1. An Error Analysis of Dimensionless Confinement Scaling Experiments
by J.G. Cordey
Euratom/UKAEA Fusion Association, Culham Science Centre, Abingdon, Oxon OX14 3DB, UK
Background, single scan experiments in the dimensionless parameters , * and * are giving rise to a whole range of scalings, particularly wrt  and * and these are also different to the scaling from the ITER database IPB98(y,2).
A sophisticated Error in Variable Analysis (J.G. Cordey et al Nucl. Fusion 45 (2005) 1078) shows how the degradation wrt  may be reduced.
The indices of b (ab) and n* (an*) of the confinement expression versus the assumed error on the loss power dP.
ITPA Milano – October 2008

2. How do errors in stored energy and power of a similar magnitude effect the single scan results for gyro-Bohm scaling?
In any dimensionless scan, for example a  scan, a key element is how well the other parameters * and * are matched. In particular how do errors in the stored energy influence the  scaling.

3. General Error Analysis
(1)
where
Consider errors in the stored energy W and input power P.
Then taking W = Wo + W, P = Po  + P.
The indices n are written in the form n = no +  n. Completing a perturbation analysis of Eq. (1) and only retaining first order terms gives
After taking the log we arrive at
For gyro-Bohm scaling the errors in W couple together.

4. Error in the determination of the  scaling from a  scan
Assume 10 = – 3.0 (gyro-Bohm scaling) and 2 = – 0.25 which is a typical value from * scans,
For a two point  scan, with an upper value of u and a lower value L.
Typical values of W/W and P/P for AUG, DIII-D and JET are 12%.
With these errors and with
and u/L = 2, 3 would have a
range of ± 1.2. This is rather extreme and assumes that the errors are correlated with the values of the stored energy etc. A more realistic approach is to sum the errors quadratically and take the square root.

5. Using the same errors for W/W and P/P of 0
Using the same errors for W/W and P/P of 0.12 we calculate the error 3 for a few examples
30 = – 0.5, u/ L = 1.6 gives 3 = ± 0.98.
2) 30 = – 0.5, u/L = 2 gives 3 = ± 0.66.
3) 30 = 0, u/ L = 3 gives 3 = ± 0.35
Hence we see that even for case 3, in which there is a large range in , the error is still quite significant.

6. Error in the determination of the * scaling
Here we take 10 = – 3.0, 2 = – 0.25, 30 = 0, which is the typical scan result from DIII-D and JET.
Using the same errors for W/W and P/P of 0.12 and assuming quadrature of errors as previously gives
In a typical * scan, in which the toroidal field B changes by a factor of 2, * has a range of Hence the error in the * index 1 = ± An error of this magnitude means that it is uncertain whether the * scaling is gyro-Bohm (10 = – 3) or Bohm (10 = – 2).
For an extended * scan with a range of 2.6, equivalent to range of 4 in B and I, , =  0.41.

7. Error in the determination of the *
Taking 10 = – 3, 20 = – 0.25 and 30 = 0, then with the same errors
For a typical range of * of 16, 2 = ± 0.14.
This error is much smaller than those of the  and * scans.

8. SUMMARY
The variability of the  scaling results is not surprising, it is due to the coupling of the errors in the stored energy for gyro-Bohm transport.
The * and * scalings require extended scans, a factor of 3 in * and 10 in *, to give a sensible result.
Suggest in future, work at ITER  and *, and construct multimachine * scans with identity pulses from each pair of devices. This would check whether one is scaling the same mode (i.e. hybrid or conventional H-mode in each device) and would also permit a large range in *.