@article{y1972_shafranov, doi = {10.1088/0029-5515/12/5/009}, url = {https://doi.org/10.1088%2F0029-5515%2F12%2F5%2F009}, year = 1972, month = {sep}, publisher = {{IOP} Publishing}, volume = {12}, number = {5}, pages = {599--601}, author = {Shafranov, V. D. and Zakharov, L. E.}, title = {Use of the virtual-casing principle in calculating the containing magnetic field in toroidal plasma systems}, journal = {Nucl. Fusion}, } @article{y1985_hirshman, author = {Hirshman, S. P. and Meier, H. K.}, title = {{Optimized Fourier representations for three-dimensional magnetic surfaces}}, journal = {Phys. Fluids}, volume = {28}, number = {5}, pages = {1387-1391}, year = {1985}, doi = {10.1063/1.864972}, URL = {https://aip.scitation.org/doi/abs/10.1063/1.864972}, eprint = {https://aip.scitation.org/doi/pdf/10.1063/1.864972} } @article{y1998_hirshman, author = {Hirshman, S. P. and Breslau, J.}, title = {{Explicit spectrally optimized Fourier series for nested magnetic surfaces}}, journal = {Phys. Plas.}, volume = {5}, number = {7}, pages = {2664-2675}, year = {1998}, doi = {10.1063/1.872954}, URL = {https://doi.org/10.1063/1.872954}, eprint = {https://doi.org/10.1063/1.872954} } @article{y2010_hirshman, title = {{BCYCLIC: A parallel block tridiagonal matrix cyclic solver}}, journal = {J. Comp. Phys.}, volume = {229}, number = {18}, pages = {6392 - 6404}, year = {2010}, issn = {0021-9991}, doi = {https://doi.org/10.1016/j.jcp.2010.04.049}, url = {http://www.sciencedirect.com/science/article/pii/S0021999110002536}, author = {Hirshman, S.P. and Perumalla, K. S. and Lynch, V. E. and Sanchez, R.}, } @article{y2011_hudson, doi = {10.1088/0741-3335/54/1/014005}, url = {https://doi.org/10.1088%2F0741-3335%2F54%2F1%2F014005}, year = 2011, month = {dec}, publisher = {{IOP} Publishing}, volume = {54}, number = {1}, pages = {014005}, author = {Hudson, S. R. and Dewar, R. L. and Hole, M. J. and McGann, M.}, title = {Non-axisymmetric, multi-region relaxed magnetohydrodynamic equilibrium solutions}, journal = {Plasma Phys. and Contr. Fusion} } @article{y2012_lazerson, doi = {10.1088/0741-3335/54/12/122002}, url = {https://doi.org/10.1088%2F0741-3335%2F54%2F12%2F122002}, year = 2012, month = {nov}, publisher = {{IOP} Publishing}, volume = {54}, number = {12}, pages = {122002}, author = {Lazerson, S. A.}, title = {{The virtual-casing principle for 3D toroidal systems}}, journal = {Plasma Phys. and Contr. Fusion} } @article{y2013_lazerson, doi = {10.1088/0741-3335/55/2/025014}, url = {https://doi.org/10.1088%2F0741-3335%2F55%2F2%2F025014}, year = 2013, month = {jan}, publisher = {{IOP} Publishing}, volume = {55}, number = {2}, pages = {025014}, author = {Lazerson, S. A. and Sakakibara, S. and Suzuki, Y.}, title = {{A magnetic diagnostic code for 3D fusion equilibria}}, journal = {Plasma Phys. and Contr. Fusion} } @article{y2015_hanson, doi = {10.1088/0741-3335/57/11/115006}, url = {https://doi.org/10.1088%2F0741-3335%2F57%2F11%2F115006}, year = 2015, month = {sep}, publisher = {{IOP} Publishing}, volume = {57}, number = {11}, pages = {115006}, author = {Hanson, J. D.}, title = {{The virtual-casing principle and Helmholtz's theorem}}, journal = {Plasma Phys. and Contr. Fusion} } @article{y2017_zhu, doi = {10.1088/1741-4326/aa8e0a}, url = {https://doi.org/10.1088%2F1741-4326%2Faa8e0a}, year = 2017, month = {nov}, publisher = {{IOP} Publishing}, volume = {58}, number = {1}, pages = {016008}, author = {Zhu, C. and Hudson, S. R. and Song, Y. and Wan, Y.}, title = {New method to design stellarator coils without the winding surface}, journal = {Nucl. Fusion} } @article{y2016_loizu, title = "Verification of the {SPEC} code in stellarator geometries", author = "Loizu, J and Hudson, S R and N{\"u}hrenberg, C", abstract = "We present the first calculations performed with the Stepped-Pressure Equilibrium Code (SPEC) in stellarator geometry. Provided a boundary magnetic surface, stellarator vacuum fields with islands are computed and verified to machine precision, for both a classical l = 2 stellarator field and a Wendelstein 7-X limiter configuration of the first experimental campaign. Beyond verification, a detailed comparison of SPEC solutions to Biot-Savart solutions for the corresponding coil currents is shown. The level of agreement is quantified, and the error is shown to be dominated by the accuracy with which the boundary representation is given. Finally, partially relaxed stellarator equilibria are computed with SPEC, and verification is presented with force-balance down to machine precision.", journal = "Phys. Plasmas", publisher = "AIP Publishing", volume = 23, number = 11, pages = "112505", month = nov, year = 2016 } @article{y2021_baillod, title = "Computation of multi-region, relaxed magnetohydrodynamic equilibria with prescribed toroidal current profile", author = "Baillod, A and Loizu, J and Qu, Z S and Kumar, A and Graves, J P", abstract = "The stepped-pressure equilibrium code (SPEC) (Hudsonet al.,Phys. Plasmas, vol. 19, issue 11, 2012, 112502) is extended to allow the computation of multi-region, relaxed magnetohydrodynamics (MRxMHD) equilibria at prescribed toroidal current profile. Toroidal currents are expressed in the framework of the MRxMHD theory, exhibiting spatial separation between pressure driven and externally driven currents. Additionally, analytical force balance derivatives at constant toroidal current are deployed in order to maintain SPEC's advantageous speed. The newly implemented capability is verified in screw pinch and classical stellarator geometries, and is applied to obtain the equilibrium$\beta$-limit of a classical stellarator without net toroidal currents. This new capability opens the possibility to study the effect of toroidal current on three-dimensional equilibria with the SPEC code.", journal = "J. Plasma Phys.", publisher = "Cambridge University Press (CUP)", volume = 87, number = 4, month = aug, year = 2021, language = "en" } @article{y2016_loizu_1, title = "Pressure-driven amplification and penetration of resonant magnetic perturbations", author = "Loizu, J and Hudson, S R and Helander, P and Lazerson, S A and Bhattacharjee, A", abstract = "We show that a resonant magnetic perturbation applied to the boundary of an ideal plasma screw-pinch equilibrium with nested surfaces can penetrate inside the resonant surface and into the core. The response is significantly amplified with increasing plasma pressure. We present a rigorous verification of nonlinear equilibrium codes against linear theory, showing excellent agreement.", journal = "Phys. Plasmas", publisher = "AIP Publishing", volume = 23, number = 5, pages = "055703", month = may, year = 2016, language = "en" } @article{y2015_loizu_1, title = "Existence of three-dimensional ideal-magnetohydrodynamic equilibria with current sheets", author = "Loizu, J and Hudson, S R and Bhattacharjee, A and Lazerson, S and Helander, P", abstract = "We consider the linear and nonlinear ideal plasma response to a boundary perturbation in a screw pinch. We demonstrate that three-dimensional, ideal-MHD equilibria with continuously nested flux-surfaces and with discontinuous rotational-transform across the resonant rational-surfaces are well defined and can be computed both perturbatively and using fully nonlinear equilibrium calculations. This rescues the possibility of constructing MHD equilibria with current sheets and continuous, smooth pressure profiles. The results predict that, even if the plasma acts as a perfectly conducting fluid, a resonant magnetic perturbation can penetrate all the way into the center of a tokamak without being shielded at the resonant surface.", journal = "Phys. Plasmas", publisher = "AIP Publishing", volume = 22, number = 9, pages = "090704", month = sep, year = 2015, language = "en" } @article{y2015_loizu, title = "Magnetic islands and singular currents at rational surfaces in three-dimensional magnetohydrodynamic equilibria", author = "Loizu, J and Hudson, S and Bhattacharjee, A and Helander, P", abstract = "Using the recently developed multiregion, relaxed MHD (MRxMHD) theory, which bridges the gap between Taylor's relaxation theory and ideal MHD, we provide a thorough analytical and numerical proof of the formation of singular currents at rational surfaces in non-axisymmetric ideal MHD equilibria. These include the force-free singular current density represented by a Dirac $\delta$-function, which presumably prevents the formation of islands, and the Pfirsch-Schl{\"u}ter 1/x singular current, which arises as a result of finite pressure gradient. An analytical model based on linearized MRxMHD is derived that can accurately (1) describe the formation of magnetic islands at resonant rational surfaces, (2) retrieve the ideal MHD limit where magnetic islands are shielded, and (3) compute the subsequent formation of singular currents. The analytical results are benchmarked against numerical simulations carried out with a fully nonlinear implementation of MRxMHD.", journal = "Phys. Plasmas", publisher = "AIP Publishing", volume = 22, number = 2, pages = "022501", month = feb, year = 2015, language = "en" } @article{y2020_loizu, title = "Direct prediction of nonlinear tearing mode saturation using a variational principle", author = "Loizu, J and Huang, Y-M and Hudson, S R and Baillod, A and Kumar, A and Qu, Z S", journal = "Phys. Plasmas", publisher = "AIP Publishing", volume = 27, number = 7, pages = "070701", month = jul, year = 2020, language = "en" } @article{y2023_loizu, title = "Nonlinear saturation of resistive tearing modes in a cylindrical tokamak with and without solving the dynamics", author = "Loizu, J and Bonfiglio, D", abstract = "We show that the saturation of resistive tearing modes in a cylindrical tokamak, as well as the corresponding island width, can be directly calculated with a magnetohydrodynamics (MHD) equilibrium code without solving the dynamics and without considering resistivity. The results are compared with initial-value resistive MHD simulations and to an analytical nonlinear theory. For small enough islands, the agreement is remarkable. For sufficiently large islands, the equilibrium calculations, which assume a flat current profile inside the island, overestimate the saturation amplitude. On the other hand, excellent agreement between nonlinear resistive MHD simulations and nonlinear theory is observed for all the considered tearing unstable equilibria.", journal = "J. Plasma Phys.", publisher = "Cambridge University Press (CUP)", volume = 89, number = 5, month = oct, year = 2023, language = "en" } @article{y2017_loizu, title = "Equilibrium \textit{$\beta$}-limits in classical stellarators", author = "Loizu, J and Hudson, S R and N{\"u}hrenberg, C and Geiger, J and Helander, P", abstract = "A numerical investigation is carried out to understand the equilibrium $\unicode[STIX]\{x1D6FD\}$-limit in a classical stellarator. The stepped-pressure equilibrium code (Hudson et al., Phys. Plasmas, vol. 19 (11), 2012) is used in order to assess whether or not magnetic islands and stochastic field-lines can emerge at high $\unicode[STIX]\{x1D6FD\}$. Two modes of operation are considered: a zero-net-current stellarator and a fixed-iota stellarator. Despite the fact that relaxation is allowed (Taylor, Rev. Mod. Phys., vol. 58 (3), 1986, pp. 741--763), the former is shown to maintain good flux surfaces up to the equilibrium $\unicode[STIX]\{x1D6FD\}$-limit predicted by ideal-magnetohydrodynamics (MHD), above which a separatrix forms. The latter, which has no ideal equilibrium $\unicode[STIX]\{x1D6FD\}$-limit, is shown to develop regions of magnetic islands and chaos at sufficiently high $\unicode[STIX]\{x1D6FD\}$, thereby providing a `non-ideal $\unicode[STIX]\{x1D6FD\}$-limit'. Perhaps surprisingly, however, the value of $\unicode[STIX]\{x1D6FD\}$ at which the Shafranov shift of the axis reaches a fraction of the minor radius follows in all cases the scaling laws predicted by ideal-MHD. We compare our results to the High-Beta-Stellarator theory of Freidberg (Ideal MHD, 2014, Cambridge University Press) and derive a new prediction for the non-ideal equilibrium $\unicode[STIX]\{x1D6FD\}$-limit above which chaos emerges.", journal = "J. Plasma Phys.", publisher = "Cambridge University Press (CUP)", volume = 83, number = 6, month = dec, year = 2017, language = "en" } @article{y2020_hudson, doi = {10.1088/1361-6587/ab9a61}, url = {https://dx.doi.org/10.1088/1361-6587/ab9a61}, year = {2020}, month = {jul}, publisher = {IOP Publishing}, volume = {62}, number = {8}, pages = {084002}, author = {S R Hudson and J Loizu and C Zhu and Z S Qu and C Nührenberg and S Lazerson and C B Smiet and M J Hole}, title = {Free-boundary MRxMHD equilibrium calculations using the stepped-pressure equilibrium code}, journal = {Plasma Physics and Controlled Fusion}, abstract = {The stepped-pressure equilibrium code (SPEC) (Hudson et al 2012 Phys. Plasmas 19, 112 502) is extended to enable free-boundary multi-region relaxed magnetohydrodynamic (MRxMHD) equilibrium calculations. The vacuum field surrounding the plasma inside an arbitrary 'computational boundary', is computed, and the virtual-casing principle is used iteratively to compute the normal field on so that the equilibrium is consistent with an externally produced magnetic field. Recent modifications to SPEC are described, such as the use of Chebyshev polynomials to describe the radial dependence of the magnetic vector potential, and a variety of free-boundary verification calculations are presented.} } @article{y2023_baillod, title = "{Equilibrium\textit{$\beta$}-limits} dependence on bootstrap current in classical stellarators", author = "Baillod, A and Loizu, J and Qu, Z S and Arbez, H P and Graves, J P", abstract = "While it is important to design stellarators with high magnetohydrodynamic stability$\beta$-limit, it is also crucial to ensure that good magnetic surfaces exist in a large range of$\beta$values. As$\beta$increases, pressure-driven currents perturb the vacuum magnetic field and often lead to the emergence of magnetic field line chaos, which can worsen the confinement and is the cause of another kind of$\beta$-limit, the so-called equilibrium$\beta$-limit. In this paper, we explore numerically the dependence of the equilibrium$\beta$-limit on the bootstrap current strength in a classical stellarator geometry using the stepped pressure equilibrium code. We develop a diagnostic to determine whether or not magnetic islands are expected to participate significantly to radial transport, and we build an analytical model to predict the expected equilibrium$\beta$-limit, which recovers the main features of the numerical results. This research opens the possibility to include additional targets in stellarator optimization functions, provides additional understanding on the existence of magnetic surfaces at large$\beta$, and is a step forward in the understanding of the equilibrium$\beta$-limit.", journal = "J. Plasma Phys.", publisher = "Cambridge University Press (CUP)", volume = 89, number = 5, month = oct, year = 2023, language = "en" } @article{y2019_loizu, title = "Multi-region relaxed magnetohydrodynamic stability of a current sheet", author = "Loizu, J and Hudson, S R", abstract = "It is shown that the resistive magnetohydrodynamic stability of a slab force-free current sheet can be calculated using the variational principle of multi-region relaxed magnetohydrodynamics and that the corresponding stability boundary is in exact agreement with linear tearing mode theory.", journal = "Phys. Plasmas", URL = {https://doi.org/10.1063/1.5091765}, publisher = "AIP Publishing", volume = 26, number = 3, pages = "030702", month = mar, year = 2019, language = "en" } @article{y2024_liu, title = "Effects of magnetic helicity on {3D} equilibria and self-organized states in {KTX} reversed field pinch", author = "Liu, Ke and Yu, Guodong and Huang, Yuhua and Mao, Wenzhe and Xie, Yidong and Nie, Xianyi and Li, Hong and Lan, Tao and Xie, Jinlin and Ding, Weixing and Liu, Wandong and Zhuang, Ge and Zhu, Caoxiang", URL = {https://doi.org/10.1088/1741-4326/ad39d9}, abstract = "Abstract The reversed field pinch (RFP) is a toroidal magnetic configuration in which plasmas can spontaneously transform into different self-organized states. Among various states, the `quasi-single-helical' (QSH) state has a dominant component for the magnetic field and significantly improves confinement. Many theoretical and experimental efforts have investigated the transitions among different states. This paper employs the multi-region relaxed magnetohydrodynamic model to study the properties of QSH and other states. The stepped-pressure equilibrium code (SPEC) is used to compute MHD equilibria for the Keda Torus eXperiment (KTX). The toroidal volume of KTX is partitioned into two subvolumes by an internal transport barrier. The geometry of this barrier is adjusted to achieve force balance across the interface, ensuring that the plasma in each subvolume is force-free and that magnetic helicity is conserved. By varying the parameters, we generate distinct self-organized states in KTX. Our findings highlight the crucial role of magnetic helicity in shaping these states. In states with low magnetic helicity in both subvolumes, the plasma exhibits axisymmetric behavior. With increasing core helicity, the plasma gradually transforms from an axisymmetric state to a double-axis helical state and finally to a single-helical-axis state. Elevated core magnetic helicity leads to a more pronounced dominant mode of the boundary magnetic field and a reduced core magnetic shear. This is consistent with previous experimental and numerical results in other RFP devices. We find a linear relationship between the plasma current and helicity in different self-organized states. Our findings suggest that KTX may enter the QSH state when the toroidal current reaches 0.72 MA. This study demonstrates that the stellarator equilibrium code SPEC unveils crucial RFP equilibrium properties, rendering it applicable to a broad range of RFP devices and other toroidal configurations.", journal = "Nucl. Fusion", publisher = "IOP Publishing", volume = 64, number = 5, pages = "056037", month = may, year = 2024, copyright = "http://creativecommons.org/licenses/by/4.0" }