Improve clarity of beta limit options

[jonmaddock]

In GitLab by @mkovari on Oct 4, 2017, 13:22

The options for how to set the g-value in the beta limit are a little opaque. (Note that this is distinct from the question of which value of beta to constrain, which is set by iculbl.) The following might be useful:

1. Reduce the two switches (iprofile and gtscale) to one switch with three or more options.
2. Explain the option used more clearly in the OUT.DAT.

Here are the switches and relevant variable:

iprofile /1/ : switch for current profile consistency:
    = 0 use input values for alphaj, rli, dnbeta (but see gtscale option);
    = 1 make these consistent with input q, q0 values (recommendation: use icurr=4 with this option) 

dnbeta /3.5/ : (Troyon-like) coefficient for beta scaling; calculated as (4.0*rli) if iprofile=1 (see also gtscale option) 

gtscale /0/ : switch for a/R scaling of dnbeta (iprofile=0 only):
    = 0 do not scale dnbeta with eps;
    = 1 scale dnbeta with eps

Note that the term "beta scaling" used above is not very helpful.

The three options listed in the Physics paper might be the way to implement this:

A reference would also be useful, as there isn't one in the physics paper!
(If $l_i$ is used it would help to explain how it is derived.)

EDIT 15/5/24
I have found the reference for $g=4 l_i$: equation 18.57 on page 584 of Fusion Plasma Physics, Stacey, 2nd ed.

[jonmaddock]

In GitLab by @mkovari on May 11, 2020, 13:54

@jmorris-uk @stuartmuldrew @schislet @ajpearcey
As I was looking at the beta value in one of Simon's outputs, I realised that I didn't understand how the beta limit is derived. I found this issue which is still open. Any volunteers to clarify this comprehensively?

[jonmaddock]

In GitLab by @stuartmuldrew on May 11, 2020, 16:55

I have had a quick look at this before. If iprofile=1 then:

alphaj = qstar/q0 - 1.0D0
rli = log(1.65D0 + 0.89D0*alphaj)
dnbeta = 4.0D0 * rli

The equation for rli is from Wesson, although there is a difference in the q used. The whole lot is documented in Hartmann and Zohm "Towards a 'Physics Design Guidelines for a DEMO Tokamak' Document". I've seen the paper version in the office but not sure I have the electronic copy.

This was all derived for large aspect ratio and does not apply for spherical tokamaks, Hartmut confirmed this when Hanni and I emailed him:

The relation q_a/q_0 = alpha_j+1 is strictly true only for 
cylindrical cross-section, large aspect ration (i.e. a periodic
straight cylinder) and current profiles of the type j=j_0(1-(r/a)^2)^alpha_j.
The Wesson fit for l_i is for this case, in which also q_a=q_cyl holds.
However, for shaped plasmas, this is no longer the case, and when
we compared to e.g. ITER, there was quite some difference already.
So, I would NOT recommend to use this relation for an ST. 

It could be a nice little project to derive a fit for shaped plasmas
in dependence of A, kappa and delta (or S=q_95/q_cyl, if it is possible
to lump kappa and delta into this one 'shape factor'). I am sure the
result would be very useful and appreciated by the community ;-)

[jonmaddock]

In GitLab by @mkovari on May 18, 2020, 12:03

This bothers me. I doubt the parametrised current profile is very accurate, and even if it is we don't know the value of alphaj with any confidence. However, the relationship between qstar, q0, alphaj and li is determied by physics that can be worked out. However the beta limit is mostly empirical, and I am concerned that dnbeta = 4.0D0 * rli could give unphysical values of beta both for small and conventional aspect ratios.

Who is our expert on the beta limit?
@stuartmuldrew @ajpearcey @jmorris-uk

[stuartmuldrew]

I have removed gtscale and added the functionality to iprofile.