Dynamics and bifurcation of passive tracers advected by a ring of point vortices on a sphere

Abstract

We consider the dynamics of a passive tracer, advected by the presence of a latitudinal ring of identical point vortices. The corresponding instantaneous motion is modeled by a one degree of freedom Hamiltonian system. Such a dynamics presents a rich variety of behaviors with respect to the number of vortices, N, and the ring’s co-latitude, θo—or, equivalently, its vertical position qo = cos θo. We carry out a complete description of the global phase portrait for the cases N = 2, 3, 4 by determining equilibrium points, their stability, and bifurcations with respect to the parameter θo, and by characterizing the separatrix skeleton. Moreover, for N ≥ 5, we prove the existence of a value of bifurcation θoN such that when θo = θoN (θo = π − θoN, respectively) the south (north, respectively) pole becomes a N-bifurcation point, i.e., a symmetric web of N centers and N saddles bifurcates from the corresponding pole.