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

Abstract: In this paper, we investigate the time evolution of multiple interacted helical vortices without swirl for the incompressible Euler equations in $\mathbb R^3$. Assuming that the initial helical symmetric vorticity is concentrated within an $\ep$ neighborhood of $N$ distinct helices with vanishing mutual distance of order $O(\frac{1}{|\ln \ep|})$, and each vortex core possesses a vorticity mass of order $O(\frac{1}{|\ln \ep|^2})$, we show that as $\ep\to 0$, the motion of the helical vortices converges to a dynamical system for positive times. Notably, in the case of two interacting helical vortices with initial mutual separation $ \frac{\rho_0}{|\ln \ep|}$, by selecting sufficiently small $\rho_0$, our analysis extends to longer timescales encompassing several periods. This result provides the first mathematical justification for the phenomena in numerical observation termed "leapfrogging of Kelvin waves" reported in e.g. [N. Hietala et al., Phys. Rev. Fluids, 2016].