An EPIC (Extended Particle-in-Cell) scheme is developed to model electrostatic drift-wave turbulence in three space dimensions. An existing two-dimensional code that treats magnetic shear and non-adiabatic effects is adapted to incorporate the ICEx algorithm, that enables instability-free numerical modelling of complex wave phenomena without a timestep restriction. Successful simulations of drift-wave interactions are presented.
1. Introduction
This paper builds on the work presented in Ref. [1 ], wherein a simulation capability was developed for electrostatic drift-wave interactions at parameters relevant to magnetic fusion experiments, specifically tokamaks. Ref.
[ I ] provides a description of the background and the motivation for a two-dimensional study of a set of equations proposed by Biskamp and Walter [2]. Their system includes dissipation, non-adiabatic "i-~" effects and also magnetic shear. Although much interest attaches to a two-dimensional model, full treatment of magnetic shear requires solving 3-D evolutionary equations, containing advective non-linearity.
There is a body of evidence from 2-D studies [ 1,3,4] that particle-mesh schemes are advantageous for modelling advective problems, particularly when because of shear it is implausible to use periodic boundary conditions in the radial direction. However generalising the Extended Particle-in-Cell (EPIC) scheme of Ref. [ 1 ] to three dimensions is not straightforward. In 2-D use was made of a trick that allowed an effectively implicit treatment of the shear wave term, which does not work quite so well in 3-D. Analytic or von Neumann analyses of schemes representing simple modifications to the code DRIFT of Ref. [ 1 ] showed the only stable scheme that could be constructed was first order, and moreover tests of the resulting code revealed a mesh-decoupling instability. Hence a major revision of DRIFT was required.
This paper describes the ICEx scheme, a modification of the ICE scheme [5] to handle cross derivatives, that appear when ICE is applied to a shear wave problem. Ref. [ 1] indicated that implementing implicit schemes in a particle-mesh context is straightforward, and this will turn out to be the case for ICEx. ICEx is an essential ingredient of the code DRIFT3 that solves problems in three-dimensional drift-wave turbulence. Section 2 sets out the precise equations solved and analytic results relevant to code testing. Section 3 outlines the scheme employed, Section 4.1 describes ICEx in detail, and the rest of Section 4 and Section 5 contain other information on the coding. Section 6 describes results from DRIFT3, and Section 7 provides a summary.