A mathematically rigorous analysis of forced axisymmetric flows in the atmosphere

This note discusses a rigorous mathematical formulation for the evolution of the Eliassen balanced vortex. It is first shown that a stable balanced vortex of finite extent can be embedded in an ambient fluid at rest, and that such a vortex exists for prescribed angular momentum and potential temperature on fluid parcels. This uses a method developed by Shutts, Booth and Norbury. This is a different way of viewing the problem from the normal methods, which analyse the stability of a prescribed vortex. The stability of the vortex depends on the presence of background rotation and on the azimuthal velocity at the boundary of the vortex being less than that in the surrounding ambient fluid. It is then shown that the evolution of this vortex under axisymmetric forcing can be written as a conservation law for a potential pseudo-density in the transformed coordinates introduced by Schubert and Hack. The stability of the vortex to non-axisymmetric perturbations is also discussed.