Abstract
Context. Magnetic helicity is an important quantity in studies of magnetized plasmas as it provides a measure of the geometrical complexity of the magnetic field in a given volume. A more detailed description of the spatial distribution of magnetic helicity is given by the field line helicity, which expresses the amount of helicity associated to individual field lines rather than in the full analysed volume. Aims. Magnetic helicity is not a gauge-invariant quantity in general, unless it is computed with respect to a reference field, yielding the so-called relative magnetic helicity. The field line helicity corresponding to the relative magnetic helicity has only been examined under specific conditions so far. This work aims to define the field line helicity corresponding to relative magnetic helicity in the most general way. In addition to its general form, we provide the expression for the relative magnetic field line helicity in a few commonly used gauges, and reproduce known results as a limit of our general formulation. Methods. By starting from the definition of relative magnetic helicity, we derived the corresponding field line helicity, and we noted the assumptions on which it is based. Results. We checked that the developed quantity reproduces relative magnetic helicity by using three different numerical simulations. For these cases we also show the morphology of field line helicity in the volume, and on the photospheric plane. As an application to solar situations, we compared the morphology of field line helicity on the photosphere with that of the connectivity-based helicity flux density in two reconstructions of an active region’s magnetic field. We discuss how the derived relative magnetic field line helicity has a wide range of applications, notably in solar physics and magnetic reconnection studies.
Highlights
Magnetic helicity is an important quantity in studies of the Sun, as well as in any other environment that can be characterized by magnetohydrodynamics (MHD)
In order to remove this limitation, the helicity can be defined with respect to a reference magnetic field. The first such relative magnetic helicity was introduced by Berger & Field (1984), and a gauge-independent version of it was later defined by Finn & Antonsen (1985). In this definition of relative magnetic helicity, the obtained helicity values do not depend on the gauges of the vector potentials of the original and reference fields as long as the normal components of the two fields coincide on the boundary of the volume
The relative magnetic field line helicity (RMFLH) can always be computed from one of the Eqs. (8)–(10), but its flux-weighted integral given by Eq (11) will be equal to the relative magnetic helicity (RMH) only if all field lines are connected to the boundary at both ends, that is, when no closed and/or ergodic field lines are present in the volume
Summary
Magnetic helicity is an important quantity in studies of the Sun, as well as in any other environment that can be characterized by magnetohydrodynamics (MHD). A more theoretical study of the properties of FLH can be found in Aly (2018) In most of these applications a special gauge for the vector potentials was used to simplify calculations, which led to inaccurate helicity budgets when the flux-weighted integral of FLH along the boundary was taken. In an alternative method that approximates helicity through its twist, Malanushenko et al (2011) compute RMH by matching the shapes of coronal loops with field lines of a linear force-free field Another method that uses the magnetic connectivity to infer a helicity-related quantity is the connectivity-based helicity flux density method (Pariat et al 2005; Dalmasse et al 2014).
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