Vlasov moments, integrable systems and singular solutions

Gibbons, John; Holm, Darryl D.; Tronci, Cesare

doi:10.1016/j.physleta.2007.08.054

Cited by 31 publications

(72 citation statements)

References 36 publications

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“…More recently, the relation between the truncated moment hierarchy of the geodesic Vlasov equation and integrable systems has been elucidated in Ref. [23]. A geometric interpretation of the Lie-Poisson structure associated with the dynamics of moments has been provided in Ref.…”

Section: Introductionmentioning

confidence: 99%

Hamiltonian fluid reductions of electromagnetic drift-kinetic equations for an arbitrary number of moments

Tassi

2015

Annals of Physics

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International audienceWe present an infinite family of Hamiltonian electromagnetic fluid models for plasmas, derived from drift-kinetic equations. An infinite hierarchy of fluid equations is obtained from a Hamiltonian drift-kinetic systems by taking moments of a generalized distribution function and using Hermite polynomials as weight functions of the velocity coordinate along the magnetic guide field. Each fluid model is then obtained by truncating the hierarchy to a finite number N + 1 of equations by means of a closure relation. We show that, for any positive N , a linear closure relation between the moment of order N + 1 and the moment of order N guarantees that the resulting fluid model possesses a Hamiltonian structure, thus respecting the Hamiltonian character of the parent drift-kinetic model. An orthogonal transformation is identified which maps the fluid moments to a new set of dynamical variables in terms of which the Poisson brackets of the fluid models become a direct sum and which unveils remarkable dynamical properties of the models in the tw

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Section: Introductionmentioning

confidence: 99%

Hamiltonian fluid reductions of electromagnetic drift-kinetic equations for an arbitrary number of moments

Tassi

2015

Annals of Physics

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“…Indeed, such expression depends explicitly onP 2 , which shows that, unlike the case of Vlasov equation [14,16], the set of functionals of the first two moments does not form a sub-algebra of the algebra of functionals of all the moments. In particular, the Jacobi identity is not respected by (38).…”

Section: Making Use Of Eq (34) the Equation Of Motion Resulting Fromentioning

confidence: 99%

Hamiltonian structure of a drift-kinetic model and Hamiltonian closures for its two-moment fluid reductions

Tassi

2014

Eur. Phys. J. D

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We address the problem of the existence of the Hamiltonian structure for an electrostatic driftkinetic model and for the related fluid models describing the evolution of the first two moments of the distribution function with respect to the parallel velocity. The drift-kinetic model, which accounts for background density and temperature gradients as well as polarization effects, is shown to possess a noncanonical Hamiltonian structure. The corresponding Poisson bracket is expressed in terms of the fluid moments and it is found that the set of functionals of the zero order moment forms a sub-algebra, thus automatically leading to a class of one-moment Hamiltonian fluid models. Fluids 26, (1983) 990 ) to describe drift waves coupled to ion-acoustic waves, is obtained and its Hamiltonian structure is provided explicitly.

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“…(1990) and Gibbons et al. (2008a, b). In plasma dynamics, the phase-space moments arise from a Taylor expansion of the Vlasov particle distribution, taken around its centroid in phase-space.…”

Section: Momentsmentioning

confidence: 99%

Multipole Vortex Blobs (MVB): Symplectic Geometry and Dynamics

Holm

Jacobs

2017

J Nonlinear Sci

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Vortex blob methods are typically characterized by a regularization length scale, below which the dynamics are trivial for isolated blobs. In this article, we observe that the dynamics need not be trivial if one is willing to consider distributional derivatives of Dirac delta functionals as valid vorticity distributions. More specifically, a new singular vortex theory is presented for regularized Euler fluid equations of ideal incompressible flow in the plane. We determine the conditions under which such regularized Euler fluid equations may admit vorticity singularities which are stronger than delta functions, e.g., derivatives of delta functions. We also describe the symplectic geometry associated with these augmented vortex structures, and we characterize the dynamics as Hamiltonian. Applications to the design of numerical methods similar to vortex blob methods are also discussed. Such findings illuminate the rich dynamics which occur below the regularization length scale and enlighten our perspective on the potential for regularized fluid models to capture multiscale phenomena.

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Vlasov moments, integrable systems and singular solutions

Cited by 31 publications

References 36 publications

Hamiltonian fluid reductions of electromagnetic drift-kinetic equations for an arbitrary number of moments

Hamiltonian fluid reductions of electromagnetic drift-kinetic equations for an arbitrary number of moments

Hamiltonian structure of a drift-kinetic model and Hamiltonian closures for its two-moment fluid reductions

Multipole Vortex Blobs (MVB): Symplectic Geometry and Dynamics

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