Beta limit for Spherical Tokamaks (ST)

[jonmaddock]

In GitLab by @mkovari on May 19, 2020, 11:02

Jack Berkery et al of Columbia looked at 5000 experimental equilibria and produced a fit of the calculated no-wall beta limit against aspect ratio, internal inductance and pressure peaking. It is for NSTX so it should be relevant to ST reactors.

where

(The "poloidal magnetic field" has not been defined - see comments.)
https://iopscience.iop.org/article/10.1088/0029-5515/55/12/123007/pdf

The dependences in this fit are plotted here:Beta_limit.pdf

The key limits of applicability are
$A<1.8$,
$l_i>0.4$

Including the wall will make things more difficult as it depends on the geometry of the wall - especially how far it is from the plasma edge. This draft paper looks at this for MAST-U.

MAST-U_globalstability_v21.docx

The most conservative assumption is the no-wall limit. When a conducting wall is present the growth rate of unstable modes just above this limit is greatly reduced. Plasma rotation and kinetic effects can then make these modes stable, or they can be actively controlled. The use of ferromagnetic material such as Eurofer in the wall also has an effect. As all this is difficult to quantify, Chris Ham suggests we could go for the no-wall limit plus some percentage, e.g. no-wall limit times 1.2. We could make this a user input, resistive_wall_factor.
$\beta_{N,max}=f_{resistive-wall} \times \beta_{N,no-wall}$

This approach still requires sensible values for $l_i$ and $p_0/\langle p \rangle $, which is a matter for another day.

Any comments @stuartmuldrew, @schislet ?

Checklist

• Run test suite
• Does it change the output? If so, give justifications.
• Does it change the baseline 2019 run? (you may need to run on Freia)
• Merge request created

[jonmaddock]

In GitLab by @stuartmuldrew on May 20, 2020, 12:32

So far I have just been taking the simplified approach of using $\beta_N=6$ as the limit for $A<2$ from the Ono & Kaita (2015) ST review article:

I have not tried to link this to the aspect ratio. All designs will be above the no-wall limit, so is it still the case that the aspect ratio dependence will remain? Alternatively, are you trying to scan down in aspect ratio from conventional values and want to capture the change?

The internal inductance can be linked to the elongation with a relation from Menard et al. (2016):

giving $\kappa_x = 3.4 - l_i$. I've not implemented this is PROCESS as I've always been a bit uncomfortable with it. It is a limit for NSTX and I haven't compared it to the data from MAST. I've also never fully been clear on $l_i$. When you talk to plasma physicists there are different definitions of internal inductance, typically referred to as $l_i(2)$ or $l_i(3)$. I have spoken with Tim Hender about them, but never convinced myself fully what we mean.

[jonmaddock]

In GitLab by @mkovari on May 20, 2020, 13:36

I am not trying the describe the full range of aspect ratio with one formula. Berkery's formula only applies to A<1.8. Is this good enough for STEP?

Technical query - is the limit on $\beta_N$ exactly the same as dnbeta?

I notice that the dashed line labelled "ST-FNSF design assumption" is well above all the data points.

I am surprised that Menard is basing FNSF parameters on NSTX, when the aspect ratio of FNSF is 4!

Frankly I am not too concerned by the definition of internal inductance. If no-one else can be bothered, why should we?
@stuartmuldrew

[jonmaddock]

In GitLab by @mkovari on May 20, 2020, 13:53

There is a somewhat sensible discussion of inductance in FusionWiki. However, many questions remain unanswered - including the definition of "poloidal field". They write,

In a tokamak, the field produced by the plasma current is the poloidal magnetic field Bθ, so only this field component enters the definition.

This is only half true - the poloidal field coils contribute to the poloidal field - funny that. Possibly they don't contribute to the integral of the field - but I would think that depends on the symmetry. In any case the inductance (i.e. self-inductance) of the plasma only depends on the field generated by the plasma by definition.

The difference between the values is certainly not negligible:

In DIII-D we note that ℓi/ℓi (3) ≈ 1.1 to 1.3.

(ITER startup studies in the DIII-D tokamak)

[jonmaddock]

In GitLab by @stuartmuldrew on May 20, 2020, 14:08

In subroutine culblm of physics.f90:

betalim = 0.01D0 * dnbeta * (plascur/1.0D6) / (rminor*bt)

So dnbeta is the limit assuming you have the right switches set. iprofile=0 to use the input value, iculbl=0 to use total beta and constraint equation 24 with fbetatry as an iteration variable to enforce it. I have just used the above combination with dnbeta=6.0 in the input file and been happy with that.

My only concern with Berkery's formula is that the no-wall limit is conservative and it's not clear if the aspect ratio dependence remains above it. It is hard to tell from the plot as there is a lack of data.

I am surprised that Menard is basing FNSF parameters on NSTX, 
when the aspect ratio of FNSF is 4!

This is the PPPL version of FNSF which is an ST with A=1.75. There are lots of designs for FNSF.

The difference between the values is certainly not negligible

For the SCENE STEP run Bhavin gave us $l_i(2)=0.4669$ and $l_i(3)=0.2741$

[jonmaddock]

In GitLab by @mkovari on May 20, 2020, 14:27

Wow - that is a big difference!

What is the aspect ratio of STEP?

Remember that I am proposing to multiply Berkery's fit by a factor.

[jonmaddock]

In GitLab by @stuartmuldrew on May 20, 2020, 14:44

That STEP SCENE run is $A=1.613$.

I have no problem using the formula with the multiplier, it is just the multiplier may also be aspect ratio dependent.

We could add another iprofile option and use $\kappa$ to calculate $l_i$ and then Berkery's fit for dnbeta. The only missing parameter is $\alpha_j$, but that is a whole load of problems in itself as to whether the current density profile is realistic, as you mentioned in a previous issue.

[jonmaddock]

In GitLab by @mkovari on May 20, 2020, 14:50

In physics.f90,

normalised_total_beta = 1.0D8*beta*rminor*bt/plascur

which is the same as dnbeta, but there is another definition,
'Normalized plasma pressure beta as defined by McDonald et al',
D.C. McDonald et al, 2007 Nuclear Fusion v47, 147:

 beta_mcdonald = 4.d0/3.d0 *rmu0 * total_plasma_internal_energy / (vol * bt**2) 

where

total_plasma_internal_energy = 1.5D0*beta*btot*btot/(2.0D0*rmu0)*vol

etc.

[jonmaddock]

In GitLab by @stuartmuldrew on Sep 21, 2020, 13:28

Dependence of the ideal ballooning beta limit on aspect ratio: https://iopscience.iop.org/article/10.1088/0029-5515/44/4/009

This appears to be used in the GA systems code GASC.

[mkovari]

Please note that normalised internal inductance is discussed at length in #1878.

[stuartmuldrew]

Section 2.4 of Emmi's paper explains how the Beta limit is set in JETTO scans: https://arxiv.org/pdf/2403.09460

This requires the pressure peaking, $F_p =p_{ax}/ &lt; p &gt;$, to set the Beta limit with $C_{\beta} = \frac{(\beta_N-3.7)F_p}{12.5-3.5F_p}$, where $C_p \le 0.5$ (see paper for explanation).

I don't think we have the pressure peaking in PROCESS, so it might be worth implementing the above instead.

[mkovari]

The formula from Emmi's paper looks good, although there is no actual justification or reference for it in the paper. PROCESS has profiles for density and temperature, so the pressure peaking factor $F_p$ can be calculated.

However, we normally keep the density and temperature profile exponents constant, so we don't fully make use of the parameter space. It would make sense to start using these as iteration variables.

[stuartmuldrew]

I've added Emmi's equation as iprofile option 6.