On the power in the Legendre modes of the solar radial magnetic field

The power in the different l modes of an expansion of the solar radial magnetic field at the surface in terms of Legendre polynomials, P(l), is calculated with the help of a solar dynamo model studied earlier. The model is of the Babcock-Leighton type, i.e., the surface eruptions of the toroidal magnetic field - through the 'tilt angle', gamma, formed by the magnetic axis of a bipolar magnetic region with the east-west line - are the sources for the poloidal field. In this paper it is assumed that the tilt angle is subject to fluctuations of the form, gamma = gamma' (sigma) + [gamma] where [gamma] is the average value and gamma' (sigma) is a random normal fluctuation with standard deviation a which is taken from Howard's observations of the distribution of tilt angles. For numerical considerations, negative values of gamma were not allowed. If this occurred, gamma was recalculated. The numerical integrations were started with a toroidal magnetic field antisymmetric across the equator, large enough to generate eruptions, and a negligible poloidal field. The fluctuations in the tilt angle destroy the antisymmetry as time increases. The power of the antisymmetric modes across the equator (i.e., odd values of l) is concentrated in frequencies, v(p), corresponding to the cycle period. The maximum power lies in the l = 3 mode with considerable power in the l = 5 mode, in broad agreement with Stenflo's results who finds a maximum power at l = 5. For the symmetric modes, there is considerable power in frequencies larger than v(p), again in broad agreement with Stenflo's power spectrum.