Point Vortex Models 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.

Relative dynamics of quantum vortices and massive cores in binary BECs

We study the motion of superfluid vortices with filled massive cores. Previous point-vortex models already pointed out the impact of the core mass on the vortex dynamical properties, but relied on an assumption that is questionable in many physical systems where the immiscibility condition is barely satisfied: the fact that the massive core always lays at the very bottom of the effective confining potential constituted by the hosting vortex. Here, we relax this assumption and present a new point-vortex model where quantum vortices are harmonically coupled to their massive cores. We thoroughly explore the new dynamical regimes offered by this improved model; we then show that the functional dependence of the system normal modes on the microscopic parameters can be correctly interpreted only within this new generalized framework. Our predictions are benchmarked against the numerical simulations of coupled Gross–Pitaevskii equations for a realistic mixture of atomic Bose–Einstein condensates.

Gaussian fluctuations around limit measures of generalized SQG point vortices

AbstractWe report on recent results on generalized surface quasi‐geostropic point vortex models. The statistical physics of these models is particularly interesting, as their mean‐field limit is a steady solution of the gSQG PDE. We present a central limit theorem‐type result for gSQG point votex models and make a comparison with results for Euler point vortex models.

Vortex collapses for the Euler and Quasi-Geostrophic models

<p style='text-indent:20px;'>This article studies point-vortex models for the Euler and surface quasi-geostrophic equations. In the case of an inviscid fluid with planar motion, the point-vortex model gives account of dynamics where the vorticity profile is sharply concentrated around some points and approximated by Dirac masses. This article contains two main theorems and also smaller propositions with several links between each other. The first main result focuses on the Euler point-vortex model, and under the non-neutral cluster hypothesis we prove a convergence result. The second result is devoted to the generalization of a classical result by Marchioro and Pulvirenti concerning the improbability of collapses and the extension of this result to the quasi-geostrophic case.</p>

Velocity statistics for point vortices in the local α-models of turbulence

ABSTRACTThe velocity fluctuations for point vortex models are studied for the α-turbulence equations, which are characterised by a fractional Laplacian relation between active scalar and the streamfunction. In particular, we focus on the local dynamics regime. The local dynamics differ from the well-studied case of 2D turbulence as it allows to consider the true thermodynamic limit, that is, to consider an infinite set of point vortices on an infinite plane keeping the density of the vortices constant. The consequence of this limit is that the results obtained are independent on the number of point vortices in the system. This limit is not defined for 2D turbulence. We show an analytical form of the probability density distribution of the velocity fluctuations for different degrees of locality. The central region of the distribution is not Gaussian, in contrast to the case of turbulence, but can be approximated with a Gaussian function in the small velocity limit. The tails of the distribution exhibit a power law behaviour and self similarity in terms of the density variable. Due to the thermodynamic limit, both the Gaussian approximation for the core and the steepness of the tails are independent on the number of point vortices, but depend on the α-model. We also show the connection between the velocity statistics for point vortices uniformly distributed, in the context of the α-model in classical turbulence, with the velocity statistics for point vortices non-uniformly distributed. Since the exponent of the power law depends just on α, we test the power law approximation obtained with the point vortex approximation, by simulating full turbulent fields for different values of α and we compute the correspondent probability density distribution for the absolute value of the velocity field. These results suggest that the local nature of the turbulent fluctuations in the ocean or in the atmosphere might be deduced from the shape of the tails of the probability density functions.

Influence of vortex street structure on the efficiency of energy separation

Regions with reduced and increased values of total enthalpy are observed in a time-averaged flow behind a bluff body. This energy redistribution takes place both in the vortex formation region and in the developed vortex wake. The present paper focuses on studying the effect of the structure of a vortex street on the intensity of energy redistribution. Two approaches are used. The first one is direct numerical simulation of the flow behind a transversely oscillating cylinder, which is known for a variety of vortex patterns in the wake. The simulations are based on a finite element solution of the Navier-Stokes equations for a compressible perfect viscous gas. The second approach is based on simplified point vortex models for infinite periodic vortex streets, which contain a finite number of vortex chains in equilibrium. It turns out that these simple models make it possible to obtain satisfactory qualitative results, particularly if a more precise approximation of velocity fields in the vortex cores (Rankine vortices) is implemented. It is shown that the effect of energy redistribution significantly depends on the vortex structure, namely the mutual arrangement of the vortices and their intensities. The estimates of the energy separation efficiency in the time-averaged flow are obtained for the general case of an arbitrary number of chains. A more detailed analysis is performed for vortex streets with 2, 3, and 4 vortex chains.

Vortex Interactions Subjected to Deformation Flows: A Review

Deformation flows are the flows incorporating shear, strain and rotational components. These flows are ubiquitous in the geophysical flows, such as the ocean and atmosphere. They appear near almost any salience, such as isolated coherent structures (vortices and jets) and various fixed obstacles (submerged obstacles and continental boundaries). Fluid structures subject to such deformation flows may exhibit drastic changes in motion. In this review paper, we focus on the motion of a small number of coherent vortices embedded in deformation flows. Problems involving isolated one and two vortices are addressed. When considering a single-vortex problem, the main focus is on the evolution of the vortex boundary and its influence on the passive scalar motion. Two vortex problems are addressed with the use of point vortex models, and the resulting stirring patterns of neighbouring scalars are studied by a combination of numerical and analytical methods from the dynamical system theory. Many dynamical effects are reviewed with emphasis on the emergence of chaotic motion of the vortex phase trajectories and the scalars in their immediate vicinity.

Collapse of generalized Euler and surface quasigeostrophic point vortices.

Point-vortex models are presented for the generalized Euler equations, which are characterized by a fractional Laplacian relation between the active scalar and the stream function. Special focus is given to the case of the surface quasigeostrophic (SQG) equations, for which the existence of finite-time singularities is still a matter of debate. Point-vortex trajectories are expressed using Nambu dynamics. The formulation is based on a noncanonical bracket and allows for a geometrical interpretation of trajectories as intersections of level sets of the Hamiltonian and Casimir. Within this setting, we focus on the collapse of solutions for the three-point-vortex model. In particular, we show that for SQG the collapse can be either self-similar or non-self-similar. Self-similarity occurs only when the Hamiltonian is zero, while non-self-similarity appears for nonzero values of the same. For both cases, collapse is allowed for any choice of circulations within a permitted interval. These results differ strikingly from the classical point-vortex model, where collapse is self-similar for any value of the Hamiltonian, but the vortex circulations must satisfy a strict relationship. Results may also shed a light on the formation of singularities in the SQG partial differential equations, where the singularity is thought to be reached only in a self-similar way.

Explicit, parallel Poisson integration of point vortices on the sphere

Point vortex models are frequently encountered in conceptual studies in geophysical fluid dynamics, but also in practical applications, for instance, in aeronautics. In spherical geometry, the motion of vortex centres is governed by a dynamical system with a known Poisson structure. We construct Poisson integration methods for these dynamics by splitting the Hamiltonian into its constituent vortex pair terms. From backward error analysis, the method is formally known to provide solutions to a modified Poisson system with the correct bracket, but with a modified Hamiltonian function. Different orderings of the pairwise interactions are considered and also used for the construction of higher order methods. The energy and momentum conservation of the splitting schemes is demonstrated for several test cases. For particular orderings of the pairwise interactions, the schemes allow scalable parallelization. This results in a linear–as opposed to quadratic–scaling of computation time with system size when scaling the number of processors accordingly.

Improving vortex models via optimal control theory

Low-order inviscid point vortex models have demonstrated success in capturing the qualitative behavior of aerodynamic forces resulting from unsteady lifting surface maneuvers. However, the quantitative agreement is often lacking for separated flows as a result of the ambiguity in the edge conditions in this fundamentally unsteady process. In this work, we develop a model reduction framework in which such models can be systematically improved with empirical results. We consider the low-order impulse matching vortex model in which, in its original form, Kutta conditions are applied at both edges to determine the strengths of single point vortices shed from each edge. Here, we relax the Kutta condition imposed at the plate׳s edges and instead seek the time rate of change of the vortex strengths that minimize the discrepancy between the model-predicted and high-fidelity simulation force histories, while the vortex positions adhere to the dynamics of the low-order model. A constrained minimization problem is constructed within an optimal control framework and solved by means of variational principles. The optimization approach is demonstrated on several unsteady wing maneuvers, including pitch-up and impulsive translation at a fixed angle of attack. Additionally, a stitching technique is introduced for extending the time interval over which the model is optimized.

Discretization of the vorticity field of a planar jet

In data assimilation, information from sensors is used to correct the state variables of a numerical model. This has been used to great advantage by the weather prediction community in the context of direct numerical simulation (DNS) models, but has seen comparatively little use in point-vortex models. This is due in large part to data-processing issues. In order to keep up with the speeds necessary for effective data assimilation, one must extract and discretize the vortex structures from velocity field data in a computationally efficient fashion—i.e., using as few discrete vortices as possible to model the measured flow. This paper describes a new strategy for accomplishing this and evaluates the results using data from a laboratory-scale vortex-dominated planar jet. Large-scale vortex structures are found using a family of variants on traditional vortex extraction methods. By augmenting these methods with simple computational topology techniques, one obtains a new method that finds the boundaries of the coherent structures in a manner that naturally follows the geometry of the flow. This strategy was evaluated in the context of two standard vortex extraction methods, vorticity thresholding and Okubo–Weiss, and tested upon velocity field data from the experimental fluid flow. The large-scale structures found in this manner were then modeled with collections of discrete vortices, and the effects of the grain size of the discretization and the parameters of the discrete vortex model were studied. The results were evaluated by comparing the instantaneous velocity field induced by the discrete vortices to that measured in the jet. These comparisons showed that the two extraction techniques were comparable in terms of sensitivity and error, suggesting that the computationally simpler vorticity thresholding method is more appropriate for applications where speed is an issue, like data assimilation. Comparisons of different discretization strategies showed that modeling each large-scale vortex structure with a single discrete vortex provided the best compromise between mean-squared error and computational effort. These results are of potential interest in any situation where one must balance accuracy and expense while extracting vortices from a snapshot of a flow field; data assimilation is only one example.

Dissipative N-point-vortex Models in the Plane

A method is presented for constructing point vortex models in the plane that dissipate the Hamiltonian function at any prescribed rate and yet conserve the level sets of the invariants of the Hamiltonian model arising from the SE (2) symmetries. The method is purely geometric in that it uses the level sets of the Hamiltonian and the invariants to construct the dissipative field and is based on elementary classical geometry in ℝ3. Extension to higher-dimensional spaces, such as the point vortex phase space, is done using exterior algebra. The method is in fact general enough to apply to any smooth finite-dimensional system with conserved quantities, and, for certain special cases, the dissipative vector field constructed can be associated with an appropriately defined double Nambu–Poisson bracket. The most interesting feature of this method is that it allows for an infinite sequence of such dissipative vector fields to be constructed by repeated application of a symmetric linear operator (matrix) at each point of the intersection of the level sets.

Explicit mean-field radius for nearly parallel vortex filaments in statistical equilibrium with applications to deep ocean convection

Deep ocean convection, under appropriate conditions, gives rise to quasi-2D vortex structures with axes parallel to the rotational axis. These vortex structures appear in axisymmetric arrays that have a characteristic radius or size. This size is dependent, not only on competition between vortex interaction and conservation of angular momentum, but on 3D effects that 2D point vortex models leave out. In this article, we propose a hypothesis that as 3D variations become more significant in these arrays, the process of interaction/angular momentum competition gives way to entropy/angular momentum competition and that this shift results in a reversal of the trend of the radius to decrease with increasing kinetic energy. We derive an explicit, closed-form, mean-field expression for the radius using a quasi-2D model for filaments with a local induction approximation (LIA). We validate the formula with Monte Carlo simulations. Both confirm that there is a reversal in the 2D point vortex contraction trend. We conclude that the proposed shift in competition does happen and that this simple LIA model is sufficient to show it.

Continuous parametric families of stationary and translating periodic point vortex configurations

The number of periodic arrangements of point vortices – point vortex streets – in two-dimensional fluid flow that are stationary is known to be finite for a generic choice of vortex circulations. When all circulations are the same in absolute value, however, stationary vortex street configurations have been associated with the zeros of certain trigonometric polynomials containing free complex parameters. The presence of these parameters may prove useful in constructing point vortex models of shear layers and wakes. In this paper it is shown that such a continuum of stationary configurations exists in a much wider class of point vortex street systems. The circulations may take on many values, not just two, providing increased flexibility in the modelling context. A simple method for computing these configurations is derived. The effects of symmetries on the solution polynomials are described, and illustrated with examples. In addition, novel translating vortex street configurations are found having arbitrary translation velocity and containing free parameters for vortex circulations ±1 and also for vortex circulations +1, −2.

Convection in Small Basins

Abstract Convection in the world's oceans often occurs in small, semienclosed basins where bottom slopes and nearby continental shelf breaks are commonplace. The evolution of convectively generated heat anomalies in such settings is studied using quasigeostrophic finite-difference and point vortex models. The displayed behaviors divide essentially into two categories: whole fluid column convection, in which bottom-slope effects are felt immediately, and partial fluid column convection, in which the topographic effects can be delayed. In both cases, topography significantly modifies the evolution of convective patches from that occurring over flat bottoms. Vertical walls induce strong self-propagation mechanisms that accelerate alongslope heat transport, while the continental shelf slope is repulsive and rejects lower-layer anticyclones. These anomalies are then “stranded,” being too far offshore to interact with the shelf break and having lost their heton partner in the interaction. Weaker deep ocean topo...

Vortex Motion and the Geometric Phase. Part II. Slowly Varying Spiral Structures

We derive formulas for the long time evolution of passive interfaces in three “canonical” incompressible, inviscid, two-dimensional flow models. The point vortex models, introduced in Part 1 [1] are (i) a “restricted” three-vortex problem, (ii) a vortex and a particle in a closed circular domain, and (iii) a particle in the flowfield of a mixing layer model undergoing a vortex pairing instability. In each configuration, it was shown in Part 1 that the passive particle exhibits a geometric or Hannay-Berry phase over long time periods induced by the slowly varying periodic background field. In this paper we show how the formula for the evolution of a passive interface driven by the dynamics of the vortices inherits this geometric phase effect. The interface wraps into a spiral formation around the “parent” vortex, with a slowly varying component induced by the farfield vorticity. The length formula for the long time growth of the slowly rotating spiral decomposes into a “dynamic” part and a “geometric” part. The dynamic part is the length in the “unperturbed” system—i.e., in the absence of the background field—and the geometric part is the contribution of the geometric phase θg for a passive particle in the flow. We derive the following simple formula for an interface along a smooth curve joining two arbitrary particles labelled A and B. Define ΔL(T) as the difference in interface lengths between the “unperturbed” system and the “perturbed” slowly varying system at the end of the long time period T. In each case, \(\Delta L = \lim _{T \to \infty } \Delta L\left( T \right) = - \int_{\xi _A }^{\xi _B } {d\left( {\xi \theta _g } \right)}\), where ξ is the radial coordinate parametrizing the interface at t = 0.

The evolution of a layer of electron vortices produced by the interaction of an intense laser pulse and plasma

The instability of the large-scale motion of a thin layer of electron vortices produced by the interaction of intense laser pulse and a uniform plasma is studied here. The standard point vortex models and vortex sheet models of the flow field involve unphysical singularities in flow field, leading to ill-posed problems. To make the problem well posed a description in terms of vortex blobs is considered. If the vortex sheet has a smooth localized core then the linear growth rates rapidly decrease to zero at high wave numbers, thus providing a natural ultraviolet cutoff. Such a model can then be further used to study the development of small-scale inhomogeneity in the magnetic field produced by the ultraintense laser beam.

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.

Stable and unstable monopolar vortices in a stratified fluid

This paper presents experiments on planar monopolar vortex structures generated in a non-rotating, stratified fluid. In order to study the dynamics of such planar vortices in the laboratory, angular momentum was generated in a specific horizontal layer of the stratified fluid, by using three different generation mechanisms. The lens-shaped monopolar vortices thus created were in some cases stable and conserved their circular symmetry, while in other cases they appeared to be unstable, leading to the formation of a multipoled vortex with a different topology. Characteristics such as cross-sectional profiles (angular velocity and vorticity) and vorticity-stream function scatter plots have been measured experimentally by using digital image processing techniques. The characteristics of the monopolar vortices are compared with analytical vortex models known from literature. Simple models, based on vertical diffusion of vorticity, are proposed to describe the monopolar vortex decay; they show reasonable agreement with the experimental results.From the multipolar structures, the tripolar vortex and a specific case of a triangular vortex, neither having been observed before in a stratified fluid, are studied in detail. A comparison with point-vortex models yields good agreement. Although these multipolar vortices appear to persist for a long while, they are found eventually to be unstable and to transform into a monopolar vortex.

Chaotic advection in point vortex models and two-dimensional turbulence

The dynamics of passively advected particles in either integrable or chaotic point vortex systems and in two-dimensional (2-D) turbulence is studied. For point vortices, it is shown that the regular or chaotic nature of the particle trajectories is not determined by the Eulerian chaoticity of the vortex motion, but rather by pure Lagrangian quantities, such as the distance of an advected particle from the vortex centers. In fact, each point vortex turns out to be surrounded by a regular island, where the advected particles are trapped and their Lagrangian Lyapunov exponent is zero, even though the vortex itself may perform a chaotic trajectory. In the field between the vortices, passive particles undergo chaotic advection with an associated positive Lyapunov exponent. For well-separated vortices, even at large times, the advected particles do not cross the boundary between the chaotic sea and the regular islands surrounding the vortices. A similar situation holds in the case of forced-dissipative 2-D turbulence, where particles trapped in the interior of the coherent structures have a null Lagrangian Lyapunov exponent, while those in the background turbulent sea move chaotically. This gives clear evidence of the important role played by chaotic advection, even in complex Eulerian flows.