Abstract
Abstract Current helicity, H c , and magnetic helicity, H m , are two main quantities used to characterize magnetic fields. For example, such quantities have been widely used to characterize solar active regions and their ejecta (magnetic clouds). It is commonly assumed that H c and H m have the same sign, but this has not been rigorously addressed beyond the simple case of linear force-free fields. We aim to answer whether H m H c ≥ 0 in general, and whether it is true over some useful set of magnetic fields. This question is addressed analytically and with numerical examples. The main focus is on cylindrically symmetric straight flux tubes, referred to as flux ropes (FRs), using the relative magnetic helicity with respect to a straight (untwisted) reference field. Counterexamples with H m H c < 0 have been found for cylindrically symmetric FRs with finite plasma pressure, and for force-free cylindrically symmetric FRs in which the poloidal field component changes direction. Our main result is a proof that H m H c ≥ 0 is true for force-free cylindrically symmetric FRs where the toroidal field and poloidal field components are each of a single sign, and the poloidal component does not exceed the toroidal component. We conclude that the conjecture that current and magnetic helicities have the same sign is not true in general, but it is true for a set of FRs of importance to coronal and heliospheric physics.
Highlights
A helicity integral measures the linking of the flux of a divergence-free field, as was originally proven in the classical paper by Moffatt (1969) in the context of a vorticity field, and later made more precise by Arnold (2014)
Helicity has become widely applied to topics as diverse as magnetohydrodynamic (MHD) turbulence, magnetic dynamos, magnetic reconnection, turbulent relaxation (e.g., Taylor relaxation), accretion disk jets, coronal mass ejections (CMEs), coronal heating, solar filaments, activeregion sigmoids, accumulation of magnetic shear at polarity inversion lines (PILs), the solar wind, and planetary magnetospheres, which are well represented in the article collections of Brown et al (1999) and Buechner & Pevtsov (2003)
We present counterexamples that show HmHc 0 does not hold in general, even for straight, cylindrically symmetric force-free flux ropes (FRs). These examples and counterexamples motivate the introduction of two further conditions, and we prove in Section 5 that force-free cylindrically symmetric FRs have the same sign of current and magnetic helicities under assumptions of no field reversals and the poloidal component not exceeding the toroidal component
Summary
A helicity integral measures the linking of the flux of a divergence-free field, as was originally proven in the classical paper by Moffatt (1969) in the context of a vorticity field, and later made more precise by Arnold (2014). Helicity provides a mathematical toolset for interpreting the handedness of magnetic fields, which, in less mathematical form, has a history dating back at least a century, since it can be found in work by this journal’s founder, George Ellery Hale, and colleagues at the Mount Wilson Solar Observatory (e.g., Hale 1908; Hale et al 1919). With these strengths, helicity has become widely applied to topics as diverse as magnetohydrodynamic (MHD) turbulence, magnetic dynamos, magnetic reconnection, turbulent relaxation (e.g., Taylor relaxation), accretion disk jets, coronal mass ejections (CMEs), coronal heating, solar filaments, activeregion sigmoids, accumulation of magnetic shear at polarity inversion lines (PILs), the solar wind, and planetary magnetospheres, which are well represented in the article collections of Brown et al (1999) and Buechner & Pevtsov (2003)
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