Point Vortices Research Articles

Transfer of passive particles in the velocity field of vortex tripole moving on a plane

Purpose of this article is to study the transport of passive particles in the velocity field of a vortex tripole with a change in the parameter that determines the speed of the configuration movement. A structure consisting of a central vortex and satellite vortices rotating around it with the opposite vorticity is understood as a tripole. We employ a system of three point vortices, the most simple mathematical representation of a vortex tripole, which may be expressed as a system of nonlinear ordinary differential equations with a parameter. Consideration is limited to a particular case of a tripole with zero total vorticity. The influence of the speed values of vortex configuration movement on the processes of passive particle transport has been studied. Methods. The study was carried out numerically using algorithms based on the dynamical systems approaches including the construction of the Poincare map and the analysis of the dynamics of marker particles. Were carried out long ´ times calculations, corresponding to hundreds and thousands of turns around the tripole center. Integrators of high orders of accuracy were used to solve the Cauchy problems, which made it possible to adequacy of the calculation result control. Results. We found that transferring passive particles is fundamentally different depending on the speed of the tripole. A vast zone of chaotic dynamics forms in the neighborhood of the vortices when the velocity is low. This zone slowly shifts along with the tripole. There are subregions of active and slow mixing inside the chaos region. The possible stages of particle dynamics are: transfer from the region to the right of the tripole to the area to the left, vigorous mixing near the vortices, and slowly drifting to the region to the left of the tripole. At a high speed of vortex configuration in the entire chaotic region, the particles are strongly mixed. The vortex tripole removes particles from the vicinity of its initial position over long distances and practically does not capture new particles along its path. In intermediate situations, both processes can be realized at varying degrees. Conclusion. Non-trivial scenarios for the transport of passive particles by a vortex tripole, which can also occur in real vortex configurations of fluids, have been discovered and described.

Tracking Point Vortices and Circulations via Advected Passive Particles: An Estimation Approach

We present a novel method for estimating the circulations and positions of point vortices in a two-dimensional (2D) environment using trajectory data of passive particles in the presence of Gaussian noise. The method comprises two algorithms: the first one calculates the vortex circulations, while the second one reconstructs the vortex trajectories. This reconstruction is done thanks to a hierarchy of optimization problems, involving the integration of systems of differential equations, over time sub-intervals all with the same amplitude defined by the autocorrelation function for the advected passive particles’ trajectories. Our findings indicate that accurately tracking the position of vortices and determining their circulations is achievable, even when passive particle trajectories are affected by noise.

Simplified model for helical vortex dynamics in the wake of an asymmetric rotor

Helical vortex systems, such as those found in the wakes of wind turbines, helicopter rotors and propellers, are subject to instabilities that lead to pairing between adjacent vortex loops. Certain modes of these instabilities can be triggered by an asymmetry in the rotor generating the vortices. In three-vortex systems, like those formed by many industrial rotors, the nonlinear vortex interactions are highly complex, introducing the need for a simple model to predict their dynamics. The current study presents a model for helical vortex systems based on an infinite strip of periodically repeating point vortices, whose motion can be computed using a single equation. This highly simplified model is shown to accurately reproduce the helical vortex dynamics predicted by a more sophisticated filament model and observed in water channel experiments on model rotors. The model is then used to investigate different types of vortex perturbations. Perturbation direction is found to have an important effect on the evolution of the instability, and displacements are observed to induce vortex pairing more quickly than circulation changes. These findings can be used to design asymmetric rotors that induce vortex breakdown more effectively, mitigating detrimental wake effects such as increased fatigue loading on downstream structures.

Negative Pell Equation and Stationary Configurations of Point Vortices on the Plane

Работа посвящена изучению модели точечных вихрей, предложенной немецким ученым Германом Гельмгольцем. Найдены необходимые и достаточные условия существования бесконечного числа неэквивалентных статических конфигураций для системы, состоящей из двух точечных вихрей интенсивности $\Gamma_1$ и произвольного числа точечных вихрей интенсивности $\Gamma_2$. Проведена классификация таких конфигураций. Впервые обнаружена связь между отрицательным диофантовым уравнением Пелля и статическими конфигурациями точечных вихрей на плоскости. Библиография: 20 названий.

Identification and dynamics prediction of a plane vortex structure based on a mathematical model of a point vortices system

The aim of the article is developing and analyse an algorithmic method for solution finding of one inverse problem of 2d vortex fluid dynamics. It is identification and prediction of the flow structure evolution of the based on the data on fluid velocity vectors in a set of reference points. Theoretical analysis of convergence and adequacy of the method is difficult due to the ill-posedness typical of inverse problems, these issues studied experimentally. Methods. The proposed method uses a mathematical model of a point vortex dynamics system for identification and prediction flow structures. The parameters of the model system are found by minimising the functional that evaluates the closeness of the original and model vectors fields at the reference points. The prediction of the vortex structure dynamics is based on the solution of the Cauchy problem for a system of ordinary differential equations with the parameters found in the first stage. Results. As a result of the calculations, we found it out: the algorithm converges to the desired minimum from a wide range of initial approximations; the algorithm converges in all cases when the identified structure consists of sufficiently distant vortices; the forecast of the development of the current gives good results with a steady flow; if the above conditions are violated, the part of successful calculations decreases, false identification and an erroneous forecast may occur; with the convergence of the method, the coordinates and circulation of the eddies of the model system are close to the characteristics of the eddies of the test configurations; the structures of the streamlines of the flows are topologically equivalent; convergence depends more on location than on the number of vectors used for identification. Conclusion. An algorithm for solving the problem of identifying and the evolution forecast of a 2d vortex flow structure is proposed when the fluid velocity vectors in a finite set of reference points are known. The method showed its high efficiency when using from 40 to 200 reference points. The results of the study make it possible to recommend the proposed algorithm for identifying flat vortex structures, which consist of vortices separated from each other.

Field evaluation of a novel holographic single-image depth reconstruction sensor

A camera-based single-image sensor is presented, that is able to measure the distance of one or multiple object points (light emitters). The sensor consists of a camera, whose lens is upgraded with a diffractive optical element (DOE). It fulfils two tasks: adding a vortex point spread function (PSF) and replication of the vortex PSFs to a predefined pattern of K spots. Both, shape and rotation of the vortex PSF is sensitive to defocus. The sensor concept is presented and its capabilities evaluated both on axis and off-axis. The achieved standard deviation of the error ranges between 8.5 μm (on-axis) and 3.5 μm (off-axis) within a measurement range of 20 mm. However, as soon as calibration and measurement position no longer match, the accuracy is limited. An analysis of the effects responsible for this are also part of the publication.

Density of states for a low-energy point vortex gas on the sphere

Point vortex systems are useful models of two-dimensional fluids. The equilibrium statistical mechanical properties of point vortices in a planar region include some unusual features, such as negative temperature states, that depend on the details of the limiting process used. The limiting form of the density of states as the vortex number increases is of central importance. Esler et al. (2013, 2015) expressed the limiting density of states for point vortices modeling low-energy fluid systems in a planar region as the inverse Fourier transform of a function determined by the spectrum of a certain differential operator. In this paper random variables related to the density of states are found for systems of point vortices on the sphere using the theory of weighted U-statistics. The limiting distributions of these statistics were determined by limits of the Fourier transforms of the probability density function; the results take a form analogous to that found in those earlier works. Two systems are investigated. The first is a neutral or weakly non-neutral point vortex gas on the sphere, using the same low-energy limiting process. The second is a lattice model where the vortex positions are fixed and the circulations are random. In both cases the limiting form of the moment generating function is found in closed form, from which the density of states can be found by inverse Fourier transform.

Shape dynamics of N point vortices on the sphere

We give a geometric account of the relative motion or the shape dynamics of N point vortices on the sphere exploiting the -symmetry of the system. The main idea is to bypass the technical difficulty of the -reduction by first lifting the dynamics from to . We then perform the -reduction using a dual pair to obtain a Lie–Poisson dynamics for the shape dynamics. This Lie–Poisson structure helps us find a family of Casimirs for the shape dynamics. We further reduce the system by -symmetry to obtain a Poisson structure for the shape dynamics involving fewer shape variables than those of the previous work by Borisov and Pavlov. As an application of the shape dynamics, we prove that the tetrahedron relative equilibria are stable when all of their circulations have the same sign, generalizing some existing results on tetrahedron relative equilibria of identical vortices.

Periodic perturbation of motion of an unbalanced circular foil in the presence of point vortices in an ideal fluid

The dynamics of a system governing the controlled motion of an unbalanced circular foil in the presence of point vortices is considered. The foil motion is controlled by periodically changing the position of the center of mass, the gyrostatic momentum, and the moment of inertia of the system. A derivation of the equations of motion based on Sedov's approach is proposed, the equations of motion are presented in the Hamiltonian form. A periodic perturbation of the known integrable case is considered.

Co-rotating and traveling vortex sheets for the 2D incompressible Euler equation

We construct co-rotating and traveling vortex sheets for 2D incompressible Euler equation, which are supported on several small closed curves. These examples represent a new type of vortex sheet solutions other than two known classes. The construction is based on Birkhoff–Rott operator, and accomplished by using implicit function theorem at point vortex solutions with suitably chosen function spaces.

Self-similar vortex configurations: Collapse, expansion, and rigid-vortex motion

The motion of point vortices that preserves the initial geometry of the vortex arrangement is an important class of vortex motion associated with N-vortex systems, referred to as the self-similar motion. While the self-similar motion of three-point vortices is well understood, larger vortex systems still need to be explored. Here, we use simple concepts from linear algebra to numerically investigate the distribution of initial conditions that lead to the self-similar motion of point vortices.

Structure transitions in arrays of point-vortices upon free space propagation

We study the free space propagation of arrays of N × N point vortices with equal unitary topological charge embedded in a Gaussian beam, initially distributed into either a square lattice or a disordered array with the same mean vortex density and separation. The far-field patterns and the near-field evolution are analyzed as a function of different control parameters, such as the size of the carrier beam, the number of vortices, the mean separation distance among them P, and a geometrical parameter S p , defined as the ratio of the size of the array to the size of the host beam. The value of S p turns out to be determinant for the final spatial distribution of the optical vortices in the far-field pattern and also plays a very relevant role in the near-field evolution. While the initial array structure is basically preserved, rotated and scaled up when , the vortices redistribute forming a circumference around the propagation axis for , depleting most of the central intensity, even if the vortex density is far lower than the previously predicted limit for depletion. In the near field, we find an expression for the rotation angle of the whole structure when preserved, as a function of the propagation distance, which depends on P, but it is around eight-fold larger than the Gouy phase shift of the host. Also, we observe the formation of a high intensity ring pattern, whose radius is independent of the size of the carrier and the initial distribution of vortices, but it is only determined by the number of vortices. We verified our numerical results with experiments, finding a very good agreement between them.

Multipole Vortex Patch Equilibria for Active Scalar Equations

We study how a general steady configuration of finitely many point vortices, with Newtonian interaction or generalized surface quasi-geostrophic interactions, can be desingularized into a steady configuration of vortex patches. The configurations can be uniformly rotating, uniformly translating, or completely stationary. Using a technique first introduced by Hmidi and Mateu [Comm. Math. Phys., 350 (2017), pp. 699--747] for vortex pairs, we reformulate the problem for the patch boundaries so that it no longer appears singular in the point vortex limit. Provided the point vortex equilibrium is nondegenerate in a natural sense, solutions can then be constructed directly using the implicit function theorem, yielding asymptotics for the shape of the patch boundaries. As an application, we construct new families of asymmetric translating and rotating pairs, as well as stationary tripoles. We also show how the techniques can be adapted for highly symmetric configurations such as regular polygons, body-centered polygons, and nested regular polygons by integrating the appropriate symmetries into the formulation of the problem.

Ensemble Equivalence for Mean Field Models and Plurisubharmonicity

We show that entropy is globally concave with respect to energy for a rich class of mean field interactions, including regularizations of the point vortex model in the plane, plasmas and self-gravitating matter in 2D, as well as the higher-dimensional logarithmic interactions appearing in conformal geometry and power laws. The proofs are based on a corresponding “microscopic” concavity result at finite N, shown by leveraging an unexpected link to Kähler geometry and plurisubharmonic functions. Under more restrictive homogeneity assumptions, strict concavity is obtained using a uniqueness result for free energy minimizers, established in a companion paper. The results imply that thermodynamic equivalence of ensembles holds for this class of mean field models. As an application, it is shown that the critical inverse negative temperatures—in the macroscopic as well as the microscopic setting—coincide with the asymptotic slope of the corresponding microcanonical entropies. Along the way, we also extend previous results on the thermodynamic equivalence of ensembles for continuous weakly positive definite interactions, concerning positive temperature states, to the general non-continuous case. In particular, singular situations are exhibited where, somewhat surprisingly, thermodynamic equivalence of ensembles fails at energy levels sufficiently close to the minimum energy level.

Infinitesimal invariance of completely Random Measures for 2D Euler Equations

We consider suitable weak solutions of 2-dimensional Euler equations on bounded domains, and show that the class of completely random measures is infinitesimally invariant for the dynamics. Space regularity of samples of these random fields falls outside of the well-posedness regime of the PDE under consideration, so it is necessary to resort to stochastic integrals with respect to the candidate invariant measure in order to give a definition of the dynamics. Our findings generalize and unify previous results on Gaussian stationary solutions of Euler equations and point vortices dynamics. We also discuss difficulties arising when attempting to produce a solution flow for Euler’s equations preserving independently scattered random measures.

Helicoids and vortices

We point out an interesting connection between fluid dynamics and minimal surface theory: When gluing helicoids into a minimal surface, the limit positions of the helicoids correspond to a ‘vortex crystal’, an equilibrium of point vortices in two-dimensional fluid that move together as a rigid body. While vortex crystals have been studied for almost 150 years, the gluing construction of minimal surfaces is relatively new. As a consequence of the connection, we obtain many new minimal surfaces and some new vortex crystals by simply comparing notes.

The vortex cosine-gaussian beam in strongly nonlocal nonlinear media

Different cosine phase factors n can make the vortex cosine-Gaussian beam(VCGB) exhibit a multi-ring structure. When the number of vortex points of the VCGB was 1, the analytical expression of the orbital angular momentum was the Dawson integral function. By the split-step Fourier method, the propagation of a VCGB in nonlocalized medium was numerically simulated. It was found that when the vortex point was asymmetric with respect to the origin, the energy flow distribution of the VCGB was asymmetric. The center of mass is move with the transmission of the beam in nonlocal medium, resulting in oblique transmission. If the sign in front of the imaginary part of the vortex point changes, the rotation direction of the soliton changes. When two VCGBs were transmitted in nonlocalized medium, they can be intertwined or transmitted in serpentine shapes. Therefore, the beam information can be encoded by changing the phase factor and the number of vortex points. The beam transmission path in the nonlocal medium can be controlled by changing the vortex position.

On the Existence of Weak Solutions for the 2D Incompressible Euler Equations with In–Out Flow and Source and Sink Points

Well-posedness for the two dimensional Euler system with given initial vorticity is known since the works of Judovi\v{c}. In this paper we show existence of solutions in the case where we allowed the fluid to enter in and exit from the boundaries and from some points of the fluid domain. In particular we derive the equations of the model as the limit when we replace the points by some small holes. To do that we extend the DiPerna-Lions theory with non-tangent velocity field on the boundary to the case of time-dependent domain, we extend the existence result for the two dimensional Euler system with in-out flow to time-dependent domain and finally we derive the system that models a fluid which is allowed to enter in and exit from the boundary and some points. The solutions are characterized by the presence of source, sink and vortex points.

Fractal structure of chaotic scattering in a simple hydrodynamic model with a point vortex embedded in a time-(quasi)periodic background flow

This paper is a study of a nonlinear dynamical system with 3/2 degrees of freedom consisting of a point vortex and an unsteady background flow. This background flow consists of two parts, a steady and a (quasi)periodic with two sinusoidal components with different frequencies and amplitudes. The considered system with a separatrix loop in the phase portrait is the simplest model of chaotic scattering of passive tracers on a topographic vortex, captured by a seamount, in the ocean. We have found that depending on variation of amplitudes of two sinusoidal components, the passive tracers enters and leaves the vortex region in portions. The frequency of escape and the size of such portions depend on the parameters of the background flow. It is shown that the structure of the fractal-like scattering function changes at low levels through the formation (disappearance) of segments according to two mechanisms, the action of which is determined by the relation between two frequencies of the perturbation.

Motion of three geostrophic Bessel vortices

This study investigates the dynamics of geostrophic vortices (described by modified Bessel functions). We focus on the three-vortex case, where the possibility of self-similar motion and general dynamics for arbitrary strengths is studied. It is found that self-similar motions are limited to rigid rotations and self-similar triple collapse is essentially impossible. For a general description, trilinear coordinates are adopted and extended to the model of geostrophic vortices. The similarities and differences with the point-vortex model are emphasized. The physical regions in the phase plane cannot be directly identified, unlike the point-vortex case where the physical regions are identified as conic sections. The boundary of the physical region approaches the vertices of the triangle in trilinear coordinates in geostrophic vortices, while making a tangent with the sides of the triangle in point vortices. For positive values of the three vortex strengths, the boundary of the physical region is a closed concave curve, whereas it is a circle or ellipse in the case of the point vortex. Importantly, we find a new type of the phase trajectory: an open curve not intersecting with the physical region boundary. We also demonstrate the physical motions and trajectories of vortices for several representative cases of vortex strengths from numerical computations.