Helicity within the vortex filament model

Area of Science:

• Fluid Dynamics
• Quantum Fluids
• Topological Fluid Mechanics

Background:

• Kinetic helicity is a fluid dynamics invariant linked to vortex line topology.
• Superfluid vorticity concentrates along filaments, but lacks a core structure, complicating helicity.
• Standard helicity definitions fail to retain quadratic invariant properties in superfluids.

Purpose of the Study:

• To redefine kinetic helicity for superfluid vortex filaments.
• To recover the quadratic invariant properties of helicity in superfluids.
• To explain the emergence of a quasi-classical helicity limit.

Main Methods:

• Utilized a Seifert framing to define a spanwise vector for vortex filaments.
• Introduced a twist component to the helicity definition.
• Calculated internal twist using examples and related it to tangential velocity.

Main Results:

• Recovered key properties of helicity as a quadratic invariant for superfluid vortex filaments.
• Demonstrated that centreline helicity alone yields ambiguous results without twist.
• Derived an internal twist angle equation similar to classical vortex tubes.

Conclusions:

• A Seifert framing approach successfully reintroduces twist, restoring helicity's invariant properties in superfluids.
• The method provides a clear understanding of how quasi-classical helicity arises from individual superfluid vortex filaments.