# Development of a Two-Dimensional Hybrid Sediment-Transport Model
## 2 Numerical Model
### Sediment-Transport Model
#### 2.2.2 Bed-Load Transport  
The governing equation for bed loads of \(k\)th size class [19] reads as follows:  
\[\frac{\partial(q_{bk}/u_{bk})}{\partial t}+\frac{\partial(\alpha_{bx}q_{bk})}{ \partial x}+\frac{\partial(\alpha_{by}q_{bk})}{\partial y}=\frac{1}{L_{b}}(q_ {b*k}-q_{bk}) \tag{11}\]  
where \(q_{bk}\) is the bed-load transport rate by volume per unit time and width (m\({}^{2}\)/s); \(q_{b*k}\) is the transport capacity (m\({}^{2}\)/s); \(L_{b}\) is the bed-load adaptation length (m); and the direction cosines of bed-load movement are \(\alpha_{bx}=u/U\) and \(\alpha_{by}=v/U\) with \(U\) the velocity magnitude (m/s) when the effect of the bed slope is ignored. To solve Equation (11), the bed-load velocity formula established by [PERSON] [43] and the bed-load transport capacity formula developed by [PERSON] et al. [41] were adopted. For the bed-load velocity, [PERSON] [43] proposed the concept of the transport stage number \(T\), which was defined as the excess bed-shear stress \(T=(U^{\prime}{}_{*}/U_{scr})^{2}-1\), with \(U^{\prime}{}_{*}=Ug^{0.5}/[18\log(4h/q_{90})]\) the effective shear velocity and \(U_{scr}\) the critical shear velocity given by the Shields diagram. Then, the bed-load velocity is estimated by  
\[u_{b}=1.5T^{0.6}\sqrt{(\rho_{s}/\rho-1)gd}. \tag{12}\]  
The bed-load transport capacity is related to the flow conditions and sediment properties. The formulas for bed-load transport capacity can be categorized into stream-power based (or velocity based) (i.e., Ref. [40]), shear-stress based (i.e., Refs. [44; 45; 46]), and probability-theory based (i.e., Ref. [38]). [PERSON] et al. [41] proposed the use of the nondimensional excess grain shear stress \(T_{k}=(n^{\prime}/n)^{3/2}\tau_{b}/\tau_{ck}-1\) for the estimation  
\[q_{b*k}=0.0053[(\frac{n^{\prime}}{n})^{3/2}\frac{\tau_{b}}{\tau_{ck}}-1]^{2.2} \cdot p_{bk}\sqrt{(\gamma_{s}/\gamma-1)gd_{k}^{3}} \tag{13}\]where \(q_{bsk}\) is the fractional bed-load transport capacity (m\({}^{2}\)/s); \(\tau_{ck}\) is the critical shear stress (Pa); \(n^{\prime}\)= \(a_{50}^{1/6}/20\), and \(n\) is the Manning's roughness coefficient (m\({}^{-1/3}\)s). Note that the hiding and exposure effect in nonuniform bed material is considered through \(\tau_{ck}\). In Equations (7) and (11), the adaptation coefficient, \(\alpha\), for suspended load, and the adaptation length, \(L_{b}\), for bed load, are critical to predicting nonequilibrium sediment transport. Physically, the adaptation length \(L_{b}\) represents the distance required for the sediment transport to move from nonequilibrium to equilibrium conditions, while the adaptation coefficient \(\alpha\) originally is related to the near-bed suspended-load concentration \(C_{b}\), with \(C_{b}=\alpha C\). To be consistent, an adaptation length for the suspended load (m) can also be defined as \(L_{s}=Uh/(\alpha\omega_{s})\). In practice, \(\alpha\) and \(L_{b}\) are case-dependent calibration parameters. The adaptation length for the bed load, \(L_{b}\), is closely related to the bed-form characteristics and can be evaluated using the characteristic lengths, such as channel width, sand-dune length, alternate-bar length, water depth, and mesh-cell length. The adaptation coefficient \(\alpha\) is in the range of [0, 1]. Based on the work of [PERSON] and [PERSON] [47], [PERSON] and [PERSON] [48] proposed the following function to evaluate \(\alpha\):  
\[\alpha=1/\{\frac{a}{h}+(1-\frac{a}{h})\exp[-1.5(\frac{a}{h})^{-1/6}\frac{ \omega_{s}}{U_{*}}]\} \tag{14}\]  
where \(a\) is related to the zero-velocity distance in the logarithmic velocity distribution, defined as \(a=33h/\exp(1+\kappa C_{h}/\sqrt{g})\). The suspended-load transport Equation (7) and the bed-load transport Equation (11) can be combined into a total-load transport equation [20; 26]. The single total-load equation is computationally more convenient since only one equation is solved, though it requires an additional fraction parameter to identify the suspended load and the bed load in the total load.