The derivation of equations for fluctuations and transport in flux-tube geometries

Abstract

A self-consistent set of equations for the fast space–time evolution of fluctuations and the slow space–time evolution of density and flows in a toroidal plasma, relevant for simulations using field-aligned coordinates in thin flux tubes, has been derived. The methodology for the derivation of these equations is outlined for a model set of equations for the plasma edge, specific to resistive ballooning modes but readily adaptable to other instabilities. The derivation proceeds by first writing the axisymmetric and fluctuating equations in the usual toroidal coordinate system. These are then transformed to the twisted coordinate flux-tube system. Most simulations which use twisted flux-tube computational grids transform to the field-aligned coordinate system first and then take averages to obtain the slow evolution. They however miss some terms since the two operations, namely, multiscale separation and coordinate transformation, do not necessarily commute, because of subsidiary assumptions on the box size. In the present formulation, all the relevant neoclassical effects such as the Pfirsch–Schlüter current and the Stringer spin-up as well as the toroidal Reynolds stress are properly included. This set of multiscale equations is appropriate for the study of the formation and evolution of transport barriers.