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

TL;DR

This paper investigates the formation of eroding vortex dipoles in axisymmetric Euler flows without swirl, demonstrating a mechanism for rapid vorticity growth up to t^{4/3} and analyzing their structure both analytically and numerically.

Contribution

It introduces a new class of eroding vortex dipoles in axisymmetric flows without swirl that achieve maximal vorticity growth, supported by asymptotic analysis and numerical verification.

Findings

Vorticity can grow as t^{4/3} in these flows.

Eroding toroidal vortex dipoles approximate solutions of Euler's equations.

Numerical results confirm the predicted structure and growth behavior.

Abstract

A review of analyses based upon anti-parallel vortex structures suggests that structurally stable vortex structures with eroding circulation may offer a path to 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 a structure in axisymmetric flow without swirl, leading to maximal growth of vorticity 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 an eroding toroidal vortex dipole which locally conserves kinetic energy. The dipole cross-section is a perturbation of the classical Sadovskii dipole having piecewise constant vorticity, which breaks the symmetry of closed streamlines. The structure of this perturbed Sadovskii dipole is analyzed asymptotically at large times, and its predicted properties are verified numerically.