Omnigenous umbilic stellarators

To better understand the dependence of the magnetic field structure in the plasma edge on the plasma boundary shape, in the context of X-point and island divertor designs, we define and develop a class of stellarators called umbilic stellarators. These equilibria are characterized by a single continuous high-curvature edge on the plasma boundary that goes around multiple times toroidally before meeting itself. We develop a technique that allows us to simultaneously optimize the plasma boundary along with a curve lying on the boundary on which we impose a high curvature while imposing omnigenity – a property of the magnetic field that ensures trapped particle confinement throughout the plasma volume. We find that umbilic stellarators naturally tend to favor piecewise omnigenity instead of omnigenity with a specific helicity. After generating omnigenous umbilic stellarators, we design coil sets for some of them and explore the field line structure in the edge and its sensitivity to small fluctuations in the plasma. Finally, using single-stage optimization, we simultaneously modify the plasma and coil shape and propose an experiment to modify an existing tokamak to a finite-beta stellarator using this technique and explore a potentially simpler way to convert a limited tokamak into a diverted stellarator.

💡 Research Summary

The paper introduces a novel class of stellarator configurations termed “umbilic stellarators” (US), designed to explore how plasma‑boundary shaping influences edge magnetic topology and particle confinement. An umbilic stellarator is defined by a single continuous high‑curvature ridge on the plasma boundary that wraps toroidally multiple times before closing on itself, reminiscent of the geometric “umbilic bracelet” whose cross‑section is a three‑cusp deltoid. This high‑curvature edge is intended to act as a continuous divertor, providing long connection lengths and a broad area for heat‑flux spreading without relying on discrete X‑points or island chains.

The authors develop a simultaneous optimization framework that treats the plasma boundary shape and the curve on which the high curvature is imposed as coupled design variables. The framework enforces a prescribed curvature along the “umbilic edge” while also imposing omnigenity—a condition ensuring that the bounce‑averaged radial drift of trapped particles vanishes throughout the volume. Omnigenity is a generalization of axisymmetric neoclassical confinement and of quasisymmetry; achieving it in a non‑axisymmetric device typically requires multi‑objective optimization.

The equilibrium calculations are performed with DESC (the “DEformed Stellarator Code”), which solves the ideal MHD force‑balance equation (j × B = ∇p) using a Fourier‑Zernike spectral representation in (ρ, θ, ζ) coordinates. The umbilic boundary is first expressed analytically (Eqs. 3.1‑3.2) with parameters controlling the major radius, minor radius, umbilic factor n (number of sides), poloidal winding m, field‑period number nFP, and several shape coefficients (r₁‑r₅). These parameters determine the toroidal periodicity of the edge, the rational rotational transform ι ≈ nFP·(m/n), and the overall three‑dimensional geometry. Because a perfectly sharp cusp cannot be represented exactly with a global Fourier basis, the authors relax the requirement to a “high curvature” condition, which is easier to enforce numerically while still producing a pronounced ridge.

Two families of equilibria are generated: vacuum (β = 0) and finite‑beta (β ≈ 0.02) cases. In both families the optimization drives the boundary rotational transform toward the target rational value, aligns the high‑curvature edge with the field‑line label α = 0, and minimizes the omnigenity and effective ripple metrics. The results reveal that umbilic stellarators naturally tend toward “piecewise omnigenity,” where different toroidal sectors achieve near‑perfect omnigenity while the global configuration may retain small deviations. This piecewise behavior reduces the overall optimization cost compared with enforcing strict global omnigenity.

Coil sets are then designed for selected equilibria. The authors construct modular coils for the vacuum case and introduce “umbilic coils” – helically wound, low‑current coils that follow the high‑curvature ridge. Field‑line tracing shows that the edge region forms a continuous divertor‑like structure with long connection lengths, and that the topology is robust to small perturbations. To test robustness, a dummy current source is placed on the magnetic axis to mimic plasma current fluctuations; the edge field‑line pattern changes only modestly, indicating resilience comparable to non‑resonant divertor concepts.

Finally, the paper proposes an experimental pathway to convert an existing tokamak (the Columbia HBT‑EP device) into a diverted stellarator using the umbilic approach. The original tokamak coil set is retained, and a single additional umbilic coil is added. By varying the direction of the umbilic coil current (co‑current or counter‑current relative to the plasma current), the authors examine the impact on the edge rotational transform (iota) and on the formation of the high‑curvature ridge. A single‑stage optimization simultaneously adjusts the plasma boundary and the umbilic coil geometry to achieve the desired omnigenity and edge topology. This strategy suggests a relatively low‑cost, low‑complexity route to retrofit existing tokamaks with stellarator‑type divertor capabilities, potentially accelerating experimental studies of edge physics in non‑axisymmetric configurations.

In summary, the work delivers a comprehensive methodology for designing stellarators with prescribed high‑curvature boundary features, demonstrates that such designs can maintain excellent neoclassical confinement (via omnigenity), provides practical coil designs, and outlines a realistic retrofit experiment. The concept of piecewise omnigenity and the use of low‑current umbilic coils represent significant innovations that could influence future stellarator divertor designs and hybrid tokamak‑stellarator research.