## 2 Material and Methods
### Description of the Test Cases
#### 2.2.2 Wind Driven Trapezoidal Closed Basin  
The wind driven circulation in a closed basin consists of a squared closed basin with a trapezoidal cross-section where the motion is induced only by a steady wind blowing in the x-direction. Figure 2 shows the geometry of the basin in blue. The basin is 10 km \(\times\) 10 km with a maximum depth of 15 m in the flat central region of 5 km width. Lateral banks have a water depth that linearly decreases from 5 m to 15 m along 2.5 km. The numerical grid is made of square cells of 100 m \(\times\) 100 m side and the vertical is discretised with 40 equidistant layers. Layer b) of Figure 2 shows vectors representing the horizontal velocity field induced by wind, and the free surface is depicted in red in Layer c). The simulation parameters are summarised in Table 3. \begin{table}
\begin{tabular}{c c c} \hline \hline
**Model Parameter** & **Variable** & **Value** \\ \hline Length, Width & \(L\), \(W\) & 1000, 50 m \\ Bed Slope & \(i_{f}\) & 0.001 \\ Number of grid spacing & \(Lm\), \(Mm\), \(Nm\) & 200, 10, 40 \\ Bottom roughness & \(z_{r}\) & 0.03 m \\ Time step & \(\Delta t\) & 1 s \\ Total simulation time & \(t_{end}\) & 86,400 s \\ Uniform water depth & \(H\) & 2.428 m \\ Hydraulic Radius & \(R_{i}\) & 2.213 m \\ Flow, Velocity & \(Q\), \(U\) & 300 m\({}^{3}\)/s, 2.47 m/s \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Parameter for the open channel flow test case. \begin{table}
\begin{tabular}{c c c} \hline \hline
**Model Parameter** & **Variable** & **Value** \\ \hline Length, Width & \(L\), \(W\) & 10,000, 10,000 m \\ Depth & \(H\) & from \(-5\) m to \(-15\) m \\ Number of grid spacing & \(Lm\), \(Mm\), \(Nm\) & 100, 100, 40 \\ Bottom roughness & \(z_{r}\) & 0.03 m \\ Time step & \(\Delta t\) & 1 s \\ Total simulation time & \(t_{end}\) & 86,400 s \\ Wind Driven Surface Stress & \(\tau_{x}^{\mathrm{s}}\) & 0.325 N/m\({}^{2}\) \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Model parameters for the wind driven trapezoidal closed basin test case. In a depth-varying cross-section, vertical profiles of velocities are quite different moving along the transverse direction. Such velocity profiles were studied in detail by [31, 32, 35, 36]. In the shallow regions close to the lateral banks, flow velocity is always positive (i.e., in the wind direction), while a reverse flow occurs in the central region. Zero-mean flow only appears at two symmetrical locations in the cross-section, corresponding to the centre of the two counter-rotating cells of the depth-averaged circulation pattern, visible in Figure 2.  
A condition that can be safely assumed to be valid in any region of the basin is a linear distribution of the shear stress along the vertical, with the only exception of the area close to the leeward and  
Figure 3: Shear stresses and velocity profiles moving from the lateral banks to the centre of the basin. (**a**) shows a profile where surface and bed shear stresses have concordant signs. This profile is associated to point A of Figure 2. The velocity profile is that of a turbulent Couette–Poiseuille flow. (**b**) shows a zero tangential stress at the bed that occurs at the transition from the lateral bank to the centre of the channel. It refers to point B of Figure 2. The velocity profile tends to a zero-gradient towards the bottom. (**c**) shows discordant shear stresses at the surface and at the bed. This condition occurs at the centre of the channel; i.e., point C of Figure 2. The velocity profile shows an inversion along the water column. Figure 2: (**a**) Geometry of the trapezoidal closed basin in blue. (**b**) Vectors in black show the presence of two counter-rotating vortices. (**c**) The free surface is depicted in red. The set-up is highlighted through the colour shading. Points A, B and C refer to the results reported in the respective panels of Figures 3 and 6.  
windward banks, where the flow is strongly three-dimensional.