Sediment transport formulas for the coastal environment

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Sediment transport in coastal waters is driven by currents and waves, which interact to produce complex patterns of sediment movement. In contrast to steady-flow conditions, wave-induced sediment transport depends on the rapid variations in near-bed velocity and turbulence during the wave cycle. Even in the absence of a mean current, nonlinear waves can generate net sediment transport because of differences between the onshore and offshore phases of the wave orbital velocity.

Sediment transport occurs as bedload, suspended load and sheet flow. The relative importance of these transport modes depends on sediment properties, bed shear stress, turbulence and bed configuration. Under breaking waves, additional turbulence and wave-induced currents strongly influence sediment suspension and transport.

Many empirical and semi-empirical formulas have been developed to estimate sediment transport rates under combined waves and currents. These formulas differ in their treatment of sediment entrainment, bed friction, wave asymmetry, sediment suspension and phase-lag effects. Their applicability depends on the hydrodynamic conditions and the sediment transport regime for which they were developed and calibrated.

This article presents a selection of commonly used formulas for noncohesive sediment transport in coastal waters, distinguishing between bedload, sheet-flow and suspended-load transport. The physical processes underlying sediment transport on the shoreface are explained in the article Shoreface profile.



Introduction

Calculating nearshore sediment transport is challenging because of the interaction between currents and waves and the influence of water-level variations (tides, wave set-up and set-down), wave breaking (turbulence and undertow), bed slope and bedforms. These processes govern different modes of sand transport, including bedload, suspended load and sheet flow [1].

Numerous sediment transport formulas have been developed, most of which depend on bed shear stress and contain empirical coefficients calibrated against laboratory or field measurements. Einstein [2] introduced a probabilistic approach to sediment transport. Bijker [3] and Bailard [4][5] developed formulas primarily tested against field data on littoral drift. Van Rijn [6][7][8] and Camenen and Larson [9][10] evaluated their formulas against a broad range of laboratory and field observations. Dibajnia and Watanabe [11][12] and Ribberink and Al Salem [13][14] focused particularly on experimental investigations of cross-shore sand transport under oscillatory flows, including sheet-flow conditions.

This article presents a selection of formulas for bedload and suspended-load transport under combined waves and currents, including formulations that account for wave asymmetry and phase-lag effects in sheet-flow transport.


Bedload transport under waves and currents

Bijker (1971) formula

One of the first sediment transport formulations that is still often used in engineering applications was proposed by Bijker [3]. It is derived from Frijlink's formula [15] for a current only with a modification of the bottom shear stress using a wave-current model. The direction of sediment fluxes is always that of the current since this formula was proposed to estimate longshore transport rate. Bedload transport [math]q_{sb}[/math] is expressed such as:

[math] q_{sb} = C_b \ d_{50} \ \sqrt{\Large \frac{\mu_c \tau_c}{\rho}} \normalsize \exp\left( -0.27 \ \Large \frac{(\rho_s-\rho) g d_{50}}{\mu_c \tau_{cw}} \normalsize \right) \qquad (1) [/math]

where [math]d_{50}[/math] is the median grain size diameter, [math]h[/math] the water depth, [math]C_b[/math] a breaking wave parameter, [math]\mu_c[/math] a ripple parameter, [math]\tau_c[/math] the total current shear stress due to current only, [math]\tau_{cw}[/math] the shear stress due to wave-current interaction, and [math]\rho_s[/math], [math]\rho[/math] the sediment and water densities, respectively.

The shear stress due to the wave-current interaction is computed following the method proposed by Bijker [3] introducing waves as a stirring factor:

[math] \tau_{cw} = \left[ 1+0.5 \Big(\xi_B \ \Large \frac{U_w}{U_c} \normalsize \Big)^2 \right] \ \tau_{cf} \qquad (2) [/math]

with [math]\xi_B = \sqrt{f_{wt}/f_{ct}}[/math] a parameter due to the wave-current interaction, [math]f_{wt}[/math] the total friction coefficient due to waves (including bedform effects), [math]U_w[/math] the peak value of the wave orbital velocity at the bottom, [math]U_c[/math] the depth-averaged current velocity and [math]\tau_{cf} [/math] the bed shear stress (including form drag).

The ripple parameter introduced by Bijker [3] is defined by the following equation:

[math] \mu_c = \left(f_c/f_{ct} \right)^{3/4} \qquad (3)[/math]

where [math]f_{ct}[/math] is the total friction coefficient due to current (including bedform effects) and [math]f_c[/math] is the skin friction coefficient due to current. The ripple factor [math]\mu_c \le 1[/math] represents the fraction of the total bed shear stress that acts on the sediment grains.

The breaking wave coefficient is defined by:

[math] C_b = 2 , \; H_w/h \lt 0.05 ; \quad C_b = 2 + 3 \ (H_w/h-0.05) / 0.35 \, , \; 0.05\lt H_w/h\lt 0.4 ; \quad C_b = 5 \, , \; 0.4\lt H_w/h . \qquad (4) [/math]

where [math]H_w[/math] is the wave height, and [math]h[/math] the water depth.

The Bijker formula treats waves primarily as a mechanism that enhances sediment mobility. It does not account for net sediment transport generated by wave asymmetry in the absence of a mean current and is therefore mainly suited to estimating current-driven transport enhanced by wave action.

Bailard and Inman (1981) formula

Bagnold [1] introduced the energetics model in which the main idea is that the sediment flux is proportional to the energy flux [math]\Omega[/math] (local rate of energy dissipation):

[math] \Omega = 0.5 \ \rho \ f_{cw} \ | \overrightarrow{u(t)} |^3 \qquad (5)[/math]

with [math]f_{cw}[/math] the friction coefficient due to the wave-current interaction, [math]\vec{u(t)}[/math] the instantaneous velocity vector, [math]\vec{u(t)} = \vec{U_c} + \vec{u_w(t)}[/math], [math]U_c[/math] the depth-averaged current velocity, and [math]u_w(t)[/math] instantaneous wave velocity over the bed.

The Bailard and Inman formula [5] is derived directly from the Bagnold model. For a horizontal bed, it can ultimately be written as a vector of sediment volume transport:

[math] \overrightarrow{q_{sb}} = \Large \frac{0.5 \ f_{cw}}{g \ (s-1)} \normalsize \left( \Large \frac{\epsilon_b}{\tan\phi} \normalsize \lt \mid\vec{u}\mid^2\vec{u}\gt \right) \qquad (6)[/math]

where [math]\epsilon_b[/math] is the bed load efficiency, [math]\phi[/math] the friction angle of the sediment, [math]s[/math] the ratio of sediment and water densities, and [math]\lt \ \gt [/math] yields an average over several periods of the wave. Because the transport rate depends nonlinearly on the instantaneous velocity, the contributions during the onshore and offshore phases of an asymmetric wave do not necessarily cancel. The formula can therefore predict net wave-induced sediment transport even in the absence of a mean current.

The bed load efficiency was found slightly different from the one given by Bagnold [16]. Bailard suggested from a calibration with field data that [math]\epsilon_b = 0.1[/math]. One difficulty for this formulation is the estimation of the friction coefficient due to the wave-current interaction as Bailard [5] did not specify any expression for this friction factor.

Van Rijn (1989) formula

The Van Rijn formula [6][8] is expressed in the same way as the Bijker formula, as a bed load formula taking into account the influence of waves as a stirring effect. The direction of sediment fluxes is also that of the current. Bedload transport can be written as follows:

[math] q_{sb} = 0.25 \ d_{50} \ d_*^{-0.3} \ (\tau_{cw}/ \rho)^{0.5} \ \left( \tau_{cw} / \tau_{cr} \ - 1 \right)^{1.5} \, , \qquad (7)[/math]

where [math]d_* = [(s-1)g/\nu^2]^{1/3}d_{50}[/math] is the dimensionless sediment diameter, [math]\tau_{cw} = 0.5 \rho f_{cw} U_{cw}^2[/math] the skin bed shear stress due to current and waves, [math]U_{cw}[/math] the peak near-bed velocity due to current and waves, and [math]f_{cw}=\alpha \beta f_c + (1-\alpha) f_w[/math] the skin friction coefficient for combined current and waves, with [math]\alpha = u_c/(u_c+U_w)[/math]. Here [math]f_c = 0.24 \, [\log_{10}(12h/k_s)]^{-2}[/math] is the skin friction coefficient due to current and [math]f_w = \exp[-6 + 5.2 (A_w/k_s)^{-0.19}][/math] (with a maximum of 0.3) the skin friction coefficient due to waves, where [math]k_s = 3 d_{90}[/math] is the grain roughness and [math]A_w=U_w T_w/(2\pi)[/math] the near-bed wave excursion. [math]u_c[/math] is the current close to the bottom as defined by Van Rijn, [math]\beta[/math] a coefficient accounting for the vertical structure of the near-bed current[8], and [math]\tau_{cr}[/math] the critical bed shear stress for inception of movement.

Van Rijn [17] updated his bedload formula. He proposed a new simplified bedload transport formula for steady flow (with or without waves):

[math] q_{sb} = 0.015 \ U_c \ h \; (\large \frac{d_{50}}{h})^{1.2} \normalsize \ \Psi^{1.5} \qquad (8) [/math]

where [math]\Psi=(U_e-U_{cr})/\sqrt{(s-1)gd_{50}}[/math] is the mobility parameter, [math]U_e=U_c+\gamma U_w[/math] the effective velocity with [math]\gamma=0.4[/math] for irregular waves and [math]\gamma=0.8[/math] for regular waves, [math]U_{cr}[/math] the critical effective velocity for inception of movement.

Ribberink (1998) formula

Ribberink [14] proposed a quasi-steady model of bed load transport where the instantaneous solid flux is assumed to be proportional to a function of the difference between the actual time-dependent bed shear stress and the critical bed shear stress (see Fig. 1). This formulation has been calibrated towards several flume data sets including wave-current interaction in a plane regime (suspended load negligible) and field data (unidirectional flows in rivers).


Figure 1: Profile of the time-dependent velocity (a) and bed shear stress (b) in the wave direction (an angle [math]\varphi[/math] exists between the current and wave directions).


The following expression for the sand transport rate was obtained:

[math] \vec{q_{sb}} = m_{Rib} \ \sqrt{(s-1) \ g \ d_{50}^3} \ \lt ( |\vec{\theta(t)}|- \theta_{cr})^{n_{Rib}} \Large \frac{\overrightarrow{\theta(t)}}{|\theta(t)|} \normalsize \gt \qquad (9) [/math]

where [math]\overrightarrow{\theta(t)} = 0.5 \ f_{cw} \ |u(t)|\overrightarrow{u(t)} \ / \ [(s-1) \ g \ d_{50}][/math] is the time-dependent Shields parameter (cf. Fig. 1) with the instantaneous velocity [math]\overrightarrow{u(t)} = \vec{U_c} + \overrightarrow{u_w(t)}[/math] and the wave-current friction factor [math]f_{cw}[/math], [math]\theta_{cr}[/math] the critical Shields parameter, [math]\lt \ \gt [/math] yields a time-averaging over several wave periods, and [math]m_{Rib}=11[/math], [math]n_{Rib}=1.65[/math] the adjusted coefficients.

In the same way as the Bailard formula, an equivalent wave-current friction coefficient has to be computed. Ribberink used the Madsen and Grant model[18] for which the friction coefficient [math]f_{cw}[/math] due to the wave and current interaction is defined such as:

[math] f_{cw}=X_v \ f_c +(1-X_v) \ f_w \qquad(10) [/math]

with [math]X_v=|U_c|/(|U_c|+U_w)[/math].

Ribberink also proposed to compute total roughness values as follows:

[math] k_{st} = {\rm max} \left( k_s;d_{50} \ [1+6 \ (\lt |\theta(t)|\gt / \theta_{cr}-1)] \right) \qquad (11)[/math]

where [math]k_s[/math] is skin roughness height.

As the formulation assumes that sediment transport responds quasi-instantaneously to the varying bed shear stress, it does not explicitly account for phase-lag effects associated with delayed sediment entrainment and settling, which can become important under intense sheet-flow conditions.

Formulas with a similar approach have been suggested by Soulsby and Damgaard [19] and Gonzalez-Rodriguez and Madsen [20].

Camenen and Larson (2005) formula

Camenen and Larson [9] developed a formula for bed load transport in a similar approach as Ribberink. They introduce an exponential function for the effect of inception of motion following the probabilistic approach introduced by Einstein [2]. The bed load transport [math]q_{sb}[/math] may be expressed as follows:

[math] q_{sbw} = a_w \ \sqrt{(s-1)g \ {d_{50}}^3} \ \sqrt{\theta_{cw,net}} \ \theta_{cw,m} \ \exp\left( -b \Large \frac{\theta_{cr}}{\theta_{cw}} \normalsize \right) , [/math]

[math] q_{sbn} = a_n \ \sqrt{(s-1)g \ {d_{50}}^3} \ \sqrt{\theta_{cn}} \ \theta_{cw,m} \ \exp\left( -b \Large \frac{\theta_{cr}}{\theta_{cw}} \normalsize \right) . \qquad (12) [/math]

where the subscripts [math]w[/math] and [math]n[/math] correspond, respectively, to the wave direction and the direction normal to the wave direction and [math]a_w[/math] [math]a_n[/math] and [math]b[/math] are empirical coefficients. [math]\theta_{cw,m}[/math] is the mean Shields parameter and [math]\theta_{cw}[/math] the maximum Shields parameter due to wave-current interaction, and [math]\theta_{cn}= \frac{1}{2} f_c (U_c \sin\varphi)^2 / ((s-1)g d_{50})[/math]. In order to simplify the calculations, the mean and maximum Shields parameter due to wave-current interaction is obtained by straightforward addition:

[math]\theta_{cw,m} = ( {\theta_c}^2 + {\theta_{w,m}}^2 + 2 \theta_{w,m} \theta_c \cos\varphi )^{1/2}[/math]

[math]\theta_{cw} = ( {\theta_c}^2 + {\theta_w}^2 + 2 \theta_w \theta_c \cos\varphi )^{1/2}[/math],

respectively, where [math]\theta_c[/math], [math]\theta_{wm}[/math], and [math]\theta_w[/math] are the current, mean wave, and maximum wave Shields number, and [math]\theta_{w,m}=0.5 \theta_w[/math] for a sinusoidal wave profile.

As [math]\theta_{cw,net}[/math] can be negative, the square root in Eq. 12 is applied as [math]{\rm sign}(\theta_{cw,net})\sqrt{|\theta_{cw,net}|}[/math]. The net Shields parameter [math]\theta_{cw,net}[/math] in Eq. 12 is given by,

[math] \theta_{cw,net} = \theta_{cw,on} - \theta_{cw,off} \qquad (13) [/math]

where [math]\theta_{cw,on}[/math] and [math]\theta_{cw,off}[/math] are the mean values of the instantaneous shear stress over the two half periods [math]T_{wc}[/math] and [math]T_{wt}[/math] ([math]T_w=T_{wc}+T_{wt}[/math], in which [math]T_w[/math] is the wave period) defined as follows (see Fig. 1),

[math] \theta_{cw,on} = \Large \frac{1}{2T_{wc}} \int_0^{T_{wc}} \frac{f_{cw} (u_w(t)+U_c\cos\varphi)^2}{(s-1)g d_{50}} \normalsize dt , [/math]

[math] \theta_{cw,off} = \Large \frac{1}{2T_{wt}} \int_{T_{wc}}^{T_w} \frac{f_{cw} (u_w(t)+U_c\cos\varphi)^2}{(s-1)g d_{50}} \normalsize dt \qquad (14) [/math]

where [math]u_w(t)[/math] is the instantaneous wave orbital velocity.

Based on comparison with an extensive data set, the following relationship is proposed for the transport coefficient [math]a_w[/math] [9][21]

[math] a_w = 6 + 6 \ X_t \qquad (15) [/math]

in which [math]X_t = \theta_c/(\theta_c + \theta_w)[/math] , where [math]\theta_c[/math] and [math]\theta_w[/math] are the Shields parameter for current and waves, respectively. The coefficient perpendicular to the waves, where only the current transports sediment, is set to [math]a_n = 12[/math], and the coefficient in the term describing initiation of motion is [math]b = 4.5[/math].


Sheet-flow transport

Sheet flow is a mode of intense sediment transport in which sand moves within a relatively thin, highly concentrated layer immediately above the bed, without the formation of distinct bedforms. The thickness of this layer is typically several to several tens of grain diameters. Sheet flow is commonly treated as a form of bedload transport, although its dynamics differ from those of individual particles moving by rolling, sliding or saltation.

Since the experiments of Dibajnia and Watanabe [11] and Ribberink and Al Salem [13], one realized that for intense sheet-flow transport, net bedload in the wave direction could be reduced or even be switched to the opposite direction of waves due to the phase-lag between sediment concentrations and fluid velocities [22]. Several authors attempt to model these effects. Phase-lag effects arise because sediment entrainment and settling require time. Sediment brought into motion during one half-cycle may remain mobile after flow reversal and be transported in the opposite direction. The resulting net transport depends on the relation between the sediment response time and the duration of the wave half-cycles (see Shoreface profile for further explanations).

Dibajnia and Watanabe (1992) formula

The sediment transport formulation of Dibajnia and Watanabe [11] is the first one to include phase-lag effects. Similar to the Bailard and Ribberink models, it breaks down the sediment transport into two half-cycles due to the presence of waves (see Fig. 2). During the first half-cycle, sediment moves in the direction of the wave, just as it moves in the opposite direction during the second half-cycle. An interesting aspect of the formula is that it takes into account a possible quantity of sand still in suspension after each half-cycle, and hence moving in the other direction. This formula enables transport under a non-linear wave to be described.


Figure 2: Bottom velocity profile in the direction of the wave propagation [11].


The solid volume flux is given by the following equation:

[math] \vec{q_s} = A_{dw} \ W_s \ d \ \Large \frac{\vec{\Gamma}}{\Gamma} \normalsize \ \Gamma^{B_{dw}} \qquad (16) [/math]

with [math]A_{dw} = 0.001[/math] and [math]B_{dw} = 0.55[/math] the calibration coefficients, and

[math] \vec{\Gamma} = \Large \frac{ T_{wc} \ \vec{u_{wc}} \ ( \Omega_c^{\ 3} + \Omega_t^{'3} ) + T_{wt} \ \vec{u_{wt}} \ ( \Omega_t^{\ 3} + \Omega_c^{'3} )} {(u_{wc} + u_{wt}) \ T_w} \normalsize , \qquad (17) [/math]

where [math]T_w[/math], [math]T_{wc}[/math], [math]T_{wt}[/math] are the period and half-periods of wave taking into account the effect of a current (cf. figure 2); [math]\Omega_c[/math], [math]\Omega_t[/math] are the amount of sand entrained and settled during the half-period [math]T_{wc}[/math] and [math]T_{wt}[/math], respectively; [math]\Omega'_c[/math], [math]\Omega'_t[/math] are the amount of suspended sand remaining from the positive and the negative half-cycle respectively. [math]u_{wc}^2[/math] and [math]u_{wt}^2[/math] the average quadratic velocities (wave + current) over each half-period expressed as:

[math] {u_{wj}}^2 = \Large \frac{2}{T_{wj}}\int_t^{t+T_{wj}} \normalsize u^2(t) \ dt + 2 \ {U_c}^2 \ \sin^2\varphi \qquad (18) [/math]

where [math]j[/math] can be [math]c[/math] or [math]t[/math], [math]u(t) = U_c \ \cos\varphi + u_w(t)[/math]. [math]u_w(t)[/math] is the instantaneous wave orbital velocity, and [math]\varphi[/math] the angle between wave direction and current direction.

If [math] \omega_j \leq \omega_{cr}[/math] then [math]\Omega_j = \omega_j \ \Large \frac{2 \ W_s \ T_{wj}}{d} \normalsize [/math] and [math]\Omega'_j=0, [/math]

If [math] \omega_j \geq \omega_{cr}[/math] then [math] \Omega_j = \omega_{cr} \Large \frac{2 \ W_s \ T_{wj}}{d} \normalsize [/math] and [math] \Omega'_j= (\omega_j-\omega_{cr}) \ \Large \frac{2 \ W_s \ T_{wj}}{d} \normalsize , \qquad (19) [/math]

with:

[math] \omega_j = \Large \frac{{u_{wj}}^2}{2 \ (s-1) \ g \ W_s \ T_{wj}} , \normalsize \qquad (20) [/math]

where [math]j[/math] can be [math]c[/math] or [math]t[/math].

[math]\omega_{cr}[/math] is a ripple parameter defined as:

[math] \omega_{cr} = 0.03 \; [/math] if [math] \; \theta_{cw(max)} \leq 0.2 ; [/math]

[math] \omega_{cr} = 1-0.97 \ [1- 6.25 \ (\theta_{cw(max)}-0.2)^2 ]^{0.5} \; [/math] if [math] \; 0.2 \lt \theta_{cw(max)} \lt 0.6 ; [/math]

[math] \omega_{cr} = 1 \; [/math] if [math] \; 0.6 \lt \theta_{cw(max)} \qquad (21) [/math]

where [math]\theta_{cw(max)}[/math] is the maximum Shields parameter due the wave-current interaction (computed following [23], pp.87-95).

Several authors developed a model based on the work of Dibajnia and Watanabe by introducing the effects of acceleration[24] and by introducing the Shields parameter in Eq.17. [25][26]. Velocity asymmetry refers to differences between the maximum onshore and offshore velocities, whereas acceleration asymmetry refers to differences in the rates of flow acceleration and deceleration during the wave cycle. Both can generate net sediment transport, but through partly different boundary-layer and sediment-response mechanisms.

Camenen and Larson (2006) formula

Figure 3: Schematic view of the instantaneous velocity and acceleration variation for a bore over a wave period and in the direction of the waves.

Following the approach proposed by Dibajnia and Watanabe, Camenen and Larson [27] introduced a parameter in Eq. 13 of the bedload formula to take into account phase-lag effects in bedload transport. This modification was eventually extended to acceleration effects [28][29]. The net Shields parameter [math]\theta_{cw,net}[/math] in Eq. 12 is then given by,

[math] \theta_{cw,net} = (1-\alpha_{pl,b})(1+\alpha_a)\theta_{cw,on} - (1+\alpha_{pl,b})(1-\alpha_a)\theta_{cw,off} \qquad (22) [/math]

in which [math]\alpha_{pl,b} = \alpha_{onshore} - \alpha_{offshore}[/math] and,

[math] \alpha_j = \Large \frac{\nu^{0.25} \ {U_{wj}}^{0.5}}{{W_s} \ {T_j}^{0.25}} \normalsize \exp\left[ - \left(\Large \frac{U_{w,crsf}}{U_{wj}} \right)^2 \normalsize \right] \qquad (23) [/math]

where [math]U_{w,crsf}[/math] is the critical velocity for inception of sheet-flow transport [27]}:

[math] U_{w,crsf} = 8.35 \ [(s-1) g \ (d_{50} \ \delta_w)^{1/2}]^{1/2} \ (1 + r_w) \qquad (24) [/math]

where [math]\delta_w = \sqrt{\nu T_w / \pi}[/math] is the Stokes boundary layer thickness [30], and [math]r_w[/math] the wave asymmetry coefficient, [math]r_w = u_{w,max}/U_w-1 ,[/math] with [math]u_{w,max}[/math] being the maximum wave velocity.

Based on the work by Watanabe and Sato [24], the coefficient [math]\alpha_a[/math] is given by [28] (see Fig. 3):

[math] \alpha_a = \Large \frac{1-R_{ac}}{1+R_{ac}} \normalsize [/math] with [math]R_{ac} = T_{ac}/T_{dc} \qquad (25)[/math]


Suspended-load

Suspended sediment transport depends on the competition between downward settling and upward turbulent mixing. This determines the vertical concentration profile. The transport rate then follows from integrating the product of sediment concentration and horizontal velocity over the water column.

Bijker (1971) formula

Bijker [3] related suspended load to bedload as the bed concentration reference value. Suspended load is then estimated using the Einstein integrals:

[math] q_{ss} = 1.83 \ q_{sb} \ \left( I_1\ln \left[ \Large \frac{33}{\delta_c} \normalsize \right] +I_2 \right) , \qquad (26) [/math]

where [math]\delta_c=100d/h[/math] is dimensionless thickness of the bed load layer.

The Einstein integrals [math]I_1[/math] and [math]I_2[/math] for the suspended load are given:

[math] I_1 = B \Large \int_{\delta_c}^{1} (\frac{1-y}{y} )^A \normalsize dy , [/math]

[math] I_2 = B \Large \int_{\delta_c}^{1} (\frac{1-y}{y} )^A \normalsize \ln y \ dy , \qquad (27) [/math]

where [math]B = 0.216 \dfrac{\delta_c^{A-1}}{(1-\delta_c)^A}[/math] is a normalization factor, [math]A=\Large \frac{W_s}{\kappa u_{*, cw}} \normalsize \; [/math] is the Rouse number determining the rate of the suspension, [math] u_{*,cw}= \sqrt{\dfrac{\tau_{cw}}{\rho}} \; [/math] is the shear velocity, [math]\kappa=0.41[/math] is the Von Karman constant, and [math]W_s[/math] the settling velocity.

Bailard (1981) formula

Following the work of Bailard and Inman [5], Bailard developed a total load formula, including a specific term for suspended load [4]:

[math] \vec{q_{ss}} = \Large \frac{0.5 \ f_{cw}}{g \ (s-1)} \normalsize \left( \Large \frac{\epsilon_s}{W_s} \normalsize \lt \mid\vec{u}\mid^3\vec{u}\gt \right) \qquad (28) [/math]

where [math]\epsilon_s[/math] is the suspended load efficiency, and [math]\lt \ \gt [/math] yields an average over several periods of the wave.

The suspended load efficiency coefficient is also slightly different from the one given by Bagnold [16]. Bailard suggested from a calibration with field data that [math]\epsilon_s = 0.02[/math].

Van Rijn (1989) formula

The Van Rijn formula [7][8] for suspended load corresponds to a resolution of the equation of concentration over depth:

[math] \Large \frac{dc}{dz} \normalsize = - \Large \frac{(1-c)^5 \ c \ W_s}{\epsilon_{scw}} \normalsize , \qquad (29)[/math]

where [math]c(z)[/math] is the mean volume concentration (time averaged) at height [math]z[/math], [math](1-c)^5[/math] corresponds to the decrease of the settling velocity due to high concentrations, and [math]\epsilon_{scw}[/math] is the mixing coefficient in case of a wave-current interaction. Then, integrating sediment fluxes over depth:

[math] q_{ss} = \int^h_{z_a} \overline{u(z)} \ c(z) \ dz \qquad (30)[/math]

where [math]h[/math] is the water depth, [math]z_a={\rm max}(k_{sct},k_{swt})[/math] the reference level, [math]k_{sct},k_{swt}[/math] total roughness values due to current and waves, respectively, and [math]\overline{u(z)}[/math] is the mean velocity (time averaged) at height [math]z[/math].

The reference concentration is estimated at the level [math]z_a[/math] based on the Van Rijn bedload formula [8]:

[math] c_a = 0.015 \ \Large \frac{d_{50}}{z_a} \normalsize \ d_{*}^{-0.3} \ (\tau_{cw} / \tau_{cr} \ -1 )^{1.5} . \qquad (31) [/math]

The sediment diffusion coefficient for a wave and current interaction is given by [7][8] :

[math] \epsilon_{scw}(z) = [\epsilon_{sc}(z)^2+\epsilon_{sw}(z)^2]^{1/2} \qquad (32) [/math]

with:

[math] \epsilon_{sc}(z) = \epsilon_{sc,max} = 0.25 \kappa \beta_s u_* h \;[/math] if [math] \; z \gt h/2[/math] ,

[math] \epsilon_{sc}(z) = \epsilon_{sc,max} \ \left[1-\left(1-2 \Large \frac{z}{h} \right)^2 \normalsize \right] \; [/math] if [math] \; z \leq h/2 \qquad (33) [/math]

where [math]\beta_s={\rm min}(1.5,1+2(W_s/u_*)^2)[/math], and [math]u_*=\sqrt{\tau_{cw}/\rho}[/math] is the shear velocity, and

[math] \epsilon_{sw}(z) = \epsilon_{sw,b} = 0.004 \ a_{br} \ d_* \ \delta_s \ U_w \; [/math] if [math] \; z \leq \delta_s , [/math]

[math] \epsilon_{sw}(z) = \epsilon_{sw,max} = 0.035 \ a_{br} \ h \Large \frac{H_w}{T_w} \normalsize \; [/math] if [math] \; z \gt h/2 , [/math]

[math] \epsilon_{sw}(z) = \epsilon_{sw,b}+(\epsilon_{sw,max}-\epsilon_{sw,b}) \ \Large \frac{z-\delta_s}{h/2-\delta_s} \normalsize \; [/math] if [math] \; \delta_s \lt z \leq h/2 \qquad (34) [/math]

with [math]\delta_s=0.3 h (H_w/h)^{0.5}[/math] is the thickness of the boundary layer, and [math]a_{br}={\rm max}(3 H_w/h-0.8,1)[/math], a coefficient. The estimation of the time-averaged velocity is based on the logarithmic velocity profile:

[math] \overline{u(z)} = U_c \ \Large \frac{\ln (30 \delta_w/k_a)}{\ln(30 h/k_a)-1} \ \frac{\ln(30z/k_{sc})}{\ln(30\delta_w/k_{sc})} \; \normalsize [/math] if [math] \; z \leq \delta_w , [/math]

[math] \overline{u(z)} = U_c \ \Large \frac{\ln(30z/k_a)}{\ln(30h/k_a)-1} \normalsize \; [/math] if [math] \; z \gt \delta_w \qquad (35) [/math]

with [math]\delta_w = 0.072 A_w (A_w/k_{sw})^{-0.25}[/math] the thickness of the wave boundary layer, [math]A_w=U_w T_w/(2\pi)[/math] the wave half-excursion.

The van Rijn approach assumes a quasi-equilibrium vertical concentration distribution. This assumption may be less appropriate under rapidly varying wave conditions, especially when sediment concentration responds with a significant time lag.

Van Rijn updated his suspended-load formula. He proposed a new simplified suspended-load transport formula for steady flow (with or without waves) [31]:

[math] q_{ss} = 0.012 \ U_c \ d_{50} \ d_*^{-0.6} \ \Psi^{2.4} \qquad (36) [/math]

where [math]\Psi[/math] is the mobility parameter defined below Eq. (8).

Camenen and Larson (2008) formula

In determining the suspended load [math]q_{ss}[/math], following the simplified approach by Madsen [32] and Madsen et al. [33], the vertical variation in the horizontal velocity was neglected and an exponential-law profile assumed for the sediment concentration. The suspended sediment load is written (components along the wave direction and perpendicular) [10]:

[math] q_{ssw} = U_{cw,net} \ c_R \Large \frac{\epsilon}{W_s} \normalsize \left[ 1 - \exp \left( -\Large \frac{W_s h}{\epsilon} \normalsize \right)\right] , [/math]

[math] q_{ssn} = U_c \sin\varphi \ c_R \Large \frac{\epsilon}{W_s} \normalsize \left[ 1 - \exp \left( -\Large \frac{W_s h}{\epsilon} \normalsize \right)\right] \qquad (37) [/math]

where [math]h[/math] is the water depth, [math]U_{cw,net}[/math] is the net mean current after a wave period, [math]c_R[/math] the reference concentration at the bottom, [math]W_s[/math] the sediment fall speed, and [math]\epsilon[/math] the sediment diffusivity. In solving the integral, the ratio [math]W_s h / \epsilon[/math] may often be assumed large, implying that the exponential term is close to zero. However, the assumption that integrating to infinity or to [math]h[/math] produces about the same result, may not be valid when strong mixing due to wave breaking is present.

The bed reference concentration is obtained from

[math] c_R = A_{cR} \ \theta_{cw,m} \ \exp \left( -4.5 \ \Large \frac{\theta_{cr}}{\theta_{cw}} \normalsize \right) \qquad (38) [/math]

in which the coefficient [math]A_{cR}[/math] is given by

[math] A_{cR} = 1.5 \ 10^{-3} \exp (-0.2 d_*) \qquad (39) [/math]

where [math]d_*=\sqrt[3]{(s-1)g/\nu^2} \ d_{50}[/math] is the dimensionless grain size.

The sediment diffusivity is related to the energy dissipation [34],

[math] \epsilon = h (D /\rho)^{1/3} \qquad (40) [/math]

in which [math]D[/math] is the total effective dissipation expressed as

[math] D = {k_b}^3 \ D_b + {k_c}^3 \ D_c + {k_w}^3 \ D_w \qquad (41) [/math]

where the energy dissipation from wave breaking ([math]D_b[/math]) and from bottom friction due to current ([math]D_c[/math]) and waves ([math]D_w[/math]) were simply added, and [math]k_b[/math], [math]k_c[/math] and [math]k_w[/math] are coefficients. The coefficient [math]k_b[/math] corresponds to an efficiency coefficient ([math]k_b=0.010[/math]), whereas [math]k_c[/math] and [math]k_w[/math] are related to the Schmidt number. Assuming a parabolic profile for the vertical sediment diffusivity, its mean value over the depth (for a steady current or waves, respectively) may be written as follows:

[math] \epsilon_j = h k_j (D_j / \rho)^{1/3} = k_j \ \kappa \ u_{*j} \ h \qquad (42) [/math]

where [math]k_j[/math] is a function of the inverse Schmidt number [math]\sigma_j[/math] or the ratio between the vertical eddy diffusivity of particles and the vertical eddy viscosity of water. [math]u_{*j}[/math] is the shear velocity due to current or waves only, with subscript [math]j[/math] taking on the values [math]c[/math] (current) or [math]w[/math] (waves), respectively. In case of a steady current, [math]k_c =\sigma_c/6 [/math] whereas for waves [math]k_w =\pi \sigma_w /3 [/math]. The following expression was developed for the inverse Schmidt number:

[math] \sigma_j = A_1 + A_2 \ \sin^{2} \left( \Large \frac{\pi}{2} \frac{W_s}{u_{*j}} \right) \; \normalsize [/math] if [math] \; W_s/u_{*j} \leq 1 , [/math]

[math] \sigma_j = 1 + (A_1+A_2-1) \ \sin^{2} \left( \Large \frac{\pi}{2} \frac{u_{*j}}{W_s} \right) \normalsize \; [/math] if [math] \; W_s/u_{*j} \gt 1 \, , \qquad (43) [/math]

where [math]j[/math] is a subscript equal to [math]c[/math] or [math]w[/math]. [math]A_{c1}=0.4[/math], [math]A_{c2}=3.5[/math], [math]A_{w1}=0.15[/math] and [math]A_{w2}=1.5[/math]. Recent measurements in large rivers showed however that [math]\sigma_c[/math] may be overestimated using Eq. 43 for large water depth [35]. For wave-current interaction, a weighted value is employed for the inverse Schmidt number:

[math] \sigma_{cw}=X_t \ \sigma_c +(1-X_t) \ \sigma_w \qquad (44) . [/math]

The net mean current is defined in a similar way to the net Shields parameter for the bed load in order to take into account a possible sediment transport due to wave asymmetry, as well as a possible phase-lag effects on the suspended concentration,

[math] U_{cw,net} = (1-\alpha_{pl,s})U_{cw,on} - (1+\alpha_{pl,s})U_{cw,off} \qquad (45) [/math]

where [math]\alpha_{pl,s}[/math] is the coefficient describing phase-lag effects on the suspended load, and [math]U_{cw,j}[/math] is the root-mean-square value of the velocity (wave+current) over the half period [math]T_{wj}[/math], where the subscript [math]j[/math] should be replaced either by [math]on[/math] (onshore) or [math]off[/math] (offshore) (see also Fig. 1) according to:


[math] U_{cw,on} = [\Large \frac{1}{T_{wc}} \int_0^{T_{wc}} \normalsize (u_w(t)+U_c \cos\varphi)^2 dt ]^{1/2},[/math]

[math] U_{cw,off} = [\Large \frac{1}{T_{wt}} \int_{T_{wc}}^{T_w} \normalsize (u_w(t)+U_c\cos\varphi)^2 dt]^{1/2}. \qquad (46) [/math]

In case of a steady current [math]U_{cw,net}=U_c[/math].

In the surf zone, the mean velocity can vary strongly over depth because of undertow. A depth-uniform velocity approximation may therefore fail to represent the actual suspended sediment flux. This is particularly important for cross-shore transport, where the vertical correlation between velocity and sediment concentration can determine the direction of net transport.


Applicability and limitations of sediment transport formulas

The sediment transport formulas presented in this article differ in their treatment of sediment mobility, wave-current interaction, wave asymmetry, turbulent mixing and phase-lag effects. No single formula is generally applicable to all coastal sediment transport conditions.

Formulas based on mean current velocity and wave-enhanced bed shear stress are mainly suited to current-driven transport, whereas formulations based on instantaneous velocity or shear stress can represent net transport generated by asymmetric waves. Under intense sheet-flow conditions, phase-lag and acceleration effects may become important and require more elaborate formulations.

Suspended-load predictions depend strongly on the estimated reference concentration, settling velocity, turbulent diffusivity and vertical velocity distribution. These quantities are particularly difficult to represent under breaking waves and rapidly varying flow conditions.

The selection of a transport formula should therefore be based on the dominant transport mechanisms, the hydrodynamic conditions and the sediment characteristics. Where possible, predictions should be evaluated against field or laboratory observations representative of the conditions being modelled.


Formula Main physical approach Important limitation
Bijker (1971[3]) Current-driven transport enhanced by waves No wave-asymmetry transport
Bailard and Inman (1981[5]) Energetics and nonlinear velocity dependence Instantaneous sediment response
Van Rijn (1989[8], 2007[17]) Sediment mobility under combined waves and currents Limited representation of intra-wave transport
Ribberink (1998[14]) Intra-wave excess Shields stress No phase-lag effects
Camenen and Larson (2005[9]) Probabilistic entrainment and wave asymmetry Empirical coefficients and no phase-lag effects
Dibajnia and Watanabe (1992[11]) Half-cycle transport with phase lag Simplified representation of sediment memory
Camenen and Larson (2006[27], 2008[36]) Phase lag and acceleration effects Empirical representation of unsteady processes


Related articles

Shoreface profile
Sand transport
Sediment deposition and erosion processes
Modelling coastal hydrodynamics
Coastal Hydrodynamics And Transport Processes
Littoral drift and shoreline modelling
Manual Sediment Transport Measurements in Rivers, Estuaries and Coastal Seas


References

  1. ↑ 1.0 1.1 Bagnold, R.A., 1963. Mechanics of marine sedimentation. In: The Sea. Vol. 3. M. N. Hill, Interscience, New York, pp. 507--528.
  2. ↑ 2.0 2.1 Einstein, H.A. 1950. The bed-load function for sediment transportation in open channel flows. Technical Bulletin No. 1026, U.S. Department of Agriculture, Soil Conservation Service, Washington, DC.
  3. ↑ 3.0 3.1 3.2 3.3 3.4 3.5 Bijker, E.W., 1971. Longshore transport computation. J. Waterways Harbors Division 97, WW4, 687--701.
  4. ↑ 4.0 4.1 Bailard, J.A., 1981. An energetic total load sediment transport model for a plane sloping beach. J. Geophysical Res. 86 (C11), 10938-10954.
  5. ↑ 5.0 5.1 5.2 5.3 5.4 Bailard, J.A., Inman, D.L., 1981. An energetic bedload model for a plane sloping beach: local transport. J. Geophysical Res. 86(C3), 2035–2043.
  6. ↑ 6.0 6.1 van Rijn, L.C., 1984. Sediment transport, part I: bed load transport. J. Hydraulic Division 110(10), 1431--1456.
  7. ↑ 7.0 7.1 7.2 van Rijn, L.C., 1984. Sediment transport, part II: suspended load transport. J. Hydraulic Division 110(11), 1613--1641.
  8. ↑ 8.0 8.1 8.2 8.3 8.4 8.5 8.6 van Rijn, L.C., 1989. Handbook sediment transport by currents and waves. Tech. Rep. H461, Delft Hydraulics Lab., The Hague.
  9. ↑ 9.0 9.1 9.2 9.3 Camenen, B., Larson, M., 2005. Bed-load transport under steady and oscillatory flow. In: Proc. Coastal Dynamics'05. ASCE, Barcelona, Spain, pp. 1--8, (CD ROM).
  10. ↑ 10.0 10.1 Camenen, B., Larson, M., 2008. A general formula for noncohesive suspended sediment transport. J. Coastal Res. 24(3), 615--627.
  11. ↑ 11.0 11.1 11.2 11.3 11.4 Dibajnia, M., Watanabe, A., 1992. Sheet flow under nonlinear waves and currents. In: Proc. 23rd Int. Conf. Coastal Eng. ASCE, Venice, Italy, pp. 2015--2029.
  12. ↑ Dibajnia, M., 1995. Sheet flow transport formula extended and applied to horizontal plane problems. Coastal Eng. J. 38(2), 178--194.
  13. ↑ 13.0 13.1 Ribberink, J.S., Al Salem, A.A., 1994. Sediment transport in oscillatory boundary layers in cases of rippled beds and sheet flow. J. Geophysical Res. 99(C6), 12707-12727.
  14. ↑ 14.0 14.1 14.2 Ribberink, J.S., 1998. Bed-load transport for steady flows and unsteady oscillatory flows. Coastal Eng. 34, 59-82.
  15. ↑ Frijlink, H.C., 1952. Discussion des formules de débit solide de Kalinske, Einstein, et MeyerPeter et Müller compte tenue des mesures récentes de transport dans les rivières néerlandaises [discussion of bedload movement formulas of Kalinske, Einstein and Meyer-Peter and Müller and their application to recent measurements of bedload movement in the rivers of Holland]. In: Comptes rendus 2eme Journée Hydraulique. Grenoble, France, pp. 98--103, (in French).
  16. ↑ 16.0 16.1 Bagnold, R.A., 1966. An approach to the sediment transport problem from general physics. Tech. Rep. 422-I, US Geol. Survey Prof. Paper.
  17. ↑ 17.0 17.1 van Rijn, L.C., 2007. Unified view of sediment transport by currents and waves. I: Initiation of motion, bed roughness, and bed-load transport. J. Hydraulic Eng. 133(6), 649—667.
  18. ↑ Madsen, O.S., Grant, W.D., 1976. Sediment transport in the coastal environment. Tech. Rep. 209, M. I. T., Cambridge, Massachusetts, USA.
  19. ↑ Soulsby, R.L., Damgaard, J.S., 2005. Bedload sediment transport in coastal waters. Coastal Eng. 52, 673—689
  20. ↑ Gonzalez-Rodriguez, D., Madsen, O.S., 2007. Seabed shear stress and bedload transport due to asymmetric and skewed waves. Coastal Eng. 54, 914—929.
  21. ↑ Camenen, B., Larson, M., 2005. A bedload sediment transport formula for the nearshore. Estuarine, Coastal and Shelf Science 63, 249--260.
  22. ↑ Dohmen-Janssen, M., 1999. Grain size influence on sediment transport in oscillatory sheet flow, phase-lags and mobile-bed effects. Ph.D. thesis, Delft Univ. of Technology, The Netherlands, ISBN 90-9012929-4.
  23. ↑ Soulsby, R.L., 1997. Dynamics of marine sands, a manual for practical applications. Thomas Telford, London, UK, ISBN 0-7277-2584 X.
  24. ↑ 24.0 24.1 Watanabe, A., Sato, S., 2004. A sheet-flow transport rate formula for asymmetric, forward-leaning waves and currents. In: Proc. 29th Int. Conf. Coastal Eng. Lisbon, Portugal, pp. 1703—1714.
  25. ↑ da Silva, P.A., Temperville, A., Seabra Santos, F., 2006. Sand transport under combined current and wave conditions: A semi-unsteady, practical model. Coastal Eng. 53, 897—913.
  26. ↑ van der A, D.A., Ribberink, J.S., van der Werf, J.J., O'Donoghue, T., Buijsrogge, R.H., Kranenburg, W.M., 2013. Practical sand transport formula for non-breaking waves and currents. Coastal Eng. 76, 26--42.
  27. ↑ 27.0 27.1 27.2 Camenen, B., Larson, M., 2006. Phase-lag effects in sheet flow transport. Coastal Eng. 53, 531—542,
  28. ↑ 28.0 28.1 Camenen, B., Larson, M., 2007. A total load formula for the nearshore. In: Proc. Coastal Sediments'07. ASCE, New Orleans, Louisiana, USA, pp. 56--67.
  29. ↑ Camenen, B., Larson, M., 2010. Discussion of "measurements of sheet flow transport in acceleration-skewed oscillatory flow and comparison with practical formulations" by D.A. van der A, T.O'Donoghue and J.S. Ribberink. Coastal Eng. 58(1), 131--134.
  30. ↑ Chan, K.W., Baird, M. H.I., Round, G.F., 1972. Behaviour of beds of dense particles in a horizontally oscillating liquid. Proc. Royal Society of London A(330), 537—559.
  31. ↑ van Rijn, L.C., 2007. Unified view of sediment transport by currents and waves. II: Suspended transport. J. Hydraulic Eng. 133(6), 668—689.
  32. ↑ Madsen, O.S., 1993. Sediment transport outside the surf zone. unpublished technical report, Waterways Experiment Station, U. S. Army Corps of Engineer, Vicksburg, Mississippi, USA.
  33. ↑ Madsen, O.S., Tajima, Y., Ebersole, B.A., 2003. Longshore sediment transport: a realistic order-of-magnitude estimate. In: Proc. Coastal Sediments'03. ASCE, Clearwater Beach, Florida, USA, pp. 1--8, cDROM.
  34. ↑ Battjes, J.A., Janssen, J. P. F.M., 1978. Energy loss and set-up due to breaking of random waves. In: Proc. 16th Int. Conf. Coastal Eng. ASCE, pp.569--587.
  35. ↑ Camenen, B., Le Coz, J., Dramais, G., Peteuil, C., Fretaud, T., Falgon, A., Dussouillez, P., Moore, S.A., 2014. A simple physically-based model for predicting sand transport dynamics in the Lower Mekong River. In: Schleiss, A.J., de Cesare, G., Franca, M.J., Pfister, M. (Eds.), River Flow, Proc. 7th Int. Conf. on Fluvial Hydraulics. Lausanne, Switzerland, pp. 2189--2197.
  36. ↑ Cite error: Invalid <ref> tag; no text was provided for refs named CL08a


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