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

2015

Annals of Physics

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

Section: Introductionmentioning

confidence: 99%

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

2015

Annals of Physics

“…Obvious examples in this context would be toroidal geometry , magnetic, temperature and density equilibrium gradients (although the latter have already been partially treated in Ref. [12]). Perhaps even more fundamental is the problem of considering Hamiltonian closures at higher-order moments.…”

Section: Discussionmentioning

confidence: 99%

“…This means that the functionals of interest will be of the kind F (P i 00 , P i 10 , P i 01 · · · P i M N , P e 00 , P e 10 , P e 01 · · · P e M N ) and will no longer depend on the infinite set of moments. The operation (12) between two such functionals, however, is not closed, in the sense that {F, G}, in general, will not be again a functional of the same kind, but it will depend also on moments of order higher than M (N ) in the parallel (perpendicular) direction. In order to obtain a closed system, the way we follow is that of modifying the Poisson bracket, imposing that such extra higher order moments be functions of the first M (N ) moments in the parallel (perpendicular) direction.…”

Section: Derivation Of the Hamiltonian Gyrofluid Modelmentioning

confidence: 99%

“…Recently, derivation procedures that preserve the Hamiltonian structure of the parent model have been applied to kinetic and drift-kinetic models [10][11][12]. A key element in these procedures consists of inserting a generic fluid closure relation in the bracket obtained from the parent kinetic model, and then imposing the constraint that the Jacobi identity be satisfied.…”

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

Hamiltonian derivation of a gyrofluid model for collisionless magnetic reconnection

2014

J. Phys.: Conf. Ser.

Abstract. We consider a simple electromagnetic gyrokinetic model for collisionless plasmas and show that it possesses a Hamiltonian structure. Subsequently, from this model we derive a two-moment gyrofluid model by means of a procedure which guarantees that the resulting gyrofluid model is also Hamiltonian. The first step in the derivation consists of imposing a generic fluid closure in the Poisson bracket of the gyrokinetic model, after expressing such bracket in terms of the gyrofluid moments. The constraint of the Jacobi identity, which every Poisson bracket has to satisfy, selects then what closures can lead to a Hamiltonian gyrofluid system. For the case at hand, it turns out that the only closures (not involving integro/differential operators or an explicit dependence on the spatial coordinates) that lead to a valid Poisson bracket are those for which the second order parallel moment, independently for each species, is proportional to the zero order moment. In particular, if one chooses an isothermal closure based on the equilibrium temperatures and derives accordingly the Hamiltonian of the system from the Hamiltonian of the parent gyrokinetic model, one recovers a known Hamiltonian gyrofluid model for collisionless reconnection. The proposed procedure, in addition to yield a gyrofluid model which automatically conserves the total energy, provides also, through the resulting Poisson bracket, a way to derive further conservation laws of the gyrofluid model, associated with the so called Casimir invariants. We show that a relation exists between Casimir invariants of the gyrofluid model and those of the gyrokinetic parent model. The application of such Hamiltonian derivation procedure to this two-moment gyrofluid model is a first step toward its application to more realistic, higher-order fluid or gyrofluid models for tokamaks. It also extends to the electromagnetic gyrokinetic case, recent applications of the same procedure to Vlasov and driftkinetic systems. IntroductionFluid models represent a widespread and effective tool for investigating important phenomena in fusion plasmas such as instabilities, turbulence and reconnection events. Indeed, fluid models offer a considerable advantage, in terms of required computational resources, with respect to kinetic models. On the other hand, compared to these, they obviously suffer from limitations in the range of scales and frequencies of the phenomena that they can describe. The derivation of sophisticated fluid models aiming at partially remedying such limitations is an active line of research since a long time. An essential part of the problem is related to the closure adopted to truncate the fluid hierarchy of equations obtained by taking moments of a parent kinetic model. A considerable effort, directed also toward applications to space plasma turbulence, has been carried out to derive fluid and gyrofluid models (see, e.g. Refs. [1][2][3][4][5][6][7][8]) by taking moments of

“…We remark that in Ref. [142], the above results were generalized to the case where the drift-kinetic distribution function is given by the sum of a perturbation with a non-uniform Maxwellian, thus leading to a Hamiltonian two-moment extension of the Charney-Hasegawa-Mima equation for drift waves.…”

Section: Two and Three-moment Drift-fluid Modelsmentioning

confidence: 94%

Hamiltonian closures in fluid models for plasmas

2017

Eur. Phys. J. D

Self 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.