Long time existence of smooth solutions for the rapidly rotating shallow-water and Euler equations
Bin Cheng and Eitan Tadmor
Abstract: 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 doi:10.1016/j.physd.2003.07.006 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 prolongs the life-span of smooth solutions for t≲ln(δ-1); here δ≪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 δ 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 days, which is the relevant time period found in NIO obesrvations.