A gyrokinetic continuum code based on the numerical Lie transform (NLT) method

Abstract We investigate the interaction of subharmonic resonances in the nonlinear quasiperiodic Mathieu equation,(1)x..+[δ+ϵ(cos⁡ω1t+cos⁡ω2t)]x+αx3=0. We assume that ϵ ≪ 1 and that the coefficient of the nonlinear term, α, is positive but not necessarily small. We utilize Lie transform perturbation theory with elliptic functions — rather than the usual trigonometric functions — to study subharmonic resonances associated with orbits in 2m : 1 resonance with a respective driver. In particular, we derive analytic expressions that place conditions on (δ, ϵ, ω1, ω2) at which subharmonic resonance bands in a Poincaré section of action space begin to overlap. These results are used in combination with Chirikov’s overlap criterion to obtain an overview of the O(ϵ) global behavior of equation (1) as a function of δ and ω2 with ω1, α, and ϵ fixed.

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ON THE REPRESENTATION OF MAPS BY LIE TRASFORMS

Istituto Lombardo - Accademia di Scienze e Lettere - Rendiconti di Scienze ◽

10.4081/scie.2012.135 ◽

2012 ◽

Author(s):

Antonio Giorgilli

Keyword(s):

Normal Form ◽

Arbitrary Dimension ◽

Lie Series ◽

Straightforward Manner ◽

Standard Map ◽

Lie Transform ◽

Normal Form Methods

The problem of representing a class of maps in a form suited for application of normal form methods is revisited. It is shown that using the methods of Lie series and of Lie transform a normal form algorithm is constructed in a straightforward manner. The examples of the Schröder–Siegel map and of the Chirikov standard map are included, with extension to arbitrary dimension.

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Guiding-centre transformation of the radiation–reaction force in a non-uniform magnetic field

Journal of Plasma Physics ◽

10.1017/s0022377815000744 ◽

2015 ◽

Vol 81 (5) ◽

Cited By ~ 8

Author(s):

E. Hirvijoki ◽

J. Decker ◽

A. J. Brizard ◽

O. Embréus

Keyword(s):

Magnetic Field ◽

Magnetic Fields ◽

Uniform Magnetic Field ◽

Reaction Force ◽

Charged Particles ◽

Collision Operator ◽

Radiation Reaction ◽

Transform Method ◽

Lie Transform ◽

Radiation Reaction Force

In this paper, we present the guiding-centre transformation of the radiation–reaction force of a classical point charge travelling in a non-uniform magnetic field. The transformation is valid as long as the gyroradius of the charged particles is much smaller than the magnetic field non-uniformity length scale, so that the guiding-centre Lie-transform method is applicable. Elimination of the gyromotion time scale from the radiation–reaction force is obtained with the Poisson-bracket formalism originally introduced by Brizard (Phys. Plasmas, vol. 11, 2004, 4429–4438), where it was used to eliminate the fast gyromotion from the Fokker–Planck collision operator. The formalism presented here is applicable to the motion of charged particles in planetary magnetic fields as well as in magnetic confinement fusion plasmas, where the corresponding so-called synchrotron radiation can be detected. Applications of the guiding-centre radiation–reaction force include tracing of charged particle orbits in complex magnetic fields as well as the kinetic description of plasma when the loss of energy and momentum due to radiation plays an important role, e.g. for runaway-electron dynamics in tokamaks.

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A many-particle approach to the gyro-kinetic theory

Journal of Plasma Physics ◽

10.1017/s0022377806006258 ◽

2007 ◽

Vol 73 (5) ◽

pp. 757-772 ◽

Cited By ~ 3

Author(s):

ALEXEY MISHCHENKO ◽

AXEL KÖNIES

Keyword(s):

Kinetic Equation ◽

Kinetic Theory ◽

First Principles ◽

Conventional Approach ◽

The Self ◽

Collision Term ◽

Lie Transform ◽

Self Consistent ◽

The Many ◽

Particle Approach

AbstractA systematic first-principles approach to the many-particle formulation of the gyro-kinetic theory is suggested. The gyro-kinetic many-particle Hamiltonian is derived using the Lie transform technique. The generalized gyro-kinetic equation is obtained following the Born–Bogoliubov–Green–Kirkwood–Yvon approach. The microscopic expression for the self-consistent potential and the polarization density is obtained. It is shown that new terms appear in the gyro-kinetic polarization that can not be derived in the conventional approach. An expression for the collision term is obtained in the Landau approximation.

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Guiding-center recursive Vlasov and Lie-transform methods in plasma physics

Journal of Plasma Physics ◽

10.1017/s0022377809007946 ◽

2009 ◽

Vol 75 (5) ◽

pp. 675-696 ◽

Cited By ~ 13

Author(s):

A. J. BRIZARD ◽

A. MISHCHENKO

Keyword(s):

Phase Space ◽

Plasma Physics ◽

Hamiltonian Dynamics ◽

Transform Methods ◽

Lie Transform ◽

Recursive Solution ◽

Space Transformation

AbstractThe gyrocenter phase-space transformation used to describe nonlinear gyrokinetic theory is rediscovered by a recursive solution of the Hamiltonian dynamics associated with the perturbed guiding-center Vlasov operator. The present work clarifies the relation between the derivation of the gyrocenter phase-space coordinates by the guiding-center recursive Vlasov method and the method of Lie-transform phase-space transformations.

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