Improved n=1 Empirical Error Field Penetration Threshold Scaling with Ohmic and L-Mode Conventional Tokamak Plasma Discharges

This paper presents an improved n=1 empirical error field penetration threshold scaling derived from an expanded database of conventional Ohmic and L-mode tokamak discharges, which offers higher fit quality and reduced uncertainty to better inform the engineering tolerances and design of future conventional tokamaks.

The Gist

Imagine a tokamak (a machine designed to create fusion energy) as a giant, invisible whirlpool of super-hot gas, held in place by powerful magnetic fields. Ideally, this whirlpool is perfectly symmetrical, like a smooth, spinning top. However, in the real world, the magnets that hold it together aren't perfect. They have tiny tilts, shifts, and imperfections. These imperfections create "error fields"—tiny, unwanted magnetic ripples that can mess up the smooth spin of the gas.

If these ripples get too strong, they can cause the whirlpool to develop a "knot" (a magnetic island) that gets stuck in place. Once stuck, the whole system can collapse, leading to a sudden shutdown called a "disruption." This is bad news for the machine and the people building it.

The Problem: How Strong is Too Strong?
Engineers need to know the exact limit: How big can these magnetic imperfections get before the machine breaks? If they set the limit too low, they have to build the machine with impossible precision, making it astronomically expensive and slow to construct. If they set it too high, the machine might crash.

For years, scientists have tried to create a "rule of thumb" (a scaling law) to predict this limit based on the size of the machine and how the gas is behaving. But the old rules were a bit shaky, like a map with blurry edges.

The Solution: A Sharper Map
This paper presents a new, upgraded "map" (an empirical scaling law) that is much clearer and more reliable. Here is how they did it, using simple analogies:

• Cleaning the Data: The researchers went through a massive database of past experiments from tokamaks around the world (like DIII-D, JET, and KSTAR). They decided to focus only on specific types of "weather" in the machine: the "Ohmic" and "L-mode" conditions. They left out the "H-mode" because that state is like a sturdy fortress—it's very hard to break, so it doesn't help us understand the weakest point of the machine. By focusing on the vulnerable states, they found the true danger zone.
• Adding New Ingredients: They added new data from two specific machines: J-TEXT (which is smaller and runs at lower currents) and more data from JET (which is huge, similar to the future ITER machine). Think of this like adding new test drives to a car safety database. You need small cars and giant trucks in the data to know how the safety rules apply to any vehicle you might build in the future.
• Better Math: They used a more sophisticated mathematical method to find the relationship between the machine's size, the magnetic field strength, the gas density, and the electric current. They found that plasma current (how much electricity is flowing through the gas) is a critical factor they hadn't fully accounted for before.

The New Findings
The new "rule of thumb" tells us that:

1. Higher density is your friend: Packing more gas into the machine makes it harder for the error fields to cause a crash.
2. Bigger machines are surprisingly resilient: Larger machines (like the future ITER) can handle bigger magnetic imperfections than we previously thought.
3. Current matters: The amount of current flowing through the plasma changes how the machine reacts to these errors.

Why This Matters for the Future
The paper specifically looks at the ITER project, a massive international fusion experiment currently under construction. Using their new, sharper map, the researchers ran millions of simulations (like running a weather forecast a million times with slightly different starting conditions).

The Result: They found that ITER is in much better shape than we thought. The "danger zone" for magnetic errors is much further away than the actual imperfections ITER is expected to have.

• The Old Map: Suggested there was a decent chance ITER might trip over its own shoelaces (get locked modes).
• The New Map: Shows that the chance of this happening is incredibly tiny (less than 1 in a million for the most likely scenario).

The Bottom Line
This paper doesn't just say "fusion is hard." It gives engineers a much more confident, precise ruler to measure the tolerances of their machines. Because the new rules show that the machines are more robust against magnetic errors, engineers might not need to build the magnets with such extreme, expensive precision. This could save time and money while keeping the machine safe.

In short: They took a blurry, confusing map of magnetic safety limits, cleaned it up with better data and smarter math, and discovered that the future of fusion power plants is safer and more achievable than we previously believed.

Technical Summary

1. Problem Statement

Tokamaks require precise axisymmetry to operate stably. However, manufacturing imperfections (coil tilts, shifts, and misalignments) create non-axisymmetric error fields (EFs). If these fields exceed a critical penetration threshold, they can trigger magnetic reconnection, form magnetic islands, and cause the islands to "lock" to the vessel wall. This leads to locked modes and subsequent plasma disruptions.

Predicting this threshold is critical for:

• Engineering Tolerances: Determining allowable coil misalignments during construction.
• Cost and Schedule: Overly strict tolerances increase construction costs and time; overly loose tolerances risk device failure.
• Future Devices: Projects like ITER and SPARC have high stored energy, making disruptions particularly dangerous.

Previous empirical scaling laws suffered from:

• Low Fit Quality: Coefficients of determination ($R^2$) as low as 0.38.
• High Uncertainty: Up to 48% uncertainty in projections for ITER.
• Inconsistent Data: Mixed inclusion of H-mode, L-mode, and Ohmic plasmas, spherical tokamaks (NSTX), and varying levels of intrinsic error field modeling across different devices.
• Limited Parameter Space: Lack of data for large major radius devices (like ITER) and specific density regimes.

2. Methodology

The authors developed an updated empirical scaling law by refining the database and the fitting methodology.

A. Database Refinement

• Regime Selection: The study restricted the dataset to conventional (non-spherical) tokamaks operating in Ohmic and L-mode regimes. H-mode was excluded because it is more resilient to error field penetration, and the authors sought to model the "most dangerous" regime. Saturated Ohmic Confinement (SOC) shots were excluded to avoid confounding density dependencies.
• Device Inclusion:
  • Added: New data from J-TEXT (providing low current/density points and a new major radius size) and an expanded dataset from JET (21 shots vs. previous 4, covering large major radius and high plasma current).
  • Excluded: NSTX (spherical tokamak) and H-mode shots.
• Standardization:
  • DIII-D: Moved from a grouped equilibrium approximation to individual equilibrium reconstructions for every shot using the SURFMN code to model intrinsic error fields more accurately.
  • J-TEXT: Used the TokaMaker code to reconstruct equilibria (as routine magnetic reconstructions were unavailable), ensuring compatibility with the GPEC (General Perturbed Equilibrium Code) used for all other devices.
  • Metric: Used the Dominant Mode Overlap ($\delta$), a coordinate-system-independent metric that accounts for plasma response, calculated via GPEC.

B. Scaling Methodology

• Regression Technique: Employed forward selection least-squares regression to identify the optimal log-linear power law.
• Parameter Selection: Expanded the parameter space beyond previous studies to include explicit dependencies on Plasma Current ($I_p$), Toroidal Field ($B_T$), Major Radius ($R_0$), Density ($n_e$), and Normalized Pressure ($\beta_n/\ell_i$).
• Validation: Used Condition Numbers to detect multicollinearity (stopping at 5 variables to keep the condition number < 20) and compared Ordinary Least Squares (OLS) with Weighted Least Squares (WLS) using Kernel Density Estimation (KDE) to avoid oversampling data-rich regions.
• Uncertainty Quantification: Assumed a baseline 10% uncertainty in locked mode identification time, propagated through Monte Carlo simulations to generate probability density functions (PDFs) and cumulative distribution functions (CDFs) for predictions.

3. Key Contributions

1. New Scaling Laws: Derived two new scaling equations (Eq. 9 for OLS and Eq. 10 for WLS) with significantly improved statistical metrics.
   • Equation 9 (OLS): $\delta \propto (\beta_n/\ell_i)^{0.25} I_p^{-0.97} R_0^{1.88} n_e^{0.77} B_T^{0.20}$
   • Equation 10 (WLS): $\delta \propto (\beta_n/\ell_i)^{0.13} I_p^{-1.01} R_0^{1.57} n_e^{0.56} B_T^{0.30}$
2. Inclusion of Explicit $I_p$ Dependence: Unlike previous scalings that relied on implicit dependencies, the new models explicitly include plasma current, revealing that lower plasma current increases resilience to error fields.
3. Shift in Toroidal Field Dependence: The scaling for $B_T$ shifted from a strong negative dependence (in previous models) to a small positive dependence, driven by the inclusion of $I_p$ and the removal of spherical tokamak data.
4. Robustness Testing: Demonstrated that the scaling is robust against the removal of individual devices (e.g., DIII-D, J-TEXT, C-MOD), whereas "minimal" device subsets (e.g., JET+KSTAR+C-MOD) led to severe overfitting.

4. Results

• Improved Fit Quality: The new scaling achieved an $R^2$ of 0.63 (OLS) and 0.66 (WLS), a significant improvement over previous values (0.38–0.58).
• Reduced Uncertainty:
  • Reduced Chi-Squared: Improved from 17.0 (2020 model) to 12.99.
  • Mean Absolute Log Error: Improved from 1.336 to 1.306 (equivalent to reducing prediction uncertainty from $\pm33.6\%$ to $\pm30.6\%$).
• Parameter Dependencies:
  • Density ($n_e$): Confirmed a positive dependence ($n_e^{0.56-0.77}$), consistent with many individual devices but distinct from some theoretical models.
  • Major Radius ($R_0$): A strong positive dependence ($R_0^{1.57-1.88}$), indicating larger devices are more resilient.
  • Plasma Current ($I_p$): A strong negative dependence ($I_p^{-0.97}$ to $-1.01$), indicating higher current makes the plasma more susceptible to locking.
• Device-Specific Outliers:
  • NSTX: The new scaling significantly underestimates NSTX's error field tolerance (predicting $\delta \sim 10^{-5}$ vs. actual $\sim 10^{-4}$), likely due to its spherical nature and high $q_{95}$ profile, confirming it should be excluded from conventional tokamak scalings.
  • C-MOD: High-density data from C-MOD was crucial for capturing non-monotonic density scaling behaviors.

5. Significance and Implications for ITER

• ITER Projection:
  • The new scaling predicts a much higher error field penetration threshold for ITER compared to previous models.
  • Most Likely Intrinsic $\delta$ for ITER: $0.38 \times 10^{-4}$.
  • Predicted Threshold (Eq. 10): $4.01 \times 10^{-4}$ (with a 1-sigma uncertainty range).
  • Locking Probability: The probability of ITER locking under its baseline scenario is calculated to be extremely low (0.0001% for the most likely intrinsic field, and 22.3% even for the maximum possible intrinsic field).
• Engineering Impact:
  • The results suggest that ITER's current engineering tolerances for coil alignment are conservative and sufficient.
  • The scaling provides a more reliable tool for designing future devices (like SPARC) and optimizing operations on existing machines by predicting the probability of mode locking based on Monte Carlo simulations of intrinsic error fields.
• Future Work: The authors suggest that adding data from JT-60SA (large radius) and ASDEX-U (high $\beta_n$) would further refine the scaling. Additionally, developing a scaling based on shots with explicit rotation measurements could further improve predictive power, as rotation is a dominant factor in error field physics.

In conclusion, this paper establishes a more robust, physics-consistent, and statistically reliable empirical scaling for n=1 error field penetration, significantly increasing confidence in the design and operation of next-generation conventional tokamaks.