Topic: Title: “Adur Estuary Sediment Characterisation – Fate and Transport of particles”

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Home >Geology homework help >Topic: Title: “Adur Estuary Sediment Characterisation – Fate and Transport of particles”

AIMS This chapter discusses the physical properties of sediments that are found in coastal environments. Sediment properties affect the modes of sediment transport and deposition that take place along coasts. Fluid properties that drive sediment transport are described with respect to shear stress and the development of turbulent fl ow. The development and geometry of the most common bedform types (ripples and dunes) are directly related to fl uid dynamics.

5.1 Introduction

The term sediment refers to both organic and inorganic loose material that can be moved by physical agents including wind, waves, currents and under gravity. The sediments found in coastal environments can be either imported from outside the region (allochthonous) or locally produced (autochthonous). Allochthonous sediments are generally derived from the breakdown and transport of rocks into smaller particles, and along coasts commonly include minerals such as quartz, and clay minerals such as illite and montmorillonite. Autochthonous sediments include materials derived from the mechanical breakdown of rocky shorelines, but more commonly consist of broken-up shells of coastal organisms and/or the chemi- cal precipitates of dissolved minerals within coastal waters and include biogenic carbonate and silica. Globally, allochthonous sediments account for about 92 per cent of sediment in the modern coastal zone. Processes by which this sediment is delivered to the coast are, in decreasing order of importance, river, glacial and wind transport and volcanic eruptions.

Coastal landforms result from patterns of erosion and deposition that take place within larger coastal sediment systems. The wide range of spatial and temporal scales evident in coastal geomorphology (Figure 1.3) indicates that in order to fully understand large-scale morphodynamic processes it is also necessary to understand smaller-scale processes. The sequence of processes that control localised sediment movement are: (1) erosional entrainment of sediment into the flow via fluid-induced stresses and forces acting on the bed; (2) transport of sediment via momentum transfer from the fluid to the sedi-

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106 INTRODUCTION TO COASTAL PROCESSES & GEOMORPHOLOGY

ment; and (3) settling or deposition of sediment back on the bed via gravity (Figure 5.1). Depending on the speed at which sediment settles to the bed rela- tive to the speed at which flow conditions change, this sequence can re-initialise in one of two ways, by re-entrainment of sediment that is at rest on the bed, or by remobilisation of sediment that has not yet completed its return to the bed. The detailed mechanics of this cycle vary, depending on the intrinsic properties of the sediment and the fluid.

Both air and water are important agents for coastal sediment transport. While many concepts presented in this chapter apply to both fluid types, they differ markedly in their density and viscosity, and hence their ability to move sediment. This chapter focuses on hydrodynamics and sediment trans- port by water. Aerodynamics and sediment transport by wind are described in Chapter 9. Sediment erosion, transport and deposition processes by water or wind are the mechanisms by which coastal landforms are developed and destroyed. Therefore sediment–fluid interactions are critical to understanding how coastal landforms respond to forcing by tides, waves and climate.

5.2 Sediment properties

5.2.1 Grain size

Grain size is the sediment property most widely measured by coastal geomor- phologists since it is important in a wide range of coastal processes. The simplest measurements of a grain’s size are the lengths of the longest, intermediate and shortest axes which are termed the a, b and c axes, respectively. By conven- tion the b-axis and c-axis are measured at right-angles to the a-axis. The axial dimensions of large grains can be measured directly with callipers, but smaller grains (of sand size or below) are measured indirectly, usually by sieving or laser granulometry. Sieving effectively measures only the b-axis length. In laser gran- ulometric techniques, the sediment sample is mixed with a circulating water source so that the grains are in motion as they move past the laser device. As a result, this technique has equal likelihood of recording any axis of the sample,

Figure 5.1 Schematic representation of the fundamental processes that together constitute sediment dynamics.

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Transport

Remobilisation__

Entrainmenl Deposition

Deposition

SEDIMENTS, BOUNDARY LAYERS AND TRANSPORT 107

so the averaged b-axis is a good approximation. Goudie (1990) describes various methods for grain-size analysis.

Grains are often classified by their b-axis length using the Udden- Wentworth Scheme (Table 5.1). Individual pebbles, cobbles and boulders are often termed clasts. Note that the class boundaries in Table 5.1 are presented both on a millimetre-scale and a phi-scale ( -scale), which is discussed below. The conversion between grain diameter D on the -scale to diameter on the millimetre-scale is

D = 2– (5.1)

and vice versa

= –log2 D (5.2)

It should be evident from both Table 5.1 and these two equations that a change from one phi class to the next involves a doubling of the axis length.

While it is common to use a single number (often the arithmetic mean) to represent the grain size of a sediment sample, this number is typically obtained from a statistical analysis of the b-axis length of all or a subsample of the indi-

Table 5.1 The Udden-Wentworth Scheme of grain size classifi cation.

mm Class terms

256 –8 128 –7 64 –6 32 –5 16 –4 8 –3 4 –2 2 –1 1 0 0.5 1 0.25 2 0.125 3

0.062 4

0.031 5 0.016 6 0.008 7

0.004 8

Boulders

Cobbles

Pebbles

Granules

very coarse coarse

Sand medium fine

� very fine

coarse mediumSilt

fine

� very fine

Clay

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108 INTRODUCTION TO COASTAL PROCESSES & GEOMORPHOLOGY

vidual grains in that sample. The procedure typically begins with the grain size measurements from a sediment sample being presented as a frequency histo- gram and cumulative-frequency curve (Figure 5.2). Typically the histogram approximates a log-normal distribution, so when it is plotted on a logarith- mic scale such as the -scale, the histogram then appears normally distrib- uted (Figure 5.2a). It is for this reason that a log transformation of grain size measurements (performed either graphically by plotting on the -scale or by applying Equation 5.2) is performed prior to calculating grain size statistics. It is important to note that large positive -values indicate finer grain sizes and large negative -values indicate coarser grain sizes.

Before the widespread availability of computers, grain size statistics were calculated by graphical means. The most widely used formulae are (Folk and Ward, 1957)

Median = 50 (5.3)

16 + 50 + 84Mean = ————–––– 3 (5.4)

84 – 16 95 – 5Sorting = ——–— + ——–— 4 6.6 (5.5)

16 + 84 – 2 50 5 + 95 – 2 50Skewness = ————–––— + ———––––— 2( 84 – 16) 2( 95 – 5) (5.6)

95 + 5Kurtosis = ——–––——– 2.44 ( 75 – 25)

(5.7)

Figure 5.2 Two methods of presenting grain size data: (a) frequency histogram and (b) cumulative-frequency curve. The example shown is for a sample of medium sand that is very well sorted and fine-skewed (cf. Tables 5.1 and 5.2).

-0 .7

5 0 0.

75 1. 5

2. 25 3

3. 75

F re

qu en

cy (%

)

Grain size (phi)

100

-0 .7

5 0 0.

75 1. 5

2. 25 3

3. 75

C um

ul at

iv e

fr eq

ue nc

y (%

) Grain size (phi)

50th percentile

Mean = 1.75 Sorting = 0.34 Skewness = 0.30

D50 = 1.63 (a) (b)

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SEDIMENTS, BOUNDARY LAYERS AND TRANSPORT 109

where, for example, 50 is the 50th percentile of the grain-size distribution plotted on the -scale (Figure 5.2b). Note that different percentiles are also used in these formulae. This is because the 16th/84th and 5th/95th percentiles approximate to the ± 2 and ± 3 standard deviations from the mean value of a normal distribution, respectively. These formulae are still widely used when the method of sediment analysis does not yield a complete grain-size distribution, because the calculation does not rely too heavily on the fine and coarse tails of the distribution. Where a significant fraction of the material is very fine (> 4 ), however, a laser granulometer or settling tube method is more appropriate for the sample. When a complete grain-size distribution is available, the statistical descriptors are calculated using the more accurate method of moment meas- ures (Pettijohn et al., 1987)

� fiM i = l

Mean = x– = ———— 100

(5.8)

——————— n

� fi(M – x –)2

i = l Sorting = = ——————–� 100 (5.9)

n fi(M – x –)3

Skewness = �—————— i = l 100 3 (5.10)

n fi(M – x –)4

Kurtosis = �—————— i = l 100

(5.11)

where n is the number of data points, fi is percentage of grains (or percentage of total weight of grains) in each size interval and M is the midpoint of each size interval in phi units.

The mean (1st-moment) is the most common value used to represent sedi- ment grain size, although if the distribution is highly skewed then the median or mode may be better representative. The Udden-Wentworth Scheme is a classification scheme based on mean grain size (Table 5.1), but there are also classification schemes for the other moment measures (Table 5.2). The sort- ing (2nd-moment) is controlled in part by the range of grain sizes present at the sediment source, as well as processes operating during transport and deposition. For example, rapidly-deposited sediment is often poorly sorted, whereas frequently reworked sediment tends to be well sorted. The skew- ness (3rd-moment) is an indicator of the symmetry of the grain size distribu- tion (Figure 5.3a). For a normal distribution, the measure of skewness is zero.

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Negative skewness means that there are more coarse grains than expected in a log-normally distributed sample, and positive skewness means that there are more fine grains than expected. Skewness can arise from the mixing of sedi- ments from different sources, but can also be indicative of sorting during trans- port and deposition. For example, beach sands are typically negatively skewed, because continual agitation by waves can resuspend the finer particles, leaving an apparent excess of coarser sizes in the bed sediment. There is often a signifi- cant shell component in beach sands that also negatively skews the distribution. The kurtosis (4th-moment) is an indicator of the peaked shape of the distri- bution (Figure 5.3b). Distributions with a high peak and low range are termed leptokurtic, whereas distributions with a subdued peak but a wider range are termed platykurtic. These moment measures in combination describe the total- ity of the grain size distribution of a sediment sample, and can often be used to distinguish its genetic origin or depositional environment (Allen, 1985). Other physical properties including grain mineralogy, shape and density also affect how individual grains behave in response to flow conditions.

Table 5.2 Descriptors for sediment sorting, skewness and kurtosis as defi ned by Folk and Ward (1957).

Sorting ( -scale) Skewness ( -scale) Kurtosis ( -scale)

< 0.35 Very well sorted

> +0.30 Strongly fine- skewed

< 0.67 Very platykurtic

0.35 to 0.50

Well sorted +0.30 to +0.10

Fine- skewed

0.37 to 0.90

Platykurtic

0.50 to 0.71

Moderately well sorted

+0.10 to –0.10

Nearly- symmetrical

0.90 to 1.11

Mesokur tic

0.71 to 1.00

Moderately sorted

-0.10 to –0.30

Coarse- skewed

1.11 to 1.50

Leptokurtic

1.00 to 2.00

Poorly sorted

< –0.30 Strongly coarse- skewed

1.50 to 3.00

Very leptokurtic

> 2.00 Very poorly sorted

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SEDIMENTS, BOUNDARY LAYERS AND TRANSPORT 111

5.2.2 Grain mass and density

The mass of a grain affects its inertia with respect to the forces applied to it by a moving fluid. A grain’s mass is equal to the product of its volume and density. The volume is clearly related to the grain size. The volume of spherical grains, for example, increases as the grain-diameter cubed. The density of a grain is its mass per unit volume, and is largely determined by mineralogy. We can distin- guish between ‘light’ and ‘heavy’ minerals, with a somewhat arbitrary dividing line at a density of 2900 kg m-3, because a similar separation often occurs in the environment. For example, alternating layers of light and dark sand seen on some beaches are usually concentrations of light and heavy mineral types, respectively. Typical ‘light’ minerals common along coasts include quartz,

Figure 5.3 Illustration of the (a) skewness and (b) kurtosis of typical coarse and fi ne sediment samples.

Coarse Fine

Positive Negative

Platykurtic

Leptokurtic

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feldspars (plagioclase and orthoclase), clay minerals (montmorillonite, kaoli- nite, illite), and forms of calcium carbonate (aragonite, calcite, dolomite). Typi- cal ‘heavy’ minerals found along coasts include garnet, hornblende, magnetite, tourmaline, zircon and others, typically derived by weathering of igneous and metamorphic rocks.

5.2.3 Grain shape and roundness

Grain shape and roundness are sometimes confused with each other, but they are different measures, and have different interpretations. Grain shape is usually determined using the length ratios of the a, b and c axes. The propor- tionality between these axial lengths allows the grains to be located on a Zingg plot (Figure 5.4), which describes the grain’s outline shape. The ratio of the b and a axes indicates the degree of grain elongation, and the ratio of the c and b axes indicates the degree of grain flattening. Another measure of the grain flat- ness is the Corey Shape Factor CSF (Corey, 1949)

Figure 5.4 Zingg’s (1935) classification of grain shape.

0 c/b ratio

0.66 1

0.66

b/ a

ra tio

Blade

Oblate (tabular)

Equant

Prolate (roller)

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SEDIMENTS, BOUNDARY LAYERS AND TRANSPORT 113

cCSF = —–— �ab

(5.12)

where CSF = 0 represents a flat disc and CSF = 1 represents a perfect sphere. The shape of the original mineral crystal largely determines the shape of a single grain, although dissolution and abrasion can have a modifying effect. Rock type and structure are also important where grains are of pebble size or larger; for example, slate rocks preferentially form blades where the blade thickness (c axis) is determined by the rock’s original cleavage. The shape of a grain influ- ences both its entrainment and settling. Flat grains are more difficult to entrain and settle more slowly than spherical grains. Sorting based on shape takes place on gravel beaches, with prolate and equant clasts predominantly found at the base of the beach face and oblate and bladed clasts found towards the top.

Roundness refers to the three-dimensional shape of the grain that considers in particular the roundness of grain corners and protrusions. Roundness can be distinguished from shape by considering the differences between a cubic box and a sphere or ball-shaped grain of the same size as the box. Both these forms have the same a, b and c axis lengths and therefore the same equant shape, but a box has corners (low roundness) whereas the sphere does not (high round- ness) (Figure 5.5). Therefore grain roundness should be considered alongside grain shape. The degree of roundness usually indicates the susceptibility of the grain to chemical weathering and/or the degree of mechanical abrasion it has experienced. Angular grains often indicate resistant minerals, or a depositional site that is close to source. Well-rounded grains indicate either easily weathered minerals, energetic environments where there is active mechanical abrasion, or a depositional site that is far from source. Grain roundness also infl uences the

Figure 5.5 Powers’ (1953) classifi cation of grain roundness for grains displaying low sphericity and high sphericity (From Tucker, 1995.) (Copyright © 1995 Blackwell Publishers, reproduced with permission.)

hi gh

-s ph

er ic

ity lo

w -s

ph er

ic ity

very-angular angular sub-angular sub-rounded rounded well-rounded

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114 INTRODUCTION TO COASTAL PROCESSES & GEOMORPHOLOGY

sediment’s friction angle and packing. Relationships between grain size, shape and roundness are explored in Case Study 5.1.

A classic location for examining the size, shape and roundness relationships of clasts is Chesil Beach, southern England. Here, a linear gravel beach backed by a lagoon (The Fleet) extends for 28 km in a northwest–southeast direction. Beach height, beach width and sediment grain size all decrease progressively towards the northwest in the direction of dominant long- shore transport (Figure 5.6). Clast lithology is mainly quartzite and flint/ chert from the cliffs behind. Although these are relatively hard rock types, the high-energy waves mean that all lithologies are rapidly abraded in the direction of longshore drift. Carr et al. (1970) describe how the quartzite and flint/chert clasts change in size and shape at various points along the

Case Study 5.1 Grain size and shape variations on a gravel beach

b- ax

is le

ng th

(c m

)

0 0 1

13 10 7

Flint/chert clasts

2 1

13 10

7 4

2 a-axis length (cm)

3 4 5 6

b- ax

is le

ng th

(c m

)

2 4

0 0 1

Quartzite clasts

2 a-axis length (cm)

3 4 5 6 7 8 9 10

0 10 km

Bridport

Isle of Portland

Weymouth

The Fleet

Figure 5.6 Plots of variation in the mean value and range of the a, b, c axis dimensions of flint/chert and quartzite clasts from different sites along Chesil Beach, southern England. (Drawn from data from Carr et al., 1970).

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SEDIMENTS, BOUNDARY LAYERS AND TRANSPORT 115

5.2.4 Sediment packing and porosity

The arrangement of deposited grains is referred to as sediment packing. Given the wide range of natural grain sizes and shapes that exists, there are many possi- ble packing arrangements. The rate of grain deposition and the direction from which they are deposited can also influence sediment packing. Many sediment grains approximate the shape of a sphere, so it is useful to consider the two packing arrangements possible for uniform spheres (see Allen, 1985). The cubic arrange- ment is where spheres sit directly on top of each other, forming a box-shaped struc- ture with vertical sides. The rhombohedral arrangement is where spheres sit in the hollows formed at the intersection of four touching spheres in the layer below them. This packing arrangement forms a rhomboid structure with sloping sides.

Sediment packing is significant because it has implications for bulk-sediment density, sediment concentration and sediment porosity, which are key geotechni- cal properties of the sediment body as a whole. The bulk-sediment density is the total sediment mass divided by the total volume of the packed sediment (i.e. grains plus void spaces). The bulk-sediment density will always be less than the density of constituent grains due to the presence of voids. The sediment concentration is the total volume of the grains divided by the total volume of the packed sediment, and can be expressed as a fraction or a percentage. The sediment porosity is the

beach, plotted in Figure 5.6. There are several important points to note. First, both quartzite and flint/chert clasts get smaller in the direction of longshore drift. This is due to abrasion by waves and during longshore transport. Second, the size range of both lithologies becomes smaller in the direction of longshore drift, in other words the clasts become more uniform. Up-drift (site 1), the quartzite clasts are around 8 cm in diam- eter and the flint/chert around 4 cm, whereas down-drift (site 13) they are all around 1–2 cm. Third, the much larger diminution of quartzite clasts over the length of the beach compared with the flint/chert clasts shows that the quartzite is more easily eroded, with the erosional products (sand- sized grains) probably transported offshore. Throughout these longshore changes in clast size, it is notable that clast shape does not change at all: all quartzite clasts remain equant and all flint/chert clasts remain bladed.

Clast transport along Chesil Beach is very episodic, both spatially and temporally (Carr, 1971). A key factor controlling longshore drift speed, and therefore the rate of change of grain size, shape and sorting, is the angle of wave approach and wave frequency. This relationship is explored in more detail in Box 5.1. West (2009) maintains an interesting website on the geol- ogy of Chesil Beach, available at www.soton.ac.uk/~imw/chespeb.htm

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116 INTRODUCTION TO COASTAL PROCESSES & GEOMORPHOLOGY

volume of void spaces contained within the sediment, which is therefore equal to 100 per cent minus the sediment concentration. For very fine sediments the bulk- sediment density and concentration are controlling factors in their ability to be entrained by waves and currents (Section 5.5.1). Moreover, the sediment porosity is an important factor in beach–groundwater dynamics (see Box 8.2).

In practice, sediment populations have a mix of grain sizes, shapes and mineralogies. This means that their properties and behaviour do not always correspond to theoretical values. In addition, densely packed natural sediments tend towards the upper end of the theoretical ranges for both bulk density and concentration (lower end for porosity), because the smaller grains can occupy the voids between the larger grains. In addition, angular grains can lock together and help support more rounded grains. Indicative values of the sedi- ment concentration and porosity for natural sands are shown in the bottom half of Table 5.3. While the packing arrangement, degree of sediment sorting and grain shape all have a significant effect on sediment concentration and porosity, mean grain size generally does not.

Table 5.3 Measured packing concentrations and porosities of some theoretical and natural materials in water (Allen, 1985). Loose packing was produced by a simple dumping of sediment onto the bed, whereas dense packing was produced by subsequently agitating the sediment. Note that the biogenic Lithothamnium sands have an elongated grain shape.

Material Concentration (%) Porosity (%)

Theoretical Spheres (cubic packing) Spheres (rhombohedral packing)

52.0 74.0

48.0 26.0

Natural (mean diameter) Quartz sands (0.27 mm) Quartz sands (1.04 mm) Lithothamnium sands (3.2 mm)

Loose Dense 55.8 65.1 54.1 62.8 58.7 70.3

Loose Dense 44.2 34.9 45.9 37.2 41.3 58.7

5.2.5 Friction angle of sediment

If one considers a pile of sediment on which grains are accumulating one by one from above, the friction angle, also known as the angle of yield, is the maximum slope angle to which the sediment mass will develop prior to the initiation of grain avalanching down the slope (Figure 5.7a). After avalanching, the new slope angle is referred to as the angle of rest, which is always less than the friction angle. On a horizontal bed, the friction angle is effectively the angle through which a grain must be rolled in order for it to change its position on the bed, rather than falling back to its original position (Figure 5.7b). Experimental data show that the friction angle depends on grain size, shape, packing arrangement and surface texture of

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SEDIMENTS, BOUNDARY LAYERS AND TRANSPORT 117

the grains. Friction angles for natural sand grains are on average 34°–37°, but indi- vidual grains can be well outside this range (Pye, 1994). Smaller angles within this range are associated with loosely-packed spherical grains and larger angles with densely-packed angular grains. Friction angles for coarser sediments can reach 45° (Allen, 1985). It is not clear why there is a size-dependence in the experimental data, since the theoretical friction angles for two different sized spheres resting on spheres of similar size are equal. It may be that larger grains tend to have larger roughness elements, increasing the surface friction.

5.3 Fluid properties

Fluid properties can explain how wind, currents and waves can apply forces and impart momentum to sediments in order to initiate and maintain grain move-

Figure 5.7 – (a) Schematic diagram showing relationships between angle of yield (friction angle) and angle of rest (repose) for a heap of natural sand grains. (b) Schematic diagram showing the role of friction angle in the initiation of sediment motion. The angle enclosed by the vertical line and the line marked P is the angle the grain must pivot through before it can change its position on the bed. This angle is equivalent to the friction angle. For a perfect sphere the pivot angle is 30o if it is resting on a bed of similar sized grains. If the bed is composed of larger or smaller grains, the pivot angle is correspondingly larger and smaller, respectively. (From Pye, 1994.) (Copyright © 1994 Blackwell Publishers, reproduced with permission.)

(a)

=34 to 37oo

Angle of yield (friction)

Sand =32 to 34oo

Angle of rest (repose)

Sand

(b)

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0.20.20.2

0.20.2

0.20.20.2 0.20.2

0.2

118 INTRODUCTION TO COASTAL PROCESSES & GEOMORPHOLOGY

ment. The fluid properties that we must consider relate to forces (including fluid mass and acceleration), and momentum (influenced by mass and veloc- ity). If we consider any horizontal plane within a body of fluid in which sedi- ment may be present, the total force that is exerted on that plane per unit area is referred to as the total stress, and it consists of a normal stress and a shear stress component. Normal stress acts downwards, at right angles to the plane, and shear stress acts tangential (sideways) to the plane. Mass, acceleration, velocity and stress are therefore the fluid properties that are directly relevant to sediment dynamics. Most fluid and sediment transport models consider a body of fluid of unit volume, so in order to account for its mass we need only to consider fluid density.

5.3.1 Fluid density

Fluid density is the mass of the fluid per unit volume. The density of pure fresh water varies with temperature and pressure, although in shallow coastal waters pressure can generally be ignored. Water density generally increases with decreasing temperature, reaching a fresh water density maximum at 4°C. It is rare for coastal waters to be purely fresh but they are generally fresher than the open ocean because of river and storm-drain runoff. Coastal waters also generally contain dissolved and suspended materials that add proportionately more to the fluid mass than to its volume, and therefore increase its density over that of fresh water. For example, open ocean waters contain dissolved salts that have a mass of c. 35 mg for every 1 kg of water. This standard volume of dissolved solutes is usually denoted as 35 parts per thousand (ppt or ‰) or 35 PSU (practical salinity units). Coastal waters typically contain less dissolved salts due to their dilution by fresh water, so salinity is generally less, in the range 0 –35 ppt. Table 5.4 lists some indicative values for water density.

Table 5.4 Typical values for fl uid density and molecular viscosity. Note the three orders of magnitude difference in density, and two orders of magnitude difference in viscosity between air and water.

Fluid type Fluid density (kg m-3)

Fluid viscosity (N s m-2)

Air At 10oC

Water Fresh (10oC) Saline (35 ppt at 10oC)

1.3

1000 1027

1.80 x 10-5

1.06 x 10-3

1.40 x 10-3

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SEDIMENTS, BOUNDARY LAYERS AND TRANSPORT 119

5.3.2 Shear stress and viscosity

Most fluids offer some inherent resistance to deformation or flow, and are there- fore referred to as viscous fluids. This resistance to deformation is provided by molecular forces within the fluid and is termed the fluid’s molecular viscosity. The molecular viscosity of water can be measured by an experiment in which a thin body of water is deformed between two smooth plates (Figure 5.8). The bottom plate is kept stationary and the top plate is moved with a velocity u. The layer of fluid in immediate contact with each plate must have the same veloc- ity as the plate, but this fluid velocity decreases with distance away from the moving plate due to internal friction between the water molecules. The result is a linear velocity-gradient within the fluid where the fluid velocity decreases from a maximum within the layer in contact with the top plate, to zero within the layer in contact with the bottom plate (Figure 5.8). At any level in the fluid the normal stress (the vertically-acting force per unit area) is equal to the prod- uct of the fluid’s mass and the gravitational acceleration. The shear stress (the tangential-force per unit area) at any level in the fluid can be written as

du = μ ——

dz (5.13)

where is the molecular viscosity of the fluid, and u and z are defined in Figure 5.8. The molecular viscosity provides the link between the shear stress and the velocity-gradient at any level in the fluid. For a given fluid volume the molecular

Figure 5.8 Experimental design used to measure the molecular viscosity of a fluid. A thin film of the fluid is sheared between two plates by moving the top plate horizontally and keeping the bottom plate still. The velocity profile through the fluid is represented by arrows whose length is proportional to the fluid velocity. (Modified from Allen, 1994.)

Force

Stationary plate

z u

Moving plate

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120 INTRODUCTION TO COASTAL PROCESSES & GEOMORPHOLOGY

viscosity varies with density, thus the viscosity of water is considerably greater than that of air, and the viscosity of ocean water is greater than that of fresh water (Table 5.4).

5.3.3 Flow velocity, acceleration, laminar and turbulent flow

We must now define specific terms related to fluid motion. The flow velocity u is the distance travelled by a fluid parcel per unit of time. Flow acceleration can be either spatial or temporal. Spatial acceleration is the change in flow veloc- ity per unit of distance x (i.e. du/dx), and temporal acceleration is the change in flow velocity per unit of time (i.e. du/dt). If the temporal (spatial) accelera- tion is constant then it produces a uniform steady flow in which the velocity is constant through time (space). If there is an increase or decrease in velocity with time (distance) then a non-uniform or unsteady flow is produced. If there are rhythmic changes in flow velocity or direction over time, then this produces an oscillatory flow (Figure 5.9). For example, if tidal flows are observed over a short enough time period (c. 10–15 minutes), they can be approximately steady. If the period of observation is increased beyond one tidal period, then tidal flows are seen to be oscillatory. Patterns and calculations of sediment transport are also related to these flow properties (Section 5.5).

Figure 5.8 shows a situation in which fluid motion, shown by the directional arrows, takes place only in the direction of flow and without lateral or verti- cal movement. This is known as laminar flow (Figure 5.10a), characterised by very thin layers (laminae) of non-mixing moving fluids. In reality, fluid flow is rarely laminar because of frictional effects that retard flow velocity, but is more commonly turbulent. Turbulent flow has fluid movement mostly in one

Figure 5.9 Schematic diagram of current velocity records used to define various flow types. Flows that are constant with respect to time are steady, and constant with respect to distance are uniform. Flows that change velocity with respect to time are unsteady or oscillatory, whereas flows that change with respect to distance are non-uniform. (Modi- fied from Leeder, 1999.)C

ur re

nt ve

lo ci

Time (Distance)

Steady (uniform)

Oscillatory

Unsteady-deceleration (non-uniform)

Unsteady-acceleration (non-uniform)

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SEDIMENTS, BOUNDARY LAYERS AND TRANSPORT 121

Figure 5.10 (a) Laminar (top panel) and turbulent (bottom panel) pipe flow visualised by a horizontal streak of dye along the centreline of flow. (Photographs by N. Johanneson and C. Lowe. Modified from van Dyke, 1982.) (Copyright © 1982 Parabolic Press, reproduced with permission.) (b) Time series of instantaneous velocity vectors in the streamwise, orthogonal and vertical directions for a turbulent tidal current. Turbulence is indicated by fluctuations in current velocity about the mean. Time series of current velocity for a steady, laminar flow would show horizontal lines without fluctuations.

Mean current speed = 45 cm s–1

Streamwise

Orhoganol

(b)

(a) Laminar

Turbulent

-30

0 10 20 30 40 50 60

C ur

re nt

ve lo

ci ty

(c m

s )

–1

Time (s)

Vertical

Orthogonal

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122 INTRODUCTION TO COASTAL PROCESSES & GEOMORPHOLOGY

direction, but with apparently random deviations laterally and vertically through the fluid body. Figure 5.10b shows a time series of flow velocity measurements from a quasi-steady turbulent flow that has a mean velocity of about 0.45 m s-1 in the principal streamwise flow direction, but which also fluctuates by periods of flow acceleration and deceleration. Mean flow velocity in the orthogonal (left– right) and vertical directions are typically zero, but at any one instant in time there is also short-lived acceleration/deceleration of the fluid in these directions too. The result is that the fluid parcel swirls downstream in a chaotic way.

In 1883, the physicist Osborne Reynolds (1842–1912) used fl ow visualisation techniques to demonstrate the nature of laminar and turbulent fl ow. He derived a parameter, now termed the Reynolds number, which delimits the conditions associated with laminar and turbulent fl ow. The Reynolds number Re in its most general form is written as

ul Re = ——

μ (5.14)

where is fluid density, u is flow velocity, l is a length scale (the pipe diameter in the case of Reynolds’ experiments) and μ is the molecular viscosity of the fluid. For pipe flow, values of Re < 500 are associated with laminar flow and values of Re > 2000 with turbulent flow. Values of Re between these extremes are associ- ated with flow that is transitional between laminar and turbulent. The Reynolds number is essentially the ratio between inertial and viscous forces acting on the fluid flow. In laminar flow the viscous forces, represented by the molecular viscosity, are sufficient to resist substantial deformation of the fluid, and so the flow remains ordered. In turbulent flow the inertial forces, represented by the fluid velocity, cause flow acceleration and deformation, so the flow becomes disorganised.

In our discussion of fluid shear stress and viscosity we stated that the tangen- tial force per unit area on any plane within the flow is called the shear stress and is proportional to the velocity gradient (Equation 5.13). In laminar flow, the proportionality coefficient is the molecular viscosity which accounts for the microscale shearing within the fluid. In the case of turbulent flow there is also shear arising from the friction between adjacent fluid eddies with differ- ing momentum. This produces an apparent viscosity that is additional to the molecular viscosity. We call this apparent viscosity due to turbulence the eddy viscosity . If Equation 5.13 defines the shear stress within laminar flow, then the shear stress in turbulent flow is defined as

du = (μ + ) ——

dz (5.15)

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SEDIMENTS, BOUNDARY LAYERS AND TRANSPORT 123

Unlike the molecular viscosity, which is constant for a given fluid, the eddy viscosity varies with the flow conditions and increases with increased turbu- lence. Representative values for eddy viscosity are generally much larger than for molecular viscosity, and indicate the greater mixing and more rapid transfer of momentum that occurs within turbulent flows. This therefore makes turbu- lent flows more energetic, and potentially more erosive, than laminar flows. The nature of the bed over which the turbulent flow is moving is also important. This factor is now examined.

5.4 Benthic boundary layers

When a fluid is in motion against a boundary such as the sea bed (but also including a coastline or sea wall), friction arises between the two. This friction initially affects only the fluid motion in direct contact with the bed, but over time these effects reach to higher elevations in the flow. The region of fluid that is closest to the bed and influenced by frictional effects is termed the benthic boundary layer. Concepts of boundary layer flow are rooted in the work of the German hydrodynamicist Ludwig Prandtl (1875–1953).

Experiments have shown that the mean horizontal velocity within the bound- ary layer increases from zero at the bed (the no-slip condition) to a maximum value at the top of the boundary layer (Figure 5.8). A plot of mean horizon- tal flow velocity at increasing elevations above the bed defines a curve that is called the velocity profile (Figure 5.11). When a fluid begins to flow over the bed the boundary layer is initially laminar and its thickness grows slowly over time, therefore with distance travelled. In most coastal settings laminar flow is short-lived, and the boundary layer develops through the transitional regime to become a turbulent boundary layer. Momentum transfer from the boundary up through the flow is slow in laminar boundary layers because the fluid viscos- ity is small (only molecular viscosity), whereas it is rapid in turbulent bound- ary layers because the viscosity is large (molecular plus eddy viscosity). The enhanced rates of momentum exchange that take place by macroscopic mixing in turbulent boundary layers explain why they can grow in thickness at much faster rates, and why they have much steeper near-bed velocity gradients than laminar boundary layers (Figure 5.11).

In our earlier discussion we described shear stress as the tangential (side- ways) component of stress acting on any plane within the fl uid. With regard to sediment dynamics the bed shear stress is the tangential component of stress occurring on the fl uid plane that is in contact with the bed. The bed shear stress is also positively related to the velocity gradient (Equation 5.13). The bed shear stress beneath a turbulent boundary layer is greater than beneath a laminar

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124 INTRODUCTION TO COASTAL PROCESSES & GEOMORPHOLOGY

boundary layer, because of the steeper velocity gradient. This means that eroded products can be lifted from the bed into the fl ow through turbulent mixing. For a more detailed discussion of benthic boundary layers the interested reader is referred to Allen (1985) and Hsu (2004). Here we focus on methods for calculat- ing bed shear stress in steady and oscillatory boundary layer models.

5.4.1 Boundary layer model for steady flow

Channelised river or tidal flows in deltas and estuaries and wind-driven water currents on the inner shelf are all approximately steady flows when observed over short time periods. In these situations the benthic boundary layer grows with distance travelled by the flow, and if permitted to become fully established, it can occupy the entire water depth. So far we have discussed the benthic boundary layer as a single entity, but in detail it consists of three layers (Figure 5.12a): • The bed layer is generally 1–10 cm thick and consists of up to two

sublayers: the buffer sublayer and viscous sublayer. The presence or absence of the viscous sublayer depends on the hydraulic characteristics of the boundary (see below).

• The logarithmic layer is where the velocity increases logarithmically with height above the bed. This layer is generally 1–2 m thick.

• The outer layer occupies about 85 per cent of the total boundary layer, which may be tens of metres in suitable water depths.

The stated thickness of each layer is only indicative and depends on the steadiness of the flow. The more steady the flow, the thicker the layers will become.

The viscous sublayer, when present, is a thin layer of laminar fl ow (< 1 cm thick) that separates the turbulent buffer sublayer from the bed (Figure 5.12b). It only exists on hydraulically-smooth beds, which are generally composed of mud (silts and clays) or very fi ne sand. Hydraulically-rough beds with higher friction are generally composed of sediments coarser than fi ne sand. Coarser grains protrude up through the viscous sublayer and disrupt the laminar fl ow, thus enabling the turbulent fl ow in the buffer sublayer to impinge directly onto

Figure 5.11 Schematic illustration of a growing boundary layer and its transformation from laminar to turbulent fl ow. Notice that the velocity profi le in the turbulent boundary layer is steeper, thus larger current velocities impinge on the bed. (From Allen, 1985.) (Copyright © 1985 George Allen & Unwin, reproduced with permission.)

Laminar Transitional Turbulent

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SEDIMENTS, BOUNDARY LAYERS AND TRANSPORT 125

the bed. Hydraulically-smooth and -rough beds can be distinguished by a vari- ant of the Reynolds number, termed the boundary Reynolds number Re*

u * k�

Re * = ——–

μ (5.16)

where u* is the shear velocity and k’ is the skin friction roughness length. These two terms are defined below. Hydraulically-smooth boundaries occur when Re* < 5, hydraulically-rough boundaries occur when Re* > 70, and transitional conditions occur between values of 5 and 70.

The velocity profile in the logarithmic layer can be described by an equation known as the Law of the Wall

u *

z u = —– ln —( zo ) (5.17) where u* is the shear velocity, is von Karman’s constant (equal to 0.4), z is

Figure 5.12 (a) Schematic current velocity profile plotted on log-linear axes and showing different layers within the boundary layer. The free-stream layer extends from the top of the boundary layer to the water surface. The vertical dimensions shown are indicative of a tidal current some 10 m deep. (Modified from Wright, 1989.) (b) Schematic diagram showing the relationship between flow and bed conditions in the presence and absence of a viscous sublayer. (Modified from Allen, 1994.)

Free-stream layer

Outer layer

Logarithmic layer

Buffer layer

Viscous sublayer (smooth-turbulent flow)

B ed

la ye

u (arbitrary scale)

102

103

104

(Top of boundary layer)

z (c

m )

(a)

Turbulence

Viscous sublayer

Turbulence

]

Hydraulically-rough bed ( > 70)Re

Hydraulically-smooth bed ( < 5)Re

(b)

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126 INTRODUCTION TO COASTAL PROCESSES & GEOMORPHOLOGY

elevation above the bed and z0 is the hydraulic bed roughness length. The shear velocity is related to the bed shear stress b by

b = u 2 *

(5.18)

The Law of the Wall allows us to estimate the bed shear stress from the veloc- ity profile. For example, consider the flow velocity measurements made at five elevations above the bed, shown in Figure 5.13. Note that the velocity is plotted on the vertical axis and the height above the bed on the horizontal axis. The data must be presented in this way so as to avoid erroneous estimates of bed shear stress and bed roughness length. A best-fit line can be drawn through the data of the form y = ax + b, where y is the velocity, x is the natural logarithm of eleva- tion above the bed, a is the gradient of the line and b is the value of y when x is 0. The bed shear stress tb can be determined from the value of using

––– u

* 1 ba = ––– = –– ––– h�

hence

b = ( a) 2 (5.19)

The bed roughness length z0 can be determined from the value of b using

b = a ln zo hence zo = e–b/a (5.20)

For the example shown in Figure 5.13 the regression coefficients a and b are 0.23 and 1.33, which yields a bed shear stress of 8.69 N m-2 and a bed roughness length of 0.0031 m.

Figure 5.13 Illustration of method for estimating bed shear stress and bed roughness using the Law of the Wall. The velocity measurements at fi ve elevations above the bed are shown as diamonds and the line of best fi t through the data is shown as the solid line.

1.4 -

1.2 -

1 -

0.8 -

0.6 -

0.4 -

0.2 -

0 -

V el

oc ity

( m

s– 1 )

–4.6 –2.3 0 2.3 Natural log. of Elevation

0.01 0.1 1 10 Elevation (m)

- - - -

y = ax + b (a = 0.23; b = 1.33)

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SEDIMENTS, BOUNDARY LAYERS AND TRANSPORT 127

While the method described in Figure 5.13 can estimate bed shear stress and bed roughness, it requires several measurements of fl ow velocity in the logarithmic layer, which is not always possible. Field investigations of fl ow over different sea bed types have yielded an alternative method for estimating bed shear stress based on a single velocity measurement

b = Cdu 2 100

(5.21)

where Cd is the fluid-drag coefficient and u100 is the mean horizontal velocity at 100 cm above the bed. Some typical values of the drag coefficient and bed roughness lengths for different sea bed types are listed in Table 5.5.

Table 5.5 Indicative values for zo and Cd under steady fl ow over different bed types (Soulsby, 1983).

Bottom type zo (mm) Cd

Mud Mud/sand Silt/sand Sand (unrippled) Sand (rippled) Sand/shell Sand/gravel Mud/sand/gravel Gravel

0.20 0.70 0.05 0.40 6.00 0.30 0.30 0.30 3.00

0.0022 0.0030 0.0016 0.0026 0.0061 0.0024 0.0024 0.0024 0.0047

5.4.2 Boundary layer model for oscillatory flow

While steady and oscillatory flows within boundary layers are generally simi- lar, a different method for estimating bed shear stress under waves is required because of the temporal variation in the boundary layer structure under waves. Figure 5.14 shows the boundary layer velocity profile at several phases of the oscillatory wave cycle. During each cycle the horizontal flow velocity acceler- ates, then decelerates, crosses zero (i.e. changes direction), then accelerates and decelerates again. Consequently, a new boundary layer grows and decays twice during each wave cycle: once on the forward stroke of the wave and once on the backward stroke. Due to the short time periods involved (generally a few seconds), the boundary layer thickness does not reach much more than 10 cm for long-period swell waves, and is considerably smaller for short-period wind waves. The relatively thin boundary layer beneath oscillatory flows means that, for a given free-stream velocity and bed roughness, the bed shear stress in oscil- latory flows is always larger than beneath steady flow.

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128 INTRODUCTION TO COASTAL PROCESSES & GEOMORPHOLOGY

Worked Example 3

The method used to estimate bed shear stress under oscillatory flow first requires a measurement of the maximum horizontal flow velocity at the top of the wave boundary layer u0. If a measurement is not available then linear wave theory can be used to predict this flow velocity (see Equation 4.13). Next we need to estimate the bed friction factor under waves fw, which can be obtained from (Nielsen, 1992)

ks 0.2

fw = exp 5.5 ––– – 6.3[ ( do ) ] (5.22) where do is the wave orbital diameter, also obtainable from linear wave theory (see Equation 4.12) and ks is the Nikuradse roughness length, which is analo- gous to the bed roughness length described for steady flows (Section 5.4.1). An empirical recipe is available to estimate the Nikuradse roughness length, but it is not straightforward. There are at least two contributors to the total rough- ness. The first is the grain (skin friction) contribution k’ which is related to the roughness of individual grains making up the bed. The second is the bedform

Figure 5.14 Schematic diagram showing current velocity profiles in oscillatory flow. During the first 8 seconds the flow is in the direction of wave travel. The current speed increases and the boundary layer grows in height in the period 0 to 4 seconds. In the period 4 to 8 seconds the current speed decreases and the boundary layer diminishes. A similar pattern occurs during the period 8 to 16 seconds, however, the flow direction has reversed. The vertical dimensions shown are indicative of a swell wave with a height of 1 m and a period of 16 s. (Modified from Sleath, 1984.)

Water surface elevation

Velocity profiles

Time (s)

Velocity

H ei

gh ta

bo ve

be d

(c m

) E

le va

tio n

(m )

0 2 4 6 8 10 12 14 16

(arbitrary scale)

-0.5

+0.5

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SEDIMENTS, BOUNDARY LAYERS AND TRANSPORT 129

contribution k’’ which is related to the roughness of any bedforms present on the bed. In most cases the total Nikuradse roughness length ks will simply be the sum of the grain and bedform roughness contributions

ks = k + k (5.23)

Experimental data suggest the following formulae for estimating each rough- ness contribution (Nielsen, 1992)

k = D (5.24)

and 2

k = 8 ––– (5.25)

where D is mean grain diameter of the bed sediment, is bedform height and is bedform spacing. Now we can estimate the maximum flow velocity at the

top of the boundary layer and the wave friction factor, we can calculate the maximum bed shear stress under waves w using

1 w = — fwu

2 o2

(5.26)

Now that we have the tools to calculate the bed shear stress under either steady or oscillatory boundary layers, we can describe the sediment dynamics under these fluid flows.

5.5 Sediment dynamics

The dynamic behaviour of sediment in a moving fluid is strongly determined by sediment grain size. For grain sizes greater than 63 μm the grains are free to behave individually and single-grain properties (i.e. the size of individual grains) are most important. For grain sizes less than 63 μm the sediment grains are cohesive due to electrostatic forces. The dynamic behaviour of cohesive sediment depends less on single-grain properties and more on bulk-sediment properties (e.g. floc size and water content) (see Section 7.4). This distinction in sediment response to fluid flow is most apparent when considering sediment entrainment and deposition mechanisms.

5.5.1 Sediment entrainment and resuspension

A cohesionless grain at rest on a bed of similar grains experiences no accelera- tion, so all of the forces acting on the grain must be in equilibrium. The forces involved are lift, drag and weight forces. The lift force arises due to the slightly

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130 INTRODUCTION TO COASTAL PROCESSES & GEOMORPHOLOGY

faster flow across the top of the grain compared to the bottom of the grain, which results in a pressure differential. The drag force arises due to the friction between the fluid and the particle, and therefore acts horizontally in the direc- tion of mean flow. The weight force arises from the mass of the grain being acted upon by gravity. In order for the grain to move it must pivot over the adjacent grain through an angle equal to the friction angle. This movement can only occur if the lift and drag forces overcome the weight force. In principle, mathematical expressions can be derived to describe the nature of these forces, which can then be solved for the critical flow conditions required to initiate grain motion. In practice, however, the drag and particularly the lift forces are difficult to quantify due to the variability of flow conditions. For this reason the flow conditions necessary to initiate sediment movement are usually predicted from data from laboratory experiments.

Based on experimental data, Shields (1936) proposed that, when grains begin to move in a steady current, a relationship exists between bed shear stress and grain size. In order to make the data performed under a range of experimental conditions comparable, the critical bed shear stress required to initiate grain motion is transformed into a non-dimensional parameter known as the Shields parameter

c c = —————gD( s – )

(5.27)

The non-dimensional grain diameter D* that appears on the horizontal axis in Figure 5.15 is given by

2 g(s – 1) 1/3 D

* = D ––––––––[ μ2 ] (5.28)

For non-dimensional grain diameters less than a value of around 10 there is an inverse relationship between grain diameter and the critical Shields parameter, whereby the bed shear stress required to initiate sediment motion increases with decreasing grain diameter (Figure 5.15). This is due to a combination of factors. Silt and clay sized grains are readily compacted to the point that indi- vidual grains do not protrude through the viscous sublayer and are therefore not exposed to turbulent flow. This corresponds to hydraulically smooth conditions (Figure 5.12b). Silt and clay grains are also cohesive and therefore experience electrostatic attraction to the bed and to each other which increases in effect as the grains become smaller. This makes these grains difficult to entrain despite their small mass. For non-dimensional grain diameters greater than a value of 10, grain cohesion factors are less important and the grains are sufficiently large

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SEDIMENTS, BOUNDARY LAYERS AND TRANSPORT 131

to break up the viscous sublayer such that turbulent flow impinges directly on the bed. This corresponds to hydraulically-rough conditions (Figure 5.12b). For these larger grains, the weight force is of overriding importance, so there is only a weakly positive relationship between critical bed shear stress and grain size (Figure 5.15).

Based on the data shown in Figure 5.15 the following expression has been proposed to predict the critical Shields parameter (and bed shear stress) required to initiate grain motion

0.30 c = –––––––– + 0.055[1 – exp(–0.02D*)]1 + 1.2D

(5.29)

It is important to note that the data used to produce Figure 5.15 are largely based on experiments involving uniform sediment grains. The poorly sorted sediments commonly found in natural environments may lead to a situation where surface grains are resting on a bed of larger or smaller grains. In the former case the pivot angle through which the grain must move is larger than the case for well sorted sediment (Figure 5.7b). In the latter case the pivot angle is smaller. There is a corresponding effect on the critical bed shear stress required to initiate motion. Equation 5.29 therefore only applies to well-sorted sediments.

Figure 5.15 Modifi ed Shields diagram showing empirical data and best fi t function (Equation 5.29) for predicting the critical Shields parameter necessary to initiate sediment motion. Equation 5.27 can be used to determine the equivalent critical bed shear stress. For the case of waves this approach yields the critical maximum bed shear stress for the wave cycle. (Modifi ed from Soulsby, 1997.)

� c

D *

� c

0.1 1 10 100 1000

0.01

0.1

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HUMAN

132 INTRODUCTION TO COASTAL PROCESSES & GEOMORPHOLOGY

The modified Shields diagram (Figure 5.15) really only predicts the move- ment of single grains from a compacted bed. In a natural environment such as an estuary, however, silt and clay grains for example display a wide range of compactions. Recently-deposited sediment is poorly compacted and has a high porosity and water content (small bulk density), whereas ‘older’ sedi- ment is more highly compacted and has a low porosity and water content (high bulk density). These two situations produce markedly different responses to an erosive tidal current (see Section 7.4.2). Poorly compacted sediments are entrained more easily than compacted sediments, but no predictive equation is yet available.

5.5.2 Modes of sediment transport

Once in motion, the transport path (or mode of transport) that a sediment grain takes is largely determined by the mass of the grain and the current speed. Here we describe sediment transport by water currents. Transport by wind currents is discussed in Section 9.2. Sediment transport in water has two main modes (Figure 5.16): • Bedload is where grains are supported by either continuous (traction)

or intermittent contact (saltation) with the bed. In the case of traction, grains slide or roll and maintain contact with the bed at all times. This is a relatively slow transport mode and is typical when weak currents are transporting sands or strong currents are transporting pebbles and boulders. In the case of saltation the grains take short hops along the bed in the direction of fl ow. Saltation is typical when moderate currents are transporting sand or very strong and turbulent currents are transporting gravel and pebbles. Saltation is the most important mechanism of sand transport by wind (see Figure 9.7).

Figure 5.16 Schematic representation of sediment transport modes showing grain paths. Note that bedload includes both saltation and traction. (Modified from Allen, 1994.)

{ { ]Bed load

S us

pe nd

ed lo

Traction Rolling

Sliding

Saltation

Suspension

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SEDIMENTS, BOUNDARY LAYERS AND TRANSPORT 133

• Suspended load is where grains are supported by turbulence within the fl uid. The grain paths of suspended load are distinguishable from saltation due to their irregularity, which arises from the grains being buffeted by turbulent eddies within the fl uid. Suspension transport is typical when moderate currents are transporting silts or strong currents are transporting sands.

Francis (1973) performed flume experiments to investigate the relationship between sediment transport mode and current velocity. His results can be summarised by introducing the concept of transport stage

u *= —

ws (5.30)

which is the ratio of the shear velocity u* to grain settling velocity ws. The former is a surrogate for all of the forces driving sediment transport, and the latter is a surrogate for all of the forces driving sediment deposition. For low , only bedload is transported. There is no clear threshold value for between bedload and suspended load transport modes. As increases, the

Sandy sediments are transported across the entire range of transport modes and, in general, the speed at which sand is transported is closely related to the transport stage (Equation 5.30). This equation shows that sand transport as bedload takes place at relatively slow speeds, and sand transport as suspended load takes place at relatively fast speeds. Gravels and pebbles, however, move almost entirely as bedload and the transport speed of individual clasts depends not only on transport stage, but also on their relationship to the background sediment mass. Based on the ratio of the size of an individual clast to the modal size of the surrounding sediment mass, there are three transport possibilities (Figure 5.17): • If the ratio is less than half, then the grain movement will be impeded

as the clast will lose its surface position and be buried by surrounding sediment; in other words, smaller clasts will be trapped within the interstices of larger ones and will not participate in sediment transport.

• If the ratio is greater than about two, then the clast will have a propensity to move faster than the surrounding sediment mass. This occurs because the clast will project further into the boundary layer, and will therefore experience larger fl uid drag forces. The preferential

Box 5.1 Life in the slow lane: The transport speeds of large clasts

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134 INTRODUCTION TO COASTAL PROCESSES & GEOMORPHOLOGY

amount of material transported as bedload declines and the proportion trav- elling as suspended load rises. Moreover, the speed at which the grains are transported also increases. For high , almost all transport is as suspended load moving at a speed that is close to the speed of the fluid. Francis’ (1973) experi- mental data are based on sand- and granule-sized material. A discussion of the factors controlling the speed of coarser sediment transport can be found in Box 5.1.

transport of such larger clasts is termed overpassing. As a result these clasts can travel further and faster than those around them.

• Clasts that are much bigger than the modal clast size and so have very large ratios may sink into the sediment (due to their large mass), become immobile, and form a lag or armour deposit. The armour develops over time as more mobile sediment is transported away leaving the largest clasts behind.

The net result of differential transport speeds is the sediment will become better sorted over time through progressive differential transport of grains of different sizes. In the context of gravel beaches, these processes may lead to segregation of the gravel deposit and the development of graded shorelines.

Fine

Entrainment

Grain size Coarse

Rejection

Background Overpassing Armour Immobile (lag)

Fr eq

ue nc

Mobile Overpassing

Transport corridor

Sink

Source

Background

Figure 5.17 Schematic diagram demonstrating the importance of the clast size distribution to the relative speeds that individual clasts are transported. (Modifi ed from Orford et al., 1991.)

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SEDIMENTS, BOUNDARY LAYERS AND TRANSPORT 135

5.5.3 Models for calculating the transport rate

The sediment transport rate q can be defined as the mass of sediment trans- ported per unit of cross-sectional area of flow per unit time. Consider a unit width of fluid that extends from the stationary bed level to the height of the highest saltating grains. The bedload transport rate qb is the product of the mass of sediment in that cross-sectional area of fluid and the grain velocity, which is equivalent to

z = a

qb = �ug(z)C(z)dz 0

(5.31)

where ug is the grain velocity as a function of height above the bed z, C is the concentration of grains as a function of height above the bed, and a is height of the bedload layer. Even though Equation 5.31 is theoretically valid, the bedload layer is typically only a few centimetres high and so it is difficult to measure grain velocities and concentrations directly. An alternative approach is there- fore required. From our discussion of transport stage we know that sediment will be transported farther if the shear velocity (and bed shear stress) increase. It is therefore reasonable to expect that the sediment transport rate is propor- tional to the bed shear stress exerted by the flow, or more precisely the excess bed shear stress above the critical amount required to initiate grain motion. Field and laboratory experiments indicate that the bedload transport rate can be predicted using an equation of the form (e.g. Meyer-Peter and Muller, 1948)

qb = A( b – c) 1.5 (5.32)

where b is bed shear stress, c is the critical bed shear stress and A is a propor- tionality coefficient that depends on sediment properties.

The suspended load transport rate can also be determined using Equation 5.31, except that integration occurs over the entire water column height. Typi- cally the only data available to predict suspended sediment transport rate is mean current velocity.

An alternative approach to the sediment transport problem was proposed by Bagnold (1963, 1966), who equated a transporting current to a machine and the sediment transport to work done by that machine. In this approach, termed the energetics approach, the work done (sediment transport rate) is propor- tional to the power of the machine (transporting current). For bedload transport Bagnold proposed

buebqb = –––––––––––tan – tan (5.33)

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136 INTRODUCTION TO COASTAL PROCESSES & GEOMORPHOLOGY

where bu is current power, eb is the bedload efficiency factor, is the friction angle of the sediment and is the slope angle of the bed. Bedload efficiency is smaller than a value of one because the current is not 100 per cent efficient – some of the power is lost due to frictional dissipation as heat and by grain–grain interaction. For suspended load transport Bagnold proposed

ouesqs = –––––––––––(ws/u) – tan (5.34)

where es is the suspended load efficiency factor. The transport efficiency factors depend on both flow conditions and sediment properties. For equivalent sedi- ment properties, waves are generally more efficient at transporting sediment than steady currents.

In order to apply any of the sediment transport models to a practical situa- tion that involves steady currents or waves, we need measurements of the bed slope, grain size, grain settling velocity and current speed. The bed shear stress is estimated using the method described in Section 5.4. Situations that involve combined waves and steady currents, however, are more complex. Nonlinear interactions between the waves and current produce a combined boundary layer that is not well described by either of the boundary layer models that we have presented. Combined wave-current boundary layer models exist, but they are beyond our scope here. In combined flows waves are mostly responsible for suspending sediment, and steady currents for transporting sediment.

5.5.4 Sediment deposition of non-cohesive sediments (large grains)

Sediment deposition involves the settling of grains towards the bed, from either the bedload or suspended load. In the case of bedload, when the bed shear stress and fluid turbulence are insufficient to keep the sediment moving, the grains cease horizontal movement and rapidly come to rest. Bed properties can be important in trapping sediment grains or inhibiting their movement. In the case of suspended load, the grains must settle a longer distance vertically through the fluid before coming to rest. Sediment will begin to settle when the upward acting fluid forces are insufficient to overcome the downward acting weight force on the grain.

When a single suspended grain starts to be deposited, it will initially experi- ence a period of acceleration towards the bed. This acceleration is short-lived, however, because the grain’s acceleration decreases to zero when it reaches its terminal (maximum) fall velocity. The grain falls through the fluid due to the downward acting weight force, but when the grain is travelling at its terminal

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SEDIMENTS, BOUNDARY LAYERS AND TRANSPORT 137

fall velocity, the weight force is matched by a combined buoyancy and fluid- drag force. In the case of a spherical grain in a stationary body of water, the balance of forces can be expressed by

4 4 1 – r3 sg = – r

3 g + – Cd r 2 w2s3 3 2

downward-acting = upward-acting + upward-acting weight force buoyancy force fluid-drag force

(5.35)

The weight force is the product of the sphere’s volume, its density and gravita- tional acceleration. The buoyancy force is the product of the sphere’s volume, water density and gravitational acceleration. The fluid-drag force is the product of the drag coefficient, water density, the sphere’s surface area and its velocity squared. The half at the front of the right-hand side of Equation 5.35 is there because the drag force is only acting on the leading hemisphere of the grain as it falls through the water.

The balance of forces in Equation 5.35 can be rearranged to yield an expres- sion for the terminal fall velocity of the sphere

––––––––––— 8gr ( s – )ws = —– ——––�3Cd (5.36)

If a spherical grain with a known diameter and density is settling in a fluid with a known density, then Equation 5.36 can be used to calculate the fall velocity, provided the drag coefficient is also known. For Reynolds numbers that are appropriate for small grains, the flow around the grain as it falls through the fluid is laminar and the drag coefficient can be estimated using

24 Cd = –––RG

(5.37)

where RG is the grain’s Reynolds number in which the grain settling velocity and the grain diameter are consistent with those used in Equation 5.14. For larger Reynolds numbers, which may relate to larger settling velocities or larger grains in a less viscous fluid, prediction of the drag coefficient is less straightforward. Equations 5.36 and 5.37 constitute Stokes Law of settling and are strictly valid only for spheres settling in a fluid with a grain Reynolds number less than 20. In practice, for grains settling in water, this restricts Stokes Law to grain sizes less than 0.15 mm diameter (i.e. very fine sand, silt or clay). Stokes Law is not valid in air, due to the air’s low viscosity and the larger density difference between sediment grains and air.

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138 INTRODUCTION TO COASTAL PROCESSES & GEOMORPHOLOGY

In the case of grains larger than very fine sand, the flow separation around the grains as they settle through the fluid complicates the drag force and leads to unpredictable behaviour. Moreover, the grain shape and roundness can modify the drag and therefore the fall velocity, the effects being larger for larger grains. For this reason the fall velocity is usually measured directly (see Lewis and McConchie, 1994, for a description), or calculated empirically. Soulsby (1997) has analysed data for typical sands and proposes the following formula to estimate the grain settling velocity

μ ————————–––– ws = ––– [ � (10.362 + 1.049 D3*) – 10.36 ]D

(5.38)

where D* is the dimensionless grain size given by Equation 5.28. The settling velocity of sand grains increases directly with grain size and grain density, and increases inversely with fluid density and fluid viscosity.

5.5.5 Sediment deposition of cohesive sediments (small grains)

We now consider the settling of cohesive sediments comprising silt and clay with grain diameters smaller than 0.062 mm or 4 . Stokes Law is appropriate for calculating the settling velocity of fine particles if they remain dispersed, but in coastal waters they generally combine to form flocs, which are composites of several particles loosely held together. Several flocs may also combine to produce an aggregate (Figure 5.18). The process of flocculation arises from the electrical charges present on individual particles. The face of a clay platelet has a slight negative charge and the edge a slight positive charge. When two platelets come into close proximity, the face of one particle and the edge of the other are electrostatically attracted. The probability of the particles coming together in fresh water is low, however, because the negatively charged faces of the two particles, which have a much larger surface area than the edges, will tend to repel the particles from one another. In the case of seawater the probability of effective attraction increases. Seawater is a strong electrolyte which helps neutralise the negatively-charged faces, thus facilitating electro- static attraction. In order for flocculation in seawater to be significant, individual particles need to be brought very close together, by either Brownian motion within a concentrated suspension, differential settling of variously sized parti- cles, and/or by fluid turbulence. Eisma (1993) discusses flocculation in some detail.

Flocs have the combined mass of their component particles, and so the effect of fl occulation is to considerably increase their fall velocity. Since collisions of individual clay particles are a necessary precursor to fl occulation,

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the more concentrated the particle suspension the greater likelihood there is for fl occulation. There is often a clear relationship between settling velocity and suspension concentration, although this relationship is not consistent from one site to another or over time. For example, Uncles et al. (2010) discuss how fl oc concentration and settling patterns vary within the Tamar estuary, southwest England. Spatially, the zone of highest suspended sediment concentration migrates up and down the estuary with tidal state, and temporally most large fl ocs (200–500 μm) settle around high and low tides when water velocities are lowest (Section 7.4). Local factors such as water biochemistry and the pres- ence/absence of organic fi lms on grains and faecal pellets are also important in fl occulation. These factors are diffi cult to account for and have hindered attempts to develop a predictive equation for the settling velocity of cohesive sediment.

5.6 Bedforms

The generic term bedform refers to any upstanding morphological feature that is composed of unconsolidated sediments. The bedforms that we are inter- ested in are those that make quasi-regular patterns on the sea floor as a result of bedload and, to a lesser extent, suspended load transport. Flow conditions control bedform morphology, and in turn bedforms locally modify both flow conditions and sediment transport modes as the bedform evolves. Bedforms increase bed roughness and therefore friction, and so reduce wave and current energy.

5.6.1 Dimensions of bedforms and relation to flow conditions

The nomenclature, flow pattern and sediment transport over a typical asym- metric bedform is shown in Figure 5.19a. As the flow rises over the stoss (or up-flow) side of the bedform it accelerates due to compression of the flow over the bedform crest. Flow acceleration increases sediment transport up the stoss side and towards the brink point of the crest. As the flow reaches the crest it separates from the bed and recirculates in the form of an eddy in the lee of the bedform. The sediment transported to the crest is piled to a slope angle greater

Figure 5.18 Indicative dimensions of a clay particle, a fl oc and an aggregate. (Modifi ed from Eisma, 1993.)

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140 INTRODUCTION TO COASTAL PROCESSES & GEOMORPHOLOGY

than its yield angle and it consequently avalanches downslope. This cycle of sediment erosion from the stoss slope and deposition on the lee slope results in the bedform migrating in the direction of flow. The relationship between the bedform and the flow is closely coupled, with small changes in one causing a corresponding change in the other.

This example applies to unidirectional fl ow, which is typical of nearshore currents outside of the surf zone. Both waves and tides, however, are oscilla- tory fl ows with succeeding forward and backward components. The pattern for oscillatory fl ow over asymmetric wave ripples is shown in Figure 5.20. In this case, the relatively strong onshore stroke of the wave forms an eddy or vortex on the lee side of the wave ripple. Provided the onshore fl ow persists, this eddy remains trapped in the lee of the ripple, but when the fl ow reverses, the eddy is thrown upwards off the bottom and a small cloud of suspended sediment generated by the eddy is ejected into the water column above the ripple (see Box 5.2 for further discussion). The sediment cloud is moved seaward by the offshore stroke of the wave. Because the offshore stroke is relatively weak, no eddy forms in the lee of the ripple. The net suspended sediment transport

Figure 5.19 Schematic diagram showing nomenclature, fl ow and sediment transport pattern over bedforms in unidirectional fl ow: (a) ripple or dune (From Tucker, 1995.) (Copyright © 1995 Blackwell Publishers, reproduced with permission.); and (b) antidune. (From Reineck and Singh, 1980.) (Copyright © 1980 Springer-Verlag, reproduced with permission.)

(a)

(b)

Water surface Flow

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during one wave cycle is therefore seaward, which produces the asymmetry in the ripple shape. If the orbital velocity-magnitude is symmetrical, then the wave ripples are symmetrical and a vortex (with associated sediment cloud) occurs on both sides of the ripple crest each wave cycle (Figure 5.20). This means that, in practice, ripple geometry can be very variable.

Figure 5.20 Schematic diagram showing suspended sediment transport by onshore asymmetric wave motion over sharp- crested ripples. See text for explanation.

�� #

"�� / ��

��

*��/ �� /��

��/ ��

�J��

��

Recent technology has improved our ability to investigate near-bed sedi- ment suspensions beneath waves and currents. One such instrument avail- able is the acoustic back-scatter sensor (ABS), which consists of a sound (acoustic) source and a receiver or hydrophone. When the acoustic signal that is emitted by the ABS encounters a sediment suspension, the signal scatters in all directions. The ABS measures the intensity of the signal that is scattered back towards the instrument. The acoustic backscatter strength is then calibrated against known suspended sediment concentrations to yield a suspension concentration ‘map’ like the one shown in Figure 5.21.

Figure 5.21 shows a slice through the suspension concentration that is parallel to the direction of wave travel, so that the wave ripple is seen in cross-section. In this example, waves were travelling from left to right, and at this particular time the seaward stroke of the wave was decelerat- ing to zero (top panel). A vortex containing high concentrations (c. 1.5 g l-1) of suspended sediment is clearly seen on the lee of the ripple crest. Above the ripple crest is also seen another vortex of suspended sediment that was ejected during an earlier wave cycle. The suspension concentra- tion in the latter is smaller (c. 0.25 g l-1) due to some deposition since ejection. Although only a few experiments have produced this type of

Box 5.2 Acoustic-visualisation of sediment suspension over wave ripples

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Most bedforms fall into the categories of ripples, dunes or antidunes. These bedform types can be distinguished based on their dimensions (Table 5.6). Ripples and dunes can be further described as either two-dimensional or three-dimensional based on the plan view of their crest line. Straight-crested bedforms are two-dimensional, and sinuous to linguoid bedforms are three- dimensional (Figure 5.22). Antidunes are low relief bedforms that have heights similar to ripples, but lengths similar to dunes. They are restricted to rapid, shallow fl ows that have a wavy water surface that matches the shape of the bedform beneath. This type of fl ow is called supercritical fl ow. Antidunes are

acoustic imagery to date (e.g. Kuhnle and Wren, 2009), the ABS has already advanced our understanding of suspension dynamics. A variant of the ABS system is the acoustic Doppler current profi ler (ADCP) which uses different sound frequencies (the Doppler shift principle) and in three dimensions, thereby allowing for spatial analysis of variations in suspended sediment concentration in response to variations in fl ow velocity and direction (e.g. Kostaschuk et al., 2005). These instruments are helping to revolutionise our understanding of fi ne-sediment transport processes.

Figure 5.21 Top panel shows the time series of oscillatory flow velocity above the ripple crest. The flow velocity history that is responsible for the suspension pattern shown in the lower panel is marked in bold. The bottom panel shows a snap-shot in time of the suspension concentrations over the ripple crest. See text for explanation. (From Villard and Osborne, 2002.) (Copyright 2002 International Association of Sedi- mentologists, reproduced with permission.)

0 0.02 0.04 0.06 0.08 0.1 0.12 Distance (m) c (g l )–1

0.5

1.5

0.05

0.1

0.15

0.2

0.25

z (m

) u

(m s

) –1

Time (s) 80 85 90 95

-0.4 -0.2

0 0.2 0.4

lee vortex ripple crest

suspension from previously ejected vortex

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so named because they have the potential to migrate in the opposite direction to the fl ow; hence the steepest slope is on the upstream side of the dune and the gentlest slope on the downstream side. Migration is achieved by scour on the upstream side by the breaking surface wave as it slowly migrates upstream (Figure 5.19b). The distinction between fl ow regimes can be determined using the Froude Number Fr

u Fr = ––––——� gh (5.39)

Subcritical flow occurs when Fr < 1 and supercritical flow occurs when Fr > 1. Ripples and dunes are the products of subcritical flow and antidunes the prod- uct of supercritical flow.

Table 5.6 Dimensions of bedforms and associated fl ow conditions (after Reineck and Singh, 1980; Sleath, 1984; Nielsen, 1992).

Unidirectional flow Oscillatory flow

Ripples Dunes Antidunes Rolling- grain ripples

Vortex ripples

Length (spacing)

0.1–0.2 m 0.6–30 m 0.1–1 m 0.02–1 m 0.02–1 m

Height < 0.06 m 0.06– 1.5 m

0.01–0.1 m A few mm A few cm

Ripple index

8–15 >15 Not applicable

> 10 4–10

Typical flow velocity

Low Moderate High Low and high

Moderate

Typical flow depth

> a few cm A few dm A few cm to dm

Up to 1–2 the wave- length

Typical grain size

0.03–0.6 mm

> 0.3 mm All sand and gravel

All sand All sand

While ripples and dunes can be distinguished solely on their dimensions, they also show morphodynamic differences. The size of ripples and dunes increases with flow velocity (and therefore bed shear stress), but they are also influenced by flow depth, sediment type and sediment supply. This means that, on mobile sandy substrates and under energetic flow conditions, dunes can be very large.

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For example, in the tidally-dominated Irish Sea, sea-floor dunes are typically 20–35 m high in water depths of 50–100 m (van Landeghem et al., 2009). With very strong tidal currents in the South China Sea, bedforms are in the region of 2–10 m high in water depths of 20–50 m (Kubicki, 2008).

For a given flow velocity and water depth, larger grain sizes yield larger ripples and dunes. This is true for ripples in sediment up to 0.6 mm diameter. Ripples do not exist beyond this grain size because here the bed is hydraulically rough for all flow velocities capable of transporting sediment, precluding the existence of a viscous sublayer at the base of the boundary layer. The implica- tion is that ripples are scaled to the bed layer thickness, whereas dunes are scaled to the entire flow depth.

5.6.2 Bedform stability fields

Flow conditions change more quickly than bedforms do, particularly dunes where a substantial volume of sediment must be moved to change their size and shape. It is not uncommon, therefore, for bedforms to be in disequilibrium with flow conditions, especially under unsteady tidal flows. The flow condi- tions under each bedform type is stable and can be identified on a bedform stability diagram (Figure 5.23a). An analogous diagram exists for oscillatory flow over wave ripples (Figure 5.23b). Only two stability fields exist in this case: one for vortex ripples and one for rolling grain (low-steepness) ripples. The former are dominated by suspended sediment transport and the latter by bedload transport.

Figure 5.22 – (a) Schematic diagram showing plan view of crest lines for two-dimensional and three-dimensional bedforms. (From Tucker, 1995.) (Copyright © 1995 Blackwell Publishers, reproduced with permission.) (b) Photo of symmetric ripples found in the inter- tidal zone. (Photo: J. Knight.)

(a) (b)

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5.6.3 Interpreting the bedform record along coasts

The flow conditions experienced at a single location change over long time scales by sea-level rise and global climate change, and on shorter time scales by interannual climate variability such as El Niño, sediment supply and anthropo- genic factors. This may mean that bedforms observed in nearshore and coastal locations today were not formed under today’s flow regime, but instead reflect the flow regime of some time in the past. The bedforms are termed relict where they do not reflect today’s conditions, and moribund where they do reflect today’s conditions, but where the bedforms are no longer active. For example, very large, relict tidal sand banks are present at 110–180 m water depth on the continental shelf in the Celtic Sea (Belderson et al., 1986). Present-day tidal currents are not strong enough at this depth to generate these bedforms, and the sand banks were formed during the last deglaciation when sea level was 100 m lower than present. Later sea-level rise drowned the sand banks, removing them from the reach of tidal processes.

Bedforms around coastal margins are composed of sediments of different physical properties, composition, source and age. The bedforms themselves form spatial patterns that reflect an episodic and dynamic history of partial reworking, reshaping, erosion and deposition in response to variations in flow conditions and sediment supply throughout the Holocene (and sometimes earlier), and sea level. Understanding these processes and patterns is a consid- erable challenge for coastal scientists. The remainder of this book focuses on

Figure 5.23 Bedform stability diagram for (a) unidirectional fl ow and (b) oscillatory fl ow beneath wind waves and swell. (Unidirectional fl ow modifi ed from Southard and Boguchwal, 1990.) (Copyright © 1990 Society for Sedimentary Petrology, reproduced with permission.) (Oscillatory fl ow modifi ed from P.A. Allen, 1997 after J.R.L. Allen, 1985.) (Copyright © 1997 Blackwell Publishers, reproduced with permission.)

(a) (b)

M ax

. o rb

ita l v

el oc

ity (m

s )

-1

M ea

n flo

w ve

lo ci

ty (m

s )

-1

Sediment grain diameter (mm)Sediment grain diameter (mm)

plane

Vortex ripples Rolling-grain ripples

Sheet flow

No sediment movement

1.20.2 0.4 0.6

0.8

1.0

0 0

0.2

0.4

0.6

0.8 1.0

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146 INTRODUCTION TO COASTAL PROCESSES & GEOMORPHOLOGY

how flow conditions lead to different coastal processes and geomorphology in different coastal environments worldwide. Interpretation of coastal processes and geomorphology in these different environments can help reconstruct flow conditions and evaluate the forcing factors behind coastal change.

SUMMARY • Sediments are typically classifi ed according to their grain size distribution

and grain shape/roundness, which are closely related to the history of the sediment, including its source and transport processes.

• Fluid fl ow results in the formation of several sublayers within the fl ow that can be recognised by the shape of the fl uid’s velocity profi le. Turbulent fl ow is typical of most fl ows in the coastal environment and arises from bed shear stress produced by fl uid movement. Shear stress can be estimated for steady fl ow using the Law of the Wall, and can be estimated for oscillatory fl ow based on the bed roughness and fl ow velocity at the top of the wave boundary layer.

• The critical bed shear stress required to entrain a sediment grain from its bed can be predicted from a modifi ed Shields diagram. For fi ne sediments the critical bed shear stress increases with decreasing grain size, due to the fact that these sediments form hydraulically smooth beds and the grains are cohesive. For coarser sediments the critical bed shear stress increases with grain size, refl ecting the importance of the weight force. The situation is more complicated in the case of poorly sorted sediments or cohesive sediments with a high water content.

• Sediment is transported as either bedload or suspended load. In bedload the grains are supported by continuous (traction) or intermittent (saltation) contact with the bed. In suspended load the grains are supported by fl uid momentum related to turbulence.

• Settling of cohesionless (sandy) sediment is governed by grain properties, whereas settling of cohesive (fi ne) sediment is governed by the suspension concentration and development of fl ocs. When deposited, sediments can accumulate and organise into bedforms which are classifi ed according to their dimensions and fl ow regimes. Current and wave ripples are small- scale bedforms; dunes and antidunes are large-scale bedforms. Ripples and dunes are the products of subcritical fl ow whereas antidunes are the product of supercritical fl ow. Ripples occur in relation to hydraulically smooth beds and dunes to hydraulically rough beds.

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