Knot spectrum of turbulence

R G Cooper,C F Barenghi,M Mesgarnezhad,A W Baggaley

doi:10.1038/s41598-019-47103-w

Abstract

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.

Highlights

The DG instability manifests itself as growing Kelvin waves - helical displacements of the vortex lines away from their initial position - and occurs if the component of the normal fluid velocity parallel to a vortex line exceeds a critical value
We have exploited the key property of quantum fluids - the discrete nature of vorticity - to quantify the topology of a small region of quantum turbulence away from boundaries in a statistical steady-state regime
We have found that the probability that a vortex loop is knotted increases with the loop’s length as for random knots studied in the context of DNA and macromolecules[8], and saturates above a characteristic length as for tumbled strings[39], despite the very different physical mechanisms of agitation of these systems