Ekman layer of rotating stratified viscous Boussinesq equations in rotation-dominant limit

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.