Global Wave Patterns in the Sun’s Magnetic Field

Abstract

When the observed pattern of solar magnetic fields is decomposed in its spherical-harmonic components and a time series analysis is performed, a resonant global wave pattern is revealed. The power spectrum indicates modes with discrete frequencies, obeying a strict parity selection rule in the case of the zonal, rotationally-symmetric modes (with spherical-harmonic order m = 0). For instance, the 22 yr resonance that dominates for the anti-symmetric modes (with odd values of the spherical-harmonic degree l) is completely absent for the symmetric modes, which instead exhibit a number of resonances having frequencies increasing with l.

A more traditional way of looking at the evolution of the zonal magnetic pattern is in the form of isocontours in latitude-time space (as in the ‘butterfly diagram’ of sunspots). We show how this pattern can to a good approximation be represented as a superposition of 14 discrete modes, each with a purely sinusoidal time variation, one mode for each value of l (= 1,2,…, 14). This corresponds to the assumption that the true, fully resolved and noise-free power spectrum consists of δ-function peaks, one for each l value.

This approach allows us to analyse the roles of the individual discrete modes in generating the well-known features in the traditional ‘butterfly diagrams’, e.g., the drift of the sunspot zones towards the equator and the prominence zones towards the poles during the course of the 11 yr cycle. It is shown that these features are accounted for entirely by the odd parity modes with the single, sinusoidal period of 22 yr. The drifts (and thus the arrow of time) are caused by the systematic phase relations between the 22 yr modes. The even modes exhibit an entirely different pattern. Since they have considerably shorter periods, they cause an undulation of the odd-mode contour lines when superposed on the anti-symmetric pattern.

The dispersion, amplitude, and phase relations of the discrete modes are given. It is indicated how they can be used in combination with spectral inversion techniques to determine the depth variation of the parameters in the governing global wave equation.

Paper dedicated to Professor Hannes Alfvén on the occasion of his 80th birthday, 30 May 1988.