Earth Surface Processes and Landforms
Earth Surf. Process. Landforms 25 , 1011±1024 (2000)
YVONNE MARTIN 1 * AND MICHAEL CHURCH 2
1 Department of Geography, The University of Calgary, Calgary, Alberta, T2N 1N4, Canada
2 Department of Geography, The University of British Columbia, Vancouver, British Columbia, V6T 1Z2, Canada
Received 1 August 1999; Revised 22 February 2000; Accepted 3 March 2000
ABSTRACT
Bagnold developed his formula for bedload transport over several decades, with the final form of the relation given in his
1980 paper. In this formula, bedload transport rate is a function of stream power above some threshold value, depth and grain size. In 1986, he presented a graph which illustrated the strength of his relation. A double-log graph of bedload transport rate, adjusted for depth and grain size, versus excess stream power was shown to collapse along a line having a slope of 1 5.
However, Bagnold based his analyses on limited data. In this paper, the formula is re-examined using a large data set in order to define the most consistent empirical representation, and dimensional analysis is performed to seek a rationalization of the formula.
Functional analysis is performed for the final version of the equation defined by Bagnold to determine if the slope of
1 5 is preserved and to assess the strength of the relation. Finally, relations between excess stream power and bedload transport are examined for a fixed slope of 1 5 to assess the performance of various depth and grain size adjustment factors. The rational scaling is found to provide the best result. Copyright # 2000 John Wiley & Sons, Ltd.
KEY WORDS: bedload transport; transport formula; stream power; depth; grain size
INTRODUCTION
R.A. Bagnold made a completely original analysis of the transport of clastic sediments by fluid currents. He focused his later efforts on transport in water and his most fundamental statement on unidirectional shear flows was made in the paper An approach to the sediment transport problem from general physics (Bagnold,
1966), in which he likened a flowing current, such as a river, to an engine able to exert power in order to effect sediment transporting work. He considered both sediments in suspension and sediment moving in traction over the bed. We are interested in the movement of bedload because, in many rivers, these sediments correspond approximately with the bed material, the movement of which modifies the river channel.
Accordingly, we focus our attention upon his bedload formulations.
In his fundamental paper, he gave the formula: i b
ˆ !
e b
= tan † … 1 † wherein i b is the rate of sediment flux by immersed weight, e b is a bedload transport efficiency factor,
!
= gdSu = u is stream power, is fluid density, g is the acceleration of gravity, d is flow depth, S is the energy gradient of the flow, u is the mean velocity of the flow, is shear stress exerted by the fluid at the bed, and tan is a friction coefficient for the bed material. The formula is derived from a straightforward analogy with simple mechanics. It represents a statement of the bulk displacement of sediment by the shearing action of the water. It simplifies the actual physics of the grain movements under the influence of the water flow to such a degree that it can be regarded as no more than a scale correlation. In that respect it is no different from
* Correspondence to: Y. Martin, Department of Geography, The University of Calgary, Calgary, Alberta, T2N 1N4, Canada.
E-mail: ymartin@ucalgary.ca
Contract/grant sponsor: Natural Sciences and Engineering Research Council of Canada
Copyright # 2000 John Wiley & Sons, Ltd.

1012 Y. MARTIN AND M. CHURCH nearly all other classical bedload sediment transport formulae, most of which depend upon a correlation between i b and .
Classical relations for `established' transport (that is, transport not too near the practical threshold of
3/2 , and Bagnold found a similar functional relation with motion) nearly always follow the correlation i b
/ stream power when he began to investigate data. So his last papers on the topic constituted a serial investigation of this correlation on an empirical basis. These empirical formulae remain interesting not only for their association with his attempt to think through the problem from first principles, but because they are relatively easily utilized. The principal unknowns are flow depth, flow velocity and energy gradient. The former two are closely correlated with river discharge. Indeed, !
can be approximated as gQS / w , in which Q is stream discharge and w is flow width (measured at the surface), provided we accept < d > = Q / wu , the
`hydraulic mean depth', as a satisfactory integral measure of the effect of the channel boundary on the flow.
Energy gradient varies modestly at a given location on a river, in comparison with the other variates, and can be approximated, to first order, by the river slope. We will see that, in practice, some summary measure of sediment properties is required as well. These quantities are relatively frequently measured and can fairly readily be approximated in theoretical studies. The final formula (Bagnold, 1980) has been relatively successful, in comparison with a number of other proposed bedload transport formulae, in estimating fluvial bedload transport (cf. Gomez and Church, 1989). It is, therefore, attractive for applications as diverse as making reconnaissance estimates of bed material transport in rivers, computational studies of river channel development, and simulations of drainage basin or landscape evolution.
Our purpose is to review Bagnold's empirical formulae for bedload transport to seek the most consistent empirical representation, and to ask whether some rationalization is possible. Achievement of these objectives should increase confidence that a formulation of bedload transport based on stream power is a viable representation of the phenomenon. An issue in any empirical investigation is the data base upon which the study is founded. Accordingly, we will begin with a brief description of Bagnold's data, and of our own.
DATA
In establishing his empirical correlation, Bagnold (1977, 1980, 1986) employed a serially expanded data set drawn from investigations on several rivers in which the observers variously employed bed traps and basket samplers (Table I). He also made reference to several well known experimental data sets from flume studies.
Notwithstanding his reassurance that `In compiling figure 1 there has been no discarding of awkward evidence . . .' (Bagnold, 1986, pp. 370±371), he in fact based his study on only a limited number of the data available in each of his sources. He offered no explanation of this circumstance.
Site
W. Mikeimin River, Sinai
Jordan River, Israel
Meshushim River, Israel
Elbow River, Alberta
East Fork River, Wyoming
Zaire River, Zaire
San Juan River, Utah
Flume (Williams)
Flume (Mantz)
Flume (Mantz)
Table I. Data used in Bagnold (1986)
Study
Schick and Inbar (1979)
Schick and Inbar (1979)
Schick and Inbar (1979)
Hollingshead (1968)
Leopold and Emmett (1976)
Peters (1971)
Leopold (pers. comm. to Bagnold)
Williams (1970)
Mantz (1980)
Mantz (1980)
Order of grain size
(mm)
30
300
300
2 5
1 2
0 6
0 32
1 1
0 04
0 018
Order of depth (m)
0 4
1 5
2 0
0 6
1 6
20
0 8
0 1
0 1
0 1
Copyright # 2000 John Wiley & Sons, Ltd.
Earth Surf. Process. Landforms 25 , 1011±1024 (2000)

BAGNOLD's EMPIRICAL BEDLOAD FORMULAE 1013
Figure1.Excess stream powervstransport rateadjusted fordepthandgrain sizeaccordingtothe original Bagnoldequation.Modifiedfrom
Bagnold (1986). There is a typographical error on the ordinate of Bagnold's original graph which has been corrected in this figure
We present his final correlation as our Figure 1. We began our study by attempting to reconstruct this plot.
The river data incorporated by Bagnold were identified easily and correspond to measurements presented in the data tables given in his 1980 paper. In cases wherein only a selection of the original data were included
(e.g. Elbow River), those points do not appear to be averages of the complete data sets and we could deduce no rational reason for his particular choice of points. Although the graph appears to indicate a remarkably strong relation, the basis for selection of the supporting data remains unknown.
In order to test the Bagnold relation more thoroughly we adopted a larger data set for analysis. Gomez and
Church (1988) compiled 410 measurements, made in various flumes and rivers, which meet a requirement of
`equilibrium transport conditions', defined to mean that flow conditions and sediment characteristics remained approximately constant throughout the measurement procedure and that certain threshold conditions were satisfied. There is, then, expected to be a relation between flow parameters and transport rate, as sediment transport is supposed to be operating at `system capacity'. This condition is implicit in all computational attempts to estimate bedload transport.
The data of Gomez and Church (1988) were obtained from measurements in gravel and coarse sand-bed channels (median grain size > 1 mm) in which no or minimal bedform development was observed. They also included 241 points in their compilation that were obtained in flume experiments reported in their collection that did not meet the strict equilibrium requirements. Some of these points were included in the present analysis as they were not separable from equilibrium results. All river points were assumed by Gomez and
Church to represent equilibrium transport conditions; it is probable that, strictly, none did.
Of the total (651) data points, two experimental data sets comprising 393 observations were eliminated because they were taken under unusual conditions. These data included 393 observations from the data set of the Institute for Hydraulic Research, University of Iowa (Mavis et al.
, 1937; Johnson, 1943), in which sediment feed and transport were independent of each other and the area of mobile bed was restricted. Many of these experiments were also run at supercritical flow. Another 100 observations from the St. Anthony Falls
Hydraulic Laboratory, (SAFHL), University of Minnesota (Paintal, 1971) were made at very low transport rates without sediment feed. In these data there is no easy way to assign a threshold transport condition.
Copyright # 2000 John Wiley & Sons, Ltd.
Earth Surf. Process. Landforms 25 , 1011±1024 (2000)

1014 Y. MARTIN AND M. CHURCH
Figure 2. Unit stream power vs unit mass transport rate. Data are from Gomez and Church (1988). Data are unadjusted for !
0 the well-known departure at low transport rates from the exponent of 1 5
, hence exhibit
A further eight observations were eliminated from the analysis as measured stream power is lower than calculated threshold stream power, although bedload transport actually occurred in these cases. In addition, three out of 130 river data points presented this conundrum. There remained 247 data points, of which 127 were derived from field observations of rivers. Of the remaining flume data, 48 observations satisfied the
`equilibrium condition' defined above, and 72 violated it in some minor way (usually meaning that width/ depth ratios were smaller than 10, or that measurements were made for only a short period). A summary of these data is given in Table II and a plot of the data is given as Figure 2. They represent the data available to us that one might reasonably expect to conform with a common flow-sediment transport correlation.
It is perhaps worth noting that the classic experimental data of G.K. Gilbert, referred to by Bagnold, are not in the Gomez-Church catalogue and are not used here because the flume was very narrow, and only bed slopes are available for the data set.
RE-EXAMINATION OF THE BAGNOLD FORMULA
The final form of the Bagnold equation is given as (Bagnold, 1980): i b
ˆ s s
ÿ i b ref
"
!
ÿ !
o
!
ÿ !
o
† ref
#
3 = 2
d = d ref
† ÿ 2 = 3 … D = D ref
† ÿ 1 = 2 … 2 wherein i b is specific bedload transport rate (dry weight; although in Bagnold's presentation it is actually the mass rate of sediment flux because he chose to ignore the acceleration of gravity in his specification of stream
Copyright # 2000 John Wiley & Sons, Ltd.
Earth Surf. Process. Landforms 25 , 1011±1024 (2000)
†

BAGNOLD's EMPIRICAL BEDLOAD FORMULAE
Table II. Data used in analyses
(a) Flume data
Run designation
(if > 1 run)
No.
data points
Velocity
(m s ÿ 1 ) Depth (m)
Environmental Research Center, University of Tsukaba (Ikeda, 1983)
16 1 05±1 75 0 084±0 313
Width (m)
4 0
Slope ( 10 3 )
Eidenossiche Technische Hochschule, Zurich (Meyer-Peter and Muller, 1948; Smart and Jaeggi, 1983)
ETH: 1
ETH: 2
ETH: 3
34
17
7
1
0
0
80±2
63±0
74±1
88
88
18
0
0
0
342±1
056±0
068±0
092
207
108
2
2
2
0
0
0
3
2
8
17±17
25±2
01±8
69
94
27
Massachusetts Institute of Technology, Cambridge (Wilcock, 1987)
MUNI
CUNI
0 5
1 0
5
2
5
4
0
0
0
0
518±0
733±0
528±0
524±0
724
826
794
729
0
0
0
0
114±0
108±0
112±0
109±0
128
111
115
112
0
0
0
0
6
6
6
6
1
3
1
1
24±2
82±4
26±4
82±3
93
91
92
30
US Geological Survey, Washington D.C. (Williams, 1970)
30 0 37±1 15 0 027±0 225 0 61±1 19
2 28±9 97
0 60±23 4
1015
D
50
(mm)
6 5
28 65
1 46
3 73
1 86
5 28
1 83
1 83
1 35
(b) River data
River study No. data points
Velocity
(m s ÿ 1 ) Depth (m) Width (m) Slope ( 10 3 ) D
50
(mm)
Clearwater River, Idaho (Jones and Seitz, 1980)
3 2 83±3 35
Snake River, Idaho (Jones and Seitz, 1980)
16 2 09±2 99
5 73±6 34
3 99±5 39
143 0±149 0
171 0±192 0
0 50±0 62
0 76±1 12
East Fork River, Wyoming (Leopold and Emmett, 1976, 1977; Emmett et al.
, 1980)
38 0 75±1 31 0 63±1 68 14 6
Elbow River, Alberta (Hollingshead, 1971; Gibbs and Neill, 1972)
19 1 62±2 39 0 63±0 84 38 7±47 4
Mountain River, South Carolina (Einstein, 1944)
37 0 466±0 582 0 088±0 129 4 33
7
0 7
45
1 48±1 63
Tanana River, Alaska (Burrows et al.
, 1981; Burrows and Harrold, 1983; Harrold and Burrows, 1983)
14 1 30±1 81 1 81±2 81 107 0±466 0 0 47±0 55
32 0
32 0
1 2
27 0
0 9
7 6
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Earth Surf. Process. Landforms 25 , 1011±1024 (2000)

1016 Y. MARTIN AND M. CHURCH power), is the specific gravity of fluid, is critical specific stream power, D s
/( is characteristic particle size, usually denoted in mixtures by s s is the specific gravity of sediment, !
is specific stream power,
D
!
ÿ ) for the conversion of immersed weight to dry weight, the latter measure being standard for presentation of most fluvial transport formulae. The following parameters were defined by Bagnold: o
50
(although Bagnold proposed a special procedure for bimodal sediments), and the subscript ref refers to some reference value obtained from a reliable data set (Bagnold used data from Williams, 1970). Gomez and
Church (1989) introduced the term
!
o
ˆ 5 75 f 0 04 … s
ÿ † g 3 = 2 … g = † 1 = 2 D 3 = 2 log … 12 d = D † … 3 wherein g is the acceleration of gravity, 0 04 is the assigned critical value of Shields' entrainment number, and:
† i b ref
ˆ 0 1 ; … !
ÿ !
o
† ref
ˆ 0 5 ; d ref
ˆ 0 1 ; D ref
ˆ 0 0011 … 4 †
The threshold stream power for bedload transport, !
o
, depends on depth and grain size, and it can also be seen to depend upon bed structure when it is realized that Shields' number varies with bed condition (Church,
1978; Church et al.
, 1998). The value of !
o critically affects the transport rate as it dictates the lowest value of stream power at which transport is detected. Bagnold presented his formulation of this term in 1980 (p. 457).
The simple stream power relation given in Equation 1 does not incorporate such a condition and it was in order to introduce it that Bagnold resorted to empiricism (see discussion in Bagnold, 1977). There are today a number of approaches to the formulation of the `threshold' or `critical' transport condition (cf. Wilcock,
1991, 1993), but we will consider here only Bagnold's formulation.
The strength of the Bagnold relation is suggested in the remarkable graph presented in Bagnold (1986) and herein (Figure 1). In this graph, excess stream power ( !
ÿ !
b
* variables given in the equation: o
) is plotted against i . The latter variable is the bedload transport rate adjusted to a common flow depth and grain size according to the scales for these i b
ˆ i b d d ref
2 = 3 D
D ref
1
2
5 †
The data collapse onto a straight line, which is the expected result if relations amongst transport, excess stream power, depth and grain size hold for the data being examined. Bagnold derived the scales by graphical analysis of limited data sets (Bagnold, 1980) after he failed to find a satisfactory rational collapse of transport data of individual rivers. These scales merit further consideration.
Bagnold (1977) originally supposed that transport intensity depends on both excess stream power and relative roughness ( d/D ) of the flow. In order to examine the functional dependence on relative roughness, he assumed a relation of form the relation: i b
/ ( !
ÿ !
o
) 3/2 for three river and two flume data sets. The excess stream power at a constant transport rate was plotted against the relative roughness. From this information Bagnold extracted
!
ÿ !
o
/ d 2 = 3
D
6 †
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Earth Surf. Process. Landforms 25 , 1011±1024 (2000)

BAGNOLD's EMPIRICAL BEDLOAD FORMULAE 1017
He then deduced the relation between bedload transport and relative roughness by taking the inverse of this relation, so that: i b
/ d
ÿ 2 = 3
D
7 †
Subsequently, Bagnold (1980) explained that, in the above-described investigation, the grain sizes ( D
50
) of the various data sets were approximately the same and, therefore, grain size should not have been included in the resultant equation. In essence, he abandoned the notion that relative roughness is a major factor influencing bedload transport (which preserved the rationality of his result). He now examined depth and grain size as two independent components. Using the experimental data of Gilbert (1914) and other data sets, and following a procedure similar to that followed to determine the (relative) depth relation, he concluded that the appropriate scaling for grain size is: i b
/ D ÿ 1 = 2 … 8 i whence one arrives at Equation 5.
Before leaving this review, we point out a simple mathematical error in the 1977 analysis, correction of which changes the result. Whilst Equation 6 is correct, the simple inversion to obtain Equation 7 is correct only if the relation between excess stream power and transport is linear. However, when the assumed relation b
/ ( !
ÿ !
o
) 3/2 is inserted into Equation 7, the outcome is: i b
/ d ÿ 1
D
7a †
†
This history demonstrates that there is some confusion regarding the appropriate scaling for depth and grain size. We therefore conducted a new investigation.
BEDLOAD SCALES
We replicated Bagnold's scale analyses for depth and grain size using data extracted from Gomez and Church
(1988). To examine depth scaling, data were separated into four groups of approximately equal grain size (1±
2 mm, 2±5 mm, 5±8 mm and 22±32 mm). The data within each group were analysed to evaluate the relation between depth and transport. In this manner, grain size is held approximately constant in each analysis. For each data set, a line with a slope of 3/2 was placed through the bivariate mean of the data of
The indicated value of i b at constant excess stream power of 1 0 kg m ÿ 1 s ÿ 1
!
ÿ !
o versus i b
(any other constant value would suffice) was noted. The relation between this transport rate and flow depth was assessed for each grain size by regression. It is reasonable to assign all of the error to the estimated bedload transport term since depth and grain size are quantities customarily measured with much greater precision.
When all of the data are analysed together (Figure 3) the slope of the line is ÿ 0 93 0 20. The only individually significant relation is for the grain size category of 1±2 mm with a slope of ÿ 0 87 0 060, but these data by themselves clearly are badly distributed. Furthermore, the numbers of data for individual size ranges remain so few that one can place no faith in the individual results. Nonetheless, we do have more data than Bagnold used in his analysis. On the basis of our results and the revised results of Bagnold's original analysis, which also indicates a slope of ÿ 1, the best relation appears to be:
.
i b
/ d ÿ 1 … 9 †
Copyright # 2000 John Wiley & Sons, Ltd.
Earth Surf. Process. Landforms 25 , 1011±1024 (2000)

1018 Y. MARTIN AND M. CHURCH
Figure 3. Depth vs transport rate at constant excess stream power. Each point represents one data set
Figure 4. Grain size vs transport rate at constant excess stream power. Each point represents one data set. The point in parentheses represents the East Fork River and is an anomaly for which an explanation could not be found
To study the relation between bedload transport and grain size, the data were separated into three groups of approximately equal depth (9±12 cm, 12±20 cm and 60±120 cm) (the large range of the last category is necessary in order to obtain sufficient data for analysis). Once again a line with a slope of 3/2 was fitted to each transport data set and the value of i at constant excess stream power of 1 0 kg m ÿ 1 s ÿ 1 b appears that the best overall representation of all the data (Figure 4) would be a relation i b
/ D determined. It
ÿ 1 , but two of the individual results are not significantly different from Bagnold's (1980) ÿ 0 5 relation. On the basis of the calculated regression slopes and Bagnold's result, there appear to be two candidate choices for the grain size/ bedload transport relation: i b
/ D ÿ 1 = 2 … 8 †
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Earth Surf. Process. Landforms 25 , 1011±1024 (2000)

BAGNOLD's EMPIRICAL BEDLOAD FORMULAE 1019 i b
/ D ÿ 1 … 8 a
However, the result with D ÿ 1/2 leaves open the appearance of an implicit scale dependence on d of the results for D , which is evident by inspection of Figure 4. Hence, the result with D ÿ 1 is preferred.
RATIONAL ANALYSIS
Bagnold's (1977) formulation was rational in the sense that it was dimensionally balanced, but he abandoned it when he discovered that available experimental data apparently do not support that formulation. His further analyses from 1980 entailed purely empirical relations (Bagnold, 1986). The question remains whether his results can be rationalized. We seek a rationalization using formal dimensional analysis, which appears not yet to have been applied to Bagnold's problem. For this purpose, we consider his 1980 formula in the dimensioned form: i b
= … !
ÿ !
o
† / … !
ÿ !
o
† 1 = 2 d x D y (1a)
†
We suppose that the following variates represent a complete set for the problem:
!
' = !
ÿ !
o
`effective' stream power of Bagnold (note that gravity is ignored in the definition of !
) i b
, sediment transport (mass flux of solids) d , flow depth
D , sediment grain size g , acceleration of gravity
, density (which is further defined below)
, fluid viscosity
[M L
[M L
ÿ 1
[M L ÿ 1
[L]
[L]
[L T
[M L
ÿ 2
ÿ 3
]
ÿ 1
]
T ÿ 1
T ÿ 1
T ÿ 1
]
]
]
Therefore f ( !
' , i b
, d , D , g , , ) = 0
We select three repeating variates that are independent (in the sense that they cannot be made rational functional expressions of each other) and that encompass all the dimensions in the problem; , g and D are selected. (We select D instead of d , the other available length scale, because we are interested in the transport of grains of some size D over some varying range of water depth, d; that is, D would be a parameter for a specific problem, whereas d remains a variate.) Then straightforward application of Buckingham's theorem yields the functional groups in: f … i b
= g 1 = 2 D 3 = 2 ; !
0 = g 1 = 2 D 3 = 2 ; D = d ; = g 1 = 2 D 3 = 2 † ˆ 0 … 10 †
We expect that, for fine sediment (sand), we would replace case. It can be re-expressed as v / ( gD ) 1/2 D , wherein entrainment of fine grains.
It is plain from inspection of the other terms that g 1/2 v d by , the sublayer thickness, or by roughness length, or by some other boundary-defining length scale. We expect
4 is the kinematic viscosity and ( gD ) z o
, the to be important only in this
1/2 has the dimensions of a velocity. So this term is a kind of particle Reynolds' number, which is known to influence the density exactly. If we adopt r
= s
ÿ f
, in which s
D 3/2 is an important scale. We have not specified is sediment density and f expression denotes sediment submerged density or `reduced sediment density', is fluid density, so that the r
, then this scale is the
Copyright # 2000 John Wiley & Sons, Ltd.
Earth Surf. Process. Landforms 25 , 1011±1024 (2000)

1020 Y. MARTIN AND M. CHURCH
Table III. Functional analyses for subsets of data for the Bagnold (1980) equation (with the mathematical error for depth scaling corrected)
Data set/subset N R 2 b 2 s.e. of slope s.e.e.
River data
Equilibrium ¯ume data
Non-equilibrium ¯ume data
All ¯ume data
River and all ¯ume data
127
48
72
120
247
0 930
0 854
0 443
0 942
0 937
1 69 0 079
1 38 0 16
1 67 0 095
1 57 0 070
1 63 0 053
0 349
0 293
0 337
0 335
0 345 x-variable
( !
ÿ !
( !
ÿ !
( !
ÿ !
( !
ÿ !
( !
ÿ !
( !
ÿ !
( !
ÿ !
0
) 1 5
0
) 1 5 d ÿ 2/3 D
0
) 1 5 d
ÿ 1
2/3
ÿ 1/2
0
0
0
0
)
)
)
)
1 5
1 5
1 5 d d d
1 5 d
ÿ 2/3 D
ÿ 1
ÿ 1
ÿ 1
D 1
D ÿ 1/2
D ÿ 1
D 1/4
Table IV. Constrained slope analysis
Adjustment R 2 s.e.e.
No adjustment
Bagnold (1977)
Bagnold (1980)
Bagnold (1977) corrected
Reanalysis in this study
Reanalysis in this study
Rational adjustment
0 656
0 853
0 859
0 892
0 818
0 216
0 899
0 412
0 269
0 264
0 231
0 300
0 623
0 224 y-intercept
4 86 10
9 82 10
ÿ 3
ÿ 3
ÿ 6 91 10 ÿ 2
ÿ 2 1 88 10
ÿ 6 58 10 ÿ 2
0 102
8 40 10 ÿ 5
Coef®cient
4 01 10
0 398
6 33 10
1 14
4 99 10
2 27 10
7 93 10
ÿ 2
ÿ 3
ÿ 3
ÿ 4
ÿ 2
Einstein bedload scale. We may manipulate it as follows: r g 1 = 2 D 3 = 2 ˆ … r gD † 3 = 2 = r
1 = 2 g
ˆ … = † 3 = 2 = r
1 = 2 g
ˆ … f u 2 = † 3 = 2 = r
1 = 2 g
ˆ … f u 3 = g †… f
= r
† 1 = 2 3 = 2 where is Shields' number. Now [ et al.
f
(1982) (when it is expressed as u i
* 3 b
/ g ] is the normalization for the non-dimensional bedload flux of Parker rather than as q b
). The balance of the final expression is composed of non-dimensional constants. The term is an important parameter for bed condition, a factor considered by
Bagnold via the adjustment of
1a is:
!
by !
o
.
We may now specify Equation 1a in the terms given in Equation 10. A particular explicit form of Equation i b
=" ˆ ‰ !
0 =" Š
3 = 2
D = d † wherein " = r g 1/2 D 3/2 , i.e. : i b
=!
0 ˆ ‰ !
0 = r g 1 = 2 D 3 = 2 Š 1 = 2 … D = d † … 11 which is very near Bagnold's formula, except that the dependence upon D is different. But we know from empirical trials that the dependence upon D is difficult to guess. Here, collecting D , we obtain D arbitrary reference scales have, of course, disappeared.
1/4 . Bagnold's
†
Copyright # 2000 John Wiley & Sons, Ltd.
Earth Surf. Process. Landforms 25 , 1011±1024 (2000)

BAGNOLD's EMPIRICAL BEDLOAD FORMULAE 1021
Figure 5.Excessstream power vstransportrate adjustedfor depthandgrainsize accordingtotherationalscalings.The datafor thelowflow non-equilibrium data from the ETH flume and the Elbow River appear individually to depart from the overall scaling. The Elbow River data were taken over three different years, during which local bed conditions may have changed substantially. Transport may have been supply-limited due to bed armour
EMPIRICAL TRIALS
Functional analysis was first performed for subsets of the data compilation of Gomez and Church (1988) in order to evaluate the 1980 Bagnold formula (Equation 2) and to determine if the slope of 1 5 is preserved.
When both variates are expected to exhibit error, the analysis is different from regression (see Mark and
Church, 1977), which is a limit case. Functional analysis optimizes the estimate of the coefficient (in this case, the slope) in the bivariate relation. Its calculation entails estimation of the ratio of errors associated with each variate.
The functional slopes for the river and flume data are near 1 5 (Table III). A slope of slightly greater than
1 5 is obtained when all data points are analysed together. These results confirm that the Bagnold correlation is functionally reasonable.
R magnitude of the estimated i
2 is high in each case, but the standard errors of estimate are about three times the b
* values.
In order to assess the performance of various depth and grain size adjustment factors, relations between excess stream power and bedload transport were assessed by performing functional analyses between
!
ÿ !
o i b and for a fixed slope of 1 5 (Table IV). Perhaps the most remarkable finding is that without any adjustment for depth and grain size the relation between excess stream power and transport is already quite strong, having a best-fit functional relation with an R 2 value of 0 66. This immediately suggests that we may be dealing with a simple scale relation in the system, but the result is not rational. It, therefore, compels a closer analysis of
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Earth Surf. Process. Landforms 25 , 1011±1024 (2000)

1022 Y. MARTIN AND M. CHURCH
Equation 10. The equation reveals that, if the particle Reynolds' number is large and the relative roughness is small ( d D ), then the transport rate does not materially depend upon grain size except insofar as grain size may affect the value of !
o
(that is, the threshold for particle entrainment). This shows why size-free correlations of sediment transport with hydraulic quantities have been employed moderately successfully.
The situation might prevail nearly universally for the transport of sands in channelled flows. The most problematic cases would occur near the threshold for motion in gravels, or in relatively steep, graveltransporting channels, when D/d may not be small.
Amongst the scaled results that we studied, the best correlation and best standard error are exhibited by the rational scaling. This result yields a coefficient of 0 0793 for the
Bagnold scaling of 1980, and the empirical reanalysis with D ÿ 1/2
!
ÿ !
o term and a y-intercept closest to the expected zero intercept. The standard error of the y-intercept is in this case 0 0154. However, the uncorrected are also reasonable.
In general, scalings with a small power of D perform better than large ones. The reason for this, in light of the rational result, can be seen to depend on the data. Values of D are quite variable, but almost always 1 0
(represented as metres), so powers of D near 1 0 produce large and variable shifts of the scaled data. Since it appears that we should be seeking a rather small dependence upon D , the disparate results obtained with very small data sets should not be a surprise.
Considering the standard error of estimate, it is evident that by far the best predictive relation is the dimensionally correct Equation 11. Cancelling the fixed constants in " , we arrive at a convenient computational form for the dimensioned sediment transport: i b
ˆ ‰ !
ÿ !
o
Š 3 = 2 ‰ D 1 = 4 = d Š‰ 1 = r
1 = 2 g 1 = 4 Š … 11 a †
The last bracket can be ignored for practical purposes in datafitting exercises, since it merely represents a coefficient 0 0139. The comparison between this value and the coefficient of 0 0793 for the rational scaling in the constrained slope analysis (Table IV) is acceptable, considering the reversal of scales and the imperfect collapse of the data. The latter is probably a result of additional factors influencing the data, such as a threshold for transport different from that calculated in the equation.
The grain and depth scalings for the rational result are applied to the dependent variate, i b
, to demonstrate sensitivity to excess stream power (Figure 5). A good deal of variation remains about the functional result.
Given the diverse sources of the data, this is perhaps not surprising. The analysis does not consider variations in bed structure that may substantially affect the threshold and level of bedload transport. Furthermore, the field data undoubtedly include a substantial measure of imprecision owing to difficulties in the measurement of bedload transport. Almost any set of bedload measurements exhibits variation of the order exhibited in
Figure 5.
CONCLUSIONS
Bagnold's final empirical bedload transport equation was a stream power correlation adjusted by an empirical scaling of depth and grain size. The equation works remarkably well over a wide range of data, but it is not a rational result. A rational result delivers superior predictive performance. Bagnold's failure to find the rational result was due to his failure to appreciate the full dependence of the transport phenomenon on governing physical constants, even though that understanding had previously been reached by H.A. Einstein.
Our analysis does not close the issue. The data available to us to investigate the depth and grain size scales are no more interpretable, despite the greater number of data points, than those available to Bagnold, and we do not find empirically the rational scaling for grain size. It remains possible that our dimensional analysis is incomplete and that still another scaling remains undiscovered. A critical test could be provided by a more thorough investigation of the grain size dependence, but that must await more adequate data.
A further major data adjustment remains unstudied. Bagnold's calculations include Equation 3 to determine !
o
, and that equation assumes a Shields number of 0 04. It is apparent that rivers exhibit various values depending on the state of the bed surface (see review in Church et al.
, 1998), and it is likely that flume
Copyright # 2000 John Wiley & Sons, Ltd.
Earth Surf. Process. Landforms 25 , 1011±1024 (2000)

BAGNOLD's EMPIRICAL BEDLOAD FORMULAE 1023 experiments operated at low rates of bedload transport do as well. Hence, it is possible that the present investigation has been made more difficult by our failure to control this factor properly. But the available data do not permit us to make any confident adjustment of the threshold stream power.
ACKNOWLEDGEMENTS
The work reported here was supported by the Natural Sciences and Engineering Research Council of Canada, through a scholarship (Y.M.) and a research grant (M.C.). Stephen Rice participated in the initial stages of this study, when the critical questions were formulated, and R.I. Ferguson discussed the rational analysis with
M.C. We thank Peter Wilcock, especially for directing our attention to the significance of the grain-size independent results, and an anonymous reviewer for comments that helped us to improve the manuscript.
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