SLAYER estimates of tearing threshold are an order of magnitude lower than DIII-D experiments

[logan-nc]

As mentioned in #164, initial tests with the gpec_slayer interface are giving estimates for the error field / RMP threshold that are unrealistically low for DIII-D.

This issue is being opened to provide a space to discuss and document what might be going on. The goal is to identify improvements and/or settings that can be used to get realistic results in existing experiments.

[logan-nc]

Here is an example n=3 calculation in DIII-D, showing the very low predicted thresholds

[logan-nc]

@parkjk mentioned in #164 that the threshold might be sensitive to inpr, but @d-burg estimated Pr in DIII-D should actually be lower than the current default by a factor of 0.1. This would decrease the predicted threshold even more (perhaps by a factor of ~0.3).

[logan-nc]

@parkjk showed this figure in the 2021 MHD workshop:

@parkjk, can you share the DIII-D runs that have good agreement with us? Perthaps attempting to replicate these results will help identify key differences in the inputs / settings.

[logan-nc]

One obvious difference is the nn toroidal mode number

[parkjk]

@d-burg : can you briefly explain the method of estimating Pr? A method Jace and I used to guess the Pr values:

where the \chi is the experimental momentum diffusivity coefficients where are normally O(1), and \eta is plasma resistivity and so \eta/\mu_0 is usually smaller than O(1), resulting Pr greater than O(1).

[d-burg]

@parkjk sure, I've been using $P_{\perp} = \tau_R / \tau_{\perp}$ and $P_{\phi} = \tau_R / \tau_{\phi}$ as defined in Fitzpatrick's "Tearing Mode Dynamics in Tokamak Plasmas". He defines $\tau_{\perp}$ as:

and $\tau_{\phi}$ as:

Here, $\chi_{\perp}$ is the energy diffusivity, and $\Xi_{\perp, i}$ is the neoclassical cross-flux-surface momentum diffusivity (which I am also calculating using his definitions).

Your and Jace's method looks more elegant than these, and the use of experimental momentum diffusivity is nice. Is it from a paper, and if so could you link it for me?

[logan-nc]

@d-burg, I like your basic definitions, but something must be funky with the details of the calculations. The rough estimates for DIII-D should be $\tau_R \approx 1.0$ and $\tau_\phi \approx \tau_E \approx 0.1$ s. This would then give $P_\phi \approx 10$ by your top level definition.

[logan-nc]

Some tau_E examples can be found in Fig. 2 of [Hawryluk NF 2015].

Both tau_e and tau_R are quoted in the abstract of [Turco NF 2010].

You can poke around google scholar to find more examples if you wish. You can also plot the results of automated taue and taur calculations directly in mdsplus.

[parkjk]

Sorry about this copy-and-paste but we define viscosity (\mu_i) in the ion momentum equation as

The definite of \mu_i is exactly same as \chi_\phi. Then we normalize the equation with \mu_0, and then

where you can see our 'P' (which is 'Pr') is the normalized /mu_i with resistivity \eta.

This is stated in Jace Waybright's physics of plasma paper doi: 10.1063/5.0185014, where we traced back the normalization from the code to the baseline theory.

[logan-nc]

I hunted around some old work and found some plots of the Prandtl number I made years ago using TRANSP to calculate the diffusivities.

You can see the Prandtl number is on the order of 0.5-5. I wouldn't trust the far edge spike, but even ignoring this you can see there can be a large change depending on where the surface is in the radial profile... if the tearing stability or EF threshold is sensitive to this then we will need a good solution here. It would be difficult to demand users all run TRANSP or some other transport code just to get this, but I am not sure if there is another way. Do you guys have any good suggestions?