The Solar Cycle is reviewed. The 11-year cycle of solar activity is characterized by the rise and fall in the numbers and surface area of sunspots. We examine a number of other solar activity indicators including the 10.7 cm radio flux, the total solar irradiance, the magnetic field, flares and coronal mass ejections, geomagnetic activity, galactic cosmic ray fluxes, and radioisotopes in tree rings and ice cores that vary in association with the sunspots. We examine the characteristics of individual solar cycles including their maxima and minima, cycle periods and amplitudes, cycle shape, and the nature of active latitudes, hemispheres, and longitudes. We examine long-term variability including the Maunder Minimum, the Gleissberg Cycle, and the Gnevyshev-Ohl Rule. Short-term variability includes the 154-day periodicity, quasi-biennial variations, and double peaked maxima. We conclude with an examination of prediction techniques for the solar cycle.

The Solar Cycle

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 \(\alpha\) 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 \(\alpha\) 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.

https://amor.cms.hu-berlin.de/~ossendrm/solardynamo.pdf

The solar dynamo

Our knowledge of the long-term evolution of solar activity and of its primary modulation, the 11-year cycle, largely depends on a single direct observational record: the visual sunspot counts that retrace the last 4 centuries, since the invention of the astronomical telescope. Currently, this activity index is available in two main forms: the International Sunspot Number initiated by R. Wolf in 1849 and the Group Number constructed more recently by Hoyt and Schatten (Sol. Phys. 179:189–219, 1998a, 181:491–512, 1998b). Unfortunately, those two series do not match by various aspects, inducing confusions and contradictions when used in crucial contemporary studies of the solar dynamo or of the solar forcing on the Earth climate. Recently, new efforts have been undertaken to diagnose and correct flaws and biases affecting both sunspot series, in the framework of a series of dedicated Sunspot Number Workshops. Here, we present a global overview of our current understanding of the sunspot number calibration.

Revisiting the Sunspot Number. A 400-Year Perspective on the Solar Cycle

The past few decades have seen dramatic progress in our understanding of solar interior dynamics, prompted by the relatively new science of helioseismology and increasingly sophisticated numerical models. As the ultimate driver of solar variability and space weather, global-scale convective motions are of particular interest from a practical as well as a theoretical perspective. Turbulent convection under the influence of rotation and stratification redistributes momentum and energy, generating differential rotation, meridional circulation, and magnetic fields through hydromagnetic dynamo processes. In the solar tachocline near the base of the convection zone, strong angular velocity shear further amplifies fields which subsequently rise to the surface to form active regions. Penetrative convection, instabilities, stratified turbulence, and waves all add to the dynamical richness of the tachocline region and pose particular modeling challenges. In this article we review observational, theoretical, and computational investigations of global-scale dynamics in the solar interior. Particular emphasis is placed on high-resolution global simulations of solar convection, highlighting what we have learned from them and how they may be improved.

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Large-Scale Dynamics of the Convection Zone and Tachocline

We construct a predictive tool based on a Babcock-Leighton-type flux-transport dynamo model of a solar cycle, run the model by updating the surface magnetic source using old cycles' data since cycle 12, and show that the model can correctly simulate the relative peaks of cycles 16-23. The simulations use the first four cycles to load the meridional circulation conveyor belt to create the Sun's memory about its past magnetic fields. Extending the simulation into the future, we predict that cycle 24 will be 30%-50% stronger than the current cycle 23. These simulations and predictions are robust for a wide range of convection zone magnetic diffusivity values between 3 × 1010 and 2 × 1011 cm2 s-1. Our model predictions are the same for three different treatments of the unknown surface magnetic source for the cycles to be predicted, namely (1) assuming some cyclic pattern, (2) incorporating zero surface source, or (3) including a surface source derived from the self-excited version of the dynamo model. Technique 3, for treating the surface source for cycles to be predicted, also shows significant skill in predicting two cycles ahead. Analyzing the evolution of magnetic field patterns over a full magnetic cycle, we show that the key to success of our prediction model lies in the formation of a seed for producing cycle n from the combination of latitudinal fields at high latitudes from three past cycles, n - 1, n - 2, and n - 3, instead of the previous cycle's polar fields. These results have many implications for both solar and stellar dynamo modeling.

Simulating and Predicting Solar Cycles Using a Flux-Transport Dynamo

Deenition The term solar cycle refers to a quasi-periodic variation with a period of about 11 years, visible in many of the Sun's observables. The solar cycle is most easily observed in indices directly related to the Sun's magnetic eld, such as sunspots. During the last decades, solar-cycle variations have also been found in many other aspects of the Sun (irradiance, surface ows, coronal shape, oscillation frequencies, etc). The modulation amplitudes may vary widely between diierent indices. It is minute in visible light, and this explains why the solar cycle is not visible to a casual naked-eye observer. Elsewhere (in the far ultraviolet, X-rays, radio frequencies) the modulation amplitude is large (Table 1). The prime cause of the solar cycle is a quasi-periodic oscillation of the solar magnetic field. Magnetic-eld indices The oldest and best known index of the solar cycle is provided by sunspots, whose discovery for Europe around 1610 is associated with the names of Fabricius, Galileo and Scheiner. The commonest sunspot index is the Z urich sunspot number R = K(10g + f), where g is the number of sunspot groups, f is the number of individual spots and K is a coeecient that corrects for diierences in the quality of the observations. Usually one derives an annual average R, shown in Fig. 1. The most striking feature of the sunspot record is the Schwabe cycle of about 11 years, discovered by Hein-rich Samuel Schwabe in 1843. By convention, the cycle that began in 1755 is assigned number 1; hence cycle 23 began in 1997. Cycle shape. A typical sunspot cycle in Fig. 1 is characterized by a sharp rise from minimum to maximum, lasting 3-6 years (on average 4.8). The duration of the rise phase is anticorrelated with the height of the maximum (Waldmeier's rule). The maximum is followed by a gradual decline lasting 5-8 years (on average 6.2). Whereas the maxima of the Z urich number are typically well-deened, this is not the case for some other solar activity indices. For example, it has been suggested that the number of large sunspots has a double-peaked maximum. Most indices vary roughly in phase with the sunspot number, and therefore the sunspot extrema are also referred to as the solar minimum and the solar maximum. On the other hand, some coronal indices are clearly out of phase with the sunspot cycle, and they peak at the sunspot …

Using Greenwich data (1879–1976) and SOON/NOAA data (1977–2002) on sunspot groups we found the following results: (i) The Sun's mean (over all the concerned cycles during 1879–1975) equatorial rotation rate (A) is significantly larger (≈0.1%) in the odd-numbered sunspot cycles (ONSCs) than in the even-numbered sunspot cycles (ENSCs). The mean rotation is significantly (≈10%) more differential in the ONSCs than in the ENSCs. North–south difference in the mean equatorial rotation rate is larger in the ONSCs than in the ENSCs. North–south difference in the mean latitude gradient of the rotation is significant in the ENSCs and insignificant in the ONSCs. (ii) The known very large decrease in A from cycle 13 to cycle 14 is confirmed. The amount of this decrease in the mean A was about 0.017 μrad s−1. Also, we find that A decreased from cycle 17 to cycle 18 by about 0.008 μrad s−1 and from cycle 21 to cycle 22 by about 0.016 μrad s−1. From cycle 13 to cycle 14 the decrease in A was more in the northern hemisphere than in the southern hemisphere, it is opposite in the later two epochs. The time gap between the consecutive drops in A is about 44 years, suggesting the existence of a `44-yr' cycle or `double Hale cycle' in A. The time gap between the two large drops, viz., from cycle 13 to cycle 14 and from cycle 21 to cycle 22, is about 90 years (Gleissberg cycle). We predict that the next drop (moderate) in A will be occurring from cycle 25 to cycle 26 and will be followed by a relatively large-amplitude `double Hale cycle' of sunspot activity. (iii) Existence of a 90-yr cycle is seen in the cycle-to-cycle variation of the latitude gradient (B). A weak 22-yr modulation in B seems to be superposed on the relatively strong 90-yr modulation. (iv) The coefficient A varies significantly only during ONSCs and the variation has maximum amplitude in the order of 0.01 μrad s−1 around activity minima. (v) There exists a good anticorrelation between the mean variation of B during the ONSCs and that during the ENSCs, suggesting the existence of a `22-yr' periodicity in B. The maximum amplitude of the variation of B is of the order of 0.05 μrad s−1 around the activity minima. (vi) It seems that the well-known Gnevyshev and Ohl rule of solar activity is applicable also to the cycle-to-cycle amplitude modulation of B from cycle 13 to cycle 20, but the cycles 12 (in the northern hemisphere, Greenwich data) and 21 (in both hemispheres, SOON/NOAA data) seem to violate this rule in B. And (vii) All the aforesaid statistically significant variations in A and B seem to be related to the approximate 179-yr cycle, 1811–1989, of variation in the Sun's motion about the center of mass of the solar system.

Long-Term Variations in the Solar Differential Rotation

We have looked for periodicities in solar differential rotation on time scales shorter than the 11-year solar cycle through the power- spectrum analysis of the differential rotation parameters determined from Mt. Wilson velocity data (1969–1994) and Greenwich sunspot group data (1879–1976). We represent the differential rotation by a set of Gegenbauer polynomials (ω(φ)= \(\bar A\) + \(\bar B\) (5sin2φ−1)+ \(\bar C\) (21sin4φ−14sin2φ+1)). For the Mt. Wilson data, we focus on observations obtained after 1981 due to the reduced instrumental noise and have binned the data into intervals of 19 days. We calculated annual averages for the sunspot data to reduce the uncertainty and corrected for outliers occuring during solar cycle minima. The power spectrum of the photospheric ‘mean rotation’ \(\bar A\) , determined from the velocity data during 1982–1994, shows peaks at the periods of 6.7–4.4 yr, 2.2 ± 0.4 yr, 1.2 ± 0.2 yr, and 243 ± 10 day with ≥99.9% confidence level, which are similar to periods found in other indicators of solar activity suggesting that they are of solar origin. However, this result has to be confirmed with other techniques and longer data sets. The 11-yr periodicity is insignificant or absent in \(\bar A\) . The power spectra of the differential rotation parameters \(\bar B\) and \(\bar C\) , determined from the same subset, show only the solar cycle period with a ≥99.9% confidence level.

Short-term periodicities of the sun's ‘mean’ and differential rotation

The solar equatorial rotation rate, determined from sunspot group data during the period 1879–2004, decreased over the last century, whereas the level of activity has increased considerably. The latitude gradient term of the solar rotation shows a significant modulation of about 79 year, which is consistent with what is expected for the existence of the Gleissberg cycle. Our analysis indicates that the level of activity will remain almost the same as the present cycle during the next few solar cycles (i.e., during the current double Hale cycle), while the length of the next double Hale cycle in sunspot activity is predicted to be longer than the current one. We find evidence for the existence of a weak linear relationship between the equatorial rotation rate and the length of sunspot cycle. Finally, we find that the length of the current cycle will be as short as that of cycle 22, indicating that the present Hale cycle may be a combination of two shorter cycles.

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Long-term variations in solar differential rotation and sunspot activity

We have studied the temporal variations in the north–south asymmetries of the differential rotation parameters A, B, and the ‘mean’ rotation rate Ā, by determining their values from Greenwich data for sunspot groups (1879–1976) in the northern and southern hemispheres, during moving time intervals of lengths 3 yr and 5 yr, successively displaced by 1 yr. The variation in the north–south asymmetry (Ā\(a \)) of Ā is similar to the variation in the asymmetry (B\(a \)) of B but with opposite sign. These variations of Ā\(a \) and B\(a \) may represent components of an anti-symmetric torsional oscillation which are in opposite phase with each other.

J. Javaraiah

Papers

The Solar Cycle

Long-Term Variations in the Solar Differential Rotation

Short-term periodicities of the sun's ‘mean’ and differential rotation

Long-term variations in solar differential rotation and sunspot activity

Periodicities in the north–south asymmetry of the solar differential rotation and surface magnetic field