Motion Of Point Vortices Research Articles

Dynamics of two vortices on a finite flat cylinder

In this work, a model that describes the motion of point vortices in an ideal incompressible fluid on a finite flat cylinder is obtained. The case of two vortices is considered in detail. It is shown that the equations of motion of vortices can be represented in Hamiltonian form and have an additional first integral. A procedure of reduction to a fixed level of the first integral is proposed. For the reduced system, phase portraits are constructed, fixed points and singularities of the system are indicated.

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.

Equations of motion for weakly compressible point vortices.

Equations of motion for compressible point vortices in the plane are obtained in the limit of small Mach number, M, using a Rayleigh–Jansen expansion and the method of Matched Asymptotic Expansions. The solution in the region between vortices is matched to solutions around each vortex core. The motion of the vortices is modified over long time scales and . Examples are given for co-rotating and co-propagating vortex pairs. The former show a correction to the rotation rate and, in general, to the centre and radius of rotation, while the latter recover the known result that the steady propagation velocity is unchanged. For unsteady configurations, the vortex solution matches to a far field in which acoustic waves are radiated.This article is part of the theme issue ‘Mathematical problems in physical fluid dynamics (part 2)’.

Vortex Motion of the Euler and Lake Equations

We start by surveying the planar point vortex motion of the Euler equations in the whole plane, half-plane and quadrant. Then, we go on to prove the non-collision property of the 2-vortex system by using the explicit form of orbits of the 2-vortex system in the half-plane. We also prove that the N-vortex system in the half-plane is nonintegrable for $$N>2$$ , which was suggested previously by numerical experiments without rigorous proof. The skew-mean-curvature (or binormal) flow in $${\mathbb {R}}^n,\;n\geqslant 3$$ with certain symmetry can be regarded as point vortex motion of these 2D lake equations. We compare point vortex motions of the Euler and lake equations. Interesting similarities between the point vortex motion in the half-plane, quadrant and the binormal motion of coaxial vortex rings, sphere product membranes are addressed. We also raise some open questions in the paper.

The Motion of a Point Vortex in Multiply-Connected Polygonal Domains

We study the motion of a single point vortex in simply- and multiply-connected polygonal domains. In the case of multiply-connected domains, the polygonal obstacles can be viewed as the cross-sections of 3D polygonal cylinders. First, we utilize conformal mappings to transfer the polygonal domains onto circular domains. Then, we employ the Schottky-Klein prime function to compute the Hamiltonian governing the point vortex motion in circular domains. We compare between the topological structures of the contour lines of the Hamiltonian in symmetric and asymmetric domains. Special attention is paid to the interaction of point vortex trajectories with the polygonal obstacles. In this context, we discuss the effect of symmetry breaking, and obstacle location and shape on the behavior of vortex motion.

Алгоритм движения точечных вихрей в ограниченной области

The problem of calculating the paths of motion of a set of point vortices in an ideal incompressible liquid in a limited area is considered. The area is supposed to be piecewise smooth. The stream function at any given time is presented as a sum: the stream functions of point vortices and the potential of a simple layer, the density of which - the density of vortexes on the border - is required to determine. Regardless of time, the stream function harmoniously extends into the exterior of the area and is identically equal to a constant at the boundary, and, therefore, is a Roben potential. Briefly, the paper describes an algorithm for calculating the Roben potential. The unknown density of vortices is sought in the form of a linear combination of a complete system of potentials, and the coefficients are determined by the values of the Roben potential, as a solution to the inverse problem; determination of the coefficients reduces to solving a system of linear equations with a Gram matrix for a linearly independent system of functions. By the function of stream by tangents determine the trajectories of movements of vortices. The results of computational experiments of the motion of several point vortices in a square are presented. A computer analysis of the movement was carried out: two, four and eight point vortices, calculated the trajectories of their movement and found stationary and periodic, stable and unstable configurations.

Getting into the vortex: On the contributions of james montaldi

James Montaldi's expertises span many areas on pure and applied mathematics. I will discuss here just one, his contributions to the motion of point vortices, specially the role of symmetries in the bifurcations and stability of equilibrium configurations in surfaces of constant curvature. This approach leads, for instance, to a very elegant proof of a classical result, the nonlinear stability of Thompson's regular heptagon in the plane. Here the plane appears 'in passing', just as the transition between positive and negative curvatures.

Integrability and Chaos in Vortex Lattice Dynamics

This paper is concerned with the problem of the interaction of vortex lattices, which is equivalent to the problem of the motion of point vortices on a torus. It is shown that the dynamics of a system of two vortices does not depend qualitatively on their strengths. Steadystate configurations are found and their stability is investigated. For two vortex lattices it is also shown that, in absolute space, vortices move along closed trajectories except for the case of a vortex pair. The problems of the motion of three and four vortex lattices with nonzero total strength are considered. For three vortices, a reduction to the level set of first integrals is performed. The nonintegrability of this problem is numerically shown. It is demonstrated that the equations of motion of four vortices on a torus admit an invariant manifold which corresponds to centrally symmetric vortex configurations. Equations of motion of four vortices on this invariant manifold and on a fixed level set of first integrals are obtained and their nonintegrability is numerically proved.

Vortex dynamics in nonlinear free surface flows

The two-dimensional motion of point vortices in an inviscid fluid with a free surface and an impenetrable bed is investigated. The work is based on forming a closed system of equations for surface variables and vortex positions using a variant of the Ablowitz, Fokas, and Musslimani formulation [M. J. Ablowitz, A. S. Fokas, and Z. H. Musslimani, J. Fluid Mech. 562, 313–343 (2006)] of the water-wave free-surface problem. The equations are approximated with a dealiased spectral method making use of a high-order approximation of the Dirichlet-Neumann operator and a high-order time-stepping scheme. Numerical simulations reveal that the combination of vortex motion and solid bottom boundary yields interesting dynamics not seen in the case of vortex motion in an infinitely deep fluid. In particular, strong deformations of the free surface, including non-symmetric surface profiles and regions of large energy concentration, are observed. Our simulations also uncover a rich variety of vortex trajectories including orbiting and nearly parallel patterns of motion. The dynamics of the free surface and of the point vortices are strongly influenced by the initial placement and polarity of the vortices. The method put forward here is flexible enough to handle a large number of vortices and may easily be extended to include the effects of varying bathymetry, stratification, and background shear currents.

Equilibrium energy spectrum of point vortex motion with remarks on ensemble choice and ergodicity

The equilibrium energy spectrum of the point vortex system is shown to be predictable as a function of the vortex interaction (Hamiltonian) energy.

A minimal-variable symplectic integrator on spheres

We construct a symplectic, globally defined, minimal-coordinate, equivariant integrator on products of 2-spheres. Examples of corresponding Hamiltonian systems, called spin systems, include the reduced free rigid body, the motion of point vortices on a sphere, and the classical Heisenberg spin chain, a spatial discretisation of the Landau-Lifschitz equation. The existence of such an integrator is remarkable, as the sphere is neither a vector space, nor a cotangent bundle, has no global coordinate chart, and its symplectic form is not even exact. Moreover, the formulation of the integrator is very simple, and resembles the geodesic midpoint method, although the latter is not symplectic.

Parametric instability of a many point-vortex system in a multi-layer flow under linear deformation

The paper deals with a dynamical system governing the motion of many point vortices located in different layers of a multi-layer flow under external deformation. The deformation consists of generally independent shear and rotational components. First, we examine the dynamics of the system’s vorticity center. We demonstrate that the vorticity center of such a multi-vortex multi-layer system behaves just like the one of two point vortices interacting in a homogeneous deformation flow. Given nonstationary shear and rotational components oscillating with different magnitudes, the vorticity center may experience parametric instability leading to its unbounded growth. However, we then show that one can shift to a moving reference frame with the origin coinciding with the position of the vorticity center. In this new reference frame, the new vorticity center always stays at the origin of coordinates, and the equations governing the vortex trajectories look exactly the same as if the vorticity center had never moved in the original reference frame. Second, we studied the relative motion of two point vortices located in different layers of a two-layer flow under linear deformation. We analyze their regular and chaotic dynamics identifying parameters resulting in effective and extensive destabilization of the vortex trajectories.

Universal statistics of point vortex turbulence

A new methodology, based on the central limit theorem, is applied to describe the statistical mechanics of two-dimensional point vortex motion in a bounded container $\mathscr{D}$, as the number of vortices $N$ tends to infinity. The key to the approach is the identification of the normal modes of the system with the eigenfunction solutions of the so-called hydrodynamic eigenvalue problem of the Laplacian in $\mathscr{D}$. The statistics of the projection of the vorticity distribution onto these eigenfunctions (‘vorticity projections’) are then investigated. The statistics are used first to obtain the density-of-states function and caloric curve for the system, generalising previous results to arbitrary (neutral) distributions of vortex circulations. Explicit expressions are then obtained for the microcanonical (i.e. fixed energy) probability density functions of the vorticity projections in a form that can be compared directly with direct numerical simulations of the dynamics. The energy spectra of the resulting flows are predicted analytically. Ensembles of simulations with $N=100$, in several conformal domains, are used to make a comprehensive validation of the theory, with good agreement found across a broad range of energies. The probability density function of the leading vorticity projection is of particular interest because it has a unimodal distribution at low energy and a bimodal distribution at high energy. This behaviour is indicative of a phase transition, known as Onsager–Kraichnan condensation in the literature, between low-energy states with no mean flow in the domain and high-energy states with a coherent mean flow. The critical temperature for the phase transition, which depends on the shape but not the size of $\mathscr{D}$, and the associated critical energy are found. Finally the accuracy and the extent of the validity of the theory, at finite $N$, are explored using a Markov chain phase-space sampling method.

The motion of point vortices on closed surfaces

We develop a mathematical framework for the dynamics of a set of point vortices on a class of differentiable surfaces conformal to the unit sphere. When the sum of the vortex circulations is non-zero, a compensating uniform vorticity field is required to satisfy the Gauss condition (that the integral of the Laplace–Beltrami operator must vanish). On variable Gaussian curvature surfaces, this results in self-induced vortex motion, a feature entirely absent on the plane, the sphere or the hyperboloid. We derive explicit equations of motion for vortices on surfaces of revolution and compute their solutions for a variety of surfaces. We also apply these equations to study the linear stability of a ring of vortices on any surface of revolution. On an ellipsoid of revolution, as few as two vortices can be unstable on oblate surfaces or sufficiently prolate ones. This extends known results for the plane, where seven vortices are marginally unstable (Thomson 1883 A treatise on the motion of vortex rings, pp. 94–108; Dritschel 1985 J. Fluid Mech. 157, 95–134 (doi:10.1017/S0022112088003088)), and the sphere, where four vortices may be unstable if sufficiently close to the equator (Polvani & Dritschel 1993 J. Fluid Mech. 255, 35–64 (doi:10.1017/S0022112093002381)).

Integrable two layer point vortex motion on the half plane

In this paper we derive the equations of motion for two-layer point vortex motion on the upper half plane. We study the invariants using symmetry, including the Hamiltonian and show that the two vortex problem is integrable. We characterize all two vortex motions for the cases where the vortex strengths are both equal, $\Gamma_{1}=\Gamma_{2}=1$ and when they are opposite $\Gamma_{1}=-\Gamma_{2}=1$. We also prove that there are no equilibria for the two vortex problem when $\Gamma_{1}=-\Gamma_{2}=1$.\ We show that there is only one relative equilibrium configuration when $\Gamma_{1}=\Gamma_{2}=1$ and the vortices are in different layers. We also make observations concerning the finite-time collapse of two vortices in the half plane. We then compare the regimes of motion for both cases (motion on the half plane) with the case of the two-layer vortex problem on the entire plane. We also study several classes of streamline topologies for two vortices in different layers. We conclude with a Hamiltonian study of integrable two-layer 3 vortex motion on the half plane by studying integrable symmetrical configurations and provide a rich class of new relative equilibria.

A computational theory for spiral point vortices in multiply connected domains with slit boundaries

A spiral point vortex, which is a point vortex with a source-sink structure, appears when we consider a two-dimensional cross section of a straight vortex line with a uniform flow in its axial direction in three-dimensional space. In the present paper, we present a computational theory for the motion of spiral point vortices in two-dimensional exterior domains that contain multiple rectilinear line segments as boundaries, called slit domains. The theory consists of the evolution equation for spiral point vortices in multiply connected domains and a numerical procedure to construct conformal mappings from multiply connected domains with circular boundaries onto given slit domains. It is applicable to many flow problems arising in biofluids and environmental flows. As an example, we apply the theory to the flow problem of finding force-enhancing equilibria consisting of two spiral point vortices in a slit domain with two parallel plates in the presence of a uniform flow, which is a two-dimensional model of a wind-turbine with vertical blades.

Fast iterative solvers for boundary value problems on a local spherical region

Boundary value problems on local spherical regions arise naturally in geophysics and oceanography when scientists model a physical quantity on large scales. Meshless methods using radial basis functions provide a simple way to construct numerical solutions with high accuracy. However, the linear systems arising from these methods are usually ill-conditioned, which poses a challenge for iterative solvers. We construct preconditioners based on an additive Schwarz method to accelerate the solution process for solving boundary value problems on local spherical regions. References D. Crowdy. Point vortex motion on the surface of a sphere with impenetrable boundaries. Physics of Fluids, 18:036602 (2006). doi:10.1063/1.2183627. A. E. Gill. Atmosphere-Ocean Dynamics, International Geophysics Series Volume 30. Academic, New York (1982). R. Kidambi and P. K. Newton. Point vortex motion on a sphere with solid boundaries. Physics of Fluids, 12:581 (2000). doi:10.1063/1.870263. Q. T. Le Gia, I. H. Sloan, and T. Tran. Overlapping additive Schwarz preconditioners for elliptic PDEs on the unit sphere. Math. Comp., 78:79--101 (2009). doi:10.1090/S0025-5718-08-02150-9. C. Muller. Spherical Harmonics, Vol. 17 of Lecture Notes in Mathematics. Springer-Verlag, Berlin (1966). M. V. Nezlin. Some remarks on coherent structures out of chaos in planetary atmospheres and oceans. Chaos, 4:109--111 (1994). doi:10.1063/1.165997. T. Tran, Q. T. Le Gia, I. H. Sloan, and E. P. Stephan. Preconditioners for pseudodifferential equations on the sphere with radial basis functions. Numer. Math., 115:141--163 (2009). doi:10.1007/s00211-009-0269-8.

A Novel Formulation of Point Vortex Dynamics on the Sphere: Geometrical and Numerical Aspects

In this paper, we present a novel Lagrangian formulation of the equations of motion for point vortices on the unit 2-sphere. We show first that no linear Lagrangian formulation exists directly on the 2-sphere but that a Lagrangian may be constructed by pulling back the dynamics to the 3-sphere by means of the Hopf fibration. We then use the isomorphism of the 3-sphere with the Lie group SU(2) to derive a variational Lie group integrator for point vortices which is symplectic, second-order, and preserves the unit-length constraint. At the end of the paper, we compare our integrator with classical fourth-order Runge--Kutta, the second-order midpoint method, and a standard Lie group Munthe-Kaas method.

Generalized Hamiltonian point vortex dynamics on arbitrary domains using the method of fundamental solutions

A new algorithm (VOR-MFS) is presented for the solution of a generalized Hamiltonian model of point vortex dynamics in an arbitrary two-dimensional computational domain. The VOR-MFS algorithm utilizes the method of fundamental solutions (MFS) to obtain an approximation to the model Hamiltonian by solution of an appropriate boundary value problem. Unlike standard point vortex methods, VOR-MFS requires knowledge only of the free-space (R2) Green’s function for the problem as opposed to the domain-adapted Green’s function, permitting solution of a much wider range of problems. VOR-MFS is first validated against a vortex image model for the case of (2D Euler) multiple vortex motion in both circular and ‘Neumann-oval’ shaped domains. It is then demonstrated that VOR-MFS can solve for quasi-geostrophic shallow water point vortex motion in the same domains. The exponential convergence of the MFS method is shown to lead to good conservation properties for each of the solutions presented.

Finite Rossby radius effects on vortex motion near a gap

This work investigates the effect of the Rossby radius of deformation on the motion of a vortex near a gap in an infinitely long barrier. A key parameter determining the behaviour of the vortex is a, the ratio of the Rossby radius of deformation to the width of the gap. Assuming quasi-geostrophic dynamics for a single-layer, reduced-gravity fluid, an integral equation is derived whose solution gives the velocity at any point in the fluid. The integral equation is solved numerically and the velocity field is integrated to give the trajectories of point vortices. Combined with the method of contour dynamics, the method can be used to compute the evolution of finite area patches of constant vorticity. The trajectories of point vortices and vortex patches are compared. The patches are initially circular and the centroids of those vortex patches that remain close to circular follow the trajectory and speed of their equivalent point vortices when appropriately normalised. The critical point vortex trajectory (the separatrix) which divides vortices that leap across the gap and those that pass through, is computed for various a. Decreasing the Rossby radius of deformation increases the tendency of vortices to pass through the gap. The effect of various background flows on both point vortex and vortex patch motion is also described.