Response to “Comment on ‘Guiding center plasma models in three dimensions’ ” [Phys. Plasmas 16, 084701 (2009)]

Sugiyama, L.

doi:10.1063/1.3206671

Cited by 7 publications

(15 citation statements)

References 18 publications

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“…Notice that the averaging process for the reduced coordinates (q, ϕ) is possible without having to deal with resonances 24 because there is no small divisor since the fast part of the dynamics involves only one angle θ, as illustrated by formula (8). This is a rather broad property.…”

Section: On Non-minimal Guiding-center Reductionsmentioning

confidence: 99%

“…At each order, one only has to write formula (5) to the n-th order, group into (N n ) q,ϕ all the terms that depend only on quantities that are already explicitly known, expand the result, and last invert the Larmor gyration operator through formula (8).…”

Section: Lie-transforming the Velocity Vector Fieldmentioning

confidence: 99%

“…To each order in the Larmor radius expansion, the algorithm consists in writing down formula (8) and making all the terms explicit. Only two special operations are involved: the Lie-derivatives in the computation of (N n ) q,ϕ , given by formula (18), and the gyro-integral dθ osc computed by (20); these are automatic operations.…”

Section: The Reduction Algorithmmentioning

confidence: 99%

“…The usual coordinate for the gyro-angle suffers from several issues, both from a mathematical and from a physical point of view [5][6][7][8][9] : it is gauge-dependent, does not exist globally for a general magnetic geometry, and induces a disagreement between the coordinates and the physical state, since it implies an anholonomic momentum. In order to avoid these issues and to give a more intrinsic framework to the theory, we consider performing the reduction using the initial physical gauge-independent coordinate for the gyro-angle.…”

Section: Introductionmentioning

confidence: 99%

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A gyro-gauge independent minimal guiding-center reduction by Lie-transforming the velocity vector field

Guillebon

Vittot

2013

Physics of Plasmas

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We introduce a gyro-gauge independent formulation of a simplified guiding-center reduction, which removes the fast time-scale from particle dynamics by Lietransforming the velocity vector field. This is close to Krylov-Bogoliubov method of averaging the equations of motion, although more geometric. At leading order, the Lie-transform consists in the generator of Larmor gyration, which can be explicitly inverted, while working with gauge-independent coordinates and operators, by using the physical gyro-angle as a (constrained) coordinate. This brings both the change of coordinates and the reduced dynamics of the minimal guiding-center reduction order by order in a Larmor radius expansion. The procedure is algorithmic and the reduction is systematically derived up to full second order, in a more straightforward way than when Lie-transforming the phase-space Lagrangian or averaging the equations of motion. The results write up some structures in the guiding-center expansion. Extensions and limitations of the method are considered.

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Section: On Non-minimal Guiding-center Reductionsmentioning

confidence: 99%

Section: Lie-transforming the Velocity Vector Fieldmentioning

confidence: 99%

Section: The Reduction Algorithmmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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A gyro-gauge independent minimal guiding-center reduction by Lie-transforming the velocity vector field

Guillebon

Vittot

2013

Physics of Plasmas

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“…Sugiyama worries that for the indicated magnetic fields it may be impossible to construct a global coordinate system with all of the constraints that are invoked in the asymptotic development of gyrokinetics. In a formal Comment, Krommes (2009) has argued in favor of the conventional asymptotics, 15 but the Response by Sugiyama (2009) shows that the issue is not yet settled.…”

Section: The Current Status Of Gyrokineticsmentioning

confidence: 99%

Nonlinear gyrokinetics: a powerful tool for the description of microturbulence in magnetized plasmas

Krommes¹

2010

Phys. Scr.

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Abstract. Gyrokinetics is the description of low-frequency dynamics in magnetized plasmas. In magnetic-confinement fusion, it provides the most fundamental basis for numerical simulations of microturbulence; there are astrophysical applications as well. In this tutorial, a sketch of the derivation of the novel dynamical system comprising the nonlinear gyrokinetic (GK) equation (GKE) and the coupled electrostatic GK Poisson equation will be given by using modern Lagrangian and Lie perturbation methods. No background in plasma physics is required in order to appreciate the logical development. The GKE describes the evolution of an ensemble of gyrocenters moving in a weakly inhomogeneous background magnetic field and in the presence of electromagnetic perturbations with wavelength of the order of the ion gyroradius. Gyrocenters move with effective drifts, which may be obtained by an averaging procedure that systematically, order by order, removes gyrophase dependence. To that end, the use of the Lagrangian differential one-form as well as the content and advantages of Lie perturbation theory will be explained.The electromagnetic fields follow via Maxwell's equations from the charge and current density of the particles. Particle and gyrocenter densities differ by an important polarization effect. That is calculated formally by a "pull-back" (a concept from differential geometry) of the gyrocenter distribution to the laboratory coordinate system. A natural truncation then leads to the closed GK dynamical system. Important properties such as GK energy conservation and fluctuation noise will be mentioned briefly, as will the possibility (and difficulties) of deriving nonlinear gyrofluid equations suitable for rapid numerical solution -although it is probably best to directly simulate the GKE. By the end of the tutorial, students should appreciate the GKE as an extremely powerful tool and will be prepared for later lectures describing its applications to physical problems.

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Gyrosymmetry: Global considerations

Burby

Qin

2012

Physics of Plasmas

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In the guiding center theory, smooth unit vectors perpendicular to the magnetic field are required to define the gyrophase. The question of global existence of these vectors is addressed using a general result from the theory of characteristic classes. It is found that there is, in certain cases, an obstruction to global existence. In these cases, the gyrophase cannot be defined globally. The implications of this fact on the basic structure of the guiding center theory are discussed. In particular, it is demonstrated that the guiding center asymptotic expansion of the equations of motion can still be performed in a globally consistent manner when a single global convention for measuring gyrophase is unavailable. The latter fact is demonstrated directly by deriving a new expression for the guiding-center Poincaré-Cartan form exhibiting no dependence on the choice of perpendicular unit vectors.

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Response to “Comment on ‘Guiding center plasma models in three dimensions’ ” [Phys. Plasmas 16, 084701 (2009)]

Abstract: A dynamical theory of electron transfer: Crossover from weak to strong electronic coupling

Cited by 7 publications

References 18 publications

A gyro-gauge independent minimal guiding-center reduction by Lie-transforming the velocity vector field

A gyro-gauge independent minimal guiding-center reduction by Lie-transforming the velocity vector field

Nonlinear gyrokinetics: a powerful tool for the description of microturbulence in magnetized plasmas

Gyrosymmetry: Global considerations

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