Leapfrogging vortex rings as scaling limit of Euler Equations

TL;DR

This paper rigorously derives the leapfrogging vortex rings phenomenon as a scaling limit of the Euler equations for an incompressible, axially symmetric fluid with concentrated vorticity, connecting microscopic initial data to macroscopic dynamics.

Contribution

It establishes the convergence of vortex ring dynamics to a known dynamical system and extends results to longer times, capturing complex overtaking interactions.

Findings

Vortex rings' motion converges to a specific dynamical system as thickness tends to zero.

Longer time behavior includes multiple overtaking events.

Provides a rigorous mathematical derivation of leapfrogging phenomenon.

Abstract

We consider an incompressible fluid with axial symmetry without swirl, assuming initial data such that the initial vorticity is very concentrated inside $N$ small disjoint rings of thickness $\varepsilon$, each one of vorticity mass and main radius of order $|\log\varepsilon|$. When $\varepsilon \to 0$, we show that, at least for small but positive times, the motion of the rings converges to a dynamical system firstly introduced in [NoDEA Nonlinear Diff. Eq. Appl. 6 (1999), 473-499]. In the special case of two vortex rings with large enough main radius, the result is improved reaching longer times, in such a way to cover the case of several overtakings between the rings, thus providing a mathematical rigorous derivation of the leapfrogging phenomenon.