## 4 Hybrid Modeling
### Numerical Simulations
 \tag{5}\]  
where \(Z_{\mathrm{b}}\) (m) = change in bed elevation, \(\gamma_{0}\) (kg/m\({}^{3}\)) = bulk weight of bed material, \(D_{\mathrm{b}}\) (kg/(m\({}^{2}\)s)) = sediment flux of deposition, \(E_{\mathrm{b}}\) (kg/(m\({}^{2}\)s)) = sediment flux of erosion, \(\tau\) (N/m\({}^{2}\)) = bed shear stress, \(\tau_{\mathrm{d}}\) and \(\tau_{\mathrm{e}}\) (N/m\({}^{2}\)) = critical stresses of deposition and erosion, \(M\) (kg/m\({}^{2}\)s) = bed scouring rate, and \(S_{\mathrm{b}}\) (kg/m\({}^{3}\)) = near-bottom sediment concentration. In light of the van [PERSON] formula [44], the bed load in the form of fine sand is entrained into the flow by imposing a concentration \(S_{\mathrm{b}}\) (kg/m\({}^{3}\)) at a reference height above the bottom (delimiting bed load and suspended load), which is dependent on current-related effective roughness height. \(S_{\mathrm{b}}\) (kg/m\({}^{3}\)) is calculated by:  
\[S_{\mathrm{b}}=0.015\rho_{\mathrm{s}}\frac{D_{50}T^{1.5}}{aD_{*}0.3} \tag{6}\]  
where \(\rho_{\mathrm{s}}\) (kg/m\({}^{3}\)) = density of sediment material, \(D_{*}\) = dimensionless particle parameter, and \(T\) = dimensionless bed shear stress. According to the van [PERSON] formula [44], the transport rate of bed load, \(q_{\mathrm{b}}\) (kg/(m\(\cdot\)s)), is expressed as  
\[q_{\mathrm{b}}=0.006\rho_{\mathrm{s}\mathrm{s}}D_{50}M_{\mathrm{s}}^{0.5}M_{ \mathrm{e}}^{0.7} \tag{7}\]  
The transport rate of suspended load, \(q_{\mathrm{s}}\) (kg/(m\(\cdot\)s)), reads  
\[q_{\mathrm{s}}=f_{\mathrm{c}}HUS_{\mathrm{b}} \tag{8}\]  
where \(M_{\mathrm{s}}\) = sediment mobility number, \(M_{\mathrm{e}}\) = excess sediment mobility number, \(f_{\mathrm{c}\mathrm{s}}\) = a shape factor, \(H\) (m) = water depth, and \(U\) (m/s) = depth-averaged velocity. The computations are performed in prototype size, with the same river length (1200 m) as in the physical model studies. Starting with a coarse mesh, global and local refinements are made to achieve a fine mesh. Several meshes of varied cell size are evaluated to ensure grid independence, which is checked through steady-state calculations. Figure 9 shows the horizontal mesh, with an enlarged view at the multigate shucice. Irrespective of local water depth, the shucice piers are treated as dry cells. The guide vanes are represented by an array of infinitely thin objects that prohibit flow exchange between the two adjacent cells. The vane crest can be below, at, or above the water surface. In the plan, the domain is covered by 15,350 cells, comprising 307 streamwise and 50 transverse. Smaller cells, 1.5 m in size, are used in the shuice to account for large velocity gradients. Boundary conditions are defined in terms of both flow and sediment, which are based on in situ measurements. At the upstream boundary, \(Q\) and \(S\) are given; at the downstream one, \(Z_{\text{d}}\) is specified (Table 1). For the riverbed and banks, a nonentry condition specifies a zero-normal gradient of sediment content. Table 3 summarizes the other model parameters, i.e., \(\omega_{\text{s}}\), \(\tau_{\text{d}}\), \(\tau_{\text{e}}\), \(M\) and \(\gamma_{0}\). To achieve numerical stability, the time step \(\Delta t\) = 0.01 s is chosen. In a coupled process, the flow, sediment, and morphology are updated from one time step to another. The simulation terminates when the solution becomes independent of iterations. For computing a steady-state case, the wall-clock time is \(\sim\)12 h.