Reference to the Original scaling law for beta limit

[medvedevsyu]

Hi All,

No-wall beta limit scaling in aspect ratio (A) is of a particular interest for me. In fact n=1 external kink modes set the limit on poloidal beta (betaP) for high bootstrap fraction low li design points. It turned out that this betaP limit converted into normalized beta (betaN) is very similar to the PROCESS original scaling law in the dependence on A. Unfortunately, I failed to find any reference to a source of this expression: betaN = 2.7*(1+5/A^3.5).

I would appreciate it very much if you could help to locate the missing reference. Thank you in advance.

Best regards,
Sergei Medvedev

[timothy-nunn]

Hi @medvedevsyu

Thank you for opening a discussion, and my apologies that we have missed it until now. I have forwarded this discussion onto the relevant people and hopefully someone will be able to provide this really soon.

Many thanks,
Timothy

[mkovari]

Hello Sergei,
Note that PROCESS allows two upper limits to be applied directly to beta_poloidal:
beta_poloidal_eps_max (default = 1.38) : maximum (eps*beta_poloidal) (constraint equation 6), eps = inverse aspect ratio
beta_poloidal_max (default = 0.19) : maximum poloidal beta (constraint equation 48).

The first limit (constraint 6) is from Operation at the tokamak equilibrium poloidal beta-limit in TFTR.

Does this help at all?
@medvedevsyu @timothy-nunn

[medvedevsyu]

Hello Michael,
Thank you for looking into that. My question was rather about ideal MHD stability limit than the equilibrium limit in poloidal beta.
Gemini tells me that the original scaling law for beta limits comes from ARIES Systems Code (ASC) and the TETRA (Tokamak Engineering Test Reactor Analysis). However I haven't managed to find the explicit expression betaN = 2.7*(1+5/A^3.5) in the related papers and reports (I also asked Peter Knight in private but got no response). Do you think it would be possible to trace a source of initial Fortran file with the corresponding expression in your repos/archives?

[mkovari]

I am afraid that my colleagues have already had a look and could not find anything. When I wrote the PROCESS papers I looked for references for everything, but couldn’t find one for this. All the best, Michael

[medvedevsyu]

I am afraid that my colleagues have already had a look and could not find anything. When I wrote the PROCESS papers I looked for references for everything, but couldn’t find one for this.

Any comments from Jonathan Maddock @jonmaddock who authored the "Initial open-source commit" with process/physics.py?

Best regards,
Sergei

[timothy-nunn]

Hi @medvedevsyu

I have had a look back at the commits we have internally that predate the initial open-source commit by over a decade. The first ever commit we have internally has the following:

if (gtscale .ne. 0) then
    dnbeta = 2.7D0 * (1.0D0 + 5.0D*eps**3.5D0)
end if

The only documentation referencing this bit of code states:

Switch gtscale determines whether the Troyon $g$ coefficient dnbeta should scale with aspect ratio (when gtscale != 0), or be fixed at the input value (gtscale = 0).

Unfortunately, this equation has been added in a period that we do not have source for. Therefore, regrettably, I do not think we will be able to find the source of this equation.

[medvedevsyu]

Thanks a lot for looking at that! Any authorship data on the initial Fortran file?

[timothy-nunn]

That first commit was made by Peter Knight, who I understand you are having difficulty contacting