Hamiltonian description of convective-cell generation

Krommes, John A.; Kolesnikov, R. A.

doi:10.1063/1.1719014

Cited by 25 publications

(37 citation statements)

References 12 publications

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“…A serious attempt to understand the wave kinetic approach was mounted by Krommes and Kim [69]; subsequently, further insights were gained by Krommes and Kolesnikov [90]. Those works together provide an interesting and nontrivial tour through a variety of useful tools, including renormalized statistical theory, Hamiltonian formulation of nonlinear PDEs, multiplescale methods, and an unusual application of wave-kinetic analysis.…”

Section: Wave Kinetics and The Calculation Of The Zonal-flow Growth Ratementioning

confidence: 99%

“…However, in a bracket formalism it is known that additional Casimir invariants can be conserved for an arbitrary Hamiltonian due to particular properties of the Poisson bracket. Using a covariant bracket formalism combined with multiple-scale analysis, Krommes and Kolesnikov [90] were able to show that for E × B advection a particular Casimir is conserved even under the interaction between short-wavelength drift waves and long-wavelength convectives cells. For the generalized Hasegawa-Mima equation, the Smolyakov-Diamond invariant is recovered.…”

Section: Wave Kinetics and The Calculation Of The Zonal-flow Growth Ratementioning

confidence: 99%

“…In the face of multiple Casimirs, that formalism is interesting and nontrivial [90], but it is sufficient here to merely consider the case of a single invariant N (discussed at length in Ref. 69).…”

Section: Wave Kinetics and The Calculation Of The Zonal-flow Growth Ratementioning

confidence: 99%

See 2 more Smart Citations

Nonlinear theoretical tools for fusion-related microturbulence: Historical evolution, and recent applications to stochastic magnetic fields, zonal-flow dynamics, and intermittency

Krommes¹

2009

J. Phys.: Conf. Ser.

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Abstract. Fusion physics poses an extremely challenging, practically complex problem that does not yield readily to simple paradigms. Nevertheless, various of the theoretical tools and conceptual advances emphasized at the KaufmanFest 2007 have motivated and/or found application to the development of fusion-related plasma turbulence theory. A brief historical commentary is given on some aspects of that specialty, with emphasis on the role (and limitations) of Hamiltonian/symplectic approaches, variational methods, oscillation-center theory, and nonlinear dynamics. It is shown how to extract a renormalized ponderomotive force from the statistical equations of plasma turbulence, and the possibility of a renormalized K-χ theorem is discussed. An unusual application of quasilinear theory to the problem of plasma equilibria in the presence of stochastic magnetic fields is described. The modern problem of zonal-flow dynamics illustrates a confluence of several techniques, including (i) the application of nonlinear-dynamics methods, especially center-manifold theory, to the problem of the transition to plasma turbulence in the face of self-generated zonal flows; and (ii) the use of Hamiltonian formalism to determine the appropriate (Casimir) invariant to be used in a novel wave-kinetic analysis of systems of interacting zonal flows and drift waves. Recent progress in the theory of intermittent chaotic statistics and the generation of coherent structures from turbulence is mentioned, and an appeal is made for some new tools to cope with these interesting and difficult problems in nonlinear plasma physics. Finally, the important influence of the intellectually stimulating research environment fostered by Prof. Allan Kaufman on the author's thinking and teaching methodology is described.

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Section: Wave Kinetics and The Calculation Of The Zonal-flow Growth Ratementioning

confidence: 99%

Section: Wave Kinetics and The Calculation Of The Zonal-flow Growth Ratementioning

confidence: 99%

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Nonlinear theoretical tools for fusion-related microturbulence: Historical evolution, and recent applications to stochastic magnetic fields, zonal-flow dynamics, and intermittency

Krommes¹

2009

J. Phys.: Conf. Ser.

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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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“…[13] Another important property of noncanonical Hamiltonian systems is the existence of families of invariants, called Casimir invariants, that characterize the dynamics.…”

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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Hamiltonian description of convective-cell generation

Cited by 25 publications

References 12 publications

Nonlinear theoretical tools for fusion-related microturbulence: Historical evolution, and recent applications to stochastic magnetic fields, zonal-flow dynamics, and intermittency

Nonlinear theoretical tools for fusion-related microturbulence: Historical evolution, and recent applications to stochastic magnetic fields, zonal-flow dynamics, and intermittency

On Hamiltonian and Action Principle Formulations of Plasma Dynamics

A compressible Hamiltonian electromagnetic gyrofluid model

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