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.
Highlights
Kinetic helicity is one of the invariants of the Euler equations that is associated with the topology of vortex lines within the fluid
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
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
Summary
Kinetic helicity is one of the invariants of the Euler equations that is associated with the topology of vortex lines within the fluid. It has become possible to realize particular cases of knotted vortex tubes produced in water and to study them under controlled conditions in the laboratory[11] This has paved the way to test assertions arising from considerations involving helicity. Even though the characteristics of superfluid 4He are different, similar knotted vortex excitations would be expected to arise Their experimental verification would be more of a challenge since it is only recently that direct visualization of quantized vortices in 4He has become possible using small tracer particles[13]. Reconnections delineate the boundaries of sharply defined intervals involving constant vortex topologies This makes superfluids an ideal setting in which to understand properties of helicity and how it relates to the topology of the superfluid vortex filaments. When the core size is much smaller than any other characteristic scale and compressiblity effects can be neglected, www.nature.com/scientificreports/
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