement of independent meters and multiple
measurements with each meter provides some confidence in the discharge estimates provided and gauging techniques
employed.
To characterize the bed texture in each reach, accumulations of sediment (bars) were identified as a basis for
sampling where they occurred; the bed material itself in the centre of the channel could not be measured directly due
to the turbulent conditions at many of the study sites. Once identified, a flexible tape was laid along the wetted edge
of the stream at low flow, at the edge of the bar and the b-axis of the surface grains under the 1 m marks on the tape
measured in a modified grid-sampling method (Kellerhals and Bray, 1971). In this way, 101 or more particles (>0·5 mm)
of the bar-edge were measured for each reach. The total sample was divided between the various coarse-sediment
deposits in a reach proportional to the relative size of the deposit; hence, larger deposits comprise a greater portion of
the total sample. Care was taken to ensure that every deposit within a reach was included in the sample to reflect any
downstream trend in sediment size. All measurements in a particular reach are combined to produce a reach-averaged
estimate of grain size. By sampling along the wetted edge of the deposit (as opposed to the surface), and because these
low-profile features tend to grade slowly into the bed, there is not thought to be a significant difference between the
material of the sample and that comprising the bed. In those reaches where no deposits occurred (e.g. step-pool
reaches) the flexible tape was laid along the wetted edge and the bed material sampled as above. No attempt was made
to differentiate between bed material forming steps and that found elsewhere in the channel.
Resistance to flow is calculated from the primary flow-variables and the mean reach bed-slope for each cross
section at each observation date, yielding 625 individual estimates of both ff and n.
In this study, grain roughness refers to the channel roughness related to the median size of gravel forming the
boundary and form roughness refers to the combined effect of all other types of roughness. Grain resistance refers to
the resistance to flow attributable to grain roughness and form resistance refers to all other direct and indirect sources
of resistance to flow. No attempt is made here to further divide form resistance into its various components (such as
form roughness generating spill resistance).
Results
Resistance to flow was calculated from hydraulic parameters measured at eight to 14 stream discharges at the 61 study
cross sections. These data yield a set of 665 individual measurements of resistance to flow in the 13 study reaches
(see the appendix). Stream discharge ranges over four orders of magnitude, from 0·0006 to 5·52 m3/s. In this flow
range relative roughness (R/D50) varies from 0·27 to 3·30 (Table II). These conditions represent large-scale
roughness (Bathurst, 1982), where large boulders protrude through the surface of the flow in much of the flow range
and even when submerged continue to affect the vertical velocity-profile. In these conditions, the Darcy–Weisbach
ff ranges from 0·29 to 12 700 and Manning’s n from 0·05 to 7·95. These flows are generally lower and the resistance
values much higher than those reported in the literature. Indeed, the only comparable data set available is that
of Millar (1999), who presents the results of several researchers across a range of stream flow from 0·7 to 8200 m3/s,
for which the Darcy–Weisbach ff ranges from about 0·2 to 0·5 in a relative roughness range of 4·0 to 200. Bathurst
(1982) reports that boulder-bed streams typically have Manning’s n values that range from 0·04 to 0·2. Marcus et al.
(1992) report n values that range from 0·056 to 0·183 for 20 cross sections in which stream discharge varies between
0·16 and 1·39 m3/s.
At the study cross sections, resistance to flow consistently decreases with increasing stream discharge (Figures 4 – 9);
the negative exponents in the power equations range from −0·29 to −1·57 (Tables III–VII). Another consistency
evident is the general good agreement between the power relation and the data at each section as calculated in the
R2 terms. For ff versus discharge R2 values range from 0·28 to 0·99, of which 80% are greater than 0·70. When
considered as a whole, the exponents of the function have a significant modal value around −0·6, with a strong
negative skew (Figure 10). The average exponent for each reach varies from −0·63 to −1·19, confirming that most of
the exponents are less than the mode of −0·6.
Copyright © 2008 John Wiley & Sons, Ltd.
Earth Surf. Process. Landforms 33, 2211–2240 (2008)
DOI: 10.1002/esp
Copyright © 2008 John Wiley & Sons, Ltd.
Lower
Uppera
Lower
Upper
Lower
Middle
Upper
Lowest
Lower
Upper
Tributary
Lower
Upper
Reach
Name
40
–
50
35
40
60
40
40
30
50
45
40
35
D16
(mm)
120
–
180
150
110
130
110
130
105
155
110
180
190
D50
(mm)
260
–
420
570
200
260
300
460
220
385
200
500
510
D84
(mm)
Surface Sediment Size
a – sediment data were not collected in Upper Frosst Creek.
Foley Creek
Chipmunk Creek
Borden Creek
Liumchen Creek
Frosst Creek
Stream
Table II. Relative roughness
0·11
–
0·10
0·14
0·03
0·08
0·08
0·09
0·08
0·08
0·06
0·15
0·17
Minimum
Depth (d) (m)
0·92
–
0·56
0·93
0·27
0·62
0·73
0·69
0·76
0·52
0·55
0·83
0·89
Maximum Relative
Roughness (d/D50)
0·28
–
0·44
0·54
0·36
0·44
0·43
0·41
0·36
0·42
0·21
0·46
0·52
Maximum
Depth (d) (m)
Observed
2·33
–
2·44
3·60
3·27
3·38
3·91
3·15
3·43
2·71
1·91
2·56
2·74
Minimum Relative
Roughness (d/D50)
0·57
–
1·15
1·17
0·73
0·59
0·83
1·08
0·65
0·66
0·47
1·43
1·31
Maximum
Depth (m)
4·75
–
6·39
7·80
6·64
4·54
7·55
8·31
6·19
4·26
4·27
7·94
6·89
Relative Roughness
(d/D50)
Bankfull
Flow resistance in steep mountain streams
2219
Earth Surf. Process. Landforms 33, 2211–2240 (2008)
DOI: 10.1002/esp
2220
D. E. Reid and E. J. Hickin
Figure 4. At-a-station hydraulic geometry and resistance to flow at cross section FST-L1 in the lower reach of Frosst Creek. This
figure is available in colour online at www.interscience.wiley.com/journal/espl
Table III. At-a-station hydraulic geometry for Frosst Creek
Manning’s n (n = tQy)
Darcy-Weisbach ff (ff = hQp)
Reach
Name
Cross
Section
Lower
Upper
Number of
Observations
h
p
R2
t
y
R2
FST-L1
FST-L2
FST-L3
FST-L4
FST-L5
0·70
0·59
0·72
0·76
0·59
0·67
0·036
0·6 to 0·74
−0·74
−1·02
−1·24
−1·00
0·82
− 0·96
0·087
−1·14 to − 0·79
0·83
0·92
0·89
0·95
0·93
0·07
0·07
0·07
0·08
0·07
0·07
0·002
0·07 to 0·08
− 0·32
− 0·46
− 0·58
− 0·47
− 0·40
− 0·44
0·044
− 0·53 to − 0·36
0·76
0·89
0·86
0·94
0·91
FST-U1
FST-U2
FST-U3
FST-U4
FST-U5
0·95
1·53
0·40
2·61
2·65
1·63
0·446
0·74 to 2·52
−1·13
−1·57
−0·98
−1·06
−1·22
−1·19
0·101
−1·39 to − 0·99
0·93
0·99
0·61
0·93
0·94
0·08
0·11
0·05
0·15
0·15
0·11
0·019
0·07 to 0·15
− 0·54
− 0·76
− 0·45
− 0·49
− 0·58
− 0·56
0·055
− 0·67 to − 0·45
0·92
0·98
0·52
0·90
0·91
12
12
12
12
12
Average
Standard error
95% confidence interval
8
8
8
8
8
Average
Standard error
95% confidence interval
Copyright © 2008 John Wiley & Sons, Ltd.
Earth Surf. Process. Landforms 33, 2211–2240 (2008)
DOI: 10.1002/esp
Flow resistance in steep mountain streams
2221
Figure 5. At-a-station hydraulic geometry and resistance to flow at cross section LCH-U3 in the upper reach of Liumchen Creek.
This figure is available in colour online at www.interscience.wiley.com/journal/espl
Table IV. At-a-station hydraulic geometry for Liumchen Creek
Darcy-Weisbach ff (ff = hQp)
Reach
Name
Cross
Section
Lower
Upper
Number of
Observations
Manning’s n (n = tQy)
h
p
R2
t
y
R2
LCH-L1
LCH-L2
LCH-L3
4·60
4·65
7·75
5·67
1·040
3·59 to 7·75
−0·66
−0·61
−0·93
− 0·74
0·100
−0·93 to −0·54
0·93
0·88
0·96
0·19
0·19
0·24
0·21
0·018
0·17 to 0·24
−0·28
−0·24
−0·42
− 0·31
0·055
−0·42 to −0·2
0·89
0·81
0·94
LCH-U1
LCH-U2
LCH-U3
9·42
20·10
3·83
11·12
4·863
1·39 to 20·84
−0·96
−1·03
−0·37
− 0·79
0·308
−1·4 to − 0·17
0·97
0·98
0·68
0·29
0·42
0·17
0·29
0·099
0·1 to 0·49
−0·44
−0·48
−0·14
− 0·36
0·147
−0·65 to − 0·06
0·96
0·97
0·52
13
13
13
Average
Standard error
95% confidence interval
14
14
14
Average
Standard error
95% confidence interval
Copyright © 2008 John Wiley & Sons, Ltd.
Earth Surf. Process. Landforms 33, 2211–2240 (2008)
DOI: 10.1002/esp
2222
D. E. Reid and E. J. Hickin
Figure 6. At-a-station hydraulic geometry and resistance to flow at cross section BRDN-U2 in the upper reach of Borden Creek.
This figure is available in colour online at www.interscience.wiley.com/journal/espl
Table V. At-a-station hydraulic geometry for Borden Creek
Darcy-Weisbach ff (ff = hQp)
Reach
Name
Cross
Section
Lower
Number of
Observations
Manning’s n (n = tQy)
h
p
R2
t
y
R2
BRDN-L1
BRDN-L2
BRDN-L3
BRDN-L4
BRDN-L5
0·92
1·25
1·21
2·33
1·92
1·53
0·258
1·01 to 2·04
−1·02
−0·50
−0·91
−1·21
−0·89
− 0·90
0·117
−1·14 to −0·67
0·97
0·92
0·97
0·99
0·97
0·08
0·10
0·09
0·13
0·13
0·11
0·010
0·09 to 0·13
−0·48
−0·19
−0·41
−0·58
− 0·40
− 0·41
0·064
−0·54 to −0·29
0·96
0·83
0·96
0·99
0·96
Middle
BRDN-M1
BRDN-M2
BRDN-M3
BRDN-M4
BRDN-M5
0·71
0·95
1·02
3·30
1·06
1·41
0·477
0·45 to 2·36
−0·67
−0·68
−0·70
−0·62
−0·63
− 0·66
0·015
−0·69 to −0·63
0·95
0·95
0·87
0·86
0·83
0·07
0·08
0·09
0·17
0·09
0·10
0·018
0·07 to 0·14
−0·28
−0·28
−0·30
−0·24
−0·25
− 0·27
0·011
−0·29 to −0·25
0·92
0·92
0·81
0·72
0·73
Upper
BRDN-U1
BRDN-U2
BRDN-U3
BRDN-U4
BRDN-U5
2·71
3·46
3·91
4·74
4·67
3·90
0·382
3·13 to 4·66
−1·14
−0·73
−0·41
−0·64
−0·31
− 0·64
0·144
−0·93 to −0·36
0·97
0·96
0·54
0·83
0·32
0·15
0·17
0·17
0·19
0·20
0·18
0·008
0·16 to 0·19
−0·53
−0·33
−0·14
−0·26
−0·10
− 0·27
0·077
−0·42 to −0·12
0·96
0·93
0·30
0·73
0·15
9
9
9
12
12
Average
Standard error
95% confidence interval
14
14
14
14
14
Average
Standard error
95% confidence interval
13
13
13
13
13
Average
Standard error
95% confidence interval
Copyright © 2008 John Wiley & Sons, Ltd.
Earth Surf. Process. Landforms 33, 2211–2240 (2008)
DOI: 10.1002/esp
Flow resistance in steep mountain streams
2223
Table VI. At-a-station hydraulic geometry for Chipmunk Creek
Darcy-Weisbach ff (ff = hQp)
Reach
Name
Cross
Section
Lowest
Number of
Observations
Manning’s n (n = tQy)
h
p
R2
t
y
R2
CHK-LL1
CHK-LL2
CHK-LL3
CHK-LL4
CHK-LL5
5·02
3·20
6·14
2·18
6·24
4·56
0·808
2·94 to 6·17
−0·55
−0·90
−1·00
−0·88
−1·03
− 0·87
0·085
−1·04 to −0·7
0·70
0·99
0·93
0·97
0·97
0·20
0·16
0·23
0·14
0·23
0·19
0·019
0·15 to 0·23
−0·21
−0·42
−0·46
−0·42
−0·48
− 0·40
0·048
−0·49 to −0·3
0·52
0·99
0·89
0·96
0·96
Lower
CHK-L1
CHK-L2
CHK-L3
CHK-L4
CHK-L5
1·93
2·06
2·26
1·88
1·27
1·88
0·165
1·55 to 2·21
−0·84
−0·85
−0·62
−0·63
−0·59
− 0·70
0·057
−0·82 to −0·59
0·97
0·99
0·75
0·68
0·86
0·12
0·13
0·13
0·12
0·10
0·12
0·006
0·11 to 0·13
−0·38
−0·39
−0·26
−0·25
−0·24
− 0·31
0·033
−0·37 to −0·24
0·95
0·99
0·65
0·53
0·77
Upper
CHK-U1
CHK-U2
CHK-U3
CHK-U4
CHK-U5
1·96
1·45
1·49
1·54
1·65
1·62
0·092
1·43 to 1·8
−0·43
−0·59
−0·64
−0·67
−0·94
− 0·65
0·083
−0·82 to −0·49
0·80
0·94
0·90
0·88
0·98
0·13
0·11
0·11
0·11
0·12
0·11
0·004
0·11 to 0·12
−0·15
−0·24
−0·26
−0·28
−0·43
− 0·27
0·045
−0·36 to −0·18
0·62
0·89
0·83
0·80
0·97
Tributary
CHK-Trib1
CHK-Trib2
CHK-Trib3
CHK-Trib4
CHK-Trib5
1·04
0·94
0·48
0·86
1·00
0·86
0·101
0·66 to 1·06
−0·54
−0·69
−0·75
−0·29
−0·89
− 0·63
0·101
−0·83 to −0·43
0·61
0·76
0·83
0·28
0·65
0·09
0·08
0·06
0·08
0·08
0·08
0·005
0·07 to 0·09
−0·25
−0·30
−0·33
−0·09
−0·41
− 0·27
0·054
−0·38 to −0·17
0·51
0·66
0·75
0·10
0·57
8
8
9
9
9
Average
Standard error
95% confidence interval
9
9
9
9
9
Average
Standard error
95% confidence interval
9
9
9
9
9
Average
Standard error
95% confidence interval
9
9
9
9
9
Average
Standard error
95% confidence interval
Table VII. At-a-station hydraulic geometry for Foley Creek
Darcy-Weisbach ff (ff = hQp)
Reach
Name
Cross
Section
Lower
Upper
Number of
Observations
Manning’s n (n = tQy)
h
p
R2
t
y
R2
FOL-L3
FOL-L4
FOL-L5
FOL-L6
FOL-L7
11·04
4·68
2·76
1·73
3·34
4·71
1·653
1·41 to 8·02
−1·01
− 0·60
− 0·41
− 0·31
− 0·83
− 0·63
0·129
− 0·89 to − 0·37
0·68
0·85
0·44
0·81
0·84
0·31
0·19
0·15
0·11
0·16
0·18
0·033
0·12 to 0·25
− 0·47
− 0·25
− 0·15
− 0·09
− 0·40
− 0·27
0·072
− 0·41 to − 0·13
0·60
0·76
0·24
0·53
0·81
FOL-U1
FOL-U2
FOL-U3
FOL-U4
FOL-U5
9·00
2·71
2·55
5·27
3·88
4·68
1·184
2·31 to 7·05
− 0·99
− 0·61
− 0·56
− 0·72
−1·05
− 0·79
0·100
− 0·99 to − 0·59
0·90
0·58
0·80
0·89
0·65
0·27
0·14
0·14
0·21
0·17
0·19
0·025
0·14 to 0·24
− 0·45
− 0·25
− 0·21
− 0·31
− 0·49
− 0·34
0·054
− 0·45 to − 0·24
0·85
0·45
0·68
0·83
0·57
12
12
12
12
12
Average
Standard error
95% confidence interval
12
12
12
12
12
Average
Standard error
95% confidence interval
Copyright © 2008 John Wiley & Sons, Ltd.
Earth Surf. Process. Landforms 33, 2211–2240 (2008)
DOI: 10.1002/esp
2224
D. E. Reid and E. J. Hickin
Figure 7. At-a-station hydraulic geometry and resistance to flow at cross section CHK-LL4 in the lowest reach of Chipmunk
Creek. This figure is available in colour online at www.interscience.wiley.com/journal/espl
Discussion
At-a-station analysis – power functions
The variation of resistance to flow with stream discharge at a cross section is well described by a general powerrelation of the form
ff = hQp
(7)
for the Darcy–Weisbach resistance factor ff where h and p are empirically derived coefficients and exponents (Figures
4–9). The remarkably strong relationship for most of the cross sections is expected for three reasons. First, the robust
form of the power function is well adapted to the variation of flow parameters with changing discharge (Reid, 2005).
Second, resistance to flow affects the primary flow variables of depth and velocity through forces such as shear stress
and variation of shear stress in the vertical velocity distribution. The power function has been shown to describe the
relations between discharge and mean depth and mean velocity well (Reid, 2005) and therefore it is not surprising that
there is good agreement between discharge and resistance to flow as described by the power function. Finally, there is
spurious correlation in the power relations. The Darcy–Weisbach resistance factor includes mean velocity in their
calculation and this value is derived from stream discharge (Q). Because both sides of the equation contain the same
variable (Q), the strength of the relationships reported as the R2 term is greater than expected in completely independent physical data. Indeed, it has been reported that correlations of 0·7 (Schlager et al., 1998) and values as high as 0·90
Copyright © 2008 John Wiley & Sons, Ltd.
Earth Surf. Process. Landforms 33, 2211–2240 (2008)
DOI: 10.1002/esp
Flow resistance in steep mountain streams
2225
Figure 8. At-a-station hydraulic geometry and resistance to flow at cross section CHK-L4 in the lower reach of Chipmunk
Creek. This figure is available in colour online at www.interscience.wiley.com/journal/espl
(Benson, 1965) are attainable in random data with spurious correlation. As a result, the R2 term should be interpreted
here as a measure of the fit, not as a measure of the explained variance with its usual statistical implications. Statistical
meaning aside, given sufficient data the power functions provide a good description of the resistance to flow change at
a cross section with varying discharge.
Despite the generally good description of the data by the power function at the individual cross sections, there exists
a large amount of variation between the sections in each study reach. Another way of examining the between-section
variation in a reach is to perform a statistical power analysis to calculate the number of cross sections required to
define a reasonably accurate reach-averaged value given the observed variation in the data (Table VIII). The technique
calculates the sample size required to determine the statistical difference between two means, given the standard
deviation of the sample. The analysis reveals that between five and 401 cross sections would be required to detect
a 10% difference in reach exponents and between 23 and 967 cross sections would be required to detect a 10%
difference in reach coefficients.
These results are important when considering the practicality of measuring the mean reach resistance using the
methods of this study. It is tempting to consider a study design where the results from a representative number of cross
sections could be averaged and applied to the entire reach. Considering that the absolute value of the resistance at any
stream discharge depends on both the exponent and the coefficient of the power function, the number of cross sections
required to predict a reasonably accurate reach mean depends on the less precise variable. As an extreme example, the
Middle Borden Creek reach is 112 m long and requires only five cross sections to accurately predict the mean
resistance exponent of the reach; this is equal to a cross section every 22·4 m. However, this same reach requires 898
cross sections to predict the mean resistance coefficient accurately, a cross section every 12 cm (Table VIII). Therefore, to
Copyright © 2008 John Wiley & Sons, Ltd.
Earth Surf. Process. Landforms 33, 2211–2240 (2008)
DOI: 10.1002/esp
2226
D. E. Reid and E. J. Hickin
Figure 9. At-a-station hydraulic geometry and resistance to flow at cross section CHK-U2 in the upper reach of Chipmunk
Creek. This figure is available in colour online at www.interscience.wiley.com/journal/espl
Table VIII. Power analysis of the reach exponents and coefficients
Darcy-Weisbach ff (ff = hQp)
Exponent (p)
Stream
Frosst Creek
Liumchen Creek
Borden Creek
Chipmunk Creek
Foley Creek
Reach
Name
Lower
Upper
Lower
Upper
Lower
Middle
Upper
Lowest
Lower
Upper
Tributary
Lower
Upper
Coefficient (h)
Reach
Average
Standard
Deviation
Number of
Sections Requireda
Reach
Average
Standard
Deviation
Number of
Sections Requireda
−0·96
−1·19
−0·74
−0·79
−0·90
−0·66
−0·64
−0·87
−0·70
−0·65
−0·63
−0·63
−0·79
0·19
0·23
0·17
0·36
0·26
0·03
0·32
0·19
0·13
0·19
0·23
0·29
0·22
65
58
87
330
133
5
401
76
53
129
204
332
126
0·67
1·63
5·67
11·12
1·53
1·41
3·90
4·56
1·88
1·62
0·86
4·71
4·68
0·08
1·00
1·80
8·26
0·58
1·07
0·85
1·81
0·37
0·21
0·23
3·70
2·65
23
589
160
868
224
898
77
247
62
27
109
967
504
a – number of sections required to detect a difference in values of 10%, α = 0·95, power = 80%.
Copyright © 2008 John Wiley & Sons, Ltd.
Earth Surf. Process. Landforms 33, 2211–2240 (2008)
DOI: 10.1002/esp
Flow resistance in steep mountain streams
2227
Figure 10. Distribution of the Darcy–Weisbach exponent (p) for the complete data set.
predict the absolute resistance value at any flow for the entire reach would require averaging the results from cross
sections placed every 12 cm; this is impractical. In fact, results from the most uniform reach, Lower Chipmunk Creek,
demonstrate that, at best, 62 cross sections would be required in 290 m of channel length; this is equal to a cross
section every 4·7 m.
From the results of the cross-section and reach analyses, it is apparent that the power equation describes well the
changes at a section but poorly defines a reach-average condition because of the section-to-section variability. A more
expedient method to predict the state of a reach is to ignore the mean state (i.e. trying to predict the hydraulic
conditions everywhere) and focus on critical or threshold conditions. By determining the critical cross section in a
reach for a defined purpose (e.g. the widest cross section when considering low-flow fisheries concerns or the
narrowest cross section when considering flooding concerns) and calculating resistance directly from the hydraulic
variables, it may be possible to predict when a critical state is reached for the entire stream reach. At-a-station
hydraulic geometry is well suited to this task.
Reach analysis – logarithmic functions
Keulegan (1938) suggested that the resistance to flow in the system (ff) can be related to the roughness length (ks)
through the equation
⎛ aR ⎞
1
= c log⎜ ⎟
ff
⎝ ks ⎠
(8)
where c and a are constants considered equal to 2·03 and 12·2 respectively (see Griffiths, 1981, p. 908, for a
discussion). Although the appropriateness of this simple, conventional model of flow resistance has been questioned in
recent studies of flow in rough channels (see Smart et al., 2002, and Katul et al., 2002) the near-vertical nature of the
Keulegan equation at low relative roughness describes well the present data (Figure 11). When the C50 value is set
to unity, and the roughness length is dependent on the median grain size alone (an estimate of the grain resistance in
the system), all of the data plot above the function. The offset between the data and the C50 = 1 curve is thought to be
an expression of the form resistance in the system (Millar, 1999). The best-fit curve for the entire data set has a
C50 = 8·6, indicating that the total resistance consists of 8·6 times as much form resistance as grain resistance.
Also evident in the logarithmic plot are the near-vertical nature of the function and the range of relative roughness
observed (Figure 11). The near-vertical domain is an expression of the sensitivity of the observed resistance to relative
Copyright © 2008 John Wiley & Sons, Ltd.
Earth Surf. Process. Landforms 33, 2211–2240 (2008)
DOI: 10.1002/esp
2228
D. E. Reid and E. J. Hickin
Figure 11. Keulegan function curve fitted to the Darcy–Weisbach resistance coefficient versus relative roughness for the entire
data set.
roughness and therefore depth of flow. In a reach where the bed material is of a constant size, flow resistance changes
quickly with the flow depth. This effect is often ignored in engineering studies and published guides to roughness
estimation, presumably because of the focus on high instream flows (bankfull) and large rivers. As relative roughness
increases above about 100, the logarithmic function flattens, reflecting the diminishing effect of changes in depth of
flow on resistance.
Much of the scatter in Figure 11 can be attributed to the physical differences among the various stream reaches
included in the study. Each reach has unique characteristics (e.g. grain-size distribution, amount of vegetation in the
channel or irregularities in the channel longitudinal profile, Figure 3) that not only produce different absolute resistance values for each depth of flow but also have a unique ratio of grain to form resistance as expressed in the varying
C50 values (Figures 12–14).
Copyright © 2008 John Wiley & Sons, Ltd.
Earth Surf. Process. Landforms 33, 2211–2240 (2008)
DOI: 10.1002/esp
Flow resistance in steep mountain streams
2229
Figure 12. Keulegan function curves fitted to the Darcy–Weisbach resistance coefficient versus relative roughness data at Frosst
and Borden Creeks.
Figure 13. Keulegan function curves fitted to the Darcy–Weisbach resistance coefficient versus relative roughness data at
Liumchen and Foley Creeks.
Copyright © 2008 John Wiley & Sons, Ltd.
Earth Surf. Process. Landforms 33, 2211–2240 (2008)
DOI: 10.1002/esp
2230
D. E. Reid and E. J. Hickin
Figure 14. Keulegan function curves fitted to the Darcy–Weisbach resistance coefficient versus relative roughness data at
Chipmunk Creek.
C50 values were calculated as the median value of the function when fitted to each data point using only the negative
limb of the function (as discussed below). This curve-fitting method was used instead of a least-squares method
because of the very steep slope of the function in the data range. The method of reporting the median C50 value of the
function fitted to each point weights each data point equally. This method of fit was checked against a least-squares fit
for the data in Lower Frosst Creek, which lacks the extreme low-relative-resistance/low-relative-roughness values of
the other reaches. In this case, the two methods produced a C50 value of 6·9.
Much of the scatter present in the entire data set has been reduced in the reach-by-reach plots (Figures 12–14). This
suggests that the form of the logarithmic function is a reasonable representation of the changing resistance in the
reach. There still remains, however, a significant degree of scatter about the functions. This is likely due to section-tosection differences in the resistance at a single stream discharge coupled with measurement error.
The C50 values for the study sites range from 5·4 to 17·1, with a mean of 9·6 (Figures 12–14), indicating that form
resistance is between five and 17 times the grain resistance. These values compare to a best-fit roughness length (ks)
value of 6·8 D50 reported by Bray (1982).
Form resistance and C values
If it is assumed that the calculated C values are an estimate of the form resistance in each reach, it is then possible to
correlate these values to direct measures of the various form-roughness components. A first step in this analysis is to
correlate the C values with the grain-sorting coefficient (σ), a measure of the ratio of the larger (D84) bed-material
fraction to the median (D50) fraction (Millar, 1999):
σ =
D84
D50
(9)
The estimate of form roughness based on the C50 value correlates well with the sorting coefficient (Figure 15(A)).
This suggests that the sorting coefficient can be used as a measure of the degree to which the larger grains protrude
from the bed and that this measure explains a significant degree of the form roughness observed in the reaches. This
Copyright © 2008 John Wiley & Sons, Ltd.
Earth Surf. Process. Landforms 33, 2211–2240 (2008)
DOI: 10.1002/esp
Flow resistance in steep mountain streams
2231
Figure 15. Form-roughness estimates (C50 and C84) versus the sorting coefficient.
result is consistent with the findings of Lee and Ferguson (2002), who use a relative submergence term (1 − 0·1ks/R)
based on the work of Thompson and Campbell (1979) to account for flow blocking by the large bed elements. In
mountain streams, the larger bed elements are a significant source of form roughness and can be considered separate
from the grain roughness (Millar, 1999). Both their spacing (roughness concentration – the proportion of the bed area
occupied by significantly projecting boulders), and their projection from the bed (relative-roughness area – the proportion of the total cross-sectional area occupied by significantly projecting boulders) are known to contribute significantly to the total resistance to flow (Bathurst, 2002; Ferro, 2003).
In contrast, a measure of the form roughness of the reach based on the D84 grain size (C84) does not correlate well
with the sorting coefficient (Figure 15(B)). One explanation for this can be found in the work of Millar (1999), who
suggests that an estimate of the grain resistance of the system based on the larger clast size overestimates the grain
roughness portion of the resistance. Hence, some of the form roughness is included in the grain roughness estimate,
perhaps explaining the poor correlation between C84 and the sorting coefficient. Alternatively, the poor correlation can
be due in part to the limited range of C84 values calculated. Limiting the range of y values (variance in the data)
necessarily decreases R2. Another interpretation of the limited C84 range is to suggest that a C84 value of 3.6 (the
median value for the entire data set) provides a reasonable first estimate of the form roughness (Figure 15(B)). This
value corresponds well to the range of values reported in the literature (e.g. Leopold et al., 1964; Limerinos, 1970;
Hey, 1979; Bray, 1979; Bathurst, 1985; Lee and Ferguson, 2002).
Surprisingly, the strong correlation between the sorting coefficient and C50 was not observed in Millar’s (1999)
work. This difference could be due to several factors. First, the 1999 study used data sets collected by four separate
authors, while this study uses a uniform protocol for data collection and analysis. That is, the different techniques used
by the various authors in the 1999 study obscures the relationship. In addition, this study is confined to a small set of
similar channel reaches and a small range of flows. This limited data set may be unique and its findings may not
extrapolate to a wider range of rivers such as those in Millar’s work.
Despite the strong correlation between C50 and the sorting coefficient there remains a significant amount of scatter
about the relationship. Our attempts to correlate C50 with a quantitative measure of the bed-profile regularity (measured as the standard deviation of the surveyed bed elevation from a best-fit line fitted to the data) or qualitative
Copyright © 2008 John Wiley & Sons, Ltd.
Earth Surf. Process. Landforms 33, 2211–2240 (2008)
DOI: 10.1002/esp
2232
D. E. Reid and E. J. Hickin
Figure 16. Form-roughness estimate (C50) versus the bed-profile regularity with all data included (A) and without two outliers (B).
measures of vegetation, large woody debris and presence of bars have been largely unsuccessful. It is only when the
two outliers of Upper Liumchen and Lower Foley Creek are ignored that the profile irregularity correlates well with
the calculated form roughness (Figure 16(B)). There is no justification, however, to ignore these outliers in the data.
There may be several reasons for the apparent lack of correlation between the form-roughness estimate and the
secondary roughness observations. First, the qualitative estimates are based on researcher observation only. There are
no quantitative measurements of these factors, so a substantial amount of error could be involved with the estimates,
obscuring a correlation. Careful quantitative measures of the secondary effects are needed to answer this question
(e.g., a measure as simple as the relative bed area covered by these secondary sources may correlate to the formresistance component). Generally, in step-pool reaches a measure of the grain-size distribution specific to the steps
may improve the correlations (Lee and Ferguson, 2002), although this morphology is uncommon in our particular data
set. Second, it is likely that these factors add only minor amounts of roughness to the system; the principal source of
roughness is derived from the large bed elements and aggregations of these elements. It is often difficult to measure
the influence of smaller, secondary factors when faced with noise or error about a primary effect.
Steady-uniform-flow assumptions
This study was initiated in part to address the practical problem of estimating flow resistance for hydraulic calculations in small steep mountain streams that are candidates for ‘run-of-the-river’ hydropower generating stations in
British Columbia. The data set we present here is very useful for this purpose and clearly indicates that very high
resistance coefficients must be used in relation to the mean bed slope if meaningful predictions of the mean flow are
to be made in these environments.
Although this engineering outcome is simple and straightforward, the scientific implications of this work are
complicated because the flows are very complex. Implicit in the derivation of all flow resistance equations is the
presence of steady uniform flow. Such flows are tolerably approximated in low-slope rivers but they certainly do not
characterize the highly turbulent mountain streams examined here. All of the resistance calculations in this study are
referred to what is essentially a locally non-existent average bed slope. The reality is that energy is consumed at nearvertical steps in the bed at a rate much higher than the reach average and at a much lower than average rate through
Copyright © 2008 John Wiley & Sons, Ltd.
Earth Surf. Process. Landforms 33, 2211–2240 (2008)
DOI: 10.1002/esp
Flow resistance in steep mountain streams
2233
Figure 17. Partial (A) and complete (B) Keulegan logarithmic functions at the lowest reach of Chipmunk Creek.
the relatively low-slope intervening sections of the reaches. As a result, flows through most of these channels have
been referred to a reach slope that is mainly steeper than the local slope and consequently the apparent resistance to
flow is extremely high. These resistance values are appropriate to predicting the mean flow in the reach from the
average bed slope but they may not reflect the real level of flow resistance encountered locally. This is a major
constraint on the science of this work. A more precise and appropriate characterization of flow resistance, however,
would have to be based on a high-resolution map of the water-surface elevation and this is simply not feasible in the
context of the present project.
The positive domain of the Keulegan function
The Keulegan equation is a logarithmic function (Equation (8)) that has a mathematical singularity at very small
values of relative roughness. Because this domain of the function has received little attention from river scientists, the
singularity has not been discussed in the literature. For example, in Figure 17(B) the ff function peaks at the singularity (at about R/D50 = 0·8) and then declines.
Although it seems likely that this positive domain of the ff/D50 relation is simply a mathematical artefact with no
physical significance, curiously there may be reasons to expect this actual behaviour at extremely low values of R/D50
in these steep mountain channels. As discharge declines and the relative roughness increases (R/D50 declines) and the
Copyright © 2008 John Wiley & Sons, Ltd.
Earth Surf. Process. Landforms 33, 2211–2240 (2008)
DOI: 10.1002/esp
2234
D. E. Reid and E. J. Hickin
channel becomes dominated by individual boulders and boulder clusters that divert the flow into sub-channels between the boulders, flow may be more hydraulically efficient than at higher flows when these obstructions are
encountered as submerged objects. Obviously, the particular resistance response at a given location will depend on the
grain-size distribution of the gravel bed and on the suite of structures formed by the larger clasts. From a management
perspective these low-flow conditions are critical to questions of fish passage and bioenergetics and deserve much
more focused attention than they have been given to date.
Conclusions
In the present study, flow resistance is calculated for a small set of steep streams in Southwest British Columbia
Canada. The hydraulics of the flow in these study reaches reflects the interaction of stream slope, bed-material size
and distribution, and stream morphology. The governing conditions in these settings are expected to be similar
throughout the mountainous regions of the world and thus the authors expect that these results are widely applicable.
Without comparative studies, however, this assumption cannot yet be tested.
Flow resistance in small, steep mountain streams is extremely high through most of the flow range for two main
reasons. First, these streams represent extremely rough environments where streamflow is forced over and between
bed material that is often of equal size to, or larger than, the depth of flow. Logs, bars, vegetation and irregularities in
the longitudinal profile of the bed (i.e. bedforms) also add to resistance in these environments (Lee and Ferguson,
2002; Curran and Wohl, 2003; MacFarlane and Wohl, 2003). Second, the uniform-flow assumptions often made in
hydraulic analyses are largely violated, as the flow is forced to wander over and between the larger bed elements. This
produces short tumbling areas of flow separated by long flat sections. In these conditions bed slope is not equal to the
water-surface or energy slope but it is easily measured in field settings so it is commonly reported in the literature.
Resistance values calculated in this analysis are those appropriate to a mean bed-slope context.
In small, steep boulder-bed streams, resistance to flow varies considerably with the depth of flow, much more so
than in large rivers. For the present data, the Darcy–Weisbach resistance factor (ff) varies over six orders of magnitude for
a mean-depth range that averages 20 cm and is less than 36 cm. Several researchers have reached this same conclusion
(Butler et al., 1978; Sargent, 1979; Jarrett, 1990; Lee and Ferguson, 2002), yet it warrants repeating. This has implications
for the development of guidelines for resistance estimation such as those of Barnes (1967), because they not only must
include a photo example of the cross section but also a curve indicating the range of resistance possible for the
section. Example photos of the various stages used to calculate the curve are required if the guide is to be widely used.
Flow resistance at a section is reasonably described by the power function. Coefficients and exponents calculated
for each cross section, however, vary widely in a reach precluding the determination of a useful reach average. A
reasonable reach average is required to predict the flow resistance at unmeasured cross sections and this does not seem
possible with the methods used in this study. Instead, focusing study on limiting or controlling cross sections and then
defining the variation of resistance with stream discharge at these sections is possible with the current methods.
Like power curves fitted to the cross-section data, the Keulegan logarithmic curves describe the reach resistance
data well. Best-fit C50 values correlate well with the measured grain-sorting measure. This result is unique in the
literature and may reflect the uniform methodology and the limited range of data examined. No simple correlations
between the calculated form roughness component of the roughness length and measures of bed irregularity, logs,
vegetation and bars were observed.
Acknowledgements
The authors wish to thank Scott Babakaiff of the B.C. Ministry of Environment for his early ideas and interest, Robert Millar of the
University of British Columbia for his initial work on the subject and critical review of this paper, field assistants Amada Denney, Emily
Huxter and Jane Bachman of Simon Fraser University for their hard work and Kaz Shimamura for his mapping help. This research
was funded in part by a Natural Sciences and Engineering Research Council (NSERC) Discovery Grant to Hickin and by research grants
to Reid from the Province of British Columbia (Ministry of Water, Land and Air Protection) and from Simon Fraser University.
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Copyright © 2008 John Wiley & Sons, Ltd.
Earth Surf. Process. Landforms 33, 2211–2240 (2008)
DOI: 10.1002/esp
Copyright © 2008 John Wiley & Sons, Ltd.
19-Jun-03
18-Jul-03
24-Jul-03
28-Jul-03
01-Aug-03
03-Sep-03
10-Sep-03
01-Oct-03
0·615
0·397
0·329
0·293
0·254
0·182
0·149
0·116
0·574
0·397
0·329
0·293
0·254
0·182
0·149
0·116
1·632
0·652
1·553
0·927
19-Jun-03
18-Jul-03
24-Jul-03
28-Jul-03
01-Aug-03
03-Sep-03
10-Sep-03
01-Oct-03
24-Oct-03
10-Nov-03
21-Nov-03
17-Dec-03
6·43
5·65
5·55
5·25
5·10
5·05
4·85
4·75
6·70
5·45
5·40
5·75
5·25
5·00
4·55
4·70
6·50
5·75
7·15
6·70
(m)
3
(m /s)
width
discharge
Date
surface
Stream
Water
Frosst creek
Appendix
0·20
0·19
0·16
0·17
0·17
0·15
0·15
0·15
0·19
0·20
0·18
0·16
0·16
0·11
0·12
0·12
0·27
0·19
0·26
0·21
(m)
depth
Mean
Darcy–
0·48
0·38
0·37
0·32
0·30
0·25
0·21
0·16
ff
1·25
2·03
2·14
2·24
2·49
1·49
1·98
3·73
0·42
0·73
0·51
0·63
1·93
2·95
2·68
3·72
4·29
5·36
8·05
13·89
XS FST-U1
0·45
0·36
0·34
0·31
0·30
0·32
0·28
0·21
0·93
0·60
0·83
0·67
XS FST-L1
(m)
velocity Weisbach
Mean
0·12
0·14
0·13
0·16
0·17
0·19
0·23
0·30
0·09
0·12
0·12
0·12
0·13
0·10
0·11
0·15
0·06
0·07
0·06
0·07
n
Manning’s
Water
7·25
6·90
7·15
7·15
7·10
6·80
6·70
6·60
7·20
6·30
6·05
6·50
5·85
5·35
5·90
5·70
6·20
4·40
6·35
5·90
(m)
width
surface
Mean
0·22
0·21
0·20
0·19
0·20
0·19
0·18
0·19
0·18
0·18
0·17
0·14
0·16
0·12
0·12
0·12
0·25
0·20
0·28
0·22
(m)
1·28
1·99
2·17
2·08
2·78
2·29
3·74
5·87
0·29
0·52
0·51
0·61
0·38
0·27
0·23
0·22
0·18
0·14
0·12
0·09
3·62
6·43
8·66
9·07
13·47
20·47
27·59
52·58
XS FST-U2
0·44
0·35
0·33
0·31
0·28
0·27
0·21
0·17
1·07
0·72
0·86
0·70
ff
Weisbach
Darcy–
XS FST-L2
(m)
depth velocity
Mean
0·17
0·22
0·25
0·26
0·31
0·38
0·44
0·62
0·10
0·12
0·12
0·12
0·14
0·12
0·15
0·19
0·05
0·06
0·06
0·07
n
Manning’s
Water
5·50
5·35
4·95
5·45
4·70
4·65
4·60
4·15
7·35
7·00
6·85
6·70
6·65
6·00
5·95
6·00
6·65
5·40
7·20
6·95
(m)
width
surface
Mean
Darcy–
0·20
0·13
0·11
0·11
0·11
0·11
0·11
0·12
0·22
0·20
0·18
0·18
0·18
0·14
0·15
0·13
0·24
0·18
0·27
0·21
(m)
0·57
0·58
0·61
0·50
0·49
0·34
0·29
0·24
ff
2·32
3·28
3·62
4·29
5·23
4·13
6·79
8·87
0·30
0·53
0·54
0·72
1·36
0·89
0·68
1·02
1·04
2·27
3·15
4·88
XS FST-U3
0·36
0·29
0·26
0·24
0·22
0·22
0·17
0·14
1·04
0·68
0·81
0·63
XS FST-L3
(m)
depth velocity Weisbach
Mean
0·10
0·08
0·06
0·08
0·08
0·12
0·14
0·17
0·13
0·15
0·16
0·17
0·19
0·16
0·21
0·24
0·05
0·06
0·07
0·07
n
Manning’s
Water
6·75
6·20
6·20
6·25
6·20
5·90
5·85
5·85
5·85
5·60
5·50
5·40
5·35
4·65
4·80
4·65
8·55
5·45
6·95
6·40
(m)
width
surface
0·27
0·22
0·22
0·20
0·21
0·17
0·18
0·17
0·21
0·19
0·18
0·19
0·18
0·15
0·16
0·15
0·23
0·20
0·27
0·25
(m)
depth
Mean
1·23
1·86
2·32
3·21
3·40
3·23
5·54
6·95
0·47
0·74
0·54
0·99
0·34
0·29
0·24
0·24
0·20
0·18
0·14
0·11
5·34
5·94
8·41
8·20
12·54
12·79
20·64
30·02
XS FST-U4
0·48
0·37
0·33
0·28
0·27
0·25
0·20
0·17
0·82
0·60
0·82
0·58
ff
Weisbach
Darcy–
XS FST-L4
(m)
velocity
Mean
0·21
0·21
0·25
0·24
0·30
0·30
0·38
0·46
0·09
0·12
0·13
0·15
0·15
0·15
0·19
0·21
0·06
0·07
0·07
0·09
n
Manning’s
Water
5·75
5·30
5·35
5·15
5·05
5·10
5·10
4·90
5·15
4·00
3·80
3·75
3·30
3·25
3·30
3·15
10·20
5·55
9·70
8·20
(m)
width
surface
Mean
0·28
0·27
0·27
0·28
0·28
0·21
0·21
0·21
0·22
0·20
0·21
0·21
0·22
0·16
0·16
0·16
0·19
0·19
0·19
0·19
(m)
depth
Mean
Darcy–
1·08
1·12
1·69
1·87
2·24
1·64
2·61
3·79
0·35
0·63
0·39
0·78
0·39
0·27
0·23
0·20
0·18
0·17
0·14
0·11
4·04
8·00
10·89
14·77
18·11
16·59
25·34
36·86
XS FST-U5
0·52
0·49
0·41
0·38
0·35
0·36
0·28
0·23
0·86
0·63
0·83
0·59
ff
Weisbach
XS FST-L5
(m)
velocity
0·18
0·25
0·29
0·34
0·38
0·35
0·43
0·52
0·09
0·09
0·11
0·12
0·13
0·10
0·13
0·16
0·05
0·07
0·05
0·07
n
Manning’s
2236
D. E. Reid and E. J. Hickin
Earth Surf. Process. Landforms 33, 2211–2240 (2008)
DOI: 10.1002/esp
0·953
0·591
0·511
0·362
0·238
0·249
0·203
0·616
5·523
1·155
4·672
2·343
2·088
0·953
0·591
0·511
0·362
0·238
0·249
0·203
0·616
5·523
1·155
4·672
3·278
2·343
2·088
08-Jul-03
18-Jul-03
23-Jul-03
30-Jul-03
04-Sep-03
10-Sep-03
01-Oct-03
10-Oct-03
24-Oct-03
11-Nov-03
21-Nov-03
09-Dec-03
17-Dec-03
08-Jul-03
18-Jul-03
23-Jul-03
30-Jul-03
04-Sep-03
10-Sep-03
01-Oct-03
10-Oct-03
24-Oct-03
11-Nov-03
21-Nov-03
06-Dec-03
09-Dec-03
17-Dec-03
Date
Stream
discharge
(m3/s)
Liumchen creek
Copyright © 2008 John Wiley & Sons, Ltd.
7·45
5·95
5·80
5·85
5·55
5·60
5·60
6·75
10·70
7·90
11·05
10·00
9·10
8·75
9·30
9·10
9·80
9·40
9·20
9·00
8·95
10·25
15·15
12·30
15·10
13·45
13·45
0·34
0·36
0·33
0·33
0·27
0·27
0·27
0·33
0·54
0·31
0·53
0·46
0·40
0·41
0·24
0·21
0·19
0·19
0·14
0·14
0·15
0·21
0·44
0·24
0·44
0·29
0·27
Water
surface Mean
width depth
(m)
(m)
Darcy–
Weisbach
ff
3·64
6·01
7·08
11·86
11·17
10·20
17·32
6·36
1·70
4·33
2·24
2·08
2·26
0·38
0·28
0·27
0·19
0·16
0·17
0·14
0·28
0·96
0·46
0·80
0·71
0·65
0·58
8·53
16·20
16·20
31·97
39·58
34·23
49·90
15·31
2·06
5·23
2·96
3·30
3·38
4·28
XS LCH-U1
0·42
0·30
0·27
0·20
0·18
0·19
0·15
0·29
0·82
0·39
0·71
0·60
0·57
XS LCH-L1
Mean
velocity
(m)
0·27
0·38
0·37
0·52
0·56
0·52
0·63
0·36
0·14
0·21
0·17
0·18
0·18
0·20
0·17
0·21
0·23
0·29
0·27
0·26
0·34
0·22
0·13
0·18
0·15
0·13
0·14
Manning’s
n
10·35
10·05
10·05
10·10
9·85
9·25
9·50
10·35
14·85
10·75
14·75
13·75
11·80
11·05
12·20
11·50
10·95
10·95
10·55
10·75
10·50
11·50
15·35
12·65
15·05
14·30
14·40
Water
surface
width
(m)
0·37
0·32
0·32
0·30
0·23
0·24
0·25
0·34
0·53
0·33
0·50
0·47
0·43
0·43
0·23
0·20
0·18
0·17
0·11
0·10
0·11
0·18
0·43
0·22
0·42
0·30
0·28
Darcy–
Weisbach
ff
5·13
7·93
7·51
11·62
7·22
17·85
9·57
4·97
1·64
3·50
1·99
2·64
2·77
0·25
0·19
0·16
0·12
0·11
0·11
0·09
0·17
0·70
0·32
0·63
0·51
0·47
0·44
21·01
32·90
44·34
73·56
73·17
77·69
119·43
40·87
3·88
11·67
4·62
6·56
7·06
8·13
XS LCH-U2
0·34
0·26
0·25
0·20
0·20
0·13
0·18
0·31
0·83
0·41
0·74
0·55
0·52
XS LCH-L2
Mean
Mean
depth velocity
(m)
(m)
0·43
0·53
0·61
0·78
0·75
0·78
0·97
0·60
0·20
0·32
0·21
0·25
0·26
0·28
0·20
0·24
0·23
0·28
0·21
0·33
0·24
0·19
0·12
0·16
0·14
0·15
0·15
Manning’s
n
8·50
6·80
5·65
5·95
4·35
4·20
4·50
7·15
16·90
10·35
16·95
15·45
12·35
12·20
16·10
15·40
14·30
14·95
14·05
14·20
13·80
16·65
18·65
16·50
18·60
18·35
18·15
0·24
0·24
0·22
0·20
0·14
0·16
0·18
0·21
0·36
0·26
0·35
0·31
0·29
0·30
0·23
0·19
0·21
0·17
0·15
0·14
0·14
0·20
0·41
0·21
0·37
0·25
0·25
Water
surface Mean
width depth
(m)
(m)
Darcy–
Weisbach
ff
0·46
0·37
0·41
0·30
0·38
0·38
0·26
0·40
0·92
0·43
0·79
0·69
0·65
0·58
4·11
6·24
4·78
8·05
3·66
3·97
9·47
4·78
1·57
5·11
2·10
2·38
2·53
3·24
9·08
11·83
18·00
23·54
29·76
23·71
35·20
14·58
2·05
4·76
2·05
2·61
3·25
XS LCH-U3
0·26
0·21
0·17
0·14
0·12
0·13
0·10
0·19
0·72
0·34
0·69
0·51
0·46
XS LCH-L3
Mean
velocity
(m)
0·18
0·22
0·19
0·24
0·15
0·16
0·26
0·19
0·12
0·20
0·14
0·14
0·15
0·16
0·26
0·29
0·37
0·41
0·45
0·39
0·48
0·33
0·14
0·19
0·14
0·14
0·16
Manning’s
n
Flow resistance in steep mountain streams
2237
Earth Surf. Process. Landforms 33, 2211–2240 (2008)
DOI: 10.1002/esp
no observations
1·562
9·20
0·444
7·75
1·421
9·25
0·563
8·15
0·464
7·80
0·628
0·576
0·334
0·297
0·233
0·084
0·085
0·078
0·117
1·840
0·639
1·535
0·738
0·487
0·713
0·334
0·297
0·233
0·084
0·085
0·078
0·117
1·840
0·639
1·535
0·738
0·487
10-Oct-03
25-Oct-03
06-Nov-03
20-Nov-03
09-Dec-03
15-Dec-03
09-Jul-03
10-Jul-03
19-Jul-03
24-Jul-03
28-Jul-03
04-Sep-03
09-Sep-03
28-Sep-03
10-Oct-03
25-Oct-03
06-Nov-03
20-Nov-03
09-Dec-03
15-Dec-03
Copyright © 2008 John Wiley & Sons, Ltd.
09-Jul-03
19-Jul-03
24-Jul-03
28-Jul-03
04-Sep-03
09-Sep-03
28-Sep-03
10-Oct-03
25-Oct-03
06-Nov-03
20-Nov-03
09-Dec-03
15-Dec-03
3·30
3·15
2·80
2·80
2·75
2·95
2·95
2·90
4·05
2·70
4·10
3·00
2·70
5·80
5·60
4·85
4·65
4·30
4·20
4·00
4·00
4·35
6·40
4·80
6·25
4·95
4·45
6·85
5·30
4·45
3·75
dry
dry
dry
0·327
0·151
0·047
0·029
0·001
0·002
0·001
09-Jul-03
19-Jul-03
24-Jul-03
28-Jul-03
04-Sep-03
09-Sep-03
28-Sep-03
(m)
3
(m /s)
width
discharge
Date
surface
Stream
Water
Borden creek
0·35
0·29
0·31
0·29
0·21
0·19
0·19
0·22
0·43
0·34
0·38
0·32
0·30
0·17
0·17
0·15
0·16
0·15
0·09
0·09
0·08
0·10
0·29
0·20
0·25
0·20
0·17
0·20
0·13
0·19
0·14
0·12
0·14
0·14
0·10
0·09
(m)
depth
Mean
Darcy–
ff
0·73
1·81
0·76
1·41
1·23
3·15
8·20
23·28
31·04
0·85
1·03
1·37
1·93
2·44
4·08
3·77
3·42
2·94
0·59
0·91
0·54
0·75
0·84
0·62
0·37
0·34
0·29
0·15
0·16
0·14
0·19
1·06
0·69
0·99
0·77
0·60
4·37
10·61
12·89
16·47
47·70
40·41
53·15
32·12
1·86
3·38
1·93
2·60
4·00
XS BRDN-U1
0·65
0·59
0·47
0·41
0·36
0·22
0·23
0·23
0·27
1·00
0·67
0·98
0·75
0·65
XS BRDN-M1
0·85
0·44
0·81
0·51
0·50
0·34
0·21
0·11
0·09
XS BRDN-L1
(m/s)
velocity Weisbach
Mean
0·19
0·29
0·32
0·36
0·58
0·53
0·61
0·48
0·13
0·17
0·13
0·15
0·18
0·08
0·08
0·09
0·11
0·13
0·15
0·15
0·14
0·13
0·07
0·08
0·06
0·07
0·08
0·07
0·11
0·07
0·10
0·09
0·14
0·23
0·37
0·42
n
Manning’s
Water
4·10
3·65
3·55
3·25
3·00
3·10
3·00
3·20
6·50
5·10
7·25
5·50
5·25
7·60
7·60
6·30
6·05
5·95
5·45
5·40
5·50
5·50
8·55
6·15
8·00
6·70
6·05
5·80
4·45
5·40
4·60
4·25
4·65
2·95
2·90
2·20
dry
dry
dry
(m)
width
surface
Mean
0·29
0·22
0·22
0·23
0·13
0·14
0·15
0·16
0·30
0·25
0·26
0·27
0·22
0·16
0·17
0·14
0·13
0·12
0·08
0·09
0·08
0·10
0·24
0·17
0·25
0·18
0·16
0·31
0·21
0·32
0·21
0·21
0·15
0·15
0·07
0·09
(m)
ff
Weisbach
Darcy–
1·01
2·31
1·10
1·66
1·81
1·77
3·14
4·03
10·21
1·37
1·79
2·22
1·98
2·39
4·79
5·90
4·85
4·95
0·61
0·98
0·86
1·09
1·23
0·59
0·41
0·38
0·32
0·21
0·20
0·18
0·23
0·96
0·50
0·81
0·49
0·42
4·28
7·10
8·07
11·83
16·87
18·81
24·23
17·14
1·74
5·33
2·17
6·15
6·84
XS BRDN-U2
0·51
0·45
0·37
0·38
0·33
0·19
0·18
0·19
0·21
0·91
0·61
0·78
0·60
0·52
XS BRDN-M2
0·87
0·48
0·84
0·57
0·53
0·47
0·34
0·22
0·15
XS BRDN-L2
(m/s)
depth velocity
Mean
0·19
0·23
0·24
0·30
0·33
0·35
0·40
0·34
0·12
0·20
0·13
0·22
0·23
0·10
0·11
0·12
0·11
0·12
0·16
0·18
0·16
0·17
0·07
0·08
0·08
0·09
0·09
0·09
0·13
0·10
0·11
0·11
0·11
0·14
0·15
0·24
n
Manning’s
Water
7·30
5·95
5·55
5·45
4·85
4·95
4·70
5·10
8·30
7·05
8·40
7·65
7·00
7·30
7·50
6·50
6·45
6·70
4·90
5·00
4·90
5·25
9·75
7·20
9·40
7·75
6·45
9·30
7·40
9·30
7·25
6·55
8·45
6·65
4·75
4·50
dry
dry
dry
(m)
width
surface
Mean
Darcy–
0·18
0·18
0·17
0·17
0·08
0·08
0·08
0·09
0·33
0·17
0·30
0·17
0·13
0·17
0·15
0·17
0·15
0·13
0·09
0·09
0·08
0·10
0·23
0·15
0·21
0·17
0·16
0·22
0·15
0·21
0·16
0·15
0·12
0·11
0·09
0·09
(m)
ff
1·02
2·37
1·05
1·65
1·76
3·40
7·14
16·84
38·54
1·36
1·31
3·96
3·58
3·92
5·14
6·08
4·87
3·80
0·71
0·86
0·75
1·20
1·40
0·55
0·32
0·31
0·25
0·21
0·22
0·20
0·27
0·68
0·53
0·61
0·57
0·52
3·21
9·55
9·72
14·74
10·88
8·66
11·55
6·91
3·85
3·43
4·43
2·87
2·84
XS BRDN-U3
0·51
0·50
0·30
0·30
0·27
0·19
0·18
0·19
0·23
0·83
0·61
0·77
0·55
0·49
XS BRDN-M3
0·75
0·41
0·73
0·50
0·47
0·31
0·20
0·12
0·08
XS BRDN-L3
(m/s)
depth velocity Weisbach
Mean
0·15
0·26
0·26
0·32
0·24
0·22
0·25
0·20
0·18
0·15
0·19
0·14
0·14
0·10
0·09
0·17
0·15
0·16
0·17
0·19
0·16
0·15
0·07
0·08
0·08
0·09
0·10
0·09
0·13
0·09
0·11
0·11
0·15
0·21
0·31
0·46
n
Manning’s
Water
6·20
6·05
5·95
5·95
5·10
5·00
4·90
5·45
7·85
6·25
8·35
6·35
5·75
7·10
7·10
6·85
6·95
6·70
6·15
6·20
6·05
6·30
6·00
5·10
5·90
5·00
4·90
9·30
6·60
9·20
7·50
7·10
10·60
9·40
8·75
8·65
3·40
3·75
2·50
(m)
width
surface
0·24
0·19
0·20
0·18
0·09
0·10
0·09
0·11
0·30
0·24
0·27
0·22
0·19
0·21
0·22
0·19
0·19
0·17
0·11
0·11
0·11
0·12
0·44
0·37
0·44
0·39
0·35
0·26
0·23
0·22
0·19
0·18
0·16
0·13
0·11
0·11
0·08
0·07
0·06
(m)
depth
Mean
ff
Weisbach
Darcy–
1·59
6·89
1·13
3·23
3·80
12·17
22·98
133
300
7 464
4 711
12 685
2·68
3·49
6·48
7·51
9·37
14·00
15·02
15·51
11·26
1·77
6·20
2·44
5·32
8·51
0·47
0·29
0·25
0·22
0·17
0·16
0·17
0·19
0·77
0·43
0·68
0·53
0·45
5·98
13·22
17·31
21·39
17·73
21·72
17·25
18·37
2·77
7·10
3·29
4·39
5·12
XS BRDN-U4
0·41
0·37
0·25
0·23
0·20
0·13
0·13
0·12
0·15
0·70
0·34
0·59
0·38
0·28
XS BRDN-M4
0·65
0·29
0·71
0·39
0·35
0·19
0·12
0·05
0·03
0·01
0·01
0·004
XS BRDN-L4
(m/s)
velocity
Mean
0·22
0·31
0·35
0·39
0·32
0·36
0·31
0·33
0·15
0·23
0·16
0·18
0·19
0·14
0·16
0·22
0·23
0·26
0·29
0·30
0·30
0·27
0·13
0·23
0·15
0·22
0·27
0·11
0·23
0·09
0·15
0·16
0·29
0·38
0·90
1·35
6·36
4·96
7·95
n
Manning’s
Water
5·35
4·30
4·70
4·55
2·20
2·05
2·25
2·85
6·45
5·05
6·30
5·35
4·55
6·50
6·55
6·35
6·10
6·05
5·15
5·05
4·75
5·60
8·15
6·00
8·20
6·50
5·55
5·40
4·80
5·50
5·10
4·90
5·20
4·65
3·85
3·50
2·10
2·10
1·90
(m)
width
surface
Mean
0·27
0·20
0·23
0·22
0·13
0·12
0·13
0·13
0·32
0·26
0·32
0·26
0·25
0·20
0·18
0·17
0·15
0·14
0·08
0·08
0·08
0·10
0·25
0·18
0·23
0·18
0·15
0·36
0·21
0·31
0·22
0·21
0·20
0·19
0·15
0·14
0·05
0·05
0·03
(m)
depth
Mean
Darcy–
ff
Weisbach
1·39
2·87
1·14
2·29
2·48
5·37
15·39
57·18
101
546
602
912
1·94
1·68
3·64
3·15
3·77
4·40
3·95
4·32
4·30
0·67
1·06
0·77
0·97
1·00
0·49
0·39
0·28
0·24
0·30
0·34
0·27
0·32
0·89
0·49
0·77
0·53
0·44
5·89
7·05
16·02
20·62
7·25
5·45
8·89
6·85
2·18
5·88
2·87
4·93
6·89
XS BRDN-U5
0·47
0·48
0·31
0·32
0·28
0·20
0·21
0·20
0·22
0·89
0·60
0·80
0·63
0·57
XS BRDN-M5
0·79
0·43
0·82
0·49
0·46
0·31
0·17
0·08
0·06
0·01
0·02
0·01
XS BRDN-L5
(m/s)
velocity
0·22
0·23
0·35
0·39
0·21
0·18
0·23
0·21
0·14
0·22
0·16
0·20
0·23
0·12
0·11
0·16
0·15
0·16
0·16
0·15
0·15
0·16
0·07
0·09
0·08
0·08
0·08
0·11
0·15
0·10
0·13
0·13
0·20
0·33
0·61
0·81
1·57
1·68
1·90
n
Manning’s
2238
D. E. Reid and E. J. Hickin
Earth Surf. Process. Landforms 33, 2211–2240 (2008)
DOI: 10.1002/esp
3·758
2·099
2·026
1·614
1·196
0·639
0·763
3·863
1·913
2·959
2·737
1·949
3·391
2·099
2·026
1·614
1·196
0·639
0·763
3·863
1·913
2·959
2·737
1·949
12-Jul-03
23-Jul-03
25-Jul-03
30-Jul-03
08-Aug-03
06-Sep-03
30-Sep-03
27-Oct-03
07-Nov-03
24-Nov-03
25-Nov-03
15-Dec-03
Copyright © 2008 John Wiley & Sons, Ltd.
12-Jul-03
23-Jul-03
25-Jul-03
30-Jul-03
08-Aug-03
06-Sep-03
30-Sep-03
27-Oct-03
07-Nov-03
24-Nov-03
25-Nov-03
15-Dec-03
18·60
17·90
17·80
17·70
17·10
15·90
16·00
20·00
19·10
19·40
19·15
18·85
18·55
16·05
14·55
14·80
14·40
13·25
13·45
16·60
14·75
16·05
14·55
14·85
(m)
(m /s)
3
Date
width
discharge
Water
surface
Stream
Foley creek
0·41
0·37
0·35
0·34
0·32
0·24
0·26
0·39
0·28
0·36
0·35
0·30
0·46
0·39
0·42
0·41
0·37
0·28
0·27
0·46
0·35
0·35
0·35
0·30
(m)
depth
Mean
Darcy–
ff
4·49
6·69
7·14
11·00
13·72
16·97
12·06
3·34
4·81
2·44
2·32
2·83
0·44
0·31
0·32
0·27
0·22
0·17
0·18
0·50
0·35
0·43
0·41
0·35
3·08
5·43
4·89
6·95
9·41
12·78
11·59
2·30
3·32
2·88
3·03
3·57
XS FOL-U1
0·44
0·33
0·33
0·26
0·22
0·17
0·21
0·51
0·37
0·52
0·53
0·44
XS FOL-L3
(m)
velocity Weisbach
Mean
0·17
0·22
0·21
0·25
0·28
0·32
0·31
0·15
0·17
0·16
0·16
0·17
0·21
0·25
0·26
0·32
0·35
0·37
0·31
0·18
0·21
0·15
0·14
0·15
n
Manning’s
Water
20·25
18·85
18·65
17·90
16·95
13·35
13·55
20·45
19·35
19·80
19·80
19·25
16·35
14·30
14·30
14·20
12·85
10·80
11·35
17·15
14·90
16·95
16·65
14·95
(m)
width
surface
Mean
0·32
0·28
0·26
0·25
0·22
0·17
0·18
0·29
0·20
0·26
0·25
0·21
0·39
0·32
0·31
0·29
0·26
0·24
0·23
0·43
0·30
0·33
0·33
0·29
(m)
2·08
2·79
2·84
3·60
3·97
7·63
5·12
2·95
3·18
2·25
2·41
2·75
0·52
0·40
0·42
0·36
0·32
0·28
0·31
0·65
0·49
0·57
0·55
0·49
1·76
2·63
2·09
2·90
3·18
3·22
2·80
1·02
1·23
1·16
1·21
1·28
XS FOL-U2
0·59
0·46
0·45
0·39
0·35
0·24
0·29
0·52
0·42
0·53
0·50
0·45
ff
Weisbach
Darcy–
XS FOL-L4
(m)
depth velocity
Mean
0·12
0·15
0·13
0·15
0·16
0·15
0·14
0·09
0·10
0·10
0·10
0·10
0·14
0·15
0·16
0·17
0·18
0·24
0·20
0·17
0·16
0·14
0·14
0·15
n
Manning’s
Water
15·45
15·35
14·75
14·70
14·25
12·85
12·60
15·65
12·95
15·05
15·05
13·05
14·95
13·05
12·95
12·65
12·25
9·90
10·05
15·00
13·35
14·60
14·50
12·90
(m)
width
surface
Mean
Darcy–
0·34
0·30
0·29
0·28
0·23
0·17
0·18
0·36
0·28
0·33
0·32
0·30
0·38
0·31
0·31
0·30
0·25
0·20
0·20
0·43
0·24
0·33
0·31
0·26
(m)
1·61
2·22
2·32
3·07
3·25
3·88
2·62
2·26
1·29
1·60
1·63
1·42
ff
0·64
0·46
0·47
0·39
0·36
0·29
0·33
0·69
0·53
0·60
0·58
0·50
1·22
2·10
1·92
2·60
2·61
3·07
2·49
1·07
1·42
1·29
1·37
1·73
XS FOL-U3
0·66
0·51
0·50
0·43
0·38
0·32
0·38
0·60
0·60
0·62
0·60
0·59
XS FOL-L5
(m)
depth velocity Weisbach
Mean
0·10
0·13
0·13
0·15
0·14
0·15
0·13
0·10
0·11
0·11
0·11
0·12
0·12
0·14
0·14
0·16
0·16
0·17
0·14
0·15
0·10
0·12
0·12
0·11
n
Manning’s
Water
13·00
10·90
10·90
10·80
9·75
9·20
9·35
13·00
11·50
12·55
12·50
11·20
16·30
14·75
14·55
14·45
13·85
11·60
11·60
18·15
15·55
17·45
17·40
15·60
(m)
width
surface
0·45
0·44
0·43
0·42
0·41
0·29
0·29
0·52
0·39
0·45
0·43
0·40
0·32
0·24
0·24
0·22
0·19
0·15
0·16
0·30
0·21
0·28
0·26
0·22
(m)
depth
Mean
1·20
1·33
1·31
1·60
1·68
2·25
1·78
1·15
1·22
1·40
1·32
1·26
0·58
0·43
0·44
0·36
0·30
0·24
0·28
0·57
0·43
0·52
0·51
0·44
1·91
3·27
3·12
4·48
6·20
7·07
5·33
2·20
2·97
2·37
2·38
2·96
XS FOL-U4
0·71
0·59
0·59
0·51
0·46
0·36
0·41
0·70
0·58
0·61
0·61
0·57
ff
Weisbach
Darcy–
XS FOL-L6
(m)
velocity
Mean
0·13
0·18
0·17
0·20
0·24
0·24
0·21
0·15
0·16
0·15
0·15
0·16
0·10
0·10
0·10
0·11
0·11
0·12
0·11
0·10
0·10
0·11
0·10
0·10
n
Manning’s
Water
17·95
16·75
15·40
16·55
15·70
15·00
14·30
17·65
16·00
17·35
17·20
16·10
15·70
13·25
13·15
12·25
10·95
8·10
8·05
17·95
13·75
16·55
15·85
14·05
(m)
width
surface
Mean
0·36
0·32
0·31
0·29
0·27
0·20
0·20
0·31
0·22
0·26
0·25
0·22
0·27
0·30
0·29
0·29
0·25
0·26
0·27
0·32
0·26
0·30
0·30
0·27
(m)
depth
Mean
Darcy–
0·66
2·05
1·91
2·64
2·61
4·94
3·88
1·37
1·71
1·57
1·71
1·89
0·53
0·39
0·43
0·34
0·29
0·22
0·27
0·72
0·55
0·65
0·63
0·55
1·84
3·13
2·38
3·66
4·73
6·29
4·01
0·86
1·07
0·91
0·95
1·07
XS FOL-U5
0·88
0·53
0·53
0·45
0·43
0·31
0·36
0·67
0·54
0·60
0·58
0·52
ff
Weisbach
XS FOL-L7
(m)
velocity
0·13
0·16
0·14
0·17
0·20
0·22
0·17
0·09
0·09
0·09
0·09
0·09
0·07
0·13
0·13
0·15
0·14
0·20
0·18
0·11
0·12
0·11
0·12
0·12
n
Manning’s
Flow resistance in steep mountain streams
2239
Earth Surf. Process. Landforms 33, 2211–2240 (2008)
DOI: 10.1002/esp
0·551
0·375
0·297
0·235
0·090
0·097
0·646
3·591
1·310
0·280
0·230
0·201
0·146
0·063
0·054
0·536
2·392
0·396
0·272
0·204
0·129
0·107
0·039
0·039
0·155
0·667
0·416
16-Jul-03
21-Jul-03
26-Jul-03
31-Jul-03
05-Sep-03
29-Sep-03
13-Oct-03
28-Oct-03
11-Nov-03
Copyright © 2008 John Wiley & Sons, Ltd.
16-Jul-03
21-Jul-03
26-Jul-03
31-Jul-03
05-Sep-03
29-Sep-03
13-Oct-03
28-Oct-03
11-Nov-03
16-Jul-03
21-Jul-03
26-Jul-03
31-Jul-03
05-Sep-03
29-Sep-03
13-Oct-03
28-Oct-03
11-Nov-03
5·10
3·55
2·40
2·25
1·70
1·90
2·45
7·95
7·05
5·00
5·10
5·35
5·35
4·75
4·65
6·65
8·50
6·45
10·50
10·40
10·15
8·50
6·65
6·65
13·00
15·95
14·60
0·13
0·17
0·21
0·20
0·11
0·12
0·15
0·17
0·15
0·19
0·15
0·13
0·13
0·09
0·09
0·21
0·42
0·20
0·19
0·17
0·15
0·18
0·12
0·12
0·16
0·29
0·21
0·25
0·570
0·375
0·297
0·235
0·090
0·097
0·646
3·591
1·152
16-Jul-03
21-Jul-03
26-Jul-03
31-Jul-03
05-Sep-03
29-Sep-03
13-Oct-03
28-Oct-03
11-Nov-03
(m)
10·65
(m)
(m3/s)
Date
depth
0·21
0·20
0·18
0·17
0·09
0·09
0·20
width
discharge
Mean
9·55
9·15
8·95
8·95
8·15
7·70
9·80
surface
Stream
Water
Chipmunk creek
Darcy–
ff
3·02
5·28
5·51
9·67
12·55
11·82
2·28
0·65
1·45
3·41
2·60
2·67
4·80
6·33
9·53
2·42
1·55
3·83
0·40
0·34
0·25
0·24
0·22
0·17
0·43
0·50
0·38
1·59
2·57
5·49
5·67
4·01
6·91
1·35
1·27
1·93
XS CHK-Trib1
0·30
0·31
0·29
0·21
0·15
0·13
0·38
0·67
0·30
XS CHK-U1
0·28
0·21
0·19
0·15
0·11
0·12
0·31
0·77
0·43
XS CHK-L1
0·28
6·90
0·21
11·67
0·18
14·36
0·16
16·98
0·12
14·95
0·14
13·20
0·32
4·92
no observation
0·43
3·43
XS CHK-LL1
(m)
velocity Weisbach
Mean
0·10
0·13
0·20
0·20
0·15
0·20
0·09
0·09
0·11
0·16
0·13
0·13
0·17
0·19
0·23
0·13
0·12
0·17
0·15
0·19
0·19
0·26
0·28
0·27
0·13
0·07
0·10
0·16
0·23
0·29
0·32
0·34
0·29
0·28
0·19
n
Manning’s
Water
5·65
5·40
5·20
4·65
3·35
3·90
5·25
6·75
6·85
5·20
5·20
4·95
4·95
3·85
3·70
5·30
8·00
5·25
10·80
9·95
9·15
8·00
6·00
6·00
11·65
14·10
12·50
8·95
7·25
7·00
5·65
5·90
4·40
4·50
8·85
(m)
width
surface
Mean
ff
XS CHK-LL2
(m)
Weisbach
Darcy–
0·16
0·14
0·12
0·12
0·07
0·08
0·11
0·16
0·14
0·19
0·16
0·15
0·14
0·10
0·10
0·23
0·38
0·18
0·19
0·17
0·17
0·17
0·14
0·14
0·18
0·34
0·23
3·23
4·72
5·84
7·85
16·48
14·53
2·61
0·80
1·51
3·74
3·46
3·78
4·97
6·92
9·17
2·09
0·98
1·67
0·31
0·27
0·21
0·20
0·17
0·12
0·26
0·60
0·43
3·09
3·58
5·33
5·70
4·14
10·88
3·18
0·84
1·39
XS CHK-Trib2
0·29
0·28
0·26
0·22
0·16
0·14
0·43
0·79
0·42
XS CHK-U2
0·27
0·22
0·19
0·17
0·11
0·11
0·30
0·74
0·45
XS CHK-L2
0·35
4·62
0·26
7·43
0·24
9·96
0·20
13·02
0·12
27·70
0·13
25·28
0·34
4·73
no observation
0·27
0·48
3·01
0·22
0·20
0·22
0·20
0·17
0·17
0·22
(m)
depth velocity
Mean
0·14
0·15
0·18
0·19
0·15
0·24
0·14
0·08
0·10
0·16
0·15
0·16
0·18
0·20
0·23
0·13
0·09
0·11
0·15
0·18
0·20
0·23
0·33
0·31
0·14
0·08
0·11
0·16
0·19
0·23
0·27
0·31
0·44
0·42
0·19
n
Manning’s
Water
6·05
5·60
5·50
5·10
3·75
4·00
5·60
7·40
6·25
6·50
6·50
6·75
6·35
5·85
5·75
7·40
9·75
6·70
9·45
8·20
8·10
7·90
5·65
5·25
10·55
14·40
12·05
8·45
7·20
7·25
7·25
6·75
6·65
8·65
9·80
9·30
(m)
width
surface
Mean
Darcy–
0·10
0·11
0·10
0·10
0·06
0·06
0·08
0·14
0·12
0·18
0·12
0·12
0·12
0·08
0·08
0·18
0·32
0·18
0·23
0·23
0·19
0·19
0·11
0·12
0·19
0·35
0·22
0·27
0·30
0·27
0·25
0·15
0·15
0·30
0·41
0·30
(m)
ff
10·37
25·16
27·66
36·22
46·03
43·41
11·97
1·23
4·53
4·37
7·05
6·41
9·40
6·77
5·86
2·51
0·89
1·20
3·74
3·46
3·78
4·97
6·92
9·17
2·09
0·98
1·67
0·43
0·32
0·23
0·21
0·17
0·16
0·33
0·65
0·56
3·09
3·58
5·33
5·70
4·14
10·88
3·18
0·84
1·39
XS CHK-Trib3
0·24
0·29
0·24
0·20
0·14
0·11
0·41
0·77
0·32
XS CHK-U3
0·26
0·20
0·19
0·16
0·15
0·16
0·32
0·71
0·49
XS CHK-L3
0·25
0·17
0·15
0·13
0·09
0·09
0·25
0·90
0·41
XS CHK-LL3
(m)
depth velocity Weisbach
Mean
0·14
0·15
0·18
0·19
0·15
0·24
0·14
0·08
0·10
0·16
0·15
0·16
0·18
0·20
0·23
0·13
0·09
0·11
0·18
0·23
0·21
0·26
0·20
0·19
0·14
0·09
0·10
0·29
0·46
0·47
0·53
0·55
0·54
0·32
0·11
0·19
n
Manning’s
Water
4·00
3·80
3·25
2·75
2·00
2·10
3·50
4·85
4·10
5·80
4·50
4·55
4·30
4·15
4·25
6·50
7·75
6·30
9·95
9·70
9·10
9·05
7·25
8·60
10·10
11·90
10·20
3·65
3·55
3·25
2·95
1·95
2·20
4·15
8·95
6·50
(m)
width
surface
0·17
0·16
0·12
0·13
0·07
0·07
0·11
0·18
0·16
0·16
0·19
0·18
0·17
0·10
0·11
0·20
0·41
0·16
0·20
0·18
0·18
0·17
0·08
0·08
0·18
0·36
0·24
0·29
0·30
0·29
0·31
0·26
0·25
0·32
0·37
0·29
(m)
depth
Mean
ff
Weisbach
Darcy–
2·30
5·68
6·62
10·01
17·79
16·81
3·12
0·79
1·96
4·37
5·44
7·04
10·18
4·89
4·60
1·87
0·65
1·12
5·13
2·53
3·70
5·33
7·63
11·89
1·84
0·90
3·00
0·40
0·33
0·33
0·31
0·30
0·26
0·41
0·76
0·62
1·05
2·03
3·65
4·24
4·17
4·37
1·41
0·61
0·73
XS CHK-Trib4
0·31
0·26
0·25
0·20
0·15
0·11
0·42
0·75
0·39
XS CHK-U4
0·27
0·21
0·18
0·15
0·15
0·15
0·35
0·84
0·53
XS CHK-L4
0·54
0·35
0·31
0·26
0·18
0·18
0·49
1·08
0·60
XS CHK-LL4
(m)
velocity
Mean
0·08
0·11
0·15
0·16
0·14
0·15
0·09
0·06
0·07
0·19
0·13
0·15
0·18
0·20
0·26
0·11
0·09
0·15
0·18
0·20
0·22
0·27
0·16
0·16
0·12
0·08
0·09
0·14
0·21
0·23
0·28
0·37
0·35
0·16
0·08
0·13
n
Manning’s
Water
6·90
6·35
6·10
5·25
3·65
3·95
6·15
7·65
7·10
5·50
4·95
4·90
4·70
4·75
5·05
6·65
7·65
6·25
9·30
8·20
8·00
7·55
5·85
6·10
10·70
13·45
11·70
8·50
8·20
7·80
7·65
7·00
7·10
9·00
12·00
9·55
(m)
width
surface
Mean
0·18
0·15
0·14
0·14
0·09
0·08
0·12
0·15
0·12
0·20
0·19
0·19
0·18
0·13
0·13
0·22
0·37
0·22
0·16
0·16
0·15
0·15
0·09
0·09
0·15
0·34
0·20
0·28
0·25
0·25
0·24
0·16
0·16
0·27
0·39
0·28
(m)
depth
Mean
Darcy–
ff
Weisbach
12·34
19·44
27·04
36·58
64·37
55·90
9·45
1·67
3·95
1·60
2·75
3·48
4·67
4·66
3·89
1·25
0·70
0·79
5·02
5·59
6·31
9·68
23·10
33·73
2·67
0·88
4·33
0·22
0·21
0·15
0·15
0·13
0·12
0·21
0·59
0·49
7·28
6·13
11·82
11·87
9·88
11·23
5·40
0·80
0·93
XS CHK-Trib5
0·26
0·24
0·22
0·17
0·10
0·08
0·37
0·84
0·29
XS CHK-U5
0·36
0·28
0·24
0·21
0·16
0·18
0·40
0·79
0·57
XS CHK-L5
0·24
0·18
0·15
0·13
0·08
0·09
0·27
0·77
0·42
XS CHK-LL5
(m)
velocity
0·23
0·20
0·28
0·28
0·23
0·25
0·18
0·07
0·08
0·19
0·20
0·21
0·26
0·38
0·46
0·14
0·09
0·18
0·10
0·14
0·15
0·18
0·16
0·15
0·09
0·08
0·08
0·32
0·39
0·46
0·53
0·66
0·62
0·28
0·12
0·18
n
Manning’s
2240
D. E. Reid and E. J. Hickin
Earth Surf. Process. Landforms 33, 2211–2240 (2008)
DOI: 10.1002/esp