Understanding the irregular variation of the solar cycle is crucial due to its significant impact on global climates and the heliosphere. Since the polar magnetic field determines the amplitude of the next solar cycle, variations in the polar field can lead to fluctuations in the solar cycle. We have explored the variability of the solar cycle at different levels of dynamo supercriticality. We observe that the variability depends on the dynamo operation regime, with the near-critical regime exhibiting more variability than the supercritical regime. Furthermore, we have explored the effects of the irregular BMR properties (emergence rate, latitude, tilt, and flux) on the polar field and the solar cycle. We find that they all produce considerable variation in the solar cycle; however, the variation due to the tilt scatter is the largest.

Solar activity shows an 11-year (quasi)periodicity with a pronounced, but limited variability of the cycle amplitudes. The properties of active region (AR) emergence play an important role in the modulation of solar cycles and are our central concern in building a model for predicting future cycle(s) in the framework of the Babcock–Leighton (BL)-type dynamo. The emergence of ARs has the property that strong cycles tend to have higher mean latitudes and lower tilt angle coefficients. Their non-linear feedbacks on the solar cycle are referred to as latitudinal quenching and tilt quenching, respectively. Meanwhile, the stochastic properties of AR emergence, e.g., rogue ARs, limit the scope of the solar cycle prediction. For physics-based prediction models of the solar cycle, we suggest that uncertainties in both the observed magnetograms assimilated as the initial field and the properties of the AR emergence should be taken into account.

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.

Babcock–Leighton process, in which the poloidal field is generated through the decay and dispersal of tilted bipolar magnetic regions (BMRs), is observed to be the major process behind the generating poloidal field in the Sun. Based on this process, the Babcock–Leighton dynamo models have been a promising tool for explaining various aspects of solar and stellar magnetic cycles. In recent years, in the toroidal to poloidal part of this dynamo loop, various nonlinear mechanisms, namely the flux loss through the magnetic buoyancy in the formation of BMRs, latitude quenching, tilt quenching, and inflows around BMRs, have been identified. While these nonlinearities tend to produce a stable magnetic cycle, the irregular properties of BMR, mainly the scatter around Joy’s law tilt, make a considerable variation in the solar cycle, including grand minima and maxima. After reviewing recent developments in these topics, I end the presentation by discussing the recent progress in making the early prediction of the solar cycle.

The induction and momentum equations of solar dynamo are simplified to a dynamic system for the convective Root-Mean-Square (rms) velocity and the rms magnetic field in the solar convection zone. The study of stable stationary points of this system gives a minor excess of the critical level of the dynamo and, accordingly, moderate magnetic field typically about 1 T (10 kG). A significantly lower rms magnetic field may be possible at some parameters of the system. The stable rms velocity is about 100 m/sec, and the characteristic magnetic times are about the half-period of solar rotation or about an average lifetime of sunspots. Relative magnetic energy is of order 5 kJ/kg that is about the kinetic energy. The unstable stationary points could be near zero magnetic fields as in periods of very lower solar activity similar to the Maunder minimum.

The tilt of the bipolar magnetic region (BMR) is crucial in the Babcock-Leighton process for the generation of the poloidal magnetic field in the Sun. We extend the work of Jha et al. (2020) and analyze the recently reported tracked BMR catalogue based on AutoTAB (Sreedevi et al. 2023) from Michelson Doppler Imager (1996–2011) and Helioseismic and Magnetic Imager (2010–2018). Using the tracked information of BMRs based on AutoTAB, we confirm that the distribution of Bmax reported by Jha et al. (2020) is not because of the BMRs are picked multiple times at the different phases of their evolution instead it is also present if we consider each BMRs only once. Moreover, we find that the slope of Joy’s law (〈γ0〉) initially increases slowly with the increase of Bmax. However, when Bmax >2.5 kG, γ0 decreases. The decrease of observed γ0 with Bmax provides a hint to a nonlinear tilt quenching in the Babcock-Leighton process.

The grand minimum in the Sun’s activity is a distinctive mode characterized by a magnetic lull that almost completely lacks the emergence of sunspots on the solar surface for an extended duration. The factors driving this transition of an otherwise magnetically active star into a quiescent phase, the processes occurring within the solar interior and across the heliosphere during this period, and the mechanisms leading to the eventual resurgence of surface magnetic activity remain enigmatic. However, there have been sustained efforts in the past few decades to unravel these mysteries by employing a combination of observation, reconstruction and simulation of solar magnetic variability. Here, we summarize recent research on the solar grand minimum and highlight some outstanding challenges – both intellectual and practical – that necessitate further investigations.

The profile of the differential rotation together with the sign of the alpha-effect determine the dynamo wave direction. In early models of the solar dynamo the dynamo wave often leads to a poleward migration of the activity belts. Flux transport by the meridional flow or the effect of the surface shear layer are possible solutions. In a model including the corona, we show that various migrations can be obtained by varying the properties of the corona. A new dynamo of Babcock-Leighton type also leads to the correct equatorward migration by the non-linear relation between flux density and rise time of the flux.

The sunspot cycle is quite variable in duration and amplitude, yet in the long term, it seems to return to solar minimum on schedule, as if guided by a clock with an average period of close to 11.05 years for the sunspot number cycle and 22.1 years for the magnetic cycle. This paper provides a brief review of the sunspot number cycle since 1750, discusses some of the processes controlling the solar dynamo, and provides clues that may add to our understanding of what controls the cadence of the solar clock.

We consider several tracers of magnetic activity that separate cycle-dependent contributions to the background solar magnetic field from those that are independent of the cycle. The main message is that background fields include two relative separate populations. The background fields with a strength up to 100 Mx cm−2 are very poorly correlated with the sunspot numbers and vary little with the phase of the cycle. In contrast, stronger magnetic fields demonstrate pronounced cyclic behaviour. Small-scale solar magnetic fields demonstrate features of fractal intermittent behaviour, which requires quantification. We investigate how the observational estimate of the solar magnetic flux density B depends on resolution D in order to obtain the scaling In BD = −k In D + a in a reasonably wide range. The quantity k demonstrates cyclic variations typical of a solar activity cycle. k depends on the magnetic flux density, i.e. the ratio of the magnetic flux to the area over which the flux is calculated, at a given instant. The quantity a demonstrates some cyclic variation, but it is much weaker than in the case of k. The scaling is typical of fractal structures. The results obtained trace small-scale action in the solar convective zone and its coexistence with the conventional large-scale solar dynamo based on differential rotation and mirror-asymmetric convection. Here we discuss the message for solar dynamo studies hidden in the above results.

We could find a new 5 year periodicity in the occurrences of peaks in sunspot activity and inferred deviations of annual Indian monsoon rainfall variations from the normal during the Maunder minimum (MM) period. This result is explained in terms of solar dynamo functioning in a different mode from normal during the MM where quadrupole field (first harmonic, 5-5.5 years) dominate over dipole field (fundamental, 11 years) causing extreme north south asymmetry in sunspot activity.

We could identify three categories of solar proton events (SPE) with distinct solar origin from an analysis of direct and indirect observations during the years 1561-2016 CE spanning 42 sunspot cycles. They are (i) 10 MeV SPE whose number of occurrences closely follow the sunspot cycles (ii) 30 MeV SPE which show secular changes with peaks near Gleissberg solar cycle minima and inferred to be associated with distinct enhancements in the efficiency of the solar dynamo and (iii) those associated with Impulsive and irregular solar activity changes such as the Carrington event of September 1859. The relevance of above results for stars exhibiting cyclic and irregular activity changes will be also discussed.

Kinematic solar dynamo models (KSDM) that use a solar like differential rotation profile usually fail to reproduce the latitudinal distribution of the toroidal magnetic fields. Usually, it is assumed that it is the larger radial shear present in the solar tachocline at higher latitudes that is the responsible for such results. We here consider variations in the shape and thickness of the tachocline and find that a better distribution of the toroidal fields is obtained when the thickness of the tachocline is $d_1=0.02R_{\odot}$, a smaller value than that conventionally used.

Mean-field and direct simulations of the hydrodynamics and hydromagnetics of the solar convection zone are discussed with the ultimate aim to understand the generation of differential rotation and magnetic fields. Various arguments constraining the values of the various turbulent diffusion coefficients are presented. It is suggested that the turbulent magnetic diffusivity is much smaller than the eddy viscosity which, in turn, is by up to a factor of ten smaller than the eddy conductivity. The magnetic field obtained from direct simulations is highly intermittent, and there is no clear systematic orientation of bipolar regions emerging from the convection zone. Various mechanisms that might cause such a field orientation are considered. Finally, the application of direct simulations to the determination of mean-field transport coefficients is emphasised.