Cooper R G, Mesgarnezhad M, Baggaley A W, Barenghi C F
School of Mathematics, Statistics and Physics Newcastle University, Newcastle upon Tyne, NE1 7RU, UK.
JQC (Joint Quantum Centre), Durham-Newcastle, UK.
Sci Rep. 2019 Jul 22;9(1):10545. doi: 10.1038/s41598-019-47103-w.
Streamlines, vortex lines and magnetic flux tubes in turbulent fluids and plasmas display a great amount of coiling, twisting and linking, raising the question as to whether their topological complexity (continually created and destroyed by reconnections) can be quantified. In superfluid helium, the discrete (quantized) nature of vorticity can be exploited to associate to each vortex loop a knot invariant called the Alexander polynomial whose degree characterizes the topology of that vortex loop. By numerically simulating the dynamics of a tangle of quantum vortex lines, we find that this quantum turbulence always contains vortex knots of very large degree which keep forming, vanishing and reforming, creating a distribution of topologies which we quantify in terms of a knot spectrum and its scaling law. We also find results analogous to those in the wider literature, demonstrating that the knotting probability of the vortex tangle grows with the vortex length, as for macromolecules, and saturates above a characteristic length, as found for tumbled strings.
湍流流体和等离子体中的流线、涡线和磁通量管呈现出大量的盘绕、扭曲和链接，这就引发了一个问题：它们的拓扑复杂性（通过重连不断产生和破坏）是否能够被量化。在超流氦中，可以利用涡度的离散（量子化）性质，为每个涡环关联一个称为亚历山大多项式的纽结不变量，其次数表征该涡环的拓扑结构。通过对一团量子涡线的动力学进行数值模拟，我们发现这种量子湍流总是包含非常高次数的涡结，这些涡结不断形成、消失和重新形成，产生了一种拓扑结构分布，我们根据纽结谱及其标度律对其进行量化。我们还发现了与更广泛文献中类似的结果，表明涡缠结的纽结概率随涡长度增加，如同大分子一样，并且在超过一个特征长度后达到饱和，如同翻滚的弦一样。
