  We study the stabilizing effect of rotational forcing in the nonlinear
setting of two-dimensional shallow-water and more general models of
compressible Euler equations. In [H. Liu and E. Tadmor, Phys. D 188 (2004), no.
3-4, 262-276] we have shown that the pressureless version of these equations
admit global smooth solution for a large set of sub-critical initial
configurations. In the present work we prove that when rotational force
dominates the pressure, it \emph{prolongs} the life-span of smooth solutions
for t < ln(1/d); here d << 1 is the ratio of the pressure gradient measured by
the inverse squared Froude number, relative to the dominant rotational forces
measured by the inverse Rossby number. Our study reveals a ``nearby''
periodic-in-time approximate solution in the small d-regime, upon which hinges
the long time existence of the exact smooth solution. These results are in
agreement with the close-to periodic dynamics observed in the ``near inertial
oscillation'' (NIO) regime which follows oceanic storms. Indeed, our results
indicate the existence of smooth, ``approximate periodic'' solution for a time
period of \emph{days}, which is the relevant time period found in NIO
obesrvations.
