Analysis of irregular behavior in the solar magnetic activity cycle

Abstract

We numerically investigate the irregular behavior of the nonlinear solar dynamo model incorporating algebraic quenching of the $\alpha$-effect and dynamical feedback via the small-scale current helicity. Numerically stable computation of quenching functions is provided by using asymptotic expansions which makes it possible to eliminate catastrophic cancellation. A parametric study is carried out to examine how the dynamo number and the magnetic contribution to the $\alpha$-effect influence the statistical and spectral characteristics of cycles. Distributions of cycle amplitudes, periods, and peak phase offsets are determined based on the long time series of the energy of the dipolar component of the toroidal magnetic field. It is shown that the inherent model dynamics generates irregular modulation without external stochastic forcing while retaining distinct cyclicity. The increasing complexity of its behavior leads to the appearance of broadband components in both energy and current helicity spectra. Comparison with the number of sunspots shows that similarity of individual cyclic sequences does not guarantee agreement between their statistical distributions: at the largest of the examined dynamo-number magnitudes the relative variability of periods and amplitudes exceeds the observed range. The interpretation of differences requires accounting for the limited observational sampling and the relationship between magnetic energy and the activity index. The stability of statistical estimates is investigated as the number of analyzed cycles increases. In the reference case, the distributions stabilize, while the local concentration of phases weakens. The variability of histograms across consecutive blocks decreases approximately in inverse proportion to the square root of the number of cycles. These results demonstrate the necessity of long-term integration to reliably estimate the cycle variability and phase coherence.