Current singularities on rational surfaces including pressure effects

Non-axisymmetric ideal magnetohydrodynamic equilibria with a continuum of nested flux surfaces and a continuous rotational transform generically exhibit singular currents on rational surfaces. These currents have two components: a surface current ($\delta$-function in radius) that prevents the formation of a magnetic island and an algebraically divergent Pfirsch-Schl\"uter current when a pressure gradient is present across the rational surface. At an adjacent flux surface of the rational surface, the traditional treatment gives the Pfirsch-Schl\"uter current density to scale as $j \sim 1/\Delta \iota$, where $\Delta \iota$ is the difference of the rotational transform between the two flux surfaces. If the distance $x$ between flux surfaces is proportional to $\Delta \iota$, the scaling relation $j \sim 1/\Delta \iota \sim 1/x$ will lead to a paradox that the Pfirsch-Schl\"uter current is not integrable. In this work, we investigate this issue by considering the pressure-gradient-driven Pfirsch-Schl\"uter current in the Hahm-Kulsrud-Taylor problem, which is a prototype for singular currents arising from resonant perturbations. We show that because of the $\delta$-function current at the resonant surface, the neighboring flux surfaces are strongly packed with $x\sim (\Delta \iota)^2$. Consequently, the Pfirsch-Schl\"uter current $j \sim 1/\sqrt{x}$, making the total current finite, thus resolving the paradox.