Knot spectrum of turbulence

Area of Science:

• Fluid Dynamics
• Quantum Mechanics
• Topology

Background:

• Turbulent fluids and plasmas exhibit complex topological structures like vortex lines.
• Quantifying the topological complexity of these structures is an open question.
• Superfluid helium offers a unique system with quantized vorticity.

Purpose of the Study:

• To quantify the topological complexity of quantum turbulence in superfluid helium.
• To investigate the formation, evolution, and distribution of vortex knots.
• To compare quantum vortex tangles with classical systems like polymers and strings.

Main Methods:

• Numerical simulation of quantum vortex line dynamics.
• Application of knot theory, specifically the Alexander polynomial, to quantify vortex topology.
• Analysis of knot spectra and scaling laws.

Main Results:

• Quantum vortex tangles in superfluid helium consistently form vortex knots of high topological degree.
• A distribution of topologies, characterized by a knot spectrum and scaling law, was identified.
• Vortex knotting probability increases with vortex length, saturating at a characteristic length.

Conclusions:

• The topological complexity of quantum turbulence can be quantified using knot invariants.
• The dynamics of quantum vortex tangles lead to complex knotting behaviors.
• Findings provide insights into the statistical properties of turbulent flows and topological structures.