### Abstract

We consider classical shallow-water equations for a rapidly rotating fluid 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 full nonlinear shallow-water equations can be decomposed as U(t, x_{1}, x_{2}) = Ū(t, x_{1}, x_{2})+W′(t, x_{1}, x_{2})+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/(f_{0}μ) 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/f_{0}→0) is proven. The vector field W′(t, x_{1}, x_{2}) 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.

Original language | English (US) |
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Pages (from-to) | 725-754 |

Number of pages | 30 |

Journal | European Journal of Mechanics, B/Fluids |

Volume | 16 |

Issue number | 5 |

State | Published - Jan 1 1997 |

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### ASJC Scopus subject areas

- Mathematical Physics
- Physics and Astronomy(all)

### Cite this

*European Journal of Mechanics, B/Fluids*,

*16*(5), 725-754.