System Of Point Vortices Research Articles

Decay of time correlations in point vortex systems

The dynamics of a large point vortex system whose initial configuration consists in uniformly distributed independent positions is investigated. Time correlations of local observables of the vortex configuration are shown to be compatible with power law decay 1/t, providing additional insight on ergodicity and mixing properties of equilibrium dynamics in point vortex models.

Zero-noise dynamics after collapse for three point vortices

The continuation of point vortex dynamics after a vortex collapse is investigated by means of a regularization procedure consisting in introducing a small stochastic diffusive term, that corresponds to a vanishing viscosity. In contrast with deterministic regularization, in which a cutoff interaction selects in the limit a single trajectory of the system after collapse, the zero-noise method produces a probability distribution supported by trajectories satisfying relevant conservation laws of the point vortex system.

Hölder regularity for collapses of point-vortices

The first part of this article studies the collapses of point-vortices for the Euler equation in the plane and for surface quasi-geostrophic equations in the general setting of α models. In these models the kernel of the Biot–Savart law is a power function of exponent . It is proved that, under a standard non-degeneracy hypothesis, the trajectories of the point-vortices have a Hölder regularity up to the time of collapse. The Hölder exponent obtained is and this exponent is proved to be optimal for all α by exhibiting an example of a three-vortex collapse. The same question is then addressed for the Euler point-vortex system in smooth bounded connected domains. It is proved that if a given point-vortex has an accumulation point in the interior of the domain as t → T, then it converges towards this point and displays the same Hölder continuity property. A partial result for point-vortices that collapse with the boundary is also established: we prove that their distance to the boundary is Hölder regular.

Axis-symmetric Onsager clustered states of point vortices in a bounded domain

We study axis-symmetric Onsager clustered states of a neutral point vortex system confined to a two-dimensional disc. Our analysis is based on the mean field of bounded point vortices in the microcanonical ensemble. The clustered vortex states are specified by the inverse temperature β and the rotation frequency ω, which are the conjugate variables of energy E and angular momentum L, respectively. The formation of the axis-symmetric clustered vortex states (azimuthal angle independent) involves the separating of vortices with opposite circulation and the clustering of vortices with the same circulation around the origin and edge. The state preserves SO(2) symmetry while breaking symmetry. We find that, near the uniform state, the rotation-free clustered state (ω = 0) emerges at particular values of L 2/E and β. At large energies, we obtain asymptotically exact vortex density distributions, whose validity condition gives rise to the lower bound of β for the rotation-free states. Noticeably, the obtained vortex density distribution near the edge at large energies provides a novel exact vortex density distribution for the corresponding chiral vortex system.

Preservation of Quadratic Invariants by Semiexplicit Symplectic Integrators for Nonseparable Hamiltonian Systems

We prove that the recently developed semiexplicit symplectic integrators for non-separable Hamiltonian systems preserve any linear and quadratic invariants possessed by the Hamiltonian systems. This is in addition to being symmetric and symplectic as shown in our previous work; hence it shares the crucial structure-preserving properties with some of the well-known symplectic Runge--Kutta methods such as the Gauss--Legendre methods. The proof follows two steps: First we show how the extended Hamiltonian system proposed by Pihajoki inherits linear and quadratic invariants in the extended phase space from the original Hamiltonian system. Then we show that this inheritance in turn implies that our integrator preserves the original linear and quadratic invariants in the original phase space. We also analyze preservation/non-preservation of these invariants by Tao's extended Hamiltonian system and the extended phase space integrators of Pihajoki and Tao. The paper concludes with numerical demonstrations of our results using a simple test case and a system of point vortices.

Kinetic theory of two-dimensional point vortices and fluctuation–dissipation theorem

We complete the kinetic theory of two-dimensional (2D) point vortices initiated in previous works. We use a simpler and more physical formalism. We consider a system of 2D point vortices submitted to a small external stochastic perturbation and determine the response of the system to the perturbation. We derive the diffusion coefficient and the drift by polarization of a test vortex. We introduce a general Fokker–Planck equation involving a diffusion term and a drift term. When the drift by polarization can be neglected, we obtain a secular dressed diffusion equation sourced by the external noise. When the external perturbation is created by a discrete collection of N point vortices, we obtain a Lenard–Balescu-like kinetic equation reducing to a Landau-like kinetic equation when collective effects are neglected. We consider a multi-species system of point vortices. We discuss the process of kinetic blocking in the single and multi-species cases. When the field vortices are at statistical equilibrium (thermal bath), we establish the proper expression of the fluctuation–dissipation theorem for 2D point vortices relating the power spectrum of the fluctuations to the response function of the system. In that case, the drift coefficient and the diffusion coefficient satisfy an Einstein-like relation and the Fokker–Planck equation reduces to a Smoluchowski-like equation. We mention the analogy between 2D point vortices and stellar systems. In particular, the drift of a point vortex in 2D hydrodynamics (Chavanis in Phys Rev E 58:R1199, 1998) is the counterpart of the Chandrasekhar dynamical friction in astrophysics. We also consider a gas of 2D Brownian point vortices described by N coupled stochastic Langevin equations and determine its mean and mesoscopic evolution. In the present paper, we treat the case of unidirectional flows, but our results can be straightforwardly generalized to axisymmetric flows.

Transition to instability of the leapfrogging vortex quartet

The point-vortex system is a system of longstanding interest in nonlinear dynamics, describing the motion of a two-dimensional inviscid fluid that is irrotational except at a discrete set of moving point vortices, at which the vorticity diverges. The leapfrogging orbit consists of two rotating pairs of like-signed vortices which, taken as a quartet, propagate at constant velocity. It is known that if the two pairs are initially widely separated, the motion is stable, while if they are closer together it becomes unstable, with this relation represented by a dimensionless parameter α defined in the text. We here demonstrate analytically that the transition from stability to instability happens at a critical value α=ϕ−2, where ϕ is the golden ratio. This value had been hypothesized based on careful numerics by Tophøj and Aref, and by the present authors using a semi-analytic argument but not previously demonstrated through exact analysis.

Improbability of Collisions of Point-Vortices in Bounded Planar Domains

In this paper, we prove that in bounded planar domains with $C^{2,\alpha}$ boundary, for almost every initial condition in the sense of the Lebesgue measure, the point-vortex system has a global solution, meaning that there is no collision between two point-vortices or with the boundary. This extends the work previously done in [C. Marchioro and M. Pulvirenti, Vortex Methods in Two-Dimensional Fluid Dynamics, Springer-Verlag, 1984] for the disk. The proof requires the construction of a regularized dynamics that approximates the real dynamics and some strong inequalities for the Green's function of the domain. The establishment of some useful estimates is discussed and the details of the proof are given in the original article [M. Donati, SIAM J. Math. Anal., 54 (2022), pp. 79--113].

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.

Vortex Pairs and Dipoles on Closed Surfaces

We set up general equations of motion for point vortex systems on closed Riemannian surfaces, allowing for the case that the sum of vorticities is not zero and there hence must be counter-vorticity present. The dynamics of global circulations which is coupled to the dynamics of the vortices is carefully taken into account. Much emphasis is put to the study of vortex pairs, having the Kimura conjecture in focus. This says that vortex pairs move, in the dipole limit, along geodesic curves, and proofs for it have previously been given by S. Boatto and J. Koiller by using Gaussian geodesic coordinates. In the present paper, we reach the same conclusion by following a slightly different route, leading directly to the geodesic equation with a reparametrized time variable. In a final section, we explain how vortex motion in planar domains can be seen as a special case of vortex motion on closed surfaces.

On the dynamics of point vortices for the two-dimensional Euler equation with Lp vorticity.

We study the evolution of solutions to the two-dimensional Euler equations whose vorticity is sharply concentrated in the Wasserstein sense around a finite number of points. Under the assumption that the vorticity is merely [Formula: see text] integrable for some [Formula: see text], we show that the evolving vortex regions remain concentrated around points, and these points are close to solutions to the Helmholtz-Kirchhoff point vortex system. This article is part of the theme issue 'Mathematical problems in physical fluid dynamics (part 2)'.

Hyperuniformity and phase enrichment in vortex and rotor assemblies

Ensembles of particles rotating in a two-dimensional fluid can exhibit chaotic dynamics yet develop signatures of hidden order. Such rotors are found in the natural world spanning vastly disparate length scales — from the rotor proteins in cellular membranes to models of atmospheric dynamics. Here we show that an initially random distribution of either driven rotors in a viscous membrane, or ideal vortices with minute perturbations, spontaneously self assemble into a distinct arrangement. Despite arising from drastically different physics, these systems share a Hamiltonian structure that sets geometrical conservation laws resulting in prominent structural states. We find that the rotationally invariant interactions isotropically suppress long-wavelength fluctuations — a hallmark of a disordered hyperuniform material. With increasing area fraction, the system orders into a hexagonal lattice. In mixtures of two co-rotating populations, the stronger population will gain order from the other and both will become phase enriched. Finally, we show that classical 2D point vortex systems arise as exact limits of the experimentally accessible microscopic membrane rotors, yielding a new system through which to study topological defects.

Geometrical Classification of Self-Similar Motion of Two-Dimensional Three Point Vortex System by Deviation Curvature on Jacobi Field

In this study, we geometrically analyze the relation between a point vortex system and deviation curvatures on the Jacobi field. First, eigenvalues of deviation curvatures are calculated from relative distances of point vortices in a three point vortex system. Afterward, based on the assumption of self-similarity, time evolutions of eigenvalues of deviation curvatures are shown. The self-similar motions of three point vortices are classified into two types, expansion and collapse, when the relative distances vary monotonously. Then, we find that the eigenvalues of self-similarity are proportional to the inverse fourth power of relative distances. The eigenvalues of the deviation curvatures monotonically convergent to zero for expansion, whereas they monotonically diverge for collapse, which indicates that the strengths of interactions between point vortices related to the time evolution of spatial geometric structure in terms of the deviation curvatures. In particular, for collapse, the collision point becomes a geometric singularity because the eigenvalues of the deviation curvature diverge. These results show that the self-similar motions of point vortices are classified by eigenvalues of the deviation curvature. Further, nonself-similar expansion is numerically analyzed. In this case, the eigenvalues of the deviation curvature are nonmonotonous but converge to zero, suggesting that the motion of the nonself-similar three point vortex system is also classified by eigenvalues of the deviation curvature.

Vortex Patches Choreography for Active Scalar Equations

This paper deals with the existence of $N$ vortex patches located at the vertex of a regular polygon with $N$ sides that rotate around the center of the polygon at a constant angular velocity. That is done for Euler and (SQG)$_\beta$ equations, with $\beta\in(0,1)$, but may be also extended to more general models. The idea is the desingularization of the Thomsom polygon for the $N$ point vortex system, that is, $N$ point vortices located at the vertex of a regular polygon with $N$ sides. The proof is based on the study of the contour dynamics equation combined with the application of the infinite dimensional Implicit Function theorem and the well--chosen of the function spaces.

Self-organizing effect of drift term against diffusion term in point vortex system evidenced by numerical simulations on PEZY-SC

We have analytically obtained a kinetic equation for a two-dimensional (2D) point vortex system with a Fokker–Planck type collision term consisting of a diffusion term and a drift term (Yatsuyanagi and Hatori 2015 Fluid Dyn. Res. 47 065506). The equation describes a mechanism of a self-organization in the system. In this paper, analytically anticipated characteristics of the collision term is evidenced by numerical simulations on PEZY-SC supercomputer system. In the previous paper, it is shown that the collision term satisfies following physically good properties: (1) charge separated, self-organized distribution typical for negative absolute temperature is achieved by the drift term, (2) when a system reaches a thermal equilibrium state, stream function ψ and vorticity ω satisfies an inequality which is important for the next property, (3) the obtained kinetic equation conserves total mean field energy and satisfies the H theorem. In this paper we will evidence the above properties numerically. The time-asymptotic distribution of the vortices corresponds to the analytically anticipated local equilibrium state in which many small regions with different local temperatures exist. Especially, the transport process by the drift term is elucidated clearly, which gives a key mechanism of the self-organization, i.e. condensation of the same-sign vortices.

Limit Theorems and Fluctuations for Point Vortices of Generalized Euler Equations

We prove a mean field limit, a law of large numbers and a central limit theorem for a system of point vortices on the 2D torus at equilibrium with positive temperature. The point vortices are formal solutions of a class of equations generalising the Euler equations, and are also known in the literature as generalised inviscid SQG. The mean-field limit is a steady solution of the equations, the CLT limit is a stationary distribution of the equations.

Vortex dynamics for 2D Euler flows with unbounded vorticity

It is well known that the dynamics of vortices in an ideal incompressible two-dimensional fluid contained in a bounded not necessarily simply connected smooth domain is described by the Kirchhoff–Routh point vortex system. In this paper, we revisit the classical problem of how well solutions to the Euler equations approximate these vortex dynamics, and extend previous rigorous results to the case where the vorticity field is unbounded. More precisely, we establish estimates for the 2-Wasserstein distance between the vorticity and the empirical measure associated with the point vortex dynamics. In particular, we derive an estimate on the order of weak convergence of the Euler solutions to the solutions of the point vortex system.

Алгоритм идентификации вихревых пятен на основе моделей точечных вихрей

An algorithm for identifying a plane vortex fluid flow from known velocity vectors at a finite set of points in the flow region is proposed. To describe the approximation of the vortex structure, a mathematical model of the system of point vortices is used. The parameters to be determined and characterizing the flow are the number of point vortices, their coordinates, and intensity. Unknown parameters are determined as a result of minimizing the functional that relates the known velocity vectors and the velocity model flows of the vortex system at the same points in the region. An implementation of the algorithm for the simplest case, one vortex patch, is presented. Computational experiments were carried out, which showed the effectiveness of the developed method.

Point vortex approximation for 2D Navier–Stokes equations driven by space-time white noise

We show that the system of point vortices, perturbed by a certain transport type noise, converges weakly to the vorticity form of 2D Navier–Stokes equations driven by the space-time white noise.

Self-similar Motions and Related Relative Equilibria in the N-point Vortex System

We study self-similar solutions of the point-vortex system. The explicit formula for self-similar solutions has been obtained for the three point-vortex problem and for a specific example of the four and five point-vortex problems. We see that the families consisting of these self-similar collapsing solutions are described by one-parameter families, and their collapse time and Hamiltonian are also expressed by functions of the same parameter. Then, the configurations at limit points of the parameter are in relative equilibria. For the many-vortex problem, we investigate the point-vortex system with the help of numerical computations. In particular, considering the case that $N - 1$ point vortices have a uniform vortex strength, we show that families of self-similar collapsing solutions continuously depend on the Hamiltonian and the self-similar solutions asymptotically approach relative equilibria as the Hamiltonian gets close to certain values. In addition, we prove the existence of relative equilibria for the four point-vortex system. We also investigate an example of seven point vortices with non-uniform vortex strengths and give numerical results for it.