论文标题
过渡到跨越涡流四重奏的不稳定
Transition to instability of the leapfrogging vortex quartet
论文作者
论文摘要
点涡流系统是一种对非线性动力学的长期兴趣的系统，描述了二维无粘性流体的运动，除了在离散的一组移动点涡流，涡度差异。跳过的轨道由两个旋转的类似涡旋的旋转对组成，它们是四重奏，以恒定速度传播。众所周知，如果两对最初被广泛分离，则运动是稳定的，而如果它们靠近，则它变得不稳定，并且该关系由文本中定义的无量纲参数$α$表示。我们在这里分析表明，从稳定性到不稳定性的过渡发生在临界值$α= ϕ^{ - 2} $上，其中$ ϕ $是黄金比率。该值是根据Tophøj和Aref的仔细数字来假设的，并且由本作者使用半分析论点，但以前没有通过精确的分析证明。
The point vortex system is a system of longstanding interest in nonlinear dynamics, describing the motion of a two-dimensional inviscid fluid that is irrotational except at a discrete set of moving point vortices, at which the vorticity diverges. The leapfrogging orbit consists of two rotating pairs of like-signed vortices which, taken as a quartet, propagate at constant velocity. It is known that if the two pairs are initially widely separated, the motion is stable, while if they are closer together it becomes unstable, with this relation represented by a dimensionless parameter $α$ defined in the text. We here demonstrate analytically that the transition from stability to instability happens at a critical value $α= ϕ^{-2}$, where $ϕ$ is the golden ratio. This value had been hypothesized based on careful numerics by Tophøj and Aref, and by the present authors using a semi-analytic argument but not previously demonstrated through exact analysis.
