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Compact quasiaxisymmetric stellarators, a near axisymmetric theory
Authors:
Wrick Sengupta,
Rogerio Jorge,
Nikita Nikulsin,
Stefan Buller,
Richard Nies,
Andrew Brown,
Amitava Bhattacharjee
Abstract:
We develop a theory of ridges in compact stellarators with quasiaxisymmetry (QA). The equilibrium with finite plasma currents and pressure is modeled by ideal magnetohydrostatics (MHS). Field lines are collimated near sharp ridges, much like X-points, making ridges attractive to divertor designs without the requirement of a rational rotational transform at the divertor. However, unlike X-points, w…
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We develop a theory of ridges in compact stellarators with quasiaxisymmetry (QA). The equilibrium with finite plasma currents and pressure is modeled by ideal magnetohydrostatics (MHS). Field lines are collimated near sharp ridges, much like X-points, making ridges attractive to divertor designs without the requirement of a rational rotational transform at the divertor. However, unlike X-points, which must cover the entire torus an integer number of times, sharp ridges are typically localized in certain parts of the flux surfaces. Motivated by recent work (Henneberg and Plunk, Phys. Rev. Research 6, L022052) on compact hybrid devices, we develop a perturbative treatment of nearly axisymmetric quasisymmetric devices by expanding in the deviation from perfect axisymmetry. As a result, we can analytically describe the key features of compact QA devices, such as the tendency for ridges to be localized on the inboard side, where the Gaussian curvature is typically negative, and the field strength is maximum. We provide comprehensive numerical evidence in support of our analytical theory.
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Submitted 2 June, 2026;
originally announced June 2026.
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On the Shafranov shift in stellarators
Authors:
Per Helander,
Nikita Nikulsin
Abstract:
As first shown by Shafranov, toroidal plasmas in magnetohydrodynamic equilibrium tend to expand in major radius when the pressure is increased. Here, an average measure of the resulting Shafranov shift is introduced, and its properties are discussed for various classes of optimised stellarator configurations. It is shown to be particularly small in quasi-helical and quasi-isodynamic stellarators w…
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As first shown by Shafranov, toroidal plasmas in magnetohydrodynamic equilibrium tend to expand in major radius when the pressure is increased. Here, an average measure of the resulting Shafranov shift is introduced, and its properties are discussed for various classes of optimised stellarator configurations. It is shown to be particularly small in quasi-helical and quasi-isodynamic stellarators with a large number of field periods, which are thus particularly robust to variations in the plasma pressure.
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Submitted 21 May, 2026;
originally announced May 2026.
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Optical analogy for stellarators: Ridges as caustics and coils as singularities
Authors:
Wrick Sengupta,
Stefan Buller,
Rogerio Jorge,
John Kappel,
Andrew Brown,
Richard Nies,
Pedro F. Gil,
Nikita Nikulsin,
Per Helander,
Amitava Bhattacharjee
Abstract:
A common feature of most numerically optimized stellarator geometries is the presence of sharp ridges on outer flux surfaces, irrespective of the rotational transform. Despite their importance, an analytical theory for their existence has been lacking. In this work, we demonstrate that ridges are not artifacts but mathematical necessities. We develop such a theory for devices with quasisymmetry (Q…
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A common feature of most numerically optimized stellarator geometries is the presence of sharp ridges on outer flux surfaces, irrespective of the rotational transform. Despite their importance, an analytical theory for their existence has been lacking. In this work, we demonstrate that ridges are not artifacts but mathematical necessities. We develop such a theory for devices with quasisymmetry (QS). We demonstrate that QS exhibits close connections with the theory of geometrical optics, following Parker's ``optical analogy" (E.N. Parker, Geophys. Astrophys. Fluid Dyn, 1989). By mapping vacuum QS to the eikonal equation of geometrical optics, we derive the conditions for ridge formation, identified as field line caustics where magnetic field lines focus. Furthermore, we prove a geometric theorem for stellarator coil design: both ridges and filamentary coils must lie on the zero-determinant manifold of the magnetic gradient tensor. This topological constraint unifies the description of plasma ridges and external coils, providing a precise criterion for identifying valid coil locations and explaining the efficacy of the magnetic gradient lengthscale (J. Kappel et al., Plasma Phys. Control. Fusion, 2024) as a coil optimization parameter. We demonstrate that as the device becomes more compact, sharp ridges naturally form on the inboard side in quasiaxisymmetry. We support our analytical theory with extensive numerical evidence.
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Submitted 20 May, 2026;
originally announced May 2026.
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Periodic Korteweg-de Vries soliton potentials generate quasisymmetric magnetic field strength in a finite plasma-beta equilibrium
Authors:
W. Sengupta,
R. Madan,
S. Buller,
N. Nikulsin,
E. J. Paul,
R. Nies,
A. A. Kaptanoglu,
S. R. Hudson,
A. Bhattacharjee
Abstract:
Quasisymmetry (QS) is a hidden symmetry of the magnetic field strength, B, that enables effective confinement of charged particles in a fully three-dimensional (3D) toroidal plasma equilibrium. Such equilibria are typically modeled by the ideal magnetohydrostatic (MHS) equation. The nonlinear, overdetermined nature of the quasisymmetric MHS equations severely complicates our understanding of the i…
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Quasisymmetry (QS) is a hidden symmetry of the magnetic field strength, B, that enables effective confinement of charged particles in a fully three-dimensional (3D) toroidal plasma equilibrium. Such equilibria are typically modeled by the ideal magnetohydrostatic (MHS) equation. The nonlinear, overdetermined nature of the quasisymmetric MHS equations severely complicates our understanding of the interplay between 3D shaping, equilibrium properties such as pressure and rotational transform, and B. Progress has been made through expansions near the magnetic axis; however, a more comprehensive theory is desirable. Using a combination of analysis and regression on a large dataset of numerically optimized quasisymmetric stellarators, we demonstrate that there is a hidden lower dimensionality of B on a magnetic flux surface with connections to the theory of periodic solitons. We show that $B$ on a flux surface is determined by three or at most four flux functions, each of which is a critical value of the derivative of B along the field line. While being consistent with the near-axis models, our results are global and hold even on the last closed flux surface.
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Submitted 8 July, 2025;
originally announced July 2025.
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A class of high-beta, large-aspect-ratio quasiaxisymmetric Palumbo-like configurations
Authors:
Andrew Brown,
Wrick Sengupta,
Nikita Nikulsin,
Amitava Bhattacharjee
Abstract:
The space of high-beta, approximately quasiaxisymmetric, large-aspect-ratio stellarator configurations is explored using an inverse coordinate approach and a quadratic polynomial ansatz for the flux function, following the method of Palumbo, extended by Hernandes and Clemente. This approach yields a system of nonlinear ODEs that, when solved, give equilibria exhibiting positive or negative triangu…
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The space of high-beta, approximately quasiaxisymmetric, large-aspect-ratio stellarator configurations is explored using an inverse coordinate approach and a quadratic polynomial ansatz for the flux function, following the method of Palumbo, extended by Hernandes and Clemente. This approach yields a system of nonlinear ODEs that, when solved, give equilibria exhibiting positive or negative triangularity, cusps, and (in an extreme limit) current singularities. It is shown that a cubic ansatz may also be used, but that polynomials of degree four or higher will lead to overdetermination.
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Submitted 22 September, 2025; v1 submitted 20 June, 2025;
originally announced June 2025.
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A Grad-Shafranov model for compact quasisymmetric stellarators
Authors:
Nikita Nikulsin,
Wrick Sengupta,
Stefan Buller,
Amitava Bhattacharjee
Abstract:
A Grad-Shafranov equation (GSE) valid for compact quasisymmetric stellarators is derived by an asymptotic expansion around a vacuum field carried to first order. We obtain an equation for the existence of flux surfaces leading up to the GSE. The flux surface label must simultaneously satisfy the existence equation and the GSE, which generally leads to an overdetermined problem. We show how the ove…
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A Grad-Shafranov equation (GSE) valid for compact quasisymmetric stellarators is derived by an asymptotic expansion around a vacuum field carried to first order. We obtain an equation for the existence of flux surfaces leading up to the GSE. The flux surface label must simultaneously satisfy the existence equation and the GSE, which generally leads to an overdetermined problem. We show how the overdetermined problem can be resolved within our model for a class of hybrid devices similar to that studied by Henneberg and Plunk (S. Henneberg and G. Plunk, PRR 2024). We are also able to solve the existence equation for flux surfaces analytically in the most general case by introducing a special coordinate system. This will enable us to carry out an optimization seeking to minimize the error in our GSE while obeying the flux surface existence equation, which will allow us to find solutions outside the class of hybrid devices. This will allow for a coarse-grained approximate search in the space of quasisymmetric equilibria that should be faster than a conventional stellarator optimization. Nevertheless, it would still be necessary to fine-tune the approximate solutions using conventional tools to obtain a more precise optimized equilibrium.
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Submitted 24 March, 2025; v1 submitted 20 January, 2025;
originally announced January 2025.
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An asymptotic Grad-Shafranov equation for quasisymmetric stellarators
Authors:
Nikita Nikulsin,
Wrick Sengupta,
Rogerio Jorge,
Amitava Bhattacharjee
Abstract:
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 be large and a simple expression can be obtained for…
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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 be large and a simple expression can be obtained for the lowest-order vacuum field. The first-order correction, which involves both vacuum and current-driven fields, is governed by a Grad-Shafranov equation and the requirement that flux surfaces exist. These two equations are not always consistent, and so this model is generally overconstrained, but special solutions exist that satisfy both equations simultaneously. One family of such solutions are the first-order near-axis solutions. Thus, the first-order near-axis model is a subset of the model presented here. Several other solutions outside the scope of the near-axis model are also found. A case study comparing one such solution to a VMEC-generated solution shows good agreement.
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Submitted 21 June, 2024; v1 submitted 12 March, 2024;
originally announced March 2024.
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Nonlinear MHD modeling of soft $β$ limits in W7-AS
Authors:
Rohan Ramasamy,
Ksenia Aleynikova,
Nikita Nikulsin,
Florian Hindenlang,
Ihor Holod,
Erika Strumberger,
Matthias Hoelzl,
the JOREK team
Abstract:
An important question for the outlook of stellarator reactors is their robustness against pressure driven modes, and the underlying mechanism behind experimentally observed soft $β$ limits. Towards building a robust answer to these questions, simulation studies are presented using a recently derived reduced nonlinear MHD model. First, the initial model implementation is extended to capture fluid c…
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An important question for the outlook of stellarator reactors is their robustness against pressure driven modes, and the underlying mechanism behind experimentally observed soft $β$ limits. Towards building a robust answer to these questions, simulation studies are presented using a recently derived reduced nonlinear MHD model. First, the initial model implementation is extended to capture fluid compression by including the influence of parallel flows. Linear benchmarks of a (2, 1) tearing mode in W7-AS geometry, and interchange modes in a finite $β$, net-zero current carrying stellarator with low magnetic shear are then used to demonstrate the modeling capabilities. Finally, a validation study is conducted on experimental reconstructions of finite $β$ W7-AS discharges. In agreement with past experimental analysis, it is shown that (i) the MHD activity is resistive, (ii) a soft $β$ limit is observed, when the plasma resistivity approaches the estimated experimental value, and (iii) low $n$ MHD activity is observed at intermediate $β$ values, particularly a nonlinearly dominant (2, 1) mode. The MHD activity is mild, explaining the soft $β$ limit, because the plasma volume remains separated into distinct sub-volumes in which field lines are ergodically confined. For the assumed transport parameters, the enhanced perpendicular transport along stochastic magnetic field lines can be overcome with the experimental heating power. The limitations in the current modeling are described, alongside an outlook for characterising soft $β$ limits in more detail in future work.
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Submitted 5 February, 2024;
originally announced February 2024.
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Asymptotic quasisymmetric high-beta 3D MHD equilibria near axisymmetry
Authors:
W. Sengupta,
N. Nikulsin,
R. Gaur,
A. Bhattacharjee
Abstract:
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 MHD equilibrium and stability of high-beta plasma in a large aspect-ratio stellarator. In particular, we show that the lowest-order description of a near-axisymmetric equilibrium vastly simplifies…
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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 MHD equilibrium and stability of high-beta plasma in a large aspect-ratio stellarator. In particular, we show that the lowest-order description of a near-axisymmetric equilibrium vastly simplifies the problem of 3D quasisymmetric MHD equilibria, which can be reduced to a standard elliptic Grad-Shafranov equation for the flux function. We show that any large aspect-ratio tokamak, deformed periodically in the vertical direction, is a stellarator with approximate volumetric QS. We discuss exact analytical solutions and numerical benchmarks. Finally, we discuss the ideal ballooning and interchange stability of some of our equilibrium configurations.
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Submitted 4 February, 2024; v1 submitted 13 December, 2023;
originally announced December 2023.
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Periodic Korteweg-de Vries soliton potentials generate quasisymmetric magnetic fields
Authors:
W. Sengupta,
N. Nikulsin,
S. Buller,
R. Madan,
E. J. Paul,
R. Nies,
A. A. Kaptanoglu,
S. R. Hudson,
A. Bhattacharjee
Abstract:
Quasisymmetry (QS) is a hidden symmetry of the magnetic field strength, B, that effectively confines charged particles 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 flux surface, which could make stellarator optimization s…
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Quasisymmetry (QS) is a hidden symmetry of the magnetic field strength, B, that effectively confines charged particles 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 flux surface, which could make stellarator optimization schemes significantly more efficient. Recent numerical breakthroughs (M. Landreman and E. Paul, Phys. Rev. Lett. 128, 035001 (2022)) have yielded configurations with excellent volumetric QS and surprisingly low magnetic shear. Given B, it may be possible to deduce an upper bound on the maximum quasisymmetric toroidal volume which depends only on the properties of B. This has been verified for the Landreman-Paul precise quasiaxisymmetric (QA) stellarator configuration. In the neighborhood of the outermost surface, we show that B approaches the form of the 1-soliton reflectionless potential (I. Gjaja and A. Bhattacharjee, Phys. Rev. Lett. 68, 2413 (1992)). The connection length diverges, indicating the possible presence of an X-point or cusp that could potentially be used as a basis for a divertor. We present a non-perturbative approach based on ensuring single-valuedness of B, which directly leads to its Painleve property and the KdV and Gardner's equations. Finally, we use an approach based on machine learning, trained on a large dataset of numerically optimized quasisymmetric stellarators. We robustly recover the KdV and Gardner's equations from the data.
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Submitted 19 April, 2026; v1 submitted 27 February, 2023;
originally announced February 2023.
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JOREK3D: An extension of the JOREK nonlinear MHD code to stellarators
Authors:
Nikita Nikulsin,
Rohan Ramasamy,
Matthias Hoelzl,
Florian Hindenlang,
Erika Strumberger,
Karl Lackner,
Sibylle Günter
Abstract:
Although the basic concept of a stellarator was known since the early days of fusion research, advances in computational technology have enabled the modelling of increasingly complicated devices, leading up to the construction of Wendelstein 7-X, which has recently shown promising results. This recent success has revived interest in the nonlinear 3D MHD modelling of stellarators in order to better…
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Although the basic concept of a stellarator was known since the early days of fusion research, advances in computational technology have enabled the modelling of increasingly complicated devices, leading up to the construction of Wendelstein 7-X, which has recently shown promising results. This recent success has revived interest in the nonlinear 3D MHD modelling of stellarators in order to better understand their performance and operational limits. This paper reports on the extension of the JOREK code to 3D geometries and on the first stellarator simulations carried out with it. The first simple simulations shown here address the classic Wendelstein 7-A stellarator using a reduced MHD model previously derived by us. The results demonstrate that stable full MHD equilibria are preserved in the reduced model: the flux surfaces do not move throughout the simulation, and closely match the flux surfaces of the full MHD equilibrium. Further, both tearing and ballooning modes were simulated, and the linear growth rates measured in JOREK are in reasonable agreement with the growth rates from the CASTOR3D linear MHD code.
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Submitted 12 May, 2022; v1 submitted 28 January, 2022;
originally announced January 2022.
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Testing of the new JOREK stellarator-capable model in the tokamak limit
Authors:
Nikita Nikulsin,
Matthias Hoelzl,
Alessandro Zocco,
Karl Lackner,
Sibylle Guenter
Abstract:
In preparation for extending the JOREK nonlinear MHD code to stellarators, a hierarchy of stellarator-capable reduced and full MHD models has been derived and tested. The derivation was presented at the EFTC 2019 conference. Continuing this line of work, we have implemented the reduced MHD model (arXiv:1907.12486) as well as an alternative model which was newly derived using a different set of pro…
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In preparation for extending the JOREK nonlinear MHD code to stellarators, a hierarchy of stellarator-capable reduced and full MHD models has been derived and tested. The derivation was presented at the EFTC 2019 conference. Continuing this line of work, we have implemented the reduced MHD model (arXiv:1907.12486) as well as an alternative model which was newly derived using a different set of projection operators for obtaining the scalar momentum equations from the full MHD vector momentum equation. With the new operators, the reduced model matches the standard JOREK reduced models for tokamaks in the tokamak limit and conserves energy exactly, while momentum conservation is less accurate than in the original model whenever field-aligned flow is present.
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Submitted 30 March, 2021; v1 submitted 27 November, 2020;
originally announced November 2020.
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The JOREK non-linear extended MHD code and applications to large-scale instabilities and their control in magnetically confined fusion plasmas
Authors:
M Hoelzl,
GTA Huijsmans,
SJP Pamela,
M Becoulet,
E Nardon,
FJ Artola,
B Nkonga,
CV Atanasiu,
V Bandaru,
A Bhole,
D Bonfiglio,
A Cathey,
O Czarny,
A Dvornova,
T Feher,
A Fil,
E Franck,
S Futatani,
M Gruca,
H Guillard,
JW Haverkort,
I Holod,
D Hu,
SK Kim,
SQ Korving
, et al. (28 additional authors not shown)
Abstract:
JOREK is a massively parallel fully implicit non-linear extended MHD code for realistic tokamak X-point plasmas. It has become a widely used versatile code for studying large-scale plasma instabilities and their control developed in an international community. This article gives a comprehensive overview of the physics models implemented, numerical methods applied for solving the equations and phys…
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JOREK is a massively parallel fully implicit non-linear extended MHD code for realistic tokamak X-point plasmas. It has become a widely used versatile code for studying large-scale plasma instabilities and their control developed in an international community. This article gives a comprehensive overview of the physics models implemented, numerical methods applied for solving the equations and physics studies performed with the code. A dedicated section highlights some of the verification work done for the code. A hierarchy of different physics models is available including a free boundary and resistive wall extension and hybrid kinetic-fluid models. The code allows for flux-surface aligned iso-parametric finite element grids in single and double X-point plasmas which can be extended to the true physical walls and uses a robust fully implicit time stepping. Particular focus is laid on plasma edge and scrape-off layer (SOL) physics as well as disruption related phenomena. Among the key results obtained with JOREK regarding plasma edge and SOL, are deep insights into the dynamics of edge localized modes (ELMs), ELM cycles, and ELM control by resonant magnetic perturbations, pellet injection, as well as by vertical magnetic kicks. Also ELM free regimes, detachment physics, the generation and transport of impurities during an ELM, and electrostatic turbulence in the pedestal region are investigated. Regarding disruptions, the focus is on the dynamics of the thermal quench and current quench triggered by massive gas injection (MGI) and shattered pellet injection (SPI), runaway electron (RE) dynamics as well as the RE interaction with MHD modes, and vertical displacement events (VDEs). Also the seeding and suppression of tearing modes (TMs), the dynamics of naturally occurring thermal quenches triggered by locked modes, and radiative collapses are being studied.
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Submitted 21 April, 2021; v1 submitted 18 November, 2020;
originally announced November 2020.
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A three-dimensional reduced MHD model consistent with full MHD
Authors:
Nikita Nikulsin,
Matthias Hoelzl,
Alessandro Zocco,
Karl Lackner,
Sibylle Günter
Abstract:
Within the context of a viscoresistive magnetohydrodynamic (MHD) model with anisotropic heat transport and cross-field mass diffusion, we introduce novel three-term representations for the magnetic field (background vacuum field, field line bending and field compression) and velocity ($\vec E\times\vec B$ flow, field-aligned flow and fluid compression), which are amenable to three-dimensional trea…
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Within the context of a viscoresistive magnetohydrodynamic (MHD) model with anisotropic heat transport and cross-field mass diffusion, we introduce novel three-term representations for the magnetic field (background vacuum field, field line bending and field compression) and velocity ($\vec E\times\vec B$ flow, field-aligned flow and fluid compression), which are amenable to three-dimensional treatment. Once the representations are inserted into the MHD equations, appropriate projection operators are applied to Faraday's law and the Navier-Stokes equation to obtain a system of scalar equations that is closed by the continuity and energy equations. If the background vacuum field is sufficiently strong and the $β$ is low, MHD waves are approximately separated by the three terms in the velocity representation, with each term containing a specific wave. Thus, by setting the appropriate term to zero, we eliminate fast magnetosonic waves, obtaining a reduced MHD model. We also show that the other two velocity terms do not compress the magnetic field, which allows us to set the field compression term to zero within the same reduced model. Dropping also the field-aligned flow, a further simplified model is obtained, leading to a fully consistent hierarchy of reduced and full MHD models for 3D plasma configurations. Finally, we discuss the conservation properties and derive the conditions under which the reduction approximation is valid. We also show that by using an ordering approach, reduced MHD equations similar to what we got from the ansatz approach can be obtained by means of a physics-based asymptotic expansion.
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Submitted 14 October, 2019; v1 submitted 29 July, 2019;
originally announced July 2019.