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

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

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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.

Original languageEnglish (US)
Article number034008
JournalPlasma Physics and Controlled Fusion
Issue number3
StatePublished - Mar 2023

All Science Journal Classification (ASJC) codes

  • Nuclear Energy and Engineering
  • Condensed Matter Physics


  • asymptotic analysis
  • boundary layer
  • current singularity
  • magnetohydrodynamic equilibrium
  • resonant magnetic perturbation


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