Group scattering of point vortices on an unbounded plane

Makarov, Viatcheslav Georgievich

doi:10.1017/jfm.2020.1035

Cited by 3 publications

(2 citation statements)

References 42 publications

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“…Starting with Gröbli [10], studies of the general three-vortex problem [1,6,7,14,16,18] have focused on formulating the equations of motion in terms of the inter-vortex separation lengths,…”

Section: The Classic Formulationmentioning

confidence: 99%

“…G j ln sin 2 θ j , (3.7) where ρ = R/R 0 is the circle radius normalized by the initial radius, R 0 , and H 0 = 1 2π γ 2 ln(2R 0 ). It has been known since Gröbli's analysis [6,10] that the special case L = 0, g 1 = 0 corresponds to self-similar motion of the vortex triangle for which the interior angles θ j are each constant in time, and numerous authors have investigated this problem; see, for example, [1,11,14,17,18]. If L = 0, then Eq.…”

Section: Constants Of Motionmentioning

confidence: 99%

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Something Old, Something New: Three Point Vortices on the Plane

Stremler

2021

Regul. Chaot. Dyn.

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The classic problem of three point vortex motion on the plane is revisited by using the interior angles of the vortex triangle, θ j , j = 1, 2, 3, as the key system variables instead of the lengths of the triangle sides, s j , as has been used classically. Similar to the classic approach, the relative vortex motion can be represented in a phase space, with the topology of the level curves characterizing the motion. In contrast to the classic approach, the alternate formulation gives a compact, consistent phase space representation and facilitates comparisons of vortex motion in a co-moving frame. This alternate formulation is used to explore the vortex behavior in the two canonical cases of equal vortex strength magnitudes, Γ

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Section: The Classic Formulationmentioning

confidence: 99%

Section: Constants Of Motionmentioning

confidence: 99%

Something Old, Something New: Three Point Vortices on the Plane

Stremler

2021

Regul. Chaot. Dyn.

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Exploring the free-energy landscape of a rotating superfluid

Cleary,

Page

2023

Chaos: An Interdisciplinary Journal of Nonlinear Science

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The equilibrium state of a superfluid in a rotating cylindrical vessel is a vortex crystal—an array of vortex lines, which is stationary in the rotating frame. Experimental realizations of this behavior typically show a sequence of transient states before the free-energy-minimizing configuration is reached. Motivated by these observations, we construct a new method for a systematic exploration of the free-energy landscape via gradient-based optimization of a scalar loss function. Our approach is inspired by the pioneering numerical work of Campbell and Ziff [Phys. Rev. B. 20, 1886 (1979)] and makes use of automatic differentiation, which crucially allows us to include entire solution trajectories in the loss. We first use the method to converge thousands of low free-energy relative equilibria in the unbounded domain for vortex numbers in the range 10≤N≤30, which reveals an extremely dense set of mostly saddle-like solutions. As part of this search, we discover new continuous families of relative equilibria, which are often global minimizers of free energy. These continuous families all consist of crystals arranged in a double-ring configuration, and we assess which state from the family is most likely to be observed experimentally by computing energy-minimizing pathways from nearby local minima—identifying a common entry point into the family. The continuous families become discrete sets of equal-energy solutions when the wall is introduced in the problem. Finally, we develop an approach to compute homoclinic orbits and use it to examine the dynamics in the vicinity of the minimizing state by converging connections for low-energy saddles.

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A new canonical reduction of three-vortex motion and its application to vortex-dipole scattering

Anurag,

Goodman,

O'Grady

2024

Physics of Fluids

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We introduce a new reduction of the motion of three point vortices in a two-dimensional ideal fluid. This proceeds in two stages: a change of variables to Jacobi coordinates and then a Nambu reduction. The new coordinates demonstrate that the dynamics evolve on a two-dimensional manifold whose topology depends on the sign of a parameter κ2 that arises in the reduction. For κ2>0, the phase space is spherical, while for κ2<0, the dynamics are confined to the upper sheet of a two-sheeted hyperboloid. We contrast this reduction with earlier reduced systems derived by Gröbli, Aref, and others in which the dynamics are determined from the pairwise distances between the vortices. The new coordinate system overcomes two related shortcomings of Gröbli's reduction that have made understanding the dynamics difficult: their lack of a standard phase plane and their singularity at all configurations in which the vortices are collinear. We apply this to two canonical problems. We first discuss the dynamics of three identical vortices and then consider the scattering of a propagating dipole by a stationary vortex. We show that the points dividing direct and exchange scattering solutions correspond to the locations of the invariant manifolds of equilibria of the reduced equations and relate changes in the scattering diagram as the circulation of one vortex is varied to bifurcations of these equilibria.

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Group scattering of point vortices on an unbounded plane

Abstract: Abstract

Cited by 3 publications

References 42 publications

Something Old, Something New: Three Point Vortices on the Plane

Something Old, Something New: Three Point Vortices on the Plane

Exploring the free-energy landscape of a rotating superfluid

A new canonical reduction of three-vortex motion and its application to vortex-dipole scattering

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