This paper investigates the variety of dynamics displayed by two identical interwoven helical vortices for different values of their helical pitch. We present simplified approaches aimed at understanding their nonlinear time evolution: in order of increasing complexity, we use models of nonlinear interaction of two point-vortex alleys, of two vortex-ring filament alleys, and of two inviscid helical filaments. Each of these inviscid models leads to a dynamical system of few degrees of freedom that can be analyzed in terms of orbits in a phase space structured by elliptical and hyperbolic points. At low pitch, the basic state with the two helical vortices in symmetric configuration is an unstable equilibrium, and it corresponds to hyperbolic points in the phase space. An initial perturbation induces different types of dynamics depending on its direction and amplitude. Features observed for increasing helical pitches in direct numerical simulations or in experiments are gradually understood as the model complexity is increased. At small pitch, leapfrog cycles are triggered by axial or azimuthal displacements of the vortices, while radial perturbations trigger overtaking; for both regimes, a periodic change in the interwaving order of the vortices occurs. Above a critical pitch value, small-amplitude perturbations trigger a new regime called fluttering, mainly characterized by a periodic evolution of the vortex radii without a change of the interwaving order; large-amplitude displacements then still allow for leapfrog dynamics far from equilibrium. The amplitudes of the radial excursions of the vortices induced by small perturbations are characterized over the whole range of helical pitches, up to the linear stabilization threshold. Mimicking viscous effects in the point-vortex alley model sheds some light on how dissipation causes overtaking to turn into leapfrog (the reverse is not observed).

中文翻译：

---

两个螺旋涡的非线性动力学：一种动力学系统方法

本文研究了两个相同的交织螺旋涡对于不同螺距值所显示的各种动力学。我们提出了旨在理解它们的非线性时间演化的简化方法：按照复杂性增加的顺序，我们使用两个点涡旋通道、两个涡旋环细丝通道和两个无粘性螺旋细丝的非线性相互作用模型。这些无粘性模型中的每一个都导致了一个具有几个自由度的动力系统，可以根据由椭圆和双曲线点构成的相空间中的轨道进行分析。在低频时，对称配置的两个螺旋涡旋的基本状态是不稳定的平衡，它对应于相空间中的双曲线点。初始扰动根据其方向和幅度引起不同类型的动力学。随着模型复杂性的增加，在直接数值模拟或实验中观察到的螺旋节距增加的特征逐渐被理解。在小间距，跃迁循环由涡旋的轴向或方位角位移触发，而径向扰动则触发超车；对于这两种状态，涡流的交错顺序都会发生周期性变化。超过临界音高值，小幅度扰动会触发一种称为颤动的新机制, 主要特征是涡旋半径的周期性演化，而交错顺序没有变化；大振幅位移仍然允许远离平衡的跳跃动力学。由小扰动引起的涡旋径向偏移的幅度在整个螺旋节距范围内表征，直到线性稳定阈值。模拟点涡巷模型中的粘性效应可以揭示耗散如何导致超车变成蛙跳（没有观察到相反的情况）。