MHD equilibrium variational principles with symmetry

Andreussi, Tommaso; Morrison, P. J.; Пегораро, Ф.

doi:10.1088/0741-3335/52/5/055001

Cited by 43 publications

(81 citation statements)

References 46 publications

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“…Notice that G (1) represents the first order generating functional, i.e. the functional (83), whereas G (2) is a second order functional.…”

Section: Dynamically Accessible Stabilitymentioning

confidence: 99%

“…[2] and its predecessor Ref. [1], we explore further ramifications of the Hamiltonian nature of ideal magnetohydrodynamics (MHD). Whereas in Refs.…”

Section: Introductionmentioning

confidence: 99%

“…Whereas in Refs. [1,2] the subject matter concerned the construction and origin of variational principles for equilibria, here we present the comprehensive approach to stability of magnetohydrodynamics (MHD) equilibria that is a direct consequence of the Hamiltonian nature of this system. The presentation organizes scattered approaches into the cohesive Hamiltonian framework, which will be seen to be useful for obtaining, interpreting, and comparing stability results Ultimately, the stability results we consider, which are a consequence of the Hamiltonian form, have their origin in two energy theorems of mechanics: Lagrange's theorem and Dirichlet's theorem (see Ref.…”

Section: Introductionmentioning

confidence: 99%

“…A main goal of this paper is to explore the consequences for stability for such different ways of enforcing constraints. The results of [1,2] will be used in Secs. II, III, and IV to construct three kinds of energy principles for stability of both static and stationary MHD equilibria.…”

Section: Introductionmentioning

confidence: 99%

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Hamiltonian magnetohydrodynamics: Lagrangian, Eulerian, and dynamically accessible stability—Theory

2013

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Stability conditions of magnetized plasma flows are obtained by exploiting the Hamiltonian structure of the magnetohydrodynamics (MHD) equations and, in particular, by using three kinds of energy principles. First, the Lagrangian variable energy principle is described and sufficient stability conditions are presented. Next, plasma flows are described in terms of Eulerian variables and the noncanonical Hamiltonian formulation of MHD is exploited. For symmetric equilibria, the energy-Casimir principle is expanded to second order and sufficient conditions for stability to symmetric perturbation are obtained. Then, dynamically accessible variations, i.e. variations that explicitly preserve invariants of the system, are introduced and the respective energy principle is considered. General criteria for stability are obtained, along with comparisons between the three different approaches.

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“…Notice that G (1) represents the first order generating functional, i.e. the functional (83), whereas G (2) is a second order functional.…”

Section: Dynamically Accessible Stabilitymentioning

confidence: 99%

“…[2] and its predecessor Ref. [1], we explore further ramifications of the Hamiltonian nature of ideal magnetohydrodynamics (MHD). Whereas in Refs.…”

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

Hamiltonian magnetohydrodynamics: Lagrangian, Eulerian, and dynamically accessible stability—Theory

2013

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“…[38,41] as well as Refs. [42][43][44][45] and references therein), which applies namely to noncanonical Hamiltonian systems. The noncanonical Hamiltonian approach is also relevant for the equilibrium statistical mechanics approach to two-dimensional turbulent flows [46].…”

Section: As Well As Sec Ii) For Noncanonicalmentioning

confidence: 99%

Hamiltonian closures in fluid models for plasmas

Tassi

2017

Eur. Phys. J. D

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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.

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