On Stability of Thomson’s Vortex <Emphasis Type="Italic">N</Emphasis>-gon in the Geostrophic Model of the Point Bessel Vortices

A stability analysis of the stationary rotation of a system of N identical point Bessel vortices lying uniformly on a circle of radius R is presented. The vortices have identical intensity Γ and length scale γ^?1 > 0. The stability of the stationary motion is interpreted as equilibrium stability of a reduced system. The quadratic part of the Hamiltonian and eigenvalues of the linearization matrix are studied. The cases for N = 2,..., 6 are studied sequentially. The case of odd N = 2?+1 ≥ 7 vortices and the case of even N = 2_n ≥ 8 vortices are considered separately. It is shown that the (2? + 1)-gon is exponentially unstable for 0 < γR<R_*(N). However, this (2? + 1)-gon is stable for γR ≥ R_*(N) in the case of the linearized problem (the eigenvalues of the linearization matrix lie on the imaginary axis). The even N = 2n ≥ 8 vortex 2n-gon is exponentially unstable for R > 0.     

Stability of the Thomson vortex polygon with evenly many vortices outside a circular domain

We consider the stability problem for the stationary rotation of a regular point vortex n-gon lying outside a circular domain. After the article of Havelock (1931), the complete solution of the problem remains unclear in the case 2 ≤ n ≤ 6. We obtain the exhaustive results for evenly many vortices n = 2, 4, 6.     

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     

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.      

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)     

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.     

Rotatingn-gon/kn-gon vortex configurations

Summary We demonstrate the existence of stationary point-vortex configurations consisting ofk vortexn-gons and a vortexkn-gon. These configurations exist only for specific values of the vortex strengths; the relative vortex strengths of such a consiguration can be uniquely expressed as functions of the radii of the polygons. Thekn-gon must be oriented so as to be fixed by any reflection fixing one of then-gons; for sufficiently smallk, we show that then-gons must be oriented in such a way that the entire configuration shares the symmetries of any of then-gons. Necessary conditions for the formal stability of general stationary point-vortex configurations set conditions on the vortex strengths. We apply these conditions to then-gon/kn-gon configurations and carry out a complete linear and formal stability analysis in the casek=n=2, showing that linearly and nonlinearly orbitally stable configurations exist.     

On the stability of discrete tripole,quadrupole, Thomson’ vortex triangle and square in a two-layer/homogeneous rotating fluid

A two-layer quasigeostrophic model is considered in the f-plane approximation. The stability of a discrete axisymmetric vortex structure is analyzed for the case when the structure consists of a central vortex of arbitrary intensity Γ and two/three identical peripheral vortices. The identical vortices, each having a unit intensity, are uniformly distributed over a circle of radius R in a single layer. The central vortex lies either in the same or in another layer. The problem has three parameters (R, Γ, α), where α is the difference between layer thicknesses. A limiting case of a homogeneous fluid is also considered.A limiting case of a homogeneous fluid is also considered.The theory of stability of steady-state motions of dynamic systems with a continuous symmetry group G is applied. The two definitions of stability used in the study are Routh stability and G-stability. The Routh stability is the stability of a one-parameter orbit of a steady-state rotation of a vortex multipole, and the G-stability is the stability of a three-parameter invariant set O _G , formed by the orbits of a continuous family of steady-state rotations of a multipole. The problem of Routh stability is reduced to the problem of stability of a family of equilibria of a Hamiltonian system. The quadratic part of the Hamiltonian and the eigenvalues of the linearization matrix are studied analytically.The cases of zero total intensity of a tripole and a quadrupole are studied separately. Also, the Routh stability of a Thomson vortex triangle and square was proved at all possible values of problem parameters. The results of theoretical analysis are sustained by numerical calculations of vortex trajectories.     

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.     

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.     

On the stability of Thomson’s vortex configurations inside a circular domain

The paper is devoted to the analysis of stability of the stationary rotation of a system of n identical point vortices located at the vertices of a regular n-gon of radius R ₀ inside a circular domain of radius R. Havelock stated (1931) that the corresponding linearized system has exponentially growing solutions for n ⩾ 7 and in the case 2 ⩽ n ≤ 6 — only if the parameter p = R ₀²/R ² is greater than a certain critical value: p _*n < p < 1. In the present paper the problem of nonlinear stability is studied for all other cases 0 < p ⩽ p _*n , n = 2, ..., 6. Necessary and sufficient conditions for stability and instability for n ≠ = 5 are formulated. A detailed proof for a vortex triangle is presented. A part of the stability conditions is substantiated by the fact that the relative Hamiltonian of the system attains a minimum on the trajectory of the stationary motion of the vortex triangle. The case where the sign of the Hamiltonian is alternating requires a special approach. The analysis uses results of KAM theory. All resonances up to and including the 4th order occurring here are enumerated and investigated. It has turned out that one of them leads to instability.     

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     

The critical cross-section of a vortex

Summary This paper discusses the problem of critical-flow cross-sections in vortex flows. It is shown that there are two different types of vortex flows, A-type and B-type vortices (say). An A-type vortex approaches its critical flow state as its cross-sectional area increases and departs from the critical state as the cross-sectional area is decreased. This property is associated with the particular dependence of total pressure and circulation on the stream function, and it holds for both subcritical and supercritical A-type vortices. On the other hand, both subcritical and supercritical B-type vortices approach their critical flow states as their cross-sectional areas are decreased and depart from their critical states for increasing cross-sectional area. As was shown by Benjamin, setting the first variation of the flow force with respect to the stream function equal to zero leads to Euler's equation of motion. The second variation also vanishes if the corresponding flow state is critical. In this case the sign of the third variation decides whether the flow is an A-type or a B-type vortex. Within the framework of inviscid-fluid flow theory the type of a vortex is preserved unless vortex breakdown occurs. Making use of the knowledge that vortex flows are controlled by two different types of critical-flow cross-sections a variety of vortex flow phenomena are investigated, including the two types of inlet vortices that are observed upstream of jet engines, the behavior of vortex valves, the flow characteristics of liquid-fuel atomizers and the bath tub vortex.     

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.     

A quantization property for static Ginzburg‐Landau vortices

For any critical point of the complex Ginzburg‐Landau functional in dimension 3, we prove that, for large coupling constants, ; if the energy of this critical point on a ball of a given radius r is relatively small compared to r log , then the ball of half‐radius contains no vortex (the modulus of the solution is larger than ½). We then show how this property can be applied to describe limiting vortices as ε → 0. © 2001 John Wiley & Sons, Inc.     

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)     

Stability and bifurcation of couette flow in the case of a narrow gap between rotating cylinders : PMM vol. 38, n 6, 1974, pp. 1025–1030

Stability and bifurcation of Couette flow between concentric rotating cylinders are investigated for the case when the ratios of their radii R and angular velocities Ω are nearly equal to unity. The limiting problem in the linear theory when R → 1 and Ω → 1 is the problem of convection stability in the layer [1]. We find that this is also correct in the case of a nonlinear problem. Below we show that solution of the problem of free convection yields the principal term of the expansion of the secondary flow (Taylor vortex) in the powers of a small parameter δ = R − 1. Therefore the results of [2, 3] can be used to provide, in the present case, a strict justification for the use of the Liapunov-Schmidt method to compute the Taylor vortices. The numerical results obtained for the critical Reynolds' number and the amplitude of the secondary flow provide a good illustration of the asymptotic passage as δ → 0.     

Instability of symmetric vortices with large charge and coupling constant

In the Ginzburg-Landau theory for superconductivity, it is well-known that there are radial symmetric vortices for any charge (vortex number) n and coupling constant λ > 0. It is powered that these vortices are unstable for large n and λ. © 1996 John Wiley & Sons, Inc.