Waldmeier Effect Research Articles

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

ABSTRACT Sunspots have been observed for over four centuries and the magnetic nature of sunspot cycles has been known for about a century; however, some of its underlying physics still remain elusive. It is known that the solar magnetic cycle involves a recycling of magnetic flux between the poloidal and toroidal components of the magnetic field, that manifests as the solar dipole and sunspots, respectively. Here, we report the discovery of a new relationship between the rise rate of the sunspot cycle and the decay rate of the solar (axial) dipole moment. This provides an extension to the Waldmeier effect in sunspot cycles and points to the existence of a causal connection between the aforementioned physical quantities, which can be succinctly stated as the decay rate of the Sun’s dipole moment is related to the rate of rise of the following sunspot cycle. We demonstrate how one may take advantage of this new relationship to predict the timing of the sunspot cycle. Our analysis indicates solar cycle 25 is expected to be a weak-moderate cycle, peaking in $2024.00_{-0.49}^{+0.68}$.

Physical link of the polar field buildup with the Waldmeier effect broadens the scope of early solar cycle prediction: Cycle 25 is likely to be slightly stronger than Cycle 24

ABSTRACT 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 buildup 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 can be made just a few years after the reversal of the previous cycle’s polar field, thereby extending the scope of the solar cycle prediction to much earlier than the usual time. Our prediction of Cycle 25 based on the rise rate of the previous polar field is 137 ± 23, which is quite close to the prediction 138 ± 26 based on the WE2 computed from the available 2 yr sunspot data of the ongoing cycle.

A prediction for the 25th solar cycle maximum amplitude

AbstractThe minimum–maximum method, belonging to the precursor class of the solar activity forecasting methods, is based on a linear relationship between relative sunspot number in the minimum and maximum epochs of solar cycles. In the present analysis, we apply a modified version of this method using data not only from the minimum year, but also from a couple of years before and after the minimum. The revised 13‐month smoothed monthly total sunspot number data set from Sunspot Index and Long‐term Solar Observations/SIDC is used. Using data for solar cycle nos. 1–24 the largest correlation coefficient () is obtained when correlating activity level 3 years before solar cycle minimum with the subsequent maximum (), independent of inclusion or exclusion of the solar cycle no. 19. For the next solar maximum of the cycle no. 25, we predict: . Our results indicate that the next solar maximum (of the cycle no. 25) will be of the similar amplitude as the previous one, or even something lower. This is in accordance with the general middle‐term lowering of the solar activity after the secular maximum in the 20th century and consistent with the Gleissberg period of the solar activity. The reliability of the “3 years before the minimum” predictor is experimentally justified by the largest CC and verified with the Student t‐test. It is satisfactorily explained with the two empirical well‐known findings: the extended solar cycle and the Waldmeier effect. Finally, we successfully reproduced the maxima of the last four solar cycles, nos. 21–25, using the 3 years before the minimum method.

Analysis of Solar Hemispheric Chromosphere Properties using the Kodaikanal Observatory Ca–K Index

Abstract The Kodaikanal Observatory has provided long-term synoptic observations of chromospheric activities in the Ca ii K line (393.34 nm) since 1907. This article investigates temporal and periodic variations of the hemispheric Ca–K-index time series in the low-latitude zone (±40°), utilizing the recently digitized photographic plates of Ca–K images from the Kodaikanal Observatory for the period of 1907–1980. We find that the temporal evolution of the Ca–K index differs from one hemisphere to another, with the solar cycle peaking at different times in the opposite hemisphere, except for cycles 14, 15, and 21, when the phase difference between the two hemispheres was not significant. The monthly averaged data show a higher activity in the northern hemisphere during solar cycles 15, 16, 18, 19, and 20, and in the southern hemisphere during cycles 14, 17, and 21. We notice an exponentially decaying distribution for each hemisphere’s Ca–K index and the whole solar disk. We explored different midterm periodicities of the measured Ca–K index using the wavelet technique, including Rieger-type and quasi-biennial oscillations on different timescales present in the time series. We find a clear manifestation of the Waldmeier effect (stronger cycles rise faster than the weaker ones) in both the hemispheres separately and the whole disk in the data. Finally, we have found the presence of the Gnevyshev gap (time interval between two cycle maxmima) in both the hemispheric data during cycles 15 to 20. Possible interpretations of our findings are discussed with the help of existing theoretical models and observations.

Average annual total sunspot area in the last 410 yr: the most probable values and limits of their uncertainties

ABSTRACT The aim of this work is to create a long (410-yr) series of average annual total sunspot areas AR – a physically based index of sunspot activity. We use telescopic observations of the AR index in 1832–1868 and 1875–2020, as well as the relationship between AR and long series of sunspot indices SN (international sunspot numbers, version 2.0) and sunspot groups GN (Svalgaard and Schatten’s version of group sunspot numbers). The Royal Greenwich Observatory series after 1976 is extended by the Kislovodsk Mountain Astronomical Station data. When reconstructing AR from SN, it is taken into account that the function AR = f(SN) has a non-linear systematic character and uncertainty associated with the heterogeneity of these indices. Therefore, in addition to modelling the most probable AR values, predictive limits of reconstruction uncertainty are determined. In the interval 1610–1699 we carried out the reconstruction on the basis of the GN series using the previously proposed decomposition in pseudo-phase space (DPS) method. The resulting series NO21y is freely available online. We show that for this series the empirical Gnevyshev–Ohl rule and Waldmeier effect are fulfilled. Wavelet analysis reveals periodicities of 8.4–13.8 yr for the main cycle (with a sharp decrease of the period before the global Maunder and Dalton minima) and a two-component Gleissberg cycle with typical periods of 50–60 and 90–110 yr.

Investigation of the Hemispheric Asymmetry in Solar Flare Index During Solar Cycle 21 – 24 from the Kandilli Observatory

The hemispheric asymmetry of the solar-flare index during 1976 – 2018 from the Kandilli Observatory is studied in this investigation. The temporal duration covers Solar Cycles 21 – 23 and almost the whole of Solar Cycle 24. Different methodologies, such as cross-correlation analysis, rescaled-range analysis, empirical mode decomposition, and date-compensated discrete Fourier transform, have been used on the hemispheric solar-flare index as well as on absolute asymmetry data to study various inherent characteristics. We observed that: i) the temporal characteristics in the northern and southern hemispheres are different during the progression of a solar cycle; ii) the T-test indicates that Solar Cycles 21 and 23 do not have any dominant hemispheric effect, whereas Solar Cycle 22 and 24 have South-dominated hemispheric characteristics; iii) the southern hemisphere is leading by ten, three, and one months during Solar Cycles 21, 22, and 24, respectively, and for Solar Cycle 23 the hemispheres are in phase; iv) anti-persistence as well as short memory-dependent characteristics are present in both the hemispheric solar-flare index and the absolute asymmetry data; v) all of the time-series data have well-known periods of 11 years and 27 days as well as short-term periods around 7 days and 14 days. Apart from those, several mid-term periodicities such as the Rieger periodicity and quasi-biennial oscillations (QBOs) are also found in both hemispheric solar-flare index as well as absolute asymmetry index data; vi) the Waldmeier effect is also validated using solar-flare-index data. These results will enrich our knowledge about the distribution of hemispheric asymmetry in solar-flare-index data and may reveal some valuable points about asymmetry behaviors.

The solar cycle: predicting the peak of solar cycle 25

Motivated by a successful prediction on the peak of solar cycle 24 (81.7, comparable to the observed 81.9, Du in Astrophys. Space Sci. 338:9, 2012), based on the logarithmic relationship between the maximum amplitude ( $R_{ \mathrm{m}}$ ) of a solar cycle and the preceding minimum $aa$ geomagnetic index ( $aa_{\mathrm{min}}$ ), we perform a prediction on the peak of the upcoming cycle 25 using the sunspot number of the new version instead. If the suggested error in $aa$ (3 nT) before 1957 is corrected, the correlation between $\ln R_{\mathrm{m}}$ and $\ln aa_{\mathrm{min}}$ ( $r=0.92$ ) is stronger than that not corrected ( $r=0.86$ ). Based on this relationship, the peak value of cycle 25 is predicted to be $R_{\mathrm{m}}(25)\simeq 151.1\pm 16.9$ , about 30% stronger than cycle 24. Employing the ‘Waldmeier effect’ that the rise time of a cycle is well anti-correlated to its amplitude, we estimated the rise time, $T_{\mathrm{a}}(25)=4.3\pm 0.2\pm 0.6$ , and the peak time of cycle 25, $2024.1 \pm 0.8 $ (years), which is during April 2023 and November 2024.

Waldmeier Effect in Stellar Cycles

Abstract One of the most robust features of the solar magnetic cycle is that the stronger cycles rise faster than the weaker ones. This is popularly known as the Waldmeier Effect, which has been known for more than 100 yr. This fundamental feature of the solar cycle has not only practical implications, e.g., in predicting the solar cycle, but also implications in understanding the solar dynamo. Here we ask whether the Waldmeier Effect exists in other Sun-like stars. To answer this question, we analyze the Ca ii H and K S-index from Mount Wilson Observatory for 21 Sun-like G–K stars. We specifically check two aspects of Waldmeier Effect, namely, (1) WE1: the anticorrelation between the rise times and the peaks and (2) WE2: the positive correlation between rise rates and amplitudes. We show that, except for HD 16160, HD 81809, HD 155886, and HD 161239, all stars considered in the analysis show WE2, while WE1 is found to be present only in some of the stars studied. Furthermore, the WE1 correlation is weaker than the WE2. Both WE1 and WE2 exist in the solar S-index as well. Similar to the solar cycles, the magnetic cycles of many stars are asymmetric about their maxima. The existence of the Waldmeier Effect and asymmetric cycles in Sun-like stars suggests that the dynamo mechanism which operates in the Sun is also operating in other stars.

The Waldmeier Effect for Two Sunspot Populations

This paper continues our studies in which we showed that sunspot groups form two populations differing by physical properties—populations of small and large groups. The known Waldmeier effect has been individually considered for these populations. Two formulations of this effect were verified: the classical one, which connects the solar activity index at the maximum of the 11-year cycle with the length of the ascending phase of a cycle, and the modified one, connecting the index value at the cycle maximum with the highest rate of the index change along the ascending branch. We used the data from the Greenwich Observatory and the Kislovodsk Mountain Astronomical Station for the solar activity cycles from 12 to 24 to construct the series of indices for the areas and the numbers of sunspot groups. It has been shown that the Waldmeier effect works more strictly for the population of large sunspot groups than for that of small groups. The results of this analysis may find an application in the dynamo theory.

Kodaikanal digitized white-light data archive (1921–2011): Analysis of various solar cycle features

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 derived. Implementation of a semi-automated method is very necessary in such studies as it minimizes the human bias in the detection procedure. Daily, monthly and yearly sunspot area variations obtained from the Kodaikanal, compared well with the Greenwich sunspot area data. We find an exponentially decaying distribution for the individual sunspot area for each of the solar cycles. Analyzing the histograms of the latitudinal distribution of the detected sunspots, we find Gaussian distributions, in both the hemispheres, with the centers at $\sim$15$^{\circ}$ latitude. The height of the Gaussian distributions are different for the two hemispheres for a particular cycle. Using our data, we show clear presence of Waldmeier effect which correlates the rise time with the cycle amplitude. Using the wavelet analysis, we explored different periodicities of different time scales present in the sunspot area times series.

The New Sunspot-Number Index and Solar-Cycle Characteristics

We revisit several characteristics of the solar cycle using the new version of the sunspot-number index. Thus, we calculated several correlations, including the recent Solar Cycles 23 and 24 in the analysis. We applied two smoothing methods to the sunspot number: i) the usual 13-month running mean and ii) a 24-month Gaussian filter. Each of these methods contains two analyses: on the one hand, we consider all of the solar cycles available, and on the other hand, only those from Solar Cycle 10 onward. It can be seen that this new version improves or yields similar results for the correlations with respect to other works using the old version of the sunspot number, except for the amplitude–descending time effect and the linear fit of the secular trend. However, employing the same methodology in the analysis and considering the same solar cycles, it can be seen that the new sunspot number, in general, does not improve the correlations with respect to the old sunspot number and, moreover, the correlations obtained with the Gaussian filter generally are stronger than those with the 13-month running mean. Furthermore, from a sinusoidal fit to the solar-maximum amplitudes of the whole series, we have obtained a periodicity of the Gleissberg cycle equal to 97.7 years ( ${\approx}\,8.9$ solar cycles) for the 13-month running mean and 99.8 years ( ${\approx}\,9.1$ solar cycles) for the Gaussian filter. Lastly, the Waldmeier effect, the modified Waldmeier effect, the amplitude–period effect, the amplitude–minimum effect, and the even–odd effect are characteristics with high correlation coefficients and significance levels; the sinusoidal fit applied to the solar-maximum amplitudes yields a lower correlation coefficient value but a high significance level; and both the amplitude–descending-time effect and secular trend of the solar activity have weaker correlation coefficients and significance levels.

A Normalized Sunspot-Area Series Starting in 1832: An Update

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 11 years and a study of the Waldmeier effect in the new version of sunspot-number and the sunspot-area series presented in this study. We conclude that while this effect is significant in the new series of sunspot number, it has a weak relationship with the sunspot-area series.

Approximation of periodicity in sunspot formation of and prediction of the 25th cycle

The empirical parameters of functions approximating solar activity cycles 8–23 are used. These parameters show the position of the cycle on the time axis (the start time) and its shape, which is characterized by the extension along the time axes and activity index. A statistical connections was found between two shape parameters of the cycle (the so-called Waldmeier effect) and between the extension of the cycle growth branch and the start time of the following cycle. A connection between the parameters of the given and future cycles has been obtained for a function approximating the “secular” variations in the cycle amplitude. The aforementioned empirical relationships can be stated in the form of three equations that contain the parameters of the current and future cycles. Solving this system, we obtain estimates for three parameters of the function approximating the next cycle. For the 25th cycle, it was found that the maximum of the smoothed Wolf number 116 is expected in March/April 2026; the duration of the activity growth branch is 4.16 years.

THE BIMODAL STRUCTURE OF THE SOLAR CYCLE

Some properties of the 11 yr solar cycle can be explained by the current solar dynamo models. However, some other features remain not well understood such as the asymmetry of the cycle, the double-peaked structure, and the "Waldmeier effect" that a stronger cycle tends to have less rise time and a shorter cycle length. We speculate that the solar cycle is governed by a bi-dynamo model forming two stochastic processes depicted by a bimodal Gaussian function with a time gap of about 2 yr, from which the above features can be reasonably explained. The first one describes the main properties of the cycle dominated by the current solar dynamo models, and the second one occurs either in the rising phase as a short weak explosive perturbation or in the declining phase as a long stochastic perturbation. The above function is the best one selected from several in terms of the Akaike information criterion. Through analyzing different distributions, one might speculate about the dominant physical process inside the convection zone. The secondary (main) process is found to be closely associated with complicated (simple) active ranges. In effect, the bi-dynamo model is a reduced form of a multi-dynamo model, which could occur from the base of the convection zone through its envelope and from low to high heliographic latitude, reflecting the active belts in the convection zone. These results are insensitive to the hemispheric asymmetry, smoothing filters, and distribution functions selected and are expected to be helpful in understanding the formation of solar and stellar cycles.

Mathematical form of the Waldmeier effect

The Waldmeier effect is considered for empirical parameters of the functions that approximate solar activity cycles nos. 8–23 according to the Zurich numbering system. The cycle amplitude is found to linearly depend on the mean rate of rise in solar activity during its rising phase. This relationship seems to be a fundamental property of 11-year cycles and may be a reason for differences in solar cycles.

Oscillator models of the solar cycle and the Waldmeier effect

AbstractWe 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 and the lack of a good correlation between the decay rate and amplitude, together with significant (≳10%) r.m.s. variation in cycle lengths and cycle amplitudes. It is found that perturbing μ alone the perturbed van der Pol oscillator does not show solar‐like behaviour. When the perturbed variable is ξ, solar‐like behaviour is displayed for perturbations with a correlation time of about 3–4 years and significant amplitude. Such studies may provide useful constraints on solar dynamo models and their parameters. (© 2013 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)

Missing solar cycle hypothesis and basic statistical regularities of solar cycles

The basic statistical properties of solar cycles, including the Gnevyshev-Ol’ rule, the Waldmeier effect, and the amplitude-period effect, are tested using data on the number of sunspot groups for 1700–1996, considering the hypothesis about a missing solar cycle in the late 18th century. The results show that the division of the long cycle of 1784–1800 into two short cycles—1784–1793 and 1793–1800—alters significantly the pattern of the solar cycles. The Gnevyshev-Ol’ cycle intensity effect becomes stronger, and almost all other statistical effects grow weaker. This change is due to the fact that the short and weak cycle of 1793–1800 is statistically very unusual and its features are very different from those of other solar cycles.

The G–O Rule and Waldmeier Effect in the Variations of the Numbers of Large and Small Sunspot Groups

We have analyzed 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 (maximum area A M<100 millionth of solar hemisphere, msh), large (100≤A M<300 msh), and big (A M≥300 msh) spot 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 (G–O rule) of solar cycles, generally described by using the Zurich sunspot number (R Z). During cycles 12 – 21 the G–O rule holds good for the variation in the number of small groups also, but it is violated by cycle pair (22, 23) as in the case of R Z. This behavior of the variations in the small groups is largely responsible for the anomalous behavior of R Z in cycle pair (22, 23). It is also found that the amplitude of an odd-numbered cycle of the number of small groups is larger than that of its immediately following even-numbered cycle. This might be called the ‘reverse G–O rule’. In the case of the number of the big groups, both cycle pairs (12, 13) and (22, 23) violated the G–O rule. In many cycles the positions of the peaks of the small, large, and big groups are different, and considerably differ with respect to the corresponding positions of the R Z peaks. In the case of cycle 23, the corresponding cycles of the small and large groups are largely symmetric/less asymmetric (the Waldmeier effect is weak/absent) with their maxima taking place two years later than that of R Z. The corresponding cycle of the big groups is more asymmetric (strong Waldmeier effect) with its maximum epoch taking place at the same time as that of R Z.

Flux-transport and mean-field dynamo theories of solar cycles

AbstractWe point out the difficulties in carrying out direct numerical simulation of the solar dynamo problem and argue that kinematic mean-field models are our best theoretical tools at present for explaining various aspects of the solar cycle in detail. The most promising kinematic mean-field model is the flux transport dynamo model, in which the toroidal field is produced by differential rotation in the tachocline, the poloidal field is produced by the Babcock–Leighton mechanism at the solar surface and the meridional circulation plays a crucial role. Depending on whether the diffusivity is high or low, either the diffusivity or the meridional circulation provides the main transport mechanism for the poloidal field to reach the bottom of the convection zone from the top. We point out that the high-diffusivity flux transport dynamo model is consistent with various aspects of observational data. The irregularities of the solar cycle are primarily produced by fluctuations in the Babcock–Leighton mechanism and in the meridional circulation. We summarize recent work on the fluctuations of meridional circulation in the flux transport dynamo, leading to explanations of such things as the Waldmeier effect.

Solar Field Mapping and Dynamo Behavior

We discuss the importance of the Sun’s large-scale magnetic field to the Sun-Planetary environment. This paper narrows its focus down to the motion and evolution of the photospheric large-scale magnetic field which affects many environments throughout this region. For this purpose we utilize a newly developed Netlogo cellular automata model. The domain of this algorithmic model is the Sun’s photosphere. Within this computational space are placed two types of entities or agents; one may refer to them as bluebirds and cardinals; the former carries outward magnetic flux and the latter carries out inward magnetic flux. One may simply call them blue and red agents. The agents provide a granularity with discrete changes not present in smooth MHD models; they undergo three processes: birth, motion, and death within the photospheric domain. We discuss these processes, as well as how we are able to develop a model that restricts its domain to the photosphere and allows the deeper layers to be considered only through boundary conditions. We show the model’s ability to mimic a number of photospheric magnetic phenomena: the solar cycle (11-year) oscillations, the Waldmeier effect, unipolar magnetic regions (e.g. sectors and coronal holes), Maunder minima, and the march/rush to the poles involving the geometry of magnetic field reversals. We also discuss why the Sun sometimes appears as a magnetic monopole, which of course requires no alteration of Maxwell’s equations.