Lagrangian formulation of neoclassical transport theory

Abstract

Neoclassical transport theory is developed in a Lagrangian formulation in contrast to the usual Eulerian development. The Lagrangian formulation is constructed from the three actions: magnetic moment, parallel invariant, and bounce-averaged poloidal flux. By averaging over the fast orbital time scales, an equation in the actions alone of the Fokker–Planck type is obtained. The coefficients give the rates of the elementary neoclassical scattering processes. This action space form of the kinetic equation contains in explicit form processes like banana diffusion and the kinematic Ware pinch. The associated fluxes can be computed by simple moments without having deviations from the local Maxwellian. All the trapped-particle contributions are of this explicit type. Another class of fluxes arise from perturbations to the Maxwellian and are termed implicit. The decomposition of the fluxes into explicit and implicit parts is a key feature of the Lagrangian formulation. These contributions correspond to distinct physical processes and have separate Onsager symmetry theorems (explicit and implicit) for their respective transport matrices. The theory does not depend on the details of the Fokker–Planck coefficients but only on some very general properties and is thus applicable—without modification of the formalism—to nonaxisymmetric and turbulent systems. This general formulation is the primary purpose of the work. To benchmark the theory, in the present paper, the tokamak transport coefficients (for the Lorentz gas) are computed and compared to the known Eulerian results, demonstrating the equivalence of the two formulations. The elementary processes responsible for the neoclassical pinch and bootstrap effects (somewhat obscured in the Eulerian picture) are identified and the physical basis for their Onsager symmetry relationship is clarified.