Weakly nonlinear dynamics in noncanonical Hamiltonian systems with applications to fluids and plasmas

Morrison, Philip; Vanneste, Jacques

doi:10.1016/j.aop.2016.02.003

Cited by 12 publications

(17 citation statements)

References 67 publications

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“…topological invariants [28], particle relabelling symmetries [39][40][41][42], reconnection based on Hamiltonian models [43][44][45], tearing modes [46], Hamiltonian closures [47,48], nonlinear waves [49,50], weakly nonlinear dynamics [51,52], the derivation of gyrofluid and hybrid fluid-kinetic models [24,31,[53][54][55], the properties of the equatorial electrojet [56], and the rapidly burgeoning field of variational integrators [57][58][59].…”

Section: Discussionmentioning

confidence: 99%

Derivation of the Hall and extended magnetohydrodynamics brackets

2016

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There are several plasma models intermediate in complexity between ideal magnetohydrodynamics (MHD) and two-fluid theory, with Hall and Extended MHD being two important examples. In this paper we investigate several aspects of these theories, with the ultimate goal of deriving the noncanonical Poisson brackets used in their Hamiltonian formulations. We present fully Lagrangian actions for each, as opposed to the fully Eulerian, or mixed Eulerian-Lagrangian, actions that have appeared previously. As an important step in this process we exhibit each theory's two advected fluxes (in analogy to ideal MHD's advected magnetic flux), discovering also that with the correct choice of gauge they have corresponding Lie-dragged potentials resembling the electromagnetic vector potential, and associated conserved helicities. Finally, using the Euler-Lagrange map, we show how to derive the noncanonical Eulerian brackets from canonical Lagrangian ones.

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Section: Discussionmentioning

confidence: 99%

Derivation of the Hall and extended magnetohydrodynamics brackets

2016

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“…Now we perform the beatification procedure, 33 a perturbative transformation that removes the functional dependence of the Poisson operator on the field variable and replaces it with a reference state. The procedure is applied to the bracket of (13) and, in preparation for the truncation procedure of section 5, the Hamiltonian for the two-dimensional Euler equation is expressed in terms of the transformed variable.…”

Section: Beatificationmentioning

confidence: 99%

A method for Hamiltonian truncation: a four-wave example

Viscondi¹,

Caldas²,

Morrison³

2016

J. Phys. A: Math. Theor.

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A method for extracting finite-dimensional Hamiltonian systems from a class of 2+1 Hamiltonian mean field theories is presented. These theories possess noncanonical Poisson brackets, which normally resist Hamiltonian truncation, but a process of beatification by coordinate transformation near a reference state is described in order to perturbatively overcome this difficulty. Two examples of four-wave truncation of Euler's equation for scalar vortex dynamics are given and compared: one a direct non-Hamiltonian truncation of the equations of motion, the other obtained by beatifying the Poisson bracket and then truncating. arXiv:1509.09247v2 [physics.flu-dyn]

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“…The rich geometry of a phase space with a defined Poisson bracket, which includes symplectic and Poisson geometry, is intellectually very interesting and allows for better insight. Moreover, it is of practical value for understanding spectra, perturbation theory, and the construction of numerical algorithms (see, e.g., [3,4,5,6]).…”

Section: Introductionmentioning

confidence: 99%

A general metriplectic framework with application to dissipative extended magnetohydrodynamics

Coquinot

Morrison

2020

J. Plasma Phys.

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General equations for conservative yet dissipative (entropy producing) extended magnetohydrodynamics are derived from two-fluid theory. Keeping all terms generates unusual cross-effects, such as thermophoresis and a current viscosity that mixes with the usual velocity viscosity. While the Poisson bracket of the ideal version of this model has already been discovered, we determine its metriplectic counterpart that describes the dissipation. This is done using a new and general thermodynamic point of view to derive dissipative brackets, a means of derivation that is natural for understanding and creating dissipative dynamics without appealing to underlying kinetic theory orderings. Finally, the formalism is used to study dissipation in the Lagrangian variable picture where, in the context of extended magnetohydrodynamics, non-local dissipative brackets naturally emerge.

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Weakly nonlinear dynamics in noncanonical Hamiltonian systems with applications to fluids and plasmas

Cited by 12 publications

References 67 publications

Derivation of the Hall and extended magnetohydrodynamics brackets

Derivation of the Hall and extended magnetohydrodynamics brackets

A method for Hamiltonian truncation: a four-wave example

A general metriplectic framework with application to dissipative extended magnetohydrodynamics

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