Growth of anti-parallel vorticity in Euler flows

In incompressible Euler flows, vorticity is intensified by line stretching, a process that can come either from the action of shear, or from advection with curvature. Focusing on the latter process, we derive some estimates on the maximal growth of vorticity in axisymmetric flow without swirl, given that vorticity support volume or kinetic energy is fixed. This leads to consideration of locally 2D anti-parallel vortex structures in three dimensions. We exhibit a class of line motions which lead to infinite vorticity in a finite time, with only a finite total line stretching. If the line is replaced by a locally 2D Euler flow, we obtain a class of models of vorticity growth which are similar to the paired vortex structures studied by Pumir and Siggia. We speculate on the mechanisms which can suppress the nonlinear effects necessary for the finite-time singularity exhibited by the moving line problem. c © 2008 Elsevier B.V. All rights reserved. PACS: 47.10.-g; 47.15.ki; 47.32.C