Long time solutions for the 2D inviscid Boussinesq equations with strong stratification

We consider the initial value problem of the 2D inviscid Boussinesq equations for stably stratified fluids. We prove the long time existence of classical solutions for large initial data in H-s(R-2) with s > 2 when the buoyancy frequency is sufficiently high. Furthermore, we consider the singular limit of the strong stratification, and show that the asymptotic profile of the long time solution is given by the corresponding linear dispersive solution.