** partially supported by Simons Foundation*

## Submitted

The Floquet exponents of periodic field lines are studied through the variations of the magnetic action on the magnetic axis, which is assumed to be elliptical. The near-axis formalism developed by Mercier, Solov’ev and Shafranov is combined with a Lagrangian approach. The on-axis Floquet exponent is shown to coincide with the on-axis…

A first-order model is derived for quasisymmetric stellarators where the vacuum field due to coils is dominant, but plasma-current-induced terms are not negligible and can contribute to magnetic differential equations, with β of the order of the ratio of induced to vacuum fields. Under these assumptions, it is proven that the aspect ratio must…

Magnetic confinement fusion designs require careful consideration for them to confine particles effectively. A possible way to do so is to impose quasisymmetry. In practice, large-scale optimizations are performed to minimize deviations from exact quasisymmetry. Unfortunately, the large number of parameters and complexity of the optimization…

## In Press

Quasisymmetry (QS), a hidden symmetry of the magnetic field strength, is known to support nested flux surfaces and provide superior particle confinement in stellarators. In this work, we study the ideal magnetohydrodynamic (MHD) equilibrium and stability of high-beta plasma in a large-aspect-ratio stellarator. In particular, we show that the…

A systematic theory of the asymptotic expansion of the magnetohydrostatics (MHS) equilibrium in the distance from the magnetic axis is developed to include arbitrary smooth currents near the magnetic axis. Compared with the vacuum and the force-free system, an additional magnetic differential equation must be solved to obtain the pressure…

*arXiv e-prints*, Feb 2023, arXiv:2302.13924

Quasisymmetry (QS) is a hidden symmetry of the magnetic field strength, B, that confines charged particles effectively in a three-dimensional toroidal plasma equilibrium. Here, we show that QS has a deep connection to the underlying symmetry that makes solitons possible. Our approach uncovers a hidden lower dimensionality of B on a magnetic…

*Journal of Plasma Physics*, Vol. 88, Jan 2022, 905880109

Quasisymmetric stellarators are an attractive class of optimised magnetic confinement configurations. The property of quasisymmetry (QS) is in practice limited to be approximate, and thus the construction requires measures that quantify the deviation from the exact property. In this paper we study three measure candidates used in the literature…

*Journal of Plasma Physics*, Vol. 88, Feb 2022

The gap between a recently developed dynamical version of relaxed magnetohydrodynamics (RxMHD) and ideal MHD (IMHD) is bridged by approximating the zero-resistivity "Ideal" Ohm's Law (IOL) constraint using an augmented Lagrangian method borrowed from optimization theory. The augmentation combines a pointwise vector Lagrange multiplier method…

Deuterium-tritium (DT) burning requires a long energy confinement times compared to collision times, so the particle distribution functions must approximate local-Maxwellians. Non-equilibrium thermodynamics is applicable, which gives relations among transport, entropy production, the collision frequency, and the deviation from a Maxwellian. The…

*Physics of Plasmas*, Vol. 29, Apr 2022, 042505

Good magnetic surfaces, as opposed to magnetic islands and chaotic field lines, are generally desirable for stellarators. In previous work, Landreman *et al.* [Phys. of Plasmas **28**, 092505 (2021)] showed that equilibria computed by the Stepped-Pressure Equilibrium Code (SPEC) [Hudson *et al.*,…

*Journal of Plasma Physics*, Vol. 87, May 2021, 905870226

Combined plasma–coil optimization approaches for designing stellarators are discussed and a new method for calculating free-boundary equilibria for multiregion relaxed magnetohydrodynmics (MRxMHD) is proposed. Four distinct categories of stellarator optimization, two of which are novel approaches, are the fixed-boundary optimization, the…

*Plasma Physics and Controlled Fusion*, Vol. 63, Feb 2021, 045006

Over the last decade, a variational principle based on a generalisation of Taylor’s relaxation, referred to as multi-region relaxed magnetohydrodynamics (MRxMHDs) has been developed. The numerical solutions of the MRxMHD equilibria have been constructed using the Stepped Pressure Equilibrium Code (SPEC) (Hudson et al 2012 Phys. Plasmas 19…

*Phys. Rev. E*, Vol. 104, Jul 2021, 015213

The *stellarator* as a concept of magnetic confinement fusion requires careful design to confine particles effectively. A design possibility is to equip the magnetic field with a property known as *quasisymmetry*. Though it is generally believed that a steady-state quasisymmetric equilibrium can only be exact locally (unless the…

*Physics of Plasmas*, Vol. 28, Sep 2021, 092510

We prove the existence of a straight-field-line coordinate system we call *generalized Boozer coordinates*. This coordinate system exists for magnetic fields with nested toroidal flux surfaces provided ∮d𝑙/𝐵(𝐣·∇𝜓)=0, where symbols have their usual meaning, and the integral is taken along closed magnetic field lines. All quasisymmetric…

Integrable or near-integrable magnetic fields are prominent in the design of plasma confinement devices. Such a field is characterized by the existence of a singular foliation consisting entirely of invariant submanifolds. A regular leaf, known as a flux surface,of this foliation must be diffeomorphic to the two-torus. In a neighborhood of a…

*Nuclear Fusion*, Vol. 61, Dec 2021, 126040

In the Wendelstein 7-X (W7-X) stellarator, the vacuum rotational transform, ι, has a flat radial profile and does not cross any major rational resonance. Nevertheless, during plasma operation the ι‐profile can be strongly modified by electron cyclotron current drive in such a way that the resulting ι-profile passes through low-order rational…

*Physics of Plasmas*, Vol. 28, Dec 2021, 122501

A formal series transformation to Birkhoff–Gustavson normal form is obtained for toroidal magnetic field configurations in the neighborhood of a magnetic axis. Bishop's rotation minimizing coordinates are used to obtain a local orthogonal frame near the axis in which the metric is diagonal, even if the curvature has zeros. We treat the cases of…

*Nuclear Fusion*, Vol. 61, Aug 2021, 096024

This paper is focused on three points: (1) overcoming obstacles to tokamak power plants may require a configuration modification as large as that of a stellarator. (2) The demonstrated reliability of the computational design of stellarators should change fusion strategy. (3) Deployment of carbon-free energy sources is mandated by the thirty…

*Journal of Plasma Physics*, Vol. 88, Oct 2020, 905880106

Adjoint methods can speed up stellarator optimisation by providing gradient information more efficiently compared with finite-difference evaluations. Adjoint methods are herein applied to vacuum magnetic fields, with objective functions targeting quasi-symmetry and a rotational transform value on a surface. To measure quasi-symmetry, a novel…

*Plasma Physics and Controlled Fusion*, Vol. 62, Nov 2020, 124004

The numerical solution of the stepped pressure equilibrium (Hudson et al 2012 Phys. Plasmas 19 112502) requires a fast and robust solver to obtain the Beltrami field in three-dimensional geometry such as stellarators. The spectral method implemented in the stepped pressure equilibrium code (SPEC) is efficient when the domain is a hollow torus,…

*Journal of Mathematical Physics*, Vol. 61, Sep 2020, 093503

Quasi-symmetry of a steady magnetic field means integrability of first-order guiding-center motion by a spatial symmetry. Here, we derive many restrictions on the possibilities for a quasi-symmetry. We also derive an analog of the Grad–Shafranov equation for the flux function in a quasi-symmetric magnetohydrostatic field.

## In Preparation

In a magnetic field, transitions between classes of guiding-centre motion can lead to cross-field diffusion and escape. We say a magnetic field is isodrastic if guiding centres make no transitions between classes of motion. Therefore, this is an important ideal for enhancing confinement. First, we present a weak formulation, based on the…

## 2024

A first-order model is derived for quasisymmetric stellarators where the vacuum field due to coils is dominant, but plasma-current-induced terms are not negligible and can contribute to magnetic differential equations, with β of the order of the ratio of induced to vacuum fields. Under these assumptions, it is proven that the aspect ratio must…

## 2022

*Physics of Plasmas*, Vol. 29, 042505

## 2021

*Journal of Physics A: Mathematical and Theoretical*, Vol. 54, Mar 2021, 125202

In a strong, inhomogeneous magnetic field, charged particle dynamics may be studied in the guiding-centre approximation, which is known to be Hamiltonian. When the magnetic field is quasisymmetric, the first-order guiding-centre (FGC) Hamiltonian structure admits a continuous symmetry, and therefore a conserved quantity in addition to the…

*Journal of Physics A: Mathematical and Theoretical*, Vol. 54, Mar 2021, 125202

In a strong, inhomogeneous magnetic field, charged particle dynamics may be studied in the guiding-centre approximation, which is known to be Hamiltonian. When the magnetic field is quasisymmetric, the first-order guiding-centre (FGC) Hamiltonian structure admits a continuous symmetry, and therefore a conserved quantity in addition to the…

*Journal of Plasma Physics*, Vol. 87, May 2021

A first-principles method to calculate the critical temperature gradient for the onset of the ion-temperature-gradient mode (ITG) in linear gyrokinetics is presented. We find that conventional notions of the connection length previously invoked in tokamak research should be revised and replaced by a generalized correlation length to explain…

*J. Plasma Phys.*, Vol. 87, 905870226

*J. Plasma Phys.*, Vol. 87, 905870403

*Chaos*, Vol. 31, Jan 2021

Invariant manifolds are of fundamental importance to the qualitative understanding of dynamical systems. In this work, we explore and extend MacKay’s converse Kolmogorov–Arnol’d–Moser condition to obtain a sufficient condition for the nonexistence of invariant surfaces that are transverse to a chosen 1D foliation. We show how useful foliations…

*Plasma Physics and Controlled Fusion*, Vol. 63, Oct 2021, 125007

The Stepped Pressure Equilibrium Code (SPEC) (Hudson et al 2012 Phys. Plasmas 19 112502) has been successful in the construction of equilibria in 3D configurations that contain a mixture of flux surfaces, islands and chaotic magnetic field lines. In this model, the plasma is sliced into sub-volumes separated by ideal interfaces, and in each…

*J. Plasma Phys.*, Vol. 87, 905870503

*J. Open Source Software*, Vol. 6, 3525

*Phys. Plasmas*, Vol. 28, 092505

## 2020

*in preparation (2020)*

*Nuclear Fusion 60, 076021 (2020)*, Vol. 60

*submitted to Plasma Physics and Controlled Fusion (2020)*

*Plasma Phys. Controlled Fusion*, Vol. 62, 124004

*Nuclear Fusion 60, 076016 (2020)*, Vol. 60

*Journal of Plasma Physics*, Vol. 86, Jan 2020, 925860101

Differential forms provide a coordinate-free way to express many quantities and relations in mathematical physics. In particular, they are useful in plasma physics. This tutorial gives a guide so that you can read the plasma physics literature that uses them and apply them yourself.

*Physics of Plasmas 27, 042509 (2020)*, Vol. 27

*to be submitted to the Journal of Computational Physics (2020)*

*in preparation (2020)*

*Physics of Plasmas 27*, Vol. 27, Jul 2020

It is shown that the variational principle of multi-region relaxed magnetohydrodynamics (MRxMHD) can be used to predict the stability and nonlinear saturation of tearing modes in strong guide field configurations without resolving the dynamics and without explicit dependence on the plasma resistivity. While the magnetic helicity is not a good…

*Plasma Physics and Controlled Fusion 62, 024004 (2020)*, Vol. 62

*Plasma Physics and Controlled Fusion 62, 044002 (2020)*, Vol. 62

*submitted (2020)*

*Plasma Phys. Controlled Fusion*, Vol. 62, 084002