Magnetic helicity in a two-flux partitioning of an ideal hydromagnetic fluid

A primitive form of magnetic helicity is constructed that (1) recovers the classical helicity of a wholly contained magnetic field, as well as the Berger-Field relative helicity of a partially contained magnetic field, and (2) generalizes the infinity of global, helicity-like invariants derived by Bhattacharjee & Dewar for a plasma approaching toroidal magnetostatic equilibrium. This construction is based on a general partitioning of an ideal hydromagnetic fluid into disjoint, infinitesimally thin, toroidal subvolumes using a two-flux description of the embedded magnetic field. Each of these toroidal subvolumes of fluid is endowed with a gauge-independent magnetic helicity conserved during its ideal Lagrangian evolution. This conservation law constitutes an equivalent statement of the frozen-in condition. The Chandrasekhar-Kendall solenoidal representation of a magnetic field is conceptually related to the two-flux description in the basic theory developed. Magnetic helicity-related phenomena in the solar corona are briefly discussed to provide an astrophysical context for this basic development, postponing proper applications to the papers to follow.