平面内大量涡的集体行为
Collective behaviour of large number of vortices in the plane
摘要：
我们研究了在大$N$的极限下，平面内$N$点涡的动力学。我们考虑{ \em相对平衡}，这是刚性旋转的格子状配置的旋涡。在几个最近的实验中观察到这些构型[Durkin和Fajans，Phys. Fluids（2000）12，289-293; Grzybowski { \em et.al} PRE（2001）64，011603]。我们表明，这些解决方案和它们的稳定性是充分的特点，通过一个相关的{ \em聚集模型}，这是最近研究的背景下，生物群[Fetecau { \em等}，Nonlinearity（2011）2681; Bertozzi { \em et.al.}，M3AS（2011）]。通过利用这种联系，我们给出了明确的解析公式的许多配置，已被观察到的实验。这些包括等强度涡流的构型;等强度$N$涡流和一个强度高得多的涡流的$N+1$构型;以及更一般的$N+K$构型。我们还给出了尚未实验研究的配置的例子，包括$N+2$配置，其中$N$涡聚集在一个椭圆内。最后，我们引入一个人工的“阻尼”的旋涡动力学，试图解释的结晶现象，经常在实际实验中观察到的。扩散打破了涡动力学的保守结构，使任何初始条件收敛到格子状的相对平衡。
英文摘要：
We investigate the dynamics of $N$ point vortices in the plane, in the limit of large $N$. We consider {\em relative equilibria}, which are rigidly rotating lattice-like configurations of vortices. These configurations were observed in several recent experiments [Durkin and Fajans, Phys. Fluids (2000) 12, 289-293; Grzybowski {\em et.al} PRE (2001)64, 011603]. We show that these solutions and their stability are fully characterized via a related {\em aggregation model} which was recently investigated in the context of biological swarms [Fetecau {\em et.al.}, Nonlinearity (2011) 2681; Bertozzi {\em et.al.}, M3AS (2011)]. By utilizing this connection, we give explicit analytic formulae for many of the configurations that have been observed experimentally. These include configurations of vortices of equal strength; the $N+1$ configurations of $N$ vortices of equal strength and one vortex of much higher strength; and more generally, $N+K$ configurations. We also give examples of configurations that have not been studied experimentally, including $N+2$ configurations where $N$ vortices aggregate inside an ellipse. Finally, we introduce an artificial ``damping'' to the vortex dynamics, in an attempt to explain the phenomenon of crystalization that is often observed in real experiments. The diffusion breaks the conservative structure of vortex dynamics so that any initial conditions converge to the lattice-like relative equilibrium.
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