  In this paper, we present an original derivation process of a non-hydrostatic
shallow water-type model which aims at approximating the incompressible Euler
and Navier-Stokes systems with free surface. The closure relations are obtained
by aminimal energy constraint instead of an asymptotic expansion. The model
slightly differs from thewell-known Green-Naghdi model and is confronted with
stationary andanalytical solutions of the Euler system corresponding to
rotationalflows. At the end of the paper, we givetime-dependent analytical
solutions for the Euler system that are alsoanalytical solutions for the
proposed model but that are not solutionsof the Green-Naghdi model. We also
give and compare analytical solutions of thetwo non-hydrostatic shallow water
models.
