Volume issue 2013 [doi 10.1016_B978-0-12-374739-6.00013-0] Sherman, D.J. -- Treatise on Geomorphology 1.13 Sediments and Sediment Transport.pdf

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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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


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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.



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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:

http://localhost/var/www/apps/conversion/tmp/scratch_1/dx.doi.org/B978-0-12-374739-6.00385-7http://localhost/var/www/apps/conversion/tmp/scratch_1/dx.doi.org/B978-0-12-374739-6.00385-7


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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.



10/24

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.



11/24

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.



12/24

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.


13/24

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



14/24

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.).

http://localhost/var/www/apps/conversion/tmp/scratch_1/dx.doi.org/B978-0-12-374739-6.00002-6http://localhost/var/www/apps/conversion/tmp/scratch_1/dx.doi.org/B978-0-12-374739-6.00002-6


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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.



18/24

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

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