Eroding dipoles and vorticity growth for Euler flows in $ \scriptstyle{\mathbb{R}}^3$ I. Axisymmetric flow without swirl

摘要

A review of analyses based upon anti-parallel vortex structures suggests thatstructurally stable vortex structures with eroding circulation may offer a pathto the study of rapid vorticity growth in solutions of Euler's equations in $\scriptstyle{\mathbb{R}}^3$. We examine here the possible formation of such astructure in axisymmetric flow without swirl, leading to maximal growth ofvorticity as $t^{4/3}$. Our study suggests that the optimizing flow giving the$t^{4/3}$ growth mimics an exact solution of Euler's equations representing aneroding toroidal vortex dipole which locally conserves kinetic energy. Thedipole cross-section is a perturbation of the classical Sadovskii dipole havingpiecewise constant vorticity, which breaks the symmetry of closed streamlines.The structure of this perturbed Sadovskii dipole is analyzed asymptotically atlarge times, and its predicted properties are verified numerically.