Nonstationary models for shallow lakes

In this paper we study the behaviour of solutions of Navier–Stokes equations with adherence to the bottom and traction by wind at the surface when the aspect ratio $\delta=\hbox{depth}/\hbox{lenght}$ of the domain tends to 0. Precisely, we prove that when wind is moderate, a variational solution converges to the solution of a vertical diffusion model. This model is either quasistationary or nonstationary, according to the characteristic time.