Potential of a Bayesian Integrated Determination of the Ion Effective Charge via Bremsstrahlung and Charge Exchange Spectroscopy in Tokamak Plasmas

Abstract

Reliable and accurate estimates of the ion effective charge <formula formulatype="inline" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex Notation="TeX">$Z_{\rm eff}$ </tex></formula> in tokamak plasmas are of key importance with respect to the impurity transport studies and the establishment of thermonuclear burn criteria. These issues are of fundamental interest to ITER and reactor operational scenarios in general. However, <formula formulatype="inline" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex Notation="TeX">$Z_{\rm eff}$</tex></formula> estimates derived from bremsstrahlung spectroscopy on the one hand and from the weighted summation of individual impurity concentrations obtained via charge exchange spectroscopy (CXS) on the other hand often are not compatible. This is a long-standing problem in fusion plasma diagnosis. A rigorous analysis of uncertainty sources and their propagation in the experimental determination of <formula formulatype="inline" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex Notation="TeX">$Z_{\rm eff}$</tex></formula> can contribute significantly to the derivation of a <formula formulatype="inline" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex Notation="TeX">$Z_{\rm eff}$</tex></formula> value with reduced uncertainty that is consistent with both the bremsstrahlung and CXS data sets. In this paper, Bayesian probability theory is used in an integrated approach as a powerful tool for an advanced error analysis in the derivation of <formula formulatype="inline" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex Notation="TeX">$Z_{\rm eff}$</tex></formula> , even in the presence of systematic errors on the data. A simple probabilistic model is proposed for the estimation of <formula formulatype="inline" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex Notation="TeX">$Z_{\rm eff}$</tex></formula> , first assuming only statistical uncertainty and then taking into account also the systematic deviations. The obtained <formula formulatype="inline" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex Notation="TeX">$Z_{\rm eff}$</tex></formula> estimates have smaller error bars than the <formula formulatype="inline" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex Notation="TeX">$Z_{\rm eff}$ </tex></formula> values derived from the individual bremsstrahlung and CXS measurements, approaching ITER requirements. The estimates are shown to be consistent with all available information. In addition, the systematic errors on the data are quantized through the requirement of data consistency between different time slices in the acquired measurements.