Using Magnetic Helicity, Topology, and Geometry to Investigate Complex Magnetic Fields

Helicity and winding fluxes as indicators of twisted flux emergence

“…The first of these quantities, the helicity flux, describes the amount of relative magnetic helicity passing through the photosphere and is used widely in both theoretical and observational studies (e.g. Berger 1984, 1988, Démoulin and Berger 2003, Prior and MacTaggart 2019, Pariat 2019). The second quantity, the winding flux, is a purely topological quantity that represents the average winding of magnetic field lines.…”

On self and mutual winding helicity

“…where A 1 and A 2 are the vector potentials of B 1 and B 2 respectively. This measure of helicity has been used extensively in solar active region applications [9,19]. Although gauge invariant decompositions of relative helicity have been proposed [8], there is no general self-mutual decomposition for relative helicity as for classical helicity in (4).…”

Magnetic winding: what is it and what is it good for?

“…If around each loop we identify a solid magnetic torus, as in [11], then since the magnetic field is tangential to the toroidal boundary, we can consider the domain to be R 3 , where the magnetic fields in the tori are 'extended by zero' outside of the tori volumes [12]. Thus equation (1.4) can lead to classical helicity formula…”

“…Consider two linked loops (a pair of closed and linked magnetic field lines) given by dx/ds = B(x) and dy/ds = B(y) (the following can be extended to n loops and also ergodic field lines, but two loops will suffice for this demonstration). Then the Gauss linking number [10] is given by Lk(x, y) = 1 4π x(s) y(σ ) dx ds • dy dσ × x − y |x − y| 3 ds dσ .…”

“…However, equation (1.1) has been used heavily in solar physics to understand the topological properties of magnetic fields in the solar atmosphere. A recent review of calculating relative helicity in solar applications can be found in [3].…”