Singular Vortex Research Articles

Structure of vortices in superfluid 3He A-like phase in uniaxially stretched aerogel

Possible vortex-core transitions in A-like phase of superfluid 3He in uniaxially stretched aerogel are investigated. Since the global anisotropy in this system induces the polar pairing state in a narrow range close to the superfluid transition in addition to the A-like and B-like phases, the polar state may occur in the core of a vortex in the A-like phase identified with the ABM pairing state, like in the case of the bulk B phase where a core including the ABM state is realized at higher pressures. We examine the core structure of a single vortex under the boundary condition compatible with the Mermin-Ho vortex in the presence of the dipole interaction. Following Salomaa and Volovik's approach, we numerically solve the Ginzburg-Landau equation for an axially symmetric vortex and, by examining its stability against nonaxisymmetric perturbations, discuss possible vortex core states. It is found that a first order transition on core states may occur on warming from an axisymmetric vortex with a nonunitary core to a singular vortex with the polar core.

Reconnection of Superfluid Vortex Bundles

Using the vortex filament model and the Gross-Pitaevskii nonlinear Schroedinger equation, we show that bundles of quantized vortex lines in He II are structurally robust and can reconnect with each other maintaining their identity. We discuss vortex stretching in superfluid turbulence and show that, during the bundle reconnection process, kelvin waves of large amplitude are generated, in agreement with the finding that helicity is produced by nearly singular vortex interactions in classical Euler flows.

On the Stability of a Singular Vortex Dynamics

In this paper we address the question of the singular vortex dynamics exhibited in [15], which generates a corner in finite time. The purpose is to prove that under some appropriate small regular perturbation the corner still remains. Our approach uses the Hasimoto transform and deals with the long range scattering properties of a Gross-Pitaevski equation with time-variable coefficients.

Theory of singular vortex solutions of the nonlinear Schrödinger equation

We present a systematic study of singular vortex solutions of the critical and supercritical two-dimensional nonlinear Schrödinger equation. In particular, we study the critical power for collapse and the asymptotic blowup profile of singular vortices.

On the possible induced charge on a graphitic nanocone at finite temperature

Electronic excitations in a graphitic monolayer (graphene) in the long-wavelength approximation are characterized by the linear dispersion law, representing a unique example of the really two-dimensional ‘ultrarelativistic’ fermionic system which in the presence of topological defects possesses rather unusual properties. A disclination that rolls up a graphitic sheet into a nanocone is described by a pointlike pseudomagnetic vortex at the apex of the cone, and the flux of the vortex is related to the deficit angle of the conical surface. A general theory of planar relativistic fermionic systems in the singular vortex background is employed, and we derive the analytical expression for the charge which is induced at finite temperature on some graphitic nanocones.

Vortex singularities of optimal strategies at the beginning of a motion in search problems on n-manifolds

Vortex singularities of optimal strategies at the beginning of a motion in search problems on n-manifolds

A numerical model for thin airfoils in unsteady compressible arbitrary motion

A numerical method based on the vortex methodology is presented in order to obtain unsteady solution of the aerodynamic coefficients of a thin airfoil in either compressible subsonic or supersonic flows. The numerical model is created through the profile discretization in uniform segments and the compressible flow vortex singularity is used. The results of the proposed model are presented as the lift and the pressure coefficient along the profile chord as a function of time. The indicial response for the unit step change of angle of attack and unit sharp-edged gust response of the profile are also obtained numerically. The results yielded by the present methodology are also compared with solutions available in the literature.

Two-layer quasi-geostrophic singular vortices embedded in a regular flow. Part 2. Steady and unsteady drift of individual vortices on a beta-plane

Drift of individual β-plane vortices confined to one layer of a two-layer fluid under the rigid-lid condition is considered. For this purpose, the theory of two-layer quasi-geostrophic singular vortices is employed. On a β-plane, any non-zonal displacement of a singular vortex results in the development of a regular flow. An individual singular β-plane vortex cannot be steady on its own: the vortex moves coexisting with a regular flow, be the drift steady or not. In this paper, both kinds of drift of a singular vortex are considered. A new steady exact solution is presented, a hybrid regular–singular modon. This hybrid modon consists of a dipole component and a circularly symmetric rider. The dipole is regular, and the rider is a superposition of the singular vortex and a regular circularly symmetric field. The unsteady drift of a singular vortex residing in one of the layers is considered under the condition that, at the initial instant, the regular field is absent. The development of barotropic and baroclinic regular β-gyres is examined. Whereas the barotropic and baroclinic modes of the singular vortex are comparable in magnitudes, the baroclinic β-gyres attenuate with time, making the trajectory of the vortex close to that of a barotropic monopole on a β-plane.

Two-layer quasi-geostrophic singular vortices embedded in a regular flow. Part 1. Invariants of motion and stability of vortex pairs

The concept of a quasi-geostrophic singular vortex is extended to several types of two-layer model: a rigid-lid two-layer, a free-surface two-layer and a $2{\textstyle{1 \over 2}}$-layer model with two active and one passive layer. Generally, a singular vortex differs from a conventional point vortex in that the intrinsic vorticity of a singular vortex, in addition to delta-function, contains an exponentially decaying term. The theory developed herein occupies an intermediate position between discrete and fully continuous multilayer models, since the regular flow and its interaction with the singular vortices are also taken into account. A system of equations describing the joint evolution of the vortices and the regular field is presented, and integrals expressing the conservation of enstrophy, energy, momentum and mass are derived. Using these integrals, the initial phases of evolution of an individual singular vortex confined to one layer and of a coaxial pair of vortices positioned in different layers of a two-layer fluid on a beta-plane are described. A valuable application of the conservation integrals is related to the stability analysis of point-vortex pairs within the $1{\textstyle{1 \over 2}}$-layer model, $2{\textstyle{1 \over 2}}$-layer model, and free-surface two-layer model on the f-plane. Such vortex pairs are shown to be nonlinearly stable with respect to any small perturbation provided its regular-flow energy and enstrophy are finite.

Two-layer quasigeostrophic potential vorticity model

We formulate the two-layer quasigeostrophic potential vortex (QGPV) model as an infinite-dimensional Hamiltonian system and apply Noether’s theorem to derive the conservation laws for these equations. The point (singular) vortex QGPV model is also analyzed as a finite-dimensional Hamiltonian system. By virtue of its point symmetries, the invariants for the finite-dimensional system are obtained. The existence of these point symmetries is used to prove the integrability of the three-vortex QGPV problem. We analyze the relative equilibrium configurations for the three-vortex, two-layer problem and study collapsing and merging (collapse in different layers) configurations. We also study two special classes of symmetrical four-vortex QGPV problems and establish their integrability and present some simulations for one of these classes of integrable QGPV four-vortex problems.

Boojums Connecting Singular and Continuous Vortices in Two-Component Bose-Einstein Condensates

A boojum is a topological defect that can form only on the surface of an ordered medium such as superfluid 3He and liquid crystals. By analogy with superfluid 3He, we numerically create boojums between two phases with different vortex structures in two-component BECs where the intercomponent interaction is spatially dependent. The detailed structure of the boojums is revealed by investigating its density distribution, effective superflow vorticity and pseudospin texture.

Hofstadter butterflies of bilayer graphene

We calculate the electronic spectrum of bilayer graphene in perpendicular magnetic fields nonperturbatively. To accommodate arbitrary displacements between the two layers, we apply a periodic gauge based on singular flux vortices of phase $2\pi$. The resulting Hofstadter-like butterfly plots show a reduced symmetry, depending on the relative position of the two layers against each other. The split of the zero-energy relativistic Landau level differs by one order of magnitude between Bernal and non-Bernal stacking.

Fast chemical reaction and multiple-scale concentration fields in singular vortices

Two species involved in a simple, fast reaction tend to become segregated in patches composed of a single of these reactants. These patches are separated by a boundary where the stoichiometric condition is satisfied and the reaction occurs, fed by diffusion. Stirred by advection, this boundary and the concentration fields within the patches may tend to present multiple-scale characteristics. Based on this segregated state, this paper aims at evaluating the temporal evolutions of the length of the boundary and diffusive flux of reactants across it, when concentrations presenting initial self-similar fluctuations are advected by a singular vortex. First the two sources of singularity, i.e., the self-similar initial conditions and the singular vortex, are considered separately. On the one hand, self-similar initial conditions are imposed to a diffusion-reaction system, for one- and two-dimensional cases. On the other hand, an imposed singular vortex advects initially on/off concentration fields, in combination with diffusion and reaction. This problem is addressed analytically, by characterizing the boundary by a box-counting dimension and the concentration fields by a Hölder exponent, and numerically, by direct numerical simulations of the advection-diffusion-reaction equations. Second, the way the two sources hang together shows that, depending on the self-similar properties of the initial concentration fields, the vortex promotes the chemical activity close to its inner smoothed-out core or close to the outer region where the boundary starts to spiral. For all the considered situations, the length of the boundary and the global reaction speed are found to evolve algebraically with time after a short transient and a good agreement is found between the analytical and numerical scaling laws.

Topological events on wave dislocation lines: birth and death of loops, and reconnection

In three-dimensional space, a wave dislocation, that is, a quantized (optical) vortex or phase singularity, is a line zero of a complex scalar wavefunction. As a ‘time’ parameter varies, the topology of the vortex can change by encounter with a line of vanishing vorticity (curl of the current associated with the wavefunction). An isolated critical point of the field intensity, sliding along the zero-vorticity line like a bead on a wire, meets the vortex as it encounters the line, and so participates in the singular event. Local expansion and gauge and coordinate transformations show that the vortex topology can change generically by the appearance or disappearance of a loop, or by the reconnection of branches of a pair of hyperbolas.

The breakdown of Alfvén’s theorem in ideal plasma flows: Necessary conditions and physical conjectures

This paper presents both rigorous results and physical theory on the breakdown of magnetic flux conservation for ideal plasmas by nonlinear effects. Our analysis is based upon an effective equation for magnetohydrodynamic (MHD) modes at length scales > ℓ , with smaller scales eliminated, as in renormalization-group methodology. We prove that flux conservation can be violated at an instant of time for an arbitrarily small length scale ℓ , and in the absence of any nonideality, but only if at least one of three necessary conditions is satisfied. These conditions are (i) nonrectifiability of advected loops, (ii) unbounded velocity or magnetic fields, (iii) singular current sheets and vortex sheets that both exist and intersect in sets of large enough dimension. This result gives analytical support to and rigorous constraints on theories of fast turbulent reconnection. Mathematically, our theorem is analogous to Onsager’s result on energy dissipation anomaly in hydrodynamic turbulence. As a physical phenomenon, the breakdown of magnetic flux conservation in ideal MHD is similar to the decay of magnetic flux through a narrow superconducting ring, by phase-slip of quantized flux lines. The effect should be observable both in numerical MHD simulations and in laboratory plasma experiments at moderately high magnetic Reynolds numbers.

Soaring interfaces, vortices and vortex systems inside the internal waves wake past the horizontally moving cylinder in a continuously stratified fluid

The flow pattern around a horizontal cylinder towed at constant velocity along isopycnic plane in a continuously stratified liquid is visualized by conventional techniques of "Vertical slit-Foucault's knife", "Maksoutov's slit-thread" and "horizontal slit-regular grating". Using sensitive high-resolution methods allows detail studying such component of stratified flow structures as soaring interfaces, singular soaring vortices and vortex systems, which arise directly inside the internal waves field past the cylinder. These flow elements having high level of vorticity are separated from the downstream wake by a strip of fluid without any small-scale inhomogeneities. Formation of singular vortex dipoles on leading edges of soaring interfaces is investigated in details in a wide range of flow parameters.

Poches de tourbillon singulières dans un fluide faiblement visqueux

In this paper, we study the singular vortex patches in the two-dimensional incompressible Navier-Stokes equations. We show, in particular, that if the initial vortex patch is C^{1+s} outside a singular set \Sigma , so the velocity is, for all time, lipschitzian outside the image of \Sigma through the viscous flow. In addition, the correponding lipschitzian norm is independant of the viscosity. This allows us to prove some results related to the inviscid limit for the geometric structures of the vortex patch.

Vortex core dynamics and singularity formations in incompressible Richtmyer-Meshkov instability

Motion of a fluid interface in Richtmyer-Meshkov instability is examined as a vortex sheet with the use of Birkhoff-Rott equation. This equation coupled with an evolution equation of the strength of the vortex sheet can describe all inviscid and incompressible fluid instabilities, i.e., Kelvin-Helmholtz, Rayleigh-Taylor, and Richtmyer-Meshkov instabilities, when Atwood numbers and initial distribution of vorticities are given. With these equations, detailed motion of a vortex core in the Richtmyer-Meshkov instability is investigated. For the Kelvin-Helmholtz and Rayleigh-Taylor instabilities, it is known that the curvature of a vortex sheet diverges at a finite time t=tc. This fact indicates that the solution loses its analyticity at tc. We show that the singularity formation also occurs in the Richtmyer-Meshkov instability and at the same time, accumulation of vorticity to some points where singularities are formed develops to the roll-up of a sheet when the sheet is regularized. We investigate motion of these accumulation points, i.e., vortex cores, and present that their trajectories and the strengths depend on the Atwood numbers.

Wind Tunnel Boundary Corrections for a Full Scale 1903 Wright Flyer Replica

A panel code is developed in order to assess and correct wind-tunnel boundary interferences during a test of a full-scale reproduction of the 1903 Wright Flyer in the Langley Full-Scale Tunnel (LFST). The LFST is a very large, low-speed wind tunnel with a 3 4-open test section. The principal concerns relating to boundary corrections are the relatively large size of the replica relative to the test-section dimensions, coupled with the lack of modern correction techniques for aeronautical testing in open-jet test sections. The Flyer is represented either by simple lifting surfaces, with the accuracy of the representation validated by direct calculations using the panel code CMARC, or by an array of simple horseshoe vortex singularities. Vortex ring panels are used in the boundary correction code. The solution starts by solving the flow in a closed test section in order to obtain the flow properties on the boundaries. An iterative method, based on the one-step method employed in adaptive wall wind-tunnel work, is then used to adapt the boundary geometry to a constant-pressure condition. Finally the boundary interference is obtained by calculating the velocities induced in the test region by the boundary singularities. Results are presented for different test-section boundary configurations: fully closed, fully open, and 3 4 open. Corrections are compared to calculations using the classical method of images.

Invariant solutions of the singular vortex in magnetohydrodynamics

We investigate properties of plasma flow, governed by an exact reduction of the ideal magnetohydrodynamics equations which was obtained in our recent article. The reduction is called a singular vortex. It gives a description of a three-dimensional nonstationary plasma flow in terms of solution of reduced system of PDEs with two independent variables. In this paper we investigate symmetry properties of the reduced system and construct its invariant solutions. Also we give a description of the physical features of plasma motion, governed by the singular vortex. Namely, we prove that in the observed class of solutions the particles trajectories as well as magnetic field lines are flat curves. The position and orientation of the plane containing the particle's trajectory depends on the initial position of the particle. We describe the shape of trajectories and magnetic field lines for the cases of stationary and self-similar plasma flow.