Closed Vortex Filament in a Cylindrical Domain: Circulation Quantization

TL;DR

This paper rigorously derives quantized circulation levels for a vortex ring in a cylindrical domain, revealing discrete values, level splitting, and a fine structure influenced by domain geometry, advancing understanding of quantum vortex dynamics.

Contribution

The paper introduces a rigorous derivation of circulation quantization and fine structure effects for vortex rings in cylindrical domains, based on a novel quantum approach.

Findings

Discrete circulation levels $\Gamma_n$ are rigorously derived.

Level splitting and corrections depend on domain geometry.

A fine structure proportional to $\hbar^2$ is identified.

Abstract

This article investigates quantum oscillations of a vortex ring with zero thickness that evolves in a cylindrical domain $V = D \times [0,L]$. The symbol $D$ denotes the planar domain which is bounded by some closed connected curve $S$. The quantization scheme of this dynamical system is based on the approach proposed by the author earlier. As result, we find the discrete values $\Gamma_n$ for circulation $\Gamma$. In contrast to the traditional approach, where such quantities are usually postulated, the values $\Gamma_n$ are deduced rigorously as the consequence of the conventional scheme of quantum theory. The model demonstrates the splitting of levels also. In particular, the levels correction values depend on the domain $V$: both the cylinder height $L$ and the form of the curve $S$ affect the final formula for the quantities $\Gamma_n$. Moreover, we prove that the basic circulation levels demonstrate a "fine structure". These anomalous terms, which are proportional to the value $\hbar^2$, are calculated in the article as well. The conclusions are compared with some results of numerical simulations by other authors.