Impurity retention by divertors. I. One dimensional models

An analytic simple fluid theory (SFT) is derived to predict when a tokamak divertor should retain impurities. Only the simplest, one dimensional (1-D), case of leakage for ions from the point of ionization, along B to points far upstream, is considered. The SFT builds directly on earlier 1-D treatments of divertor retention of impurities. It is found essential to introduce cross-field leakage into the SFT, associated with the two dimensionality of the actual divertor situation, otherwise the upstream regions suffer catastrophic, and unphysical, impurity accumulation. Without this correction, divertor leakage is predicted to occur even under rather cold (collisional) divertor operation. With this correction, appreciable leakage is predicted to occur only for divertor temperatures that are so high as to be unlikely to occur, at least for the case where impurity neutrals sputtered from the target plate are ionized on their first pass through the divertor plasma fan-the 'shallow injection' case. Thus, previous analytic prescriptions for divertor retention are too pessimistic. For 'deep injection' cases, as can occur with recycling gases such as neon, or with wall sputtered sources, a prescription is found for the plasma temperature above which impurity leakage occurs. The most critical factor governing retention is the location at which the impurity neutrals are first ionized. The predictions of the SFT are compared with results using the Monte Carlo impurity transport code DIVIMP (Divertor Impurity). Close agreement is found for plasma conditions that are strongly collisional, but for weaker collisionality the SFT is found to overestimate leakage