The standard one-dimensional formulation of impurity transport has proved to be highly questionable for impurities with medium/high Z, and is unsuitable for tungsten. This allows to reveal an inconsistent interpretation of the coupling between particle transport and ionization/recombination processes. A discrete two-dimensional (2D) Fokker–Planck–Kolmogorov equation with charge and radius variables is proposed to study impurity equilibrium and transport in tokamak and stellarator plasmas. The impurity behaviour is consistently considered in terms of a 2D Markovian stochastic process, which has a complex structure according to probabilistic laws that include particle transport as part of a more general impurity charge state transport. It is found that a self-consistent 2D Markovian impurity equilibrium has the consequence that particle transport corresponding to the stationary state is governed by ionization and recombination processes, which are statistically independent of the particle motion. These generalised 2D coronal equilibrium conditions, usually observed for heavy impurities in experiments, turn out to be the stationary solution of the discrete 2D equation introduced and are justified in a discrete 2D grid model of impurity equilibrium. A complete analysis of the simplest discrete 2D equilibrium cell provides the basis for a general solution, while the grid model is reduced to a set of such local equilibrium cells using a pseudo-state technique.Modelling of the density profiles of carbon, argon and tungsten impurities is carried out by TICS (transport of impurity charge states) code developed using available experimental data. It is found that the introduced equilibrium function, which is a discrete profile of the ratios of reduced ionization and recombination rates, systematically predicts the central accumulation, equilibrium and radial transport of impurities in H- and L-mode and ion transport barrier plasmas, in agreement with the experimental data.
Modification of local plasma parameters by impurity injection