ANALYSIS OF SOME SHALLOW WATER PROBLEMS WITH RIGID-LID HYPOTHESIS

We present in this paper the analysis and the comparison of two two-dimensional 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 variation of the surface in a weak formulation of a usual viscous shallow water model in depth-mean velocity formulation (noted SWFS for shallow water with free surface). We establish some existence results for this model and we propose a numerical resolution method. The second model we consider is an adaptation of the lake equations proposed by Levermore, Oliver and Tit,(4) in which we take into account the viscous effects, in order to compare the two approaches. For the numerical resolution, we apply the curl operator on these equations and we propose a numerical algorithm to solve this problem, that we note SWLV (shallow water for the lake with viscosity). We present finally some comparative results in an idealized configuration between the SWRL, SWLV and SWFS models (in the case where the rigid-lid approximation seems to be reasonable).