Dynamics and leapfrogging phenomena of multiple helical vortices for 3D incompressible Euler equations

TL;DR

This paper rigorously analyzes the evolution of multiple helical vortices in 3D incompressible Euler equations, demonstrating convergence to a dynamical system and providing the first mathematical proof of the leapfrogging phenomenon of Kelvin waves.

Contribution

It introduces a new mathematical framework for the evolution of helical vortices, including the leapfrogging phenomenon, under general initial conditions in 3D Euler equations.

Findings

Convergence of vortex motion to a derived dynamical system as vortex core size shrinks.

Mathematical justification of leapfrogging of Kelvin waves.

Extension of analysis to multiple periods for two vortices with small initial separation.

Abstract

In this paper, we investigate the time evolution of helical vortices without swirl for the incompressible Euler equations in $\mathbb R^3$ under general initial assumptions. Assume the initial helical vorticity is sharply concentrated in $N$ distinct $\ep$-neighborhoods, whose mutual distances vanish as $O(1/|\ln \ep|)$, and each vortex core possesses vorticity mass of order $1/|\ln \ep|^{1+b}$ for an arbitrary fixed $b\in\mathbb R$. We prove that as $\ep\to 0$, the motion of these helical vortices converges uniformly to a dynamical system derived herein over a time interval of order $1/|\ln\varepsilon|^{1-b}$. In the particular case $b=-1$, our results establish the evolution counterpart for interacting vortex helices constructed in [I. Guerra, M. Musso, Ann. Inst. H. Poincar\'e C Anal. Non Lin\'aire, 2025]. Notably, for two interacting helical vortices with initial mutual distance $ \rho_0/|\ln \ep|$, by choosing $\rho_0$ sufficiently small, our analysis extends to timescales covering multiple periods. This result provides the first mathematical justification for the numerically observed phenomenon termed ``leapfrogging of Kelvin waves" reported in [N. Hietala et al., Phys. Rev. Fluids, 2016].