Closed vortex filament in a cylindrical domain: Circulation quantization

This article investigates quantum oscillations of a vortex ring with zero thickness that evolves in a cylindrical domain V = D x [ 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 [Small oscillations of a vortex ring: Hamiltonian formalism and quantization, Eur. J. Mech. B/Fluids 92, 100-106 (2022); arXiv:Math-ph/2112.04859v1]. 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 2, are calculated in the article as well. The conclusions are compared with some results of numerical simulations by other authors Intermittency of velocity circulation in quantum turbulence, Phys. Rev. X 11, 011053 (2021).

Automated summary

This article investigates the quantum oscillations of a vortex ring with zero thickness in a cylindrical domain. The quantization scheme is based on the author's previous approach. The study finds the discrete values of circulation as a result of the conventional scheme of quantum theory. The model demonstrates the splitting of levels, and the levels correction values depend on the height of the cylinder and the shape of the curve. Additionally, the study proves the existence of fine structure in the basic circulation levels. The conclusions are compared with numerical simulations by other authors.