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Non-axisymmetric MHD equilibria with an arbitrary number of field periods

This paper derives closed-form non-axisymmetric magnetohydrodynamic equilibria with nested toroidal flux surfaces for an arbitrary number of field periods $N \geq 1$ , featuring non-constant pressure and specific rotational transforms, while also constructing a matched exterior vacuum field for reflection-symmetric cases to serve as benchmarks for 3D equilibrium codes.

Original author: Opal Issan, Lucas José, Matt Landreman, Jonathan Citrin, Brendan Tracey

Published 2026-11-06

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CC BY 4.0

Original author: Opal Issan, Lucas José, Matt Landreman, Jonathan Citrin, Brendan Tracey

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the author. For technical accuracy, refer to the original paper. Read full disclaimer

Inside the heart of a fusion reactor, a superheated gas called plasma is trapped by invisible magnetic cages. To keep this energy-dense fuel from touching the walls and cooling down, the magnetic field must form perfect, nested tubes that loop endlessly around a central axis, much like the layers of an onion. For decades, scientists have known how to build these cages when they are perfectly round and symmetrical, like a simple donut. However, the most promising designs for future power plants are not perfect donuts; they are twisted, three-dimensional shapes with multiple lobes that repeat around the circle. These complex structures, known as stellarators, are designed to hold the plasma more stably, but calculating the exact magnetic fields needed to make them work has been a monumental challenge. Without precise mathematical blueprints, engineers must rely on computer simulations that can sometimes hide errors or miss subtle instabilities.

For a long time, the only exact mathematical descriptions of these twisted magnetic cages were limited to very simple shapes with just one or two repeating sections. This left a gap in our knowledge for the more complex designs that real-world reactors actually use. A team of researchers has now filled that gap by discovering a new set of exact mathematical solutions. They have derived closed-form equations that describe stable, three-dimensional magnetic equilibria for any number of repeating sections, from a single twist to ten or more. These solutions are not just approximations or computer guesses; they are precise formulas that satisfy the fundamental laws of physics governing how magnetic fields and hot gas interact.

The researchers found that these new magnetic configurations can exist in three distinct families, each with its own unique geometry. The first family describes a shape where the magnetic field lines do not twist around the torus at all as they travel; they simply loop straight around. This specific type of equilibrium had been proven to exist mathematically decades ago, but until now, no one had an explicit formula to describe it for complex shapes. The team's work provides that missing formula, showing exactly how the magnetic field and the pressure of the plasma must vary to keep the system stable. The other two families describe shapes where the magnetic field lines do twist, completing a specific number of turns around the short way of the torus for every trip around the long way. These twisted configurations are crucial because they mimic the behavior of the magnetic fields in actual stellarator designs.

What makes these findings particularly valuable is that they work for any integer number of repeating sections. While previous exact solutions were restricted to shapes with only one or two repeating segments, these new formulas apply to the complex, multi-lobed designs used in major experimental reactors. The researchers demonstrated that these magnetic cages can hold a non-uniform pressure of plasma, meaning the gas is denser and hotter in some areas than others, just as it would be in a real machine. They also showed that the magnetic field lines in these solutions are perfectly closed, meaning a particle traveling along a field line will eventually return to its starting point without drifting away, a critical requirement for keeping the plasma contained.

To ensure these mathematical shapes are not just theoretical curiosities, the team verified that the magnetic surfaces they describe are smooth and nested, with no gaps or tears. They also checked that the plasma pressure and magnetic forces balance perfectly at every point, preventing the gas from collapsing or exploding outward. The study further explored the stability of these shapes, revealing that while they are mathematically perfect, they are inherently unstable to a specific type of disturbance known as interchange instability. This finding is not a failure of the solution but a vital piece of information; it tells engineers exactly where the system is vulnerable, allowing them to design active controls to stabilize it.

Perhaps most importantly, the researchers constructed a matching magnetic field for the empty space outside the plasma cage. In a real reactor, the magnetic field inside the plasma must connect seamlessly to the field generated by external magnets in the vacuum outside. The team derived an exact formula for this exterior field, creating a complete, self-consistent model that bridges the gap between the hot plasma and the vacuum chamber. This provides a rigorous benchmark for the computer codes that engineers use to design fusion reactors. By comparing their complex simulations against these exact formulas, scientists can now verify that their software is calculating the magnetic fields correctly, ensuring that the designs for future power plants are built on a foundation of mathematical certainty rather than approximation.

The work represents a significant step forward in the quest for fusion energy. By providing the first exact, closed-form solutions for these complex, multi-period magnetic cages, the researchers have given the fusion community a new set of tools to test and refine their designs. These formulas act as a gold standard, a known truth against which all other calculations can be measured. As the world moves closer to building the first fusion power plants, having such precise mathematical anchors ensures that the magnetic cages holding the star-like fire will be as robust and reliable as the science that created them.

Technical Summary: Non-axisymmetric MHD Equilibria with an Arbitrary Number of Field Periods

Problem Statement
Recent advances have yielded several closed-form, non-axisymmetric magnetic fields with nested toroidal flux surfaces, serving as exact benchmarks for verifying three-dimensional magnetohydrodynamic (MHD) equilibrium codes. However, these existing solutions are restricted to at most two field periods ( $N \le 2$ ). In contrast, practical stellarator designs typically utilize $N \ge 3$ (e.g., NCSX with $N=3$ , Wendelstein 7-X with $N=5$ ). The absence of analytical solutions for $N \ge 3$ limits the ability to rigorously test equilibrium solvers such as VMEC, DESC, GVEC, and SPEC in regimes relevant to modern fusion devices. Furthermore, while the existence of smooth non-axisymmetric equilibria with non-constant pressure was conjectured by Grad and later proven by Lortz (for $N$ -periodic, reflection-symmetric, $\iota=0$ cases) and Gómez-Serrano et al. (for discrete symmetry with $\iota=\infty$ ), these proofs did not provide explicit closed-form expressions.

Methodology
The authors derive closed-form non-axisymmetric MHD equilibria for an arbitrary integer number of field periods $N \ge 1$ . The derivation relies on a specific coordinate transformation ansatz that maps field-aligned curvilinear coordinates $(\rho, \zeta, \eta)$ to Cartesian coordinates $(x, y, z)$ .

Coordinate Ansatz: The mapping is constructed by perturbing an axisymmetric map ( $x+iy = \rho e^{i\zeta}, z=\eta$ ) with an $N$ -periodic vertical oscillation in $z$ . To satisfy the force-balance and divergence-free equations, the complex coordinate $x+iy$ requires a three-harmonic Fourier expansion in the field-aligned toroidal angle $\zeta$ :
$x + iy = C_1(\rho) e^{im_1\zeta} + (C_2(\rho) + i\eta C_4(\rho)) e^{im_2\zeta} - C_3(\rho) e^{im_3\zeta}$
$z = \eta + C_5(\rho) \sin(N\zeta)$
The toroidal mode numbers $(m_1, m_2, m_3)$ are spaced by $N$ and depend on an integer shift index $k \in \{1, 0, -1\}$ .
Reduction of Equations: This ansatz reduces the three-dimensional nonlinear MHD partial differential equations to a sequence of separable first-order linear ordinary differential equations and algebraic relations for the auxiliary functions $C_1(\rho)$ through $C_5(\rho)$ .
Solution Families: Solving this system yields three distinct families of equilibria, categorized by the index $k$ :
- Family 1 ( $k=1$ ): Valid for $N \ge 1$ . Features zero rotational transform ( $\iota = 0$ ).
- Family 2 ( $k=0$ ): Valid for $N \ge 2$ . Features an $\iota = 0$ branch (for $N \ge 2$ ) and an integer rotational transform branch $\iota = \pm N$ (for $N \ge 3$ ).
- Family 3 ( $k=-1$ ): Valid for $N \ge 3$ . Features $\iota = \pm N$ .

Key Results
The derived solutions possess the following properties:

Nested Flux Surfaces: All families exhibit nested toroidal flux surfaces with non-constant plasma pressure $p(\psi) = p_0 - 2\psi$ .
Rotational Transform:
- Family 1 provides explicit closed-form realizations of Lortz's $\iota=0$ equilibria for any $N \ge 1$ .
- Families 2 and 3 provide solutions with integer rotational transform $\iota = \pm N$ for $N \ge 3$ .
Symmetries: The equilibria satisfy discrete $N$ -fold rotational symmetry. Family 1 (with specific parameter choices) also satisfies reflection symmetry across a poloidal plane, while all families satisfy stellarator symmetry (invariance under $\pi$ -rotation about the $x$ -axis).
Stability: All three families are found to be ideal-MHD interchange unstable, consistent with the lack of magnetic shear or a stabilizing magnetic well ( $d^2V/d\Phi_t^2 = 0$ ).
Geometric Characteristics:
- Family 1 ( $\iota=0$ ) maintains nested surfaces for a wide range of parameters.
- Family 2's $\iota = \pm N$ branch is restricted to narrow cross-sections and breaks down at small boundary flux values.
- Family 3's $\iota = \pm N$ branch supports much broader cross-sections, larger plasma volumes, and higher volume-averaged beta ( $\beta_V$ ) compared to Family 2.
Exterior Vacuum Field: For the reflection-symmetric $\iota=0$ equilibria (Family 1 with $\rho_a = \rho_0$ ), the authors construct a matched exterior vacuum field. This field is curl-free, divergence-free, and continuous across the plasma boundary, providing a complete benchmark for free-boundary equilibrium codes.

Significance and Claims
The paper claims to provide the first closed-form non-axisymmetric MHD equilibria for an arbitrary number of field periods $N \ge 3$ . This extends the variety of exact benchmarks available for verifying 3D equilibrium codes beyond the previously limited $N \le 2$ regime.

Specifically, the work:

Offers an explicit closed-form realization of the non-axisymmetric $\iota=0$ equilibria whose existence was previously only proven via fixed-point iteration (Lortz).
Provides the first analytical benchmarks for stellarators with $N \ge 3$ field periods, covering both zero and integer rotational transform cases.
Enables the testing of free-boundary equilibrium solvers through the construction of a matched exterior vacuum field for the $\iota=0$ case.

The authors note that while these solutions are distinct from previous closed-form vacuum or constant-pressure solutions, they represent a specific class of exact solutions. They suggest that the discovery of these families implies that further classes of exact three-dimensional solutions may exist, governed by a more general mathematical theory yet to be fully understood.

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