On Stability of the Thomson’s Vortex N-gon in the Geostrophic Model of the Point Vortices in Two-layer Fluid

Abstract

A two-layer quasigeostrophic model is considered. The stability analysis of the stationary rotation of a system of N identical point vortices lying uniformly on a circle of radius R in one of the layers is presented. The vortices have identical intensity and length scale is $$\gamma ^{-1}>0$$ . The problem has three parameters: N, $$\gamma R$$ and $$\beta $$ , where $$\beta $$ is the ratio of the fluid layer thicknesses. The stability of the stationary rotation is interpreted as orbital stability. The instability of the stationary rotation is instability of system reduced equilibrium. The quadratic part of the Hamiltonian and eigenvalues of the linearization matrix are studied. The parameter space $$(N,\gamma R,\beta )$$ is divided on three parts: $$\varvec{A}$$ is the domain of stability in an exact nonlinear setting, $$\varvec{B}$$ is the linear stability domain, where the stability problem requires the nonlinear analysis, and $$\varvec{C}$$ is the instability domain. The case $$\varvec{A}$$ takes place for $$N=2,3,4$$ for all possible values of parameters $$\gamma R$$ and $$\beta $$ . In the case of $$N=5$$ , we have two domains: $$\varvec{A}$$ and $$\varvec{B}$$ . In the case $$N=6$$ , part $$\varvec{B}$$ is curve, which divides the space of parameters $$(\gamma R, \beta )$$ into the domains: $$\varvec{A}$$ and $$\varvec{C}$$ . In the case of $$N=7$$ , there are all three domains: $$\varvec{A}$$ , $$\varvec{B}$$ and $$\varvec{C}$$ . The instability domain $$\varvec{C}$$ takes place always if $$N=2n\geqslant 8$$ . In the case of $$N=2\ell +1\geqslant 9$$ , there are two domains: $$\varvec{B}$$ and $$\varvec{C}$$ . The results of research are presented in two versions: for parameter $$\beta $$ and parameter $$\alpha $$ , where $$\alpha $$ is the difference between layer thicknesses. A number of statements about the stability of the Thomson N-gon is obtained for the systems of interacting particles with the general Hamiltonian depending only on distances between the particles. The results of theoretical analysis are confirmed by numerical calculations of the vortex trajectories.