Impurity charge state transport in tokamak and stellarator plasmas

Abstract

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.

Nonlinear Problems of Equilibrium Charge State Transport in Hot Plasmas

“…Moreover, the concept of impurity CS transport allows one to draw an unexpected conclusion about the determining role of atomic processes in impurity transport [17]. Clear evidence of this interpretation can be revealed in experiments and related modelling practice.…”

“…They are: -coronal equilibrium (CE) of heavy impurities indeed found in the early studies of Cr, Ni, and Mo impurities on the TFR tokamak [22,23], and then confirmed by modelling practice for W on the ASDEX Upgrade tokamak [14,15]. This meant that the equilibrium assumed by the DCM between 1D particle transport and 1D balance of ionization-recombination processes could not be observed; -a noticeable drop in the diffusion coefficient D at the central impurity accumulation and a sharp difference between transport in the plasma core from that in the rest peripheral plasma (see, e.g., [24]) (that occurs due to the largest size in the central equilibrium cell (CEC) [17]); -the coupling found between central accumulation of impurities and diffusion coefficient D: the greater the accumulation of the impurity in the plasma core, the lower D [17,25], and vice versa, that is, the "flat" distribution is systematically associated with an increase D to anomalous values (in the conventional comparison with neoclassical predictions) from the core to the plasma edge; -two characteristic maxima on the total density profiles: impurity accumulation takes place simultaneously at the edge and in the plasma core, as found, e.g., for W in JET [1,17].…”

“…Meanwhile, the imposition of ionization-recombination on the particle transport results in a fundamental nonlinearity of transport analysis. So, the analysis of nonlinear equations of even a simplest 2D system of four charge states (two in space and two in charge called the equilibrium cell (EC)) -leads to a transcendental equation [17]. The proposed general solution can be developed based on a number of reducing 2D schemes and the use of the pseudo-state technique.…”

“…In this paper, we analyze in detail the CS equilibrium and its main dimensionless scale, constant, λ, which turns out to be common to all emerging nonlinear problems and possible reducing schemes for a given GM [17]. So, a direct relationship is established between λ, the standard transport coefficients and the impurity confinement time p…”

“…A new approach to this problem was based on the concept of impurity CS transport [17][18][19][20][21]. First of all, it was required to make a terminology distinction between particle and CS transports, which proved useful for clarifying the structure and probabilistic nature of transport.…”