Vortices in stably-stratified rapidly rotating Boussinesq convection

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.