The Evolution of Multipole Moments for the Global Solar Magnetic Field

Abstract

Based on synoptic magnetic maps, we use spherical harmonic functions to decompose the global magnetic field of the Sun. Through the expansion coefficients, the power spectrum $C_{l}$, and its zonal component $C_{l}^{Z}$ and sectorial component $C_{l}^{S}$ for order $l$ are defined. Through two sampled synoptic maps, it is easy to show that the multipole moments in solar active years are much stronger than those in the quiet years. We decomposed the synoptic maps for Carrington rotations (CRs) 1625 to 2224, i.e., corresponding Solar Cycles 21 – 24, and studied the evolution of multipole moments. Except for the monopole, the power spectra $C_{l}$ for most moments have an obvious 11-year period. The dipole and octupole in Solar Cycle 22 are much stronger than in the other three cycles. The zonal components of dipole and octupole are opposite to the phases of most modes, while the sectorial components of them are consistent with most phases. In addition, compared to the phase of the yearly sunspot number, the phases of most multipoles are relatively delayed, except for the zonal modes of the quadrupole and 32-pole ($l=5$). We find that the zonal mode of $l=5$ is very strong, which leads ahead of the phase of the yearly sunspot number for about 10 CRs. Additionally, unlike the other modes, it has a strong 5.5-year period. Then, the zonal mode of $l = 5$ is mainly induced by the magnetic field of the sunspots, which is further confirmed in this work.