# Development of a Two-Dimensional Hybrid Sediment-Transport Model
## 2 Numerical Model
### Sediment-Transport Model
#### 2.2.1. _Suspended-Load Transport_  
For suspended loads, the governing equation for the \(k\)th size class reads [19]  
\[ \begin{array}{c}\frac{\partial(hC_{k}/\beta_{kl})}{\partial t}+\frac{\partial (hC_{k})}{\partial x}+\frac{\partial(hC_{k})}{\partial y}=\frac{\partial}{ \partial x}[h(\varepsilon_{s}\frac{\partial(C_{k})}{\partial x}+D_{ssk})]+ \frac{\partial}{\partial y}[h(\varepsilon_{s}\frac{\partial(C_{k})}{\partial y }+D_{syk})]\\ +\alpha\omega_{sk}(C_{*k}-C_{k})\ (k=1,2,\ldots,n_{s})\end{array} \] (7a) where \[C\] is the depth-averaged suspended-load concentration by volume (m3 /m 3 ), the concentration by mass is therefore \[\rho_{s}C_{k}\ (\text{kg/m}^{3})\] with \[\rho_{s}\] the sediment density (kg /m 3 ); \[C_{*k}\] is the suspended-load transport capacity; \[\omega_{sk}\] is the settling velocity (m /s); \[\alpha\] is the adaptation coefficient for the suspended load; \[n_{s}\] is the number of size classes; the correction factor \[\beta_{sk}=(\int\limits_{z_{b}+\delta}^{\eta}u_{s}C_{k}dz)/(U_{s}\int\limits_{z _{b}+\delta}^{\eta}C_{k}dz)=\overline{U_{s}}/U\] (7b) represents the lag between the flow and the suspended-load transport ( \[\leq\] 1 ) with \[\delta\] the thickness of the bed-load zone near the bed, \[\rho_{s}C_{k}\ (\text{kg/m}^{3})\]0 the velocity of the suspended load (m /s), and \[\rho_{s}C_{k}\ (\text{kg/m}^{3})\]1 the velocity magnitude (m /s). The suspended-load diffusivity coefficient is \[\rho_{s}C_{k}\ (\text{kg/m}^{3})\]2, with the Schmidt number \[\rho_{s}C_{k}\ (\text{kg/m}^{3})\]3 and the dispersion terms \[\rho_{s}C_{k}\ (\text{kg/m}^{3})\]4 for the nonuniform distributions of flow velocity and sediment concentration over flow depth, with small \[\rho_{s}C_{k}\ (\text{kg/m}^{3})\]5, \[\rho_{s}C_{k}\ (\text{kg/m}^{3})\]6, and \[\rho_{s}C_{k}\ (\text{kg/m}^{3})\]7 in 3D. In this study, the dispersion terms were ignored so that \[\rho_{s}C_{k}\ (\text{kg/m}^{3})\]8 and \[\rho_{s}C_{k}\ (\text{kg/m}^{3})\]9. In Equation (7), there is one calibration parameter, the adaptation coefficient \(\alpha\) for the suspended load, and three input variables, including the settling velocity \(\omega_{s}\), the suspended-load transport capacity C, and the correction factor \(\beta_{s}\), to be determined by semiempirical formulas. As mentioned previously, there are so many formulas available with their own advantages, limitations, and application ranges, this study does not focus on the comparisons of those formulas for \"best picks\"; we selected appropriate formulas that provided reasonable results during the validation process. For the settling velocity, [PERSON]'s formula [18] for naturally-worn sediment particles was used, which assumes a combination of drag force in the laminar region and the turbulent region. \[\rho_{s}\]0  
where \(d\) is the particle diameter (m) and \(\
u\) is the kinematical viscosity (m2/s). [PERSON] [37] and [PERSON] [38] also proposed similar formulas as Equation (8). A variety of formulas have been developed for the suspended-load transport capacity. For example, [PERSON] [39] determined the suspended-load transport rate by integrating the local sediment concentration over the suspended-load zone; [PERSON] [18], based on the energy balance of sediment-laden flow, derived the relation between suspended-load transport capacity C - and parameter \(U^{3}/(gR\omega_{s})\); and [PERSON]'s formula [40] was based onthe stream power concept. The formula proposed by [PERSON] et al. [41], based on the stream power concept, was selected:  
\[\rho_{s}\]1  
where \(q_{s*k}\) is the fractional suspended transport capacity (m\({}^{2}\)/s); \(\gamma_{s}\) and \(\gamma\) are the specific weight (\(N\)) of sediment and water, respectively; \(p_{bk}\) is the fraction of \(k\)th size class; and \(\tau\) is the shear stress of entire cross section (Pa); \(\tau_{ck}=0.03(\gamma_{s}-\gamma)d_{k}(p_{hk}/p_{ek})^{0.6}\) with the hiding probability \(p_{hk}=\sum\limits_{j}p_{bj}d_{j}/(d_{j}+d_{k})\) and the exposure probability \(p_{ek}=\sum\limits_{j}p_{bj}d_{k}/(d_{j}+d_{k})\).