Polar Photospheric Magnetic Field Evolution and Global Flux Transport

The Sun’s polar magnetic fields are of paramount importance in structuring the heliosphere and in seeding the activity cycle, but they are difficult to observe from the ecliptic plane. However, they are formed by observable active-region magnetic-flux decay and transport processes. We give an observational description of photospheric flux transport and polar field evolution using the >40\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$>40$\end{document} yr series (1974 – 2017) of full-disk line-of-sight photospheric magnetograms from the NSO Kitt Peak Vacuum Telescope (KPVT) 512-channel and Spectromagnetograph (SPMG) and the Synoptic Optical Long-term Investigation of the Sun (SOLIS) Vector Spectro-Magnetograph (VSM). An analytical technique introduced by Durrant, Turner, and Wilson (Solar Physics, 222, 345, 2004). is adapted to analyze time series of full-disk magnetograms to estimate global photospheric flux transport self-consistently. The major hemispheric flux changes, due to cancellation or transport of leading-polarity flux across the equator, occur around activity maximum in each cycle, when peak levels of flux emerge ≥10∘\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\ge 10^{\circ }$\end{document} from the equator. This highlights the efficiency of flux cancellation and transport within each hemisphere, allowing peak levels of leading-polarity flux to reach the equator at this time, letting peak trailing-polarity flux surge poleward. Polar flux evolution follows the hemispheric flux closely each cycle, but with temporal variations dampened by lower-latitude flux cancellation. Major net hemispheric flux change ceased early during Cycle 23, producing weakened polar fields. Cycle 24 was distinguished by large hemispheric flux fluctuations forming a three-part structure: (i) dominant northern activity and north polar flux decrease toward zero; (ii) dominant southern activity peak, swift south polar field decrease, polarity reversal, and growth while the north pole was stalled; and (iii) southern activity decline, north again dominant, north polar flux growth with reversed polarity. Recent global photospheric field evolution and polar field changes are described using Global Oscillations Network Group (GONG, 2006 – present) and Solar Dynamics Observatory Helioseismic and Magnetic Imager (SDO/HMI) line-of-sight magnetograms.


Introduction
The Sun's polar fields, the magnetic fields located at the heliographic poles of the Sun, have a large-scale unipolar distribution covering the latitude range from about ±60 • to ±90 • during most phases of the solar cycle, except while they reverse magnetic polarity. Their magnetic configuration is relatively simple with predominantly near-vertical field lines, but they are highly structured, and the kG facular fields that dominate the poles are small, about ≈ 5 across, as observed from (near) Earth (Tsuneta et al., 2008a), and sparsely distributed: the overall polar field is only of order 5 -10 G. The polar fields are therefore much weaker than active region (AR) fields, and they contain less magnetic flux than a major AR. Nevertheless, they have far-reaching importance because of their unipolarity over large spatial scales, and because of their role in the solar activity cycle (Petrie, 2015). The Sun's polar magnetic fields dominate the global structure of the corona and heliosphere. Most polar magnetic flux does not connect back to the Sun, unlike AR flux, which is generally closed. This, combined with their large spatial extent, means that the polar fields supply most of the interplanetary mean field and channel most of the fast solar wind, except when the polar fields are reversing polarity, which occurs approximately every 11 years during solar activity maximum. The Earth spends most of the solar cycle magnetically connected to the polar coronal holes (Luhmann et al., 2009). Quantitative measurements of the polar fields are therefore extremely important for heliophysics.
Although polar field observations have been taken for several decades covering several activity cycles, several long-standing challenges undermine their measurement (Petrie, 2015). These near-vertical fields correspond to transverse field orientations as seen from Earth, where the polar fields are observed with a large (> 80 • ) viewing angle even under optimal conditions, and are not visible at all for six months at a time. The overall polar field is weak due to the sparse distribution of these ≈ 5 facular fields, and these structures are only resolved by high-resolution magnetographs. The Zeeman effect makes these transverse signals much harder to observe than the longitudinal ones; typically, sensitivity to transverse fields is one order of magnitude lower (Del Toro Iniesta and Martínez Pillet, 2012).
High-resolution polar field observations have emphasized dominant kilogauss structures associated with faculae (Homann, Kneer, and Makarov, 1997;Okunev and Kneer, 2004;Blanco Rodríguez et al., 2007). The Solar Optical Telescope Spectro-Polarimeter (SOT/SP: Tsuneta et al., 2008b) on the Hinode spacecraft (Kosugi et al., 2007) has provided detailed vector field maps every March and September since 2006 (Tsuneta et al., 2008a;Ito et al., 2010;Shiota et al., 2012). Based on such maps, Tsuneta et al. (2008a) presented polar field maps dominated by nearly-vertical magnetic fields of facular scale and kilogauss strength, whose magnetic polarity matched (and defined) the overall polar field polarity. As Cycle 24 progressed, the decline of the polar magnetic flux was associated with reductions of both the number and size of large flux concentrations, and the appearance of opposite-polarity concentrations, beginning at low latitudes (Shiota et al., 2012). At each pole, while the polar field was close to full strength, the larger concentrations, specifically those larger than 10 18 Mx, were of one magnetic polarity, matching the overall polarity of the polar region. The smaller flux concentrations, in sharp contrast, were approximately flux-balanced, as with low-latitude quiet-Sun flux concentrations.
As well as the Hinode/SOT/SP measurements, but less frequent, high-resolution polar vector field measurements have also been taken from the ground with, e.g., the Tenerife Infrared Polarimeter (TIP II: Collados et al., 2007) at the 70 cm Vacuum Tower Telescope (VTT) using the Fe I infrared (IR) lines near 1.56 micron. Compared to the visible spectral line used by Hinode/SOT/SP, this IR spectral line is twice as sensitive to line-of-sight (LOS) fields, and four times as sensitive to transverse fields (Pastor Yabar, Martínez González, and Collados, 2018). Petrie (2017) did indeed find some signs of low signal in SOT/SP vector magnetograms close to the poles, but one can see a similar fall-off of signal in Pastor Yabar, Martínez González, and Collados (2018) (e.g., their Figure 13), perhaps because of the restriction to (1 ) spatial resolution by seeing effects. Petrie (2022) found that the SOT/SP vector magnetograms recorded lower polar radial magnetic fluxes than those from Synoptic Optical Long-term Investigation of the Sun (SOLIS) Vector Spectro-Magnetograph (VSM) LOS magnetograms, but also discovered that polar fluxes based on LOS magnetograms derived from SP Stokes data were significantly higher than their VSM counterparts (and consistent with radial interplanetary field measurements), because of superior spatial resolution and longer observation time. With its adaptive optics, the Daniel K. Inouye Solar Telescope's (DKIST) 4 m mirror will allow its spectro-polarimeters to observe all Stokes polarization parameters with superior spatial resolution, albeit with limited spatio-temporal coverage.
Measurements of the full field vector are further complicated by difficulties of detection and interpretation of the signal for the transverse field component (Harvey, 1969). The Zeeman effect for linear polarization is proportional to the square of the transverse field strength, and is therefore weaker than the Zeeman effect for circular polarization, which has linear dependence on the longitudinal field strength. It is due to this quadratic dependence that the orientation of the transverse field suffers from the so-called 180 • ambiguity in azimuth that must be resolved (by using further assumptions) so that the data can be used for most scientific applications. Of course, these challenges are combined with the fact mentioned above, that sensitivity to transverse fields with the Zeeman effect is typically one order of magnitude lower than to longitudinal signals.
The usual way around the difficulty of detecting transverse field components at the poles is to focus instead on the more detectable longitudinal component. The predominantly nearvertical structure allows one to estimate the strength of the polar field vector from the longitudinal measurement alone by dividing it by the cosine of the heliocentric angle ρ, the angle between the LOS and the local normal. Evidence that the photospheric field is nearly vertical, including the photospheric polar field, comes from numerous sources. Svalgaard, Duvall, and Scherrer (1978), analyzed low-latitude field measurements from the Wilcox Solar Observatory (WSO) by binning them according to heliocentric angle and plotting against the cosine of the heliocentric angle cos ρ, separately for east and west hemispheres, and also separating the fields by their sign during their passage across central meridian. The result was a rhombus-shaped plot representing a linear decrease of average field strength for decreasing cos ρ, consistent with the line-of-sight projection of a radially-directed field. Petrie and Patrikeeva (2009) instead studied the evolution of bundles of line-of-sight flux using time series of SOLIS/VSM magnetograms, exploiting the solar rotation to plot the line-ofsight flux at this surface location as a function of cos ρ, and found the best-fitting linear combination of projected radial and azimuthal vector components to the observed variation of line-of-sight flux against cos ρ. They found that a vast majority of fields of significant strength exhibited a ρ-dependence consistent with a quasi-static vector rotating with the Sun, and most of those were approximately radially-directed. The distribution of tilt angles was peaked within a couple of degrees of vertical, with a standard deviation of about 12 • . Petrie and Patrikeeva (2009) then analyzed the polar fields by exploiting the tilt angle B 0 of the solar rotation axis with respect to our LOS. Observations of line-of-sight flux were collected from fixed locations on the solar disk at central meridian, eliminating the effects of center-tolimb variation on the observations, while the heliographic latitudes at these locations varied as known functions of B 0 . The photospheric line-of-sight fields were well-defined functions of latitude at both poles, increasing in strength monotonically between ±46 • and ±80 • . Solving the resulting linear equations for the radial and meridional field components again yielded a nearly radially-directed photospheric field at the poles. (SOLIS/VSM line-of-sight chromospheric data, in contrast, were not close to being radially-directed in general.) Vector measurements have also shown that the dominant polar field is predominantly radially directed (Tsuneta et al., 2008a;Ito et al., 2010;Petrie, 2017). Petrie (2017) again found a tilt distribution peaked near 0 • with width around 10 • . Wang and Sheeley (1992) showed that this radial-field assumption results in the best coronal field models extrapolated from synoptic maps for the radial field component B r , derived from photospheric observations of the line-of-sight magnetic field component B l , i.e., B r = B l / cos ρ.
The usual and arguably best way to study polar field evolution, and its relation to global flux transport over several solar activity cycles is to apply the above radial field assumption to full-disk LOS magnetograms from long-term synoptic observing programs, which have been observing routinely for several decades. In the first synoptic program at Mt. Wilson, Howard (1967) began the first daily full-disk synoptic magnetogram observations from which full-surface synoptic magnetograms were constructed (Howard, 1989). Similar programs began at the NSO Kitt Peak Vacuum Telescope (KPVT: Livingston et al., 1976;Harvey et al., 1980) and at WSO (Svalgaard, Duvall, and Scherrer, 1978) in the 1970s. The NSO and WSO programs continue to the present day. Successive instrument upgrades occurred at NSO (Keller, Harvey, and Giampapa, 2003), where the KPVT 512-channel (1974KPVT 512-channel ( -1993 and Spectro-Magnetograph (1992 instruments were succeeded by the SOLIS/VSM. The NSO also adapted the Global Oscillations Network Group (GONG) instrument to produce science-worthy longitudinal magnetograms (2006( -present Hill, 2018. The GONG instrument was initially designed to measure Doppler velocities. Magnetograms were added later using a quarter-wave plate. Space-borne full-disk synoptic magnetogram programs, the Solar and Heliospheric Observatory's Michelson Doppler Imager (SOHO/MDI, 1996-2011Scherrer et al., 1995) and Solar Dynamics Observatory's Helioseismic and Magnetic Imager (SDO/HMI, 2010 -present;Scherrer et al., 2012) have provided continuous full-disk observations from outside the Earth's atmosphere. The VSM and HMI instruments produce both LOS and vector full-disk magnetograms but, for reasons of sensitivity discussed above, this paper focuses on LOS magnetograms.
In this paper, we focus mainly on the nearly continuous > 40 yr data series formed by the KPVT 512-channel and SPMG data and the SOLIS/VSM data in Section 3, before analyzing the GONG and HMI data in Section 4. The KPVT 512-channel and SPMG were spatially-scanning slit spectrographs, as was their successor the SOLIS/VSM, prioritizing simultaneous measurement of the spectral line with good wavelength resolution and spectropolarimetric sensitivity, whereas GONG and HMI are wavelength-scanning narrow-band filtergraphs, where a high-speed simultaneous snapshot of a spatial field of view is the most important consideration.
The polar fields are formed and changed by the poleward transport of decayed activeregion flux from low latitudes, processes that we can observe more easily from (near) Earth with the above synoptic observing programs. Solar photospheric magnetic flux has for many years been understood to emerge from the interior in the form of ARs, mostly bipolar in structure. The flux emergence occurs in cycles of period ≈ 11 years on average, though cycle lengths can be several years shorter or longer than this canonical value. Cycle minima also vary in length from less than a year to a few years, and cycle amplitudes, measured in terms of, e.g., sunspot number, can vary significantly, by a factor of two or more. This AR emergence occurs at low latitudes (±30 • latitude), whereupon the magnetic flux is dispersed across the solar surface by the diffusion-like effect of supergranular convective flows, as well as the large-scale differential rotation and meridional flow patterns (Wang, Nash, and Sheeley, 1989). The patterns of flux emergence are key to the cyclical interaction of the ARs and the polar fields. Hale's polarity law says that the bipoles in the north or south solar hemisphere during a given activity cycle have leading/trailing magnetic flux of matching polarity, with opposite polarity in the north and south hemispheres and with polarity reversing every cycle, strong statistical patterns that are rarely violated. Furthermore, the leading-polarity flux of the magnetic bipoles is on average located at a slightly lower latitude than the trailing-polarity flux, a bias referred to as Joy's law (Hale et al., 1919). Thus the bipoles appear tilted with respect to the equator, a phenomenon referred to as Joy's law tilt. Joy's law causes the trailing polarity of a bipolar AR to dominate that portion of the bipole's flux that is dispersed to polar latitudes. Meanwhile, the corresponding leading-polarity flux interacts across or near the equator with the leading-polarity flux in the opposite hemisphere and, being of opposite polarity, cancels with it. This cancellation of flux at or near the equator causes the net flux of each hemisphere to change, and subsequent poleward flux transport ultimately leads to polar field changes, with global consequences. This paper describes these processes and changes quantitatively and in some detail, based on observations from NSO's KPVT 512-channel and SPMG and SOLIS/VSM instruments in Section 3, and GONG and HMI observations in Section 4. We first describe in Section 2 the theoretical technique used in the analysis.

Photospheric Flux Transport Theory
We summarize briefly the theory of Durrant, Turner, and Wilson (2004), before describing how we modify it for application in this paper, (i) to analyze the magnetic flux in the timelatitude data format used here, rather than synoptic maps; (ii) to derive the meridional flow speed self-consistently from the magnetogram data, rather than imposing an external model profile; and (iii) to handle complications with interpreting results for AR magnetic fields.
Following Durrant, Turner, and Wilson (2004), the net magnetic flux crossing a surface S enclosed by a curve C is where B is the magnetic vector. If the surface S is static, then the rate of change of is The MHD induction equation is, where v is the plasma velocity, and η is the magnetic diffusivity. Because ∇ · B = 0, and, by Stokes' theorem, If the surface S is defined as the solar photospheric area (r = R) contained within a circle C of constant colatitude θ = θ C , then and the photospheric bulk flow velocity (excluding turbulent diffusion, which is represented by the diffusivity η) can be resolved into meridional and zonal components, where v mf is the meridional flow speed and v dr is the differential rotation speed. The rate of change of radial magnetic flux through S (the circle at θ = θ C , r = R) is, We expect the contribution from B θ to be small compared to the contribution from B r because weak phostospheric fields are generally nearly vertical, and contributions to B θ from strong closed-loop footpoints should approximately cancel. Equation 8 therefore reduces to Equation 9 can be applied latitude-by-latitude (or colatitude-by-colatitude), computing flux change as a function of latitude. Here, in practice, Equation 9 is integrated slightly differently in the two hemispheres, with limits of integration in colatitude θ from 0 to θ C in the north and from θ C to π in the south. In this paper, we refer to the term on the left-hand side of Equation 9 as the rate of change of magnetic flux and to the two terms of the integral on the right-hand side as the meridional flow advection and the supergranular diffusion. This analysis method was originally applied by Durrant, Turner, and Wilson (2004) to individual full-surface synoptic maps for B r , constructed from NSO KPVT line-of-sight magnetograms, which included active-region magnetic fields as well as polar field data, with observational gaps filled map-by-map by spatial interpolation. In this paper, the method is adapted in the following key respects.
• The calculation is applied to the long-term data series (> 40 yr of KPVT and SOLIS magnetograms) as a whole, rather than to individual synoptic maps. As Section 3 discusses more fully, the magnetogram data are processed from level 2 full-disk line-of-sight magnetograms to form time-latitude maps of radial magnetic flux density, where each temporal pixel spans a single full Carrington rotation (CR, about 27.27 days). Because the model is linear, Equation 5 can be applied to a circle C of constant latitude θ = θ C by computing the terms of Equation 9 for each temporal column of the time-latitude map. • The v mf B r meridional flow term of Equation 9 was constructed by Durrant, Turner, and Wilson (2004) by imposing a separate, empirically-derived model profile for v mf combined with the B r values from the synoptic maps. The rate of change ∂ /∂t of radial magnetic flux was then derived from Equation 9 as the sum as this meridional flow advection term and the supergranular diffusion term. Here we turn this calculation around in the following way. We first calculate ∂ /∂t and ∂B r /∂θ directly from the time-latitude map for B r , and then derive the v mf B r meridional flow advection term using Equation 9, self-consistently without using a separate model. • Durrant, Turner, and Wilson (2004) included ARs (strong, bipolar magnetic field concentrations) in their analysis, although meridional flow measurements are generally derived with active-region pixels excluded from the calculation (e.g. González Hernández et al., 2008;Ulrich, 2010). AR magnetic fields affect the Doppler velocity measurements that form the basis of meridional flow calculations, and these effects are poorly understood and difficult to account for reliably. The safest course of action is therefore simply to exclude ARs from the analysis. Here we work with magnetograms rather than Dopplergrams, but we face an analogous problem. Equation 9 does not account for AR flux emergence. Outside ARs, flux emergence occurs in the form of small, ephemeral bipoles whose flux contributions are too small, brief and random to influence flux calculations with Equation 9 over the length scales of interest. AR flux emergence, however, could affect the results of Equation 9 in ways that are difficult to determine. To address this problem, we perform parallel calculations, cases including and excluding AR fields. This will allow us to study flux transport independent of AR emergence, as well as to assess the importance of strong AR fields to the different components of Equation 9.
Regarding the last item above, we include the flux-transport analysis with AR fields as well as the one without, because most of the analysis concerns the flux change ∂ ∂t or the flux itself, where the inclusion of AR fields is straightforward and unaffected by the problems cited above. We also include the analysis of the other terms with as well as without AR fields for completeness and because the comparison, notwithstanding the problems cited above, is illuminating regarding the influence of AR fields.
As Durrant, Turner, and Wilson (2004) point out, ∂B r /∂t is expected to be small at polar latitudes because the area of integration is small. In addition, during most phases of the cycle, the exception being polar reversal, we expect ∂B r /∂θ ≈ 0 whenever the large-scale structure of the polar field is approximately axisymmetric, when the polar cap is well formed. At such times, B r takes a slowly-varying non-zero value at the pole, and ∂ /∂t ≈ 0 for C at a polar latitude. Under these circumstances, from Equation 9, one expects v mf ≈ 0 at the pole. These conditions are not enforced here, but they seem likely to arise for physical reasons. On the other hand, there is no reason to expect from Equation 9 alone that v mf should vanish at the equator. Antisymmetric meridional flow profiles crossing zero at the equator have, however, been derived from Doppler and helioseismic observations and used in modeling work, but we do not enforce this condition here and it will not necessarily arise naturally. Figure 1 shows time-latitude plots, "butterfly diagrams", of the average radial magnetic flux density B r , plotted on a regular CR-sine(latitude) grid with one column of pixels per CR and 180 rows of pixels uniformly spaced in sine(latitude). The map spans the years 1974 -2017, beginning before the onset of Cycle 21 and ending during the declining phase of Cycle 24. The top panel shows average B r with all fields included, including AR fields, defined as fields stronger than 100 G at the VSM's spatial resolution, and the bottom panel with weak fields only, i.e., excluding AR fields. It is well known that weak and strong fields respond differently to near-surface flows (Schrijver, 1989). To produce the weak-field plot in the bottom panel of Figure 1, all fields of strength |B r | = |B l |/ cos(ρ) above a threshold of 100 G were excluded from the original level 2 full-disk magnetograms before these magnetograms were included in the butterfly diagram. All subsequent calculations were applied to both the all-field case (top panel) and the weak-field case (bottom panel) to study the different roles of strong and weak fields in global magnetic flux transport.

Radial Photospheric Magnetic Flux in Time and Latitude
Compared to butterfly diagrams published in the past (Petrie, 2012(Petrie, , 2013Petrie and Ettinger, 2017), those in Figure 1 have been constructed a little differently. As before, the approximately daily full-disk observations were divided through by the cosine of the heliocentric angle cos ρ to convert the observed line-of-sight field component B l to the estimated radial field component B r , assuming that the magnetic vector is radial. The data within 30 • of the central meridian were averaged in longitude, and stacked along the time axis to form the butterfly pattern. Next, the unobserved and poorly observed polar fields were spatiotemporally interpolated as described in Petrie (2012). So far, the process follows Petrie (2012Petrie ( , 2013. Next, departing from Petrie (2012) and Petrie (2013), the data were interpolated onto a regular daily grid. Next, a regular CR grid was constructed. The daily data were averaged over each CR and stacked along the CR axis. We expect a CR of daily observations to sum to approximately zero because of the solenoidal condition ∇ · B = 0 on the magnetic field B, which, integrating over the photosphere, implies phot B r = 0 for the radial component B r . Accordingly, at this stage, magnetic monopoles were removed in the following way.
There was a small minority (≈ 5) of CRs whose monopoles were approximately a gauss, much larger than the others. These CRs were discarded. For the remaining CRs, the positive and negative magnetic fluxes were calculated and their balance eliminated by rescaling the positive and negative fields. This procedure kept relative magnetic field adjustments uniform and preserved neutral lines. Scaling factors were close to 1, and the largest adjustments were confined to strong fields. The differences between the butterfly diagrams before and after monopole removal are very minor. These adjustments were applied separately in the cases including and excluding AR fields.
The poleward transfer of decayed active-region flux can be seen in the butterfly diagrams in Figure 1 in the form of oblique plumes, surges of polarity-biased flux traveling poleward at speeds of order 10 m s −1 . In each hemisphere, these poleward surges have a pronounced polarity bias over a given cycle, matching the prevailing polarity of the trailing bipolar flux in that hemisphere. The flux traveling poleward accumulates at each pole, forming a large unipolar "cap" of relatively low flux density, ≈ 5 -10 G, typically extending from each pole to about ±60 • latitude.
Each activity cycle begins with a fully-formed polar cap in each hemisphere, whose polarity is determined by the trailing bipole flux of that hemisphere during the preceding cycle. With the onset of the new cycle, the poleward flux surges, dominated by the new trailing magnetic polarity, weaken the polar fields and then reverse their polarity to match this new trailing polarity in each hemisphere. This polar magnetic polarity reversal tends to occur around the time of activity maximum in each hemisphere, when the flux emergence and transport are at their height and the poleward surges are strongest. As Figure 1 shows, polar reversal can occur simultaneously in the two hemispheres or with several years of lag between them. They can also occur at different rates in the two hemispheres, and with different degrees of complexity. However, Figure 1 shows that the stronger of the few cycles that have been fully observed with full-disk magnetographs, Cycles 21 and 22, have exhibited more north-south symmetry in their activity and polar cycles than the weaker Cycles 23 and 24. As an activity cycle declines after the polar field reversal, the polar fields continue to strengthen and form caps of polarity matching the trailing flux and poleward surges. The cycle then ends with full polar caps of reversed magnetic polarity. The strength of the resulting polar fields depends on both the strength of the polarity bias of the poleward surges during the cycle, and on the strength of the pre-existing polar fields before the cycle began, i.e., on the initial polar field flux and on the quantity of net flux added to it. For example, weak and/or mixed-polarity poleward surges acting on a strong pre-existing polar field may result in a weak polar field -this appears to have occurred during Cycle 23, besides other factors that produced the weak polar fields during that cycle, in particular changes in flux emergence patterns that resulted in a reduced mean Joy's law tilt (Petrie, 2015).
An observed polar field reversal was first reported by Babcock (1959), who first associated this phenomenon with the activity cycle observed at low latitudes, establishing that the polar reversal occurs in each hemisphere shortly after the sunspot activity maximum. Babcock (1961) postulated a poleward meridional flow. Leighton (1964) introduced a diffusion model for magnetic-field dispersal by supergranular flows (Leighton, Noyes, and Simon, 1962). The meridional flows were observed by Duvall (1979) and Howard and Labonte (1980) and others. Based on these observational results, the magnetic flux is usually modeled to be advected kinematically poleward by these meridional flows, as well as advection by differential rotation and dispersal by supergranular diffusion (e.g. Wang, Nash, and Sheeley, 1989;Schrijver and De Rosa, 2003;Arge et al., 2010;Upton and Hathaway, 2014). However, there is some evidence that the strongest fields may partially resist the effects of the near-surface flows (Schrijver, 1989). The supergranular flows expand the flux distributions in longitude and latitude, while the differential rotation stretches them in longitude. Both of these flow patterns weaken the flux density, which may make the flux easier for the near-surface flows to advect. The meridional flows move the expanding flux patterns slowly poleward, though they are no match for the supergranular diffusion wherever the effect of the latter is directed equatorward: η takes values of a few hundreds of km 2 s −1 versus meridional flow speeds of a few tens of m s −1 .
Cameron and Schüssler (2015) found observational support for the conjecture of Babcock (1961) that the poloidal flux associated with the polar fields is the seed field for the toroidal field in the interior that produces the subsequent activity cycle. They showed that the net toroidal flux generated by differential rotation in the convection zone in each hemisphere can be estimated from the observed surface magnetic flux distribution, and that this quantity is comparable to the unsigned magnetic flux in NSO synoptic magnetograms. In a reference frame rotating at the equatorial angular velocity at the equator, the kinematic induction equation can be reduced, using some simplifying assumptions and an application of Stokes' theorem, to an expression for the toroidal flux time derivative in terms of the surface magnetic-flux density and the (model) differential rotation profile, so that the toroidal flux itself can be estimated from a known initial value. The mathematical method is similar to that of Durrant, Turner, and Wilson (2004) outlined in Section 2 and applied below, but the assumptions and details, e.g., geometry, are different.
There has been some investigation into how much the eventual effect of a decaying bipolar AR on the polar field depends on the latitude of the region (Giovanelli, 1985;Wang and Sheeley, 1991;Cameron et al., 2013). A decaying bipolar AR, located close to the equator, may undergo flux diffusion across the equator, which, if the two polarities of the bipole are well separated in latitude and mostly located in opposite hemispheres, could produce a flux imbalance in each hemisphere. This is because subsequent poleward meridional transport could then allow much of this flux to escape cancellation, and preserve the flux imbalance in each hemisphere all the way to the pole. Because the magnetic flux contained in a single AR is comparable to the total flux of a full-strength polar cap, the possible effects of well-separated polarities of a bipolar AR on the poles would be very significant in such a scenario.
It is curious, therefore, that the major polar field changes so far observed, and the poleward surges of magnetic flux associated with them, have not been associated with ARs close to the equator, but only with major activity complexes, clusters of ARs, during phases of high activity. Petrie and Ettinger (2017) concluded that this phenomenon does not play a major role in the polar field reversals that they studied. Petrie and Ettinger (2017) tracked one of the example equatorial active regions pointed out by Cameron et al. (2013), whose polarities lay mostly on opposite sides of the equator. The region might therefore have been expected to produce a positive/negative change in the north/south polar field, but, according to subsequent observations, this is not what occurred. Instead, the region's positive and negative fluxes appear to have dispersed and canceled with each other and with neighboring flux systems. The region had no significant effect on the northern and southern high-latitude net fluxes. Indeed, while the region had mostly positive/negative north/south-hemispheric flux, the northern and southern polar fluxes in fact became negative and positive, respectively. Studying four solar cycles in detail, Petrie and Ettinger (2017) could find no example of an equatorial active region producing a significant polar field change, and concluded instead that, while polar field reversals may have been dominated by only a few ARs during some cycles, large, long-lived complexes at the usual active latitudes (about ±10 • to ±30 • ) were responsible for this phenomenon rather than regions straddling the equator. Regarding the overall flux evolution during solar cycles, Whitbread, Yeates, and Muñoz-Jaramillo (2018) applied a 2D surface flux transport model to Cycles 21, 22, and 23, and concluded that, although the top ≈ 10% of contributing ARs tend to define sudden large variations in the axial dipole moment, the cumulative contribution of many weaker regions cannot be ignored. Figure 1 shows no example of a major poleward surge of magnetic flux originating close to the equator and having a visible impact on the polar field. Note that such an influential poleward surge of flux occurring during the declining-phase would be necessarily detectable and visible in Figure 1 as a long oblique plume stemming from an equatorial latitude and extending all the way to the polar regions, accompanied by a significant response of the polar flux to its arrival. No such behavior is present in Figure 1. There is, however, evidence of declining-phase active regions stalling the polar field reversal of Cycle 23 (Petrie, 2012), resulting in ≈ 40% weaker polar fields after the Cycle 23 polar reversal compared to before. In Figure 1, one does see in plumes emanate from the active latitudes during the declining phase of Cycle 23, well correlated in time to this stalling of the polar fields' growth during this time (Petrie, 2012), but, as we will see, this does not diminish the temporal association between the Cycle 23 activity rise and maximum phases and the major Cycle 23 magnetic flux changes. We will return to this topic in a more quantitative manner later.
At this point we can say that the largest polar field changes, those most clearly visible in Figure 1, have occurred relatively early during the activity cycle, when the major activity is located quite distant from the equator, rather than later in the cycle when activity tends to occur closer to the equator. The top butterfly diagram in Figure 1 indicates that large ARs emerging at relatively high latitudes, ±10 • -±30 • , as tends to occur at the height of activity maximum, can have a major influence on the polar fields. It may only take a small number of such regions to dominate the polar reversal in some cases (Petrie and Ettinger, 2017). In simulations with high diffusivity (e.g. Durrant and Wilson, 2003) high-latitude ARs contribute significantly to polar field reversals.
The four vertical solid lines in each panel of Figure 1 indicate the start times of the four Solar Cycles 21 -24, and the four vertical dashed lines indicate their times of activity maximum. 1 Comparing the times of major polar field change to these cycle start and maximum times, in almost all cases the polar reversals occur close in time to the activity maxima: generally within a year or two, before or after. The Cycle 22 south polar reversal occurred about two years later than the Cycle 22 maximum, a large time lag compared to the other cases shown in Figure 1. However, this polar reversal is associated with a giant poleward surge of negative magnetic flux which appears to have originated from a moderately high active latitude, between −30 • and −20 • , and originating a year or so after the published cycle maximum date. We will revisit this poleward surge in Section 3.4.
Note the unique hemispheric asymmetry pattern of the magnetic flux emergence during Cycle 24: the northern hemispheric activity began before the southern, whereupon the south quickly peaked and was briefly dominant for a few years, and finally the south declined as quickly as it ascended, leaving the north to be again the more active hemisphere during the decline of Cycle 24. Such asymmetric activity had profound effects on the flux transport, hemispheric flux changes, and polar field changes, as we will see. . This term describes the rate of change of radial magnetic flux at each time and latitude between that latitude and the pole in each hemisphere. The values are necessarily equal and opposite on each side of the equator, because both the total magnetic flux and its time-derivative must be zero throughout due to the solenoidal condition on the field phot B r = 0.

Magnetic Flux Change in Time and Latitude
The differences between the plots including (top panel) and excluding (bottom panel) ARs are quite modest, because averaged over a CR the net magnetic flux of an AR is small due to its two polarities almost canceling each other. Nevertheless, one can still see some signature of the butterfly wings in the top plot. This is absent from the bottom plot, which is dominated by nearly-vertical streaks. The equatorial evolution is more complex in the top plot because of flux emergence at very low-latitude tending to occur during the declining phase of the cycle, some of which straddle the equator. The strongest nearly-vertical streaks, however, generally occur relatively early during each cycle, when the ARs emerge in larger numbers and at higher latitudes.
That the largest poleward flux surges, and the associated polar field changes described in Section 3.1, are associated with the maximum phase of the activity cycle is consistent with the following result of Petrie (2012): the mean latitudinal centroid separation of the positive and negative AR fluxes in each hemisphere is generally large during this phase of the cycle, and the product of this centroid separation and the mean unsigned AR flux is well correlated with the mean poleward surge flux density in each hemisphere, which in turn is well correlated with the polar field change. Petrie (2012) thus says that the centroid separation and the AR flux both need to be large for major polar flux change, explaining why activity maximum, with its complexes of mutually-canceling Joy's law tilted bipoles, produces the largest polar field changes.
At all times, the flux changes are small at the poles because the area of integration for the flux is small and the polar fields are relatively weak. The top and bottom panels of Figure 2, as in Figure 1, are almost identical poleward of the active latitudes. Strong, AR-strength fields are not found at high latitudes because they are weakened by the near-surface flows, particularly supergranular diffusion and differential rotation, before there is time for them to be dispersed by supergranular diffusion and meridional flow to higher latitudes.
The key patterns shown in Figure 2 are explained by the flux transport or cancellation at the equator. When, as during Cycle 21, the leading magnetic polarity in the north is positive and the leading polarity in the south is negative, the north's positive leading flux decreases due to transport across the equator or cancellation with negative flux near the equator, and likewise for the south's negative leading flux. Meanwhile, each hemisphere's trailing-polarity flux is relatively unharmed by these processes, and may even increase due to transport of the other hemisphere's leading-polarity flux across the equator. The net effect of positive/negative leading flux in the northern/southern hemisphere is therefore a negative/positive flux change, which is what happened during Cycle 21 according to both panels of Figure 2. This pattern alternates cycle by cycle as the leading/trailing magnetic polarities reverse each cycle in each hemisphere. Cycle 24 stands apart from Cycles 21 -23 in having a distinct three-part structure with a dominant positive/negative flux change in the north/south bracketed by two smaller negative/positive changes. This pattern is due to the significant hemispheric asymmetry of the magnetic flux emergence during Cycle 24: the northern hemispheric activity began before the southern, then the south was dominant for a few years, and finally the north was again the more active hemisphere during the decline of Cycle 24. Such asymmetric activity enabled more trailing-polarity flux than usual to cross the equator and affect the hemispheric flux changes during Cycle 24.  Figure 3 shows time-latitude plots of the supergranular diffusion term of Equation 8, including (top panel) and excluding (bottom panel) AR fields. The amplitude of the diffusion is proportional to the diffusivity η which is a free parameter. Here we set η = 600 km s −1 , a typical value (e.g. Wang, Nash, and Sheeley, 1989).

Supergranular Diffusion in Time and Latitude
The linear, constant-η model for diffusion is an approximation of a more complex reality where the effectiveness of supergranular flows in dispersing magnetic fields is known to be dependent on the strength of the magnetic field, implying a nonlinear diffusion model. A nonlinear, non-constant-η diffusion model could be explored by going back to the fulldisk magnetograms and applying the spatial derivatives image-by-image, with diffusivity varying as a function of field strength (Schrijver, 1989), but we do not pursue this here. A simple constant-η model is sufficient for our present purposes.
The ARs are very prominent in the top panel of Figure 3 because their field strengths and spatial derivatives are large. At active latitudes the butterfly wings are, therefore, very pronounced, as are the AR polarity biases that alternate from cycle to cycle: in both hemispheres, regions of predominantly positive ∂B r /∂θ were located systematically northward of regions of predominantly negative ∂B r ∂θ during Cycles 21 and 23, and systematically southward during Cycles 22 and 24.
The bottom panel of Figure 3 shows the signatures of decayed AR flux at both active and high latitudes. At active latitudes the decayed AR flux follows the patterns of the AR flux in the top panel, and shares their cycle-by-cycle hemispheric polarity biases. Regarding the high latitudes, the polar field diffusion patterns are very organized and unipolar. They alternate from cycle to cycle, with opposing polarities at the two poles during each cycle. Comparing Figure 3 to Figure 1, the diffusion term, proportional to ∂B r /∂θ , is almost always of opposite sign to B r . This is easiest to explain for the poles, because the polar fields usually have a 'topknot' structure, i.e., a flux distribution with maximum strength close to the pole itself and with decreasing strength as one moves away from the pole. The supergranular diffusion, being proportional to ∂B r /∂θ , therefore tends to act away from the pole, weakening positive/negative polar fields via a negative/positive contribution to Equation 8. This is consistent with standard flux-transport modeling, where supergranular diffusion tends to spread the polar field and meridional flow tends to concentrate the field (Wang, Nash, and Sheeley, 1989). Therefore, the diffusion term tends to be positive/negative at a negative/positive pole, a rule that extends to AR and decayed AR flux at lower latitudes. Wherever positive/negative flux tends to lie north/south of negative flux in Figure 1, in Figure 3 the diffusion term, being proportional to ∂B r /∂θ , is generally negative/positive. This is why Figure 3 resembles Figure 1 with reversed magnetic polarity. However, at the poles themselves ∂B r /∂θ tends to be small because B r is close to its local maximum strength, and the large-scale effects of diffusion are small. Figure 4 shows the time-latitude distribution of the residual meridional flow advection, the first term on the right-hand side of Equation 8, which is the difference between the timechange of flux and the diffusion term. The meridional flux-transport term collects all changes that are unexplained by linear diffusion, including the global background meridional flow and the local inflows that are usually found around ARs (e.g. González Hernández et al., 2008). As in the previous figures, in Figure 4 the ARs are included in the top panel and excluded from the bottom panel.

Meridional Flow Advection in Time and Latitude
The butterfly pattern of the ARs is prominent in the top panel of Figure 4. In this panel, the polarity pattern of the ARs is opposite to the corresponding pattern in Figure 3, because it describes inflows around ARs, as opposed to outward magnetic flux dispersal by supergranular diffusion. Outside ARs, the polarity pattern of the top panel of Figure 4 resembles that of Figure 1 because the background meridional flow patterns, away from ARs, are directed almost entirely poleward. The polarity of the meridional flux-transport term outside ARs is therefore determined by the magnetic polarity of the flux alone. There is no indication of any additional latitudinal meridional flow cell at high latitudes in Figure 4.
The bottom panel of Figure 4 shows the meridional flux-transport patterns excluding ARs at all latitudes. This plot is dominated by nearly-vertical streaks similar to those in Figure 2. High-latitude plumes indicating poleward surges of magnetic flux in Figure 1 are seen again here in Figure 4; in the bottom panel these plumes extend back to active latitudes, indicating their sources to be the sites of AR decay, consistent with standard flux-transport theory. Most of the major flux surges that ultimately reach polar latitudes have maximum strength at latitudes typical of AR emergence during the maximum phases of the activity cycle, ±10 • -30 • , and not close to the equator. This indicates that most of the polaritybiased magnetic flux carried poleward, producing polar field reversals, comes from ARs emerging at the more typical latitudes for the maximum phase of the cycle rather than near the equator. Comparison of Figure 4 with Figure 1 shows that the poleward surges that are most important in driving polar field changes can indeed be seen to originate close to height of the activity cycle.
In particular, the giant poleward surge associated with the Cycle 22 south polar reversal, discussed in Section 3.1, appears very prominent in the lower panel of Figure 4. Whereas in Figure 1 this surge appeared to originate a year or so after cycle maximum, and between −30 • and −20 • latitude, in Figure 4 this strong plume can be traced back to a latitude of origin of about −10 • , with a start time close to activity maximum. Overall, the lower panel of Figure 4 presents a consistent pattern of major poleward flux surges associated with polar field reversal originating close in time to cycle maxima.
Near ARs, the supergranular diffusion and meridional flow mostly cancel each other because AR flux is nearly balanced in general, so the flux-change term ∂ /∂t of Equation 8 has relatively little contribution from ARs. This implies that the effects of most diffusion effects occurring within ARs are localized by inflows.
Of course, the results in Figure   The trans-equatorial sum of the magnetic flux change is also plotted as a dotted line. This sum represents the change of the monopole, which is much smaller than the terms of Equation 8. Theoretically, of course, the monopole must be zero, but a small monopole error results from the processing of the data, including the numerical time-and latitude-derivatives. However, this error does not significantly affect the results.

Trans-Equatorial Diffusion
The flux-change curves for the cases including and excluding AR fields are very alike, reflecting the near-balance of positive and negative AR flux and the relatively small contribution of ARs to trans-equatorial flux transport. The flux transport across the equator is mostly, almost entirely, in the form of weak fields. Note that this result is independent of the choice of magnetic diffusivity value η. Note also that the largest time-changes of magnetic flux during a given cycle tend to begin close in time to the activity maximum of that cycle. In the case of Cycle 23, the largest flux changes occurred a couple of years before the published date of activity maximum. This is because Cycle 23 had a double-peaked structure, with the maximum monthly sunspot number occurring during the second peak. The largest flux changes of Cycle 23 are associated with the initial peak. The remaining flux changes of Cycle 23 were relatively modest, mixed in sign, and almost canceling each other out. This lack of further major net flux change may be linked to the stalling of the polar fields' development during the decline of Cycle 23, and to the weakness of the polar fields since the Cycle 23 polar reversal.
For meridional flow advection and supergranular diffusion, the story is quite different: the dashed and dot-dashed curves in the top and bottom panels of Figure 5 differ signif-icantly from each other, showing that ARs have large contributions to these terms during active phases of the solar cycle. However, during these times the contributions of ARs to trans-equatorial meridional flow advection and diffusion mostly cancel each other, leaving a simpler cycle-dependent flux-change pattern that alternates sign each cycle (but see the discussion of the AR results in Section 3.4 -these results should not be over-interpreted). However, even in the simpler case excluding ARs, the trans-equatorial flux change and diffusion oppose each other much of the time. The resulting flux-change curves with and without ARs have quasi-periodic appearance due to the alternating leading-and trailing-polarity flux surges in each hemisphere, with a period close to but not exactly a year. More important is the clear cycle-dependent pattern of peaks: predominantly negative during Cycles 21 and 23 and positive during Cycles 22 and 24. However, the cycles's peak structures differ greatly from each other: note the relative weakness of the Cycle 23 peaks relative to Cycles 21 and 22, and the three-part structure of Cycle 24 alluded to above in the discussion of Figure 2, with its principal positive peak bracketed by two smaller negative troughs.  Figure 1 shows that ±60 • latitude roughly corresponds to the low-latitude boundaries of the unipolar bodies of flux at the poles. We therefore adopt the latitudes ±60 • to define the latitude range of the polar field in calculations of polar magnetic flux in Figure 6. Over most of the > 40 yr data set, all three curves in each panel of Figure 6 follow each other quite closely in both hemispheres. This demonstrates the close relationship between magnetic-flux transport across the equator and flux changes at the poles.

Time-Change of Hemispheric and Polar Magnetic Flux
The deviations between the curves that do occur dampen out within a year or so. These deviations are due to flux surges (or cancelation) of alternating sign occurring in quick succession across the equator, with the opposing-polarity surges of flux subsequently canceling before reaching ±60 • latitude. The clearest example of this is the three-part structure of Cycle 24, discussed in Section 3.1 regarding Figure 3. During Cycle 24, the northern hemisphere began to be active before the south, then there followed a period of a few years (mid-2013 -2015) when the south was more active than the north, whose activity level reached a plateau, and finally, in the third phase, the southern activity declined below northern levels while the north continued its plateau of low-level activity for a few years more. This behavior resulted in a complex flux-transport pattern of negative net flux transport into the north, then net-positive, then net-negative, a pattern very evident in the bottom panel of Figure 6. These wild swings in equatorial flux transport, unique in this data set to Cycle 24, were due to the changing hemispheric asymmetry of the activity which enabled more trailing-polarity flux to cross the equator in both directions and affect hemispheric flux balances than was possible during the previous, more hemispherically-balanced cycles.
Trailing-polarity flux also had some effect during the declining phase of Cycle 22 in 1993 -1995. Furthermore, the trans-equatorial flux transport during the declining phase of Cycle 23 featured a mix of polarities, whose cause, AR emergence with near-zero mean Joy's law tilt, was described at length in Petrie (2012) and Petrie (2015). As in Figure 5, in Figure 6 one can see the same concentration of the main Cycle 23 hemispheric flux changes in the early phase of the cycle, and Figure 6 shows that the high-latitude flux changes followed this pattern with a time lag much less than a year early in the cycle and by a year or Figure 6 Plots of net magnetic-flux transport (time-change) against time for high-latitude fields (poleward of ±60 • , solid red lines), and for the full hemisphere including all fields (dotted blue lines) and only weak fields (|B r | < 100 G, dashed green lines). The top panel shows the curves for the northern hemisphere, the middle panel for the southern hemisphere, and the bottom panel for the difference between the two hemispheres. The four vertical solid lines in each panel indicate the start times of the four Solar Cycles 21 -24, and the four vertical dashed lines indicate their times of activity maximum. so around cycle maximum. High-latitude flux changes during the decline of Cycle 23 were smaller and of mixed sign, with important results for the polar fields, as we will see in the next section.
However, the large swings in trans-equatorial flux transport seen during Cycle 24 are unprecedented since regular synoptic observations began. The serial asymmetric flux transport of Cycle 24 was evidently more efficient in driving hemispheric flux change than the more attritional symmetric flux cancellation of Cycles 21 -23. Figure 6 also reaffirms that the hemispheric flux changes of greatest amplitude during a given cycle have start times close to the activity maximum of that cycle. The largest highlatitude flux changes then follow these hemispheric flux changes in their turn. Thus, in gen- eral, we can associate these major polar field changes with maximum phases of the solar activity cycle. Figure 7 shows the net magnetic flux poleward of ±60 • latitude (solid lines), and of the full hemisphere including (dotted lines) and excluding (dashed lines) AR fields, for the northern hemisphere (top panel), the southern hemisphere (middle panel), and the difference between them (bottom panel). The curves in Figure 6 are therefore the time-derivatives of the corresponding curves in Figure 7.

Hemispheric and Polar Magnetic Flux Comparison
The polar flux curves follow the behavior of the polar field curves discussed in Petrie (2013) and Petrie (2015). In Cycles 21 and 22, the polar fields were 40 -50% stronger than they were in Cycles 23 and 24, and this is also true of the fluxes shown in Figure 7: about 3 × 10 22 Mx compared to about 2 × 10 22 Mx.
Note the temporal relationship in Figure 7 between the major hemispheric and highlatitude flux changes and the activity maxima. In particular, the high-latitude fluxes change magnetic polarity within a year or two of activity maxima, sometimes occurring earlier and sometimes later than the published activity maximum date. The Cycle 22 southern polar reversal stands out as having lagged the activity maximum by a couple of years, but recall from Figure 4 in Section 3.4 that even this polar reversal can be linked to the activity maximum by a particularly prominent negative poleward surge that began around activity maximum and appeared decisive in reversing the polar flux polarity from positive to negative. Signatures of this behavior can also be seen in Figure 6 in Section 3.6, where the major Cycle 22 southern hemispheric flux changes began shortly after activity maximum, and the high-latitude flux changes followed those.
As described in Petrie (2012), the change in polar field strength seems to have taken place during the declining phase of Cycle 23, when the ARs' Joy's law tilts lost their hemispheric biases. This is reflected in the relatively short, well-defined, but modest Cycle 23 peak in Figure 6, ending around 2001, followed by several years of mixed-polarity changes. The corresponding behavior in Figure 7 (Petrie, 2015).
The similarity of the curves in each panel of Figure 7, as in Figure 6, demonstrates the importance of trans-equatorial flux transport in defining the polar fluxes. The contribution of ARs to the net hemispheric magnetic flux is generally small because ARs are nearly fluxbalanced. Furthermore, the low-latitude weak polarity biases, associated with decayed AR flux, cause only temporary deviations of the hemispheric net flux from the polar flux. The largest such deviations occur at the end of the time series, during Cycle 24, whose three-part structure, discussed above, caused significant net magnetic flux imbalances and variations of both polarities in each hemisphere. In particular, the solid curve deviates from the dotted and dashed curves in the northern hemisphere plot than in the southern hemisphere plot around the Cycle 24 polar reversal (around 2013 -2016) in Figure 7. The north and the south dotted/dashed curves are, of course, mirror images of each other, because the hemispheric net fluxes must be equal and opposite at all times, whereas the north and south solid curves are strikingly different from each other during the Cycle 24 polar reversal. The key difference between the two hemispheres during this period is the stark contrast discussed above between the north and south flux emergence (and transport) patterns.
The hemispheric net flux imbalances produced by the asymmetric flux emergence patterns and subsequent flux transport occurred at low latitudes, and caused the polar fields to evolve quite differently, in the north-south asymmetric fashion described above, but these differences finally disappeared due to magnetic diffusion and flux cancellation during the declining phase of Cycle 24. Indeed, by the end of the time series, all curves are again close to each other in each panel of Figure 7. Ultimately, the north and south polar fluxes arrived at similar values after activity Cycle 24 had completed, as they must when the low latitudes host little net flux, but they evolved very differently towards these similar flux values.
The Cycle 24 polar flux reached a value close to its maximum for the cycle when the VSM ceased observations in October 2017. This value was similar to the maximum polar flux of Cycle 23, between 5% and 10% higher. Babcock-Leighton dynamo theory leads one to expect solar activity cycle amplitudes to be driven mostly by the polar field strength at the beginning of the cycle (e.g. Cameron and Schüssler, 2015). If this is correct, then we should expect Cycle 25 to be of similar strength to Cycle 24, perhaps a little stronger. To examine the progress of Cycle 25 so far, in the next section we use data from other full-disk magnetographs that have been observing continuously until the present.

Photospheric Flux Transport in GONG and HMI Magnetograms
The KPVT and SOLIS/VSM data described in Section 3 cover 1974 -2017, and the continuity was broken by the relocation of the VSM from Tucson, AZ to Big Bear, CA. The VSM has not yet returned to normal operations, as of the time of writing, but two wellknown synoptic magnetograph programs that do continue to the present day are the NSO's ground-based GONG and the space-borne SDO/HMI. Unlike the KPVT 512-channel and SPMG and SOLIS/VSM spectro-magnetographs, the GONG and HMI instruments are filtergraphs, which prioritize high temporal cadence over spectral resolution and do not have the dynamic range or sensitivity to magnetic fields that the VSM offers. They do, however, pick up most photospheric fields and can be used to study the global solar field over time.
For our purposes, science-grade GONG magnetograms have covered the end of solar Cycle 23 and its extended minimum, all of Cycle 24, and the ascent of Cycle 25, and HMI magnetograms have caught Cycle 24 and the ascent of Cycle 25. Figure 8 shows butterfly diagrams of GONG (left panels) and HMI (right panels), including (top panels) and excluding (bottom panels) ARs. Where the plots in Figure 8 overlap with the corresponding plots in Figure 1, the qualitative agreement is reasonable: the main features and patterns are reproduced, including the three-part structure of Cycle 24: the early dominance of the northern hemisphere, followed by the strong and brief dominance of the southern hemisphere while the north stalled, and finally the sharp decline of the south and the north's dominance during the declining phase. Accompanying this behavior of the activity, the north polar field can again be seen to have begun its reversal earlier than the southern polar field, but stalled around zero for several years and reversed polarity multiple times, while the south swiftly completed its reversal thanks to a giant poleward surge in 2014 -2015, after which the north finally completed its reversal in 2016. All of this behavior is exhibited by the GONG and HMI data in Figure 8 as with the VSM data in Figure 1.
The differences between these GONG and HMI data and the VSM data shown in Section 3 become more obvious when their flux properties are analyzed in more detail. The GONG and HMI data are more difficult to analyze in this way because their sensitivity to weak fields is lower and they are more prone to saturation and zero-point errors. Efforts to address such errors are underway, and pending corrections we do not show GONG or HMI flux-transport analyses here.
As with Figure 7 for the KPVT and SOLIS data, Figure 9 shows for the GONG and HMI data the net hemispheric and polar magnetic fluxes, and the differences between them. Although the GONG and HMI results are qualitatively similar to each other and to the corresponding SOLIS results, the match is far from perfect. They do agree, however, that the divergence between the polar flux (solid curves) and the hemispheric flux (dotted and dashed curves) was much greater in the northern than in the southern hemisphere. This is because of the hemispheric asymmetry in the flux emergence and transport patterns as already discussed regarding Figure 7.
It seems worth noting that, judging from Figures 8 and 9, Cycle 25 began more hemispherically symmetric than Cycle 24, but recently the polar fields changed asymmetrically, weakening more quickly in the south than in the north. However, a sustained negativepolarity poleward surge in the north and a new negative-polarity poleward surge in the south suggest that the hemispheres might become more even in the near future.

Figure 9
Plots of net magnetic flux against time for high-latitude fields (poleward of ±60 • , solid lines), and for the full hemisphere, including all fields (dotted lines) and only weak fields (|B r | < 100 G, dashed lines), derived from NSO GONG (2006 -present, left panels) and NASA SDO/HMI (2010 -present, right panels) line-of-sight images. The top panel shows the curves for the northern hemisphere, the middle panel for the southern hemisphere, and the bottom panel for the difference between the two hemispheres. Figure 10 compares the KPVT/SOLIS, GONG and HMI polar fluxes over time on a single scale. Because these quantities represent the common latitude range from ±60 • to ±90 • , one can also compare the measured mean polar field strengths (radial flux densities) using the same curves and referring to the second axis on the right of the plot.
At all times for which more than one data set is available, the shapes of the curves are very similar, but they do differ in amplitude and in their details. For example, the curves all describe a triple polar reversal at the north pole between 2012 and 2016, but the U-shaped swings of the three curves around the beginning of 2015 are different: the GONG curve has the deepest swing and the HMI curve the shallowest, with SOLIS/VSM in between. (In contrast, the n-shaped swings of the three curves a year or two earlier are very similar.) Generally the SOLIS curves have the largest amplitude and the GONG and HMI curves are comparable: sometimes GONG shows a slightly stronger field and sometimes HMI.

Conclusion
This quantitative study of photospheric flux transport, based direct analysis of full-disk magnetograms, complements existing flux-transport modeling. It confirms and extends some past results: global hemispheric and polar flux changes were described in detail, empirical links between flux emergence and transport patterns and polar flux changes were established, and key distinct properties of the activity cycles were brought out, with particular emphasis on Cycle 24.
The largest hemispheric and polar flux changes occur relatively early during the solar cycle, in response to the maximum phase of the activity, when active-region flux emergence is at its height and when the ARs are emerging in large numbers at relatively high latitudes, ±10 • -±30 • . The hemispheric flux changes are driven by flux cancellation or transport across the equator. The flux of leading sunspot polarity in a given hemisphere decreases due to cancellation with leading-polarity flux of the opposite hemisphere, or unopposed diffusion across the equator. Meanwhile, the trailing-polarity flux is less harmed by these processes, and so the hemispheric flux change during a given cycle matches the trailing sunspot polarity in that hemisphere during that cycle.
The supergranular diffusion is directed downward along magnetic field gradients, and so generally has the opposite sign to the magnetic field polarity. It therefore tends to disperse the polar field, whereas the meridional flow tends to concentrate it towards the pole. Likewise, the diffusion tends to disperse active-region flux, whereas the meridional flow tends to have inflows around the ARs, concentrating their flux. Even the analysis that excludes the AR fields themselves shows this last pattern at active latitudes.
The meridional flow advection has maximum strength at latitudes typical of active-region emergence, ±10 • -±30 • , implying that most of the decayed active-region flux that is responsible for hemispheric and polar flux changes originates from latitudes typical of solar maximum flux emergence rather than near the equator. This is consistent with the timing of the greatest hemispheric and polar flux changes close to the activity maximum.
Comparing the hemispheric flux changes over time, including and excluding strong (active-region) fields, shows that trans-equatorial flux transport is due almost entirely to weak, decayed flux. Strong active-region flux is too polarity-balanced to have much overall impact on hemispheric flux changes. This result is independent of the details of the supergranular diffusion and meridional flow: it is based on straightforward flux calculations alone. That this equatorial cancellation of leading-polarity flux peaks early in the cycle, when the flux emergence rate is at its height but occurs ≥ 10 • away from the equator in each hemisphere, emphasizes the importance of the transport and cancellation processes within each hemisphere in allowing peak levels of leading-polarity flux to reach the equator and trailing-polarity flux to reach each pole at this phase of the cycle, when the activity is close to neither the equator nor (of course) the poles. In contrast, when the activity is close to the equator during the declining phase of the cycle, the associated hemispheric flux changes are relatively small.
Although the flux changes are dominated by the transport of weak fields, the strong fields' contributions to the transport terms associated with supergranular diffusion and meridional flow are large during active phases of the cycle. However, the contributions to these terms oppose each other and their net effect is generally small. One can question the validity of the linear diffusion model for strong fields, but a nonlinear model would have lower diffusivity for strong than for weak fields (Schrijver, 1989), and would therefore not significantly change this result.
The net polar flux evolution follows the hemispheric changes closely, with some departures dampened within a year or so by cancellation of flux on its way poleward. The good agreement between full-hemispheric and polar fluxes over time demonstrates the close link between trans-equatorial flux transport and polar field change.
Major net hemispheric flux change ceased early during Cycle 23, producing weakened polar fields, with broad consequences for heliospheric physics. Declining-phase active regions are known to have played a role in stalling the polar fields' development during the decline of Cycle 23, and their contribution appears here as a series of modest hemispheric flux changes of mixed magnetic polarity, producing little change in the polar fluxes.
The recent Cycle 24 stands out from its three predecessors in having particularly large deviations between hemispheric and polar flux variations in the northern hemisphere: the polar flux stalled close to zero for three years, while the active latitudes sent several alternatingpolarity surges of decayed flux poleward, changing and reversing the polarity of the polar flux several times, before ultimately mutually canceling at the pole. While the north stalled and reversed polarity multiple times, the south reversed quite straightforwardly, before the north completed its polarity reversal, giving Cycle 24 a distinctive three-part structure. At Cycle 24 minimum, the polar fluxes settled at similar values, having evolved towards these values very differently.
The full-disk line-of-sight magnetograms from the GONG and HMI filter-magnetographs provide flux patterns qualitatively similar to those from the SOLIS/VSM, where the temporal coverage overlaps. The GONG and HMI flux changes follow the same general patterns described above, but the details of the changes are quite different. The discrepancies between the data sets require more attention, and emphasize the need for both long-term observations with full-disk spectro-magnetographs of excellent sensitivity, and much more detailed understanding of the differences among magnetogram data sets. Efforts to resume regular operations with the SOLIS/VSM, and attempts to improve and cross-calibrate GONG and HMI data, are ongoing. agreement with the National Science Foundation. SOLIS/VSM vector magnetograms are produced cooperatively by NSF/NSO and NASA/LWS. NSO/Kitt Peak 512-channel and SPMG data used here were produced cooperatively by NSF/NOAO, NASA/GSFC, and NOAA/SEL. This work utilizes data obtained by the NSO Integrated Synoptic Program (NISP) Global Oscillations Network Group (GONG), managed by the National Solar Observatory, which is operated by AURA, Inc. under a cooperative agreement with the National Science Foundation. The data were acquired by instruments operated by the Big Bear Solar Observatory, High Altitude Observatory, Learmonth Solar Observatory, Udaipur Solar Observatory, Instituto de Astrofísica de Canarias, and Cerro Tololo Interamerican Observatory. SDO is a mission for NASA's Living With a Star program.

Author contributions
The author confirms sole responsibility for the following: study conception and design, data collection, analysis and interpretation of results, and manuscript preparation.

Declarations
Competing interests The authors declare no competing interests.