## Transport Equation

The transport equation for the suspended load is given by

 ${\displaystyle {\frac {\partial (hC)}{\partial t}}+{\frac {\partial (u_{j}hC)}{\partial x_{j}}}={\frac {\partial }{\partial x_{j}}}{\biggl [}\nu _{s}h{\frac {\partial C}{\partial x_{j}}}{\biggr ]}+E_{b}-D_{b}}$ (1)

## Bed Change Equation

If the advection-diffusion (A-D) equation is selected to simulate the sediment transport and mixing, the change in the water depth is calculated by the sediment continuity equation

 ${\displaystyle (1-p'_{m}){\frac {\partial \zeta }{\partial t}}={\frac {\partial q_{b*j}}{\partial x_{j}}}+{\frac {\partial }{\partial x_{j}}}{\biggl [}D_{s}|q_{t*}|{\frac {\partial \zeta }{\partial x_{j}}}{\biggr ]}+E_{b}-D_{b}}$ (2)

where ${\displaystyle p'_{m}}$ is the sediment porosity, and ${\displaystyle D_{s}}$ is a bedslope coefficient.

## Pick-up and Deposition Rate

The AD model calculates the bed level change due to suspended load from the difference between pick-up rate and deposition rate in

 ${\displaystyle {\frac {\partial h}{\partial t}}={\frac {1}{1-p}}\left({\frac {\partial q_{bx}}{\partial x}}+{\frac {\partial q_{by}}{\partial y}}+P-D\right)}$ (3)

where qbx = bed load transport rate parallel to the x-axis, qby = bed load transport rate parallel to the y-axis, and p = porosity of sediment.

The pick-up rate and the deposition rate are also applied as the bottom boundary condition in

 ${\displaystyle {\frac {\partial (Cd)}{\partial t}}+{\frac {\partial (Cq_{x})}{\partial x}}+{\frac {\partial (Cq_{y})}{\partial y}}={\frac {\partial }{\partial x}}\left(K_{x}d{\frac {\partial C}{\partial x}}\right)+{\frac {\partial }{\partial y}}\left(K_{y}d{\frac {\partial C}{\partial y}}\right)+P-D}$ (4)

in which

C = depth-averaged sediment concentration
d = total depth (= h+η)
h = still-water depth
η = deviation of the water-surface elevation from the still-water level
t = time
qx = flow per unit width parallel to the x-axis (= ud)
qy = flow per unit width parallel to the y-axis (= vd)
u = depth-averaged current velocity parallel to the x-axis
v = depth-averaged current velocity parallel to the y-axis
Kx = sediment diffusion coefficient for the x direction
Ky = sediment diffusion coefficient for the y direction
P = sediment pick-up rate (upward sediment flux)
D = sediment deposition rate (downward sediment flux)

The boundary conditions are specified at an arbitrary level ${\displaystyle \alpha }$ above the mean bed level:

 ${\displaystyle P=\left.-\epsilon {\frac {\partial c}{\partial z}}\right|_{z=a}=c_{a}w_{f}}$ (5)
 ${\displaystyle D=c_{0}w_{f}}$ (6)

where c = equilibrium concentration of suspended sediment at a given elevation, and z = vertical coordinate. Both ca and c0 are reference concentrations defined at z = a. Because the upward flux of sediment depends on the bed shear stress, ca is determined from the bed shear stress calculated from the local hydrodynamic conditions. Representation of ca within CMS-M2D is dependent on selection of either the van Rijn or Lund-CIRP models. The downward sediment flux depends on the concentration in the upper water column; therefore, c0 is specified from solution of Equation 4.

Assuming that the suspended concentration is in equilibrium ${\displaystyle (\partial c/\partial t=0,\partial u/\partial x=\partial v/\partial y=0,\partial c/\partial x=\partial c/\partial y=0)}$ , then the basic equation for suspended sediment concentration can be written:

 ${\displaystyle cw_{f}+\epsilon {\frac {\partial c}{\partial z}}=0}$ (7)

This equation can be solved analytically by applying an appropriate mixing coefficient ${\displaystyle \epsilon }$, and the vertical profile of suspended concentration is obtained in the following form:

 ${\displaystyle c(z)=c_{0}F(z)}$ (8)

where F(z) is a function of the vertical concentration distribution. The relationship between the reference concentration c0 and the depth-averaged concentration C is:

 ${\displaystyle C={\frac {1}{d-a}}\int _{a}^{d}c_{0}F(z)dz}$ (9)

Thus, c0 may be written in the following form by introducing a conversion function ${\displaystyle \beta _{d}}$:

 ${\displaystyle c_{0}={\frac {C}{{\frac {1}{d-a}}\int _{a}^{d}F(z)dz}}={\frac {C}{\beta _{d}}}}$ (10)

The bed level change due to suspended load is based upon the difference of the two types of reference concentration:

 ${\displaystyle P-D=(c_{a}-c_{0})w_{f}}$ (11)

This equation implies that erosion occurs if ca > c0, and accretion occurs if ca < c0.

In the AD model, three methods of specifying ca and c0 (that is, ${\displaystyle \beta _{d}}$) are implemented. Two of the methods are based on the van Rijn formula (van Rijn 1985) and one on the Lund-CIRP formula (Camenen and Larson 2006). Table 1 summarizes general features of the methods.

van Rijn formula

The reference concentration ca in the van Rijn formula is given by:

 ${\displaystyle c_{a}=0.015{\frac {d_{50}}{a}}\left({\frac {\tau _{s,max}-\tau _{cr}}{\tau _{cr}}}\right)^{1.5}d_{*}^{-0.3}}$ (12)

where a = reference height and ${\displaystyle \tau _{s,max}}$ = maximum bed shear stress given by

 ${\displaystyle \tau _{b,max}={\sqrt {(\tau _{m}+tau_{w}cos(\phi ))^{2}+(\tau _{w}sin(\phi ))^{2}}}}$ (13)

Table 1
Features of Calculation of Pick-up and Deposition Rates
Exponential ProfileEq. 10Eq. 24 and 25Fast computation.
Tends to overestimate sediment transport rate.
Can be used for some tests.
Van Rijn ProfileEq. 10Solves Eq. 5 numerically with Eq. 23 (Runge-Kutta 4th)Requires substantial computing time. Provides the same results as van Rijn (1985).
Lund-CIRP ProfileEq. 70Eq. 24 and 72Fast computation.
Newly developed sediment transport formula.

The current-related shear stress is calculated from:

 ${\displaystyle \tau _{c}={\frac {\rho _{w}\kappa ^{2}}{\left[1+ln(k_{s}^{'}/30d)\right]^{2}}}\ U_{c}^{2}}$ (14)

The wave-related shear stress is given as:

 ${\displaystyle \tau _{w}=\rho _{w}{\frac {f_{w}}{2}}\ U_{w}^{2}}$ (15)
 ${\displaystyle f_{w}=exp\lfloor 5.5(A_{w}/k_{s}^{'})^{-0.2}-6.3\rfloor }$ (16)

where ks' = roughness height defined as:

 ${\displaystyle k_{s}^{'}=k_{sd}+k_{ss}}$ (17)

in which ksd and kss are calculated from

 ${\displaystyle k_{sd}=2.5d_{50}}$ (18)

and

 ${\displaystyle k_{ss}=5d_{50}\theta _{i}}$ (19)

respectively.

Bed concentration in the van Rijn model is defined at the height a as:

 ${\displaystyle a=max(0.5H_{r},0.01d)}$ (20)

where Hr = ripple height. If ripples are present, the total roughness height is modified as:

 ${\displaystyle k_{s}=k_{s}^{'}+k_{sf}}$ (21)

where, ${\displaystyle k_{sf}=20(H_{r}^{2}/L_{r})}$ is specified for the van Rijn formula, and ${\displaystyle k_{sf}=7.5(H_{r}^{2}/L_{r})}$ is specified for the Lund-CIRP formula. The ripple dimension is obtained by selecting the largest ripple height for the case of current or waves

${\displaystyle L_{r}=1000d_{50}}$
 ${\displaystyle H_{r}={\frac {L-r}{7}}}$ (22)
${\displaystyle H_{r}=0.22A_{w}}$
${\displaystyle L_{r}=1.25A_{w}\quad \quad \quad \quad \psi _{w}<10}$

${\displaystyle H_{r}=2.8\ x\ 10^{-13}(250-\psi _{w})^{5}A_{w}}$

 ${\displaystyle L_{r}=1.4\ x\ 10^{-6}(250-\phi _{w})^{2}.5A_{w}\quad \quad 10\leq \psi _{w}<250}$ (23)

${\displaystyle H_{r}=0}$
${\displaystyle L_{r}=0\quad \quad \quad \psi _{w}>250}$

The conversion parameter ${\displaystyle \beta _{d}}$ to determine ${\displaystyle c_{0}}$ is obtained from the vertical mixing coefficient. Van Rijn (1985) proposed a distribution of the mixing coefficients for only current or waves according to Figure 1. The current-related mixing coefficient is given by:

 ${\displaystyle \epsilon _{c}=\epsilon _{c,max}-\left(1-{\frac {2z}{h}}\right)^{\vartheta }{\text{where}}\ z<0.5h}$ (24)

 ${\displaystyle \epsilon _{c}=\epsilon _{c,max}=0.25\kappa u_{*c}h\ {\text{where}}\ z>0.5h}$ (25)

where u*c = current-related bed shear velocity expressed as:

 ${\displaystyle u_{*c}={\frac {\kappa }{\left[-1+ln\left({\frac {30d}{k_{s}}}\right)\right]}}\ U_{c}}$ (26)

and ${\displaystyle \vartheta }$ is a coefficient obtained from:

 ${\displaystyle \vartheta =-0.25{\frac {U_{w}}{|U|}}+2\quad {\text{ where }}\quad 0\leq {\frac {U_{w}}{|U|}}\leq 4}$ (27)
 ${\displaystyle \vartheta =1\quad {\text{ where }}\quad {\frac {U_{w}}{|U|}}>4}$ (28)

Figure 1. Vertical distributions of mixing coefficient due to current and waves

The wave-related mixing coefficient is:

${\displaystyle \ \epsilon _{w}=\epsilon _{w,bed}\ \ \ \ z\leq \delta }$
 ${\displaystyle \epsilon _{w}=\epsilon _{w,bed}+(\epsilon _{w,max}-\epsilon _{w,bed})\left({\frac {z-\delta }{0.5h-\delta }}\right){\text{ where }}\delta (29)
${\displaystyle \ \epsilon _{w}=\epsilon _{w,max}\ \ \ {\text{ where }}z>0.5h}$

where

${\displaystyle \epsilon _{w,bed}=0.00065D_{*}\alpha _{br}\delta u_{w}}$
 ${\displaystyle \epsilon _{w,max}=0.035\alpha _{br}{\frac {hH}{T}}}$ (30)
${\displaystyle \alpha _{br}=5\left({\frac {H}{h}}\right)-2\ \ \ {\text{ where }}H/h\geq 0.6}$
 ${\displaystyle \alpha _{br}=1\ \ \ \ {\text{ where }}H/h<0.6}$ (31)

in which the parameter ${\displaystyle \delta }$ = height from the bed given by ${\displaystyle \delta =3H_{r},H=}$ significant wave height, and T = significant wave period. If waves and current coexist, the combined mixing coefficient is given by:

 ${\displaystyle \epsilon _{cw}={\sqrt {\epsilon _{c}^{2}+\epsilon _{w}^{2}}}}$ (32)

By applying the expression for ${\displaystyle \epsilon _{cw}}$, the concentration profile can be derived from Equation 5, and the conversion parameter ${\displaystyle \beta _{d}}$ calculated. In the AD-model based on the van Rijn equations, two methods are implemented to calculate ${\displaystyle \beta _{d}}$. One is based on an exponential profile employing a depth-averaged mixing coefficient ${\displaystyle {\overline {\epsilon }}_{cw}}$, and the other uses the original van Rijn profile, where the ${\displaystyle \beta _{d}}$ is obtained by numerical integration of Equation 5. Assuming an exponential profile of suspended sediment concentration, ${\displaystyle \beta _{d}}$ is obtained analytically as:

 ${\displaystyle \beta _{d}={\frac {1}{d-a}}{\frac {{\overline {\epsilon }}_{cw}}{w_{f}}}\left[1-exp\left(-{\frac {w_{f}}{{\overline {\epsilon }}_{cw}}}(d-a)\right)\right]}$ (33)

where

 ${\displaystyle {\overline {\epsilon }}_{cw}={\frac {1}{d-a}}\int _{a}^{d}\epsilon _{cw}dz}$ (34)

Lund-CIRP formula

Reference concentration and sediment diffusivity calculated by the Lund- CIRP formula may also be applied in the AD-model. The formulas used for ${\displaystyle c_{R}{\text{ and }}\epsilon }$ are presented in the section describing the Lund-CIRP formula.

## Boundary Conditions

There are three types of boundary conditions in the sediment transport: Wet-dry, Outflow and Inflow.

1. Wet-dry interface.

The interface between wet and dry cells has a zero-flux boundary condition. Both the advective and diffusive fluxes are set to zero at the wet-dry interfaces. Note that avalanching may still occur between wet-dry cells.

2. Outflow Boundary Condition

Outflow boundaries are assigned a zero-gradient boundary condition and sediments are allowed to be transported freely out of the domain.

3. Inflow Boundary Condition

When flow is entering the domain, it is necessary to specify the sediment concentration. In CMS-Flow, the inflow sediment concentration is set to the equilibrium sediment concentation. For some cases, it is desired to reduce the amount of sediment entering from the boundary such as in locations where the sediment source is limited (i.e. coral reefs). The inflow equilibrium sediment concentration may be adjusted by multiplying by a loading scaling factor and is specified by the Advanced Card:
   NET_LOADING_FACTOR <white space> #


## References

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Camenen, B., and Larson, M. (2007). “A unified sediment transport formulation for coastal inlet application”. Technical Report ERDC-CHL CR-07-01. Vicksburg, MS: U.S. Army Engineer Research and Development Center, U.S.A

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