Guiding center plasma models in three dimensions

Abstract

Guiding center plasma models describe the fast charged particle gyration around magnetic field lines by an angle coordinate, defined relative to local orthogonal coordinate axes (ê1,ê2,b̂=B∕B) at each guiding center location. In three dimensions (3D), unlike uniform straight two-dimensional (2D) fields, geometrical effects make the small gyroradius expansion nonuniform in velocity phase space in first order O(ρi∕L). At second order, Hamiltonian and Lagrangian solutions may be undefined even when good magnetic flux surfaces exist; existence requires the magnetic field torsion τ=b̂⋅∇×b̂=0 and τg≡b̂⋅(∇ê1)⋅ê2=0, unless the magnetic field has a 2D symmetry, such as toroidal axisymmetry. Keeping complete 3D geometrical effects also requires the magnetic vector potential term to appear in the electric field at the same order as the electrostatic potential. These problems express properties of magnetic vector potentials, Lagrangians, and the curvature of manifolds, and have analogies to attempts to connect small scale Lagrangian theories to higher dimensional, large scale ones in the grand unification theories of physics.

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

“…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.…”

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

“…The paper 4 argued that while the NCL formulation is formally valid to all orders in simple magnetic fields, it breaks down at second order ⑀ 2 when the magnetic field is sufficiently complex and 3D, that is, fully 3D with finite field line torsion. The reason was that the parameter R could not be defined, where R ϵٌ͑ê 1 ͒ · ê 2 measures the rotation of the magnetic coordinate system that defines the fast gyroangle, for unit coordinate axes ͑ê 1 , ê 2 , b ͒ with b ϵ B / B, ê 1 · b =0, and ê 2 ϵ b ϫ ê 1 .…”

“…4 Because the lowest order ͑⑀ 0 ͒ GC orbits follow B exactly, only the component b · R appears. If good flux surfaces exist, defined by some function of space, then b · R can always be defined as the negative of the geodesic torsion of field lines on the surface ͓Eq.…”

“…4 The total R appears in the Lagrangian. Its definition requires consideration of open 3D neighborhoods, not just the field line or flux surface, since a vector identity based on local orthogonality of the coordinates requires the definition of both ٌê 2 and ٌê 1 ٌ͑ê 1 · ê 2 =−ٌê 2 · ê 1 , Sec.…”

Comment on “Guiding center plasma models in three dimensions” [Phys. Plasmas 15, 092112 (2008)]

“…Recently Sugiyama 1 argued that the asymptotic expansion for guiding-center ͑GC͒ motion, which underlies all modern nonlinear gyrokinetic ͑GK͒ theory, 2 "may be undefined even when good magnetic flux surfaces exist," the problem supposedly arising for three-dimensional ͑3D͒ magnetic fields B with nonzero torsion b · ١ ϫ b ͑b Џ B / ͉B͉͒. One implication is that for such fields the magnetic moment c annot be shown to be adiabatically conserved beyond first order in the expansion in magnetic inhomogeneity ⑀.…”