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Various approaches to forecasting the solar cycle based on solar dynamo models are considered. The importance of separating predictions of catastrophic events such as the Maunder minimum, predictions of general trends in solar activity between Grand minima, and forecasts of the next cycle, based on current knowledge of previous cycles is noted. The role of fluctuations of dynamo drivers for the forecast is emphasized. The polar magnetic field in the current cycle is considered as a natural predictor of the amplitude of the following cycle. [ABSTRACT FROM AUTHOR]

We investigate the frequency of all (X-ray flare events higher than class B1.0), B, C, M and X-class flares, respectively, derived from the National Geophysical Data Center (NGDC) list of solar flares between May 1983 and September 2014, which corresponds to the two complete solar cycles (SCs) 22 and 23 as well as the rise and maximum phases of SC 24. Analysis shows that the temporal behavior for these various class flares is quite different. The main findings of this study, confirmed by using the Hinode flare catalog where possible, are as follows. (1) The B-class flares are in complete antiphase with all, C, M and X-class flares. (2) While, there is a small decreasing trend in the peak values of the smoothed monthly C-class flare numbers from SC 22 to 24, the occurrence rate of M and X-class flares dropped by almost half and two-thirds, respectively, during SC 23 and remained almost the same during SC 24. This class-dependent temporal behavior provides support for dynamo models that involve the coexistence of a deep global and a superficial local dynamo. [ABSTRACT FROM AUTHOR]

In our earlier works (Ivanov and Miletsky, 2012, 2014), we demonstrated that (1) the evolution of average sunspot latitudes can be described by a universal latitude curve, the shape of which does not depend on the cycle amplitude and (2) at the cycle decline phase, these latitudes correlate well with the current level of solar activity. In this work, we demonstrate that these features of the latitude evolution of the cycle, as well as the empirical Waldmeier rule, can be described by a simple model of a convective α-ω dynamo in a thin spherical layer with the addition of some nonlinearities. [ABSTRACT FROM AUTHOR]

Using different proxies of solar activity, we have studied the following features of the solar cycle: i) The linear correlation between the amplitude of cycle and its decay rate, ii) the linear correlation between the amplitude of cycle $n$ and the decay rate of cycle $(n - 1)$, and iii) the anti-correlation between the amplitude of cycle $n$ and the period of cycle $(n - 1)$. Features ii) and iii) are very useful because they provide precursors for future cycles. We have reproduced these features using a flux-transport dynamo model with stochastic fluctuations in the Babcock-Leighton $\alpha$ effect and in the meridional circulation. Only when we introduce fluctuations in meridional circulation, are we able to reproduce different observed features of the solar cycle. We discuss the possible reasons for these correlations. [ABSTRACT FROM AUTHOR]

Based on an analysis of the observational data for solar cycles 12-23 (Royal Greenwich Observatory-USAF/NOAA Sunspot Data), we have studied various parameters of the 'Maunder butterflies.' Based on the observational data for cycles 16-23, we have found that BT/ Land S depend linearly on each other, where B is the mean magnetic field of the cycle, T is the cycle duration, S is the cycle strength, and L is the mean sunspot latitude in the cycle (the arithmetic mean of the absolute values of the mean latitudes in the north and south). The connection of the observed quantities with the α- ω-dynamo theory is discussed. [ABSTRACT FROM AUTHOR]

Context. The large-scale magnetic field in the Sun varies with a period of approximately 22 years, although the amplitude of the cycle is subject to long-term modulation with recurrent phases of significantly reduced magnetic activity. It is believed that a hydromagnetic dynamo is responsible for producing this large-scale field, although this dynamo process is not well understood. Aims. Within the framework of mean-field dynamo theory, our aim is to investigate how competing mechanisms for poloidal field regeneration (namely a time-delayed Babcock-Leighton surface α-effect and an interface-type α-effect), can lead to the modulation of magnetic activity in a deep-seated solar dynamo model. Methods. We solve the standard αΩ dynamo equations in one spatial dimension, including source terms corresponding to both of the competing α-effects in the evolution equation for the poloidal field. This system is solved using two different methods. In addition to solving the one-dimensional partial differential equations directly, using numerical techniques, we also use a local approximation to reduce the governing equations to a set of coupled ordinary differential equations (ODEs), which are studied using a combination of analytical and numerical methods. Results. In the ODE model, it is straightforward to find parameters such that a series of bifurcations can be identified as the time delay is increased, with the dynamo transitioning from periodic states to chaotic states via multiply periodic solutions. Similar transitions can be observed in the full model, with the chaotically modulated solutions exhibiting solar-like behaviour. Conclusions. Competing α-effects could explain the observed modulation in the solar cycle. [ABSTRACT FROM AUTHOR]

Abstract: The common methodologies used to predict the smooth sunspot number (SSN) at peak (Rmax) and the rise time (Tr) for a cycle are noted. The estimates based on geomagnetic precursors give the best prediction of Rmax for five SSN cycles (20–24). In particular, an empirical technique invoking three-cycle quasi-periodicity (TCQP) in Ap index has made accurate predictions of Rmax and Tr for two consecutive SSN cycles (23 and 24). The dynamo theories are unable to account for TCQP. If it endures in the 21st century the Sun shall enter a Dalton-like grand minimum. It was a period of global cooling. The current status of the ascending phase of cycle 24 is described and the delayed reversal of the solar polar field reversal in the southern hemisphere in September 2013 is noted. [Copyright &y& Elsevier]

We construct a series of kinematic axisymmetric mean-field dynamo models operating in the αΩ, α²Ω and α² regimes, all using the full α-tensor extracted from a global magnetohydrodynamical simulation of solar convection producing large-scale magnetic fields undergoing solar-like cyclic polarity reversals. We also include an internal differential rotation profile produced in a purely hydrodynamical parent simulation of solar convection, and a simple meridional flow profile described by a single cell per meridional quadrant. An α²Ω mean-field model, presumably closest to the mode of dynamo action characterizing the MHD simulation, produces a spatiotemporal evolution of magnetic fields that share some striking similarities with the zonally-averaged toroidal component extracted from the simulation. Comparison with α² and αΩ mean-field models operating in the same parameter regimes indicates that much of the complexity observed in the spatiotemporal evolution of the large-scale magnetic field in the simulation can be traced to the turbulent electromotive force. Oscillating α²solutions are readily produced, and show some similarities with the observed solar cycle, including a deep-seated toroidal component concentrated at low latitudes and migrating equatorward in the course of the solar cycle. Various numerical experiments performed using the mean-field models reveal that turbulent pumping plays an important role in setting the global characteristics of the magnetic cycles. [ABSTRACT FROM AUTHOR]

The well known Waldmeier rule and its modifications applied to forecasting of the solar activity are discussed. The maximum annual mean rate of the increase in the magnetic flux observed on the growth branch of the cycle is proposed as a predicting (diagnosing) parameter for the amplitude of the cycle currently in progress. The maximum Wolf number of the current (24th) cycle is estimated to reach 104 (±12) or more, and to occur in 2013 or later. The yearly average Wolf numbers observed during the 24th cycle are presented. [ABSTRACT FROM AUTHOR]

ABSTRACT We develop and subsequently explore the solution space of a simple flux transport dynamo model that incorporates a time-dependent large-scale meridional circulation. Based on recent observations, we prescribed an analytical form for the amplitude of this circulation and study its impact in the evolution of the magnetic field. We find that cyclic variations in the amplitude and frequency of the meridional flow affect the strength of the solar cycle. Variations in the amplitude of the fluctuations influence the shape of the solar cycle but are only relevant to the cycle's strength variations when they occur at a frequency different from or out of phase of the solar cycle's. [ABSTRACT FROM AUTHOR]

We use a truncated dynamo model and observations of the sunspot number to infer cycle-to-cycle variations of several dynamo parameters. A correlation study between model parameters and solar cycle observables (period, amplitude, and rising time) is done, and our results are compared with the results of other authors. Using the strongest correlations found, we determined a series of relationships that can be used for prediction purposes. This technique indicates that solar cycle 24 will have a sunspot maximum of approximately 84 around June of 2013. In the context of the used model, most of the variability found in the solar cycle can be attributed to changes in the strength of the meridional circulation. The results also point to a common origin for the source of variation of the meridional flow and differential rotation and/or the α-effect. [ABSTRACT FROM AUTHOR]

The paper presents a study of a solar dynamo model operating in the bulk of the convection zone with the toroidal magnetic field flux concentrated in the subsurface rotational shear layer. We explore how this type of dynamo may depend on spatial variations of turbulent parameters and on the differential rotation near the surface. The mean-field dynamo model takes into account the evolution of magnetic helicity and describes its nonlinear feedback on the generation of large-scale magnetic field by the α-effect. We compare the magnetic cycle characteristics predicted by the model, including the cycle asymmetry (associated with the growth and decay times) and the duration--amplitude relation (Waldmeier's effects), with the observed sunspot cycle properties. We show that the model qualitatively reproduces the basic properties of the solar cycles. [ABSTRACT FROM AUTHOR]

During the last few years the Sun and solar wind have shown a behavior that was so unexpected that the phenomena was described as 'the strange solar minimum'. It has been speculated that the 23/24 solar cycle minimum may have indicated the onset of a Maunder-Minimum-type Grand Minimum. Here we review what is known from 1500 years of proxy data about Maunder-type Grand Minima and the minima of the cyclic Centennial Gleissberg variations. We generate criteria that distinguish between the two types of event. Applying these criteria to the observed solar terrestrial data we conclude that the unexpected behavior began well before the solar cycle 23/24 minimum. The data do not support the Maunder Minimum conjecture. However, the behavior can be understood as a minimum of the Centennial Gleissberg Cycle that previously minimized in the beginning of the 20th century. We conclude that the Centennial Gleissberg Cycle is a persistent variation that has been present 80% of the time during the last 1500 years and should be explained by solar dynamo theory. [ABSTRACT FROM AUTHOR]

Abstract: This review provides an introduction to the generation and evolution of the Sun’s magnetic field, summarising both observational evidence and theoretical models. The eleven year solar cycle, which is well known from a variety of observed quantities, strongly supports the idea of a large-scale solar dynamo. Current theoretical ideas on the location and mechanism of this dynamo are presented. The solar cycle influences the behaviour of the global coronal magnetic field and it is the eruptions of this field that can impact on the Earth’s environment. These global coronal variations can be modelled to a surprising degree of accuracy. Recent high resolution observations of the Sun’s magnetic field in quiet regions, away from sunspots, show that there is a continual evolution of a small-scale magnetic field, presumably produced by small-scale dynamo action in the solar interior. Sunspots, a natural consequence of the large-scale dynamo, emerge, evolve and disperse over a period of several days. Numerical simulations can help to determine the physical processes governing the emergence of sunspots. We discuss the interaction of these emerging fields with the pre-existing coronal field, resulting in a variety of dynamic phenomena. [Copyright &y& Elsevier]

The unexpected development of cycle 24 emphasizes the need for a better way to model future solar activity. In this article, we analyze the accumulation of spotless days during individual cycles from 1798 - 2010. The analysis shows that spotless days do not disappear abruptly in the transition toward an active Sun. A comparison with past cycles indicates that the ongoing accumulation of spotless days is comparable to that of cycle 5 near the Dalton minimum and to that of cycles 12, 14, and 15. It also suggests that the ongoing cycle has as much as 20±8 spotless days left, from July 2010, before it reaches the next solar maximum. The last spotless day is predicted to be in December 2012, with an uncertainty of 11 months. This trend may serve as input to the solar dynamo theories. [ABSTRACT FROM AUTHOR]

The behavior of dynamo waves in a two-layer medium is investigated in terms of the Parker dynamo model. The solar cycle duration is shown to depend on the ratio of turbulent diffusivities in the layers. Meridional circulation has been incorporated into the Parker system. An increase in the intensity of meridional flows is shown to decelerate the propagation of dynamo waves. The minimum of solar magnetic activity can occur not only in the case of intense meridional circulation in both layers but also when a difference in physical characteristics arises between the layers and the meridional flows are moderate. [ABSTRACT FROM AUTHOR]

Abstract: The correlation between geomagnetic activity and the sunspot number in the 11-year solar cycle exhibits long-term variations due to the varying time lag between the sunspot-related and non-sunspot related geomagnetic activity, and the varying relative amplitude of the respective geomagnetic activity peaks. As the sunspot-related and non-sunspot related geomagnetic activity peaks are caused by different solar agents, related to the solar toroidal and poloidal fields, respectively, we use their variations to derive the parameters of the solar dynamo transforming the poloidal field into toroidal field and back. We find that in the last 12 cycles the solar surface meridional circulation varied between 5 and 20m/s (averaged over latitude and over the sunspot cycle), the deep circulation varied between 2.5 and 5.5m/s, and the diffusivity in the whole of the convection zone was ∼108 m2/s. In the last 12 cycles solar dynamo has been operating in moderately diffusion dominated regime in the bulk of the convection zone. This means that a part of the poloidal field generated at the surface is advected by the meridional circulation all the way to the poles, down to the tachocline and equatorward to sunspot latitudes, while another part is diffused directly to the tachocline at midlatitudes, “short-circuiting” the meridional circulation. The sunspot maximum is the superposition of the two surges of toroidal field generated by these two parts of the poloidal field, which is the explanation of the double peaks and the Gnevyshev gap in sunspot maximum. Near the tachocline, dynamo has been operating in diffusion dominated regime in which diffusion is more important than advection, so with increasing speed of the deep circulation the time for diffusive decay of the poloidal field decreases, and more toroidal field is generated leading to a higher sunspot maximum. During the Maunder minimum the dynamo was operating in advection dominated regime near the tachocline, with the transition from diffusion dominated to advection dominated regime caused by a sharp drop in the surface meridional circulation which is in general the most important factor modulating the amplitude of the sunspot cycle. [Copyright &y& Elsevier]

Abstract: In this work we use a low order dynamo model and study under which conditions can it reproduce solar grand minima. We begin by building the phase space of a proxy for the toroidal component of the solar magnetic field and we develop a model, derived from mean field dynamo theory, that gives the time evolution of the toroidal field. This model is characterized by a non-linear oscillator whose coefficients retain most of the physics behind dynamo theory. In the derivation of the model we also include stochastic oscillations in the effect. We found evidences that stochastic fluctuations in effect can trigger grand minima episodes in this model under some considerations. We also explore other ways of creating grand minima by looking into the physical mechanisms that compose the coefficients of the oscillator. The balance between meridional circulation and magnetic diffusivity as well as the field intensification by buoyancy driven instabilities, might have a crucial role in inducing grand minima. [ABSTRACT FROM AUTHOR]

Abstract: We investigate the dynamo underpinning of solar cycle precursor schemes based on direct or indirect measures of the solar surface magnetic field. We do so for various types of mean-field-like kinematic axisymmetric dynamo models, where amplitude fluctuations are driven by zero-mean stochastic forcing of the dynamo number controlling the strength of the poloidal source term. In all stochastically forced models considered, the surface poloidal magnetic field is found to have precursor value only if it feeds back into the dynamo loop, which suggests that accurate determination of the magnetic flux budget of the solar polar fields may hold the key to dynamo model-based cycle forecasting. [ABSTRACT FROM AUTHOR]

Observations relevant to current models of the solar dynamo are presented, with emphasis on the history of solar magnetic activity and on the location and nature of the solar tachocline. The problems encountered when direct numerical simulation is used to analyse the solar cycle are discussed, and recent progress is reviewed. Mean field dynamo theory is still the basis of most theories of the solar dynamo, so a discussion of its fundamental principles and its underlying assumptions is given. The role of magnetic helicity is discussed. Some of the most popular models based on mean field theory are reviewed briefly. Dynamo models based on severe truncations of the full MHD equations are discussed. [ABSTRACT FROM AUTHOR]

Two systems of Lorenz-type equations modelling solar magnetic activity are studied: Firstly a low order dynamic system in which the toroidal and poloidal fields are represented by x- and y-coordinates respectively, and the hydrodynamical information is given by the z coordinate. Secondly a complex generalization of the three ordinary differential equations studied by Lorenz. By studying the Poincaré map we give numerical evidence that the flow has an attractor with fractal structure. The period is defined as the time needed for a point on a hyperplane to return to the hyperplane again. The periods are distributed in an interval. For large values of the Dynamo number there is a long tail toward long periods and other interesting comet-like features. These general relations found for periods can further be physically interpreted with improved helioseismic estimates of the parameters used by the dynamical systems. Solar Dynamic Observatory is expected to offer such improved measurements. [ABSTRACT FROM AUTHOR]

Abstract: The differential rotation of the solar corona has been analyzed using as the input data the brightness of the coronal green line Fe xiv 530.3nm for more than five activity cycles. It is found that the character of rotation of the solar corona changes during the activity cycle. Approximately at the middle of the descending branch the differential rotation is weakly pronounced, while the greatest differential gradient is observed at the ascending branch and, occasionally, at the maximum of the cycle. An explanation of this difference has been suggested. The total rotation rate of the corona can be represented as a superposition of two rotation modes (components) – the fast and slow ones. The synodic period of the fast mode near the equator is about 27 days, increasing slightly with latitude. The synodic period of the slow mode exceeds 30 days. The changing relative fraction of these two modes results in variation of the latitude dependence of the observed rotation rate during the activity cycle. The characteristics of two principal types of differential rotation of the solar corona have been determined. The first type consists of the fast mode alone and is established approximately at the middle of the descending branch of the cycle. The second type is the sum of both modes with the fast mode dominating at low latitudes and the slow mode at high latitudes. The results obtained can be used for in-depth study of interaction of the velocity field and dynamo mechanism in the Sun and stars. [Copyright &y& Elsevier]

The running cross-correlation coefficient between solar-cycle amplitudes and rise times at a certain cycle lag is found to vary in time, when using the smoothed monthly-mean sunspot group numbers available for 1610 – 1995. It may be negative or positive for different periods of time. The Waldmeier effect (in which the rise times decrease with amplitude) is also found to be very weak for some cycles. This result represents an observational constraint on solar-dynamo models and can help us better understand the long-term evolution of solar activity. [ABSTRACT FROM AUTHOR]

Together with the main 11-year cycle, solar activity also displays intracycle periodicities. A simple nonlinear model that describes the 11-year solar cycle with subperiodicities can be derived from the usual α – ω dynamo theory in the form of a Van der Pol equation with a forcing term. In this paper the results obtained from the Van der Pol oscillator describing the amplitude modulations and periodicities observed from the data set of the global daily coronal emission of the Fe xiv line at 530.3 nm are presented. [ABSTRACT FROM AUTHOR]

Abstract: It is well accepted that the solar cycle originates from a magnetohydrodynamics dynamo deep inside the Sun. Many dynamo models have long been proposed based on a lot of observational constraints. In this paper, using 342 NSO/Kitt Peak solar synoptic charts we study the solar cycle phases in different solar latitudinal zones to set further constraints. Our results can be summarized as follows: (1) The variability of solar polar regions’ area has a correlation with total unsigned magnetic flux in advance of 5years. (2) The high-latitude region mainly appears unipolar in the whole solar cycle and its flux peak time lags sunspot cycle for 3years. (3) For the activity belt, it is not surprised that its phase be the same as sunspot’s. (4) The flux peak time of the low-latitude region shifts forward with an average gradient of 32.2day/deg. These typical characteristics may provide some hints for constructing an actual solar dynamo. [Copyright &y& Elsevier]

Abstract: Current helicity data for solar active region provide the unique observational information for so-called α-effect which plays a key role in astrophysical dynamos producing large-scale magnetic fields of celestial bodies. We discuss problems arising in comparison between helicity data and predictions of solar dynamo theory. In particular, a significant excess of active regions with the “wrong” sign of helicity is seen to occur just at the beginning of the 11-year solar cycle. We attribute the excess partially to additional current helicity production from the twisting of rising magnetic flux tubes. We demonstrate that the excess is much less pronounced in the twist data. Future perspectives of helicity observations are discussed. [Copyright &y& Elsevier]

Secular variations of solar activity (Gleissberg and Suess cycles) have approximately 80 – 130 and 200 year periods. They are manifested in both observed and proxy data. Here, we show that the basic dynamic features of the Schwabe cycle (asymmetry of its growth and decay phases) and secular cycles (multi-frequency structure and irregular Grand-extremes), as well as a connection between them, can be described by parameter tuning of the electromechanical “toy” dynamo system which has been widely used to model the inversions of the geomagnetic field. An amplitude-frequency diagram for the model magnetic flux has the same shape as the directly observed and reconstructed sunspot area indices. [ABSTRACT FROM AUTHOR]

We discuss the current status of solar dynamo theory and describe the dynamo model developed by our group. The toroidal magnetic field is generated in the tachocline by the strong differential rotation and rises to the solar surface due to magnetic buoyancy to create active regions. The decay of these active regions at the surface gives rise to the poloidal magnetic field by the Babcock-Leighton mechanism. This poloidal field is advected by the meridional circulation first to high latitudes and then down below to the tachocline. Dynamo models based on these ideas match different aspects of observational data reasonably well. [ABSTRACT FROM AUTHOR]

The solar dynamo continues to pose a challenge to observers and theoreticians. Observations of the solar surface reveal a magnetic field with a complex, hierarchical structure consisting of widely different scales. Systematic features such as the solar cycle, the butterfly diagram, and Hale's polarity laws point to the existence of a deep-rooted large-scale magnetic field. At the other end of the scale are magnetic elements and small-scale mixed-polarity magnetic fields. In order to explain these phenomena, dynamo theory provides all the necessary ingredients including the α effect, magnetic field amplification by differential rotation, magnetic pumping, turbulent diffusion, magnetic buoyancy, flux storage, stochastic variations and nonlinear dynamics. Due to advances in helioseismology, observations of stellar magnetic fields and computer capabilities, significant progress has been made in our understanding of these and other aspects such as the role of the tachocline, convective plumes and magnetic helicity conservation. However, remaining uncertainties about the nature of the deep-seated toroidal magnetic field and the α effect, and the forbidding range of length scales of the magnetic field and the flow have thus far prevented the formulation of a coherent model for the solar dynamo. A preliminary evaluation of the various dynamo models that have been proposed seems to favor a buoyancy-driven or distributed scenario. The viewpoint proposed here is that progress in understanding the solar dynamo and explaining the observations can be achieved only through a combination of approaches including local numerical experiments and global mean-field modeling. [ABSTRACT FROM AUTHOR]

Measurements of the general magnetic field of the Sun seen as a star were performed over last 45 years by the CrAO and five other observatories (1968–2012, nearly 23 thousand daily strengths B ). Analysis of the B time series showed that the most substantial long-term period of the field variation is the Hale’s cycle 22 years, which cannot be explained by dynamo theory. It reveals a saw-edged profile, indicating perhaps a cosmic origin of the cycle. [ABSTRACT FROM AUTHOR]

The level of solar magnetic activity, as exemplified by the number of sunspots and by energetic events in the corona, varies on a wide range of timescales. Most prominent is the 11-year solar cycle, which is significantly modulated on longer timescales. Drawing from dynamo theory, together with the empirical results of past solar activity and similar phenomena for solar-like stars, we show that the variability of the solar cycle can be essentially understood in terms of a weakly nonlinear limit cycle affected by random noise. In contrast to ad hoc “toy models” for the solar cycle, this leads to a generic normal-form model, whose parameters are all constrained by observations. The model reproduces the characteristics of the variable solar activity on timescales between decades and millennia, including the occurrence and statistics of extended periods of very low activity (grand minima). Comparison with results obtained with a Babcock–Leighton-type dynamo model confirm the validity of the normal-mode approach. [ABSTRACT FROM AUTHOR]

The variable magnetic field of the solar photosphere exhibits periodic reversals as a result of dynamo activity occurring within the solar interior. We decompose the surface field as observed by both the Wilcox Solar Observatory and the Michelson Doppler Imager into its harmonic constituents, and present the time evolution of the mode coefficients for the past three sunspot cycles. The interplay between the various modes is then interpreted from the perspective of general dynamo theory, where the coupling between the primary and secondary families of modes is found to correlate with large-scale polarity reversals for many examples of cyclic dynamos. Mean-field dynamos based on the solar parameter regime are then used to explore how such couplings may result in the various long-term trends in the surface magnetic field observed to occur in the solar case. [ABSTRACT FROM AUTHOR]