Multipole Vortex Blobs (MVB): Symplectic Geometry and Dynamics

Holm, Darryl D.; Jacobs, Henry O.

doi:10.1007/s00332-017-9367-4

Cited by 6 publications

(6 citation statements)

References 45 publications

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“…See Theorem 8.1. The Kimura conjecture and other matters related to dipole motion have been discussed also in Smith (2011), Smith and Nagem (2013), Rodrigues et al (2018), Koiller (2020), Krishnamurthy et al (2021), and from slightly different points of view in Chorin (1973), Kulik et al (2010), Jacobs (2017), andCawte et al (2019). It may be remarked that Kimura's conjecture is counterintuitive.…”

Section: Introductionmentioning

confidence: 88%

Vortex Pairs and Dipoles on Closed Surfaces

Gustafsson

2022

J Nonlinear Sci

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We set up general equations of motion for point vortex systems on closed Riemannian surfaces, allowing for the case that the sum of vorticities is not zero and there hence must be counter-vorticity present. The dynamics of global circulations which is coupled to the dynamics of the vortices is carefully taken into account. Much emphasis is put to the study of vortex pairs, having the Kimura conjecture in focus. This says that vortex pairs move, in the dipole limit, along geodesic curves, and proofs for it have previously been given by S. Boatto and J. Koiller by using Gaussian geodesic coordinates. In the present paper, we reach the same conclusion by following a slightly different route, leading directly to the geodesic equation with a reparametrized time variable. In a final section, we explain how vortex motion in planar domains can be seen as a special case of vortex motion on closed surfaces.

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Section: Introductionmentioning

confidence: 88%

Vortex Pairs and Dipoles on Closed Surfaces

Gustafsson

2022

J Nonlinear Sci

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“…Here, we evaluate v h (z, t) using ( 10) after obtaining the final position of each vortex by solving (13) via the conservative schemes. Also, recall that the exact solution v (z, t) to (3a) − (3e) is available from (25). We numerically approximate the double integral over Ω(T ) in ( 28) via an 8 th order Gaussian quadrature in polar coordinates.…”

Section: Comparison Of Theoretical and Numerical Spatial Convergencementioning

confidence: 99%

“…A brief survey of different vortex methods in the literature can be found in [24]. Also, a more recent vortex method based on a new singular vortex theory for regularized Euler fluid equations of ideal incompressible flow in the plane can be found in [25].…”

Section: Introductionmentioning

confidence: 99%

Conservative Integrators for Vortex Blob Methods

Gormezano,

Nave,

Wan

2021

Preprint

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Conservative symmetric second-order one-step integrators are derived using the Discrete Multiplier Method for a family of vortex-blob models approximating the incompressible Euler's equations on the plane. Conservative properties and second order convergence are proved. A rational function approximation was used to approximate the exponential integral that appears in the Hamiltonian. Numerical experiments are shown to verify the conservative property of these integrators, their second-order accuracy, and as well as the resulting spatial and temporal accuracy of the vortex blob method. Moreover, the derived implicit conservative integrators are shown to be better at preserving conserved quantities than standard higher-order explicit integrators on comparable computation times.

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“…In contrast, point vortices in Euler flow define a symplectic momentum map[36] which also generalises to higher-order derivatives,[24,13,12].…”

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confidence: 99%

Nonlinear dispersion in wave-current interactions

Holm

Ruiao

2022

JGM

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Via a sequence of approximations of the Lagrangian in Hamilton's principle for dispersive nonlinear gravity waves we derive a hierarchy of Hamiltonian models for describing wave-current interaction (WCI) in nonlinear dispersive wave dynamics on free surfaces. A subclass of these WCI Hamiltonians admits emergent singular solutions for certain initial conditions. These singular solutions are identified with a singular momentum map for left action of the diffeomorphisms on a semidirect-product Lie algebra. This semidirect-product Lie algebra comprises vector fields representing horizontal current velocity acting on scalar functions representing wave elevation. We use computational simulations to demonstrate the dynamical interactions of the emergent wavefront trains which are admitted by this special subclass of Hamiltonians for a variety of initial conditions.In this paper, we investigate:(1) A variety of localised initial current configurations in still water whose subsequent propagation generates surface-elevation dynamics on an initially flat surface; and(2) The release of initially confined configurations of surface elevation in still water that generate dynamically interacting fronts of localised currents and wave trains. The results of these simulations show intricate wave-current interaction patterns whose structures are similar to those seen, for example, in Synthetic Aperture Radar (SAR) images taken from the space shuttle.

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Multipole Vortex Blobs (MVB): Symplectic Geometry and Dynamics

Cited by 6 publications

References 45 publications

Vortex Pairs and Dipoles on Closed Surfaces

Vortex Pairs and Dipoles on Closed Surfaces

Conservative Integrators for Vortex Blob Methods

Nonlinear dispersion in wave-current interactions

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