Spatial Scales and Locality of Magnetic Helicity: Part 1

TL;DR

This paper introduces a multiresolution wavelet method to decompose magnetic helicity across spatial scales, enabling detailed analysis of its distribution, transport, and conservation in resistive MHD models, with applications to solar magnetic phenomena.

Contribution

It develops a wavelet-based approach for local, scale-dependent decomposition of magnetic helicity, extending previous correlation-based methods and allowing analysis of inhomogeneous magnetic fields.

Findings

Wavelet decomposition provides local additivity of helicity.

A linear energy-topology conservation law is demonstrated.

Application reveals scale-dependent helicity variations in solar magnetic fields.

Abstract

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.There exists no consistent systematic means of decomposing helicity over varying spatial scales and in localised regions. 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. 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.