The solar corona exhibits a rich variety of magnetic structures. In this regime, the magnetic field is able to reconnect, and releases energy even though the electrical resistivity in the surrounding plasma is small. The field line helicity (FLH) can be used to study the topology of magnetic field during relaxation. It is calculated as the vector potential integrated along the magnetic field lines, and can be interpreted as the average winding of magnetic flux along these field lines. Based on 3D numerical simulations, past studies [1,2] have hypothesised that the evolution of FLH behaves at leading order as advection of a 2D scalar field f by an unknown velocity field. In this study [3], we search for the simplest unstirred pattern that may be reached under incompressible advection, and compare the optimal unstirred state with the FLH of the relaxed state obtained from 3D simulations. We test two approaches: a variational method with minimal constraints, and a magnetic relaxation scheme where the effective velocity is determined explicitly by the pattern of f. Both methods achieve similar convergence for simple test cases. However, the magnetic relaxation method guarantees a monotonic decrease in the Dirichlet seminorm of f, and is numerically more robust. We therefore apply the latter method to two complex mixed patterns modelled on the FLH of 3D magnetic braids. We find that the unstirred state has a small number of large-scale regions determined by its initial topology. Interestingly, the effective velocity field used for unstirring also has the same large-scale topology as f. Our result supports the idea that advection is an important principle for FLH evolution.
References:
[1] Russell et al. (2015). Phys. Plasmas 22 (3), 032106.
[2] Yeates (2020). CISM International Centre for Mechanical Sciences, pp. 117–143. Springer International Publishing.
[3] Chen et al. (2021). J. Fluid Mech. (2021), vol. 911, A30.
