Vortex filamentation: its onset and its role on axisymmetrization and merger

Vortex filamentation in two-dimensional flows is revisited. Attention is centered on its onset and its role on the axisymmetrization of elongated vortices and on the merger of vortex pairs. The objective is to test two generally accepted hypotheses; namely, that filamentation occurs because a saddle stagnation point enters the vortex and that filaments cause both axisymmetrization and merger. These hypotheses are based on the analysis of the Eulerian flow geometry, i.e., the set of stagnation points and associated streamlines of the instantaneous velocity field. Here, on the contrary, filamentation is described and quantified by analyzing the Lagrangian flow geometry, i.e., the set of hyperbolic trajectories and associated manifolds of the time evolving velocity field. This dynamical-systems approach is applied to the numerical representation of the velocity field obtained by solving the Euler equations with a vortex-in-cell model. Filamentation is found to occur because a stable manifold of a hyperbolic trajectory enters the vortex, and it is unimportant whether the hyperbolic trajectory or the saddle point are inside the vortex or whether they are outside. It is also found that filamentation, although an important part of both axisymmetrization and merger, is not the cause of these processes.