# Development of a Two-Dimensional Hybrid Sediment-Transport Model
## 2 Numerical Model
### Sediment-Transport Model
#### 2.2.3 Bed Changes and Sorting  
The source terms on the right-hand side (RHS) of Equations (7a) and (11) represent the net exchange of sediment between the bed and the flowing water, including the suspended load in the water column and the bed load in the near-bed zone. Therefore, the bed changes can be calculated by  
\[(1-p^{\prime}_{m})\frac{\partial z_{lk}}{\partial t}=\alpha\omega_{sk}(C_{k} -C_{*k})+\frac{1}{L_{b}}(q_{bk}-q_{b*k}) \tag{15}\]  
where \(p^{\prime}_{m}\) is the sediment porosity, and \(z_{lk}\) represents the bed elevation (m) contributed by \(k\)th size class.  
In the vertical direction, the bed can be divided into at least three layers from top to bottom: the mixing layer (also called the active layer), the immediate subsurface layer, and the bottom subsurface layer. The bottom subsurface layer can be further divided into more sublayers. The exchanges of sediment between the water column and the bed leads to bed sorting with dynamic variations of sediment composition in each bed layer.  
As is widely accepted, sediment exchange occurs at the mixing layer. The composition of the mixing layer can be found using a mass-balance approach (e.g., [19; 25; 26; 49]). This study used the equation proposed by [PERSON] [19] for the mixing layer.  
\[\frac{\partial(A_{m}p_{bk})}{\partial t}=\frac{\partial z_{bk}}{\partial t}+p_ {bk}^{*}(\frac{\partial A_{m}}{\partial t}-\frac{\partial z_{b}}{\partial t}) \tag{16}\]  
where \(A_{m}\) is the thickness of the mixing layer and \(p_{bk}\) is the fraction of the \(k\)th size class in the mixing layer. When \(\frac{\partial A_{m}}{\partial t}-\frac{\partial z_{b}}{\partial t}\geq 0\), bed changes remain in the current-mixing layer, so \(p_{bk}^{*}=p_{bk}\); when \(\frac{\partial A_{m}}{\partial t}-\frac{\partial z_{b}}{\partial t}<0\), bed changes penetrate to the immediate subsurface layer, so \(p_{bk}^{*}=p_{ik}\) (the fraction of \(k\)th size class in the immediate subsurface layer).  
The thickness of the bed layers (the mixing layer, the immediate subsurface layer, and the bottom subsurface layer) are user-input variables for the simulation. For the mixing layer, a minimum thickness, \(A_{\text{min}}\), was set to maintain the active sediment exchanges in the mixing layer. This minimum thickness of the mixing layer, \(A_{\text{min}}\), can be either user-specified or determined by \(d_{50}\) and the sand dune height \(\Delta_{s}\)[20].  
\[A_{\text{min}}=\max(2d_{50},0.5\Delta_{s}) \tag{17}\]  
The mixing-layer thickness changes little during the computation, though the division of each layer may move upward with net deposition or downward with net erosion. In such cases, the composition changes of the mixing layer may induce corresponding changes in the other subsurface layers. For example, for the immediate subsurface layer, one can obtain [20]  
\[\frac{\partial(A_{i}p_{ik})}{\partial t}=-p_{kk}^{*}(\frac{\partial A_{m}}{ \partial t}-\frac{\partial z_{b}}{\partial t}) \tag{18}\]  
where \(A_{i}\) is the thickness of the immediate subsurface layer.  
For the nonerodible bed with only net depositions allowed, both erosions and the bed-layer thickness need to be limited.