Energy Bounds from Relative Magnetic Helicity in Spherical Shells

TY - JOUR

T1 - Energy Bounds from Relative Magnetic Helicity in Spherical Shells

AU - Yeates, Anthony R.

AU - Hornig, Gunnar

N1 - © 2026. The Author(s).

PY - 2026/3/1

Y1 - 2026/3/1

N2 - 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.

AB - 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.

KW - Solar magnetic fields

U2 - 10.3847/1538-4357/ae30dd

DO - 10.3847/1538-4357/ae30dd

M3 - Article

SN - 0004-637X

VL - 999

JO - The Astrophysical Journal

JF - The Astrophysical Journal

IS - 1

M1 - 46

ER -