## 2 Pollutant Transport by Shallow Water Flow  
Pollutant transport by shallow water flow can be modeled by shallow water equations coupled with the advection-diffusion equation. The system of shallow water equations is obtained by integrating the Navier-Stokes equations over the depth of the water. This system of equations is used to describe the flow of water in cases where the depth is much smaller than the horizontal scale of motion. These equations are used to model flows in shallow coastal zone, lakes, and rivers, among other phenomena. They are also used in simulating river flooding. Whenever shallow water transports non-reactive pollutants, the shallow water equations are coupled with the advection-diffusion equation, with the resulting global system written in a conservative form, being as follows:  
\[ \begin{cases}\partial_{t}h+\partial_{x}(q_{x})+\partial_{y}(q_{y})=0\\ \partial_{t}(q_{x})+\partial_{x}\big{(}\frac{q_{x}^{2}}{h}+\frac{1}{2}gl^{2} \big{)}+\partial_{y}\big{(}\frac{q_{x}q_{y}}{h}\big{)}+gh\partial_{x}z+gh\tau_ {x}-F_{x}=0\\ \partial_{t}(q_{y})+\partial_{x}\big{(}\frac{q_{x}q_{y}}{h}\big{)}+\partial_{ y}\big{(}\frac{q_{y}^{2}}{h}+\frac{1}{2}gh^{2}\big{)}+gh\partial_{y}z+gh\tau_{y}-F_{y}=0\\ \partial_{t}(q_{c})+\partial_{x}\Big{[}\frac{q_{x}q_{c}}{h}+hK_{xx}\partial_ {x}\big{(}\frac{q_{c}}{h}\big{)}\Big{]}+\partial_{y}\Big{[}\frac{q_{y}q_{c}}{h }+hK_{yy}\partial_{y}\big{(}\frac{q_{c}}{h}\big{)}\Big{]}=0\end{cases}  \tag{1}\]  
where \(q_{x}=hu\) and \(q_{y}=hv\) are the flow discharge, \(u\) and \(v\) being the average velocities in the horizontal \(x\), \(y\) plane. Gravity is taken along the \(z\) axis with acceleration \(g\). We define \(z=z(x,y)\) as the bottom of the river and \(h(x,y)+z(x,y)\) as the elevation of the free surface, where \(h\) is the water height. The quantities \(F_{x}\) and \(F_{y}\) are forces that represent the Coriolis acceleration or the drag force due to wind, and \(\tau_{x}\) and \(\tau_{y}\) are the bottom frictional stresses defined by [PERSON]'s law:  
\[\tau_{x}=\frac{\eta^{2}q_{x}\sqrt{q_{x}^{2}+q_{y}^{2}}}{h^{\frac{10}{3}}}, \hskip 14.226378 pt\tau_{y}=\frac{\eta^{2}q_{y}\sqrt{q_{x}^{2}+q_{y}^{2}}}{h^{ \frac{10}{3}}} \tag{2}\]  
where \(\eta\) is the Manning coefficient. In the advection-diffusion equation, \(q_{c}\) represents the concentration of the pollutant and \(K_{xx}\) and \(K_{yy}\) are the elements of the diagonal diffusion tensor of the pollutant. It should be noted that the diffusion coefficients depend on the flow velocity, the water depth, the wind and also the vertical turbulence. In this paper, the calculation of these terms is not considered. For more details on how to calculate the diffusion tensor elements in the context of shallow water flows, refer to the work of [PERSON] [1]. We simplify the system of Equation (1) by rewriting it as:  
\[\partial_{t}\mathbf{Q}+\mathcal{F}(\mathbf{Q})+\mathcal{G}(\mathbf{Q})=\mathbf{ S}(\mathbf{Q}) \tag{3}\]  
with  
\[\mathbf{Q}=\left[ \begin{array}{c}h\\ q_{x}\\ q_{y}\\ q_{c}\end{array} \right],\,\mathcal{F}(\mathbf{Q})=\partial_{x}\left[ \begin{array} []{c}q_{x}\\ \frac{q_{x}^{2}}{h}+\frac{1}{2}gh^{2}\\ \frac{q_{x}q_{y}}{h}\\ \frac{q_{x}q_{c}}{h}+hK_{xx}\partial_{x}\big{(}\frac{q_{c}}{h}\big{)}\end{array}  \right],\,\mathcal{G}(\mathbf{Q})=\partial_{y}\left[ \begin{array}{c}q_{y}\\ \frac{q_{y}^{2}y}{h}\\ \frac{q_{y}^{2}y}{h}+\frac{1}{2}gh^{2}\\ \frac{q_{y}q_{c}}{h}+hK_{yy}\partial_{y}\big{(}\frac{q_{c}}{h}\big{)}\end{array} \right] \tag{4}\]  
The source term \(\mathbf{S}(\mathbf{Q})\), including differential and scalar terms, is expressed as follows:  
\[\mathbf{S}(\mathbf{Q})=\mathcal{S}_{b}(\mathbf{Q})+\mathbf{S}_{f}(\mathbf{Q})+ \mathbf{S}_{s}(\mathbf{Q}) \tag{5}\]  
where \(\mathbf{S}_{b}(\mathbf{Q})\) is the bed-slope term, \(\mathbf{S}_{f}(\mathbf{Q})\) is the bottom friction term, and \(\mathbf{S}_{s}(\mathbf{Q})\) is the other scalar forces described in the system (1).