Hamiltonian fluid closures of the Vlasov-Ampère equations: From water-bags to <i>N</i> moment models

Perin, Maxime; Chandre, Cristel; Morrison, Philip; Tassi, Emanuele

doi:10.1063/1.4930097

Cited by 9 publications

(7 citation statements)

References 20 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…In Ref. [127] it was shown, on the other hand, that there exist a family of closures of the Vlasov-Ampère system, those associated with the so called water-bag distribution functions, which lead to Hamiltonian fluid models for an arbitrary number of moments.…”

Section: A Four-moment Model Derived From the Vlasov-ampère Systemmentioning

confidence: 99%

Hamiltonian closures in fluid models for plasmas

Tassi

2017

Eur. Phys. J. D

View full text Add to dashboard Cite

This article reviews recent activity on the Hamiltonian formulation of fluid models for plasmas in the non-dissipative limit, with emphasis on the relations between the fluid closures adopted for the different models and the Hamiltonian structures. The review focuses on results obtained during the last decade, but a few classical results are also described, in order to illustrate connections with the most recent developments. With the hope of making the review accessible not only to specialists in the field, an introduction to the mathematical tools applied in the Hamiltonian formalism for continuum models is provided. Subsequently, we review the Hamiltonian formulation of models based on the magnetohydrodynamics description, including those based on the adiabatic and double adiabatic closure. It is shown how Dirac's theory of constrained Hamiltonian systems can be applied to impose the incompressibility closure on a magnetohydrodynamic model and how an extended version of barotropic magnetohydrodynamics, accounting for two-fluid effects, is amenable to a Hamiltonian formulation. Hamiltonian reduced fluid models, valid in the presence of a strong magnetic field, are also reviewed. In particular, reduced magnetohydrodynamics and models assuming cold ions and different closures for the electron fluid are discussed. Hamiltonian models relaxing the cold-ion assumption are then introduced. These include models where finite Larmor radius effects are added by means of the gyromap technique, and gyrofluid models.Numerical simulations of Hamiltonian reduced fluid models investigating the phenomenon of magnetic reconnection are illustrated. The last part of the review concerns recent results based on the derivation of closures preserving a Hamiltonian structure, based on the Hamiltonian structure of parent kinetic models. Identification of such closures for fluid models derived from kinetic systems based on the Vlasov and drift-kinetic equations are presented, and connections with previously discussed fluid models are pointed out.

show abstract

Section: A Four-moment Model Derived From the Vlasov-ampère Systemmentioning

confidence: 99%

Hamiltonian closures in fluid models for plasmas

Tassi

2017

Eur. Phys. J. D

View full text Add to dashboard Cite

show abstract

“…As shown in the previous section, Hamiltonian models with three moments derived from the drift-kinetic equation are those whose third and fourth order reduced moments, namely S 3 and S 4 , satisfy equations ( 14) and (15) respectively. However, in order for S 4 , which corresponds to the kurtosis of the distribution function, to stay bounded as the particle density ρ tends to 0, one has to impose  = 0 in equation (15).…”

Section: Correspondence With Water-bagsmentioning

confidence: 99%

“…The computation of the constraints given by equations ( 14) and (15), which are necessary and sufficient conditions in order for Bracket (12) to satisfy the Jacobi identity, is done by using Mathematica ® . In this appendix, we provide the detailed code along with the procedure required in order to obtain these constraints.…”

Section: Appendix Computation Of Jmentioning

confidence: 99%

Hamiltonian fluid reductions of drift-kinetic equations and the link with water-bags

Perin¹,

Chandre²,

Tassi³

2016

J. Phys. A: Math. Theor.

Self Cite

View full text Add to dashboard Cite

Hamiltonian models for the first three moments of the drift-kinetic distribution function, namely the density, the fluid velocity and the parallel pressure, are derived from the Hamiltonian structure of the drift-kinetic equations. The link with the water-bag closure is established, showing that, unlike the one-dimensional Vlasov equations, these solutions are the only Hamiltonian fluid reductions for the drift-kinetic equation. These models are discussed through their equations of motion and their Casimir invariants.

show abstract

“…As a simple and analytically tractable example, we study the water-bag model of a one-dimensional charged-beam system. [12][13][14][15] A series of water-bag models with different numbers of water-bags define the hierarchical Vlasov sub-algebra.…”

Section: Introductionmentioning

confidence: 99%

Hierarchical foliation of one-dimensional Vlasov–Poisson system

Maekaku

Yoshida

2022

Physics of Plasmas

View full text Add to dashboard Cite

We elucidate the intermediate of the macroscopic fluid model and the microscopic kinetic model by studying the Poisson algebraic structure of the one-dimensional Vlasov–Poisson system. The water-bag model helps formulating the hierarchy of sub-algebras, which interpolates the gap between the fluid and kinetic models. By analyzing the embedding of the sub-manifold of an intermediate hierarchy in a more microscopic hierarchy, we characterize the microscopic effect as the symmetry breaking pertinent to a macroscopic invariant.

show abstract

Hamiltonian fluid closures of the Vlasov-Ampère equations: From water-bags to N moment models

Cited by 9 publications

References 20 publications

Hamiltonian closures in fluid models for plasmas

Hamiltonian closures in fluid models for plasmas

Hamiltonian fluid reductions of drift-kinetic equations and the link with water-bags

Hierarchical foliation of one-dimensional Vlasov–Poisson system

Contact Info

Product

Resources

About