Jacob Halpern

We demonstrate the design of a flexible, university-scale hybrid tokamak-stellarator experiment based on an axisymmetric array of planar high-temperature superconducting (HTS) dipole coils. Because the coil array has few geometric degrees of freedom, we use single-stage optimization of the coil currents initialized from two-stage solutions to obtain mutually consistent equilibria and coil sets within estimated engineering limits. We find that the field error sets a minimum coil-plasma distance which bounds the axisymmetric envelope available to the boundary, within which rotational transform, volume, coil current, and quasi-symmetry (QS) error trade off against one another. Tighter current limits delocalize the non-axisymmetric shaping and raise QS error at fixed transform. From this single coil array we obtain a broad range of equilibria—quasi-axisymmetric vacuum stellarators with edge ι up to 0.2, finite- β hybrids with realistic profiles reaching on-axis ι ≈ 1 and vacuum transform relevant for MHD stabilization, and strongly shaped tokamaks with elongation κ ≈ 1.7 and triangularity δ ≈ ± 0.6 —all at peak pointwise coil forces within estimated HTS tolerances. We show the same array can additionally correct TF coil ripple, reducing the number of TF coils required compared to the equivalent tokamak. These results establish the design as a promising platform for hybrid tokamak-stellarator research.

This work presents the compact experimental negative triangularity (NT) reactor (CENTAUR), a low overnight cost, high-field tokamak, breakeven reactor design, achieving a predicted total fusion power of 40 MW and scientific energy gain of 1.3. Ballooning stability calculations confirm that the device’s pedestal is within the first stability regime, which is consistent with the expected ELM-free operation associated with NT plasmas. The geometry of the NT divertor allows for high fraction of radiated power (13.5 % ) between the separatrix and plasma facing components. Heat transport modeling based on simulations of the edge region show heat loads into plasma facing components well below material limits. The magnet system employs rare-earth barium copper oxide high-temperature superconductors in 18 toroidal field coils, an hourglass-shaped central solenoid, and six poloidal field coils to support high-field ( B 0 = 10.9 T) plasma confinement, shaping, and current drive. Neutronics analysis shows that a 12 cm B 4 C shield keeps superconducting magnet heating below the 33 K quench limit during 10 s, 40 MW DT pulses. With this shielding, the modeled fluence indicates HTS components can survive more than ten times the 3000-pulse design lifetime. Iteration of economic analysis in tandem with the technical design process allows CENTAUR to achieve its overnight cost goal of $2 billion US determined using a custom costing model that predicts a total overnight cost of 1.6 B ± 0.2 B US.

Non-axisymmetric magnetic field coils have been designed to provide efficient error field (EF) correction and suppress edge localized modes in SPARC—a compact high-field tokamak that is presently under construction at Commonwealth Fusion Systems. These designs utilize the Generalized Perturbed Equilibrium Code’s (GPEC’s) representation of the multi-modal, non-axisymmetric plasma response to optimize the geometric coupling between EF correction coil arrays and the desired core or edge plasma response. Error field correction coils are designed to couple to the plasma-amplified kink that dominates the drive of core resonances. The maximum allowable EF is projected to SPARC using an empirical scaling that is consistent with linear and nonlinear magnetohydrodynamics (MHD) modeling expectations. Asymmetric construction and assembly tolerances are then balanced against the corresponding kA-turns needed for correction to levels below the allowable limit. These physics-driven coil designs provide confidence in our ability to operate SPARC in new high field tokamak regimes without EF induced locked modes.

The National Spherical Torus Experiment Upgrade (NSTX-U) is preparing to resume operation, representing a crucial step toward realizing compact, cost-effective fusion pilot plants (FPPs). In advance of this, extensive modeling and data analysis have been conducted to advance the physics basis for low-aspect-ratio, high-performance plasma regimes, focusing on three core objectives: confinement and stability, power and particle handling, and steady-state operation. Significant progress has been made in understanding the electron temperature flattening in high-β plasmas, which is shown to be driven by a complex interplay of MHD instabilities (e.g., non-resonant infernal modes), fast-ion-driven Alfvén eigenmodes, and electron and ion-scale micro-instabilities, particularly Kinetic Ballooning Modes (KBMs), whose destabilization is strongly dependent on parallel magnetic field fluctuations (δB _‖ ). Furthermore, a new Gyrokinetic Critical Pedestal model was developed, accurately predicting pedestal structure by identifying KBMs as the primary stability limit, offering a critical constraint for future high-confinement scenarios. To address the challenge of high heat flux, novel liquid lithium plasma-facing components were modeled. The analysis confirmed that lithium vapor shielding is a self-regulating mechanism for heat mitigation, while also emphasizing that strong main ion parallel flow is essential to minimize core lithium contamination. Finally, progress toward steady-state operation was anchored by developing the required physics basis and control tools. This includes predictive modeling for Reversed Magnetic Shear sustainment, demonstrating that magnetic island-induced bootstrap current reduction is negligible in STs, and advancing real-time control and disruption avoidance capabilities. The development of high-speed surrogate models (e.g., MMMNet) provides computationally efficient tools vital for non-inductive scenario optimization and integrated, low-disruptivity operations planned for NSTX-U.

The prediction of perturbed equilibrium models for tokamaks with small non-axisymmetric fields is strongly dependent on which reference frame for axisymmetry is assumed. This assumption directly affects the applied field spectrum on the plasma, with the incorrect choice resulting in an incorrect prediction of the plasma response. We use fully 3D equilibria generated by VMEC to determine the correct reference frame when calculating error fields in perturbed equilibrium codes subject to n = 1 misalignments in several coil sets. We analyze a case of independently offset toroidal field (TF) coils and central solenoid (CS)/poloidal field (PF) coils in the SPARC tokamak and find that the appropriate reference frame can be well approximated by the centroid of the TF coil set. We also consider the case of NSTX-U with an independent centerpost consisting of the inner legs of the TF coils and CS, and find that the reference frame can be well approximated by the radial location of the TF coil inboard legs at the midplane. We determine the correct frame by analyzing the shifted magnetic axis in the nonlinear equilibria, and use our findings to generalize to 3D fields from misalignments which modify the reference frame and those which impact the plasma response. We also analyze the magnetic field line displacement to identify where the linearized MHD theory begins to break down, and compare that to typical coil tolerances in existing tokamaks. We find that linear theory is valid for existing tolerances, validating the use of perturbative codes to set these tolerances, but with a sufficiently small margin to be of note for future devices with relative tolerances larger than approximately 1% of the minor radius. This study enables engineers to confidently use 3D perturbative models for determining assembly tolerances by providing insight into the correct applications of the theory.

Access to the plasma chamber in a stellarator reactor is essential for maintenance and diagnostics. However, the complex geometry of stellarator coils, often characterized by their strong twisting, can severely limit the space available for access ports. This study introduces a novel optimization approach in which access ports are represented as closed curves on the plasma boundary. By carefully selecting a set of objectives and penalties related to the access port, we demonstrate the first stellarator coil optimization explicitly targeting improved access port size. The trade-off between magnetic field quality and port size is analyzed through the Pareto front of their respective objectives. The optimal location of a port is explained using a current potential approach. Finally, we show that additional shaping coils, such as windowpane coils, can enable the crossing of the Pareto front to achieve superior configurations.

This Corrigendum corrects and clarifies several inaccuracies in section 1A in the original article (Kaptanoglu et al 2025 Nucl. Fusion 65 046029).

In this work, we utilize new coil objectives for stellarator optimization with autodifferentiation, including pointwise and net coil–coil forces and torques. We use these methods to perform the first large-scale optimization of planar dipole coil arrays, since arrays of small and geometrically simple coils have been proposed to partially produce the 3D magnetic fields for stellarators, generate advantageous magnetic field perturbations in tokamaks, and provide active, real-time control capabilities. We perform an ablation study to show that minimizing the orientation and location of each coil may be essential to get coil forces, coil torques, and field errors to tolerable levels. We conclude with solutions for three reactor-scale quasi-symmetric stellarators by jointly optimizing nonplanar TF coils and planar coil arrays.

Bijective point transformations were recently used to derive the classical space-charge-limited current (SCLC) in one-dimensional (1D) nonplanar devices for electrons emitted into vacuum with nonzero monoenergetic initial velocity. Using these transformations, we first derive a canonical form of SCLC for a relativistic diode with zero initial velocity that holds for any orthogonal 1D geometry and recovers the previously derived planar result. We extend this result to derive a canonical form of SCLC that accounts for nonzero monoenergetic initial velocity and relativistic effects, while recovering SCLC for nonrelativistic diodes with zero and nonzero initial velocity and the relativistic diode with zero initial velocity in appropriate limits. We then use appropriate bijective point transformations to convert from the canonical solution to concentric cylindrical and spherical coordinates. This equation has no closed form solution and must be numerically integrated. The relativistic effects of initial velocity do not become significant until the Lorentz factor γ0≳1.1; for lower γ0, nonrelativistic SCLC gives a reasonable approximation. In the ultra-relativistic limit, Jr,SCLC/JSCLC∝V1/2, where Jr,SCLC and JSCLC are the SCLC for the relativistic diode with general initial velocity and nonrelativistic diode with zero initial velocity, respectively. These asymptotic equations match the exact solutions for sufficiently large γ0 and V. This analysis provides an exact, numerical solution for SCLC for nonzero monoenergetic initial velocity that incorporates relativistic effects for any 1D orthogonal geometry.

Understanding space-charge-limited current density (SCLCD) is fundamentally and practically important for characterizing many high-power and high-current vacuum devices. Despite this, no analytic equations for SCLCD with nonzero monoenergetic initial velocity have been derived for nonplanar diodes from first principles. Obtaining analytic equations for SCLCD for nonplanar geometries is often complicated by the nonlinearity of the problem and over constrained boundary conditions. In this Letter, we use the canonical coordinates obtained by identifying Lie-point symmetries to linearize the governing differential equations to derive SCLCD for any orthogonal diode. Using this method, we derive exact analytic equations for SCLCD with a monoenergetic injection velocity for one-dimensional cylindrical, spherical, tip-to-tip (t-t), and tip-to-plate (t-p) diodes. We specifically demonstrate that the correction factor from zero initial velocity to monoenergetic emission depends only on the initial kinetic and electric potential energies and not on the diode geometry and that SCLCD is universal when plotted as a function of the canonical gap size. We also show that SCLCD for a t-p diode is a factor of four larger than a t-t diode independent of injection velocity. The results reduce to previously derived results for zero initial velocity using variational calculus and conformal mapping.