An analytical solution of the gyrokinetic equation for the calculation of neoclassical effects

TL;DR

This paper derives an analytical solution to the gyrokinetic equation for collisional plasmas with magnetic islands, enabling better understanding of neoclassical effects when island width approaches ion Larmor radius.

Contribution

It introduces an analytical method to solve the gyrokinetic equation in the presence of magnetic islands, extending previous drift-kinetic approaches to new regimes.

Findings

Analytical solution for gyrokinetic equation with magnetic islands.

Enhanced understanding of neoclassical effects in specific plasma regimes.

Applicable to plasmas where island width is comparable to ion Larmor radius.

Abstract

The purpose of this document is to find an analytical solution for the gyrokinetic equation under specific, simplificative hypotheses. The case I am considering is that of a collisional plasma in the presence of a chain of magnetic islands. The presence of the magnetic islands causes the onset of perturbative fields, in particular an electrostatic field, with a gradient length-scale comparable with the island's width. When the island's width w becomes comparable with the ion Larmor radius rho_i , the drift-kinetic equation is inadequate to treat the transport and the calculation of the neoclassical effects. Nevertheless, I'm going to solve the equation with the methods described by S. P. Hirshman and D. J. Sigmar in the review paper "Neoclassical transport of impurities in tokamak plasmas", which was developed to solve the drift-kinetic equation in different regimes of collisionality. I'm going to remind first the drift-kinetic theory, which was largely used to study classical and neoclassical transport in magnetized plasmas. Then I'm moving to the gyrokinetic theory, which brings to a more complicated kinetic equation, and I'm going to solve it by applying the approach used previously by Hirshman & Sigmar.