Vortices and Polynomials

The relationship between point vortex dynamics and the properties of polynomials with roots at the vortex positions is discussed. Classical polynomials, such as the Hermite polynomials, have roots that describe the equilibria of identical vortices on the line. Stationary and uniformly translating vortex configurations with vortices of the same strength but positive or negative orientation are given by the zeros of the Adler-Moser polynomials, which arise in the description of rational solutions of the Korteweg-de Vries equation. For quadupole background flow, vortex configurations are given by the zeros of polynomials expressed as wronskians of Hermite polynomials. Further new solutions are found in this case using the special polynomials arising the in the description of rational solutions of the fourth Painleve equation.Comment: 17 pages, minor revisions and references adde

Recent progress in the relative equilibria of point vortices — In memoriam Hassan Aref

AbstractHassan Aref, who sadly passed away in 2011, was one of the world's leading researchers in the dynamics and equilibria of point vortices. We review two problems on the subject of point vortex relative equilibria in which he was engaged at the time of his death: bilinear relative equilibria and the geometry of the three-vortex problem as it relates to equilibria. A set of point vortices is in relative equilibrium if it is at most rotating rigidly around the center of vorticity, and the configuration is bilinear if the vortices are placed on two orthogonal lines in the co-rotating frame. A very complete characterisation of the bilinear case can be obtained when one of the lines contains only two vortices. The classic three-vortex problem can be viewed anew by considering the dy- namics of the circle circumscribing the vortex triangle and the interior angles of that triangle. This approach leads naturally to the observation that the equilateral triangle is the only equilibrium configuration for three point vortices, regardless of their strength values

Dynamics of Perturbed Relative Equilibria of Point Vortices on the Sphere or Plane

Stable assemblages of localized vortices exist which have particle-like properties, such as mass, and which can interact with one another when they closely approach. In this article I calculate the mass of these localized states and numerically investigate some aspects of their interactions.Comment: 14 pages, 3 figure

Point vortices on a sphere: Stability of relative equilibria

In this paper we analyze the dynamics of N point vortices moving on a sphere from the point of view of geometric mechanics. The formalism is developed for the general case of N vortices, and the details are worked out for the (integrable) case of three vortices. The system under consideration is SO(3) invariant; the associated momentum map generated by this SO(3) symmetry is equivariant and corresponds to the moment of vorticity. Poisson reduction corresponding to this symmetry is performed; the quotient space is constructed and its Poisson bracket structure and symplectic leaves are found explicitly. The stability of relative equilibria is analyzed by the energy-momentum method. Explicit criteria for stability of different configurations with generic and nongeneric momenta are obtained. In each case a group of transformations is specified, modulo which one has stability in the original (unreduced) phase space. Special attention is given to the distinction between the cases when the relative equilibrium is a nongreat circle equilateral triangle and when the vortices line up on a great circle

Point vortices and classical orthogonal polynomials

Stationary equilibria of point vortices with arbitrary choice of circulations in a background flow are studied. Differential equations satisfied by generating polynomials of vortex configurations are derived. It is shown that these equations can be reduced to a single one. It is found that polynomials that are Wronskians of classical orthogonal polynomials solve the latter equation. As a consequence vortex equilibria at a certain choice of background flows can be described with the help of Wronskians of classical orthogonal polynomials.Comment: 20 pages, 12 figure

A transformation between stationary point vortex equilibria

A new transformation between stationary point vortex equilibria in the unbounded plane is presented.Given a point vortex equilibrium involving only vortices with negative circulation normalized to−1 and vortices with positive circulations that are either integers, or half-integers, the transformation produces a new equilibrium with a free complex parameter that appears as an integration constant.When iterated the transformation can produce infinite hierarchies of equilibria, or finite sequences that terminate after a finite number of iterations,each iteration generating equilibria with increasing numbers of point vortices and free parameters. In particular, starting from an isolated point vortex as a seed equilibrium, we recover two known infinite hierarchies of equilibria corresponding to the Adler–Moser polynomials and a class of polynomials found, using very different methods, by Loutsenko[J. Phys. A: Math. Gen. 37, (2004)]. For the latter polynomials the existence of such a transformation appears to be new. The new transformation therefore unifies a wide range of disparate results in the literature on point vortex equilibria

blocking events and the stability of the polar vortex

The present study investigates non-linear dynamics of atmospheric flow phenomena on different scales as interactions of vortices. Thereby, we apply the idealised, two-dimensional concept of point vortices considering two important issues in atmospheric dynamics. First, we propose this not widely spread concept in meteorology to explain blocked weather situations using a three-point vortex equilibrium. Here, a steady state is given if the zonal mean flow is identical to the opposed translational velocity of the vortex system. We apply this concept exemplarily to two major blocked events establishing a new pattern recognition technique based on the kinematic vorticity number to determine the circulations and positions of the interacting vortices. By using reanalysis data, we demonstrate that the velocity of the tripole in a westward direction is almost equal to the westerly flow explaining the steady state of blocked events. Second, we introduce a novel idea to transfer a stability analysis of a vortex equilibrium to the stability of the polar vortex concerning its interaction with the quasi-biennial oscillation (QBO). Here, the point vortex system is built as a polygon ring of vortices around a central vortex. On this way we confirm observations that perturbations of the polar vortex during the QBO east phase lead to instability, whereas the polar vortex remains stable in QBO west phases. Thus, by applying point vortex theory to challenging problems in atmospheric dynamics we show an alternative, discrete view of synoptic and planetary scale motion

Determination of stable branches of relative equilibria of the NN-vortex problem on the sphere

We consider the $N$-vortex problem on the sphere assuming that all vorticities have equal strength. We investigate relative equilibria (RE) consisting of $n$ latitudinal rings which are uniformly rotating about the vertical axis with angular velocity $\omega$. Each such ring contains $m$ vortices placed at the vertices of a concentric regular polygon and we allow the presence of additional vortices at the poles. We develop a framework to prove existence and orbital stability of branches of RE of this type parametrised by $\omega$. Such framework is implemented to rigorously determine and prove stability of segments of branches using computer-assisted proofs. This approach circumvents the analytical complexities that arise when the number of rings $n\geq 2$ and allows us to give several new rigorous results. We exemplify our method providing new contributions consisting in the determination of enclosures and proofs of stability of several equilibria and RE for $5\leq N\leq 12$.Comment: 60 page