Geometric Characterization of Atmospheric Islands Formed by Two Point Vortices

Generically, in a system with more than three point vortices, there exist regions of stability around each vortex, even if the system is chaotic. These regions are usually called stability islands, and they have a morphology that is hard to characterize. In a system of two or three point vortices, these stability islands are better named vortex atmospheres or atmospheric islands, since the whole system is regular. In this work, we present an analytical study to characterize these atmospheres in two point vortex systems for arbitrary values of the circulations Γ1 and Γ2 in the infinite two-dimensional plane x,y∈R2—the simplest scenario—by studying the dynamics of passive particles in these environments. We use the trajectories of passive particles to find the stagnation points of these systems, the special trajectories that partition R2 in different regions and the analytical expressions that define the boundary of the atmospheric islands. In order to characterize the geometry of these atmospheres, we compute their perimeter and area as a function of γ=Γ1Γ1+Γ2, if Γ1+Γ2≠0. The case Γ1+Γ2=0 is treated separately, as the perimeter and area of the atmospheres do not depend on the circulations. Furthermore, in this latter case, we observe that the atmospheric island has a very similar morphology to an ellipse, only differing from the ellipse that best approximates the atmosphere by a relative error of ≈3.76‰ in area.