A jetlet hierarchy for ideal fluid dynamics

Cotter, Colin J.; Holm, Darryl D.; Jacobs, Henry O.; Meier, David M.

doi:10.1088/1751-8113/47/35/352001

Cited by 7 publications

(12 citation statements)

References 12 publications

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“…Then, we analyze the asymptotic dynamics of this merged state and show that it coincides with the dynamics of a single 1-jetlet particle. This improves the claim in [CHJM14] which showed the convergence to a 1-jetlet state without explicitly considering the dynamics.…”

Section: Introductionsupporting

confidence: 76%

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Weak Dual Pairs and Jetlet Methods for Ideal Incompressible Fluid Models in $$n \ge 2$$ n ≥ 2 Dimensions

et al. 2016

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We review the role of dual pairs in mechanics and use them to derive particlelike solutions to regularized incompressible fluid systems. In our case we have a dual pair resulting from the action of diffeomorphisms on point particles (essentially by moving the points). We then augment our dual pair by considering the action of diffeomorphisms on Taylor series, also known as jets. The augmented weak dual pairs induce a hierarchy of particle-like solutions and conservation laws with particles carrying a copy of a jet group. We call these augmented particles jetlets. The jet groups serve as finite-dimensional models of the diffeomorphism group itself, and so the jetlet particles serve as a finite-dimensional model of the self-similarity exhibited by ideal incompressible fluids. The conservation law associated to jetlet solutions is shown to be a shadow of Kelvin's circulation theorem. Finally, we study the dynamics of infinite time particle mergers. We prove that two merging particles at the zeroth level in the hierarchy yield dynamics which asymptotically approach that of a single particle in the first level in the hierarchy. This merging behavior is then verified numerically as well as the exchange of angular momentum which must occur during a near collision of two particles. The resulting particle-like solutions suggest a new class of meshless methods which work in dimensions n ≥ 2 and which exhibit a shadow of Kelvin's circulation theorem. More broadly, this provides one of the first finite-dimensional models of selfsimilarity in ideal fluids.

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Section: Introductionsupporting

confidence: 76%

“…First order particle-like solutions. In this section we revisit the first order particlelike solutions of [CHJM14] and discuss their weak dual pair, before extending the treatment to the higher levels of the hierarchy in the subsequent section. Let SL(n) denote the Lie group of n × n matrices with unit determinant.…”

Section: Particle-like Solutionsmentioning

confidence: 99%

Weak Dual Pairs and Jetlet Methods for Ideal Incompressible Fluid Models in $$n \ge 2$$ n ≥ 2 Dimensions

et al. 2016

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“…In [48], one of the kernel parameters in [46], which controls the compressibility of the u, was taken to the incompressible limit. This allowed a realization of the particle methods described in [47].…”

Section: Fluid Mechanicsmentioning

confidence: 99%

“…This allowed a realization of the particle methods described in [47]. The constructions of [48] are the same as presented in this article, but with Diff(M ) replaced by the group of volume-preserving diffeomorphisms of R d . Velocity fields induced by first order jet-particles are visualized in Figure 8.…”

Section: Fluid Mechanicsmentioning

confidence: 99%

Reduction by Lie Group Symmetries in Diffeomorphic Image Registration and Deformation Modelling

Sommer

Jacobs

2015

Symmetry

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Abstract:We survey the role of reduction by symmetry in the large deformation diffeomorphic metric mapping framework for registration of a variety of data types (landmarks, curves, surfaces, images and higher-order derivative data). Particle relabelling symmetry allows the equations of motion to be reduced to the Lie algebra allowing the equations to be written purely in terms of the Eulerian velocity field. As a second use of symmetry, the infinite dimensional problem of finding correspondences between objects can be reduced for a range of concrete data types, resulting in compact representations of shape and spatial structure. Using reduction by symmetry, we describe these models in a common theoretical framework that draws on links between the registration problem and geometric mechanics. We outline these constructions and further cases where reduction by symmetry promises new approaches to the registration of complex data types.

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“…In contrast, point vortices in Euler flow define a symplectic momentum map[34] which also generalises to higher-order derivatives,[22,13,12]. 3 See[38] for the definitive discussion of dual pairs.…”

mentioning

confidence: 99%

Nonlinear dispersion in wave-current interactions

Holm¹,

Ruiao²

2021

Preprint

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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 semidirectproduct 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 particular, 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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A jetlet hierarchy for ideal fluid dynamics

Cited by 7 publications

References 12 publications

Weak Dual Pairs and Jetlet Methods for Ideal Incompressible Fluid Models in $$n \ge 2$$ n ≥ 2 Dimensions

Weak Dual Pairs and Jetlet Methods for Ideal Incompressible Fluid Models in $$n \ge 2$$ n ≥ 2 Dimensions

Reduction by Lie Group Symmetries in Diffeomorphic Image Registration and Deformation Modelling

Nonlinear dispersion in wave-current interactions

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