University of Oulu

Usoskin, I., Kovaltsov, G. & Kiviaho, W. Robustness of Solar-Cycle Empirical Rules Across Different Series Including an Updated Active-Day Fraction (ADF) Sunspot Group Series. Sol Phys 296, 13 (2021).

Robustness of solar-cycle empirical rules across different series including an updated active-day fraction (ADF) sunspot group series

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Author: Usoskin, Ilya1,2; Kovaltsov, Gennady3; Kiviaho, Wilma1,4
Organizations: 1University of Oulu, Oulu, Finland
2St. Petersburg State University, St. Petersburg, Russia
3A.F. Ioffe Physical-Technical Institute, St. Petersburg, Russia
4University of Turku, Turku, Finland
Format: article
Version: accepted version
Access: open
Online Access: PDF Full Text (PDF, 0.5 MB)
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Language: English
Published: Springer Nature, 2021
Publish Date: 2022-01-11


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 different sunspot (group) series for the period 1749 – 1996, using four classical and revised international sunspot-number and group sunspot-number series. We also provide an update of the sunspot-group series based on the active-day fraction (ADF) method, using the new database of solar observations. We show that the Waldmeier rule is robust and independent of the exact sunspot (group) series: its classical and n+1 (relating the length of nth cycle to the magnitude of (n+1)th cycle) formulations are significant or highly significant for all series, while its simplified formulation (relating the magnitude of a cycle to its full length) is insignificant for all series. The Gnevyshev–Ohl rule was found robust for all analyzed series for Solar Cycles 8 – 21, but unstable across the Dalton minimum and before it.

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Series: Solar physics
ISSN: 0038-0938
ISSN-E: 1573-093X
ISSN-L: 0038-0938
Volume: 296
Article number: 13
DOI: 10.1007/s11207-020-01750-9
Type of Publication: A1 Journal article – refereed
Field of Science: 115 Astronomy and space science
Funding: This work was partially supported by the Academy of Finland (project No. 321882 ESPERA) and by the Russian Science Foundation (RSF project No. 20-67-46016).
Academy of Finland Grant Number: 321882
Detailed Information: 321882 (Academy of Finland Funding decision)
Copyright information: © The Author(s), under exclusive licence to Springer Nature B.V. part of Springer Nature 2021. This is a post-peer-review, pre-copyedit version of an article published in Sol Phys. The final authenticated version is available online at