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In this paper, we investigate the rotation-dominant three-scale singular limit of the initial-value problem to the inviscid rotating stratified Boussinesq equations. We first prove a dispersive decay estimate for the linear propagator of that limit regime, which requires novel techniques involving an elaborate frequency cut-off since the phase function is degenerate on both horizontal and vertical directions in some areas. Using the dispersive decay estimate, we then investigate rigorously the rotation-dominant limit of local strong solutions to Boussinesq equations with ill-prepared initial data. The limiting system is shown to be the two-dimensional incompressible Euler equations coupled with a two-dimensional transport equation but in three spatial dimensions. [ABSTRACT FROM AUTHOR]

The asymptotics of weak solutions to the Boussinesq equations with no-slip boundary and moderately ill-prepared data is investigated in rotation-dominant limit regime as the Rossby number, the Froude number and the vertical viscosity tend to zero simultaneously. The new ingredient of this paper is to give a first proof of the three scale singular limit coupled with Ekman boundary layer by introducing an asymptotic profile to the original system. [ABSTRACT FROM AUTHOR]

Singular limits of the rotating stratified Boussinesq equations are considered as the Froude number and Rossby number tend to zero at different rates. The limiting dynamics are validated for the rotation-dominant and the stratification-dominant cases, respectively. Furthermore, the convergence rate estimates are obtained. [ABSTRACT FROM AUTHOR]