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.     

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.     

Dynamics near relative equilibria: Nongeneric momenta at a 1:1 group-reduced resonance

An interesting situation occurs when the linearized dynamics of the shape of a formally stable Hamiltonian relative equilibrium at nongeneric momentum 1:1 resonates with a frequency of the relative equilibrium's generator. In this case some of the shape variables couple to the group variables to first order in the momentum perturbation, and the first order perturbation theory implies that the relative equilibrium slowly changes orientation in the same way that a charged particle with magnetic moment moves on a sphere under the influence of a radial magnetic monopole. In the course of showing this a normal form is constructed for linearizations of relative equilibria and for Hamiltonians near group orbits of relative equilibria. Received August 27, 1998; in final form February 20, 1999     

Relative equilibria of symmetric n-body problems on a sphere: Inverse and direct results

In this paper, the author proves that the existence of a class of relative equilibria for SO(3)-symmetric n-body problems on a sphere implies that these problems are in fact O(3)-symmetric in a time-reversing sense. Besides this “inverse” result, the author also proves a set of “direct” results, in which the existence of certain symmetric relative equilibria are deduced purely from symmetry considerations of the n-body problems on a sphere. The author then formulates an explicit method for the reduction of this class of symmetric Hamiltonians, which leads to symplectic shape or relative variables for the further study of the relative equilibria and equilibria. This class of problems includes the point vortex problem on a sphere, and the author applies the main results in this paper to that problem. © 1998 John Wiley & Sons, Inc.     

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.      

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.      

Interaction between Kirchhoff vortices and point vortices in an ideal fluid

We consider the interaction of two vortex patches (elliptic Kirchhoff vortices) which move in an unbounded volume of an ideal incompressible fluid. A moment second-order model is used to describe the interaction. The case of integrability of a Kirchhoff vortex and a point vortex by the variable separation method is qualitatively analyzed. A new case of integrability of two Kirchhoff vortices is found. A reduced form of equations for two Kirchhoff vortices is proposed and used to analyze their regular and chaotic behavior.     

Relative Equilibria of Molecules

Summary. We describe a method for finding the families of relative equilibria of molecules that bifurcate from an equilibrium point as the angular momentum is increased from 0 . Relative equilibria are steady rotations about a stationary axis during which the shape of the molecule remains constant. We show that the bifurcating families correspond bijectively to the critical points of a function h on the two-sphere which is invariant under an action of the symmetry group of the equilibrium point. From this it follows that for each rotation axis of the equilibrium configuration there is a bifurcating family of relative equilibria for which the molecule rotates about that axis. In addition, for each reflection plane there is a family of relative equilibria for which the molecule rotates about an axis perpendicular to the plane. We also show that if the equilibrium is nondegenerate and stable, then the minima, maxima, and saddle points of h correspond respectively to relative equilibria which are (orbitally) Liapounov stable, linearly stable, and linearly unstable. The stabilities of the bifurcating branches of relative equilibria are computed explicitly for XY ₂ , X ₃ , and XY ₄ molecules. These existence and stability results are corollaries of more general theorems on relative equilibria of G -invariant Hamiltonian systems that bifurcate from equilibria with finite isotropy subgroups as the momentum is varied. In the general case, the function h is defined on the Lie algebra dual {\frak g} ^* and the bifurcating relative equilibria correspond to critical points of the restrictions of h to the coadjoint orbits in {\frak g} ^* . Received June 9, 1997; second revision received December 15, 1997; final revision received January 19, 1998     

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.     

Vortex dynamics of the full time‐dependent Ginzburg‐Landau equations

In the Ginzburg‐Landau model for superconductivity a large Ginzburg‐Landau parameter κ corresponds to the formation of tight, stable vortices. These vortices are located exactly where an applied magnetic field pierces the superconducting bulk, and each vortex induces a quantized supercurrent about the vortex. The energy of large‐κ solutions blows up near each vortex which brings about difficulties in analysis. Rigorous asymptotic static theory has previously established the existence of a finite number of the vortices, and these vortices are located precisely at the critical points of the renormalized energy (the free energy less the vortex self‐induction energy). A rigorous study of the full time‐dependent Ginzburg‐Landau equations under the classical Lorentz gauge is done under the asymptotic limit κ → ∞. Under slow times the vortices remain pinned to their initial configuration. Under a fast time of order κ the vortices move according to a steepest descent of the renormalized energy. © 2002 John Wiley & Sons, Inc.     

Stationary Configurations of Point Vortices on a Cylinder

In this paper we study the problem of constructing and classifying stationary equilibria of point vortices on a cylindrical surface. Introducing polynomials with roots at vortex positions, we derive an ordinary differential equation satisfied by the polynomials. We prove that this equation can be used to find any stationary configuration. The multivortex systems containing point vortices with circulation Γ₁ and Γ₂ (Γ₂ = ?μΓ₁) are considered in detail. All stationary configurations with the number of point vortices less than five are constructed. Several theorems on existence of polynomial solutions of the ordinary differential equation under consideration are proved. The values of the parameters of the mathematical model for which there exists an infinite number of nonequivalent vortex configurations on a cylindrical surface are found. New point vortex configurations are obtained.     

Deformations of symplectic vortices

We prove a gluing theorem for a symplectic vortex on a compact complex curve and a collection of holomorphic sphere bubbles. Using the theorem we show that the moduli space of regular strongly stable symplectic vortices on a fixed curve with varying markings has the structure of a stratified-smooth topological orbifold. In addition, we show that the moduli space has a non-canonical C ¹-orbifold structure.     

Openness of momentum maps and persistence of extremal relative equilibria

We prove that for every proper Hamiltonian action of a Lie group G in finite dimensions the momentum map is locally G-open relative to its image (i.e. images of G-invariant open sets are open). As an application we deduce that in a Hamiltonian system with continuous Hamiltonian symmetries, extremal relative equilibria persist for every perturbation of the value of the momentum map, provided the isotropy subgroup of this value is compact. We also demonstrate how this persistence result applies to an example of ellipsoidal figures of rotating fluid. We also provide an example with plane point vortices which shows how the compactness assumption is related to persistence.     

On the equilibrium position of Ginzburg Landau vortices

We study a few problems related to superconducting vortices. Our main concern is the stable equilibrium distribution of them. Our starting point is the asymptotic form of the Ginzburg Landau energy functional with a large Ginzburg Landau parameter. We consider in particular the interaction of the vortices with an applied magnetic field, and the effects of impurities on the vortex locations.     

Dipole and Multipole Flows with Point Vortices and Vortex Sheets

An exact method is presented for obtaining uniformly translating distributions of vorticity in a two-dimensional ideal fluid, or equivalently, stationary distributions in the presence of a uniform background flow. These distributions are generalizations of the well-known vortex dipole and consist of a collection of point vortices and an equal number of bounded vortex sheets. Both the vorticity density of the vortex sheets and the velocity field of the fluid are expressed in terms of a simple rational function in which the point vortex positions and strengths appear as parameters. The vortex sheets lie on heteroclinic streamlines of the flow. Dipoles and multipoles that move parallel to a straight fluid boundary are also obtained. By setting the translation velocity to zero, equilibrium configurations of point vortices and vortex sheets are found.     

Relative equilibrium and collapse configurations of four point vortices

Relative equilibrium configurations of point vortices in the plane can be related to a system of polynomial equations in the vortex positions and circulations. For systems of four vortices the solution set to this system is proved to be finite, so long as a number of polynomial expressions in the vortex circulations are nonzero, and the number of relative equilibrium configurations is thereby shown to have an upper bound of 56. A sharper upper bound is found for the special case of vanishing total circulation. The polynomial system is simple enough to allow the complete set of relative equilibrium configurations to be found numerically when the circulations are chosen appropriately. Collapse configurations of four vortices are also considered; while finiteness is not proved, the approach provides an effective computational method that yields all configurations with a given ratio of velocity to position.      

Point vortices and classical orthogonal polynomials

Stationary equilibria of point vortices in the plane and on the cylinder in the presence of a background flow are studied. Vortex systems with an arbitrary choice of circulations are considered. 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.     

Coupled motion of a rigid body and point vortices on a two-dimensional spherical surface

The paper is concerned with a class of problems which involves the dynamical interaction of a rigid body with point vortices on the surface of a two-dimensional sphere. The general approach to the 2D hydrodynamics is further developed. The problem of motion of a dynamically symmetric circular body interacting with a single vortex is shown to be integrable. Mass vortices on S ² are introduced and the related issues (such as equations of motion, integrability, partial solutions, etc.) are discussed. This paper is a natural progression of the author’s previous research on interaction of rigid bodies and point vortices in a plane.