  Relative magnetic helicity is a gauge invariant quantity suitable for the
study of the magnetic helicity content of heliospheric plasmas. Relative
magnetic helicity can be decomposed uniquely into two gauge invariant
quantities, the magnetic helicity of the non-potential component of the field,
and a complementary volume-threading helicity. Recent analysis of numerical
experiments simulating the generation of solar eruptions have shown that the
ratio of the non-potential helicity to the total relative helicity is a clear
marker of the eruptivity of the magnetic system, and that the high value of
that quantity could be a sufficient condition for the onset of the instability
generating the eruptions. The present study introduces the first analytical
examination of the time variations of these non-potential and volume-threading
helicities. The validity of the analytical formulas derived are confirmed with
analysis of three-dimensional (3D) magnetohydrodynamics (MHD) simulations of
solar coronal dynamics. Both the analytical investigation, and the numerical
application show that, unlike magnetic helicity, the non-potential and the
volume-threading helicities are not conserved quantities, even in the ideal MHD
regime. A term corresponding to the transformation between the non-potential
and volume-threading helicities frequently dominates their dynamics. This
finding has an important consequence for their estimation in the solar corona:
unlike with relative helicity, their volume coronal evolution cannot be
ascertained by the flux of these quantities through the volume's boundaries.
Only techniques extrapolating the 3D coronal field will enable both the proper
study of the non-potential and volume-threading helicities, and the
observational analysis of helicity-based solar-eruptivity proxies.
