Grain‐Size, Sediment‐Transport Regime, and Channel Slope in Alluvial Rivers

Author(s): W. Brian Dade and Peter F. Friend

Source: The Journal of Geology, Vol. 106, No. 6 (November 1998), pp. 661-676
Published by: The University of Chicago Press
Stable URL: http://www.jstor.org/stable/10.1086/516052 .
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 Grain-Size, Sediment-Transport Regime, and Channel Slope
in Alluvial Rivers 1

W. Brian Dade and Peter F. Friend 2

Institute of Theoretical Geophysics, Department of Earth Sciences and Department of Applied Mathematics
and Theoretical Physics, University of Cambridge, Cambridge CB2 3EQ, UK

ABSTRACT
The general relationship between channel morphology and the grain size of sediment in the channel bed is an impor-
tant but poorly known aspect of alluvial rivers. An analysis of an equation for total sediment flux in the limits of
suspension-, bedload-, and mixed-modes of transport indicates distinct, steady-state regimes of channel morphology.
Such regimes are readily seen in published data for modern alluvial rivers by way of a conventional Shields plot or
a plot of channel slope as a function of relative grain size d/h and the ratio w s /u , where d and w s are, respectively,
*
mean diameter and fall speed of bed sediments, and h and u are, respectively, mean depth and friction velocity of
*
the flow. With slope and mode of transport in an alluvial river constrained by grain size and channel depth alone,
estimates of discharge and sediment flux follow directly. Introduction of the sediment flux relationship into conven-
tional diffusion models for the evolution of an alluvial system provides nominal estimates of the response time for
channel adjustment to some external changes. For some major modern rivers, this time of response along the entire
length of channel is in the range 10 3 –10 5 yr, underscoring the potential for complicated, long-time interaction of large
alluvial systems with, for example, climatic variability.

Introduction
The form of an alluvial river channel reflects the versal properties. The processes that result in such
range of environmental factors that determine the a pattern can include the adjustment of channel
erosion, transportation, and deposition of uncon- morphology to a state in which sediment flux
solidated debris by the river itself. An understand- through a given reach of a river, averaged over a
ing of the interaction of these factors is important suitable interval of time, is continuous. Under such
for the analysis of the response of modern rivers to conditions neither net erosion nor net deposition
human perturbation and to the interpretation of an- takes place, and the river is said to be ‘‘graded’’ ‘‘in
cient fluvial deposits. At present, however, the rela- grade’’ (Mackin 1948; Hack 1960), or ‘‘in regime’’
tionships between the hydraulic and sedimentary (Lacey 1930). This concept can be traced (in the En-
controls of the form of an alluvial channel are ex- glish-language literature) back to the ideas of G. K.
pressed in terms of empirical laws that may or may Gilbert, which are summarized by Baker and Pyne
not be readily applied to new settings. An example (1978), Chorley and Beckinsale (1980), and Leopold
of such an approach is the classification of rivers (1980).
with plan forms characterized by single-thread (me- Despite its pedigree, the notion of the graded
andering and straight) or multiple-thread (braided) stream has not met with universal acceptance. One
channels (Leopold and Wolman 1957). Most of possibility is that real alluvial systems never per-
these schemes proceed, implicitly or explicitly, fectly achieve in time the dynamic equilibrium rep-
from the assumption that a ‘‘self-forming,’’ alluvial resented by the ideally graded channel. Important
river achieves a channel pattern with specific, uni- efforts to understand the evolution of an alluvial
system include the consideration of the mathemat-
1
ical behavior of the system of equations that de-
Manuscript received December 31, 1997; accepted July 7,
1998. scribe the spatial distribution of momentum and
2
Department of Earth Sciences, University of Cambridge, the continuity of water and sediment transport in
Cambridge CB2 3EQ, UK. a stream flow (Ribberink and van der Sande 1985).

[The Journal of Geology, 1998, volume 106, p. 661–675]  1998 by The University of Chicago. All rights reserved. 0022-1376/98/10602-0004$01.00

661

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662 W. B. DADE AND P. F. FRIEND

More recently, C. Paola and colleagues have made Table 1. Summary of Notation
important advances in their consideration of a gen-
Symbol Units Definition
eral system of equations that describe the transport
of mass and momentum transport in a gravel-bed- A depth-averaged concentration of sus-
ded stream as it relates to the evolution of alluvial pended sediment
basin fill (Paola et al. 1992), downstream distribu- AD L2 area of drainage basin
B mobility parameter
tion of grain size (Paola and Seal 1995), and the in- C volumetric concentration of sediment
ference of channel slope in ancient braided streams D grain size parameter, (gRd 3 /ν 2) 1/3
(Paola and Mohrig 1996). K L 2 T ⫺1 diffusion coefficient
L L length of channel in drainage basin
In this paper we reconsider the general hypothe- P Rouse parameter, w s /κu*
sis that within a graded stream the total sediment R relative excess density of sediment parti-
flux q T (or, equivalently, sediment discharge per cles, (ρ p ⫺ ρ)/ρ
Re flow Reynolds number, uh/ν
width of channel) is constant in time and does not Re grain Reynolds number, u d/ν
vary in the downstream (x) direction. This latter Q* L 3 T ⫺1 total water discharge, uhw*
condition is given mathematically as TR T basin response time
CD quadratic drag coefficient
d L median grain size of channel bed mate-
rial
∂q T /∂x ⫽ 0. (1) g LT ⫺2 acceleration due to gravity
h L channel depth
q L 2 T ⫺1 water flux, Q/w
In this regard our results follow the ideas of Parker qb L 2 T ⫺1 bedload sediment flux
(1978, 1979) for self-formed, gravel-bed rivers. In qs L 2 T ⫺1 suspended sediment flux
qT L 2 T ⫺1 total sediment flux
our consideration of the ramifications of eq. 1, how- s channel slope
ever, we make the important distinction between u L T ⫺1 average flow speed
alluvial systems characterized by regimes of pre- u L T ⫺1 friction velocity
w* L channel width
dominantly bedload, mixed-load, or suspended-sed- ws LT ⫺1
settling velocity of individual, sedimen-
iment transport. By considering mass flux per chan- tary particles
nel width, we avoid complications associated with x L downstream spatial coordinate
z L vertical spatial coordinate
converging channels and increasing discharge in α grouping of coefficients κ, c D and γ s
the downstream direction. These complications χb fraction of total sediment flux in bed-
represent an important aspect of alluvial-river load
⑀b bedload transport efficiency or friction
form, of course, but are secondary to the main ad- factor
vance from previous studies. We find that sedimen- ⑀s suspended-load transport efficiency
tological control is reflected in an approximately factor
η L bed elevation above arbitrary datum
constant value of the Shields parameter for each of κ von Karman’s constant
the three different transport regimes. A key as- ν L 2 T ⫺1 kinematic viscosity of transporting fluid
sumption underlying this result is that the grain ρ M L ⫺3 density
θ Shields parameter sh/Rd
size of material making up a river bed is strongly τ0 M L T bed shear stress, ρghs
⫺1 2

correlated with (and thus in equilibrium with) the ζ relative depth, h/z b
caliber of the sediment load in transport through a ψ dimensionless sediment transport
ω L 2 T ⫺1 specific stream power, qs
graded reach. subscripts on C, u and θ
In the following analysis we first summarize a b bedload
basis for distinguishing the different modes of sedi- i index for b, m or s
m mixed load
ment transport, and then propose a general equa- o stationary bed
tion for sediment transport that accommodates the s suspended load
different transport regimes and is based on ener-
getic constraints of stream flow. We then derive Note M—mass; L—length; T—time.
predictive relationships that emerge from this
equation for conditions of continuous sediment
Analysis
flux in the different sediment-transport regimes.
These relationships are used to reanalyze existing Modes of Sediment Transport. Sediment trans-
data for sedimentological controls of channel slope port associated with local hydraulic conditions oc-
and depth. Last, we consider several, specific, appli- curs in two distinct modes (e.g., Middleton and
cations of our findings and summarize the key Southard 1984). Sedimentary particles that roll,
findings of our analysis. A summary of the notation slide, skip, and hop along the bottom make up the
used in this paper appears in table 1. bedload. Such material is transported primarily

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Journal of Geology CHANNEL SLOPE IN ALLUVIAL RIVERS 663

near the bed and is in contact with the channel bed and sediment concentration, respectively, of the
for a considerable fraction of the time. In contrast, bedload layer.
particles supported by the turbulence of the mean From eq. 2 the relative, depth-averaged concen-
flow reside primarily in the water column and tration of sediment A in a flow of depth h is given
spend little time in contact with the bed. Such ma- in terms of P and the relative depth ζ ⫽ h/z b by the
terial makes up the suspended load. Sediment for expression
which the transport mode is intermediate in char-
h
acter contributes to a third transport regime, that of
mixed load. An additional mode of transport, wash
A ⫽ (h ⫺ z b ) ⫺1 冮
zb
(z/z b )⫺P dz
(4)
load, is recognized by some workers and comprises
⫽ (ζ 1⫺P
⫺ 1)/{(1 ⫺ P)(ζ ⫺ 1)}.
extremely small particles that pass through a river
system relatively independently of local flow con-
ditions. We suggest that the removal of such mate- From this result we note that the ratio of the depth-
rial from an alluvial system would have little affect integrated mass of sediment in suspension trans-
on channel morphology. Accordingly, we do not port to that in bedload transport is given by (ζ 1⫺P
consider wash load in this analysis of total trans- ⫺ 1)/(1 ⫺ P), and thus the fraction χ b of the total
port q T. sediment load which travels in the bedload is given
The distinctions between suspended-, mixed-, by
and bedload transport are arbitrary, and in an indi-
vidual river all three modes can occur to varying χ b ⫽ (1 ⫺ P)/(ζ1⫺P ⫺ P). (5)
degrees. An objective basis for the distinction of the
different transport modes can nevertheless be given In figure 1, values of χ b calculated from eq. 5 are
as follows. In a channel flow that is neither eroding shown as a function of the ratio w s /u* and relative
nor depositing sediment, the vertical distribution depth ζ. From these calculations we note that for
of material in the flow represents a balance be- conditions in which values of the ratio w s /u are
*
tween the effects of the downward settling and up- much less than unity, χ b becomes small and mate-
ward turbulent diffusion of individual particles. In rial in transport is predominantly in turbulent sus-
such a setting, the volumetric concentration C(z) pension. Under such conditions the material in
at elevation z above the bed relative to a near-bed transport is well mixed throughout the flow and so
reference concentration C b at z b is given approxi- the degree of partitioning between bedload and sus-
mately by: pended load is related to the relative depth ζ. For
conditions in which w s /u* is much greater than
unity, on the other hand, χ b approaches unity and
C(z)/C b ⫽ (z/z b )⫺P, (2)
so the mobile material travels predominantly in the

where the Rouse parameter P ⫽ w s /ku , w s is the

*
average settling velocity of the material available
for transport and κ ⫽ 0.4 is von Karman’s constant.
The friction velocity u is a characteristic velocity
*
and is related to the bed shear stress τ o of a channel
flow of water with density ρ by the equation

τ o ⬅ ρu 2*. (3)

Equation 2 is a simplified form of a more complete

expression given by Middleton and Southard
(1984), and details concerning the its derivation and
physical significance can be found there. An analy-
sis similar to that which follows but using the more
complicated expression given by Middleton and
Figure 1. Fraction χ b of total load travelling as bedload
Southard, which accommodates a vanishing sedi- as a function of w s /u and relative depth ζ. Solid lines
ment concentration at the free surface of the chan- *
indicate calculations from eq. 5. Dashed lines indicate
nel flow, requires numerical calculations yet adds criteria given in eq. 6 for the distinction of flows charac-
little to the key result. We assume the reference terised by predominantly suspended-load, mixed-load or
parameters z b and C b to correspond to the thickness bedload transport.

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664 W. B. DADE AND P. F. FRIEND

bedload. Intermediate conditions for which w s /u w s ⫽ (gRd) 1/2. (9)

*
⬇ 1 correspond to mixed-load transport in which
the fluxes of suspended and bedload material are We note that the choice of the inertial drag coeffi-
approximately equal in magnitude. cient approaching unity as the ratio w s d/ν becomes
For the sake of definiteness in this analysis, we very large is one of mathematical expedience,
adopt the following criteria for the distinction of which also describes well the settling behavior of
bedload, mixed-load, and suspended-load transport: prolate- and irregularly-shaped particles. In the
course of our analysis we consider the settling be-
w s /u* ⱕ 0.3 haviors given by eqs. 8 and 9 for the end-member
cases of suspended- and bedload transport.
predominantly suspended-load transport
An Equation for Sediment Transport. In steady
0.3 ⬍ w s /u* ⬍ 3 and uniform channel flow, there is an approximate
balance between the downslope weight of the flow
predominantly mixed-load transport
and the boundary shear stress τo (cf. eq. 3). This bal-
w s /u* ⱖ 3 ance is expressed as
predominantly bedload transport. (6)
u 2 ⬅ c D u 2 ⫽ ghs, (10)
*
These criteria are indicated in figure 1. From this
where c D is a dimensionless drag coefficient, which
plot we see that the dominance of suspension trans-
is typically in the range 10 ⫺3 –10 ⫺2, and h and s are
port corresponds to conditions for which less than
respectively the depth and slope of the channel.
approximately 10–20% of the total sediment load
In such a flow there is also a balance between
is bedload. The dominance of bedload transport
the rate of production of turbulent kinetic energy
corresponds to conditions for which more than ap-
(TKE) associated with shear in the mean flow and
proximately 80–90% of the total sediment load is
the rate of dissipation of TKE due to viscous forces.
bedload.
In the case of a sediment-transporting flow, energy
In applying these criteria to observations from
is also lost owing to the mobilization and transport
rivers, we adopt an analytical expression for the set-
of debris. This balance can provide a basis for a sedi-
tling velocity w s of water-borne transport of debris
ment-transport equation (Bagnold 1966). We pro-
given in terms of the kinematic viscosity of water
pose the following equation that describes this rela-
v, the acceleration due to gravity g, and the average
tionship in terms of the mean flow speed u and
size d and the relative excess density R of individ-
channel depth h, the critical shear stress for the on-
ual grains in transport. This expression is given by
set of sediment motion ρu 2 cr, the flux q s ⬅ uhC b A
*
and fall speed w s of suspended sediment, and the
w s ⫽ (ν/d ){(81 ⫹ D 3 ) ⫺ 9}, (7) flux of bedload material q b ⬅ u b z b C b in the near-
bed transport layer:
where the dimensionless grain size parameter D ⫽
(gRd 3 /ν 2 ) 1/3 and R ⫽ (ρ s ⫺ ρ)/ρ, where ρ s is the den- u(u 2* ⫺ u 2*cr ) ⫽ ⑀ ⫺1
s gRq s w s /u ⫹ ⑀b gRq b.
⫺1
(11)
sity of individual grains submerged in water with
density ρ. Equation 7 reflects a force balance be- I II III
tween the submerged weight of a particle and the Term I in eq. 11 represents the rate of TKE produc-
drag force associated with its settling motion de- tion through the interaction of the mean flow and
scribed in terms of the quadratic drag coefficient the shear stress in excess of that required to mobi-
24ν/w s d ⫹ 1. Equation 7 is valid for all grain sizes lize sediment making up the channel bed. Term II
to within a factor of about 2. We note in particular represents the rate at which energy is consumed in
that for water-borne debris with the density of maintaining the suspended load, and term III repre-
quartz and a grain size much smaller than about 1 sents the rate at which energy is consumed in
mm, this expression yields values of w s which cor- maintaining the bedload. The coefficient ε s repre-
respond to Stokes’ equation sents a dimensionless ‘‘efficiency parameter’’ that
must be determined empirically, and typically
w s ⫽ gRd 2 /18ν. (8) takes on a value in the range 10 ⫺2 –10 ⫺1 (Bagnold
1966). The value of ε s reflects the small fraction of
For sediment with similar density but for which d TKE production available for the maintenance of
⬎ 1 mm, on the other hand, eq. 7 yields the inertial the suspended load. The majority of TKE goes into
settling law a cascade of turbulent eddies and is ultimately lost

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Journal of Geology CHANNEL SLOPE IN ALLUVIAL RIVERS 665

to viscous dissipation. Because this constraint ap- suspended-load transport for which q T ⬇ q s and B
plies only to the suspended load, the coefficient ⑀ s ⬇ α/P. Equation 13 thus approaches the limiting
is applied only to Term II. The coefficient ⑀ b, on the form given by
other hand, is an empirical friction term that re-
lates the weight of the bedload to the downstream- q T ⫽ αqs(θ ⫺ θ cr )/PRθ. (15)
directed shear stress of the transporting flow. It is
expected to be of the order of unity. Upon substitution of the definitions of P and α, eq.
The balance indicated in eq. 11 reflects the as- 15 is equivalently given as
sumption that sediment transport in a natural river
is driven by the channel flow and represents work q T ⫽ (⑀ s qs/R) (u/w s ) (θ ⫺ θ cr )/θ. (16)
performed by the flow. The dichotomy between
suspended- and bedload transport is related to the In conditions of high-stage transport for which θ ⬎⬎
dynamics that control the respective transport θ cr, eq. 16 reduces to Bagnold’s formula for sus-
modes. Given unlimited supply of fine material, pended-load transport rate (Bagnold 1966). Substi-
sediment transport in a suspended-load river is lim- tution of eq. 8 and subsequent rearrangement indi-
ited by the TKE available throughout the flow. Sed- cate that for the transport of fine-grained material
iment transport in a bedload-river, on the other eq. 16 can be expressed as
hand, is limited solely by the stress conditions in
the immediate vicinity of the bed. We will revisit q T /ν ⫽ (18⑀ s /c D )θ(θ ⫺ θ cr ), (17)
this point in the discussion.
Upon substitution of eq. 10 and the dimen-
and, under conditions of high-stage transport,
sionless Shields parameter θ ⬅ u 2* /gRd, eq. 11 can
equivalently given by
be rearranged to give
q T /ν ⫽ (18⑀ s /c D )θ 2. (18)
b q b ⫽ qs(θ ⫺ θ cr )/Rθ.
α ⫺1 q s P ⫹ ⑀⫺1 (12)

In eq. 12, q ⬅ uh is the specific discharge of water, This transport equation can also be expressed in
or discharge per unit of width of the channel, P is terms of the dimensionless transport parameter ψ
the Rouse parameter defined in the previous sec- ⫽ q T /(gRd 3 ) 1/2 and the grain-size parameter D and
tion and the dimensionless coefficient α ⫺1 ⫽ given by
κc 1/2
D ⑀ s conveniently combines the parameters κ,
⫺1

c D, and ⑀ s. Upon substitution of the definition for ψ ⫽ (18⑀ s /c D )θ 2 D ⫺3/2. (19)

total sediment flux q T ⬅ q s ⫹ q b and subsequent
rearrangement, eq. 12 yields an expression for total Equations 18 and 19 are different but equivalent di-
transport flux given by mensionless forms of a sediment transport equa-
tion comparable to expressions found in the sedi-
q T ⫽ Bqs (θ ⫺ θcr )/Rθ, (13) mentological and engineering literature. Equation
19, for example, is similar to many of the empirical
where formulae listed by Sleath (1984) for suspended- and
total-load transport of fine-grained material. For
B ⫽ {α ⫺1 P ⫺ χ b (α ⫺1 P ⫺ ⑀⫺1 ⫺1
b )} . (14) nominal values of c D ⫽ 0.005 and ⑀ s ⫽ 0.03, the pre-
factor in parentheses is expected to take on a value
Equation 13 represents a general equation for total of the order 10 2.
sediment flux and is analogous to the expression Under conditions for which P is much greater
derived by Bagnold (1966). From his extensive study than unity q T ⬇ q b, the ratio χ b approaches unity
of sediment transport we assume the general valid- and thus B ⬇ ⑀ b. Upon substitution of the relation-
ity of eq. 13, and now consider its asymptotic be- ship qs ⫽ uhs ⫽ (ghs) 3/2 /gc D 1/2, eq. 13 yields a gen-
havior for conditions in which the relative depth ζ eral expression for the dimensionless bedload
is large and in which the Rouse parameter P be- transport rate given by
comes, respectively, very large (corresponding to
predominantly bedload transport), very small (cor- ψ ⫽ (⑀ b R/c 1/2
D )(θ
3/2
⫺ θ cr θ 1/2 ). (20)
responding to predominantly suspended transport),
and intermediate in value. For c D ⫽ 0.005, R ⫽ 1.6, and ⑀ b of order unity, the
For example, under conditions for which P is prefactor (⑀ b R/c 1/2
D ) in eq. 20 takes on a value of or-
much less than unity, eqs. 13 and 14 correspond to der 10. This result thus corresponds well to the em-

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666 W. B. DADE AND P. F. FRIEND

pirical law reported by Meyer-Peter and Müller In the case of mixed-load transport, the parame-
(1948) for bedload transport and given by ters P, α, and B are all near unity so that

ψ ⫽ 8.5(θ ⫺ 0.047) 3/2. (21) q T ⬇ qs (25)

Under conditions of mixed-load transport for if, as is expected in this case, θ ⬎⬎ θ cr. Under such
which P approaches unity, the parameter B in eq. conditions there are no explicit constraints on the
b ⫹ α ln(ζ)}.
13 approaches the value {1 ⫹ ln(ζ)}/{⑀ ⫺1 ⫺1 Shields parameter θ. We note, however, that in the
For the nominal parameter values c D ⫽ 0.005, ⑀ s ⫽ case of coarse debris w s ⫽ (gRd) 1/2 (c.f. eq. 9), so that
0.03 and ⑀ b ⫽ 1, this result indicates that B is also P ⬇ κ ⫺1θ ⫺1/2. Thus under conditions of mixed-load
of order unity if ζ is large. transport of coarse sand and gravel for which P ⬇
Sedimentological Constraints on the Graded 1, the Shields parameter is also expected to take on
Stream. We consider the constraints imposed by a value θ m of the order of unity, and to reflect transi-
eq. 1 on eqs. 17 and 20 for a graded stream in which tional behavior between θ b and θ s.
neither deposition nor erosion occurs. In the case Summary of the Analysis. We have proposed a
of high-stage transport of suspended sediment, for general equation for sediment transport analogous
example, eq. 17 indicates that for ∂q T ⫽ 0 (either to Bagnold’s formula for total sediment load. The
in time or space), the Shields parameter θ must ap- constraint that sediment flux is steady and non-di-
proach a constant value θ s. Noting that the flow vergent in an alluvial channel corresponds to a
Reynolds number Re ⬅ q/ν and assuming that the state of dynamic equilibrium in which neither net
drag coefficient c D is constant, this limiting value deposition nor erosion occurs. This constraint for
is given by a ‘‘graded’’ channel requires that the Shields param-
eter θ approach a constant value for each of the re-
θ s ⫽ (c D ReC s /18⑀ s ) 1/2 (22) gimes of predominantly suspended-, mixed-, and
bedload transport. The parameter θ ⫽ sh/Rd, where
s and h are, respectively, channel slope and depth,
for the transport of fine-grained material in suspen-
and R and d are, respectively, relative excess den-
sion. Taking again the values c D ⫽ 0.005 and ⑀ s ⫽
sity and average grain size of the individual parti-
0.03 and the nominal values Re ⫽ 10 7 and c s ⫽ 10 ⫺3,
cles in transport. To the degree to which the mate-
θ s is estimated to be of the order of 10. The value
rial making up the bed of an alluvial river in grade
of this estimate is not necessarily universal but is
is correlated with the debris in transport through a
related to, among other things, the Reynolds num-
given reach, this result indicates a direct relation-
ber of a channel flow and the capacity-limited con-
ship between the sediment characteristics and the
centration of suspended sediment carried by a
depth-slope product of a channel.
stream in approximate grade. The condition θ s ⬇
The different transport regimes are distin-
10 does not reflect limitations imposed by the sup-
guished by the criteria given in eq. 6. In general,
ply of fine material in an alluvial basin, and so prob-
we predict that for predominantly suspended-load
ably represents a maximal value.
transport of fine-grained material for which the ra-
In the case of predominantly bedload transport,
tio w s /u* is very small, the Shields parameter ap-
the condition ∂q T ⫽ 0 applied to eq. 20 requires that
proaches the value θ s given by eq. 22. The value of
θ s is seen from eq. 22 to be dependent on the Rey-
∂D/∂θ ⫽ (D/θ)(θ ⫺ θ cr /3)/(θ ⫺ θ cr ), (23) nolds number of the flow and the equilibrium con-
centration of sediment in suspended transport. It is
if c D and ⑀ b are constants. If grain size is also insen- expected to be of the order of 10 for most natural
sitive to transport conditions then ∂D/∂θ ⫽ 0, and rivers.
eq. 23 indicates that the Shields’ parameter ap- In the case of predominantly bedload transport
proaches a constant value θ b given by for which w s /u becomes much greater than unity,
*
the Shields parameter is predicted to take on a
θ b ⫽ θ cr /3. (24) value θ b on the order of the critical value θ cr ⬇ 0.04
for the mobilization of coarse sand and gravel. This
Sediment transport does not occur, of course, for relationship, given in eq. 24, reflects a relative in-
values of θ less than the critical value θ cr, The limit sensitivity of grain size to changes in transport con-
to θ b, then, is θ cr which is typically taken to be in ditions. Equivalently, it reflects a condition of in-
the range 0.03–0.06 for gravel-sized material. variant specific stream power ω ⬅ qs. Under

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Journal of Geology CHANNEL SLOPE IN ALLUVIAL RIVERS 667

conditions of mixed-load transport for which w s /

u* is near unity, the Shields parameter is expected
to take on a value θ m, which is also near unity.

Comparison of the Analysis with Data from

Alluvial Rivers
Summary of the Data. We compare our analysis
with extensive observations of Leopold and Wol-
man (1957), Schumm (1968), Chitale (1970), and
Andrews (1984). Additional data for sand-bed rivers
from Church and Rood (1983) are also included. In
each of these sources the data appear in concise
summary tables. The data comprise over 120 cases Figure 2. Shields parameter θ as a function of particle
of channel slope, depth, width, discharge and grain Reynolds number Re and mode of sediment transport
*
size of the bed material of alluvial rivers. A wide in alluvial rivers. The data, the sources of which are de-
range of transport conditions for perennial streams scribed in the text, are represented by the symbols and
is represented. The values of mean annual dis- have been partitioned into regimes of channel flow char-
charge are in the range 10 ⫺1 ⫺ 10 5 m 3 s ⫺1, and the acterized by predominantly suspended-load, mixed-load
range of Reynolds numbers of the channel flows are and bedload transport. Solid lines indicate regimes char-
in the range 10 4 ⫺ 10 8. Estimates of channel slope acterized by no sediment motion (NM), ripples and dunes
(R/D) and upper-stage plane bed (UP) after Allen (1982).
range from 10 ⫺5 to 10 ⫺2, and estimates of the aver-
age grain size of bed material range from 10 µm to
30 cm. Among the largest rivers considered are the
Mississippi and the Indus Rivers each with a mean ‘‘Shields plot’’ and is a conventional presentation
annual discharge in excess of 10 4 m 3 s ⫺1. Bed mate- for sediment-transport phenomena (cf. Middleton
rials in each of these rivers are composed of me- and Southard 1984). It plots the Shields parameter
dium sand. Among the smallest flows considered θ ⫽ sh/Rd as a function of the particle Reynolds
are the well-studied Watts Branch near Rockville, number Re ⫽ u d/ν for natural alluvial rivers.
* *
Maryland, with a mean annual discharge of 10 ⫺1 m 3 The data have been partitioned into classes distin-
s ⫺1, and many small streams throughout the conti- guished by the criteria given by eq. 6, which corre-
nental US. In general these smaller streams have spond to rivers dominated by suspended-, mixed-,
beds of gravel and coarse sand. and bedload transport. Indicated in this plot are the
The data represent either single or multiple ob- flow regimes that correspond approximately to no
servations of channel properties. In cases of multi- sediment motion (NM), undulatory bedforms (i.e.,
ple reports for a single river, each entry usually rep- ripples and dunes, R/D) and upper-stage plane beds
resents a distinct reach in a relatively large river. (UP) in laboratory flows (Allen 1982). The consider-
Average values are used, however, where multiple able scatter in the data reflects the wide-ranging
entries were recorded for a single reach (as in the sources of error in the observations associated the
data listed by Church and Rood 1983). In general different techniques and measures of discharge,
the discharges used in our calculations correspond channel geometry and grain size employed in the
to mean annual values or, in a few cases, values of original studies. The data nevertheless yield well-
bank-full discharge. Estimates of slope correspond defined mean estimates (⫾1 standard error) of the
to the values measured in the field unless only map Shields parameter θ s ⫽ 12 ⫾ 2, θ m ⫽ 1.9 ⫾ 0.2 and
estimates were given. Estimates of the median θ b ⫽ 0.04 ⫾ 0.003 for alluvial rivers dominated re-
grain size of channel material reported in both data spectively by suspended-, mixed-, and bedload
sets reflect a range of techniques with different lev- transport. This is in good agreement with the pre-
els of accuracy and precision. Estimates of bed shear dictions from our analysis that θ s ⬇ 10, θ m ⬇ 1, and
stress are made from eq. 10 with no correction for θ b ⬇ 0.04 for graded alluvial systems.
the drag associated with bedforms. There has been
no screening of the data other than to eliminate a
Discussion
few redundant cases between the data sets.
Sedimentological Constraints on the Longitudinal Given the numerous sources of scatter in the ex-
Profile of Graded Channels. Figure 2 is known as a isting data (figure 2 and mentioned above), we con-

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668 W. B. DADE AND P. F. FRIEND

sider the good agreement between observation and competence of the overlying flow. This distinction
prediction for alluvial streams remarkable. The im- represents a fundamental difference between rivers
plication is that the values of θ s, θ m and θ b that dominated by suspension transport of fine-grained
emerge from our analysis and are confirmed by ob- material and rivers dominated by bedload transport
servation represent quasi-universal values. This is of coarse debris, and justifies the partitioning of
intriguing. The description of bedload rivers in sediment-transport terms in eq. 11.
terms of a constant value of θ b that is universally Constraints on the Interpretation of Laboratory Ana-
related to θ cr for coarse debris has been recognized logs. Experiments that involve laboratory-scale
before (Kellerhals 1967; Parker 1979), but we know ‘‘alluvial channels’’ have long been an important
of no physical reason why the values of θ m and θ s, component of the study of the evolution and form
in particular, should be strictly universal for sandy of river channels (e.g., Leopold and Wolman 1957;
rivers as they appear to be in figure 2. There are Wolman and Brush 1961; Schumm and Kahn 1972).
several complications, moreover, which we have In light of the analysis given here, however, we
not considered in our analysis. These include the point out an important limitation of this approach.
effects of increasing channel width and total dis- In figure 3a is a Shields plot with the flume data of
charge in the downstream direction or, more tech- Leopold and Wolman (1957). The data for natural
nically, the constraint imposed on the average bed alluvial rivers shown in figure 2 are included for
shear stress in an active channel with stable (non- comparison. The experimental systems character-
eroding) banks (Parker 1978, 1979). These compli- ized by mixed-mode transport do not achieve the
cations do not appear to undermine the general pre- same equilibrium values of the Shields parameter
dictions of our simple analysis, however. θ m as do the prototype rivers. In fact the form of all
Additional implications and applications of our laboratory-scale channels appear to be near critical
analysis include consideration of the importance of conditions for sediment transport. The longitudi-
limitations due to flow competence and capacity nal profiles of such systems are thus interpreted to
for fluvial sediment transport, consideration of the be competence-limited, as are the natural alluvial
importance of laboratory analogues for natural, al- systems dominated by bedload transport.
luvial systems, and application of our analysis to This difference between experimental and natu-
the interpretation of spatial pattern and temporal ral channels is due in part to the relatively small
evolution of ancient and modern rivers. To pursue Reynolds numbers of laboratory-scale flows (figure
some of these points we propose that alluvial rivers 3b). Channel flows for which Re is less than about
are in a state of quasi-equilibrium, although we ex- 10 4 do not exhibit fully developed turbulence
amine this hypothesis as well. (Schlichting 1979). Another potentially important
Competence vs. Capacity in Graded Rivers. The factor is the physical limitation to the ratio d/h in
concepts of competence and capacity are often in- small-scale flows. In short, well-developed turbu-
voked by engineers and geomorphologists to de- lent suspension of the material in transport is sim-
scribe the ability of a river to transport sediment. ply not to be expected in the flume studies of Leo-
Competence is a measure of the largest grain size pold and Wolman (1957). These and similar
that can be mobilized by a channel flow, while ca- experiments appear to provide at best a basis for the
pacity is a measure of the sediment discharge of a study of natural rivers dominated by bedload trans-
stream in the absence of supply limitations. Equa- port and whose equilibrium form is limited by the
tions 18 and 22 indicate that the limiting value of competence of the channel flow. A useful labora-
θ s for rivers dominated by suspended sediment tory analogy for natural, fine-grained rivers domi-
transport is determined by the limiting value of nated by suspended-load transport, on the other
sediment discharge in such rivers. The value of θ s hand, seems problematic. It may be that some as-
is much greater than the value required for the mo- pects of channel form are independent of the mode
bilization of the fine-grained sediment. Compe- of sediment transport. To our knowledge this has
tence, therefore, appears to have little relevance, not been established, however, and as we have
and the graded profile of a fine-grained river is de- shown here, it is certainly not the case that the lon-
termined rather by the capacity of the channel gitudinal profile of a natural, alluvial channel is in-
flow. In contrast, the Shields parameter for streams sensitive to the mode of sediment transport.
dominated by bedload transport is near the critical Constraints on Downstream Fining. Gradual
value required for the mobilization of average sedi- downstream fining of material in gravel-bed rivers
ment in the bed. The graded profile of a bedload is well studied and is understood to reflect the in-
river is thus closely related to the mobility of the teraction of abrasion and size-selective transport of
debris making up the channel bed, and thus to the individual clasts (e.g., Paola and Seal 1995). The

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Journal of Geology CHANNEL SLOPE IN ALLUVIAL RIVERS 669

Shown in figure 4a are the same data as in figure

2, but now as a plot of stream power ω as a function
of the grain size parameter (gR 3 d 3 /c D ) 1/2 and the ra-
tio w s /u . The solid lines indicate the proportional-
*
ity ω ⬀ (gR 3 d 3 /c D ) 1/2 for suspension-, mixed-, and
bedload-dominated regimes.
In general the longitudinal profiles of natural riv-
ers are concave owing to, among other things, some
combination of the effects of subsidence and a
downstream increase in mass flux (Sinha and Par-
ker 1996). In addition, natural rivers are dissipa-
tive—as the energy of a stream is lost to heat in
the downstream direction, stream power and the
capacity to carry material also diminishes. The re-
lationships in figure 4a suggest that the natural pro-
cess of debouchment, in which ω diminishes in the
downstream direction, is systematically associated
with a downstream decrease in grain size if the
channel is in approximate grade. This is seen as
well in the single case of downstream fining in the
bedload-dominated reach of the Allt Dubhaig of
Scotland, as shown in figure 4b. The pattern of
downstream fining in this locality is clearly associ-
ated with a decrease in specific stream power in
such a way that the Shields parameter remains ap-
proximately constant about the critical value for
gravel.
The explanation for downstream fining given
Figure 3. a. Shields parameter θ as a function of particle
Reynolds number Re and mode of sediment transport
above is related to an ultimate cause for the pat-
* tern. The proximal mechanisms by which a pattern
in laboratory experiments and alluvial rivers. The data
from laboratory experiments were reported by Leopold of downstream fining develops is not readily evi-
and Wolman (1957). River data same as in figure 2 for dent from the analysis but is likely to include a pro-
comparison. b. Shields parameter as a function of flow cess of selective sorting similar to that considered
Reynolds number Re and mode of sediment transport. by Paola and Seal (1995). In any event, we suggest
Laboratory and river data as in 3a. that a downstream decrease in grain size in an indi-
vidual river will be predictable and gradual as long
analysis given here, however, provides a comple- as the stream remains in the same transport re-
mentary explanation. We note that nominal flow gime. In many rivers, however, downstream
depth h can be equivalently expressed as q/u changes in channel properties include abrupt tran-
which, upon substitution of eq. 10 and some re- sitions in grain size and slope (Yatsu 1955; Sam-
arrangement, yields the relationship h ⫽ (c D q 2 /gs) 1/3. brook Smith and Ferguson 1995). The nature of this
Thus the nominally constant values of θ s, θ m, and θ b step-like change in channel characteristics has re-
can be rearranged to yield the equivalent relation- mained an unresolved question regarding natural,
ship for specific stream power ω ⫽ qs given by alluvial rivers (Parker 1996). Our analysis provides
a basis for the interpretation of this phenomenon.
ω ⫽ (θ 3i /c D ) 1/2(gR 3 d 3 ) 1/2, (26) Abrupt Downstream Changes in Channel Slope. In
figure 5 is a plot of channel slope s as a function of
where the subscript i indicates the appropriate pa- the ratio of grain size to flow depth d/h. The data
rameter value for systems which are dominated by for natural rivers in figure 2 are distinguished as be-
suspension (i ⫽ s), mixed-load (m), and bedload (b) fore regarding channels dominated by suspended-,
transport. Equation 26 suggests that in a graded mixed-, and bedload transport. The different classes
channel the stream power ω is proportional to grain of transport, earlier distinguished by different val-
size d raised to the power 3/2. The value of the co- ues of the parameter θ ⫽ sh/Rd appear as distinct,
efficient of proportionality is dependent on the linear trends in figure 5, where s ⬇ θ i Rd/h.
value of θ i and c D. We propose that an abrupt transition from gravel

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670 W. B. DADE AND P. F. FRIEND

Figure 4. a. Specific stream power

ω as a function of the grains size pa-
rameter (gR 3 d 3 /c D ) 1/2 and mode of
sediment transport. Data as in figure
2. b. Specific stream power ω and the
Shields parameter θ as functions of
grain size in the uppermost 3-km
reach of the Allt Dubhaig, Scotland.
Data represented by symbols from
Ferguson and Ashworth (1991). The
solid lines indicate the relationship
ω ⬀ d 3/2. The dashed line in (b) indi-
cates the critical value of the Shields
parameter θπ ⫽ 0.04.

to sand that is accompanied by an abrupt change in

the slope in many individual rivers reflects a
change in the mode of sediment transport from pre-
dominantly bedload to incipient suspension during
the natural course of downstream debouchment
and fining. This view is consistent with the obser-
vations reported by Sambrook Smith and Ferguson
(1995) for the Allt Dubhaig, indicated in figure 5.
A river in approximate grade accommodates a
change in transport mode during the course of
downstream fining with a change in channel form
that corresponds to a jump from θ b to θ m. To the
degree to which the flux of water in an alluvial
channel remains continuous in the downstream di-
rection, the jump from θ b to θ m requires a quantum,
Figure 5. Channel slope s as a function of relative grain
size d/h and mode of sediment transport. Data as in fig- interdependent adjustment of both slope and the
ure 2. The solid lines indicate the relationship s ⬀ d/h. grain size of material making up the bed.
U → D indicates the character of the abrupt upstream- Alluvial Response to Perturbation. In each of the
downstream transition from gravel to sand at about 3 km topics discussed above, we implicitly assumed that
on the Allt Dubhaig, Scotland, as reported by Sambrook a river is allowed to achieve in time a state of undis-
Smith and Ferguson (1995). turbed quasi-equilibrium. Externally imposed

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Journal of Geology CHANNEL SLOPE IN ALLUVIAL RIVERS 671

changes to river channels, natural or otherwise, of- namic equilibrium of a graded profile is derived as
ten occur, however. By externally imposed change, follows. The sediment continuity equation states
we refer, for example, to climatically determined that the rate of change of bed elevation η above an
change in discharge, tectonic change of slope, and arbitrary datum equals the sum of the effects of lo-
the introduction of anomalously-sized sediment cal subsidence S(x) and the divergence in sediment
due to local bank failure or the emergence of a new flux (e.g., Paola et al. 1992). In mathematical terms
source of debris. An understanding of the effects of this relationship is given by
such changes presents a difficult challenge. The
analysis provides a quantitative basis for the predic- ∂η/∂t ⫽ ⫺C o⫺1∂q T /∂x ⫹ S(x), (27)
tion of the response of an alluvial channel to exter-
nally imposed disturbances. where Co is the volumetric concentration of solids
Channel response is most easily interpreted with in the stationary bed. Note that the condition spec-
reference to figure 4a. Given the availability of all ified by eq. 1 corresponds to a state in which the
grain sizes, for example, a change of stream power river neither aggrades nor degrades and there is neg-
ω in a graded stream will elicit a complementary ligible tectonic movement.
change in bed grain-size d to the effect that a new Substitution of eq. 13 into eq. 27 yields an ex-
equilibrium profile is achieved from among the pression for suspension and mixed load transport
quasi-universal states indicated by the data for the for which θ ⬎⬎ θ cr given by
respective regimes of suspension-, mixed-, and bed-
load-dominated regimes. Conversely, an externally ∂η/∂t ⫽ ⫺C ⫺1
0 ∂(Bqs)/∂x ⫹ S(x). (28)
imposed change in grain size will elicit appropriate
changes in slope and discharge to the effect that a Equation 28 and similar expressions provide a basis
new, equilibrium stream power is achieved. The di- for the analysis of the evolution of the sedimentary
rection of the changes is not necessarily unique. For fill of an alluvial basin in terms of a diffusion equa-
instance, a change in discharge is likely to be asso- tion (e.g., Paola et al. 1992). Upon substituting into
ciated with a change in depth h. This is likely to eq. 28 the relationship s ⫽ ⫺∂η/∂x and assuming
be associated with complementary changes in s and that specific discharge q is constant in time and
d if a constant value of θ i is maintained. Thus an space and that there is no subsidence, one obtains
increase in discharge of a bedload stream, say, could the expression
be associated with the emplacement of coarser sed-
iments on the channel bed or a reduction in the ∂η/∂t ⫽ K ∂ 2η/∂x 2, (29)
channel gradient through scour so that θ ultimately
remains constant at θ b. If the increase in discharge where K ⫽ (Bq/R). Equation 29 is a form of the lin-
is sufficient to elicit a change in regime from bed- ear diffusion equation for which K is the effective
load to predominantly mixed-load or suspension- diffusion coefficient. In the application of eq. 29 to
load transport, however, then channel slope and alluvial systems, boundary conditions are usually
grain size will co-evolve to meet the new θ m or θ s given in terms of a source flux condition and a far-
constraint. field value of η ⫽ 0 at base level. The characteristic
An important limitation to this analysis arises time T R required for the evolution of a system de-
due to the effects of a critical condition for sedi- scribed by eq. 29 to the quasi-steady, graded chan-
ment mobility. If material is introduced to a chan- nel with overall length L is given by
nel for which the existing conditions correspond to
a Shields parameter θ for the new material which T R ⫽ L 2 /K. (30)
is less than the critical value θ cr for mobility, then
that sediment cannot be reworked by the existing The response time of an alluvial system defined in
mean flow. The introduced debris remains in the this way is thus seen to be proportional to the
channel as a stranded deposit until one or more ex- square of its linear dimension, and inversely pro-
treme events redistribute it downstream (cf. Friend portional to the specific discharge q and the relative
1993). mobility of sediment embodied in the parameter B
The Response Time of an Alluvial System. The re- given by eq. 14. We note that our choice of K in eqs.
sponse of a channel to changes in discharge, re- 29 and 30 reflects one possible scheme of lineariza-
gional slope or sediment supply is, of course, not tion of the behavior of an alluvial system. It does
necessarily instantaneous. An estimate of the time not, for example, accommodate downstream in-
required for an alluvial system to achieve the dy- creases in specific discharge q. In addition, this pa-

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672 W. B. DADE AND P. F. FRIEND

Table 2. Response Times of Selected Rivers

Q AD L q h d TR
River (m 3 s ⫺1) (km 2) (km) (m 2 s ⫺1) (m) (cm) w s /u (ka)
*
Mississippi 4.3 ⫻ 10 4 3.2 ⫻ 10 6 6710 28 18.3 .04 0.55 65
Brahmaputra 2.5 ⫻ 10 4 10 6 2900 2.6 1.5 .03 1.65 85
Indus 7.1 ⫻ 10 3 10 6 3100 7.2 4.9 .018 0.31 21
Savannah (SE USA) 850 2.7 ⫻ 10 4 530 8.0 5.2 .08 1.33 2.4
North Platte, midwest USA) 65 8.3 ⫻ 10 4 1000 .4 4.9 .018 0.26 74
Cheyenne (midwest USA) 3.6 1.9 ⫻ 10 4 420 .13 0.24 .005 0.034 5.5

Note. Values for Q, A D, L, q, h and d from Leopold and Wolman (1957), Chitale (1970) and Leopold (1994). Channel length L of
Savannah, North Platte and Cheyenne Rivers estimated from the empirical relationship L ⫽ 1.16A 0.6 D (where L and A D are expressed
in km and km 2, respectively) modified from the expression given by Leopold et al. (1964). T R calculated from eq. 30 with ⑀ s ⫽ 0.03.

rameterization corresponds only to rivers domi- discussion we address the application of our analy-
nated by suspended- or mixed-load transport. sis to the interpretation of ancient alluvial deposits.
Finally, in the following exercise we assign values To do so we assume that from exposure(s) a field
to K based on current properties of apparently geologist can estimate for a paleochannel a repre-
nearly graded rivers. Accordingly, calculations sentative grain size d of bed material, channel
based on these properties overestimate the re- width w and depth h. In modern alluvial rivers
sponse time T R. The values of T R given below are there is a clear relationship between grain size and
meant only to be comparative. the ratio w S /u*, as is indicated by the data shown
Listed in table 2 are nominal response times cal- in figure 6. Using this empirical relationship one
culated from eq. 30 and estimated properties of se- can use grain size alone to ascertain whether the
lected rivers. These values of T R range from thou- ancient system is likely to represent a channel
sands of years to tens-of-thousands of years. Note dominated by suspension-, mixed-, or bedload
that these times correspond to the approximate in- transport. Having made this distinction, one can es-
terval required for the evolution of a channel over timate paleoslope from figure 5 and stream power
its entire length given present hydrological and sed- from figure 4a. With estimates of s, ω, and w one
imentological conditions, and not over a reach of then can calculate total discharge Q, and calcula-
local extent. These calculations indicate, even tions for sediment transport q T follow from eqs. 18
given our caveat regarding the likely overestima- or 25.
tion of T R, that some of the largest alluvial systems We give two specific, hypothetical examples of
are only now emerging from the effects of the last this analysis. There are important caveats, how-
glaciation that ended approximately 10 ka ago. This ever, attached to the application of our analysis of
result leads us to suggest that some of the scatter modern, graded streams to the ancient. The very
in the data shown in figure 2–5 reflects the fact that existence of the deposits indicates that the ancient
many individual rivers are only now approaching
the dynamic equilibrium associated with Holocene
conditions. This result may also explain in part
why specific sediment yields from glaciated ter-
rains are in general considerably greater than from
unglaciated basins of similar area and relatively
low stream discharge (Church et al. 1989). Put sim-
ply, glaciated basins of low stream discharge are
still adjusting to the hydrological conditions only
‘‘recently’’ imposed by the Holocene climate. An-
other ramification of the magnitude of these calcu-
lated response times is that the full extent of the
adjustment of alluvial rivers to human-induced en-
vironmental change will take considerable time to
observe. The predictive relationships given here for Figure 6. The ratio w s /u as a function of grain size d
the ultimate equilibrium forms expected of per- *
in alluvial rivers. Data sources are summarized in the
turbed rivers may help in addressing this problem. text. Solid line indicates an approximate relationship be-
Mass Flux in Ancient Rivers. As a final topic of tween ws/u and d.
*

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Journal of Geology CHANNEL SLOPE IN ALLUVIAL RIVERS 673

channel flow was alluviating and thus not in equi- tween suspended-load, mixed-load, and bedload
librium per eq. 1. The estimates of total discharge rivers.
and sediment yield given below correspond rather We then proposed a general expression for sedi-
to nominal values for the analogous channel in ment transport analogous to that derived by Bag-
grade. In addition, the estimates of sediment flux nold (1966). The basis for this expression is an
given here do not include the wash load. energetic balance for the total sediment flux,
Case A: d ⫽ 0.08 cm; h ⫽ 5 m; w ⫽ 100 m— partitioned into the suspended load and bedload,
From figure 6 we estimate that w s /u*, was of the and the incorporation of semi-empirical efficiency
order of unity, which implies that the ancient river factors. It yields expressions that correspond well
was dominated by mixed-load transport. We esti- to existing formulae for suspended-load and bed-
mate from figure 4a that stream power ω ⬇ 2 ⫻ 10 ⫺3 load transport for the limits in which the value of
m 2 s ⫺1, and from figure 5 that paleoslope was 3 ⫻ w s /u is much less than unity and much greater
*
10 ⫺4. Together with the estimate for channel width than unity, respectively.
w, these results imply that specific discharge q ⬇ In each of the limiting regimes of suspended-
7 m 2 s ⫺1 and total discharge Q ⬇ 700 m 3 s ⫺1. From load, mixed-load, and bedload transport, our analy-
eq. 25 we estimate that the sediment flux q r ⬇ ω, sis indicates that the condition that sediment flux
and so total sediment yield was approximately 0.2 be continuous in a graded channel requires that a
m 3 s ⫺1, or equivalently about 17 ton a ⫺1. The Savan- distinctive value of the Shields parameter be main-
nah River (SE USA) is the modern setting on which tained in that channel. When data from over 120
this example is based. published cases were plotted on a Shields plot (fig-
Case B: d ⫽ 5 cm; h ⫽ 0.5 m; w ⫽ 50 m—From ure 2), the constant values of the Shields parameter
figure 6 we estimate that w s /u* was of the order of for the different regimes are clear, in spite of consid-
10, which implies that the ancient river was domi- erable scatter. This suggests that the respective val-
nated by bedload transport. We estimate from fig- ues for the Shields parameter for each of the three
ure 4a that stream power ω ⬇ 7 ⫻ 10 ⫺4 m 2 s ⫺1, and transport modes may be universal. This important
from figure 5 that paleoslope was 2 ⫻ 10 ⫺3. To- result provides a powerful basis for the interpreta-
gether with the estimate for channel width w, these tion of river sedimentation patterns. It deserves
results imply that specific discharge q ⬇ 0.4 m 2 s ⫺1 closer scrutiny. Several points of discussion
and total discharge Q ⬇ 20 m 3 s ⫺1. During mean emerged from this analysis.
flow conditions there would have been only small 1) In an alluvial river characterized by sus-
amounts of sediment transport through this bed- pended-sediment transport, sediment flux and the
load-dominated system. The Wind River (Wyo- longitudinal profile of the channel reflects con-
ming, USA) is the modern alluvial environment on straint imposed by the capacity of the flow to trans-
which this example is based. port fine material, whereas in a river dominated by
bedload transport, sediment flux and channel pro-
file reflect constraints imposed by the competence
Summary and Conclusions
of the flow.
We have established quantitative relationships be- 2) Laboratory-scale, alluvial systems exhibit lon-
tween the calibre of the bed sediment and the longi- gitudinal profiles characteristic of bedload chan-
tudinal profile of a stream channel in approximate nels.
grade. We first established criteria to distinguish 3) A gradual pattern of fining in channel deposits
rivers characterized by suspended-sediment trans- can be related to the gradual reduction of specific
port, mixed-load transport, and bedload transport. stream power in the downstream direction.
The ratio of the representative settling velocity w s 4) An abrupt downstream change in grain size
of the material making up a channel bed and the and channel profile can be related to a change in
friction velocity u of the channel flow is the key the mode of sediment transport.
*
parameter in this approach. In eqs. 2–5 we exam- 5) The change in channel profile associated with
ined the quantitative relationship between w s /u* an externally imposed change in discharge (e.g., cli-
and the vertical distribution of the debris in trans- matic), slope (tectonic), or grain size (local supply)
port in a stream flow. The calculated fraction of to- can be interpreted in terms of the complementary
tal load that travels as bedload, shown in figure 1 changes required of the equilibrium state within
as a function of the ratio w s /u , exhibits a step-like the existing transport mode, or non-unique changes
*
decrease for flow conditions that correspond to w s / associated with a change in transport mode.
u* ⬇ 1. This is the basis of our exact, although arbi- 6) The quantitative description of the evolution
trary, criteria given in eq. 6 for the distinction be- of an alluvial system can be reduced to a form of

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674 W. B. DADE AND P. F. FRIEND

the diffusion equation. Consequently the system slope, total discharge, and sediment yield of the
exhibits a characteristic response time to approach system can be made.
the equilibrium or graded state considered here.
Nominal values of this response time are in the
ACKNOWLEDGMENTS
range of thousands to tens-of-thousands of years.
Some large alluvial basins or basins with relatively C. Paola, R. Slingerland, and P. Talling offered help-
low discharge may only now be approaching a form ful comments which served to clarify our thinking
that reflects Holocene conditions. and to improve the text. M. Church and T. Hoey
7) If the cross-sectional geometry and the grain generously provided data. W. B. Dade gratefully ac-
size of the fill of an ancient alluvial channel can be knowledges support from the Leverhulme Founda-
measured from its deposits, nominal estimates of tion and E. F. Dade.

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