Phase relationships among different types of solar magnetic fields at high and low latitudes

1 Introduction

The solar magnetic field is the primary driver of both solar activity and solar variability. The solar disk presents a complex, dynamic tapestry of magnetic structures that vary dramatically in strength, spatial scale, and lifetime. Understanding the distinct characteristics, origins, and cyclic evolution of these magnetic fields is fundamental to uncovering the physics of solar magnetism, improving space weather prediction, and comprehending stellar magnetic activity in general (Mackay and Yeates, 2012; Hathaway, 2015; Bellot Rubio and Orozco Suarez, 2019; Charbonneau, 2020; Karak, 2023).

Thanks to long-term high-resolution observations and extensive research, it is now established that there are four primary types of magnetic field structures on the solar surface: sunspot magnetic fields (within active regions), ephemeral-region magnetic fields (Harvey and Martin, 1973; Martin, 1990; Harvey, 1992), network magnetic fields (Leighton et al., 1962), and intranetwork magnetic fields (Livingston and Harvey, 1975). Among these, sunspot magnetic fields are large-scale features, whereas the other three types are considered small-scale magnetic elements (Parnell et al., 2009; Shiota et al., 2012; Jin and Wang, 2015a; 2015b; Li et al., 2022; Rempel et al., 2023; Trelles Arjona et al., 2023; Yang et al., 2024a; Yang et al., 2024b; Yang et al., 2025).

Sunspots are the most intense and well-organized magnetic features visible on the solar disk, with field strengths typically ranging from a few hundred to over 3,000 G. They generally appear as bipolar or multipolar magnetic complexes, though isolated unipolar spots are also observed. Key metrics for tracking the strength and phase of the solar cycle include their number, total area, total magnetic flux, and latitudinal distribution. As the primary markers of solar activity, sunspots are governed by two fundamental laws: Sporer’s Law (manifested in the sunspot butterfly diagram) and Hale’s Polarity Law (which defines the full 22-year magnetic cycle) (Hathaway, 2015).

Ephemeral regions are short-lived, small-scale active regions that emerge within the quiet Sun, outside of established sunspot activity belts. When the magnetic flux of an ephemeral region exceeds approximately 1 0 20 Mx, it can develop into a pore or a small sunspot (Martin, 1990; Wang et al., 2012). Thus, ephemeral regions are considered underdeveloped or proto-active regions. (Martin, 2018; Martin, 2024) argued that active and ephemeral regions are the same magnetic bipoles. Like sunspots, the number and latitudinal distribution of ephemeral regions follow the solar cycle at low latitudes; however, ephemeral regions can emerge at much higher latitudes, extending into polar regions (Harvey, 1992; Li et al., 2008). The precise nature of the solar-cycle dependence for high-latitude ephemeral regions remains an open question.

Network and intranetwork magnetic fields are ubiquitous small-scale magnetic features that cover the quiet Sun. They are significantly weaker in magnetic flux but far more numerous than ephemeral-region fields. Intranetwork fields are even weaker than network fields, representing the smallest and faintest magnetic elements currently distinguishable (Livingston and Harvey, 1975; Lites et al., 1996; Bellot Rubio and Orozco Suárez, 2019).

The prevailing view holds that, at low latitudes, network magnetic fields are in anti-phase with the solar cycle, while intranetwork fields exhibit little to no solar-cycle dependence (Meunier, 2003; Jin et al., 2011; Jin and Wang, 2014; Shiota et al., 2012; Korpi-Lagg et al., 2022). At high latitudes, network fields also appear to be in anti-phase with the cycle (Shiota et al., 2012; Trelles Arjona et al., 2023), whereas the solar-cycle relationship for high-latitude intranetwork fields remains unclear, with studies reporting conflicting or inconclusive results (Hagenaar et al., 2003; Buehler et al., 2013; Shiota et al., 2012; Faurobert and Ricort, 2021; Korpi-Lagg et al., 2022; Petrie, 2023; Trelles Arjona et al., 2023). The solar-cycle behavior of small-scale magnetic fields, particularly at high latitudes, therefore warrants further investigation.

In this study, we utilize synoptic magnetograms to investigate the solar-cycle phase relationships of full-surface magnetic fields. By combining the various solar-cycle phase relationships determined from observations, we then infer the composition of magnetic fields in the synoptic magnetograms.

2 Data

The data used in this study consist of synoptic magnetograms covering Carrington Rotations (CRs) 1,625 to 2,007, spanning from 19 February 1975, to 25 September 2003. These observations were acquired with the vacuum telescope at the National Solar Observatory/Kitt Peak and are shown in Figure 1. Although the latitude values in the figure are negative in the Southern Hemisphere, next, we distinguish only between high and low latitudes, regardless of hemisphere. For each synoptic magnetogram, the radial magnetic field strength was measured at 180 equally spaced sine-latitude intervals and 360 equally spaced longitude intervals.

The solar surface is conventionally divided into high- and low-latitude regions at approximately 55 ° (Yang et al., 2024b). For the purposes of this analysis, we define low latitudes as 0 - 50 ° and high latitudes as 60 ° - 90 °. Chen et al. (2026) demonstrated that magnetic elements on the solar disk can be classified into strong-field and weak-field groups at both low ( 0 ° − 50 ° ) and high ( 60 ° − 90 ° ) latitudes, based on a distinguishing field strength of about 5.29 G. This threshold represents the intersection point of the two correlation coefficient curves at low and high latitudes. The phase relationship between magnetic field strength and the solar activity cycle varies continuously with field strength, implying that the threshold is not unique. A modest change in the threshold would introduce only a small fraction of anti-phase values, which would likely leave the derived phase unchanged. A substantial change, by contrast, would include a large fraction of anti-phase values and could thus alter the phase. The latitude-based division is a conventional rather than a physically motivated one; accordingly, the 5.29 G value is merely a working threshold. Given that different instruments have varying resolutions and that the observed magnetic field values are in fact magnetic flux, this threshold is instrument-dependent. Magnetic elements with a field strength not exceeding 5.29 G are assigned to the weak-field group (WG), while those with a field strength greater than 5.29 G are assigned to the strong-field group (SG). The SG elements exhibit an in-phase relationship with the solar cycle at low latitudes, transitioning to an anti-phase relationship at high latitudes. The WG elements show the opposite latitudinal dependence. All these relationships are statistically significant at the p < 0.001 level.

Daily magnetic field strengths were computed separately for the low- and high-latitude bands. For each day, the quantity is defined as the total unsigned field strength of all magnetic elements within a given latitude band of the synoptic maps, excluding pixels with zero field strength. The original synoptic magnetogram values are used directly, without additional processing. The results for the SG and WG are shown in Figure 2 (low latitudes) and Figure 3 (high latitudes). Overall, at both high and low latitudes, the SG values are several times larger than the WG values. As illustrated in these figures, the daily magnetic field intensity of the SG at low latitudes ( S G low ) is in anti-phase with that of the WG at low latitudes ( W G low ); similarly, the daily magnetic field intensity of the SG at high latitudes ( S G high ) is in anti-phase with that of the WG at high latitudes ( W G high ). In the following section, we investigate the precise phase relationships among these four daily magnetic field intensity time series. For brevity, we denote the four time series as S G low, W G low, S G high, and W G high in the following analysis. They are the daily total unsigned flux for SG/WG at low/high latitudes, which are summarized in Table 1.

TABLE 1. Characteristics of the four derived quantities *.

*Defined as the daily total unsigned flux of the SG, or WG, within low or high latitudes.

3 Results

3.1 Phase relationship between S G low and W G low

S G low and W G low are paired on a daily basis. We begin by calculating the correlation coefficient (CC) between the original time series S G low and W G low, corresponding to a phase shift of 0 days. To examine how the correlation varies with phase lag, we systematically shift one series relative to the other. For a shift of 1 day, the first data point is removed from one series and the last data point from the other, after which the CC is recalculated. This process is iterated, increasing the shift by 1 day each time (following the method of Li et al. (2016)), yielding a sequence of CC values as a function of phase shift. The results are plotted in Figure 4, where positive shifts indicate that W G low leads S G low, and negative shifts indicate that S G low leads W G low. As shown in the figure, the CC reaches a minimum at zero shift, confirming that the two original series are in perfect anti-phase alignment. Two distinct peaks appear at shifts of approximately −1776 days and +1886 days. The larger peak at ∼ −1776 days indicates that S G low leads W G low by roughly half a solar cycle.

We also tested the sensitivity of our results to the threshold value by repeating the analysis with 4.5 G and 6.0 G. As shown in Figure 4, the phase relationships are essentially unchanged, confirming the robustness of our conclusions.

To verify that the correlation coefficient (CC) reaches its minimum precisely at zero phase shift, we performed a non-overlapping block bootstrap analysis (Xu et al., 2017). The block length is set to 10 × N 1 / 4, where N = 10,447 is the length of the data set. In each bootstrap iteration, we draw approximately 63 % (i.e., 1 − 1 / e) of the total number of blocks uniformly at random with replacement (for details, see Xu et al., 2017). The procedure was repeated 3,000 times. The distribution of the phase shifts corresponding to the minimum CC from each iteration is shown in Figure 5. As demonstrated, the minimum CC consistently occurs at a shift of 0 days in all 3,000 trials, confirming that the two original time series are statistically in perfect anti-phase alignment. The distribution of the two peak CC values obtained from each iteration is presented in Figure 6. In nearly every bootstrap realization (e.g., > 99% of cases), the peak CC for negative shifts exceeds that for positive shifts. This result indicates that the two peaks are significantly different from each other, and thus the S G low series statistically significantly leads the W G low series by approximately half a solar cycle.

3.2 Phase relationship between S G high and W G high

Following the methodology outlined in the previous subsection, we performed an analogous phase-shift correlation analysis for the high-latitude magnetic field groups, replacing S G low and W G low with S G high and W G high. The results are presented in Figures 7–9, which correspond sequentially to Figures 4–6. Consistent with the convention adopted earlier, a positive phase shift indicates that W G high leads S G high, while a negative shift indicates that S G high leads W G high. Figure 8 shows that the phase shift is zero in all bootstrap iterations, confirming that the original time series S G high and W G high are in perfect anti-phase alignment with statistical significance at high latitudes. Figure 9 shows that only 3 out of the 3,000 bootstrap realizations fall below the diagonal line, while the remaining 2,997 lie above it. This indicates that the two half-cycle correlation peaks are significantly different, and thus we conclude that the W G high series leads the S G high series by approximately half a solar cycle, with a statistical significance exceeding the 99.9 % confidence level.

We have also tested the sensitivity of our conclusions to the exact value of the threshold. As shown in Figure 7, using 4.5 G or 6.0 G instead of 5.29 G does not alter any of the reported phase relationships, indicating that our findings are robust across a reasonable range of threshold values.

3.3 Phase relationship between S G high and S G low

We applied the same correlation analysis to examine the phase relationship between the strong-field components at different latitudes, now replacing S G high and W G high with S G high and S G low. The results are shown in Figure 10, where a positive phase shift indicates that S G high leads S G low. The correlation coefficient exhibits two peaks at phase shifts of approximately ± 1800 days. The larger peak occurs at negative shifts, indicating that S G low leads S G high by roughly half a solar cycle.

To assess the statistical significance of this lead–lag relationship, we performed the same block bootstrap procedure described in the previous subsection. The results are presented in Figure 11. As the figure shows, nearly all bootstrap realizations (e.g., > 99.9%) fall in the lower-right quadrant relative to the diagonal, confirming that the S G low series leads the S G high series by approximately half a solar cycle with a statistical significance exceeding the 99.9 % confidence level.

To verify the robustness of our findings against the choice of threshold, we repeated the analysis using two alternative cutoff values: 4.5 G and 6.0 G. The results, shown in Figure 10, are essentially indistinguishable from those obtained with 5.29 G, confirming that our phase relationships are stable across a reasonable range of thresholds.

4 Conclusions and discussion

In this study, we use synoptic magnetograms from Carrington Rotations 1,625 to 2,007 to investigate the phase relationships among different groups of magnetic elements. At both low and high latitudes, magnetic elements with strengths exceeding 5.29 G are found to be in strong anti-correlation with zero phase offset relative to those below this threshold.

At low latitudes, strong magnetic elements (> 5.29 G) are in phase with the solar activity cycle and consist primarily of active regions and ephemeral regions. Active regions occupy the upper end of the magnetic flux distribution, while ephemeral regions lie at the lower end. Weak magnetic elements (< 5.29 G) at low latitudes are dominated by network magnetic fields, which are in anti-phase with the solar cycle. Jin et al. (2011) and Jin and Wang (2015a) obtained similar results using data from different instruments and time intervals, further confirming the phase relationship reported here. The transition between these two regimes is gradual; hence the 5.29 G threshold should be viewed as a rough boundary rather than a sharp physical division.

In order to test the robustness of our results against the choice of threshold, we repeated the entire analysis using thresholds of 4.5 G and 6 G. The phase relationships reported in this study - specifically, the half-cycle leads between the four time series - were found to be stable across all thresholds tested. The amplitude of the daily field strength varies with the threshold, but the phase offsets do not. Figure 3 of Chen et al. (2026) illustrates that the phase relationship between strong and weak magnetic fields with the solar cycle at both high and low latitudes is sensitive to the threshold value. The 5.29 G threshold used in this study is therefore not chosen arbitrarily; it is guided by the behavior shown in that figure. Moreover, since the observed quantities are magnetic fluxes, which are inherently resolution-dependent, this specific value should not be overemphasized. The fact that the phase relationships we report are robust across both latitude bands suggests that the underlying patterns are real and not artifacts of the chosen cutoff.

At high latitudes, where sunspots are absent, ephemeral regions are known to be outnumbered by network and intranetwork magnetic elements (Harvey, 1992; Li et al., 2008; 2022; Jin et al., 2011; Jin and Wang, 2015a; Jin and Wang, 2015b). We therefore infer that, at high latitudes, the strong magnetic elements (> 5.29 G) are predominantly composed of network magnetic fields, which are in anti-phase with the solar cycle - consistent with Shiota et al. (2012). Conversely, the weak magnetic elements (< 5.29 G) are most likely dominated by intranetwork magnetic fields (horizontal components), which are in phase with the solar cycle, in agreement with Trelles Arjona et al. (2023).

The apparent area of a magnetic feature decreases with increasing latitude, as it is approximately proportional to the cosine of the latitude. Consequently, the projection effect at high latitudes makes the observed magnetic fields appear more concentrated (Shiota et al., 2012). This explains why network magnetic fields appear stronger at high latitudes than at low latitudes. What is measured at high latitudes is primarily the horizontal component of the magnetic field. Intranetwork magnetic fields are characterized by a high degree of inclination; as a result, the magnitude of their vertical magnetic flux density is nearly one order of magnitude lower than that of their horizontal component (Jin and Wang, 2015b). Owing to these two factors - the projection effect and the high inclination - the intranetwork magnetic flux at high latitudes exceeds that at low latitudes by tens of times, reaching values comparable to the network field flux in the low-latitude region. This is why intranetwork fields are detectable at high latitudes but typically not at low latitudes. This interpretation, however, remains a plausible speculation. It cannot be ruled out that the high-latitude weak magnetic elements (< 5.29 G) represent a new type of magnetic element, distinct from the currently known low-latitude magnetic field populations.

The cyclic behavior of the high-latitude weak magnetic elements (WG, < 5.29 G), which we attribute primarily to intranetwork fields, has direct implications for local dynamo scenarios. Local or small-scale dynamo models, operating purely on granular or supergranular scales, generally produce magnetic fields that are either stochastic or only weakly modulated by the 11-year activity cycle (Vogler and Schussler, 2007; Rempel, 2014). In such models, the magnetic energy in the intranetwork component is supplied by near-surface turbulence and is largely independent of the global toroidal field reservoir. Our observation that the high-latitude WG exhibits a clear, phase-locked in-phase relationship with the sunspot cycle is therefore difficult to reconcile with a purely local dynamo origin. Instead, it favors a scenario in which a substantial fraction of the intranetwork fields at high latitudes are not generated in situ, but rather represent the horizontal or highly inclined remnants of active-region flux that have been transported poleward and subsequently fragmented (Jin and Wang, 2015a; Jin and Wang, 2015b). This does not rule out the existence of a local turbulent dynamo—which is almost certainly operating at all latitudes—but it suggests that its contribution to the observable vertical flux at high latitudes is secondary to that of the large-scale, cycle-modulated flux transport. In this sense, our results support a hierarchical dynamo picture, where the global BL dynamo provides the seed or background field upon which local turbulent processes act, rather than two entirely independent dynamo systems.

Around solar maximum, trailing-polarity magnetic flux from decaying active regions is transported poleward by meridional flow. This flux gradually cancels and eventually reverses the pre-existing polar field. The newly established polar field reaches its maximum strength around solar minimum - a process known as polar field reversal. This well-established physical mechanism explains the key result presented here: strong magnetic fields at low latitudes lead strong magnetic fields at high latitudes by approximately half a solar cycle. In other words, the observed phase relationship lends empirical support to this physical mechanism.

A methodological caveat must be addressed here. Cross-correlating time series that share the same dominant periodicity - the ∼11-year solar cycle - naturally produces correlation peaks at half-cycle offsets, regardless of whether a physical causal link exists. This is a well-known mathematical consequence of periodic signals. In our analysis, both the positive and negative half-cycle offsets do produce correlation peaks, which is expected given the periodic nature of the data. Crucially, however, our bootstrap analysis reveals that the two opposite offsets are statistically significantly different from each other (see Figures 6, 9, 11): one direction consistently yields a higher correlation coefficient than the other.

A similar temporal offset is observed at low latitudes: the magnetic fields associated with sunspots and ephemeral regions lead the network magnetic elements by approximately half a solar cycle. This finding is consistent with the conclusion stated earlier - that strong magnetic fields at low latitudes precede those at high latitudes by roughly half a cycle - since the network field remains in phase and continues to accumulate at both high and low latitudes.

The physical interpretation of these half-cycle phase offsets can be understood within the framework of the flux-transport dynamo. The ∼5–6 years lag between low-latitude strong fields and high-latitude strong fields is quantitatively consistent with the meridional flow transit time from the sunspot belt to the polar regions, as predicted by Dikpati and Charbonneau (1999) and Charbonneau (2020). This agreement suggests that the observed phase offset directly traces the propagation of magnetic flux from active regions to the poles - a key process in the solar cycle dynamo. More importantly, this observed lag provides a stringent observational constraint on the dynamo-relevant transport parameters.

At low latitudes, the offset between strong fields (active regions + ephemeral regions) and weak fields (network) can be interpreted as the characteristic timescale over which magnetic flux from emerging active regions is gradually redistributed into the network through dispersion and cancellation processes (Hagenaar et al., 2003; Jin et al., 2011). Thus, the systematic phase offsets we observe across both latitude bands are not merely statistical correlations, but rather observational signatures of the mass and magnetic flux transport processes that couple different components of the solar magnetic field over the solar cycle.

It cannot be ruled out that this lead-lag relationship at low latitudes is related to the evolution and re-emergence of leading-polarity flux from decaying active regions. That is, trailing-polarity flux drives the high-latitude effect, while leading-polarity flux drives the low-latitude effect. Some network fields may originate from decaying sunspots. However, this study did not find any significant short-term phase delay (shorter than several months) between sunspots and network fields. This suggests that network fields produced by decaying sunspots represent only a very small fraction of the overall network field population.

At low latitudes, the positive and negative correlations obtained in this study could in principle be interpreted as a mathematical complementary effect - if the sum of the two time series (representing the strengths of strong and weak magnetic elements) remained constant at any given moment. However, the correlation coefficient between the daily total strengths of the strong and weak magnetic elements is −0.6246; when number counts are used instead of strength, the coefficient is −0.7244. Neither value is close to −1.0. Thus, the observed positive and negative correlations cannot be attributed to a mathematical complementary effect. This is unsurprising, given that magnetic elements with zero strength are excluded from the analysis. At high latitudes, the corresponding correlation coefficients are −0.6887 and −0.8417, respectively, again indicating that a complementary effect is not at play. This conclusion is further supported by the fact that, at both high and low latitudes, the strong-field strengths are several times larger than the weak-field strengths. Given this substantial amplitude disparity, the opposite phases alone cannot yield a compensatory effect. As shown in Figures 8, 10 of Chen et al. (2026), the time series correlate with the solar cycle when all magnetic elements are considered, demonstrating that the series do not remain constant. Jin et al. (2011), using daily magnetic maps and excluding the intranetwork field, reached a similar conclusion and likewise ruled out the possibility of a mathematical complementary effect.

Taken together, the phase relationships between the SG and WG at both low and high latitudes are supported not only by our internal consistency checks but also by independent literature results obtained with different instruments, giving us confidence that our findings are robust. Our observed half-cycle lag of approximately 5–6 years is consistent with the flux transport dynamo model predictions of Dikpati and Charbonneau (1999) and Charbonneau (2020), which give a meridional flow transit time from sunspot belt to poles of about 5–6 years.

Magnetic field measurements at high latitudes are subject to certain uncertainties, including projection effects, limb darkening, and instrument resolution, among others. Even so, the distinct solar-cycle behaviors of different types of magnetic elements in these regions can still be discerned, and the results obtained are therefore considered reliable (Chen et al., 2026). However, the specific types of magnetic elements that give rise to these cyclic behaviors remain speculative. Their fundamental nature thus awaits confirmation through dedicated face-to-face observational campaigns at high latitudes in the future.

Although the phase offsets we have found are robust, their interpretation in terms of a single transport mechanism must be qualified by two considerations. First, the observational record spans only about 2.5 solar cycles, which limits our ability to distinguish between periodic and secular variations in the lag times. Second, the classification of the SG and WG at low and high latitudes may introduce systematic offsets if the discrimination thresholds vary over the data set. Nevertheless, the consistency of the half-cycle lag across the SG and WG makes a purely instrumental or classificational origin unlikely. Future observational studies using other data sets (e.g., MDI, HMI, and SOLIS) will be critical for testing whether these phase relationships persist in higher-resolution data and in different phases of the solar cycle.

In summary, while we recognize that the periodic nature of the solar cycle contributes to the cross-correlation structure, the directional asymmetry revealed by the bootstrap analysis, the quantitative consistency with dynamo model predictions, and the robustness across thresholds and latitudes collectively argue that the phase offsets reported here reflect real physical couplings among different magnetic field populations. From a dynamo-theoretic perspective, the phase relationships established in this study serve as a discriminating tool among competing models. The half-cycle lag we observe is naturally reproduced by flux-transport and BL-type dynamos that include a finite meridional circulation timescale. Moreover, the presence of a cycle-modulated signal in the intranetwork component favors models that couple the small-scale field to the large-scale cycle.

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Footnotes

Edited by: Xinhua Zhao, Chinese Academy of Sciences (CAS), China

Reviewed by: Debi Prasad Choudhary, California State University, Northridge, United States

Chunlan Jin, Chinese Academy of Sciences (CAS), China