Resonances and regularity for Boussinesq equations

We consider classical Boussinesq equations for a rotating stably stratified fluid with large N₀ = N/F, where N₀ describes the stratification, Ω₀ = 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, x₁, x₂, x₃) = Ũ(t, x₁, x₂, x₃) + W^g(t, x₁, x₂, x₃) + 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, x₁, x₂, x₃) 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 → ∞).