Comparison with standard calculations

NEMEC simulations for some selected W7-AS shots were carried out [66] using a method similar to the one employed here, where an attempt was made to find the equilibrium pressure profile that symmetrized the Thomson electron pressure profile and simultaneously matched the kinetic energy content to the measured diamagnetic energy. This was severely time consuming however, since the procedure required several NEMEC calculations with manual intervention between iterations to determine the direction of descent. We now compare these calculations to results obtained from our interpretive procedure for a few experimental shots. Note that matching of the plasma energy content was not enforced for these interpretations. Also, the ion pressure shown in figures 4.12, 4.13 and 4.14 is simply the deficit between the converged equilibrium pressure and the fit to the Thomson ${p_{\mathrm{e}}}$, and as such may differ from the physical ion pressure. This is not a difficulty, however, since our interpretive scheme does not rely on the absolute magnitude of ${p_{\mathrm{e}}}$ being correct to give the right equilibrium pressure profile, but only that the topology of ${p_{\mathrm{e}}}(R)$ is correct.

Figure: Comparison of interpreted and standard equilibrium simulation for shot# 31114: [a] interpreted equilibrium parameters, [b] Thomson ${p_{\mathrm{e}}}$ data (points) with smoothing spline fit versus $R$, [c] interpreted ${p_{\mathrm{eq}}}$ (solid) with that from standard shot simulation (dashed) versus $s$, [d] interpreted flux surface geometry (solid) and standard simulation (dashed) in the Thomson plane, [e] residual ${p_{\mathrm{e}}}$ asymmetry for interpreted (solid) and standard (dashed) equilibria, [f] interpreted ${p_{\mathrm{eq}}}$ with Thomson ${p_{\mathrm{e}}}$ fit and $p_{\mathrm{i}}$ versus $R$.

Figure: Comparison of interpreted and standard equilibrium simulation for shot# 31119: [a] interpreted equilibrium parameters, [b] Thomson ${p_{\mathrm{e}}}$ data (points) with smoothing spline fit versus $R$, [c] interpreted ${p_{\mathrm{eq}}}$ (solid) with that from standard shot simulation (dashed) versus $s$, [d] interpreted flux surface geometry (solid) and standard simulation (dashed) in the Thomson plane, [e] residual ${p_{\mathrm{e}}}$ asymmetry for interpreted (solid) and standard (dashed) equilibria, [f] interpreted ${p_{\mathrm{eq}}}$ with Thomson ${p_{\mathrm{e}}}$ fit and $p_{\mathrm{i}}$ versus $R$.

Figure: Comparison of interpreted and standard equilibrium simulation for shot# 31909: [a] interpreted equilibrium parameters, [b] Thomson ${p_{\mathrm{e}}}$ data (points) with smoothing spline fit versus $R$, [c] interpreted ${p_{\mathrm{eq}}}$ (solid) with that from standard shot simulation (dashed) versus $s$, [d] interpreted flux surface geometry (solid) and standard simulation (dashed) in the Thomson plane, [e] residual ${p_{\mathrm{e}}}$ asymmetry for interpreted (solid) and standard (dashed) equilibria, [f] interpreted ${p_{\mathrm{eq}}}$ with Thomson ${p_{\mathrm{e}}}$ fit and $p_{\mathrm{i}}$ versus $R$.

Shot # 31114 (figure 4.12) was a high-$\beta$ discharge whose magnetic axis suffers a large Shafranov shift. The Thomson ${p_{\mathrm{e}}}(R)$ data is well described by the smoothing spline and exhibits no obvious unmatched features on the inboard or outboard side. The agreement between the FP flux surfaces and those from the standard calculation is good. The interpreted pressure profile differs only slightly from that used in the standard calculation; however, the final asymmetry in ${p_{\mathrm{e}}}(R)$ is better for the interpretive method. The ion and electron pressures are nearly equal, which is consistent with the usual high-temperature high-density assumption.

Shot # 31119 (figure 4.13) was a medium-$\beta$ discharge whose ${p_{\mathrm{e}}}(R)$ fit already appears somewhat asymmetric to the eye in that the inboard side exhibits a marked shoulder that is not reflected in the more or less uniform fall-off on the outboard side. This may indicate that the experimental errors (which are used in determining the optimum spline fit) are not representative of the true deviations in this case. As a result of the slightly skewed ${p_{\mathrm{e}}}$ fit, the electron and ion pressures peak at different points, however, flux surface agreement is good and the interpreted $p(s)$ is close to that in the standard calculation. The final asymmetry is similar in both cases, demonstrating that even with a less than ideal ${p_{\mathrm{e}}}(R)$ fit, the interpretive method produces results which are at least as good as the standard approach.

Shot # 31909 (figure 4.14) is another high-$\beta$ case whose spline fit to the ${p_{\mathrm{e}}}$ profile is reasonably good. The resulting ion pressure is somewhat more peaked than expected, but again the interpretive method produces a close match to the standard profile with slightly improved final asymmetry in ${p_{\mathrm{e}}}(R)$.

In summary, both fitting methods yield comparable flux geometry results, but the FP interpretive method generally gives a slightly superior overall asymmetry in ${p_{\mathrm{e}}}(R)$ and is far less cumbersome to perform. The interpretation for these cases took less than 10 seconds on a 300 MHz UltraSPARC workstation whereas the standard calculations required several NEMEC equilibrium calculations, a matter of several hours of computer time. As with the predictive FP reconstructions, the interpretation was performed by a C++ code specially written for the purpose.