On the formation of Moore curvature singularities in vortex sheets

Moore (1979) demonstrated that the cumulative influence of small nonlinear effects on the evolution of a slightly perturbed vortex sheet is such that a curvature singularity can develop at a large, but finite, time. By means of an analytical continuation of the problem into the complex spatial plane , we find a consistent asymptotic solution to the problem posed by Moore. O ur solution includes the shape of the vortex sheet as the curvature singula rity forms. Analytic results are confirmed by comparison with numerical sol utions. Further, for a wide class of initial conditions (including perturba tions of finite amplitude), we demonstrate that 3/2-power singularities can spontaneously form at t = 0+ in the complex plane. We show that these sing ularities propagate around the complex plane. If two singularities collide on the real axis, then a point of infinite curvature develops on the vortex sheet. For such an occurrence we give an asymptotic description of the vor tex-sheet shape at times close to singularity formation.