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Helicity within the vortex filament model

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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.


Vortex ring with a Kelvin mode of m = 3.The blue tubes denote the vortex and the red strips on panels (a–c) denote the normal unit vectors where the ratio of the Kelvin wave amplitude to the vortex ring radius (R) is A/R = 0.0975, 0.1025, and 0.200, respectively. In the right panel (d) the green strips denote the spanwise vector when A/R = 0.200, but it turns out to be rather insensitive to the amplitude of the Kelvin wave.
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f3: Vortex ring with a Kelvin mode of m = 3.The blue tubes denote the vortex and the red strips on panels (a–c) denote the normal unit vectors where the ratio of the Kelvin wave amplitude to the vortex ring radius (R) is A/R = 0.0975, 0.1025, and 0.200, respectively. In the right panel (d) the green strips denote the spanwise vector when A/R = 0.200, but it turns out to be rather insensitive to the amplitude of the Kelvin wave.

Mentions: To determine the internal twist angle (up to a constant), and, therefore, also the spanwise vector, N, we can either use Eq. (13) for the direction of the constant phase, or alternatively use Eq. (16) for the classical case. Figure 2b illustrates the internal twist angle θ when using the Seifert frame, Eq. (13), for the case of m = 3 at various Kelvin wave amplitudes. At the critical amplitude, Ac, the angle θ has a jump of π at the azimuthal locations , i = 1, …, m. This jump compensates the jump of −π in the normal and binormal around the tangent, thus ensuring that the N vector varies smoothly as we move along the length of the vortex. The behaviour of the unit vectors and N is illustrated in Fig. 3 for few different Kelvin wave amplitudes and with m = 3. We note that in this case, the normal and binormal vectors make three complete rotations around the tangent only for amplitudes above Ac.


Helicity within the vortex filament model
Vortex ring with a Kelvin mode of m = 3.The blue tubes denote the vortex and the red strips on panels (a–c) denote the normal unit vectors where the ratio of the Kelvin wave amplitude to the vortex ring radius (R) is A/R = 0.0975, 0.1025, and 0.200, respectively. In the right panel (d) the green strips denote the spanwise vector when A/R = 0.200, but it turns out to be rather insensitive to the amplitude of the Kelvin wave.
© Copyright Policy - open-access
Related In: Results  -  Collection

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

f3: Vortex ring with a Kelvin mode of m = 3.The blue tubes denote the vortex and the red strips on panels (a–c) denote the normal unit vectors where the ratio of the Kelvin wave amplitude to the vortex ring radius (R) is A/R = 0.0975, 0.1025, and 0.200, respectively. In the right panel (d) the green strips denote the spanwise vector when A/R = 0.200, but it turns out to be rather insensitive to the amplitude of the Kelvin wave.
Mentions: To determine the internal twist angle (up to a constant), and, therefore, also the spanwise vector, N, we can either use Eq. (13) for the direction of the constant phase, or alternatively use Eq. (16) for the classical case. Figure 2b illustrates the internal twist angle θ when using the Seifert frame, Eq. (13), for the case of m = 3 at various Kelvin wave amplitudes. At the critical amplitude, Ac, the angle θ has a jump of π at the azimuthal locations , i = 1, …, m. This jump compensates the jump of −π in the normal and binormal around the tangent, thus ensuring that the N vector varies smoothly as we move along the length of the vortex. The behaviour of the unit vectors and N is illustrated in Fig. 3 for few different Kelvin wave amplitudes and with m = 3. We note that in this case, the normal and binormal vectors make three complete rotations around the tangent only for amplitudes above Ac.

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.