System Of Point Vortices Research Articles

Stochastic properties of a system of point vortices

The dynamic and stochastic properties of a system of point vortices occurring at the vertices of a regular N-gon have been investigated. The time of reversibility of the phase trajectories was determined and their stability was studied. It is established that such a system exhibits stochastic properties only in the presence of a local instability, which is possible for N≥8.

A Method for Assimilation of Lagrangian Data

Difficulties in the assimilation of Lagrangian data arise because the state of the prognostic model is generally described in terms of Eulerian variables computed on a fixed grid in space, as a result there is no direct connection between the model variables and Lagrangian observations that carry time-integrated information. A method is presented for assimilating Lagrangian tracer positions, observed at discrete times, directly into the model. The idea is to augment the model with tracer advection equations and to track the correlations between the flow and the tracers via the extended Kalman filter. The augmented model state vector includes tracer coordinates and is updated through the correlations to the observed tracers. The technique is tested for point vortex flows: an NF point vortex system with a Gaussian noise term is modeled by its deterministic counterpart. Positions of ND tracer particles are observed at regular time intervals and assimilated into the model. Numerical experiments demonstrate successful system tracking for (NF, ND) = (2, 1), (4, 2), provided the observations are reasonably frequent and accurate and the system noise level is not too high. The performance of the filter strongly depends on initial tracer positions (drifter launch locations). Analysis of this dependence shows that the good launch locations are separated from the bad ones by Lagrangian flow structures (separatrices or invariant manifolds of the velocity field). The method is compared to an alternative indirect approach, where the flow velocity, estimated from two (or more) consecutive drifter observations, is assimilated directly into the model.

Point-vortex cluster formation in the plane and on the sphere: an energy bifurcation condition.

The dynamics of a system of point vortices is considered in the plane and on the sphere. Particular attention is given to the formation of vortex clusters and to global vortex dynamics, especially in the spherical case. For integrable systems and systems with given symmetries, we show the existence of a critical energy above or below which (depending on the geometry of the surface) the system splits into clusters and vortex dynamics is confined to a particular region. The case of nonidentical vortices is of particular interest since we observe quite different global dynamics depending on the energy and the initial conditions. Furthermore we identify all the relative equilibria configurations as critical points of the reduced energy and we give an instability criterion to deduce instability for certain configurations.

Geometrical theory of two-dimensional hydrodynamics with special reference to a system of point vortices

Two-dimensional hydrodynamics is formulated geometrically with a metric defined in terms of stream function and vorticity for flows of an inviscid incompressible fluid in unbounded space. Based on this, the vorticity equation is derived as a geodesic equation over a group of diffeomorphisms of a fluid at rest at infinity with vorticity having compact supports, which is also rewritten in the form of a Hamilton's equation with a Lie–Poisson bracket. In particular, this formulation is applied to a flow induced by a finite number ( N) of point vortices. It is shown that the geodesic equation reduces to the well-known Hamiltonian system of N point vortices. This is considered as a dynamics of an internal space of finite degrees of freedom, as against the former background dynamical system described by a group of diffeomorphisms of infinite dimensions and Riemannian geometry. It is shown that the dynamical trajectories of the system of N degrees of freedom governed by Hamiltonian system of equations are the projections of the geodesics of ( N+1)-dimensional Finslerian space associated with a fundamental function obtained from the original Lagrangian of N point vortices. The Finsler geometry provides us curvature tensors and equation of geodesic deviation. As an example, the Finslerian geometry is applied to a system of a vortex pair, in which sum of two vortex strengths vanishes and the pair makes steady translational motion with a constant velocity. The scalar curvature of the motion is found to be zero, and its relation to geodesic deviation is considered.

Rain, power laws, and advection.

Localized rain events have been found to follow power-law size and duration distributions over several decades, suggesting parallels between precipitation and seismic activity [O. Peters, Phys. Rev. Lett. 88, 018701 (2002)]]. Similar power laws are generated by treating rain as a passive tracer undergoing advection in a velocity field generated by a two-dimensional system of point vortices.

Topological fluid mechanics of point vortex motions

Topological techniques are used to study the motions of systems of point vortices in the infinite plane, in singly periodic arrays, and in doubly periodic lattices. Restricting to three vortices with zero net circulation, the symmetries are used to reduce each system to a 1 degree-of-freedom Hamiltonian. The phase portrait of the reduced system is subdivided into regimes using the separatrix motions, and a braid representing the topology of all vortex motions in each regime is computed. This braid also describes the isotopy class of the advection homeomorphism induced by the vortex motion. The Thurston–Nielsen theory is then used to analyze these isotopy classes, and in certain cases strong implications about the chaotic dynamics of the advection can be drawn. This points to an important mechanism by which the topological kinematics of large scale, two-dimensional fluid motions generate chaotic advection.

Identification of regions of fastest mixing in a system of point vortices

Identification of regions of fastest mixing in a system of point vortices

Identification of regions of fastest mixing in a system of point vortices

AbstractThis paper describes a numerical method for efficiently identifying the regions of fastest mixing of a passive dye in a flow due to a system of point vortices. Results obtained from computations are presented for systems of three and four point vortices, both in the unbounded domain and inside a circular cylinder. The flow is two‐dimensional and the fluid is incompressible. The regions where mixing is possible are found by studying the largest Lagrangian Lyapunov exponent distribution with respect to various initial positions of tracer particles. The regions of fastest mixing are then identified from the Lyapunov exponent distribution at small times. The results of the method are verified by quantifying the mixing by using a traditional box counting technique. The technique is then applied to several different initial configurations of vortices and some interesting results are obtained. Some numerical findings about the nature of the exponents computed are also discussed. Copyright © 2002 John Wiley Sons, Ltd.

Local Stretching Maps: an Application for an Advection Problem in an Arbitrary Velocity Field

The local stirring properties of a passive fluid domain with arbitrary borders in known velocity field are discussed. Analytical solution for local stretching permits to single out an exponential coefficient that describes stretching of the domain studied and is analogous to the largest Lyapunov's exponent used in chaotic dynamics. This coefficient exists in all solutions; it does not depend on the shape of the contour, and is determined by the gradients of the velocity field components only. Another local mechanism of stirring is determined by integral characteristics of the flow and the shape of the contour under consideration. Construction of maps for local stretching values in fixed moments allows to analyze informatively an evolution of regions, in which an intensive stirring takes place. The stirring process is explored in a sample of an advection problem of a passive impurity in the velocity field induced by a system of point vortices moved periodically. This interaction regime generates a chaotic motion of passive fluid particles. Local stretching maps show that the regions of chaotic motion of fluid particles and of intensive stirring do not coincide. Chaotic region has a zone of weak stirring, in which contours are transported from one intensive stretching zone to another without any deformation.

Kinetic theory of point vortices: diffusion coefficient and systematic drift.

We develop a kinetic theory for point vortices in two-dimensional hydrodynamics. Using standard projection operator techniques, we derive a Fokker-Planck equation describing the relaxation of a "test" vortex in a bath of "field" vortices at statistical equilibrium. The relaxation is due to the combined effect of a diffusion and a drift. The drift is shown to be responsible for the organization of point vortices at negative temperatures. A description that goes beyond the thermal bath approximation is attempted. A new kinetic equation is obtained which respects all conservation laws of the point vortex system and satisfies a H theorem. Close to equilibrium, this equation reduces to the ordinary Fokker-Planck equation.

Perturbed Motion of a Centrally Symmetric Point Vortex System

The types of motion in a system of point vortices initially located at the vertices of a 2n-sided regular polygon, produced by a spatially periodic perturbation, in the presence of which the vortices are positioned at the corners of two regular concentric n-sided polygons, are described.

Relative equilibria of point vortices on the sphere

We prove the existence of many different symmetry types of relative equilibria for systems of identical point vortices on a non-rotating sphere. The proofs use the rotational symmetry group SO(3) and the resulting conservation laws, the time-reversing reflectional symmetries in O(3), and the finite symmetry group of permutations of identical vortices. Results include both global existence theorems and local results on bifurcations from equilibria. A more detailed study is made of relative equilibria which consist of two parallel rings with n vortices in each rotating about a common axis. The paper ends with discussions of the bifurcation diagrams for systems of 3–6 identical vortices.

On a dynamical system related to fluid mechanics

We introduce a dynamical system strictly related to fluid mechanics and similar to the classical N point vortex system. In the first part we analyze the qualitative behavior of the time evolution and in particular we show the properties of collapse and chaoticity. In the second part of the paper we investigate the relation of the dynamical system with a system of N concentrated large enough smoke rings in an incompressible and inviscid fluid, with axial symmetry and without swirl. We prove the rigorous connection between the two models at time zero for any N. The extension of the same result to any time is obtained only for a smoke ring alone, while for the general case it is just a matter of conjecture.

TRANSPORT BY COHERENT BAROTROPIC VORTICES

▪ Abstract This article reviews the transport properties of coherent vortices in rotating barotropic flows. It is shown that vortices induce regular Lagrangian motion inside their cores and are highly impermeable to inward and outward particle fluxes. Passive tracers can be trapped inside vortex cores for long times and are transported by the vortex motion over large distances. Absolute dispersion in vortex-dominated flows is discussed by studying particle dynamics in 2D turbulence, point-vortex systems, and subsurface float trajectories in the ocean. Finally, it is shown that anticyclonic coherent vortices in cyclonically rotating reference frames can concentrate heavy impurities (e.g. dust grains) in their cores. This process may play an important role in the formation of planetesimals in the early solar nebula.

On stagnation points and streamline topology in vortex flows

The problem of locating stagnation points in the flow produced by a system of N interacting point vortices in two dimensions is considered. The general solution follows from an 1864 theorem by Siebeck, that the stagnation points are the foci of a certain plane curve of class N−1 that has all lines connecting vortices pairwise as tangents. The case N=3, for which Siebeck's curve is a conic, is considered in some detail. It is shown that the classification of the type of conic coincides with the known classification of regimes of motion for the three vortices. A similarity result for the triangular coordinates of the stagnation point in a flow produced by three vortices with sum of strengths zero is found. Using topological arguments the distinct streamline patterns for flow about three vortices are also determined. Partial results are given for two special sets of vortex strengths on the changes between these patterns as the motion evolves. The analysis requires a number of unfamiliar mathematical tools which are explained.

On the Inviscid Limit for a Fluid with a Concentrated Vorticity

We study the time evolution of a viscous incompressible fluid in ℝ2 when the initial vorticity is sharply concentrated in N regions of diameter e. We prove that in the zero viscosity limit it converges as e vanishes to the inviscid fluid and in particular to the point vortex system.

Lagrangian dynamics in high-dimensional point-vortex systems

We study the Lagrangian dynamics of systems of N point vortices and passive particles in a two-dimensional, doubly periodic domain. The probability distribution function of vortex velocity, pN, has a slow-velocity Gaussian component and a significant high-velocity tail caused by close vortex pairs. In the limit for N→∞, pN tends to a Gaussian. However, the form of the single-vortex velocity causes very slow convergence with N; for N≈106 the non-Gaussian high-velocity tails still play a significant role. At finite N, the Gaussian component is well modeled by an Ornstein-Uhlenbeck (OU) stochastic process with variance σN=N ln N/2π. Considering in detail the case N=100, we show that at short times the velocity autocorrelation is dominated by the Gaussian component and displays an exponential decay with a short Lagrangian decorrelation time. The close pairs have a long correlation time and cause nonergodicity over at least the time of the integration. Due to close vortex dipoles the absolute dispersion differs significantly from the OU prediction, and shows evidence of long-time anomalous dispersion. We discuss the mathematical form of a new stochastic model for the Lagrangian dynamics, consisting of an OU model combined with long-lived close same-sign vortices engaged in rapid rotation and long-lived close dipoles engaged in ballistic motion. From a dynamical-systems perspective this work indicates that systems of dimension O(100) can have behavior which is a combination of both low-dimensional behavior, i.e., close pairs, and extremely high-dimensional behavior described by traditional stochastic processes.

Extended Kalman filtering for vortex systems. Part II: Rankine vortices and observing-system design

Abstract Point-vortex systems with both regular and chaotic motion can be tracked by a combination of Lagrangian observations of vortex position and of Eulerian observations of fluid velocity at a few fixed points. If only Eulerian observations are available, singularities in the tracking can develop due to nonlinearity in the observing functions on two occasions: when point vortices pass very close to an observing station, and when the observed speed is small. In Part II of this two-part paper, the vorticity concentrations are approximated by Rankine vortices with finite core in solid-body rotation, while a cut-off criterion is imposed on the Eulerian observations; this removes the singularities and hence results in satisfactory tracking when the observations are judiciously chosen. The main result is that a number of observations comparable to that of the distinct vorticity concentrations suffices for very accurate tracking. The key elements for successful tracking of Rankine-vortex systems using nonlinear, Eulerian observations are: distribution of available stations, threshold criteria for the selection of stations used in each update, and the observing frequency. Simple analysis provides the optimal estimate for these key elements, together with the optimal size of the estimated Rankine vortices.

Extended Kalman filtering for vortex systems. Part 1: Methodology and point vortices

Planetary flows-atmospheric and oceanic-are approximately two-dimensional and dominated by coherent concentrations of vorticity. Data assimilation attempts to determine optimally the current state of a fluid system from a limited number of current and past observations. In this two-part paper, an advanced method of data assimilation, the extended Kalman filter, is applied to the Lagrangian representation of a two-dimensional flow in terms of vortex systems. Smaller scales of motion are approximated here by stochastic forcing of the vortices. In Part I, the systems studied have either two point vortices, leading to regular motion or four point vortices and chaotic motion, in the absence of stochastic forcing. Numerical experiments are performed in the presence or absence of stochastic forcing. Point-vortex systems with both regular and chaotic motion can be tracked by a combination of Lagrangian observations of vortex positions and of Eulerian observations of fluid velocity at a few fixed points. Dynamically, the usual extended Kalman filter tends to yield insufficient gain if stochastic forcing is absent, whether the underlying system is regular or chaotic. Statistically, the type and accuracy of observations are the key factors in achieving a sufficiently accurate flow description. A simple analysis of the update mechanism supports the numerical results and also provides geometrical insight into them. In Part II, tracking of Rankine vortices with a finite core area is investigated and the results are used for observing-system design.

Vortex dynamics in the studies of looping in tropical cyclone tracks

This paper proposes criteria for predicting the tendency of looping in tropical cyclone tracks using the approach of vortex dynamics. We model the asymmetric structure of a cyclone by a system of vortex patches. The evolution of such system of vortices is simulated by the method of contour dynamics. A new set of exact analytic formulas for contour dynamics calculations is derived, which is shown to be more computationally effective. Based on point-vortex models, we derive analytic formulas for the criteria of looping in a cyclone track. From numerical experiments, the simulated trajectories obtained from the point-vortex system and vortex patch system agree quite well. Hence, the looping criteria obtained from the point-vortex system can be applied by forecasters to stay alert for tendency of looping in a cyclone track. To demonstrate the applicability of the proposed criteria, the trajectory of Typhoon Yancy (9012), whose field data are available from ‘TCM-90’, is simulated. The case study shows that the asymmetric structure similar to the pattern of a beta gyre is responsible for its recurvature when Yancy landed Fujian Province, China on 20 August 1990.