  We study the Boussinesq approximation for rapidly rotating stably-stratified
fluids in a three dimensional infinite layer with either stress-free or
periodic boundary conditions in the vertical direction. For initial conditions
satisfying a certain quasi-geostrophic smallness condition, we use dispersive
estimates and the large rotation limit to prove global-in-time existence of
solutions. We then use self-similar variable techniques to show that the
barotropic vorticity converges to an Oseen vortex, while other components decay
to zero. We finally use algebraically weighted spaces to determine leading
order asymptotics. In particular we show that the barotropic vorticity
approaches the Oseen vortex with algebraic rate while the barotropic vertical
velocity and thermal fluctuations go to zero as Gaussians whose amplitudes
oscillate in opposite phase of each other while decaying with an algebraic
rate.
