TY - JOUR

T1 - Structure of pressure-gradient-driven current singularity in ideal magnetohydrodynamic equilibrium

AU - Huang, Yi Min

AU - Zhou, Yao

AU - Loizu, Joaquim

AU - Hudson, Stuart

AU - Bhattacharjee, Amitava

N1 - Funding Information:
We thank Dr Greg Hammett for beneficial discussion. This research was supported by the U.S. Department of Energy under Contract No. DE-AC02-09CH11466 and by a grant from the Simons Foundation/SFARI (560651, A B). Part of this work has been carried out within the framework of the EUROfusion Consortium and has received funding from the Euratom research and training programme 2014–2018 and 2019–2020 under Grant Agreement No. 633053. The views and opinions expressed herein do not necessarily reflect those of the European Commission. Y Z was sponsored by Shanghai Pujiang Program under Grant No. 21PJ1408600. Part of the numerical calculations were performed with computers at the National Energy Research Scientific Computing Center.
Funding Information:
We thank Dr Greg Hammett for beneficial discussion. This research was supported by the U.S. Department of Energy under Contract No. DE-AC02-09CH11466 and by a grant from the Simons Foundation/SFARI (560651, A B). Part of this work has been carried out within the framework of the EUROfusion Consortium and has received funding from the Euratom research and training programme 2014-2018 and 2019-2020 under Grant Agreement No. 633053. The views and opinions expressed herein do not necessarily reflect those of the European Commission. Y Z was sponsored by Shanghai Pujiang Program under Grant No. 21PJ1408600. Part of the numerical calculations were performed with computers at the National Energy Research Scientific Computing Center.
Publisher Copyright:
© 2023 The Author(s). Published by IOP Publishing Ltd.

PY - 2023/3

Y1 - 2023/3

N2 - Singular currents typically appear on rational surfaces in non-axisymmetric ideal magnetohydrodynamic (MHD) equilibria with a continuum of nested flux surfaces and a continuous rotational transition. These currents have two components: a surface current (Dirac δ-function in flux surface labeling) that prevents the formation of magnetic islands, and an algebraically divergent Pfirsch-Schlüter current density when a pressure gradient is present across the rational surface. On flux surfaces adjacent to the rational surface, the traditional treatment gives the Pfirsch-Schlüter current density scaling as J ∼ 1 / Δ ι , where Δ ι is the difference of the rotational transform relative to the rational surface. If the distance s between flux surfaces is proportional to Δ ι , the scaling relation J ∼ 1 / Δ ι ∼ 1 / s will lead to a paradox that the Pfirsch-Schlüter current is not integrable. In this work, we investigate this issue by considering the pressure-gradient-driven singular current in the Hahm-Kulsrud-Taylor problem, which is a prototype for singular currents arising from resonant magnetic perturbations. We show that not only the Pfirsch-Schlüter current density but also the diamagnetic current density are divergent as ∼ 1 / Δ ι . However, due to the formation of a Dirac δ-function current sheet at the rational surface, the neighboring flux surfaces are strongly packed with s ∼ ( Δ ι ) 2 . Consequently, the singular current density J ∼ 1 / s , making the total current finite, thus resolving the paradox. Furthermore, the strong packing of flux surfaces causes a steepening of the pressure gradient near the rational surface, with ∇ p ∼ d p / d s ∼ 1 / s . In general non-axisymmetric MHD equilibrium, contrary to Grad’s conjecture that the pressure profile is flat around densely distributed rational surfaces, our result suggests a pressure profile that densely steepens around them.

AB - Singular currents typically appear on rational surfaces in non-axisymmetric ideal magnetohydrodynamic (MHD) equilibria with a continuum of nested flux surfaces and a continuous rotational transition. These currents have two components: a surface current (Dirac δ-function in flux surface labeling) that prevents the formation of magnetic islands, and an algebraically divergent Pfirsch-Schlüter current density when a pressure gradient is present across the rational surface. On flux surfaces adjacent to the rational surface, the traditional treatment gives the Pfirsch-Schlüter current density scaling as J ∼ 1 / Δ ι , where Δ ι is the difference of the rotational transform relative to the rational surface. If the distance s between flux surfaces is proportional to Δ ι , the scaling relation J ∼ 1 / Δ ι ∼ 1 / s will lead to a paradox that the Pfirsch-Schlüter current is not integrable. In this work, we investigate this issue by considering the pressure-gradient-driven singular current in the Hahm-Kulsrud-Taylor problem, which is a prototype for singular currents arising from resonant magnetic perturbations. We show that not only the Pfirsch-Schlüter current density but also the diamagnetic current density are divergent as ∼ 1 / Δ ι . However, due to the formation of a Dirac δ-function current sheet at the rational surface, the neighboring flux surfaces are strongly packed with s ∼ ( Δ ι ) 2 . Consequently, the singular current density J ∼ 1 / s , making the total current finite, thus resolving the paradox. Furthermore, the strong packing of flux surfaces causes a steepening of the pressure gradient near the rational surface, with ∇ p ∼ d p / d s ∼ 1 / s . In general non-axisymmetric MHD equilibrium, contrary to Grad’s conjecture that the pressure profile is flat around densely distributed rational surfaces, our result suggests a pressure profile that densely steepens around them.

KW - asymptotic analysis

KW - boundary layer

KW - current singularity

KW - magnetohydrodynamic equilibrium

KW - resonant magnetic perturbation

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UR - http://www.scopus.com/inward/citedby.url?scp=85148663924&partnerID=8YFLogxK

U2 - 10.1088/1361-6587/acb382

DO - 10.1088/1361-6587/acb382

M3 - Article

AN - SCOPUS:85148663924

SN - 0741-3335

VL - 65

JO - Plasma Physics and Controlled Fusion

JF - Plasma Physics and Controlled Fusion

IS - 3

M1 - 034008

ER -