# Constructing current singularity in a 3D line-tied plasma

## TL;DR

This study investigates the potential formation of current singularities in 3D line-tied plasmas using a novel numerical method, extending previous 2D results and exploring the conditions under which singularities may occur.

## Contribution

It applies a variational Lagrangian method to 3D line-tied plasmas, providing new insights into current singularity formation and extending prior 2D analyses.

## Key findings

- Linear solutions are smooth for arbitrary system length.
- Finite amplitude solutions can become pathological in long systems.
- Scaling suggests nonlinear solutions may become singular at finite length.

## Abstract

We revisit Parker's conjecture of current singularity formation in 3D line-tied plasmas using a recently developed numerical method, variational integration for ideal magnetohydrodynamics in Lagrangian labeling. With the frozen-in equation built-in, the method is free of artificial reconnection, and hence it is arguably an optimal tool for studying current singularity formation. Using this method, the formation of current singularity has previously been confirmed in the Hahm--Kulsrud--Taylor problem in 2D. In this paper, we extend this problem to 3D line-tied geometry. The linear solution, which is singular in 2D, is found to be smooth for arbitrary system length. However, with finite amplitude, the linear solution can become pathological when the system is sufficiently long. The nonlinear solutions turn out to be smooth for short systems. Nonetheless, the scaling of peak current density versus system length suggests that the nonlinear solution may become singular at finite length. With the results in hand, we can neither confirm nor rule out this possibility conclusively, since we cannot obtain solutions with system length near the extrapolated critical value.