Magnetic helicity is a quantity that underpins many theories of magnetic relaxation in electrically conducting fluids, both laminar and turbulent. Although much theoretical effort has been expended on magnetic fields that are everywhere tangent to their domain boundaries, many applications, both in astrophysics and laboratories, actually involve magnetic fields that are line-tied to the boundary, i.e., with a non-trivial normal component on the boundary. This modification of the boundary condition requires a modification of magnetic helicity, whose suitable replacement is called relative magnetic helicity. In this work, we investigate rigorously the behaviour of relative magnetic helicity under turbulent relaxation. In particular, we specify the normal component of the magnetic field on the boundary and consider the ideal magnetohydrodynamic limit of resistivity tending to zero in order to model the turbulent evolution in the sense of Onsager’s theory of turbulence. We show that relative magnetic helicity is conserved in this distinguished limit and that, for constant viscosity, the magnetic field relaxes in a weak, time-averaged sense to a magnetohydrostatic equilibrium.
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