Sand transport
Contents
General characteristics
Sand can be transported by gravity, wind, wave, tide and densitydriven currents (currentrelated transport), by the oscillatory water motion itself (waverelated transport) as caused by the deformation of short waves under the influence of decreasing water depth (wave asymmetry) or by a combination of currents and short waves. In rivers the gravityinduced flow generally is steady or quasisteady generating bed load and suspended load transport of particles in conditions with an alluvial river bed. A typical feature of sediment transport along an alluvial bed is the generation of bed forms from smallscale ripples (order 0.1 m) up to largescale dunes (order 100 m). The adjustment of largescale bed forms such as dunes and sand waves may lead to nonsteady effects (hysteresis effects) as it takes time for these largescale features to adjust to changed flow conditions (flood waves).
In the lower reaches of the river (estuary or tidal river) the influence of the tidal motion may introduce nonsteady effects with varying current velocities and water levels on a diurnal or semidiurnal time scale. Furthermore, densityinduced flow may be generated due to the interaction of fresh river water and saline sea water (salt wedge). In coastal waters, sediment transport processes are strongly affected by highfrequency waves introducing oscillatory motions acting on the particles. The highfrequency (short) waves generally act as sediment stirring agents; net sediment transport is due to the mean current.
Field experience over a long period of time in the coastal zone has led to the notion that storm waves cause sediments to move offshore while fairweather waves and swell return the sediments shorewards. During conditions with low nonbreaking waves, onshoredirected transport processes related to waveasymmetry and waveinduced streaming are dominant, usually resulting in accretion processes in the beach zone. During highenergy conditions with breaking waves (storm cycles), the beach and dune zone of the coast are attacked severely by the incoming waves, usually resulting in erosion processes.
Definitions
Sand transport is herein defined as the transport of particles with sizes in the range of 0.05 to 2 mm as found in the bed of rivers, estuaries and coastal waters. The two main modes of sand transport are bedload transport and suspended load transport.
Bedload transport is defined to consist of gliding, rolling and saltating particles in close contact with the bed and is dominated by flowinduced drag forces and by gravity forces acting on the particles. The suspended load transport is the irregular motion of the particles through the water column induced by turbulenceinduced drag forces on the particles. Detailed information is presented by Van Rijn ^{[1]}. The definition of bedload transport is not universally agreed upon. Sheet flow transport at high bedshear stresses may be considered as a type of bedload transport, but it may also be seen as suspended load transport. Some regard bedload transport as occurring in the region where concentrations are so high that graingrain interactions are important, and grains are not supported purely by fluid forces.
Suspended load transport can be determined by depthintegration of the product of sand concentration and fluid velocity from the top of the bedload layer to the water surface. Herein, the net (averaged over the wave period) total sediment transport in coastal waters is defined as the vectorial sum of net the bed load [math]q_b[/math] and net suspended load [math]q_s[/math] transport rates:
[math]q_{tot}=q_b+q_s[/math].
For practical reasons, suspended transport in coastal waters will be subdivided into currentrelated and waverelated transport components. Thus, the suspended sand transport is represented as the vectorial sum of the currentrelated ([math]q_{s,c}[/math] in current direction) and the waverelated ([math]q_{s,w}[/math] in wave direction) transport components, as follows:
[math]\vec q_s=\vec q_{s,c} + \vec q_{s,w}=\int \vec{v} cdz + \int \lt (\vec V  \vec v)(Cc)\gt dz , \qquad (1) [/math]
in which:
[math]q_{s,c}[/math] = timeaveraged currentrelated suspended sediment transport rate,
[math]q_{s,w}[/math] = timeaveraged waverelated suspended sediment transport rate (oscillating component),
[math]v[/math] = timeaveraged velocity, [math]V[/math] = instantaneous velocity,
[math]C[/math] = instantaneous concentration, [math]c[/math] = timeaveraged concentration,
[math]\lt f\gt [/math] represents averaging over time, [math] \int f dz[/math] represents the integral from the top of the bedload layer to the water surface.
The precise definition of the lower limit of integration is of essential importance for accurate determination of the suspended transport rates. Furthermore, the velocity and concentration profiles must be known.
The currentrelated suspended transport component [math]q_{s,c}[/math] is defined as the advective transport of sediment particles by the timeaveraged (mean) current velocities (longshore currents, rip currents, undertow currents); this component therefore represents the transport of sediment carried by the steady flow. In the case of waves superimposed on the current both the current velocities and the sediment concentrations will be affected by the wave motion. It is known that the wave motion reduces the current velocities near the bed while, in contrast, the nearbed concentrations are strongly increased due to the stirring action of the waves. These effects are included in the currentrelated transport. The waverelated suspended sediment transport [math]q_{s,w}[/math] is defined as the transport of sediment particles by the highfrequency and lowfrequency oscillating fluid components (crossshore orbital motion). The suspended transport vector can be combined with the bed load transport vector to obtain the total transport vector: [math]q_{tot}[/math].
In this note only the currentrelated bed load and suspended load transport components are considered. Usually, these components are dominant in river and tidal flows and also in wavedriven longshore flows.
Sand transport in steady river flow
Basic characteristics
The transport of bed material particles may take place either as bedload or as bedload plus suspended load, depending on the size of the bed material particles and the flow conditions. The suspended load may also contain some wash load (usually, clay and silt particles smaller than 0.05 mm), which is generally defined as the portion of suspended load which is governed by the upstream supply rate and not by the composition and properties of the bed material. The wash load is mainly determined by land surface erosion (rainfall, no vegetation) and not by channel bed erosion. Although in natural conditions there is no sharp division between bedload transport and suspended load transport, it is necessary to define a layer with bedload transport for mathematical representation.
When the value of the bedshear velocity just exceeds the critical value for initiation of motion, the particles will be rolling and sliding or both, in continuous contact with the bed. For increasing values of the bedshear velocity, the particles will be moving along the bed by more or less regular jumps, which is called saltation. When the value of the bedshear velocity exceeds the fall velocity of the particles, the sediment particles can be lifted to a level at which the upward turbulent forces will be comparable with or of higher order than the submerged weight of the particles and as result the particles may go in suspension. This corresponds to the socalled sheet flow regime where the bed shear velocity [math]u_*[/math] is greater than [math]\sqrt{ (1(1/s)) g d_{50}}[/math], where [math]s = \rho_s/\rho_w[/math] = relative density, [math]\rho_s[/math] = sediment density, [math]\rho_w[/math] = fluid density, [math]g[/math] = acceleration of gravity, [math]d_{50} [/math] = median sediment diameter. In terms of the dimensionless bedshear stress (the Shields parameter) [math]\theta = \tau_b / [(\rho_s\rho_w) g d_{50}][/math], this is equivalent to the condition [math]\theta \gt 1/s \approx 0.4[/math] ^{[2]}.
The sediment transport in a steady uniform current over an alluvial bed is assumed to be equal to the transport capacity defined as the quantity of sediment that can be carried by the flow without net erosion or deposition, given sufficient availability of bed material (no armour layer). In general, a river flood wave is a relatively slow process with a time scale of a few days. Consequently, the sediment transport process in river flow can be represented as a quasisteady process. Therefore, the available bedload transport formulas and suspended load transport formulas can be applied for transport rate predictions. Flume and field data show that the sand transport rate is most strongly related to the depthaveraged velocity. The power of velocity is approximately 3 – 4 in the sheet flow regime.
The bed material in natural conditions consists of nonuniform sediment particles. The effect of the nonuniformity of the sediments will result in selective transport processes (grain sorting). Grain sorting is related to the selective movement of sediment particles in a mixture near incipient motion at low bedshear stresses and during generalized transport at higher shear stresses. Sorting effects can only be represented by taking the fullsize composition of the bed material, which may vary horizontally and vertically, into account.
Initiation of motion
Particle movement will occur when the instantaneous fluid force on a particle is just larger than the instantaneous resisting force related to the submerged particle weight and the friction coefficient. The degree of exposure of a grain with respect to surrounding grains (hiding of smaller particles resting or moving between the larger particles) obviously is an important parameter determining the forces at initiation of motion. Cohesive forces are important when the bed consists of appreciable amounts of clay and silt particles. The driving forces are strongly related to the local nearbed velocities. In turbulent flow conditions the velocities are fluctuating in space and time. This makes together with the randomness of both particle size, shape and position that initiation of motion is not merely a deterministic phenomenon but a stochastic process as well. The fluid forces acting on a sediment particle resting on a horizontal bed consist of skin friction forces and pressure forces. The skin friction force acts on the surface of the particles by viscous shear. The pressure force consisting of a drag and a lift force is generated by pressure differences along the surface of the particle. These forces per unit bed surface area can be reformulated in a timeaveraged bedshear stress.
Initiation of motion in steady flow is defined to occur when the dimensionless bedshear stress [math]\theta[/math] is larger than a threshold value [math]\theta_{cr}[/math]. Thus, [math]\theta \gt \theta_{cr}[/math] , with:
[math]\theta = \tau_b / [(\rho_s\rho_w) g d_{50}][/math] = Shields parameter, [math]d_{50}[/math] = medium grain size, [math]\tau_b [/math] = bedshear stress,
[math]\rho_s[/math] = sediment density, [math]\rho_w[/math] = fluid density, [math]d_{50} [/math] = median sediment diameter.
The [math]\theta_{cr}[/math] factor depends on the hydraulic conditions near the bed, the particle shape and the particle position relative to the other particles. The hydraulic conditions near the bed can be expressed by the Reynolds number
[math]Re_* = u_* d/\nu[/math], where
[math]u_*[/math] = bed shear velocity , [math]d[/math] = grain diameter, [math]\nu [/math] = kinematic viscosity [math](\sim 10^{6}).[/math] Thus: [math]\theta_{cr}= F(Re_*)[/math] .
Many experiments have been performed to determine the [math]\theta_{cr}[/math] values as a function of [math]Re_*[/math]. The experimental results of Shields ^{[3]} related to a flat bed surface are most widely used to represent the critical conditions for initiation of motion (see Figure 1). The curve represents a critical stage at which only a minor part (say 1% to 10%) of the bed surface is moving.
Initiation of motion in combined steady and oscillatory flow (wave motion) can also be expressed in terms of the Shields parameter or as function of a dimensionless particle size [math]D_*[/math]^{[1]}.
The [math]D_*[/math]parameter is defined as:
[math]D_*= [u_*d/(\nu \sqrt{\theta}]^{2/3} = d [(s1) g / \nu^2]^{1/3}[/math] , with:
[math]g[/math] = acceleration of gravity, [math]s = \rho_s/\rho_w[/math] = relative density.
A simple expression for initiation of motion (movement of particles along the bed) is given by Soulsby^{[4]}:
[math] \theta_{cr,motion} = 0.3/(1+1.2 D_*) + 0.055 [1  \exp(0.02 D_*)] . \qquad (2) [/math] .
A simple expression for initiation of suspension (particles moving in suspension) is given by:
[math] \theta_{cr,susp} = 0.3/(1+D_*) + 0.1 [1  \exp(0.05 D_*)] . \qquad (3) [/math] .
Equations (2) and (3) are shown in Figure 2.
Both equations can be used to compute the critical depthaveraged velocity for initiation of motion and suspension, as follows:
[math]U_{cr,motion} = 5.75 [\log(2h/d_{50})] [\theta_{cr,motion} (s1) g d_{50}]^{0.5} , \qquad (4) [/math]
[math]U_{cr,susp} = 5.75 [\log(2h/d_{50})] [\theta_{cr, susp } (s1) g d_{50}]^{0.5} , \qquad (5) [/math]
with: [math]U[/math] = depthaveraged velocity, [math]C = 5.75 \sqrt{g} \log(4h/d_{90})[/math] = Chézy coefficient,
[math]h[/math] = water depth, [math]d_{90} \approx 2d_{50}[/math] = 90% particle size.
Simple approximation formulas (10% accurate) are:
[math]U_{cr,motion} = 0.19 (d_{50})^{0.1} \log(2h/d_{50})[/math] for [math]0.0001\lt d_{50} \lt 0.0005 \; m ; [/math]
[math]U_{cr,motion} = 8.5 (d_{50})^{0.6} \log(2h/d_{50})[/math] for [math]0.0005\lt d_{50} \lt 0.002 \; m ; \qquad (6) [/math]
[math]U_{cr,susp} = 2.8 (h/d_{50})^{0.1} [(s1) g d_{50}]^{0.5}[/math] for [math]0.0001\lt d_{50} \lt 0.002 \; m . \qquad (7) [/math]
Figure 3 shows the critical depthaveraged velocities at initiation of motion and suspension for sediment with [math]d_{50}[/math] between 0.1 and 2 mm based on Equations (4) and (5).
Bed load transport
The transport of particles by rolling, sliding and saltating is known as bedload transport. For example, Bagnold^{[5]} defines the bedload transport as that in which the successive contacts of the particles with the bed are strictly limited by the effect of gravity, while the suspendedload transport is defined as that in which the excess weight of the particles is supported by random successions of upward impulses imported by turbulent eddies. Einstein^{[6]}, however, has a somewhat different approach. He defines the bedload transport as the transport of sediment particles in a thin layer of 2 particle diameters thick just above the bed by sliding, rolling and sometimes by making jumps with a longitudinal distance of a few particle diameters. The bed layer is considered as a layer in which the mixing due to turbulence is so small that it cannot influence the sediment particles, and therefore suspension of particles is impossible in the bedload layer. Further, Einstein assumes that the average distance travelled by any bedload particle (as a series of successive movements) is a constant distance of 100 particle diameters, independent of the flow condition, the transport rate and the bed composition. In the view of Einstein, the saltating particles belong to the suspension mode of transport, because the jump lengths of saltating particles are considerably larger than a few grain diameters.
The first reliable empirical bed load transport formula was presented by MeyerPeter and Mueller^{[7]}. They performed flume experiments with uniform particles and with particle mixtures. Based on data analysis, the following simple formula for bedload transport [math]q_b[/math] was obtained, which is frequently used:
[math]\Large\frac{q_b}{\sqrt{g (s1) d_{50}^3}}\normalsize = m \, (\theta  \theta_{cr,motion})^n , [/math]
where [math]s=\rho_s / \rho_w[/math] and for [math]\theta_{cr,motion}[/math] Eq. (2) can be used. For the dimensionless parameters [math]m[/math] and [math]n[/math] they advised the values [math]m=8[/math] and [math]n=1.5[/math]. In other studies a best fit to experimental data was found with values in the range [math]4\lt m\lt 12[/math] and [math]1.4\lt n\lt 1.65[/math].
Einstein^{[6]} introduced statistical methods to represent the turbulent behaviour of the flow. Einstein gave a detailed but complicated statistical description of the particle motion in which the exchange probability of a particle is related to the hydrodynamic lift force and particle weight. Einstein proposed [math]d_{35}[/math] as the effective diameter for particle mixtures and [math]d_{65}[/math] as the effective diameter for grain roughness.
Bagnold^{[8]} introduced an energy concept and related the sediment transport rate to the work done by the fluid.
Engelund and Hansen^{[9]} presented a simple and reliable formula for the total load transport [math]q_{total}[/math] in rivers:
[math]\Large\frac{q_{total}}{\sqrt{g (s1) d_{50}^3}}\normalsize = m \, \theta^{\, n} , [/math]
with recommended values [math]n=2.5[/math] and [math]m=0.04 / c_D[/math]. The drag coefficient , which is defined as [math]c_D=\tau_b/(\rho u^2)[/math], is related to the Chezy coefficient [math]C \approx 5.75 \sqrt{g} \, \log_{10}(4 h / d_{90})[/math] by [math]c_D=g/C^2[/math]. This formula includes suspended load transport, but ignores phase lag effects of sediment suspension and settling. These phase lag effects can be important in nonsteady flows. Therefore, in nonsteady flows the EngelundHansen formula is not as accurate as total load formulas in which suspended load is computed with the formulas presented in the next sections.
Van Rijn^{[10]} solved the equations of motions of an individual bedload particle and computed the saltation characteristics and the particle velocity as a function of the flow conditions and the particle diameter for plane bed conditions. The results of sensitivity computations show that the bed load transport is only weakly affected by particle diameter. A 25%variation of the particle diameter ([math]d_{50} = 0.8 \pm 0.2 \; mm[/math]) results in a 10%variation of the transport rate.
Bed load transport [math]q_b[/math] can be determined from the measured bed form dimensions and the measured bed form migration velocity using echo sounding results. The bed load transport is by definition: mass [math]m_s[/math] per unit area of bed times migration velocity [math]c[/math]: [math] q_b= c m_s [/math]. The mass [math]M_s[/math] of a triangular bed form with length [math]L[/math], height [math]\Delta H[/math], sediment density [math]\rho_s (\approx 2650 kg/m^3) [/math] and porosity [math]p (\approx 0.4)[/math] is [math]M_s = 0.5 \rho_s (1p) L \Delta H [/math]. The mass per unit area is the mass [math]M_s[/math] divided by the length [math]L[/math] giving [math]m_s = 0.5 \rho_s (1p) \Delta H[/math]. Bed load transport is thus given by:
[math]q_b = 0.5 \rho_s (1p)c \Delta H [/math].
As bed forms are not fully triangular, a more general expression is:
[math]q_b= \alpha \rho_s (1p) c \Delta H [/math], with [math]\alpha = 0.5  0.7[/math].
Suspended load transport
When the value of the bedshear velocity exceeds the particle fall velocity, the particles can be lifted to a level at which the upward turbulent forces will be comparable to or higher than the submerged particle weight resulting in random particle trajectories due to turbulent velocity fluctuations. The particle velocity in longitudinal direction is almost equal to the fluid velocity. Usually, the behaviour of the suspended sediment particles is described in terms of the sediment concentration, which is the solid volume per unit fluid volume or the solid mass per unit fluid volume.
Observations show that the suspended sediment concentrations decrease with distance up from the bed. The rate of decrease depends on the ratio of the fall velocity [math]w_s[/math] and the bedshear velocity [math]u_*[/math]. The depthintegrated suspendedload transport [math]q_{s,c}[/math] is herein defined as the integration of the product of velocity [math]u[/math] and concentration [math]c[/math] from the edge of the bedload layer, [math]z=a[/math], to the water surface, [math]z=h[/math]. This definition requires the determination of the velocity profile, concentration profile and a known concentration [math]c_a[/math] close to the bed ([math]z=a[/math]), see Figure 4. These latter parameters are referred to as the reference concentration and the reference level.
Sometimes, the suspended load transport is given as a mean volumetric concentration defined as the ratio of the volumetric suspended load transport (= sediment discharge) and the flow discharge [math]q[/math]: [math]c_{mean}=q_{s,c}/q[/math]. The mean concentration [math]c_{mean}[/math] is approximately equal to the depthaveraged concentration for fine sediments (mud).
The concentration can be expressed as a weight concentration [math]c_g[/math] in kg/m3 or as a dimensionless volume concentration [math]c_v[/math]. Sometimes the volume concentration is expressed as a volume percentage after multiplying with 100%.
Some rivers carry very high concentrations of fine sediments (particles < 0.05 mm), usually referred to as the wash load. Experience shows that the presence of fines enhances the suspended sand transport rate because the fluid viscosity and density are increased by the fine sediments. As a result the fall velocity of the suspended sand particles will be reduced with respect to that in clear water and hence the suspended sand transport capacity of the flow will increase.
The sediment concentration distribution over the water depth can be described as a diffusion process, which yields for steady, uniform flow ^{[1]}:
[math]c w_s + \varepsilon_s dc/dz = 0 , \qquad (8) [/math]
with [math]c[/math] = sand concentration, [math]w_s[/math] = particle fall velocity, [math]\varepsilon_s[/math] = sediment diffusivity coefficient.
Using a parabolic sediment diffusivity coefficient over the depth, the concentration profile can be expressed by the Rouse profile:
[math]c(z)/c_a = [((hz)/z) (a/(ha))]^{w_s/\kappa u_*} , \qquad (9) [/math]
with: [math]h[/math] = water depth, [math]a[/math] = reference level, [math]z[/math] = height above bed, [math]c_a[/math] = reference concentration,
[math]w_s[/math] = particle fall velocity, [math]u_*[/math]= bedshear velocity, [math]\kappa[/math] = Von Karmann coefficient (=0.4).
The relative importance of the suspended load transport is determined by the Rouse number [math]Z=w_s / \kappa u_*[/math]. The following values can be used:
[math]Z=5[/math]: suspended sediment in nearbed layer ([math]z\lt 0.1h[/math]),
[math]Z=2[/math]: suspended sediment up to mid of water depth ([math]z\lt 0.5h[/math]),
[math]Z=1[/math]: suspended sediment up to water surface ([math]z\lt h[/math]),
[math]Z=0.1[/math]: suspended sediment almost uniformly distributed over water depth.
Sand transport in nonsteady (tidal) flow
In nonsteady flow the actual sediment transport rate may be smaller (underload) or larger (overload) than the transport capacity resulting in net erosion or deposition assuming sufficient availability of bed material (no armour layers).
Bedload transport in nonsteady flow can be modelled by a formula similar to steady flow because the adjustment of the transport of sediment particles close to the bed proceeds rapidly to the new hydraulic conditions.
Suspended load transport, however, does not have such a behaviour because it takes time (time lag effects) to transport the particles upwards and downwards over the depth and therefore it is necessary to model the vertical convectiondiffusion process.
Effect of time lag
Tidal flow is characterized by a daily ebb and flood cycle with a time scale of 6 to 12 hours (semidiurnal or diurnal tide) and by a neapspring cycle with a time scale of about 14 days. Sediment concentration measurements in tidal flow over a fine sand bed (0.05 to 0.3 mm) show a continuous adjustment of the concentrations to the flow velocities with a lag period in the range of 0 to 60 minutes
The basic transport process in tidal flow is shown in Figure 5. Sediment particles go into suspension when the current velocity exceeds a critical value. In accelerating flow there always is a net vertical upward transport of sediment particles due to turbulencerelated diffusive processes, which continues as long as the sediment transport capacity exceeds the actual transport rate. The time lag period [math]\Delta T_1[/math] is the time period between the time of maximum flow and the time at which the transport capacity is equal to the actual transport rate. After this latter time there is a net downward sediment transport because settling dominates yielding smaller concentrations and transport rates. In case of very fine sediments (silt) or a large depth, the settling process can continue during the slack water period giving a large time lag [math]\Delta T_2[/math] which is defined as the period between the time of zero transport capacity and the start of a new erosion cycle. Figure 5 shows that the suspended sediment transport during decelerating flow is always larger than during accelerating flow.
Time lag effects can be neglected for sediments larger than about 0.3 mm and hence a quasisteady approach based on the available sediment transport formulas can be applied.
Effect of salinity stratification
In a stratified estuary a highdensity salt wedge exists in the nearbed region resulting in relatively high nearbed densities and relatively low nearsurface densities. Stratified flow will result in damping of turbulence because turbulence energy is consumed in the mixing of heavier fluid from a lower level to a higher level against the action of gravity. The usual method to account for the salinityrelated stratification effect on the velocity and concentration profiles is the reduction of the fluid mixing coefficient, by introducing a damping factor related to the Richardsonnumber [math]Ri[/math], as follows: [math]\varepsilon_f = \phi \; \varepsilon_{f,0}[/math] with [math]\varepsilon_{f,0}[/math] = fluid mixing coefficient in fresh water, [math]\phi = F(Ri) [/math] = damping factor (< 1), [math]Ri[/math] = local Richardson number. The [math]\phi[/math]factor can be represented by a function given by MunkAnderson^{[11]}: [math] \phi = (1 + 3.3 Ri)^{1.5}[/math].
The simple sand transport formulas do not give realistic results when the salinityrelated damping effect is significant (vertical density gradient in stratified flow).
Effect of mud
In most tidal basins the sediment bed consists of a mixture of sand and mud. The sandmud mixture generally behaves as a mixture with cohesive properties when the mud fraction (all sediments < 0.05 mm) is dominant (> 0.3) and as a noncohesive mixture when the sand fraction is dominant (> 0.7). The distinction between noncohesive mixtures and cohesive mixtures can be related to a critical mud content ([math]p_{mud,cr}[/math]). Most important is the value of the clayfraction (sediments < 0.005 mm) in the mixture. Cohesive properties become dominant when the clayfraction is larger than about 5% to 10%. Assuming a claymud ratio of 1/2 to 1/4 for natural mud beds, the critical mud content will be about [math]p_{mud,cr}= 0.2  0.4 .[/math] If the mud content is below the critical value ([math]p_{mud} \lt p_{mud,cr}[/math]), the sandmud mixture can be assumed to be homogeneous with depth and to have noncohesive properties. Furthermore, the erosion of the sand particles is the dominant erosion mechanism. The mud particles will be washed out together with the sand particles. Laboratory and field observations^{[1]} have shown that the erosion or pickup process of the sand particles is slowed down by the presence of the mud particles. This behaviour can be quite well modelled by increasing the critical bedshear stress for initiation of motion of the sand particles. Herein, it is assumed that: [math] \tau_{b,cr,sand} = (1+p_{mud})^{\beta} \tau_{b,cr,Shields}[/math] with [math]\beta=3[/math] based on analysis of field data.
Sand transport in nonsteady coastal flows
Sand transport in a coastal environment generally occurs under the combined influence of a variety of hydrodynamic processes such as winds, waves and currents.
Sand can be transported by wind, wave, tide and densitydriven currents (currentrelated or advective transport), by the oscillatory water motion itself (waverelated or oscillating transport) as caused by the deformation of short waves under the influence of decreasing water depth (wave asymmetry), or by a combination of currents and short waves. The waves generally act as sediment stirring agents; the sediments are transported by the mean current. Lowfrequency waves interacting with short waves may also contribute to the sediment transport process. Windblown sand represents another basic transport process in the beachdune zone.
In frictiondominated deeper water outside the breaker (surf) zone the transport process is generally concentrated in a layer close to the sea bed; bedload transport (bed form migration) and suspended transport may be equally important. Bed load type transport dominates in areas where the mean currents are relatively weak in comparison with the wave motion (small ratio of depthaveraged velocity and peak orbital velocity). Suspension of sediments can be caused by ripplerelated vortices. The suspended load transport becomes increasingly important with increasing strength of the tide and winddriven mean current, due to the turbulencerelated mixing capacity of the mean current (shearing in boundary layer). By this mechanism the sediments are mixed up from the bedload layer to the upper layers of the flow.
In the surf zone of sandy beaches the transport is generally dominated by waves through wave breaking and the associated waveinduced currents in the longshore and crossshore directions. The longshore transport in the surf zone is also known as the longshore drift. The breaking process together with the nearbed waveinduced oscillatory water motion can bring relatively large quantities of sand into suspension (stirring) which is then transported as suspended load by net (wavecycle averaged) currents such as tide, wind and density (salinity)driven currents. The concentrations are generally maximum near the plunging point and decrease sharply on both sides of this location.
The nature of the sea bed (plane or rippled bed) has a fundamental role in the transport of sediments by waves and currents. The configuration of the sea bed controls the nearbed velocity profile, the shear stresses and the turbulence and, thereby, the mixing and transport of the sediment particles. For example, the presence of ripples reduces nearbed velocities, but it enhances bedshear stresses, turbulence and the entrainment of sediment particles, resulting in larger overall suspension levels. Several types of bed forms can be identified, depending on the type of wavecurrent motion and the bed material composition. Focusing on fine sand in the range of 0.1 to 0.3 mm, there is a sequence starting with the generation of rolling grain ripples, to vortex ripples and, finally, to upper plane bed with sheet flow for increasing bedshear. Rolling grain ripples are low relief ripples that are formed just beyond the stage of initiation of motion. These ripples are transformed into more pronounced vortex ripples due to the generation of sedimentladen vortices formed in the lee of the ripple crests under increasing wave motion. The vortex ripples are washed out under large storm waves (in shallow water) resulting in plane bed sheet flow characterised by a thin layer of large sediment concentrations.
Simple general formulas for sand transport in rivers, estuaries and coastal waters
Bed load transport
Van Rijn ^{[12]}^{[1]}^{[13]} proposed a simplified formula for bedload transport [math]q_b[/math] for combined steady current and waves, which reads as:
[math]q_b = \alpha_b \rho_s U h (d_{50} / h )^{1.2} M_e^{1.5} , \qquad (10) [/math]
where
[math]\alpha_b = 0.015[/math], [math]M_e = (U_e  U_{cr})/[(s1)gd_{50}]^{0.5}[/math] ,
[math]U_e = U+\gamma U_w[/math] , with [math]\gamma=0.4[/math] for irregular waves and [math]\gamma=0.8[/math] for regular waves,
[math]U[/math] = wave and depthaveraged flow velocity, [math]U_w = \pi H_s / [T_p \sinh(kh)][/math] = peak orbital velocity, [math]H_s[/math] = significant wave height,
[math]U_{cr} = \beta U_{cr,c} +(1\beta) U_{cr,w}[/math] = critical velocity, with [math]\beta = U / (U+U_w) [/math] ,
[math]U_{cr,c}[/math] = critical velocity for currents based on Shields ^{[1]}, [math]U_{cr,w}[/math] = critical velocity for waves ^{[1]},
[math]U_{cr,c}= 0.19 d_{50}^{0.1} \log(4 h / d_{90})[/math] for [math]0.0001\lt d_{50}\lt 0.0005 \; m, [/math]
[math] U_{cr,c} = 8.5 d_{50}^{0.6} \log (4 h / d_{90})[/math] for [math]0.0005\lt d_{50}\lt 0.002 \; m[/math]
[math]U_{cr,w} = 0.24 [(s1)g]^{0.66} d_{50}^{0.33} T_p^{0.33}[/math] for [math]0.0001\lt d_{50}\lt 0.0005 \; m,[/math]
[math]U_{cr,w} = 0.95 [(s1)g]^{0.57} d_{50}^{0.43} T_p^{0.14}[/math] for [math]0.0005\lt d_{50}\lt 0.002 \; m.[/math]
The inaccuracies are largest (underprediction) for relatively low [math]U_e[/math] velocities (<0.5 m/s) close to the critical velocities. Equation (6.1) describes the net bed load transport in currentdominated conditions (longshore flows). It cannot be used to compute the net crossshore bedload transport in the inner surf and swash zone. For these complicated conditions the full intrawave method should be used, or other formulas such as given in Sediment transport formulas for the coastal environment.
Suspended load transport
The simplified suspended load transport formula for steady flow proposed by Van Rijn^{[14]} was extended to coastal flow (waves) and reads, as (see also Soulsby^{[4]}):
[math] q_s = \alpha_s \rho_s U d_{50} M_e^{\eta} D_*^{0.6} , \qquad (11) [/math]
with the same symbol definitions as for Eq. (10).
According to the detailed TR2004 results ^{[15]}, the best matching values are [math]\alpha_s = 0.008[/math] and [math]\eta = 2.4 .[/math]
Equation (11) defines the currentrelated suspended transport [math]q_s[/math] which is the transport of sediment by the mean current including the effect of wave stirring on the sediment load. The suspended transport of very fine sediments (<0.1 mm) is somewhat underestimated.
Equation (11) is based on the assumption that suspended load transport occurs for [math]U \gt U_{cr,motion}[/math], with [math]U_{cr,motion}[/math] based on Equation (4). It is more physical to assume that suspended load transport occurs for [math]U \gt U_{cr,susp}[/math] with [math]U_{cr,susp}[/math] based on Equation (5). Using this latter approach, the suspended load transport is best represented by the parameters [math]\alpha_s=0.015[/math] and [math]\eta=2[/math].
Figure 6 shows both approaches for a water depth of [math]h= 5 \; m[/math], [math]d_{50} = 0.00025 \; m[/math]. Measured transport rates (green trendline) based on the dataset given by Van Rijn ^{[15]} are also shown. Results are similar except for very small velocities < 0.5 m.
Sand transport computations for combined wave plus current conditions in water depth of 5 m
The detailed TR2004 model ^{[13]}^{[15]} was used to compute the total sand transport rates (bed load plus suspended load transport) for a depth of 5 m and a median size of [math]d_{50} = 0.00025 \; m \; (d_{10} = 0.000125 \; m, \; d_{90} = 0.0005 \; m).[/math] The wave height was varied in the range of 0 to 3 m and wave periods in the range of 5 to 8 s. The wave direction is assumed to be normal to the coast, whereas the current is assumed to be parallel to the coast.
The total load transport (bed load + suspended load) results of the TR2004 model are shown in Figure 7 together with earlier results of the TR1993 model ^{[1]} for the same parameter range. Some measured data of rivers and estuaries are also shown. The total sand transport based on the simplified formulae (Equations (10) and (11)) is also shown in Figure 7. Table 1 shows some values based on these equations with [math]\alpha_b = 0.015 , \alpha_s = 0.008[/math] for [math]h = 5 \; m, v= 1 \; m/s,[/math] [math]H_s = 0, 0.5, 1, 2, 3 \; m/s .[/math] It is noted that the transport rates are approximately equal for [math]H_s = 0  0.5 \; m[/math] in the velocity range 1  2 m/s. Small waves of [math]H_s = 0.5 \; m[/math] in a depth of 5 m have almost no effect on the sediment transport rate for velocities larger than about 1 m/s, because the currentrelated mixing is dominant. The TR2004 model yields slightly smaller values than the TR1993 model for the case without waves ([math]H_s = 0 \; m[/math]). The TR2004 model yields considerably smaller (up to factor 3) total load transport rates for a steady current with high waves ([math]H_s = 3\; m[/math]) compared to the results of the TR1993 model. This is mainly caused by the inclusion of a damping factor acting on the waverelated nearbed diffusivity in the upper regime with storm waves. The TR2004 results for steady flow (without waves) show reasonable agreement with measured values (data from major rivers and estuaries with depth of about 5 m and sediment size of about 0.00025 m) over the full velocity range from 0.6  2 m/s.
The results of the TR2004 show that the total transport varies with [math]U^5[/math] for [math]H_s = 0 \; m,[/math] [math]U^{2.5}[/math] for [math]H_s = 1\; m[/math] and [math]U^2[/math] for [math]H_s = 3\; m [/math] The transport rate varies with [math]H_s^3[/math] for [math]U = 0.5 \; m/s ,[/math] with [math]H_s^{1.5}[/math] for [math]U = 1 \; m/s[/math] and with [math]H_s[/math] for [math]U = 2 \; m/s .[/math] The total transport rate ([math]q_{tot}=q_b+q_s[/math]) based on the simplified method generally are within a factor of 2 of the more detailed TR2004 model. The simplified method tend to underpredict for low and high velocities.
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References
 ↑ ^{1.0} ^{1.1} ^{1.2} ^{1.3} ^{1.4} ^{1.5} ^{1.6} ^{1.7} Van Rijn, L.C., 1993, 2012. Principles of sediment transport in rivers, estuaries and coastal seas. Aqua Publications, Amsterdam, The Netherlands (WWW.AQUAPUBLICATIONS.NL)
 ↑ Ancey, C. and Recking, A. 2023. Scaling behavior of bedload transport: what if Bagnold was right? EarthScience Reviews 246, 104571
 ↑ ^{3.0} ^{3.1} Shields, A., 1936. Anwendung der Ähnlichkeitsmechanik und der Turbulenz Forschung auf die Geschiebebewegung. Mitt. der Preuss. Versuchsamst. für Wasserbau und Schiffbau, Heft 26, Berlin, Deutschland Cite error: Invalid
<ref>
tag; name "Shields" defined multiple times with different content  ↑ ^{4.0} ^{4.1} Soulsby, R., 1997. Dynamics of marine sands. Thomas Telford, UK
 ↑ Bagnold, R.A., 1956. The Flow of Cohesionless Grains in Fluids. Proc. Royal Soc. Philos.Trans., London, Vol. 249.
 ↑ ^{6.0} ^{6.1} Einstein, H.A., 1950. The BedLoad Function for Sediment Transportation in Open Channel Flow. Technical Bulletin No. 1026, U.S. Dep. of Agriculture, Washington, D.C.
 ↑ MeyerPeter, E. and Mueller, R., 1948. Formulas for BedLoad Transport. Sec. Int. IAHR congress, Stockholm, Sweden.
 ↑ Bagnold, R.A., 1966. An Approach to the Sediment Transport Problem from General Physics. Geological Survey Prof. Paper 422I, Washington.
 ↑ Engelund, F. and Hansen, E., 1967. A Monograph on Sediment Transport in Alluvial Streams. Teknisk Forlag, Copenhagen, Denmark.
 ↑ Van Rijn, L.C., 1984a. Sediment Transport, Part I: Bed Load Transport. Journal of Hydraulic Engineering, ASCE, Vol. 110, No. 10
 ↑ Munk, W.H. and Anderson, E.R., 1948. Notes on the theory of the thermocline. Journal of Marine Research, Vol. 3, p 276295.
 ↑ Van Rijn, L.C., 1984c. Sediment Transport, Part III: Bed Forms and Alluvial Roughness. Journal of Hydraulic Engineering, ASCE, Vol. 110, No. 12.
 ↑ ^{13.0} ^{13.1} Van Rijn, L.C., 2007. Unified view of sediment transport by currents and waves, I: Initiation of motion, bed roughness, and bedload transport. Journal of Hydraulic Engineering, 133(6), p 649667.
 ↑ Van Rijn, L.C., 1984b. Sediment Transport, Part II: Suspended Load Transport. Journal of Hydraulic Engineering, ASCE, Vol. 110, No. 11.
 ↑ ^{15.0} ^{15.1} ^{15.2} Van Rijn, L.C., 2007. Unified view of sediment transport by currents and waves, II: Suspended transport. Journal of Hydraulic Engineering, 133(6), p 668389.
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