Global modes constituting the solar magnetic cycle. 3: `Shapes' and `sizes' of the sunspot cycles and maintenance of MHD spectrum by energy cascade

Abstract

The `sunspot occurrence probability' defined in Paper 1 is used to determine the Legendre-Fourier (LF) terms in the `rate of emergence of toroidal magnetic flux, Q(theta, t), above the photosphere per unit latitude interval, per unit time'. Assuming that the magnetic flux tubes whose emergence yields solar activity are produced by interference of global MHD waves in the Sun, we determine how the amplitudes and phases of the LF terms in the toroidal magnetic field BPhi, representing the waves, will be related to those of the LF terms in Q(theta, t). The set of LF terms in `Q' that represents the set of waves whose interference produces most of the observed sunspot activity is l = 1, 3, ..., 13; nu = n nu*, n = 1, 3, 5), where nu* = 1/21.4/yr. However, among the `shapes' of sunspot cycles modeled using various sets of the computed LF terms the best agreement with the observed shape, for each cycle, is given by the set (l = 3 or l = 3, 5; and n = 1, 3 or n = 1, 3, 5). The sets of terms: (l = 1, 3, 5, 7; n = 1), (l = 1, 3, 5, 7; n = 3), )l = 9, 11, 13, 15; n = 1) and (l = 9, 11, 13, 15; n = 3) seem to represent four modes of global MHD oscillation. Correlations between the amplitudes (and phases) of LF terms in different modes suggest possible existence of cascade of energy from constituent MHD waves of lower l and n to those of higher l and n. The spectrum of the MHD waves trapped in the Sun may be maintained by the combined effect of this energy cascade and the loss of energy in the form of the emerging flux tubes. The primary energy input into the spectrum may be occurring in the mode (l = 1, 3, 5, 7; n = 1). As expected from the above phenomenological model, the size of a sunspot cycle and its excess over the previous cycle are well correlated (e.g., approximately 90%) to the phase-changes of the two most dominant oscillation modes during the previous one or two cycles. These correlations may provide a physical basis to forecast the cycle sizes.