Relative equilibrium and collapse configurations of heterogeneous vortex triple rings

Configurations of point vortices in a 2D fluid that are placed at the vertices of concentric regular m-gons, i.e. point vortex rings, are considered. The number of configurations that rotate uniformly-relative equilibria-is shown to be finite in the case of three rings with arbitrary circulations, subject to a few circulation constraints. Similar finiteness results for collapse configurations of three rings are also obtained, and an effective computational method is described.     

Fixed equilibria of point vortices in symmetric multiply connected domains

The paper provides symmetric fixed configurations of point vortices in multiply connected domains in the unit circle with many circular obstacles. When the circular domain is invariant with respect to rotation around the origin by a degree of 2π/M, a regular M-polygonal ring configuration of identical point vortices becomes a fixed equilibrium. On the other hand, when we assume a special symmetry, called the folding symmetry, on the circular domain, we find a fixed equilibrium in which M point vortices with the positive unit strength and M point vortices with the negative unit strength are arranged alternately at the vertices of a 2M-polygon. We also investigate the stability of these fixed equilibria and their bifurcation for a special circular domain with the rotational symmetry as well as the folding symmetry. Furthermore, we discuss fixed equilibria in non-circular multiply connected domains with the same symmetries. We give sufficient conditions for the conformal mappings, by which fixed equilibria in the circular domains are mapped to those in the general multiply connected domains. Some examples of such conformal mappings are also provided.     

The stability of stationary rotation of a regular vortex polygon

This paper is devoted to the Lord Kelvin's (1878) problem on stability of the stationary rotation of the system of n equal vortices located in the vertices of a regular n-gon. During the last decades this problem again became actual in connection with the investigation of point vortices in liquid helium and electron columns in plasma physics. This regime is described by the explicit solution of the Kirchhoff equations. The corresponding eigenvalue problem for the linearization matrix can be also decided explicitly. This was used in the works of Thomson (1883) and Havelock (1931) to obtain exhaustive results on the linear stability. Kurakin (1994) proved that for n/=8 it is unstable. We also present the general theory of stationary motions of a dynamical system with symmetry group. The definitions of stability and instability are necessary to modify in the specific case of stationary regimes. We do not assume that the system is conservative. Thus, the results can be applied not only to various stationary regimes of an ideal fluid flows but, for instance, also to motions of viscous fluids. (c) 2002 American Institute of Physics.     

Evaluation of a phase shifting method for vortex localization in optical vortex interferometry

The optical vortex interferometer uses a regular lattice of the optical vortices. In the previous papers we showed that changes in vortex lattice geometry can be related to physical quantities of the object being measured. The accuracy of such a measurement depends strongly on the precision of vortex point localization (vortex points are points where wavefront phase is undetermined). In this paper we compared the measurements of object wave tilt calculated from different localization methods applied to real interferograms and evaluate various localization methods.     

Irregular mixing due to a vortex pair interacting with a fixed vortex

The paper examines scalar advection caused by a point–vortex pair encountering a fixed point vortex in a uniform flow. The interaction produces two types of vortex motion. First is unbounded as the pair moves unrestrictedly after encountering the fixed vortex. The scalar exchanging between the pair's bubble and fixed vortex's neighbourhood is numerically estimated. Second is bounded as the pair's vortices periodically oscillate about the fixed vortex. The pair's periodic motion perturbs scalar motion causing a portion of scalar trajectories to manifest chaotic behaviour. We analyse scalar transport using Poincaré sections, which reveal regular and chaotic transport regions.     

Crystallized vortex crystals

Numerical simulations of the equations of motion of 300 charged particles confined to a plane with an additional magnetic field orthogonal to the plane reproduce recently observed self-organization of non-neutral plasmas into a small number of interacting vortices. In the presence of damping we observe crystallized vortices, i.e. vortices with regular internal structure. We also observe crystallized vortex crystals, i.e. geometric patterns of crystallized vortices. Fractal vortex arrangements are investigated and found to be stable. Our results are relevant for quantum dots and artificial atoms. Received: 24 February 1998 / Revised: 4 March 1998 / Accepted: 4 May 1998     

Self-dual Chern–Simons Vortices on Bounded Domains

In this Letter we consider the Abelian Chern–Simons vortices on a bounded simply connected domain. We establish the existence of solutions for the self-duality equations. We prove the uniqueness of solutions when all the vortex points are equal and the domain is star-shaped. We also show the radial symmetry of solutions on balls centered at the vortex point.     

基于特征值方法的旋翼尾迹稳定性分析

给出一个旋翼尾迹线性化稳定性分析的方法.在该方法中,尾迹涡线被离散为直线涡段,尾迹的扰动归结为涡元端点的扰动,考虑了桨尖涡的自诱导和涡线的互诱导以及桨尖涡与桨叶的实际干扰.使用该方法,分别以UH-1H和AH-1G模型旋翼为例,对悬停和前飞状态的旋翼尾迹的稳定性进行计算和分析.结果表明:旋翼尾迹运动存在大于0的特征值,是内在不稳定的,且最大发散率随波数变化呈现出一定规律性;前飞与悬停状态不同,其最大发散率减小,不稳定性减弱.     

Magnetic effects on the coalescence of Kelvin-Helmholtz vortices

We simulate the coalescence process of MHD-scale Kelvin-Helmholtz vortices with the electron inertial effects taken into account. Reconnection of highly stretched magnetic field lines within a rolled-up vortex destroys the vortex itself and the coalescence process, which is well known in ordinary fluid dynamics, is seen to be inhibited. When the magnetic field is initially antiparallel across the shear layer, on the other hand, multiple vortices are seen to coalesce continuously because another type of magnetic reconnection prevents the vortex decay. This type of reconnection at the hyperbolic point also changes the field line connectivity and thus leads to large-scale plasma mixing across the shear layer.     

On the vortex formation at the moving front of lightweight granular particles

The formation of vortices at a moving front of lightweight granular particles is investigated experimentally. The particles used in this study are made of polystyrene foam with three different diameters of nearly uniform size. Pairs of vortices are found to emerge at the moving front at regular intervals, thereby forming a wavy pattern. Once the vortices are produced, the flow velocity tends to increase. A simple analysis suggests the existence of a velocity boundary layer at the moving front, whose thickness increases with increasing particle diameter. The frontal radius of each vortex pair is about the size of this boundary layer; when the radius exceeds this size, the front tends to bifurcate into a train of vortices with the size of the boundary layer. The formation of twin vortices leads to a reduction in the air drag force exerted on the system, and thereby the system attains a higher flow velocity, i.e., a higher conversion rate of gravitational potential energy to the kinetic energy of the particle motion. The higher conversion rate of potential energy thus feeds back to the development of the vortex motion, resulting in the twin vortex formation.     

Landau levels and the Thomas-fermi structure of rapidly rotating bose-einstein condensates

We show that, within mean-field theory, the density profile of a rapidly rotating harmonically trapped Bose-Einstein condensate is of the Thomas-Fermi form as long as the number of vortices is much larger than unity. Two forms of the condensate wave function are explored: (i) the lowest Landau level (LLL) wave function with a regular lattice of vortices multiplied by a slowly varying envelope function, which gives rise to components in higher Landau levels; (ii) the LLL wave function with a nonuniform vortex lattice. From variational calculations, we find it most favorable energetically to retain the LLL form of the wave function but to allow the vortices to deviate slightly from a regular lattice. The predicted distortions of the lattice are small, but in accord with recent measurements at lower rates of rotation.     

Stability of giant vortices in quantum liquids

We show how giant vortices can be stabilized for strong external potentials in Bose-Einstein condensates. We illustrate the formation of these vortices thanks to the Ginzburg-Landau dissipative dynamics for two typical potentials in two spatial dimensions. The giant vortex stability is studied for the particular case of a rotating cylindrical hard wall. Due to axial symmetry the minimization of the perturbed energy is simplified into a one dimensional relaxation dynamics. Solving this 1D minimization problem, we observe that giant vortices are either never stable, or only stable in a finite frequency range. Finally we obtain the marginal curve for the minimum frequency needed to observe a giant vortex.     

Vortex deconfinement in the XY model with a magnetic field

We study vortex unbinding for the classical two-dimensional XY model in a magnetic field on square and triangular lattices. A renormalization group analysis combined with duality in the model shows that at high temperature and high field, the vortices unbind as the magnetic field is lowered in a two-step process: strings of overturned spins first proliferate and then vortices unbind. The transitions are highly continuous but are not of the Kosterlitz-Thouless type. The unbound vortex fixed point is shown to inherit properties of the underlying lattice, in particular containing a set of nodal lines that reflect the lattice symmetry.     

Nonequilibrium effects of anisotropic compression applied to vortex lattices in bose-einstein condensates

We have studied the dynamics of large vortex lattices in a dilute-gas Bose-Einstein condensate. While undisturbed lattices have a regular hexagonal structure, large-amplitude quadrupolar shape oscillations of the condensate are shown to induce a wealth of nonequilibrium lattice dynamics. When exciting an m=-2 mode, we observe shifting of lattice planes, changes of lattice structure, and sheetlike structures in which individual vortices appear to have merged. Excitation of an m=+2 mode dissolves the regular lattice, leading to randomly arranged but still strictly parallel vortex lines.     

Eigenstates of the linearlized Ginzburg–Landau equation for mesoscopic multiply-connected superconductor

Multiply-connected mesoscopic superconductors have rich structures of vortex systems that result from interference of order parameter. We studied magnetic field dependence of transition temperatures and vortex arrangements of finite sized honeycomb superconducting networks with 6-fold rotational symmetries. Near and above the lower critical field, vortices locate at center of the network. As increasing the field, vortices form a hexagon or hexagonal multi-shell structure. In higher field, order parameter damps exponentially from the central point of the network to the edge of the network.     

Vortex lattice generation using interferometric techniques based on lateral shearing

Interferometric techniques that use lateral shearing for generation of a regular network of optical vortices are reported. These techniques are simple, robust, and less sensitive to vibrations compared to the existing interferometric methods of vortex lattice generation. Vortex lattice is created using three-wave interference of plane as well as spherical waves. Theory of vortex lattice formation using phasor approach is provided. The location of vortices in the interference field is described in terms of spatial frequencies of the interfering waves. The presence of the vortices in the interference field is confirmed experimentally by producing fork fringes, as vortex signatures. Results of simulation studies are presented to support experimentally recorded interferograms.     

Passive tracer dynamics in 4 point-vortex flow

The advection of passive tracers in a system of 4 identical point vortices is studied when the motion of the vortices is chaotic. The phenomenon of vortex-pairing has been observed and statistics of the pairing time is computed. The distribution exhibits a power-law tail with exponent ∼ 3.6 implying finite average pairing time. This exponents is in agreement with its computed analytical estimate of 3.5. Tracer motion is studied for a chosen initial condition of the vortex system. Accessible phase space is investigated. The size of the cores around the vortices is well approximated by the minimum inter-vortex distance and stickiness to these cores is observed. We investigate the origin of stickiness which we link to the phenomenon of vortex pairing and jumps of tracers between cores. Motion within the core is considered and fluctuations are shown to scale with tracer-vortex distance r as r ⁶. No outward or inward diffusion of tracers are observed. This investigation allows the separation of the accessible phase space in four distinct regions, each with its own specific properties: the region within the cores, the reunion of the periphery of all cores, the region where vortex motion is restricted and finally the far-field region. We speculate that the stickiness to the cores induced by vortex-pairings influences the long-time behavior of tracers and their anomalous diffusion. Received 28 September 2000 and Received in final form 9 February 2001     

Traces of vortices in superfluid helium droplets

We report on the observation of vortices in superfluid 4He droplets produced in the expansion of liquid He. The vortices were traced by introducing Ag atoms, which clustered along the vortex lines, into the droplets. The Ag clusters were subsequently surface-deposited and imaged via electron microscopy. The prevalence of elongated track-shaped deposits shows that vortices are present in droplets larger than about 300 nm and that their lifetime exceeds a few milliseconds. We discuss the possible formation mechanisms and the stability of the vortices.     

Index theoretic characterization of d-wave superconductors in the vortex state

We study analytically the low energy spectrum of a lattice d-wave superconductor in the vortex lattice state. For an inversion symmetric hc/2e vortex lattice and in the presence of particle-hole symmetry we prove an index theorem that imposes a lower bound on the number of zero-energy modes. Generic cases are constructed in which this bound exceeds the number of zero modes of an equivalent lattice of hc/e vortices, despite the identical point group symmetries. The quasiparticle spectrum around the zero modes is doubly degenerate and exhibits a Dirac-like dispersion, with velocities that become universal functions of Delta(0)/t in the limit of low magnetic field. For weak particle-hole symmetry breaking, the gapped state can be characterized by a topological quantum number, related to spin-Hall conductivity, which generally differs in the cases of the hc/2e and hc/e vortex lattices.     

Kinetic theory of point vortices in two dimensions: analytical results and numerical simulations

We develop the kinetic theory of point vortices in two-dimensional hydrodynamics and illustrate the main results of the theory with numerical simulations. We first consider the evolution of the system “as a whole” and show that the evolution of the vorticity profile is due to resonances between different orbits of the point vortices. The evolution stops when the profile of angular velocity becomes monotonic even if the system has not reached the statistical equilibrium state (Boltzmann distribution). In that case, the system remains blocked in a quasi stationary state with a non standard distribution. We also study the relaxation of a test vortex in a steady bath of field vortices. The relaxation of the test vortex is described by a Fokker-Planck equation involving a diffusion term and a drift term. The diffusion coefficient, which is proportional to the density of field vortices and inversely proportional to the shear, usually decreases rapidly with the distance. The drift is proportional to the gradient of the density profile of the field vortices and is connected to the diffusion coefficient by a generalized Einstein relation. We study the evolution of the tail of the distribution function of the test vortex and show that it has a front structure. We also study how the temporal auto-correlation function of the position of the test vortex decreases with time and find that it usually exhibits an algebraic behavior with an exponent that we compute analytically. We mention analogies with other systems with long-range interactions.