Self-similar collapse of 2D and 3D vortex filament models

In this article, a very simple toy model for a candidate blow-up solution of the Euler equation by Boratav and Pelz (vortex dodecapole) is investigated. In this model, vortex tubes are replaced with straight vortex filaments of infinitesimal thickness, and the entire motion is monitored by tracing the motion of a representative point on one vortex filament. It is demonstrated that this model permits a self-similar collapse solution which provides the time dependence of the length scale as (t (c) - t)(1/2), (t < t (c)), where the collapse time t (c) depends on the initial configuration. From the conservation of circulation, this time dependence implies that vorticity omega scales as (t (c) - t) (-1), which agrees with the one observed in the direct numerical (pseudo spectral) simulations of the vortex dodecapole. Finally, possible modification of the model is considered.