GLOBAL SPLITTING AND REGULARITY OF ROTATING SHALLOW-WATER EQUATIONS

We consider classical shallow-water equations for a rapidly rotating f luid layer, f(0) being the Coriolis parameter with periodic or no-flux boundary conditions. The Poincare/Kelvin linear propagator describes fast oscillating waves for the linearized system. Solutions of the ful l nonlinear shallow-water equations can be decomposed as U (t, x(1), x (2)) = (U) over tilde (t, x(1), x(2)) + W' (t, x(1), x(2)) + x where ( U) over tilde is a solution of the quasigeostrophic equation. We show that the remainder r is uniformly (in initial data and spatial periods which are not resonant) estimated from above by a majorant of order 1 /(f(0) mu) where mu is the Lebesgue measure of almost resonant aspect ratios. The existence on a long time interval T of regular solutions to classical shallow-water equations with general initial data and asp ects ratio (T --> +infinity, as 1/f(0) --> 0) is proven. The vector f ield W' (t, x(1), x(2)) describes the rapidly oscillating ageostrophic component with phase-locked turbulence. This component is exactly sol ved (for generic aspect ratios and symmetric initial data equivalent t o no-flux boundary conditions) in terms of Poincare-Kelvin waves with phase shifts explicitly determined from the nonlinear quasigeostrophic equations.