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To investigate the differential rotation of sunspot groups using the Greenwich data, we propose an approach based on a statistical analysis of the histograms of particular longitudinal velocities in different latitude intervals. The general statistical velocity distributions for all such intervals are shown to be described by two rather than one normal distribution, so that two fundamental rotation modes exist simultaneously: fast and slow. The differentiality of rotation for the modes is the same: the coefficient at sin2 in Faye’s law is 2.87-2.88 deg/day, while the equatorial rotation rates differ significantly, 0.27 deg/day. On the other hand, an analysis of the longitudinal velocities for the previously revealed two differing populations of sunspot groups has shown that small short-lived groups (SSGs) are associated with the fast rotation mode, while large long-lived groups (LLGs) are associated with both fast and slow modes. The results obtained not only suggest a real physical difference between the two populations of sunspots but also give new empirical data for the development of a dynamo theory, in particular, for the theory of a spatially distributed dynamo. [ABSTRACT FROM AUTHOR]

We investigate the dynamics of solar activity using a nonlinear one-dimensional dynamo model and a phenomenological equation for the evolution ofWolf 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 behaviour 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 yr 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 per cent correlation over all intervals of observation (about 270 yr). 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 dependences are in good agreement with observations. [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]

The paper by Popova et al. presents an oversimplified mathematical model of solar activity with a claim of predicting/postdicting it for several millennia ahead/backwards. The work contains several flaws devaluating the results: (1) the method is unreliable from the point of view of signal processing (it is impossible to make harmonic predictions for thousands of years based on only 35 years of data) and lacks quality control; (2) the result of post-diction apparently contradicts the observational data. (3) theoretical speculations make little sense. To summarize, a multi-harmonic mathematical model, hardly related to full solar dynamo theory, is presented, which is not applicable to realistic solar conditions because of the significant chaotic/stochastic intrinsic component and strong non-stationarity of solar activity. The obtained result is apparently inconsistent with the data in the past and thus cannot be trusted for the future predictions. [ABSTRACT FROM AUTHOR]

Solar activity slowly and irregularly decreases from the first spotless day (FSD) in the declining phase of the old sunspot cycle and systematically, but also in an irregular way, increases to the new cycle maximum after the last spotless day (LSD). The time interval between the first and the last spotless day can be called the passive interval (PI), while the time interval from the last spotless day to the first one after the new cycle maximum is the related active interval (AI). Minima of solar cycles are inside PIs, while maxima are inside AIs. In this article, we study the properties of passive and active intervals to determine the relation between them. We have found that some properties of PIs, and related AIs, differ significantly between two group of solar cycles; this has allowed us to classify Cycles 8 - 15 as passive cycles, and Cycles 17 - 23 as active ones. We conclude that the solar activity in the PI declining phase (a descending phase of the previous cycle) determines the strength of the approaching maximum in the case of active cycles, while the activity of the PI rising phase (a phase of the ongoing cycle early growth) determines the strength of passive cycles. This can have implications for solar dynamo models. Our approach indicates the important role of solar activity during the declining and the rising phases of the solar-cycle minimum. [ABSTRACT FROM AUTHOR]

Aims. The Sun shows strong variability in its magnetic activity, from Grand minima to Grand maxima, but the nature of the variability is not fully understood, mostly because of the insufficient length of the directly observed solar activity records and of uncertainties related to long-term reconstructions. Here we present a new adjustment-free reconstruction of solar activity over three millennia and study its different modes. Methods. We present a new adjustment-free, physical reconstruction of solar activity over the past three millennia, using the latest verified carbon cycle, 14C production, and archeomagnetic field models. This great improvement allowed us to study different modes of solar activity at an unprecedented level of details. Results. The distribution of solar activity is clearly bi-modal, implying the existence of distinct modes of activity. The main regular activity mode corresponds to moderate activity that varies in a relatively narrow band between sunspot numbers 20 and 67. The existence of a separate Grand minimum mode with reduced solar activity, which cannot be explained by random fluctuations of the regular mode, is confirmed at a high confidence level. The possible existence of a separate Grand maximum mode is also suggested, but the statistics is too low to reach a confident conclusion. Conclusions. The Sun is shown to operate in distinct modes - a main general mode, a Grand minimum mode corresponding to an inactive Sun, and a possible Grand maximum mode corresponding to an unusually active Sun. These results provide important constraints for both dynamo models of Sun-like stars and investigations of possible solar influence on Earth's climate. [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]

The spatial (latitude) distribution of sunspots is studied, including its dependence on solar activity. It is shown that the latitude distributions of sunspots for a given year can be approximately described by the normal law, with its variance being a linear function of the current level of solar activity. Thus, an increase in activity is accompanied by an expansion of the zone of solar activity, in good agreement with earlier results. As the solar activity increases, the width of the zone of sunspot generation and the latitude maximum of the sunspot density grow somewhat more slowly than the number of sunspots, in agreement with observations. The results obtained can be used to reconstruct the spatial distributions of sunspots in the past, interpret the magnetic activity of stars, and address the requirements of the dynamo theory in the form of constraints imposed on models of cyclicity. [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]

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]

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]

Photospheric ephemeral regions (EPRs) cover the Sun like a magnetic carpet. From this, we update the Babcock – Leighton solar dynamo. Rather than sunspot fields appearing in the photosphere de novo from eruptions originating in the deep interior, we consider that sunspots form directly in the photosphere by a rapid accumulation of like-sign field from EPRs. This would only occur during special circumstances: locations and times when the temperature structure is highly superadiabatic and contains a large subsurface horizontal magnetic field (only present in the Sun’s lower latitudes). When these conditions are met, superadiabatic percolation occurs, wherein an inflow and downflow of gas scours the surface of EPRs to form active regions. When these conditions are not met, magnetic elements undergo normal percolation, wherein magnetic elements move about the photosphere in Brownian-type motions. Cellular automata (CA) models are developed that allow these processes to be calculated and thereby both small-scale and large-scale models of magnetic motions can be obtained. The small-scale model is compared with active region development and Hinode observations. The large-scale CA model offers a solar dynamo, which suggests that fields from decaying bipolar magnetic regions (BMRs) drift on the photosphere driven by subsurface magnetic forces. These models are related to observations and are shown to support Waldmeier’s findings of an inverse relationship between solar cycle length and cycle size. Evidence for significant amounts of deep magnetic activity could disprove the model presented here, but recent helioseismic observations of “butterfly patterns” at depth are likely just a reflection of surface activity. Their existence seems to support the contention made here that the field and flow separate, allowing cool, relatively field-free downdrafts to descend with little field into the nether worlds of the solar interior. There they heat by compression to form a hot solar-type Santa Ana wind deep below active regions. [ABSTRACT FROM AUTHOR]

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]

The north – south (N – S) asymmetry of solar activity is investigated by using the data on coronal green-line brightness and total number and total area of sunspots over the period of 1939 – 2001. Typical time variations of the N – S asymmetry are found to be consonant in these indices. Quasi-biennial oscillations (QBO) of solar activity are well recognizable in the N – S asymmetry of the examined indices. Moreover, the QBO are much better manifested in the N – S asymmetry of the individual indices than in the original (N plus S) indices. The time variations of relative QBO power are synchronous for the N – S asymmetry of various solar activity indices whereas such a synchronization is weaker for the indices themselves. It is revealed that the relative QBO power found in the N – S asymmetry of the studied indices has a negative correlation with the value of the N – S asymmetry itself. The findings indicate that the N – S asymmetry should be regarded as a fundamental phenomenon of solar activity similarly manifested in different activity indices. These findings should be taken into account when any dynamo theory of solar activity is constructed. [ABSTRACT FROM AUTHOR]

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]

We calculate helicities of solar active regions based on the idea that poloidal flux lines get wrapped around a toroidal flux tube rising through the convection zone, thereby giving rise to the helicity. We use our solar dynamo model based on the Babcock-Leighton α-effect to study how helicity varies with latitude and time. [ABSTRACT FROM AUTHOR]

Sunspot drawings made by Galileo Galilei in 1612 are used to derive the law of differential rotation at that time. The main interest of the work is during the time of observations, just at the beginning of telescopic observations and some decades before the Maunder Minimum (1645 – 1715), a period where the sunspots almost disappeared from the solar surface. For this purpose we have carried out careful corrections of the different sources of errors derived from the observing technique. By comparing with other results of the same century, a significant difference is only detected by comparing with data corresponding to the deep Maunder Minimum (Paris Observatory drawings). The characteristics of the solar differential rotation, and extrapolating the behavior of solar activity, did not differ before or after the Maunder Minimum. We also include an analysis of hitherto ignored sunspot drawings by N. Bion made in October and November 1672. [ABSTRACT FROM AUTHOR]

We describe the properties of the Sun, those of its Active Regions (Centres of Activity, ARs or CAs) and the 11 and 22-year cycles as observed via the variable numbers of sunspots. We describe the variations with time of the solar irradiance and of the flux of ejected magnetised plasma. We discuss the probable cause of solar variability. Planetary influences are ruled out; the variability is intrinsic and is described by the solar dynamo. The dynamo is characterised by internal toroidal and more superficial poloidal fields, interchanging and alternating in a 22-year periodicity. From these two components in the solar magnetic fields emanate two possible scenarios for the Sun-climate interaction. Solar irradiance variations are related to those in the solar toroidal magnetic fields. The fraction of the solar irradiance that reaches the Earth's ground level and low troposphere is emitted by the solar photosphere. That fraction does not significantly vary since the quiet photosphere does not significantly vary during the cycle. The variable part of the solar radiation flux is mainly emitted by the chromospheric parts of the CAs. That radiation component does not reach the Earth's troposphere since it is absorbed in the higher, stratospheric terrestrial layers. Tropospheric solar-driven variations should therefore be due to stratosphere–troposphere coupling. The Group Sunspot number R Gs is a proxy for the variable irradiance component and for the toroidal field variations. Ejected solar plasma clouds such as the Coronal Mass Ejections (CMEs) and plasma ejected from Ephemeral Solar Regions and from the polar facular regions are related to variations in the poloidal magnetic fields. On the average they have their maximum intensity about a year after the maximum number of spots: we call this interval the Energetic Emissions Delay. These gas clouds fill the heliosphere with magnetised plasma. Thus, by emitting magnetised plasma, the Sun influences the Earth's atmosphere indirectly, by heliospheric modulation of the component of the galactic cosmic radiation (CR) that reaches tropospheric levels. Modulation is only important for cosmic ray particles with energies below about 50 GeV. Cosmic ray ionisation plays a minor role at ground level but it is the predominant ionising agent in higher atmospheric layers, already above a few kilometres. The amplitudes of the CR variations depend on those of the solar cycle. The atmospheric rate of ionisation varies with CR-intensity. A current hypothesis is that the variable ionisation may affect the degree of cloudiness. Cosmogenic radionuclides such as 10Be are proxies for this influence and for the poloidal field variations. The R G and cosmogenic radionuclide proxies, although loosely correlated, refer to the two different aspects of the solar dynamo with their different terrestrial effects; they do not reach maximum intensity simultaneously and should therefore neither be confused nor be interchanged. Cases have occurred in which the one varied strongly while the other did hardly or not at all. The explanation must be intrinsic in dynamo theory. There is a significant solar signal in the troposphere; it depends on latitude and longitude. A physical investigation of the cause of the Sun-climate relationship based on one unique Δ T(time)-curve, assumed valid for the whole Earth's surface, is therefore basically incorrect. Never during the past 10,000 years has the Sun been as active in ejecting magnetised plasma as during the past few decades. Estimates suggest that the level of solar activity may recently have passed its maximum and that it may decrease in coming decades. Solar activity is variable with six well-determined quasi-periodicities. Attempts to theoretically describe the solar dynamo have so far succeeded only in explaining the qualitative aspects. They fail in a numerical description and notably in one that would permit one to forecast solar activity with acceptable precision. This is so because the solar dynamo is a non-linear system that occasionally shows phase catastrophes. It is a quasi-periodic engine with the properties of deterministic chaos. “The future of such a chaotic system is intrinsically unpredictable”. [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]