Analysis of some shallow water problems with rigid-lid hypothesis

We present in this paper the analysis and the comparison of two two-dimensi onal geophysical flow problems using a rigid-lid approximation (i.e. we do not take into account the variation of surface elevation xi). A first rigid -lid shallow water model (noted SWRL) is obtained by neglecting the variati on of the surface in a weak formulation of a usual viscous shallow water mo del in depth-mean velocity formulation (noted SWFS for shallow water with f ree surface). We establish some existence results for this model and we pro pose a numerical resolution method. The second model we consider is an adap tation of the lake equations proposed by Levermore, Oliver and Tit,(4) in w hich we take into account the viscous effects, in order to compare the two approaches. For the numerical resolution, we apply the curl operator on the se equations and we propose a numerical algorithm to solve this problem, th at we note SWLV (shallow water for the lake with viscosity). We present fin ally some comparative results in an idealized configuration between the SWR L, SWLV and SWFS models (in the case where the rigid-lid approximation seem s to be reasonable).