Helicity within the vortex filament model

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.

Helicity in superfluids: Existence and the classical limit

“…The mapping between superfluid flow and Euler flow makes it tempting to conclude that classical helicity is conserved in superfluids just as in Euler flows. However, numerical simulations show that the expression for helicity in Euler flows: H Euler = d 3 x u · ω ω ω , ω ω ω = ∇ × u is not conserved in superfluid flows [9,19,21]. H Euler evaluated for singular vortex lines has two contributions: (a) the Gauss linking integral for pairs of vortex lines, giving the linking between them, and (b) the Gauss linking integral evaluated for each vortex line and itself giving its writhe [32].…”

“…it is a Seifert surface [20,[65][66][67] for the vortex lines in the superfluid. This relation between linking and writhing of vortex lines and the twisting of phase isosurfaces has been used in superfluid simulations [9,68] to calculate the centerline helicity (linking and writhing of vortex lines), and was elaborated on in recent efforts to define a superfluid helicity [21,22].…”

Dual cascade and dissipation mechanisms in helical quantum turbulence

“…Its role in quantum turbulence is less clear. Efforts focus on determining if it is conserved by studying simple configurations of reconnecting vortex knots [14][15][16][17][18]. Numerical evidence indicates that in this case helicity is transferred from large to small scales [14,16], and that reconnection or the transfer of helicity can excite non-linear interacting Kelvin waves [17,19], which eventually may lead to a loss of helicity by sound emission.…”

Crossover from interaction to driven regimes in quantum vortex reconnections

“…By altering the topology of the flow [28], reconnections also seem to redistribute its helicity [29,30], although the precise definition of helicity in superfluids is currently debated [30][31][32], and the effects of reconnections [33][34][35][36] on its geometric ingredients (link, writhe and twist) are still discussed. In the low-temperature limit, losses due to viscosity or mutual friction are negligible, and reconnections are the ultimate mechanism for the dissipation of the incompressible kinetic energy of the superfluid via sound radiation at the reconnecting event [37,38] followed by further sound emission by the Kelvin wave cas-cade [39][40][41] which follows the relaxation of the reconnection cusps.…”