The canonical undestanding of stellar convection has recently been put under doubt due to helioseismic results and global 3D convection simulations. This “convective conundrum” is manifested by much higher velocity amplitudes in simulations at large scales in comparison to helioseismic results, and the difficulty in reproducing the solar differential rotation and dynamo with global 3D simulations. Here some aspects of this conundrum are discussed from the viewpoint of hydrodynamic Cartesian 3D simulations targeted at testing the rotational influence and surface forcing on deep convection. More specifically, the dominant scale of convection and the depths of the convection zone and the weakly subadiabatic – yet convecting – Deardorff zone are discussed in detail.

The magnetic network is a typical magnetic structure of the quiet Sun. Investigating its cycle dependence is crucial for understanding its evolution. We aim to identify and analyze the spatial scales of the magnetic network within magnetic power spectra derived from high-resolution Solar and Heliospheric Observatory (SOHO)/Michelson Doppler Imager (MDI) and Solar Dynamics Observatory (SDO)/ Helioseismic and Magnetic Imager (HMI) synoptic magnetograms. The data sets cover the entirety of solar cycles 23, 24, and part of cycle 25. We find that the identified magnetic network sizes identified range from 26 Mm to 41 Mm. There seems to be no obvious dependence on the solar cycle, and the sizes are distributed uniformly within the identification range.

In this contribution, I present a selected overview of optical interferometry imaging results that brought insights on stellar activity and mass loss in evolved stars. I briefly introduce the STELLIM project that aims to characterize stellar surfaces and circumstellar environments by producing fast and reliable interferometric images.

The study of the differential rotation in the chromosphere of the Sun is of significant importance as it provides valuable insights into the rotational behaviour of the solar atmosphere at higher altitudes and the coupling mechanism between the various layers of the solar atmosphere. In this work, we employed the image correlation technique, explicitly focusing on plages, intending to estimate the chromospheric differential rotation. For this purpose, we have utilized Ca ii K spectroheliograms (1907 – 2007) from the Kodaikanal Solar Observatory (KoSO), recently calibrated with a better technique to ensure accuracy. Our analysis indicates that plages in the chromosphere exhibit faster rotation and a smaller latitudinal gradient when compared to the rotation rate obtained through sunspot tracking. Furthermore, we investigate the temporal analysis of the chromospheric differential rotation parameters across various solar cycles.

The Sun’s meridional circulation is a crucial component for understanding the Sun’s dynamo and its interior dynamics. However, the determination of meridional circulation is affected by a systematic center-to-limb (CtoL) effect, which introduces systematic errors 5–10 times stronger than the meridional-flow-induced travel-time shifts in deep-flow measurements. Recently, it was found that the CtoL effect has a significant acoustic-frequency dependence, while flow-induced travel-time shifts show little frequency dependence (Chen & Zhao 2018). This discovery forms the basis for designing a new method to remove the CtoL effect. We therefore propose a frequency-dependent approach to measure the CtoL effect and the flow-induced signals in the Fourier domain. In this work, we present this new method and compare time–distance measurements in different frequency bands with those obtained by previous time-domain methods. The results demonstrate consistency with conventional time-domain fitting methods in the dominant frequency range, promising the potential for conducting meridional flow inversion across a broader frequency spectrum.

We have measured zonal and meridional components of subsurface flows up to a depth of 30 Mm below the solar surface by applying the technique of ring diagram on Dopplergrams which are constructed from the spherical harmonic (SH) coefficients. The SH coefficients are obtained from the Helioseismic and Magnetic Imager (HMI) full-disk Dopplergrams. We find a good agreement and some differences between the flows obtained in this study with those from the traditional methods using direct Dopplergrams.

We present a new method to measure the rotational height gradient in the solar photosphere. The method is inspired from differential interferometric techniques, we applied it to spectroscopic observations in the FeI 630.15 nm obtained at the solar telescope THEMIS which is equipped with an efficient adaptive optics system. The spectroscopic data was used to obtain images of the granulation at different line cords formed at different heights in the photosphere. Cross-correlation allows us to measure small systematic shifts between similar images. When observations are performed out of the center of the solar disk, the perspective effect gives rise to a radial shift between images formed at different heights. The measurement of this shift provides us with their formation-height difference. At the center of the disk the perspective effect vanishes but we measured a systematic retrograde shift along the east/west direction of the images formed at higher heights. The measured shifts are proportional to the formation height of the images. We interpret these findings as the evidence of a decrease of the rotational velocity with height in the low photosphere of the Sun and we give an estimate of this gradient.

Helioseismology has discovered a thin layer beneath the solar surface where the rotation rate increases rapidly with depth. The normalized rotational shear in the upper 10 Mm of the layer is constant with latitude. Differential rotation theory explains such a rotational state by a radial-type anisotropy of the near-surface convection and a short correlation time of convective turbulence compared to the rotation period. The shear layer is the main driver of the global meridional circulation.

We present a theoretical model of the near-surface shear layer (NSSL) of the Sun. Convection cells deeper down are affected by the Sun’s rotation, but this is not the case in a layer just below the solar surface due to the smallness of the convection cells there. Based on this idea, we show that the thermal wind balance equation (the basic equation in the theory of the meridional circulation which holds inside the convection zone) can be solved to obtain the structure of the NSSL, matching observational data remarkably well.

Just like the Sun, other stars also exhibit differential rotation. Currently, the rotation profile of a star that hosts a transiting planet can be estimated if during a transits, the planet occults a spot on the photosphere of the star, causing slight variations in its light curve. By detecting the same spot during a later transit, the stellar rotation period at that latitude is determined. Here, we present the results of differential rotation for 48 stars, 13 from the spot transit mapping method, while the remaining 35 stars from other techniques. The results show that the differential rotation is correlated with the stellar mean rotation period for fast rotating stars and strongly anti-correlated for slow rotators. The transition occuring at rotation period of 5 days. On the other hand, the differential shear increases with effective temperature for fast rotating stars, but the correlation is lost for the slow rotators.

We investigate flare activity using photometric data obtained with the Transiting Exoplanet Survey Satellite (TESS). Long-term seasonal period analysis was applied on our APT (Automatic Photoelectric Telescopes af Fairborn observatory, Arizona) time series to study changes in the rotational period. We also looked for activity cycle-like changes with short-term Fourier-transform. We also studied the phase and frequency distribution of hand-selected flares on the available TESS data. The MUlti-SIte Continuous Spectroscopy (MUSICOS) campaign was designed in 1998 to achieve high-resolution, multi-wavelength spectroscopic observations from many sites around the globe, which meant that uninterrupted phase coverage of EI Eri became available. We use these data to reconstruct successive surface-temperature maps of the star in order to study the changes of starspots on a very short timescale. We applied our multi-line Doppler imaging code to reconstruct four consecutive Doppler images. These images were also used to measure surface differential rotation with our cross-correlation technique.

Tidal forces in close binaries and multiple systems that contain magnetically active component are supposed to influence the operation of magnetic dynamo. Through synchronization the tidal effect of a close companion helps maintain fast rotation, thus supporting an efficient dynamo. At the same time, it can also suppress the differential rotation of the convection zone, or even force the formation of active longitudes at certain phases fixed to the orbit. V815 Her is a four-star system consisting of two close binaries orbiting each other, one of which contains an active G-type main-sequence star. Therefore, the system offers an excellent opportunity to investigate the influence of gravitational effects on solar-type magnetic activity using different methods.

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.

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.

In most of the Babcock–Leighton (BL) type solar dynamo models, the toroidal magnetic field is assumed to be generated in the tachocline. However, magnetic activity of fully convective stars and MHD simulations of global stellar convection have recently raised serious doubts about the importance of the tachocline in the generation of the toroidal field. We have developed a BL-type dynamo model operating in the bulk of the convection zone, and are extending this model to solar-type stars. In this study, we aim at exploring how the starspot properties affect the stellar magnetic cycle. Observations show that faster rotating stars tend to have stronger magnetic activity and shorter magnetic cycles. By considering the higher latitudes and larger tilt angles of starspots for faster rotators, our simulations reproduce observations that faster rotating stars have shorter magnetic cycle and stronger activity.

Planetary influence on a stellar convective shell can result in a periodic modulation of stellar dynamo drivers. Similar modulation can arise in stellar binary systems. Using the Parker low-mode dynamo model we investigate the properties of nonlinear parametric resonance. This model is a system of four ordinary differential equations and, in the first approximation, describes the processes of generation and oscillation of large-scale magnetic fields in stellar systems. In the absence of nonlinear suppression effects, the problem, by analogy with a system of harmonic oscillations, allows an asymptotic selection of multiple resonant frequencies. Despite the fact that at first glance at these frequencies it is reasonable to expect an increase in the amplitude, the behavior of the system can be just the opposite. All this stuff deserves a systematic analysis of swing excitation in the dynamo sistems in comparison with classical swing excitation in the framework of the Mathieu equation.

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.

During minima of solar activity, it is possible to estimate the influence of convection zone turbulence on the magnetic flux tubes forming active regions (ARs), because the toroidal field of the old cycle weakens, and the new toroidal field is still weak. We analyzed ARs of solar minima between 23-24 and 24-25 solar cycles. ARs were classified as regular, irregular and unipolar spots. Regular ARs follow the empirical laws consistent with the Babcock–Leighton dynamo theory. We found that regular ARs dominate by flux and by number during the solar minima. Irregular ARs are mainly represented by bipolar structures of deformed orientation and contribute only one-third in the total flux and one-third in the total number. Very complex multipolar ARs are extremely rare. So, during solar minima the global dynamo still guides the formation of ARs, whereas the turbulence only slightly affects the toroidal flux tubes orientation.

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.

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.