Asymmetric Distribution of the Solar Photospheric Magnetic Field Values

Magnetic field couples the solar interior, the solar atmosphere, throughout the heliosphere; it is responsible for the many solar activities, heating of the chromosphere and corona, and acceleration of the solar wind (e.g., Wiegelmann et al., 2014; Komm et al., 2015; Cheung et al., 2017); besides, its characteristics are essential for interpreting the solar dynamo and solar cycle (Brun & Browning, 2017; Charbonneau, 2020). Therefore, understanding the spatial and temporal evolution of the solar magnetic field is of vital importance. Hitherto, only the photospheric magnetic field can be measured in detail routinely with high-resolution space-borne instruments (Lagg et al., 2017), e.g., the Michelson Doppler Imager onboard the Solar and Heliospheric Observatory (MDI/SOHO; Scherrer et al. 1995) and the Helioseismic and Magnetic Imager onboard the Solar Dynamics Observatory (HMI/SDO; Schou et al. 2011; Scherrer et al. 2011; Hoeksema et al. 2014). The interpretation and modeling of magnetic field on other parts of the solar atmosphere are mainly dependent on the photospheric measurements (Lagg et al., 2017). In this study, we will investigate the asymmetric distribution of the solar photospheric magnetic field values.

The solar surface is overspread with ubiquitous magnetic fields coming from the quiet Sun and active regions (Sheeley, 1966; Harvey, 1971; Jin et al., 2011). The quiet Sun photospheric magnetic field is usually divided into kilogauss magnetic network and internetwork magnetic field with weaker strength; the former is situated in the profiles of supergranular cells where the horizontal flows become downdrafts, and the latter is small-scale flux concentrations scattered on the solar surface (Bellot Rubio & Orozco Suárez, 2019). The distribution of the magnetic field values is a fundamental property of the Sun. Getachew et al. (2019a, b) studied the asymmetry of solar weak photospheric magnetic field values, using synoptic chart data from several sources, including the Wilcox Solar Observatory, Mount Wilson Observatory, Kitt Peak Vacuum Telescope, SOHO/MDI, SOLIS/VSM, and SDO/HMI. In their study, ‘asymmetry’ is defined as the nonzero field value of the fitted Gaussian distribution maximum. They fitted the distribution with a parameterized Gaussian function, and then the asymmetry is revealed, mainly when the spatial resolution of the synoptic chart is considerably reduced, such as $360\times 180$ pixels or even $120\times 48$ pixels which are much lower than the original resolution (e.g., $3600\times 1440$ pixels for HMI). They showed that the lower the resolution, the more prominent the calculated asymmetry; according to which, they inferred that it is the supergranulation that brings about the asymmetry.

There are several shortcomings in estimating the asymmetry with highly resolution-reduced synoptic charts. First, high-spatial-resolution magnetograms and synoptic charts resolve quiet Sun small-scale magnetic fields much better, such as internetwork, network, etc. There is no reason to give up the most essential advantage of observation, i.e., high-resolution. For a chart of 3600$\times$1440 (1080) pixels, the spatial resolution is about $1.68^{\prime\prime}\times 1.33^{\prime\prime}\ (1.78^{\prime\prime})$ (the physical spatial resolution on the solar surface is in the magnitude of ${\sim}10^{3}$ km). When the spatial resolution decreases to around ${\sim}3^{\prime\prime}$, most of the magnetic features are canceled and the magnetograms look very vague (Bellot Rubio & Orozco Suárez, 2019). It will be unable to resolve many of the adjacent magnetic elements with opposite polarities. For charts of $120\times 48$ pixels, the spatial resolution is about $50.3^{\prime\prime}\times 40.0^{\prime\prime}$, which is much larger than ${\sim}3^{\prime\prime}\times 3^{\prime\prime}$. Second, the number of pixels in the distribution decreases substantially when a chart is downsized, which affects the fitting result. From $3600\times 1440$ to $120\times 48$ pixels, the number of pixels decreases by nearly 3 orders of magnitude. In the distribution of a downsized chart, the probability of adjacent bins vary drastically (e.g., Fig. 2 in Getachew et al. 2019a). The parameters of the fitted Gaussian model are unlikely reliable in judging the asymmetry. Last, the noise of measurements (a few Gauss) should be taken into consideration, because they dominate the location parameter of a Gaussian function during the fitting of magnetic field values, especially when the distribution is narrow.

In this study, we show that the distribution of solar photospheric magnetic field values is much better fitted with the Lorentzian function, rather than the Gaussian function. The asymmetry is clearly revealed after merely excluding the noise. We also uncovered the temporal characteristic of the asymmetry variation. This paper is organized as follows. In Section 2, we introduce the synoptic chart data. Section 3 introduces the method and Section 4 the analyses and results. In the final Section 5, we conclude the paper and discuss the results.

Due to the substantial instrumental differences between various ground and space-based measurements, we only make use of MDI/SOHO and HMI/SDO high-resolution synoptic charts that are coming from space-borne observations. The HMI started observation with similar yet upgraded instruments when the MDI was reaching its end of life, i.e., the HMI is a successor to the MDI (Scherrer et al., 2011). Synoptic charts from both of them are constructed from zero-level offset corrected magnetograms (Liu et al., 2004). The HMI (MDI) charts are 3600 points in Carrington longitude by 1440 (1080) points equally spaced in sine latitude. The inferred radial synoptic charts provide the imputed radial component of the magnetic field over the entire solar disk. Each pixel on a synoptic chart is generated using near-central-meridian data from 20 magnetograms. A detailed description and comparison of the charts from the two instruments can be found in Liu et al. (2012). In this study, we used synoptic charts of MDI span from Carrington Rotation (CR) 1909 to CR-2096 (from May 6, 1996 to Dec 10, 2010; 188 charts in total), and that of HMI from CR-2097 to CR-2232 (from May 6, 2010 to Jun 18, 2020; 136 charts in total). In all, there are 324 synoptic charts, spanning 24 years which covers more than two solar cycles.

Examples of HMI and MDI synoptic charts are depicted in Figure 1. In the Figure, the CR number of the HMI chart shown here (CR-2172) is the same as in Getachew et al. (2019a), and the CR number (CR-1950) of MDI is chosen so that it is roughly amid solar maximum and minimum (similar to the case of CR-2172). The two subfigures share the same colorbar, and the colormap is hmimag from the SunPy (Community et al., 2020) open-source software. The magnetic field values measured by MDI is greater than that of HMI by a factor of 1.40 (Liu et al., 2012), and that is why the magnetic fields in the lower panel of Figure 1 seems stronger.

To estimate the asymmetry of solar photospheric magnetic fields (denoted as ‘$\mathbb{A}$’ afterward), we model the magnetic field value distribution of each synoptic chart with a three-parameter Lorentzian function (or Cauchy-Lorentz distribution). The Lorentzian function is written as

where $x_{0}$ is the location parameter, specifying the location where the distribution reaches the peak. It is interpreted as the solar photospheric magnetic asymmetry $\mathbb{A}$ in this study. Besides, $I$ is the height of the peak, and $\gamma$ is half of the full width at half maximum (FWHM). Since only in the special case when $I=1/(\pi\gamma)$, the integration of this function equals to 1, the Lorentzian function is normally not a probability density function. We use the astropy.modeling package to do the fitting. The model parameters and their uncertainties are estimated with the Levenberg-Marquardt algorithm and the least-squares statistic. The significance of an obtained asymmetric value $\mathbb{A}$  is quantified with the same method as in Getachew et al. (2019a) (with different degrees of freedom).

Figure 2 shows the distributions and model fittings for the two exemplary synoptic charts in Figure 1. Specifically, the magnetic field values in the synoptic charts of HMI CR-2172 and MDI CR-1950 are fitted with both the Lorentzian model and Gaussian model, respectively. The Lorentzian function is an example of a fat-tailed distribution that exhibits a large kurtosis relative to that of a Gaussian function. The result indicates that the Lorentzian model is much more suitable than the Gaussian model in describing the distribution of magnetic field values, both at the center and the tails. It should reflect the property of the evolution of magnetic fields from strong magnetic fields in active regions decaying into weaker fields.

As exhibited in the modeled Gaussian distribution in Figure 2, the major part of the distribution is concentrated within about $\pm 5$ $\mathrm{Mx\ cm^{-2}}$ (for HMI) or $\pm 10$ $\mathrm{Mx\ cm^{-2}}$ (for MDI), and it plunges to zero in the tails very quickly. It is obvious that the Gaussian model describes the distributions poorly; especially, it seriously understates the tails of the distribution of magnetic field values. For the HMI and MDI line-of-sight magnetic field synoptic charts, the noise levels are 2.3 and 5.0 $\mathrm{Mx\ cm^{-2}}$, respectively, and they are quite uniform over the entire chart (Liu et al., 2012).

The magnetic field values larger than tens of Gauss are totally ignored. Thus the asymmetry is almost completely dominated by the very weak magnetic fields mixed with indistinguishable noise. Therefore, not surprisingly, the calculated asymmetry with the Gaussian model is not dependent on the upper range of magnetic field values from 10 to hundreds Gauss (Liu et al., 2004; Getachew et al., 2019a). In this study, in the process of estimating the $\mathbb{A}$, we will exclude the very weak field values which are unresolvable from noise.

In this section, first, we calculate the asymmetry values $\mathbb{A}$ of each synoptic chart in three different cases. Then, we calculate $\mathbb{A}$ after excluding polar regions and high-latitude areas; the temporal characteristics of the obtained $\mathbb{A}$ are explored. In addition, we attempt to determine the reversal times of polar field polarities of solar cycles 23 and 24 from the reversal of $\mathbb{A}$.

In this research, we investigated the asymmetry of solar photospheric magnetic field values and its temporal variations with data of synoptic charts spanning the recent 24 years (covering solar cycles 23 and 24). Note that the synoptic charts are constructed with magnetograms that are all zero-level offset corrected (Liu et al., 2004, 2012).

The distributions of magnetic field values in the synoptic charts are found to be Lorentzian rather than Gaussian. Besides an obvious better fitting at the central part of the distribution, especially, the Lorentzian function better represents the stronger magnetic fields at the tails of the distribution, which probably reflects the gradual evolution process of magnetic fields from strong fields to quiet Sun magnetic fields.

The asymmetry of distribution is uncovered by the Lorentzian function fitting, despite that it is somewhat concealed by the symmetrically distributed noise and very weak fields. Further, after excluding the noise, as well as weak fields that are indistinguishable from the noise, the asymmetry is clearly revealed. We argue that it is the stronger small-scale magnetic fields rather than supergranulations that bring about the asymmetry. In a resolution-reduced synoptic chart, the magnetic field value of each pixel is coming from the average of values of dozens or even thousands (depending on the resolution of the resulted chart) of pixels from the original chart; consequently, the strong magnetic fields are likely averaged out to the range within $\pm$10 G, which leads to the calculated asymmetry in previous studies (Getachew et al., 2019a, b). The magnetic fields of supergranulation mainly reside on the cell boundaries and thus can be resolved only with high-resolution observations (Rincon & Rieutord, 2018). Even if the asymmetry is caused by the supergranulation, it can only be detected with high-resolution observations.

We obtained asymmetry values in several circumstances (see sections 4.1 and 4.3), and also studied their periodicity. Taken all of the $\mathbb{A}$ values into consideration, almost all of them show the half-year and/or annual periods; additionally, some of them show a period approximately half of the solar cycle. We infer that the annual (and half-year) period is caused by the variation of the B₀ angle, and the asymmetry is modulated by the solar cycle, i.e., the evolution of the global magnetic field. More details are discussed in the following paragraph. These characteristics are not revealed at all in previous studies.

The various features of the asymmetry values are a result of the emergence and transportation of surface magnetic flux. The typical butterfly diagram (sunspot areas as a function of latitude and time, i.e., Spörer’s law) manifests that active regions emerge at higher latitudes at the beginning of each cycle (within $\pm 40$ degree); as the cycle progresses, the central emergent latitudes drift towards the equator. The emerged flux at active latitudes usually consists of bipolar active regions with a leading and a following part; the flux of these two opposite polarity parts are approximately equal. Between the two hemispheres, the polarities of the leading (or following) parts are opposite; and also, from one solar cycle to another, of the same hemisphere, the polarities of the leading and following parts change alternatively (Hale’s polarity law, Hale & Nicholson, 1925). The leading polarity is usually more compact in morphology and intense in magnetism. Statistically, the leading polarity spots of the emerged bipolar active regions are at a lower latitude than the following polarity; the latitudinally averaged tilt angles increase with latitude, (and the overall averaged title angle is about 5–6 degrees) (Joy’s law, Hale et al., 1919). The emerged flux is dispersed by the differential rotation, the turbulent diffusion, and the meridional flow. Both polarities of the bipolar regions are being carried poleward. The dispersed flux from the leading polarities in one hemisphere, since it is closer to the equator, is more likely drifting toward the equator to meet and cancel with the opposite-polarity dispersed flux of the leading polarity of the other hemisphere. On the contrary, the following polarities at higher latitudes have a higher probability of reaching the pole instead of being canceled. Also, The leading polarity, while being carried poleward, is more likely to meet and cancel with the following polarity from the same bipole. Some of these characteristics are well displayed in the magnetic butterfly diagram in Figure 6 and the asymmetry values.

The calculation of the three $\mathbb{A}$40 is restricted to the activity belt ($\pm$40 degree). Though there are lots of flux emergence during solar maxima, the emerged bipolar regions are approximately flux-balanced, and the amplitudes of $\mathbb{A}$40 are small (compared with $\mathbb{A}$90N/S). Surprisingly, during solar minimum, even when there are very few or no sunspots and the flux emergence is sparse, $\mathbb{A}$40N and $\mathbb{A}$40S still systematically deviate from zero, and the three generally agree well with each other. During times other than the solar minimum, $\mathbb{A}$40N and $\mathbb{A}$40S are anti-correlated sometimes, which could be caused by cross-equator flux cancellation events; during such processes, the total flux of leading polarities on both hemispheres decreases, and thus the asymmetries $\mathbb{A}$40 will be dominated by the following polarities (Cameron et al., 2013). The magnitude of the $\mathbb{A}$40 variation seems independent of the solar cycle. It is not clear where does this asymmetry comes from. The magnetic elements that cause the asymmetry in $\mathbb{A}$40 seems to come from a source independent of the global solar dynamo. We speculate that it is probably related with magnetic flux coming from a local turbulent dynamo (e.g., Jin et al., 2011). Another probability is that the $\mathbb{A}$40 variation is caused by the cross-equatorial flux plumes (Cameron et al., 2013), as well as the variation of the B₀ angle. That is to say, the asymmetries brought about by cross-equatorial flux plumes are further modulated by the annual variation of the solar axis inclination, which is noteworthy even in the $\mathbb{A}$40F.

When high-latitudes and polar regions are taken into consideration, the asymmetries $\mathbb{A}$90N/S are dominated by the poleward polarity surges and polar fields. The polar fields are often located at $\pm$60 degrees poleward, and they are almost unipolar and relatively weak and have large spatial scales (Petrie & Ettinger, 2017). Thus the amplitudes of $\mathbb{A}$90N/S are substantially larger. The variation of the polar field strength lags the $\sim$22-year magnetic activity cycle by $\sim$11-year in time phase. During solar minima, the magnetic activity in the solar surface is at a low level, while the strength of polar fields reaches the peak, so the amplitudes of A90N/S also reach the peak. Each of the poles is unable to be observed during half a year periodically, and this effect is shown in the annual variation of $\mathbb{A}$90 and in Figure 6.

As shown in Figure 5 (a) and Figure 6, during the late declining phase of the cycle approaching solar minima, the 0.5/1-year periodicity is most noticeable in $\mathbb{A}$90. This is because, during this time, the poleward polarity surges become less frequent and finally cease, and then the asymmetries $\mathbb{A}$90 will be dominated by high-latitude regions and polar fields. On the contrary, during the time when the polarity surges are strong and frequent, the 0.5/1-year modulation is partially masked by the lower-latitude streams, and thus it is not that discernible.

We assert that the reversal of the hemispheric $\mathbb{A}$ can be another indicator of the polar field reversal. Especially, we show that the polar reversal of the N- and S-hemisphere of solar cycle 24 may have happened contemporarily, both in March 2014. It is well known that the Sun’s global dipole magnetic field reverses polarity at each solar maximum during the 11-year solar cycle. Polarity reversal is essential for the Babcock-Leighton model of the solar cycle (Babcock, 1961; Leighton, 1969). Since emergent active regions located in low latitudes are typically bipolar and flux-balanced while large spatial-scale high-latitude fields and polar fields are usually unipolar (Petrie & Ettinger, 2017), the hemispheric asymmetry $\mathbb{A}$90N/S mainly reflect the asymmetry of magnetic fields at high latitudes and polar regions which reverse their signs during solar maxima. At the beginning of each cycle, the sign of the following-polarity of emergent flux is opposite to the polar fields. As the cycle progresses and reaches the maximum, the poleward surges of this flux reverse the sign of the magnetic field at high-latitude and polar regions. These surges then continue to build up new polar fields during the declining phase of the cycle(Hathaway, 2015).

It is a controversial issue when did the polar field of the two hemispheres reversed in solar cycles 23 and 24; different groups of researchers reported different results (for a list of reversal times from previous research, see, e.g., Pishkalo, 2019). For solar cycle 24, in previous studies, the reversal time of the N-hemisphere varies from Jun 2012 to Sep 2015, and that of the S-hemisphere varies from Nov 2013 to Jun 2014 (e.g., Sun et al., 2015; Petrie & Ettinger, 2017; Janardhan et al., 2018). The reversal time of the S-hemisphere determined here agrees well with that of Sun et al. (2015). The so-called triple or multiple reversals in the N-hemisphere in cycle 24 as indicated in, e.g., Pishkalo (2019) and Janardhan et al. (2018), is well shown in the variation of asymmetry values. For cycle 23, the polar reversal time of the S-hemisphere determined with asymmetry values agrees with Wang et al. (2002); furthermore, it can be assured that the reversal of the N-hemisphere was completed prior to the S-hemisphere despite that the determination of the exact date is impeded by missing values. More research can be done to investigate the polar field reversal with the asymmetry of magnetic field values. For instance, calculating the asymmetry of distribution with daily magnetograms, restricting the calculations within high latitudes and polar regions, or calculating the asymmetry in various latitudes ranges, etc.