  The work addresses a singular limit for a rotating compressible Euler system
in the low Mach number and low Rossby number regime. Based on the concept of
dissipative measure-valued solution, the quasi-geostrophic system is identified
as the limit problem in case of ill-prepared initial data. The ill-prepared
initial data will cause rapidly oscillating acoustic waves. Using dispersive
estimates of Strichartz type, the effect of the acoustic waves in the
asymptotic limit is eliminated.
