High-dimensional heteroclinic and homoclinic connections in odd point-vortex ring on a sphere

Abstract

We consider the motion of the N -vortex points that are equally spaced along a line of latitude on sphere with fixed pole vortices, called "N -ring". We are especially interested in the case when the number of the vortex points is odd. Since the eigenvalues that determine the stability of the odd Nring are double, each of the unstable and stable manifolds corresponding to them is two-dimensional. Hence, it is generally difficult to describe the global structure of the manifolds. In this article, based on the linear stability analysis, we propose a projection method to observe the structure of the iso-surface of the Hamiltonian, in which the orbit of the vortex points evolves. Applying the projection method to the motion of the 3-ring and 5-ring, we characterize the complex evolution of the unstable odd N -ring from the topological structure of the iso-surface of the Hamiltonian.

Three ways to solve the Poisson equation on a sphere with Gaussian forcing

“…More recent work has therefore shifted to vortex blobs; a popular species of blob is the ''Gaussian vortex" in which the vorticity falls as an exponential of the square of distance from the center of the vortex [31,3]. Point vortex methods have been applied to flow on the surface of a sphere [19,2,14,17,15,16,13,18,20,[24][25][26][27][28][29][32][33][34][35][36][37]39]. However, extensions of Gaussian vortex methods to the sphere, as appropriate for adaptive modeling of atmospheric and oceanic flows, are handicapped by the lack of explicit solutions for the Poisson equation with Gaussian forcing.…”

Chaotic Motion of the N-Vortex Problem on a Sphere: I. Saddle-Centers in Two-Degree-of-Freedom Hamiltonians

“…[39]. First, we review the results of the linear stability analysis of the N -ring [37,38]. The N -ring configuration at the line of latitude θ 0 is represented by…”

Invariant dynamical systems embedded in the N-vortex problem on a sphere with pole vortices

“…The motion of the N-ring has been investigated in particular, since the ring configuration of the vortex structure is often observed in the numerical research of the atmospheric phenomena [4,14,16]; The linear and nonlinear stability analysis of the N-ring with and without the pole vortices were given [1,2,3,8,17]. The unstable motion of the perturbed N-ring was investigated [17,18].…”

“…Therefore we sometimes reduce the N-vortex system to low-dimensional systems by assuming a certain symmetry, and then study them as embedded subsystems. For instance, in the papers [17] and [18], the N-vortex system was successfully reduced to the integrable two-dimensional systems, with which the existence of the periodic, the heteroclinic and the homoclinic orbits and their stability were investigated. Thus the reduced systems help us understand the dynamics of the large number of the N-vortex points.…”

“…In the article, we give a reduction method of the N-vortex system to invariant dynamical systems. In §2, the linear stability analysis of the N-ring [17,18] is reviewed. It is required to characterize the invariant systems in the following sections.…”