A striking feature of the solar cycle is that at the beginning, sunspots appear around mid-latitudes, and over time the latitudes of emergences migrate towards the equator. The maximum level of activity varies from cycle to cycle. For strong cycles, the activity begins early and at higher latitudes with wider sunspot distributions than for weak cycles. The activity and the width of sunspot belts increase rapidly and begin to decline when the belts are still at high latitudes. However, in the late stages of the cycles, the level of activity, and properties of the butterfly wings all have the same statistical properties independent of the peak strength of the cycles. We have modelled these features using Babcock–Leighton type dynamo model and shown that the toroidal flux loss from the solar interior due to magnetic buoyancy is an essential nonlinearity that leads to all the cycles decline in the same way.

The inherent stochastic and nonlinear nature of the solar dynamo makes the strength of the solar cycles vary in a wide range, making it difficult to predict the strength of an upcoming solar cycle. Recently, our work has shown that by using the observed correlation of the polar field rise rate with the peak of polar field at cycle minimum and amplitude of following cycle, an early prediction can be made. In a follow-up study, we perform SFT simulations to explore the robustness of this correlation against variation of meridional flow speed, and against stochastic fluctuations of BMR tilt properties that give rise to anti-Joy and anti-Hale type anomalous BMRs. The results suggest that the observed correlation is a robust feature of the solar cycle and can be utilized for a reliable prediction of peak strength of a cycle at least 2 to 3 years earlier than the minimum.

The most promising model for explaining the origin of solar magnetism is the flux transport dynamo model, in which the toroidal field is produced by differential rotation in the tachocline, the poloidal field is produced by the Babcock–Leighton mechanism at the solar surface and the meridional circulation plays a crucial role. After discussing how this model explains the regular periodic features of the solar cycle, we come to the questions of what causes irregularities of solar cycles and whether we can predict future cycles. Only if the diffusivity within the convection zone is sufficiently high, the polar field at the sunspot minimum is correlated with strength of the next cycle. This is in conformity with the limited available observational data.

The winding number problem (Lévy (1940)) concerns the net angle through which the route of a random walk winds about the origin. We consider the problem of finding the winding number for a walk with finite step sizes; the eigenfunction method (Roberts and Ursell (1960)) is shown to be inapplicable because the probability distribution for a sequence of steps of different length depends on the order in which those steps are taken. In the diffusion limit, however, commutivity is restored. We derive the winding number distribution for a diffusion process, starting from a point displaced from the origin, and consider its asymptotic form. An important difference between the finite step and diffusion distributions is that the former possesses finite moments while the latter does not. We compute numerically the finite step distributions for 20000 particles undergoing N = 100000 steps, and compare the results with the diffusion distribution. Even for small winding numbers, perceptible differences between the two distributions appear even for N as large as 100000.