Filamentation near Hill's vortex

TL;DR

This paper proves that small outward perturbations near Hill's vortex grow linearly over time in the axisymmetric incompressible Euler equations, confirming previous numerical simulations.

Contribution

It introduces a rigorous proof of linear filamentation near Hill's vortex, combining recent stability results with a dynamical bootstrapping scheme.

Findings

Small perturbations grow linearly over time

Confirms numerical simulations from 1986

Provides rigorous mathematical proof of filamentation

Abstract

For the axisymmetric incompressible Euler equations, we prove linear in time filamentation near Hill's vortex: there exists an arbitrary small outward perturbation growing linearly for all times. This is based on combining the recent nonlinear orbital stability obtained in [13] with a dynamical bootstrapping scheme for particle trajectories. These results rigorously confirm numerical simulations by Pozrikidis [45] in 1986.