Field linkage and magnetic helicity density

Magnetic helicity is a fundamental property of magnetic fields that measures the amount of linkage and twist of field lines within a given volume. Since it is exactly conserved in ideal MHD and highly conserved for high magnetic Reynolds numbers in general \citep{Woltjer1958,Taylor1974}, helicity is an important factor when attempting to understand how magnetic fields are generated and evolve \citep[e.g.][]{Brandenburg2005,Chatterjee2011,Pipin2019}.  Until recently, this could only be measured for the Sun \citep[e.g. reviews by][]{Demoulin2007,Demoulin2009}. We can, however, now map all three components of the large-scale magnetic field at the surfaces of stars using the spectropolarimetric technique of Zeeman-Doppler imaging \citep{Semel1989}. 

These magnetic field maps now exist for a large enough sample of stars that trends with stellar mass and rotation period have become apparent \citep{donati2009}. In particular, it appears that magnetic fields show different strengths and topologies in the mass ranges above and below $\sim$ 0.5 M$_\odot$, which \textbf{is believed to correspond to the onset of the transition from partially to fully convective interiors}. Rapidly-rotating stars in the mass range above $\sim$ 0.5 M$_\odot$ tend to have fields that are predominantly toroidal \citep{Donati2008b}. The stronger the toroidal field, the more likely it is to be axisymmetric \citep{See2015}. In the mass range below $\sim$ 0.5 M$_\odot$, stars show predominantly axisymmetric poloidal fields. For the lowest masses, however, a bimodal behaviour is found, such that stars may have strong, predominantly axisymmetric poloidal fields, or much weaker, non-axisymmetric poloidal fields \citep{Donati2008b,Morin2008b,donati2009,Morin2010}.

This difference in magnetic fields in stars that are partially or fully convective is also apparent in their photospheric helicity densities. Using observations of 51 stars, \citet{Lund2020} found that the helicity density scales with the toroidal energy according to $|\langle{h\,}\rangle|$ $\propto$ $\langle{\rm{B_{tor}}^2_{}\,\rangle}^{0.86\,\pm\,0.04}$. The scaling with the poloidal energy is more complex, however, revealing two groups with different behaviours. Specifically, stars less massive than $\sim$ 0.5 M$_\odot$ appear to have an excess of poloidal energy when compared to more massive stars with similar helicity densities. It appears that stars with different internal structures and different total magnetic energies may nonetheless generate magnetic fields with the same helicity density at their surfaces. The aim of this paper is to explore the nature of this division and the types of flux linkage that support the measured helicity densities. In order to do that, we have developed a novel method of visualising the linkages of different field components across the surfaces of stars.