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.