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Magnetic helicity, $H$, measures magnetic linkages in a volume. The earlytheoretical 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 fluxthreads the boundary, $\hat{\boldsymbol{n}}\cdot\boldsymbol{B}|_\mathcal{S}=0$.Berger and Field (1984) and Finn and Antonsen (1985) extended the definition ofhelicity to relative helicity, $\mathcal{H}$, for magnetically open systemswhere magnetic flux may thread the boundary. Berger (1999,2003) expressed thisrelative helicity as two gauge invariant terms that describe the self helicityof magnetic field that closes inside $\mathcal{V}$ and the mutual helicitybetween the magnetic field that threads the boundary $\mathcal{S}$ and themagnetic field that closes inside $\mathcal{V}$. The total magnetic field thatpermeates $\mathcal{V}$ entangles magnetic fields that are produced by currentsources $\boldsymbol{J}$ in $\mathcal{V}$ with magnetic fields that areproduced by current sources $\boldsymbol{J}^*$ in $\mathcal{V}^*$. Building onthis fact, we extend Berger's expressions for relative magnetic helicity toeight gauge invariant quantities that simultaneously characterize both of theseself 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. Wearrange these eight terms into novel expressions for internal and externalhelicity (self) and internal-external helicity (mutual) based on their domainof origin. The implications of these linkages for interpreting magnetic energyis discussed and new boundary observables are proposed for tracking theevolution of the field that threads the boundary.

Disentangling the Entangled Linkages of Relative Magnetic Helicity