scalar potential given by the gauge condition (see ). Choosing the gauge so that the scalar potential vanishes, , the time evolution of magnetic helicity in a volume is given by: The dot product in the integrand of the first term is zero since is orthogonal to the cross product , and the second term can be integrated by parts to give where the second term is a surface integral over the boundary surface of the closed system. The dot product in the integrand of the first term is zero because is orthogonal to The second term also vanishes because motions inside the closed system cannot affect the vector potential outside, so that at the boundary surface since the magnetic vector potential is a continuous function. Therefore, and magnetic helicity is ideally conserved. In all situations where magnetic helicity is gauge invariant, magnetic helicity is ideally conserved without the need of the specific gauge choice Magnetic helicity remains conserved in a good approximation even with a small but finite resistivity, in which case magnetic reconnection dissipates energy. Inverse transfer property Small-scale helical structures tend to form larger and larger magnetic structures. This can be called an inverse transfer in Fourier space, as opposed to the (direct) energy cascade in threedimensional turbulent hydrodynamical flows. The possibility of such an inverse transfer has first been proposed by Uriel Frisch and collaborators and has been verified through many numerical experiments. As a consequence, the presence of magnetic helicity is a possibility to explain the existence and sustainment of large-scale magnetic structures in the Universe. An argument for this inverse transfer taken from is repeated here, which is based on the so-called "realizability condition" on the magnetic helicity Fourier spectrum (where is the Fourier coefficient at the wavevector of the magnetic field , and similarly for , the star denoting the complex conjugate). The "realizability condition" corresponds to an application of Cauchy-Schwarz inequality, which yields: , with the magnetic energy spectrum. To obtain this inequality, the fact that (with the solenoidal part of the Fourier transformed magnetic vector potential, orthogonal to the wavevector in Fourier space) has been used, since . The factor 2 is not present in the paper since the magnetic helicity is defined there alternatively as . One can then imagine an initial situation with no velocity field and a magnetic field only present at two wavevectors and . We assume