Geomorphology 41 Ž2001. 369–390
www.elsevier.comrlocatergeomorph
Annual regime of bedforms, roughness and flow resistance,
Lillooet River, British Columbia, BC
Mariette
¨ T.H. Prent a,) , Edward J. Hickin b,1
b
a
Department of Geography, Simon Fraser UniÕersity, Burnaby, BC, Canada V5A 1S6
Departments of Geography and Earth Sciences, Simon Fraser UniÕersity, Burnaby, BC, Canada V5A 1S6
Received 14 March 2000; received in revised form 7 March 2001; accepted 9 March 2001
Abstract
A field study to examine the statistical character of dune morphology and the correlation among dune morphology,
discharge, and flow resistance was conducted in a meandering reach of Lillooet River, near Pemberton, British Columbia,
Canada. The field season spanned the 1995r1996 hydrologic year with sample day discharge events ranging between 33 and
425 m3 sy1. Surveys of the bed morphology along the thalweg in two dune fields ŽA and B. were completed using an echo
sounder with chart recorder that enabled the measurement of more than 4000 dunes. Dune height ranged between 0.08 and
0.96 m, length between 2.01 and 20.99 m, and steepness between 0.02 and 0.10. Histograms of each dune shape Žheight,
length, steepness. sample most often displayed positive skewness and non-Gaussian distributions ŽGamma, Beta, and
Weibull.; median sample height and length histograms displayed positive skewness; and steepness was nearly Gaussian.
Histograms of all dimensionless dunes Ži.e., measurement divided by average measurement of sample. were Gaussian and
slightly leptokurtic. Neither the height nor length of dunes measured in this investigation were successfully predicted by the
empirical models of Allen wAllen, J.R.L., 1984. Developments in Sedimentology. Sedimentary Structures: Their Character
and Physical Basis, 2nd edn. Elsevier, New York, vol. 30 ŽA and B., 1256 pp.x, Fredsøe wJ. Hydraul. Div., Am. Soc. Civ.
Eng. 108ŽHY8. Ž1982. 932.x, or Yalin wJ. Hydraul. Div., Am. Soc. Civ. Eng. 90ŽHY5. Ž1964. 105.x. Least-squares regression
models for dune–height relations produced here are similar to models published by other field researchers; regression
models for dune length only conform to those developed elsewhere if the discharge of the study rivers was similar. The
energy gradient in dune field A varied within a smaller range than in field B, enabling dune size to become more fully
equilibrated with respect to flow environment. Although the average Froude numbers were much less than critical, dunes
appeared to wash out towards a plane bed as discharge increased due to a change from a bedload to suspended-load
dominated sediment-transport regime. Flow resistance increased most rapidly during changes in base flow and at the
beginning and end of the seasonal flood; resistance tended to be smaller in field A than B, reflecting local differences in
energy gradient. Flow resistance increased until a dune steepness of 0.070 was attained and then decreased. The steepness
value was considered to be coincident with kolk generation wDyer, K.R., 1986. Coastal and Estuarine Sediment Dynamics.
)
Corresponding author. Current address: PARISH Geomorphic Ltd., 10 Mountainview Road S., Suite 207, Georgetown, ON, Canada
L7G 4J9. Fax: q1-905-877-4143.
E-mail addresses: mprent@parishgeomorphic.com ŽM.T.H. Prent., hickin@sfu.ca ŽE.J. Hickin..
1
Fax: q1-604-291-4198.
0169-555Xr01r$ - see front matter q 2001 Elsevier Science B.V. All rights reserved.
PII: S 0 1 6 9 - 5 5 5 X Ž 0 1 . 0 0 0 6 8 - X
370
M.T.H. Prent, E.J. Hickin r Geomorphology 41 (2001) 369–390
Wiley, NY, 342 pp.x, suggesting that macroturbulent flow structures play an important role in defining the roughness of
dunes on a channel bed. q 2001 Elsevier Science B.V. All rights reserved.
Keywords: Bedform; Dune; Flow resistance; Roughness
1. Introduction
Bedforms have long been recognized as exerting a
significant and interactive control on the hydraulics
and geomorphology of alluvial river channels. Much
of what we understand about the nature of
bedformrflow interaction has been learned from the
many excellent flume studies undertaken since 1950
Žfor examples, see the review by Graf, 1971.. There
remains a surprising paucity of bedform studies in
natural rivers, however, even though it is widely
recognized that further understanding of bedform
behavior in unsteady natural flows must be based on
field studies ŽAllen, 1983; Bridge, 1987; Gabel,
1993.. Field studies which examine the connection
between channel roughness and flow resistance in
natural settings are similarly under-represented in the
literature ŽGee, 1975..
The size and shape of bedforms are a function of
the forces that are exerted on the channel boundary
by the flow, and of the character of the sediment in
which a channel is formed ŽSimons and Richardson,
1960; Bridge, 1987; McLean, 1990.. Discharge is
often used as a general measure of flow vigor because it implicitly encompasses all of the stresses
that are at work within a channel. Allen Ž1974., on
the other hand, suggested that a more direct expression of flow energy is given by flow velocity. Another view Že.g., Yalin, 1964; Simons and Richardson, 1966. is that water depth is the limiting factor of
dune growth and is therefore best used to examine
the response of dunes to their flow environment.
Velocity gradient Žvelocityrwater depth. provides
yet another measure of the flow environment. Regardless of the flow variable chosen, dune size is
expected to increase with flow volume.
Simons and Richardson Ž1966. stated that a change
in water-surface slope can alter the channel-bed configuration even when water depth remains constant.
This statement was confirmed by Ikeda and Iseya
Ž1980. who observed a spatial variation in dune
length even though water depth remained unchanged
in the downstream direction of a section of Teshio
River. Ikeda and Iseya attributed the variation to the
role of flow velocity or water-surface gradient in
shaping dunes and, for this reason, recommended
that analyses of bedform response to flows should
involve quantities that include these variables. Because stream power and boundary shear stress include velocity and gradient, they express the force
that water exerts on a channel bed and therefore
describe the flow environment more comprehensively than either discharge or water depth. Therefore, they are often used to relate bedform type to the
flow environment Že.g., in bedform stability fields..
Through their analyses, Ikeda and Iseya Ž1980. found
that the relation between dune length and stream
power was better defined than between dune length
and water depth.
As flowing water begins to mould the substrate
materials into bedforms, they begin to interfere with
streamlines of the flow. This interference can cause a
dissipation of flow energy at the channel bed as
macroturbulent flow structures develop. The roughness exerted by bedforms is commonly quantified
through calculations of Manning’s N or the Darcy–
Weisbach resistance coefficient Ž ff .. Another measure of the roughness of the channel bed is relative
smoothness, a descriptor of the relative influence of
a bedform on the water column in relation to water
depth. Previous research has focused on the relation
between dune size and roughnessrflow resistance
showing that the initial development of bedforms
from a plane surface causes the largest increase in
flow resistance than at any other stage of bedform
development ŽSimons and Richardson, 1966.. Little
research has investigated the relation between flow
resistance and measures of flow that include discharge, stream power, or tractive force.
It is the purpose of this paper to describe the
statistical character of bedforms and to examine the
correlations among discharge, flow resistance, and
bed morphology in a natural channel. Data and
results will be compared to previously published
M.T.H. Prent, E.J. Hickin r Geomorphology 41 (2001) 369–390
findings to identify common trends in field data and
to evaluate the appropriateness of empirical and theoretically derived predictive relations. The basic data
have been obtained by monitoring flow and bed
morphology in the Lillooet River, near Pemberton in
southwestern Canada. The study took place over a
complete hydrologic year in which mean monthly
discharges varied by an order of magnitude.
371
2. Field setting
Lillooet River originates from Lillooet Glacier
and drains an area of 3150 km2 before flowing into
Lillooet Lake ŽFig. 1.. At Pemberton, Lillooet River
has a meandering river planform; channel boundary
materials vary in composition with large areas dominated by sand. The two dune fields that were exam-
Fig. 1. Map of study area.
372
M.T.H. Prent, E.J. Hickin r Geomorphology 41 (2001) 369–390
ined, labelled A and B in Fig. 1, have respective
lengths of 154 and 247 m.
Approximately 16% of the drainage area is
glacierized. Melting snow and ice, during the spring
and summer, account for the seasonal flood Žthe
freshet. that characterizes the annual hydrograph.
Rain-on-snow precipitation events in autumn cause
sharp increases in the mean daily discharge for several days at a time. The remainder of the year is
characterized by sustained periods of modest flow.
Based on a 72-year record of mean monthly discharge events, the discharge ratio Žhighestrlowest
mean monthly discharge. of flow in Lillooet River is
10.5; the mean annual discharge is 125 m3 sy1
ŽWater Survey of Canada Hydrometric station
08MG005; Environment Canada, 1991..
The 1995r1996 annual hydrograph ŽFig. 2. shows
a 5-month-long freshet ŽMay–September. and several high-magnitude autumnal flow events in October and November. Flows less than 100 m3 sy1
occupy the channel during approximately 50% of the
annual hydrograph. Sample day discharges ranged
between 33.1 and 425 m3 sy1 ŽTable 1..
The flow of Lillooet River, especially in spring
and summer, is influenced by the diurnal regime of
snow and glacier melt. In summer, the diurnal variation of hourly discharge ranged from 7% to 35% of
the mean daily discharge, while in winter the variation was less than 10%.
The range of average Froude numbers in dune
fields A Ž FA . and B Ž FB . was narrow Ž FA : 0.13–0.29;
FB : 0.12–0.36.. All values indicate that the flow
environment was subcritical on each sample day
ŽTable 1.. The entire range of Froude numbers Ž Fr .
occurred when discharge was smaller than 125 m3
sy1 in FA ; when discharge was ) 125 m3 sy1 , Fr
enveloped a narrower range Ž Fr s 0.17–0.21.. In
FB , the Froude number tended to increase with increasing discharge.
As evident in Fig. 1, dune fields A and B are
situated only 630 m apart and are separated by a
sharp meander bend. The water-surface gradient
measured in field B tends to be steeper than in field
A during any given sample day. As discharge increases, the water-surface gradient in FA remains
essentially constant; whereas in FB , the gradient
increases continuously. The range of stream power
associated with sample day discharge events is greater
in FB than in FA and encompasses several orders of
magnitude Ž FA : 46–1592 W my1 ; FB : 74–3830 W
my1 ..
3. Methods
3.1. Data collection and processing
Systematic data collection began in May 1995 and
ended in July 1996. Sampling was conducted weekly
during the freshet and biweekly during the remainder
of the year.
Fig. 2. Hydrograph of Lillooet River at ŽWSC 08MG005. during the field season. Circles indicate sample days.
M.T.H. Prent, E.J. Hickin r Geomorphology 41 (2001) 369–390
373
Table 1
Flow environment for dune field A ŽA. and B ŽB. during each sample day
Date
Q Žm3 s -1 .
D Žm.
a.m.
(A)
May 12, 1995
May 18, 1995
May 24, 1995
May 29, 1995
Jun 3, 1995
Jun 8, 1995
Jun 15, 1995
Jun 22, 1995
Jul 1, 1995
Jul 6, 1995
Jul 12, 1995
Jul 19, 1995
Aug 5, 1995
Aug 11, 1995
Aug 16, 1995
Aug 27, 1995
Aug 30, 1995
Sep 16, 1995
Oct 2, 1995
Oct 23, 1995
Nov 6, 1995
Nov 20, 1995
Dec 6, 1995
Dec 16, 1995
Jan 9, 1996
Feb 20, 1996
Mar 5, 1996
Mar 10, 1996
Mar 19, 1996
Apr 2, 1996
Apr 16, 1996
Apr 30, 1996
May 10, 1996
May 27, 1996
Jun 3, 1996
Jun 17, 1996
Jul 15, 1996
148
213
210
329
293
273
215
327
401
319
255
343
287
192
155
105
110
161
61
64.5
40.8
123
79
56.3
52.9
58.2
33.1
53.6
58.7
44.4
116
67.7
55.6
172
253
148
425
2.74
3.74
3.52
2.93
2.53
3.94
3.93
3.55
3.24
3.80
3.51
2.48
2.78
1.69
1.83
2.48
1.23
1.44
1.26
2.77
1.74
1.59
1.44
1.50
1.07
1.31
1.28
1.03
1.60
1.28
1.15
2.82
2.92
2.49
4.61
(B)
May 24, 1995
May 29, 1995
Jun 3, 1995
Jun 8, 1995
Jun 15, 1995
Jun 22, 1995
Jul 1, 1995
Jul 6, 1995
Jul 12, 1995
Jul 19, 1995
Aug 5, 1995
Aug 11, 1995
210
329
293
273
215
327
401
319
255
343
287
192
2.68
3.57
3.26
3.02
2.63
3.55
4.12
3.05
2.91
3.74
3.55
2.65
V Žm.
p.m.
3.53
3.03
2.57
3.26
4.13
3.58
3.00
3.63
3.50
2.69
2.69
1.64
2.15
2.53
1.73
2.94
1.66
1.44
1.43
1.45
1.17
1.06
1.84
1.33
1.16
2.55
3.15
2.49
2.71
3.25
2.83
2.98
2.61
3.46
3.71
3.13
2.74
3.55
3.4
2.72
a.m.
1.10
1.17
1.12
1.25
1.12
1.13
1.38
1.21
1.08
1.22
1.10
1.12
0.86
0.95
0.97
1.02
0.92
0.75
0.60
0.73
0.81
0.63
0.68
0.69
0.61
0.71
0.84
0.81
1.16
0.90
0.91
0.89
1.20
0.92
1.24
1.18
1.39
1.37
1.38
1.22
1.38
1.44
1.54
1.31
1.36
1.23
1.09
p.m.
1.24
1.21
1.11
1.36
1.31
1.20
1.17
1.28
1.10
1.04
0.89
0.98
0.83
1.00
0.63
0.68
0.84
0.70
0.69
0.72
0.56
0.79
1.01
0.87
0.91
0.98
1.11
0.92
1.16
1.52
1.57
1.39
1.23
1.42
1.60
1.97
1.39
1.43
1.28
1.06
Gradient Ž%.
Froude
a.m.
a.m.
0.03
0.03
0.03
0.04
0.03
0.03
0.04
0.04
0.03
0.04
0.03
0.03
0.02
0.03
0.03
0.03
0.03
0.02
0.01
0.02
0.02
0.01
0.02
0.02
0.01
0.02
0.02
0.02
0.03
0.03
0.03
0.03
0.04
0.03
0.04
0.07
0.09
0.08
0.08
0.08
0.09
0.09
0.09
0.08
0.08
0.08
0.07
p.m.
0.04
0.04
0.03
0.04
0.04
0.04
0.03
0.04
0.03
0.03
0.03
0.03
0.02
0.03
0.01
0.02
0.02
0.02
0.02
0.02
0.01
0.02
0.03
0.02
0.03
0.03
0.03
0.03
0.07
0.09
0.09
0.09
0.08
0.09
0.09
0.10
0.09
0.09
0.08
0.07
0.21
0.19
0.19
0.23
0.23
0.18
0.22
0.20
0.19
0.20
0.19
0.23
0.17
0.23
0.23
0.21
0.27
0.20
0.17
0.14
0.19
0.16
0.18
0.18
0.19
0.20
0.24
0.26
0.29
0.25
0.27
0.17
0.22
0.19
0.18
0.23
0.23
0.24
0.25
0.24
0.23
0.23
0.28
0.25
0.22
0.21
0.21
p.m.
0.21
0.22
0.22
0.24
0.21
0.20
0.22
0.21
0.19
0.20
0.17
0.24
0.18
0.20
0.15
0.13
0.21
0.18
0.18
0.19
0.17
0.25
0.24
0.24
0.27
0.20
0.20
0.19
0.23
0.27
0.30
0.26
0.24
0.24
0.27
0.36
0.27
0.24
0.22
0.20
M.T.H. Prent, E.J. Hickin r Geomorphology 41 (2001) 369–390
374
Table 1 Ž continued .
Date
(B)
Aug 16, 1995
Aug 27, 1995
Aug 30, 1995
Sep 16, 1995
Oct 2, 1995
Oct 23, 1995
Nov 6, 1995
Nov 20, 1995
Dec 6, 1995
Dec 16, 1995
Jan 9, 1996
Feb 20, 1996
Mar 5, 1996
Mar 10, 1996
Mar 19, 1996
Apr 2, 1996
Apr 16, 1996
Apr 30, 1996
May 10, 1996
May 27, 1996
Jun 3, 1996
Jun 17, 1996
Jul 15, 1996
Q Žm3 s -1 .
155
105
110
161
61
64.5
40.8
123
79
56.3
52.9
58.2
33.1
53.6
58.7
44.4
116
67.7
55.6
172
253
148
425
D Žm.
V Žm.
Gradient Ž%.
Froude
a.m.
p.m.
a.m.
p.m.
a.m.
p.m.
a.m.
p.m.
2.22
2.02
2.13
2.56
1.53
1.14
1
2.19
1.71
1.45
1.26
1.19
0.98
1.28
1.47
1.26
2.14
1.56
1.44
2.75
3.11
2.43
4.25
2.26
1.05
0.77
0.77
0.94
0.64
0.94
1.00
0.90
0.78
0.66
0.71
0.81
0.57
0.67
0.69
0.59
0.86
0.71
0.65
0.98
1.23
0.93
1.54
1.03
0.07
0.04
0.04
0.06
0.03
0.06
0.00
0.05
0.04
0.03
0.04
0.05
0.02
0.03
0.04
0.03
0.05
0.04
0.03
0.06
0.08
0.06
0.09
0.06
0.00
0.03
0.06
0.00
0.00
0.00
0.05
0.04
0.03
0.03
0.04
0.03
0.00
0.04
0.02
0.05
0.03
0.03
0.06
0.08
0.06
0.09
0.23
0.17
0.17
0.19
0.17
0.28
0.22
0.19
0.19
0.17
0.20
0.24
0.18
0.19
0.18
0.17
0.19
0.18
0.17
0.19
0.22
0.19
0.24
0.19
0.18
0.15
0.16
0.21
0.20
2.64
2.65
2.18
1.75
1.59
1.48
1.28
0.92
1.42
1.28
2.23
1.76
1.48
2.79
3.11
2.38
4.2
Data were collected on all stages of the hydrograph, from 33.1 to 425 m3 sy1 ; water depth ranged
between 0.92 and 4.61 m ŽTable 1.. Most data were
collected on the falling limb of the diurnal hydrograph, several hours after the crest.
Echo-sounder survey transects for measuring bedform arrays were located in the channel thalweg of
the dune fields ŽFig. 1.. Equilibration of dunes to
changing flow conditions in the thalweg is likely
more complete than elsewhere in the channel. Thus,
correlations between dune size and flow intensity
here should be well defined ŽPretious and Blench,
1951; Harbor, 1998..
The echo-sounder system employed a 208 coneangle transducer mounted on the transom of the
survey boat; data were recorded by an onboard chart
recorder. Two longitudinal transects, surveyed 5 h
apart, were obtained for each dune field on most
sample days, yielding a total of 130 transects of the
bed configuration. Channel position during surveys
0.62
0.91
0.90
0.76
0.60
0.61
0.75
0.60
0.71
0.58
0.82
0.63
0.63
0.97
1.23
0.95
1.56
0.12
0.18
0.19
0.16
0.18
0.15
0.17
0.18
0.22
0.20
0.24
was recorded on the chart recorder in relation to
surveyed markers situated on the banks of the channel. A detailed discussion of the survey methodology
and verification is available in Prent Ž1998..
The bed configuration recorded on the sonar charts
was traced, scanned, and subsequently imported into
a graphics program to facilitate scale adjustments
and data processing ŽFig. 3.. The crest and trough
points of each bedform were digitized and used to
calculate bedform height and length. More than 4000
bedforms were measured, each of which falls within
the small, medium, and large dune classes of the
SEPM classification scheme ŽAshley, 1990.. Only
bedforms that were larger than 0.05 = water depth
were deemed large enough to be reliably detected.
Median values of dune height, length, and steepness
were used to specify the dune size in each sample
Ži.e., one run Ža.m. or p.m.. through a field.. All
dune measurements were changed to a dimensionless
quantity Ži.e., individual dune height, length, and
M.T.H. Prent, E.J. Hickin r Geomorphology 41 (2001) 369–390
Fig. 3. ŽA. Each of the sonar charts Ži. collected in the bedform fields was traced Žii., scanned and scaled Žiii. to common depth and distance scales for each dune field. ŽB.
Schematic diagram of a dune, illustrating the operational definitions of dune height and length, measured for each individual dune along the profile Žiii..
375
M.T.H. Prent, E.J. Hickin r Geomorphology 41 (2001) 369–390
376
steepness divided by the average for each sample. in
order to compare distributions throughout the hydrologic year.
Samples of the boundary materials in each dune
field were collected with a dredge once during lowflow conditions. Five samples, collected at equal
Table 2
Sample size and measures of median dune height Ž H ., length Ž L., steepness Ž HrL. and associated median absolute deviation ŽMAD. for
dune fields A ŽA. and B ŽB.
Date
Number
a.m.
(A)
May 12, 1995
May 18, 1995
May 24, 1995
May 29, 1995
Jun 3, 1995
Jun 8, 1995
Jun 15, 1995
Jun 22, 1995
Jul 1, 1995
Jul 6, 1995
Jul 12, 1995
Jul 19, 1995
Aug 5, 1995
Aug 11, 1995
Aug 16, 1995
Aug 27, 1995
Aug 30, 1995
Sep 16, 1995
Oct 2, 1995
Oct 23, 1995
Nov 6, 1995
Nov 20, 1995
Dec 6, 1995
Dec 16, 1995
Jan 9, 1996
Feb 20, 1996
Mar 5, 1996
Mar 10, 1996
Mar 19, 1996
Apr 2, 1996
Apr 16, 1996
Apr 30, 1996
May 10, 1996
May 27, 1996
Jun 3, 1996
Jun 17, 1996
Jul 15, 1996
19
14
18
12
15
18
19
14
10
15
19
17
13
27
21
34
29
19
47
38
36
24
37
29
55
29
10
46
37
50
33
46
37
20
17
24
7
(B)
May 18, 1995
May 24, 1995
May 29, 1995
Jun 3, 1995
Jun 8, 1995
Jun 15, 1995
33
28
22
22
27
39
H Žm.
p.m.
17
16
22
7
13
15
17
17
15
26
25
22
17
29
17
16
39
32
23
43
13
35
28
53
44
15
17
27
28
18
24
29
40
a.m.
0.42
0.77
0.58
0.61
0.62
0.65
0.48
0.68
0.45
0.45
0.37
0.47
0.79
0.36
0.44
0.25
0.29
0.44
0.14
0.13
0.13
0.34
0.17
0.13
0.08
0.13
0.15
0.09
0.11
0.10
0.29
0.14
0.09
0.43
0.51
0.31
0.96
0.28
0.63
0.67
0.65
0.49
0.35
p.m.
0.48
0.61
0.42
0.76
0.90
0.62
0.41
0.51
0.68
0.32
0.34
0.17
0.24
0.26
0.13
0.32
0.18
0.11
0.11
0.12
0.11
0.09
0.28
0.12
0.10
0.59
0.54
0.29
0.56
0.48
0.48
0.49
0.36
H MAD
L Žm.
a.m.
a.m.
0.13
0.19
0.09
0.29
0.20
0.33
0.12
0.22
0.23
0.16
0.11
0.15
0.07
0.06
0.10
0.04
0.06
0.14
0.04
0.03
0.04
0.07
0.03
0.04
0.02
0.03
0.05
0.02
0.03
0.03
0.10
0.03
0.01
0.06
0.12
0.08
0.18
0.09
0.13
0.24
0.25
0.23
0.10
p.m.
0.24
0.16
0.09
0.32
0.20
0.15
0.10
0.12
0.25
0.11
0.10
0.06
0.08
0.11
0.03
0.09
0.05
0.03
0.03
0.02
0.03
0.02
0.10
0.04
0.02
0.19
0.21
0.08
0.14
0.15
0.22
0.18
0.09
5.82
9.93
6.71
9.42
9.78
8.67
6.49
10.99
7.13
7.52
5.02
7.36
10.34
4.88
6.45
3.94
4.59
6.92
2.38
2.92
2.78
5.61
2.82
2.96
2.07
3.49
5.02
2.06
2.33
2.63
4.02
2.40
2.47
6.41
8.13
4.60
10.90
6.11
8.04
9.28
9.27
7.67
5.43
L MAD
p.m.
5.83
8.81
6.18
10.05
10.98
8.20
6.66
7.86
8.45
5.04
5.24
4.23
5.12
4.30
3.83
3.79
2.91
2.64
2.72
2.84
6.34
2.95
4.21
2.40
2.26
7.52
7.49
4.46
7.92
9.69
7.10
7.56
5.23
a.m.
1.12
2.10
1.51
3.32
2.22
2.31
1.88
1.98
4.47
1.62
2.45
3.76
1.66
0.97
1.26
0.76
0.75
1.59
0.49
0.92
0.82
1.05
0.78
0.97
0.59
0.89
2.15
0.50
0.61
0.56
0.86
0.72
0.75
1.48
1.84
0.90
2.86
1.61
1.56
4.15
3.01
2.37
1.20
p.m.
2.13
1.68
0.95
2.99
2.78
0.66
2.26
2.87
1.63
1.38
0.84
1.04
1.21
1.52
1.24
1.17
0.71
0.70
0.83
0.77
2.59
1.06
1.15
0.53
0.63
2.38
2.06
1.34
1.42
3.11
2.44
1.88
1.26
HrL Žm.
HrL MAD
a.m.
a.m.
0.07
0.07
0.08
0.06
0.08
0.08
0.07
0.07
0.05
0.07
0.07
0.08
0.08
0.07
0.07
0.07
0.07
0.07
0.05
0.04
0.05
0.07
0.06
0.04
0.04
0.04
0.02
0.05
0.05
0.04
0.07
0.06
0.04
0.06
0.08
0.07
0.10
0.05
0.07
0.06
0.07
0.06
0.08
p.m.
0.06
0.07
0.07
0.08
0.08
0.07
0.06
0.06
0.05
0.07
0.06
0.04
0.06
0.08
0.03
0.07
0.06
0.05
0.04
0.04
0.02
0.03
0.07
0.06
0.04
0.08
0.07
0.07
0.07
0.06
0.06
0.06
0.07
0.01
0.01
0.02
0.01
0.03
0.01
0.01
0.02
0.02
0.02
0.02
0.02
0.02
0.02
0.01
0.02
0.01
0.01
0.01
0.01
0.01
0.01
0.01
0.01
0.02
0.01
0.01
0.01
0.01
0.01
0.02
0.01
0.01
0.02
0.01
0.02
0.01
0.01
0.02
0.02
0.01
0.01
0.02
p.m.
0.02
0.02
0.02
0.01
0.01
0.02
0.01
0.02
0.02
0.02
0.02
0.01
0.02
0.02
0.01
0.03
0.02
0.01
0.01
0.01
0.01
0.01
0.02
0.01
0.01
0.01
0.02
0.01
0.01
0.02
0.02
0.02
0.01
M.T.H. Prent, E.J. Hickin r Geomorphology 41 (2001) 369–390
377
Table 2 Ž continued .
Date
(B)
Jun 22, 1995
Jul 1, 1995
Jul 6, 1995
Jul 12, 1995
Jul 19, 1995
Aug 5, 1995
Aug 11, 1995
Aug 16, 1995
Aug 27, 1995
Aug 30, 1995
Sep 16, 1995
Oct 2, 1995
Oct 23, 1995
Nov 20, 1995
Dec 6, 1995
Dec 16, 1995
Jan 9, 1996
Feb 20, 1996
Mar 5, 1996
Mar 10, 1996
Mar 19, 1996
Apr 2, 1996
Apr 16, 1996
Apr 30, 1996
May 10, 1996
May 27, 1996
Jun 3, 1996
Jun 17, 1996
Jul 15, 1996
Number
H Žm.
H MAD
L Žm.
L MAD
HrL Žm.
HrL MAD
a.m.
p.m.
a.m.
p.m.
a.m.
p.m.
a.m.
p.m.
a.m.
p.m.
a.m.
p.m.
a.m.
p.m.
21
17
23
27
19
27
42
52
47
45
38
64
54
32
55
45
42
58
44
69
56
28
31
44
38
26
21
41
13
25
13
26
36
28
28
34
50
0.57
0.62
0.42
0.50
0.41
0.53
0.32
0.18
0.33
0.33
0.33
0.17
0.12
0.47
0.20
0.14
0.10
0.14
0.10
0.12
0.12
0.09
0.39
0.14
0.12
0.40
0.65
0.28
0.63
0.37
0.48
0.43
0.40
0.37
0.44
0.44
0.28
0.24
0.26
0.19
0.13
0.17
0.18
0.11
0.05
0.11
0.10
0.16
0.04
0.03
0.13
0.08
0.04
0.03
0.03
0.02
0.03
0.03
0.02
0.10
0.03
0.02
0.12
0.13
0.07
0.22
0.11
0.23
0.20
0.12
0.12
0.16
0.11
0.06
7.67
10.39
8.24
6.54
8.33
8.05
5.05
3.27
4.45
4.59
5.43
3.36
3.06
6.94
3.60
3.00
2.65
2.88
3.07
2.75
2.78
3.60
5.19
3.40
2.70
7.04
8.93
4.58
11.73
6.36
11.63
8.18
5.37
6.50
7.64
5.69
4.41
2.29
3.25
2.84
1.33
2.65
2.37
1.29
1.09
1.03
1.18
2.17
0.75
1.15
1.92
0.95
0.71
0.76
0.53
0.79
0.67
0.58
0.90
1.29
0.70
0.70
1.82
1.73
1.29
3.49
2.46
4.64
3.54
1.76
2.34
3.06
1.05
0.90
0.07
0.07
0.05
0.08
0.06
0.07
0.07
0.06
0.06
0.07
0.07
0.06
0.05
0.07
0.06
0.05
0.04
0.05
0.03
0.04
0.05
0.02
0.08
0.05
0.05
0.06
0.07
0.06
0.05
0.07
0.05
0.06
0.07
0.06
0.07
0.07
0.07
0.01
0.01
0.01
0.01
0.01
0.01
0.01
0.02
0.02
0.01
0.01
0.01
0.01
0.02
0.02
0.02
0.01
0.01
0.01
0.01
0.01
0.01
0.02
0.01
0.01
0.02
0.01
0.01
0.02
0.02
0.01
0.02
0.01
0.01
0.01
0.02
0.02
41
55
34
62
37
29
61
40
58
42
33
47
56
31
19
42
9
0.27
0.23
0.30
0.24
0.14
0.11
0.15
0.14
0.20
0.09
0.39
0.16
0.14
0.62
0.45
0.33
0.61
0.09
0.10
0.11
0.07
0.04
0.03
0.04
0.05
0.06
0.02
0.10
0.03
0.04
0.19
0.16
0.10
0.27
intervals within each dune field, were combined; and
two subsamples were taken for grain-size analysis.
The samples were air dried, weighed, and sieved at
whole phi intervals. The graphic method ŽFolk and
Ward, 1957. was used to determine the mean grain
size and sorting.
At the upstream and downstream limits of each
dune field, the wetted width of the channel was
determined in relation to monumented survey markers on each bank. These widths were averaged and,
in conjunction with the average water depth calculated from sonar charts and the mean daily discharge
ŽWater Survey of Canada Hydrometric Station
08MG005., were used to calculate the average flow
velocity over each dune field.
The water-surface gradient was measured by surveying the water-surface elevation in relation to a
4.74
3.98
6.63
3.70
2.40
2.76
3.08
2.76
3.04
2.61
5.32
3.06
3.03
6.57
7.31
4.91
14.62
1.24
1.34
1.45
0.84
0.82
0.71
0.68
0.71
0.66
0.71
0.95
0.71
0.72
1.44
1.72
1.23
3.11
0.06
0.06
0.05
0.07
0.05
0.04
0.05
0.06
0.06
0.03
0.08
0.05
0.04
0.09
0.06
0.06
0.04
0.01
0.01
0.01
0.02
0.02
0.01
0.01
0.03
0.02
0.01
0.02
0.01
0.01
0.01
0.01
0.01
0.01
local datum established at the boundaries of each
dune field ŽTable 1.. These gradient data, were locally quite variable and could be subject to minor
measurement error ŽPrent, 1998.. To generalize the
data, it was assumed that water-surface gradient was,
in part, a function of flow velocity. As such, a
velocity–gradient rating curve was created Ž R 2 s 0.9.
and used to generalize the gradient for each dune
sample.
3.2. DeriÕed hydraulic measurements
The state of the flow is described here using
conventional measures, including
Õ
Froude number:
Fr s
Ž 1.
gD
'
stream power:
V s g QS
Ž 2.
378
M.T.H. Prent, E.J. Hickin r Geomorphology 41 (2001) 369–390
t s r gDS
Ž 3.
Õ
and velocity gradient:
SÕ s
Ž 4.
D
where Õ s mean velocity, g s gravitational acceleration, D s mean depth of flow taken as an approximation of hydraulic radius, g s specific weight of
water, Q s discharge, S s water-surface slope, and
r s density of water.
boundary shear stress:
3.3. Measures of flow resistance and channel
smoothness
Flow resistance is specified here using conventional measures, including
D 0.67S 0.5
Manning’s roughness factor:
ns
Ž 5.
Õ
and the Darcy–Weisbach friction factor:
8 gDS
ff s
Ž 6.
Õ2
The roughness state of the channel is expressed as an
index of relative smoothness Ž R S .:
D
RS s
Ž 7.
H
where H is the median height of the bedforms.
4. Results
4.1. ObserÕations
When discharge was less than 100 m3 sy1 , bedforms were visible from the survey boat since turbidity of the water was low. The dune fronts were
sinuous and generally oriented transverse to the flow;
they did not extend across the full width of the
channel. Spurs jutted out from the avalanche face,
and superimposition of ripples on dunes was common. Avalanching of bedform fronts was evident,
and a thin veneer of sediment was in transport over
the stoss side of the bedforms. Volcanic tuff and
organics were often nestled against the lee side of
the dunes. Based on grain-size analyses of the sediment samples, the bed material within the dune fields
was moderately sorted coarse sand Žmedian size FA :
0.52 mm, FB : 0.6 mm.. At low stage, a lateral bar
within each dune field and a point bar upstream of
the dune fields were exposed; the lateral bars did not
intersect the survey run.
When the turbidity of the flow and water depth
were high, visibility in the water column was limited
to - 5 cm. Although not visible, the existence of
Fig. 4. Frequency distributions of dimensionless dune height, length, and steepness for single sonar samples. Each of the distributions shown
occurred for dune height, length, and steepness samples.
M.T.H. Prent, E.J. Hickin r Geomorphology 41 (2001) 369–390
379
bedforms on the channel bed could be inferred from
the presence of boils on the water surface ŽColeman,
1969; Kostaschuk and Church, 1993; Babakaiff and
Hickin, 1996.. Boils were larger and more vigorous
at high stages and were more vigorous in dune field
A than B.
4.2. Within single sonar samples
The number of bedforms of each sample varied
throughout the field season and was inversely related
to flow magnitude Že.g., fewer bedforms at higher
flows. ŽTable 2.. Regardless of the number of bedforms recorded, each of the longitudinal profiles was
characterized by non-uniformity in bedform size and
shape.
Histograms of dimensionless dune height, length,
and steepness ŽFig. 4. are mainly characterized by
positive skewness and non-Gaussian distributions
ŽGamma, Beta, and Weibull.; while a smaller group
are characterized by negative skewness, leptokurtic,
bimodal, or simple Gaussian distributions.
Variability or dispersion about the typical dune
morphology is expressed here in terms of a coefficient of variation ŽCV. for median values ŽTable 2..
For example, the CV for the median dune length of a
sample is given by:
CVMA D s 100 = MADrL
Ž 8.
where MAD is the median absolute deviation of a
sample. Variance in dune shape increased as discharge increased in Lillooet River.
4.3. Annual regime of dune geometry distribution
statistics
For each sample, the median bedform height,
length, and steepness values were calculated and
used to represent the sample in subsequent analyses
ŽTable 2.. The median height Ž H . of the bedforms
ranged from 0.08 to 0.96 m Ž FA . and from 0.09 to
0.67 m Ž FB .; the range of median length Ž L. was
2.01–20.99 m Ž FA . and 2.40–14.62 m Ž FB .; the
range of steepness Ž HrL. was 0.02–0.10 Ž FA . and
0.02–0.09 Ž FB .. A time series plot of the data
showed that dunes tend to be largest in the
springrsummer, coincident with the freshet, and
smallest in winter ŽFig. 5..
Fig. 5. Time series of median dune height ŽA., length ŽB., and
steepness ŽC. for dune field A, relative to the 1995r1996 hydrograph. The time series of field B dunes are similar to those shown
here. Y-axis represents month, beginning with May.
Histograms of the combined dimensionless data
of all measured dunes Ži.e., ) 4000. shows that dune
height and length resemble a Gaussian distribution
with slight positive skewness. The histogram of di-
380
M.T.H. Prent, E.J. Hickin r Geomorphology 41 (2001) 369–390
Fig. 6. Histogram of the median dune-shape variables for the combined field A and B data sets.
Fig. 7. Relations among dune shape variables for fields A Žshaded circles. and B Žopen circles. data.
mensionless dune steepness is essentially Gaussian
and slightly leptokurtic. The frequency distributions
of median Žof each sample. dune height and length
exhibit a clear positive skewness, while dune steepness approaches a Gaussian distribution ŽFig. 6.. In
FB , both height and length display elements of bimodality. Even with similar hydrologic regimes, the
general characteristics of dune shape varies between
dune fields.
A strong, positive correlation exists between median dune height and length Ž R 2 ) 0.8; Fig. 7, Table
3.; dune length Ž L. is approximately 12 or 17 times
dune height Ž H .. Dune steepness Ž HrL. increases
rapidly in a nearly unique relation with dune height
or length, appears to attain an upper limit, and then
continues to increase at a reduced rate, accompanied
by an increase in data scatter ŽFig. 7.. This change Ža
discontinuity or threshold. occurs when dunes attain
a height of 0.25 m or a length of 4 m. The two linear
regression models that predict dune height as a function of dune length were significantly different Ž p -
0.001. as determined through hypothesis testing ŽTable 3..
4.4. Dune shape and the flow
4.4.1. Dune height and length
Relations between the dune shape variables and
several measures of flow are generally linear, although dune height and steepness in field B appear
to approach an upper limit ŽFig. 8..
Table 3
Simple linear regression models to predict dune height or length
Žin metres. for each dune field. All paired models were significantly different Ž p- 0.001.
Equation
n
S.E.E.
R2
HA s 0.083 LA y0.095
H B s 0.058 L B q0.012
HA s 0.215DA y0.163
H B s 0.178 D B y0.091
LA s 2.523 DA y0.656
L B s 2.853 D B y1.238
61
62
61
62
61
62
0.056
0.081
0.098
0.088
1.131
1.153
0.94
0.79
0.82
0.75
0.82
0.82
M.T.H. Prent, E.J. Hickin r Geomorphology 41 (2001) 369–390
381
Fig. 8. The relation of fields A Žshaded circles. and B Žopen circles. to discharge ŽA. and velocity ŽB..
Linear least-squares regression of Lillooet River
data yields a predictive equation for dune height as a
function of water depth Ž R 2 ) 0.75; Table 3.. Similar linear least-squares regression analyses of the
dune length data were completed Ž R 2 ) 0.82; Table
3.. In general, the scatter of dune height and length
increases with discharge and water depth but remains
nearly constant for increasing flow velocity. The
paired regression models for both dune height and
length were significantly different Ž p - 0.001. as
determined through hypothesis testing ŽTable 3..
The pattern of data scatter evident when shape
variables are plotted vs. stream power or shear stress
varies between the two dune fields ŽFig. 9A.. In
dune field A, both dune height and length increase
rapidly in a strong linear trend with respect to stream
power. In field B, dune height appears to approach
an upper limit. The linear rate of change of dune
length is more gradual in field B than in field A.
In Fig. 9B, a distinct relation between velocity
gradient and dune size is evident. That is, during
low-flow events when the velocity gradient in each
382
M.T.H. Prent, E.J. Hickin r Geomorphology 41 (2001) 369–390
Fig. 9. The relation of fields A Žshaded circles. and B Žopen circles. dune-shape to stream power ŽA. and to velocity gradient ŽB.. The
relation between dune shape and shear stress resembles the patterns depicted in ŽA..
Fig. 10. The relations of Manning N ŽA. and relative smoothness ŽB. to dune-shape Ž FA : shaded circles; FB : open circles.. The relation
between Darcy Weisbach’s ff and dune-shape is similar to that in ŽA..
M.T.H. Prent, E.J. Hickin r Geomorphology 41 (2001) 369–390
383
Fig. 11. The relation of roughness to discharge ŽA., water depth ŽB., flow velocity ŽC., stream power ŽD. and velocity gradient ŽE.. The
relation between roughness and shear stress resembles that of ŽD.. Ž FA : shaded circles; FB : open circles..
384
M.T.H. Prent, E.J. Hickin r Geomorphology 41 (2001) 369–390
dune field was steepest Ži.e., VrD ) 0.55., dune size
appeared to be limited. As discharge increased, the
velocity gradient decreased and dune size was no
longer limited. The smallest dunes occurred throughout the full range of velocity gradient observed
during the field season.
4.4.2. Dune steepness
Relations between dune steepness and the various
measures of flow Ži.e., discharge, velocity, shear
stress, and stream power. are distinctly curvilinear
and vary between the dune fields. In Figs. 8 and 9,
dune steepness appears to approach an upper limit of
0.085. In both fields, the upper limit is attained when
stream power is approximately 500 W my1 . Once
attained, dune steepness decreases gradually as
stream power continues to increase. Due to the range
of stream power calculated for dune field A Župper
limit of 1592 W my1 compared to 3830 W my1 in
field B. the trend is truncated abruptly for that data
set; while in field B, steepness continues to decrease
gradually after 500 W my1 . Dune steepness decreases linearly with increasing velocity gradient although the relation clearly is associated with considerable data scatter.
4.5. Flow resistance, bedforms, and the flow
The relation between flow resistance and dune
geometry in Lillooet River is shown graphically in
Fig. 10A. As expected, flow resistance increases
rapidly until a threshold dune size is attained Ži.e.,
H s 0.25 m; L s 5.5 m.. The rate of change of
roughness decreases after this threshold has been
exceeded and remains nearly constant in field A and
increases slightly in field B. Manning’s N appears to
approach a limit of 0.037 in field A and a limit of
0.055 in field B. Darcy–Weisbach’s ff approaches
limits of 0.07 Ž FA . and 0.14 Ž FB .. Manning’s N
varies directly with dune steepness.
The relation of flow resistance to dune steepness
differs from the relations to dune height and length.
That is, flow resistance is low and remains nearly
constant when dune steepness is less than 0.04 and
then increases rapidly with further increases in steepness.
Relations between the relative smoothness of the
channel bed and dune geometry variables ŽFig. 10B.
show a trend similar to those depicted in Fig. 10A.
Specifically, the smoothness of the channel bed decreases rapidly until dunes attain a height of 0.25 m
or a length of 5.5 m. As dunes become larger, the
relative smoothness of the channel bed decreases
more gradually. Relative smoothness varies inversely
with dune steepness.
The relations in Fig. 11 reveal a curvilinear response of flow resistance to increasing flow energy.
Flow resistance increases most rapidly until discharge reaches about 125 m3 sy1 , velocity is 1 mrs,
water depth is 2.5 m, stream power is - 500 W
my1 , and tractive force is - 10 Nrm2 . Once these
limits are exceeded, however, the rate of change in
Manning’s N, in Darcy–Weisbach’s ff, and in relative smoothness decreases to nearly constant values.
Both N and ff continue to increase, albeit at a
reduced rate in FB , with increasing flow depth.
5. Discussion
5.1. Bedform geometry statistics
The variability of bedform size and shape reported here is similar to that recorded by most other
field researchers Že.g., Coleman, 1969; Neill, 1969;
Gabel 1993.. Some patterns of bedform shape are
similar to those illustrated by Gabel Ž1993. who
interpreted them as reflecting dune superimposition
andror various phases of the dune creation–destruction processes described by Ashley Ž1990.. The large
proportion of small dunes in any sample is expected
given that both dune creation and destruction processes create small bedforms; large bedforms occur
less frequently.
The most common dune height or length frequency distribution types identified in the literature
include Rayleigh, Weibull, Exponential, and Gaussian ŽAnnambhotla et al., 1972; Shen and Cheong,
1977; Mehrdad, 1989.. In addition to those identified
in the literature, Lillooet River dunes were also
represented by bimodal, Gamma, Beta, and negatively skewed frequency distributions. Histograms of
each dune variable sample pair, for any given sample
day, were rarely characterized by the same shape.
Given the variance in dune size observed on the
sonar charts and the processes of dune formation and
M.T.H. Prent, E.J. Hickin r Geomorphology 41 (2001) 369–390
propagation, variance in the frequency distributions
between samples on a sample day was not unexpected.
Frequency distributions of the median dune shape
of samples collected during the annual hydrograph
clearly showed a spatial difference between dune
fields. even though the flow regime in these areas of
the river was similar. The water-surface gradient did
vary between dune fields, however; and since this is
an important determinant of the forces acting on the
bed, this difference likely accounts for much of the
spatial variation in frequency distributions.
The response of dunes to changes in discharge is
sometimes regarded as a stochastic process, leading
Raudkivi Ž1982. to suggest that it is best described
by the statistical variance of a sample of dunes.
Although variance is often assumed to be a measure
of disequilibrium in the processrresponse system,
sets of dunes equilibrated to uniform steady flow are
also characterized by morphological variance ŽAllen,
1983.. Variance in dune shape, which clearly increases as discharge increases on Lillooett River, is
comparable to other studies although the range of
variance is larger than that reported by Gabel Ž1993..
The difference is likely a function of the large range
in discharge magnitude that was sampled Ži.e., over
an order of magnitude. since high rates of dune
creation are associated with larger flows ŽAllen,
1983..
In the literature, dune length is sometimes expressed as a ratio to dune height since these measures of dune shape often exhibit a strong linear
relation Žas they do in this study.. The data from this
study were compared to previously published results
and theoretical or flume based models ŽTable 4.. The
comparison showed that Yalin’s Ž1964. theoretical
model tended to overpredict field measured dune
LrH ratios.
Dune steepness in Lillooet River increases from
initiation until a threshold height Ž0.25 m. or length
Ž6 m. is attained ŽFig. 7.. Once this limit is exceeded, any further increase in length is associated
with a gradual decrease in dune steepness while an
increase in height is associated with only a slight
increase in dune steepness. Thus, dune length appears to be the main control of dune steepness.
5.2. Dune shape and the flow
Relations between Lillooet River dune height or
length and discharge or depth and velocity are nearly
linear, a finding also reported by Jackson Ž1976.,
Gabel Ž1993., and Babakaiff and Hickin Ž1996..
Fredsøe Ž1982. indicated that dune height will grow
until a limiting stage of height:water depth is attained. As discharge and water depth increase, the
scatter of dune height or length increases and is
attributable to flow history and the stage of bedform
development Ži.e., creation, destruction. for each
sample. While water depth is considered to be the
limiting factor of dune height development, only the
height of dunes in field B appears to reach an upper
limit that is independent of discharge.
5.3. Empirical relations
Several published empirical models to predict
dune geometry in both steady and unsteady flow
Table 4
Ranges of dune height, length, and lengthrheight ratios have been documented for various field investigations
Dune height Žm.
Dune length Žm.
Dune L: H
Author, river
0.08–0.96
0.15–1.22
0.15–9.0
1.09–2.42
0.29–2.08
2.01–20.99
2.40–21.34
2.5–550
28.7–38.1
6.9–54.1
12–17
6.5–12.5
10–400
15.4–27.8
18–20 Žfreshet.,
25–28 Žnon-freshet.
15–20
30
Prent Ž1998. —Lillooet River, Canada.
Neill Ž1969. —Red Deer River, Canada
Harbor Ž1998. —Lower Mississippi, US
Kostaschuk and Villard Ž1996. —Fraser River, Canada
Kostaschuck et al. Ž1989. —Fraser River, Canada
0.097–0.195
2–4.05
385
Gabel Ž1993. —Calamus River, US
Yalin Ž1964. —theoretical
386
M.T.H. Prent, E.J. Hickin r Geomorphology 41 (2001) 369–390
environments Že.g., Fredsøe 1979, 1982; Yalin 1964,
1977. deserve some attention here ŽTable 5.. Yalin’s
model suggests that dune height Ž H . cannot exceed
0.167D and that dune length Ž L. is approximated by
2p D. Based on theoretical considerations of shear
stress and bedloadrsuspended load, Fredsøe Ž1982.
argues that HrD approaches a constant of 0.285 at
large bed shear stress.
In this study, dune height for each dune field was
predicted as a function of water depth ŽTable 3..
Both estimates fall between those predicted by the
Yalin and Fredsøe models ŽTable 5.. Comparison to
other field studies shows that Lillooet River HrD
ratios overlap slightly with those reported by Gabel
and by Neill but do not overlap with Allen’s Ž1984.
predictions ŽTable 5.. Because the general field setting of Lillooet River is not unlike Gabel’s field site
Žwhere Yalin’s model over predicted HrD ., the
poor correspondence here with Gabel’s results and
the relatively close agreement of HrD with Yalin’s
model was not anticipated and remains unexplained.
Lillooet River dune length was predicted as L s
2.522 D y 0.657 Ž FA . and as L s 2.853 D y 1.235
Ž FB .. Similar relations published by Yalin and Gabel
over predict the length of Lillooet River dunes ŽTable 5.. The LrD ratios overlap the range reported by
Ikeda and Iseya from Teshio River and are smaller
than the range reported by Jackson from the lower
Wabash River. The similarity of Lillooet and Teshio
results likely reflects the similarity of discharge magnitude and discharge ratio in the two studies, although the range of discharge in Teshio River is
larger than in Lillooet River.
5.4. Role of water-surface slope
The pattern of dune morphology with respect to
stream power or shear stress ŽFig. 9A. for fields A
and B data differs, one from the other. We attribute
the difference to local variation in water-surface
gradient Ža function of channel planform and largescale channel bed morphology., which encompasses
a smaller range in field A than in B. In consequence,
the rate of change in tractive force with increasing
flow is smaller in field A, allowing dunes in this
reach to become more fully equilibrated to flow
conditions and to develop the strong linear relation
observed in Fig. 9A. The results from this study,
therefore, confirm that water-surface gradient is an
important factor in determining bedform size as suggested by Ikeda and Iseya Ž1980. and by Simons and
Richardson Ž1966..
In Lillooet River, as discharge increased, the velocity gradient decreased and dunes became larger
ŽFig. 9B.. This result is likely a function of the
increasing influence that water depth Ži.e., through
boundary tractive force. exerts on bed form size.
5.5. Transition between upper and lower regime
bedforms
When dune steepness is plotted vs. various measures of flow such as discharge, velocity, shear
stress, and stream power, a distinctly curvilinear
relation is apparent ŽFigs. 8 and 9. that varies in
detail between the two dune fields. Peak values
coincide with a discharge of 125 m3 sy1 , water
Table 5
Various empirical models relating dune height Ž H . and length Ž L. to water depth Ž D . have been developed based on field of flume data and
theoretical work. Models predict dune height in metres
Dune height
Dune length
Author, river
H s 0.05 y 0.24 D
H s 0.21 y 0.38 D
H s 0.14 y 0.60 D
H s 0.2 D
L s 1.29 y 4.70 D
L s 6.42 D y 0.27
L s 1.3 y 7D
L s 4 y 9D
L s 0.5 y 3.0 D
L s 5D
L s 2p D
Prent Ž1998. —Lillooet River, Canada.
Gabel Ž1993. —Calamus River, US
Neill Ž1969. —Red Deer River, Canada
Jackson Ž1976. —Lower Wabash River, US
Ikeda and Iseya Ž1980. —Teshio River, Japan
Yalin Ž1964. —flume and field based
Yalin Ž1977. —theoretical
Allen Ž1984. —field based
Fredsøe Ž1982. —theoretical
H s 0.167D
H G 30 D
H s 0.285D
M.T.H. Prent, E.J. Hickin r Geomorphology 41 (2001) 369–390
depth of 2.5 m, flow velocity of 1.1 m sy1 , and
stream power of 500 W my1 . The upper limit of
steepness attained Ži.e., 0.085. is higher than the
maximum dune steepness proposed by Haque and
Mahmood Ž1986. on the basis of an analytical study
of bedform steepness.
The curvilinear trend of dune steepness in relation
to the changing flow is noted elsewhere in the
literature Že.g., Yalin and Karahan, 1979; Fredsøe,
1982.. Typically, as flow energy increases, dune
length increases more rapidly than dune height, which
causes bedforms to flatten. In the expected sequence
of bedform development that accompanies an increase in flow energy in natural channels, dunes
become larger and begin to wash out at Froude
numbers smaller than unity in a transition to upperregime bedforms ŽEngelund and Fredsøe, 1982..
Karim Ž1995. noted that the transition between lowerand upper-regime bedforms may occur at a mean
Froude number as low as 0.55. Froude numbers
calculated for flow events that occupied the channel
during sampling days encompass a narrow range
with an upper limit of 0.29 and 0.36, respectively, in
dune fields A and B. Because the numbers represent
averages, a Froude number at any point within the
channel could be either higher or lower. Neill Ž1969.
also observed an apparent washing out of bedforms
in the Lower Red Deer River in Alberta for a similar
range of Froude numbers and discharge examined
here. Statistical analyses of relations involving dune
height, length, and steepness and Froude number
indicate that dune size and shape are independent of
Froude number on Lillooet River.
Johns et al. Ž1990. Žas mentioned in Harbor,
1998. suggested that the loss of dune mass to suspended load can flatten the shape of a dune as flow
magnitude increases. Indeed, Fredsøe Ž1979, 1982.
suggested that, as the tractive force increases, a
larger proportion of bedload becomes suspended load
and, consequently, only a small part of bedload will
be carried past, and settle on, the dune front. Furthermore, the shear exerted on a dune reaches a maximum at the dune crest ŽFredsøe, 1979.. Thus, as the
flow energy Ži.e., tractive force. increases, the dune
crest is eroded while dune length continues to increase, causing a decrease in dune steepness.
In an examination of bedform stability fields,
Allen Ž1983. observed that lower-stage plane bed
387
and dune fields can overlap when the dune
heightrwater depth ratio is as low as 0.27. For
Lillooet River data, the average heightrdepth ratio is
seldom greater than 0.20 and does not exceed 0.24.
It is likely that locally the ratio will be larger. If the
trend in dune steepness with respect to stream power
does indicate a gradual washing out of the dunes,
then the channel bed configuration is changing to a
lower-regime plane bed with sediment movement
rather than an upper-regime plane bed.
5.6. Flow resistance elements and dunes
It is widely recognized that flow resistance varies
with the shape and size of bedforms. The most
marked increase in boundary roughness occurs when
dunes begin to form on a previously flat or rippled
channel bed ŽLeopold et al., 1964; Simons and
Richardson, 1966; Engelund and Fredsøe, 1982.. In
Lillooet River, roughness increases Žrelative smoothness decreased. rapidly with increasing dune size
during low-flow conditions until a threshold height
Ž0.25 m. or length Ž5.5 m. is attained. Boundary
roughness increases more slowly once the threshold
dune size is exceeded. In contrast, during low-flow
conditions, flow resistance is low but increases
rapidly when dune steepness exceeds 0.04. The response of flow resistance to dune steepness is to be
expected since flowlines are increasingly disrupted
as dune steepness increases, thus progressively draining energy from the mean flow.
5.7. Flow resistance elements and the flow
While the relation between flow resistance and
dune size has been described and discussed in the
literature, the relation between flow resistance and
discharge, in contrast, has received much less attention and is not as well understood ŽGee, 1975..
Results of this study show that flow resistance increases rapidly until a limiting discharge Ži.e., 125
m3 sy1 . is attained. Within this same range of flow,
and ending at the same limiting discharge, dune size
increases rapidly in both dune fields. After the
threshold flow parameters are exceeded, flow resistance and relative smoothness fluctuate about a nearly
constant value while dune height and length continue
to increase, although at different rates so that dune
388
M.T.H. Prent, E.J. Hickin r Geomorphology 41 (2001) 369–390
steepness begins to decrease. The decrease in dune
steepness must reduce flowline distortion and therefore reduced, in turn, the resistance to flow offered
by the dunes.
The flume studies of Simons and Richardson
Ž1966. suggested that flow resistance increases with
an increase in flow depth only when the sediment in
transport is larger than 0.3 mm. They suggested
further, based on field studies, that at large depths
flow resistance may decrease when the sediment is
coarser than 0.3 mm, even if dune size continues to
increase. Results obtained by Robbins Ž1976. show
that, as discharge increased, Manning’s N decreased.
The decrease in flow resistance Žwith increasing
depth or discharge. is due to the reduced interference
the increasing dune size exerts on the channel flow.
The present data do not indicate, however, that the
channel bed becomes smoother as the various flow
parameters increase; the relative roughness of the
bed remains nearly constant as the flow deepens
ŽFig. 11b.. Water depth increases as a linear function
of discharge in both dune fields.
It is not clear whether this puzzling result simply
reflects the fact that AroughnessB is a complex property not well captured by simple ratios of dune
length and height or whether some sort of mean flow
discontinuity is also affecting flow resistance.
For example, Dyer Ž1986. concluded from a review of the literature that macroturbulent flow structures Žkolks. caused by flow separation at dune
fronts occur when dune steepness G 0.070. Once
peak steepness Ž; 0.085. of dunes in this study is
attained Žat discharges 125 m3 sy1 , water depth s
2.5 m, flow velocitys 1.1 m sy1 , stream powers
500 W my1 , tractive force s 10 N my2 ., dune steepness begins to decrease, while flow resistance remains sensibly constant. If flow separation occurs
when steepness exceeds 0.070, we might speculate
that the trend observed in Fig. 10 reflects the onset
of eddy shedding from the dunes.
Flow resistance quantities are based on absolute
values of the energy gradient and therefore do not
necessarily reflect the true roughness that dunes exert in the flow environment. Specifically, for a given
stream power, dunes are largest and steepest in field
A, but flow resistance is greater in field B where the
energy gradient is largest. Regardless of absolute
value, however, flow resistance and relative rough-
ness increase most rapidly during base flow and at
the beginning and end of the seasonal flood Ži.e.,
discharge is - 125 m3 sy1 . when dune size also
increases most rapidly. Any further increase in dune
size or flow Že.g., seasonal flood. has little effect on
flow resistance and the relative smoothness of the
channel bed since these values fluctuate around a
nearly constant average, while dune height and length
continue to increase and dune steepness decreases.
The decrease in dune steepness must reduce flow
line distortion and therefore reduce, in turn, the
resistance to flow offered by the dunes.
The trends observed in Fig. 11 have not been
reported by other field researchers. Indeed, in a study
conducted on the Mississippi River where the range
of discharge is similar to that encompassed in this
study, Robbins Ž1976. observed a decrease in Manning’s N with increasing discharge as suggested by
Leopold et al. Ž1964.. Harbor Ž1998. observed only a
linear change Žboth positive and negative. for the
Friction Factor in relation to increasing discharge in
another section of the Mississippi River.
6. Summary and conclusions
An examination of channel bed configuration over
an order of magnitude seasonal range of discharge
on Lillooet River in British Columbia provides insight into bedform characteristics, the relation between bedforms and the flow environment, and the
flow resistance that these bedforms exert in a natural
channel.
Each of the channel bed surveys are characterized
by non-uniformity in bedform shape and size. Frequency distributions of each dune Žheight, length, or
steepness. varied between samples on any day and
between dune fields. Most often the distributions
exhibited positive skewness ŽWeibull, Gamma, Beta.,
a reflection of the creation–destruction process that
leads to a preponderance of small rather than large
dunes in any sample.
Neither the height nor length of dunes measured
in this investigation are successfully predicted by the
empirical models of Allen, Fredsøe’s or Yalin’s
Ž1964.. Least-squares regression models for dune–
height relations produced here are similar to models
published by other field researchers, but regression
M.T.H. Prent, E.J. Hickin r Geomorphology 41 (2001) 369–390
models for dune length only conform to those developed elsewhere if the discharge of the study rivers
was similar.
Although the average Froude number of the flow
environments in both dune fields was much less than
critical, the dunes appeared to wash out towards a
lower-regime plane bed with sediment movement.
The process responsible for the phase change is
attributed to increased flow competence causing a
shift from dominant bedload to suspended sediment
load regime.
Various relations examined in this study indicate
that the energy gradient is an important control in
determining the size that bedforms can attain in
otherwise similar flow regimes. Results indicate that
when the energy gradient changes little, potential
dune size may be realized more quickly than in a
flow environment where the range in energy gradient
is large.
Flow resistance increases rapidly during low flow
conditions until a limiting dune height Ž0.25 m. or
length Ž5.5 m. is attained. Any further increase in
dune size causes only a gradual increase in boundary
roughness. Flow resistance is affected by dune steepness: resistance increases rapidly when steepness
exceeds 0.04 due to the increased bedform interference with the flow.
Flow resistance and relative roughness of bedforms on the channel bed increase most rapidly until
a threshold discharge Ž125 m3 sy1 . is attained. In
general, flow resistance increases most rapidly during changes in base flow and at the beginning and
end of the seasonal flood. Any further increase in
flow Žor dune size. has little effect on flow resistance.
Flow resistance variation is not clearly explained
by changes in bedform geometry and may simply
reflect an inadequate measure of AroughnessB adopted
in this study; or it may, perhaps, reflect the confounding influence of flow-structure changes due to
macroturbulence production.
Acknowledgements
This study is a project partly funded by the Natural Sciences and Engineering Research Council of
Canada and by the Department of Geography at
389
Simon Fraser University. The Water Survey of
Canada provided discharge data for hydrometric station 08MG005. Field assistance was provided mainly
by Alan Paige but also by Grant Burns, Rene LeClerc,
Csaba Ekes, and Shirley McCuaig.
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