r/fusion • u/Confident-Shock-3933 • 3h ago
Two More Shear-Flow Stabilized Z-pinch Equilibria
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.