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7/26/2019 Volume issue 2013 [doi 10.1016_B978-0-12-374739-6.00013-0] Sherman, D.J. -- Treatise on Geomorphology 1.13
1/24
1.13 Sediments and Sediment Transport
DJ Sherman and L Davis, University of Alabama, Tuscaloosa, AL, USASL Namikas, Louisiana State University, Baton Rouge, LA, USA
r 2013 Elsevier Inc. All rights reserved.
1.13.1 Introduction 2341.13.2 Key Concepts 235
1.13.2.1 The Froude Number 2351.13.2.2 The Reynolds Number 2351.13.2.3 The Prandlt and von Karman Boundary-Layer Concepts 2351.13.2.4 Nikuradses Sand Grain Roughness 2361.13.2.5 The Rouse Number 2371.13.3 The Properties of Sediment 2381.13.3.1 Particle Size and Its Measurement 2381.13.3.1.1 Particle-size scales 2381.13.3.1.2 Particle-size measurement 2401.13.3.2 Particle Shape 2411.13.3.2.1 Sphericity 2411.13.3.2.2 Roundness 2431.13.3.3 Sediment Size Distributions 2431.13.4 Initiation of Sediment Motion 245
1.13.4.1 The Hjulstrom Curve 2451.13.4.2 The Shields Curve 2461.13.4.3 Bagnolds (1936)Equation 2481.13.5 Sediment Transport 2481.13.5.1 Grove Karl Gilbert 2481.13.5.2 Ralph Alger Bagnold 2511.13.5.3 Douglas Lamar Inman 2531.13.6 Conclusions 253References 253
GlossaryCapacity The total amount of suspended and bed
sediment a stream is capable of transporting. It is
determined by the available unit stream power and
bed-shear stress distributed across the width of a channel
cross-section. It differs from the total load of a channel
as the load refers to what the stream is actually carrying,
which is dependent on the amount of sediment
supplied from upstream, and this is usually less than the
capacity.
Competence The largest caliber of sediment a stream is
capable of entraining and transporting. Competence is
proportional to flow velocity.
Form ratio The mathematical relationship betweenstream channel width and depth, usually expressed as mean
depth/width. Form ratio is often calculated in order to
determine channel cross-sectional area and/or channel
capacity.
Law of the wall A deterministic model to describe the
rate of change of fluid velocity in the stress region of a
turbulent boundary layer. The model underpins the use of
measured velocity profiles to estimate shear velocity and
shear stress.
Mixing length A theoretical construct that represents the
scale of eddies that transfer fluid momentum within a
turbulent boundary layer between the surface and the top of
the boundary layer, where the free-stream velocity is
attained. The assumption of a characteristic mixing length is
fundamental to the law of the wall.
Phi-scale The phi-scale is widely used to express the size
of sediment particles or populations. A phi (j) value is the
negative base-2 logarithm of the grain size in millimeters.
Roughness length A scaling parameter used to represent
the magnitude of the influence of a surface on an adjacent
fluid flow. It is commonly expressed as a function of the
grain size of bed sediment. The roughness length is
Sherman, D.J., Davis, L., Namikas, S.L., 2013. Sediments and sediment
transport. In: Shroder, J. (Editor in chief), Orme, A.R., Sack, D. (Eds.),
Treatise on Geomorphology. Academic Press, San Diego, CA, vol. 1, The
Foundations of Geomorphology, pp. 233256.
Treatise on Geomorphology, Volume 1 http://dx.doi.org/10.1016/B978-0-12-374739-6.00013-0 233
7/26/2019 Volume issue 2013 [doi 10.1016_B978-0-12-374739-6.00013-0] Sherman, D.J. -- Treatise on Geomorphology 1.13
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sometimes interpreted physically as the height above the
bed at which a fluid flow becomes zero.
Sediment budget A sediment budget comprising the
entire suite of sources and sinks of clastic material that
affect a given location. A positive sediment budget produces
net deposition at that location, whereas a negative budget
results in net erosion and a balanced budget is associated
with no net change.
Settling velocity The rate at which a sediment particle will
fall through a quiescent fluid after the gravity force is
balanced by the drag force so that there is no further
acceleration. Settling velocity, also termed as fall or terminal
velocity, is used as an indicator of the hydrodynamic or
aerodynamic behavior of a particle.
Abstract
Sediment transport is one of the most basic and important processes responsible for shaping the Earths surface, and is thus
of fundamental interest to geomorphologists. Existing landforms are sculpted and altered by the erosion of weathered
sediments, and the subsequent deposition of those materials produces new suites of landforms at other locations. The
purpose of this chapter is to review the development of some key concepts and techniques in sediment transport that have
become part of the repertoire of modern geomorphology. This body of knowledge has grown out of contributions from
many scientific disciplines, including, but not limited to, engineering, geography, geology, geomorphology, hydraulics,
physics, oceanography, and sedimentology. Herein, the authors aim to highlight the especially important advances.
The chapter begins with introductions to key supporting
concepts, mostly drawn from work in fluid mechanics con-
ducted between the mid-nineteenth and mid-twentieth cen-
turies, which were of a nature to change fundamentally the
way that we conceive of the physics of sediment transport.
These include the dimensionless numbers developed by
William Froude and Osborne Reynolds, which remain widely
used to characterize the nature of flows and to establish dy-
namic similitude in models; the boundary-layer theory and
law of the wall developed by Ludwig Prandtl and his student
Theodore von Karman, which permeate studies of sediment
transport across nearly all environments; the characterization
of the roughness length of sediment surfaces developed by
Johann Nikuradse; and the dimensionless parameter de-veloped by Hunter Rouse that is used to characterize and
normalize profiles of suspended sediment concentration.
The remainder of the chapter addresses three themes rep-
resenting major subcomponents of sediment transport: (1)
developments in the characterization and measurement of the
size and form of sediments and sediment populations; (2)
major contributions to our understanding of the initiation of
sediment motion, focusing on the contributions of Filip
Hjulstrom, Albert Shields, and Ralph Alger Bagnold; and (3)
major contributions to the study and modeling of sediment
transport in various environments, including Bagnolds classic
aeolian transport model, Grove Karl Gilberts work in fluvial
systems, and Douglas Lamar Inman studies of sediment
transport in coastal environments.
1.13.1 Introduction
A rich heritage of research and discovery concerning sediment
and sediment transport is relevant to geomorphology. This
work directly underpins much of process geomorphology and
is also fundamental to many environmental interpretation
and reconstruction studies. The generation of sediments by
weathering and the subsequent erosion of those sediments
lead to the reshaping of landforms. Similarly, the deposition
of transported sediments leads to the formation and evolution
of a different suite of landforms. Furthermore, the nature of
sediment deposits provides insight to the process environment
that is associated with their transport and deposition. For
these reasons, among others, understanding the fundamentals
of sediments and sediment transport provides the geo-
morphologist with powerful tools for modeling and inter-
preting landform evolution. The purpose of this chapter is to
review the development of some of the key concepts and
techniques that have become part of the repertoire of modern
geomorphology. Such classic work comes to us from many
scientific disciplines, including, but not limited to, engin-eering, geography, geology, geomorphology, physics, ocean-
ography, and sedimentology.
Numerous books have been written concerning particular
aspects of sediment and sediment transport, but there is in-
sufficient space in this chapter to detail all of the important
contributions of even the last century or so. Therefore, the
attention is focused on a selection of key publications or-
ganized by three themes: (1) developments in measuring and
characterizing sediments; (2) major contributions to the study
of the initiation of motion, and (3) major contributions to the
study of sediment transport. In each of these sections, a se-
lection of developments is detailed in their historical context
to provide what is hoped to be a deeper appreciation of their
background. Shorter, more general introductions to sup-porting concepts that contributed each advance, generally
from fluid mechanics, are also included. These concepts were
of a nature to change fundamentally the way in which the
physics of sediment transport is conceived.
In the section characterizing sediments, the classic works of
Wentworth, Wadell, Krumbein, and Folk and Ward are the
focus. For the initiation of motion, the studies by Hjulstrom,
Shields, and Bagnold are discussed. In the section on sediment
transport, the work of Gilbert, Bagnold, and Inman are
234 Sediments and Sediment Transport
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considered. Our coverage is not intended to be comprehen-
sive, but, hopefully, not idiosyncratic. For fuller consideration
of sediments and sediment transport, the reader is referred,
topically, to the following. There are many excellent treatments
of sediment properties, including the classic text byKrumbein
and Pettijohn (1938)and later works by Carver (1971),Folk
(1980), and Tucker (2001). Similar information also appears
in general texts on sedimentology and sedimentary petrology.
For treatments of motion initiation and sediment transport invarious environments, there are again many excellent com-
pendia, including books by Bagnold (1941), Allen (1982),
Graf (1984), andJulien (2010). TheTreatise on Geomorphology
also includes several volumes of direct relevance to the prin-
ciples reviewed in this chapter, notably Volume 9, Fluvial
Geomorphology (Ellen Wohl, Editor); Volume 10, Coastal
Geomorphology (Douglas Sherman, Editor); Chapter 11.1,
and Volume 14, Methods in Geomorphology (Adam Switzer
and David Kennedy, Editors). Also, in Chapter 1.2 of this
volume, The Foundations of Geomorphology, Antony Orme
discusses these principles during geomorphologys formative
years from the Renaissance to the early nineteenth century.
1.13.2 Key Concepts
Much of what we understand concerning sediment transport is
based on a series of fundamental concepts in fluid mechanics.
These reflect ideas explored in the seventeenth and eighteenth
centuries that were advanced significantly during the late
nineteenth and early twentieth centuries. In this chapter, we
examine a particular subset of advances that is believed to be
of special relevance to modern geomorphologists concerned
with, especially, sand transport. These include important
developments by William Froude (1866: the Froude number);
Osborne Reynolds (1883: the Reynolds number); Theodore
von Karman and Ludwig Prandtl (early twentieth century:
boundary-layer theory and the law of the wall); JohannNikuradse (1933: equivalent sand grain roughness); and
Hunter Rouse (1938: the Rouse number).
1.13.2.1 The Froude Number
William Froude (18101879) was an English hydrodynamicist
and naval architect with a degree in mathematics from Oxford.
His major contribution to the study of sediment transport in
geomorphology lies in the dimensionless number that bears
his name, although the relation was proposed earlier by Jean-
Baptiste Belanger (Chanson, 2009). The Froude number (Fr)
can be expressed in several forms, but most generally as:
Fr Vffiffiffiffiffi
gLp 1
where V is a characteristic velocity, gis the gravitational con-
stant, andL is a characteristic length (Graf, 1984). The Froude
number can be interpreted as the ratio of inertial to gravi-
tational forces, or as the ratio of mean flow velocity to the
celerity of a shallow water surface wave.
In the context of open channel flow,Vrepresents the flow
velocity averaged over the entire channel cross-section andLis
the hydraulic depth (the cross-sectional channel area divided
by the surface width). In rivers, the Froude number provides
one approach to distinguish between flow regimes. A Froude
numbero1 indicates subcritical or tranquil flow. In this state,
flow velocity is smaller than that of a wave propagating on the
surface and gravitational forces are dominant. For Fr41, the
flow is termed supercritical or rapid, and the inertial forces are
dominant. The Froude number is also useful in establishing
similitude between model and prototype in laboratory studies.
1.13.2.2 The Reynolds Number
Osborne Reynolds (18421912) was an Irish-born, Cambridge-
educated mathematician and engineer. Virtually, his entire
professional career was spent as a Professor of Engineering at
Owens College. The author of more than 70 scholarly publi-
cations on topics ranging from fluid mechanics to naval
architecture, and from thermodynamics to civil engineering,
Reynolds many achievements led him to be elected a fellow of
the Royal Society in 1877 (Jackson, 1995).
Reynolds accomplishments in the realm of fluid mech-
anics include development of the useful concept that has
come to be known as Reynolds-averaging, in which turbulentflows are characterized through decomposition into mean and
fluctuating components. But he is best known for his studies
of flow in pipes and the quantification of conditions associ-
ated with the transition from laminar to turbulent flow, as
characterized by the well-known Reynolds number (Re):
Re VLn
2
where n is kinematic viscosity (Reynolds, 1883). This dimen-
sionless number represents the ratio of inertial to viscous
forces. At small values (Re o2300 in pipe flow), viscosity is
dominant and flow will be laminar. At high values (Re 44000
for pipe flow), stronger inertial forces will produce turbulentflows. A transitional zone exists between the laminar and
turbulent regimes in which either flow condition may prevail
depending on additional factors like surface roughness.
In the original studies, the characteristic length scale (L)
was the pipe diameter, but in later practice, it varied with
application. In the case of open channel flow, for example,
hydraulic depth is generally used. For particles settling in a
fluid, the particle diameter is used for L (and the resulting
quantity is termed the particle Reynolds number). Along with
the Froude number, the Reynolds number provides a key tool
for determining whether dynamic similitude exists between
model and prototype flows (e.g., Middleton and Wilcock,
1994).
1.13.2.3 The Prandlt and von Karman Boundary-LayerConcepts
Every modern textbook on fluid dynamics or mechanics will
include a discussion of boundary-layer concepts based on the
work of Ludwig Prandtl (18751953) and his student,
Theodore von Karman (18811963). The motivation for the
work was the desire to quantify shear stresses across the sur-
faces of aircraft wings. Because the NavierStokes equations
Sediments and Sediment Transport 235
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were intractable, such quantification was impossible before
Prandtls short (10 min) presentation at the Third Inter-
national Mathematics Congress in Heidelberg in 1904 (see
discussion inAnderson, 2005), when he postulated the pres-
ence of a boundary layer within which the flow is influenced
by friction. In the free stream above the boundary layer,
frictional effects are negligible. At the base of the boundary
layer there was hypothesized a no-slip condition where
flow velocity became zero. These concepts revolutionizedthe study of flow across a surface. Within a few decades, the
boundary-layer theory found its way into sediment-transport
applications (discussed below), especially with regard to the
development of the law of the wall and the shear velocity
concept. Both of these developments are related through,
among other concepts, the theories of mixing lengths.
For laminar flow in a boundary layer, the change in velocity
from zero at the surface to the free-stream velocity at the top of
the boundary layer is caused by vertical momentum transport
associated with molecular motion along a mean free path. For
turbulent flows, it is assumed that small (relative to boundary-
layer thickness) parcels of fluid eddies may behave in an
analogous manner while conserving a characteristic mo-
mentum (or other physical property). The distance throughwhich momentum is conserved is the mixing length. In a
turbulent boundary layer, the mean velocity,u, at an elevation,
y, above the surface is an average of the velocities of the slower
moving eddies arriving from one mixing length, l, below that
elevation and faster moving eddies arriving from one mixing
length above that elevation, as depicted inFigure 1. This is the
key element in Prandtls momentumtransport theory
(Prandtl, 1926, as cited inVennard and Street, 1982):
trl2 dudy
23
wheret is shear stress andr is the fluid density. He also noted
thatl is a function of distance from the boundary: l ky, withk an empirical constant.
Theodore von Karman (18811963) was a student of
Prandtl at the University of Gottingen, and his PhD, written
about the behavior of solids, was awarded in 1908. His
interests turned to fluid mechanics almost by accident, ac-
cording to his biographer (Dryden, 1965). However, he was
soon appointed director of the Aerodynamics Institute at the
University of Aachen in 1912, where he honed his interests in
boundary layers. In 1930, he presented his Similarity Theory
for mixing length, arguing that the structure of turbulent ed-
dies is similar at all elevations in the boundary layer, except
that their dimensions scale with elevation, so that there is azone of constant shear stress (von Karman, 1930, cited in
Duncan et al., 1970). From his arguments:
lk dudy
d2u
dy2
u= dudy, 4
whereu
is the shear velocity, defined as uffiffiffiffiffiffiffiffit=r
p .
This formulation led to the law of the wall:
uzuk
ln z
z0
5
wherek (now known as the von Karman constant)0.4, andz0 is the surface roughness length. From this relationship,
shear velocity can be estimated using the slope, m, of a log-
linear velocity profile: ukm.
The law of the wall and its applications are used extensively
in process geomorphology, especially for deriving estimates of
shear velocity. The latter is a critical parameter for estimating
the threshold condition for the movement of sediments, and
also is a common element in models of sediment-transport
rates.
Both Prandtl and von Karman continued to make funda-
mental contributions to the field of aerodynamics; the former
for Nazi Germany and the latter for the allies. Von Karman
immigrated to USA in 1930 to take up a position at the
California Institute of Technology, where he later helped toestablish the Jet Propulsion Laboratory. Both earned many
honors during their lifetimes, but perhaps without recognizing
that their boundary-layer theories would underpin one
element of the discipline of geomorphology.
1.13.2.4 Nikuradses Sand Grain Roughness
Johann Nikuradse (18941979) was another of the
notable students who studied at the University of Gottingen
under Ludwig Prandtl. Nikuradse completed his PhD in 1923
and continued at Gottingen for another decade, conducting
extensive research on the nature of flow in pipes and channels
of various types with Prandtl (Hager and Liiv, 2008). His most
important contribution to sediment transport derives from hisclassic paper on the nature of turbulent flow in rough pipes
(Nikuradse, 1933).
In a painstaking series of experiments, Nikuradse affixed
uniform coatings of sand grains to the interior of pipes using
thin lacquer. The sand was sieved to a very narrow size range
to produce a uniform surface roughness. He then measured
the influence of various surface roughness lengths (different
grain sizes) on flows across a wide range of Reynolds numbers,
to determine a resistance or friction factor.
y
u
l1(du/dy)
u+ l1(du/dy)
u
u
y1
l1
l1
Figure 1 Schematic of the mixing length concept. The fluid speedu
at elevation y1 is the average of the speeds arriving with eddies from
a mixing length above and below.
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Nikuradse found that at low Reynolds numbers, the fric-
tion factor is independent of grain size (surface roughness)
and it decreases as Reynolds numbers increase (Figure 2). This
results from the roughness elements remaining within the
thicker laminar sublayer. As the Reynolds number is increased,
a transitional zone is entered in which the roughness elements
are of approximately the same size as the laminar sublayer and
the friction factor increases with Re. Beyond the transitional
zone, the roughness elements protrude through the laminar
layer and influence the outer flow directly. In this region,which is characteristic of many natural sediment-transport
situations, the friction factor becomes a constant that is in-
dependent of the Reynolds number and controlled by the
surface roughness length (grain size).
Following Nikuradses work, a number of simple ex-
pressions have been proposed to relate surface roughness
length,z0, to grain size, d, in studies of sediment transport in
various environments. For example,Bagnold (1941)suggested
z0d/30, whereasEinstein (1950)used z0d65/30 (d65is thediameter in a grain-size population at which 65% of the grains
are finer). In many cases, the drag imparted on a moving fluid
by surface grains (also referred to as skin drag) is mainly de-
termined by the size of the sediments that comprise that
surface. Other surface irregularities, including bedforms (formdrag) and vegetation, will also contribute to the total drag and
these latter factors may be far more significant, especially in
natural environments.
Rouse (1991) remarked on Nikuradses unusual experi-
mental approach, in which individual measurements were
immediately plotted and subsequently discarded if they devi-
ated significantly from the general trend. Nonetheless, Yang
and Joseph (2009) recently suggested that Nikuradses work
remains the gold standard for experimental studies of flow in
rough pipes, and Hager and Liiv (2008) concluded that
Nikuradses contribution to hydraulic engineering will sur-
vive. According toOswatitsch and Wieghardt (1987), the re-
ports on those experiments were the last substantive pieces of
research Nikuradse published, as he left the Kaiser Wilhelm
Institut after he tried unsuccessfully, with the help of a Nazi
Party official, to replace Prandtl as director.
1.13.2.5 The Rouse Number
Hunter Rouse (19061996) was a pioneer in ythe appli-
cation of fluid mechanics to hydraulics, fusing theory and
experimental techniques to form the basis for modern en-
gineering hydraulics as recognized in the text of his award by
the American Society of Civil Engineers of the John Frits Medal
in 1991 (Mutel and Ettema, 2010: 229). Among his many
accomplishments was the recognition and quantification of a
characteristic, vertical concentration profile for suspended
sediments, leading to the development of what is now referred
to as the Rouse number.
Rouse (1938) was interested in relationships between tur-
bulence and suspended sediments in water. His reasoning
began with recognition of the basic relationship between
vertical velocities associated with turbulent eddies and thesettling velocity of transported sediments, and the vertical
velocity profile as described by the law of the wall. Using a
blender-like apparatus to suspend particles via vertical oscil-
lation, he was able to produce and measure vertical concen-
tration profiles with four different sediment sizes, ranging
from 0.03 to 0.25 mm in diameter. Results are presented in his
Figure 4, depicting the linear relationship between elevation
and the log of concentration. There was a distinct profile for
each of the grain sizes, but each of the slopes,S, followed the
1.0
0.9
0.8
log(10
0)
0.7
0.6
0.5
0.4
0.3
0.22.6 2.8 3.0 3.2 3.4 3.6 3.8 4.0 4.2 4.4 4.6
log Re
4.8 5.0 5.2 5.4 5.6 5.8 6.0
1.1
15k
======
30.660126252507
Figure 2 Nikuradses friction factor (l) as a function of Reynolds number (Re) for various relative surface roughness lengths (tpipe radiusand kgrain diameter). Reproduced from Nikuradse, J., 1933. Stromungsgesetze in Rauhen Rohren. Forschung auf dem Gebiete desIngenieurwesens, Forschungsheft 361, VDI Verlag, Berlin, Germany (English Translation: Laws of Flow in Rough Pipes). Technical Memorandum
1292, National Advisory Committee for Aeronautics, Washington, DC, 1950.
Sediments and Sediment Transport 237
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relationship S2.3(e/ws), where 2.3 is the ln to log10 con-version, ebu0l, where b is a constant of proportionality usually assumed to be 1, u0 is the mean velocity associatedwith turbulent fluctuations, and ws is the sediment fall (set-
tling) velocity. The inverse of the term e/ws is now recognized
as the Rouse number,P, but with e parameterized as ku.
The Rouse number is used to normalize suspended sedi-
ment concentrations under different flow conditions and with
different grain sizes, to a characteristic form the Rouseprofile:
CsCa
zhzazahz p
a 6
where cs is a reference concentration at elevation z above the
bed, ca is the concentration at elevation za, h is water depth,
and a is a constant of proportionality (bin Rouse, 1938) that
varies from approximately 1.0 for low concentrations of fine
sediments to approximately 10 for medium sands (e.g., Dyer,
1986).Rose and Thorne (2001)foundb to range from 0.90 to
2.38 with only relatively small changes in grain size, but
showing a general increase with decreasing shear velocity. TheRouse profile has been widely used in fluvial and coastal en-
vironments. A few examples of river applications include
studies by Li et al. (1998), Duan and Julien (2005), Waeles
et al. (2007), Wiele et al. (2007), Davy and Lague (2009),
Shugar et al. (2010), andBouchez et al. (2011). In beach and
nearshore work, the Rouse profile concept has been used by
Beach and Sternberg (1992),Hardisty et al. (1993),Osborne
and Greenwood (1993), Vincent and Osborne (1995), Bass
et al. (2002), Vitorino et al. (2002), Nielson and Teakle
(2004), Masselink et al. (2005),van Rijn (2007), orPacheco
et al. (2011). In estuaries and marshes, it has been employed
by, among many others,Geyer (1993),Murphy and Voulgaris
(2006),Winterwerp et al. (2009),Shi (2010), andChant et al.
(2011). There has also been more limited application in ae-olian studies: in an apparently independent derivation by
Sundborg (1955)and byUdo and Mano (2011) for sand. A
broader literature has occurred for suspended dust, including
work by Anderson (1986), Tsoar and Pye (1987), Scott
(1994), and Duran et al. (2011). Related applications of
Rouses work have also been applied to gravity flows and to
transport processes on other planets. Such has been the im-
portance of the Rouse profile to the study of sediment trans-
port that it places among the leading innovations discussed in
this section.
1.13.3 The Properties of Sediment
The fundamental properties of a sediment particle, especially
with regard to potential transport, are size, shape, and com-
position. A population of mixed-size particles, typically found
in nature, is usually described in terms of the statistical or
graphical mean, sorting, skewness, and kurtosis of an appro-
priate sample. In this section, several of the key individuals
and papers that led in the development of: (1) methods of
estimating particle size using manual, mechanical, and visual
analyses, emphasizing methods used for sand-size particles or
larger; (2) descriptive and quantitative approaches developed
to categorize particle sizes; (3) methods of characterizing
particle shape; and (4) methods of describing particle popu-
lations are reviewed.
Sediment characteristics can yield a variety of information
about deposition and transport processes, sediment source
areas, and can help reconstruct environmental conditions. But
in order to interpret sediment characteristics and their geo-
morphic, geologic, and environmental significance, it is firstnecessary to describe sediment in some way that allows con-
clusions and comparisons to be made. Before the nineteenth
century, most geologists and physical geographers used indi-
vidually developed techniques and nomenclatures to describe
sediment, which, in addition to creating a great deal of con-
fusion, all but excluded the possibility of comparing data and
results between investigators. Some of the major impediments
to the development of standard sediment characterizations
include debate about which characteristics are the most
meaningful, what nomenclature should be used, and the
cumbersome and time-consuming nature of some of the
measurement techniques that hinder reproducibility. Of
the many ways that sediment can be characterized, several
measures or descriptors have survived the passage of time orhave been so seminal that they form the basis for the tech-
niques utilized today. These are the focus of the following
discussion.
1.13.3.1 Particle Size and Its Measurement
1.13.3.1.1 Particle-size scalesMajor headway in characterizing sediment occurred in the
early twentieth century as investigators began seeking standard
techniques and nomenclature. In 1922, Chester Wentworth
(18911969) of the State University of Iowa published a
named grade scale for clastic sediments, which as the
UddenWentworth scale (Figure 3) became the universalstandard for describing grain size in sediments and sedi-
mentary rocks (Blair and McPherson, 1999). Wentworths
(1922c) scheme clarified and improved an existing classifi-
cation scheme developed byUdden (1914). During the later
nineteenth century, many scientists had devised schemes that
divided particles into classes based on the diameter of their
intermediate axis, which was used because it was found to
control how particles pass through, and are thus separated by,
sieve openings. Sieving was then the most widely used tech-
nique for particle-size analysis. However, the differing prac-
tices and preferences of those who developed these schemes,
and the names they assigned, restricted comparative studies of
sediments.
Wentworth modified Uddens classification scheme by re-naming some of the clast grades, including a boulder class
beginning at 256 mm instead of 16 mm, reassigning Uddens
large (128256 mm) and medium boulders (64128 mm) to
cobble gravel (64256 mm), renaming particle classes be-
tween 4 and 64 mm as pebble gravel, introducing granule
gravel for medium gravel (24 mm), renaming fine gravel
(12 mm) as very coarse sand, and describing the four silt
classes (1/161/256 mm) collectively as silt, and coarse to fine
clay simply into clay (finer than 1/256 mm) ( Figure 3). The
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affix gravel was later dropped but is still used informally, as is
shingle in Britain and elsewhere, for particles in the granule,
pebble, and cobble range.
Although Wentworth renamed the clast grades recognized
by Udden, this revised scheme became more widely used be-
cause Wentworth chose class names based on a survey of 28
geologists in the US Geological Survey. Another reason for its
acceptance lay in the geometric progression of Uddens clas-
sification, which Wentworth confirmed. Because the intervalsbetween progressive classes in the UddenWentworth scale
maintain a constant ratio of 1:2 (defined by fractions or
decimals), the scheme preserves equal weighting between fine
and coarse particle sizes when size data are graphically de-
picted, making graphical displays of data easier to interpret. To
make this scale more mathematically versatile, Krumbein
(1934, 1938) converted it to whole numbers by taking the
negative logarithm to the base-2 of the intermediate particle
axis in millimeter (f log2 dmm). This phi-scale (f) nor-malizes the particle-size distribution, making it easier to de-
scribe and analyze. It has become customary for the phi-scale
to be converted from base-2 to base-10 logarithms because of
the latters wider application.
1.13.3.1.2 Particle-size measurementThere are many ways to measure the diameter of individual
sediment particles or the size statistics of grain populations
(see Switzer,Chapter 14.19). The most common approach for
the analysis of sand-sized particles has been mechanical siev-
ing. One of the shortcomings of sieving is that for non-
spherical particles, it is the intermediate axis that controls the
ability of a particle to pass through a sieve opening. This axis
may or may not be representative of the hydrodynamic or
aerodynamic behavior of a grain. Thus, it has been argued
that the settling velocity of a particle is a more fundamental
dynamic property than any geometrically defined measure of
size (Syvitski et al., 1991: 45). Several approaches have been
adopted to measure the settling velocities of sediments, butthe use of settling tubes (fall columns) is most common.
A century-long tradition of particle sizing has used con-
tinuous-weighing settling tubes (Oden, 1915, cited inGibbs,
1972), althoughKrumbein (1932, citingJarilow, 1913) noted
that the principles of grain settling through water were dis-
cussed as early as 400 BC. Those principles are relatively
simple. The equilibrium rate at which a single particle will fall
through a given column of water or air is a function of its size,
shape, and density. That equilibrium rate will be obtained if
the length of fall is sufficient to cause the accelerating force of
gravity to be offset by the resisting force of the fluid. The
resulting rate is termed the fall, settling, or terminal velocity of
the grain for a particular medium.
A quantitative relationship for terminal velocity of smallspheres was first proposed byStokes (1851), and was used to
estimate a hydraulic equivalent grain diameter by Schone
(1867, cited inKrumbein, 1932). The latter was based on the
understanding that natural grains are not spherical, and will
thus behave in a manner not exactly described by Stokes law.
But, according toKrumbein (1932: 108109, andFigure 11), it
wasOdens (1915)work that set the stage for modern settling-
tube designs by introducing a balance to weigh the sediments
accumulating on the pan as they fell through the water. It was
early recognized that the Stokes equation would not work for
larger (e.g., sand sized) particles. Gibbs et al. (1971) intro-
duced a more general empirical relationship that was valid for
a range of fluid densities and viscosities, and spherical grains
with diameters from 0.1 mm to 6 mm over a range of densities:
ws3Z 9Z2 gr2rfrsrf0:015476 0:19481r0:5
rf0:011607 0:14881r7
wherewsis the fall velocity (centimeter per second) of a sphere
of radiusr(centimeter),Z is dynamic viscosity (poise), gis the
gravity constant (centimeter per second), and rf and rs are
fluid and sediment densities (gram per cubic centimeter). For
nonspherical grains, eqn [7] predicts fall velocities faster than
those observed. Baba and Komar (1981) and de Lange et al.
(1997), for example, found that there were differences of 15%
or more between grain diameters calculated from fall velocity
and those found by sieving the same sand samples. Several
empirical relationships have been proposed to equate settling
tube and sieve-derived grain sizes. One of the most commonly
used (because of its simplicity) is theBaba and Komar (1981)
conversion (using centimeter per second):
wm 0:997ws0:913 8
where wm is the fall velocity measured in a settling tube. A
more accurate expression (determined empirically) is that of
Jimenez and Madsen (2003), simplified from the work of
Dietrich (1982):
w wsffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffis 1gdnp A B
S
19
wherewis a dimensionless fall velocity, sisrs/r,dnis nominal
grain diameter (diameter of a sphere of volume equivalent to
that of the grain being considered), A and B are empiricalconstants, and S
is a modification of Madsen and Grants
(1976) fluid-sediment parameter:
Sdn4u
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiS 1gdn
p 10
with u representing kinematic viscosity. By curve fitting,
Jimenez and Madsen (2003) found that for typical natural
sand grains,dnd/0.9 (whered is particle diameter found viasieving), A0.954, and B5.12. Sieving is still the mostcommon method for quantifying grain size, whereas fall vel-
ocity is increasingly important in geomorphological appli-
cations (e.g., Deans parameter for beach morphodynamics
(Wright and Short, 1984) and the Rouse number, describedearlier). Therefore, the above conversion factors remain valu-
able tools for understanding the dynamic behavior of sedi-
ments and sediment transport.
Relatively few studies have been made of the terminal
velocity of sand grains falling through air. Some examples
include Bagnolds (1935) study where he found that the
aerodynamically equivalent diameter, de, of a sphere was
0.750.85 of the sieve diameter. More recently, for natural
sand grains in air, Cui et al. (1983) found that:
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wm 1:10w0:9s 11
Malcolm and Raupach (1991)found a simple expression,
similar toBagnolds (1935), de0.9d, andChen and Fryrear(2001)presented similar data graphically.
1.13.3.2 Particle Shape
Many approaches have been used to describe the geometric
form of sediment particles, to the degree that there is general
confusion about what is meant by the seemingly inter-
changeable terms of form, shape, and morphology. This ne-
cessitates some clarification of what is meant by these terms.
In a recent review,Blott and Pye (2008) define particle shape
as the broad- and medium-scale components of morphology
and surface texture as characterized by small-scale, particle-
surface features. Furthermore, they define shape in terms of
form, roundness, sphericity, and irregularity. The major re-
search works that led to the standard definition of form,
roundness, and sphericity (the three most prevalent measures
of particle shape), are discussed below.
Particle form is important for determining particle settlingvelocity and entrainment potential. It is characterized using
ratios of a particles three linear axes: length (L), breadth (I),
and thickness (S), where L is the longest dimension, I is the
longest dimension perpendicular to L, and S is the longest
dimension perpendicular to both L and I (Krumbein, 1941;
Sneed and Folk, 1958). These axes have been notated in other
ways, including D0, D00 , and D00 0(Wentworth, 1922a,b), and a,b, and c (Zingg, 1935). Wentworth (1922a) made an early
attempt in characterizing particle form by developing a flat-
ness index, expressed as (L I)/2S. However, it seems the ul-timate goal of many early particle-form characterization efforts
was to move beyond form indices and devise a singular
graphical tool that could be used to describe particle morph-
ology easily. One of the first of such efforts was a diagram ofpebble shape created by Zingg (1935). This diagram was
divided into four quadrants that consisted of different shape
classes: disc-shaped, spherical, bladed, and rod-like. Each class
was separated based on 2/3 ratios of breadth to length (I/Lor
b/a) and thickness to breadth (S/I or c/b) (Figure 4, from
Krumbein, 1941). However, this early effort was quite limited
in that it only represented four possible shapes, under-
representing rod-like particles, and overrepresenting bladed
particles. To accommodate these three-dimensional shapes,
Sneed and Folk (1958) developed a triangular plot with 10
form categories by dividing the S/L ratio into three parts
(delineated by 0.3, 0.5, and 0.7), and the L/Iand L/S ratios
into two parts (delineated by 0.33 and 0.67) ( Figure 5, from
Blott and Pye, 2008). The advantage of the Sneed and Folktriangle over the Zingg diagram is that it represents form more
as a continuum and avoids unequal distributions of one shape
over another.
1.13.3.2.1 SphericitySphericity of sediment particles is significant in that it can be
used to determine sediment-transport distance and the po-
tential for particles to remain transported in suspension
(Bunte and Abt, 2001). Sphericity can be defined in several
ways and was once used interchangeably with roundness.
Hakon Wadell of the University of Chicago was among the
first to distinguish between sphericity and roundness (Wadell,
1932,1933). He defined sphericity as the ratio of the surface
area of a particle to its volume: the smaller the ratio, the closer
the form to a sphere. Because this ratio, c, was difficult to
measure, the actual ratio was refined as follows:
c ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
Volume of the particle
Volume of a sphere that can circumscribe the particle
3s12
Wadells measurement of sphericity required the following
steps: (1) measurement of the volume of the particle (pebbles
or larger); (2) measurement of the particle0s longest diameter;(3) calculation of the diameter of a sphere having the same
volume as the pebble or the nominal diameter; and (4) cal-
culation of the ratio expressed above. Because this procedure
was time consuming, a simpler method was developed by
Krumbein (1941), in which the long (a), intermediate (b), and
short axes (c) are measured, and the b/a ratio and c/b ratio
calculated and used to read a sphericity value from a
chart (Figure 6; Krumbein, 1941). These ratios were latersimplified byPye and Pye (1943)as follows:
c bca2
1=313
Values of sphericity as measured with Krumbeins techni-
que, called intercept sphericity, range from 0 to 1, with 1
being a perfect sphere and 0 representing platy or elongated
shapes. Using graduate student labor,Krumbein (1941)tested
I
Disk-shaped
(oblale spheroid)
III
Bladed
(triaxial)
I
b/a
00
2/3
2/3 Ic/b
IV
Rod-like
(prolate
spheroid)
II
Spherical
Figure 4 Zingg classification of pebble shapes taken fromKrumbein
(1941). Reproduced from Figure 4 in Krumbein, W.C., 1941.Measurement and geological significance of shape and roundness of
sedimentary particles. Journal of Sedimentary Petrology 11(2), 6472.
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his approach against Wadells method and found good cor-
respondence for average sphericity. Thus, in addition to sim-plifying sphericity measurements, Krumbein effectively
married Wadells definition of sphericity with Zinggs (1935)
graphical classification of pebble shape.
For estimating transportability (suspension potential and
settling velocity), two other measures of sphericity are now
commonly used: theCorey (1949)shape factor and theSneed
and Folk (1958) effective settling velocity. The Corey shape
factor is calculated by:
C cab0:5 14
Particles that have been transported far tend to approach aCorey shape-factor of 1 (a perfect sphere), with 0 being the least
spheroidal shape. The Sneed and Folk effective settling velocity,
a measure of compactness, is designed to capture the tendency
for platy particles to settle more slowly than particles shaped
otherwise (Bunte and Abt, 2001). It is calculated as follows:
S ca
15
Compact
1.0
0.9
C
CP
P
VP VB
0.
5
VE
B E
CB CE
0.8
0.7
0.6
0.5S/L
0.4
0.3
Bladed
Platy Elongated(LI)/(LS)
0.2
0.1
0.0
0.0
0.1
0.2
0.3
0.4
0.6
0.7
0.8
0.9
1.0
Figure 5 Triangular plot for particle size analysis bySneed and Folk (1958). Reproduced from Figure 2 in Blott, S.J., Pye, K., 2008. Particle
shape: a review and new methods of characterization and classification. Sedimentology 55, 3163.
1
0.9
0.8
0.7
0.6
0.5b/a
0.4
0.3
0.2
0.1
0.1 0.2 0.3 0.4
c/b
0.5 0.6 0.7 0.8 0.9 10
0.3
0.4
0.5 0.6 0.7 0.8 0.9
0.8
0.6
0.4
Figure 6 Chart for determining intercept sphericity developed by
Krumbein (1941). Reproduced from Figure 5 in Krumbein, W.C., 1941.
Measurement and geological significance of shape and roundness
of sedimentary particles. Journal of Sedimentary Petrology 11(2),
6472.
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1.13.3.2.2 RoundnessParticle roundness describes not how circular a particle is, but
how curved its corners and edges are. Roundness is commonly
used to discern the travel distance of particles, with rounder
particles assumed to have travelled farther and thereby become
rounder as their edges are abraded during transport (not always
a valid assumption).Wadell (1932,1933,1935)was the first todevelop a technique for measuring roundness, which he de-
fined from the ratio of the curvature of particle corners and
edges to the curvature of the particle as a whole. His method
was arduous, requiring the projection of an image of the par-
ticle from which to measure the radii of all corners and the
maximum inscribed circle within the outline. Roundness (P)
was calculated as follows, withrmean size of radii that can befitted into corners (cornersn), and Rradius of the max-imum inscribed circle (Bunte and Abt, 2001):
P SrnnR 16
Krumbein (1941)developed a chart with drawings of peb-
bles that had been assigned Wadells original roundness values
(Figure 7; Bunte and Abt, 2001: 91). Krumbeins chart allows
for visual analysis of roundness by comparing a sample to the
drawn images in the chart and reading the corresponding
roundness value under the matching image. Roundness values
range from very angular (0.1) to very smooth (0.9).
1.13.3.3 Sediment Size Distributions
Descriptive statistics are used to interpret particle-size distri-
butions in order to understand what, if anything, these data
may indicate about transport distance and duration, trans-
portation mode, and perhaps transport potential. Statistics
used include the mean (a measure of central tendency), the
standard deviation (SD, the range of values or sorting co-
efficient), skewness (the symmetry of a distribution), and
kurtosis (the peakedness of a distribution). Two main cat-
egories of techniques are used to derive these descriptivestatistics: the graphic method (percentile approach) and the
moment method (frequency distribution approach). These
techniques were developed for sediments earlier in the twen-
tieth century (e.g., Trask, 1932; Krumbein, 1936; Inman,
1952; Folk and Ward, 1957). Table 1 (fromBunte and Abt,
2001) provides an excellent summary of the different methods
most commonly used to determine these statistics by the
above methods. Krumbein and Pettijohn (1938) and Bunte
and Abt (2001) discussed the full suite of these techniques.
This discussion of the principles, assumptions, and differences
between the graphical and moment methods, is based mostly
on the latter.
Graphic and moment methods are applied in different
ways depending on whether the data are in millimeter orjunits. Graphic methods can be applied to particle-size data
measured in millimeter, using a geometric approach, and j
units using an arithmetic approach. The moment method
can be applied to particle-size data measured in junits and in
log-transformed millimeter. Many of the techniques applied
in these methods assume a normal or Gaussian distribution
to the data. Grain-size distributions are generally log-normal,
thus requiring some transformation from size data in milli-
meter. Thejtransformation is one such example. The geometric
Roundness = 0.1 0.2 0.3 0.4
0.90.80.70.6
0.4
0.4
0.50.5
0.40.3
Broken pebbles
0.5
Figure 7 Chart for visual analysis of pebble roundness with Wadells original roundness values developed byKrumbein (1941). Reproduced
from Figure 5 in Krumbein, W.C., 1941. Measurement and geological significance of shape and roundness of sedimentary particles. Journal of
Sedimentary Petrology 11(2), 6472.
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Table 1 Summary of methods used for computing particle size distribution mean, standard deviation (sorting), skewness, and kurtosis
Distribution parameter Graphic methods
Geometric approaches Mixed approach Arithmetic approaches
Particles sizes in millimeter Particle sizes in f-units
nth root computation Log computation Trask (1932) Inman (1952) Folk and Ward (1
Mean (central value) Root of percentile product Log of percentile product Arithmetic mean of 2 or more percentiles ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiD16D84
p log
D16D842
D25D752
f16f842
fmf16f50f84
3
Sorting (standard deviation) Root of percentile ratio Log of percentile ratio Root of percentile ratio Standard deviation Weighted percent
ffiffiffiffiffiffiffiD84D16
s logD84=D162 ffiffiffiffiffiffiffiD25D75r f84
f16
2 sff84
f16
4 f95
6
Skewness (symmetry) Mean/sorting (Fredle Index) Mean/sorting Mean/mean Mean1median/
sorting
Meanmedian sor
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiD16D84D75=D25
s logD16D84
logD75=D25D25D75
D250
fmf50sf
f16f842f502f84f16
Kurtosis (peakedness) Theoretically: sorting/sorting Theoretically: sorting/sorting Sorting/sorting Mean-sorting/sorting Sorting/sorting ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiD16=D84D75=D25
s logD16=D84
logD75=D25D75D25
2D90D100:5f95f5 sf
sf
f95f52:44f75f25
Source: Reproduced from Bunte, K., Abt, S.R., 2001 Sampling surface and subsurface particle-size distributions in wadeable gravel- and cobble-bed streams for analyses in sediment transpo
Report RMRS-GTR-74, US Forest Service, 428 pp.
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approach differs from the arithmetic approach in how the
mean is determined, which affects other statistics that use the
mean in their derivation. The geometric mean (mg) is calcu-
lated from the nth root of the product ofn numbers:
mgffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
46 93p
6 17
Graphic methods calculate the statistics of particle-size datausing a few percentile values that are derived from a cumu-
lative frequency distribution plotted on arithmetic or prob-
ability paper. This approach was common before the
introduction of personal computers. More recently, computers
have been used to determine percentile values by using linear
interpolation of percentile values between adjacentj or log-
transformed mm size classes of cumulative frequency distri-
butions:
fx x2x1 yxy1
y2y1
x1 18
where y2 is the cumulative percent frequency just below thecumulative frequency of interest, y1 is the cumulative percent
frequency just above the cumulative frequency of interest (yx),
x2 is the j unit associated with y2, and x1 is the j unit asso-
ciated withy1.
In cases where a sediment population is not log-normally
distributed, the accuracy of the calculated distribution par-
ameters is increased by using a larger number of graphically
obtained j data. Inman (1952) and Folk and Ward (1957)
used j50, j16, and j84 percentiles, which represent71 SD
from the mean, and the j5 and j95percentiles, which repre-
sent72 SDs from the mean, to calculate mean, SD, skewness,
and kurtosis. If j units are used to calculate the arithmetic
mean from percentiles, and this mean is converted to milli-
meter, then it is equal to the geometric mean.The moment method requires the percentage or the
absolute frequency of all particle-size classes, from fine to
coarse, to be known, and that the size classes be equidistant. It
uses the percentage or absolute frequency of the size classes to
calculate the four moments (roughly speaking shapes created
by the distribution of data points in data space) that corres-
pond to the mean, SD, skewness, and kurtosis. It is not
suitable for use in situations where the percentage or the ab-
solute frequency of all size classes is not known, such as in the
case of having an unsieved component of the sample in the
receiving pan. It has also been shown to over predict SD values
if the sediment is only sieved in a few large sieve classes (Folk,
1966). With widely available software, the derivation of par-
ticle-size statistics using the moment method has becomethe standard approach for producing sediment-population
statistics.
1.13.4 Initiation of Sediment Motion
Fundamental to the accurate prediction of sediment-transport
rates is the specification of a threshold condition for the
initiation of grain movement. Here, three of the most
notable approaches to this problem, the Hjulstrom (1935)
and Shields (1936) curves for fluvial systems and Bagnolds
(1936) equation for wind-blown sand are reviewed. Each
of these developments relied, to different degrees, on advances
in understanding boundary-layer dynamics, described earlier.
1.13.4.1 The Hjulstrom Curve
Filip Hjulstrom (19021982) was a Swedish geomorphologist
whose study of fluvial processes led to his defining dissertation
on the morphological activities of rivers that, among other
things, linked flow conditions quantitatively to sediment-
transport processes. The dissertation is remarkable in several
contexts. It is a very early exercise in whatHjulstrom (1935:
221) termed physiogeographical and geological dynamics
that anticipatedStrahlers (1952: 937) dynamic-quantitative
geomorphology. Hjulstrom emphasized that his dissertation
was ywritten with the conviction that the knowledge of
forces at work on the land-surfaces of the earth is quite as
important in geomorphology and physiography as the results
brought about by these forces (Hjulstrom, 1935: 221). His
work inspired a succession of process-oriented geomorph-ologists who were his students at the (then) Geographical
Institute at the University of Uppsala. Beyond the 5 years of
field measurements made on the River Fyris, his dissertation
presented a state-of-the-art review of fluid mechanics and
sediment dynamics. He displayed a firm grasp of con-
temporary boundary-layer theory, citing Nikuradse, von Kar-
man, Prandtl, and Leighly (1932, 1934) and used that
knowledge as a starting point for his development of the
Hjulstrom curve.
Hjulstrom completed many laboratory experiments on the
behavior of suspended sediments (using a salt mixture as
surrogate). However, his seminal contribution on the curves
of erosion and deposition of a uniform material is based on
an assessment of the experiments of others. He set out toanalyze those findings to relate the conditions of erosion,
transportation, and deposition of different size sediments to
flow velocity. He recognized that this approach has, to a cer-
tain degree, been considered antiquated and out-of-date
(Hjulstrom, 1935: 293), but noted that other approaches,
such as those employing concepts of critical tractive force, had
not been successful. He believed that his velocity-based ap-
proach would be successful, and it was.
Although the specification of the Hjulstrom curve itself
was a major accomplishment, it was only possible because
of some foundational work. First, he rationalized different
representations of flow velocity. Some earlier studies had
reported depth-averaged mean flow; others had reported
bottom velocity or surface velocity. Hjulstrom chose to usethe mean flow velocity for his study and corrected bottom
velocities to the mean by increasing their values by 40%,
and he decreased surface velocities by 20% for the same pur-
pose. These corrections were made based on his understand-
ing of the logarithmic distribution of velocities above the
bed. He also recognized that flow depth had an effect on
potential transport conditions so he made velocity adjust-
ments for flow depths less than approximately 0.3 m by
adding 0.2 ms1. His next challenge was rationalizing the
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different visual observations of flow conditions to allow a
consistent comparison. He did this by careful reading of
the respective reports, although in some cases clast-size in-
formation was minimally provided, and the transport obser-
vations somewhat vague.
Figure 8 is a reproduction of his now classic Figure 18, in
which velocity (centimeter per second) and particle size (milli-
meter) were depicted logarithmically (Hjulstrom, 1935: 298).
He explained that the vague nature of the transport data is whythe threshold velocity curve was drawn as a band rather than a
line (Hjulstrom, 1935: 296), although most modern represen-
tations of this diagram reproduce the erosiontransportation
boundary as a line rather than a zone (e.g., Schubert, 2006;
Weiss and Bahlburg, 2006; Callow and Smettem, 2007). Fur-
thermore, it is perhaps inevitable that later reproductions of the
curve do not include the parallel straight lines, representing
Hjulstroms interpretation of the erosiontransportation
sedimentation regimes for coarse particles fromOwens (1908)
equation.
The Hjulstrom curve is still largely used as he had origin-
ally intended, but in applications that might have surprised
him. For example, Weiss and Bahlburg (2006) used it in
their investigation of tsunami sedimentation. Callow andSmettem (2007) used it to demonstrate the effects of vege-
tation change on flow velocities and the resulting changes in
sedimentological regime.Abhyankar and Beebe (2007)used it
to explain the settling (and patterning) of cells onto substrate.
Pipan et al. (2010) used the curve to explain possible bias in
sampling because of favorable transport of particular sizes of
copepods. And, of course it is still used to study sediment
transport in fluvial systems and is included in most intro-
ductory physical geography and geology textbooks.
1.13.4.2 The Shields Curve
Albert Shields (190874) was an American engineer who
produced one of the landmark concepts in fluvial geo-
morphology the Shields Curve almost accidentally. Ac-
cording to Kennedy (1995), Shields was deflected from a
planned course of study because of financial constraints when
beginning studies for a Doctor of Engineering degree at the
Technischen Hochschule Berlin in late 1933. A project con-cerning bedload transport was made available to him at
minimal cost, and he accepted that as his dissertation topic.
He was given access to a flume and other laboratory facilities
at the Prussian Research Institute (PRI), and provided with
some technical support staff. Using data from his experiments
as well as those from his predecessors at PRI, he produced, in
1936, his dissertation,Anwendung der Ahnlichkeitmechanik und
der Turbulenzforschung auf die Geschiebebewegung(Application
of similarity principles and turbulence research to bed-load
movement). The work was in four parts, the second of which
concerned the initiation of bedload motion.
The description of the development of the Shields cri-
terion requires only 11 pages of text (in the translated ver-
sion). Using similarity arguments and dimensional analysis,he efficiently laid out the basis for his reasoning. First, the
resistance force, K0, of the grain is proportional to the grain
weight: a2(g1 g)a1d3, where a1 is the influence of grainshape on porosity, a2is the influence of grain shape on bed-
friction coefficient, g is the specific weight of the fluid, g1 is
specific weight of the grain, and d is mean grain diameter.
Against the resistance force, he balanced the effective force of
the flow: za3d2g(uc
2/2g), where z is the grain resistance co-
efficient at a critical velocity uc, and a3 is decisive grain area
Erosion
Velocityincm/sek
Sedimentation
Size of particles in mm
1000
500
300
200
100
50
3020
10
5
32
1
0.5
0.30.2
0.10.001
0.002
0.003
0.005
0.01
0.02
0.03
0.05
0.1
0.2
0.3
0.5
1 2 3 5 1 0
20
30
50
100
200
300
500
Transpo
rtatio
n
Figure 8 Reproduction ofHjulstroms (1935) Figure 18; the classic loglog plot of grain size and flow velocity. Reproduced from Hjulstrom, F.,
1935. Studies of the morphological activity of rivers as illustrated by the River Fyris. Bulletin of the Geological Institute of Uppsala 25, 221527.
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(another shape term). Following the work of Nikuradse and
based on the law of the wall, Shields argued:
ucvfa4 vdu
19
where v
is shear (friction) velocity (his symbology has been
kept here for coherence with his classic representation of data,
Figure 9),fa4is another grain shape function, d(in this case) is
grain roughness length, and u is kinematic viscosity. For ap-
plicability in the flume experiments, he defined shear velocity
in terms of the characteristics of the channel:
vffiffiffiffiffiffiffiffi
gRSp
ffiffiffiffiffiffiffiffit=r
p 20
whereRis hydraulic radius,Sis slope,tis shear stress, and ris
fluid density. Shields then redefined the grain resistance co-efficient as:
zfa45 vdu
21
where again the subscripta indicates grain shape coefficients.
The fluid forcing of the grain could then be rewritten as:
a3d2gRSfa6
vdu
22
Shields then manipulated these relationships, along with
several derivations based on the law of the wall, and
argued that the balance of driving (the two left-hand termsbelow) versus resisting forces (the two right-hand terms) at the
initiation of motion must be:
gRS
g1gd t0g1gd
fa vdu
fa1 d
d
23
where d is the boundary-layer thickness (dC(u/v) (C isChezys C; see Orme, Chapter 1.2, this volume). These rela-
tionships set the backdrop for Shields flume experiments and
results, including the classicShields (1936)curve (Figure 9).
Unlike most reconstructions of this diagram, the original de-
picts the curve as a shaded area rather than a distinct line.
Shields included data from several sources, and described ex-
istence regimes for bedforms and saltation.Shields data and his interpretations came very close to
being lost to the research community. He left Germany shortly
after defending his dissertation (Kennedy, 1995) and put bed-
load transport behind him, finding employment designing
corrugated-box machinery and winning more than 200 patents.
It was the chance discovery of Shields dissertation by Hunter
Rouse, during a visit to PRI where he had once studied, that led
to the introduction of the work to the fluvial community. Rouse
obtained and studied Shields work, brought it to USA where he
Figure 9 Reproduction ofShields (1936)diagram relating sediment characteristics and fluid and flow characteristics with resulting transport
conditions. Reproduced from Shields, A., 1936. Anwendung Der Aenlichkeitsmechanik und Der Turbulenzforschung Auf Die Geschiebebewegung.
Mitteilungen der Preussischen Versuchsanstalt fur Wasserbau und Schiffbau, Berlin, Germany. California Institute of Technology, Pasadena,
(English translation: Ott, W.P., van Uchelen, J.C.).
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had it translated by two Soil Conservation Service employees,
W.P. Ott and J.C. van Uchelen.Guo (2002) noted the possi-
bility that Rouse saved what might have been the only copy of
Shields dissertation to escape destruction during World War II.
However, Kennedy (1995) reported that Shields himself had
purchased one copy. Presumably, without the intervention of
Rouse, that one copy would be resting in an attic somewhere. It
was not until a round of correspondence between Rouse and
Shields that the latter had any indication that his research wasplaying a fundamental role in the study of sediment discharge
in fluvial systems (Kennedy, 1995).
1.13.4.3 Bagnolds (1936)Equation
Ralph Alger Bagnold (18961990) made numerous funda-
mental contributions to the study of sediments and sand
transport. Trained as an engineer, he traveled extensively in the
deserts of North Africa, sponsored early on by the Royal
Geographical Society. He began publishing, in 1931, a se-
quence of papers concerning first his expeditions (Bagnold,
1931, 1933) and then changing abruptly to focus on wind-
blown sand and desert dunes (Bagnold, 1935,1936,1937a,b,
1938), although his earlier works did include abundant ob-servations of dunes, ripples, and the behavior of sand. Most of
the results published in this latter set of articles were repro-
duced and expanded on in his classic book on The Physics of
Blown Sand and Desert Dunes(Bagnold, 1941). Here, one of his
most enduring contributions, an equation to predict the ini-
tiation of the motion of sand by wind is detailed.
In a series of wind-tunnel studies,Bagnold (1936)carefully
described the behavior of a sand surface as wind speed is
slowly increased from an initially slow flow. His observations
(Bagnold, 1936: 600) included the progression of motions
from the occurrence of sporadic transport disturbances to that
of ya steady sand flow. In particular, he noted the difficulty
in establishing a specific threshold wind speed, but did define
different threshold conditions for static and dynamic sur-faces, with the latter requiring a lower wind speed for the
initiation of motion. Later in that same paper, he first for-
malized his threshold equations in terms of wind speed and
shear velocity (Bagnold, 1936: 607). He began with Jeffreys
(1929) equation for threshold velocity, rewritten as:
u2tA rsrr
gd 24
where A is a constant (from Jeffreys, 1929, A(1/3 1/9p2)1.43). Bagnold recognized (as did Jeffreys) that aproblem with eqn [24] was that velocity is not constant with
elevation above the bed. Jeffreys (1929): 274specified a vel-
ocity at the top of the grain a value impractical to measure.
Bagnold argued that a better representative velocity could be
estimated using the law of the wall to extrapolate the log-
linear velocity profile down to the height ofk0, his focus at3-mm elevation:
utAlog30k0
d
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffirsrr
gd
r 25
where A0.43, as determined from his wind-tunnel experi-ments (note that inBagnold, 1937, this changes to A 0.47).
This equation is intended to predict the dynamic threshold of
motion, whereby sand transport, once begun, will continue.
Bagnold (1936) also provided the first threshold shear stress,
utmodel, written to parallel Jeffreys term:
u2tA0 rsrr
gd 26
And no value is given for A0. In Bagnold (1937), thisequation is combined with eqn [25] using the law of the wall
to obtain:
ut0:475:75
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffirsrr
gd
r 27
and the first term to the right of the equality sign reduces to
0.082. This represents the first value given for Bagnolds A
used for estimating the dynamic (afterward termed impact in
Bagnold, 1941) threshold shear velocity.
The familiar form of Bagnolds threshold shear velocity
equation first appears inBagnold (1941: 86):
utAffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
rsrr
gd
r 28
Based on his wind-tunnel experiments, he established two
values forA. Where the shear stress is entirely grain borne, he
specifiedA 0.08 (rounded down from 0.082) as the impactthreshold value. Where the shear stress is entirely wind borne,
he specified A0.1. The total number of experiments thatBagnold conducted to determine the threshold shear velocities
(fluid and impact) cannot be determined from reading the
series of his publications. It could be as few as three or four. It
is also difficult to determine exactly what part of his deriv-
ations can be credited to the work ofHjulstrom (1935) or
Shields (1936). Both are cited inBagnolds (1941)chapter on
Threshold Speed and Grain Size, but it is unclear to what
degree the earlier works influenced his findings, if at all.
1.13.5 Sediment Transport
The developments discussed earlier, along with a host of re-
lated concepts, are of interest to the geomorphologist mainly
as they pertain to sediment transport. This is because it is
sediment transport that has the potential to shape landforms
by either erosion or deposition. From the rich literature de-
scribing the results of laboratory, field, and modeling research,
we have chosen here to focus on the key advances made bythree scientists whose contributions represent landmarks
within the respective fields of fluvial, aeolian, and coastal
geomorphology: Grove Karl Gilbert, Ralph Alger Bagnold, and
Douglas Lamar Inman.
1.13.5.1 Grove Karl Gilbert
With the publication in 1914 of his US Geological Survey
Report on The Transportation of Debris by Running Water,
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Grove Karl Gilbert (18431918) produced one of the most
cited works from the geomorphology of the twentieth century
(Leopold, 1980). To date, it remains a foundation for con-
temporary fluvial geomorphology, contributing toward a bet-
ter understanding of the mechanics of fluvial systems, the role
of channel slope in system-scale processes, human impacts in
river systems, and sediment transport. The last theme is the
focus here.
Gilberts (1914) report summarized the methods andfindings of a series of flume experiments that he conducted
with Edward Charles Murphy. This study marked a return to
active research for Gilbert, after spending some time in a
largely administrative position as head of the Appalachian
Division of the US Geological Survey (Bourgeois, 1998). The
bulk of Gilberts field career was spent working in the
American West. He was a member of two of the four federal
government survey groups (King, Powell, Wheeler, and Hay-
den surveys), the latter three of which survived to be merged
into the US Geological Survey in 1879. Gilbert initiated his
experiences in the American West as a member of the
Wheeler Survey from 1871 to 1874, where the goal was to
conduct a geographical survey west of the 100th meridian for
military and engineering purposes. Gilbert was then invitedto join the Powell Survey of the Rocky Mountain region.
Gilberts work as part of these surveys, and then as head of
the short-lived Great Basin Division of the US Geological
Survey, resulted in many physiographic, structural, geo-
physical, and sediment studies. His flume studies of sediment
transport had their roots in issues that arose during his earlier
field work with the Powell Survey and the US Geological
Survey. Later, Gilbert had been tasked with investigating
issues related to hydraulic mining waste in California rivers,
specifically the problems of the transport capacity of im-
pacted streams (Gilbert, 1914). With this assignment came an
opportunity to conduct experiments that would allow the
largely qualitative and deductive ideas about sediment
transport that Gilbert had published in 1877 to be tested,and the results used to help understand sediment dynamics
and system responses observed in streams impaired by
hydraulic mining debris.
The experiments took place in a flume that had been
constructed for the project at the University of California,
Berkeley, USA, between 1907 and 1909 (Figure 10, from
Gilbert, 1914). The role that Edward Murphy played in
the research is not generally discussed, but in the preface to
the 1914 report, Gilbert made clear that Murphy played a
large role and by todays standards would most definitely
have shared authorship. Gilbert had to leave the research
project for some time due to illness. Although this was
mentioned in the preface, Gilbert did not disclose the nature
of the illness. In his youth, Gilbert was twice called up formilitary service in the Civil War but never drafted. This
event, among others, has been cited as evidence that Gilbert
had a lifelong battle with poor health (Pyne, 1980; Bour-
geois, 1998). In light of these issues, Gilberts field activities
in the American West are all the more impressive given the
rigorous and physically demanding nature of the work car-
ried out by the various surveys.
The design of the flume experiments was all of Gilberts
origin, but as a result of his illness, Murphy conducted most of
the experiments on his own and wrote a report of the results,
which he submitted to Gilbert on his return. Gilbert used
Murphys report to make his much cited and respected 1914
report and made clear in its preface the substantial role that
Murphy had in the research:
It will readily be understood from this account that I am respon-
sible for the planning of the experimental work as well as for the
discussion of results here contained, while Mr. Murphy is respon-
sible for the experimental work. It must not be understood, how-
ever, that in assuming responsibility for the discussion I also claim
sole credit for what is novel in the generalizations. Many conclu-
sions were reached by us jointly during our association, and others
were developed by Mr. Murphy in his report. These have been in-
corporated in the present report, so far as they appear to be sus-tained by the more elaborate analysis, and specific credit is given
only where I find it practicable to quote from Mr. Murphys
manuscript (G.K.Gilbert, 1914: 9).
One can only speculate as to why Murphy was not made
coauthor or his name not associated with the study despite his
contributions beyond that of a technician. Murphys role in
such an important and impressionable work should be
acknowledged.
Figure 10 Gilberts flume constructed on the campus of the
University of California Berkeley. Reproduced Gilbert, G.K., 1914. The
transportation of debris by running water. Professional Paper 86, US
Geological Survey.
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As for the study itself, Gilberts experiments investigated
three main aspects of sediment transport. The first of these was
competence, where he endeavored to describe the relation-
ships between size-dependent thresholds of entrainment and
the maximum size of sediment that could be transported.
A second focus was capacity, specifically the maximum weight
of load that could be transported for given flow conditions
such as stream energy, channel shape, and particle size. A
third component was to investigate bedform developmentand geometries and their relationship to sediment transport.
The experiments consisted of measuring the slopes at
which sediment transport occurred under controlled dis-
charge, sediment load, and width conditions. The experiments
were conducted on two different types of beds: plastic beds,
which consisted of sediment, and rigid beds, which consisted
of the planed wood forming the base of the flumes. Unlike
most flumes constructed today, the slope of Gilberts flume
was fixed, and sediment, which consisted of pre-sieved sands
and gravels of uniform size, was manually fed into the top of
the flume before each run. For each experiment, sediment of a
specific size was fed into a stream of a fixed width and dis-
charge. Once a slope developed from the aggradation of
introduced sediment and sediment transport began down thisslope, the slope was measured. The sediment that accrued at
the lower end of the flume was collected and weighed as a
measure of the amount of sediment transported during the
experiment.
The experiments resulted in three equations that explained
stream capacity, C:
Cb1Ssn 29aCb3Qko 29bCb4Ffp 29c
Each equation explains how capacity varies with a change
in one of the controlling variables: slope (S), discharge (Q), orsediment fineness (F), whereas the other two controlling
variables are held constant. The Greek letterss,k, andj stand
for the threshold values ofS , Q, and F, respectively, at which
sediment transport begins to occur. The exponents n,o, andp
can vary with changes in mean velocity and the form factor
R, which is the hydraulic mean depth over width. Although
presented as independent in the above equations, Gilbert
acknowledged interdependency between the variables:
In eqn [10] b1, s, andn are constant so long as Q and Fhold the
same values; they do not vary with variation in S. But when the
values ofQ and Fare changed, those ofb1, s, andn are modified
(Gilbert, 1914: 186).
To produce a final, overall equation for stream com-
petence, Gilbert combined eqns [29] and added additional
terms to account for the effects of the form ratio, which was
difficult to control in the experiments. This yielded:
CbSsnQkoFfp 1 mm 1
R
p
Rm 30
The variablep is the optimum form ratio or the value ofR
that equals the maximum value of competence. It varies with
changes in all the other controls. The variable m is meant to
account for flow resistance from the channel sides. The distri-
bution of values forRwas quite different from those of the other
controlling variables. Instead of increasing from zero to infinity
like the other variables, Gilbert found that the sensitivity of
capacity to changes in R increased to a finite maximum, which
he calledp, the optimum form ratio, and then decreased to zero.
Gilbert also developed a system of equations based on
eqns [29] and [30] that show trends of change that occur withchanges in the four independent variables (S, Q,R, and F):
31
=f1(Q, F, R)
=f2(S, F, R)
=f3(S, Q, R)
=f4(S, Q, F)
n = f5(Q, F, R)
o = f6(S, F, R)
p = f7(S, Q, R)
m = f8(S, Q, F)
Despite the novelty and important implications ofGilberts
1914report, it had its limitations.Clifford (2008)pointed out
that Gilbert was aware of the role of fluid motion as an im-
portant factor in sediment transport since he qualitatively
discussed the role of turbulence in sediment transport in his1877 (Gilbert, 1877) work and his peers were publishing on
it, but he did not address turbulence outright in his 1914
report.Leopold (1980) questioned how meaningful some of
results were, given Gilberts use of a flume that required
sediment be fed into it at the start of each experiment, making
sediment an independent variable in the experiment. In con-
trast, experiments using sediment-recirculating flumes treat
sediment as a dependent variable, and the difference between
dependent or independent variables can affect the interpret-
ations of results. Leopold also worried about Gilberts slope
measurements, which came not from the slope of the water
surface but from the slope of the debris bed, which was
usually graded before measurement by scraping from crests
into adjacent hollows (Gilbert, 1914: 25).Gilbert himself discussed two specific limitations to his
study. One was his measurement of depth, which was ren-
dered uncertain because the gauge rod interfered with flow
conditions. The other was how transferrable his results were to
natural streams. He believed that the relations he found would
hold true for streams of similar slope, form ratio, and fineness
to that used in his experiments, but he realized that this would
include a very limited number of streams. He was concerned
that the range of discharges and channel shapes experie