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Leapfrogging vortex rings as scaling limit of Euler Equations

Paolo Buttà, Guido Cavallaro, Carlo Marchioro

TL;DR

This work provides a rigorous derivation of the leapfrogging dynamics for multiple coaxial vortex rings in axisymmetric Euler flows. By letting the ring thickness $\varepsilon$ tend to zero, the authors show that the rings’ centers converge to trajectories solving a 2D point-vortex system augmented by an axial drift, capturing both inter-ring interactions and the self-induced drift. The analysis hinges on precise concentration and localization estimates: energy and inertia considerations bound inter-ring coupling, while a two-stage iterative scheme propagates confinement in time. In the two-ring, large-radius regime, the convergence time can be extended to encompass several leapfrogging events, providing a rigorous link between the Euler dynamics and the classical leapfrogging phenomenon observed in experiments and simulations.

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.

Leapfrogging vortex rings as scaling limit of Euler Equations

TL;DR

This work provides a rigorous derivation of the leapfrogging dynamics for multiple coaxial vortex rings in axisymmetric Euler flows. By letting the ring thickness tend to zero, the authors show that the rings’ centers converge to trajectories solving a 2D point-vortex system augmented by an axial drift, capturing both inter-ring interactions and the self-induced drift. The analysis hinges on precise concentration and localization estimates: energy and inertia considerations bound inter-ring coupling, while a two-stage iterative scheme propagates confinement in time. In the two-ring, large-radius regime, the convergence time can be extended to encompass several leapfrogging events, providing a rigorous link between the Euler dynamics and the classical leapfrogging phenomenon observed in experiments and simulations.

Abstract

We consider an incompressible fluid with axial symmetry without swirl, assuming initial data such that the initial vorticity is very concentrated inside small disjoint rings of thickness , each one of vorticity mass and main radius of order . When , 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.
Paper Structure (9 sections, 10 theorems, 198 equations, 1 figure)

This paper contains 9 sections, 10 theorems, 198 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Notation and main result
  3. Concentration estimates
  4. Energy estimates
  5. Mass concentration and bound on the moment of inertia
  6. Iterative procedure
  7. Localization of vortices support
  8. Conclusion of the proof of Theorem \ref{['thm:1']}
  9. An example of leapfrogging vortex rings

Key Result

Theorem 2.1

Assume the initial condition $\omega_\varepsilon(x,0)$ verifies Eqs. in, initial, ai, and Mgamma. Then, for any fixed (independent of $\varepsilon$) $\varrho>0$ such that the closed disks $\overline{\Sigma(\zeta^i|2\varrho)}$ are mutually disjointed, there exists $T_\varrho \in (0,T^*)$ such that fo

Figures (1)

  • Figure 7.1: Phase portrait of the dynamical system describing the motion of the relative position $x$ between the two rings.

Theorems & Definitions (21)

  • Theorem 2.1
  • Lemma 3.1
  • proof
  • Proposition 3.2
  • proof
  • Theorem 3.3
  • proof
  • Theorem 3.4
  • proof
  • Lemma 4.1
  • ...and 11 more