Orbit-averaged guiding-center Fokker–Planck operator for numerical applications

Abstract

A guiding-center Fokker–Planck operator is derived in a coordinate system that is well suited for the implementation in a numerical code. This differential operator is transformed such that it can commute with the orbit-averaging operation. Thus, in the low-collisionality approximation, a three-dimensional Fokker–Planck evolution equation for the orbit-averaged distribution function in a space of invariants is obtained. This transformation is applied to a collision operator with nonuniform isotropic field particles. Explicit neoclassical collisional transport diffusion and convection coefficients are derived, and analytical expressions are obtained in the thin orbit approximation. To illustrate this formalism and validate our results, the bootstrap current is analytically calculated in the Lorentz limit.

Radiation reaction induced non-monotonic features in runaway electron distributions

“…(2.7) are gyro-averaged projections of their guiding-center push-forwarded particle phase-space counterparts. For a detailed definition of the guiding-center friction and diffusion coefficients, we refer to (Brizard 2004;Decker et al 2010;Hirvijoki et al 2013). The general expressions are non-trivial but, in the limit of a uniform plasma, the test-particle operator assuming isotropic background particle distributions becomes diagonal with reasonably simple non-zero components…”

“…It can, however, be eliminated using guiding-center Lie-transform perturbation methods. The transformation of the Hamiltonian equations of motion is one of the classical results in modern plasma physics (see Littlejohn 1983;Cary & Brizard 2009), and the FokkerPlanck collision operator has been considered in (Brizard 2004;Decker et al 2010;Hirvijoki et al 2013). The final step necessary to formulate our problem, the transformation of the RRforce, was given recently in .…”

Monte Carlo method and High Performance Computing for solving Fokker–Planck equation of minority plasma particles

“…An interested reader may consult Ref. (Decker et al 2010) for more details. It is clear that the increased efficiency of the computational effort comes at the price of mathematical complexity.…”

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

“…Radial transport coefficents including neoclassical effects could then be explicitly derived. Such operator could then be readily implemented in a 3-D orbit-averaged guiding-center kinetic code [19]. The relativistic guiding-center Lagrangian one-form for the guiding-center phase-space…”