The magnetic helicity spectrum from solar vector magnetograms

The gauge-invariant (or relative) magnetic helicity is often measured to characterize the degree of magnetic complexity of active regions. However, magnetic helicity is expected to have different signs on different length scales that can be identified with the large- and small-scale fields used in dynamo theory. To address this, it is important to determine magnetic helicity spectra as functions of wavenumber. These spectra are defined such that the integral over all wavenumbers gives the usual magnetic helicity density in a particular patch of interest. Using vector magnetograms from the Helioseismic and Magnetic Imager on board the Solar Dynamics Observatory for active region NOAA 11515, which was on the southern hemisphere, we show that the magnetic helicity spectrum has positive sign on scales below 30 Mm, but negative sign on larger scales. This active region was rather complex and its magnetic helicity was within 26% of its theoretical maximum value. This is much more than that of NOAA 11158, which was also rather complex, but only within 5% of its theoretical maximum value. Since the contribution of larger length scales turned out to be important in the case of NOAA 11515, its total magnetic helicity is dominated by the negative values from large length scales, which explains the unusual sign for the southern hemisphere. Measuring magnetic helicity spectra with DKIST may become an important tool to learn about the workings of the underlying dynamo.