An analytical study of the self-induced inviscid dynamics of two-dimensional uniform vortices

The self-induced dynamics of a uniform planar vortex in an isochoric, inviscid fluid is analytically investigated, by writing a new nonlinear integrodifferential problem which describes the time evolution of the Schwarz function of its boundary. In order to overcome the difficulties related to the nonlinear nature of the problem, an approximate solution of the above problem is proposed in terms of a hierarchy of linear integral equations. In particular, the solution at the first order is here investigated and applied to a class of uniform vortices, the kinematics of which is well known. A rich mathematical phenomenology has been found behind the approximate dynamics. In particular, the motion of certain branch points and the consequent changes in the algebraic structure of the Schwarz function appear to be its key features. The approximate solutions are finally compared with the contour dynamics simulations of the vortex motion. The agreement is satisfactory up to times of the order of a quarter of the eddy turnover time, while it becomes more and more qualitative at later times. Eventually, the physical meaning of the approximate solutions is lost, the corresponding vortex boundaries becoming non-simple curves. © 2012 Springer-Verlag Wien.