Middleton G.V. (Ed.) Encyclopedia of Sediments and Sedimentary Rocks
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SEDIMENT TRANSPORT
BY
L'NIDIREC TIONAL WATER ELOWS
615
surface
0,8
0,6
0,4
0.2
-S-....O
0
0,1 0,2 0,3 0,4 0,5 0,6 0,7 0,8 0,9 1,0
RELATIVE CONCENTRATION,
C
bed
0,00001
0,0001
0,001
c
0,01
Figure
SlO Al
V^^riation
of
relative concentration
of
suspended sediment with distance from
the
bed as
a
function
of z
according
(o the
Ruuse
eiju.ilion (<iller
ASC^E,
1975). The distribution
oi
suspended sediment becomes more uniform throughoul
the
fbvv depth as exponent
z
decreases,
that
is as
settling velocity (grain size) decreases and/or as shear velocity (near-bed turbulence intensity} increases,
Bi
Comparison between
the
Rouse equation
,ind
data from
an
upper stage plane
bed
(from Bennett
ft.?/,,
1998). Predicted suspended sedimenl concentration
is
less than
observed
in ihe
upper part
of the
flow.
\tilume
of
grains
per
unit
bed
area divided
by the
mean
thickness
of the
bed-load layer.
i>,,
given using Bridge
and
Dominic's (1984) model
as
T,,
- I,
(Eq.29)
The mean grain concentration
ean be
assumed
to
occur
at a
distance from
the bed of
half
the
thiekness
of the
bed-load
layer,
or
half
the
equivalent roughness height.
An alternative
to the
eonvection-diffusion approaeh
to
calculating suspended sediment eoneentration
in
water tlow
is
that based
on the
mechanics
of
two-phase flows (e.g..
McTigue.
I9S1: Cao ef af. 1995;
Greimann
ct af, 1999;
Greimann
and
Holly. 2001).
The
two-phase flow approach
is
useful
for
simulating interactions between grains,
and the
efTects
of
grain inertia, Interaetions between particles lead
to
an increase
in the
ability
of the How to
suspend particles.
Particle interactions
are
only important near
the bed
where
local grain concentrations exceed about
0.1.
Also, particle
inertia
is
only important
for
large grains near
the bed.
Unlbrtunately. this approach
is not
fully developed,
and
cannot
yet be
applied easily
to
real rivers.
Evaluation
of
sediment transport rate models
There
are
many different models
for
sediment transport rate
in
steady, uniform flows. Some
are for bed
load only, whereas
others predict
bed
load
and
suspended load either separately
or together (total load). Important differences
in
these rnodels
depend
on
whether
or not
they consider individual sediment
size-density fraetions.
the
sediment available
in the bed. and
the effeels
of
turbulent variations
in
lluid drac
and
lift
on bed
load:
the way in
whieh
How
over
bed
forms sueh
as
ripples
and
dunes
is
treated;
the
deseription
of the
vertical variation
in
flow velocity, turbulence intensity,
and
diffusivities
of
fluid
momentum
and
suspended sediment,
and; how
moving
sediment interacts with
and
modifies
the
water flow. Most
sediment transport models
are
both theoretical
and
empirical,
but
it is
desirable
to
minimize
the use of
empirical faetors that
limit
the
general applieability
of a
model. There have been
many comparisons
of the
performances
of the
various
sediment transport models (e.g..
Graf, 1971;
ASCE.
1975;
Yaiin, 1977; Chang. 1988; Gomez
and
Church. 1989).
Variation
of
sediment transport rate
in
time
and
space
Sediment transport rate varies over
a
large range
of
time
and
space scales, assoeiated
for
example with Iluid turbulenee.
the
migration
of bed
waves (such
as
ripples, dunes,
and
bars),
the
break-up
of
armored
or
partially cemented beds, bank slumps,
seasonal floods, flow di\ersions,
and
earthquake-induced
ehanges
in
sediment supply.
A
characteristic feature
of
unsteady, nonuniform flows
is
that sediment transport rate
commonly varies ineongruently with temporal
and
spatial
changes
in
water discharge;
i.e..
there
is a lag or
hysteresis. This
may
be due to
variations
in
available sediment supply,
or due
to
the lag of
sediment
and
water
(low
behind
a
faster moving
flood wave. Wash load
is
never welt correlated with water
discharge,
and
commonly reaches
a
peak during rising
discharge because
of an
overland flow souree.
The lag of
bed-load transport rate behind
a
spatial change
in
flow
characteristics
is
very small (order
of
grain diameters):
however,
for
suspended sediment this spatial
lag is on the
order
of
flow depth.
616
SCDIMFNl IRANSPORT BY UNIDIRFCTIONAI. WATER FLOWS
Sorting and abrasion of grains during transport
The fact thtit bed load and suspended loud travel at ditTerent
speeds and in dilTerent parts o\' a turbulent flow leads to the
possibility of
,sortittg.
or physical separation of grains wiih
different settling velocities (determined partly by grain size,
shape and density). Figure Sll indicates approximately the
range of grain sizes (for a given grain shape and density) that
can be transported as bed load for a partieular bed shear stress.
The actual grain size range in the bed load depends on what is
available for transport, the threshold of entrainment or
suspension tor individual grain fractions, and turbulent
fluctuations in bed shear stress (Bridge and Bennett. 1992).
Nevertheless, Figure Sll explains in a general way why
intermittently suspended load exists, and why it requires
temporal variations in bed shear stress. It is now possible to
calculate theoretically the range of sizes of grains with a
specific shape and density that will form the bed load and
intermltleiitiy suspended load, given a distribution of available
sediment (Bridge and Bennett, 1992: Bennett and Bridge.
1995).
In general, the mean and standard deviation of grain
size in the bed load increases as the bed shear stress increases
(Figure Sll). The size distribution of suspended sediment is
more difficult to speeity. Although the coarse end of the
distribution is controlled by bed shear stress, the overall
distribution is controlled by sediment availability.
Collisions between near-bed grains in transport results in
attrition {abrasioti) of the grains (see Attritt'oti. Fluvial).
Experimental studies of abrasion of gravel during river
transport indicate that grain size is reduced and grain shape
is modified, depending on the hardness of the grains and
distance of transport. Abrasion o\' sand in water is negligible
{in contrast to that in air flows).
Bed shear
stress:
•=0
Pa
10"
Suspended
load
Grain size range of
inlermittently suspended load
Bed load
.ii^.W''"' No movernent
Grain size range ot bed load
1 \ r r
10-'
10° 10'' 102
Grain size ; D rnm
Figure S11 Approximate prediction of grain-size range of bed lo^id
and inlermiltently-suspended lojd where bed shear stress is allowerl to
vary between limits
i,,n,,v,
and
x,,^,.,^.
The mean grain size and ranf>e ot'
sizes increases as instant,Tneou5 bed shear stress increases. The
prediction is only approximate because the thresholds ot' motion and
suspension of individual grains in a mixture of different-sized grains
are nol necessarily exactly as shown. Threshold curves are based on
sediment ot constant grain size and density, and time-averaged bed
shear stress.
Erosion and deposition
Erosion and deposition are due to ehanges in seditnent
transport rate in space (mainly) and in time (minor), as
expressed in the equation of conservation of sediment mass
(sediment eontinuity equation). A one-dimensional version of
this equation is
-C/,dh/c\t = dijd.x + diju., d/
(Eq. 30)
where C/, is the volume concentration of sediment in the bed
(1-porosity), /; is bed height, t is time, /\ is the downstreatn
sediment transport rate by volume.
-V
is downstream distance.
and w,v is downstream velocity of the sediment. In uniform
flow, d/v/dv =
O,
but an increase in sediment transport rate in
time (d/,/"«vd'>Ol would result in erosion. Such erosion would
involve entrainment of a layer of bed grains atiioutiting to an
average thickness not exceeding a few grain diameters. In other
words, erosion and deposition in uniform, unsteady flows is
normally tiegligible. In contrast., rates of erosion and deposi-
tion in nonunilbrtii flows ean be substantial, depending on the
magnitude of diyld.x. Erosion occurs in spatially accelerating
flows where d/\/dv is positive, and deposition occurs in
spatially decelerating flows where
dijd.\
is negative.
It is very common in nature for relatively long periods of
little net erosion or deposition to be punctuated by relatively
short periods (e.g., during seasonal Hoods) when there is
marked erosion followed by tnarked deposition or vvVt' versa
(i.e.,
the terms in equation 30 vary over time). This explains
why sedimentary strata are normally bounded by planes that
record erosion or nondcposition. If d/,Mv is negative and does
not vary very tnuch in space, a layer of sediment will be
deposited that varies little in thickness laterally. If
dijd,\
is
negative but changes markedly in space, sediment will be
deposited as mounds or ridges (e.g., ripples, dunes, and bars).
Similarly, a surfaee of erosion will tend to be planar if d;\/d.v
is positive and does not vary very much in space, whereas it
will tend to be strongly curved (e.g.. trough-shaped or spoon-
shaped) if d/v/dv is positive and changes locally. In reality,
d/\/d.v varies in space on various different scales, such that
there are dilTerent scales of variation in the geometry of strata
and erosion sut"faees superimposed.
Deposition rates and erosion rates depend on the titne scale
and space scale of measurement. Vertieal deposition rates
averaged over a few years and over bed waves like ripples,
dunes and bars, can range frotn fractions of a miliitneter per
year to tens of meters per year (Bridge and Leeder, 1979;
Sadler, 1981: Enos, 1991). However, as many deposits will be
completely or partially eroded before final burial, deposition
rates over progressively longer time periods tend to decrease.
Deposition rates and erosion rates averaged over millions ol'
years arc typically on the order of 0.1mm per year. The
deposits with the best
ptvsi-rvatiint
potetitia! were deposited in
the topographically lowest areas (e.g., deepest parts of
channels) and those that subsided rapidly.
Sediment sorting and orientation during deposition
Application of equation 30 to individual grain fractions leads
to prediction of sorlitig in deposited sediment. Rational
prediction of the sorting of deposits requires knowledge of:
(1) the sediment types available for transport; (2) the time and
space variations in fiow and sediment transport rates of
iiidi\itlua! uraiti fractions, and: (3) the flow and sediment
StDIMENT TRANSPORT UY UNIDIRECTIONAI. WATER ELOWS
617
iransport over bed Ibrms such as ripples, dunes and bars. Such
prediction has been attempted in only very limited cases (e.g..
Bennett and Bridge. 1995: Cui elal.. 1996), using seditnent
routing tnodels as discussed below. In general, when erosion
occurs, the coarsest and/or detisest grains are most likely to
remain in the bed. The large, itiitnobile grains that are left on
ati eroded bed surface are eommonly relerred to as a lag de-
posit. Tiie term artitoritig refers to tlie situation where there arc
sufficient large immobile (or only inlermittetitly mobile) grains
on an eroding bed to protect the potentially mobile grains
underneath from entrainment (see Artijor). Armored beds
eotnmonly occur in gravel-bed rivers during falling flow stages
after floods. The devcloptnetit of armoring results in an
increase iti bed t"oughness and a decrease in sediment tt"ansport
rate(Gotiiez.
!99_'^;
Hassan and Church. 2000). The increase in
bed roughness is partly to do with increasing bed sediment size,
and partly due to the developtnent of bed structures such as
pebble clusters. When deposition occurs, the coarsest and/or
dctiscst grain fractions in the bed-load are preferentially
deposited, bttt the texture of the deposited sediment is closely
related to that of the bed-load frotn which il was derived. It is
cotiimo[i for the deposits associated with flood conditions
(high bed shear stress) to be eoarser grained than those
associated with fallitig and low (low stages (low bed shear
stress),
because mean grain size of sediment in transport and
being deposited is directly related to bed shear stress. Thus,
layers of sediment deposited during floods tend to have ftne-
grained tops, unless bed annoring occurred.
Mean graiti size of sediment in the bed surface and in
iransporl eotnmonly deereases exponetitially iti the direction of
tratisport. This is mainly due to downstream decrease in the
bed shear stress and turbulence intensity o\' the transporting
and depositing flow, such that the eoarsest bed load grains are
progressively lost in the downstream direction (review in
Bridge. 200.1). Downstream reductioti in size due to progtcs-
sive abrasion associated with grain collisions is of minor
importance, as only the larger and softer gravel grains suffer
appreciable abrasion. Intertnittently suspended grains may be
deposited wiih bed kxid graitis, either as a relatively fine-
grained surficial layer, or by infiltrating the pore spaces
between the larger grains in the bed (thus reducing the degree
ol'sorling). Most of the suspended load is not deposited with
bed load, but accumulates in places where turbulence ititetisity
is low (e.g., flood basins, lakes, and ponds). Suspended
sediment can be sorted by selective deposition aeeording to
settling velocity.
Hcavyttiitii'ittls (e.g.. iron cotnpounds. gold) in the bed load
and suspended load are commotiK fmer grained thati associated
light minerals (see Heavy
.Mitierals).
Heavy mitierais may be
concctitrated in the bed where a flow that is powerl'ul enough to
entrain and transport all grain fractions decelerates and
deposits some of them. The relatively small heavy minerals
become protected from rc-entrainment by the larger light
tninerals. atid some may infiltrate into the pore spaces between
the larger grains in the bed. Subsequent erosion may remove the
large light minerals but not the small heavies. Sueh conditions
are found in association with various types ol bed forms.
Grains tend to be orictited during transport on the bed
immediately prior lo deposition. On plane beds, ellipsoidal or
rod-shaped grains tend to be oriented with their long axes
approximately parallel to flow, and dise- and tabular-shaped
pebbles tend to be oriented with their maxitnutn projection
planes dippitig upstream (ituhricatioit. q.w). However, these
characteristic orientations are modifled uhere bed forms are
present.
Grain size, shape, grain otientation and packing (fabric) in
seditnetits influence potvsity atiti pertveahilitw with the most
important controls beitig mean grain size, size sorting, and
fabric (Brayshaw etai. 1996; see
Fahrie.
Pofo,sity.
Pefttieahilitv),
The spatial variatioti of porosity and pertneability varies
grcally with spatial variation in texture and fabric (i.e., with
stratifieation).
Nature of erosion
The amount of erosion at a poinl is limited by two main
factors. The lirst is bed armoring, such that the progressive
accumulation of the heaviest grains on the bed protects the
underlying seditnent from further erosion. The olher lactor is
the generation during vertical erosion of a progressively larger
depression in the sediment bed. Sueh a depression leads to
expansion and deceleration of the flow, thereby leading to
cessation of vertical erosion, or at least to relocation of the
zone of erosion.
Surfaces of erosion arc recognizable by truncation of
underlying strata and by a single overlying layer of the
coarsest and/or densest grains from the underlying material
Ihat could not be entrained (i.e., a lag deposii). Features
indicative of local erosion include eurrenien:scent.t. horseshoe-
shaped scours around the upstream margins of immobile
obstacles to the flow (e.g., large grains, tree trunks). The
eroded material is commonly deposited on the lee side as a
ridge of seditnent or satid shitdow. Other erosional features
inelude elongate, flow-parallel channels called gutterca.sts. and
latgc-scalc features sueh as channels.
Erosion of cohesive tnud depends on the degree and nature
of consolidation of the mud. Soft mud that is entrained as
mud-sized particles travels in suspension. However, various
bed features tnay be eroded into the tnud by the Ilow aeeording
to the bed shear stress (Allen. 1982; see
.Seout;
.Seour
Mark.^):
straight or tiieatideritig Itingitudiital grooves (ttid ridges: /lute
tihtrks:
traftsver,-ie
ridge tiiarks. Soft mud is also eroded and
molded by various hard objects (tools) earried by the flow.
Tool
tiiarks are eomtnonly long grooves of varying width atid
depth due to objeets (e.g.. shells, bits of wood, shale chips)
sliding and rolling along the bed and eroding into the mud.
There arc also marks due to intermittent contacts of objects
with the bed (e.g., bounec. skip atid prod tnarks). These and
the current-formed tnarks are normally preserved as coarser
sediment eovers the mud more-or-less itntnediately following
tnark formation (e.g..
.sole
tnarks on the base o^ sandstone
strata).
Cotisolidated (cotnpacted) tnud is eotnmonly eroded as
pellets or flat chips of sand or gravel that then travel as bed
load. Such relatively large grains may occur as lag deposits.
Erosion marks in consolidated mud include stecp-walled
gutters and potholes.
Sediment routing models
The only rational uay of analyzing and predicting bed-
elevation changes and sediment sorting during erosion,
transport, and deposition is with .seditnetu routing models
(reviewed by Bridge, 2003). Sediment routing models are used
to deal with problems such as bed erosion and armoring
downstream of dams, and reservoir sedimentation. Sediment
routitig models normally reqitire tt"eatnient of unsteady.
618
StDIME.NT TRANSPORT
BY
UNIDIRECTIONAL WATER ELOWS
gradually varied flow aeting upon movable, heterogeneous
sediment beds. A realistic sediment routing model requires
specificalioti of; (1) seditnent types available for transport; (2)
the tnean and turbulent fluctuating values of fluid force acting
upon sediment grains, and how these vary in titne and space;
(3) the interaction between fluid forces and indi\ idual sediment
fractions, resulting in their entrainment and transport as bed-
load or suspended load: (4) erosion and deposition as
determined by a sediment continuity equation applied to
ditTerent grain fractions; (5) the concept of an active bed layer
to which seditnent cati be added or be removed; and (6) an
accounting sehetne to tecord the nature of deposited sediment
or armored, eroded surfaces. Many sedimeni routing tnodels
do not have all of these desirable features, and contain tnatiy
arbitrary correction factors. Exatnples of the use of seditnent
routing models lo predict the nature of sediment transport,
erosion and deposition, the development or armor layers, and
the sorting of sediment by size and density are given by Vogel
eral. (1992). Bennett and Bridge (1995). and sec Nutneriad
Models and Sitiitdatioii.
Sediment transport and bed forms
One of the most striking and important characteristics of
sediment transport over a bed of eohesionless sediment is the
development of bed Ibrms such as ripples, dunes and bars.
These are itnportant to seditnentary geologists because they are
associated with certain kinds of seditnentary structures, and to
engineers because they influence flow resistance and sediment
transport rate. The origin, geometry, flow and seditnentary
processes of bed forms are reviewed by Allen (1982). Bridge
(2003) and see Sttrfdee Fomts.
John S. Bridge
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Komar. P.D., 1996. Entrainment
of
sediments tVom deposits
of
mixed
grain sizes
and
densities.
In
Carling.
P.A.. and
Dawson.
M.R.
(eds.).
Advanees
in
Fluvial Dynamics
and
Sirati^raphv. Wiley:
pp.
127-181.
McManus.
J.. 19SS.
Grain size determination
and
interpretation.
In Tucker.
M.
(ed.). TcchniijuesinSedinientoIo^v, Blackuell: pp.
63-
85.
McTigne.
D.F.. 1981.
Mixture theory
for
suspended sediment
Iransport. Journalof the Hydraulic Division. ASCE. 107:
659 673.
Miller.
M.C.,
McCave. I.N..'and Komar.
P.D.. 1977.
Threshold
of
sediment motion
in
unidirectional currents. Sedimentoloi'v.
24:
507-528.
Nelson.
J.M.. and
Smith.
.I.D.,
19S9.
Mechanics
of
How over ripples
and dunes. Jmirnal of Geophysical Reseurch.
94:
SI46-RI62.
Parker.
G..
1990. Surface-based bedload transport relation
lor
gnivel
rivers,
.foitmal of Hydraulic Research. 28:
417 436.
Sadler. P.M.. 198!. Sediment accumulation rates
and
the completeness
of stratigraphic sections. Journalof Geology.
89:
569-584.
Van Rijn. L.C.. 1984a. Sediment transport, part
I: bed
load transport.
.lournalof HydraulU Engineering, ASCE. 110: 1431-1456.
Van Rijn,
E.C.,
1984b. Sediment transport, part
II:
suspended
load transport. Journal
of
Hydraulic Emiineerin^, ASCE.
110:
I6I3-164I.
Van Rijn.
L.C..
1984c. Sediment transport, part
IM: bed
forms
and
alluvial roughness. Joitrnal
<>l
Hyihaalic Engineering, .4SCE.
110:
1733
1754.
Vogel.
K.. Van
Nickcrk.
A..
Slingerland,
R.. and
Bridge.
J.S.. 1992.
Routing
of
heterogeneous size-density sediments over
a
moveable
bed: model vcritlcation
and
testing. Journal
of
Hydraulic Engineer-
ing.
ASCE. 118:
2fi3-
279.
Wiberg.
P.L.. and
Smith.
J.D.. 1987.
Calculations
of
critical shear
stress
lor
motion
of
unitbrm
and
heterogeneous sediments. Water
Resources Researeh.
11:
1471- !48t).
SEDIMENT TRANSPORT BY WAVES
619
Wiberg. P.L.. iiLid Smith. J.D.. 19K9. Model Ibr culculaling bed load
transport of sedimenl. .hniriialof flyihuitlicEii^itieeriinr. .XSCE. 115:
101 123.
Wilcock. P.R.. ;ind McArdell. B.W.. 1497. Partial transport ofa siiiid-
gravel sediment.
Waier
Resaunes Research. 33: 2.13-245.
Yalin. M.S.. 1977. Mechunics i>f Sedimenl Traiisporl. 2iid edn.
Pergamon Press.
"I'alin. M.S.. and Kiirahan. E.. 1979. Inception of sedimenl transpori.
./oiiriiah'lIhc llvihuulici Division, .ASCL. 105; [4}} 1443.
Cross-references
Alirilion (Abrasion). Mtivial
Armor
Cross-stratilication
Lrosion and Sediment Yield
I abric. Porosily and Permeability
riow Resistance
(.irain Sellliiig
Cirain Si/c and Sliiipc
Cirain Threshold
(iiiltcrs and Ciiiller Crisis
I
lea\y Minerals
linbricalion and Fiow-Oricntcd Clasts
Parling Lineations and Current Ocsccnts
Pliysics ol' Sediment Transpori: The Coniribuiions ot" R.A. Bagnold
Clanar and Parallel Lamination
Ripple. Ripple Mark. Ripple Slruclure
Scour. Scotir Marks
Stirliicc F-orms
Tool Marks
Tracers lor Sediment Movemenl
SEDIMENT TRANSPORT BY WAVES
Water waves
Water waves are periodic nndtihtlions of the air water
interface (Figure S12) deiined by their heli^ht (//), miveleni^th
(/,). period of oseillation (71 and
.speed
of propagation (C), The
water moves in qiiiisi-eireular or elliptical orbits: the orbital
diameter (d,J equals the wave height at the air water iiilerface
and deeays with increasing water depth. At depths >L/2 the
oscillation is essentially dissipated; at depths <L/2, the
oscillation interacts with the bed to form an oscillatory
houndary layer. Waves are found whereever wind blows over
a significant body of water (e.g., the deep ocean, the
continental
shelf,
the shoreface. estuaries, lakes, etc.). They
can be pwgre.ssive (e.g.. wind waves) or standing (e.g.. a lake
seiche] and either actively forced
{,\ea)
or /)•[•(' (propagate
freely: e.g.. .swell). Swift (1976) defined hydraulic provinces
stretching from the deep ocean to the coastline. In the
innermost zone {sfwrejacc). both the hydrodynamics and
sediment transport are controlled by waves, while further
offshore [continentalsheff) waves play a major role in the en-
trainment of sediment (Cacchione and Drake, 1990), although
the transport may be dominated by other Hows. Modern wave
theory dates back to the classic work of Stokes (1847) and a
recent theoretical ireatinenl of the hydrodynamics of water
waves is given by Le Mehautc (1976: especially Part 3). Smith
(1977).
Sleath (1984). Grant and Madsen(1979; 1982), Nielsen
(1992).
Fredsoe and Deigaard (1992) and Van Rijn (1993)
review the theory of sediment transport under waves.
Water waves are classified by their period (frequency) of
oscillation and Ihe force generating them as well as the force
restoring equilibrium of the water surface (Kinsman. 1975).
The most important waves outside the zone of wave breaking
(surfrone) are
f^ravitv
niivcs (forced by wind and reslored by
gravity), with periods ranging from I-25 s. Their magnitude
depends upon Ihe wind speed, duration of time the wind blows
and Ihc fetch length (length of open water over which the wind
blows).
A modulation of wave height is common and gravity
waves propagate as groups of large waves separated by several
smaller waves (Figure SI3). This modulation forces a
secondary wave, the ^roup-htnindlong
wave,
which propagates
at Ihe group speed and has a period equal to the group period.
Waves with periods of -25s to several minutes or longer
(frequencies of 0.0(H-0.04 Hz) are infhtgravity waves (called
.surf heat by Munk, 1949). They are generally small (<lcm
high) in the deep ocean, but increase in size shoreward and are
important to water circulation and sediment transport within
the .surf zone (Bowen and Huntley. 1984). In intermediate
water depths, infragravity energy results from wave wave
interaetions and consists ofa mixture of forced 'dnd free wave
motions (Herbers
etaf.
1995). Several meehanisms assoeiated
with wave breaking in the surf zone have heen proposed for
their generation including: (i) release ofthe group hound
fong
Direction of Wave Propagation
— L-
Figure S12 The surface form of deep water and shoaling surface gravity waves, the orbital motion beneath the surface, the inleraclion [)f the
wave with a rippled bed and the characteristic wave and ripple properties
i modi tied
alter Clifton, 1984). i = wave length; H=vjA\je height;
4,
= orbital diameter; /j = w.iler depth; /. = ripple wavelength;
</
= ripple height; /f = ripple cresl-to-trough dislance.
620
SFDIMFNT TRANSPORT BY WAVES
wave propagation
individual waves
wave packet
envelope
Figure S13 The surface profiles
of
high frequency surface gravity waves (solid line}, the wave envelope Idotted line) and the furced group bound
lon;^
wave (dashed
linel.
The horizontal near-bed velocities under the long wave (rest and trough (u,
and (/,)
and
the
rekitive sediment
concentrations istippled) are also shown,
as is the
water depth
(/i).
Note
the
coupling
of
large sediment concentrations indue ed
by ihe
large
gravity waves with
the
upwave (offshore) velocity under the trough
of
the long wave; there
is
oflen
a
small phase
lag in
this relationship.
wave (List. 1992);
(ii)
periodic shift
in the
position
of
wave
breaking (Symonds
etaf,.
1982);
(iii)
persistence
of
groupiness
into
the
surf zone causing
a
rise
and
fall
of
water
at the
shoreline (Watson
and
Peregrine, 1992). Nearshore infragrav-
ity energy
can
take several forms including standing
and
progressive edge wares (trapped
to the
shoreline
by
refraction)
and long leaky waves (reliected seaward without trapping;
Huntley. 1976).
An
important characteristic
of
infragravity
waves
in the
surf zone
is
that they increase
in
height
as the
offshore wave height increases.
In
contrast, gravity waves
are
.saturated
in
shallow water
and
thus decrease
in
importance
proportionally during storms. Shear
waves
are a
subset ofthe
injragravity field (periods range from several tens
of
seconds
to
several tens
of
minutes)
and
they form within
the
surfzotw
through instabilities
in the
longsfmre current,
The\
appear
as
large Iliictiiations
in the
current (Bowcn
and
Holman,
1989;
Oltman-Shay
etaf.
1989)
and can
contribute significantly
to
both
the
cross-shore
and
alongshore transport
of
sediment
(e.g.. Aagaard
and
Greenwood. 1995).
Wave propagation
Propagating gravitywaves transfer energy
and
momentum,
and
as they move into shallow water they
.slioal.
With increased
frictional drag,
the
orbital motions
are
deformed, wave speed
and length decreases
and
height increases.
The
specific
energy
density (according
to
linear theory. E=l/8pigH') increases
until
the
wave breaks. Energy
may
also
be
re-distributcd
laterally (parallel
lo the
wave crests) through
the
processes
of
refraction
and
diffraction. Further,
the
simple near-sinusoidal
shape, characteristic
of
deep water, beeomes increasing
non-
symmetrical about
the
horizontal
and
vertical axes (wave
skewne.ss
and
asymmetry). These non-linearities (Figure
S14)
are critical
to the net
transport
by
waves
as
they introduee
asymmetries
in
the oseillatory veloeities.
In
very shallow water.
waves become unstable
and
break when ////(5:0.4
1.0. The
wave form
may be
destroyed
in a
plunging breaker, producing
a large rolfer
vorie.x.
or the
wave
may
propagate further
as a
spillingbrcaf<er
or
surffiore with reduced height.
The
surfzone
is
a
complex hydrodynamic environmeni. where interactions
with .secondary waves
and
quasi-steady currents
of
various
origins occur. Ultimately wave etiergy
is
dissipated
in the
reversing currents
of the
.swash
on the
beach face:
the
uprush
and backwash induee high frequency, reversing sediment
transport similar
to
that
of
waves. Transport
is
complicated
Wave Skewness
Surf ate profile
Wave Asymmetry
Surface profile
Figure S14 Surface gravity wave profiles
tor
nonsymmetric waves. The
s^ewerfsurface profile denotes a lack
of
symmetry with respect
to the
horizontal axis; oscillatory velocities will increase under
ihe
crest
relative
to
those under the Irough, although they will last
for
a shorter
period
of
time. The asymmetrical surface profile denotes
a
lack
of
symmetry with respect
to the
vertical axis, resulting
in
marked
differences
in
accelerations between
Ihe
ieading and trailing faces
of
the wave.
by extremely small water depths, large Froude Numbers, large
pressure gradients near bore fronts
and the
infiltration
or
exfiltration
of
water from
the
fwach face water table
{for a
review
see
Butt
and
Russell. 2000). With propagation over
a
sedimentary
bed. a
distinct
wave
boundary layer develops
and
decays each half-wave eycle. passing from laminar (Re<IO"^)
to
turbulent (Re>IO'')
and
back
to
laminar
How
(Jensen
etaf..
1989).
This layer
may be
small (-10cm)
but the
velocity
gradients
and
thus
the
instantaneous stresses
may be
large.
Theoretically, gravity waves propagating over
a
permeable
bed
induce
an
oscillating flow even within the bed.
as a
result ofthe
spatial variation
in
pressures over
a
single wavelength. Flows
normal
to the bed
surface
are
also predicted; flow
is out
ofthe
ENT TRANSPORT BY WAVES 621
VELOCITY
Acceleration Deceleration
Stage
PROFILE
At
R
Tluming & Falloiii
Figure S15 Sequence of development nl the Iluid-granular bound.uy layer under stresses associated with flow acceleralion and deceleration
Lindor waves (modified after Conley and Inman, 1992). Note Ihat the straking-roil ing Iransition marks Ihe change from laminar to turhulent flow
and that the boundary layer exfjands upward dramatically during the deceleration phase.
bed and into the fluid iinjcetion) under the wave trough, and
out of the fltiid and into the bed (suction) under the crest,
resulting in a "ventilation" of the boundary layer. Kor detailed
reviews of wave and cotnbined-flow boundat"y layers see Grant
and Madsen (1986) and Sleath (1990).
Bed stress and modes of transport
The tolal bed stress under waves can be divided into two
components: (i) form drag: (ii) skin /riction. The latter is the
stress primarily responsible for transport (the e/Jectivestress)
and is defined using the quadratic stress law as:
<! = \t>,fi'l (Eq. 1)
where
r"",'.
= skin friction {sf) stress under waves (ir),
/>,
= fluid
density. / = a wave friction factor, and
ii,,.
= maximum
horizontal oscillatory velocity al the bed. The skin friction
is parameter
for
number) under waves is defined as:
(Eq.
2)
"crest
high concentration
where />, = particle density and D = particle diameter. Theore-
tically, the cffi'din' stresses imposed on the bed are periodic
and decay to zero at the flow reversal. Once the
threshiiUI
for
the initiation of moiion (sec Grain Thrcsliok/) is exceeded,
particles will roll back-and-forth in place. As flow accelerates,
a fliii(/'iiniini/ar boundary layer may develop, with three
distinct phases (l-'igure S15): (i)
stretikiiiii:
as flow starts, grains
roll and may separate by density into streaks or Uinthila-
slmpetl patterns under laminar conditions; (ii) niilini;: conliiui-
ing acceleration causes
slwetfUnv.
with sediment ejected upward
into the fluid sporadieally. In this phase also, stream-wise
periodicities in the sediment may form transition
ripples:
(iii)
plitniin}^:
as flow decelerates, the sheetfiow layer may lift ofl"
the bed. lbrcing sediment up through the boundary layer and
inio the free stream, where it ultimately diffuses and settles.
The change tVom streakini; to
roi/iiii^
marks the transition from
laminar to turbulent How and also the transition from heil-loail
to cotiibined bed and suspeiuled load.
Sediment transport modes are directly related to hedrou^h-
ness (Fieurc S12) and distinct resimes of flow occur similar to
deposition from
vortex A
mild
Figure S16 Lee vortex generation and ejection over a steep rippled
bed during passage of a successive wave crest and trcjugh
lu = mtiximum oscillatory velocity; A =
e\ecW(\
vortex). Note that the
larger vortex produced during the passage of the gravity
w.ive
crest
may be advected in a direction o[)posite to that of the wave advance.
The opposite would be true for the smaller vortex formed under the
wave trough, resulting in a net transport in ,i direction o|)posite to that
of the wave propagation direction (modified after Van
Rijn,
1993).
that of quasi-steady flow (Clifton
(-/(//..
1976: Davidson-Arnolt
and Greenwood. 1976: see Bars, Littoral; Siirfaee Forms). Two
distinct regimes occur under near-symmetric waves: (i) a ripple
regime (Shields mii}ihers<0.^): and (ii) a sheetJIow regime
(Ribberink and Al-Salem. 1995). where bedforms are erased
{Shields numhers>O.^-\.Q). Theoretically, sheet/low is sup-
ported by inter-granular forees and sediment-flow interaction.
Mass concentrations exceed
--20(1
kg m ' and the sheetflon-
layer increases in thickness with decreasing grain size. In
eontrasl, suspended lead is supported by Reynolds stresses, and
several mechanisms are responsible for the eiitrainment and
vertical mi.xing of sediment:
{'\]
flow
sepurcnion and vortcxi^en-
eration in the lee of bedform crests, and the release of these
vortiees from the bed as flows reverse (Figure SI6). The vortex
622
SEDIMFNT TRANSPORT BY WAVES
Figure S17 Time series ot near-bed horizontal velocity (u; z = 0,OJ m), suspended sediment lontentration (c; z = 0.04ml and net sediment
transport iu<t measured in the field under !iho^ling "groupy" waves over a bed with a mean grain diameter of 130}im. The dashed line in the
upper panel is the filtered long wave moiion; a similar filter is applied to the concentrations in the center panel. Note the "episodic" nature of
sediment
suspension,
ihe response to individual waves and wave
groups,
the good correlalion between suspension "episodes" and ihe long wave
motion,
and the small overall down-wave (onshorej net sediment transport.
carries scdimenl away from the bed and then, rapidly unwinds,
diffusing sediment outward, resulting in sediment settling,
creating a strongly epi.sodie pattern of suspension transport
(Figure S17). A distinct phase lug between the horizontal
currents and sediment concentration at any elevation is
created. Thae phase lags determine both the rate and direction
of the net suspension transport, and may result in a net
transport directed opposite to that of the oscillatory current
initiating the sediment motion. Bedform asymmetry will cause
significant ditTerences in the size and intensities of the vortices
associated with the two directions of motion; this inequality
may also lead to complex net tratisport patterns; (ii) turbulence
generated through interaction between the fluid and the
bed. and between the vortices and the surrounding fluid
induces a vertical diffusion of sediment; (iii) ventilation of the
wave boundary layer may also contribute to the suspended
load.
The relative importance of bed-load and suspendedhiad is an
open question lef Komar, 1978; Sternberg etal.. 1989) and
almosi certainly depends on the level of applied stress. In deep
water, sediment will move as bed-load
']Wi,t
above the threshold
and bedforms will develop: their inigralion may play a major
role in the net sediment transport (e.g.. Traykovski etal.. 1999)
even though suspension is enhanced by bedfunns. As depth
decreases and waves become asymmetric and break, suspen-
sion load usually dominates, at least lor the net transport (e.g.,
Aagaard etaL, 2002). Time-averaged sediment concentrations
do increase continuously frotn the top of ihe suspended load
layer to the immobile particles at the bed. and follow a
logarithmic law similar to that in quasi-steady flows (Osborne
and Greenwood, 1993). However, as the granular layer
accelerates and decelerates during each half-wave cycle,
sediment may move from bed-load into suspended
h>ad
and
back again. In practice, the two modes cannot be separated
even with fast response optical and acoustic sensors (White.
1998).
Transport rates
Net sediment transport
rales
are a product of the sediment mass
in motion and the velocity with which the mass moves,
integrated over space and time:
T --,.
{Eq. 3)
uhere / = immersed weight transport rate across a unit width.
/>,
= mass density of the solid, p, = mass density of the water.
^ = acceleration due to gravity. A'= sediment volume concen-
tration. (/= horizontal sediment velocity. : = elevation,
z.,
= elevation at the top of the mobile sediment, and
7-some measure of time (usually the wave period). Theory
indicates that the heal net .sediment transport is a function of
the balance between periurbations of the symmetric oscillatory
velocity Held and gravity acting on a slope (e.g., Bagnold, 1963:
TRANSPORT BY WAVES
b2i
Bowen, 1980; Bailard, 1981). It is rare that oscillatory
motion is perfectly symmetrical in nature, and rare to find a
perfectly horizontal, flat bed where gravity has no effeet. Thus,
there is always a secondary stress siiperitnposed on the
symmetrical oscillatory stress. This may be induced by
streaming in the wave boundary layer (mass transport), by
skewne.'is
and/or asymmetry of the waveforrn (Figure SI3), or
by some other process in shallow water (e.g.. iindertons. long-
shore eurrents. drift vcloeities associated with infragravity
wares, etc.). Differences exist between the onshore and off-
shore velocities as shoaling occurs: generally, larger velocities
under the crest will entrain larger volumes of sediment and
larger sizes than the smaller velocities under the trough. Since
transport rates have been related to the velocity raised to
sotne exponent (values from 3 to 7 have been suggested:
Dyer and Souisby. 1988: Sleath. 1995). small variations in the
on-olTshore velocities can be very important to sediment
transport.
A simple concept of waves "stirring" and eurrents
"tran.sporting" seditnent, has often been used to model
transport under waves (Bagnold. 1963). The shoreface is
viewed as a balance between the cross-shore cotnponent of
mass
tran.spori
associated with the incident waves, which is
landward and a seaward directed mean current (e.g.. undertow:
Roelvink and Stive, 1988). The oscillation itself may be
responsible for a significant part of the transport, at least for
the suspended load owing to the nature ot" the flu.x eoupling
between velocity and sediment concentration (Jaffe et al..
1985).
especially over bedforms. Outside the surf zone,
oscillatory transport associated with ihc incident wave band
is directed onshore, while an offshore transport is associated
with the group-forced bound long wave. Sediment transport at
low frequencies results from the increased sediment concentra-
lions due to the passage of groups of large waves and transport
direeted offshore by the currents associated with the trough of
the long wave (see Figure S13). Outside the surf zone, the nei
sediment tran.sport rate is generally positive (i.e., sediment
moves in the direction of wave propagation), except on very
steep beaches sloping upward in the direction of wave
propagation, where gravity may reverse the transport (Fredsoe
and Dcigaard. 1992). On plane. horizoiUal beds (he itnportant
mechanisms contributing to this positive flux are: wave skew-
ness and asymmetry, wave drift and streaming in the wave
boundary layer. In contrast, transport by second order waves,
such as the group hound long wave, is directed up-wave or
offshore. Greater sediment concentrations in the water column
are associated with the larger waves in a group which coincide
with the trough (upwave or offshore flow) of the long wave,
Shi and Larsen (1984) demonstrated this theoretically, but it
has also been well documented in the field (e.g.. Osborne and
Greenwood, 1992). Infragravity waves influence transport in
the surf zone, especially in the case of deposition in periodie
bar forms (e.g.. Bauer and Greenwood, 1990): it is assumed
that the transport is induced either by the second order drift
velocities associated with standing infragravity waves or by the
primary orbital motions.
However, in all instances the direction and rate of transport
is strongly influenced by the pha.se eoupling between the
horizontal velocities in the wave-current boundary layer and
the concentrations of sediment produced by the ejection of
separation vortices from bottom roughness. At times the phase
lag may eause the suspended toad to be transported up wave or
offshore (see Figure S17).
Cross-shore and longshore sediment transport
The wave orbiial motion and eross-shore mean eurrents within
and immediately above the
wave
boundary layer dominate the
eross-shore transport. Complex nonlinear interactions occur
between the oscillatory currents, the mean currents and the
srnall-scale bed roughness (e.g., vortex ripples; see Surface
Forms; Bars. Littoral). Assuming a no-slip condition between
the fluid and sediment grains, the time-averaged local net
sediment transport rate
<qy,>,i
is given by:
(Eq. 4)
where 7= wave period, / = time, z = elevation. /; = water
depth, u' = instantaneous velocity and e' = instantaneous mass
concentration. In terms of the steady, periodic and random
components, this may be written as:
d~
where <>=timc average, <i(',-,jX£'',_-,^> = transport by
time-averaged, quasi-steady, mean eurrents, and ^ and r are
the periodic and random components respectively. The latter is
generally considered to be small where the periodic component
is large. According to Jaffe etal. (1985) the term u',:_t,e',:j, is a
measure of the eorrelation between fluctuations in velocity and
sediment concentration (typically nonzero) and represents the
time-varying sediment flux due to waves. A convenient way of
computing this quaniily is through cross-spectral analysis
(Huntley and Hancs. 1987). as the cospcctruin (the real part of
the cross-spectriirn) yields the cross product of \elocity and
concentration as a function of frequency. The local net
oscillatory suspended sediment transport rate may thus be
computed from:
. 6)
where
C,,,(/)
= the cospectral variance al a given frequency
(/). A/'=the cospectral resolution and
f^the
cospeetral
frequency range. The cospectrum reveals the relative contribu-
tion of different frequeney bands to both the direction and rate
of suspended sediment transport and can be used to identity
transport due to wind waves, infragravity waves (such as the
group-bound long wave, edge waves, etc.) and far infragravity
waves such as shear waves.
Osborne and Greenwood (1992) found good agree-
ment between the quantities <^^>« and
«q^>,,,-\-
<i/s>,,s,).
While this procedure incorporates some of the
nonlinear effects in a shoaling, irregular wave field, others are
omitted. For example, part of the transport associated with
wave .skcwness
is iticluded in the cospectral estimates through
the coupling between larger onshore (or offshore) current
velocities and larger sediment concentrations. However, other
non-linear effects such as waveasymmetry or the higher order
moments of the velocity field (e.g.. Guza and Thornton. 1985)
are not accounted for.
624
SEDIMENT TRANSPORT BY WAVF.S
Time-averaged oscillatory currents
Stokes second order theory suggests a Lagrangian water
motion (wavedrift) occurs under shoaling waves, being largest
at the surface and decreasing exponentially with depth.
Longuet-Higgins (195.1) defined a Eulerian mass transport
velocity, with a vertical structure which varied with the wave
number (K=2n/L). This mass transport was directed down-
wave near the bed and at the surface, and directed up-wave at
intermediate depths. At the bed it was;
5 fnH
C
{Eq. 7)
where
u/,
^ velocity at the bed, overbar = time-averaged
mean, //==wave height. f=wave celerity, /(^water depth
and
h.-
= wave number
(2K/L).
It has been suggested that it
is this current that maintains the onshore sediment flux to
create the upwardly sloping shoreface profile in balance uith
gravity.
As waves break, secondary waves and quasi-.steady eurrents
are eommon. Transport becotnes complex and the net
transport vector will depend upon the interaction of several
mechanisms, resulting in both
.shore-parallel
as well as eross-
shore motion. Generally, onshore transport still results from
the .skewed and asymmetric breaking waves and bores.
although turbulence induced by the breaking wave at the
surface becomes itnportant. Offshore transport may be
indueed by: (i) undertow: (ii) rip currents: (iii) oscillatoty and
drift velocities associated with secondary waves, such as edge
wares or leaky waves or shear waves. Within the surf zone,
pressure gradient driven, time-averaged mean flows directed
seaward (undertows) are caused by water level setup at the
shoreline. IVare setup results from the onshore flux of
momentum due to waves (radiation stress) and is combined
with a
nui.ss flu.x
of water directed onshore by the waves plus
any applied windstress.
When waves break at an oblique angle to the shoreline, the
alongshore component of the radiation
stre.ss
creates a shore-
parallel current or longshore current. Pressure gradients due to
setup ditTcrentials along shore, as well as the shore-parallel
component of the wind
.stre.ss
can enhanee this forcing. The
volume of sediment transported alongshore by these quasi-
steady
longshore
currents is termed
littoral
drift and essentially
confined to the surfzone. Transport reaches its maximutn at a
cross-shore position between that of the maximutn current
velocity and the breakei' line and decreases rapidly offshore;
there is a subsidiary peak of transport in the swash zone. Rip
currents are discrete, narrow, high velocity jeis of offshore-
directed flow across the surfzone. They are often associated
with bathymetric forms such as cross-shore oriented depres-
sions (ripchannels) or breaks in a quasi-shore parallel bar. but
are found also on uniformly sloping beaches. Rip currents are
not generally steady state phenotnena, but are capable of
transporting large volumes of both coarse bed-toad, especially
in migrating tnegaripples (e.g.. Gruszczynski etal., 1993). and
suspended load.
Size fractionation
An itnportant characteristic of waves shoaling across the
shoreface are bed-sediments of similar density which increase
in size frotn deep to shallow water, in association with
increasing loeal slope (e.g.. Guiflen and Hoekstra, 1996). On
the upper shoreface grain-size distributions are more variable,
reflecting local variations in bathymetry and the rapid changes
in hydrodynamics associated with wave breaking and complex
neai'shore circulation, especially in the presence of bars (see
Bar, Littoral). Suggested mechanisms for size-sorting include:
(i) onshore coarse bed-load transport by mass transport within
the wave boundary layer, and an offshore movctnent of fines
by the return flow at mid-water depths; (ii) a near-bed velocity
asymmetry (amplitude and duration) induced by wave shoal-
ing, enabling all sediment sizes to move landward under the
wave crests, but only finer sediments move offshore under the
trough; (iii) setup-induced undertow and/or rip currents, which
continue offshore from the breaker zone: and (iv) a net
transport induced by group-forced long waves. Greenwood
and Xu (2001) review sizc-fractionation processes and show
that under shoaling waves over a near-horizontal, rippled bed,
a distinct horizontal fractionation of sediment by size occurs
by suspension transport alone.
Transport models
Models of sediment transport under waves depend upon the
spatial and temporal scales of integration used in equation 3:
(i) at the individual wave cycle-storm cycle scale (10"" lO"
years);
these models incorporate the fundamental physical
processes as far as possible; (ii) at the decadal time scale (lO' to
10^
years): these may be titne-integrated. physical-based
tnodels. but more usually are highly parameterized models or
morphological models (see List and Terwindt. 1995 for
examples): and (iii) at the geologic time scale (>10' years);
these tnodels are based either on simple geometric rules (e.g..
the "Bruun Rule") and sediment conservation (e.g.. Cowell
etal.. 1999) or diffusion rules (Kauffman etal... 1992); a more
detailed process response approach has been attempted
recently by Storms etal. (2002). Physically based models are
developed on the basis of the physics of individual wave cycles
and then integrated to Ihe scale of storms. However, it is clear
that integrating small-scale processes to larger spatial and
temporal scales is probletnatic, as it involves interactions
between the wave climate, sea level, availability of sediment
and the shoreface morphology (slope). In this complex
geomorphic system, non-linear processes are often driven by
stochastic boundary conditions.
Two approaches to wave-scale modeling have been
adopted; (i) the energetics approach of Bagnold
(1963;
1966;
Bailard, 1981); (ii) the applied stress approach (Granl and
Madsen. 1982), Souisby (1997) gives an extensive review of a
large range ol' models for suspended and bed-load transport
in both waves and combined flows and Schoonees and
Theron (1995) evaluated len different time-dependent sedi-
ment transport models and concluded that the energetics
approach was generally the best. However, this model does
not consider the threshold of motion, assumes an
instantaneous response of the bed and does not consider the
increased turbulence associated with wave breaking. One
major difficulty encountered in (he modeling of suspended
seditnent transport is the existence of phase lags between
the near-bed horizontal veloeity and sediment concentration
noted earlier.
Longshore transport under quasi-steady, wave-forced cur-
rents In the surfzone has been relatively well understood and
modeled for the last three decades (for a review see Komar,
1998).
The basic energy flux approach of R.A. Bagnold is