Summary: We study the variable bathymetry shallow water equations in a channel with a prescribed Flather condition at the open ocean-channel boundary denoted \(\sum_\ell\). The flow in the vicinity of this boundary is influenced by the combined effect of the tide and a vortex created by a singular punctual source. Using the Crank-Nicholson scheme, a semi-discrete model in time is first obtained. The numerical discrete model is then built with the \(P_1^{NC}-P_1\) mixed finite element method. We show the existence and uniqueness of the solution of the discrete model \((\mathbf{u}_h, \eta_h)\) in the appropriate space \(( \mathcal{V}_h\times{\mathcal{Q}_h}_{/Ker \mathbb{B}_h^t} )\). When the flow is only influenced by the tide, the evolution in time of the elevation of the free surface at different points of the channel predicted by the model is in agreement with the experimental results observed in the Vridi channel. Furthermore, we show that the elevation of the free surface at a given point of the channel increases with the circulation of the vortex.
For the entire collection see [Zbl 1515.35013].