POLAR NETWORK INDEX AS A MAGNETIC PROXY FOR THE SOLAR CYCLE STUDIES

We first compare our PNI values with the WSO polar field measurements for the period of 1976–1990 in Figure 2. We find that they have a good correlation (the linear correlation coefficient ∼0.95). The red symbols correspond to the solar minima. It is seen that the polar field tends to be stronger during solar minima. The slope b of the linear fit provides the calibration to obtain the magnetic field from PNI. The calibrated data give the polar fields for the period when no direct polar field measurement exists in any data archive. This new estimated polar field has been used as an input to the dynamo simulation. To estimate the error bars for the calibrated PNI, we calculate the standard deviation (σ) and then errors ($\sigma /\sqrt{n}$) and compounded it with the 95% confidence bounds on the fitted slope.

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Muñoz-Jaramillo et al. (2012) generated polar flux data by validating and calibrating MWO polar faculae count data against MDI magnetograms and WSO polar field measurements. These data have been used to validate our KKL PNI data. The yearly averaged data of MWO Polar faculae include only two months of data of each year (August–September for northern hemisphere and February–March for the southern hemisphere; Muñoz-Jaramillo et al. 2012) in which solar poles are most visible from the Earth. Figure 3(a) shows an overplot of PNI and MWO facular counts for the years 1909–1990 covering solar cycles 15–21 for both the northern and southern hemispheres. Though both facular measurements and PNI are strictly positive quantities, they have been given signs (which reverse when they reach a minima) to match the polarity believed to be present at each pole during each solar cycle. Both indexes are found to be in very good agreement with a correlation coefficient of 0.91. Only around 1945 do we notice a significant divergence between them. This was a period when there was a dearth of usable spectroheliogram plates to construct PNI (see figure in http://kso.iiap.res.in/data), probably leading to higher averaging errors.

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The sunspot area (SA) data, downloaded from http://solarscience.msfc.nasa.gov/greenwch/, are plotted in Figure 3(b). A correlation plot of the PNI amplitude at a sunspot minimum and the amplitude of the next sunspot cycle are shown in Figure 3(c). A similar correlation plot using the MWO facular count was given in Figure 2(b) of Jiang et al. (2007), where two points were outliers. They corresponded to the two weakest cycles 16 and 20 of the last century, which were preceded by not-too-low facular counts in the preceding minima (around 1923 and 1964, respectively). However, the A(t) index (Makarov et al. 2001) was low during these two minima; see Figure 2(a) of Jiang et al. (2007). We now find the point for cycle 16 to be an outlier when we use PNI for the correlation plot. Although the low value of A(t) during the preceding minimum (around 1923) might suggest a weak polar field, the fact that neither the MWO facular count nor the PNI happened to be too low during this minimum probably indicates that the polar field was not too weak during this minimum. So we come to the conclusion that at least once in the past century a not-too-weak polar field during a sunspot minimum was followed by a very weak sunspot cycle. Otherwise, we find a good correlation between the polar field at the minimum (as indicated by PNI) and the strength of the next sunspot cycle, except at the points 18 and 21 in Figure 3(c). It may be noted that there was a dearth of spectroheliogram plates during the minima preceding cycles 18 and 21 (around 1943 and 1975). Therefore, it is possible that the poor correlations seen for 18 and 21 may be due to errors in the values of PNI. If we exclude points 16, 18, and 21, the remaining data points in Figure 3(c) give a correlation coefficient 0.96.

In addition to the WSO polar field data, the polar faculae counts as recorded from the NAOJ for the interval of 1954 March to 1996 March have been used for comparison. The data have been downloaded from http://solarwww.mtk.nao.ac.jp/en/db_faculae.html. The correlation coefficients between the two data sets are only about 0.65 and 0.33 for the northern and southern hemispheres, respectively. In other words, PNI that constructed are in good agreement with the MWO faculae counts, but not with the NOAJ faculae counts. Furthermore, Makarov et al. (1989) manually counted the polar bright points above 50° in both hemispheres for the period 1940–1957 (with a data gap between 1947 and 1948) from KKL Ca-K spectroheliograms. They found an anti-correlation between polar bright points and sunspot number for the two cycles and suggested that Ca-K polar bright points can be used as proxies to predict the characteristics of the next solar cycle. We have compared these data with our PNI and found very good correlation between them.

The hemispheric asymmetry of MWO faculae counts at a sunspot minimum is found to have a good correlation (correlation coefficient = 0.67) with the hemispheric asymmetry of the next sunspot cycle (Goel & Choudhuri 2009). We explore whether the hemispheric asymmetry of PNI at a sunspot minimum also has a similar good correlation with the hemispheric asymmetry of the next cycle. We have computed the hemispheric asymmetry of PNI of the nthe cycle using the formula

$$\begin{equation} \mathrm{ PNI_{AS} = \frac{PNI_N - PNI_S}{PNI_N+PNI_S}}, \end{equation} \tag{ 1 }$$

where PNI_N and PNI_S are the north and south PNI during the minimum at the end of the nth cycle. In a similar way, the hemispheric asymmetry of the SA for the (n + 1)th cycle is taken as follows:

$$\begin{equation*} \mathrm{SA_{AS} = \frac{SA_N - SA_S}{SA_N+SA_S}}\protect, \protect \end{equation*}$$

where SA_N and SA_S are the north and south SA during the (n + 1)th solar cycle.

Figure 4 plots the PNI asymmetry of the nth cycle against the SA asymmetry of the (n + 1)th cycle, showing a good correlation with a correlation coefficient around 0.78. We conclude that the hemispheric asymmetry of PNI during a sunspot minimum is a good predictor for the hemispheric asymmetry of the next cycle. However, the value of SA_AS for the (n + 1)th cycle is found to be statistically lower than the value of PNI_AS for the nth cycle. A similar result was also found for MWO facular counts (Goel & Choudhuri 2009). Since both the PNI and the faculae number are proxies for the polar field, we reach the following conclusion. Although the hemispheric asymmetry of the polar field at the sunspot minimum determines (in a statistical sense) the asymmetry of the next cycle, the value of the hemispheric asymmetry of the next cycle turns out to be lower. During a dynamo cycle, the diffusive decay in the solar convection zone probably tries to reduce the asymmetry introduced into the polar field at the solar minimum (Goel & Choudhuri 2009).

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In the flux transport dynamo model, the poloidal field is generated by the Babcock–Leighton mechanism. It has been proposed that the fluctuations in this mechanism give rise to the irregularities of the sunspot cycle (Choudhuri et al. 2007; Jiang et al. 2007). However, of late, we have become aware of a second source of irregularities in the flux transport dynamo: fluctuations in the meridional circulation (Karak 2010; Choudhuri & Karak 2012) which may complicate things further and may be the reason why the correlation between the polar field at the sunspot minimum and the sunspot number of the next cycle is not as good as expected. One of the jobs of the flux transport dynamo model is to model the actual sunspot cycles. In order to do this, information about the fluctuations has to be fed to the theoretical model from observational data.

We now present a simple dynamo calculation to demonstrate that the PNI can be used very conveniently to model the fluctuations in the Babcock–Leighton mechanism. The calculations are based on the Surya code. The early results obtained with this code were presented in Nandy & Choudhuri (2002). We use the same parameters as were used in Karak & Choudhuri (2011), except v₀ which is taken here to be 25.3 m s⁻¹, to get the correct cycle periods. If this dynamo code is run uninterrupted, we obtain periodic solutions in which the polar field produced at the end of each cycle would have equal strength. We take the point of view that it is the fluctuations of the Babcock–Leighton mechanism which primarily cause the variations of the polar field from cycle to cycle. In order to model the actual cycles, we need to feed the information about the poloidal field in each sunspot minimum. A method for this has been proposed by Choudhuri et al. (2007) which was further extended by correcting the poloidal field in two hemispheres separately by Goel & Choudhuri (2009), where the MWO polar faculae counts of the two poles of the Sun were used.

Here, we use the polar field data as derived from the PNI during solar minimum as a measure of the Sun's poloidal field. Due to the limited and noisy nature of the data, we calculate the average over five years around the solar minimum to estimate the strength of the polar field during that minimum. We compute this value for each cycle and for each hemisphere separately. So, corresponding to each cycle, we get one value for the northern hemisphere and one for the southern hemisphere. We rescale these values by their mean to determine the strengths of the polar field γ_N = [0.62, 1.41, 0.74, 1.00, 1.63, 1.07, 0.78, 1.30], γ_S = [0.63, 1.21, 0.95, 1.00, 1.41, 0.55, 0.71, 1.26] at the ends of cycles 14–21 for the northern and southern hemispheres, respectively. Note that due to insufficient observations in cycle 17, the value of PNI becomes very low and can produce incorrect results. Therefore, for this cycle, we take γ_N = γ_S = 1.0, i.e., the average polar field. We stop the dynamo code at each sunspot minimum and "correct" the value of the poloidal field exactly the way it was done in Goel & Choudhuri (2009). The newly generated poloidal field at the solar minimum remains mostly above 0.8 R_☉. Therefore, correcting this poloidal field above 0.8 R_☉ by multiplying with γ_N in the northern hemisphere and with γ_S in the southern hemisphere captures the effect of fluctuations in the Babcock–Leighton process to a great extent. After such a correction at a minimum, the dynamo code runs until the next minimum when it is stopped again and the poloidal field is corrected. From this dynamo calculation, we obtain the strength of sunspot activity as a function of time by counting the number of eruptions (assumed to take place whenever the toroidal field above the base of the convection zone exceeds a critical value). Figure 5 displays a plot of the observed SA data for the two hemispheres, along with the theoretically obtained sunspot activity strength suitably scaled to match the SA data. We note that cycles 16 and 21 are not modeled well, which we would expect since the data points for these cycles were outliers in Figure 3(c). Most of the other cycles are modeled reasonably well, demonstrating that PNI is indeed a useful index for reconstructing the past history of solar activity.

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