Journal of Nonlinear Science

, Volume 22, Issue 5, pp 849–885

# Bilinear Relative Equilibria of Identical Point Vortices

Article

## Abstract

A new class of bilinear relative equilibria of identical point vortices in which the vortices are constrained to be on two perpendicular lines, conveniently taken to be the x- and y-axes of a Cartesian coordinate system, is introduced and studied. In the general problem we have m vortices on the y-axis and n on the x-axis. We define generating polynomials q(z) and p(z), respectively, for each set of vortices. A second-order, linear ODE for p(z) given q(z) is derived. Several results relating the general solution of the ODE to relative equilibrium configurations are established. Our strongest result, obtained using Sturm’s comparison theorem, is that if p(z) satisfies the ODE for a given q(z) with its imaginary zeros symmetric relative to the x-axis, then it must have at least nm+2 simple, real zeros. For m=2 this provides a complete characterization of all zeros, and we study this case in some detail. In particular, we show that, given q(z)=z 2+η 2, where η is real, there is a unique p(z) of degree n, and a unique value of η 2=A n , such that the zeros of q(z) and p(z) form a relative equilibrium of n+2 point vortices. We show that $$A_{n} \approx\frac{2}{3}n + \frac{1}{2}$$, as n→∞, where the coefficient of n is determined analytically, the next-order term numerically. The paper includes extensive numerical documentation on this family of relative equilibria.

## Keywords

Ideal fluids Vortex dynamics Point vortices Relative equilibria Polynomials

## Mathematics Subject Classification

76B47 34M99 34B24 15A15

## Notes

### Acknowledgements

H.A. thanks Peter Clarkson for correspondence regarding the integrable ODE mentioned in Sect. 6, and he thanks Kevin O’Neil for several comments regarding issues relevant to this work. This work was supported in part by a Niels Bohr Visiting Professorship at the Technical University of Denmark sponsored by the Danish National Research Foundation. Peter Beelen was partially supported by DNRF (Denmark) and NSFC (China), grant No. 11061130539.

## References

1. Aref, H.: On the equilibrium and stability of a row of point vortices. J. Fluid Mech. 290, 167–181 (1995). doi:
2. Aref, H., Vainchtein, D.L.: Asymmetric equilibrium patterns of point vortices. Nature 392, 769–770 (1998). doi:
3. Aref, H., van Buren, M.: Vortex triple rings. Phys. Fluids 17, 057104 (2005), 21 pp. doi:
4. Aref, H., Newton, P.K., Stremler, M.A., Tokieda, T., Vainchtein, D.L.: Vortex crystals. Adv. Appl. Mech. 39, 1–79 (2003). doi:
5. Calogero, F., Perelomov, A.M.: Asymptotic density of the zeros of Hermite polynomials of diverging order, and related properties of certain singular integral operators. Lett. Nuovo Cimento 23, 650–652 (1978). doi:
6. Clarkson, P.A.: Vortices and polynomials. Stud. Appl. Math. 123, 37–62 (2009). doi:
7. Dirksen, T., Aref, H.: Close pairs of relative equilibria for identical point vortices. Phys. Fluids Lett. 23, 051706 (2011). doi:
8. Havelock, T.H.: Stability of motion of rectilinear vortices in ring formation. Philos. Mag. Ser. 7 11, 617–633 (1931). doi: Google Scholar
9. Ince, E.L.: Ordinary Differential Equations. General Publishing Co., Toronto (1926); republished by Dover Publications, Inc., New York. ISBN 0486603490 Google Scholar
10. Muir, T.: A Treatise on the Theory of Determinants. Constable & Co., London (1960). Revised and enlarged by W.H. Metzler; republished by Dover Publications, New York Google Scholar
11. O’Neil, K.A.: Minimal polynomial systems for point vortex equilibria. Physica D 219, 69–79 (2006). doi:
12. O’Neil, K.A.: Relative equilibrium and collapse configurations of four point vortices. Regul. Chaotic Dyn. 12, 117–126 (2007). doi:
13. O’Neil, K.A.: Central configurations of identical masses lying along curves. Phys. Rev. E 79, 066601 (2009). doi:
14. O’Neil, K.A.: Collapse and concentration of vortex sheets in two-dimensional flow. Theor. Comput. Fluid Dyn. 24, 39–44 (2010). doi:
15. Saffman, P.G.: Vortex Dynamics. Cambridge University Press, Cambridge (1992). ISBN 052142058X
16. Stieltjes, T.J.: Sur certains polynômes qui verifient une équation différentielle. Acta Math. 6–7, 321–326 (1885). doi:
17. Szegö, G.: Orthogonal Polynomials, 4th edn. Colloquium Publications, vol. XXIII. American Mathematical Society, Providence (1975). ISBN 0821810235