Nonlinear Stability of a Latitudinal Ring of Point-Vortices on a Nonrotating Sphere

Abstract

We study the nonlinear stability of relative equilibria of configurations of identical point-vortices on the surface of a sphere. In particular, we study how the stability changes as a function of the colatitude θ and of the number of vortices N. By using the integrals of motion, we view the system in a suitable corotating frame where the polygonal vortex configuration is at rest. Then after a sufficient criterion due to Dirichlet, the stability ranges are the θ-intervals for which the Hessian of the Hamiltonian-evaluated at the equilibrium configuration-is positive or negative definite. We find that the stability intervals coincide with those for linear stability determined by

Latitudinal point vortex rings on the spheroid

“…But the case of N = 7 (called the Thomson heptagon) was noticed as a peculiar configuration dividing stability and instability, which eventually turns out to be neutrally stable after several decades (see Boatto & Cabral (2003) for details). The ring of point vortices problem is generalized later and is investigated analogously for various underlying surfaces such as a sphere (Polvani & Dritschel 1993;Boatto & Cabral 2003;Boatto & Simo 2004;Kurakin 2004;Boatto & Simo 2008) or a cylinder (Souliere & Tokieda 2002;Montaldi et al 2003). In the same spirit, here we study the stability of a vortex ring on the spheroid as an application and also a concrete example of previous mathematical formulation.…”

“…Here, we briefly sketch the criterion of stability for the reader (details are in Boatto & Cabral (2003) or Boatto & Simo (2004)). Via the change of variables…”

“…We are primarily interested in the difference with the spherical case, and particular attention is given to the effect of eccentricity on the stability of vortex rings. Computations are performed using MAPLE and results are compared with those of the spherical case in Boatto & Cabral (2003) and Boatto & Simo (2004). For N = 2 to N = 6 this procedure is relatively simple and easily computable.…”

“…This problem originated from the classical work of Thomson (1883), in which the stability of point vortices on the edge of a regular n-gon is analysed in the plane. This study was subsequently generalized and has been applied to various underlying surfaces such as a cylinder (Souliere & Tokieda 2002;Montaldi et al 2003) and a sphere (Polvani & Dritschel 1993;Boatto & Cabral 2003;Boatto & Simo 2004;Kurakin 2004;Boatto & Simo 2008). The stability of the point vortex ring on a spheroid will be analysed by comparing it with the case of a sphere.…”

Relative periodic solutions of the <inline-formula><tex-math id="M1">\begin{document}$ n $\end{document}</tex-math></inline-formula>-vortex problem on the sphere

“…The nonlinear stability of this polygonal relative equilibrium has been established in [20] and [4]. In [20], a constant background vorticity is imposed on the sphere so that the total vorticity adds to zero, equation (1), while [4] considers an additional vortex fixed at one of the poles.…”