Sediment Transport Overview

For sand transport, the wash-load (i.e. sediment transport which does not contribute to the bed-material) can be assumed to be zero, and therefore, the total-load transport is equal to the sum of the bed- and the suspended-load transports: Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle q_{tk*} = q_{sk*} + q_{bk*}} .

There are currently three sediment transport models available in CMS:

(1) Equilibrium total load

(2) Equilibrium bed load plus non-equilibrium suspended load, and

(3) Non-equilibrium total-load.

The first two models are single-size sediment transport models and are only available with the explicit time-stepping schemes. The third is multiple-sized sediment transport model and is available with both the explicit and implicit time-stepping schemes.

Equilibrium Total-load Transport Model

In this model, both the bed load and suspended load are assumed to be in equilibrium. The bed change is solved using a simple mass balance equation known as the Exner equation.

Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \frac{\rho_s(1-\rho_m^')}{f_{morph}} \frac{\partial z_b}{\partial t} = \frac{\partial q_{t*j}}{\partial x_j} + \frac {\partial}{\partial x_j} \left(D_s \mid q_{t*}\mid\frac{\partial z_b}{\partial x_j} \right) } (2-42)

for Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle j = 1,2; k=1,2,.....,N} , where N is the number of sediment size classes and

t = time [s]

h = total water depth [m]

Because the model assumes that both the sediment transport is equilibri-um, it only recommended for coarse grids with resolutions larger than 50-100 m where the assumption of equilibrium sediment transport is more appropriate. As mentioned above the equilibrium total-load sediment transport model is a single-size sediment transport model and is only available with the explicit time-stepping scheme. For more information on the equilibrium sediment transport model, the reader is referred to Buttolph et al. (2007).

Equilibrium Bed-load plus Nonequilibrium Suspended Load Transport Model

Calculations of suspended load and bed load are conducted separately. The bed load is assumed to be in equilibrium and is included in the bed change equation while the suspended load is solved through the solution of an advection-diffusion equation. Actually the advection diffusion equation is a non-equilibrium formulation, but because the bed load is assumed to be in equilibrium, this model is referred to the "Equilibrium A-D" model.

Suspended-load Transport Equation

The transport equation for the suspended load is given by

Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \frac {\partial (hC)}{\partial t} + \frac {\partial (hV_j C)}{\partial x_j} = \frac{\partial}{\partial x_j} \left(v_s h \frac{\partial C}{\partial x_j}\right) + E_b - D_b } (2-43)

where

t = time[s]

h = water depth [m]

The entrainment and deposition functions are calculated as

Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle E_b = -\varepsilon_s \frac{\partial c}{\partial z} \left|_{z=a}\right. = \omega_s c_{a*} } (2-44)

where

z = vertical coordinate from the bed [m]

c = local sediment concentration [kg/m³]

Bed Change Equation

The bed change is calculated as

Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \frac {\rho_s(1 - p_m^')}{f_{morph}} \frac{\partial z_b}{\partial t} = \frac {\partial q_{b*j}}{\partial x_j} + \frac{\partial}{\partial x_j} \left(D_s\mid q_t*\mid \frac{\partial z_b}{\partial x_j} \right) + E_b - D_b } (2-46)

where

As mentioned above the equilibrium total-load sediment transport model is a single-size sediment transport model and is only available with the explicit time-stepping scheme. For more information on the equilibrium sediment transport model, the reader is referred to Buttolph et al. (2007).

Nonequilibrium Total-Load Transport Model

Total-load Transport Equation

The single-sized sediment transport model described in Sánchez and Wu (2011a) was extended to multiple-sized sediments within CMS by Sánchez and Wu (2011b). In this model, the sediment transport is separated into current- and wave-related transports. The transport due to currents includes the stirring effect of waves; and the wave-related transport includes the transport due to asymmetric oscillatory wave motion as well as steady contributions by Stokes drift, surface roller, and undertow. The current-related bed and suspended transports are combined into a single total-load transport equation, thus reducing the computational costs and simplifying the bed change computation. The two-dimensional horizontal (2DH) transport equation for the current-related total load is

Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \frac {\partial}{\partial t}\left (\frac {hC_{tk}}{\beta_{tk}} \right) + \frac {\partial(hV_j C_{tk})} {\partial x_j} = \frac {\partial}{\partial x_j} \left[ v_s h \frac {\partial (r_{sk}C_{tk})} {\partial x_j} \right] + \alpha_t \omega_{sk} (C_{t*k} - C_{tk}) (2-47) }

for j=1,2; k=1,2,......N , where N is the number of sediment size classes and

t = time [s]

h = water depth [m]

Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle c{tk}} = depth-averaged total-load sediment mass concentration for size class k defined as Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle c_{tk} = q_{tk} /(Uh)} in which Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle q_{tk}} is the total-load mass transport [kg/m³]

Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \beta_{tk}} = total-load correction factor described in the Total-Load Cor-rection Factor section [-]

Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle r_{sk}} = fraction of suspended load in total load for size class k and is described in Fraction of Suspended Sediments section [-]

Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle v_s} = horizontal sediment mixing coefficient described in the Hori-zontal Sediment Mixing Coefficient section [m²/s]

Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \alpha_t} = total-load adaptation coefficient described in the Adaptation Coefficient section [-]

The above equation may be applied to single-sized sediment transport by using a single sediment size class (i.e. N=1). The bed composition, however, does not vary when using a single sediment size class. The sediment mass concentrations are used rather than volume concentrations in order to avoid precision errors at low concentrations.

Fraction of Suspended Sediments

In order to solve the system of equations for sediment transport implicitly, the fraction of suspended sediments must be determined explicitly. This is done by assuming

Adaptation Coefficient

The total-load adaptation coefficient, Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \alpha_t } , is an important parameter in the sediment transport model. There are many variations of this parameter in literature (e.g. Lin 1984, Gallappatti and Vreugdenhil 1985, and Armanini and di Silvio 1986). CMS uses a total-load adaptation coefficient Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \alpha_t } that is related to the total-load adaptation length Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle L_t} and time Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle T_t} by

where

Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \omega_s} = sediment fall velocity corresponding to the transport grain size for single-sized sediment transport or the median grain size for multiple-sized sediment transport [m/s]

U = depth-averaged current velocity [m/s]

h = water depth [m]

The adaptation length (time) is a characteristic distance (time) for sedi-ment to adjust from non-equilibrium to equilibrium transport. Because the total load is a combination of the bed and suspended loads, the associated adaptation length may be calculated as Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle L_t = r_s L_s + (1-r_s )L_b} or Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle L_t = max(L_s , L_b )} , where L_s and L_b are the suspended- and bed-load adaptation lengths. L_s is defined as

Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \frac{1}{\alpha} = \frac {\delta}{h} + \left(1 - \frac {\partial}{h} \right) exp \left[-1.5 \left( \frac{\delta}{h} \right)^{- 1/6} \frac{\omega_s}{u_*} \right] } (2-52)

Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle T_s = \frac {h}{u_*} exp \left [ \begin{align} &(1.57 - 20.12 u_r )\omega_*^3 + (326.832 u_r^{2.2047} - 0.2)\omega_*^2 \\ &+(0.1385 ln u_r - 6.4061)\omega_* + (0.5467 u_r - 2.1963) \end{align} \right ] } (2-53)

The bed-load adaptation length, Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle L_b} , is generally related to the dimension of bed forms such as sand dunes. Large bed forms are generally proportional to the water depth and therefore the bed load adaptation length can be estimated as Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle L_b = a_b h} in which Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle a_b} is an empirical coefficient on the order of 5-10. Fang (2003) found that Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle L_b} of approximately two or three times the grid resolution works well for field applications. Although limited guidance exists on methods to estimate Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle L_b} , the determination of Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle L_b} is still empirical and in the developmental stage. For a detailed discussion of the adaptation length, the reader is referred to Wu (2007). In general, it is recommended that the adaptation length be calibrated with field data in order to achieve the best and most reliable results.

Total-Load Correction Factor

The correction factor, Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \beta_{tk}} , accounts for the vertical distribution of the suspended sediment concentration and velocity profiles, as well as the fact that bed load travels a slower velocity than the depth-averaged current velocity (see Figure 2.3). By definition, Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \beta_{tk}} is the ratio of the depth-averaged total-load and flow velocities.

fig_2_3.png

Figure 2.3. Schematic of sediment and current vertical profiles.

In a combined bed load and suspended load model, the correction factor is given by

where Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle u_{bk}} is the bed load velocity and Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \beta_{sk}} is the suspended load correction factor and is defined as the ratio of the depth-averaged sediment and flow velocities. Since most sediment is transported near the bed, both the total and suspended load correction factors (Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \beta_{tk}} and Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \beta_{sk}} ) are usually less than 1 and typically in the range of 0.3 and 0.7, respectively. By assuming logarithmic current velocity and exponential suspended sediment concentration profiles, an explicit expression for the suspended load correction factor Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \beta_{sk}} may be obtained as (Sánchez and Wu 2011)

Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \beta_{sk} = \frac{\int_{a}^{h} uc_k dz} {U\int_{a}^{h} c_k dz} = \frac {E_1 (\phi_k A)- E_1 (\phi_k) + ln(A/Z)e^{-\phi_k A} -ln(1/Z)e^{-\phi_k}} {e^{-\phi_k A}[ln(1/Z) -1][1-e^{-\phi_k (1-A)}]} } (2-55)

where Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \phi_k = \omega_{sk}h/ \epsilon , A = a/h , z = z_a/h ,} in which Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \omega_{sk}} is the sediment fall velocity for size class k, Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \epsilon} is the vertical mixing coefficient, a is a reference height for the suspended load, h is the total water depth, Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle z_a} is the apparent roughness length, and Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle E_1} is the exponential integral. The equation can be further simplified by assuming that the reference height is proportional to the roughness height (e.g.Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle a=30z_a} ), so that Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \beta_{sk} = \beta_{sk}(z,\phi_k ) } . Figure 2.4 shows a comparison of the suspended load correction factor based on the logarithmic velocity with exponential and Rouse suspended sediment concentration profiles.

Figure 2.4. Suspend load correction factors based on the logarithmic velocity profile and (a) exponential and (b) Rouse suspended sediment profile. The Rouse number is Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle r = \omega_s /(\kappa u_*)} .

The bed load velocity, Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle u_{bk}} , is calculated using the van Rijn (1984a) formula with re-calibrated coefficients from Wu et al. (2006)

Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle u_{bk} = 1.64 \left(\frac {\tau_b^'}{\tau_{crk}} - 1 \right)^{0.5} \sqrt {(s-1)gd_k}} (2-56)

where s is the specific gravity, g is the gravitational constant, Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle d_{50}} is the median grain size diameter, Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \tau_b^'} is the bed shear stress related to the grain roughness and is determined by Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \tau_b^' = (n^' / n)^{3/2}\tau_b} where Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle n^' = d_{50}^{1/6} / 20} is the Manning’s coefficient corresponding to the grain roughness and Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \tau_{crk}} is the critical bed shear stress.

Bed Change Equation

The fractional bed change is calculated as

Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \frac {\rho_s(1 - \rho_m^')}{f_{morph}} \left(\frac {\partial z_b}{\partial t} \right)_k = \alpha_t \omega_{sk} (C_{tk} - C_{tk*}) + \frac{\partial}{\partial x_j} \left(D_s \mid q_{bk} \mid \frac {\partial z_b}{\partial x_j} \right) } (2-57)

where

Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle C_{tk}} = depth-averaged total-load sediment mass concentration for size class k defined as Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle C_{tk} = q_{tk} /(Uh)} in which Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle q_{tk}} is the total-load mass transport [kg/m³]

Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \alpha_t} = total-load adaptation coefficient described in the Adaptation Coefficient section [-]

The sediment density is required in the previous equation (2-51) since mass concentrations are used. For a detailed derivation of the above equation, the reader is referred to Sanchez and Wu (2011a). The total bed change is calculated as the sum of the bed change for all size classes

Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \frac {\partial z_b}{\partial t} = \sum_k \left(\frac {\partial z_b}{\partial t} \right)_k } (2-58)

The purpose of the morphologic acceleration factor Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle f_{morph}} is to speed-up the bed change so that the simulation time Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle t_{sim}} represents approximately the change that would occur in Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle t{morph} = f_{morph}t_{sim} } . This factor should be used with caution and only for idealized cases or time periods which are periodic (mainly tidal). If time-varying winds or waves are important pro-cesses for driving sediment transport, then it is recommended to use re-duced or representative wind and wave conditions. Since the CMS runs relatively fast, it is generally recommended to not use the morphologic acceleration factor when validating the sediment transport model using hindcast measurements. If good initial and boundary conditions are available should be available and therefore it. The morphologic acceleration factor is useful however when simulating idealized cases or analyzing project alternatives.

Bed material sorting and layering

Bed sorting is the process in which the bed material changes composition (fraction of each grain size class). The bed is descritized into multiple lay-ers each with a uniform bed composition. The fraction of each size class is then calculated and stored in each layer. The sorting of sediments is then calculating using the mixing or active layer concept (Hirano 1971; Karim and Kennedy 1982; and Wu 1991). The active layer is the top layer of the bed which exchanges material directly with the sediment transport.

The temporal variation of the bed-material gradation in the first (mixing or active) layer is calculated as (Wu 2007)

Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \frac {\partial (\delta_1 p_{1k})}{\partial t} = \left(\frac {\partial z_b} {\partial t} \right)_k + p^*_{1k} \left(\frac{\partial \delta_1}{\partial t} - \frac {\partial z_b}{\partial t} \right) } (2-59)

Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \frac {\partial (\delta_2 p_{2k})} {\partial t} = - p_{1k}^* \left(\frac {\partial \delta_1} {\partial t} - \frac {\partial z_b} {\partial t} \right) } (2-60)

where Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \delta_2} is the thickness of the second layer, and <\math>p_{2k}</math> is the fraction of the Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle k^{th}} sediment size in the second layer. It is noted that there is no material exchanged between the sediment layers below the second layer.

The sediment transport, bed change, and bed gradation equations are solved simultaneously (coupled), but are decoupled from the flow calculation at a given time step. To illustrate the bed layering process, Figure 2.5 shows an example of the temporal evolution of 7 bed layers during erosional and depositional regimes.

Figure 2.5. Schematic showing an example bed layer evolution. Colors indicate layer number and not bed composition.

Mixing Layer Thickness

The mixing layer thickness is calculated as

where Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle z_{hb} } is the elevation of the hard bottom. A hard bottom is a nonerodable bed surface such as bed rock or a coastal structure.

Equilibrium Concentrations and Transport Rates

In order to close the system of equations describing the sediment transport, bed change, and bed sorting equations, the fractional equilibri-um depth-averaged total-load concentration Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle C_{tk*}} must be estimated from an empirical formula. The depth-averaged equilibrium concentration is defined as

For convenience, Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle C_{tk*} } is written in general form as

where Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle p_{1k}} is the fraction of the sediment size k in the first (top) bed layer and Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle C_{tk}^*} is the potential equilibrium total-load concentration. The potential concentration Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle C_{tk}^*} can be interpreted as the equilibrium concentration for uniform sediment of size Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle d_k} . Equation 2-57 above is essential for the coupling of sediment transport, bed change, and bed sorting equations.

Equilibrium Transport and Concentration Formulas

Lund-CIRP

Camenen and Larson (2005, 2007, 2008) developed general sediment transport formulas for bed and suspended loads under combined waves and currents. These are referred to as the Lund-CIRP transport formulas. The general transport formulas can be used for both symmetric and asymmetric waves, but for simplicity the waves are assumed to be symmetric. The current-related bed- and suspended-load transport with wave stirring is given by

Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \frac {q_b^*}{\sqrt {(s-1)gd_{50}^3}} = f_b \rho_s 12\sqrt {\Theta_c}\Theta_{cw,m} exp \left(-4.5 \frac{\Theta_{cr}} {\Theta_{cw}} \right) } (2-65)

Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \frac {q_b^*}{\sqrt {(s-1)gd_{50}^3}} = f_s \rho_s C_R U \frac{\epsilon}{\omega_s} \left[ 1 - exp \left(- \frac {\omega_s h}{\epsilon} \right) \right] } (2-66)

where

The reference concentration is given by

Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle c_R = \rho_s A_{cR}\Theta_{CW,m}exp\left(-4.5\frac{\Theta_{cr}}{\Theta_{cw}} \right)} (2-67)

where the coefficient Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle A_{cR}} is determined by the following relationship

The current-related shear stress is calculated as

where Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \kappa} is the von Karman constant (0.4), Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle h} is the total water depth, and Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle z_0} is the roughness length calculated as Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle z_0 = k_s / 30} , where Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle k_s} is the total bed Nikuradse roughness. The total bed roughness is assumed to be a linear sum of the grain-related roughness Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle k_{s,d}} , form-drag (ripple) roughness Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle k_{s,r}} , and sediment-related roughness Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle k_{s,s}} . Bed forms are also separated into current and wave-related bed forms. The current- and wave-related total roughness is then

where the subscript c|w indicates either the current (c) or wave (w) related component. The grain-related roughness is estimated as Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle k_{sg} = 2d_{50}} .

The ripple roughness is calculated as (Soulsby 1997)

The wave-related ripple height and length are calculated using the expressions proposed by van Rijn (1984b, 1989)

Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle H_{r,c} = \left\{ \begin{align} &0.22 A_w &\text{for} \psi_w < 10 \\ &2.8 \times 10^{-13}(250 - \psi_w)^5 A_w &\text{for} 10\leq \psi_w < 250 \\ &0 &\text{for} 250 \leq \psi_w \end{align} \right. } (2-75)

Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle L_{r,w} = \left\{ \begin{align} &1.25 A_w &\text{for} \psi_w < 10 \\ &1.4 \times 10^{-6}(250 - \psi_w)^{2.5} A_w &\text{for} 10 \leq \psi_w < 250 \\ &0 \quad &\text{for} 250 \leq \psi_w \end{align} \right. } (2-76)

in which Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle u_w} is the peak bottom orbital velocity and Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle T} is the wave period. For random waves, Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle u_w = \sqrt u_{rms}} and Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle T = T_p} are used as representative values. The wave mobility parameter,Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \psi_w} , is defined as

For the Lund-CIRP sediment transport equations, a wave bottom shear stress is calculated as

Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle f_w = \left\{ \begin{align} &exp(5.21 r^{-0.19} - 6.0) \text{for} r > 1.57 \\ &0.3 \quad \quad\quad \quad\quad \quad\quad \quad \text{for} r\leq 1.57 \end{align} \right. } (2-80)

The current- and wave-related sediment roughnesses are estimated as

Equation 2-73 above must be solved simultaneously with the expressions for the bottom shear stress because the roughness depends on the stress. The exact solution is approximated using explicit polynomial fits in order to avoid time-consuming iterations in calculating the bed shear stress.

The critical Shields parameter is estimated using the formula proposed by Soulsby (1997)

The sediment fall velocity is calculated using the formula by Soulsby (1997)

in which Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle k_b, k_c, \text{and} k_w } are coefficients, Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle D_b} is the wave breaking dissipation, and Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle D_c \text{and} D_w} are the bottom friction dissipation due to currents and waves, respectively. The dissipation from bottom friction due to current, Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle D_c} , and the dissipation from bottom friction due to waves, Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle D_w} , are expressed as

where the subscript c|w indicates either the current (c) or wave (w) relat-ed component, and Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle u*_{c\mid w}} and Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \tau_{c\mid w} } are the current- or wave-related bed shear velocity and stress, respectively. The coefficient Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle k_b} =0.017, and Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle k_c \text{and} k_w} are function of the Schmidt number:

where Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \sigma_{c\mid w}} is either the current or wave-related Schmidt number calculated from the following relationships (Camenen and Larson 2008):

Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \sigma_c = \left\{ \begin{align} &0.7 + 3.6 sin^{2.5} \left(\frac {\pi \omega_s}{2u*_c} \right) & \text{for} \frac {\omega_s}{u*_c} \leq 1 \\ &1 + 3.3 sin^{2.5} \left(\frac{\pi u*_c}{2\omega_s} \right) & \text{for} \frac {\omega_s}{u*_c} > 1 \end{align} \right. }

(2-89)

Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \sigma_w = \left\{ \begin{align} &0.09 + 1.4 sin^{2.5}\left(\frac{\pi\omega_s}{2u*_w}\right) & \text{for} \frac{\omega_s}{u*_w}\leq 1 \\ &1 + 0.49 sin^{2.5}\left(\frac{\pi u*_w}{2 \omega_s}\right) & \text{for} \frac{\omega_s}{u*_w}>1 \end{align} \right. }

(2-90)

In the case of coexisting or combined waves and currents, the wave-current Schmidt number is estimated as

(2-91)

For multiple-sized (nonuniform) sediments, the fractional equilibrium sediment transport rates are calculated as

Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \frac{q_{bk*}}{\sqrt{(s-1)gd_k^3}} = f_b \zeta_k^{\zeta-1}p_{bk}\rho_s 12 \sqrt{\Theta_c}\Theta_{cw,m}exp \left(-4.5 \frac{\Theta_{crk}}{\Theta_{cw}} \right) }

(2-92)

Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \frac{q_{bk*}}{\sqrt{(s-1)gd_k^3}} = f_s \zeta_k^{\zeta-1}p{bk}\rho_s c_{Rk}U\frac{\epsilon_k}{\omega_{sk}}\left[1 - exp \left( -\frac{\omega_{sk}h}{\epsilon} \right) \right] }

(2-93)

where the subscript k indicates variables which are calculated based only on the sediment size class k. Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \zeta_k} is the hiding and exposure coefficient de-scribed in Hiding and Exposure.

Van Rijn

The van Rijn (1984a,b) sediment transport equations for bed load and suspended load are used with the recalibrated coefficients of van Rijn (2007a,b), as given by

Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle q_b^* = f_b \rho_s 0.015Uh \left(\frac{U_e - U_{cr}}{\sqrt{(s-1)gd_{50}}} \right)^{1.5} \left( \frac{d_{50}}{h} \right)^{1.2} } (2-94)

Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle q_s^* = f_s \rho_s 0.012Ud_{50} \left(\frac{U_e - U_{cr}}{\sqrt{(s-1)gd_{50}}} \right)^{2.4}d_*^{-0.6} } (2-95)

where

s = sediment specific gravity or relative density [-]

g = gravitational constant (9.81 m/²)

U = depth-averaged current velocity [m/s]

The effective depth-averaged velocity is calculated as Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle U_e = U + \gamma u_w} with Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \gamma} =0.4 for random waves and Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \gamma} =0.8 for regular waves. Here Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle u_w} is the bottom wave orbital velocity based on linear wave theory and the signifi-cant wave height The critical depth-averaged velocity is estimated as Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle U_{cr} = \beta U_{crc} + (1 - \beta)u_{crw}} where Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \beta} is a blending factor and Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle U_{crc}} and Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle u_{crw}} are the critical velocities for currents and waves, respectively. As described in van Rijn (2007), the critical velocity for currents is

Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle U_{cr} = \left\{ \begin{align} &0.19 d_{50}^{0.1} log_{10} \left(\frac{4h}{d_{90}}\right), &for 0.1 \leq d_{50} \leq 0.5 mm\\ &8.5 d_{50}^{0.6} log_{10} \left(\frac{4h}{d_{90}}\right), &for 0.5 \leq d_{50} \leq 2.0 mm \end{align} \right. } (2-96)

where Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle d_{90}} is the sediment grain size in meters of 90^th percentile. The critical velocity for waves is based on Komar and Miller (1975),

Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle U_{crw} = \left\{ \begin{align} &0.24[(s-1)g]^{0.66} d_{50}^{0.33}T_p^{0.33}, &for 0.1 \leq d_{50} \leq 0.5 mm \\ &0.95[(s-1)g]^{0.57} d_{50}^{0.43}T_p^{0.14}, &for 0.5 \leq d_{50} \leq 2.0 mm \end{align} \right. } (2-97)

where Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle T_p} is the peak wave period. According to van Rijn (2007), the bed load transport formula predicts transport rates by a factor of 2 for velocities higher than 0.6 m/s, but underpredicts transport rates by a factor of 2-3 for velocities close to the initiation of motion.

The near-bed equilibrium sediment concentration for the van Rijn (formula is given by (van Rijn 1985)

Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle c_{a*} = f_s \rho_s 0.015 \frac {d_{50}}{a} \left(\frac {\tau^'_{max} - \tau_{cr}} {\tau_{cr}} \right)^{1.5} d_*^{-0.3} } (2-98)

The van Rijn formula (1984a,b, 2007a,b) were originally proposed for well-sorted sediments. The sediment availability is included by multiplication of transport rateswith the fraction of the sediment size class in the upper bed layer. The hiding and exposure is considered by in a correction factor which multiples to the critical velocity. When applied to multiple-sized sediments, the fractional equilibrium transport rate is calculated as

Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle q_{bk*} = f_b \rho_s p _{1k}0.015Uh \left(\frac{U_e - \zeta_k^{1/2}U_{crk}}{\sqrt{(s-1)gd_k}} \right)^{1.5} \left(\frac {d_k}{h} \right)^{1.2} } (2-100)

Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle q_{sk*} = f_s \rho_s p_{1k}0.012Uh \left(\frac{U_e - \zeta_k^{1/2}U_{crk}}{\sqrt{(s-1)gd_k}} \right)^{2.4} \left(\frac{d_k}{h} \right) d_{*k}^{-0.6} } (2-101)

Soulsby-van Rijn

Soulsby (1997) proposed the following equation for the total load sediment transport rate under action of combined current and waves,

Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle q_{b*} = f_b \rho_s 0.005Uh \left(\frac{U_e - U_{cr}}{\sqrt{(s-1)gd_{50}}}\right)^{2.4} \left(\frac{d_{50}}{h} \right)^{1.2}} (2-102)

Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle q_{s*} = f_b \rho_s 0.012Uh \left(\frac{U_e - U_{cr}}{\sqrt{(s-1)gd_{50}}}\right)^{2.4} \left(\frac{d_{50}}{h} \right)d_*^{-0.6}} (2-103)

where

U = depth-averaged current velocity [m/s]

Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle U_e} = effective velocity defined as Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle = \sqrt{U^2 + \frac{0.018}{C_d}u^2_{rms}}} [m/s]

Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle u{rms}} = peak bottom wave orbital velocity based on the root-mean-squared wave height [m/s]

The coefficients A_sb and A_ss are related to the bed and suspended loads, respectively, and given by

Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle A_{sb} = f_b \frac{0.005h(d_{50}/h)^{1.2}} {[(s-1)gd_{50}]^{1.2}}, A_{ss} = f_s \frac{0.012d_{50}d_*^{-0.6}} {[(s-1)gd_{50}]^{1.2}} } (2-104)

Note that the bed and suspended load scaling factors (Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle f_b} and Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle f_s} ) are in-cluded in the calculations for the coefficients Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle A_{sb}} and Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle A_{ss}} .

The current drag coefficient is calculated as

The Soulsby-van Rijn formula is modified for multiple-sized sediments similarly to the van Rijn formula in the previous section with the equation

Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle q_{bk*} = f_b \rho_s p_{1k}0.005Uh \left(\frac{U_e - \zeta_k^{1/2}U_{crk} } {\sqrt{(s-1)gd_k}} \right)^{2.4} \left(\frac{d_k}{h} \right)^{1.2} (1 - \alpha_{sv}tan \beta) } (2-106)

Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle q_{sk*} = f_b \rho_s p_{1k}0.012Uh \left(\frac{U_e - \zeta_k^{1/2}U_{crk} } {\sqrt{(s-1)gd_k}} \right)^{2.4} \left(\frac{d_k}{h}\right) d_{*k}^{-0.6} \left(1 - \frac{e_s}{\omega_s}tan \beta \right) } (2-107)

where

The subscript k indicates that the value is calculated based only on the size class k and not the median grain size. The availability of sediment fractions is included through Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle p_{1k}} , while hiding and exposure of grain sizes is accounted for by modifying the critical velocity. it is noted that the Soulsby-van Rijn (1997) formula are very similar to the van Rijn (1984ab, 207ab) except for the definition of the effective velocity and the recalibration of the bed-load formula coefficients in 2007a. The proposed changes should be verified with measurements or numerical simulations for multiple-sized sediment transport.

Watanabe

The equilibrium total load sediment transport rate is determined by Watanabe (1987) as

where

g = gravitational constant (9.81 m/s²)

The critical shear stress is determined from the Shields diagram. The maximum bed shear stress Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \tau_{bmax}} is calculated as (Soulsby 1997)

where r is the relative roughness defined as Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle r = A_w / k_sg} . Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle A_w} is the semi-orbital excursion defined as Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle A_w = u_w T /(2\pi)} .

Hiding and Exposure

When the bed material is composed of multiple grain sizes, larger grains have a greater probability of being exposed to the flow while smaller particles have a greater probability of being hidden to the flow. Figure 2.6 shows an example of a sediment grain Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle d_k} being exposed to the flow by an exposure height Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \Delta_e} , and sediment grain Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle d_j} being hidden by Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle d_k} .

Figure 2.6. Schematic of the exposure height of bed sediment grains

For the van Rijn (Reference(s)), Soulsby-van Rijn (Reference) and Watanabe (Reference) transport formulas, the hiding and exposure mechanism is considered by correcting the critical shear stress or velocity using a hiding and exposure correction function,Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \zeta_k} . For the Lund-CIRP transport formula, an alternate approach is required due to the way in which the Shields number and grain size are included in the formulation; thus, the hiding and exposure correction function is directly used to multiply the transport rate. Two methods are used to calculate Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \zeta_k} , depending on whether the sediment transport model is run with a single sediment size or with multiple sediment sizes and the methods are described in the following sections.

Single-sized sediment transport

In some applications, the coastal bed material is dominated by a single sediment size with patches of other sediment sizes or materials (e.g. shell hash) that may not contribute significantly to morphology change in the areas of interest; however, they may modify the sediment transport through hiding and exposure. For example, it is possible for the bed material to consist of mostly uniform sand with patches of shell fragments (bimodal distribution) in some regions. The shell material is difficult to model numerically because it is usually poorly sorted and its hydraulic properties are unknown. For such regions, sediment transport models commonly estimate excessive erosion due to the absence of the hiding effect of the coarser shell material (e.g. Cayocca 2001). A better and more physical plausible approach is to use the local bed composition along with a correction function for hiding and exposure to account for the effects of the uniform sand with the patches of coarser shell material. For single-sized sediment transport, the correction function for hiding and exposure is calculated following Parker et al. (1982) and others as,

where i is an empirical coefficient between 0.5-1.0. The aforementioned sediment transport equations are implemented by using the transport grain size Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle d_k} rather than the bed material Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle d_{50}} . A single and constant transport size Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle d_k} is used, while the bed material Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle d_{50}} varies spatially. The spatial distribution of can be obtained from field measurement data and for simplicity is assumed constant during the model simulation time. This is a significant assumption and may not be reasonable for some applications. However, this method provides a simple conceptual mechanism for considering an important process in the proposed single-sized sediment transport model. The approach has been successfully applied to Shinnecock Inlet, NY to simulate morphology change at a coastal inlet (Sánchez and Wu 2011a). A more accurate and complex approach is to simulate the transport and sorting of multiple sediments.

Multiple-sized sediment transport

The hiding and exposure of the each sediment size class is considered by modifying the critical shields parameter Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \Theta_{ck}} for each sediment size class based on Wu et al. 2000

where m is an empirical coefficient that varies for each transport formula, approximately equal to 0.6-1.0. Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle P_{ek}} and Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle P_{hk}} are the total hiding and exposure probabilities and are calculated as

Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle P_{hk} = \sum^N_{j=1}p_{1j}\frac {d_j}{l_k + d_j} \quad P_{ek} = \sum^N_{j=1}p_{1j}\frac{d_k}{d_k + d_j} } (2-114)

where N is the number of grain size classes.

Horizontal Sediment Mixing Coefficient

The horizontal sediment mixing coefficient, v_s, represents the combined effects of turbulent diffusion and dispersion due to nonuniform vertical profiles. In CMS, the horizontal sediment mixing coefficient is assumed to be proportional to the turbulent eddy viscosity as

Boundary Conditions

At the interface between wet and dry cells, the sediment transport rate is set to zero. The inflow boundary condition requires a given sediment concentration at the boundary. However, for most coastal applications, the actual sediment concentration is not available and the model implements the equilibrium concentration capacity (Dirichlet boundary condition). CMS also requires the size distribution of the inflow transport. For stability reasons, the inflow size distribution is assumed to be equal to the initial size distribution of the bed at the boundary. Inflow sediment transport rates are specified either as a total sediment transport rate, Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle Q_{sed,tot}} in kg/s, or as a fractional sediment transport rate,Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle Q_{sed,k}} in kg/s, for each sediment size class. If only the total sediment transport rate is specified, then the fractional sediment transport rate is calculated as Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle Q{sed,k} = p_{1k}Q_{sed.tot}} where Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle p_{1k}} is the bed material fraction of the first layer at the boundary. The fractional sediment transport Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle q_{tk,B} } at boundary cell B in kg/m/s is then calculated along the cell string according to

where Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle q^*_{tk,B}} is the potential sediment transport rate at boundary cell B and Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \Delta l_f} is the inflow cell face width. If the flow is directed outward of the do-main, a zero-gradient boundary condition is used for sediment concentration (Neumann boundary condition).

Salinity Transport

Transport Equation

CMS calculates the salinity transport based on the following 2DH salinity conservation equation

Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \frac {\partial(hC_{sal})} {\partial t} + \frac {\partial(hV_j C_{sal})} {\partial x_j} = \frac{\partial}{\partial x_f} \left(v_{sal}h \frac{\partial C_{sal}} {\partial x_j} \right) + S_{sal} } (2-117)

where

h = water depth [m]

Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle v_{sal}} = horizontal mixing coefficient Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle v_{sal} = v_t / \sigma_s \quad [m^2/s]}

Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle S_{sal} } = source/sink term due to precipitation, evaporation, and structures (e.g. culverts) [ppt m/s]

Equation (2-106) represents the horizontal fluxes of salt in water bodies and is balanced by exchanges of salt via diffusive fluxes. Major processes that contribute to the salinity are: freshwater inflows from rivers, vertical fluxes of freshwater by precipitation and evaporation at the water surface, and groundwater fluxes. These processes can be specified as the surface and bottom boundary conditions in the equation.

Boundary Conditions

The boundary condition for the salinity transport equation is dependent on the flow direction. At cell faces between wet and dry cells, a zero-flux boundary condition is applied. The cell face velocity Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle U_f} is zero between wet and dry cells and this eliminates advection transport. In addition, the salinity horizontal mixing coefficient at the cell face, Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \bar v_{sal,f}} , is set to zero and this eliminates the diffusive flux. If the flow is directed inward of the modeling domain, then a user-specified salinity concentration is pre-scribed (Direchlet boundary condition). If the flow is directed outward of the modeling domain, then a zero-gradient boundary condition is applied (Neumann boundary condition). Overall comments for this section: 1) A figure would be very HELPFUL showing inward and outward directed flow with the modeling domain and 2) Direchlet and Neumann BCs should be explained with reference citations.

Surface Roller

As a wave transitions from non-breaking to fully-breaking, some of the energy is converted into momentum that goes into the aerated region of the water column known. This phenomenon is known as the surface roller. Under the assumption that the surface roller moves in the mean wave direction, the evolution and dissipation of the surface roller energy is calculated by a steady-steate energy balance equation (Stive and De Vriend 1994, Ruessink et al. 2001)

where

The surface roller speed is calculated using the long-wave approximation Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle c = \sqrt{gh}} , where h is the local water depth. The surface roller dissipation is approximated as

where Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \beta_D} is the surface roller dissipation coefficient approximately equal to 0.05-0.1. The surface roller contribution to the wave stresses, Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle R_{ij} } , is given by

One effect of the surface roller is to push the peak alongshore current ve-locity closer to shore. Another side effect of the surface roller is to improve model stability (Sánchez et al. 2011a). The influence of the surface roller on the mean water surface elevation is relatively minor (Sánchez et al. 2011a).

Sediment Transport 1

Contents

Sediment Transport Overview

Equilibrium Total-load Transport Model

Equilibrium Bed-load plus Nonequilibrium Suspended Load Transport Model

Bed Change Equation

Nonequilibrium Total-Load Transport Model

Fraction of Suspended Sediments

Adaptation Coefficient

Total-Load Correction Factor

Bed Change Equation

Bed material sorting and layering

Mixing Layer Thickness

Equilibrium Concentrations and Transport Rates

Equilibrium Transport and Concentration Formulas

Lund-CIRP

Van Rijn

Soulsby-van Rijn

Watanabe

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Sediment Transport 1

Sediment Transport Overview

Equilibrium Total-load Transport Model

Equilibrium Bed-load plus Nonequilibrium Suspended Load Transport Model

Bed Change Equation

Nonequilibrium Total-Load Transport Model

Fraction of Suspended Sediments

Adaptation Coefficient

Total-Load Correction Factor

Bed Change Equation

Bed material sorting and layering

Mixing Layer Thickness

Equilibrium Concentrations and Transport Rates

Equilibrium Transport and Concentration Formulas

Lund-CIRP

Van Rijn

Soulsby-van Rijn

Watanabe

Navigation menu

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