The motion of an incompressible fluid confined to a shallow basin with
a slightly varying bottom topography is considered. Coriolis force,
surface wind and pressure stresses, together with bottom and
lateral friction stresses are taken into account. We introduce
appropriate scalings into a three-dimensional anisotropic eddy
viscosity model; after averaging on the vertical direction and
considering some asymptotic assumptions, we obtain a two-dimensional
model, which approximates the three-dimensional model at the second
order with respect to the ratio between the vertical scale and the
longitudinal scale. The derived model is shown to be symmetrizable
through a suitable change of variables. Finally, we propose some
numerical tests with the aim to validate the proposed model.