Toward a live homogeneous database of solar active regions based on SOHO/MDI and SDO/HMI synoptic magnetograms. II. parameters for solar cycle variability

After removing repeat ARs in our database, we calculate the axial dipole fields of ARs, including the initial dipole field ($D_{i}$) and final dipole field ($D_{f}$). Each parameter is determined using two distinct methods: one based on the actual configuration of ARs, termed the AR method, and the other based on the BMR approximation, referred to as the BMR method. For the BMR approximation, we approximate each polarity of AR as a finite-sized Gaussian-like polarity patch (van Ballegooijen et al., 1998; Jiang et al., 2014b). Figure 2 shows an example of an AR and its BMR approximation.

The axial dipole field of an AR is the coefficient of the $Y^{0}_{1}$ term in the spherical harmonic expansion of its radial magnetic fields. Mathematically, this is expressed as:

where $\lambda$ and $\varphi$ denote latitude and longitude, respectively, and $B(\lambda,\varphi)$ is the radial magnetic field distribution of the AR. When applying the BMR approximation to an AR, Equation (3) can be simplified as:

where $\Phi$ represents the unsigned flux of one polarity of the BMR. $d_{\lambda}$ is the latitude difference of the BMR. It can be calculated using the latitude of the positive polarity minus the latitude of the negative polarity, or as $d_{\lambda}=d\sin\alpha$, where $d$ is the polarity separation, and $\alpha$ is the tilt angle (Wang & Sheeley, 1991; Yeates et al., 2023).

Equations (3) and (4) are used to quantify the axial dipole field of an AR when it emerges on the solar surface, referred to as the initial dipole field ($D_{i}$). After a BMR emergence, its axial dipole field changes due to meridional flow and supergranular diffusion. The ratio ($f_{\infty}$) between the dipole field at the end of a BMR’s evolution (final dipole field, $D_{f}$) and $D_{i}$ follows a Gaussian relation about the BMR emergence latitude $\lambda_{0}$. The relation is expressed as $f_{\infty}\propto\exp({-a\lambda_{0}^{2}})$, where $a$ is a constant (Jiang et al., 2014a). Petrovay et al. (2020) provide an algebraic expression for $f_{\infty}$, and thus provide an algebraic method to calculate the final dipole field of a BMR ($D_{f}^{B}$). The method is

where $f_{\infty}$ is given by

In Equation (6), $\lambda_{0}$ represents the emergence latitude of the BMR, and $n$ describes the approximated polar field profile in solar minimum, i.e. $B(\lambda)\propto\sin^{n}(\lambda)$. The value of $n$ is set to 7 in our calculations, following the value of Wang et al. (2009). $\lambda_{R}^{B}$ is the dynamo effectivity range, calculated by:

Here, $\eta$ is the supergranular diffusivity, $\Delta u$ is the divergence of the meridional flow at the equator, i.e., $\frac{1}{R_{\odot}}\left.\frac{\mathrm{d}u}{\mathrm{d}\lambda}\right|_{\lambda=0}$, and $\sigma_{0}$ is the initial Gaussian width of the BMR.

However, Equation (5) just applies to BMRs. For ARs with arbitrary configurations, Wang et al. (2021) provides a more general method expressed as

where $A_{0}$ is a constant, set to 0.21 according to Wang et al. (2021). In this equation, the dynamo effectivity range $\lambda_{R}$ is defined as

This is different from $\lambda_{R}^{B}$ in Equation (6), for the BMR method. The form of Equation (8) has been slightly adjusted from that presented in Wang et al. (2021) to improve its conciseness and clarity.

In the computation of final dipole fields ($D_{f}$), the term $\frac{\eta}{R_{\odot}^{2}\Delta u}$ is present in both the BMR method and the AR method. The meridional flow (u) and supergranular diffusivity ($\eta$) are less constrained by observations (Petrovay et al., 2020; Jiang et al., 2023). A comprehensive discussion on the observations of $\Delta u$ and $\eta$ is provided in Jiang et al. (2023). $\Delta u$ and $\eta$ are sometimes optimized with magnetogram observations when employed in Surface Flux Transport (SFT) models (Lemerle et al., 2015; Whitbread et al., 2017). According to the BL mechanism, the solar axial dipole field is built up through the emergence and evolution of ARs (Babcock, 1961; Leighton, 1964). Consequently, the cumulative final dipole fields ($cD_{f}$) in one solar cycle, defined as the sum of all ARs’ $D_{f}$, should be equal to the difference between the axial dipole fields at the end-of-cycle solar minimum and that at the start-of-cycle minimum. Since the AR method is considered more accurate than the BMR method for ARs with real configurations (Wang et al., 2021), we utilize the $D_{f}$ calculated by the AR method to optimize $\lambda_{R}$, which is equivalent to optimizing $\frac{\eta}{R_{\odot}^{2}\Delta u}$. We also refer to $\lambda_{R}$ as the transport parameter.

We conduct a comparison between the $cD_{f}$ and the solar dipole fields obtained from the MDI and HMI synoptic magnetograms. The MDI maps cover the time range from CR 1909 to CR 2096, roughly corresponding to cycle 23, while the HMI maps cover the time range from CR 2097 to CR 2225, roughly corresponding to cycle 24. To calculate the solar dipole field, we use the synoptic magnetograms with the polar field corrected. The correction is performed by interpolation using relatively well-observed data during September/March for many years (Sun et al., 2011; Sun, 2018; Luo et al., 2023). The dipole field for the beginning or the end of a cycle is determined by averaging the dipole fields calculated in six CRs near this time.

The comparison is depicted in Figure 3. As $\lambda_{R}$ increases, the absolute values of $cD_{f}$ for both the MDI and HMI maps also increase. The optimal value of $\lambda_{R}$ for the MDI maps is $5^{\circ}$, while for the HMI maps, it is $7^{\circ}$. The transport parameter $\lambda_{R}$ used in the SFT simulation based on HMI observations by Yeates (2020) is $6.35^{\circ}$, which is only slightly smaller than the parameter we use. The minor discrepancy may be attributed to the differences in the magnetic field distributions of ARs in our database and that of Yeates (2020).

To minimize the overall deviation between the $cD_{f}$ and the solar dipole fields obtained from both the MDI and HMI maps, the optimal $\lambda_{R}$ is also $5^{\circ}$. This value is not the smallest among the transport parameters used in past research, such as $4^{\circ}$ in Lemerle et al. (2015) and $4.36^{\circ}$ in Wang (2017). The total deviation for the optimal $\lambda_{R}$ is still significant. Factors contributing to this deviation may include the detection of new ARs in the ACs, limitations in synoptic magnetogram observations, and limitations in polar field observations. Although our detection method, incorporating the repeat-AR-removal module, can eliminate some repeated regions and detect new ARs in the ACs, it may still miss some repeat regions or erroneously remove newly emerged flux. Synoptic magnetograms lack observations of the far-side solar disk and have relatively low time resolution. The limitations in synoptic magnetogram observations may result in not all detected ARs being in their fully emerged phase, leading to the potential loss of some emerged AR flux. Additionally, synoptic maps fail to capture some short-lived ARs, particularly those emerging on the far-side of the sun. Considering the small tilt angle of the Sun’s rotation axis relative to the ecliptic and the projection effect of magnetic field observations, measurements of the polar magnetic field are poor. Magnetic field observations near polar regions often exhibit a high noise level and contain missing values. Interpolation is applied to obtain missing data (Sun et al., 2011; Luo et al., 2023), thereby affecting the polar field. As a result, the solar dipole field from the synoptic maps may also deviate from the actual value.

We use $\lambda_{R}=5^{\circ}$ to calculate $D_{f}$, and meanwhile, we compare the results of $D_{f}$ calculated with $\lambda_{R}=7^{\circ}$ to show the impact of $\lambda_{R}$. In addition, we also provide the processing programs along with the database and anyone can calculate the parameter $D_{f}$ with another $\lambda_{R}$ if necessary.

With the optimal transport parameter, $\lambda_{R}=5^{\circ}$, we apply the calculation methods described in Section 3.1 to AR detection results to calculate dipole fields for each AR. First, we balance the magnetic flux of each AR by increasing the flux of the weak polarity because a part of the flux of the more dispersed polarity, usually the following polarity, is missed by the detection. The method is the same as Wang et al. (2020) and Jiang et al. (2019). Then, we get the magnetic field distribution, $B(sin\lambda,\varphi)$, used for the AR method, directly from AR detection of the maps. Flux $\Phi$, latitude difference $d_{\lambda}$, and initial Gaussian width $\sigma_{0}$ used in the BMR method are also calculated from the AR detection results.

To align with the results from MDI maps, the AR flux detected in HMI synoptic magnetograms needs to be multiplied by a factor of 1.36 (Wang et al., 2023). Since equations of both AR and BMR methods are linear in terms of magnetic fields, we also calibrate the AR dipole fields in HMI maps by applying a factor of 1.36. Comparing the parameters of the same AR detected in MDI and HMI maps during the overlap period, as shown in Figure 4, the factor works well for all parameters.

First, we systematically compare the initial dipole field ($D_{i}$) and final dipole field ($D_{f}$) of ARs with real configurations to illustrate the impact of flux transport on each AR and all ARs in a solar cycle. Figure 5 shows the distribution of $D_{i}$ and $D_{f}$. Both $D_{i}$ and $D_{f}$ exhibit concentration around zero, with the distribution of $D_{f}$ being more centered than $D_{i}$. $D_{i}$ tends to be negative during cycle 23 and positive in cycle 24, while $D_{f}$ shows a more symmetric distribution around zero for both cycles. Although some ARs exhibit significant $D_{i}$ and $D_{f}$, the majority have relatively small $D_{i}$ or $D_{f}$. In two cycles, 1709 ARs, approximately 68% of all ARs, have an absolute $D_{i}$ smaller than 0.01 G, which is less than 1% of the total dipole field. The corresponding number for $D_{f}$ is 2002, accounting for about 80%. While certain ARs can notably influence the solar dipole field, the impact of most ARs is limited.

Figure 6 shows the comparison between $D_{i}$ and $D_{f}$. For some ARs, $D_{f}$ is larger than $D_{i}$, smaller than $D_{i}$, or even exhibits a different sign (referred to as changing-sign ARs), as denoted by the red dots. Figure 2 is an example of changing-sign ARs, with $D_{i}$ of -0.0064 G and $D_{f}$ of 0.0296 G. The changing-sign ARs are first reported by Jiang et al. (2019). They will be further discussed in Section 4.2. Among ARs with the same sign for $D_{f}$ and $D_{i}$, there are 898 ARs for cycle 23 and 687 ARs for cycle 24 whose axial dipole field decreases (absolute value of $D_{f}$ smaller than that of $D_{i}$). Conversely, there are 403 and 281 ARs for cycles 23 and 24, respectively, whose axial dipole field increases. These changes in dipole fields contribute to the difference between the distributions of $D_{f}$ and $D_{i}$, as illustrated in Figure 5. As the axial dipole fields of most ARs decrease, the distribution of $D_{f}$ is more concentrated around zero compared to $D_{i}$. In both cycles, the number of $D_{f}$ in bins close to zero surpasses that of $D_{i}$. Additionally, as flux transport increases the dipole field of certain ARs, the instances with absolute value greater than 0.1 G for $D_{f}$ are also more than those for $D_{i}$, as shown in Figure 5.

When $\lambda_{R}$ is increased from $5^{\circ}$ to $7^{\circ}$, the number of ARs with $D_{f}$ smaller than 0.01G decreases to 1809, accounting for approximately 72.5% of all ARs. In cycles 23 and 24, the number of ARs with dipole field increasing rises from 684 to 991, and for ARs with dipole field decreasing reduces from 1585 to 1374. The contribution of most ARs to the solar dipole field is still small and the number of ARs with dipole fields decreasing still exceeds those with dipole fields increasing. The impact of $\lambda_{R}$ is limited.

As for the cumulative dipole fields, defined as the sum of all dipole fields, the absolute value of the cumulative final dipole field ($cD_{f}$) in each cycle is smaller than that of the cumulative initial dipole field ($cD_{i}$). The values of $cD_{f}$ are -3.91 G for cycle 23 and 1.63 G for cycle 24, while the values of $cD_{i}$ are -11.85 G for cycle 23 and 4.70 G for cycle 24. However, the $cD_{f}$ is influenced by the transport parameter $\lambda_{R}$, increasing with $\lambda_{R}$. Consequently, $cD_{f}$ might surpass $cD_{i}$ for a larger $\lambda_{R}$. Nevertheless, $\lambda_{R}$ is optimized to minimize the deviation between $cD_{f}$ and the solar axial dipole field of each cycle. The dipole field observations of MDI and HMI are -3.88 G and 2.54 G, respectively and the corresponding $cD_{i}$ values for the MDI and HMI time range are -11.69 G and 3.34 G. These indicate that the absolute values of $cD_{i}$ are also greater than those of the axial dipole fields of cycles 23 and 24. Therefore, $cD_{f}$ will consistently be smaller than $cD_{i}$ as long as $\lambda_{R}$ is appropriately chosen. The result is consistent with the finding of cycle 21 by Wang & Sheeley (1991). The fact that the absolute values of $cD_{f}$ are smaller than that of $cD_{i}$ indicates that the solar axial dipole field primarily originates from flux emergence. The net effect of flux transport is to reduce the solar dipole field that is obtained during flux emergence. In the jargon of the BL-type mechanism, the generation of the poloidal flux ($\alpha$-effect) mainly happens in the flux emergence, while the surface flux transport mainly decreases rather than generates the poloidal flux.

The BMR approximation of AR is widely used in AR emergence and evolution research. Here we compare the initial dipole field ($D_{i}$) and final dipole field ($D_{f}$) calculated using the AR method with those calculated using the BMR method ($D_{i}^{B}$ and $D_{f}^{B}$) to test the performance of the BMR approximation. The results are shown in Figure 7. For the initial dipole field, the values of the BMR approximation ($D_{i}^{B}$) are highly consistent with that of the AR method ($D_{i}$). This finding is consistent with the research of Yeates (2020). Furthermore, the cumulative initial dipole field of BMR approximation ($cD_{i}^{B}$) and that of AR ($cD_{i}$) are also nearly the same. For example, $cD_{i}^{B}$ is -11.854 G in cycle 23, while $cD_{i}$ is -11.850 G. It shows that the BMR approximation of the initial dipole field proposed by Wang & Sheeley (1991) (Equation (4)), is in line with the AR method (Equation (3)). The BMR approximation is more efficient than the AR method and is therefore recommended for use.

However, the real contribution of ARs to the polar field at the solar minimum is $D_{f}$, not $D_{i}$. For $D_{f}$, the BMR approximation yields less satisfactory results. The result aligns with the findings of Wang et al. (2020) but with more data than it. Although the deviations between $D_{f}^{B}$ and $D_{f}$ for many ARs are small, there are instances where the deviations are significant, exceeding 0.1 G, a magnitude comparable to the $D_{f}$ of these ARs.

Additionally, the BMR approximation sometimes incorrectly predicts the sign of the final dipole field for certain ARs. Given that $D_{i}$ is nearly equal to $D_{i}^{B}$, and the signs of $D_{f}^{B}$ and $D_{i}^{B}$ are the same (as seen from Equations (5 - 6)), the sign of $D_{f}^{B}$ aligns with $D_{i}$. Consequently, ARs with different signs of $D_{f}$ and $D_{f}^{B}$ also exhibit different signs of $D_{f}$ and $D_{i}$. These ARs are referred to as changing-sign ARs in Section 4.1. Among 2952 ARs in cycles 23, 24, and part of 25, there are 266 changing-sign ARs in total. Similarly, Yeates (2020) identifies 53 ARs out of 1090 ARs where $D_{f}$ obtained with the BMR method and the AR method have different signs. The change in sign is typically attributed to the complex configurations of these ARs, which significantly differ from the BMR model (Jiang et al., 2019). For instance, the AR shown in Figure 2 changes the sign of the dipole field due to the small section of positive polarity located at about $-5^{\circ}$ latitude, which is ignored by BMR approximation. Just like the small positive polarity, these regions emerging near the equator can induce significant variations in both the AR’s dipole field and that of the Sun. They should be precisely considered in research, not be approximated.

Some changing-sign ARs have a notable impact on the solar polar field; for example, $D_{f}$ for the AR labeled with a red circle in Figure 7 is 0.49 G, whereas the BMR approximation $D_{f}^{B}$ of it is -0.11 G. The deviation between the BMR approximation and the AR method of the AR is about 0.6 G, about 15% of the total dipole field of cycle 23. To accurately predict the contribution of these changing-sign ARs to the polar field, it is recommended to employ their final dipole field ($D_{f}$) while considering their real configurations. The BMR approximation is prone to incorrectly predicting the dipole field of these ARs.

Moreover, the cumulative values of $D_{f}^{B}$ and $D_{f}$ exhibit more significant differences. The $cD_{f}^{B}$ and $cD_{f}$ values are -7.95 G and -3.91 G for cycle 23, and 3.85 G and 1.63 G for cycle 24. The $cD_{f}^{B}$ value for each cycle is approximately twice the $cD_{f}$ value. As the BMR approximation displays a substantial deviation from the AR method, predictions relying on the BMR approximation, as exemplified by Pal & Nandy (2023), may be affected by this deviation, raising concerns about their reliability. For the BMR approximation, $D_{f}^{B}/D_{i}^{B}$ is proportional to $exp{\left(-\frac{\lambda_{0}^{2}}{2(\lambda_{R}^{B})^{2}}\right)}$, as shown in Equation (6), where $\lambda_{R}^{B}=\sqrt{\frac{\eta}{R_{\odot}^{2}\Delta u}+\sigma_{0}^{2}}$. Here, $\sigma_{0}$ represents the initial Gaussian width of the BMR. In our analysis, we take half the polarity separation as $\sigma_{0}$ for the BMR. However, if we set $\sigma_{0}$ as $0^{\circ}$, the cumulative values of $D_{f}^{B}$ become more similar to those of $D_{f}$, resulting in -4.39 G for cycle 23 and 2.56 G for cycle 24. The impact of $\sigma_{0}$ on the final dipole field is also discussed by Iijima et al. (2019). The $\sigma_{0}$ of BMRs influences their $D_{f}^{B}$ but is challenging to determine accurately. The AR method proposed by Wang et al. (2021) (Equation (8)), which does not require an initial Gaussian width $\sigma_{0}$, can calculate the AR final dipole field more accurately. This shows the advantage of the AR method.

When $\lambda_{R}=7^{\circ}$, the number of changing-sign ARs decreases to 150. The corresponding $D_{f}$ and $D_{f}^{B}$ for the AR labeled in Figure 7 become 0.33 G and -0.08 G, respectively. There is still a large deviation of 0.41 G, approximately 8% of the total dipole field. Significant deviation persists between $cD_{f}^{B}$ and $cD_{f}$. For instance, $cD_{f}$ for cycle 23 is -8.05 G, while $cD_{f}^{B}$ is -11.12 G. Adjusting the initial Gaussian width $\sigma_{0}$ to 0 improves the BMR approximation, yielding $cD_{f}^{B}$ of -8.55 G, closer to $cD_{f}$. When $\lambda_{R}=7^{\circ}$, although the changing-sign ARs are relatively strongly influenced, other results only weakly change. The BMR approximation remains poor for the final dipole field, and setting $\sigma_{0}$ to 0 works better than employing half the polarity separation.

In summary, although the BMR approximation performs well for the initial dipole field, it yields inferior results for the final dipole field, which describes the true contribution of an AR to the polar field in the solar minimum. When measuring an AR’s contribution to the polar field, its final dipole field should be utilized while taking into account its real configuration, i.e., the parameter $D_{f}$ in our database.

As shown in Section 4.1, the $D_{f}$ of most ARs are small. To research the impact of numerous ARs with small $D_{f}$, we rank ARs according to the absolute values of their $D_{f}$ and calculate the cumulative final dipole field for different numbers of top ARs ($cD_{f,n}$). The analysis is conducted with the AR method ($cD_{f,n}$) and the BMR method ($cD_{f,n}^{B}$). The variations of $cD_{f,n}$ and $cD_{f,n}^{B}$ with the number of ARs (n) are presented in Figure 8. For $cD_{f,n}$ calculated with the AR method, its deviation with the cumulative final dipole field of all ARs ($cD_{f}$) in each cycle firstly increases due to certain rogue ARs with contributions opposite to those of other ARs. Then, the deviation gradually decreases under the effect of plenty of ARs with small $D_{f}$. When the AR number increases to about 500, the $cD_{f,n}$ of two cycles both becomes almost equal to the $cD_{f}$. The deviations are 3.3% for cycle 23 and 1.2% for cycle 24. If a similar deviation threshold is set for cycle 24, the AR number decreases to about 350, roughly one-third of all ARs (1059). The 500 ARs also constitute about one-third of all ARs in cycle 23, during which the total number of ARs is 1436. The smallest absolute values of $D_{f}$ in the top one-third ARs are 0.0028 G and 0.0023 G for cycles 23 and 24, respectively.

For $cD_{f,n}^{B}$ calculated with the BMR method, the orange line in Figure 8, its deviations with the cumulative final dipole field for all BMRs, $cD_{f}^{B}$, gradually decreases for each cycle with the increase of BMR number without the initial increase for the deviation between $cD_{f,n}$ and $cD_{f}$. It shows that these ARs with opposite contributions, which cause the initial increase for the deviation between $cD_{f,n}$ and $cD_{f}$, are largely a result of their complex configurations that significantly differ from the BMR. As the number of BMRs increases to around 500, the $cD_{f,n}^{B}$ values for cycles 23 and 24 also become nearly equivalent to the $cD_{f}^{B}$.

When $\lambda_{R}=7^{\circ}$, the initial increase in the variation of deviation between $cD_{f,n}$ and $cD_{f}$ with the number of ARs (n) disappears in cycle 24 but persists in cycle 23. This further supports the presence of rogue ARs in cycle 23. The deviations between $cD_{f,n}$ of the top 500 ARs and $cD_{f}$ are 5.1% and 3% for cycles 23 and 24, respectively. When applying the top 350 ARs, the top one-third ARs of cycle 24, the deviation for cycle 24 is 8%. Although these values are larger than those for $\lambda_{R}=5^{\circ}$, they are still deemed acceptable. The top 500 ARs, or the top one-third ARs, remain sufficient for reconstructing the axial dipole fields during the solar minimum. As for $cD_{f,n}^{B}$, there is no significant change.

Our results align with the findings of Nagy et al. (2020) and Whitbread et al. (2018), but our study requires a smaller number of ARs, approximately 500 ARs compared to their 750 ARs. These studies reveal that while several rogue ARs can induce noticeable variations in the solar polar field, the collective impact of numerous ARs with smaller contributions to the polar field cannot be disregarded. Notably, not all ARs are essential, and approximately the top 500 ARs, or the top one-third of all ARs, are sufficient for building up the polar field during the solar minimum. Hofer et al. (2024) also emphasizes the importance of small ARs on the polar field variation.