Turbulent magnetic decay controlled by two conserved quantities

The decay of a turbulent magnetic field is slower with helicity than without. Furthermore, the magnetic correlation length grows faster for a helical than a nonhelical field. Both helical and nonhelical decay laws involve conserved quantities: the mean magnetic helicity density and the Hosking integral. Using direct numerical simulations in a triply periodic domain, we show quantitatively that in the fractionally helical case the mean magnetic energy density and correlation length are approximately given by the maximum of the values for the purely helical and purely nonhelical cases. The time of switch-over from one to the other decay law can be obtained on dimensional grounds and is approximately given by $I_\mathrm{H}^{1/2}I_\mathrm{M}^{-3/2}$, where $I_\mathrm{H}$ is the Hosking integral and $I_\mathrm{M}$ is the mean magnetic helicity density. An earlier approach based on the decay time is found to agree with our new result and suggests that the Hosking integral exceeds naive estimates by the square of the same resistivity-dependent factor by which also the turbulent decay time exceeds the Alfv\'en time. In the presence of an applied magnetic field, the mean magnetic helicity density is known to be not conserved, and we show that then also the Hosking integral is not conserved.