Flux surface geometry (3D) recovery

Figure 3.9: Radial behaviour of some leading order flux surface Fourier coefficients for the equilibrium shown

Figure 3.9 shows the flux surface geometry at $\phi =0$ of a high-$\beta$ equilibrium from the database together with some of its leading Fourier coefficients versus the radial flux coordinate. Their smooth radial behaviour is quite apparent, a result of both the spectral minimization and the energy principle applied in NEMEC, which yields smooth flux surfaces since higher order Fourier contributions are penalized and in any case make little contribution to the overall energy [22]. In order to reconstruct the flux surface geometry we can again exploit this smoothness and treat each Fourier coefficient of $R$ and $Z$ as an independent 1D profile quantity, leading to a large number of separate regressions. The model used to recover individual Fourier coefficients $y$ is thus:

$$y = \sum_{k=M_{\mathrm{min}}}^M { \sum_{i=0 \atop j=i}^{N} {a_{ijk} X_i X_j \rho^k} }$$ (3.5)

where $M_{\mathrm{min}}=0$ for $m=0$ and $M_{\mathrm{min}}=1$ for $m>0$: this is necessary to ensure that the flux surface volume vanishes as $s \rightarrow 0$.

Note that the success of this procedure is by no means guaranteed. Firstly, if the recovery of the individual ${R_{\mathrm{mn}}}$ and ${Z_{\mathrm{mn}}}$ is sufficiently poor, then the FP-reconstructed flux surface geometry will show a correspondingly large deviation from that of NEMEC and the procedure will be too inaccurate to be of use. The sum of the individual recovery errors of the Fourier coefficients provides an upper bound to the final error in the flux surface geometry, since in general, the errors will have different correlations. Also, modelling will fail in any case if the Fourier coefficient representation is not unique. We anticipate, however, that the Fourier decomposition will be unique, since it already accounts for the symmetry of the system and furthermore satisfies the spectral minimization condition.