Luo & Tan (2024) fitted the 323-year continuous record of annual mean sunspot numbers (ASN) and identified a centennial modulation of solar activity with a best-fit period of 104.0 years superposed on a slowly increasing secular trend, concluding that the Sun must possess a global physical mechanism acting on a scale larger than the 11-year Schwabe cycle, while stating explicitly that this mechanism is not understood and that the 323-year record cannot establish whether the centennial component and the secular trend are persistent, stationary properties of the Sun. The centennial modulation is the well-known Gleissberg cycle, a quasi-periodic band of roughly 60-140 years; pre-existing radiocarbon evidence shows the band persists across the last ~12,000 years, but the most recent authoritative reviews (2023) describe it as intermittent, barely significant, and lacking clock-like phase locking, and competing explanations - intrinsic nonlinear dynamo behavior, stochastic realization noise, and planetary-tidal synchronization of the dynamo - remain mutually exclusive and unsettled. The open core survives intact: no physical origin of the centennial modulation has been identified or generally accepted, and the stationarity of the centennial component and secular trend on millennial timescales has not been established.

Background

Solar activity, traced since 1700 by the annual mean sunspot number (ASN, the yearly average of the relative number of sunspots and sunspot groups visible on the solar disk), varies primarily with the ~11-year Schwabe cycle. Superposed on this basic rhythm are longer modulations: extended episodes of very weak activity (grand minima, e.g., the Maunder Minimum of 1645-1715), and an amplitude modulation on a timescale of roughly one century. The centennial modulation was first isolated by W. Gleissberg in 1939 from smoothed sunspot amplitudes and is called the Gleissberg or century/secular cycle; published period estimates span a wide band (about 78-104 years in direct sunspot data, with some analyses extending the characteristic range to 60-150 years), and it is associated with the recurrence of extended activity minima (Feynman & Ruzmaikin 2014). Because the 323-year continuous ASN record contains only about three full century cycles, and because the centennial signal is intermittent, its physical status is difficult to settle from sunspot data alone.

Two further ingredients frame the question. First, solar activity can be reconstructed much further back in time using cosmogenic isotopes: radioactive nuclides such as 14^{14}C in tree rings and 10^{10}Be in ice cores are produced by galactic cosmic rays whose flux reaching Earth is modulated by the heliospheric magnetic field, itself controlled by solar activity, so these proxies extend the record to roughly twelve millennia (albeit with carbon-cycle damping for 14^{14}C and contamination by geomagnetic-field variations). Second, the physical engine of the activity cycle is the solar dynamo, the magnetohydrodynamic process by which the Sun's differential rotation and convective motions regenerate and reverse the large-scale magnetic field roughly every 11 years (22 years for the full magnetic, or Hale, cycle).

Luo & Tan (2024) performed wavelet analysis (a time-frequency decomposition showing how the periodic content of a series varies over time) and empirical-function fitting of the ASN series and obtained the century function R=78.84+23.86sin⁡(2πx/104.0−0.5385π)+0.06028xR = 78.84 + 23.86\sin(2\pi x/104.0 - 0.5385\pi) + 0.06028x (where xx represents the years after 1700 AD and RR the annual mean sunspot number corresponding to that year), i.e., a 104.0-year sinusoid plus a weakly increasing linear term, and used phase-similar extrapolation (the assumption that Schwabe cycles at the same phase of the century cycle have similar parameters) to predict solar cycles 25 and 26. They concluded that the existence of this century cycle implies a global solar mechanism larger in scale than the Schwabe cycle, while stating that this mechanism is not yet understood, and that the 323-year record is too short to distinguish genuine quasi-periodicity from an artifact of incomplete sampling; to settle this, they recommend the forward extension of the sunspot number sequence and, separately, the analysis of long-term cosmogenic-isotope data (especially 14^{14}C), which has the potential to extend the time series to thousands of years in length.

The question they pose is the long-standing open problem of the origin of the Gleissberg cycle, and it is genuinely contested in three directions. A mean-field dynamo model (a dynamo model in which the turbulent convective motions are averaged into large-scale transport coefficients) with magnetic feedback on angular-momentum transport can produce century-scale modulation (described in the review by Biswas, Karak, Usoskin & Weisshaar 2023), and dynamo reviews (Karak 2023) list magnetic feedback on the flow, stochastic forcing, and time delays in dynamo processes as candidate causes. Cameron & Schüssler (2019) argued the opposite: using a generic model of a noisy, weakly nonlinear limit cycle (a self-sustained oscillator whose amplitude and phase wander under stochastic disturbances) calibrated to observations, they showed that all spectral peaks beyond the 11-year band, including the Gleissberg peak, are statistically consistent with realization noise - random amplitude and phase fluctuations of the basic cycle - so the centennial 'cycle' may not be an intrinsic periodicity at all. A third line (Stefani et al. 2020, extended 2025) produces Gleissberg-band frequencies as nonlinear beats in a planetary-tidal synchronization model of the dynamo, although the phase analyses reported by Biswas et al. (2023) reject clock-like external synchronization of the solar-cycle phase at high significance, a conclusion the proponents of the tidal scenario continue to dispute. Cosmogenic-isotope studies add a key constraint: the ~88-year Gleissberg cycle is detectable in 14^{14}C over the last ~12,000 years (Peristykh & Damon 2003), yet the most recent reviews describe the centennial band as intermittent and less pronounced through the Holocene, with no clock-like phase locking, and the late-20th-century activity level as exceptional over the previous 11,000 years (Solanki et al. 2004) - which bears directly on whether the secular increase fitted by Luo & Tan is a stationary trend.

Problem Statement

Identify the physical origin of the centennial (Gleissberg-band) modulation of solar activity - the ~104-year century component that Luo & Tan (2024) fitted to the 323-year annual sunspot-number record together with its weak secular increasing term - and determine whether that centennial component and secular trend are persistent, stationary properties of the Sun on millennial timescales. Concretely, the question comprises two facets: (1) whether the centennial modulation is an intrinsic, global physical mechanism of the Sun operating on a scale larger than the ~11-year Schwabe cycle (as Luo & Tan infer), a stochastic modulation/realization noise of the basic cycle, or another process such as external synchronization - and, if intrinsic or externally driven, identify that global physical mechanism; and (2) whether the centennial component and the secular trend, fitted from only about three full century cycles of direct observation, persist (in period and amplitude) through the ~twelve millennia covered by cosmogenic-isotope proxies (14^{14}C, 10^{10}Be), or are artifacts of the short, intermittent telescopic record - including whether the fitted linear increase reflects a genuine long-term trend or the recovery into the exceptional modern grand maximum. In both facets the object is the centennial (Gleissberg-band) modulation as a physical phenomenon: the fitted 104.0-year component and secular term are the specific telescopic instantiation whose millennial persistence is tested in facet (2), and the fitted period lies within the broader 60-140-year band consensus rather than replacing it. Secondary facets noted in the source - the influence of the intermittent ~56-year component, the effect of the early low-quality 'anomaly' sunspot data on the fitted century function, and the forecast uncertainty from having only two historical phase-analogues per cycle - are part of the same objective insofar as they affect the century-cycle fit and its interpretation, but the open core consists of (1) and (2).

References

  1. Luo, Pei-Xin; Tan, Bao-Lin (2024). Long-term Evolution of Solar Activity and Prediction of the Following Solar Cycles. Research in Astronomy and Astrophysics 24(3), 035016. DOI: 10.1088/1674-4527/ad1ed2. https://doi.org/10.1088/1674-4527/ad1ed2 (preprint: arXiv:2402.13173, https://arxiv.org/abs/2402.13173)
  2. Peristykh, Alexei N.; Damon, Paul E. (2003). Persistence of the Gleissberg 88-year solar cycle over the last ~12,000 years: Evidence from cosmogenic isotopes. Journal of Geophysical Research: Space Physics 108(A1), 1003. DOI: 10.1029/2002JA009390. https://doi.org/10.1029/2002JA009390
  3. Cameron, R. H.; Schuessler, M. (2019). Solar activity: periodicities beyond 11 years are consistent with random forcing. Astronomy & Astrophysics 625, A28. DOI: 10.1051/0004-6361/201935290. https://doi.org/10.1051/0004-6361/201935290
  4. Stefani, F.; Giesecke, A.; Seilmayer, R.; Stepanov, R.; Weier, T. (2020). Schwabe, Gleissberg, Suess-de Vries: Towards a consistent model of planetary synchronization of solar cycles. Magnetohydrodynamics 56(2-3), 269-280. DOI: 10.22364/mhd.56.2-3.18. https://doi.org/10.22364/mhd.56.2-3.18 (preprint: arXiv:1910.10383, https://arxiv.org/abs/1910.10383)
  5. Stefani, F.; Horstmann, G. M.; Mamatsashvili, G.; Weier, T. (2025). Adding further pieces to the synchronization puzzle: QBO, bimodality, and phase jumps. arXiv:2503.22337. https://arxiv.org/abs/2503.22337
  6. Karak, Bidya Binay (2023). Models for the long-term variations of solar activity. Living Reviews in Solar Physics 20(1), 3. DOI: 10.1007/s41116-023-00037-y. https://doi.org/10.1007/s41116-023-00037-y
  7. Biswas, Akash; Karak, Bidya Binay; Usoskin, Ilya; Weisshaar, Eckhard (2023). Long-Term Modulation of Solar Cycles. Space Science Reviews 219(3), 19. DOI: 10.1007/s11214-023-00968-w. https://doi.org/10.1007/s11214-023-00968-w
  8. Usoskin, Ilya G. (2017). A history of solar activity over millennia. Living Reviews in Solar Physics 14(1), 3. DOI: 10.1007/s41116-017-0006-9. https://doi.org/10.1007/s41116-017-0006-9
  9. Solanki, S. K.; Usoskin, I. G.; Kromer, B.; Schuessler, M.; Beer, J. (2004). Unusual activity of the Sun during recent decades compared to the previous 11,000 years. Nature 431, 1084-1087. DOI: 10.1038/nature02995. https://doi.org/10.1038/nature02995
  10. Brehm, N.; Bayliss, A.; Christl, M.; Synal, H.-A.; Adolphi, F.; Beer, J.; Kromer, B.; Muscheler, R.; Solanki, S. K.; Usoskin, I.; Bleicher, N.; Bollhalder, S.; Tyers, C.; Wacker, L. (2021). Eleven-year solar cycles over the last millennium revealed by radiocarbon in tree rings. Nature Geoscience 14(1), 10-15. DOI: 10.1038/s41561-020-00674-0. https://doi.org/10.1038/s41561-020-00674-0
  11. Feynman, J.; Ruzmaikin, A. (2014). The Centennial Gleissberg Cycle and its association with extended minima. Journal of Geophysical Research: Space Physics 119, 6027-6041. DOI: 10.1002/2013JA019478. https://doi.org/10.1002/2013JA019478
  12. Rodríguez, José-Víctor; Sánchez Carrasco, Víctor Manuel; Rodríguez-Rodríguez, Ignacio; Pérez Aparicio, Alejandro Jesús; Vaquero, José Manuel (2024). Are we crossing a minimum of the Gleissberg centennial cycle? Multivariate machine learning-based prediction of the sunspot number using different proxies of solar activity and spectral analysis. Advances in Space Research 74(8), 4173-4188. DOI: 10.1016/j.asr.2024.08.033. https://doi.org/10.1016/j.asr.2024.08.033
  13. Karak, Bidya Binay (2026). Solar cycle prediction: challenges, progress, and future perspectives. Reviews of Modern Plasma Physics 10(1), 11. DOI: 10.1007/s41614-026-00220-2. https://doi.org/10.1007/s41614-026-00220-2