Solar-Terrestrial Physics Solar-Terrestrial Physics Solar-Terrestrial Physics 2500-0535 122579 10.12737/stp-122202601 tiqmuz Results of current research Results of current research Results of current research Waldmeier law for sunspot cycles: Statistical significance and implications for dynamo Waldmeier law for sunspot cycles: Statistical significance and implications for dynamo Кичатинов Леонид Леонидович Kitchatinov Leonid Leonidovich kit@iszf.irk.ru

доктор физико-математических наук;

doctor of physical and mathematical sciences;

 Институт солнечно-земной физики СО РАН Иркутск Россия Institute of Solar-Terrestrial Physics SB RAS Irkutsk Russian Federation 12 2 4 7 25 02 2026 06 04 2026 https://naukaru.ru/en/nauka/article/122579/view

Waldmeier law for the relation between the rise phase duration and amplitude of solar cycles is an indicator for strength of nonlinearity in the solar dynamo. The paper compares the Waldmeier law parameters for sunspot number and area in solar cycles 12–25. Uncertainty in dates of cycles’ maxima and minima is fixed by smoothing with the Gaussian filter. The ratio of sunspot number to area is shown to differ significantly between rise and decline phases of solar activity. The high correlation for Waldmeier law in sunspot number is an artifact of the Wolf number definition and does not imply any strong nonlinearity in the dynamo mechanism. The relative low correlation and significance level of about 70 % in Waldmeier law for sunspot area indicates a weak nonlinearity in the solar dynamo and agrees with evidences from observations of stellar rotation.

Waldmeier law for the relation between the rise phase duration and amplitude of solar cycles is an indicator for strength of nonlinearity in the solar dynamo. The paper compares the Waldmeier law parameters for sunspot number and area in solar cycles 12–25. Uncertainty in dates of cycles’ maxima and minima is fixed by smoothing with the Gaussian filter. The ratio of sunspot number to area is shown to differ significantly between rise and decline phases of solar activity. The high correlation for Waldmeier law in sunspot number is an artifact of the Wolf number definition and does not imply any strong nonlinearity in the dynamo mechanism. The relative low correlation and significance level of about 70 % in Waldmeier law for sunspot area indicates a weak nonlinearity in the solar dynamo and agrees with evidences from observations of stellar rotation.

 solar activity sunspot number sunspot area dynamo nonlinearity solar activity sunspot number sunspot area dynamo nonlinearity This work is financially supported by the Ministry of Science and High Education of the Russian Federation This work is financially supported by the Ministry of Science and High Education of the Russian Federation

Badalyan O.G., Obridko V.N. North-South asymmetry of the sunspot indeces and its quasi-biennual oscillations. New Astronomy. 2011, vol. 16, iss. 6, pp. 357–365. https://doi.org/10.1016/j.newast.2011.01.005. Badalyan O.G., Obridko V.N. North-South asymmetry of the sunspot indeces and its quasi-biennual oscillations. New Astronomy. 2011, vol. 16, iss. 6, pp. 357–365. https://doi.org/10.1016/j.newast.2011.01.005. Cameron R.H., Schussler M. Understandiang solar cycle variability. Astrophys. J. 2017, vol. 843, iss. 2, id. 111. https://doi.org/10.3847/1538-4357/aa767a. Cameron R.H., Schussler M. Understandiang solar cycle variability. Astrophys. J. 2017, vol. 843, iss. 2, id. 111. https://doi.org/10.3847/1538-4357/aa767a. Carrasco V.M.S., Vaquero J.M., Gallego M.C., Sanchez-Bajo F. A normalized sunspot-area series starting in 1832: An update. Solar Phys. 2016, vol. 291, iss. 9-10, pp. 2931–2940. https://doi.org/10.1007/s11207-016-0943-9. Carrasco V.M.S., Vaquero J.M., Gallego M.C., Sanchez-Bajo F. A normalized sunspot-area series starting in 1832: An update. Solar Phys. 2016, vol. 291, iss. 9-10, pp. 2931–2940. https://doi.org/10.1007/s11207-016-0943-9. Charbonneau P. Dynamo models of the solar cycle. Living Rev. Solar Phys. 2020, vol. 17, iss. 1, id. 4. https://doi.org/10.1007/s41116-020-00025-6. Charbonneau P. Dynamo models of the solar cycle. Living Rev. Solar Phys. 2020, vol. 17, iss. 1, id. 4. https://doi.org/10.1007/s41116-020-00025-6. Charbonneau P., Sokoloff D. Evolution of solar and stellar dynamo theory. Space Sci. Rev. 2023, vol. 219, iss. 5, id. 35. https://doi.org/10.1007/s11214-023-00980-0. Charbonneau P., Sokoloff D. Evolution of solar and stellar dynamo theory. Space Sci. Rev. 2023, vol. 219, iss. 5, id. 35. https://doi.org/10.1007/s11214-023-00980-0. Dasi-Espuig M., Solanki S.K., Krivova N.A., et al. Sunspot group tilt angles and the strength of the solar cycle. Astron. Astrophys. 2010, vol. 518, id. A7. https://doi.org/10.1051/0004-6361/201014301 Dasi-Espuig M., Solanki S.K., Krivova N.A., et al. Sunspot group tilt angles and the strength of the solar cycle. Astron. Astrophys. 2010, vol. 518, id. A7. https://doi.org/10.1051/0004-6361/201014301 Dikpati M., Gilman P.A., de Toma G. The Waldmeier effect: An artifact of the definition of Wolf sunspot number? Astrophys. J. Lett. 2008, vol. 673, iss. 1, id.: L99. https://doi.org/10.1086/527360. Dikpati M., Gilman P.A., de Toma G. The Waldmeier effect: An artifact of the definition of Wolf sunspot number? Astrophys. J. Lett. 2008, vol. 673, iss. 1, id.: L99. https://doi.org/10.1086/527360. Garg S., Karak B.B., Egeland R., et al. Waldmeier effect in stellar cycles. Astrophys. J. 2019, vol. 886, iss. 2, id. 132. https://doi.org/10.3847/1538-4357/ab4a17. Garg S., Karak B.B., Egeland R., et al. Waldmeier effect in stellar cycles. Astrophys. J. 2019, vol. 886, iss. 2, id. 132. https://doi.org/10.3847/1538-4357/ab4a17. Hathaway D.H. The solar cycle. Living Rev. in Solar Phys. 2015, vol. 12, no. 1, id. 4. https://doi.org/10.1007/lrsp-2015-4. Hathaway D.H. The solar cycle. Living Rev. in Solar Phys. 2015, vol. 12, no. 1, id. 4. https://doi.org/10.1007/lrsp-2015-4. Jiang J., Cameron R.H., Schmitt D., Isik E. Modeling solar cycles 15 to 21 using a flux transport dynamo. Astron. Astrophys. 2013, vol. 553, id. A128. https://doi.org/10.1051/0004-6361/201321145. Jiang J., Cameron R.H., Schmitt D., Isik E. Modeling solar cycles 15 to 21 using a flux transport dynamo. Astron. Astrophys. 2013, vol. 553, id. A128. https://doi.org/10.1051/0004-6361/201321145. Karak B.B., Choudhuri A.R. The Waldmeier effect and the flux-transport solar dynamo. Monthly Notices of the Royal Astronomical Society. 2011, vol. 410, iss. 3, pp. 1503–1512. https://doi.org/10.1111/j.1365-2966.2010.17531.x. Karak B.B., Choudhuri A.R. The Waldmeier effect and the flux-transport solar dynamo. Monthly Notices of the Royal Astronomical Society. 2011, vol. 410, iss. 3, pp. 1503–1512. https://doi.org/10.1111/j.1365-2966.2010.17531.x. Kitchatinov L.L. Flux tubes forming instability near the base of the rotating convection zone: A possible explanation for the low latitudes of sunspots. Astrophys. J. 2020, vol. 893, iss. 2, id. 131. https://doi.org/10.3847/1538-4357/ab7fa8. Kitchatinov L.L. Flux tubes forming instability near the base of the rotating convection zone: A possible explanation for the low latitudes of sunspots. Astrophys. J. 2020, vol. 893, iss. 2, id. 131. https://doi.org/10.3847/1538-4357/ab7fa8. Kitchatinov L.L., Olemskoy S.V. Does the Babcock-Leighton mechanism operate on the Sun? Astron. Lett. 2011, vol. 37, iss. 9, pp. 656–658. https://doi.org/10.1134/S0320010811080031. Kitchatinov L.L., Olemskoy S.V. Does the Babcock-Leighton mechanism operate on the Sun? Astron. Lett. 2011, vol. 37, iss. 9, pp. 656–658. https://doi.org/10.1134/S0320010811080031. Kitchatinov L.L., Nepomnyashchikh A.A. A joined model for solar dynamo and differential rotation. Astron. Lett. 2017a, vol. 43, iss. 5, pp. 332–343. https://doi.org/10.1134/S106377371704003X. Kitchatinov L.L., Nepomnyashchikh A.A. A joined model for solar dynamo and differential rotation. Astron. Lett. 2017a, vol. 43, iss. 5, pp. 332–343. https://doi.org/10.1134/S106377371704003X. Kitchatinov L., Nepomnyashchikh A. How supercritical are stellar dynamos, or why do old main-sequence dwarfs not obey gyrochronology? Monthly Not. Royal Astron. Soc. 2017b, vol. 470, iss. 3, pp. 3124–3130. https://doi.org/10.1093/mnras/stx1473. Kitchatinov L., Nepomnyashchikh A. How supercritical are stellar dynamos, or why do old main-sequence dwarfs not obey gyrochronology? Monthly Not. Royal Astron. Soc. 2017b, vol. 470, iss. 3, pp. 3124–3130. https://doi.org/10.1093/mnras/stx1473. Metcalfe T.S., Egeland R., van Saders J. Stellar evidence that the solar dynamo may be in transition. Astrophys. J. Lett. 2016, vol. 826, iss. 1, id. L2. https://doi.org/10.3847/2041-8205/826/1/L2. Metcalfe T.S., Egeland R., van Saders J. Stellar evidence that the solar dynamo may be in transition. Astrophys. J. Lett. 2016, vol. 826, iss. 1, id. L2. https://doi.org/10.3847/2041-8205/826/1/L2. Moss D., Sokoloff D., Usoskin I. Solar grand minima and random fluctuations in dynamo parameters. Solar Phys. 2008, vol. 250, iss. 2, id. 221. https://doi.org/10.1007/s11207-008-9202-z. Moss D., Sokoloff D., Usoskin I. Solar grand minima and random fluctuations in dynamo parameters. Solar Phys. 2008, vol. 250, iss. 2, id. 221. https://doi.org/10.1007/s11207-008-9202-z. Nagovitsyn Yu.A., Pevtsov A.A. On the presence of two populations of sunspots. Astrophys. J. 2016, vol. 833, iss. 1, id. 94. https://doi.org/10.3847/1538-4357/833/1/94. Nagovitsyn Yu.A., Pevtsov A.A. On the presence of two populations of sunspots. Astrophys. J. 2016, vol. 833, iss. 1, id. 94. https://doi.org/10.3847/1538-4357/833/1/94. Nagovitsyn Yu.A., Osipova A.A. Average annual total sunspot area in the last 410 yr: the most probable values and limits of their uncertainties. Monthly Not. Royal Astron. Soc. 2021, vol. 505, iss. 1, pp. 1206–1212. https://doi.org/10.1093/mnras/stab1328. Nagovitsyn Yu.A., Osipova A.A. Average annual total sunspot area in the last 410 yr: the most probable values and limits of their uncertainties. Monthly Not. Royal Astron. Soc. 2021, vol. 505, iss. 1, pp. 1206–1212. https://doi.org/10.1093/mnras/stab1328. Nagovitsyn Yu.A., Pevtsov A.A., Livingston W.C. On a possible explanation of the long-term decrease in sunspot field strength. Astrophys. J. Lett. 2012, vol. 758, iss. 1, id. L20. https://doi.org/10.1088/2041-8205/758/1/L20. Nagovitsyn Yu.A., Pevtsov A.A., Livingston W.C. On a possible explanation of the long-term decrease in sunspot field strength. Astrophys. J. Lett. 2012, vol. 758, iss. 1, id. L20. https://doi.org/10.1088/2041-8205/758/1/L20. Nagovitsyn Yu.A., Pevtsov A.A., Osipova A.A., et al. Two populations of sunspots and secular variations of their characteristics. Astron. Lett. 2016, vol. 42, iss. 10, pp. 703–712. https://doi.org/10.1134/S1063773716090048. Nagovitsyn Yu.A., Pevtsov A.A., Osipova A.A., et al. Two populations of sunspots and secular variations of their characteristics. Astron. Lett. 2016, vol. 42, iss. 10, pp. 703–712. https://doi.org/10.1134/S1063773716090048. Olemskoy S.V., Choudhuri A.R., Kitchatinov L.L. Fluctuations in the alpha-effect and grand solar minima. Astron. Rep. 2013, vol. 57, iss. 6, pp. 458–468. https://doi.org/10.1134/S1063772913050065. Olemskoy S.V., Choudhuri A.R., Kitchatinov L.L. Fluctuations in the alpha-effect and grand solar minima. Astron. Rep. 2013, vol. 57, iss. 6, pp. 458–468. https://doi.org/10.1134/S1063772913050065. Osipova A.A., Nagovitsyn Yu.A. The Waldmeier effect for two sunspot populations. Geomagnetism and Aeronomy. 2017, vol. 57, iss. 8, pp. 1092–1100. https://doi.org/10.1134/S0016793217080199. Osipova A.A., Nagovitsyn Yu.A. The Waldmeier effect for two sunspot populations. Geomagnetism and Aeronomy. 2017, vol. 57, iss. 8, pp. 1092–1100. https://doi.org/10.1134/S0016793217080199. Pipin V.V., Kosovichev A.G. The asymmetry of sunspot cycles and Waldmeier relation as a result of nonlinear surface-shear shaped dynamo. Astrophys. J. 2011, vol. 741, iss. 1, id. 1. https://doi.org/10.1088/0004-637X/741/1/1. Pipin V.V., Kosovichev A.G. The asymmetry of sunspot cycles and Waldmeier relation as a result of nonlinear surface-shear shaped dynamo. Astrophys. J. 2011, vol. 741, iss. 1, id. 1. https://doi.org/10.1088/0004-637X/741/1/1. Sreedevi A., Jha B.K., Karak B.B., Banerjee D. Analysis of BMR tilt from Auto TAB catalog: Hinting towards thin flux tube model? Astrophys. J. 2024, vol. 966, iss. 1, id. 112. https://doi.org/10.3847/1538-4357/ad34b8. Sreedevi A., Jha B.K., Karak B.B., Banerjee D. Analysis of BMR tilt from Auto TAB catalog: Hinting towards thin flux tube model? Astrophys. J. 2024, vol. 966, iss. 1, id. 112. https://doi.org/10.3847/1538-4357/ad34b8. Tlatov A.G., Pevtsov A.A. Bimodal distribution of magnetic fields and areas of sunspots. Solar Phys. 2014, vol. 289, iss. 4, pp. 1143–1152. https://doi.org/10.1007/s11207-013-0382-9. Tlatov A.G., Pevtsov A.A. Bimodal distribution of magnetic fields and areas of sunspots. Solar Phys. 2014, vol. 289, iss. 4, pp. 1143–1152. https://doi.org/10.1007/s11207-013-0382-9. van Saders J.L., Ceillier T., Metcalfe T.S., et al. Weakend magnetic braking as the origin of anomalously rapid rotation in old field stars. Nature. 2016, vol. 529, iss. 7585, pp. 181–184. https://doi.org/10.1038/nature16168. van Saders J.L., Ceillier T., Metcalfe T.S., et al. Weakend magnetic braking as the origin of anomalously rapid rotation in old field stars. Nature. 2016, vol. 529, iss. 7585, pp. 181–184. https://doi.org/10.1038/nature16168. Vitinsky Yu.I., Kopecky M., Kuklin G.V. Statistics of Sunspot Activity. Moscow, Nauka, 1986, 296 p. [In Rissian]. Vitinsky Yu.I., Kopecky M., Kuklin G.V. Statistics of Sunspot Activity. Moscow, Nauka, 1986, 296 p. [In Rissian]. Waldmeier M. Neue Eigenshaften der Sonnenfleckenkurve. Astron. Mitt. Zurich. 1935, vol. 14. pp. 105–136. Waldmeier M. Neue Eigenshaften der Sonnenfleckenkurve. Astron. Mitt. Zurich. 1935, vol. 14. pp. 105–136. URL: http://solarcyclescience.com/AR_Database/sunspot_area.txt (accessed December 12, 2025). URL: http://solarcyclescience.com/AR_Database/sunspot_area.txt (accessed December 12, 2025). URL: http://www.wdcb.ru/stp/data/solar.act/sunspot/SILSO/ver2/SN_m/SN_m_tot_V2.0.txt (accessed December 12, 2025). URL: http://www.wdcb.ru/stp/data/solar.act/sunspot/SILSO/ver2/SN_m/SN_m_tot_V2.0.txt (accessed December 12, 2025).