  We consider the two-dimensional shallow water model derived by Levermore and
Sammartino (Nonlinearity 14,2001), describing the motion of an incompressible
fluid, confined in a shallow basin, with varying bottom topography. We
construct the approximate inertial manifolds for the associated dynamical
system and estimate its order. Finally, considering the whole domain R^2 and
under suitable conditions on the time dependent forcing term, we prove the L^2
asymptotic decay of the weak solutions.
