Nonlinear anisotropic equilibrium reconstruction in axisymmetric magnetic mirrors

This paper presents a novel nonlinear anisotropic equilibrium reconstruction method utilizing a new basis set and constrained Bayesian optimization to accurately infer sloshing ions in high-$\beta$ axisymmetric magnetic mirrors with limited diagnostics, offering a robust tool for future fusion power plants.

The Gist

Imagine you are trying to figure out what's inside a sealed, glowing black box. You can't open it, and you can't see inside. All you have are a few tiny sensors on the outside that tell you how much the box is pushing against the walls and how hot it feels in a few specific spots.

This is the challenge scientists face with fusion plasma—the super-hot, electrically charged gas inside a magnetic bottle (like the WHAM experiment described in this paper). The goal is to build a fusion reactor, but to do that, we need to know exactly what the plasma is doing inside.

Here is a simple breakdown of what this paper does, using everyday analogies:

1. The Problem: The "Blind" Magnetic Mirror

Think of a magnetic mirror as a magnetic funnel. It uses strong magnets to trap hot gas (plasma) in the middle, bouncing it back and forth like a ping-pong ball between two walls.

• The Old Way: For decades, scientists tried to guess what was inside by assuming the gas was calm and uniform, like a bowl of lukewarm soup. They used simple math to fit their sensors' data to this "soup" model.
• The Reality: In high-performance experiments, the plasma isn't soup. It's more like a roiling storm. The particles are moving wildly, bouncing off the magnetic walls, and creating "sloshing" waves of energy. The old "soup" models couldn't handle this chaos, especially when the plasma gets very energetic (high "beta").

2. The Solution: A New "Recipe" and a Smart Chef

The authors created a new way to reconstruct the picture of the plasma. They did two main things:

A. The New Recipe (The Kinetic Basis)

Instead of assuming the plasma is a uniform soup, they created a new mathematical "recipe" based on how individual particles actually behave.

• The Analogy: Imagine trying to describe a crowd at a concert.
  • Old Method: "The crowd is a solid block of people."
  • New Method: "The crowd has a group of people jumping up and down (sloshing ions) while others are standing still."
• This new recipe accounts for the "sloshing ions"—fast-moving particles that bounce back and forth, creating peaks of pressure that the old models missed.

B. The Smart Chef (Machine Learning)

Reconstructing the plasma is like trying to bake a cake where you can only taste a crumb from the edge and see the color of the frosting. You have to guess the ingredients (temperature, density, pressure) to match what you see.

• The Old Way: The chef would guess, bake, taste, get it wrong, guess again, and bake again. This took forever and often got stuck making a "bad cake" (a local minimum) that looked okay but wasn't the best.
• The New Way (Bayesian Optimization): The authors used a Machine Learning Chef. This chef doesn't just guess randomly. It builds a "map" of all possible cakes. It learns from every tiny taste test to figure out exactly which ingredients are needed.
  • Uncertainty Quantification: The best part? The chef doesn't just say, "Here is the cake." It says, "Here is the cake, and I am 95% sure this is the right recipe." It tells you how confident it is in its answer.

3. The Experiment: Testing the Theory

The team tested this new method on data from the WHAM (Wisconsin High-Temperature Superconducting Axisymmetric Mirror) experiment.

• The Test: They took real data from the machine's sensors (magnetic loops and Thomson scattering, which is like a laser radar for plasma density).
• The Result: The new method successfully reconstructed the plasma state.
  • In some shots (high density), the plasma was indeed like the "soup" (gas dynamic).
  • In other shots (moderate density), the new method detected the "sloshing ions"—the wild, bouncing particles that the old models would have missed.
• The "Smoking Gun": They even had to rule out a suspect. Could the "sloshing" be caused by fast electrons instead of ions? They ran a simulation with fast electrons and found it didn't fit the data. The "sloshing" was definitely caused by the heavy ions.

4. Why This Matters

This is a big deal for the future of fusion energy.

• Fewer Sensors Needed: Future fusion power plants will be huge and expensive. We won't be able to stick thousands of sensors inside them. This new method allows us to get a clear picture of the plasma using very few sensors.
• Safety and Efficiency: By knowing exactly how the plasma is behaving (and how confident we are in that knowledge), we can run the reactor more efficiently and avoid dangerous instabilities.
• The Future: The authors plan to make the "Smart Chef" even smarter by using multi-objective optimization (balancing many different goals at once) and adding more physics (like how the plasma flows).

Summary

In short, this paper is about teaching a computer to "see" inside a fusion reactor when we can't look directly. By using a better mathematical model for how particles move and a smart machine-learning algorithm to fit the data, they can accurately map the invisible, chaotic storm of plasma inside a magnetic mirror, even with very limited data. It's like solving a complex jigsaw puzzle with only a few pieces, but having a super-smart assistant that knows exactly where every piece fits.

Technical Summary

1. Problem Statement

Magnetic equilibrium reconstruction is essential for interpreting diagnostic data in fusion experiments. While well-established for tokamaks with isotropic pressure and low plasma beta ($\beta$), applying these techniques to axisymmetric magnetic mirrors presents unique challenges:

• High-$\beta$ and Anisotropy: Mirror plasmas often operate at high $\beta$ with highly anisotropic pressure tensors ($p_\parallel \neq p_\perp$) due to neutral beam injection (NBI) creating "sloshing ions."
• Instability Constraints: Valid equilibria must satisfy specific stability conditions (firehose and mirror instabilities) which are easily violated if the pressure profiles are not physically consistent.
• Diagnostic Limitations: Devices like the Wisconsin High-Temperature Superconducting Axisymmetric Magnetic Mirror (WHAM) have limited diagnostic capabilities (few flux loops and Thomson scattering points), making linear reconstruction methods prone to ambiguity and error.
• Computational Cost: Traditional iterative methods for solving the Grad-Shafranov (GS) equation with kinetic basis functions are computationally expensive and struggle with uncertainty quantification.

The core problem is how to accurately reconstruct the 2D plasma equilibrium (flux, pressure, density) in high-$\beta$, anisotropic mirror plasmas using sparse data, while quantifying uncertainty and ensuring physical stability.

2. Methodology

The authors developed a comprehensive framework combining a kinetic solver, a novel physics-based basis set, and machine learning optimization.

A. The Pleiades Solver

• Green's Function Approach: The authors use Pleiades, a free-boundary Grad-Shafranov solver written in Python/Fortran. It solves the GS equation ($\Delta^*\psi = -\mu_0 R J_\phi$) using a Green's function formulation.
• Iterative Scheme: The solver iteratively calculates the magnetic flux ($\psi$) and field ($B$) by interpolating pressure profiles ($p_\parallel, p_\perp$) onto the vacuum fields, calculating the diamagnetic current, and updating the flux until convergence.
• Boundary Conditions: It handles perfectly conducting walls via a pre-computed Singular Value Decomposition (SVD) of the Green's function, allowing for rapid enforcement of boundary conditions.

B. Novel Kinetic Basis Functions

Instead of using ad-hoc polynomial fits, the authors derived a semi-analytic kinetic basis based on the Fokker-Planck equation for NBI-fueled plasmas:

• Distribution Function: Modeled as a sum of Legendre functions (Egedal et al. formulation) to describe hot sloshing ions.
• Pressure/Density Integration: The basis integrates the distribution function to generate self-consistent $p_\parallel(\psi, B)$, $p_\perp(\psi, B)$, and density $n(\psi, B)$ profiles.
• Physical Advantages: This basis naturally includes ion-ion scattering (smoothing discontinuities that trigger mirror instabilities) and self-consistently links pressure and density, allowing both to constrain the reconstruction.

C. Machine Learning Optimization (SCBO)

To fit the reconstruction parameters to experimental data, the authors employed Scalable Constrained Bayesian Optimization (SCBO):

• Objective Function: Minimizes $\chi^2$ between synthetic diagnostic outputs (from the solver) and actual experimental measurements.
• Constraints: SCBO enforces physical constraints (e.g., $\beta < 1$, stability conditions against firehose/mirror instabilities). If a candidate parameter set violates these, the optimizer handles it gracefully without crashing.
• Uncertainty Quantification: The Gaussian Process (GP) surrogate model used in SCBO provides confidence intervals for the reconstructed parameters (e.g., $\beta$, stored energy) with minimal computational overhead.

3. Key Contributions

1. First Kinetic Reconstruction in Mirrors: This is the first application of fully kinetic basis functions (derived from Fokker-Planck solutions) to magnetic equilibrium reconstruction in fusion experiments.
2. SCBO for Plasma Physics: Demonstrated the efficacy of Scalable Constrained Bayesian Optimization for high-dimensional, non-linear plasma equilibrium fitting, offering robustness against local minima and built-in uncertainty quantification.
3. Self-Consistent Basis: Introduced a pressure/density basis that inherently satisfies the parallel force balance and stability criteria, reducing the risk of unphysical solutions.
4. Diagnostic Efficiency: Showed that accurate reconstructions of key metrics ($\langle \beta \rangle$, $W_{tot}$, $\langle E_i \rangle$) are possible with the sparse diagnostic suite available in WHAM.

4. Results

A. Verification with Synthetic Data

The method was tested on synthetic data generated by CQL3D-m/Pleiades simulations:

• Maxwellian Case: Correctly identified gas-dynamic plasmas, though slightly underestimated ion energy due to grid mismatches.
• Hybrid Case (Short NBI pulse): Successfully tracked the transition from gas-dynamic to kinetic sloshing ions over time.
• Kinetic Case (Steady-state NBI): Accurately reconstructed total stored energy ($W_{tot}$) and identified the dominance of fast ions.
• Accuracy: The reconstruction measured ion energy, plasma beta, and stored energy within ~20% of the ground truth values.

B. Application to WHAM Experiments

The technique was applied to two sets of WHAM experiments (Shot 250305121–43 "High-Density" and 250306045–64 "Moderate-Density"):

• Sloshing Ion Detection:
  • Moderate-Density Shots: The reconstruction indicated a significant population of sloshing ions ($W_{fast}/W_{tot} \approx 0.17$). The pressure profiles showed distinct peaks at the fast ion turning points ($Z \approx \pm 0.4$ m).
  • High-Density Shots: The reconstruction suggested a predominantly gas-dynamic plasma with negligible sloshing ions.
• Differentiation from Fast Electrons: The authors tested whether the signals could be caused by sloshing electrons (from Electron Cyclotron Heating). Using a fast-electron basis function, the $\chi^2$ fit was poor, and the model predicted zero hot electron pressure. This confirmed that the observed pressure peaking was due to sloshing ions, not electrons.
• Uncertainty: The SCBO method provided joint confidence regions for the fit parameters, demonstrating that the distinction between the two plasma regimes was statistically significant.

5. Significance and Future Work

• Impact on Fusion Research: This work provides a critical tool for diagnosing high-performance, high-$\beta$ mirror plasmas, which are candidates for compact fusion reactors. It proves that kinetic effects can be inferred even with limited diagnostics.
• Generalizability: The SCBO and kinetic basis approach is not limited to mirrors; it can be adapted to toroidal devices (tokamaks/stellarators) where non-Maxwellian distributions and limited diagnostics are common.
• Future Directions:
  • Multi-objective Optimization: Implementing algorithms (e.g., MORBO) to treat each diagnostic signal as an individual objective for better Pareto front analysis.
  • Flow Inclusion: Incorporating parallel and perpendicular flows, which may become significant in specific regions (expanders).
  • Wall Effects: Adapting solvers to handle 3D resistive walls in future mirror designs.
  • Fast Electron Basis: Developing more rigorous semi-analytic bases for fast electrons to fully decouple ion and electron contributions in future, better-diagnosed experiments.

In summary, this paper establishes a robust, physics-informed, machine-learning-accelerated framework for reconstructing complex, anisotropic plasma equilibria, successfully identifying sloshing ion populations in the WHAM experiment where traditional methods might fail.