Geometric Symmetries in Superfluid Vortex Dynamics

TL;DR

This paper explores how geometric symmetries in superfluid vortex lines influence their dynamics, especially the behavior of Kelvin waves, leading to local cascades in wavenumber space and revealing broader implications for systems with geometric degrees of freedom.

Contribution

It uncovers the role of geometric symmetries in superfluid vortex dynamics and their impact on Kelvin-wave cascades, highlighting the importance of geometric considerations in fluid systems.

Findings

Geometric symmetry leads to local Kelvin-wave cascades.

Constants of motion include linear and angular momenta.

Broader implications for systems with geometric degrees of freedom.

Abstract

Dynamics of quantized vortex lines in a superfluid feature symmetries associated with the geometric character of the complex-valued field, $w(z)=x(z)+iy(z)$, describing the instant shape of the line. Along with a natural set of Noether's constants of motion, which---apart from their rather specific expressions in terms of $w(z)$---are nothing but components of the total linear and angular momenta of the fluid, the geometric symmetry brings about crucial consequences for kinetics of distortion waves on the vortex lines---the Kelvin waves. It is the geometric symmetry that renders Kelvin-wave cascade local in the wavenumber space. Similar considerations apply to other systems with purely geometric degrees of freedom.