Non-dimensional confinement scaling in similar negative triangularity plasmas on the DIII-D and TCV tokamaks

This paper presents similarity experiments on the DIII-D and TCV tokamaks demonstrating that negative triangularity plasmas exhibit weak confinement improvement with increasing collisionality and a machine size scaling behavior intermediate between Bohm and gyro-Bohm, which aligns with engineering scaling observations.

The Gist

The Big Picture: Building a Better Fusion Reactor

Imagine scientists are trying to build a star on Earth (a fusion reactor) to provide limitless clean energy. To do this, they need to trap super-hot gas (plasma) inside a magnetic bottle. The biggest challenge is keeping that heat inside long enough for the atoms to fuse.

For decades, scientists have focused on a specific shape for this magnetic bottle called "Positive Triangularity" (shaped like a D). However, a new shape called Negative Triangularity (shaped like a "D" that's been squished inward, or a "C") is showing promise. It keeps heat just as well but has a superpower: it naturally avoids violent eruptions (called ELMs) that can damage the reactor walls.

The Problem: We know Negative Triangularity works on small machines today, but we don't know how it will behave in a giant future reactor. Will the heat escape faster? Slower? We need to know the "rules of the road" before we build the car.

The Experiment: The "Scale Model" Test

To figure out the rules without building a massive reactor, the scientists used two existing tokamaks (fusion machines): DIII-D (a large machine in California) and TCV (a smaller, flexible machine in Switzerland).

They treated these machines like scale models of a future reactor. Instead of just changing random knobs, they used a special mathematical method called Non-Dimensional Scaling.

The Analogy: The Recipe
Imagine you have a recipe for a cake.

• Engineering Scaling is like saying, "If I double the flour, I need to double the sugar." This is useful, but it doesn't tell you why the cake rises.
• Non-Dimensional Scaling is like looking at the ratios. It asks, "Is the cake rising because of the ratio of flour to eggs, or the ratio of heat to pan size?"

By keeping the "ratios" (the physics) the same and only changing one variable at a time, the scientists could predict how a giant reactor would behave based on data from these two smaller machines.

The Two Main Tests

The team ran two specific types of experiments to see how the "Negative Triangularity" shape handles heat.

1. The Size Test (The "Gyroradius" Scan)

The Concept: Imagine a dancer spinning on a stage. The "gyroradius" is the size of the circle they spin in.

• In a small machine, the dancer spins in a circle that takes up a huge chunk of the stage.
• In a giant reactor, the dancer spins in a tiny circle relative to the massive stage.

The Experiment: The scientists tried to simulate this "tiny circle" effect on their machines by changing the magnetic field strength.

• On the Big Machine (DIII-D): They found that the heat confinement behaved like Bohm diffusion.
  • Analogy: Imagine trying to walk through a crowded room where everyone is bumping into you randomly. The bigger the crowd (the machine), the harder it is to get through. The heat escapes relatively easily.
• On the Small Machine (TCV): They found the heat behaved like Gyro-Bohm diffusion.
  • Analogy: This is like a dance floor where everyone is moving in a coordinated pattern. The heat stays trapped much better.

Why the difference? The scientists suspect it's because the "crowd" (the plasma density and impurities) was different in the two rooms. The ions (heavy particles) in the big machine were struggling more than the electrons (light particles).

2. The "Traffic Jam" Test (The Collisionality Scan)

The Concept: "Collisionality" is how often the particles in the plasma bump into each other.

• Low Collisionality: Particles are like cars on a highway at 3 AM. They rarely bump into each other and zoom past.
• High Collisionality: Particles are like cars in rush hour traffic. They are constantly bumping, slowing down, and getting stuck.

The Experiment: They changed the magnetic field to see if "bumping" helped or hurt the heat retention.

• The Result: On both machines, they found that more bumps actually helped keep the heat in.
• Analogy: It turns out that in this specific "Negative Triangularity" shape, the traffic jams (collisions) actually stop the heat from escaping. It's like a crowd of people holding hands in a circle; if they bump into each other, they stay together better than if they are all running freely.

The Takeaway: What Does This Mean?

1. The Shape Works: Negative Triangularity is a viable shape for future reactors. It keeps heat well without needing complex fixes for wall damage.
2. We Have a Map: By using these "scale model" experiments, the scientists have created a mathematical map. They can now predict how a reactor 10 times bigger than DIII-D will perform.
3. The "Secret Sauce": The experiments showed that the way heat moves depends heavily on the specific mix of particles and how often they bump into each other.
4. Next Steps: While there are still some uncertainties (because the machines aren't quite as big as the future reactors), these results give computer models a solid reality check. Now, scientists can trust their simulations to design the next generation of fusion power plants with much higher confidence.

In a nutshell: The scientists proved that this specific "squished D" shape is a strong contender for future fusion energy. They used two different-sized machines to figure out the physics rules, ensuring that when we finally build the giant reactor, we won't be flying blind.

Technical Summary

1. Problem Statement

Negative Triangularity (NT) plasma configurations are emerging as a promising solution for future fusion reactors due to their ability to sustain high confinement and pressure without an edge pedestal, thereby eliminating the need for Edge Localized Mode (ELM) mitigation. However, designing a reactor based on NT shaping requires accurate predictions of energy confinement time under reactor-relevant conditions.

Current prediction methods rely on engineering scaling laws derived from regression analysis of experimental data. This approach faces significant challenges for NT plasmas:

• Data Scarcity: Very few devices can produce NT plasmas, limiting the dataset size and the variation of independent parameters required for robust regression.
• Uncertainty: Small datasets lead to large uncertainties in scaling coefficients and potential "Simpson paradox" errors (incorrect dependencies).
• Model Limitations: First-principles models are difficult to validate without experimental benchmarks, making extrapolation to reactor conditions risky.

The paper addresses the need for a more robust, physics-based approach to determine how energy confinement scales with non-dimensional variables (normalized Larmor radius, collisionality, etc.) rather than engineering parameters.

2. Methodology

The authors conducted the first systematic, cross-machine similarity experiments on the DIII-D and TCV tokamaks, the only two devices capable of creating strongly shaped NT plasmas.

• Experimental Design:
  • Configuration: Lower Single Null (LSN) plasmas with large, up-down symmetric negative triangularity ($d_{avg} \approx -0.5$).
  • Shape Matching: The separatrix shapes were closely matched between the two devices to ensure geometric similarity.
  • Non-Dimensional Scaling: Experiments were designed to vary one non-dimensional parameter at a time while keeping others fixed, based on the relations in Table 1 of the paper.
    • $\rho^*$ Scan (Normalized Larmor Radius): Varied by changing magnetic field ($B_T$), plasma current ($I_P$), density ($n$), and temperature ($T$).
    • $\nu^*$ Scan (Normalized Collisionality): Varied by changing $B_T$, $I_P$, and auxiliary power at fixed density.
  • Operational Constraints: Experiments focused on beam-heated plasmas (no RF heating) to avoid coupling issues at low fields. Safety factor ($q_{95}$) was fixed at 2.8, and on-axis $q$ was maintained near unity via sawteeth.
  • Analysis: Energy confinement time ($\tau_E$) was calculated using the TRANSP code with power balance analysis, accounting for impurity content and fast-ion diffusion.

3. Key Contributions

• First Cross-Machine NT Scaling: This is the first study to perform non-dimensional scaling experiments for NT plasmas across two different tokamaks (DIII-D and TCV).
• Validation of Similarity: The study successfully demonstrated that NT plasmas on different machines can be made "similar" by matching separatrix shapes and scaling discharge parameters according to non-dimensional theory.
• Decoupling Transport Channels: The experiments allowed for the separation of ion and electron transport behaviors, revealing distinct scaling laws for each species.
• Benchmarking for Reactors: The results provide a quantitative path to extrapolate NT confinement to future reactors and offer a critical benchmark for validating first-principles gyro-kinetic transport solvers.

4. Key Results

A. Normalized Larmor Radius ($\rho^*$) Scaling

• DIII-D Results:
  • Achieved a 40% variation in $\rho^*$.
  • Found Bohm scaling for total energy confinement ($B_T \tau_E \propto \rho^{*-2}$).
  • Species Separation: The Bohm scaling is driven primarily by ion transport, while electron transport follows favorable gyro-Bohm scaling ($\rho^{*-1}$).
  • Engineering Scaling: A large dataset regression confirmed these findings, yielding $B_T \tau_E \propto \rho^{*-1.97}$.
• TCV Results:
  • Achieved a 50% variation in $\rho^*$.
  • Found gyro-Bohm scaling for both ions and electrons ($B_T \tau_E \propto \rho^{*-1}$).
  • Cross-Machine Comparison: A direct cross-machine $\rho^*$ scan was attempted but failed to match profiles perfectly due to TCV plasmas rotating faster and having higher density peaking than DIII-D.
• Discrepancy Analysis: The difference in scaling (Bohm on DIII-D vs. Gyro-Bohm on TCV) is hypothesized to be caused by differences in normalized pressure ($\beta$) and collisionality ($\nu^*$), as well as higher impurity content ($Z_{eff}$) in TCV plasmas.

B. Normalized Collisionality ($\nu^*$) Scaling

• DIII-D Results: Showed a near-collisionless dependence ($B_T \tau_E \propto \nu^{*0.04}$), suggesting confinement is largely independent of collisionality in this regime.
• TCV Results: Showed a weak but positive dependence ($B_T \tau_E \propto \nu^{*0.09}$).
• Interpretation: The positive dependence on TCV (where collisionality is higher) suggests that increasing collisionality stabilizes micro-instabilities, slightly improving confinement. This aligns with previous non-linear gyro-kinetic simulations predicting collisionality quenching of turbulence.

C. Comparison with Existing Laws

• The NT engineering scaling differs from the standard ITER97-L scaling (for positive triangularity) by showing a less pronounced dependence on plasma density and a stronger dependence on plasma current.
• In non-dimensional units, NT confinement behaves similarly to L-mode positive triangularity regarding weak collisionality dependence but exhibits distinct $\rho^*$ scaling behaviors depending on the specific machine regime.

5. Significance

• Reactor Extrapolation: The results provide a physics-based foundation for predicting the performance of NT-based fusion reactors. The weak dependence on collisionality and the specific $\rho^*$ scaling are critical inputs for reactor design.
• Physics Understanding: The study confirms that NT plasmas can sustain high confinement without an edge pedestal, but the transport mechanisms (ion vs. electron) are sensitive to machine size, collisionality, and impurity levels.
• Future Validation: The data serves as a crucial benchmark for validating first-principles transport models. Once these models can reproduce the observed Bohm vs. Gyro-Bohm transitions and collisionality dependencies, confidence in extrapolating NT performance to reactor scales will be significantly improved.

In conclusion, this work establishes that while NT plasmas offer a viable path to high-performance reactors, their confinement scaling is complex and machine-dependent, necessitating the use of non-dimensional analysis to bridge the gap between current experiments and future fusion devices.