Description
Reliable magnetic equilibrium reconstruction is key for the interpretation of experimental data and for physics modelling. Uncertainties of equilibrium quantities are frequently not provided although essential for the validation and quantification of derived physical quantities. A Monte-Carlo approach applicable to a free-boundary equilibrium reconstruction is suitable to provide uncertainties on any scalar or profile quantity or shape configuration.
A Monte-Carlo approach is applied to the Integrated Data Equilibrium
(IDE) code. IDE couples a kinetic free-boundary Grad-Shafranov solver with a solver for the current diffusion equation [1,2]. A comprehensive set of external and internal measurements available to constrain the equilibrium is thereby complemented by a flux-surface averaged current density provided by the current diffusion. A covariance matrix is evaluated from the response function, coupling the coefficients of the equilibrium base functions with the set of measurements, and from the uncertainties of all measured and modeled constraints. Within a Monte-Carlo sampling approach the equilibrium coefficients are sampled from a multivariate normal distribution using this covariance matrix. A corresponding sample of the equilibrium is reconstructed from this sample of coefficients. Any equilibrium quantity of interest can be evaluated from this equilibrium sample. Estimates of the equilibrium quantities and their uncertainties are given by the mean and standard deviation of the samples.
A set of scalar equilibrium quantities comprise typically the geometry coordinates of the magnetic axis, the saddle points, dedicated points on the separatrix, the strike-line positions, volume-averaged coordinates, shape coordinates such as upper and lower triangularity, plasma current, energy content, plasma self inductivity, various beta values, dedicated $q$-values, the distance between the 1$^\text{st}$ and 2$^\text{nd}$ X-points defining the equilibrium category, and the reliability to distinguish the alternative divertor configurations SN, XD, HFS-SF-, LFS-SF-, and SF+ of the new upper divertor at ASDEX Upgrade. The set of profile quantities comprises, e.g., the current density profile, and the $q$- and magnetic shear profiles. The list of physical quantities can easily be extended as with the applied Monte-Carlo method the uncertainty of any magnetic quantity can be estimated.
[1] R. Fischer et al., Fusion Sci. Technol., 69:526-536, 2016
[2] R. Fischer et al., Nucl. Fusion, 59:056010, 2019