Axis-symmetric Onsager Clustered States of Point Vortices in a Bounded Domain

Abstract

We study axis-symmetric Onsager clustered states of a neutral point vortex system confined to a two-dimensional disc. Our analysis is based on the mean field of bounded point vortices in the microcanonical ensemble. The clustered vortex states are specified by the inverse temperature $\beta$ and the rotation frequency $\omega$, which are the conjugate variables of energy $E$ and angular momentum $L$, respectively. The formation of the axis-symmetric clustered vortex states (azimuthal angle independent) involves the separating of vortices with opposite circulation and the clustering of vortices with same circulation around origin and edge. The state preserves $\mathrm{SO}(2)$ symmetry while breaks ${\mathbb{Z}}_2$ symmetry. We find that, near the uniform state ($E=0$), the rotation free state ($\omega =0$) emerges at particular values of $L^2/E$ and $\beta$. At large energies, we obtain asymptotically exact vortex density distributions, whose validity condition gives rise the lower bound of $\beta$ for the rotation free states. Noticeably, the obtained vortex density distribution near the edge at large energies provides a novel exact vortex density distribution for the corresponding chiral vortex system.

6 pages, 4 figures

References in corpus (9)

• Inverse Energy Cascade in Forced 2D Quantum Turbulence
• Emergence of order from turbulence in an isolated planar superfluid
• Classical and quantum regimes of two-dimensional turbulence in trapped Bose-Einstein condensates
• Spectral energy transport in two-dimensional quantum vortex dynamics
• Onset of vortex clustering and inverse energy cascade in dissipative quantum fluids
• Turbulent relaxation to equilibrium in a two-dimensional quantum vortex gas
• The true mechanism of spontaneous order from turbulence in two-dimensional superfluid manifolds
• Phase separation of quantized vortices in two-component miscible Bose-Einstein condensates in a two-dimensional box potential
• Chiral Edge Modes in Helmholtz-Onsager Vortex Systems

Cited by in corpus (2)

• Conformal-invariance of 2D quantum turbulence in an exciton-polariton fluid of light
• Onsager vortex clusters on a sphere