Recent studies by Willoughby (1992 and 1995) discuss the motion of a vortex on a beta plane. This linear barotropic model has been converted to a baroclinic model with two lower layers, where the vortex and environmental flows are given, and an upper inert layer. The upper and
lower vortices rotate cyclonically about each other. Depending upon the choice of layer depths, layer densities, and vortex structure these loops expand, contract or continue looping at about the same diameter.
Even with substantial vertical shear of the environment wind, solutions exist in which the upper and lower vortices reach an equilibrium separation and propagate together at very nearly the same speeds. The beta gyres that are due to the advection of planetary vorticity or environment potential vorticity adjust to compensate for the environment shear flow to allow these near equilibrium solutions of a linear vortex tracking model.