Solar Physics (2006) 233: 3–27 DOI: 10.1007/s11207-006-0010-z C  Springer 2006 BASIC PROPERTIES OF MUTUAL MAGNETIC HELICITY P. DEMOULIN and E. PARIAT Observatoire de Paris, LESIA, FRE 2461 (CNRS), F-92195 Meudon Principal Cedex, France (e-mails: pascal.demoulin@obspm.fr; etienne.pariat@obspm.fr) and M. A. BERGER Department of Mathematics, University College London (e-mail: m.berger@ucl.ac.uk) (Received 15 June 2005; accepted 27 September 2005) Abstract. We derive the magnetic helicity for configurations formed by flux tubes contained fully or only partially in the spatial domain considered (called closed and open configurations, respectively). In both cases, magnetic helicity is computed as the sum of mutual helicity over all possible pairs of magnetic flux tubes weighted by their magnetic fluxes. We emphasize that these mutual helicities have properties which are not those of mutual inductances in classical circuit theory. For closed configurations, the mutual helicity of two closed flux tubes is their relative winding around each other (known as the Gauss linkage number). For open configurations, the magnetic helicity is derived directly from the geometry of the interlaced flux tubes so it can be computed without reference to a ground state (such as a potential field). We derive the explicit expression in the case of a planar and spherical boundary. The magnetic helicity has two parts. The first one is given only by the relative positions of the flux tubes on the boundary. It is the only part if all flux tubes are arch-shaped. The second part counts the integer number of turns each pair of flux tubes wind about each other. This provides a general method to compute the magnetic helicity with discrete or continuous distributions of magnetic field. The method sets closed and open configurations on an equal level within the same theoretical framework. 1. Introduction Magnetic helicity quantifies how the magnetic field is sheared and/or twisted com- pared to its lowest energy state (potential field). Observations of sheared, and even helical, magnetic structures in the photosphere, corona and solar wind have attracted considerable attention, with the consequent interest in magnetic helicity studies (see reviews in Brown, Canfield, and Pevtsov, 1999; Berger, 2003). Stressed magnetic fields are often observed in association to flares, eruptive filaments, and coronal mass ejections (CMEs), but the precise role of magnetic helicity in such activity events still needs to be clarified. Magnetic helicity plays a key role in magnetohydrodynamics (MHD) because it is almost preserved on a timescale less than the global diffusion timescale (Berger, 1984). Its conservation defines a constraint to the magnetic field evolution; in particular a stressed magnetic field with finite total helicity cannot relax to a potential