Additivity of relative magnetic helicity in finite volumes

Abstract

Context. Relative magnetic helicity is conserved by magneto-hydrodynamic evolution even in the presence of moderate resistivity. For that reason, it is often invoked as the most relevant constraint on the dynamical evolution of plasmas in complex systems, such as solar and stellar dynamos, photospheric flux emergence, solar eruptions, and relaxation processes in laboratory plasmas. However, such studies often indirectly imply that relative magnetic helicity in a given spatial domain can be algebraically split into the helicity contributions of the composing subvolumes, in other words that it is an additive quantity. A limited number of very specific applications have shown that this is not the case. Aims. Progress in understanding the nonadditivity of relative magnetic helicity requires removal of restrictive assumptions in favor of a general formalism that can be used in both theoretical investigations and numerical applications. Methods. We derive the analytical gauge-invariant expression for the partition of relative magnetic helicity between contiguous finite volumes, without any assumptions on either the shape of the volumes and interface, or the employed gauge. Results. We prove the nonadditivity of relative magnetic helicity in finite volumes in the most general, gauge-invariant formalism, and verify this numerically. We adopt more restrictive assumptions to derive known specific approximations, which yields a unified view of the additivity issue. As an example, the case of a flux rope embedded in a potential field shows that the nonadditivity term in the partition equation is, in general, non-negligible. Conclusions. The nonadditivity of relative magnetic helicity can potentially be a serious impediment to the application of relative helicity conservation as a constraint on the complex dynamics of magnetized plasmas. The relative helicity partition formula can be applied to numerical simulations to precisely quantify the effect of nonadditivity on global helicity budgets of complex physical processes.