Poisson brackets for fluids and plasmas

Morrison, Philip

doi:10.1063/1.33633

Cited by 213 publications

(346 citation statements)

References 3 publications

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“…When L is inserted into the Poisson bracket an expression for the operator that generates rotations is obtained. The Hamiltonian with these invariants and another, the position of the center of mass that generates Galilean boosts, together with the Poisson bracket, are a realization of the algebra of the ten parameter Galilean group (see [19]). …”

Section: Lagrangian Stability a General Hamiltonian Form Relabementioning

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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Section: Lagrangian Stability a General Hamiltonian Form Relabementioning

confidence: 99%

“…Isothermal processes (γ = 1) have U = κ ln(ρ). The MHD model can be generalized by altering the Hamiltonian in many physically meaningful ways: for example, an anisotropic pressure tensor can be treated as in [19,22] by assuming U depends on B = |B| with…”

Section: B Mhd and The Lagrange-euler Mapmentioning

confidence: 99%

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

2013

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“…For example, one could make a special choice for U that gives rise to /? = K{s)p^, and then use this relation to eliminate the entropy per unit mass s in terms of the pressure p. Another possibility would be to allow |5| dependence in U and in this way obtain the anisotropic pressure tensor of the CGL equations [19,1].…”

Section: Mhd Actionmentioning

confidence: 99%

“…The HAP formulation of this parent model gives rise to that of the most commonly used ideal plasma models such as the magnetohydrodynamics (MHD), the Vlasov description, BBGKY hierarchy, etc. (for my own reviews see [1,2,3,4]). …”

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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“…This occurs, in particular, at points where the Poisson bracket changes rank. [12] The existence of non-trivial (e.g. 2D) equilibrium states is essential for physical applications.…”

Section: Introductionmentioning

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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Poisson brackets for fluids and plasmas

Cited by 213 publications

References 3 publications

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

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

On Hamiltonian and Action Principle Formulations of Plasma Dynamics

A compressible Hamiltonian electromagnetic gyrofluid model

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