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Reconnection scaling in quantum fluids
Contributed by Katepalli R. Sreenivasan, December 7, 2018 (sent for review September 24, 2018; reviewed by Carlo F. Barenghi, Davide Proment, and Makoto Tsubota)

Significance
Superfluid helium exhibits topological defects in the form of line-like objects called quantum vortices. Reconnection occurs when two vortices collide and recoil by exchanging tails. We observe such a reconnection for nearly isolated conditions and find that the intervortex separation for a certain range scales closely as the square root of the time after reconnection and that the prefactor in the square-root law shows an analytical dependence on the reconnection angle. Reconnection is important because it provides a mechanism for energy dissipation which otherwise does not occur in the zero-temperature limit. The kinematics of reconnections are similar in systems of classical vortices, cosmic strings, magnetic flux tubes in plasmas, liquid crystals, and even DNA.
Abstract
Fundamental to classical and quantum vortices, superconductors, magnetic flux tubes, liquid crystals, cosmic strings, and DNA is the phenomenon of reconnection of line-like singularities. We visualize reconnection of quantum vortices in superfluid 4He, using submicrometer frozen air tracers. Compared with previous work, the fluid was almost at rest, leading to fewer, straighter, and slower-moving vortices. For distances that are large compared with vortex diameter but small compared with those from other nonparticipating vortices and solid boundaries (called here the intermediate asymptotic region), we find a robust 1/2-power scaling of the intervortex separation with time and characterize the influence of the intervortex angle on the evolution of the recoiling vortices. The agreement of the experimental data with the analytical and numerical models suggests that the dynamics of reconnection of long straight vortices can be described by self-similar solutions of the local induction approximation or Biot–Savart equations. Reconnection dynamics for straight vortices in the intermediate asymptotic region are substantially different from those in a vortex tangle or on distances of the order of the vortex diameter.
Reconnections are collisions of two line-like topological defects which subsequently recombine by exchanging each other’s tails (1). These changes in topological configuration occur in magnetic flux tubes (2), cosmic strings (3), polymers, liquid crystals (4), superconductors (5), and DNA (6) and are common in quantum (7) as well as classical vortices (8). Reconnections are crucial in redistributing and dissipating energy (kinetic or magnetic) in turbulence, astrophysical plasmas, and fusion devices (2).
Here, we study reconnection of quantized vortices in superfluid 4He. Superfluid helium can be modeled as a mixture of a viscous normal component and an inviscid superfluid component. Vorticity in the superfluid fraction is constrained to quantum vortices (9), which are phase singularities and topological defects in the order parameter describing the superfluid. The circulation of the quantized vortices is constrained to be the quantum of circulation
The first person to address reconnection analytically was Crow (18) with his work on trailing vortices (in classical fluids). Schwarz (19) introduced numerical techniques based on the Biot–Savart equation and studied the problem quantitatively. Other works (20, 21) are based on the nonlinear Schrödinger equation, also known as the Gross–Pitaevksii (GP) equation. In classical fluids, reconnection has been studied experimentally (22, 23) and numerically (24, 25). The first experimental study of reconnection in superfluid helium was by Bewley et al. (7), who used a technique previously developed in ref. 26 to visualize the vortex position by means of micrometer-sized solid hydrogen tracers trapped on vortex cores. Later studies by Paoletti et al. (27, 28) included some 20,000 reconnection events and found that the intervortex distance δ scales as
Compared with these previous experimental studies we use a visualization technique that uses frozen air tracers which are much finer, described briefly in Materials and Methods and in detail in ref. 33. The technique, previously used to visualize kelvin waves (34), uses submicrometer particles injected directly in the superfluid phase, permitting the visualization of longer and straighter vortices. While the studies by Paoletti et al. (27, 28) inferred the reconnection events indirectly from rapid separations of particle pairs, we observe reconnection events when many particles are visible on both vortices along their length. However, the number of events is smaller by a factor of 1,000 (20 events vs. 20,000 events) for two reasons. First, we worked in a quiet system, at a temperature around 1.9 K, in which there was no imposed heat flux (velocity of the normal component
When two straight vortices approach each other, they locally reconfigure into antiparallel configuration (19, 23, 35, 36). However, on the larger separation scale considered here, two approaching straight vortices are always in a parallel configuration and there exists a self-similar evolution for which the vortex asymptotes form a global angle θ (Fig. 1). This self-similar evolution is what we observe in our experimental visualizations (Fig. 1). We first note that the intervortex distance δ follows the 1/2 scaling,
(Left) The first frame of images shows the experimentally observed vortices within
The intervortex distance δ as a function of time after reconnection for 20 reconnection events. Each point represents a measurement. The trend is in broad agreement with the
If we insist on fits of the type used in refs. 27 and 28, we find that the distribution of c (Fig. 3) peaks near 0, with a median value of −0.10 s−1 and an average of −0.15 s−1. This means that the deviation from the average is less than 10%. Moreover, all values of c are negative. This fact could be due to boundary conditions: The 1/2 scaling breaks down when the vortex comes to rest as the cusp relaxes into a straight vortex and when the boundary conditions progressively slow down its motion. This contrasts from refs. 27 and 28 in which the distribution peaked at 0 and was much broader with a slightly larger number of positive values than negative values (Fig. 3, Inset). This difference is most likely the result of very different conditions of the system: quiet in our case, turbulent in refs. 27 and 28. Finally, deviations from the 1/2 scaling of the intermediate asymptotic regime could be due to the curvature and/or the presence of neighboring vortices.
Normalized frequency distribution of the parameter c representing the deviation from the 1/2-power scaling,
Although the statistics are limited, another important distinction is the distribution of the prefactor A. As is clear from Fig. 4, we observe four instances in which
Normalized frequency distribution of the prefactor A of the scaling
The surprising fact that in previous experiments there were no reported reconnections with A > 3 might also be caused by their technique for measuring the separation distance δ between particle pairs. If the separation was too rapid, the algorithm might not have detected the particles as a pair. Moreover, the bigger particles compared with the present ones might have slowed down the separation between vortices due to drag, reducing the effective A. We cannot observe the prefactor A before reconnection because of the way in which the vortices are arranged: The observations are for straight vortices approaching each other in the direction orthogonal to the field of view, so it is not possible to see the intervortex distance δ before reconnection. The reason for the difference from simulations in ref. 29 may well be that they were performed just for the reconnection of vortex rings.
Let us consider the localized induction approximation (LIA) for the vortices, first derived by Da Rios (37) and rediscovered multiple times (38, 39). The Da Rios equation for the position of the vortex s, also related to the elastica (40⇓–42) originally solved by Euler, is
As already stated, our data were obtained around 1.9 K, at which there is still 40% of normal component, with the attendant damping effect due to mutual friction (44). To account for this effect, it is possible to include in the Da Rios equation the extra term
Plot of the prefactor A as a function of the angle of reconnection. The circles represent the present data while the square represents the data in ref. 34. The dashed line represents the analytical relationship
In conclusion, we have examined the reconnection of almost straight reconnecting vortices. On a global scale these vortices reconnect in a parallel configuration and display a self-similar evolution. The local rearrangement of vortices at the reconnection “point” is not visible on this scale, and a locally antiparallel configuration is by no means precluded. To observe the local rearrangement into antiparallel configuration and deviations from the self-similar solution of LIA and Biot–Savart equations, it is necessary to make observations at much smaller scales on the order of the vortex core.
Our results are that deviations from the 1/2-power scaling of the intervortex distance as a function of reconnecting time are much smaller than in previous pulsed counterflow experiments (27, 28). We have shown that the relationship between the angle of reconnection and the prefactor A follows the analytical formula deduced from the self-similar solution of LIA. We also observed reconnection events with A > 3 (corresponding to θ < 50○), higher than any of the events in the pulsed counterflow experiments. More broadly, in the intermediate asymptotic regime for which reconnecting vortices are almost straight, the behavior of reconnection is substantially different from that during the decay of a vortex tangle, where the vortices are curved and close together.
Materials and Methods
The experimental setup and visualization method is the same as in ref. 34 and is described in detail in ref. 33. It consists of an optical cryostat filled with liquid helium in which we inject a dilute mixture of atmospheric air in helium gas. The atmospheric air freezes into tracer particles that get trapped on the quantum vortices due to a Bernoulli pressure gradient. The particle size was estimated to be on the order of 0.5 μm, sometimes as small as few hundred nanometers, based on both optical (48) and terminal (33) velocity considerations. The particles are illuminated with a laser sheet 1 cm wide and a thickness with a full width at half maximum (FWHM) of 150 μm generated with a 4-mW laser. The imaging setup consists of a Princeton Instruments Pro-EM, EM-CCD low-light camera running at a frame rate of 30–100 frames per second and a resolution of 512 × 512–512 × 128, on which is mounted a macro-lens (Micro-Nikkor 105 mm f/2.8 lens). The 8.2-mm × 8.2-mm field of view is inside a glass cell, 2 cm2, that is used to stabilize the system. Note that no particular procedure was used to generate the vortices, which were present from transition or created by parasitic heat flux. However, we cooled on the order of a few hundred microkelvins per second and waited on the order of at least tens of minutes in the superfluid state to obtain the long, straight, and relaxed vortices in which we were interested. The original movies can be downloaded from https://zenodo.org/record/2543528#.XEId3lVKhph and questions concerning them can be addressed to the corresponding author.
Acknowledgments
We thank David P. Meichle, Nicholas T. Ouellette, Sahand Hormoz, Matthew S. Paoletti, and Cecilia Rorai for fruitful discussions and comments. This work was partially supported by the National Science Foundation Grant NSF-DMR1407472.
Footnotes
- ↵1To whom correspondence should be addressed. Email: katepalli.sreenivasan{at}nyu.edu.
Author contributions: E.F., K.R.S., and D.P.L. designed research; E.F., K.R.S., and D.P.L. performed research; E.F. contributed new reagents/analytic tools; E.F. analyzed data; and E.F., K.R.S., and D.P.L. wrote the paper.
Reviewers: C.F.B., Newcastle University; D.P., University of East Anglia; and M.T., Osaka City University.
The authors declare no conflict of interest.
Data deposition: Movies related to this work have been deposited on Zenodo (https://zenodo.org/record/2543528#.XEId3lVKhph).
- Copyright © 2019 the Author(s). Published by PNAS.
This open access article is distributed under Creative Commons Attribution-NonCommercial-NoDerivatives License 4.0 (CC BY-NC-ND).
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