Global splitting and regularity of rotating shallow-water equations

We consider classical shallow-water equations for a rapidly rotating fluid layer, f₀ 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 full nonlinear shallow-water equations can be decomposed as U(t, x₁, x₂) = Ū(t, x₁, x₂)+W′(t, x₁, x₂)+r where Ū 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/(f₀μ) where μ 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 aspects ratio (T*→+∞, as 1/f₀→0) is proven. The vector field W′(t, x₁, x₂) describes the rapidly oscillating ageostrophic component with phase-locked turbulence. This component is exactly solved (for generic aspect ratios and symmetric initial data equivalent to no-flux boundary conditions) in terms of Poincare-Kelvin waves with phase shifts explicitly determined from the nonlinear quasigeostrophic equations.