Hamiltonian formulation of reduced magnetohydrodynamics

Morrison, P.J.; Hazeltine, R. D.

doi:10.1063/1.864718

Cited by 137 publications

(167 citation statements)

References 22 publications

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“…The rigid body in a gravitational field is an example of finite dimension. An example of infinite dimension, which was given in the context of reduced magnetohydrodynamics (RMHD) (Morrison and Hazeltine, 1984;Zeitlin, 1992), but which also occurs in fluid mechanics, is the semidirect product extension of the noncanonical bracket for the 2D Euler fluid. For this example the bracket of Eq.…”

Section: Semidirect Product Reductionsmentioning

confidence: 99%

Hamiltonian description of the ideal fluid

1998

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The Hamiltonian viewpoint of fluid mechanical systems with few and infinite number of degrees of freedom is described. Rudimentary concepts of finite-degree-of-freedom Hamiltonian dynamics are reviewed, in the context of the passive advection of a scalar or tracer field by a fluid. The notions of integrability, invariant-tori, chaos, overlap criteria, and invariant-tori breakup are described in this context. Preparatory to the introduction of field theories, systems with an infinite number of degrees of freedom, elements of functional calculus and action principles of mechanics are reviewed. The action principle for the ideal compressible fluid is described in terms of Lagrangian or material variables. Hamiltonian systems in terms of noncanonical variables are presented, including several examples of Eulerian or inviscid fluid dynamics. Lie group theory sufficient for the treatment of reduction is reviewed. The reduction from Lagrangian to Eulerian variables is treated along with Clebsch variable decompositions. Stability in the canonical and noncanonical Hamiltonian contexts is described. Sufficient conditions for stability, such as Rayleigh-like criteria, are seen to be only sufficient in the general case because of the existence of negative-energy modes, which are possessed by interesting fluid equilibria. Linearly stable equilibria with negative energy modes are argued to be unstable when nonlinearity or dissipation is added. The energy-Casimir method is discussed and a variant of it that depends upon the notion of dynamical accessibility is described. The energy content of a perturbation about a general fluid equilibrium is calculated using three methods.[S0034-6861(98)00102-0] CONTENTS

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Section: Semidirect Product Reductionsmentioning

confidence: 99%

Hamiltonian description of the ideal fluid

1998

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“…Perhaps the most important of these properties is the existence of Casimir functionals, which are invariants of the motion that specify a foliation of phase space. [23,32] In the presence of an ignorable coordinate, these invariants usually appear in infinite families and become useful to construct variational principles for studying the equilibrium and stability of a system. In this section we assume ∂ z = 0: the generalization to the case of helical symmetry is straightforward and is described in Ref.…”

Section: Invariantsmentioning

confidence: 99%

“…In this section we assume ∂ z = 0: the generalization to the case of helical symmetry is straightforward and is described in Ref. [32].…”

Section: Invariantsmentioning

confidence: 99%

A compressible Hamiltonian electromagnetic gyrofluid model

Waelbroeck

Tassi

2012

Communications in Nonlinear Science and Numerical Simulation

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A Lie-Poisson bracket is presented for a four-field gyrofluid model with compressible ions and magnetic field curvature, thereby showing the model to be Hamiltonian. In particular, in addition to commonly adopted magnetic curvature terms present in the continuity equations, analogous terms must be retained also in the momentum equations, in order to have a Lie-Poisson structure.The corresponding Casimir invariants are presented, and shown to be associated to four Lagrangian invariants, that get advected by appropriate "velocity" fields during the dynamics. This differs from a cold ion limit, in which the Lie-Poisson bracket transforms into the sum of direct and semidirect products, leading to only three Lagrangian invariants.

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“…One of the early examples is reduced MHD [6], but many models for many purposes have been obtained over the years, e.g. [7,8,9,10,11,12,13,14,15,16]. Sometimes models like these arise from systematic asymptotic expansion, while in other cases they have been proposed in a more ad hoc manner.…”

Section: Introductionmentioning

confidence: 99%

On Hamiltonian and Action Principle Formulations of Plasma Dynamics

Morrison

Eliasson²,

Shukła³

2009

AIP Conference Proceedings

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Abstract.A general discussion of Hamiltonian and action principle formulations for fliud and plasma models is given. A procedure, based on Hamilton's principle of mechanics but adapted for continua, for the construction of action principles for fliud and kinetic models is given. The transformation from action principles in terms of the Lagrangian variable description to the Eulerian variable description in terms of noncanonical Poisson brackets is described. Two examples are developed: ideal MHD and Braginskii's fliud model with gyroviscosity.

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Hamiltonian formulation of reduced magnetohydrodynamics

Cited by 137 publications

References 22 publications

Hamiltonian description of the ideal fluid

Hamiltonian description of the ideal fluid

A compressible Hamiltonian electromagnetic gyrofluid model

On Hamiltonian and Action Principle Formulations of Plasma Dynamics

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