Long-time existence of smooth solutions for the rapidly rotating shallow-water and Euler equations

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 [Phys. D, 188 ( 2004), pp. 262-276] Liu and Tadmor have shown that the pressureless version of these equations admit a global smooth solution for a large set of subcritical initial configurations. In the present work we prove that when rotational force dominates the pressure, it prolongs the lifespan of smooth solutions for t less than or similar to ln(delta(-1)); here delta << 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 a smooth, "approximate periodic" solution for a time period of days, which is the relevant time period found in NIO obesrvations.