Investigation of Toroidal Rotation Effects on Spherical Torus Equilibria using the Fast Spectral Solver VEQ-R

TL;DR

This paper introduces VEQ-R, a fast, rotation-aware fixed-boundary equilibrium solver based on a 12-parameter shifted Chebyshev spectral expansion and a Matrix-Kernel acceleration to solve the Generalized Grad-Shafranov equation in real time. By employing an inverse coordinate formulation and high-order spectral moments, VEQ-R captures non-rigid flux-surface distortions under strong toroidal rotation, including sonic regimes near $M\sim1$, while maintaining high geometric fidelity to high-resolution benchmarks. Benchmarking against a high-fidelity Finite Difference Model demonstrates sub-percent accuracy for core quantities and a few percent for the $q$-profile, with convergence times around $5$ ms on a single core—facilitating real-time control and large-scale parameter studies. Physics investigations reveal rotation-driven centrifugal flux compression significantly lowers the core safety factor $q_0$ toward unity, induces strong asymmetries between the low- and high-field sides, and highlights a delicate balance between local confinement gains and global MHD stability in spherical tokamaks. The work also outlines a path to extending the framework to multi-fluid MHD for advanced fusion scenarios, underscoring its potential for next-generation devices and control systems.

Abstract

Standard reduced models often fail to adequately describe the complex geometric response of tokamak plasmas to strong toroidal rotation. In this work, we present VEQ-R, a computationally efficient spectral solver designed to calculate fixed-boundary equilibria with arbitrary toroidal flow. In contrast to computationally intensive grid-based codes, our model employs a 12-parameter shifted Chebyshev spectral expansion to explicitly resolve radial variations in high-order shaping profiles--such as dynamic elongation and triangularity. This capability allows the solver to accurately capture differential flux surface distortions (non-rigid effects) even in challenging sonic regimes ($M \sim 1.0$). By synergizing this compact variational formulation with a novel ``Matrix-Kernel'' acceleration technique, we transform the problem into pre-computed algebraic matrix operations. This approach achieves convergence in approximately 5 ms, maintaining exceptional geometric fidelity compared to high-resolution benchmarks while balancing speed and accuracy. Our analysis reveals that rotation-induced flux compression leads to a monotonic decrease in the core safety factor $q_0$, pushing it dangerously close to unity--a structural deformation mechanism effectively captured by this approximate yet robust solver.

Figures (10)

• Figure 1: Input physical profiles used for benchmarking the solver. (a) Ion temperature profile $T(\rho)$; (b) Toroidal angular velocity profile $\Omega(\rho)$; (c) The self-consistently calculated squared Mach number profile $M^2(\psi)$ derived from inputs (a) and (b). Note that the initialization ensures thermodynamic consistency where the Mach number peaks at the magnetic axis.
• Figure 2: Evolution of spectral geometric coefficients with increasing axis Mach number $M_{axis}$. (a) Shafranov shift coefficients ($h$); (b) Elongation coefficients ($\kappa$); (c) Triangularity/Distortion coefficients ($\delta$). The distinct monotonic trends of the high-order harmonic terms (red lines, e.g., $C_3$) confirm that the 12-parameter VEQ-R model successfully captures the "non-rigid" geometric distortions excited by centrifugal forces.
• Figure 3: Benchmark results under static conditions ($M=0$). Left: Comparison of magnetic flux surfaces between the proposed VEQ-R solver (blue solid lines) and the high-resolution FDM reference (red dashed lines), showing excellent geometric agreement. Right: Comparison of midplane profiles for toroidal current density $J_\phi$, pressure $P$, poloidal current function $F$, and safety factor $q$.
• Figure 4: Quantitative error analysis for the static case ($M=0$). The subplots display the radial distribution of relative errors for (top-left) toroidal current density $J_\phi$, (top-right) pressure $P$, (bottom-left) poloidal current function $F$, and (bottom-right) safety factor $q$. The mean relative error for each quantity is indicated, demonstrating high accuracy in the absence of rotation.
• Figure 5: Benchmark results under transonic rotation conditions ($M_{axis}^2=1.0$). Left: Flux surface comparison showing the significant outward Shafranov shift and geometric compression on the low-field side driven by centrifugal forces. Right: Comparison of key physical profiles ($J_\phi, P, F, q$), illustrating the VEQ-R solver's capability to accurately reconstruct the asymmetric current pile-up and pressure shifts.