## 3 Results and Discussion
### Wind Driven Circulation in a Closed Basin  
The circulation in a trapezoidal closed basin was considered in order to provide an assessment of the behaviour of the numerical models corresponding to different ratios between surface and bottom stresses, as depicted in Figure 3. Particular attention is drawn by velocity profiles in points A, B and C of Figure 2 since it is possible to cast an analogy with Couette-Poiseuille turbulent flows and the present circulation. \begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**ROMS** & \(\mathbf{\tau_{b}^{x}}\) & **Relative Error (\%)** & **Delft3D** & \(\mathbf{\tau_{b}^{x}}\) & **Relative Error (\%)** \\ \hline \(k-\epsilon\) & 22.095 & 1.759 & \(k-\epsilon\) & 24.471 & 12.702 \\ \(k-\omega\) & 22.411 & 3.215 & \(k-L\) & 21.537 & 0.811 \\ \(k-kl\) & 20.450 & 5.817 & _Algebraic_ & 21.951 & 1.096 \\ \(M\)Y25 & 16.988 & 21.761 & & & \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Bottom tangential stresses \(\tau_{b}^{x}\) for Delft3D and ROMS at the varying of turbulence models. The relative error refers to theoretical value equal to 21.713 \(\mathrm{N/m^{2}}\). More than three significant digits were employed in the calculations. Figure 5: Comparison of x-component tangential stress profiles for the open channel flow. The profiles are compared with the model output values at the bottom. Profiles are computed since they are not part of the output of the models. Figure 6 shows the vertical profiles of the longitudinal velocity computed in such points. Panel (a) shows the behaviour of the profiles along the lateral banks, i.e., point A. Panel (b) shows that in point B velocities tend to zero with significantly different profiles. This happens since zero tangential stresses are reached in different areas depending on the turbulence closure model adopted. This is confirmed by Figure 7 where tangential stresses are shown for the \(k-\epsilon\) closures for both models (other turbulence models are omitted for the sake of brevity). As far as panel (c) of Figure 6 is concerned, it is clear that algebraic and \(k-L\) model of Delft3D cannot reproduce correctly the flow inversion in point C. Indeed, this panel shows that only two-equation turbulence models can follow the strong gradients expected in this region, providing a positive velocity at the surface and a reversal in the central region. Figure 6: Comparison of x-component velocity profiles for the trapezoidal closed basin. Panels (**a–c**) correspond to the respective panels of Figure 3.  
Despite similarities in the magnitude of the bed shear stresses depicted in Figure 7, the patterns show differences that can be of importance for sediment transport. The activation of bed transport occurs when shear stresses overcome a threshold. Activation of bed transport, therefore, can differ from point to point. Therefore, it is important to observe that the different patterns of bed shear stress determine different positions of the points in which the value of the bed stress is zero. Figure 8 shows the x-component velocity profiles for every turbulence closure taken in exam in accordance of zero shear-stress at the bottom. The position in the basin of these points changes as a function of the turbulence closure model. It is evident that the behaviour of the algebraic and \(k-L\) closures does not manage to capture strong gradients in the velocity profiles. Figure 8 also reports the analytical solution obtained by [32]. They introduced a bi-logarithmic model to describe the velocity profile:  
\[u(z)=Au_{s}^{*}\ln\left[1+\frac{z}{z_{os}}\right]+Bu_{b}^{*}\ln\left[1-\frac{z }{(z_{ob}+H)}\right]+C, \tag{36}\]  
where \(z_{os}\) and \(z_{ob}\) are the reference levels at the free surface and at the bottom; \(H\) is the total depth; and \(u_{s}^{*}\) and \(u_{b}^{*}\) are the friction velocities, respectively, for the free surface and the bottom.