The Waldmeier Effect is the observation that the rise time of a sunspot cycle varies inversely with the cycle amplitude: strong cycles rise to their maximum faster than weak cycles. The shape of the cycle and thus the rise time does not depend on the scale factor of the sunspot number and can thus be used to verify the constancy of the scale factor with time as already noted by Wolfer (1902) and Waldmeier (1978). We extend their analysis until the present using the new SILSO sunspot number (version 2) and group number and confirm that the scale factors have not varied significantly the past 250 years. The effect is also found in sunspot areas, in an EUV (and F10.7) proxy (the daily range of a geomagnetic variation), and in Cosmic Ray Modulation. The result is that solar activity reached similar high values in every one of the (17th?) 18th, 19th, and 20th centuries, supporting the finding that there has been no modern Grand Maximum.
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 th…
We discuss two aspects of the Waldmeier Effect, namely (1) the rise times of sunspot cycles are anti-correlated to their strengths (WE1) and (2) the rates of rise of the cycles are correlated to their strengths (WE2). From analysis of four different data sets we conclude that both WE1 and WE2 exist in all the data sets. We study these effects theoretically by introducing suitable stochastic fluctuations in our regular solar dynamo model.
We study the shape of sunspot cycles using the Wolf sunspot numbers and group sunspot numbers of solar cycles 1-23. We determine the most typical "model" cycles and the most asymmetric cycles, and test the validity of the two Waldmeier rules: the anti-correlation between cycle height and the length of its ascending phase (rule 1), and between cycle height and the length of the preceding cycle (rule 2). We applied the principal component analysis to sunspot cycles and studied the first two components, which describe the average cycle shape and cycle asymmetry, respectively. We also calculated t…
The long-term variability of the sunspot cycle, as recorded by the Wolf numbers, are imprinted in different kinds of statistical relations which relate the cycle amplitudes, duration and shapes. This subject always gets a special attention because it is important for the solar activity forecast. We discuss statistical properties of the mean-field dynamo model with the fluctuating α-effect. Also, we estimate dynamical properties of the model for the long and short time-scale and compare it with the dynamics of the sunspot numbers data sets.
We have analysed the combined Greenwich and Solar Optical Observing Network (SOON) sunspot group data during the period of 1874-2011 and determined variations in the annual numbers (counts) of the small, large and big sunspot groups (these classifications are made on the basis of the maximum areas of the sunspot groups). We found that the amplitude of an even-numbered cycle of the number of large groups is smaller than that of its immediately following odd-numbered cycle. This is consistent with the well known Gnevyshev and Ohl rule or G-O rule of solar cycles, generally described by using the…
The linear relationship between the maximum amplitudes (R$_{max}$) of sunspot cycles and preceding minima (R$_{min}$) is one of the precursor methods used to predict the amplitude of the upcoming solar cycle. In the recent past this method has been subjected to severe criticism. In this communication we show that this simple method is reliable and can profitably be used for prediction purposes. With the 13-month smoothed R$_{min}$ of 1.8 at the beginning, it is predicted that the R$_{max}$ of the ongoing cycle will be around 85$\pm$17, suggesting that Cycle 24 may be of moderate strength. Base…
Temporal variations in the Suns internal velocity field with a periodicity of about 11 years have been observed over the last four decades. The period of these torsional oscillations and their latitudinal propagation roughly coincides with the period and equatorward propagation of sunspots which originate from a magnetohydrodynamic dynamo mechanism operating in the Suns interior. While the solar differential rotation plays an important role in this dynamo mechanism by inducting the toroidal component of magnetic field, the impact of torsional oscillations on the dynamo mechanism and hence the …
The main purpose of this study is the determination of solar minimum date of the new sunspot cycle No 24. It is provided by using of four types of mean daily data values for the period Jan 01. 2006 - Dec 31. 2009: (1) the solar radioindex F10.7; (2) the International sunspot number Ri; (3) the total solar irradiance index (TSI), and (4) the daily number of X-ray flares of classes from "B" to "X" from the soft X-ray GOES satellite channel (0.1 - 0.8 nm). It is found that the mean starting moment of the upward solar activity tendency (the mean solar minimum) is Nov. 06th, 2008. So, the solar cyc…
We study the possibility to reproduce the statistical relations of the sunspot activity cycle, like the so-called Waldmeier relations, the cycle period - amplitude and the cycle rise rate - amplitude relations, by means of the mean field dynamo models with the fluctuating α-effect. The dynamo model includes the long-term fluctuations of the α-effect and two types of the nonlinear feedback of the mean-field on the α-effect including the algebraic quenching and the dynamic quenching due to the magnetic helicity generation. We found that the models are able to reproduce qualitatively and quantita…
More than 70 years ago it was recognised that ionospheric F2-layer critical frequencies $foF2$ had a strong relationship to sunspot number. Using historic datasets from the Slough and Washington ionosondes, we evaluate the best statistical fits of $foF2$ to sunspot numbers (at each Universal Time [UT] separately) in order to search for drifts and abrupt changes in the fit residuals over Solar Cycles 17 - 21. Polynomial fits are made both with and without allowance for the white-light facular area, which has been reported as being associated with cycle-to-cycle changes in the sunspot number - $…
We confirm that the evidence for the Waldmeier effect WE1 (the anti-correlation between rise times of sunspot cycles and their strengths) and the related effect WE2 (the correlation between rise rates of cycles and their strengths) is found in different kinds of sunspot data. We explore whether these effects can be explained theoretically on the basis of the flux transport dynamo models of sunspot cycles. Two sources of irregularities of sunspot cycles are included in our model: fluctuations in the poloidal field generation process and fluctuations in the meridional circulation. We find WE2 to…
We study the behaviour of the van der Pol oscillator when either its damping parameter $μ$ or its nonlinearity parameter $ξ$ is subject to additive or multiplicative random noise. Assuming various power law exponents for the relation between the oscillating variable and the sunspot number, for each case we map the parameter plane defined by the amplitude and the correlation time of the perturbation and mark the parameter regime where the sunspot number displays solar-like behaviour. Solar-like behaviour is defined here as a good correlation between the rise rate and cycle amplitude {\it and} t…
Empirical rules of solar cycle evolution form important observational constraints for the solar dynamo theory. This includes the Waldmeier rule relating the magnitude of a solar cycle to the length of its ascending phase, and the Gnevyshev--Ohl rule clustering cycles to pairs of an even-numbered cycle followed by a stronger odd-numbered cycle. These rules were established as based on the "classical" Wolf sunspot number series, which has been essentially revisited recently, with several revised sets released by the research community. Here we test the robustness of these empirical rules for dif…
We review the evidence for the argument that Rudolf Wolf's calibration of the Sunspot Number is likely to be correct and that Max Waldmeier introduced an upwards jump in the sunspot number in 1945. The combined effect of these adjustments suggests that there has been no secular change in the sunspot number since coming out of the Maunder Minimum ~1700.
A new normalized sunspot-area series has been reconstructed from the series obtained by the Royal Greenwich Observatory and other contemporary institutions for the period 1874 - 2008 and the area series compiled by De la Rue, Stewart, and Loewy from 1832 to 1868. Since the two sets of series do not overlap in time, we used as a link between them the new version of sunspot index number (Version 2) published by SILSO (Sunspot Index and Long-term Solar Observations). We also present a spectral analysis of the normalized area series in search of periodicities beyond the well-known solar cycle of 1…
Waldmeier [1971] found a very tight relationship between the F10.7 solar radio flux and the sunspot number and suggested using the flux for an objective calibration of the sunspot number. He suggested that if this relationship changed later on, the sunspot number should be re-calibrated, assuming that the calibration must have drifted with time. I repeat his analysis using data up to the present and it is, indeed, clear that the relationship has changed significantly. This could be due to a drift of the calibration or to a secular change in the visibility of sunspots, or both.
The increasing interest in understanding stellar magnetic activity cycles is a strong motivation for the development of parameterized starspot models which can be constrained observationally. In this work we develop an empirical tool for the stochastic reconstruction of sunspot cycles, using the average solar properties as a reference. The synthetic sunspot cycle is compared with the sunspot data extracted from the National Geophysical Data Center, in particular using the Kolmogorov-Smirnov test. This tool yields synthetic spot group records, including date, area, latitude, longitude, rotation…
Waldmeier in 1947 introduced a weighting (on a scale from 1 to 5) of the sunspot count made at Zurich and its auxiliary station Locarno, whereby larger spots were counted more than once. This counting method inflates the relative sunspot number over that which corresponds to the scale set by Wolfer and Brunner. Svalgaard re-counted some 60,000 sunspots on drawings from the reference station Locarno and determined that the number of sunspots reported were 'over counted' by 44% on average, leading to an inflation (measured by a weight factor) in excess of 1.2 for high solar activity. In a double…
The Swiss Federal Observatory, which had been founded in 1863 by Rudolf Wolf, was dissolved in connection with the retirement of Max Waldmeier in 1979. The determination of the Zurich sunpot number, which had been a cornerstone activity of the observatory, was then discontinued by ETH Zurich. A smooth transition of the responsibility for the sunspot number from Zurich to Brussels could however be achieved in 1980, through which it was possible to avoid a discontinuity in this important time series. Here we describe the circumstances that led to the termination in Zurich, how Brussels was chose…
Long-lived (>20 days) sunspot groups extracted from the Greenwich Photo- heliographic Results (GPR) are examined for evidence of decadal change. The problem of identifying sunspot groups which are observed on consecutive solar rotations (recurrent sunspot groups) is tackled by first constructing manually an example dataset of recurrent sunspot groups and then using machine learning to generalise this subset to the whole GPR. The resulting dataset of recurrent sunspot groups is verified against previous work by A. Maunder and other Royal Greenwich Observatory (RGO) compilers. Recurrent groups a…
Prediction of the solar cycle is challenging but essential because it drives space weather. Several predictions with varying amplitudes of the ongoing Cycle~25 have been made. We show that an aspect of the Waldmeier effect (WE2), i.e., a strong positive correlation between the rise rate and the amplitude of the cycle, has a physical link with the build-up of the previous cycle's polar field after its reversal. We find that the rise rate of the polar field is highly correlated with the rise rate and the amplitude of the next solar cycle. Thus, the prediction of the amplitude of the solar cycle …
The recalibration of the sunspot number series, the primary long-term record of the solar cycle, requires the recovery of the entire collection of raw sunspot counts collected by the Zurich Observatory for the production of this index between 1849 and 1980. Here, we report about the major progresses accomplished recently in the construction of this global digital sunspot number database, and we derive global statistics of all the individual observers and professional observatories who provided sunspot data over more than 130 years. First, we can announce the full recovery of long-lost source-d…
Discovery of a relation between the decay rate of the Sun's magnetic dipole and the growth rate of the following sunspot cycle: a new precursor for solar cycle prediction
The sunspot number varies roughly periodically with time. However the individual cycle durations and the amplitudes are found to vary in an irregular manner. It is observed that the stronger cycles are having shorter rise times and vice versa. This leads to an important effect know as the Waldmeier effect. Another important feature of the solar cycle irregularity are the grand minima during which the activity level is strongly reduced. We explore whether these solar cycle irregularities can be studied with the help of the flux transport dynamo model of the solar cycle. We show that with a suit…
The long-standing disparity between the sunspot number record and the Hoyt and Schatten (1998, H&S) Group Sunspot Number series was initially resolved by the Clette et al. (2014) revision of the sunspot number and the group number series. The revisions resulted in a flurry of dissenting group number series while the revised sunspot number series was generally accepted. Thus, the disparity persisted and confusion reigned, with the choice of solar activity dataset continuing to be a free parameter. A number of workshops and follow-up collaborative efforts by the community have not yet brought cl…
Long-term sunspot observations are key to understand and predict the solar activities and its effects on the space weather.Consistent observations which are crucial for long-term variations studies,are generally not available due to upgradation/modifications of observatories over the course of time. We present the data for a period of 90 years acquired from persistent observation at the Kodaikanal observatory in India. We use an advanced semi-automated algorithm to detect the sunspots form each calibrated white-light image. Area, longitude and latitude of each of the detected sunspots are deri…
As we are heading towards the next solar cycle, presumably with a relatively small amplitude, it is of significant interest to reconstruct and describe the past grand minima on the basis of actual observations of the time. The Dalton Minimum is often considered one of the grand minima captured in the coverage of telescopic observations. Nevertheless, the reconstructions of the sunspot group number vary significantly, and the existing butterfly diagrams have a large data gap during the period. This is partially because most long-term observations have remained unexplored in historical archives.…
The method of active day fraction (ADF) was proposed recently to calibrate different solar observers to the standard observational conditions. The result of the calibration may depend on the overall level of solar activity during the observational period. This dependency is studied quantitatively using data of the Royal Greenwich Observatory, by formally calibrating synthetic pseudo-observers to the full reference dataset. It is shown that the sunspot group number is precisely estimated by the ADF method for periods of moderate activity, may be slightly underestimated by 0.5\,--\,1.5 groups ($…
We use sunspot group observations from the Royal Greenwich Observatory (RGO) to investigate the effects of intercalibrating data from observers with different visual acuities. The tests are made by counting the number of groups $R_B$ above a variable cut-off threshold of observed total whole-spot area (uncorrected for foreshortening) to simulate what a lower acuity observer would have seen. The synthesised annual means of $R_B$ are then re-scaled to the observed RGO group number $R_A$ using a variety of regression techniques. It is found that a very high correlation between $R_A$ and $R_B$ ($r…
Group sunspot number (GSN) series constitute the longest instrumental astronomical database providing information on solar activity. It is a compilation of observations by many individual observers, and their inter-calibration has usually been performed using linear rescaling. There are multiple published series that show different long-term trends for solar activity. We aim at producing a GSN series, with a non-linear non-parametric calibration. The only underlying assumptions are that the differences between the various series are due to different acuity thresholds of the observers, and that…
The distribution of the sunspot group size (area) and its dependence on the level of solar activity is studied. It is shown that the fraction of small groups is not constant but decreases with the level of solar activity so that high solar activity is largely defined by big groups. We study the possible influence of solar activity on the ability of a realistic observer to see and report the daily number of sunspot groups. It is shown that the relation between the number of sunspot groups as seen by different observers with different observational acuity thresholds is strongly non-linear and ca…
Results. The PCA confirms the existence of the Gnevyshev gap (GG) for solar cycles at about 40% from the start of the cycle. The temporal evolution of sunspot area data for even cycles shows that the GG exists at least at the 95% confidence level for all sizes of sunspots. On the other hand, the GG is shorter and statistically insignificant for the odd cycles of aerial sunspot data. Furthermore, the analysis of sunspot area sizes for even and odd cycles of SC12-SC23 shows that the greatest difference is at 4.2-4.6 years, where even cycles have a far smaller total area than odd cycles. The aver…
The sunspot number is analyzed by using detailed sunspot data, including aspects of observability, sunspot sizes, and proper identification of sunspot groups as discrete entities of the solar activity. The tests show that besides the subjective factors there are also objective causes of the ambiguities in the series of sunspot numbers. To introduce an alternative activity measure the physical meaning of the sunspot number has to be reconsidered. It contains two components whose numbers are governed by different physical mechanisms, this is one source of the ambiguity. This article suggests an …
The 11-year solar cycle is the dominant pattern of solar activity reflecting the oscillatory dynamo mechanism in the Sun. Solar cycles were directly observed since 1700, while indirect proxies suggest their existence over a much longer period of time but generally without resolving individual cycles and their continuity. Here we reconstruct individual cycles for the last millennium using recent 14C data and state-of-the-art models. Starting with the 14C production rate determined from the so far most precise measurements of radiocarbon content in tree rings, solar activity is reconstructed in …
We show that the Active-Day-Fraction calibration method (Willamo et al. [2017]) fails for Gustav Spörer's sunspot group observations. Spörer was labeled a 'perfect observer' on account of his 'observational threshold SS area' being determined to be equal to zero, based on the assumption that the observer can see and report all the groups with the area larger than SS, while missing all smaller groups. So, Spörer could apparently, according to the ADF calibration method, see and report all groups, regardless of size and should never miss any. This suggests a very direct test: compute the yearly …
We compare four sunspot-number data sequences against geomagnetic and terrestrial auroral observations. The comparisons are made for the original SIDC composite of Wolf-Zurich-International sunspot number [$R_{ISNv1}$], the group sunspot number [$R_{G}$] by Hoyt and Schatten (Solar Phys., 1998), the new "backbone" group sunspot number [$R_{BB}$] by Svalgaard and Schatten (Solar Phys., 2016), and the "corrected" sunspot number [$R_{C}$] by Lockwood at al. (J.G.R., 2014). Each sunspot number is fitted with terrestrial observations, or parameters derived from terrestrial observations to be linear…
We report progress on the ongoing recalibration of the Wolf sunspot number (SN) and Group sunspot number (GN) following the release of version 2.0 of SN in 2015. This report constitutes both an update of the efforts reported in the 2016 Topical Issue of Solar Physics and a summary of work by the International Space Science Institute (ISSI) International Team formed in 2017 to develop optimal SN and GN re-construction methods while continuing to expand the historical sunspot number database. Significant progress has been made on the database side while more work is needed to bring the various p…
Photographic images are valuable data resources for studying long term changes in the solar magnetic field and its influence on the Earth's climate and weather. We digitized more than 100 years of white light images stored in photographic plates and films that are available at Kodaikanal observatory starting from 1904. The digitized images were calibrated for relative plate density and aligned in such a way that the solar north is in upward direction. A semi-automated sunspot detection technique was used to identify the sunspots on the digitized images. In addition to describing the calibratio…
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 (1998a,b). Unfortunately, those two series do not match by various aspects, inducing confusions and contradictions when used in c…
The Sunspot Number, created by R.Wolf in 1849, provides a direct long-term record of solar activity from 1700 to the present. In spite of its central role in multiple studies of the solar dynamo and of the past Sun-Earth relations, it was never submitted to a global critical revision. However, various discrepancies with other solar indices recently motivated a full re-calibration of this series. Based on various diagnostics and corrections established in the framework of several Sunspot Number Workshops and described in Clette et al. 2014, we assembled all corrections in order to produce a new…
New sunspot data composites, some of which are radically different in the character of their long-term variation, are evaluated over the interval 1845-2014. The method commonly used to calibrate historic sunspot data, relative to modern-day data, is "daisy-chaining", whereby calibration is passed from one data subset to the neighbouring one, usually using regressions of the data subsets for the intervals of their overlap. Recent studies have illustrated serious pitfalls in these regressions and the resulting errors can be compounded by their repeated use as the data sequence is extended back i…
Oscillating dynamo models with a time-nonlocal $α$ effect
Leibniz-Institut für Astrophysik Potsdam (AIP)（波茨坦阿斯特罗物理研究所）；University of Potsdam（波茨坦大学）；INAF, Catania Astrophysical Observatory（意大利国家天体物理研究所，卡塔尼亚天文台）
Sunspot area data play an important role in the studies of solar activity and its long-term variations. In order to reveal real long-term solar variations precise homogeneous sunspot area databases should be used. However, the measured areas may be burdened with systematic deviations, which may vary in time. Thus, there is a need to investigate the long-term variation of sunspot area datasets and to determine the time-dependent cross-calibration factors. In this study, we investigate the time-dependent differences between the available long-term sunspot databases. Using the results, we estimat…
Although the sunspot-number series have existed since the mid-19th century, they are still the subject of intense debate, with the largest uncertainty being related to the "calibration" of the visual acuity of individual observers in the past. Daisy-chain regression methods are applied to inter-calibrate the observers which may lead to significant bias and error accumulation. Here we present a novel method to calibrate the visual acuity of the key observers to the reference data set of Royal Greenwich Observatory sunspot groups for the period 1900-1976, using the statistics of the active-day f…
