A similarity approach is utilized to investigate a simple axisymmetric steady-state model of the convergence region of a laboratory vortex. The resulting simplified set of equations are solved for a range of swirl angles by varying the tangential or radial velocity component at the outer rim. By increasing the swirl angle the flow is found to go from a one cell to a two cell configuration, i.e., the vertical velocity changes from everywhere positive to negative in the vicinity of the axis. Correspondingly the vertical vorticity maximum moves from the axis outward toward the radius of maximum tangential velocity, making the flow barotropically unstable with respect to unsymmetrical perturbations.