CMSFlow:Equilibrium Bed load plus AD Suspended load
Equilibrium Bed load plus AdvectionDiffusion Suspened load Transport model
Transport Equation
The transport equation for the suspended load is given by

(1) 
Bed Change Equation
If the advectiondiffusion (AD) equation is selected to simulate the sediment transport and mixing, the change in the water depth is calculated by the sediment continuity equation

(2) 
where is the sediment porosity, and is a bedslope coefficient.
Pickup and Deposition Rate
The AD model calculates the bed level change due to suspended load from the difference between pickup rate and deposition rate in
(3) 
where q_{bx} = bed load transport rate parallel to the xaxis, q_{by} = bed load transport rate parallel to the yaxis, and p = porosity of sediment.
The pickup rate and the deposition rate are also applied as the bottom boundary condition in
(4) 
in which
 C = depthaveraged sediment concentration
 d = total depth (= h+η)
 h = stillwater depth
 η = deviation of the watersurface elevation from the stillwater level
 t = time
 q_{x} = flow per unit width parallel to the xaxis (= ud)
 q_{y} = flow per unit width parallel to the yaxis (= vd)
 u = depthaveraged current velocity parallel to the xaxis
 v = depthaveraged current velocity parallel to the yaxis
 K_{x} = sediment diffusion coefficient for the x direction
 K_{y} = sediment diffusion coefficient for the y direction
 P = sediment pickup rate (upward sediment flux)
 D = sediment deposition rate (downward sediment flux)
The boundary conditions are specified at an arbitrary level above the mean bed level:
(5) 
(6) 
where c = equilibrium concentration of suspended sediment at a given elevation, and z = vertical coordinate. Both c_{a} and c_{0} are reference concentrations defined at z = a. Because the upward flux of sediment depends on the bed shear stress, c_{a} is determined from the bed shear stress calculated from the local hydrodynamic conditions. Representation of c_{a} within CMSM2D is dependent on selection of either the van Rijn or LundCIRP models. The downward sediment flux depends on the concentration in the upper water column; therefore, c_{0} is specified from solution of Equation 4.
Assuming that the suspended concentration is in equilibrium , then the basic equation for suspended sediment concentration can be written:
(7) 
This equation can be solved analytically by applying an appropriate mixing coefficient , and the vertical profile of suspended concentration is obtained in the following form:
(8) 
where F(z) is a function of the vertical concentration distribution. The relationship between the reference concentration c_{0} and the depthaveraged concentration C is:
(9) 
Thus, c_{0} may be written in the following form by introducing a conversion function :
(10) 
The bed level change due to suspended load is based upon the difference of
the two types of reference concentration:
(11) 
This equation implies that erosion occurs if c_{a} > c_{0}, and accretion occurs if c_{a} < c_{0}.
In the AD model, three methods of specifying c_{a} and c_{0} (that is, ) are implemented. Two of the methods are based on the van Rijn formula (van Rijn 1985) and one on the LundCIRP formula (Camenen and Larson 2006). Table 1 summarizes general features of the methods.
van Rijn formula
The reference concentration c_{a} in the van Rijn formula is given by:
(12) 
where a = reference height and = maximum bed shear stress given by
(13) 
Table 1 Features of Calculation of Pickup and Deposition Rates 


Method  Reference Concentration  Conversion Parameter  Comments 
Exponential Profile  Eq. 10  Eq. 24 and 25  Fast computation. Tends to overestimate sediment transport rate. Can be used for some tests. 
Van Rijn Profile  Eq. 10  Solves Eq. 5 numerically with Eq. 23 (RungeKutta 4^{th})  Requires substantial computing time. Provides the same results as van Rijn (1985). 
LundCIRP Profile  Eq. 70  Eq. 24 and 72  Fast computation. Newly developed sediment transport formula. 
The currentrelated shear stress is calculated from:
(14) 
The waverelated shear stress is given as:
(15) 
(16) 
where k_{s}^{'} = roughness height defined as:
(17) 
in which k_{sd} and k_{ss} are calculated from
(18) 
and
(19) 
respectively.
Bed concentration in the van Rijn model is defined at the height a as:
(20) 
where H_{r} = ripple height. If ripples are present, the total roughness height is modified as:
(21) 
where, is specified for the van Rijn formula, and is specified for the LundCIRP formula. The ripple dimension is obtained by selecting the largest ripple height for the case of current or waves
(22) 
(23) 
The conversion parameter to determine 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 currentrelated mixing coefficient is given by:
(24) 
(25) 
where u_{*c} = currentrelated bed shear velocity expressed as:
(26) 
and is a coefficient obtained from:
(27) 
(28) 
Figure 1. Vertical distributions of mixing coefficient due to current and waves
The waverelated mixing coefficient is:
(29) 
where
(30) 
(31) 
in which the parameter = height from the bed given by significant wave height, and T = significant wave period. If waves and current coexist, the combined mixing coefficient is given by:
(32) 
By applying the expression for , the concentration profile can be derived from Equation 5, and the conversion parameter calculated. In the ADmodel based on the van Rijn equations, two methods are implemented to calculate . One is based on an exponential profile employing a depthaveraged mixing coefficient , and the other uses the original van Rijn profile, where the is obtained by numerical integration of Equation 5. Assuming an exponential profile of suspended sediment concentration, is obtained analytically as:
(33) 
where
(34) 
LundCIRP formula
Reference concentration and sediment diffusivity calculated by the Lund CIRP formula may also be applied in the ADmodel. The formulas used for are presented in the section describing the LundCIRP formula.
Boundary Conditions
There are three types of boundary conditions in the sediment transport: Wetdry, Outflow and Inflow.
1. Wetdry interface.
 The interface between wet and dry cells has a zeroflux boundary condition. Both the advective and diffusive fluxes are set to zero at the wetdry interfaces. Note that avalanching may still occur between wetdry cells.
2. Outflow Boundary Condition
 Outflow boundaries are assigned a zerogradient 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 CMSFlow, 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> #
 where # is the loading factor in dimensionless units.
References
Buttolph, A. M., C. W. Reed, N. C. Kraus, N. Ono, M. Larson, B. Camenen, H. Hanson, T. Wamsley, and A. K. Zundel. (2006). “Twodimensional depthaveraged circulation model CMSM2D: Version 3.0, Report 2: Sediment transport and morphology change.” Coastal and Hydraulics Laboratory Technical Report ERDC/CHL TR069. Vicksburg, MS: U.S. Army Engineer Research and Development Center, U.S.A.
Camenen, B., and Larson, M. (2007). “A unified sediment transport formulation for coastal inlet application”. Technical Report ERDCCHL CR0701. Vicksburg, MS: U.S. Army Engineer Research and Development Center, U.S.A
Soulsby, R. L. (1997). "Dynamics of marine sands, a manual for practical applications". H. R. Wallingford, UK: Thomas Telford.
Watanabe, A. (1987). “3dimensional numerical model of beach evolution”. Proc. Coastal Sediments ’87, ASCE, 802817.
Wu, W. (2004).“Depthaveraged 2D numerical modeling of unsteady flow and nonuniform sediment transport in open channels”. J. Hydraulic Eng., ASCE, 135(10), 1013–1024.
van Rijn, L. C. (1985). “Flume experiments of sedimentation in channels by currents and waves.” Report S 347II, Delft Hydraulics laboratory, Deflt, Netherlands.
Zhu, J. (1991). “A low diffusive and oscillationfree convection scheme”. Com. App. Num. Meth., 7, 225232.
Zundel, A. K. (2000). “Surfacewater modeling system reference manual”. Brigham Young University, Environmental Modeling Research Laboratory, Provo, UT.
External Links
 Aug 2006 TwoDimensional DepthAveraged Circulation Model CMSM2D: Version 3.0, Report 2, Sediment Transport and Morphology Change [1]
 Aug 2008 CMSWave: A Nearshore Spectral Wave Processes Model for Coastal Inlets and Navigation Projects [2]