Self-consistent modelling and qualitative comparison of mildly relativistic runaway electron dynamics with a closed flux surface formation model during tokamak startup

This paper presents a self-consistent, reduced-kinetic model for mildly relativistic runaway electrons integrated into the DYON predictive code (DYON-RE), which successfully reproduces KSTAR ohmic startup observations and offers a framework for designing runaway-free startup scenarios for future devices like ITER.

The Gist

Imagine a tokamak (a doughnut-shaped nuclear fusion reactor) trying to start up like a car engine. It needs to turn a cold, empty vacuum into a hot, swirling ball of plasma. But there's a dangerous side effect: sometimes, a few electrons get kicked so hard they turn into "runaway" particles, zooming around at nearly the speed of light. If too many of these runaway electrons form, they can act like a high-powered laser beam, melting the reactor's walls and shutting down the experiment.

This paper is about building a better map to predict when and how these runaway electrons appear during that tricky "start-up" phase. The authors, working with the KSTAR fusion reactor in South Korea, developed a new model called DYON-RE.

Here is the breakdown of their work using simple analogies:

1. The Problem: The "Speed of Light" Mistake

In the past, scientists tried to predict these runaway electrons by assuming they were already traveling at the speed of light the moment they started running away.

• The Analogy: Imagine a race car driver. Old models assumed that as soon as the driver stepped on the gas, the car was instantly doing 200 mph.
• The Reality: In the early stages of the reactor's startup, the electrons are "mildly relativistic." They are fast, but they haven't hit top speed yet. They are more like a car accelerating from 0 to 60 mph.
• The Fix: The authors created a new model that accounts for this acceleration phase. By realizing the electrons aren't instantly at top speed, their model stops overestimating how much dangerous current these electrons create. It's like realizing the car is only going 40 mph, not 200, which changes how much damage it might do.

2. The Challenge: The "Open vs. Closed" Trap

During startup, the magnetic fields that hold the plasma in place are changing shape.

• The Analogy: Think of the magnetic field as a fence.
  • Open Field: At the very beginning, the fence has gaps. If a runaway electron tries to run, it hits a gap and escapes (like a dog running out of an open gate).
  • Closed Field: As the reactor heats up, the fence closes up into a perfect circle (a closed flux surface). Now, the runaway electron is trapped inside a cage and can't escape.
• The Old Way: Previous models treated the fence as either always open or always closed, or they used a blurry average of the two.
• The New Way: The DYON-RE model is like a smart security system that knows exactly when the fence is closing. It tracks the electrons separately: those running in the "open field" (where they get lost quickly) and those trapped in the "closed field" (where they build up). This is crucial because the moment the fence closes is when the danger really starts to build up.

3. The Experiment: Watching the "Radiation Thermometer"

The team tested their new model against real data from the KSTAR reactor. They couldn't see the runaway electrons directly, so they looked for clues.

• The Analogy: Imagine trying to figure out if a room is full of people by listening to the noise level.
• The Clue: They used a tool called Electron Cyclotron Emission (ECE), which acts like a "radiation thermometer." When runaway electrons get excited, they emit radiation that makes this thermometer read a very high temperature.
• The Result: They looked at two different startup attempts:
  1. The "Runaway Rich" Shot: The reactor had a lot of runaway electrons. The model predicted this, and the "thermometer" showed a massive spike in temperature, just like the model said.
  2. The "Runaway Scarce" Shot: The reactor had very few runaway electrons. The model predicted this too, and the thermometer stayed relatively calm, with only small, rhythmic "bursts" (like a heartbeat) instead of a massive spike.

4. The Secret Ingredient: The Walls

One of the paper's key findings is that the reactor's walls play a bigger role than previously thought.

• The Analogy: Imagine trying to fill a bucket with a hose (gas injection). If the bucket has a hidden leak (the walls absorbing gas), you need to turn the hose on harder to get the same amount of water.
• The Discovery: The researchers found that even when they used the exact same gas settings, the reactor behaved differently because the "walls" were acting differently (absorbing or releasing gas at different rates). To make their model work, they had to adjust for these wall conditions. Without accounting for the walls, the model couldn't predict the electron density correctly.

Summary

The paper doesn't claim to have solved the runaway electron problem forever, but it has built a better, more realistic simulator.

• It stops assuming electrons are instantly at top speed.
• It tracks exactly when the magnetic "fence" closes to trap them.
• It successfully predicts the "temperature spikes" seen in real experiments.

This gives scientists a more reliable tool to design future reactors (like ITER) so they can start up safely without accidentally creating a beam of electrons that could damage the machine.

Technical Summary

1. Problem Statement

Runaway Electrons (REs) are a critical concern in tokamaks, traditionally associated with disruptions. However, REs can also form during the startup phase (specifically during burn-through and early current ramp-up), potentially hindering plasma initiation or damaging the device.

Existing models for startup REs suffer from several limitations:

• Velocity Assumption: Many models assume runaway electrons travel at the speed of light ($v=c$). This significantly overestimates runaway current density when electrons are in the "mildly relativistic" regime (accelerated but not yet ultra-relativistic).
• Magnetic Configuration: Models often fail to distinguish between open field lines (early startup) and closed flux surfaces (formed later). The transition between these configurations drastically alters RE confinement, which previous single-population models could not capture accurately.
• Validation: There is a lack of self-consistent validation against experimental data, particularly regarding the qualitative behavior of REs (e.g., kinetic instabilities) rather than just total current.

2. Methodology

The authors developed a new predictive framework integrated into the DYON plasma initiation code, referred to as DYON-RE.

A. The Mildly Relativistic Runaway Model

• Reduced-Kinetic Approach: Instead of assuming $v=c$, the model calculates the mean velocity ($\beta = v/c$) of the RE population.
• Multi-Fluid to Single-Fluid Reduction: The authors first formulated a multi-fluid model tracking REs born at different time steps. They then reduced this to a single-fluid form by assuming the mean velocity characterizes the overall current evolution. This preserves accuracy while maintaining numerical efficiency.
• Boundary Definition: The model defines the runaway boundary not just at the critical momentum ($p_c$) but at a region-based boundary ($p_V \approx 1.3 p_c$), which better aligns with kinetic simulations of the Dreicer generation rate.
• Effective Fields: The model incorporates an effective critical electric field and effective Dreicer field to account for inelastic collisions (ionization) with neutral particles and impurities, which are abundant during startup.

B. Dual Magnetic Configuration (CFSF Model)

The model explicitly separates RE populations into two groups based on the magnetic topology:

1. Open Field Lines ($n_{op}^{RE}$): REs are lost primarily via parallel streaming along field lines.
2. Closed Flux Surfaces ($n_{cl}^{RE}$): REs are confined radially, with losses governed by Rechester-Rosenbluth diffusion.

• Transition: The model dynamically tracks the formation of Closed Flux Surfaces (CFSF). As the plasma current rises and a local closed region forms (even before a global closed surface exists), REs born in this region experience significantly improved confinement.

C. Self-Consistent Coupling

The RE model is coupled self-consistently with:

• Electromagnetic Circuit: The runaway current is subtracted from the total plasma current in the circuit equation to calculate the resistive electric field accurately.
• Plasma Initiation: The code simulates the full burn-through process, including gas ionization, recycling, and impurity influx.

D. Validation Tools

To compare with KSTAR experiments, the authors used:

• KIAT (Kinetic Instability Analysis Tool): To predict the onset of kinetic instabilities (whistler waves) driven by REs.
• SYNO (Synthetic Non-thermal ECE Reconstruction): To reconstruct the radiative temperature ($T_{rad}$) expected from REs and compare it with Electron Cyclotron Emission (ECE) diagnostics.

3. Key Contributions

1. Mildly Relativistic Correction: Demonstrated that assuming $v=c$ overestimates startup RE currents. The new model accounts for the actual velocity distribution, providing a more accurate current density prediction.
2. Dual-Topology Transport: Introduced a model that distinguishes between open and closed magnetic configurations. This captures the "seed" confinement effect where REs born in local closed flux surfaces survive longer, significantly impacting the total RE population during the transition phase.
3. Qualitative Validation Framework: Established a method to validate RE models not just by current magnitude (which is hard to measure directly in KSTAR) but by comparing ECE radiation characteristics (sawtooth patterns, temperature spikes) and kinetic instability thresholds.
4. Wall Condition Sensitivity: Highlighted that accurate prediction of startup REs requires precise modeling of wall conditions (recycling coefficients) and pre-fill gas pressure, as these dictate the electron density evolution which drives RE generation.

4. Results

The model was tested against two KSTAR ohmic discharges:

• Shot #26031 (RE-Rich):
  • Observation: High HXR signals and a rapid spike in ECE radiative temperature ($>12$ keV), indicating strong kinetic instability.
  • DYON-RE Prediction: Successfully reproduced the plasma current and density. It predicted a high initial RE current (~24 kA) due to seed confinement in local closed flux surfaces. The model showed that the RE density was sufficient to drive strong kinetic instabilities ($\gamma_{kin} \gg \gamma_{damp}$), matching the observed ECE behavior.
• Shot #27340 (RE-Scarce):
  • Observation: Lower HXR signals and "sawtooth-like" bursting in ECE temperature, suggesting marginal instability.
  • DYON-RE Prediction: Predicted a lower RE current (~16 kA). The model showed that the kinetic growth rate was only slightly above the damping rate, consistent with the observed bursting behavior.
• Comparison: Without the dual-configuration model (using simple interpolation), the predicted initial RE current was only ~1 kA, failing to explain the experimental observations. The new model successfully explained the differences between the two shots as a result of varying wall conditions (recycling) affecting density evolution, rather than just pre-fill pressure.

5. Significance

• Predictive Capability for Future Devices: The DYON-RE code provides a robust tool for designing runaway-free startup scenarios for future reactors like ITER and CPD (Compact Fusion Reactor).
• Safety and Operation: By understanding that REs can form during startup and persist due to local closed flux surfaces, operators can better mitigate risks of magnet quenching or component damage.
• Physics Insight: The work clarifies that the "mildly relativistic" nature of startup REs and the specific magnetic topology during the CFSF transition are critical factors often overlooked in previous simplified models.
• Experimental Guidance: The study emphasizes that controlling wall conditions (recycling) is as crucial as gas puffing for managing startup REs, suggesting that predictive simulations must account for time-varying recycling coefficients to be reliable.

In conclusion, this paper presents a significant advancement in tokamak startup physics by integrating a physically rigorous, mildly relativistic RE model with dynamic magnetic topology changes, validated qualitatively against KSTAR experimental data.