Statistical analysis of the equilibrium configurations of the W7-X stellarator using function parameterization

The W7-X stellarator, under construction at IPP-Greifswald, is being designed to demonstrate the steady state capability of fusion devices. Due to the pulse length involved, real time monitoring and control of the discharges is a crucial issue in steady state operations. For W7-X, we have planned a sequence of in-depth analyses of the magnetic configurations which, ultimately, will lead to a proper understanding of plasma equilibrium, stability and transport. It should also provide insight into the parameterization of the various plasma-related quantities which is important from the point of view of real time study. The first step in our sequence of analyses involved a study of the vacuum configuration, including the detectable magnetic islands, of W7-X. We now proceed to the scenario at finite beta considering full magnetohydrodynamic (MHD) equilibria based on vmec2000 calculations. A database of order 10000 equilibria was calculated on the same parameter space for the coil current ratios. The parameters which were varied randomly and independently consist of the external coil current ratios (6), the parameters of the profiles (as functions of normalised toroidal flux) of plasma pressure and toroidal current (4+4) and the plasma size (aeff) which is required to vary the plasma volume. A statistical analysis, using Function Parametrization (FP), was performed on a sample of well-converged equilibria. The plasma parameters were varied to allow a good FP for the expected values in W7-X, i.e. volume-averaged <β> up to 5% and toroidal net-current of up to ±50 kA for a mean field strength of about 2 T throughout the database. The profiles were chosen as a sequence of polynomials with the property that the addition of a higher order polynomial would not change the lower order volume-averaged moments of the resulting profile. The aim of this was to try to avoid cross correlations in the independent input parameters for the database generation. However, some restrictions had to be incorporated in order to maximize the number of converged vmec runs with randomly generated inputs. Therefore, the peaking factor ss βaxis/<β> for the pressure profile was restricted to avoid values of βaxis > 12%. This is reasonable in view of experimental scenarios where large peaking factors are usually connected with low values of <β>. Additionally, the central current density was limited in order to avoid very small values of central iota. This was to ensure a reasonable vmec-convergence. We will present in detail the regression models that were employed and a statistical description of the recovery accuracy of plasma information from this database such as the profiles of the rotational transform and the differential volume V' as well as those of the leading order Fourier coefficients for the geometry and the magnetic field strength. (author)