On the Helicity of Open Magnetic Fields

Abstract

We reconsider the topological interpretation of magnetic helicity for magnetic fields in open domains, and relate this to the relative helicity. Specifically, our domains stretch between two parallel planes, and each of these ends may be magnetically open. It is demonstrated that, while the magnetic helicity is gaugedependent, its value in any gauge may be physically interpreted as the average winding number among all pairs of field lines with respect to some orthonormal frame field. In fact, the choice of gauge is equivalent to the choice of reference field in the relative helicity, meaning that the magnetic helicity is no less physically meaningful. We prove that a particular gauge always measures the winding with respect to a fixed frame, and propose that this is normally the best choice. For periodic fields, this choice is equivalent to measuring relative helicity with respect to a potential reference field. But for aperiodic fields, we show that the potential field can be twisted. We prove by construction that there always exists a possible untwisted reference field.

Relative magnetic helicity as a diagnostic of solar eruptivity

“…For relative magnetic helicity to be gauge invariant, the reference field must have the same distribution of the normal component of the studied field B along the surface. A classical choice, adopted here, is to use the potential field B p as the reference field (see Prior & Yeates 2014, for a possible different class of reference field). As in Valori et al (2012), we use here the definition of relative magnetic helicity from Finn & Antonsen (1985):…”

Evolution of field line helicity during magnetic reconnection

“…It is therefore desirable to restrict oneself to gauges that measure winding with respect to a fixed basis, which makes A gauge invariant for a given frame. Furthermore, it has been shown that the choice of basis used to measure winding of field lines corresponds directly to a choice of reference field for relative helicity, 7 so there is an equivalence between the relative helicity and winding interpretations.…”

“…(9), hence the U terms cancel for every field line. Physically, this restriction ensures that A measures winding with respect to a time-independent frame, eliminating nonphysical changes, 7 or equivalently that A is a field line helicity for relative helicity with a time-independent reference field. 31 The W terms correspond to a net voltage drop along the field line.…”

“…3 More precisely, total magnetic helicity (the volume integral ofà ÁB whereà is the vector potential and B ¼ r Âà is the magnetic field) is an ideal MHD invariant for magnetically closed domains, 4 which is readily extended to magnetically open domains either as total relative magnetic helicity 5,6 or by an appropriate restriction of the gauge of the vector potential. 7 Ideal invariance holds because ideal evolutions neither create nor destroy magnetic helicity and are unable to transport helicity across the magnetic field. In non-ideal MHD, exact conservation of magnetic helicity breaks down but magnetic reconnection at high magnetic Reynolds number nonetheless conserves total (relative) magnetic helicity very well.…”

Magnetic Helicity Estimations in Models and Observations of the Solar Magnetic Field. III. Twist Number Method

“…However, magnetic helicity does have a local density per elementary flux tube, namely, the field line helicity defined as the integral of A along a magnetic field line (Yeates & Hornig 2014Russell et al 2015). Besides the relative magnetic helicity in Equation (1), there are some other expressions and interpretations of the magnetic helicity (Jensen & Chu 1984;Hornig 2006;Low 2006Low , 2011Longcope & Malanushenko 2008;Prior & Yeates 2014). Here, we only focus on the relative magnetic helicity.…”