A method for Hamiltonian truncation: a four-wave example

Viscondi, Thiago F.; Caldas, Iberê L.; Morrison, Philip

doi:10.1088/1751-8113/49/16/165501

Cited by 8 publications

(11 citation statements)

References 60 publications

(182 reference statements)

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“…The reader is directed to the analysis by Abdelhamid et al [43] that drew extensively upon the HAP approach (for e.g., to construct nonlinear wave solutions), and thereby arrived at the energy and helicity spectra of XMHD. We also point out the recent beatification procedure of Viscondi et al [44] as an elegant alternative, that explicitly relies on the Hamiltonian formulation.…”

Section: Introductionmentioning

confidence: 98%

On the structure and statistical theory of turbulence of extended magnetohydrodynamics

2017

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Recent progress regarding the noncanonical Hamiltonian formulation of extended magnetohydrodynamics (XMHD), a model with Hall drift and electron inertia, is summarized. The advantages of the Hamiltonian approach are invoked to study some general properties of XMHD turbulence, and to compare them against their ideal MHD counterparts. For instance, the helicity flux transfer rates for XMHD are computed, and Liouville's theorem for this model is also verified. The latter is used, in conjunction with the absolute equilibrium states, to arrive at the spectra for the invariants, and to determine the direction of the cascades, e.g., generalizations of the well-known ideal MHD inverse cascade of magnetic helicity. After a similar analysis is conducted for XMHD by inspecting second order structure functions and absolute equilibrium states, a couple of interesting results emerge. When cross helicity is taken to be ignorable, the inverse cascade of injected magnetic helicity also occurs in the Hall MHD range-this is shown to be consistent with previous results in the literature. In contrast, in the inertial MHD range, viz at scales smaller than the electron skin depth, all spectral quantities are expected to undergo direct cascading. The consequences and relevance of our results in space and astrophysical plasmas are also briefly discussed.

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Section: Introductionmentioning

confidence: 98%

On the structure and statistical theory of turbulence of extended magnetohydrodynamics

2017

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“…The beatification transformations considered in this paper flattened the Poisson bracket to first order. In recent work this has been generalized by flattening to second order [69], which is necessary to obtain consistent dynamics if one expands about a state that is not an equilibrium state. Also, this higher order beatification can be used to address the accuracy problems mentioned above.…”

Section: Discussionmentioning

confidence: 99%

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

Morrison

Vanneste

2016

Annals of Physics

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A method, called beatification, is presented for rapidly extracting weakly nonlinear Hamiltonian systems that describe the dynamics near equilibria for systems possessing Hamiltonian form in terms of noncanonical Poisson brackets. The procedure applies to systems like fluids and plasmas in terms of Eulerian variables that have such noncanonical Poisson brackets, i.e., brackets with nonstandard and possibly degenerate form. A collection of examples of both finite and infinite dimensions is presented.

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“…In particular, Squire et al (2013) considered the construction of a reduced bracket for the gyrokinetic Vlasov-Poisson equations by using this method. Another context is that of 'beatification' (Morrison & Vanneste 2016;Viscondi, Caldas & Morrison 2016) where one uses transformations to perturbatively remove nonlinearity from the Poisson bracket and place it in the Hamiltonian functional.…”

Section: Functional Transformation Of a Hamiltonian Bracketmentioning

confidence: 99%

Lifting of the Vlasov–Maxwell bracket by Lie-transform method

et al. 2016

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The Vlasov-Maxwell equations possess a Hamiltonian structure expressed in terms of a Hamiltonian functional and a functional bracket. In the present paper, the transformation ("lift") of the Vlasov-Maxwell bracket induced by the dynamical reduction of singleparticle dynamics is investigated when the reduction is carried out by Lie-transform perturbation methods. The ultimate goal of this work is to derive explicit Hamiltonian formulations for the guiding-center and gyrokinetic Vlasov-Maxwell equations that have important applications in our understanding of turbulent magnetized plasmas.where the Jacobian J is shown explicitly, and K = |p| 2 /(2m) denotes the kinetic energy of a particle of mass m and charge e (summation over particle species is implied whenever applicable). The Vlasov-Maxwell bracket is a bilinear operator on arbitrary functionals F [f, E, B] and G[f, E, B]:where the antisymmetric Poisson matrix components J αβ (z) ≡ {z α , z β } satisfy the Liouville identities 1 J ∂ ∂z α J J αβ ≡ 0.(3.6)The divergence form (3.5) implies the phase-space integral identitywhere a sequence of phase-space coordinate transformations and integrations by parts lead to the relations (4.2).J ≡ T −ǫ J 0 1 − ǫ d · G 1 − ǫ 2 d · G 2 − 1 2 G 1 · dG 1 + · · · ≡ T −ǫ J 0 J ǫ , (6.8)where J ǫ is defined by the identity T −ǫ (d 6 z 0 ) ≡ J ǫ d 6 Z.

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A method for Hamiltonian truncation: a four-wave example

Cited by 8 publications

References 60 publications

On the structure and statistical theory of turbulence of extended magnetohydrodynamics

On the structure and statistical theory of turbulence of extended magnetohydrodynamics

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

Lifting of the Vlasov–Maxwell bracket by Lie-transform method

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