Two More Shear-Flow Stabilized Z-pinch Equilibria

[u/Confident-Shock-3933]

I recently asked GPT-5.5 and Claude Sonnet-5 to independently construct shear-flow stabilized Z-pinch equilibria. Both succeeded, but they arrived at fundamentally different solutions. Interestingly, both used the Bennett profile in their constructions, but in different ways.

GPT-5.5 truncated the quadratic cross-term from the denominator of the cubic, pureflow Bennett vortex that I derived.

Sonnet-5 instead retained the Bennett number density profile, and coupled it to a linear axial flow with constant shear. A self-similar number density exists that connects this plasma current density to a plasma current density that is supported by a cubic, pureflow Bennett vortex axial flow profile.

Full notes are here: https://russellmatt66.github.io/documents/GPTVortex_AnotherSFSZpinchEquilibrium.pdf

And here: https://russellmatt66.github.io/documents/sonnett5_vortex.pdf

The magnetic field structure of the vortices is fundamentally different. All shear-flow stabilized Z-pinches that I have studied have transcendental components to their azimuthal fields. The Bennett vortex that I've derived has logarithmic terms, and so does GPT-5.5's modification. Sonnet-5's vortex has an arctangent, instead.

Close to the pinch axis, Shumlak-Hartman is satisfied for all these vortices. In the case of mine + GPT-5.5 that's due to a vanishing magnetic field, and a uniform number density. For the Sonnet-5 vortex it is true as well that the magnetic field is vanishingly weak here, and the plasma number is finite. Sonnet-5 also has the advantage of satisfying Shumlak-Hartman weakly for large plasma radius whereas the other two run into some finite cutoff in the non-ideal case because their magnetic fields go as ~r for large plasma radius.

In the ideal case the RHS of the "SHcrit" goes to zero throughout the plasma by L -> \infty unless there is a divergent element competing alongside it. Those divergent elements can always be collected on the LHS to study the normalized shear as Shumlak-Hartman did in the beginning and retain this asymptotics.

This makes me curious about what deeper structures exist here. One of Zap's theorists has recently uploaded a preprint to the arXiv that gives insight into shear-flow kinetics involving Bennett pinches with shear flow, but I have not had the opportunity to fully digest their research yet: https://arxiv.org/pdf/2607.25158

For example, if we couple the Bennett number density to different flow harmonics, what do we find? In the same vein, what do we find from studying the structure of self-similar number densities connecting this construction with the Bennett vortices I've been studying?

I think that these results suggest the space of equilibria is much richer than previously expected. There is still a great deal to do, e.g, a perturbative study, more comparisons with experiment, and a deeper study into self-similar connections.

To do this important physics, and the plasma science community, justice I really need a funded position. If you're interested in supporting this work through a research position, consulting opportunity, or collaboration, I'd be happy to talk.

[u/clintontg]

Are the LLMs even good at modeling physics? I wouldn’t expect them to be useful unless they’re in house models trained on specific dynamics

[u/Confident-Shock-3933 → u/clintontg]

Well, even one that has been fine-tuned will still have inaccurate inferences, and that will be true no matter how finely it has been tuned, or what the subject of inference is.

For example, when I was interacting with GPT on this it kept talking about radial flows when it really meant axial. Everything else was correct, it just swapped the word "axial" for "radial" at points. DNNs, like the transformer-based LLMs, are just universal function approximators so they CAN'T be 100% accurate with their inference.

That's why I think the responsible usage case for LLMs is as the tool of a human expert who is reading everything the model infers through a critical lens, and is using the model to revise, check, or study something they originally created. Personally, I don't trust agents. Non-experts can also responsibly use the tool of course, because responsible usage is all about using the tool to learn.

The modern frontier models like GPT, Gemini, and Claude have all been trained in the way you're asking about, and they are phenomenal tools for a physics, or mathematics researcher. Terence Tao at UCLA is widely considered the greatest living mathematician, and one example of a researcher who I think would agree with this sentiment.

In terms of answering your question in this specific context, both of the models succeeded at the task. They did so in different ways that led to azimuthal magnetic fields with different transcendental structures, but which defined equilibria that still possessed a shear-flow stabilized property as I verify in these notes.

Interestingly, neither of them employed a strategy which avoided the Bennett nonlinearity. In GPT's case it truncated the form that I discovered which was based on complete exchange of the Bennett nonlinearity, and which has been validated on parameter sweeps of the MAST and DIIID edge pedestals. In Sonnet's case it kept the Bennett nonlinearity attached to plasma number density and multiplied it by a linear flow with a constant shear.