4 North-South asymmetry

In Fig. 3 the derived smoothed monthly mean northern and southern Sunspot Numbers for the period 1975-2000 are shown. From the figure it is obvious that the activity during the solar cycle is not symmetric for both hemispheres. For instance, cycle 21 shows a clear phase shift between the northern and the southern hemispheric activity. The existence of a N-S asymmetry has been established and analyzed in several studies for a variety of solar activity phenomena (e.g., flares, prominences, bursts, coronal mass ejections, long-lived filaments, etc.). However, the physical causes are still not satisfactorily interpreted. The analyses of N-S asymmetries in the appearance of sunspots have mostly been made on the basis of sunspot areas and sunspot group numbers (Newton & Milsom 1955; Waldmeier 1957, 1971; White & Trotter 1977; Yallop & Hohenkerk 1980; Vizoso & Ballester 1990; Carbonell et al. 1993; Oliver & Ballester 1994; Li et al. 2000, 2001, 2002). Studies based on relative sunspot numbers have been carried out by Swinson et al. (1986) who used data provided by Koyama (1985) for the period 1947-1983.

In the following, we study the relation between Sunspot Numbers and areas, the significance of the N-S asymmetry excesses as a function of the solar cycle, and dominant rotational periods for the northern and southern hemisphere.

4.1 Sunspot Numbers - Sunspot areas

In Fig. 4, we plot the smoothed monthly mean sunspot areas (data are taken from the Royal Greenwich Observatory) together with the derived Sunspot Numbers, for the whole Sun as well as separately for the northern and southern hemisphere. The figure clearly reveals that considering the activity of the northern and southern hemisphere separately (areas as well as Sunspot Numbers), different information is provided than when considering the whole disk. For instance, in the hemispheric indices, the activity gaps during the maximum phase, the so-called Gnevyshev gaps (Gnevyshev 1963), are clearly visible, whereas they are often smeared out when considering the total activity. Furthermore, it can be clearly seen that the northern and southern hemisphere do not reach their maximum simultaneously but there may be a shift of up to several years (see in particular cycle 22). Thus, the time as well as the height of the maximum and the Gnevyshev gap in the total activity can be understood as a superposition of both hemispheres, which provide the primary physical information on solar activity.

For the cross-correlation coefficients between the monthly mean sunspot areas and Sunspot Numbers, we obtain 0.90, 0.91 and 0.94 for the northern, the southern and the total component, respectively, indicating a good correspondence between both activity indices. However, the relationship is far from being one-to-one. Especially during the maximum phase, significant differences between sunspot numbers and areas appear (see Fig. 4). In principle, sunspot areas are a more direct physical parameter, being closely related to the magnetic field. However, the reliable measurement of sunspot areas is not an easy task, and the results derived by different techniques and different observatories may differ by an order of magnitude (Pettauer & Brandt 1997). This poses in particular problems for mid- and long-term investigations of solar activity. Furthermore, from an ongoing study we obtained that, for instance, the hemispheric occurrence of H$\alpha$ flares is more closely related to the Sunspot Numbers than to the sunspot areas, which emphasizes the high physical relevance of Sunspot Numbers (Temmer et al., in preparation).

Figure 4: Smoothed monthly mean Sunspot Numbers and sunspot areas for the total Sun (top panel), the northern (middle panel) and the southern (bottom panel) hemisphere, respectively. Thin (thick) lines indicate hemispheric sunspot areas (Sunspot Numbers).

4.2 N-S Asymmetry: Excess

Figure 5 shows the cumulative monthly mean northern and southern Sunspot Numbers, separately plotted for solar cycles 21, 22 and the rising phase of the current cycle 23. In order to assess the significance of the activity excess of the northern or southern hemisphere, respectively, we applied the paired Student's t-test. The test statistics $\hat{t}$ is defined by

$$\hat{t} = \frac{\bar{D}}{s_{\bar{D}}} = \frac{(\Sigma D_{\rm ... ...\Sigma D_{\rm i}^{2} - (\Sigma D_{\rm i})^{2}/n}{n(n - 1)}}} ,$$ (3)

where $D_{\rm i}$ is the difference of paired values (here, daily $R_{\rm n}$, $R_{\rm s}$), $\bar{D}$ the mean of a number of ndifferences and $s_{\bar{D}}$ the respective standard deviation with n-1 degrees of freedom. Since we want to test the significance of the monthly value, n is given by the number of days of the considered month. The calculated test value, $\hat{t}$, based on the degrees of freedom is compared to the corresponding $\hat{t}_{n-1, \alpha}$ given in statistical tables on a preselected error probability $\alpha$. We have chosen $\alpha=0.05$, i.e. if $\hat{t}>\hat{t}_{n-1, \alpha}$, the difference between the paired values is statistically significant at a 95% level. Thus for each month, the paired Student's t-test is utilized to determine the significance of the difference between the northern and southern Sunspot Numbers.

Figure 5: The cumulative monthly mean Sunspot Numbers for the northern (thick line) and southern (thin line) hemisphere, respectively, are presented separately for solar cycle 21, 22 and the rising phase of solar cycle 23. Crosses (circles) indicate an excess of the northern (southern) hemisphere for the respective month on a 95% significance level. Triangles indicate the maximum of the solar cycle.

The results of this test are indicated in the graph of the cumulative Sunspot Numbers (Fig. 5). The calculated $\hat{t}$ values with 95% significance are overplotted at each specific month signed as crosses (circles) for the northern (southern) hemisphere to represent that the excess of the flagged hemisphere is highly significant. This representation shows that during solar cycle 21 the excess of the southern hemisphere has significant excesses predominantly at the end of the cycle, whereas the northern hemisphere is more active at the beginning and the maximum phase. For solar cycle 22, both hemispheres show a similar amount of activity excesses during the ascending phase, and almost no predominance of one hemisphere over the other during the maximum phase. The distinct excess of the southern activity is exclusively built up during the declining phase. The current solar cycle 23 is only covered by its rising phase with a slight excess of northern activity. Similar results for solar cycle 21 based on cumulative counts of soft X-ray flares are reported by Garcia (1990), and for solar cycle 21 and 22 on the basis of H$\alpha$ flares by Temmer et al. (2001).

In Table 2 the outcome of the Student's t-test is summarized, listing the percentages of significant months in relation to the total number of months during the specific solar cycles. In each cycle, about 60% of all months reveal a highly significant N-S asymmetry. For solar cycle 21, the percentage of months with significant activity excess is slightly higher for the northern than for the southern hemisphere. For solar cycle 22, the southern hemisphere covers almost twice as much significant months than the northern (cf. Table 2). Regarding the total activity during the cycle, we obtain a slight excess of the southern hemisphere over the northern hemisphere with about 50.9% during solar cycle 21 and a distinct excess of 53.6% during solar cycle 22 (cf. Fig. 5).

Swinson et al. (1986) who analyzed Sunspot Numbers for the period 1947 until 1983 almost completely including solar cycles 19, 20 and 21, show good agreement with our results, concluding that the northern hemisphere peaks in its activity excess about two years after solar minimum. These authors report that this peak is greater during even cycles which points out a relation to the 22 year solar magnetic cycle. Balthasar & Schüssler (1983, 1984) interpreted the distribution of daily relative sunspot numbers as a kind of solar memory with preferred hemispheres that alternate with the 22 year magnetic cycle. Contrary to that, Newton & Milsom (1955) and White & Trotter (1977), by analyzing sunspot areas, did not find a systematic change in activity between both hemispheres, i.e. no evidence for a dependence on the 22 year solar magnetic cycle.

 

 

Table 2: The percentage of months (T) with 95% significant N-S asymmetry with respect to the total number of months for solar cycles 21, 22 and the rising phase of solar cycle 23 is given. Additionally, the significant months are subdivided into the northern (N) and southern (S) hemisphere.

Cycle No.	$T(\%)$	$N(\%)$	$S(\%)$
21	59.3	30.9	28.4
22	63.2	22.2	41.0
23	60.3	35.8	24.5

4.3 Rotational periods

Various periodicities have been detected in solar activity time series. The most prominent are the 11 year solar cycle and the 27 day Bartels rotation (Bartels 1934). In the present study, we are interested in periods related to the solar rotation and in their behavior with regard to the N-S occurrence. For this purpose, we analyze power spectra and autocorrelation functions from daily Sunspot Numbers of the northern hemisphere, the southern hemisphere and the total solar disk.

For the power spectrum analysis, we have adopted the Lomb-Scargle periodogram technique (Lomb 1976; Scargle 1982) modified by Horne & Baliunas (1986). By this method, the power spectral density (PSD) is calculated normalized by the total variance of the data. This periodogram technique is particularly useful in order to assess the statistical confidence of a frequency identified in the periodogram by computing the false alarm probability (FAP). In our case, the relevant time series are prepared from the daily Sunspot Numbers, which are not independent of each other but correlated with a typical correlation time of about 7 days (Oliver & Ballester 1995). Thus, the statistical significance of a peak of height z in the periodogram has to be tested for the case that the data are statistically correlated. Therefore, the PSD has to be normalized by a correction factor k, which should be determined empirically (described below). Then the FAP can be derived by the following equation

$$FAP = 1 - [1 - \exp(-z_{m})]^{N},$$ (4)

where N denotes the number of independent frequencies in the time series, z_m=z/k is the derived normalized power, z is the Scargle power and k the normalization factor due to event correlation (Scargle 1982; Horne & Baliunas 1986; Bai & Cliver 1990; Bai 1992).

Figure 6: The cumulative distribution function of the power values derived from the Lomb-Scargle periodogram for the total solar disk. The y-axis shows the cumulative number of frequencies whose power exceeds a height z and the x-axis the values of power, z. The solid line shows the best fit to the distribution for z<7.

The number of independent frequencies is given by the spectral window investigated and the value of the independent Fourier spacing, $\Delta f_{\rm ifs}=1/\tau$, where $\tau$ is the time span of the data (Scargle 1982). Here, we are interested in periods related to the Sun's rotation, for which we have chosen the spectral window [386,463] nHz, i.e. [25,30] days. Considering a time span from January 1975 to December 2000 we have $\tau=9497$ days, hence $\Delta f_{\rm ifs}=1.22$ nHz. However, de Jager (1987) has shown by Monte-Carlo simulations that the Fourier powers taken at intervals of one-third of the independent Fourier spacing are still statistically independent, i.e. $\frac{\Delta f_{\rm ifs}}{3}=0.41$ nHz. Thus we accepted the number N=190 as the number of independent frequencies in the chosen spectral window.

We briefly describe the method used to calculate the normalization factor k. The normalization factor k is derived from the cumulative distribution function of the Scargle power values zfor the 190 independent frequencies (shown for the Sunspot Numbers of the total solar disk in Fig. 6). The distribution can be well fitted to the equation $y=190\exp(-z/k)$ for power values z<7, which gives the normalization factor k (indicated in Fig. 6 as solid line). For the time series of the total solar disk we obtain k=3.85, thus the power spectrum is normalized once more by dividing the Scargle power by 3.85. For the northern hemisphere we obtain k=6.63 and for the southern hemisphere k=5.18. Further detailed descriptions of this procedure can be found in Bai & Cliver (1990), Oliver & Ballester (1995) and Zieba et al. (2001).

Figure 7: Periodograms derived from the daily Sunspot Numbers of the total solar disk (top panel), the northern hemisphere (middle panel) and the southern hemisphere (bottom panel), from the time span 1975-2000. The dashed lines indicate various FAP levels.

The normalized PSD for the total and the hemispheric Sunspot Numbers are represented in Fig. 7. The FAP levels calculated by Eq. (4) are indicated as dashed lines for the probabilities of 50%, 40%, 20% and 10%. As can be seen, the total Sunspot Numbers show only one peak above the 50% significance level, namely at 27.0 days with a FAP value of 38%. The northern Sunspot Numbers show also one peak above the 50% significance level at 27.0 days with a FAP value of 19%. For the southern Sunspot Numbers we get one peak at 28.2 days which lies above the 50% significance level with a FAP value of 12%. Thus, for the northern hemisphere the power is mainly concentrated at $\sim$27 days, whereas for the southern hemisphere it is mainly concentrated at $\sim$28 days. These rotational properties manifest a strong asymmetry with respect to the solar equator. It is worth noting that the spectral power of the peaks is lower for the total Sunspot Numbers than for the hemispheric ones. This phenomenon probably arises from the fact that in the case of the total Sunspot Numbers, the components of both hemispheres, which are obviously not in phase, overlap and result in a lower PSD than for the hemispheric Sunspot Numbers. Since there is no enhanced PSD at 27 days for the time series of the southern Sunspot Numbers, the signal of the 27 day Bartels rotation found for the total Sunspot Numbers is a consequence of the northern component, in which the 27 days peak is very strong. On the other hand, the enhanced PSD at $\sim$28 days, found for the southern Sunspot Numbers, is not powerful enough to be reflected as a significant period in the PSD of the total Sunspot Numbers (see Fig. 7, top panel).

In Fig. 8, we show the autocorrelation function derived from the northern, the southern and the total daily Sunspot Numbers, up to a time lag of 500 days. Distinct differences appear for the behavior of the northern and the southern hemisphere. The northern Sunspot Numbers reveal a stable periodicity of about 27 days, which can be followed for up to 15 periods. Contrary to that, the signal from the southern hemisphere is strongly attenuated after 3 periods, and shows a distinctly less regular periodicity. The autocorrelation function of the total Sunspot Numbers reveals an intermediate behavior, resulting from the superposition of the northern and southern components.

Figure 8: Autocorrelation function of the daily Sunspot Numbers for the period 1975-2000, plotted up to a time lag of 500 days. The top line indicates the autocorrelation function for the total disk, the thick line for the northern and the thin line for the southern hemisphere. The 27 day Bartels rotation up to 15 periods is overplotted by dotted lines. (For better representation, the autocorrelation function of the southern Sunspot Numbers is shifted downwards by a factor 0.1.)