Experimental reconstructions

Figure: Raw signals for shot 43359 from the nmul program. Key to signals: [a] $\cos(m=0)$, [b] $\cos(m=1)$, [c] $\cos(m=0)$, [d] $\phi =0^\circ$ and $36^\circ$ diamagnetic signals, [e] modular coil current, [f] special coil current, [g] toroidal coil current, [h] vertical coil current and [i] Ohmic and plasma currents.

As alluded to in section 3.4, the majority of the available magnetic diagnostics on W7-AS were unusable in our analysis due to lack of compensation for drifts in the power supply and/or failing sampling electronics. Figure 3.19 of the raw magnetic signals for shot 43359 illustrates some of these problems. The moments from the poloidal field measurements [a]-[c] clearly suffer from non-zero signal drift, which is a result of a constant offset voltage across the integration electronics. If the drift were reasonably linear, as in the case of [a], then this could easily be subtracted off after sampling. However, for [b] and [c], this drift is clearly not purely linear and thus introduces varying degrees of uncertainty in the signal magnitude. Here, this is more serious for [c] due to the lower signal strength. Possible explanations (aside from malfunctioning electronics) may be variations in the (supposedly constant) vacuum field due to drifts in the coil currents [e]-[h] originating in the power supply. If these were measured sufficiently accurately, they could be compensated for during post-processing using a series of calibration shots. The jumps in [e]-[g] indicate that the currents are roughly constant within the signal bit resolution (24.4 A), however the magnetic diagnostics are sensitive to field variations in this range. We remark that the overall signal quality of the $B_\theta$ cosine moments is otherwise quite acceptable at half-field (i.e. for $I_{\mathrm{mod}}$ and $I_{\mathrm{s}}$ near 15 kA each, where the main magnetic field is roughly 1.25 T) but degrades sharply at full field (for $I_{\mathrm{mod}}$ and $I_{\mathrm{s}}$ near 30 kA, where the field is roughly 2.5 T) as shown in figure 3.20.

Figure: Raw signals for shot 43664 from the nmul program. Key to signals: [a] $\cos(m=0)$, [b] $\cos(m=1)$, [c] $\cos(m=0)$, [d] $\phi =0^\circ$ and $36^\circ$ diamagnetic signals, [e] modular coil current, [f] special coil current, [g] toroidal coil current, [h] vertical coil current and [i] Ohmic and plasma currents.

Figure 3.21: The compensated versus uncompensated diamagnetic signal

The diamagnetic signals [d] were unaffected by these considerations, however, and we demonstrate the FP procedure and show the typical accuracy achievable using only the vacuum field information and a diamagnetic signal to measure the plasma energy content and thus the first principal component of the foregoing section. Of the two possibilities, we choose the diamagnetic coil located near $\phi =0$ over that near $\phi =36^\circ$ (both shown in fig. 2.6), as the latter lacks hardware compensation. However, the compensation itself could also prove problematic, since the compensation coil in question also picks up a signal from the plasma itself, thus subtraction of the compensation signal systematically reduces the measured diamagnetic signal and gives erroneously low estimates of $W_{\mathrm{p}}$. Figure 3.21 demonstrates this effect by plotting the uncompensated flux signal from the diamagnetic coil against the compensated signal for each case in the database -- with very slight scatter due to weak configurational dependence, the compensated signal is consistently 13% smaller than the uncompensated one. This presents no difficulty to FP, however, since the plasma contribution to the compensation coil signals can be simulated using DIAGNO and the over-compensation effect can be accurately predicted. Reconstructions can thus still be performed directly with the experimental signal.

FP reconstructions using experimental data from the diamagnetic coil are compared to standard NEMEC equilibrium calculations in three cases. Note that this indicates the most pessimistic level of accuracy achievable in magnetic reconstructions, since it assumes knowledge only of the integral of the pressure profile and not its shape. It essentially leads to a single average profile form on all recovered equilibria and to discrepancies in the flux surface geometry when the profile in question differs strongly from the database average (see below). A fully quadratic model in the FP predictors is used: in the notation of section 3.5, we have $X_0=1, X_1=I_{\mathrm{s}}/I_{\mathrm{mod}}, X_2 = I_{\mathrm{tor}}/I_{\mathrm{m... ...mathrm{mod}}, X_4 = {Z_{\mathrm{lim}}} \mbox{ and } X_5 = {\Phi_{\mathrm{dia}}}$, the diamagnetic flux. Note that since the vacuum predictors in the database are chosen independently of one another and the plasma information is itself independent of the vacuum parameters, the usual PCA step in the FP procedure is unnecessary and is omitted.

Figure 3.22: Shot# 31114: [a] discharge parameters, [c] standard equilibrium pressure profile, [e] average flux surface difference (mm) versus $s$; [b],[d],[f] FP (solid) and standard (dashed) flux surfaces for $\phi=0^\circ,18^\circ \mbox{ and } 36^\circ$ respectively.

Figure 3.23: Shot# 31119. [a] discharge parameters, [c] standard equilibrium pressure profile, [e] average flux surface difference (mm) versus $s$; [b],[d],[f] FP (solid) and standard (dashed) flux surfaces for $\phi=0^\circ,18^\circ \mbox{ and } 36^\circ$ respectively.

Figure 3.24: Shot# 31901. [a] discharge parameters, [c] standard equilibrium pressure profile, [e] average flux surface difference (mm) versus $s$; [b],[d],[f] FP (solid) and standard (dashed) flux surfaces for $\phi=0^\circ,18^\circ \mbox{ and } 36^\circ$ respectively.

We compare FP reconstructions to standard equilibrium reconstructions using Thomson data for three cases in figs. 3.22, 3.23 and 3.24. The key to the plots is as follows: equilibrium parameters in the top left, $p(s)$ profile in the top middle, root-mean-square (RMS) difference between FP and standard NEMEC surfaces as a function of $s$ in the poloidal sections at $\phi = 0,18 \mbox{ and } 36^\circ$ shown in the lower 3 plots.

The first case (fig. 3.22) is shot 31114, a high $\beta$ discharge with a fairly linear pressure profile in $s$ that is close to the average profile shape in the FP database. The agreement of the flux surfaces is moderate near the axis but improves steadily towards the boundary. This is to be expected, since the Shafranov shift ($\Delta(s)$) of the flux surfaces is dependent on the local $\beta$ value (and thus on the shape of the pressure profile), which is poorly diagnosed by external magnetic measurements towards the plasma core. Agreement with integral quantities $W_{\mathrm{p}}$, ${B_{\mathrm{axis}}}$ etc. is good, however, and the plasma boundary matches that in the standard calculation.

Shot 31119 (fig. 3.23) has a lower $\beta$ and thus the deviation of the flux surfaces from the well recovered vacuum configuration is not as severe as for 31114. The pressure profile is linear in $s$ and similar to the database average. Flux surfaces agree well towards the boundary and are still reasonably close in the vicinity of the magnetic axis. Scalar parameters agree well also.

The limitation of having no profile information is most apparent for the high-$\beta$ shot 31901 (fig. 3.24), where the peaked pressure profile differs strongly from the database average. This results in a higher Shafranov shift for inner flux surfaces ( $\Delta(s) \propto \beta(s)$) and thus a large discrepancy between FP flux surfaces and the standard calculations towards the centre of the plasma. Again, boundary agreement is within a few mm and scalar parameters are well recovered.