Feasibility of a Flexible, Hybrid Tokamak-Stellarator Experiment using an Axisymmetric Dipole Coil Array

This paper presents the design of a flexible, university-scale hybrid tokamak-stellarator experiment utilizing an axisymmetric array of planar HTS dipole coils, which enables the generation of a wide range of equilibria—from quasi-axisymmetric stellarators to strongly shaped tokamaks—while maintaining engineering feasibility and reducing the required number of toroidal field coils.

The Gist

Imagine trying to build a perfect, invisible cage out of magnetic fields to hold a super-hot ball of plasma (the fuel for fusion energy). Scientists usually have two main ways to build this cage:

1. The Tokamak: Like a donut-shaped ring. It's simple and holds heat well, but it needs a massive electric current flowing inside the plasma itself to work. This is risky because if that current gets unstable, the whole thing can crash (a "disruption").
2. The Stellarator: Like a twisted, knotted pretzel. It uses complex, 3D-shaped magnets outside the plasma to hold it. It's very stable, but those magnets are incredibly hard to build, expensive, and difficult to design.

The New Idea: A "Hybrid" with a Twist
This paper proposes a clever middle-ground experiment. Instead of building unique, complex magnets for every shape, the researchers designed a flexible "Lego set" of magnets.

The "Lego" Analogy
Imagine a circular track (the vacuum vessel). Instead of placing a few huge, custom-shaped magnets around it, they placed a grid of many small, flat, rectangular magnets (dipole coils) all around the track.

• The Magic: Because there are so many of them, they can turn the current on or off in different patterns.
• The Result: By changing the electricity flowing through these magnets, they can instantly reshape the magnetic cage. One moment, it looks like a simple donut (Tokamak); the next, it looks like a twisted pretzel (Stellarator).

The Challenge: The "Tightrope" Walk
The paper explains that this grid of magnets is very rigid; they can't move the magnets around, they can only change the electricity. This makes the math very hard.

• The Trade-off: Think of the plasma as a balloon inside a rigid box. If you want the balloon to be very twisted (high "rotational transform" for stability), you have to push it closer to the walls. But if it gets too close, the magnets have to work too hard (too much electric current) and might break.
• The Solution: The researchers used a super-computer to find the "sweet spot." They found that no matter how they twisted the balloon, it always had to stay within a specific, invisible "envelope" or boundary. Inside this boundary, they could trade off between how twisted the shape is, how much space the plasma has, and how hard the magnets have to work.

What They Actually Built (On Paper)
Using this design, they showed they could create:

• Stellarators: Twisted shapes that are stable without needing a dangerous internal current.
• Tokamaks: Donut shapes that are highly stretched and squashed (like a peanut) to improve performance.
• Hybrids: A mix of both, where the magnets provide just enough twist to stop the Tokamak from crashing, but not so much that it becomes a complex Stellarator.

Bonus Superpowers
The paper highlights two extra tricks this "Lego set" can do:

1. Smoothing the Bumps: In standard Tokamaks, the gaps between the big magnets create "ripples" in the magnetic field that let heat escape. This new array of small magnets can act like a "filler" to smooth out those ripples, meaning you could use fewer big magnets.
2. Shaping the Plasma: By turning the magnets on in a specific way, they can act like standard shaping coils, allowing them to create plasma shapes that are usually very hard to achieve, like "negative triangularity" (a shape that looks a bit like a D turned upside down).

The Bottom Line
The paper doesn't claim they built the machine yet. Instead, they proved that the design is feasible. They showed that with a fixed grid of magnets and a smart computer algorithm, you can create a wide variety of stable fusion shapes without breaking the magnets. It's a flexible, university-scale platform that could help scientists study how to make fusion energy safer and more efficient.

Technical Summary

Technical Summary: Feasibility of a Flexible, Hybrid Tokamak-Stellarator Experiment using an Axisymmetric Dipole Coil Array

Problem Statement
Tokamaks offer favorable confinement but suffer from current-driven instabilities and disruptions due to their reliance on toroidal plasma current. Stellarators avoid these issues through 3D magnetic field shaping without plasma current but face challenges in coil complexity, manufacturing cost, and optimization over large parameter spaces. Hybrid approaches aim to combine the benefits of both, yet traditional hybrid designs often require complex, non-planar coils or lack the flexibility to explore diverse equilibrium configurations. This paper addresses the feasibility of a university-scale hybrid experiment based on an axisymmetric array of planar High-Temperature Superconducting (HTS) dipole coils. The primary challenge is that the coil geometry in this design is highly constrained (determined largely by the vacuum vessel shape and discrete coil counts), rendering traditional two-stage optimization (separate plasma and coil optimization) insufficient for finding mutually consistent equilibria within realistic engineering limits.

Methodology
The authors propose a design based on the geometry of the HBT-EP experiment at Columbia University (major radius $R_0 \approx 1.0$ m). The system consists of standard toroidal field (TF) coils, a central solenoid, poloidal field coils, and a novel array of planar dipole coils mounted on an axisymmetric vacuum vessel. The dipole coils are arranged in a grid ($N_\theta$ poloidal $\times$ $N_\phi$ toroidal) and are initialized as rounded rectangles tangent to the vessel.

To overcome the limitations of the constrained coil geometry, the authors employ a single-stage optimization approach. Unlike standard two-stage methods, this algorithm simultaneously optimizes the plasma equilibrium surface and the dipole coil currents.

1. Initialization: The process begins with a "hot-start" using a two-stage approach. Stage-I optimizes a quasi-axisymmetric (QA) vacuum equilibrium with constraints on volume, distance to the vessel, and curvature. Stage-II then finds a coil set and currents that minimize field error for that equilibrium.
2. Single-Stage Refinement: The resulting configuration serves as an initial condition for a single-stage routine. This routine minimizes the quasi-symmetry (QS) error while satisfying constraints on rotational transform ($\iota$), plasma volume, and coil currents (represented by a high-order $p$-norm to approximate maximum current).
3. Engineering Constraints: The optimization incorporates realistic HTS constraints, specifically a maximum pointwise force per unit length ($|dF/dl|$) well below the typical HTS tolerance of $4\text{--}8 \times 10^5$ N m$^{-1}$. The study also investigates "sparse" configurations where outboard coils are removed to improve diagnostic access.
4. Finite-$\beta$ and Tokamak Modes: The authors extend the analysis to finite-$\beta$ equilibria using VMEC with realistic HBT-EP pressure and current profiles. Additionally, they model the dipole array's utility in tokamak mode, specifically for correcting TF coil ripple and creating shaped equilibria (triangularity) using the dipole currents as steady-state poloidal field (PF) coils.

Key Contributions and Results

• Optimization Strategy: The paper demonstrates that for highly constrained coil geometries, single-stage optimization is necessary to achieve consistent equilibria. The authors show that the single-stage method converges robustly from varied initial conditions due to the linear relationship between dipole currents and the magnetic field.
• Trade-off Envelope: A central finding is that field error and coil current thresholds jointly define a minimum and maximum coil-plasma distance. This confines the plasma boundary to a roughly fixed axisymmetric envelope. Within this envelope, rotational transform, plasma volume, coil current, and QS error must trade off against one another.
• Vacuum Stellarator Performance: The design achieves quasi-axisymmetric vacuum equilibria with rotational transform up to $\iota \approx 0.15$ (for $V=0.3$ m$^3$) and low QS error. Peak pointwise coil forces reach approximately $2.75 \times 10^5$ N m$^{-1}$, remaining safely below estimated HTS limits.
• Finite-$\beta$ Hybrid Operation: The array supports finite-$\beta$ configurations with realistic profiles, achieving on-axis $\iota \approx 1$ (typical of tokamaks) while maintaining an edge vacuum transform fraction ($f_{ERT}$) of $\approx 0.25$, which is sufficient to stabilize current-driven MHD instabilities. The inclusion of plasma current increases coil currents by $\approx 30\%$ and field error by $\approx 0.1\%$, but the QA quality remains largely preserved.
• Sparsity: Removing the outboard midplane coils (creating a sparse array) degrades QS error by only 10–20% while significantly improving diagnostic access, demonstrating the design's robustness to geometric modifications.
• Tokamak Versatility: The same array can correct TF coil ripple, allowing for a reduction in the number of TF coils (from 20 to 12–16) while maintaining ripple parameters below 1%. Furthermore, by aligning currents in the toroidal direction, the array acts as PF coils to create strongly shaped tokamak equilibria with elongation $\kappa \approx 1.7$ and triangularity $\delta \approx \pm 0.6$.

Significance and Claims
The paper claims to establish the design as a promising platform for hybrid tokamak-stellarator research. Its primary significance lies in its flexibility: a single, fixed coil array can generate a broad spectrum of equilibria, ranging from vacuum stellarators to current-carrying hybrids and strongly shaped tokamaks. This allows for the study of how edge geometry influences exhaust physics and how vacuum rotational transform affects MHD stability across different operating modes.

The authors are modest about the current scope, noting that the design relies on idealized filamentary coil models and analytic force proxies. They acknowledge that detailed finite-element analysis of support structures and the inclusion of finite-build coil models are necessary next steps. However, the results suggest that the engineering headroom (forces well below HTS limits) allows for future exploration of more complex configurations, such as quasi-helical or quasi-isodynamic equilibria, potentially at higher aspect ratios. The work provides a foundational framework for a university-scale experiment capable of bridging the gap between tokamak and stellarator physics.