Magnetic helicity evolution in a periodic domain with imposed field

In helical hydromagnetic turbulence with an imposed magnetic field (which is constant in space and time) the magnetic helicity of the field within a periodic domain is no longer an invariant of the ideal equations. Alternatively, there is a generalized magnetic helicity that is an invariant of the ideal equations. It is shown that this quantity is not gauge invariant and that it can therefore not be used in practice. Instead, the evolution equation of the magnetic helicity of the field describing the deviation from the imposed field is shown to be a useful tool. It is demonstrated that this tool can determine steady state quenching of the alpha-effect. A simple three-scale model is derived to describe the evolution of the magnetic helicity and to predict its sign as a function of the imposed field strength. The results of the model agree favorably with simulations.