Skip to main content
Log in

Identification of Gleissberg Cycles and a Rising Trend in a 315-Year-Long Series of Sunspot Numbers

  • Published:
Solar Physics Aims and scope Submit manuscript


We show in this short note that the method of singular spectrum analysis (SSA) is able to clearly extract a strong, clean, and clear component from the longest available sunspot (International Sunspot Number, ISN) time series (1700 – 2015) that cannot be an artifact of the method and that can be safely identified as the Gleissberg cycle. This is not a small component, as it accounts for 13% of the total variance of the total original signal. Almost three and a half clear Gleissberg cycles are identified in the sunspot number series. Four extended solar minima (XSM) are determined by SSA, the latest around 2000 (Cycle 23/24 minimum). Several authors have argued in favor of a double-peaked structure for the Gleissberg cycle, with one peak between 55 and 59 years and another between 88 and 97 years. We find no evidence of the former: solar activity contains an important component that has undergone clear oscillations of \(\approx90\) years over the past three centuries, with some small but systematic longer-term evolution of “instantaneous” period and amplitude. Half of the variance of solar activity on these time scales can be satisfactorily reproduced as the sum of a monotonous multi-secular increase, a \(\approx90\)-year Gleissberg cycle, and a double-peaked (\(\approx10.0\) and 11.0 years) Schwabe cycle (the sum amounts to 46% of the total variance of the signal). The Gleissberg-cycle component definitely needs to be addressed when attempting to build dynamo models of solar activity. The first SSA component offers evidence of an increasing long-term trend in sunspot numbers, which is compatible with the existence of the modern grand maximum.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Subscribe and save

Springer+ Basic
EUR 32.99 /Month
  • Get 10 units per month
  • Download Article/Chapter or Ebook
  • 1 Unit = 1 Article or 1 Chapter
  • Cancel anytime
Subscribe now

Buy Now

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Figure 1
Figure 2
Figure 3
Figure 4
Figure 5

Similar content being viewed by others


  • Clette, F., Svalgaard, L., Vaquero, J.M., Cliver, E.W.: 2014, Revisiting the sunspot number. Space Sci. Rev. 186, 35. DOI .

    Article  ADS  Google Scholar 

  • Cole, T.W.: 1973, Periodicities in solar activity. Solar Phys. 30, 103.

    Article  ADS  Google Scholar 

  • Feynman, J., Fougere, P.F.: 1984, Eighty-eight year cycle in solar terrestrial phenomena confirmed. J. Geophys. Res. 89, 3023.

    Article  ADS  Google Scholar 

  • Feynman, J., Ruzmaikin, A.: 2011, The Sun’s strange behaviour: Maunder minimum or Gleissberg cycle? Solar Phys. 273, 351.

    Article  ADS  Google Scholar 

  • Feynman, J., Ruzmaikin, A.: 2013, The centennial Gleissberg cycle and its association with extended minima. J. Geophys. Res. Space Phys. 119, 6027.

    Article  ADS  Google Scholar 

  • Garcia, A., Mouradian, Z.: 1998, The Gleissberg cycle of minima. Solar Phys. 180, 495.

    Article  ADS  Google Scholar 

  • Gleissberg, W.: 1939, A long-periodic fluctuation of the sunspot numbers. Observatory 62, 158.

    ADS  Google Scholar 

  • Golub, G., Kahan, W.: 1965, Calculating the singular values and pseudoinverse of a matrix. J. Soc. Ind. Appl. Math., Ser. B, Numer. Anal. 2, 205.

    Article  MathSciNet  MATH  Google Scholar 

  • Gao, P.X.: 2016, Long-term trend of sunspot numbers. Astrophys. J. 830, 140. DOI .

    Article  ADS  Google Scholar 

  • Hathaway, D.H., Wilson, R.M., Reichmann, E.J.: 1999, A synthesis of solar cycle prediction techniques. J. Geophys. Res. 104, 22375.

    Article  ADS  Google Scholar 

  • Hathaway, D.H.: 2015, The solar cycle. Living Rev. Solar Phys. 12, 4. DOI .

    Article  ADS  Google Scholar 

  • Hoyt, D.V., Schatten, K.H.: 1998, Group sunspot numbers: A new solar activity reconstruction. Solar Phys. 181, 491. DOI .

    Article  ADS  Google Scholar 

  • Kittler, J., Young, P.C.: 1973, A new approach to feature selection based on the Karhunen–Loeve expansion. Pattern Recognit. 5, 335.

    Article  MathSciNet  Google Scholar 

  • McCracken, K.G., Beer, J., Steinhilber, F., Abreu, J.: 2013, A phenomenological study of the cosmic ray variations over the past 9400 years, and their implications regarding solar activity and the solar dynamo. Solar Phys. 286, 609. DOI .

    Article  ADS  Google Scholar 

  • Nagovitsyn, Yu.A.: 1997, A nonlinear mathematical model for the solar cyclicity and prospects for reconstructing the solar activity in the past. Astron. Lett. 23, 742.

    ADS  Google Scholar 

  • Nagovitsyn, Yu.A.: 2001, Solar activity during the last two millennia: Solar patrol in ancient and medieval China. Geomagn. Aeron. 41, 680.

    Google Scholar 

  • Ogurtsov, M.G., Nagovitsyn, Y.A., Kocharov, G.E., Jungner, H.: 2002, Long-period cycles of the Sun’s activity recorded in direct solar data and proxies. Solar Phys. 211, 371.

    Article  ADS  Google Scholar 

  • Otaola, G.A., Zenteno, G.: 1983, On the existence of long term periodicities in solar activity. Solar Phys. 89, 209.

    Article  ADS  Google Scholar 

  • Rozelot, J.P.: 1994, On the stability of the 11-year solar cycle period (and a few others). Solar Phys. 149, 149.

    Article  ADS  Google Scholar 

  • Solanki, S.K., Usoskin, I.G., Kromer, B., Schüssler, M., Beer, J.: 2004, Unusual activity of the Sun during recent decades compared to the previous 11,000 years. Nature 431, 1084. DOI .

    Article  ADS  Google Scholar 

  • Svalgaard, L., Schatten, K.H.: 2016, Reconstruction of the sunspot group number: The backbone method. Solar Phys. 291, 2653. DOI .

    Article  ADS  Google Scholar 

  • Usoskin, I.G., Kovaltsov, G.A., Lockwood, M., Mursula, K., Owens, M.J., Solanki, S.K.: 2016, A new calibrated sunspot group series since 1749: Statistics of active day fractions. Solar Phys. 291, 2685. DOI .

    Article  ADS  Google Scholar 

  • Vautard, R., Ghil, M.: 1989, Singular spectrum analysis in nonlinear dynamics, with applications to paleoclimatic time series. Physica D 35, 395.

    Article  ADS  MathSciNet  MATH  Google Scholar 

  • Vautard, R., Yiou, P., Ghil, M.: 1992, Singular-spectrum analysis: A toolkit for short, noisy chaotic signals. Physica D 58, 95.

    Article  ADS  Google Scholar 

  • Waldmeier, M.: 1957, Der lange Sonnenzyklus. Z. Astrophys. 43, 149.

    ADS  Google Scholar 

  • Wittmann, A.: 1978, The sunspot cycle before the Maunder minimum. Astron. Astrophys. 66, 93.

    ADS  Google Scholar 

Download references


We thank A. Shapoval for useful discussions of the SSA and principal component analysis (PCA). We thank the anonymous referee for useful remarks. This is IPGP contribution NS 3822.

Author information

Authors and Affiliations


Corresponding author

Correspondence to Vincent Courtillot.

Ethics declarations

Disclosure of Potential Conflicts of Interest

The authors declare that they have no conflicts of interest.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Le Mouël, JL., Lopes, F. & Courtillot, V. Identification of Gleissberg Cycles and a Rising Trend in a 315-Year-Long Series of Sunspot Numbers. Sol Phys 292, 43 (2017).

Download citation

  • Received:

  • Accepted:

  • Published:

  • DOI: