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

This paper introduces VEQ-R, a computationally efficient spectral solver that utilizes a compact Chebyshev expansion and a novel "Matrix-Kernel" technique to rapidly and accurately model fixed-boundary spherical torus equilibria with strong toroidal rotation, revealing that rotation-induced flux compression significantly reduces the core safety factor.

The Gist

Imagine you are trying to predict the shape of a giant, swirling, super-hot balloon made of magnetic fields and plasma. This is what scientists do when they study tokamaks, the doughnut-shaped machines designed to harness nuclear fusion (the power of the sun).

Usually, these balloons are relatively calm. But in the most advanced experiments, we spin them incredibly fast using powerful beams of particles. This is like spinning a pizza dough so fast that the centrifugal force tries to fling the dough outward.

Here is the problem: When you spin a plasma this fast, it doesn't just move as a solid block. It squishes, stretches, and warps in complex, weird ways. Traditional computer programs used to predict these shapes are like slow, heavy calculators. They can get the answer right, but they take minutes or even hours to crunch the numbers. That's too slow for a real-time control system that needs to make decisions in milliseconds to keep the reactor safe.

This paper introduces a new tool called VEQ-R. Think of it as a super-fast, high-speed camera that can predict the shape of that spinning plasma balloon in the blink of an eye (about 5 milliseconds).

Here is how they did it, broken down into simple concepts:

1. The "Rigid" vs. "Jelly" Problem

Old, fast computer models treated the spinning plasma like a rigid metal ring. If you spun it, the whole ring just shifted to one side.
But real plasma is more like jelly. When you spin jelly, the center might stay put while the edges stretch out, or the middle might get squished while the sides bulge. This is called a "non-rigid" effect.

• The Innovation: VEQ-R doesn't treat the plasma as a rigid ring. It uses a special mathematical "skeleton" (a 12-parameter spectral expansion) that allows the jelly to stretch, squash, and warp realistically, just like a real spinning balloon.

2. The "Matrix-Kernel" Magic Trick

Usually, calculating how a spinning jelly deforms requires doing millions of tiny math steps over and over again. It's like trying to count every grain of sand on a beach by picking them up one by one.

• The Innovation: The authors created a "Matrix-Kernel" technique. Imagine instead of counting grains of sand one by one, you have a pre-made map that tells you exactly where every grain is based on a few key numbers. They did all the hard math before the computer even started running. Now, the computer just looks up the answer in a pre-calculated table.
• The Result: This turns a task that takes minutes into one that takes 5 milliseconds. That's a 1,000x speedup!

3. The Dangerous "Squeezing" Effect

Using this new fast tool, the scientists discovered something surprising and potentially dangerous about spinning plasmas.

• The Analogy: Imagine a spinning top. As it spins faster, the centrifugal force pushes everything to the outside edge. In the plasma, this pushes the pressure against the "low-field" side (the outer wall of the doughnut).
• The Discovery: This pushing force is so strong that it compresses the magnetic "cage" holding the plasma. This compression forces the electric current in the center of the plasma to bunch up tightly.
• The Risk: When that current bunches up too much, a safety number called the safety factor ($q_0$) drops dangerously close to 1. In the world of fusion, a value near 1 is like a house of cards waiting to collapse. It can trigger a "sawtooth crash," where the core of the plasma suddenly dumps its energy, potentially damaging the machine.

4. Why This Matters

• Speed: Because VEQ-R is so fast, it can be used in the real-time control systems of future fusion reactors. It can tell the reactor's computer, "Hey, you're spinning too fast, the shape is getting dangerous, slow down!" instantly.
• Accuracy: It's not just fast; it's accurate. The paper proves that VEQ-R matches the results of the slow, super-accurate models almost perfectly, even when the plasma is spinning at "sonic" speeds (as fast as sound).
• Spherical Tori: This is especially important for "Spherical Tokamaks" (which look more like cored apples than doughnuts). These machines spin faster and are more sensitive to these shape changes. VEQ-R helps us understand how to keep them stable.

The Bottom Line

The authors built a fast, flexible, and accurate digital twin for spinning fusion plasmas. They found that while spinning helps control some problems, it creates a new danger: it squeezes the plasma so hard that it risks collapsing its own magnetic cage. Thanks to their new "Matrix-Kernel" trick, we can now predict and prevent these collapses in real-time, bringing us one step closer to clean, limitless fusion energy.

Technical Summary

1. Problem Statement

High-performance tokamak scenarios, particularly in Spherical Tori (STs), are characterized by strong toroidal rotation driven by Neutral Beam Injection (NBI). This rotation introduces significant centrifugal forces that fundamentally alter plasma equilibrium by:

• Shifting the plasma pressure toward the low-field side (LFS).
• Inducing non-rigid geometric distortions (dynamic changes in elongation and triangularity).
• Breaking the assumption that pressure is solely a flux function.

The Challenge: Existing high-fidelity equilibrium solvers (e.g., EFIT, FLOW, FINESSE) rely on grid-based methods or complex inverse coordinate iterations. While accurate, they require seconds to minutes to converge, making them unsuitable for real-time Plasma Control Systems (PCS), massive transport simulations, or rapid parameter scans. Conversely, traditional reduced models (variational moment methods) are fast but rely on low-order "rigid" geometric approximations that fail to capture the complex, non-rigid distortions caused by sonic rotation ($M \sim 1.0$).

2. Methodology: The VEQ-R Solver

The authors propose VEQ-R (VEquilibrium with Rotation), a computationally efficient spectral solver designed to calculate fixed-boundary equilibria with arbitrary toroidal flow.

A. Physical Model

• Governing Equation: The Generalized Grad-Shafranov (GGS) equation, which incorporates inertial forces.
• Thermodynamics: The model assumes rigid toroidal rotation and constant temperature on flux surfaces ($T=T(\psi)$). The total pressure is derived from a Bernoulli-type expression where density stratifies due to centrifugal forces, leading to a pressure profile $P(R, \psi) = P_0(\psi) \exp\left(\frac{M^2}{2}(\frac{R^2}{R_0^2}-1)\right)$.
• Input Strategy: The solver accepts independent inputs for reference pressure $P_0(\psi)$, ion temperature $T(\psi)$, and angular velocity $\Omega(\psi)$, self-consistently calculating the Mach number profile to ensure thermodynamic consistency.

B. Numerical Implementation

• Inverse Coordinate Formulation: The solver maps a computational domain $(\rho, \theta)$ to the physical domain $(R, Z)$ using a generalized angle $\bar{\theta}$ to capture asymmetric flux surfaces.
• 12-Parameter Spectral Expansion:
  • Geometric functions (Shafranov shift $h$, elongation $\kappa$, triangularity $s_1$) and poloidal flux $\psi$ are expanded using Shifted Chebyshev Polynomials.
  • Unlike low-order models, this formulation explicitly includes high-order terms (up to $l=2$ or $3$) to resolve "non-rigid" distortions where flux surfaces stretch and compress dynamically.
• Matrix-Kernel Acceleration:
  • Instead of performing runtime numerical integration for the variational functional, the authors pre-calculate projection matrices.
  • The non-linear problem is transformed into highly optimized algebraic matrix operations.
  • This eliminates loop operations, reducing the residual evaluation time to the microsecond scale.
• Solver Strategy: A hybrid approach using Broyden's quasi-Newton method combined with Truncated SVD Regularization to handle the near-singularity of the Jacobian matrix in sonic regimes ($M > 1.0$).

3. Key Contributions

1. Algorithmic Innovation: Development of the "Matrix-Kernel" technique, which transforms complex variational integrations into pre-computed algebraic matrix multiplications, achieving a speedup factor of $\sim 1000$ compared to high-resolution Finite Difference Method (FDM) codes.
2. High-Fidelity Reduced Model: Introduction of a 12-degree-of-freedom spectral expansion that captures high-order geometric harmonics (dynamic triangularity and elongation), overcoming the "rigid profile" limitation of previous algebraic models.
3. Real-Time Capability: The solver converges in approximately 5 ms on a single CPU core (MATLAB), making it viable for real-time control applications and massive parameter scans.
4. Robustness in Sonic Regimes: The solver successfully handles extreme conditions ($M \sim 1.0$) where centrifugal forces cause severe geometric compression and thermodynamic asymmetry, maintaining stability through SVD regularization.

4. Results and Validation

Benchmarking

• Static Case ($M=0$): VEQ-R results were compared against a high-resolution FDM solver (513 $\times$ 513 grid). The magnetic axis position error was $< 1$ mm, and relative errors for pressure and current profiles were $< 1\%$ in the core.
• Transonic Case ($M \approx 1.0$): In the sonic regime, VEQ-R accurately reproduced the non-linear Shafranov shift (discrepancy $< 0.8\%$ of minor radius) and the asymmetric pile-up of pressure and current on the low-field side.
• Efficiency: VEQ-R achieved convergence in ~5 ms, compared to 1–200 seconds for existing high-fidelity codes (Feng et al., FLOW, FINESSE).

Physics Findings

• Flux Compression & Safety Factor: Strong rotation compresses flux surfaces against the LFS boundary. This forces a concentration of core toroidal current density, causing the core safety factor ($q_0$) to drop monotonically toward unity. This indicates a heightened risk of $m/n=1/1$ internal kink modes and sawtooth crashes in rotating plasmas.
• Thermodynamic Decoupling: While temperature remains a flux function ($T=T(\psi)$), the density profile becomes strongly stratified toward the LFS due to centrifugal potential. Consequently, isobaric surfaces no longer coincide with isothermal surfaces, creating complex non-aligned gradients.
• Aspect Ratio Sensitivity: Spherical Tori (low aspect ratio $A \approx 1.3-1.85$) are significantly more susceptible to rotation-induced deformation than conventional tokamaks. The normalized Shafranov shift nearly doubles in the ST regime.
• Beta Saturation: While rotation increases local pressure peaking, the global normalized beta ($\beta_N$) shows a slight decline. The centrifugal "hold" on the mass distribution acts as a geometric constraint that resists the paramagnetic current peaking usually driven by high pressure, leading to a non-linear saturation of the safety factor drop at high $\beta_N$.

5. Significance

The VEQ-R solver bridges the critical gap between computational speed and physical fidelity in plasma equilibrium reconstruction.

• Operational Impact: Its millisecond-scale convergence enables deployment in real-time Plasma Control Systems (PCS) for next-generation devices, allowing for active control of rotation-induced instabilities.
• Scientific Insight: It reveals that strong rotation, while beneficial for turbulence suppression, poses a structural risk by lowering the safety factor threshold for internal kink modes.
• Future Applications: The framework is designed to be extended to multi-fluid MHD solvers, which will be essential for analyzing advanced fusion fuels (e.g., p-11B) where species separation and differential rotation play a dominant role.

In summary, VEQ-R provides a robust, fast, and physically accurate tool for modeling the complex interplay between strong toroidal rotation and plasma equilibrium, offering new insights into the stability limits of Spherical Tori and future fusion reactors.