Differential formulation of the gyrokinetic Landau operator

TL;DR

This paper explores the potential for a differential formulation of the gyrokinetic Landau collision operator, highlighting challenges in reducing the system to five dimensions and suggesting the necessity of gyroangle dependence in potential functions.

Contribution

It investigates the feasibility of a differential form of the gyrokinetic Landau operator and discusses the implications of gyroangle dependence in potential functions.

Findings

Differential formulation is possible in gyrokinetic phase space.

Reducing to five dimensions via gyroaveraging is challenging.

Gyroangle dependence of potential functions is likely necessary.

Abstract

Subsequent to the recent rigorous derivation of an energetically consistent gyrokinetic collision operator in the so-called Landau representation, this paper investigates the possibility of finding a differential formulation of the gyrokinetic Landau collision operator. It is observed that, while a differential formulation is possible in the gyrokinetic phase space, reduction of the resulting system of partial differential equations to five dimensions via gyroaveraging poses a challenge. Based on the present work, it is likely that the gyrocentre analogues of the Rosenbluth-MacDonald-Judd potential functions must be kept gyroangle dependent.