David Dritschel

The paper deals with the calculation of the internal singularities of the Schwarz function corresponding to the boundary of a planar vortex patch during its self-induced motion in an inviscid, isochoric fluid. The vortex boundary is approximated by a simple, time-dependent map onto the unit circle, whose coefficients are obtained by fitting to the boundary computed in a contour dynamics numerical simulation of the motion. At any given time, the branch points of the Schwarz function are calculated, and from them, the generally curved shape of the internal branch cut, together with the jump of the Schwarz function across it. The knowledge of the internal singularities enables the calculation of the Schwarz function at any point inside the vortex, so that it is possible to check the validity of the map during the motion by comparing left and right hand sides of the evolution equation of the Schwarz function. Our procedure yields explicit functional forms of the analytic continuations of the velocity and its conjugate on the vortex boundary. It also opens a new way to understand the relation between the time evolution of the shape of a vortex patch during its motion, and the corresponding changes in the singular set of its Schwarz function.

This paper examines the stability and nonlinear evolution of configurations of equalstrength point vortices equally spaced on a ring of constant radius, with or without a central vortex, in the three-dimensional quasi-geostrophic compressible atmosphere model. While the ring lies at constant height, the central vortex can be at a different height and also have a different strength to the vortices on the ring. All such configurations are relative equilibria, in the sense that they steadily rotate about the $z$ axis. Here, the domains of stability for two or more vortices on a ring with an additional central vortex are determined. For a compressible atmosphere, the problem also depends on the density scale height $H$, the vertical scale over which the background density varies by a factor $e$. Decreasing $H$ while holding other parameters fixed generally stabilises a configuration. Nonlinear simulations of the dynamics verify the linear analysis and reveal potentially chaotic dynamics for configurations having four or more vortices in total. The simulations also reveal the existence of staggered ring configurations, and oscillations between single and double ring configurations. The results are consistent with the observations of ring configurations of polar vortices seen at both of Jupiter’s poles [1].