PHASE RELATIONSHIPS OF SOLAR HEMISPHERIC TOROIDAL AND POLOIDAL CYCLES

Several solar phenomena exhibit hemispheric asymmetries and variations. Most of the relevant literature focuses on the variation in the amplitude of the asymmetry. Various periods have been found: 3.7 years (Vizoso & Ballester 1990); periods between 9 and 12 years (Chang 2008); 43.25, 8.65, and 1.44 years (Ballester et al. 2005); and a timescale of 12 cycles (Li et al. 2009). Signatures ofsolar hemispheric asymmetry have been claimed in solar wind speed (Zieger & Mursula 1998) and cosmic rays (Krymsky et al. 2009).

Ballester et al. (2005) did not find an 11 year period in the normalized north–south asymmetry index. This timescale has to be studied in a different manner, that is, by examining the phase lags of the hemispheric cycles. Earlier investigations of this behavior (Waldmeier 1957, 1971; Li et al. 2009; Zolotova et al. 2009) indicated that these phase lags exhibit long-term variation.

In our previous paper (Muraközy & Ludmány 2012, henceforth Paper I), the phase lags of the hemispheric cycles were examined in Cycles 12–23 using different methods, and a characteristic behavior was detected: the same hemisphere leads in four consecutive cycles and in the next four consecutive cycles the other hemispheric cycle leads. This characteristic timescale is reminiscent of that published by Gleissberg (1939).

The present work was motivated by the question of whether this variation was also working before the Greenwich era, which began with Cycle 12. Another aspect was raised by the recent work of Svalgaard & Kamide (2013), who examined the hemispheric phase lags of the polarity reversals of the poloidal field. They investigated the time interval of 1945–2011; this feature can also be examined over a longer time interval in comparison with the phase lags of the hemispheric cycles.

The investigation of Paper I was based on the Greenwich Photoheliographic Results (henceforth GPR; Royal Observatory 1874) and the Debrecen Photoheliographic Data (henceforth DPD; Győri et al. 2011). In the present study, which extends our earlier work, sunspot data for the previous cycles have been gathered from the observations of Johann Caspar Staudacher for the period 1749–1798 (Arlt 2009) and Samuel Heinrich Schwabe for the period 1825–1867 (Arlt et al. 2013). The observations of Staudacher were sparse (Figure 1), and only a few observations were made in certain years.

Zoom In Zoom Out Reset image size

After two and a half cycles without any data, Schwabe carried out a long, continuous series of observations covering 43 years. He conducted observations more regularly than Staudacher and identified the sunspot groups, although sometimes he considered two groups as a single one if they were at the same longitude but at different latitudes (Arlt et al. 2013). His observations provide position and area data as well as identifying numbers of the groups, making it possible to track the development of certain sunspot groups.

The GPR covers Cycles 12–20 and provides position and area data for sunspot groups. The DPD covers the time interval since Cycle 21 up to now and contains position and area data for not only sunspot groups but also for all observable individual sunspots. Both catalogs present the data on a daily basis.

As Figure 1 shows, hemispheric sunspot data are not available in electronic form for Cycle 11, between the Schwabe data and the start of GPR, but this gap can be filled by using the data of Spörer (Spörer 1874, 1878), which have been read into the computer manually. This data set is based on the observations of Carrington and Spörer between 1854 and 1878 for Cycles 10 and 11. Spörer took the observed sunspot groups into account once and weighted them by their area summarized in five Carrington rotations. He denoted the obtained data as hemispheric frequencies.

Considering the differences between the data sets of different observational periods, the input data are somewhat different. In the time intervals of GPR and DPD, the monthly sums of sunspot groups are used. Meanwhile, in the time intervals of Staudacher's and Schwabe's observations, the monthly sums of sunspots and the monthly sums of sunspot areas are considered. No calibrations have been made between these data sets because, on the one hand, there is no overlap between the data sets of Staudacher and Schwabe, while, on the other hand, all cycles were considered to be separate entities. Only the north–south differences were targeted within each cycle and the strengths of the cycles were not compared to each other. Because of Spörer's weighting method, his data set is not directly comparable to those of Schwabe and the GPR.

The present work uses the monthly values of the number of sunspots (${N}_{\mathrm{SS}}$)—which is not the well-known International sunspot Number (ISSN)—and the sunspot group number (N_G), which consider all sunspots and sunspot groups, respectively, as often as they were observable, instead of the sunspot group number (SGN) used in Paper I.

As noted above, there are missing days in the periods of Staudacher and Schwabe. The approximately true profiles of these cycles can only be reconstructed by using some reasonable substitutions to fill the gaps. The monthly sums of sunspots have been calculated in such a way that the monthly mean value of the observed days was applied for the missing days and these daily values have been summed up for the month (middle panels of Figures 2 and 3). As can be seen in the uppermost panels of Figures 2 and 3, the strengths of these modified cycles fit the ISSN cycles. The modified hemispheric ${N}_{\mathrm{SS}}$ values are plotted in Figure 4. The hemispheric minima between the cycles are denoted by vertical dashed and solid lines for the northern and southern hemispheres, respectively. The minimum is determined as the time of the lowest monthly value of the inter-cycle profile by smoothing the cycle profiles with a 21-month window. In order to describe each individual cycle as a whole, the centers of weight of both hemispheric cycle profiles have been computed using the original, unsmoothed cycle profiles. The positions of the centers of weight (CW_N and CW_S) are plotted for all hemispheric cycles in Figure 4. Their time difference is the measure of the hemispheric phase lags.

Zoom In Zoom Out Reset image size

Zoom In Zoom Out Reset image size

Zoom In Zoom Out Reset image size

Figure 5 shows the same diagrams for Cycles 12–24, i.e., the GPR-DPD era. As Cycle 24 is incomplete at the time of this work, its center of weight has not been considered. Figure 5 is similar to Figure 1 of Paper I, except for the input data. In Paper I, the monthly values of SGN were comupted by counting all of the sunspot groups only once in a month. This cannot be done by using the other data sets, so that, although an intercalibration cannot be carried out, at least the types of input data can be as consistent as possible.

Zoom In Zoom Out Reset image size

Figure 6 has been plotted for Cycles 10 and 11 by using Spörer's data. There are no smoothings on these hemispheric profiles because the area weighted hemispheric sunspot data are summarized over five Carrington rotations. The centers of weight are calculated as in the case of ${N}_{\mathrm{SS}}$ and N_G.

Zoom In Zoom Out Reset image size

To eliminate the uncertainty in the determination of sunspot groups, the hemispheric monthly umbral area of sunspots (${A}_{\mathrm{SS}}$) or sunspot groups (A_G) have been calculated. Figure 7 shows ${A}_{\mathrm{SS}}$ based on the data of Schwabe smoothed with a 21-month window, while Figure 8 depicts A_G using the data of GPR and DPD smoothed with an 11-month window. The hemispheric umbral area is measured in millionth of solar hemispheres (MSH). There is no such plot for the Staudacher era because his data contain the daily sum of the area of sunspots, which does not allow us to distinguish between the hemispheres.

Zoom In Zoom Out Reset image size

Zoom In Zoom Out Reset image size

The centers of weight and the hemispheric phase lags are also determined from the ${A}_{\mathrm{SS}}$ and A_G profile. They are plotted in the same way as in the case of N_G (second panel of Figure 9).

Zoom In Zoom Out Reset image size

The long-term variation of the hemispheric phase lags can be studied using different methods. One such method uses the difference between the averages of the normalized asymmetry index in the ascending and descending phases:

$$\begin{eqnarray}&&{\rm{\Delta }}{AI}={{AI}}_{{\rm{A}}}-{{AI}}_{{\rm{D}}}={\left(\displaystyle \frac{{N}_{{\rm{N}}}-{N}_{{\rm{S}}}}{{N}_{{\rm{N}}}+{N}_{{\rm{S}}}}\right)}_{{\rm{A}}}-{\left(\displaystyle \frac{{N}_{{\rm{N}}}-{N}_{{\rm{S}}}}{{N}_{{\rm{N}}}+{N}_{{\rm{S}}}}\right)}_{{\rm{D}}}\end{eqnarray} \tag{ 1 }$$

where N_N and N_S are the monthly group numbers of the northern and southern hemispheres, respectively, and the indices A and D denote the ascending and descending phases, respectively. It can be seen that if this difference is positive, then the northern hemispheric cycle leads (third panel of Figure 9 where the vertical axis is reverted in order to compare them more easily).

Another method is the study of the difference between the maxima of the hemispheric latitudinal distributions of active regions (bottom panel of Figure 9). This method exploits the shift toward the equator of the active belt during the solar cycle. This means that the bulge of the latitudinal distribution of the activity is closer to the equator in the leading hemisphere, i.e., ${\theta }_{{\rm{N}}}$ – ${\theta }_{{\rm{S}}}$ (where θ denotes the latitude) is negative if the northern hemisphere leads in time. The bars of the bottom panel of Figure 9 are calculated by averaging the absolute values of these diferences over the cycles.

The differences between the time coordinates of the centers of weight of the hemispheric cycle profiles are plotted in the first and second panels of Figure 9 using the monthly number of sunspots or sunspot groups and the monthly value of sunspot area, respectively. Similar to the result in Paper I, the hemispheric phase lags alternating by four cycles are also recognizable during the GPR and DPD era using the four different methods described above. The case is different during the pre-Greenwich era beacuse there is no uniform pattern in that period. Cycles 1, 4, and 9 do not fit into the 4+4 alternation when using the first method.

Since the ascending phase of Cycle 7 is missing and the descending phase of Cycle 24 is not complete as yet, these cycles are disregarded in these studies. The presented methods do not allow us to determine the real hemispheric centers of weight without full coverage of the cycles, and thus these cycles are missing from Figure 9.

When examining the ascending phase of Cycle 24 (Figures 5 and 8), it can be seen that this phase is similar to the case of Cycle 16. The ascending phase of Cycle 24 might indicate northern leading because of the northern predominance of the activity. However, the ascending phase of Cycle 16 also exhibited northern predominance, but examination of the entire hemispheric cycle profiles showed southern leading. This means that the real phase lag can only be determined after the full cycle is complete. Svalgaard & Kamide (2013) also formulated a cautious expectation on this phase lag.

There are no area data for the Staudacher era as described above or for the Spörer period. By studying the N–S phase shift using the area data, we can conclude that Cycle 9 is also an exception to the rule.

The asymmetry index method provides fairly similar results. When ${\rm{\Delta }}{AI}$ in Equation (1) is positive/negative, then the northern/southern hemisphere leads during the cycle. It can be clearly seen, with a reversed vertical axis, in the third panel of Figure 9, that this was the case during the Greeenwich and DPD eras. During the pre-Greenwich period, Cycles 2 and 10 do not fit into the 4+4 alternation when using this third method.

The fourth panel of Figure 9 shows the hemispheric phase lags obtained from the hemispheric Spörer diagrams. It can be seen that the 4+4 alternation can be pointed out on the GPR and DPD data, but cannot be clearly observed in the pre-Greenwich age because Cycles 3 and 10 are exceptions to the 4+4 alternation.

It can be seen that the different methods result in different behavioral patterns in the pre-Greenwich cycles. In order to determine the authenticity of each cycle, we let the authenticity of the cycles mean the number of right cases from all of the investigatable methods. It can be seen in Table 1 (created using Figure 9) that there are four cycles with two-thirds, two cycles with one-half, and another two cycles with one authenticity. In spite of the low coverages of the eight full pre-Greenwich cycles, there are six cycles with authenticity higher than one-half. Obviously, the results of these kinds of investigations will be better and more reliable if the sunspot data sets are full or almost full. This is why long-term databases are so important.

Table 1.  Authenticities of the Cycles

Cycle	N–S phase shift ${N}_{\mathrm{SS},{\rm{G}}}$	N–S phase shift ${A}_{\mathrm{SS},{\rm{G}}}$	${\rm{\Delta }}{AI}$	${{\rm{\Theta }}}_{{\rm{N}}}-{{\rm{\Theta }}}_{{\rm{S}}}$	Authenticity
1	–	no data	+	+	2/3
2	+	no data	–	+	2/3
3	+	no data	+	–	2/3
4	–	no data	+	+	2/3
8	+	+	+	+	1
9	–	–	+	+	1/2
10	+	+	–	–	1/2
11	+	no data	no data	no data	1

Note. +/–mean right/wrong results on the basis of Figure 9.

Download table as:  ASCIITypeset image

However, Cycle 10 shows half authenticity in this study using reconstructed data; it cannot be disregarded that this cycle and Cycle 11 fit the 4+4 rule in the work of Waldmeier (1971) using the Zürich data (see Paper I).

Svalgaard & Kamide (2013) investigated the asymmetries of hemispheric activity cycles in connection with the timings of polar field reversals by examining the supersynoptic maps of the Mount Wilson Observatory starting in 1970. They did not study any long-term variations or regularities in this relationship because of the short time interval, but the backward extension of the set of the times of the polarity reversals makes it possible. Similar to the long-term sunspot studies and all long-term investigations, the work with these data also has to compromise with the broad variety of sources and types of observations.

The most suitable set of dates has been published by Makarov & Sivaraman (1986). Their procedure is based on the method of McIntosh (1972), who reconstructed the large-scale surface magnetic field distribution by using H-alpha synoptic charts. Large regions of radial magnetic fields of opposite polarities are separated by borderlines indicated by filament bands of mainly east–west direction. It is a century-old finding (Fényi 1908) that these filaments migrate toward the poles, and thus by tracking the poleward migration of these borderlines, the time of the polarity reversal can be determined.

Makarov & Sivaraman (1986) have compiled a set of reversal dates from different sources covering the period 1870–1981. The Kodaikanal H-alpha and Ca ii K spectroheliograms cover the period 1904–1964; their reliability in identifying opposite polarity regions has been checked by comparing them to magnetograms. After 1964, magnetograms were used. The period 1870–1903 was covered by using the limb filament observations of Ricco (1914). After 1981, I used the reversal dates published by Svalgaard & Kamide (2013) and Alvestad (2015) for Cycle 24.

The upper panel of Figure 10 shows the N–S differences between the polarity reversal dates of the poloidal magnetic field while, for the sake of comparison, the lower panel shows the phase lags of the hemispheric cycles. In those cases (Cycles 12, 14, 16, 19, 20, and 24) where there were two or more polarity changes, I have taken into account the dates of the final reversals because the magnetic field can strongly vary around the time of polarity reversal.

Zoom In Zoom Out Reset image size

As can be seen in Table 1 of Makarov & Sivaraman (1986) the northern and southern polarity reversals took place simultaneously in Cycles 11 and 13. The time differences are zero in these cycles and these results neither contradict nor corroborate the examined long-term variation, but the pattern of the other cycles is conspicuous. The variation of the poloidal polarity reversals in the upper panel of Figure 10 seems to fit the regularity of the variation of the hemispheric phase lags by 4+4 cycles. Cycle 20 is the only exception to that regularity.

The similarity between the toroidal and poloidal phase lags is remarkable merely by visual inspection, but their comparison in Figure 11 is even more informative as it shows the diagram of the relationship between these two kinds of phase lags. Two regression lines are indicated: the steeper one disregards the dot of the non-fitting Cycle 20, and the less steep line takes it into account. Apparently, the hemispheric poloidal fields sense the statuses of the hemispheric toroidal fields, and therefore their phase relationships correspond to those of the toroidal fields. This corroborates and generalizes the existence of the phase-lag variation of 4+4 cycles during Cycles 12–23.

Zoom In Zoom Out Reset image size

The study published in Paper I has been extended both temporally and physically.

The phase lags of the hemispheric cycles have been examined in the pre-Greenwich era: eight additional cycles were more or less suitably covered by the necessary sunspot data. The results show that the phase-lag variation by 4+4 cycles can be more or less recognized, though with certain exceptions. Therefore, it cannot be stated for sure that this variation was at work in the pre-Greenwich cycles. It may either have been absent or its existence cannot be pointed out because of the decreasing observational coverage. Otherwise, there are six cycles with two-thirds or more and just two cycles with one-half authenticity during the pre-Greenwich times.

An objective physical cause may be the uncertain status of Cycle 4 around 1790, where a cycle may have been lost (Usoskin et al. 2001, 2009), a statement that is debated by Krivova et al. (2002) and Zolotova & Ponyavin (2011). This cycle, as a single entity, fits unambiguously into the set of 4+4 cycles but the next documented cycle of the group does not. The next group contains two fitting and one unfitting cycle. It cannot be excluded that the case of the "missing cycle" temporarily distorted this long-term variation, similar to the Gnevyshev-Ohl rule. As can be seen in the middle panel of Figure 2, the so-called lost cycle can be observable in the northern hemispheric activity using the modified sunspot data, while in the original data of Staudacher (uppermost panel of this figure) it cannot be observed. The bottom panel may strengthen the existence of the lost cycle because the mean hemispheric latitudes rise after 1792, similar to the beginning of a new cycle, and decrease after 1794; however, the southern activity continuously decreases after the maximum of Cycle 4.

Regardless, the Schwabe cycle itself also has stochastic features, and even extended solar minima. Any regularities can only work in restricted time intervals.

A more convincing corroboration of the phase-lag variations of 4+4 cycles is obtained by the other extension of the study, i.e., the examination of the differences between the polarity reversals of the poloidal field in the GPR-DPD era. Figure 10 shows these differences in comparison to the hemispheric phase lags. The two column diagrams are fairly similar with a single exception: Cycle 20. This implies that the regularity of the 4+4 cycles in the phase lags is a more general feature of the solar dynamo and involves both toroidal and poloidal process.

The two topologies are continuously alternating by being transformed into each other, but these diagrams may raise a "chicken-and-egg" type of question. It should be noted that the columns of the poloidal diagram belong to those of the toroidal one; for instance, the phase lag between the reversals of hemispheric poloidal fields denoted by 18 happened around the maximum of Cycle 18. Thus, the presented alternation may mean that this specific temporal feature of the solar cycle is ruled by the long-term behavior of the hemispheric toroidal fields. The temporally leading hemispheric cycle is able to initiate an earlier polar reversal than the opposite hemispheric cycle. The presented variation of the 4+4 cycles should be the evolutional property of the toroidal field.

It would be premature to speculate about any underlying mechanisms. Relevant phase relations have previously been targeted in different ways. Stix (1976) as well as Schlichenmaier & Stix (1995) theoretically examined phase relations between ${B}_{\varphi }$ and B_r fields. Apparently, an unknown agent has yet to be identified that might be responsible for this long-term behavioral pattern needing long-term memory.

The research leading to these results has received funding from the European Community's Seventh Framework Programme (FP7/2010-2013) under grant agreement No. 284461. The Staudacher and Schwabe data are courtesy of Rainer Arlt. Thanks are due to András Ludmány for reading and discussing the manuscript. The author is deeply indebted to those people for the inspiration who asked the following question "What guarantees that this variation will be continued before the GPR-era and after Cycle 23?" in several conversations.