Vortices in stably-stratified rapidly rotating Boussinesq convection

TL;DR

This paper investigates the behavior of vortices in rapidly rotating, stably-stratified fluids using the Boussinesq approximation, proving global existence of solutions and describing their asymptotic vortex structures.

Contribution

It provides a rigorous analysis of vortex formation and decay in rotating stratified fluids, including convergence to an Oseen vortex and detailed asymptotics.

Findings

Barotropic vorticity converges to an Oseen vortex with algebraic rate.

Vertical velocity and thermal fluctuations decay as Gaussians with oscillating amplitudes.

Global-in-time existence of solutions under certain initial conditions.

Abstract

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.