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

Guillebon, Loïc de; Vittot, Michel

doi:10.1063/1.4817020

Cited by 3 publications

(31 citation statements)

References 26 publications

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“…where following Littlejohn's notations, osc = 1 − avg is the projector onto gyro-fluctuations, with avg the complementary projector onto gyro-averages: [12], for instance. This lets some freedom in the procedure and suggests to impose stronger requirements for the reduction.…”

Section: The Hierarchy Of Requirementsmentioning

confidence: 99%

“…In a previous work [12], we proposed a guiding-center reduction which avoided to introduce a gyro-gauge. The idea was to Lie transform directly the equations of motion instead of Lie transforming the Lagrangian, as is usually done [13,14].…”

Section: Introductionmentioning

confidence: 99%

“…It was already present in [12] when Lie transforming the equations of motion, but it is more involved to deal with for the full guiding-center reduction, because the coordinate c will be changed as well, and not only derivatives or vector fields are involved, but also differential forms.…”

Section: Introductionmentioning

confidence: 99%

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Gyro-gauge-independent formulation of the guiding-center reduction to arbitrary order in the Larmor radius

Guillebon¹,

Vittot²

2013

Plasma Phys. Control. Fusion

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The guiding-center reduction is studied using gyro-gauge-independent coordinates. The Lagrangian 1-form of charged particle dynamics is Lie transformed without introducing a gyro-gauge, but using directly the unit vector of the component of the velocity perpendicular to the magnetic field as the coordinate corresponding to Larmor gyration.The reduction is shown to provide a maximal reduction for the Lagrangian and to work to all orders in the Larmor radius, following exactly the same procedure as when working with the standard gauge-dependent coordinate.The gauge-dependence is removed from the coordinate system by using a constrained variable for the gyro-angle. The closed 1-form dθ is replaced by a more general non-closed 1-form, which is equal to dθ in the gauge-dependent case. The gauge vector is replaced by a more general connection in the definition of the gradient, which behaves as a covariant derivative, in perfect agreement with the circle-bundle picture. This explains some results of previous works, whose gauge-independent expressions did not correspond to a gauge fixing but indeed correspond to a connection fixing.In addition, some general results are obtained for the guiding-center reduction. The expansion is polynomial in the cotangent of the pitch-angle as an effect of the structure of the Lagrangian, preserved by Lie derivatives. The induction for the reduction is shown to rely on the inversion of a matrix which is the same for all orders higher than three. It is inverted and explicit induction relations are obtained to go to arbitrary order in the perturbation expansion. The Hamiltonian and symplectic representations of the guiding-center reduction are recovered, but conditions for the symplectic representation at each order are emphasized.

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Section: The Hierarchy Of Requirementsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Gyro-gauge-independent formulation of the guiding-center reduction to arbitrary order in the Larmor radius

Guillebon¹,

Vittot²

2013

Plasma Phys. Control. Fusion

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“…A first illustration is provided by the guiding-center transformation. For the gyro-angle c, the transformation is connection-independent 18,19 . For the coordinate θ, the transformationθ = ··e G 2 e G 1 θ is gauge dependent, but in such a way as to make the induced transformationc = ··e G 2 e G 1 c for c gauge independent, with G n the vector field generating the n-th order transformation.…”

Section: Intrinsic Counterpart Of the Gauge Arbitrarinessmentioning

confidence: 99%

“…shows thatφ is also singular in p sin ϕ = 0. At higher orders, this singularity is expected to affect all the coordinates, because it is related to the order in the cotangent of the pitch-angle compared to the order in the Larmor radius, as is emphasized in Refs [17][18][19]…”

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confidence: 99%

On an intrinsic approach of the guiding-center anholonomy and gyro-gauge arbitrariness

Guillebon

Vittot

2013

Physics of Plasmas

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In guiding center theory, the standard gyro-angle coordinate is associated with gyro-gauge dependence, the global existence problem for unit vectors perpendicular to the magnetic field, and the notion of anholonomy, which is the failure of the gyro-angle to return to its original value after being transported around a loop in configuration space. We analyse these three intriguing topics through the lens of a recently proposed, global, gauge-independent gyro-angle. This coordinate is constrained and therefore necessitates the use of a covariant derivative. It also highlights the intrinsic meaning and physical content of gyro-gauge freedom and anholonomy. There are, in fact, many possible covariant derivatives compatible with the intrinsic gyro-angle, and each possibility corresponds to a different notion of gyro-angle transport. This observation sheds new light on Littlejohn's notion of gyro-angle transport and suggests a new derivation of the recently discovered global existence condition for unit vectors perpendicular to the magnetic field. We also discuss the relationship between Cartesian position-momentum coordinates and the intrinsic gyro-angle. V C 2013 AIP Publishing LLC. [http://dx.

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A very general electromagnetic gyrokinetic formalism

McMillan

Sharma

2016

Physics of Plasmas

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We derive a gyrokinetic formalism which is very generally valid: the ordering allows both large inhomogeneities in plasma flow and magnetic field at long wavelength, like typical drift-kinetic theories, as well as fluctuations at the gyro-scale. The underlying approach is to order the vorticity to be small, and to assert that the timescales in the local plasma frame are long compared to the gyrofrequency. Unlike most other derivations, we do not treat the long and short wavelength components of the fluctuating fields separately; the single-field description defines the particle motion and their interaction with the electromagnetic field at small-scale, the system-scale, and intermediate length scales in a unified fashion. As in earlier literature, the work consists of identifying a coordinate system where the gyroangle-dependent terms are small, and using a near-unity transform to systematically find a set of coordinates where the gyroangle dependence vanishes.We derive a gyrokinetic Lagrangian which is valid where the vorticity |∇ × (E × B/B)| is small compared to the gyrofrequency Ω, and the magnetic field scale length is long compared to the gyroradius; we also require that time variation be slow in an appropriately chosen reference frame. This appears to be a minimum set of constraints on a gyrokinetic theory, and is substantially more general than earlier approaches. It is the general-geometry electromagnetic extension of Ref [1] (which is an electrostatic formalism with a homogeneous background magnetic field). This approach also does not require a separate treatment of

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

Cited by 3 publications

References 26 publications

Gyro-gauge-independent formulation of the guiding-center reduction to arbitrary order in the Larmor radius

Gyro-gauge-independent formulation of the guiding-center reduction to arbitrary order in the Larmor radius

On an intrinsic approach of the guiding-center anholonomy and gyro-gauge arbitrariness

A very general electromagnetic gyrokinetic formalism

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