TWO NOVEL PARAMETERS TO EVALUATE THE GLOBAL COMPLEXITY OF THE SUN'S MAGNETIC FIELD AND TRACK THE SOLAR CYCLE

Since the amplitude of the sunspot number at solar maximum is highly solar-cycle dependent and the properties (e.g., density, temperature, etc.) of the solar wind also change from cycle to cycle, these factors are limited in their ability to quantify the global variance of the Sun's magnetic field across solar cycles. An arguably more robust measure, whose cycle dependence has not been recorded to change since space data have been available, is the reversal of the solar polar field polarity as measured through the inclination of the HCS. The HCS is always relatively flat at solar minimum (Figure 1(a)) and very tilted at solar maximum, when the Sun's magnetic field is turning over (Cliver & Ling 2001; Hoeksema 1995; Figure 1(b)). This invariant cycle dependence of the HCS leads us to introduce a pair of new parameters that are independent of the sunspot number but are very efficient at tracking the complexity of the solar magnetic field as manifested in the HCS.

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To mathematically describe the different shapes of the HCS at different solar activity phases, we introduce two parameters: the latitudinal standard deviation (SD) and the integrated slope (SL) of the HCS. We define the SD as the standard deviation of the absolute values of the HCS' latitudes on the source surface for one Carrington rotation (CR; Equation (1)); SL is the total absolute value of the slope between two neighboring simulated points of the HCS on the source surface for one CR, or, in other words, the total absolute value of the derivative of the HCS' latitude with respect to longitude (Equation (2)). The definition of these two parameters can be analytically described as

$$\begin{equation} {\rm SD}=\sqrt{\sum {(|\theta _{i}|-\overline{\theta })}^{2}} \end{equation} \tag{ 1 }$$

$$\begin{equation} {\rm SL}=\sum \left| \frac{\partial \theta _{i}}{\partial \phi _{i}}\right|, \end{equation} \tag{ 2 }$$

where θ_i (ϕ_i) is heliographic latitude (longitude) of an HCS simulation point on the synoptic map, and $\overline{\theta }$ is the average of θ_i over the entire CR. In this study, we calculate these two parameters from the curve of the HCS on the synoptic Carrington maps provided by the potential field source surface (PFSS) model (Wang & Sheeley 1992) at the source surface of 2.5 R_s (Figure 1).

An example of how SD and SL describe the shape of the HCS is shown in Figure 1. In this figure, the polarity inversion line at 2.5 R_s is calculated from photospheric field observations as a boundary condition with a PFSS model (i.e., Wang & Sheeley 1990) and is shown by a red solid line, which can be used as an indicator of the HCS. As described above, the cycle dependence of the HCS is obvious: at solar minimum, very small values of SD and SL indicate that the HCS is more or less a flat sheet around the ecliptic plane (Figure 1(a)), whereas at solar maximum the HCS is tilted to high latitude (>60°) and its increased waviness corresponds to much higher values of SD and SL (Figure 1(b)).

A similar parameter based on the HCS calculated by the PFSS model, i.e., the maximum extent in latitude reached by HCS in one CR, or, in short, the maximum tilt angle, has also been used to track solar cycle variation (Cliver & Ling 2001). However, this parameter has limitations compared to SD and SL. For example, since the magnetic field lines are not computed above 70° by the PFSS model, the maximum tilt angle of the HCS above 70° is ill-defined. This limitation is especially problematic at solar maxima when the HCS commonly reaches latitudes > 70°. SD and SL, depending upon the behavior of the HCS throughout the CR, continue to provide meaningful distinctions from rotation to rotation even when the maximum tilt angle has essentially saturated at 70°. Also, the maximum tilt angle cannot evaluate the three-dimensional shape of the HCS, as it measures only a single number per rotation, i.e., the maximum latitudinal extent. On the other hand, SD and SL provide a global overall estimation about the inclination of the HCS and the complexity of its shape, which yields more information. This is illustrated in Figures 1(c) (CR2094) and 1(d) (CR2029): the two CRs have the same maximum tilt angle, but they have very different HCS shapes—CR2094's curve is more gradual, characterized by a relatively smaller SL but larger SD (Figure 1(c)) and CR2029 is much wavier, resulting in a relatively larger SL but smaller SD (Figure 1(d)).

A comparison of SD and SL with monthly sunspot numbers is shown in Figure 2, where the sunspot numbers are plotted versus time from 1975 January to 2012 December. Monthly sunspot numbers (http://sidc.oma.be/sunspot-data/) are depicted by gray shading and SD (SL) is plotted in red (blue). This figure shows that both SD and SL vary with time in a similar way as the sunspot number does, and follow the same 11 yr cycle. However, quantitatively, these two parameters do not follow the sunspot number consistently, for two reasons. First, while the range of sunspot numbers can change significantly from cycle to cycle, the values of SD and SL oscillate within a constant range. Second, starting from the descending and minimum phases of cycle 23 (from 2003), the sunspot number showed an unusually slowly decreasing slope until reaching a very deep minimum in 2009. During this descending phase, SD and SL showed larger deviations from the sunspot number and stayed at higher levels than any same phase in the previous cycles shown in Figure 2, implying that the global magnetic field of the Sun was behaving differently than in previous cycles. This difference indeed is associated with the unusual solar minimum in 2008–2010, when the HCS was much more tilted and low-latitude coronal holes were more common than in the previous minima. The dramatic difference in SD and SL in the descending phase of cycle 23 makes this cycle stand out from the previous cycles, and since this difference occurred a few years before the sunspot number anomaly became apparent, it can be considered an early warning that the minimum between cycle 23 and 24 would be abnormal and—possibly—that cycle 24 will also be magnetically different.

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After 260 spotless days in 2009 (Zięba & Nieckarz 2012), sunspots gradually returned. In 2012, sunspots slowly climbed to about half of their historic maximum value from previous cycles. However, unlike the sunspot number, SD and SL increased at a similar rate as they did in previous solar cycles, and returned to their normal historic maximum levels in 2011, while the sunspot number was still only halfway to its peak reached in cycles 21–23. Again, this distinction between the sunspot number and the SD and SL parameters may indicate that cycle 24 will be magnetically different than the cycles of recent history. Also, the fact that SD and SL have reached their usual maximum values seems to suggest that the sunspot number also has reached the maximum for this cycle. Thus, SD and SL might also give an early warning of the cycle maximum, although we only have three cycles to support this statement and more would be needed.

Since the sunspot number is qualitatively periodic but quantitatively highly cycle dependent, as shown in Figure 2, we calculate the normalized sunspot number in each cycle and compare it with SD and SL (Figure 3). The normalized sunspot number is defined as the sunspot number divided by its maximum value in each cycle and is plotted versus the years after the beginning of each cycle in Figure 3. Note that for cycle 24, we use the maximum monthly sunspot number so far, which is 96.7 recorded in November 2011. The start date of each cycle that we use in this paper is from Owens et al. (2011), who defined the start date as the moment when the average latitude of sunspots sharply increases (Table 1). Figure 3(a) shows that in the recent four cycles (cycles 21–24), the normalized sunspot numbers all reach 0.8 (horizontal dashed line in Figure 3(a)) at about 2.5 yr (vertical dashed lines) after the beginning of the cycle. Therefore, if we define the solar maximum phase as the period when the normalized sunspot number is greater than 0.8, then the solar maxima in the four cycles all began about 2.5 yr after the start dates of the cycles.

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Similarly, after 2.5 yr from the start date of the cycles, SD also reaches its maximum level (SD > 20, marked by the horizontal dashed line in Figure 3(b)). However, since SL is nosier than SD throughout all the cycles, there is no clear threshold of SL consistent with the onset of solar maximum (Figure 3(c)). Instead of using SL itself, we average SL every 10 months and find that the smoothed SL reaches a high level of 363 2.5 yr after the start date of each cycle (Figure 3(d)). These high values of SD and SL (SD > 20 and SL > 363) indicate that the HCS is highly tilted and wavy, which implies that the polar field is about to flip. This time period also marks the solar maximum. The coincidence of the high values of SD and SL and the maximum level of the normalized sunspot number suggests that SD and SL can be used as proxies for indicating the onset of solar maximum, in terms of magnetic field complexity. The onset dates of the recent four solar maxima determined from Figure 3 are tabulated in Table 1.

Note that the advantage of SD and SL in capturing the magnetic anomalous cycle is clearly illustrated in Figure 3. In Figure 3(a), even though the normalized sunspot numbers show a prolonged (unusual) solar minimum of cycle 23, this anomaly was not clear until the long minimum was underway. In contrast, the descending and minimum phases of cycle 23 clearly stand out from the other cycles in terms of the enhanced values of SD and SL in Figures 3(b)–(d). The enhanced SD and SL indicate that the HCS was much more tilted and extended to much higher latitude than it was in the previous cycles, implying that the global three-dimensional morphology of the Sun's magnetic field at this time was definitely different from past normal cycles. Indeed, this is evident by the higher rate of appearance of low-latitude coronal holes and pseudostreamer structures at that time (Zhao & Fisk 2011; Zhao et al. 2013). Also, note that the anomalous behavior of SD and SL was apparent already in 2003, while the anomaly of the minimum became obvious many years later. Thus, SD and SL seem to be giving early warnings of the anomalies of the HCS and the magnetic field.

It is also interesting to note that during the ascending phase (2009–2011) of cycle 24, when the sunspot number was so low compared with the ascending phases in the previous cycles (Figure 2), the SD and SL behaved similarly to the other three cycles (Figures 3(b)–(d)). This fact implies that if cycle 24 is unusual (Ramesh & Lakshmi 2012), it may become apparent in the SD and SL during the descending or minimum phase of cycle 24, in a way that will not be apparent in the sunspot number.