www.elsevier.com/locate/geomorph

Geomorphology 62 (2004) 35–45

Optimal alluvial channel width under a bank stability constraint

Brett C. Eatona,1, Robert G. Millarb,*

aDepartment of Geography, The University of British Columbia, Vancouver, British Columbia, Canada V6T 1Z2bDepartment of Civil Engineering, The University of British Columbia, Vancouver, British Columbia, Canada V6T 1Z4

Received 15 September 2003; received in revised form 11 February 2004; accepted 12 February 2004

Available online 21 April 2004

Abstract

To properly predict alluvial channel width using rational regime models, an analysis of bank stability must be

included in the model. When bank stability is not considered, optimizations assuming maximum sediment transport

capacity (MTC) typically under-predict alluvial channel width for natural and laboratory streams. Such discrepancies

between regime model predictions and observed channel widths have been used to argue that optimizations such as

MTC do not describe the behaviour of alluvial systems. However, rational regime models that explicitly consider bank

stability exhibit no such bias and can predict alluvial channel widths quite accurately. We present an analysis of both

laboratory and natural alluvial channels, using both kinds of models, and demonstrate the importance of bank stability

in constraining optimization solutions. We also identify a scale effect, whereby the effect of vegetation on bank

strength declines as the absolute scale of the system increases. We argue that comparisons of alluvial channel widths

against predictions from rational regime models unconstrained by bank stability are inappropriate, because they

introduce a known and quantifiable bias (toward under-prediction by the model) due to the absence of a bank stability

constraint.

D 2004 Elsevier B.V. All rights reserved.

Keywords: Alluvial channel width; Rational regime model; Bank stability; Maximum sediment transport capacity; Bank vegetation

1. Introduction term, regime, is generally preferred. Blench (1969, p. 1)

Dynamic equilibrium and adjustment of alluvial

rivers in sensu Mackin (1948) is a widely applied

concept in fluvial geomorphology and river engineer-

ing. In hydraulic engineering literature, the equivalent

0169-555X/$ - see front matter D 2004 Elsevier B.V. All rights reserved.

doi:10.1016/j.geomorph.2004.02.003

* Corresponding author. Tel.: +1-604-822-2775; fax: 1+604-

822-6901.

E-mail addresses: beaton@geog.ubc.ca (B.C. Eaton),

millar@civil.ubc.ca (R.G. Millar).1 Fax: +1-604-822-6150.

states:

The fundamental fact of river science, pure and

applied, is that regime channels tend to adjust

themselves to average widths, depths, slopes and

meander sizes that depend on: (i) the sequence of

water discharges imposed on them, (ii) the

sequence of sediment discharges acquired by

them from catchment erosion, erosion of their

own boundaries, or other sources, and (iii) the

liability of their cohesive banks to erosion or

deposition.

B.C. Eaton, R.G. Millar / Geomorphology 62 (2004) 35–4536

Clearly, while all alluvial rivers may not necessarily

be in equilibrium (Stevens et al., 1975), for many

rivers, regime remains a reasonable approximation

and provides a basic framework for the interpretation

and analysis of river behaviour. For instance, predicting

changes in alluvial channel morphology in response to,

for example, climate change, instream alterations, and

landuse changes is becoming an increasingly important

part of environmental management. However, the

geomorphic tools available to do this are primarily

qualitative constructs (Lane, 1955; Schumm, 1971;

Kellerhals and Church, 1989; Montgomery and Buf-

fington, 1998), or empirical regime equations (Leopold

and Maddock, 1953; Leopold and Wolman, 1957; Hey

and Thorne, 1986). While these tools provide some

guidance, they afford little insight into the fundamental

controls on hydraulic geometry. In contrast, rational

regime models (Chang, 1979, 1980; White et al., 1982;

Millar and Quick, 1993) (which are based on the

solution of equations describing flow resistance, sedi-

ment transporting capacity, and sometimes bank sta-

bility) can provide a more robust theoretical framework

for examining potential changes in response to various

governing conditions (Ferguson, 1986).

However, the rational regime approach has attracted

widespread criticism (Griffiths, 1984; Ferguson, 1986;

Darby and Thorne, 1995; Mosselman, 2000), in part

because use of an ‘‘extremal hypothesis’’ gives the

appearance of insufficient physical basis. Recent work

(Eaton et al., 2004) presents a more physically based

explanation for the necessary extremal hypotheses

used in rational regime models by relating morpho-

logical stability to adjustments of the flow resistance

for the fluvial system. In this paper, we address recent

criticism that observed channel widths are invariably

greater than predicted by rational regime models

(Griffiths and Carson, 2000; Valentine et al., 2001;

Shields et al., 2003). This criticism is based on

optimization models that are not constrained with

respect to bank strength. We believe that the analyses

performed by Griffiths and Carson (2000), Valentine

et al. (2001), and others (Chang, 1979; White et al.,

1982) are incomplete because they do not explicitly

incorporate the strength of the channel banks relative

to the channel bed. Therefore, their regime predic-

tions are inappropriate since they ignore the funda-

mental constraint of relative bank strength—an issue

identified by Griffiths and Carson (2000) as impor-

tant, but not incorporated into their analytical regime

model.

The confusion regarding the applicability of ra-

tional regime models is well illustrated in the follow-

ing quote from Griffiths and Carson (2000, p. 122):

‘‘comparison with actual channel widths. . .show that

natural channels will invariably exceed the width that

maximizes bedload transport capacity because of the

huge shear stresses on the channel sides.’’ Their

analysis indicates that the observed channel widths

of natural rivers in New Zealand are two to four times

the width predicted by a rational regime model that

applies the maximum sediment transport capacity

(MTC) hypothesis. Valentine et al. (2001) make sim-

ilar observations after comparing the channel widths

obtained from their laboratory experiments to widths

predicted using the MTC model presented by White et

al. (1982). Based on this inconsistency between the

predicted and observed alluvial channel widths, Grif-

fiths and Carson (2000, p. 123) conclude, ‘‘it is thus

clear that the notion of ‘equilibrium’ channels in nature

attaining the condition of maximum transport, as

sometimes entertained in the past, is invalid.’’ We

are in agreement with the first quote by Griffiths and

Carson, which is also consistent with recent observa-

tions by Valentine et al. (2001). However, we strongly

disagree with the subsequent conclusion in the second

quote. The inherent limitation in the models used by

Griffiths and Carson (2000), White et al. (1982), and

others is that the stability of the banks is not included

as an explicit constraint on the optimum solution.

When channel banks are relatively strong, as they are

for most sand-bed streams with cohesive banks or

when the banks are strongly influenced by vegetation,

this limitation does not significantly affect the model

accuracy. We show that omitting the bank stability

constraint results in a bias in width prediction in that

the calculated equilibrium widths are wider than ob-

served. This bias is greatest in channels where the bed

and banks are composed on noncohesive alluvium of

similar erodibility, and accurate prediction of channel

width using regime models based on the MTC hy-

pothesis is possible only by including a bank stability

constraint.

The basis for our argument is not new. Henderson

(1966, p. 451) points out that analytical regime

theory requires the simultaneous solution of three

fundamental relations: (i) a sediment transport equa-

B.C. Eaton, R.G. Millar / Geomorphology 62 (2004) 35–45 37

tion; (ii) a flow resistance equation; and (iii) a bank

stability criterion. The requirement for a bank stabil-

ity criterion is frequently overlooked (e.g., Shields et

al., 2003). If the discharge (Q), sediment transport

rate (Qb), and sediment calibre (D) are specified, the

channel geometry [comprising slope (S), hydraulic

radius (R) or depth (d), and wetted perimeter (P) or

width (W)] can be calculated. However, the bank

stability criterion does not impose a unique value,

merely a lower limit on the width/depth ratio (W/d).

Thus, a range of possible solutions exist, necessitat-

ing the application of what we term optimality

theory. Various extremal hypotheses have been pro-

posed, including minimum stream power (Chang,

1979), minimum unit stream power (Yang, 1976),

maximum sediment transport capacity (White et al.,

1982), and maximum friction factor (Davies and

Sutherland, 1983). Interestingly, these hypotheses

can be demonstrated to be equivalent under certain

circumstances (White et al., 1982; Davies and

Sutherland, 1983) and thus seem to reflect various

facets of some more general optimality theory that

adapts a one dimensional model (rational regime

theory) to the description of a three-dimensional

reality.

While the use of extremal hypotheses has been

criticised by Griffiths (1984), others (Chang, 1984;

Song and Yang, 1986) have refuted his analysis.

Chang (1984) pointed out that the wide-channel

approximation employed by Griffiths resulted in un-

realistic solution curves and that this prevented the

successful application of optimality theory. This point

seems largely to have been ignored by subsequent

critics of optimality theory (e.g., Mosselman, 2000).

More fundamentally, the proponents of regime theory

have been unable to propose a physical basis for the

extremal hypotheses (see Ferguson, 1986 for a thor-

ough review). As Eaton et al. (2004) demonstrate, this

obstacle may be overcome by considering the flow

resistance at the scale of the fluvial system. In our

model, we use the maximum sediment transport

capacity hypothesis because it is computationally

convenient.

Our objectives are to (i) investigate the role of bank

stability in constraining MTC solutions; and (ii)

demonstrate that optimal MTC solutions are valid

when applied to laboratory and natural channels,

when the bank strength is properly considered.

2. Theoretical model

The model used to examine the predicted behav-

iour of alluvial systems is based on that presented by

Millar and Quick (1993). The model assumes a

trapezoidal cross section with banks at an angle hfrom the horizontal. The mean flow velocity is calcu-

lated using Manning’s equation

u ¼ R2=3S1=2

nð1Þ

where u is the mean flow velocity (m/s), R is the

hydraulic radius (m), S is the energy slope, and n is the

Manning roughness coefficient. When testing the

model against existing alluvial channels, n is calculat-

ed based on the reported channel dimensions and flow

properties. Discharge Q is given by the product uA,

where A is the cross-sectional area of the trapezoid.

Bank stability and bed load transport are evaluated

by distributing the shear stress between the bed and the

banks. The proportion of the shear force acting on the

bank (SFbank), and the mean bed and bank shear stress

values are estimated using the following equations

(Knight, 1981; Knight et al., 1984; Flintham and

Carling, 1988):

logSFbank ¼�1:4026logPbed

Pbank

þ 1:5

� �þ 2:247 ð2Þ

sbankcYoS

¼ SFbank

100

ðW þ PbedÞsinh4Yo

� �ð3Þ

sbedcYoS

¼ 1� SFbank

100

� �W

2Pbed

þ 0:5

� �ð4Þ

where s is the shear stress, P is the wetted perimeter, Yois the maximum water depth, and W is the width of the

channel. The subscripts are self-evident. The stability

of the bank is assessed by comparing the value sbankcalculated from Eq. (3) with a modified USBR bank

stability criterion (after Millar and Quick, 1993), based

on the bank sediment calibre (D50 bank) and bank

friction angle (/V):

sbankðcs � cÞD50bank

Vc tan/V

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1� sin2h

sin2/V

sð5Þ

B.C. Eaton, R.G. Millar / Geomorphology 62 (2004) 35–4538

The coefficient c is dependent upon the properties

of the unconsolidated, noncohesive sediment where

bank strength is unmodified by bank vegetation. The

coefficient is defined as c = s*c/tan/, where s*c is thecritical dimensionless shear stress for bed material of

the same calibre, and / is the angle of repose. The

value of /V ranges from a lower bound of /V=/,where the bank sediment is unaffected by bank

vegetation or interstitial cohesive sediment, up to a

maximum value approaching 90j, which corresponds

to a nonerodible bank (Millar and Quick, 1993). The

value of / varies with grain size and shape (Hender-

son, 1966, p. 420), ranging from a minimum of about

25j for fine sand to about 40j for subrounded gravel.

Setting s*cc 0.04, c will thus vary between about

0.086–0.048. In this paper, we use c = 0.069 for

laboratory channels in sand (/ = 30j) and 0.048 for

natural gravel rivers (/ = 40j).Sediment transport is estimated using the equations

presented by Parker (1990) that are based on a single

characteristic grain diameter. The advantage of this

approach is that there is no threshold for sediment

transport. Parker’s model specifies three different

equations for the dimensionless sediment transport

rate G as a function of the ratio of the dimensionless

shear stress to a reference dimensionless shear stress

(s*/sr*), evaluated for the median bed particle diam-

eter (D50). The equations are:

Gðsbed*=sr*Þ ¼

5474 1� 0:853

sbed*=sr*

� �4:5

sbed*=sr* > 1:59

expð14:2ðsbed*=sr*� 1Þ � 9:28ðsbed*=sr*� 1Þ2Þ 1Vsbed*=sr*V1:59

ðsbed*=sr*Þ14:2 sbed*=sr* < 1

8>>>>><>>>>>:

ð6Þ

Fig. 1. Solution curves are shown for an unconstrained MTC model

(equivalent to the limit /V! 90j) and for constrained MTC models

assuming values of /V ranging from 40j to 60j. Q is arbitrarily

specified to be 100 m3/s, S to be 0.003 m/m, and D50 to be 32 mm.

The ‘‘optimum’’ solutions correspond to the local maxima.

where sbed� ¼ sbed=ðcs � cÞD50. This can be rendered

in dimensional form using Eq. (7),

qb ¼ G0:0025ðsbed=qÞ3=2

gðs� 1Þ

!ð7Þ

where g is the acceleration of gravity, s is the specific

sediment weight, qb is the volumetric transport rate

per unit width (m3/s/m), sbed is the shear stress on the

bed, and q is the density of water. The total sediment

transport rate for the entire channel, Qb, is then the

product of the active width [Pbed in Eqs. (2)–(4)] and

qb.

Using this formulation of rational regime theory, the

role of bank strength in constraining the optimization

can be demonstrated by varying /V for a hypotheticalgravel river. Solution curves are generated by solving

Eqs. (1)–(7) over a range of channel widths, for fixed

values of Q (100 m3/s), S (0.003), and D50 (32 mm)

assuming various relative bank strength values (/V)(Fig. 1). Unconstrained MTC models, such as those

proposed by White et al. (1982), are equivalent to our

model in the limit /V! 90j, that is with nonerodible

banks. As the relative bank strength increases, the

sediment transport optimum shifts toward narrower

channels with higher transport rates, and the solution

curves for the constrained regime model with erodible

banks tend toward the unconstrained MTC regime

B.C. Eaton, R.G. Millar / Geomorphology 62 (2004) 35–45 39

model (i.e., /V! 90j). The widest optimum occurs

when the banks and bed are of equal erodibility, which

in our model corresponds to the condition where

/V=/. Our analysis indicates that when bank stability

is not explicitly considered, predictions of alluvial

channel width assuming MTC should be valid only

for alluvial channels with highly resistant boundaries

and that channels with erodible banks should be wider

than those predicted by unconstrained MTC models.

For /V= 40j, which describes a channel formed in

noncohesive coarse gravel with little bank vegetation,

the predicted optimumwidth in this case is almost three

times that predicted by the unconstrained MTC model

(Fig. 1). This result demonstrates that appropriate

application of rational regime theory must consider

bank stability.

Based on Fig. 1, we propose two hypotheses: (i)

that alluvial channels with erodible banks found in

nature and generated in the laboratory will be wider

than predicted by imposing MTC without explicitly

constraining the solution by bank stability; and (ii)

that, as bank strength increases, the observed width

will narrow and converge toward the unconstrained

MTC solution. We test these two hypotheses in the

following section.

Fig. 2. Widths predicted by both constrained (/V= 30j, closed

symbols) and unconstrained (open symbols) MTC regime models

are plotted against the observed widths for straight (Wolman and

Brush, 1961) and meandering (Schumm and Khan, 1972) laboratory

channels. Channels where the sediment transport rate was too low to

be accurately measured were excluded from this analysis.

3. Analysis

3.1. Laboratory experiments

We test the first of our hypotheses using experi-

mental data from self-formed laboratory channels.

Wolman and Brush (1961) used a tilting stream table

16 m long and 1.2 m wide, in which straight, self-

formed channels were developed for a range of dis-

charges (0.1–7.9 L/s) and bed slopes (0.00131–

0.0137). They adjusted the sediment feed rate to match

the sediment transport out of the system. The experi-

ments were run until the channel reached an equilib-

rium form where both channel shape and the rate of

bed load transport became constant; the time to reach

equilibrium ranged from 8 to 54 h. Wolman and Brush

used two different sediment types: one with a charac-

teristic grain diameter of 0.67 mm and the other with a

grain diameter of 2.0 mm. In another set of experi-

ments, Schumm and Khan (1972) developed mean-

dering channels in flume 30 m long and 7 m wide

using sediment with a median grain diameter of 0.7

mm. Discharge was held constant (4.3 L/s) and slope

varied (0.0026–0.013), generating thalweg sinuosity

up to 1.25. Sediment was fed at the upstream end of the

flume at a rate such that no bed degradation occurred at

the head of the flume. Experiments were stopped after

a stable channel form developed, which occurred

between 2 and 24 h (occurring most rapidly for the

steepest slopes).

Two sets of calculations are presented here. The

first set of width calculations represent the uncon-

strained MTC model by setting h = 45j and invoking

no bank stability constraint (simulating the analysis by

White et al., 1982 and Valentine et al., 2001). In the

second set, the widths are recalculated using our

model by specifying /V =/ = 30j. In our model, the

bank angle is allowed to adjust, thereby producing a

stable bank configuration for all solutions.

The results are indeed consistent with our first

hypothesis (Fig. 2). Using the unconstrained model,

the observed widths are consistently on the order of

twice the calculated width. Similar observations are

B.C. Eaton, R.G. Millar / Geomo40

reported by Valentine et al. (2001) and Griffiths and

Carson (2000). The performance of the uncon-

strained model is particularly poor for the Schumm

and Khan (1972) experiments, where the variability

is imparted by changing channel slope for a constant

Q. Because bank stability is ignored, the uncon-

strained model predicts a similar W/d ratio for all

of the channels; and as S (and therefore u) increases,

the predicted width actually decreases slightly, while

the observed width increases. For the Wolman and

Brush (1961) channels, where discharge imparts

much of the variability in channel width, predicted

widths increase with the observed widths in a more

consistent fashion but are all below the line of

perfect agreement.

When the model is reformulated using an appro-

priate value of /V (30j), then the calculated widths

more closely agree with the observed width and

scatter about the line of perfect agreement (Fig. 2).

Furthermore, our calculated sideslope for the straight

channels average 22j, which agrees well with Wol-

man and Brush’s measured average value of 25j (they

report values ranging from 18j to 30j). Strikingly,Schumm and Khan’s (1972) meandering channels,

which are so poorly described by the unconstrained

model, are nearly perfectly described by the bank-

stability constrained model and exhibit very little

scatter about the line of perfect agreement. Thus, this

analysis supports our first hypothesis.

Fig. 3. Widths predicted by (A) an unconstrained MTC regime model and (

for gravel bed channels (Hey and Thorne, 1986). Channels with weak ban

(vegetation Types 3 and 4) are grouped together, for clarity. The value of

3.2. Natural channels

One can also examine the effect of bank strength

on the accuracy of regime model predictions using

data from natural channels. The relation between /Vand bank vegetation for channels in noncohesive

alluvium is investigated by Millar and Quick (1993),

who found a systematic variation in characteristic

bank strength with vegetation density, as classified

by Hey and Thorne (1986) for their data set of natural

gravel bed streams. They find that /Vcan be related to

vegetation type and density, ranging systematically

with vegetation density from an average mean value

of about 40j for vegetation Type 1 (grass with no

trees or bushes) to 65j for vegetation Type 4 (more

than 50% trees and bushes).

Channel widths for Hey and Thorne’s data set are

calculated using the unconstrained MTC model, again

setting h = 45j and imposing no bank stability con-

straint. As in the case laboratory data shown in Fig. 2,

the observed widths are nearly all greater than the

widths calculated using the unconstrained MTC mod-

el (Fig. 3a). Furthermore, the degree by which the

width is under-predicted as defined by the ratio Wpred/

Wobs varies systematically from an average value of

0.45 for vegetation Type 1 channels to 0.69 for Type 4

channels (Table 1). This is evident in Fig. 3a, where

there is a clear pattern of greater deviation for Types 1

and 2 versus Types 3 and 4 despite substantial scatter

rphology 62 (2004) 35–45

B) a constrained MTC model are plotted against the observed widths

k vegetation (vegetation Types 1 and 2) and strong bank vegetation

/Vapplied to each vegetation class is listed in Table 1.

Table 1

Centroids of predicted and observed widths for different vegetation types

Vegetation type Observed Unconstrained MTC model Bank strength constrained model

Wobserved (m) Wpredicted (m) Wpred/Wobs /V Wpredicted (m) Wpred/Wobs

1 32.3 14.7 0.45 40j 34.8 1.07

2 22.4 11.5 0.51 45j 21.8 0.97

3 27.0 15.9 0.59 49j 26.1 0.96

4 20.2 13.9 0.69 55j 20.8 1.03

B.C. Eaton, R.G. Millar / Geomorphology 62 (2004) 35–45 41

in the data. These results are consistent with Fig. 1

and our second hypothesis and demonstrate that

channels with progressively stronger (more densely

vegetated) banks tend to converge toward the uncon-

strained MTC predicted values.

Constrained MTC models were also fit to the

data. The value of /V for each vegetation class was

iterated until the average observed widths predicted

by the model corresponded approximately to the

average widths (Table 1). The values of /V increaseprogressively with increasing vegetation density, as

expected, although the data for individual channels

are scattered about the line of perfect agreement (Fig.

3b). Since the vegetation classes used by Hey and

Thorne (1986) are rather general, they embrace a

range of relative bank strengths, and the deviation of

/V for any individual channel from the mean /Vvalue for the vegetation class may be largely respon-

sible for the scatter in Fig. 3b.

3.3. Bank vegetation and channel scaling

The effect of vegetation density appears to be most

prominent for smaller channels (Fig. 3a). For channels

with observed widths less than about 30 m, the data

are well stratified, with Type 3 and 4 channels plotting

closest to the line of perfect agreement and Types 1

and 2 plotting farther below. This stratification is not

evident for larger channels, suggesting a significant

influence of channel scale on bank vegetation influ-

ences. The parameter Qbf1/2 is used as an index of

channel scale in preference to width because, unlike

width, Qbf1/2 is independent of bank vegetation effects.

Many empirical regime studies have demonstrated

that W scales in proportion to Qbf1/2 (e.g., Hey and

Thorne, 1986). The degree to which observed channel

width differs from that predicted using unconstrained

MTC models (/V! 90j)—given by the ratio Wpred/

Wobs—is plotted against Qbf1/2 (Fig. 4). For a given

vegetation density, a consistent scale dependence of

vegetation-induced bank strength is apparent. The

scatter in these plots is undoubtedly the result of the

general nature of the vegetation classes: More infor-

mation on the vegetation—such as the species, age,

stem density, and rooting depth—could probably be

used to substantially reduce this scatter. Linear regres-

sions relating the ratio Wpred/Wobs and Qbf1/2 were fit to

the data and are shown in Fig. 4. The regression slope

coefficients and their statistical significance levels are

reported in Table 2.

The fitted curves (Fig. 4) illustrate the systematic

effect of increasing under-prediction of channel width

as the bank vegetation becomes less dense, which is

also evident in Fig. 3 and Table 1. The smallest Type

4 channels are rather well described by the uncon-

strained model (Wpred/Wobsf 1), while the smallest

Type 3 channels are somewhat less well described

(Wpred/Wobsf 0.8), and the smallest Type 2 channels

are even less well described (Wpred/Wobsf 0.6). All

of the Type 1 channels are poorly described (Wpred/

Wobsf 0.45). The regression curves also reveal an

increase in the degree of under-prediction with in-

creasing discharge for Types 2, 3, and 4 and imply

that vegetation-induced bank stability depends on

channel scale, as well as vegetation density. For Type

1, which exhibits little or no vegetation-related bank

stability, there is no scale dependency. The regression

line coincides with a horizontal line representing a

mean Wpred/Wobs ratio of 0.45 (see Table 1), implying

that the observed channels are about twice as wide as

those predicted by unconstrained MTC models. This

reference line is also shown on the plots for Types 2,

3, and 4. The intersection of the regression line with

this reference line represents the scale at which

vegetation-related bank stability nominally disap-

pears. This intercept varies systematically with vege-

tation type. The vegetation effect vanishes at about

Qbf1/2 = 10 (Q = 100 m3/s) for Type 2 channels, at

Fig. 4. Degree of under-prediction for unconstrained MTC models (given by the ratio Wpred/Wobs) plotted against Q bf1/2 for gravel bed channels,

classified according to bank vegetation density (Hey and Thorne, 1986). Linear regressions of Wpred/Wobs on Q bf1/2 for each vegetation type are

shown (solid line), as is the mean degree of under-prediction in the absence of a vegetation effect (horizontal dashed line). Slope coefficients and

P values for the regressions are presented in Table 2.

B.C. Eaton, R.G. Millar / Geomorphology 62 (2004) 35–4542

about Qbf1/2 = 15 (225 m3/s) for Type 3, and at Qbf

1/

2 = 20 (400 m3/s) for Type 4.

Clearly, then, properly constrained regime models

can predict alluvial channel widths, provided the bank

friction angle is known. This is a simple matter for

laboratory channels developed in noncohesive sedi-

Table 2

Regression slope coefficientsa

Vegetation type Regression slope

a P n

1 � 0.0001 0.957 13

2 � 0.0249 < 0.001 16

3 � 0.0290 < 0.001 13

4 � 0.0254 0.015 20

a The fitted regression model is: Wpred/Wobs = a(Qbf)1/2 + b;

where P refers to the probability that the slope is not different

from zero.

ments, where /V is similar to the angle of repose for

the sediment. Similarly, natural channels that are

either sparsely vegetated (e.g., Hey and Thorne’s Type

1 channels) or large enough that vegetation does not

significantly affect bank strength can be well de-

scribed by assuming a friction angle near the angle

of repose as long as bank sediment is not strengthened

by appreciable silt or clay content. For smaller chan-

nels, bank strength can be expected to vary with the

density, age, rooting depth, and type of vegetation.

4. Discussion

Our analysis of laboratory and natural channels

supports our hypotheses that (i) alluvial channels

with erodible banks will be wider than predicted

by an unconstrained MTC model; and (ii) as bank

B.C. Eaton, R.G. Millar / Geomorphology 62 (2004) 35–45 43

strength increases, the observed channel width will

narrow and converge toward the unconstrained MTC

solution. Thus, the bias identified by Griffiths and

Carson (2000), and by Valentine et al. (2001),

disappears once a reasonable bank stability constraint

is applied. The analysis in this paper demonstrates

that a good agreement between predicted and ob-

served channel width can be obtained for both

laboratory and vegetated natural channels only when

appropriate values of bank strength are used. Thus,

channels developed in laboratory experiments using

noncohesive alluvium typically exhibit channels

about twice as wide as predicted by an unconstrained

optimization. When a bank stability constraint is

added to the model, assuming a bank friction angle

on the order of 30j, the observed laboratory channel

widths are well predicted. We hope that this analysis

will clarify the role of bank stability in constraining

the optimum solutions.

Analysis of natural channels is more complicated,

because their banks tend to be reinforced by bank

vegetation, and they are characterized by friction

angles that can be much different than that for non-

cohesive sediment. The relation between /Vand bank

vegetation density is further confounded by channel

scaling effects. The scale dependency of vegetation-

induced bank strength is evident when the uncon-

strained MTC predicted widths are compared with the

observed channel widths for each vegetation class,

because the degree of under-prediction varies system-

atically with vegetation density and channel scale

(Fig. 4). Despite this confounding scale effect, good

agreement between the predicted and observed chan-

nel widths for Hey and Thorne’s data can be obtained

by applying friction angles ranging from 40j to 55jbased on the vegetation type class, although signifi-

cant scatter persists (Fig. 3b; Table 1).

Importantly, our analysis shows that for a given set

of Q, S, and D50, there is a range of solutions that

depends upon the strength of the banks, which we

characterize using the friction angle /V. Changes in

bank strength, as a consequence of riparian logging

for instance, can cause channel widening (for exam-

ple, the analysis of Slesse Creek in Millar, 2000). We

interpret this as equivalent to a reduction of /V, and as

an adjustment to a wider optimum width. Fig. 1

indicates that—if the channel slope remains constant

during this width adjustment—the sediment transport

capacity will be reduced. In fact, in British Columbia,

aggradation of the channel bed often accompanies

widening following riparian logging, as the channel

no longer has the capacity to transport its former load,

nor to immediately remove the additional sediment

mobilized from the flood plain during widening.

Ultimately, an increase in channel gradient would be

required to regain its original, pre-disturbance trans-

porting capacity.

The analysis and interpretation presented herein are

supported by a recent study. Gran and Paola (2001)

reported changes in channel width in a laboratory

braided channel (formed in sand) as a consequence of

increasing vegetation density. They grew alfalfa on

the surface of existing braided channels and braided

plains at relatively low water discharges, thereby

imparting some degree of additional strength to the

banks and braided plain. Different vegetation densities

were applied, ranging from 1.2 to 9.2 stems/cm2. A

continuous reduction in braid intensity and total active

channel width resulted from this progressive increase

in vegetation density. Because the experiments were

conducted with constant Q and Qb, and the channel

patterns did not reportedly aggrade or degrade, the

reduction in braid intensity and channel width implies

that the increase in sediment transport intensity for

each channel has occurred with increasing vegetation

density. This is entirely consistent with the behaviour

shown in Fig. 1, where increasing bank strength

implies increasing sediment transport intensity at the

optimum channel width, and demonstrates that our

argument relating vegetation density to bank strength

is a reasonable one. Unfortunately, because Gran and

Paola (2001) performed their experiments using a

braided channel, rational regime theory cannot be

used to predict the observed channel dimensions.

However, the transition from a braided to a wandering

pattern observed by Gran and Paola (2001) is pre-

dicted using the theory developed by Millar (2000),

which is based on the rational regime approach

investigated in this paper.

In a recent paper, Griffiths and Carson (2000)

suggested that MTC optimization does not in fact

occur in their study streams, which they attribute in

part to the very high shear stresses that would be

exerted on the stream banks in such a channel.

Similarly, Shields et al. (2003) dismissed the potential

for channel restoration design based on optimality

B.C. Eaton, R.G. Millar / Geomorphology 62 (2004) 35–4544

theory because of differences between observed and

calculated channel dimensions. We argue that this

discrepancy between modeled and observed channel

widths should not be interpreted as evidence that

MTC optimization does not occur, or that optimality

theory is not capable of describing alluvial systems.

Rather, it implies that their models used to determine

the optimal MTC channel dimensions were not ap-

propriately constrained by bank stability. While our

analysis is not intended to demonstrate that regime-

based optimality theory is entirely sufficient to predict

alluvial channel dimensions, it does show that the

criticisms based on models not constrained by bank

strength are unfounded. Further research should focus

more clearly on understanding the nature of bank

strength in alluvial streams. Specifically, appropriate

methods of estimating bank strength as a function of

alluvial stratigraphy and vegetative cover are needed,

as is a better understanding of the potential scale

relation between vegetation and bank strength.

5. Conclusions

A rational regime model has been formulated to

demonstrate the role of bank stability in constraining

the optimum solution, assuming maximum sediment

transport capacity. The unconstrained version is con-

sistent with earlier work by White et al. (1982) and

Griffiths and Carson (2000) and is shown to grossly

under-predict the regime channel widths for laborato-

ry data of Wolman and Brush (1961) and Schumm

and Khan (1972). This is consistent with recent

analysis by Valentine et al. (2001). When the model

is formulated to explicitly account for bank stability,

then and only then can we expect a reasonable

agreement between modeled and observed laboratory

channel widths.

Based on natural river data from Hey and Thorne

(1986), the degree of under-prediction by the uncon-

strained MTC model clearly decreases as bank vege-

tation density (and by inference, bank strength)

increases. The variation in bank strength as a function

of vegetation density is known to exhibit a depen-

dence on channel scale (Zimmerman et al., 1967), a

point which is evident based on the analysis presented

in Fig. 4. Nevertheless, the average bank strength

required to collapse the data for Hey and Thorne’s

four vegetation types to the line of perfect agreement

(Fig. 3) varies systematically with vegetation density.

This behaviour is consistent with our theory, which

predicts that as bank strength increases, channel

widths will converge toward the predictions from

unconstrained MTC optimization models (e.g., White

et al., 1982).

Our results also show that for a given value of Q, S,

and D50, a range of channel geometries can form,

depending upon the strength of the bank material, and

that there is not one single optimum. Thus, use of

unconstrained MTC models is tantamount to assum-

ing that all stream banks are highly resistant to erosion

and results in a predictable and consistent under-

prediction of alluvial channel widths, especially when

banks are composed of highly erodible, noncohesive

sand, or gravel.

Acknowledgements

This work was funded in part by the Natural

Sciences and Engineering Research Council of

Canada, through B. Eaton’s graduate scholarship

and R. Millar’s operating grant. We thank Dr. M.

Church for thoughtful comments on the paper.

References

Blench, T., 1969. Mobile Bed Fluviology. University of Alberta

Press, Edmonton. 164 pp.

Chang, H.H., 1979. Minimum stream power and river channel pat-

terns. J. Hydrol. 41, 303–327.

Chang, H.H., 1980. Geometry of gravel streams. J. Hydraul. Div.

106 (9), 1443–1456.

Chang, H.H., 1984. Comment on ‘‘Extremal hypotheses for river

regime: an illusion of progress’’ by George A. Griffiths. Water

Resour. Res. 20 (11), 1767–1768.

Darby, S.E., Thorne, C.R., 1995. Effect of bank stability on geo-

metry of gravel rivers. J. Hydraul. Eng. 121 (4), 382–384.

Davies, T.R.H., Sutherland, A.J., 1983. Extremal hypotheses for

river behavior. Water Resour. Res. 19 (1), 141–148.

Eaton, B.C., Church, M., Millar, R.G., in press. Rational regime

models of alluvial channel morphology and response. Earth

Surf. Processes Landf. 29, doi:10.1002/esp.1062.

Ferguson, R.I., 1986. Hydraulics and hydraulic geometry. Prog.

Phys. Geogr. 10, 1–31.

Flintham, T.P., Carling, P.A., 1988. The prediction of mean bed and

wall boundary shear in uniform and compositely roughened

channels. Int. Conf. On River Regime. Wiley, Chichester, UK,

pp. 267–287.

B.C. Eaton, R.G. Millar / Geomorphology 62 (2004) 35–45 45

Gran, K., Paola, C., 2001. Riparian vegetation controls on braided

stream dynamics. Water Resour. Res. 37 (12), 3275–3283.

Griffiths, G.A., 1984. Extremal hypotheses for river regime: an

illusion of progress. Water Resour. Res. 20 (1), 113–118.

Griffiths, G.A., Carson, M.A., 2000. Channel width for maximum

bedload transport capacity in gravel-bed rivers, South Island,

New Zealand. J. Hydrol., N.Z. 39 (2), 107–126.

Henderson, F.M., 1966. Open Channel Flow. MacMillan, New

York. 522 pp.

Hey, R.D., Thorne, C.R., 1986. Stable channels with mobile gravel

beds. J. Hydraul. Div. 112 (8), 671–689.

Kellerhals, R., Church, M., 1989. The morphology of large rivers:

characterization and management. In: Dodge, D.P. (Ed.), Pro-

ceedings of the International Large River Symposium. Can. Spec.

Publ. Fish. Aquat. Sci., vol. 106. Ottawa, Ontario, pp. 31–48.

Knight, D.W., 1981. Boundary shear in smooth and rough channels.

J. Hydraul. Div. 107 (7), 839–851.

Knight, D.W., Demetrious, J.D., Hamed, M.E., 1984. Boundary

shear in smooth rectangular channels. J. Hydraul. Div. 110

(4), 405–422.

Lane, E.W., 1955. The design of stable channels. Trans. ASCE 120,

1234–1279.

Leopold, L.B., Maddock, T., 1953. The hydraulic geometry of

stream channels and some physiographic implications. USGS

Prof. Paper, vol. 252. U.S. Geological Society, Washington, DC.

Leopold, L.B., Wolman, M.G., 1957. River channel patterns; braid-

ed meandering and straight. USGS Prof. Paper, vol. 282-B. US.

Geological Society, Washington, DC.

Mackin, J.H.R., 1948. Concept of a graded river. Bull. Geol. Soc.

Am. 59 (5), 463–512.

Millar, R.G., 2000. Influences of bank vegetation on alluvial chan-

nel patterns. Water Resour. Res. 36 (4), 1109–1118.

Millar, R.G., Quick, M.C., 1993. Effect of bank stability on geo-

metry of gravel rivers. J. Hydraul. Eng. 119 (12), 1343–1363.

Montgomery, D.R., Buffington, J.M., 1998. Channel processes,

classification and response. In: Naiman, R., Bilby, R.

(Eds.), River Ecology and Management. Springer, New

York, pp. 13–42.

Mosselman, E., 2000. River width adjustment: 1. Processes and

mechanisms—discussion. J. Hydraul. Eng. 126 (2), 159–160.

Parker, G., 1990. Surface-based bedload transport relation for gra-

vel rivers. J. Hydraul. Res. 28 (4), 417–436.

Schumm, S.A., 1971. Fluvial geomorphology, channel adjustment

and river metamorphosis. In: Shen, H.W. (Ed.), River Mecha-

nics (I). H.W. Shen, Fort Collins. 22 pp.

Schumm, S.A., Khan, H.R., 1972. Experimental study of channel

patterns. Geol. Soc. Amer. Bull. 83, 1755–1770.

Shields, F.D., Copeland, R.R., Klingeman, P.C., Doyle, M.W., Si-

mon, A., 2003. Design for stream restoration. J. Hydraul. Eng.

129 (8), 575–584.

Song, C.C.S., Yang, C.T., 1986. Comment on ‘‘Extremal hypothe-

ses for river regime: an illusion of progress’’ by George A.

Griffiths. Water Resour. Res. 22 (6), 993–994.

Stevens, M.A., Simons, D.B., Richardson, E.V., 1975. Nonequilib-

rium river form. J. Hydraul. Div. 101 (HY5), 557–566.

Valentine, E.M., Benson, I.A., Nalluri, C., Bathurst, J.C., 2001.

Regime theory and the stability of straight channels with bank-

ful and overbank flow. J. Hydraul. Res. 39 (3), 259–268.

White, W.R., Bettess, R., Paris, E., 1982. Analytical approach to

river regime. J. Hydraul. Div. 108 (HY10), 1179–1193.

Wolman, M.G., Brush, L.M., 1961. Factors controlling the size and

shape of stream channels in coarse noncohesive sands. USGS

Prof. Paper, vol. 282-G. U.S. Geological Survey, Washington,

DC.

Yang, C.T., 1976. Minimum unit stream power and fluvial hydrau-

lics. J. Hydraul. Div. 102 (HY7), 769–784.

Zimmerman, R.G., Goodlet, J.C., Coner, G.H., 1967. The influence

of vegetation on the channel form of small streams. IAHS Publ.

75, 255–275.