[Paper Review] Stability of Thomson's Configurations of Vortices on a Sphere

А. В. Борисов, Alexander A. Kilin|ArXiv.org|Mar 31, 2005

Stability and Controllability of Differential EquationsEngineering2 references19 citations

TL;DR

This paper investigates the linear and nonlinear stability of Thomson's vortex configurations—regular N-gons of equal-strength vortices—on a spherical surface. Using Hamiltonian mechanics and Birkhoff normalization, it proves that such configurations are linearly stable for N ≤ 7 but become unstable as N increases beyond 7, particularly near the equator where instability arises due to positive real parts in the growth rates of perturbations.

ABSTRACT

In this work stability of polygonal configurations on a plane and sphere is investigated. The conditions of linear stability are obtained. A nonlinear analysis of the problem is made with the help of Birkhoff normalization. Some problems are also formulated.

Motivation & Objective

• To analyze the linear and nonlinear stability of symmetric vortex configurations on a sphere, analogous to Thomson’s classical planar configurations.
• To determine the conditions under which regular N-gonal vortex arrangements on a sphere remain stable under small perturbations.
• To extend the classical planar stability results (N ≤ 7 stable) to the spherical geometry using Hamiltonian dynamics and normal form techniques.
• To investigate the role of latitude (θ₀) and vortex count (N) in determining stability boundaries, particularly near the equator.
• To formulate open problems related to nonlinear stability and resonance effects in the vortex system.

Proposed method

• Formulates the dynamics of N point vortices on a sphere using a Hamiltonian system with Poisson brackets and a logarithmic interaction potential.
• Derives the linearized equations of motion around the symmetric N-gon configuration by introducing small perturbations δθᵢ and δφᵢ.
• Expresses the linearized system in terms of an N×N matrix A with elements depending on angular differences, which is diagonalized via Fourier transformation.
• Computes the eigenfrequencies Ωₘ from the matrix spectrum, identifying zero-frequency modes corresponding to rigid rotation and determining stability from the sign of the real part.
• Applies Birkhoff normalization to analyze nonlinear stability, focusing on resonance monomials and the structure of the normal form Hamiltonian.
• Identifies resonance conditions via inequalities mᵢ ≤ ∑ⱼ≠ᵢ lⱼ, which prevent complete elimination of monomials and signal potential instability.

Experimental results

Research questions

• RQ1 For which values of N are Thomson’s vortex configurations on a sphere linearly stable?
• RQ2 How does the latitude θ₀ of the vortex ring affect the stability of the configuration?
• RQ3 What is the role of the equator in the onset of instability for large N?
• RQ4 Can nonlinear stability be established beyond linear analysis using Birkhoff normalization?
• RQ5 What are the resonance conditions that prevent complete normalization and may lead to instability?

Key findings

• Thomson’s vortex configurations on a sphere are linearly stable for N ≤ 7, with instability emerging for N > 7 due to positive real parts in the growth rates of perturbations.
• The critical threshold for instability occurs when the radicand in the frequency expression (10) becomes negative, which happens for N ≥ 7 regardless of θ₀, though the instability is most pronounced near the equator.
• For N = 7, the maximum value of m(N−m) is 12.25, and the radicand becomes negative when cos²θ₀ > (12.25 − 6)/12.25 ≈ 0.5, corresponding to θ₀ > arccos(√0.5) ≈ 45°, indicating instability near the equator.
• The zero-frequency mode at m = 0 corresponds to rigid rotation of the entire configuration, which must be factored out to assess relative stability.
• Nonlinear stability analysis via Birkhoff normalization reveals that resonance monomials—those satisfying mᵢ ≤ ∑ⱼ≠ᵢ lⱼ—cannot be eliminated and may lead to instability in higher-order terms.
• The presence of two-dimensional Jordan blocks in the linearized system for k = 2 zero frequencies leads to a normal form Hamiltonian that includes resonant monomials M_{l,m} with specific constraints on indices.