Stability of leapfrogging vortex pairs: A semi-analytic approach

We investigate the stability of a one-parameter family of periodic solutions of the four-vortex problem known as “leapfrogging” orbits. These solutions, which consist of two pairs of identical yet oppositely signed vortices, were known to W. Gröbli [Ph.D. thesis, Georg-August-Universität Göttingen, 1877] and A. E. H. Love [Proc. London Math. Soc. 1, 185 (1883)] and can be parameterized by a dimensionless parameter $\alpha$ related to the geometry of the initial configuration. Simulations by Acheson [Eur. J. Phys. 21, 269 (2000)] and numerical Floquet analysis by Tophøj and Aref [Phys. Fluids 25, 014107 (2013)] both indicate, to many digits, that the bifurcation occurs when $1/\alpha ={\varphi }^2$, where $\varphi$ is the golden ratio. This study aims to explain the origin of this remarkable value. Using a trick from the gravitational two-body problem, we change variables to render the Floquet problem in an explicit form that is more amenable to analysis. We then implement G. W. Hill's [Acta Math. 8, 1 (1886)] method of harmonic balance to high order using computer algebra to construct a rapidly converging sequence of asymptotic approximations to the bifurcation value, confirming the value found earlier.

Figure 1

(a) Opposite-signed vortices move in parallel along straight lines. (b) Like-signed vortices move along a circular path.

Figure 2

Motion in physical ($z$) space. Average motion is from left to right. Markers are given every half-period.

Figure 3

Motion in physical ($z$) space. (a) This solution features several bouts of walkabout motion including one extended period of three consecutive walkabout “dances.” (b) In this solution the last period of walkabout is braided as the two negative (blue) vortices take turns orbiting the tightly bound pair of positive (red) vortices. (c) A leapfrogging motion that transitions to walkabout motion before disintegrating. (d) A leapfrogging motion that disintegrates without a walkabout stage.

Figure 4

Level sets of the one-degree-of-freedom Hamiltonian Eq. (5) in the $X-Y$ plane, including the critical energy level $H=h_{\mathrm{c}}$ (bold) and the separatrix at $H=h_{\mathrm{s}}=\frac{1}{2}$ (dashed). Unbounded orbits not shown. The center at the origin corresponds $h=0$ in Eq. (5) and to the limiting physical state in which the pairs of like-vorticity are at an infinitesimal distance and rotate with a divergent frequency as described by the original Hamiltonian. Stable orbits foliate the area between this point and the critical energy level.

Figure 5

(a) The trace of the monodromy matrix as a function of the energy $h$. (b) The periodic orbit at $h=\frac{1}{8}$.