Axis-symmetric Onsager clustered states of point vortices in a bounded domain

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 the same circulation around the origin and edge. The state preserves SO(2) symmetry while breaking Z(2) symmetry. We find that, near the uniform state, the rotation-free clustered 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 to 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.