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

Journal Article · 15 February 2023 · Plasma Physics and Controlled Fusion

DOI: https://doi.org/10.1088/1361-6587/acb382 · OSTI ID:1926781

Huang, Yi-Min ; Zhou, Yao ; Loizu, Joaquim; Hudson, Stuart ; Bhattacharjee, Amitava

Abstract 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 $J\sim 1/\mathrm{\Delta }\iota$ , where $\mathrm{\Delta }\iota$ is the difference of the rotational transform relative to the rational surface. If the distance s between flux surfaces is proportional to $\mathrm{\Delta }\iota$ , the scaling relation $J\sim 1/\mathrm{\Delta }\iota \sim 1/s$ 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 $\sim 1/\mathrm{\Delta }\iota$ . However, due to the formation of a Dirac δ -function current sheet at the rational surface, the neighboring flux surfaces are strongly packed with $s\sim (\mathrm{\Delta }\iota {)}^2$ . Consequently, the singular current density $J\sim 1/\sqrt{s}$ , 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 $\mathrm{\nabla }p\sim \mathrm{d}p/\mathrm{d}s\sim 1/\sqrt{s}$ . 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.

• Research Organization: Princeton Plasma Physics Laboratory (PPPL), Princeton, NJ (United States)
• Sponsoring Organization: Euratom Research and Training Programme; Shanghai Pujiang Program; Simons Foundation; USDOE
• Grant/Contract Number: AC02-09CH11466
• OSTI ID: 1926781
• Journal Information: Plasma Physics and Controlled Fusion, Journal Name: Plasma Physics and Controlled Fusion Journal Issue: 3 Vol. 65; ISSN 0741-3335
• Publisher: IOP PublishingCopyright Statement
• Country of Publication: United Kingdom
• Language: English