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Helicity within the vortex filament model

View Article: PubMed Central - PubMed

ABSTRACT

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.

No MeSH data available.


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Vortex reconnection, when the initial configuration is a Hopf link with ring radii of 1 mm.This simulation was computed using a large core size of a0 = 0.0025 mm as well as a large value of the mutual friction, α = 1. The left panel (a) shows the time development of the linking (L, green-dashed), writhe (W, blue), and torsion part of the twist (Ttors, black), plus their sum (red). The inset illustrates that the jump in the linking is compensated by the jump in the writhe. Panels (b–g) illustrate the vortex configurations both before (upper ones) and after (lower ones) the reconnection, which occurs at 4.737 s. The green stripe is the direction of the normal vector while the yellow stripe indicates the direction of the constant phase.
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f5: Vortex reconnection, when the initial configuration is a Hopf link with ring radii of 1 mm.This simulation was computed using a large core size of a0 = 0.0025 mm as well as a large value of the mutual friction, α = 1. The left panel (a) shows the time development of the linking (L, green-dashed), writhe (W, blue), and torsion part of the twist (Ttors, black), plus their sum (red). The inset illustrates that the jump in the linking is compensated by the jump in the writhe. Panels (b–g) illustrate the vortex configurations both before (upper ones) and after (lower ones) the reconnection, which occurs at 4.737 s. The green stripe is the direction of the normal vector while the yellow stripe indicates the direction of the constant phase.

Mentions: Figure 5 illustrates the reconnection of two vortex rings with initial radii of 1 mm, and which are initially linked (in the form of a Hopf link). The left-most panel of Fig. 5 illustrates the time development of the different helicity components, while the other panels show the behaviour of the normal vector and the spanwise direction when using the Seifert frame. A characteristic feature is that before reconnection, both vortices host one inflection point. Additionally, one may notice that at the instant of reconnection the linking is converted to writhe, which then decays due to mutual friction, as shown more clearly in the inset of Fig. 5a.


Helicity within the vortex filament model
Vortex reconnection, when the initial configuration is a Hopf link with ring radii of 1 mm.This simulation was computed using a large core size of a0 = 0.0025 mm as well as a large value of the mutual friction, α = 1. The left panel (a) shows the time development of the linking (L, green-dashed), writhe (W, blue), and torsion part of the twist (Ttors, black), plus their sum (red). The inset illustrates that the jump in the linking is compensated by the jump in the writhe. Panels (b–g) illustrate the vortex configurations both before (upper ones) and after (lower ones) the reconnection, which occurs at 4.737 s. The green stripe is the direction of the normal vector while the yellow stripe indicates the direction of the constant phase.
© Copyright Policy - open-access
Related In: Results  -  Collection

License
Show All Figures
getmorefigures.php?uid=PMC5121624&req=5

f5: Vortex reconnection, when the initial configuration is a Hopf link with ring radii of 1 mm.This simulation was computed using a large core size of a0 = 0.0025 mm as well as a large value of the mutual friction, α = 1. The left panel (a) shows the time development of the linking (L, green-dashed), writhe (W, blue), and torsion part of the twist (Ttors, black), plus their sum (red). The inset illustrates that the jump in the linking is compensated by the jump in the writhe. Panels (b–g) illustrate the vortex configurations both before (upper ones) and after (lower ones) the reconnection, which occurs at 4.737 s. The green stripe is the direction of the normal vector while the yellow stripe indicates the direction of the constant phase.
Mentions: Figure 5 illustrates the reconnection of two vortex rings with initial radii of 1 mm, and which are initially linked (in the form of a Hopf link). The left-most panel of Fig. 5 illustrates the time development of the different helicity components, while the other panels show the behaviour of the normal vector and the spanwise direction when using the Seifert frame. A characteristic feature is that before reconnection, both vortices host one inflection point. Additionally, one may notice that at the instant of reconnection the linking is converted to writhe, which then decays due to mutual friction, as shown more clearly in the inset of Fig. 5a.

View Article: PubMed Central - PubMed

ABSTRACT

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.

No MeSH data available.


Related in: MedlinePlus