To attain this model asymptotic analysis has been applied, but not in the original domain as generally is done in the case of fluids [1], and the surface has not been supposed to be flat [2], we have preferred to use the technique in the same way as in [3].

A change of variable from a reference domain (independent of the parameter and time) to the original domain is defined to move the dependence on epsilon from the domain to the functions. Then, the asymptotic technique is employed to study what happens when epsilon becomes small. We assume that the solution to the problem in the reference domain allows an expansion in powers of epsilon, replacing this expansion in the equations and identifying the terms multiplied by the same power of epsilon; the equations obtained are used to determine them. Finally, an approximation to the solution in the reference domain is built and we arrive at a model in the original domain.

Formal asymptotic analysis allows us to obtain a shallow water model, without making a priori assumptions, that generalizes the classic shallow water model, providing a horizontal velocity with explicit dependence on z if any of the two first components of the vorticity is not zero. This fact represents an interesting novelty respect to the other shallow water models that can be found in the literature.

The comparisons with exact solutions to Euler equations prove that our model always achieves better results, if the solution depends on z, than the classical shallow water model, whose errors can be very big for values of z away from the medium depth. We have approximated the vorticity by a polynomial in z and we have seen that the greater the degree of this polynomial is, more accurate solutions we obtain.

Numerical experiments confirm that our model gives the same precision for all z (if the degree of precision used to approximate the vorticity is large enough) than the classical model for the averaged velocity, so we can consider our model as an improvement of the classical model.

1

J.-F. Gerbeau, B. Perthame, "Derivation of viscous Saint-Venant system for laminar shallow water; numerical validation", Discrete and Continuous Dynamical Systems-Series B, 1(1), 89-102, 2001. doi:10.3934/dcdsb.2001.1.89

2

P. Azérad, F. Guillén, "Mathematical justification of the hydrostatic approximation in the primitive equations of geophysical fluid dynamics", SIAM J. Math. Anal., 33(4), 847-859, 2001. doi:10.1137/S0036141000375962

3

P.G. Ciarlet, "Mathematical Elasticity. Volume II: Theory of Plates", North-Holland, 1997.

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