Relative Equilibria of the (1+N)-Vortex Problem

We examine existence and stability of relative equilibria of the n-vortex problem specialized to the case where N vortices have small and equal circulation and one vortex has large circulation. As the small circulation tends to zero, the weak vortices tend to a circle centered on the strong vortex. A special potential function of this limiting problem can be used to characterize orbits and stability. Whenever a critical point of this function is nondegenerate, we prove that the orbit can be continued via the Implicit Function Theorem, and its linear stability is determined by the eigenvalues of the Hessian matrix of the potential. For N≥3 there are at least three distinct families of critical points associated to the limiting problem. Assuming nondegeneracy, one of these families continues to a linearly stable class of relative equilibria with small and large circulation of the same sign. This class becomes unstable as the small circulation passes through zero and changes sign. Another family of critical points which is always nondegenerate continues to a configuration with small vortices arranged in an N-gon about the strong central vortex. This class of relative equilibria is linearly unstable regardless of the sign of the small circulation when N≥4. Numerical results suggest that the third family of critical points of the limiting problem also continues to a linearly unstable class of solutions of the full problem independent of the sign of the small circulation. Thus there is evidence that linearly stable relative equilibria exist when the large and small circulation strengths are of the same sign, but that no such solutions exist when they have opposite signs. The results of this paper are in contrast to those of the analogous celestial mechanics problem, for which the N-gon is the only relative equilibrium for N sufficiently large, and is linearly stable if and only if N≥7.     

Multiplicity results for the two-vortex Chern-Simons Higgs model on the two-sphere

We consider a Ginzburg-Landau type functional on S ² with a 6 ^th order potential and the corresponding selfduality equations. We study the limiting behavior in the two vortex case when a coupling parameter tends to zero. This two vortex case is a limiting case for the Moser inequality, and we correspondingly detect a rich and varied asymptotic behavior depending on the position of the vortices. We exploit analogies with the Nirenberg problem for the prescribed Gauss curvature equation on S ². Received: December 3, 1997     

A further note on the force discrepancy for wing theory in Euler flow

Uniform steady potential flow past a wing aligned at a small angle to the flow direction is considered. The standard approach is to model this by a vortex sheet, approximated by a finite distribution of horseshoe vortices. In the limit as the span of the horseshoe vortices tends to zero, an integral distribution of infinitesimal horseshoe vortices over the vortex sheet is obtained. The contribution to the force on the wing due to the presence of one of the infinitesimal horseshoe vortices in the distribution is focused upon. Most of the algebra in the force calculation is evaluated using Maple software and is given in the appendices. As in the two previous papers by the authors on wing theory in Euler flow [E Chadwick, A slender-wing theory in potential flow, Proc. R. Soc. A461 (2005) 415–432, and E Chadwick and A Hatam, The physical interpretation of the lift discrepancy in Lanchester-Prandtl lifting wing theory for Euler flow, leading to the proposal of an alternative model in Oseen flow, Proc. R. Soc. A463 (2007) 2257–2275], it is shown that the normal force is half that expected. In this further note, in addition it is demonstrated that the axial force is infinite. The implications and reasons for these results are discussed.     

On the motion of two-dimensional vortices with mass

Summary The Helmholtz-Kirchhoff ODEs governing the planar motion ofN point vortices in an ideal, incompressible fluid are extended to the case where the fluid has impurities. In this case the resulting ODEs have an additional inertia-type term, so the point vortices are termed massive. Using an electromagnetic analogy, these equations also determine the behavior of columns of charges in an external magnetic field. Using the symmetries, we reduce the four degrees of freedom system of two “massive” vortices totwo degrees of freedom. We exhibit an integrable case and a nonintegrable one, according to choices of parameters. Nonintegrability is verified using a recent result obtained independently by Lerman and by Mielke, Holmes, and O'Reilly. Finally, we discuss the behavior of solutions as the masses of the vortices tend to zero, using for initial conditions a point of the trajectory of the Helmholtz-Kirchhoff equations.     

Dynamics of planar vortex clusters with binaries

We study an N -vortex problem having J of them forming a cluster, which means the distances between the vortices in the cluster is much smaller by O (ε) than the distances, O (ℓ), to the N – J vortices outside of the cluster. With the strengths of N vortices being of the same order, the velocity and time scales for the motion of the J vortices relative to those of the N – J vortices are O (ε^–1) and O (ε²) respectively. We show that this two-time and two-length scale problem can be converted to a standard two-time scale problem and then the leading order solution of the N -vortex problem can be uncoupled to two problems, one for the motion of J vortices in the cluster relative to the center of the cluster and one for the motion of the N – J vortex plus the center of the cluster. For N = 3 and J = 2, the 3-vortex problem is uncoupled to two binary vortices problems in the length scales ℓ and ℓε respectively. When perturbed in the scale ℓ, say by a fourth vortex even of finite strength, the binary problem becomes a 3-vortex problem, admitting periodic solutions. Since 3-vortex problems are solvable, the uncoupling enables us to solve 3-cluster problems having at most three vortices in each cluster. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Existence of a Weak Solution in L p to the Vortex-Wave System

The vortex-wave system is a coupling of the two-dimensional vorticity equation with the point-vortex system. It is a model for the motion of a finite number of concentrated vortices moving in a distributed vorticity background. In this article, we prove existence of a weak solution to this system with an initial background vorticity in L ^p , p>2, up to the time of first collision of point vortices.     

The influence of turbulence on a columnar vortex with axial flow

The interaction between a columnar vortex and external turbulence is investigated numerically. A q -vortex is immersed in an initially isotropic homogeneous turbulence field, which itself is produced numerically by a direct numerical simulation of decaying turbulence. The formation of turbulent eddies around the columnar vortex and the vortex-core deformations are studied in detail by visualizing the flow field. In the less-stable case with q = –1.5, small thin spiral structures are formed inside the vortex core. In the unstable case with q = –0.45, the linear unstable modes grow until the columnar vortex make one turn. Its growth rate agrees with that of the linear analysis of Mayer and Powell[1]. After two turns of the vortex, the secondary instability is excited, which causes collapse of the columnar q -vortex and the sudden appearance of many fine scale vortices. (© 2006 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

On the resistanceless statement of the two-dimensional Neumann-Kelvin problem for a surface-piercing tandem

A new set of supplementary conditions is proposed for the two-dimensionalNeumann-Kelvin problem describing the steady-state forward motionof a surface-piercing tandem in an infinite-depth fluid. Thisproblem is shown to be uniquely solvable for almost every valueof the forward speed U. The velocity potential solving the problemcorresponds to a flow about the tandem providing no resistance(wave and spray resistance vanish simultaneously). On the otherhand, for exceptional values of U examples of non-uniqueness(trapped modes) are constructed using the inverse procedurerecently applied by McIver (J. Fluid Mech. 1996) to the problemof time-harmonic water waves. For the proposed statement ofthe Neumann-Kelvin problem the inverse method involves the investigationof streamlines generated by two vortices placed in the freesurface. The spacing of vortices delivering trapped modes dependson U.     

Chaotic Motion of the N-Vortex Problem on a Sphere: I. Saddle-Centers in Two-Degree-of-Freedom Hamiltonians

We study the motion of N point vortices with N∈ℕ on a sphere in the presence of fixed pole vortices, which are governed by a Hamiltonian dynamical system with N degrees of freedom. Special attention is paid to the evolution of their polygonal ring configuration called the N -ring, in which they are equally spaced along a line of latitude of the sphere. When the number of the point vortices is N=5n or 6n with n∈ℕ, the system is reduced to a two-degree-of-freedom Hamiltonian with some saddle-center equilibria, one of which corresponds to the unstable N-ring. Using a Melnikov-type method applicable to two-degree-of-freedom Hamiltonian systems with saddle-center equilibria and a numerical method to compute stable and unstable manifolds, we show numerically that there exist transverse homoclinic orbits to unstable periodic orbits in the neighborhood of the saddle-centers and hence chaotic motions occur. Especially, the evolution of the unstable N-ring is shown to be chaotic.      

Stationary corner vortex configurations

Given a stable configuration of point vortices for steady two dimensional inviscid, incompressible fluid flow in a domainD, it is shown that there is another stable configuration of stationary point vortices inD with vortices near the original vortices plus additional vortices near any of the convex corners ofD. It follows that there are steady flows which have a finite sequence, of arbitrary length, of vortices of alternating sign descending into any convex corner ofD. Several computed examples are given.     

Central Units,Class Sums and Characters of the Symmetric Group

In the search for central units of a group algebra, we look at the class sums of the group algebra of the symmetric group S _n in characteristic zero, and we show that they are units in very special instances.     

Euler configurations and quasi-polynomial systems

Consider the problem of three point vortices (also called Helmholtz’ vortices) on a plane, with arbitrarily given vorticities. The interaction between vortices is proportional to 1/r, where r is the distance between two vortices. The problem has 2 equilateral and at most 3 collinear normalized relative equilibria. This 3 is the optimal upper bound. Our main result is that the above standard statements remain unchanged if we consider an interaction proportional to r ^b, for any b < 0. For 0 < b < 1, the optimal upper bound becomes 5. For positive vorticities and any b < 1, there are exactly 3 collinear normalized relative equilibria. The case b = −2 of this last statement is the well-known theorem due to Euler: in the Newtonian 3-body problem, for any choice of the 3 masses, there are 3 Euler configurations (also known as the 3 Euler points). These small upper bounds strengthen the belief of Kushnirenko and Khovanskii [18]: real varieties defined by simple systems should have a simple topology. We indicate some hard conjectures about the configurations of relative equilibrium and suggest they could be attacked within the quasi-polynomial framework.     

Zero cycles on some involution varieties

The group of zero cycles on involution varieties corresponding to central simple algebras of index at most two with an orthogonal involution is computed. This computation generalizes the Karpenko-Swan theorem on zero cycles on projective nonsingular quadrics. Bibliography:10 titles. Translated fromZapiski Nauchnykh Seminarov POMI, Vol. 227, 1995, pp. 93–105.     

Vortices and pinning effects for the Ginzburg‐Landau model in multiply connected domains

We consider the two‐dimensional Ginzburg‐Landau model with magnetic field for a superconductor with a multiply connected cross section. We study energy minimizers in the London limit as the Ginzburg‐Landau parameter κ = 1/? → ∞ to determine the number and asymptotic location of vortices. We show that the holes act as pinning sites, acquiring nonzero winding for bounded fields and attracting all vortices away from the interior for fields up to a critical value h_ex = O(|1n?|). At the critical level the pinning effect breaks down, and vortices appear in the interior of the superconductor at locations that we identify explicitly in terms of the solutions of an elliptic boundary value problem. The method involves sharp upper and lower energy estimates, and a careful analysis of the limiting problem that captures the interaction between the vortices and the holes. © 2005 Wiley Periodicals, Inc.     

Homostrophic vortex interaction under external strain,in a coupled QG-SQG model

The interaction between two co-rotating vortices, embedded in a steady external strain field, is studied in a coupled Quasi-Geostrophic — Surface Quasi-Geostrophic (hereafter referred to as QG-SQG) model. One vortex is an anomaly of surface density, and the other is an anomaly of internal potential vorticity. The equilibria of singular point vortices and their stability are presented first. The number and form of the equilibria are determined as a function of two parameters: the external strain rate and the vertical separation between the vortices. A curve is determined analytically which separates the domain of existence of one saddle-point, and that of one neutral point and two saddle-points. Then, a Contour-Advective Semi-Lagrangian (hereafter referred to as CASL) numerical model of the coupled QG-SQG equations is used to simulate the time-evolution of a sphere of uniform potential vorticity, with radius R at depth −2H interacting with a disk of uniform density anomaly, with radius R, at the surface. In the absence of external strain, distant vortices co-rotate, while closer vortices align vertically, either completely or partially (depending on their initial distance). With strain, a fourth regime appears in which vortices are strongly elongated and drift away from their common center, irreversibly. An analysis of the vertical tilt and of the horizontal deformation of the internal vortex in the regimes of partial or complete alignment is used to quantify the three-dimensional deformation of the internal vortex in time. A similar analysis is performed to understand the deformation of the surface vortex.     

Abelian groups with normal endomorphism rings

A ring is said to be normal if all of its idempotents are central. It is proved that a mixed group A with a normal endomorphism ring contains a pure fully invariant subgroup G ⊕ B, the endomorphism ring of a group G is commutative, and a subgroup B is not always distinguished by a direct summand in A. We describe separable, coperiodic, and other groups with normal endomorphism rings. Also we consider Abelian groups in which the square of the Lie bracket of any two endomorphisms is the zero endomorphism. It is proved that every central invariant subgroup of a group is fully invariant iff the endomorphism ring of the group is commutative.     

Growth for the Central Polynomials

We study the growth of the central polynomials for the infinite dimensional Grassmann algebra G, and for the algebra M_k(F) of the k × k matrices, both over a field F of characteristic zero.     

Equilibrium configurations of point vortices on a sphere

Configurations of point vortices on the sphere are considered in which all vortex velocities are zero. A sharp upper bound for the number of equilibria lying on a great circle is found, valid for generic circulations, and some unusual equilibrium configurations with a free real parameter are described. Equilibria of rings (vortices evenly spaced along circles of latitude) are also discussed. All equilibrium configurations of four vortices are determined.