5.2 Accuracy and Internal Constraints
5.2.1 Statistical Errors
ρpol
1 2 3 4
[MA/m2 ] a)jtor
ρpol
1 2 3 4 b)q
0.0 0.2 0.4 0.6 0.8 1.0
ρpol
0.0 0.5 1.0 1.5
[105 Pa]
c)p
1.0 1.5 2.0
R [m]
−0.5 0.0 0.5 1.0
z[m]
d) 1×MSE @ 0.2◦ 1×MSE @ 1.0◦
Figure 5.2: Same as figure 5.1, but with one MSE channel at the location indicated on the right-hand plot. The MSE data were applied with an uncertainty of σMSE = 0.2°
and σMSE = 1.0° (blue and red, respectively).
Ip =
∫
jtor dA W = 3
2
∫
pdV (5.8)
The effect of a single channel of MSE data from the core is displayed in figure 5.2, with an uncertainty of the MSE data of σMSE = 0.2° and σMSE = 1.0°. To avoid confusion it has to be stressed that σ here refers to the uncertainty of the MSE data, not the σ component of the MSE spectrum. As discussed in section 4.1, the MSE diagnostic strongly constrains Bpol, which directly improves the accuracy of the q-profile through the additional data. The exact radial location of the MSE data can be seen by the contraction in the uncertainty band. Both pressure and current density profile are also less uncertain, although as an integral measurement, one MSE channel alone cannot properly constrain them.
Note that the q-profile and its uncertainty are virtually identical at σMSE = 0.2° and at σMSE = 1.0°. In either case it is a very strong constraint for q and it must therefore be ensured that systematic errors do not affect the measurement as this would have imme-diate consequences for the reconstruction. A discussion of the influence of systematic MSE errors is given below.
5.2 Accuracy and Internal Constraints
ρpol
1 2 3 4
[MA/m2 ] a)jtor
ρpol
1 2 3 4 b)q
0.0 0.2 0.4 0.6 0.8 1.0
ρpol
0.0 0.5 1.0 1.5
[105 Pa]
c)p
1.0 1.5 2.0
R [m]
−0.5 0.0 0.5 1.0
z[m]
d) 10×MSE
Figure 5.3: Same as figure 5.1, but with ten MSE channels at the locations indicated on the right-hand plot.
It should also be noted that DCN polarimetry produces similar results, but is not as radially resolved as MSE as it picks up contributions from several points along its line of sight.
Finally, using a full set of ten MSE channels yields the results shown in figure 5.3.
The q-profile is now well-known along the whole minor radius with the exception of the volume right around the magnetic axis. The various integral current measurements of the MSE also help constrain the toroidal current density much more strongly. Even though a single measurement only yields the total current enclosed by a flux surface, the set of integral measurements combines to constrain the overall current density profile.
The pressure profile, however, is still afflicted with large error bars. That is because even if the toroidal current profile is well-known, there still remains some leeway to shift current density between p′ and f f′. This can be improved by supplying core pressure information, for example using the fast ion pressure calculated by TRANSP on top of the measured kinetic pressure.
As mentioned before, the correct values of regularisation through curvature and amp-litude constraints are not clear-cut. In figure 5.4 on the following page the case with ten MSE channels is shown with varying strength of the regularisation. It is immediately clear that the black case with regularisation increased tenfold has little credibility: it suggests that the current and pressure profiles are known with less than about 15% of
ρpol
1 2 3 4
[MA/m2 ] a)jtor
ρpol
1 2 3 4 b)q
0.0 0.2 0.4 0.6 0.8 1.0
ρpol
0.0 0.5 1.0 1.5
[105 Pa]
c)p
1.0 1.5 2.0
R [m]
−0.5 0.0 0.5 1.0
z[m]
d) default regularisation 0.1×regularisation 10×regularisation
Figure 5.4: Same as figure 5.1, but with ten MSE channels at the locations indicated on the right-hand plot and different values of regularisation.
uncertainty on the magnetic axis and virtually no uncertainty outside mid-radius, even though none of the experimentally available information is constraining the equilibrium in that regard.
In contrast, the uncertainties of the red case where regularisation has been weakened by a factor of ten are significantly larger than in the blue reference case with default regularisation. Despite this, the resulting profiles are unchanged inside the uncertainty of the reference case. Only jtor approaches the reference uncertainty at ρpol ≈ 0.5 as it starts oscillating compared to the reference case. The red pressure profile develops a step just outside mid-radius, but otherwise remains very close to the reference case. This step is not supported by any experimental evidence and simply an artefact of the reduced regularisation. In q, the uncertainties remain unchanged outside of ρpol ≈ 0.3 and the overall result is largely identical to the reference case — which is to be expected since it is strongly constrained by the MSE information. In summary, despite the profiles now having the option to be an order of magnitude steeper, the changes compared to the blue reference case are small.
Consequently, the regularisation of the blue case has been used for the subsequent ana-lyses of this work. It is a reasonable compromise between artificially reducing the uncer-tainties towards unrealistic accuracy and increasing them needlessly without affecting the actual result, or even introducing baseless artefacts. It has to be noted that for
5.2 Accuracy and Internal Constraints
ρpol
1 2 3 4
[MA/m2 ] a)jtor
ρpol
1 2 3 4 b)q
0.0 0.2 0.4 0.6 0.8 1.0
ρpol
0.0 0.5 1.0 1.5
[105 Pa]
c)p
1.0 1.5 2.0
R [m]
−0.5 0.0 0.5 1.0
z[m]
d) 10×MSE ±0.0◦ 10×MSE −0.3◦ 10×MSE + 0.3◦
Figure 5.5: Same as figure 5.1, but with ten MSE channels shifted by ±0.3° to gauge the impact of systematic errors.
analyses in the next chapter the q-profile is of main importance and, as shown here, it and its uncertainty does not change much due to the regularisation.