Evolution of the Singularities of the Schwarz Function Corresponding to the Motion of a Vortex Patch in the Two-dimensional Euler Equations

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.