Formulating the geometric information from the spatial to flux transformation along a plasma chord

Figure 5.7: First 10 PCA eigenvalues of the $R(\rho ^2)$ transformation

Figure 5.8: Pressure recovery RMSE about the database mean profile with one standard deviation lines (left axis) and percentage error $\varepsilon$ indicated as crosses (right axis) for the 5 PC model
Figure 5.9: Pressure recovery RMSE about the database mean profile with one standard deviation lines (left axis) and percentage error $\varepsilon$ indicated as crosses (right axis) for the 7 PC model

We attempt to remedy the shortcomings of the previous analysis by adopting an alternative approach to $p(s)$ recovery which is reminiscent of a zero-iteration version of the interpretive procedure used in chapter 4. We include the finest possible changes in the geometry by performing a PCA of the $R(s)$ transformation at many points along an arbitrarily chosen line of sight through the magnetic axis (for convenience we use the $Z$=0 line in the $\phi =0$ plane) for each database equilibrium and use the PCs to model the pressure as above.

Each NEMEC equilibrium in the database was computed on a grid of 55 $s$ points, thus we can compute $R(s)$ on a maximum of 109 points along a line of sight which passes through the magnetic axis. However, if we simply use the values of $R$ on the $s$-grid as predictors, we are also including knowledge of the normalized toroidal flux. We rigorously exclude any incorporation of the flux information in the predictors by defining a purely geometric flux surface label $\rho^2$ as follows:

$$\rho^2(s) = \left[ \frac{R_\mathrm{out}(s) - R_\mathrm{in}(s)}{R_\mathrm{out}(s=1) - R_\mathrm{in}(s=1)} \right]^2$$ (5.5)

where $\rho^2$ has similar radial behaviour to $s$ but is dependent only on the geometry. We interpolate the values of $R$ onto $\rho^2$ and perform a PCA to condense the information into a few PCs. We stress that these PCs pertain only to the flux surface geometry and not the flux itself.

The PCA eigenvalues are shown in figure 5.7. There are far fewer independently varying components here than were in the PCA of the skeleton set of Fourier coefficients, and we can retain virtually all significant PCs for use in a fully quadratic model. This ensures that we capture the finest geometry variations. Note that here, we must also include the current ratios $I_{\mathrm{tor}}/I_{\mathrm{mod}}$ and $I_{\mathrm{vert}}/I_{\mathrm{mod}}$ in the regression model, since $R(\rho ^2)$ along the chosen single line of sight depends strongly on these parameters as well as $p(s)$. At face value, this seems to contradict our assertion that only geometric information is necessary, but it actually represents no difficulty since as stated already, the vacuum parameters are decorrelated from $p(s)$ by construction, and thus convey no direct information regarding the pressure. Furthermore, these current ratios are also recoverable well from geometric information alone, using PCs of the skeleton Fourier coefficients. They are necessary here only because they cannot be determined from the geometry along a single line of sight, which we are limited to because of model size constraints. We also remark that although ${Z_{\mathrm{lim}}}$ is normally required to fix the size of the plasma, it is superfluous here due to the use of the spatial predictors and the fact that the elongation of the plasma cannot be varied to compensate for a change in ${Z_{\mathrm{lim}}}$ in such a way as to preserve the $R(s)$ transformation.

Figures 5.8 and 5.9 show $p(s)$ recovery error (presented on the mean database profile with one standard deviation indicated by solid lines and crosses indicating percentage error) for models comprising the baseline parameters plus 5 and 7 PCs of the spatial information. Recoveries with 4 PCs or less are quite poor, and the fact that the less significant PCs contribute much of the improvement indicates that although the pressure can be determined from the geometry, it is an ill-posed inverse problem. Regardless of how many extra PCs are added, the error in $p(s)$ in the last few channels ($s \geq 0.8$) remains at the same level as for 8 PCs, nevertheless the RMSE is very small and we regard $p(s)$ to be well determined.

This exercise constitutes a proof-of-principle which is interesting for its own sake, but it is unlikely to be of use on an actual experiment, since the accuracy with which the flux surfaces must be specified in order for the procedure to work is extremely high. We found that for good recovery of $p(s)$, it is necessary to include the $5^\mathrm{th}$ PC in models and its eigenvalue can be seem from figure 5.7 to correspond to variations in the range of 3 mm. This probably far exceeds the precision attainable on diagnostics which enable the geometry to be inferred from measurements of flux surface quantities, e.g.  ${p_{\mathrm{e}}}$ measurements from Thomson scattering. The $p(s)$ profile can still be deduced from the experimentally inferred topology, however this is more suited to an interpretive scheme such as that presented in chapter 4 than via the predictive means that we have considered here.

We emphasize that our results are neither limited to a single configuration nor to a narrow range of $p(s)$ profiles, as the database covers a wide range of the operational space of W7-AS. The only restriction is to zero net toroidal current equilibria, which is the normal operating regime for W7-AS. It is an open question as to how the situation would deteriorate in the presence of finite net toroidal current, since this cannot be tested on the current database. It is possible that the presence of two independent profiles might introduce a degeneracy which the procedure would be unable to cope with.