Hamiltonian formalism of extended magnetohydrodynamics

Abdelhamid, Hamdi M.; Kawazura, Yohei; Yoshida, Zensho

doi:10.1088/1751-8113/48/23/235502

Cited by 56 publications

(118 citation statements)

References 29 publications

(59 reference statements)

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“…The next step is to render the above equations dimensionless. This is done by normalizing everything in terms of Alfvénic units, and the reader is directed to Abdelhamid et al [11], Lingam et al [31] for further details. We also introduce the dynamical variable…”

Section: Preliminary Model Considerationsmentioning

confidence: 99%

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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: Preliminary Model Considerationsmentioning

confidence: 99%

“…We are now in a position to present the Hamiltonian formulation of XMHD. A detailed derivation of this structure can be found in D'Avignon et al [10], Abdelhamid et al [11], Lingam et al [12], and a recent overview was provided in Lingam et al [31].…”

Section: The Hamiltonian Structure Of Xmhdmentioning

confidence: 99%

On the structure and statistical theory of turbulence of extended magnetohydrodynamics

2017

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“…Notice that, the details of derivation of the Poisson bracket and the proof of the Jacobi identity were given in [12].…”

Section: Arxiv:151101599v1 [Physicsplasm-ph] 5 Nov 2015mentioning

confidence: 99%

Nonlinear Alfvén waves in extended magnetohydrodynamics

Abdelhamid

Yoshida

2016

Physics of Plasmas

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Large-amplitude Alfvén waves are observed in various systems in space and laboratories, demonstrating an interesting property that the wave shapes are stable even in the nonlinear regime. The ideal magnetohydrodynamics (MHD) model predicts that an Alfvén wave keeps an arbitrary shape constant when it propagates on a homogeneous ambient magnetic field. However, such arbitrariness is an artifact of the idealized model that omits the dispersive effects. Only special wave forms, consisting of two component sinusoidal functions, can maintain the shape; we derive fully nonlinear Alfvén waves by an extended MHD model that includes both the Hall and electron inertia effects. Interestingly, these "small-scale effects" change the picture completely; the large-scale component of the wave cannot be independent of the small scale component, and the coexistence of them forbids the large scale component to have a free wave form. This is a manifestation of the nonlinearitydispersion interplay, which is somewhat different from that of solitons.PACS numbers: 52.35. Bj, 52.30.Cv, 52.35.Mw, 47.10.Df Alfvén waves are the most typical electromagnetic phenomena in magnetized plasmas. In particular, nonlinear Alfvén waves deeply influence various plasma regimes in laboratory as well as in space, which have a crucial role in plasma heating [1,2], turbulence [3][4][5], reconnection[6], etc. As an interesting property of the Alfvén waves, the amplitudes as well as wave forms are totally arbitrary when they propagate on a homogenous ambient magnetic field [7,8]. In fact, we often observe large-amplitude Alfvén waves in orderly propagation (for example [9]). To put it in theoretical language, the set A of Alfvén waves after an appropriate transformation (see [10]), is a closed linear subspace, i.e., every linear combination of the members of A gives solution to the fully nonlinear wave equation. Needless to say, the set of general solutions to a linear equation is, by definition, a linear subspace. However, it is remarkable that the nonlinear MHD equation has such a linear subspace A of solutions.Here we investigate the underlying mechanism producing such solutions in the context of a more accurate framework, generalized MHD. When we take into account dispersion effects (we consider both ion and electron inertial effects [11,12]), the wave forms are no longer arbitrary (remember that the ideal MHD model is dispersion free). Yet, we find that the generalized MHD system has a linear subspace of nonlinear solutions. The Casimir invariants of the system is the root cause of this interesting property [13].We start by reviewing how the nonlinear Alfvén waves are created in ideal MHD; we put the problem in the perspective of Hamiltonian mechanics. We then formulate the generalized MHD system in a Hamiltonian form. Via constructing equilibrium solutions (so-called Beltrami equilibriums) on Casimir leaves, we derive nonlinear wave solutions. The dispersion relation is exactly that of the linear theory, while the wave amplitude may be arbitrarily ...

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“…The extended MHD model treated in Ref. [80] assumes a barotropic closure, thus showing that the Hamiltonian character of barotropic MHD survives an extension accouting for electron and ion scales. In normalized Alfvén units, with lenghts normalized by L, the barotropic extended MHD model reads…”

Section: Barotropic Extended Mhdmentioning

confidence: 96%

“…[79] and its Hamiltonian formulation was presented in Ref. [80]. The extended MHD model treated in Ref.…”

Section: Barotropic Extended Mhdmentioning

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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Hamiltonian formalism of extended magnetohydrodynamics

Cited by 56 publications

References 29 publications

On the structure and statistical theory of turbulence of extended magnetohydrodynamics

On the structure and statistical theory of turbulence of extended magnetohydrodynamics

Nonlinear Alfvén waves in extended magnetohydrodynamics

Hamiltonian closures in fluid models for plasmas

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