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spec_refs.bib
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@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{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"
}