Point vortex dynamics: A classical mathematics playground

Aref, Hassan

doi:10.1063/1.2425103

Cited by 199 publications

(219 citation statements)

References 44 publications

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“…Twenty years later, Kirchhoff presented the general Hamiltonian structure of N point vortices and Gro¨bli analysed in detail the motion of three-point vortices (Kirchhoff, 1876;Gro¨bli, 1877). Since then, numerous papers have been published on the motion of point vortices, see for example the works of Novikov (1975), Newton (2013) or Aref (2007). In the following, let v be a solenoidal vector field, that is, 9×v00, and denote v09)v the vorticity vector.…”

Section: Point Vortex Theorymentioning

confidence: 99%

Applications of point vortex equilibria: blocking events and the stability of the polar vortex

Müller¹,

Névir²,

Schielicke³

et al. 2015

Tellus A: Dynamic Meteorology and Oceanography

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A B S T R A C TThe present study investigates non-linear dynamics of atmospheric flow phenomena on different scales as interactions of vortices. Thereby, we apply the idealised, two-dimensional concept of point vortices considering two important issues in atmospheric dynamics. First, we propose this not widely spread concept in meteorology to explain blocked weather situations using a three-point vortex equilibrium. Here, a steady state is given if the zonal mean flow is identical to the opposed translational velocity of the vortex system. We apply this concept exemplarily to two major blocked events establishing a new pattern recognition technique based on the kinematic vorticity number to determine the circulations and positions of the interacting vortices. By using reanalysis data, we demonstrate that the velocity of the tripole in a westward direction is almost equal to the westerly flow explaining the steady state of blocked events. Second, we introduce a novel idea to transfer a stability analysis of a vortex equilibrium to the stability of the polar vortex concerning its interaction with the quasi-biennial oscillation (QBO). Here, the point vortex system is built as a polygon ring of vortices around a central vortex. On this way we confirm observations that perturbations of the polar vortex during the QBO east phase lead to instability, whereas the polar vortex remains stable in QBO west phases. Thus, by applying point vortex theory to challenging problems in atmospheric dynamics we show an alternative, discrete view of synoptic and planetary scale motion.

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Section: Point Vortex Theorymentioning

confidence: 99%

Applications of point vortex equilibria: blocking events and the stability of the polar vortex

Müller¹,

Névir²,

Schielicke³

et al. 2015

Tellus A: Dynamic Meteorology and Oceanography

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“…To end this section, a qualitative study of the convection of passive point vortices in an ideal incompressible fluid is presented [164][165][166][167]. A system of N point vortices or tracers is described by a singular vorticity field …”

Section: Vortex Dynamicmentioning

confidence: 99%

A review of some geometric integrators

Razafindralandy

Hamdouni

Chhay

2018

Adv. Model. and Simul. in Eng. Sci.

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Some of the most important geometric integrators for both ordinary and partial differential equations are reviewed and illustrated with examples in mechanics. The class of Hamiltonian differential systems is recalled and its symplectic structure is highlighted. The associated natural geometric integrators, known as symplectic integrators, are then presented. In particular, their ability to numerically reproduce first integrals with a bounded error over a long time interval is shown. The extension to partial differential Hamiltonian systems and to multisymplectic integrators is presented afterwards. Next, the class of Lagrangian systems is described. It is highlighted that the variational structure carries both the dynamics (Euler-Lagrange equations) and the conservation laws (Noether's theorem). Integrators preserving the variational structure are constructed by mimicking the calculus of variation at the discrete level. We show that this approach leads to numerical schemes which preserve exactly the energy of the system. After that, the Lie group of local symmetries of partial differential equations is recalled. A construction of Lie-symmetry-preserving numerical scheme is then exposed. This is done via the moving frame method. Applications to Burgers equation are shown. The last part is devoted to the Discrete Exterior Calculus, which is a structure-preserving integrator based on differential geometry and exterior calculus. The efficiency of the approach is demonstrated on fluid flow problems with a passive scalar advection.

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“…16 Our proof of the tripole stability will be based on the invariance of the quantities H, P x , and P y that, to within constant factors, can be written as H ¼ Hðr AB ; r AC ; r BC Þ ¼ 2 ln r AB þ 2 ln r AC À ln r BC ; (19)…”

Section: Unperturbed Circulationsmentioning

confidence: 99%

Stability of point-vortex multipoles revisited

Kizner

2011

Physics of Fluids

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The point-vortex tripoles and pentapoles with zero total circulation are considered in the rigid-lid barotropic, equivalent-barotropic, and quasigeostrophic two-layer models. A tripole is assembled by three symmetrically arranged collinear vortices, while a pentapole by five vortices, of which one is located at the center of a square and four in the vertices of the square. The vortices on the sides, termed satellite vortices, are equal in strength and opposite in sign to the central vortex. To fulfill the zero-total-circulation condition, the central vortex is taken to be twice as strong as each of the satellite vortices in a tripole and four times as strong in a pentapole. In the two-layer model, two cases are distinguished, namely, the flat multipoles whose vortices are all located in the same layer and the carousel multipoles whose central vortex and satellite vortices reside in different layers. In all the models, the tripoles are shown to be nonlinearly stable and pentapoles, generally, unstable. Carousel pentapoles comparable in their size with the Rossby radius, and smaller, are exceptional in that they are stable to centrally symmetric perturbations (and, presumably, to arbitrary perturbations). The simple proof of the tripole stability is based on the fact that among the possible three-vortex configurations with zero total circulation characterized by the same (fixed) value of the Hamiltonian, there exists only one tripole, and, within the iso-Hamiltonian sheet, the squared linear momentum vanishes at this unique tripole only. This approach, being in essence universal for all models, works only with tripoles. For instance, a quadrupole cannot be treated in such a way, because there is a continuum of configurations of four vortices with zero total circulation on which the squared impulse vanishes. Dealing with pentapoles, we consider the perturbations that do not violate the central symmetry of the vortex configuration, fix the angular momentum, and examine the second derivatives of the Hamiltonian on the iso-momentum sheet.

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Point vortex dynamics: A classical mathematics playground

Cited by 199 publications

References 44 publications

Applications of point vortex equilibria: blocking events and the stability of the polar vortex

Applications of point vortex equilibria: blocking events and the stability of the polar vortex

A review of some geometric integrators

Stability of point-vortex multipoles revisited

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