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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]

A dispersive decay estimate is proven for the linear propagator of the three-scale rotating stratified Boussinesq equations in the stratification-dominant regime. Because the phase function of that propagator is both singular and degenerate, obtaining the estimate requires novel techniques involving cutting phase-space shells into several pieces and using different methods in each region. Using the decay estimate, we prove the long-time existence of solutions to the initial-value problem for the nonlinear equations and obtain a rate of convergence to the limit system and intermediate asymptotics. [ABSTRACT FROM AUTHOR]