Disentangling the Entangled Linkages of Relative Magnetic Helicity

Magnetic helicity, $H$, measures magnetic linkages in a volume. The early theoretical development of helicity focused on magnetically closed systems in $\mathcal{V}$ bounded by $\mathcal{S}$. For magnetically closed systems, $\mathcal{V}\in\mathbb{R}^3=\mathcal{V}+\mathcal{V}^*$, no magnetic flux threads the boundary, $\hat{\boldsymbol{n}}\cdot\boldsymbol{B}|_\mathcal{S}=0$. Berger and Field (1984) and Finn and Antonsen (1985) extended the definition of helicity to relative helicity, $\mathcal{H}$, for magnetically open systems where magnetic flux may thread the boundary. Berger (1999,2003) expressed this relative helicity as two gauge invariant terms that describe the self helicity of magnetic field that closes inside $\mathcal{V}$ and the mutual helicity between the magnetic field that threads the boundary $\mathcal{S}$ and the magnetic field that closes inside $\mathcal{V}$. The total magnetic field that permeates $\mathcal{V}$ entangles magnetic fields that are produced by current sources $\boldsymbol{J}$ in $\mathcal{V}$ with magnetic fields that are produced by current sources $\boldsymbol{J}^*$ in $\mathcal{V}^*$. Building on this fact, we extend Berger's expressions for relative magnetic helicity to eight gauge invariant quantities that simultaneously characterize both of these self and mutual helicities and attribute their origins to currents $\boldsymbol{J}$ in $\mathcal{V}$ and/or $\boldsymbol{J}^*$ in $\mathcal{V}^*$, thereby disentangling the domain of origin for these entangled linkages. We arrange these eight terms into novel expressions for internal and external helicity (self) and internal-external helicity (mutual) based on their domain of origin. The implications of these linkages for interpreting magnetic energy is discussed and new boundary observables are proposed for tracking the evolution of the field that threads the boundary.