Issue 
A&A
Volume 618, October 2018



Article Number  A89  
Number of page(s)  5  
Section  The Sun  
DOI  https://doi.org/10.1051/00046361/201833532  
Published online  16 October 2018 
Origin of the hemispheric asymmetry of solar activity
MaxPlanckInstitut für Sonnensystemforschung, JustusvonLiebigWeg 3, 37077, Göttingen, Germany
email: msch@mps.mpg.de
Received:
30
May
2018
Accepted:
13
July
2018
The frequency spectrum of the hemispheric asymmetry of solar activity shows enhanced power for the period ranges around 8.5 years and between 30 and 50 years. This can be understood as the sum and beat periods of the superposition of two dynamo modes: a dipolar mode with a (magnetic) period of about 22 years and a quadrupolar mode with a period between 13 and 15 years. An updated Babcock–Leightontype dynamo model with weak driving as indicated by stellar observations shows an excited dipole mode and a damped quadrupole mode in the correct range of periods. Random excitation of the quadrupole by stochastic fluctuations of the source term for the poloidal field leads to a time evolution of activity and asymmetry that is consistent with the observational results.
Key words: Sun: activity / Sun: magnetic fields / dynamo
© ESO 2018
1. Introduction
The various manifestations of solar magnetic activity, such as sunspots, prominences, and flares, typically are distributed unevenly between the northern and southern hemisphere of the Sun (cf. Norton et al. 2014; Hathaway 2015; Deng et al. 2016; and references therein). Normally, this hemispheric asymmetry does not exceed a level of about 20% (Norton & Gallagher 2010). However, during the Maunder minimum in the second half of the 17th century nearly all of the few sunspots observed during this time appeared in the southern hemisphere (Ribes & NesmeRibes 1993; Vaquero et al. 2015). Various studies demonstrated that the asymmetry has systematic components that cannot be explained by random fluctuations of flux emergence alone (e.g. Carbonell et al. 1993, 2007; Deng et al. 2016). It has been suggested that there is also a systematic variation of the phase shift of the activity cycle between the hemispheres (e.g. Zolotova et al. 2010; Norton & Gallagher 2010; Muraközy & Ludmány 2012; McIntosh et al. 2013).
A number of studies investigate the hemispheric asymmetry by way of frequency analysis (e.g. Deng et al. 2016; and references therein). Ballester et al. (2005) demonstrated that the commonly used normalised asymmetry parameter, (N − S)/(N + S), where N and S represent the quantity under consideration in the northern and southern hemisphere, respectively, is not a sensible choice for this kind of analysis: the denominator introduces a contamination of the power spectrum by the strong 11yr periodicity. Ballester et al. (2005) instead carry out a frequency analysis of the unnormalised asymmetry, A_{N}A_{S}, of the monthly hemispheric sunspot areas between 1874 and 2004. They use the dataset compiled by D. Hathaway, based upon the Greenwich Photoheliographic Results and the USAF/SOON data, and find three significant periods with falsealarm probabilities below 0.5%: 43.25, 8.65, and 1.44 years. Very similar periods (among others) are also found by Knaack et al. (2004), who use the same dataset, while Deng et al. (2016) reported periods of 51.3 and 8.7 years. Studying the hemispheric asymmetry of filaments between 1919 and 1989, Duchlev & Dermendjiev (1996) find periods of 35 and 8.75 years, although they considered only the former to be statistically significant. On the other hand, Chang (2009) suggests that only the period around nine years in the asymmetry of sunspot areas is significant while other periodicities may not. Since the length of the various data series does not exceed about 150 years, the frequency resolution for the longer periods is rather low. Thus we may conclude from these studies that there is evidence for a short period (around 9 years) and a long period (between 35 and 50 years) in the data for the unnormalised asymmetry, while the 11yr cycle does not significantly appear.
In this paper, we show that the periods found in the observational data for the absolute hemispheric asymmetry occur naturally as the beat period and the sum period of a mixedmode dynamo solution comprised of a dipole mode with a (magnetic) period of about 22 years and a quadrupole mode with a period between 13 and 15 years. We also find that periods in this range are reproduced by the updated Babcock–Leighton dynamo model of Cameron & Schüssler (2017a) in the case of weak dynamo driving as suggested by stellar observations. While the dipole mode is permanently excited, the quadrupole is subcritical and only occasionally kicks in through random fluctuations of the poloidal source term.
This paper is structured as follows. In Sect. 2, we present a simple model of superposed harmonic oscillations to illustrate the origin of the various periodicities. From the observed periods of the hemispheric asymmetry, we determine the periods of the antisymmetric (dipole) mode and the symmetric (quadrupole) mode. Section. 3 gives the corresponding results obtained with the updated Babcock–Leighton model. We summarise our conclusions in Sect. 4.
2. Hemispheric aymmetry by superposition of symmetric and antisymmetric modes
As a simple illustration of the possible origin of the various periods detected in the sunspot area data (full disk, hemispheric, and asymmetry), we have considered the superposition of two harmonic oscillations with different frequencies. They are taken to represent two dynamo modes for the toroidal field, B_{ϕ}: one mode is antisymmetric with respect to the equator (dipole parity, frequency ω_{D}), the other mode is symmetric (quadrupole parity, frequency ω_{Q}). Since we are only interested in the frequencies resulting from the superposition, we set the amplitudes of the modes to be equal and normalise them to unity. Taking the activity index to be proportional to , that is, the square of the superposed modes, we have for the indices in the northern hemisphere, A_{N}, and in the southern hemisphere, A_{S}, respectively:
The indices for the full disk and for the (absolute) asymmetry are then given by the sum and the difference, respectively, of the hemispheric signals, viz.
where ω_{Q}−ω_{D} ≡ ω_{b} is the beat frequency and ω_{Q} + ω_{D} ≡ ω_{s} is the sum frequency. It is clear from Eqs. (1) and (2) that the frequencies appearing in the various quantities are different: while only the double frequencies, 2ω_{D} and 2ω_{Q}, show up in the fulldisk index, the absolute asymmetry is governed solely by the beat and the sum frequencies. The hemispheric indices are affected by all four of these frequencies. In terms of periods, we have
for the beat period and
for the period corresponding to the sum frequency, where P_{D} = 2π/ω_{D} and P_{Q} = 2π/ω_{Q}, respectively, are the periods of the dipole and quadrupole dynamo modes.
Consistent with the expectation from this simple model, the analysis of the absolute asymmetry of hemispheric sunspot areas by Ballester et al. (2005) results in two dominant periods, 43.25 years and 8.65 years, while the 11yr cycle period (dominated by the dipole) does not appear. Tentatively identifying these two observed periods with the beat and sum periods, we can use Eqs. (3) and (4) determine the dipole period, P_{D}, and the quadrupole period, P_{Q}. With P_{b} = 43.25 years and P_{s} = 8.65 years we obtain P_{D} = 21.6 years and P_{Q} = 14.4 years. If we take into account the limited frequency resolution and assume a range between 30 years and 50 years for the for the longer (beat) period, we find P_{D} ≈ 21. . .24 years and P_{Q} ≈ 13. . .15 years. The obtained dipole period is consistent with the 11yr activity cycle. This suggests that our simple model is a viable representation of the asymmetry data, suggesting the presence of a solar quadrupole dynamo mode with a period between 13 and 15 years. Incidentally, MuñozJaramillo et al. (2013) show that considering both the dipole and quadrupole moments of the poloidal field during cycle minima improves the predictive power for the amplitude of the subsequent cycle (see also Goel & Choudhuri 2009).
3. Dynamo model
Hemispheric asymmetry in dynamo models has been studied by various authors (reviewed by Norton et al. 2014; Brun et al. 2015). There are two main approaches that have been followed: nonlinear effects and stochastic fluctuations. Nonlinearity in the dynamo equations can lead to the coupling of symmetric (even) and antisymmetric (odd) modes, strong hemispheric asymmetry, and the occurence of extended “grand minima” (e.g. Kleeorin & Ruzmaikin 1984; Tobias 1997; Brooke et al. 1998; Hotta & Yokoyama 2010; Weiss & Tobias 2016). Alternatively, stochastic fluctuations of model ingredients (such as meanfield αeffect or meridional flow speed) can also lead to hemispheric asymmetry as well as to the (temporal) excitation of higher eigenmodes and mixedmode solutions (e.g. Hoyng et al. 1994; Olemskoy & Kitchatinov 2013; Belucz & Dikpati 2013; Passos et al. 2014). Combinations of both effects have been studied as well (e.g. Moss et al. 1992; Schmitt et al. 1996; Mininni&Gómez 2002; Charbonneau 2007; Moss & Sokoloff 2017). For example, Sokoloff et al. (2010) and Usoskin et al. (2009) find that random fluctuations of the dynamo excitation in a simple Parkertype dynamo wave model can lead to substantial mixing between dipole and quadrupole modes, particularly so during episodes of low dynamo amplitude akin to grand minima of solar activity. Global 3DMHD simulations exhibit features that are similar to the results provided by the more idealised approaches (Norton et al. 2014; Brun et al. 2015; Käpylä et al. 2016).
Dominance of the dipole mode and relatively weak hemispheric asymmetry can be provided by sufficiently strong hemispheric coupling via turbulent diffusion, crossequator flows, or crossequator cancellation of toroidal flux (Norton et al. 2014; Cameron & Schüssler 2016). Moreover, observational gyrochronology of solarlike stars (van Saders et al. 2016; Metcalfe et al. 2016) indicates weak excitation of the solar dynamo. In this case, there is the possibility that only the lowest (dipole) dynamo mode is excited while the quadrupole mode is linearly damped.
As an illustration for the stochastic excitation of a mixedmode dynamo solution that is consistent with the observed features of the hemispheric asymmetry, we show results from the updated Babcock–Leighton model of Cameron & Schüssler (2017a) with stochastic fluctuations of the poloidal field source (see Sect. 3 in Cameron & Schüssler 2017b). The parameters for this case were chosen according to the following observational constraints: (a) excited dipole mode with a period of about 22 years, (b) phase difference between polar radial field and subsurface toroidal flux of about 90°, (c) no linearly excited quadrupolar mode. In such a case, linearly damped quadrupole modes can be stochastically excited by the fluctuations of the poloidal field source, so that a mixedmode solution occasionally develops. The parameters for the case discussed here were: η_{R}_{⊙} = 150 km^{2} s^{−1} and η_{0} = 50 km^{2} s^{−1} for the turbulent magnetic diffusivities in the nearsurface layers and in the bulk of the convection zone, respectively; α_{0} = 1.3 m s^{−1} for the average poloidal source level, and σ* = 0.046 for the fluctuation level. The source fluctuations are local in latitude (in steps of 1 deg) and in time (in steps of 1 day), governed by a Wiener process with a variance of 1 radian^{−1} after 11 years. This corresponds to a RMS fluctuation of the source term of about 5% (integrated over 11 years and one radian). We used the nearsurface meridional flow as determined by Hathaway & Rightmire (2011) and V_{0} = 1.7 m s^{−1} for the amplitude of the equatorward meridional return flow affecting the toroidal flux in the convection zone. The critical integrated flux for the cutoff nonlinearity in the poloidal source term was 10^{24} Mx.
Linear analysis shows that for these parameters we have an excited oscillating antisymmetric (dipolar) mode with a period of 22.4 years (α/α_{crit} = 1.06) and a damped symmetric (quadrupolar) mode with a period of 13.5 years and a damping time of 13 years (α/α_{crit} = 0.12). In addition, there is a symmetric stationary mode with a damping time of about 10 years. Figure 1 shows the spatiotemporal structure of these linear modes. Figure 2 shows the symmetric and antisymmetric components of a simulation of the nonlinear case with fluctuations. The quadrupolar (symmetric) modes are occasionally excited owing to the random fluctuations of the poloidal source term. In addition, there are variations of the dynamo amplitude. The ratio of the RMS values of the toroidal field variable between the quadrupole and the dipole modes is about 0.18. Owing to the nonlinearity, the oscillation periods become somewhat variable and their mean values differ from their linear counterparts: 20.8 years for the dipole and 14.7 years for the oscillatory quadrupole. According to Eqs. (3) and (4), the linear periods lead to a beat period P_{b} = 34 years and a sum period P_{s} = 8.4 years while the nonlinear periods give P_{b} = 50.1 years and P_{s} = 8.6 years. Hence, the beat period is much more sensitive to period variations than the sum period. We therefore expect periodic signals in the absolute asymmetry (for instance of the toroidal flux taken as an activity indicator) around 8.5 years (P_{s}) and in the range 30–50 years (P_{b}). Such signals can in fact be seen in Fig. 3, which shows power spectra of the absolute hemispheric asymmetry of the hemispheric toroidal flux integrated between 0° and ±40° latitude (red curve) from the Babcock–Leighton model in comparison with that of the observed sunspot area (blue curve). This is consistent with the expectation from the simple model described in Sect. 2. The widths of the peaks results from the variability of the (nonlinear) periods, the damping of the quadrupole mode, as well as from realisation noise. The dynamo model serves only as an illustration of the proposed mechanism behind the hemispheric asymmetry; we have made no attempt to finetune the parameters in order to have a precise agreement of the model result with the observed peak at 43 years. It suffices to state that the updated Babcock–Leighton model with low excitation and stochastic fluctuations of the poloidal source yields results that are consistent with the observed features.
Fig. 1.
Spatiotemporal structure of the first linear eigenmodes obtained with the updated Babcock–Leighton model (Cameron & Schüssler 2017a). Shown are latitudetime diagrams of the azimuthally averaged radial surface field (left panels) and the radially integrated toroidal flux (right panels). The three modes shown are an excited oscillatory dipolar mode (top panels), a damped oscillatory quadrupolar mode (middle panels) and a damped stationary quadrupolar mode (bottom panels). The quantities are normalised to their individual maxima in all cases. 

Open with DEXTER 
Fig. 2.
Nonlinear case with random fluctuations of the poloidal source term. Shown are timelatitude diagrams of the antisymmetric (dipolar, upper panel) and symmetric (quadrupolar, lower panel) components of the radially integrated toroidal magnetic flux. The linearly damped oscillatory and stationary quadrupolar modes are occasionally excited. 

Open with DEXTER 
Fig. 3.
Power spectra of the absolute hemispheric asymmetry. The blue curve corresponds to the absolute asymmetry of the sunspot areas from the Greenwich/SOON data. The red curve gives the average power spectrum for the difference of the unsigned toroidal fluxes integrated between the equator and ±30/40° latitude). The latter is based on 1000 realizations of 500 year each from the Babcock–Leighton model with source fluctuations (Cameron & Schüssler 2017b). The pink shading indicates ±1 standard deviation from the mean. 

Open with DEXTER 
We may also consider the phase lag between the hemispheres. From the sunspot record, some authors suggest a periodicity of the phase lag of eight cycles, that is, about 90 years (e.g. Zolotova et al. 2010; Norton&Gallagher 2010; Muraközy & Ludmány 2012), or even twelve cycles (e.g. Zhang&Feng 2015). Some caution seems to be in order in view of the fact that the time series used are not much longer than the inferred periods. Using a crosscorrelation method, McIntosh et al. (2013) find longterm variations with the hemispheres alternating in phase shift for intervals between 30 and 60 years since 1874. On the other hand, on the basis of the simple model in Sect. 2, we expect a periodicity of the phase lag with the beat period, P_{b}. We have applied a method similar to that used by McIntosh et al. (2013) to the results from the updated Babcock–Leighton model, simulated with the same parameters as described above. Using the hemispheric toroidal flux integrated from the equator to ±40° latitude, we considered 20yr segments from 50 simulations covering 10 000 year each. Removing the mean of the signals and applying a Hann window to the segments, we then calculated the crosscovariance between the northern and southern hemisphere signals and determined the time lag between the hemispheres by considering the maximum of the crosscovariance. A typical example for the temporal variation of the lag is shown in Fig. 4. It suggest a modulation with a period around 30 years. This is confirmed by the power spectrum shown in Fig. 5. The curve gives the mean power spectrum for 1000 realizations of 500 years length each while the shading indicates ±1 standard deviation. Comparison with Fig. 3 shows that the main power appears in the range of the beat period of 3050 years, but there is considerable power at longer periods as well, indicating longerterm modulations. This is also obvious from Fig. 4, where the sign of the lag often remains the same for intervals exceeding the beat period.
Fig. 4.
Phase lag (time shift) between the toroidal fluxes in the northern and southern hemispheres for a 1000yr stretch of a simulation on the basis of the updated Babcock–Leighton model. The lag was determined by using the crosscovariance between segments with a length of 20 years. 

Open with DEXTER 
Fig. 5.
Average power spectrum of the time shift between the hemispheres from 20 realizations of the updated Babcock–Leighton dynamo of 10 000 years length each. The shading indicates ±1 standard deviations. 

Open with DEXTER 
4. Conclusions
We have shown that the observed power spectrum of the absolute hemispheric asymmetry of solar activity can be naturally explained by the superposition of an excited dipolar mode (toroidal field antisymmetric with respect to the equator) with a magnetic period of about 22 years and a linearly damped, but randomly excited quadrupolar mode (toroidal field symmetric with respect to the equator) with a period between 13 and 15 years. The updated Babcock–Leighton dynamo model of Cameron & Schüssler (2017a) with weak excitation reproduces these conditions and yields a time evolution of the magnetic field and its asymmetry that is consistent with the observations.
References
 Ballester, J. L., Oliver, R., & Carbonell, M. 2005, A&A, 431, L5 [NASA ADS] [CrossRef] [EDP Sciences] [Google Scholar]
 Belucz, B., & Dikpati, M. 2013, ApJ, 779, 4 [NASA ADS] [CrossRef] [Google Scholar]
 Brooke, J. M., Pelt, J., Tavakol, R., & Tworkowski, A. 1998, A&A, 332, 339 [NASA ADS] [Google Scholar]
 Brun, A. S., Browning, M. K., Dikpati, M., Hotta, H., & Strugarek, A. 2015, Space Sci. Rev., 196, 101 [NASA ADS] [CrossRef] [Google Scholar]
 Cameron, R. H., & Schüssler, M. 2016, A&A, 591, A46 [NASA ADS] [CrossRef] [EDP Sciences] [Google Scholar]
 Cameron, R. H., & Schüssler, M. 2017a, A&A, 599, A52 [NASA ADS] [CrossRef] [EDP Sciences] [Google Scholar]
 Cameron, R. H., & Schüssler, M. 2017b, ApJ, 843, 111 [NASA ADS] [CrossRef] [Google Scholar]
 Carbonell, M., Oliver, R., & Ballester, J. L. 1993, A&A, 274, 497 [NASA ADS] [Google Scholar]
 Carbonell, M., Terradas, J., Oliver, R., & Ballester, J. L. 2007, A&A, 476, 951 [NASA ADS] [CrossRef] [EDP Sciences] [Google Scholar]
 Chang, H.Y. 2009, New Astron., 14, 133 [NASA ADS] [CrossRef] [Google Scholar]
 Charbonneau, P. 2007, Adv. Space Res., 40, 899 [NASA ADS] [CrossRef] [Google Scholar]
 Deng, L. H., Xiang, Y. Y., Qu, Z. N., & An, J. M. 2016, AJ, 151, 70 [NASA ADS] [CrossRef] [Google Scholar]
 Duchlev, P. I., & Dermendjiev, V. N. 1996, Sol. Phys., 168, 205 [NASA ADS] [CrossRef] [Google Scholar]
 Goel, A., & Choudhuri, A. R. 2009, Res. Astron. Astrophys., 9, 115 [NASA ADS] [CrossRef] [Google Scholar]
 Hathaway, D. H. 2015, Liv. Rev. Sol. Phys., 12, 12, http://link.springer.com/article/10.1007/lrsp [Google Scholar]
 Hathaway, D. H., & Rightmire, L. 2011, ApJ, 729, 80 [NASA ADS] [CrossRef] [Google Scholar]
 Hotta, H., & Yokoyama, T. 2010, ApJ, 714, L308 [NASA ADS] [CrossRef] [Google Scholar]
 Hoyng, P., Schmitt, D., & Teuben, L. J. W. 1994, A&A, 289, 265 [NASA ADS] [Google Scholar]
 Käpylä, M. J., Käpylä, P. J., Olspert, N., et al. 2016, A&A, 589, A56 [NASA ADS] [CrossRef] [EDP Sciences] [Google Scholar]
 Kleeorin, N. I., & Ruzmaikin, A. A. 1984, Astron. Nachr., 305, 265 [NASA ADS] [CrossRef] [Google Scholar]
 Knaack, R., Stenflo, J. O., & Berdyugina, S. V. 2004, A&A, 418, L17 [NASA ADS] [CrossRef] [EDP Sciences] [Google Scholar]
 McIntosh, S. W., Leamon, R. J., Gurman, J. B., et al. 2013, ApJ, 765, 146 [NASA ADS] [CrossRef] [Google Scholar]
 Metcalfe, T. S., Egeland, R., & van Saders, J. 2016, ApJ, 826, L2 [NASA ADS] [CrossRef] [Google Scholar]
 Mininni, P. D., & Gómez, D. O. 2002, ApJ, 573, 454 [NASA ADS] [CrossRef] [Google Scholar]
 Moss, D. L., & Sokoloff, D. D. 2017, Astron. Rep., 61, 878 [NASA ADS] [CrossRef] [Google Scholar]
 Moss, D., Brandenburg, A., Tavakol, R., & Tuominen, I. 1992, A&A, 265, 843 [NASA ADS] [Google Scholar]
 MuñozJaramillo, A., Balmaceda, L. A., & DeLuca, E. E. 2013, Phys. Rev. Lett., 111, 041106 [NASA ADS] [CrossRef] [Google Scholar]
 Muraközy, J., & Ludmány, A. 2012, MNRAS, 419, 3624 [NASA ADS] [CrossRef] [Google Scholar]
 Norton, A. A., & Gallagher, J. C. 2010, Sol. Phys., 261, 193 [NASA ADS] [CrossRef] [Google Scholar]
 Norton, A. A., Charbonneau, P., & Passos, D. 2014, Space Sci. Rev., 186, 251 [NASA ADS] [CrossRef] [Google Scholar]
 Olemskoy, S. V., & Kitchatinov, L. L. 2013, ApJ, 777, 71 [NASA ADS] [CrossRef] [Google Scholar]
 Passos, D., Nandy, D., Hazra, S., & Lopes, I. 2014, A&A, 563, A18 [NASA ADS] [CrossRef] [EDP Sciences] [Google Scholar]
 Ribes, J. C., & NesmeRibes, E. 1993, A&A, 276, 549 [NASA ADS] [Google Scholar]
 Schmitt, D., Schüssler, M., & FerrizMas, A. 1996, A&A, 311, L1 [NASA ADS] [Google Scholar]
 Sokoloff, D., Arlt, R., Moss, D., Saar, S. H., & Usoskin, I. 2010, in Solar and Stellar Variability: Impact on Earth and Planets, eds. A. G. Kosovichev, A. H. Andrei, & J.P. Rozelot, IAU Symp., 264, 119 [NASA ADS] [Google Scholar]
 Tobias, S. M. 1997, A&A, 322, 1007 [NASA ADS] [Google Scholar]
 Usoskin, I. G., Sokoloff, D., & Moss, D. 2009, Sol. Phys., 254, 345 [NASA ADS] [CrossRef] [Google Scholar]
 van Saders, J. L., Ceillier, T., Metcalfe, T. S., et al. 2016, Nature, 529, 181 [NASA ADS] [CrossRef] [PubMed] [Google Scholar]
 Vaquero, J. M., Nogales, J. M., & SánchezBajo, F. 2015, Adv. Space Res., 55, 1546 [NASA ADS] [CrossRef] [Google Scholar]
 Weiss, N. O., & Tobias, S. M. 2016, MNRAS, 456, 2654 [NASA ADS] [CrossRef] [Google Scholar]
 Zhang, J., & Feng, W. 2015, AJ, 150, 74 [NASA ADS] [CrossRef] [Google Scholar]
 Zolotova, N. V., Ponyavin, D. I., Arlt, R., & Tuominen, I. 2010, Astron. Nachr., 331, 765 [NASA ADS] [CrossRef] [Google Scholar]
All Figures
Fig. 1.
Spatiotemporal structure of the first linear eigenmodes obtained with the updated Babcock–Leighton model (Cameron & Schüssler 2017a). Shown are latitudetime diagrams of the azimuthally averaged radial surface field (left panels) and the radially integrated toroidal flux (right panels). The three modes shown are an excited oscillatory dipolar mode (top panels), a damped oscillatory quadrupolar mode (middle panels) and a damped stationary quadrupolar mode (bottom panels). The quantities are normalised to their individual maxima in all cases. 

Open with DEXTER  
In the text 
Fig. 2.
Nonlinear case with random fluctuations of the poloidal source term. Shown are timelatitude diagrams of the antisymmetric (dipolar, upper panel) and symmetric (quadrupolar, lower panel) components of the radially integrated toroidal magnetic flux. The linearly damped oscillatory and stationary quadrupolar modes are occasionally excited. 

Open with DEXTER  
In the text 
Fig. 3.
Power spectra of the absolute hemispheric asymmetry. The blue curve corresponds to the absolute asymmetry of the sunspot areas from the Greenwich/SOON data. The red curve gives the average power spectrum for the difference of the unsigned toroidal fluxes integrated between the equator and ±30/40° latitude). The latter is based on 1000 realizations of 500 year each from the Babcock–Leighton model with source fluctuations (Cameron & Schüssler 2017b). The pink shading indicates ±1 standard deviation from the mean. 

Open with DEXTER  
In the text 
Fig. 4.
Phase lag (time shift) between the toroidal fluxes in the northern and southern hemispheres for a 1000yr stretch of a simulation on the basis of the updated Babcock–Leighton model. The lag was determined by using the crosscovariance between segments with a length of 20 years. 

Open with DEXTER  
In the text 
Fig. 5.
Average power spectrum of the time shift between the hemispheres from 20 realizations of the updated Babcock–Leighton dynamo of 10 000 years length each. The shading indicates ±1 standard deviations. 

Open with DEXTER  
In the text 
Current usage metrics show cumulative count of Article Views (fulltext article views including HTML views, PDF and ePub downloads, according to the available data) and Abstracts Views on Vision4Press platform.
Data correspond to usage on the plateform after 2015. The current usage metrics is available 4896 hours after online publication and is updated daily on week days.
Initial download of the metrics may take a while.