We investigate the dynamics of solar activity using a nonlinear
one-dimensional dynamo model and a phenomenological equation for the evolution
of Wolf numbers. This system of equations is solved numerically. We take into
account the algebraic and dynamic nonlinearities of the alpha effect. The
dynamic nonlinearity is related to the evolution of a small-scale magnetic
helicity, and it leads to a complicated behavior of solar activity. The
evolution equation for the Wolf number is based on a mechanism of formation of
magnetic spots as a result of the negative effective magnetic pressure
instability (NEMPI). This phenomenon was predicted 25 years ago and has been
investigated intensively in recent years through direct numerical simulations
and mean-field simulations. The evolution equation for the Wolf number includes
the production and decay of sunspots. Comparison between the results of
numerical simulations and observational data of Wolf numbers shows a 70 %
correlation over all intervals of observation (about 270 years). We determine
the dependence of the maximum value of the Wolf number versus the period of the
cycle and the asymmetry of the solar cycles versus the amplitude of the cycle.
These dependencies are in good agreement with observations.Comment: 9 pages, 13 figures, final revised paper for MNRA
