### Abstract

We consider classical Boussinesq equations for a rotating stably stratified fluid with large N_{0} = N/F, where N_{0} describes the stratification, Ω_{0} = NΩ is the rate of rotation, and N is a large scaling parameter. Solutions of full nonlinear Boussinesq equations have a decomposition of the form U(t, x_{1}, x_{2}, x_{3}) = Ũ(t, x_{1}, x_{2}, x_{3}) + W^{g}(t, x_{1}, x_{2}, x_{3}) + r, where Ũ is a solution of the quasigeostrophic equation and r is a remainder, which is uniformly bounded above by a majorant of the order of 1/N. The vector field W^{g}(t, x_{1}, x_{2}, x_{3}) describes the rapidly oscillating gravity wave component The "amplitude" of this component describes the propagation of slow waves, and it satisfies a linear equation with coefficients determined by the quasigeostrophic component found from the nonlinear quasigeostrophic equations. The control of the error r based on estimates related to small denominators, for generic values of parameters, is used to prove the existence, on a long time interval T*, of regular solutions to classical Boussinesq equations with general initial data (T* → + ∞ as N → ∞).

Original language | English (US) |
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Pages (from-to) | 417-428 |

Number of pages | 12 |

Journal | Russian Journal of Mathematical Physics |

Volume | 4 |

Issue number | 4 |

State | Published - Dec 1 1996 |

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### ASJC Scopus subject areas

- Statistical and Nonlinear Physics
- Mathematical Physics

### Cite this

*Russian Journal of Mathematical Physics*,

*4*(4), 417-428.