Conservation of writhe helicity under anti-parallel reconnection

Laing, Christian; Ricca, Renzo L.; Sumners, D. W.

doi:10.1038/srep09224

Cited by 71 publications

(87 citation statements)

References 39 publications

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“…However, in other works, the total torsion is also included in the definition of centreline helicity as in ref. 29. In our view, it would seem more appropriate to include the total torsion in the definition of the centreline helicity since it is a quantity that is fully determined by the instantaneous configuration of the vortex and has the desirable property that it is an integral quantity although not necessarily conserved under a diffeomorphism that produces an inflection point.…”

Section: Discussionmentioning

confidence: 99%

Helicity within the vortex filament model

Hänninen

Hietala

Salman

2016

Sci Rep

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Kinetic helicity is one of the invariants of the Euler equations that is associated with the topology of vortex lines within the fluid. In superfluids, the vorticity is concentrated along vortex filaments. In this setting, helicity would be expected to acquire its simplest form. However, the lack of a core structure for vortex filaments appears to result in a helicity that does not retain its key attribute as a quadratic invariant. By defining a spanwise vector to the vortex through the use of a Seifert framing, we are able to introduce twist and henceforth recover the key properties of helicity. We present several examples for calculating internal twist to illustrate why the centreline helicity alone will lead to ambiguous results if a twist contribution is not introduced. Our choice of the spanwise vector can be expressed in terms of the tangential component of velocity along the filament. Since the tangential velocity does not alter the configuration of the vortex at later times, we are able to recover a similar equation for the internal twist angle to that of classical vortex tubes. Our results allow us to explain how a quasi-classical limit of helicity emerges from helicity considerations for individual superfluid vortex filaments.

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Section: Discussionmentioning

confidence: 99%

Helicity within the vortex filament model

Hänninen

Hietala

Salman

2016

Sci Rep

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“…On one hand laboratory experiments on the production and evolution of vortex knots in water show [12] that the centerline helicity of a vortex filament remains essentially conserved throughout the spontaneous reconnection of the interacting vortices. This result is mirrored by the mathematical analysis of conservation of writhe and total torsion (for definitions, see Sec.III here below) under the assumption of antiparallel reconnection of the interacting strands [13]. On the other hand recent numerical results [14], based on a linearized model of interacting Burgers-type vortices brought together by an ambient irrotational strain field, show that the initial helicity associated with the skewed geometry is eliminated during the process.…”

Section: Introductionmentioning

confidence: 93%

“…In recent months a number of remarkable results based on experimental observations [12], mathematical analysis [13] and theoretical and numerical work [14] have provided contradictory information as regards helicity transfer through reconnection. On one hand laboratory experiments on the production and evolution of vortex knots in water show [12] that the centerline helicity of a vortex filament remains essentially conserved throughout the spontaneous reconnection of the interacting vortices.…”

Section: Introductionmentioning

confidence: 99%

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Helicity conservation under quantum reconnection of vortex rings

Zuccher

Ricca

2015

Phys. Rev. E

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Here we show that under quantum reconnection, simulated by using the three-dimensional GrossPitaevskii equation, self-helicity of a system of two interacting vortex rings remains conserved. By resolving the fine structure of the vortex cores, we demonstrate that total length of the vortex system reaches a maximum at the reconnection time, while both writhe helicity and twist helicity remain separately unchanged throughout the process. Self-helicity is computed by two independent methods, and topological information is based on the extraction and analysis of geometric quantities such as writhe, total torsion and intrinsic twist of the reconnecting vortex rings.

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“…After each reconnection event, helices with a range of length scales are produced on the reconnected vortices. If one assumes a perfectly antiparallel reconnection without any spatial cutoff, this process is expected to exactly conserve helicity 12,25 . However, in the GPE, helical distortions on the scale of the healing length are radiated away as sound waves (Supplementary Movie 4).…”

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confidence: 99%

How superfluid vortex knots untie

2016

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Knots and links often occur in physical systems, including shaken strands of rope 1 and DNA (ref. 2), as well as the more subtle structure of vortices in fluids 3 and magnetic fields in plasmas 4 . Theories of fluid flows without dissipation predict these tangled structures persist 5 , constraining the evolution of the flow much like a knot tied in a shoelace. This constraint gives rise to a conserved quantity known as helicity 6,7 , o ering both fundamental insights and enticing possibilities for controlling complex flows. However, even small amounts of dissipation allow knots to untie by means of 'cutand-splice' operations known as reconnections 3,4,8-11 . Despite the potentially fundamental role of these reconnections in understanding helicity-and the stability of knotted fields more generally-their e ect is known only for a handful of simple knots 12 . Here we study the evolution of 322 elemental knots and links in the Gross-Pitaevskii model for a superfluid, and find that they universally untie. We observe that the centreline helicity is partially preserved even as the knots untie, a remnant of the perfect helicity conservation predicted for idealized fluids. Moreover, we find that the topological pathways of untying knots have simple descriptions in terms of minimal two-dimensional knot diagrams, and tend to concentrate in states which are twisted in only one direction. These results have direct analogies to previous studies of simple knots in several systems, including DNA recombination 2 and classical fluids 3,12 . This similarity in the geometric and topological evolution suggests there are universal aspects in the behaviour of knots in dissipative fields.Tying a knot has long been a metaphor for creating stability, and for good reason: untangling even a common knotted string requires either scissors or a complicated series of moves. This persistence has important consequences for filamentous physical structures such as DNA, the behaviour of which is altered by knots and links 9,13 . An analogous effect can be seen in physical fields, for example, magnetic fields in plasmas or vortices in fluid flow; in both cases knots never untie in idealized models, giving rise to new conserved quantities 6,14 . At the same time, there are numerous examples in which forcing real (non-ideal) physical systems causes them to become knotted: vortices in classical or superfluid turbulence 15,16 , magnetic fields in the solar corona 4 , and defects in condensed matter phases 10 . This presents a conundrum: why doesn't everything get stuck in a tangled web, much like headphone cords in a pocket 1 ?In all of these systems, 'reconnection events' allow fields to untangle by cutting and splicing together nearby lines/structures ( Fig. 1a; refs 3,4,8-11). As a result, the balance of knottedness, and its fundamental role as a constraint on the evolution of physical systems, depends critically on understanding if and how these mechanisms cause knots to untie.Previous studies of the evolution of knotted fields have been restricted ...

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Conservation of writhe helicity under anti-parallel reconnection

Cited by 71 publications

References 39 publications

Helicity within the vortex filament model

Helicity within the vortex filament model

Helicity conservation under quantum reconnection of vortex rings

How superfluid vortex knots untie

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