Omnigenous umbilic stellarators

TL;DR

This work introduces umbilic stellarators, a class with a continuously high-curvature edge on the plasma boundary that wraps toroidally before reconnecting, enabling edge-region control relevant to X-point and island divertor designs. The authors develop a DESC-based method to simultaneously optimize the boundary and a designated umbilic edge curve while enforcing omnigenity, and they demonstrate vacuum and finite-$\beta$ omnigenous equilibria (US131 and US252) with detailed edge-field analyses and coil designs. They further explore a practical path to convert a tokamak into a diverted stellarator by adding low-current umbilic coils (co-$I$ and counter-$I$ configurations) and perform single-stage optimizations to achieve robust edge structure with small normal-field errors and acceptable neoclassical transport. The findings indicate that edge ridges can serve as divertor-like regions and that piecewise omnigenity can arise without a fixed helicity, offering new avenues for divertor physics and tokamak-to-stellarator transitions, with implications for boundary stability and edge confinement. $\iota$ on the boundary can be tuned via shaping and coil currents, suggesting practical control of edge topology in fusion devices.

Abstract

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.

Figures (16)

• Figure 1: An umbilic boundary design with $n_{\rm{FP}}=4, n=3, m=2, \varrho = 0.344$. The rest of the parameters $r_j=0$. In (a) we show the full shape and in (b) we present a single field period from the parametrization. A point $p_i \in \{0, 1, 2\}$ corresponding to the sharp edge at each toroidal cross section connects to$\mod(p_i+m, 3)$ after a single field period. The sharp edge defined by $R_{\mathrm{US}}(\theta= 0, \zeta), Z_{\mathrm{US}}(\theta=0, \zeta)$ meets itself after $n = 3$ toroidal turns.
• Figure 2: Contour plot of the flux surface average force balance error $|F|$ plotted on the cross section $\zeta = 0$ of a vacuum DESC equilibrium created using a US131 boundary with $\rho = 0.1, r_1=0.133, r_5=0.1$. The actual boundary (in solid black) is obtained after fitting a Fourier-Zernike series to the true umbilic boundary(in dashed blue).The resolution of the DESC equilibrium is $L=M=N=14$ and the resolution of the umbilic curve $N_{\mathrm{u}} = 12$.
• Figure 3: US131 optimization results showing a significant increase in magnetic axis torsion in figure (a) from the initial (dotted) to the optimized (solid) boundary. This leads to an increase in the rotational transform, as shown in figure (b) such that $\iota(\rho = 1)$ approaches $1/3$. The shaping also leads the fieldlines to align with the umbilic edge in the optimized equilibrium as shown in figure (c).
• Figure 4: Output from the vacuum US131 optimization showing Boozer plots on the boundary and the effective ripple. Figure (b) does not have a specific helicity but still achieves a low ripple transport.
• Figure 5: Coils and $\bm{B}_{\mathrm{out}}\cdot\hat{\bm{n}}$ error plotted on the plasma boundary. The $\mathrm{max}|(\bm{B}_{\mathrm{out}}\cdot\hat{\bm{n}})/\bm{B}_{\mathrm{out}}| \leq 1\%$. Only one group of unique coils are plotted. On the upper right we plot the Poincaré cross-section at $\zeta=0$ along with the fixed-boundary solution(in red). The gray curves mark a favorable position to place a divertor. In the lower right part we plot the $\bm{B}_{\mathrm{out}}\cdot\hat{\bm{n}}$ error on a 2D plane along with the umbilic edge(dashed purple).