Spatial scales and locality of magnetic helicity

Abstract

Context. Magnetic helicity is approximately conserved in resistive MHD models. It quantifies the entanglement of the magnetic field within the plasma. The transport and removal of helicity is crucial in both the dynamo in the solar interior and active region evolution in the solar corona. This transport typically leads to highly inhomogeneous distributions of entanglement.Aims. There exists no consistent systematic means of decomposing helicity over varying spatial scales and in localised regions. Spectral helicity decompositions can be used in periodic domains and is fruitful for the analysis of homogeneous phenomena. This paper aims to develop methods for analysing the evolution of magnetic field topology in non-homogeneous systems. Methods. We apply a multiresolution wavelet decomposition to the magnetic field and demonstrate how it can be applied to various quantities associated with magnetic helicity, including the field line helicity. We use a geometrical definition of helicity which allows these quantities to be calculated for fields with arbitrary boundary conditions. Results. It is shown that the multiresolution decomposition of helicity has the crucial property of local additivity and demonstrate a general linear energy-topology conservation law which is a significant generalisation of the two point correlation decomposition used in the analysis of homogeneous turbulence and periodic fields. The localisation property of the wavelet representation is shown to characterise inhomogeneous distributions which a Fourier representation cannot. Using an analytic representation of a resistive braided field relaxation we demonstrate a clear correlation between the variations in energy at various length scales and the variations in helicity at the same spatial scales. Its application to helicity flows in a surface flux transport model show how various contributions to the global helicity input from active region field evolution and polar field development are naturally separated by this representation. Conclusions. The multiresolution wavelet decomposition can be used to analyse the evolution of helicity in magnetic fields in a manner which is consistently additive. This method has the advantage over more established spectral methods in that it clearly characterises the inhomogeneous nature of helicity flows where spectral methods cannot. Further its applicability in aperiodic models significantly increase the range of potential applications.

Spherical winding and helicity

“…Equation (39) links winding helicity to a quantity defined purely from the B itself without using a vector potential. It also gives H W (B) a computational advantage, recently exploited in the wavelet analysis by Prior et al [32]. Such a geometrical interpretation for H W (B) generalises the case of gauge-invariant, closedfield helicity as the average flux-weighted pairwise field line linking [1,29].…”

“…with dχ ′ dt similarly defined in x-centred winding basis as in equation (29). Note that both the Green's function G(x, x ′ ) and the winding helicity density H(B; x, x ′ ) are symmetric about x and x ′ , but neither equation ( 31) nor (32) are. To restore this symmetry, we further define their arithmetic average as the pairwise winding rate of curves x and x ′ , namely, 1 2 (dω/dt + dω ′ /dt).…”

Magnetic Helicity Flux Oscillations in the Atmospheres of Flaring and Nonflaring Active Regions

“…A recent theoretical paper, based on multiresolution wavelet decomposition, by Prior et al (2020) reported that spatial scales of magnetic field helicity is consistently additive. This theory is supported by, or at least is consistent with, Korsós et al (2020b) and Soós et al (2022) in their observation-based helicity analyses.…”

On the differences in the periodic behaviour of magnetic helicity flux in flaring active regions with and without X-class events

“…Recently, Prior et al (2020) showed that the multiresolution wavelet decomposition is a useful tool to analyze the magnetic helicity. Based on their theoretical work, Korsós et al (2020) investigated the dynamic evolution of emergence (EM), shearing (SH), and total (T) magnetic helicity flux terms using a wavelet analysis in the case of three flaring and three non-flaring ARs.…”