Systematic comparison of VMEC and HINT equilibrium calculations for finite-beta LHD plasmas

This paper systematically compares VMEC and HINT equilibrium calculations for Large Helical Device plasmas, revealing that while both codes agree at low beta, they diverge at higher beta values as HINT captures edge stochasticity and flux surface breaking that the nested-flux-surface assumption of VMEC cannot represent.

The Gist

Imagine trying to bake a perfect, round cake inside a very strange, twisted oven. In the world of fusion energy, scientists use machines called stellarators (like the Large Helical Device, or LHD) to trap super-hot plasma. To keep this plasma stable, they need to calculate exactly how the magnetic "walls" holding it in should look.

This paper compares two different "bakers" (computer programs) trying to figure out the shape of these magnetic walls when the plasma gets very hot and pressurized.

The Two Bakers: VMEC and HINT

1. VMEC (The Strict Architect): This program is like an architect who insists that every layer of the cake must be a perfect, smooth, nested onion. It assumes that the magnetic walls never break or touch each other. It's great for simple, low-pressure situations, but it has a blind spot: it refuses to believe that the walls can ever get messy or broken.
2. HINT (The Realistic Observer): This program is like a scientist who watches the cake actually bake. It doesn't assume the layers are perfect. Instead, it lets the physics happen naturally. If the heat gets too high, it allows the magnetic walls to get wobbly, break apart, or turn into a chaotic mess.

The Experiment: Turning Up the Heat

The researchers tested these two programs on the LHD machine with three different shapes of the magnetic "oven" (some shifted inward, some outward). They slowly increased the pressure of the plasma (the "heat" of the cake) from 0% to 5%.

What happened at low pressure?
When the plasma was cool and calm, both bakers agreed. The magnetic walls stayed smooth and nested, just like the Strict Architect (VMEC) predicted. Everything was fine.

What happened when the heat turned up?
Once the pressure crossed a certain "critical point," the two bakers started to disagree.

• VMEC kept drawing perfect, smooth, expanding onion layers. It thought the plasma was just getting bigger and rounder.
• HINT saw something different. It noticed that the magnetic walls were starting to get "stochastic."

The "Stochastic" Mess: A Creative Analogy

Think of the magnetic field lines as a bundle of spaghetti.

• In a perfect state (low pressure), the spaghetti strands are neatly bundled and run parallel to each other.
• As the pressure rises, the Pfirsch-Schlüter current (a type of electrical current that naturally forms in the plasma) acts like a chaotic hand mixing the spaghetti.
• Eventually, the strands start to overlap and tangle. This is called magnetic islands and stochasticity. The neat "onion layers" break apart.

Because HINT allows for this tangle, it sees the "magnetic cage" shrinking. The chaotic mixing at the edge of the plasma makes the effective volume smaller. VMEC, however, is still drawing the perfect, expanding onion, so it thinks the volume is getting bigger.

The Key Findings

1. The "Tipping Point": There is a specific pressure level where the neat onion layers break. Once you pass this point, VMEC is no longer accurate because it can't see the broken walls.
2. Shape Matters: The "tipping point" happens sooner (at lower pressure) if the machine is shifted outward.
   • Analogy: Imagine the outward-shifted machine is like a wobbly table. It's easier to knock over (create chaos) than a sturdy, inward-shifted table. The outward shape creates more "ripples" in the magnetic field, making the spaghetti tangle faster.
3. Volume Loss: In the outward-shifted and standard configurations, as the pressure gets very high, the actual volume of the plasma (according to the realistic HINT model) starts to shrink because the magnetic walls break down. VMEC misses this entirely, thinking the volume keeps growing.

The Bottom Line

This paper shows that for high-pressure fusion plasmas, we can't just rely on the "perfect onion" model (VMEC). We need the "realistic observer" (HINT) to see when the magnetic walls are breaking apart and turning chaotic. This is especially true for machines shifted outward, where the magnetic field is more sensitive to these messy, 3D effects. The study confirms that as we push for higher energy, the assumption of perfect, smooth magnetic layers becomes less and less valid.

Technical Summary

Technical Summary: Systematic Comparison of VMEC and HINT Equilibrium Calculations for Finite-β LHD Plasmas

Problem Statement
In stellarator physics, the calculation of three-dimensional magnetohydrodynamic (MHD) equilibrium is frequently performed using the VMEC solver. VMEC relies on the variational principle and strictly assumes the existence of perfectly nested flux surfaces. However, in finite-β plasmas, increased plasma pressure can alter the magnetic field topology, generating magnetic islands and stochastic regions. These phenomena, particularly in zero net-current devices where they are driven by Pfirsch-Schlüter (PS) currents, can compromise the validity of the nested flux surface assumption. When magnetic islands overlap, the last closed flux surface (LCFS) breaks, shrinking the enclosed plasma volume—a limitation observed to be more severe than the Shafranov shift. While the HINT code solves dynamic equations for magnetic fields and pressure without assuming nested surfaces, a systematic comparison between VMEC and HINT across various configurations and beta values in the Large Helical Device (LHD) has been necessary to clarify the influence of these assumptions.

Methodology
The study conducted a systematic comparison of equilibrium calculations between VMEC and HINT for LHD plasmas. The investigation focused on three specific vacuum magnetic-axis configurations: $R_{ax}^V = 3.53$ m (inward-shifted), $3.60$ m (standard), and $3.85$ m (outward-shifted). The analysis covered on-axis beta values ($\beta_0$) ranging from $0.0\%$ to $5.0\%$ in increments of $0.5\%$.

Both codes utilized an initial pressure profile defined as $p = p_0(1-s)$, where $s$ is the normalized toroidal flux. The study compared three primary metrics:

1. The position of the magnetic axis ($\langle R_{ax} \rangle$).
2. The rotational transform on the axis ($\iota/2\pi$).
3. The volume enclosed by the clear LCFS.

The HINT code, which employs a relaxation method to resolve magnetic islands and stochastic regions, was contrasted against VMEC, which enforces the nested flux surface constraint.

Key Results

• Low-β Regime: For low values of $\beta_0$, VMEC and HINT produced consistent equilibrium solutions. This indicates that the assumption of nested flux surfaces remains largely valid in this regime, and the 3D MHD responses are comparable between the two codes.
• Critical $\beta_0$ and Divergence: A configuration-dependent critical $\beta_0$ was identified. Beyond this threshold, the solutions from the two codes began to diverge. In the HINT calculations, the magnetic axis exhibited a larger Shafranov shift and a higher rotational transform compared to VMEC.
• Flux Surface Breaking and Volume: In HINT, the volume enclosed by the clear LCFS decreased at higher beta values for all three configurations. This reduction is attributed to the evolution of stochastic magnetic fields near the plasma edge, which breaks the flux surfaces. Conversely, VMEC, constrained by its nested surface assumption, showed a monotonic increase in enclosed volume as $\beta_0$ increased, failing to represent the flux surface breaking.
• Configuration Dependence: The critical $\beta_0$ value decreased from inward-shifted to outward-shifted configurations. The outward-shifted configuration ($R_{ax}^V = 3.85$ m) was found to be the most sensitive to 3D equilibrium responses. This sensitivity arises because the helical ripple increases with $R_{ax}^V$, driving stronger PS currents, while the magnetic shear decreases. The combination of stronger PS currents and lower shear leads to larger magnetic islands and earlier stochasticity at lower beta values.

Significance and Claims
The paper establishes that 3D equilibrium responses in LHD plasmas become increasingly significant as the configuration shifts from inward to outward. The study demonstrates that the nested flux surface assumption, central to VMEC, is compromised at finite beta due to PS current-driven perturbations and resulting edge stochasticity.

The authors claim that while VMEC and HINT are comparable for low-$\beta_0$ configurations, HINT provides a more physically complete description at higher beta by resolving the breaking of flux surfaces and the associated reduction in plasma volume. The divergence between the codes highlights that the limitation of the nested flux surface assumption is not uniform but depends heavily on the vacuum magnetic configuration, specifically the interplay between toroidicity (driving PS currents) and magnetic shear. The results underscore the importance of using codes capable of resolving stochasticity, such as HINT, when analyzing high-beta equilibria in stellarators, particularly in outward-shifted configurations where the critical beta threshold is lower.