Energy-Momentum Stability of Icosahedral Configurations of Point Vortices on a Sphere

Abstract

We investigate the nonlinear stability of the icosahedral relative equilibrium configuration of point vortices on a sphere. The relative equilibrium problem is formulated as a problem of finding the nullspace of the configuration matrix that encodes the geometry of the icosahedron, as in Jamaloodeen and Newton (Proc. Royal Soc. A, Math. Phys. Eng. Sci. 462(2075):3277, 2006). The seven-dimensional nullspace of the configuration matrix, A, associated with the icosahedral geometry gives rise to a basis set of vortex strengths for which the icosahedron stays in relative formation, and we use these values to form the augmented Hamiltonian governing the stability. We choose the basis set made up of (i) one element with equal strength vortices on every vertex of the icosahedron (the uniform icosahedron); (ii) six elements made up of equal and opposite antipodal pairs. We start by proving nonlinear stability of the antipodal vortex pair (by direct methods). Following the methods laid out in Simo et al. (Arch. Ration. Mech. Anal. 115(1):15–59, 1991) and Pekarsky and Marsden (J. Math. Phys. 39(11):5894–5907, 1998) and more generally in Marsden and Ratiu (Introduction to Mechanics and Symmetry, 1999), we then combine our knowledge of the nullspace structure of A with the structure of the underlying Hamiltonian, and analyze the stability of the icosahedron using the energy-momentum method. Because the parameter space is large, we focus on the physically motivated and important case obtained by combining the basis elements into (i) the uniform icosahedron; (ii) a von Kármán vortex street configuration of equal and opposite staggered, evenly spaced latitudinal rows equidistant from the equator (Chamoun et al. in Phys. Fluids 21:116603, 2009), and (iii) the North Pole–South Pole equal and opposite vortex pair. Stability boundaries in a three-parameter space are calculated for linear combinations of these grouped basis configurations.

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Notes

  1. 1.

    A symmetry induced basis is a basis of the invariant subspaces of the dihedral group D 5 (symmetry group of regular 5-gon). As such, it is the biggest symmetry subgroup of the configuration under investigation; i.e., the configuration will not change if we do cyclic rotations by an angle \(\frac{2\pi}{5}\) about the z-axis or flip about any plane which goes through the z-axis and any vertex.

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Acknowledgements

We dedicate this paper to the memory of Jerry Marsden, whose work in Hamiltonian mechanics and stability theory laid the groundwork for much that is described in this paper. Support from the National Science Foundation, grant NSF-DMS-0804629, is greatly appreciated.

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Correspondence to Paul K. Newton.

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Communicated by Anthony Bloch.

Appendix

Appendix

Matrix C can be written as

$$\mathbf{C}=\left ( \begin{array}{c@{\quad}c@{\quad}c@{\quad}c} C_1 & C_2 & C_3 & C_4\\ C_2 & C_5 & C_6 & C_7\\ C_3 & C_6 & C_{8} & C_{9}\\ C_4 & C_8 & C_{9} & C_{10}\\ \end{array} \right ), $$

where C i ,i=1,…,10 are 2×2 matrices:

Matrix D can be written as

$$\mathbf{D}=\left ( \begin{array}{c@{\quad}c@{\quad}c@{\quad}c@{\quad}c} D_1 & D_2 & D_3 & D_4 & D_5\\ D_2 & D_6 & D_7 & D_8 & D_9\\ D_3 & D_7 & D_{10} & D_{11} & D_{12}\\ D_4 & D_8 & D_{11} & D_{13} & D_{14}\\ D_5 & D_9 & D_{12} & D_{14} & D_{15} \end{array} \right ), $$

where D i ,i=1,…,15 are 2×2 matrices:

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Newton, P.K., Ostrovskyi, V. Energy-Momentum Stability of Icosahedral Configurations of Point Vortices on a Sphere. J Nonlinear Sci 22, 499–515 (2012). https://doi.org/10.1007/s00332-012-9142-5

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Keywords

  • Stability of point vortices
  • Relative equilibria
  • Point vortex equilibria
  • Energy-momentum method

Mathematics Subject Classification

  • 76B47
  • 70H14
  • 70E50
  • 37N10
  • 37J25