New Stationary $N$-Symmetric States of Three Point Vortices in a Two-Layer Rotating Fluid

Within the framework of the two-layer quasi-geostrophic model on a rotating plane, the motions for a special case of three point vortices with intensities $\big(\kappa_1^1, \kappa_2^1, \kappa_2^2\big)=(4,1,-2)$ are considered (here, the subscript denotes the layer number: 1 is the upper layer, 2 is the lower layer, and the superscript denotes the vortex number in the layer). Thus, it is assumed that one cyclonic vortex is located in the upper layer, and two vortices, cyclonic and anticyclonic, are located in the lower layer. It is shown that in the general case each vortex performs periodic motions in such a way that every half-period the vortex structure takes a collinear state. In this case, over time, the trajectory of each vortex completely fills a certain ring region around the vorticity center. However, among the continuum of these quasi-ordered trajectories, one can always find a family of closed periodic solutions, both purely circular (preserving the collinear structure) and more complex, so-called $N$-modal ($N$-symmetric) stationary solutions. In this paper, these solutions are constructed and their main properties are described.