NPG Nonlinear Processes in Geophysics NPG Nonlin. Processes Geophys. 1607-7946 Copernicus Publications Göttingen, Germany 10.5194/npg-21-797-2014 Can irregularities of solar proxies help understand quasi-biennial solar variations? Shapoval A. https://orcid.org/0000-0001-5340-1930 1 3 4 Le Mouël J. L. 2 Shnirman M. 1 2 Courtillot V. 2 IEPT RAS, Profsoyuznaya str. 84/32, 117 997 Moscow, Russia IPGP, 1 rue Jussieu, 75005 Paris, France Financial University, Leningradsky pr. 49, 125 167 Moscow, Russia National Research University Higher School of Economics, 20 Myasnitskaya Ulitsa, 101 000 Moscow, Russia 01 08 2014 21 4 797 813 Copyright: © 2014 A. Shapoval et al. 2014 This work is licensed under the Creative Commons Attribution 3.0 Unported License. To view a copy of this licence, visit https://creativecommons.org/licenses/by/3.0/ This article is available from https://npg.copernicus.org/articles/21/797/2014/npg-21-797-2014.html The full text article is available as a PDF file from https://npg.copernicus.org/articles/21/797/2014/npg-21-797-2014.pdf

We define, calculate and analyze irregularity indices &lambda;<sub>ISSN</sub> of daily series of the International Sunspot Number ISSN as a function of increasing smoothing from <i>N</i> = 162 to 648 days. The irregularity indices λ are computed within 4-year sliding windows, with embedding dimensions <i>m</i> = 1 and 2. &lambda;<sub>ISSN</sub> displays Schwabe cycles with ~5.5-year variations ("half Schwabe variations" HSV). The mean of &lambda;<sub>ISSN</sub> undergoes a downward step and the amplitude of its variations strongly decreases around 1930. We observe changes in the ratio <i>R</i> of the mean amplitude of λ peaks at solar cycle minima with respect to peaks at solar maxima as a function of date, embedding dimension and, importantly, smoothing parameter <i>N</i>. We identify two distinct regimes, called Q1 and Q2, defined mainly by the evolution of <i>R</i> as a function of <i>N</i>: Q1, with increasing HSV behavior and <i>R</i> value as <i>N</i> is increased, occurs before 1915–1930; and Q2, with decreasing HSV behavior and <i>R</i> value as <i>N</i> is increased, occurs after ~1975. We attempt to account for these observations with an autoregressive (order 1) model with Poissonian noise and a mean modulated by two sine waves of periods <i>T</i><sub>1</sub> and <i>T</i><sub>2</sub> (<i>T</i><sub>1</sub> = 11 years, and intermediate <i>T</i><sub>2</sub> is tuned to mimic quasi-biennial oscillations QBO). The model can generate both Q1 and Q2 regimes. When <i>m</i> = 1, HSV appears in the absence of <i>T</i><sub>2</sub> variations. When <i>m</i> = 2, Q1 occurs when <i>T</i><sub>2</sub> variations are present, whereas Q2 occurs when <i>T</i><sub>2</sub> variations are suppressed. We propose that the HSV behavior of the irregularity index of ISSN may be linked to the presence of strong QBO before 1915–1930, a transition and their disappearance around 1975, corresponding to a change in regime of solar activity.

 References Bartels, J.: Twenty-Seven Day Recurrences in Terrestrial-Magnetic and Solar Activity, 1923–1933, Terr. Magnet. Atmos. Elect., 39, 201–202, 1934. Bergé, P., Pomeau, Y., and Vidal, C.: L'Ordre dans le Chaos, Hermann, Paris, France, 353 pp., 1984. Bershadskii, A.: New dynamics of the Sun convection zone and global warming, arXiv:0805.2108v1 [astro-ph.SR], 2008. Bershadskii, A.: Chaotic mean wind in turbulent thermal convection and long-term correlations in solar activity, arXiv:0908.4008v4 [astro-ph.SR], 2009. Blanter, E. M., Shnirman, M. G., and Le Mouël, J.-L.: Solar variability: Evolution of correlation properties, J. Atmos. Solar-Terr. Phys. 67, 521–534, 2005. Blanter, E. M., Le Mouël, J.-L., Perrier, F., and Shnirman, M. G.: Short-term correlation of solar activity and sunspot: evidence of lifetime increase, Solar Phys. 237, 329–350, 2006. Charbonneau, P., Beaubien, G., and St-Jean, C.: Fluctuations in Babcok-Leighton dynamos: II. Revisiting the Gnevyshev-Ohl rule, Astrophys. J., 658, 657–662, 2007. Choudhuri, A. R. and Karak, B. B.: Origin of Grand Minima in Sunspot Cycles, Phys. Rev. Lett., 109, 171103–171106, 2012. Ding, R. and Li, J.: Nonlinear finite-time Lyapunov exponent and predictability, Phys. Lett. A 364, 396–400, 2007. Duhau, S. and de Jager, C.: The Solar Dynamo and its Phase Transitions during the Last Millenium, Solar Phys. 250, 1–15, 2008. Eckmann, J.-P. and Ruelle, D.: Ergodic theory of chaos and strange attractors, Rev. Modern Phys., 57, 617–656, 1985. Fraser, A. M. and Swinney, H. L.: Independent coordinates for strange attractors from mutual information, Phys. Rev. A, 33, 1134–1140, 1986. Greenkorn, R. A.: 2009, Analysis of Sunspot Activity Cycles, Solar Phys., 255, 301–323, <a href="http://dx.doi.org/10.1007/s11207-009-9331-z">https://doi.org/10.1007/s11207-009-9331-z</a>, 2009. Howe, R.: Solar interior rotation and its variation, Living Rev. Solar Phys., 6, 1–91, 2009. Hoyt, D. V. and Schatten, K. H.: Group Sunspot Numbers: A new Solar Activity Reconstruction, Solar Phys., 181, 491–512, 1998. Ivanov, E. V., Obridko, V. N., and Shelting, B. D.: Quasi-biennial oscillations of the solar magnetic fields, in: Solar variability: from core to outer frontiers, The 10th European Solar Physics Meeting, 9–14 September 2002, Prague, Czech Republic, edited by: Wilson, A., ESA SP-506 v. 2, ESA Publications Division, Noordwijk, 847–850, 2002. Kantz, H.: A robust method to estimate the maximal Lyapunov exponent of a time series, Phys. Lett. A, 185, 77–87, 1994. Kitchatinov, L. L. and Olemskoi, S. V.: Active longitudes of the sun: The rotation period and statistical significance, Astron. Lett., 31, 280–284, 2005. Kudela, K., Rybak, J., Antalova, A., and Storini, M.: Time evolution of low-frequency periodicities in cosmic ray intensity, Solar Phys. 205, 165–175, 2002. Lawrence, J. K., Cadavid, A. C., and Ruzmaikin, A. A.: Turbulent and chaotic dynamics underlying solar magnetic variability, Astron. Astrophys., 455, 366–375, 1995. Lawrence, J. K., Cadavid, A. C., and Ruzmaikin, A. A.: Rotational Quasi-Periodicities and the Sun-Heliosphere Connection, Solar Phys., 252, 179–193, 2008. Le Mouël, J.-L., Shnirman, M. G., and Blanter, E. M.: The 27-Day Signal in Sunspot Number Series and the Solar Dynamo, Solar Phys., 246, 295–307, 2007. Li, Q.-X. and Li, K.-J.: 2007, Low dimensional chaos from the group sunspot numbers, Chinese J. Astron. Astrophys., 7, 435–440, 2007. Lockwood, M.: Long-term variations in the magnetic fields of the sun and the heliosphere: their origin, effects and implications, J. Geophys. Res., 106, 16021–16038, 2001. Love, J. J. and Rigler, E. J.: Sunspot random walk and 22-year variation, Geophys. Res. Lett., 39, L10103–L10108, 2012. Macek, W. M., Bruno, R., and Consolini, G.: Testing for multifractality of the slow solar wind, Adv. Space Res. 37, 461–466, 2006. Mavromichalaki, H., Preka-Papadema, P., Petropoulos, B., Tsagouri, I., Georgakopoulos, S., and Polygiannakis, J.: Low- and high-frequency spectral behavior of cosmic-ray intensity for the period 1953–1996, Ann. Geophys., 21, 1681–1689, <a href="http://dx.doi.org/10.5194/angeo-21-1681-2003">https://doi.org/10.5194/angeo-21-1681-2003</a>, 2003. Mayr, H. G. and Schatten, K. H.: Nonlinear oscillators in space physics, J. Atmos. Sol.-Terr. Phys., 74, 44–50, 2012. McIntosh, P. S., Thompson, R. J., and Willock, E. C.: A 600-day periodicity in solar coronal holes, Nature, 360, 322–324, 1992. Mursula, K., Zieger, B., and Vilppola, J. H.: Mid-term quasi-periodicities in geomagnetic activity during the last 15 solar cycles: connection to solar dynamo strength, Solar Phys., 212, 201–207, 2003. Obridko, V. N. and Shelting, B. D.: Occurrence of the 1.3-year periodicity in the large-scale solar magnetic field for 8 solar cycles, Adv. Space Res., 40, 1006–1014, 2007. Oseledets, V. I.: A multiplicative ergodic theorem. Lyapunov characteristic numbers for dynamical systems, Trans. Moscow Math. Soc., 19, 197–231, 1968. Ostryakov, V. N. and Usoskin, I. G.: On the dimension of solar attractor, Solar Phys., 127, 405–412, 1990. Pesnell, W. D.: Solar Cycle Predictions (Invited Review), Solar Phys., 281, 507–532, 2012. Price, C. P., Prichard, D., and Hogenson, E. A.: Do the sunspot numbers form a chaotic set?, J. Geophys. Res., 97, 19113–19120, 1992. Rosenstein, M. T., Collings, J. J., and De Luca, C. J.: A practical method for calculating largest Lyapunov exponents from small data sets, Physica D, 65, 117–134, 1993. Rouillard, A. and Lockwood, M.: Oscillations in the open solar magnetic flux with a period of 1.68 years: imprint on galactic cosmic rays and implications for heliospheric shielding, Ann. Geophys., 22, 4381–4395, <a href="http://dx.doi.org/10.5194/angeo-22-4381-2004">https://doi.org/10.5194/angeo-22-4381-2004</a>, 2004. Ruzmaikin, A., Feynman, J., Kosacheva, V.: On Long-Term Dynamics of the Solar Cycle, in: The solar cycle; Proceedings of the National Solar Observatory/Sacramento Peak 12th Summer Workshop 27, edited by: Harvey, K. L., ASP Conference Series, San Francisco, 547–556, 1992. Sello, S.: Solar cycle forecasting: A nonlinear dynamics approach, Astron. Astrophys., 377, 312–320, 2001. Shapoval, A., Le Mouël, J.-L., Courtillot, V., and Shnirman, M.: Two regimes in the regularity of sunspot numbers, Astrophys. J., 779, 108–116, 2013. Shapoval, A., Le Mouël, J.-L., Courtillot, V., and Shnirman, M.: submitted, Is a sudden increase of irregularity of sunspot numbers a precursor of a return to low solar activity?, J. Geophys. Res., in review, 2014. SIDC-team: World Data Center for the Sunspot Index, Royal Observatory of Belgium, Monthly Report on the International Sunspot Number, online catalogue of the sunspot index: <a href="http://www.sidc.be/sunspot-data/">http://www.sidc.be/sunspot-data/</a>, 1850–2005, 2005. Spiegel, E. A. and Wolf, A.: Chaos and the Solar Cycle, in: Chaotic Phenomena in Astrophysics, vol. 497, edited by: Buchier, J.-R. and Eichhorn, H., Ann. N.Y. Acad. Sci., New York, 55–60, 1987. Svalgaard, L., Updating the Historical Sunspot Record, arXiv:1003.4666 [astro-ph.SR], 2010. Svalgaard, L.: How well do we know the sunspot number? Comparative Magnetic Minima: Characterizing quiet times in the Sun and Stars, Proceedings of the International Astronomical Union, IAU Symposium 286, Mendoza, Argentina, 27–33, 2012. Takens, F.: Detecting strange attractors in turbulence, edited by: Rand, D. A., and Young, L. S., Springer, Berlin, 366 – 381, 1981. Valdes-Galicia, J. F., Perez-Enrizuez, R., and Otaola, J. A.: The cosmic ray 1.68 year variation: a clue to understand the nature of the solar cycle?, Solar Phys., 167, 409–417, 1996. Vecchio, A., Laurenza, M., Carbone, V., and Storini, M.: Quasi-biennial modulation of solar neutrino flux and solar and galactic comsic rays by solar cyclic activity, Astrophys. J. Lett., 709, L1–L5, 2010. Vecchio, A., Laurenza, M., Meduri, D., Carbone, V., and Storini, M.: The dynamics of the solar magnetic field: polarity reversals, butterfly diagrams, and quasi-biennial oscillations, Astrophys. J., 749, 27–36, <a href="http://dx.doi.org/10.1088/0004-637X/749/1/27">https://doi.org/10.1088/0004-637X/749/1/27</a>, 2012. Wolf, A., Swift, J. B., Swinney, H. L., and Vastano, J. A.: Determining Lyapunov Exponent from a Time Series, Physica D, 16, 285–317, 1985. Zhang, Q.: A nonlinear prediction of the smoothed monthly sunspot numbers, Astron. Astrophys. 310, 646–650, 1996.