c stab : dependency on radial misalignment and over-

stabilisation

The fit for c_stab obtained considering the geometric corrections still presents a considerable scatter over the whole database of discharges. However, once the geometric effects have been accounted in the fit for cstab, the only two other pos- sible reasons for the discrepancy in its values are that:

3) the ratio jECCD/jBS, in case of small deposition profile discharges W_dsat >1 is far away too high to clearly be able to isolate the value of c_stab indepen- dently from it

4) the value of radial misalignment xmis which is considered for the fit is es- timated with a high uncertainty and small changes of its value can largely affect the evaluation of c_stab

In ASDEX Upgrade, as mentioned in chapter 4, the radial misalignment is ro- bustly measured since the linear change in the toroidal magnetic field BT is di-

rectly proportional to the shift in the radial position of the ECCD beam and, therefore, point 4 is not expected to be relevant. On the other hand, indication that point 3 may influence the determination of c_stab is provided by noticing that for ASDEX Upgrade figure 5.5 and 5.6 show that:

• the discharges with lowest c_stabare those characterised by a value of jECCD jBS > 1 given a misalignment of about 2-3 cm (see table A.1 in the Appendix)

• the discharge with highest value of c_stab is #17977 which is only marginally stabilised during the experiment and is characterised by jECCD

jBS =0.7, very poor initial alignment xmis =0.05 m and a large deposition profile W_dsat <1 (figure A.1 in the Appendix). This indicates that for the discharges with smaller misalignment xmis <0.05 cm and W_d >1 the marginal value jECCD_jBS will lie well below 0.7.

In general, for JT-60U discharges the misalignment is a much more unknown quantity than in ASDEX Upgrade and for the fitting of c_staban initial value xmis= 2 cm has been considered. However, the very rapid decay of the mode as ECCD is switched on indicates that the misalignment can be even smaller. In addition, in figure 5.5 and 5.6 some other aspects need to be considered:

• the two series of (2,1) NTM stabilisation experiment differ very much in terms of results for c_stab although among these there are two discharges which are only marginally stabilised (#46367 and #47796) since the ratio

jECCD

jBS has been explicitly lowered in order to study ECCD marginal power requirements

5.5. cstab: comparison between ASDEX Upgrade and JT-60U discharges 97

• one of the (3,2) NTM discharges from JT-60U has a lower c_stab value com- pared to the other (3,2) NTM discharges and is characterised during the stabilisation by a drop of the ECRH power which affects the length of sta- bilisation (figure A.5 in the Appendix)

The first point is an indication that these discharges may be characterised by a different value of misalignment x_mis (e.g assuming 2 cm for all of them may be too simplistic). The second point, instead, suggests that also in the JT-60U case, although not as much as in the ASDEX Upgrade case, over-stabilisation for the (3,2) NTM discharges plays a role in the determination of c_stab. Given these ar- guments, in order to improve the discrepancy of c_stab among the discharges, it is reasonable to assume:

• a misalignment x_mis≈0 cm for the JT-60U discharges which have c_stab>0.8 in figure 5.6

• a ratiojECCD

jBS =0.5 for the ASDEX Upgrade discharges which have cstab<0.5 in figure 5.6

The fit for c_stabis carried out again under these conditions and the result is plotted in figure 5.9. By assuming perfect alignment for the JT-60U discharges with initial

c 1.2 0 0.5 1 1.5 2 2.5 0 0.25 0.5 0.75 1 1.25 1.5 1.75 2 j ECCD/j BS stab (2,1) NT JT−60U (3,2) NTM JT−60U (2,1) NTM AUG (3,2) NTM AUG 0 0.2 0.4 0.6 0.8 1 0 0.25 0.5 0.75 1 1.25 1.5 1.75 2 x mis0/d c stab j

ECCD/j BS set to marginal value ~0.5

x

mis0 set to 0 cm

Figure 5.9: Fitting results for cstabobtained by considering for the calculation of the effi-

ciency functionηECCDthe island width measured on the HFS Wsat,HFSand an asymmetry

factor γ=0.5 and by accounting for over-stabilisation in case of ASDEX Upgrade and smaller misalignment for group C for JT-60U discharges.

high values of c_stab, stabilisation efficiency improves and c_stabresults to be smaller by a factor 0.7-0.8. In addition, by considering the effect of over-stabilisation for the ASDEX Upgrade discharges, the points with c_stab <0.5 shift up leading to

a smaller scatter in the ASDEX Upgrade database. A large discrepancy still re- mains between the (2,1) NTM discharges of JT-60U: these are, however, the most difficult discharges to analyse as mode coupling is present and one could even argue that for the first set of discharges the (3,2) was stabilised and for the sec- ond set the (2,1) was stabilised due to different resonance location of the ECCD injection. Last but no least, the geometry effects, the assumptions on misalign- ment and over-stabilisation have been treated here quite schematically whereas for each discharge these effects should be weighted differently. In general, some important observations related to the fitting c_stab are the following:

• at ASDEX Upgrade the initial misalignment can initially be larger than at JT-60U (xmis0/d≤1.0).

• at JT-60U it has to be less than half of the saturated island width x_mis/d≤0.6 cm to be stabilising in agreement with ref. [Nagasaki(2003)]

• for ASDEX Upgrade there is a systematical difference between small depo- sition profiles and large deposition profile discharges which translates into a different behavior in time of the mode stabilisation (Mirnov signal)

• the value of the marginal island width Wmarg is consistent with Wd or Wb

model for all the discharges except for the (2,1) NTM discharges of JT-60U. However these are characterised by the co-existance of (2,1) and (3,2) NTM, therefore the Modified Rutherford equation may be incomplete (an addi- tional term related to the coupling should be included).

• the ratio jECCD/jBS which marginally stabilises the NTM is ∼0.3÷0.6 for Wsat/Wmarg≈2−3 in both ASDEX Upgrade and JT-60U.

• the mean value of c_stab calculated over the whole database is:

Chapter 6

Error analysis for

c

_sat

and

c

_stab

In chapter 5, the fitting coefficients csat and cstab have been calculated for a set of NTM stabilisation discharges from ASDEX Upgrade and JT-60U. The value of these fitting coefficients depends on many plasma parameters which are cor- related among themselves and are deteriorated with measurement uncertainty. Therefore, the question arises on how consistently one can give an estimate of the uncertainty on csatand cstab. In order to approach this problem, an error analysis using a probabilistic method which generalises the standard error propagation technique is developed. The main focus is to investigate how do the measure- ments on the experimental quantities affect the evaluation of csat and cstab in the Modified Rutherford equation. The numerical tool which has been developed is able to give together with the fitting coefficients of the Modified Rutherford equation also their standard deviation in the most consistent way.

6.1

Error analysis for the Modified Rutherford Equa-

tion using a probabilistic approach

To determine the error on csatand cstab a probabilistic approach is applied which makes use of the Bayesian Probability Theory (BPT) [Goldstein(2007)]. The Bayes Theorem states that a posterior probability distribution of the event c given the event B can be calculated by multiplying the prior probability distribution P(c) by the likelihood P(B|c) that event B will occur if c is true and its formula is:

P(c|B) = P(B|c)P(c)

P(B) (6.1)

The likelihood is assumed to be a normal distribution having the form: P(Bi|A) = 1 √ 2πσBi exp− 1 2( B−_Bi σ_Bi )2 (6.2)

In this way, the joint probability can be calculated as: p(B|A) =

∏

i

p(Bi|A) (6.3)

This method applied to the Modified Rutherford equation has several advantages compared to a standard gaussian error propagation since the coefficients csatand c_stabhave complex dependencies in the equation 2.49:

csat =csat(Wsat, Lq, jBS, jECCD...) cstab =cstab(WECCD, Lq, jBS, jECCD...)

It also makes possible to consider the physics constraints and the correlations present in the model and to make a sensitivity study on the model to identify which are the most influencing quantities affecting their determination.

6.2

Application to the Modified Rutherford equation