i_confinement_time ITPA20-IL Scaling equation fix

[OceanNuclear]

Description

Closes #4017
reference [^23] is dereferenced into [^22].

Checklist

I confirm that I have completed the following checks:

• My changes follow the PROCESS style guide
• I have justified any large differences in the regression tests caused by this pull request in the comments.
• I have added new tests where appropriate for the changes I have made.
• If I have had to change any existing unit or integration tests, I have justified this change in the pull request comments.
• If I have made documentation changes, I have checked they render correctly.
• I have added documentation for my change, if appropriate.

[codecov-commenter]

Codecov Report

✅ All modified and coverable lines are covered by tests.
✅ Project coverage is 49.44%. Comparing base (6980eb9) to head (2d82030).
⚠️ Report is 12 commits behind head on main.

Additional details and impacted files

@@            Coverage Diff             @@
##             main    #4158      +/-   ##
==========================================
+ Coverage   47.97%   49.44%   +1.47%     
==========================================
  Files         143      149       +6     
  Lines       30125    29796     -329     
==========================================
+ Hits        14451    14734     +283     
+ Misses      15674    15062     -612     

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[OceanNuclear]

Before the PR, the documentation page looks like:

The top reference ([22]) shows this equation:

matching the top equation. It also shows the bottom equation:

However, the bottom reference ([23]) shows this equation:

But the [3] here is actually citing the first paper ([22]). This also explains the index rounding issue mentioned in #4017.

[OceanNuclear]

According to reference [22], the values (and the errors) for the top equation is:

and for the bottom equation is:

[OceanNuclear]

Regarding making the roundings more consistent, these are the values (α) and their uncertainties (σ) as recorded in the first paper ([22]):

ITPA20 scaling (H-mode)	$\alpha_0$	$\alpha_I$	$\alpha_B$	$\alpha_n$	$\alpha_P$	$\alpha_R$	$\alpha_{1+\delta}$	$\alpha_\kappa$	$\alpha_\epsilon$	$\alpha_M$
Mean	0.0534	0.976	0.218	0.2442	-0.6687	1.71	0.362	0.799	0.354	0.195
σ	0.0013	0.015	0.013	0.009	0.0063	0.023	0.033	0.027	0.031	0.016
Mean + 0.5 σ	0.05405	0.9835	0.2245	0.2487	-0.66555	1.7215	0.3785	0.8125	0.3695	0.203
Mean – 0.5 σ	0.05275	0.9685	0.2115	0.2397	-0.67185	1.6985	0.3455	0.7855	0.3385	0.187
Best rounding	0.053	0.98	0.22	0.24	-0.67	1.7	0.36	0.8	0.35	0.2

ITPA20-IL scaling (H-mode)	$\alpha_0$	$\alpha_I$	$\alpha_B$	$\alpha_n$	$\alpha_P$	$\alpha_R$	$\alpha_{1+\delta}$	$\alpha_\kappa$	$\alpha_\epsilon$	$\alpha_M$
Mean	0.067	1.291	-0.134	0.1473	-0.6442	1.194	0.56	0.673	0	0.302
σ	0.0021	0.017	0.017	0.0088	0.0063	0.027	0.032	0.039	0.033	0.016
Mean + 0.5 σ	0.06805	1.2995	-0.1255	0.1517	-0.64105	1.2075	0.576	0.6925	0.0165	0.31
Mean – 0.5 σ	0.06595	1.2825	-0.1425	0.1429	-0.64735	1.1805	0.544	0.6535	-0.0165	0.294
Best rounding	0.067	1.29	-0.13	0.15	-0.644	1.2	0.56	0.67	0	0.3

[OceanNuclear]

@stuartmuldrew can you identify what the (+0059 - 0.032) is doing in the first term of the top equation, and the (+0.030 - 0.018) in the first term of the bottom equation? I can't see these being the upper and lower bounds of that constant

[timothy-nunn]

@jonmaddock have you approved this as a maintainer as ready to be merged? Also are you managing this PR and should I put you as the assignee?

[stuartmuldrew]

@stuartmuldrew can you identify what the (+0059 - 0.032) is doing in the first term of the top equation, and the (+0.030 - 0.018) in the first term of the bottom equation? I can't see these being the upper and lower bounds of that constant

My interpretation from the equation and the table is that is the error on the constant at the start. Admittedly, that is a pretty large upper bound.

[OceanNuclear]

Thanks @stuartmuldrew. I think those values contradict with the actual upper and lower bound I calculated using the values given in the table (tabulated in the previous comment). Perhaps the (+0.059 - 0.032) and (+0.030 - 0.018) will remain a mystery forever. It doesn't affect anything above anyways.

As for the failing regression test, that's due to the actual constants being changed (due to correction in number of digits rounded). Shall we change the regression test so that it passes then? @timothy-nunn

[jonmaddock]

I don't understand why certain coefficients are being rounded to a lower accuracy: can you explain?

[OceanNuclear]

@stuartmuldrew Asked for a more consistent rounding in the issue #4017. The reduced accuracy reflects the lower confidence/higher uncertainty.