New Plasma Solver Accurately Models Complex Behaviour at Sonic Speeds

Tokamak plasmas exhibiting strong toroidal rotation present a significant challenge to conventional equilibrium modelling. Xingyu Li, Huasheng Xie from ENN Science and Technology Development Co., Ltd., and Lai Wei and Zhengxiong Wang from the Key Laboratory of Materials Modification by Laser, Ion, and Electron Beams of the Ministry of Education, School of Physics, Dalian University of Technology, have addressed this issue through the development of VEQ-R, a fast spectral solver for calculating fixed-boundary equilibria with arbitrary toroidal flow. This research is significant because it moves beyond the limitations of standard reduced models by employing a 12-parameter shifted Chebyshev spectral expansion to accurately resolve radial variations in plasma shaping profiles, even in sonic regimes. The team’s innovative “Matrix-Kernel” acceleration technique achieves rapid convergence, approximately 5ms, while maintaining geometric fidelity, and their analysis demonstrates that rotation-induced flux compression can dangerously reduce the core safety factor, a structural deformation mechanism effectively captured by this robust solver.

These plasmas, crucial for achieving high-performance fusion energy, exhibit complex geometric responses to rotation that standard modelling techniques often fail to capture.

VEQ-R addresses this limitation by employing a 12-parameter shifted Chebyshev spectral expansion, allowing it to explicitly resolve radial variations in plasma shaping profiles, including dynamic elongation and triangularity, with exceptional fidelity. This innovative approach enables the solver to accurately simulate differential flux distortions, even in challenging conditions where plasma rotation approaches sonic speeds.
The research transforms the complex problem of plasma equilibrium calculation into pre-computed algebraic matrix operations, achieving convergence in approximately 5 milliseconds. This represents a substantial speed increase compared to traditional high-resolution benchmarks, while maintaining geometric accuracy essential for predicting plasma stability.

By synergizing a compact variational formulation with a novel “Matrix-Kernel” acceleration technique, VEQ-R offers a pathway towards real-time plasma control and advanced simulations. The solver’s capabilities extend beyond simply tracking plasma movement; it reveals how rotation-induced flux compression leads to a monotonic decrease in the core safety factor, a critical parameter indicating potential structural instabilities.

Analysis using VEQ-R demonstrates that strong toroidal rotation compresses the magnetic flux surfaces, driving the core safety factor dangerously close to unity. This phenomenon, a structural deformation mechanism, is effectively captured by the solver’s robust yet approximate approach. The work builds upon earlier variational moment methods by incorporating high-order geometric approximations and a spectral formulation, moving beyond the limitations of low-order models.

This advancement is particularly relevant for spherical tokamaks and next-step fusion devices where core Mach numbers can exceed 0.5, demanding accurate modelling of rotational effects on plasma equilibrium. VEQ-R’s speed and accuracy promise to unlock new possibilities for millisecond-scale applications, including real-time Plasma Control Systems and massive integrated transport simulations.

The solver’s reliance on pre-calculated projection matrices, rather than runtime numerical integration, is a key innovation. This “Matrix-Kernel” scheme significantly reduces computational demands, enabling rapid equilibrium calculations without sacrificing geometric fidelity. The research highlights the importance of accurately modelling the interplay between plasma rotation, magnetic geometry, and stability, a crucial step towards realising practical fusion energy.

Convergence to a fixed-boundary equilibrium was consistently achieved in approximately 5 milliseconds. This rapid solution time stems from the implementation of a novel “Matrix-Kernel” acceleration technique, transforming the problem into pre-computed algebraic matrix operations. The solver demonstrates exceptional geometric fidelity when benchmarked against high-resolution codes, balancing computational speed with accuracy.

Analysis of the resulting equilibria reveals a monotonic decrease in core safety factor, q, with increasing toroidal rotation. Specifically, the safety factor is driven dangerously close to unity, a condition indicative of potential structural deformation, and this behaviour is effectively captured by the model.

The research employs a 12-parameter shifted Chebyshev spectral expansion to resolve radial variations in high-order shaping profiles, including dynamic elongation and triangularity. This approach allows accurate capture of differential flux distortions, even in sonic regimes where conventional methods struggle. The input strategy accepts independent inputs for reference pressure and boundary conditions.

Chebyshev spectral expansion and Matrix-Kernel technique for rapid tokamak equilibrium reconstruction

A 12-parameter shifted Chebyshev spectral expansion forms the core of VEQ-R, a computationally efficient solver designed to calculate fixed-boundary tokamak equilibria with arbitrary toroidal flow. This approach explicitly resolves radial variations in high-order shaping profiles, including dynamic elongation and triangularity, offering a significant advancement over lower-order models that often assume rigid profiles.

By employing spectral methods, the research circumvents the limitations of computationally intensive grid-based solvers like EFIT, which rely on Picard iterations and struggle with millisecond-scale applications. The choice of Chebyshev polynomials provides efficient and accurate representation of functions, particularly well-suited for capturing the complex geometric response of plasmas.

To further accelerate calculations, the study introduces a novel “Matrix-Kernel” technique that transforms the problem into pre-computed algebraic matrix operations. This innovation bypasses runtime numerical integration of the variational functional, instead utilising pre-calculated projection matrices to dramatically reduce computational demands.

The solver is implemented within a vectorized MATLAB environment, enabling rapid execution and facilitating parameter scans for massive integrated transport simulations. This prioritisation of speed and ease of use distinguishes VEQ-R from high-fidelity inverse coordinate solvers that pursue numerical exactness at the expense of computation time.

The methodology explicitly incorporates high-order triangularity terms to capture subtle geometric responses to centrifugal forces, moving beyond basic displacement and elongation parameters. The research builds upon previous variational moment methods, pioneered by Lao et al., by extending the spectral moment framework to explicitly include toroidal flow effects.

This allows for accurate capture of differential flux distortions, even in challenging sonic regimes where strong centrifugal forces induce complex, non-rigid geometric distortions. Consequently, the solver achieves convergence in approximately 5ms on a single core, representing a nearly 1000-fold speedup compared to high-resolution finite difference codes.

The Bigger Picture

The longstanding challenge of accurately modelling plasma behaviour within tokamak fusion reactors has just encountered a significant, if subtle, advance. For decades, simulations have struggled to reconcile computational speed with the geometric complexity introduced by plasmas in motion. Existing models often rely on simplifications that, while easing the processing burden, obscure crucial details of how rotating plasmas distort and deform.

This new solver, VEQ-R, doesn’t eliminate the need for approximation entirely, but it shifts the balance decisively towards fidelity. What distinguishes this work is not simply faster computation, though a convergence time of around 5 milliseconds is impressive, but the method employed to achieve it. By leveraging spectral expansions and a clever “Matrix-Kernel” acceleration technique, the researchers have effectively traded memory for speed, pre-computing much of the necessary algebra.

This allows for a more nuanced representation of the plasma’s shape, particularly the subtle changes in safety factor induced by rotation. The finding that rotation can drive this factor dangerously low, potentially triggering structural instabilities, is a direct consequence of this improved geometric resolution.

However, VEQ-R remains a fixed-boundary model, meaning it doesn’t account for the plasma pushing back against the confining magnetic fields. This is a deliberate simplification, but one that limits its ability to predict truly disruptive events. Furthermore, the model’s reliance on Chebyshev polynomials, while efficient, may not be optimal for all plasma configurations.

Future work will undoubtedly focus on extending this approach to free-boundary equilibria and exploring alternative spectral basis functions. More broadly, this represents a step towards bridging the gap between predictive modelling and real-time control, offering the tantalising prospect of actively stabilising plasmas and sustaining fusion reactions.