Formation of current singularity in a topologically constrained plasma

Recently a variational integrator for ideal magnetohydrodynamics in Lagrangian labeling has been developed. Its built-in frozen-in equation makes it optimal for studying current sheet formation. We use this scheme to study the Hahm-Kulsrud-Taylor problem, which considers the response of a 2D plasma magnetized by a sheared field under sinusoidal boundary forcing. We obtain an equilibrium solution that preserves the magnetic topology of the initial field exactly, with a fluid mapping that is non-differentiable. Unlike previous studies that examine the current density output, we identify a singular current sheet from the fluid mapping. These results are benchmarked with a constrained Grad-Shafranov solver. The same signature of current singularity can be found in other cases with more complex magnetic topologies.

Figure 1

Equilibrium field line configuration in the vicinity of the neutral line, and the entire domain (inset). The field lines appear equally spaced along $y=0$ near the neutral line.

Figure 2

Numerical solutions of $x(x_0,0)$ (a), $B_y(x,0)$ (inset of a), and $\partial y/\partial y_0{|}_{(x_0,0)}$ (b) for different resolutions (dotted lines). The converged parts agree with the results obtained with a constrained Grad-Shafranov solver (dashed lines). Near the neutral line, $x(x_0,0)$ and $\partial y/\partial y_0{|}_{(x_0,0)}$ show $x_0^2$ and $x_0^{-1}$ power laws, respectively, while $B_y(x,0)$ approaches a finite constant. The solutions do not converge for the few vertices closest to the neutral line. In the inset of (b), the final versus initial distance to $(0,0.5)$ for the vertices on the neutral line, i. e.,  $0.5-y(0,y_0)$ vs. $0.5-y_0$ for different resolutions are shown to not converge. Solutions with higher resolutions are shown in part.