Bed-Slope Term

The Finite-Volume discretization of the bed-slope term is

1)	$\begin{array}{l}\displaystyle S_{bki} = \frac{w_i}{A^W} \sum _f \kappa _{bkf} \left\| q_{bkf} \right\| \nabla _{\bot} z_{bf}^W L_f^W\end{array}$

where

$\begin{array}{l}w_{i}\end{array}$ : weights applied to each subarea [-]

$\begin{array}{l}A^{W}\end{array}$ : cell wet area [-]

$\begin{array}{l}κ_{bkf}\end{array}$ : non-dimensional bed-slope coefficient at face f [-]

$\begin{array}{l}|q_{bkf}|\end{array}$ : bed-load mass transport magnitude at face [M/L/T]

$\begin{array}{l}∇_⊥{z_{bf}}^W\end{array}$ : face-normal gradient of wet elevations at face f [L]

$\begin{array}{l}{L_{f}}^{W}\end{array}$ : length of wet portion of face f [L]

The be-slope term is computed within a subcell by applying depth weighting in order to capture the effect in which deeper parts of cell are expected to experience larger bed change compared to less deep portions of the cell. The weights are computed as

2)	$\begin{array}{l}\displaystyle w_i = \frac{h_i^b}{\sum_k h_k^b}\end{array}$

where

$\begin{array}{l}h_{i}\end{array}$ : subarea water depth [L]

$\begin{array}{l}b\end{array}$ : nondimensional empirical coefficient (b ≥ 0)

A coefficient b = 0 represents applying the same slope-induced bed change to each subregion.