Energy Bounds from Relative Magnetic Helicity in Spherical Shells

TL;DR

This paper derives new energy bounds from relative magnetic helicity in spherical shells, providing insights into solar corona dynamics and introducing the concept of unsigned helicity as a new invariant.

Contribution

It presents an intrinsic expression for relative helicity in spherical shells and proves new lower bounds for magnetic energy, including a novel unsigned helicity invariant.

Findings

Non-zero relative helicity implies lower bounds for magnetic energy.

Derived a stronger energy bound using spatial decomposition of helicity.

Illustrated bounds with analytical and data-driven solar corona models.

Abstract

Relative magnetic helicity is commonly used in solar physics to avoid the well known gauge ambiguity of standard magnetic helicity in magnetically open domains. But its physical interpretation is difficult owing to the invocation of a reference field. For the specific case of spherical shell domains (with potential reference field), relative helicity may be written intrinsically in terms of the magnetic field alone, without the need to calculate the reference field or its vector potential. We use this intrinsic expression to prove that non-zero relative helicity implies lower bounds for both magnetic energy and free magnetic energy, generalizing the important Arnol'd inequality known for closed-field magnetic helicity. Further, we derive a stronger energy bound by spatially decomposing the relative helicity over a magnetic partition of the domain to obtain a new ideal invariant which we call unsigned helicity. The bounds are illustrated with analytical linear force-free fields (that maximize relative helicity for given boundary conditions) as well as a non-potential data-driven model of the solar corona. These bounds confirm that both relative helicity and the unsigned helicity can influence the dynamics in the solar corona.