Time evolution of vortex rings with large radius and very concentrated vorticity

TL;DR

This paper analyzes how vortex rings with large radii and concentrated vorticity evolve over time, showing convergence to a point vortex model under specific conditions, extending previous power-law assumptions.

Contribution

It proves the convergence of the vorticity field to the point vortex model for large-radius vortex rings with concentrated vorticity, generalizing prior power-law results.

Findings

Vorticity converges to the point vortex model as epsilon approaches zero.

Convergence holds for large radii with logarithmic relation to epsilon.

Extends previous models by relaxing power-law assumptions.

Abstract

We study the time evolution of an incompressible fluid with axial symmetry without swirl when the vorticity is sharply concentrated on $N$ annuli of radii $\approx$ $r_0$ and thickness $\epsilon$. We prove that when $r_0= |\log \epsilon|^\alpha, \,\, \alpha>2$, the vorticity field of the fluid converges as $\epsilon \to 0$ to the point vortex model, at least for a small but positive time. This result generalizes a previous paper that assumed a power law for the relation between $r_0$ and $\epsilon$.