Magnetohydrodynamical equilibria with current singularities and continuous rotational transform

TL;DR

This paper revisits the HKT problem using boundary-layer techniques to analyze resonant magnetic perturbations, demonstrating the validity of RDR's approach and showing the rotational transform remains continuous.

Contribution

It applies RDR's boundary-layer method to the HKT problem, clarifies the matching procedure, and confirms the continuous rotational transform contrary to recent claims.

Findings

Boundary-layer solution matches numerical results closely

DC current singularity appears at the resonant surface

Rotational transform remains continuous in the solution

Abstract

We revisit the Hahm-Kulsrud-Taylor (HKT) problem, a classic prototype problem for studying resonant magnetic perturbations and 3D magnetohydrodynamical equilibria. We employ the boundary-layer techniques developed by Rosenbluth, Dagazian, and Rutherford (RDR) for the internal $m=1$ kink instability, while addressing the subtle difference in the matching procedure for the HKT problem. Pedagogically, the essence of RDR's approach becomes more transparent in the simplified slab geometry of the HKT problem. We then compare the boundary-layer solution, which yields a "DC" current singularity at the resonant surface, to the numerical solution obtained using a flux-preserving Grad-Shafranov solver. The remarkable agreement between the solutions demonstrates the validity and universality of RDR's approach. In addition, we show that RDR's approach consistently preserves the rotational transform, which hence stays continuous, contrary to a recent claim that RDR's solution contains a discontinuity in the rotational transform.